LMFDB - Embedded newform 1134.3.b.c.323.1

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			323.1
Character	$\chi$	$=$	1134.323
Dual form			1134.3.b.c.323.24

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} -2.00000 q^{4} -7.74628i q^{5} -2.64575 q^{7} +2.82843i q^{8} -10.9549 q^{10} +11.5558i q^{11} +19.3170 q^{13} +3.74166i q^{14} +4.00000 q^{16} +18.8486i q^{17} -18.8740 q^{19} +15.4926i q^{20} +16.3424 q^{22} +30.2089i q^{23} -35.0049 q^{25} -27.3183i q^{26} +5.29150 q^{28} +57.1188i q^{29} -10.7982 q^{31} -5.65685i q^{32} +26.6559 q^{34} +20.4947i q^{35} +6.77073 q^{37} +26.6918i q^{38} +21.9098 q^{40} -60.7722i q^{41} +16.3205 q^{43} -23.1117i q^{44} +42.7218 q^{46} +19.3608i q^{47} +7.00000 q^{49} +49.5043i q^{50} -38.6340 q^{52} +35.4340i q^{53} +89.5147 q^{55} -7.48331i q^{56} +80.7782 q^{58} -9.68197i q^{59} +90.2805 q^{61} +15.2709i q^{62} -8.00000 q^{64} -149.635i q^{65} -58.9477 q^{67} -37.6972i q^{68} +28.9839 q^{70} +43.4041i q^{71} +36.4770 q^{73} -9.57526i q^{74} +37.7479 q^{76} -30.5739i q^{77} -45.1650 q^{79} -30.9851i q^{80} -85.9449 q^{82} +56.7235i q^{83} +146.007 q^{85} -23.0806i q^{86} -32.6848 q^{88} -52.3173i q^{89} -51.1079 q^{91} -60.4178i q^{92} +27.3803 q^{94} +146.203i q^{95} +14.1419 q^{97} -9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 7.74628i	− 1.54926i	−0.632417	−	0.774628i	$-0.717937\pi$
$5$	− 7.74628i	− 1.54926i	0.632417	−	0.774628i	$-0.282063\pi$
$6$	0	0
$7$	−2.64575	−0.377964
$7$	−2.64575	−0.377964
$8$	2.82843i	0.353553i
$9$	0	0
$10$	−10.9549	−1.09549
$11$	11.5558i	1.05053i	0.850939	+	0.525265i	$0.176034\pi$
$11$	11.5558i	1.05053i	−0.850939	+	0.525265i	$0.823966\pi$
$12$	0	0
$13$	19.3170	1.48592	0.742961	−	0.669335i	$-0.233421\pi$
$13$	19.3170	1.48592	0.742961	+	0.669335i	$0.233421\pi$
$14$	3.74166i	0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	18.8486i	1.10874i	0.832270	+	0.554371i	$0.187041\pi$
$17$	18.8486i	1.10874i	−0.832270	+	0.554371i	$0.812959\pi$
$18$	0	0
$19$	−18.8740	−0.993366	−0.496683	−	0.867932i	$-0.665449\pi$
$19$	−18.8740	−0.993366	−0.496683	+	0.867932i	$0.665449\pi$
$20$	15.4926i	0.774628i
$21$	0	0
$22$	16.3424	0.742837
$23$	30.2089i	1.31343i	0.754139	+	0.656715i	$0.228055\pi$
$23$	30.2089i	1.31343i	−0.754139	+	0.656715i	$0.771945\pi$
$24$	0	0
$25$	−35.0049	−1.40019
$26$	− 27.3183i	− 1.05071i
$27$	0	0
$28$	5.29150	0.188982
$29$	57.1188i	1.96961i	0.173650	+	0.984807i	$0.444444\pi$
$29$	57.1188i	1.96961i	−0.173650	+	0.984807i	$0.555556\pi$
$30$	0	0
$31$	−10.7982	−0.348328	−0.174164	−	0.984717i	$-0.555722\pi$
$31$	−10.7982	−0.348328	−0.174164	+	0.984717i	$0.555722\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	26.6559	0.783998
$35$	20.4947i	0.585564i
$36$	0	0
$37$	6.77073	0.182993	0.0914963	−	0.995805i	$-0.470835\pi$
$37$	6.77073	0.182993	0.0914963	+	0.995805i	$0.470835\pi$
$38$	26.6918i	0.702416i
$39$	0	0
$40$	21.9098	0.547745
$41$	− 60.7722i	− 1.48225i	−0.671367	−	0.741125i	$-0.734293\pi$
$41$	− 60.7722i	− 1.48225i	0.671367	−	0.741125i	$-0.265707\pi$
$42$	0	0
$43$	16.3205	0.379546	0.189773	−	0.981828i	$-0.439225\pi$
$43$	16.3205	0.379546	0.189773	+	0.981828i	$0.439225\pi$
$44$	− 23.1117i	− 0.525265i
$45$	0	0
$46$	42.7218	0.928736
$47$	19.3608i	0.411932i	0.978559	+	0.205966i	$0.0660337\pi$
$47$	19.3608i	0.411932i	−0.978559	+	0.205966i	$0.933966\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	49.5043i	0.990087i
$51$	0	0
$52$	−38.6340	−0.742961
$53$	35.4340i	0.668567i	0.942473	+	0.334283i	$0.108494\pi$
$53$	35.4340i	0.668567i	−0.942473	+	0.334283i	$0.891506\pi$
$54$	0	0
$55$	89.5147	1.62754
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	80.7782	1.39273
$59$	− 9.68197i	− 0.164101i	−0.996628	−	0.0820506i	$-0.973853\pi$
$59$	− 9.68197i	− 0.164101i	0.996628	−	0.0820506i	$-0.0261469\pi$
$60$	0	0
$61$	90.2805	1.48001	0.740004	−	0.672602i	$-0.234824\pi$
$61$	90.2805	1.48001	0.740004	+	0.672602i	$0.234824\pi$
$62$	15.2709i	0.246305i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 149.635i	− 2.30207i
$66$	0	0
$67$	−58.9477	−0.879817	−0.439908	−	0.898043i	$-0.644989\pi$
$67$	−58.9477	−0.879817	−0.439908	+	0.898043i	$0.644989\pi$
$68$	− 37.6972i	− 0.554371i
$69$	0	0
$70$	28.9839	0.414056
$71$	43.4041i	0.611325i	0.952140	+	0.305662i	$0.0988779\pi$
$71$	43.4041i	0.611325i	−0.952140	+	0.305662i	$0.901122\pi$
$72$	0	0
$73$	36.4770	0.499685	0.249842	−	0.968287i	$-0.419621\pi$
$73$	36.4770	0.499685	0.249842	+	0.968287i	$0.419621\pi$
$74$	− 9.57526i	− 0.129395i
$75$	0	0
$76$	37.7479	0.496683
$77$	− 30.5739i	− 0.397063i
$78$	0	0
$79$	−45.1650	−0.571709	−0.285855	−	0.958273i	$-0.592277\pi$
$79$	−45.1650	−0.571709	−0.285855	+	0.958273i	$0.592277\pi$
$80$	− 30.9851i	− 0.387314i
$81$	0	0
$82$	−85.9449	−1.04811
$83$	56.7235i	0.683416i	0.939806	+	0.341708i	$0.111005\pi$
$83$	56.7235i	0.683416i	−0.939806	+	0.341708i	$0.888995\pi$
$84$	0	0
$85$	146.007	1.71772
$86$	− 23.0806i	− 0.268380i
$87$	0	0
$88$	−32.6848	−0.371418
$89$	− 52.3173i	− 0.587835i	−0.955831	−	0.293917i	$-0.905041\pi$
$89$	− 52.3173i	− 0.587835i	0.955831	−	0.293917i	$-0.0949591\pi$
$90$	0	0
$91$	−51.1079	−0.561626
$92$	− 60.4178i	− 0.656715i
$93$	0	0
$94$	27.3803	0.291280
$95$	146.203i	1.53898i
$96$	0	0
$97$	14.1419	0.145793	0.0728966	−	0.997340i	$-0.476776\pi$
$97$	14.1419	0.145793	0.0728966	+	0.997340i	$0.476776\pi$
$98$	− 9.89949i	− 0.101015i
$99$	0	0
$100$	70.0097	0.700097
$101$	12.6059i	0.124811i	0.998051	+	0.0624055i	$0.0198772\pi$
$101$	12.6059i	0.124811i	−0.998051	+	0.0624055i	$0.980123\pi$
$102$	0	0
$103$	−133.299	−1.29416	−0.647080	−	0.762422i	$-0.724010\pi$
$103$	−133.299	−1.29416	−0.647080	+	0.762422i	$0.724010\pi$
$104$	54.6367i	0.525353i
$105$	0	0
$106$	50.1113	0.472748
$107$	− 8.07684i	− 0.0754845i	−0.999288	−	0.0377422i	$-0.987983\pi$
$107$	− 8.07684i	− 0.0754845i	0.999288	−	0.0377422i	$-0.0120166\pi$
$108$	0	0
$109$	160.039	1.46825	0.734123	−	0.679016i	$-0.237593\pi$
$109$	160.039	1.46825	0.734123	+	0.679016i	$0.237593\pi$
$110$	− 126.593i	− 1.15084i
$111$	0	0
$112$	−10.5830	−0.0944911
$113$	64.7680i	0.573168i	0.958055	+	0.286584i	$0.0925198\pi$
$113$	64.7680i	0.573168i	−0.958055	+	0.286584i	$0.907480\pi$
$114$	0	0
$115$	234.007	2.03484
$116$	− 114.238i	− 0.984807i
$117$	0	0
$118$	−13.6924	−0.116037
$119$	− 49.8687i	− 0.419065i
$120$	0	0
$121$	−12.5372	−0.103613
$122$	− 127.676i	− 1.04652i
$123$	0	0
$124$	21.5963	0.174164
$125$	77.5004i	0.620003i
$126$	0	0
$127$	−1.32557	−0.0104376	−0.00521879	−	0.999986i	$-0.501661\pi$
$127$	−1.32557	−0.0104376	−0.00521879	+	0.999986i	$0.501661\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	−211.615	−1.62781
$131$	4.49309i	0.0342984i	0.999853	+	0.0171492i	$0.00545902\pi$
$131$	4.49309i	0.0342984i	−0.999853	+	0.0171492i	$0.994541\pi$
$132$	0	0
$133$	49.9358	0.375457
$134$	83.3646i	0.622124i
$135$	0	0
$136$	−53.3119	−0.391999
$137$	218.770i	1.59686i	0.602086	+	0.798431i	$0.294336\pi$
$137$	218.770i	1.59686i	−0.602086	+	0.798431i	$0.705664\pi$
$138$	0	0
$139$	87.4784	0.629341	0.314670	−	0.949201i	$-0.398106\pi$
$139$	87.4784	0.629341	0.314670	+	0.949201i	$0.398106\pi$
$140$	− 40.9895i	− 0.292782i
$141$	0	0
$142$	61.3826	0.432272
$143$	223.224i	1.56101i
$144$	0	0
$145$	442.458	3.05144
$146$	− 51.5863i	− 0.353331i
$147$	0	0
$148$	−13.5415	−0.0914963
$149$	211.945i	1.42245i	0.702963	+	0.711226i	$0.251860\pi$
$149$	211.945i	1.42245i	−0.702963	+	0.711226i	$0.748140\pi$
$150$	0	0
$151$	188.984	1.25155	0.625775	−	0.780004i	$-0.284783\pi$
$151$	188.984	1.25155	0.625775	+	0.780004i	$0.284783\pi$
$152$	− 53.3836i	− 0.351208i
$153$	0	0
$154$	−43.2380	−0.280766
$155$	83.6456i	0.539649i
$156$	0	0
$157$	1.31276	0.00836154	0.00418077	−	0.999991i	$-0.498669\pi$
$157$	1.31276	0.00836154	0.00418077	+	0.999991i	$0.498669\pi$
$158$	63.8730i	0.404259i
$159$	0	0
$160$	−43.8196	−0.273872
$161$	− 79.9252i	− 0.496430i
$162$	0	0
$163$	−187.224	−1.14861	−0.574307	−	0.818640i	$-0.694728\pi$
$163$	−187.224	−1.14861	−0.574307	+	0.818640i	$0.694728\pi$
$164$	121.544i	0.741125i
$165$	0	0
$166$	80.2192	0.483248
$167$	− 106.521i	− 0.637853i	−0.947779	−	0.318926i	$-0.896678\pi$
$167$	− 106.521i	− 0.637853i	0.947779	−	0.318926i	$-0.103322\pi$
$168$	0	0
$169$	204.146	1.20796
$170$	− 206.484i	− 1.21461i
$171$	0	0
$172$	−32.6410	−0.189773
$173$	− 259.950i	− 1.50260i	−0.659961	−	0.751300i	$-0.729427\pi$
$173$	− 259.950i	− 1.50260i	0.659961	−	0.751300i	$-0.270573\pi$
$174$	0	0
$175$	92.6141	0.529224
$176$	46.2233i	0.262633i
$177$	0	0
$178$	−73.9878	−0.415662
$179$	− 27.0801i	− 0.151285i	−0.997135	−	0.0756426i	$-0.975899\pi$
$179$	− 27.0801i	− 0.151285i	0.997135	−	0.0756426i	$-0.0241008\pi$
$180$	0	0
$181$	−164.901	−0.911058	−0.455529	−	0.890221i	$-0.650550\pi$
$181$	−164.901	−0.911058	−0.455529	+	0.890221i	$0.650550\pi$
$182$	72.2775i	0.397129i
$183$	0	0
$184$	−85.4437	−0.464368
$185$	− 52.4480i	− 0.283502i
$186$	0	0
$187$	−217.811	−1.16477
$188$	− 38.7217i	− 0.205966i
$189$	0	0
$190$	206.762	1.08822
$191$	− 6.54668i	− 0.0342758i	−0.999853	−	0.0171379i	$-0.994545\pi$
$191$	− 6.54668i	− 0.0342758i	0.999853	−	0.0171379i	$-0.00545543\pi$
$192$	0	0
$193$	210.507	1.09071	0.545355	−	0.838205i	$-0.316395\pi$
$193$	210.507	1.09071	0.545355	+	0.838205i	$0.316395\pi$
$194$	− 19.9997i	− 0.103091i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	51.9722i	0.263818i	0.991262	+	0.131909i	$0.0421107\pi$
$197$	51.9722i	0.263818i	−0.991262	+	0.131909i	$0.957889\pi$
$198$	0	0
$199$	−375.071	−1.88478	−0.942389	−	0.334518i	$-0.891427\pi$
$199$	−375.071	−1.88478	−0.942389	+	0.334518i	$0.891427\pi$
$200$	− 99.0087i	− 0.495043i
$201$	0	0
$202$	17.8275	0.0882547
$203$	− 151.122i	− 0.744444i
$204$	0	0
$205$	−470.759	−2.29638
$206$	188.513i	0.915109i
$207$	0	0
$208$	77.2679	0.371480
$209$	− 218.104i	− 1.04356i
$210$	0	0
$211$	−35.3099	−0.167345	−0.0836726	−	0.996493i	$-0.526665\pi$
$211$	−35.3099	−0.167345	−0.0836726	+	0.996493i	$0.526665\pi$
$212$	− 70.8681i	− 0.334283i
$213$	0	0
$214$	−11.4224	−0.0533756
$215$	− 126.423i	− 0.588014i
$216$	0	0
$217$	28.5693	0.131656
$218$	− 226.329i	− 1.03821i
$219$	0	0
$220$	−179.029	−0.813770
$221$	364.098i	1.64750i
$222$	0	0
$223$	5.17718	0.0232161	0.0116080	−	0.999933i	$-0.496305\pi$
$223$	5.17718	0.0232161	0.0116080	+	0.999933i	$0.496305\pi$
$224$	14.9666i	0.0668153i
$225$	0	0
$226$	91.5958	0.405291
$227$	434.624i	1.91464i	0.289027	+	0.957321i	$0.406668\pi$
$227$	434.624i	1.91464i	−0.289027	+	0.957321i	$0.593332\pi$
$228$	0	0
$229$	184.642	0.806297	0.403148	−	0.915135i	$-0.367916\pi$
$229$	184.642	0.806297	0.403148	+	0.915135i	$0.367916\pi$
$230$	− 330.935i	− 1.43885i
$231$	0	0
$232$	−161.556	−0.696364
$233$	− 258.706i	− 1.11033i	−0.831741	−	0.555164i	$-0.812656\pi$
$233$	− 258.706i	− 1.11033i	0.831741	−	0.555164i	$-0.187344\pi$
$234$	0	0
$235$	149.974	0.638189
$236$	19.3639i	0.0820506i
$237$	0	0
$238$	−70.5250	−0.296324
$239$	− 317.321i	− 1.32770i	−0.747865	−	0.663851i	$-0.768921\pi$
$239$	− 317.321i	− 1.32770i	0.747865	−	0.663851i	$-0.231079\pi$
$240$	0	0
$241$	−275.479	−1.14307	−0.571533	−	0.820579i	$-0.693651\pi$
$241$	−275.479	−1.14307	−0.571533	+	0.820579i	$0.693651\pi$
$242$	17.7303i	0.0732657i
$243$	0	0
$244$	−180.561	−0.740004
$245$	− 54.2240i	− 0.221322i
$246$	0	0
$247$	−364.588	−1.47606
$248$	− 30.5418i	− 0.123152i
$249$	0	0
$250$	109.602	0.438408
$251$	84.0847i	0.334999i	0.985872	+	0.167499i	$0.0535692\pi$
$251$	84.0847i	0.334999i	−0.985872	+	0.167499i	$0.946431\pi$
$252$	0	0
$253$	−349.089	−1.37980
$254$	1.87464i	0.00738049i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 7.79301i	− 0.0303230i	−0.999885	−	0.0151615i	$-0.995174\pi$
$257$	− 7.79301i	− 0.0303230i	0.999885	−	0.0151615i	$-0.00482624\pi$
$258$	0	0
$259$	−17.9137	−0.0691647
$260$	299.270i	1.15104i
$261$	0	0
$262$	6.35418	0.0242526
$263$	− 298.208i	− 1.13387i	−0.823763	−	0.566935i	$-0.808129\pi$
$263$	− 298.208i	− 1.13387i	0.823763	−	0.566935i	$-0.191871\pi$
$264$	0	0
$265$	274.482	1.03578
$266$	− 70.6199i	− 0.265488i
$267$	0	0
$268$	117.895	0.439908
$269$	15.2957i	0.0568615i	0.999596	+	0.0284307i	$0.00905100\pi$
$269$	15.2957i	0.0568615i	−0.999596	+	0.0284307i	$0.990949\pi$
$270$	0	0
$271$	401.934	1.48315	0.741575	−	0.670870i	$-0.234079\pi$
$271$	401.934	1.48315	0.741575	+	0.670870i	$0.234079\pi$
$272$	75.3944i	0.277185i
$273$	0	0
$274$	309.388	1.12915
$275$	− 404.510i	− 1.47095i
$276$	0	0
$277$	−335.563	−1.21142	−0.605709	−	0.795686i	$-0.707110\pi$
$277$	−335.563	−1.21142	−0.605709	+	0.795686i	$0.707110\pi$
$278$	− 123.713i	− 0.445011i
$279$	0	0
$280$	−57.9679	−0.207028
$281$	341.669i	1.21591i	0.793973	+	0.607953i	$0.208009\pi$
$281$	341.669i	1.21591i	−0.793973	+	0.607953i	$0.791991\pi$
$282$	0	0
$283$	393.110	1.38908	0.694541	−	0.719453i	$-0.255607\pi$
$283$	393.110	1.38908	0.694541	+	0.719453i	$0.255607\pi$
$284$	− 86.8081i	− 0.305662i
$285$	0	0
$286$	315.686	1.10380
$287$	160.788i	0.560238i
$288$	0	0
$289$	−66.2697	−0.229307
$290$	− 625.731i	− 2.15769i
$291$	0	0
$292$	−72.9540	−0.249842
$293$	461.623i	1.57550i	0.615993	+	0.787752i	$0.288755\pi$
$293$	461.623i	1.57550i	−0.615993	+	0.787752i	$0.711245\pi$
$294$	0	0
$295$	−74.9992	−0.254235
$296$	19.1505i	0.0646977i
$297$	0	0
$298$	299.736	1.00583
$299$	583.545i	1.95165i
$300$	0	0
$301$	−43.1799	−0.143455
$302$	− 267.264i	− 0.884979i
$303$	0	0
$304$	−75.4958	−0.248342
$305$	− 699.338i	− 2.29291i
$306$	0	0
$307$	−401.560	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$307$	−401.560	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$308$	61.1477i	0.198532i
$309$	0	0
$310$	118.293	0.381589
$311$	304.361i	0.978654i	0.872100	+	0.489327i	$0.162758\pi$
$311$	304.361i	0.978654i	−0.872100	+	0.489327i	$0.837242\pi$
$312$	0	0
$313$	588.646	1.88066	0.940329	−	0.340265i	$-0.110517\pi$
$313$	588.646	1.88066	0.940329	+	0.340265i	$0.110517\pi$
$314$	− 1.85653i	− 0.00591250i
$315$	0	0
$316$	90.3301	0.285855
$317$	197.678i	0.623590i	0.950149	+	0.311795i	$0.100930\pi$
$317$	197.678i	0.623590i	−0.950149	+	0.311795i	$0.899070\pi$
$318$	0	0
$319$	−660.056	−2.06914
$320$	61.9702i	0.193657i
$321$	0	0
$322$	−113.031	−0.351029
$323$	− 355.748i	− 1.10139i
$324$	0	0
$325$	−676.188	−2.08058
$326$	264.775i	0.812193i
$327$	0	0
$328$	171.890	0.524054
$329$	− 51.2239i	− 0.155696i
$330$	0	0
$331$	−149.384	−0.451312	−0.225656	−	0.974207i	$-0.572453\pi$
$331$	−149.384	−0.451312	−0.225656	+	0.974207i	$0.572453\pi$
$332$	− 113.447i	− 0.341708i
$333$	0	0
$334$	−150.644	−0.451030
$335$	456.625i	1.36306i
$336$	0	0
$337$	−287.008	−0.851654	−0.425827	−	0.904805i	$-0.640017\pi$
$337$	−287.008	−0.851654	−0.425827	+	0.904805i	$0.640017\pi$
$338$	− 288.706i	− 0.854159i
$339$	0	0
$340$	−292.013	−0.858862
$341$	− 124.782i	− 0.365929i
$342$	0	0
$343$	−18.5203	−0.0539949
$344$	46.1613i	0.134190i
$345$	0	0
$346$	−367.624	−1.06250
$347$	143.660i	0.414005i	0.978340	+	0.207002i	$0.0663708\pi$
$347$	143.660i	0.414005i	−0.978340	+	0.207002i	$0.933629\pi$
$348$	0	0
$349$	−254.201	−0.728370	−0.364185	−	0.931327i	$-0.618652\pi$
$349$	−254.201	−0.728370	−0.364185	+	0.931327i	$0.618652\pi$
$350$	− 130.976i	− 0.374218i
$351$	0	0
$352$	65.3696	0.185709
$353$	64.2134i	0.181908i	0.995855	+	0.0909539i	$0.0289916\pi$
$353$	64.2134i	0.181908i	−0.995855	+	0.0909539i	$0.971008\pi$
$354$	0	0
$355$	336.220	0.947099
$356$	104.635i	0.293917i
$357$	0	0
$358$	−38.2970	−0.106975
$359$	452.646i	1.26085i	0.776250	+	0.630426i	$0.217120\pi$
$359$	452.646i	1.26085i	−0.776250	+	0.630426i	$0.782880\pi$
$360$	0	0
$361$	−4.77384	−0.0132239
$362$	233.206i	0.644215i
$363$	0	0
$364$	102.216	0.280813
$365$	− 282.561i	− 0.774140i
$366$	0	0
$367$	252.250	0.687330	0.343665	−	0.939092i	$-0.388332\pi$
$367$	252.250	0.687330	0.343665	+	0.939092i	$0.388332\pi$
$368$	120.836i	0.328358i
$369$	0	0
$370$	−74.1726	−0.200467
$371$	− 93.7497i	− 0.252695i
$372$	0	0
$373$	−209.064	−0.560494	−0.280247	−	0.959928i	$-0.590416\pi$
$373$	−209.064	−0.560494	−0.280247	+	0.959928i	$0.590416\pi$
$374$	308.032i	0.823614i
$375$	0	0
$376$	−54.7607	−0.145640
$377$	1103.36i	2.92669i
$378$	0	0
$379$	−259.675	−0.685158	−0.342579	−	0.939489i	$-0.611301\pi$
$379$	−259.675	−0.685158	−0.342579	+	0.939489i	$0.611301\pi$
$380$	− 292.406i	− 0.769489i
$381$	0	0
$382$	−9.25841	−0.0242367
$383$	424.127i	1.10738i	0.832723	+	0.553690i	$0.186781\pi$
$383$	424.127i	1.10738i	−0.832723	+	0.553690i	$0.813219\pi$
$384$	0	0
$385$	−236.834	−0.615152
$386$	− 297.702i	− 0.771248i
$387$	0	0
$388$	−28.2839	−0.0728966
$389$	− 667.457i	− 1.71583i	−0.513794	−	0.857913i	$-0.671761\pi$
$389$	− 667.457i	− 1.71583i	0.513794	−	0.857913i	$-0.328239\pi$
$390$	0	0
$391$	−569.395	−1.45625
$392$	19.7990i	0.0505076i
$393$	0	0
$394$	73.4998	0.186548
$395$	349.861i	0.885724i
$396$	0	0
$397$	−146.230	−0.368339	−0.184169	−	0.982895i	$-0.558959\pi$
$397$	−146.230	−0.368339	−0.184169	+	0.982895i	$0.558959\pi$
$398$	530.431i	1.33274i
$399$	0	0
$400$	−140.019	−0.350049
$401$	266.900i	0.665585i	0.943000	+	0.332793i	$0.107991\pi$
$401$	266.900i	0.665585i	−0.943000	+	0.332793i	$0.892009\pi$
$402$	0	0
$403$	−208.588	−0.517588
$404$	− 25.2118i	− 0.0624055i
$405$	0	0
$406$	−213.719	−0.526402
$407$	78.2414i	0.192239i
$408$	0	0
$409$	−97.1741	−0.237589	−0.118795	−	0.992919i	$-0.537903\pi$
$409$	−97.1741	−0.237589	−0.118795	+	0.992919i	$0.537903\pi$
$410$	665.753i	1.62379i
$411$	0	0
$412$	266.597	0.647080
$413$	25.6161i	0.0620244i
$414$	0	0
$415$	439.396	1.05879
$416$	− 109.273i	− 0.262676i
$417$	0	0
$418$	−308.446	−0.737909
$419$	372.552i	0.889146i	0.895743	+	0.444573i	$0.146645\pi$
$419$	372.552i	0.889146i	−0.895743	+	0.444573i	$0.853355\pi$
$420$	0	0
$421$	380.449	0.903679	0.451840	−	0.892099i	$-0.350768\pi$
$421$	380.449	0.903679	0.451840	+	0.892099i	$0.350768\pi$
$422$	49.9357i	0.118331i
$423$	0	0
$424$	−100.223	−0.236374
$425$	− 659.792i	− 1.55245i
$426$	0	0
$427$	−238.860	−0.559390
$428$	16.1537i	0.0377422i
$429$	0	0
$430$	−178.789	−0.415789
$431$	588.231i	1.36480i	0.730977	+	0.682402i	$0.239065\pi$
$431$	588.231i	1.36480i	−0.730977	+	0.682402i	$0.760935\pi$
$432$	0	0
$433$	263.825	0.609295	0.304647	−	0.952465i	$-0.401461\pi$
$433$	263.825	0.609295	0.304647	+	0.952465i	$0.401461\pi$
$434$	− 40.4030i	− 0.0930945i
$435$	0	0
$436$	−320.078	−0.734123
$437$	− 570.161i	− 1.30472i
$438$	0	0
$439$	−438.976	−0.999945	−0.499973	−	0.866041i	$-0.666657\pi$
$439$	−438.976	−0.999945	−0.499973	+	0.866041i	$0.666657\pi$
$440$	253.186i	0.575422i
$441$	0	0
$442$	514.912	1.16496
$443$	− 371.016i	− 0.837509i	−0.908100	−	0.418754i	$-0.862467\pi$
$443$	− 371.016i	− 0.837509i	0.908100	−	0.418754i	$-0.137533\pi$
$444$	0	0
$445$	−405.265	−0.910707
$446$	− 7.32164i	− 0.0164162i
$447$	0	0
$448$	21.1660	0.0472456
$449$	− 134.719i	− 0.300043i	−0.988683	−	0.150021i	$-0.952066\pi$
$449$	− 134.719i	− 0.300043i	0.988683	−	0.150021i	$-0.0479342\pi$
$450$	0	0
$451$	702.273	1.55715
$452$	− 129.536i	− 0.286584i
$453$	0	0
$454$	614.651	1.35386
$455$	395.896i	0.870102i
$456$	0	0
$457$	258.543	0.565739	0.282870	−	0.959158i	$-0.408714\pi$
$457$	258.543	0.565739	0.282870	+	0.959158i	$0.408714\pi$
$458$	− 261.123i	− 0.570138i
$459$	0	0
$460$	−468.013	−1.01742
$461$	− 20.0571i	− 0.0435078i	−0.999763	−	0.0217539i	$-0.993075\pi$
$461$	− 20.0571i	− 0.0435078i	0.999763	−	0.0217539i	$-0.00692503\pi$
$462$	0	0
$463$	622.238	1.34393	0.671963	−	0.740585i	$-0.265451\pi$
$463$	622.238	1.34393	0.671963	+	0.740585i	$0.265451\pi$
$464$	228.475i	0.492404i
$465$	0	0
$466$	−365.866	−0.785120
$467$	433.794i	0.928895i	0.885601	+	0.464448i	$0.153747\pi$
$467$	433.794i	0.928895i	−0.885601	+	0.464448i	$0.846253\pi$
$468$	0	0
$469$	155.961	0.332539
$470$	− 212.096i	− 0.451268i
$471$	0	0
$472$	27.3847	0.0580185
$473$	188.597i	0.398725i
$474$	0	0
$475$	660.680	1.39091
$476$	99.7374i	0.209532i
$477$	0	0
$478$	−448.759	−0.938827
$479$	80.0446i	0.167108i	0.996503	+	0.0835539i	$0.0266271\pi$
$479$	80.0446i	0.167108i	−0.996503	+	0.0835539i	$0.973373\pi$
$480$	0	0
$481$	130.790	0.271913
$482$	389.586i	0.808270i
$483$	0	0
$484$	25.0744	0.0518067
$485$	− 109.547i	− 0.225871i
$486$	0	0
$487$	−658.966	−1.35311	−0.676556	−	0.736391i	$-0.736529\pi$
$487$	−658.966	−1.35311	−0.676556	+	0.736391i	$0.736529\pi$
$488$	255.352i	0.523262i
$489$	0	0
$490$	−76.6843	−0.156498
$491$	− 284.769i	− 0.579977i	−0.957030	−	0.289988i	$-0.906349\pi$
$491$	− 284.769i	− 0.579977i	0.957030	−	0.289988i	$-0.0936514\pi$
$492$	0	0
$493$	−1076.61	−2.18379
$494$	515.605i	1.04373i
$495$	0	0
$496$	−43.1927	−0.0870820
$497$	− 114.836i	− 0.231059i
$498$	0	0
$499$	−638.811	−1.28018	−0.640091	−	0.768299i	$-0.721103\pi$
$499$	−638.811	−1.28018	−0.640091	+	0.768299i	$0.721103\pi$
$500$	− 155.001i	− 0.310002i
$501$	0	0
$502$	118.914	0.236880
$503$	− 202.125i	− 0.401838i	−0.979608	−	0.200919i	$-0.935607\pi$
$503$	− 202.125i	− 0.401838i	0.979608	−	0.200919i	$-0.0643929\pi$
$504$	0	0
$505$	97.6489	0.193364
$506$	493.686i	0.975665i
$507$	0	0
$508$	2.65115	0.00521879
$509$	311.286i	0.611564i	0.952102	+	0.305782i	$0.0989179\pi$
$509$	311.286i	0.611564i	−0.952102	+	0.305782i	$0.901082\pi$
$510$	0	0
$511$	−96.5091	−0.188863
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−11.0210	−0.0214416
$515$	1032.57i	2.00499i
$516$	0	0
$517$	−223.730	−0.432747
$518$	25.3337i	0.0489068i
$519$	0	0
$520$	423.231	0.813906
$521$	− 723.521i	− 1.38872i	−0.719629	−	0.694358i	$-0.755688\pi$
$521$	− 723.521i	− 1.38872i	0.719629	−	0.694358i	$-0.244312\pi$
$522$	0	0
$523$	995.501	1.90344	0.951722	−	0.306961i	$-0.0993121\pi$
$523$	995.501	1.90344	0.951722	+	0.306961i	$0.0993121\pi$
$524$	− 8.98617i	− 0.0171492i
$525$	0	0
$526$	−421.729	−0.801767
$527$	− 203.530i	− 0.386205i
$528$	0	0
$529$	−383.578	−0.725100
$530$	− 388.176i	− 0.732408i
$531$	0	0
$532$	−99.8716	−0.187729
$533$	− 1173.94i	− 2.20251i
$534$	0	0
$535$	−62.5655	−0.116945
$536$	− 166.729i	− 0.311062i
$537$	0	0
$538$	21.6314	0.0402071
$539$	80.8908i	0.150076i
$540$	0	0
$541$	−447.086	−0.826407	−0.413204	−	0.910639i	$-0.635590\pi$
$541$	−447.086	−0.826407	−0.413204	+	0.910639i	$0.635590\pi$
$542$	− 568.420i	− 1.04875i
$543$	0	0
$544$	106.624	0.196000
$545$	− 1239.71i	− 2.27469i
$546$	0	0
$547$	1029.23	1.88159	0.940795	−	0.338977i	$-0.110081\pi$
$547$	1029.23	1.88159	0.940795	+	0.338977i	$0.110081\pi$
$548$	− 437.540i	− 0.798431i
$549$	0	0
$550$	−572.064	−1.04012
$551$	− 1078.06i	− 1.95655i
$552$	0	0
$553$	119.495	0.216086
$554$	474.557i	0.856602i
$555$	0	0
$556$	−174.957	−0.314670
$557$	− 618.337i	− 1.11012i	−0.831810	−	0.555061i	$-0.812695\pi$
$557$	− 618.337i	− 1.11012i	0.831810	−	0.555061i	$-0.187305\pi$
$558$	0	0
$559$	315.262	0.563976
$560$	81.9789i	0.146391i
$561$	0	0
$562$	483.194	0.859775
$563$	6.89457i	0.0122461i	0.999981	+	0.00612306i	$0.00194904\pi$
$563$	6.89457i	0.0122461i	−0.999981	+	0.00612306i	$0.998051\pi$
$564$	0	0
$565$	501.711	0.887984
$566$	− 555.942i	− 0.982230i
$567$	0	0
$568$	−122.765	−0.216136
$569$	627.136i	1.10217i	0.834448	+	0.551086i	$0.185786\pi$
$569$	627.136i	1.10217i	−0.834448	+	0.551086i	$0.814214\pi$
$570$	0	0
$571$	−584.524	−1.02368	−0.511842	−	0.859080i	$-0.671037\pi$
$571$	−584.524	−1.02368	−0.511842	+	0.859080i	$0.671037\pi$
$572$	− 446.448i	− 0.780503i
$573$	0	0
$574$	227.389	0.396148
$575$	− 1057.46i	− 1.83906i
$576$	0	0
$577$	113.503	0.196713	0.0983563	−	0.995151i	$-0.468642\pi$
$577$	113.503	0.196713	0.0983563	+	0.995151i	$0.468642\pi$
$578$	93.7195i	0.162144i
$579$	0	0
$580$	−884.917	−1.52572
$581$	− 150.076i	− 0.258307i
$582$	0	0
$583$	−409.470	−0.702350
$584$	103.173i	0.176665i
$585$	0	0
$586$	652.833	1.11405
$587$	− 131.018i	− 0.223199i	−0.993753	−	0.111600i	$-0.964403\pi$
$587$	− 131.018i	− 0.223199i	0.993753	−	0.111600i	$-0.0355974\pi$
$588$	0	0
$589$	203.804	0.346017
$590$	106.065i	0.179771i
$591$	0	0
$592$	27.0829	0.0457482
$593$	718.635i	1.21186i	0.795517	+	0.605931i	$0.207199\pi$
$593$	718.635i	1.21186i	−0.795517	+	0.605931i	$0.792801\pi$
$594$	0	0
$595$	−386.297	−0.649239
$596$	− 423.891i	− 0.711226i
$597$	0	0
$598$	825.257	1.38003
$599$	− 1056.67i	− 1.76405i	−0.471200	−	0.882027i	$-0.656179\pi$
$599$	− 1056.67i	− 1.76405i	0.471200	−	0.882027i	$-0.343821\pi$
$600$	0	0
$601$	−588.233	−0.978757	−0.489379	−	0.872071i	$-0.662776\pi$
$601$	−588.233	−0.978757	−0.489379	+	0.872071i	$0.662776\pi$
$602$	61.0657i	0.101438i
$603$	0	0
$604$	−377.968	−0.625775
$605$	97.1168i	0.160524i
$606$	0	0
$607$	996.317	1.64138	0.820690	−	0.571374i	$-0.193589\pi$
$607$	996.317	1.64138	0.820690	+	0.571374i	$0.193589\pi$
$608$	106.767i	0.175604i
$609$	0	0
$610$	−989.013	−1.62133
$611$	373.993i	0.612099i
$612$	0	0
$613$	−334.457	−0.545607	−0.272803	−	0.962070i	$-0.587951\pi$
$613$	−334.457	−0.545607	−0.272803	+	0.962070i	$0.587951\pi$
$614$	567.892i	0.924905i
$615$	0	0
$616$	86.4759	0.140383
$617$	− 191.882i	− 0.310991i	−0.987837	−	0.155496i	$-0.950303\pi$
$617$	− 191.882i	− 0.310991i	0.987837	−	0.155496i	$-0.0496975\pi$
$618$	0	0
$619$	874.358	1.41253	0.706267	−	0.707946i	$-0.250378\pi$
$619$	874.358	1.41253	0.706267	+	0.707946i	$0.250378\pi$
$620$	− 167.291i	− 0.269824i
$621$	0	0
$622$	430.432	0.692013
$623$	138.419i	0.222181i
$624$	0	0
$625$	−274.782	−0.439651
$626$	− 832.471i	− 1.32983i
$627$	0	0
$628$	−2.62552	−0.00418077
$629$	127.619i	0.202891i
$630$	0	0
$631$	515.900	0.817592	0.408796	−	0.912626i	$-0.365949\pi$
$631$	515.900	0.817592	0.408796	+	0.912626i	$0.365949\pi$
$632$	− 127.746i	− 0.202130i
$633$	0	0
$634$	279.559	0.440944
$635$	10.2683i	0.0161705i
$636$	0	0
$637$	135.219	0.212275
$638$	933.459i	1.46310i
$639$	0	0
$640$	87.6392	0.136936
$641$	605.857i	0.945175i	0.881284	+	0.472588i	$0.156680\pi$
$641$	605.857i	0.945175i	−0.881284	+	0.472588i	$0.843320\pi$
$642$	0	0
$643$	218.570	0.339922	0.169961	−	0.985451i	$-0.445636\pi$
$643$	218.570	0.339922	0.169961	+	0.985451i	$0.445636\pi$
$644$	159.850i	0.248215i
$645$	0	0
$646$	−503.103	−0.778797
$647$	− 500.886i	− 0.774167i	−0.922045	−	0.387084i	$-0.873482\pi$
$647$	− 500.886i	− 0.774167i	0.922045	−	0.387084i	$-0.126518\pi$
$648$	0	0
$649$	111.883	0.172393
$650$	956.274i	1.47119i
$651$	0	0
$652$	374.448	0.574307
$653$	− 848.574i	− 1.29950i	−0.760148	−	0.649750i	$-0.774873\pi$
$653$	− 848.574i	− 1.29950i	0.760148	−	0.649750i	$-0.225127\pi$
$654$	0	0
$655$	34.8047	0.0531369
$656$	− 243.089i	− 0.370562i
$657$	0	0
$658$	−72.4416	−0.110094
$659$	270.621i	0.410654i	0.978693	+	0.205327i	$0.0658258\pi$
$659$	270.621i	0.410654i	−0.978693	+	0.205327i	$0.934174\pi$
$660$	0	0
$661$	−952.799	−1.44145	−0.720726	−	0.693221i	$-0.756191\pi$
$661$	−952.799	−1.44145	−0.720726	+	0.693221i	$0.756191\pi$
$662$	211.261i	0.319126i
$663$	0	0
$664$	−160.438	−0.241624
$665$	− 386.817i	− 0.581679i
$666$	0	0
$667$	−1725.50	−2.58695
$668$	213.043i	0.318926i
$669$	0	0
$670$	645.766	0.963830
$671$	1043.27i	1.55479i
$672$	0	0
$673$	69.8556	0.103797	0.0518987	−	0.998652i	$-0.483473\pi$
$673$	69.8556	0.103797	0.0518987	+	0.998652i	$0.483473\pi$
$674$	405.890i	0.602211i
$675$	0	0
$676$	−408.292	−0.603982
$677$	150.717i	0.222625i	0.993785	+	0.111312i	$0.0355054\pi$
$677$	150.717i	0.222625i	−0.993785	+	0.111312i	$0.964495\pi$
$678$	0	0
$679$	−37.4160	−0.0551046
$680$	412.969i	0.607307i
$681$	0	0
$682$	−176.468	−0.258751
$683$	− 87.2405i	− 0.127731i	−0.997959	−	0.0638656i	$-0.979657\pi$
$683$	− 87.2405i	− 0.127731i	0.997959	−	0.0638656i	$-0.0203429\pi$
$684$	0	0
$685$	1694.65	2.47395
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	65.2819	0.0948865
$689$	684.479i	0.993438i
$690$	0	0
$691$	699.912	1.01290	0.506449	−	0.862270i	$-0.330958\pi$
$691$	699.912	1.01290	0.506449	+	0.862270i	$0.330958\pi$
$692$	519.899i	0.751300i
$693$	0	0
$694$	203.165	0.292746
$695$	− 677.632i	− 0.975010i
$696$	0	0
$697$	1145.47	1.64343
$698$	359.495i	0.515035i
$699$	0	0
$700$	−185.228	−0.264612
$701$	− 760.013i	− 1.08418i	−0.840319	−	0.542092i	$-0.817632\pi$
$701$	− 760.013i	− 1.08418i	0.840319	−	0.542092i	$-0.182368\pi$
$702$	0	0
$703$	−127.790	−0.181779
$704$	− 92.4466i	− 0.131316i
$705$	0	0
$706$	90.8115	0.128628
$707$	− 33.3521i	− 0.0471741i
$708$	0	0
$709$	494.571	0.697561	0.348780	−	0.937204i	$-0.386596\pi$
$709$	494.571	0.697561	0.348780	+	0.937204i	$0.386596\pi$
$710$	− 475.487i	− 0.669700i
$711$	0	0
$712$	147.976	0.207831
$713$	− 326.201i	− 0.457504i
$714$	0	0
$715$	1729.15	2.41840
$716$	54.1601i	0.0756426i
$717$	0	0
$718$	640.137	0.891556
$719$	1035.58i	1.44031i	0.693812	+	0.720156i	$0.255930\pi$
$719$	1035.58i	1.44031i	−0.693812	+	0.720156i	$0.744070\pi$
$720$	0	0
$721$	352.675	0.489147
$722$	6.75123i	0.00935074i
$723$	0	0
$724$	329.803	0.455529
$725$	− 1999.44i	− 2.75784i
$726$	0	0
$727$	−1144.96	−1.57491	−0.787455	−	0.616372i	$-0.788602\pi$
$727$	−1144.96	−1.57491	−0.787455	+	0.616372i	$0.788602\pi$
$728$	− 144.555i	− 0.198565i
$729$	0	0
$730$	−399.602	−0.547399
$731$	307.618i	0.420818i
$732$	0	0
$733$	1251.72	1.70767	0.853834	−	0.520545i	$-0.174271\pi$
$733$	1251.72	1.70767	0.853834	+	0.520545i	$0.174271\pi$
$734$	− 356.735i	− 0.486016i
$735$	0	0
$736$	170.887	0.232184
$737$	− 681.190i	− 0.924274i
$738$	0	0
$739$	−490.046	−0.663121	−0.331560	−	0.943434i	$-0.607575\pi$
$739$	−490.046	−0.663121	−0.331560	+	0.943434i	$0.607575\pi$
$740$	104.896i	0.141751i
$741$	0	0
$742$	−132.582	−0.178682
$743$	− 760.621i	− 1.02372i	−0.859070	−	0.511858i	$-0.828957\pi$
$743$	− 760.621i	− 1.02372i	0.859070	−	0.511858i	$-0.171043\pi$
$744$	0	0
$745$	1641.79	2.20374
$746$	295.662i	0.396329i
$747$	0	0
$748$	435.622	0.582383
$749$	21.3693i	0.0285305i
$750$	0	0
$751$	1321.81	1.76007	0.880037	−	0.474906i	$-0.157518\pi$
$751$	1321.81	1.76007	0.880037	+	0.474906i	$0.157518\pi$
$752$	77.4433i	0.102983i
$753$	0	0
$754$	1560.39	2.06948
$755$	− 1463.92i	− 1.93897i
$756$	0	0
$757$	−465.423	−0.614825	−0.307413	−	0.951576i	$-0.599463\pi$
$757$	−465.423	−0.614825	−0.307413	+	0.951576i	$0.599463\pi$
$758$	367.236i	0.484480i
$759$	0	0
$760$	−413.524	−0.544111
$761$	− 255.654i	− 0.335945i	−0.985792	−	0.167972i	$-0.946278\pi$
$761$	− 255.654i	− 0.335945i	0.985792	−	0.167972i	$-0.0537219\pi$
$762$	0	0
$763$	−423.423	−0.554945
$764$	13.0934i	0.0171379i
$765$	0	0
$766$	599.806	0.783036
$767$	− 187.026i	− 0.243842i
$768$	0	0
$769$	721.799	0.938621	0.469310	−	0.883033i	$-0.344503\pi$
$769$	721.799	0.938621	0.469310	+	0.883033i	$0.344503\pi$
$770$	334.933i	0.434978i
$771$	0	0
$772$	−421.014	−0.545355
$773$	910.437i	1.17780i	0.808207	+	0.588898i	$0.200438\pi$
$773$	910.437i	1.17780i	−0.808207	+	0.588898i	$0.799562\pi$
$774$	0	0
$775$	377.988	0.487727
$776$	39.9994i	0.0515456i
$777$	0	0
$778$	−943.926	−1.21327
$779$	1147.01i	1.47242i
$780$	0	0
$781$	−501.570	−0.642215
$782$	805.247i	1.02973i
$783$	0	0
$784$	28.0000	0.0357143
$785$	− 10.1690i	− 0.0129542i
$786$	0	0
$787$	69.0850	0.0877827	0.0438914	−	0.999036i	$-0.486024\pi$
$787$	69.0850	0.0877827	0.0438914	+	0.999036i	$0.486024\pi$
$788$	− 103.944i	− 0.131909i
$789$	0	0
$790$	494.778	0.626301
$791$	− 171.360i	− 0.216637i
$792$	0	0
$793$	1743.95	2.19918
$794$	206.801i	0.260455i
$795$	0	0
$796$	750.142	0.942389
$797$	146.955i	0.184385i	0.995741	+	0.0921926i	$0.0293875\pi$
$797$	146.955i	0.184385i	−0.995741	+	0.0921926i	$0.970612\pi$
$798$	0	0
$799$	−364.924	−0.456726
$800$	198.017i	0.247522i
$801$	0	0
$802$	377.453	0.470640
$803$	421.522i	0.524934i
$804$	0	0
$805$	−619.123	−0.769097
$806$	294.988i	0.365990i
$807$	0	0
$808$	−35.6549	−0.0441274
$809$	814.298i	1.00655i	0.864127	+	0.503274i	$0.167872\pi$
$809$	814.298i	1.00655i	−0.864127	+	0.503274i	$0.832128\pi$
$810$	0	0
$811$	296.389	0.365461	0.182731	−	0.983163i	$-0.441506\pi$
$811$	296.389	0.365461	0.182731	+	0.983163i	$0.441506\pi$
$812$	302.244i	0.372222i
$813$	0	0
$814$	110.650	0.135934
$815$	1450.29i	1.77950i
$816$	0	0
$817$	−308.032	−0.377028
$818$	137.425i	0.168001i
$819$	0	0
$820$	941.517	1.14819
$821$	− 220.314i	− 0.268348i	−0.990958	−	0.134174i	$-0.957162\pi$
$821$	− 220.314i	− 0.268348i	0.990958	−	0.134174i	$-0.0428381\pi$
$822$	0	0
$823$	121.059	0.147095	0.0735476	−	0.997292i	$-0.476568\pi$
$823$	121.059	0.147095	0.0735476	+	0.997292i	$0.476568\pi$
$824$	− 377.025i	− 0.457555i
$825$	0	0
$826$	36.2266	0.0438579
$827$	792.775i	0.958615i	0.877647	+	0.479308i	$0.159112\pi$
$827$	792.775i	0.958615i	−0.877647	+	0.479308i	$0.840888\pi$
$828$	0	0
$829$	361.158	0.435655	0.217828	−	0.975987i	$-0.430103\pi$
$829$	361.158	0.435655	0.217828	+	0.975987i	$0.430103\pi$
$830$	− 621.400i	− 0.748675i
$831$	0	0
$832$	−154.536	−0.185740
$833$	131.940i	0.158392i
$834$	0	0
$835$	−825.145	−0.988197
$836$	436.208i	0.521780i
$837$	0	0
$838$	526.869	0.628721
$839$	74.4834i	0.0887764i	0.999014	+	0.0443882i	$0.0141339\pi$
$839$	74.4834i	0.0887764i	−0.999014	+	0.0443882i	$0.985866\pi$
$840$	0	0
$841$	−2421.56	−2.87938
$842$	− 538.036i	− 0.638998i
$843$	0	0
$844$	70.6197	0.0836726
$845$	− 1581.37i	− 1.87144i
$846$	0	0
$847$	33.1704	0.0391622
$848$	141.736i	0.167142i
$849$	0	0
$850$	−933.087	−1.09775
$851$	204.536i	0.240348i
$852$	0	0
$853$	547.490	0.641840	0.320920	−	0.947106i	$-0.396008\pi$
$853$	547.490	0.641840	0.320920	+	0.947106i	$0.396008\pi$
$854$	337.799i	0.395549i
$855$	0	0
$856$	22.8448	0.0266878
$857$	− 199.118i	− 0.232343i	−0.993229	−	0.116172i	$-0.962938\pi$
$857$	− 199.118i	− 0.232343i	0.993229	−	0.116172i	$-0.0370623\pi$
$858$	0	0
$859$	−772.474	−0.899272	−0.449636	−	0.893212i	$-0.648446\pi$
$859$	−772.474	−0.899272	−0.449636	+	0.893212i	$0.648446\pi$
$860$	252.846i	0.294007i
$861$	0	0
$862$	831.884	0.965063
$863$	731.096i	0.847157i	0.905859	+	0.423578i	$0.139226\pi$
$863$	731.096i	0.847157i	−0.905859	+	0.423578i	$0.860774\pi$
$864$	0	0
$865$	−2013.64	−2.32791
$866$	− 373.104i	− 0.430836i
$867$	0	0
$868$	−57.1385	−0.0658278
$869$	− 521.919i	− 0.600598i
$870$	0	0
$871$	−1138.69	−1.30734
$872$	452.658i	0.519103i
$873$	0	0
$874$	−806.330	−0.922574
$875$	− 205.047i	− 0.234339i
$876$	0	0
$877$	−520.430	−0.593421	−0.296710	−	0.954968i	$-0.595890\pi$
$877$	−520.430	−0.593421	−0.296710	+	0.954968i	$0.595890\pi$
$878$	620.806i	0.707068i
$879$	0	0
$880$	358.059	0.406885
$881$	− 667.717i	− 0.757908i	−0.925415	−	0.378954i	$-0.876284\pi$
$881$	− 667.717i	− 0.757908i	0.925415	−	0.378954i	$-0.123716\pi$
$882$	0	0
$883$	−133.934	−0.151680	−0.0758402	−	0.997120i	$-0.524164\pi$
$883$	−133.934	−0.151680	−0.0758402	+	0.997120i	$0.524164\pi$
$884$	− 728.196i	− 0.823751i
$885$	0	0
$886$	−524.696	−0.592208
$887$	− 822.425i	− 0.927198i	−0.886045	−	0.463599i	$-0.846558\pi$
$887$	− 822.425i	− 0.927198i	0.886045	−	0.463599i	$-0.153442\pi$
$888$	0	0
$889$	3.50714	0.00394504
$890$	573.131i	0.643967i
$891$	0	0
$892$	−10.3544	−0.0116080
$893$	− 365.415i	− 0.409200i
$894$	0	0
$895$	−209.770	−0.234380
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	−190.522	−0.212162
$899$	− 616.778i	− 0.686072i
$900$	0	0
$901$	−667.882	−0.741268
$902$	− 993.165i	− 1.10107i
$903$	0	0
$904$	−183.192	−0.202646
$905$	1277.37i	1.41146i
$906$	0	0
$907$	1082.24	1.19321	0.596606	−	0.802534i	$-0.296516\pi$
$907$	1082.24	1.19321	0.596606	+	0.802534i	$0.296516\pi$
$908$	− 869.247i	− 0.957321i
$909$	0	0
$910$	559.882	0.615255
$911$	− 711.803i	− 0.781342i	−0.920530	−	0.390671i	$-0.872243\pi$
$911$	− 711.803i	− 0.781342i	0.920530	−	0.390671i	$-0.127757\pi$
$912$	0	0
$913$	−655.487	−0.717949
$914$	− 365.635i	− 0.400038i
$915$	0	0
$916$	−369.284	−0.403148
$917$	− 11.8876i	− 0.0129636i
$918$	0	0
$919$	1577.09	1.71609	0.858045	−	0.513575i	$-0.171679\pi$
$919$	1577.09	1.71609	0.858045	+	0.513575i	$0.171679\pi$
$920$	661.871i	0.719425i
$921$	0	0
$922$	−28.3650	−0.0307647
$923$	838.436i	0.908381i
$924$	0	0
$925$	−237.008	−0.256225
$926$	− 879.977i	− 0.950299i
$927$	0	0
$928$	323.113	0.348182
$929$	801.283i	0.862522i	0.902227	+	0.431261i	$0.141931\pi$
$929$	801.283i	0.862522i	−0.902227	+	0.431261i	$0.858069\pi$
$930$	0	0
$931$	−132.118	−0.141909
$932$	517.413i	0.555164i
$933$	0	0
$934$	613.478	0.656828
$935$	1687.23i	1.80452i
$936$	0	0
$937$	−776.731	−0.828955	−0.414478	−	0.910060i	$-0.636036\pi$
$937$	−776.731	−0.828955	−0.414478	+	0.910060i	$0.636036\pi$
$938$	− 220.562i	− 0.235141i
$939$	0	0
$940$	−299.949	−0.319094
$941$	246.742i	0.262212i	0.991368	+	0.131106i	$0.0418529\pi$
$941$	246.742i	0.262212i	−0.991368	+	0.131106i	$0.958147\pi$
$942$	0	0
$943$	1835.86	1.94683
$944$	− 38.7279i	− 0.0410253i
$945$	0	0
$946$	266.716	0.281941
$947$	120.513i	0.127258i	0.997974	+	0.0636288i	$0.0202674\pi$
$947$	120.513i	0.127258i	−0.997974	+	0.0636288i	$0.979733\pi$
$948$	0	0
$949$	704.625	0.742493
$950$	− 934.343i	− 0.983518i
$951$	0	0
$952$	141.050	0.148162
$953$	− 1774.27i	− 1.86177i	−0.365306	−	0.930887i	$-0.619036\pi$
$953$	− 1774.27i	− 1.86177i	0.365306	−	0.930887i	$-0.380964\pi$
$954$	0	0
$955$	−50.7124	−0.0531020
$956$	634.641i	0.663851i
$957$	0	0
$958$	113.200	0.118163
$959$	− 578.811i	− 0.603557i
$960$	0	0
$961$	−844.400	−0.878668
$962$	− 184.965i	− 0.192271i
$963$	0	0
$964$	550.958	0.571533
$965$	− 1630.65i	− 1.68979i
$966$	0	0
$967$	1359.41	1.40580	0.702899	−	0.711289i	$-0.251888\pi$
$967$	1359.41	1.40580	0.702899	+	0.711289i	$0.251888\pi$
$968$	− 35.4606i	− 0.0366329i
$969$	0	0
$970$	−154.923	−0.159715
$971$	− 535.351i	− 0.551340i	−0.961252	−	0.275670i	$-0.911100\pi$
$971$	− 535.351i	− 0.551340i	0.961252	−	0.275670i	$-0.0888997\pi$
$972$	0	0
$973$	−231.446	−0.237869
$974$	931.919i	0.956795i
$975$	0	0
$976$	361.122	0.370002
$977$	− 940.284i	− 0.962420i	−0.876605	−	0.481210i	$-0.840197\pi$
$977$	− 940.284i	− 0.962420i	0.876605	−	0.481210i	$-0.159803\pi$
$978$	0	0
$979$	604.570	0.617538
$980$	108.448i	0.110661i
$981$	0	0
$982$	−402.724	−0.410106
$983$	443.512i	0.451182i	0.974222	+	0.225591i	$0.0724313\pi$
$983$	443.512i	0.451182i	−0.974222	+	0.225591i	$0.927569\pi$
$984$	0	0
$985$	402.591	0.408722
$986$	1522.56i	1.54417i
$987$	0	0
$988$	729.176	0.738032
$989$	493.024i	0.498507i
$990$	0	0
$991$	1030.06	1.03941	0.519707	−	0.854344i	$-0.326041\pi$
$991$	1030.06	1.03941	0.519707	+	0.854344i	$0.326041\pi$
$992$	61.0836i	0.0615762i
$993$	0	0
$994$	−162.403	−0.163383
$995$	2905.40i	2.92000i
$996$	0	0
$997$	−1041.76	−1.04489	−0.522446	−	0.852672i	$-0.674980\pi$
$997$	−1041.76	−1.04489	−0.522446	+	0.852672i	$0.674980\pi$
$998$	903.415i	0.905225i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1134.3.b.c.323.1		24
3.2	odd	2	inner	1134.3.b.c.323.24		24
9.2	odd	6		378.3.q.a.71.7		24
9.4	even	3		378.3.q.a.197.7		24
9.5	odd	6		126.3.q.a.29.3	✓	24
9.7	even	3		126.3.q.a.113.3	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.q.a.29.3	✓	24	9.5	odd	6
126.3.q.a.113.3	yes	24	9.7	even	3
378.3.q.a.71.7		24	9.2	odd	6
378.3.q.a.197.7		24	9.4	even	3
1134.3.b.c.323.1		24	1.1	even	1	trivial
1134.3.b.c.323.24		24	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
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)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -7.74628i q^{5} -2.64575 q^{7} +2.82843i q^{8} -10.9549 q^{10} +11.5558i q^{11} +19.3170 q^{13} +3.74166i q^{14} +4.00000 q^{16} +18.8486i q^{17} -18.8740 q^{19} +15.4926i q^{20} +16.3424 q^{22} +30.2089i q^{23} -35.0049 q^{25} -27.3183i q^{26} +5.29150 q^{28} +57.1188i q^{29} -10.7982 q^{31} -5.65685i q^{32} +26.6559 q^{34} +20.4947i q^{35} +6.77073 q^{37} +26.6918i q^{38} +21.9098 q^{40} -60.7722i q^{41} +16.3205 q^{43} -23.1117i q^{44} +42.7218 q^{46} +19.3608i q^{47} +7.00000 q^{49} +49.5043i q^{50} -38.6340 q^{52} +35.4340i q^{53} +89.5147 q^{55} -7.48331i q^{56} +80.7782 q^{58} -9.68197i q^{59} +90.2805 q^{61} +15.2709i q^{62} -8.00000 q^{64} -149.635i q^{65} -58.9477 q^{67} -37.6972i q^{68} +28.9839 q^{70} +43.4041i q^{71} +36.4770 q^{73} -9.57526i q^{74} +37.7479 q^{76} -30.5739i q^{77} -45.1650 q^{79} -30.9851i q^{80} -85.9449 q^{82} +56.7235i q^{83} +146.007 q^{85} -23.0806i q^{86} -32.6848 q^{88} -52.3173i q^{89} -51.1079 q^{91} -60.4178i q^{92} +27.3803 q^{94} +146.203i q^{95} +14.1419 q^{97} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 48 q^{4} + 96 q^{16} + 24 q^{19} - 48 q^{22} - 144 q^{25} + 120 q^{31} + 96 q^{34} - 168 q^{37} - 120 q^{43} + 168 q^{49} + 264 q^{55} - 192 q^{64} - 144 q^{67} + 24 q^{73} - 48 q^{76} - 24 q^{79}+ \cdots - 96 q^{97}+O(q^{100}) \) 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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