LMFDB - Embedded newform 2952.2.j.c.2377.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2952,2,Mod(2377,2952)] 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("2952.2377"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2952, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2952 = 2^{3} \cdot 3^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2952.j (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$23.5718386767$
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:	$\Z[a_1, \ldots, a_{11}]$
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			2377.4
Root			$0.629384i$ of defining polynomial
Character	$\chi$	$=$	2952.2377
Dual form			2952.2.j.c.2377.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.890084 q^{5} +1.91894i q^{7} +0.349282i q^{11} +2.26822i q^{13} +5.65685i q^{17} +3.17771i q^{19} -7.20775 q^{23} -4.20775 q^{25} -7.92508i q^{29} -7.20775 q^{31} +1.70802i q^{35} -0.890084 q^{37} +(-5.98792 - 2.26822i) q^{41} +3.20775 q^{43} -7.96347i q^{47} +3.31767 q^{49} -0.871095i q^{53} +0.310890i q^{55} -6.76809 q^{59} +12.7681 q^{61} +2.01891i q^{65} +4.18716i q^{67} -6.56634i q^{71} -12.0978 q^{73} -0.670251 q^{77} +7.96347i q^{79} -13.9758 q^{83} +5.03507i q^{85} +15.2284i q^{89} -4.35258 q^{91} +2.82843i q^{95} -17.2473i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{5} - 8 q^{23} + 10 q^{25} - 8 q^{31} - 4 q^{37} + 2 q^{41} - 16 q^{43} - 14 q^{49} + 36 q^{61} - 36 q^{73} - 8 q^{83} - 56 q^{91}+O(q^{100}) $ 6 * q + 4 * q^5 - 8 * q^23 + 10 * q^25 - 8 * q^31 - 4 * q^37 + 2 * q^41 - 16 * q^43 - 14 * q^49 + 36 * q^61 - 36 * q^73 - 8 * q^83 - 56 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2952\mathbb{Z}\right)^\times$.

$n$	$1441$	$1477$	$2215$	$2297$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		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	$0.118126\pi$
$7$			1.91894i			0.725291i	−0.931927	+	0.362646i	$0.881874\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			0.349282i			0.105312i	0.998613	+	0.0526562i	$0.0167688\pi$
$11$			0.349282i			0.105312i	−0.998613	+	0.0526562i	$0.983231\pi$
$12$		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
$15$		0			0
$16$		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
$21$		0			0
$22$		0			0
$23$	−7.20775			−1.50292			−0.751460	−	0.659779i	$-0.770650\pi$
$23$	−7.20775			−1.50292			−0.751460	+	0.659779i	$0.770650\pi$
$24$		0			0
$25$	−4.20775			−0.841550
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	7.92508i		−	1.47165i	−0.677172	−	0.735825i	$-0.736795\pi$
$29$		−	7.92508i		−	1.47165i	0.677172	−	0.735825i	$-0.263205\pi$
$30$		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
$33$		0			0
$34$		0			0
$35$			1.70802i			0.288708i
$36$		0			0
$37$	−0.890084			−0.146329			−0.0731644	−	0.997320i	$-0.523310\pi$
$37$	−0.890084			−0.146329			−0.0731644	+	0.997320i	$0.523310\pi$
$38$		0			0
$39$		0			0
$40$		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	$-0.421347\pi$
$43$	3.20775			0.489177			0.244589	+	0.969627i	$0.421347\pi$
$44$		0			0
$45$		0			0
$46$		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
$51$		0			0
$52$		0			0
$53$		−	0.871095i		−	0.119654i	−0.998209	−	0.0598270i	$-0.980945\pi$
$53$		−	0.871095i		−	0.119654i	0.998209	−	0.0598270i	$-0.0190549\pi$
$54$		0			0
$55$			0.310890i			0.0419204i
$56$		0			0
$57$		0			0
$58$		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	$-0.195419\pi$
$61$	12.7681			1.63479			0.817393	+	0.576081i	$0.195419\pi$
$62$		0			0
$63$		0			0
$64$		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
$69$		0			0
$70$		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
$75$		0			0
$76$		0			0
$77$	−0.670251			−0.0763822
$78$		0			0
$79$			7.96347i			0.895960i	0.894044	+	0.447980i	$0.147856\pi$
$79$			7.96347i			0.895960i	−0.894044	+	0.447980i	$0.852144\pi$
$80$		0			0
$81$		0			0
$82$		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
$87$		0			0
$88$		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
$93$		0			0
$94$		0			0
$95$			2.82843i			0.290191i
$96$		0			0
$97$		−	17.2473i		−	1.75120i	−0.483040	−	0.875598i	$-0.660468\pi$
$97$		−	17.2473i		−	1.75120i	0.483040	−	0.875598i	$-0.339532\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			11.0644i			1.10095i	0.834852	+	0.550474i	$0.185553\pi$
$101$			11.0644i			1.10095i	−0.834852	+	0.550474i	$0.814447\pi$
$102$		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
$105$		0			0
$106$		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
$111$		0			0
$112$		0			0
$113$	8.53750			0.803140			0.401570	−	0.915828i	$-0.368465\pi$
$113$	8.53750			0.803140			0.401570	+	0.915828i	$0.368465\pi$
$114$		0			0
$115$	−6.41550			−0.598249
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−10.8552			−0.995092
$120$		0			0
$121$	10.8780			0.988909
$122$		0			0
$123$		0			0
$124$		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
$129$		0			0
$130$		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
$135$		0			0
$136$		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	$-0.768694\pi$
$139$	−17.6233			−1.49478			−0.747392	+	0.664383i	$0.768694\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.792249			−0.0662512
$144$		0			0
$145$		−	7.05398i		−	0.585801i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			4.78576i			0.392065i	0.980597	+	0.196032i	$0.0628058\pi$
$149$			4.78576i			0.392065i	−0.980597	+	0.196032i	$0.937194\pi$
$150$		0			0
$151$			15.6392i			1.27270i	0.771399	+	0.636351i	$0.219557\pi$
$151$			15.6392i			1.27270i	−0.771399	+	0.636351i	$0.780443\pi$
$152$		0			0
$153$		0			0
$154$		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
$159$		0			0
$160$		0			0
$161$		−	13.8312i		−	1.09005i
$162$		0			0
$163$	0.439665			0.0344372			0.0172186	−	0.999852i	$-0.494519\pi$
$163$	0.439665			0.0344372			0.0172186	+	0.999852i	$0.494519\pi$
$164$		0			0
$165$		0			0
$166$		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
$171$		0			0
$172$		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
$177$		0			0
$178$		0			0
$179$		−	4.18716i		−	0.312963i	−0.987681	−	0.156482i	$-0.949985\pi$
$179$		−	4.18716i		−	0.312963i	0.987681	−	0.156482i	$-0.0500152\pi$
$180$		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
$183$		0			0
$184$		0			0
$185$	−0.792249			−0.0582473
$186$		0			0
$187$	−1.97584			−0.144487
$188$		0			0
$189$		0			0
$190$		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	$-0.963958\pi$
$193$		−	3.13932i		−	0.225973i	0.993597	−	0.112987i	$-0.0360417\pi$
$194$		0			0
$195$		0			0
$196$		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	$-0.853467\pi$
$199$		−	12.5341i		−	0.888518i	0.895898	−	0.444259i	$-0.146533\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	15.2078			1.06737
$204$		0			0
$205$	−5.32975	−	2.01891i	−0.372246	−	0.141007i
$206$		0			0
$207$		0			0
$208$		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
$213$		0			0
$214$		0			0
$215$	2.85517			0.194721
$216$		0			0
$217$		−	13.8312i		−	0.938926i
$218$		0			0
$219$		0			0
$220$		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
$225$		0			0
$226$		0			0
$227$			19.1046i			1.26802i	0.773325	+	0.634010i	$0.218592\pi$
$227$			19.1046i			1.26802i	−0.773325	+	0.634010i	$0.781408\pi$
$228$		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
$231$		0			0
$232$		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
$237$		0			0
$238$		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
$243$		0			0
$244$		0			0
$245$	2.95300			0.188660
$246$		0			0
$247$	−7.20775			−0.458618
$248$		0			0
$249$		0			0
$250$		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
$255$		0			0
$256$		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
$261$		0			0
$262$		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
$267$		0			0
$268$		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
$273$		0			0
$274$		0			0
$275$		−	1.46969i		−	0.0886257i
$276$		0			0
$277$	9.08575			0.545910			0.272955	−	0.962027i	$-0.411999\pi$
$277$	9.08575			0.545910			0.272955	+	0.962027i	$0.411999\pi$
$278$		0			0
$279$		0			0
$280$		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	$-0.675485\pi$
$283$	−17.6233			−1.04759			−0.523797	+	0.851843i	$0.675485\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.35258	−	11.4905i	0.256925	−	0.678260i
$288$		0			0
$289$	−15.0000			−0.882353
$290$		0			0
$291$		0			0
$292$		0			0
$293$			1.64644i			0.0961862i	0.998843	+	0.0480931i	$0.0153144\pi$
$293$			1.64644i			0.0961862i	−0.998843	+	0.0480931i	$0.984686\pi$
$294$		0			0
$295$	−6.02416			−0.350740
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	16.3488i		−	0.945475i
$300$		0			0
$301$			6.15548i			0.354796i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	11.3647			0.650739
$306$		0			0
$307$	−31.9517			−1.82358			−0.911789	−	0.410659i	$-0.865299\pi$
$307$	−31.9517			−1.82358			−0.911789	+	0.410659i	$0.865299\pi$
$308$		0			0
$309$		0			0
$310$		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	$-0.901126\pi$
$313$		−	10.8151i		−	0.611304i	0.952143	−	0.305652i	$-0.0988745\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	18.6170i		−	1.04564i	−0.852445	−	0.522818i	$-0.824881\pi$
$317$		−	18.6170i		−	1.04564i	0.852445	−	0.522818i	$-0.175119\pi$
$318$		0			0
$319$	2.76809			0.154983
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−17.9758			−1.00020
$324$		0			0
$325$		−	9.54412i		−	0.529412i
$326$		0			0
$327$		0			0
$328$		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
$333$		0			0
$334$		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
$339$		0			0
$340$		0			0
$341$		−	2.51754i		−	0.136332i
$342$		0			0
$343$			19.7990i			1.06904i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			6.73887i			0.361762i	0.983505	+	0.180881i	$0.0578948\pi$
$347$			6.73887i			0.361762i	−0.983505	+	0.180881i	$0.942105\pi$
$348$		0			0
$349$	28.6461			1.53339			0.766695	−	0.642012i	$-0.221900\pi$
$349$	28.6461			1.53339			0.766695	+	0.642012i	$0.221900\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−34.3177			−1.82655			−0.913273	−	0.407349i	$-0.866453\pi$
$353$	−34.3177			−1.82655			−0.913273	+	0.407349i	$0.866453\pi$
$354$		0			0
$355$		−	5.84459i		−	0.310199i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.97584			−0.104281			−0.0521403	−	0.998640i	$-0.516604\pi$
$359$	−1.97584			−0.104281			−0.0521403	+	0.998640i	$0.516604\pi$
$360$		0			0
$361$	8.90217			0.468535
$362$		0			0
$363$		0			0
$364$		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
$369$		0			0
$370$		0			0
$371$	1.67158			0.0867841
$372$		0			0
$373$	6.00000			0.310668			0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000			0.310668			0.155334	+	0.987862i	$0.450355\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	17.9758			0.925803
$378$		0			0
$379$	1.23191			0.0632792			0.0316396	−	0.999499i	$-0.489927\pi$
$379$	1.23191			0.0632792			0.0316396	+	0.999499i	$0.489927\pi$
$380$		0			0
$381$		0			0
$382$		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
$387$		0			0
$388$		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
$393$		0			0
$394$		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
$399$		0			0
$400$		0			0
$401$	−4.29350			−0.214407			−0.107204	−	0.994237i	$-0.534190\pi$
$401$	−4.29350			−0.214407			−0.107204	+	0.994237i	$0.534190\pi$
$402$		0			0
$403$		−	16.3488i		−	0.814391i
$404$		0			0
$405$		0			0
$406$		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
$411$		0			0
$412$		0			0
$413$		−	12.9876i		−	0.639076i
$414$		0			0
$415$	−12.4397			−0.610639
$416$		0			0
$417$		0			0
$418$		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
$423$		0			0
$424$		0			0
$425$		−	23.8026i		−	1.15460i
$426$		0			0
$427$			24.5012i			1.18570i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5.23191			−0.252012			−0.126006	−	0.992029i	$-0.540216\pi$
$431$	−5.23191			−0.252012			−0.126006	+	0.992029i	$0.540216\pi$
$432$		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
$435$		0			0
$436$		0			0
$437$		−	22.9041i		−	1.09565i
$438$		0			0
$439$		−	18.4677i		−	0.881413i	−0.897651	−	0.440707i	$-0.854728\pi$
$439$		−	18.4677i		−	0.881413i	0.897651	−	0.440707i	$-0.145272\pi$
$440$		0			0
$441$		0			0
$442$		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
$447$		0			0
$448$		0			0
$449$	8.32842			0.393042			0.196521	−	0.980500i	$-0.437036\pi$
$449$	8.32842			0.393042			0.196521	+	0.980500i	$0.437036\pi$
$450$		0			0
$451$	0.792249	−	2.09147i	0.0373056	−	0.0984835i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.87416			−0.181624
$456$		0			0
$457$			12.0891i			0.565502i	0.959193	+	0.282751i	$0.0912470\pi$
$457$			12.0891i			0.565502i	−0.959193	+	0.282751i	$0.908753\pi$
$458$		0			0
$459$		0			0
$460$		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	$0.100351\pi$
$463$			13.3436i			0.620130i	−0.950715	+	0.310065i	$0.899649\pi$
$464$		0			0
$465$		0			0
$466$		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
$471$		0			0
$472$		0			0
$473$			1.12041i			0.0515165i
$474$		0			0
$475$		−	13.3710i		−	0.613504i
$476$		0			0
$477$		0			0
$478$		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
$483$		0			0
$484$		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
$489$		0			0
$490$		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
$495$		0			0
$496$		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
$501$		0			0
$502$		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
$507$		0			0
$508$		0			0
$509$			2.76685i			0.122639i	0.998118	+	0.0613193i	$0.0195308\pi$
$509$			2.76685i			0.122639i	−0.998118	+	0.0613193i	$0.980469\pi$
$510$		0			0
$511$		−	23.2150i		−	1.02697i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.95167			0.174132
$516$		0			0
$517$	2.78150			0.122330
$518$		0			0
$519$		0			0
$520$		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	$-0.546662\pi$
$523$	−6.68100			−0.292140			−0.146070	+	0.989274i	$0.546662\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	40.7732i		−	1.77611i
$528$		0			0
$529$	28.9517			1.25877
$530$		0			0
$531$		0			0
$532$		0			0
$533$	5.14483	−	13.5819i	0.222847	−	0.588299i
$534$		0			0
$535$	−6.72933			−0.290934
$536$		0			0
$537$		0			0
$538$		0			0
$539$			1.15880i			0.0499131i
$540$		0			0
$541$	22.6219			0.972593			0.486296	−	0.873794i	$-0.338348\pi$
$541$	22.6219			0.972593			0.486296	+	0.873794i	$0.338348\pi$
$542$		0			0
$543$		0			0
$544$		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
$549$		0			0
$550$		0			0
$551$	25.1836			1.07286
$552$		0			0
$553$	−15.2814			−0.649832
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	22.6548i		−	0.959916i	−0.877292	−	0.479958i	$-0.840652\pi$
$557$		−	22.6548i		−	0.959916i	0.877292	−	0.479958i	$-0.159348\pi$
$558$		0			0
$559$			7.27589i			0.307737i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			14.9559i			0.630315i	0.949039	+	0.315157i	$0.102057\pi$
$563$			14.9559i			0.630315i	−0.949039	+	0.315157i	$0.897943\pi$
$564$		0			0
$565$	7.59909			0.319696
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.0978			0.507168			0.253584	−	0.967313i	$-0.418391\pi$
$569$	12.0978			0.507168			0.253584	+	0.967313i	$0.418391\pi$
$570$		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
$573$		0			0
$574$		0			0
$575$	30.3284			1.26478
$576$		0			0
$577$		−	2.51754i		−	0.104806i	−0.998626	−	0.0524032i	$-0.983312\pi$
$577$		−	2.51754i		−	0.104806i	0.998626	−	0.0524032i	$-0.0166881\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	26.8188i		−	1.11263i
$582$		0			0
$583$	0.304258			0.0126011
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	11.8629i		−	0.489635i	−0.969569	−	0.244818i	$-0.921272\pi$
$587$		−	11.8629i		−	0.489635i	0.969569	−	0.244818i	$-0.0787281\pi$
$588$		0			0
$589$		−	22.9041i		−	0.943748i
$590$		0			0
$591$		0			0
$592$		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
$597$		0			0
$598$		0			0
$599$	38.7198			1.58205			0.791023	−	0.611786i	$-0.209549\pi$
$599$	38.7198			1.58205			0.791023	+	0.611786i	$0.209549\pi$
$600$		0			0
$601$			37.8558i			1.54417i	0.635519	+	0.772085i	$0.280786\pi$
$601$			37.8558i			1.54417i	−0.635519	+	0.772085i	$0.719214\pi$
$602$		0			0
$603$		0			0
$604$		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
$609$		0			0
$610$		0			0
$611$	18.0629			0.730747
$612$		0			0
$613$	−13.7211			−0.554190			−0.277095	−	0.960843i	$-0.589372\pi$
$613$	−13.7211			−0.554190			−0.277095	+	0.960843i	$0.589372\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.12067			−0.0451165			−0.0225582	−	0.999746i	$-0.507181\pi$
$617$	−1.12067			−0.0451165			−0.0225582	+	0.999746i	$0.507181\pi$
$618$		0			0
$619$	23.6474			0.950470			0.475235	−	0.879859i	$-0.342363\pi$
$619$	23.6474			0.950470			0.475235	+	0.879859i	$0.342363\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−29.2223			−1.17077
$624$		0			0
$625$	13.7439			0.549757
$626$		0			0
$627$		0			0
$628$		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
$633$		0			0
$634$		0			0
$635$	−8.17416			−0.324382
$636$		0			0
$637$			7.52521i			0.298160i
$638$		0			0
$639$		0			0
$640$		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
$645$		0			0
$646$		0			0
$647$	12.8310			0.504439			0.252219	−	0.967670i	$-0.418840\pi$
$647$	12.8310			0.504439			0.252219	+	0.967670i	$0.418840\pi$
$648$		0			0
$649$		−	2.36397i		−	0.0927939i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			35.6424i			1.39479i	0.716685	+	0.697397i	$0.245659\pi$
$653$			35.6424i			1.39479i	−0.716685	+	0.697397i	$0.754341\pi$
$654$		0			0
$655$	10.6810			0.417341
$656$		0			0
$657$		0			0
$658$		0			0
$659$			19.4497i			0.757653i	0.925468	+	0.378827i	$0.123672\pi$
$659$			19.4497i			0.757653i	−0.925468	+	0.378827i	$0.876328\pi$
$660$		0			0
$661$	−29.7211			−1.15602			−0.578008	−	0.816031i	$-0.696170\pi$
$661$	−29.7211			−1.15602			−0.578008	+	0.816031i	$0.696170\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5.42758			−0.210473
$666$		0			0
$667$			57.1220i			2.21177i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			4.45966i			0.172163i
$672$		0			0
$673$		−	35.1163i		−	1.35364i	−0.736151	−	0.676818i	$-0.763359\pi$
$673$		−	35.1163i		−	1.35364i	0.736151	−	0.676818i	$-0.236641\pi$
$674$		0			0
$675$		0			0
$676$		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
$681$		0			0
$682$		0			0
$683$		−	12.7492i		−	0.487836i	−0.969796	−	0.243918i	$-0.921567\pi$
$683$		−	12.7492i		−	0.487836i	0.969796	−	0.243918i	$-0.0784327\pi$
$684$		0			0
$685$		−	6.27863i		−	0.239894i
$686$		0			0
$687$		0			0
$688$		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
$693$		0			0
$694$		0			0
$695$	−15.6862			−0.595010
$696$		0			0
$697$	12.8310	−	33.8728i	0.486009	−	1.28302i
$698$		0			0
$699$		0			0
$700$		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
$705$		0			0
$706$		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
$711$		0			0
$712$		0			0
$713$	51.9517			1.94561
$714$		0			0
$715$	−0.705168			−0.0263718
$716$		0			0
$717$		0			0
$718$		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
$723$		0			0
$724$		0			0
$725$			33.3467i			1.23847i
$726$		0			0
$727$		−	13.4752i		−	0.499767i	−0.968276	−	0.249884i	$-0.919608\pi$
$727$		−	13.4752i		−	0.499767i	0.968276	−	0.249884i	$-0.0803924\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			18.1458i			0.671146i
$732$		0			0
$733$	−26.3526			−0.973355			−0.486677	−	0.873582i	$-0.661791\pi$
$733$	−26.3526			−0.973355			−0.486677	+	0.873582i	$0.661791\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.46250			−0.0538719
$738$		0			0
$739$	−10.4155			−0.383140			−0.191570	−	0.981479i	$-0.561358\pi$
$739$	−10.4155			−0.383140			−0.191570	+	0.981479i	$0.561358\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	30.4155			1.11584			0.557918	−	0.829896i	$-0.311600\pi$
$743$	30.4155			1.11584			0.557918	+	0.829896i	$0.311600\pi$
$744$		0			0
$745$			4.25973i			0.156064i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	14.5078i		−	0.530105i
$750$		0			0
$751$			52.6087i			1.91972i	0.280481	+	0.959860i	$0.409506\pi$
$751$			52.6087i			1.91972i	−0.280481	+	0.959860i	$0.590494\pi$
$752$		0			0
$753$		0			0
$754$		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
$759$		0			0
$760$		0			0
$761$	−26.7090			−0.968201			−0.484100	−	0.875012i	$-0.660853\pi$
$761$	−26.7090			−0.968201			−0.484100	+	0.875012i	$0.660853\pi$
$762$		0			0
$763$	−15.2078			−0.550558
$764$		0			0
$765$		0			0
$766$		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
$771$		0			0
$772$		0			0
$773$		−	13.7355i		−	0.494032i	−0.969011	−	0.247016i	$-0.920550\pi$
$773$		−	13.7355i		−	0.494032i	0.969011	−	0.247016i	$-0.0794499\pi$
$774$		0			0
$775$	30.3284			1.08943
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7.20775	−	19.0279i	0.258244	−	0.681744i
$780$		0			0
$781$	2.29350			0.0820680
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	18.3677i		−	0.655571i
$786$		0			0
$787$	−13.5845			−0.484235			−0.242118	−	0.970247i	$-0.577842\pi$
$787$	−13.5845			−0.484235			−0.242118	+	0.970247i	$0.577842\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			16.3830i			0.582511i
$792$		0			0
$793$			28.9609i			1.02843i
$794$		0			0
$795$		0			0
$796$		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
$801$		0			0
$802$		0			0
$803$		−	4.22555i		−	0.149117i
$804$		0			0
$805$		−	12.3110i		−	0.433905i
$806$		0			0
$807$		0			0
$808$		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	$-0.732318\pi$
$811$	−37.9758			−1.33351			−0.666756	+	0.745276i	$0.732318\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.391339			0.0137080
$816$		0			0
$817$			10.1933i			0.356618i
$818$		0			0
$819$		0			0
$820$		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	$-0.677754\pi$
$823$		−	48.6599i		−	1.69618i	0.529855	−	0.848088i	$-0.322246\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	22.2781i		−	0.774687i	−0.921936	−	0.387343i	$-0.873393\pi$
$827$		−	22.2781i		−	0.774687i	0.921936	−	0.387343i	$-0.126607\pi$
$828$		0			0
$829$	−46.1124			−1.60155			−0.800775	−	0.598965i	$-0.795579\pi$
$829$	−46.1124			−1.60155			−0.800775	+	0.598965i	$0.795579\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			18.7676i			0.650257i
$834$		0			0
$835$			8.36213i			0.289383i
$836$		0			0
$837$		0			0
$838$		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
$843$		0			0
$844$		0			0
$845$	6.99176			0.240524
$846$		0			0
$847$			20.8742i			0.717247i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.41550			0.219921
$852$		0			0
$853$	−6.83100			−0.233889			−0.116945	−	0.993138i	$-0.537310\pi$
$853$	−6.83100			−0.233889			−0.116945	+	0.993138i	$0.537310\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	11.2185			0.383217			0.191608	−	0.981471i	$-0.438630\pi$
$857$	11.2185			0.383217			0.191608	+	0.981471i	$0.438630\pi$
$858$		0			0
$859$	−14.8552			−0.506852			−0.253426	−	0.967355i	$-0.581557\pi$
$859$	−14.8552			−0.506852			−0.253426	+	0.967355i	$0.581557\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	27.6474			0.941129			0.470564	−	0.882366i	$-0.344050\pi$
$863$	27.6474			0.941129			0.470564	+	0.882366i	$0.344050\pi$
$864$		0			0
$865$	1.78017			0.0605275
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.78150			−0.0943558
$870$		0			0
$871$	−9.49742			−0.321808
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	15.7270i		−	0.531670i
$876$		0			0
$877$	9.59909			0.324138			0.162069	−	0.986779i	$-0.448183\pi$
$877$	9.59909			0.324138			0.162069	+	0.986779i	$0.448183\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	7.09651			0.239087			0.119544	−	0.992829i	$-0.461857\pi$
$881$	7.09651			0.239087			0.119544	+	0.992829i	$0.461857\pi$
$882$		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
$885$		0			0
$886$		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
$891$		0			0
$892$		0			0
$893$	25.3056			0.846819
$894$		0			0
$895$		−	3.72693i		−	0.124577i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			57.1220i			1.90512i
$900$		0			0
$901$	4.92766			0.164164
$902$		0			0
$903$		0			0
$904$		0			0
$905$			0.775347i			0.0257734i
$906$		0			0
$907$	6.85517			0.227622			0.113811	−	0.993502i	$-0.463694\pi$
$907$	6.85517			0.227622			0.113811	+	0.993502i	$0.463694\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−9.88876			−0.327629			−0.163815	−	0.986491i	$-0.552380\pi$
$911$	−9.88876			−0.327629			−0.163815	+	0.986491i	$0.552380\pi$
$912$		0			0
$913$		−	4.88151i		−	0.161554i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			23.0273i			0.760428i
$918$		0			0
$919$		−	38.5092i		−	1.27030i	−0.772388	−	0.635151i	$-0.780938\pi$
$919$		−	38.5092i		−	1.27030i	0.772388	−	0.635151i	$-0.219062\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	14.8939			0.490239
$924$		0			0
$925$	3.74525			0.123143
$926$		0			0
$927$		0			0
$928$		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
$933$		0			0
$934$		0			0
$935$	−1.75866			−0.0575143
$936$		0			0
$937$		−	7.12233i		−	0.232676i	−0.993210	−	0.116338i	$-0.962884\pi$
$937$		−	7.12233i		−	0.232676i	0.993210	−	0.116338i	$-0.0371156\pi$
$938$		0			0
$939$		0			0
$940$		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
$945$		0			0
$946$		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
$951$		0			0
$952$		0			0
$953$	−21.8780			−0.708698			−0.354349	−	0.935113i	$-0.615298\pi$
$953$	−21.8780			−0.708698			−0.354349	+	0.935113i	$0.615298\pi$
$954$		0			0
$955$			17.6813i			0.572154i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	13.5362			0.437106
$960$		0			0
$961$	20.9517			0.675860
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	2.79426i		−	0.0899503i
$966$		0			0
$967$		−	18.8127i		−	0.604976i	−0.953153	−	0.302488i	$-0.902183\pi$
$967$		−	18.8127i		−	0.604976i	0.953153	−	0.302488i	$-0.0978173\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	28.4458i		−	0.912870i	−0.889757	−	0.456435i	$-0.849126\pi$
$971$		−	28.4458i		−	0.912870i	0.889757	−	0.456435i	$-0.150874\pi$
$972$		0			0
$973$		−	33.8180i		−	1.08415i
$974$		0			0
$975$		0			0
$976$		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
$981$		0			0
$982$		0			0
$983$	18.9422			0.604164			0.302082	−	0.953282i	$-0.402318\pi$
$983$	18.9422			0.604164			0.302082	+	0.953282i	$0.402318\pi$
$984$		0			0
$985$	−12.1473			−0.387047
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−23.1207			−0.735195
$990$		0			0
$991$		−	16.9048i		−	0.536998i	−0.963280	−	0.268499i	$-0.913472\pi$
$991$		−	16.9048i		−	0.536998i	0.963280	−	0.268499i	$-0.0865275\pi$
$992$		0			0
$993$		0			0
$994$		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
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2952.2.j.c.2377.4		6
3.2	odd	2		328.2.d.b.81.6	yes	6
12.11	even	2		656.2.d.f.81.1		6
24.5	odd	2		2624.2.d.o.2049.1		6
24.11	even	2		2624.2.d.n.2049.6		6
41.40	even	2	inner	2952.2.j.c.2377.3		6
123.122	odd	2		328.2.d.b.81.1	✓	6
492.491	even	2		656.2.d.f.81.6		6
984.245	odd	2		2624.2.d.o.2049.6		6
984.491	even	2		2624.2.d.n.2049.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
328.2.d.b.81.1	✓	6	123.122	odd	2
328.2.d.b.81.6	yes	6	3.2	odd	2
656.2.d.f.81.1		6	12.11	even	2
656.2.d.f.81.6		6	492.491	even	2
2624.2.d.n.2049.1		6	984.491	even	2
2624.2.d.n.2049.6		6	24.11	even	2
2624.2.d.o.2049.1		6	24.5	odd	2
2624.2.d.o.2049.6		6	984.245	odd	2
2952.2.j.c.2377.3		6	41.40	even	2	inner
2952.2.j.c.2377.4		6	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:	\( N \)	\(=\)	\( 2952 = 2^{3} \cdot 3^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2952.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.890084 q^{5} +1.91894i q^{7} +0.349282i q^{11} +2.26822i q^{13} +5.65685i q^{17} +3.17771i q^{19} -7.20775 q^{23} -4.20775 q^{25} -7.92508i q^{29} -7.20775 q^{31} +1.70802i q^{35} -0.890084 q^{37} +(-5.98792 - 2.26822i) q^{41} +3.20775 q^{43} -7.96347i q^{47} +3.31767 q^{49} -0.871095i q^{53} +0.310890i q^{55} -6.76809 q^{59} +12.7681 q^{61} +2.01891i q^{65} +4.18716i q^{67} -6.56634i q^{71} -12.0978 q^{73} -0.670251 q^{77} +7.96347i q^{79} -13.9758 q^{83} +5.03507i q^{85} +15.2284i q^{89} -4.35258 q^{91} +2.82843i q^{95} -17.2473i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{5} - 8 q^{23} + 10 q^{25} - 8 q^{31} - 4 q^{37} + 2 q^{41} - 16 q^{43} - 14 q^{49} + 36 q^{61} - 36 q^{73} - 8 q^{83} - 56 q^{91}+O(q^{100}) \) 6 * q + 4 * q^5 - 8 * q^23 + 10 * q^25 - 8 * q^31 - 4 * q^37 + 2 * q^41 - 16 * q^43 - 14 * q^49 + 36 * q^61 - 36 * q^73 - 8 * q^83 - 56 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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