LMFDB - Embedded newform 2952.2.j.d.2377.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2952,2,Mod(2377,2952)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	no
Analytic conductor:	$23.5718386767$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.265727878144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4 $ x^8 + 15x^6 + 67x^4 + 77*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 984)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2377.3
Root			$-1.29051i$ of defining polynomial
Character	$\chi$	$=$	2952.2377
Dual form			2952.2.j.d.2377.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.29051 q^{5} -1.04406i q^{7} +O(q^{10})$	$q-2.29051 q^{5} -1.04406i q^{7} -5.59383i q^{11} -2.59383i q^{13} -2.25926i q^{17} +5.17486i q^{19} -3.92840 q^{23} +0.246456 q^{25} +0.987200i q^{29} +10.5535 q^{31} +2.39143i q^{35} -1.75354 q^{37} +(-5.84028 - 2.62509i) q^{41} -9.15640 q^{43} -3.22800i q^{47} +5.90994 q^{49} +6.88434i q^{53} +12.8127i q^{55} -1.61229 q^{59} -1.88062 q^{61} +5.94120i q^{65} +14.8881i q^{67} +4.92840i q^{71} -2.76869 q^{73} -5.84028 q^{77} -3.50709i q^{79} +4.89714 q^{83} +5.17486i q^{85} -2.45023i q^{89} -2.70811 q^{91} -11.8531i q^{95} +0.374913i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 10 q^{5}+O(q^{10}) $ 8 * q - 10 * q^5	$ 8 q - 10 q^{5} + 20 q^{23} + 2 q^{25} + 8 q^{31} - 14 q^{37} - 12 q^{41} + 6 q^{43} - 32 q^{49} - 6 q^{59} - 22 q^{61} + 64 q^{73} - 12 q^{77} - 22 q^{83} - 12 q^{91}+O(q^{100}) $ 8 * q - 10 * q^5 + 20 * q^23 + 2 * q^25 + 8 * q^31 - 14 * q^37 - 12 * q^41 + 6 * q^43 - 32 * q^49 - 6 * q^59 - 22 * q^61 + 64 * q^73 - 12 * q^77 - 22 * q^83 - 12 * q^91

Display 100 coefficients 10 coefficients

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
$5$	−2.29051			−1.02435			−0.512175	−	0.858881i	$-0.671160\pi$
$5$	−2.29051			−1.02435			−0.512175	+	0.858881i	$0.671160\pi$
$6$		0			0
$7$		−	1.04406i		−	0.394617i	−0.980341	−	0.197309i	$-0.936780\pi$
$7$		−	1.04406i		−	0.394617i	0.980341	−	0.197309i	$-0.0632201\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.59383i		−	1.68660i	−0.537441	−	0.843301i	$-0.680609\pi$
$11$		−	5.59383i		−	1.68660i	0.537441	−	0.843301i	$-0.319391\pi$
$12$		0			0
$13$		−	2.59383i		−	0.719399i	−0.933068	−	0.359699i	$-0.882879\pi$
$13$		−	2.59383i		−	0.719399i	0.933068	−	0.359699i	$-0.117121\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.25926i		−	0.547950i	−0.961737	−	0.273975i	$-0.911661\pi$
$17$		−	2.25926i		−	0.547950i	0.961737	−	0.273975i	$-0.0883385\pi$
$18$		0			0
$19$			5.17486i			1.18719i	0.804763	+	0.593597i	$0.202293\pi$
$19$			5.17486i			1.18719i	−0.804763	+	0.593597i	$0.797707\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.92840			−0.819128			−0.409564	−	0.912281i	$-0.634319\pi$
$23$	−3.92840			−0.819128			−0.409564	+	0.912281i	$0.634319\pi$
$24$		0			0
$25$	0.246456			0.0492912
$26$		0			0
$27$		0			0
$28$		0			0
$29$			0.987200i			0.183319i	0.995790	+	0.0916593i	$0.0292171\pi$
$29$			0.987200i			0.183319i	−0.995790	+	0.0916593i	$0.970783\pi$
$30$		0			0
$31$	10.5535			1.89546			0.947731	−	0.319070i	$-0.103370\pi$
$31$	10.5535			1.89546			0.947731	+	0.319070i	$0.103370\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			2.39143i			0.404226i
$36$		0			0
$37$	−1.75354			−0.288281			−0.144140	−	0.989557i	$-0.546042\pi$
$37$	−1.75354			−0.288281			−0.144140	+	0.989557i	$0.546042\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.84028	−	2.62509i	−0.912099	−	0.409970i
$41$	−5.84028	−	2.62509i	−0.912099	−	0.409970i
$42$		0			0
$43$	−9.15640			−1.39634			−0.698169	−	0.715933i	$-0.746001\pi$
$43$	−9.15640			−1.39634			−0.698169	+	0.715933i	$0.746001\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	3.22800i		−	0.470852i	−0.971892	−	0.235426i	$-0.924352\pi$
$47$		−	3.22800i		−	0.470852i	0.971892	−	0.235426i	$-0.0756485\pi$
$48$		0			0
$49$	5.90994			0.844277
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.88434i			0.945637i	0.881160	+	0.472819i	$0.156763\pi$
$53$			6.88434i			0.945637i	−0.881160	+	0.472819i	$0.843237\pi$
$54$		0			0
$55$			12.8127i			1.72767i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.61229			−0.209902			−0.104951	−	0.994477i	$-0.533469\pi$
$59$	−1.61229			−0.209902			−0.104951	+	0.994477i	$0.533469\pi$
$60$		0			0
$61$	−1.88062			−0.240789			−0.120395	−	0.992726i	$-0.538416\pi$
$61$	−1.88062			−0.240789			−0.120395	+	0.992726i	$0.538416\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			5.94120i			0.736915i
$66$		0			0
$67$			14.8881i			1.81887i	0.415850	+	0.909433i	$0.363484\pi$
$67$			14.8881i			1.81887i	−0.415850	+	0.909433i	$0.636516\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			4.92840i			0.584894i	0.956282	+	0.292447i	$0.0944694\pi$
$71$			4.92840i			0.584894i	−0.956282	+	0.292447i	$0.905531\pi$
$72$		0			0
$73$	−2.76869			−0.324050			−0.162025	−	0.986787i	$-0.551803\pi$
$73$	−2.76869			−0.324050			−0.162025	+	0.986787i	$0.551803\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.84028			−0.665562
$78$		0			0
$79$		−	3.50709i		−	0.394578i	−0.980345	−	0.197289i	$-0.936786\pi$
$79$		−	3.50709i		−	0.394578i	0.980345	−	0.197289i	$-0.0632138\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.89714			0.537531			0.268766	−	0.963206i	$-0.413384\pi$
$83$	4.89714			0.537531			0.268766	+	0.963206i	$0.413384\pi$
$84$		0			0
$85$			5.17486i			0.561292i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	2.45023i		−	0.259724i	−0.991532	−	0.129862i	$-0.958547\pi$
$89$		−	2.45023i		−	0.259724i	0.991532	−	0.129862i	$-0.0414534\pi$
$90$		0			0
$91$	−2.70811			−0.283887
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	11.8531i		−	1.21610i
$96$		0			0
$97$			0.374913i			0.0380666i	0.999819	+	0.0190333i	$0.00605886\pi$
$97$			0.374913i			0.0380666i	−0.999819	+	0.0190333i	$0.993941\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	1.20240i		−	0.119643i	−0.998209	−	0.0598215i	$-0.980947\pi$
$101$		−	1.20240i		−	0.119643i	0.998209	−	0.0598215i	$-0.0190532\pi$
$102$		0			0
$103$	−18.9449			−1.86670			−0.933349	−	0.358970i	$-0.883128\pi$
$103$	−18.9449			−1.86670			−0.933349	+	0.358970i	$0.883128\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.8995			−1.15037			−0.575183	−	0.818025i	$-0.695069\pi$
$107$	−11.8995			−1.15037			−0.575183	+	0.818025i	$0.695069\pi$
$108$		0			0
$109$		−	6.20612i		−	0.594438i	−0.954809	−	0.297219i	$-0.903941\pi$
$109$		−	6.20612i		−	0.594438i	0.954809	−	0.297219i	$-0.0960592\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−11.8659			−1.11625			−0.558124	−	0.829757i	$-0.688479\pi$
$113$	−11.8659			−1.11625			−0.558124	+	0.829757i	$0.688479\pi$
$114$		0			0
$115$	8.99806			0.839073
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.35879			−0.216230
$120$		0			0
$121$	−20.2909			−1.84463
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.8881			0.973858
$126$		0			0
$127$	6.00372			0.532744			0.266372	−	0.963870i	$-0.414175\pi$
$127$	6.00372			0.532744			0.266372	+	0.963870i	$0.414175\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−18.9398			−1.65478			−0.827390	−	0.561628i	$-0.810175\pi$
$131$	−18.9398			−1.65478			−0.827390	+	0.561628i	$0.810175\pi$
$132$		0			0
$133$	5.40285			0.468487
$134$		0			0
$135$		0			0
$136$		0			0
$137$			13.8554i			1.18375i	0.806030	+	0.591874i	$0.201612\pi$
$137$			13.8554i			1.18375i	−0.806030	+	0.591874i	$0.798388\pi$
$138$		0			0
$139$	−10.8843			−0.923198			−0.461599	−	0.887089i	$-0.652724\pi$
$139$	−10.8843			−0.923198			−0.461599	+	0.887089i	$0.652724\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−14.5094			−1.21334
$144$		0			0
$145$		−	2.26120i		−	0.187782i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			8.66915i			0.710204i	0.934828	+	0.355102i	$0.115554\pi$
$149$			8.66915i			0.710204i	−0.934828	+	0.355102i	$0.884446\pi$
$150$		0			0
$151$		−	20.9307i		−	1.70332i	−0.524095	−	0.851660i	$-0.675596\pi$
$151$		−	20.9307i		−	1.70332i	0.524095	−	0.851660i	$-0.324404\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−24.1729			−1.94162
$156$		0			0
$157$			1.39337i			0.111203i	0.998453	+	0.0556016i	$0.0177077\pi$
$157$			1.39337i			0.111203i	−0.998453	+	0.0556016i	$0.982292\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			4.10148i			0.323242i
$162$		0			0
$163$	−21.4805			−1.68248			−0.841242	−	0.540659i	$-0.818175\pi$
$163$	−21.4805			−1.68248			−0.841242	+	0.540659i	$0.818175\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			15.1033i			1.16873i	0.811493	+	0.584363i	$0.198655\pi$
$167$			15.1033i			1.16873i	−0.811493	+	0.584363i	$0.801345\pi$
$168$		0			0
$169$	6.27206			0.482466
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.86588			0.445975			0.222987	−	0.974821i	$-0.428419\pi$
$173$	5.86588			0.445975			0.222987	+	0.974821i	$0.428419\pi$
$174$		0			0
$175$		−	0.257314i		−	0.0194511i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			19.7027i			1.47265i	0.676627	+	0.736326i	$0.263441\pi$
$179$			19.7027i			1.47265i	−0.676627	+	0.736326i	$0.736559\pi$
$180$		0			0
$181$		−	9.57537i		−	0.711732i	−0.934537	−	0.355866i	$-0.884186\pi$
$181$		−	9.57537i		−	0.711732i	0.934537	−	0.355866i	$-0.115814\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.01652			0.295300
$186$		0			0
$187$	−12.6379			−0.924174
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.6066i		−	0.912184i	−0.889933	−	0.456092i	$-0.849249\pi$
$191$		−	12.6066i		−	0.912184i	0.889933	−	0.456092i	$-0.150751\pi$
$192$		0			0
$193$		−	24.9862i		−	1.79855i	−0.437386	−	0.899274i	$-0.644096\pi$
$193$		−	24.9862i		−	1.79855i	0.437386	−	0.899274i	$-0.355904\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	7.79760			0.555556			0.277778	−	0.960645i	$-0.410402\pi$
$197$	7.79760			0.555556			0.277778	+	0.960645i	$0.410402\pi$
$198$		0			0
$199$			12.9009i			0.914518i	0.889334	+	0.457259i	$0.151169\pi$
$199$			12.9009i			0.914518i	−0.889334	+	0.457259i	$0.848831\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.03069			0.0723406
$204$		0			0
$205$	13.3773	+	6.01280i	0.934308	+	0.419952i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	28.9473			2.00232
$210$		0			0
$211$		−	2.81811i		−	0.194006i	−0.995284	−	0.0970032i	$-0.969074\pi$
$211$		−	2.81811i		−	0.194006i	0.995284	−	0.0970032i	$-0.0309257\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	20.9729			1.43034
$216$		0			0
$217$		−	11.0185i		−	0.747982i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.86012			−0.394194
$222$		0			0
$223$	−4.88806			−0.327329			−0.163664	−	0.986516i	$-0.552331\pi$
$223$	−4.88806			−0.327329			−0.163664	+	0.986516i	$0.552331\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			12.9321i			0.858335i	0.903225	+	0.429167i	$0.141193\pi$
$227$			12.9321i			0.858335i	−0.903225	+	0.429167i	$0.858807\pi$
$228$		0			0
$229$			8.48863i			0.560944i	0.959862	+	0.280472i	$0.0904911\pi$
$229$			8.48863i			0.560944i	−0.959862	+	0.280472i	$0.909509\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			20.1933i			1.32291i	0.749986	+	0.661454i	$0.230060\pi$
$233$			20.1933i			1.32291i	−0.749986	+	0.661454i	$0.769940\pi$
$234$		0			0
$235$			7.39377i			0.482317i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	12.5686i		−	0.812997i	−0.913651	−	0.406499i	$-0.866750\pi$
$239$		−	12.5686i		−	0.812997i	0.913651	−	0.406499i	$-0.133250\pi$
$240$		0			0
$241$	−0.291892			−0.0188024			−0.00940119	−	0.999956i	$-0.502993\pi$
$241$	−0.291892			−0.0188024			−0.00940119	+	0.999956i	$0.502993\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−13.5368			−0.864835
$246$		0			0
$247$	13.4227			0.854065
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.02560			−0.127855			−0.0639273	−	0.997955i	$-0.520363\pi$
$251$	−2.02560			−0.127855			−0.0639273	+	0.997955i	$0.520363\pi$
$252$		0			0
$253$			21.9748i			1.38154i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			15.1729i			0.946460i	0.880939	+	0.473230i	$0.156912\pi$
$257$			15.1729i			0.946460i	−0.880939	+	0.473230i	$0.843088\pi$
$258$		0			0
$259$			1.83080i			0.113761i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			22.8867i			1.41125i	0.708584	+	0.705627i	$0.249334\pi$
$263$			22.8867i			1.41125i	−0.708584	+	0.705627i	$0.750666\pi$
$264$		0			0
$265$		−	15.7687i		−	0.968663i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−25.5828			−1.55981			−0.779906	−	0.625897i	$-0.784733\pi$
$269$	−25.5828			−1.55981			−0.779906	+	0.625897i	$0.784733\pi$
$270$		0			0
$271$	−14.8275			−0.900706			−0.450353	−	0.892851i	$-0.648702\pi$
$271$	−14.8275			−0.900706			−0.450353	+	0.892851i	$0.648702\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	1.37863i		−	0.0831346i
$276$		0			0
$277$	24.1033			1.44822			0.724112	−	0.689682i	$-0.242250\pi$
$277$	24.1033			1.44822			0.724112	+	0.689682i	$0.242250\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	27.2942i		−	1.62824i	−0.580698	−	0.814119i	$-0.697220\pi$
$281$		−	27.2942i		−	1.62824i	0.580698	−	0.814119i	$-0.302780\pi$
$282$		0			0
$283$	−8.01514			−0.476451			−0.238225	−	0.971210i	$-0.576566\pi$
$283$	−8.01514			−0.476451			−0.238225	+	0.971210i	$0.576566\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.74074	+	6.09760i	−0.161781	+	0.359930i
$288$		0			0
$289$	11.8958			0.699751
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	15.2536i		−	0.891125i	−0.895251	−	0.445562i	$-0.853004\pi$
$293$		−	15.2536i		−	0.891125i	0.895251	−	0.445562i	$-0.146996\pi$
$294$		0			0
$295$	3.69297			0.215013
$296$		0			0
$297$		0			0
$298$		0			0
$299$			10.1896i			0.589280i
$300$		0			0
$301$			9.55981i			0.551019i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.30760			0.246652
$306$		0			0
$307$	−19.1772			−1.09450			−0.547250	−	0.836969i	$-0.684325\pi$
$307$	−19.1772			−1.09450			−0.547250	+	0.836969i	$0.684325\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			31.0871i			1.76279i	0.472380	+	0.881395i	$0.343395\pi$
$311$			31.0871i			1.76279i	−0.472380	+	0.881395i	$0.656605\pi$
$312$		0			0
$313$		−	18.0004i		−	1.01744i	−0.860931	−	0.508721i	$-0.830118\pi$
$313$		−	18.0004i		−	1.01744i	0.860931	−	0.508721i	$-0.169882\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	19.1681i		−	1.07659i	−0.842757	−	0.538294i	$-0.819069\pi$
$317$		−	19.1681i		−	1.07659i	0.842757	−	0.538294i	$-0.180931\pi$
$318$		0			0
$319$	5.52223			0.309185
$320$		0			0
$321$		0			0
$322$		0			0
$323$	11.6913			0.650523
$324$		0			0
$325$		−	0.639264i		−	0.0354600i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.37022			−0.185806
$330$		0			0
$331$			12.6695i			0.696381i	0.937424	+	0.348191i	$0.113204\pi$
$331$			12.6695i			0.696381i	−0.937424	+	0.348191i	$0.886796\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	34.1013i		−	1.86315i
$336$		0			0
$337$	−1.21520			−0.0661960			−0.0330980	−	0.999452i	$-0.510537\pi$
$337$	−1.21520			−0.0661960			−0.0330980	+	0.999452i	$0.510537\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	59.0344i		−	3.19689i
$342$		0			0
$343$		−	13.4787i		−	0.727783i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			2.06594i			0.110905i	0.998461	+	0.0554527i	$0.0176602\pi$
$347$			2.06594i			0.110905i	−0.998461	+	0.0554527i	$0.982340\pi$
$348$		0			0
$349$	7.79994			0.417521			0.208761	−	0.977967i	$-0.433057\pi$
$349$	7.79994			0.417521			0.208761	+	0.977967i	$0.433057\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	34.1809			1.81927			0.909633	−	0.415412i	$-0.136363\pi$
$353$	34.1809			1.81927			0.909633	+	0.415412i	$0.136363\pi$
$354$		0			0
$355$		−	11.2886i		−	0.599135i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	15.4327			0.814509			0.407254	−	0.913315i	$-0.366486\pi$
$359$	15.4327			0.814509			0.407254	+	0.913315i	$0.366486\pi$
$360$		0			0
$361$	−7.77914			−0.409429
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.34171			0.331940
$366$		0			0
$367$	18.8370			0.983282			0.491641	−	0.870798i	$-0.336397\pi$
$367$	18.8370			0.983282			0.491641	+	0.870798i	$0.336397\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	7.18766			0.373165
$372$		0			0
$373$	3.13080			0.162107			0.0810533	−	0.996710i	$-0.474172\pi$
$373$	3.13080			0.162107			0.0810533	+	0.996710i	$0.474172\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.56063			0.131879
$378$		0			0
$379$	−20.6712			−1.06181			−0.530904	−	0.847432i	$-0.678147\pi$
$379$	−20.6712			−1.06181			−0.530904	+	0.847432i	$0.678147\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			6.05354i			0.309321i	0.987968	+	0.154661i	$0.0494284\pi$
$383$			6.05354i			0.309321i	−0.987968	+	0.154661i	$0.950572\pi$
$384$		0			0
$385$	13.3773			0.681768
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8.25360			0.418474			0.209237	−	0.977865i	$-0.432902\pi$
$389$	8.25360			0.418474			0.209237	+	0.977865i	$0.432902\pi$
$390$		0			0
$391$			8.87526i			0.448841i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			8.03304i			0.404186i
$396$		0			0
$397$		−	29.1454i		−	1.46276i	−0.681968	−	0.731382i	$-0.738876\pi$
$397$		−	29.1454i		−	1.46276i	0.681968	−	0.731382i	$-0.261124\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.36788			0.118246			0.0591230	−	0.998251i	$-0.481170\pi$
$401$	2.36788			0.118246			0.0591230	+	0.998251i	$0.481170\pi$
$402$		0			0
$403$		−	27.3739i		−	1.36359i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			9.80903i			0.486215i
$408$		0			0
$409$	26.6880			1.31964			0.659818	−	0.751425i	$-0.270633\pi$
$409$	26.6880			1.31964			0.659818	+	0.751425i	$0.270633\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			1.68332i			0.0828309i
$414$		0			0
$415$	−11.2170			−0.550620
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−19.1541			−0.935737			−0.467868	−	0.883798i	$-0.654978\pi$
$419$	−19.1541			−0.935737			−0.467868	+	0.883798i	$0.654978\pi$
$420$		0			0
$421$			39.7744i			1.93849i	0.246102	+	0.969244i	$0.420850\pi$
$421$			39.7744i			1.93849i	−0.246102	+	0.969244i	$0.579150\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	0.556807i		−	0.0270091i
$426$		0			0
$427$			1.96348i			0.0950195i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	19.6668			0.947317			0.473658	−	0.880709i	$-0.342933\pi$
$431$	19.6668			0.947317			0.473658	+	0.880709i	$0.342933\pi$
$432$		0			0
$433$	11.4170			0.548667			0.274334	−	0.961635i	$-0.411543\pi$
$433$	11.4170			0.548667			0.274334	+	0.961635i	$0.411543\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	20.3289i		−	0.972464i
$438$		0			0
$439$		−	4.38663i		−	0.209363i	−0.994506	−	0.104681i	$-0.966618\pi$
$439$		−	4.38663i		−	0.209363i	0.994506	−	0.104681i	$-0.0333822\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−32.4879			−1.54355			−0.771775	−	0.635896i	$-0.780631\pi$
$443$	−32.4879			−1.54355			−0.771775	+	0.635896i	$0.780631\pi$
$444$		0			0
$445$			5.61229i			0.266048i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−16.0700			−0.758388			−0.379194	−	0.925317i	$-0.623799\pi$
$449$	−16.0700			−0.758388			−0.379194	+	0.925317i	$0.623799\pi$
$450$		0			0
$451$	−14.6843	+	32.6695i	−0.691456	+	1.53835i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.20296			0.290799
$456$		0			0
$457$			24.7203i			1.15637i	0.815906	+	0.578184i	$0.196239\pi$
$457$			24.7203i			1.15637i	−0.815906	+	0.578184i	$0.803761\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−19.1326			−0.891093			−0.445546	−	0.895259i	$-0.646991\pi$
$461$	−19.1326			−0.891093			−0.445546	+	0.895259i	$0.646991\pi$
$462$		0			0
$463$		−	22.9634i		−	1.06720i	−0.845737	−	0.533599i	$-0.820839\pi$
$463$		−	22.9634i		−	1.06720i	0.845737	−	0.533599i	$-0.179161\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−28.2492			−1.30722			−0.653608	−	0.756833i	$-0.726746\pi$
$467$	−28.2492			−1.30722			−0.653608	+	0.756833i	$0.726746\pi$
$468$		0			0
$469$	15.5440			0.717756
$470$		0			0
$471$		0			0
$472$		0			0
$473$			51.2193i			2.35507i
$474$		0			0
$475$			1.27537i			0.0585181i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	4.24314i		−	0.193874i	−0.995291	−	0.0969369i	$-0.969095\pi$
$479$		−	4.24314i		−	0.193874i	0.995291	−	0.0969369i	$-0.0309045\pi$
$480$		0			0
$481$			4.54839i			0.207389i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	0.858743i		−	0.0389935i
$486$		0			0
$487$	10.7270			0.486085			0.243043	−	0.970016i	$-0.421854\pi$
$487$	10.7270			0.486085			0.243043	+	0.970016i	$0.421854\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	33.2725			1.50156			0.750782	−	0.660550i	$-0.229677\pi$
$491$	33.2725			1.50156			0.750782	+	0.660550i	$0.229677\pi$
$492$		0			0
$493$	2.23034			0.100449
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.14554			0.230809
$498$		0			0
$499$		−	20.1013i		−	0.899859i	−0.893064	−	0.449929i	$-0.851449\pi$
$499$		−	20.1013i		−	0.899859i	0.893064	−	0.449929i	$-0.148551\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	30.4012i		−	1.35552i	−0.735282	−	0.677761i	$-0.762950\pi$
$503$		−	30.4012i		−	1.35552i	0.735282	−	0.677761i	$-0.237050\pi$
$504$		0			0
$505$			2.75411i			0.122556i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	7.59851i		−	0.336798i	−0.985719	−	0.168399i	$-0.946140\pi$
$509$		−	7.59851i		−	0.336798i	0.985719	−	0.168399i	$-0.0538597\pi$
$510$		0			0
$511$			2.89067i			0.127876i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	43.3936			1.91215
$516$		0			0
$517$	−18.0569			−0.794140
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	6.73605i		−	0.295112i	−0.989054	−	0.147556i	$-0.952859\pi$
$521$		−	6.73605i		−	0.295112i	0.989054	−	0.147556i	$-0.0471406\pi$
$522$		0			0
$523$	6.21795			0.271892			0.135946	−	0.990716i	$-0.456593\pi$
$523$	6.21795			0.271892			0.135946	+	0.990716i	$0.456593\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	23.8430i		−	1.03862i
$528$		0			0
$529$	−7.56767			−0.329029
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−6.80903	+	15.1487i	−0.294932	+	0.656163i
$534$		0			0
$535$	27.2559			1.17838
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	33.0592i		−	1.42396i
$540$		0			0
$541$	−14.2798			−0.613935			−0.306967	−	0.951720i	$-0.599314\pi$
$541$	−14.2798			−0.613935			−0.306967	+	0.951720i	$0.599314\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			14.2152i			0.608912i
$546$		0			0
$547$			33.1934i			1.41925i	0.704581	+	0.709624i	$0.251135\pi$
$547$			33.1934i			1.41925i	−0.704581	+	0.709624i	$0.748865\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−5.10862			−0.217635
$552$		0			0
$553$	−3.66161			−0.155707
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	36.7889i		−	1.55880i	−0.626529	−	0.779398i	$-0.715525\pi$
$557$		−	36.7889i		−	1.55880i	0.626529	−	0.779398i	$-0.284475\pi$
$558$		0			0
$559$			23.7501i			1.00452i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			0.794848i			0.0334989i	0.999860	+	0.0167494i	$0.00533176\pi$
$563$			0.794848i			0.0334989i	−0.999860	+	0.0167494i	$0.994668\pi$
$564$		0			0
$565$	27.1790			1.14343
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8.03936			0.337028			0.168514	−	0.985699i	$-0.446103\pi$
$569$	8.03936			0.337028			0.168514	+	0.985699i	$0.446103\pi$
$570$		0			0
$571$		−	10.1688i		−	0.425551i	−0.977101	−	0.212775i	$-0.931750\pi$
$571$		−	10.1688i		−	0.425551i	0.977101	−	0.212775i	$-0.0682503\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.968177			−0.0403758
$576$		0			0
$577$			43.1400i			1.79594i	0.440054	+	0.897971i	$0.354959\pi$
$577$			43.1400i			1.79594i	−0.440054	+	0.897971i	$0.645041\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	5.11290i		−	0.212119i
$582$		0			0
$583$	38.5098			1.59491
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	11.2831i		−	0.465702i	−0.972512	−	0.232851i	$-0.925194\pi$
$587$		−	11.2831i		−	0.465702i	0.972512	−	0.232851i	$-0.0748055\pi$
$588$		0			0
$589$			54.6128i			2.25028i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	7.71126i		−	0.316664i	−0.987386	−	0.158332i	$-0.949388\pi$
$593$		−	7.71126i		−	0.316664i	0.987386	−	0.158332i	$-0.0506116\pi$
$594$		0			0
$595$	5.40285			0.221495
$596$		0			0
$597$		0			0
$598$		0			0
$599$	24.4772			1.00011			0.500056	−	0.865993i	$-0.333313\pi$
$599$	24.4772			1.00011			0.500056	+	0.865993i	$0.333313\pi$
$600$		0			0
$601$		−	30.7950i		−	1.25615i	−0.778151	−	0.628077i	$-0.783842\pi$
$601$		−	30.7950i		−	1.25615i	0.778151	−	0.628077i	$-0.216158\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	46.4766			1.88954
$606$		0			0
$607$	−26.7667			−1.08643			−0.543214	−	0.839594i	$-0.682793\pi$
$607$	−26.7667			−1.08643			−0.543214	+	0.839594i	$0.682793\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.37287			−0.338730
$612$		0			0
$613$	11.0237			0.445241			0.222621	−	0.974905i	$-0.428539\pi$
$613$	11.0237			0.445241			0.222621	+	0.974905i	$0.428539\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−25.1417			−1.01216			−0.506082	−	0.862485i	$-0.668907\pi$
$617$	−25.1417			−1.01216			−0.506082	+	0.862485i	$0.668907\pi$
$618$		0			0
$619$	8.59617			0.345509			0.172755	−	0.984965i	$-0.444733\pi$
$619$	8.59617			0.345509			0.172755	+	0.984965i	$0.444733\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.55818			−0.102491
$624$		0			0
$625$	−26.1715			−1.04686
$626$		0			0
$627$		0			0
$628$		0			0
$629$			3.96170i			0.157963i
$630$		0			0
$631$	−28.6389			−1.14009			−0.570047	−	0.821612i	$-0.693075\pi$
$631$	−28.6389			−1.14009			−0.570047	+	0.821612i	$0.693075\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−13.7516			−0.545716
$636$		0			0
$637$		−	15.3294i		−	0.607372i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	34.2382i		−	1.35233i	−0.736752	−	0.676163i	$-0.763641\pi$
$641$		−	34.2382i		−	1.35233i	0.736752	−	0.676163i	$-0.236359\pi$
$642$		0			0
$643$			7.54977i			0.297734i	0.988857	+	0.148867i	$0.0475626\pi$
$643$			7.54977i			0.297734i	−0.988857	+	0.148867i	$0.952437\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.11994			−0.0440294			−0.0220147	−	0.999758i	$-0.507008\pi$
$647$	−1.11994			−0.0440294			−0.0220147	+	0.999758i	$0.507008\pi$
$648$		0			0
$649$			9.01886i			0.354021i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	12.8440i		−	0.502625i	−0.967906	−	0.251312i	$-0.919138\pi$
$653$		−	12.8440i		−	0.502625i	0.967906	−	0.251312i	$-0.0808621\pi$
$654$		0			0
$655$	43.3819			1.69507
$656$		0			0
$657$		0			0
$658$		0			0
$659$			3.24274i			0.126319i	0.998003	+	0.0631596i	$0.0201177\pi$
$659$			3.24274i			0.126319i	−0.998003	+	0.0631596i	$0.979882\pi$
$660$		0			0
$661$	8.98378			0.349428			0.174714	−	0.984619i	$-0.444100\pi$
$661$	8.98378			0.349428			0.174714	+	0.984619i	$0.444100\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−12.3753			−0.479894
$666$		0			0
$667$		−	3.87812i		−	0.150161i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			10.5199i			0.406116i
$672$		0			0
$673$		−	11.7318i		−	0.452226i	−0.974101	−	0.226113i	$-0.927398\pi$
$673$		−	11.7318i		−	0.452226i	0.974101	−	0.226113i	$-0.0726019\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−11.1359			−0.427987			−0.213994	−	0.976835i	$-0.568647\pi$
$677$	−11.1359			−0.427987			−0.213994	+	0.976835i	$0.568647\pi$
$678$		0			0
$679$	0.391431			0.0150217
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	19.7553i		−	0.755916i	−0.925823	−	0.377958i	$-0.876626\pi$
$683$		−	19.7553i		−	0.755916i	0.925823	−	0.377958i	$-0.123374\pi$
$684$		0			0
$685$		−	31.7360i		−	1.21257i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	17.8568			0.680290
$690$		0			0
$691$		−	9.30566i		−	0.354004i	−0.984211	−	0.177002i	$-0.943360\pi$
$691$		−	9.30566i		−	0.354004i	0.984211	−	0.177002i	$-0.0566399\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	24.9307			0.945677
$696$		0			0
$697$	−5.93074	+	13.1947i	−0.224643	+	0.499785i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	0.168797			0.00637538			0.00318769	−	0.999995i	$-0.498985\pi$
$701$	0.168797			0.00637538			0.00318769	+	0.999995i	$0.498985\pi$
$702$		0			0
$703$		−	9.07434i		−	0.342245i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.25537			−0.0472132
$708$		0			0
$709$		−	29.2331i		−	1.09787i	−0.835864	−	0.548936i	$-0.815033\pi$
$709$		−	29.2331i		−	1.09787i	0.835864	−	0.548936i	$-0.184967\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−41.4583			−1.55263
$714$		0			0
$715$	33.2341			1.24288
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	29.7879i		−	1.11090i	−0.831549	−	0.555451i	$-0.812545\pi$
$719$		−	29.7879i		−	1.11090i	0.831549	−	0.555451i	$-0.187455\pi$
$720$		0			0
$721$			19.7796i			0.736631i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			0.243301i			0.00903598i
$726$		0			0
$727$		−	45.0193i		−	1.66967i	−0.550499	−	0.834836i	$-0.685562\pi$
$727$		−	45.0193i		−	1.66967i	0.550499	−	0.834836i	$-0.314438\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			20.6866i			0.765123i
$732$		0			0
$733$	−30.6938			−1.13370			−0.566850	−	0.823821i	$-0.691838\pi$
$733$	−30.6938			−1.13370			−0.566850	+	0.823821i	$0.691838\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	83.2813			3.06770
$738$		0			0
$739$	−17.1638			−0.631382			−0.315691	−	0.948862i	$-0.602236\pi$
$739$	−17.1638			−0.631382			−0.315691	+	0.948862i	$0.602236\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	44.7325			1.64107			0.820537	−	0.571593i	$-0.193674\pi$
$743$	44.7325			1.64107			0.820537	+	0.571593i	$0.193674\pi$
$744$		0			0
$745$		−	19.8568i		−	0.727497i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			12.4238i			0.453954i
$750$		0			0
$751$			39.9336i			1.45720i	0.684941	+	0.728599i	$0.259828\pi$
$751$			39.9336i			1.45720i	−0.684941	+	0.728599i	$0.740172\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			47.9422i			1.74479i
$756$		0			0
$757$		−	10.7199i		−	0.389623i	−0.980841	−	0.194811i	$-0.937590\pi$
$757$		−	10.7199i		−	0.389623i	0.980841	−	0.194811i	$-0.0624095\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	10.7754			0.390609			0.195304	−	0.980743i	$-0.437431\pi$
$761$	10.7754			0.390609			0.195304	+	0.980743i	$0.437431\pi$
$762$		0			0
$763$	−6.47955			−0.234575
$764$		0			0
$765$		0			0
$766$		0			0
$767$			4.18200i			0.151003i
$768$		0			0
$769$	−22.6329			−0.816164			−0.408082	−	0.912945i	$-0.633802\pi$
$769$	−22.6329			−0.816164			−0.408082	+	0.912945i	$0.633802\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			31.5178i			1.13362i	0.823849	+	0.566809i	$0.191822\pi$
$773$			31.5178i			1.13362i	−0.823849	+	0.566809i	$0.808178\pi$
$774$		0			0
$775$	2.60097			0.0934295
$776$		0			0
$777$		0			0
$778$		0			0
$779$	13.5845	−	30.2226i	0.486713	−	1.08284i
$780$		0			0
$781$	27.5686			0.986483
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	3.19154i		−	0.113911i
$786$		0			0
$787$	−16.4578			−0.586656			−0.293328	−	0.956012i	$-0.594763\pi$
$787$	−16.4578			−0.586656			−0.293328	+	0.956012i	$0.594763\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			12.3887i			0.440491i
$792$		0			0
$793$			4.87802i			0.173223i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	15.0615			0.533507			0.266754	−	0.963765i	$-0.414049\pi$
$797$	15.0615			0.533507			0.266754	+	0.963765i	$0.414049\pi$
$798$		0			0
$799$	−7.29287			−0.258003
$800$		0			0
$801$		0			0
$802$		0			0
$803$			15.4875i			0.546544i
$804$		0			0
$805$		−	9.39450i		−	0.331113i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1.43137i			0.0503243i	0.999683	+	0.0251622i	$0.00801021\pi$
$809$			1.43137i			0.0503243i	−0.999683	+	0.0251622i	$0.991990\pi$
$810$		0			0
$811$	−35.9345			−1.26183			−0.630915	−	0.775852i	$-0.717320\pi$
$811$	−35.9345			−1.26183			−0.630915	+	0.775852i	$0.717320\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	49.2014			1.72345
$816$		0			0
$817$		−	47.3830i		−	1.65772i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	40.2214			1.40374			0.701868	−	0.712307i	$-0.252350\pi$
$821$	40.2214			1.40374			0.701868	+	0.712307i	$0.252350\pi$
$822$		0			0
$823$		−	37.8642i		−	1.31986i	−0.751325	−	0.659932i	$-0.770585\pi$
$823$		−	37.8642i		−	1.31986i	0.751325	−	0.659932i	$-0.229415\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	27.4298i		−	0.953829i	−0.878950	−	0.476914i	$-0.841755\pi$
$827$		−	27.4298i		−	0.953829i	0.878950	−	0.476914i	$-0.158245\pi$
$828$		0			0
$829$	−18.0461			−0.626768			−0.313384	−	0.949626i	$-0.601463\pi$
$829$	−18.0461			−0.626768			−0.313384	+	0.949626i	$0.601463\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	13.3521i		−	0.462622i
$834$		0			0
$835$		−	34.5942i		−	1.19718i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			34.6749i			1.19711i	0.801081	+	0.598555i	$0.204258\pi$
$839$			34.6749i			1.19711i	−0.801081	+	0.598555i	$0.795742\pi$
$840$		0			0
$841$	28.0254			0.966394
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−14.3662			−0.494213
$846$		0			0
$847$			21.1849i			0.727922i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.88862			0.236139
$852$		0			0
$853$	−40.4366			−1.38452			−0.692261	−	0.721647i	$-0.743385\pi$
$853$	−40.4366			−1.38452			−0.692261	+	0.721647i	$0.743385\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−27.1763			−0.928326			−0.464163	−	0.885750i	$-0.653645\pi$
$857$	−27.1763			−0.928326			−0.464163	+	0.885750i	$0.653645\pi$
$858$		0			0
$859$	45.9762			1.56869			0.784343	−	0.620327i	$-0.213000\pi$
$859$	45.9762			1.56869			0.784343	+	0.620327i	$0.213000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−21.9796			−0.748194			−0.374097	−	0.927390i	$-0.622047\pi$
$863$	−21.9796			−0.748194			−0.374097	+	0.927390i	$0.622047\pi$
$864$		0			0
$865$	−13.4359			−0.456834
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−19.6181			−0.665497
$870$		0			0
$871$	38.6171			1.30849
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	11.3678i		−	0.384301i
$876$		0			0
$877$	11.9393			0.403160			0.201580	−	0.979472i	$-0.435392\pi$
$877$	11.9393			0.403160			0.201580	+	0.979472i	$0.435392\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.11096			−0.0711201			−0.0355601	−	0.999368i	$-0.511322\pi$
$881$	−2.11096			−0.0711201			−0.0355601	+	0.999368i	$0.511322\pi$
$882$		0			0
$883$		−	31.9475i		−	1.07512i	−0.843226	−	0.537560i	$-0.819346\pi$
$883$		−	31.9475i		−	1.07512i	0.843226	−	0.537560i	$-0.180654\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	4.72657i		−	0.158703i	−0.996847	−	0.0793513i	$-0.974715\pi$
$887$		−	4.72657i		−	0.158703i	0.996847	−	0.0793513i	$-0.0252849\pi$
$888$		0			0
$889$		−	6.26823i		−	0.210230i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	16.7044			0.558992
$894$		0			0
$895$		−	45.1294i		−	1.50851i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			10.4184i			0.347473i
$900$		0			0
$901$	15.5535			0.518162
$902$		0			0
$903$		0			0
$904$		0			0
$905$			21.9325i			0.729062i
$906$		0			0
$907$	51.2974			1.70330			0.851651	−	0.524110i	$-0.175602\pi$
$907$	51.2974			1.70330			0.851651	+	0.524110i	$0.175602\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−40.7898			−1.35143			−0.675713	−	0.737165i	$-0.736164\pi$
$911$	−40.7898			−1.35143			−0.675713	+	0.737165i	$0.736164\pi$
$912$		0			0
$913$		−	27.3938i		−	0.906602i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			19.7743i			0.653004i
$918$		0			0
$919$		−	41.8828i		−	1.38159i	−0.723053	−	0.690793i	$-0.757262\pi$
$919$		−	41.8828i		−	1.38159i	0.723053	−	0.690793i	$-0.242738\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12.7834			0.420772
$924$		0			0
$925$	−0.432171			−0.0142097
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	42.3018i		−	1.38788i	−0.720035	−	0.693938i	$-0.755874\pi$
$929$		−	42.3018i		−	1.38788i	0.720035	−	0.693938i	$-0.244126\pi$
$930$		0			0
$931$			30.5831i			1.00232i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	28.9473			0.946677
$936$		0			0
$937$			17.3383i			0.566417i	0.959058	+	0.283209i	$0.0913989\pi$
$937$			17.3383i			0.566417i	−0.959058	+	0.283209i	$0.908601\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−48.4739			−1.58020			−0.790102	−	0.612976i	$-0.789972\pi$
$941$	−48.4739			−1.58020			−0.790102	+	0.612976i	$0.789972\pi$
$942$		0			0
$943$	22.9430	+	10.3124i	0.747126	+	0.335818i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	37.7078			1.22534			0.612670	−	0.790339i	$-0.290095\pi$
$947$	37.7078			1.22534			0.612670	+	0.790339i	$0.290095\pi$
$948$		0			0
$949$			7.18149i			0.233121i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	9.06098			0.293514			0.146757	−	0.989173i	$-0.453116\pi$
$953$	9.06098			0.293514			0.146757	+	0.989173i	$0.453116\pi$
$954$		0			0
$955$			28.8757i			0.934394i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.4659			0.467127
$960$		0			0
$961$	80.3761			2.59278
$962$		0			0
$963$		0			0
$964$		0			0
$965$			57.2313i			1.84234i
$966$		0			0
$967$			48.0702i			1.54583i	0.634508	+	0.772916i	$0.281203\pi$
$967$			48.0702i			1.54583i	−0.634508	+	0.772916i	$0.718797\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			32.7359i			1.05054i	0.850934	+	0.525272i	$0.176037\pi$
$971$			32.7359i			1.05054i	−0.850934	+	0.525272i	$0.823963\pi$
$972$		0			0
$973$			11.3639i			0.364310i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			38.2601i			1.22405i	0.790839	+	0.612024i	$0.209644\pi$
$977$			38.2601i			1.22405i	−0.790839	+	0.612024i	$0.790356\pi$
$978$		0			0
$979$	−13.7062			−0.438051
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−43.2157			−1.37837			−0.689183	−	0.724587i	$-0.742031\pi$
$983$	−43.2157			−1.37837			−0.689183	+	0.724587i	$0.742031\pi$
$984$		0			0
$985$	−17.8605			−0.569084
$986$		0			0
$987$		0			0
$988$		0			0
$989$	35.9700			1.14378
$990$		0			0
$991$		−	5.02630i		−	0.159666i	−0.996808	−	0.0798328i	$-0.974561\pi$
$991$		−	5.02630i		−	0.159666i	0.996808	−	0.0798328i	$-0.0254386\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	29.5496i		−	0.936785i
$996$		0			0
$997$		−	43.9767i		−	1.39276i	−0.717675	−	0.696378i	$-0.754794\pi$
$997$		−	43.9767i		−	1.39276i	0.717675	−	0.696378i	$-0.245206\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2952.2.j.d.2377.3		8
3.2	odd	2		984.2.j.b.409.7	yes	8
12.11	even	2		1968.2.j.f.1393.3		8
41.40	even	2	inner	2952.2.j.d.2377.4		8
123.122	odd	2		984.2.j.b.409.3	✓	8
492.491	even	2		1968.2.j.f.1393.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
984.2.j.b.409.3	✓	8	123.122	odd	2
984.2.j.b.409.7	yes	8	3.2	odd	2
1968.2.j.f.1393.3		8	12.11	even	2
1968.2.j.f.1393.7		8	492.491	even	2
2952.2.j.d.2377.3		8	1.1	even	1	trivial
2952.2.j.d.2377.4		8	41.40	even	2	inner

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

\(f(q)\)	\(=\)	\(q-2.29051 q^{5} -1.04406i q^{7} +O(q^{10})\)	\(q-2.29051 q^{5} -1.04406i q^{7} -5.59383i q^{11} -2.59383i q^{13} -2.25926i q^{17} +5.17486i q^{19} -3.92840 q^{23} +0.246456 q^{25} +0.987200i q^{29} +10.5535 q^{31} +2.39143i q^{35} -1.75354 q^{37} +(-5.84028 - 2.62509i) q^{41} -9.15640 q^{43} -3.22800i q^{47} +5.90994 q^{49} +6.88434i q^{53} +12.8127i q^{55} -1.61229 q^{59} -1.88062 q^{61} +5.94120i q^{65} +14.8881i q^{67} +4.92840i q^{71} -2.76869 q^{73} -5.84028 q^{77} -3.50709i q^{79} +4.89714 q^{83} +5.17486i q^{85} -2.45023i q^{89} -2.70811 q^{91} -11.8531i q^{95} +0.374913i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 10 q^{5}+O(q^{10}) \) 8 * q - 10 * q^5	\( 8 q - 10 q^{5} + 20 q^{23} + 2 q^{25} + 8 q^{31} - 14 q^{37} - 12 q^{41} + 6 q^{43} - 32 q^{49} - 6 q^{59} - 22 q^{61} + 64 q^{73} - 12 q^{77} - 22 q^{83} - 12 q^{91}+O(q^{100}) \) 8 * q - 10 * q^5 + 20 * q^23 + 2 * q^25 + 8 * q^31 - 14 * q^37 - 12 * q^41 + 6 * q^43 - 32 * q^49 - 6 * q^59 - 22 * q^61 + 64 * q^73 - 12 * q^77 - 22 * q^83 - 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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