LMFDB - Embedded newform 6174.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6174,2,Mod(1,6174)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6174, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6174.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6174 = 2 \cdot 3^{2} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6174.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$49.2996382079$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2058)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	6174.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -0.554958 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -0.554958 q^{5} +1.00000 q^{8} -0.554958 q^{10} -4.54288 q^{11} -5.96077 q^{13} +1.00000 q^{16} +7.18598 q^{17} -2.30798 q^{19} -0.554958 q^{20} -4.54288 q^{22} +3.80194 q^{23} -4.69202 q^{25} -5.96077 q^{26} +10.1588 q^{29} -3.02715 q^{31} +1.00000 q^{32} +7.18598 q^{34} -0.0217703 q^{37} -2.30798 q^{38} -0.554958 q^{40} +1.77479 q^{41} +4.93900 q^{43} -4.54288 q^{44} +3.80194 q^{46} -6.94869 q^{47} -4.69202 q^{50} -5.96077 q^{52} +5.63102 q^{53} +2.52111 q^{55} +10.1588 q^{58} +13.2959 q^{59} +11.7235 q^{61} -3.02715 q^{62} +1.00000 q^{64} +3.30798 q^{65} +13.2959 q^{67} +7.18598 q^{68} +0.423272 q^{71} -14.3327 q^{73} -0.0217703 q^{74} -2.30798 q^{76} +14.2446 q^{79} -0.554958 q^{80} +1.77479 q^{82} +2.41789 q^{83} -3.98792 q^{85} +4.93900 q^{86} -4.54288 q^{88} +4.83877 q^{89} +3.80194 q^{92} -6.94869 q^{94} +1.28083 q^{95} -11.9051 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{8} - 2 q^{10} + 5 q^{11} - 5 q^{13} + 3 q^{16} + 7 q^{17} - 12 q^{19} - 2 q^{20} + 5 q^{22} + 7 q^{23} - 9 q^{25} - 5 q^{26} + 22 q^{29} - 3 q^{31} + 3 q^{32} + 7 q^{34} + 3 q^{37} - 12 q^{38} - 2 q^{40} + 7 q^{41} + 5 q^{43} + 5 q^{44} + 7 q^{46} + 11 q^{47} - 9 q^{50} - 5 q^{52} + 2 q^{53} - 8 q^{55} + 22 q^{58} + 26 q^{59} + 5 q^{61} - 3 q^{62} + 3 q^{64} + 15 q^{65} + 26 q^{67} + 7 q^{68} + 4 q^{71} - q^{73} + 3 q^{74} - 12 q^{76} - 3 q^{79} - 2 q^{80} + 7 q^{82} + 13 q^{83} + 7 q^{85} + 5 q^{86} + 5 q^{88} - 18 q^{89} + 7 q^{92} + 11 q^{94} + 15 q^{95} - 10 q^{97}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^8 - 2 * q^10 + 5 * q^11 - 5 * q^13 + 3 * q^16 + 7 * q^17 - 12 * q^19 - 2 * q^20 + 5 * q^22 + 7 * q^23 - 9 * q^25 - 5 * q^26 + 22 * q^29 - 3 * q^31 + 3 * q^32 + 7 * q^34 + 3 * q^37 - 12 * q^38 - 2 * q^40 + 7 * q^41 + 5 * q^43 + 5 * q^44 + 7 * q^46 + 11 * q^47 - 9 * q^50 - 5 * q^52 + 2 * q^53 - 8 * q^55 + 22 * q^58 + 26 * q^59 + 5 * q^61 - 3 * q^62 + 3 * q^64 + 15 * q^65 + 26 * q^67 + 7 * q^68 + 4 * q^71 - q^73 + 3 * q^74 - 12 * q^76 - 3 * q^79 - 2 * q^80 + 7 * q^82 + 13 * q^83 + 7 * q^85 + 5 * q^86 + 5 * q^88 - 18 * q^89 + 7 * q^92 + 11 * q^94 + 15 * q^95 - 10 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−0.554958	−0.248185	−0.124092	−	0.992271i	$-0.539602\pi$
$5$	−0.554958	−0.248185	−0.124092	+	0.992271i	$0.539602\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	−0.554958	−0.175493
$11$	−4.54288	−1.36973	−0.684864	−	0.728671i	$-0.740139\pi$
$11$	−4.54288	−1.36973	−0.684864	+	0.728671i	$0.740139\pi$
$12$	0	0
$13$	−5.96077	−1.65322	−0.826610	−	0.562775i	$-0.809734\pi$
$13$	−5.96077	−1.65322	−0.826610	+	0.562775i	$0.809734\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	7.18598	1.74286	0.871428	−	0.490523i	$-0.163194\pi$
$17$	7.18598	1.74286	0.871428	+	0.490523i	$0.163194\pi$
$18$	0	0
$19$	−2.30798	−0.529487	−0.264743	−	0.964319i	$-0.585287\pi$
$19$	−2.30798	−0.529487	−0.264743	+	0.964319i	$0.585287\pi$
$20$	−0.554958	−0.124092
$21$	0	0
$22$	−4.54288	−0.968545
$23$	3.80194	0.792759	0.396379	−	0.918087i	$-0.370266\pi$
$23$	3.80194	0.792759	0.396379	+	0.918087i	$0.370266\pi$
$24$	0	0
$25$	−4.69202	−0.938404
$26$	−5.96077	−1.16900
$27$	0	0
$28$	0	0
$29$	10.1588	1.88645	0.943224	−	0.332157i	$-0.107776\pi$
$29$	10.1588	1.88645	0.943224	+	0.332157i	$0.107776\pi$
$30$	0	0
$31$	−3.02715	−0.543692	−0.271846	−	0.962341i	$-0.587634\pi$
$31$	−3.02715	−0.543692	−0.271846	+	0.962341i	$0.587634\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.18598	1.23239
$35$	0	0
$36$	0	0
$37$	−0.0217703	−0.00357901	−0.00178950	−	0.999998i	$-0.500570\pi$
$37$	−0.0217703	−0.00357901	−0.00178950	+	0.999998i	$0.500570\pi$
$38$	−2.30798	−0.374404
$39$	0	0
$40$	−0.554958	−0.0877466
$41$	1.77479	0.277176	0.138588	−	0.990350i	$-0.455744\pi$
$41$	1.77479	0.277176	0.138588	+	0.990350i	$0.455744\pi$
$42$	0	0
$43$	4.93900	0.753191	0.376595	−	0.926378i	$-0.377095\pi$
$43$	4.93900	0.753191	0.376595	+	0.926378i	$0.377095\pi$
$44$	−4.54288	−0.684864
$45$	0	0
$46$	3.80194	0.560565
$47$	−6.94869	−1.01357	−0.506785	−	0.862072i	$-0.669166\pi$
$47$	−6.94869	−1.01357	−0.506785	+	0.862072i	$0.669166\pi$
$48$	0	0
$49$	0	0
$50$	−4.69202	−0.663552
$51$	0	0
$52$	−5.96077	−0.826610
$53$	5.63102	0.773480	0.386740	−	0.922189i	$-0.373601\pi$
$53$	5.63102	0.773480	0.386740	+	0.922189i	$0.373601\pi$
$54$	0	0
$55$	2.52111	0.339946
$56$	0	0
$57$	0	0
$58$	10.1588	1.33392
$59$	13.2959	1.73098	0.865489	−	0.500928i	$-0.167008\pi$
$59$	13.2959	1.73098	0.865489	+	0.500928i	$0.167008\pi$
$60$	0	0
$61$	11.7235	1.50104	0.750519	−	0.660849i	$-0.229804\pi$
$61$	11.7235	1.50104	0.750519	+	0.660849i	$0.229804\pi$
$62$	−3.02715	−0.384448
$63$	0	0
$64$	1.00000	0.125000
$65$	3.30798	0.410304
$66$	0	0
$67$	13.2959	1.62435	0.812176	−	0.583412i	$-0.198283\pi$
$67$	13.2959	1.62435	0.812176	+	0.583412i	$0.198283\pi$
$68$	7.18598	0.871428
$69$	0	0
$70$	0	0
$71$	0.423272	0.0502331	0.0251165	−	0.999685i	$-0.492004\pi$
$71$	0.423272	0.0502331	0.0251165	+	0.999685i	$0.492004\pi$
$72$	0	0
$73$	−14.3327	−1.67752	−0.838760	−	0.544502i	$-0.816719\pi$
$73$	−14.3327	−1.67752	−0.838760	+	0.544502i	$0.816719\pi$
$74$	−0.0217703	−0.00253074
$75$	0	0
$76$	−2.30798	−0.264743
$77$	0	0
$78$	0	0
$79$	14.2446	1.60264	0.801321	−	0.598235i	$-0.204131\pi$
$79$	14.2446	1.60264	0.801321	+	0.598235i	$0.204131\pi$
$80$	−0.554958	−0.0620462
$81$	0	0
$82$	1.77479	0.195993
$83$	2.41789	0.265398	0.132699	−	0.991156i	$-0.457636\pi$
$83$	2.41789	0.265398	0.132699	+	0.991156i	$0.457636\pi$
$84$	0	0
$85$	−3.98792	−0.432550
$86$	4.93900	0.532586
$87$	0	0
$88$	−4.54288	−0.484272
$89$	4.83877	0.512909	0.256454	−	0.966556i	$-0.417446\pi$
$89$	4.83877	0.512909	0.256454	+	0.966556i	$0.417446\pi$
$90$	0	0
$91$	0	0
$92$	3.80194	0.396379
$93$	0	0
$94$	−6.94869	−0.716703
$95$	1.28083	0.131411
$96$	0	0
$97$	−11.9051	−1.20878	−0.604392	−	0.796687i	$-0.706584\pi$
$97$	−11.9051	−1.20878	−0.604392	+	0.796687i	$0.706584\pi$
$98$	0	0
$99$	0	0
$100$	−4.69202	−0.469202
$101$	−16.6189	−1.65365	−0.826823	−	0.562462i	$-0.809854\pi$
$101$	−16.6189	−1.65365	−0.826823	+	0.562462i	$0.809854\pi$
$102$	0	0
$103$	12.1075	1.19299	0.596495	−	0.802617i	$-0.296560\pi$
$103$	12.1075	1.19299	0.596495	+	0.802617i	$0.296560\pi$
$104$	−5.96077	−0.584502
$105$	0	0
$106$	5.63102	0.546933
$107$	15.0465	1.45460	0.727301	−	0.686318i	$-0.240774\pi$
$107$	15.0465	1.45460	0.727301	+	0.686318i	$0.240774\pi$
$108$	0	0
$109$	−0.521106	−0.0499129	−0.0249565	−	0.999689i	$-0.507945\pi$
$109$	−0.521106	−0.0499129	−0.0249565	+	0.999689i	$0.507945\pi$
$110$	2.52111	0.240378
$111$	0	0
$112$	0	0
$113$	5.41119	0.509042	0.254521	−	0.967067i	$-0.418082\pi$
$113$	5.41119	0.509042	0.254521	+	0.967067i	$0.418082\pi$
$114$	0	0
$115$	−2.10992	−0.196751
$116$	10.1588	0.943224
$117$	0	0
$118$	13.2959	1.22399
$119$	0	0
$120$	0	0
$121$	9.63773	0.876157
$122$	11.7235	1.06139
$123$	0	0
$124$	−3.02715	−0.271846
$125$	5.37867	0.481083
$126$	0	0
$127$	9.80731	0.870258	0.435129	−	0.900368i	$-0.356703\pi$
$127$	9.80731	0.870258	0.435129	+	0.900368i	$0.356703\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.30798	0.290129
$131$	0.188374	0.0164583	0.00822914	−	0.999966i	$-0.497381\pi$
$131$	0.188374	0.0164583	0.00822914	+	0.999966i	$0.497381\pi$
$132$	0	0
$133$	0	0
$134$	13.2959	1.14859
$135$	0	0
$136$	7.18598	0.616193
$137$	−2.76271	−0.236034	−0.118017	−	0.993012i	$-0.537654\pi$
$137$	−2.76271	−0.236034	−0.118017	+	0.993012i	$0.537654\pi$
$138$	0	0
$139$	−3.45712	−0.293229	−0.146615	−	0.989194i	$-0.546838\pi$
$139$	−3.45712	−0.293229	−0.146615	+	0.989194i	$0.546838\pi$
$140$	0	0
$141$	0	0
$142$	0.423272	0.0355202
$143$	27.0790	2.26446
$144$	0	0
$145$	−5.63773	−0.468188
$146$	−14.3327	−1.18619
$147$	0	0
$148$	−0.0217703	−0.00178950
$149$	−9.74632	−0.798449	−0.399225	−	0.916853i	$-0.630721\pi$
$149$	−9.74632	−0.798449	−0.399225	+	0.916853i	$0.630721\pi$
$150$	0	0
$151$	−3.81700	−0.310623	−0.155312	−	0.987866i	$-0.549638\pi$
$151$	−3.81700	−0.310623	−0.155312	+	0.987866i	$0.549638\pi$
$152$	−2.30798	−0.187202
$153$	0	0
$154$	0	0
$155$	1.67994	0.134936
$156$	0	0
$157$	−3.12737	−0.249592	−0.124796	−	0.992182i	$-0.539828\pi$
$157$	−3.12737	−0.249592	−0.124796	+	0.992182i	$0.539828\pi$
$158$	14.2446	1.13324
$159$	0	0
$160$	−0.554958	−0.0438733
$161$	0	0
$162$	0	0
$163$	13.2905	1.04099	0.520497	−	0.853864i	$-0.325747\pi$
$163$	13.2905	1.04099	0.520497	+	0.853864i	$0.325747\pi$
$164$	1.77479	0.138588
$165$	0	0
$166$	2.41789	0.187665
$167$	3.17629	0.245789	0.122894	−	0.992420i	$-0.460782\pi$
$167$	3.17629	0.245789	0.122894	+	0.992420i	$0.460782\pi$
$168$	0	0
$169$	22.5308	1.73314
$170$	−3.98792	−0.305859
$171$	0	0
$172$	4.93900	0.376595
$173$	13.9511	1.06068	0.530341	−	0.847785i	$-0.322064\pi$
$173$	13.9511	1.06068	0.530341	+	0.847785i	$0.322064\pi$
$174$	0	0
$175$	0	0
$176$	−4.54288	−0.342432
$177$	0	0
$178$	4.83877	0.362681
$179$	3.35690	0.250906	0.125453	−	0.992100i	$-0.459962\pi$
$179$	3.35690	0.250906	0.125453	+	0.992100i	$0.459962\pi$
$180$	0	0
$181$	23.4306	1.74158	0.870790	−	0.491655i	$-0.163608\pi$
$181$	23.4306	1.74158	0.870790	+	0.491655i	$0.163608\pi$
$182$	0	0
$183$	0	0
$184$	3.80194	0.280283
$185$	0.0120816	0.000888256 0
$186$	0	0
$187$	−32.6450	−2.38724
$188$	−6.94869	−0.506785
$189$	0	0
$190$	1.28083	0.0929213
$191$	−3.71917	−0.269110	−0.134555	−	0.990906i	$-0.542960\pi$
$191$	−3.71917	−0.269110	−0.134555	+	0.990906i	$0.542960\pi$
$192$	0	0
$193$	−12.7603	−0.918508	−0.459254	−	0.888305i	$-0.651883\pi$
$193$	−12.7603	−0.918508	−0.459254	+	0.888305i	$0.651883\pi$
$194$	−11.9051	−0.854740
$195$	0	0
$196$	0	0
$197$	−10.8901	−0.775886	−0.387943	−	0.921683i	$-0.626814\pi$
$197$	−10.8901	−0.775886	−0.387943	+	0.921683i	$0.626814\pi$
$198$	0	0
$199$	0.841166	0.0596287	0.0298144	−	0.999555i	$-0.490508\pi$
$199$	0.841166	0.0596287	0.0298144	+	0.999555i	$0.490508\pi$
$200$	−4.69202	−0.331776
$201$	0	0
$202$	−16.6189	−1.16930
$203$	0	0
$204$	0	0
$205$	−0.984935	−0.0687908
$206$	12.1075	0.843571
$207$	0	0
$208$	−5.96077	−0.413305
$209$	10.4849	0.725253
$210$	0	0
$211$	−14.6571	−1.00904	−0.504518	−	0.863401i	$-0.668330\pi$
$211$	−14.6571	−1.00904	−0.504518	+	0.863401i	$0.668330\pi$
$212$	5.63102	0.386740
$213$	0	0
$214$	15.0465	1.02856
$215$	−2.74094	−0.186930
$216$	0	0
$217$	0	0
$218$	−0.521106	−0.0352938
$219$	0	0
$220$	2.52111	0.169973
$221$	−42.8340	−2.88133
$222$	0	0
$223$	−21.9530	−1.47008	−0.735041	−	0.678023i	$-0.762837\pi$
$223$	−21.9530	−1.47008	−0.735041	+	0.678023i	$0.762837\pi$
$224$	0	0
$225$	0	0
$226$	5.41119	0.359947
$227$	−6.94869	−0.461201	−0.230600	−	0.973049i	$-0.574069\pi$
$227$	−6.94869	−0.461201	−0.230600	+	0.973049i	$0.574069\pi$
$228$	0	0
$229$	−5.72886	−0.378574	−0.189287	−	0.981922i	$-0.560618\pi$
$229$	−5.72886	−0.378574	−0.189287	+	0.981922i	$0.560618\pi$
$230$	−2.10992	−0.139124
$231$	0	0
$232$	10.1588	0.666960
$233$	14.8073	0.970059	0.485030	−	0.874498i	$-0.338809\pi$
$233$	14.8073	0.970059	0.485030	+	0.874498i	$0.338809\pi$
$234$	0	0
$235$	3.85623	0.251553
$236$	13.2959	0.865489
$237$	0	0
$238$	0	0
$239$	11.9444	0.772618	0.386309	−	0.922370i	$-0.373750\pi$
$239$	11.9444	0.772618	0.386309	+	0.922370i	$0.373750\pi$
$240$	0	0
$241$	0.625646	0.0403014	0.0201507	−	0.999797i	$-0.493585\pi$
$241$	0.625646	0.0403014	0.0201507	+	0.999797i	$0.493585\pi$
$242$	9.63773	0.619537
$243$	0	0
$244$	11.7235	0.750519
$245$	0	0
$246$	0	0
$247$	13.7573	0.875358
$248$	−3.02715	−0.192224
$249$	0	0
$250$	5.37867	0.340177
$251$	23.9608	1.51239	0.756195	−	0.654346i	$-0.227056\pi$
$251$	23.9608	1.51239	0.756195	+	0.654346i	$0.227056\pi$
$252$	0	0
$253$	−17.2717	−1.08586
$254$	9.80731	0.615366
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.55927	0.596291	0.298145	−	0.954520i	$-0.403632\pi$
$257$	9.55927	0.596291	0.298145	+	0.954520i	$0.403632\pi$
$258$	0	0
$259$	0	0
$260$	3.30798	0.205152
$261$	0	0
$262$	0.188374	0.0116378
$263$	−7.36658	−0.454243	−0.227122	−	0.973866i	$-0.572931\pi$
$263$	−7.36658	−0.454243	−0.227122	+	0.973866i	$0.572931\pi$
$264$	0	0
$265$	−3.12498	−0.191966
$266$	0	0
$267$	0	0
$268$	13.2959	0.812176
$269$	−16.0640	−0.979438	−0.489719	−	0.871880i	$-0.662901\pi$
$269$	−16.0640	−0.979438	−0.489719	+	0.871880i	$0.662901\pi$
$270$	0	0
$271$	2.51035	0.152493	0.0762465	−	0.997089i	$-0.475706\pi$
$271$	2.51035	0.152493	0.0762465	+	0.997089i	$0.475706\pi$
$272$	7.18598	0.435714
$273$	0	0
$274$	−2.76271	−0.166901
$275$	21.3153	1.28536
$276$	0	0
$277$	−5.91723	−0.355532	−0.177766	−	0.984073i	$-0.556887\pi$
$277$	−5.91723	−0.355532	−0.177766	+	0.984073i	$0.556887\pi$
$278$	−3.45712	−0.207344
$279$	0	0
$280$	0	0
$281$	19.7409	1.17765	0.588823	−	0.808262i	$-0.299592\pi$
$281$	19.7409	1.17765	0.588823	+	0.808262i	$0.299592\pi$
$282$	0	0
$283$	−7.23191	−0.429893	−0.214946	−	0.976626i	$-0.568958\pi$
$283$	−7.23191	−0.429893	−0.214946	+	0.976626i	$0.568958\pi$
$284$	0.423272	0.0251165
$285$	0	0
$286$	27.0790	1.60122
$287$	0	0
$288$	0	0
$289$	34.6383	2.03755
$290$	−5.63773	−0.331059
$291$	0	0
$292$	−14.3327	−0.838760
$293$	24.1250	1.40940	0.704698	−	0.709507i	$-0.251082\pi$
$293$	24.1250	1.40940	0.704698	+	0.709507i	$0.251082\pi$
$294$	0	0
$295$	−7.37867	−0.429603
$296$	−0.0217703	−0.00126537
$297$	0	0
$298$	−9.74632	−0.564589
$299$	−22.6625	−1.31061
$300$	0	0
$301$	0	0
$302$	−3.81700	−0.219644
$303$	0	0
$304$	−2.30798	−0.132372
$305$	−6.50604	−0.372535
$306$	0	0
$307$	−26.5730	−1.51660	−0.758301	−	0.651905i	$-0.773970\pi$
$307$	−26.5730	−1.51660	−0.758301	+	0.651905i	$0.773970\pi$
$308$	0	0
$309$	0	0
$310$	1.67994	0.0954142
$311$	−25.1250	−1.42471	−0.712354	−	0.701821i	$-0.752371\pi$
$311$	−25.1250	−1.42471	−0.712354	+	0.701821i	$0.752371\pi$
$312$	0	0
$313$	−7.19029	−0.406419	−0.203210	−	0.979135i	$-0.565137\pi$
$313$	−7.19029	−0.406419	−0.203210	+	0.979135i	$0.565137\pi$
$314$	−3.12737	−0.176488
$315$	0	0
$316$	14.2446	0.801321
$317$	−6.05429	−0.340043	−0.170022	−	0.985440i	$-0.554384\pi$
$317$	−6.05429	−0.340043	−0.170022	+	0.985440i	$0.554384\pi$
$318$	0	0
$319$	−46.1503	−2.58392
$320$	−0.554958	−0.0310231
$321$	0	0
$322$	0	0
$323$	−16.5851	−0.922819
$324$	0	0
$325$	27.9681	1.55139
$326$	13.2905	0.736094
$327$	0	0
$328$	1.77479	0.0979964
$329$	0	0
$330$	0	0
$331$	3.13706	0.172429	0.0862143	−	0.996277i	$-0.472523\pi$
$331$	3.13706	0.172429	0.0862143	+	0.996277i	$0.472523\pi$
$332$	2.41789	0.132699
$333$	0	0
$334$	3.17629	0.173799
$335$	−7.37867	−0.403140
$336$	0	0
$337$	−15.8649	−0.864214	−0.432107	−	0.901822i	$-0.642230\pi$
$337$	−15.8649	−0.864214	−0.432107	+	0.901822i	$0.642230\pi$
$338$	22.5308	1.22551
$339$	0	0
$340$	−3.98792	−0.216275
$341$	13.7520	0.744710
$342$	0	0
$343$	0	0
$344$	4.93900	0.266293
$345$	0	0
$346$	13.9511	0.750015
$347$	−11.9541	−0.641728	−0.320864	−	0.947125i	$-0.603973\pi$
$347$	−11.9541	−0.641728	−0.320864	+	0.947125i	$0.603973\pi$
$348$	0	0
$349$	−12.3381	−0.660444	−0.330222	−	0.943903i	$-0.607124\pi$
$349$	−12.3381	−0.660444	−0.330222	+	0.943903i	$0.607124\pi$
$350$	0	0
$351$	0	0
$352$	−4.54288	−0.242136
$353$	−15.7700	−0.839353	−0.419676	−	0.907674i	$-0.637856\pi$
$353$	−15.7700	−0.839353	−0.419676	+	0.907674i	$0.637856\pi$
$354$	0	0
$355$	−0.234898	−0.0124671
$356$	4.83877	0.256454
$357$	0	0
$358$	3.35690	0.177417
$359$	3.15883	0.166717	0.0833584	−	0.996520i	$-0.473435\pi$
$359$	3.15883	0.166717	0.0833584	+	0.996520i	$0.473435\pi$
$360$	0	0
$361$	−13.6732	−0.719644
$362$	23.4306	1.23148
$363$	0	0
$364$	0	0
$365$	7.95407	0.416335
$366$	0	0
$367$	−9.42221	−0.491835	−0.245918	−	0.969291i	$-0.579089\pi$
$367$	−9.42221	−0.491835	−0.245918	+	0.969291i	$0.579089\pi$
$368$	3.80194	0.198190
$369$	0	0
$370$	0.0120816	0.000628092 0
$371$	0	0
$372$	0	0
$373$	37.4010	1.93655	0.968276	−	0.249884i	$-0.0803923\pi$
$373$	37.4010	1.93655	0.968276	+	0.249884i	$0.0803923\pi$
$374$	−32.6450	−1.68803
$375$	0	0
$376$	−6.94869	−0.358351
$377$	−60.5545	−3.11871
$378$	0	0
$379$	24.5851	1.26285	0.631426	−	0.775436i	$-0.282470\pi$
$379$	24.5851	1.26285	0.631426	+	0.775436i	$0.282470\pi$
$380$	1.28083	0.0657053
$381$	0	0
$382$	−3.71917	−0.190289
$383$	14.3099	0.731202	0.365601	−	0.930772i	$-0.380864\pi$
$383$	14.3099	0.731202	0.365601	+	0.930772i	$0.380864\pi$
$384$	0	0
$385$	0	0
$386$	−12.7603	−0.649483
$387$	0	0
$388$	−11.9051	−0.604392
$389$	−5.25667	−0.266524	−0.133262	−	0.991081i	$-0.542545\pi$
$389$	−5.25667	−0.266524	−0.133262	+	0.991081i	$0.542545\pi$
$390$	0	0
$391$	27.3207	1.38166
$392$	0	0
$393$	0	0
$394$	−10.8901	−0.548634
$395$	−7.90515	−0.397751
$396$	0	0
$397$	32.0422	1.60815	0.804076	−	0.594526i	$-0.202660\pi$
$397$	32.0422	1.60815	0.804076	+	0.594526i	$0.202660\pi$
$398$	0.841166	0.0421639
$399$	0	0
$400$	−4.69202	−0.234601
$401$	−16.1642	−0.807202	−0.403601	−	0.914935i	$-0.632242\pi$
$401$	−16.1642	−0.807202	−0.403601	+	0.914935i	$0.632242\pi$
$402$	0	0
$403$	18.0441	0.898842
$404$	−16.6189	−0.826823
$405$	0	0
$406$	0	0
$407$	0.0988996	0.00490227
$408$	0	0
$409$	34.0194	1.68215	0.841075	−	0.540919i	$-0.181923\pi$
$409$	34.0194	1.68215	0.841075	+	0.540919i	$0.181923\pi$
$410$	−0.984935	−0.0486424
$411$	0	0
$412$	12.1075	0.596495
$413$	0	0
$414$	0	0
$415$	−1.34183	−0.0658679
$416$	−5.96077	−0.292251
$417$	0	0
$418$	10.4849	0.512831
$419$	−15.8616	−0.774890	−0.387445	−	0.921893i	$-0.626642\pi$
$419$	−15.8616	−0.774890	−0.387445	+	0.921893i	$0.626642\pi$
$420$	0	0
$421$	−24.3696	−1.18770	−0.593850	−	0.804576i	$-0.702393\pi$
$421$	−24.3696	−1.18770	−0.593850	+	0.804576i	$0.702393\pi$
$422$	−14.6571	−0.713497
$423$	0	0
$424$	5.63102	0.273467
$425$	−33.7168	−1.63550
$426$	0	0
$427$	0	0
$428$	15.0465	0.727301
$429$	0	0
$430$	−2.74094	−0.132180
$431$	6.00431	0.289218	0.144609	−	0.989489i	$-0.453808\pi$
$431$	6.00431	0.289218	0.144609	+	0.989489i	$0.453808\pi$
$432$	0	0
$433$	−14.4233	−0.693138	−0.346569	−	0.938024i	$-0.612653\pi$
$433$	−14.4233	−0.693138	−0.346569	+	0.938024i	$0.612653\pi$
$434$	0	0
$435$	0	0
$436$	−0.521106	−0.0249565
$437$	−8.77479	−0.419755
$438$	0	0
$439$	18.1280	0.865201	0.432600	−	0.901586i	$-0.357596\pi$
$439$	18.1280	0.865201	0.432600	+	0.901586i	$0.357596\pi$
$440$	2.52111	0.120189
$441$	0	0
$442$	−42.8340	−2.03741
$443$	18.8213	0.894228	0.447114	−	0.894477i	$-0.352452\pi$
$443$	18.8213	0.894228	0.447114	+	0.894477i	$0.352452\pi$
$444$	0	0
$445$	−2.68532	−0.127296
$446$	−21.9530	−1.03950
$447$	0	0
$448$	0	0
$449$	−9.24027	−0.436076	−0.218038	−	0.975940i	$-0.569966\pi$
$449$	−9.24027	−0.436076	−0.218038	+	0.975940i	$0.569966\pi$
$450$	0	0
$451$	−8.06265	−0.379656
$452$	5.41119	0.254521
$453$	0	0
$454$	−6.94869	−0.326118
$455$	0	0
$456$	0	0
$457$	−21.0978	−0.986915	−0.493458	−	0.869770i	$-0.664267\pi$
$457$	−21.0978	−0.986915	−0.493458	+	0.869770i	$0.664267\pi$
$458$	−5.72886	−0.267692
$459$	0	0
$460$	−2.10992	−0.0983754
$461$	23.0320	1.07271	0.536355	−	0.843993i	$-0.319801\pi$
$461$	23.0320	1.07271	0.536355	+	0.843993i	$0.319801\pi$
$462$	0	0
$463$	0.0814412	0.00378489	0.00189245	−	0.999998i	$-0.499398\pi$
$463$	0.0814412	0.00378489	0.00189245	+	0.999998i	$0.499398\pi$
$464$	10.1588	0.471612
$465$	0	0
$466$	14.8073	0.685936
$467$	23.7101	1.09717	0.548586	−	0.836094i	$-0.315166\pi$
$467$	23.7101	1.09717	0.548586	+	0.836094i	$0.315166\pi$
$468$	0	0
$469$	0	0
$470$	3.85623	0.177875
$471$	0	0
$472$	13.2959	0.611993
$473$	−22.4373	−1.03167
$474$	0	0
$475$	10.8291	0.496872
$476$	0	0
$477$	0	0
$478$	11.9444	0.546323
$479$	10.6722	0.487624	0.243812	−	0.969823i	$-0.421602\pi$
$479$	10.6722	0.487624	0.243812	+	0.969823i	$0.421602\pi$
$480$	0	0
$481$	0.129768	0.00591689
$482$	0.625646	0.0284974
$483$	0	0
$484$	9.63773	0.438079
$485$	6.60686	0.300002
$486$	0	0
$487$	35.2078	1.59542	0.797708	−	0.603044i	$-0.206046\pi$
$487$	35.2078	1.59542	0.797708	+	0.603044i	$0.206046\pi$
$488$	11.7235	0.530697
$489$	0	0
$490$	0	0
$491$	−16.7399	−0.755460	−0.377730	−	0.925916i	$-0.623295\pi$
$491$	−16.7399	−0.755460	−0.377730	+	0.925916i	$0.623295\pi$
$492$	0	0
$493$	73.0012	3.28781
$494$	13.7573	0.618972
$495$	0	0
$496$	−3.02715	−0.135923
$497$	0	0
$498$	0	0
$499$	32.9898	1.47683	0.738414	−	0.674348i	$-0.235575\pi$
$499$	32.9898	1.47683	0.738414	+	0.674348i	$0.235575\pi$
$500$	5.37867	0.240541
$501$	0	0
$502$	23.9608	1.06942
$503$	29.4282	1.31214	0.656069	−	0.754701i	$-0.272218\pi$
$503$	29.4282	1.31214	0.656069	+	0.754701i	$0.272218\pi$
$504$	0	0
$505$	9.22282	0.410410
$506$	−17.2717	−0.767822
$507$	0	0
$508$	9.80731	0.435129
$509$	−5.38644	−0.238750	−0.119375	−	0.992849i	$-0.538089\pi$
$509$	−5.38644	−0.238750	−0.119375	+	0.992849i	$0.538089\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	9.55927	0.421641
$515$	−6.71917	−0.296082
$516$	0	0
$517$	31.5670	1.38832
$518$	0	0
$519$	0	0
$520$	3.30798	0.145064
$521$	−3.94810	−0.172969	−0.0864847	−	0.996253i	$-0.527563\pi$
$521$	−3.94810	−0.172969	−0.0864847	+	0.996253i	$0.527563\pi$
$522$	0	0
$523$	37.0756	1.62120	0.810601	−	0.585599i	$-0.199140\pi$
$523$	37.0756	1.62120	0.810601	+	0.585599i	$0.199140\pi$
$524$	0.188374	0.00822914
$525$	0	0
$526$	−7.36658	−0.321198
$527$	−21.7530	−0.947576
$528$	0	0
$529$	−8.54527	−0.371533
$530$	−3.12498	−0.135741
$531$	0	0
$532$	0	0
$533$	−10.5791	−0.458233
$534$	0	0
$535$	−8.35019	−0.361010
$536$	13.2959	0.574295
$537$	0	0
$538$	−16.0640	−0.692567
$539$	0	0
$540$	0	0
$541$	−22.7385	−0.977606	−0.488803	−	0.872394i	$-0.662566\pi$
$541$	−22.7385	−0.977606	−0.488803	+	0.872394i	$0.662566\pi$
$542$	2.51035	0.107829
$543$	0	0
$544$	7.18598	0.308096
$545$	0.289192	0.0123876
$546$	0	0
$547$	31.1704	1.33275	0.666376	−	0.745616i	$-0.267845\pi$
$547$	31.1704	1.33275	0.666376	+	0.745616i	$0.267845\pi$
$548$	−2.76271	−0.118017
$549$	0	0
$550$	21.3153	0.908886
$551$	−23.4464	−0.998849
$552$	0	0
$553$	0	0
$554$	−5.91723	−0.251399
$555$	0	0
$556$	−3.45712	−0.146615
$557$	19.5502	0.828367	0.414184	−	0.910193i	$-0.364067\pi$
$557$	19.5502	0.828367	0.414184	+	0.910193i	$0.364067\pi$
$558$	0	0
$559$	−29.4403	−1.24519
$560$	0	0
$561$	0	0
$562$	19.7409	0.832721
$563$	46.8394	1.97404	0.987022	−	0.160586i	$-0.0513382\pi$
$563$	46.8394	1.97404	0.987022	+	0.160586i	$0.0513382\pi$
$564$	0	0
$565$	−3.00298	−0.126336
$566$	−7.23191	−0.303980
$567$	0	0
$568$	0.423272	0.0177601
$569$	33.6728	1.41164	0.705818	−	0.708393i	$-0.250580\pi$
$569$	33.6728	1.41164	0.705818	+	0.708393i	$0.250580\pi$
$570$	0	0
$571$	−17.7409	−0.742435	−0.371218	−	0.928546i	$-0.621060\pi$
$571$	−17.7409	−0.742435	−0.371218	+	0.928546i	$0.621060\pi$
$572$	27.0790	1.13223
$573$	0	0
$574$	0	0
$575$	−17.8388	−0.743928
$576$	0	0
$577$	19.2849	0.802840	0.401420	−	0.915894i	$-0.368517\pi$
$577$	19.2849	0.802840	0.401420	+	0.915894i	$0.368517\pi$
$578$	34.6383	1.44076
$579$	0	0
$580$	−5.63773	−0.234094
$581$	0	0
$582$	0	0
$583$	−25.5810	−1.05946
$584$	−14.3327	−0.593093
$585$	0	0
$586$	24.1250	0.996594
$587$	−20.8122	−0.859012	−0.429506	−	0.903064i	$-0.641312\pi$
$587$	−20.8122	−0.859012	−0.429506	+	0.903064i	$0.641312\pi$
$588$	0	0
$589$	6.98659	0.287877
$590$	−7.37867	−0.303775
$591$	0	0
$592$	−0.0217703	−0.000894752 0
$593$	−11.6256	−0.477408	−0.238704	−	0.971092i	$-0.576723\pi$
$593$	−11.6256	−0.477408	−0.238704	+	0.971092i	$0.576723\pi$
$594$	0	0
$595$	0	0
$596$	−9.74632	−0.399225
$597$	0	0
$598$	−22.6625	−0.926738
$599$	−3.57109	−0.145911	−0.0729554	−	0.997335i	$-0.523243\pi$
$599$	−3.57109	−0.145911	−0.0729554	+	0.997335i	$0.523243\pi$
$600$	0	0
$601$	14.6455	0.597402	0.298701	−	0.954347i	$-0.403447\pi$
$601$	14.6455	0.597402	0.298701	+	0.954347i	$0.403447\pi$
$602$	0	0
$603$	0	0
$604$	−3.81700	−0.155312
$605$	−5.34854	−0.217449
$606$	0	0
$607$	−38.5864	−1.56617	−0.783087	−	0.621912i	$-0.786356\pi$
$607$	−38.5864	−1.56617	−0.783087	+	0.621912i	$0.786356\pi$
$608$	−2.30798	−0.0936009
$609$	0	0
$610$	−6.50604	−0.263422
$611$	41.4196	1.67566
$612$	0	0
$613$	−12.0285	−0.485826	−0.242913	−	0.970048i	$-0.578103\pi$
$613$	−12.0285	−0.485826	−0.242913	+	0.970048i	$0.578103\pi$
$614$	−26.5730	−1.07240
$615$	0	0
$616$	0	0
$617$	−32.7356	−1.31788	−0.658942	−	0.752194i	$-0.728996\pi$
$617$	−32.7356	−1.31788	−0.658942	+	0.752194i	$0.728996\pi$
$618$	0	0
$619$	−33.8745	−1.36153	−0.680766	−	0.732501i	$-0.738353\pi$
$619$	−33.8745	−1.36153	−0.680766	+	0.732501i	$0.738353\pi$
$620$	1.67994	0.0674680
$621$	0	0
$622$	−25.1250	−1.00742
$623$	0	0
$624$	0	0
$625$	20.4752	0.819007
$626$	−7.19029	−0.287382
$627$	0	0
$628$	−3.12737	−0.124796
$629$	−0.156441	−0.00623770
$630$	0	0
$631$	12.6890	0.505143	0.252571	−	0.967578i	$-0.418724\pi$
$631$	12.6890	0.505143	0.252571	+	0.967578i	$0.418724\pi$
$632$	14.2446	0.566619
$633$	0	0
$634$	−6.05429	−0.240447
$635$	−5.44265	−0.215985
$636$	0	0
$637$	0	0
$638$	−46.1503	−1.82711
$639$	0	0
$640$	−0.554958	−0.0219366
$641$	25.0304	0.988641	0.494321	−	0.869280i	$-0.335417\pi$
$641$	25.0304	0.988641	0.494321	+	0.869280i	$0.335417\pi$
$642$	0	0
$643$	13.4644	0.530985	0.265492	−	0.964113i	$-0.414465\pi$
$643$	13.4644	0.530985	0.265492	+	0.964113i	$0.414465\pi$
$644$	0	0
$645$	0	0
$646$	−16.5851	−0.652532
$647$	−19.6088	−0.770901	−0.385450	−	0.922729i	$-0.625954\pi$
$647$	−19.6088	−0.770901	−0.385450	+	0.922729i	$0.625954\pi$
$648$	0	0
$649$	−60.4016	−2.37097
$650$	27.9681	1.09700
$651$	0	0
$652$	13.2905	0.520497
$653$	−8.25475	−0.323033	−0.161517	−	0.986870i	$-0.551639\pi$
$653$	−8.25475	−0.323033	−0.161517	+	0.986870i	$0.551639\pi$
$654$	0	0
$655$	−0.104539	−0.00408469
$656$	1.77479	0.0692939
$657$	0	0
$658$	0	0
$659$	−11.0164	−0.429138	−0.214569	−	0.976709i	$-0.568835\pi$
$659$	−11.0164	−0.429138	−0.214569	+	0.976709i	$0.568835\pi$
$660$	0	0
$661$	−4.86054	−0.189053	−0.0945266	−	0.995522i	$-0.530134\pi$
$661$	−4.86054	−0.189053	−0.0945266	+	0.995522i	$0.530134\pi$
$662$	3.13706	0.121925
$663$	0	0
$664$	2.41789	0.0938325
$665$	0	0
$666$	0	0
$667$	38.6233	1.49550
$668$	3.17629	0.122894
$669$	0	0
$670$	−7.37867	−0.285063
$671$	−53.2583	−2.05601
$672$	0	0
$673$	−45.5918	−1.75743	−0.878717	−	0.477343i	$-0.841600\pi$
$673$	−45.5918	−1.75743	−0.878717	+	0.477343i	$0.841600\pi$
$674$	−15.8649	−0.611091
$675$	0	0
$676$	22.5308	0.866569
$677$	8.55688	0.328868	0.164434	−	0.986388i	$-0.447420\pi$
$677$	8.55688	0.328868	0.164434	+	0.986388i	$0.447420\pi$
$678$	0	0
$679$	0	0
$680$	−3.98792	−0.152930
$681$	0	0
$682$	13.7520	0.526590
$683$	−6.18731	−0.236751	−0.118375	−	0.992969i	$-0.537769\pi$
$683$	−6.18731	−0.236751	−0.118375	+	0.992969i	$0.537769\pi$
$684$	0	0
$685$	1.53319	0.0585801
$686$	0	0
$687$	0	0
$688$	4.93900	0.188298
$689$	−33.5652	−1.27873
$690$	0	0
$691$	10.8702	0.413523	0.206762	−	0.978391i	$-0.433708\pi$
$691$	10.8702	0.413523	0.206762	+	0.978391i	$0.433708\pi$
$692$	13.9511	0.530341
$693$	0	0
$694$	−11.9541	−0.453770
$695$	1.91856	0.0727751
$696$	0	0
$697$	12.7536	0.483077
$698$	−12.3381	−0.467004
$699$	0	0
$700$	0	0
$701$	19.2198	0.725923	0.362962	−	0.931804i	$-0.381766\pi$
$701$	19.2198	0.725923	0.362962	+	0.931804i	$0.381766\pi$
$702$	0	0
$703$	0.0502453	0.00189504
$704$	−4.54288	−0.171216
$705$	0	0
$706$	−15.7700	−0.593512
$707$	0	0
$708$	0	0
$709$	−19.9571	−0.749503	−0.374751	−	0.927125i	$-0.622272\pi$
$709$	−19.9571	−0.749503	−0.374751	+	0.927125i	$0.622272\pi$
$710$	−0.234898	−0.00881557
$711$	0	0
$712$	4.83877	0.181341
$713$	−11.5090	−0.431016
$714$	0	0
$715$	−15.0277	−0.562006
$716$	3.35690	0.125453
$717$	0	0
$718$	3.15883	0.117887
$719$	−6.60281	−0.246243	−0.123122	−	0.992392i	$-0.539291\pi$
$719$	−6.60281	−0.246243	−0.123122	+	0.992392i	$0.539291\pi$
$720$	0	0
$721$	0	0
$722$	−13.6732	−0.508865
$723$	0	0
$724$	23.4306	0.870790
$725$	−47.6655	−1.77025
$726$	0	0
$727$	34.3889	1.27542	0.637708	−	0.770278i	$-0.279883\pi$
$727$	34.3889	1.27542	0.637708	+	0.770278i	$0.279883\pi$
$728$	0	0
$729$	0	0
$730$	7.95407	0.294393
$731$	35.4916	1.31270
$732$	0	0
$733$	4.64071	0.171409	0.0857043	−	0.996321i	$-0.472686\pi$
$733$	4.64071	0.171409	0.0857043	+	0.996321i	$0.472686\pi$
$734$	−9.42221	−0.347780
$735$	0	0
$736$	3.80194	0.140141
$737$	−60.4016	−2.22492
$738$	0	0
$739$	−33.8786	−1.24624	−0.623122	−	0.782125i	$-0.714136\pi$
$739$	−33.8786	−1.24624	−0.623122	+	0.782125i	$0.714136\pi$
$740$	0.0120816	0.000444128 0
$741$	0	0
$742$	0	0
$743$	23.1631	0.849773	0.424887	−	0.905247i	$-0.360314\pi$
$743$	23.1631	0.849773	0.424887	+	0.905247i	$0.360314\pi$
$744$	0	0
$745$	5.40880	0.198163
$746$	37.4010	1.36935
$747$	0	0
$748$	−32.6450	−1.19362
$749$	0	0
$750$	0	0
$751$	−17.6160	−0.642815	−0.321408	−	0.946941i	$-0.604156\pi$
$751$	−17.6160	−0.642815	−0.321408	+	0.946941i	$0.604156\pi$
$752$	−6.94869	−0.253393
$753$	0	0
$754$	−60.5545	−2.20526
$755$	2.11828	0.0770920
$756$	0	0
$757$	37.7622	1.37249	0.686246	−	0.727370i	$-0.259257\pi$
$757$	37.7622	1.37249	0.686246	+	0.727370i	$0.259257\pi$
$758$	24.5851	0.892971
$759$	0	0
$760$	1.28083	0.0464606
$761$	−49.7211	−1.80239	−0.901194	−	0.433416i	$-0.857308\pi$
$761$	−49.7211	−1.80239	−0.901194	+	0.433416i	$0.857308\pi$
$762$	0	0
$763$	0	0
$764$	−3.71917	−0.134555
$765$	0	0
$766$	14.3099	0.517038
$767$	−79.2538	−2.86169
$768$	0	0
$769$	25.2573	0.910800	0.455400	−	0.890287i	$-0.349496\pi$
$769$	25.2573	0.910800	0.455400	+	0.890287i	$0.349496\pi$
$770$	0	0
$771$	0	0
$772$	−12.7603	−0.459254
$773$	−24.6786	−0.887628	−0.443814	−	0.896119i	$-0.646375\pi$
$773$	−24.6786	−0.887628	−0.443814	+	0.896119i	$0.646375\pi$
$774$	0	0
$775$	14.2034	0.510203
$776$	−11.9051	−0.427370
$777$	0	0
$778$	−5.25667	−0.188461
$779$	−4.09618	−0.146761
$780$	0	0
$781$	−1.92287	−0.0688057
$782$	27.3207	0.976984
$783$	0	0
$784$	0	0
$785$	1.73556	0.0619449
$786$	0	0
$787$	5.07474	0.180895	0.0904474	−	0.995901i	$-0.471170\pi$
$787$	5.07474	0.180895	0.0904474	+	0.995901i	$0.471170\pi$
$788$	−10.8901	−0.387943
$789$	0	0
$790$	−7.90515	−0.281253
$791$	0	0
$792$	0	0
$793$	−69.8810	−2.48155
$794$	32.0422	1.13714
$795$	0	0
$796$	0.841166	0.0298144
$797$	42.5023	1.50551	0.752755	−	0.658301i	$-0.228725\pi$
$797$	42.5023	1.50551	0.752755	+	0.658301i	$0.228725\pi$
$798$	0	0
$799$	−49.9332	−1.76651
$800$	−4.69202	−0.165888
$801$	0	0
$802$	−16.1642	−0.570778
$803$	65.1118	2.29775
$804$	0	0
$805$	0	0
$806$	18.0441	0.635577
$807$	0	0
$808$	−16.6189	−0.584652
$809$	13.7754	0.484317	0.242158	−	0.970237i	$-0.422145\pi$
$809$	13.7754	0.484317	0.242158	+	0.970237i	$0.422145\pi$
$810$	0	0
$811$	28.9028	1.01491	0.507457	−	0.861677i	$-0.330586\pi$
$811$	28.9028	1.01491	0.507457	+	0.861677i	$0.330586\pi$
$812$	0	0
$813$	0	0
$814$	0.0988996	0.00346643
$815$	−7.37568	−0.258359
$816$	0	0
$817$	−11.3991	−0.398804
$818$	34.0194	1.18946
$819$	0	0
$820$	−0.984935	−0.0343954
$821$	−54.0364	−1.88588	−0.942941	−	0.332960i	$-0.891953\pi$
$821$	−54.0364	−1.88588	−0.942941	+	0.332960i	$0.891953\pi$
$822$	0	0
$823$	30.4580	1.06170	0.530849	−	0.847466i	$-0.321873\pi$
$823$	30.4580	1.06170	0.530849	+	0.847466i	$0.321873\pi$
$824$	12.1075	0.421786
$825$	0	0
$826$	0	0
$827$	37.3840	1.29997	0.649985	−	0.759947i	$-0.274775\pi$
$827$	37.3840	1.29997	0.649985	+	0.759947i	$0.274775\pi$
$828$	0	0
$829$	−43.6450	−1.51585	−0.757927	−	0.652339i	$-0.773788\pi$
$829$	−43.6450	−1.51585	−0.757927	+	0.652339i	$0.773788\pi$
$830$	−1.34183	−0.0465756
$831$	0	0
$832$	−5.96077	−0.206653
$833$	0	0
$834$	0	0
$835$	−1.76271	−0.0610011
$836$	10.4849	0.362627
$837$	0	0
$838$	−15.8616	−0.547930
$839$	−20.0640	−0.692686	−0.346343	−	0.938108i	$-0.612577\pi$
$839$	−20.0640	−0.692686	−0.346343	+	0.938108i	$0.612577\pi$
$840$	0	0
$841$	74.2019	2.55869
$842$	−24.3696	−0.839831
$843$	0	0
$844$	−14.6571	−0.504518
$845$	−12.5036	−0.430139
$846$	0	0
$847$	0	0
$848$	5.63102	0.193370
$849$	0	0
$850$	−33.7168	−1.15648
$851$	−0.0827692	−0.00283729
$852$	0	0
$853$	−30.2543	−1.03589	−0.517943	−	0.855415i	$-0.673302\pi$
$853$	−30.2543	−1.03589	−0.517943	+	0.855415i	$0.673302\pi$
$854$	0	0
$855$	0	0
$856$	15.0465	0.514280
$857$	−11.7909	−0.402770	−0.201385	−	0.979512i	$-0.564544\pi$
$857$	−11.7909	−0.402770	−0.201385	+	0.979512i	$0.564544\pi$
$858$	0	0
$859$	0.821315	0.0280229	0.0140115	−	0.999902i	$-0.495540\pi$
$859$	0.821315	0.0280229	0.0140115	+	0.999902i	$0.495540\pi$
$860$	−2.74094	−0.0934652
$861$	0	0
$862$	6.00431	0.204508
$863$	40.5459	1.38020	0.690099	−	0.723715i	$-0.257567\pi$
$863$	40.5459	1.38020	0.690099	+	0.723715i	$0.257567\pi$
$864$	0	0
$865$	−7.74227	−0.263245
$866$	−14.4233	−0.490123
$867$	0	0
$868$	0	0
$869$	−64.7114	−2.19518
$870$	0	0
$871$	−79.2538	−2.68541
$872$	−0.521106	−0.0176469
$873$	0	0
$874$	−8.77479	−0.296812
$875$	0	0
$876$	0	0
$877$	−50.7569	−1.71394	−0.856969	−	0.515369i	$-0.827655\pi$
$877$	−50.7569	−1.71394	−0.856969	+	0.515369i	$0.827655\pi$
$878$	18.1280	0.611789
$879$	0	0
$880$	2.52111	0.0849865
$881$	1.44504	0.0486847	0.0243423	−	0.999704i	$-0.492251\pi$
$881$	1.44504	0.0486847	0.0243423	+	0.999704i	$0.492251\pi$
$882$	0	0
$883$	−55.8133	−1.87827	−0.939133	−	0.343553i	$-0.888369\pi$
$883$	−55.8133	−1.87827	−0.939133	+	0.343553i	$0.888369\pi$
$884$	−42.8340	−1.44066
$885$	0	0
$886$	18.8213	0.632314
$887$	−11.5244	−0.386950	−0.193475	−	0.981105i	$-0.561976\pi$
$887$	−11.5244	−0.386950	−0.193475	+	0.981105i	$0.561976\pi$
$888$	0	0
$889$	0	0
$890$	−2.68532	−0.0900120
$891$	0	0
$892$	−21.9530	−0.735041
$893$	16.0374	0.536672
$894$	0	0
$895$	−1.86294	−0.0622711
$896$	0	0
$897$	0	0
$898$	−9.24027	−0.308352
$899$	−30.7523	−1.02565
$900$	0	0
$901$	40.4644	1.34807
$902$	−8.06265	−0.268457
$903$	0	0
$904$	5.41119	0.179974
$905$	−13.0030	−0.432234
$906$	0	0
$907$	−37.7633	−1.25391	−0.626955	−	0.779056i	$-0.715699\pi$
$907$	−37.7633	−1.25391	−0.626955	+	0.779056i	$0.715699\pi$
$908$	−6.94869	−0.230600
$909$	0	0
$910$	0	0
$911$	−44.6510	−1.47935	−0.739677	−	0.672962i	$-0.765022\pi$
$911$	−44.6510	−1.47935	−0.739677	+	0.672962i	$0.765022\pi$
$912$	0	0
$913$	−10.9842	−0.363524
$914$	−21.0978	−0.697854
$915$	0	0
$916$	−5.72886	−0.189287
$917$	0	0
$918$	0	0
$919$	−38.9874	−1.28608	−0.643039	−	0.765834i	$-0.722327\pi$
$919$	−38.9874	−1.28608	−0.643039	+	0.765834i	$0.722327\pi$
$920$	−2.10992	−0.0695619
$921$	0	0
$922$	23.0320	0.758520
$923$	−2.52303	−0.0830464
$924$	0	0
$925$	0.102147	0.00335856
$926$	0.0814412	0.00267632
$927$	0	0
$928$	10.1588	0.333480
$929$	39.2529	1.28785	0.643924	−	0.765090i	$-0.277305\pi$
$929$	39.2529	1.28785	0.643924	+	0.765090i	$0.277305\pi$
$930$	0	0
$931$	0	0
$932$	14.8073	0.485030
$933$	0	0
$934$	23.7101	0.775817
$935$	18.1166	0.592477
$936$	0	0
$937$	−28.5007	−0.931076	−0.465538	−	0.885028i	$-0.654139\pi$
$937$	−28.5007	−0.931076	−0.465538	+	0.885028i	$0.654139\pi$
$938$	0	0
$939$	0	0
$940$	3.85623	0.125776
$941$	49.4626	1.61244	0.806218	−	0.591619i	$-0.201511\pi$
$941$	49.4626	1.61244	0.806218	+	0.591619i	$0.201511\pi$
$942$	0	0
$943$	6.74764	0.219734
$944$	13.2959	0.432745
$945$	0	0
$946$	−22.4373	−0.729499
$947$	−27.5198	−0.894273	−0.447136	−	0.894466i	$-0.647556\pi$
$947$	−27.5198	−0.894273	−0.447136	+	0.894466i	$0.647556\pi$
$948$	0	0
$949$	85.4341	2.77331
$950$	10.8291	0.351342
$951$	0	0
$952$	0	0
$953$	−31.4196	−1.01778	−0.508890	−	0.860832i	$-0.669944\pi$
$953$	−31.4196	−1.01778	−0.508890	+	0.860832i	$0.669944\pi$
$954$	0	0
$955$	2.06398	0.0667889
$956$	11.9444	0.386309
$957$	0	0
$958$	10.6722	0.344802
$959$	0	0
$960$	0	0
$961$	−21.8364	−0.704399
$962$	0.129768	0.00418387
$963$	0	0
$964$	0.625646	0.0201507
$965$	7.08144	0.227960
$966$	0	0
$967$	2.02848	0.0652314	0.0326157	−	0.999468i	$-0.489616\pi$
$967$	2.02848	0.0652314	0.0326157	+	0.999468i	$0.489616\pi$
$968$	9.63773	0.309768
$969$	0	0
$970$	6.60686	0.212133
$971$	37.4849	1.20295	0.601473	−	0.798893i	$-0.294581\pi$
$971$	37.4849	1.20295	0.601473	+	0.798893i	$0.294581\pi$
$972$	0	0
$973$	0	0
$974$	35.2078	1.12813
$975$	0	0
$976$	11.7235	0.375259
$977$	−0.351519	−0.0112461	−0.00562305	−	0.999984i	$-0.501790\pi$
$977$	−0.351519	−0.0112461	−0.00562305	+	0.999984i	$0.501790\pi$
$978$	0	0
$979$	−21.9820	−0.702546
$980$	0	0
$981$	0	0
$982$	−16.7399	−0.534191
$983$	43.5719	1.38973	0.694865	−	0.719141i	$-0.255464\pi$
$983$	43.5719	1.38973	0.694865	+	0.719141i	$0.255464\pi$
$984$	0	0
$985$	6.04354	0.192563
$986$	73.0012	2.32483
$987$	0	0
$988$	13.7573	0.437679
$989$	18.7778	0.597098
$990$	0	0
$991$	−18.3744	−0.583681	−0.291840	−	0.956467i	$-0.594268\pi$
$991$	−18.3744	−0.583681	−0.291840	+	0.956467i	$0.594268\pi$
$992$	−3.02715	−0.0961120
$993$	0	0
$994$	0	0
$995$	−0.466812	−0.0147989
$996$	0	0
$997$	35.9259	1.13778	0.568892	−	0.822413i	$-0.307372\pi$
$997$	35.9259	1.13778	0.568892	+	0.822413i	$0.307372\pi$
$998$	32.9898	1.04428
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6174.2.a.j.1.2		3
3.2	odd	2		2058.2.a.f.1.2	yes	3
7.6	odd	2		6174.2.a.o.1.2		3
21.2	odd	6		2058.2.e.g.361.2		6
21.5	even	6		2058.2.e.l.361.2		6
21.11	odd	6		2058.2.e.g.667.2		6
21.17	even	6		2058.2.e.l.667.2		6
21.20	even	2		2058.2.a.a.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2058.2.a.a.1.2	✓	3	21.20	even	2
2058.2.a.f.1.2	yes	3	3.2	odd	2
2058.2.e.g.361.2		6	21.2	odd	6
2058.2.e.g.667.2		6	21.11	odd	6
2058.2.e.l.361.2		6	21.5	even	6
2058.2.e.l.667.2		6	21.17	even	6
6174.2.a.j.1.2		3	1.1	even	1	trivial
6174.2.a.o.1.2		3	7.6	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6174 = 2 \cdot 3^{2} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6174.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.554958 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.554958 q^{5} +1.00000 q^{8} -0.554958 q^{10} -4.54288 q^{11} -5.96077 q^{13} +1.00000 q^{16} +7.18598 q^{17} -2.30798 q^{19} -0.554958 q^{20} -4.54288 q^{22} +3.80194 q^{23} -4.69202 q^{25} -5.96077 q^{26} +10.1588 q^{29} -3.02715 q^{31} +1.00000 q^{32} +7.18598 q^{34} -0.0217703 q^{37} -2.30798 q^{38} -0.554958 q^{40} +1.77479 q^{41} +4.93900 q^{43} -4.54288 q^{44} +3.80194 q^{46} -6.94869 q^{47} -4.69202 q^{50} -5.96077 q^{52} +5.63102 q^{53} +2.52111 q^{55} +10.1588 q^{58} +13.2959 q^{59} +11.7235 q^{61} -3.02715 q^{62} +1.00000 q^{64} +3.30798 q^{65} +13.2959 q^{67} +7.18598 q^{68} +0.423272 q^{71} -14.3327 q^{73} -0.0217703 q^{74} -2.30798 q^{76} +14.2446 q^{79} -0.554958 q^{80} +1.77479 q^{82} +2.41789 q^{83} -3.98792 q^{85} +4.93900 q^{86} -4.54288 q^{88} +4.83877 q^{89} +3.80194 q^{92} -6.94869 q^{94} +1.28083 q^{95} -11.9051 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{8} - 2 q^{10} + 5 q^{11} - 5 q^{13} + 3 q^{16} + 7 q^{17} - 12 q^{19} - 2 q^{20} + 5 q^{22} + 7 q^{23} - 9 q^{25} - 5 q^{26} + 22 q^{29} - 3 q^{31} + 3 q^{32} + 7 q^{34} + 3 q^{37} - 12 q^{38} - 2 q^{40} + 7 q^{41} + 5 q^{43} + 5 q^{44} + 7 q^{46} + 11 q^{47} - 9 q^{50} - 5 q^{52} + 2 q^{53} - 8 q^{55} + 22 q^{58} + 26 q^{59} + 5 q^{61} - 3 q^{62} + 3 q^{64} + 15 q^{65} + 26 q^{67} + 7 q^{68} + 4 q^{71} - q^{73} + 3 q^{74} - 12 q^{76} - 3 q^{79} - 2 q^{80} + 7 q^{82} + 13 q^{83} + 7 q^{85} + 5 q^{86} + 5 q^{88} - 18 q^{89} + 7 q^{92} + 11 q^{94} + 15 q^{95} - 10 q^{97}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^8 - 2 * q^10 + 5 * q^11 - 5 * q^13 + 3 * q^16 + 7 * q^17 - 12 * q^19 - 2 * q^20 + 5 * q^22 + 7 * q^23 - 9 * q^25 - 5 * q^26 + 22 * q^29 - 3 * q^31 + 3 * q^32 + 7 * q^34 + 3 * q^37 - 12 * q^38 - 2 * q^40 + 7 * q^41 + 5 * q^43 + 5 * q^44 + 7 * q^46 + 11 * q^47 - 9 * q^50 - 5 * q^52 + 2 * q^53 - 8 * q^55 + 22 * q^58 + 26 * q^59 + 5 * q^61 - 3 * q^62 + 3 * q^64 + 15 * q^65 + 26 * q^67 + 7 * q^68 + 4 * q^71 - q^73 + 3 * q^74 - 12 * q^76 - 3 * q^79 - 2 * q^80 + 7 * q^82 + 13 * q^83 + 7 * q^85 + 5 * q^86 + 5 * q^88 - 18 * q^89 + 7 * q^92 + 11 * q^94 + 15 * q^95 - 10 * q^97

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