LMFDB - Embedded newform 1134.3.b.c.323.19

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			323.19
Character	$\chi$	$=$	1134.323
Dual form			1134.3.b.c.323.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.41421i q^{2} -2.00000 q^{4} +2.65693i q^{5} -2.64575 q^{7} -2.82843i q^{8} -3.75747 q^{10} +2.55284i q^{11} -19.4066 q^{13} -3.74166i q^{14} +4.00000 q^{16} -28.6298i q^{17} +10.6664 q^{19} -5.31386i q^{20} -3.61026 q^{22} +7.71876i q^{23} +17.9407 q^{25} -27.4451i q^{26} +5.29150 q^{28} +2.18771i q^{29} +27.5863 q^{31} +5.65685i q^{32} +40.4887 q^{34} -7.02958i q^{35} -37.5014 q^{37} +15.0845i q^{38} +7.51493 q^{40} -72.3179i q^{41} +37.6131 q^{43} -5.10568i q^{44} -10.9160 q^{46} +43.8450i q^{47} +7.00000 q^{49} +25.3720i q^{50} +38.8132 q^{52} +24.2591i q^{53} -6.78272 q^{55} +7.48331i q^{56} -3.09389 q^{58} -80.2698i q^{59} -12.3127 q^{61} +39.0129i q^{62} -8.00000 q^{64} -51.5619i q^{65} +74.0584 q^{67} +57.2596i q^{68} +9.94132 q^{70} -98.3829i q^{71} +93.6849 q^{73} -53.0350i q^{74} -21.3328 q^{76} -6.75418i q^{77} -94.0671 q^{79} +10.6277i q^{80} +102.273 q^{82} -41.3070i q^{83} +76.0674 q^{85} +53.1929i q^{86} +7.22052 q^{88} +75.2547i q^{89} +51.3450 q^{91} -15.4375i q^{92} -62.0062 q^{94} +28.3398i q^{95} +164.132 q^{97} +9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.65693i	0.531386i	0.964058	+	0.265693i	$0.0856007\pi$
$5$	2.65693i	0.531386i	−0.964058	+	0.265693i	$0.914399\pi$
$6$	0	0
$7$	−2.64575	−0.377964
$7$	−2.64575	−0.377964
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	−3.75747	−0.375747
$11$	2.55284i	0.232076i	0.993245	+	0.116038i	$0.0370195\pi$
$11$	2.55284i	0.232076i	−0.993245	+	0.116038i	$0.962981\pi$
$12$	0	0
$13$	−19.4066	−1.49281	−0.746407	−	0.665490i	$-0.768223\pi$
$13$	−19.4066	−1.49281	−0.746407	+	0.665490i	$0.768223\pi$
$14$	− 3.74166i	− 0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 28.6298i	− 1.68411i	−0.539394	−	0.842053i	$-0.681347\pi$
$17$	− 28.6298i	− 1.68411i	0.539394	−	0.842053i	$-0.318653\pi$
$18$	0	0
$19$	10.6664	0.561388	0.280694	−	0.959797i	$-0.409435\pi$
$19$	10.6664	0.561388	0.280694	+	0.959797i	$0.409435\pi$
$20$	− 5.31386i	− 0.265693i
$21$	0	0
$22$	−3.61026	−0.164103
$23$	7.71876i	0.335598i	0.985821	+	0.167799i	$0.0536660\pi$
$23$	7.71876i	0.335598i	−0.985821	+	0.167799i	$0.946334\pi$
$24$	0	0
$25$	17.9407	0.717629
$26$	− 27.4451i	− 1.05558i
$27$	0	0
$28$	5.29150	0.188982
$29$	2.18771i	0.0754383i	0.999288	+	0.0377192i	$0.0120092\pi$
$29$	2.18771i	0.0754383i	−0.999288	+	0.0377192i	$0.987991\pi$
$30$	0	0
$31$	27.5863	0.889881	0.444940	−	0.895560i	$-0.353225\pi$
$31$	27.5863	0.889881	0.444940	+	0.895560i	$0.353225\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	40.4887	1.19084
$35$	− 7.02958i	− 0.200845i
$36$	0	0
$37$	−37.5014	−1.01355	−0.506775	−	0.862078i	$-0.669163\pi$
$37$	−37.5014	−1.01355	−0.506775	+	0.862078i	$0.669163\pi$
$38$	15.0845i	0.396962i
$39$	0	0
$40$	7.51493	0.187873
$41$	− 72.3179i	− 1.76385i	−0.471388	−	0.881926i	$-0.656247\pi$
$41$	− 72.3179i	− 1.76385i	0.471388	−	0.881926i	$-0.343753\pi$
$42$	0	0
$43$	37.6131	0.874723	0.437361	−	0.899286i	$-0.355913\pi$
$43$	37.6131	0.874723	0.437361	+	0.899286i	$0.355913\pi$
$44$	− 5.10568i	− 0.116038i
$45$	0	0
$46$	−10.9160	−0.237304
$47$	43.8450i	0.932873i	0.884555	+	0.466436i	$0.154462\pi$
$47$	43.8450i	0.932873i	−0.884555	+	0.466436i	$0.845538\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	25.3720i	0.507440i
$51$	0	0
$52$	38.8132	0.746407
$53$	24.2591i	0.457718i	0.973460	+	0.228859i	$0.0734995\pi$
$53$	24.2591i	0.457718i	−0.973460	+	0.228859i	$0.926501\pi$
$54$	0	0
$55$	−6.78272	−0.123322
$56$	7.48331i	0.133631i
$57$	0	0
$58$	−3.09389	−0.0533430
$59$	− 80.2698i	− 1.36051i	−0.732978	−	0.680253i	$-0.761870\pi$
$59$	− 80.2698i	− 1.36051i	0.732978	−	0.680253i	$-0.238130\pi$
$60$	0	0
$61$	−12.3127	−0.201847	−0.100924	−	0.994894i	$-0.532180\pi$
$61$	−12.3127	−0.201847	−0.100924	+	0.994894i	$0.532180\pi$
$62$	39.0129i	0.629241i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 51.5619i	− 0.793260i
$66$	0	0
$67$	74.0584	1.10535	0.552675	−	0.833397i	$-0.313607\pi$
$67$	74.0584	1.10535	0.552675	+	0.833397i	$0.313607\pi$
$68$	57.2596i	0.842053i
$69$	0	0
$70$	9.94132	0.142019
$71$	− 98.3829i	− 1.38567i	−0.721094	−	0.692837i	$-0.756360\pi$
$71$	− 98.3829i	− 1.38567i	0.721094	−	0.692837i	$-0.243640\pi$
$72$	0	0
$73$	93.6849	1.28336	0.641678	−	0.766974i	$-0.278239\pi$
$73$	93.6849	1.28336	0.641678	+	0.766974i	$0.278239\pi$
$74$	− 53.0350i	− 0.716689i
$75$	0	0
$76$	−21.3328	−0.280694
$77$	− 6.75418i	− 0.0877166i
$78$	0	0
$79$	−94.0671	−1.19072	−0.595362	−	0.803458i	$-0.702991\pi$
$79$	−94.0671	−1.19072	−0.595362	+	0.803458i	$0.702991\pi$
$80$	10.6277i	0.132846i
$81$	0	0
$82$	102.273	1.24723
$83$	− 41.3070i	− 0.497674i	−0.968545	−	0.248837i	$-0.919952\pi$
$83$	− 41.3070i	− 0.497674i	0.968545	−	0.248837i	$-0.0800484\pi$
$84$	0	0
$85$	76.0674	0.894911
$86$	53.1929i	0.618522i
$87$	0	0
$88$	7.22052	0.0820514
$89$	75.2547i	0.845558i	0.906233	+	0.422779i	$0.138945\pi$
$89$	75.2547i	0.845558i	−0.906233	+	0.422779i	$0.861055\pi$
$90$	0	0
$91$	51.3450	0.564231
$92$	− 15.4375i	− 0.167799i
$93$	0	0
$94$	−62.0062	−0.659641
$95$	28.3398i	0.298314i
$96$	0	0
$97$	164.132	1.69208	0.846041	−	0.533118i	$-0.178980\pi$
$97$	164.132	1.69208	0.846041	+	0.533118i	$0.178980\pi$
$98$	9.89949i	0.101015i
$99$	0	0
$100$	−35.8814	−0.358814
$101$	− 17.0973i	− 0.169280i	−0.996412	−	0.0846399i	$-0.973026\pi$
$101$	− 17.0973i	− 0.169280i	0.996412	−	0.0846399i	$-0.0269740\pi$
$102$	0	0
$103$	139.689	1.35621	0.678104	−	0.734966i	$-0.262802\pi$
$103$	139.689	1.35621	0.678104	+	0.734966i	$0.262802\pi$
$104$	54.8901i	0.527789i
$105$	0	0
$106$	−34.3075	−0.323655
$107$	24.3628i	0.227689i	0.993499	+	0.113845i	$0.0363166\pi$
$107$	24.3628i	0.227689i	−0.993499	+	0.113845i	$0.963683\pi$
$108$	0	0
$109$	30.8295	0.282839	0.141420	−	0.989950i	$-0.454833\pi$
$109$	30.8295	0.282839	0.141420	+	0.989950i	$0.454833\pi$
$110$	− 9.59221i	− 0.0872019i
$111$	0	0
$112$	−10.5830	−0.0944911
$113$	− 45.7926i	− 0.405244i	−0.979257	−	0.202622i	$-0.935054\pi$
$113$	− 45.7926i	− 0.405244i	0.979257	−	0.202622i	$-0.0649463\pi$
$114$	0	0
$115$	−20.5082	−0.178332
$116$	− 4.37542i	− 0.0377192i
$117$	0	0
$118$	113.519	0.962023
$119$	75.7474i	0.636533i
$120$	0	0
$121$	114.483	0.946141
$122$	− 17.4128i	− 0.142727i
$123$	0	0
$124$	−55.1726	−0.444940
$125$	114.090i	0.912724i
$126$	0	0
$127$	187.439	1.47590	0.737951	−	0.674855i	$-0.235794\pi$
$127$	187.439	1.47590	0.737951	+	0.674855i	$0.235794\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	72.9196	0.560920
$131$	− 27.8116i	− 0.212302i	−0.994350	−	0.106151i	$-0.966147\pi$
$131$	− 27.8116i	− 0.212302i	0.994350	−	0.106151i	$-0.0338527\pi$
$132$	0	0
$133$	−28.2206	−0.212185
$134$	104.734i	0.781600i
$135$	0	0
$136$	−80.9773	−0.595422
$137$	− 200.257i	− 1.46173i	−0.682523	−	0.730864i	$-0.739117\pi$
$137$	− 200.257i	− 1.46173i	0.682523	−	0.730864i	$-0.260883\pi$
$138$	0	0
$139$	−28.9683	−0.208405	−0.104202	−	0.994556i	$-0.533229\pi$
$139$	−28.9683	−0.208405	−0.104202	+	0.994556i	$0.533229\pi$
$140$	14.0592i	0.100423i
$141$	0	0
$142$	139.134	0.979820
$143$	− 49.5419i	− 0.346447i
$144$	0	0
$145$	−5.81260	−0.0400869
$146$	132.491i	0.907469i
$147$	0	0
$148$	75.0028	0.506775
$149$	175.734i	1.17943i	0.807613	+	0.589713i	$0.200759\pi$
$149$	175.734i	1.17943i	−0.807613	+	0.589713i	$0.799241\pi$
$150$	0	0
$151$	−260.354	−1.72420	−0.862099	−	0.506740i	$-0.830851\pi$
$151$	−260.354	−1.72420	−0.862099	+	0.506740i	$0.830851\pi$
$152$	− 30.1691i	− 0.198481i
$153$	0	0
$154$	9.55185	0.0620250
$155$	73.2949i	0.472870i
$156$	0	0
$157$	−282.552	−1.79970	−0.899848	−	0.436204i	$-0.856323\pi$
$157$	−282.552	−1.79970	−0.899848	+	0.436204i	$0.856323\pi$
$158$	− 133.031i	− 0.841968i
$159$	0	0
$160$	−15.0299	−0.0939366
$161$	− 20.4219i	− 0.126844i
$162$	0	0
$163$	−12.6900	−0.0778525	−0.0389262	−	0.999242i	$-0.512394\pi$
$163$	−12.6900	−0.0778525	−0.0389262	+	0.999242i	$0.512394\pi$
$164$	144.636i	0.881926i
$165$	0	0
$166$	58.4169	0.351909
$167$	− 211.643i	− 1.26732i	−0.773610	−	0.633662i	$-0.781551\pi$
$167$	− 211.643i	− 1.26732i	0.773610	−	0.633662i	$-0.218449\pi$
$168$	0	0
$169$	207.615	1.22849
$170$	107.576i	0.632797i
$171$	0	0
$172$	−75.2262	−0.437361
$173$	− 116.489i	− 0.673346i	−0.941622	−	0.336673i	$-0.890698\pi$
$173$	− 116.489i	− 0.673346i	0.941622	−	0.336673i	$-0.109302\pi$
$174$	0	0
$175$	−47.4667	−0.271238
$176$	10.2114i	0.0580191i
$177$	0	0
$178$	−106.426	−0.597900
$179$	− 272.774i	− 1.52388i	−0.647650	−	0.761938i	$-0.724248\pi$
$179$	− 272.774i	− 1.52388i	0.647650	−	0.761938i	$-0.275752\pi$
$180$	0	0
$181$	−27.1230	−0.149851	−0.0749255	−	0.997189i	$-0.523872\pi$
$181$	−27.1230	−0.149851	−0.0749255	+	0.997189i	$0.523872\pi$
$182$	72.6128i	0.398971i
$183$	0	0
$184$	21.8319	0.118652
$185$	− 99.6385i	− 0.538587i
$186$	0	0
$187$	73.0873	0.390841
$188$	− 87.6900i	− 0.466436i
$189$	0	0
$190$	−40.0786	−0.210940
$191$	− 227.057i	− 1.18878i	−0.804178	−	0.594389i	$-0.797394\pi$
$191$	− 227.057i	− 1.18878i	0.804178	−	0.594389i	$-0.202606\pi$
$192$	0	0
$193$	239.682	1.24187	0.620937	−	0.783861i	$-0.286752\pi$
$193$	239.682	1.24187	0.620937	+	0.783861i	$0.286752\pi$
$194$	232.118i	1.19648i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	139.283i	0.707022i	0.935430	+	0.353511i	$0.115012\pi$
$197$	139.283i	0.707022i	−0.935430	+	0.353511i	$0.884988\pi$
$198$	0	0
$199$	71.5320	0.359457	0.179729	−	0.983716i	$-0.442478\pi$
$199$	71.5320	0.359457	0.179729	+	0.983716i	$0.442478\pi$
$200$	− 50.7440i	− 0.253720i
$201$	0	0
$202$	24.1792	0.119699
$203$	− 5.78814i	− 0.0285130i
$204$	0	0
$205$	192.144	0.937286
$206$	197.551i	0.958983i
$207$	0	0
$208$	−77.6263	−0.373204
$209$	27.2296i	0.130285i
$210$	0	0
$211$	−312.541	−1.48124	−0.740619	−	0.671925i	$-0.765468\pi$
$211$	−312.541	−1.48124	−0.740619	+	0.671925i	$0.765468\pi$
$212$	− 48.5181i	− 0.228859i
$213$	0	0
$214$	−34.4541	−0.161001
$215$	99.9353i	0.464815i
$216$	0	0
$217$	−72.9865	−0.336343
$218$	43.5995i	0.199998i
$219$	0	0
$220$	13.5654	0.0616611
$221$	555.607i	2.51406i
$222$	0	0
$223$	113.249	0.507843	0.253921	−	0.967225i	$-0.418280\pi$
$223$	113.249	0.507843	0.253921	+	0.967225i	$0.418280\pi$
$224$	− 14.9666i	− 0.0668153i
$225$	0	0
$226$	64.7605	0.286551
$227$	259.305i	1.14231i	0.820842	+	0.571156i	$0.193505\pi$
$227$	259.305i	1.14231i	−0.820842	+	0.571156i	$0.806495\pi$
$228$	0	0
$229$	283.602	1.23844	0.619219	−	0.785218i	$-0.287449\pi$
$229$	283.602	1.23844	0.619219	+	0.785218i	$0.287449\pi$
$230$	− 29.0030i	− 0.126100i
$231$	0	0
$232$	6.18778	0.0266715
$233$	122.039i	0.523773i	0.965099	+	0.261887i	$0.0843447\pi$
$233$	122.039i	0.523773i	−0.965099	+	0.261887i	$0.915655\pi$
$234$	0	0
$235$	−116.493	−0.495715
$236$	160.540i	0.680253i
$237$	0	0
$238$	−107.123	−0.450096
$239$	− 111.310i	− 0.465734i	−0.972509	−	0.232867i	$-0.925189\pi$
$239$	− 111.310i	− 0.465734i	0.972509	−	0.232867i	$-0.0748107\pi$
$240$	0	0
$241$	−103.101	−0.427805	−0.213902	−	0.976855i	$-0.568617\pi$
$241$	−103.101	−0.427805	−0.213902	+	0.976855i	$0.568617\pi$
$242$	161.903i	0.669022i
$243$	0	0
$244$	24.6254	0.100924
$245$	18.5985i	0.0759123i
$246$	0	0
$247$	−206.998	−0.838048
$248$	− 78.0259i	− 0.314620i
$249$	0	0
$250$	−161.348	−0.645393
$251$	− 426.514i	− 1.69926i	−0.527380	−	0.849629i	$-0.676826\pi$
$251$	− 426.514i	− 1.69926i	0.527380	−	0.849629i	$-0.323174\pi$
$252$	0	0
$253$	−19.7047	−0.0778844
$254$	265.079i	1.04362i
$255$	0	0
$256$	16.0000	0.0625000
$257$	42.6633i	0.166005i	0.996549	+	0.0830026i	$0.0264510\pi$
$257$	42.6633i	0.166005i	−0.996549	+	0.0830026i	$0.973549\pi$
$258$	0	0
$259$	99.2193	0.383086
$260$	103.124i	0.396630i
$261$	0	0
$262$	39.3315	0.150120
$263$	− 206.177i	− 0.783945i	−0.919977	−	0.391972i	$-0.871793\pi$
$263$	− 206.177i	− 0.783945i	0.919977	−	0.391972i	$-0.128207\pi$
$264$	0	0
$265$	−64.4546	−0.243225
$266$	− 39.9099i	− 0.150037i
$267$	0	0
$268$	−148.117	−0.552675
$269$	− 452.410i	− 1.68182i	−0.541174	−	0.840910i	$-0.682020\pi$
$269$	− 452.410i	− 1.68182i	0.541174	−	0.840910i	$-0.317980\pi$
$270$	0	0
$271$	−213.136	−0.786478	−0.393239	−	0.919436i	$-0.628646\pi$
$271$	−213.136	−0.786478	−0.393239	+	0.919436i	$0.628646\pi$
$272$	− 114.519i	− 0.421027i
$273$	0	0
$274$	283.206	1.03360
$275$	45.7998i	0.166545i
$276$	0	0
$277$	−101.158	−0.365190	−0.182595	−	0.983188i	$-0.558450\pi$
$277$	−101.158	−0.365190	−0.182595	+	0.983188i	$0.558450\pi$
$278$	− 40.9673i	− 0.147365i
$279$	0	0
$280$	−19.8826	−0.0710094
$281$	283.930i	1.01043i	0.862995	+	0.505213i	$0.168586\pi$
$281$	283.930i	1.01043i	−0.862995	+	0.505213i	$0.831414\pi$
$282$	0	0
$283$	−361.801	−1.27845	−0.639225	−	0.769020i	$-0.720745\pi$
$283$	−361.801	−1.27845	−0.639225	+	0.769020i	$0.720745\pi$
$284$	196.766i	0.692837i
$285$	0	0
$286$	70.0628	0.244975
$287$	191.335i	0.666673i
$288$	0	0
$289$	−530.666	−1.83622
$290$	− 8.22025i	− 0.0283457i
$291$	0	0
$292$	−187.370	−0.641678
$293$	− 18.7899i	− 0.0641293i	−0.999486	−	0.0320647i	$-0.989792\pi$
$293$	− 18.7899i	− 0.0641293i	0.999486	−	0.0320647i	$-0.0102082\pi$
$294$	0	0
$295$	213.271	0.722953
$296$	106.070i	0.358344i
$297$	0	0
$298$	−248.526	−0.833980
$299$	− 149.795i	− 0.500986i
$300$	0	0
$301$	−99.5148	−0.330614
$302$	− 368.196i	− 1.21919i
$303$	0	0
$304$	42.6655	0.140347
$305$	− 32.7139i	− 0.107259i
$306$	0	0
$307$	158.613	0.516656	0.258328	−	0.966057i	$-0.416828\pi$
$307$	158.613	0.516656	0.258328	+	0.966057i	$0.416828\pi$
$308$	13.5084i	0.0438583i
$309$	0	0
$310$	−103.655	−0.334370
$311$	− 161.240i	− 0.518456i	−0.965816	−	0.259228i	$-0.916532\pi$
$311$	− 161.240i	− 0.518456i	0.965816	−	0.259228i	$-0.0834680\pi$
$312$	0	0
$313$	595.578	1.90280	0.951402	−	0.307951i	$-0.0996433\pi$
$313$	595.578	1.90280	0.951402	+	0.307951i	$0.0996433\pi$
$314$	− 399.589i	− 1.27258i
$315$	0	0
$316$	188.134	0.595362
$317$	436.397i	1.37665i	0.725404	+	0.688324i	$0.241653\pi$
$317$	436.397i	1.37665i	−0.725404	+	0.688324i	$0.758347\pi$
$318$	0	0
$319$	−5.58488	−0.0175075
$320$	− 21.2554i	− 0.0664232i
$321$	0	0
$322$	28.8809	0.0896924
$323$	− 305.376i	− 0.945438i
$324$	0	0
$325$	−348.168	−1.07129
$326$	− 17.9463i	− 0.0550500i
$327$	0	0
$328$	−204.546	−0.623616
$329$	− 116.003i	− 0.352593i
$330$	0	0
$331$	−547.791	−1.65496	−0.827478	−	0.561498i	$-0.810225\pi$
$331$	−547.791	−1.65496	−0.827478	+	0.561498i	$0.810225\pi$
$332$	82.6139i	0.248837i
$333$	0	0
$334$	299.309	0.896134
$335$	196.768i	0.587367i
$336$	0	0
$337$	−623.784	−1.85099	−0.925495	−	0.378759i	$-0.876351\pi$
$337$	−623.784	−1.85099	−0.925495	+	0.378759i	$0.876351\pi$
$338$	293.613i	0.868676i
$339$	0	0
$340$	−152.135	−0.447455
$341$	70.4234i	0.206520i
$342$	0	0
$343$	−18.5203	−0.0539949
$344$	− 106.386i	− 0.309261i
$345$	0	0
$346$	164.740	0.476127
$347$	− 366.491i	− 1.05617i	−0.849192	−	0.528085i	$-0.822910\pi$
$347$	− 366.491i	− 1.05617i	0.849192	−	0.528085i	$-0.177090\pi$
$348$	0	0
$349$	−489.287	−1.40197	−0.700985	−	0.713176i	$-0.747256\pi$
$349$	−489.287	−1.40197	−0.700985	+	0.713176i	$0.747256\pi$
$350$	− 67.1280i	− 0.191794i
$351$	0	0
$352$	−14.4410	−0.0410257
$353$	− 351.925i	− 0.996956i	−0.866902	−	0.498478i	$-0.833893\pi$
$353$	− 351.925i	− 0.996956i	0.866902	−	0.498478i	$-0.166107\pi$
$354$	0	0
$355$	261.396	0.736328
$356$	− 150.509i	− 0.422779i
$357$	0	0
$358$	385.761	1.07754
$359$	− 7.11763i	− 0.0198263i	−0.999951	−	0.00991313i	$-0.996845\pi$
$359$	− 7.11763i	− 0.0198263i	0.999951	−	0.00991313i	$-0.00315550\pi$
$360$	0	0
$361$	−247.228	−0.684843
$362$	− 38.3578i	− 0.105961i
$363$	0	0
$364$	−102.690	−0.282115
$365$	248.914i	0.681957i
$366$	0	0
$367$	127.245	0.346717	0.173359	−	0.984859i	$-0.444538\pi$
$367$	127.245	0.346717	0.173359	+	0.984859i	$0.444538\pi$
$368$	30.8750i	0.0838995i
$369$	0	0
$370$	140.910	0.380838
$371$	− 64.1834i	− 0.173001i
$372$	0	0
$373$	−59.4804	−0.159465	−0.0797324	−	0.996816i	$-0.525407\pi$
$373$	−59.4804	−0.159465	−0.0797324	+	0.996816i	$0.525407\pi$
$374$	103.361i	0.276367i
$375$	0	0
$376$	124.012	0.329820
$377$	− 42.4560i	− 0.112615i
$378$	0	0
$379$	512.562	1.35241	0.676204	−	0.736715i	$-0.263624\pi$
$379$	512.562	1.35241	0.676204	+	0.736715i	$0.263624\pi$
$380$	− 56.6796i	− 0.149157i
$381$	0	0
$382$	321.106	0.840593
$383$	150.044i	0.391760i	0.980628	+	0.195880i	$0.0627563\pi$
$383$	150.044i	0.391760i	−0.980628	+	0.195880i	$0.937244\pi$
$384$	0	0
$385$	17.9454	0.0466114
$386$	338.961i	0.878137i
$387$	0	0
$388$	−328.264	−0.846041
$389$	− 584.004i	− 1.50130i	−0.660703	−	0.750648i	$-0.729742\pi$
$389$	− 584.004i	− 1.50130i	0.660703	−	0.750648i	$-0.270258\pi$
$390$	0	0
$391$	220.987	0.565183
$392$	− 19.7990i	− 0.0505076i
$393$	0	0
$394$	−196.976	−0.499940
$395$	− 249.930i	− 0.632733i
$396$	0	0
$397$	321.479	0.809770	0.404885	−	0.914368i	$-0.367312\pi$
$397$	321.479	0.809770	0.404885	+	0.914368i	$0.367312\pi$
$398$	101.162i	0.254175i
$399$	0	0
$400$	71.7629	0.179407
$401$	− 334.393i	− 0.833897i	−0.908930	−	0.416949i	$-0.863099\pi$
$401$	− 334.393i	− 0.833897i	0.908930	−	0.416949i	$-0.136901\pi$
$402$	0	0
$403$	−535.356	−1.32843
$404$	34.1945i	0.0846399i
$405$	0	0
$406$	8.18567	0.0201617
$407$	− 95.7350i	− 0.235221i
$408$	0	0
$409$	−46.0745	−0.112652	−0.0563258	−	0.998412i	$-0.517939\pi$
$409$	−46.0745	−0.112652	−0.0563258	+	0.998412i	$0.517939\pi$
$410$	271.732i	0.662761i
$411$	0	0
$412$	−279.379	−0.678104
$413$	212.374i	0.514223i
$414$	0	0
$415$	109.750	0.264457
$416$	− 109.780i	− 0.263895i
$417$	0	0
$418$	−38.5084	−0.0921254
$419$	− 632.268i	− 1.50899i	−0.656304	−	0.754496i	$-0.727881\pi$
$419$	− 632.268i	− 1.50899i	0.656304	−	0.754496i	$-0.272119\pi$
$420$	0	0
$421$	−556.228	−1.32121	−0.660604	−	0.750735i	$-0.729700\pi$
$421$	−556.228	−1.32121	−0.660604	+	0.750735i	$0.729700\pi$
$422$	− 442.000i	− 1.04739i
$423$	0	0
$424$	68.6150	0.161828
$425$	− 513.640i	− 1.20856i
$426$	0	0
$427$	32.5763	0.0762911
$428$	− 48.7255i	− 0.113845i
$429$	0	0
$430$	−141.330	−0.328674
$431$	374.687i	0.869344i	0.900589	+	0.434672i	$0.143136\pi$
$431$	374.687i	0.869344i	−0.900589	+	0.434672i	$0.856864\pi$
$432$	0	0
$433$	176.315	0.407195	0.203598	−	0.979055i	$-0.434737\pi$
$433$	176.315	0.407195	0.203598	+	0.979055i	$0.434737\pi$
$434$	− 103.219i	− 0.237831i
$435$	0	0
$436$	−61.6590	−0.141420
$437$	82.3312i	0.188401i
$438$	0	0
$439$	−359.588	−0.819107	−0.409554	−	0.912286i	$-0.634316\pi$
$439$	−359.588	−0.819107	−0.409554	+	0.912286i	$0.634316\pi$
$440$	19.1844i	0.0436009i
$441$	0	0
$442$	−785.747	−1.77771
$443$	− 281.707i	− 0.635907i	−0.948106	−	0.317954i	$-0.897004\pi$
$443$	− 281.707i	− 0.635907i	0.948106	−	0.317954i	$-0.102996\pi$
$444$	0	0
$445$	−199.946	−0.449318
$446$	160.158i	0.359099i
$447$	0	0
$448$	21.1660	0.0472456
$449$	− 110.602i	− 0.246329i	−0.992386	−	0.123164i	$-0.960696\pi$
$449$	− 110.602i	− 0.246329i	0.992386	−	0.123164i	$-0.0393042\pi$
$450$	0	0
$451$	184.616	0.409348
$452$	91.5852i	0.202622i
$453$	0	0
$454$	−366.712	−0.807736
$455$	136.420i	0.299824i
$456$	0	0
$457$	−662.985	−1.45073	−0.725366	−	0.688363i	$-0.758330\pi$
$457$	−662.985	−1.45073	−0.725366	+	0.688363i	$0.758330\pi$
$458$	401.074i	0.875708i
$459$	0	0
$460$	41.0164	0.0891660
$461$	− 706.958i	− 1.53353i	−0.641928	−	0.766765i	$-0.721865\pi$
$461$	− 706.958i	− 1.53353i	0.641928	−	0.766765i	$-0.278135\pi$
$462$	0	0
$463$	510.344	1.10225	0.551127	−	0.834421i	$-0.314198\pi$
$463$	510.344	1.10225	0.551127	+	0.834421i	$0.314198\pi$
$464$	8.75085i	0.0188596i
$465$	0	0
$466$	−172.589	−0.370364
$467$	204.177i	0.437211i	0.975813	+	0.218605i	$0.0701507\pi$
$467$	204.177i	0.437211i	−0.975813	+	0.218605i	$0.929849\pi$
$468$	0	0
$469$	−195.940	−0.417783
$470$	− 164.746i	− 0.350524i
$471$	0	0
$472$	−227.037	−0.481011
$473$	96.0202i	0.203002i
$474$	0	0
$475$	191.363	0.402869
$476$	− 151.495i	− 0.318266i
$477$	0	0
$478$	157.417	0.329324
$479$	− 2.32831i	− 0.00486076i	−0.999997	−	0.00243038i	$-0.999226\pi$
$479$	− 2.32831i	− 0.00486076i	0.999997	−	0.00243038i	$-0.000773615\pi$
$480$	0	0
$481$	727.774	1.51304
$482$	− 145.807i	− 0.302504i
$483$	0	0
$484$	−228.966	−0.473070
$485$	436.087i	0.899148i
$486$	0	0
$487$	−700.930	−1.43928	−0.719641	−	0.694347i	$-0.755693\pi$
$487$	−700.930	−1.43928	−0.719641	+	0.694347i	$0.755693\pi$
$488$	34.8255i	0.0713637i
$489$	0	0
$490$	−26.3023	−0.0536781
$491$	900.521i	1.83406i	0.398824	+	0.917028i	$0.369419\pi$
$491$	900.521i	1.83406i	−0.398824	+	0.917028i	$0.630581\pi$
$492$	0	0
$493$	62.6338	0.127046
$494$	− 292.739i	− 0.592590i
$495$	0	0
$496$	110.345	0.222470
$497$	260.297i	0.523736i
$498$	0	0
$499$	−437.378	−0.876508	−0.438254	−	0.898851i	$-0.644403\pi$
$499$	−437.378	−0.876508	−0.438254	+	0.898851i	$0.644403\pi$
$500$	− 228.181i	− 0.456362i
$501$	0	0
$502$	603.182	1.20156
$503$	359.340i	0.714394i	0.934029	+	0.357197i	$0.116268\pi$
$503$	359.340i	0.714394i	−0.934029	+	0.357197i	$0.883732\pi$
$504$	0	0
$505$	45.4262	0.0899529
$506$	− 27.8667i	− 0.0550726i
$507$	0	0
$508$	−374.879	−0.737951
$509$	− 650.254i	− 1.27751i	−0.769409	−	0.638757i	$-0.779449\pi$
$509$	− 650.254i	− 1.27751i	0.769409	−	0.638757i	$-0.220551\pi$
$510$	0	0
$511$	−247.867	−0.485063
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	−60.3351	−0.117383
$515$	371.145i	0.720669i
$516$	0	0
$517$	−111.929	−0.216498
$518$	140.317i	0.270883i
$519$	0	0
$520$	−145.839	−0.280460
$521$	952.688i	1.82858i	0.405065	+	0.914288i	$0.367249\pi$
$521$	952.688i	1.82858i	−0.405065	+	0.914288i	$0.632751\pi$
$522$	0	0
$523$	515.678	0.985999	0.493000	−	0.870030i	$-0.335900\pi$
$523$	515.678	0.985999	0.493000	+	0.870030i	$0.335900\pi$
$524$	55.6231i	0.106151i
$525$	0	0
$526$	291.579	0.554333
$527$	− 789.791i	− 1.49865i
$528$	0	0
$529$	469.421	0.887374
$530$	− 91.1526i	− 0.171986i
$531$	0	0
$532$	56.4412	0.106092
$533$	1403.44i	2.63310i
$534$	0	0
$535$	−64.7301	−0.120991
$536$	− 209.469i	− 0.390800i
$537$	0	0
$538$	639.804	1.18923
$539$	17.8699i	0.0331538i
$540$	0	0
$541$	−299.942	−0.554422	−0.277211	−	0.960809i	$-0.589410\pi$
$541$	−299.942	−0.554422	−0.277211	+	0.960809i	$0.589410\pi$
$542$	− 301.419i	− 0.556124i
$543$	0	0
$544$	161.955	0.297711
$545$	81.9118i	0.150297i
$546$	0	0
$547$	−72.1611	−0.131922	−0.0659608	−	0.997822i	$-0.521011\pi$
$547$	−72.1611	−0.131922	−0.0659608	+	0.997822i	$0.521011\pi$
$548$	400.514i	0.730864i
$549$	0	0
$550$	−64.7707	−0.117765
$551$	23.3350i	0.0423502i
$552$	0	0
$553$	248.878	0.450051
$554$	− 143.058i	− 0.258228i
$555$	0	0
$556$	57.9366	0.104202
$557$	− 472.989i	− 0.849172i	−0.905388	−	0.424586i	$-0.860420\pi$
$557$	− 472.989i	− 0.849172i	0.905388	−	0.424586i	$-0.139580\pi$
$558$	0	0
$559$	−729.941	−1.30580
$560$	− 28.1183i	− 0.0502113i
$561$	0	0
$562$	−401.537	−0.714479
$563$	217.801i	0.386858i	0.981114	+	0.193429i	$0.0619610\pi$
$563$	217.801i	0.386858i	−0.981114	+	0.193429i	$0.938039\pi$
$564$	0	0
$565$	121.668	0.215341
$566$	− 511.664i	− 0.904001i
$567$	0	0
$568$	−278.269	−0.489910
$569$	486.959i	0.855815i	0.903823	+	0.427907i	$0.140749\pi$
$569$	486.959i	0.855815i	−0.903823	+	0.427907i	$0.859251\pi$
$570$	0	0
$571$	−519.344	−0.909534	−0.454767	−	0.890611i	$-0.650277\pi$
$571$	−519.344	−0.909534	−0.454767	+	0.890611i	$0.650277\pi$
$572$	99.0838i	0.173223i
$573$	0	0
$574$	−270.589	−0.471409
$575$	138.480i	0.240835i
$576$	0	0
$577$	139.881	0.242428	0.121214	−	0.992626i	$-0.461321\pi$
$577$	139.881	0.242428	0.121214	+	0.992626i	$0.461321\pi$
$578$	− 750.476i	− 1.29840i
$579$	0	0
$580$	11.6252	0.0200434
$581$	109.288i	0.188103i
$582$	0	0
$583$	−61.9295	−0.106226
$584$	− 264.981i	− 0.453735i
$585$	0	0
$586$	26.5729	0.0453463
$587$	338.886i	0.577319i	0.957432	+	0.288660i	$0.0932096\pi$
$587$	338.886i	0.577319i	−0.957432	+	0.288660i	$0.906790\pi$
$588$	0	0
$589$	294.246	0.499569
$590$	301.611i	0.511205i
$591$	0	0
$592$	−150.006	−0.253388
$593$	− 111.970i	− 0.188820i	−0.995533	−	0.0944101i	$-0.969904\pi$
$593$	− 111.970i	− 0.188820i	0.995533	−	0.0944101i	$-0.0300965\pi$
$594$	0	0
$595$	−201.255	−0.338244
$596$	− 351.469i	− 0.589713i
$597$	0	0
$598$	211.842	0.354250
$599$	− 507.663i	− 0.847517i	−0.905775	−	0.423758i	$-0.860711\pi$
$599$	− 507.663i	− 0.847517i	0.905775	−	0.423758i	$-0.139289\pi$
$600$	0	0
$601$	99.4004	0.165392	0.0826959	−	0.996575i	$-0.473647\pi$
$601$	99.4004	0.165392	0.0826959	+	0.996575i	$0.473647\pi$
$602$	− 140.735i	− 0.233779i
$603$	0	0
$604$	520.708	0.862099
$605$	304.173i	0.502766i
$606$	0	0
$607$	488.841	0.805340	0.402670	−	0.915345i	$-0.368082\pi$
$607$	488.841	0.805340	0.402670	+	0.915345i	$0.368082\pi$
$608$	60.3382i	0.0992404i
$609$	0	0
$610$	46.2645	0.0758434
$611$	− 850.882i	− 1.39261i
$612$	0	0
$613$	91.0276	0.148495	0.0742476	−	0.997240i	$-0.476344\pi$
$613$	91.0276	0.148495	0.0742476	+	0.997240i	$0.476344\pi$
$614$	224.313i	0.365331i
$615$	0	0
$616$	−19.1037	−0.0310125
$617$	683.188i	1.10727i	0.832758	+	0.553637i	$0.186760\pi$
$617$	683.188i	1.10727i	−0.832758	+	0.553637i	$0.813240\pi$
$618$	0	0
$619$	455.958	0.736604	0.368302	−	0.929706i	$-0.379939\pi$
$619$	455.958	0.736604	0.368302	+	0.929706i	$0.379939\pi$
$620$	− 146.590i	− 0.236435i
$621$	0	0
$622$	228.027	0.366603
$623$	− 199.105i	− 0.319591i
$624$	0	0
$625$	145.388	0.232620
$626$	842.274i	1.34549i
$627$	0	0
$628$	565.104	0.899848
$629$	1073.66i	1.70693i
$630$	0	0
$631$	857.311	1.35865	0.679327	−	0.733836i	$-0.262272\pi$
$631$	857.311	1.35865	0.679327	+	0.733836i	$0.262272\pi$
$632$	266.062i	0.420984i
$633$	0	0
$634$	−617.159	−0.973437
$635$	498.013i	0.784273i
$636$	0	0
$637$	−135.846	−0.213259
$638$	− 7.89821i	− 0.0123796i
$639$	0	0
$640$	30.0597	0.0469683
$641$	1004.08i	1.56643i	0.621753	+	0.783213i	$0.286421\pi$
$641$	1004.08i	1.56643i	−0.621753	+	0.783213i	$0.713579\pi$
$642$	0	0
$643$	708.873	1.10245	0.551223	−	0.834358i	$-0.314161\pi$
$643$	708.873	1.10245	0.551223	+	0.834358i	$0.314161\pi$
$644$	40.8438i	0.0634221i
$645$	0	0
$646$	431.868	0.668526
$647$	− 906.254i	− 1.40070i	−0.713799	−	0.700351i	$-0.753027\pi$
$647$	− 906.254i	− 1.40070i	0.713799	−	0.700351i	$-0.246973\pi$
$648$	0	0
$649$	204.916	0.315741
$650$	− 492.384i	− 0.757514i
$651$	0	0
$652$	25.3799	0.0389262
$653$	1187.33i	1.81828i	0.416496	+	0.909138i	$0.363258\pi$
$653$	1187.33i	1.81828i	−0.416496	+	0.909138i	$0.636742\pi$
$654$	0	0
$655$	73.8934	0.112814
$656$	− 289.272i	− 0.440963i
$657$	0	0
$658$	164.053	0.249321
$659$	121.389i	0.184202i	0.995750	+	0.0921010i	$0.0293583\pi$
$659$	121.389i	0.184202i	−0.995750	+	0.0921010i	$0.970642\pi$
$660$	0	0
$661$	471.341	0.713072	0.356536	−	0.934282i	$-0.383958\pi$
$661$	471.341	0.713072	0.356536	+	0.934282i	$0.383958\pi$
$662$	− 774.693i	− 1.17023i
$663$	0	0
$664$	−116.834	−0.175954
$665$	− 74.9801i	− 0.112752i
$666$	0	0
$667$	−16.8864	−0.0253170
$668$	423.286i	0.633662i
$669$	0	0
$670$	−278.272	−0.415331
$671$	− 31.4323i	− 0.0468440i
$672$	0	0
$673$	526.376	0.782133	0.391067	−	0.920362i	$-0.372106\pi$
$673$	526.376	0.782133	0.391067	+	0.920362i	$0.372106\pi$
$674$	− 882.163i	− 1.30885i
$675$	0	0
$676$	−415.231	−0.614247
$677$	659.913i	0.974760i	0.873190	+	0.487380i	$0.162047\pi$
$677$	659.913i	0.974760i	−0.873190	+	0.487380i	$0.837953\pi$
$678$	0	0
$679$	−434.252	−0.639547
$680$	− 215.151i	− 0.316399i
$681$	0	0
$682$	−99.5938	−0.146032
$683$	412.168i	0.603467i	0.953392	+	0.301734i	$0.0975654\pi$
$683$	412.168i	0.603467i	−0.953392	+	0.301734i	$0.902435\pi$
$684$	0	0
$685$	532.068	0.776742
$686$	− 26.1916i	− 0.0381802i
$687$	0	0
$688$	150.452	0.218681
$689$	− 470.785i	− 0.683288i
$690$	0	0
$691$	729.601	1.05586	0.527931	−	0.849287i	$-0.322968\pi$
$691$	729.601	1.05586	0.527931	+	0.849287i	$0.322968\pi$
$692$	232.978i	0.336673i
$693$	0	0
$694$	518.296	0.746825
$695$	− 76.9667i	− 0.110743i
$696$	0	0
$697$	−2070.45	−2.97052
$698$	− 691.957i	− 0.991342i
$699$	0	0
$700$	94.9334	0.135619
$701$	56.3677i	0.0804104i	0.999191	+	0.0402052i	$0.0128012\pi$
$701$	56.3677i	0.0804104i	−0.999191	+	0.0402052i	$0.987199\pi$
$702$	0	0
$703$	−400.004	−0.568996
$704$	− 20.4227i	− 0.0290095i
$705$	0	0
$706$	497.698	0.704954
$707$	45.2351i	0.0639817i
$708$	0	0
$709$	782.114	1.10312	0.551561	−	0.834134i	$-0.314032\pi$
$709$	782.114	1.10312	0.551561	+	0.834134i	$0.314032\pi$
$710$	369.670i	0.520662i
$711$	0	0
$712$	212.852	0.298950
$713$	212.932i	0.298642i
$714$	0	0
$715$	131.629	0.184097
$716$	545.548i	0.761938i
$717$	0	0
$718$	10.0658	0.0140193
$719$	− 131.717i	− 0.183194i	−0.995796	−	0.0915971i	$-0.970803\pi$
$719$	− 131.717i	− 0.183194i	0.995796	−	0.0915971i	$-0.0291972\pi$
$720$	0	0
$721$	−369.583	−0.512598
$722$	− 349.634i	− 0.484257i
$723$	0	0
$724$	54.2461	0.0749255
$725$	39.2491i	0.0541367i
$726$	0	0
$727$	705.382	0.970263	0.485132	−	0.874441i	$-0.338772\pi$
$727$	705.382	0.970263	0.485132	+	0.874441i	$0.338772\pi$
$728$	− 145.226i	− 0.199486i
$729$	0	0
$730$	−352.018	−0.482216
$731$	− 1076.86i	− 1.47313i
$732$	0	0
$733$	246.524	0.336322	0.168161	−	0.985760i	$-0.446217\pi$
$733$	246.524	0.336322	0.168161	+	0.985760i	$0.446217\pi$
$734$	179.952i	0.245166i
$735$	0	0
$736$	−43.6639	−0.0593259
$737$	189.059i	0.256525i
$738$	0	0
$739$	573.403	0.775918	0.387959	−	0.921677i	$-0.373180\pi$
$739$	573.403	0.775918	0.387959	+	0.921677i	$0.373180\pi$
$740$	199.277i	0.269293i
$741$	0	0
$742$	90.7691	0.122330
$743$	283.466i	0.381515i	0.981637	+	0.190758i	$0.0610945\pi$
$743$	283.466i	0.381515i	−0.981637	+	0.190758i	$0.938906\pi$
$744$	0	0
$745$	−466.914	−0.626730
$746$	− 84.1180i	− 0.112759i
$747$	0	0
$748$	−146.175	−0.195421
$749$	− 64.4578i	− 0.0860585i
$750$	0	0
$751$	1044.99	1.39147	0.695735	−	0.718299i	$-0.255079\pi$
$751$	1044.99	1.39147	0.695735	+	0.718299i	$0.255079\pi$
$752$	175.380i	0.233218i
$753$	0	0
$754$	60.0419	0.0796311
$755$	− 691.742i	− 0.916215i
$756$	0	0
$757$	595.724	0.786954	0.393477	−	0.919334i	$-0.371272\pi$
$757$	595.724	0.786954	0.393477	+	0.919334i	$0.371272\pi$
$758$	724.873i	0.956296i
$759$	0	0
$760$	80.1571	0.105470
$761$	833.349i	1.09507i	0.836783	+	0.547535i	$0.184434\pi$
$761$	833.349i	1.09507i	−0.836783	+	0.547535i	$0.815566\pi$
$762$	0	0
$763$	−81.5672	−0.106903
$764$	454.113i	0.594389i
$765$	0	0
$766$	−212.194	−0.277016
$767$	1557.76i	2.03098i
$768$	0	0
$769$	316.693	0.411824	0.205912	−	0.978570i	$-0.433984\pi$
$769$	316.693	0.411824	0.205912	+	0.978570i	$0.433984\pi$
$770$	25.3786i	0.0329592i
$771$	0	0
$772$	−479.363	−0.620937
$773$	− 367.509i	− 0.475432i	−0.971335	−	0.237716i	$-0.923601\pi$
$773$	− 367.509i	− 0.475432i	0.971335	−	0.237716i	$-0.0763987\pi$
$774$	0	0
$775$	494.918	0.638604
$776$	− 464.235i	− 0.598241i
$777$	0	0
$778$	825.906	1.06158
$779$	− 771.371i	− 0.990206i
$780$	0	0
$781$	251.156	0.321582
$782$	312.522i	0.399645i
$783$	0	0
$784$	28.0000	0.0357143
$785$	− 750.721i	− 0.956333i
$786$	0	0
$787$	−289.146	−0.367403	−0.183702	−	0.982982i	$-0.558808\pi$
$787$	−289.146	−0.367403	−0.183702	+	0.982982i	$0.558808\pi$
$788$	− 278.567i	− 0.353511i
$789$	0	0
$790$	353.454	0.447410
$791$	121.156i	0.153168i
$792$	0	0
$793$	238.947	0.301320
$794$	454.639i	0.572594i
$795$	0	0
$796$	−143.064	−0.179729
$797$	880.592i	1.10488i	0.833551	+	0.552442i	$0.186304\pi$
$797$	880.592i	1.10488i	−0.833551	+	0.552442i	$0.813696\pi$
$798$	0	0
$799$	1255.27	1.57106
$800$	101.488i	0.126860i
$801$	0	0
$802$	472.903	0.589655
$803$	239.163i	0.297836i
$804$	0	0
$805$	54.2596	0.0674032
$806$	− 757.108i	− 0.939340i
$807$	0	0
$808$	−48.3583	−0.0598494
$809$	776.794i	0.960190i	0.877217	+	0.480095i	$0.159398\pi$
$809$	776.794i	0.960190i	−0.877217	+	0.480095i	$0.840602\pi$
$810$	0	0
$811$	−985.397	−1.21504	−0.607520	−	0.794305i	$-0.707835\pi$
$811$	−985.397	−1.21504	−0.607520	+	0.794305i	$0.707835\pi$
$812$	11.5763i	0.0142565i
$813$	0	0
$814$	135.390	0.166326
$815$	− 33.7163i	− 0.0413697i
$816$	0	0
$817$	401.195	0.491059
$818$	− 65.1592i	− 0.0796567i
$819$	0	0
$820$	−384.287	−0.468643
$821$	− 288.288i	− 0.351142i	−0.984467	−	0.175571i	$-0.943823\pi$
$821$	− 288.288i	− 0.351142i	0.984467	−	0.175571i	$-0.0561772\pi$
$822$	0	0
$823$	559.148	0.679402	0.339701	−	0.940533i	$-0.389674\pi$
$823$	559.148	0.679402	0.339701	+	0.940533i	$0.389674\pi$
$824$	− 395.101i	− 0.479492i
$825$	0	0
$826$	−300.342	−0.363610
$827$	− 402.564i	− 0.486776i	−0.969929	−	0.243388i	$-0.921741\pi$
$827$	− 402.564i	− 0.486776i	0.969929	−	0.243388i	$-0.0782589\pi$
$828$	0	0
$829$	123.079	0.148466	0.0742332	−	0.997241i	$-0.476349\pi$
$829$	123.079	0.148466	0.0742332	+	0.997241i	$0.476349\pi$
$830$	155.210i	0.186999i
$831$	0	0
$832$	155.253	0.186602
$833$	− 200.409i	− 0.240587i
$834$	0	0
$835$	562.321	0.673439
$836$	− 54.4591i	− 0.0651425i
$837$	0	0
$838$	894.162	1.06702
$839$	− 1576.04i	− 1.87848i	−0.343267	−	0.939238i	$-0.611534\pi$
$839$	− 1576.04i	− 1.87848i	0.343267	−	0.939238i	$-0.388466\pi$
$840$	0	0
$841$	836.214	0.994309
$842$	− 786.625i	− 0.934235i
$843$	0	0
$844$	625.083	0.740619
$845$	551.620i	0.652804i
$846$	0	0
$847$	−302.894	−0.357608
$848$	97.0362i	0.114429i
$849$	0	0
$850$	726.396	0.854584
$851$	− 289.464i	− 0.340146i
$852$	0	0
$853$	−52.3942	−0.0614235	−0.0307117	−	0.999528i	$-0.509777\pi$
$853$	−52.3942	−0.0614235	−0.0307117	+	0.999528i	$0.509777\pi$
$854$	46.0698i	0.0539459i
$855$	0	0
$856$	68.9083	0.0805003
$857$	82.7472i	0.0965545i	0.998834	+	0.0482773i	$0.0153731\pi$
$857$	82.7472i	0.0965545i	−0.998834	+	0.0482773i	$0.984627\pi$
$858$	0	0
$859$	113.970	0.132677	0.0663387	−	0.997797i	$-0.478868\pi$
$859$	113.970	0.132677	0.0663387	+	0.997797i	$0.478868\pi$
$860$	− 199.871i	− 0.232408i
$861$	0	0
$862$	−529.888	−0.614719
$863$	864.215i	1.00141i	0.865619	+	0.500704i	$0.166925\pi$
$863$	864.215i	1.00141i	−0.865619	+	0.500704i	$0.833075\pi$
$864$	0	0
$865$	309.503	0.357807
$866$	249.348i	0.287930i
$867$	0	0
$868$	145.973	0.168172
$869$	− 240.138i	− 0.276339i
$870$	0	0
$871$	−1437.22	−1.65008
$872$	− 87.1990i	− 0.0999989i
$873$	0	0
$874$	−116.434	−0.133220
$875$	− 301.855i	− 0.344977i
$876$	0	0
$877$	−631.737	−0.720339	−0.360169	−	0.932887i	$-0.617281\pi$
$877$	−631.737	−0.720339	−0.360169	+	0.932887i	$0.617281\pi$
$878$	− 508.534i	− 0.579196i
$879$	0	0
$880$	−27.1309	−0.0308305
$881$	293.562i	0.333215i	0.986023	+	0.166607i	$0.0532812\pi$
$881$	293.562i	0.333215i	−0.986023	+	0.166607i	$0.946719\pi$
$882$	0	0
$883$	−483.341	−0.547385	−0.273693	−	0.961817i	$-0.588245\pi$
$883$	−483.341	−0.547385	−0.273693	+	0.961817i	$0.588245\pi$
$884$	− 1111.21i	− 1.25703i
$885$	0	0
$886$	398.394	0.449654
$887$	− 947.090i	− 1.06774i	−0.845565	−	0.533872i	$-0.820736\pi$
$887$	− 947.090i	− 1.06774i	0.845565	−	0.533872i	$-0.179264\pi$
$888$	0	0
$889$	−495.918	−0.557838
$890$	− 282.767i	− 0.317716i
$891$	0	0
$892$	−226.498	−0.253921
$893$	467.668i	0.523704i
$894$	0	0
$895$	724.741	0.809767
$896$	29.9333i	0.0334077i
$897$	0	0
$898$	156.414	0.174181
$899$	60.3509i	0.0671311i
$900$	0	0
$901$	694.532	0.770846
$902$	261.087i	0.289453i
$903$	0	0
$904$	−129.521	−0.143275
$905$	− 72.0640i	− 0.0796288i
$906$	0	0
$907$	−99.6415	−0.109858	−0.0549292	−	0.998490i	$-0.517493\pi$
$907$	−99.6415	−0.109858	−0.0549292	+	0.998490i	$0.517493\pi$
$908$	− 518.610i	− 0.571156i
$909$	0	0
$910$	−192.927	−0.212008
$911$	− 1461.99i	− 1.60482i	−0.596775	−	0.802409i	$-0.703551\pi$
$911$	− 1461.99i	− 1.60482i	0.596775	−	0.802409i	$-0.296449\pi$
$912$	0	0
$913$	105.450	0.115498
$914$	− 937.602i	− 1.02582i
$915$	0	0
$916$	−567.205	−0.619219
$917$	73.5825i	0.0802426i
$918$	0	0
$919$	−1025.26	−1.11562	−0.557811	−	0.829968i	$-0.688359\pi$
$919$	−1025.26	−1.11562	−0.557811	+	0.829968i	$0.688359\pi$
$920$	58.0059i	0.0630499i
$921$	0	0
$922$	999.789	1.08437
$923$	1909.28i	2.06855i
$924$	0	0
$925$	−672.802	−0.727353
$926$	721.735i	0.779412i
$927$	0	0
$928$	−12.3756	−0.0133357
$929$	1172.50i	1.26211i	0.775740	+	0.631053i	$0.217377\pi$
$929$	1172.50i	1.26211i	−0.775740	+	0.631053i	$0.782623\pi$
$930$	0	0
$931$	74.6647	0.0801983
$932$	− 244.078i	− 0.261887i
$933$	0	0
$934$	−288.750	−0.309155
$935$	194.188i	0.207688i
$936$	0	0
$937$	989.490	1.05602	0.528009	−	0.849239i	$-0.322939\pi$
$937$	989.490	1.05602	0.528009	+	0.849239i	$0.322939\pi$
$938$	− 277.101i	− 0.295417i
$939$	0	0
$940$	232.986	0.247858
$941$	240.400i	0.255472i	0.991808	+	0.127736i	$0.0407711\pi$
$941$	240.400i	0.255472i	−0.991808	+	0.127736i	$0.959229\pi$
$942$	0	0
$943$	558.204	0.591945
$944$	− 321.079i	− 0.340126i
$945$	0	0
$946$	−135.793	−0.143544
$947$	− 1195.46i	− 1.26236i	−0.775636	−	0.631181i	$-0.782571\pi$
$947$	− 1195.46i	− 1.26236i	0.775636	−	0.631181i	$-0.217429\pi$
$948$	0	0
$949$	−1818.10	−1.91581
$950$	270.628i	0.284871i
$951$	0	0
$952$	214.246	0.225048
$953$	261.104i	0.273981i	0.990572	+	0.136991i	$0.0437430\pi$
$953$	261.104i	0.273981i	−0.990572	+	0.136991i	$0.956257\pi$
$954$	0	0
$955$	603.273	0.631700
$956$	222.621i	0.232867i
$957$	0	0
$958$	3.29272	0.00343708
$959$	529.830i	0.552481i
$960$	0	0
$961$	−199.996	−0.208112
$962$	1029.23i	1.06988i
$963$	0	0
$964$	206.202	0.213902
$965$	636.817i	0.659914i
$966$	0	0
$967$	−1105.62	−1.14335	−0.571677	−	0.820479i	$-0.693707\pi$
$967$	−1105.62	−1.14335	−0.571677	+	0.820479i	$0.693707\pi$
$968$	− 323.807i	− 0.334511i
$969$	0	0
$970$	−616.720	−0.635794
$971$	− 139.155i	− 0.143311i	−0.997429	−	0.0716554i	$-0.977172\pi$
$971$	− 139.155i	− 0.143311i	0.997429	−	0.0716554i	$-0.0228282\pi$
$972$	0	0
$973$	76.6429	0.0787696
$974$	− 991.265i	− 1.01773i
$975$	0	0
$976$	−49.2507	−0.0504618
$977$	1117.15i	1.14345i	0.820445	+	0.571725i	$0.193726\pi$
$977$	1117.15i	1.14345i	−0.820445	+	0.571725i	$0.806274\pi$
$978$	0	0
$979$	−192.113	−0.196234
$980$	− 37.1970i	− 0.0379561i
$981$	0	0
$982$	−1273.53	−1.29687
$983$	− 1861.31i	− 1.89350i	−0.321969	−	0.946750i	$-0.604345\pi$
$983$	− 1861.31i	− 1.89350i	0.321969	−	0.946750i	$-0.395655\pi$
$984$	0	0
$985$	−370.066	−0.375701
$986$	88.5776i	0.0898353i
$987$	0	0
$988$	413.996	0.419024
$989$	290.326i	0.293555i
$990$	0	0
$991$	−73.7864	−0.0744565	−0.0372283	−	0.999307i	$-0.511853\pi$
$991$	−73.7864	−0.0744565	−0.0372283	+	0.999307i	$0.511853\pi$
$992$	156.052i	0.157310i
$993$	0	0
$994$	−368.115	−0.370337
$995$	190.055i	0.191011i
$996$	0	0
$997$	−1192.09	−1.19568	−0.597838	−	0.801617i	$-0.703973\pi$
$997$	−1192.09	−1.19568	−0.597838	+	0.801617i	$0.703973\pi$
$998$	− 618.545i	− 0.619785i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1134.3.b.c.323.19		24
3.2	odd	2	inner	1134.3.b.c.323.6		24
9.2	odd	6		126.3.q.a.113.6	yes	24
9.4	even	3		126.3.q.a.29.6	✓	24
9.5	odd	6		378.3.q.a.197.10		24
9.7	even	3		378.3.q.a.71.10		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.q.a.29.6	✓	24	9.4	even	3
126.3.q.a.113.6	yes	24	9.2	odd	6
378.3.q.a.71.10		24	9.7	even	3
378.3.q.a.197.10		24	9.5	odd	6
1134.3.b.c.323.6		24	3.2	odd	2	inner
1134.3.b.c.323.19		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)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +2.65693i q^{5} -2.64575 q^{7} -2.82843i q^{8} -3.75747 q^{10} +2.55284i q^{11} -19.4066 q^{13} -3.74166i q^{14} +4.00000 q^{16} -28.6298i q^{17} +10.6664 q^{19} -5.31386i q^{20} -3.61026 q^{22} +7.71876i q^{23} +17.9407 q^{25} -27.4451i q^{26} +5.29150 q^{28} +2.18771i q^{29} +27.5863 q^{31} +5.65685i q^{32} +40.4887 q^{34} -7.02958i q^{35} -37.5014 q^{37} +15.0845i q^{38} +7.51493 q^{40} -72.3179i q^{41} +37.6131 q^{43} -5.10568i q^{44} -10.9160 q^{46} +43.8450i q^{47} +7.00000 q^{49} +25.3720i q^{50} +38.8132 q^{52} +24.2591i q^{53} -6.78272 q^{55} +7.48331i q^{56} -3.09389 q^{58} -80.2698i q^{59} -12.3127 q^{61} +39.0129i q^{62} -8.00000 q^{64} -51.5619i q^{65} +74.0584 q^{67} +57.2596i q^{68} +9.94132 q^{70} -98.3829i q^{71} +93.6849 q^{73} -53.0350i q^{74} -21.3328 q^{76} -6.75418i q^{77} -94.0671 q^{79} +10.6277i q^{80} +102.273 q^{82} -41.3070i q^{83} +76.0674 q^{85} +53.1929i q^{86} +7.22052 q^{88} +75.2547i q^{89} +51.3450 q^{91} -15.4375i q^{92} -62.0062 q^{94} +28.3398i q^{95} +164.132 q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 48 q^{4} + 96 q^{16} + 24 q^{19} - 48 q^{22} - 144 q^{25} + 120 q^{31} + 96 q^{34} - 168 q^{37} - 120 q^{43} + 168 q^{49} + 264 q^{55} - 192 q^{64} - 144 q^{67} + 24 q^{73} - 48 q^{76} - 24 q^{79}+ \cdots - 96 q^{97}+O(q^{100}) \) 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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