LMFDB - Embedded newform 1216.3.g.e.417.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(417,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1216.417");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.g (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:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.4
Character	$\chi$	$=$	1216.417
Dual form			1216.3.g.e.417.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.07504 q^{3} -5.51741i q^{5} -8.74188 q^{7} +16.7560 q^{9} +O(q^{10})$	$q-5.07504 q^{3} -5.51741i q^{5} -8.74188 q^{7} +16.7560 q^{9} -0.103239i q^{11} -7.16700 q^{13} +28.0010i q^{15} +24.4109 q^{17} +(7.40930 - 17.4958i) q^{19} +44.3654 q^{21} +10.9002 q^{23} -5.44176 q^{25} -39.3621 q^{27} +12.9795 q^{29} +40.6764i q^{31} +0.523940i q^{33} +48.2325i q^{35} +42.1034 q^{37} +36.3728 q^{39} -1.13034i q^{41} -63.9047i q^{43} -92.4497i q^{45} +17.7063 q^{47} +27.4204 q^{49} -123.886 q^{51} -52.4515 q^{53} -0.569609 q^{55} +(-37.6025 + 88.7917i) q^{57} +117.338 q^{59} -26.0056i q^{61} -146.479 q^{63} +39.5432i q^{65} -67.7026 q^{67} -55.3191 q^{69} +92.2133i q^{71} -20.5023 q^{73} +27.6171 q^{75} +0.902499i q^{77} +72.3781i q^{79} +48.9599 q^{81} -69.7748i q^{83} -134.685i q^{85} -65.8716 q^{87} -8.48968i q^{89} +62.6530 q^{91} -206.434i q^{93} +(-96.5313 - 40.8801i) q^{95} -130.846i q^{97} -1.72987i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 264 q^{9}+O(q^{10}) $ 48 * q + 264 * q^9	$ 48 q + 264 q^{9} + 24 q^{17} - 160 q^{25} + 328 q^{49} + 8 q^{57} - 472 q^{73} - 736 q^{81}+O(q^{100}) $ 48 * q + 264 * q^9 + 24 * q^17 - 160 * q^25 + 328 * q^49 + 8 * q^57 - 472 * q^73 - 736 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−5.07504			−1.69168			−0.845840	−	0.533437i	$-0.820900\pi$
$3$	−5.07504			−1.69168			−0.845840	+	0.533437i	$0.820900\pi$
$4$		0			0
$5$		−	5.51741i		−	1.10348i	−0.834016	−	0.551741i	$-0.813964\pi$
$5$		−	5.51741i		−	1.10348i	0.834016	−	0.551741i	$-0.186036\pi$
$6$		0			0
$7$	−8.74188			−1.24884			−0.624420	−	0.781089i	$-0.714665\pi$
$7$	−8.74188			−1.24884			−0.624420	+	0.781089i	$0.714665\pi$
$8$		0			0
$9$	16.7560			1.86178
$10$		0			0
$11$		−	0.103239i		−	0.00938532i	−0.999989	−	0.00469266i	$-0.998506\pi$
$11$		−	0.103239i		−	0.00938532i	0.999989	−	0.00469266i	$-0.00149373\pi$
$12$		0			0
$13$	−7.16700			−0.551308			−0.275654	−	0.961257i	$-0.588894\pi$
$13$	−7.16700			−0.551308			−0.275654	+	0.961257i	$0.588894\pi$
$14$		0			0
$15$			28.0010i			1.86674i
$16$		0			0
$17$	24.4109			1.43594			0.717969	−	0.696075i	$-0.245072\pi$
$17$	24.4109			1.43594			0.717969	+	0.696075i	$0.245072\pi$
$18$		0			0
$19$	7.40930	−	17.4958i	0.389963	−	0.920830i
$19$	7.40930	−	17.4958i	0.389963	−	0.920830i
$20$		0			0
$21$	44.3654			2.11264
$22$		0			0
$23$	10.9002			0.473923			0.236961	−	0.971519i	$-0.423848\pi$
$23$	10.9002			0.473923			0.236961	+	0.971519i	$0.423848\pi$
$24$		0			0
$25$	−5.44176			−0.217670
$26$		0			0
$27$	−39.3621			−1.45785
$28$		0			0
$29$	12.9795			0.447570			0.223785	−	0.974639i	$-0.428159\pi$
$29$	12.9795			0.447570			0.223785	+	0.974639i	$0.428159\pi$
$30$		0			0
$31$			40.6764i			1.31214i	0.754699	+	0.656072i	$0.227783\pi$
$31$			40.6764i			1.31214i	−0.754699	+	0.656072i	$0.772217\pi$
$32$		0			0
$33$			0.523940i			0.0158770i
$34$		0			0
$35$			48.2325i			1.37807i
$36$		0			0
$37$	42.1034			1.13793			0.568965	−	0.822362i	$-0.307344\pi$
$37$	42.1034			1.13793			0.568965	+	0.822362i	$0.307344\pi$
$38$		0			0
$39$	36.3728			0.932636
$40$		0			0
$41$		−	1.13034i		−	0.0275693i	−0.999905	−	0.0137847i	$-0.995612\pi$
$41$		−	1.13034i		−	0.0275693i	0.999905	−	0.0137847i	$-0.00438793\pi$
$42$		0			0
$43$		−	63.9047i		−	1.48616i	−0.669205	−	0.743078i	$-0.733365\pi$
$43$		−	63.9047i		−	1.48616i	0.669205	−	0.743078i	$-0.266635\pi$
$44$		0			0
$45$		−	92.4497i		−	2.05444i
$46$		0			0
$47$	17.7063			0.376729			0.188365	−	0.982099i	$-0.439681\pi$
$47$	17.7063			0.376729			0.188365	+	0.982099i	$0.439681\pi$
$48$		0			0
$49$	27.4204			0.559600
$50$		0			0
$51$	−123.886			−2.42915
$52$		0			0
$53$	−52.4515			−0.989651			−0.494825	−	0.868992i	$-0.664768\pi$
$53$	−52.4515			−0.989651			−0.494825	+	0.868992i	$0.664768\pi$
$54$		0			0
$55$	−0.569609			−0.0103565
$56$		0			0
$57$	−37.6025	+	88.7917i	−0.659693	+	1.55775i
$58$		0			0
$59$	117.338			1.98878			0.994392	−	0.105753i	$-0.0337253\pi$
$59$	117.338			1.98878			0.994392	+	0.105753i	$0.0337253\pi$
$60$		0			0
$61$		−	26.0056i		−	0.426320i	−0.977017	−	0.213160i	$-0.931624\pi$
$61$		−	26.0056i		−	0.426320i	0.977017	−	0.213160i	$-0.0683756\pi$
$62$		0			0
$63$	−146.479			−2.32506
$64$		0			0
$65$			39.5432i			0.608357i
$66$		0			0
$67$	−67.7026			−1.01049			−0.505243	−	0.862977i	$-0.668597\pi$
$67$	−67.7026			−1.01049			−0.505243	+	0.862977i	$0.668597\pi$
$68$		0			0
$69$	−55.3191			−0.801726
$70$		0			0
$71$			92.2133i			1.29878i	0.760456	+	0.649389i	$0.224975\pi$
$71$			92.2133i			1.29878i	−0.760456	+	0.649389i	$0.775025\pi$
$72$		0			0
$73$	−20.5023			−0.280853			−0.140427	−	0.990091i	$-0.544847\pi$
$73$	−20.5023			−0.280853			−0.140427	+	0.990091i	$0.544847\pi$
$74$		0			0
$75$	27.6171			0.368229
$76$		0			0
$77$			0.902499i			0.0117208i
$78$		0			0
$79$			72.3781i			0.916178i	0.888906	+	0.458089i	$0.151466\pi$
$79$			72.3781i			0.916178i	−0.888906	+	0.458089i	$0.848534\pi$
$80$		0			0
$81$	48.9599			0.604443
$82$		0			0
$83$		−	69.7748i		−	0.840660i	−0.907371	−	0.420330i	$-0.861914\pi$
$83$		−	69.7748i		−	0.840660i	0.907371	−	0.420330i	$-0.138086\pi$
$84$		0			0
$85$		−	134.685i		−	1.58453i
$86$		0			0
$87$	−65.8716			−0.757145
$88$		0			0
$89$		−	8.48968i		−	0.0953896i	−0.998862	−	0.0476948i	$-0.984813\pi$
$89$		−	8.48968i		−	0.0953896i	0.998862	−	0.0476948i	$-0.0151875\pi$
$90$		0			0
$91$	62.6530			0.688495
$92$		0			0
$93$		−	206.434i		−	2.21973i
$94$		0			0
$95$	−96.5313	−	40.8801i	−1.01612	−	0.430317i
$96$		0			0
$97$		−	130.846i		−	1.34893i	−0.738309	−	0.674463i	$-0.764375\pi$
$97$		−	130.846i		−	1.34893i	0.738309	−	0.674463i	$-0.235625\pi$
$98$		0			0
$99$		−	1.72987i		−	0.0174734i
$100$		0			0
$101$			94.7330i			0.937950i	0.883211	+	0.468975i	$0.155377\pi$
$101$			94.7330i			0.937950i	−0.883211	+	0.468975i	$0.844623\pi$
$102$		0			0
$103$		−	100.957i		−	0.980167i	−0.871676	−	0.490083i	$-0.836966\pi$
$103$		−	100.957i		−	0.980167i	0.871676	−	0.490083i	$-0.163034\pi$
$104$		0			0
$105$		−	244.782i		−	2.33125i
$106$		0			0
$107$	48.4316			0.452632			0.226316	−	0.974054i	$-0.427332\pi$
$107$	48.4316			0.452632			0.226316	+	0.974054i	$0.427332\pi$
$108$		0			0
$109$	−194.697			−1.78621			−0.893107	−	0.449845i	$-0.851479\pi$
$109$	−194.697			−1.78621			−0.893107	+	0.449845i	$0.851479\pi$
$110$		0			0
$111$	−213.676			−1.92501
$112$		0			0
$113$		−	193.464i		−	1.71207i	−0.516919	−	0.856034i	$-0.672921\pi$
$113$		−	193.464i		−	1.71207i	0.516919	−	0.856034i	$-0.327079\pi$
$114$		0			0
$115$		−	60.1410i		−	0.522965i
$116$		0			0
$117$	−120.090			−1.02641
$118$		0			0
$119$	−213.397			−1.79326
$120$		0			0
$121$	120.989			0.999912
$122$		0			0
$123$			5.73653i			0.0466385i
$124$		0			0
$125$		−	107.911i		−	0.863286i
$126$		0			0
$127$		−	233.080i		−	1.83528i	−0.397416	−	0.917638i	$-0.630093\pi$
$127$		−	233.080i		−	1.83528i	0.397416	−	0.917638i	$-0.369907\pi$
$128$		0			0
$129$			324.319i			2.51410i
$130$		0			0
$131$			144.179i			1.10060i	0.834967	+	0.550300i	$0.185487\pi$
$131$			144.179i			1.10060i	−0.834967	+	0.550300i	$0.814513\pi$
$132$		0			0
$133$	−64.7712	+	152.946i	−0.487002	+	1.14997i
$134$		0			0
$135$			217.176i			1.60871i
$136$		0			0
$137$	141.122			1.03009			0.515043	−	0.857164i	$-0.327776\pi$
$137$	141.122			1.03009			0.515043	+	0.857164i	$0.327776\pi$
$138$		0			0
$139$		−	250.346i		−	1.80105i	−0.434804	−	0.900525i	$-0.643182\pi$
$139$		−	250.346i		−	1.80105i	0.434804	−	0.900525i	$-0.356818\pi$
$140$		0			0
$141$	−89.8601			−0.637305
$142$		0			0
$143$			0.739910i			0.00517420i
$144$		0			0
$145$		−	71.6134i		−	0.493885i
$146$		0			0
$147$	−139.160			−0.946664
$148$		0			0
$149$		−	25.8510i		−	0.173497i	−0.996230	−	0.0867484i	$-0.972352\pi$
$149$		−	25.8510i		−	0.173497i	0.996230	−	0.0867484i	$-0.0276476\pi$
$150$		0			0
$151$		−	182.920i		−	1.21139i	−0.795697	−	0.605695i	$-0.792895\pi$
$151$		−	182.920i		−	1.21139i	0.795697	−	0.605695i	$-0.207105\pi$
$152$		0			0
$153$	409.030			2.67340
$154$		0			0
$155$	224.428			1.44793
$156$		0			0
$157$			303.651i			1.93408i	0.254618	+	0.967042i	$0.418050\pi$
$157$			303.651i			1.93408i	−0.254618	+	0.967042i	$0.581950\pi$
$158$		0			0
$159$	266.193			1.67417
$160$		0			0
$161$	−95.2885			−0.591854
$162$		0			0
$163$			114.570i			0.702883i	0.936210	+	0.351442i	$0.114308\pi$
$163$			114.570i			0.702883i	−0.936210	+	0.351442i	$0.885692\pi$
$164$		0			0
$165$	2.89079			0.0175199
$166$		0			0
$167$		−	193.042i		−	1.15594i	−0.816058	−	0.577970i	$-0.803845\pi$
$167$		−	193.042i		−	1.15594i	0.816058	−	0.577970i	$-0.196155\pi$
$168$		0			0
$169$	−117.634			−0.696060
$170$		0			0
$171$	124.150	−	293.159i	0.726026	−	1.71438i
$172$		0			0
$173$	−248.002			−1.43354			−0.716769	−	0.697311i	$-0.754380\pi$
$173$	−248.002			−1.43354			−0.716769	+	0.697311i	$0.754380\pi$
$174$		0			0
$175$	47.5712			0.271836
$176$		0			0
$177$	−595.496			−3.36439
$178$		0			0
$179$	−213.534			−1.19293			−0.596464	−	0.802640i	$-0.703428\pi$
$179$	−213.534			−1.19293			−0.596464	+	0.802640i	$0.703428\pi$
$180$		0			0
$181$	140.186			0.774507			0.387253	−	0.921973i	$-0.373424\pi$
$181$	140.186			0.774507			0.387253	+	0.921973i	$0.373424\pi$
$182$		0			0
$183$			131.979i			0.721198i
$184$		0			0
$185$		−	232.302i		−	1.25568i
$186$		0			0
$187$		−	2.52015i		−	0.0134767i
$188$		0			0
$189$	344.098			1.82063
$190$		0			0
$191$	−171.684			−0.898867			−0.449434	−	0.893314i	$-0.648374\pi$
$191$	−171.684			−0.898867			−0.449434	+	0.893314i	$0.648374\pi$
$192$		0			0
$193$			157.845i			0.817848i	0.912568	+	0.408924i	$0.134096\pi$
$193$			157.845i			0.817848i	−0.912568	+	0.408924i	$0.865904\pi$
$194$		0			0
$195$		−	200.683i		−	1.02915i
$196$		0			0
$197$			5.76133i			0.0292453i	0.999893	+	0.0146227i	$0.00465471\pi$
$197$			5.76133i			0.0292453i	−0.999893	+	0.0146227i	$0.995345\pi$
$198$		0			0
$199$	212.623			1.06846			0.534229	−	0.845340i	$-0.320602\pi$
$199$	212.623			1.06846			0.534229	+	0.845340i	$0.320602\pi$
$200$		0			0
$201$	343.593			1.70942
$202$		0			0
$203$	−113.466			−0.558943
$204$		0			0
$205$	−6.23656			−0.0304222
$206$		0			0
$207$	182.644			0.882340
$208$		0			0
$209$	−1.80624	−	0.764926i	−0.00864229	−	0.00365993i
$210$		0			0
$211$	−4.05652			−0.0192252			−0.00961260	−	0.999954i	$-0.503060\pi$
$211$	−4.05652			−0.0192252			−0.00961260	+	0.999954i	$0.503060\pi$
$212$		0			0
$213$		−	467.986i		−	2.19712i
$214$		0			0
$215$	−352.588			−1.63994
$216$		0			0
$217$		−	355.588i		−	1.63866i
$218$		0			0
$219$	104.050			0.475114
$220$		0			0
$221$	−174.953			−0.791643
$222$		0			0
$223$			80.1901i			0.359597i	0.983703	+	0.179798i	$0.0575446\pi$
$223$			80.1901i			0.359597i	−0.983703	+	0.179798i	$0.942455\pi$
$224$		0			0
$225$	−91.1822			−0.405254
$226$		0			0
$227$	−205.254			−0.904201			−0.452101	−	0.891967i	$-0.649325\pi$
$227$	−205.254			−0.904201			−0.452101	+	0.891967i	$0.649325\pi$
$228$		0			0
$229$			13.6000i			0.0593886i	0.999559	+	0.0296943i	$0.00945338\pi$
$229$			13.6000i			0.0593886i	−0.999559	+	0.0296943i	$0.990547\pi$
$230$		0			0
$231$		−	4.58021i		−	0.0198278i
$232$		0			0
$233$	215.348			0.924240			0.462120	−	0.886817i	$-0.347089\pi$
$233$	215.348			0.924240			0.462120	+	0.886817i	$0.347089\pi$
$234$		0			0
$235$		−	97.6927i		−	0.415714i
$236$		0			0
$237$		−	367.321i		−	1.54988i
$238$		0			0
$239$	255.829			1.07041			0.535207	−	0.844721i	$-0.320234\pi$
$239$	255.829			1.07041			0.535207	+	0.844721i	$0.320234\pi$
$240$		0			0
$241$			101.170i			0.419793i	0.977724	+	0.209896i	$0.0673126\pi$
$241$			101.170i			0.419793i	−0.977724	+	0.209896i	$0.932687\pi$
$242$		0			0
$243$	105.785			0.435331
$244$		0			0
$245$		−	151.290i		−	0.617508i
$246$		0			0
$247$	−53.1025	+	125.392i	−0.214990	+	0.507661i
$248$		0			0
$249$			354.110i			1.42213i
$250$		0			0
$251$			321.296i			1.28006i	0.768348	+	0.640032i	$0.221079\pi$
$251$			321.296i			1.28006i	−0.768348	+	0.640032i	$0.778921\pi$
$252$		0			0
$253$		−	1.12532i		−	0.00444792i
$254$		0			0
$255$			683.532i			2.68052i
$256$		0			0
$257$			400.913i			1.55997i	0.625795	+	0.779987i	$0.284774\pi$
$257$			400.913i			1.55997i	−0.625795	+	0.779987i	$0.715226\pi$
$258$		0			0
$259$	−368.063			−1.42109
$260$		0			0
$261$	217.485			0.833277
$262$		0			0
$263$	−472.797			−1.79771			−0.898854	−	0.438248i	$-0.855599\pi$
$263$	−472.797			−1.79771			−0.898854	+	0.438248i	$0.855599\pi$
$264$		0			0
$265$			289.396i			1.09206i
$266$		0			0
$267$			43.0854i			0.161369i
$268$		0			0
$269$	417.786			1.55311			0.776555	−	0.630050i	$-0.216966\pi$
$269$	417.786			1.55311			0.776555	+	0.630050i	$0.216966\pi$
$270$		0			0
$271$	193.575			0.714298			0.357149	−	0.934047i	$-0.383749\pi$
$271$	193.575			0.714298			0.357149	+	0.934047i	$0.383749\pi$
$272$		0			0
$273$	−317.966			−1.16471
$274$		0			0
$275$			0.561800i			0.00204291i
$276$		0			0
$277$		−	15.5682i		−	0.0562027i	−0.999605	−	0.0281014i	$-0.991054\pi$
$277$		−	15.5682i		−	0.0562027i	0.999605	−	0.0281014i	$-0.00894612\pi$
$278$		0			0
$279$			681.575i			2.44292i
$280$		0			0
$281$			5.98179i			0.0212875i	0.999943	+	0.0106438i	$0.00338808\pi$
$281$			5.98179i			0.0212875i	−0.999943	+	0.0106438i	$0.996612\pi$
$282$		0			0
$283$			52.8002i			0.186573i	0.995639	+	0.0932866i	$0.0297373\pi$
$283$			52.8002i			0.186573i	−0.995639	+	0.0932866i	$0.970263\pi$
$284$		0			0
$285$	489.900	+	207.468i	1.71895	+	0.727959i
$286$		0			0
$287$			9.88132i			0.0344297i
$288$		0			0
$289$	306.894			1.06192
$290$		0			0
$291$			664.048i			2.28195i
$292$		0			0
$293$	−284.326			−0.970395			−0.485197	−	0.874405i	$-0.661252\pi$
$293$	−284.326			−0.970395			−0.485197	+	0.874405i	$0.661252\pi$
$294$		0			0
$295$		−	647.403i		−	2.19459i
$296$		0			0
$297$			4.06368i			0.0136824i
$298$		0			0
$299$	−78.1219			−0.261277
$300$		0			0
$301$			558.647i			1.85597i
$302$		0			0
$303$		−	480.773i		−	1.58671i
$304$		0			0
$305$	−143.483			−0.470437
$306$		0			0
$307$	108.449			0.353253			0.176627	−	0.984278i	$-0.443482\pi$
$307$	108.449			0.353253			0.176627	+	0.984278i	$0.443482\pi$
$308$		0			0
$309$			512.361i			1.65813i
$310$		0			0
$311$	404.664			1.30117			0.650585	−	0.759433i	$-0.274524\pi$
$311$	404.664			1.30117			0.650585	+	0.759433i	$0.274524\pi$
$312$		0			0
$313$	−112.046			−0.357974			−0.178987	−	0.983851i	$-0.557282\pi$
$313$	−112.046			−0.357974			−0.178987	+	0.983851i	$0.557282\pi$
$314$		0			0
$315$			808.184i			2.56566i
$316$		0			0
$317$	−191.095			−0.602824			−0.301412	−	0.953494i	$-0.597458\pi$
$317$	−191.095			−0.602824			−0.301412	+	0.953494i	$0.597458\pi$
$318$		0			0
$319$		−	1.33999i		−	0.00420059i
$320$		0			0
$321$	−245.792			−0.765708
$322$		0			0
$323$	180.868	−	427.088i	0.559963	−	1.32225i
$324$		0			0
$325$	39.0011			0.120003
$326$		0			0
$327$	988.096			3.02170
$328$		0			0
$329$	−154.786			−0.470475
$330$		0			0
$331$	143.451			0.433388			0.216694	−	0.976240i	$-0.430473\pi$
$331$	143.451			0.433388			0.216694	+	0.976240i	$0.430473\pi$
$332$		0			0
$333$	705.485			2.11857
$334$		0			0
$335$			373.543i			1.11505i
$336$		0			0
$337$		−	67.8835i		−	0.201435i	−0.994915	−	0.100717i	$-0.967886\pi$
$337$		−	67.8835i		−	0.201435i	0.994915	−	0.100717i	$-0.0321138\pi$
$338$		0			0
$339$			981.836i			2.89627i
$340$		0			0
$341$	4.19938			0.0123149
$342$		0			0
$343$	188.646			0.549989
$344$		0			0
$345$			305.218i			0.884689i
$346$		0			0
$347$		−	70.8633i		−	0.204217i	−0.994773	−	0.102108i	$-0.967441\pi$
$347$		−	70.8633i		−	0.204217i	0.994773	−	0.102108i	$-0.0325589\pi$
$348$		0			0
$349$		−	231.425i		−	0.663110i	−0.943436	−	0.331555i	$-0.892427\pi$
$349$		−	231.425i		−	0.663110i	0.943436	−	0.331555i	$-0.107573\pi$
$350$		0			0
$351$	282.108			0.803726
$352$		0			0
$353$	77.6586			0.219996			0.109998	−	0.993932i	$-0.464916\pi$
$353$	77.6586			0.219996			0.109998	+	0.993932i	$0.464916\pi$
$354$		0			0
$355$	508.778			1.43318
$356$		0			0
$357$	1083.00			3.03361
$358$		0			0
$359$	−265.945			−0.740794			−0.370397	−	0.928873i	$-0.620778\pi$
$359$	−265.945			−0.740794			−0.370397	+	0.928873i	$0.620778\pi$
$360$		0			0
$361$	−251.204	−	259.263i	−0.695857	−	0.718180i
$362$		0			0
$363$	−614.026			−1.69153
$364$		0			0
$365$			113.119i			0.309916i
$366$		0			0
$367$	21.2136			0.0578028			0.0289014	−	0.999582i	$-0.490799\pi$
$367$	21.2136			0.0578028			0.0289014	+	0.999582i	$0.490799\pi$
$368$		0			0
$369$		−	18.9400i		−	0.0513280i
$370$		0			0
$371$	458.525			1.23592
$372$		0			0
$373$	−724.367			−1.94200			−0.971001	−	0.239075i	$-0.923156\pi$
$373$	−724.367			−1.94200			−0.971001	+	0.239075i	$0.923156\pi$
$374$		0			0
$375$			547.651i			1.46040i
$376$		0			0
$377$	−93.0243			−0.246749
$378$		0			0
$379$	257.591			0.679659			0.339830	−	0.940487i	$-0.389631\pi$
$379$	257.591			0.679659			0.339830	+	0.940487i	$0.389631\pi$
$380$		0			0
$381$			1182.89i			3.10470i
$382$		0			0
$383$		−	649.965i		−	1.69704i	−0.529166	−	0.848518i	$-0.677495\pi$
$383$		−	649.965i		−	1.69704i	0.529166	−	0.848518i	$-0.322505\pi$
$384$		0			0
$385$	4.97945			0.0129336
$386$		0			0
$387$		−	1070.79i		−	2.76689i
$388$		0			0
$389$		−	129.573i		−	0.333093i	−0.986034	−	0.166547i	$-0.946738\pi$
$389$		−	129.573i		−	0.333093i	0.986034	−	0.166547i	$-0.0532616\pi$
$390$		0			0
$391$	266.085			0.680524
$392$		0			0
$393$		−	731.713i		−	1.86186i
$394$		0			0
$395$	399.339			1.01098
$396$		0			0
$397$			367.454i			0.925576i	0.886469	+	0.462788i	$0.153151\pi$
$397$			367.454i			0.925576i	−0.886469	+	0.462788i	$0.846849\pi$
$398$		0			0
$399$	328.716	−	776.206i	0.823851	−	1.94538i
$400$		0			0
$401$		−	401.016i		−	1.00004i	−0.866014	−	0.500020i	$-0.833326\pi$
$401$		−	401.016i		−	1.00004i	0.866014	−	0.500020i	$-0.166674\pi$
$402$		0			0
$403$		−	291.528i		−	0.723394i
$404$		0			0
$405$		−	270.131i		−	0.666991i
$406$		0			0
$407$		−	4.34669i		−	0.0106798i
$408$		0			0
$409$			563.440i			1.37760i	0.724950	+	0.688801i	$0.241863\pi$
$409$			563.440i			1.37760i	−0.724950	+	0.688801i	$0.758137\pi$
$410$		0			0
$411$	−716.199			−1.74258
$412$		0			0
$413$	−1025.76			−2.48367
$414$		0			0
$415$	−384.976			−0.927653
$416$		0			0
$417$			1270.52i			3.04680i
$418$		0			0
$419$		−	260.123i		−	0.620819i	−0.950603	−	0.310410i	$-0.899534\pi$
$419$		−	260.123i		−	0.620819i	0.950603	−	0.310410i	$-0.100466\pi$
$420$		0			0
$421$	−261.196			−0.620417			−0.310209	−	0.950669i	$-0.600399\pi$
$421$	−261.196			−0.620417			−0.310209	+	0.950669i	$0.600399\pi$
$422$		0			0
$423$	296.687			0.701387
$424$		0			0
$425$	−132.839			−0.312561
$426$		0			0
$427$			227.337i			0.532406i
$428$		0			0
$429$		−	3.75507i		−	0.00875308i
$430$		0			0
$431$		−	460.860i		−	1.06928i	−0.845080	−	0.534640i	$-0.820447\pi$
$431$		−	460.860i		−	1.06928i	0.845080	−	0.534640i	$-0.179553\pi$
$432$		0			0
$433$		−	572.695i		−	1.32262i	−0.750112	−	0.661311i	$-0.770000\pi$
$433$		−	572.695i		−	1.32262i	0.750112	−	0.661311i	$-0.230000\pi$
$434$		0			0
$435$			363.441i			0.835496i
$436$		0			0
$437$	80.7631	−	190.708i	0.184813	−	0.436403i
$438$		0			0
$439$		−	110.661i		−	0.252075i	−0.992025	−	0.126038i	$-0.959774\pi$
$439$		−	110.661i		−	0.252075i	0.992025	−	0.126038i	$-0.0402260\pi$
$440$		0			0
$441$	459.457			1.04185
$442$		0			0
$443$		−	709.613i		−	1.60184i	−0.598774	−	0.800918i	$-0.704345\pi$
$443$		−	709.613i		−	1.60184i	0.598774	−	0.800918i	$-0.295655\pi$
$444$		0			0
$445$	−46.8410			−0.105261
$446$		0			0
$447$			131.195i			0.293501i
$448$		0			0
$449$		−	309.663i		−	0.689672i	−0.938663	−	0.344836i	$-0.887934\pi$
$449$		−	309.663i		−	0.689672i	0.938663	−	0.344836i	$-0.112066\pi$
$450$		0			0
$451$	−0.116695			−0.000258747
$452$		0			0
$453$			928.326i			2.04928i
$454$		0			0
$455$		−	345.682i		−	0.759741i
$456$		0			0
$457$	230.432			0.504228			0.252114	−	0.967698i	$-0.418874\pi$
$457$	230.432			0.504228			0.252114	+	0.967698i	$0.418874\pi$
$458$		0			0
$459$	−960.865			−2.09339
$460$		0			0
$461$		−	560.586i		−	1.21602i	−0.793928	−	0.608011i	$-0.791968\pi$
$461$		−	560.586i		−	1.21602i	0.793928	−	0.608011i	$-0.208032\pi$
$462$		0			0
$463$	−702.709			−1.51773			−0.758865	−	0.651248i	$-0.774246\pi$
$463$	−702.709			−1.51773			−0.758865	+	0.651248i	$0.774246\pi$
$464$		0			0
$465$	−1138.98			−2.44943
$466$		0			0
$467$			38.1953i			0.0817887i	0.999163	+	0.0408943i	$0.0130207\pi$
$467$			38.1953i			0.0817887i	−0.999163	+	0.0408943i	$0.986979\pi$
$468$		0			0
$469$	591.848			1.26194
$470$		0			0
$471$		−	1541.04i		−	3.27185i
$472$		0			0
$473$	−6.59743			−0.0139481
$474$		0			0
$475$	−40.3197	+	95.2079i	−0.0848835	+	0.200438i
$476$		0			0
$477$	−878.878			−1.84251
$478$		0			0
$479$	438.004			0.914413			0.457206	−	0.889361i	$-0.348850\pi$
$479$	438.004			0.914413			0.457206	+	0.889361i	$0.348850\pi$
$480$		0			0
$481$	−301.755			−0.627349
$482$		0			0
$483$	483.593			1.00123
$484$		0			0
$485$	−721.930			−1.48851
$486$		0			0
$487$			57.0979i			0.117244i	0.998280	+	0.0586220i	$0.0186707\pi$
$487$			57.0979i			0.117244i	−0.998280	+	0.0586220i	$0.981329\pi$
$488$		0			0
$489$		−	581.447i		−	1.18905i
$490$		0			0
$491$		−	208.671i		−	0.424993i	−0.977162	−	0.212496i	$-0.931841\pi$
$491$		−	208.671i		−	0.424993i	0.977162	−	0.212496i	$-0.0681593\pi$
$492$		0			0
$493$	316.843			0.642683
$494$		0			0
$495$	−9.54437			−0.0192816
$496$		0			0
$497$		−	806.117i		−	1.62197i
$498$		0			0
$499$		−	158.463i		−	0.317561i	−0.987314	−	0.158781i	$-0.949244\pi$
$499$		−	158.463i		−	0.317561i	0.987314	−	0.158781i	$-0.0507562\pi$
$500$		0			0
$501$			979.696i			1.95548i
$502$		0			0
$503$	−374.731			−0.744992			−0.372496	−	0.928034i	$-0.621498\pi$
$503$	−374.731			−0.744992			−0.372496	+	0.928034i	$0.621498\pi$
$504$		0			0
$505$	522.680			1.03501
$506$		0			0
$507$	596.998			1.17751
$508$		0			0
$509$	426.947			0.838795			0.419398	−	0.907803i	$-0.362241\pi$
$509$	426.947			0.838795			0.419398	+	0.907803i	$0.362241\pi$
$510$		0			0
$511$	179.229			0.350741
$512$		0			0
$513$	−291.645	+	688.670i	−0.568510	+	1.34244i
$514$		0			0
$515$	−557.022			−1.08160
$516$		0			0
$517$		−	1.82797i		−	0.00353573i
$518$		0			0
$519$	1258.62			2.42509
$520$		0			0
$521$		−	534.084i		−	1.02511i	−0.858654	−	0.512556i	$-0.828699\pi$
$521$		−	534.084i		−	1.02511i	0.858654	−	0.512556i	$-0.171301\pi$
$522$		0			0
$523$	203.523			0.389146			0.194573	−	0.980888i	$-0.437668\pi$
$523$	203.523			0.389146			0.194573	+	0.980888i	$0.437668\pi$
$524$		0			0
$525$	−241.426			−0.459859
$526$		0			0
$527$			992.950i			1.88416i
$528$		0			0
$529$	−410.185			−0.775397
$530$		0			0
$531$	1966.12			3.70268
$532$		0			0
$533$			8.10116i			0.0151992i
$534$		0			0
$535$		−	267.217i		−	0.499471i
$536$		0			0
$537$	1083.69			2.01805
$538$		0			0
$539$		−	2.83084i		−	0.00525203i
$540$		0			0
$541$			607.359i			1.12266i	0.827592	+	0.561330i	$0.189710\pi$
$541$			607.359i			1.12266i	−0.827592	+	0.561330i	$0.810290\pi$
$542$		0			0
$543$	−711.448			−1.31022
$544$		0			0
$545$			1074.22i			1.97105i
$546$		0			0
$547$	389.263			0.711632			0.355816	−	0.934556i	$-0.384203\pi$
$547$	389.263			0.711632			0.355816	+	0.934556i	$0.384203\pi$
$548$		0			0
$549$		−	435.749i		−	0.793715i
$550$		0			0
$551$	96.1693	−	227.087i	0.174536	−	0.412136i
$552$		0			0
$553$		−	632.720i		−	1.14416i
$554$		0			0
$555$			1178.94i			2.12422i
$556$		0			0
$557$		−	568.180i		−	1.02007i	−0.860153	−	0.510036i	$-0.829632\pi$
$557$		−	568.180i		−	1.02007i	0.860153	−	0.510036i	$-0.170368\pi$
$558$		0			0
$559$			458.005i			0.819329i
$560$		0			0
$561$			12.7899i			0.0227983i
$562$		0			0
$563$	556.694			0.988799			0.494399	−	0.869235i	$-0.335388\pi$
$563$	556.694			0.988799			0.494399	+	0.869235i	$0.335388\pi$
$564$		0			0
$565$	−1067.42			−1.88924
$566$		0			0
$567$	−428.001			−0.754852
$568$		0			0
$569$			106.338i			0.186886i	0.995625	+	0.0934430i	$0.0297873\pi$
$569$			106.338i			0.186886i	−0.995625	+	0.0934430i	$0.970213\pi$
$570$		0			0
$571$			918.783i			1.60908i	0.593901	+	0.804538i	$0.297587\pi$
$571$			918.783i			1.60908i	−0.593901	+	0.804538i	$0.702413\pi$
$572$		0			0
$573$	871.301			1.52060
$574$		0			0
$575$	−59.3165			−0.103159
$576$		0			0
$577$	−823.280			−1.42683			−0.713414	−	0.700743i	$-0.752852\pi$
$577$	−823.280			−1.42683			−0.713414	+	0.700743i	$0.752852\pi$
$578$		0			0
$579$		−	801.068i		−	1.38354i
$580$		0			0
$581$			609.963i			1.04985i
$582$		0			0
$583$			5.41502i			0.00928819i
$584$		0			0
$585$			662.587i			1.13263i
$586$		0			0
$587$		−	809.611i		−	1.37923i	−0.724174	−	0.689617i	$-0.757779\pi$
$587$		−	809.611i		−	1.37923i	0.724174	−	0.689617i	$-0.242221\pi$
$588$		0			0
$589$	711.666	+	301.384i	1.20826	+	0.511688i
$590$		0			0
$591$		−	29.2390i		−	0.0494737i
$592$		0			0
$593$	−581.819			−0.981144			−0.490572	−	0.871401i	$-0.663212\pi$
$593$	−581.819			−0.981144			−0.490572	+	0.871401i	$0.663212\pi$
$594$		0			0
$595$			1177.40i			1.97882i
$596$		0			0
$597$	−1079.07			−1.80749
$598$		0			0
$599$		−	9.59418i		−	0.0160170i	−0.999968	−	0.00800849i	$-0.997451\pi$
$599$		−	9.59418i		−	0.0160170i	0.999968	−	0.00800849i	$-0.00254921\pi$
$600$		0			0
$601$		−	629.588i		−	1.04757i	−0.851851	−	0.523784i	$-0.824520\pi$
$601$		−	629.588i		−	1.04757i	0.851851	−	0.523784i	$-0.175480\pi$
$602$		0			0
$603$	−1134.43			−1.88130
$604$		0			0
$605$		−	667.547i		−	1.10338i
$606$		0			0
$607$		−	872.028i		−	1.43662i	−0.695723	−	0.718310i	$-0.744916\pi$
$607$		−	872.028i		−	1.43662i	0.695723	−	0.718310i	$-0.255084\pi$
$608$		0			0
$609$	575.842			0.945553
$610$		0			0
$611$	−126.901			−0.207694
$612$		0			0
$613$			180.485i			0.294428i	0.989105	+	0.147214i	$0.0470307\pi$
$613$			180.485i			0.294428i	−0.989105	+	0.147214i	$0.952969\pi$
$614$		0			0
$615$	31.6508			0.0514647
$616$		0			0
$617$	198.869			0.322316			0.161158	−	0.986929i	$-0.448477\pi$
$617$	198.869			0.322316			0.161158	+	0.986929i	$0.448477\pi$
$618$		0			0
$619$			156.095i			0.252174i	0.992019	+	0.126087i	$0.0402418\pi$
$619$			156.095i			0.252174i	−0.992019	+	0.126087i	$0.959758\pi$
$620$		0			0
$621$	−429.055			−0.690911
$622$		0			0
$623$			74.2157i			0.119126i
$624$		0			0
$625$	−731.431			−1.17029
$626$		0			0
$627$	9.16673	+	3.88203i	0.0146200	+	0.00619143i
$628$		0			0
$629$	1027.78			1.63400
$630$		0			0
$631$	−1023.23			−1.62161			−0.810804	−	0.585318i	$-0.800970\pi$
$631$	−1023.23			−1.62161			−0.810804	+	0.585318i	$0.800970\pi$
$632$		0			0
$633$	20.5870			0.0325229
$634$		0			0
$635$	−1286.00			−2.02519
$636$		0			0
$637$	−196.522			−0.308512
$638$		0			0
$639$			1545.13i			2.41804i
$640$		0			0
$641$		−	139.425i		−	0.217512i	−0.994068	−	0.108756i	$-0.965313\pi$
$641$		−	139.425i		−	0.217512i	0.994068	−	0.108756i	$-0.0346867\pi$
$642$		0			0
$643$			0.119878i			0.000186436i	1.00000		9.32179e-5i	$2.96722e-5\pi$
$643$			0.119878i			0.000186436i	−1.00000		9.32179e-5i	$0.999970\pi$
$644$		0			0
$645$	1789.40			2.77426
$646$		0			0
$647$	−820.768			−1.26857			−0.634287	−	0.773097i	$-0.718706\pi$
$647$	−820.768			−1.26857			−0.634287	+	0.773097i	$0.718706\pi$
$648$		0			0
$649$		−	12.1138i		−	0.0186654i
$650$		0			0
$651$			1804.63i			2.77208i
$652$		0			0
$653$		−	66.8769i		−	0.102415i	−0.998688	−	0.0512074i	$-0.983693\pi$
$653$		−	66.8769i		−	0.102415i	0.998688	−	0.0512074i	$-0.0163070\pi$
$654$		0			0
$655$	795.492			1.21449
$656$		0			0
$657$	−343.537			−0.522887
$658$		0			0
$659$	−810.587			−1.23003			−0.615013	−	0.788517i	$-0.710849\pi$
$659$	−810.587			−1.23003			−0.615013	+	0.788517i	$0.710849\pi$
$660$		0			0
$661$	1079.30			1.63282			0.816411	−	0.577471i	$-0.195960\pi$
$661$	1079.30			1.63282			0.816411	+	0.577471i	$0.195960\pi$
$662$		0			0
$663$	887.894			1.33921
$664$		0			0
$665$	843.865	+	357.369i	1.26897	+	0.537397i
$666$		0			0
$667$	141.480			0.212114
$668$		0			0
$669$		−	406.968i		−	0.608323i
$670$		0			0
$671$	−2.68477			−0.00400115
$672$		0			0
$673$			747.743i			1.11106i	0.831497	+	0.555529i	$0.187484\pi$
$673$			747.743i			1.11106i	−0.831497	+	0.555529i	$0.812516\pi$
$674$		0			0
$675$	214.199			0.317332
$676$		0			0
$677$	−695.960			−1.02801			−0.514003	−	0.857788i	$-0.671838\pi$
$677$	−695.960			−1.02801			−0.514003	+	0.857788i	$0.671838\pi$
$678$		0			0
$679$			1143.84i			1.68459i
$680$		0			0
$681$	1041.67			1.52962
$682$		0			0
$683$	532.703			0.779946			0.389973	−	0.920826i	$-0.372484\pi$
$683$	532.703			0.779946			0.389973	+	0.920826i	$0.372484\pi$
$684$		0			0
$685$		−	778.626i		−	1.13668i
$686$		0			0
$687$		−	69.0205i		−	0.100466i
$688$		0			0
$689$	375.920			0.545602
$690$		0			0
$691$		−	281.522i		−	0.407413i	−0.979032	−	0.203706i	$-0.934701\pi$
$691$		−	281.522i		−	0.407413i	0.979032	−	0.203706i	$-0.0652988\pi$
$692$		0			0
$693$			15.1223i			0.0218215i
$694$		0			0
$695$	−1381.26			−1.98743
$696$		0			0
$697$		−	27.5927i		−	0.0395878i
$698$		0			0
$699$	−1092.90			−1.56352
$700$		0			0
$701$		−	1219.64i		−	1.73986i	−0.493176	−	0.869929i	$-0.664164\pi$
$701$		−	1219.64i		−	1.73986i	0.493176	−	0.869929i	$-0.335836\pi$
$702$		0			0
$703$	311.957	−	736.632i	0.443751	−	1.04784i
$704$		0			0
$705$			495.794i			0.703254i
$706$		0			0
$707$		−	828.144i		−	1.17135i
$708$		0			0
$709$		−	1212.52i		−	1.71019i	−0.518471	−	0.855095i	$-0.673499\pi$
$709$		−	1212.52i		−	1.71019i	0.518471	−	0.855095i	$-0.326501\pi$
$710$		0			0
$711$			1212.77i			1.70572i
$712$		0			0
$713$			443.383i			0.621855i
$714$		0			0
$715$	4.08239			0.00570963
$716$		0			0
$717$	−1298.34			−1.81080
$718$		0			0
$719$	−317.916			−0.442164			−0.221082	−	0.975255i	$-0.570959\pi$
$719$	−317.916			−0.442164			−0.221082	+	0.975255i	$0.570959\pi$
$720$		0			0
$721$			882.555i			1.22407i
$722$		0			0
$723$		−	513.442i		−	0.710154i
$724$		0			0
$725$	−70.6315			−0.0974228
$726$		0			0
$727$	968.731			1.33250			0.666252	−	0.745726i	$-0.267897\pi$
$727$	968.731			1.33250			0.666252	+	0.745726i	$0.267897\pi$
$728$		0			0
$729$	−977.504			−1.34088
$730$		0			0
$731$		−	1559.97i		−	2.13403i
$732$		0			0
$733$			639.217i			0.872056i	0.899933	+	0.436028i	$0.143615\pi$
$733$			639.217i			0.872056i	−0.899933	+	0.436028i	$0.856385\pi$
$734$		0			0
$735$			767.800i			1.04463i
$736$		0			0
$737$			6.98952i			0.00948374i
$738$		0			0
$739$			1114.79i			1.50851i	0.656580	+	0.754256i	$0.272002\pi$
$739$			1114.79i			1.50851i	−0.656580	+	0.754256i	$0.727998\pi$
$740$		0			0
$741$	269.497	−	636.370i	0.363694	−	0.858799i
$742$		0			0
$743$			790.850i			1.06440i	0.846618	+	0.532201i	$0.178635\pi$
$743$			790.850i			1.06440i	−0.846618	+	0.532201i	$0.821365\pi$
$744$		0			0
$745$	−142.631			−0.191450
$746$		0			0
$747$		−	1169.15i		−	1.56512i
$748$		0			0
$749$	−423.383			−0.565265
$750$		0			0
$751$		−	3.95691i		−	0.00526885i	−0.999997	−	0.00263443i	$-0.999161\pi$
$751$		−	3.95691i		−	0.00526885i	0.999997	−	0.00263443i	$-0.000838565\pi$
$752$		0			0
$753$		−	1630.59i		−	2.16546i
$754$		0			0
$755$	−1009.24			−1.33675
$756$		0			0
$757$		−	270.891i		−	0.357848i	−0.983863	−	0.178924i	$-0.942738\pi$
$757$		−	270.891i		−	0.357848i	0.983863	−	0.178924i	$-0.0572616\pi$
$758$		0			0
$759$			5.71106i			0.00752445i
$760$		0			0
$761$	341.236			0.448404			0.224202	−	0.974543i	$-0.428022\pi$
$761$	341.236			0.448404			0.224202	+	0.974543i	$0.428022\pi$
$762$		0			0
$763$	1702.02			2.23069
$764$		0			0
$765$		−	2256.78i		−	2.95005i
$766$		0			0
$767$	−840.963			−1.09643
$768$		0			0
$769$	616.420			0.801587			0.400793	−	0.916168i	$-0.368735\pi$
$769$	616.420			0.801587			0.400793	+	0.916168i	$0.368735\pi$
$770$		0			0
$771$		−	2034.65i		−	2.63898i
$772$		0			0
$773$	−1106.79			−1.43182			−0.715908	−	0.698194i	$-0.753987\pi$
$773$	−1106.79			−1.43182			−0.715908	+	0.698194i	$0.753987\pi$
$774$		0			0
$775$		−	221.352i		−	0.285615i
$776$		0			0
$777$	1867.93			2.40403
$778$		0			0
$779$	−19.7762	−	8.37505i	−0.0253867	−	0.0107510i
$780$		0			0
$781$	9.51997			0.0121895
$782$		0			0
$783$	−510.901			−0.652492
$784$		0			0
$785$	1675.37			2.13422
$786$		0			0
$787$	−1201.49			−1.52667			−0.763335	−	0.646002i	$-0.776440\pi$
$787$	−1201.49			−1.52667			−0.763335	+	0.646002i	$0.776440\pi$
$788$		0			0
$789$	2399.46			3.04115
$790$		0			0
$791$			1691.24i			2.13810i
$792$		0			0
$793$			186.382i			0.235034i
$794$		0			0
$795$		−	1468.70i		−	1.84742i
$796$		0			0
$797$	889.940			1.11661			0.558306	−	0.829635i	$-0.311451\pi$
$797$	889.940			1.11661			0.558306	+	0.829635i	$0.311451\pi$
$798$		0			0
$799$	432.227			0.540960
$800$		0			0
$801$		−	142.253i		−	0.177594i
$802$		0			0
$803$			2.11663i			0.00263590i
$804$		0			0
$805$			525.745i			0.653099i
$806$		0			0
$807$	−2120.28			−2.62736
$808$		0			0
$809$	−881.557			−1.08969			−0.544844	−	0.838537i	$-0.683411\pi$
$809$	−881.557			−1.08969			−0.544844	+	0.838537i	$0.683411\pi$
$810$		0			0
$811$	−1251.06			−1.54261			−0.771305	−	0.636466i	$-0.780396\pi$
$811$	−1251.06			−1.54261			−0.771305	+	0.636466i	$0.780396\pi$
$812$		0			0
$813$	−982.400			−1.20836
$814$		0			0
$815$	632.129			0.775618
$816$		0			0
$817$	−1118.06	−	473.489i	−1.36850	−	0.579546i
$818$		0			0
$819$	1049.81			1.28183
$820$		0			0
$821$		−	1171.84i		−	1.42734i	−0.700484	−	0.713668i	$-0.747032\pi$
$821$		−	1171.84i		−	1.42734i	0.700484	−	0.713668i	$-0.252968\pi$
$822$		0			0
$823$	1287.88			1.56485			0.782427	−	0.622742i	$-0.213981\pi$
$823$	1287.88			1.56485			0.782427	+	0.622742i	$0.213981\pi$
$824$		0			0
$825$		−	2.85115i		−	0.00345594i
$826$		0			0
$827$	−12.5777			−0.0152088			−0.00760440	−	0.999971i	$-0.502421\pi$
$827$	−12.5777			−0.0152088			−0.00760440	+	0.999971i	$0.502421\pi$
$828$		0			0
$829$	399.573			0.481994			0.240997	−	0.970526i	$-0.422526\pi$
$829$	399.573			0.481994			0.240997	+	0.970526i	$0.422526\pi$
$830$		0			0
$831$			79.0090i			0.0950770i
$832$		0			0
$833$	669.358			0.803551
$834$		0			0
$835$	−1065.09			−1.27556
$836$		0			0
$837$		−	1601.11i		−	1.91291i
$838$		0			0
$839$			1372.17i			1.63548i	0.575589	+	0.817739i	$0.304773\pi$
$839$			1372.17i			1.63548i	−0.575589	+	0.817739i	$0.695227\pi$
$840$		0			0
$841$	−672.532			−0.799681
$842$		0			0
$843$		−	30.3578i		−	0.0360116i
$844$		0			0
$845$			649.035i			0.768089i
$846$		0			0
$847$	−1057.67			−1.24873
$848$		0			0
$849$		−	267.963i		−	0.315622i
$850$		0			0
$851$	458.937			0.539291
$852$		0			0
$853$			238.338i			0.279412i	0.990193	+	0.139706i	$0.0446157\pi$
$853$			238.338i			0.279412i	−0.990193	+	0.139706i	$0.955384\pi$
$854$		0			0
$855$	−1617.48	−	684.988i	−1.89179	−	0.801156i
$856$		0			0
$857$		−	794.657i		−	0.927254i	−0.886030	−	0.463627i	$-0.846548\pi$
$857$		−	794.657i		−	0.927254i	0.886030	−	0.463627i	$-0.153452\pi$
$858$		0			0
$859$		−	424.661i		−	0.494367i	−0.968969	−	0.247184i	$-0.920495\pi$
$859$		−	424.661i		−	0.494367i	0.968969	−	0.247184i	$-0.0795051\pi$
$860$		0			0
$861$		−	50.1481i		−	0.0582440i
$862$		0			0
$863$		−	715.215i		−	0.828755i	−0.910105	−	0.414377i	$-0.863999\pi$
$863$		−	715.215i		−	0.828755i	0.910105	−	0.414377i	$-0.136001\pi$
$864$		0			0
$865$			1368.33i			1.58188i
$866$		0			0
$867$	−1557.50			−1.79642
$868$		0			0
$869$	7.47220			0.00859862
$870$		0			0
$871$	485.224			0.557089
$872$		0			0
$873$		−	2192.45i		−	2.51140i
$874$		0			0
$875$			943.342i			1.07811i
$876$		0			0
$877$	444.385			0.506710			0.253355	−	0.967373i	$-0.418466\pi$
$877$	444.385			0.506710			0.253355	+	0.967373i	$0.418466\pi$
$878$		0			0
$879$	1442.96			1.64160
$880$		0			0
$881$	931.618			1.05746			0.528728	−	0.848792i	$-0.322669\pi$
$881$	931.618			1.05746			0.528728	+	0.848792i	$0.322669\pi$
$882$		0			0
$883$			1085.52i			1.22935i	0.788781	+	0.614675i	$0.210713\pi$
$883$			1085.52i			1.22935i	−0.788781	+	0.614675i	$0.789287\pi$
$884$		0			0
$885$			3285.59i			3.71254i
$886$		0			0
$887$		−	175.875i		−	0.198281i	−0.995073	−	0.0991403i	$-0.968391\pi$
$887$		−	175.875i		−	0.198281i	0.995073	−	0.0991403i	$-0.0316092\pi$
$888$		0			0
$889$			2037.56i			2.29197i
$890$		0			0
$891$		−	5.05454i		−	0.00567289i
$892$		0			0
$893$	131.191	−	309.785i	0.146911	−	0.346904i
$894$		0			0
$895$			1178.15i			1.31637i
$896$		0			0
$897$	396.472			0.441997
$898$		0			0
$899$			527.961i			0.587276i
$900$		0			0
$901$	−1280.39			−1.42108
$902$		0			0
$903$		−	2835.16i		−	3.13971i
$904$		0			0
$905$		−	773.461i		−	0.854653i
$906$		0			0
$907$	−145.717			−0.160658			−0.0803289	−	0.996768i	$-0.525597\pi$
$907$	−145.717			−0.160658			−0.0803289	+	0.996768i	$0.525597\pi$
$908$		0			0
$909$			1587.35i			1.74626i
$910$		0			0
$911$		−	1028.87i		−	1.12939i	−0.825301	−	0.564693i	$-0.808995\pi$
$911$		−	1028.87i		−	1.12939i	0.825301	−	0.564693i	$-0.191005\pi$
$912$		0			0
$913$	−7.20345			−0.00788987
$914$		0			0
$915$	728.183			0.795828
$916$		0			0
$917$		−	1260.39i		−	1.37447i
$918$		0			0
$919$	−291.356			−0.317036			−0.158518	−	0.987356i	$-0.550672\pi$
$919$	−291.356			−0.317036			−0.158518	+	0.987356i	$0.550672\pi$
$920$		0			0
$921$	−550.381			−0.597591
$922$		0			0
$923$		−	660.893i		−	0.716027i
$924$		0			0
$925$	−229.117			−0.247694
$926$		0			0
$927$		−	1691.64i		−	1.82485i
$928$		0			0
$929$	1427.31			1.53640			0.768198	−	0.640212i	$-0.221154\pi$
$929$	1427.31			1.53640			0.768198	+	0.640212i	$0.221154\pi$
$930$		0			0
$931$	203.166	−	479.741i	0.218224	−	0.515297i
$932$		0			0
$933$	−2053.68			−2.20116
$934$		0			0
$935$	−13.9047			−0.0148713
$936$		0			0
$937$	1215.98			1.29773			0.648867	−	0.760902i	$-0.275243\pi$
$937$	1215.98			1.29773			0.648867	+	0.760902i	$0.275243\pi$
$938$		0			0
$939$	568.638			0.605578
$940$		0			0
$941$	68.4668			0.0727596			0.0363798	−	0.999338i	$-0.488417\pi$
$941$	68.4668			0.0727596			0.0363798	+	0.999338i	$0.488417\pi$
$942$		0			0
$943$		−	12.3210i		−	0.0130657i
$944$		0			0
$945$		−	1898.53i		−	2.00903i
$946$		0			0
$947$			1667.83i			1.76117i	0.473888	+	0.880585i	$0.342850\pi$
$947$			1667.83i			1.76117i	−0.473888	+	0.880585i	$0.657150\pi$
$948$		0			0
$949$	146.940			0.154837
$950$		0			0
$951$	969.815			1.01978
$952$		0			0
$953$			1414.36i			1.48411i	0.670339	+	0.742055i	$0.266149\pi$
$953$			1414.36i			1.48411i	−0.670339	+	0.742055i	$0.733851\pi$
$954$		0			0
$955$			947.248i			0.991883i
$956$		0			0
$957$			6.80049i			0.00710605i
$958$		0			0
$959$	−1233.67			−1.28641
$960$		0			0
$961$	−693.573			−0.721720
$962$		0			0
$963$	811.521			0.842701
$964$		0			0
$965$	870.893			0.902480
$966$		0			0
$967$	−395.648			−0.409150			−0.204575	−	0.978851i	$-0.565581\pi$
$967$	−395.648			−0.409150			−0.204575	+	0.978851i	$0.565581\pi$
$968$		0			0
$969$	−917.912	+	2167.49i	−0.947278	+	2.23683i
$970$		0			0
$971$	136.128			0.140194			0.0700970	−	0.997540i	$-0.477669\pi$
$971$	136.128			0.140194			0.0700970	+	0.997540i	$0.477669\pi$
$972$		0			0
$973$			2188.49i			2.24922i
$974$		0			0
$975$	−197.932			−0.203007
$976$		0			0
$977$		−	1270.95i		−	1.30087i	−0.759564	−	0.650433i	$-0.774588\pi$
$977$		−	1270.95i		−	1.30087i	0.759564	−	0.650433i	$-0.225412\pi$
$978$		0			0
$979$	−0.876462			−0.000895262
$980$		0			0
$981$	−3262.35			−3.32553
$982$		0			0
$983$			12.7076i			0.0129274i	0.999979	+	0.00646368i	$0.00205747\pi$
$983$			12.7076i			0.0129274i	−0.999979	+	0.00646368i	$0.997943\pi$
$984$		0			0
$985$	31.7876			0.0322717
$986$		0			0
$987$	785.546			0.795892
$988$		0			0
$989$		−	696.576i		−	0.704323i
$990$		0			0
$991$			133.052i			0.134260i	0.997744	+	0.0671299i	$0.0213842\pi$
$991$			133.052i			0.134260i	−0.997744	+	0.0671299i	$0.978616\pi$
$992$		0			0
$993$	−728.022			−0.733154
$994$		0			0
$995$		−	1173.13i		−	1.17902i
$996$		0			0
$997$		−	568.469i		−	0.570180i	−0.958501	−	0.285090i	$-0.907977\pi$
$997$		−	568.469i		−	0.570180i	0.958501	−	0.285090i	$-0.0920235\pi$
$998$		0			0
$999$	−1657.28			−1.65894

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.g.e.417.4	yes	48
4.3	odd	2	inner	1216.3.g.e.417.48	yes	48
8.3	odd	2	inner	1216.3.g.e.417.3	yes	48
8.5	even	2	inner	1216.3.g.e.417.47	yes	48
19.18	odd	2	inner	1216.3.g.e.417.46	yes	48
76.75	even	2	inner	1216.3.g.e.417.2	yes	48
152.37	odd	2	inner	1216.3.g.e.417.1	✓	48
152.75	even	2	inner	1216.3.g.e.417.45	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.3.g.e.417.1	✓	48	152.37	odd	2	inner
1216.3.g.e.417.2	yes	48	76.75	even	2	inner
1216.3.g.e.417.3	yes	48	8.3	odd	2	inner
1216.3.g.e.417.4	yes	48	1.1	even	1	trivial
1216.3.g.e.417.45	yes	48	152.75	even	2	inner
1216.3.g.e.417.46	yes	48	19.18	odd	2	inner
1216.3.g.e.417.47	yes	48	8.5	even	2	inner
1216.3.g.e.417.48	yes	48	4.3	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.07504 q^{3} -5.51741i q^{5} -8.74188 q^{7} +16.7560 q^{9} +O(q^{10})\)	\(q-5.07504 q^{3} -5.51741i q^{5} -8.74188 q^{7} +16.7560 q^{9} -0.103239i q^{11} -7.16700 q^{13} +28.0010i q^{15} +24.4109 q^{17} +(7.40930 - 17.4958i) q^{19} +44.3654 q^{21} +10.9002 q^{23} -5.44176 q^{25} -39.3621 q^{27} +12.9795 q^{29} +40.6764i q^{31} +0.523940i q^{33} +48.2325i q^{35} +42.1034 q^{37} +36.3728 q^{39} -1.13034i q^{41} -63.9047i q^{43} -92.4497i q^{45} +17.7063 q^{47} +27.4204 q^{49} -123.886 q^{51} -52.4515 q^{53} -0.569609 q^{55} +(-37.6025 + 88.7917i) q^{57} +117.338 q^{59} -26.0056i q^{61} -146.479 q^{63} +39.5432i q^{65} -67.7026 q^{67} -55.3191 q^{69} +92.2133i q^{71} -20.5023 q^{73} +27.6171 q^{75} +0.902499i q^{77} +72.3781i q^{79} +48.9599 q^{81} -69.7748i q^{83} -134.685i q^{85} -65.8716 q^{87} -8.48968i q^{89} +62.6530 q^{91} -206.434i q^{93} +(-96.5313 - 40.8801i) q^{95} -130.846i q^{97} -1.72987i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 264 q^{9}+O(q^{10}) \) 48 * q + 264 * q^9	\( 48 q + 264 q^{9} + 24 q^{17} - 160 q^{25} + 328 q^{49} + 8 q^{57} - 472 q^{73} - 736 q^{81}+O(q^{100}) \) 48 * q + 264 * q^9 + 24 * q^17 - 160 * q^25 + 328 * q^49 + 8 * q^57 - 472 * q^73 - 736 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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