LMFDB - Embedded newform 3312.2.a.u.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 207)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-3.41421 q^{5} +0.585786 q^{7} +O(q^{10})$	$q-3.41421 q^{5} +0.585786 q^{7} +2.82843 q^{11} -4.58579 q^{17} -2.24264 q^{19} +1.00000 q^{23} +6.65685 q^{25} +8.48528 q^{29} +8.48528 q^{31} -2.00000 q^{35} +0.828427 q^{37} -9.65685 q^{41} +10.2426 q^{43} -11.6569 q^{47} -6.65685 q^{49} -9.07107 q^{53} -9.65685 q^{55} -3.65685 q^{59} +4.82843 q^{61} -11.4142 q^{67} -2.34315 q^{71} -9.31371 q^{73} +1.65685 q^{77} +11.8995 q^{79} +1.17157 q^{83} +15.6569 q^{85} +1.07107 q^{89} +7.65685 q^{95} -10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q - 4 * q^5 + 4 * q^7	$ 2 q - 4 q^{5} + 4 q^{7} - 12 q^{17} + 4 q^{19} + 2 q^{23} + 2 q^{25} - 4 q^{35} - 4 q^{37} - 8 q^{41} + 12 q^{43} - 12 q^{47} - 2 q^{49} - 4 q^{53} - 8 q^{55} + 4 q^{59} + 4 q^{61} - 20 q^{67} - 16 q^{71} + 4 q^{73} - 8 q^{77} + 4 q^{79} + 8 q^{83} + 20 q^{85} - 12 q^{89} + 4 q^{95} - 20 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 + 4 * q^7 - 12 * q^17 + 4 * q^19 + 2 * q^23 + 2 * q^25 - 4 * q^35 - 4 * q^37 - 8 * q^41 + 12 * q^43 - 12 * q^47 - 2 * q^49 - 4 * q^53 - 8 * q^55 + 4 * q^59 + 4 * q^61 - 20 * q^67 - 16 * q^71 + 4 * q^73 - 8 * q^77 + 4 * q^79 + 8 * q^83 + 20 * q^85 - 12 * q^89 + 4 * q^95 - 20 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	0	0
$7$	0.585786	0.221406	0.110703	−	0.993854i	$-0.464690\pi$
$7$	0.585786	0.221406	0.110703	+	0.993854i	$0.464690\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.82843	0.852803	0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.58579	−1.11222	−0.556108	−	0.831110i	$-0.687706\pi$
$17$	−4.58579	−1.11222	−0.556108	+	0.831110i	$0.687706\pi$
$18$	0	0
$19$	−2.24264	−0.514497	−0.257249	−	0.966345i	$-0.582816\pi$
$19$	−2.24264	−0.514497	−0.257249	+	0.966345i	$0.582816\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.48528	1.57568	0.787839	−	0.615882i	$-0.211200\pi$
$29$	8.48528	1.57568	0.787839	+	0.615882i	$0.211200\pi$
$30$	0	0
$31$	8.48528	1.52400	0.762001	−	0.647576i	$-0.224217\pi$
$31$	8.48528	1.52400	0.762001	+	0.647576i	$0.224217\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	0.828427	0.136193	0.0680963	−	0.997679i	$-0.478307\pi$
$37$	0.828427	0.136193	0.0680963	+	0.997679i	$0.478307\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.65685	−1.50815	−0.754074	−	0.656790i	$-0.771914\pi$
$41$	−9.65685	−1.50815	−0.754074	+	0.656790i	$0.771914\pi$
$42$	0	0
$43$	10.2426	1.56199	0.780994	−	0.624538i	$-0.214713\pi$
$43$	10.2426	1.56199	0.780994	+	0.624538i	$0.214713\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.6569	−1.70033	−0.850163	−	0.526519i	$-0.823497\pi$
$47$	−11.6569	−1.70033	−0.850163	+	0.526519i	$0.823497\pi$
$48$	0	0
$49$	−6.65685	−0.950979
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.07107	−1.24601	−0.623003	−	0.782219i	$-0.714088\pi$
$53$	−9.07107	−1.24601	−0.623003	+	0.782219i	$0.714088\pi$
$54$	0	0
$55$	−9.65685	−1.30213
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.65685	−0.476082	−0.238041	−	0.971255i	$-0.576505\pi$
$59$	−3.65685	−0.476082	−0.238041	+	0.971255i	$0.576505\pi$
$60$	0	0
$61$	4.82843	0.618217	0.309108	−	0.951027i	$-0.399969\pi$
$61$	4.82843	0.618217	0.309108	+	0.951027i	$0.399969\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.4142	−1.39447	−0.697234	−	0.716844i	$-0.745586\pi$
$67$	−11.4142	−1.39447	−0.697234	+	0.716844i	$0.745586\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.34315	−0.278080	−0.139040	−	0.990287i	$-0.544402\pi$
$71$	−2.34315	−0.278080	−0.139040	+	0.990287i	$0.544402\pi$
$72$	0	0
$73$	−9.31371	−1.09009	−0.545044	−	0.838408i	$-0.683487\pi$
$73$	−9.31371	−1.09009	−0.545044	+	0.838408i	$0.683487\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.65685	0.188816
$78$	0	0
$79$	11.8995	1.33880	0.669399	−	0.742903i	$-0.266552\pi$
$79$	11.8995	1.33880	0.669399	+	0.742903i	$0.266552\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.17157	0.128597	0.0642984	−	0.997931i	$-0.479519\pi$
$83$	1.17157	0.128597	0.0642984	+	0.997931i	$0.479519\pi$
$84$	0	0
$85$	15.6569	1.69822
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.07107	0.113533	0.0567665	−	0.998387i	$-0.481921\pi$
$89$	1.07107	0.113533	0.0567665	+	0.998387i	$0.481921\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.65685	0.785577
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.3137	−1.12576	−0.562878	−	0.826540i	$-0.690306\pi$
$101$	−11.3137	−1.12576	−0.562878	+	0.826540i	$0.690306\pi$
$102$	0	0
$103$	0.585786	0.0577193	0.0288596	−	0.999583i	$-0.490812\pi$
$103$	0.585786	0.0577193	0.0288596	+	0.999583i	$0.490812\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.48528	0.820303	0.410152	−	0.912017i	$-0.365476\pi$
$107$	8.48528	0.820303	0.410152	+	0.912017i	$0.365476\pi$
$108$	0	0
$109$	14.4853	1.38744	0.693719	−	0.720246i	$-0.255971\pi$
$109$	14.4853	1.38744	0.693719	+	0.720246i	$0.255971\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.5858	−1.18397	−0.591986	−	0.805949i	$-0.701656\pi$
$113$	−12.5858	−1.18397	−0.591986	+	0.805949i	$0.701656\pi$
$114$	0	0
$115$	−3.41421	−0.318377
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.68629	−0.246252
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−12.4853	−1.10789	−0.553945	−	0.832553i	$-0.686878\pi$
$127$	−12.4853	−1.10789	−0.553945	+	0.832553i	$0.686878\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−16.9706	−1.48272	−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706	−1.48272	−0.741362	+	0.671105i	$0.765820\pi$
$132$	0	0
$133$	−1.31371	−0.113913
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.41421	−0.291696	−0.145848	−	0.989307i	$-0.546591\pi$
$137$	−3.41421	−0.291696	−0.145848	+	0.989307i	$0.546591\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−28.9706	−2.40587
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.4142	1.59047	0.795237	−	0.606298i	$-0.207346\pi$
$149$	19.4142	1.59047	0.795237	+	0.606298i	$0.207346\pi$
$150$	0	0
$151$	−1.65685	−0.134833	−0.0674164	−	0.997725i	$-0.521476\pi$
$151$	−1.65685	−0.134833	−0.0674164	+	0.997725i	$0.521476\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−28.9706	−2.32697
$156$	0	0
$157$	−18.4853	−1.47529	−0.737643	−	0.675191i	$-0.764061\pi$
$157$	−18.4853	−1.47529	−0.737643	+	0.675191i	$0.764061\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.585786	0.0461664
$162$	0	0
$163$	−13.1716	−1.03168	−0.515839	−	0.856686i	$-0.672520\pi$
$163$	−13.1716	−1.03168	−0.515839	+	0.856686i	$0.672520\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.65685	0.437741	0.218870	−	0.975754i	$-0.429763\pi$
$167$	5.65685	0.437741	0.218870	+	0.975754i	$0.429763\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−5.17157	−0.393187	−0.196594	−	0.980485i	$-0.562988\pi$
$173$	−5.17157	−0.393187	−0.196594	+	0.980485i	$0.562988\pi$
$174$	0	0
$175$	3.89949	0.294774
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	−16.8284	−1.25085	−0.625424	−	0.780285i	$-0.715074\pi$
$181$	−16.8284	−1.25085	−0.625424	+	0.780285i	$0.715074\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.82843	−0.207950
$186$	0	0
$187$	−12.9706	−0.948501
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−23.3137	−1.68692	−0.843460	−	0.537191i	$-0.819485\pi$
$191$	−23.3137	−1.68692	−0.843460	+	0.537191i	$0.819485\pi$
$192$	0	0
$193$	9.65685	0.695116	0.347558	−	0.937659i	$-0.387011\pi$
$193$	9.65685	0.695116	0.347558	+	0.937659i	$0.387011\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.8284	1.05648	0.528241	−	0.849095i	$-0.322852\pi$
$197$	14.8284	1.05648	0.528241	+	0.849095i	$0.322852\pi$
$198$	0	0
$199$	−6.24264	−0.442529	−0.221265	−	0.975214i	$-0.571018\pi$
$199$	−6.24264	−0.442529	−0.221265	+	0.975214i	$0.571018\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.97056	0.348865
$204$	0	0
$205$	32.9706	2.30276
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.34315	−0.438765
$210$	0	0
$211$	−4.48528	−0.308780	−0.154390	−	0.988010i	$-0.549341\pi$
$211$	−4.48528	−0.308780	−0.154390	+	0.988010i	$0.549341\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−34.9706	−2.38497
$216$	0	0
$217$	4.97056	0.337424
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−20.9706	−1.40429	−0.702146	−	0.712033i	$-0.747775\pi$
$223$	−20.9706	−1.40429	−0.702146	+	0.712033i	$0.747775\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.1421	1.46963	0.734813	−	0.678270i	$-0.237270\pi$
$227$	22.1421	1.46963	0.734813	+	0.678270i	$0.237270\pi$
$228$	0	0
$229$	−2.48528	−0.164232	−0.0821160	−	0.996623i	$-0.526168\pi$
$229$	−2.48528	−0.164232	−0.0821160	+	0.996623i	$0.526168\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.3431	−0.677602	−0.338801	−	0.940858i	$-0.610021\pi$
$233$	−10.3431	−0.677602	−0.338801	+	0.940858i	$0.610021\pi$
$234$	0	0
$235$	39.7990	2.59620
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.31371	0.214346	0.107173	−	0.994240i	$-0.465820\pi$
$239$	3.31371	0.214346	0.107173	+	0.994240i	$0.465820\pi$
$240$	0	0
$241$	10.9706	0.706676	0.353338	−	0.935496i	$-0.385047\pi$
$241$	10.9706	0.706676	0.353338	+	0.935496i	$0.385047\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	22.7279	1.45203
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3.51472	−0.221847	−0.110924	−	0.993829i	$-0.535381\pi$
$251$	−3.51472	−0.221847	−0.110924	+	0.993829i	$0.535381\pi$
$252$	0	0
$253$	2.82843	0.177822
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.6569	−1.10140	−0.550702	−	0.834702i	$-0.685640\pi$
$257$	−17.6569	−1.10140	−0.550702	+	0.834702i	$0.685640\pi$
$258$	0	0
$259$	0.485281	0.0301539
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0.686292	0.0423185	0.0211593	−	0.999776i	$-0.493264\pi$
$263$	0.686292	0.0423185	0.0211593	+	0.999776i	$0.493264\pi$
$264$	0	0
$265$	30.9706	1.90251
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.1421	0.618377	0.309188	−	0.951001i	$-0.399943\pi$
$269$	10.1421	0.618377	0.309188	+	0.951001i	$0.399943\pi$
$270$	0	0
$271$	−4.48528	−0.272461	−0.136231	−	0.990677i	$-0.543499\pi$
$271$	−4.48528	−0.272461	−0.136231	+	0.990677i	$0.543499\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.8284	1.13540
$276$	0	0
$277$	−3.65685	−0.219719	−0.109860	−	0.993947i	$-0.535040\pi$
$277$	−3.65685	−0.219719	−0.109860	+	0.993947i	$0.535040\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.4142	−0.680915	−0.340457	−	0.940260i	$-0.610582\pi$
$281$	−11.4142	−0.680915	−0.340457	+	0.940260i	$0.610582\pi$
$282$	0	0
$283$	27.2132	1.61766	0.808829	−	0.588045i	$-0.200102\pi$
$283$	27.2132	1.61766	0.808829	+	0.588045i	$0.200102\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.65685	−0.333914
$288$	0	0
$289$	4.02944	0.237026
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4.58579	−0.267905	−0.133952	−	0.990988i	$-0.542767\pi$
$293$	−4.58579	−0.267905	−0.133952	+	0.990988i	$0.542767\pi$
$294$	0	0
$295$	12.4853	0.726921
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	6.00000	0.345834
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−16.4853	−0.943944
$306$	0	0
$307$	−5.17157	−0.295157	−0.147579	−	0.989050i	$-0.547148\pi$
$307$	−5.17157	−0.295157	−0.147579	+	0.989050i	$0.547148\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.3137	0.754951	0.377476	−	0.926020i	$-0.376792\pi$
$311$	13.3137	0.754951	0.377476	+	0.926020i	$0.376792\pi$
$312$	0	0
$313$	−21.3137	−1.20472	−0.602361	−	0.798224i	$-0.705773\pi$
$313$	−21.3137	−1.20472	−0.602361	+	0.798224i	$0.705773\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−25.4558	−1.42974	−0.714871	−	0.699256i	$-0.753515\pi$
$317$	−25.4558	−1.42974	−0.714871	+	0.699256i	$0.753515\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10.2843	0.572232
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.82843	−0.376463
$330$	0	0
$331$	1.17157	0.0643955	0.0321977	−	0.999482i	$-0.489749\pi$
$331$	1.17157	0.0643955	0.0321977	+	0.999482i	$0.489749\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	38.9706	2.12919
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	24.0000	1.29967
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.68629	0.144208	0.0721038	−	0.997397i	$-0.477029\pi$
$347$	2.68629	0.144208	0.0721038	+	0.997397i	$0.477029\pi$
$348$	0	0
$349$	12.9706	0.694298	0.347149	−	0.937810i	$-0.387150\pi$
$349$	12.9706	0.694298	0.347149	+	0.937810i	$0.387150\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−34.1421	−1.81720	−0.908601	−	0.417665i	$-0.862849\pi$
$353$	−34.1421	−1.81720	−0.908601	+	0.417665i	$0.862849\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.2843	−1.28167	−0.640837	−	0.767677i	$-0.721413\pi$
$359$	−24.2843	−1.28167	−0.640837	+	0.767677i	$0.721413\pi$
$360$	0	0
$361$	−13.9706	−0.735293
$362$	0	0
$363$	0	0
$364$	0	0
$365$	31.7990	1.66444
$366$	0	0
$367$	17.5563	0.916434	0.458217	−	0.888840i	$-0.348488\pi$
$367$	17.5563	0.916434	0.458217	+	0.888840i	$0.348488\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.31371	−0.275874
$372$	0	0
$373$	−10.4853	−0.542907	−0.271454	−	0.962452i	$-0.587504\pi$
$373$	−10.4853	−0.542907	−0.271454	+	0.962452i	$0.587504\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−3.41421	−0.175376	−0.0876882	−	0.996148i	$-0.527948\pi$
$379$	−3.41421	−0.175376	−0.0876882	+	0.996148i	$0.527948\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10.3431	0.528510	0.264255	−	0.964453i	$-0.414874\pi$
$383$	10.3431	0.528510	0.264255	+	0.964453i	$0.414874\pi$
$384$	0	0
$385$	−5.65685	−0.288300
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−15.8995	−0.806136	−0.403068	−	0.915170i	$-0.632056\pi$
$389$	−15.8995	−0.806136	−0.403068	+	0.915170i	$0.632056\pi$
$390$	0	0
$391$	−4.58579	−0.231913
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−40.6274	−2.04419
$396$	0	0
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.0416	0.900956	0.450478	−	0.892788i	$-0.351254\pi$
$401$	18.0416	0.900956	0.450478	+	0.892788i	$0.351254\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.34315	0.116145
$408$	0	0
$409$	25.6569	1.26865	0.634325	−	0.773067i	$-0.281278\pi$
$409$	25.6569	1.26865	0.634325	+	0.773067i	$0.281278\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.14214	−0.105408
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.51472	0.171705	0.0858526	−	0.996308i	$-0.472639\pi$
$419$	3.51472	0.171705	0.0858526	+	0.996308i	$0.472639\pi$
$420$	0	0
$421$	11.4558	0.558324	0.279162	−	0.960244i	$-0.409943\pi$
$421$	11.4558	0.558324	0.279162	+	0.960244i	$0.409943\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−30.5269	−1.48077
$426$	0	0
$427$	2.82843	0.136877
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.3137	1.50833	0.754164	−	0.656686i	$-0.228042\pi$
$431$	31.3137	1.50833	0.754164	+	0.656686i	$0.228042\pi$
$432$	0	0
$433$	7.65685	0.367965	0.183982	−	0.982930i	$-0.441101\pi$
$433$	7.65685	0.367965	0.183982	+	0.982930i	$0.441101\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.24264	−0.107280
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	−3.65685	−0.173352
$446$	0	0
$447$	0	0
$448$	0	0
$449$	34.6274	1.63417	0.817084	−	0.576518i	$-0.195589\pi$
$449$	34.6274	1.63417	0.817084	+	0.576518i	$0.195589\pi$
$450$	0	0
$451$	−27.3137	−1.28615
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.02944	0.0481550	0.0240775	−	0.999710i	$-0.492335\pi$
$457$	1.02944	0.0481550	0.0240775	+	0.999710i	$0.492335\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14.8284	0.690629	0.345314	−	0.938487i	$-0.387772\pi$
$461$	14.8284	0.690629	0.345314	+	0.938487i	$0.387772\pi$
$462$	0	0
$463$	24.9706	1.16048	0.580240	−	0.814445i	$-0.302959\pi$
$463$	24.9706	1.16048	0.580240	+	0.814445i	$0.302959\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	34.1421	1.57991	0.789955	−	0.613165i	$-0.210104\pi$
$467$	34.1421	1.57991	0.789955	+	0.613165i	$0.210104\pi$
$468$	0	0
$469$	−6.68629	−0.308744
$470$	0	0
$471$	0	0
$472$	0	0
$473$	28.9706	1.33207
$474$	0	0
$475$	−14.9289	−0.684986
$476$	0	0
$477$	0	0
$478$	0	0
$479$	39.3137	1.79629	0.898145	−	0.439700i	$-0.144915\pi$
$479$	39.3137	1.79629	0.898145	+	0.439700i	$0.144915\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	34.1421	1.55031
$486$	0	0
$487$	24.9706	1.13152	0.565762	−	0.824569i	$-0.308582\pi$
$487$	24.9706	1.13152	0.565762	+	0.824569i	$0.308582\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	29.3137	1.32291	0.661455	−	0.749985i	$-0.269939\pi$
$491$	29.3137	1.32291	0.661455	+	0.749985i	$0.269939\pi$
$492$	0	0
$493$	−38.9117	−1.75249
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.37258	−0.0615688
$498$	0	0
$499$	7.51472	0.336405	0.168203	−	0.985752i	$-0.446204\pi$
$499$	7.51472	0.336405	0.168203	+	0.985752i	$0.446204\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.34315	0.282827	0.141413	−	0.989951i	$-0.454835\pi$
$503$	6.34315	0.282827	0.141413	+	0.989951i	$0.454835\pi$
$504$	0	0
$505$	38.6274	1.71890
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.85786	0.436942	0.218471	−	0.975843i	$-0.429893\pi$
$509$	9.85786	0.436942	0.218471	+	0.975843i	$0.429893\pi$
$510$	0	0
$511$	−5.45584	−0.241352
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.00000	−0.0881305
$516$	0	0
$517$	−32.9706	−1.45004
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6.72792	0.294756	0.147378	−	0.989080i	$-0.452917\pi$
$521$	6.72792	0.294756	0.147378	+	0.989080i	$0.452917\pi$
$522$	0	0
$523$	−23.6985	−1.03626	−0.518131	−	0.855301i	$-0.673372\pi$
$523$	−23.6985	−1.03626	−0.518131	+	0.855301i	$0.673372\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−38.9117	−1.69502
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−28.9706	−1.25251
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−18.8284	−0.810998
$540$	0	0
$541$	−28.9706	−1.24554	−0.622771	−	0.782404i	$-0.713993\pi$
$541$	−28.9706	−1.24554	−0.622771	+	0.782404i	$0.713993\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−49.4558	−2.11846
$546$	0	0
$547$	4.48528	0.191777	0.0958884	−	0.995392i	$-0.469431\pi$
$547$	4.48528	0.191777	0.0958884	+	0.995392i	$0.469431\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−19.0294	−0.810681
$552$	0	0
$553$	6.97056	0.296418
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.7574	0.582918	0.291459	−	0.956583i	$-0.405859\pi$
$557$	13.7574	0.582918	0.291459	+	0.956583i	$0.405859\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	26.8284	1.13068	0.565342	−	0.824857i	$-0.308744\pi$
$563$	26.8284	1.13068	0.565342	+	0.824857i	$0.308744\pi$
$564$	0	0
$565$	42.9706	1.80779
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11.4142	−0.478509	−0.239254	−	0.970957i	$-0.576903\pi$
$569$	−11.4142	−0.478509	−0.239254	+	0.970957i	$0.576903\pi$
$570$	0	0
$571$	−10.2426	−0.428641	−0.214321	−	0.976763i	$-0.568754\pi$
$571$	−10.2426	−0.428641	−0.214321	+	0.976763i	$0.568754\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.65685	0.277610
$576$	0	0
$577$	24.2843	1.01097	0.505484	−	0.862836i	$-0.331314\pi$
$577$	24.2843	1.01097	0.505484	+	0.862836i	$0.331314\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.686292	0.0284722
$582$	0	0
$583$	−25.6569	−1.06260
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.9706	1.69104	0.845518	−	0.533947i	$-0.179292\pi$
$587$	40.9706	1.69104	0.845518	+	0.533947i	$0.179292\pi$
$588$	0	0
$589$	−19.0294	−0.784094
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−23.3137	−0.957379	−0.478690	−	0.877984i	$-0.658888\pi$
$593$	−23.3137	−0.957379	−0.478690	+	0.877984i	$0.658888\pi$
$594$	0	0
$595$	9.17157	0.375998
$596$	0	0
$597$	0	0
$598$	0	0
$599$	27.3137	1.11601	0.558004	−	0.829838i	$-0.311567\pi$
$599$	27.3137	1.11601	0.558004	+	0.829838i	$0.311567\pi$
$600$	0	0
$601$	−9.65685	−0.393911	−0.196956	−	0.980412i	$-0.563105\pi$
$601$	−9.65685	−0.393911	−0.196956	+	0.980412i	$0.563105\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.2426	0.416423
$606$	0	0
$607$	25.6569	1.04138	0.520690	−	0.853746i	$-0.325675\pi$
$607$	25.6569	1.04138	0.520690	+	0.853746i	$0.325675\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−11.4558	−0.462697	−0.231349	−	0.972871i	$-0.574314\pi$
$613$	−11.4558	−0.462697	−0.231349	+	0.972871i	$0.574314\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.44365	0.0983777	0.0491888	−	0.998789i	$-0.484336\pi$
$617$	2.44365	0.0983777	0.0491888	+	0.998789i	$0.484336\pi$
$618$	0	0
$619$	13.7574	0.552955	0.276477	−	0.961020i	$-0.410833\pi$
$619$	13.7574	0.552955	0.276477	+	0.961020i	$0.410833\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.627417	0.0251369
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.79899	−0.151476
$630$	0	0
$631$	−28.8701	−1.14930	−0.574649	−	0.818400i	$-0.694862\pi$
$631$	−28.8701	−1.14930	−0.574649	+	0.818400i	$0.694862\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	42.6274	1.69162
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17.0711	0.674267	0.337133	−	0.941457i	$-0.390543\pi$
$641$	17.0711	0.674267	0.337133	+	0.941457i	$0.390543\pi$
$642$	0	0
$643$	−28.3848	−1.11939	−0.559693	−	0.828700i	$-0.689081\pi$
$643$	−28.3848	−1.11939	−0.559693	+	0.828700i	$0.689081\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.00000	0.235884	0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000	0.235884	0.117942	+	0.993020i	$0.462370\pi$
$648$	0	0
$649$	−10.3431	−0.406004
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.3431	0.404759	0.202379	−	0.979307i	$-0.435133\pi$
$653$	10.3431	0.404759	0.202379	+	0.979307i	$0.435133\pi$
$654$	0	0
$655$	57.9411	2.26395
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.85786	0.228190	0.114095	−	0.993470i	$-0.463603\pi$
$659$	5.85786	0.228190	0.114095	+	0.993470i	$0.463603\pi$
$660$	0	0
$661$	−40.1421	−1.56135	−0.780674	−	0.624938i	$-0.785124\pi$
$661$	−40.1421	−1.56135	−0.780674	+	0.624938i	$0.785124\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.48528	0.173932
$666$	0	0
$667$	8.48528	0.328551
$668$	0	0
$669$	0	0
$670$	0	0
$671$	13.6569	0.527217
$672$	0	0
$673$	−7.02944	−0.270965	−0.135482	−	0.990780i	$-0.543258\pi$
$673$	−7.02944	−0.270965	−0.135482	+	0.990780i	$0.543258\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	23.6985	0.910807	0.455403	−	0.890285i	$-0.349495\pi$
$677$	23.6985	0.910807	0.455403	+	0.890285i	$0.349495\pi$
$678$	0	0
$679$	−5.85786	−0.224804
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−21.6569	−0.828676	−0.414338	−	0.910123i	$-0.635987\pi$
$683$	−21.6569	−0.828676	−0.414338	+	0.910123i	$0.635987\pi$
$684$	0	0
$685$	11.6569	0.445386
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	17.1716	0.653237	0.326619	−	0.945156i	$-0.394091\pi$
$691$	17.1716	0.653237	0.326619	+	0.945156i	$0.394091\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13.6569	−0.518034
$696$	0	0
$697$	44.2843	1.67739
$698$	0	0
$699$	0	0
$700$	0	0
$701$	15.8995	0.600516	0.300258	−	0.953858i	$-0.402927\pi$
$701$	15.8995	0.600516	0.300258	+	0.953858i	$0.402927\pi$
$702$	0	0
$703$	−1.85786	−0.0700707
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.62742	−0.249250
$708$	0	0
$709$	9.79899	0.368009	0.184004	−	0.982925i	$-0.441094\pi$
$709$	9.79899	0.368009	0.184004	+	0.982925i	$0.441094\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.48528	0.317776
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	18.6863	0.696881	0.348441	−	0.937331i	$-0.386711\pi$
$719$	18.6863	0.696881	0.348441	+	0.937331i	$0.386711\pi$
$720$	0	0
$721$	0.343146	0.0127794
$722$	0	0
$723$	0	0
$724$	0	0
$725$	56.4853	2.09781
$726$	0	0
$727$	−46.2426	−1.71504	−0.857522	−	0.514447i	$-0.827997\pi$
$727$	−46.2426	−1.71504	−0.857522	+	0.514447i	$0.827997\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−46.9706	−1.73727
$732$	0	0
$733$	29.7990	1.10065	0.550325	−	0.834950i	$-0.314504\pi$
$733$	29.7990	1.10065	0.550325	+	0.834950i	$0.314504\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−32.2843	−1.18921
$738$	0	0
$739$	−20.6863	−0.760958	−0.380479	−	0.924790i	$-0.624241\pi$
$739$	−20.6863	−0.760958	−0.380479	+	0.924790i	$0.624241\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−40.9706	−1.50306	−0.751532	−	0.659697i	$-0.770685\pi$
$743$	−40.9706	−1.50306	−0.751532	+	0.659697i	$0.770685\pi$
$744$	0	0
$745$	−66.2843	−2.42847
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.97056	0.181620
$750$	0	0
$751$	−49.3553	−1.80100	−0.900501	−	0.434854i	$-0.856800\pi$
$751$	−49.3553	−1.80100	−0.900501	+	0.434854i	$0.856800\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.65685	0.205874
$756$	0	0
$757$	−4.14214	−0.150548	−0.0752742	−	0.997163i	$-0.523983\pi$
$757$	−4.14214	−0.150548	−0.0752742	+	0.997163i	$0.523983\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.79899	−0.282713	−0.141357	−	0.989959i	$-0.545146\pi$
$761$	−7.79899	−0.282713	−0.141357	+	0.989959i	$0.545146\pi$
$762$	0	0
$763$	8.48528	0.307188
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1.02944	−0.0371225	−0.0185612	−	0.999828i	$-0.505909\pi$
$769$	−1.02944	−0.0371225	−0.0185612	+	0.999828i	$0.505909\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	28.3848	1.02093	0.510465	−	0.859899i	$-0.329473\pi$
$773$	28.3848	1.02093	0.510465	+	0.859899i	$0.329473\pi$
$774$	0	0
$775$	56.4853	2.02901
$776$	0	0
$777$	0	0
$778$	0	0
$779$	21.6569	0.775937
$780$	0	0
$781$	−6.62742	−0.237148
$782$	0	0
$783$	0	0
$784$	0	0
$785$	63.1127	2.25259
$786$	0	0
$787$	−31.6985	−1.12993	−0.564964	−	0.825115i	$-0.691110\pi$
$787$	−31.6985	−1.12993	−0.564964	+	0.825115i	$0.691110\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7.37258	−0.262139
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.9289	0.812184	0.406092	−	0.913832i	$-0.366891\pi$
$797$	22.9289	0.812184	0.406092	+	0.913832i	$0.366891\pi$
$798$	0	0
$799$	53.4558	1.89113
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−26.3431	−0.929629
$804$	0	0
$805$	−2.00000	−0.0704907
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−43.1127	−1.51576	−0.757881	−	0.652393i	$-0.773765\pi$
$809$	−43.1127	−1.51576	−0.757881	+	0.652393i	$0.773765\pi$
$810$	0	0
$811$	16.7696	0.588859	0.294429	−	0.955673i	$-0.404870\pi$
$811$	16.7696	0.588859	0.294429	+	0.955673i	$0.404870\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	44.9706	1.57525
$816$	0	0
$817$	−22.9706	−0.803638
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−11.3137	−0.394851	−0.197426	−	0.980318i	$-0.563258\pi$
$821$	−11.3137	−0.394851	−0.197426	+	0.980318i	$0.563258\pi$
$822$	0	0
$823$	−32.4853	−1.13237	−0.566183	−	0.824280i	$-0.691580\pi$
$823$	−32.4853	−1.13237	−0.566183	+	0.824280i	$0.691580\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−31.7990	−1.10576	−0.552880	−	0.833261i	$-0.686471\pi$
$827$	−31.7990	−1.10576	−0.552880	+	0.833261i	$0.686471\pi$
$828$	0	0
$829$	19.0294	0.660920	0.330460	−	0.943820i	$-0.392796\pi$
$829$	19.0294	0.660920	0.330460	+	0.943820i	$0.392796\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	30.5269	1.05769
$834$	0	0
$835$	−19.3137	−0.668378
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−19.0294	−0.656969	−0.328485	−	0.944509i	$-0.606538\pi$
$839$	−19.0294	−0.656969	−0.328485	+	0.944509i	$0.606538\pi$
$840$	0	0
$841$	43.0000	1.48276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	44.3848	1.52688
$846$	0	0
$847$	−1.75736	−0.0603836
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.828427	0.0283981
$852$	0	0
$853$	−15.9411	−0.545814	−0.272907	−	0.962040i	$-0.587985\pi$
$853$	−15.9411	−0.545814	−0.272907	+	0.962040i	$0.587985\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.3431	−0.626590	−0.313295	−	0.949656i	$-0.601433\pi$
$857$	−18.3431	−0.626590	−0.313295	+	0.949656i	$0.601433\pi$
$858$	0	0
$859$	−48.9706	−1.67085	−0.835427	−	0.549601i	$-0.814780\pi$
$859$	−48.9706	−1.67085	−0.835427	+	0.549601i	$0.814780\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	17.6569	0.600351
$866$	0	0
$867$	0	0
$868$	0	0
$869$	33.6569	1.14173
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.31371	−0.112024
$876$	0	0
$877$	−32.3431	−1.09215	−0.546075	−	0.837736i	$-0.683879\pi$
$877$	−32.3431	−1.09215	−0.546075	+	0.837736i	$0.683879\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.07107	−0.0360852	−0.0180426	−	0.999837i	$-0.505743\pi$
$881$	−1.07107	−0.0360852	−0.0180426	+	0.999837i	$0.505743\pi$
$882$	0	0
$883$	28.4853	0.958606	0.479303	−	0.877649i	$-0.340889\pi$
$883$	28.4853	0.958606	0.479303	+	0.877649i	$0.340889\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.6569	0.391399	0.195699	−	0.980664i	$-0.437302\pi$
$887$	11.6569	0.391399	0.195699	+	0.980664i	$0.437302\pi$
$888$	0	0
$889$	−7.31371	−0.245294
$890$	0	0
$891$	0	0
$892$	0	0
$893$	26.1421	0.874813
$894$	0	0
$895$	61.4558	2.05424
$896$	0	0
$897$	0	0
$898$	0	0
$899$	72.0000	2.40133
$900$	0	0
$901$	41.5980	1.38583
$902$	0	0
$903$	0	0
$904$	0	0
$905$	57.4558	1.90990
$906$	0	0
$907$	−23.6985	−0.786895	−0.393448	−	0.919347i	$-0.628718\pi$
$907$	−23.6985	−0.786895	−0.393448	+	0.919347i	$0.628718\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−58.6274	−1.94241	−0.971206	−	0.238239i	$-0.923430\pi$
$911$	−58.6274	−1.94241	−0.971206	+	0.238239i	$0.923430\pi$
$912$	0	0
$913$	3.31371	0.109668
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.94113	−0.328285
$918$	0	0
$919$	−39.2132	−1.29352	−0.646762	−	0.762692i	$-0.723877\pi$
$919$	−39.2132	−1.29352	−0.646762	+	0.762692i	$0.723877\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	5.51472	0.181323
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−0.686292	−0.0225165	−0.0112582	−	0.999937i	$-0.503584\pi$
$929$	−0.686292	−0.0225165	−0.0112582	+	0.999937i	$0.503584\pi$
$930$	0	0
$931$	14.9289	0.489276
$932$	0	0
$933$	0	0
$934$	0	0
$935$	44.2843	1.44825
$936$	0	0
$937$	20.3431	0.664582	0.332291	−	0.943177i	$-0.392178\pi$
$937$	20.3431	0.664582	0.332291	+	0.943177i	$0.392178\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−12.3848	−0.403732	−0.201866	−	0.979413i	$-0.564701\pi$
$941$	−12.3848	−0.403732	−0.201866	+	0.979413i	$0.564701\pi$
$942$	0	0
$943$	−9.65685	−0.314470
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.3137	0.822585	0.411292	−	0.911503i	$-0.365077\pi$
$947$	25.3137	0.822585	0.411292	+	0.911503i	$0.365077\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−51.0122	−1.65245	−0.826224	−	0.563342i	$-0.809515\pi$
$953$	−51.0122	−1.65245	−0.826224	+	0.563342i	$0.809515\pi$
$954$	0	0
$955$	79.5980	2.57573
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−32.9706	−1.06136
$966$	0	0
$967$	32.4853	1.04466	0.522328	−	0.852745i	$-0.325064\pi$
$967$	32.4853	1.04466	0.522328	+	0.852745i	$0.325064\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−55.1127	−1.76865	−0.884325	−	0.466871i	$-0.845381\pi$
$971$	−55.1127	−1.76865	−0.884325	+	0.466871i	$0.845381\pi$
$972$	0	0
$973$	2.34315	0.0751178
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.0416	0.577203	0.288601	−	0.957449i	$-0.406810\pi$
$977$	18.0416	0.577203	0.288601	+	0.957449i	$0.406810\pi$
$978$	0	0
$979$	3.02944	0.0968212
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0.686292	0.0218893	0.0109446	−	0.999940i	$-0.496516\pi$
$983$	0.686292	0.0218893	0.0109446	+	0.999940i	$0.496516\pi$
$984$	0	0
$985$	−50.6274	−1.61312
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10.2426	0.325697
$990$	0	0
$991$	25.4558	0.808632	0.404316	−	0.914619i	$-0.367510\pi$
$991$	25.4558	0.808632	0.404316	+	0.914619i	$0.367510\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	21.3137	0.675690
$996$	0	0
$997$	36.9706	1.17087	0.585435	−	0.810720i	$-0.300924\pi$
$997$	36.9706	1.17087	0.585435	+	0.810720i	$0.300924\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3312.2.a.u.1.1		2
3.2	odd	2		3312.2.a.be.1.2		2
4.3	odd	2		207.2.a.b.1.2	✓	2
12.11	even	2		207.2.a.e.1.1	yes	2
20.19	odd	2		5175.2.a.bo.1.1		2
60.59	even	2		5175.2.a.bc.1.2		2
92.91	even	2		4761.2.a.k.1.2		2
276.275	odd	2		4761.2.a.z.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.2.a.b.1.2	✓	2	4.3	odd	2
207.2.a.e.1.1	yes	2	12.11	even	2
3312.2.a.u.1.1		2	1.1	even	1	trivial
3312.2.a.be.1.2		2	3.2	odd	2
4761.2.a.k.1.2		2	92.91	even	2
4761.2.a.z.1.1		2	276.275	odd	2
5175.2.a.bc.1.2		2	60.59	even	2
5175.2.a.bo.1.1		2	20.19	odd	2

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)

\(f(q)\)	\(=\)	\(q-3.41421 q^{5} +0.585786 q^{7} +O(q^{10})\)	\(q-3.41421 q^{5} +0.585786 q^{7} +2.82843 q^{11} -4.58579 q^{17} -2.24264 q^{19} +1.00000 q^{23} +6.65685 q^{25} +8.48528 q^{29} +8.48528 q^{31} -2.00000 q^{35} +0.828427 q^{37} -9.65685 q^{41} +10.2426 q^{43} -11.6569 q^{47} -6.65685 q^{49} -9.07107 q^{53} -9.65685 q^{55} -3.65685 q^{59} +4.82843 q^{61} -11.4142 q^{67} -2.34315 q^{71} -9.31371 q^{73} +1.65685 q^{77} +11.8995 q^{79} +1.17157 q^{83} +15.6569 q^{85} +1.07107 q^{89} +7.65685 q^{95} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q - 4 * q^5 + 4 * q^7	\( 2 q - 4 q^{5} + 4 q^{7} - 12 q^{17} + 4 q^{19} + 2 q^{23} + 2 q^{25} - 4 q^{35} - 4 q^{37} - 8 q^{41} + 12 q^{43} - 12 q^{47} - 2 q^{49} - 4 q^{53} - 8 q^{55} + 4 q^{59} + 4 q^{61} - 20 q^{67} - 16 q^{71} + 4 q^{73} - 8 q^{77} + 4 q^{79} + 8 q^{83} + 20 q^{85} - 12 q^{89} + 4 q^{95} - 20 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 + 4 * q^7 - 12 * q^17 + 4 * q^19 + 2 * q^23 + 2 * q^25 - 4 * q^35 - 4 * q^37 - 8 * q^41 + 12 * q^43 - 12 * q^47 - 2 * q^49 - 4 * q^53 - 8 * q^55 + 4 * q^59 + 4 * q^61 - 20 * q^67 - 16 * q^71 + 4 * q^73 - 8 * q^77 + 4 * q^79 + 8 * q^83 + 20 * q^85 - 12 * q^89 + 4 * q^95 - 20 * q^97

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