LMFDB - Embedded newform 76.8.e.a.45.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,8,Mod(45,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("76.45");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	76.e (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:	$23.7412619368$
Analytic rank:	$0$
Dimension:	$22$
Relative dimension:	$11$ over $\Q(\zeta_{3})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			45.9
Character	$\chi$	$=$	76.45
Dual form			76.8.e.a.49.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(20.3841 + 35.3063i) q^{3} +(172.055 + 298.008i) q^{5} +763.824 q^{7} +(262.475 - 454.621i) q^{9} +O(q^{10})$	\(q+(20.3841 + 35.3063i) q^{3} +(172.055 + 298.008i) q^{5} +763.824 q^{7} +(262.475 - 454.621i) q^{9} +7897.42 q^{11} +(-2125.23 + 3681.00i) q^{13} +(-7014.37 + 12149.2i) q^{15} +(-9333.93 - 16166.8i) q^{17} +(26381.1 - 14068.1i) q^{19} +(15569.9 + 26967.8i) q^{21} +(-52061.5 + 90173.2i) q^{23} +(-20143.2 + 34889.1i) q^{25} +110561. q^{27} +(57126.3 - 98945.6i) q^{29} -254738. q^{31} +(160982. + 278829. i) q^{33} +(131420. + 227625. i) q^{35} -269808. q^{37} -173283. q^{39} +(355506. + 615754. i) q^{41} +(377944. + 654618. i) q^{43} +180641. q^{45} +(142495. - 246809. i) q^{47} -240116. q^{49} +(380528. - 659094. i) q^{51} +(-111042. + 192331. i) q^{53} +(1.35879e6 + 2.35349e6i) q^{55} +(1.03445e6 + 644653. i) q^{57} +(-1.18510e6 - 2.05264e6i) q^{59} +(-1.00808e6 + 1.74604e6i) q^{61} +(200485. - 347250. i) q^{63} -1.46262e6 q^{65} +(1.18150e6 - 2.04642e6i) q^{67} -4.24491e6 q^{69} +(-2.86418e6 - 4.96090e6i) q^{71} +(-1.26377e6 - 2.18891e6i) q^{73} -1.64241e6 q^{75} +6.03224e6 q^{77} +(-1.01818e6 - 1.76354e6i) q^{79} +(1.67966e6 + 2.90926e6i) q^{81} -1.00053e6 q^{83} +(3.21190e6 - 5.56317e6i) q^{85} +4.65788e6 q^{87} +(1.55510e6 - 2.69352e6i) q^{89} +(-1.62330e6 + 2.81163e6i) q^{91} +(-5.19262e6 - 8.99387e6i) q^{93} +(8.73139e6 + 5.44128e6i) q^{95} +(-278642. - 482622. i) q^{97} +(2.07288e6 - 3.59033e6i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9}+O(q^{10}) $ 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9	$ 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9} + 472 q^{11} - 567 q^{13} + 2995 q^{15} + 5589 q^{17} + 80912 q^{19} + 44412 q^{21} - 15425 q^{23} - 32806 q^{25} + 50290 q^{27} - 18919 q^{29} + 150296 q^{31} + 314618 q^{33} + 92808 q^{35} + 350100 q^{37} + 948810 q^{39} + 698891 q^{41} + 402545 q^{43} + 1477508 q^{45} - 653621 q^{47} - 1938490 q^{49} - 1386401 q^{51} - 106763 q^{53} + 414508 q^{55} + 1267563 q^{57} + 3136737 q^{59} + 2004581 q^{61} + 1465000 q^{63} - 7397638 q^{65} + 4344391 q^{67} + 1732238 q^{69} - 133823 q^{71} - 8349685 q^{73} - 12136824 q^{75} + 9147480 q^{77} - 94679 q^{79} - 838595 q^{81} - 2884080 q^{83} - 1421409 q^{85} - 31740598 q^{87} - 7039347 q^{89} + 1520096 q^{91} - 1993628 q^{93} + 1707587 q^{95} + 13308115 q^{97} + 6011488 q^{99}+O(q^{100}) $ 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9 + 472 * q^11 - 567 * q^13 + 2995 * q^15 + 5589 * q^17 + 80912 * q^19 + 44412 * q^21 - 15425 * q^23 - 32806 * q^25 + 50290 * q^27 - 18919 * q^29 + 150296 * q^31 + 314618 * q^33 + 92808 * q^35 + 350100 * q^37 + 948810 * q^39 + 698891 * q^41 + 402545 * q^43 + 1477508 * q^45 - 653621 * q^47 - 1938490 * q^49 - 1386401 * q^51 - 106763 * q^53 + 414508 * q^55 + 1267563 * q^57 + 3136737 * q^59 + 2004581 * q^61 + 1465000 * q^63 - 7397638 * q^65 + 4344391 * q^67 + 1732238 * q^69 - 133823 * q^71 - 8349685 * q^73 - 12136824 * q^75 + 9147480 * q^77 - 94679 * q^79 - 838595 * q^81 - 2884080 * q^83 - 1421409 * q^85 - 31740598 * q^87 - 7039347 * q^89 + 1520096 * q^91 - 1993628 * q^93 + 1707587 * q^95 + 13308115 * q^97 + 6011488 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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
$3$	20.3841	+	35.3063i	0.435881	+	0.754967i	0.997367	−	0.0725186i	$-0.0231037\pi$
$3$	20.3841	+	35.3063i	0.435881	+	0.754967i	−0.561486	+	0.827486i	$0.689770\pi$
$4$		0			0
$5$	172.055	+	298.008i	0.615562	+	1.06618i	0.990286	+	0.139048i	$0.0444043\pi$
$5$	172.055	+	298.008i	0.615562	+	1.06618i	−0.374724	+	0.927137i	$0.622262\pi$
$6$		0			0
$7$	763.824			0.841686			0.420843	−	0.907134i	$-0.361734\pi$
$7$	763.824			0.841686			0.420843	+	0.907134i	$0.361734\pi$
$8$		0			0
$9$	262.475	−	454.621i	0.120016	−	0.207874i
$10$		0			0
$11$	7897.42			1.78900			0.894501	−	0.447066i	$-0.147531\pi$
$11$	7897.42			1.78900			0.894501	+	0.447066i	$0.147531\pi$
$12$		0			0
$13$	−2125.23	+	3681.00i	−0.268289	+	0.464691i	−0.968420	−	0.249324i	$-0.919792\pi$
$13$	−2125.23	+	3681.00i	−0.268289	+	0.464691i	0.700131	+	0.714014i	$0.253125\pi$
$14$		0			0
$15$	−7014.37	+	12149.2i	−0.536623	+	0.929458i
$16$		0			0
$17$	−9333.93	−	16166.8i	−0.460780	−	0.798094i	0.538220	−	0.842804i	$-0.319097\pi$
$17$	−9333.93	−	16166.8i	−0.460780	−	0.798094i	−0.999000	+	0.0447103i	$0.985764\pi$
$18$		0			0
$19$	26381.1	−	14068.1i	0.882378	−	0.470541i
$19$	26381.1	−	14068.1i	0.882378	−	0.470541i
$20$		0			0
$21$	15569.9	+	26967.8i	0.366874	+	0.635445i
$22$		0			0
$23$	−52061.5	+	90173.2i	−0.892215	+	1.54536i	−0.0550002	+	0.998486i	$0.517516\pi$
$23$	−52061.5	+	90173.2i	−0.892215	+	1.54536i	−0.837214	+	0.546875i	$0.815817\pi$
$24$		0			0
$25$	−20143.2	+	34889.1i	−0.257833	+	0.446580i
$26$		0			0
$27$	110561.			1.08101
$28$		0			0
$29$	57126.3	−	98945.6i	0.434954	−	0.753362i	−0.562338	−	0.826907i	$-0.690098\pi$
$29$	57126.3	−	98945.6i	0.434954	−	0.753362i	0.997292	+	0.0735454i	$0.0234314\pi$
$30$		0			0
$31$	−254738.			−1.53578			−0.767889	−	0.640583i	$-0.778693\pi$
$31$	−254738.			−1.53578			−0.767889	+	0.640583i	$0.778693\pi$
$32$		0			0
$33$	160982.	+	278829.i	0.779791	+	1.35064i
$34$		0			0
$35$	131420.	+	227625.i	0.518110	+	0.897392i
$36$		0			0
$37$	−269808.			−0.875686			−0.437843	−	0.899051i	$-0.644257\pi$
$37$	−269808.			−0.875686			−0.437843	+	0.899051i	$0.644257\pi$
$38$		0			0
$39$	−173283.			−0.467768
$40$		0			0
$41$	355506.	+	615754.i	0.805570	+	1.39529i	0.915906	+	0.401394i	$0.131474\pi$
$41$	355506.	+	615754.i	0.805570	+	1.39529i	−0.110336	+	0.993894i	$0.535193\pi$
$42$		0			0
$43$	377944.	+	654618.i	0.724916	+	1.25559i	0.959008	+	0.283378i	$0.0914550\pi$
$43$	377944.	+	654618.i	0.724916	+	1.25559i	−0.234092	+	0.972214i	$0.575212\pi$
$44$		0			0
$45$	180641.			0.295510
$46$		0			0
$47$	142495.	−	246809.i	0.200197	−	0.346752i	−0.748394	−	0.663254i	$-0.769175\pi$
$47$	142495.	−	246809.i	0.200197	−	0.346752i	0.948592	+	0.316502i	$0.102508\pi$
$48$		0			0
$49$	−240116.			−0.291565
$50$		0			0
$51$	380528.	−	659094.i	0.401690	−	0.695747i
$52$		0			0
$53$	−111042.	+	192331.i	−0.102453	+	0.177453i	−0.912695	−	0.408642i	$-0.866002\pi$
$53$	−111042.	+	192331.i	−0.102453	+	0.177453i	0.810242	+	0.586096i	$0.199336\pi$
$54$		0			0
$55$	1.35879e6	+	2.35349e6i	1.10124	+	1.90741i
$56$		0			0
$57$	1.03445e6	+	644653.i	0.739855	+	0.461067i
$58$		0			0
$59$	−1.18510e6	−	2.05264e6i	−0.751227	−	1.30116i	−0.947228	−	0.320560i	$-0.896129\pi$
$59$	−1.18510e6	−	2.05264e6i	−0.751227	−	1.30116i	0.196001	−	0.980604i	$-0.437204\pi$
$60$		0			0
$61$	−1.00808e6	+	1.74604e6i	−0.568643	+	0.984918i	0.428058	+	0.903751i	$0.359198\pi$
$61$	−1.00808e6	+	1.74604e6i	−0.568643	+	0.984918i	−0.996701	+	0.0811670i	$0.974135\pi$
$62$		0			0
$63$	200485.	−	347250.i	0.101016	−	0.174965i
$64$		0			0
$65$	−1.46262e6			−0.660595
$66$		0			0
$67$	1.18150e6	−	2.04642e6i	0.479923	−	0.831251i	−0.519812	−	0.854281i	$-0.673998\pi$
$67$	1.18150e6	−	2.04642e6i	0.479923	−	0.831251i	0.999735	+	0.0230300i	$0.00733131\pi$
$68$		0			0
$69$	−4.24491e6			−1.55560
$70$		0			0
$71$	−2.86418e6	−	4.96090e6i	−0.949720	−	1.64496i	−0.746013	−	0.665931i	$-0.768034\pi$
$71$	−2.86418e6	−	4.96090e6i	−0.949720	−	1.64496i	−0.203707	−	0.979032i	$-0.565299\pi$
$72$		0			0
$73$	−1.26377e6	−	2.18891e6i	−0.380222	−	0.658563i	0.610872	−	0.791729i	$-0.290819\pi$
$73$	−1.26377e6	−	2.18891e6i	−0.380222	−	0.658563i	−0.991094	+	0.133166i	$0.957486\pi$
$74$		0			0
$75$	−1.64241e6			−0.449538
$76$		0			0
$77$	6.03224e6			1.50578
$78$		0			0
$79$	−1.01818e6	−	1.76354e6i	−0.232343	−	0.402430i	0.726154	−	0.687532i	$-0.241306\pi$
$79$	−1.01818e6	−	1.76354e6i	−0.232343	−	0.402430i	−0.958497	+	0.285102i	$0.907973\pi$
$80$		0			0
$81$	1.67966e6	+	2.90926e6i	0.351176	+	0.608255i
$82$		0			0
$83$	−1.00053e6			−0.192069			−0.0960345	−	0.995378i	$-0.530616\pi$
$83$	−1.00053e6			−0.192069			−0.0960345	+	0.995378i	$0.530616\pi$
$84$		0			0
$85$	3.21190e6	−	5.56317e6i	0.567277	−	0.982553i
$86$		0			0
$87$	4.65788e6			0.758352
$88$		0			0
$89$	1.55510e6	−	2.69352e6i	0.233827	−	0.405000i	−0.725104	−	0.688639i	$-0.758208\pi$
$89$	1.55510e6	−	2.69352e6i	0.233827	−	0.405000i	0.958931	+	0.283639i	$0.0915418\pi$
$90$		0			0
$91$	−1.62330e6	+	2.81163e6i	−0.225815	+	0.391123i
$92$		0			0
$93$	−5.19262e6	−	8.99387e6i	−0.669416	−	1.15946i
$94$		0			0
$95$	8.73139e6	+	5.44128e6i	1.04484	+	0.651131i
$96$		0			0
$97$	−278642.	−	482622.i	−0.0309989	−	0.0536916i	0.850110	−	0.526606i	$-0.176535\pi$
$97$	−278642.	−	482622.i	−0.0309989	−	0.0536916i	−0.881109	+	0.472914i	$0.843202\pi$
$98$		0			0
$99$	2.07288e6	−	3.59033e6i	0.214709	−	0.371887i
$100$		0			0
$101$	5.20859e6	−	9.02155e6i	0.503032	−	0.871277i	−0.496962	−	0.867772i	$-0.665551\pi$
$101$	5.20859e6	−	9.02155e6i	0.503032	−	0.871277i	0.999994	−	0.00350465i	$-0.00111557\pi$
$102$		0			0
$103$	−1.42567e7			−1.28555			−0.642776	−	0.766054i	$-0.722217\pi$
$103$	−1.42567e7			−1.28555			−0.642776	+	0.766054i	$0.722217\pi$
$104$		0			0
$105$	−5.35774e6	+	9.27988e6i	−0.451668	+	0.782312i
$106$		0			0
$107$	−1.55135e7			−1.22424			−0.612121	−	0.790764i	$-0.709683\pi$
$107$	−1.55135e7			−1.22424			−0.612121	+	0.790764i	$0.709683\pi$
$108$		0			0
$109$	2.13010e6	+	3.68944e6i	0.157546	+	0.272877i	0.933983	−	0.357317i	$-0.116308\pi$
$109$	2.13010e6	+	3.68944e6i	0.157546	+	0.272877i	−0.776437	+	0.630195i	$0.782975\pi$
$110$		0			0
$111$	−5.49979e6	−	9.52592e6i	−0.381695	−	0.661114i
$112$		0			0
$113$	2.30464e7			1.50255			0.751273	−	0.659991i	$-0.229440\pi$
$113$	2.30464e7			1.50255			0.751273	+	0.659991i	$0.229440\pi$
$114$		0			0
$115$	−3.58297e7			−2.19685
$116$		0			0
$117$	1.11564e6	+	1.93234e6i	0.0643981	+	0.111541i
$118$		0			0
$119$	−7.12948e6	−	1.23486e7i	−0.387832	−	0.671744i
$120$		0			0
$121$	4.28821e7			2.20053
$122$		0			0
$123$	−1.44933e7	+	2.51032e7i	−0.702265	+	1.21636i
$124$		0			0
$125$	1.30206e7			0.596275
$126$		0			0
$127$	−1.43040e6	+	2.47753e6i	−0.0619647	+	0.107326i	−0.895344	−	0.445376i	$-0.853070\pi$
$127$	−1.43040e6	+	2.47753e6i	−0.0619647	+	0.107326i	0.833379	+	0.552702i	$0.186403\pi$
$128$		0			0
$129$	−1.54081e7	+	2.66876e7i	−0.631954	+	1.09458i
$130$		0			0
$131$	1.66732e7	+	2.88789e7i	0.647992	+	1.12236i	0.983602	+	0.180354i	$0.0577244\pi$
$131$	1.66732e7	+	2.88789e7i	0.647992	+	1.12236i	−0.335609	+	0.942001i	$0.608942\pi$
$132$		0			0
$133$	2.01505e7	−	1.07455e7i	0.742685	−	0.396048i
$134$		0			0
$135$	1.90226e7	+	3.29482e7i	0.665430	+	1.15256i
$136$		0			0
$137$	−1.75793e7	+	3.04482e7i	−0.584089	+	1.01167i	0.410899	+	0.911681i	$0.365215\pi$
$137$	−1.75793e7	+	3.04482e7i	−0.584089	+	1.01167i	−0.994988	+	0.0999912i	$0.968119\pi$
$138$		0			0
$139$	1.11587e7	−	1.93274e7i	0.352421	−	0.610411i	−0.634252	−	0.773126i	$-0.718692\pi$
$139$	1.11587e7	−	1.93274e7i	0.352421	−	0.610411i	0.986673	+	0.162715i	$0.0520252\pi$
$140$		0			0
$141$	1.16186e7			0.349049
$142$		0			0
$143$	−1.67838e7	+	2.90704e7i	−0.479970	+	0.831332i
$144$		0			0
$145$	3.93154e7			1.07096
$146$		0			0
$147$	−4.89456e6	−	8.47763e6i	−0.127088	−	0.220122i
$148$		0			0
$149$	−1.07619e7	−	1.86402e7i	−0.266525	−	0.461635i	0.701437	−	0.712731i	$-0.252542\pi$
$149$	−1.07619e7	−	1.86402e7i	−0.266525	−	0.461635i	−0.967962	+	0.251097i	$0.919209\pi$
$150$		0			0
$151$	2.11138e7			0.499054			0.249527	−	0.968368i	$-0.419725\pi$
$151$	2.11138e7			0.499054			0.249527	+	0.968368i	$0.419725\pi$
$152$		0			0
$153$	−9.79971e6			−0.221204
$154$		0			0
$155$	−4.38289e7	−	7.59140e7i	−0.945366	−	1.63742i
$156$		0			0
$157$	9.37055e6	+	1.62303e7i	0.193249	+	0.334716i	0.946325	−	0.323217i	$-0.104764\pi$
$157$	9.37055e6	+	1.62303e7i	0.193249	+	0.334716i	−0.753076	+	0.657933i	$0.771431\pi$
$158$		0			0
$159$	−9.05401e6			−0.178629
$160$		0			0
$161$	−3.97658e7	+	6.88764e7i	−0.750964	+	1.30071i
$162$		0			0
$163$	5.64707e7			1.02133			0.510665	−	0.859780i	$-0.329399\pi$
$163$	5.64707e7			1.02133			0.510665	+	0.859780i	$0.329399\pi$
$164$		0			0
$165$	−5.53954e7	+	9.59477e7i	−0.960020	+	1.66280i
$166$		0			0
$167$	1.96978e7	−	3.41175e7i	0.327272	−	0.566852i	−0.654697	−	0.755891i	$-0.727204\pi$
$167$	1.96978e7	−	3.41175e7i	0.327272	−	0.566852i	0.981970	+	0.189039i	$0.0605372\pi$
$168$		0			0
$169$	2.23411e7	+	3.86959e7i	0.356042	+	0.616682i
$170$		0			0
$171$	528732.	−	1.56859e7i	0.00808628	−	0.239896i
$172$		0			0
$173$	−9.18329e6	−	1.59059e7i	−0.134846	−	0.233560i	0.790693	−	0.612213i	$-0.209721\pi$
$173$	−9.18329e6	−	1.59059e7i	−0.134846	−	0.233560i	−0.925538	+	0.378654i	$0.876387\pi$
$174$		0			0
$175$	−1.53859e7	+	2.66491e7i	−0.217014	+	0.375880i
$176$		0			0
$177$	4.83142e7	−	8.36827e7i	0.654891	−	1.13430i
$178$		0			0
$179$	2.85565e7			0.372151			0.186076	−	0.982535i	$-0.440423\pi$
$179$	2.85565e7			0.372151			0.186076	+	0.982535i	$0.440423\pi$
$180$		0			0
$181$	6.54109e7	−	1.13295e8i	0.819927	−	1.42015i	−0.0858088	−	0.996312i	$-0.527347\pi$
$181$	6.54109e7	−	1.13295e8i	0.819927	−	1.42015i	0.905736	−	0.423843i	$-0.139319\pi$
$182$		0			0
$183$	−8.21951e7			−0.991442
$184$		0			0
$185$	−4.64217e7	−	8.04048e7i	−0.539039	−	0.933643i
$186$		0			0
$187$	−7.37140e7	−	1.27676e8i	−0.824336	−	1.42779i
$188$		0			0
$189$	8.44495e7			0.909872
$190$		0			0
$191$	−1.80507e8			−1.87447			−0.937235	−	0.348699i	$-0.886623\pi$
$191$	−1.80507e8			−1.87447			−0.937235	+	0.348699i	$0.886623\pi$
$192$		0			0
$193$	3.89076e7	+	6.73900e7i	0.389569	+	0.674753i	0.992392	−	0.123122i	$-0.0392907\pi$
$193$	3.89076e7	+	6.73900e7i	0.389569	+	0.674753i	−0.602823	+	0.797875i	$0.705957\pi$
$194$		0			0
$195$	−2.98142e7	−	5.16398e7i	−0.287940	−	0.498727i
$196$		0			0
$197$	−1.00103e8			−0.932862			−0.466431	−	0.884558i	$-0.654460\pi$
$197$	−1.00103e8			−0.932862			−0.466431	+	0.884558i	$0.654460\pi$
$198$		0			0
$199$	4.11667e7	−	7.13028e7i	0.370305	−	0.641388i	−0.619307	−	0.785149i	$-0.712586\pi$
$199$	4.11667e7	−	7.13028e7i	0.370305	−	0.641388i	0.989612	+	0.143761i	$0.0459197\pi$
$200$		0			0
$201$	9.63352e7			0.836756
$202$		0			0
$203$	4.36344e7	−	7.55770e7i	0.366094	−	0.634094i
$204$		0			0
$205$	−1.22333e8	+	2.11887e8i	−0.991756	+	1.71777i
$206$		0			0
$207$	2.73297e7	+	4.73365e7i	0.214160	+	0.370937i
$208$		0			0
$209$	2.08342e8	−	1.11102e8i	1.57858	−	0.841799i
$210$		0			0
$211$	−8.52366e7	−	1.47634e8i	−0.624651	−	1.08193i	−0.988608	−	0.150512i	$-0.951908\pi$
$211$	−8.52366e7	−	1.47634e8i	−0.624651	−	1.08193i	0.363957	−	0.931416i	$-0.381425\pi$
$212$		0			0
$213$	1.16767e8	−	2.02247e8i	0.827929	−	1.43402i
$214$		0			0
$215$	−1.30054e8	+	2.25260e8i	−0.892462	+	1.54579i
$216$		0			0
$217$	−1.94575e8			−1.29264
$218$		0			0
$219$	5.15215e7	−	8.92379e7i	0.331462	−	0.574110i
$220$		0			0
$221$	7.93468e7			0.494489
$222$		0			0
$223$	2.26304e7	+	3.91969e7i	0.136655	+	0.236693i	0.926228	−	0.376963i	$-0.123032\pi$
$223$	2.26304e7	+	3.91969e7i	0.136655	+	0.236693i	−0.789574	+	0.613656i	$0.789698\pi$
$224$		0			0
$225$	1.05742e7	+	1.83150e7i	0.0618883	+	0.107194i
$226$		0			0
$227$	−5.05766e7			−0.286985			−0.143492	−	0.989651i	$-0.545833\pi$
$227$	−5.05766e7			−0.286985			−0.143492	+	0.989651i	$0.545833\pi$
$228$		0			0
$229$	1.90300e8			1.04716			0.523582	−	0.851975i	$-0.324595\pi$
$229$	1.90300e8			1.04716			0.523582	+	0.851975i	$0.324595\pi$
$230$		0			0
$231$	1.22962e8	+	2.12976e8i	0.656339	+	1.13681i
$232$		0			0
$233$	−6.73168e7	−	1.16596e8i	−0.348641	−	0.603863i	0.637368	−	0.770560i	$-0.280023\pi$
$233$	−6.73168e7	−	1.16596e8i	−0.348641	−	0.603863i	−0.986008	+	0.166697i	$0.946690\pi$
$234$		0			0
$235$	9.80681e7			0.492936
$236$		0			0
$237$	4.15094e7	−	7.18964e7i	0.202548	−	0.350823i
$238$		0			0
$239$	−1.54361e7			−0.0731382			−0.0365691	−	0.999331i	$-0.511643\pi$
$239$	−1.54361e7			−0.0731382			−0.0365691	+	0.999331i	$0.511643\pi$
$240$		0			0
$241$	8.15573e6	−	1.41261e7i	0.0375321	−	0.0650075i	−0.846649	−	0.532151i	$-0.821384\pi$
$241$	8.15573e6	−	1.41261e7i	0.0375321	−	0.0650075i	0.884181	+	0.467144i	$0.154717\pi$
$242$		0			0
$243$	5.24220e7	−	9.07976e7i	0.234364	−	0.405931i
$244$		0			0
$245$	−4.13132e7	−	7.15565e7i	−0.179476	−	0.310862i
$246$		0			0
$247$	−4.28106e6	+	1.27007e8i	−0.0180764	+	0.536274i
$248$		0			0
$249$	−2.03949e7	−	3.53251e7i	−0.0837191	−	0.145006i
$250$		0			0
$251$	−1.43578e8	+	2.48684e8i	−0.573100	+	0.992638i	0.423146	+	0.906062i	$0.360926\pi$
$251$	−1.43578e8	+	2.48684e8i	−0.573100	+	0.992638i	−0.996245	+	0.0865759i	$0.972407\pi$
$252$		0			0
$253$	−4.11152e8	+	7.12136e8i	−1.59617	+	2.76465i
$254$		0			0
$255$	2.61887e8			0.989060
$256$		0			0
$257$	2.40727e7	−	4.16952e7i	0.0884626	−	0.153222i	−0.818399	−	0.574651i	$-0.805138\pi$
$257$	2.40727e7	−	4.16952e7i	0.0884626	−	0.153222i	0.906861	+	0.421429i	$0.138471\pi$
$258$		0			0
$259$	−2.06086e8			−0.737052
$260$		0			0
$261$	−2.99885e7	−	5.19416e7i	−0.104403	−	0.180831i
$262$		0			0
$263$	−1.11604e8	−	1.93303e8i	−0.378298	−	0.655231i	0.612517	−	0.790457i	$-0.290157\pi$
$263$	−1.11604e8	−	1.93303e8i	−0.378298	−	0.655231i	−0.990815	+	0.135227i	$0.956824\pi$
$264$		0			0
$265$	−7.64216e7			−0.252264
$266$		0			0
$267$	1.26798e8			0.407682
$268$		0			0
$269$	−2.87382e7	−	4.97759e7i	−0.0900173	−	0.155914i	0.817501	−	0.575927i	$-0.195359\pi$
$269$	−2.87382e7	−	4.97759e7i	−0.0900173	−	0.155914i	−0.907518	+	0.420013i	$0.862026\pi$
$270$		0			0
$271$	344314.	+	596369.i	0.00105090	+	0.00182022i	0.866550	−	0.499090i	$-0.166332\pi$
$271$	344314.	+	596369.i	0.00105090	+	0.00182022i	−0.865499	+	0.500910i	$0.832999\pi$
$272$		0			0
$273$	−1.32358e8			−0.393714
$274$		0			0
$275$	−1.59079e8	+	2.75534e8i	−0.461264	+	0.798933i
$276$		0			0
$277$	4.27655e8			1.20897			0.604483	−	0.796618i	$-0.293380\pi$
$277$	4.27655e8			1.20897			0.604483	+	0.796618i	$0.293380\pi$
$278$		0			0
$279$	−6.68625e7	+	1.15809e8i	−0.184318	+	0.319248i
$280$		0			0
$281$	7.75581e7	−	1.34335e8i	0.208523	−	0.361173i	−0.742726	−	0.669595i	$-0.766468\pi$
$281$	7.75581e7	−	1.34335e8i	0.208523	−	0.361173i	0.951250	+	0.308422i	$0.0998009\pi$
$282$		0			0
$283$	2.37217e8	+	4.10871e8i	0.622147	+	1.07759i	0.989085	+	0.147344i	$0.0470725\pi$
$283$	2.37217e8	+	4.10871e8i	0.622147	+	1.07759i	−0.366939	+	0.930245i	$0.619594\pi$
$284$		0			0
$285$	−1.41298e7	+	4.19189e8i	−0.0361558	+	1.07264i
$286$		0			0
$287$	2.71544e8	+	4.70328e8i	0.678037	+	1.17439i
$288$		0			0
$289$	3.09248e7	−	5.35633e7i	0.0753640	−	0.130534i
$290$		0			0
$291$	1.13598e7	−	1.96757e7i	0.0270236	−	0.0468063i
$292$		0			0
$293$	−3.69999e8			−0.859338			−0.429669	−	0.902986i	$-0.641370\pi$
$293$	−3.69999e8			−0.859338			−0.429669	+	0.902986i	$0.641370\pi$
$294$		0			0
$295$	4.07803e8	−	7.06335e8i	0.924853	−	1.60189i
$296$		0			0
$297$	8.73150e8			1.93393
$298$		0			0
$299$	−2.21285e8	−	3.83277e8i	−0.478743	−	0.829207i
$300$		0			0
$301$	2.88683e8	+	5.00013e8i	0.610152	+	1.05681i
$302$		0			0
$303$	4.24690e8			0.877048
$304$		0			0
$305$	−6.93779e8			−1.40014
$306$		0			0
$307$	3.93752e8	+	6.81998e8i	0.776673	+	1.34524i	0.933849	+	0.357666i	$0.116428\pi$
$307$	3.93752e8	+	6.81998e8i	0.776673	+	1.34524i	−0.157176	+	0.987571i	$0.550239\pi$
$308$		0			0
$309$	−2.90611e8	−	5.03353e8i	−0.560347	−	0.970550i
$310$		0			0
$311$	−3.56302e8			−0.671672			−0.335836	−	0.941920i	$-0.609019\pi$
$311$	−3.56302e8			−0.671672			−0.335836	+	0.941920i	$0.609019\pi$
$312$		0			0
$313$	−3.79419e7	+	6.57172e7i	−0.0699380	+	0.121136i	−0.898874	−	0.438208i	$-0.855613\pi$
$313$	−3.79419e7	+	6.57172e7i	−0.0699380	+	0.121136i	0.828936	+	0.559344i	$0.188947\pi$
$314$		0			0
$315$	1.37978e8			0.248726
$316$		0			0
$317$	−1.64310e8	+	2.84594e8i	−0.289706	+	0.501786i	−0.973740	−	0.227665i	$-0.926891\pi$
$317$	−1.64310e8	+	2.84594e8i	−0.289706	+	0.501786i	0.684033	+	0.729451i	$0.260224\pi$
$318$		0			0
$319$	4.51150e8	−	7.81415e8i	0.778133	−	1.34777i
$320$		0			0
$321$	−3.16229e8	−	5.47725e8i	−0.533623	−	0.924262i
$322$		0			0
$323$	−4.73676e8	−	2.95188e8i	−0.782118	−	0.487405i
$324$		0			0
$325$	−8.56177e7	−	1.48294e8i	−0.138348	−	0.239625i
$326$		0			0
$327$	−8.68404e7	+	1.50412e8i	−0.137342	+	0.237884i
$328$		0			0
$329$	1.08841e8	−	1.88519e8i	0.168503	−	0.291856i
$330$		0			0
$331$	−2.39307e8			−0.362709			−0.181354	−	0.983418i	$-0.558048\pi$
$331$	−2.39307e8			−0.362709			−0.181354	+	0.983418i	$0.558048\pi$
$332$		0			0
$333$	−7.08179e7	+	1.22660e8i	−0.105096	+	0.182032i
$334$		0			0
$335$	8.13130e8			1.18169
$336$		0			0
$337$	−1.21762e8	−	2.10899e8i	−0.173304	−	0.300171i	0.766269	−	0.642520i	$-0.222111\pi$
$337$	−1.21762e8	−	2.10899e8i	−0.173304	−	0.300171i	−0.939573	+	0.342349i	$0.888778\pi$
$338$		0			0
$339$	4.69780e8	+	8.13683e8i	0.654931	+	1.13437i
$340$		0			0
$341$	−2.01178e9			−2.74751
$342$		0			0
$343$	−8.12448e8			−1.08709
$344$		0			0
$345$	−7.30358e8	−	1.26502e9i	−0.957566	−	1.65855i
$346$		0			0
$347$	−5.83780e8	−	1.01114e9i	−0.750061	−	1.29914i	−0.947793	−	0.318887i	$-0.896691\pi$
$347$	−5.83780e8	−	1.01114e9i	−0.750061	−	1.29914i	0.197732	−	0.980256i	$-0.436642\pi$
$348$		0			0
$349$	2.42418e8			0.305264			0.152632	−	0.988283i	$-0.451225\pi$
$349$	2.42418e8			0.305264			0.152632	+	0.988283i	$0.451225\pi$
$350$		0			0
$351$	−2.34968e8	+	4.06976e8i	−0.290024	+	0.502336i
$352$		0			0
$353$	−5.36964e8			−0.649731			−0.324865	−	0.945760i	$-0.605319\pi$
$353$	−5.36964e8			−0.649731			−0.324865	+	0.945760i	$0.605319\pi$
$354$		0			0
$355$	9.85591e8	−	1.70709e9i	1.16922	−	2.02515i
$356$		0			0
$357$	2.90656e8	−	5.03431e8i	0.338097	−	0.585601i
$358$		0			0
$359$	−2.13560e8	−	3.69896e8i	−0.243606	−	0.421938i	0.718133	−	0.695906i	$-0.244997\pi$
$359$	−2.13560e8	−	3.69896e8i	−0.243606	−	0.421938i	−0.961739	+	0.273968i	$0.911664\pi$
$360$		0			0
$361$	4.98049e8	−	7.42263e8i	0.557182	−	0.830391i
$362$		0			0
$363$	8.74113e8	+	1.51401e9i	0.959168	+	1.66133i
$364$		0			0
$365$	4.34874e8	−	7.53224e8i	0.468100	−	0.810773i
$366$		0			0
$367$	1.14235e8	−	1.97861e8i	0.120634	−	0.208943i	−0.799384	−	0.600820i	$-0.794841\pi$
$367$	1.14235e8	−	1.97861e8i	0.120634	−	0.208943i	0.920018	+	0.391877i	$0.128174\pi$
$368$		0			0
$369$	3.73246e8			0.386726
$370$		0			0
$371$	−8.48169e7	+	1.46907e8i	−0.0862330	+	0.149360i
$372$		0			0
$373$	−2.22173e8			−0.221672			−0.110836	−	0.993839i	$-0.535353\pi$
$373$	−2.22173e8			−0.221672			−0.110836	+	0.993839i	$0.535353\pi$
$374$		0			0
$375$	2.65414e8	+	4.59710e8i	0.259905	+	0.450168i
$376$		0			0
$377$	2.42812e8	+	4.20564e8i	0.233387	+	0.404238i
$378$		0			0
$379$	−4.44798e8			−0.419687			−0.209843	−	0.977735i	$-0.567295\pi$
$379$	−4.44798e8			−0.419687			−0.209843	+	0.977735i	$0.567295\pi$
$380$		0			0
$381$	−1.16630e8			−0.108037
$382$		0			0
$383$	−3.31050e7	−	5.73395e7i	−0.0301091	−	0.0521504i	0.850578	−	0.525849i	$-0.176252\pi$
$383$	−3.31050e7	−	5.73395e7i	−0.0301091	−	0.0521504i	−0.880687	+	0.473698i	$0.842919\pi$
$384$		0			0
$385$	1.03788e9	+	1.79765e9i	0.926899	+	1.60544i
$386$		0			0
$387$	3.96804e8			0.348007
$388$		0			0
$389$	−5.39474e8	+	9.34397e8i	−0.464673	+	0.804836i	−0.999187	−	0.0403231i	$-0.987161\pi$
$389$	−5.39474e8	+	9.34397e8i	−0.464673	+	0.804836i	0.534514	+	0.845160i	$0.320495\pi$
$390$		0			0
$391$	1.94375e9			1.64446
$392$		0			0
$393$	−6.79738e8	+	1.17734e9i	−0.564895	+	0.978426i
$394$		0			0
$395$	3.50366e8	−	6.06851e8i	0.286043	−	0.495441i
$396$		0			0
$397$	−4.36185e8	−	7.55495e8i	−0.349868	−	0.605989i	0.636358	−	0.771394i	$-0.280440\pi$
$397$	−4.36185e8	−	7.55495e8i	−0.349868	−	0.605989i	−0.986226	+	0.165405i	$0.947107\pi$
$398$		0			0
$399$	7.90136e8	+	4.92401e8i	0.622725	+	0.388073i
$400$		0			0
$401$	−5.83386e8	−	1.01045e9i	−0.451805	−	0.782549i	0.546694	−	0.837333i	$-0.315886\pi$
$401$	−5.83386e8	−	1.01045e9i	−0.451805	−	0.782549i	−0.998498	+	0.0547842i	$0.982553\pi$
$402$		0			0
$403$	5.41376e8	−	9.37691e8i	0.412033	−	0.713661i
$404$		0			0
$405$	−5.77989e8	+	1.00111e9i	−0.432341	+	0.748837i
$406$		0			0
$407$	−2.13078e9			−1.56660
$408$		0			0
$409$	1.26686e7	−	2.19427e7i	0.00915583	−	0.0158584i	−0.861411	−	0.507908i	$-0.830419\pi$
$409$	1.26686e7	−	2.19427e7i	0.00915583	−	0.0158584i	0.870567	+	0.492050i	$0.163752\pi$
$410$		0			0
$411$	−1.43335e9			−1.01837
$412$		0			0
$413$	−9.05204e8	−	1.56786e9i	−0.632297	−	1.09517i
$414$		0			0
$415$	−1.72146e8	−	2.98166e8i	−0.118230	−	0.204781i
$416$		0			0
$417$	9.09841e8			0.614454
$418$		0			0
$419$	2.27777e9			1.51273			0.756364	−	0.654151i	$-0.226974\pi$
$419$	2.27777e9			1.51273			0.756364	+	0.654151i	$0.226974\pi$
$420$		0			0
$421$	6.34147e8	+	1.09838e9i	0.414193	+	0.717403i	0.995343	−	0.0963929i	$-0.0307305\pi$
$421$	6.34147e8	+	1.09838e9i	0.414193	+	0.717403i	−0.581150	+	0.813796i	$0.697397\pi$
$422$		0			0
$423$	−7.48031e7	−	1.29563e8i	−0.0480539	−	0.0832317i
$424$		0			0
$425$	7.52061e8			0.475217
$426$		0			0
$427$	−7.69994e8	+	1.33367e9i	−0.478619	+	0.828992i
$428$		0			0
$429$	−1.36849e9			−0.836838
$430$		0			0
$431$	−7.45935e8	+	1.29200e9i	−0.448777	+	0.777305i	−0.998307	−	0.0581694i	$-0.981474\pi$
$431$	−7.45935e8	+	1.29200e9i	−0.448777	+	0.777305i	0.549530	+	0.835474i	$0.314807\pi$
$432$		0			0
$433$	−2.28240e8	+	3.95324e8i	−0.135109	+	0.234016i	−0.925639	−	0.378407i	$-0.876472\pi$
$433$	−2.28240e8	+	3.95324e8i	−0.135109	+	0.234016i	0.790530	+	0.612423i	$0.209805\pi$
$434$		0			0
$435$	8.01410e8	+	1.38808e9i	0.466812	+	0.808543i
$436$		0			0
$437$	−1.04873e8	+	3.11127e9i	−0.0601144	+	1.78342i
$438$		0			0
$439$	−1.05633e8	−	1.82961e8i	−0.0595898	−	0.103213i	0.834692	−	0.550718i	$-0.185646\pi$
$439$	−1.05633e8	−	1.82961e8i	−0.0595898	−	0.103213i	−0.894281	+	0.447505i	$0.852313\pi$
$440$		0			0
$441$	−6.30246e7	+	1.09162e8i	−0.0349925	+	0.0606088i
$442$		0			0
$443$	4.72858e8	−	8.19014e8i	0.258415	−	0.447588i	−0.707402	−	0.706811i	$-0.750133\pi$
$443$	4.72858e8	−	8.19014e8i	0.258415	−	0.447588i	0.965818	+	0.259223i	$0.0834665\pi$
$444$		0			0
$445$	1.07025e9			0.575740
$446$		0			0
$447$	4.38745e8	−	7.59928e8i	0.232346	−	0.402435i
$448$		0			0
$449$	−1.59343e9			−0.830753			−0.415376	−	0.909650i	$-0.636350\pi$
$449$	−1.59343e9			−0.830753			−0.415376	+	0.909650i	$0.636350\pi$
$450$		0			0
$451$	2.80758e9	+	4.86287e9i	1.44117	+	2.49617i
$452$		0			0
$453$	4.30387e8	+	7.45451e8i	0.217528	+	0.376769i
$454$		0			0
$455$	−1.11718e9			−0.556013
$456$		0			0
$457$	1.49871e9			0.734533			0.367266	−	0.930116i	$-0.380294\pi$
$457$	1.49871e9			0.734533			0.367266	+	0.930116i	$0.380294\pi$
$458$		0			0
$459$	−1.03197e9	−	1.78743e9i	−0.498109	−	0.862749i
$460$		0			0
$461$	−8.40723e8	−	1.45618e9i	−0.399668	−	0.692246i	0.594016	−	0.804453i	$-0.297541\pi$
$461$	−8.40723e8	−	1.45618e9i	−0.399668	−	0.692246i	−0.993685	+	0.112207i	$0.964208\pi$
$462$		0			0
$463$	−2.39074e9			−1.11943			−0.559717	−	0.828684i	$-0.689090\pi$
$463$	−2.39074e9			−1.11943			−0.559717	+	0.828684i	$0.689090\pi$
$464$		0			0
$465$	1.78683e9	−	3.09488e9i	0.824134	−	1.42744i
$466$		0			0
$467$	−3.94773e8			−0.179365			−0.0896827	−	0.995970i	$-0.528585\pi$
$467$	−3.94773e8			−0.179365			−0.0896827	+	0.995970i	$0.528585\pi$
$468$		0			0
$469$	9.02457e8	−	1.56310e9i	0.403944	−	0.699652i
$470$		0			0
$471$	−3.82021e8	+	6.61679e8i	−0.168467	+	0.291793i
$472$		0			0
$473$	2.98478e9	+	5.16980e9i	1.29688	+	2.24626i
$474$		0			0
$475$	−4.05766e7	+	1.20379e9i	−0.0173719	+	0.515374i
$476$		0			0
$477$	5.82918e7	+	1.00964e8i	0.0245920	+	0.0425945i
$478$		0			0
$479$	1.31983e9	−	2.28602e9i	0.548713	−	0.950398i	−0.449650	−	0.893205i	$-0.648451\pi$
$479$	1.31983e9	−	2.28602e9i	0.548713	−	0.950398i	0.998363	−	0.0571936i	$-0.0182152\pi$
$480$		0			0
$481$	5.73402e8	−	9.93162e8i	0.234937	−	0.406923i
$482$		0			0
$483$	−3.24236e9			−1.30932
$484$		0			0
$485$	9.58835e7	−	1.66075e8i	0.0381634	−	0.0661010i
$486$		0			0
$487$	−4.67689e9			−1.83487			−0.917436	−	0.397884i	$-0.869745\pi$
$487$	−4.67689e9			−1.83487			−0.917436	+	0.397884i	$0.869745\pi$
$488$		0			0
$489$	1.15110e9	+	1.99377e9i	0.445178	+	0.771071i
$490$		0			0
$491$	−4.40204e8	−	7.62455e8i	−0.167830	−	0.290689i	0.769827	−	0.638253i	$-0.220342\pi$
$491$	−4.40204e8	−	7.62455e8i	−0.167830	−	0.290689i	−0.937657	+	0.347563i	$0.887009\pi$
$492$		0			0
$493$	−2.13285e9			−0.801672
$494$		0			0
$495$	1.42659e9			0.528667
$496$		0			0
$497$	−2.18773e9	−	3.78925e9i	−0.799366	−	1.38454i
$498$		0			0
$499$	1.18972e9	+	2.06065e9i	0.428639	+	0.742425i	0.996753	−	0.0805254i	$-0.0256598\pi$
$499$	1.18972e9	+	2.06065e9i	0.428639	+	0.742425i	−0.568113	+	0.822950i	$0.692326\pi$
$500$		0			0
$501$	1.60609e9			0.570607
$502$		0			0
$503$	1.37727e9	−	2.38550e9i	0.482537	−	0.835779i	−0.517262	−	0.855827i	$-0.673049\pi$
$503$	1.37727e9	−	2.38550e9i	0.482537	−	0.835779i	0.999799	+	0.0200484i	$0.00638203\pi$
$504$		0			0
$505$	3.58465e9			1.23859
$506$		0			0
$507$	−9.10807e8	+	1.57756e9i	−0.310383	+	0.537600i
$508$		0			0
$509$	−5.97659e8	+	1.03518e9i	−0.200882	+	0.347938i	−0.948813	−	0.315839i	$-0.897714\pi$
$509$	−5.97659e8	+	1.03518e9i	−0.200882	+	0.347938i	0.747931	+	0.663777i	$0.231048\pi$
$510$		0			0
$511$	−9.65295e8	−	1.67194e9i	−0.320027	−	0.554303i
$512$		0			0
$513$	2.91673e9	−	1.55539e9i	0.953861	−	0.508661i
$514$		0			0
$515$	−2.45294e9	−	4.24861e9i	−0.791337	−	1.37064i
$516$		0			0
$517$	1.12535e9	−	1.94916e9i	0.358154	−	0.620340i
$518$		0			0
$519$	3.74387e8	−	6.48457e8i	0.117553	−	0.203608i
$520$		0			0
$521$	5.77984e9			1.79054			0.895269	−	0.445526i	$-0.146983\pi$
$521$	5.77984e9			1.79054			0.895269	+	0.445526i	$0.146983\pi$
$522$		0			0
$523$	−2.15665e9	+	3.73543e9i	−0.659211	+	1.14179i	0.321609	+	0.946872i	$0.395776\pi$
$523$	−2.15665e9	+	3.73543e9i	−0.659211	+	1.14179i	−0.980820	+	0.194914i	$0.937557\pi$
$524$		0			0
$525$	−1.25451e9			−0.378370
$526$		0			0
$527$	2.37771e9	+	4.11831e9i	0.707655	+	1.22569i
$528$		0			0
$529$	−3.71839e9	−	6.44044e9i	−1.09209	−	1.89156i
$530$		0			0
$531$	−1.24423e9			−0.360637
$532$		0			0
$533$	−3.02212e9			−0.864503
$534$		0			0
$535$	−2.66917e9	−	4.62315e9i	−0.753596	−	1.30527i
$536$		0			0
$537$	5.82099e8	+	1.00823e9i	0.162214	+	0.280962i
$538$		0			0
$539$	−1.89630e9			−0.521611
$540$		0			0
$541$	−2.58977e9	+	4.48560e9i	−0.703186	+	1.21795i	0.264157	+	0.964480i	$0.414906\pi$
$541$	−2.58977e9	+	4.48560e9i	−0.703186	+	1.21795i	−0.967342	+	0.253473i	$0.918427\pi$
$542$		0			0
$543$	5.33337e9			1.42956
$544$		0			0
$545$	−7.32987e8	+	1.26957e9i	−0.193958	+	0.335946i
$546$		0			0
$547$	2.30165e9	−	3.98657e9i	0.601289	−	1.04146i	−0.391338	−	0.920247i	$-0.627988\pi$
$547$	2.30165e9	−	3.98657e9i	0.601289	−	1.04146i	0.992626	−	0.121215i	$-0.0386791\pi$
$548$		0			0
$549$	5.29191e8	+	9.16586e8i	0.136493	+	0.236412i
$550$		0			0
$551$	1.15075e8	−	3.41395e9i	0.0293057	−	0.869414i
$552$		0			0
$553$	−7.77710e8	−	1.34703e9i	−0.195560	−	0.338720i
$554$		0			0
$555$	1.89253e9	−	3.27796e9i	0.469913	−	0.813914i
$556$		0			0
$557$	7.39358e8	−	1.28061e9i	0.181285	−	0.313995i	−0.761033	−	0.648713i	$-0.775308\pi$
$557$	7.39358e8	−	1.28061e9i	0.181285	−	0.313995i	0.942318	+	0.334718i	$0.108641\pi$
$558$		0			0
$559$	−3.21286e9			−0.777949
$560$		0			0
$561$	3.00519e9	−	5.20514e9i	0.718624	−	1.24469i
$562$		0			0
$563$	−2.10479e9			−0.497084			−0.248542	−	0.968621i	$-0.579951\pi$
$563$	−2.10479e9			−0.497084			−0.248542	+	0.968621i	$0.579951\pi$
$564$		0			0
$565$	3.96524e9	+	6.86799e9i	0.924911	+	1.60199i
$566$		0			0
$567$	1.28297e9	+	2.22216e9i	0.295580	+	0.511959i
$568$		0			0
$569$	−8.07912e9			−1.83853			−0.919266	−	0.393637i	$-0.871217\pi$
$569$	−8.07912e9			−1.83853			−0.919266	+	0.393637i	$0.871217\pi$
$570$		0			0
$571$	−5.56191e9			−1.25025			−0.625126	−	0.780524i	$-0.714952\pi$
$571$	−5.56191e9			−1.25025			−0.625126	+	0.780524i	$0.714952\pi$
$572$		0			0
$573$	−3.67949e9	−	6.37306e9i	−0.817045	−	1.41516i
$574$		0			0
$575$	−2.09737e9	−	3.63276e9i	−0.460085	−	0.796891i
$576$		0			0
$577$	4.65106e9			1.00795			0.503973	−	0.863720i	$-0.331871\pi$
$577$	4.65106e9			1.00795			0.503973	+	0.863720i	$0.331871\pi$
$578$		0			0
$579$	−1.58620e9	+	2.74737e9i	−0.339611	+	0.588223i
$580$		0			0
$581$	−7.64229e8			−0.161662
$582$		0			0
$583$	−8.76949e8	+	1.51892e9i	−0.183288	+	0.317464i
$584$		0			0
$585$	−3.83902e8	+	6.64938e8i	−0.0792820	+	0.137320i
$586$		0			0
$587$	−1.12889e9	−	1.95529e9i	−0.230365	−	0.399004i	0.727550	−	0.686054i	$-0.240659\pi$
$587$	−1.12889e9	−	1.95529e9i	−0.230365	−	0.399004i	−0.957916	+	0.287050i	$0.907325\pi$
$588$		0			0
$589$	−6.72027e9	+	3.58368e9i	−1.35514	+	0.722647i
$590$		0			0
$591$	−2.04052e9	−	3.53429e9i	−0.406616	−	0.704280i
$592$		0			0
$593$	3.83822e9	−	6.64799e9i	0.755855	−	1.30918i	−0.189093	−	0.981959i	$-0.560555\pi$
$593$	3.83822e9	−	6.64799e9i	0.755855	−	1.30918i	0.944948	−	0.327220i	$-0.106112\pi$
$594$		0			0
$595$	2.45332e9	−	4.24928e9i	0.477469	−	0.827000i
$596$		0			0
$597$	3.35658e9			0.645636
$598$		0			0
$599$	−4.77718e9	+	8.27432e9i	−0.908192	+	1.57303i	−0.0916174	+	0.995794i	$0.529204\pi$
$599$	−4.77718e9	+	8.27432e9i	−0.908192	+	1.57303i	−0.816574	+	0.577240i	$0.804130\pi$
$600$		0			0
$601$	5.21987e9			0.980841			0.490421	−	0.871486i	$-0.336843\pi$
$601$	5.21987e9			0.980841			0.490421	+	0.871486i	$0.336843\pi$
$602$		0			0
$603$	−6.20229e8	−	1.07427e9i	−0.115197	−	0.199527i
$604$		0			0
$605$	7.37807e9	+	1.27792e10i	1.35456	+	2.34617i
$606$		0			0
$607$	1.03564e10			1.87953			0.939766	−	0.341818i	$-0.111043\pi$
$607$	1.03564e10			1.87953			0.939766	+	0.341818i	$0.111043\pi$
$608$		0			0
$609$	3.55780e9			0.638294
$610$		0			0
$611$	6.05670e8	+	1.04905e9i	0.107422	+	0.186060i
$612$		0			0
$613$	2.26571e9	+	3.92433e9i	0.397277	+	0.688103i	0.993389	−	0.114798i	$-0.0366220\pi$
$613$	2.26571e9	+	3.92433e9i	0.397277	+	0.688103i	−0.596112	+	0.802901i	$0.703289\pi$
$614$		0			0
$615$	−9.97460e9			−1.72915
$616$		0			0
$617$	−3.40556e9	+	5.89860e9i	−0.583701	+	1.01100i	0.411335	+	0.911484i	$0.365062\pi$
$617$	−3.40556e9	+	5.89860e9i	−0.583701	+	1.01100i	−0.995036	+	0.0995151i	$0.968271\pi$
$618$		0			0
$619$	4.17695e9			0.707852			0.353926	−	0.935273i	$-0.384847\pi$
$619$	4.17695e9			0.707852			0.353926	+	0.935273i	$0.384847\pi$
$620$		0			0
$621$	−5.75600e9	+	9.96968e9i	−0.964495	+	1.67055i
$622$		0			0
$623$	1.18782e9	−	2.05737e9i	0.196809	−	0.340883i
$624$		0			0
$625$	3.81395e9	+	6.60595e9i	0.624877	+	1.08232i
$626$		0			0
$627$	8.16947e9	+	5.09109e9i	1.32360	+	0.824849i
$628$		0			0
$629$	2.51837e9	+	4.36194e9i	0.403498	+	0.698880i
$630$		0			0
$631$	5.09597e9	−	8.82648e9i	0.807466	−	1.39857i	−0.107148	−	0.994243i	$-0.534172\pi$
$631$	5.09597e9	−	8.82648e9i	0.807466	−	1.39857i	0.914614	−	0.404328i	$-0.132495\pi$
$632$		0			0
$633$	3.47495e9	−	6.01878e9i	0.544547	−	0.943182i
$634$		0			0
$635$	−9.84429e8			−0.152572
$636$		0			0
$637$	5.10301e8	−	8.83868e8i	0.0782238	−	0.135488i
$638$		0			0
$639$	−3.00710e9			−0.455927
$640$		0			0
$641$	4.70675e9	+	8.15234e9i	0.705860	+	1.22258i	0.966380	+	0.257117i	$0.0827724\pi$
$641$	4.70675e9	+	8.15234e9i	0.705860	+	1.22258i	−0.260521	+	0.965468i	$0.583894\pi$
$642$		0			0
$643$	1.72052e9	+	2.98003e9i	0.255224	+	0.442061i	0.964956	−	0.262411i	$-0.0845175\pi$
$643$	1.72052e9	+	2.98003e9i	0.255224	+	0.442061i	−0.709732	+	0.704471i	$0.751184\pi$
$644$		0			0
$645$	−1.06042e10			−1.55603
$646$		0			0
$647$	1.18237e10			1.71628			0.858140	−	0.513416i	$-0.171620\pi$
$647$	1.18237e10			1.71628			0.858140	+	0.513416i	$0.171620\pi$
$648$		0			0
$649$	−9.35919e9	−	1.62106e10i	−1.34395	−	2.32778i
$650$		0			0
$651$	−3.96624e9	−	6.86973e9i	−0.563438	−	0.975903i
$652$		0			0
$653$	2.26605e9			0.318473			0.159237	−	0.987240i	$-0.449097\pi$
$653$	2.26605e9			0.318473			0.159237	+	0.987240i	$0.449097\pi$
$654$		0			0
$655$	−5.73741e9	+	9.93749e9i	−0.797759	+	1.38176i
$656$		0			0
$657$	−1.32683e9			−0.182531
$658$		0			0
$659$	2.34562e9	−	4.06273e9i	0.319270	−	0.552992i	−0.661066	−	0.750328i	$-0.729896\pi$
$659$	2.34562e9	−	4.06273e9i	0.319270	−	0.552992i	0.980336	+	0.197336i	$0.0632290\pi$
$660$		0			0
$661$	4.10580e9	−	7.11145e9i	0.552958	−	0.957752i	−0.445101	−	0.895480i	$-0.646832\pi$
$661$	4.10580e9	−	7.11145e9i	0.552958	−	0.957752i	0.998059	−	0.0622714i	$-0.0198344\pi$
$662$		0			0
$663$	1.61742e9	+	2.80145e9i	0.215538	+	0.373323i
$664$		0			0
$665$	6.66924e9	+	4.15617e9i	0.879429	+	0.548047i
$666$		0			0
$667$	5.94816e9	+	1.03025e10i	0.776144	+	1.34432i
$668$		0			0
$669$	−9.22600e8	+	1.59799e9i	−0.119130	+	0.206340i
$670$		0			0
$671$	−7.96121e9	+	1.37892e10i	−1.01730	+	1.76202i
$672$		0			0
$673$	−1.29691e10			−1.64005			−0.820024	−	0.572329i	$-0.806040\pi$
$673$	−1.29691e10			−1.64005			−0.820024	+	0.572329i	$0.806040\pi$
$674$		0			0
$675$	−2.22706e9	+	3.85739e9i	−0.278721	+	0.482758i
$676$		0			0
$677$	−2.30963e9			−0.286076			−0.143038	−	0.989717i	$-0.545687\pi$
$677$	−2.30963e9			−0.286076			−0.143038	+	0.989717i	$0.545687\pi$
$678$		0			0
$679$	−2.12834e8	−	3.68638e8i	−0.0260913	−	0.0451915i
$680$		0			0
$681$	−1.03096e9	−	1.78567e9i	−0.125091	−	0.216664i
$682$		0			0
$683$	−9.30069e9			−1.11697			−0.558487	−	0.829514i	$-0.688618\pi$
$683$	−9.30069e9			−1.11697			−0.558487	+	0.829514i	$0.688618\pi$
$684$		0			0
$685$	−1.20984e10			−1.43817
$686$		0			0
$687$	3.87910e9	+	6.71880e9i	0.456439	+	0.790575i
$688$		0			0
$689$	−4.71981e8	−	8.17494e8i	−0.0549739	−	0.0952176i
$690$		0			0
$691$	−4.16697e9			−0.480449			−0.240224	−	0.970717i	$-0.577221\pi$
$691$	−4.16697e9			−0.480449			−0.240224	+	0.970717i	$0.577221\pi$
$692$		0			0
$693$	1.58331e9	−	2.74238e9i	0.180718	−	0.313012i
$694$		0			0
$695$	7.67963e9			0.867748
$696$		0			0
$697$	6.63653e9	−	1.14948e10i	0.742381	−	1.28584i
$698$		0			0
$699$	2.74439e9	−	4.75342e9i	0.303931	−	0.526425i
$700$		0			0
$701$	1.34218e9	+	2.32472e9i	0.147162	+	0.254893i	0.930178	−	0.367110i	$-0.119653\pi$
$701$	1.34218e9	+	2.32472e9i	0.147162	+	0.254893i	−0.783015	+	0.622002i	$0.786319\pi$
$702$		0			0
$703$	−7.11781e9	+	3.79568e9i	−0.772686	+	0.412046i
$704$		0			0
$705$	1.99903e9	+	3.46243e9i	0.214861	+	0.372150i
$706$		0			0
$707$	3.97845e9	−	6.89087e9i	0.423395	−	0.733341i
$708$		0			0
$709$	−6.40028e9	+	1.10856e10i	−0.674431	+	1.16815i	0.302204	+	0.953243i	$0.402278\pi$
$709$	−6.40028e9	+	1.10856e10i	−0.674431	+	1.16815i	−0.976635	+	0.214905i	$0.931056\pi$
$710$		0			0
$711$	−1.06899e9			−0.111540
$712$		0			0
$713$	1.32621e10	−	2.29706e10i	1.37024	−	2.37333i
$714$		0			0
$715$	−1.15509e10			−1.18181
$716$		0			0
$717$	−3.14651e8	−	5.44992e8i	−0.0318795	−	0.0552170i
$718$		0			0
$719$	−3.87954e7	−	6.71956e7i	−0.00389250	−	0.00674201i	0.864073	−	0.503367i	$-0.167906\pi$
$719$	−3.87954e7	−	6.71956e7i	−0.00389250	−	0.00674201i	−0.867965	+	0.496625i	$0.834572\pi$
$720$		0			0
$721$	−1.08896e10			−1.08203
$722$		0			0
$723$	6.64989e8			0.0654381
$724$		0			0
$725$	2.30141e9	+	3.98617e9i	0.224291	+	0.388483i
$726$		0			0
$727$	−1.07271e9	−	1.85799e9i	−0.103541	−	0.179338i	0.809600	−	0.586982i	$-0.199684\pi$
$727$	−1.07271e9	−	1.85799e9i	−0.103541	−	0.179338i	−0.913141	+	0.407644i	$0.866351\pi$
$728$		0			0
$729$	1.16212e10			1.11097
$730$		0			0
$731$	7.05541e9	−	1.22203e10i	0.668054	−	1.15710i
$732$		0			0
$733$	1.44935e10			1.35928			0.679641	−	0.733545i	$-0.262136\pi$
$733$	1.44935e10			1.35928			0.679641	+	0.733545i	$0.262136\pi$
$734$		0			0
$735$	1.68427e9	−	2.91723e9i	0.156461	−	0.270998i
$736$		0			0
$737$	9.33079e9	−	1.61614e10i	0.858583	−	1.48711i
$738$		0			0
$739$	−3.36673e9	−	5.83134e9i	−0.306869	−	0.531512i	0.670807	−	0.741632i	$-0.265948\pi$
$739$	−3.36673e9	−	5.83134e9i	−0.306869	−	0.531512i	−0.977676	+	0.210120i	$0.932615\pi$
$740$		0			0
$741$	−4.57140e9	+	2.43777e9i	−0.412748	+	0.220104i
$742$		0			0
$743$	7.89679e9	+	1.36776e10i	0.706301	+	1.22335i	0.966220	+	0.257719i	$0.0829708\pi$
$743$	7.89679e9	+	1.36776e10i	0.706301	+	1.22335i	−0.259919	+	0.965630i	$0.583696\pi$
$744$		0			0
$745$	3.70328e9	−	6.41427e9i	0.328125	−	0.568330i
$746$		0			0
$747$	−2.62615e8	+	4.54862e8i	−0.0230514	+	0.0399261i
$748$		0			0
$749$	−1.18496e10			−1.03043
$750$		0			0
$751$	6.28316e9	−	1.08827e10i	0.541300	−	0.937559i	−0.457530	−	0.889194i	$-0.651266\pi$
$751$	6.28316e9	−	1.08827e10i	0.541300	−	0.937559i	0.998830	−	0.0483648i	$-0.0154010\pi$
$752$		0			0
$753$	−1.17068e10			−0.999212
$754$		0			0
$755$	3.63273e9	+	6.29208e9i	0.307199	+	0.532083i
$756$		0			0
$757$	9.66258e9	+	1.67361e10i	0.809576	+	1.40223i	0.913158	+	0.407606i	$0.133636\pi$
$757$	9.66258e9	+	1.67361e10i	0.809576	+	1.40223i	−0.103582	+	0.994621i	$0.533030\pi$
$758$		0			0
$759$	−3.35239e10			−2.78297
$760$		0			0
$761$	1.32887e10			1.09304			0.546520	−	0.837446i	$-0.315952\pi$
$761$	1.32887e10			1.09304			0.546520	+	0.837446i	$0.315952\pi$
$762$		0			0
$763$	1.62702e9	+	2.81808e9i	0.132604	+	0.229677i
$764$		0			0
$765$	−1.68609e9	−	2.92039e9i	−0.136165	−	0.235844i
$766$		0			0
$767$	1.00744e10			0.806184
$768$		0			0
$769$	−8.41494e9	+	1.45751e10i	−0.667281	+	1.15577i	0.311380	+	0.950285i	$0.399209\pi$
$769$	−8.41494e9	+	1.45751e10i	−0.667281	+	1.15577i	−0.978661	+	0.205480i	$0.934125\pi$
$770$		0			0
$771$	1.96281e9			0.154236
$772$		0			0
$773$	2.09688e9	−	3.63191e9i	0.163285	−	0.282818i	−0.772760	−	0.634698i	$-0.781124\pi$
$773$	2.09688e9	−	3.63191e9i	0.163285	−	0.282818i	0.936045	+	0.351880i	$0.114458\pi$
$774$		0			0
$775$	5.13125e9	−	8.88758e9i	0.395974	−	0.685848i
$776$		0			0
$777$	−4.20087e9	−	7.27612e9i	−0.321267	−	0.556450i
$778$		0			0
$779$	1.80411e10	+	1.12430e10i	1.36736	+	0.852117i
$780$		0			0
$781$	−2.26196e10	−	3.91783e10i	−1.69905	−	2.94284i
$782$		0			0
$783$	6.31597e9	−	1.09396e10i	0.470190	−	0.814393i
$784$		0			0
$785$	−3.22450e9	+	5.58499e9i	−0.237913	+	0.412077i
$786$		0			0
$787$	−1.29627e10			−0.947949			−0.473974	−	0.880539i	$-0.657181\pi$
$787$	−1.29627e10			−0.947949			−0.473974	+	0.880539i	$0.657181\pi$
$788$		0			0
$789$	4.54989e9	−	7.88064e9i	0.329785	−	0.571205i
$790$		0			0
$791$	1.76034e10			1.26467
$792$		0			0
$793$	−4.28478e9	−	7.42146e9i	−0.305122	−	0.528486i
$794$		0			0
$795$	−1.55779e9	−	2.69817e9i	−0.109957	−	0.190451i
$796$		0			0
$797$	−2.05545e10			−1.43814			−0.719072	−	0.694935i	$-0.755433\pi$
$797$	−2.05545e10			−1.43814			−0.719072	+	0.694935i	$0.755433\pi$
$798$		0			0
$799$	−5.32017e9			−0.368988
$800$		0			0
$801$	−8.16353e8	−	1.41396e9i	−0.0561260	−	0.0972131i
$802$		0			0
$803$	−9.98049e9	−	1.72867e10i	−0.680217	−	1.17817i
$804$		0			0
$805$	−2.73676e10			−1.84906
$806$		0			0
$807$	1.17160e9	−	2.02928e9i	0.0784736	−	0.135920i
$808$		0			0
$809$	−8.55722e9			−0.568215			−0.284108	−	0.958792i	$-0.591697\pi$
$809$	−8.55722e9			−0.568215			−0.284108	+	0.958792i	$0.591697\pi$
$810$		0			0
$811$	−3.06340e9	+	5.30596e9i	−0.201665	+	0.349294i	−0.949065	−	0.315080i	$-0.897969\pi$
$811$	−3.06340e9	+	5.30596e9i	−0.201665	+	0.349294i	0.747400	+	0.664374i	$0.231302\pi$
$812$		0			0
$813$	−1.40371e7	+	2.43129e7i	−0.000916136	+	0.00158679i
$814$		0			0
$815$	9.71605e9	+	1.68287e10i	0.628692	+	1.08893i
$816$		0			0
$817$	1.91798e10	+	1.19526e10i	1.23046	+	0.766804i
$818$		0			0
$819$	8.52151e8	+	1.47597e9i	0.0542029	+	0.0938823i
$820$		0			0
$821$	−1.45124e9	+	2.51363e9i	−0.0915248	+	0.158526i	−0.908153	−	0.418639i	$-0.862507\pi$
$821$	−1.45124e9	+	2.51363e9i	−0.0915248	+	0.158526i	0.816628	+	0.577164i	$0.195841\pi$
$822$		0			0
$823$	−1.53722e9	+	2.66254e9i	−0.0961251	+	0.166494i	−0.910078	−	0.414438i	$-0.863978\pi$
$823$	−1.53722e9	+	2.66254e9i	−0.0961251	+	0.166494i	0.813953	+	0.580931i	$0.197312\pi$
$824$		0			0
$825$	−1.29708e10			−0.804224
$826$		0			0
$827$	9.13705e9	−	1.58258e10i	0.561742	−	0.972965i	−0.435603	−	0.900139i	$-0.643465\pi$
$827$	9.13705e9	−	1.58258e10i	0.561742	−	0.972965i	0.997345	−	0.0728262i	$-0.0232018\pi$
$828$		0			0
$829$	9.01460e9			0.549548			0.274774	−	0.961509i	$-0.411397\pi$
$829$	9.01460e9			0.549548			0.274774	+	0.961509i	$0.411397\pi$
$830$		0			0
$831$	8.71737e9	+	1.50989e10i	0.526965	+	0.912730i
$832$		0			0
$833$	2.24123e9	+	3.88193e9i	0.134347	+	0.232696i
$834$		0			0
$835$	1.35564e10			0.805826
$836$		0			0
$837$	−2.81642e10			−1.66019
$838$		0			0
$839$	−7.54610e9	−	1.30702e10i	−0.441119	−	0.764041i	0.556654	−	0.830745i	$-0.312085\pi$
$839$	−7.54610e9	−	1.30702e10i	−0.441119	−	0.764041i	−0.997773	+	0.0667039i	$0.978752\pi$
$840$		0			0
$841$	2.09811e9	+	3.63403e9i	0.121630	+	0.210670i
$842$		0			0
$843$	6.32381e9			0.363565
$844$		0			0
$845$	−7.68779e9	+	1.33156e10i	−0.438332	+	0.759213i
$846$		0			0
$847$	3.27543e10			1.85215
$848$		0			0
$849$	−9.67091e9	+	1.67505e10i	−0.542363	+	0.939401i
$850$		0			0
$851$	1.40466e10	−	2.43294e10i	0.781300	−	1.35325i
$852$		0			0
$853$	5.67500e9	+	9.82938e9i	0.313072	+	0.542256i	0.979026	−	0.203736i	$-0.0653086\pi$
$853$	5.67500e9	+	9.82938e9i	0.313072	+	0.542256i	−0.665954	+	0.745993i	$0.731975\pi$
$854$		0			0
$855$	4.76549e9	−	2.54127e9i	0.260751	−	0.139049i
$856$		0			0
$857$	1.66231e10	+	2.87920e10i	0.902150	+	1.56257i	0.824698	+	0.565573i	$0.191345\pi$
$857$	1.66231e10	+	2.87920e10i	0.902150	+	1.56257i	0.0774512	+	0.996996i	$0.475322\pi$
$858$		0			0
$859$	−1.50363e9	+	2.60437e9i	−0.0809404	+	0.140193i	−0.903654	−	0.428263i	$-0.859126\pi$
$859$	−1.50363e9	+	2.60437e9i	−0.0809404	+	0.140193i	0.822714	+	0.568456i	$0.192459\pi$
$860$		0			0
$861$	−1.10704e10	+	1.91744e10i	−0.591086	+	1.02379i
$862$		0			0
$863$	4.49472e9			0.238048			0.119024	−	0.992891i	$-0.462023\pi$
$863$	4.49472e9			0.238048			0.119024	+	0.992891i	$0.462023\pi$
$864$		0			0
$865$	3.16006e9	−	5.47338e9i	0.166012	−	0.287541i
$866$		0			0
$867$	2.52150e9			0.131399
$868$		0			0
$869$	−8.04100e9	−	1.39274e10i	−0.415662	−	0.719948i
$870$		0			0
$871$	5.02190e9	+	8.69819e9i	0.257516	+	0.446031i
$872$		0			0
$873$	−2.92547e8			−0.0148815
$874$		0			0
$875$	9.94546e9			0.501876
$876$		0			0
$877$	−4.18672e8	−	7.25162e8i	−0.0209593	−	0.0363025i	0.855356	−	0.518041i	$-0.173339\pi$
$877$	−4.18672e8	−	7.25162e8i	−0.0209593	−	0.0363025i	−0.876315	+	0.481739i	$0.840005\pi$
$878$		0			0
$879$	−7.54211e9	−	1.30633e10i	−0.374569	−	0.648772i
$880$		0			0
$881$	−3.15343e10			−1.55370			−0.776850	−	0.629686i	$-0.783183\pi$
$881$	−3.15343e10			−1.55370			−0.776850	+	0.629686i	$0.783183\pi$
$882$		0			0
$883$	−1.55490e10	+	2.69317e10i	−0.760046	+	1.31644i	0.182780	+	0.983154i	$0.441490\pi$
$883$	−1.55490e10	+	2.69317e10i	−0.760046	+	1.31644i	−0.942826	+	0.333285i	$0.891843\pi$
$884$		0			0
$885$	3.32508e10			1.61250
$886$		0			0
$887$	−1.04311e10	+	1.80672e10i	−0.501878	+	0.869278i	0.498120	+	0.867108i	$0.334024\pi$
$887$	−1.04311e10	+	1.80672e10i	−0.501878	+	0.869278i	−0.999998	+	0.00216966i	$0.999309\pi$
$888$		0			0
$889$	−1.09257e9	+	1.89239e9i	−0.0521548	+	0.0903348i
$890$		0			0
$891$	1.32650e10	+	2.29757e10i	0.628255	+	1.08817i
$892$		0			0
$893$	2.87044e8	−	8.51574e9i	0.0134886	−	0.400168i
$894$		0			0
$895$	4.91328e9	+	8.51006e9i	0.229082	+	0.396782i
$896$		0			0
$897$	9.02139e9	−	1.56255e10i	0.417350	−	0.722871i
$898$		0			0
$899$	−1.45523e10	+	2.52052e10i	−0.667992	+	1.15700i
$900$		0			0
$901$	4.14585e9			0.188833
$902$		0			0
$903$	−1.17691e10	+	2.03846e10i	−0.531907	+	0.921289i
$904$		0			0
$905$	4.50170e10			2.01886
$906$		0			0
$907$	3.03278e9	+	5.25293e9i	0.134963	+	0.233763i	0.925583	−	0.378544i	$-0.123575\pi$
$907$	3.03278e9	+	5.25293e9i	0.134963	+	0.233763i	−0.790620	+	0.612307i	$0.790242\pi$
$908$		0			0
$909$	−2.73425e9	−	4.73587e9i	−0.120744	−	0.209135i
$910$		0			0
$911$	4.13646e9			0.181265			0.0906326	−	0.995884i	$-0.471111\pi$
$911$	4.13646e9			0.181265			0.0906326	+	0.995884i	$0.471111\pi$
$912$		0			0
$913$	−7.90161e9			−0.343612
$914$		0			0
$915$	−1.41421e10	−	2.44948e10i	−0.610294	−	1.05706i
$916$		0			0
$917$	1.27354e10	+	2.20584e10i	0.545406	+	0.944671i
$918$		0			0
$919$	3.44886e8			0.0146579			0.00732895	−	0.999973i	$-0.497667\pi$
$919$	3.44886e8			0.0146579			0.00732895	+	0.999973i	$0.497667\pi$
$920$		0			0
$921$	−1.60526e10	+	2.78039e10i	−0.677073	+	1.17273i
$922$		0			0
$923$	2.43481e10			1.01920
$924$		0			0
$925$	5.43479e9	−	9.41334e9i	0.225781	−	0.391064i
$926$		0			0
$927$	−3.74204e9	+	6.48140e9i	−0.154287	+	0.267233i
$928$		0			0
$929$	−1.47597e10	−	2.55645e10i	−0.603978	−	1.04612i	−0.992212	−	0.124560i	$-0.960248\pi$
$929$	−1.47597e10	−	2.55645e10i	−0.603978	−	1.04612i	0.388234	−	0.921561i	$-0.373085\pi$
$930$		0			0
$931$	−6.33453e9	+	3.37798e9i	−0.257271	+	0.137193i
$932$		0			0
$933$	−7.26291e9	−	1.25797e10i	−0.292769	−	0.507090i
$934$		0			0
$935$	2.53657e10	−	4.39347e10i	1.01486	−	1.75779i
$936$		0			0
$937$	3.46858e9	−	6.00776e9i	0.137741	−	0.238574i	−0.788900	−	0.614521i	$-0.789349\pi$
$937$	3.46858e9	−	6.00776e9i	0.137741	−	0.238574i	0.926641	+	0.375947i	$0.122683\pi$
$938$		0			0
$939$	−3.09365e9			−0.121939
$940$		0			0
$941$	−3.66311e9	+	6.34469e9i	−0.143313	+	0.248226i	−0.928742	−	0.370726i	$-0.879109\pi$
$941$	−3.66311e9	+	6.34469e9i	−0.143313	+	0.248226i	0.785429	+	0.618951i	$0.212442\pi$
$942$		0			0
$943$	−7.40327e10			−2.87496
$944$		0			0
$945$	1.45299e10	+	2.51666e10i	0.560083	+	0.970092i
$946$		0			0
$947$	1.03267e10	+	1.78864e10i	0.395128	+	0.684382i	0.993118	−	0.117122i	$-0.0373669\pi$
$947$	1.03267e10	+	1.78864e10i	0.395128	+	0.684382i	−0.597990	+	0.801504i	$0.704034\pi$
$948$		0			0
$949$	1.07432e10			0.408037
$950$		0			0
$951$	−1.33973e10			−0.505109
$952$		0			0
$953$	4.69049e9	+	8.12417e9i	0.175547	+	0.304056i	0.940350	−	0.340207i	$-0.110497\pi$
$953$	4.69049e9	+	8.12417e9i	0.175547	+	0.304056i	−0.764803	+	0.644264i	$0.777164\pi$
$954$		0			0
$955$	−3.10572e10	−	5.37926e10i	−1.15385	−	1.99853i
$956$		0			0
$957$	3.67852e10			1.35669
$958$		0			0
$959$	−1.34275e10	+	2.32571e10i	−0.491619	+	0.851510i
$960$		0			0
$961$	3.73790e10			1.35861
$962$		0			0
$963$	−4.07192e9	+	7.05276e9i	−0.146929	+	0.254488i
$964$		0			0
$965$	−1.33885e10	+	2.31895e10i	−0.479607	+	0.830704i
$966$		0			0
$967$	1.57077e10	+	2.72065e10i	0.558624	+	0.967566i	0.997612	+	0.0690723i	$0.0220039\pi$
$967$	1.57077e10	+	2.72065e10i	0.558624	+	0.967566i	−0.438987	+	0.898493i	$0.644663\pi$
$968$		0			0
$969$	7.66537e8	−	2.27409e10i	0.0270645	−	0.802924i
$970$		0			0
$971$	1.31990e10	+	2.28613e10i	0.462672	+	0.801372i	0.999093	−	0.0425786i	$-0.0135573\pi$
$971$	1.31990e10	+	2.28613e10i	0.462672	+	0.801372i	−0.536421	+	0.843951i	$0.680224\pi$
$972$		0			0
$973$	8.52328e9	−	1.47627e10i	0.296628	−	0.513774i
$974$		0			0
$975$	3.49048e9	−	6.04570e9i	0.120606	−	0.208896i
$976$		0			0
$977$	1.07059e10			0.367276			0.183638	−	0.982994i	$-0.441213\pi$
$977$	1.07059e10			0.367276			0.183638	+	0.982994i	$0.441213\pi$
$978$		0			0
$979$	1.22813e10	−	2.12718e10i	0.418317	−	0.724546i
$980$		0			0
$981$	2.23639e9			0.0756322
$982$		0			0
$983$	5.30088e9	+	9.18139e9i	0.177996	+	0.308298i	0.941194	−	0.337866i	$-0.109705\pi$
$983$	5.30088e9	+	9.18139e9i	0.177996	+	0.308298i	−0.763198	+	0.646165i	$0.776372\pi$
$984$		0			0
$985$	−1.72233e10	−	2.98316e10i	−0.574234	−	0.994603i
$986$		0			0
$987$	8.87455e9			0.293789
$988$		0			0
$989$	−7.87054e10			−2.58712
$990$		0			0
$991$	−2.42320e10	−	4.19711e10i	−0.790919	−	1.36991i	−0.925398	−	0.378996i	$-0.876269\pi$
$991$	−2.42320e10	−	4.19711e10i	−0.790919	−	1.36991i	0.134479	−	0.990916i	$-0.457064\pi$
$992$		0			0
$993$	−4.87807e9	−	8.44907e9i	−0.158098	−	0.273833i
$994$		0			0
$995$	2.83317e10			0.911783
$996$		0			0
$997$	2.47652e10	−	4.28946e10i	0.791423	−	1.37078i	−0.133663	−	0.991027i	$-0.542674\pi$
$997$	2.47652e10	−	4.28946e10i	0.791423	−	1.37078i	0.925086	−	0.379758i	$-0.123993\pi$
$998$		0			0
$999$	−2.98303e10			−0.946627

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.e.a.45.9	✓	22
19.11	even	3	inner	76.8.e.a.49.9	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.e.a.45.9	✓	22	1.1	even	1	trivial
76.8.e.a.49.9	yes	22	19.11	even	3	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:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	76.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(20.3841 + 35.3063i) q^{3} +(172.055 + 298.008i) q^{5} +763.824 q^{7} +(262.475 - 454.621i) q^{9} +O(q^{10})\)	\(q+(20.3841 + 35.3063i) q^{3} +(172.055 + 298.008i) q^{5} +763.824 q^{7} +(262.475 - 454.621i) q^{9} +7897.42 q^{11} +(-2125.23 + 3681.00i) q^{13} +(-7014.37 + 12149.2i) q^{15} +(-9333.93 - 16166.8i) q^{17} +(26381.1 - 14068.1i) q^{19} +(15569.9 + 26967.8i) q^{21} +(-52061.5 + 90173.2i) q^{23} +(-20143.2 + 34889.1i) q^{25} +110561. q^{27} +(57126.3 - 98945.6i) q^{29} -254738. q^{31} +(160982. + 278829. i) q^{33} +(131420. + 227625. i) q^{35} -269808. q^{37} -173283. q^{39} +(355506. + 615754. i) q^{41} +(377944. + 654618. i) q^{43} +180641. q^{45} +(142495. - 246809. i) q^{47} -240116. q^{49} +(380528. - 659094. i) q^{51} +(-111042. + 192331. i) q^{53} +(1.35879e6 + 2.35349e6i) q^{55} +(1.03445e6 + 644653. i) q^{57} +(-1.18510e6 - 2.05264e6i) q^{59} +(-1.00808e6 + 1.74604e6i) q^{61} +(200485. - 347250. i) q^{63} -1.46262e6 q^{65} +(1.18150e6 - 2.04642e6i) q^{67} -4.24491e6 q^{69} +(-2.86418e6 - 4.96090e6i) q^{71} +(-1.26377e6 - 2.18891e6i) q^{73} -1.64241e6 q^{75} +6.03224e6 q^{77} +(-1.01818e6 - 1.76354e6i) q^{79} +(1.67966e6 + 2.90926e6i) q^{81} -1.00053e6 q^{83} +(3.21190e6 - 5.56317e6i) q^{85} +4.65788e6 q^{87} +(1.55510e6 - 2.69352e6i) q^{89} +(-1.62330e6 + 2.81163e6i) q^{91} +(-5.19262e6 - 8.99387e6i) q^{93} +(8.73139e6 + 5.44128e6i) q^{95} +(-278642. - 482622. i) q^{97} +(2.07288e6 - 3.59033e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9}+O(q^{10}) \) 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9	\( 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9} + 472 q^{11} - 567 q^{13} + 2995 q^{15} + 5589 q^{17} + 80912 q^{19} + 44412 q^{21} - 15425 q^{23} - 32806 q^{25} + 50290 q^{27} - 18919 q^{29} + 150296 q^{31} + 314618 q^{33} + 92808 q^{35} + 350100 q^{37} + 948810 q^{39} + 698891 q^{41} + 402545 q^{43} + 1477508 q^{45} - 653621 q^{47} - 1938490 q^{49} - 1386401 q^{51} - 106763 q^{53} + 414508 q^{55} + 1267563 q^{57} + 3136737 q^{59} + 2004581 q^{61} + 1465000 q^{63} - 7397638 q^{65} + 4344391 q^{67} + 1732238 q^{69} - 133823 q^{71} - 8349685 q^{73} - 12136824 q^{75} + 9147480 q^{77} - 94679 q^{79} - 838595 q^{81} - 2884080 q^{83} - 1421409 q^{85} - 31740598 q^{87} - 7039347 q^{89} + 1520096 q^{91} - 1993628 q^{93} + 1707587 q^{95} + 13308115 q^{97} + 6011488 q^{99}+O(q^{100}) \) 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9 + 472 * q^11 - 567 * q^13 + 2995 * q^15 + 5589 * q^17 + 80912 * q^19 + 44412 * q^21 - 15425 * q^23 - 32806 * q^25 + 50290 * q^27 - 18919 * q^29 + 150296 * q^31 + 314618 * q^33 + 92808 * q^35 + 350100 * q^37 + 948810 * q^39 + 698891 * q^41 + 402545 * q^43 + 1477508 * q^45 - 653621 * q^47 - 1938490 * q^49 - 1386401 * q^51 - 106763 * q^53 + 414508 * q^55 + 1267563 * q^57 + 3136737 * q^59 + 2004581 * q^61 + 1465000 * q^63 - 7397638 * q^65 + 4344391 * q^67 + 1732238 * q^69 - 133823 * q^71 - 8349685 * q^73 - 12136824 * q^75 + 9147480 * q^77 - 94679 * q^79 - 838595 * q^81 - 2884080 * q^83 - 1421409 * q^85 - 31740598 * q^87 - 7039347 * q^89 + 1520096 * q^91 - 1993628 * q^93 + 1707587 * q^95 + 13308115 * q^97 + 6011488 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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