LMFDB - Embedded newform 3654.2.a.bc.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3654,2,Mod(1,3654)] 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("3654.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3654, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3654 = 2 \cdot 3^{2} \cdot 7 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3654.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,0,3,-5,0,-3,3,0,-5,-1,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$29.1773368986$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 406)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.12489$ of defining polynomial
Character	$\chi$	$=$	3654.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.12489 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.12489 q^{10} -5.76491 q^{11} +5.12489 q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.60975 q^{17} -2.96972 q^{19} +1.12489 q^{20} -5.76491 q^{22} -7.28005 q^{23} -3.73463 q^{25} +5.12489 q^{26} -1.00000 q^{28} -1.00000 q^{29} +4.15516 q^{31} +1.00000 q^{32} -5.60975 q^{34} -1.12489 q^{35} -2.24977 q^{37} -2.96972 q^{38} +1.12489 q^{40} -7.67030 q^{41} +0.734633 q^{43} -5.76491 q^{44} -7.28005 q^{46} +1.12489 q^{47} +1.00000 q^{49} -3.73463 q^{50} +5.12489 q^{52} +1.76491 q^{53} -6.48486 q^{55} -1.00000 q^{56} -1.00000 q^{58} -6.39025 q^{59} +2.24977 q^{61} +4.15516 q^{62} +1.00000 q^{64} +5.76491 q^{65} -15.5298 q^{67} -5.60975 q^{68} -1.12489 q^{70} +2.31032 q^{71} -2.39025 q^{73} -2.24977 q^{74} -2.96972 q^{76} +5.76491 q^{77} -0.545414 q^{79} +1.12489 q^{80} -7.67030 q^{82} +18.1093 q^{83} -6.31032 q^{85} +0.734633 q^{86} -5.76491 q^{88} +1.29942 q^{89} -5.12489 q^{91} -7.28005 q^{92} +1.12489 q^{94} -3.34060 q^{95} +8.16984 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 5 q^{5} - 3 q^{7} + 3 q^{8} - 5 q^{10} - q^{11} + 7 q^{13} - 3 q^{14} + 3 q^{16} - 8 q^{17} - 8 q^{19} - 5 q^{20} - q^{22} - 6 q^{23} + 6 q^{25} + 7 q^{26} - 3 q^{28} - 3 q^{29}+ \cdots + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 5 * q^5 - 3 * q^7 + 3 * q^8 - 5 * q^10 - q^11 + 7 * q^13 - 3 * q^14 + 3 * q^16 - 8 * q^17 - 8 * q^19 - 5 * q^20 - q^22 - 6 * q^23 + 6 * q^25 + 7 * q^26 - 3 * q^28 - 3 * q^29 + 5 * q^31 + 3 * q^32 - 8 * q^34 + 5 * q^35 + 10 * q^37 - 8 * q^38 - 5 * q^40 - 16 * q^41 - 15 * q^43 - q^44 - 6 * q^46 - 5 * q^47 + 3 * q^49 + 6 * q^50 + 7 * q^52 - 11 * q^53 - 19 * q^55 - 3 * q^56 - 3 * q^58 - 28 * q^59 - 10 * q^61 + 5 * q^62 + 3 * q^64 + q^65 - 14 * q^67 - 8 * q^68 + 5 * q^70 - 8 * q^71 - 16 * q^73 + 10 * q^74 - 8 * q^76 + q^77 - 3 * q^79 - 5 * q^80 - 16 * q^82 + 12 * q^83 - 4 * q^85 - 15 * q^86 - q^88 + 10 * q^89 - 7 * q^91 - 6 * q^92 - 5 * q^94 + 4 * q^95 - 16 * q^97 + 3 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.12489	0.503064	0.251532	−	0.967849i	$-0.419066\pi$
$5$	1.12489	0.503064	0.251532	+	0.967849i	$0.419066\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	1.12489	0.355720
$11$	−5.76491	−1.73819	−0.869093	−	0.494649i	$-0.835297\pi$
$11$	−5.76491	−1.73819	−0.869093	+	0.494649i	$0.835297\pi$
$12$	0	0
$13$	5.12489	1.42139	0.710694	−	0.703502i	$-0.248381\pi$
$13$	5.12489	1.42139	0.710694	+	0.703502i	$0.248381\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.60975	−1.36056	−0.680282	−	0.732951i	$-0.738143\pi$
$17$	−5.60975	−1.36056	−0.680282	+	0.732951i	$0.738143\pi$
$18$	0	0
$19$	−2.96972	−0.681301	−0.340651	−	0.940190i	$-0.610647\pi$
$19$	−2.96972	−0.681301	−0.340651	+	0.940190i	$0.610647\pi$
$20$	1.12489	0.251532
$21$	0	0
$22$	−5.76491	−1.22908
$23$	−7.28005	−1.51799	−0.758997	−	0.651094i	$-0.774310\pi$
$23$	−7.28005	−1.51799	−0.758997	+	0.651094i	$0.774310\pi$
$24$	0	0
$25$	−3.73463	−0.746927
$26$	5.12489	1.00507
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	4.15516	0.746289	0.373145	−	0.927773i	$-0.378280\pi$
$31$	4.15516	0.746289	0.373145	+	0.927773i	$0.378280\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.60975	−0.962064
$35$	−1.12489	−0.190140
$36$	0	0
$37$	−2.24977	−0.369860	−0.184930	−	0.982752i	$-0.559206\pi$
$37$	−2.24977	−0.369860	−0.184930	+	0.982752i	$0.559206\pi$
$38$	−2.96972	−0.481753
$39$	0	0
$40$	1.12489	0.177860
$41$	−7.67030	−1.19790	−0.598950	−	0.800787i	$-0.704415\pi$
$41$	−7.67030	−1.19790	−0.598950	+	0.800787i	$0.704415\pi$
$42$	0	0
$43$	0.734633	0.112030	0.0560152	−	0.998430i	$-0.482160\pi$
$43$	0.734633	0.112030	0.0560152	+	0.998430i	$0.482160\pi$
$44$	−5.76491	−0.869093
$45$	0	0
$46$	−7.28005	−1.07338
$47$	1.12489	0.164081	0.0820407	−	0.996629i	$-0.473856\pi$
$47$	1.12489	0.164081	0.0820407	+	0.996629i	$0.473856\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.73463	−0.528157
$51$	0	0
$52$	5.12489	0.710694
$53$	1.76491	0.242429	0.121214	−	0.992626i	$-0.461321\pi$
$53$	1.76491	0.242429	0.121214	+	0.992626i	$0.461321\pi$
$54$	0	0
$55$	−6.48486	−0.874419
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−1.00000	−0.131306
$59$	−6.39025	−0.831940	−0.415970	−	0.909378i	$-0.636558\pi$
$59$	−6.39025	−0.831940	−0.415970	+	0.909378i	$0.636558\pi$
$60$	0	0
$61$	2.24977	0.288054	0.144027	−	0.989574i	$-0.453995\pi$
$61$	2.24977	0.288054	0.144027	+	0.989574i	$0.453995\pi$
$62$	4.15516	0.527706
$63$	0	0
$64$	1.00000	0.125000
$65$	5.76491	0.715049
$66$	0	0
$67$	−15.5298	−1.89727	−0.948635	−	0.316374i	$-0.897535\pi$
$67$	−15.5298	−1.89727	−0.948635	+	0.316374i	$0.897535\pi$
$68$	−5.60975	−0.680282
$69$	0	0
$70$	−1.12489	−0.134450
$71$	2.31032	0.274185	0.137092	−	0.990558i	$-0.456224\pi$
$71$	2.31032	0.274185	0.137092	+	0.990558i	$0.456224\pi$
$72$	0	0
$73$	−2.39025	−0.279758	−0.139879	−	0.990169i	$-0.544671\pi$
$73$	−2.39025	−0.279758	−0.139879	+	0.990169i	$0.544671\pi$
$74$	−2.24977	−0.261531
$75$	0	0
$76$	−2.96972	−0.340651
$77$	5.76491	0.656972
$78$	0	0
$79$	−0.545414	−0.0613639	−0.0306819	−	0.999529i	$-0.509768\pi$
$79$	−0.545414	−0.0613639	−0.0306819	+	0.999529i	$0.509768\pi$
$80$	1.12489	0.125766
$81$	0	0
$82$	−7.67030	−0.847043
$83$	18.1093	1.98775	0.993876	−	0.110498i	$-0.0352445\pi$
$83$	18.1093	1.98775	0.993876	+	0.110498i	$0.0352445\pi$
$84$	0	0
$85$	−6.31032	−0.684451
$86$	0.734633	0.0792175
$87$	0	0
$88$	−5.76491	−0.614541
$89$	1.29942	0.137739	0.0688694	−	0.997626i	$-0.478061\pi$
$89$	1.29942	0.137739	0.0688694	+	0.997626i	$0.478061\pi$
$90$	0	0
$91$	−5.12489	−0.537234
$92$	−7.28005	−0.758997
$93$	0	0
$94$	1.12489	0.116023
$95$	−3.34060	−0.342738
$96$	0	0
$97$	8.16984	0.829522	0.414761	−	0.909930i	$-0.363865\pi$
$97$	8.16984	0.829522	0.414761	+	0.909930i	$0.363865\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−3.73463	−0.373463
$101$	−15.2195	−1.51440	−0.757198	−	0.653185i	$-0.773432\pi$
$101$	−15.2195	−1.51440	−0.757198	+	0.653185i	$0.773432\pi$
$102$	0	0
$103$	−3.21949	−0.317226	−0.158613	−	0.987341i	$-0.550702\pi$
$103$	−3.21949	−0.317226	−0.158613	+	0.987341i	$0.550702\pi$
$104$	5.12489	0.502536
$105$	0	0
$106$	1.76491	0.171423
$107$	−3.52982	−0.341240	−0.170620	−	0.985337i	$-0.554577\pi$
$107$	−3.52982	−0.341240	−0.170620	+	0.985337i	$0.554577\pi$
$108$	0	0
$109$	6.85574	0.656661	0.328330	−	0.944563i	$-0.393514\pi$
$109$	6.85574	0.656661	0.328330	+	0.944563i	$0.393514\pi$
$110$	−6.48486	−0.618307
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	18.8099	1.76948	0.884742	−	0.466082i	$-0.154335\pi$
$113$	18.8099	1.76948	0.884742	+	0.466082i	$0.154335\pi$
$114$	0	0
$115$	−8.18922	−0.763649
$116$	−1.00000	−0.0928477
$117$	0	0
$118$	−6.39025	−0.588270
$119$	5.60975	0.514245
$120$	0	0
$121$	22.2342	2.02129
$122$	2.24977	0.203685
$123$	0	0
$124$	4.15516	0.373145
$125$	−9.82546	−0.878816
$126$	0	0
$127$	−19.2195	−1.70545	−0.852727	−	0.522357i	$-0.825053\pi$
$127$	−19.2195	−1.70545	−0.852727	+	0.522357i	$0.825053\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	5.76491	0.505616
$131$	18.0294	1.57523	0.787616	−	0.616166i	$-0.211315\pi$
$131$	18.0294	1.57523	0.787616	+	0.616166i	$0.211315\pi$
$132$	0	0
$133$	2.96972	0.257508
$134$	−15.5298	−1.34157
$135$	0	0
$136$	−5.60975	−0.481032
$137$	13.8401	1.18244	0.591222	−	0.806509i	$-0.298646\pi$
$137$	13.8401	1.18244	0.591222	+	0.806509i	$0.298646\pi$
$138$	0	0
$139$	−9.13957	−0.775208	−0.387604	−	0.921826i	$-0.626697\pi$
$139$	−9.13957	−0.775208	−0.387604	+	0.921826i	$0.626697\pi$
$140$	−1.12489	−0.0950702
$141$	0	0
$142$	2.31032	0.193878
$143$	−29.5445	−2.47063
$144$	0	0
$145$	−1.12489	−0.0934166
$146$	−2.39025	−0.197819
$147$	0	0
$148$	−2.24977	−0.184930
$149$	−4.98440	−0.408338	−0.204169	−	0.978936i	$-0.565449\pi$
$149$	−4.98440	−0.408338	−0.204169	+	0.978936i	$0.565449\pi$
$150$	0	0
$151$	−9.34060	−0.760127	−0.380064	−	0.924960i	$-0.624098\pi$
$151$	−9.34060	−0.760127	−0.380064	+	0.924960i	$0.624098\pi$
$152$	−2.96972	−0.240876
$153$	0	0
$154$	5.76491	0.464550
$155$	4.67408	0.375431
$156$	0	0
$157$	−6.06055	−0.483685	−0.241842	−	0.970316i	$-0.577752\pi$
$157$	−6.06055	−0.483685	−0.241842	+	0.970316i	$0.577752\pi$
$158$	−0.545414	−0.0433908
$159$	0	0
$160$	1.12489	0.0889300
$161$	7.28005	0.573748
$162$	0	0
$163$	−4.67408	−0.366102	−0.183051	−	0.983103i	$-0.558597\pi$
$163$	−4.67408	−0.366102	−0.183051	+	0.983103i	$0.558597\pi$
$164$	−7.67030	−0.598950
$165$	0	0
$166$	18.1093	1.40555
$167$	−22.4390	−1.73638	−0.868191	−	0.496231i	$-0.834717\pi$
$167$	−22.4390	−1.73638	−0.868191	+	0.496231i	$0.834717\pi$
$168$	0	0
$169$	13.2645	1.02034
$170$	−6.31032	−0.483980
$171$	0	0
$172$	0.734633	0.0560152
$173$	−16.8898	−1.28411	−0.642054	−	0.766660i	$-0.721917\pi$
$173$	−16.8898	−1.28411	−0.642054	+	0.766660i	$0.721917\pi$
$174$	0	0
$175$	3.73463	0.282312
$176$	−5.76491	−0.434546
$177$	0	0
$178$	1.29942	0.0973960
$179$	−12.4995	−0.934260	−0.467130	−	0.884189i	$-0.654712\pi$
$179$	−12.4995	−0.934260	−0.467130	+	0.884189i	$0.654712\pi$
$180$	0	0
$181$	22.1240	1.64446	0.822231	−	0.569154i	$-0.192729\pi$
$181$	22.1240	1.64446	0.822231	+	0.569154i	$0.192729\pi$
$182$	−5.12489	−0.379882
$183$	0	0
$184$	−7.28005	−0.536692
$185$	−2.53073	−0.186063
$186$	0	0
$187$	32.3397	2.36491
$188$	1.12489	0.0820407
$189$	0	0
$190$	−3.34060	−0.242353
$191$	−3.21949	−0.232954	−0.116477	−	0.993193i	$-0.537160\pi$
$191$	−3.21949	−0.232954	−0.116477	+	0.993193i	$0.537160\pi$
$192$	0	0
$193$	−8.24977	−0.593831	−0.296916	−	0.954904i	$-0.595958\pi$
$193$	−8.24977	−0.593831	−0.296916	+	0.954904i	$0.595958\pi$
$194$	8.16984	0.586560
$195$	0	0
$196$	1.00000	0.0714286
$197$	−7.15894	−0.510054	−0.255027	−	0.966934i	$-0.582084\pi$
$197$	−7.15894	−0.510054	−0.255027	+	0.966934i	$0.582084\pi$
$198$	0	0
$199$	−11.3406	−0.803914	−0.401957	−	0.915659i	$-0.631670\pi$
$199$	−11.3406	−0.803914	−0.401957	+	0.915659i	$0.631670\pi$
$200$	−3.73463	−0.264078
$201$	0	0
$202$	−15.2195	−1.07084
$203$	1.00000	0.0701862
$204$	0	0
$205$	−8.62821	−0.602620
$206$	−3.21949	−0.224313
$207$	0	0
$208$	5.12489	0.355347
$209$	17.1202	1.18423
$210$	0	0
$211$	8.85574	0.609654	0.304827	−	0.952408i	$-0.401401\pi$
$211$	8.85574	0.609654	0.304827	+	0.952408i	$0.401401\pi$
$212$	1.76491	0.121214
$213$	0	0
$214$	−3.52982	−0.241293
$215$	0.826378	0.0563585
$216$	0	0
$217$	−4.15516	−0.282071
$218$	6.85574	0.464329
$219$	0	0
$220$	−6.48486	−0.437209
$221$	−28.7493	−1.93389
$222$	0	0
$223$	3.21949	0.215593	0.107797	−	0.994173i	$-0.465620\pi$
$223$	3.21949	0.215593	0.107797	+	0.994173i	$0.465620\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	18.8099	1.25121
$227$	−0.140482	−0.00932410	−0.00466205	−	0.999989i	$-0.501484\pi$
$227$	−0.140482	−0.00932410	−0.00466205	+	0.999989i	$0.501484\pi$
$228$	0	0
$229$	−16.0294	−1.05925	−0.529625	−	0.848232i	$-0.677667\pi$
$229$	−16.0294	−1.05925	−0.529625	+	0.848232i	$0.677667\pi$
$230$	−8.18922	−0.539981
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	17.6438	1.15588	0.577942	−	0.816078i	$-0.303856\pi$
$233$	17.6438	1.15588	0.577942	+	0.816078i	$0.303856\pi$
$234$	0	0
$235$	1.26537	0.0825435
$236$	−6.39025	−0.415970
$237$	0	0
$238$	5.60975	0.363626
$239$	14.4390	0.933981	0.466990	−	0.884262i	$-0.345338\pi$
$239$	14.4390	0.933981	0.466990	+	0.884262i	$0.345338\pi$
$240$	0	0
$241$	17.1736	1.10625	0.553125	−	0.833098i	$-0.313435\pi$
$241$	17.1736	1.10625	0.553125	+	0.833098i	$0.313435\pi$
$242$	22.2342	1.42927
$243$	0	0
$244$	2.24977	0.144027
$245$	1.12489	0.0718663
$246$	0	0
$247$	−15.2195	−0.968393
$248$	4.15516	0.263853
$249$	0	0
$250$	−9.82546	−0.621417
$251$	−16.3250	−1.03043	−0.515213	−	0.857062i	$-0.672287\pi$
$251$	−16.3250	−1.03043	−0.515213	+	0.857062i	$0.672287\pi$
$252$	0	0
$253$	41.9688	2.63856
$254$	−19.2195	−1.20594
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.5142	−0.780616	−0.390308	−	0.920684i	$-0.627631\pi$
$257$	−12.5142	−0.780616	−0.390308	+	0.920684i	$0.627631\pi$
$258$	0	0
$259$	2.24977	0.139794
$260$	5.76491	0.357524
$261$	0	0
$262$	18.0294	1.11386
$263$	28.5748	1.76200	0.880998	−	0.473120i	$-0.156872\pi$
$263$	28.5748	1.76200	0.880998	+	0.473120i	$0.156872\pi$
$264$	0	0
$265$	1.98532	0.121957
$266$	2.96972	0.182085
$267$	0	0
$268$	−15.5298	−0.948635
$269$	13.2800	0.809699	0.404849	−	0.914383i	$-0.367324\pi$
$269$	13.2800	0.809699	0.404849	+	0.914383i	$0.367324\pi$
$270$	0	0
$271$	−17.9348	−1.08946	−0.544729	−	0.838612i	$-0.683368\pi$
$271$	−17.9348	−1.08946	−0.544729	+	0.838612i	$0.683368\pi$
$272$	−5.60975	−0.340141
$273$	0	0
$274$	13.8401	0.836113
$275$	21.5298	1.29830
$276$	0	0
$277$	21.2195	1.27496	0.637478	−	0.770469i	$-0.279978\pi$
$277$	21.2195	1.27496	0.637478	+	0.770469i	$0.279978\pi$
$278$	−9.13957	−0.548155
$279$	0	0
$280$	−1.12489	−0.0672248
$281$	31.7943	1.89669	0.948344	−	0.317245i	$-0.102758\pi$
$281$	31.7943	1.89669	0.948344	+	0.317245i	$0.102758\pi$
$282$	0	0
$283$	9.79897	0.582488	0.291244	−	0.956649i	$-0.405931\pi$
$283$	9.79897	0.582488	0.291244	+	0.956649i	$0.405931\pi$
$284$	2.31032	0.137092
$285$	0	0
$286$	−29.5445	−1.74700
$287$	7.67030	0.452763
$288$	0	0
$289$	14.4693	0.851133
$290$	−1.12489	−0.0660555
$291$	0	0
$292$	−2.39025	−0.139879
$293$	−5.09083	−0.297409	−0.148705	−	0.988882i	$-0.547510\pi$
$293$	−5.09083	−0.297409	−0.148705	+	0.988882i	$0.547510\pi$
$294$	0	0
$295$	−7.18830	−0.418519
$296$	−2.24977	−0.130765
$297$	0	0
$298$	−4.98440	−0.288739
$299$	−37.3094	−2.15766
$300$	0	0
$301$	−0.734633	−0.0423435
$302$	−9.34060	−0.537491
$303$	0	0
$304$	−2.96972	−0.170325
$305$	2.53073	0.144909
$306$	0	0
$307$	−27.5445	−1.57205	−0.786024	−	0.618196i	$-0.787864\pi$
$307$	−27.5445	−1.57205	−0.786024	+	0.618196i	$0.787864\pi$
$308$	5.76491	0.328486
$309$	0	0
$310$	4.67408	0.265470
$311$	4.70058	0.266545	0.133273	−	0.991079i	$-0.457451\pi$
$311$	4.70058	0.266545	0.133273	+	0.991079i	$0.457451\pi$
$312$	0	0
$313$	26.7346	1.51113	0.755565	−	0.655073i	$-0.227362\pi$
$313$	26.7346	1.51113	0.755565	+	0.655073i	$0.227362\pi$
$314$	−6.06055	−0.342017
$315$	0	0
$316$	−0.545414	−0.0306819
$317$	−0.0605522	−0.00340095	−0.00170047	−	0.999999i	$-0.500541\pi$
$317$	−0.0605522	−0.00340095	−0.00170047	+	0.999999i	$0.500541\pi$
$318$	0	0
$319$	5.76491	0.322773
$320$	1.12489	0.0628830
$321$	0	0
$322$	7.28005	0.405701
$323$	16.6594	0.926954
$324$	0	0
$325$	−19.1396	−1.06167
$326$	−4.67408	−0.258873
$327$	0	0
$328$	−7.67030	−0.423521
$329$	−1.12489	−0.0620169
$330$	0	0
$331$	−34.8245	−1.91413	−0.957065	−	0.289873i	$-0.906387\pi$
$331$	−34.8245	−1.91413	−0.957065	+	0.289873i	$0.906387\pi$
$332$	18.1093	0.993876
$333$	0	0
$334$	−22.4390	−1.22781
$335$	−17.4693	−0.954448
$336$	0	0
$337$	−3.93945	−0.214595	−0.107298	−	0.994227i	$-0.534220\pi$
$337$	−3.93945	−0.214595	−0.107298	+	0.994227i	$0.534220\pi$
$338$	13.2645	0.721491
$339$	0	0
$340$	−6.31032	−0.342225
$341$	−23.9541	−1.29719
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0.734633	0.0396087
$345$	0	0
$346$	−16.8898	−0.908001
$347$	0.310323	0.0166590	0.00832951	−	0.999965i	$-0.497349\pi$
$347$	0.310323	0.0166590	0.00832951	+	0.999965i	$0.497349\pi$
$348$	0	0
$349$	26.0946	1.39681	0.698406	−	0.715702i	$-0.253893\pi$
$349$	26.0946	1.39681	0.698406	+	0.715702i	$0.253893\pi$
$350$	3.73463	0.199625
$351$	0	0
$352$	−5.76491	−0.307271
$353$	−19.1202	−1.01766	−0.508832	−	0.860866i	$-0.669923\pi$
$353$	−19.1202	−1.01766	−0.508832	+	0.860866i	$0.669923\pi$
$354$	0	0
$355$	2.59885	0.137933
$356$	1.29942	0.0688694
$357$	0	0
$358$	−12.4995	−0.660621
$359$	6.98440	0.368623	0.184311	−	0.982868i	$-0.440995\pi$
$359$	6.98440	0.368623	0.184311	+	0.982868i	$0.440995\pi$
$360$	0	0
$361$	−10.1807	−0.535828
$362$	22.1240	1.16281
$363$	0	0
$364$	−5.12489	−0.268617
$365$	−2.68876	−0.140736
$366$	0	0
$367$	11.9201	0.622223	0.311111	−	0.950373i	$-0.399299\pi$
$367$	11.9201	0.622223	0.311111	+	0.950373i	$0.399299\pi$
$368$	−7.28005	−0.379499
$369$	0	0
$370$	−2.53073	−0.131567
$371$	−1.76491	−0.0916295
$372$	0	0
$373$	−24.0147	−1.24343	−0.621716	−	0.783242i	$-0.713564\pi$
$373$	−24.0147	−1.24343	−0.621716	+	0.783242i	$0.713564\pi$
$374$	32.3397	1.67225
$375$	0	0
$376$	1.12489	0.0580115
$377$	−5.12489	−0.263945
$378$	0	0
$379$	8.37844	0.430371	0.215186	−	0.976573i	$-0.430964\pi$
$379$	8.37844	0.430371	0.215186	+	0.976573i	$0.430964\pi$
$380$	−3.34060	−0.171369
$381$	0	0
$382$	−3.21949	−0.164724
$383$	−4.96972	−0.253941	−0.126971	−	0.991906i	$-0.540525\pi$
$383$	−4.96972	−0.253941	−0.126971	+	0.991906i	$0.540525\pi$
$384$	0	0
$385$	6.48486	0.330499
$386$	−8.24977	−0.419902
$387$	0	0
$388$	8.16984	0.414761
$389$	−34.2791	−1.73802	−0.869010	−	0.494794i	$-0.835244\pi$
$389$	−34.2791	−1.73802	−0.869010	+	0.494794i	$0.835244\pi$
$390$	0	0
$391$	40.8392	2.06533
$392$	1.00000	0.0505076
$393$	0	0
$394$	−7.15894	−0.360662
$395$	−0.613528	−0.0308700
$396$	0	0
$397$	−35.0937	−1.76130	−0.880651	−	0.473766i	$-0.842894\pi$
$397$	−35.0937	−1.76130	−0.880651	+	0.473766i	$0.842894\pi$
$398$	−11.3406	−0.568453
$399$	0	0
$400$	−3.73463	−0.186732
$401$	26.7640	1.33653	0.668265	−	0.743923i	$-0.267037\pi$
$401$	26.7640	1.33653	0.668265	+	0.743923i	$0.267037\pi$
$402$	0	0
$403$	21.2947	1.06077
$404$	−15.2195	−0.757198
$405$	0	0
$406$	1.00000	0.0496292
$407$	12.9697	0.642885
$408$	0	0
$409$	−20.9503	−1.03593	−0.517964	−	0.855402i	$-0.673310\pi$
$409$	−20.9503	−1.03593	−0.517964	+	0.855402i	$0.673310\pi$
$410$	−8.62821	−0.426117
$411$	0	0
$412$	−3.21949	−0.158613
$413$	6.39025	0.314444
$414$	0	0
$415$	20.3709	0.999967
$416$	5.12489	0.251268
$417$	0	0
$418$	17.1202	0.837376
$419$	−32.0487	−1.56568	−0.782842	−	0.622221i	$-0.786231\pi$
$419$	−32.0487	−1.56568	−0.782842	+	0.622221i	$0.786231\pi$
$420$	0	0
$421$	28.1892	1.37386	0.686929	−	0.726724i	$-0.258958\pi$
$421$	28.1892	1.37386	0.686929	+	0.726724i	$0.258958\pi$
$422$	8.85574	0.431091
$423$	0	0
$424$	1.76491	0.0857116
$425$	20.9503	1.01624
$426$	0	0
$427$	−2.24977	−0.108874
$428$	−3.52982	−0.170620
$429$	0	0
$430$	0.826378	0.0398515
$431$	14.0606	0.677273	0.338636	−	0.940917i	$-0.390034\pi$
$431$	14.0606	0.677273	0.338636	+	0.940917i	$0.390034\pi$
$432$	0	0
$433$	−32.2380	−1.54926	−0.774629	−	0.632416i	$-0.782063\pi$
$433$	−32.2380	−1.54926	−0.774629	+	0.632416i	$0.782063\pi$
$434$	−4.15516	−0.199454
$435$	0	0
$436$	6.85574	0.328330
$437$	21.6197	1.03421
$438$	0	0
$439$	−14.0899	−0.672475	−0.336237	−	0.941777i	$-0.609154\pi$
$439$	−14.0899	−0.672475	−0.336237	+	0.941777i	$0.609154\pi$
$440$	−6.48486	−0.309154
$441$	0	0
$442$	−28.7493	−1.36747
$443$	−15.5904	−0.740721	−0.370360	−	0.928888i	$-0.620766\pi$
$443$	−15.5904	−0.740721	−0.370360	+	0.928888i	$0.620766\pi$
$444$	0	0
$445$	1.46170	0.0692914
$446$	3.21949	0.152447
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−23.0908	−1.08972	−0.544862	−	0.838526i	$-0.683418\pi$
$449$	−23.0908	−1.08972	−0.544862	+	0.838526i	$0.683418\pi$
$450$	0	0
$451$	44.2186	2.08217
$452$	18.8099	0.884742
$453$	0	0
$454$	−0.140482	−0.00659314
$455$	−5.76491	−0.270263
$456$	0	0
$457$	−19.5298	−0.913566	−0.456783	−	0.889578i	$-0.650998\pi$
$457$	−19.5298	−0.913566	−0.456783	+	0.889578i	$0.650998\pi$
$458$	−16.0294	−0.749003
$459$	0	0
$460$	−8.18922	−0.381824
$461$	5.28005	0.245916	0.122958	−	0.992412i	$-0.460762\pi$
$461$	5.28005	0.245916	0.122958	+	0.992412i	$0.460762\pi$
$462$	0	0
$463$	14.3103	0.665057	0.332529	−	0.943093i	$-0.392098\pi$
$463$	14.3103	0.665057	0.332529	+	0.943093i	$0.392098\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	17.6438	0.817333
$467$	34.7640	1.60869	0.804343	−	0.594165i	$-0.202518\pi$
$467$	34.7640	1.60869	0.804343	+	0.594165i	$0.202518\pi$
$468$	0	0
$469$	15.5298	0.717100
$470$	1.26537	0.0583670
$471$	0	0
$472$	−6.39025	−0.294135
$473$	−4.23509	−0.194730
$474$	0	0
$475$	11.0908	0.508882
$476$	5.60975	0.257122
$477$	0	0
$478$	14.4390	0.660424
$479$	−1.74553	−0.0797554	−0.0398777	−	0.999205i	$-0.512697\pi$
$479$	−1.74553	−0.0797554	−0.0398777	+	0.999205i	$0.512697\pi$
$480$	0	0
$481$	−11.5298	−0.525714
$482$	17.1736	0.782237
$483$	0	0
$484$	22.2342	1.01064
$485$	9.19014	0.417303
$486$	0	0
$487$	28.4995	1.29144	0.645719	−	0.763575i	$-0.276558\pi$
$487$	28.4995	1.29144	0.645719	+	0.763575i	$0.276558\pi$
$488$	2.24977	0.101842
$489$	0	0
$490$	1.12489	0.0508171
$491$	7.29473	0.329206	0.164603	−	0.986360i	$-0.447366\pi$
$491$	7.29473	0.329206	0.164603	+	0.986360i	$0.447366\pi$
$492$	0	0
$493$	5.60975	0.252650
$494$	−15.2195	−0.684757
$495$	0	0
$496$	4.15516	0.186572
$497$	−2.31032	−0.103632
$498$	0	0
$499$	7.52982	0.337081	0.168540	−	0.985695i	$-0.446095\pi$
$499$	7.52982	0.337081	0.168540	+	0.985695i	$0.446095\pi$
$500$	−9.82546	−0.439408
$501$	0	0
$502$	−16.3250	−0.728621
$503$	−36.5630	−1.63026	−0.815131	−	0.579277i	$-0.803335\pi$
$503$	−36.5630	−1.63026	−0.815131	+	0.579277i	$0.803335\pi$
$504$	0	0
$505$	−17.1202	−0.761838
$506$	41.9688	1.86574
$507$	0	0
$508$	−19.2195	−0.852727
$509$	5.12489	0.227157	0.113578	−	0.993529i	$-0.463769\pi$
$509$	5.12489	0.227157	0.113578	+	0.993529i	$0.463769\pi$
$510$	0	0
$511$	2.39025	0.105739
$512$	1.00000	0.0441942
$513$	0	0
$514$	−12.5142	−0.551979
$515$	−3.62156	−0.159585
$516$	0	0
$517$	−6.48486	−0.285204
$518$	2.24977	0.0988493
$519$	0	0
$520$	5.76491	0.252808
$521$	18.6959	0.819081	0.409541	−	0.912292i	$-0.365689\pi$
$521$	18.6959	0.819081	0.409541	+	0.912292i	$0.365689\pi$
$522$	0	0
$523$	3.48108	0.152217	0.0761085	−	0.997100i	$-0.475750\pi$
$523$	3.48108	0.152217	0.0761085	+	0.997100i	$0.475750\pi$
$524$	18.0294	0.787616
$525$	0	0
$526$	28.5748	1.24592
$527$	−23.3094	−1.01537
$528$	0	0
$529$	29.9991	1.30431
$530$	1.98532	0.0862368
$531$	0	0
$532$	2.96972	0.128754
$533$	−39.3094	−1.70268
$534$	0	0
$535$	−3.97064	−0.171666
$536$	−15.5298	−0.670786
$537$	0	0
$538$	13.2800	0.572543
$539$	−5.76491	−0.248312
$540$	0	0
$541$	9.65092	0.414926	0.207463	−	0.978243i	$-0.433479\pi$
$541$	9.65092	0.414926	0.207463	+	0.978243i	$0.433479\pi$
$542$	−17.9348	−0.770363
$543$	0	0
$544$	−5.60975	−0.240516
$545$	7.71192	0.330342
$546$	0	0
$547$	15.2489	0.651994	0.325997	−	0.945371i	$-0.394300\pi$
$547$	15.2489	0.651994	0.325997	+	0.945371i	$0.394300\pi$
$548$	13.8401	0.591222
$549$	0	0
$550$	21.5298	0.918035
$551$	2.96972	0.126514
$552$	0	0
$553$	0.545414	0.0231934
$554$	21.2195	0.901530
$555$	0	0
$556$	−9.13957	−0.387604
$557$	44.4390	1.88294	0.941470	−	0.337096i	$-0.109445\pi$
$557$	44.4390	1.88294	0.941470	+	0.337096i	$0.109445\pi$
$558$	0	0
$559$	3.76491	0.159239
$560$	−1.12489	−0.0475351
$561$	0	0
$562$	31.7943	1.34116
$563$	28.4461	1.19886	0.599430	−	0.800427i	$-0.295394\pi$
$563$	28.4461	1.19886	0.599430	+	0.800427i	$0.295394\pi$
$564$	0	0
$565$	21.1589	0.890163
$566$	9.79897	0.411881
$567$	0	0
$568$	2.31032	0.0969390
$569$	−35.4986	−1.48818	−0.744090	−	0.668080i	$-0.767117\pi$
$569$	−35.4986	−1.48818	−0.744090	+	0.668080i	$0.767117\pi$
$570$	0	0
$571$	−22.3709	−0.936192	−0.468096	−	0.883678i	$-0.655060\pi$
$571$	−22.3709	−0.936192	−0.468096	+	0.883678i	$0.655060\pi$
$572$	−29.5445	−1.23532
$573$	0	0
$574$	7.67030	0.320152
$575$	27.1883	1.13383
$576$	0	0
$577$	−34.5407	−1.43795	−0.718974	−	0.695037i	$-0.755388\pi$
$577$	−34.5407	−1.43795	−0.718974	+	0.695037i	$0.755388\pi$
$578$	14.4693	0.601842
$579$	0	0
$580$	−1.12489	−0.0467083
$581$	−18.1093	−0.751300
$582$	0	0
$583$	−10.1745	−0.421386
$584$	−2.39025	−0.0989094
$585$	0	0
$586$	−5.09083	−0.210300
$587$	40.6987	1.67982	0.839908	−	0.542728i	$-0.182609\pi$
$587$	40.6987	1.67982	0.839908	+	0.542728i	$0.182609\pi$
$588$	0	0
$589$	−12.3397	−0.508448
$590$	−7.18830	−0.295938
$591$	0	0
$592$	−2.24977	−0.0924650
$593$	24.6741	1.01324	0.506622	−	0.862169i	$-0.330894\pi$
$593$	24.6741	1.01324	0.506622	+	0.862169i	$0.330894\pi$
$594$	0	0
$595$	6.31032	0.258698
$596$	−4.98440	−0.204169
$597$	0	0
$598$	−37.3094	−1.52570
$599$	−13.0156	−0.531803	−0.265901	−	0.964000i	$-0.585670\pi$
$599$	−13.0156	−0.531803	−0.265901	+	0.964000i	$0.585670\pi$
$600$	0	0
$601$	6.88979	0.281041	0.140520	−	0.990078i	$-0.455122\pi$
$601$	6.88979	0.281041	0.140520	+	0.990078i	$0.455122\pi$
$602$	−0.734633	−0.0299414
$603$	0	0
$604$	−9.34060	−0.380064
$605$	25.0109	1.01684
$606$	0	0
$607$	−39.2148	−1.59168	−0.795840	−	0.605507i	$-0.792970\pi$
$607$	−39.2148	−1.59168	−0.795840	+	0.605507i	$0.792970\pi$
$608$	−2.96972	−0.120438
$609$	0	0
$610$	2.53073	0.102466
$611$	5.76491	0.233223
$612$	0	0
$613$	−13.4839	−0.544611	−0.272306	−	0.962211i	$-0.587786\pi$
$613$	−13.4839	−0.544611	−0.272306	+	0.962211i	$0.587786\pi$
$614$	−27.5445	−1.11161
$615$	0	0
$616$	5.76491	0.232275
$617$	10.4702	0.421514	0.210757	−	0.977539i	$-0.432407\pi$
$617$	10.4702	0.421514	0.210757	+	0.977539i	$0.432407\pi$
$618$	0	0
$619$	−5.64380	−0.226844	−0.113422	−	0.993547i	$-0.536181\pi$
$619$	−5.64380	−0.226844	−0.113422	+	0.993547i	$0.536181\pi$
$620$	4.67408	0.187716
$621$	0	0
$622$	4.70058	0.188476
$623$	−1.29942	−0.0520603
$624$	0	0
$625$	7.62065	0.304826
$626$	26.7346	1.06853
$627$	0	0
$628$	−6.06055	−0.241842
$629$	12.6206	0.503218
$630$	0	0
$631$	38.4390	1.53023	0.765116	−	0.643892i	$-0.222682\pi$
$631$	38.4390	1.53023	0.765116	+	0.643892i	$0.222682\pi$
$632$	−0.545414	−0.0216954
$633$	0	0
$634$	−0.0605522	−0.00240483
$635$	−21.6197	−0.857953
$636$	0	0
$637$	5.12489	0.203055
$638$	5.76491	0.228235
$639$	0	0
$640$	1.12489	0.0444650
$641$	−15.4693	−0.610999	−0.305500	−	0.952192i	$-0.598823\pi$
$641$	−15.4693	−0.610999	−0.305500	+	0.952192i	$0.598823\pi$
$642$	0	0
$643$	−16.2909	−0.642452	−0.321226	−	0.947003i	$-0.604095\pi$
$643$	−16.2909	−0.642452	−0.321226	+	0.947003i	$0.604095\pi$
$644$	7.28005	0.286874
$645$	0	0
$646$	16.6594	0.655455
$647$	−6.84106	−0.268950	−0.134475	−	0.990917i	$-0.542935\pi$
$647$	−6.84106	−0.268950	−0.134475	+	0.990917i	$0.542935\pi$
$648$	0	0
$649$	36.8392	1.44607
$650$	−19.1396	−0.750716
$651$	0	0
$652$	−4.67408	−0.183051
$653$	33.4087	1.30738	0.653692	−	0.756761i	$-0.273219\pi$
$653$	33.4087	1.30738	0.653692	+	0.756761i	$0.273219\pi$
$654$	0	0
$655$	20.2810	0.792443
$656$	−7.67030	−0.299475
$657$	0	0
$658$	−1.12489	−0.0438526
$659$	−13.1736	−0.513171	−0.256586	−	0.966521i	$-0.582598\pi$
$659$	−13.1736	−0.513171	−0.256586	+	0.966521i	$0.582598\pi$
$660$	0	0
$661$	−24.7687	−0.963390	−0.481695	−	0.876339i	$-0.659979\pi$
$661$	−24.7687	−0.963390	−0.481695	+	0.876339i	$0.659979\pi$
$662$	−34.8245	−1.35349
$663$	0	0
$664$	18.1093	0.702777
$665$	3.34060	0.129543
$666$	0	0
$667$	7.28005	0.281885
$668$	−22.4390	−0.868191
$669$	0	0
$670$	−17.4693	−0.674897
$671$	−12.9697	−0.500691
$672$	0	0
$673$	8.35620	0.322108	0.161054	−	0.986946i	$-0.448511\pi$
$673$	8.35620	0.322108	0.161054	+	0.986946i	$0.448511\pi$
$674$	−3.93945	−0.151742
$675$	0	0
$676$	13.2645	0.510171
$677$	−9.12019	−0.350517	−0.175259	−	0.984522i	$-0.556076\pi$
$677$	−9.12019	−0.350517	−0.175259	+	0.984522i	$0.556076\pi$
$678$	0	0
$679$	−8.16984	−0.313530
$680$	−6.31032	−0.241990
$681$	0	0
$682$	−23.9541	−0.917251
$683$	18.3179	0.700914	0.350457	−	0.936579i	$-0.386026\pi$
$683$	18.3179	0.700914	0.350457	+	0.936579i	$0.386026\pi$
$684$	0	0
$685$	15.5686	0.594845
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	0.734633	0.0280076
$689$	9.04496	0.344585
$690$	0	0
$691$	27.7290	1.05486	0.527431	−	0.849598i	$-0.323155\pi$
$691$	27.7290	1.05486	0.527431	+	0.849598i	$0.323155\pi$
$692$	−16.8898	−0.642054
$693$	0	0
$694$	0.310323	0.0117797
$695$	−10.2810	−0.389979
$696$	0	0
$697$	43.0284	1.62982
$698$	26.0946	0.987696
$699$	0	0
$700$	3.73463	0.141156
$701$	28.5142	1.07697	0.538484	−	0.842636i	$-0.318997\pi$
$701$	28.5142	1.07697	0.538484	+	0.842636i	$0.318997\pi$
$702$	0	0
$703$	6.68120	0.251986
$704$	−5.76491	−0.217273
$705$	0	0
$706$	−19.1202	−0.719598
$707$	15.2195	0.572388
$708$	0	0
$709$	16.3250	0.613098	0.306549	−	0.951855i	$-0.400826\pi$
$709$	16.3250	0.613098	0.306549	+	0.951855i	$0.400826\pi$
$710$	2.59885	0.0975331
$711$	0	0
$712$	1.29942	0.0486980
$713$	−30.2498	−1.13286
$714$	0	0
$715$	−33.2342	−1.24289
$716$	−12.4995	−0.467130
$717$	0	0
$718$	6.98440	0.260656
$719$	53.3094	1.98811	0.994053	−	0.108900i	$-0.0347329\pi$
$719$	53.3094	1.98811	0.994053	+	0.108900i	$0.0347329\pi$
$720$	0	0
$721$	3.21949	0.119900
$722$	−10.1807	−0.378888
$723$	0	0
$724$	22.1240	0.822231
$725$	3.73463	0.138701
$726$	0	0
$727$	−18.7999	−0.697249	−0.348625	−	0.937262i	$-0.613351\pi$
$727$	−18.7999	−0.697249	−0.348625	+	0.937262i	$0.613351\pi$
$728$	−5.12489	−0.189941
$729$	0	0
$730$	−2.68876	−0.0995155
$731$	−4.12110	−0.152425
$732$	0	0
$733$	−6.74931	−0.249292	−0.124646	−	0.992201i	$-0.539779\pi$
$733$	−6.74931	−0.249292	−0.124646	+	0.992201i	$0.539779\pi$
$734$	11.9201	0.439978
$735$	0	0
$736$	−7.28005	−0.268346
$737$	89.5280	3.29781
$738$	0	0
$739$	−22.7952	−0.838534	−0.419267	−	0.907863i	$-0.637713\pi$
$739$	−22.7952	−0.838534	−0.419267	+	0.907863i	$0.637713\pi$
$740$	−2.53073	−0.0930316
$741$	0	0
$742$	−1.76491	−0.0647918
$743$	−14.5601	−0.534158	−0.267079	−	0.963675i	$-0.586058\pi$
$743$	−14.5601	−0.534158	−0.267079	+	0.963675i	$0.586058\pi$
$744$	0	0
$745$	−5.60688	−0.205420
$746$	−24.0147	−0.879240
$747$	0	0
$748$	32.3397	1.18246
$749$	3.52982	0.128977
$750$	0	0
$751$	6.93853	0.253191	0.126595	−	0.991954i	$-0.459595\pi$
$751$	6.93853	0.253191	0.126595	+	0.991954i	$0.459595\pi$
$752$	1.12489	0.0410204
$753$	0	0
$754$	−5.12489	−0.186637
$755$	−10.5071	−0.382393
$756$	0	0
$757$	15.5979	0.566916	0.283458	−	0.958985i	$-0.408518\pi$
$757$	15.5979	0.566916	0.283458	+	0.958985i	$0.408518\pi$
$758$	8.37844	0.304319
$759$	0	0
$760$	−3.34060	−0.121176
$761$	27.2413	0.987496	0.493748	−	0.869605i	$-0.335627\pi$
$761$	27.2413	0.987496	0.493748	+	0.869605i	$0.335627\pi$
$762$	0	0
$763$	−6.85574	−0.248194
$764$	−3.21949	−0.116477
$765$	0	0
$766$	−4.96972	−0.179563
$767$	−32.7493	−1.18251
$768$	0	0
$769$	−23.2682	−0.839074	−0.419537	−	0.907738i	$-0.637808\pi$
$769$	−23.2682	−0.839074	−0.419537	+	0.907738i	$0.637808\pi$
$770$	6.48486	0.233698
$771$	0	0
$772$	−8.24977	−0.296916
$773$	−23.0596	−0.829397	−0.414699	−	0.909959i	$-0.636113\pi$
$773$	−23.0596	−0.829397	−0.414699	+	0.909959i	$0.636113\pi$
$774$	0	0
$775$	−15.5180	−0.557423
$776$	8.16984	0.293280
$777$	0	0
$778$	−34.2791	−1.22897
$779$	22.7787	0.816131
$780$	0	0
$781$	−13.3188	−0.476584
$782$	40.8392	1.46041
$783$	0	0
$784$	1.00000	0.0357143
$785$	−6.81743	−0.243324
$786$	0	0
$787$	23.0109	0.820250	0.410125	−	0.912029i	$-0.365485\pi$
$787$	23.0109	0.820250	0.410125	+	0.912029i	$0.365485\pi$
$788$	−7.15894	−0.255027
$789$	0	0
$790$	−0.613528	−0.0218284
$791$	−18.8099	−0.668802
$792$	0	0
$793$	11.5298	0.409436
$794$	−35.0937	−1.24543
$795$	0	0
$796$	−11.3406	−0.401957
$797$	−26.5601	−0.940807	−0.470403	−	0.882451i	$-0.655892\pi$
$797$	−26.5601	−0.940807	−0.470403	+	0.882451i	$0.655892\pi$
$798$	0	0
$799$	−6.31032	−0.223243
$800$	−3.73463	−0.132039
$801$	0	0
$802$	26.7640	0.945069
$803$	13.7796	0.486271
$804$	0	0
$805$	8.18922	0.288632
$806$	21.2947	0.750075
$807$	0	0
$808$	−15.2195	−0.535420
$809$	30.0975	1.05817	0.529085	−	0.848569i	$-0.322535\pi$
$809$	30.0975	1.05817	0.529085	+	0.848569i	$0.322535\pi$
$810$	0	0
$811$	−11.0790	−0.389037	−0.194518	−	0.980899i	$-0.562314\pi$
$811$	−11.0790	−0.389037	−0.194518	+	0.980899i	$0.562314\pi$
$812$	1.00000	0.0350931
$813$	0	0
$814$	12.9697	0.454589
$815$	−5.25781	−0.184173
$816$	0	0
$817$	−2.18166	−0.0763265
$818$	−20.9503	−0.732512
$819$	0	0
$820$	−8.62821	−0.301310
$821$	36.3544	1.26878	0.634388	−	0.773015i	$-0.281252\pi$
$821$	36.3544	1.26878	0.634388	+	0.773015i	$0.281252\pi$
$822$	0	0
$823$	7.18074	0.250305	0.125152	−	0.992138i	$-0.460058\pi$
$823$	7.18074	0.250305	0.125152	+	0.992138i	$0.460058\pi$
$824$	−3.21949	−0.112156
$825$	0	0
$826$	6.39025	0.222345
$827$	20.7952	0.723119	0.361560	−	0.932349i	$-0.382244\pi$
$827$	20.7952	0.723119	0.361560	+	0.932349i	$0.382244\pi$
$828$	0	0
$829$	22.8411	0.793303	0.396651	−	0.917969i	$-0.370172\pi$
$829$	22.8411	0.793303	0.396651	+	0.917969i	$0.370172\pi$
$830$	20.3709	0.707083
$831$	0	0
$832$	5.12489	0.177673
$833$	−5.60975	−0.194366
$834$	0	0
$835$	−25.2413	−0.873511
$836$	17.1202	0.592114
$837$	0	0
$838$	−32.0487	−1.10711
$839$	36.0634	1.24505	0.622524	−	0.782601i	$-0.286107\pi$
$839$	36.0634	1.24505	0.622524	+	0.782601i	$0.286107\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	28.1892	0.971465
$843$	0	0
$844$	8.85574	0.304827
$845$	14.9210	0.513298
$846$	0	0
$847$	−22.2342	−0.763975
$848$	1.76491	0.0606072
$849$	0	0
$850$	20.9503	0.718591
$851$	16.3784	0.561446
$852$	0	0
$853$	24.1892	0.828223	0.414112	−	0.910226i	$-0.364092\pi$
$853$	24.1892	0.828223	0.414112	+	0.910226i	$0.364092\pi$
$854$	−2.24977	−0.0769856
$855$	0	0
$856$	−3.52982	−0.120647
$857$	−29.6732	−1.01362	−0.506808	−	0.862059i	$-0.669175\pi$
$857$	−29.6732	−1.01362	−0.506808	+	0.862059i	$0.669175\pi$
$858$	0	0
$859$	25.9835	0.886545	0.443273	−	0.896387i	$-0.353817\pi$
$859$	25.9835	0.886545	0.443273	+	0.896387i	$0.353817\pi$
$860$	0.826378	0.0281792
$861$	0	0
$862$	14.0606	0.478904
$863$	−5.84014	−0.198801	−0.0994004	−	0.995048i	$-0.531692\pi$
$863$	−5.84014	−0.198801	−0.0994004	+	0.995048i	$0.531692\pi$
$864$	0	0
$865$	−18.9991	−0.645988
$866$	−32.2380	−1.09549
$867$	0	0
$868$	−4.15516	−0.141035
$869$	3.14426	0.106662
$870$	0	0
$871$	−79.5885	−2.69675
$872$	6.85574	0.232165
$873$	0	0
$874$	21.6197	0.731298
$875$	9.82546	0.332161
$876$	0	0
$877$	−1.22661	−0.0414198	−0.0207099	−	0.999786i	$-0.506593\pi$
$877$	−1.22661	−0.0414198	−0.0207099	+	0.999786i	$0.506593\pi$
$878$	−14.0899	−0.475511
$879$	0	0
$880$	−6.48486	−0.218605
$881$	−11.0790	−0.373261	−0.186631	−	0.982430i	$-0.559757\pi$
$881$	−11.0790	−0.373261	−0.186631	+	0.982430i	$0.559757\pi$
$882$	0	0
$883$	35.0303	1.17886	0.589431	−	0.807818i	$-0.299352\pi$
$883$	35.0303	1.17886	0.589431	+	0.807818i	$0.299352\pi$
$884$	−28.7493	−0.966944
$885$	0	0
$886$	−15.5904	−0.523769
$887$	20.6253	0.692531	0.346266	−	0.938137i	$-0.387450\pi$
$887$	20.6253	0.692531	0.346266	+	0.938137i	$0.387450\pi$
$888$	0	0
$889$	19.2195	0.644601
$890$	1.46170	0.0489964
$891$	0	0
$892$	3.21949	0.107797
$893$	−3.34060	−0.111789
$894$	0	0
$895$	−14.0606	−0.469992
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−23.0908	−0.770551
$899$	−4.15516	−0.138582
$900$	0	0
$901$	−9.90069	−0.329840
$902$	44.2186	1.47232
$903$	0	0
$904$	18.8099	0.625607
$905$	24.8869	0.827270
$906$	0	0
$907$	−6.34908	−0.210818	−0.105409	−	0.994429i	$-0.533615\pi$
$907$	−6.34908	−0.210818	−0.105409	+	0.994429i	$0.533615\pi$
$908$	−0.140482	−0.00466205
$909$	0	0
$910$	−5.76491	−0.191105
$911$	−19.1349	−0.633966	−0.316983	−	0.948431i	$-0.602670\pi$
$911$	−19.1349	−0.633966	−0.316983	+	0.948431i	$0.602670\pi$
$912$	0	0
$913$	−104.398	−3.45508
$914$	−19.5298	−0.645989
$915$	0	0
$916$	−16.0294	−0.529625
$917$	−18.0294	−0.595382
$918$	0	0
$919$	9.71904	0.320601	0.160301	−	0.987068i	$-0.448754\pi$
$919$	9.71904	0.320601	0.160301	+	0.987068i	$0.448754\pi$
$920$	−8.18922	−0.269991
$921$	0	0
$922$	5.28005	0.173889
$923$	11.8401	0.389723
$924$	0	0
$925$	8.40207	0.276258
$926$	14.3103	0.470266
$927$	0	0
$928$	−1.00000	−0.0328266
$929$	49.1807	1.61357	0.806784	−	0.590847i	$-0.201206\pi$
$929$	49.1807	1.61357	0.806784	+	0.590847i	$0.201206\pi$
$930$	0	0
$931$	−2.96972	−0.0973288
$932$	17.6438	0.577942
$933$	0	0
$934$	34.7640	1.13751
$935$	36.3784	1.18970
$936$	0	0
$937$	26.4995	0.865702	0.432851	−	0.901465i	$-0.357508\pi$
$937$	26.4995	0.865702	0.432851	+	0.901465i	$0.357508\pi$
$938$	15.5298	0.507067
$939$	0	0
$940$	1.26537	0.0412717
$941$	51.5251	1.67967	0.839835	−	0.542841i	$-0.182651\pi$
$941$	51.5251	1.67967	0.839835	+	0.542841i	$0.182651\pi$
$942$	0	0
$943$	55.8401	1.81841
$944$	−6.39025	−0.207985
$945$	0	0
$946$	−4.23509	−0.137695
$947$	−49.7337	−1.61613	−0.808064	−	0.589094i	$-0.799485\pi$
$947$	−49.7337	−1.61613	−0.808064	+	0.589094i	$0.799485\pi$
$948$	0	0
$949$	−12.2498	−0.397644
$950$	11.0908	0.359834
$951$	0	0
$952$	5.60975	0.181813
$953$	32.1727	1.04218	0.521088	−	0.853503i	$-0.325526\pi$
$953$	32.1727	1.04218	0.521088	+	0.853503i	$0.325526\pi$
$954$	0	0
$955$	−3.62156	−0.117191
$956$	14.4390	0.466990
$957$	0	0
$958$	−1.74553	−0.0563956
$959$	−13.8401	−0.446921
$960$	0	0
$961$	−13.7346	−0.443053
$962$	−11.5298	−0.371736
$963$	0	0
$964$	17.1736	0.553125
$965$	−9.28005	−0.298735
$966$	0	0
$967$	−57.7337	−1.85659	−0.928296	−	0.371843i	$-0.878726\pi$
$967$	−57.7337	−1.85659	−0.928296	+	0.371843i	$0.878726\pi$
$968$	22.2342	0.714633
$969$	0	0
$970$	9.19014	0.295077
$971$	2.15046	0.0690117	0.0345058	−	0.999404i	$-0.489014\pi$
$971$	2.15046	0.0690117	0.0345058	+	0.999404i	$0.489014\pi$
$972$	0	0
$973$	9.13957	0.293001
$974$	28.4995	0.913184
$975$	0	0
$976$	2.24977	0.0720134
$977$	−58.1122	−1.85917	−0.929586	−	0.368605i	$-0.879836\pi$
$977$	−58.1122	−1.85917	−0.929586	+	0.368605i	$0.879836\pi$
$978$	0	0
$979$	−7.49106	−0.239415
$980$	1.12489	0.0359331
$981$	0	0
$982$	7.29473	0.232784
$983$	13.2460	0.422481	0.211241	−	0.977434i	$-0.432250\pi$
$983$	13.2460	0.422481	0.211241	+	0.977434i	$0.432250\pi$
$984$	0	0
$985$	−8.05299	−0.256590
$986$	5.60975	0.178651
$987$	0	0
$988$	−15.2195	−0.484197
$989$	−5.34816	−0.170062
$990$	0	0
$991$	22.5895	0.717578	0.358789	−	0.933419i	$-0.383190\pi$
$991$	22.5895	0.717578	0.358789	+	0.933419i	$0.383190\pi$
$992$	4.15516	0.131927
$993$	0	0
$994$	−2.31032	−0.0732790
$995$	−12.7569	−0.404420
$996$	0	0
$997$	−17.8089	−0.564015	−0.282008	−	0.959412i	$-0.591000\pi$
$997$	−17.8089	−0.564015	−0.282008	+	0.959412i	$0.591000\pi$
$998$	7.52982	0.238352
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3654.2.a.bc.1.3		3
3.2	odd	2		406.2.a.f.1.2	✓	3
12.11	even	2		3248.2.a.u.1.2		3
21.20	even	2		2842.2.a.n.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
406.2.a.f.1.2	✓	3	3.2	odd	2
2842.2.a.n.1.2		3	21.20	even	2
3248.2.a.u.1.2		3	12.11	even	2
3654.2.a.bc.1.3		3	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:	\( N \)	\(=\)	\( 3654 = 2 \cdot 3^{2} \cdot 7 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3654.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.12489 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.12489 q^{10} -5.76491 q^{11} +5.12489 q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.60975 q^{17} -2.96972 q^{19} +1.12489 q^{20} -5.76491 q^{22} -7.28005 q^{23} -3.73463 q^{25} +5.12489 q^{26} -1.00000 q^{28} -1.00000 q^{29} +4.15516 q^{31} +1.00000 q^{32} -5.60975 q^{34} -1.12489 q^{35} -2.24977 q^{37} -2.96972 q^{38} +1.12489 q^{40} -7.67030 q^{41} +0.734633 q^{43} -5.76491 q^{44} -7.28005 q^{46} +1.12489 q^{47} +1.00000 q^{49} -3.73463 q^{50} +5.12489 q^{52} +1.76491 q^{53} -6.48486 q^{55} -1.00000 q^{56} -1.00000 q^{58} -6.39025 q^{59} +2.24977 q^{61} +4.15516 q^{62} +1.00000 q^{64} +5.76491 q^{65} -15.5298 q^{67} -5.60975 q^{68} -1.12489 q^{70} +2.31032 q^{71} -2.39025 q^{73} -2.24977 q^{74} -2.96972 q^{76} +5.76491 q^{77} -0.545414 q^{79} +1.12489 q^{80} -7.67030 q^{82} +18.1093 q^{83} -6.31032 q^{85} +0.734633 q^{86} -5.76491 q^{88} +1.29942 q^{89} -5.12489 q^{91} -7.28005 q^{92} +1.12489 q^{94} -3.34060 q^{95} +8.16984 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 5 q^{5} - 3 q^{7} + 3 q^{8} - 5 q^{10} - q^{11} + 7 q^{13} - 3 q^{14} + 3 q^{16} - 8 q^{17} - 8 q^{19} - 5 q^{20} - q^{22} - 6 q^{23} + 6 q^{25} + 7 q^{26} - 3 q^{28} - 3 q^{29}+ \cdots + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 5 * q^5 - 3 * q^7 + 3 * q^8 - 5 * q^10 - q^11 + 7 * q^13 - 3 * q^14 + 3 * q^16 - 8 * q^17 - 8 * q^19 - 5 * q^20 - q^22 - 6 * q^23 + 6 * q^25 + 7 * q^26 - 3 * q^28 - 3 * q^29 + 5 * q^31 + 3 * q^32 - 8 * q^34 + 5 * q^35 + 10 * q^37 - 8 * q^38 - 5 * q^40 - 16 * q^41 - 15 * q^43 - q^44 - 6 * q^46 - 5 * q^47 + 3 * q^49 + 6 * q^50 + 7 * q^52 - 11 * q^53 - 19 * q^55 - 3 * q^56 - 3 * q^58 - 28 * q^59 - 10 * q^61 + 5 * q^62 + 3 * q^64 + q^65 - 14 * q^67 - 8 * q^68 + 5 * q^70 - 8 * q^71 - 16 * q^73 + 10 * q^74 - 8 * q^76 + q^77 - 3 * q^79 - 5 * q^80 - 16 * q^82 + 12 * q^83 - 4 * q^85 - 15 * q^86 - q^88 + 10 * q^89 - 7 * q^91 - 6 * q^92 - 5 * q^94 + 4 * q^95 - 16 * q^97 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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