LMFDB - Embedded newform 84.12.k.b.5.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,12,Mod(5,84)]

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, 3, 5]))

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

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

chi := DirichletCharacter("84.5");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	84.k (of order $6$, 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:	$56$
Relative dimension:	$28$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			5.17
Character	$\chi$	$=$	84.5
Dual form			84.12.k.b.17.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(91.2256 + 410.883i) q^{3} +(3956.77 - 6853.32i) q^{5} +(-29669.5 - 33121.7i) q^{7} +(-160503. + 74966.1i) q^{9} +O(q^{10})$	\(q+(91.2256 + 410.883i) q^{3} +(3956.77 - 6853.32i) q^{5} +(-29669.5 - 33121.7i) q^{7} +(-160503. + 74966.1i) q^{9} +(-363724. + 209996. i) q^{11} +1.44373e6i q^{13} +(3.17687e6 + 1.00057e6i) q^{15} +(287593. + 498126. i) q^{17} +(6.18865e6 + 3.57302e6i) q^{19} +(1.09025e7 - 1.52122e7i) q^{21} +(870326. + 502483. i) q^{23} +(-6.89792e6 - 1.19475e7i) q^{25} +(-4.54443e7 - 5.91091e7i) q^{27} -7.51861e7i q^{29} +(1.49459e8 - 8.62903e7i) q^{31} +(-1.19465e8 - 1.30291e8i) q^{33} +(-3.44389e8 + 7.22798e7i) q^{35} +(2.71093e8 - 4.69547e8i) q^{37} +(-5.93206e8 + 1.31706e8i) q^{39} +5.49836e8 q^{41} +5.27391e8 q^{43} +(-1.21305e8 + 1.39660e9i) q^{45} +(1.34305e9 - 2.32623e9i) q^{47} +(-2.16767e8 + 1.96541e9i) q^{49} +(-1.78436e8 + 1.63609e8i) q^{51} +(3.08162e9 - 1.77918e9i) q^{53} +3.32362e9i q^{55} +(-9.03529e8 + 2.86876e9i) q^{57} +(2.78649e9 + 4.82635e9i) q^{59} +(-3.54241e9 - 2.04521e9i) q^{61} +(7.24504e9 + 3.09192e9i) q^{63} +(9.89437e9 + 5.71252e9i) q^{65} +(-7.59665e9 - 1.31578e10i) q^{67} +(-1.27066e8 + 4.03441e8i) q^{69} -1.95220e10i q^{71} +(-5.80661e9 + 3.35245e9i) q^{73} +(4.27978e9 - 3.92416e9i) q^{75} +(1.77469e10 + 5.81667e9i) q^{77} +(1.82341e10 - 3.15824e10i) q^{79} +(2.01412e10 - 2.40645e10i) q^{81} +4.79805e10 q^{83} +4.55176e9 q^{85} +(3.08927e10 - 6.85890e9i) q^{87} +(-1.32245e10 + 2.29055e10i) q^{89} +(4.78189e10 - 4.28349e10i) q^{91} +(4.90897e10 + 5.35383e10i) q^{93} +(4.89740e10 - 2.82752e10i) q^{95} +7.77493e10i q^{97} +(4.26361e10 - 6.09719e10i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 729 q^{3} - 27194 q^{7} - 172347 q^{9}+O(q^{10}) $ 56 * q - 729 * q^3 - 27194 * q^7 - 172347 * q^9	$ 56 q - 729 q^{3} - 27194 q^{7} - 172347 q^{9} - 4853058 q^{15} + 28700520 q^{19} - 11325429 q^{21} - 316601194 q^{25} - 1368416388 q^{31} + 40874949 q^{33} - 87435712 q^{37} + 1177474410 q^{39} - 3055078348 q^{43} + 4109921793 q^{45} - 742582522 q^{49} - 694793715 q^{51} + 14605100370 q^{57} + 72584834058 q^{61} - 7310837811 q^{63} + 6131679148 q^{67} - 74402605464 q^{73} - 161115157854 q^{75} + 52181713528 q^{79} + 44948282337 q^{81} + 4658488716 q^{85} + 243101263104 q^{87} - 85311757146 q^{91} - 256628211777 q^{93} + 157345775874 q^{99}+O(q^{100}) $ 56 * q - 729 * q^3 - 27194 * q^7 - 172347 * q^9 - 4853058 * q^15 + 28700520 * q^19 - 11325429 * q^21 - 316601194 * q^25 - 1368416388 * q^31 + 40874949 * q^33 - 87435712 * q^37 + 1177474410 * q^39 - 3055078348 * q^43 + 4109921793 * q^45 - 742582522 * q^49 - 694793715 * q^51 + 14605100370 * q^57 + 72584834058 * q^61 - 7310837811 * q^63 + 6131679148 * q^67 - 74402605464 * q^73 - 161115157854 * q^75 + 52181713528 * q^79 + 44948282337 * q^81 + 4658488716 * q^85 + 243101263104 * q^87 - 85311757146 * q^91 - 256628211777 * q^93 + 157345775874 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$e\left(\frac{5}{6}\right)$

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
$3$	91.2256	+	410.883i	0.216745	+	0.976228i
$3$	91.2256	+	410.883i	0.216745	+	0.976228i
$4$		0			0
$5$	3956.77	−	6853.32i	0.566246	−	0.980767i	−0.430686	−	0.902502i	$-0.641728\pi$
$5$	3956.77	−	6853.32i	0.566246	−	0.980767i	0.996933	−	0.0782655i	$-0.0249382\pi$
$6$		0			0
$7$	−29669.5	−	33121.7i	−0.667223	−	0.744858i
$7$	−29669.5	−	33121.7i	−0.667223	−	0.744858i
$8$		0			0
$9$	−160503.	+	74966.1i	−0.906043	+	0.423186i
$10$		0			0
$11$	−363724.	+	209996.i	−0.680945	+	0.393144i	−0.800211	−	0.599719i	$-0.795279\pi$
$11$	−363724.	+	209996.i	−0.680945	+	0.393144i	0.119266	+	0.992862i	$0.461946\pi$
$12$		0			0
$13$			1.44373e6i			1.07845i	0.842163	+	0.539224i	$0.181282\pi$
$13$			1.44373e6i			1.07845i	−0.842163	+	0.539224i	$0.818718\pi$
$14$		0			0
$15$	3.17687e6	+	1.00057e6i	1.08018	+	0.340209i
$16$		0			0
$17$	287593.	+	498126.i	0.0491258	+	0.0850884i	0.889543	−	0.456852i	$-0.151023\pi$
$17$	287593.	+	498126.i	0.0491258	+	0.0850884i	−0.840417	+	0.541940i	$0.817690\pi$
$18$		0			0
$19$	6.18865e6	+	3.57302e6i	0.573391	+	0.331047i	0.758503	−	0.651670i	$-0.225931\pi$
$19$	6.18865e6	+	3.57302e6i	0.573391	+	0.331047i	−0.185112	+	0.982718i	$0.559265\pi$
$20$		0			0
$21$	1.09025e7	−	1.52122e7i	0.582534	−	0.812807i
$22$		0			0
$23$	870326.	+	502483.i	0.0281954	+	0.0162786i	0.514031	−	0.857771i	$-0.328151\pi$
$23$	870326.	+	502483.i	0.0281954	+	0.0162786i	−0.485836	+	0.874050i	$0.661485\pi$
$24$		0			0
$25$	−6.89792e6	−	1.19475e7i	−0.141269	−	0.244686i
$26$		0			0
$27$	−4.54443e7	−	5.91091e7i	−0.609506	−	0.792781i
$28$		0			0
$29$		−	7.51861e7i		−	0.680689i	−0.940301	−	0.340344i	$-0.889456\pi$
$29$		−	7.51861e7i		−	0.680689i	0.940301	−	0.340344i	$-0.110544\pi$
$30$		0			0
$31$	1.49459e8	−	8.62903e7i	0.937633	−	0.541343i	0.0484158	−	0.998827i	$-0.484583\pi$
$31$	1.49459e8	−	8.62903e7i	0.937633	−	0.541343i	0.889218	+	0.457484i	$0.151249\pi$
$32$		0			0
$33$	−1.19465e8	−	1.30291e8i	−0.531390	−	0.579546i
$34$		0			0
$35$	−3.44389e8	+	7.22798e7i	−1.10834	+	0.232618i
$36$		0			0
$37$	2.71093e8	−	4.69547e8i	0.642701	−	1.11319i	−0.342126	−	0.939654i	$-0.611147\pi$
$37$	2.71093e8	−	4.69547e8i	0.642701	−	1.11319i	0.984827	−	0.173537i	$-0.0555197\pi$
$38$		0			0
$39$	−5.93206e8	+	1.31706e8i	−1.05281	+	0.233748i
$40$		0			0
$41$	5.49836e8			0.741177			0.370589	−	0.928797i	$-0.379156\pi$
$41$	5.49836e8			0.741177			0.370589	+	0.928797i	$0.379156\pi$
$42$		0			0
$43$	5.27391e8			0.547087			0.273544	−	0.961860i	$-0.411804\pi$
$43$	5.27391e8			0.547087			0.273544	+	0.961860i	$0.411804\pi$
$44$		0			0
$45$	−1.21305e8	+	1.39660e9i	−0.0979966	+	1.12824i
$46$		0			0
$47$	1.34305e9	−	2.32623e9i	0.854189	−	1.47950i	−0.0232065	−	0.999731i	$-0.507388\pi$
$47$	1.34305e9	−	2.32623e9i	0.854189	−	1.47950i	0.877395	−	0.479768i	$-0.159279\pi$
$48$		0			0
$49$	−2.16767e8	+	1.96541e9i	−0.109626	+	0.993973i
$50$		0			0
$51$	−1.78436e8	+	1.63609e8i	−0.0724179	+	0.0664005i
$52$		0			0
$53$	3.08162e9	−	1.77918e9i	1.01219	−	0.584388i	0.100358	−	0.994951i	$-0.468001\pi$
$53$	3.08162e9	−	1.77918e9i	1.01219	−	0.584388i	0.911832	+	0.410563i	$0.134668\pi$
$54$		0			0
$55$			3.32362e9i			0.890465i
$56$		0			0
$57$	−9.03529e8	+	2.86876e9i	−0.198898	+	0.631513i
$58$		0			0
$59$	2.78649e9	+	4.82635e9i	0.507425	+	0.878886i	0.999963	+	0.00859497i	$0.00273590\pi$
$59$	2.78649e9	+	4.82635e9i	0.507425	+	0.878886i	−0.492538	+	0.870291i	$0.663931\pi$
$60$		0			0
$61$	−3.54241e9	−	2.04521e9i	−0.537012	−	0.310044i	0.206855	−	0.978372i	$-0.433677\pi$
$61$	−3.54241e9	−	2.04521e9i	−0.537012	−	0.310044i	−0.743867	+	0.668327i	$0.767011\pi$
$62$		0			0
$63$	7.24504e9	+	3.09192e9i	0.919746	+	0.392514i
$64$		0			0
$65$	9.89437e9	+	5.71252e9i	1.05771	+	0.610667i
$66$		0			0
$67$	−7.59665e9	−	1.31578e10i	−0.687402	−	1.19062i	−0.972675	−	0.232169i	$-0.925418\pi$
$67$	−7.59665e9	−	1.31578e10i	−0.687402	−	1.19062i	0.285273	−	0.958446i	$-0.407916\pi$
$68$		0			0
$69$	−1.27066e8	+	4.03441e8i	−0.00978044	+	0.0310535i
$70$		0			0
$71$		−	1.95220e10i		−	1.28412i	−0.766656	−	0.642058i	$-0.778081\pi$
$71$		−	1.95220e10i		−	1.28412i	0.766656	−	0.642058i	$-0.221919\pi$
$72$		0			0
$73$	−5.80661e9	+	3.35245e9i	−0.327829	+	0.189272i	−0.654877	−	0.755736i	$-0.727279\pi$
$73$	−5.80661e9	+	3.35245e9i	−0.327829	+	0.189272i	0.327048	+	0.945008i	$0.393946\pi$
$74$		0			0
$75$	4.27978e9	−	3.92416e9i	0.208250	−	0.190946i
$76$		0			0
$77$	1.77469e10	+	5.81667e9i	0.747179	+	0.244893i
$78$		0			0
$79$	1.82341e10	−	3.15824e10i	0.666707	−	1.15477i	−0.312113	−	0.950045i	$-0.601037\pi$
$79$	1.82341e10	−	3.15824e10i	0.666707	−	1.15477i	0.978820	−	0.204725i	$-0.0656300\pi$
$80$		0			0
$81$	2.01412e10	−	2.40645e10i	0.641828	−	0.766849i
$82$		0			0
$83$	4.79805e10			1.33701			0.668506	−	0.743707i	$-0.266934\pi$
$83$	4.79805e10			1.33701			0.668506	+	0.743707i	$0.266934\pi$
$84$		0			0
$85$	4.55176e9			0.111269
$86$		0			0
$87$	3.08927e10	−	6.85890e9i	0.664508	−	0.147536i
$88$		0			0
$89$	−1.32245e10	+	2.29055e10i	−0.251035	+	0.434805i	−0.963811	−	0.266587i	$-0.914104\pi$
$89$	−1.32245e10	+	2.29055e10i	−0.251035	+	0.434805i	0.712776	+	0.701391i	$0.247437\pi$
$90$		0			0
$91$	4.78189e10	−	4.28349e10i	0.803290	−	0.719565i
$92$		0			0
$93$	4.90897e10	+	5.35383e10i	0.731702	+	0.798011i
$94$		0			0
$95$	4.89740e10	−	2.82752e10i	0.649361	−	0.374909i
$96$		0			0
$97$			7.77493e10i			0.919288i	0.888103	+	0.459644i	$0.152023\pi$
$97$			7.77493e10i			0.919288i	−0.888103	+	0.459644i	$0.847977\pi$
$98$		0			0
$99$	4.26361e10	−	6.09719e10i	0.450593	−	0.644371i
$100$		0			0
$101$	1.89580e10	+	3.28362e10i	0.179484	+	0.310875i	0.941704	−	0.336443i	$-0.109224\pi$
$101$	1.89580e10	+	3.28362e10i	0.179484	+	0.310875i	−0.762220	+	0.647318i	$0.775891\pi$
$102$		0			0
$103$	−7.08160e10	−	4.08856e10i	−0.601903	−	0.347509i	0.167887	−	0.985806i	$-0.446306\pi$
$103$	−7.08160e10	−	4.08856e10i	−0.601903	−	0.347509i	−0.769790	+	0.638297i	$0.779639\pi$
$104$		0			0
$105$	−6.11156e10	−	1.34910e11i	−0.467316	−	1.03158i
$106$		0			0
$107$	2.02018e11	+	1.16635e11i	1.39245	+	0.803932i	0.993586	−	0.113077i	$-0.0360706\pi$
$107$	2.02018e11	+	1.16635e11i	1.39245	+	0.803932i	0.398866	+	0.917009i	$0.369404\pi$
$108$		0			0
$109$	−6.27041e10	−	1.08607e11i	−0.390346	−	0.676099i	0.602149	−	0.798384i	$-0.294311\pi$
$109$	−6.27041e10	−	1.08607e11i	−0.390346	−	0.676099i	−0.992495	+	0.122284i	$0.960978\pi$
$110$		0			0
$111$	2.17660e11	+	6.85529e10i	1.22603	+	0.386144i
$112$		0			0
$113$			1.28230e11i			0.654726i	0.944899	+	0.327363i	$0.106160\pi$
$113$			1.28230e11i			0.654726i	−0.944899	+	0.327363i	$0.893840\pi$
$114$		0			0
$115$	6.88735e9	−	3.97641e9i	0.0319311	−	0.0184354i
$116$		0			0
$117$	−1.08231e11	−	2.31723e11i	−0.456384	−	0.977119i
$118$		0			0
$119$	7.96603e9	−	2.43047e10i	0.0306009	−	0.0933647i
$120$		0			0
$121$	−5.44591e10	+	9.43260e10i	−0.190876	+	0.330607i
$122$		0			0
$123$	5.01591e10	+	2.25918e11i	0.160647	+	0.723558i
$124$		0			0
$125$	2.77229e11			0.812519
$126$		0			0
$127$	4.43714e11			1.19174			0.595871	−	0.803080i	$-0.296807\pi$
$127$	4.43714e11			1.19174			0.595871	+	0.803080i	$0.296807\pi$
$128$		0			0
$129$	4.81116e10	+	2.16696e11i	0.118579	+	0.534082i
$130$		0			0
$131$	4.97204e10	−	8.61183e10i	0.112601	−	0.195031i	−0.804217	−	0.594336i	$-0.797415\pi$
$131$	4.97204e10	−	8.61183e10i	0.112601	−	0.195031i	0.916818	+	0.399305i	$0.130748\pi$
$132$		0			0
$133$	−6.52697e10	−	3.10988e11i	−0.135997	−	0.647977i
$134$		0			0
$135$	−5.84905e11	+	7.75633e10i	−1.12266	+	0.148875i
$136$		0			0
$137$	−3.04125e11	+	1.75587e11i	−0.538380	+	0.310834i	−0.744422	−	0.667709i	$-0.767275\pi$
$137$	−3.04125e11	+	1.75587e11i	−0.538380	+	0.310834i	0.206042	+	0.978543i	$0.433942\pi$
$138$		0			0
$139$			7.65471e11i			1.25126i	0.780120	+	0.625630i	$0.215158\pi$
$139$			7.65471e11i			1.25126i	−0.780120	+	0.625630i	$0.784842\pi$
$140$		0			0
$141$	1.07833e12	+	3.39625e11i	1.62947	+	0.513209i
$142$		0			0
$143$	−3.03179e11	−	5.25121e11i	−0.423985	−	0.734363i
$144$		0			0
$145$	−5.15274e11	−	2.97494e11i	−0.667597	−	0.385437i
$146$		0			0
$147$	−8.27328e11	+	9.02296e10i	−0.994105	+	0.108419i
$148$		0			0
$149$	−7.93584e11	−	4.58176e11i	−0.885255	−	0.511102i	−0.0128678	−	0.999917i	$-0.504096\pi$
$149$	−7.93584e11	−	4.58176e11i	−0.885255	−	0.511102i	−0.872388	+	0.488815i	$0.837429\pi$
$150$		0			0
$151$	−8.53418e11	−	1.47816e12i	−0.884684	−	1.53232i	−0.846075	−	0.533064i	$-0.821041\pi$
$151$	−8.53418e11	−	1.47816e12i	−0.884684	−	1.53232i	−0.0386094	−	0.999254i	$-0.512293\pi$
$152$		0			0
$153$	−8.35021e10	−	5.83909e10i	−0.0805183	−	0.0563044i
$154$		0			0
$155$		−	1.36572e12i		−	1.22613i
$156$		0			0
$157$	3.01966e11	−	1.74340e11i	0.252644	−	0.145864i	−0.368330	−	0.929695i	$-0.620070\pi$
$157$	3.01966e11	−	1.74340e11i	0.252644	−	0.145864i	0.620974	+	0.783831i	$0.286737\pi$
$158$		0			0
$159$	1.01216e12	+	1.10388e12i	0.789884	+	0.861465i
$160$		0			0
$161$	−9.17905e9	−	4.37351e10i	−0.00668737	−	0.0318631i
$162$		0			0
$163$	7.02496e10	−	1.21676e11i	0.0478202	−	0.0828271i	−0.841125	−	0.540842i	$-0.818106\pi$
$163$	7.02496e10	−	1.21676e11i	0.0478202	−	0.0828271i	0.888945	+	0.458014i	$0.151439\pi$
$164$		0			0
$165$	−1.36562e12	+	3.03199e11i	−0.869297	+	0.193004i
$166$		0			0
$167$	−6.81957e11			−0.406272			−0.203136	−	0.979151i	$-0.565113\pi$
$167$	−6.81957e11			−0.406272			−0.203136	+	0.979151i	$0.565113\pi$
$168$		0			0
$169$	−2.92209e11			−0.163048
$170$		0			0
$171$	−1.26115e12	−	1.09541e11i	−0.659611	−	0.0572923i
$172$		0			0
$173$	−1.30594e12	+	2.26196e12i	−0.640724	+	1.10977i	0.344547	+	0.938769i	$0.388033\pi$
$173$	−1.30594e12	+	2.26196e12i	−0.640724	+	1.10977i	−0.985271	+	0.170998i	$0.945301\pi$
$174$		0			0
$175$	−1.91065e11	+	5.82949e11i	−0.0879979	+	0.268486i
$176$		0			0
$177$	−1.72886e12	+	1.58521e12i	−0.748011	+	0.685857i
$178$		0			0
$179$	−2.89309e12	+	1.67033e12i	−1.17671	+	0.679376i	−0.955252	−	0.295794i	$-0.904416\pi$
$179$	−2.89309e12	+	1.67033e12i	−1.17671	+	0.679376i	−0.221461	+	0.975169i	$0.571083\pi$
$180$		0			0
$181$			4.27923e12i			1.63732i	0.574278	+	0.818660i	$0.305283\pi$
$181$			4.27923e12i			1.63732i	−0.574278	+	0.818660i	$0.694717\pi$
$182$		0			0
$183$	5.17184e11	−	1.64209e12i	0.186279	−	0.591447i
$184$		0			0
$185$	−2.14530e12	−	3.71578e12i	−0.727854	−	1.26068i
$186$		0			0
$187$	−2.09209e11	−	1.20787e11i	−0.0669039	−	0.0386270i
$188$		0			0
$189$	−6.09484e11	+	3.25893e12i	−0.183832	+	0.982958i
$190$		0			0
$191$	3.88256e12	+	2.24160e12i	1.10519	+	0.638079i	0.937578	−	0.347775i	$-0.113063\pi$
$191$	3.88256e12	+	2.24160e12i	1.10519	+	0.638079i	0.167607	+	0.985854i	$0.446396\pi$
$192$		0			0
$193$	−5.94509e11	−	1.02972e12i	−0.159806	−	0.276792i	0.774993	−	0.631970i	$-0.217754\pi$
$193$	−5.94509e11	−	1.02972e12i	−0.159806	−	0.276792i	−0.934799	+	0.355178i	$0.884420\pi$
$194$		0			0
$195$	−1.44456e12	+	4.58656e12i	−0.366897	+	1.16492i
$196$		0			0
$197$			1.85082e12i			0.444426i	0.974998	+	0.222213i	$0.0713280\pi$
$197$			1.85082e12i			0.444426i	−0.974998	+	0.222213i	$0.928672\pi$
$198$		0			0
$199$	4.82368e12	−	2.78495e12i	1.09569	−	0.632595i	0.160602	−	0.987019i	$-0.448656\pi$
$199$	4.82368e12	−	2.78495e12i	1.09569	−	0.632595i	0.935085	+	0.354424i	$0.115323\pi$
$200$		0			0
$201$	4.71330e12	−	4.32166e12i	1.01332	−	0.929122i
$202$		0			0
$203$	−2.49029e12	+	2.23073e12i	−0.507016	+	0.454171i
$204$		0			0
$205$	2.17557e12	−	3.76820e12i	0.419689	−	0.726922i
$206$		0			0
$207$	−1.77359e11	−	1.54050e10i	−0.0324352	−	0.00281724i
$208$		0			0
$209$	−3.00128e12			−0.520597
$210$		0			0
$211$	−9.18196e12			−1.51141			−0.755704	−	0.654913i	$-0.772705\pi$
$211$	−9.18196e12			−1.51141			−0.755704	+	0.654913i	$0.772705\pi$
$212$		0			0
$213$	8.02127e12	−	1.78091e12i	1.25359	−	0.278326i
$214$		0			0
$215$	2.08676e12	−	3.61438e12i	0.309786	−	0.536565i
$216$		0			0
$217$	−7.29246e12	−	2.39015e12i	−1.02883	−	0.337207i
$218$		0			0
$219$	−1.90717e12	−	2.08001e12i	−0.255828	−	0.279012i
$220$		0			0
$221$	−7.19162e11	+	4.15208e11i	−0.0917633	+	0.0529796i
$222$		0			0
$223$		−	1.29829e13i		−	1.57651i	−0.615350	−	0.788254i	$-0.710985\pi$
$223$		−	1.29829e13i		−	1.57651i	0.615350	−	0.788254i	$-0.289015\pi$
$224$		0			0
$225$	2.00280e12	+	1.40050e12i	0.231544	+	0.161913i
$226$		0			0
$227$	−3.01090e12	−	5.21503e12i	−0.331554	−	0.574268i	0.651263	−	0.758852i	$-0.274240\pi$
$227$	−3.01090e12	−	5.21503e12i	−0.331554	−	0.574268i	−0.982817	+	0.184584i	$0.940906\pi$
$228$		0			0
$229$	2.84036e12	+	1.63988e12i	0.298043	+	0.172075i	0.641563	−	0.767070i	$-0.278286\pi$
$229$	2.84036e12	+	1.63988e12i	0.298043	+	0.172075i	−0.343521	+	0.939145i	$0.611620\pi$
$230$		0			0
$231$	−7.70998e11	+	7.82254e12i	−0.0771236	+	0.782496i
$232$		0			0
$233$	4.92836e12	+	2.84539e12i	0.470159	+	0.271446i	0.716306	−	0.697786i	$-0.245831\pi$
$233$	4.92836e12	+	2.84539e12i	0.470159	+	0.271446i	−0.246147	+	0.969232i	$0.579165\pi$
$234$		0			0
$235$	−1.06283e13	−	1.84087e13i	−0.967362	−	1.67552i
$236$		0			0
$237$	1.46401e13	+	4.61096e12i	1.27182	+	0.400567i
$238$		0			0
$239$			2.66309e12i			0.220901i	0.993882	+	0.110450i	$0.0352293\pi$
$239$			2.66309e12i			0.220901i	−0.993882	+	0.110450i	$0.964771\pi$
$240$		0			0
$241$	1.73035e13	−	9.99020e12i	1.37101	−	0.791553i	0.379955	−	0.925005i	$-0.375939\pi$
$241$	1.73035e13	−	9.99020e12i	1.37101	−	0.791553i	0.991055	+	0.133451i	$0.0426060\pi$
$242$		0			0
$243$	1.17251e13	+	6.08039e12i	0.887733	+	0.460359i
$244$		0			0
$245$	1.26119e13	+	9.26224e12i	0.912780	+	0.670351i
$246$		0			0
$247$	−5.15849e12	+	8.93476e12i	−0.357017	+	0.618372i
$248$		0			0
$249$	4.37705e12	+	1.97144e13i	0.289791	+	1.30523i
$250$		0			0
$251$	3.05056e13			1.93274			0.966372	−	0.257147i	$-0.0827825\pi$
$251$	3.05056e13			1.93274			0.966372	+	0.257147i	$0.0827825\pi$
$252$		0			0
$253$	−4.22078e11			−0.0255994
$254$		0			0
$255$	4.15237e11	+	1.87024e12i	0.0241171	+	0.108624i
$256$		0			0
$257$	8.70202e12	−	1.50723e13i	0.484159	−	0.838588i	−0.515675	−	0.856784i	$-0.672459\pi$
$257$	8.70202e12	−	1.50723e13i	0.484159	−	0.838588i	0.999834	+	0.0181959i	$0.00579225\pi$
$258$		0			0
$259$	−2.35954e13	+	4.95217e12i	−1.25799	+	0.264026i
$260$		0			0
$261$	5.63641e12	+	1.20676e13i	0.288058	+	0.616733i
$262$		0			0
$263$	2.02789e13	−	1.17081e13i	0.993777	−	0.573757i	0.0873756	−	0.996175i	$-0.472152\pi$
$263$	2.02789e13	−	1.17081e13i	0.993777	−	0.573757i	0.906401	+	0.422418i	$0.138819\pi$
$264$		0			0
$265$		−	2.81591e13i		−	1.32363i
$266$		0			0
$267$	−1.06179e13	−	3.34415e12i	−0.478879	−	0.150825i
$268$		0			0
$269$	1.78031e12	+	3.08358e12i	0.0770651	+	0.133481i	0.901982	−	0.431773i	$-0.142112\pi$
$269$	1.78031e12	+	3.08358e12i	0.0770651	+	0.133481i	−0.824917	+	0.565253i	$0.808778\pi$
$270$		0			0
$271$	2.05407e12	+	1.18592e12i	0.0853659	+	0.0492860i	0.542075	−	0.840330i	$-0.317639\pi$
$271$	2.05407e12	+	1.18592e12i	0.0853659	+	0.0492860i	−0.456709	+	0.889616i	$0.650972\pi$
$272$		0			0
$273$	2.19624e13	+	1.57404e13i	0.876569	+	0.628232i
$274$		0			0
$275$	5.01788e12	+	2.89707e12i	0.192393	+	0.111078i
$276$		0			0
$277$	−1.64107e13	−	2.84241e13i	−0.604627	−	1.04725i	−0.992110	−	0.125369i	$-0.959989\pi$
$277$	−1.64107e13	−	2.84241e13i	−0.604627	−	1.04725i	0.387483	−	0.921877i	$-0.373345\pi$
$278$		0			0
$279$	−1.75198e13	+	2.50542e13i	−0.620448	+	0.887273i
$280$		0			0
$281$			2.66708e13i			0.908135i	0.890967	+	0.454068i	$0.150028\pi$
$281$			2.66708e13i			0.908135i	−0.890967	+	0.454068i	$0.849972\pi$
$282$		0			0
$283$	−2.46384e13	+	1.42250e13i	−0.806838	+	0.465828i	−0.845857	−	0.533410i	$-0.820910\pi$
$283$	−2.46384e13	+	1.42250e13i	−0.806838	+	0.465828i	0.0390186	+	0.999238i	$0.487577\pi$
$284$		0			0
$285$	1.60855e13	+	1.75432e13i	0.506742	+	0.552665i
$286$		0			0
$287$	−1.63134e13	−	1.82115e13i	−0.494531	−	0.552072i
$288$		0			0
$289$	1.69705e13	−	2.93938e13i	0.495173	−	0.857665i
$290$		0			0
$291$	−3.19459e13	+	7.09272e12i	−0.897435	+	0.199251i
$292$		0			0
$293$	2.94747e12			0.0797403			0.0398701	−	0.999205i	$-0.487306\pi$
$293$	2.94747e12			0.0797403			0.0398701	+	0.999205i	$0.487306\pi$
$294$		0			0
$295$	4.41020e13			1.14931
$296$		0			0
$297$	2.89418e13	+	1.19563e13i	0.726717	+	0.300217i
$298$		0			0
$299$	−7.25452e11	+	1.25652e12i	−0.0175556	+	0.0304073i
$300$		0			0
$301$	−1.56474e13	−	1.74681e13i	−0.365029	−	0.407502i
$302$		0			0
$303$	−1.17624e13	+	1.07850e13i	−0.264583	+	0.242598i
$304$		0			0
$305$	−2.80329e13	+	1.61848e13i	−0.608162	+	0.351123i
$306$		0			0
$307$		−	3.64743e13i		−	0.763354i	−0.924296	−	0.381677i	$-0.875347\pi$
$307$		−	3.64743e13i		−	0.763354i	0.924296	−	0.381677i	$-0.124653\pi$
$308$		0			0
$309$	1.03390e13	−	3.28269e13i	0.208788	−	0.662915i
$310$		0			0
$311$	−4.45806e13	−	7.72159e13i	−0.868889	−	1.50496i	−0.863134	−	0.504976i	$-0.831501\pi$
$311$	−4.45806e13	−	7.72159e13i	−0.868889	−	1.50496i	−0.00575498	−	0.999983i	$-0.501832\pi$
$312$		0			0
$313$	−6.08237e13	−	3.51166e13i	−1.14440	−	0.660721i	−0.196885	−	0.980427i	$-0.563083\pi$
$313$	−6.08237e13	−	3.51166e13i	−1.14440	−	0.660721i	−0.947517	+	0.319706i	$0.896416\pi$
$314$		0			0
$315$	4.98568e13	−	3.74186e13i	0.905767	−	0.679797i
$316$		0			0
$317$	3.47943e13	+	2.00885e13i	0.610496	+	0.352470i	0.773159	−	0.634212i	$-0.218675\pi$
$317$	3.47943e13	+	2.00885e13i	0.610496	+	0.352470i	−0.162664	+	0.986682i	$0.552009\pi$
$318$		0			0
$319$	1.57888e13	+	2.73470e13i	0.267609	+	0.463512i
$320$		0			0
$321$	−2.94943e13	+	9.36461e13i	−0.483014	+	1.53360i
$322$		0			0
$323$			4.11030e12i			0.0650519i
$324$		0			0
$325$	1.72491e13	−	9.95877e12i	0.263881	−	0.152352i
$326$		0			0
$327$	3.89044e13	−	3.56717e13i	0.575422	−	0.527608i
$328$		0			0
$329$	−1.16896e14	+	2.45340e13i	−1.67195	+	0.350907i
$330$		0			0
$331$	1.33765e13	−	2.31688e13i	0.185050	−	0.320515i	−0.758544	−	0.651622i	$-0.774089\pi$
$331$	1.33765e13	−	2.31688e13i	0.185050	−	0.320515i	0.943593	+	0.331107i	$0.107422\pi$
$332$		0			0
$333$	−8.31110e12	+	9.56865e13i	−0.111228	+	1.28058i
$334$		0			0
$335$	−1.20233e14			−1.55696
$336$		0			0
$337$	−7.15665e13			−0.896902			−0.448451	−	0.893807i	$-0.648024\pi$
$337$	−7.15665e13			−0.896902			−0.448451	+	0.893807i	$0.648024\pi$
$338$		0			0
$339$	−5.26877e13	+	1.16979e13i	−0.639162	+	0.141909i
$340$		0			0
$341$	−3.62412e13	+	6.27717e13i	−0.425651	+	0.737250i
$342$		0			0
$343$	7.15291e13	−	5.11330e13i	0.813514	−	0.581546i
$344$		0			0
$345$	2.26214e12	+	2.46714e12i	0.0249181	+	0.0271762i
$346$		0			0
$347$	2.89174e13	−	1.66955e13i	0.308566	−	0.178151i	−0.337719	−	0.941247i	$-0.609655\pi$
$347$	2.89174e13	−	1.66955e13i	0.308566	−	0.178151i	0.646285	+	0.763097i	$0.276322\pi$
$348$		0			0
$349$			1.12717e14i			1.16533i	0.812712	+	0.582665i	$0.197990\pi$
$349$			1.12717e14i			1.16533i	−0.812712	+	0.582665i	$0.802010\pi$
$350$		0			0
$351$	8.53378e13	−	6.56094e13i	0.854973	−	0.657321i
$352$		0			0
$353$	9.09345e13	+	1.57503e14i	0.883014	+	1.52943i	0.847973	+	0.530040i	$0.177823\pi$
$353$	9.09345e13	+	1.57503e14i	0.883014	+	1.52943i	0.0350413	+	0.999386i	$0.488844\pi$
$354$		0			0
$355$	−1.33791e14	−	7.72441e13i	−1.25942	−	0.727126i
$356$		0			0
$357$	1.07131e13	+	1.05589e12i	0.0977778	+	0.00963708i
$358$		0			0
$359$	−1.39713e14	−	8.06631e13i	−1.23656	−	0.713930i	−0.268172	−	0.963371i	$-0.586420\pi$
$359$	−1.39713e14	−	8.06631e13i	−1.23656	−	0.713930i	−0.968390	+	0.249441i	$0.919753\pi$
$360$		0			0
$361$	−3.27122e13	−	5.66592e13i	−0.280815	−	0.486386i
$362$		0			0
$363$	−4.37250e13	−	1.37714e13i	−0.364119	−	0.114681i
$364$		0			0
$365$			5.30594e13i			0.428698i
$366$		0			0
$367$	−1.08295e14	+	6.25240e13i	−0.849071	+	0.490212i	−0.860337	−	0.509725i	$-0.829747\pi$
$367$	−1.08295e14	+	6.25240e13i	−0.849071	+	0.490212i	0.0112660	+	0.999937i	$0.496414\pi$
$368$		0			0
$369$	−8.82502e13	+	4.12191e13i	−0.671538	+	0.313656i
$370$		0			0
$371$	−1.50360e14	−	4.92813e13i	−1.11064	−	0.364020i
$372$		0			0
$373$	−4.05260e13	+	7.01932e13i	−0.290627	+	0.503380i	−0.973958	−	0.226728i	$-0.927197\pi$
$373$	−4.05260e13	+	7.01932e13i	−0.290627	+	0.503380i	0.683331	+	0.730108i	$0.260530\pi$
$374$		0			0
$375$	2.52904e13	+	1.13909e14i	0.176110	+	0.793204i
$376$		0			0
$377$	1.08549e14			0.734087
$378$		0			0
$379$	−5.43437e13			−0.356971			−0.178486	−	0.983942i	$-0.557120\pi$
$379$	−5.43437e13			−0.356971			−0.178486	+	0.983942i	$0.557120\pi$
$380$		0			0
$381$	4.04780e13	+	1.82314e14i	0.258305	+	1.16341i
$382$		0			0
$383$	9.36135e13	−	1.62143e14i	0.580424	−	1.00532i	−0.415005	−	0.909819i	$-0.636220\pi$
$383$	9.36135e13	−	1.62143e14i	0.580424	−	1.00532i	0.995429	−	0.0955047i	$-0.0304465\pi$
$384$		0			0
$385$	1.10084e14	−	9.86102e13i	0.663270	−	0.594139i
$386$		0			0
$387$	−8.46478e13	+	3.95365e13i	−0.495684	+	0.231519i
$388$		0			0
$389$	1.51837e14	−	8.76630e13i	0.864279	−	0.498992i	−0.00116383	−	0.999999i	$-0.500370\pi$
$389$	1.51837e14	−	8.76630e13i	0.864279	−	0.498992i	0.865443	+	0.501008i	$0.167037\pi$
$390$		0			0
$391$			5.78043e11i			0.00319880i
$392$		0			0
$393$	3.99203e13	+	1.25731e13i	0.214800	+	0.0676524i
$394$		0			0
$395$	−1.44296e14	−	2.49928e14i	−0.755040	−	1.30777i
$396$		0			0
$397$	1.24898e14	+	7.21100e13i	0.635635	+	0.366984i	0.782931	−	0.622108i	$-0.213724\pi$
$397$	1.24898e14	+	7.21100e13i	0.635635	+	0.366984i	−0.147296	+	0.989092i	$0.547057\pi$
$398$		0			0
$399$	1.21825e14	−	5.51883e13i	0.603097	−	0.273210i
$400$		0			0
$401$	3.18153e14	+	1.83686e14i	1.53229	+	0.884670i	0.999256	+	0.0385794i	$0.0122832\pi$
$401$	3.18153e14	+	1.83686e14i	1.53229	+	0.884670i	0.533038	+	0.846091i	$0.321050\pi$
$402$		0			0
$403$	1.24580e14	+	2.15779e14i	0.583810	+	1.01119i
$404$		0			0
$405$	−8.52278e13	−	2.33252e14i	−0.388668	−	1.06371i
$406$		0			0
$407$			2.27714e14i			1.01070i
$408$		0			0
$409$	1.96559e13	−	1.13484e13i	0.0849211	−	0.0490292i	−0.456938	−	0.889498i	$-0.651054\pi$
$409$	1.96559e13	−	1.13484e13i	0.0849211	−	0.0490292i	0.541859	+	0.840469i	$0.317721\pi$
$410$		0			0
$411$	−9.98895e13	−	1.08942e14i	−0.420136	−	0.458210i
$412$		0			0
$413$	7.71829e13	−	2.35489e14i	0.316079	−	0.964373i
$414$		0			0
$415$	1.89848e14	−	3.28826e14i	0.757078	−	1.31130i
$416$		0			0
$417$	−3.14519e14	+	6.98305e13i	−1.22151	+	0.271205i
$418$		0			0
$419$	−1.69917e14			−0.642776			−0.321388	−	0.946948i	$-0.604149\pi$
$419$	−1.69917e14			−0.642776			−0.321388	+	0.946948i	$0.604149\pi$
$420$		0			0
$421$	3.36021e14			1.23827			0.619134	−	0.785286i	$-0.287484\pi$
$421$	3.36021e14			1.23827			0.619134	+	0.785286i	$0.287484\pi$
$422$		0			0
$423$	−4.11748e13	+	4.74050e14i	−0.147829	+	1.70197i
$424$		0			0
$425$	3.96759e12	−	6.87207e12i	0.0138799	−	0.0240408i
$426$		0			0
$427$	3.73606e13	+	1.78011e14i	0.127368	+	0.606866i
$428$		0			0
$429$	1.88106e14	−	1.72475e14i	0.625009	−	0.573076i
$430$		0			0
$431$	1.97920e14	−	1.14269e14i	0.641011	−	0.370088i	−0.143993	−	0.989579i	$-0.545994\pi$
$431$	1.97920e14	−	1.14269e14i	0.641011	−	0.370088i	0.785004	+	0.619491i	$0.212661\pi$
$432$		0			0
$433$		−	3.27684e14i		−	1.03460i	−0.855805	−	0.517299i	$-0.826938\pi$
$433$		−	3.27684e14i		−	1.03460i	0.855805	−	0.517299i	$-0.173062\pi$
$434$		0			0
$435$	7.52290e13	−	2.38857e14i	0.231576	−	0.735269i
$436$		0			0
$437$	3.59076e12	+	6.21938e12i	0.0107780	+	0.0186680i
$438$		0			0
$439$	−3.48291e14	−	2.01086e14i	−1.01950	−	0.588609i	−0.105542	−	0.994415i	$-0.533658\pi$
$439$	−3.48291e14	−	2.01086e14i	−1.01950	−	0.588609i	−0.913959	+	0.405806i	$0.866991\pi$
$440$		0			0
$441$	−1.12547e14	−	3.31704e14i	−0.321309	−	0.946974i
$442$		0			0
$443$	5.19138e14	+	2.99724e14i	1.44565	+	0.834644i	0.998218	−	0.0596758i	$-0.0190067\pi$
$443$	5.19138e14	+	2.99724e14i	1.44565	+	0.834644i	0.447428	+	0.894320i	$0.352340\pi$
$444$		0			0
$445$	1.04652e14	+	1.81263e14i	0.284295	+	0.492413i
$446$		0			0
$447$	1.15862e14	−	3.67868e14i	0.307078	−	0.974990i
$448$		0			0
$449$			5.03298e14i			1.30158i	0.759258	+	0.650790i	$0.225562\pi$
$449$			5.03298e14i			1.30158i	−0.759258	+	0.650790i	$0.774438\pi$
$450$		0			0
$451$	−1.99989e14	+	1.15463e14i	−0.504701	+	0.291389i
$452$		0			0
$453$	5.29498e14	−	4.85501e14i	1.30414	−	1.19578i
$454$		0			0
$455$	−1.04353e14	−	4.97206e14i	−0.250866	−	1.19529i
$456$		0			0
$457$	2.94369e13	−	5.09862e13i	0.0690801	−	0.119650i	−0.829417	−	0.558631i	$-0.811327\pi$
$457$	2.94369e13	−	5.09862e13i	0.0690801	−	0.119650i	0.898497	+	0.438980i	$0.144660\pi$
$458$		0			0
$459$	1.63743e13	−	3.96363e13i	0.0375140	−	0.0908079i
$460$		0			0
$461$	3.78994e14			0.847767			0.423884	−	0.905717i	$-0.360667\pi$
$461$	3.78994e14			0.847767			0.423884	+	0.905717i	$0.360667\pi$
$462$		0			0
$463$	−6.98091e14			−1.52481			−0.762406	−	0.647099i	$-0.775982\pi$
$463$	−6.98091e14			−1.52481			−0.762406	+	0.647099i	$0.775982\pi$
$464$		0			0
$465$	5.61152e14	−	1.24589e14i	1.19699	−	0.265759i
$466$		0			0
$467$	6.08338e13	−	1.05367e14i	0.126737	−	0.219514i	−0.795674	−	0.605725i	$-0.792883\pi$
$467$	6.08338e13	−	1.05367e14i	0.126737	−	0.219514i	0.922410	+	0.386211i	$0.126216\pi$
$468$		0			0
$469$	−2.10419e14	+	6.41999e14i	−0.428189	+	1.30642i
$470$		0			0
$471$	9.91804e13	+	1.08168e14i	0.197156	+	0.215023i
$472$		0			0
$473$	−1.91825e14	+	1.10750e14i	−0.372536	+	0.215084i
$474$		0			0
$475$		−	9.85855e13i		−	0.187068i
$476$		0			0
$477$	−3.61231e14	+	5.16580e14i	−0.669783	+	0.957826i
$478$		0			0
$479$	2.74582e14	+	4.75589e14i	0.497538	+	0.861761i	0.999996	−	0.00284086i	$-0.000904274\pi$
$479$	2.74582e14	+	4.75589e14i	0.497538	+	0.861761i	−0.502458	+	0.864601i	$0.667571\pi$
$480$		0			0
$481$	6.77902e14	+	3.91387e14i	1.20052	+	0.693120i
$482$		0			0
$483$	1.71326e13	−	7.76127e12i	0.0296562	−	0.0134346i
$484$		0			0
$485$	5.32841e14	+	3.07636e14i	0.901608	+	0.520544i
$486$		0			0
$487$	5.03806e14	+	8.72617e14i	0.833401	+	1.44349i	0.895326	+	0.445412i	$0.146943\pi$
$487$	5.03806e14	+	8.72617e14i	0.833401	+	1.44349i	−0.0619251	+	0.998081i	$0.519724\pi$
$488$		0			0
$489$	5.64031e13	+	1.77644e13i	0.0912230	+	0.0287311i
$490$		0			0
$491$		−	1.54729e14i		−	0.244694i	−0.992487	−	0.122347i	$-0.960958\pi$
$491$		−	1.54729e14i		−	0.244694i	0.992487	−	0.122347i	$-0.0390421\pi$
$492$		0			0
$493$	3.74522e13	−	2.16230e13i	0.0579187	−	0.0334394i
$494$		0			0
$495$	−2.49159e14	−	5.33450e14i	−0.376832	−	0.806799i
$496$		0			0
$497$	−6.46603e14	+	5.79209e14i	−0.956484	+	0.856792i
$498$		0			0
$499$	−1.46837e14	+	2.54330e14i	−0.212463	+	0.367997i	−0.952485	−	0.304586i	$-0.901482\pi$
$499$	−1.46837e14	+	2.54330e14i	−0.212463	+	0.367997i	0.740022	+	0.672583i	$0.234815\pi$
$500$		0			0
$501$	−6.22119e13	−	2.80205e14i	−0.0880575	−	0.396614i
$502$		0			0
$503$	−9.55898e14			−1.32369			−0.661847	−	0.749639i	$-0.730227\pi$
$503$	−9.55898e14			−1.32369			−0.661847	+	0.749639i	$0.730227\pi$
$504$		0			0
$505$	3.00049e14			0.406528
$506$		0			0
$507$	−2.66569e13	−	1.20064e14i	−0.0353400	−	0.159172i
$508$		0			0
$509$	4.79530e14	−	8.30570e14i	0.622111	−	1.07753i	−0.366981	−	0.930228i	$-0.619609\pi$
$509$	4.79530e14	−	8.30570e14i	0.622111	−	1.07753i	0.989092	−	0.147299i	$-0.0470580\pi$
$510$		0			0
$511$	2.83318e14	+	9.28593e13i	0.359716	+	0.117899i
$512$		0			0
$513$	−7.00408e13	−	5.28178e14i	−0.0870373	−	0.656349i
$514$		0			0
$515$	−5.60404e14	+	3.23550e14i	−0.681650	+	0.393551i
$516$		0			0
$517$			1.12814e15i			1.34328i
$518$		0			0
$519$	−1.04854e15	−	3.30242e14i	−1.22226	−	0.384956i
$520$		0			0
$521$	−6.89512e14	−	1.19427e15i	−0.786927	−	1.36300i	−0.927841	−	0.372976i	$-0.878337\pi$
$521$	−6.89512e14	−	1.19427e15i	−0.786927	−	1.36300i	0.140914	−	0.990022i	$-0.454996\pi$
$522$		0			0
$523$	2.78174e14	+	1.60604e14i	0.310854	+	0.179472i	0.647309	−	0.762228i	$-0.275895\pi$
$523$	2.78174e14	+	1.60604e14i	0.310854	+	0.179472i	−0.336455	+	0.941700i	$0.609228\pi$
$524$		0			0
$525$	−2.56954e14	−	2.53256e13i	−0.281176	−	0.0277130i
$526$		0			0
$527$	8.59669e13	+	4.96330e13i	0.0921240	+	0.0531878i
$528$		0			0
$529$	−4.75900e14	−	8.24283e14i	−0.499470	−	0.865107i
$530$		0			0
$531$	−8.09052e14	−	5.65750e14i	−0.831681	−	0.581573i
$532$		0			0
$533$			7.93817e14i			0.799321i
$534$		0			0
$535$	1.59868e15	−	9.22998e14i	1.57694	−	0.910447i
$536$		0			0
$537$	−9.50233e14	−	1.03635e15i	−0.918273	−	1.00149i
$538$		0			0
$539$	−3.33885e14	−	7.60386e14i	−0.316125	−	0.719940i
$540$		0			0
$541$	−1.48344e14	+	2.56939e14i	−0.137621	+	0.238367i	−0.926596	−	0.376059i	$-0.877279\pi$
$541$	−1.48344e14	+	2.56939e14i	−0.137621	+	0.238367i	0.788975	+	0.614426i	$0.210612\pi$
$542$		0			0
$543$	−1.75826e15	+	3.90375e14i	−1.59840	+	0.354882i
$544$		0			0
$545$	−9.92421e14			−0.884128
$546$		0			0
$547$	1.12665e15			0.983690			0.491845	−	0.870683i	$-0.336323\pi$
$547$	1.12665e15			0.983690			0.491845	+	0.870683i	$0.336323\pi$
$548$		0			0
$549$	7.21887e14	+	6.27014e13i	0.617762	+	0.0536574i
$550$		0			0
$551$	2.68641e14	−	4.65300e14i	0.225340	−	0.390301i
$552$		0			0
$553$	−1.58706e15	+	3.33089e14i	−1.30498	+	0.273888i
$554$		0			0
$555$	1.33104e15	−	1.22044e15i	1.07295	−	0.983799i
$556$		0			0
$557$	−1.62113e15	+	9.35958e14i	−1.28119	+	0.739695i	−0.977066	−	0.212937i	$-0.931697\pi$
$557$	−1.62113e15	+	9.35958e14i	−1.28119	+	0.739695i	−0.304124	+	0.952633i	$0.598364\pi$
$558$		0			0
$559$			7.61413e14i			0.590005i
$560$		0			0
$561$	3.05441e13	−	9.69793e13i	0.0232077	−	0.0736857i
$562$		0			0
$563$	−2.82652e14	−	4.89568e14i	−0.210599	−	0.364768i	0.741303	−	0.671170i	$-0.234208\pi$
$563$	−2.82652e14	−	4.89568e14i	−0.210599	−	0.364768i	−0.951902	+	0.306402i	$0.900875\pi$
$564$		0			0
$565$	8.78804e14	+	5.07378e14i	0.642134	+	0.370736i
$566$		0			0
$567$	−1.39464e15	+	4.68711e13i	−0.999436	+	0.0335891i
$568$		0			0
$569$	1.01682e15	+	5.87063e14i	0.714706	+	0.412636i	0.812801	−	0.582541i	$-0.197942\pi$
$569$	1.01682e15	+	5.87063e14i	0.714706	+	0.412636i	−0.0980950	+	0.995177i	$0.531275\pi$
$570$		0			0
$571$	1.05854e15	+	1.83345e15i	0.729809	+	1.26407i	0.956964	+	0.290208i	$0.0937246\pi$
$571$	1.05854e15	+	1.83345e15i	0.729809	+	1.26407i	−0.227154	+	0.973859i	$0.572942\pi$
$572$		0			0
$573$	−5.66846e14	+	1.79977e15i	−0.383367	+	1.21721i
$574$		0			0
$575$		−	1.38643e13i		−	0.00919869i
$576$		0			0
$577$	−1.45340e15	+	8.39121e14i	−0.946059	+	0.546208i	−0.891855	−	0.452322i	$-0.850596\pi$
$577$	−1.45340e15	+	8.39121e14i	−0.946059	+	0.546208i	−0.0542048	+	0.998530i	$0.517262\pi$
$578$		0			0
$579$	3.68860e14	−	3.38211e14i	0.235575	−	0.216001i
$580$		0			0
$581$	−1.42356e15	−	1.58920e15i	−0.892086	−	0.995884i
$582$		0			0
$583$	−7.47240e14	+	1.29426e15i	−0.459497	+	0.795873i
$584$		0			0
$585$	−2.01632e15	−	1.75133e14i	−1.21675	−	0.105684i
$586$		0			0
$587$	9.88936e14			0.585677			0.292839	−	0.956162i	$-0.405400\pi$
$587$	9.88936e14			0.585677			0.292839	+	0.956162i	$0.405400\pi$
$588$		0			0
$589$	1.23327e15			0.716841
$590$		0			0
$591$	−7.60469e14	+	1.68842e14i	−0.433861	+	0.0963272i
$592$		0			0
$593$	1.29844e15	−	2.24897e15i	0.727147	−	1.25946i	−0.230937	−	0.972969i	$-0.574179\pi$
$593$	1.29844e15	−	2.24897e15i	0.727147	−	1.25946i	0.958084	−	0.286487i	$-0.0924876\pi$
$594$		0			0
$595$	−1.35048e14	−	1.50762e14i	−0.0742414	−	0.0828797i
$596$		0			0
$597$	1.58433e15	+	1.72791e15i	0.855042	+	0.932528i
$598$		0			0
$599$	−2.58250e15	+	1.49101e15i	−1.36834	+	0.790010i	−0.990716	−	0.135950i	$-0.956591\pi$
$599$	−2.58250e15	+	1.49101e15i	−1.36834	+	0.790010i	−0.377622	+	0.925960i	$0.623258\pi$
$600$		0			0
$601$			2.60328e15i			1.35429i	0.735849	+	0.677145i	$0.236783\pi$
$601$			2.60328e15i			1.35429i	−0.735849	+	0.677145i	$0.763217\pi$
$602$		0			0
$603$	2.20567e15	+	1.54237e15i	1.12667	+	0.787850i
$604$		0			0
$605$	4.30964e14	+	7.46452e14i	0.216166	+	0.374410i
$606$		0			0
$607$	−2.24583e15	−	1.29663e15i	−1.10621	−	0.638673i	−0.168367	−	0.985724i	$-0.553849\pi$
$607$	−2.24583e15	−	1.29663e15i	−1.10621	−	0.638673i	−0.937846	+	0.347052i	$0.887183\pi$
$608$		0			0
$609$	−1.14375e15	−	8.19719e14i	−0.553268	−	0.396524i
$610$		0			0
$611$	3.35846e15	+	1.93901e15i	1.59556	+	0.921198i
$612$		0			0
$613$	−8.43272e14	−	1.46059e15i	−0.393491	−	0.681546i	0.599416	−	0.800437i	$-0.295399\pi$
$613$	−8.43272e14	−	1.46059e15i	−0.393491	−	0.681546i	−0.992907	+	0.118891i	$0.962066\pi$
$614$		0			0
$615$	1.74676e15	+	5.50150e14i	0.800608	+	0.252155i
$616$		0			0
$617$		−	1.01872e15i		−	0.458655i	−0.973349	−	0.229328i	$-0.926347\pi$
$617$		−	1.01872e15i		−	0.458655i	0.973349	−	0.229328i	$-0.0736528\pi$
$618$		0			0
$619$	−8.90995e13	+	5.14416e13i	−0.0394073	+	0.0227518i	−0.519574	−	0.854425i	$-0.673909\pi$
$619$	−8.90995e13	+	5.14416e13i	−0.0394073	+	0.0227518i	0.480167	+	0.877177i	$0.340576\pi$
$620$		0			0
$621$	−9.85002e12	−	7.42791e13i	−0.00427990	−	0.0322747i
$622$		0			0
$623$	1.15103e15	−	2.41577e14i	0.491364	−	0.103127i
$624$		0			0
$625$	1.43374e15	−	2.48332e15i	0.601355	−	1.04158i
$626$		0			0
$627$	−2.73793e14	−	1.23317e15i	−0.112837	−	0.508221i
$628$		0			0
$629$	3.11858e14			0.126293
$630$		0			0
$631$	2.41852e15			0.962473			0.481237	−	0.876591i	$-0.340188\pi$
$631$	2.41852e15			0.962473			0.481237	+	0.876591i	$0.340188\pi$
$632$		0			0
$633$	−8.37630e14	−	3.77271e15i	−0.327591	−	1.47548i
$634$		0			0
$635$	1.75567e15	−	3.04091e15i	0.674819	−	1.16882i
$636$		0			0
$637$	−2.83753e15	−	3.12954e14i	−1.07195	−	0.118226i
$638$		0			0
$639$	1.46349e15	+	3.13334e15i	0.543420	+	1.16346i
$640$		0			0
$641$	−3.53679e15	+	2.04197e15i	−1.29089	+	0.745297i	−0.978812	−	0.204759i	$-0.934359\pi$
$641$	−3.53679e15	+	2.04197e15i	−1.29089	+	0.745297i	−0.312080	+	0.950056i	$0.601026\pi$
$642$		0			0
$643$		−	2.12552e15i		−	0.762613i	−0.924449	−	0.381307i	$-0.875474\pi$
$643$		−	2.12552e15i		−	0.762613i	0.924449	−	0.381307i	$-0.124526\pi$
$644$		0			0
$645$	1.67545e15	+	5.27692e14i	0.590955	+	0.186124i
$646$		0			0
$647$	2.14842e15	+	3.72118e15i	0.744983	+	1.29035i	0.950203	+	0.311632i	$0.100876\pi$
$647$	2.14842e15	+	3.72118e15i	0.744983	+	1.29035i	−0.205220	+	0.978716i	$0.565791\pi$
$648$		0			0
$649$	−2.02703e15	−	1.17031e15i	−0.691057	−	0.398982i
$650$		0			0
$651$	3.16813e14	−	3.21439e15i	0.106196	−	1.07747i
$652$		0			0
$653$	−1.34746e15	−	7.77954e14i	−0.444111	−	0.256408i	0.261229	−	0.965277i	$-0.415872\pi$
$653$	−1.34746e15	−	7.77954e14i	−0.444111	−	0.256408i	−0.705340	+	0.708869i	$0.749206\pi$
$654$		0			0
$655$	−3.93464e14	−	6.81500e14i	−0.127520	−	0.220871i
$656$		0			0
$657$	6.80657e14	−	9.73376e14i	0.216930	−	0.310221i
$658$		0			0
$659$			5.17044e15i			1.62053i	0.586062	+	0.810266i	$0.300678\pi$
$659$			5.17044e15i			1.62053i	−0.586062	+	0.810266i	$0.699322\pi$
$660$		0			0
$661$	3.01184e15	−	1.73889e15i	0.928375	−	0.535998i	0.0420781	−	0.999114i	$-0.486602\pi$
$661$	3.01184e15	−	1.73889e15i	0.928375	−	0.535998i	0.886297	+	0.463116i	$0.153269\pi$
$662$		0			0
$663$	−2.36208e14	−	2.57614e14i	−0.0716094	−	0.0780989i
$664$		0			0
$665$	−2.38956e15	−	7.83193e14i	−0.712522	−	0.233534i
$666$		0			0
$667$	3.77797e13	−	6.54364e13i	0.0110807	−	0.0191923i
$668$		0			0
$669$	5.33447e15	−	1.18438e15i	1.53903	−	0.341701i
$670$		0			0
$671$	1.71794e15			0.487568
$672$		0			0
$673$	−5.19492e14			−0.145043			−0.0725213	−	0.997367i	$-0.523105\pi$
$673$	−5.19492e14			−0.145043			−0.0725213	+	0.997367i	$0.523105\pi$
$674$		0			0
$675$	−3.92737e14	+	9.50677e14i	−0.107878	+	0.261133i
$676$		0			0
$677$	3.09956e15	−	5.36860e15i	0.837651	−	1.45085i	−0.0542031	−	0.998530i	$-0.517262\pi$
$677$	3.09956e15	−	5.36860e15i	0.837651	−	1.45085i	0.891854	−	0.452324i	$-0.149405\pi$
$678$		0			0
$679$	2.57519e15	−	2.30678e15i	0.684739	−	0.613371i
$680$		0			0
$681$	1.86810e15	−	1.71287e15i	0.488754	−	0.448142i
$682$		0			0
$683$	−6.86275e14	+	3.96221e14i	−0.176679	+	0.102006i	−0.585731	−	0.810505i	$-0.699193\pi$
$683$	−6.86275e14	+	3.96221e14i	−0.176679	+	0.102006i	0.409053	+	0.912511i	$0.365859\pi$
$684$		0			0
$685$			2.77902e15i			0.704034i
$686$		0			0
$687$	−4.14687e14	+	1.31666e15i	−0.103385	+	0.328254i
$688$		0			0
$689$	2.56866e15	+	4.44904e15i	0.630232	+	1.09159i
$690$		0			0
$691$	−1.57810e15	−	9.11118e14i	−0.381071	−	0.220011i	0.297213	−	0.954811i	$-0.403943\pi$
$691$	−1.57810e15	−	9.11118e14i	−0.381071	−	0.220011i	−0.678284	+	0.734800i	$0.737276\pi$
$692$		0			0
$693$	−3.28449e15	+	3.96826e14i	−0.780611	+	0.0943121i
$694$		0			0
$695$	5.24602e15	+	3.02879e15i	1.22719	+	0.708521i
$696$		0			0
$697$	1.58129e14	+	2.73888e14i	0.0364109	+	0.0630656i
$698$		0			0
$699$	−7.19530e14	+	2.28455e15i	−0.163089	+	0.517817i
$700$		0			0
$701$		−	5.75889e15i		−	1.28496i	−0.766302	−	0.642481i	$-0.777905\pi$
$701$		−	5.75889e15i		−	1.28496i	0.766302	−	0.642481i	$-0.222095\pi$
$702$		0			0
$703$	3.35540e15	−	1.93724e15i	0.737038	−	0.425529i
$704$		0			0
$705$	6.59425e15	−	6.04632e15i	1.42602	−	1.30753i
$706$		0			0
$707$	5.25117e14	−	1.60216e15i	0.111802	−	0.341113i
$708$		0			0
$709$	1.72464e15	−	2.98717e15i	0.361530	−	0.626189i	−0.626682	−	0.779275i	$-0.715588\pi$
$709$	1.72464e15	−	2.98717e15i	0.361530	−	0.626189i	0.988213	+	0.153086i	$0.0489210\pi$
$710$		0			0
$711$	−5.59015e14	+	6.43599e15i	−0.115383	+	1.32841i
$712$		0			0
$713$	1.73437e14			0.0352493
$714$		0			0
$715$	−4.79843e15			−0.960319
$716$		0			0
$717$	−1.09422e15	+	2.42942e14i	−0.215650	+	0.0478792i
$718$		0			0
$719$	1.63595e15	−	2.83355e15i	0.317513	−	0.549949i	−0.662455	−	0.749101i	$-0.730486\pi$
$719$	1.63595e15	−	2.83355e15i	0.317513	−	0.549949i	0.979969	+	0.199153i	$0.0638189\pi$
$720$		0			0
$721$	7.46874e14	+	3.55860e15i	0.142759	+	0.680198i
$722$		0			0
$723$	5.68333e15	+	6.19836e15i	1.06990	+	1.16685i
$724$		0			0
$725$	−8.98290e14	+	5.18628e14i	−0.166555	+	0.0961605i
$726$		0			0
$727$		−	6.23590e15i		−	1.13883i	−0.822049	−	0.569417i	$-0.807169\pi$
$727$		−	6.23590e15i		−	1.13883i	0.822049	−	0.569417i	$-0.192831\pi$
$728$		0			0
$729$	−1.42870e15	+	5.37233e15i	−0.257004	+	0.966410i
$730$		0			0
$731$	1.51674e14	+	2.62707e14i	0.0268761	+	0.0465508i
$732$		0			0
$733$	−2.42939e15	−	1.40261e15i	−0.424058	−	0.244830i	0.272754	−	0.962084i	$-0.412065\pi$
$733$	−2.42939e15	−	1.40261e15i	−0.424058	−	0.244830i	−0.696812	+	0.717254i	$0.745399\pi$
$734$		0			0
$735$	−2.65517e15	+	6.02696e15i	−0.456575	+	1.03638i
$736$		0			0
$737$	5.52617e15	+	3.19053e15i	0.936166	+	0.540496i
$738$		0			0
$739$	−2.43363e15	−	4.21517e15i	−0.406172	−	0.703511i	0.588285	−	0.808654i	$-0.299803\pi$
$739$	−2.43363e15	−	4.21517e15i	−0.406172	−	0.703511i	−0.994457	+	0.105143i	$0.966470\pi$
$740$		0			0
$741$	−4.14173e15	−	1.30446e15i	−0.681054	−	0.214501i
$742$		0			0
$743$			5.39615e15i			0.874270i	0.899396	+	0.437135i	$0.144007\pi$
$743$			5.39615e15i			0.874270i	−0.899396	+	0.437135i	$0.855993\pi$
$744$		0			0
$745$	−6.28005e15	+	3.62579e15i	−1.00254	+	0.578820i
$746$		0			0
$747$	−7.70101e15	+	3.59691e15i	−1.21139	+	0.565805i
$748$		0			0
$749$	−2.13063e15	−	1.01517e16i	−0.330261	−	1.57358i
$750$		0			0
$751$	5.36130e15	−	9.28604e15i	0.818936	−	1.41844i	−0.0875305	−	0.996162i	$-0.527898\pi$
$751$	5.36130e15	−	9.28604e15i	0.818936	−	1.41844i	0.906467	−	0.422277i	$-0.138769\pi$
$752$		0			0
$753$	2.78289e15	+	1.25342e16i	0.418913	+	1.88680i
$754$		0			0
$755$	−1.35071e16			−2.00380
$756$		0			0
$757$	−1.27253e16			−1.86055			−0.930277	−	0.366858i	$-0.880433\pi$
$757$	−1.27253e16			−1.86055			−0.930277	+	0.366858i	$0.880433\pi$
$758$		0			0
$759$	−3.85043e13	−	1.73425e14i	−0.00554855	−	0.0249908i
$760$		0			0
$761$	3.35419e15	−	5.80963e15i	0.476400	−	0.825150i	−0.523234	−	0.852189i	$-0.675275\pi$
$761$	3.35419e15	−	5.80963e15i	0.476400	−	0.825150i	0.999634	+	0.0270393i	$0.00860794\pi$
$762$		0			0
$763$	−1.73684e15	+	5.29917e15i	−0.243150	+	0.741862i
$764$		0			0
$765$	−7.30570e14	+	3.41227e14i	−0.100815	+	0.0470875i
$766$		0			0
$767$	−6.96796e15	+	4.02296e15i	−0.947832	+	0.547231i
$768$		0			0
$769$		−	2.36172e15i		−	0.316689i	−0.987384	−	0.158344i	$-0.949384\pi$
$769$		−	2.36172e15i		−	0.316689i	0.987384	−	0.158344i	$-0.0506156\pi$
$770$		0			0
$771$	6.98682e15	+	2.20053e15i	0.923593	+	0.290890i
$772$		0			0
$773$	1.77222e15	+	3.06958e15i	0.230957	+	0.400029i	0.958090	−	0.286468i	$-0.0924811\pi$
$773$	1.77222e15	+	3.06958e15i	0.230957	+	0.400029i	−0.727133	+	0.686496i	$0.759148\pi$
$774$		0			0
$775$	−2.06191e15	−	1.19045e15i	−0.264918	−	0.152950i
$776$		0			0
$777$	−4.18727e15	−	9.24319e15i	−0.530414	−	1.17086i
$778$		0			0
$779$	3.40274e15	+	1.96457e15i	0.424984	+	0.245365i
$780$		0			0
$781$	4.09955e15	+	7.10063e15i	0.504842	+	0.874412i
$782$		0			0
$783$	−4.44418e15	+	3.41678e15i	−0.539637	+	0.414884i
$784$		0			0
$785$		−	2.75929e15i		−	0.330380i
$786$		0			0
$787$	−1.88415e15	+	1.08781e15i	−0.222461	+	0.128438i	−0.607089	−	0.794634i	$-0.707663\pi$
$787$	−1.88415e15	+	1.08781e15i	−0.222461	+	0.128438i	0.384628	+	0.923072i	$0.374330\pi$
$788$		0			0
$789$	6.66060e15	+	7.26420e15i	0.775514	+	0.845794i
$790$		0			0
$791$	4.24721e15	−	3.80453e15i	0.487678	−	0.436848i
$792$		0			0
$793$	2.95274e15	−	5.11429e15i	0.334366	−	0.579139i
$794$		0			0
$795$	1.15701e16	−	2.56883e15i	1.29217	−	0.286891i
$796$		0			0
$797$	−8.98808e15			−0.990025			−0.495013	−	0.868886i	$-0.664837\pi$
$797$	−8.98808e15			−0.990025			−0.495013	+	0.868886i	$0.664837\pi$
$798$		0			0
$799$	1.54501e15			0.167851
$800$		0			0
$801$	4.05432e14	−	4.66778e15i	0.0434450	−	0.500186i
$802$		0			0
$803$	1.40800e15	−	2.43873e15i	0.148822	−	0.257768i
$804$		0			0
$805$	−3.36050e14	−	1.10142e14i	−0.0350369	−	0.0114836i
$806$		0			0
$807$	−1.10458e15	+	1.01280e15i	−0.113604	+	0.104164i
$808$		0			0
$809$	−1.27846e16	+	7.38118e15i	−1.29709	+	0.748874i	−0.979900	−	0.199489i	$-0.936072\pi$
$809$	−1.27846e16	+	7.38118e15i	−1.29709	+	0.748874i	−0.317188	+	0.948363i	$0.602739\pi$
$810$		0			0
$811$		−	1.66586e16i		−	1.66734i	−0.552261	−	0.833671i	$-0.686235\pi$
$811$		−	1.66586e16i		−	1.66734i	0.552261	−	0.833671i	$-0.313765\pi$
$812$		0			0
$813$	−2.99890e14	+	9.52169e14i	−0.0296117	+	0.0940191i
$814$		0			0
$815$	−5.55922e14	−	9.62885e14i	−0.0541561	−	0.0938010i
$816$		0			0
$817$	3.26384e15	+	1.88438e15i	0.313695	+	0.181112i
$818$		0			0
$819$	−4.46391e15	+	1.04599e16i	−0.423305	+	0.991898i
$820$		0			0
$821$	−1.41291e16	−	8.15745e15i	−1.32199	−	0.763250i	−0.337943	−	0.941167i	$-0.609731\pi$
$821$	−1.41291e16	−	8.15745e15i	−1.32199	−	0.763250i	−0.984046	+	0.177916i	$0.943064\pi$
$822$		0			0
$823$	6.67192e14	+	1.15561e15i	0.0615959	+	0.106687i	0.895179	−	0.445707i	$-0.147048\pi$
$823$	6.67192e14	+	1.15561e15i	0.0615959	+	0.106687i	−0.833583	+	0.552394i	$0.813714\pi$
$824$		0			0
$825$	−7.32599e14	+	2.32605e15i	−0.0667375	+	0.211896i
$826$		0			0
$827$			1.54809e15i			0.139160i	0.997576	+	0.0695801i	$0.0221659\pi$
$827$			1.54809e15i			0.139160i	−0.997576	+	0.0695801i	$0.977834\pi$
$828$		0			0
$829$	−1.13497e16	+	6.55276e15i	−1.00678	+	0.581265i	−0.910247	−	0.414065i	$-0.864109\pi$
$829$	−1.13497e16	+	6.55276e15i	−1.00678	+	0.581265i	−0.0965332	+	0.995330i	$0.530775\pi$
$830$		0			0
$831$	1.01819e16	−	9.33588e15i	0.891300	−	0.817240i
$832$		0			0
$833$	−1.04136e15	+	4.57261e14i	−0.0899610	+	0.0395018i
$834$		0			0
$835$	−2.69834e15	+	4.67367e15i	−0.230050	+	0.398458i
$836$		0			0
$837$	−1.18926e16	−	4.91299e15i	−1.00066	−	0.413386i
$838$		0			0
$839$	−1.57899e16			−1.31126			−0.655631	−	0.755082i	$-0.727597\pi$
$839$	−1.57899e16			−1.31126			−0.655631	+	0.755082i	$0.727597\pi$
$840$		0			0
$841$	6.54756e15			0.536663
$842$		0			0
$843$	−1.09586e16	+	2.43306e15i	−0.886547	+	0.196834i
$844$		0			0
$845$	−1.15620e15	+	2.00260e15i	−0.0923255	+	0.159912i
$846$		0			0
$847$	4.74001e15	−	9.94827e14i	0.373612	−	0.0784131i
$848$		0			0
$849$	−8.09244e15	−	8.82580e15i	−0.629633	−	0.686692i
$850$		0			0
$851$	4.71879e14	−	2.72439e14i	0.0362425	−	0.0209246i
$852$		0			0
$853$		−	1.86206e16i		−	1.41180i	−0.708309	−	0.705902i	$-0.750542\pi$
$853$		−	1.86206e16i		−	1.41180i	0.708309	−	0.705902i	$-0.249458\pi$
$854$		0			0
$855$	−5.74079e15	+	8.20964e15i	−0.429693	+	0.614484i
$856$		0			0
$857$	5.37825e15	+	9.31540e15i	0.397417	+	0.688346i	0.993406	−	0.114646i	$-0.0365733\pi$
$857$	5.37825e15	+	9.31540e15i	0.397417	+	0.688346i	−0.595989	+	0.802992i	$0.703240\pi$
$858$		0			0
$859$	−1.21534e16	−	7.01678e15i	−0.886617	−	0.511888i	−0.0137823	−	0.999905i	$-0.504387\pi$
$859$	−1.21534e16	−	7.01678e15i	−0.886617	−	0.511888i	−0.872834	+	0.488017i	$0.837721\pi$
$860$		0			0
$861$	5.99460e15	−	8.36424e15i	0.431761	−	0.602434i
$862$		0			0
$863$	1.97250e15	+	1.13882e15i	0.140267	+	0.0809834i	0.568491	−	0.822689i	$-0.307527\pi$
$863$	1.97250e15	+	1.13882e15i	0.140267	+	0.0809834i	−0.428224	+	0.903673i	$0.640861\pi$
$864$		0			0
$865$	1.03346e16	+	1.79001e16i	0.725615	+	1.25680i
$866$		0			0
$867$	1.36256e16	+	4.29143e15i	0.944604	+	0.297507i
$868$		0			0
$869$			1.53163e16i			1.04845i
$870$		0			0
$871$	1.89964e16	−	1.09676e16i	1.28402	−	0.741327i
$872$		0			0
$873$	−5.82856e15	−	1.24790e16i	−0.389030	−	0.832915i
$874$		0			0
$875$	−8.22525e15	−	9.18230e15i	−0.542132	−	0.605211i
$876$		0			0
$877$	−4.45135e15	+	7.70996e15i	−0.289730	+	0.501827i	−0.973745	−	0.227640i	$-0.926899\pi$
$877$	−4.45135e15	+	7.70996e15i	−0.289730	+	0.501827i	0.684015	+	0.729468i	$0.260232\pi$
$878$		0			0
$879$	2.68885e14	+	1.21107e15i	0.0172833	+	0.0778447i
$880$		0			0
$881$	2.28068e16			1.44776			0.723881	−	0.689925i	$-0.242356\pi$
$881$	2.28068e16			1.44776			0.723881	+	0.689925i	$0.242356\pi$
$882$		0			0
$883$	1.40349e16			0.879883			0.439942	−	0.898026i	$-0.354999\pi$
$883$	1.40349e16			0.879883			0.439942	+	0.898026i	$0.354999\pi$
$884$		0			0
$885$	4.02323e15	+	1.81208e16i	0.249108	+	1.12199i
$886$		0			0
$887$	−1.05437e16	+	1.82622e16i	−0.644783	+	1.11680i	0.339568	+	0.940581i	$0.389719\pi$
$887$	−1.05437e16	+	1.82622e16i	−0.644783	+	1.11680i	−0.984352	+	0.176216i	$0.943614\pi$
$888$		0			0
$889$	−1.31648e16	−	1.46965e16i	−0.795158	−	0.887679i
$890$		0			0
$891$	−2.27239e15	+	1.29824e16i	−0.135567	+	0.774512i
$892$		0			0
$893$	1.66233e16	−	9.59748e15i	0.979569	−	0.565554i
$894$		0			0
$895$			2.64364e16i			1.53878i
$896$		0			0
$897$	−5.82462e14	−	1.83449e14i	−0.0334895	−	0.0105477i
$898$		0			0
$899$	−6.48783e15	−	1.12372e16i	−0.368486	−	0.638237i
$900$		0			0
$901$	1.77251e15	+	1.02336e15i	0.0994493	+	0.0574171i
$902$		0			0
$903$	5.74990e15	−	8.02281e15i	0.318697	−	0.444676i
$904$		0			0
$905$	2.93269e16	+	1.69319e16i	1.60583	+	0.927127i
$906$		0			0
$907$	−2.70935e15	−	4.69274e15i	−0.146563	−	0.253855i	0.783392	−	0.621528i	$-0.213488\pi$
$907$	−2.70935e15	−	4.69274e15i	−0.146563	−	0.253855i	−0.929955	+	0.367673i	$0.880155\pi$
$908$		0			0
$909$	−5.50442e15	−	3.84910e15i	−0.294178	−	0.205711i
$910$		0			0
$911$		−	2.06305e16i		−	1.08933i	−0.838654	−	0.544664i	$-0.816657\pi$
$911$		−	2.06305e16i		−	1.08933i	0.838654	−	0.544664i	$-0.183343\pi$
$912$		0			0
$913$	−1.74517e16	+	1.00757e16i	−0.910432	+	0.525638i
$914$		0			0
$915$	−9.20739e15	−	1.00418e16i	−0.474592	−	0.517601i
$916$		0			0
$917$	−4.32757e15	+	9.08263e14i	−0.220400	+	0.0462573i
$918$		0			0
$919$	−3.73749e15	+	6.47352e15i	−0.188081	+	0.325766i	−0.944610	−	0.328194i	$-0.893560\pi$
$919$	−3.73749e15	+	6.47352e15i	−0.188081	+	0.325766i	0.756530	+	0.653960i	$0.226893\pi$
$920$		0			0
$921$	1.49867e16	−	3.32739e15i	0.745207	−	0.165453i
$922$		0			0
$923$	2.81846e16			1.38485
$924$		0			0
$925$	−7.47992e15			−0.363176
$926$		0			0
$927$	1.44312e16	+	1.25346e15i	0.692411	+	0.0601412i
$928$		0			0
$929$	1.07530e16	−	1.86248e16i	0.509852	−	0.883089i	−0.490083	−	0.871676i	$-0.663034\pi$
$929$	1.07530e16	−	1.86248e16i	0.509852	−	0.883089i	0.999935	−	0.0114133i	$-0.00363304\pi$
$930$		0			0
$931$	−8.36394e15	+	1.13887e16i	−0.391911	+	0.533644i
$932$		0			0
$933$	2.76598e16	−	2.53615e16i	1.28086	−	1.17443i
$934$		0			0
$935$	−1.65558e15	+	9.55851e14i	−0.0757682	+	0.0437448i
$936$		0			0
$937$			2.14239e16i			0.969017i	0.874786	+	0.484509i	$0.161002\pi$
$937$			2.14239e16i			0.969017i	−0.874786	+	0.484509i	$0.838998\pi$
$938$		0			0
$939$	8.88012e15	−	2.81949e16i	0.396971	−	1.26041i
$940$		0			0
$941$	6.58944e15	+	1.14132e16i	0.291142	+	0.504273i	0.974080	−	0.226203i	$-0.0726314\pi$
$941$	6.58944e15	+	1.14132e16i	0.291142	+	0.504273i	−0.682938	+	0.730476i	$0.739298\pi$
$942$		0			0
$943$	4.78536e14	+	2.76283e14i	0.0208978	+	0.0120654i
$944$		0			0
$945$	1.99229e16	+	1.70718e16i	0.859958	+	0.736893i
$946$		0			0
$947$	8.23381e15	+	4.75379e15i	0.351298	+	0.202822i	0.665257	−	0.746614i	$-0.268322\pi$
$947$	8.23381e15	+	4.75379e15i	0.351298	+	0.202822i	−0.313959	+	0.949437i	$0.601655\pi$
$948$		0			0
$949$	−4.84004e15	−	8.38320e15i	−0.204120	−	0.353546i
$950$		0			0
$951$	−5.07990e15	+	1.61290e16i	−0.211769	+	0.672379i
$952$		0			0
$953$			2.42402e16i			0.998909i	0.866340	+	0.499454i	$0.166466\pi$
$953$			2.42402e16i			0.998909i	−0.866340	+	0.499454i	$0.833534\pi$
$954$		0			0
$955$	3.07248e16	−	1.77390e16i	1.25161	−	0.722620i
$956$		0			0
$957$	−9.79607e15	+	8.98209e15i	−0.394490	+	0.361711i
$958$		0			0
$959$	1.48390e16	+	4.86357e15i	0.590747	+	0.193621i
$960$		0			0
$961$	2.18778e15	−	3.78935e15i	0.0861044	−	0.149137i
$962$		0			0
$963$	−4.11682e16	−	3.57577e15i	−1.60183	−	0.139132i
$964$		0			0
$965$	−9.40933e15			−0.361958
$966$		0			0
$967$	1.31942e16			0.501809			0.250904	−	0.968012i	$-0.419272\pi$
$967$	1.31942e16			0.501809			0.250904	+	0.968012i	$0.419272\pi$
$968$		0			0
$969$	−1.68885e15	+	3.74965e14i	−0.0635055	+	0.0140997i
$970$		0			0
$971$	9.34749e15	−	1.61903e16i	0.347528	−	0.601936i	−0.638282	−	0.769803i	$-0.720355\pi$
$971$	9.34749e15	−	1.61903e16i	0.347528	−	0.601936i	0.985810	+	0.167867i	$0.0536879\pi$
$972$		0			0
$973$	2.53537e16	−	2.27111e16i	0.932010	−	0.834869i
$974$		0			0
$975$	5.66545e15	+	6.17886e15i	0.205925	+	0.224586i
$976$		0			0
$977$	7.06726e15	−	4.08028e15i	0.253998	−	0.146646i	−0.367595	−	0.929986i	$-0.619819\pi$
$977$	7.06726e15	−	4.08028e15i	0.253998	−	0.146646i	0.621594	+	0.783340i	$0.286486\pi$
$978$		0			0
$979$		−	1.11084e16i		−	0.394771i
$980$		0			0
$981$	1.82060e16	+	1.27310e16i	0.639786	+	0.447386i
$982$		0			0
$983$	−1.74611e16	−	3.02436e16i	−0.606776	−	1.05097i	−0.991768	−	0.128047i	$-0.959129\pi$
$983$	−1.74611e16	−	3.02436e16i	−0.606776	−	1.05097i	0.384992	−	0.922920i	$-0.374204\pi$
$984$		0			0
$985$	1.26842e16	+	7.32325e15i	0.435878	+	0.251654i
$986$		0			0
$987$	−2.07446e16	−	4.57926e16i	−0.704952	−	1.55615i
$988$		0			0
$989$	4.59002e14	+	2.65005e14i	0.0154254	+	0.00890583i
$990$		0			0
$991$	−1.67476e16	−	2.90077e16i	−0.556605	−	0.964068i	−0.997777	−	0.0666452i	$-0.978770\pi$
$991$	−1.67476e16	−	2.90077e16i	−0.556605	−	0.964068i	0.441172	−	0.897423i	$-0.354563\pi$
$992$		0			0
$993$	1.07399e16	+	3.38259e15i	0.353005	+	0.111180i
$994$		0			0
$995$		−	4.40776e16i		−	1.43282i
$996$		0			0
$997$	−1.49053e16	+	8.60555e15i	−0.479199	+	0.276666i	−0.720083	−	0.693888i	$-0.755896\pi$
$997$	−1.49053e16	+	8.60555e15i	−0.479199	+	0.276666i	0.240884	+	0.970554i	$0.422563\pi$
$998$		0			0
$999$	−4.00741e16	+	5.31416e15i	−1.27425	+	0.168976i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.k.b.5.17	yes	56
3.2	odd	2	inner	84.12.k.b.5.6	✓	56
7.3	odd	6	inner	84.12.k.b.17.6	yes	56
21.17	even	6	inner	84.12.k.b.17.17	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.k.b.5.6	✓	56	3.2	odd	2	inner
84.12.k.b.5.17	yes	56	1.1	even	1	trivial
84.12.k.b.17.6	yes	56	7.3	odd	6	inner
84.12.k.b.17.17	yes	56	21.17	even	6	inner

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.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(91.2256 + 410.883i) q^{3} +(3956.77 - 6853.32i) q^{5} +(-29669.5 - 33121.7i) q^{7} +(-160503. + 74966.1i) q^{9} +O(q^{10})\)	\(q+(91.2256 + 410.883i) q^{3} +(3956.77 - 6853.32i) q^{5} +(-29669.5 - 33121.7i) q^{7} +(-160503. + 74966.1i) q^{9} +(-363724. + 209996. i) q^{11} +1.44373e6i q^{13} +(3.17687e6 + 1.00057e6i) q^{15} +(287593. + 498126. i) q^{17} +(6.18865e6 + 3.57302e6i) q^{19} +(1.09025e7 - 1.52122e7i) q^{21} +(870326. + 502483. i) q^{23} +(-6.89792e6 - 1.19475e7i) q^{25} +(-4.54443e7 - 5.91091e7i) q^{27} -7.51861e7i q^{29} +(1.49459e8 - 8.62903e7i) q^{31} +(-1.19465e8 - 1.30291e8i) q^{33} +(-3.44389e8 + 7.22798e7i) q^{35} +(2.71093e8 - 4.69547e8i) q^{37} +(-5.93206e8 + 1.31706e8i) q^{39} +5.49836e8 q^{41} +5.27391e8 q^{43} +(-1.21305e8 + 1.39660e9i) q^{45} +(1.34305e9 - 2.32623e9i) q^{47} +(-2.16767e8 + 1.96541e9i) q^{49} +(-1.78436e8 + 1.63609e8i) q^{51} +(3.08162e9 - 1.77918e9i) q^{53} +3.32362e9i q^{55} +(-9.03529e8 + 2.86876e9i) q^{57} +(2.78649e9 + 4.82635e9i) q^{59} +(-3.54241e9 - 2.04521e9i) q^{61} +(7.24504e9 + 3.09192e9i) q^{63} +(9.89437e9 + 5.71252e9i) q^{65} +(-7.59665e9 - 1.31578e10i) q^{67} +(-1.27066e8 + 4.03441e8i) q^{69} -1.95220e10i q^{71} +(-5.80661e9 + 3.35245e9i) q^{73} +(4.27978e9 - 3.92416e9i) q^{75} +(1.77469e10 + 5.81667e9i) q^{77} +(1.82341e10 - 3.15824e10i) q^{79} +(2.01412e10 - 2.40645e10i) q^{81} +4.79805e10 q^{83} +4.55176e9 q^{85} +(3.08927e10 - 6.85890e9i) q^{87} +(-1.32245e10 + 2.29055e10i) q^{89} +(4.78189e10 - 4.28349e10i) q^{91} +(4.90897e10 + 5.35383e10i) q^{93} +(4.89740e10 - 2.82752e10i) q^{95} +7.77493e10i q^{97} +(4.26361e10 - 6.09719e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 729 q^{3} - 27194 q^{7} - 172347 q^{9}+O(q^{10}) \) 56 * q - 729 * q^3 - 27194 * q^7 - 172347 * q^9	\( 56 q - 729 q^{3} - 27194 q^{7} - 172347 q^{9} - 4853058 q^{15} + 28700520 q^{19} - 11325429 q^{21} - 316601194 q^{25} - 1368416388 q^{31} + 40874949 q^{33} - 87435712 q^{37} + 1177474410 q^{39} - 3055078348 q^{43} + 4109921793 q^{45} - 742582522 q^{49} - 694793715 q^{51} + 14605100370 q^{57} + 72584834058 q^{61} - 7310837811 q^{63} + 6131679148 q^{67} - 74402605464 q^{73} - 161115157854 q^{75} + 52181713528 q^{79} + 44948282337 q^{81} + 4658488716 q^{85} + 243101263104 q^{87} - 85311757146 q^{91} - 256628211777 q^{93} + 157345775874 q^{99}+O(q^{100}) \) 56 * q - 729 * q^3 - 27194 * q^7 - 172347 * q^9 - 4853058 * q^15 + 28700520 * q^19 - 11325429 * q^21 - 316601194 * q^25 - 1368416388 * q^31 + 40874949 * q^33 - 87435712 * q^37 + 1177474410 * q^39 - 3055078348 * q^43 + 4109921793 * q^45 - 742582522 * q^49 - 694793715 * q^51 + 14605100370 * q^57 + 72584834058 * q^61 - 7310837811 * q^63 + 6131679148 * q^67 - 74402605464 * q^73 - 161115157854 * q^75 + 52181713528 * q^79 + 44948282337 * q^81 + 4658488716 * q^85 + 243101263104 * q^87 - 85311757146 * q^91 - 256628211777 * q^93 + 157345775874 * q^99

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