LMFDB - Embedded newform 507.4.b.h.337.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(337,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.b (of order $2$, degree $1$, not 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:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 54x^{6} + 889x^{4} + 4584x^{2} + 5776 $ x^8 + 54x^6 + 889x^4 + 4584*x^2 + 5776
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 13^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.8
Root			$4.33039i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.h.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.33039i q^{2} -3.00000 q^{3} -20.4131 q^{4} -16.4131i q^{5} -15.9912i q^{6} -9.67968i q^{7} -66.1667i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+5.33039i q^{2} -3.00000 q^{3} -20.4131 q^{4} -16.4131i q^{5} -15.9912i q^{6} -9.67968i q^{7} -66.1667i q^{8} +9.00000 q^{9} +87.4882 q^{10} -27.5882i q^{11} +61.2393 q^{12} +51.5965 q^{14} +49.2393i q^{15} +189.390 q^{16} -107.928 q^{17} +47.9735i q^{18} -2.24723i q^{19} +335.042i q^{20} +29.0391i q^{21} +147.056 q^{22} -41.8090 q^{23} +198.500i q^{24} -144.390 q^{25} -27.0000 q^{27} +197.592i q^{28} +61.6213 q^{29} -262.465 q^{30} +191.932i q^{31} +480.187i q^{32} +82.7645i q^{33} -575.300i q^{34} -158.874 q^{35} -183.718 q^{36} -98.4236i q^{37} +11.9786 q^{38} -1086.00 q^{40} -30.7452i q^{41} -154.790 q^{42} -238.325 q^{43} +563.160i q^{44} -147.718i q^{45} -222.858i q^{46} +511.482i q^{47} -568.169 q^{48} +249.304 q^{49} -769.653i q^{50} +323.785 q^{51} +492.825 q^{53} -143.921i q^{54} -452.807 q^{55} -640.472 q^{56} +6.74170i q^{57} +328.466i q^{58} -484.179i q^{59} -1005.13i q^{60} -444.021 q^{61} -1023.07 q^{62} -87.1172i q^{63} -1044.47 q^{64} -441.167 q^{66} +190.114i q^{67} +2203.15 q^{68} +125.427 q^{69} -846.858i q^{70} +484.785i q^{71} -595.500i q^{72} +957.780i q^{73} +524.636 q^{74} +433.169 q^{75} +45.8729i q^{76} -267.045 q^{77} -375.216 q^{79} -3108.47i q^{80} +81.0000 q^{81} +163.884 q^{82} -715.765i q^{83} -592.777i q^{84} +1771.43i q^{85} -1270.37i q^{86} -184.864 q^{87} -1825.42 q^{88} +1038.15i q^{89} +787.394 q^{90} +853.451 q^{92} -575.796i q^{93} -2726.40 q^{94} -36.8840 q^{95} -1440.56i q^{96} +65.5636i q^{97} +1328.89i q^{98} -248.293i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9}+O(q^{10}) $ 8 * q - 24 * q^3 - 44 * q^4 + 72 * q^9	$ 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9} + 124 q^{10} + 132 q^{12} + 80 q^{14} + 244 q^{16} - 196 q^{17} + 440 q^{22} - 208 q^{23} + 116 q^{25} - 216 q^{27} + 388 q^{29} - 372 q^{30} + 176 q^{35} - 396 q^{36} - 664 q^{38} - 1996 q^{40} - 240 q^{42} - 900 q^{43} - 732 q^{48} - 2140 q^{49} + 588 q^{51} + 524 q^{53} + 408 q^{55} - 4328 q^{56} - 1856 q^{61} - 5560 q^{62} - 2052 q^{64} - 1320 q^{66} + 3572 q^{68} + 624 q^{69} + 2316 q^{74} - 348 q^{75} - 5016 q^{77} - 1492 q^{79} + 648 q^{81} - 3468 q^{82} - 1164 q^{87} - 6120 q^{88} + 1116 q^{90} + 664 q^{92} - 1544 q^{94} - 4408 q^{95}+O(q^{100}) $ 8 * q - 24 * q^3 - 44 * q^4 + 72 * q^9 + 124 * q^10 + 132 * q^12 + 80 * q^14 + 244 * q^16 - 196 * q^17 + 440 * q^22 - 208 * q^23 + 116 * q^25 - 216 * q^27 + 388 * q^29 - 372 * q^30 + 176 * q^35 - 396 * q^36 - 664 * q^38 - 1996 * q^40 - 240 * q^42 - 900 * q^43 - 732 * q^48 - 2140 * q^49 + 588 * q^51 + 524 * q^53 + 408 * q^55 - 4328 * q^56 - 1856 * q^61 - 5560 * q^62 - 2052 * q^64 - 1320 * q^66 + 3572 * q^68 + 624 * q^69 + 2316 * q^74 - 348 * q^75 - 5016 * q^77 - 1492 * q^79 + 648 * q^81 - 3468 * q^82 - 1164 * q^87 - 6120 * q^88 + 1116 * q^90 + 664 * q^92 - 1544 * q^94 - 4408 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$1$	$-1$

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 $

$2$	5.33039i	1.88458i	0.334800	+	0.942289i	$0.391331\pi$
$2$	5.33039i	1.88458i	−0.334800	+	0.942289i	$0.608669\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−20.4131	−2.55164
$5$	− 16.4131i	− 1.46803i	−0.679132	−	0.734016i	$-0.737644\pi$
$5$	− 16.4131i	− 1.46803i	0.679132	−	0.734016i	$-0.262356\pi$
$6$	− 15.9912i	− 1.08806i
$7$	− 9.67968i	− 0.522654i	−0.965250	−	0.261327i	$-0.915840\pi$
$7$	− 9.67968i	− 0.522654i	0.965250	−	0.261327i	$-0.0841601\pi$
$8$	− 66.1667i	− 2.92418i
$9$	9.00000	0.333333
$10$	87.4882	2.76662
$11$	− 27.5882i	− 0.756195i	−0.925766	−	0.378098i	$-0.876578\pi$
$11$	− 27.5882i	− 0.756195i	0.925766	−	0.378098i	$-0.123422\pi$
$12$	61.2393	1.47319
$13$	0	0
$13$	0	0
$14$	51.5965	0.984982
$15$	49.2393i	0.847568i
$16$	189.390	2.95921
$17$	−107.928	−1.53979	−0.769895	−	0.638171i	$-0.779691\pi$
$17$	−107.928	−1.53979	−0.769895	+	0.638171i	$0.779691\pi$
$18$	47.9735i	0.628193i
$19$	− 2.24723i	− 0.0271342i	−0.999908	−	0.0135671i	$-0.995681\pi$
$19$	− 2.24723i	− 0.0271342i	0.999908	−	0.0135671i	$-0.00431868\pi$
$20$	335.042i	3.74588i
$21$	29.0391i	0.301754i
$22$	147.056	1.42511
$23$	−41.8090	−0.379034	−0.189517	−	0.981877i	$-0.560692\pi$
$23$	−41.8090	−0.379034	−0.189517	+	0.981877i	$0.560692\pi$
$24$	198.500i	1.68828i
$25$	−144.390	−1.15512
$26$	0	0
$27$	−27.0000	−0.192450
$28$	197.592i	1.33362i
$29$	61.6213	0.394579	0.197289	−	0.980345i	$-0.436786\pi$
$29$	61.6213	0.394579	0.197289	+	0.980345i	$0.436786\pi$
$30$	−262.465	−1.59731
$31$	191.932i	1.11200i	0.831182	+	0.556000i	$0.187665\pi$
$31$	191.932i	1.11200i	−0.831182	+	0.556000i	$0.812335\pi$
$32$	480.187i	2.65269i
$33$	82.7645i	0.436589i
$34$	− 575.300i	− 2.90185i
$35$	−158.874	−0.767272
$36$	−183.718	−0.850545
$37$	− 98.4236i	− 0.437317i	−0.975801	−	0.218659i	$-0.929832\pi$
$37$	− 98.4236i	− 0.437317i	0.975801	−	0.218659i	$-0.0701681\pi$
$38$	11.9786	0.0511366
$39$	0	0
$40$	−1086.00	−4.29279
$41$	− 30.7452i	− 0.117112i	−0.998284	−	0.0585561i	$-0.981350\pi$
$41$	− 30.7452i	− 0.117112i	0.998284	−	0.0585561i	$-0.0186496\pi$
$42$	−154.790	−0.568680
$43$	−238.325	−0.845216	−0.422608	−	0.906313i	$-0.638885\pi$
$43$	−238.325	−0.845216	−0.422608	+	0.906313i	$0.638885\pi$
$44$	563.160i	1.92953i
$45$	− 147.718i	− 0.489344i
$46$	− 222.858i	− 0.714319i
$47$	511.482i	1.58739i	0.608316	+	0.793695i	$0.291845\pi$
$47$	511.482i	1.58739i	−0.608316	+	0.793695i	$0.708155\pi$
$48$	−568.169	−1.70850
$49$	249.304	0.726833
$50$	− 769.653i	− 2.17691i
$51$	323.785	0.888998
$52$	0	0
$53$	492.825	1.27726	0.638630	−	0.769514i	$-0.279502\pi$
$53$	492.825	1.27726	0.638630	+	0.769514i	$0.279502\pi$
$54$	− 143.921i	− 0.362687i
$55$	−452.807	−1.11012
$56$	−640.472	−1.52833
$57$	6.74170i	0.0156660i
$58$	328.466i	0.743615i
$59$	− 484.179i	− 1.06838i	−0.845363	−	0.534192i	$-0.820616\pi$
$59$	− 484.179i	− 1.06838i	0.845363	−	0.534192i	$-0.179384\pi$
$60$	− 1005.13i	− 2.16269i
$61$	−444.021	−0.931985	−0.465993	−	0.884789i	$-0.654303\pi$
$61$	−444.021	−0.931985	−0.465993	+	0.884789i	$0.654303\pi$
$62$	−1023.07	−2.09565
$63$	− 87.1172i	− 0.174218i
$64$	−1044.47	−2.03998
$65$	0	0
$66$	−441.167	−0.822787
$67$	190.114i	0.346658i	0.984864	+	0.173329i	$0.0554524\pi$
$67$	190.114i	0.346658i	−0.984864	+	0.173329i	$0.944548\pi$
$68$	2203.15	3.92898
$69$	125.427	0.218835
$70$	− 846.858i	− 1.44598i
$71$	484.785i	0.810329i	0.914244	+	0.405164i	$0.132786\pi$
$71$	484.785i	0.810329i	−0.914244	+	0.405164i	$0.867214\pi$
$72$	− 595.500i	− 0.974727i
$73$	957.780i	1.53561i	0.640683	+	0.767806i	$0.278651\pi$
$73$	957.780i	1.53561i	−0.640683	+	0.767806i	$0.721349\pi$
$74$	524.636	0.824159
$75$	433.169	0.666907
$76$	45.8729i	0.0692367i
$77$	−267.045	−0.395228
$78$	0	0
$79$	−375.216	−0.534368	−0.267184	−	0.963646i	$-0.586093\pi$
$79$	−375.216	−0.534368	−0.267184	+	0.963646i	$0.586093\pi$
$80$	− 3108.47i	− 4.34422i
$81$	81.0000	0.111111
$82$	163.884	0.220707
$83$	− 715.765i	− 0.946571i	−0.880909	−	0.473286i	$-0.843068\pi$
$83$	− 715.765i	− 0.946571i	0.880909	−	0.473286i	$-0.156932\pi$
$84$	− 592.777i	− 0.769967i
$85$	1771.43i	2.26046i
$86$	− 1270.37i	− 1.59288i
$87$	−184.864	−0.227810
$88$	−1825.42	−2.21125
$89$	1038.15i	1.23645i	0.786002	+	0.618224i	$0.212148\pi$
$89$	1038.15i	1.23645i	−0.786002	+	0.618224i	$0.787852\pi$
$90$	787.394	0.922207
$91$	0	0
$92$	853.451	0.967157
$93$	− 575.796i	− 0.642013i
$94$	−2726.40	−2.99156
$95$	−36.8840	−0.0398339
$96$	− 1440.56i	− 1.53153i
$97$	65.5636i	0.0686286i	0.999411	+	0.0343143i	$0.0109247\pi$
$97$	65.5636i	0.0686286i	−0.999411	+	0.0343143i	$0.989075\pi$
$98$	1328.89i	1.36977i
$99$	− 248.293i	− 0.252065i
$100$	2947.44	2.94744
$101$	−531.798	−0.523920	−0.261960	−	0.965079i	$-0.584369\pi$
$101$	−531.798	−0.523920	−0.261960	+	0.965079i	$0.584369\pi$
$102$	1725.90i	1.67539i
$103$	735.984	0.704064	0.352032	−	0.935988i	$-0.385491\pi$
$103$	735.984	0.704064	0.352032	+	0.935988i	$0.385491\pi$
$104$	0	0
$105$	476.621	0.442985
$106$	2626.95i	2.40710i
$107$	−783.265	−0.707673	−0.353837	−	0.935307i	$-0.615123\pi$
$107$	−783.265	−0.707673	−0.353837	+	0.935307i	$0.615123\pi$
$108$	551.153	0.491063
$109$	− 532.339i	− 0.467788i	−0.972262	−	0.233894i	$-0.924853\pi$
$109$	− 532.339i	− 0.467788i	0.972262	−	0.233894i	$-0.0751468\pi$
$110$	− 2413.64i	− 2.09210i
$111$	295.271i	0.252485i
$112$	− 1833.23i	− 1.54664i
$113$	−180.589	−0.150340	−0.0751699	−	0.997171i	$-0.523950\pi$
$113$	−180.589	−0.150340	−0.0751699	+	0.997171i	$0.523950\pi$
$114$	−35.9359	−0.0295237
$115$	686.215i	0.556434i
$116$	−1257.88	−1.00682
$117$	0	0
$118$	2580.86	2.01346
$119$	1044.71i	0.804777i
$120$	3258.00	2.47844
$121$	569.893	0.428169
$122$	− 2366.81i	− 1.75640i
$123$	92.2357i	0.0676147i
$124$	− 3917.92i	− 2.83742i
$125$	318.242i	0.227716i
$126$	464.369	0.328327
$127$	−1431.63	−1.00029	−0.500146	−	0.865941i	$-0.666720\pi$
$127$	−1431.63	−1.00029	−0.500146	+	0.865941i	$0.666720\pi$
$128$	− 1725.94i	− 1.19182i
$129$	714.976	0.487986
$130$	0	0
$131$	2067.32	1.37880	0.689400	−	0.724381i	$-0.257874\pi$
$131$	2067.32	1.37880	0.689400	+	0.724381i	$0.257874\pi$
$132$	− 1689.48i	− 1.11402i
$133$	−21.7525	−0.0141818
$134$	−1013.38	−0.653304
$135$	443.153i	0.282523i
$136$	7141.25i	4.50262i
$137$	387.512i	0.241660i	0.992673	+	0.120830i	$0.0385555\pi$
$137$	387.512i	0.241660i	−0.992673	+	0.120830i	$0.961444\pi$
$138$	668.575i	0.412412i
$139$	752.568	0.459223	0.229611	−	0.973282i	$-0.426254\pi$
$139$	752.568	0.459223	0.229611	+	0.973282i	$0.426254\pi$
$140$	3243.10	1.95780
$141$	− 1534.45i	− 0.916480i
$142$	−2584.09	−1.52713
$143$	0	0
$144$	1704.51	0.986404
$145$	− 1011.40i	− 0.579254i
$146$	−5105.34	−2.89398
$147$	−747.911	−0.419637
$148$	2009.13i	1.11587i
$149$	− 2636.72i	− 1.44972i	−0.688895	−	0.724862i	$-0.741904\pi$
$149$	− 2636.72i	− 1.44972i	0.688895	−	0.724862i	$-0.258096\pi$
$150$	2308.96i	1.25684i
$151$	3332.42i	1.79595i	0.440046	+	0.897975i	$0.354962\pi$
$151$	3332.42i	1.79595i	−0.440046	+	0.897975i	$0.645038\pi$
$152$	−148.692	−0.0793454
$153$	−971.354	−0.513263
$154$	− 1423.45i	− 0.744839i
$155$	3150.20	1.63245
$156$	0	0
$157$	−1625.26	−0.826179	−0.413089	−	0.910690i	$-0.635550\pi$
$157$	−1625.26	−0.826179	−0.413089	+	0.910690i	$0.635550\pi$
$158$	− 2000.05i	− 1.00706i
$159$	−1478.48	−0.737426
$160$	7881.36	3.89423
$161$	404.698i	0.198104i
$162$	431.762i	0.209398i
$163$	1835.37i	0.881944i	0.897521	+	0.440972i	$0.145366\pi$
$163$	1835.37i	0.881944i	−0.897521	+	0.440972i	$0.854634\pi$
$164$	627.605i	0.298828i
$165$	1358.42	0.640927
$166$	3815.31	1.78389
$167$	− 1945.00i	− 0.901248i	−0.892714	−	0.450624i	$-0.851202\pi$
$167$	− 1945.00i	− 0.901248i	0.892714	−	0.450624i	$-0.148798\pi$
$168$	1921.42	0.882384
$169$	0	0
$170$	−9442.44	−4.26001
$171$	− 20.2251i	− 0.00904474i
$172$	4864.96	2.15668
$173$	−2531.63	−1.11258	−0.556289	−	0.830989i	$-0.687775\pi$
$173$	−2531.63	−1.11258	−0.556289	+	0.830989i	$0.687775\pi$
$174$	− 985.397i	− 0.429326i
$175$	1397.65i	0.603726i
$176$	− 5224.91i	− 2.23774i
$177$	1452.54i	0.616832i
$178$	−5533.76	−2.33018
$179$	−4263.01	−1.78007	−0.890035	−	0.455892i	$-0.849320\pi$
$179$	−4263.01	−1.78007	−0.890035	+	0.455892i	$0.849320\pi$
$180$	3015.38i	1.24863i
$181$	−3944.61	−1.61989	−0.809946	−	0.586504i	$-0.800504\pi$
$181$	−3944.61	−1.61989	−0.809946	+	0.586504i	$0.800504\pi$
$182$	0	0
$183$	1332.06	0.538082
$184$	2766.36i	1.10836i
$185$	−1615.43	−0.641995
$186$	3069.22	1.20992
$187$	2977.54i	1.16438i
$188$	− 10440.9i	− 4.05044i
$189$	261.351i	0.100585i
$190$	− 196.606i	− 0.0750701i
$191$	−214.109	−0.0811119	−0.0405559	−	0.999177i	$-0.512913\pi$
$191$	−214.109	−0.0811119	−0.0405559	+	0.999177i	$0.512913\pi$
$192$	3133.42	1.17779
$193$	1207.19i	0.450234i	0.974332	+	0.225117i	$0.0722764\pi$
$193$	1207.19i	0.450234i	−0.974332	+	0.225117i	$0.927724\pi$
$194$	−349.480	−0.129336
$195$	0	0
$196$	−5089.06	−1.85461
$197$	− 927.631i	− 0.335487i	−0.985831	−	0.167744i	$-0.946352\pi$
$197$	− 927.631i	− 0.335487i	0.985831	−	0.167744i	$-0.0536481\pi$
$198$	1323.50	0.475036
$199$	−478.951	−0.170613	−0.0853064	−	0.996355i	$-0.527187\pi$
$199$	−478.951	−0.170613	−0.0853064	+	0.996355i	$0.527187\pi$
$200$	9553.77i	3.37777i
$201$	− 570.341i	− 0.200143i
$202$	− 2834.69i	− 0.987368i
$203$	− 596.474i	− 0.206228i
$204$	−6609.44	−2.26840
$205$	−504.624	−0.171924
$206$	3923.08i	1.32686i
$207$	−376.281	−0.126345
$208$	0	0
$209$	−61.9970	−0.0205188
$210$	2540.58i	0.834840i
$211$	−1450.95	−0.473402	−0.236701	−	0.971583i	$-0.576066\pi$
$211$	−1450.95	−0.473402	−0.236701	+	0.971583i	$0.576066\pi$
$212$	−10060.1	−3.25910
$213$	− 1454.35i	− 0.467843i
$214$	− 4175.11i	− 1.33367i
$215$	3911.66i	1.24080i
$216$	1786.50i	0.562759i
$217$	1857.84	0.581191
$218$	2837.58	0.881583
$219$	− 2873.34i	− 0.886586i
$220$	9243.19	2.83262
$221$	0	0
$222$	−1573.91	−0.475828
$223$	− 2059.79i	− 0.618536i	−0.950975	−	0.309268i	$-0.899916\pi$
$223$	− 2059.79i	− 0.618536i	0.950975	−	0.309268i	$-0.100084\pi$
$224$	4648.06	1.38644
$225$	−1299.51	−0.385039
$226$	− 962.612i	− 0.283327i
$227$	4482.46i	1.31062i	0.755359	+	0.655311i	$0.227463\pi$
$227$	4482.46i	1.31062i	−0.755359	+	0.655311i	$0.772537\pi$
$228$	− 137.619i	− 0.0399738i
$229$	1630.39i	0.470477i	0.971938	+	0.235239i	$0.0755872\pi$
$229$	1630.39i	0.470477i	−0.971938	+	0.235239i	$0.924413\pi$
$230$	−3657.80	−1.04864
$231$	801.134	0.228185
$232$	− 4077.27i	− 1.15382i
$233$	1903.69	0.535258	0.267629	−	0.963522i	$-0.413760\pi$
$233$	1903.69	0.535258	0.267629	+	0.963522i	$0.413760\pi$
$234$	0	0
$235$	8395.00	2.33034
$236$	9883.58i	2.72613i
$237$	1125.65	0.308517
$238$	−5568.72	−1.51667
$239$	3763.79i	1.01866i	0.860572	+	0.509328i	$0.170106\pi$
$239$	3763.79i	1.01866i	−0.860572	+	0.509328i	$0.829894\pi$
$240$	9325.40i	2.50813i
$241$	3614.74i	0.966166i	0.875575	+	0.483083i	$0.160483\pi$
$241$	3614.74i	0.966166i	−0.875575	+	0.483083i	$0.839517\pi$
$242$	3037.75i	0.806918i
$243$	−243.000	−0.0641500
$244$	9063.85	2.37809
$245$	− 4091.84i	− 1.06701i
$246$	−491.652	−0.127425
$247$	0	0
$248$	12699.5	3.25169
$249$	2147.30i	0.546503i
$250$	−1696.36	−0.429148
$251$	5729.77	1.44088	0.720438	−	0.693520i	$-0.243941\pi$
$251$	5729.77	1.44088	0.720438	+	0.693520i	$0.243941\pi$
$252$	1778.33i	0.444541i
$253$	1153.43i	0.286624i
$254$	− 7631.18i	− 1.88513i
$255$	− 5314.30i	− 1.30508i
$256$	844.191	0.206101
$257$	5525.79	1.34120	0.670602	−	0.741818i	$-0.266036\pi$
$257$	5525.79	1.34120	0.670602	+	0.741818i	$0.266036\pi$
$258$	3811.10i	0.919647i
$259$	−952.709	−0.228565
$260$	0	0
$261$	554.591	0.131526
$262$	11019.6i	2.59846i
$263$	5223.21	1.22463	0.612313	−	0.790615i	$-0.290239\pi$
$263$	5223.21	1.22463	0.612313	+	0.790615i	$0.290239\pi$
$264$	5476.25	1.27667
$265$	− 8088.78i	− 1.87506i
$266$	− 115.949i	− 0.0267267i
$267$	− 3114.46i	− 0.713864i
$268$	− 3880.81i	− 0.884545i
$269$	7203.88	1.63282	0.816410	−	0.577473i	$-0.195961\pi$
$269$	7203.88	1.63282	0.816410	+	0.577473i	$0.195961\pi$
$270$	−2362.18	−0.532436
$271$	8577.69i	1.92272i	0.275293	+	0.961360i	$0.411225\pi$
$271$	8577.69i	1.92272i	−0.275293	+	0.961360i	$0.588775\pi$
$272$	−20440.5	−4.55656
$273$	0	0
$274$	−2065.59	−0.455427
$275$	3983.44i	0.873493i
$276$	−2560.35	−0.558388
$277$	−7169.19	−1.55507	−0.777536	−	0.628838i	$-0.783531\pi$
$277$	−7169.19	−1.55507	−0.777536	+	0.628838i	$0.783531\pi$
$278$	4011.48i	0.865442i
$279$	1727.39i	0.370667i
$280$	10512.1i	2.24364i
$281$	− 849.157i	− 0.180272i	−0.995929	−	0.0901360i	$-0.971270\pi$
$281$	− 849.157i	− 0.180272i	0.995929	−	0.0901360i	$-0.0287302\pi$
$282$	8179.20	1.72718
$283$	−1115.37	−0.234283	−0.117141	−	0.993115i	$-0.537373\pi$
$283$	−1115.37	−0.234283	−0.117141	+	0.993115i	$0.537373\pi$
$284$	− 9895.95i	− 2.06766i
$285$	110.652	0.0229981
$286$	0	0
$287$	−297.604	−0.0612091
$288$	4321.69i	0.884229i
$289$	6735.49	1.37095
$290$	5391.14	1.09165
$291$	− 196.691i	− 0.0396227i
$292$	− 19551.2i	− 3.91832i
$293$	1863.53i	0.371565i	0.982591	+	0.185782i	$0.0594819\pi$
$293$	1863.53i	0.371565i	−0.982591	+	0.185782i	$0.940518\pi$
$294$	− 3986.66i	− 0.790839i
$295$	−7946.87	−1.56842
$296$	−6512.36	−1.27879
$297$	744.880i	0.145530i
$298$	14054.8	2.73212
$299$	0	0
$300$	−8842.31	−1.70170
$301$	2306.91i	0.441755i
$302$	−17763.1	−3.38461
$303$	1595.39	0.302485
$304$	− 425.602i	− 0.0802959i
$305$	7287.76i	1.36818i
$306$	− 5177.70i	− 0.967285i
$307$	6387.50i	1.18747i	0.804660	+	0.593736i	$0.202348\pi$
$307$	6387.50i	1.18747i	−0.804660	+	0.593736i	$0.797652\pi$
$308$	5451.21	1.00848
$309$	−2207.95	−0.406492
$310$	16791.8i	3.07648i
$311$	3492.59	0.636806	0.318403	−	0.947955i	$-0.396853\pi$
$311$	3492.59	0.636806	0.318403	+	0.947955i	$0.396853\pi$
$312$	0	0
$313$	−5912.01	−1.06762	−0.533812	−	0.845603i	$-0.679241\pi$
$313$	−5912.01	−1.06762	−0.533812	+	0.845603i	$0.679241\pi$
$314$	− 8663.29i	− 1.55700i
$315$	−1429.86	−0.255757
$316$	7659.31	1.36351
$317$	− 1677.54i	− 0.297224i	−0.988896	−	0.148612i	$-0.952519\pi$
$317$	− 1677.54i	− 0.297224i	0.988896	−	0.148612i	$-0.0474805\pi$
$318$	− 7880.86i	− 1.38974i
$319$	− 1700.02i	− 0.298378i
$320$	17143.0i	2.99476i
$321$	2349.79	0.408575
$322$	−2157.20	−0.373342
$323$	242.540i	0.0417810i
$324$	−1653.46	−0.283515
$325$	0	0
$326$	−9783.22	−1.66209
$327$	1597.02i	0.270077i
$328$	−2034.31	−0.342457
$329$	4950.98	0.829655
$330$	7240.92i	1.20788i
$331$	2010.31i	0.333827i	0.985972	+	0.166913i	$0.0533800\pi$
$331$	2010.31i	0.333827i	−0.985972	+	0.166913i	$0.946620\pi$
$332$	14611.0i	2.41531i
$333$	− 885.812i	− 0.145772i
$334$	10367.6	1.69847
$335$	3120.35	0.508905
$336$	5499.69i	0.892955i
$337$	−7139.24	−1.15400	−0.577002	−	0.816743i	$-0.695778\pi$
$337$	−7139.24	−1.15400	−0.577002	+	0.816743i	$0.695778\pi$
$338$	0	0
$339$	541.768	0.0867988
$340$	− 36160.5i	− 5.76787i
$341$	5295.05	0.840889
$342$	107.808	0.0170455
$343$	− 5733.31i	− 0.902536i
$344$	15769.2i	2.47156i
$345$	− 2058.64i	− 0.321257i
$346$	− 13494.6i	− 2.09674i
$347$	1.13990	0.000176349 0	8.81743e−5	−	1.00000i	$-0.499972\pi$
$347$	1.13990	0.000176349 0	8.81743e−5		1.00000i	$0.499972\pi$
$348$	3773.64	0.581289
$349$	− 12199.1i	− 1.87107i	−0.353235	−	0.935535i	$-0.614918\pi$
$349$	− 12199.1i	− 1.87107i	0.353235	−	0.935535i	$-0.385082\pi$
$350$	−7450.00	−1.13777
$351$	0	0
$352$	13247.5	2.00595
$353$	− 10892.3i	− 1.64232i	−0.570698	−	0.821160i	$-0.693327\pi$
$353$	− 10892.3i	− 1.64232i	0.570698	−	0.821160i	$-0.306673\pi$
$354$	−7742.59	−1.16247
$355$	7956.81	1.18959
$356$	− 21191.9i	− 3.15497i
$357$	− 3134.13i	− 0.464638i
$358$	− 22723.5i	− 3.35468i
$359$	3525.78i	0.518339i	0.965832	+	0.259169i	$0.0834488\pi$
$359$	3525.78i	0.518339i	−0.965832	+	0.259169i	$0.916551\pi$
$360$	−9773.99	−1.43093
$361$	6853.95	0.999264
$362$	− 21026.3i	− 3.05281i
$363$	−1709.68	−0.247204
$364$	0	0
$365$	15720.1	2.25433
$366$	7100.42i	1.01406i
$367$	2383.75	0.339049	0.169525	−	0.985526i	$-0.445777\pi$
$367$	2383.75	0.339049	0.169525	+	0.985526i	$0.445777\pi$
$368$	−7918.19	−1.12164
$369$	− 276.707i	− 0.0390374i
$370$	− 8610.90i	− 1.20989i
$371$	− 4770.39i	− 0.667564i
$372$	11753.8i	1.63818i
$373$	−13282.2	−1.84377	−0.921885	−	0.387463i	$-0.873352\pi$
$373$	−13282.2	−1.84377	−0.921885	+	0.387463i	$0.873352\pi$
$374$	−15871.5	−2.19437
$375$	− 954.727i	− 0.131472i
$376$	33843.1	4.64181
$377$	0	0
$378$	−1393.11	−0.189560
$379$	− 4436.73i	− 0.601318i	−0.953732	−	0.300659i	$-0.902793\pi$
$379$	− 4436.73i	− 0.601318i	0.953732	−	0.300659i	$-0.0972065\pi$
$380$	752.917	0.101642
$381$	4294.90	0.577519
$382$	− 1141.28i	− 0.152862i
$383$	− 810.412i	− 0.108120i	−0.998538	−	0.0540602i	$-0.982784\pi$
$383$	− 810.412i	− 0.108120i	0.998538	−	0.0540602i	$-0.0172163\pi$
$384$	5177.83i	0.688100i
$385$	4383.03i	0.580207i
$386$	−6434.78	−0.848501
$387$	−2144.93	−0.281739
$388$	− 1338.35i	− 0.175115i
$389$	−3463.79	−0.451469	−0.225734	−	0.974189i	$-0.572478\pi$
$389$	−3463.79	−0.451469	−0.225734	+	0.974189i	$0.572478\pi$
$390$	0	0
$391$	4512.37	0.583633
$392$	− 16495.6i	− 2.12539i
$393$	−6201.96	−0.796050
$394$	4944.64	0.632252
$395$	6158.45i	0.784469i
$396$	5068.44i	0.643178i
$397$	− 425.405i	− 0.0537796i	−0.999638	−	0.0268898i	$-0.991440\pi$
$397$	− 425.405i	− 0.0537796i	0.999638	−	0.0268898i	$-0.00856031\pi$
$398$	− 2553.00i	− 0.321533i
$399$	65.2575	0.00818787
$400$	−27345.9	−3.41823
$401$	1186.85i	0.147801i	0.997266	+	0.0739007i	$0.0235448\pi$
$401$	1186.85i	0.147801i	−0.997266	+	0.0739007i	$0.976455\pi$
$402$	3040.14	0.377185
$403$	0	0
$404$	10855.6	1.33685
$405$	− 1329.46i	− 0.163115i
$406$	3179.44	0.388653
$407$	−2715.33	−0.330697
$408$	− 21423.7i	− 2.59959i
$409$	8007.42i	0.968071i	0.875048	+	0.484036i	$0.160830\pi$
$409$	8007.42i	0.968071i	−0.875048	+	0.484036i	$0.839170\pi$
$410$	− 2689.84i	− 0.324005i
$411$	− 1162.54i	− 0.139522i
$412$	−15023.7	−1.79652
$413$	−4686.70	−0.558395
$414$	− 2005.73i	− 0.238106i
$415$	−11747.9	−1.38960
$416$	0	0
$417$	−2257.70	−0.265132
$418$	− 330.468i	− 0.0386692i
$419$	6832.46	0.796629	0.398314	−	0.917249i	$-0.369595\pi$
$419$	6832.46	0.796629	0.398314	+	0.917249i	$0.369595\pi$
$420$	−9729.30	−1.13034
$421$	10739.6i	1.24326i	0.783309	+	0.621632i	$0.213530\pi$
$421$	10739.6i	1.24326i	−0.783309	+	0.621632i	$0.786470\pi$
$422$	− 7734.16i	− 0.892163i
$423$	4603.34i	0.529130i
$424$	− 32608.6i	− 3.73494i
$425$	15583.7	1.77864
$426$	7752.28	0.881688
$427$	4297.99i	0.487106i
$428$	15988.9	1.80573
$429$	0	0
$430$	−20850.7	−2.33839
$431$	5214.45i	0.582763i	0.956607	+	0.291382i	$0.0941150\pi$
$431$	5214.45i	0.582763i	−0.956607	+	0.291382i	$0.905885\pi$
$432$	−5113.52	−0.569501
$433$	−8642.24	−0.959168	−0.479584	−	0.877496i	$-0.659212\pi$
$433$	−8642.24	−0.959168	−0.479584	+	0.877496i	$0.659212\pi$
$434$	9903.02i	1.09530i
$435$	3034.19i	0.334432i
$436$	10866.7i	1.19362i
$437$	93.9545i	0.0102848i
$438$	15316.0	1.67084
$439$	13026.2	1.41619	0.708097	−	0.706116i	$-0.249554\pi$
$439$	13026.2	1.41619	0.708097	+	0.706116i	$0.249554\pi$
$440$	29960.7i	3.24619i
$441$	2243.73	0.242278
$442$	0	0
$443$	−11533.0	−1.23690	−0.618450	−	0.785824i	$-0.712239\pi$
$443$	−11533.0	−1.23690	−0.618450	+	0.785824i	$0.712239\pi$
$444$	− 6027.39i	− 0.644250i
$445$	17039.3	1.81515
$446$	10979.5	1.16568
$447$	7910.17i	0.836998i
$448$	10110.2i	1.06621i
$449$	− 9882.75i	− 1.03874i	−0.854548	−	0.519372i	$-0.826166\pi$
$449$	− 9882.75i	− 1.03874i	0.854548	−	0.519372i	$-0.173834\pi$
$450$	− 6926.88i	− 0.725636i
$451$	−848.204	−0.0885596
$452$	3686.38	0.383613
$453$	− 9997.26i	− 1.03689i
$454$	−23893.3	−2.46997
$455$	0	0
$456$	446.075	0.0458101
$457$	− 15628.1i	− 1.59967i	−0.600218	−	0.799836i	$-0.704920\pi$
$457$	− 15628.1i	− 1.59967i	0.600218	−	0.799836i	$-0.295080\pi$
$458$	−8690.63	−0.886652
$459$	2914.06	0.296333
$460$	− 14007.8i	− 1.41982i
$461$	− 7747.46i	− 0.782723i	−0.920237	−	0.391361i	$-0.872004\pi$
$461$	− 7747.46i	− 0.782723i	0.920237	−	0.391361i	$-0.127996\pi$
$462$	4270.36i	0.430033i
$463$	333.422i	0.0334675i	0.999860	+	0.0167337i	$0.00532676\pi$
$463$	333.422i	0.0334675i	−0.999860	+	0.0167337i	$0.994673\pi$
$464$	11670.4	1.16764
$465$	−9450.59	−0.942496
$466$	10147.4i	1.00874i
$467$	−8198.33	−0.812363	−0.406182	−	0.913792i	$-0.633140\pi$
$467$	−8198.33	−0.812363	−0.406182	+	0.913792i	$0.633140\pi$
$468$	0	0
$469$	1840.24	0.181182
$470$	44748.7i	4.39171i
$471$	4875.79	0.476995
$472$	−32036.5	−3.12415
$473$	6574.96i	0.639148i
$474$	6000.14i	0.581425i
$475$	324.477i	0.0313432i
$476$	− 21325.8i	− 2.05350i
$477$	4435.43	0.425753
$478$	−20062.5	−1.91974
$479$	6435.88i	0.613910i	0.951724	+	0.306955i	$0.0993101\pi$
$479$	6435.88i	0.613910i	−0.951724	+	0.306955i	$0.900690\pi$
$480$	−23644.1	−2.24833
$481$	0	0
$482$	−19268.0	−1.82082
$483$	− 1214.09i	− 0.114375i
$484$	−11633.3	−1.09253
$485$	1076.10	0.100749
$486$	− 1295.29i	− 0.120896i
$487$	8095.37i	0.753257i	0.926364	+	0.376629i	$0.122917\pi$
$487$	8095.37i	0.753257i	−0.926364	+	0.376629i	$0.877083\pi$
$488$	29379.4i	2.72529i
$489$	− 5506.10i	− 0.509191i
$490$	21811.1	2.01087
$491$	−5116.46	−0.470270	−0.235135	−	0.971963i	$-0.575553\pi$
$491$	−5116.46	−0.470270	−0.235135	+	0.971963i	$0.575553\pi$
$492$	− 1882.82i	− 0.172528i
$493$	−6650.67	−0.607568
$494$	0	0
$495$	−4075.26	−0.370039
$496$	36349.9i	3.29064i
$497$	4692.56	0.423521
$498$	−11445.9	−1.02993
$499$	− 18050.7i	− 1.61936i	−0.586870	−	0.809682i	$-0.699640\pi$
$499$	− 18050.7i	− 1.61936i	0.586870	−	0.809682i	$-0.300360\pi$
$500$	− 6496.31i	− 0.581047i
$501$	5834.99i	0.520336i
$502$	30541.9i	2.71544i
$503$	10531.1	0.933512	0.466756	−	0.884386i	$-0.345423\pi$
$503$	10531.1	0.933512	0.466756	+	0.884386i	$0.345423\pi$
$504$	−5764.25	−0.509445
$505$	8728.45i	0.769131i
$506$	−6148.25	−0.540165
$507$	0	0
$508$	29224.1	2.55238
$509$	− 1963.31i	− 0.170967i	−0.996340	−	0.0854834i	$-0.972757\pi$
$509$	− 1963.31i	− 0.170967i	0.996340	−	0.0854834i	$-0.0272435\pi$
$510$	28327.3	2.45952
$511$	9271.00	0.802593
$512$	− 9307.69i	− 0.803409i
$513$	60.6753i	0.00522198i
$514$	29454.6i	2.52760i
$515$	− 12079.8i	− 1.03359i
$516$	−14594.9	−1.24516
$517$	14110.9	1.20038
$518$	− 5078.31i	− 0.430750i
$519$	7594.89	0.642348
$520$	0	0
$521$	−7044.93	−0.592407	−0.296203	−	0.955125i	$-0.595721\pi$
$521$	−7044.93	−0.592407	−0.296203	+	0.955125i	$0.595721\pi$
$522$	2956.19i	0.247872i
$523$	−3213.29	−0.268657	−0.134328	−	0.990937i	$-0.542888\pi$
$523$	−3213.29	−0.268657	−0.134328	+	0.990937i	$0.542888\pi$
$524$	−42200.4	−3.51819
$525$	− 4192.94i	− 0.348561i
$526$	27841.7i	2.30790i
$527$	− 20714.9i	− 1.71225i
$528$	15674.7i	1.29196i
$529$	−10419.0	−0.856333
$530$	43116.4	3.53369
$531$	− 4357.61i	− 0.356128i
$532$	444.036	0.0361868
$533$	0	0
$534$	16601.3	1.34533
$535$	12855.8i	1.03889i
$536$	12579.2	1.01369
$537$	12789.0	1.02772
$538$	38399.5i	3.07718i
$539$	− 6877.83i	− 0.549627i
$540$	− 9046.13i	− 0.720895i
$541$	− 11251.4i	− 0.894150i	−0.894497	−	0.447075i	$-0.852466\pi$
$541$	− 11251.4i	− 0.894150i	0.894497	−	0.447075i	$-0.147534\pi$
$542$	−45722.4	−3.62352
$543$	11833.8	0.935245
$544$	− 51825.8i	− 4.08458i
$545$	−8737.33	−0.686727
$546$	0	0
$547$	1533.54	0.119871	0.0599353	−	0.998202i	$-0.480911\pi$
$547$	1533.54	0.119871	0.0599353	+	0.998202i	$0.480911\pi$
$548$	− 7910.31i	− 0.616628i
$549$	−3996.19	−0.310662
$550$	−21233.3	−1.64617
$551$	− 138.477i	− 0.0107066i
$552$	− 8299.08i	− 0.639914i
$553$	3631.97i	0.279289i
$554$	− 38214.6i	− 2.93066i
$555$	4846.30	0.370656
$556$	−15362.2	−1.17177
$557$	− 16845.7i	− 1.28146i	−0.767766	−	0.640731i	$-0.778631\pi$
$557$	− 16845.7i	− 1.28146i	0.767766	−	0.640731i	$-0.221369\pi$
$558$	−9207.65	−0.698550
$559$	0	0
$560$	−30089.0	−2.27052
$561$	− 8932.62i	− 0.672256i
$562$	4526.34	0.339737
$563$	20820.1	1.55855	0.779273	−	0.626685i	$-0.215589\pi$
$563$	20820.1	1.55855	0.779273	+	0.626685i	$0.215589\pi$
$564$	31322.8i	2.33852i
$565$	2964.03i	0.220704i
$566$	− 5945.37i	− 0.441524i
$567$	− 784.054i	− 0.0580726i
$568$	32076.6	2.36955
$569$	−23636.6	−1.74147	−0.870735	−	0.491752i	$-0.836357\pi$
$569$	−23636.6	−1.74147	−0.870735	+	0.491752i	$0.836357\pi$
$570$	589.819i	0.0433417i
$571$	26955.1	1.97554	0.987771	−	0.155913i	$-0.0498319\pi$
$571$	26955.1	1.97554	0.987771	+	0.155913i	$0.0498319\pi$
$572$	0	0
$573$	642.326	0.0468300
$574$	− 1586.35i	− 0.115353i
$575$	6036.78	0.437828
$576$	−9400.25	−0.679995
$577$	23499.8i	1.69551i	0.530388	+	0.847755i	$0.322046\pi$
$577$	23499.8i	1.69551i	−0.530388	+	0.847755i	$0.677954\pi$
$578$	35902.8i	2.58367i
$579$	− 3621.56i	− 0.259943i
$580$	20645.7i	1.47805i
$581$	−6928.38	−0.494729
$582$	1048.44	0.0746721
$583$	− 13596.1i	− 0.965857i
$584$	63373.1	4.49040
$585$	0	0
$586$	−9933.33	−0.700243
$587$	4637.50i	0.326082i	0.986619	+	0.163041i	$0.0521303\pi$
$587$	4637.50i	0.326082i	−0.986619	+	0.163041i	$0.947870\pi$
$588$	15267.2	1.07076
$589$	431.316	0.0301733
$590$	− 42359.9i	− 2.95582i
$591$	2782.89i	0.193694i
$592$	− 18640.4i	− 1.29411i
$593$	− 12633.5i	− 0.874869i	−0.899250	−	0.437434i	$-0.855887\pi$
$593$	− 12633.5i	− 0.874869i	0.899250	−	0.437434i	$-0.144113\pi$
$594$	−3970.51	−0.274262
$595$	17146.9	1.18144
$596$	53823.7i	3.69917i
$597$	1436.85	0.0985033
$598$	0	0
$599$	−18757.1	−1.27946	−0.639730	−	0.768600i	$-0.720954\pi$
$599$	−18757.1	−1.27946	−0.639730	+	0.768600i	$0.720954\pi$
$600$	− 28661.3i	− 1.95016i
$601$	−3632.98	−0.246576	−0.123288	−	0.992371i	$-0.539344\pi$
$601$	−3632.98	−0.246576	−0.123288	+	0.992371i	$0.539344\pi$
$602$	−12296.8	−0.832522
$603$	1711.02i	0.115553i
$604$	− 68025.0i	− 4.58261i
$605$	− 9353.71i	− 0.628566i
$606$	8504.08i	0.570057i
$607$	−12700.0	−0.849219	−0.424610	−	0.905377i	$-0.639589\pi$
$607$	−12700.0	−0.849219	−0.424610	+	0.905377i	$0.639589\pi$
$608$	1079.09	0.0719786
$609$	1789.42i	0.119066i
$610$	−38846.6	−2.57845
$611$	0	0
$612$	19828.3	1.30966
$613$	21640.1i	1.42584i	0.701248	+	0.712918i	$0.252627\pi$
$613$	21640.1i	1.42584i	−0.701248	+	0.712918i	$0.747373\pi$
$614$	−34047.9	−2.23788
$615$	1513.87	0.0992605
$616$	17669.5i	1.15572i
$617$	− 16541.7i	− 1.07933i	−0.841881	−	0.539663i	$-0.818552\pi$
$617$	− 16541.7i	− 1.07933i	0.841881	−	0.539663i	$-0.181448\pi$
$618$	− 11769.2i	− 0.766066i
$619$	21138.9i	1.37261i	0.727316	+	0.686303i	$0.240767\pi$
$619$	21138.9i	1.37261i	−0.727316	+	0.686303i	$0.759233\pi$
$620$	−64305.2	−4.16542
$621$	1128.84	0.0729451
$622$	18616.9i	1.20011i
$623$	10049.0	0.646235
$624$	0	0
$625$	−12825.4	−0.820823
$626$	− 31513.3i	− 2.01202i
$627$	185.991	0.0118465
$628$	33176.6	2.10811
$629$	10622.7i	0.673377i
$630$	− 7621.73i	− 0.481995i
$631$	− 5489.80i	− 0.346348i	−0.984891	−	0.173174i	$-0.944598\pi$
$631$	− 5489.80i	− 0.346348i	0.984891	−	0.173174i	$-0.0554023\pi$
$632$	24826.8i	1.56259i
$633$	4352.86	0.273319
$634$	8941.94	0.560141
$635$	23497.5i	1.46846i
$636$	30180.3	1.88164
$637$	0	0
$638$	9061.76	0.562318
$639$	4363.06i	0.270110i
$640$	−28328.1	−1.74963
$641$	−4297.04	−0.264778	−0.132389	−	0.991198i	$-0.542265\pi$
$641$	−4297.04	−0.264778	−0.132389	+	0.991198i	$0.542265\pi$
$642$	12525.3i	0.769993i
$643$	25696.9i	1.57603i	0.615655	+	0.788016i	$0.288892\pi$
$643$	25696.9i	1.57603i	−0.615655	+	0.788016i	$0.711108\pi$
$644$	− 8261.14i	− 0.505488i
$645$	− 11735.0i	− 0.716378i
$646$	−1292.83	−0.0787396
$647$	−2174.98	−0.132160	−0.0660798	−	0.997814i	$-0.521049\pi$
$647$	−2174.98	−0.132160	−0.0660798	+	0.997814i	$0.521049\pi$
$648$	− 5359.50i	− 0.324909i
$649$	−13357.6	−0.807907
$650$	0	0
$651$	−5573.52	−0.335551
$652$	− 37465.5i	− 2.25040i
$653$	−15454.5	−0.926160	−0.463080	−	0.886316i	$-0.653256\pi$
$653$	−15454.5	−0.926160	−0.463080	+	0.886316i	$0.653256\pi$
$654$	−8512.73	−0.508982
$655$	− 33931.1i	− 2.02412i
$656$	− 5822.82i	− 0.346560i
$657$	8620.02i	0.511870i
$658$	26390.7i	1.56355i
$659$	−3148.77	−0.186129	−0.0930643	−	0.995660i	$-0.529666\pi$
$659$	−3148.77	−0.186129	−0.0930643	+	0.995660i	$0.529666\pi$
$660$	−27729.6	−1.63541
$661$	2099.70i	0.123553i	0.998090	+	0.0617767i	$0.0196767\pi$
$661$	2099.70i	0.123553i	−0.998090	+	0.0617767i	$0.980323\pi$
$662$	−10715.7	−0.629122
$663$	0	0
$664$	−47359.8	−2.76795
$665$	357.026i	0.0208193i
$666$	4721.73	0.274720
$667$	−2576.32	−0.149559
$668$	39703.4i	2.29966i
$669$	6179.36i	0.357112i
$670$	16632.7i	0.959071i
$671$	12249.7i	0.704763i
$672$	−13944.2	−0.800460
$673$	−30970.8	−1.77390	−0.886950	−	0.461865i	$-0.847181\pi$
$673$	−30970.8	−1.77390	−0.886950	+	0.461865i	$0.847181\pi$
$674$	− 38055.0i	− 2.17481i
$675$	3898.52	0.222302
$676$	0	0
$677$	14640.6	0.831141	0.415570	−	0.909561i	$-0.363582\pi$
$677$	14640.6	0.831141	0.415570	+	0.909561i	$0.363582\pi$
$678$	2887.83i	0.163579i
$679$	634.635	0.0358690
$680$	117210.	6.60999
$681$	− 13447.4i	− 0.756688i
$682$	28224.7i	1.58472i
$683$	6685.83i	0.374563i	0.982306	+	0.187281i	$0.0599676\pi$
$683$	6685.83i	0.374563i	−0.982306	+	0.187281i	$0.940032\pi$
$684$	412.856i	0.0230789i
$685$	6360.27	0.354764
$686$	30560.8	1.70090
$687$	− 4891.18i	− 0.271630i
$688$	−45136.3	−2.50117
$689$	0	0
$690$	10973.4	0.605434
$691$	− 30194.1i	− 1.66228i	−0.556062	−	0.831141i	$-0.687688\pi$
$691$	− 30194.1i	− 1.66228i	0.556062	−	0.831141i	$-0.312312\pi$
$692$	51678.4	2.83890
$693$	−2403.40	−0.131743
$694$	6.07611i	0 0.000332343i
$695$	− 12352.0i	− 0.674154i
$696$	12231.8i	0.666158i
$697$	3318.28i	0.180328i
$698$	65026.0	3.52618
$699$	−5711.08	−0.309031
$700$	− 28530.3i	− 1.54049i
$701$	30300.9	1.63260	0.816298	−	0.577631i	$-0.196023\pi$
$701$	30300.9	1.63260	0.816298	+	0.577631i	$0.196023\pi$
$702$	0	0
$703$	−221.181	−0.0118663
$704$	28815.1i	1.54263i
$705$	−25185.0	−1.34542
$706$	58060.3	3.09508
$707$	5147.64i	0.273829i
$708$	− 29650.8i	− 1.57393i
$709$	− 26123.2i	− 1.38375i	−0.722017	−	0.691875i	$-0.756785\pi$
$709$	− 26123.2i	− 1.38375i	0.722017	−	0.691875i	$-0.243215\pi$
$710$	42412.9i	2.24187i
$711$	−3376.94	−0.178123
$712$	68691.1	3.61560
$713$	− 8024.48i	− 0.421486i
$714$	16706.2	0.875647
$715$	0	0
$716$	87021.3	4.54209
$717$	− 11291.4i	− 0.588122i
$718$	−18793.8	−0.976850
$719$	−19325.7	−1.00240	−0.501200	−	0.865331i	$-0.667108\pi$
$719$	−19325.7	−1.00240	−0.501200	+	0.865331i	$0.667108\pi$
$720$	− 27976.2i	− 1.44807i
$721$	− 7124.09i	− 0.367982i
$722$	36534.2i	1.88319i
$723$	− 10844.2i	− 0.557816i
$724$	80521.7	4.13338
$725$	−8897.47	−0.455784
$726$	− 9113.26i	− 0.465875i
$727$	−26065.8	−1.32975	−0.664875	−	0.746954i	$-0.731515\pi$
$727$	−26065.8	−1.32975	−0.664875	+	0.746954i	$0.731515\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	83794.4i	4.24845i
$731$	25722.0	1.30145
$732$	−27191.5	−1.37299
$733$	1055.45i	0.0531843i	0.999646	+	0.0265921i	$0.00846554\pi$
$733$	1055.45i	0.0531843i	−0.999646	+	0.0265921i	$0.991534\pi$
$734$	12706.4i	0.638965i
$735$	12275.5i	0.616041i
$736$	− 20076.2i	− 1.00546i
$737$	5244.89	0.262141
$738$	1474.96	0.0735690
$739$	− 9410.40i	− 0.468426i	−0.972185	−	0.234213i	$-0.924749\pi$
$739$	− 9410.40i	− 0.468426i	0.972185	−	0.234213i	$-0.0752514\pi$
$740$	32976.0	1.63814
$741$	0	0
$742$	25428.1	1.25808
$743$	− 7523.70i	− 0.371491i	−0.982598	−	0.185746i	$-0.940530\pi$
$743$	− 7523.70i	− 0.371491i	0.982598	−	0.185746i	$-0.0594700\pi$
$744$	−38098.5	−1.87736
$745$	−43276.8	−2.12824
$746$	− 70799.4i	− 3.47473i
$747$	− 6441.89i	− 0.315524i
$748$	− 60780.8i	− 2.97108i
$749$	7581.76i	0.369868i
$750$	5089.07	0.247769
$751$	12984.1	0.630886	0.315443	−	0.948945i	$-0.397847\pi$
$751$	12984.1	0.630886	0.315443	+	0.948945i	$0.397847\pi$
$752$	96869.3i	4.69742i
$753$	−17189.3	−0.831890
$754$	0	0
$755$	54695.3	2.63651
$756$	− 5334.99i	− 0.256656i
$757$	−27934.6	−1.34122	−0.670609	−	0.741811i	$-0.733967\pi$
$757$	−27934.6	−1.34122	−0.670609	+	0.741811i	$0.733967\pi$
$758$	23649.5	1.13323
$759$	− 3460.30i	− 0.165482i
$760$	2440.49i	0.116482i
$761$	15519.3i	0.739255i	0.929180	+	0.369627i	$0.120515\pi$
$761$	15519.3i	0.739255i	−0.929180	+	0.369627i	$0.879485\pi$
$762$	22893.5i	1.08838i
$763$	−5152.88	−0.244491
$764$	4370.62	0.206968
$765$	15942.9i	0.753487i
$766$	4319.82	0.203761
$767$	0	0
$768$	−2532.57	−0.118993
$769$	12885.2i	0.604228i	0.953272	+	0.302114i	$0.0976924\pi$
$769$	12885.2i	0.604228i	−0.953272	+	0.302114i	$0.902308\pi$
$770$	−23363.3	−1.09345
$771$	−16577.4	−0.774344
$772$	− 24642.4i	− 1.14883i
$773$	5892.04i	0.274155i	0.990560	+	0.137078i	$0.0437710\pi$
$773$	5892.04i	0.274155i	−0.990560	+	0.137078i	$0.956229\pi$
$774$	− 11433.3i	− 0.530958i
$775$	− 27713.0i	− 1.28449i
$776$	4338.12	0.200682
$777$	2858.13	0.131962
$778$	− 18463.4i	− 0.850828i
$779$	−69.0916	−0.00317775
$780$	0	0
$781$	13374.3	0.612766
$782$	24052.7i	1.09990i
$783$	−1663.77	−0.0759367
$784$	47215.5	2.15085
$785$	26675.6i	1.21286i
$786$	− 33058.9i	− 1.50022i
$787$	21020.4i	0.952091i	0.879421	+	0.476045i	$0.157930\pi$
$787$	21020.4i	0.952091i	−0.879421	+	0.476045i	$0.842070\pi$
$788$	18935.8i	0.856042i
$789$	−15669.6	−0.707038
$790$	−32827.0	−1.47839
$791$	1748.05i	0.0785757i
$792$	−16428.7	−0.737084
$793$	0	0
$794$	2267.58	0.101352
$795$	24266.4i	1.08256i
$796$	9776.87	0.435342
$797$	−31355.6	−1.39356	−0.696782	−	0.717283i	$-0.745386\pi$
$797$	−31355.6	−1.39356	−0.696782	+	0.717283i	$0.745386\pi$
$798$	347.848i	0.0154307i
$799$	− 55203.3i	− 2.44425i
$800$	− 69334.0i	− 3.06416i
$801$	9343.37i	0.412150i
$802$	−6326.37	−0.278543
$803$	26423.4	1.16122
$804$	11642.4i	0.510692i
$805$	6642.34	0.290822
$806$	0	0
$807$	−21611.7	−0.942709
$808$	35187.3i	1.53204i
$809$	−18132.5	−0.788017	−0.394009	−	0.919107i	$-0.628912\pi$
$809$	−18132.5	−0.788017	−0.394009	+	0.919107i	$0.628912\pi$
$810$	7086.55	0.307402
$811$	− 24755.3i	− 1.07186i	−0.844263	−	0.535928i	$-0.819962\pi$
$811$	− 24755.3i	− 1.07186i	0.844263	−	0.535928i	$-0.180038\pi$
$812$	12175.9i	0.526219i
$813$	− 25733.1i	− 1.11008i
$814$	− 14473.8i	− 0.623225i
$815$	30124.0	1.29472
$816$	61321.4	2.63073
$817$	535.572i	0.0229343i
$818$	−42682.7	−1.82441
$819$	0	0
$820$	10300.9	0.438688
$821$	4082.65i	0.173551i	0.996228	+	0.0867755i	$0.0276563\pi$
$821$	4082.65i	0.173551i	−0.996228	+	0.0867755i	$0.972344\pi$
$822$	6196.77	0.262941
$823$	34327.0	1.45391	0.726954	−	0.686687i	$-0.240935\pi$
$823$	34327.0	1.45391	0.726954	+	0.686687i	$0.240935\pi$
$824$	− 48697.6i	− 2.05881i
$825$	− 11950.3i	− 0.504312i
$826$	− 24981.9i	− 1.05234i
$827$	− 3228.87i	− 0.135767i	−0.997693	−	0.0678833i	$-0.978375\pi$
$827$	− 3228.87i	− 0.135767i	0.997693	−	0.0678833i	$-0.0216245\pi$
$828$	7681.06	0.322386
$829$	10452.4	0.437908	0.218954	−	0.975735i	$-0.429735\pi$
$829$	10452.4	0.437908	0.218954	+	0.975735i	$0.429735\pi$
$830$	− 62621.0i	− 2.61880i
$831$	21507.6	0.897821
$832$	0	0
$833$	−26906.9	−1.11917
$834$	− 12034.5i	− 0.499663i
$835$	−31923.4	−1.32306
$836$	1265.55	0.0523564
$837$	− 5182.16i	− 0.214004i
$838$	36419.7i	1.50131i
$839$	28289.0i	1.16406i	0.813169	+	0.582028i	$0.197741\pi$
$839$	28289.0i	1.16406i	−0.813169	+	0.582028i	$0.802259\pi$
$840$	− 31536.4i	− 1.29537i
$841$	−20591.8	−0.844308
$842$	−57246.1	−2.34303
$843$	2547.47i	0.104080i
$844$	29618.5	1.20795
$845$	0	0
$846$	−24537.6	−0.997187
$847$	− 5516.39i	− 0.223784i
$848$	93335.9	3.77968
$849$	3346.12	0.135263
$850$	83067.2i	3.35198i
$851$	4114.99i	0.165758i
$852$	29687.8i	1.19377i
$853$	26631.8i	1.06900i	0.845170	+	0.534498i	$0.179499\pi$
$853$	26631.8i	1.06900i	−0.845170	+	0.534498i	$0.820501\pi$
$854$	−22910.0	−0.917989
$855$	−331.956	−0.0132780
$856$	51826.0i	2.06936i
$857$	−11796.7	−0.470209	−0.235104	−	0.971970i	$-0.575543\pi$
$857$	−11796.7	−0.470209	−0.235104	+	0.971970i	$0.575543\pi$
$858$	0	0
$859$	−22672.8	−0.900567	−0.450283	−	0.892886i	$-0.648677\pi$
$859$	−22672.8	−0.900567	−0.450283	+	0.892886i	$0.648677\pi$
$860$	− 79849.0i	− 3.16608i
$861$	892.812	0.0353391
$862$	−27795.0	−1.09826
$863$	21421.1i	0.844940i	0.906377	+	0.422470i	$0.138837\pi$
$863$	21421.1i	0.844940i	−0.906377	+	0.422470i	$0.861163\pi$
$864$	− 12965.1i	− 0.510510i
$865$	41551.9i	1.63330i
$866$	− 46066.6i	− 1.80763i
$867$	−20206.5	−0.791520
$868$	−37924.3	−1.48299
$869$	10351.5i	0.404086i
$870$	−16173.4	−0.630264
$871$	0	0
$872$	−35223.1	−1.36790
$873$	590.072i	0.0228762i
$874$	−500.814	−0.0193825
$875$	3080.48	0.119016
$876$	58653.7i	2.26224i
$877$	− 5155.20i	− 0.198493i	−0.995063	−	0.0992466i	$-0.968357\pi$
$877$	− 5155.20i	− 0.198493i	0.995063	−	0.0992466i	$-0.0316433\pi$
$878$	69435.0i	2.66893i
$879$	− 5590.58i	− 0.214523i
$880$	−85756.9	−3.28507
$881$	−23692.2	−0.906027	−0.453013	−	0.891504i	$-0.649651\pi$
$881$	−23692.2	−0.906027	−0.453013	+	0.891504i	$0.649651\pi$
$882$	11960.0i	0.456591i
$883$	14591.5	0.556108	0.278054	−	0.960565i	$-0.410311\pi$
$883$	14591.5	0.556108	0.278054	+	0.960565i	$0.410311\pi$
$884$	0	0
$885$	23840.6	0.905529
$886$	− 61475.2i	− 2.33104i
$887$	−9722.26	−0.368029	−0.184014	−	0.982924i	$-0.558909\pi$
$887$	−9722.26	−0.368029	−0.184014	+	0.982924i	$0.558909\pi$
$888$	19537.1	0.738312
$889$	13857.8i	0.522806i
$890$	90826.1i	3.42078i
$891$	− 2234.64i	− 0.0840217i
$892$	42046.6i	1.57828i
$893$	1149.42	0.0430726
$894$	−42164.3	−1.57739
$895$	69969.2i	2.61320i
$896$	−16706.6	−0.622911
$897$	0	0
$898$	52678.9	1.95759
$899$	11827.1i	0.438771i
$900$	26526.9	0.982479
$901$	−53189.7	−1.96671
$902$	− 4521.26i	− 0.166898i
$903$	− 6920.74i	− 0.255048i
$904$	11949.0i	0.439621i
$905$	64743.2i	2.37805i
$906$	53289.3	1.95411
$907$	−11799.0	−0.431951	−0.215975	−	0.976399i	$-0.569293\pi$
$907$	−11799.0	−0.431951	−0.215975	+	0.976399i	$0.569293\pi$
$908$	− 91500.9i	− 3.34423i
$909$	−4786.18	−0.174640
$910$	0	0
$911$	−43012.4	−1.56429	−0.782143	−	0.623099i	$-0.785873\pi$
$911$	−43012.4	−1.56429	−0.782143	+	0.623099i	$0.785873\pi$
$912$	1276.81i	0.0463589i
$913$	−19746.6	−0.715793
$914$	83303.8	3.01471
$915$	− 21863.3i	− 0.789921i
$916$	− 33281.3i	− 1.20049i
$917$	− 20011.0i	− 0.720635i
$918$	15533.1i	0.558462i
$919$	−4951.41	−0.177728	−0.0888639	−	0.996044i	$-0.528324\pi$
$919$	−4951.41	−0.177728	−0.0888639	+	0.996044i	$0.528324\pi$
$920$	45404.5	1.62711
$921$	− 19162.5i	− 0.685587i
$922$	41297.0	1.47510
$923$	0	0
$924$	−16353.6	−0.582245
$925$	14211.3i	0.505152i
$926$	−1777.27	−0.0630721
$927$	6623.85	0.234688
$928$	29589.8i	1.04669i
$929$	− 8934.86i	− 0.315547i	−0.987475	−	0.157774i	$-0.949568\pi$
$929$	− 8934.86i	− 0.315547i	0.987475	−	0.157774i	$-0.0504316\pi$
$930$	− 50375.4i	− 1.77621i
$931$	− 560.243i	− 0.0197221i
$932$	−38860.2	−1.36578
$933$	−10477.8	−0.367660
$934$	− 43700.3i	− 1.53096i
$935$	48870.6	1.70935
$936$	0	0
$937$	−13182.8	−0.459620	−0.229810	−	0.973235i	$-0.573811\pi$
$937$	−13182.8	−0.459620	−0.229810	+	0.973235i	$0.573811\pi$
$938$	9809.20i	0.341452i
$939$	17736.0	0.616393
$940$	−171368.	−5.94618
$941$	− 21693.7i	− 0.751536i	−0.926714	−	0.375768i	$-0.877379\pi$
$941$	− 21693.7i	− 0.751536i	0.926714	−	0.375768i	$-0.122621\pi$
$942$	25989.9i	0.898934i
$943$	1285.43i	0.0443895i
$944$	− 91698.4i	− 3.16158i
$945$	4289.59	0.147662
$946$	−35047.1	−1.20452
$947$	49790.0i	1.70851i	0.519856	+	0.854254i	$0.325986\pi$
$947$	49790.0i	1.70851i	−0.519856	+	0.854254i	$0.674014\pi$
$948$	−22977.9	−0.787224
$949$	0	0
$950$	−1729.59	−0.0590687
$951$	5032.61i	0.171602i
$952$	69125.0	2.35331
$953$	−4217.93	−0.143371	−0.0716853	−	0.997427i	$-0.522838\pi$
$953$	−4217.93	−0.143371	−0.0716853	+	0.997427i	$0.522838\pi$
$954$	23642.6i	0.802365i
$955$	3514.19i	0.119075i
$956$	− 76830.5i	− 2.59924i
$957$	5100.05i	0.172269i
$958$	−34305.8	−1.15696
$959$	3750.99	0.126304
$960$	− 51429.0i	− 1.72903i
$961$	−7046.87	−0.236544
$962$	0	0
$963$	−7049.38	−0.235891
$964$	− 73788.1i	− 2.46530i
$965$	19813.7	0.660958
$966$	6471.60	0.215549
$967$	− 40927.9i	− 1.36107i	−0.732717	−	0.680534i	$-0.761748\pi$
$967$	− 40927.9i	− 1.36107i	0.732717	−	0.680534i	$-0.238252\pi$
$968$	− 37707.9i	− 1.25204i
$969$	− 727.619i	− 0.0241223i
$970$	5736.04i	0.189869i
$971$	−17114.8	−0.565645	−0.282822	−	0.959172i	$-0.591271\pi$
$971$	−17114.8	−0.565645	−0.282822	+	0.959172i	$0.591271\pi$
$972$	4960.38	0.163688
$973$	− 7284.62i	− 0.240015i
$974$	−43151.5	−1.41957
$975$	0	0
$976$	−84093.0	−2.75794
$977$	− 118.470i	− 0.00387940i	−0.999998	−	0.00193970i	$-0.999383\pi$
$977$	− 118.470i	− 0.00387940i	0.999998	−	0.00193970i	$-0.000617427\pi$
$978$	29349.7	0.959610
$979$	28640.7	0.934996
$980$	83527.2i	2.72263i
$981$	− 4791.05i	− 0.155929i
$982$	− 27272.8i	− 0.886261i
$983$	− 26002.8i	− 0.843705i	−0.906665	−	0.421852i	$-0.861380\pi$
$983$	− 26002.8i	− 0.843705i	0.906665	−	0.421852i	$-0.138620\pi$
$984$	6102.93	0.197718
$985$	−15225.3	−0.492506
$986$	− 35450.7i	− 1.14501i
$987$	−14853.0	−0.479002
$988$	0	0
$989$	9964.14	0.320365
$990$	− 21722.8i	− 0.697368i
$991$	−16062.0	−0.514860	−0.257430	−	0.966297i	$-0.582876\pi$
$991$	−16062.0	−0.514860	−0.257430	+	0.966297i	$0.582876\pi$
$992$	−92163.3	−2.94979
$993$	− 6030.93i	− 0.192735i
$994$	25013.2i	0.798159i
$995$	7861.07i	0.250465i
$996$	− 43832.9i	− 1.39448i
$997$	1361.54	0.0432503	0.0216251	−	0.999766i	$-0.493116\pi$
$997$	1361.54	0.0432503	0.0216251	+	0.999766i	$0.493116\pi$
$998$	96217.5	3.05182
$999$	2657.44i	0.0841617i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.h.337.8		8
13.2	odd	12		39.4.e.c.22.1	yes	8
13.5	odd	4		507.4.a.m.1.4		4
13.6	odd	12		39.4.e.c.16.1	✓	8
13.8	odd	4		507.4.a.i.1.1		4
13.12	even	2	inner	507.4.b.h.337.1		8
39.2	even	12		117.4.g.e.100.4		8
39.5	even	4		1521.4.a.v.1.1		4
39.8	even	4		1521.4.a.bb.1.4		4
39.32	even	12		117.4.g.e.55.4		8
52.15	even	12		624.4.q.i.529.1		8
52.19	even	12		624.4.q.i.289.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.e.c.16.1	✓	8	13.6	odd	12
39.4.e.c.22.1	yes	8	13.2	odd	12
117.4.g.e.55.4		8	39.32	even	12
117.4.g.e.100.4		8	39.2	even	12
507.4.a.i.1.1		4	13.8	odd	4
507.4.a.m.1.4		4	13.5	odd	4
507.4.b.h.337.1		8	13.12	even	2	inner
507.4.b.h.337.8		8	1.1	even	1	trivial
624.4.q.i.289.1		8	52.19	even	12
624.4.q.i.529.1		8	52.15	even	12
1521.4.a.v.1.1		4	39.5	even	4
1521.4.a.bb.1.4		4	39.8	even	4

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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.33039i q^{2} -3.00000 q^{3} -20.4131 q^{4} -16.4131i q^{5} -15.9912i q^{6} -9.67968i q^{7} -66.1667i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+5.33039i q^{2} -3.00000 q^{3} -20.4131 q^{4} -16.4131i q^{5} -15.9912i q^{6} -9.67968i q^{7} -66.1667i q^{8} +9.00000 q^{9} +87.4882 q^{10} -27.5882i q^{11} +61.2393 q^{12} +51.5965 q^{14} +49.2393i q^{15} +189.390 q^{16} -107.928 q^{17} +47.9735i q^{18} -2.24723i q^{19} +335.042i q^{20} +29.0391i q^{21} +147.056 q^{22} -41.8090 q^{23} +198.500i q^{24} -144.390 q^{25} -27.0000 q^{27} +197.592i q^{28} +61.6213 q^{29} -262.465 q^{30} +191.932i q^{31} +480.187i q^{32} +82.7645i q^{33} -575.300i q^{34} -158.874 q^{35} -183.718 q^{36} -98.4236i q^{37} +11.9786 q^{38} -1086.00 q^{40} -30.7452i q^{41} -154.790 q^{42} -238.325 q^{43} +563.160i q^{44} -147.718i q^{45} -222.858i q^{46} +511.482i q^{47} -568.169 q^{48} +249.304 q^{49} -769.653i q^{50} +323.785 q^{51} +492.825 q^{53} -143.921i q^{54} -452.807 q^{55} -640.472 q^{56} +6.74170i q^{57} +328.466i q^{58} -484.179i q^{59} -1005.13i q^{60} -444.021 q^{61} -1023.07 q^{62} -87.1172i q^{63} -1044.47 q^{64} -441.167 q^{66} +190.114i q^{67} +2203.15 q^{68} +125.427 q^{69} -846.858i q^{70} +484.785i q^{71} -595.500i q^{72} +957.780i q^{73} +524.636 q^{74} +433.169 q^{75} +45.8729i q^{76} -267.045 q^{77} -375.216 q^{79} -3108.47i q^{80} +81.0000 q^{81} +163.884 q^{82} -715.765i q^{83} -592.777i q^{84} +1771.43i q^{85} -1270.37i q^{86} -184.864 q^{87} -1825.42 q^{88} +1038.15i q^{89} +787.394 q^{90} +853.451 q^{92} -575.796i q^{93} -2726.40 q^{94} -36.8840 q^{95} -1440.56i q^{96} +65.5636i q^{97} +1328.89i q^{98} -248.293i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9}+O(q^{10}) \) 8 * q - 24 * q^3 - 44 * q^4 + 72 * q^9	\( 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9} + 124 q^{10} + 132 q^{12} + 80 q^{14} + 244 q^{16} - 196 q^{17} + 440 q^{22} - 208 q^{23} + 116 q^{25} - 216 q^{27} + 388 q^{29} - 372 q^{30} + 176 q^{35} - 396 q^{36} - 664 q^{38} - 1996 q^{40} - 240 q^{42} - 900 q^{43} - 732 q^{48} - 2140 q^{49} + 588 q^{51} + 524 q^{53} + 408 q^{55} - 4328 q^{56} - 1856 q^{61} - 5560 q^{62} - 2052 q^{64} - 1320 q^{66} + 3572 q^{68} + 624 q^{69} + 2316 q^{74} - 348 q^{75} - 5016 q^{77} - 1492 q^{79} + 648 q^{81} - 3468 q^{82} - 1164 q^{87} - 6120 q^{88} + 1116 q^{90} + 664 q^{92} - 1544 q^{94} - 4408 q^{95}+O(q^{100}) \) 8 * q - 24 * q^3 - 44 * q^4 + 72 * q^9 + 124 * q^10 + 132 * q^12 + 80 * q^14 + 244 * q^16 - 196 * q^17 + 440 * q^22 - 208 * q^23 + 116 * q^25 - 216 * q^27 + 388 * q^29 - 372 * q^30 + 176 * q^35 - 396 * q^36 - 664 * q^38 - 1996 * q^40 - 240 * q^42 - 900 * q^43 - 732 * q^48 - 2140 * q^49 + 588 * q^51 + 524 * q^53 + 408 * q^55 - 4328 * q^56 - 1856 * q^61 - 5560 * q^62 - 2052 * q^64 - 1320 * q^66 + 3572 * q^68 + 624 * q^69 + 2316 * q^74 - 348 * q^75 - 5016 * q^77 - 1492 * q^79 + 648 * q^81 - 3468 * q^82 - 1164 * q^87 - 6120 * q^88 + 1116 * q^90 + 664 * q^92 - 1544 * q^94 - 4408 * q^95

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