LMFDB - Embedded newform 1240.2.a.m.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1240,2,Mod(1,1240)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1240.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1240 = 2^{3} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1240.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-1,0,6,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.90144985064$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 19x^{4} + 15x^{3} + 98x^{2} - 44x - 108 $ x^6 - x^5 - 19x^4 + 15x^3 + 98x^2 - 44x - 108
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.36730$ of defining polynomial
Character	$\chi$	$=$	1240.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.36730 q^{3} +1.00000 q^{5} +0.985130 q^{7} +2.60412 q^{9} +5.54333 q^{11} -4.06881 q^{13} +2.36730 q^{15} -1.17603 q^{17} +7.21627 q^{19} +2.33210 q^{21} -7.30978 q^{23} +1.00000 q^{25} -0.937162 q^{27} +5.10364 q^{29} +1.00000 q^{31} +13.1228 q^{33} +0.985130 q^{35} -5.43822 q^{37} -9.63211 q^{39} +0.0935165 q^{41} +2.25971 q^{43} +2.60412 q^{45} -3.31723 q^{47} -6.02952 q^{49} -2.78402 q^{51} +1.60296 q^{53} +5.54333 q^{55} +17.0831 q^{57} +0.294339 q^{59} +12.1703 q^{61} +2.56540 q^{63} -4.06881 q^{65} +6.27495 q^{67} -17.3045 q^{69} +6.48166 q^{71} -1.42124 q^{73} +2.36730 q^{75} +5.46090 q^{77} -9.03697 q^{79} -10.0309 q^{81} -16.9074 q^{83} -1.17603 q^{85} +12.0819 q^{87} -12.5401 q^{89} -4.00831 q^{91} +2.36730 q^{93} +7.21627 q^{95} +9.58240 q^{97} +14.4355 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + 6 q^{5} - 2 q^{7} + 21 q^{9} + 8 q^{11} + 4 q^{13} - q^{15} + 3 q^{17} - 5 q^{19} + 8 q^{21} - 12 q^{23} + 6 q^{25} - 7 q^{27} - 4 q^{29} + 6 q^{31} + 14 q^{33} - 2 q^{35} + 3 q^{37} + 10 q^{39}+ \cdots - 40 q^{99}+O(q^{100}) $ 6 * q - q^3 + 6 * q^5 - 2 * q^7 + 21 * q^9 + 8 * q^11 + 4 * q^13 - q^15 + 3 * q^17 - 5 * q^19 + 8 * q^21 - 12 * q^23 + 6 * q^25 - 7 * q^27 - 4 * q^29 + 6 * q^31 + 14 * q^33 - 2 * q^35 + 3 * q^37 + 10 * q^39 + 17 * q^41 - 17 * q^43 + 21 * q^45 - 6 * q^47 + 34 * q^49 + 23 * q^51 + 9 * q^53 + 8 * q^55 - q^57 + 39 * q^59 + 2 * q^61 - 16 * q^63 + 4 * q^65 + 12 * q^67 - 8 * q^69 + 21 * q^71 + 43 * q^73 - q^75 + 20 * q^77 - 2 * q^79 + 50 * q^81 - 5 * q^83 + 3 * q^85 - 44 * q^87 + 6 * q^89 + 24 * q^91 - q^93 - 5 * q^95 - 12 * q^97 - 40 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	2.36730	1.36676	0.683381	−	0.730061i	$-0.260509\pi$
$3$	2.36730	1.36676	0.683381	+	0.730061i	$0.260509\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.985130	0.372344	0.186172	−	0.982517i	$-0.440392\pi$
$7$	0.985130	0.372344	0.186172	+	0.982517i	$0.440392\pi$
$8$	0	0
$9$	2.60412	0.868041
$10$	0	0
$11$	5.54333	1.67138	0.835689	−	0.549203i	$-0.185069\pi$
$11$	5.54333	1.67138	0.835689	+	0.549203i	$0.185069\pi$
$12$	0	0
$13$	−4.06881	−1.12848	−0.564242	−	0.825609i	$-0.690832\pi$
$13$	−4.06881	−1.12848	−0.564242	+	0.825609i	$0.690832\pi$
$14$	0	0
$15$	2.36730	0.611235
$16$	0	0
$17$	−1.17603	−0.285229	−0.142615	−	0.989778i	$-0.545551\pi$
$17$	−1.17603	−0.285229	−0.142615	+	0.989778i	$0.545551\pi$
$18$	0	0
$19$	7.21627	1.65553	0.827763	−	0.561079i	$-0.189613\pi$
$19$	7.21627	1.65553	0.827763	+	0.561079i	$0.189613\pi$
$20$	0	0
$21$	2.33210	0.508906
$22$	0	0
$23$	−7.30978	−1.52420	−0.762098	−	0.647462i	$-0.775830\pi$
$23$	−7.30978	−1.52420	−0.762098	+	0.647462i	$0.775830\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.937162	−0.180357
$28$	0	0
$29$	5.10364	0.947722	0.473861	−	0.880600i	$-0.342860\pi$
$29$	5.10364	0.947722	0.473861	+	0.880600i	$0.342860\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	13.1228	2.28438
$34$	0	0
$35$	0.985130	0.166517
$36$	0	0
$37$	−5.43822	−0.894037	−0.447019	−	0.894525i	$-0.647514\pi$
$37$	−5.43822	−0.894037	−0.447019	+	0.894525i	$0.647514\pi$
$38$	0	0
$39$	−9.63211	−1.54237
$40$	0	0
$41$	0.0935165	0.0146048	0.00730241	−	0.999973i	$-0.497676\pi$
$41$	0.0935165	0.0146048	0.00730241	+	0.999973i	$0.497676\pi$
$42$	0	0
$43$	2.25971	0.344603	0.172301	−	0.985044i	$-0.444880\pi$
$43$	2.25971	0.344603	0.172301	+	0.985044i	$0.444880\pi$
$44$	0	0
$45$	2.60412	0.388200
$46$	0	0
$47$	−3.31723	−0.483868	−0.241934	−	0.970293i	$-0.577782\pi$
$47$	−3.31723	−0.483868	−0.241934	+	0.970293i	$0.577782\pi$
$48$	0	0
$49$	−6.02952	−0.861360
$50$	0	0
$51$	−2.78402	−0.389841
$52$	0	0
$53$	1.60296	0.220183	0.110092	−	0.993921i	$-0.464886\pi$
$53$	1.60296	0.220183	0.110092	+	0.993921i	$0.464886\pi$
$54$	0	0
$55$	5.54333	0.747463
$56$	0	0
$57$	17.0831	2.26271
$58$	0	0
$59$	0.294339	0.0383197	0.0191599	−	0.999816i	$-0.493901\pi$
$59$	0.294339	0.0383197	0.0191599	+	0.999816i	$0.493901\pi$
$60$	0	0
$61$	12.1703	1.55825	0.779127	−	0.626866i	$-0.215663\pi$
$61$	12.1703	1.55825	0.779127	+	0.626866i	$0.215663\pi$
$62$	0	0
$63$	2.56540	0.323210
$64$	0	0
$65$	−4.06881	−0.504674
$66$	0	0
$67$	6.27495	0.766607	0.383304	−	0.923622i	$-0.374786\pi$
$67$	6.27495	0.766607	0.383304	+	0.923622i	$0.374786\pi$
$68$	0	0
$69$	−17.3045	−2.08321
$70$	0	0
$71$	6.48166	0.769232	0.384616	−	0.923077i	$-0.374334\pi$
$71$	6.48166	0.769232	0.384616	+	0.923077i	$0.374334\pi$
$72$	0	0
$73$	−1.42124	−0.166344	−0.0831720	−	0.996535i	$-0.526505\pi$
$73$	−1.42124	−0.166344	−0.0831720	+	0.996535i	$0.526505\pi$
$74$	0	0
$75$	2.36730	0.273353
$76$	0	0
$77$	5.46090	0.622328
$78$	0	0
$79$	−9.03697	−1.01674	−0.508369	−	0.861139i	$-0.669751\pi$
$79$	−9.03697	−1.01674	−0.508369	+	0.861139i	$0.669751\pi$
$80$	0	0
$81$	−10.0309	−1.11455
$82$	0	0
$83$	−16.9074	−1.85583	−0.927916	−	0.372790i	$-0.878401\pi$
$83$	−16.9074	−1.85583	−0.927916	+	0.372790i	$0.878401\pi$
$84$	0	0
$85$	−1.17603	−0.127558
$86$	0	0
$87$	12.0819	1.29531
$88$	0	0
$89$	−12.5401	−1.32925	−0.664625	−	0.747177i	$-0.731409\pi$
$89$	−12.5401	−1.32925	−0.664625	+	0.747177i	$0.731409\pi$
$90$	0	0
$91$	−4.00831	−0.420185
$92$	0	0
$93$	2.36730	0.245478
$94$	0	0
$95$	7.21627	0.740373
$96$	0	0
$97$	9.58240	0.972946	0.486473	−	0.873696i	$-0.338283\pi$
$97$	9.58240	0.972946	0.486473	+	0.873696i	$0.338283\pi$
$98$	0	0
$99$	14.4355	1.45082
$100$	0	0
$101$	6.45192	0.641990	0.320995	−	0.947081i	$-0.395983\pi$
$101$	6.45192	0.641990	0.320995	+	0.947081i	$0.395983\pi$
$102$	0	0
$103$	−7.02952	−0.692639	−0.346320	−	0.938117i	$-0.612569\pi$
$103$	−7.02952	−0.692639	−0.346320	+	0.938117i	$0.612569\pi$
$104$	0	0
$105$	2.33210	0.227590
$106$	0	0
$107$	9.85101	0.952333	0.476167	−	0.879355i	$-0.342026\pi$
$107$	9.85101	0.952333	0.476167	+	0.879355i	$0.342026\pi$
$108$	0	0
$109$	−14.0080	−1.34173	−0.670863	−	0.741582i	$-0.734076\pi$
$109$	−14.0080	−1.34173	−0.670863	+	0.741582i	$0.734076\pi$
$110$	0	0
$111$	−12.8739	−1.22194
$112$	0	0
$113$	8.77069	0.825077	0.412538	−	0.910940i	$-0.364642\pi$
$113$	8.77069	0.825077	0.412538	+	0.910940i	$0.364642\pi$
$114$	0	0
$115$	−7.30978	−0.681641
$116$	0	0
$117$	−10.5957	−0.979571
$118$	0	0
$119$	−1.15854	−0.106204
$120$	0	0
$121$	19.7286	1.79350
$122$	0	0
$123$	0.221382	0.0199613
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−17.8871	−1.58722	−0.793612	−	0.608425i	$-0.791802\pi$
$127$	−17.8871	−1.58722	−0.793612	+	0.608425i	$0.791802\pi$
$128$	0	0
$129$	5.34942	0.470990
$130$	0	0
$131$	−4.86420	−0.424988	−0.212494	−	0.977162i	$-0.568159\pi$
$131$	−4.86420	−0.424988	−0.212494	+	0.977162i	$0.568159\pi$
$132$	0	0
$133$	7.10896	0.616425
$134$	0	0
$135$	−0.937162	−0.0806580
$136$	0	0
$137$	3.12633	0.267100	0.133550	−	0.991042i	$-0.457362\pi$
$137$	3.12633	0.267100	0.133550	+	0.991042i	$0.457362\pi$
$138$	0	0
$139$	−6.26963	−0.531783	−0.265892	−	0.964003i	$-0.585666\pi$
$139$	−6.26963	−0.531783	−0.265892	+	0.964003i	$0.585666\pi$
$140$	0	0
$141$	−7.85289	−0.661333
$142$	0	0
$143$	−22.5548	−1.88612
$144$	0	0
$145$	5.10364	0.423834
$146$	0	0
$147$	−14.2737	−1.17727
$148$	0	0
$149$	−8.91391	−0.730256	−0.365128	−	0.930957i	$-0.618975\pi$
$149$	−8.91391	−0.730256	−0.365128	+	0.930957i	$0.618975\pi$
$150$	0	0
$151$	6.59727	0.536879	0.268439	−	0.963297i	$-0.413492\pi$
$151$	6.59727	0.536879	0.268439	+	0.963297i	$0.413492\pi$
$152$	0	0
$153$	−3.06253	−0.247591
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	18.8922	1.50776	0.753880	−	0.657012i	$-0.228180\pi$
$157$	18.8922	1.50776	0.753880	+	0.657012i	$0.228180\pi$
$158$	0	0
$159$	3.79468	0.300938
$160$	0	0
$161$	−7.20109	−0.567525
$162$	0	0
$163$	−20.5476	−1.60941	−0.804705	−	0.593675i	$-0.797677\pi$
$163$	−20.5476	−1.60941	−0.804705	+	0.593675i	$0.797677\pi$
$164$	0	0
$165$	13.1228	1.02160
$166$	0	0
$167$	−23.7319	−1.83643	−0.918215	−	0.396082i	$-0.870370\pi$
$167$	−23.7319	−1.83643	−0.918215	+	0.396082i	$0.870370\pi$
$168$	0	0
$169$	3.55522	0.273478
$170$	0	0
$171$	18.7920	1.43706
$172$	0	0
$173$	16.9219	1.28655	0.643275	−	0.765635i	$-0.277575\pi$
$173$	16.9219	1.28655	0.643275	+	0.765635i	$0.277575\pi$
$174$	0	0
$175$	0.985130	0.0744688
$176$	0	0
$177$	0.696790	0.0523740
$178$	0	0
$179$	17.4267	1.30253	0.651267	−	0.758849i	$-0.274238\pi$
$179$	17.4267	1.30253	0.651267	+	0.758849i	$0.274238\pi$
$180$	0	0
$181$	4.75903	0.353736	0.176868	−	0.984235i	$-0.443403\pi$
$181$	4.75903	0.353736	0.176868	+	0.984235i	$0.443403\pi$
$182$	0	0
$183$	28.8109	2.12976
$184$	0	0
$185$	−5.43822	−0.399826
$186$	0	0
$187$	−6.51913	−0.476726
$188$	0	0
$189$	−0.923226	−0.0671548
$190$	0	0
$191$	12.0286	0.870361	0.435181	−	0.900343i	$-0.356684\pi$
$191$	12.0286	0.870361	0.435181	+	0.900343i	$0.356684\pi$
$192$	0	0
$193$	−11.9703	−0.861638	−0.430819	−	0.902438i	$-0.641775\pi$
$193$	−11.9703	−0.861638	−0.430819	+	0.902438i	$0.641775\pi$
$194$	0	0
$195$	−9.63211	−0.689769
$196$	0	0
$197$	8.58050	0.611335	0.305668	−	0.952138i	$-0.401120\pi$
$197$	8.58050	0.611335	0.305668	+	0.952138i	$0.401120\pi$
$198$	0	0
$199$	−11.8013	−0.836573	−0.418286	−	0.908315i	$-0.637369\pi$
$199$	−11.8013	−0.836573	−0.418286	+	0.908315i	$0.637369\pi$
$200$	0	0
$201$	14.8547	1.04777
$202$	0	0
$203$	5.02775	0.352879
$204$	0	0
$205$	0.0935165	0.00653148
$206$	0	0
$207$	−19.0356	−1.32306
$208$	0	0
$209$	40.0022	2.76701
$210$	0	0
$211$	−14.1650	−0.975161	−0.487580	−	0.873078i	$-0.662120\pi$
$211$	−14.1650	−0.975161	−0.487580	+	0.873078i	$0.662120\pi$
$212$	0	0
$213$	15.3441	1.05136
$214$	0	0
$215$	2.25971	0.154111
$216$	0	0
$217$	0.985130	0.0668750
$218$	0	0
$219$	−3.36451	−0.227353
$220$	0	0
$221$	4.78505	0.321877
$222$	0	0
$223$	−2.37262	−0.158883	−0.0794413	−	0.996840i	$-0.525314\pi$
$223$	−2.37262	−0.158883	−0.0794413	+	0.996840i	$0.525314\pi$
$224$	0	0
$225$	2.60412	0.173608
$226$	0	0
$227$	29.8943	1.98415	0.992077	−	0.125630i	$-0.0400954\pi$
$227$	29.8943	1.98415	0.992077	+	0.125630i	$0.0400954\pi$
$228$	0	0
$229$	−20.1903	−1.33421	−0.667107	−	0.744962i	$-0.732467\pi$
$229$	−20.1903	−1.33421	−0.667107	+	0.744962i	$0.732467\pi$
$230$	0	0
$231$	12.9276	0.850575
$232$	0	0
$233$	−9.33952	−0.611853	−0.305926	−	0.952055i	$-0.598966\pi$
$233$	−9.33952	−0.611853	−0.305926	+	0.952055i	$0.598966\pi$
$234$	0	0
$235$	−3.31723	−0.216392
$236$	0	0
$237$	−21.3932	−1.38964
$238$	0	0
$239$	−26.5702	−1.71868	−0.859340	−	0.511404i	$-0.829125\pi$
$239$	−26.5702	−1.71868	−0.859340	+	0.511404i	$0.829125\pi$
$240$	0	0
$241$	12.0879	0.778651	0.389326	−	0.921100i	$-0.372708\pi$
$241$	12.0879	0.778651	0.389326	+	0.921100i	$0.372708\pi$
$242$	0	0
$243$	−20.9347	−1.34296
$244$	0	0
$245$	−6.02952	−0.385212
$246$	0	0
$247$	−29.3616	−1.86824
$248$	0	0
$249$	−40.0250	−2.53648
$250$	0	0
$251$	17.7604	1.12103	0.560513	−	0.828146i	$-0.310604\pi$
$251$	17.7604	1.12103	0.560513	+	0.828146i	$0.310604\pi$
$252$	0	0
$253$	−40.5206	−2.54751
$254$	0	0
$255$	−2.78402	−0.174342
$256$	0	0
$257$	−0.631221	−0.0393745	−0.0196873	−	0.999806i	$-0.506267\pi$
$257$	−0.631221	−0.0393745	−0.0196873	+	0.999806i	$0.506267\pi$
$258$	0	0
$259$	−5.35735	−0.332890
$260$	0	0
$261$	13.2905	0.822662
$262$	0	0
$263$	11.8054	0.727951	0.363975	−	0.931409i	$-0.381419\pi$
$263$	11.8054	0.727951	0.363975	+	0.931409i	$0.381419\pi$
$264$	0	0
$265$	1.60296	0.0984689
$266$	0	0
$267$	−29.6863	−1.81677
$268$	0	0
$269$	−7.44393	−0.453864	−0.226932	−	0.973911i	$-0.572870\pi$
$269$	−7.44393	−0.453864	−0.226932	+	0.973911i	$0.572870\pi$
$270$	0	0
$271$	−6.55308	−0.398071	−0.199036	−	0.979992i	$-0.563781\pi$
$271$	−6.55308	−0.398071	−0.199036	+	0.979992i	$0.563781\pi$
$272$	0	0
$273$	−9.48888	−0.574293
$274$	0	0
$275$	5.54333	0.334276
$276$	0	0
$277$	6.18448	0.371589	0.185795	−	0.982589i	$-0.440514\pi$
$277$	6.18448	0.371589	0.185795	+	0.982589i	$0.440514\pi$
$278$	0	0
$279$	2.60412	0.155905
$280$	0	0
$281$	−5.54092	−0.330544	−0.165272	−	0.986248i	$-0.552850\pi$
$281$	−5.54092	−0.330544	−0.165272	+	0.986248i	$0.552850\pi$
$282$	0	0
$283$	21.6103	1.28460	0.642299	−	0.766454i	$-0.277981\pi$
$283$	21.6103	1.28460	0.642299	+	0.766454i	$0.277981\pi$
$284$	0	0
$285$	17.0831	1.01191
$286$	0	0
$287$	0.0921259	0.00543802
$288$	0	0
$289$	−15.6170	−0.918644
$290$	0	0
$291$	22.6845	1.32979
$292$	0	0
$293$	−3.82098	−0.223224	−0.111612	−	0.993752i	$-0.535601\pi$
$293$	−3.82098	−0.223224	−0.111612	+	0.993752i	$0.535601\pi$
$294$	0	0
$295$	0.294339	0.0171371
$296$	0	0
$297$	−5.19500	−0.301444
$298$	0	0
$299$	29.7421	1.72003
$300$	0	0
$301$	2.22611	0.128311
$302$	0	0
$303$	15.2737	0.877448
$304$	0	0
$305$	12.1703	0.696872
$306$	0	0
$307$	−32.6735	−1.86478	−0.932389	−	0.361457i	$-0.882279\pi$
$307$	−32.6735	−1.86478	−0.932389	+	0.361457i	$0.882279\pi$
$308$	0	0
$309$	−16.6410	−0.946673
$310$	0	0
$311$	−29.1422	−1.65250	−0.826251	−	0.563303i	$-0.809531\pi$
$311$	−29.1422	−1.65250	−0.826251	+	0.563303i	$0.809531\pi$
$312$	0	0
$313$	6.53765	0.369530	0.184765	−	0.982783i	$-0.440848\pi$
$313$	6.53765	0.369530	0.184765	+	0.982783i	$0.440848\pi$
$314$	0	0
$315$	2.56540	0.144544
$316$	0	0
$317$	13.4685	0.756465	0.378232	−	0.925711i	$-0.376532\pi$
$317$	13.4685	0.756465	0.378232	+	0.925711i	$0.376532\pi$
$318$	0	0
$319$	28.2912	1.58400
$320$	0	0
$321$	23.3203	1.30161
$322$	0	0
$323$	−8.48656	−0.472205
$324$	0	0
$325$	−4.06881	−0.225697
$326$	0	0
$327$	−33.1612	−1.83382
$328$	0	0
$329$	−3.26790	−0.180165
$330$	0	0
$331$	−31.4525	−1.72879	−0.864394	−	0.502816i	$-0.832297\pi$
$331$	−31.4525	−1.72879	−0.864394	+	0.502816i	$0.832297\pi$
$332$	0	0
$333$	−14.1618	−0.776061
$334$	0	0
$335$	6.27495	0.342837
$336$	0	0
$337$	−25.7087	−1.40044	−0.700221	−	0.713926i	$-0.746915\pi$
$337$	−25.7087	−1.40044	−0.700221	+	0.713926i	$0.746915\pi$
$338$	0	0
$339$	20.7629	1.12768
$340$	0	0
$341$	5.54333	0.300188
$342$	0	0
$343$	−12.8358	−0.693066
$344$	0	0
$345$	−17.3045	−0.931641
$346$	0	0
$347$	−23.3800	−1.25510	−0.627552	−	0.778575i	$-0.715943\pi$
$347$	−23.3800	−1.25510	−0.627552	+	0.778575i	$0.715943\pi$
$348$	0	0
$349$	−27.3696	−1.46506	−0.732532	−	0.680733i	$-0.761661\pi$
$349$	−27.3696	−1.46506	−0.732532	+	0.680733i	$0.761661\pi$
$350$	0	0
$351$	3.81313	0.203530
$352$	0	0
$353$	−4.35420	−0.231751	−0.115875	−	0.993264i	$-0.536967\pi$
$353$	−4.35420	−0.231751	−0.115875	+	0.993264i	$0.536967\pi$
$354$	0	0
$355$	6.48166	0.344011
$356$	0	0
$357$	−2.74262	−0.145155
$358$	0	0
$359$	30.7835	1.62469	0.812345	−	0.583177i	$-0.198191\pi$
$359$	30.7835	1.62469	0.812345	+	0.583177i	$0.198191\pi$
$360$	0	0
$361$	33.0745	1.74076
$362$	0	0
$363$	46.7035	2.45130
$364$	0	0
$365$	−1.42124	−0.0743913
$366$	0	0
$367$	5.35665	0.279615	0.139807	−	0.990179i	$-0.455352\pi$
$367$	5.35665	0.279615	0.139807	+	0.990179i	$0.455352\pi$
$368$	0	0
$369$	0.243529	0.0126776
$370$	0	0
$371$	1.57912	0.0819839
$372$	0	0
$373$	−35.6351	−1.84512	−0.922558	−	0.385859i	$-0.873905\pi$
$373$	−35.6351	−1.84512	−0.922558	+	0.385859i	$0.873905\pi$
$374$	0	0
$375$	2.36730	0.122247
$376$	0	0
$377$	−20.7657	−1.06949
$378$	0	0
$379$	−7.12193	−0.365829	−0.182914	−	0.983129i	$-0.558553\pi$
$379$	−7.12193	−0.365829	−0.182914	+	0.983129i	$0.558553\pi$
$380$	0	0
$381$	−42.3442	−2.16936
$382$	0	0
$383$	21.9837	1.12331	0.561656	−	0.827371i	$-0.310164\pi$
$383$	21.9837	1.12331	0.561656	+	0.827371i	$0.310164\pi$
$384$	0	0
$385$	5.46090	0.278313
$386$	0	0
$387$	5.88457	0.299129
$388$	0	0
$389$	8.49500	0.430714	0.215357	−	0.976535i	$-0.430909\pi$
$389$	8.49500	0.430714	0.215357	+	0.976535i	$0.430909\pi$
$390$	0	0
$391$	8.59653	0.434745
$392$	0	0
$393$	−11.5150	−0.580857
$394$	0	0
$395$	−9.03697	−0.454699
$396$	0	0
$397$	3.50316	0.175818	0.0879092	−	0.996128i	$-0.471981\pi$
$397$	3.50316	0.175818	0.0879092	+	0.996128i	$0.471981\pi$
$398$	0	0
$399$	16.8291	0.842507
$400$	0	0
$401$	22.1260	1.10492	0.552459	−	0.833540i	$-0.313690\pi$
$401$	22.1260	1.10492	0.552459	+	0.833540i	$0.313690\pi$
$402$	0	0
$403$	−4.06881	−0.202682
$404$	0	0
$405$	−10.0309	−0.498440
$406$	0	0
$407$	−30.1459	−1.49427
$408$	0	0
$409$	39.2014	1.93838	0.969192	−	0.246308i	$-0.0792174\pi$
$409$	39.2014	1.93838	0.969192	+	0.246308i	$0.0792174\pi$
$410$	0	0
$411$	7.40097	0.365063
$412$	0	0
$413$	0.289962	0.0142681
$414$	0	0
$415$	−16.9074	−0.829953
$416$	0	0
$417$	−14.8421	−0.726822
$418$	0	0
$419$	−1.07894	−0.0527094	−0.0263547	−	0.999653i	$-0.508390\pi$
$419$	−1.07894	−0.0527094	−0.0263547	+	0.999653i	$0.508390\pi$
$420$	0	0
$421$	−11.1898	−0.545356	−0.272678	−	0.962105i	$-0.587909\pi$
$421$	−11.1898	−0.545356	−0.272678	+	0.962105i	$0.587909\pi$
$422$	0	0
$423$	−8.63848	−0.420017
$424$	0	0
$425$	−1.17603	−0.0570459
$426$	0	0
$427$	11.9894	0.580207
$428$	0	0
$429$	−53.3940	−2.57789
$430$	0	0
$431$	0.456098	0.0219695	0.0109847	−	0.999940i	$-0.496503\pi$
$431$	0.456098	0.0219695	0.0109847	+	0.999940i	$0.496503\pi$
$432$	0	0
$433$	32.4769	1.56074	0.780371	−	0.625317i	$-0.215030\pi$
$433$	32.4769	1.56074	0.780371	+	0.625317i	$0.215030\pi$
$434$	0	0
$435$	12.0819	0.579281
$436$	0	0
$437$	−52.7493	−2.52334
$438$	0	0
$439$	1.14026	0.0544217	0.0272109	−	0.999630i	$-0.491337\pi$
$439$	1.14026	0.0544217	0.0272109	+	0.999630i	$0.491337\pi$
$440$	0	0
$441$	−15.7016	−0.747696
$442$	0	0
$443$	−25.8488	−1.22811	−0.614057	−	0.789262i	$-0.710463\pi$
$443$	−25.8488	−1.22811	−0.614057	+	0.789262i	$0.710463\pi$
$444$	0	0
$445$	−12.5401	−0.594459
$446$	0	0
$447$	−21.1019	−0.998086
$448$	0	0
$449$	19.4353	0.917207	0.458603	−	0.888641i	$-0.348350\pi$
$449$	19.4353	0.917207	0.458603	+	0.888641i	$0.348350\pi$
$450$	0	0
$451$	0.518393	0.0244102
$452$	0	0
$453$	15.6177	0.733786
$454$	0	0
$455$	−4.00831	−0.187912
$456$	0	0
$457$	14.2862	0.668282	0.334141	−	0.942523i	$-0.391554\pi$
$457$	14.2862	0.668282	0.334141	+	0.942523i	$0.391554\pi$
$458$	0	0
$459$	1.10213	0.0514431
$460$	0	0
$461$	−13.4177	−0.624923	−0.312461	−	0.949930i	$-0.601153\pi$
$461$	−13.4177	−0.624923	−0.312461	+	0.949930i	$0.601153\pi$
$462$	0	0
$463$	−6.83204	−0.317512	−0.158756	−	0.987318i	$-0.550748\pi$
$463$	−6.83204	−0.317512	−0.158756	+	0.987318i	$0.550748\pi$
$464$	0	0
$465$	2.36730	0.109781
$466$	0	0
$467$	5.64299	0.261126	0.130563	−	0.991440i	$-0.458321\pi$
$467$	5.64299	0.261126	0.130563	+	0.991440i	$0.458321\pi$
$468$	0	0
$469$	6.18164	0.285442
$470$	0	0
$471$	44.7235	2.06075
$472$	0	0
$473$	12.5263	0.575962
$474$	0	0
$475$	7.21627	0.331105
$476$	0	0
$477$	4.17430	0.191128
$478$	0	0
$479$	−8.17987	−0.373748	−0.186874	−	0.982384i	$-0.559836\pi$
$479$	−8.17987	−0.373748	−0.186874	+	0.982384i	$0.559836\pi$
$480$	0	0
$481$	22.1271	1.00891
$482$	0	0
$483$	−17.0472	−0.775672
$484$	0	0
$485$	9.58240	0.435115
$486$	0	0
$487$	35.1782	1.59408	0.797038	−	0.603929i	$-0.206399\pi$
$487$	35.1782	1.59408	0.797038	+	0.603929i	$0.206399\pi$
$488$	0	0
$489$	−48.6423	−2.19968
$490$	0	0
$491$	1.43574	0.0647942	0.0323971	−	0.999475i	$-0.489686\pi$
$491$	1.43574	0.0647942	0.0323971	+	0.999475i	$0.489686\pi$
$492$	0	0
$493$	−6.00204	−0.270318
$494$	0	0
$495$	14.4355	0.648828
$496$	0	0
$497$	6.38528	0.286419
$498$	0	0
$499$	39.8021	1.78179	0.890893	−	0.454213i	$-0.150079\pi$
$499$	39.8021	1.78179	0.890893	+	0.454213i	$0.150079\pi$
$500$	0	0
$501$	−56.1806	−2.50996
$502$	0	0
$503$	27.1288	1.20961	0.604807	−	0.796372i	$-0.293250\pi$
$503$	27.1288	1.20961	0.604807	+	0.796372i	$0.293250\pi$
$504$	0	0
$505$	6.45192	0.287107
$506$	0	0
$507$	8.41627	0.373780
$508$	0	0
$509$	11.9357	0.529039	0.264519	−	0.964380i	$-0.414787\pi$
$509$	11.9357	0.529039	0.264519	+	0.964380i	$0.414787\pi$
$510$	0	0
$511$	−1.40011	−0.0619372
$512$	0	0
$513$	−6.76281	−0.298585
$514$	0	0
$515$	−7.02952	−0.309758
$516$	0	0
$517$	−18.3885	−0.808726
$518$	0	0
$519$	40.0593	1.75841
$520$	0	0
$521$	2.94950	0.129220	0.0646100	−	0.997911i	$-0.479420\pi$
$521$	2.94950	0.129220	0.0646100	+	0.997911i	$0.479420\pi$
$522$	0	0
$523$	−15.7057	−0.686763	−0.343382	−	0.939196i	$-0.611572\pi$
$523$	−15.7057	−0.686763	−0.343382	+	0.939196i	$0.611572\pi$
$524$	0	0
$525$	2.33210	0.101781
$526$	0	0
$527$	−1.17603	−0.0512287
$528$	0	0
$529$	30.4329	1.32317
$530$	0	0
$531$	0.766495	0.0332631
$532$	0	0
$533$	−0.380501	−0.0164813
$534$	0	0
$535$	9.85101	0.425896
$536$	0	0
$537$	41.2543	1.78025
$538$	0	0
$539$	−33.4236	−1.43966
$540$	0	0
$541$	8.80080	0.378376	0.189188	−	0.981941i	$-0.439414\pi$
$541$	8.80080	0.378376	0.189188	+	0.981941i	$0.439414\pi$
$542$	0	0
$543$	11.2661	0.483473
$544$	0	0
$545$	−14.0080	−0.600038
$546$	0	0
$547$	14.6926	0.628212	0.314106	−	0.949388i	$-0.398295\pi$
$547$	14.6926	0.628212	0.314106	+	0.949388i	$0.398295\pi$
$548$	0	0
$549$	31.6931	1.35263
$550$	0	0
$551$	36.8292	1.56898
$552$	0	0
$553$	−8.90259	−0.378576
$554$	0	0
$555$	−12.8739	−0.546467
$556$	0	0
$557$	19.4660	0.824802	0.412401	−	0.911002i	$-0.364690\pi$
$557$	19.4660	0.824802	0.412401	+	0.911002i	$0.364690\pi$
$558$	0	0
$559$	−9.19434	−0.388879
$560$	0	0
$561$	−15.4328	−0.651572
$562$	0	0
$563$	24.3805	1.02752	0.513758	−	0.857935i	$-0.328253\pi$
$563$	24.3805	1.02752	0.513758	+	0.857935i	$0.328253\pi$
$564$	0	0
$565$	8.77069	0.368986
$566$	0	0
$567$	−9.88175	−0.414995
$568$	0	0
$569$	−27.7112	−1.16171	−0.580857	−	0.814006i	$-0.697282\pi$
$569$	−27.7112	−1.16171	−0.580857	+	0.814006i	$0.697282\pi$
$570$	0	0
$571$	−38.1619	−1.59703	−0.798513	−	0.601978i	$-0.794379\pi$
$571$	−38.1619	−1.59703	−0.798513	+	0.601978i	$0.794379\pi$
$572$	0	0
$573$	28.4754	1.18958
$574$	0	0
$575$	−7.30978	−0.304839
$576$	0	0
$577$	0.287423	0.0119656	0.00598279	−	0.999982i	$-0.498096\pi$
$577$	0.287423	0.0119656	0.00598279	+	0.999982i	$0.498096\pi$
$578$	0	0
$579$	−28.3372	−1.17766
$580$	0	0
$581$	−16.6560	−0.691008
$582$	0	0
$583$	8.88573	0.368009
$584$	0	0
$585$	−10.5957	−0.438077
$586$	0	0
$587$	16.1557	0.666817	0.333409	−	0.942782i	$-0.391801\pi$
$587$	16.1557	0.666817	0.333409	+	0.942782i	$0.391801\pi$
$588$	0	0
$589$	7.21627	0.297341
$590$	0	0
$591$	20.3126	0.835550
$592$	0	0
$593$	31.1709	1.28003	0.640017	−	0.768361i	$-0.278927\pi$
$593$	31.1709	1.28003	0.640017	+	0.768361i	$0.278927\pi$
$594$	0	0
$595$	−1.15854	−0.0474957
$596$	0	0
$597$	−27.9373	−1.14340
$598$	0	0
$599$	33.0516	1.35045	0.675226	−	0.737611i	$-0.264046\pi$
$599$	33.0516	1.35045	0.675226	+	0.737611i	$0.264046\pi$
$600$	0	0
$601$	−1.59108	−0.0649014	−0.0324507	−	0.999473i	$-0.510331\pi$
$601$	−1.59108	−0.0649014	−0.0324507	+	0.999473i	$0.510331\pi$
$602$	0	0
$603$	16.3407	0.665447
$604$	0	0
$605$	19.7286	0.802080
$606$	0	0
$607$	0.476139	0.0193259	0.00966295	−	0.999953i	$-0.496924\pi$
$607$	0.476139	0.0193259	0.00966295	+	0.999953i	$0.496924\pi$
$608$	0	0
$609$	11.9022	0.482302
$610$	0	0
$611$	13.4972	0.546038
$612$	0	0
$613$	−8.06261	−0.325646	−0.162823	−	0.986655i	$-0.552060\pi$
$613$	−8.06261	−0.325646	−0.162823	+	0.986655i	$0.552060\pi$
$614$	0	0
$615$	0.221382	0.00892698
$616$	0	0
$617$	18.0955	0.728497	0.364248	−	0.931302i	$-0.381326\pi$
$617$	18.0955	0.728497	0.364248	+	0.931302i	$0.381326\pi$
$618$	0	0
$619$	33.3665	1.34111	0.670556	−	0.741859i	$-0.266056\pi$
$619$	33.3665	1.34111	0.670556	+	0.741859i	$0.266056\pi$
$620$	0	0
$621$	6.85045	0.274899
$622$	0	0
$623$	−12.3537	−0.494939
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	94.6973	3.78184
$628$	0	0
$629$	6.39551	0.255006
$630$	0	0
$631$	4.67166	0.185976	0.0929879	−	0.995667i	$-0.470358\pi$
$631$	4.67166	0.185976	0.0929879	+	0.995667i	$0.470358\pi$
$632$	0	0
$633$	−33.5329	−1.33281
$634$	0	0
$635$	−17.8871	−0.709828
$636$	0	0
$637$	24.5330	0.972032
$638$	0	0
$639$	16.8790	0.667725
$640$	0	0
$641$	−9.24646	−0.365213	−0.182607	−	0.983186i	$-0.558453\pi$
$641$	−9.24646	−0.365213	−0.182607	+	0.983186i	$0.558453\pi$
$642$	0	0
$643$	45.0412	1.77625	0.888127	−	0.459599i	$-0.152007\pi$
$643$	45.0412	1.77625	0.888127	+	0.459599i	$0.152007\pi$
$644$	0	0
$645$	5.34942	0.210633
$646$	0	0
$647$	35.4197	1.39249	0.696245	−	0.717804i	$-0.254853\pi$
$647$	35.4197	1.39249	0.696245	+	0.717804i	$0.254853\pi$
$648$	0	0
$649$	1.63162	0.0640467
$650$	0	0
$651$	2.33210	0.0914022
$652$	0	0
$653$	21.7581	0.851462	0.425731	−	0.904850i	$-0.360017\pi$
$653$	21.7581	0.851462	0.425731	+	0.904850i	$0.360017\pi$
$654$	0	0
$655$	−4.86420	−0.190060
$656$	0	0
$657$	−3.70109	−0.144393
$658$	0	0
$659$	12.7766	0.497705	0.248852	−	0.968541i	$-0.419947\pi$
$659$	12.7766	0.497705	0.248852	+	0.968541i	$0.419947\pi$
$660$	0	0
$661$	−16.4047	−0.638068	−0.319034	−	0.947743i	$-0.603358\pi$
$661$	−16.4047	−0.638068	−0.319034	+	0.947743i	$0.603358\pi$
$662$	0	0
$663$	11.3277	0.439930
$664$	0	0
$665$	7.10896	0.275674
$666$	0	0
$667$	−37.3065	−1.44451
$668$	0	0
$669$	−5.61671	−0.217155
$670$	0	0
$671$	67.4643	2.60443
$672$	0	0
$673$	−23.6401	−0.911258	−0.455629	−	0.890170i	$-0.650586\pi$
$673$	−23.6401	−0.911258	−0.455629	+	0.890170i	$0.650586\pi$
$674$	0	0
$675$	−0.937162	−0.0360714
$676$	0	0
$677$	−30.6129	−1.17655	−0.588275	−	0.808661i	$-0.700193\pi$
$677$	−30.6129	−1.17655	−0.588275	+	0.808661i	$0.700193\pi$
$678$	0	0
$679$	9.43991	0.362271
$680$	0	0
$681$	70.7689	2.71187
$682$	0	0
$683$	12.0562	0.461319	0.230660	−	0.973035i	$-0.425912\pi$
$683$	12.0562	0.461319	0.230660	+	0.973035i	$0.425912\pi$
$684$	0	0
$685$	3.12633	0.119451
$686$	0	0
$687$	−47.7966	−1.82355
$688$	0	0
$689$	−6.52213	−0.248473
$690$	0	0
$691$	−8.38794	−0.319092	−0.159546	−	0.987190i	$-0.551003\pi$
$691$	−8.38794	−0.319092	−0.159546	+	0.987190i	$0.551003\pi$
$692$	0	0
$693$	14.2209	0.540206
$694$	0	0
$695$	−6.26963	−0.237821
$696$	0	0
$697$	−0.109978	−0.00416573
$698$	0	0
$699$	−22.1095	−0.836257
$700$	0	0
$701$	31.0951	1.17445	0.587224	−	0.809425i	$-0.300221\pi$
$701$	31.0951	1.17445	0.587224	+	0.809425i	$0.300221\pi$
$702$	0	0
$703$	−39.2436	−1.48010
$704$	0	0
$705$	−7.85289	−0.295757
$706$	0	0
$707$	6.35598	0.239041
$708$	0	0
$709$	19.3786	0.727778	0.363889	−	0.931442i	$-0.381449\pi$
$709$	19.3786	0.727778	0.363889	+	0.931442i	$0.381449\pi$
$710$	0	0
$711$	−23.5334	−0.882570
$712$	0	0
$713$	−7.30978	−0.273754
$714$	0	0
$715$	−22.5548	−0.843501
$716$	0	0
$717$	−62.8996	−2.34903
$718$	0	0
$719$	−39.5478	−1.47488	−0.737441	−	0.675411i	$-0.763966\pi$
$719$	−39.5478	−1.47488	−0.737441	+	0.675411i	$0.763966\pi$
$720$	0	0
$721$	−6.92499	−0.257900
$722$	0	0
$723$	28.6158	1.06423
$724$	0	0
$725$	5.10364	0.189544
$726$	0	0
$727$	17.5323	0.650238	0.325119	−	0.945673i	$-0.394596\pi$
$727$	17.5323	0.650238	0.325119	+	0.945673i	$0.394596\pi$
$728$	0	0
$729$	−19.4661	−0.720966
$730$	0	0
$731$	−2.65749	−0.0982909
$732$	0	0
$733$	35.0182	1.29342	0.646712	−	0.762734i	$-0.276143\pi$
$733$	35.0182	1.29342	0.646712	+	0.762734i	$0.276143\pi$
$734$	0	0
$735$	−14.2737	−0.526493
$736$	0	0
$737$	34.7842	1.28129
$738$	0	0
$739$	15.8284	0.582255	0.291128	−	0.956684i	$-0.405970\pi$
$739$	15.8284	0.582255	0.291128	+	0.956684i	$0.405970\pi$
$740$	0	0
$741$	−69.5078	−2.55343
$742$	0	0
$743$	27.6782	1.01542	0.507708	−	0.861529i	$-0.330493\pi$
$743$	27.6782	1.01542	0.507708	+	0.861529i	$0.330493\pi$
$744$	0	0
$745$	−8.91391	−0.326580
$746$	0	0
$747$	−44.0290	−1.61094
$748$	0	0
$749$	9.70453	0.354596
$750$	0	0
$751$	25.9078	0.945389	0.472694	−	0.881226i	$-0.343281\pi$
$751$	25.9078	0.945389	0.472694	+	0.881226i	$0.343281\pi$
$752$	0	0
$753$	42.0442	1.53218
$754$	0	0
$755$	6.59727	0.240099
$756$	0	0
$757$	10.0908	0.366755	0.183378	−	0.983043i	$-0.441297\pi$
$757$	10.0908	0.366755	0.183378	+	0.983043i	$0.441297\pi$
$758$	0	0
$759$	−95.9245	−3.48184
$760$	0	0
$761$	14.1305	0.512229	0.256114	−	0.966646i	$-0.417558\pi$
$761$	14.1305	0.512229	0.256114	+	0.966646i	$0.417558\pi$
$762$	0	0
$763$	−13.7997	−0.499583
$764$	0	0
$765$	−3.06253	−0.110726
$766$	0	0
$767$	−1.19761	−0.0432432
$768$	0	0
$769$	48.1352	1.73580	0.867900	−	0.496739i	$-0.165469\pi$
$769$	48.1352	1.73580	0.867900	+	0.496739i	$0.165469\pi$
$770$	0	0
$771$	−1.49429	−0.0538156
$772$	0	0
$773$	15.3836	0.553310	0.276655	−	0.960969i	$-0.410774\pi$
$773$	15.3836	0.553310	0.276655	+	0.960969i	$0.410774\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−12.6825	−0.454981
$778$	0	0
$779$	0.674840	0.0241787
$780$	0	0
$781$	35.9300	1.28568
$782$	0	0
$783$	−4.78294	−0.170928
$784$	0	0
$785$	18.8922	0.674291
$786$	0	0
$787$	−49.7918	−1.77489	−0.887444	−	0.460917i	$-0.847521\pi$
$787$	−49.7918	−1.77489	−0.887444	+	0.460917i	$0.847521\pi$
$788$	0	0
$789$	27.9469	0.994936
$790$	0	0
$791$	8.64027	0.307213
$792$	0	0
$793$	−49.5188	−1.75847
$794$	0	0
$795$	3.79468	0.134584
$796$	0	0
$797$	49.8424	1.76551	0.882755	−	0.469834i	$-0.155686\pi$
$797$	49.8424	1.76551	0.882755	+	0.469834i	$0.155686\pi$
$798$	0	0
$799$	3.90117	0.138013
$800$	0	0
$801$	−32.6560	−1.15384
$802$	0	0
$803$	−7.87842	−0.278024
$804$	0	0
$805$	−7.20109	−0.253805
$806$	0	0
$807$	−17.6220	−0.620325
$808$	0	0
$809$	−6.04065	−0.212378	−0.106189	−	0.994346i	$-0.533865\pi$
$809$	−6.04065	−0.212378	−0.106189	+	0.994346i	$0.533865\pi$
$810$	0	0
$811$	−51.5479	−1.81009	−0.905046	−	0.425313i	$-0.860164\pi$
$811$	−51.5479	−1.81009	−0.905046	+	0.425313i	$0.860164\pi$
$812$	0	0
$813$	−15.5131	−0.544069
$814$	0	0
$815$	−20.5476	−0.719750
$816$	0	0
$817$	16.3067	0.570499
$818$	0	0
$819$	−10.4381	−0.364738
$820$	0	0
$821$	31.9355	1.11456	0.557278	−	0.830326i	$-0.311846\pi$
$821$	31.9355	1.11456	0.557278	+	0.830326i	$0.311846\pi$
$822$	0	0
$823$	36.3667	1.26766	0.633831	−	0.773472i	$-0.281481\pi$
$823$	36.3667	1.26766	0.633831	+	0.773472i	$0.281481\pi$
$824$	0	0
$825$	13.1228	0.456876
$826$	0	0
$827$	3.23122	0.112361	0.0561803	−	0.998421i	$-0.482108\pi$
$827$	3.23122	0.112361	0.0561803	+	0.998421i	$0.482108\pi$
$828$	0	0
$829$	8.95101	0.310881	0.155441	−	0.987845i	$-0.450320\pi$
$829$	8.95101	0.310881	0.155441	+	0.987845i	$0.450320\pi$
$830$	0	0
$831$	14.6405	0.507874
$832$	0	0
$833$	7.09090	0.245685
$834$	0	0
$835$	−23.7319	−0.821277
$836$	0	0
$837$	−0.937162	−0.0323930
$838$	0	0
$839$	54.9886	1.89842	0.949209	−	0.314647i	$-0.101886\pi$
$839$	54.9886	1.89842	0.949209	+	0.314647i	$0.101886\pi$
$840$	0	0
$841$	−2.95284	−0.101822
$842$	0	0
$843$	−13.1170	−0.451775
$844$	0	0
$845$	3.55522	0.122303
$846$	0	0
$847$	19.4352	0.667801
$848$	0	0
$849$	51.1581	1.75574
$850$	0	0
$851$	39.7522	1.36269
$852$	0	0
$853$	17.3518	0.594116	0.297058	−	0.954859i	$-0.403995\pi$
$853$	17.3518	0.594116	0.297058	+	0.954859i	$0.403995\pi$
$854$	0	0
$855$	18.7920	0.642674
$856$	0	0
$857$	13.6870	0.467539	0.233770	−	0.972292i	$-0.424894\pi$
$857$	13.6870	0.467539	0.233770	+	0.972292i	$0.424894\pi$
$858$	0	0
$859$	−50.2602	−1.71485	−0.857427	−	0.514605i	$-0.827938\pi$
$859$	−50.2602	−1.71485	−0.857427	+	0.514605i	$0.827938\pi$
$860$	0	0
$861$	0.218090	0.00743249
$862$	0	0
$863$	6.65178	0.226429	0.113215	−	0.993571i	$-0.463885\pi$
$863$	6.65178	0.226429	0.113215	+	0.993571i	$0.463885\pi$
$864$	0	0
$865$	16.9219	0.575363
$866$	0	0
$867$	−36.9701	−1.25557
$868$	0	0
$869$	−50.0949	−1.69935
$870$	0	0
$871$	−25.5316	−0.865105
$872$	0	0
$873$	24.9538	0.844557
$874$	0	0
$875$	0.985130	0.0333035
$876$	0	0
$877$	8.33808	0.281557	0.140778	−	0.990041i	$-0.455039\pi$
$877$	8.33808	0.281557	0.140778	+	0.990041i	$0.455039\pi$
$878$	0	0
$879$	−9.04543	−0.305095
$880$	0	0
$881$	46.3790	1.56255	0.781275	−	0.624187i	$-0.214570\pi$
$881$	46.3790	1.56255	0.781275	+	0.624187i	$0.214570\pi$
$882$	0	0
$883$	−36.5638	−1.23047	−0.615236	−	0.788343i	$-0.710939\pi$
$883$	−36.5638	−1.23047	−0.615236	+	0.788343i	$0.710939\pi$
$884$	0	0
$885$	0.696790	0.0234223
$886$	0	0
$887$	22.3750	0.751280	0.375640	−	0.926766i	$-0.377423\pi$
$887$	22.3750	0.751280	0.375640	+	0.926766i	$0.377423\pi$
$888$	0	0
$889$	−17.6211	−0.590993
$890$	0	0
$891$	−55.6047	−1.86283
$892$	0	0
$893$	−23.9380	−0.801055
$894$	0	0
$895$	17.4267	0.582510
$896$	0	0
$897$	70.4086	2.35087
$898$	0	0
$899$	5.10364	0.170216
$900$	0	0
$901$	−1.88513	−0.0628027
$902$	0	0
$903$	5.26988	0.175370
$904$	0	0
$905$	4.75903	0.158195
$906$	0	0
$907$	−21.3451	−0.708753	−0.354377	−	0.935103i	$-0.615307\pi$
$907$	−21.3451	−0.708753	−0.354377	+	0.935103i	$0.615307\pi$
$908$	0	0
$909$	16.8016	0.557274
$910$	0	0
$911$	47.3928	1.57019	0.785097	−	0.619373i	$-0.212613\pi$
$911$	47.3928	1.57019	0.785097	+	0.619373i	$0.212613\pi$
$912$	0	0
$913$	−93.7235	−3.10180
$914$	0	0
$915$	28.8109	0.952459
$916$	0	0
$917$	−4.79187	−0.158242
$918$	0	0
$919$	−54.7372	−1.80561	−0.902807	−	0.430045i	$-0.858498\pi$
$919$	−54.7372	−1.80561	−0.902807	+	0.430045i	$0.858498\pi$
$920$	0	0
$921$	−77.3482	−2.54871
$922$	0	0
$923$	−26.3726	−0.868066
$924$	0	0
$925$	−5.43822	−0.178807
$926$	0	0
$927$	−18.3057	−0.601239
$928$	0	0
$929$	−57.5495	−1.88814	−0.944069	−	0.329747i	$-0.893036\pi$
$929$	−57.5495	−1.88814	−0.944069	+	0.329747i	$0.893036\pi$
$930$	0	0
$931$	−43.5106	−1.42600
$932$	0	0
$933$	−68.9884	−2.25858
$934$	0	0
$935$	−6.51913	−0.213198
$936$	0	0
$937$	15.8831	0.518878	0.259439	−	0.965760i	$-0.416462\pi$
$937$	15.8831	0.518878	0.259439	+	0.965760i	$0.416462\pi$
$938$	0	0
$939$	15.4766	0.505060
$940$	0	0
$941$	−11.9126	−0.388341	−0.194170	−	0.980968i	$-0.562201\pi$
$941$	−11.9126	−0.388341	−0.194170	+	0.980968i	$0.562201\pi$
$942$	0	0
$943$	−0.683586	−0.0222606
$944$	0	0
$945$	−0.923226	−0.0300325
$946$	0	0
$947$	13.2974	0.432108	0.216054	−	0.976381i	$-0.430681\pi$
$947$	13.2974	0.432108	0.216054	+	0.976381i	$0.430681\pi$
$948$	0	0
$949$	5.78277	0.187717
$950$	0	0
$951$	31.8839	1.03391
$952$	0	0
$953$	−36.3936	−1.17890	−0.589452	−	0.807803i	$-0.700656\pi$
$953$	−36.3936	−1.17890	−0.589452	+	0.807803i	$0.700656\pi$
$954$	0	0
$955$	12.0286	0.389237
$956$	0	0
$957$	66.9738	2.16496
$958$	0	0
$959$	3.07984	0.0994533
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	25.6532	0.826664
$964$	0	0
$965$	−11.9703	−0.385336
$966$	0	0
$967$	23.8574	0.767201	0.383600	−	0.923499i	$-0.374684\pi$
$967$	23.8574	0.767201	0.383600	+	0.923499i	$0.374684\pi$
$968$	0	0
$969$	−20.0902	−0.645392
$970$	0	0
$971$	−6.71681	−0.215553	−0.107776	−	0.994175i	$-0.534373\pi$
$971$	−6.71681	−0.215553	−0.107776	+	0.994175i	$0.534373\pi$
$972$	0	0
$973$	−6.17640	−0.198006
$974$	0	0
$975$	−9.63211	−0.308474
$976$	0	0
$977$	28.5668	0.913933	0.456966	−	0.889484i	$-0.348936\pi$
$977$	28.5668	0.913933	0.456966	+	0.889484i	$0.348936\pi$
$978$	0	0
$979$	−69.5141	−2.22168
$980$	0	0
$981$	−36.4786	−1.16467
$982$	0	0
$983$	−24.4066	−0.778451	−0.389225	−	0.921143i	$-0.627257\pi$
$983$	−24.4066	−0.778451	−0.389225	+	0.921143i	$0.627257\pi$
$984$	0	0
$985$	8.58050	0.273397
$986$	0	0
$987$	−7.73612	−0.246243
$988$	0	0
$989$	−16.5180	−0.525242
$990$	0	0
$991$	8.11038	0.257635	0.128817	−	0.991668i	$-0.458882\pi$
$991$	8.11038	0.257635	0.128817	+	0.991668i	$0.458882\pi$
$992$	0	0
$993$	−74.4577	−2.36284
$994$	0	0
$995$	−11.8013	−0.374127
$996$	0	0
$997$	−32.4400	−1.02738	−0.513692	−	0.857974i	$-0.671723\pi$
$997$	−32.4400	−1.02738	−0.513692	+	0.857974i	$0.671723\pi$
$998$	0	0
$999$	5.09649	0.161246

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1240.2.a.m.1.5	✓	6
4.3	odd	2		2480.2.a.ba.1.2		6
5.4	even	2		6200.2.a.w.1.2		6
8.3	odd	2		9920.2.a.co.1.5		6
8.5	even	2		9920.2.a.cp.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1240.2.a.m.1.5	✓	6	1.1	even	1	trivial
2480.2.a.ba.1.2		6	4.3	odd	2
6200.2.a.w.1.2		6	5.4	even	2
9920.2.a.co.1.5		6	8.3	odd	2
9920.2.a.cp.1.2		6	8.5	even	2

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:	\( N \)	\(=\)	\( 1240 = 2^{3} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1240.a (trivial)

\(f(q)\)	\(=\)	\(q+2.36730 q^{3} +1.00000 q^{5} +0.985130 q^{7} +2.60412 q^{9} +5.54333 q^{11} -4.06881 q^{13} +2.36730 q^{15} -1.17603 q^{17} +7.21627 q^{19} +2.33210 q^{21} -7.30978 q^{23} +1.00000 q^{25} -0.937162 q^{27} +5.10364 q^{29} +1.00000 q^{31} +13.1228 q^{33} +0.985130 q^{35} -5.43822 q^{37} -9.63211 q^{39} +0.0935165 q^{41} +2.25971 q^{43} +2.60412 q^{45} -3.31723 q^{47} -6.02952 q^{49} -2.78402 q^{51} +1.60296 q^{53} +5.54333 q^{55} +17.0831 q^{57} +0.294339 q^{59} +12.1703 q^{61} +2.56540 q^{63} -4.06881 q^{65} +6.27495 q^{67} -17.3045 q^{69} +6.48166 q^{71} -1.42124 q^{73} +2.36730 q^{75} +5.46090 q^{77} -9.03697 q^{79} -10.0309 q^{81} -16.9074 q^{83} -1.17603 q^{85} +12.0819 q^{87} -12.5401 q^{89} -4.00831 q^{91} +2.36730 q^{93} +7.21627 q^{95} +9.58240 q^{97} +14.4355 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + 6 q^{5} - 2 q^{7} + 21 q^{9} + 8 q^{11} + 4 q^{13} - q^{15} + 3 q^{17} - 5 q^{19} + 8 q^{21} - 12 q^{23} + 6 q^{25} - 7 q^{27} - 4 q^{29} + 6 q^{31} + 14 q^{33} - 2 q^{35} + 3 q^{37} + 10 q^{39}+ \cdots - 40 q^{99}+O(q^{100}) \) 6 * q - q^3 + 6 * q^5 - 2 * q^7 + 21 * q^9 + 8 * q^11 + 4 * q^13 - q^15 + 3 * q^17 - 5 * q^19 + 8 * q^21 - 12 * q^23 + 6 * q^25 - 7 * q^27 - 4 * q^29 + 6 * q^31 + 14 * q^33 - 2 * q^35 + 3 * q^37 + 10 * q^39 + 17 * q^41 - 17 * q^43 + 21 * q^45 - 6 * q^47 + 34 * q^49 + 23 * q^51 + 9 * q^53 + 8 * q^55 - q^57 + 39 * q^59 + 2 * q^61 - 16 * q^63 + 4 * q^65 + 12 * q^67 - 8 * q^69 + 21 * q^71 + 43 * q^73 - q^75 + 20 * q^77 - 2 * q^79 + 50 * q^81 - 5 * q^83 + 3 * q^85 - 44 * q^87 + 6 * q^89 + 24 * q^91 - q^93 - 5 * q^95 - 12 * q^97 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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