LMFDB - Embedded newform 1134.3.b.c.323.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1134,3,Mod(323,1134)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1134, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1134.323"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1134 = 2 \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1134.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$30.8992619785$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			323.14
Character	$\chi$	$=$	1134.323
Dual form			1134.3.b.c.323.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.41421i q^{2} -2.00000 q^{4} -8.59478i q^{5} +2.64575 q^{7} -2.82843i q^{8} +12.1548 q^{10} +14.3608i q^{11} +2.41973 q^{13} +3.74166i q^{14} +4.00000 q^{16} +4.27995i q^{17} +20.7972 q^{19} +17.1896i q^{20} -20.3092 q^{22} +37.6353i q^{23} -48.8702 q^{25} +3.42201i q^{26} -5.29150 q^{28} -43.3995i q^{29} +45.5815 q^{31} +5.65685i q^{32} -6.05276 q^{34} -22.7396i q^{35} +4.78176 q^{37} +29.4116i q^{38} -24.3097 q^{40} -2.97891i q^{41} +24.4298 q^{43} -28.7215i q^{44} -53.2244 q^{46} -1.53301i q^{47} +7.00000 q^{49} -69.1129i q^{50} -4.83946 q^{52} -90.8320i q^{53} +123.427 q^{55} -7.48331i q^{56} +61.3761 q^{58} -17.1433i q^{59} +53.5618 q^{61} +64.4620i q^{62} -8.00000 q^{64} -20.7970i q^{65} +101.014 q^{67} -8.55990i q^{68} +32.1587 q^{70} -127.047i q^{71} -2.38851 q^{73} +6.76242i q^{74} -41.5943 q^{76} +37.9950i q^{77} -48.8195 q^{79} -34.3791i q^{80} +4.21282 q^{82} -85.8416i q^{83} +36.7852 q^{85} +34.5489i q^{86} +40.6184 q^{88} +13.8255i q^{89} +6.40200 q^{91} -75.2706i q^{92} +2.16800 q^{94} -178.747i q^{95} -3.88170 q^{97} +9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 48 q^{4} + 96 q^{16} + 24 q^{19} - 48 q^{22} - 144 q^{25} + 120 q^{31} + 96 q^{34} - 168 q^{37} - 120 q^{43} + 168 q^{49} + 264 q^{55} - 192 q^{64} - 144 q^{67} + 24 q^{73} - 48 q^{76} - 24 q^{79}+ \cdots - 96 q^{97}+O(q^{100}) $ 24 * q - 48 * q^4 + 96 * q^16 + 24 * q^19 - 48 * q^22 - 144 * q^25 + 120 * q^31 + 96 * q^34 - 168 * q^37 - 120 * q^43 + 168 * q^49 + 264 * q^55 - 192 * q^64 - 144 * q^67 + 24 * q^73 - 48 * q^76 - 24 * q^79 - 288 * q^82 + 312 * q^85 + 96 * q^88 - 168 * q^91 - 96 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times$.

$n$	$325$	$407$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.41421i	0.707107i
$2$	1.41421i	0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	− 8.59478i	− 1.71896i	−0.511173	−	0.859478i	$-0.670789\pi$
$5$	− 8.59478i	− 1.71896i	0.511173	−	0.859478i	$-0.329211\pi$
$6$	0	0
$7$	2.64575	0.377964
$7$	2.64575	0.377964
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	12.1548	1.21548
$11$	14.3608i	1.30552i	0.757563	+	0.652762i	$0.226390\pi$
$11$	14.3608i	1.30552i	−0.757563	+	0.652762i	$0.773610\pi$
$12$	0	0
$13$	2.41973	0.186133	0.0930665	−	0.995660i	$-0.470333\pi$
$13$	2.41973	0.186133	0.0930665	+	0.995660i	$0.470333\pi$
$14$	3.74166i	0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	4.27995i	0.251762i	0.992045	+	0.125881i	$0.0401757\pi$
$17$	4.27995i	0.251762i	−0.992045	+	0.125881i	$0.959824\pi$
$18$	0	0
$19$	20.7972	1.09459	0.547294	−	0.836940i	$-0.315658\pi$
$19$	20.7972	1.09459	0.547294	+	0.836940i	$0.315658\pi$
$20$	17.1896i	0.859478i
$21$	0	0
$22$	−20.3092	−0.923144
$23$	37.6353i	1.63632i	0.574992	+	0.818159i	$0.305005\pi$
$23$	37.6353i	1.63632i	−0.574992	+	0.818159i	$0.694995\pi$
$24$	0	0
$25$	−48.8702	−1.95481
$26$	3.42201i	0.131616i
$27$	0	0
$28$	−5.29150	−0.188982
$29$	− 43.3995i	− 1.49653i	−0.663398	−	0.748267i	$-0.730886\pi$
$29$	− 43.3995i	− 1.49653i	0.663398	−	0.748267i	$-0.269114\pi$
$30$	0	0
$31$	45.5815	1.47037	0.735186	−	0.677866i	$-0.237095\pi$
$31$	45.5815	1.47037	0.735186	+	0.677866i	$0.237095\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	−6.05276	−0.178022
$35$	− 22.7396i	− 0.649704i
$36$	0	0
$37$	4.78176	0.129237	0.0646183	−	0.997910i	$-0.479417\pi$
$37$	4.78176	0.129237	0.0646183	+	0.997910i	$0.479417\pi$
$38$	29.4116i	0.773991i
$39$	0	0
$40$	−24.3097	−0.607742
$41$	− 2.97891i	− 0.0726565i	−0.999340	−	0.0363282i	$-0.988434\pi$
$41$	− 2.97891i	− 0.0726565i	0.999340	−	0.0363282i	$-0.0115662\pi$
$42$	0	0
$43$	24.4298	0.568134	0.284067	−	0.958804i	$-0.408316\pi$
$43$	24.4298	0.568134	0.284067	+	0.958804i	$0.408316\pi$
$44$	− 28.7215i	− 0.652762i
$45$	0	0
$46$	−53.2244	−1.15705
$47$	− 1.53301i	− 0.0326172i	−0.999867	−	0.0163086i	$-0.994809\pi$
$47$	− 1.53301i	− 0.0326172i	0.999867	−	0.0163086i	$-0.00519143\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	− 69.1129i	− 1.38226i
$51$	0	0
$52$	−4.83946	−0.0930665
$53$	− 90.8320i	− 1.71381i	−0.515473	−	0.856906i	$-0.672384\pi$
$53$	− 90.8320i	− 1.71381i	0.515473	−	0.856906i	$-0.327616\pi$
$54$	0	0
$55$	123.427	2.24414
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	61.3761	1.05821
$59$	− 17.1433i	− 0.290565i	−0.989390	−	0.145283i	$-0.953591\pi$
$59$	− 17.1433i	− 0.290565i	0.989390	−	0.145283i	$-0.0464091\pi$
$60$	0	0
$61$	53.5618	0.878063	0.439031	−	0.898472i	$-0.355322\pi$
$61$	53.5618	0.878063	0.439031	+	0.898472i	$0.355322\pi$
$62$	64.4620i	1.03971i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 20.7970i	− 0.319954i
$66$	0	0
$67$	101.014	1.50768	0.753838	−	0.657060i	$-0.228200\pi$
$67$	101.014	1.50768	0.753838	+	0.657060i	$0.228200\pi$
$68$	− 8.55990i	− 0.125881i
$69$	0	0
$70$	32.1587	0.459410
$71$	− 127.047i	− 1.78940i	−0.446669	−	0.894699i	$-0.647390\pi$
$71$	− 127.047i	− 1.78940i	0.446669	−	0.894699i	$-0.352610\pi$
$72$	0	0
$73$	−2.38851	−0.0327193	−0.0163596	−	0.999866i	$-0.505208\pi$
$73$	−2.38851	−0.0327193	−0.0163596	+	0.999866i	$0.505208\pi$
$74$	6.76242i	0.0913841i
$75$	0	0
$76$	−41.5943	−0.547294
$77$	37.9950i	0.493441i
$78$	0	0
$79$	−48.8195	−0.617968	−0.308984	−	0.951067i	$-0.599989\pi$
$79$	−48.8195	−0.617968	−0.308984	+	0.951067i	$0.599989\pi$
$80$	− 34.3791i	− 0.429739i
$81$	0	0
$82$	4.21282	0.0513759
$83$	− 85.8416i	− 1.03424i	−0.855914	−	0.517118i	$-0.827005\pi$
$83$	− 85.8416i	− 1.03424i	0.855914	−	0.517118i	$-0.172995\pi$
$84$	0	0
$85$	36.7852	0.432767
$86$	34.5489i	0.401731i
$87$	0	0
$88$	40.6184	0.461572
$89$	13.8255i	0.155343i	0.996979	+	0.0776715i	$0.0247485\pi$
$89$	13.8255i	0.155343i	−0.996979	+	0.0776715i	$0.975251\pi$
$90$	0	0
$91$	6.40200	0.0703516
$92$	− 75.2706i	− 0.818159i
$93$	0	0
$94$	2.16800	0.0230639
$95$	− 178.747i	− 1.88155i
$96$	0	0
$97$	−3.88170	−0.0400175	−0.0200088	−	0.999800i	$-0.506369\pi$
$97$	−3.88170	−0.0400175	−0.0200088	+	0.999800i	$0.506369\pi$
$98$	9.89949i	0.101015i
$99$	0	0
$100$	97.7404	0.977404
$101$	− 91.5770i	− 0.906703i	−0.891332	−	0.453351i	$-0.850228\pi$
$101$	− 91.5770i	− 0.906703i	0.891332	−	0.453351i	$-0.149772\pi$
$102$	0	0
$103$	41.1343	0.399362	0.199681	−	0.979861i	$-0.436009\pi$
$103$	41.1343	0.399362	0.199681	+	0.979861i	$0.436009\pi$
$104$	− 6.84403i	− 0.0658079i
$105$	0	0
$106$	128.456	1.21185
$107$	149.508i	1.39727i	0.715477	+	0.698637i	$0.246210\pi$
$107$	149.508i	1.39727i	−0.715477	+	0.698637i	$0.753790\pi$
$108$	0	0
$109$	96.1264	0.881894	0.440947	−	0.897533i	$-0.354643\pi$
$109$	96.1264	0.881894	0.440947	+	0.897533i	$0.354643\pi$
$110$	174.553i	1.58684i
$111$	0	0
$112$	10.5830	0.0944911
$113$	− 44.6953i	− 0.395534i	−0.980249	−	0.197767i	$-0.936631\pi$
$113$	− 44.6953i	− 0.395534i	0.980249	−	0.197767i	$-0.0633689\pi$
$114$	0	0
$115$	323.467	2.81276
$116$	86.7990i	0.748267i
$117$	0	0
$118$	24.2443	0.205461
$119$	11.3237i	0.0951570i
$120$	0	0
$121$	−85.2313	−0.704391
$122$	75.7479i	0.620884i
$123$	0	0
$124$	−91.1630	−0.735186
$125$	205.159i	1.64127i
$126$	0	0
$127$	−27.4655	−0.216264	−0.108132	−	0.994137i	$-0.534487\pi$
$127$	−27.4655	−0.216264	−0.108132	+	0.994137i	$0.534487\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	29.4114	0.226242
$131$	148.389i	1.13274i	0.824150	+	0.566371i	$0.191653\pi$
$131$	148.389i	1.13274i	−0.824150	+	0.566371i	$0.808347\pi$
$132$	0	0
$133$	55.0241	0.413715
$134$	142.856i	1.06609i
$135$	0	0
$136$	12.1055	0.0890112
$137$	236.774i	1.72828i	0.503254	+	0.864138i	$0.332136\pi$
$137$	236.774i	1.72828i	−0.503254	+	0.864138i	$0.667864\pi$
$138$	0	0
$139$	−42.6164	−0.306593	−0.153296	−	0.988180i	$-0.548989\pi$
$139$	−42.6164	−0.306593	−0.153296	+	0.988180i	$0.548989\pi$
$140$	45.4793i	0.324852i
$141$	0	0
$142$	179.672	1.26530
$143$	34.7491i	0.243001i
$144$	0	0
$145$	−373.009	−2.57247
$146$	− 3.37786i	− 0.0231360i
$147$	0	0
$148$	−9.56351	−0.0646183
$149$	− 95.7252i	− 0.642451i	−0.947003	−	0.321225i	$-0.895905\pi$
$149$	− 95.7252i	− 0.642451i	0.947003	−	0.321225i	$-0.104095\pi$
$150$	0	0
$151$	−99.5708	−0.659409	−0.329705	−	0.944084i	$-0.606949\pi$
$151$	−99.5708	−0.659409	−0.329705	+	0.944084i	$0.606949\pi$
$152$	− 58.8233i	− 0.386995i
$153$	0	0
$154$	−53.7330	−0.348916
$155$	− 391.763i	− 2.52750i
$156$	0	0
$157$	−9.49278	−0.0604636	−0.0302318	−	0.999543i	$-0.509625\pi$
$157$	−9.49278	−0.0604636	−0.0302318	+	0.999543i	$0.509625\pi$
$158$	− 69.0412i	− 0.436969i
$159$	0	0
$160$	48.6194	0.303871
$161$	99.5737i	0.618470i
$162$	0	0
$163$	77.1330	0.473209	0.236604	−	0.971606i	$-0.423965\pi$
$163$	77.1330	0.473209	0.236604	+	0.971606i	$0.423965\pi$
$164$	5.95783i	0.0363282i
$165$	0	0
$166$	121.398	0.731315
$167$	216.493i	1.29637i	0.761485	+	0.648183i	$0.224471\pi$
$167$	216.493i	1.29637i	−0.761485	+	0.648183i	$0.775529\pi$
$168$	0	0
$169$	−163.145	−0.965355
$170$	52.0221i	0.306013i
$171$	0	0
$172$	−48.8595	−0.284067
$173$	118.169i	0.683058i	0.939871	+	0.341529i	$0.110945\pi$
$173$	118.169i	0.683058i	−0.939871	+	0.341529i	$0.889055\pi$
$174$	0	0
$175$	−129.298	−0.738848
$176$	57.4430i	0.326381i
$177$	0	0
$178$	−19.5523	−0.109844
$179$	66.3443i	0.370638i	0.982678	+	0.185319i	$0.0593319\pi$
$179$	66.3443i	0.370638i	−0.982678	+	0.185319i	$0.940668\pi$
$180$	0	0
$181$	−129.221	−0.713928	−0.356964	−	0.934118i	$-0.616188\pi$
$181$	−129.221	−0.713928	−0.356964	+	0.934118i	$0.616188\pi$
$182$	9.05380i	0.0497461i
$183$	0	0
$184$	106.449	0.578526
$185$	− 41.0981i	− 0.222152i
$186$	0	0
$187$	−61.4633	−0.328681
$188$	3.06602i	0.0163086i
$189$	0	0
$190$	252.786	1.33046
$191$	− 17.2249i	− 0.0901826i	−0.998983	−	0.0450913i	$-0.985642\pi$
$191$	− 17.2249i	− 0.0901826i	0.998983	−	0.0450913i	$-0.0143579\pi$
$192$	0	0
$193$	263.692	1.36628	0.683141	−	0.730287i	$-0.260614\pi$
$193$	263.692	1.36628	0.683141	+	0.730287i	$0.260614\pi$
$194$	− 5.48955i	− 0.0282967i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	− 383.896i	− 1.94871i	−0.225010	−	0.974356i	$-0.572242\pi$
$197$	− 383.896i	− 1.94871i	0.225010	−	0.974356i	$-0.427758\pi$
$198$	0	0
$199$	324.904	1.63268	0.816342	−	0.577569i	$-0.195999\pi$
$199$	324.904	1.63268	0.816342	+	0.577569i	$0.195999\pi$
$200$	138.226i	0.691129i
$201$	0	0
$202$	129.509	0.641136
$203$	− 114.824i	− 0.565637i
$204$	0	0
$205$	−25.6031	−0.124893
$206$	58.1727i	0.282392i
$207$	0	0
$208$	9.67891	0.0465332
$209$	298.663i	1.42901i
$210$	0	0
$211$	−248.456	−1.17752	−0.588759	−	0.808309i	$-0.700383\pi$
$211$	−248.456	−1.17752	−0.588759	+	0.808309i	$0.700383\pi$
$212$	181.664i	0.856906i
$213$	0	0
$214$	−211.437	−0.988021
$215$	− 209.968i	− 0.976597i
$216$	0	0
$217$	120.597	0.555748
$218$	135.943i	0.623593i
$219$	0	0
$220$	−246.855	−1.12207
$221$	10.3563i	0.0468612i
$222$	0	0
$223$	85.5137	0.383470	0.191735	−	0.981447i	$-0.438589\pi$
$223$	85.5137	0.383470	0.191735	+	0.981447i	$0.438589\pi$
$224$	14.9666i	0.0668153i
$225$	0	0
$226$	63.2088	0.279685
$227$	− 110.129i	− 0.485148i	−0.970133	−	0.242574i	$-0.922008\pi$
$227$	− 110.129i	− 0.485148i	0.970133	−	0.242574i	$-0.0779918\pi$
$228$	0	0
$229$	301.578	1.31693	0.658467	−	0.752610i	$-0.271205\pi$
$229$	301.578	1.31693	0.658467	+	0.752610i	$0.271205\pi$
$230$	457.451i	1.98892i
$231$	0	0
$232$	−122.752	−0.529105
$233$	− 175.577i	− 0.753550i	−0.926305	−	0.376775i	$-0.877033\pi$
$233$	− 175.577i	− 0.753550i	0.926305	−	0.376775i	$-0.122967\pi$
$234$	0	0
$235$	−13.1759	−0.0560676
$236$	34.2867i	0.145283i
$237$	0	0
$238$	−16.0141	−0.0672861
$239$	22.7242i	0.0950805i	0.998869	+	0.0475403i	$0.0151382\pi$
$239$	22.7242i	0.0950805i	−0.998869	+	0.0475403i	$0.984862\pi$
$240$	0	0
$241$	67.6076	0.280530	0.140265	−	0.990114i	$-0.455205\pi$
$241$	67.6076	0.280530	0.140265	+	0.990114i	$0.455205\pi$
$242$	− 120.535i	− 0.498080i
$243$	0	0
$244$	−107.124	−0.439031
$245$	− 60.1634i	− 0.245565i
$246$	0	0
$247$	50.3235	0.203739
$248$	− 128.924i	− 0.519855i
$249$	0	0
$250$	−290.138	−1.16055
$251$	226.850i	0.903785i	0.892072	+	0.451892i	$0.149251\pi$
$251$	226.850i	0.903785i	−0.892072	+	0.451892i	$0.850749\pi$
$252$	0	0
$253$	−540.471	−2.13625
$254$	− 38.8421i	− 0.152922i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 439.815i	− 1.71134i	−0.517519	−	0.855672i	$-0.673144\pi$
$257$	− 439.815i	− 1.71134i	0.517519	−	0.855672i	$-0.326856\pi$
$258$	0	0
$259$	12.6513	0.0488469
$260$	41.5941i	0.159977i
$261$	0	0
$262$	−209.854	−0.800969
$263$	296.816i	1.12858i	0.825578	+	0.564288i	$0.190849\pi$
$263$	296.816i	1.12858i	−0.825578	+	0.564288i	$0.809151\pi$
$264$	0	0
$265$	−780.681	−2.94596
$266$	77.8159i	0.292541i
$267$	0	0
$268$	−202.029	−0.753838
$269$	− 402.596i	− 1.49664i	−0.663337	−	0.748321i	$-0.730860\pi$
$269$	− 402.596i	− 1.49664i	0.663337	−	0.748321i	$-0.269140\pi$
$270$	0	0
$271$	483.422	1.78385	0.891923	−	0.452187i	$-0.149356\pi$
$271$	483.422	1.78385	0.891923	+	0.452187i	$0.149356\pi$
$272$	17.1198i	0.0629404i
$273$	0	0
$274$	−334.849	−1.22208
$275$	− 701.813i	− 2.55205i
$276$	0	0
$277$	171.722	0.619933	0.309967	−	0.950747i	$-0.399682\pi$
$277$	171.722	0.619933	0.309967	+	0.950747i	$0.399682\pi$
$278$	− 60.2687i	− 0.216794i
$279$	0	0
$280$	−64.3174	−0.229705
$281$	81.5746i	0.290301i	0.989410	+	0.145151i	$0.0463666\pi$
$281$	81.5746i	0.290301i	−0.989410	+	0.145151i	$0.953633\pi$
$282$	0	0
$283$	183.279	0.647629	0.323815	−	0.946121i	$-0.395035\pi$
$283$	183.279	0.647629	0.323815	+	0.946121i	$0.395035\pi$
$284$	254.095i	0.894699i
$285$	0	0
$286$	−49.1427	−0.171828
$287$	− 7.88147i	− 0.0274616i
$288$	0	0
$289$	270.682	0.936616
$290$	− 527.514i	− 1.81901i
$291$	0	0
$292$	4.77701	0.0163596
$293$	378.172i	1.29069i	0.763892	+	0.645344i	$0.223286\pi$
$293$	378.172i	1.29069i	−0.763892	+	0.645344i	$0.776714\pi$
$294$	0	0
$295$	−147.343	−0.499468
$296$	− 13.5248i	− 0.0456921i
$297$	0	0
$298$	135.376	0.454281
$299$	91.0672i	0.304573i
$300$	0	0
$301$	64.6351	0.214734
$302$	− 140.814i	− 0.466273i
$303$	0	0
$304$	83.1887	0.273647
$305$	− 460.352i	− 1.50935i
$306$	0	0
$307$	−313.447	−1.02100	−0.510499	−	0.859878i	$-0.670539\pi$
$307$	−313.447	−1.02100	−0.510499	+	0.859878i	$0.670539\pi$
$308$	− 75.9900i	− 0.246721i
$309$	0	0
$310$	554.036	1.78721
$311$	354.444i	1.13969i	0.821751	+	0.569846i	$0.192997\pi$
$311$	354.444i	1.13969i	−0.821751	+	0.569846i	$0.807003\pi$
$312$	0	0
$313$	−260.599	−0.832584	−0.416292	−	0.909231i	$-0.636671\pi$
$313$	−260.599	−0.832584	−0.416292	+	0.909231i	$0.636671\pi$
$314$	− 13.4248i	− 0.0427542i
$315$	0	0
$316$	97.6390	0.308984
$317$	149.312i	0.471016i	0.971872	+	0.235508i	$0.0756754\pi$
$317$	149.312i	0.471016i	−0.971872	+	0.235508i	$0.924325\pi$
$318$	0	0
$319$	623.249	1.95376
$320$	68.7582i	0.214869i
$321$	0	0
$322$	−140.818	−0.437324
$323$	89.0108i	0.275575i
$324$	0	0
$325$	−118.253	−0.363854
$326$	109.083i	0.334609i
$327$	0	0
$328$	−8.42564	−0.0256879
$329$	− 4.05596i	− 0.0123282i
$330$	0	0
$331$	−376.618	−1.13782	−0.568909	−	0.822400i	$-0.692634\pi$
$331$	−376.618	−1.13782	−0.568909	+	0.822400i	$0.692634\pi$
$332$	171.683i	0.517118i
$333$	0	0
$334$	−306.167	−0.916669
$335$	− 868.195i	− 2.59163i
$336$	0	0
$337$	−60.9156	−0.180758	−0.0903792	−	0.995907i	$-0.528808\pi$
$337$	−60.9156	−0.180758	−0.0903792	+	0.995907i	$0.528808\pi$
$338$	− 230.722i	− 0.682609i
$339$	0	0
$340$	−73.5704	−0.216384
$341$	654.585i	1.91960i
$342$	0	0
$343$	18.5203	0.0539949
$344$	− 69.0978i	− 0.200866i
$345$	0	0
$346$	−167.116	−0.482995
$347$	− 199.846i	− 0.575924i	−0.957642	−	0.287962i	$-0.907022\pi$
$347$	− 199.846i	− 0.575924i	0.957642	−	0.287962i	$-0.0929777\pi$
$348$	0	0
$349$	590.653	1.69242	0.846208	−	0.532852i	$-0.178880\pi$
$349$	590.653	1.69242	0.846208	+	0.532852i	$0.178880\pi$
$350$	− 182.855i	− 0.522444i
$351$	0	0
$352$	−81.2367	−0.230786
$353$	167.974i	0.475846i	0.971284	+	0.237923i	$0.0764666\pi$
$353$	167.974i	0.475846i	−0.971284	+	0.237923i	$0.923533\pi$
$354$	0	0
$355$	−1091.94	−3.07590
$356$	− 27.6511i	− 0.0776715i
$357$	0	0
$358$	−93.8250	−0.262081
$359$	− 374.424i	− 1.04296i	−0.853262	−	0.521482i	$-0.825379\pi$
$359$	− 374.424i	− 1.04296i	0.853262	−	0.521482i	$-0.174621\pi$
$360$	0	0
$361$	71.5223	0.198123
$362$	− 182.746i	− 0.504823i
$363$	0	0
$364$	−12.8040	−0.0351758
$365$	20.5287i	0.0562429i
$366$	0	0
$367$	−452.977	−1.23427	−0.617135	−	0.786858i	$-0.711707\pi$
$367$	−452.977	−1.23427	−0.617135	+	0.786858i	$0.711707\pi$
$368$	150.541i	0.409079i
$369$	0	0
$370$	58.1215	0.157085
$371$	− 240.319i	− 0.647760i
$372$	0	0
$373$	104.212	0.279387	0.139694	−	0.990195i	$-0.455388\pi$
$373$	104.212	0.279387	0.139694	+	0.990195i	$0.455388\pi$
$374$	− 86.9222i	− 0.232412i
$375$	0	0
$376$	−4.33601	−0.0115319
$377$	− 105.015i	− 0.278554i
$378$	0	0
$379$	169.966	0.448458	0.224229	−	0.974536i	$-0.428014\pi$
$379$	169.966	0.448458	0.224229	+	0.974536i	$0.428014\pi$
$380$	357.494i	0.940774i
$381$	0	0
$382$	24.3597	0.0637687
$383$	− 500.621i	− 1.30710i	−0.756882	−	0.653552i	$-0.773278\pi$
$383$	− 500.621i	− 1.30710i	0.756882	−	0.653552i	$-0.226722\pi$
$384$	0	0
$385$	326.558	0.848204
$386$	372.917i	0.966107i
$387$	0	0
$388$	7.76340	0.0200088
$389$	− 64.8688i	− 0.166758i	−0.996518	−	0.0833789i	$-0.973429\pi$
$389$	− 64.8688i	− 0.166758i	0.996518	−	0.0833789i	$-0.0265712\pi$
$390$	0	0
$391$	−161.077	−0.411962
$392$	− 19.7990i	− 0.0505076i
$393$	0	0
$394$	542.912	1.37795
$395$	419.593i	1.06226i
$396$	0	0
$397$	−156.916	−0.395254	−0.197627	−	0.980277i	$-0.563323\pi$
$397$	−156.916	−0.395254	−0.197627	+	0.980277i	$0.563323\pi$
$398$	459.484i	1.15448i
$399$	0	0
$400$	−195.481	−0.488702
$401$	470.283i	1.17278i	0.810030	+	0.586388i	$0.199451\pi$
$401$	470.283i	1.17278i	−0.810030	+	0.586388i	$0.800549\pi$
$402$	0	0
$403$	110.295	0.273685
$404$	183.154i	0.453351i
$405$	0	0
$406$	162.386	0.399965
$407$	68.6696i	0.168721i
$408$	0	0
$409$	−290.388	−0.709996	−0.354998	−	0.934867i	$-0.615518\pi$
$409$	−290.388	−0.709996	−0.354998	+	0.934867i	$0.615518\pi$
$410$	− 36.2083i	− 0.0883128i
$411$	0	0
$412$	−82.2687	−0.199681
$413$	− 45.3570i	− 0.109823i
$414$	0	0
$415$	−737.789	−1.77781
$416$	13.6881i	0.0329040i
$417$	0	0
$418$	−422.373	−1.01046
$419$	− 556.113i	− 1.32724i	−0.748071	−	0.663619i	$-0.769020\pi$
$419$	− 556.113i	− 1.32724i	0.748071	−	0.663619i	$-0.230980\pi$
$420$	0	0
$421$	239.708	0.569377	0.284688	−	0.958620i	$-0.408110\pi$
$421$	239.708	0.569377	0.284688	+	0.958620i	$0.408110\pi$
$422$	− 351.370i	− 0.832631i
$423$	0	0
$424$	−256.912	−0.605924
$425$	− 209.162i	− 0.492146i
$426$	0	0
$427$	141.711	0.331876
$428$	− 299.016i	− 0.698637i
$429$	0	0
$430$	296.940	0.690558
$431$	200.137i	0.464354i	0.972674	+	0.232177i	$0.0745849\pi$
$431$	200.137i	0.464354i	−0.972674	+	0.232177i	$0.925415\pi$
$432$	0	0
$433$	−268.223	−0.619453	−0.309727	−	0.950826i	$-0.600238\pi$
$433$	−268.223	−0.619453	−0.309727	+	0.950826i	$0.600238\pi$
$434$	170.550i	0.392973i
$435$	0	0
$436$	−192.253	−0.440947
$437$	782.708i	1.79109i
$438$	0	0
$439$	−502.840	−1.14542	−0.572711	−	0.819757i	$-0.694108\pi$
$439$	−502.840	−1.14542	−0.572711	+	0.819757i	$0.694108\pi$
$440$	− 349.106i	− 0.793422i
$441$	0	0
$442$	−14.6460	−0.0331358
$443$	− 45.1666i	− 0.101956i	−0.998700	−	0.0509781i	$-0.983766\pi$
$443$	− 45.1666i	− 0.101956i	0.998700	−	0.0509781i	$-0.0162339\pi$
$444$	0	0
$445$	118.827	0.267028
$446$	120.935i	0.271154i
$447$	0	0
$448$	−21.1660	−0.0472456
$449$	689.707i	1.53610i	0.640392	+	0.768048i	$0.278772\pi$
$449$	689.707i	1.53610i	−0.640392	+	0.768048i	$0.721228\pi$
$450$	0	0
$451$	42.7795	0.0948547
$452$	89.3907i	0.197767i
$453$	0	0
$454$	155.746	0.343052
$455$	− 55.0238i	− 0.120931i
$456$	0	0
$457$	313.310	0.685581	0.342790	−	0.939412i	$-0.388628\pi$
$457$	313.310	0.685581	0.342790	+	0.939412i	$0.388628\pi$
$458$	426.495i	0.931212i
$459$	0	0
$460$	−646.934	−1.40638
$461$	413.145i	0.896194i	0.893985	+	0.448097i	$0.147898\pi$
$461$	413.145i	0.896194i	−0.893985	+	0.448097i	$0.852102\pi$
$462$	0	0
$463$	271.125	0.585582	0.292791	−	0.956176i	$-0.405416\pi$
$463$	271.125	0.585582	0.292791	+	0.956176i	$0.405416\pi$
$464$	− 173.598i	− 0.374133i
$465$	0	0
$466$	248.304	0.532841
$467$	− 304.556i	− 0.652155i	−0.945343	−	0.326078i	$-0.894273\pi$
$467$	− 304.556i	− 0.652155i	0.945343	−	0.326078i	$-0.105727\pi$
$468$	0	0
$469$	267.259	0.569848
$470$	− 18.6335i	− 0.0396458i
$471$	0	0
$472$	−48.4887	−0.102730
$473$	350.830i	0.741712i
$474$	0	0
$475$	−1016.36	−2.13971
$476$	− 22.6474i	− 0.0475785i
$477$	0	0
$478$	−32.1369	−0.0672321
$479$	146.318i	0.305465i	0.988268	+	0.152733i	$0.0488073\pi$
$479$	146.318i	0.305465i	−0.988268	+	0.152733i	$0.951193\pi$
$480$	0	0
$481$	11.5706	0.0240552
$482$	95.6116i	0.198364i
$483$	0	0
$484$	170.463	0.352196
$485$	33.3623i	0.0687883i
$486$	0	0
$487$	−422.034	−0.866599	−0.433300	−	0.901250i	$-0.642651\pi$
$487$	−422.034	−0.866599	−0.433300	+	0.901250i	$0.642651\pi$
$488$	− 151.496i	− 0.310442i
$489$	0	0
$490$	85.0839	0.173641
$491$	426.496i	0.868627i	0.900762	+	0.434314i	$0.143009\pi$
$491$	426.496i	0.868627i	−0.900762	+	0.434314i	$0.856991\pi$
$492$	0	0
$493$	185.748	0.376770
$494$	71.1682i	0.144065i
$495$	0	0
$496$	182.326	0.367593
$497$	− 336.136i	− 0.676329i
$498$	0	0
$499$	−311.469	−0.624187	−0.312094	−	0.950051i	$-0.601030\pi$
$499$	−311.469	−0.624187	−0.312094	+	0.950051i	$0.601030\pi$
$500$	− 410.318i	− 0.820636i
$501$	0	0
$502$	−320.814	−0.639072
$503$	44.9656i	0.0893949i	0.999001	+	0.0446975i	$0.0142324\pi$
$503$	44.9656i	0.0893949i	−0.999001	+	0.0446975i	$0.985768\pi$
$504$	0	0
$505$	−787.084	−1.55858
$506$	− 764.342i	− 1.51056i
$507$	0	0
$508$	54.9310	0.108132
$509$	75.6433i	0.148612i	0.997236	+	0.0743058i	$0.0236741\pi$
$509$	75.6433i	0.148612i	−0.997236	+	0.0743058i	$0.976326\pi$
$510$	0	0
$511$	−6.31939	−0.0123667
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	621.993	1.21010
$515$	− 353.540i	− 0.686486i
$516$	0	0
$517$	22.0152	0.0425826
$518$	17.8917i	0.0345399i
$519$	0	0
$520$	−58.8229	−0.113121
$521$	393.511i	0.755300i	0.925948	+	0.377650i	$0.123268\pi$
$521$	393.511i	0.755300i	−0.925948	+	0.377650i	$0.876732\pi$
$522$	0	0
$523$	−140.208	−0.268083	−0.134042	−	0.990976i	$-0.542796\pi$
$523$	−140.208	−0.268083	−0.134042	+	0.990976i	$0.542796\pi$
$524$	− 296.778i	− 0.566371i
$525$	0	0
$526$	−419.761	−0.798024
$527$	195.087i	0.370183i
$528$	0	0
$529$	−887.416	−1.67753
$530$	− 1104.05i	− 2.08311i
$531$	0	0
$532$	−110.048	−0.206858
$533$	− 7.20816i	− 0.0135238i
$534$	0	0
$535$	1284.99	2.40185
$536$	− 285.712i	− 0.533044i
$537$	0	0
$538$	569.357	1.05829
$539$	100.525i	0.186503i
$540$	0	0
$541$	−299.982	−0.554495	−0.277248	−	0.960798i	$-0.589422\pi$
$541$	−299.982	−0.554495	−0.277248	+	0.960798i	$0.589422\pi$
$542$	683.662i	1.26137i
$543$	0	0
$544$	−24.2110	−0.0445056
$545$	− 826.185i	− 1.51594i
$546$	0	0
$547$	−456.382	−0.834337	−0.417169	−	0.908829i	$-0.636978\pi$
$547$	−456.382	−0.834337	−0.417169	+	0.908829i	$0.636978\pi$
$548$	− 473.548i	− 0.864138i
$549$	0	0
$550$	992.513	1.80457
$551$	− 902.586i	− 1.63809i
$552$	0	0
$553$	−129.164	−0.233570
$554$	242.851i	0.438359i
$555$	0	0
$556$	85.2328	0.153296
$557$	696.230i	1.24996i	0.780639	+	0.624982i	$0.214894\pi$
$557$	696.230i	1.24996i	−0.780639	+	0.624982i	$0.785106\pi$
$558$	0	0
$559$	59.1134	0.105748
$560$	− 90.9586i	− 0.162426i
$561$	0	0
$562$	−115.364	−0.205274
$563$	452.950i	0.804530i	0.915523	+	0.402265i	$0.131777\pi$
$563$	452.950i	0.804530i	−0.915523	+	0.402265i	$0.868223\pi$
$564$	0	0
$565$	−384.146	−0.679905
$566$	259.196i	0.457943i
$567$	0	0
$568$	−359.344	−0.632648
$569$	− 146.183i	− 0.256912i	−0.991715	−	0.128456i	$-0.958998\pi$
$569$	− 146.183i	− 0.256912i	0.991715	−	0.128456i	$-0.0410021\pi$
$570$	0	0
$571$	−642.563	−1.12533	−0.562665	−	0.826685i	$-0.690224\pi$
$571$	−642.563	−1.12533	−0.562665	+	0.826685i	$0.690224\pi$
$572$	− 69.4983i	− 0.121500i
$573$	0	0
$574$	11.1461	0.0194183
$575$	− 1839.24i	− 3.19869i
$576$	0	0
$577$	221.339	0.383604	0.191802	−	0.981434i	$-0.438567\pi$
$577$	221.339	0.383604	0.191802	+	0.981434i	$0.438567\pi$
$578$	382.802i	0.662288i
$579$	0	0
$580$	746.018	1.28624
$581$	− 227.116i	− 0.390904i
$582$	0	0
$583$	1304.42	2.23742
$584$	6.75571i	0.0115680i
$585$	0	0
$586$	−534.816	−0.912654
$587$	72.2440i	0.123073i	0.998105	+	0.0615366i	$0.0196001\pi$
$587$	72.2440i	0.123073i	−0.998105	+	0.0615366i	$0.980400\pi$
$588$	0	0
$589$	947.966	1.60945
$590$	− 208.375i	− 0.353177i
$591$	0	0
$592$	19.1270	0.0323092
$593$	− 232.191i	− 0.391552i	−0.980649	−	0.195776i	$-0.937277\pi$
$593$	− 232.191i	− 0.391552i	0.980649	−	0.195776i	$-0.0627226\pi$
$594$	0	0
$595$	97.3245	0.163571
$596$	191.450i	0.321225i
$597$	0	0
$598$	−128.788	−0.215365
$599$	− 402.549i	− 0.672035i	−0.941856	−	0.336018i	$-0.890920\pi$
$599$	− 402.549i	− 0.672035i	0.941856	−	0.336018i	$-0.109080\pi$
$600$	0	0
$601$	−360.906	−0.600509	−0.300255	−	0.953859i	$-0.597072\pi$
$601$	−360.906	−0.600509	−0.300255	+	0.953859i	$0.597072\pi$
$602$	91.4078i	0.151840i
$603$	0	0
$604$	199.142	0.329705
$605$	732.544i	1.21082i
$606$	0	0
$607$	−1172.54	−1.93170	−0.965851	−	0.259098i	$-0.916575\pi$
$607$	−1172.54	−1.93170	−0.965851	+	0.259098i	$0.916575\pi$
$608$	117.647i	0.193498i
$609$	0	0
$610$	651.036	1.06727
$611$	− 3.70947i	− 0.00607114i
$612$	0	0
$613$	1042.01	1.69985	0.849926	−	0.526903i	$-0.176647\pi$
$613$	1042.01	1.69985	0.849926	+	0.526903i	$0.176647\pi$
$614$	− 443.281i	− 0.721955i
$615$	0	0
$616$	107.466	0.174458
$617$	261.914i	0.424495i	0.977216	+	0.212248i	$0.0680784\pi$
$617$	261.914i	0.424495i	−0.977216	+	0.212248i	$0.931922\pi$
$618$	0	0
$619$	707.768	1.14340	0.571702	−	0.820461i	$-0.306283\pi$
$619$	707.768	1.14340	0.571702	+	0.820461i	$0.306283\pi$
$620$	783.526i	1.26375i
$621$	0	0
$622$	−501.260	−0.805884
$623$	36.5789i	0.0587141i
$624$	0	0
$625$	541.540	0.866464
$626$	− 368.543i	− 0.588726i
$627$	0	0
$628$	18.9856	0.0302318
$629$	20.4657i	0.0325368i
$630$	0	0
$631$	−40.4979	−0.0641805	−0.0320902	−	0.999485i	$-0.510216\pi$
$631$	−40.4979	−0.0641805	−0.0320902	+	0.999485i	$0.510216\pi$
$632$	138.082i	0.218485i
$633$	0	0
$634$	−211.159	−0.333059
$635$	236.060i	0.371748i
$636$	0	0
$637$	16.9381	0.0265904
$638$	881.408i	1.38152i
$639$	0	0
$640$	−97.2388	−0.151936
$641$	− 570.913i	− 0.890660i	−0.895367	−	0.445330i	$-0.853086\pi$
$641$	− 570.913i	− 0.890660i	0.895367	−	0.445330i	$-0.146914\pi$
$642$	0	0
$643$	−176.883	−0.275090	−0.137545	−	0.990496i	$-0.543921\pi$
$643$	−176.883	−0.275090	−0.137545	+	0.990496i	$0.543921\pi$
$644$	− 199.147i	− 0.309235i
$645$	0	0
$646$	−125.880	−0.194861
$647$	506.340i	0.782597i	0.920264	+	0.391298i	$0.127974\pi$
$647$	506.340i	0.782597i	−0.920264	+	0.391298i	$0.872026\pi$
$648$	0	0
$649$	246.191	0.379339
$650$	− 167.234i	− 0.257284i
$651$	0	0
$652$	−154.266	−0.236604
$653$	85.6822i	0.131213i	0.997846	+	0.0656066i	$0.0208982\pi$
$653$	85.6822i	0.131213i	−0.997846	+	0.0656066i	$0.979102\pi$
$654$	0	0
$655$	1275.37	1.94713
$656$	− 11.9157i	− 0.0181641i
$657$	0	0
$658$	5.73600	0.00871732
$659$	1156.69i	1.75522i	0.479377	+	0.877609i	$0.340863\pi$
$659$	1156.69i	1.75522i	−0.479377	+	0.877609i	$0.659137\pi$
$660$	0	0
$661$	−921.751	−1.39448	−0.697240	−	0.716838i	$-0.745589\pi$
$661$	−921.751	−1.39448	−0.697240	+	0.716838i	$0.745589\pi$
$662$	− 532.618i	− 0.804559i
$663$	0	0
$664$	−242.797	−0.365658
$665$	− 472.920i	− 0.711158i
$666$	0	0
$667$	1633.35	2.44880
$668$	− 432.986i	− 0.648183i
$669$	0	0
$670$	1227.81	1.83256
$671$	769.188i	1.14633i
$672$	0	0
$673$	−839.808	−1.24786	−0.623929	−	0.781481i	$-0.714464\pi$
$673$	−839.808	−1.24786	−0.623929	+	0.781481i	$0.714464\pi$
$674$	− 86.1477i	− 0.127816i
$675$	0	0
$676$	326.290	0.482677
$677$	1002.01i	1.48007i	0.672567	+	0.740036i	$0.265192\pi$
$677$	1002.01i	1.48007i	−0.672567	+	0.740036i	$0.734808\pi$
$678$	0	0
$679$	−10.2700	−0.0151252
$680$	− 104.044i	− 0.153006i
$681$	0	0
$682$	−925.723	−1.35736
$683$	− 434.891i	− 0.636737i	−0.947967	−	0.318368i	$-0.896865\pi$
$683$	− 434.891i	− 0.636737i	0.947967	−	0.318368i	$-0.103135\pi$
$684$	0	0
$685$	2035.02	2.97083
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	97.7190	0.142033
$689$	− 219.789i	− 0.318997i
$690$	0	0
$691$	−250.078	−0.361907	−0.180953	−	0.983492i	$-0.557918\pi$
$691$	−250.078	−0.361907	−0.180953	+	0.983492i	$0.557918\pi$
$692$	− 236.338i	− 0.341529i
$693$	0	0
$694$	282.624	0.407240
$695$	366.279i	0.527020i
$696$	0	0
$697$	12.7496	0.0182921
$698$	835.310i	1.19672i
$699$	0	0
$700$	258.597	0.369424
$701$	− 81.6624i	− 0.116494i	−0.998302	−	0.0582471i	$-0.981449\pi$
$701$	− 81.6624i	− 0.116494i	0.998302	−	0.0582471i	$-0.0185511\pi$
$702$	0	0
$703$	99.4470	0.141461
$704$	− 114.886i	− 0.163190i
$705$	0	0
$706$	−237.551	−0.336474
$707$	− 242.290i	− 0.342701i
$708$	0	0
$709$	−599.279	−0.845245	−0.422623	−	0.906306i	$-0.638890\pi$
$709$	−599.279	−0.845245	−0.422623	+	0.906306i	$0.638890\pi$
$710$	− 1544.24i	− 2.17499i
$711$	0	0
$712$	39.1045	0.0549221
$713$	1715.47i	2.40599i
$714$	0	0
$715$	298.661	0.417708
$716$	− 132.689i	− 0.185319i
$717$	0	0
$718$	529.516	0.737487
$719$	− 253.130i	− 0.352058i	−0.984385	−	0.176029i	$-0.943675\pi$
$719$	− 253.130i	− 0.352058i	0.984385	−	0.176029i	$-0.0563253\pi$
$720$	0	0
$721$	108.831	0.150945
$722$	101.148i	0.140094i
$723$	0	0
$724$	258.442	0.356964
$725$	2120.94i	2.92543i
$726$	0	0
$727$	−781.360	−1.07477	−0.537386	−	0.843336i	$-0.680588\pi$
$727$	−781.360	−1.07477	−0.537386	+	0.843336i	$0.680588\pi$
$728$	− 18.1076i	− 0.0248731i
$729$	0	0
$730$	−29.0319	−0.0397698
$731$	104.558i	0.143034i
$732$	0	0
$733$	351.718	0.479834	0.239917	−	0.970793i	$-0.422880\pi$
$733$	351.718	0.479834	0.239917	+	0.970793i	$0.422880\pi$
$734$	− 640.606i	− 0.872760i
$735$	0	0
$736$	−212.897	−0.289263
$737$	1450.64i	1.96831i
$738$	0	0
$739$	189.901	0.256971	0.128485	−	0.991711i	$-0.458988\pi$
$739$	189.901	0.256971	0.128485	+	0.991711i	$0.458988\pi$
$740$	82.1962i	0.111076i
$741$	0	0
$742$	339.862	0.458035
$743$	924.808i	1.24470i	0.782741	+	0.622348i	$0.213821\pi$
$743$	924.808i	1.24470i	−0.782741	+	0.622348i	$0.786179\pi$
$744$	0	0
$745$	−822.736	−1.10434
$746$	147.377i	0.197557i
$747$	0	0
$748$	122.927	0.164340
$749$	395.562i	0.528120i
$750$	0	0
$751$	−382.695	−0.509580	−0.254790	−	0.966996i	$-0.582006\pi$
$751$	−382.695	−0.509580	−0.254790	+	0.966996i	$0.582006\pi$
$752$	− 6.13204i	− 0.00815431i
$753$	0	0
$754$	148.514	0.196968
$755$	855.789i	1.13349i
$756$	0	0
$757$	346.761	0.458072	0.229036	−	0.973418i	$-0.426443\pi$
$757$	346.761	0.458072	0.229036	+	0.973418i	$0.426443\pi$
$758$	240.368i	0.317108i
$759$	0	0
$760$	−505.573	−0.665228
$761$	351.644i	0.462082i	0.972944	+	0.231041i	$0.0742131\pi$
$761$	351.644i	0.462082i	−0.972944	+	0.231041i	$0.925787\pi$
$762$	0	0
$763$	254.327	0.333325
$764$	34.4498i	0.0450913i
$765$	0	0
$766$	707.985	0.924262
$767$	− 41.4822i	− 0.0540837i
$768$	0	0
$769$	1477.86	1.92180	0.960899	−	0.276900i	$-0.0893070\pi$
$769$	1477.86	1.92180	0.960899	+	0.276900i	$0.0893070\pi$
$770$	461.823i	0.599771i
$771$	0	0
$772$	−527.385	−0.683141
$773$	892.237i	1.15425i	0.816655	+	0.577126i	$0.195826\pi$
$773$	892.237i	1.15425i	−0.816655	+	0.577126i	$0.804174\pi$
$774$	0	0
$775$	−2227.58	−2.87429
$776$	10.9791i	0.0141483i
$777$	0	0
$778$	91.7383	0.117916
$779$	− 61.9530i	− 0.0795289i
$780$	0	0
$781$	1824.50	2.33610
$782$	− 227.798i	− 0.291301i
$783$	0	0
$784$	28.0000	0.0357143
$785$	81.5883i	0.103934i
$786$	0	0
$787$	955.161	1.21367	0.606837	−	0.794826i	$-0.292438\pi$
$787$	955.161	1.21367	0.606837	+	0.794826i	$0.292438\pi$
$788$	767.793i	0.974356i
$789$	0	0
$790$	−593.393	−0.751131
$791$	− 118.253i	− 0.149498i
$792$	0	0
$793$	129.605	0.163436
$794$	− 221.912i	− 0.279487i
$795$	0	0
$796$	−649.808	−0.816342
$797$	− 1144.39i	− 1.43587i	−0.696110	−	0.717935i	$-0.745087\pi$
$797$	− 1144.39i	− 1.43587i	0.696110	−	0.717935i	$-0.254913\pi$
$798$	0	0
$799$	6.56121	0.00821177
$800$	− 276.451i	− 0.345564i
$801$	0	0
$802$	−665.081	−0.829278
$803$	− 34.3008i	− 0.0427158i
$804$	0	0
$805$	855.813	1.06312
$806$	155.981i	0.193524i
$807$	0	0
$808$	−259.019	−0.320568
$809$	− 331.844i	− 0.410190i	−0.978742	−	0.205095i	$-0.934250\pi$
$809$	− 331.844i	− 0.410190i	0.978742	−	0.205095i	$-0.0657504\pi$
$810$	0	0
$811$	−1101.69	−1.35843	−0.679216	−	0.733938i	$-0.737680\pi$
$811$	−1101.69	−1.35843	−0.679216	+	0.733938i	$0.737680\pi$
$812$	229.648i	0.282818i
$813$	0	0
$814$	−97.1135	−0.119304
$815$	− 662.941i	− 0.813425i
$816$	0	0
$817$	508.070	0.621872
$818$	− 410.671i	− 0.502043i
$819$	0	0
$820$	51.2062	0.0624466
$821$	− 1494.95i	− 1.82089i	−0.413632	−	0.910444i	$-0.635740\pi$
$821$	− 1494.95i	− 1.82089i	0.413632	−	0.910444i	$-0.364260\pi$
$822$	0	0
$823$	935.165	1.13629	0.568144	−	0.822929i	$-0.307662\pi$
$823$	935.165	1.13629	0.568144	+	0.822929i	$0.307662\pi$
$824$	− 116.345i	− 0.141196i
$825$	0	0
$826$	64.1445	0.0776568
$827$	62.4043i	0.0754586i	0.999288	+	0.0377293i	$0.0120125\pi$
$827$	62.4043i	0.0754586i	−0.999288	+	0.0377293i	$0.987988\pi$
$828$	0	0
$829$	−1447.62	−1.74622	−0.873110	−	0.487524i	$-0.837900\pi$
$829$	−1447.62	−1.74622	−0.873110	+	0.487524i	$0.837900\pi$
$830$	− 1043.39i	− 1.25710i
$831$	0	0
$832$	−19.3578	−0.0232666
$833$	29.9596i	0.0359660i
$834$	0	0
$835$	1860.71	2.22839
$836$	− 597.326i	− 0.714505i
$837$	0	0
$838$	786.462	0.938499
$839$	118.138i	0.140808i	0.997519	+	0.0704038i	$0.0224288\pi$
$839$	118.138i	0.140808i	−0.997519	+	0.0704038i	$0.977571\pi$
$840$	0	0
$841$	−1042.51	−1.23961
$842$	338.998i	0.402610i
$843$	0	0
$844$	496.912	0.588759
$845$	1402.19i	1.65940i
$846$	0	0
$847$	−225.501	−0.266235
$848$	− 363.328i	− 0.428453i
$849$	0	0
$850$	295.800	0.348000
$851$	179.963i	0.211472i
$852$	0	0
$853$	1507.43	1.76721	0.883606	−	0.468232i	$-0.155109\pi$
$853$	1507.43	1.76721	0.883606	+	0.468232i	$0.155109\pi$
$854$	200.410i	0.234672i
$855$	0	0
$856$	422.873	0.494011
$857$	1110.43i	1.29572i	0.761759	+	0.647860i	$0.224336\pi$
$857$	1110.43i	1.29572i	−0.761759	+	0.647860i	$0.775664\pi$
$858$	0	0
$859$	−1120.66	−1.30461	−0.652303	−	0.757959i	$-0.726197\pi$
$859$	−1120.66	−1.30461	−0.652303	+	0.757959i	$0.726197\pi$
$860$	419.937i	0.488298i
$861$	0	0
$862$	−283.036	−0.328348
$863$	307.811i	0.356676i	0.983969	+	0.178338i	$0.0570720\pi$
$863$	307.811i	0.356676i	−0.983969	+	0.178338i	$0.942928\pi$
$864$	0	0
$865$	1015.64	1.17415
$866$	− 379.325i	− 0.438020i
$867$	0	0
$868$	−241.195	−0.277874
$869$	− 701.085i	− 0.806772i
$870$	0	0
$871$	244.427	0.280628
$872$	− 271.887i	− 0.311797i
$873$	0	0
$874$	−1106.92	−1.26649
$875$	542.799i	0.620342i
$876$	0	0
$877$	1448.16	1.65126	0.825630	−	0.564211i	$-0.190820\pi$
$877$	1448.16	1.65126	0.825630	+	0.564211i	$0.190820\pi$
$878$	− 711.123i	− 0.809936i
$879$	0	0
$880$	493.710	0.561034
$881$	− 499.464i	− 0.566928i	−0.958983	−	0.283464i	$-0.908516\pi$
$881$	− 499.464i	− 0.566928i	0.958983	−	0.283464i	$-0.0914837\pi$
$882$	0	0
$883$	1302.89	1.47553	0.737766	−	0.675057i	$-0.235881\pi$
$883$	1302.89	1.47553	0.737766	+	0.675057i	$0.235881\pi$
$884$	− 20.7126i	− 0.0234306i
$885$	0	0
$886$	63.8752	0.0720939
$887$	− 1354.65i	− 1.52722i	−0.645675	−	0.763612i	$-0.723424\pi$
$887$	− 1354.65i	− 1.52722i	0.645675	−	0.763612i	$-0.276576\pi$
$888$	0	0
$889$	−72.6669	−0.0817400
$890$	168.047i	0.188817i
$891$	0	0
$892$	−171.027	−0.191735
$893$	− 31.8823i	− 0.0357024i
$894$	0	0
$895$	570.214	0.637111
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	−975.393	−1.08618
$899$	− 1978.21i	− 2.20046i
$900$	0	0
$901$	388.756	0.431472
$902$	60.4993i	0.0670724i
$903$	0	0
$904$	−126.418	−0.139842
$905$	1110.62i	1.22721i
$906$	0	0
$907$	770.907	0.849953	0.424976	−	0.905204i	$-0.360282\pi$
$907$	770.907	0.849953	0.424976	+	0.905204i	$0.360282\pi$
$908$	220.257i	0.242574i
$909$	0	0
$910$	77.8153	0.0855114
$911$	− 1045.50i	− 1.14764i	−0.818980	−	0.573822i	$-0.805460\pi$
$911$	− 1045.50i	− 1.14764i	0.818980	−	0.573822i	$-0.194540\pi$
$912$	0	0
$913$	1232.75	1.35022
$914$	443.088i	0.484779i
$915$	0	0
$916$	−603.155	−0.658467
$917$	392.601i	0.428136i
$918$	0	0
$919$	238.491	0.259512	0.129756	−	0.991546i	$-0.458581\pi$
$919$	238.491	0.259512	0.129756	+	0.991546i	$0.458581\pi$
$920$	− 914.903i	− 0.994460i
$921$	0	0
$922$	−584.276	−0.633705
$923$	− 307.420i	− 0.333066i
$924$	0	0
$925$	−233.685	−0.252633
$926$	383.428i	0.414069i
$927$	0	0
$928$	245.505	0.264552
$929$	− 1131.84i	− 1.21834i	−0.793039	−	0.609171i	$-0.791502\pi$
$929$	− 1131.84i	− 1.21834i	0.793039	−	0.609171i	$-0.208498\pi$
$930$	0	0
$931$	145.580	0.156370
$932$	351.155i	0.376775i
$933$	0	0
$934$	430.708	0.461143
$935$	528.263i	0.564988i
$936$	0	0
$937$	295.116	0.314958	0.157479	−	0.987522i	$-0.449663\pi$
$937$	295.116	0.314958	0.157479	+	0.987522i	$0.449663\pi$
$938$	377.961i	0.402943i
$939$	0	0
$940$	26.3518	0.0280338
$941$	− 1.75599i	− 0.00186609i	−1.00000		0.000933044i	$-0.999703\pi$
$941$	− 1.75599i	− 0.00186609i	1.00000		0.000933044i	$-0.000296997\pi$
$942$	0	0
$943$	112.112	0.118889
$944$	− 68.5734i	− 0.0726413i
$945$	0	0
$946$	−496.148	−0.524470
$947$	717.709i	0.757876i	0.925422	+	0.378938i	$0.123711\pi$
$947$	717.709i	0.757876i	−0.925422	+	0.378938i	$0.876289\pi$
$948$	0	0
$949$	−5.77954	−0.00609013
$950$	− 1437.35i	− 1.51300i
$951$	0	0
$952$	32.0282	0.0336431
$953$	299.394i	0.314160i	0.987586	+	0.157080i	$0.0502080\pi$
$953$	299.394i	0.314160i	−0.987586	+	0.157080i	$0.949792\pi$
$954$	0	0
$955$	−148.044	−0.155020
$956$	− 45.4485i	− 0.0475403i
$957$	0	0
$958$	−206.925	−0.215996
$959$	626.445i	0.653227i
$960$	0	0
$961$	1116.67	1.16199
$962$	16.3632i	0.0170096i
$963$	0	0
$964$	−135.215	−0.140265
$965$	− 2266.38i	− 2.34858i
$966$	0	0
$967$	−222.369	−0.229958	−0.114979	−	0.993368i	$-0.536680\pi$
$967$	−222.369	−0.229958	−0.114979	+	0.993368i	$0.536680\pi$
$968$	241.071i	0.249040i
$969$	0	0
$970$	−47.1815	−0.0486407
$971$	1425.20i	1.46777i	0.679276	+	0.733883i	$0.262294\pi$
$971$	1425.20i	1.46777i	−0.679276	+	0.733883i	$0.737706\pi$
$972$	0	0
$973$	−112.752	−0.115881
$974$	− 596.846i	− 0.612778i
$975$	0	0
$976$	214.247	0.219516
$977$	− 1469.41i	− 1.50400i	−0.659165	−	0.751999i	$-0.729090\pi$
$977$	− 1469.41i	− 1.50400i	0.659165	−	0.751999i	$-0.270910\pi$
$978$	0	0
$979$	−198.545	−0.202804
$980$	120.327i	0.122783i
$981$	0	0
$982$	−603.156	−0.614212
$983$	− 303.973i	− 0.309230i	−0.987975	−	0.154615i	$-0.950586\pi$
$983$	− 303.973i	− 0.309230i	0.987975	−	0.154615i	$-0.0494137\pi$
$984$	0	0
$985$	−3299.50	−3.34975
$986$	262.687i	0.266417i
$987$	0	0
$988$	−100.647	−0.101869
$989$	919.421i	0.929647i
$990$	0	0
$991$	−235.302	−0.237439	−0.118719	−	0.992928i	$-0.537879\pi$
$991$	−235.302	−0.237439	−0.118719	+	0.992928i	$0.537879\pi$
$992$	257.848i	0.259927i
$993$	0	0
$994$	475.367	0.478237
$995$	− 2792.48i	− 2.80651i
$996$	0	0
$997$	131.171	0.131566	0.0657829	−	0.997834i	$-0.479046\pi$
$997$	131.171	0.131566	0.0657829	+	0.997834i	$0.479046\pi$
$998$	− 440.484i	− 0.441367i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1134.3.b.c.323.14		24
3.2	odd	2	inner	1134.3.b.c.323.11		24
9.2	odd	6		378.3.q.a.71.2		24
9.4	even	3		378.3.q.a.197.2		24
9.5	odd	6		126.3.q.a.29.7	✓	24
9.7	even	3		126.3.q.a.113.7	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.q.a.29.7	✓	24	9.5	odd	6
126.3.q.a.113.7	yes	24	9.7	even	3
378.3.q.a.71.2		24	9.2	odd	6
378.3.q.a.197.2		24	9.4	even	3
1134.3.b.c.323.11		24	3.2	odd	2	inner
1134.3.b.c.323.14		24	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1134.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -8.59478i q^{5} +2.64575 q^{7} -2.82843i q^{8} +12.1548 q^{10} +14.3608i q^{11} +2.41973 q^{13} +3.74166i q^{14} +4.00000 q^{16} +4.27995i q^{17} +20.7972 q^{19} +17.1896i q^{20} -20.3092 q^{22} +37.6353i q^{23} -48.8702 q^{25} +3.42201i q^{26} -5.29150 q^{28} -43.3995i q^{29} +45.5815 q^{31} +5.65685i q^{32} -6.05276 q^{34} -22.7396i q^{35} +4.78176 q^{37} +29.4116i q^{38} -24.3097 q^{40} -2.97891i q^{41} +24.4298 q^{43} -28.7215i q^{44} -53.2244 q^{46} -1.53301i q^{47} +7.00000 q^{49} -69.1129i q^{50} -4.83946 q^{52} -90.8320i q^{53} +123.427 q^{55} -7.48331i q^{56} +61.3761 q^{58} -17.1433i q^{59} +53.5618 q^{61} +64.4620i q^{62} -8.00000 q^{64} -20.7970i q^{65} +101.014 q^{67} -8.55990i q^{68} +32.1587 q^{70} -127.047i q^{71} -2.38851 q^{73} +6.76242i q^{74} -41.5943 q^{76} +37.9950i q^{77} -48.8195 q^{79} -34.3791i q^{80} +4.21282 q^{82} -85.8416i q^{83} +36.7852 q^{85} +34.5489i q^{86} +40.6184 q^{88} +13.8255i q^{89} +6.40200 q^{91} -75.2706i q^{92} +2.16800 q^{94} -178.747i q^{95} -3.88170 q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 48 q^{4} + 96 q^{16} + 24 q^{19} - 48 q^{22} - 144 q^{25} + 120 q^{31} + 96 q^{34} - 168 q^{37} - 120 q^{43} + 168 q^{49} + 264 q^{55} - 192 q^{64} - 144 q^{67} + 24 q^{73} - 48 q^{76} - 24 q^{79}+ \cdots - 96 q^{97}+O(q^{100}) \) 24 * q - 48 * q^4 + 96 * q^16 + 24 * q^19 - 48 * q^22 - 144 * q^25 + 120 * q^31 + 96 * q^34 - 168 * q^37 - 120 * q^43 + 168 * q^49 + 264 * q^55 - 192 * q^64 - 144 * q^67 + 24 * q^73 - 48 * q^76 - 24 * q^79 - 288 * q^82 + 312 * q^85 + 96 * q^88 - 168 * q^91 - 96 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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