LMFDB - Embedded newform 2624.2.d.o.2049.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2624,2,Mod(2049,2624)] 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("2624.2049"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2624 = 2^{6} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2624.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,4,0,0,0,-6,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,10,0,0,0,0, 0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$20.9527454904$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1229312.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 10x^{4} + 24x^{2} + 8 $ x^6 + 10x^4 + 24x^2 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 328)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2049.6
Root			$0.629384i$ of defining polynomial
Character	$\chi$	$=$	2624.2049
Dual form			2624.2.d.o.2049.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.17771i q^{3} +0.890084 q^{5} -1.91894i q^{7} -7.09783 q^{9} -0.349282i q^{11} +2.26822i q^{13} +2.82843i q^{15} +5.65685i q^{17} +3.17771i q^{19} +6.09783 q^{21} +7.20775 q^{23} -4.20775 q^{25} -13.0217i q^{27} +7.92508i q^{29} -7.20775 q^{31} +1.10992 q^{33} -1.70802i q^{35} +0.890084 q^{37} -7.20775 q^{39} +(5.98792 - 2.26822i) q^{41} -3.20775 q^{43} -6.31767 q^{45} -7.96347i q^{47} +3.31767 q^{49} -17.9758 q^{51} +0.871095i q^{53} -0.310890i q^{55} -10.0978 q^{57} -6.76809 q^{59} -12.7681 q^{61} +13.6203i q^{63} +2.01891i q^{65} +4.18716i q^{67} +22.9041i q^{69} -6.56634i q^{71} -12.0978 q^{73} -13.3710i q^{75} -0.670251 q^{77} -7.96347i q^{79} +20.0858 q^{81} -13.9758 q^{83} +5.03507i q^{85} -25.1836 q^{87} +15.2284i q^{89} +4.35258 q^{91} -22.9041i q^{93} +2.82843i q^{95} +17.2473i q^{97} +2.47915i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{5} - 6 q^{9} + 8 q^{23} + 10 q^{25} - 8 q^{31} + 8 q^{33} + 4 q^{37} - 8 q^{39} - 2 q^{41} + 16 q^{43} - 4 q^{45} - 14 q^{49} - 32 q^{51} - 24 q^{57} - 36 q^{61} - 36 q^{73} + 46 q^{81} - 8 q^{83}+ \cdots + 56 q^{91}+O(q^{100}) $ 6 * q + 4 * q^5 - 6 * q^9 + 8 * q^23 + 10 * q^25 - 8 * q^31 + 8 * q^33 + 4 * q^37 - 8 * q^39 - 2 * q^41 + 16 * q^43 - 4 * q^45 - 14 * q^49 - 32 * q^51 - 24 * q^57 - 36 * q^61 - 36 * q^73 + 46 * q^81 - 8 * q^83 - 40 * q^87 + 56 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2624\mathbb{Z}\right)^\times$.

$n$	$129$	$575$	$1477$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			3.17771i			1.83465i	0.398138	+	0.917326i	$0.369657\pi$
$3$			3.17771i			1.83465i	−0.398138	+	0.917326i	$0.630343\pi$
$4$		0			0
$5$	0.890084			0.398058			0.199029	−	0.979994i	$-0.436221\pi$
$5$	0.890084			0.398058			0.199029	+	0.979994i	$0.436221\pi$
$6$		0			0
$7$		−	1.91894i		−	0.725291i	−0.931927	−	0.362646i	$-0.881874\pi$
$7$		−	1.91894i		−	0.725291i	0.931927	−	0.362646i	$-0.118126\pi$
$8$		0			0
$9$	−7.09783			−2.36594
$10$		0			0
$11$		−	0.349282i		−	0.105312i	−0.998613	−	0.0526562i	$-0.983231\pi$
$11$		−	0.349282i		−	0.105312i	0.998613	−	0.0526562i	$-0.0167688\pi$
$12$		0			0
$13$			2.26822i			0.629092i	0.949242	+	0.314546i	$0.101852\pi$
$13$			2.26822i			0.629092i	−0.949242	+	0.314546i	$0.898148\pi$
$14$		0			0
$15$			2.82843i			0.730297i
$16$		0			0
$17$			5.65685i			1.37199i	0.727607	+	0.685994i	$0.240633\pi$
$17$			5.65685i			1.37199i	−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$			3.17771i			0.729016i	0.931200	+	0.364508i	$0.118763\pi$
$19$			3.17771i			0.729016i	−0.931200	+	0.364508i	$0.881237\pi$
$20$		0			0
$21$	6.09783			1.33066
$22$		0			0
$23$	7.20775			1.50292			0.751460	−	0.659779i	$-0.229350\pi$
$23$	7.20775			1.50292			0.751460	+	0.659779i	$0.229350\pi$
$24$		0			0
$25$	−4.20775			−0.841550
$26$		0			0
$27$		−	13.0217i		−	2.50603i
$28$		0			0
$29$			7.92508i			1.47165i	0.677172	+	0.735825i	$0.263205\pi$
$29$			7.92508i			1.47165i	−0.677172	+	0.735825i	$0.736795\pi$
$30$		0			0
$31$	−7.20775			−1.29455			−0.647275	−	0.762256i	$-0.724092\pi$
$31$	−7.20775			−1.29455			−0.647275	+	0.762256i	$0.724092\pi$
$32$		0			0
$33$	1.10992			0.193212
$34$		0			0
$35$		−	1.70802i		−	0.288708i
$36$		0			0
$37$	0.890084			0.146329			0.0731644	−	0.997320i	$-0.476690\pi$
$37$	0.890084			0.146329			0.0731644	+	0.997320i	$0.476690\pi$
$38$		0			0
$39$	−7.20775			−1.15416
$40$		0			0
$41$	5.98792	−	2.26822i	0.935156	−	0.354237i
$41$	5.98792	−	2.26822i	0.935156	−	0.354237i
$42$		0			0
$43$	−3.20775			−0.489177			−0.244589	−	0.969627i	$-0.578653\pi$
$43$	−3.20775			−0.489177			−0.244589	+	0.969627i	$0.578653\pi$
$44$		0			0
$45$	−6.31767			−0.941782
$46$		0			0
$47$		−	7.96347i		−	1.16159i	−0.814049	−	0.580796i	$-0.802742\pi$
$47$		−	7.96347i		−	1.16159i	0.814049	−	0.580796i	$-0.197258\pi$
$48$		0			0
$49$	3.31767			0.473952
$50$		0			0
$51$	−17.9758			−2.51712
$52$		0			0
$53$			0.871095i			0.119654i	0.998209	+	0.0598270i	$0.0190549\pi$
$53$			0.871095i			0.119654i	−0.998209	+	0.0598270i	$0.980945\pi$
$54$		0			0
$55$		−	0.310890i		−	0.0419204i
$56$		0			0
$57$	−10.0978			−1.33749
$58$		0			0
$59$	−6.76809			−0.881130			−0.440565	−	0.897721i	$-0.645222\pi$
$59$	−6.76809			−0.881130			−0.440565	+	0.897721i	$0.645222\pi$
$60$		0			0
$61$	−12.7681			−1.63479			−0.817393	−	0.576081i	$-0.804581\pi$
$61$	−12.7681			−1.63479			−0.817393	+	0.576081i	$0.804581\pi$
$62$		0			0
$63$			13.6203i			1.71600i
$64$		0			0
$65$			2.01891i			0.250415i
$66$		0			0
$67$			4.18716i			0.511543i	0.966737	+	0.255772i	$0.0823295\pi$
$67$			4.18716i			0.511543i	−0.966737	+	0.255772i	$0.917670\pi$
$68$		0			0
$69$			22.9041i			2.75733i
$70$		0			0
$71$		−	6.56634i		−	0.779281i	−0.920967	−	0.389641i	$-0.872599\pi$
$71$		−	6.56634i		−	0.779281i	0.920967	−	0.389641i	$-0.127401\pi$
$72$		0			0
$73$	−12.0978			−1.41594			−0.707972	−	0.706240i	$-0.750390\pi$
$73$	−12.0978			−1.41594			−0.707972	+	0.706240i	$0.750390\pi$
$74$		0			0
$75$		−	13.3710i		−	1.54395i
$76$		0			0
$77$	−0.670251			−0.0763822
$78$		0			0
$79$		−	7.96347i		−	0.895960i	−0.894044	−	0.447980i	$-0.852144\pi$
$79$		−	7.96347i		−	0.895960i	0.894044	−	0.447980i	$-0.147856\pi$
$80$		0			0
$81$	20.0858			2.23175
$82$		0			0
$83$	−13.9758			−1.53405			−0.767024	−	0.641619i	$-0.778263\pi$
$83$	−13.9758			−1.53405			−0.767024	+	0.641619i	$0.778263\pi$
$84$		0			0
$85$			5.03507i			0.546130i
$86$		0			0
$87$	−25.1836			−2.69996
$88$		0			0
$89$			15.2284i			1.61420i	0.590412	+	0.807102i	$0.298965\pi$
$89$			15.2284i			1.61420i	−0.590412	+	0.807102i	$0.701035\pi$
$90$		0			0
$91$	4.35258			0.456275
$92$		0			0
$93$		−	22.9041i		−	2.37505i
$94$		0			0
$95$			2.82843i			0.290191i
$96$		0			0
$97$			17.2473i			1.75120i	0.483040	+	0.875598i	$0.339532\pi$
$97$			17.2473i			1.75120i	−0.483040	+	0.875598i	$0.660468\pi$
$98$		0			0
$99$			2.47915i			0.249163i
$100$		0			0
$101$		−	11.0644i		−	1.10095i	−0.834852	−	0.550474i	$-0.814447\pi$
$101$		−	11.0644i		−	1.10095i	0.834852	−	0.550474i	$-0.185553\pi$
$102$		0			0
$103$	4.43967			0.437453			0.218727	−	0.975786i	$-0.429810\pi$
$103$	4.43967			0.437453			0.218727	+	0.975786i	$0.429810\pi$
$104$		0			0
$105$	5.42758			0.529678
$106$		0			0
$107$	−7.56033			−0.730885			−0.365443	−	0.930834i	$-0.619082\pi$
$107$	−7.56033			−0.730885			−0.365443	+	0.930834i	$0.619082\pi$
$108$		0			0
$109$			7.92508i			0.759085i	0.925174	+	0.379542i	$0.123919\pi$
$109$			7.92508i			0.759085i	−0.925174	+	0.379542i	$0.876081\pi$
$110$		0			0
$111$			2.82843i			0.268462i
$112$		0			0
$113$	−8.53750			−0.803140			−0.401570	−	0.915828i	$-0.631535\pi$
$113$	−8.53750			−0.803140			−0.401570	+	0.915828i	$0.631535\pi$
$114$		0			0
$115$	6.41550			0.598249
$116$		0			0
$117$		−	16.0995i		−	1.48840i
$118$		0			0
$119$	10.8552			0.995092
$120$		0			0
$121$	10.8780			0.988909
$122$		0			0
$123$	7.20775	+	19.0279i	0.649901	+	1.71568i
$124$		0			0
$125$	−8.19567			−0.733043
$126$		0			0
$127$	−9.18359			−0.814912			−0.407456	−	0.913225i	$-0.633584\pi$
$127$	−9.18359			−0.814912			−0.407456	+	0.913225i	$0.633584\pi$
$128$		0			0
$129$		−	10.1933i		−	0.897470i
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	6.09783			0.528749
$134$		0			0
$135$		−	11.5904i		−	0.997545i
$136$		0			0
$137$		−	7.05398i		−	0.602662i	−0.953520	−	0.301331i	$-0.902569\pi$
$137$		−	7.05398i		−	0.602662i	0.953520	−	0.301331i	$-0.0974310\pi$
$138$		0			0
$139$	17.6233			1.49478			0.747392	−	0.664383i	$-0.231306\pi$
$139$	17.6233			1.49478			0.747392	+	0.664383i	$0.231306\pi$
$140$		0			0
$141$	25.3056			2.13111
$142$		0			0
$143$	0.792249			0.0662512
$144$		0			0
$145$			7.05398i			0.585801i
$146$		0			0
$147$			10.5426i			0.869537i
$148$		0			0
$149$		−	4.78576i		−	0.392065i	−0.980597	−	0.196032i	$-0.937194\pi$
$149$		−	4.78576i		−	0.392065i	0.980597	−	0.196032i	$-0.0628058\pi$
$150$		0			0
$151$		−	15.6392i		−	1.27270i	−0.771399	−	0.636351i	$-0.780443\pi$
$151$		−	15.6392i		−	1.27270i	0.771399	−	0.636351i	$-0.219557\pi$
$152$		0			0
$153$		−	40.1514i		−	3.24605i
$154$		0			0
$155$	−6.41550			−0.515306
$156$		0			0
$157$		−	20.6359i		−	1.64693i	−0.567371	−	0.823463i	$-0.692039\pi$
$157$		−	20.6359i		−	1.64693i	0.567371	−	0.823463i	$-0.307961\pi$
$158$		0			0
$159$	−2.76809			−0.219523
$160$		0			0
$161$		−	13.8312i		−	1.09005i
$162$		0			0
$163$	−0.439665			−0.0344372			−0.0172186	−	0.999852i	$-0.505481\pi$
$163$	−0.439665			−0.0344372			−0.0172186	+	0.999852i	$0.505481\pi$
$164$		0			0
$165$	0.987918			0.0769093
$166$		0			0
$167$			9.39477i			0.726989i	0.931596	+	0.363494i	$0.118416\pi$
$167$			9.39477i			0.726989i	−0.931596	+	0.363494i	$0.881584\pi$
$168$		0			0
$169$	7.85517			0.604244
$170$		0			0
$171$		−	22.5549i		−	1.72481i
$172$		0			0
$173$	2.00000			0.152057			0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000			0.152057			0.0760286	+	0.997106i	$0.475776\pi$
$174$		0			0
$175$			8.07442i			0.610369i
$176$		0			0
$177$		−	21.5070i		−	1.61657i
$178$		0			0
$179$			4.18716i			0.312963i	0.987681	+	0.156482i	$0.0500152\pi$
$179$			4.18716i			0.312963i	−0.987681	+	0.156482i	$0.949985\pi$
$180$		0			0
$181$			0.871095i			0.0647480i	0.999476	+	0.0323740i	$0.0103068\pi$
$181$			0.871095i			0.0647480i	−0.999476	+	0.0323740i	$0.989693\pi$
$182$		0			0
$183$		−	40.5733i		−	2.99926i
$184$		0			0
$185$	0.792249			0.0582473
$186$		0			0
$187$	1.97584			0.144487
$188$		0			0
$189$	−24.9879			−1.81760
$190$		0			0
$191$			19.8648i			1.43737i	0.695338	+	0.718683i	$0.255255\pi$
$191$			19.8648i			1.43737i	−0.695338	+	0.718683i	$0.744745\pi$
$192$		0			0
$193$			3.13932i			0.225973i	0.993597	+	0.112987i	$0.0360417\pi$
$193$			3.13932i			0.225973i	−0.993597	+	0.112987i	$0.963958\pi$
$194$		0			0
$195$	−6.41550			−0.459424
$196$		0			0
$197$	−13.6474			−0.972338			−0.486169	−	0.873865i	$-0.661606\pi$
$197$	−13.6474			−0.972338			−0.486169	+	0.873865i	$0.661606\pi$
$198$		0			0
$199$			12.5341i			0.888518i	0.895898	+	0.444259i	$0.146533\pi$
$199$			12.5341i			0.888518i	−0.895898	+	0.444259i	$0.853467\pi$
$200$		0			0
$201$	−13.3056			−0.938504
$202$		0			0
$203$	15.2078			1.06737
$204$		0			0
$205$	5.32975	−	2.01891i	0.372246	−	0.141007i
$206$		0			0
$207$	−51.1594			−3.55583
$208$		0			0
$209$	1.10992			0.0767745
$210$		0			0
$211$			17.3966i			1.19763i	0.800886	+	0.598817i	$0.204362\pi$
$211$			17.3966i			1.19763i	−0.800886	+	0.598817i	$0.795638\pi$
$212$		0			0
$213$	20.8659			1.42971
$214$		0			0
$215$	−2.85517			−0.194721
$216$		0			0
$217$			13.8312i			0.938926i
$218$		0			0
$219$		−	38.4434i		−	2.59776i
$220$		0			0
$221$	−12.8310			−0.863107
$222$		0			0
$223$	3.56033			0.238418			0.119209	−	0.992869i	$-0.461964\pi$
$223$	3.56033			0.238418			0.119209	+	0.992869i	$0.461964\pi$
$224$		0			0
$225$	29.8659			1.99106
$226$		0			0
$227$		−	19.1046i		−	1.26802i	−0.773325	−	0.634010i	$-0.781408\pi$
$227$		−	19.1046i		−	1.26802i	0.773325	−	0.634010i	$-0.218592\pi$
$228$		0			0
$229$			26.5695i			1.75576i	0.478880	+	0.877881i	$0.341043\pi$
$229$			26.5695i			1.75576i	−0.478880	+	0.877881i	$0.658957\pi$
$230$		0			0
$231$		−	2.12986i		−	0.140135i
$232$		0			0
$233$			16.4719i			1.07911i	0.841950	+	0.539556i	$0.181408\pi$
$233$			16.4719i			1.07911i	−0.841950	+	0.539556i	$0.818592\pi$
$234$		0			0
$235$		−	7.08815i		−	0.462380i
$236$		0			0
$237$	25.3056			1.64377
$238$		0			0
$239$			14.3189i			0.926212i	0.886303	+	0.463106i	$0.153265\pi$
$239$			14.3189i			0.926212i	−0.886303	+	0.463106i	$0.846735\pi$
$240$		0			0
$241$	0.768086			0.0494768			0.0247384	−	0.999694i	$-0.492125\pi$
$241$	0.768086			0.0494768			0.0247384	+	0.999694i	$0.492125\pi$
$242$		0			0
$243$			24.7615i			1.58845i
$244$		0			0
$245$	2.95300			0.188660
$246$		0			0
$247$	−7.20775			−0.458618
$248$		0			0
$249$		−	44.4111i		−	2.81444i
$250$		0			0
$251$	−23.9517			−1.51182			−0.755908	−	0.654678i	$-0.772804\pi$
$251$	−23.9517			−1.51182			−0.755908	+	0.654678i	$0.772804\pi$
$252$		0			0
$253$		−	2.51754i		−	0.158276i
$254$		0			0
$255$	−16.0000			−1.00196
$256$		0			0
$257$		−	2.51754i		−	0.157040i	−0.996913	−	0.0785198i	$-0.974981\pi$
$257$		−	2.51754i		−	0.157040i	0.996913	−	0.0785198i	$-0.0250194\pi$
$258$		0			0
$259$		−	1.70802i		−	0.106131i
$260$		0			0
$261$		−	56.2509i		−	3.48184i
$262$		0			0
$263$			21.9946i			1.35625i	0.734947	+	0.678124i	$0.237207\pi$
$263$			21.9946i			1.35625i	−0.734947	+	0.678124i	$0.762793\pi$
$264$		0			0
$265$			0.775347i			0.0476292i
$266$		0			0
$267$	−48.3913			−2.96150
$268$		0			0
$269$	23.1836			1.41353			0.706764	−	0.707449i	$-0.250154\pi$
$269$	23.1836			1.41353			0.706764	+	0.707449i	$0.250154\pi$
$270$		0			0
$271$	7.20775			0.437840			0.218920	−	0.975743i	$-0.429747\pi$
$271$	7.20775			0.437840			0.218920	+	0.975743i	$0.429747\pi$
$272$		0			0
$273$			13.8312i			0.837105i
$274$		0			0
$275$			1.46969i			0.0886257i
$276$		0			0
$277$	−9.08575			−0.545910			−0.272955	−	0.962027i	$-0.588001\pi$
$277$	−9.08575			−0.545910			−0.272955	+	0.962027i	$0.588001\pi$
$278$		0			0
$279$	51.1594			3.06283
$280$		0			0
$281$			17.7459i			1.05863i	0.848425	+	0.529316i	$0.177551\pi$
$281$			17.7459i			1.05863i	−0.848425	+	0.529316i	$0.822449\pi$
$282$		0			0
$283$	17.6233			1.04759			0.523797	−	0.851843i	$-0.324515\pi$
$283$	17.6233			1.04759			0.523797	+	0.851843i	$0.324515\pi$
$284$		0			0
$285$	−8.98792			−0.532398
$286$		0			0
$287$	−4.35258	−	11.4905i	−0.256925	−	0.678260i
$288$		0			0
$289$	−15.0000			−0.882353
$290$		0			0
$291$	−54.8068			−3.21283
$292$		0			0
$293$		−	1.64644i		−	0.0961862i	−0.998843	−	0.0480931i	$-0.984686\pi$
$293$		−	1.64644i		−	0.0961862i	0.998843	−	0.0480931i	$-0.0153144\pi$
$294$		0			0
$295$	−6.02416			−0.350740
$296$		0			0
$297$	−4.54825			−0.263916
$298$		0			0
$299$			16.3488i			0.945475i
$300$		0			0
$301$			6.15548i			0.354796i
$302$		0			0
$303$	35.1594			2.01986
$304$		0			0
$305$	−11.3647			−0.650739
$306$		0			0
$307$	31.9517			1.82358			0.911789	−	0.410659i	$-0.134701\pi$
$307$	31.9517			1.82358			0.911789	+	0.410659i	$0.134701\pi$
$308$		0			0
$309$			14.1080i			0.802574i
$310$		0			0
$311$		−	30.4799i		−	1.72836i	−0.503184	−	0.864179i	$-0.667838\pi$
$311$		−	30.4799i		−	1.72836i	0.503184	−	0.864179i	$-0.332162\pi$
$312$		0			0
$313$			10.8151i			0.611304i	0.952143	+	0.305652i	$0.0988745\pi$
$313$			10.8151i			0.611304i	−0.952143	+	0.305652i	$0.901126\pi$
$314$		0			0
$315$			12.1232i			0.683066i
$316$		0			0
$317$			18.6170i			1.04564i	0.852445	+	0.522818i	$0.175119\pi$
$317$			18.6170i			1.04564i	−0.852445	+	0.522818i	$0.824881\pi$
$318$		0			0
$319$	2.76809			0.154983
$320$		0			0
$321$		−	24.0245i		−	1.34092i
$322$		0			0
$323$	−17.9758			−1.00020
$324$		0			0
$325$		−	9.54412i		−	0.529412i
$326$		0			0
$327$	−25.1836			−1.39266
$328$		0			0
$329$	−15.2814			−0.842492
$330$		0			0
$331$		−	3.14354i		−	0.172784i	−0.996261	−	0.0863922i	$-0.972466\pi$
$331$		−	3.14354i		−	0.172784i	0.996261	−	0.0863922i	$-0.0275338\pi$
$332$		0			0
$333$	−6.31767			−0.346206
$334$		0			0
$335$			3.72693i			0.203624i
$336$		0			0
$337$	9.43834			0.514139			0.257069	−	0.966393i	$-0.417243\pi$
$337$	9.43834			0.514139			0.257069	+	0.966393i	$0.417243\pi$
$338$		0			0
$339$		−	27.1297i		−	1.47348i
$340$		0			0
$341$			2.51754i			0.136332i
$342$		0			0
$343$		−	19.7990i		−	1.06904i
$344$		0			0
$345$			20.3866i			1.09758i
$346$		0			0
$347$		−	6.73887i		−	0.361762i	−0.983505	−	0.180881i	$-0.942105\pi$
$347$		−	6.73887i		−	0.361762i	0.983505	−	0.180881i	$-0.0578948\pi$
$348$		0			0
$349$	−28.6461			−1.53339			−0.766695	−	0.642012i	$-0.778100\pi$
$349$	−28.6461			−1.53339			−0.766695	+	0.642012i	$0.778100\pi$
$350$		0			0
$351$	29.5362			1.57652
$352$		0			0
$353$	34.3177			1.82655			0.913273	−	0.407349i	$-0.133547\pi$
$353$	34.3177			1.82655			0.913273	+	0.407349i	$0.133547\pi$
$354$		0			0
$355$		−	5.84459i		−	0.310199i
$356$		0			0
$357$			34.4946i			1.82565i
$358$		0			0
$359$	1.97584			0.104281			0.0521403	−	0.998640i	$-0.483396\pi$
$359$	1.97584			0.104281			0.0521403	+	0.998640i	$0.483396\pi$
$360$		0			0
$361$	8.90217			0.468535
$362$		0			0
$363$			34.5671i			1.81430i
$364$		0			0
$365$	−10.7681			−0.563627
$366$		0			0
$367$	28.0388			1.46361			0.731806	−	0.681513i	$-0.238678\pi$
$367$	28.0388			1.46361			0.731806	+	0.681513i	$0.238678\pi$
$368$		0			0
$369$	−42.5013	+	16.0995i	−2.21253	+	0.838105i
$370$		0			0
$371$	1.67158			0.0867841
$372$		0			0
$373$	−6.00000			−0.310668			−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000			−0.310668			−0.155334	+	0.987862i	$0.549645\pi$
$374$		0			0
$375$		−	26.0435i		−	1.34488i
$376$		0			0
$377$	−17.9758			−0.925803
$378$		0			0
$379$	−1.23191			−0.0632792			−0.0316396	−	0.999499i	$-0.510073\pi$
$379$	−1.23191			−0.0632792			−0.0316396	+	0.999499i	$0.510073\pi$
$380$		0			0
$381$		−	29.1828i		−	1.49508i
$382$		0			0
$383$		−	6.45539i		−	0.329855i	−0.986306	−	0.164927i	$-0.947261\pi$
$383$		−	6.45539i		−	0.329855i	0.986306	−	0.164927i	$-0.0527390\pi$
$384$		0			0
$385$	−0.596580			−0.0304045
$386$		0			0
$387$	22.7681			1.15737
$388$		0			0
$389$	13.9517			0.707378			0.353689	−	0.935363i	$-0.384927\pi$
$389$	13.9517			0.707378			0.353689	+	0.935363i	$0.384927\pi$
$390$		0			0
$391$			40.7732i			2.06199i
$392$		0			0
$393$			38.1325i			1.92353i
$394$		0			0
$395$		−	7.08815i		−	0.356644i
$396$		0			0
$397$			28.0350i			1.40703i	0.710678	+	0.703517i	$0.248388\pi$
$397$			28.0350i			1.40703i	−0.710678	+	0.703517i	$0.751612\pi$
$398$		0			0
$399$			19.3771i			0.970071i
$400$		0			0
$401$	4.29350			0.214407			0.107204	−	0.994237i	$-0.465810\pi$
$401$	4.29350			0.214407			0.107204	+	0.994237i	$0.465810\pi$
$402$		0			0
$403$		−	16.3488i		−	0.814391i
$404$		0			0
$405$	17.8780			0.888365
$406$		0			0
$407$		−	0.310890i		−	0.0154103i
$408$		0			0
$409$	1.92633			0.0952508			0.0476254	−	0.998865i	$-0.484835\pi$
$409$	1.92633			0.0952508			0.0476254	+	0.998865i	$0.484835\pi$
$410$		0			0
$411$	22.4155			1.10568
$412$		0			0
$413$			12.9876i			0.639076i
$414$		0			0
$415$	−12.4397			−0.610639
$416$		0			0
$417$			56.0016i			2.74241i
$418$		0			0
$419$	30.7681			1.50312			0.751560	−	0.659665i	$-0.229302\pi$
$419$	30.7681			1.50312			0.751560	+	0.659665i	$0.229302\pi$
$420$		0			0
$421$			13.0833i			0.637641i	0.947815	+	0.318821i	$0.103287\pi$
$421$			13.0833i			0.637641i	−0.947815	+	0.318821i	$0.896713\pi$
$422$		0			0
$423$			56.5234i			2.74826i
$424$		0			0
$425$		−	23.8026i		−	1.15460i
$426$		0			0
$427$			24.5012i			1.18570i
$428$		0			0
$429$			2.51754i			0.121548i
$430$		0			0
$431$	5.23191			0.252012			0.126006	−	0.992029i	$-0.459784\pi$
$431$	5.23191			0.252012			0.126006	+	0.992029i	$0.459784\pi$
$432$		0			0
$433$	−8.85517			−0.425552			−0.212776	−	0.977101i	$-0.568250\pi$
$433$	−8.85517			−0.425552			−0.212776	+	0.977101i	$0.568250\pi$
$434$		0			0
$435$	−22.4155			−1.07474
$436$		0			0
$437$			22.9041i			1.09565i
$438$		0			0
$439$			18.4677i			0.881413i	0.897651	+	0.440707i	$0.145272\pi$
$439$			18.4677i			0.881413i	−0.897651	+	0.440707i	$0.854728\pi$
$440$		0			0
$441$	−23.5483			−1.12135
$442$		0			0
$443$	20.3913			0.968822			0.484411	−	0.874841i	$-0.339034\pi$
$443$	20.3913			0.968822			0.484411	+	0.874841i	$0.339034\pi$
$444$		0			0
$445$			13.5545i			0.642546i
$446$		0			0
$447$	15.2078			0.719302
$448$		0			0
$449$	−8.32842			−0.393042			−0.196521	−	0.980500i	$-0.562964\pi$
$449$	−8.32842			−0.393042			−0.196521	+	0.980500i	$0.562964\pi$
$450$		0			0
$451$	−0.792249	−	2.09147i	−0.0373056	−	0.0984835i
$452$		0			0
$453$	49.6969			2.33497
$454$		0			0
$455$	3.87416			0.181624
$456$		0			0
$457$		−	12.0891i		−	0.565502i	−0.959193	−	0.282751i	$-0.908753\pi$
$457$		−	12.0891i		−	0.565502i	0.959193	−	0.282751i	$-0.0912470\pi$
$458$		0			0
$459$	73.6620			3.43825
$460$		0			0
$461$	4.45042			0.207277			0.103638	−	0.994615i	$-0.466952\pi$
$461$	4.45042			0.207277			0.103638	+	0.994615i	$0.466952\pi$
$462$		0			0
$463$		−	13.3436i		−	0.620130i	−0.950715	−	0.310065i	$-0.899649\pi$
$463$		−	13.3436i		−	0.620130i	0.950715	−	0.310065i	$-0.100351\pi$
$464$		0			0
$465$		−	20.3866i		−	0.945406i
$466$		0			0
$467$	−5.67158			−0.262449			−0.131225	−	0.991353i	$-0.541891\pi$
$467$	−5.67158			−0.262449			−0.131225	+	0.991353i	$0.541891\pi$
$468$		0			0
$469$	8.03492			0.371018
$470$		0			0
$471$	65.5749			3.02153
$472$		0			0
$473$			1.12041i			0.0515165i
$474$		0			0
$475$		−	13.3710i		−	0.613504i
$476$		0			0
$477$		−	6.18289i		−	0.283095i
$478$		0			0
$479$		−	3.31607i		−	0.151515i	−0.997126	−	0.0757575i	$-0.975863\pi$
$479$		−	3.31607i		−	0.151515i	0.997126	−	0.0757575i	$-0.0241375\pi$
$480$		0			0
$481$			2.01891i			0.0920543i
$482$		0			0
$483$	43.9517			1.99987
$484$		0			0
$485$			15.3515i			0.697077i
$486$		0			0
$487$	−9.09651			−0.412202			−0.206101	−	0.978531i	$-0.566078\pi$
$487$	−9.09651			−0.412202			−0.206101	+	0.978531i	$0.566078\pi$
$488$		0			0
$489$		−	1.39713i		−	0.0631803i
$490$		0			0
$491$	−25.6233			−1.15636			−0.578181	−	0.815909i	$-0.696237\pi$
$491$	−25.6233			−1.15636			−0.578181	+	0.815909i	$0.696237\pi$
$492$		0			0
$493$	−44.8310			−2.01909
$494$		0			0
$495$			2.20665i			0.0991814i
$496$		0			0
$497$	−12.6004			−0.565206
$498$		0			0
$499$			34.5330i			1.54591i	0.634462	+	0.772954i	$0.281222\pi$
$499$			34.5330i			1.54591i	−0.634462	+	0.772954i	$0.718778\pi$
$500$		0			0
$501$	−29.8538			−1.33377
$502$		0			0
$503$			20.0989i			0.896165i	0.893992	+	0.448083i	$0.147893\pi$
$503$			20.0989i			0.896165i	−0.893992	+	0.448083i	$0.852107\pi$
$504$		0			0
$505$		−	9.84824i		−	0.438241i
$506$		0			0
$507$			24.9614i			1.10858i
$508$		0			0
$509$		−	2.76685i		−	0.122639i	−0.998118	−	0.0613193i	$-0.980469\pi$
$509$		−	2.76685i		−	0.122639i	0.998118	−	0.0613193i	$-0.0195308\pi$
$510$		0			0
$511$			23.2150i			1.02697i
$512$		0			0
$513$	41.3793			1.82694
$514$		0			0
$515$	3.95167			0.174132
$516$		0			0
$517$	−2.78150			−0.122330
$518$		0			0
$519$			6.35542i			0.278972i
$520$		0			0
$521$		−	13.0559i		−	0.571989i	−0.958231	−	0.285995i	$-0.907676\pi$
$521$		−	13.0559i		−	0.571989i	0.958231	−	0.285995i	$-0.0923240\pi$
$522$		0			0
$523$	6.68100			0.292140			0.146070	−	0.989274i	$-0.453338\pi$
$523$	6.68100			0.292140			0.146070	+	0.989274i	$0.453338\pi$
$524$		0			0
$525$	−25.6582			−1.11981
$526$		0			0
$527$		−	40.7732i		−	1.77611i
$528$		0			0
$529$	28.9517			1.25877
$530$		0			0
$531$	48.0388			2.08470
$532$		0			0
$533$	5.14483	+	13.5819i	0.222847	+	0.588299i
$534$		0			0
$535$	−6.72933			−0.290934
$536$		0			0
$537$	−13.3056			−0.574178
$538$		0			0
$539$		−	1.15880i		−	0.0499131i
$540$		0			0
$541$	−22.6219			−0.972593			−0.486296	−	0.873794i	$-0.661652\pi$
$541$	−22.6219			−0.972593			−0.486296	+	0.873794i	$0.661652\pi$
$542$		0			0
$543$	−2.76809			−0.118790
$544$		0			0
$545$			7.05398i			0.302159i
$546$		0			0
$547$			6.31703i			0.270097i	0.990839	+	0.135048i	$0.0431190\pi$
$547$			6.31703i			0.270097i	−0.990839	+	0.135048i	$0.956881\pi$
$548$		0			0
$549$	90.6258			3.86781
$550$		0			0
$551$	−25.1836			−1.07286
$552$		0			0
$553$	−15.2814			−0.649832
$554$		0			0
$555$			2.51754i			0.106863i
$556$		0			0
$557$			22.6548i			0.959916i	0.877292	+	0.479958i	$0.159348\pi$
$557$			22.6548i			0.959916i	−0.877292	+	0.479958i	$0.840652\pi$
$558$		0			0
$559$		−	7.27589i		−	0.307737i
$560$		0			0
$561$			6.27863i			0.265084i
$562$		0			0
$563$		−	14.9559i		−	0.630315i	−0.949039	−	0.315157i	$-0.897943\pi$
$563$		−	14.9559i		−	0.630315i	0.949039	−	0.315157i	$-0.102057\pi$
$564$		0			0
$565$	−7.59909			−0.319696
$566$		0			0
$567$		−	38.5434i		−	1.61867i
$568$		0			0
$569$	−12.0978			−0.507168			−0.253584	−	0.967313i	$-0.581609\pi$
$569$	−12.0978			−0.507168			−0.253584	+	0.967313i	$0.581609\pi$
$570$		0			0
$571$			18.1416i			0.759201i	0.925151	+	0.379600i	$0.123938\pi$
$571$			18.1416i			0.759201i	−0.925151	+	0.379600i	$0.876062\pi$
$572$		0			0
$573$	−63.1245			−2.63706
$574$		0			0
$575$	−30.3284			−1.26478
$576$		0			0
$577$			2.51754i			0.104806i	0.998626	+	0.0524032i	$0.0166881\pi$
$577$			2.51754i			0.104806i	−0.998626	+	0.0524032i	$0.983312\pi$
$578$		0			0
$579$	−9.97584			−0.414582
$580$		0			0
$581$			26.8188i			1.11263i
$582$		0			0
$583$	0.304258			0.0126011
$584$		0			0
$585$		−	14.3299i		−	0.592467i
$586$		0			0
$587$			11.8629i			0.489635i	0.969569	+	0.244818i	$0.0787281\pi$
$587$			11.8629i			0.489635i	−0.969569	+	0.244818i	$0.921272\pi$
$588$		0			0
$589$		−	22.9041i		−	0.943748i
$590$		0			0
$591$		−	43.3675i		−	1.78390i
$592$		0			0
$593$			32.9743i			1.35409i	0.735941	+	0.677046i	$0.236740\pi$
$593$			32.9743i			1.35409i	−0.735941	+	0.677046i	$0.763260\pi$
$594$		0			0
$595$	9.66201			0.396104
$596$		0			0
$597$	−39.8297			−1.63012
$598$		0			0
$599$	−38.7198			−1.58205			−0.791023	−	0.611786i	$-0.790451\pi$
$599$	−38.7198			−1.58205			−0.791023	+	0.611786i	$0.790451\pi$
$600$		0			0
$601$		−	37.8558i		−	1.54417i	−0.635519	−	0.772085i	$-0.719214\pi$
$601$		−	37.8558i		−	1.54417i	0.635519	−	0.772085i	$-0.280786\pi$
$602$		0			0
$603$		−	29.7198i		−	1.21028i
$604$		0			0
$605$	9.68233			0.393643
$606$		0			0
$607$	34.7681			1.41119			0.705597	−	0.708614i	$-0.250679\pi$
$607$	34.7681			1.41119			0.705597	+	0.708614i	$0.250679\pi$
$608$		0			0
$609$			48.3258i			1.95826i
$610$		0			0
$611$	18.0629			0.730747
$612$		0			0
$613$	13.7211			0.554190			0.277095	−	0.960843i	$-0.410628\pi$
$613$	13.7211			0.554190			0.277095	+	0.960843i	$0.410628\pi$
$614$		0			0
$615$	6.41550	+	16.9364i	0.258698	+	0.682941i
$616$		0			0
$617$	1.12067			0.0451165			0.0225582	−	0.999746i	$-0.492819\pi$
$617$	1.12067			0.0451165			0.0225582	+	0.999746i	$0.492819\pi$
$618$		0			0
$619$	−23.6474			−0.950470			−0.475235	−	0.879859i	$-0.657637\pi$
$619$	−23.6474			−0.950470			−0.475235	+	0.879859i	$0.657637\pi$
$620$		0			0
$621$		−	93.8574i		−	3.76637i
$622$		0			0
$623$	29.2223			1.17077
$624$		0			0
$625$	13.7439			0.549757
$626$		0			0
$627$			3.52699i			0.140854i
$628$		0			0
$629$			5.03507i			0.200762i
$630$		0			0
$631$	5.53617			0.220392			0.110196	−	0.993910i	$-0.464852\pi$
$631$	5.53617			0.220392			0.110196	+	0.993910i	$0.464852\pi$
$632$		0			0
$633$	−55.2814			−2.19724
$634$		0			0
$635$	−8.17416			−0.324382
$636$		0			0
$637$			7.52521i			0.298160i
$638$		0			0
$639$			46.6068i			1.84374i
$640$		0			0
$641$		−	10.9686i		−	0.433235i	−0.976256	−	0.216618i	$-0.930497\pi$
$641$		−	10.9686i		−	0.433235i	0.976256	−	0.216618i	$-0.0695025\pi$
$642$		0			0
$643$		−	47.0012i		−	1.85355i	−0.375619	−	0.926774i	$-0.622570\pi$
$643$		−	47.0012i		−	1.85355i	0.375619	−	0.926774i	$-0.377430\pi$
$644$		0			0
$645$		−	9.07289i		−	0.357245i
$646$		0			0
$647$	−12.8310			−0.504439			−0.252219	−	0.967670i	$-0.581160\pi$
$647$	−12.8310			−0.504439			−0.252219	+	0.967670i	$0.581160\pi$
$648$		0			0
$649$			2.36397i			0.0927939i
$650$		0			0
$651$	−43.9517			−1.72260
$652$		0			0
$653$		−	35.6424i		−	1.39479i	−0.716685	−	0.697397i	$-0.754341\pi$
$653$		−	35.6424i		−	1.39479i	0.716685	−	0.697397i	$-0.245659\pi$
$654$		0			0
$655$	10.6810			0.417341
$656$		0			0
$657$	85.8684			3.35005
$658$		0			0
$659$		−	19.4497i		−	0.757653i	−0.925468	−	0.378827i	$-0.876328\pi$
$659$		−	19.4497i		−	0.757653i	0.925468	−	0.378827i	$-0.123672\pi$
$660$		0			0
$661$	29.7211			1.15602			0.578008	−	0.816031i	$-0.303830\pi$
$661$	29.7211			1.15602			0.578008	+	0.816031i	$0.303830\pi$
$662$		0			0
$663$		−	40.7732i		−	1.58350i
$664$		0			0
$665$	5.42758			0.210473
$666$		0			0
$667$			57.1220i			2.21177i
$668$		0			0
$669$			11.3137i			0.437413i
$670$		0			0
$671$			4.45966i			0.172163i
$672$		0			0
$673$			35.1163i			1.35364i	0.736151	+	0.676818i	$0.236641\pi$
$673$			35.1163i			1.35364i	−0.736151	+	0.676818i	$0.763359\pi$
$674$		0			0
$675$			54.7922i			2.10895i
$676$		0			0
$677$	10.8659			0.417611			0.208806	−	0.977957i	$-0.433042\pi$
$677$	10.8659			0.417611			0.208806	+	0.977957i	$0.433042\pi$
$678$		0			0
$679$	33.0965			1.27013
$680$		0			0
$681$	60.7090			2.32637
$682$		0			0
$683$			12.7492i			0.487836i	0.969796	+	0.243918i	$0.0784327\pi$
$683$			12.7492i			0.487836i	−0.969796	+	0.243918i	$0.921567\pi$
$684$		0			0
$685$		−	6.27863i		−	0.239894i
$686$		0			0
$687$	−84.4301			−3.22121
$688$		0			0
$689$	−1.97584			−0.0752734
$690$		0			0
$691$		−	31.6619i		−	1.20448i	−0.798317	−	0.602238i	$-0.794276\pi$
$691$		−	31.6619i		−	1.20448i	0.798317	−	0.602238i	$-0.205724\pi$
$692$		0			0
$693$	4.75733			0.180716
$694$		0			0
$695$	15.6862			0.595010
$696$		0			0
$697$	12.8310	+	33.8728i	0.486009	+	1.28302i
$698$		0			0
$699$	−52.3430			−1.97979
$700$		0			0
$701$	16.1849			0.611296			0.305648	−	0.952145i	$-0.401127\pi$
$701$	16.1849			0.611296			0.305648	+	0.952145i	$0.401127\pi$
$702$		0			0
$703$			2.82843i			0.106676i
$704$		0			0
$705$	22.5241			0.848306
$706$		0			0
$707$	−21.2319			−0.798508
$708$		0			0
$709$		−	50.5940i		−	1.90010i	−0.312100	−	0.950049i	$-0.601032\pi$
$709$		−	50.5940i		−	1.90010i	0.312100	−	0.950049i	$-0.398968\pi$
$710$		0			0
$711$			56.5234i			2.11979i
$712$		0			0
$713$	−51.9517			−1.94561
$714$		0			0
$715$	0.705168			0.0263718
$716$		0			0
$717$	−45.5013			−1.69928
$718$		0			0
$719$		−	26.4543i		−	0.986579i	−0.869865	−	0.493290i	$-0.835794\pi$
$719$		−	26.4543i		−	0.986579i	0.869865	−	0.493290i	$-0.164206\pi$
$720$		0			0
$721$		−	8.51945i		−	0.317281i
$722$		0			0
$723$			2.44075i			0.0907726i
$724$		0			0
$725$		−	33.3467i		−	1.23847i
$726$		0			0
$727$			13.4752i			0.499767i	0.968276	+	0.249884i	$0.0803924\pi$
$727$			13.4752i			0.499767i	−0.968276	+	0.249884i	$0.919608\pi$
$728$		0			0
$729$	−18.4276			−0.682503
$730$		0			0
$731$		−	18.1458i		−	0.671146i
$732$		0			0
$733$	26.3526			0.973355			0.486677	−	0.873582i	$-0.338209\pi$
$733$	26.3526			0.973355			0.486677	+	0.873582i	$0.338209\pi$
$734$		0			0
$735$			9.38378i			0.346126i
$736$		0			0
$737$	1.46250			0.0538719
$738$		0			0
$739$	10.4155			0.383140			0.191570	−	0.981479i	$-0.438642\pi$
$739$	10.4155			0.383140			0.191570	+	0.981479i	$0.438642\pi$
$740$		0			0
$741$		−	22.9041i		−	0.841404i
$742$		0			0
$743$	−30.4155			−1.11584			−0.557918	−	0.829896i	$-0.688400\pi$
$743$	−30.4155			−1.11584			−0.557918	+	0.829896i	$0.688400\pi$
$744$		0			0
$745$		−	4.25973i		−	0.156064i
$746$		0			0
$747$	99.1982			3.62947
$748$		0			0
$749$			14.5078i			0.530105i
$750$		0			0
$751$		−	52.6087i		−	1.91972i	−0.280481	−	0.959860i	$-0.590494\pi$
$751$		−	52.6087i		−	1.91972i	0.280481	−	0.959860i	$-0.409506\pi$
$752$		0			0
$753$		−	76.1114i		−	2.77366i
$754$		0			0
$755$		−	13.9202i		−	0.506609i
$756$		0			0
$757$			26.5695i			0.965684i	0.875707	+	0.482842i	$0.160396\pi$
$757$			26.5695i			0.965684i	−0.875707	+	0.482842i	$0.839604\pi$
$758$		0			0
$759$	8.00000			0.290382
$760$		0			0
$761$	26.7090			0.968201			0.484100	−	0.875012i	$-0.339147\pi$
$761$	26.7090			0.968201			0.484100	+	0.875012i	$0.339147\pi$
$762$		0			0
$763$	15.2078			0.550558
$764$		0			0
$765$		−	35.7381i		−	1.29211i
$766$		0			0
$767$		−	15.3515i		−	0.554311i
$768$		0			0
$769$	30.8310			1.11179			0.555897	−	0.831251i	$-0.312375\pi$
$769$	30.8310			1.11179			0.555897	+	0.831251i	$0.312375\pi$
$770$		0			0
$771$	8.00000			0.288113
$772$		0			0
$773$			13.7355i			0.494032i	0.969011	+	0.247016i	$0.0794499\pi$
$773$			13.7355i			0.494032i	−0.969011	+	0.247016i	$0.920550\pi$
$774$		0			0
$775$	30.3284			1.08943
$776$		0			0
$777$	5.42758			0.194713
$778$		0			0
$779$	7.20775	+	19.0279i	0.258244	+	0.681744i
$780$		0			0
$781$	−2.29350			−0.0820680
$782$		0			0
$783$	103.198			3.68800
$784$		0			0
$785$		−	18.3677i		−	0.655571i
$786$		0			0
$787$	13.5845			0.484235			0.242118	−	0.970247i	$-0.422158\pi$
$787$	13.5845			0.484235			0.242118	+	0.970247i	$0.422158\pi$
$788$		0			0
$789$	−69.8926			−2.48824
$790$		0			0
$791$			16.3830i			0.582511i
$792$		0			0
$793$		−	28.9609i		−	1.02843i
$794$		0			0
$795$	−2.46383			−0.0873830
$796$		0			0
$797$	−16.7198			−0.592244			−0.296122	−	0.955150i	$-0.595694\pi$
$797$	−16.7198			−0.592244			−0.296122	+	0.955150i	$0.595694\pi$
$798$		0			0
$799$	45.0482			1.59369
$800$		0			0
$801$		−	108.088i		−	3.81912i
$802$		0			0
$803$			4.22555i			0.149117i
$804$		0			0
$805$		−	12.3110i		−	0.433905i
$806$		0			0
$807$			73.6707i			2.59333i
$808$		0			0
$809$			34.7165i			1.22057i	0.792183	+	0.610283i	$0.208944\pi$
$809$			34.7165i			1.22057i	−0.792183	+	0.610283i	$0.791056\pi$
$810$		0			0
$811$	37.9758			1.33351			0.666756	−	0.745276i	$-0.267682\pi$
$811$	37.9758			1.33351			0.666756	+	0.745276i	$0.267682\pi$
$812$		0			0
$813$			22.9041i			0.803283i
$814$		0			0
$815$	−0.391339			−0.0137080
$816$		0			0
$817$		−	10.1933i		−	0.356618i
$818$		0			0
$819$	−30.8939			−1.07952
$820$		0			0
$821$	−22.2306			−0.775853			−0.387926	−	0.921690i	$-0.626809\pi$
$821$	−22.2306			−0.775853			−0.387926	+	0.921690i	$0.626809\pi$
$822$		0			0
$823$			48.6599i			1.69618i	0.529855	+	0.848088i	$0.322246\pi$
$823$			48.6599i			1.69618i	−0.529855	+	0.848088i	$0.677754\pi$
$824$		0			0
$825$	−4.67025			−0.162597
$826$		0			0
$827$			22.2781i			0.774687i	0.921936	+	0.387343i	$0.126607\pi$
$827$			22.2781i			0.774687i	−0.921936	+	0.387343i	$0.873393\pi$
$828$		0			0
$829$	46.1124			1.60155			0.800775	−	0.598965i	$-0.204421\pi$
$829$	46.1124			1.60155			0.800775	+	0.598965i	$0.204421\pi$
$830$		0			0
$831$		−	28.8719i		−	1.00155i
$832$		0			0
$833$			18.7676i			0.650257i
$834$		0			0
$835$			8.36213i			0.289383i
$836$		0			0
$837$			93.8574i			3.24418i
$838$		0			0
$839$		−	1.91894i		−	0.0662492i	−0.999451	−	0.0331246i	$-0.989454\pi$
$839$		−	1.91894i		−	0.0662492i	0.999451	−	0.0331246i	$-0.0105458\pi$
$840$		0			0
$841$	−33.8068			−1.16575
$842$		0			0
$843$	−56.3913			−1.94222
$844$		0			0
$845$	6.99176			0.240524
$846$		0			0
$847$		−	20.8742i		−	0.717247i
$848$		0			0
$849$			56.0016i			1.92197i
$850$		0			0
$851$	6.41550			0.219921
$852$		0			0
$853$	6.83100			0.233889			0.116945	−	0.993138i	$-0.462690\pi$
$853$	6.83100			0.233889			0.116945	+	0.993138i	$0.462690\pi$
$854$		0			0
$855$		−	20.0757i		−	0.686575i
$856$		0			0
$857$	−11.2185			−0.383217			−0.191608	−	0.981471i	$-0.561370\pi$
$857$	−11.2185			−0.383217			−0.191608	+	0.981471i	$0.561370\pi$
$858$		0			0
$859$	14.8552			0.506852			0.253426	−	0.967355i	$-0.418443\pi$
$859$	14.8552			0.506852			0.253426	+	0.967355i	$0.418443\pi$
$860$		0			0
$861$	36.5133	−	13.8312i	1.24437	−	0.471368i
$862$		0			0
$863$	−27.6474			−0.941129			−0.470564	−	0.882366i	$-0.655950\pi$
$863$	−27.6474			−0.941129			−0.470564	+	0.882366i	$0.655950\pi$
$864$		0			0
$865$	1.78017			0.0605275
$866$		0			0
$867$		−	47.6656i		−	1.61881i
$868$		0			0
$869$	−2.78150			−0.0943558
$870$		0			0
$871$	−9.49742			−0.321808
$872$		0			0
$873$		−	122.418i		−	4.14323i
$874$		0			0
$875$			15.7270i			0.531670i
$876$		0			0
$877$	−9.59909			−0.324138			−0.162069	−	0.986779i	$-0.551817\pi$
$877$	−9.59909			−0.324138			−0.162069	+	0.986779i	$0.551817\pi$
$878$		0			0
$879$	5.23191			0.176468
$880$		0			0
$881$	−7.09651			−0.239087			−0.119544	−	0.992829i	$-0.538143\pi$
$881$	−7.09651			−0.239087			−0.119544	+	0.992829i	$0.538143\pi$
$882$		0			0
$883$			5.97196i			0.200973i	0.994938	+	0.100486i	$0.0320399\pi$
$883$			5.97196i			0.200973i	−0.994938	+	0.100486i	$0.967960\pi$
$884$		0			0
$885$		−	19.1430i		−	0.643486i
$886$		0			0
$887$		−	2.81744i		−	0.0946003i	−0.998881	−	0.0473002i	$-0.984938\pi$
$887$		−	2.81744i		−	0.0946003i	0.998881	−	0.0473002i	$-0.0150617\pi$
$888$		0			0
$889$			17.6228i			0.591048i
$890$		0			0
$891$		−	7.01559i		−	0.235031i
$892$		0			0
$893$	25.3056			0.846819
$894$		0			0
$895$			3.72693i			0.124577i
$896$		0			0
$897$	−51.9517			−1.73462
$898$		0			0
$899$		−	57.1220i		−	1.90512i
$900$		0			0
$901$	−4.92766			−0.164164
$902$		0			0
$903$	−19.5603			−0.650927
$904$		0			0
$905$			0.775347i			0.0257734i
$906$		0			0
$907$	−6.85517			−0.227622			−0.113811	−	0.993502i	$-0.536306\pi$
$907$	−6.85517			−0.227622			−0.113811	+	0.993502i	$0.536306\pi$
$908$		0			0
$909$			78.5332i			2.60478i
$910$		0			0
$911$	9.88876			0.327629			0.163815	−	0.986491i	$-0.447620\pi$
$911$	9.88876			0.327629			0.163815	+	0.986491i	$0.447620\pi$
$912$		0			0
$913$			4.88151i			0.161554i
$914$		0			0
$915$		−	36.1136i		−	1.19388i
$916$		0			0
$917$		−	23.0273i		−	0.760428i
$918$		0			0
$919$			38.5092i			1.27030i	0.772388	+	0.635151i	$0.219062\pi$
$919$			38.5092i			1.27030i	−0.772388	+	0.635151i	$0.780938\pi$
$920$		0			0
$921$			101.533i			3.34563i
$922$		0			0
$923$	14.8939			0.490239
$924$		0			0
$925$	−3.74525			−0.123143
$926$		0			0
$927$	−31.5120			−1.03499
$928$		0			0
$929$		−	45.1865i		−	1.48252i	−0.671218	−	0.741260i	$-0.734228\pi$
$929$		−	45.1865i		−	1.48252i	0.671218	−	0.741260i	$-0.265772\pi$
$930$		0			0
$931$			10.5426i			0.345519i
$932$		0			0
$933$	96.8563			3.17093
$934$		0			0
$935$	1.75866			0.0575143
$936$		0			0
$937$			7.12233i			0.232676i	0.993210	+	0.116338i	$0.0371156\pi$
$937$			7.12233i			0.232676i	−0.993210	+	0.116338i	$0.962884\pi$
$938$		0			0
$939$	−34.3672			−1.12153
$940$		0			0
$941$	29.0723			0.947731			0.473866	−	0.880597i	$-0.342858\pi$
$941$	29.0723			0.947731			0.473866	+	0.880597i	$0.342858\pi$
$942$		0			0
$943$	43.1594	−	16.3488i	1.40546	−	0.532390i
$944$		0			0
$945$	−22.2413			−0.723511
$946$		0			0
$947$	−19.2078			−0.624168			−0.312084	−	0.950055i	$-0.601027\pi$
$947$	−19.2078			−0.624168			−0.312084	+	0.950055i	$0.601027\pi$
$948$		0			0
$949$		−	27.4406i		−	0.890759i
$950$		0			0
$951$	−59.1594			−1.91838
$952$		0			0
$953$	21.8780			0.708698			0.354349	−	0.935113i	$-0.384702\pi$
$953$	21.8780			0.708698			0.354349	+	0.935113i	$0.384702\pi$
$954$		0			0
$955$			17.6813i			0.572154i
$956$		0			0
$957$			8.79617i			0.284340i
$958$		0			0
$959$	−13.5362			−0.437106
$960$		0			0
$961$	20.9517			0.675860
$962$		0			0
$963$	53.6620			1.72923
$964$		0			0
$965$			2.79426i			0.0899503i
$966$		0			0
$967$			18.8127i			0.604976i	0.953153	+	0.302488i	$0.0978173\pi$
$967$			18.8127i			0.604976i	−0.953153	+	0.302488i	$0.902183\pi$
$968$		0			0
$969$		−	57.1220i		−	1.83502i
$970$		0			0
$971$			28.4458i			0.912870i	0.889757	+	0.456435i	$0.150874\pi$
$971$			28.4458i			0.912870i	−0.889757	+	0.456435i	$0.849126\pi$
$972$		0			0
$973$		−	33.8180i		−	1.08415i
$974$		0			0
$975$	30.3284			0.971287
$976$		0			0
$977$			15.9733i			0.511031i	0.966805	+	0.255516i	$0.0822452\pi$
$977$			15.9733i			0.511031i	−0.966805	+	0.255516i	$0.917755\pi$
$978$		0			0
$979$	5.31900			0.169996
$980$		0			0
$981$		−	56.2509i		−	1.79595i
$982$		0			0
$983$	−18.9422			−0.604164			−0.302082	−	0.953282i	$-0.597682\pi$
$983$	−18.9422			−0.604164			−0.302082	+	0.953282i	$0.597682\pi$
$984$		0			0
$985$	−12.1473			−0.387047
$986$		0			0
$987$		−	48.5599i		−	1.54568i
$988$		0			0
$989$	−23.1207			−0.735195
$990$		0			0
$991$			16.9048i			0.536998i	0.963280	+	0.268499i	$0.0865275\pi$
$991$			16.9048i			0.536998i	−0.963280	+	0.268499i	$0.913472\pi$
$992$		0			0
$993$	9.98925			0.316999
$994$		0			0
$995$			11.1564i			0.353681i
$996$		0			0
$997$		−	26.7914i		−	0.848492i	−0.905547	−	0.424246i	$-0.860539\pi$
$997$		−	26.7914i		−	0.848492i	0.905547	−	0.424246i	$-0.139461\pi$
$998$		0			0
$999$		−	11.5904i		−	0.366705i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2624.2.d.o.2049.6		6
4.3	odd	2		2624.2.d.n.2049.1		6
8.3	odd	2		656.2.d.f.81.6		6
8.5	even	2		328.2.d.b.81.1	✓	6
24.5	odd	2		2952.2.j.c.2377.3		6
41.40	even	2	inner	2624.2.d.o.2049.1		6
164.163	odd	2		2624.2.d.n.2049.6		6
328.163	odd	2		656.2.d.f.81.1		6
328.245	even	2		328.2.d.b.81.6	yes	6
984.245	odd	2		2952.2.j.c.2377.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
328.2.d.b.81.1	✓	6	8.5	even	2
328.2.d.b.81.6	yes	6	328.245	even	2
656.2.d.f.81.1		6	328.163	odd	2
656.2.d.f.81.6		6	8.3	odd	2
2624.2.d.n.2049.1		6	4.3	odd	2
2624.2.d.n.2049.6		6	164.163	odd	2
2624.2.d.o.2049.1		6	41.40	even	2	inner
2624.2.d.o.2049.6		6	1.1	even	1	trivial
2952.2.j.c.2377.3		6	24.5	odd	2
2952.2.j.c.2377.4		6	984.245	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2624 = 2^{6} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2624.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.17771i q^{3} +0.890084 q^{5} -1.91894i q^{7} -7.09783 q^{9} -0.349282i q^{11} +2.26822i q^{13} +2.82843i q^{15} +5.65685i q^{17} +3.17771i q^{19} +6.09783 q^{21} +7.20775 q^{23} -4.20775 q^{25} -13.0217i q^{27} +7.92508i q^{29} -7.20775 q^{31} +1.10992 q^{33} -1.70802i q^{35} +0.890084 q^{37} -7.20775 q^{39} +(5.98792 - 2.26822i) q^{41} -3.20775 q^{43} -6.31767 q^{45} -7.96347i q^{47} +3.31767 q^{49} -17.9758 q^{51} +0.871095i q^{53} -0.310890i q^{55} -10.0978 q^{57} -6.76809 q^{59} -12.7681 q^{61} +13.6203i q^{63} +2.01891i q^{65} +4.18716i q^{67} +22.9041i q^{69} -6.56634i q^{71} -12.0978 q^{73} -13.3710i q^{75} -0.670251 q^{77} -7.96347i q^{79} +20.0858 q^{81} -13.9758 q^{83} +5.03507i q^{85} -25.1836 q^{87} +15.2284i q^{89} +4.35258 q^{91} -22.9041i q^{93} +2.82843i q^{95} +17.2473i q^{97} +2.47915i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{5} - 6 q^{9} + 8 q^{23} + 10 q^{25} - 8 q^{31} + 8 q^{33} + 4 q^{37} - 8 q^{39} - 2 q^{41} + 16 q^{43} - 4 q^{45} - 14 q^{49} - 32 q^{51} - 24 q^{57} - 36 q^{61} - 36 q^{73} + 46 q^{81} - 8 q^{83}+ \cdots + 56 q^{91}+O(q^{100}) \) 6 * q + 4 * q^5 - 6 * q^9 + 8 * q^23 + 10 * q^25 - 8 * q^31 + 8 * q^33 + 4 * q^37 - 8 * q^39 - 2 * q^41 + 16 * q^43 - 4 * q^45 - 14 * q^49 - 32 * q^51 - 24 * q^57 - 36 * q^61 - 36 * q^73 + 46 * q^81 - 8 * q^83 - 40 * q^87 + 56 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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