LMFDB - Embedded newform 2268.2.f.b.1133.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,2,Mod(1133,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2268.1133");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.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:	$18.1100711784$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 $ x^16 - 3x^14 - 9x^12 - 9x^10 + 225x^8 - 81x^6 - 729x^4 - 2187*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1133.4
Root			$-1.69483 - 0.357142i$ of defining polynomial
Character	$\chi$	$=$	2268.1133
Dual form			2268.2.f.b.1133.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.42488 q^{5} +(-2.62893 + 0.297883i) q^{7} +O(q^{10})$	$q-2.42488 q^{5} +(-2.62893 + 0.297883i) q^{7} -2.42118i q^{11} -5.46836i q^{13} +2.58069 q^{17} -0.402708i q^{19} +3.54372i q^{23} +0.880040 q^{25} +7.29521i q^{29} -4.20001i q^{31} +(6.37484 - 0.722330i) q^{35} -3.19360 q^{37} -8.07849 q^{41} +8.45146 q^{43} +4.51537 q^{47} +(6.82253 - 1.56622i) q^{49} +14.0288i q^{53} +5.87106i q^{55} +0.155809 q^{59} +11.8112i q^{61} +13.2601i q^{65} -5.07364 q^{67} +8.73987i q^{71} -8.80274i q^{73} +(0.721227 + 6.36510i) q^{77} -11.3315 q^{79} -15.0187 q^{83} -6.25786 q^{85} -15.6668 q^{89} +(1.62893 + 14.3759i) q^{91} +0.976519i q^{95} -5.74710i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{7}+O(q^{10}) $ 16 * q + 2 * q^7	$ 16 q + 2 q^{7} + 16 q^{25} + 4 q^{37} - 8 q^{43} + 10 q^{49} - 28 q^{67} - 40 q^{79} - 12 q^{85} - 18 q^{91}+O(q^{100}) $ 16 * q + 2 * q^7 + 16 * q^25 + 4 * q^37 - 8 * q^43 + 10 * q^49 - 28 * q^67 - 40 * q^79 - 12 * q^85 - 18 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.42488			−1.08444			−0.542220	−	0.840237i	$-0.682416\pi$
$5$	−2.42488			−1.08444			−0.542220	+	0.840237i	$0.682416\pi$
$6$		0			0
$7$	−2.62893	+	0.297883i	−0.993642	+	0.112589i
$7$	−2.62893	+	0.297883i	−0.993642	+	0.112589i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	2.42118i		−	0.730013i	−0.931005	−	0.365006i	$-0.881067\pi$
$11$		−	2.42118i		−	0.730013i	0.931005	−	0.365006i	$-0.118933\pi$
$12$		0			0
$13$		−	5.46836i		−	1.51665i	−0.651877	−	0.758325i	$-0.726018\pi$
$13$		−	5.46836i		−	1.51665i	0.651877	−	0.758325i	$-0.273982\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.58069			0.625909			0.312954	−	0.949768i	$-0.398681\pi$
$17$	2.58069			0.625909			0.312954	+	0.949768i	$0.398681\pi$
$18$		0			0
$19$		−	0.402708i		−	0.0923876i	−0.998932	−	0.0461938i	$-0.985291\pi$
$19$		−	0.402708i		−	0.0923876i	0.998932	−	0.0461938i	$-0.0147092\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.54372i			0.738916i	0.929247	+	0.369458i	$0.120457\pi$
$23$			3.54372i			0.738916i	−0.929247	+	0.369458i	$0.879543\pi$
$24$		0			0
$25$	0.880040			0.176008
$26$		0			0
$27$		0			0
$28$		0			0
$29$			7.29521i			1.35469i	0.735667	+	0.677343i	$0.236869\pi$
$29$			7.29521i			1.35469i	−0.735667	+	0.677343i	$0.763131\pi$
$30$		0			0
$31$		−	4.20001i		−	0.754345i	−0.926143	−	0.377172i	$-0.876896\pi$
$31$		−	4.20001i		−	0.754345i	0.926143	−	0.377172i	$-0.123104\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.37484	−	0.722330i	1.07754	−	0.122096i
$36$		0			0
$37$	−3.19360			−0.525025			−0.262513	−	0.964929i	$-0.584551\pi$
$37$	−3.19360			−0.525025			−0.262513	+	0.964929i	$0.584551\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.07849			−1.26165			−0.630824	−	0.775926i	$-0.717283\pi$
$41$	−8.07849			−1.26165			−0.630824	+	0.775926i	$0.717283\pi$
$42$		0			0
$43$	8.45146			1.28884			0.644418	−	0.764674i	$-0.277100\pi$
$43$	8.45146			1.28884			0.644418	+	0.764674i	$0.277100\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.51537			0.658635			0.329317	−	0.944219i	$-0.393181\pi$
$47$	4.51537			0.658635			0.329317	+	0.944219i	$0.393181\pi$
$48$		0			0
$49$	6.82253	−	1.56622i	0.974647	−	0.223746i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			14.0288i			1.92701i	0.267690	+	0.963505i	$0.413740\pi$
$53$			14.0288i			1.92701i	−0.267690	+	0.963505i	$0.586260\pi$
$54$		0			0
$55$			5.87106i			0.791654i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.155809			0.0202846			0.0101423	−	0.999949i	$-0.496772\pi$
$59$	0.155809			0.0202846			0.0101423	+	0.999949i	$0.496772\pi$
$60$		0			0
$61$			11.8112i			1.51227i	0.654415	+	0.756136i	$0.272915\pi$
$61$			11.8112i			1.51227i	−0.654415	+	0.756136i	$0.727085\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			13.2601i			1.64471i
$66$		0			0
$67$	−5.07364			−0.619844			−0.309922	−	0.950762i	$-0.600303\pi$
$67$	−5.07364			−0.619844			−0.309922	+	0.950762i	$0.600303\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			8.73987i			1.03723i	0.855007	+	0.518616i	$0.173552\pi$
$71$			8.73987i			1.03723i	−0.855007	+	0.518616i	$0.826448\pi$
$72$		0			0
$73$		−	8.80274i		−	1.03028i	−0.857105	−	0.515141i	$-0.827739\pi$
$73$		−	8.80274i		−	1.03028i	0.857105	−	0.515141i	$-0.172261\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.721227	+	6.36510i	0.0821914	+	0.725371i
$78$		0			0
$79$	−11.3315			−1.27489			−0.637447	−	0.770494i	$-0.720009\pi$
$79$	−11.3315			−1.27489			−0.637447	+	0.770494i	$0.720009\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−15.0187			−1.64852			−0.824260	−	0.566211i	$-0.808409\pi$
$83$	−15.0187			−1.64852			−0.824260	+	0.566211i	$0.808409\pi$
$84$		0			0
$85$	−6.25786			−0.678760
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−15.6668			−1.66068			−0.830338	−	0.557260i	$-0.811852\pi$
$89$	−15.6668			−1.66068			−0.830338	+	0.557260i	$0.811852\pi$
$90$		0			0
$91$	1.62893	+	14.3759i	0.170758	+	1.50701i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			0.976519i			0.100189i
$96$		0			0
$97$		−	5.74710i		−	0.583529i	−0.956490	−	0.291765i	$-0.905758\pi$
$97$		−	5.74710i		−	0.583529i	0.956490	−	0.291765i	$-0.0942424\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.67675			0.962873			0.481436	−	0.876481i	$-0.340115\pi$
$101$	9.67675			0.962873			0.481436	+	0.876481i	$0.340115\pi$
$102$		0			0
$103$			19.2676i			1.89850i	0.314526	+	0.949249i	$0.398155\pi$
$103$			19.2676i			1.89850i	−0.314526	+	0.949249i	$0.601845\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1.09086i		−	0.105458i	−0.998609	−	0.0527288i	$-0.983208\pi$
$107$		−	1.09086i		−	0.105458i	0.998609	−	0.0527288i	$-0.0167919\pi$
$108$		0			0
$109$	2.31356			0.221599			0.110800	−	0.993843i	$-0.464659\pi$
$109$	2.31356			0.221599			0.110800	+	0.993843i	$0.464659\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			16.0351i			1.50845i	0.656614	+	0.754227i	$0.271988\pi$
$113$			16.0351i			1.50845i	−0.656614	+	0.754227i	$0.728012\pi$
$114$		0			0
$115$		−	8.59309i		−	0.801309i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.78444	+	0.768742i	−0.621929	+	0.0704705i
$120$		0			0
$121$	5.13790			0.467082
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.99041			0.893569
$126$		0			0
$127$	−3.06425			−0.271909			−0.135954	−	0.990715i	$-0.543410\pi$
$127$	−3.06425			−0.271909			−0.135954	+	0.990715i	$0.543410\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	11.4630			1.00153			0.500765	−	0.865584i	$-0.333052\pi$
$131$	11.4630			1.00153			0.500765	+	0.865584i	$0.333052\pi$
$132$		0			0
$133$	0.119960	+	1.05869i	0.0104018	+	0.0918002i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1.97456i		−	0.168698i	−0.996436	−	0.0843488i	$-0.973119\pi$
$137$		−	1.97456i		−	0.168698i	0.996436	−	0.0843488i	$-0.0268810\pi$
$138$		0			0
$139$			6.21002i			0.526727i	0.964697	+	0.263364i	$0.0848319\pi$
$139$			6.21002i			0.526727i	−0.964697	+	0.263364i	$0.915168\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−13.2399			−1.10717
$144$		0			0
$145$		−	17.6900i		−	1.46907i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			12.1692i			0.996943i	0.866906	+	0.498472i	$0.166105\pi$
$149$			12.1692i			0.996943i	−0.866906	+	0.498472i	$0.833895\pi$
$150$		0			0
$151$	10.6357			0.865519			0.432759	−	0.901509i	$-0.357540\pi$
$151$	10.6357			0.865519			0.432759	+	0.901509i	$0.357540\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			10.1845i			0.818041i
$156$		0			0
$157$			3.53516i			0.282136i	0.990000	+	0.141068i	$0.0450537\pi$
$157$			3.53516i			0.282136i	−0.990000	+	0.141068i	$0.954946\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.05561	−	9.31618i	−0.0831939	−	0.734218i
$162$		0			0
$163$	6.33150			0.495921			0.247961	−	0.968770i	$-0.420240\pi$
$163$	6.33150			0.495921			0.247961	+	0.968770i	$0.420240\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	16.7956			1.29968			0.649840	−	0.760071i	$-0.274836\pi$
$167$	16.7956			1.29968			0.649840	+	0.760071i	$0.274836\pi$
$168$		0			0
$169$	−16.9029			−1.30022
$170$		0			0
$171$		0			0
$172$		0			0
$173$	16.6170			1.26337			0.631684	−	0.775226i	$-0.282364\pi$
$173$	16.6170			1.26337			0.631684	+	0.775226i	$0.282364\pi$
$174$		0			0
$175$	−2.31356	+	0.262149i	−0.174889	+	0.0198166i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	14.5587i		−	1.08817i	−0.839030	−	0.544086i	$-0.816877\pi$
$179$		−	14.5587i		−	1.08817i	0.839030	−	0.544086i	$-0.183123\pi$
$180$		0			0
$181$			4.02355i			0.299068i	0.988757	+	0.149534i	$0.0477774\pi$
$181$			4.02355i			0.299068i	−0.988757	+	0.149534i	$0.952223\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.74410			0.569358
$186$		0			0
$187$		−	6.24830i		−	0.456921i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	8.41775i		−	0.609087i	−0.952498	−	0.304543i	$-0.901496\pi$
$191$		−	8.41775i		−	0.609087i	0.952498	−	0.304543i	$-0.0985039\pi$
$192$		0			0
$193$	−8.63567			−0.621609			−0.310805	−	0.950474i	$-0.600599\pi$
$193$	−8.63567			−0.621609			−0.310805	+	0.950474i	$0.600599\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			23.3303i			1.66221i	0.556112	+	0.831107i	$0.312292\pi$
$197$			23.3303i			1.66221i	−0.556112	+	0.831107i	$0.687708\pi$
$198$		0			0
$199$		−	14.1383i		−	1.00223i	−0.865379	−	0.501117i	$-0.832923\pi$
$199$		−	14.1383i		−	1.00223i	0.865379	−	0.501117i	$-0.167077\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.17312	−	19.1786i	−0.152523	−	1.34607i
$204$		0			0
$205$	19.5894			1.36818
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.975028			−0.0674441
$210$		0			0
$211$	13.5157			0.930460			0.465230	−	0.885190i	$-0.345972\pi$
$211$	13.5157			0.930460			0.465230	+	0.885190i	$0.345972\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−20.4938			−1.39766
$216$		0			0
$217$	1.25111	+	11.0415i	0.0849310	+	0.749548i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	14.1121i		−	0.949284i
$222$		0			0
$223$		−	15.4792i		−	1.03657i	−0.855209	−	0.518283i	$-0.826572\pi$
$223$		−	15.4792i		−	1.03657i	0.855209	−	0.518283i	$-0.173428\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−15.0167			−0.996694			−0.498347	−	0.866978i	$-0.666059\pi$
$227$	−15.0167			−0.996694			−0.498347	+	0.866978i	$0.666059\pi$
$228$		0			0
$229$			19.5935i			1.29478i	0.762160	+	0.647389i	$0.224139\pi$
$229$			19.5935i			1.29478i	−0.762160	+	0.647389i	$0.775861\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			19.5471i			1.28057i	0.768136	+	0.640287i	$0.221185\pi$
$233$			19.5471i			1.28057i	−0.768136	+	0.640287i	$0.778815\pi$
$234$		0			0
$235$	−10.9492			−0.714249
$236$		0			0
$237$		0			0
$238$		0			0
$239$			8.50102i			0.549886i	0.961461	+	0.274943i	$0.0886589\pi$
$239$			8.50102i			0.549886i	−0.961461	+	0.274943i	$0.911341\pi$
$240$		0			0
$241$		−	8.33094i		−	0.536643i	−0.963329	−	0.268321i	$-0.913531\pi$
$241$		−	8.33094i		−	0.536643i	0.963329	−	0.268321i	$-0.0864689\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−16.5438	+	3.79791i	−1.05695	+	0.242639i
$246$		0			0
$247$	−2.20215			−0.140120
$248$		0			0
$249$		0			0
$250$		0			0
$251$	13.5763			0.856928			0.428464	−	0.903559i	$-0.359055\pi$
$251$	13.5763			0.856928			0.428464	+	0.903559i	$0.359055\pi$
$252$		0			0
$253$	8.57997			0.539418
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.98060			−0.373059			−0.186530	−	0.982449i	$-0.559724\pi$
$257$	−5.98060			−0.373059			−0.186530	+	0.982449i	$0.559724\pi$
$258$		0			0
$259$	8.39575	−	0.951319i	0.521687	−	0.0591121i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			3.36602i			0.207558i	0.994600	+	0.103779i	$0.0330934\pi$
$263$			3.36602i			0.207558i	−0.994600	+	0.103779i	$0.966907\pi$
$264$		0			0
$265$		−	34.0183i		−	2.08973i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.2822			−0.748862			−0.374431	−	0.927255i	$-0.622162\pi$
$269$	−12.2822			−0.748862			−0.374431	+	0.927255i	$0.622162\pi$
$270$		0			0
$271$		−	29.4244i		−	1.78741i	−0.448659	−	0.893703i	$-0.648098\pi$
$271$		−	29.4244i		−	1.78741i	0.448659	−	0.893703i	$-0.351902\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	2.13073i		−	0.128488i
$276$		0			0
$277$	−9.20215			−0.552904			−0.276452	−	0.961028i	$-0.589159\pi$
$277$	−9.20215			−0.552904			−0.276452	+	0.961028i	$0.589159\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	1.21536i		−	0.0725026i	−0.999343	−	0.0362513i	$-0.988458\pi$
$281$		−	1.21536i		−	0.0725026i	0.999343	−	0.0362513i	$-0.0115417\pi$
$282$		0			0
$283$		−	12.2139i		−	0.726043i	−0.931781	−	0.363021i	$-0.881745\pi$
$283$		−	12.2139i		−	0.726043i	0.931781	−	0.363021i	$-0.118255\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	21.2378	−	2.40644i	1.25363	−	0.142048i
$288$		0			0
$289$	−10.3400			−0.608238
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.30165			−0.368146			−0.184073	−	0.982913i	$-0.558928\pi$
$293$	−6.30165			−0.368146			−0.184073	+	0.982913i	$0.558928\pi$
$294$		0			0
$295$	−0.377817			−0.0219974
$296$		0			0
$297$		0			0
$298$		0			0
$299$	19.3783			1.12068
$300$		0			0
$301$	−22.2183	+	2.51754i	−1.28064	+	0.145109i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	28.6408i		−	1.63997i
$306$		0			0
$307$			28.7264i			1.63950i	0.572721	+	0.819750i	$0.305888\pi$
$307$			28.7264i			1.63950i	−0.572721	+	0.819750i	$0.694112\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−17.7325			−1.00552			−0.502758	−	0.864427i	$-0.667681\pi$
$311$	−17.7325			−1.00552			−0.502758	+	0.864427i	$0.667681\pi$
$312$		0			0
$313$			27.0808i			1.53070i	0.643616	+	0.765348i	$0.277433\pi$
$313$			27.0808i			1.53070i	−0.643616	+	0.765348i	$0.722567\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			5.37385i			0.301825i	0.988547	+	0.150913i	$0.0482212\pi$
$317$			5.37385i			0.301825i	−0.988547	+	0.150913i	$0.951779\pi$
$318$		0			0
$319$	17.6630			0.988938
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1.03926i		−	0.0578262i
$324$		0			0
$325$		−	4.81237i		−	0.266942i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−11.8706	+	1.34505i	−0.654447	+	0.0741551i
$330$		0			0
$331$	−7.44207			−0.409053			−0.204527	−	0.978861i	$-0.565565\pi$
$331$	−7.44207			−0.409053			−0.204527	+	0.978861i	$0.565565\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	12.3030			0.672183
$336$		0			0
$337$	10.6357			0.579362			0.289681	−	0.957123i	$-0.406451\pi$
$337$	10.6357			0.579362			0.289681	+	0.957123i	$0.406451\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−10.1690			−0.550681
$342$		0			0
$343$	−17.4694	+	6.14981i	−0.943259	+	0.332058i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	1.89189i		−	0.101562i	−0.998710	−	0.0507809i	$-0.983829\pi$
$347$		−	1.89189i		−	0.101562i	0.998710	−	0.0507809i	$-0.0161710\pi$
$348$		0			0
$349$			18.6028i			0.995785i	0.867239	+	0.497892i	$0.165893\pi$
$349$			18.6028i			0.995785i	−0.867239	+	0.497892i	$0.834107\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.89010			0.207049			0.103525	−	0.994627i	$-0.466988\pi$
$353$	3.89010			0.207049			0.103525	+	0.994627i	$0.466988\pi$
$354$		0			0
$355$		−	21.1931i		−	1.12481i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	8.62553i		−	0.455238i	−0.973750	−	0.227619i	$-0.926906\pi$
$359$		−	8.62553i		−	0.455238i	0.973750	−	0.227619i	$-0.0730940\pi$
$360$		0			0
$361$	18.8378			0.991465
$362$		0			0
$363$		0			0
$364$		0			0
$365$			21.3456i			1.11728i
$366$		0			0
$367$			18.3252i			0.956568i	0.878205	+	0.478284i	$0.158741\pi$
$367$			18.3252i			0.956568i	−0.878205	+	0.478284i	$0.841259\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.17895	−	36.8808i	−0.216960	−	1.91476i
$372$		0			0
$373$	−7.69583			−0.398475			−0.199237	−	0.979951i	$-0.563846\pi$
$373$	−7.69583			−0.398475			−0.199237	+	0.979951i	$0.563846\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	39.8928			2.05458
$378$		0			0
$379$	−7.52510			−0.386539			−0.193269	−	0.981146i	$-0.561909\pi$
$379$	−7.52510			−0.386539			−0.193269	+	0.981146i	$0.561909\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.40934			0.174209			0.0871047	−	0.996199i	$-0.472239\pi$
$383$	3.40934			0.174209			0.0871047	+	0.996199i	$0.472239\pi$
$384$		0			0
$385$	−1.74889	−	15.4346i	−0.0891316	−	0.786621i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	9.62356i		−	0.487934i	−0.969784	−	0.243967i	$-0.921551\pi$
$389$		−	9.62356i		−	0.487934i	0.969784	−	0.243967i	$-0.0784489\pi$
$390$		0			0
$391$			9.14523i			0.462494i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	27.4775			1.38254
$396$		0			0
$397$			23.4807i			1.17846i	0.807964	+	0.589232i	$0.200570\pi$
$397$			23.4807i			1.17846i	−0.807964	+	0.589232i	$0.799430\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	25.6365i		−	1.28023i	−0.768281	−	0.640113i	$-0.778887\pi$
$401$		−	25.6365i		−	1.28023i	0.768281	−	0.640113i	$-0.221113\pi$
$402$		0			0
$403$	−22.9672			−1.14408
$404$		0			0
$405$		0			0
$406$		0			0
$407$			7.73228i			0.383275i
$408$		0			0
$409$			15.4321i			0.763066i	0.924355	+	0.381533i	$0.124604\pi$
$409$			15.4321i			0.763066i	−0.924355	+	0.381533i	$0.875396\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.409610	+	0.0464127i	−0.0201556	+	0.00228382i
$414$		0			0
$415$	36.4186			1.78772
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−14.2129			−0.694343			−0.347172	−	0.937802i	$-0.612858\pi$
$419$	−14.2129			−0.694343			−0.347172	+	0.937802i	$0.612858\pi$
$420$		0			0
$421$	1.69993			0.0828495			0.0414247	−	0.999142i	$-0.486810\pi$
$421$	1.69993			0.0828495			0.0414247	+	0.999142i	$0.486810\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.27111			0.110165
$426$		0			0
$427$	−3.51836	−	31.0509i	−0.170265	−	1.50266i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	32.6628i		−	1.57331i	−0.617392	−	0.786656i	$-0.711811\pi$
$431$		−	32.6628i		−	1.57331i	0.617392	−	0.786656i	$-0.288189\pi$
$432$		0			0
$433$			13.6919i			0.657992i	0.944331	+	0.328996i	$0.106710\pi$
$433$			13.6919i			0.657992i	−0.944331	+	0.328996i	$0.893290\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.42708			0.0682667
$438$		0			0
$439$			8.82352i			0.421124i	0.977581	+	0.210562i	$0.0675293\pi$
$439$			8.82352i			0.421124i	−0.977581	+	0.210562i	$0.932471\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	24.6600i		−	1.17163i	−0.810444	−	0.585816i	$-0.800774\pi$
$443$		−	24.6600i		−	1.17163i	0.810444	−	0.585816i	$-0.199226\pi$
$444$		0			0
$445$	37.9901			1.80090
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	4.61306i		−	0.217704i	−0.994058	−	0.108852i	$-0.965283\pi$
$449$		−	4.61306i		−	0.217704i	0.994058	−	0.108852i	$-0.0347174\pi$
$450$		0			0
$451$			19.5595i			0.921019i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.94996	−	34.8599i	−0.185177	−	1.63426i
$456$		0			0
$457$	7.71871			0.361066			0.180533	−	0.983569i	$-0.442218\pi$
$457$	7.71871			0.361066			0.180533	+	0.983569i	$0.442218\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−18.5724			−0.865004			−0.432502	−	0.901633i	$-0.642369\pi$
$461$	−18.5724			−0.865004			−0.432502	+	0.901633i	$0.642369\pi$
$462$		0			0
$463$	−35.4092			−1.64561			−0.822804	−	0.568326i	$-0.807591\pi$
$463$	−35.4092			−1.64561			−0.822804	+	0.568326i	$0.807591\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	15.8433			0.733141			0.366571	−	0.930390i	$-0.380532\pi$
$467$	15.8433			0.733141			0.366571	+	0.930390i	$0.380532\pi$
$468$		0			0
$469$	13.3382	−	1.51135i	0.615903	−	0.0697877i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	20.4625i		−	0.940866i
$474$		0			0
$475$		−	0.354399i		−	0.0162610i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	26.5025			1.21093			0.605465	−	0.795872i	$-0.292987\pi$
$479$	26.5025			1.21093			0.605465	+	0.795872i	$0.292987\pi$
$480$		0			0
$481$			17.4638i			0.796279i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			13.9360i			0.632802i
$486$		0			0
$487$	−7.00939			−0.317626			−0.158813	−	0.987309i	$-0.550767\pi$
$487$	−7.00939			−0.317626			−0.158813	+	0.987309i	$0.550767\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	18.9957i		−	0.857264i	−0.903479	−	0.428632i	$-0.858996\pi$
$491$		−	18.9957i		−	0.857264i	0.903479	−	0.428632i	$-0.141004\pi$
$492$		0			0
$493$			18.8267i			0.847910i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.60346	−	22.9765i	−0.116781	−	1.03064i
$498$		0			0
$499$	1.25786			0.0563094			0.0281547	−	0.999604i	$-0.491037\pi$
$499$	1.25786			0.0563094			0.0281547	+	0.999604i	$0.491037\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−37.9507			−1.69214			−0.846070	−	0.533072i	$-0.821037\pi$
$503$	−37.9507			−1.69214			−0.846070	+	0.533072i	$0.821037\pi$
$504$		0			0
$505$	−23.4650			−1.04418
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5.80942			−0.257498			−0.128749	−	0.991677i	$-0.541096\pi$
$509$	−5.80942			−0.257498			−0.128749	+	0.991677i	$0.541096\pi$
$510$		0			0
$511$	2.62218	+	23.1418i	0.115999	+	1.02373i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	46.7217i		−	2.05881i
$516$		0			0
$517$		−	10.9325i		−	0.480812i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−13.6999			−0.600203			−0.300102	−	0.953907i	$-0.597021\pi$
$521$	−13.6999			−0.600203			−0.300102	+	0.953907i	$0.597021\pi$
$522$		0			0
$523$		−	24.4881i		−	1.07079i	−0.844602	−	0.535394i	$-0.820163\pi$
$523$		−	24.4881i		−	1.07079i	0.844602	−	0.535394i	$-0.179837\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	10.8389i		−	0.472151i
$528$		0			0
$529$	10.4421			0.454003
$530$		0			0
$531$		0			0
$532$		0			0
$533$			44.1760i			1.91348i
$534$		0			0
$535$			2.64521i			0.114362i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.79211	−	16.5186i	−0.163338	−	0.711505i
$540$		0			0
$541$	−37.1608			−1.59767			−0.798833	−	0.601552i	$-0.794549\pi$
$541$	−37.1608			−1.59767			−0.798833	+	0.601552i	$0.794549\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−5.61011			−0.240311
$546$		0			0
$547$	6.63652			0.283757			0.141878	−	0.989884i	$-0.454686\pi$
$547$	6.63652			0.283757			0.141878	+	0.989884i	$0.454686\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.93784			0.125156
$552$		0			0
$553$	29.7897	−	3.37546i	1.26679	−	0.143539i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	24.6188i		−	1.04313i	−0.853211	−	0.521565i	$-0.825348\pi$
$557$		−	24.6188i		−	1.04313i	0.853211	−	0.521565i	$-0.174652\pi$
$558$		0			0
$559$		−	46.2156i		−	1.95471i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−12.5711			−0.529808			−0.264904	−	0.964275i	$-0.585340\pi$
$563$	−12.5711			−0.529808			−0.264904	+	0.964275i	$0.585340\pi$
$564$		0			0
$565$		−	38.8831i		−	1.63583i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			14.8809i			0.623838i	0.950109	+	0.311919i	$0.100972\pi$
$569$			14.8809i			0.623838i	−0.950109	+	0.311919i	$0.899028\pi$
$570$		0			0
$571$	−16.9115			−0.707723			−0.353861	−	0.935298i	$-0.615132\pi$
$571$	−16.9115			−0.707723			−0.353861	+	0.935298i	$0.615132\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			3.11861i			0.130055i
$576$		0			0
$577$			20.9013i			0.870133i	0.900398	+	0.435067i	$0.143275\pi$
$577$			20.9013i			0.870133i	−0.900398	+	0.435067i	$0.856725\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	39.4832	−	4.47382i	1.63804	−	0.185605i
$582$		0			0
$583$	33.9663			1.40674
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−29.7085			−1.22620			−0.613100	−	0.790006i	$-0.710078\pi$
$587$	−29.7085			−1.22620			−0.613100	+	0.790006i	$0.710078\pi$
$588$		0			0
$589$	−1.69138			−0.0696921
$590$		0			0
$591$		0			0
$592$		0			0
$593$	11.7921			0.484241			0.242121	−	0.970246i	$-0.422157\pi$
$593$	11.7921			0.484241			0.242121	+	0.970246i	$0.422157\pi$
$594$		0			0
$595$	16.4515	−	1.86411i	0.674444	−	0.0764210i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	32.0996i		−	1.31156i	−0.754954	−	0.655778i	$-0.772341\pi$
$599$		−	32.0996i		−	1.31156i	0.754954	−	0.655778i	$-0.227659\pi$
$600$		0			0
$601$		−	18.6642i		−	0.761327i	−0.924714	−	0.380664i	$-0.875696\pi$
$601$		−	18.6642i		−	0.761327i	0.924714	−	0.380664i	$-0.124304\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−12.4588			−0.506522
$606$		0			0
$607$		−	13.8684i		−	0.562900i	−0.959576	−	0.281450i	$-0.909185\pi$
$607$		−	13.8684i		−	0.562900i	0.959576	−	0.281450i	$-0.0908154\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	24.6917i		−	0.998918i
$612$		0			0
$613$	−20.9021			−0.844227			−0.422114	−	0.906543i	$-0.638712\pi$
$613$	−20.9021			−0.844227			−0.422114	+	0.906543i	$0.638712\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			17.6558i			0.710797i	0.934715	+	0.355399i	$0.115655\pi$
$617$			17.6558i			0.710797i	−0.934715	+	0.355399i	$0.884345\pi$
$618$		0			0
$619$			5.52857i			0.222212i	0.993809	+	0.111106i	$0.0354393\pi$
$619$			5.52857i			0.222212i	−0.993809	+	0.111106i	$0.964561\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	41.1869	−	4.66686i	1.65012	−	0.186974i
$624$		0			0
$625$	−28.6257			−1.14503
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−8.24169			−0.328618
$630$		0			0
$631$	24.3544			0.969533			0.484766	−	0.874644i	$-0.338905\pi$
$631$	24.3544			0.969533			0.484766	+	0.874644i	$0.338905\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7.43045			0.294868
$636$		0			0
$637$	−8.56467	−	37.3080i	−0.339345	−	1.47820i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	29.2340i		−	1.15467i	−0.816506	−	0.577337i	$-0.804092\pi$
$641$		−	29.2340i		−	1.15467i	0.816506	−	0.577337i	$-0.195908\pi$
$642$		0			0
$643$			40.0144i			1.57802i	0.614383	+	0.789008i	$0.289405\pi$
$643$			40.0144i			1.57802i	−0.614383	+	0.789008i	$0.710595\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	7.56553			0.297432			0.148716	−	0.988880i	$-0.452486\pi$
$647$	7.56553			0.297432			0.148716	+	0.988880i	$0.452486\pi$
$648$		0			0
$649$		−	0.377240i		−	0.0148080i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			39.1877i			1.53353i	0.641927	+	0.766766i	$0.278135\pi$
$653$			39.1877i			1.53353i	−0.641927	+	0.766766i	$0.721865\pi$
$654$		0			0
$655$	−27.7965			−1.08610
$656$		0			0
$657$		0			0
$658$		0			0
$659$			26.3957i			1.02823i	0.857721	+	0.514115i	$0.171880\pi$
$659$			26.3957i			1.02823i	−0.857721	+	0.514115i	$0.828120\pi$
$660$		0			0
$661$			31.5812i			1.22837i	0.789163	+	0.614184i	$0.210515\pi$
$661$			31.5812i			1.22837i	−0.789163	+	0.614184i	$0.789485\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.290888	−	2.56720i	−0.0112802	−	0.0995517i
$666$		0			0
$667$	−25.8522			−1.00100
$668$		0			0
$669$		0			0
$670$		0			0
$671$	28.5971			1.10398
$672$		0			0
$673$	14.4335			0.556371			0.278186	−	0.960527i	$-0.410267\pi$
$673$	14.4335			0.556371			0.278186	+	0.960527i	$0.410267\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−34.4198			−1.32286			−0.661430	−	0.750006i	$-0.730050\pi$
$677$	−34.4198			−1.32286			−0.661430	+	0.750006i	$0.730050\pi$
$678$		0			0
$679$	1.71196	+	15.1087i	0.0656990	+	0.579819i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			19.0172i			0.727674i	0.931463	+	0.363837i	$0.118533\pi$
$683$			19.0172i			0.727674i	−0.931463	+	0.363837i	$0.881467\pi$
$684$		0			0
$685$			4.78806i			0.182942i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	76.7147			2.92260
$690$		0			0
$691$		−	19.3753i		−	0.737070i	−0.929614	−	0.368535i	$-0.879859\pi$
$691$		−	19.3753i		−	0.737070i	0.929614	−	0.368535i	$-0.120141\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	15.0586i		−	0.571204i
$696$		0			0
$697$	−20.8481			−0.789676
$698$		0			0
$699$		0			0
$700$		0			0
$701$			23.0297i			0.869819i	0.900474	+	0.434910i	$0.143220\pi$
$701$			23.0297i			0.869819i	−0.900474	+	0.434910i	$0.856780\pi$
$702$		0			0
$703$			1.28609i			0.0485058i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−25.4395	+	2.88254i	−0.956750	+	0.108409i
$708$		0			0
$709$	−40.8238			−1.53317			−0.766585	−	0.642143i	$-0.778045\pi$
$709$	−40.8238			−1.53317			−0.766585	+	0.642143i	$0.778045\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	14.8837			0.557398
$714$		0			0
$715$	32.1051			1.20066
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−42.3156			−1.57811			−0.789054	−	0.614324i	$-0.789429\pi$
$719$	−42.3156			−1.57811			−0.789054	+	0.614324i	$0.789429\pi$
$720$		0			0
$721$	−5.73950	−	50.6533i	−0.213750	−	1.88643i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			6.42008i			0.238436i
$726$		0			0
$727$			5.29724i			0.196464i	0.995164	+	0.0982318i	$0.0313187\pi$
$727$			5.29724i			0.196464i	−0.995164	+	0.0982318i	$0.968681\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	21.8106			0.806694
$732$		0			0
$733$		−	18.6499i		−	0.688852i	−0.938814	−	0.344426i	$-0.888074\pi$
$733$		−	18.6499i		−	0.688852i	0.938814	−	0.344426i	$-0.111926\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			12.2842i			0.452494i
$738$		0			0
$739$	41.1837			1.51497			0.757483	−	0.652855i	$-0.226429\pi$
$739$	41.1837			1.51497			0.757483	+	0.652855i	$0.226429\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			23.7674i			0.871940i	0.899961	+	0.435970i	$0.143595\pi$
$743$			23.7674i			0.871940i	−0.899961	+	0.435970i	$0.856405\pi$
$744$		0			0
$745$		−	29.5089i		−	1.08112i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.324949	+	2.86780i	0.0118734	+	0.104787i
$750$		0			0
$751$	18.1230			0.661318			0.330659	−	0.943750i	$-0.392729\pi$
$751$	18.1230			0.661318			0.330659	+	0.943750i	$0.392729\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−25.7902			−0.938603
$756$		0			0
$757$	37.1059			1.34864			0.674319	−	0.738440i	$-0.264437\pi$
$757$	37.1059			1.34864			0.674319	+	0.738440i	$0.264437\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	11.8107			0.428139			0.214070	−	0.976818i	$-0.431328\pi$
$761$	11.8107			0.428139			0.214070	+	0.976818i	$0.431328\pi$
$762$		0			0
$763$	−6.08219	+	0.689170i	−0.220190	+	0.0249496i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	0.852017i		−	0.0307646i
$768$		0			0
$769$			3.62594i			0.130755i	0.997861	+	0.0653773i	$0.0208251\pi$
$769$			3.62594i			0.130755i	−0.997861	+	0.0653773i	$0.979175\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.1111			−0.435605			−0.217802	−	0.975993i	$-0.569889\pi$
$773$	−12.1111			−0.435605			−0.217802	+	0.975993i	$0.569889\pi$
$774$		0			0
$775$		−	3.69618i		−	0.132771i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			3.25327i			0.116561i
$780$		0			0
$781$	21.1608			0.757192
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	8.57233i		−	0.305960i
$786$		0			0
$787$		−	20.3222i		−	0.724406i	−0.932099	−	0.362203i	$-0.882025\pi$
$787$		−	20.3222i		−	0.724406i	0.932099	−	0.362203i	$-0.117975\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−4.77657	−	42.1551i	−0.169835	−	1.49886i
$792$		0			0
$793$	64.5880			2.29359
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−33.7945			−1.19706			−0.598532	−	0.801099i	$-0.704249\pi$
$797$	−33.7945			−1.19706			−0.598532	+	0.801099i	$0.704249\pi$
$798$		0			0
$799$	11.6528			0.412245
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−21.3130			−0.752119
$804$		0			0
$805$	2.55973	+	22.5906i	0.0902187	+	0.796214i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			4.84175i			0.170227i	0.996371	+	0.0851134i	$0.0271253\pi$
$809$			4.84175i			0.170227i	−0.996371	+	0.0851134i	$0.972875\pi$
$810$		0			0
$811$			0.493486i			0.0173286i	0.999962	+	0.00866432i	$0.00275797\pi$
$811$			0.493486i			0.0173286i	−0.999962	+	0.00866432i	$0.997242\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−15.3531			−0.537797
$816$		0			0
$817$		−	3.40347i		−	0.119072i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			30.8225i			1.07571i	0.843036	+	0.537857i	$0.180766\pi$
$821$			30.8225i			1.07571i	−0.843036	+	0.537857i	$0.819234\pi$
$822$		0			0
$823$	−15.8850			−0.553716			−0.276858	−	0.960911i	$-0.589293\pi$
$823$	−15.8850			−0.553716			−0.276858	+	0.960911i	$0.589293\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			35.6756i			1.24056i	0.784379	+	0.620282i	$0.212982\pi$
$827$			35.6756i			1.24056i	−0.784379	+	0.620282i	$0.787018\pi$
$828$		0			0
$829$		−	3.32551i		−	0.115500i	−0.998331	−	0.0577498i	$-0.981607\pi$
$829$		−	3.32551i		−	0.115500i	0.998331	−	0.0577498i	$-0.0183926\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	17.6068	−	4.04194i	0.610040	−	0.140045i
$834$		0			0
$835$	−40.7273			−1.40942
$836$		0			0
$837$		0			0
$838$		0			0
$839$	41.9378			1.44785			0.723926	−	0.689877i	$-0.242336\pi$
$839$	41.9378			1.44785			0.723926	+	0.689877i	$0.242336\pi$
$840$		0			0
$841$	−24.2201			−0.835175
$842$		0			0
$843$		0			0
$844$		0			0
$845$	40.9875			1.41001
$846$		0			0
$847$	−13.5072	+	1.53049i	−0.464112	+	0.0525883i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	11.3172i		−	0.387949i
$852$		0			0
$853$			14.4719i			0.495507i	0.968823	+	0.247754i	$0.0796924\pi$
$853$			14.4719i			0.495507i	−0.968823	+	0.247754i	$0.920308\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−47.4657			−1.62140			−0.810699	−	0.585463i	$-0.800913\pi$
$857$	−47.4657			−1.62140			−0.810699	+	0.585463i	$0.800913\pi$
$858$		0			0
$859$		−	6.00809i		−	0.204993i	−0.994733	−	0.102497i	$-0.967317\pi$
$859$		−	6.00809i		−	0.204993i	0.994733	−	0.102497i	$-0.0326831\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	31.2998i		−	1.06546i	−0.846286	−	0.532729i	$-0.821166\pi$
$863$		−	31.2998i		−	1.06546i	0.846286	−	0.532729i	$-0.178834\pi$
$864$		0			0
$865$	−40.2942			−1.37004
$866$		0			0
$867$		0			0
$868$		0			0
$869$			27.4356i			0.930688i
$870$		0			0
$871$			27.7445i			0.940086i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−26.2641	+	2.97597i	−0.887887	+	0.100606i
$876$		0			0
$877$	41.3764			1.39718			0.698591	−	0.715521i	$-0.253811\pi$
$877$	41.3764			1.39718			0.698591	+	0.715521i	$0.253811\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−39.7350			−1.33870			−0.669352	−	0.742945i	$-0.733428\pi$
$881$	−39.7350			−1.33870			−0.669352	+	0.742945i	$0.733428\pi$
$882$		0			0
$883$	−48.1324			−1.61978			−0.809892	−	0.586579i	$-0.800474\pi$
$883$	−48.1324			−1.61978			−0.809892	+	0.586579i	$0.800474\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−28.1122			−0.943916			−0.471958	−	0.881621i	$-0.656453\pi$
$887$	−28.1122			−0.943916			−0.471958	+	0.881621i	$0.656453\pi$
$888$		0			0
$889$	8.05571	−	0.912788i	0.270180	−	0.0306139i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	1.81838i		−	0.0608497i
$894$		0			0
$895$			35.3032i			1.18006i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	30.6400			1.02190
$900$		0			0
$901$			36.2041i			1.20613i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	9.75663i		−	0.324321i
$906$		0			0
$907$	59.4859			1.97520			0.987599	−	0.156997i	$-0.0501812\pi$
$907$	59.4859			1.97520			0.987599	+	0.156997i	$0.0501812\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	29.3874i		−	0.973648i	−0.873500	−	0.486824i	$-0.838155\pi$
$911$		−	29.3874i		−	0.973648i	0.873500	−	0.486824i	$-0.161845\pi$
$912$		0			0
$913$			36.3630i			1.20344i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−30.1355	+	3.41464i	−0.995161	+	0.112761i
$918$		0			0
$919$	−41.5987			−1.37222			−0.686108	−	0.727500i	$-0.740682\pi$
$919$	−41.5987			−1.37222			−0.686108	+	0.727500i	$0.740682\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	47.7927			1.57312
$924$		0			0
$925$	−2.81050			−0.0924086
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2.43228			0.0798004			0.0399002	−	0.999204i	$-0.487296\pi$
$929$	2.43228			0.0798004			0.0399002	+	0.999204i	$0.487296\pi$
$930$		0			0
$931$	−0.630731	−	2.74749i	−0.0206714	−	0.0900453i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			15.1514i			0.495503i
$936$		0			0
$937$			7.29837i			0.238427i	0.992869	+	0.119214i	$0.0380374\pi$
$937$			7.29837i			0.238427i	−0.992869	+	0.119214i	$0.961963\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−42.4735			−1.38460			−0.692298	−	0.721612i	$-0.743402\pi$
$941$	−42.4735			−1.38460			−0.692298	+	0.721612i	$0.743402\pi$
$942$		0			0
$943$		−	28.6279i		−	0.932252i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			54.6311i			1.77527i	0.460546	+	0.887636i	$0.347654\pi$
$947$			54.6311i			1.77527i	−0.460546	+	0.887636i	$0.652346\pi$
$948$		0			0
$949$	−48.1365			−1.56258
$950$		0			0
$951$		0			0
$952$		0			0
$953$			17.1877i			0.556764i	0.960470	+	0.278382i	$0.0897981\pi$
$953$			17.1877i			0.556764i	−0.960470	+	0.278382i	$0.910202\pi$
$954$		0			0
$955$			20.4120i			0.660518i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	0.588186	+	5.19097i	0.0189935	+	0.167625i
$960$		0			0
$961$	13.3599			0.430964
$962$		0			0
$963$		0			0
$964$		0			0
$965$	20.9405			0.674098
$966$		0			0
$967$	22.3091			0.717413			0.358706	−	0.933450i	$-0.383218\pi$
$967$	22.3091			0.717413			0.358706	+	0.933450i	$0.383218\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	54.2481			1.74091			0.870453	−	0.492252i	$-0.163826\pi$
$971$	54.2481			1.74091			0.870453	+	0.492252i	$0.163826\pi$
$972$		0			0
$973$	−1.84986	−	16.3257i	−0.0593037	−	0.523378i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	41.2154i		−	1.31860i	−0.751881	−	0.659299i	$-0.770853\pi$
$977$		−	41.2154i		−	1.31860i	0.751881	−	0.659299i	$-0.229147\pi$
$978$		0			0
$979$			37.9321i			1.21231i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18.4394			−0.588125			−0.294062	−	0.955786i	$-0.595007\pi$
$983$	−18.4394			−0.588125			−0.294062	+	0.955786i	$0.595007\pi$
$984$		0			0
$985$		−	56.5731i		−	1.80257i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			29.9496i			0.952341i
$990$		0			0
$991$	−3.72231			−0.118243			−0.0591216	−	0.998251i	$-0.518830\pi$
$991$	−3.72231			−0.118243			−0.0591216	+	0.998251i	$0.518830\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			34.2836i			1.08686i
$996$		0			0
$997$			0.475330i			0.0150539i	0.999972	+	0.00752693i	$0.00239592\pi$
$997$			0.475330i			0.0150539i	−0.999972	+	0.00752693i	$0.997604\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.f.b.1133.4		16
3.2	odd	2	inner	2268.2.f.b.1133.14		16
7.6	odd	2	inner	2268.2.f.b.1133.13		16
9.2	odd	6		252.2.x.a.41.2	✓	16
9.4	even	3		252.2.x.a.209.7	yes	16
9.5	odd	6		756.2.x.a.629.2		16
9.7	even	3		756.2.x.a.125.7		16
21.20	even	2	inner	2268.2.f.b.1133.3		16
36.7	odd	6		3024.2.cc.c.881.7		16
36.11	even	6		1008.2.cc.c.545.7		16
36.23	even	6		3024.2.cc.c.2897.2		16
36.31	odd	6		1008.2.cc.c.209.2		16
63.2	odd	6		1764.2.bm.b.1697.7		16
63.4	even	3		1764.2.bm.b.1685.2		16
63.5	even	6		5292.2.w.a.521.7		16
63.11	odd	6		1764.2.w.a.509.4		16
63.13	odd	6		252.2.x.a.209.2	yes	16
63.16	even	3		5292.2.bm.b.2285.2		16
63.20	even	6		252.2.x.a.41.7	yes	16
63.23	odd	6		5292.2.w.a.521.2		16
63.25	even	3		5292.2.w.a.1097.7		16
63.31	odd	6		1764.2.bm.b.1685.7		16
63.32	odd	6		5292.2.bm.b.4625.7		16
63.34	odd	6		756.2.x.a.125.2		16
63.38	even	6		1764.2.w.a.509.5		16
63.40	odd	6		1764.2.w.a.1109.4		16
63.41	even	6		756.2.x.a.629.7		16
63.47	even	6		1764.2.bm.b.1697.2		16
63.52	odd	6		5292.2.w.a.1097.2		16
63.58	even	3		1764.2.w.a.1109.5		16
63.59	even	6		5292.2.bm.b.4625.2		16
63.61	odd	6		5292.2.bm.b.2285.7		16
252.83	odd	6		1008.2.cc.c.545.2		16
252.139	even	6		1008.2.cc.c.209.7		16
252.167	odd	6		3024.2.cc.c.2897.7		16
252.223	even	6		3024.2.cc.c.881.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.x.a.41.2	✓	16	9.2	odd	6
252.2.x.a.41.7	yes	16	63.20	even	6
252.2.x.a.209.2	yes	16	63.13	odd	6
252.2.x.a.209.7	yes	16	9.4	even	3
756.2.x.a.125.2		16	63.34	odd	6
756.2.x.a.125.7		16	9.7	even	3
756.2.x.a.629.2		16	9.5	odd	6
756.2.x.a.629.7		16	63.41	even	6
1008.2.cc.c.209.2		16	36.31	odd	6
1008.2.cc.c.209.7		16	252.139	even	6
1008.2.cc.c.545.2		16	252.83	odd	6
1008.2.cc.c.545.7		16	36.11	even	6
1764.2.w.a.509.4		16	63.11	odd	6
1764.2.w.a.509.5		16	63.38	even	6
1764.2.w.a.1109.4		16	63.40	odd	6
1764.2.w.a.1109.5		16	63.58	even	3
1764.2.bm.b.1685.2		16	63.4	even	3
1764.2.bm.b.1685.7		16	63.31	odd	6
1764.2.bm.b.1697.2		16	63.47	even	6
1764.2.bm.b.1697.7		16	63.2	odd	6
2268.2.f.b.1133.3		16	21.20	even	2	inner
2268.2.f.b.1133.4		16	1.1	even	1	trivial
2268.2.f.b.1133.13		16	7.6	odd	2	inner
2268.2.f.b.1133.14		16	3.2	odd	2	inner
3024.2.cc.c.881.2		16	252.223	even	6
3024.2.cc.c.881.7		16	36.7	odd	6
3024.2.cc.c.2897.2		16	36.23	even	6
3024.2.cc.c.2897.7		16	252.167	odd	6
5292.2.w.a.521.2		16	63.23	odd	6
5292.2.w.a.521.7		16	63.5	even	6
5292.2.w.a.1097.2		16	63.52	odd	6
5292.2.w.a.1097.7		16	63.25	even	3
5292.2.bm.b.2285.2		16	63.16	even	3
5292.2.bm.b.2285.7		16	63.61	odd	6
5292.2.bm.b.4625.2		16	63.59	even	6
5292.2.bm.b.4625.7		16	63.32	odd	6

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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.42488 q^{5} +(-2.62893 + 0.297883i) q^{7} +O(q^{10})\)	\(q-2.42488 q^{5} +(-2.62893 + 0.297883i) q^{7} -2.42118i q^{11} -5.46836i q^{13} +2.58069 q^{17} -0.402708i q^{19} +3.54372i q^{23} +0.880040 q^{25} +7.29521i q^{29} -4.20001i q^{31} +(6.37484 - 0.722330i) q^{35} -3.19360 q^{37} -8.07849 q^{41} +8.45146 q^{43} +4.51537 q^{47} +(6.82253 - 1.56622i) q^{49} +14.0288i q^{53} +5.87106i q^{55} +0.155809 q^{59} +11.8112i q^{61} +13.2601i q^{65} -5.07364 q^{67} +8.73987i q^{71} -8.80274i q^{73} +(0.721227 + 6.36510i) q^{77} -11.3315 q^{79} -15.0187 q^{83} -6.25786 q^{85} -15.6668 q^{89} +(1.62893 + 14.3759i) q^{91} +0.976519i q^{95} -5.74710i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{7}+O(q^{10}) \) 16 * q + 2 * q^7	\( 16 q + 2 q^{7} + 16 q^{25} + 4 q^{37} - 8 q^{43} + 10 q^{49} - 28 q^{67} - 40 q^{79} - 12 q^{85} - 18 q^{91}+O(q^{100}) \) 16 * q + 2 * q^7 + 16 * q^25 + 4 * q^37 - 8 * q^43 + 10 * q^49 - 28 * q^67 - 40 * q^79 - 12 * q^85 - 18 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more