LMFDB - Embedded newform 4140.2.f.d.829.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(829,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("4140.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.f (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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.8
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.d.829.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.06710 + 1.96502i) q^{5} -3.96013i q^{7} +0.632117 q^{11} +2.50635i q^{13} -5.58129i q^{17} -5.75876 q^{19} +1.00000i q^{23} +(-2.72259 - 4.19374i) q^{25} +4.05521 q^{29} -0.852012 q^{31} +(7.78172 + 4.22585i) q^{35} +8.51300i q^{37} +6.20651 q^{41} -7.38322i q^{43} +5.59349i q^{47} -8.68259 q^{49} +12.3519i q^{53} +(-0.674532 + 1.24212i) q^{55} -11.4310 q^{59} -14.7790 q^{61} +(-4.92503 - 2.67453i) q^{65} +0.719943i q^{67} +5.38501 q^{71} +10.6695i q^{73} -2.50326i q^{77} -12.0268 q^{79} +1.99379i q^{83} +(10.9673 + 5.95579i) q^{85} -6.30598 q^{89} +9.92548 q^{91} +(6.14518 - 11.3161i) q^{95} +0.585493i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 8 q^{19} - 8 q^{25} - 12 q^{31} - 28 q^{49} - 16 q^{55} - 16 q^{61} + 8 q^{79} + 12 q^{85} - 16 q^{91}+O(q^{100}) $ 24 * q + 8 * q^19 - 8 * q^25 - 12 * q^31 - 28 * q^49 - 16 * q^55 - 16 * q^61 + 8 * q^79 + 12 * q^85 - 16 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	−1.06710	+	1.96502i	−0.477222	+	0.878783i
$5$	−1.06710	+	1.96502i	−0.477222	+	0.878783i
$6$		0			0
$7$		−	3.96013i		−	1.49679i	−0.663255	−	0.748393i	$-0.730826\pi$
$7$		−	3.96013i		−	1.49679i	0.663255	−	0.748393i	$-0.269174\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.632117			0.190590			0.0952952	−	0.995449i	$-0.469621\pi$
$11$	0.632117			0.190590			0.0952952	+	0.995449i	$0.469621\pi$
$12$		0			0
$13$			2.50635i			0.695137i	0.937655	+	0.347569i	$0.112993\pi$
$13$			2.50635i			0.695137i	−0.937655	+	0.347569i	$0.887007\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.58129i		−	1.35366i	−0.736139	−	0.676830i	$-0.763353\pi$
$17$		−	5.58129i		−	1.35366i	0.736139	−	0.676830i	$-0.236647\pi$
$18$		0			0
$19$	−5.75876			−1.32115			−0.660576	−	0.750759i	$-0.729688\pi$
$19$	−5.75876			−1.32115			−0.660576	+	0.750759i	$0.729688\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	−2.72259	−	4.19374i	−0.544518	−	0.838749i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.05521			0.753034			0.376517	−	0.926410i	$-0.377122\pi$
$29$	4.05521			0.753034			0.376517	+	0.926410i	$0.377122\pi$
$30$		0			0
$31$	−0.852012			−0.153026			−0.0765129	−	0.997069i	$-0.524379\pi$
$31$	−0.852012			−0.153026			−0.0765129	+	0.997069i	$0.524379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	7.78172	+	4.22585i	1.31535	+	0.714300i
$36$		0			0
$37$			8.51300i			1.39953i	0.714374	+	0.699765i	$0.246712\pi$
$37$			8.51300i			1.39953i	−0.714374	+	0.699765i	$0.753288\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.20651			0.969295			0.484647	−	0.874710i	$-0.338948\pi$
$41$	6.20651			0.969295			0.484647	+	0.874710i	$0.338948\pi$
$42$		0			0
$43$		−	7.38322i		−	1.12593i	−0.826481	−	0.562965i	$-0.809660\pi$
$43$		−	7.38322i		−	1.12593i	0.826481	−	0.562965i	$-0.190340\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			5.59349i			0.815894i	0.913006	+	0.407947i	$0.133755\pi$
$47$			5.59349i			0.815894i	−0.913006	+	0.407947i	$0.866245\pi$
$48$		0			0
$49$	−8.68259			−1.24037
$50$		0			0
$51$		0			0
$52$		0			0
$53$			12.3519i			1.69667i	0.529461	+	0.848334i	$0.322394\pi$
$53$			12.3519i			1.69667i	−0.529461	+	0.848334i	$0.677606\pi$
$54$		0			0
$55$	−0.674532	+	1.24212i	−0.0909539	+	0.167488i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11.4310			−1.48819			−0.744096	−	0.668072i	$-0.767120\pi$
$59$	−11.4310			−1.48819			−0.744096	+	0.668072i	$0.767120\pi$
$60$		0			0
$61$	−14.7790			−1.89226			−0.946130	−	0.323787i	$-0.895044\pi$
$61$	−14.7790			−1.89226			−0.946130	+	0.323787i	$0.895044\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.92503	−	2.67453i	−0.610875	−	0.331735i
$66$		0			0
$67$			0.719943i			0.0879551i	0.999033	+	0.0439775i	$0.0140030\pi$
$67$			0.719943i			0.0879551i	−0.999033	+	0.0439775i	$0.985997\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.38501			0.639083			0.319542	−	0.947572i	$-0.396471\pi$
$71$	5.38501			0.639083			0.319542	+	0.947572i	$0.396471\pi$
$72$		0			0
$73$			10.6695i			1.24877i	0.781118	+	0.624383i	$0.214650\pi$
$73$			10.6695i			1.24877i	−0.781118	+	0.624383i	$0.785350\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.50326i		−	0.285273i
$78$		0			0
$79$	−12.0268			−1.35312			−0.676559	−	0.736388i	$-0.736530\pi$
$79$	−12.0268			−1.35312			−0.676559	+	0.736388i	$0.736530\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			1.99379i			0.218847i	0.993995	+	0.109424i	$0.0349005\pi$
$83$			1.99379i			0.218847i	−0.993995	+	0.109424i	$0.965100\pi$
$84$		0			0
$85$	10.9673	+	5.95579i	1.18957	+	0.645997i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.30598			−0.668432			−0.334216	−	0.942496i	$-0.608472\pi$
$89$	−6.30598			−0.668432			−0.334216	+	0.942496i	$0.608472\pi$
$90$		0			0
$91$	9.92548			1.04047
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.14518	−	11.3161i	0.630482	−	1.16101i
$96$		0			0
$97$			0.585493i			0.0594478i	0.999558	+	0.0297239i	$0.00946281\pi$
$97$			0.585493i			0.0594478i	−0.999558	+	0.0297239i	$0.990537\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	19.4840			1.93874			0.969368	−	0.245614i	$-0.0789897\pi$
$101$	19.4840			1.93874			0.969368	+	0.245614i	$0.0789897\pi$
$102$		0			0
$103$		−	6.86329i		−	0.676260i	−0.941099	−	0.338130i	$-0.890206\pi$
$103$		−	6.86329i		−	0.676260i	0.941099	−	0.338130i	$-0.109794\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	9.77096i		−	0.944595i	−0.881439	−	0.472297i	$-0.843425\pi$
$107$		−	9.77096i		−	0.944595i	0.881439	−	0.472297i	$-0.156575\pi$
$108$		0			0
$109$	0.762905			0.0730730			0.0365365	−	0.999332i	$-0.488367\pi$
$109$	0.762905			0.0730730			0.0365365	+	0.999332i	$0.488367\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	4.33952i		−	0.408228i	−0.978947	−	0.204114i	$-0.934569\pi$
$113$		−	4.33952i		−	0.408228i	0.978947	−	0.204114i	$-0.0654313\pi$
$114$		0			0
$115$	−1.96502	−	1.06710i	−0.183239	−	0.0995077i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−22.1026			−2.02614
$120$		0			0
$121$	−10.6004			−0.963675
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.1461	−	0.874794i	0.996934	−	0.0782439i
$126$		0			0
$127$			8.14070i			0.722370i	0.932494	+	0.361185i	$0.117628\pi$
$127$			8.14070i			0.722370i	−0.932494	+	0.361185i	$0.882372\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−12.6804			−1.10789			−0.553946	−	0.832553i	$-0.686879\pi$
$131$	−12.6804			−1.10789			−0.553946	+	0.832553i	$0.686879\pi$
$132$		0			0
$133$			22.8054i			1.97748i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			1.51434i			0.129379i	0.997905	+	0.0646896i	$0.0206057\pi$
$137$			1.51434i			0.129379i	−0.997905	+	0.0646896i	$0.979394\pi$
$138$		0			0
$139$	0.414154			0.0351281			0.0175640	−	0.999846i	$-0.494409\pi$
$139$	0.414154			0.0351281			0.0175640	+	0.999846i	$0.494409\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			1.58431i			0.132486i
$144$		0			0
$145$	−4.32732	+	7.96856i	−0.359364	+	0.661753i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−18.4315			−1.50996			−0.754982	−	0.655745i	$-0.772355\pi$
$149$	−18.4315			−1.50996			−0.754982	+	0.655745i	$0.772355\pi$
$150$		0			0
$151$	−7.77717			−0.632897			−0.316449	−	0.948610i	$-0.602491\pi$
$151$	−7.77717			−0.632897			−0.316449	+	0.948610i	$0.602491\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.909183	−	1.67422i	0.0730273	−	0.134476i
$156$		0			0
$157$			13.6685i			1.09086i	0.838155	+	0.545432i	$0.183634\pi$
$157$			13.6685i			1.09086i	−0.838155	+	0.545432i	$0.816366\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.96013			0.312102
$162$		0			0
$163$			23.5276i			1.84283i	0.388585	+	0.921413i	$0.372964\pi$
$163$			23.5276i			1.84283i	−0.388585	+	0.921413i	$0.627036\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	18.7427i		−	1.45035i	−0.688563	−	0.725177i	$-0.741758\pi$
$167$		−	18.7427i		−	1.45035i	0.688563	−	0.725177i	$-0.258242\pi$
$168$		0			0
$169$	6.71819			0.516784
$170$		0			0
$171$		0			0
$172$		0			0
$173$			12.4393i			0.945742i	0.881132	+	0.472871i	$0.156782\pi$
$173$			12.4393i			0.945742i	−0.881132	+	0.472871i	$0.843218\pi$
$174$		0			0
$175$	−16.6078	+	10.7818i	−1.25543	+	0.815028i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.97298			−0.670672			−0.335336	−	0.942099i	$-0.608850\pi$
$179$	−8.97298			−0.670672			−0.335336	+	0.942099i	$0.608850\pi$
$180$		0			0
$181$	5.68186			0.422329			0.211165	−	0.977451i	$-0.432274\pi$
$181$	5.68186			0.422329			0.211165	+	0.977451i	$0.432274\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−16.7282	−	9.08423i	−1.22988	−	0.667886i
$186$		0			0
$187$		−	3.52802i		−	0.257995i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.5612			−0.764182			−0.382091	−	0.924125i	$-0.624796\pi$
$191$	−10.5612			−0.764182			−0.382091	+	0.924125i	$0.624796\pi$
$192$		0			0
$193$		−	3.59916i		−	0.259073i	−0.991575	−	0.129537i	$-0.958651\pi$
$193$		−	3.59916i		−	0.259073i	0.991575	−	0.129537i	$-0.0413489\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			18.2896i			1.30308i	0.758614	+	0.651540i	$0.225877\pi$
$197$			18.2896i			1.30308i	−0.758614	+	0.651540i	$0.774123\pi$
$198$		0			0
$199$	−21.2131			−1.50376			−0.751879	−	0.659301i	$-0.770852\pi$
$199$	−21.2131			−1.50376			−0.751879	+	0.659301i	$0.770852\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	16.0591i		−	1.12713i
$204$		0			0
$205$	−6.62298	+	12.1959i	−0.462569	+	0.851800i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.64021			−0.251799
$210$		0			0
$211$	−4.26181			−0.293395			−0.146698	−	0.989181i	$-0.546864\pi$
$211$	−4.26181			−0.293395			−0.146698	+	0.989181i	$0.546864\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	14.5082	+	7.87864i	0.989448	+	0.537319i
$216$		0			0
$217$			3.37407i			0.229047i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	13.9887			0.940980
$222$		0			0
$223$		−	16.5506i		−	1.10831i	−0.832413	−	0.554155i	$-0.813041\pi$
$223$		−	16.5506i		−	1.10831i	0.832413	−	0.554155i	$-0.186959\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			12.5241i			0.831251i	0.909536	+	0.415625i	$0.136437\pi$
$227$			12.5241i			0.831251i	−0.909536	+	0.415625i	$0.863563\pi$
$228$		0			0
$229$	−6.21503			−0.410701			−0.205350	−	0.978689i	$-0.565833\pi$
$229$	−6.21503			−0.410701			−0.205350	+	0.978689i	$0.565833\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			6.05930i			0.396958i	0.980105	+	0.198479i	$0.0636002\pi$
$233$			6.05930i			0.396958i	−0.980105	+	0.198479i	$0.936400\pi$
$234$		0			0
$235$	−10.9913	−	5.96881i	−0.716993	−	0.389362i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	21.1468			1.36787			0.683936	−	0.729542i	$-0.260267\pi$
$239$	21.1468			1.36787			0.683936	+	0.729542i	$0.260267\pi$
$240$		0			0
$241$	−28.6397			−1.84485			−0.922423	−	0.386180i	$-0.873794\pi$
$241$	−28.6397			−1.84485			−0.922423	+	0.386180i	$0.873794\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.26520	−	17.0615i	0.591932	−	1.09002i
$246$		0			0
$247$		−	14.4335i		−	0.918382i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−20.1089			−1.26926			−0.634631	−	0.772815i	$-0.718848\pi$
$251$	−20.1089			−1.26926			−0.634631	+	0.772815i	$0.718848\pi$
$252$		0			0
$253$			0.632117i			0.0397408i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			18.1051i			1.12937i	0.825308	+	0.564683i	$0.191002\pi$
$257$			18.1051i			1.12937i	−0.825308	+	0.564683i	$0.808998\pi$
$258$		0			0
$259$	33.7126			2.09480
$260$		0			0
$261$		0			0
$262$		0			0
$263$			5.91089i			0.364481i	0.983254	+	0.182241i	$0.0583350\pi$
$263$			5.91089i			0.364481i	−0.983254	+	0.182241i	$0.941665\pi$
$264$		0			0
$265$	−24.2718	−	13.1808i	−1.49100	−	0.809688i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	9.06792			0.552881			0.276440	−	0.961031i	$-0.410845\pi$
$269$	9.06792			0.552881			0.276440	+	0.961031i	$0.410845\pi$
$270$		0			0
$271$	−7.62266			−0.463044			−0.231522	−	0.972830i	$-0.574370\pi$
$271$	−7.62266			−0.463044			−0.231522	+	0.972830i	$0.574370\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.72100	−	2.65094i	−0.103780	−	0.159857i
$276$		0			0
$277$		−	21.9421i		−	1.31837i	−0.751980	−	0.659186i	$-0.770901\pi$
$277$		−	21.9421i		−	1.31837i	0.751980	−	0.659186i	$-0.229099\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−15.7141			−0.937424			−0.468712	−	0.883351i	$-0.655282\pi$
$281$	−15.7141			−0.937424			−0.468712	+	0.883351i	$0.655282\pi$
$282$		0			0
$283$			1.95159i			0.116010i	0.998316	+	0.0580049i	$0.0184739\pi$
$283$			1.95159i			0.116010i	−0.998316	+	0.0580049i	$0.981526\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	24.5786i		−	1.45083i
$288$		0			0
$289$	−14.1507			−0.832397
$290$		0			0
$291$		0			0
$292$		0			0
$293$			27.7742i			1.62258i	0.584642	+	0.811292i	$0.301235\pi$
$293$			27.7742i			1.62258i	−0.584642	+	0.811292i	$0.698765\pi$
$294$		0			0
$295$	12.1981	−	22.4622i	0.710198	−	1.30780i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.50635			−0.144946
$300$		0			0
$301$	−29.2385			−1.68528
$302$		0			0
$303$		0			0
$304$		0			0
$305$	15.7707	−	29.0411i	0.903028	−	1.66289i
$306$		0			0
$307$			11.4925i			0.655912i	0.944693	+	0.327956i	$0.106360\pi$
$307$			11.4925i			0.655912i	−0.944693	+	0.327956i	$0.893640\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−32.7811			−1.85885			−0.929424	−	0.369014i	$-0.879696\pi$
$311$	−32.7811			−1.85885			−0.929424	+	0.369014i	$0.879696\pi$
$312$		0			0
$313$		−	15.7825i		−	0.892079i	−0.895013	−	0.446040i	$-0.852834\pi$
$313$		−	15.7825i		−	0.892079i	0.895013	−	0.446040i	$-0.147166\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	24.2612i		−	1.36264i	−0.731984	−	0.681321i	$-0.761406\pi$
$317$		−	24.2612i		−	1.36264i	0.731984	−	0.681321i	$-0.238594\pi$
$318$		0			0
$319$	2.56337			0.143521
$320$		0			0
$321$		0			0
$322$		0			0
$323$			32.1413i			1.78839i
$324$		0			0
$325$	10.5110	−	6.82378i	0.583046	−	0.378515i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	22.1509			1.22122
$330$		0			0
$331$	7.37500			0.405367			0.202683	−	0.979244i	$-0.435034\pi$
$331$	7.37500			0.405367			0.202683	+	0.979244i	$0.435034\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.41470	−	0.768252i	−0.0772934	−	0.0419741i
$336$		0			0
$337$			29.3276i			1.59758i	0.601611	+	0.798789i	$0.294526\pi$
$337$			29.3276i			1.59758i	−0.601611	+	0.798789i	$0.705474\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.538571			−0.0291652
$342$		0			0
$343$			6.66328i			0.359783i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			2.92590i			0.157070i	0.996911	+	0.0785352i	$0.0250243\pi$
$347$			2.92590i			0.157070i	−0.996911	+	0.0785352i	$0.974976\pi$
$348$		0			0
$349$	−30.2310			−1.61823			−0.809115	−	0.587650i	$-0.800053\pi$
$349$	−30.2310			−1.61823			−0.809115	+	0.587650i	$0.800053\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	16.1413i		−	0.859115i	−0.903039	−	0.429558i	$-0.858670\pi$
$353$		−	16.1413i		−	0.859115i	0.903039	−	0.429558i	$-0.141330\pi$
$354$		0			0
$355$	−5.74635	+	10.5816i	−0.304984	+	0.561615i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	21.8644			1.15396			0.576980	−	0.816759i	$-0.304231\pi$
$359$	21.8644			1.15396			0.576980	+	0.816759i	$0.304231\pi$
$360$		0			0
$361$	14.1634			0.745441
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−20.9657	−	11.3854i	−1.09739	−	0.595939i
$366$		0			0
$367$		−	5.14275i		−	0.268449i	−0.990951	−	0.134225i	$-0.957146\pi$
$367$		−	5.14275i		−	0.268449i	0.990951	−	0.134225i	$-0.0428544\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	48.9152			2.53955
$372$		0			0
$373$		−	6.85935i		−	0.355164i	−0.984106	−	0.177582i	$-0.943173\pi$
$373$		−	6.85935i		−	0.355164i	0.984106	−	0.177582i	$-0.0568275\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			10.1638i			0.523462i
$378$		0			0
$379$	33.4282			1.71709			0.858545	−	0.512739i	$-0.171369\pi$
$379$	33.4282			1.71709			0.858545	+	0.512739i	$0.171369\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			19.1557i			0.978812i	0.872056	+	0.489406i	$0.162786\pi$
$383$			19.1557i			0.978812i	−0.872056	+	0.489406i	$0.837214\pi$
$384$		0			0
$385$	4.91895	+	2.67123i	0.250693	+	0.136139i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−17.1962			−0.871882			−0.435941	−	0.899975i	$-0.643584\pi$
$389$	−17.1962			−0.871882			−0.435941	+	0.899975i	$0.643584\pi$
$390$		0			0
$391$	5.58129			0.282258
$392$		0			0
$393$		0			0
$394$		0			0
$395$	12.8338	−	23.6328i	0.645738	−	1.18910i
$396$		0			0
$397$		−	23.9509i		−	1.20206i	−0.799226	−	0.601031i	$-0.794757\pi$
$397$		−	23.9509i		−	1.20206i	0.799226	−	0.601031i	$-0.205243\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5.61069			0.280185			0.140092	−	0.990138i	$-0.455260\pi$
$401$	5.61069			0.280185			0.140092	+	0.990138i	$0.455260\pi$
$402$		0			0
$403$		−	2.13544i		−	0.106374i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			5.38121i			0.266737i
$408$		0			0
$409$	−28.1149			−1.39019			−0.695096	−	0.718917i	$-0.744638\pi$
$409$	−28.1149			−1.39019			−0.695096	+	0.718917i	$0.744638\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			45.2683i			2.22751i
$414$		0			0
$415$	−3.91784	−	2.12758i	−0.192319	−	0.104439i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.75937			−0.232510			−0.116255	−	0.993219i	$-0.537089\pi$
$419$	−4.75937			−0.232510			−0.116255	+	0.993219i	$0.537089\pi$
$420$		0			0
$421$	−16.7449			−0.816095			−0.408048	−	0.912961i	$-0.633790\pi$
$421$	−16.7449			−0.816095			−0.408048	+	0.912961i	$0.633790\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−23.4065	+	15.1956i	−1.13538	+	0.737093i
$426$		0			0
$427$			58.5268i			2.83231i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−11.2542			−0.542095			−0.271047	−	0.962566i	$-0.587370\pi$
$431$	−11.2542			−0.542095			−0.271047	+	0.962566i	$0.587370\pi$
$432$		0			0
$433$		−	9.66867i		−	0.464647i	−0.972639	−	0.232323i	$-0.925367\pi$
$433$		−	9.66867i		−	0.464647i	0.972639	−	0.232323i	$-0.0746328\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	5.75876i		−	0.275479i
$438$		0			0
$439$	14.2986			0.682433			0.341216	−	0.939985i	$-0.389161\pi$
$439$	14.2986			0.682433			0.341216	+	0.939985i	$0.389161\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	28.1209i		−	1.33607i	−0.744132	−	0.668033i	$-0.767137\pi$
$443$		−	28.1209i		−	1.33607i	0.744132	−	0.668033i	$-0.232863\pi$
$444$		0			0
$445$	6.72911	−	12.3914i	0.318991	−	0.587407i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	8.40234			0.396531			0.198266	−	0.980148i	$-0.436469\pi$
$449$	8.40234			0.396531			0.198266	+	0.980148i	$0.436469\pi$
$450$		0			0
$451$	3.92324			0.184738
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−10.5915	+	19.5037i	−0.496536	+	0.914349i
$456$		0			0
$457$			0.747155i			0.0349504i	0.999847	+	0.0174752i	$0.00556282\pi$
$457$			0.747155i			0.0349504i	−0.999847	+	0.0174752i	$0.994437\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.92909			0.462444			0.231222	−	0.972901i	$-0.425728\pi$
$461$	9.92909			0.462444			0.231222	+	0.972901i	$0.425728\pi$
$462$		0			0
$463$			6.05649i			0.281469i	0.990047	+	0.140734i	$0.0449464\pi$
$463$			6.05649i			0.281469i	−0.990047	+	0.140734i	$0.955054\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	12.3607i		−	0.571984i	−0.958232	−	0.285992i	$-0.907677\pi$
$467$		−	12.3607i		−	0.571984i	0.958232	−	0.285992i	$-0.0923231\pi$
$468$		0			0
$469$	2.85107			0.131650
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	4.66706i		−	0.214591i
$474$		0			0
$475$	15.6788	+	24.1508i	0.719391	+	1.10811i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	30.0536			1.37318			0.686591	−	0.727044i	$-0.259106\pi$
$479$	30.0536			1.37318			0.686591	+	0.727044i	$0.259106\pi$
$480$		0			0
$481$	−21.3366			−0.972865
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.15050	−	0.624780i	−0.0522417	−	0.0283698i
$486$		0			0
$487$		−	36.3158i		−	1.64563i	−0.568312	−	0.822813i	$-0.692403\pi$
$487$		−	36.3158i		−	1.64563i	0.568312	−	0.822813i	$-0.307597\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	30.5741			1.37979			0.689896	−	0.723909i	$-0.257656\pi$
$491$	30.5741			1.37979			0.689896	+	0.723909i	$0.257656\pi$
$492$		0			0
$493$		−	22.6333i		−	1.01935i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	21.3253i		−	0.956571i
$498$		0			0
$499$	−19.1175			−0.855819			−0.427909	−	0.903822i	$-0.640750\pi$
$499$	−19.1175			−0.855819			−0.427909	+	0.903822i	$0.640750\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			19.7346i			0.879924i	0.898016	+	0.439962i	$0.145008\pi$
$503$			19.7346i			0.879924i	−0.898016	+	0.439962i	$0.854992\pi$
$504$		0			0
$505$	−20.7914	+	38.2865i	−0.925207	+	1.70373i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	33.4190			1.48127			0.740636	−	0.671906i	$-0.234524\pi$
$509$	33.4190			1.48127			0.740636	+	0.671906i	$0.234524\pi$
$510$		0			0
$511$	42.2524			1.86914
$512$		0			0
$513$		0			0
$514$		0			0
$515$	13.4865	+	7.32382i	0.594286	+	0.322726i
$516$		0			0
$517$			3.53574i			0.155501i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−12.7444			−0.558343			−0.279172	−	0.960241i	$-0.590060\pi$
$521$	−12.7444			−0.558343			−0.279172	+	0.960241i	$0.590060\pi$
$522$		0			0
$523$			1.99335i			0.0871631i	0.999050	+	0.0435815i	$0.0138768\pi$
$523$			1.99335i			0.0871631i	−0.999050	+	0.0435815i	$0.986123\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			4.75532i			0.207145i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$			15.5557i			0.673793i
$534$		0			0
$535$	19.2001	+	10.4266i	0.830094	+	0.450781i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5.48841			−0.236403
$540$		0			0
$541$	38.0217			1.63468			0.817340	−	0.576156i	$-0.195448\pi$
$541$	38.0217			1.63468			0.817340	+	0.576156i	$0.195448\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−0.814096	+	1.49912i	−0.0348720	+	0.0642153i
$546$		0			0
$547$		−	4.14065i		−	0.177042i	−0.996074	−	0.0885208i	$-0.971786\pi$
$547$		−	4.14065i		−	0.177042i	0.996074	−	0.0885208i	$-0.0282140\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−23.3530			−0.994872
$552$		0			0
$553$			47.6276i			2.02533i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	6.26855i		−	0.265607i	−0.991142	−	0.132804i	$-0.957602\pi$
$557$		−	6.26855i		−	0.265607i	0.991142	−	0.132804i	$-0.0423979\pi$
$558$		0			0
$559$	18.5050			0.782677
$560$		0			0
$561$		0			0
$562$		0			0
$563$			45.8001i			1.93025i	0.261798	+	0.965123i	$0.415685\pi$
$563$			45.8001i			1.93025i	−0.261798	+	0.965123i	$0.584315\pi$
$564$		0			0
$565$	8.52723	+	4.63070i	0.358743	+	0.194815i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−39.3833			−1.65103			−0.825517	−	0.564377i	$-0.809116\pi$
$569$	−39.3833			−1.65103			−0.825517	+	0.564377i	$0.809116\pi$
$570$		0			0
$571$	26.6622			1.11578			0.557889	−	0.829916i	$-0.311612\pi$
$571$	26.6622			1.11578			0.557889	+	0.829916i	$0.311612\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.19374	−	2.72259i	0.174891	−	0.113540i
$576$		0			0
$577$			34.8227i			1.44969i	0.688914	+	0.724843i	$0.258088\pi$
$577$			34.8227i			1.44969i	−0.688914	+	0.724843i	$0.741912\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.89566			0.327567
$582$		0			0
$583$			7.80786i			0.323369i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			30.9195i			1.27619i	0.769959	+	0.638093i	$0.220276\pi$
$587$			30.9195i			1.27619i	−0.769959	+	0.638093i	$0.779724\pi$
$588$		0			0
$589$	4.90654			0.202170
$590$		0			0
$591$		0			0
$592$		0			0
$593$			14.0424i			0.576654i	0.957532	+	0.288327i	$0.0930990\pi$
$593$			14.0424i			0.576654i	−0.957532	+	0.288327i	$0.906901\pi$
$594$		0			0
$595$	23.5857	−	43.4320i	0.966919	−	1.78054i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−0.787601			−0.0321805			−0.0160903	−	0.999871i	$-0.505122\pi$
$599$	−0.787601			−0.0321805			−0.0160903	+	0.999871i	$0.505122\pi$
$600$		0			0
$601$	7.26851			0.296489			0.148244	−	0.988951i	$-0.452638\pi$
$601$	7.26851			0.296489			0.148244	+	0.988951i	$0.452638\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11.3117	−	20.8300i	0.459887	−	0.846861i
$606$		0			0
$607$		−	17.7055i		−	0.718645i	−0.933213	−	0.359323i	$-0.883008\pi$
$607$		−	17.7055i		−	0.718645i	0.933213	−	0.359323i	$-0.116992\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−14.0193			−0.567158
$612$		0			0
$613$			22.5129i			0.909286i	0.890674	+	0.454643i	$0.150233\pi$
$613$			22.5129i			0.909286i	−0.890674	+	0.454643i	$0.849767\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	27.3604i		−	1.10149i	−0.834674	−	0.550744i	$-0.814344\pi$
$617$		−	27.3604i		−	1.10149i	0.834674	−	0.550744i	$-0.185656\pi$
$618$		0			0
$619$	3.75297			0.150845			0.0754223	−	0.997152i	$-0.475970\pi$
$619$	3.75297			0.150845			0.0754223	+	0.997152i	$0.475970\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			24.9725i			1.00050i
$624$		0			0
$625$	−10.1750	+	22.8357i	−0.407000	+	0.913428i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	47.5135			1.89449
$630$		0			0
$631$	12.8149			0.510154			0.255077	−	0.966921i	$-0.417899\pi$
$631$	12.8149			0.510154			0.255077	+	0.966921i	$0.417899\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−15.9966	−	8.68695i	−0.634807	−	0.344731i
$636$		0			0
$637$		−	21.7616i		−	0.862228i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−27.4409			−1.08385			−0.541925	−	0.840427i	$-0.682304\pi$
$641$	−27.4409			−1.08385			−0.541925	+	0.840427i	$0.682304\pi$
$642$		0			0
$643$		−	11.4109i		−	0.450003i	−0.974358	−	0.225002i	$-0.927761\pi$
$643$		−	11.4109i		−	0.450003i	0.974358	−	0.225002i	$-0.0722388\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	31.3223i		−	1.23141i	−0.787977	−	0.615704i	$-0.788872\pi$
$647$		−	31.3223i		−	1.23141i	0.787977	−	0.615704i	$-0.211128\pi$
$648$		0			0
$649$	−7.22574			−0.283635
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	5.28776i		−	0.206926i	−0.994633	−	0.103463i	$-0.967008\pi$
$653$		−	5.28776i		−	0.206926i	0.994633	−	0.103463i	$-0.0329923\pi$
$654$		0			0
$655$	13.5313	−	24.9172i	0.528710	−	0.973596i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	45.2167			1.76139			0.880696	−	0.473682i	$-0.157076\pi$
$659$	45.2167			1.76139			0.880696	+	0.473682i	$0.157076\pi$
$660$		0			0
$661$	14.6915			0.571431			0.285716	−	0.958314i	$-0.407769\pi$
$661$	14.6915			0.571431			0.285716	+	0.958314i	$0.407769\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−44.8131	−	24.3357i	−1.73778	−	0.943698i
$666$		0			0
$667$			4.05521i			0.157018i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9.34207			−0.360647
$672$		0			0
$673$		−	15.6295i		−	0.602472i	−0.953550	−	0.301236i	$-0.902601\pi$
$673$		−	15.6295i		−	0.602472i	0.953550	−	0.301236i	$-0.0973993\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	30.7946i		−	1.18353i	−0.806109	−	0.591766i	$-0.798431\pi$
$677$		−	30.7946i		−	1.18353i	0.806109	−	0.591766i	$-0.201569\pi$
$678$		0			0
$679$	2.31863			0.0889807
$680$		0			0
$681$		0			0
$682$		0			0
$683$			27.6880i			1.05945i	0.848169	+	0.529726i	$0.177705\pi$
$683$			27.6880i			1.05945i	−0.848169	+	0.529726i	$0.822295\pi$
$684$		0			0
$685$	−2.97571	−	1.61596i	−0.113696	−	0.0617426i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−30.9583			−1.17942
$690$		0			0
$691$	49.0186			1.86475			0.932377	−	0.361488i	$-0.117731\pi$
$691$	49.0186			1.86475			0.932377	+	0.361488i	$0.117731\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.441944	+	0.813820i	−0.0167639	+	0.0308699i
$696$		0			0
$697$		−	34.6403i		−	1.31210i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−23.5229			−0.888448			−0.444224	−	0.895916i	$-0.646521\pi$
$701$	−23.5229			−0.888448			−0.444224	+	0.895916i	$0.646521\pi$
$702$		0			0
$703$		−	49.0244i		−	1.84899i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	77.1593i		−	2.90187i
$708$		0			0
$709$	12.4830			0.468811			0.234405	−	0.972139i	$-0.424686\pi$
$709$	12.4830			0.468811			0.234405	+	0.972139i	$0.424686\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	0.852012i		−	0.0319081i
$714$		0			0
$715$	−3.11319	−	1.69062i	−0.116427	−	0.0632255i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	32.6257			1.21673			0.608367	−	0.793656i	$-0.291825\pi$
$719$	32.6257			1.21673			0.608367	+	0.793656i	$0.291825\pi$
$720$		0			0
$721$	−27.1795			−1.01222
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−11.0407	−	17.0065i	−0.410041	−	0.631606i
$726$		0			0
$727$		−	11.8490i		−	0.439453i	−0.975561	−	0.219727i	$-0.929483\pi$
$727$		−	11.8490i		−	0.439453i	0.975561	−	0.219727i	$-0.0705166\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−41.2079			−1.52413
$732$		0			0
$733$			12.2824i			0.453662i	0.973934	+	0.226831i	$0.0728365\pi$
$733$			12.2824i			0.453662i	−0.973934	+	0.226831i	$0.927164\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			0.455088i			0.0167634i
$738$		0			0
$739$	−2.16279			−0.0795595			−0.0397798	−	0.999208i	$-0.512666\pi$
$739$	−2.16279			−0.0795595			−0.0397798	+	0.999208i	$0.512666\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			43.0351i			1.57880i	0.613877	+	0.789402i	$0.289609\pi$
$743$			43.0351i			1.57880i	−0.613877	+	0.789402i	$0.710391\pi$
$744$		0			0
$745$	19.6682	−	36.2182i	0.720588	−	1.32693i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−38.6942			−1.41386
$750$		0			0
$751$	−14.8480			−0.541813			−0.270906	−	0.962606i	$-0.587323\pi$
$751$	−14.8480			−0.541813			−0.270906	+	0.962606i	$0.587323\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.29903	−	15.2823i	0.302033	−	0.556179i
$756$		0			0
$757$		−	15.3742i		−	0.558785i	−0.960177	−	0.279393i	$-0.909867\pi$
$757$		−	15.3742i		−	0.558785i	0.960177	−	0.279393i	$-0.0901331\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	49.2070			1.78375			0.891876	−	0.452280i	$-0.149389\pi$
$761$	49.2070			1.78375			0.891876	+	0.452280i	$0.149389\pi$
$762$		0			0
$763$		−	3.02120i		−	0.109375i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	28.6502i		−	1.03450i
$768$		0			0
$769$	32.7693			1.18169			0.590845	−	0.806785i	$-0.298794\pi$
$769$	32.7693			1.18169			0.590845	+	0.806785i	$0.298794\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	15.0139i		−	0.540012i	−0.962859	−	0.270006i	$-0.912974\pi$
$773$		−	15.0139i		−	0.540012i	0.962859	−	0.270006i	$-0.0870257\pi$
$774$		0			0
$775$	2.31968	+	3.57312i	0.0833254	+	0.128350i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−35.7419			−1.28059
$780$		0			0
$781$	3.40395			0.121803
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−26.8588	−	14.5856i	−0.958632	−	0.520584i
$786$		0			0
$787$		−	32.3577i		−	1.15343i	−0.816946	−	0.576714i	$-0.804335\pi$
$787$		−	32.3577i		−	1.15343i	0.816946	−	0.576714i	$-0.195665\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−17.1850			−0.611030
$792$		0			0
$793$		−	37.0415i		−	1.31538i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			2.01914i			0.0715217i	0.999360	+	0.0357609i	$0.0113855\pi$
$797$			2.01914i			0.0715217i	−0.999360	+	0.0357609i	$0.988615\pi$
$798$		0			0
$799$	31.2188			1.10444
$800$		0			0
$801$		0			0
$802$		0			0
$803$			6.74435i			0.238003i
$804$		0			0
$805$	−4.22585	+	7.78172i	−0.148942	+	0.274270i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	40.6376			1.42874			0.714371	−	0.699767i	$-0.246713\pi$
$809$	40.6376			1.42874			0.714371	+	0.699767i	$0.246713\pi$
$810$		0			0
$811$	−54.1537			−1.90159			−0.950797	−	0.309814i	$-0.899733\pi$
$811$	−54.1537			−1.90159			−0.950797	+	0.309814i	$0.899733\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−46.2322	−	25.1063i	−1.61944	−	0.879437i
$816$		0			0
$817$			42.5182i			1.48752i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	31.3214			1.09313			0.546563	−	0.837418i	$-0.315936\pi$
$821$	31.3214			1.09313			0.546563	+	0.837418i	$0.315936\pi$
$822$		0			0
$823$			16.6203i			0.579346i	0.957126	+	0.289673i	$0.0935466\pi$
$823$			16.6203i			0.579346i	−0.957126	+	0.289673i	$0.906453\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			27.1647i			0.944609i	0.881436	+	0.472304i	$0.156578\pi$
$827$			27.1647i			0.944609i	−0.881436	+	0.472304i	$0.843422\pi$
$828$		0			0
$829$	−21.9455			−0.762200			−0.381100	−	0.924534i	$-0.624455\pi$
$829$	−21.9455			−0.762200			−0.381100	+	0.924534i	$0.624455\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			48.4600i			1.67904i
$834$		0			0
$835$	36.8297	+	20.0003i	1.27455	+	0.692140i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	40.7077			1.40539			0.702694	−	0.711493i	$-0.251981\pi$
$839$	40.7077			1.40539			0.702694	+	0.711493i	$0.251981\pi$
$840$		0			0
$841$	−12.5553			−0.432940
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−7.16899	+	13.2014i	−0.246621	+	0.454141i
$846$		0			0
$847$			41.9790i			1.44242i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.51300			−0.291822
$852$		0			0
$853$		−	3.15887i		−	0.108158i	−0.998537	−	0.0540788i	$-0.982778\pi$
$853$		−	3.15887i		−	0.108158i	0.998537	−	0.0540788i	$-0.0172222\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			36.8578i			1.25904i	0.776985	+	0.629519i	$0.216748\pi$
$857$			36.8578i			1.25904i	−0.776985	+	0.629519i	$0.783252\pi$
$858$		0			0
$859$	−24.5381			−0.837228			−0.418614	−	0.908164i	$-0.637484\pi$
$859$	−24.5381			−0.837228			−0.418614	+	0.908164i	$0.637484\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	0.915713i		−	0.0311712i	−0.999879	−	0.0155856i	$-0.995039\pi$
$863$		−	0.915713i		−	0.0311712i	0.999879	−	0.0155856i	$-0.00496126\pi$
$864$		0			0
$865$	−24.4434	−	13.2740i	−0.831101	−	0.451329i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.60233			−0.257891
$870$		0			0
$871$	−1.80443			−0.0611409
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.46429	−	44.1398i	−0.117114	−	1.49220i
$876$		0			0
$877$		−	45.3238i		−	1.53047i	−0.643748	−	0.765237i	$-0.722622\pi$
$877$		−	45.3238i		−	1.53047i	0.643748	−	0.765237i	$-0.277378\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14.9074			−0.502242			−0.251121	−	0.967956i	$-0.580799\pi$
$881$	−14.9074			−0.502242			−0.251121	+	0.967956i	$0.580799\pi$
$882$		0			0
$883$		−	47.5043i		−	1.59865i	−0.600900	−	0.799324i	$-0.705191\pi$
$883$		−	47.5043i		−	1.59865i	0.600900	−	0.799324i	$-0.294809\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	9.07718i		−	0.304782i	−0.988320	−	0.152391i	$-0.951303\pi$
$887$		−	9.07718i		−	0.304782i	0.988320	−	0.152391i	$-0.0486973\pi$
$888$		0			0
$889$	32.2382			1.08123
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	32.2116i		−	1.07792i
$894$		0			0
$895$	9.57508	−	17.6321i	0.320060	−	0.589375i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.45509			−0.115234
$900$		0			0
$901$	68.9397			2.29671
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.06311	+	11.1650i	−0.201545	+	0.371136i
$906$		0			0
$907$		−	9.03237i		−	0.299915i	−0.988692	−	0.149958i	$-0.952086\pi$
$907$		−	9.03237i		−	0.299915i	0.988692	−	0.149958i	$-0.0479137\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	51.9368			1.72074			0.860372	−	0.509667i	$-0.170231\pi$
$911$	51.9368			1.72074			0.860372	+	0.509667i	$0.170231\pi$
$912$		0			0
$913$			1.26031i			0.0417101i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			50.2160i			1.65828i
$918$		0			0
$919$	8.30811			0.274059			0.137030	−	0.990567i	$-0.456244\pi$
$919$	8.30811			0.274059			0.137030	+	0.990567i	$0.456244\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			13.4967i			0.444251i
$924$		0			0
$925$	35.7014	−	23.1774i	1.17385	−	0.762069i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14.0455			0.460816			0.230408	−	0.973094i	$-0.425994\pi$
$929$	14.0455			0.460816			0.230408	+	0.973094i	$0.425994\pi$
$930$		0			0
$931$	50.0010			1.63872
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.93263	+	3.76476i	0.226721	+	0.123121i
$936$		0			0
$937$		−	42.7645i		−	1.39706i	−0.715583	−	0.698528i	$-0.753839\pi$
$937$		−	42.7645i		−	1.39706i	0.715583	−	0.698528i	$-0.246161\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−28.8836			−0.941577			−0.470789	−	0.882246i	$-0.656031\pi$
$941$	−28.8836			−0.941577			−0.470789	+	0.882246i	$0.656031\pi$
$942$		0			0
$943$			6.20651i			0.202112i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			35.4439i			1.15177i	0.817530	+	0.575886i	$0.195343\pi$
$947$			35.4439i			1.15177i	−0.817530	+	0.575886i	$0.804657\pi$
$948$		0			0
$949$	−26.7415			−0.868065
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	33.5601i		−	1.08712i	−0.839371	−	0.543559i	$-0.817076\pi$
$953$		−	33.5601i		−	1.08712i	0.839371	−	0.543559i	$-0.182924\pi$
$954$		0			0
$955$	11.2699	−	20.7530i	0.364684	−	0.671550i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5.99699			0.193653
$960$		0			0
$961$	−30.2741			−0.976583
$962$		0			0
$963$		0			0
$964$		0			0
$965$	7.07241	+	3.84066i	0.227669	+	0.123635i
$966$		0			0
$967$		−	57.9430i		−	1.86332i	−0.363329	−	0.931661i	$-0.618360\pi$
$967$		−	57.9430i		−	1.86332i	0.363329	−	0.931661i	$-0.381640\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.73497			0.0877694			0.0438847	−	0.999037i	$-0.486027\pi$
$971$	2.73497			0.0877694			0.0438847	+	0.999037i	$0.486027\pi$
$972$		0			0
$973$		−	1.64010i		−	0.0525792i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			21.7481i			0.695782i	0.937535	+	0.347891i	$0.113102\pi$
$977$			21.7481i			0.695782i	−0.937535	+	0.347891i	$0.886898\pi$
$978$		0			0
$979$	−3.98611			−0.127397
$980$		0			0
$981$		0			0
$982$		0			0
$983$			14.1002i			0.449726i	0.974390	+	0.224863i	$0.0721935\pi$
$983$			14.1002i			0.449726i	−0.974390	+	0.224863i	$0.927806\pi$
$984$		0			0
$985$	−35.9394	−	19.5169i	−1.14513	−	0.621859i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.38322			0.234773
$990$		0			0
$991$	6.03850			0.191819			0.0959096	−	0.995390i	$-0.469424\pi$
$991$	6.03850			0.191819			0.0959096	+	0.995390i	$0.469424\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	22.6365	−	41.6842i	0.717627	−	1.32148i
$996$		0			0
$997$			39.9276i			1.26452i	0.774756	+	0.632261i	$0.217873\pi$
$997$			39.9276i			1.26452i	−0.774756	+	0.632261i	$0.782127\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.f.d.829.8	yes	24
3.2	odd	2	inner	4140.2.f.d.829.17	yes	24
5.4	even	2	inner	4140.2.f.d.829.7	✓	24
15.14	odd	2	inner	4140.2.f.d.829.18	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.f.d.829.7	✓	24	5.4	even	2	inner
4140.2.f.d.829.8	yes	24	1.1	even	1	trivial
4140.2.f.d.829.17	yes	24	3.2	odd	2	inner
4140.2.f.d.829.18	yes	24	15.14	odd	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.06710 + 1.96502i) q^{5} -3.96013i q^{7} +0.632117 q^{11} +2.50635i q^{13} -5.58129i q^{17} -5.75876 q^{19} +1.00000i q^{23} +(-2.72259 - 4.19374i) q^{25} +4.05521 q^{29} -0.852012 q^{31} +(7.78172 + 4.22585i) q^{35} +8.51300i q^{37} +6.20651 q^{41} -7.38322i q^{43} +5.59349i q^{47} -8.68259 q^{49} +12.3519i q^{53} +(-0.674532 + 1.24212i) q^{55} -11.4310 q^{59} -14.7790 q^{61} +(-4.92503 - 2.67453i) q^{65} +0.719943i q^{67} +5.38501 q^{71} +10.6695i q^{73} -2.50326i q^{77} -12.0268 q^{79} +1.99379i q^{83} +(10.9673 + 5.95579i) q^{85} -6.30598 q^{89} +9.92548 q^{91} +(6.14518 - 11.3161i) q^{95} +0.585493i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8 q^{19} - 8 q^{25} - 12 q^{31} - 28 q^{49} - 16 q^{55} - 16 q^{61} + 8 q^{79} + 12 q^{85} - 16 q^{91}+O(q^{100}) \) 24 * q + 8 * q^19 - 8 * q^25 - 12 * q^31 - 28 * q^49 - 16 * q^55 - 16 * q^61 + 8 * q^79 + 12 * q^85 - 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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