LMFDB - Embedded newform 2952.2.j.d.2377.6

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			2377.6
Root			$-0.233455i$ of defining polynomial
Character	$\chi$	$=$	2952.2377
Dual form			2952.2.j.d.2377.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.766545 q^{5} +4.17895i q^{7} +O(q^{10})$	$q-0.766545 q^{5} +4.17895i q^{7} -1.38801i q^{11} -4.38801i q^{13} -6.33351i q^{17} +4.85492i q^{19} +1.44251 q^{23} -4.41241 q^{25} -4.92110i q^{29} +5.26954 q^{31} -3.20336i q^{35} -6.41241 q^{37} +(5.80042 + 2.71204i) q^{41} +11.8761 q^{43} -12.4336i q^{47} -10.4637 q^{49} +1.62146i q^{53} +1.06397i q^{55} -5.63314 q^{59} +11.4580 q^{61} +3.36360i q^{65} -11.2150i q^{67} +0.442508i q^{71} +14.2429 q^{73} +5.80042 q^{77} +12.8248i q^{79} -7.54256 q^{83} +4.85492i q^{85} +12.5670i q^{89} +18.3373 q^{91} -3.72151i q^{95} -0.287956i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.766545			−0.342809			−0.171405	−	0.985201i	$-0.554831\pi$
$5$	−0.766545			−0.342809			−0.171405	+	0.985201i	$0.554831\pi$
$6$		0			0
$7$			4.17895i			1.57950i	0.613431	+	0.789748i	$0.289789\pi$
$7$			4.17895i			1.57950i	−0.613431	+	0.789748i	$0.710211\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.38801i		−	0.418500i	−0.977862	−	0.209250i	$-0.932898\pi$
$11$		−	1.38801i		−	0.418500i	0.977862	−	0.209250i	$-0.0671022\pi$
$12$		0			0
$13$		−	4.38801i		−	1.21701i	−0.793548	−	0.608507i	$-0.791769\pi$
$13$		−	4.38801i		−	1.21701i	0.793548	−	0.608507i	$-0.208231\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	6.33351i		−	1.53610i	−0.640389	−	0.768050i	$-0.721227\pi$
$17$		−	6.33351i		−	1.53610i	0.640389	−	0.768050i	$-0.278773\pi$
$18$		0			0
$19$			4.85492i			1.11379i	0.830581	+	0.556897i	$0.188008\pi$
$19$			4.85492i			1.11379i	−0.830581	+	0.556897i	$0.811992\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.44251			0.300784			0.150392	−	0.988626i	$-0.451946\pi$
$23$	1.44251			0.300784			0.150392	+	0.988626i	$0.451946\pi$
$24$		0			0
$25$	−4.41241			−0.882482
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.92110i		−	0.913825i	−0.889512	−	0.456912i	$-0.848955\pi$
$29$		−	4.92110i		−	0.913825i	0.889512	−	0.456912i	$-0.151045\pi$
$30$		0			0
$31$	5.26954			0.946437			0.473218	−	0.880945i	$-0.343092\pi$
$31$	5.26954			0.946437			0.473218	+	0.880945i	$0.343092\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	3.20336i		−	0.541466i
$36$		0			0
$37$	−6.41241			−1.05419			−0.527097	−	0.849805i	$-0.676719\pi$
$37$	−6.41241			−1.05419			−0.527097	+	0.849805i	$0.676719\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.80042	+	2.71204i	0.905873	+	0.423550i
$41$	5.80042	+	2.71204i	0.905873	+	0.423550i
$42$		0			0
$43$	11.8761			1.81108			0.905541	−	0.424258i	$-0.139465\pi$
$43$	11.8761			1.81108			0.905541	+	0.424258i	$0.139465\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	12.4336i		−	1.81362i	−0.421539	−	0.906810i	$-0.638510\pi$
$47$		−	12.4336i		−	1.81362i	0.421539	−	0.906810i	$-0.361490\pi$
$48$		0			0
$49$	−10.4637			−1.49481
$50$		0			0
$51$		0			0
$52$		0			0
$53$			1.62146i			0.222725i	0.993780	+	0.111362i	$0.0355214\pi$
$53$			1.62146i			0.222725i	−0.993780	+	0.111362i	$0.964479\pi$
$54$		0			0
$55$			1.06397i			0.143466i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.63314			−0.733372			−0.366686	−	0.930345i	$-0.619508\pi$
$59$	−5.63314			−0.733372			−0.366686	+	0.930345i	$0.619508\pi$
$60$		0			0
$61$	11.4580			1.46704			0.733521	−	0.679667i	$-0.237876\pi$
$61$	11.4580			1.46704			0.733521	+	0.679667i	$0.237876\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			3.36360i			0.417204i
$66$		0			0
$67$		−	11.2150i		−	1.37013i	−0.728480	−	0.685067i	$-0.759773\pi$
$67$		−	11.2150i		−	1.37013i	0.728480	−	0.685067i	$-0.240227\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			0.442508i			0.0525160i	0.999655	+	0.0262580i	$0.00835914\pi$
$71$			0.442508i			0.0525160i	−0.999655	+	0.0262580i	$0.991641\pi$
$72$		0			0
$73$	14.2429			1.66701			0.833504	−	0.552513i	$-0.186331\pi$
$73$	14.2429			1.66701			0.833504	+	0.552513i	$0.186331\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.80042			0.661019
$78$		0			0
$79$			12.8248i			1.44290i	0.692464	+	0.721452i	$0.256525\pi$
$79$			12.8248i			1.44290i	−0.692464	+	0.721452i	$0.743475\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.54256			−0.827903			−0.413952	−	0.910299i	$-0.635852\pi$
$83$	−7.54256			−0.827903			−0.413952	+	0.910299i	$0.635852\pi$
$84$		0			0
$85$			4.85492i			0.526590i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			12.5670i			1.33210i	0.745909	+	0.666048i	$0.232015\pi$
$89$			12.5670i			1.33210i	−0.745909	+	0.666048i	$0.767985\pi$
$90$		0			0
$91$	18.3373			1.92227
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	3.72151i		−	0.381819i
$96$		0			0
$97$		−	0.287956i		−	0.0292375i	−0.999893	−	0.0146188i	$-0.995347\pi$
$97$		−	0.287956i		−	0.0292375i	0.999893	−	0.0146188i	$-0.00465346\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	6.59136i		−	0.655865i	−0.944701	−	0.327933i	$-0.893648\pi$
$101$		−	6.59136i		−	0.655865i	0.944701	−	0.327933i	$-0.106352\pi$
$102$		0			0
$103$	−14.4729			−1.42606			−0.713028	−	0.701135i	$-0.752677\pi$
$103$	−14.4729			−1.42606			−0.713028	+	0.701135i	$0.752677\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	18.2768			1.76688			0.883442	−	0.468540i	$-0.155220\pi$
$107$	18.2768			1.76688			0.883442	+	0.468540i	$0.155220\pi$
$108$		0			0
$109$			3.24513i			0.310827i	0.987849	+	0.155414i	$0.0496711\pi$
$109$			3.24513i			0.310827i	−0.987849	+	0.155414i	$0.950329\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.64261			0.718956			0.359478	−	0.933154i	$-0.382955\pi$
$113$	7.64261			0.718956			0.359478	+	0.933154i	$0.382955\pi$
$114$		0			0
$115$	−1.10575			−0.103111
$116$		0			0
$117$		0			0
$118$		0			0
$119$	26.4674			2.42626
$120$		0			0
$121$	9.07344			0.824858
$122$		0			0
$123$		0			0
$124$		0			0
$125$	7.21503			0.645332
$126$		0			0
$127$	10.8365			0.961584			0.480792	−	0.876835i	$-0.340349\pi$
$127$	10.8365			0.961584			0.480792	+	0.876835i	$0.340349\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.9343			1.13008			0.565039	−	0.825064i	$-0.308861\pi$
$131$	12.9343			1.13008			0.565039	+	0.825064i	$0.308861\pi$
$132$		0			0
$133$	−20.2885			−1.75923
$134$		0			0
$135$		0			0
$136$		0			0
$137$			19.4557i			1.66222i	0.556111	+	0.831108i	$0.312293\pi$
$137$			19.4557i			1.66222i	−0.556111	+	0.831108i	$0.687707\pi$
$138$		0			0
$139$	−2.37854			−0.201745			−0.100872	−	0.994899i	$-0.532163\pi$
$139$	−2.37854			−0.201745			−0.100872	+	0.994899i	$0.532163\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.09058			−0.509320
$144$		0			0
$145$			3.77224i			0.313268i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	11.8910i		−	0.974148i	−0.873361	−	0.487074i	$-0.838064\pi$
$149$		−	11.8910i		−	0.974148i	0.873361	−	0.487074i	$-0.161936\pi$
$150$		0			0
$151$		−	2.17674i		−	0.177141i	−0.996070	−	0.0885704i	$-0.971770\pi$
$151$		−	2.17674i		−	0.177141i	0.996070	−	0.0885704i	$-0.0282298\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.03934			−0.324447
$156$		0			0
$157$		−	12.3091i		−	0.982373i	−0.871054	−	0.491187i	$-0.836563\pi$
$157$		−	12.3091i		−	0.982373i	0.871054	−	0.491187i	$-0.163437\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			6.02817i			0.475087i
$162$		0			0
$163$	11.7437			0.919838			0.459919	−	0.887961i	$-0.347878\pi$
$163$	11.7437			0.919838			0.459919	+	0.887961i	$0.347878\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			0.297425i			0.0230154i	0.999934	+	0.0115077i	$0.00366310\pi$
$167$			0.297425i			0.0230154i	−0.999934	+	0.0115077i	$0.996337\pi$
$168$		0			0
$169$	−6.25460			−0.481123
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−13.6426			−1.03723			−0.518614	−	0.855008i	$-0.673552\pi$
$173$	−13.6426			−1.03723			−0.518614	+	0.855008i	$0.673552\pi$
$174$		0			0
$175$		−	18.4393i		−	1.39388i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	12.2568i		−	0.916117i	−0.888922	−	0.458059i	$-0.848545\pi$
$179$		−	12.2568i		−	0.916117i	0.888922	−	0.458059i	$-0.151455\pi$
$180$		0			0
$181$		−	8.40915i		−	0.625047i	−0.949910	−	0.312524i	$-0.898826\pi$
$181$		−	8.40915i		−	0.625047i	0.949910	−	0.312524i	$-0.101174\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.91540			0.361387
$186$		0			0
$187$	−8.79095			−0.642858
$188$		0			0
$189$		0			0
$190$		0			0
$191$			1.69090i			0.122349i	0.998127	+	0.0611745i	$0.0194846\pi$
$191$			1.69090i			0.122349i	−0.998127	+	0.0611745i	$0.980515\pi$
$192$		0			0
$193$		−	21.4896i		−	1.54686i	−0.633884	−	0.773428i	$-0.718540\pi$
$193$		−	21.4896i		−	1.54686i	0.633884	−	0.773428i	$-0.281460\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	15.5914			1.11084			0.555419	−	0.831570i	$-0.312558\pi$
$197$	15.5914			1.11084			0.555419	+	0.831570i	$0.312558\pi$
$198$		0			0
$199$		−	5.29394i		−	0.375277i	−0.982238	−	0.187639i	$-0.939917\pi$
$199$		−	5.29394i		−	0.375277i	0.982238	−	0.187639i	$-0.0600834\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	20.5650			1.44338
$204$		0			0
$205$	−4.44628	−	2.07890i	−0.310542	−	0.145197i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.73866			0.466123
$210$		0			0
$211$		−	24.6581i		−	1.69753i	−0.528770	−	0.848765i	$-0.677346\pi$
$211$		−	24.6581i		−	1.69753i	0.528770	−	0.848765i	$-0.322654\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−9.10354			−0.620856
$216$		0			0
$217$			22.0211i			1.49489i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−27.7915			−1.86946
$222$		0			0
$223$	−1.21503			−0.0813647			−0.0406824	−	0.999172i	$-0.512953\pi$
$223$	−1.21503			−0.0813647			−0.0406824	+	0.999172i	$0.512953\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	12.3940i		−	0.822618i	−0.911496	−	0.411309i	$-0.865072\pi$
$227$		−	12.3940i		−	0.822618i	0.911496	−	0.411309i	$-0.134928\pi$
$228$		0			0
$229$		−	6.80367i		−	0.449599i	−0.974405	−	0.224800i	$-0.927827\pi$
$229$		−	6.80367i		−	0.449599i	0.974405	−	0.224800i	$-0.0721727\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	21.1662i		−	1.38665i	−0.720627	−	0.693323i	$-0.756146\pi$
$233$		−	21.1662i		−	1.38665i	0.720627	−	0.693323i	$-0.243854\pi$
$234$		0			0
$235$			9.53088i			0.621726i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	14.3858i		−	0.930540i	−0.885169	−	0.465270i	$-0.845957\pi$
$239$		−	14.3858i		−	0.930540i	0.885169	−	0.465270i	$-0.154043\pi$
$240$		0			0
$241$	−21.3373			−1.37445			−0.687227	−	0.726442i	$-0.741172\pi$
$241$	−21.3373			−1.37445			−0.687227	+	0.726442i	$0.741172\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8.02086			0.512434
$246$		0			0
$247$	21.3034			1.35550
$248$		0			0
$249$		0			0
$250$		0			0
$251$	5.84219			0.368756			0.184378	−	0.982855i	$-0.440973\pi$
$251$	5.84219			0.368756			0.184378	+	0.982855i	$0.440973\pi$
$252$		0			0
$253$		−	2.00221i		−	0.125878i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			4.96066i			0.309438i	0.987959	+	0.154719i	$0.0494472\pi$
$257$			4.96066i			0.309438i	−0.987959	+	0.154719i	$0.950553\pi$
$258$		0			0
$259$		−	26.7972i		−	1.66509i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			3.35570i			0.206921i	0.994634	+	0.103461i	$0.0329916\pi$
$263$			3.35570i			0.206921i	−0.994634	+	0.103461i	$0.967008\pi$
$264$		0			0
$265$		−	1.24292i		−	0.0763522i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−17.2638			−1.05259			−0.526297	−	0.850301i	$-0.676420\pi$
$269$	−17.2638			−1.05259			−0.526297	+	0.850301i	$0.676420\pi$
$270$		0			0
$271$	−7.12068			−0.432551			−0.216275	−	0.976332i	$-0.569391\pi$
$271$	−7.12068			−0.432551			−0.216275	+	0.976332i	$0.569391\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			6.12445i			0.369318i
$276$		0			0
$277$	8.70258			0.522887			0.261444	−	0.965219i	$-0.415801\pi$
$277$	8.70258			0.522887			0.261444	+	0.965219i	$0.415801\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			30.6030i			1.82562i	0.408380	+	0.912812i	$0.366094\pi$
$281$			30.6030i			1.82562i	−0.408380	+	0.912812i	$0.633906\pi$
$282$		0			0
$283$	13.6553			0.811725			0.405863	−	0.913934i	$-0.366971\pi$
$283$	13.6553			0.811725			0.405863	+	0.913934i	$0.366971\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−11.3335	+	24.2397i	−0.668996	+	1.43082i
$288$		0			0
$289$	−23.1133			−1.35961
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	8.27575i		−	0.483474i	−0.970342	−	0.241737i	$-0.922283\pi$
$293$		−	8.27575i		−	0.483474i	0.970342	−	0.241737i	$-0.0777172\pi$
$294$		0			0
$295$	4.31806			0.251407
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	6.32973i		−	0.366058i
$300$		0			0
$301$			49.6295i			2.86060i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.78304			−0.502915
$306$		0			0
$307$	8.58915			0.490209			0.245104	−	0.969497i	$-0.421178\pi$
$307$	8.58915			0.490209			0.245104	+	0.969497i	$0.421178\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			13.0528i			0.740157i	0.929001	+	0.370078i	$0.120669\pi$
$311$			13.0528i			0.740157i	−0.929001	+	0.370078i	$0.879331\pi$
$312$		0			0
$313$		−	9.83998i		−	0.556189i	−0.960554	−	0.278094i	$-0.910297\pi$
$313$		−	9.83998i		−	0.556189i	0.960554	−	0.278094i	$-0.0897028\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			0.168441i			0.00946057i	0.999989	+	0.00473028i	$0.00150570\pi$
$317$			0.168441i			0.00946057i	−0.999989	+	0.00473028i	$0.998494\pi$
$318$		0			0
$319$	−6.83051			−0.382435
$320$		0			0
$321$		0			0
$322$		0			0
$323$	30.7486			1.71090
$324$		0			0
$325$			19.3617i			1.07399i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	51.9593			2.86461
$330$		0			0
$331$			11.9490i			0.656776i	0.944543	+	0.328388i	$0.106505\pi$
$331$			11.9490i			0.656776i	−0.944543	+	0.328388i	$0.893495\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			8.59683i			0.469695i
$336$		0			0
$337$	10.5125			0.572650			0.286325	−	0.958133i	$-0.407566\pi$
$337$	10.5125			0.572650			0.286325	+	0.958133i	$0.407566\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	7.31415i		−	0.396083i
$342$		0			0
$343$		−	14.4745i		−	0.781547i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			7.49974i			0.402607i	0.979529	+	0.201303i	$0.0645177\pi$
$347$			7.49974i			0.402607i	−0.979529	+	0.201303i	$0.935482\pi$
$348$		0			0
$349$	−2.14287			−0.114705			−0.0573527	−	0.998354i	$-0.518266\pi$
$349$	−2.14287			−0.114705			−0.0573527	+	0.998354i	$0.518266\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	11.2473			0.598636			0.299318	−	0.954153i	$-0.403241\pi$
$353$	11.2473			0.598636			0.299318	+	0.954153i	$0.403241\pi$
$354$		0			0
$355$		−	0.339202i		−	0.0180030i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−25.7592			−1.35952			−0.679758	−	0.733436i	$-0.737915\pi$
$359$	−25.7592			−1.35952			−0.679758	+	0.733436i	$0.737915\pi$
$360$		0			0
$361$	−4.57022			−0.240538
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−10.9178			−0.571466
$366$		0			0
$367$	−25.4769			−1.32988			−0.664942	−	0.746895i	$-0.731544\pi$
$367$	−25.4769			−1.32988			−0.664942	+	0.746895i	$0.731544\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.77601			−0.351793
$372$		0			0
$373$	−10.0339			−0.519534			−0.259767	−	0.965671i	$-0.583646\pi$
$373$	−10.0339			−0.519534			−0.259767	+	0.965671i	$0.583646\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−21.5938			−1.11214
$378$		0			0
$379$	22.3796			1.14956			0.574781	−	0.818308i	$-0.305087\pi$
$379$	22.3796			1.14956			0.574781	+	0.818308i	$0.305087\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			27.4186i			1.40103i	0.713640	+	0.700513i	$0.247045\pi$
$383$			27.4186i			1.40103i	−0.713640	+	0.700513i	$0.752955\pi$
$384$		0			0
$385$	−4.44628			−0.226603
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.2757			−0.774511			−0.387256	−	0.921972i	$-0.626577\pi$
$389$	−15.2757			−0.774511			−0.387256	+	0.921972i	$0.626577\pi$
$390$		0			0
$391$		−	9.13613i		−	0.462034i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	9.83080i		−	0.494641i
$396$		0			0
$397$			6.77578i			0.340067i	0.985438	+	0.170033i	$0.0543876\pi$
$397$			6.77578i			0.340067i	−0.985438	+	0.170033i	$0.945612\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−35.2250			−1.75905			−0.879527	−	0.475849i	$-0.842141\pi$
$401$	−35.2250			−1.75905			−0.879527	+	0.475849i	$0.842141\pi$
$402$		0			0
$403$		−	23.1228i		−	1.15183i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			8.90047i			0.441180i
$408$		0			0
$409$	13.0722			0.646377			0.323188	−	0.946335i	$-0.395245\pi$
$409$	13.0722			0.646377			0.323188	+	0.946335i	$0.395245\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	23.5406i		−	1.15836i
$414$		0			0
$415$	5.78171			0.283813
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−15.8582			−0.774722			−0.387361	−	0.921928i	$-0.626613\pi$
$419$	−15.8582			−0.774722			−0.387361	+	0.921928i	$0.626613\pi$
$420$		0			0
$421$			18.6770i			0.910261i	0.890425	+	0.455131i	$0.150407\pi$
$421$			18.6770i			0.910261i	−0.890425	+	0.455131i	$0.849593\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			27.9460i			1.35558i
$426$		0			0
$427$			47.8823i			2.31719i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	40.6252			1.95685			0.978424	−	0.206606i	$-0.0662417\pi$
$431$	40.6252			1.95685			0.978424	+	0.206606i	$0.0662417\pi$
$432$		0			0
$433$	4.36116			0.209584			0.104792	−	0.994494i	$-0.466582\pi$
$433$	4.36116			0.209584			0.104792	+	0.994494i	$0.466582\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			7.00326i			0.335011i
$438$		0			0
$439$			6.33246i			0.302232i	0.988516	+	0.151116i	$0.0482867\pi$
$439$			6.33246i			0.302232i	−0.988516	+	0.151116i	$0.951713\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−8.92929			−0.424243			−0.212122	−	0.977243i	$-0.568037\pi$
$443$	−8.92929			−0.424243			−0.212122	+	0.977243i	$0.568037\pi$
$444$		0			0
$445$		−	9.63314i		−	0.456655i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−39.8731			−1.88173			−0.940864	−	0.338785i	$-0.889984\pi$
$449$	−39.8731			−1.88173			−0.940864	+	0.338785i	$0.889984\pi$
$450$		0			0
$451$	3.76433	−	8.05101i	0.177256	−	0.379107i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−14.0563			−0.658972
$456$		0			0
$457$		−	12.2066i		−	0.571001i	−0.958379	−	0.285501i	$-0.907840\pi$
$457$		−	12.2066i		−	0.571001i	0.958379	−	0.285501i	$-0.0921599\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−0.696879			−0.0324569			−0.0162284	−	0.999868i	$-0.505166\pi$
$461$	−0.696879			−0.0324569			−0.0162284	+	0.999868i	$0.505166\pi$
$462$		0			0
$463$			29.4940i			1.37070i	0.728212	+	0.685352i	$0.240352\pi$
$463$			29.4940i			1.37070i	−0.728212	+	0.685352i	$0.759648\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	21.9866			1.01742			0.508710	−	0.860938i	$-0.330123\pi$
$467$	21.9866			1.01742			0.508710	+	0.860938i	$0.330123\pi$
$468$		0			0
$469$	46.8671			2.16412
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	16.4841i		−	0.757938i
$474$		0			0
$475$		−	21.4219i		−	0.982903i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	33.0889i		−	1.51187i	−0.654647	−	0.755935i	$-0.727183\pi$
$479$		−	33.0889i		−	1.51187i	0.654647	−	0.755935i	$-0.272817\pi$
$480$		0			0
$481$			28.1377i			1.28297i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			0.220731i			0.0100229i
$486$		0			0
$487$	−27.1561			−1.23056			−0.615280	−	0.788308i	$-0.710957\pi$
$487$	−27.1561			−1.23056			−0.615280	+	0.788308i	$0.710957\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7.09458			−0.320174			−0.160087	−	0.987103i	$-0.551177\pi$
$491$	−7.09458			−0.320174			−0.160087	+	0.987103i	$0.551177\pi$
$492$		0			0
$493$	−31.1678			−1.40373
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.84922			−0.0829488
$498$		0			0
$499$		−	5.40317i		−	0.241879i	−0.992660	−	0.120940i	$-0.961409\pi$
$499$		−	5.40317i		−	0.241879i	0.992660	−	0.120940i	$-0.0385907\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			23.1421i			1.03186i	0.856632	+	0.515928i	$0.172553\pi$
$503$			23.1421i			1.03186i	−0.856632	+	0.515928i	$0.827447\pi$
$504$		0			0
$505$			5.05258i			0.224837i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	34.8565i		−	1.54499i	−0.635023	−	0.772493i	$-0.719009\pi$
$509$		−	34.8565i		−	1.54499i	0.635023	−	0.772493i	$-0.280991\pi$
$510$		0			0
$511$			59.5205i			2.63303i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	11.0941			0.488865
$516$		0			0
$517$	−17.2579			−0.759000
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	39.9137i		−	1.74865i	−0.485340	−	0.874326i	$-0.661304\pi$
$521$		−	39.9137i		−	1.74865i	0.485340	−	0.874326i	$-0.338696\pi$
$522$		0			0
$523$	39.6290			1.73286			0.866428	−	0.499303i	$-0.166410\pi$
$523$	39.6290			1.73286			0.866428	+	0.499303i	$0.166410\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	33.3746i		−	1.45382i
$528$		0			0
$529$	−20.9192			−0.909529
$530$		0			0
$531$		0			0
$532$		0			0
$533$	11.9005	−	25.4523i	0.515466	−	1.10246i
$534$		0			0
$535$	−14.0100			−0.605704
$536$		0			0
$537$		0			0
$538$		0			0
$539$			14.5236i			0.625577i
$540$		0			0
$541$	29.5829			1.27187			0.635935	−	0.771743i	$-0.280615\pi$
$541$	29.5829			1.27187			0.635935	+	0.771743i	$0.280615\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	2.48754i		−	0.106555i
$546$		0			0
$547$			22.2101i			0.949635i	0.880084	+	0.474817i	$0.157486\pi$
$547$			22.2101i			0.949635i	−0.880084	+	0.474817i	$0.842514\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	23.8915			1.01781
$552$		0			0
$553$	−53.5943			−2.27906
$554$		0			0
$555$		0			0
$556$		0			0
$557$			25.5090i			1.08085i	0.841392	+	0.540425i	$0.181737\pi$
$557$			25.5090i			1.08085i	−0.841392	+	0.540425i	$0.818263\pi$
$558$		0			0
$559$		−	52.1122i		−	2.20411i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			36.5501i			1.54040i	0.637801	+	0.770202i	$0.279844\pi$
$563$			36.5501i			1.54040i	−0.637801	+	0.770202i	$0.720156\pi$
$564$		0			0
$565$	−5.85840			−0.246465
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−44.0683			−1.84744			−0.923719	−	0.383071i	$-0.874866\pi$
$569$	−44.0683			−1.84744			−0.923719	+	0.383071i	$0.874866\pi$
$570$		0			0
$571$			13.0428i			0.545825i	0.962039	+	0.272913i	$0.0879870\pi$
$571$			13.0428i			0.545825i	−0.962039	+	0.272913i	$0.912013\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.36493			−0.265436
$576$		0			0
$577$		−	34.3699i		−	1.43084i	−0.698696	−	0.715418i	$-0.746236\pi$
$577$		−	34.3699i		−	1.43084i	0.698696	−	0.715418i	$-0.253764\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	31.5200i		−	1.30767i
$582$		0			0
$583$	2.25060			0.0932103
$584$		0			0
$585$		0			0
$586$		0			0
$587$			0.0935525i			0.00386132i	0.999998	+	0.00193066i	$0.000614549\pi$
$587$			0.0935525i			0.00386132i	−0.999998	+	0.00193066i	$0.999385\pi$
$588$		0			0
$589$			25.5832i			1.05414i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			3.96420i			0.162790i	0.996682	+	0.0813952i	$0.0259376\pi$
$593$			3.96420i			0.162790i	−0.996682	+	0.0813952i	$0.974062\pi$
$594$		0			0
$595$	−20.2885			−0.831746
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−41.4202			−1.69238			−0.846191	−	0.532879i	$-0.821110\pi$
$599$	−41.4202			−1.69238			−0.846191	+	0.532879i	$0.821110\pi$
$600$		0			0
$601$			6.61123i			0.269678i	0.990868	+	0.134839i	$0.0430517\pi$
$601$			6.61123i			0.269678i	−0.990868	+	0.134839i	$0.956948\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.95520			−0.282769
$606$		0			0
$607$	0.348670			0.0141521			0.00707605	−	0.999975i	$-0.497748\pi$
$607$	0.348670			0.0141521			0.00707605	+	0.999975i	$0.497748\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−54.5585			−2.20720
$612$		0			0
$613$	−6.94794			−0.280625			−0.140312	−	0.990107i	$-0.544811\pi$
$613$	−6.94794			−0.280625			−0.140312	+	0.990107i	$0.544811\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.06071			0.0829612			0.0414806	−	0.999139i	$-0.486793\pi$
$617$	2.06071			0.0829612			0.0414806	+	0.999139i	$0.486793\pi$
$618$		0			0
$619$	−16.1222			−0.648008			−0.324004	−	0.946056i	$-0.605029\pi$
$619$	−16.1222			−0.648008			−0.324004	+	0.946056i	$0.605029\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−52.5167			−2.10404
$624$		0			0
$625$	16.5314			0.661256
$626$		0			0
$627$		0			0
$628$		0			0
$629$			40.6130i			1.61935i
$630$		0			0
$631$	15.5140			0.617603			0.308802	−	0.951126i	$-0.400072\pi$
$631$	15.5140			0.617603			0.308802	+	0.951126i	$0.400072\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8.30666			−0.329640
$636$		0			0
$637$			45.9146i			1.81920i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	17.3348i		−	0.684683i	−0.939576	−	0.342341i	$-0.888780\pi$
$641$		−	17.3348i		−	0.684683i	0.939576	−	0.342341i	$-0.111220\pi$
$642$		0			0
$643$			2.56696i			0.101231i	0.998718	+	0.0506155i	$0.0161183\pi$
$643$			2.56696i			0.101231i	−0.998718	+	0.0506155i	$0.983882\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.99297			−0.0783519			−0.0391759	−	0.999232i	$-0.512473\pi$
$647$	−1.99297			−0.0783519			−0.0391759	+	0.999232i	$0.512473\pi$
$648$		0			0
$649$			7.81884i			0.306916i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			6.03608i			0.236210i	0.993001	+	0.118105i	$0.0376820\pi$
$653$			6.03608i			0.236210i	−0.993001	+	0.118105i	$0.962318\pi$
$654$		0			0
$655$	−9.91475			−0.387401
$656$		0			0
$657$		0			0
$658$		0			0
$659$			6.24891i			0.243423i	0.992566	+	0.121711i	$0.0388382\pi$
$659$			6.24891i			0.243423i	−0.992566	+	0.121711i	$0.961162\pi$
$660$		0			0
$661$	36.6209			1.42439			0.712195	−	0.701982i	$-0.247701\pi$
$661$	36.6209			1.42439			0.712195	+	0.701982i	$0.247701\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	15.5520			0.603082
$666$		0			0
$667$		−	7.09872i		−	0.274864i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	15.9037i		−	0.613956i
$672$		0			0
$673$		−	27.2852i		−	1.05177i	−0.850556	−	0.525884i	$-0.823735\pi$
$673$		−	27.2852i		−	1.05177i	0.850556	−	0.525884i	$-0.176265\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−25.3734			−0.975177			−0.487589	−	0.873073i	$-0.662123\pi$
$677$	−25.3734			−0.975177			−0.487589	+	0.873073i	$0.662123\pi$
$678$		0			0
$679$	1.20336			0.0461806
$680$		0			0
$681$		0			0
$682$		0			0
$683$			19.1432i			0.732493i	0.930518	+	0.366246i	$0.119357\pi$
$683$			19.1432i			0.732493i	−0.930518	+	0.366246i	$0.880643\pi$
$684$		0			0
$685$		−	14.9137i		−	0.569823i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7.11498			0.271059
$690$		0			0
$691$		−	13.8888i		−	0.528354i	−0.964474	−	0.264177i	$-0.914900\pi$
$691$		−	13.8888i		−	0.528354i	0.964474	−	0.264177i	$-0.0851004\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.82326			0.0691601
$696$		0			0
$697$	17.1767	−	36.7370i	0.650616	−	1.39151i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.04282			0.114926			0.0574629	−	0.998348i	$-0.481699\pi$
$701$	3.04282			0.114926			0.0574629	+	0.998348i	$0.481699\pi$
$702$		0			0
$703$		−	31.1317i		−	1.17415i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	27.5450			1.03594
$708$		0			0
$709$			40.9737i			1.53880i	0.638768	+	0.769399i	$0.279444\pi$
$709$			40.9737i			1.53880i	−0.638768	+	0.769399i	$0.720556\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	7.60135			0.284673
$714$		0			0
$715$	4.66871			0.174600
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	31.9017i		−	1.18973i	−0.803824	−	0.594867i	$-0.797205\pi$
$719$		−	31.9017i		−	1.18973i	0.803824	−	0.594867i	$-0.202795\pi$
$720$		0			0
$721$		−	60.4815i		−	2.25245i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			21.7139i			0.806434i
$726$		0			0
$727$			0.341182i			0.0126537i	0.999980	+	0.00632686i	$0.00201392\pi$
$727$			0.341182i			0.0126537i	−0.999980	+	0.00632686i	$0.997986\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	75.2171i		−	2.78201i
$732$		0			0
$733$	24.3619			0.899828			0.449914	−	0.893072i	$-0.351455\pi$
$733$	24.3619			0.899828			0.449914	+	0.893072i	$0.351455\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−15.5665			−0.573401
$738$		0			0
$739$	−5.79693			−0.213243			−0.106622	−	0.994300i	$-0.534003\pi$
$739$	−5.79693			−0.213243			−0.106622	+	0.994300i	$0.534003\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6.41113			0.235202			0.117601	−	0.993061i	$-0.462480\pi$
$743$	6.41113			0.235202			0.117601	+	0.993061i	$0.462480\pi$
$744$		0			0
$745$			9.11498i			0.333947i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			76.3779i			2.79079i
$750$		0			0
$751$		−	5.58841i		−	0.203924i	−0.994788	−	0.101962i	$-0.967488\pi$
$751$		−	5.58841i		−	0.203924i	0.994788	−	0.101962i	$-0.0325120\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1.66857i			0.0607255i
$756$		0			0
$757$			26.0466i			0.946680i	0.880880	+	0.473340i	$0.156952\pi$
$757$			26.0466i			0.946680i	−0.880880	+	0.473340i	$0.843048\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.73372			0.0990973			0.0495486	−	0.998772i	$-0.484222\pi$
$761$	2.73372			0.0990973			0.0495486	+	0.998772i	$0.484222\pi$
$762$		0			0
$763$	−13.5613			−0.490951
$764$		0			0
$765$		0			0
$766$		0			0
$767$			24.7183i			0.892525i
$768$		0			0
$769$	−4.54505			−0.163899			−0.0819494	−	0.996636i	$-0.526115\pi$
$769$	−4.54505			−0.163899			−0.0819494	+	0.996636i	$0.526115\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			7.54139i			0.271245i	0.990761	+	0.135623i	$0.0433034\pi$
$773$			7.54139i			0.271245i	−0.990761	+	0.135623i	$0.956697\pi$
$774$		0			0
$775$	−23.2513			−0.835213
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−13.1667	+	28.1605i	−0.471748	+	1.00896i
$780$		0			0
$781$	0.614204			0.0219779
$782$		0			0
$783$		0			0
$784$		0			0
$785$			9.43548i			0.336767i
$786$		0			0
$787$	20.1364			0.717784			0.358892	−	0.933379i	$-0.383155\pi$
$787$	20.1364			0.717784			0.358892	+	0.933379i	$0.383155\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			31.9381i			1.13559i
$792$		0			0
$793$		−	50.2776i		−	1.78541i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−21.2106			−0.751318			−0.375659	−	0.926758i	$-0.622584\pi$
$797$	−21.2106			−0.751318			−0.375659	+	0.926758i	$0.622584\pi$
$798$		0			0
$799$	−78.7480			−2.78590
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	19.7693i		−	0.697642i
$804$		0			0
$805$		−	4.62087i		−	0.162864i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	28.3858i		−	0.997991i	−0.866604	−	0.498996i	$-0.833702\pi$
$809$		−	28.3858i		−	0.997991i	0.866604	−	0.498996i	$-0.166298\pi$
$810$		0			0
$811$	−17.6598			−0.620118			−0.310059	−	0.950717i	$-0.600349\pi$
$811$	−17.6598			−0.620118			−0.310059	+	0.950717i	$0.600349\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.00208			−0.315329
$816$		0			0
$817$			57.6573i			2.01717i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−40.7865			−1.42346			−0.711730	−	0.702454i	$-0.752088\pi$
$821$	−40.7865			−1.42346			−0.711730	+	0.702454i	$0.752088\pi$
$822$		0			0
$823$			36.7880i			1.28235i	0.767395	+	0.641174i	$0.221552\pi$
$823$			36.7880i			1.28235i	−0.767395	+	0.641174i	$0.778448\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			16.4401i			0.571677i	0.958278	+	0.285839i	$0.0922721\pi$
$827$			16.4401i			0.571677i	−0.958278	+	0.285839i	$0.907728\pi$
$828$		0			0
$829$	−44.4460			−1.54367			−0.771837	−	0.635820i	$-0.780662\pi$
$829$	−44.4460			−1.54367			−0.771837	+	0.635820i	$0.780662\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			66.2716i			2.29618i
$834$		0			0
$835$		−	0.227989i		−	0.00788989i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			3.54307i			0.122321i	0.998128	+	0.0611603i	$0.0194801\pi$
$839$			3.54307i			0.122321i	−0.998128	+	0.0611603i	$0.980520\pi$
$840$		0			0
$841$	4.78281			0.164924
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.79443			0.164934
$846$		0			0
$847$			37.9175i			1.30286i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−9.24995			−0.317084
$852$		0			0
$853$	−17.2387			−0.590241			−0.295121	−	0.955460i	$-0.595360\pi$
$853$	−17.2387			−0.590241			−0.295121	+	0.955460i	$0.595360\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	16.6605			0.569112			0.284556	−	0.958659i	$-0.408154\pi$
$857$	16.6605			0.569112			0.284556	+	0.958659i	$0.408154\pi$
$858$		0			0
$859$	48.5729			1.65729			0.828644	−	0.559776i	$-0.189113\pi$
$859$	48.5729			1.65729			0.828644	+	0.559776i	$0.189113\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.873106			−0.0297209			−0.0148604	−	0.999890i	$-0.504730\pi$
$863$	−0.873106			−0.0297209			−0.0148604	+	0.999890i	$0.504730\pi$
$864$		0			0
$865$	10.4577			0.355572
$866$		0			0
$867$		0			0
$868$		0			0
$869$	17.8009			0.603855
$870$		0			0
$871$	−49.2116			−1.66747
$872$		0			0
$873$		0			0
$874$		0			0
$875$			30.1513i			1.01930i
$876$		0			0
$877$	−7.46935			−0.252222			−0.126111	−	0.992016i	$-0.540250\pi$
$877$	−7.46935			−0.252222			−0.126111	+	0.992016i	$0.540250\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	44.6257			1.50348			0.751740	−	0.659460i	$-0.229215\pi$
$881$	44.6257			1.50348			0.751740	+	0.659460i	$0.229215\pi$
$882$		0			0
$883$		−	31.2627i		−	1.05207i	−0.850462	−	0.526036i	$-0.823678\pi$
$883$		−	31.2627i		−	1.05207i	0.850462	−	0.526036i	$-0.176322\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	5.31613i		−	0.178498i	−0.996009	−	0.0892491i	$-0.971553\pi$
$887$		−	5.31613i		−	0.178498i	0.996009	−	0.0892491i	$-0.0284467\pi$
$888$		0			0
$889$			45.2852i			1.51882i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	60.3639			2.02000
$894$		0			0
$895$			9.39540i			0.314054i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	25.9319i		−	0.864877i
$900$		0			0
$901$	10.2695			0.342128
$902$		0			0
$903$		0			0
$904$		0			0
$905$			6.44599i			0.214272i
$906$		0			0
$907$	−18.8112			−0.624614			−0.312307	−	0.949981i	$-0.601102\pi$
$907$	−18.8112			−0.624614			−0.312307	+	0.949981i	$0.601102\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−45.5803			−1.51014			−0.755072	−	0.655642i	$-0.772398\pi$
$911$	−45.5803			−1.51014			−0.755072	+	0.655642i	$0.772398\pi$
$912$		0			0
$913$			10.4691i			0.346477i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			54.0520i			1.78495i
$918$		0			0
$919$		−	4.56719i		−	0.150658i	−0.997159	−	0.0753289i	$-0.975999\pi$
$919$		−	4.56719i		−	0.150658i	0.997159	−	0.0753289i	$-0.0240007\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.94173			0.0639127
$924$		0			0
$925$	28.2942			0.930307
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1.10028i		−	0.0360991i	−0.999837	−	0.0180495i	$-0.994254\pi$
$929$		−	1.10028i		−	0.0360991i	0.999837	−	0.0180495i	$-0.00574566\pi$
$930$		0			0
$931$		−	50.8002i		−	1.66491i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.73866			0.220378
$936$		0			0
$937$		−	23.7820i		−	0.776924i	−0.921465	−	0.388462i	$-0.873006\pi$
$937$		−	23.7820i		−	0.776924i	0.921465	−	0.388462i	$-0.126994\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	50.0967			1.63310			0.816552	−	0.577272i	$-0.195883\pi$
$941$	50.0967			1.63310			0.816552	+	0.577272i	$0.195883\pi$
$942$		0			0
$943$	8.36714	+	3.91214i	0.272472	+	0.127397i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	57.6640			1.87383			0.936915	−	0.349558i	$-0.113668\pi$
$947$	57.6640			1.87383			0.936915	+	0.349558i	$0.113668\pi$
$948$		0			0
$949$		−	62.4980i		−	2.02877i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−14.7456			−0.477658			−0.238829	−	0.971062i	$-0.576764\pi$
$953$	−14.7456			−0.477658			−0.238829	+	0.971062i	$0.576764\pi$
$954$		0			0
$955$		−	1.29615i		−	0.0419424i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−81.3047			−2.62546
$960$		0			0
$961$	−3.23199			−0.104258
$962$		0			0
$963$		0			0
$964$		0			0
$965$			16.4728i			0.530277i
$966$		0			0
$967$			40.8795i			1.31460i	0.753630	+	0.657299i	$0.228301\pi$
$967$			40.8795i			1.31460i	−0.753630	+	0.657299i	$0.771699\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			29.2887i			0.939919i	0.882688	+	0.469960i	$0.155732\pi$
$971$			29.2887i			0.939919i	−0.882688	+	0.469960i	$0.844268\pi$
$972$		0			0
$973$		−	9.93980i		−	0.318655i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			26.0135i			0.832245i	0.909309	+	0.416122i	$0.136611\pi$
$977$			26.0135i			0.832245i	−0.909309	+	0.416122i	$0.863389\pi$
$978$		0			0
$979$	17.4430			0.557481
$980$		0			0
$981$		0			0
$982$		0			0
$983$	52.7288			1.68179			0.840893	−	0.541201i	$-0.182030\pi$
$983$	52.7288			1.68179			0.840893	+	0.541201i	$0.182030\pi$
$984$		0			0
$985$	−11.9515			−0.380806
$986$		0			0
$987$		0			0
$988$		0			0
$989$	17.1313			0.544744
$990$		0			0
$991$		−	2.14584i		−	0.0681650i	−0.999419	−	0.0340825i	$-0.989149\pi$
$991$		−	2.14584i		−	0.0681650i	0.999419	−	0.0340825i	$-0.0108509\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			4.05804i			0.128649i
$996$		0			0
$997$		−	37.6434i		−	1.19218i	−0.802919	−	0.596089i	$-0.796721\pi$
$997$		−	37.6434i		−	1.19218i	0.802919	−	0.596089i	$-0.203279\pi$
$998$		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.2377.6		8
3.2	odd	2		984.2.j.b.409.2	✓	8
12.11	even	2		1968.2.j.f.1393.6		8
41.40	even	2	inner	2952.2.j.d.2377.5		8
123.122	odd	2		984.2.j.b.409.6	yes	8
492.491	even	2		1968.2.j.f.1393.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
984.2.j.b.409.2	✓	8	3.2	odd	2
984.2.j.b.409.6	yes	8	123.122	odd	2
1968.2.j.f.1393.2		8	492.491	even	2
1968.2.j.f.1393.6		8	12.11	even	2
2952.2.j.d.2377.5		8	41.40	even	2	inner
2952.2.j.d.2377.6		8	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.766545 q^{5} +4.17895i q^{7} +O(q^{10})\)	\(q-0.766545 q^{5} +4.17895i q^{7} -1.38801i q^{11} -4.38801i q^{13} -6.33351i q^{17} +4.85492i q^{19} +1.44251 q^{23} -4.41241 q^{25} -4.92110i q^{29} +5.26954 q^{31} -3.20336i q^{35} -6.41241 q^{37} +(5.80042 + 2.71204i) q^{41} +11.8761 q^{43} -12.4336i q^{47} -10.4637 q^{49} +1.62146i q^{53} +1.06397i q^{55} -5.63314 q^{59} +11.4580 q^{61} +3.36360i q^{65} -11.2150i q^{67} +0.442508i q^{71} +14.2429 q^{73} +5.80042 q^{77} +12.8248i q^{79} -7.54256 q^{83} +4.85492i q^{85} +12.5670i q^{89} +18.3373 q^{91} -3.72151i q^{95} -0.287956i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 10 q^{5}+O(q^{10}) \) 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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