LMFDB - Embedded newform 1656.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1656,2,Mod(1,1656)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1656, 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("1656.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1656 = 2^{3} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1656.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:	$13.2232265747$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 552)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	1656.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.35026 q^{5} -2.96239 q^{7} +O(q^{10})$	$q-3.35026 q^{5} -2.96239 q^{7} -1.61213 q^{11} +2.00000 q^{13} -4.96239 q^{17} -1.35026 q^{19} +1.00000 q^{23} +6.22425 q^{25} +7.92478 q^{29} +5.92478 q^{31} +9.92478 q^{35} -2.31265 q^{37} +1.22425 q^{41} -4.57452 q^{43} -1.92478 q^{47} +1.77575 q^{49} +4.12601 q^{53} +5.40105 q^{55} -2.70052 q^{59} +14.3127 q^{61} -6.70052 q^{65} +4.57452 q^{67} +16.6253 q^{71} +2.00000 q^{73} +4.77575 q^{77} -5.03761 q^{79} +15.0132 q^{83} +16.6253 q^{85} +5.73813 q^{89} -5.92478 q^{91} +4.52373 q^{95} +12.7005 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{7}+O(q^{10}) $ 3 * q + 2 * q^7	$ 3 q + 2 q^{7} - 4 q^{11} + 6 q^{13} - 4 q^{17} + 6 q^{19} + 3 q^{23} + 17 q^{25} + 2 q^{29} - 4 q^{31} + 8 q^{35} + 14 q^{37} + 2 q^{41} - 2 q^{43} + 16 q^{47} + 7 q^{49} + 4 q^{53} - 24 q^{55} + 12 q^{59} + 22 q^{61} + 2 q^{67} + 8 q^{71} + 6 q^{73} + 16 q^{77} - 26 q^{79} + 4 q^{83} + 8 q^{85} + 8 q^{89} + 4 q^{91} + 32 q^{95} + 18 q^{97}+O(q^{100}) $ 3 * q + 2 * q^7 - 4 * q^11 + 6 * q^13 - 4 * q^17 + 6 * q^19 + 3 * q^23 + 17 * q^25 + 2 * q^29 - 4 * q^31 + 8 * q^35 + 14 * q^37 + 2 * q^41 - 2 * q^43 + 16 * q^47 + 7 * q^49 + 4 * q^53 - 24 * q^55 + 12 * q^59 + 22 * q^61 + 2 * q^67 + 8 * q^71 + 6 * q^73 + 16 * q^77 - 26 * q^79 + 4 * q^83 + 8 * q^85 + 8 * q^89 + 4 * q^91 + 32 * q^95 + 18 * 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
$5$	−3.35026	−1.49828	−0.749141	−	0.662410i	$-0.769534\pi$
$5$	−3.35026	−1.49828	−0.749141	+	0.662410i	$0.769534\pi$
$6$	0	0
$7$	−2.96239	−1.11968	−0.559839	−	0.828602i	$-0.689137\pi$
$7$	−2.96239	−1.11968	−0.559839	+	0.828602i	$0.689137\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.61213	−0.486075	−0.243037	−	0.970017i	$-0.578144\pi$
$11$	−1.61213	−0.486075	−0.243037	+	0.970017i	$0.578144\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.96239	−1.20356	−0.601778	−	0.798663i	$-0.705541\pi$
$17$	−4.96239	−1.20356	−0.601778	+	0.798663i	$0.705541\pi$
$18$	0	0
$19$	−1.35026	−0.309771	−0.154886	−	0.987932i	$-0.549501\pi$
$19$	−1.35026	−0.309771	−0.154886	+	0.987932i	$0.549501\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	6.22425	1.24485
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.92478	1.47159	0.735797	−	0.677202i	$-0.236808\pi$
$29$	7.92478	1.47159	0.735797	+	0.677202i	$0.236808\pi$
$30$	0	0
$31$	5.92478	1.06412	0.532061	−	0.846706i	$-0.321418\pi$
$31$	5.92478	1.06412	0.532061	+	0.846706i	$0.321418\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9.92478	1.67759
$36$	0	0
$37$	−2.31265	−0.380197	−0.190099	−	0.981765i	$-0.560881\pi$
$37$	−2.31265	−0.380197	−0.190099	+	0.981765i	$0.560881\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.22425	0.191196	0.0955982	−	0.995420i	$-0.469524\pi$
$41$	1.22425	0.191196	0.0955982	+	0.995420i	$0.469524\pi$
$42$	0	0
$43$	−4.57452	−0.697607	−0.348804	−	0.937196i	$-0.613412\pi$
$43$	−4.57452	−0.697607	−0.348804	+	0.937196i	$0.613412\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.92478	−0.280758	−0.140379	−	0.990098i	$-0.544832\pi$
$47$	−1.92478	−0.280758	−0.140379	+	0.990098i	$0.544832\pi$
$48$	0	0
$49$	1.77575	0.253678
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.12601	0.566751	0.283375	−	0.959009i	$-0.408546\pi$
$53$	4.12601	0.566751	0.283375	+	0.959009i	$0.408546\pi$
$54$	0	0
$55$	5.40105	0.728277
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.70052	−0.351578	−0.175789	−	0.984428i	$-0.556248\pi$
$59$	−2.70052	−0.351578	−0.175789	+	0.984428i	$0.556248\pi$
$60$	0	0
$61$	14.3127	1.83255	0.916274	−	0.400553i	$-0.131182\pi$
$61$	14.3127	1.83255	0.916274	+	0.400553i	$0.131182\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.70052	−0.831098
$66$	0	0
$67$	4.57452	0.558866	0.279433	−	0.960165i	$-0.409854\pi$
$67$	4.57452	0.558866	0.279433	+	0.960165i	$0.409854\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	16.6253	1.97306	0.986530	−	0.163580i	$-0.0523041\pi$
$71$	16.6253	1.97306	0.986530	+	0.163580i	$0.0523041\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.77575	0.544247
$78$	0	0
$79$	−5.03761	−0.566776	−0.283388	−	0.959005i	$-0.591458\pi$
$79$	−5.03761	−0.566776	−0.283388	+	0.959005i	$0.591458\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.0132	1.64791	0.823955	−	0.566655i	$-0.191763\pi$
$83$	15.0132	1.64791	0.823955	+	0.566655i	$0.191763\pi$
$84$	0	0
$85$	16.6253	1.80327
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.73813	0.608241	0.304121	−	0.952634i	$-0.401637\pi$
$89$	5.73813	0.608241	0.304121	+	0.952634i	$0.401637\pi$
$90$	0	0
$91$	−5.92478	−0.621085
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.52373	0.464125
$96$	0	0
$97$	12.7005	1.28954	0.644771	−	0.764375i	$-0.276953\pi$
$97$	12.7005	1.28954	0.644771	+	0.764375i	$0.276953\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.4010	1.53246	0.766231	−	0.642566i	$-0.222130\pi$
$101$	15.4010	1.53246	0.766231	+	0.642566i	$0.222130\pi$
$102$	0	0
$103$	−13.6629	−1.34625	−0.673123	−	0.739530i	$-0.735048\pi$
$103$	−13.6629	−1.34625	−0.673123	+	0.739530i	$0.735048\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.2374	−1.76308	−0.881539	−	0.472111i	$-0.843492\pi$
$107$	−18.2374	−1.76308	−0.881539	+	0.472111i	$0.843492\pi$
$108$	0	0
$109$	−19.4617	−1.86409	−0.932045	−	0.362341i	$-0.881977\pi$
$109$	−19.4617	−1.86409	−0.932045	+	0.362341i	$0.881977\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.66291	−0.720866	−0.360433	−	0.932785i	$-0.617371\pi$
$113$	−7.66291	−0.720866	−0.360433	+	0.932785i	$0.617371\pi$
$114$	0	0
$115$	−3.35026	−0.312414
$116$	0	0
$117$	0	0
$118$	0	0
$119$	14.7005	1.34759
$120$	0	0
$121$	−8.40105	−0.763732
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−4.10157	−0.366856
$126$	0	0
$127$	−15.4763	−1.37330	−0.686648	−	0.726990i	$-0.740919\pi$
$127$	−15.4763	−1.37330	−0.686648	+	0.726990i	$0.740919\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−0.775746	−0.0677773	−0.0338886	−	0.999426i	$-0.510789\pi$
$131$	−0.775746	−0.0677773	−0.0338886	+	0.999426i	$0.510789\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.96239	0.765709	0.382854	−	0.923809i	$-0.374941\pi$
$137$	8.96239	0.765709	0.382854	+	0.923809i	$0.374941\pi$
$138$	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
$143$	−3.22425	−0.269626
$144$	0	0
$145$	−26.5501	−2.20486
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.0508	0.823392	0.411696	−	0.911321i	$-0.364936\pi$
$149$	10.0508	0.823392	0.411696	+	0.911321i	$0.364936\pi$
$150$	0	0
$151$	11.8496	0.964303	0.482152	−	0.876088i	$-0.339855\pi$
$151$	11.8496	0.964303	0.482152	+	0.876088i	$0.339855\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−19.8496	−1.59435
$156$	0	0
$157$	22.9380	1.83065	0.915324	−	0.402718i	$-0.131935\pi$
$157$	22.9380	1.83065	0.915324	+	0.402718i	$0.131935\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.96239	−0.233469
$162$	0	0
$163$	6.70052	0.524826	0.262413	−	0.964956i	$-0.415482\pi$
$163$	6.70052	0.524826	0.262413	+	0.964956i	$0.415482\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−24.6253	−1.90556	−0.952781	−	0.303657i	$-0.901792\pi$
$167$	−24.6253	−1.90556	−0.952781	+	0.303657i	$0.901792\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	23.9248	1.81897	0.909484	−	0.415740i	$-0.136477\pi$
$173$	23.9248	1.81897	0.909484	+	0.415740i	$0.136477\pi$
$174$	0	0
$175$	−18.4387	−1.39383
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−19.3258	−1.44448	−0.722240	−	0.691643i	$-0.756887\pi$
$179$	−19.3258	−1.44448	−0.722240	+	0.691643i	$0.756887\pi$
$180$	0	0
$181$	20.2374	1.50424	0.752118	−	0.659028i	$-0.229032\pi$
$181$	20.2374	1.50424	0.752118	+	0.659028i	$0.229032\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.74798	0.569643
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.14903	0.662001	0.331000	−	0.943631i	$-0.392614\pi$
$191$	9.14903	0.662001	0.331000	+	0.943631i	$0.392614\pi$
$192$	0	0
$193$	−6.77575	−0.487729	−0.243864	−	0.969809i	$-0.578415\pi$
$193$	−6.77575	−0.487729	−0.243864	+	0.969809i	$0.578415\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	24.5501	1.74912	0.874560	−	0.484917i	$-0.161150\pi$
$197$	24.5501	1.74912	0.874560	+	0.484917i	$0.161150\pi$
$198$	0	0
$199$	−15.3357	−1.08712	−0.543559	−	0.839371i	$-0.682923\pi$
$199$	−15.3357	−1.08712	−0.543559	+	0.839371i	$0.682923\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−23.4763	−1.64771
$204$	0	0
$205$	−4.10157	−0.286466
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.17679	0.150572
$210$	0	0
$211$	−5.14903	−0.354474	−0.177237	−	0.984168i	$-0.556716\pi$
$211$	−5.14903	−0.354474	−0.177237	+	0.984168i	$0.556716\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	15.3258	1.04521
$216$	0	0
$217$	−17.5515	−1.19147
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9.92478	−0.667613
$222$	0	0
$223$	9.55149	0.639615	0.319808	−	0.947482i	$-0.396382\pi$
$223$	9.55149	0.639615	0.319808	+	0.947482i	$0.396382\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.68735	−0.244738	−0.122369	−	0.992485i	$-0.539049\pi$
$227$	−3.68735	−0.244738	−0.122369	+	0.992485i	$0.539049\pi$
$228$	0	0
$229$	12.2374	0.808672	0.404336	−	0.914611i	$-0.367503\pi$
$229$	12.2374	0.808672	0.404336	+	0.914611i	$0.367503\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.40105	0.222810	0.111405	−	0.993775i	$-0.464465\pi$
$233$	3.40105	0.222810	0.111405	+	0.993775i	$0.464465\pi$
$234$	0	0
$235$	6.44851	0.420654
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.55149	0.100358	0.0501789	−	0.998740i	$-0.484021\pi$
$239$	1.55149	0.100358	0.0501789	+	0.998740i	$0.484021\pi$
$240$	0	0
$241$	20.7005	1.33344	0.666719	−	0.745309i	$-0.267698\pi$
$241$	20.7005	1.33344	0.666719	+	0.745309i	$0.267698\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.94921	−0.380081
$246$	0	0
$247$	−2.70052	−0.171830
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.46310	−0.534186	−0.267093	−	0.963671i	$-0.586063\pi$
$251$	−8.46310	−0.534186	−0.267093	+	0.963671i	$0.586063\pi$
$252$	0	0
$253$	−1.61213	−0.101354
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.6253	−1.16181	−0.580907	−	0.813970i	$-0.697302\pi$
$257$	−18.6253	−1.16181	−0.580907	+	0.813970i	$0.697302\pi$
$258$	0	0
$259$	6.85097	0.425699
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0.523730	0.0322946	0.0161473	−	0.999870i	$-0.494860\pi$
$263$	0.523730	0.0322946	0.0161473	+	0.999870i	$0.494860\pi$
$264$	0	0
$265$	−13.8232	−0.849153
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−0.700523	−0.0427117	−0.0213558	−	0.999772i	$-0.506798\pi$
$269$	−0.700523	−0.0427117	−0.0213558	+	0.999772i	$0.506798\pi$
$270$	0	0
$271$	−10.0752	−0.612026	−0.306013	−	0.952027i	$-0.598995\pi$
$271$	−10.0752	−0.612026	−0.306013	+	0.952027i	$0.598995\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.0343	−0.605090
$276$	0	0
$277$	−22.4749	−1.35038	−0.675192	−	0.737642i	$-0.735939\pi$
$277$	−22.4749	−1.35038	−0.675192	+	0.737642i	$0.735939\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.8134	−0.704726	−0.352363	−	0.935863i	$-0.614622\pi$
$281$	−11.8134	−0.704726	−0.352363	+	0.935863i	$0.614622\pi$
$282$	0	0
$283$	29.1998	1.73575	0.867874	−	0.496784i	$-0.165486\pi$
$283$	29.1998	1.73575	0.867874	+	0.496784i	$0.165486\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.62672	−0.214078
$288$	0	0
$289$	7.62530	0.448547
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.27504	−0.541854	−0.270927	−	0.962600i	$-0.587330\pi$
$293$	−9.27504	−0.541854	−0.270927	+	0.962600i	$0.587330\pi$
$294$	0	0
$295$	9.04746	0.526764
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	13.5515	0.781095
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−47.9511	−2.74567
$306$	0	0
$307$	31.9511	1.82355	0.911774	−	0.410693i	$-0.134713\pi$
$307$	31.9511	1.82355	0.911774	+	0.410693i	$0.134713\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.1490	−0.745613	−0.372807	−	0.927909i	$-0.621605\pi$
$311$	−13.1490	−0.745613	−0.372807	+	0.927909i	$0.621605\pi$
$312$	0	0
$313$	24.0263	1.35805	0.679025	−	0.734115i	$-0.262403\pi$
$313$	24.0263	1.35805	0.679025	+	0.734115i	$0.262403\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.9248	−1.56841	−0.784206	−	0.620501i	$-0.786929\pi$
$317$	−27.9248	−1.56841	−0.784206	+	0.620501i	$0.786929\pi$
$318$	0	0
$319$	−12.7757	−0.715304
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.70052	0.372827
$324$	0	0
$325$	12.4485	0.690519
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.70194	0.314358
$330$	0	0
$331$	6.70052	0.368294	0.184147	−	0.982899i	$-0.441048\pi$
$331$	6.70052	0.368294	0.184147	+	0.982899i	$0.441048\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−15.3258	−0.837339
$336$	0	0
$337$	−10.3733	−0.565069	−0.282534	−	0.959257i	$-0.591175\pi$
$337$	−10.3733	−0.565069	−0.282534	+	0.959257i	$0.591175\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9.55149	−0.517242
$342$	0	0
$343$	15.4763	0.835640
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.7743	0.954176	0.477088	−	0.878855i	$-0.341692\pi$
$347$	17.7743	0.954176	0.477088	+	0.878855i	$0.341692\pi$
$348$	0	0
$349$	18.6253	0.996989	0.498495	−	0.866893i	$-0.333886\pi$
$349$	18.6253	0.996989	0.498495	+	0.866893i	$0.333886\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.7743	0.839583	0.419791	−	0.907621i	$-0.362103\pi$
$353$	15.7743	0.839583	0.419791	+	0.907621i	$0.362103\pi$
$354$	0	0
$355$	−55.6991	−2.95620
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.29948	0.279696	0.139848	−	0.990173i	$-0.455339\pi$
$359$	5.29948	0.279696	0.139848	+	0.990173i	$0.455339\pi$
$360$	0	0
$361$	−17.1768	−0.904042
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.70052	−0.350721
$366$	0	0
$367$	4.51388	0.235623	0.117811	−	0.993036i	$-0.462412\pi$
$367$	4.51388	0.235623	0.117811	+	0.993036i	$0.462412\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−12.2228	−0.634578
$372$	0	0
$373$	13.6873	0.708704	0.354352	−	0.935112i	$-0.384701\pi$
$373$	13.6873	0.708704	0.354352	+	0.935112i	$0.384701\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	15.8496	0.816294
$378$	0	0
$379$	30.3488	1.55892	0.779458	−	0.626455i	$-0.215495\pi$
$379$	30.3488	1.55892	0.779458	+	0.626455i	$0.215495\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−28.3733	−1.44981	−0.724904	−	0.688850i	$-0.758116\pi$
$383$	−28.3733	−1.44981	−0.724904	+	0.688850i	$0.758116\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−19.4518	−0.986247	−0.493124	−	0.869959i	$-0.664145\pi$
$389$	−19.4518	−0.986247	−0.493124	+	0.869959i	$0.664145\pi$
$390$	0	0
$391$	−4.96239	−0.250959
$392$	0	0
$393$	0	0
$394$	0	0
$395$	16.8773	0.849190
$396$	0	0
$397$	−28.8021	−1.44554	−0.722768	−	0.691091i	$-0.757130\pi$
$397$	−28.8021	−1.44554	−0.722768	+	0.691091i	$0.757130\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.1378	1.20538	0.602691	−	0.797974i	$-0.294095\pi$
$401$	24.1378	1.20538	0.602691	+	0.797974i	$0.294095\pi$
$402$	0	0
$403$	11.8496	0.590268
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.72829	0.184804
$408$	0	0
$409$	−28.1768	−1.39325	−0.696626	−	0.717434i	$-0.745316\pi$
$409$	−28.1768	−1.39325	−0.696626	+	0.717434i	$0.745316\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	−50.2981	−2.46903
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.4617	1.04847	0.524236	−	0.851573i	$-0.324351\pi$
$419$	21.4617	1.04847	0.524236	+	0.851573i	$0.324351\pi$
$420$	0	0
$421$	13.1636	0.641556	0.320778	−	0.947154i	$-0.396056\pi$
$421$	13.1636	0.641556	0.320778	+	0.947154i	$0.396056\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−30.8872	−1.49825
$426$	0	0
$427$	−42.3996	−2.05186
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.5223	1.03669	0.518347	−	0.855171i	$-0.326548\pi$
$431$	21.5223	1.03669	0.518347	+	0.855171i	$0.326548\pi$
$432$	0	0
$433$	−2.77575	−0.133394	−0.0666969	−	0.997773i	$-0.521246\pi$
$433$	−2.77575	−0.133394	−0.0666969	+	0.997773i	$0.521246\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.35026	−0.0645918
$438$	0	0
$439$	3.84955	0.183729	0.0918646	−	0.995772i	$-0.470717\pi$
$439$	3.84955	0.183729	0.0918646	+	0.995772i	$0.470717\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−26.9525	−1.28055	−0.640277	−	0.768144i	$-0.721180\pi$
$443$	−26.9525	−1.28055	−0.640277	+	0.768144i	$0.721180\pi$
$444$	0	0
$445$	−19.2243	−0.911317
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.07381	−0.0506761	−0.0253381	−	0.999679i	$-0.508066\pi$
$449$	−1.07381	−0.0506761	−0.0253381	+	0.999679i	$0.508066\pi$
$450$	0	0
$451$	−1.97365	−0.0929357
$452$	0	0
$453$	0	0
$454$	0	0
$455$	19.8496	0.930561
$456$	0	0
$457$	7.29948	0.341455	0.170728	−	0.985318i	$-0.445388\pi$
$457$	7.29948	0.341455	0.170728	+	0.985318i	$0.445388\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5.59754	−0.260703	−0.130352	−	0.991468i	$-0.541611\pi$
$461$	−5.59754	−0.260703	−0.130352	+	0.991468i	$0.541611\pi$
$462$	0	0
$463$	−3.84955	−0.178904	−0.0894520	−	0.995991i	$-0.528512\pi$
$463$	−3.84955	−0.178904	−0.0894520	+	0.995991i	$0.528512\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.68735	0.170630	0.0853151	−	0.996354i	$-0.472810\pi$
$467$	3.68735	0.170630	0.0853151	+	0.996354i	$0.472810\pi$
$468$	0	0
$469$	−13.5515	−0.625750
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.37470	0.339089
$474$	0	0
$475$	−8.40437	−0.385619
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.59895	0.118749	0.0593746	−	0.998236i	$-0.481089\pi$
$479$	2.59895	0.118749	0.0593746	+	0.998236i	$0.481089\pi$
$480$	0	0
$481$	−4.62530	−0.210896
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−42.5501	−1.93210
$486$	0	0
$487$	33.2506	1.50673	0.753364	−	0.657603i	$-0.228430\pi$
$487$	33.2506	1.50673	0.753364	+	0.657603i	$0.228430\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	13.9248	0.628416	0.314208	−	0.949354i	$-0.398261\pi$
$491$	13.9248	0.628416	0.314208	+	0.949354i	$0.398261\pi$
$492$	0	0
$493$	−39.3258	−1.77115
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−49.2506	−2.20919
$498$	0	0
$499$	19.5975	0.877306	0.438653	−	0.898656i	$-0.355456\pi$
$499$	19.5975	0.877306	0.438653	+	0.898656i	$0.355456\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−28.4749	−1.26963	−0.634816	−	0.772664i	$-0.718924\pi$
$503$	−28.4749	−1.26963	−0.634816	+	0.772664i	$0.718924\pi$
$504$	0	0
$505$	−51.5975	−2.29606
$506$	0	0
$507$	0	0
$508$	0	0
$509$	39.6239	1.75630	0.878149	−	0.478387i	$-0.158778\pi$
$509$	39.6239	1.75630	0.878149	+	0.478387i	$0.158778\pi$
$510$	0	0
$511$	−5.92478	−0.262097
$512$	0	0
$513$	0	0
$514$	0	0
$515$	45.7743	2.01706
$516$	0	0
$517$	3.10299	0.136469
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.2130	0.797926	0.398963	−	0.916967i	$-0.369370\pi$
$521$	18.2130	0.797926	0.398963	+	0.916967i	$0.369370\pi$
$522$	0	0
$523$	−24.8265	−1.08559	−0.542794	−	0.839866i	$-0.682634\pi$
$523$	−24.8265	−1.08559	−0.542794	+	0.839866i	$0.682634\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−29.4010	−1.28073
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.44851	0.106057
$534$	0	0
$535$	61.1002	2.64159
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.86273	−0.123306
$540$	0	0
$541$	6.47486	0.278376	0.139188	−	0.990266i	$-0.455551\pi$
$541$	6.47486	0.278376	0.139188	+	0.990266i	$0.455551\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	65.2017	2.79294
$546$	0	0
$547$	2.55008	0.109033	0.0545167	−	0.998513i	$-0.482638\pi$
$547$	2.55008	0.109033	0.0545167	+	0.998513i	$0.482638\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−10.7005	−0.455858
$552$	0	0
$553$	14.9234	0.634606
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.75131	0.201319	0.100660	−	0.994921i	$-0.467905\pi$
$557$	4.75131	0.201319	0.100660	+	0.994921i	$0.467905\pi$
$558$	0	0
$559$	−9.14903	−0.386963
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−19.6873	−0.829723	−0.414861	−	0.909885i	$-0.636170\pi$
$563$	−19.6873	−0.829723	−0.414861	+	0.909885i	$0.636170\pi$
$564$	0	0
$565$	25.6728	1.08006
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.48612	0.0623013	0.0311507	−	0.999515i	$-0.490083\pi$
$569$	1.48612	0.0623013	0.0311507	+	0.999515i	$0.490083\pi$
$570$	0	0
$571$	−11.1246	−0.465550	−0.232775	−	0.972531i	$-0.574781\pi$
$571$	−11.1246	−0.465550	−0.232775	+	0.972531i	$0.574781\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.22425	0.259569
$576$	0	0
$577$	0.176793	0.00736000	0.00368000	−	0.999993i	$-0.498829\pi$
$577$	0.176793	0.00736000	0.00368000	+	0.999993i	$0.498829\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−44.4749	−1.84513
$582$	0	0
$583$	−6.65165	−0.275483
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.6728	0.564335	0.282168	−	0.959365i	$-0.408947\pi$
$587$	13.6728	0.564335	0.282168	+	0.959365i	$0.408947\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−21.2243	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$593$	−21.2243	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$594$	0	0
$595$	−49.2506	−2.01908
$596$	0	0
$597$	0	0
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	40.1768	1.63885	0.819423	−	0.573190i	$-0.194294\pi$
$601$	40.1768	1.63885	0.819423	+	0.573190i	$0.194294\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	28.1457	1.14429
$606$	0	0
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.84955	−0.155736
$612$	0	0
$613$	14.3127	0.578083	0.289041	−	0.957317i	$-0.406664\pi$
$613$	14.3127	0.578083	0.289041	+	0.957317i	$0.406664\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.6907	0.832975	0.416488	−	0.909141i	$-0.363261\pi$
$617$	20.6907	0.832975	0.416488	+	0.909141i	$0.363261\pi$
$618$	0	0
$619$	32.3028	1.29836	0.649180	−	0.760635i	$-0.275112\pi$
$619$	32.3028	1.29836	0.649180	+	0.760635i	$0.275112\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−16.9986	−0.681034
$624$	0	0
$625$	−17.3799	−0.695197
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.4763	0.457589
$630$	0	0
$631$	3.58769	0.142824	0.0714118	−	0.997447i	$-0.477250\pi$
$631$	3.58769	0.142824	0.0714118	+	0.997447i	$0.477250\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	51.8496	2.05759
$636$	0	0
$637$	3.55149	0.140715
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.71179	−0.146607	−0.0733034	−	0.997310i	$-0.523354\pi$
$641$	−3.71179	−0.146607	−0.0733034	+	0.997310i	$0.523354\pi$
$642$	0	0
$643$	43.1246	1.70067	0.850334	−	0.526243i	$-0.176400\pi$
$643$	43.1246	1.70067	0.850334	+	0.526243i	$0.176400\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28.1016	1.10479	0.552393	−	0.833584i	$-0.313715\pi$
$647$	28.1016	1.10479	0.552393	+	0.833584i	$0.313715\pi$
$648$	0	0
$649$	4.35359	0.170893
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	0	0
$655$	2.59895	0.101549
$656$	0	0
$657$	0	0
$658$	0	0
$659$	33.8350	1.31802	0.659012	−	0.752133i	$-0.270975\pi$
$659$	33.8350	1.31802	0.659012	+	0.752133i	$0.270975\pi$
$660$	0	0
$661$	16.6107	0.646082	0.323041	−	0.946385i	$-0.395295\pi$
$661$	16.6107	0.646082	0.323041	+	0.946385i	$0.395295\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−13.4010	−0.519670
$666$	0	0
$667$	7.92478	0.306849
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−23.0738	−0.890754
$672$	0	0
$673$	8.17679	0.315192	0.157596	−	0.987504i	$-0.449626\pi$
$673$	8.17679	0.315192	0.157596	+	0.987504i	$0.449626\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−46.8989	−1.80247	−0.901236	−	0.433328i	$-0.857339\pi$
$677$	−46.8989	−1.80247	−0.901236	+	0.433328i	$0.857339\pi$
$678$	0	0
$679$	−37.6239	−1.44387
$680$	0	0
$681$	0	0
$682$	0	0
$683$	4.92619	0.188495	0.0942477	−	0.995549i	$-0.469955\pi$
$683$	4.92619	0.188495	0.0942477	+	0.995549i	$0.469955\pi$
$684$	0	0
$685$	−30.0263	−1.14725
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.25202	0.314377
$690$	0	0
$691$	−36.1016	−1.37337	−0.686684	−	0.726956i	$-0.740934\pi$
$691$	−36.1016	−1.37337	−0.686684	+	0.726956i	$0.740934\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.4010	0.508331
$696$	0	0
$697$	−6.07522	−0.230115
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−46.2736	−1.74773	−0.873865	−	0.486168i	$-0.838394\pi$
$701$	−46.2736	−1.74773	−0.873865	+	0.486168i	$0.838394\pi$
$702$	0	0
$703$	3.12268	0.117774
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−45.6239	−1.71586
$708$	0	0
$709$	26.5647	0.997657	0.498828	−	0.866701i	$-0.333764\pi$
$709$	26.5647	0.997657	0.498828	+	0.866701i	$0.333764\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.92478	0.221885
$714$	0	0
$715$	10.8021	0.403975
$716$	0	0
$717$	0	0
$718$	0	0
$719$	26.5501	0.990151	0.495075	−	0.868850i	$-0.335140\pi$
$719$	26.5501	0.990151	0.495075	+	0.868850i	$0.335140\pi$
$720$	0	0
$721$	40.4749	1.50736
$722$	0	0
$723$	0	0
$724$	0	0
$725$	49.3258	1.83192
$726$	0	0
$727$	18.4387	0.683852	0.341926	−	0.939727i	$-0.388921\pi$
$727$	18.4387	0.683852	0.341926	+	0.939727i	$0.388921\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	22.7005	0.839609
$732$	0	0
$733$	20.1162	0.743007	0.371504	−	0.928431i	$-0.378842\pi$
$733$	20.1162	0.743007	0.371504	+	0.928431i	$0.378842\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.37470	−0.271651
$738$	0	0
$739$	−34.6516	−1.27468	−0.637341	−	0.770582i	$-0.719966\pi$
$739$	−34.6516	−1.27468	−0.637341	+	0.770582i	$0.719966\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	3.32582	0.122013	0.0610063	−	0.998137i	$-0.480569\pi$
$743$	3.32582	0.122013	0.0610063	+	0.998137i	$0.480569\pi$
$744$	0	0
$745$	−33.6728	−1.23367
$746$	0	0
$747$	0	0
$748$	0	0
$749$	54.0263	1.97408
$750$	0	0
$751$	28.1114	1.02580	0.512900	−	0.858448i	$-0.328571\pi$
$751$	28.1114	1.02580	0.512900	+	0.858448i	$0.328571\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−39.6991	−1.44480
$756$	0	0
$757$	42.6859	1.55145	0.775723	−	0.631073i	$-0.217385\pi$
$757$	42.6859	1.55145	0.775723	+	0.631073i	$0.217385\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	23.7743	0.861819	0.430909	−	0.902395i	$-0.358193\pi$
$761$	23.7743	0.861819	0.430909	+	0.902395i	$0.358193\pi$
$762$	0	0
$763$	57.6531	2.08718
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.40105	−0.195021
$768$	0	0
$769$	−33.2243	−1.19810	−0.599049	−	0.800713i	$-0.704454\pi$
$769$	−33.2243	−1.19810	−0.599049	+	0.800713i	$0.704454\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−4.27645	−0.153813	−0.0769067	−	0.997038i	$-0.524504\pi$
$773$	−4.27645	−0.153813	−0.0769067	+	0.997038i	$0.524504\pi$
$774$	0	0
$775$	36.8773	1.32467
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.65306	−0.0592271
$780$	0	0
$781$	−26.8021	−0.959054
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−76.8481	−2.74283
$786$	0	0
$787$	−19.0230	−0.678098	−0.339049	−	0.940769i	$-0.610105\pi$
$787$	−19.0230	−0.678098	−0.339049	+	0.940769i	$0.610105\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	22.7005	0.807138
$792$	0	0
$793$	28.6253	1.01651
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.35026	0.260360	0.130180	−	0.991490i	$-0.458445\pi$
$797$	7.35026	0.260360	0.130180	+	0.991490i	$0.458445\pi$
$798$	0	0
$799$	9.55149	0.337908
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.22425	−0.113781
$804$	0	0
$805$	9.92478	0.349802
$806$	0	0
$807$	0	0
$808$	0	0
$809$	26.3733	0.927235	0.463618	−	0.886035i	$-0.346551\pi$
$809$	26.3733	0.927235	0.463618	+	0.886035i	$0.346551\pi$
$810$	0	0
$811$	−8.99859	−0.315983	−0.157992	−	0.987440i	$-0.550502\pi$
$811$	−8.99859	−0.315983	−0.157992	+	0.987440i	$0.550502\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−22.4485	−0.786337
$816$	0	0
$817$	6.17679	0.216099
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−9.10299	−0.317696	−0.158848	−	0.987303i	$-0.550778\pi$
$821$	−9.10299	−0.317696	−0.158848	+	0.987303i	$0.550778\pi$
$822$	0	0
$823$	−33.7743	−1.17730	−0.588650	−	0.808388i	$-0.700340\pi$
$823$	−33.7743	−1.17730	−0.588650	+	0.808388i	$0.700340\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−23.4156	−0.814241	−0.407121	−	0.913374i	$-0.633467\pi$
$827$	−23.4156	−0.814241	−0.407121	+	0.913374i	$0.633467\pi$
$828$	0	0
$829$	−10.7757	−0.374257	−0.187129	−	0.982335i	$-0.559918\pi$
$829$	−10.7757	−0.374257	−0.187129	+	0.982335i	$0.559918\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−8.81194	−0.305316
$834$	0	0
$835$	82.5012	2.85507
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.92478	0.204546	0.102273	−	0.994756i	$-0.467388\pi$
$839$	5.92478	0.204546	0.102273	+	0.994756i	$0.467388\pi$
$840$	0	0
$841$	33.8021	1.16559
$842$	0	0
$843$	0	0
$844$	0	0
$845$	30.1524	1.03727
$846$	0	0
$847$	24.8872	0.855133
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.31265	−0.0792766
$852$	0	0
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.2995	−0.861204	−0.430602	−	0.902542i	$-0.641699\pi$
$863$	−25.2995	−0.861204	−0.430602	+	0.902542i	$0.641699\pi$
$864$	0	0
$865$	−80.1543	−2.72533
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8.12127	0.275495
$870$	0	0
$871$	9.14903	0.310003
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.1504	0.410760
$876$	0	0
$877$	30.6253	1.03414	0.517071	−	0.855942i	$-0.327022\pi$
$877$	30.6253	1.03414	0.517071	+	0.855942i	$0.327022\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	44.1866	1.48869	0.744343	−	0.667798i	$-0.232763\pi$
$881$	44.1866	1.48869	0.744343	+	0.667798i	$0.232763\pi$
$882$	0	0
$883$	−6.70052	−0.225491	−0.112745	−	0.993624i	$-0.535964\pi$
$883$	−6.70052	−0.225491	−0.112745	+	0.993624i	$0.535964\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.5764	1.09381	0.546905	−	0.837195i	$-0.315806\pi$
$887$	32.5764	1.09381	0.546905	+	0.837195i	$0.315806\pi$
$888$	0	0
$889$	45.8467	1.53765
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.59895	0.0869706
$894$	0	0
$895$	64.7466	2.16424
$896$	0	0
$897$	0	0
$898$	0	0
$899$	46.9525	1.56595
$900$	0	0
$901$	−20.4749	−0.682116
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−67.8007	−2.25377
$906$	0	0
$907$	−1.75272	−0.0581982	−0.0290991	−	0.999577i	$-0.509264\pi$
$907$	−1.75272	−0.0581982	−0.0290991	+	0.999577i	$0.509264\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	9.67276	0.320473	0.160236	−	0.987079i	$-0.448774\pi$
$911$	9.67276	0.320473	0.160236	+	0.987079i	$0.448774\pi$
$912$	0	0
$913$	−24.2031	−0.801007
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.29806	0.0758887
$918$	0	0
$919$	25.4109	0.838228	0.419114	−	0.907934i	$-0.362341\pi$
$919$	25.4109	0.838228	0.419114	+	0.907934i	$0.362341\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	33.2506	1.09446
$924$	0	0
$925$	−14.3945	−0.473289
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−24.0263	−0.788279	−0.394139	−	0.919051i	$-0.628957\pi$
$929$	−24.0263	−0.788279	−0.394139	+	0.919051i	$0.628957\pi$
$930$	0	0
$931$	−2.39772	−0.0785822
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−26.8021	−0.876522
$936$	0	0
$937$	−49.8496	−1.62851	−0.814257	−	0.580505i	$-0.802855\pi$
$937$	−49.8496	−1.62851	−0.814257	+	0.580505i	$0.802855\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	30.2012	0.984532	0.492266	−	0.870445i	$-0.336169\pi$
$941$	30.2012	0.984532	0.492266	+	0.870445i	$0.336169\pi$
$942$	0	0
$943$	1.22425	0.0398672
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10.8218	0.351661	0.175830	−	0.984420i	$-0.443739\pi$
$947$	10.8218	0.351661	0.175830	+	0.984420i	$0.443739\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	33.6893	1.09130	0.545651	−	0.838012i	$-0.316282\pi$
$953$	33.6893	1.09130	0.545651	+	0.838012i	$0.316282\pi$
$954$	0	0
$955$	−30.6516	−0.991864
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−26.5501	−0.857347
$960$	0	0
$961$	4.10299	0.132354
$962$	0	0
$963$	0	0
$964$	0	0
$965$	22.7005	0.730756
$966$	0	0
$967$	43.0249	1.38359	0.691794	−	0.722095i	$-0.256820\pi$
$967$	43.0249	1.38359	0.691794	+	0.722095i	$0.256820\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−42.0381	−1.34907	−0.674534	−	0.738244i	$-0.735655\pi$
$971$	−42.0381	−1.34907	−0.674534	+	0.738244i	$0.735655\pi$
$972$	0	0
$973$	11.8496	0.379879
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−11.1881	−0.357938	−0.178969	−	0.983855i	$-0.557276\pi$
$977$	−11.1881	−0.357938	−0.178969	+	0.983855i	$0.557276\pi$
$978$	0	0
$979$	−9.25060	−0.295651
$980$	0	0
$981$	0	0
$982$	0	0
$983$	18.9234	0.603562	0.301781	−	0.953377i	$-0.402419\pi$
$983$	18.9234	0.603562	0.301781	+	0.953377i	$0.402419\pi$
$984$	0	0
$985$	−82.2492	−2.62068
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.57452	−0.145461
$990$	0	0
$991$	−62.1279	−1.97356	−0.986779	−	0.162071i	$-0.948183\pi$
$991$	−62.1279	−1.97356	−0.986779	+	0.162071i	$0.948183\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	51.3785	1.62881
$996$	0	0
$997$	−55.8759	−1.76961	−0.884804	−	0.465964i	$-0.845708\pi$
$997$	−55.8759	−1.76961	−0.884804	+	0.465964i	$0.845708\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1656.2.a.n.1.1		3
3.2	odd	2		552.2.a.g.1.3	✓	3
4.3	odd	2		3312.2.a.bf.1.1		3
12.11	even	2		1104.2.a.o.1.3		3
24.5	odd	2		4416.2.a.bp.1.1		3
24.11	even	2		4416.2.a.bs.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.a.g.1.3	✓	3	3.2	odd	2
1104.2.a.o.1.3		3	12.11	even	2
1656.2.a.n.1.1		3	1.1	even	1	trivial
3312.2.a.bf.1.1		3	4.3	odd	2
4416.2.a.bp.1.1		3	24.5	odd	2
4416.2.a.bs.1.1		3	24.11	even	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:	\( N \)	\(=\)	\( 1656 = 2^{3} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1656.a (trivial)

\(f(q)\)	\(=\)	\(q-3.35026 q^{5} -2.96239 q^{7} +O(q^{10})\)	\(q-3.35026 q^{5} -2.96239 q^{7} -1.61213 q^{11} +2.00000 q^{13} -4.96239 q^{17} -1.35026 q^{19} +1.00000 q^{23} +6.22425 q^{25} +7.92478 q^{29} +5.92478 q^{31} +9.92478 q^{35} -2.31265 q^{37} +1.22425 q^{41} -4.57452 q^{43} -1.92478 q^{47} +1.77575 q^{49} +4.12601 q^{53} +5.40105 q^{55} -2.70052 q^{59} +14.3127 q^{61} -6.70052 q^{65} +4.57452 q^{67} +16.6253 q^{71} +2.00000 q^{73} +4.77575 q^{77} -5.03761 q^{79} +15.0132 q^{83} +16.6253 q^{85} +5.73813 q^{89} -5.92478 q^{91} +4.52373 q^{95} +12.7005 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{7}+O(q^{10}) \) 3 * q + 2 * q^7	\( 3 q + 2 q^{7} - 4 q^{11} + 6 q^{13} - 4 q^{17} + 6 q^{19} + 3 q^{23} + 17 q^{25} + 2 q^{29} - 4 q^{31} + 8 q^{35} + 14 q^{37} + 2 q^{41} - 2 q^{43} + 16 q^{47} + 7 q^{49} + 4 q^{53} - 24 q^{55} + 12 q^{59} + 22 q^{61} + 2 q^{67} + 8 q^{71} + 6 q^{73} + 16 q^{77} - 26 q^{79} + 4 q^{83} + 8 q^{85} + 8 q^{89} + 4 q^{91} + 32 q^{95} + 18 q^{97}+O(q^{100}) \) 3 * q + 2 * q^7 - 4 * q^11 + 6 * q^13 - 4 * q^17 + 6 * q^19 + 3 * q^23 + 17 * q^25 + 2 * q^29 - 4 * q^31 + 8 * q^35 + 14 * q^37 + 2 * q^41 - 2 * q^43 + 16 * q^47 + 7 * q^49 + 4 * q^53 - 24 * q^55 + 12 * q^59 + 22 * q^61 + 2 * q^67 + 8 * q^71 + 6 * q^73 + 16 * q^77 - 26 * q^79 + 4 * q^83 + 8 * q^85 + 8 * q^89 + 4 * q^91 + 32 * q^95 + 18 * q^97

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