LMFDB - Embedded newform 3312.2.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3312,2,Mod(1,3312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3312, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3312.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3312 = 2^{4} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3312.a (trivial)

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:	yes
Analytic conductor:	$26.4464531494$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.44688.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 6x + 2 $ x^4 - 2x^3 - 5x^2 + 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1656)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.27733$ of defining polynomial
Character	$\chi$	$=$	3312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.64575 q^{5} +1.01109 q^{7} +5.21151 q^{11} +2.55467 q^{13} +3.56576 q^{17} -3.09108 q^{19} +1.00000 q^{23} -2.29150 q^{25} +1.44533 q^{29} -1.44533 q^{31} -1.66401 q^{35} +6.10217 q^{37} +4.73683 q^{41} -11.3319 q^{43} -0.0221841 q^{47} -5.97769 q^{49} +7.66794 q^{53} -8.57685 q^{55} +3.28535 q^{59} +7.21151 q^{61} -4.20435 q^{65} -11.4919 q^{67} -0.736834 q^{71} +11.6862 q^{73} +5.26932 q^{77} +2.12043 q^{79} +9.21151 q^{83} -5.86836 q^{85} -7.96660 q^{89} +2.58301 q^{91} +5.08715 q^{95} -7.31369 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 2 q^{7} - 2 q^{11} + 4 q^{13} + 2 q^{17} - 8 q^{19} + 4 q^{23} + 12 q^{25} + 12 q^{29} - 12 q^{31} + 12 q^{35} + 14 q^{37} + 4 q^{41} - 4 q^{43} + 12 q^{47} + 28 q^{49} + 8 q^{53} - 16 q^{55}+ \cdots + 4 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 2 * q^7 - 2 * q^11 + 4 * q^13 + 2 * q^17 - 8 * q^19 + 4 * q^23 + 12 * q^25 + 12 * q^29 - 12 * q^31 + 12 * q^35 + 14 * q^37 + 4 * q^41 - 4 * q^43 + 12 * q^47 + 28 * q^49 + 8 * q^53 - 16 * q^55 + 16 * q^59 + 6 * q^61 + 4 * q^65 - 8 * q^67 + 12 * q^71 + 16 * q^73 + 12 * q^77 - 10 * q^79 + 14 * q^83 + 16 * q^85 + 14 * q^89 - 32 * q^91 + 20 * q^95 + 4 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.64575	−0.736002	−0.368001	−	0.929825i	$-0.619958\pi$
$5$	−1.64575	−0.736002	−0.368001	+	0.929825i	$0.619958\pi$
$6$	0	0
$7$	1.01109	0.382157	0.191078	−	0.981575i	$-0.438802\pi$
$7$	1.01109	0.382157	0.191078	+	0.981575i	$0.438802\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.21151	1.57133	0.785665	−	0.618652i	$-0.212321\pi$
$11$	5.21151	1.57133	0.785665	+	0.618652i	$0.212321\pi$
$12$	0	0
$13$	2.55467	0.708538	0.354269	−	0.935144i	$-0.384730\pi$
$13$	2.55467	0.708538	0.354269	+	0.935144i	$0.384730\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.56576	0.864824	0.432412	−	0.901676i	$-0.357663\pi$
$17$	3.56576	0.864824	0.432412	+	0.901676i	$0.357663\pi$
$18$	0	0
$19$	−3.09108	−0.709143	−0.354571	−	0.935029i	$-0.615373\pi$
$19$	−3.09108	−0.709143	−0.354571	+	0.935029i	$0.615373\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.29150	−0.458301
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.44533	0.268391	0.134196	−	0.990955i	$-0.457155\pi$
$29$	1.44533	0.268391	0.134196	+	0.990955i	$0.457155\pi$
$30$	0	0
$31$	−1.44533	−0.259589	−0.129795	−	0.991541i	$-0.541432\pi$
$31$	−1.44533	−0.259589	−0.129795	+	0.991541i	$0.541432\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.66401	−0.281268
$36$	0	0
$37$	6.10217	1.00319	0.501596	−	0.865102i	$-0.332747\pi$
$37$	6.10217	1.00319	0.501596	+	0.865102i	$0.332747\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.73683	0.739769	0.369885	−	0.929078i	$-0.379397\pi$
$41$	4.73683	0.739769	0.369885	+	0.929078i	$0.379397\pi$
$42$	0	0
$43$	−11.3319	−1.72810	−0.864052	−	0.503402i	$-0.832082\pi$
$43$	−11.3319	−1.72810	−0.864052	+	0.503402i	$0.832082\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.0221841	−0.00323589	−0.00161794	−	0.999999i	$-0.500515\pi$
$47$	−0.0221841	−0.00323589	−0.00161794	+	0.999999i	$0.500515\pi$
$48$	0	0
$49$	−5.97769	−0.853956
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.66794	1.05327	0.526636	−	0.850091i	$-0.323453\pi$
$53$	7.66794	1.05327	0.526636	+	0.850091i	$0.323453\pi$
$54$	0	0
$55$	−8.57685	−1.15650
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.28535	0.427716	0.213858	−	0.976865i	$-0.431397\pi$
$59$	3.28535	0.427716	0.213858	+	0.976865i	$0.431397\pi$
$60$	0	0
$61$	7.21151	0.923340	0.461670	−	0.887052i	$-0.347251\pi$
$61$	7.21151	0.923340	0.461670	+	0.887052i	$0.347251\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.20435	−0.521485
$66$	0	0
$67$	−11.4919	−1.40396	−0.701981	−	0.712196i	$-0.747701\pi$
$67$	−11.4919	−1.40396	−0.701981	+	0.712196i	$0.747701\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.736834	−0.0874461	−0.0437230	−	0.999044i	$-0.513922\pi$
$71$	−0.736834	−0.0874461	−0.0437230	+	0.999044i	$0.513922\pi$
$72$	0	0
$73$	11.6862	1.36777	0.683883	−	0.729592i	$-0.260290\pi$
$73$	11.6862	1.36777	0.683883	+	0.729592i	$0.260290\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.26932	0.600495
$78$	0	0
$79$	2.12043	0.238567	0.119283	−	0.992860i	$-0.461940\pi$
$79$	2.12043	0.238567	0.119283	+	0.992860i	$0.461940\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.21151	1.01109	0.505547	−	0.862799i	$-0.331291\pi$
$83$	9.21151	1.01109	0.505547	+	0.862799i	$0.331291\pi$
$84$	0	0
$85$	−5.86836	−0.636513
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.96660	−0.844458	−0.422229	−	0.906489i	$-0.638752\pi$
$89$	−7.96660	−0.844458	−0.422229	+	0.906489i	$0.638752\pi$
$90$	0	0
$91$	2.58301	0.270773
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.08715	0.521931
$96$	0	0
$97$	−7.31369	−0.742592	−0.371296	−	0.928514i	$-0.621087\pi$
$97$	−7.31369	−0.742592	−0.371296	+	0.928514i	$0.621087\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.42315	0.340616	0.170308	−	0.985391i	$-0.445524\pi$
$101$	3.42315	0.340616	0.170308	+	0.985391i	$0.445524\pi$
$102$	0	0
$103$	7.43412	0.732505	0.366253	−	0.930515i	$-0.380641\pi$
$103$	7.43412	0.732505	0.366253	+	0.930515i	$0.380641\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.5252	−1.01751	−0.508755	−	0.860912i	$-0.669894\pi$
$107$	−10.5252	−1.01751	−0.508755	+	0.860912i	$0.669894\pi$
$108$	0	0
$109$	17.7945	1.70441	0.852203	−	0.523212i	$-0.175266\pi$
$109$	17.7945	1.70441	0.852203	+	0.523212i	$0.175266\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.835079	0.0785576	0.0392788	−	0.999228i	$-0.487494\pi$
$113$	0.835079	0.0785576	0.0392788	+	0.999228i	$0.487494\pi$
$114$	0	0
$115$	−1.64575	−0.153467
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.60531	0.330498
$120$	0	0
$121$	16.1599	1.46908
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	8.60519	0.763587	0.381794	−	0.924248i	$-0.375307\pi$
$127$	8.60519	0.763587	0.381794	+	0.924248i	$0.375307\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.3137	1.16322	0.581611	−	0.813467i	$-0.302423\pi$
$131$	13.3137	1.16322	0.581611	+	0.813467i	$0.302423\pi$
$132$	0	0
$133$	−3.12537	−0.271004
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.8066	1.35045	0.675225	−	0.737612i	$-0.264046\pi$
$137$	15.8066	1.35045	0.675225	+	0.737612i	$0.264046\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13.3137	1.11335
$144$	0	0
$145$	−2.37866	−0.197537
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.9088	1.63099	0.815496	−	0.578763i	$-0.196464\pi$
$149$	19.9088	1.63099	0.815496	+	0.578763i	$0.196464\pi$
$150$	0	0
$151$	−10.0505	−0.817900	−0.408950	−	0.912557i	$-0.634105\pi$
$151$	−10.0505	−0.817900	−0.408950	+	0.912557i	$0.634105\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.37866	0.191058
$156$	0	0
$157$	4.48083	0.357609	0.178805	−	0.983885i	$-0.442777\pi$
$157$	4.48083	0.357609	0.178805	+	0.983885i	$0.442777\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.01109	0.0796852
$162$	0	0
$163$	20.1376	1.57729	0.788647	−	0.614846i	$-0.210782\pi$
$163$	20.1376	1.57729	0.788647	+	0.614846i	$0.210782\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.9999	−1.16073	−0.580363	−	0.814358i	$-0.697089\pi$
$167$	−14.9999	−1.16073	−0.580363	+	0.814358i	$0.697089\pi$
$168$	0	0
$169$	−6.47367	−0.497974
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.29138	−0.478325	−0.239162	−	0.970980i	$-0.576873\pi$
$173$	−6.29138	−0.478325	−0.239162	+	0.970980i	$0.576873\pi$
$174$	0	0
$175$	−2.31692	−0.175143
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.2914	1.21767	0.608837	−	0.793295i	$-0.291636\pi$
$179$	16.2914	1.21767	0.608837	+	0.793295i	$0.291636\pi$
$180$	0	0
$181$	12.5252	0.930991	0.465495	−	0.885050i	$-0.345876\pi$
$181$	12.5252	0.930991	0.465495	+	0.885050i	$0.345876\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−10.0427	−0.738351
$186$	0	0
$187$	18.5830	1.35892
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.2693	0.960134	0.480067	−	0.877232i	$-0.340612\pi$
$191$	13.2693	0.960134	0.480067	+	0.877232i	$0.340612\pi$
$192$	0	0
$193$	7.50402	0.540152	0.270076	−	0.962839i	$-0.412951\pi$
$193$	7.50402	0.540152	0.270076	+	0.962839i	$0.412951\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.75902	−0.196572	−0.0982859	−	0.995158i	$-0.531336\pi$
$197$	−2.75902	−0.196572	−0.0982859	+	0.995158i	$0.531336\pi$
$198$	0	0
$199$	−15.2741	−1.08275	−0.541377	−	0.840780i	$-0.682097\pi$
$199$	−15.2741	−1.08275	−0.541377	+	0.840780i	$0.682097\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.46136	0.102568
$204$	0	0
$205$	−7.79565	−0.544472
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−16.1092	−1.11430
$210$	0	0
$211$	−7.33587	−0.505022	−0.252511	−	0.967594i	$-0.581256\pi$
$211$	−7.33587	−0.505022	−0.252511	+	0.967594i	$0.581256\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	18.6496	1.27189
$216$	0	0
$217$	−1.46136	−0.0992038
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.10934	0.612760
$222$	0	0
$223$	11.1155	0.744348	0.372174	−	0.928163i	$-0.378612\pi$
$223$	11.1155	0.744348	0.372174	+	0.928163i	$0.378612\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.1082	−1.13551	−0.567756	−	0.823197i	$-0.692188\pi$
$227$	−17.1082	−1.13551	−0.567756	+	0.823197i	$0.692188\pi$
$228$	0	0
$229$	−13.7945	−0.911567	−0.455784	−	0.890091i	$-0.650641\pi$
$229$	−13.7945	−0.911567	−0.455784	+	0.890091i	$0.650641\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.95563	−0.259142	−0.129571	−	0.991570i	$-0.541360\pi$
$233$	−3.95563	−0.259142	−0.129571	+	0.991570i	$0.541360\pi$
$234$	0	0
$235$	0.0365096	0.00238162
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.04437	0.261608	0.130804	−	0.991408i	$-0.458244\pi$
$239$	4.04437	0.261608	0.130804	+	0.991408i	$0.458244\pi$
$240$	0	0
$241$	−12.3787	−0.797379	−0.398690	−	0.917086i	$-0.630535\pi$
$241$	−12.3787	−0.797379	−0.398690	+	0.917086i	$0.630535\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	9.83780	0.628514
$246$	0	0
$247$	−7.89669	−0.502454
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.6852	1.17940	0.589699	−	0.807623i	$-0.299246\pi$
$251$	18.6852	1.17940	0.589699	+	0.807623i	$0.299246\pi$
$252$	0	0
$253$	5.21151	0.327645
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.4169	−1.27357	−0.636785	−	0.771042i	$-0.719736\pi$
$257$	−20.4169	−1.27357	−0.636785	+	0.771042i	$0.719736\pi$
$258$	0	0
$259$	6.16986	0.383376
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.10934	−0.561706	−0.280853	−	0.959751i	$-0.590617\pi$
$263$	−9.10934	−0.561706	−0.280853	+	0.959751i	$0.590617\pi$
$264$	0	0
$265$	−12.6195	−0.775211
$266$	0	0
$267$	0	0
$268$	0	0
$269$	19.5607	1.19264	0.596318	−	0.802748i	$-0.296630\pi$
$269$	19.5607	1.19264	0.596318	+	0.802748i	$0.296630\pi$
$270$	0	0
$271$	−2.02218	−0.122839	−0.0614195	−	0.998112i	$-0.519563\pi$
$271$	−2.02218	−0.122839	−0.0614195	+	0.998112i	$0.519563\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−11.9422	−0.720141
$276$	0	0
$277$	−24.4230	−1.46744	−0.733719	−	0.679453i	$-0.762217\pi$
$277$	−24.4230	−1.46744	−0.733719	+	0.679453i	$0.762217\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	13.4403	0.801779	0.400890	−	0.916126i	$-0.368701\pi$
$281$	13.4403	0.801779	0.400890	+	0.916126i	$0.368701\pi$
$282$	0	0
$283$	7.13545	0.424159	0.212079	−	0.977252i	$-0.431976\pi$
$283$	7.13545	0.424159	0.212079	+	0.977252i	$0.431976\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.78938	0.282708
$288$	0	0
$289$	−4.28535	−0.252079
$290$	0	0
$291$	0	0
$292$	0	0
$293$	28.2731	1.65173	0.825867	−	0.563865i	$-0.190686\pi$
$293$	28.2731	1.65173	0.825867	+	0.563865i	$0.190686\pi$
$294$	0	0
$295$	−5.40687	−0.314800
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.55467	0.147740
$300$	0	0
$301$	−11.4576	−0.660407
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−11.8684	−0.679580
$306$	0	0
$307$	5.07283	0.289522	0.144761	−	0.989467i	$-0.453759\pi$
$307$	5.07283	0.289522	0.144761	+	0.989467i	$0.453759\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−29.9250	−1.69689	−0.848446	−	0.529281i	$-0.822462\pi$
$311$	−29.9250	−1.69689	−0.848446	+	0.529281i	$0.822462\pi$
$312$	0	0
$313$	7.35806	0.415902	0.207951	−	0.978139i	$-0.433321\pi$
$313$	7.35806	0.415902	0.207951	+	0.978139i	$0.433321\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.3563	1.76115	0.880574	−	0.473909i	$-0.157157\pi$
$317$	31.3563	1.76115	0.880574	+	0.473909i	$0.157157\pi$
$318$	0	0
$319$	7.53236	0.421731
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−11.0221	−0.613284
$324$	0	0
$325$	−5.85403	−0.324723
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.0224302	−0.00123662
$330$	0	0
$331$	25.8745	1.42219	0.711096	−	0.703095i	$-0.248199\pi$
$331$	25.8745	1.42219	0.711096	+	0.703095i	$0.248199\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	18.9128	1.03332
$336$	0	0
$337$	−29.7811	−1.62228	−0.811139	−	0.584853i	$-0.801152\pi$
$337$	−29.7811	−1.62228	−0.811139	+	0.584853i	$0.801152\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−7.53236	−0.407900
$342$	0	0
$343$	−13.1216	−0.708502
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.2347	−0.656793	−0.328397	−	0.944540i	$-0.606508\pi$
$347$	−12.2347	−0.656793	−0.328397	+	0.944540i	$0.606508\pi$
$348$	0	0
$349$	−21.0221	−1.12529	−0.562643	−	0.826700i	$-0.690215\pi$
$349$	−21.0221	−1.12529	−0.562643	+	0.826700i	$0.690215\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.91285	−0.261484	−0.130742	−	0.991416i	$-0.541736\pi$
$353$	−4.91285	−0.261484	−0.130742	+	0.991416i	$0.541736\pi$
$354$	0	0
$355$	1.21265	0.0643605
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8.15998	−0.430667	−0.215334	−	0.976541i	$-0.569084\pi$
$359$	−8.15998	−0.430667	−0.215334	+	0.976541i	$0.569084\pi$
$360$	0	0
$361$	−9.44521	−0.497116
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−19.2326	−1.00668
$366$	0	0
$367$	24.7034	1.28951	0.644754	−	0.764390i	$-0.276960\pi$
$367$	24.7034	1.28951	0.644754	+	0.764390i	$0.276960\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.75299	0.402515
$372$	0	0
$373$	4.16087	0.215442	0.107721	−	0.994181i	$-0.465645\pi$
$373$	4.16087	0.215442	0.107721	+	0.994181i	$0.465645\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.69234	0.190165
$378$	0	0
$379$	−11.1276	−0.571586	−0.285793	−	0.958291i	$-0.592257\pi$
$379$	−11.1276	−0.571586	−0.285793	+	0.958291i	$0.592257\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.89066	−0.147706	−0.0738530	−	0.997269i	$-0.523530\pi$
$383$	−2.89066	−0.147706	−0.0738530	+	0.997269i	$0.523530\pi$
$384$	0	0
$385$	−8.67199	−0.441965
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−17.7266	−0.898776	−0.449388	−	0.893337i	$-0.648358\pi$
$389$	−17.7266	−0.898776	−0.449388	+	0.893337i	$0.648358\pi$
$390$	0	0
$391$	3.56576	0.180328
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3.48970	−0.175586
$396$	0	0
$397$	7.47367	0.375093	0.187546	−	0.982256i	$-0.439947\pi$
$397$	7.47367	0.375093	0.187546	+	0.982256i	$0.439947\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.56576	0.178066	0.0890328	−	0.996029i	$-0.471622\pi$
$401$	3.56576	0.178066	0.0890328	+	0.996029i	$0.471622\pi$
$402$	0	0
$403$	−3.69234	−0.183929
$404$	0	0
$405$	0	0
$406$	0	0
$407$	31.8016	1.57634
$408$	0	0
$409$	24.7144	1.22205	0.611024	−	0.791612i	$-0.290758\pi$
$409$	24.7144	1.22205	0.611024	+	0.791612i	$0.290758\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.32179	0.163455
$414$	0	0
$415$	−15.1599	−0.744168
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.15282	−0.154025	−0.0770126	−	0.997030i	$-0.524538\pi$
$419$	−3.15282	−0.154025	−0.0770126	+	0.997030i	$0.524538\pi$
$420$	0	0
$421$	−10.0578	−0.490187	−0.245094	−	0.969499i	$-0.578819\pi$
$421$	−10.0578	−0.490187	−0.245094	+	0.969499i	$0.578819\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−8.17095	−0.396349
$426$	0	0
$427$	7.29150	0.352861
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.9494	−0.623749	−0.311874	−	0.950123i	$-0.600957\pi$
$431$	−12.9494	−0.623749	−0.311874	+	0.950123i	$0.600957\pi$
$432$	0	0
$433$	10.5243	0.505766	0.252883	−	0.967497i	$-0.418621\pi$
$433$	10.5243	0.505766	0.252883	+	0.967497i	$0.418621\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.09108	−0.147867
$438$	0	0
$439$	18.6274	0.889036	0.444518	−	0.895770i	$-0.353375\pi$
$439$	18.6274	0.889036	0.444518	+	0.895770i	$0.353375\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.69234	0.175428	0.0877142	−	0.996146i	$-0.472044\pi$
$443$	3.69234	0.175428	0.0877142	+	0.996146i	$0.472044\pi$
$444$	0	0
$445$	13.1110	0.621523
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−17.1012	−0.807054	−0.403527	−	0.914968i	$-0.632216\pi$
$449$	−17.1012	−0.807054	−0.403527	+	0.914968i	$0.632216\pi$
$450$	0	0
$451$	24.6861	1.16242
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.25098	−0.199289
$456$	0	0
$457$	−35.3137	−1.65190	−0.825952	−	0.563740i	$-0.809362\pi$
$457$	−35.3137	−1.65190	−0.825952	+	0.563740i	$0.809362\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	29.7794	1.38696	0.693482	−	0.720474i	$-0.256076\pi$
$461$	29.7794	1.38696	0.693482	+	0.720474i	$0.256076\pi$
$462$	0	0
$463$	6.17431	0.286944	0.143472	−	0.989654i	$-0.454173\pi$
$463$	6.17431	0.286944	0.143472	+	0.989654i	$0.454173\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	38.1732	1.76644	0.883222	−	0.468955i	$-0.155369\pi$
$467$	38.1732	1.76644	0.883222	+	0.468955i	$0.155369\pi$
$468$	0	0
$469$	−11.6194	−0.536534
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−59.0566	−2.71542
$474$	0	0
$475$	7.08322	0.325001
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.80168	−0.219394	−0.109697	−	0.993965i	$-0.534988\pi$
$479$	−4.80168	−0.219394	−0.109697	+	0.993965i	$0.534988\pi$
$480$	0	0
$481$	15.5890	0.710799
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.0365	0.546550
$486$	0	0
$487$	25.2104	1.14239	0.571196	−	0.820814i	$-0.306480\pi$
$487$	25.2104	1.14239	0.571196	+	0.820814i	$0.306480\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−11.4453	−0.516521	−0.258260	−	0.966075i	$-0.583149\pi$
$491$	−11.4453	−0.516521	−0.258260	+	0.966075i	$0.583149\pi$
$492$	0	0
$493$	5.15371	0.232111
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.745007	−0.0334181
$498$	0	0
$499$	−30.7083	−1.37469	−0.687345	−	0.726331i	$-0.741224\pi$
$499$	−30.7083	−1.37469	−0.687345	+	0.726331i	$0.741224\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−31.2247	−1.39224	−0.696120	−	0.717925i	$-0.745092\pi$
$503$	−31.2247	−1.39224	−0.696120	+	0.717925i	$0.745092\pi$
$504$	0	0
$505$	−5.63365	−0.250694
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−35.7207	−1.58329	−0.791646	−	0.610981i	$-0.790775\pi$
$509$	−35.7207	−1.58329	−0.791646	+	0.610981i	$0.790775\pi$
$510$	0	0
$511$	11.8158	0.522701
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−12.2347	−0.539126
$516$	0	0
$517$	−0.115613	−0.00508465
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−25.6446	−1.12351	−0.561756	−	0.827303i	$-0.689874\pi$
$521$	−25.6446	−1.12351	−0.561756	+	0.827303i	$0.689874\pi$
$522$	0	0
$523$	−6.53026	−0.285548	−0.142774	−	0.989755i	$-0.545602\pi$
$523$	−6.53026	−0.285548	−0.142774	+	0.989755i	$0.545602\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.15371	−0.224499
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12.1010	0.524154
$534$	0	0
$535$	17.3219	0.748889
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−31.1528	−1.34185
$540$	0	0
$541$	4.61336	0.198344	0.0991720	−	0.995070i	$-0.468381\pi$
$541$	4.61336	0.198344	0.0991720	+	0.995070i	$0.468381\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−29.2854	−1.25445
$546$	0	0
$547$	−17.6112	−0.753001	−0.376501	−	0.926416i	$-0.622873\pi$
$547$	−17.6112	−0.753001	−0.376501	+	0.926416i	$0.622873\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.46764	−0.190328
$552$	0	0
$553$	2.14395	0.0911700
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.44140	0.0610742	0.0305371	−	0.999534i	$-0.490278\pi$
$557$	1.44140	0.0610742	0.0305371	+	0.999534i	$0.490278\pi$
$558$	0	0
$559$	−28.9494	−1.22443
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8.84718	−0.372864	−0.186432	−	0.982468i	$-0.559692\pi$
$563$	−8.84718	−0.372864	−0.186432	+	0.982468i	$0.559692\pi$
$564$	0	0
$565$	−1.37433	−0.0578186
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.47861	0.187753	0.0938765	−	0.995584i	$-0.470074\pi$
$569$	4.47861	0.187753	0.0938765	+	0.995584i	$0.470074\pi$
$570$	0	0
$571$	7.49192	0.313527	0.156764	−	0.987636i	$-0.449894\pi$
$571$	7.49192	0.313527	0.156764	+	0.987636i	$0.449894\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.29150	−0.0955623
$576$	0	0
$577$	14.2265	0.592258	0.296129	−	0.955148i	$-0.404304\pi$
$577$	14.2265	0.592258	0.296129	+	0.955148i	$0.404304\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	9.31369	0.386397
$582$	0	0
$583$	39.9615	1.65504
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.94131	−0.0801263	−0.0400631	−	0.999197i	$-0.512756\pi$
$587$	−1.94131	−0.0801263	−0.0400631	+	0.999197i	$0.512756\pi$
$588$	0	0
$589$	4.46764	0.184086
$590$	0	0
$591$	0	0
$592$	0	0
$593$	40.1659	1.64942	0.824708	−	0.565559i	$-0.191340\pi$
$593$	40.1659	1.64942	0.824708	+	0.565559i	$0.191340\pi$
$594$	0	0
$595$	−5.93345	−0.243248
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−17.4170	−0.711639	−0.355820	−	0.934555i	$-0.615798\pi$
$599$	−17.4170	−0.711639	−0.355820	+	0.934555i	$0.615798\pi$
$600$	0	0
$601$	43.3925	1.77002	0.885009	−	0.465573i	$-0.154152\pi$
$601$	43.3925	1.77002	0.885009	+	0.465573i	$0.154152\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−26.5951	−1.08124
$606$	0	0
$607$	20.2692	0.822701	0.411351	−	0.911477i	$-0.365057\pi$
$607$	20.2692	0.822701	0.411351	+	0.911477i	$0.365057\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.0566731	−0.00229275
$612$	0	0
$613$	11.6792	0.471716	0.235858	−	0.971787i	$-0.424210\pi$
$613$	11.6792	0.471716	0.235858	+	0.971787i	$0.424210\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−43.6589	−1.75764	−0.878821	−	0.477151i	$-0.841670\pi$
$617$	−43.6589	−1.75764	−0.878821	+	0.477151i	$0.841670\pi$
$618$	0	0
$619$	10.1193	0.406729	0.203364	−	0.979103i	$-0.434812\pi$
$619$	10.1193	0.406729	0.203364	+	0.979103i	$0.434812\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8.05497	−0.322715
$624$	0	0
$625$	−8.29150	−0.331660
$626$	0	0
$627$	0	0
$628$	0	0
$629$	21.7589	0.867584
$630$	0	0
$631$	−42.8188	−1.70459	−0.852295	−	0.523062i	$-0.824790\pi$
$631$	−42.8188	−1.70459	−0.852295	+	0.523062i	$0.824790\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−14.1620	−0.562002
$636$	0	0
$637$	−15.2710	−0.605060
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.13646	0.242376	0.121188	−	0.992630i	$-0.461330\pi$
$641$	6.13646	0.242376	0.121188	+	0.992630i	$0.461330\pi$
$642$	0	0
$643$	−35.3319	−1.39336	−0.696678	−	0.717384i	$-0.745339\pi$
$643$	−35.3319	−1.39336	−0.696678	+	0.717384i	$0.745339\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.5403	0.571640	0.285820	−	0.958283i	$-0.407734\pi$
$647$	14.5403	0.571640	0.285820	+	0.958283i	$0.407734\pi$
$648$	0	0
$649$	17.1216	0.672083
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20.3136	−0.794931	−0.397466	−	0.917617i	$-0.630110\pi$
$653$	−20.3136	−0.794931	−0.397466	+	0.917617i	$0.630110\pi$
$654$	0	0
$655$	−21.9110	−0.856134
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−28.9605	−1.12814	−0.564071	−	0.825726i	$-0.690766\pi$
$659$	−28.9605	−1.12814	−0.564071	+	0.825726i	$0.690766\pi$
$660$	0	0
$661$	−17.1669	−0.667715	−0.333857	−	0.942624i	$-0.608350\pi$
$661$	−17.1669	−0.667715	−0.333857	+	0.942624i	$0.608350\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.14358	0.199459
$666$	0	0
$667$	1.44533	0.0559635
$668$	0	0
$669$	0	0
$670$	0	0
$671$	37.5829	1.45087
$672$	0	0
$673$	15.3359	0.591154	0.295577	−	0.955319i	$-0.404488\pi$
$673$	15.3359	0.591154	0.295577	+	0.955319i	$0.404488\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.79970	0.338200	0.169100	−	0.985599i	$-0.445914\pi$
$677$	8.79970	0.338200	0.169100	+	0.985599i	$0.445914\pi$
$678$	0	0
$679$	−7.39481	−0.283787
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−16.6861	−0.638475	−0.319237	−	0.947675i	$-0.603427\pi$
$683$	−16.6861	−0.638475	−0.319237	+	0.947675i	$0.603427\pi$
$684$	0	0
$685$	−26.0138	−0.993935
$686$	0	0
$687$	0	0
$688$	0	0
$689$	19.5890	0.746283
$690$	0	0
$691$	−2.97154	−0.113043	−0.0565214	−	0.998401i	$-0.518001\pi$
$691$	−2.97154	−0.113043	−0.0565214	+	0.998401i	$0.518001\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.58301	0.249708
$696$	0	0
$697$	16.8904	0.639770
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.8216	1.08858	0.544289	−	0.838898i	$-0.316799\pi$
$701$	28.8216	1.08858	0.544289	+	0.838898i	$0.316799\pi$
$702$	0	0
$703$	−18.8623	−0.711406
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.46112	0.130169
$708$	0	0
$709$	29.6345	1.11295	0.556474	−	0.830865i	$-0.312154\pi$
$709$	29.6345	1.11295	0.556474	+	0.830865i	$0.312154\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.44533	−0.0541281
$714$	0	0
$715$	−21.9110	−0.819426
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20.0870	−0.749120	−0.374560	−	0.927203i	$-0.622206\pi$
$719$	−20.0870	−0.749120	−0.374560	+	0.927203i	$0.622206\pi$
$720$	0	0
$721$	7.51658	0.279932
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3.31198	−0.123004
$726$	0	0
$727$	19.8227	0.735181	0.367591	−	0.929988i	$-0.380183\pi$
$727$	19.8227	0.735181	0.367591	+	0.929988i	$0.380183\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−40.4070	−1.49451
$732$	0	0
$733$	−43.6343	−1.61167	−0.805835	−	0.592141i	$-0.798283\pi$
$733$	−43.6343	−1.61167	−0.805835	+	0.592141i	$0.798283\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−59.8903	−2.20609
$738$	0	0
$739$	−35.4611	−1.30446	−0.652229	−	0.758022i	$-0.726166\pi$
$739$	−35.4611	−1.30446	−0.652229	+	0.758022i	$0.726166\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21.9431	0.805014	0.402507	−	0.915417i	$-0.368139\pi$
$743$	21.9431	0.805014	0.402507	+	0.915417i	$0.368139\pi$
$744$	0	0
$745$	−32.7649	−1.20041
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−10.6419	−0.388848
$750$	0	0
$751$	44.6445	1.62910	0.814550	−	0.580093i	$-0.196984\pi$
$751$	44.6445	1.62910	0.814550	+	0.580093i	$0.196984\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16.5407	0.601976
$756$	0	0
$757$	−1.85346	−0.0673650	−0.0336825	−	0.999433i	$-0.510724\pi$
$757$	−1.85346	−0.0673650	−0.0336825	+	0.999433i	$0.510724\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.93345	−0.0700874	−0.0350437	−	0.999386i	$-0.511157\pi$
$761$	−1.93345	−0.0700874	−0.0350437	+	0.999386i	$0.511157\pi$
$762$	0	0
$763$	17.9919	0.651350
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.39298	0.303053
$768$	0	0
$769$	37.8523	1.36499	0.682495	−	0.730890i	$-0.260895\pi$
$769$	37.8523	1.36499	0.682495	+	0.730890i	$0.260895\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	27.5568	0.991148	0.495574	−	0.868566i	$-0.334958\pi$
$773$	27.5568	0.991148	0.495574	+	0.868566i	$0.334958\pi$
$774$	0	0
$775$	3.31198	0.118970
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−14.6419	−0.524602
$780$	0	0
$781$	−3.84002	−0.137407
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.37433	−0.263201
$786$	0	0
$787$	−28.3380	−1.01014	−0.505070	−	0.863079i	$-0.668533\pi$
$787$	−28.3380	−1.01014	−0.505070	+	0.863079i	$0.668533\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.844342	0.0300213
$792$	0	0
$793$	18.4230	0.654221
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.73918	0.132449	0.0662243	−	0.997805i	$-0.478905\pi$
$797$	3.73918	0.132449	0.0662243	+	0.997805i	$0.478905\pi$
$798$	0	0
$799$	−0.0791033	−0.00279847
$800$	0	0
$801$	0	0
$802$	0	0
$803$	60.9027	2.14921
$804$	0	0
$805$	−1.66401	−0.0586485
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−42.0222	−1.47742	−0.738711	−	0.674023i	$-0.764565\pi$
$809$	−42.0222	−1.47742	−0.738711	+	0.674023i	$0.764565\pi$
$810$	0	0
$811$	−28.8382	−1.01265	−0.506323	−	0.862344i	$-0.668996\pi$
$811$	−28.8382	−1.01265	−0.506323	+	0.862344i	$0.668996\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−33.1414	−1.16089
$816$	0	0
$817$	35.0280	1.22547
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0.269195	0.00939499	0.00469749	−	0.999989i	$-0.498505\pi$
$821$	0.269195	0.00939499	0.00469749	+	0.999989i	$0.498505\pi$
$822$	0	0
$823$	−43.4007	−1.51285	−0.756427	−	0.654078i	$-0.773057\pi$
$823$	−43.4007	−1.51285	−0.756427	+	0.654078i	$0.773057\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−48.7439	−1.69499	−0.847495	−	0.530803i	$-0.821890\pi$
$827$	−48.7439	−1.69499	−0.847495	+	0.530803i	$0.821890\pi$
$828$	0	0
$829$	46.1758	1.60375	0.801875	−	0.597491i	$-0.203836\pi$
$829$	46.1758	1.60375	0.801875	+	0.597491i	$0.203836\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−21.3150	−0.738522
$834$	0	0
$835$	24.6861	0.854297
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−54.7571	−1.89042	−0.945212	−	0.326457i	$-0.894145\pi$
$839$	−54.7571	−1.89042	−0.945212	+	0.326457i	$0.894145\pi$
$840$	0	0
$841$	−26.9110	−0.927966
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10.6540	0.366510
$846$	0	0
$847$	16.3391	0.561418
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.10217	0.209180
$852$	0	0
$853$	29.7811	1.01968	0.509842	−	0.860268i	$-0.329704\pi$
$853$	29.7811	1.01968	0.509842	+	0.860268i	$0.329704\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.42930	0.185461	0.0927307	−	0.995691i	$-0.470440\pi$
$857$	5.42930	0.185461	0.0927307	+	0.995691i	$0.470440\pi$
$858$	0	0
$859$	−7.32801	−0.250029	−0.125014	−	0.992155i	$-0.539898\pi$
$859$	−7.32801	−0.250029	−0.125014	+	0.992155i	$0.539898\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	17.5484	0.597354	0.298677	−	0.954354i	$-0.403455\pi$
$863$	17.5484	0.597354	0.298677	+	0.954354i	$0.403455\pi$
$864$	0	0
$865$	10.3540	0.352048
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11.0506	0.374867
$870$	0	0
$871$	−29.3581	−0.994760
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.1331	0.410174
$876$	0	0
$877$	2.86062	0.0965963	0.0482981	−	0.998833i	$-0.484620\pi$
$877$	2.86062	0.0965963	0.0482981	+	0.998833i	$0.484620\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−55.1425	−1.85780	−0.928899	−	0.370334i	$-0.879243\pi$
$881$	−55.1425	−1.85780	−0.928899	+	0.370334i	$0.879243\pi$
$882$	0	0
$883$	13.8622	0.466500	0.233250	−	0.972417i	$-0.425064\pi$
$883$	13.8622	0.466500	0.233250	+	0.972417i	$0.425064\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.6049	−0.826153	−0.413077	−	0.910696i	$-0.635546\pi$
$887$	−24.6049	−0.826153	−0.413077	+	0.910696i	$0.635546\pi$
$888$	0	0
$889$	8.70064	0.291810
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.0685730	0.00229471
$894$	0	0
$895$	−26.8116	−0.896212
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.08898	−0.0696715
$900$	0	0
$901$	27.3420	0.910895
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−20.6134	−0.685211
$906$	0	0
$907$	32.2571	1.07108	0.535540	−	0.844510i	$-0.320108\pi$
$907$	32.2571	1.07108	0.535540	+	0.844510i	$0.320108\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−26.2041	−0.868181	−0.434090	−	0.900869i	$-0.642930\pi$
$911$	−26.2041	−0.868181	−0.434090	+	0.900869i	$0.642930\pi$
$912$	0	0
$913$	48.0059	1.58876
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.4614	0.444533
$918$	0	0
$919$	−27.9584	−0.922263	−0.461132	−	0.887332i	$-0.652556\pi$
$919$	−27.9584	−0.922263	−0.461132	+	0.887332i	$0.652556\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.88237	−0.0619589
$924$	0	0
$925$	−13.9831	−0.459763
$926$	0	0
$927$	0	0
$928$	0	0
$929$	46.7810	1.53483	0.767417	−	0.641149i	$-0.221542\pi$
$929$	46.7810	1.53483	0.767417	+	0.641149i	$0.221542\pi$
$930$	0	0
$931$	18.4775	0.605577
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−30.5830	−1.00017
$936$	0	0
$937$	−14.8017	−0.483550	−0.241775	−	0.970332i	$-0.577730\pi$
$937$	−14.8017	−0.483550	−0.241775	+	0.970332i	$0.577730\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26.5951	−0.866976	−0.433488	−	0.901159i	$-0.642717\pi$
$941$	−26.5951	−0.866976	−0.433488	+	0.901159i	$0.642717\pi$
$942$	0	0
$943$	4.73683	0.154253
$944$	0	0
$945$	0	0
$946$	0	0
$947$	57.0200	1.85290	0.926451	−	0.376415i	$-0.122843\pi$
$947$	57.0200	1.85290	0.926451	+	0.376415i	$0.122843\pi$
$948$	0	0
$949$	29.8543	0.969113
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36.1266	1.17025	0.585127	−	0.810941i	$-0.301045\pi$
$953$	36.1266	1.17025	0.585127	+	0.810941i	$0.301045\pi$
$954$	0	0
$955$	−21.8380	−0.706661
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.9819	0.516084
$960$	0	0
$961$	−28.9110	−0.932613
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−12.3498	−0.397553
$966$	0	0
$967$	10.2470	0.329522	0.164761	−	0.986334i	$-0.447315\pi$
$967$	10.2470	0.329522	0.164761	+	0.986334i	$0.447315\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−8.37954	−0.268912	−0.134456	−	0.990920i	$-0.542929\pi$
$971$	−8.37954	−0.268912	−0.134456	+	0.990920i	$0.542929\pi$
$972$	0	0
$973$	−4.04437	−0.129656
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.9280	−1.62933	−0.814666	−	0.579931i	$-0.803080\pi$
$977$	−50.9280	−1.62933	−0.814666	+	0.579931i	$0.803080\pi$
$978$	0	0
$979$	−41.5180	−1.32692
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−17.9857	−0.573654	−0.286827	−	0.957982i	$-0.592600\pi$
$983$	−17.9857	−0.573654	−0.286827	+	0.957982i	$0.592600\pi$
$984$	0	0
$985$	4.54066	0.144677
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.3319	−0.360335
$990$	0	0
$991$	−47.0931	−1.49596	−0.747980	−	0.663721i	$-0.768976\pi$
$991$	−47.0931	−1.49596	−0.747980	+	0.663721i	$0.768976\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	25.1374	0.796910
$996$	0	0
$997$	14.4391	0.457289	0.228645	−	0.973510i	$-0.426571\pi$
$997$	14.4391	0.457289	0.228645	+	0.973510i	$0.426571\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3312.2.a.bh.1.2		4
3.2	odd	2		3312.2.a.bg.1.4		4
4.3	odd	2		1656.2.a.p.1.1	yes	4
12.11	even	2		1656.2.a.o.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1656.2.a.o.1.3	✓	4	12.11	even	2
1656.2.a.p.1.1	yes	4	4.3	odd	2
3312.2.a.bg.1.4		4	3.2	odd	2
3312.2.a.bh.1.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3312.a (trivial)

\(f(q)\)	\(=\)	\(q-1.64575 q^{5} +1.01109 q^{7} +5.21151 q^{11} +2.55467 q^{13} +3.56576 q^{17} -3.09108 q^{19} +1.00000 q^{23} -2.29150 q^{25} +1.44533 q^{29} -1.44533 q^{31} -1.66401 q^{35} +6.10217 q^{37} +4.73683 q^{41} -11.3319 q^{43} -0.0221841 q^{47} -5.97769 q^{49} +7.66794 q^{53} -8.57685 q^{55} +3.28535 q^{59} +7.21151 q^{61} -4.20435 q^{65} -11.4919 q^{67} -0.736834 q^{71} +11.6862 q^{73} +5.26932 q^{77} +2.12043 q^{79} +9.21151 q^{83} -5.86836 q^{85} -7.96660 q^{89} +2.58301 q^{91} +5.08715 q^{95} -7.31369 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 2 q^{7} - 2 q^{11} + 4 q^{13} + 2 q^{17} - 8 q^{19} + 4 q^{23} + 12 q^{25} + 12 q^{29} - 12 q^{31} + 12 q^{35} + 14 q^{37} + 4 q^{41} - 4 q^{43} + 12 q^{47} + 28 q^{49} + 8 q^{53} - 16 q^{55}+ \cdots + 4 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 2 * q^7 - 2 * q^11 + 4 * q^13 + 2 * q^17 - 8 * q^19 + 4 * q^23 + 12 * q^25 + 12 * q^29 - 12 * q^31 + 12 * q^35 + 14 * q^37 + 4 * q^41 - 4 * q^43 + 12 * q^47 + 28 * q^49 + 8 * q^53 - 16 * q^55 + 16 * q^59 + 6 * q^61 + 4 * q^65 - 8 * q^67 + 12 * q^71 + 16 * q^73 + 12 * q^77 - 10 * q^79 + 14 * q^83 + 16 * q^85 + 14 * q^89 - 32 * q^91 + 20 * q^95 + 4 * q^97

Introduction

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