LMFDB - Embedded newform 1216.3.g.e.417.29

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			417.29
Character	$\chi$	$=$	1216.417
Dual form			1216.3.g.e.417.32

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73941 q^{3} +5.26292i q^{5} +4.69464 q^{7} -1.49561 q^{9} +O(q^{10})$	$q+2.73941 q^{3} +5.26292i q^{5} +4.69464 q^{7} -1.49561 q^{9} -10.2686i q^{11} +17.6737 q^{13} +14.4173i q^{15} +25.2750 q^{17} +(-11.7565 + 14.9259i) q^{19} +12.8606 q^{21} +10.4985 q^{23} -2.69834 q^{25} -28.7518 q^{27} -27.2666 q^{29} -34.2018i q^{31} -28.1299i q^{33} +24.7075i q^{35} +69.6637 q^{37} +48.4156 q^{39} +45.1232i q^{41} +7.45023i q^{43} -7.87130i q^{45} +45.8506 q^{47} -26.9604 q^{49} +69.2387 q^{51} +18.3757 q^{53} +54.0427 q^{55} +(-32.2060 + 40.8883i) q^{57} +71.0173 q^{59} +75.5544i q^{61} -7.02137 q^{63} +93.0153i q^{65} -96.9126 q^{67} +28.7597 q^{69} -19.5780i q^{71} -62.7616 q^{73} -7.39187 q^{75} -48.2072i q^{77} +62.4377i q^{79} -65.3026 q^{81} +101.658i q^{83} +133.020i q^{85} -74.6944 q^{87} -73.5179i q^{89} +82.9716 q^{91} -93.6928i q^{93} +(-78.5541 - 61.8737i) q^{95} +107.607i q^{97} +15.3578i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.73941			0.913138			0.456569	−	0.889688i	$-0.349078\pi$
$3$	2.73941			0.913138			0.456569	+	0.889688i	$0.349078\pi$
$4$		0			0
$5$			5.26292i			1.05258i	0.850304	+	0.526292i	$0.176418\pi$
$5$			5.26292i			1.05258i	−0.850304	+	0.526292i	$0.823582\pi$
$6$		0			0
$7$	4.69464			0.670662			0.335331	−	0.942100i	$-0.391152\pi$
$7$	4.69464			0.670662			0.335331	+	0.942100i	$0.391152\pi$
$8$		0			0
$9$	−1.49561			−0.166179
$10$		0			0
$11$		−	10.2686i		−	0.933507i	−0.884388	−	0.466753i	$-0.845424\pi$
$11$		−	10.2686i		−	0.933507i	0.884388	−	0.466753i	$-0.154576\pi$
$12$		0			0
$13$	17.6737			1.35952			0.679758	−	0.733437i	$-0.262085\pi$
$13$	17.6737			1.35952			0.679758	+	0.733437i	$0.262085\pi$
$14$		0			0
$15$			14.4173i			0.961155i
$16$		0			0
$17$	25.2750			1.48676			0.743382	−	0.668867i	$-0.233220\pi$
$17$	25.2750			1.48676			0.743382	+	0.668867i	$0.233220\pi$
$18$		0			0
$19$	−11.7565	+	14.9259i	−0.618765	+	0.785576i
$19$	−11.7565	+	14.9259i	−0.618765	+	0.785576i
$20$		0			0
$21$	12.8606			0.612407
$22$		0			0
$23$	10.4985			0.456455			0.228228	−	0.973608i	$-0.426707\pi$
$23$	10.4985			0.456455			0.228228	+	0.973608i	$0.426707\pi$
$24$		0			0
$25$	−2.69834			−0.107934
$26$		0			0
$27$	−28.7518			−1.06488
$28$		0			0
$29$	−27.2666			−0.940226			−0.470113	−	0.882606i	$-0.655787\pi$
$29$	−27.2666			−0.940226			−0.470113	+	0.882606i	$0.655787\pi$
$30$		0			0
$31$		−	34.2018i		−	1.10328i	−0.834081	−	0.551642i	$-0.814002\pi$
$31$		−	34.2018i		−	1.10328i	0.834081	−	0.551642i	$-0.185998\pi$
$32$		0			0
$33$		−	28.1299i		−	0.852420i
$34$		0			0
$35$			24.7075i			0.705929i
$36$		0			0
$37$	69.6637			1.88280			0.941402	−	0.337287i	$-0.109509\pi$
$37$	69.6637			1.88280			0.941402	+	0.337287i	$0.109509\pi$
$38$		0			0
$39$	48.4156			1.24142
$40$		0			0
$41$			45.1232i			1.10057i	0.834978	+	0.550283i	$0.185480\pi$
$41$			45.1232i			1.10057i	−0.834978	+	0.550283i	$0.814520\pi$
$42$		0			0
$43$			7.45023i			0.173261i	0.996241	+	0.0866306i	$0.0276100\pi$
$43$			7.45023i			0.173261i	−0.996241	+	0.0866306i	$0.972390\pi$
$44$		0			0
$45$		−	7.87130i		−	0.174918i
$46$		0			0
$47$	45.8506			0.975545			0.487772	−	0.872971i	$-0.337810\pi$
$47$	45.8506			0.975545			0.487772	+	0.872971i	$0.337810\pi$
$48$		0			0
$49$	−26.9604			−0.550212
$50$		0			0
$51$	69.2387			1.35762
$52$		0			0
$53$	18.3757			0.346711			0.173356	−	0.984859i	$-0.444539\pi$
$53$	18.3757			0.346711			0.173356	+	0.984859i	$0.444539\pi$
$54$		0			0
$55$	54.0427			0.982594
$56$		0			0
$57$	−32.2060	+	40.8883i	−0.565018	+	0.717339i
$58$		0			0
$59$	71.0173			1.20368			0.601841	−	0.798616i	$-0.294434\pi$
$59$	71.0173			1.20368			0.601841	+	0.798616i	$0.294434\pi$
$60$		0			0
$61$			75.5544i			1.23860i	0.785156	+	0.619298i	$0.212583\pi$
$61$			75.5544i			1.23860i	−0.785156	+	0.619298i	$0.787417\pi$
$62$		0			0
$63$	−7.02137			−0.111450
$64$		0			0
$65$			93.0153i			1.43100i
$66$		0			0
$67$	−96.9126			−1.44646			−0.723228	−	0.690609i	$-0.757343\pi$
$67$	−96.9126			−1.44646			−0.723228	+	0.690609i	$0.757343\pi$
$68$		0			0
$69$	28.7597			0.416807
$70$		0			0
$71$		−	19.5780i		−	0.275746i	−0.990450	−	0.137873i	$-0.955973\pi$
$71$		−	19.5780i		−	0.275746i	0.990450	−	0.137873i	$-0.0440265\pi$
$72$		0			0
$73$	−62.7616			−0.859748			−0.429874	−	0.902889i	$-0.641442\pi$
$73$	−62.7616			−0.859748			−0.429874	+	0.902889i	$0.641442\pi$
$74$		0			0
$75$	−7.39187			−0.0985583
$76$		0			0
$77$		−	48.2072i		−	0.626068i
$78$		0			0
$79$			62.4377i			0.790350i	0.918606	+	0.395175i	$0.129316\pi$
$79$			62.4377i			0.790350i	−0.918606	+	0.395175i	$0.870684\pi$
$80$		0			0
$81$	−65.3026			−0.806205
$82$		0			0
$83$			101.658i			1.22480i	0.790549	+	0.612399i	$0.209795\pi$
$83$			101.658i			1.22480i	−0.790549	+	0.612399i	$0.790205\pi$
$84$		0			0
$85$			133.020i			1.56495i
$86$		0			0
$87$	−74.6944			−0.858556
$88$		0			0
$89$		−	73.5179i		−	0.826043i	−0.910721	−	0.413022i	$-0.864473\pi$
$89$		−	73.5179i		−	0.826043i	0.910721	−	0.413022i	$-0.135527\pi$
$90$		0			0
$91$	82.9716			0.911776
$92$		0			0
$93$		−	93.6928i		−	1.00745i
$94$		0			0
$95$	−78.5541	−	61.8737i	−0.826885	−	0.651303i
$96$		0			0
$97$			107.607i			1.10935i	0.832068	+	0.554674i	$0.187157\pi$
$97$			107.607i			1.10935i	−0.832068	+	0.554674i	$0.812843\pi$
$98$		0			0
$99$			15.3578i			0.155130i
$100$		0			0
$101$			160.350i			1.58763i	0.608160	+	0.793814i	$0.291908\pi$
$101$			160.350i			1.58763i	−0.608160	+	0.793814i	$0.708092\pi$
$102$		0			0
$103$		−	189.735i		−	1.84209i	−0.389456	−	0.921045i	$-0.627337\pi$
$103$		−	189.735i		−	1.84209i	0.389456	−	0.921045i	$-0.372663\pi$
$104$		0			0
$105$			67.6841i			0.644610i
$106$		0			0
$107$	148.162			1.38469			0.692347	−	0.721565i	$-0.256577\pi$
$107$	148.162			1.38469			0.692347	+	0.721565i	$0.256577\pi$
$108$		0			0
$109$	50.2608			0.461108			0.230554	−	0.973060i	$-0.425946\pi$
$109$	50.2608			0.461108			0.230554	+	0.973060i	$0.425946\pi$
$110$		0			0
$111$	190.838			1.71926
$112$		0			0
$113$		−	186.687i		−	1.65210i	−0.563600	−	0.826048i	$-0.690584\pi$
$113$		−	186.687i		−	1.65210i	0.563600	−	0.826048i	$-0.309416\pi$
$114$		0			0
$115$			55.2526i			0.480458i
$116$		0			0
$117$	−26.4330			−0.225923
$118$		0			0
$119$	118.657			0.997117
$120$		0			0
$121$	15.5564			0.128566
$122$		0			0
$123$			123.611i			1.00497i
$124$		0			0
$125$			117.372i			0.938975i
$126$		0			0
$127$		−	24.5981i		−	0.193686i	−0.995300	−	0.0968430i	$-0.969126\pi$
$127$		−	24.5981i		−	0.193686i	0.995300	−	0.0968430i	$-0.0308745\pi$
$128$		0			0
$129$			20.4093i			0.158211i
$130$		0			0
$131$			65.8406i			0.502600i	0.967909	+	0.251300i	$0.0808581\pi$
$131$			65.8406i			0.502600i	−0.967909	+	0.251300i	$0.919142\pi$
$132$		0			0
$133$	−55.1927	+	70.0719i	−0.414983	+	0.526856i
$134$		0			0
$135$		−	151.319i		−	1.12088i
$136$		0			0
$137$	−201.126			−1.46807			−0.734037	−	0.679109i	$-0.762366\pi$
$137$	−201.126			−1.46807			−0.734037	+	0.679109i	$0.762366\pi$
$138$		0			0
$139$		−	102.084i		−	0.734417i	−0.930139	−	0.367209i	$-0.880314\pi$
$139$		−	102.084i		−	0.734417i	0.930139	−	0.367209i	$-0.119686\pi$
$140$		0			0
$141$	125.604			0.890807
$142$		0			0
$143$		−	181.484i		−	1.26912i
$144$		0			0
$145$		−	143.502i		−	0.989667i
$146$		0			0
$147$	−73.8556			−0.502419
$148$		0			0
$149$			85.2086i			0.571870i	0.958249	+	0.285935i	$0.0923041\pi$
$149$			85.2086i			0.571870i	−0.958249	+	0.285935i	$0.907696\pi$
$150$		0			0
$151$		−	30.9703i		−	0.205101i	−0.994728	−	0.102551i	$-0.967300\pi$
$151$		−	30.9703i		−	0.205101i	0.994728	−	0.102551i	$-0.0327003\pi$
$152$		0			0
$153$	−37.8016			−0.247070
$154$		0			0
$155$	180.001			1.16130
$156$		0			0
$157$			127.135i			0.809777i	0.914366	+	0.404888i	$0.132690\pi$
$157$			127.135i			0.809777i	−0.914366	+	0.404888i	$0.867310\pi$
$158$		0			0
$159$	50.3386			0.316595
$160$		0			0
$161$	49.2865			0.306127
$162$		0			0
$163$		−	295.514i		−	1.81297i	−0.422240	−	0.906484i	$-0.638756\pi$
$163$		−	295.514i		−	1.81297i	0.422240	−	0.906484i	$-0.361244\pi$
$164$		0			0
$165$	148.045			0.897244
$166$		0			0
$167$			89.2035i			0.534153i	0.963675	+	0.267076i	$0.0860576\pi$
$167$			89.2035i			0.534153i	−0.963675	+	0.267076i	$0.913942\pi$
$168$		0			0
$169$	143.360			0.848281
$170$		0			0
$171$	17.5832	−	22.3235i	0.102826	−	0.130547i
$172$		0			0
$173$	−29.8953			−0.172805			−0.0864027	−	0.996260i	$-0.527537\pi$
$173$	−29.8953			−0.172805			−0.0864027	+	0.996260i	$0.527537\pi$
$174$		0			0
$175$	−12.6677			−0.0723871
$176$		0			0
$177$	194.546			1.09913
$178$		0			0
$179$	18.6560			0.104224			0.0521119	−	0.998641i	$-0.483405\pi$
$179$	18.6560			0.104224			0.0521119	+	0.998641i	$0.483405\pi$
$180$		0			0
$181$	9.73288			0.0537728			0.0268864	−	0.999638i	$-0.491441\pi$
$181$	9.73288			0.0537728			0.0268864	+	0.999638i	$0.491441\pi$
$182$		0			0
$183$			206.975i			1.13101i
$184$		0			0
$185$			366.635i			1.98181i
$186$		0			0
$187$		−	259.538i		−	1.38790i
$188$		0			0
$189$	−134.979			−0.714177
$190$		0			0
$191$	50.1333			0.262478			0.131239	−	0.991351i	$-0.458104\pi$
$191$	50.1333			0.262478			0.131239	+	0.991351i	$0.458104\pi$
$192$		0			0
$193$		−	128.499i		−	0.665797i	−0.942963	−	0.332899i	$-0.891973\pi$
$193$		−	128.499i		−	0.665797i	0.942963	−	0.332899i	$-0.108027\pi$
$194$		0			0
$195$			254.807i			1.30670i
$196$		0			0
$197$			147.060i			0.746496i	0.927732	+	0.373248i	$0.121756\pi$
$197$			147.060i			0.746496i	−0.927732	+	0.373248i	$0.878244\pi$
$198$		0			0
$199$	105.724			0.531276			0.265638	−	0.964073i	$-0.414417\pi$
$199$	105.724			0.531276			0.265638	+	0.964073i	$0.414417\pi$
$200$		0			0
$201$	−265.484			−1.32081
$202$		0			0
$203$	−128.007			−0.630574
$204$		0			0
$205$	−237.480			−1.15844
$206$		0			0
$207$	−15.7017			−0.0758535
$208$		0			0
$209$	153.268	+	120.723i	0.733340	+	0.577621i
$210$		0			0
$211$	−301.385			−1.42836			−0.714181	−	0.699961i	$-0.753201\pi$
$211$	−301.385			−1.42836			−0.714181	+	0.699961i	$0.753201\pi$
$212$		0			0
$213$		−	53.6321i		−	0.251794i
$214$		0			0
$215$	−39.2100			−0.182372
$216$		0			0
$217$		−	160.565i		−	0.739931i
$218$		0			0
$219$	−171.930			−0.785069
$220$		0			0
$221$	446.703			2.02128
$222$		0			0
$223$		−	399.952i		−	1.79351i	−0.442531	−	0.896753i	$-0.645919\pi$
$223$		−	399.952i		−	1.79351i	0.442531	−	0.896753i	$-0.354081\pi$
$224$		0			0
$225$	4.03568			0.0179364
$226$		0			0
$227$	5.58217			0.0245910			0.0122955	−	0.999924i	$-0.496086\pi$
$227$	5.58217			0.0245910			0.0122955	+	0.999924i	$0.496086\pi$
$228$		0			0
$229$		−	378.433i		−	1.65254i	−0.563272	−	0.826272i	$-0.690457\pi$
$229$		−	378.433i		−	1.65254i	0.563272	−	0.826272i	$-0.309543\pi$
$230$		0			0
$231$		−	132.060i		−	0.571686i
$232$		0			0
$233$	−375.953			−1.61353			−0.806766	−	0.590871i	$-0.798784\pi$
$233$	−375.953			−1.61353			−0.806766	+	0.590871i	$0.798784\pi$
$234$		0			0
$235$			241.308i			1.02684i
$236$		0			0
$237$			171.043i			0.721699i
$238$		0			0
$239$	−54.6197			−0.228534			−0.114267	−	0.993450i	$-0.536452\pi$
$239$	−54.6197			−0.228534			−0.114267	+	0.993450i	$0.536452\pi$
$240$		0			0
$241$		−	138.818i		−	0.576008i	−0.957629	−	0.288004i	$-0.907008\pi$
$241$		−	138.818i		−	0.576008i	0.957629	−	0.288004i	$-0.0929916\pi$
$242$		0			0
$243$	79.8756			0.328706
$244$		0			0
$245$		−	141.890i		−	0.579144i
$246$		0			0
$247$	−207.781	+	263.797i	−0.841221	+	1.06800i
$248$		0			0
$249$			278.484i			1.11841i
$250$		0			0
$251$			222.994i			0.888424i	0.895922	+	0.444212i	$0.146516\pi$
$251$			222.994i			0.888424i	−0.895922	+	0.444212i	$0.853484\pi$
$252$		0			0
$253$		−	107.804i		−	0.426104i
$254$		0			0
$255$			364.398i			1.42901i
$256$		0			0
$257$		−	173.190i		−	0.673891i	−0.941524	−	0.336945i	$-0.890606\pi$
$257$		−	173.190i		−	0.673891i	0.941524	−	0.336945i	$-0.109394\pi$
$258$		0			0
$259$	327.046			1.26273
$260$		0			0
$261$	40.7802			0.156246
$262$		0			0
$263$	209.372			0.796090			0.398045	−	0.917366i	$-0.369689\pi$
$263$	209.372			0.796090			0.398045	+	0.917366i	$0.369689\pi$
$264$		0			0
$265$			96.7098i			0.364943i
$266$		0			0
$267$		−	201.396i		−	0.754292i
$268$		0			0
$269$	78.6403			0.292343			0.146172	−	0.989259i	$-0.453305\pi$
$269$	78.6403			0.292343			0.146172	+	0.989259i	$0.453305\pi$
$270$		0			0
$271$	110.916			0.409285			0.204642	−	0.978837i	$-0.434397\pi$
$271$	110.916			0.409285			0.204642	+	0.978837i	$0.434397\pi$
$272$		0			0
$273$	227.293			0.832577
$274$		0			0
$275$			27.7081i			0.100757i
$276$		0			0
$277$		−	357.833i		−	1.29181i	−0.763416	−	0.645907i	$-0.776479\pi$
$277$		−	357.833i		−	1.29181i	0.763416	−	0.645907i	$-0.223521\pi$
$278$		0			0
$279$			51.1527i			0.183343i
$280$		0			0
$281$		−	183.167i		−	0.651839i	−0.945398	−	0.325919i	$-0.894326\pi$
$281$		−	183.167i		−	0.651839i	0.945398	−	0.325919i	$-0.105674\pi$
$282$		0			0
$283$		−	158.895i		−	0.561465i	−0.959786	−	0.280732i	$-0.909423\pi$
$283$		−	158.895i		−	0.561465i	0.959786	−	0.280732i	$-0.0905773\pi$
$284$		0			0
$285$	−215.192	−	169.498i	−0.755060	−	0.594729i
$286$		0			0
$287$			211.837i			0.738109i
$288$		0			0
$289$	349.826			1.21047
$290$		0			0
$291$			294.779i			1.01299i
$292$		0			0
$293$	−128.281			−0.437817			−0.218909	−	0.975745i	$-0.570250\pi$
$293$	−128.281			−0.437817			−0.218909	+	0.975745i	$0.570250\pi$
$294$		0			0
$295$			373.758i			1.26698i
$296$		0			0
$297$			295.240i			0.994075i
$298$		0			0
$299$	185.547			0.620558
$300$		0			0
$301$			34.9761i			0.116200i
$302$		0			0
$303$			439.266i			1.44972i
$304$		0			0
$305$	−397.637			−1.30373
$306$		0			0
$307$	−259.621			−0.845670			−0.422835	−	0.906207i	$-0.638965\pi$
$307$	−259.621			−0.845670			−0.422835	+	0.906207i	$0.638965\pi$
$308$		0			0
$309$		−	519.763i		−	1.68208i
$310$		0			0
$311$	−556.165			−1.78831			−0.894157	−	0.447754i	$-0.852224\pi$
$311$	−556.165			−1.78831			−0.894157	+	0.447754i	$0.852224\pi$
$312$		0			0
$313$	197.344			0.630492			0.315246	−	0.949010i	$-0.397913\pi$
$313$	197.344			0.630492			0.315246	+	0.949010i	$0.397913\pi$
$314$		0			0
$315$		−	36.9529i		−	0.117311i
$316$		0			0
$317$	−46.0689			−0.145328			−0.0726639	−	0.997356i	$-0.523150\pi$
$317$	−46.0689			−0.145328			−0.0726639	+	0.997356i	$0.523150\pi$
$318$		0			0
$319$			279.989i			0.877707i
$320$		0			0
$321$	405.877			1.26442
$322$		0			0
$323$	−297.147	+	377.253i	−0.919958	+	1.16797i
$324$		0			0
$325$	−47.6897			−0.146737
$326$		0			0
$327$	137.685			0.421055
$328$		0			0
$329$	215.252			0.654261
$330$		0			0
$331$	202.375			0.611405			0.305703	−	0.952127i	$-0.401109\pi$
$331$	202.375			0.611405			0.305703	+	0.952127i	$0.401109\pi$
$332$		0			0
$333$	−104.190			−0.312883
$334$		0			0
$335$		−	510.043i		−	1.52252i
$336$		0			0
$337$			20.5252i			0.0609056i	0.999536	+	0.0304528i	$0.00969493\pi$
$337$			20.5252i			0.0609056i	−0.999536	+	0.0304528i	$0.990305\pi$
$338$		0			0
$339$		−	511.412i		−	1.50859i
$340$		0			0
$341$	−351.203			−1.02992
$342$		0			0
$343$	−356.606			−1.03967
$344$		0			0
$345$			151.360i			0.438724i
$346$		0			0
$347$		−	197.014i		−	0.567763i	−0.958859	−	0.283881i	$-0.908378\pi$
$347$		−	197.014i		−	0.567763i	0.958859	−	0.283881i	$-0.0916222\pi$
$348$		0			0
$349$			219.206i			0.628098i	0.949407	+	0.314049i	$0.101686\pi$
$349$			219.206i			0.628098i	−0.949407	+	0.314049i	$0.898314\pi$
$350$		0			0
$351$	−508.151			−1.44772
$352$		0			0
$353$	218.242			0.618249			0.309124	−	0.951022i	$-0.399964\pi$
$353$	218.242			0.618249			0.309124	+	0.951022i	$0.399964\pi$
$354$		0			0
$355$	103.037			0.290246
$356$		0			0
$357$	325.050			0.910506
$358$		0			0
$359$	−67.2437			−0.187308			−0.0936541	−	0.995605i	$-0.529855\pi$
$359$	−67.2437			−0.187308			−0.0936541	+	0.995605i	$0.529855\pi$
$360$		0			0
$361$	−84.5676	−	350.955i	−0.234259	−	0.972174i
$362$		0			0
$363$	42.6155			0.117398
$364$		0			0
$365$		−	330.310i		−	0.904958i
$366$		0			0
$367$	234.617			0.639283			0.319642	−	0.947538i	$-0.396437\pi$
$367$	234.617			0.639283			0.319642	+	0.947538i	$0.396437\pi$
$368$		0			0
$369$		−	67.4869i		−	0.182891i
$370$		0			0
$371$	86.2672			0.232526
$372$		0			0
$373$	−393.307			−1.05444			−0.527221	−	0.849728i	$-0.676766\pi$
$373$	−393.307			−1.05444			−0.527221	+	0.849728i	$0.676766\pi$
$374$		0			0
$375$			321.530i			0.857414i
$376$		0			0
$377$	−481.901			−1.27825
$378$		0			0
$379$	−459.442			−1.21225			−0.606124	−	0.795370i	$-0.707277\pi$
$379$	−459.442			−1.21225			−0.606124	+	0.795370i	$0.707277\pi$
$380$		0			0
$381$		−	67.3844i		−	0.176862i
$382$		0			0
$383$			87.1105i			0.227442i	0.993513	+	0.113721i	$0.0362771\pi$
$383$			87.1105i			0.227442i	−0.993513	+	0.113721i	$0.963723\pi$
$384$		0			0
$385$	253.711			0.658989
$386$		0			0
$387$		−	11.1427i		−	0.0287924i
$388$		0			0
$389$			317.883i			0.817179i	0.912718	+	0.408589i	$0.133979\pi$
$389$			317.883i			0.817179i	−0.912718	+	0.408589i	$0.866021\pi$
$390$		0			0
$391$	265.349			0.678642
$392$		0			0
$393$			180.365i			0.458943i
$394$		0			0
$395$	−328.605			−0.831910
$396$		0			0
$397$			338.098i			0.851632i	0.904810	+	0.425816i	$0.140013\pi$
$397$			338.098i			0.851632i	−0.904810	+	0.425816i	$0.859987\pi$
$398$		0			0
$399$	−151.196	+	191.956i	−0.378936	+	0.481092i
$400$		0			0
$401$		−	798.720i		−	1.99182i	−0.0903539	−	0.995910i	$-0.528800\pi$
$401$		−	798.720i		−	1.99182i	0.0903539	−	0.995910i	$-0.471200\pi$
$402$		0			0
$403$		−	604.472i		−	1.49993i
$404$		0			0
$405$		−	343.683i		−	0.848599i
$406$		0			0
$407$		−	715.347i		−	1.75761i
$408$		0			0
$409$			237.012i			0.579492i	0.957104	+	0.289746i	$0.0935708\pi$
$409$			237.012i			0.579492i	−0.957104	+	0.289746i	$0.906429\pi$
$410$		0			0
$411$	−550.968			−1.34055
$412$		0			0
$413$	333.400			0.807265
$414$		0			0
$415$	−535.019			−1.28920
$416$		0			0
$417$		−	279.650i		−	0.670624i
$418$		0			0
$419$		−	124.585i		−	0.297338i	−0.988887	−	0.148669i	$-0.952501\pi$
$419$		−	124.585i		−	0.297338i	0.988887	−	0.148669i	$-0.0474988\pi$
$420$		0			0
$421$	−448.383			−1.06504			−0.532522	−	0.846416i	$-0.678756\pi$
$421$	−448.383			−1.06504			−0.532522	+	0.846416i	$0.678756\pi$
$422$		0			0
$423$	−68.5748			−0.162115
$424$		0			0
$425$	−68.2006			−0.160472
$426$		0			0
$427$			354.700i			0.830680i
$428$		0			0
$429$		−	497.159i		−	1.15888i
$430$		0			0
$431$			174.115i			0.403979i	0.979388	+	0.201989i	$0.0647406\pi$
$431$			174.115i			0.403979i	−0.979388	+	0.201989i	$0.935259\pi$
$432$		0			0
$433$		−	461.148i		−	1.06501i	−0.846428	−	0.532503i	$-0.821251\pi$
$433$		−	461.148i		−	1.06501i	0.846428	−	0.532503i	$-0.178749\pi$
$434$		0			0
$435$		−	393.111i		−	0.903703i
$436$		0			0
$437$	−123.426	+	156.700i	−0.282439	+	0.358580i
$438$		0			0
$439$		−	435.302i		−	0.991577i	−0.868443	−	0.495788i	$-0.834879\pi$
$439$		−	435.302i		−	0.991577i	0.868443	−	0.495788i	$-0.165121\pi$
$440$		0			0
$441$	40.3223			0.0914338
$442$		0			0
$443$		−	160.935i		−	0.363283i	−0.983365	−	0.181642i	$-0.941859\pi$
$443$		−	160.935i		−	0.363283i	0.983365	−	0.181642i	$-0.0581411\pi$
$444$		0			0
$445$	386.919			0.869480
$446$		0			0
$447$			233.421i			0.522196i
$448$		0			0
$449$			171.555i			0.382082i	0.981582	+	0.191041i	$0.0611864\pi$
$449$			171.555i			0.382082i	−0.981582	+	0.191041i	$0.938814\pi$
$450$		0			0
$451$	463.351			1.02739
$452$		0			0
$453$		−	84.8403i		−	0.187285i
$454$		0			0
$455$			436.673i			0.959721i
$456$		0			0
$457$	−361.665			−0.791390			−0.395695	−	0.918382i	$-0.629496\pi$
$457$	−361.665			−0.791390			−0.395695	+	0.918382i	$0.629496\pi$
$458$		0			0
$459$	−726.702			−1.58323
$460$		0			0
$461$		−	406.221i		−	0.881174i	−0.897710	−	0.440587i	$-0.854770\pi$
$461$		−	406.221i		−	0.881174i	0.897710	−	0.440587i	$-0.145230\pi$
$462$		0			0
$463$	−815.644			−1.76165			−0.880825	−	0.473441i	$-0.843012\pi$
$463$	−815.644			−1.76165			−0.880825	+	0.473441i	$0.843012\pi$
$464$		0			0
$465$	493.098			1.06043
$466$		0			0
$467$		−	711.741i		−	1.52407i	−0.647536	−	0.762035i	$-0.724200\pi$
$467$		−	711.741i		−	1.52407i	0.647536	−	0.762035i	$-0.275800\pi$
$468$		0			0
$469$	−454.969			−0.970084
$470$		0			0
$471$			348.275i			0.739438i
$472$		0			0
$473$	76.5032			0.161740
$474$		0			0
$475$	31.7232	−	40.2753i	0.0667856	−	0.0847901i
$476$		0			0
$477$	−27.4829			−0.0576162
$478$		0			0
$479$	752.130			1.57021			0.785104	−	0.619364i	$-0.212610\pi$
$479$	752.130			1.57021			0.785104	+	0.619364i	$0.212610\pi$
$480$		0			0
$481$	1231.22			2.55970
$482$		0			0
$483$	135.016			0.279537
$484$		0			0
$485$	−566.326			−1.16768
$486$		0			0
$487$		−	725.897i		−	1.49055i	−0.666758	−	0.745274i	$-0.732319\pi$
$487$		−	725.897i		−	1.49055i	0.666758	−	0.745274i	$-0.267681\pi$
$488$		0			0
$489$		−	809.534i		−	1.65549i
$490$		0			0
$491$		−	732.543i		−	1.49194i	−0.665979	−	0.745971i	$-0.731986\pi$
$491$		−	732.543i		−	1.49194i	0.665979	−	0.745971i	$-0.268014\pi$
$492$		0			0
$493$	−689.162			−1.39790
$494$		0			0
$495$	−80.8270			−0.163287
$496$		0			0
$497$		−	91.9114i		−	0.184932i
$498$		0			0
$499$			110.238i			0.220919i	0.993881	+	0.110459i	$0.0352322\pi$
$499$			110.238i			0.220919i	−0.993881	+	0.110459i	$0.964768\pi$
$500$		0			0
$501$			244.365i			0.487755i
$502$		0			0
$503$	−259.861			−0.516622			−0.258311	−	0.966062i	$-0.583166\pi$
$503$	−259.861			−0.516622			−0.258311	+	0.966062i	$0.583166\pi$
$504$		0			0
$505$	−843.912			−1.67111
$506$		0			0
$507$	392.721			0.774598
$508$		0			0
$509$	−914.579			−1.79681			−0.898407	−	0.439163i	$-0.855275\pi$
$509$	−914.579			−1.79681			−0.898407	+	0.439163i	$0.855275\pi$
$510$		0			0
$511$	−294.643			−0.576601
$512$		0			0
$513$	338.022	−	429.148i	0.658912	−	0.836546i
$514$		0			0
$515$	998.562			1.93896
$516$		0			0
$517$		−	470.820i		−	0.910677i
$518$		0			0
$519$	−81.8956			−0.157795
$520$		0			0
$521$			1019.80i			1.95739i	0.205318	+	0.978695i	$0.434177\pi$
$521$			1019.80i			1.95739i	−0.205318	+	0.978695i	$0.565823\pi$
$522$		0			0
$523$	448.157			0.856897			0.428449	−	0.903566i	$-0.359060\pi$
$523$	448.157			0.856897			0.428449	+	0.903566i	$0.359060\pi$
$524$		0			0
$525$	−34.7022			−0.0660994
$526$		0			0
$527$		−	864.450i		−	1.64032i
$528$		0			0
$529$	−418.782			−0.791648
$530$		0			0
$531$	−106.214			−0.200027
$532$		0			0
$533$			797.494i			1.49624i
$534$		0			0
$535$			779.766i			1.45751i
$536$		0			0
$537$	51.1066			0.0951706
$538$		0			0
$539$			276.845i			0.513626i
$540$		0			0
$541$		−	500.797i		−	0.925687i	−0.886440	−	0.462843i	$-0.846829\pi$
$541$		−	500.797i		−	0.925687i	0.886440	−	0.462843i	$-0.153171\pi$
$542$		0			0
$543$	26.6624			0.0491020
$544$		0			0
$545$			264.519i			0.485355i
$546$		0			0
$547$	−886.809			−1.62122			−0.810612	−	0.585584i	$-0.800865\pi$
$547$	−886.809			−1.62122			−0.810612	+	0.585584i	$0.800865\pi$
$548$		0			0
$549$		−	113.000i		−	0.205829i
$550$		0			0
$551$	320.560	−	406.979i	0.581779	−	0.738619i
$552$		0			0
$553$			293.122i			0.530058i
$554$		0			0
$555$			1004.36i			1.80967i
$556$		0			0
$557$			403.210i			0.723896i	0.932198	+	0.361948i	$0.117888\pi$
$557$			403.210i			0.723896i	−0.932198	+	0.361948i	$0.882112\pi$
$558$		0			0
$559$			131.673i			0.235551i
$560$		0			0
$561$		−	710.982i		−	1.26735i
$562$		0			0
$563$	−494.180			−0.877762			−0.438881	−	0.898545i	$-0.644625\pi$
$563$	−494.180			−0.877762			−0.438881	+	0.898545i	$0.644625\pi$
$564$		0			0
$565$	982.518			1.73897
$566$		0			0
$567$	−306.572			−0.540691
$568$		0			0
$569$			588.191i			1.03373i	0.856068	+	0.516864i	$0.172901\pi$
$569$			588.191i			1.03373i	−0.856068	+	0.516864i	$0.827099\pi$
$570$		0			0
$571$		−	185.627i		−	0.325092i	−0.986701	−	0.162546i	$-0.948029\pi$
$571$		−	185.627i		−	0.325092i	0.986701	−	0.162546i	$-0.0519706\pi$
$572$		0			0
$573$	137.336			0.239679
$574$		0			0
$575$	−28.3285			−0.0492669
$576$		0			0
$577$	−128.772			−0.223174			−0.111587	−	0.993755i	$-0.535593\pi$
$577$	−128.772			−0.223174			−0.111587	+	0.993755i	$0.535593\pi$
$578$		0			0
$579$		−	352.011i		−	0.607965i
$580$		0			0
$581$			477.248i			0.821426i
$582$		0			0
$583$		−	188.692i		−	0.323657i
$584$		0			0
$585$		−	139.115i		−	0.237803i
$586$		0			0
$587$			867.915i			1.47856i	0.673397	+	0.739281i	$0.264834\pi$
$587$			867.915i			1.47856i	−0.673397	+	0.739281i	$0.735166\pi$
$588$		0			0
$589$	510.494	+	402.095i	0.866713	+	0.682673i
$590$		0			0
$591$			402.857i			0.681654i
$592$		0			0
$593$	452.273			0.762687			0.381344	−	0.924433i	$-0.375462\pi$
$593$	452.273			0.762687			0.381344	+	0.924433i	$0.375462\pi$
$594$		0			0
$595$			624.482i			1.04955i
$596$		0			0
$597$	289.622			0.485128
$598$		0			0
$599$			733.094i			1.22386i	0.790910	+	0.611932i	$0.209607\pi$
$599$			733.094i			1.22386i	−0.790910	+	0.611932i	$0.790393\pi$
$600$		0			0
$601$			488.835i			0.813370i	0.913568	+	0.406685i	$0.133315\pi$
$601$			488.835i			0.813370i	−0.913568	+	0.406685i	$0.866685\pi$
$602$		0			0
$603$	144.944			0.240371
$604$		0			0
$605$			81.8723i			0.135326i
$606$		0			0
$607$			303.240i			0.499572i	0.968301	+	0.249786i	$0.0803603\pi$
$607$			303.240i			0.499572i	−0.968301	+	0.249786i	$0.919640\pi$
$608$		0			0
$609$	−350.663			−0.575801
$610$		0			0
$611$	810.350			1.32627
$612$		0			0
$613$		−	570.220i		−	0.930212i	−0.885255	−	0.465106i	$-0.846016\pi$
$613$		−	570.220i		−	0.930212i	0.885255	−	0.465106i	$-0.153984\pi$
$614$		0			0
$615$	−650.556			−1.05781
$616$		0			0
$617$	−67.8845			−0.110024			−0.0550118	−	0.998486i	$-0.517520\pi$
$617$	−67.8845			−0.110024			−0.0550118	+	0.998486i	$0.517520\pi$
$618$		0			0
$619$			584.121i			0.943652i	0.881692	+	0.471826i	$0.156405\pi$
$619$			584.121i			0.943652i	−0.881692	+	0.471826i	$0.843595\pi$
$620$		0			0
$621$	−301.850			−0.486071
$622$		0			0
$623$		−	345.140i		−	0.553996i
$624$		0			0
$625$	−685.178			−1.09628
$626$		0			0
$627$	419.865	+	330.710i	0.669641	+	0.527448i
$628$		0			0
$629$	1760.75			2.79929
$630$		0			0
$631$	−566.835			−0.898311			−0.449156	−	0.893453i	$-0.648275\pi$
$631$	−566.835			−0.898311			−0.449156	+	0.893453i	$0.648275\pi$
$632$		0			0
$633$	−825.617			−1.30429
$634$		0			0
$635$	129.458			0.203871
$636$		0			0
$637$	−476.490			−0.748021
$638$		0			0
$639$			29.2811i			0.0458233i
$640$		0			0
$641$			714.270i			1.11431i	0.830410	+	0.557153i	$0.188106\pi$
$641$			714.270i			1.11431i	−0.830410	+	0.557153i	$0.811894\pi$
$642$		0			0
$643$		−	483.851i		−	0.752490i	−0.926520	−	0.376245i	$-0.877215\pi$
$643$		−	483.851i		−	0.752490i	0.926520	−	0.376245i	$-0.122785\pi$
$644$		0			0
$645$	−107.412			−0.166531
$646$		0			0
$647$	680.997			1.05255			0.526273	−	0.850316i	$-0.323589\pi$
$647$	680.997			1.05255			0.526273	+	0.850316i	$0.323589\pi$
$648$		0			0
$649$		−	729.246i		−	1.12365i
$650$		0			0
$651$		−	439.854i		−	0.675659i
$652$		0			0
$653$			932.232i			1.42761i	0.700342	+	0.713807i	$0.253031\pi$
$653$			932.232i			1.42761i	−0.700342	+	0.713807i	$0.746969\pi$
$654$		0			0
$655$	−346.514			−0.529029
$656$		0			0
$657$	93.8672			0.142872
$658$		0			0
$659$	−465.128			−0.705808			−0.352904	−	0.935659i	$-0.614806\pi$
$659$	−465.128			−0.705808			−0.352904	+	0.935659i	$0.614806\pi$
$660$		0			0
$661$	−961.648			−1.45484			−0.727419	−	0.686193i	$-0.759280\pi$
$661$	−961.648			−1.45484			−0.727419	+	0.686193i	$0.759280\pi$
$662$		0			0
$663$	1223.70			1.84571
$664$		0			0
$665$	−368.783	−	290.475i	−0.554561	−	0.436804i
$666$		0			0
$667$	−286.257			−0.429171
$668$		0			0
$669$		−	1095.63i		−	1.63772i
$670$		0			0
$671$	775.835			1.15624
$672$		0			0
$673$		−	310.790i		−	0.461798i	−0.972978	−	0.230899i	$-0.925833\pi$
$673$		−	310.790i		−	0.461798i	0.972978	−	0.230899i	$-0.0741667\pi$
$674$		0			0
$675$	77.5823			0.114937
$676$		0			0
$677$	999.473			1.47633			0.738163	−	0.674623i	$-0.235694\pi$
$677$	999.473			1.47633			0.738163	+	0.674623i	$0.235694\pi$
$678$		0			0
$679$			505.175i			0.743998i
$680$		0			0
$681$	15.2919			0.0224550
$682$		0			0
$683$	225.474			0.330123			0.165061	−	0.986283i	$-0.447218\pi$
$683$	225.474			0.330123			0.165061	+	0.986283i	$0.447218\pi$
$684$		0			0
$685$		−	1058.51i		−	1.54527i
$686$		0			0
$687$		−	1036.68i		−	1.50900i
$688$		0			0
$689$	324.766			0.471359
$690$		0			0
$691$		−	257.168i		−	0.372168i	−0.982534	−	0.186084i	$-0.940420\pi$
$691$		−	257.168i		−	0.372168i	0.982534	−	0.186084i	$-0.0595796\pi$
$692$		0			0
$693$			72.0994i			0.104040i
$694$		0			0
$695$	537.260			0.773036
$696$		0			0
$697$			1140.49i			1.63628i
$698$		0			0
$699$	−1029.89			−1.47338
$700$		0			0
$701$		−	75.7758i		−	0.108097i	−0.998538	−	0.0540484i	$-0.982787\pi$
$701$		−	75.7758i		−	0.108097i	0.998538	−	0.0540484i	$-0.0172125\pi$
$702$		0			0
$703$	−819.004	+	1039.80i	−1.16501	+	1.47909i
$704$		0			0
$705$			661.043i			0.937649i
$706$		0			0
$707$			752.787i			1.06476i
$708$		0			0
$709$		−	396.211i		−	0.558831i	−0.960170	−	0.279415i	$-0.909859\pi$
$709$		−	396.211i		−	0.558831i	0.960170	−	0.279415i	$-0.0901407\pi$
$710$		0			0
$711$		−	93.3827i		−	0.131340i
$712$		0			0
$713$		−	359.066i		−	0.503600i
$714$		0			0
$715$	955.134			1.33585
$716$		0			0
$717$	−149.626			−0.208683
$718$		0			0
$719$	238.120			0.331182			0.165591	−	0.986194i	$-0.447047\pi$
$719$	238.120			0.331182			0.165591	+	0.986194i	$0.447047\pi$
$720$		0			0
$721$		−	890.738i		−	1.23542i
$722$		0			0
$723$		−	380.280i		−	0.525974i
$724$		0			0
$725$	73.5745			0.101482
$726$		0			0
$727$	987.494			1.35831			0.679157	−	0.733993i	$-0.262346\pi$
$727$	987.494			1.35831			0.679157	+	0.733993i	$0.262346\pi$
$728$		0			0
$729$	806.536			1.10636
$730$		0			0
$731$			188.305i			0.257599i
$732$		0			0
$733$			892.138i			1.21711i	0.793514	+	0.608553i	$0.208250\pi$
$733$			892.138i			1.21711i	−0.793514	+	0.608553i	$0.791750\pi$
$734$		0			0
$735$		−	388.696i		−	0.528839i
$736$		0			0
$737$			995.154i			1.35028i
$738$		0			0
$739$			1196.05i			1.61847i	0.587486	+	0.809235i	$0.300118\pi$
$739$			1196.05i			1.61847i	−0.587486	+	0.809235i	$0.699882\pi$
$740$		0			0
$741$	−569.199	+	722.648i	−0.768150	+	0.975233i
$742$		0			0
$743$		−	991.010i		−	1.33380i	−0.745149	−	0.666898i	$-0.767622\pi$
$743$		−	991.010i		−	1.33380i	0.745149	−	0.666898i	$-0.232378\pi$
$744$		0			0
$745$	−448.446			−0.601941
$746$		0			0
$747$		−	152.041i		−	0.203536i
$748$		0			0
$749$	695.568			0.928662
$750$		0			0
$751$			287.993i			0.383480i	0.981446	+	0.191740i	$0.0614130\pi$
$751$			287.993i			0.383480i	−0.981446	+	0.191740i	$0.938587\pi$
$752$		0			0
$753$			610.874i			0.811254i
$754$		0			0
$755$	162.994			0.215886
$756$		0			0
$757$			844.099i			1.11506i	0.830157	+	0.557529i	$0.188251\pi$
$757$			844.099i			1.11506i	−0.830157	+	0.557529i	$0.811749\pi$
$758$		0			0
$759$		−	295.321i		−	0.389092i
$760$		0			0
$761$	448.650			0.589553			0.294777	−	0.955566i	$-0.404755\pi$
$761$	448.650			0.589553			0.294777	+	0.955566i	$0.404755\pi$
$762$		0			0
$763$	235.956			0.309248
$764$		0			0
$765$		−	198.947i		−	0.260062i
$766$		0			0
$767$	1255.14			1.63642
$768$		0			0
$769$	701.512			0.912239			0.456120	−	0.889919i	$-0.349239\pi$
$769$	701.512			0.912239			0.456120	+	0.889919i	$0.349239\pi$
$770$		0			0
$771$		−	474.439i		−	0.615355i
$772$		0			0
$773$	1099.51			1.42240			0.711198	−	0.702992i	$-0.248153\pi$
$773$	1099.51			1.42240			0.711198	+	0.702992i	$0.248153\pi$
$774$		0			0
$775$			92.2881i			0.119081i
$776$		0			0
$777$	895.914			1.15304
$778$		0			0
$779$	−673.507	−	530.493i	−0.864579	−	0.680992i
$780$		0			0
$781$	−201.038			−0.257410
$782$		0			0
$783$	783.963			1.00123
$784$		0			0
$785$	−669.101			−0.852358
$786$		0			0
$787$	420.402			0.534183			0.267092	−	0.963671i	$-0.413937\pi$
$787$	420.402			0.534183			0.267092	+	0.963671i	$0.413937\pi$
$788$		0			0
$789$	573.556			0.726940
$790$		0			0
$791$		−	876.427i		−	1.10800i
$792$		0			0
$793$			1335.32i			1.68389i
$794$		0			0
$795$			264.928i			0.333243i
$796$		0			0
$797$	−32.5902			−0.0408911			−0.0204455	−	0.999791i	$-0.506508\pi$
$797$	−32.5902			−0.0408911			−0.0204455	+	0.999791i	$0.506508\pi$
$798$		0			0
$799$	1158.87			1.45041
$800$		0			0
$801$			109.954i			0.137271i
$802$		0			0
$803$			644.472i			0.802581i
$804$		0			0
$805$			259.391i			0.322225i
$806$		0			0
$807$	215.428			0.266950
$808$		0			0
$809$	−972.332			−1.20189			−0.600947	−	0.799289i	$-0.705210\pi$
$809$	−972.332			−1.20189			−0.600947	+	0.799289i	$0.705210\pi$
$810$		0			0
$811$	−992.060			−1.22325			−0.611627	−	0.791146i	$-0.709485\pi$
$811$	−992.060			−1.22325			−0.611627	+	0.791146i	$0.709485\pi$
$812$		0			0
$813$	303.845			0.373733
$814$		0			0
$815$	1555.27			1.90830
$816$		0			0
$817$	−111.202	−	87.5889i	−0.136110	−	0.107208i
$818$		0			0
$819$	−124.093			−0.151518
$820$		0			0
$821$			386.695i			0.471005i	0.971874	+	0.235502i	$0.0756736\pi$
$821$			386.695i			0.471005i	−0.971874	+	0.235502i	$0.924326\pi$
$822$		0			0
$823$	−624.294			−0.758559			−0.379279	−	0.925282i	$-0.623828\pi$
$823$	−624.294			−0.758559			−0.379279	+	0.925282i	$0.623828\pi$
$824$		0			0
$825$			75.9040i			0.0920048i
$826$		0			0
$827$	−184.998			−0.223698			−0.111849	−	0.993725i	$-0.535677\pi$
$827$	−184.998			−0.223698			−0.111849	+	0.993725i	$0.535677\pi$
$828$		0			0
$829$	1489.30			1.79651			0.898253	−	0.439478i	$-0.144837\pi$
$829$	1489.30			1.79651			0.898253	+	0.439478i	$0.144837\pi$
$830$		0			0
$831$		−	980.251i		−	1.17960i
$832$		0			0
$833$	−681.424			−0.818036
$834$		0			0
$835$	−469.471			−0.562241
$836$		0			0
$837$			983.364i			1.17487i
$838$		0			0
$839$			1103.27i			1.31498i	0.753465	+	0.657489i	$0.228381\pi$
$839$			1103.27i			1.31498i	−0.753465	+	0.657489i	$0.771619\pi$
$840$		0			0
$841$	−97.5349			−0.115975
$842$		0			0
$843$		−	501.769i		−	0.595219i
$844$		0			0
$845$			754.490i			0.892887i
$846$		0			0
$847$	73.0318			0.0862241
$848$		0			0
$849$		−	435.278i		−	0.512695i
$850$		0			0
$851$	731.363			0.859416
$852$		0			0
$853$		−	1005.86i		−	1.17920i	−0.807696	−	0.589599i	$-0.799286\pi$
$853$		−	1005.86i		−	1.17920i	0.807696	−	0.589599i	$-0.200714\pi$
$854$		0			0
$855$	117.487	+	92.5392i	0.137411	+	0.108233i
$856$		0			0
$857$		−	1160.47i		−	1.35411i	−0.735935	−	0.677053i	$-0.763257\pi$
$857$		−	1160.47i		−	1.35411i	0.735935	−	0.677053i	$-0.236743\pi$
$858$		0			0
$859$		−	944.049i		−	1.09901i	−0.835491	−	0.549505i	$-0.814816\pi$
$859$		−	944.049i		−	1.09901i	0.835491	−	0.549505i	$-0.185184\pi$
$860$		0			0
$861$			580.310i			0.673995i
$862$		0			0
$863$		−	1345.70i		−	1.55932i	−0.626201	−	0.779662i	$-0.715391\pi$
$863$		−	1345.70i		−	1.55932i	0.626201	−	0.779662i	$-0.284609\pi$
$864$		0			0
$865$		−	157.337i		−	0.181892i
$866$		0			0
$867$	958.317			1.10533
$868$		0			0
$869$	641.146			0.737797
$870$		0			0
$871$	−1712.80			−1.96648
$872$		0			0
$873$		−	160.938i		−	0.184351i
$874$		0			0
$875$			551.018i			0.629735i
$876$		0			0
$877$	371.035			0.423073			0.211536	−	0.977370i	$-0.432153\pi$
$877$	371.035			0.423073			0.211536	+	0.977370i	$0.432153\pi$
$878$		0			0
$879$	−351.413			−0.399788
$880$		0			0
$881$	−862.919			−0.979476			−0.489738	−	0.871870i	$-0.662908\pi$
$881$	−862.919			−0.979476			−0.489738	+	0.871870i	$0.662908\pi$
$882$		0			0
$883$		−	445.613i		−	0.504658i	−0.967641	−	0.252329i	$-0.918803\pi$
$883$		−	445.613i		−	0.504658i	0.967641	−	0.252329i	$-0.0811965\pi$
$884$		0			0
$885$			1023.88i			1.15692i
$886$		0			0
$887$		−	833.493i		−	0.939676i	−0.882753	−	0.469838i	$-0.844312\pi$
$887$		−	833.493i		−	0.939676i	0.882753	−	0.469838i	$-0.155688\pi$
$888$		0			0
$889$		−	115.479i		−	0.129898i
$890$		0			0
$891$			670.565i			0.752598i
$892$		0			0
$893$	−539.044	+	684.363i	−0.603633	+	0.766364i
$894$		0			0
$895$			98.1853i			0.109704i
$896$		0			0
$897$	508.289			0.566655
$898$		0			0
$899$			932.565i			1.03734i
$900$		0			0
$901$	464.446			0.515478
$902$		0			0
$903$			95.8140i			0.106106i
$904$		0			0
$905$			51.2234i			0.0566004i
$906$		0			0
$907$	440.320			0.485469			0.242734	−	0.970093i	$-0.421956\pi$
$907$	440.320			0.485469			0.242734	+	0.970093i	$0.421956\pi$
$908$		0			0
$909$		−	239.822i		−	0.263831i
$910$		0			0
$911$			525.421i			0.576752i	0.957517	+	0.288376i	$0.0931153\pi$
$911$			525.421i			0.576752i	−0.957517	+	0.288376i	$0.906885\pi$
$912$		0			0
$913$	1043.88			1.14336
$914$		0			0
$915$	−1089.29			−1.19048
$916$		0			0
$917$			309.098i			0.337075i
$918$		0			0
$919$	−285.119			−0.310249			−0.155124	−	0.987895i	$-0.549578\pi$
$919$	−285.119			−0.310249			−0.155124	+	0.987895i	$0.549578\pi$
$920$		0			0
$921$	−711.208			−0.772213
$922$		0			0
$923$		−	346.015i		−	0.374881i
$924$		0			0
$925$	−187.977			−0.203218
$926$		0			0
$927$			283.771i			0.306117i
$928$		0			0
$929$	−97.2881			−0.104724			−0.0523618	−	0.998628i	$-0.516675\pi$
$929$	−97.2881			−0.104724			−0.0523618	+	0.998628i	$0.516675\pi$
$930$		0			0
$931$	316.961	−	402.409i	0.340452	−	0.432233i
$932$		0			0
$933$	−1523.57			−1.63298
$934$		0			0
$935$	1365.93			1.46089
$936$		0			0
$937$	−40.8602			−0.0436075			−0.0218037	−	0.999762i	$-0.506941\pi$
$937$	−40.8602			−0.0436075			−0.0218037	+	0.999762i	$0.506941\pi$
$938$		0			0
$939$	540.607			0.575726
$940$		0			0
$941$	1426.18			1.51560			0.757800	−	0.652487i	$-0.226274\pi$
$941$	1426.18			1.51560			0.757800	+	0.652487i	$0.226274\pi$
$942$		0			0
$943$			473.725i			0.502360i
$944$		0			0
$945$		−	710.386i		−	0.751731i
$946$		0			0
$947$			328.560i			0.346948i	0.984838	+	0.173474i	$0.0554992\pi$
$947$			328.560i			0.346948i	−0.984838	+	0.173474i	$0.944501\pi$
$948$		0			0
$949$	−1109.23			−1.16884
$950$		0			0
$951$	−126.202			−0.132704
$952$		0			0
$953$		−	133.225i		−	0.139796i	−0.997554	−	0.0698979i	$-0.977733\pi$
$953$		−	133.225i		−	0.139796i	0.997554	−	0.0698979i	$-0.0222674\pi$
$954$		0			0
$955$			263.848i			0.276280i
$956$		0			0
$957$			767.005i			0.801468i
$958$		0			0
$959$	−944.214			−0.984582
$960$		0			0
$961$	−208.762			−0.217234
$962$		0			0
$963$	−221.593			−0.230107
$964$		0			0
$965$	676.279			0.700808
$966$		0			0
$967$	1580.53			1.63447			0.817236	−	0.576303i	$-0.195505\pi$
$967$	1580.53			1.63447			0.817236	+	0.576303i	$0.195505\pi$
$968$		0			0
$969$	−814.007	+	1033.45i	−0.840049	+	1.06651i
$970$		0			0
$971$	691.676			0.712333			0.356167	−	0.934422i	$-0.384084\pi$
$971$	691.676			0.712333			0.356167	+	0.934422i	$0.384084\pi$
$972$		0			0
$973$		−	479.247i		−	0.492546i
$974$		0			0
$975$	−130.642			−0.133992
$976$		0			0
$977$		−	287.830i		−	0.294606i	−0.989091	−	0.147303i	$-0.952941\pi$
$977$		−	287.830i		−	0.294606i	0.989091	−	0.147303i	$-0.0470592\pi$
$978$		0			0
$979$	−754.923			−0.771117
$980$		0			0
$981$	−75.1708			−0.0766267
$982$		0			0
$983$		−	1925.56i		−	1.95886i	−0.201794	−	0.979428i	$-0.564677\pi$
$983$		−	1925.56i		−	1.95886i	0.201794	−	0.979428i	$-0.435323\pi$
$984$		0			0
$985$	−773.964			−0.785750
$986$		0			0
$987$	589.664			0.597431
$988$		0			0
$989$			78.2160i			0.0790860i
$990$		0			0
$991$		−	1310.31i		−	1.32221i	−0.750292	−	0.661107i	$-0.770087\pi$
$991$		−	1310.31i		−	1.32221i	0.750292	−	0.661107i	$-0.229913\pi$
$992$		0			0
$993$	554.389			0.558297
$994$		0			0
$995$			556.417i			0.559213i
$996$		0			0
$997$		−	911.840i		−	0.914583i	−0.889317	−	0.457292i	$-0.848820\pi$
$997$		−	911.840i		−	0.914583i	0.889317	−	0.457292i	$-0.151180\pi$
$998$		0			0
$999$	−2002.96			−2.00496

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.g.e.417.29	yes	48
4.3	odd	2	inner	1216.3.g.e.417.17	✓	48
8.3	odd	2	inner	1216.3.g.e.417.30	yes	48
8.5	even	2	inner	1216.3.g.e.417.18	yes	48
19.18	odd	2	inner	1216.3.g.e.417.19	yes	48
76.75	even	2	inner	1216.3.g.e.417.31	yes	48
152.37	odd	2	inner	1216.3.g.e.417.32	yes	48
152.75	even	2	inner	1216.3.g.e.417.20	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.3.g.e.417.17	✓	48	4.3	odd	2	inner
1216.3.g.e.417.18	yes	48	8.5	even	2	inner
1216.3.g.e.417.19	yes	48	19.18	odd	2	inner
1216.3.g.e.417.20	yes	48	152.75	even	2	inner
1216.3.g.e.417.29	yes	48	1.1	even	1	trivial
1216.3.g.e.417.30	yes	48	8.3	odd	2	inner
1216.3.g.e.417.31	yes	48	76.75	even	2	inner
1216.3.g.e.417.32	yes	48	152.37	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)

\(f(q)\)	\(=\)	\(q+2.73941 q^{3} +5.26292i q^{5} +4.69464 q^{7} -1.49561 q^{9} +O(q^{10})\)	\(q+2.73941 q^{3} +5.26292i q^{5} +4.69464 q^{7} -1.49561 q^{9} -10.2686i q^{11} +17.6737 q^{13} +14.4173i q^{15} +25.2750 q^{17} +(-11.7565 + 14.9259i) q^{19} +12.8606 q^{21} +10.4985 q^{23} -2.69834 q^{25} -28.7518 q^{27} -27.2666 q^{29} -34.2018i q^{31} -28.1299i q^{33} +24.7075i q^{35} +69.6637 q^{37} +48.4156 q^{39} +45.1232i q^{41} +7.45023i q^{43} -7.87130i q^{45} +45.8506 q^{47} -26.9604 q^{49} +69.2387 q^{51} +18.3757 q^{53} +54.0427 q^{55} +(-32.2060 + 40.8883i) q^{57} +71.0173 q^{59} +75.5544i q^{61} -7.02137 q^{63} +93.0153i q^{65} -96.9126 q^{67} +28.7597 q^{69} -19.5780i q^{71} -62.7616 q^{73} -7.39187 q^{75} -48.2072i q^{77} +62.4377i q^{79} -65.3026 q^{81} +101.658i q^{83} +133.020i q^{85} -74.6944 q^{87} -73.5179i q^{89} +82.9716 q^{91} -93.6928i q^{93} +(-78.5541 - 61.8737i) q^{95} +107.607i q^{97} +15.3578i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 264 q^{9}+O(q^{10}) \) 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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