LMFDB - Embedded newform 84.12.i.b.25.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,12,Mod(25,84)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("84.25");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	84.i (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$64.5408271670$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 581500324 x^{14} - 481772282104 x^{13} + \cdots + 79\!\cdots\!77 $ x^16 - 581500324x^14 - 481772282104x^13 + 132272376701859942x^12 + 189765040124256066208x^11 - 14668309022748191106194508x^10 - 25034330150368988089577197656x^9 + 804524023197576594568815873370179x^8 + 1124828921377792402289540847268007968x^7 - 19980431516277460534216470290766973558668x^6 + 592727132887361668148704829618109324342008x^5 + 215459975341822734071937522290785139860745873306x^4 - 372150170517680072482976606267991582419751904690704x^3 - 31534379943215460069637664338468383174583531082850128x^2 + 257934680050232001165280545701170549413209663811915341424x + 79978257504936591964642108900330900905737463001503418421877
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{42}\cdot 3^{15}\cdot 7^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			25.8
Root			$-11427.1 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	84.25
Dual form			84.12.i.b.37.8

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-121.500 + 210.444i) q^{3} +(5578.82 + 9662.79i) q^{5} +(24922.9 + 36826.3i) q^{7} +(-29524.5 - 51137.9i) q^{9} +O(q^{10})$	\(q+(-121.500 + 210.444i) q^{3} +(5578.82 + 9662.79i) q^{5} +(24922.9 + 36826.3i) q^{7} +(-29524.5 - 51137.9i) q^{9} +(-196201. + 339830. i) q^{11} -2.28291e6 q^{13} -2.71130e6 q^{15} +(-2.79207e6 + 4.83601e6i) q^{17} +(6.40868e6 + 1.11002e7i) q^{19} +(-1.07780e7 + 770481. i) q^{21} +(-1.12939e7 - 1.95616e7i) q^{23} +(-3.78323e7 + 6.55275e7i) q^{25} +1.43489e7 q^{27} +1.65025e8 q^{29} +(9.99385e7 - 1.73099e8i) q^{31} +(-4.76768e7 - 8.25786e7i) q^{33} +(-2.16805e8 + 4.46272e8i) q^{35} +(2.10302e8 + 3.64253e8i) q^{37} +(2.77373e8 - 4.80425e8i) q^{39} -7.64804e8 q^{41} -2.38223e8 q^{43} +(3.29423e8 - 5.70578e8i) q^{45} +(-9.33471e7 - 1.61682e8i) q^{47} +(-7.35026e8 + 1.83564e9i) q^{49} +(-6.78474e8 - 1.17515e9i) q^{51} +(1.62451e9 - 2.81374e9i) q^{53} -4.37827e9 q^{55} -3.11462e9 q^{57} +(6.84439e7 - 1.18548e8i) q^{59} +(1.63005e9 + 2.82334e9i) q^{61} +(1.14739e9 - 2.36178e9i) q^{63} +(-1.27359e10 - 2.20593e10i) q^{65} +(-7.66329e9 + 1.32732e10i) q^{67} +5.48883e9 q^{69} +1.53915e10 q^{71} +(1.23489e10 - 2.13889e10i) q^{73} +(-9.19325e9 - 1.59232e10i) q^{75} +(-1.74046e10 + 1.24419e9i) q^{77} +(1.05790e10 + 1.83233e10i) q^{79} +(-1.74339e9 + 3.01964e9i) q^{81} -5.62969e10 q^{83} -6.23059e10 q^{85} +(-2.00505e10 + 3.47285e10i) q^{87} +(-1.87036e10 - 3.23955e10i) q^{89} +(-5.68967e10 - 8.40711e10i) q^{91} +(2.42851e10 + 4.20630e10i) q^{93} +(-7.15057e10 + 1.23851e11i) q^{95} -7.72068e10 q^{97} +2.31709e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 1944 q^{3} - 2156 q^{5} + 50512 q^{7} - 472392 q^{9}+O(q^{10}) $ 16 * q - 1944 * q^3 - 2156 * q^5 + 50512 * q^7 - 472392 * q^9	$ 16 q - 1944 q^{3} - 2156 q^{5} + 50512 q^{7} - 472392 q^{9} - 222796 q^{11} + 2703176 q^{13} + 1047816 q^{15} + 5114600 q^{17} + 6910556 q^{19} - 18340668 q^{21} - 51387712 q^{23} - 191456372 q^{25} + 229582512 q^{27} + 118854616 q^{29} + 164659160 q^{31} - 54139428 q^{33} + 55239344 q^{35} + 75658364 q^{37} - 328435884 q^{39} - 1815568608 q^{41} + 10754408 q^{43} - 127309644 q^{45} - 1034359464 q^{47} + 4123496848 q^{49} + 1242847800 q^{51} - 665159988 q^{53} - 1264543896 q^{55} - 3358530216 q^{57} + 1040514580 q^{59} - 14391208024 q^{61} + 1474099236 q^{63} - 20938150200 q^{65} - 33307097284 q^{67} + 24974428032 q^{69} + 65848902896 q^{71} + 17709749204 q^{73} - 46523898396 q^{75} + 8594484604 q^{77} - 26626784032 q^{79} - 27894275208 q^{81} - 210306955048 q^{83} - 25867402032 q^{85} - 14440835844 q^{87} - 55951560072 q^{89} + 66078280292 q^{91} + 40012175880 q^{93} + 106810047392 q^{95} - 156216030712 q^{97} + 26311762008 q^{99}+O(q^{100}) $ 16 * q - 1944 * q^3 - 2156 * q^5 + 50512 * q^7 - 472392 * q^9 - 222796 * q^11 + 2703176 * q^13 + 1047816 * q^15 + 5114600 * q^17 + 6910556 * q^19 - 18340668 * q^21 - 51387712 * q^23 - 191456372 * q^25 + 229582512 * q^27 + 118854616 * q^29 + 164659160 * q^31 - 54139428 * q^33 + 55239344 * q^35 + 75658364 * q^37 - 328435884 * q^39 - 1815568608 * q^41 + 10754408 * q^43 - 127309644 * q^45 - 1034359464 * q^47 + 4123496848 * q^49 + 1242847800 * q^51 - 665159988 * q^53 - 1264543896 * q^55 - 3358530216 * q^57 + 1040514580 * q^59 - 14391208024 * q^61 + 1474099236 * q^63 - 20938150200 * q^65 - 33307097284 * q^67 + 24974428032 * q^69 + 65848902896 * q^71 + 17709749204 * q^73 - 46523898396 * q^75 + 8594484604 * q^77 - 26626784032 * q^79 - 27894275208 * q^81 - 210306955048 * q^83 - 25867402032 * q^85 - 14440835844 * q^87 - 55951560072 * q^89 + 66078280292 * q^91 + 40012175880 * q^93 + 106810047392 * q^95 - 156216030712 * q^97 + 26311762008 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−121.500	+	210.444i	−0.288675	+	0.500000i
$3$	−121.500	+	210.444i	−0.288675	+	0.500000i
$4$		0			0
$5$	5578.82	+	9662.79i	0.798375	+	1.38283i	0.920674	+	0.390333i	$0.127640\pi$
$5$	5578.82	+	9662.79i	0.798375	+	1.38283i	−0.122299	+	0.992493i	$0.539027\pi$
$6$		0			0
$7$	24922.9	+	36826.3i	0.560479	+	0.828169i
$7$	24922.9	+	36826.3i	0.560479	+	0.828169i
$8$		0			0
$9$	−29524.5	−	51137.9i	−0.166667	−	0.288675i
$10$		0			0
$11$	−196201.	+	339830.i	−0.367317	+	0.636211i	−0.989145	−	0.146942i	$-0.953057\pi$
$11$	−196201.	+	339830.i	−0.367317	+	0.636211i	0.621828	+	0.783154i	$0.286390\pi$
$12$		0			0
$13$	−2.28291e6			−1.70530			−0.852649	−	0.522484i	$-0.825005\pi$
$13$	−2.28291e6			−1.70530			−0.852649	+	0.522484i	$0.825005\pi$
$14$		0			0
$15$	−2.71130e6			−0.921884
$16$		0			0
$17$	−2.79207e6	+	4.83601e6i	−0.476934	+	0.826073i	−0.999651	−	0.0264331i	$-0.991585\pi$
$17$	−2.79207e6	+	4.83601e6i	−0.476934	+	0.826073i	0.522717	+	0.852506i	$0.324918\pi$
$18$		0			0
$19$	6.40868e6	+	1.11002e7i	0.593778	+	1.02845i	0.993718	+	0.111912i	$0.0356975\pi$
$19$	6.40868e6	+	1.11002e7i	0.593778	+	1.02845i	−0.399940	+	0.916541i	$0.630969\pi$
$20$		0			0
$21$	−1.07780e7	+	770481.i	−0.575881	+	0.0411676i
$22$		0			0
$23$	−1.12939e7	−	1.95616e7i	−0.365881	−	0.633725i	0.623036	−	0.782193i	$-0.285899\pi$
$23$	−1.12939e7	−	1.95616e7i	−0.365881	−	0.633725i	−0.988917	+	0.148468i	$0.952566\pi$
$24$		0			0
$25$	−3.78323e7	+	6.55275e7i	−0.774806	+	1.34200i
$26$		0			0
$27$	1.43489e7			0.192450
$28$		0			0
$29$	1.65025e8			1.49403			0.747016	−	0.664806i	$-0.231486\pi$
$29$	1.65025e8			1.49403			0.747016	+	0.664806i	$0.231486\pi$
$30$		0			0
$31$	9.99385e7	−	1.73099e8i	0.626966	−	1.08594i	−0.361192	−	0.932492i	$-0.617630\pi$
$31$	9.99385e7	−	1.73099e8i	0.626966	−	1.08594i	0.988157	−	0.153445i	$-0.0490367\pi$
$32$		0			0
$33$	−4.76768e7	−	8.25786e7i	−0.212070	−	0.367317i
$34$		0			0
$35$	−2.16805e8	+	4.46272e8i	−0.697741	+	1.43623i
$36$		0			0
$37$	2.10302e8	+	3.64253e8i	0.498579	+	0.863563i	0.999999	−	0.00164058i	$-0.000522215\pi$
$37$	2.10302e8	+	3.64253e8i	0.498579	+	0.863563i	−0.501420	+	0.865204i	$0.667189\pi$
$38$		0			0
$39$	2.77373e8	−	4.80425e8i	0.492277	−	0.852649i
$40$		0			0
$41$	−7.64804e8			−1.03095			−0.515476	−	0.856904i	$-0.672385\pi$
$41$	−7.64804e8			−1.03095			−0.515476	+	0.856904i	$0.672385\pi$
$42$		0			0
$43$	−2.38223e8			−0.247120			−0.123560	−	0.992337i	$-0.539431\pi$
$43$	−2.38223e8			−0.247120			−0.123560	+	0.992337i	$0.539431\pi$
$44$		0			0
$45$	3.29423e8	−	5.70578e8i	0.266125	−	0.460942i
$46$		0			0
$47$	−9.33471e7	−	1.61682e8i	−0.0593694	−	0.102831i	0.834813	−	0.550533i	$-0.185576\pi$
$47$	−9.33471e7	−	1.61682e8i	−0.0593694	−	0.102831i	−0.894183	+	0.447703i	$0.852242\pi$
$48$		0			0
$49$	−7.35026e8	+	1.83564e9i	−0.371727	+	0.928342i
$50$		0			0
$51$	−6.78474e8	−	1.17515e9i	−0.275358	−	0.476934i
$52$		0			0
$53$	1.62451e9	−	2.81374e9i	0.533587	−	0.924200i	−0.465643	−	0.884973i	$-0.654177\pi$
$53$	1.62451e9	−	2.81374e9i	0.533587	−	0.924200i	0.999230	−	0.0392276i	$-0.0124897\pi$
$54$		0			0
$55$	−4.37827e9			−1.17303
$56$		0			0
$57$	−3.11462e9			−0.685635
$58$		0			0
$59$	6.84439e7	−	1.18548e8i	0.0124637	−	0.0215878i	−0.859726	−	0.510755i	$-0.829366\pi$
$59$	6.84439e7	−	1.18548e8i	0.0124637	−	0.0215878i	0.872190	+	0.489167i	$0.162699\pi$
$60$		0			0
$61$	1.63005e9	+	2.82334e9i	0.247109	+	0.428005i	0.962722	−	0.270492i	$-0.0871862\pi$
$61$	1.63005e9	+	2.82334e9i	0.247109	+	0.428005i	−0.715614	+	0.698496i	$0.753853\pi$
$62$		0			0
$63$	1.14739e9	−	2.36178e9i	0.145659	−	0.299824i
$64$		0			0
$65$	−1.27359e10	−	2.20593e10i	−1.36147	−	2.35813i
$66$		0			0
$67$	−7.66329e9	+	1.32732e10i	−0.693432	+	1.20106i	0.277275	+	0.960791i	$0.410569\pi$
$67$	−7.66329e9	+	1.32732e10i	−0.693432	+	1.20106i	−0.970707	+	0.240268i	$0.922765\pi$
$68$		0			0
$69$	5.48883e9			0.422483
$70$		0			0
$71$	1.53915e10			1.01242			0.506210	−	0.862410i	$-0.331046\pi$
$71$	1.53915e10			1.01242			0.506210	+	0.862410i	$0.331046\pi$
$72$		0			0
$73$	1.23489e10	−	2.13889e10i	0.697192	−	1.20757i	−0.272244	−	0.962228i	$-0.587766\pi$
$73$	1.23489e10	−	2.13889e10i	0.697192	−	1.20757i	0.969436	−	0.245344i	$-0.0789008\pi$
$74$		0			0
$75$	−9.19325e9	−	1.59232e10i	−0.447334	−	0.774806i
$76$		0			0
$77$	−1.74046e10	+	1.24419e9i	−0.732764	+	0.0523826i
$78$		0			0
$79$	1.05790e10	+	1.83233e10i	0.386807	+	0.669970i	0.992018	−	0.126095i	$-0.0402445\pi$
$79$	1.05790e10	+	1.83233e10i	0.386807	+	0.669970i	−0.605211	+	0.796065i	$0.706911\pi$
$80$		0			0
$81$	−1.74339e9	+	3.01964e9i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−5.62969e10			−1.56875			−0.784377	−	0.620284i	$-0.787017\pi$
$83$	−5.62969e10			−1.56875			−0.784377	+	0.620284i	$0.787017\pi$
$84$		0			0
$85$	−6.23059e10			−1.52309
$86$		0			0
$87$	−2.00505e10	+	3.47285e10i	−0.431290	+	0.747016i
$88$		0			0
$89$	−1.87036e10	−	3.23955e10i	−0.355041	−	0.614950i	0.632084	−	0.774900i	$-0.282200\pi$
$89$	−1.87036e10	−	3.23955e10i	−0.355041	−	0.614950i	−0.987125	+	0.159950i	$0.948867\pi$
$90$		0			0
$91$	−5.68967e10	−	8.40711e10i	−0.955783	−	1.41227i
$92$		0			0
$93$	2.42851e10	+	4.20630e10i	0.361979	+	0.626966i
$94$		0			0
$95$	−7.15057e10	+	1.23851e11i	−0.948115	+	1.64218i
$96$		0			0
$97$	−7.72068e10			−0.912874			−0.456437	−	0.889756i	$-0.650875\pi$
$97$	−7.72068e10			−0.912874			−0.456437	+	0.889756i	$0.650875\pi$
$98$		0			0
$99$	2.31709e10			0.244878
$100$		0			0
$101$	6.27863e10	−	1.08749e11i	0.594426	−	1.02958i	−0.399202	−	0.916863i	$-0.630713\pi$
$101$	6.27863e10	−	1.08749e11i	0.594426	−	1.02958i	0.993628	−	0.112712i	$-0.0359539\pi$
$102$		0			0
$103$	−9.56009e10	−	1.65586e11i	−0.812563	−	1.40740i	−0.911064	−	0.412264i	$-0.864738\pi$
$103$	−9.56009e10	−	1.65586e11i	−0.812563	−	1.40740i	0.0985011	−	0.995137i	$-0.468595\pi$
$104$		0			0
$105$	−6.75735e10	−	9.98473e10i	−0.516696	−	0.763476i
$106$		0			0
$107$	2.85938e10	+	4.95259e10i	0.197088	+	0.341367i	0.947583	−	0.319509i	$-0.103518\pi$
$107$	2.85938e10	+	4.95259e10i	0.197088	+	0.341367i	−0.750495	+	0.660876i	$0.770185\pi$
$108$		0			0
$109$	1.21619e11	−	2.10650e11i	0.757102	−	1.31134i	−0.187220	−	0.982318i	$-0.559948\pi$
$109$	1.21619e11	−	2.10650e11i	0.757102	−	1.31134i	0.944322	−	0.329022i	$-0.106719\pi$
$110$		0			0
$111$	−1.02207e11			−0.575709
$112$		0			0
$113$	4.16630e10			0.212726			0.106363	−	0.994327i	$-0.466080\pi$
$113$	4.16630e10			0.212726			0.106363	+	0.994327i	$0.466080\pi$
$114$		0			0
$115$	1.26013e11	−	2.18261e11i	0.584221	−	1.01190i
$116$		0			0
$117$	6.74018e10	+	1.16743e11i	0.284216	+	0.492277i
$118$		0			0
$119$	−2.47679e11	+	1.77057e10i	−0.951439	+	0.0680149i
$120$		0			0
$121$	6.56664e10	+	1.13738e11i	0.230157	+	0.398643i
$122$		0			0
$123$	9.29237e10	−	1.60948e11i	0.297610	−	0.515476i
$124$		0			0
$125$	−2.99432e11			−0.877592
$126$		0			0
$127$	−3.71377e10			−0.0997457			−0.0498728	−	0.998756i	$-0.515882\pi$
$127$	−3.71377e10			−0.0997457			−0.0498728	+	0.998756i	$0.515882\pi$
$128$		0			0
$129$	2.89441e10	−	5.01327e10i	0.0713374	−	0.123560i
$130$		0			0
$131$	−1.98899e11	−	3.44503e11i	−0.450443	−	0.780191i	0.547970	−	0.836498i	$-0.315401\pi$
$131$	−1.98899e11	−	3.44503e11i	−0.450443	−	0.780191i	−0.998414	+	0.0563070i	$0.982067\pi$
$132$		0			0
$133$	−2.49055e11	+	5.12656e11i	−0.518933	+	1.06817i
$134$		0			0
$135$	8.00499e10	+	1.38651e11i	0.153647	+	0.266125i
$136$		0			0
$137$	−4.18742e11	+	7.25282e11i	−0.741282	+	1.28394i	0.210630	+	0.977566i	$0.432448\pi$
$137$	−4.18742e11	+	7.25282e11i	−0.741282	+	1.28394i	−0.951912	+	0.306372i	$0.900885\pi$
$138$		0			0
$139$	−6.68822e11			−1.09327			−0.546637	−	0.837370i	$-0.684092\pi$
$139$	−6.68822e11			−1.09327			−0.546637	+	0.837370i	$0.684092\pi$
$140$		0			0
$141$	4.53667e10			0.0685539
$142$		0			0
$143$	4.47908e11	−	7.75800e11i	0.626384	−	1.08493i
$144$		0			0
$145$	9.20642e11	+	1.59460e12i	1.19280	+	2.06599i
$146$		0			0
$147$	−2.96993e11	−	3.77712e11i	−0.356863	−	0.453853i
$148$		0			0
$149$	3.37890e11	+	5.85243e11i	0.376922	+	0.652847i	0.990613	−	0.136699i	$-0.0436493\pi$
$149$	3.37890e11	+	5.85243e11i	0.376922	+	0.652847i	−0.613691	+	0.789546i	$0.710316\pi$
$150$		0			0
$151$	2.19258e11	−	3.79765e11i	0.227291	−	0.393679i	−0.729714	−	0.683753i	$-0.760347\pi$
$151$	2.19258e11	−	3.79765e11i	0.227291	−	0.393679i	0.957004	+	0.290074i	$0.0936800\pi$
$152$		0			0
$153$	3.29738e11			0.317956
$154$		0			0
$155$	2.23015e12			2.00221
$156$		0			0
$157$	1.47959e11	−	2.56272e11i	0.123792	−	0.214414i	−0.797468	−	0.603361i	$-0.793828\pi$
$157$	1.47959e11	−	2.56272e11i	0.123792	−	0.214414i	0.921260	+	0.388947i	$0.127161\pi$
$158$		0			0
$159$	3.94756e11	+	6.83738e11i	0.308067	+	0.533587i
$160$		0			0
$161$	4.38905e11	−	9.03443e11i	0.319763	−	0.658201i
$162$		0			0
$163$	1.12083e12	+	1.94133e12i	0.762971	+	1.32150i	0.941313	+	0.337535i	$0.109593\pi$
$163$	1.12083e12	+	1.94133e12i	0.762971	+	1.32150i	−0.178342	+	0.983968i	$0.557073\pi$
$164$		0			0
$165$	5.31960e11	−	9.21381e11i	0.338623	−	0.586513i
$166$		0			0
$167$	−2.62678e11			−0.156489			−0.0782445	−	0.996934i	$-0.524931\pi$
$167$	−2.62678e11			−0.156489			−0.0782445	+	0.996934i	$0.524931\pi$
$168$		0			0
$169$	3.41951e12			1.90804
$170$		0			0
$171$	3.78426e11	−	6.55453e11i	0.197926	−	0.342818i
$172$		0			0
$173$	−9.72304e11	−	1.68408e12i	−0.477033	−	0.826245i	0.522621	−	0.852565i	$-0.324954\pi$
$173$	−9.72304e11	−	1.68408e12i	−0.477033	−	0.826245i	−0.999654	+	0.0263201i	$0.991621\pi$
$174$		0			0
$175$	−3.35602e12	+	2.39910e11i	−1.54567	+	0.110494i
$176$		0			0
$177$	1.66319e10	+	2.88072e10i	0.00719594	+	0.0124637i
$178$		0			0
$179$	−2.23898e12	+	3.87802e12i	−0.910663	+	1.57732i	−0.0975338	+	0.995232i	$0.531095\pi$
$179$	−2.23898e12	+	3.87802e12i	−0.910663	+	1.57732i	−0.813129	+	0.582083i	$0.802238\pi$
$180$		0			0
$181$	4.68844e12			1.79389			0.896945	−	0.442142i	$-0.145781\pi$
$181$	4.68844e12			1.79389			0.896945	+	0.442142i	$0.145781\pi$
$182$		0			0
$183$	−7.92206e11			−0.285337
$184$		0			0
$185$	−2.34647e12	+	4.06421e12i	−0.796105	+	1.37889i
$186$		0			0
$187$	−1.09561e12	−	1.89766e12i	−0.350371	−	0.606861i
$188$		0			0
$189$	3.57616e11	+	5.28417e11i	0.107864	+	0.159381i
$190$		0			0
$191$	1.93575e12	+	3.35282e12i	0.551019	+	0.954393i	0.998201	+	0.0599500i	$0.0190941\pi$
$191$	1.93575e12	+	3.35282e12i	0.551019	+	0.954393i	−0.447182	+	0.894443i	$0.647573\pi$
$192$		0			0
$193$	1.46950e12	−	2.54524e12i	0.395005	−	0.684169i	−0.598097	−	0.801424i	$-0.704076\pi$
$193$	1.46950e12	−	2.54524e12i	0.395005	−	0.684169i	0.993102	+	0.117255i	$0.0374094\pi$
$194$		0			0
$195$	6.18966e12			1.57209
$196$		0			0
$197$	−3.10385e12			−0.745309			−0.372654	−	0.927970i	$-0.621552\pi$
$197$	−3.10385e12			−0.745309			−0.372654	+	0.927970i	$0.621552\pi$
$198$		0			0
$199$	1.08230e12	−	1.87460e12i	0.245842	−	0.425810i	−0.716526	−	0.697560i	$-0.754269\pi$
$199$	1.08230e12	−	1.87460e12i	0.245842	−	0.425810i	0.962368	+	0.271750i	$0.0876023\pi$
$200$		0			0
$201$	−1.86218e12	−	3.22539e12i	−0.400353	−	0.693432i
$202$		0			0
$203$	4.11289e12	+	6.07725e12i	0.837373	+	1.23731i
$204$		0			0
$205$	−4.26670e12	−	7.39014e12i	−0.823087	−	1.42563i
$206$		0			0
$207$	−6.66893e11	+	1.15509e12i	−0.121960	+	0.211242i
$208$		0			0
$209$	−5.02955e12			−0.872418
$210$		0			0
$211$	2.54746e12			0.419328			0.209664	−	0.977773i	$-0.432763\pi$
$211$	2.54746e12			0.419328			0.209664	+	0.977773i	$0.432763\pi$
$212$		0			0
$213$	−1.87007e12	+	3.23906e12i	−0.292260	+	0.506210i
$214$		0			0
$215$	−1.32900e12	−	2.30190e12i	−0.197294	−	0.341724i
$216$		0			0
$217$	8.86534e12	−	6.33751e11i	1.25074	−	0.0894108i
$218$		0			0
$219$	3.00078e12	+	5.19751e12i	0.402524	+	0.697192i
$220$		0			0
$221$	6.37405e12	−	1.10402e13i	0.813314	−	1.40870i
$222$		0			0
$223$	−4.37566e12			−0.531333			−0.265667	−	0.964065i	$-0.585592\pi$
$223$	−4.37566e12			−0.531333			−0.265667	+	0.964065i	$0.585592\pi$
$224$		0			0
$225$	4.46792e12			0.516537
$226$		0			0
$227$	−6.86393e12	+	1.18887e13i	−0.755841	+	1.30915i	0.189114	+	0.981955i	$0.439438\pi$
$227$	−6.86393e12	+	1.18887e13i	−0.755841	+	1.30915i	−0.944955	+	0.327200i	$0.893895\pi$
$228$		0			0
$229$	5.79914e11	+	1.00444e12i	0.0608511	+	0.105397i	0.894846	−	0.446375i	$-0.147285\pi$
$229$	5.79914e11	+	1.00444e12i	0.0608511	+	0.105397i	−0.833995	+	0.551772i	$0.813952\pi$
$230$		0			0
$231$	1.85282e12	−	3.81386e12i	0.185339	−	0.381503i
$232$		0			0
$233$	−2.23260e12	−	3.86698e12i	−0.212987	−	0.368904i	0.739661	−	0.672980i	$-0.234986\pi$
$233$	−2.23260e12	−	3.86698e12i	−0.212987	−	0.368904i	−0.952648	+	0.304075i	$0.901653\pi$
$234$		0			0
$235$	1.04153e12	−	1.80399e12i	0.0947981	−	0.164195i
$236$		0			0
$237$	−5.14138e12			−0.446647
$238$		0			0
$239$	−2.16228e13			−1.79359			−0.896795	−	0.442447i	$-0.854111\pi$
$239$	−2.16228e13			−1.79359			−0.896795	+	0.442447i	$0.854111\pi$
$240$		0			0
$241$	−9.67406e12	+	1.67560e13i	−0.766505	+	1.32763i	0.172942	+	0.984932i	$0.444673\pi$
$241$	−9.67406e12	+	1.67560e13i	−0.766505	+	1.32763i	−0.939447	+	0.342694i	$0.888661\pi$
$242$		0			0
$243$	−4.23644e11	−	7.33773e11i	−0.0320750	−	0.0555556i
$244$		0			0
$245$	−2.18379e13	+	3.13827e12i	−1.58051	+	0.227131i
$246$		0			0
$247$	−1.46304e13	−	2.53407e13i	−1.01257	−	1.75382i
$248$		0			0
$249$	6.84007e12	−	1.18474e13i	0.452860	−	0.784377i
$250$		0			0
$251$	−1.43958e13			−0.912075			−0.456037	−	0.889961i	$-0.650732\pi$
$251$	−1.43958e13			−0.912075			−0.456037	+	0.889961i	$0.650732\pi$
$252$		0			0
$253$	8.86347e12			0.537577
$254$		0			0
$255$	7.57016e12	−	1.31119e13i	0.439678	−	0.761544i
$256$		0			0
$257$	−8.96043e12	−	1.55199e13i	−0.498536	−	0.863490i	0.501463	−	0.865179i	$-0.332795\pi$
$257$	−8.96043e12	−	1.55199e13i	−0.498536	−	0.863490i	−0.999999	+	0.00168967i	$0.999462\pi$
$258$		0			0
$259$	−8.17278e12	+	1.68229e13i	−0.435734	+	0.896916i
$260$		0			0
$261$	−4.87227e12	−	8.43902e12i	−0.249005	−	0.431290i
$262$		0			0
$263$	−3.34897e12	+	5.80059e12i	−0.164117	+	0.284260i	−0.936342	−	0.351091i	$-0.885811\pi$
$263$	−3.34897e12	+	5.80059e12i	−0.164117	+	0.284260i	0.772224	+	0.635350i	$0.219144\pi$
$264$		0			0
$265$	3.62514e13			1.70401
$266$		0			0
$267$	9.08993e12			0.409967
$268$		0			0
$269$	−1.73220e13	+	3.00027e13i	−0.749828	+	1.29874i	0.198077	+	0.980187i	$0.436531\pi$
$269$	−1.73220e13	+	3.00027e13i	−0.749828	+	1.29874i	−0.947905	+	0.318554i	$0.896803\pi$
$270$		0			0
$271$	−7.52159e12	−	1.30278e13i	−0.312593	−	0.541426i	0.666330	−	0.745657i	$-0.267864\pi$
$271$	−7.52159e12	−	1.30278e13i	−0.312593	−	0.541426i	−0.978923	+	0.204231i	$0.934531\pi$
$272$		0			0
$273$	2.46052e13	−	1.75894e12i	0.982048	−	0.0702030i
$274$		0			0
$275$	−1.48454e13	−	2.57131e13i	−0.569198	−	0.985880i
$276$		0			0
$277$	−2.61416e12	+	4.52785e12i	−0.0963148	+	0.166822i	−0.910157	−	0.414264i	$-0.864039\pi$
$277$	−2.61416e12	+	4.52785e12i	−0.0963148	+	0.166822i	0.813842	+	0.581087i	$0.197372\pi$
$278$		0			0
$279$	−1.18025e13			−0.417977
$280$		0			0
$281$	−1.40636e13			−0.478863			−0.239431	−	0.970913i	$-0.576961\pi$
$281$	−1.40636e13			−0.478863			−0.239431	+	0.970913i	$0.576961\pi$
$282$		0			0
$283$	9.20998e12	−	1.59522e13i	0.301601	−	0.522389i	−0.674897	−	0.737911i	$-0.735812\pi$
$283$	9.20998e12	−	1.59522e13i	0.301601	−	0.522389i	0.976499	+	0.215523i	$0.0691455\pi$
$284$		0			0
$285$	−1.73759e13	−	3.00959e13i	−0.547394	−	0.948115i
$286$		0			0
$287$	−1.90611e13	−	2.81649e13i	−0.577827	−	0.853803i
$288$		0			0
$289$	1.54459e12	+	2.67531e12i	0.0450687	+	0.0780613i
$290$		0			0
$291$	9.38063e12	−	1.62477e13i	0.263524	−	0.456437i
$292$		0			0
$293$	2.56171e13			0.693040			0.346520	−	0.938043i	$-0.387363\pi$
$293$	2.56171e13			0.693040			0.346520	+	0.938043i	$0.387363\pi$
$294$		0			0
$295$	1.52734e12			0.0398030
$296$		0			0
$297$	−2.81527e12	+	4.87618e12i	−0.0706901	+	0.122439i
$298$		0			0
$299$	2.57829e13	+	4.46573e13i	0.623937	+	1.08069i
$300$		0			0
$301$	−5.93721e12	−	8.77288e12i	−0.138505	−	0.204657i
$302$		0			0
$303$	1.52571e13	+	2.64260e13i	0.343192	+	0.594426i
$304$		0			0
$305$	−1.81875e13	+	3.15018e13i	−0.394571	+	0.683417i
$306$		0			0
$307$	4.44656e13			0.930600			0.465300	−	0.885153i	$-0.345946\pi$
$307$	4.44656e13			0.930600			0.465300	+	0.885153i	$0.345946\pi$
$308$		0			0
$309$	4.64620e13			0.938267
$310$		0			0
$311$	−4.91286e13	+	8.50933e13i	−0.957530	+	1.65849i	−0.229061	+	0.973412i	$0.573566\pi$
$311$	−4.91286e13	+	8.50933e13i	−0.957530	+	1.65849i	−0.728469	+	0.685079i	$0.759768\pi$
$312$		0			0
$313$	−1.19408e13	−	2.06821e13i	−0.224667	−	0.389135i	0.731552	−	0.681785i	$-0.238796\pi$
$313$	−1.19408e13	−	2.06821e13i	−0.224667	−	0.389135i	−0.956220	+	0.292650i	$0.905463\pi$
$314$		0			0
$315$	2.92225e13	−	2.08901e12i	0.530895	−	0.0379518i
$316$		0			0
$317$	4.97132e13	+	8.61057e13i	0.872259	+	1.51080i	0.859654	+	0.510877i	$0.170679\pi$
$317$	4.97132e13	+	8.61057e13i	0.872259	+	1.51080i	0.0126053	+	0.999921i	$0.495988\pi$
$318$		0			0
$319$	−3.23779e13	+	5.60802e13i	−0.548783	+	0.950520i
$320$		0			0
$321$	−1.38966e13			−0.227578
$322$		0			0
$323$	−7.15741e13			−1.13277
$324$		0			0
$325$	8.63677e13	−	1.49593e14i	1.32127	−	2.28851i
$326$		0			0
$327$	2.95533e13	+	5.11879e13i	0.437113	+	0.757102i
$328$		0			0
$329$	3.62767e12	−	7.46721e12i	0.0518860	−	0.106802i
$330$		0			0
$331$	5.27130e12	+	9.13017e12i	0.0729229	+	0.126306i	0.900181	−	0.435516i	$-0.143434\pi$
$331$	5.27130e12	+	9.13017e12i	0.0729229	+	0.126306i	−0.827258	+	0.561822i	$0.810101\pi$
$332$		0			0
$333$	1.24181e13	−	2.15088e13i	0.166193	−	0.287854i
$334$		0			0
$335$	−1.71008e14			−2.21447
$336$		0			0
$337$	1.57113e14			1.96900			0.984502	−	0.175375i	$-0.0561139\pi$
$337$	1.57113e14			1.96900			0.984502	+	0.175375i	$0.0561139\pi$
$338$		0			0
$339$	−5.06206e12	+	8.76775e12i	−0.0614086	+	0.106363i
$340$		0			0
$341$	3.92160e13	+	6.79241e13i	0.460590	+	0.797765i
$342$		0			0
$343$	−8.59186e13	+	1.86810e13i	−0.977169	+	0.212463i
$344$		0			0
$345$	3.06212e13	+	5.30374e13i	0.337300	+	0.584221i
$346$		0			0
$347$	−1.53483e10	+	2.65840e10i	−0.000163775	+	0.000283666i	−0.866107	−	0.499858i	$-0.833385\pi$
$347$	−1.53483e10	+	2.65840e10i	−0.000163775	+	0.000283666i	0.865944	+	0.500142i	$0.166719\pi$
$348$		0			0
$349$	1.00590e14			1.03995			0.519976	−	0.854181i	$-0.325941\pi$
$349$	1.00590e14			1.03995			0.519976	+	0.854181i	$0.325941\pi$
$350$		0			0
$351$	−3.27573e13			−0.328185
$352$		0			0
$353$	9.13814e13	−	1.58277e14i	0.887354	−	1.53694i	0.0443634	−	0.999015i	$-0.485874\pi$
$353$	9.13814e13	−	1.58277e14i	0.887354	−	1.53694i	0.842991	−	0.537928i	$-0.180793\pi$
$354$		0			0
$355$	8.58665e13	+	1.48725e14i	0.808291	+	1.40000i
$356$		0			0
$357$	2.63670e13	−	5.42739e13i	0.240649	−	0.495354i
$358$		0			0
$359$	3.57145e13	+	6.18593e13i	0.316100	+	0.547502i	0.979671	−	0.200612i	$-0.0642930\pi$
$359$	3.57145e13	+	6.18593e13i	0.316100	+	0.547502i	−0.663570	+	0.748114i	$0.730960\pi$
$360$		0			0
$361$	−2.38973e13	+	4.13913e13i	−0.205144	+	0.355319i
$362$		0			0
$363$	−3.19139e13			−0.265762
$364$		0			0
$365$	2.75569e14			2.22648
$366$		0			0
$367$	3.49074e13	−	6.04615e13i	0.273687	−	0.474040i	−0.696116	−	0.717930i	$-0.745090\pi$
$367$	3.49074e13	−	6.04615e13i	0.273687	−	0.474040i	0.969803	+	0.243889i	$0.0784233\pi$
$368$		0			0
$369$	2.25804e13	+	3.91105e13i	0.171825	+	0.297610i
$370$		0			0
$371$	1.44107e14	−	1.03017e13i	1.06446	−	0.0760942i
$372$		0			0
$373$	−8.26603e13	−	1.43172e14i	−0.592787	−	1.02674i	−0.993855	−	0.110689i	$-0.964694\pi$
$373$	−8.26603e13	−	1.43172e14i	−0.592787	−	1.02674i	0.401068	−	0.916048i	$-0.368639\pi$
$374$		0			0
$375$	3.63809e13	−	6.30136e13i	0.253339	−	0.438796i
$376$		0			0
$377$	−3.76736e14			−2.54777
$378$		0			0
$379$	3.93971e13			0.258791			0.129395	−	0.991593i	$-0.458696\pi$
$379$	3.93971e13			0.258791			0.129395	+	0.991593i	$0.458696\pi$
$380$		0			0
$381$	4.51223e12	−	7.81540e12i	0.0287941	−	0.0498728i
$382$		0			0
$383$	−3.88422e13	−	6.72766e13i	−0.240830	−	0.417130i	0.720121	−	0.693848i	$-0.244086\pi$
$383$	−3.88422e13	−	6.72766e13i	−0.240830	−	0.417130i	−0.960951	+	0.276719i	$0.910753\pi$
$384$		0			0
$385$	−1.09119e14	−	1.61235e14i	−0.657456	−	0.971464i
$386$		0			0
$387$	7.03342e12	+	1.21822e13i	0.0411867	+	0.0713374i
$388$		0			0
$389$	5.89738e13	−	1.02146e14i	0.335688	−	0.581429i	−0.647928	−	0.761701i	$-0.724364\pi$
$389$	5.89738e13	−	1.02146e14i	0.335688	−	0.581429i	0.983617	+	0.180272i	$0.0576977\pi$
$390$		0			0
$391$	1.26133e14			0.698004
$392$		0			0
$393$	9.66649e13			0.520127
$394$		0			0
$395$	−1.18036e14	+	2.04445e14i	−0.617635	+	1.06977i
$396$		0			0
$397$	1.37279e14	+	2.37774e14i	0.698643	+	1.21009i	0.968937	+	0.247308i	$0.0795459\pi$
$397$	1.37279e14	+	2.37774e14i	0.698643	+	1.21009i	−0.270294	+	0.962778i	$0.587121\pi$
$398$		0			0
$399$	−7.76253e13	−	1.14700e14i	−0.384284	−	0.567822i
$400$		0			0
$401$	9.60090e13	+	1.66292e14i	0.462400	+	0.800900i	0.999080	−	0.0428855i	$-0.0136551\pi$
$401$	9.60090e13	+	1.66292e14i	0.462400	+	0.800900i	−0.536680	+	0.843786i	$0.680322\pi$
$402$		0			0
$403$	−2.28151e14	+	3.95168e14i	−1.06916	+	1.85184i
$404$		0			0
$405$	−3.89043e13			−0.177417
$406$		0			0
$407$	−1.65045e14			−0.732545
$408$		0			0
$409$	−1.84891e14	+	3.20240e14i	−0.798798	+	1.38356i	0.121602	+	0.992579i	$0.461197\pi$
$409$	−1.84891e14	+	3.20240e14i	−0.798798	+	1.38356i	−0.920399	+	0.390979i	$0.872136\pi$
$410$		0			0
$411$	−1.01754e14	−	1.76244e14i	−0.427979	−	0.741282i
$412$		0			0
$413$	6.07151e12	−	4.34030e11i	0.0248640	−	0.00177744i
$414$		0			0
$415$	−3.14070e14	−	5.43985e14i	−1.25245	−	2.16931i
$416$		0			0
$417$	8.12618e13	−	1.40750e14i	0.315601	−	0.546637i
$418$		0			0
$419$	3.40515e13			0.128813			0.0644064	−	0.997924i	$-0.479485\pi$
$419$	3.40515e13			0.128813			0.0644064	+	0.997924i	$0.479485\pi$
$420$		0			0
$421$	−2.32682e14			−0.857456			−0.428728	−	0.903434i	$-0.641038\pi$
$421$	−2.32682e14			−0.857456			−0.428728	+	0.903434i	$0.641038\pi$
$422$		0			0
$423$	−5.51205e12	+	9.54716e12i	−0.0197898	+	0.0342769i
$424$		0			0
$425$	−2.11261e14	−	3.65915e14i	−0.739062	−	1.28009i
$426$		0			0
$427$	−6.33474e13	+	1.30395e14i	−0.215961	+	0.444535i
$428$		0			0
$429$	1.08842e14	+	1.88519e14i	0.361643	+	0.626384i
$430$		0			0
$431$	−2.06089e14	+	3.56956e14i	−0.667467	+	1.15609i	0.311144	+	0.950363i	$0.399288\pi$
$431$	−2.06089e14	+	3.56956e14i	−0.667467	+	1.15609i	−0.978610	+	0.205723i	$0.934045\pi$
$432$		0			0
$433$	3.16336e14			0.998770			0.499385	−	0.866380i	$-0.333559\pi$
$433$	3.16336e14			0.998770			0.499385	+	0.866380i	$0.333559\pi$
$434$		0			0
$435$	−4.47432e14			−1.37732
$436$		0			0
$437$	1.44758e14	−	2.50728e14i	0.434504	−	0.752584i
$438$		0			0
$439$	2.45107e14	+	4.24537e14i	0.717464	+	1.24268i	0.962002	+	0.273044i	$0.0880305\pi$
$439$	2.45107e14	+	4.24537e14i	0.717464	+	1.24268i	−0.244538	+	0.969640i	$0.578636\pi$
$440$		0			0
$441$	1.15572e14	−	1.66085e13i	0.329944	−	0.0474153i
$442$		0			0
$443$	−2.50000e14	−	4.33012e14i	−0.696176	−	1.20581i	−0.969783	−	0.243971i	$-0.921550\pi$
$443$	−2.50000e14	−	4.33012e14i	−0.696176	−	1.20581i	0.273606	−	0.961842i	$-0.411783\pi$
$444$		0			0
$445$	2.08687e14	−	3.61457e14i	0.566912	−	0.981921i
$446$		0			0
$447$	−1.64215e14			−0.435232
$448$		0			0
$449$	1.51607e13			0.0392071			0.0196036	−	0.999808i	$-0.493760\pi$
$449$	1.51607e13			0.0392071			0.0196036	+	0.999808i	$0.493760\pi$
$450$		0			0
$451$	1.50055e14	−	2.59903e14i	0.378686	−	0.655904i
$452$		0			0
$453$	5.32796e13	+	9.22829e13i	0.131226	+	0.227291i
$454$		0			0
$455$	4.94945e14	−	1.01880e15i	1.18986	−	2.44921i
$456$		0			0
$457$	−3.11245e14	−	5.39093e14i	−0.730405	−	1.26510i	−0.956710	−	0.291042i	$-0.905998\pi$
$457$	−3.11245e14	−	5.39093e14i	−0.730405	−	1.26510i	0.226306	−	0.974056i	$-0.427335\pi$
$458$		0			0
$459$	−4.00632e13	+	6.93915e13i	−0.0917859	+	0.158978i
$460$		0			0
$461$	8.14992e14			1.82305			0.911524	−	0.411246i	$-0.134906\pi$
$461$	8.14992e14			1.82305			0.911524	+	0.411246i	$0.134906\pi$
$462$		0			0
$463$	7.98210e13			0.174350			0.0871749	−	0.996193i	$-0.472216\pi$
$463$	7.98210e13			0.174350			0.0871749	+	0.996193i	$0.472216\pi$
$464$		0			0
$465$	−2.70964e14	+	4.69323e14i	−0.577990	+	1.00111i
$466$		0			0
$467$	2.26295e14	+	3.91955e14i	0.471447	+	0.816570i	0.999466	−	0.0326623i	$-0.0103986\pi$
$467$	2.26295e14	+	3.91955e14i	0.471447	+	0.816570i	−0.528020	+	0.849232i	$0.677065\pi$
$468$		0			0
$469$	−6.79794e14	+	4.85960e13i	−1.38333	+	0.0988895i
$470$		0			0
$471$	3.59540e13	+	6.22742e13i	0.0714714	+	0.123792i
$472$		0			0
$473$	4.67396e13	−	8.09553e13i	0.0907713	−	0.157220i
$474$		0			0
$475$	−9.69821e14			−1.84025
$476$		0			0
$477$	−1.91852e14			−0.355725
$478$		0			0
$479$	−3.51335e14	+	6.08530e14i	−0.636614	+	1.10265i	0.349557	+	0.936915i	$0.386332\pi$
$479$	−3.51335e14	+	6.08530e14i	−0.636614	+	1.10265i	−0.986171	+	0.165732i	$0.947001\pi$
$480$		0			0
$481$	−4.80100e14	−	8.31558e14i	−0.850225	−	1.47263i
$482$		0			0
$483$	1.36797e14	+	2.02133e14i	0.236793	+	0.349888i
$484$		0			0
$485$	−4.30723e14	−	7.46033e14i	−0.728816	−	1.26235i
$486$		0			0
$487$	−2.97188e14	+	5.14744e14i	−0.491611	+	0.851495i	−0.999953	−	0.00965994i	$-0.996925\pi$
$487$	−2.97188e14	+	5.14744e14i	−0.491611	+	0.851495i	0.508342	+	0.861155i	$0.330258\pi$
$488$		0			0
$489$	−5.44723e14			−0.881002
$490$		0			0
$491$	7.12724e14			1.12713			0.563564	−	0.826073i	$-0.309430\pi$
$491$	7.12724e14			1.12713			0.563564	+	0.826073i	$0.309430\pi$
$492$		0			0
$493$	−4.60761e14	+	7.98062e14i	−0.712554	+	1.23418i
$494$		0			0
$495$	1.29266e14	+	2.23896e14i	0.195504	+	0.338623i
$496$		0			0
$497$	3.83601e14	+	5.66813e14i	0.567440	+	0.838455i
$498$		0			0
$499$	−4.67371e14	−	8.09510e14i	−0.676252	−	1.17130i	−0.976101	−	0.217316i	$-0.930270\pi$
$499$	−4.67371e14	−	8.09510e14i	−0.676252	−	1.17130i	0.299849	−	0.953987i	$-0.403064\pi$
$500$		0			0
$501$	3.19154e13	−	5.52791e13i	0.0451745	−	0.0782445i
$502$		0			0
$503$	−7.21102e14			−0.998556			−0.499278	−	0.866442i	$-0.666401\pi$
$503$	−7.21102e14			−0.998556			−0.499278	+	0.866442i	$0.666401\pi$
$504$		0			0
$505$	1.40109e15			1.89830
$506$		0			0
$507$	−4.15471e14	+	7.19617e14i	−0.550804	+	0.954020i
$508$		0			0
$509$	−4.57952e13	−	7.93196e13i	−0.0594117	−	0.102904i	0.834790	−	0.550569i	$-0.185589\pi$
$509$	−4.57952e13	−	7.93196e13i	−0.0594117	−	0.102904i	−0.894201	+	0.447665i	$0.852256\pi$
$510$		0			0
$511$	1.09544e15	−	7.83094e13i	1.39083	−	0.0994257i
$512$		0			0
$513$	9.19576e13	+	1.59275e14i	0.114273	+	0.197926i
$514$		0			0
$515$	1.06668e15	−	1.84754e15i	1.29746	−	2.24727i
$516$		0			0
$517$	7.32591e13			0.0872295
$518$		0			0
$519$	4.72540e14			0.550830
$520$		0			0
$521$	−7.21411e14	+	1.24952e15i	−0.823332	+	1.42605i	0.0798550	+	0.996806i	$0.474554\pi$
$521$	−7.21411e14	+	1.24952e15i	−0.823332	+	1.42605i	−0.903187	+	0.429247i	$0.858779\pi$
$522$		0			0
$523$	6.15443e14	+	1.06598e15i	0.687746	+	1.19121i	0.972565	+	0.232630i	$0.0747332\pi$
$523$	6.15443e14	+	1.06598e15i	0.687746	+	1.19121i	−0.284819	+	0.958581i	$0.591934\pi$
$524$		0			0
$525$	3.57269e14	−	7.35405e14i	0.390949	−	0.804730i
$526$		0			0
$527$	5.58072e14	+	9.66608e14i	0.598042	+	1.03584i
$528$		0			0
$529$	2.21301e14	−	3.83305e14i	0.232262	−	0.402289i
$530$		0			0
$531$	−8.08308e12			−0.00830916
$532$		0			0
$533$	1.74598e15			1.75808
$534$		0			0
$535$	−3.19039e14	+	5.52592e14i	−0.314701	+	0.545078i
$536$		0			0
$537$	−5.44071e14	−	9.42359e14i	−0.525772	−	0.910663i
$538$		0			0
$539$	−4.79591e14	−	6.09937e14i	−0.454080	−	0.577493i
$540$		0			0
$541$	2.37856e14	+	4.11979e14i	0.220663	+	0.382200i	0.955010	−	0.296575i	$-0.0958445\pi$
$541$	2.37856e14	+	4.11979e14i	0.220663	+	0.382200i	−0.734346	+	0.678775i	$0.762511\pi$
$542$		0			0
$543$	−5.69645e14	+	9.86654e14i	−0.517852	+	0.896945i
$544$		0			0
$545$	2.71395e15			2.41781
$546$		0			0
$547$	−2.16232e14			−0.188795			−0.0943975	−	0.995535i	$-0.530092\pi$
$547$	−2.16232e14			−0.188795			−0.0943975	+	0.995535i	$0.530092\pi$
$548$		0			0
$549$	9.62531e13	−	1.66715e14i	0.0823696	−	0.142668i
$550$		0			0
$551$	1.05759e15	+	1.83180e15i	0.887123	+	1.53654i
$552$		0			0
$553$	−4.11122e14	+	8.46255e14i	−0.338051	+	0.695846i
$554$		0			0
$555$	−5.70192e14	−	9.87602e14i	−0.459632	−	0.796105i
$556$		0			0
$557$	−1.41948e14	+	2.45861e14i	−0.112183	+	0.194306i	−0.916650	−	0.399691i	$-0.869118\pi$
$557$	−1.41948e14	+	2.45861e14i	−0.112183	+	0.194306i	0.804467	+	0.593997i	$0.202451\pi$
$558$		0			0
$559$	5.43842e14			0.421413
$560$		0			0
$561$	5.32468e14			0.404574
$562$		0			0
$563$	1.12143e15	−	1.94238e15i	0.835559	−	1.44723i	−0.0580164	−	0.998316i	$-0.518478\pi$
$563$	1.12143e15	−	1.94238e15i	0.835559	−	1.44723i	0.893575	−	0.448914i	$-0.148189\pi$
$564$		0			0
$565$	2.32430e14	+	4.02581e14i	0.169835	+	0.294162i
$566$		0			0
$567$	−1.54653e14	+	1.10556e13i	−0.110828	+	0.00792271i
$568$		0			0
$569$	−3.11079e14	−	5.38804e14i	−0.218652	−	0.378716i	0.735744	−	0.677259i	$-0.236833\pi$
$569$	−3.11079e14	−	5.38804e14i	−0.218652	−	0.378716i	−0.954396	+	0.298544i	$0.903499\pi$
$570$		0			0
$571$	−1.28739e15	+	2.22982e15i	−0.887585	+	1.53734i	−0.0448633	+	0.998993i	$0.514285\pi$
$571$	−1.28739e15	+	2.22982e15i	−0.887585	+	1.53734i	−0.842722	+	0.538349i	$0.819048\pi$
$572$		0			0
$573$	−9.40776e14			−0.636262
$574$		0			0
$575$	1.70909e15			1.13395
$576$		0			0
$577$	−1.24489e15	+	2.15621e15i	−0.810332	+	1.40354i	0.102300	+	0.994754i	$0.467380\pi$
$577$	−1.24489e15	+	2.15621e15i	−0.810332	+	1.40354i	−0.912632	+	0.408783i	$0.865953\pi$
$578$		0			0
$579$	3.57087e14	+	6.18493e14i	0.228056	+	0.395005i
$580$		0			0
$581$	−1.40308e15	−	2.07321e15i	−0.879253	−	1.29919i
$582$		0			0
$583$	6.37460e14	+	1.10411e15i	0.391991	+	0.678948i
$584$		0			0
$585$	−7.52044e14	+	1.30258e15i	−0.453822	+	0.786043i
$586$		0			0
$587$	−2.39872e15			−1.42059			−0.710297	−	0.703902i	$-0.751440\pi$
$587$	−2.39872e15			−1.42059			−0.710297	+	0.703902i	$0.751440\pi$
$588$		0			0
$589$	2.56190e15			1.48911
$590$		0			0
$591$	3.77117e14	−	6.53187e14i	0.215152	−	0.372654i
$592$		0			0
$593$	4.74884e13	+	8.22524e13i	0.0265942	+	0.0460625i	0.879016	−	0.476792i	$-0.158200\pi$
$593$	4.74884e13	+	8.22524e13i	0.0265942	+	0.0460625i	−0.852422	+	0.522854i	$0.824867\pi$
$594$		0			0
$595$	−1.55284e15	−	2.29449e15i	−0.853658	−	1.26137i
$596$		0			0
$597$	2.62999e14	+	4.55527e14i	0.141937	+	0.245842i
$598$		0			0
$599$	8.09261e14	−	1.40168e15i	0.428787	−	0.742680i	−0.567979	−	0.823043i	$-0.692275\pi$
$599$	8.09261e14	−	1.40168e15i	0.428787	−	0.742680i	0.996766	+	0.0803629i	$0.0256079\pi$
$600$		0			0
$601$	−6.48361e14			−0.337293			−0.168646	−	0.985677i	$-0.553940\pi$
$601$	−6.48361e14			−0.337293			−0.168646	+	0.985677i	$0.553940\pi$
$602$		0			0
$603$	9.05019e14			0.462288
$604$		0			0
$605$	−7.32682e14	+	1.26904e15i	−0.367503	+	0.636534i
$606$		0			0
$607$	−8.47944e14	−	1.46868e15i	−0.417666	−	0.723419i	0.578038	−	0.816010i	$-0.303819\pi$
$607$	−8.47944e14	−	1.46868e15i	−0.417666	−	0.723419i	−0.995704	+	0.0925906i	$0.970485\pi$
$608$		0			0
$609$	−1.77864e15	+	1.27148e14i	−0.860384	+	0.0615057i
$610$		0			0
$611$	2.13103e14	+	3.69105e14i	0.101243	+	0.175357i
$612$		0			0
$613$	−6.00792e14	+	1.04060e15i	−0.280344	+	0.485570i	−0.971469	−	0.237165i	$-0.923782\pi$
$613$	−6.00792e14	+	1.04060e15i	−0.280344	+	0.485570i	0.691125	+	0.722735i	$0.257115\pi$
$614$		0			0
$615$	2.07362e15			0.950419
$616$		0			0
$617$	2.76415e15			1.24450			0.622248	−	0.782820i	$-0.286219\pi$
$617$	2.76415e15			1.24450			0.622248	+	0.782820i	$0.286219\pi$
$618$		0			0
$619$	1.21861e15	−	2.11070e15i	0.538972	−	0.933528i	−0.459987	−	0.887926i	$-0.652146\pi$
$619$	1.21861e15	−	2.11070e15i	0.538972	−	0.933528i	0.998960	−	0.0456021i	$-0.0145206\pi$
$620$		0			0
$621$	−1.62055e14	−	2.80687e14i	−0.0704139	−	0.121960i
$622$		0			0
$623$	7.26860e14	−	1.49617e15i	0.310289	−	0.638700i
$624$		0			0
$625$	1.76807e14	+	3.06239e14i	0.0741582	+	0.128446i
$626$		0			0
$627$	6.11090e14	−	1.05844e15i	0.251845	−	0.436209i
$628$		0			0
$629$	−2.34871e15			−0.951155
$630$		0			0
$631$	1.10045e15			0.437935			0.218968	−	0.975732i	$-0.429731\pi$
$631$	1.10045e15			0.437935			0.218968	+	0.975732i	$0.429731\pi$
$632$		0			0
$633$	−3.09517e14	+	5.36098e14i	−0.121050	+	0.209664i
$634$		0			0
$635$	−2.07184e14	−	3.58853e14i	−0.0796345	−	0.137931i
$636$		0			0
$637$	1.67800e15	−	4.19059e15i	0.633906	−	1.58310i
$638$		0			0
$639$	−4.54427e14	−	7.87091e14i	−0.168737	−	0.292260i
$640$		0			0
$641$	−1.47446e15	+	2.55384e15i	−0.538163	+	0.932126i	0.460840	+	0.887483i	$0.347548\pi$
$641$	−1.47446e15	+	2.55384e15i	−0.538163	+	0.932126i	−0.999003	+	0.0446427i	$0.985785\pi$
$642$		0			0
$643$	−2.86149e15			−1.02667			−0.513337	−	0.858187i	$-0.671591\pi$
$643$	−2.86149e15			−1.02667			−0.513337	+	0.858187i	$0.671591\pi$
$644$		0			0
$645$	6.45896e14			0.227816
$646$		0			0
$647$	−5.03509e14	+	8.72103e14i	−0.174596	+	0.302409i	−0.940021	−	0.341116i	$-0.889195\pi$
$647$	−5.03509e14	+	8.72103e14i	−0.174596	+	0.302409i	0.765426	+	0.643524i	$0.222529\pi$
$648$		0			0
$649$	2.68575e13	+	4.65185e13i	0.00915628	+	0.0158591i
$650$		0			0
$651$	−9.43770e14	+	1.94266e15i	−0.316352	+	0.651180i
$652$		0			0
$653$	−1.08161e15	−	1.87340e15i	−0.356489	−	0.617458i	0.630882	−	0.775878i	$-0.282693\pi$
$653$	−1.08161e15	−	1.87340e15i	−0.356489	−	0.617458i	−0.987372	+	0.158421i	$0.949360\pi$
$654$		0			0
$655$	2.21924e15	−	3.84384e15i	0.719246	−	1.24577i
$656$		0			0
$657$	−1.45838e15			−0.464795
$658$		0			0
$659$	2.09934e15			0.657981			0.328991	−	0.944333i	$-0.393292\pi$
$659$	2.09934e15			0.657981			0.328991	+	0.944333i	$0.393292\pi$
$660$		0			0
$661$	−1.85165e15	+	3.20716e15i	−0.570758	+	0.988581i	0.425731	+	0.904850i	$0.360017\pi$
$661$	−1.85165e15	+	3.20716e15i	−0.570758	+	0.988581i	−0.996488	+	0.0837314i	$0.973316\pi$
$662$		0			0
$663$	1.54889e15	+	2.68276e15i	0.469567	+	0.813314i
$664$		0			0
$665$	−6.34312e15	+	4.53447e14i	−1.89140	+	0.135209i
$666$		0			0
$667$	−1.86377e15	−	3.22814e15i	−0.546638	−	0.946805i
$668$		0			0
$669$	5.31643e14	−	9.20833e14i	0.153383	−	0.265667i
$670$		0			0
$671$	−1.27927e15			−0.363069
$672$		0			0
$673$	1.46499e15			0.409027			0.204513	−	0.978864i	$-0.434439\pi$
$673$	1.46499e15			0.409027			0.204513	+	0.978864i	$0.434439\pi$
$674$		0			0
$675$	−5.42852e14	+	9.40248e14i	−0.149111	+	0.258269i
$676$		0			0
$677$	9.92626e14	+	1.71928e15i	0.268255	+	0.464631i	0.968411	−	0.249358i	$-0.0802197\pi$
$677$	9.92626e14	+	1.71928e15i	0.268255	+	0.464631i	−0.700156	+	0.713990i	$0.746886\pi$
$678$		0			0
$679$	−1.92422e15	−	2.84324e15i	−0.511647	−	0.756014i
$680$		0			0
$681$	−1.66793e15	−	2.88895e15i	−0.436385	−	0.755841i
$682$		0			0
$683$	−2.26798e15	+	3.92825e15i	−0.583882	+	1.01131i	0.411132	+	0.911576i	$0.365134\pi$
$683$	−2.26798e15	+	3.92825e15i	−0.583882	+	1.01131i	−0.995014	+	0.0997375i	$0.968200\pi$
$684$		0			0
$685$	−9.34433e15			−2.36728
$686$		0			0
$687$	−2.81838e14			−0.0702648
$688$		0			0
$689$	−3.70861e15	+	6.42350e15i	−0.909925	+	1.57604i
$690$		0			0
$691$	1.08860e15	+	1.88552e15i	0.262869	+	0.455303i	0.967003	−	0.254764i	$-0.0819978\pi$
$691$	1.08860e15	+	1.88552e15i	0.262869	+	0.455303i	−0.704134	+	0.710067i	$0.748664\pi$
$692$		0			0
$693$	5.77486e14	+	8.53299e14i	0.137249	+	0.202800i
$694$		0			0
$695$	−3.73123e15	−	6.46268e15i	−0.872842	−	1.51181i
$696$		0			0
$697$	2.13539e15	−	3.69860e15i	0.491696	−	0.851642i
$698$		0			0
$699$	1.08504e15			0.245936
$700$		0			0
$701$	1.04536e15			0.233247			0.116623	−	0.993176i	$-0.462793\pi$
$701$	1.04536e15			0.233247			0.116623	+	0.993176i	$0.462793\pi$
$702$		0			0
$703$	−2.69551e15	+	4.66877e15i	−0.592090	+	1.02553i
$704$		0			0
$705$	2.53092e14	+	4.38369e14i	0.0547317	+	0.0947981i
$706$		0			0
$707$	5.56964e15	−	3.98154e14i	1.18583	−	0.0847703i
$708$		0			0
$709$	1.92041e15	+	3.32625e15i	0.402569	+	0.697270i	0.994035	−	0.109060i	$-0.0347841\pi$
$709$	1.92041e15	+	3.32625e15i	0.402569	+	0.697270i	−0.591466	+	0.806330i	$0.701451\pi$
$710$		0			0
$711$	6.24678e14	−	1.08197e15i	0.128936	−	0.223323i
$712$		0			0
$713$	−4.51478e15			−0.917580
$714$		0			0
$715$	9.99519e15			2.00036
$716$		0			0
$717$	2.62717e15	−	4.55039e15i	0.517765	−	0.896795i
$718$		0			0
$719$	−1.91066e15	−	3.30937e15i	−0.370830	−	0.642297i	0.618863	−	0.785499i	$-0.287594\pi$
$719$	−1.91066e15	−	3.30937e15i	−0.370830	−	0.642297i	−0.989694	+	0.143202i	$0.954260\pi$
$720$		0			0
$721$	3.71525e15	−	7.64750e15i	0.710141	−	1.46176i
$722$		0			0
$723$	−2.35080e15	−	4.07170e15i	−0.442542	−	0.766505i
$724$		0			0
$725$	−6.24326e15	+	1.08136e16i	−1.15758	+	2.00499i
$726$		0			0
$727$	1.10567e14			0.0201923			0.0100962	−	0.999949i	$-0.496786\pi$
$727$	1.10567e14			0.0201923			0.0100962	+	0.999949i	$0.496786\pi$
$728$		0			0
$729$	2.05891e14			0.0370370
$730$		0			0
$731$	6.65137e14	−	1.15205e15i	0.117860	−	0.204139i
$732$		0			0
$733$	−2.31452e15	−	4.00887e15i	−0.404007	−	0.699761i	0.590198	−	0.807258i	$-0.299050\pi$
$733$	−2.31452e15	−	4.00887e15i	−0.404007	−	0.699761i	−0.994205	+	0.107497i	$0.965716\pi$
$734$		0			0
$735$	1.99288e15	−	4.97697e15i	0.342689	−	0.855824i
$736$		0			0
$737$	−3.00708e15	−	5.20842e15i	−0.509418	−	0.882338i
$738$		0			0
$739$	3.10450e15	−	5.37716e15i	0.518141	−	0.897446i	−0.481637	−	0.876371i	$-0.659958\pi$
$739$	3.10450e15	−	5.37716e15i	0.518141	−	0.897446i	0.999778	−	0.0210753i	$-0.00670898\pi$
$740$		0			0
$741$	7.11039e15			1.16921
$742$		0			0
$743$	−3.31622e15			−0.537286			−0.268643	−	0.963240i	$-0.586575\pi$
$743$	−3.31622e15			−0.537286			−0.268643	+	0.963240i	$0.586575\pi$
$744$		0			0
$745$	−3.77005e15	+	6.52992e15i	−0.601850	+	1.04243i
$746$		0			0
$747$	1.66214e15	+	2.87891e15i	0.261459	+	0.452860i
$748$		0			0
$749$	−1.11122e15	+	2.28733e15i	−0.172246	+	0.354551i
$750$		0			0
$751$	3.72462e15	+	6.45123e15i	0.568934	+	0.985423i	0.996672	+	0.0815203i	$0.0259775\pi$
$751$	3.72462e15	+	6.45123e15i	0.568934	+	0.985423i	−0.427737	+	0.903903i	$0.640689\pi$
$752$		0			0
$753$	1.74909e15	−	3.02951e15i	0.263293	−	0.456037i
$754$		0			0
$755$	4.89279e15			0.725852
$756$		0			0
$757$	−9.51588e15			−1.39130			−0.695651	−	0.718380i	$-0.744884\pi$
$757$	−9.51588e15			−1.39130			−0.695651	+	0.718380i	$0.744884\pi$
$758$		0			0
$759$	−1.07691e15	+	1.86527e15i	−0.155185	+	0.268789i
$760$		0			0
$761$	8.70949e14	+	1.50853e15i	0.123702	+	0.214258i	0.921225	−	0.389031i	$-0.127190\pi$
$761$	8.70949e14	+	1.50853e15i	0.123702	+	0.214258i	−0.797523	+	0.603289i	$0.793857\pi$
$762$		0			0
$763$	1.07885e16	−	7.71233e14i	1.51035	−	0.107969i
$764$		0			0
$765$	1.83955e15	+	3.18619e15i	0.253848	+	0.439678i
$766$		0			0
$767$	−1.56251e14	+	2.70635e14i	−0.0212544	+	0.0368137i
$768$		0			0
$769$	8.11684e15			1.08841			0.544205	−	0.838953i	$-0.316832\pi$
$769$	8.11684e15			1.08841			0.544205	+	0.838953i	$0.316832\pi$
$770$		0			0
$771$	4.35477e15			0.575660
$772$		0			0
$773$	3.78929e15	−	6.56324e15i	0.493822	−	0.855325i	−0.506152	−	0.862444i	$-0.668933\pi$
$773$	3.78929e15	−	6.56324e15i	0.493822	−	0.855325i	0.999975	+	0.00711869i	$0.00226597\pi$
$774$		0			0
$775$	7.56181e15	+	1.30974e16i	0.971553	+	1.68278i
$776$		0			0
$777$	−2.54729e15	−	3.76389e15i	−0.322673	−	0.476784i
$778$		0			0
$779$	−4.90138e15	−	8.48944e15i	−0.612157	−	1.06029i
$780$		0			0
$781$	−3.01983e15	+	5.23049e15i	−0.371879	+	0.644113i
$782$		0			0
$783$	2.36792e15			0.287527
$784$		0			0
$785$	3.30174e15			0.395330
$786$		0			0
$787$	−6.45454e15	+	1.11796e16i	−0.762086	+	1.31997i	0.179687	+	0.983724i	$0.442492\pi$
$787$	−6.45454e15	+	1.11796e16i	−0.762086	+	1.31997i	−0.941773	+	0.336248i	$0.890842\pi$
$788$		0			0
$789$	−8.13800e14	−	1.40954e15i	−0.0947533	−	0.164117i
$790$		0			0
$791$	1.03836e15	+	1.53430e15i	0.119228	+	0.176173i
$792$		0			0
$793$	−3.72127e15	−	6.44542e15i	−0.421394	−	0.729876i
$794$		0			0
$795$	−4.40454e15	+	7.62889e15i	−0.491906	+	0.852006i
$796$		0			0
$797$	−1.34695e14			−0.0148365			−0.00741823	−	0.999972i	$-0.502361\pi$
$797$	−1.34695e14			−0.0148365			−0.00741823	+	0.999972i	$0.502361\pi$
$798$		0			0
$799$	1.04253e15			0.113261
$800$		0			0
$801$	−1.10443e15	+	1.91292e15i	−0.118347	+	0.204983i
$802$		0			0
$803$	4.84572e15	+	8.39304e15i	0.512181	+	0.887123i
$804$		0			0
$805$	1.11784e16	−	7.99100e14i	1.16547	−	0.0833151i
$806$		0			0
$807$	−4.20926e15	−	7.29065e15i	−0.432913	−	0.749828i
$808$		0			0
$809$	−4.02674e15	+	6.97452e15i	−0.408542	+	0.707616i	−0.994727	−	0.102562i	$-0.967296\pi$
$809$	−4.02674e15	+	6.97452e15i	−0.408542	+	0.707616i	0.586184	+	0.810178i	$0.300629\pi$
$810$		0			0
$811$	1.14139e16			1.14240			0.571201	−	0.820810i	$-0.306478\pi$
$811$	1.14139e16			1.14240			0.571201	+	0.820810i	$0.306478\pi$
$812$		0			0
$813$	3.65549e15			0.360951
$814$		0			0
$815$	−1.25058e16	+	2.16607e16i	−1.21827	+	2.11011i
$816$		0			0
$817$	−1.52670e15	−	2.64432e15i	−0.146734	−	0.254151i
$818$		0			0
$819$	−2.61938e15	+	5.39174e15i	−0.248391	+	0.511290i
$820$		0			0
$821$	−1.00153e15	−	1.73471e15i	−0.0937083	−	0.162308i	0.815360	−	0.578954i	$-0.196539\pi$
$821$	−1.00153e15	−	1.73471e15i	−0.0937083	−	0.162308i	−0.909069	+	0.416646i	$0.863205\pi$
$822$		0			0
$823$	2.61011e15	−	4.52084e15i	0.240968	−	0.417369i	−0.720022	−	0.693951i	$-0.755868\pi$
$823$	2.61011e15	−	4.52084e15i	0.240968	−	0.417369i	0.960990	+	0.276582i	$0.0892018\pi$
$824$		0			0
$825$	7.21489e15			0.657253
$826$		0			0
$827$	1.05282e16			0.946395			0.473198	−	0.880956i	$-0.343100\pi$
$827$	1.05282e16			0.946395			0.473198	+	0.880956i	$0.343100\pi$
$828$		0			0
$829$	−4.64529e14	+	8.04589e14i	−0.0412063	+	0.0713714i	−0.885893	−	0.463890i	$-0.846453\pi$
$829$	−4.64529e14	+	8.04589e14i	−0.0412063	+	0.0713714i	0.844687	+	0.535261i	$0.179787\pi$
$830$		0			0
$831$	−6.35240e14	−	1.10027e15i	−0.0556074	−	0.0963148i
$832$		0			0
$833$	−6.82491e15	−	8.67983e15i	−0.589589	−	0.749831i
$834$		0			0
$835$	−1.46543e15	−	2.53821e15i	−0.124937	−	0.216397i
$836$		0			0
$837$	1.43401e15	−	2.48378e15i	0.120660	−	0.208989i
$838$		0			0
$839$	−5.65480e15			−0.469598			−0.234799	−	0.972044i	$-0.575443\pi$
$839$	−5.65480e15			−0.469598			−0.234799	+	0.972044i	$0.575443\pi$
$840$		0			0
$841$	1.50326e16			1.23213
$842$		0			0
$843$	1.70872e15	−	2.95960e15i	0.138236	−	0.239431i
$844$		0			0
$845$	1.90768e16	+	3.30421e16i	1.52333	+	2.63849i
$846$		0			0
$847$	−2.55194e15	+	5.25292e15i	−0.201146	+	0.414040i
$848$		0			0
$849$	2.23803e15	+	3.87637e15i	0.174130	+	0.301601i
$850$		0			0
$851$	4.75025e15	−	8.22767e15i	0.364841	−	0.631924i
$852$		0			0
$853$	−1.37095e16			−1.03945			−0.519725	−	0.854334i	$-0.673965\pi$
$853$	−1.37095e16			−1.03945			−0.519725	+	0.854334i	$0.673965\pi$
$854$		0			0
$855$	8.44468e15			0.632076
$856$		0			0
$857$	−1.18745e16	+	2.05672e16i	−0.877446	+	1.51978i	−0.0233113	+	0.999728i	$0.507421\pi$
$857$	−1.18745e16	+	2.05672e16i	−0.877446	+	1.51978i	−0.854134	+	0.520052i	$0.825912\pi$
$858$		0			0
$859$	−6.15402e15	−	1.06591e16i	−0.448948	−	0.777601i	0.549370	−	0.835579i	$-0.314868\pi$
$859$	−6.15402e15	−	1.06591e16i	−0.448948	−	0.777601i	−0.998318	+	0.0579783i	$0.981535\pi$
$860$		0			0
$861$	8.24306e15	−	5.89267e14i	0.593706	−	0.0424419i
$862$		0			0
$863$	−9.88641e14	−	1.71238e15i	−0.0703039	−	0.121770i	0.828731	−	0.559648i	$-0.189064\pi$
$863$	−9.88641e14	−	1.71238e15i	−0.0703039	−	0.121770i	−0.899034	+	0.437878i	$0.855730\pi$
$864$		0			0
$865$	1.08486e16	−	1.87903e16i	0.761702	−	1.31931i
$866$		0			0
$867$	−7.50671e14			−0.0520409
$868$		0			0
$869$	−8.30241e15			−0.568323
$870$		0			0
$871$	1.74946e16	−	3.03015e16i	1.18251	−	2.04816i
$872$		0			0
$873$	2.27949e15	+	3.94820e15i	0.152146	+	0.263524i
$874$		0			0
$875$	−7.46270e15	−	1.10270e16i	−0.491871	−	0.726794i
$876$		0			0
$877$	−8.28766e15	−	1.43546e16i	−0.539429	−	0.934318i	−0.998935	−	0.0461432i	$-0.985307\pi$
$877$	−8.28766e15	−	1.43546e16i	−0.539429	−	0.934318i	0.459506	−	0.888175i	$-0.348026\pi$
$878$		0			0
$879$	−3.11248e15	+	5.39097e15i	−0.200063	+	0.346520i
$880$		0			0
$881$	4.16808e15			0.264587			0.132293	−	0.991211i	$-0.457766\pi$
$881$	4.16808e15			0.264587			0.132293	+	0.991211i	$0.457766\pi$
$882$		0			0
$883$	9.69534e15			0.607826			0.303913	−	0.952700i	$-0.401707\pi$
$883$	9.69534e15			0.607826			0.303913	+	0.952700i	$0.401707\pi$
$884$		0			0
$885$	−1.85572e14	+	3.21420e14i	−0.0114901	+	0.0199015i
$886$		0			0
$887$	−1.04631e16	−	1.81226e16i	−0.639852	−	1.10826i	−0.985465	−	0.169879i	$-0.945662\pi$
$887$	−1.04631e16	−	1.81226e16i	−0.639852	−	1.10826i	0.345613	−	0.938377i	$-0.387671\pi$
$888$		0			0
$889$	−9.25578e14	−	1.36764e15i	−0.0559053	−	0.0826063i
$890$		0			0
$891$	−6.84109e14	−	1.18491e15i	−0.0408130	−	0.0706901i
$892$		0			0
$893$	1.19646e15	−	2.07234e15i	0.0705044	−	0.122117i
$894$		0			0
$895$	−4.99633e16			−2.90820
$896$		0			0
$897$	−1.25305e16			−0.720460
$898$		0			0
$899$	1.64923e16	−	2.85655e16i	0.936706	−	1.62242i
$900$		0			0
$901$	9.07151e15	+	1.57123e16i	0.508971	+	0.881564i
$902$		0			0
$903$	2.56757e15	−	1.83546e14i	0.142312	−	0.0101733i
$904$		0			0
$905$	2.61559e16	+	4.53034e16i	1.43220	+	2.48064i
$906$		0			0
$907$	−9.10092e15	+	1.57633e16i	−0.492317	+	0.852719i	−0.999961	−	0.00884840i	$-0.997183\pi$
$907$	−9.10092e15	+	1.57633e16i	−0.492317	+	0.852719i	0.507643	+	0.861567i	$0.330517\pi$
$908$		0			0
$909$	−7.41494e15			−0.396284
$910$		0			0
$911$	−6.24694e15			−0.329850			−0.164925	−	0.986306i	$-0.552738\pi$
$911$	−6.24694e15			−0.329850			−0.164925	+	0.986306i	$0.552738\pi$
$912$		0			0
$913$	1.10455e16	−	1.91313e16i	0.576230	−	0.998059i
$914$		0			0
$915$	−4.41957e15	−	7.65493e15i	−0.227806	−	0.394571i
$916$		0			0
$917$	7.72964e15	−	1.59107e16i	0.393666	−	0.810324i
$918$		0			0
$919$	9.67523e15	+	1.67580e16i	0.486884	+	0.843309i	0.999886	−	0.0150789i	$-0.00479996\pi$
$919$	9.67523e15	+	1.67580e16i	0.486884	+	0.843309i	−0.513002	+	0.858387i	$0.671467\pi$
$920$		0			0
$921$	−5.40257e15	+	9.35753e15i	−0.268641	+	0.465300i
$922$		0			0
$923$	−3.51375e16			−1.72648
$924$		0			0
$925$	−3.18248e16			−1.54521
$926$		0			0
$927$	−5.64514e15	+	9.77766e15i	−0.270854	+	0.469134i
$928$		0			0
$929$	−1.36668e16	−	2.36716e16i	−0.648009	−	1.12238i	−0.983598	−	0.180375i	$-0.942269\pi$
$929$	−1.36668e16	−	2.36716e16i	−0.648009	−	1.12238i	0.335589	−	0.942008i	$-0.391065\pi$
$930$		0			0
$931$	−2.50864e16	+	3.60509e15i	−1.17548	+	0.168925i
$932$		0			0
$933$	−1.19383e16	−	2.06777e16i	−0.552830	−	0.957530i
$934$		0			0
$935$	1.22245e16	−	2.11734e16i	0.559456	−	0.969005i
$936$		0			0
$937$	2.80386e16			1.26820			0.634100	−	0.773251i	$-0.281371\pi$
$937$	2.80386e16			1.26820			0.634100	+	0.773251i	$0.281371\pi$
$938$		0			0
$939$	5.80323e15			0.259423
$940$		0			0
$941$	1.56608e16	−	2.71253e16i	0.691944	−	1.19848i	−0.279255	−	0.960217i	$-0.590088\pi$
$941$	1.56608e16	−	2.71253e16i	0.691944	−	1.19848i	0.971200	−	0.238266i	$-0.0765791\pi$
$942$		0			0
$943$	8.63761e15	+	1.49608e16i	0.377206	+	0.653341i
$944$		0			0
$945$	−3.11091e15	+	6.40351e15i	−0.134280	+	0.276403i
$946$		0			0
$947$	−4.36126e15	−	7.55392e15i	−0.186075	−	0.322291i	0.757864	−	0.652413i	$-0.226243\pi$
$947$	−4.36126e15	−	7.55392e15i	−0.186075	−	0.322291i	−0.943938	+	0.330123i	$0.892910\pi$
$948$		0			0
$949$	−2.81914e16	+	4.88290e16i	−1.18892	+	2.05927i
$950$		0			0
$951$	−2.41606e16			−1.00720
$952$		0			0
$953$	−6.72272e15			−0.277034			−0.138517	−	0.990360i	$-0.544234\pi$
$953$	−6.72272e15			−0.277034			−0.138517	+	0.990360i	$0.544234\pi$
$954$		0			0
$955$	−2.15984e16	+	3.74096e16i	−0.879840	+	1.52393i
$956$		0			0
$957$	−7.86784e15	−	1.36275e16i	−0.316840	−	0.548783i
$958$		0			0
$959$	−3.71457e16	+	2.65541e15i	−1.47879	+	0.105713i
$960$		0			0
$961$	−7.27119e15	−	1.25941e16i	−0.286172	−	0.495664i
$962$		0			0
$963$	1.68843e15	−	2.92445e15i	0.0656961	−	0.113789i
$964$		0			0
$965$	3.27922e16			1.26145
$966$		0			0
$967$	7.36720e15			0.280193			0.140096	−	0.990138i	$-0.455259\pi$
$967$	7.36720e15			0.280193			0.140096	+	0.990138i	$0.455259\pi$
$968$		0			0
$969$	8.69625e15	−	1.50623e16i	0.327003	−	0.566385i
$970$		0			0
$971$	−1.65266e16	−	2.86249e16i	−0.614436	−	1.06424i	−0.990483	−	0.137634i	$-0.956050\pi$
$971$	−1.65266e16	−	2.86249e16i	−0.614436	−	1.06424i	0.376047	−	0.926601i	$-0.377283\pi$
$972$		0			0
$973$	−1.66690e16	−	2.46302e16i	−0.612757	−	0.905415i
$974$		0			0
$975$	2.09874e16	+	3.63512e16i	0.762838	+	1.32127i
$976$		0			0
$977$	8.22861e14	−	1.42524e15i	0.0295738	−	0.0512233i	−0.850860	−	0.525393i	$-0.823918\pi$
$977$	8.22861e14	−	1.42524e15i	0.0295738	−	0.0512233i	0.880433	+	0.474170i	$0.157252\pi$
$978$		0			0
$979$	1.46786e16			0.521651
$980$		0			0
$981$	−1.43629e16			−0.504735
$982$		0			0
$983$	4.50985e15	−	7.81128e15i	0.156717	−	0.271443i	−0.776966	−	0.629543i	$-0.783242\pi$
$983$	4.50985e15	−	7.81128e15i	0.156717	−	0.271443i	0.933683	+	0.358100i	$0.116575\pi$
$984$		0			0
$985$	−1.73158e16	−	2.99918e16i	−0.595036	−	1.03063i
$986$		0			0
$987$	1.13067e15	+	1.67069e15i	0.0384230	+	0.0567742i
$988$		0			0
$989$	2.69047e15	+	4.66003e15i	0.0904166	+	0.156606i
$990$		0			0
$991$	−2.84665e15	+	4.93055e15i	−0.0946083	+	0.163866i	−0.909445	−	0.415824i	$-0.863493\pi$
$991$	−2.84665e15	+	4.93055e15i	−0.0946083	+	0.163866i	0.814837	+	0.579691i	$0.196827\pi$
$992$		0			0
$993$	−2.56185e15			−0.0842042
$994$		0			0
$995$	2.41518e16			0.785095
$996$		0			0
$997$	−3.63428e15	+	6.29476e15i	−0.116841	+	0.202375i	−0.918514	−	0.395388i	$-0.870610\pi$
$997$	−3.63428e15	+	6.29476e15i	−0.116841	+	0.202375i	0.801673	+	0.597763i	$0.203943\pi$
$998$		0			0
$999$	3.01760e15	+	5.22664e15i	0.0959515	+	0.166193i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.i.b.25.8	✓	16
3.2	odd	2		252.12.k.d.109.1		16
7.2	even	3	inner	84.12.i.b.37.8	yes	16
21.2	odd	6		252.12.k.d.37.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.i.b.25.8	✓	16	1.1	even	1	trivial
84.12.i.b.37.8	yes	16	7.2	even	3	inner
252.12.k.d.37.1		16	21.2	odd	6
252.12.k.d.109.1		16	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	84.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-121.500 + 210.444i) q^{3} +(5578.82 + 9662.79i) q^{5} +(24922.9 + 36826.3i) q^{7} +(-29524.5 - 51137.9i) q^{9} +O(q^{10})\)	\(q+(-121.500 + 210.444i) q^{3} +(5578.82 + 9662.79i) q^{5} +(24922.9 + 36826.3i) q^{7} +(-29524.5 - 51137.9i) q^{9} +(-196201. + 339830. i) q^{11} -2.28291e6 q^{13} -2.71130e6 q^{15} +(-2.79207e6 + 4.83601e6i) q^{17} +(6.40868e6 + 1.11002e7i) q^{19} +(-1.07780e7 + 770481. i) q^{21} +(-1.12939e7 - 1.95616e7i) q^{23} +(-3.78323e7 + 6.55275e7i) q^{25} +1.43489e7 q^{27} +1.65025e8 q^{29} +(9.99385e7 - 1.73099e8i) q^{31} +(-4.76768e7 - 8.25786e7i) q^{33} +(-2.16805e8 + 4.46272e8i) q^{35} +(2.10302e8 + 3.64253e8i) q^{37} +(2.77373e8 - 4.80425e8i) q^{39} -7.64804e8 q^{41} -2.38223e8 q^{43} +(3.29423e8 - 5.70578e8i) q^{45} +(-9.33471e7 - 1.61682e8i) q^{47} +(-7.35026e8 + 1.83564e9i) q^{49} +(-6.78474e8 - 1.17515e9i) q^{51} +(1.62451e9 - 2.81374e9i) q^{53} -4.37827e9 q^{55} -3.11462e9 q^{57} +(6.84439e7 - 1.18548e8i) q^{59} +(1.63005e9 + 2.82334e9i) q^{61} +(1.14739e9 - 2.36178e9i) q^{63} +(-1.27359e10 - 2.20593e10i) q^{65} +(-7.66329e9 + 1.32732e10i) q^{67} +5.48883e9 q^{69} +1.53915e10 q^{71} +(1.23489e10 - 2.13889e10i) q^{73} +(-9.19325e9 - 1.59232e10i) q^{75} +(-1.74046e10 + 1.24419e9i) q^{77} +(1.05790e10 + 1.83233e10i) q^{79} +(-1.74339e9 + 3.01964e9i) q^{81} -5.62969e10 q^{83} -6.23059e10 q^{85} +(-2.00505e10 + 3.47285e10i) q^{87} +(-1.87036e10 - 3.23955e10i) q^{89} +(-5.68967e10 - 8.40711e10i) q^{91} +(2.42851e10 + 4.20630e10i) q^{93} +(-7.15057e10 + 1.23851e11i) q^{95} -7.72068e10 q^{97} +2.31709e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 1944 q^{3} - 2156 q^{5} + 50512 q^{7} - 472392 q^{9}+O(q^{10}) \) 16 * q - 1944 * q^3 - 2156 * q^5 + 50512 * q^7 - 472392 * q^9	\( 16 q - 1944 q^{3} - 2156 q^{5} + 50512 q^{7} - 472392 q^{9} - 222796 q^{11} + 2703176 q^{13} + 1047816 q^{15} + 5114600 q^{17} + 6910556 q^{19} - 18340668 q^{21} - 51387712 q^{23} - 191456372 q^{25} + 229582512 q^{27} + 118854616 q^{29} + 164659160 q^{31} - 54139428 q^{33} + 55239344 q^{35} + 75658364 q^{37} - 328435884 q^{39} - 1815568608 q^{41} + 10754408 q^{43} - 127309644 q^{45} - 1034359464 q^{47} + 4123496848 q^{49} + 1242847800 q^{51} - 665159988 q^{53} - 1264543896 q^{55} - 3358530216 q^{57} + 1040514580 q^{59} - 14391208024 q^{61} + 1474099236 q^{63} - 20938150200 q^{65} - 33307097284 q^{67} + 24974428032 q^{69} + 65848902896 q^{71} + 17709749204 q^{73} - 46523898396 q^{75} + 8594484604 q^{77} - 26626784032 q^{79} - 27894275208 q^{81} - 210306955048 q^{83} - 25867402032 q^{85} - 14440835844 q^{87} - 55951560072 q^{89} + 66078280292 q^{91} + 40012175880 q^{93} + 106810047392 q^{95} - 156216030712 q^{97} + 26311762008 q^{99}+O(q^{100}) \) 16 * q - 1944 * q^3 - 2156 * q^5 + 50512 * q^7 - 472392 * q^9 - 222796 * q^11 + 2703176 * q^13 + 1047816 * q^15 + 5114600 * q^17 + 6910556 * q^19 - 18340668 * q^21 - 51387712 * q^23 - 191456372 * q^25 + 229582512 * q^27 + 118854616 * q^29 + 164659160 * q^31 - 54139428 * q^33 + 55239344 * q^35 + 75658364 * q^37 - 328435884 * q^39 - 1815568608 * q^41 + 10754408 * q^43 - 127309644 * q^45 - 1034359464 * q^47 + 4123496848 * q^49 + 1242847800 * q^51 - 665159988 * q^53 - 1264543896 * q^55 - 3358530216 * q^57 + 1040514580 * q^59 - 14391208024 * q^61 + 1474099236 * q^63 - 20938150200 * q^65 - 33307097284 * q^67 + 24974428032 * q^69 + 65848902896 * q^71 + 17709749204 * q^73 - 46523898396 * q^75 + 8594484604 * q^77 - 26626784032 * q^79 - 27894275208 * q^81 - 210306955048 * q^83 - 25867402032 * q^85 - 14440835844 * q^87 - 55951560072 * q^89 + 66078280292 * q^91 + 40012175880 * q^93 + 106810047392 * q^95 - 156216030712 * q^97 + 26311762008 * q^99

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