LMFDB - Embedded newform 1134.3.b.c.323.21

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			323.21
Character	$\chi$	$=$	1134.323
Dual form			1134.3.b.c.323.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.41421i q^{2} -2.00000 q^{4} +4.16679i q^{5} +2.64575 q^{7} -2.82843i q^{8} -5.89274 q^{10} -6.47398i q^{11} -14.7202 q^{13} +3.74166i q^{14} +4.00000 q^{16} -18.6466i q^{17} +15.6987 q^{19} -8.33359i q^{20} +9.15559 q^{22} -19.7621i q^{23} +7.63784 q^{25} -20.8175i q^{26} -5.29150 q^{28} -10.7471i q^{29} -22.3015 q^{31} +5.65685i q^{32} +26.3702 q^{34} +11.0243i q^{35} +70.8586 q^{37} +22.2013i q^{38} +11.7855 q^{40} +8.41935i q^{41} -49.8973 q^{43} +12.9480i q^{44} +27.9479 q^{46} -72.1830i q^{47} +7.00000 q^{49} +10.8015i q^{50} +29.4404 q^{52} -80.3093i q^{53} +26.9757 q^{55} -7.48331i q^{56} +15.1987 q^{58} +26.4245i q^{59} -65.5805 q^{61} -31.5391i q^{62} -8.00000 q^{64} -61.3360i q^{65} +23.8077 q^{67} +37.2931i q^{68} -15.5907 q^{70} +111.926i q^{71} +13.3109 q^{73} +100.209i q^{74} -31.3973 q^{76} -17.1285i q^{77} +119.751 q^{79} +16.6672i q^{80} -11.9068 q^{82} +11.6297i q^{83} +77.6964 q^{85} -70.5654i q^{86} -18.3112 q^{88} -83.5516i q^{89} -38.9460 q^{91} +39.5242i q^{92} +102.082 q^{94} +65.4131i q^{95} +168.123 q^{97} +9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	4.16679i	0.833359i	0.909054	+	0.416679i	$0.136806\pi$
$5$	4.16679i	0.833359i	−0.909054	+	0.416679i	$0.863194\pi$
$6$	0	0
$7$	2.64575	0.377964
$7$	2.64575	0.377964
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	−5.89274	−0.589274
$11$	− 6.47398i	− 0.588544i	−0.955722	−	0.294272i	$-0.904923\pi$
$11$	− 6.47398i	− 0.588544i	0.955722	−	0.294272i	$-0.0950771\pi$
$12$	0	0
$13$	−14.7202	−1.13232	−0.566162	−	0.824294i	$-0.691572\pi$
$13$	−14.7202	−1.13232	−0.566162	+	0.824294i	$0.691572\pi$
$14$	3.74166i	0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 18.6466i	− 1.09686i	−0.836197	−	0.548428i	$-0.815226\pi$
$17$	− 18.6466i	− 1.09686i	0.836197	−	0.548428i	$-0.184774\pi$
$18$	0	0
$19$	15.6987	0.826245	0.413123	−	0.910675i	$-0.364438\pi$
$19$	15.6987	0.826245	0.413123	+	0.910675i	$0.364438\pi$
$20$	− 8.33359i	− 0.416679i
$21$	0	0
$22$	9.15559	0.416163
$23$	− 19.7621i	− 0.859223i	−0.903014	−	0.429611i	$-0.858651\pi$
$23$	− 19.7621i	− 0.859223i	0.903014	−	0.429611i	$-0.141349\pi$
$24$	0	0
$25$	7.63784	0.305513
$26$	− 20.8175i	− 0.800674i
$27$	0	0
$28$	−5.29150	−0.188982
$29$	− 10.7471i	− 0.370589i	−0.982683	−	0.185294i	$-0.940676\pi$
$29$	− 10.7471i	− 0.370589i	0.982683	−	0.185294i	$-0.0593239\pi$
$30$	0	0
$31$	−22.3015	−0.719403	−0.359702	−	0.933067i	$-0.617122\pi$
$31$	−22.3015	−0.719403	−0.359702	+	0.933067i	$0.617122\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	26.3702	0.775595
$35$	11.0243i	0.314980i
$36$	0	0
$37$	70.8586	1.91510	0.957548	−	0.288273i	$-0.0930810\pi$
$37$	70.8586	1.91510	0.957548	+	0.288273i	$0.0930810\pi$
$38$	22.2013i	0.584244i
$39$	0	0
$40$	11.7855	0.294637
$41$	8.41935i	0.205350i	0.994715	+	0.102675i	$0.0327402\pi$
$41$	8.41935i	0.205350i	−0.994715	+	0.102675i	$0.967260\pi$
$42$	0	0
$43$	−49.8973	−1.16040	−0.580201	−	0.814474i	$-0.697026\pi$
$43$	−49.8973	−1.16040	−0.580201	+	0.814474i	$0.697026\pi$
$44$	12.9480i	0.294272i
$45$	0	0
$46$	27.9479	0.607562
$47$	− 72.1830i	− 1.53581i	−0.640564	−	0.767904i	$-0.721300\pi$
$47$	− 72.1830i	− 1.53581i	0.640564	−	0.767904i	$-0.278700\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	10.8015i	0.216031i
$51$	0	0
$52$	29.4404	0.566162
$53$	− 80.3093i	− 1.51527i	−0.652679	−	0.757635i	$-0.726355\pi$
$53$	− 80.3093i	− 1.51527i	0.652679	−	0.757635i	$-0.273645\pi$
$54$	0	0
$55$	26.9757	0.490468
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	15.1987	0.262046
$59$	26.4245i	0.447872i	0.974604	+	0.223936i	$0.0718907\pi$
$59$	26.4245i	0.447872i	−0.974604	+	0.223936i	$0.928109\pi$
$60$	0	0
$61$	−65.5805	−1.07509	−0.537545	−	0.843235i	$-0.680648\pi$
$61$	−65.5805	−1.07509	−0.537545	+	0.843235i	$0.680648\pi$
$62$	− 31.5391i	− 0.508695i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 61.3360i	− 0.943631i
$66$	0	0
$67$	23.8077	0.355338	0.177669	−	0.984090i	$-0.443144\pi$
$67$	23.8077	0.355338	0.177669	+	0.984090i	$0.443144\pi$
$68$	37.2931i	0.548428i
$69$	0	0
$70$	−15.5907	−0.222724
$71$	111.926i	1.57643i	0.615403	+	0.788213i	$0.288993\pi$
$71$	111.926i	1.57643i	−0.615403	+	0.788213i	$0.711007\pi$
$72$	0	0
$73$	13.3109	0.182342	0.0911709	−	0.995835i	$-0.470939\pi$
$73$	13.3109	0.182342	0.0911709	+	0.995835i	$0.470939\pi$
$74$	100.209i	1.35418i
$75$	0	0
$76$	−31.3973	−0.413123
$77$	− 17.1285i	− 0.222449i
$78$	0	0
$79$	119.751	1.51583	0.757917	−	0.652351i	$-0.226217\pi$
$79$	119.751	1.51583	0.757917	+	0.652351i	$0.226217\pi$
$80$	16.6672i	0.208340i
$81$	0	0
$82$	−11.9068	−0.145204
$83$	11.6297i	0.140117i	0.997543	+	0.0700586i	$0.0223186\pi$
$83$	11.6297i	0.140117i	−0.997543	+	0.0700586i	$0.977681\pi$
$84$	0	0
$85$	77.6964	0.914075
$86$	− 70.5654i	− 0.820528i
$87$	0	0
$88$	−18.3112	−0.208082
$89$	− 83.5516i	− 0.938782i	−0.882990	−	0.469391i	$-0.844473\pi$
$89$	− 83.5516i	− 0.938782i	0.882990	−	0.469391i	$-0.155527\pi$
$90$	0	0
$91$	−38.9460	−0.427978
$92$	39.5242i	0.429611i
$93$	0	0
$94$	102.082	1.08598
$95$	65.4131i	0.688559i
$96$	0	0
$97$	168.123	1.73323	0.866613	−	0.498981i	$-0.166292\pi$
$97$	168.123	1.73323	0.866613	+	0.498981i	$0.166292\pi$
$98$	9.89949i	0.101015i
$99$	0	0
$100$	−15.2757	−0.152757
$101$	− 22.0497i	− 0.218313i	−0.994025	−	0.109157i	$-0.965185\pi$
$101$	− 22.0497i	− 0.218313i	0.994025	−	0.109157i	$-0.0348150\pi$
$102$	0	0
$103$	126.512	1.22828	0.614138	−	0.789199i	$-0.289504\pi$
$103$	126.512	1.22828	0.614138	+	0.789199i	$0.289504\pi$
$104$	41.6350i	0.400337i
$105$	0	0
$106$	113.574	1.07146
$107$	39.7026i	0.371053i	0.982639	+	0.185526i	$0.0593990\pi$
$107$	39.7026i	0.371053i	−0.982639	+	0.185526i	$0.940601\pi$
$108$	0	0
$109$	93.2221	0.855249	0.427624	−	0.903957i	$-0.359351\pi$
$109$	93.2221	0.855249	0.427624	+	0.903957i	$0.359351\pi$
$110$	38.1495i	0.346813i
$111$	0	0
$112$	10.5830	0.0944911
$113$	− 80.4321i	− 0.711788i	−0.934526	−	0.355894i	$-0.884176\pi$
$113$	− 80.4321i	− 0.711788i	0.934526	−	0.355894i	$-0.115824\pi$
$114$	0	0
$115$	82.3447	0.716041
$116$	21.4941i	0.185294i
$117$	0	0
$118$	−37.3698	−0.316693
$119$	− 49.3342i	− 0.414573i
$120$	0	0
$121$	79.0876	0.653616
$122$	− 92.7448i	− 0.760203i
$123$	0	0
$124$	44.6030	0.359702
$125$	135.995i	1.08796i
$126$	0	0
$127$	−22.5391	−0.177473	−0.0887367	−	0.996055i	$-0.528283\pi$
$127$	−22.5391	−0.177473	−0.0887367	+	0.996055i	$0.528283\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	86.7423	0.667248
$131$	− 237.078i	− 1.80976i	−0.425672	−	0.904878i	$-0.639962\pi$
$131$	− 237.078i	− 1.80976i	0.425672	−	0.904878i	$-0.360038\pi$
$132$	0	0
$133$	41.5348	0.312291
$134$	33.6691i	0.251262i
$135$	0	0
$136$	−52.7405	−0.387798
$137$	193.993i	1.41601i	0.706209	+	0.708004i	$0.250404\pi$
$137$	193.993i	1.41601i	−0.706209	+	0.708004i	$0.749596\pi$
$138$	0	0
$139$	−158.923	−1.14333	−0.571667	−	0.820486i	$-0.693703\pi$
$139$	−158.923	−1.14333	−0.571667	+	0.820486i	$0.693703\pi$
$140$	− 22.0486i	− 0.157490i
$141$	0	0
$142$	−158.288	−1.11470
$143$	95.2983i	0.666422i
$144$	0	0
$145$	44.7808	0.308833
$146$	18.8245i	0.128935i
$147$	0	0
$148$	−141.717	−0.957548
$149$	− 101.858i	− 0.683612i	−0.939770	−	0.341806i	$-0.888961\pi$
$149$	− 101.858i	− 0.683612i	0.939770	−	0.341806i	$-0.111039\pi$
$150$	0	0
$151$	−140.462	−0.930213	−0.465106	−	0.885255i	$-0.653984\pi$
$151$	−140.462	−0.930213	−0.465106	+	0.885255i	$0.653984\pi$
$152$	− 44.4025i	− 0.292122i
$153$	0	0
$154$	24.2234	0.157295
$155$	− 92.9257i	− 0.599521i
$156$	0	0
$157$	248.908	1.58540	0.792700	−	0.609612i	$-0.208675\pi$
$157$	248.908	1.58540	0.792700	+	0.609612i	$0.208675\pi$
$158$	169.353i	1.07186i
$159$	0	0
$160$	−23.5709	−0.147318
$161$	− 52.2857i	− 0.324756i
$162$	0	0
$163$	168.569	1.03417	0.517083	−	0.855935i	$-0.327018\pi$
$163$	168.569	1.03417	0.517083	+	0.855935i	$0.327018\pi$
$164$	− 16.8387i	− 0.102675i
$165$	0	0
$166$	−16.4469	−0.0990778
$167$	− 291.087i	− 1.74304i	−0.490362	−	0.871519i	$-0.663135\pi$
$167$	− 291.087i	− 1.74304i	0.490362	−	0.871519i	$-0.336865\pi$
$168$	0	0
$169$	47.6844	0.282156
$170$	109.879i	0.646349i
$171$	0	0
$172$	99.7945	0.580201
$173$	138.322i	0.799552i	0.916613	+	0.399776i	$0.130912\pi$
$173$	138.322i	0.799552i	−0.916613	+	0.399776i	$0.869088\pi$
$174$	0	0
$175$	20.2078	0.115473
$176$	− 25.8959i	− 0.147136i
$177$	0	0
$178$	118.160	0.663819
$179$	− 304.532i	− 1.70130i	−0.525735	−	0.850649i	$-0.676210\pi$
$179$	− 304.532i	− 1.70130i	0.525735	−	0.850649i	$-0.323790\pi$
$180$	0	0
$181$	−260.720	−1.44044	−0.720221	−	0.693745i	$-0.755960\pi$
$181$	−260.720	−1.44044	−0.720221	+	0.693745i	$0.755960\pi$
$182$	− 55.0780i	− 0.302626i
$183$	0	0
$184$	−55.8957	−0.303781
$185$	295.253i	1.59596i
$186$	0	0
$187$	−120.718	−0.645548
$188$	144.366i	0.767904i
$189$	0	0
$190$	−92.5081	−0.486885
$191$	57.7251i	0.302226i	0.988517	+	0.151113i	$0.0482857\pi$
$191$	57.7251i	0.302226i	−0.988517	+	0.151113i	$0.951714\pi$
$192$	0	0
$193$	89.5108	0.463786	0.231893	−	0.972741i	$-0.425508\pi$
$193$	89.5108	0.463786	0.231893	+	0.972741i	$0.425508\pi$
$194$	237.762i	1.22558i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	1.87709i	0.00952838i	0.999989	+	0.00476419i	$0.00151650\pi$
$197$	1.87709i	0.00952838i	−0.999989	+	0.00476419i	$0.998484\pi$
$198$	0	0
$199$	−80.1938	−0.402984	−0.201492	−	0.979490i	$-0.564579\pi$
$199$	−80.1938	−0.402984	−0.201492	+	0.979490i	$0.564579\pi$
$200$	− 21.6031i	− 0.108015i
$201$	0	0
$202$	31.1829	0.154371
$203$	− 28.4341i	− 0.140069i
$204$	0	0
$205$	−35.0817	−0.171130
$206$	178.916i	0.868523i
$207$	0	0
$208$	−58.8808	−0.283081
$209$	− 101.633i	− 0.486282i
$210$	0	0
$211$	−161.110	−0.763556	−0.381778	−	0.924254i	$-0.624688\pi$
$211$	−161.110	−0.763556	−0.381778	+	0.924254i	$0.624688\pi$
$212$	160.619i	0.757635i
$213$	0	0
$214$	−56.1480	−0.262374
$215$	− 207.912i	− 0.967031i
$216$	0	0
$217$	−59.0042	−0.271909
$218$	131.836i	0.604752i
$219$	0	0
$220$	−53.9515	−0.245234
$221$	274.481i	1.24200i
$222$	0	0
$223$	−20.4977	−0.0919180	−0.0459590	−	0.998943i	$-0.514634\pi$
$223$	−20.4977	−0.0919180	−0.0459590	+	0.998943i	$0.514634\pi$
$224$	14.9666i	0.0668153i
$225$	0	0
$226$	113.748	0.503310
$227$	− 150.821i	− 0.664411i	−0.943207	−	0.332206i	$-0.892207\pi$
$227$	− 150.821i	− 0.664411i	0.943207	−	0.332206i	$-0.107793\pi$
$228$	0	0
$229$	−297.761	−1.30027	−0.650133	−	0.759821i	$-0.725287\pi$
$229$	−297.761	−1.30027	−0.650133	+	0.759821i	$0.725287\pi$
$230$	116.453i	0.506317i
$231$	0	0
$232$	−30.3973	−0.131023
$233$	− 386.415i	− 1.65843i	−0.558928	−	0.829216i	$-0.688787\pi$
$233$	− 386.415i	− 1.65843i	0.558928	−	0.829216i	$-0.311213\pi$
$234$	0	0
$235$	300.772	1.27988
$236$	− 52.8489i	− 0.223936i
$237$	0	0
$238$	69.7691	0.293147
$239$	207.987i	0.870239i	0.900373	+	0.435119i	$0.143294\pi$
$239$	207.987i	0.870239i	−0.900373	+	0.435119i	$0.856706\pi$
$240$	0	0
$241$	323.466	1.34218	0.671092	−	0.741374i	$-0.265826\pi$
$241$	323.466	1.34218	0.671092	+	0.741374i	$0.265826\pi$
$242$	111.847i	0.462176i
$243$	0	0
$244$	131.161	0.537545
$245$	29.1676i	0.119051i
$246$	0	0
$247$	−231.088	−0.935577
$248$	63.0782i	0.254347i
$249$	0	0
$250$	−192.326	−0.769304
$251$	288.081i	1.14773i	0.818948	+	0.573867i	$0.194558\pi$
$251$	288.081i	1.14773i	−0.818948	+	0.573867i	$0.805442\pi$
$252$	0	0
$253$	−127.940	−0.505690
$254$	− 31.8751i	− 0.125493i
$255$	0	0
$256$	16.0000	0.0625000
$257$	128.960i	0.501789i	0.968014	+	0.250895i	$0.0807248\pi$
$257$	128.960i	0.501789i	−0.968014	+	0.250895i	$0.919275\pi$
$258$	0	0
$259$	187.474	0.723838
$260$	122.672i	0.471816i
$261$	0	0
$262$	335.279	1.27969
$263$	− 31.6833i	− 0.120469i	−0.998184	−	0.0602344i	$-0.980815\pi$
$263$	− 31.6833i	− 0.120469i	0.998184	−	0.0602344i	$-0.0191848\pi$
$264$	0	0
$265$	334.632	1.26276
$266$	58.7390i	0.220823i
$267$	0	0
$268$	−47.6154	−0.177669
$269$	154.678i	0.575011i	0.957779	+	0.287505i	$0.0928260\pi$
$269$	154.678i	0.575011i	−0.957779	+	0.287505i	$0.907174\pi$
$270$	0	0
$271$	−170.350	−0.628598	−0.314299	−	0.949324i	$-0.601769\pi$
$271$	−170.350	−0.628598	−0.314299	+	0.949324i	$0.601769\pi$
$272$	− 74.5863i	− 0.274214i
$273$	0	0
$274$	−274.348	−1.00127
$275$	− 49.4472i	− 0.179808i
$276$	0	0
$277$	−527.777	−1.90533	−0.952665	−	0.304021i	$-0.901671\pi$
$277$	−527.777	−1.90533	−0.952665	+	0.304021i	$0.901671\pi$
$278$	− 224.752i	− 0.808459i
$279$	0	0
$280$	31.1814	0.111362
$281$	− 403.617i	− 1.43636i	−0.695859	−	0.718179i	$-0.744976\pi$
$281$	− 403.617i	− 1.43636i	0.695859	−	0.718179i	$-0.255024\pi$
$282$	0	0
$283$	−221.962	−0.784317	−0.392158	−	0.919898i	$-0.628272\pi$
$283$	−221.962	−0.784317	−0.392158	+	0.919898i	$0.628272\pi$
$284$	− 223.852i	− 0.788213i
$285$	0	0
$286$	−134.772	−0.471231
$287$	22.2755i	0.0776150i
$288$	0	0
$289$	−58.6945	−0.203095
$290$	63.3297i	0.218378i
$291$	0	0
$292$	−26.6219	−0.0911709
$293$	261.631i	0.892937i	0.894799	+	0.446469i	$0.147319\pi$
$293$	261.631i	0.892937i	−0.894799	+	0.446469i	$0.852681\pi$
$294$	0	0
$295$	−110.105	−0.373238
$296$	− 200.418i	− 0.677089i
$297$	0	0
$298$	144.049	0.483387
$299$	290.902i	0.972918i
$300$	0	0
$301$	−132.016	−0.438591
$302$	− 198.643i	− 0.657760i
$303$	0	0
$304$	62.7947	0.206561
$305$	− 273.260i	− 0.895935i
$306$	0	0
$307$	−137.772	−0.448769	−0.224384	−	0.974501i	$-0.572037\pi$
$307$	−137.772	−0.448769	−0.224384	+	0.974501i	$0.572037\pi$
$308$	34.2571i	0.111224i
$309$	0	0
$310$	131.417	0.423925
$311$	− 342.939i	− 1.10270i	−0.834275	−	0.551349i	$-0.814113\pi$
$311$	− 342.939i	− 1.10270i	0.834275	−	0.551349i	$-0.185887\pi$
$312$	0	0
$313$	−352.257	−1.12542	−0.562711	−	0.826654i	$-0.690242\pi$
$313$	−352.257	−1.12542	−0.562711	+	0.826654i	$0.690242\pi$
$314$	352.009i	1.12105i
$315$	0	0
$316$	−239.502	−0.757917
$317$	147.888i	0.466522i	0.972414	+	0.233261i	$0.0749397\pi$
$317$	147.888i	0.466522i	−0.972414	+	0.233261i	$0.925060\pi$
$318$	0	0
$319$	−69.5764	−0.218108
$320$	− 33.3343i	− 0.104170i
$321$	0	0
$322$	73.9431	0.229637
$323$	− 292.726i	− 0.906273i
$324$	0	0
$325$	−112.430	−0.345940
$326$	238.393i	0.731266i
$327$	0	0
$328$	23.8135	0.0726022
$329$	− 190.978i	− 0.580481i
$330$	0	0
$331$	560.056	1.69201	0.846006	−	0.533173i	$-0.179001\pi$
$331$	560.056	1.69201	0.846006	+	0.533173i	$0.179001\pi$
$332$	− 23.2594i	− 0.0700586i
$333$	0	0
$334$	411.660	1.23251
$335$	99.2017i	0.296124i
$336$	0	0
$337$	−23.9200	−0.0709794	−0.0354897	−	0.999370i	$-0.511299\pi$
$337$	−23.9200	−0.0709794	−0.0354897	+	0.999370i	$0.511299\pi$
$338$	67.4359i	0.199515i
$339$	0	0
$340$	−155.393	−0.457038
$341$	144.380i	0.423400i
$342$	0	0
$343$	18.5203	0.0539949
$344$	141.131i	0.410264i
$345$	0	0
$346$	−195.617	−0.565368
$347$	− 242.058i	− 0.697572i	−0.937202	−	0.348786i	$-0.886594\pi$
$347$	− 242.058i	− 0.697572i	0.937202	−	0.348786i	$-0.113406\pi$
$348$	0	0
$349$	35.9525	0.103016	0.0515079	−	0.998673i	$-0.483597\pi$
$349$	35.9525	0.103016	0.0515079	+	0.998673i	$0.483597\pi$
$350$	28.5782i	0.0816519i
$351$	0	0
$352$	36.6224	0.104041
$353$	307.258i	0.870420i	0.900329	+	0.435210i	$0.143326\pi$
$353$	307.258i	0.870420i	−0.900329	+	0.435210i	$0.856674\pi$
$354$	0	0
$355$	−466.373	−1.31373
$356$	167.103i	0.469391i
$357$	0	0
$358$	430.674	1.20300
$359$	− 529.049i	− 1.47368i	−0.676070	−	0.736838i	$-0.736318\pi$
$359$	− 529.049i	− 1.47368i	0.676070	−	0.736838i	$-0.263682\pi$
$360$	0	0
$361$	−114.552	−0.317318
$362$	− 368.714i	− 1.01855i
$363$	0	0
$364$	77.8920	0.213989
$365$	55.4640i	0.151956i
$366$	0	0
$367$	−339.069	−0.923893	−0.461946	−	0.886908i	$-0.652849\pi$
$367$	−339.069	−0.923893	−0.461946	+	0.886908i	$0.652849\pi$
$368$	− 79.0485i	− 0.214806i
$369$	0	0
$370$	−417.551	−1.12852
$371$	− 212.478i	− 0.572718i
$372$	0	0
$373$	494.789	1.32651	0.663255	−	0.748393i	$-0.269174\pi$
$373$	494.789	1.32651	0.663255	+	0.748393i	$0.269174\pi$
$374$	− 170.720i	− 0.456472i
$375$	0	0
$376$	−204.164	−0.542990
$377$	158.199i	0.419626i
$378$	0	0
$379$	−14.4703	−0.0381802	−0.0190901	−	0.999818i	$-0.506077\pi$
$379$	−14.4703	−0.0381802	−0.0190901	+	0.999818i	$0.506077\pi$
$380$	− 130.826i	− 0.344279i
$381$	0	0
$382$	−81.6356	−0.213706
$383$	− 298.793i	− 0.780138i	−0.920786	−	0.390069i	$-0.872451\pi$
$383$	− 298.793i	− 0.780138i	0.920786	−	0.390069i	$-0.127549\pi$
$384$	0	0
$385$	71.3711	0.185380
$386$	126.587i	0.327947i
$387$	0	0
$388$	−336.246	−0.866613
$389$	− 88.9149i	− 0.228573i	−0.993448	−	0.114287i	$-0.963542\pi$
$389$	− 88.9149i	− 0.228573i	0.993448	−	0.114287i	$-0.0364582\pi$
$390$	0	0
$391$	−368.496	−0.942444
$392$	− 19.7990i	− 0.0505076i
$393$	0	0
$394$	−2.65461	−0.00673758
$395$	498.977i	1.26323i
$396$	0	0
$397$	−537.580	−1.35411	−0.677053	−	0.735935i	$-0.736743\pi$
$397$	−537.580	−1.35411	−0.677053	+	0.735935i	$0.736743\pi$
$398$	− 113.411i	− 0.284953i
$399$	0	0
$400$	30.5513	0.0763784
$401$	− 174.661i	− 0.435563i	−0.975998	−	0.217782i	$-0.930118\pi$
$401$	− 174.661i	− 0.435563i	0.975998	−	0.217782i	$-0.0698820\pi$
$402$	0	0
$403$	328.283	0.814597
$404$	44.0993i	0.109157i
$405$	0	0
$406$	40.2119	0.0990440
$407$	− 458.737i	− 1.12712i
$408$	0	0
$409$	215.263	0.526315	0.263158	−	0.964753i	$-0.415236\pi$
$409$	215.263	0.526315	0.263158	+	0.964753i	$0.415236\pi$
$410$	− 49.6130i	− 0.121007i
$411$	0	0
$412$	−253.025	−0.614138
$413$	69.9125i	0.169280i
$414$	0	0
$415$	−48.4587	−0.116768
$416$	− 83.2700i	− 0.200168i
$417$	0	0
$418$	143.731	0.343853
$419$	− 233.582i	− 0.557475i	−0.960367	−	0.278738i	$-0.910084\pi$
$419$	− 233.582i	− 0.557475i	0.960367	−	0.278738i	$-0.0899160\pi$
$420$	0	0
$421$	−814.328	−1.93427	−0.967136	−	0.254261i	$-0.918168\pi$
$421$	−814.328	−1.93427	−0.967136	+	0.254261i	$0.918168\pi$
$422$	− 227.844i	− 0.539915i
$423$	0	0
$424$	−227.149	−0.535729
$425$	− 142.419i	− 0.335105i
$426$	0	0
$427$	−173.510	−0.406346
$428$	− 79.4052i	− 0.185526i
$429$	0	0
$430$	294.031	0.683794
$431$	317.979i	0.737770i	0.929475	+	0.368885i	$0.120260\pi$
$431$	317.979i	0.737770i	−0.929475	+	0.368885i	$0.879740\pi$
$432$	0	0
$433$	−344.299	−0.795148	−0.397574	−	0.917570i	$-0.630148\pi$
$433$	−344.299	−0.795148	−0.397574	+	0.917570i	$0.630148\pi$
$434$	− 83.4446i	− 0.192269i
$435$	0	0
$436$	−186.444	−0.427624
$437$	− 310.239i	− 0.709929i
$438$	0	0
$439$	−271.364	−0.618141	−0.309071	−	0.951039i	$-0.600018\pi$
$439$	−271.364	−0.618141	−0.309071	+	0.951039i	$0.600018\pi$
$440$	− 76.2989i	− 0.173407i
$441$	0	0
$442$	−388.175	−0.878224
$443$	− 189.796i	− 0.428433i	−0.976786	−	0.214217i	$-0.931280\pi$
$443$	− 189.796i	− 0.428433i	0.976786	−	0.214217i	$-0.0687199\pi$
$444$	0	0
$445$	348.142	0.782342
$446$	− 28.9881i	− 0.0649958i
$447$	0	0
$448$	−21.1660	−0.0472456
$449$	− 601.830i	− 1.34038i	−0.742190	−	0.670190i	$-0.766213\pi$
$449$	− 601.830i	− 1.34038i	0.742190	−	0.670190i	$-0.233787\pi$
$450$	0	0
$451$	54.5067	0.120858
$452$	160.864i	0.355894i
$453$	0	0
$454$	213.294	0.469810
$455$	− 162.280i	− 0.356659i
$456$	0	0
$457$	−177.083	−0.387490	−0.193745	−	0.981052i	$-0.562063\pi$
$457$	−177.083	−0.387490	−0.193745	+	0.981052i	$0.562063\pi$
$458$	− 421.097i	− 0.919426i
$459$	0	0
$460$	−164.689	−0.358020
$461$	200.233i	0.434345i	0.976133	+	0.217173i	$0.0696834\pi$
$461$	200.233i	0.434345i	−0.976133	+	0.217173i	$0.930317\pi$
$462$	0	0
$463$	474.507	1.02485	0.512427	−	0.858731i	$-0.328747\pi$
$463$	474.507	1.02485	0.512427	+	0.858731i	$0.328747\pi$
$464$	− 42.9883i	− 0.0926472i
$465$	0	0
$466$	546.473	1.17269
$467$	− 253.035i	− 0.541832i	−0.962603	−	0.270916i	$-0.912673\pi$
$467$	− 253.035i	− 0.541832i	0.962603	−	0.270916i	$-0.0873265\pi$
$468$	0	0
$469$	62.9892	0.134305
$470$	425.355i	0.905012i
$471$	0	0
$472$	74.7396	0.158347
$473$	323.034i	0.682947i
$474$	0	0
$475$	119.904	0.252429
$476$	98.6684i	0.207286i
$477$	0	0
$478$	−294.138	−0.615352
$479$	− 731.000i	− 1.52610i	−0.646342	−	0.763048i	$-0.723702\pi$
$479$	− 731.000i	− 1.52610i	0.646342	−	0.763048i	$-0.276298\pi$
$480$	0	0
$481$	−1043.05	−2.16851
$482$	457.450i	0.949067i
$483$	0	0
$484$	−158.175	−0.326808
$485$	700.533i	1.44440i
$486$	0	0
$487$	638.705	1.31151	0.655754	−	0.754974i	$-0.272351\pi$
$487$	638.705	1.31151	0.655754	+	0.754974i	$0.272351\pi$
$488$	185.490i	0.380102i
$489$	0	0
$490$	−41.2491	−0.0841819
$491$	− 562.708i	− 1.14604i	−0.819540	−	0.573022i	$-0.805771\pi$
$491$	− 562.708i	− 1.14604i	0.819540	−	0.573022i	$-0.194229\pi$
$492$	0	0
$493$	−200.396	−0.406483
$494$	− 326.807i	− 0.661553i
$495$	0	0
$496$	−89.2060	−0.179851
$497$	296.129i	0.595833i
$498$	0	0
$499$	758.182	1.51940	0.759701	−	0.650272i	$-0.225345\pi$
$499$	758.182	1.51940	0.759701	+	0.650272i	$0.225345\pi$
$500$	− 271.990i	− 0.543980i
$501$	0	0
$502$	−407.409	−0.811571
$503$	367.332i	0.730283i	0.930952	+	0.365142i	$0.118979\pi$
$503$	367.332i	0.730283i	−0.930952	+	0.365142i	$0.881021\pi$
$504$	0	0
$505$	91.8764	0.181933
$506$	− 180.934i	− 0.357577i
$507$	0	0
$508$	45.0782	0.0887367
$509$	294.338i	0.578267i	0.957289	+	0.289134i	$0.0933672\pi$
$509$	294.338i	0.578267i	−0.957289	+	0.289134i	$0.906633\pi$
$510$	0	0
$511$	35.2175	0.0689187
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	−182.377	−0.354819
$515$	527.151i	1.02359i
$516$	0	0
$517$	−467.312	−0.903891
$518$	265.129i	0.511831i
$519$	0	0
$520$	−173.485	−0.333624
$521$	53.1458i	0.102007i	0.998698	+	0.0510037i	$0.0162420\pi$
$521$	53.1458i	0.102007i	−0.998698	+	0.0510037i	$0.983758\pi$
$522$	0	0
$523$	−776.112	−1.48396	−0.741981	−	0.670421i	$-0.766113\pi$
$523$	−776.112	−1.48396	−0.741981	+	0.670421i	$0.766113\pi$
$524$	474.156i	0.904878i
$525$	0	0
$526$	44.8070	0.0851843
$527$	415.846i	0.789083i
$528$	0	0
$529$	138.459	0.261736
$530$	473.241i	0.892908i
$531$	0	0
$532$	−83.0695	−0.156146
$533$	− 123.935i	− 0.232523i
$534$	0	0
$535$	−165.433	−0.309220
$536$	− 67.3383i	− 0.125631i
$537$	0	0
$538$	−218.748	−0.406594
$539$	− 45.3179i	− 0.0840777i
$540$	0	0
$541$	414.383	0.765958	0.382979	−	0.923757i	$-0.374898\pi$
$541$	414.383	0.765958	0.382979	+	0.923757i	$0.374898\pi$
$542$	− 240.911i	− 0.444486i
$543$	0	0
$544$	105.481	0.193899
$545$	388.437i	0.712729i
$546$	0	0
$547$	−389.189	−0.711498	−0.355749	−	0.934582i	$-0.615774\pi$
$547$	−389.189	−0.711498	−0.355749	+	0.934582i	$0.615774\pi$
$548$	− 387.986i	− 0.708004i
$549$	0	0
$550$	69.9289	0.127143
$551$	− 168.715i	− 0.306197i
$552$	0	0
$553$	316.831	0.572932
$554$	− 746.389i	− 1.34727i
$555$	0	0
$556$	317.847	0.571667
$557$	752.828i	1.35158i	0.737096	+	0.675788i	$0.236197\pi$
$557$	752.828i	1.35158i	−0.737096	+	0.675788i	$0.763803\pi$
$558$	0	0
$559$	734.498	1.31395
$560$	44.0972i	0.0787450i
$561$	0	0
$562$	570.800	1.01566
$563$	663.796i	1.17903i	0.807756	+	0.589517i	$0.200682\pi$
$563$	663.796i	1.17903i	−0.807756	+	0.589517i	$0.799318\pi$
$564$	0	0
$565$	335.144	0.593175
$566$	− 313.901i	− 0.554596i
$567$	0	0
$568$	316.575	0.557351
$569$	572.869i	1.00680i	0.864054	+	0.503400i	$0.167918\pi$
$569$	572.869i	1.00680i	−0.864054	+	0.503400i	$0.832082\pi$
$570$	0	0
$571$	359.195	0.629063	0.314531	−	0.949247i	$-0.398153\pi$
$571$	359.195	0.629063	0.314531	+	0.949247i	$0.398153\pi$
$572$	− 190.597i	− 0.333211i
$573$	0	0
$574$	−31.5023	−0.0548821
$575$	− 150.940i	− 0.262504i
$576$	0	0
$577$	327.352	0.567334	0.283667	−	0.958923i	$-0.408449\pi$
$577$	327.352	0.567334	0.283667	+	0.958923i	$0.408449\pi$
$578$	− 83.0066i	− 0.143610i
$579$	0	0
$580$	−89.5617	−0.154417
$581$	30.7694i	0.0529593i
$582$	0	0
$583$	−519.921	−0.891803
$584$	− 37.6490i	− 0.0644675i
$585$	0	0
$586$	−370.002	−0.631402
$587$	811.391i	1.38227i	0.722727	+	0.691134i	$0.242888\pi$
$587$	811.391i	1.38227i	−0.722727	+	0.691134i	$0.757112\pi$
$588$	0	0
$589$	−350.104	−0.594404
$590$	− 155.712i	− 0.263919i
$591$	0	0
$592$	283.434	0.478774
$593$	1078.74i	1.81912i	0.415576	+	0.909559i	$0.363580\pi$
$593$	1078.74i	1.81912i	−0.415576	+	0.909559i	$0.636420\pi$
$594$	0	0
$595$	205.565	0.345488
$596$	203.716i	0.341806i
$597$	0	0
$598$	−411.398	−0.687957
$599$	− 109.727i	− 0.183184i	−0.995797	−	0.0915918i	$-0.970805\pi$
$599$	− 109.727i	− 0.183184i	0.995797	−	0.0915918i	$-0.0291955\pi$
$600$	0	0
$601$	804.332	1.33832	0.669161	−	0.743117i	$-0.266654\pi$
$601$	804.332	1.33832	0.669161	+	0.743117i	$0.266654\pi$
$602$	− 186.698i	− 0.310130i
$603$	0	0
$604$	280.924	0.465106
$605$	329.541i	0.544697i
$606$	0	0
$607$	−219.091	−0.360940	−0.180470	−	0.983580i	$-0.557762\pi$
$607$	−219.091	−0.360940	−0.180470	+	0.983580i	$0.557762\pi$
$608$	88.8051i	0.146061i
$609$	0	0
$610$	386.448	0.633522
$611$	1062.55i	1.73903i
$612$	0	0
$613$	897.371	1.46390	0.731951	−	0.681358i	$-0.238610\pi$
$613$	897.371	1.46390	0.731951	+	0.681358i	$0.238610\pi$
$614$	− 194.839i	− 0.317327i
$615$	0	0
$616$	−48.4468	−0.0786475
$617$	713.146i	1.15583i	0.816097	+	0.577915i	$0.196133\pi$
$617$	713.146i	1.15583i	−0.816097	+	0.577915i	$0.803867\pi$
$618$	0	0
$619$	−421.950	−0.681664	−0.340832	−	0.940124i	$-0.610709\pi$
$619$	−421.950	−0.681664	−0.340832	+	0.940124i	$0.610709\pi$
$620$	185.851i	0.299760i
$621$	0	0
$622$	484.989	0.779725
$623$	− 221.057i	− 0.354826i
$624$	0	0
$625$	−375.718	−0.601148
$626$	− 498.167i	− 0.795794i
$627$	0	0
$628$	−497.815	−0.792700
$629$	− 1321.27i	− 2.10059i
$630$	0	0
$631$	−601.037	−0.952515	−0.476257	−	0.879306i	$-0.658007\pi$
$631$	−601.037	−0.952515	−0.476257	+	0.879306i	$0.658007\pi$
$632$	− 338.707i	− 0.535928i
$633$	0	0
$634$	−209.145	−0.329881
$635$	− 93.9158i	− 0.147899i
$636$	0	0
$637$	−103.041	−0.161760
$638$	− 98.3958i	− 0.154225i
$639$	0	0
$640$	47.1419	0.0736592
$641$	− 934.197i	− 1.45740i	−0.684830	−	0.728702i	$-0.740124\pi$
$641$	− 934.197i	− 1.45740i	0.684830	−	0.728702i	$-0.259876\pi$
$642$	0	0
$643$	−20.0083	−0.0311172	−0.0155586	−	0.999879i	$-0.504953\pi$
$643$	−20.0083	−0.0311172	−0.0155586	+	0.999879i	$0.504953\pi$
$644$	104.571i	0.162378i
$645$	0	0
$646$	413.977	0.640832
$647$	− 724.799i	− 1.12025i	−0.828410	−	0.560123i	$-0.810754\pi$
$647$	− 724.799i	− 1.12025i	0.828410	−	0.560123i	$-0.189246\pi$
$648$	0	0
$649$	171.071	0.263592
$650$	− 159.001i	− 0.244617i
$651$	0	0
$652$	−337.138	−0.517083
$653$	− 78.7248i	− 0.120559i	−0.998182	−	0.0602793i	$-0.980801\pi$
$653$	− 78.7248i	− 0.120559i	0.998182	−	0.0602793i	$-0.0191991\pi$
$654$	0	0
$655$	987.855	1.50818
$656$	33.6774i	0.0513375i
$657$	0	0
$658$	270.084	0.410462
$659$	− 949.596i	− 1.44097i	−0.693473	−	0.720483i	$-0.743920\pi$
$659$	− 949.596i	− 1.44097i	0.693473	−	0.720483i	$-0.256080\pi$
$660$	0	0
$661$	384.317	0.581418	0.290709	−	0.956812i	$-0.406109\pi$
$661$	384.317	0.581418	0.290709	+	0.956812i	$0.406109\pi$
$662$	792.039i	1.19643i
$663$	0	0
$664$	32.8938	0.0495389
$665$	173.067i	0.260251i
$666$	0	0
$667$	−212.385	−0.318418
$668$	582.175i	0.871519i
$669$	0	0
$670$	−140.292	−0.209392
$671$	424.567i	0.632737i
$672$	0	0
$673$	199.887	0.297009	0.148505	−	0.988912i	$-0.452554\pi$
$673$	199.887	0.297009	0.148505	+	0.988912i	$0.452554\pi$
$674$	− 33.8281i	− 0.0501900i
$675$	0	0
$676$	−95.3688	−0.141078
$677$	988.989i	1.46084i	0.682998	+	0.730420i	$0.260676\pi$
$677$	988.989i	1.46084i	−0.682998	+	0.730420i	$0.739324\pi$
$678$	0	0
$679$	444.811	0.655098
$680$	− 219.759i	− 0.323174i
$681$	0	0
$682$	−204.183	−0.299389
$683$	− 472.371i	− 0.691612i	−0.938306	−	0.345806i	$-0.887606\pi$
$683$	− 472.371i	− 0.691612i	0.938306	−	0.345806i	$-0.112394\pi$
$684$	0	0
$685$	−808.329	−1.18004
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	−199.589	−0.290100
$689$	1182.17i	1.71578i
$690$	0	0
$691$	759.183	1.09867	0.549336	−	0.835601i	$-0.314881\pi$
$691$	759.183	1.09867	0.549336	+	0.835601i	$0.314881\pi$
$692$	− 276.645i	− 0.399776i
$693$	0	0
$694$	342.321	0.493258
$695$	− 662.201i	− 0.952807i
$696$	0	0
$697$	156.992	0.225240
$698$	50.8445i	0.0728432i
$699$	0	0
$700$	−40.4156	−0.0577366
$701$	590.449i	0.842295i	0.906992	+	0.421147i	$0.138372\pi$
$701$	590.449i	0.842295i	−0.906992	+	0.421147i	$0.861628\pi$
$702$	0	0
$703$	1112.38	1.58234
$704$	51.7919i	0.0735680i
$705$	0	0
$706$	−434.529	−0.615480
$707$	− 58.3379i	− 0.0825147i
$708$	0	0
$709$	750.407	1.05840	0.529201	−	0.848496i	$-0.322492\pi$
$709$	750.407	1.05840	0.529201	+	0.848496i	$0.322492\pi$
$710$	− 659.552i	− 0.928946i
$711$	0	0
$712$	−236.320	−0.331910
$713$	440.725i	0.618128i
$714$	0	0
$715$	−397.088	−0.555368
$716$	609.064i	0.850649i
$717$	0	0
$718$	748.189	1.04205
$719$	− 528.232i	− 0.734676i	−0.930088	−	0.367338i	$-0.880269\pi$
$719$	− 528.232i	− 0.734676i	0.930088	−	0.367338i	$-0.119731\pi$
$720$	0	0
$721$	334.721	0.464245
$722$	− 162.001i	− 0.224378i
$723$	0	0
$724$	521.440	0.720221
$725$	− 82.0844i	− 0.113220i
$726$	0	0
$727$	−675.799	−0.929572	−0.464786	−	0.885423i	$-0.653869\pi$
$727$	−675.799	−0.929572	−0.464786	+	0.885423i	$0.653869\pi$
$728$	110.156i	0.151313i
$729$	0	0
$730$	−78.4379	−0.107449
$731$	930.413i	1.27279i
$732$	0	0
$733$	−167.940	−0.229114	−0.114557	−	0.993417i	$-0.536545\pi$
$733$	−167.940	−0.229114	−0.114557	+	0.993417i	$0.536545\pi$
$734$	− 479.515i	− 0.653291i
$735$	0	0
$736$	111.791	0.151891
$737$	− 154.130i	− 0.209132i
$738$	0	0
$739$	−832.894	−1.12706	−0.563528	−	0.826097i	$-0.690556\pi$
$739$	−832.894	−1.12706	−0.563528	+	0.826097i	$0.690556\pi$
$740$	− 590.506i	− 0.797981i
$741$	0	0
$742$	300.490	0.404973
$743$	806.883i	1.08598i	0.839739	+	0.542990i	$0.182708\pi$
$743$	806.883i	1.08598i	−0.839739	+	0.542990i	$0.817292\pi$
$744$	0	0
$745$	424.422	0.569694
$746$	699.737i	0.937985i
$747$	0	0
$748$	241.435	0.322774
$749$	105.043i	0.140245i
$750$	0	0
$751$	407.737	0.542925	0.271462	−	0.962449i	$-0.412493\pi$
$751$	407.737	0.542925	0.271462	+	0.962449i	$0.412493\pi$
$752$	− 288.732i	− 0.383952i
$753$	0	0
$754$	−223.727	−0.296721
$755$	− 585.277i	− 0.775201i
$756$	0	0
$757$	709.798	0.937646	0.468823	−	0.883292i	$-0.344678\pi$
$757$	709.798	0.937646	0.468823	+	0.883292i	$0.344678\pi$
$758$	− 20.4641i	− 0.0269975i
$759$	0	0
$760$	185.016	0.243442
$761$	− 524.925i	− 0.689784i	−0.938642	−	0.344892i	$-0.887916\pi$
$761$	− 524.925i	− 0.689784i	0.938642	−	0.344892i	$-0.112084\pi$
$762$	0	0
$763$	246.643	0.323254
$764$	− 115.450i	− 0.151113i
$765$	0	0
$766$	422.557	0.551641
$767$	− 388.973i	− 0.507136i
$768$	0	0
$769$	−328.774	−0.427534	−0.213767	−	0.976885i	$-0.568573\pi$
$769$	−328.774	−0.427534	−0.213767	+	0.976885i	$0.568573\pi$
$770$	100.934i	0.131083i
$771$	0	0
$772$	−179.022	−0.231893
$773$	343.076i	0.443824i	0.975067	+	0.221912i	$0.0712298\pi$
$773$	343.076i	0.443824i	−0.975067	+	0.221912i	$0.928770\pi$
$774$	0	0
$775$	−170.335	−0.219787
$776$	− 475.523i	− 0.612788i
$777$	0	0
$778$	125.745	0.161626
$779$	132.173i	0.169670i
$780$	0	0
$781$	724.608	0.927796
$782$	− 521.132i	− 0.666409i
$783$	0	0
$784$	28.0000	0.0357143
$785$	1037.15i	1.32121i
$786$	0	0
$787$	330.465	0.419905	0.209952	−	0.977712i	$-0.432669\pi$
$787$	330.465	0.419905	0.209952	+	0.977712i	$0.432669\pi$
$788$	− 3.75418i	− 0.00476419i
$789$	0	0
$790$	−705.660	−0.893241
$791$	− 212.803i	− 0.269031i
$792$	0	0
$793$	965.358	1.21735
$794$	− 760.253i	− 0.957497i
$795$	0	0
$796$	160.388	0.201492
$797$	230.862i	0.289664i	0.989456	+	0.144832i	$0.0462642\pi$
$797$	230.862i	0.289664i	−0.989456	+	0.144832i	$0.953736\pi$
$798$	0	0
$799$	−1345.97	−1.68456
$800$	43.2061i	0.0540077i
$801$	0	0
$802$	247.008	0.307990
$803$	− 86.1748i	− 0.107316i
$804$	0	0
$805$	217.864	0.270638
$806$	464.262i	0.576007i
$807$	0	0
$808$	−62.3658	−0.0771855
$809$	1138.49i	1.40729i	0.710554	+	0.703643i	$0.248444\pi$
$809$	1138.49i	1.40729i	−0.710554	+	0.703643i	$0.751556\pi$
$810$	0	0
$811$	1346.08	1.65977	0.829887	−	0.557931i	$-0.188405\pi$
$811$	1346.08	1.65977	0.829887	+	0.557931i	$0.188405\pi$
$812$	56.8682i	0.0700347i
$813$	0	0
$814$	648.752	0.796993
$815$	702.392i	0.861831i
$816$	0	0
$817$	−783.320	−0.958777
$818$	304.428i	0.372161i
$819$	0	0
$820$	70.1634	0.0855651
$821$	479.519i	0.584067i	0.956408	+	0.292034i	$0.0943320\pi$
$821$	479.519i	0.584067i	−0.956408	+	0.292034i	$0.905668\pi$
$822$	0	0
$823$	1246.20	1.51422	0.757109	−	0.653289i	$-0.226611\pi$
$823$	1246.20	1.51422	0.757109	+	0.653289i	$0.226611\pi$
$824$	− 357.831i	− 0.434261i
$825$	0	0
$826$	−98.8713	−0.119699
$827$	− 464.446i	− 0.561603i	−0.959766	−	0.280802i	$-0.909400\pi$
$827$	− 464.446i	− 0.561603i	0.959766	−	0.280802i	$-0.0906003\pi$
$828$	0	0
$829$	479.856	0.578837	0.289418	−	0.957203i	$-0.406538\pi$
$829$	479.856	0.578837	0.289418	+	0.957203i	$0.406538\pi$
$830$	− 68.5309i	− 0.0825673i
$831$	0	0
$832$	117.762	0.141540
$833$	− 130.526i	− 0.156694i
$834$	0	0
$835$	1212.90	1.45258
$836$	203.266i	0.243141i
$837$	0	0
$838$	330.335	0.394195
$839$	− 287.604i	− 0.342794i	−0.985202	−	0.171397i	$-0.945172\pi$
$839$	− 287.604i	− 0.342794i	0.985202	−	0.171397i	$-0.0548280\pi$
$840$	0	0
$841$	725.500	0.862664
$842$	− 1151.63i	− 1.36774i
$843$	0	0
$844$	322.220	0.381778
$845$	198.691i	0.235137i
$846$	0	0
$847$	209.246	0.247044
$848$	− 321.237i	− 0.378817i
$849$	0	0
$850$	201.411	0.236955
$851$	− 1400.32i	− 1.64549i
$852$	0	0
$853$	−679.964	−0.797144	−0.398572	−	0.917137i	$-0.630494\pi$
$853$	−679.964	−0.797144	−0.398572	+	0.917137i	$0.630494\pi$
$854$	− 245.380i	− 0.287330i
$855$	0	0
$856$	112.296	0.131187
$857$	837.228i	0.976929i	0.872584	+	0.488464i	$0.162443\pi$
$857$	837.228i	0.976929i	−0.872584	+	0.488464i	$0.837557\pi$
$858$	0	0
$859$	346.622	0.403518	0.201759	−	0.979435i	$-0.435334\pi$
$859$	346.622	0.403518	0.201759	+	0.979435i	$0.435334\pi$
$860$	415.823i	0.483515i
$861$	0	0
$862$	−449.690	−0.521682
$863$	62.3236i	0.0722174i	0.999348	+	0.0361087i	$0.0114962\pi$
$863$	62.3236i	0.0722174i	−0.999348	+	0.0361087i	$0.988504\pi$
$864$	0	0
$865$	−576.361	−0.666313
$866$	− 486.912i	− 0.562255i
$867$	0	0
$868$	118.008	0.135954
$869$	− 775.265i	− 0.892135i
$870$	0	0
$871$	−350.454	−0.402358
$872$	− 263.672i	− 0.302376i
$873$	0	0
$874$	438.744	0.501995
$875$	359.809i	0.411211i
$876$	0	0
$877$	−1690.15	−1.92720	−0.963598	−	0.267354i	$-0.913851\pi$
$877$	−1690.15	−1.92720	−0.963598	+	0.267354i	$0.913851\pi$
$878$	− 383.767i	− 0.437092i
$879$	0	0
$880$	107.903	0.122617
$881$	1480.43i	1.68040i	0.542280	+	0.840198i	$0.317561\pi$
$881$	1480.43i	1.68040i	−0.542280	+	0.840198i	$0.682439\pi$
$882$	0	0
$883$	−688.171	−0.779356	−0.389678	−	0.920951i	$-0.627414\pi$
$883$	−688.171	−0.779356	−0.389678	+	0.920951i	$0.627414\pi$
$884$	− 548.963i	− 0.620998i
$885$	0	0
$886$	268.412	0.302948
$887$	625.526i	0.705216i	0.935771	+	0.352608i	$0.114705\pi$
$887$	625.526i	0.705216i	−0.935771	+	0.352608i	$0.885295\pi$
$888$	0	0
$889$	−59.6329	−0.0670786
$890$	492.347i	0.553199i
$891$	0	0
$892$	40.9954	0.0459590
$893$	− 1133.18i	− 1.26896i
$894$	0	0
$895$	1268.92	1.41779
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	851.116	0.947791
$899$	239.676i	0.266603i
$900$	0	0
$901$	−1497.49	−1.66203
$902$	77.0842i	0.0854592i
$903$	0	0
$904$	−227.496	−0.251655
$905$	− 1086.37i	− 1.20041i
$906$	0	0
$907$	1007.97	1.11133	0.555664	−	0.831407i	$-0.312464\pi$
$907$	1007.97	1.11133	0.555664	+	0.831407i	$0.312464\pi$
$908$	301.643i	0.332206i
$909$	0	0
$910$	229.498	0.252196
$911$	− 383.228i	− 0.420668i	−0.977630	−	0.210334i	$-0.932545\pi$
$911$	− 383.228i	− 0.420668i	0.977630	−	0.210334i	$-0.0674551\pi$
$912$	0	0
$913$	75.2906	0.0824651
$914$	− 250.433i	− 0.273997i
$915$	0	0
$916$	595.521	0.650133
$917$	− 627.249i	− 0.684023i
$918$	0	0
$919$	−769.956	−0.837819	−0.418910	−	0.908028i	$-0.637588\pi$
$919$	−769.956	−0.837819	−0.418910	+	0.908028i	$0.637588\pi$
$920$	− 232.906i	− 0.253159i
$921$	0	0
$922$	−283.172	−0.307128
$923$	− 1647.58i	− 1.78502i
$924$	0	0
$925$	541.206	0.585088
$926$	671.055i	0.724681i
$927$	0	0
$928$	60.7946	0.0655115
$929$	− 373.067i	− 0.401579i	−0.979634	−	0.200789i	$-0.935649\pi$
$929$	− 373.067i	− 0.401579i	0.979634	−	0.200789i	$-0.0643507\pi$
$930$	0	0
$931$	109.891	0.118035
$932$	772.829i	0.829216i
$933$	0	0
$934$	357.846	0.383133
$935$	− 503.005i	− 0.537973i
$936$	0	0
$937$	−987.512	−1.05391	−0.526954	−	0.849894i	$-0.676666\pi$
$937$	−987.512	−1.05391	−0.526954	+	0.849894i	$0.676666\pi$
$938$	89.0802i	0.0949682i
$939$	0	0
$940$	−601.543	−0.639940
$941$	1360.67i	1.44598i	0.690857	+	0.722991i	$0.257233\pi$
$941$	1360.67i	1.44598i	−0.690857	+	0.722991i	$0.742767\pi$
$942$	0	0
$943$	166.384	0.176441
$944$	105.698i	0.111968i
$945$	0	0
$946$	−456.839	−0.482917
$947$	1552.86i	1.63977i	0.572529	+	0.819884i	$0.305962\pi$
$947$	1552.86i	1.63977i	−0.572529	+	0.819884i	$0.694038\pi$
$948$	0	0
$949$	−195.940	−0.206470
$950$	169.570i	0.178494i
$951$	0	0
$952$	−139.538	−0.146574
$953$	− 756.704i	− 0.794023i	−0.917814	−	0.397012i	$-0.870047\pi$
$953$	− 756.704i	− 0.794023i	0.917814	−	0.397012i	$-0.129953\pi$
$954$	0	0
$955$	−240.529	−0.251862
$956$	− 415.974i	− 0.435119i
$957$	0	0
$958$	1033.79	1.07911
$959$	513.257i	0.535200i
$960$	0	0
$961$	−463.643	−0.482459
$962$	− 1475.10i	− 1.53337i
$963$	0	0
$964$	−646.932	−0.671092
$965$	372.973i	0.386500i
$966$	0	0
$967$	−99.3994	−0.102792	−0.0513958	−	0.998678i	$-0.516367\pi$
$967$	−99.3994	−0.102792	−0.0513958	+	0.998678i	$0.516367\pi$
$968$	− 223.693i	− 0.231088i
$969$	0	0
$970$	−990.704	−1.02134
$971$	− 24.0392i	− 0.0247571i	−0.999923	−	0.0123786i	$-0.996060\pi$
$971$	− 24.0392i	− 0.0247571i	0.999923	−	0.0123786i	$-0.00394032\pi$
$972$	0	0
$973$	−420.472	−0.432140
$974$	903.265i	0.927377i
$975$	0	0
$976$	−262.322	−0.268772
$977$	− 1271.78i	− 1.30172i	−0.759199	−	0.650859i	$-0.774409\pi$
$977$	− 1271.78i	− 1.30172i	0.759199	−	0.650859i	$-0.225591\pi$
$978$	0	0
$979$	−540.912	−0.552514
$980$	− 58.3351i	− 0.0595256i
$981$	0	0
$982$	795.789	0.810376
$983$	1017.31i	1.03490i	0.855712	+	0.517452i	$0.173119\pi$
$983$	1017.31i	1.03490i	−0.855712	+	0.517452i	$0.826881\pi$
$984$	0	0
$985$	−7.82145	−0.00794056
$986$	− 283.403i	− 0.287427i
$987$	0	0
$988$	462.175	0.467789
$989$	986.076i	0.997043i
$990$	0	0
$991$	−1748.26	−1.76414	−0.882070	−	0.471118i	$-0.843851\pi$
$991$	−1748.26	−1.76414	−0.882070	+	0.471118i	$0.843851\pi$
$992$	− 126.156i	− 0.127174i
$993$	0	0
$994$	−418.790	−0.421317
$995$	− 334.151i	− 0.335830i
$996$	0	0
$997$	247.516	0.248260	0.124130	−	0.992266i	$-0.460386\pi$
$997$	247.516	0.248260	0.124130	+	0.992266i	$0.460386\pi$
$998$	1072.23i	1.07438i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1134.3.b.c.323.21		24
3.2	odd	2	inner	1134.3.b.c.323.4		24
9.2	odd	6		378.3.q.a.71.5		24
9.4	even	3		378.3.q.a.197.5		24
9.5	odd	6		126.3.q.a.29.8	✓	24
9.7	even	3		126.3.q.a.113.8	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.q.a.29.8	✓	24	9.5	odd	6
126.3.q.a.113.8	yes	24	9.7	even	3
378.3.q.a.71.5		24	9.2	odd	6
378.3.q.a.197.5		24	9.4	even	3
1134.3.b.c.323.4		24	3.2	odd	2	inner
1134.3.b.c.323.21		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)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +4.16679i q^{5} +2.64575 q^{7} -2.82843i q^{8} -5.89274 q^{10} -6.47398i q^{11} -14.7202 q^{13} +3.74166i q^{14} +4.00000 q^{16} -18.6466i q^{17} +15.6987 q^{19} -8.33359i q^{20} +9.15559 q^{22} -19.7621i q^{23} +7.63784 q^{25} -20.8175i q^{26} -5.29150 q^{28} -10.7471i q^{29} -22.3015 q^{31} +5.65685i q^{32} +26.3702 q^{34} +11.0243i q^{35} +70.8586 q^{37} +22.2013i q^{38} +11.7855 q^{40} +8.41935i q^{41} -49.8973 q^{43} +12.9480i q^{44} +27.9479 q^{46} -72.1830i q^{47} +7.00000 q^{49} +10.8015i q^{50} +29.4404 q^{52} -80.3093i q^{53} +26.9757 q^{55} -7.48331i q^{56} +15.1987 q^{58} +26.4245i q^{59} -65.5805 q^{61} -31.5391i q^{62} -8.00000 q^{64} -61.3360i q^{65} +23.8077 q^{67} +37.2931i q^{68} -15.5907 q^{70} +111.926i q^{71} +13.3109 q^{73} +100.209i q^{74} -31.3973 q^{76} -17.1285i q^{77} +119.751 q^{79} +16.6672i q^{80} -11.9068 q^{82} +11.6297i q^{83} +77.6964 q^{85} -70.5654i q^{86} -18.3112 q^{88} -83.5516i q^{89} -38.9460 q^{91} +39.5242i q^{92} +102.082 q^{94} +65.4131i q^{95} +168.123 q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 48 q^{4} + 96 q^{16} + 24 q^{19} - 48 q^{22} - 144 q^{25} + 120 q^{31} + 96 q^{34} - 168 q^{37} - 120 q^{43} + 168 q^{49} + 264 q^{55} - 192 q^{64} - 144 q^{67} + 24 q^{73} - 48 q^{76} - 24 q^{79}+ \cdots - 96 q^{97}+O(q^{100}) \) 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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