LMFDB - Embedded newform 4356.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4356,2,Mod(1,4356)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4356, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4356.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4356 = 2^{2} \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4356.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:	$34.7828351205$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 132)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4356.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{5} -0.236068 q^{7} +O(q^{10})$	$q+1.61803 q^{5} -0.236068 q^{7} +4.23607 q^{13} -0.145898 q^{17} +6.85410 q^{19} +5.00000 q^{23} -2.38197 q^{25} +2.00000 q^{29} +3.38197 q^{31} -0.381966 q^{35} +2.23607 q^{37} -8.70820 q^{41} +2.52786 q^{43} -9.56231 q^{47} -6.94427 q^{49} -6.61803 q^{53} -11.7984 q^{59} +10.8541 q^{61} +6.85410 q^{65} -4.38197 q^{67} +14.5623 q^{71} +15.7082 q^{73} +8.70820 q^{79} -1.76393 q^{83} -0.236068 q^{85} +5.18034 q^{89} -1.00000 q^{91} +11.0902 q^{95} +15.0344 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q + q^5 + 4 * q^7	$ 2 q + q^{5} + 4 q^{7} + 4 q^{13} - 7 q^{17} + 7 q^{19} + 10 q^{23} - 7 q^{25} + 4 q^{29} + 9 q^{31} - 3 q^{35} - 4 q^{41} + 14 q^{43} + q^{47} + 4 q^{49} - 11 q^{53} + q^{59} + 15 q^{61} + 7 q^{65} - 11 q^{67} + 9 q^{71} + 18 q^{73} + 4 q^{79} - 8 q^{83} + 4 q^{85} - 12 q^{89} - 2 q^{91} + 11 q^{95} + q^{97}+O(q^{100}) $ 2 * q + q^5 + 4 * q^7 + 4 * q^13 - 7 * q^17 + 7 * q^19 + 10 * q^23 - 7 * q^25 + 4 * q^29 + 9 * q^31 - 3 * q^35 - 4 * q^41 + 14 * q^43 + q^47 + 4 * q^49 - 11 * q^53 + q^59 + 15 * q^61 + 7 * q^65 - 11 * q^67 + 9 * q^71 + 18 * q^73 + 4 * q^79 - 8 * q^83 + 4 * q^85 - 12 * q^89 - 2 * q^91 + 11 * q^95 + 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
$5$	1.61803	0.723607	0.361803	−	0.932254i	$-0.382161\pi$
$5$	1.61803	0.723607	0.361803	+	0.932254i	$0.382161\pi$
$6$	0	0
$7$	−0.236068	−0.0892253	−0.0446127	−	0.999004i	$-0.514205\pi$
$7$	−0.236068	−0.0892253	−0.0446127	+	0.999004i	$0.514205\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.23607	1.17487	0.587437	−	0.809270i	$-0.300137\pi$
$13$	4.23607	1.17487	0.587437	+	0.809270i	$0.300137\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.145898	−0.0353855	−0.0176927	−	0.999843i	$-0.505632\pi$
$17$	−0.145898	−0.0353855	−0.0176927	+	0.999843i	$0.505632\pi$
$18$	0	0
$19$	6.85410	1.57244	0.786219	−	0.617947i	$-0.212036\pi$
$19$	6.85410	1.57244	0.786219	+	0.617947i	$0.212036\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	0	0
$25$	−2.38197	−0.476393
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	3.38197	0.607419	0.303710	−	0.952765i	$-0.401775\pi$
$31$	3.38197	0.607419	0.303710	+	0.952765i	$0.401775\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.381966	−0.0645640
$36$	0	0
$37$	2.23607	0.367607	0.183804	−	0.982963i	$-0.441159\pi$
$37$	2.23607	0.367607	0.183804	+	0.982963i	$0.441159\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.70820	−1.35999	−0.679996	−	0.733215i	$-0.738019\pi$
$41$	−8.70820	−1.35999	−0.679996	+	0.733215i	$0.738019\pi$
$42$	0	0
$43$	2.52786	0.385496	0.192748	−	0.981248i	$-0.438260\pi$
$43$	2.52786	0.385496	0.192748	+	0.981248i	$0.438260\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.56231	−1.39481	−0.697403	−	0.716679i	$-0.745661\pi$
$47$	−9.56231	−1.39481	−0.697403	+	0.716679i	$0.745661\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.61803	−0.909057	−0.454528	−	0.890732i	$-0.650192\pi$
$53$	−6.61803	−0.909057	−0.454528	+	0.890732i	$0.650192\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.7984	−1.53602	−0.768009	−	0.640439i	$-0.778752\pi$
$59$	−11.7984	−1.53602	−0.768009	+	0.640439i	$0.778752\pi$
$60$	0	0
$61$	10.8541	1.38973	0.694863	−	0.719142i	$-0.255465\pi$
$61$	10.8541	1.38973	0.694863	+	0.719142i	$0.255465\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.85410	0.850147
$66$	0	0
$67$	−4.38197	−0.535342	−0.267671	−	0.963510i	$-0.586254\pi$
$67$	−4.38197	−0.535342	−0.267671	+	0.963510i	$0.586254\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.5623	1.72823	0.864114	−	0.503296i	$-0.167880\pi$
$71$	14.5623	1.72823	0.864114	+	0.503296i	$0.167880\pi$
$72$	0	0
$73$	15.7082	1.83851	0.919253	−	0.393667i	$-0.128794\pi$
$73$	15.7082	1.83851	0.919253	+	0.393667i	$0.128794\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.70820	0.979749	0.489875	−	0.871793i	$-0.337043\pi$
$79$	8.70820	0.979749	0.489875	+	0.871793i	$0.337043\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.76393	−0.193617	−0.0968083	−	0.995303i	$-0.530863\pi$
$83$	−1.76393	−0.193617	−0.0968083	+	0.995303i	$0.530863\pi$
$84$	0	0
$85$	−0.236068	−0.0256052
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.18034	0.549115	0.274557	−	0.961571i	$-0.411469\pi$
$89$	5.18034	0.549115	0.274557	+	0.961571i	$0.411469\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	11.0902	1.13783
$96$	0	0
$97$	15.0344	1.52652	0.763258	−	0.646094i	$-0.223598\pi$
$97$	15.0344	1.52652	0.763258	+	0.646094i	$0.223598\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.2361	1.61555	0.807775	−	0.589492i	$-0.200672\pi$
$101$	16.2361	1.61555	0.807775	+	0.589492i	$0.200672\pi$
$102$	0	0
$103$	−9.41641	−0.927826	−0.463913	−	0.885881i	$-0.653555\pi$
$103$	−9.41641	−0.927826	−0.463913	+	0.885881i	$0.653555\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.6525	−1.51318	−0.756591	−	0.653888i	$-0.773137\pi$
$107$	−15.6525	−1.51318	−0.756591	+	0.653888i	$0.773137\pi$
$108$	0	0
$109$	−8.94427	−0.856706	−0.428353	−	0.903612i	$-0.640906\pi$
$109$	−8.94427	−0.856706	−0.428353	+	0.903612i	$0.640906\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.94427	0.747334	0.373667	−	0.927563i	$-0.378100\pi$
$113$	7.94427	0.747334	0.373667	+	0.927563i	$0.378100\pi$
$114$	0	0
$115$	8.09017	0.754412
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.0344419	0.00315728
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.9443	−1.06833
$126$	0	0
$127$	−1.70820	−0.151579	−0.0757893	−	0.997124i	$-0.524148\pi$
$127$	−1.70820	−0.151579	−0.0757893	+	0.997124i	$0.524148\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.14590	−0.187488	−0.0937440	−	0.995596i	$-0.529884\pi$
$131$	−2.14590	−0.187488	−0.0937440	+	0.995596i	$0.529884\pi$
$132$	0	0
$133$	−1.61803	−0.140301
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.2361	1.21627	0.608135	−	0.793834i	$-0.291918\pi$
$137$	14.2361	1.21627	0.608135	+	0.793834i	$0.291918\pi$
$138$	0	0
$139$	−11.5623	−0.980702	−0.490351	−	0.871525i	$-0.663131\pi$
$139$	−11.5623	−0.980702	−0.490351	+	0.871525i	$0.663131\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	3.23607	0.268741
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22.2361	1.82165	0.910825	−	0.412793i	$-0.135447\pi$
$149$	22.2361	1.82165	0.910825	+	0.412793i	$0.135447\pi$
$150$	0	0
$151$	−4.47214	−0.363937	−0.181969	−	0.983304i	$-0.558247\pi$
$151$	−4.47214	−0.363937	−0.181969	+	0.983304i	$0.558247\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.47214	0.439533
$156$	0	0
$157$	−2.29180	−0.182905	−0.0914526	−	0.995809i	$-0.529151\pi$
$157$	−2.29180	−0.182905	−0.0914526	+	0.995809i	$0.529151\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.18034	−0.0930238
$162$	0	0
$163$	−12.6180	−0.988321	−0.494160	−	0.869371i	$-0.664524\pi$
$163$	−12.6180	−0.988321	−0.494160	+	0.869371i	$0.664524\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.56231	0.353042	0.176521	−	0.984297i	$-0.443516\pi$
$167$	4.56231	0.353042	0.176521	+	0.984297i	$0.443516\pi$
$168$	0	0
$169$	4.94427	0.380329
$170$	0	0
$171$	0	0
$172$	0	0
$173$	25.3262	1.92552	0.962759	−	0.270361i	$-0.0871430\pi$
$173$	25.3262	1.92552	0.962759	+	0.270361i	$0.0871430\pi$
$174$	0	0
$175$	0.562306	0.0425063
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.7082	−1.09934	−0.549671	−	0.835381i	$-0.685247\pi$
$179$	−14.7082	−1.09934	−0.549671	+	0.835381i	$0.685247\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.61803	0.266003
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.52786	0.327625	0.163812	−	0.986492i	$-0.447621\pi$
$191$	4.52786	0.327625	0.163812	+	0.986492i	$0.447621\pi$
$192$	0	0
$193$	6.79837	0.489358	0.244679	−	0.969604i	$-0.421317\pi$
$193$	6.79837	0.489358	0.244679	+	0.969604i	$0.421317\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.67376	0.261745	0.130872	−	0.991399i	$-0.458222\pi$
$197$	3.67376	0.261745	0.130872	+	0.991399i	$0.458222\pi$
$198$	0	0
$199$	17.9443	1.27204	0.636018	−	0.771674i	$-0.280580\pi$
$199$	17.9443	1.27204	0.636018	+	0.771674i	$0.280580\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.472136	−0.0331374
$204$	0	0
$205$	−14.0902	−0.984100
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	8.90983	0.613378	0.306689	−	0.951810i	$-0.400779\pi$
$211$	8.90983	0.613378	0.306689	+	0.951810i	$0.400779\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.09017	0.278947
$216$	0	0
$217$	−0.798374	−0.0541972
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.618034	−0.0415735
$222$	0	0
$223$	2.05573	0.137662	0.0688309	−	0.997628i	$-0.478073\pi$
$223$	2.05573	0.137662	0.0688309	+	0.997628i	$0.478073\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.47214	−0.230454	−0.115227	−	0.993339i	$-0.536760\pi$
$227$	−3.47214	−0.230454	−0.115227	+	0.993339i	$0.536760\pi$
$228$	0	0
$229$	−17.4164	−1.15091	−0.575454	−	0.817834i	$-0.695175\pi$
$229$	−17.4164	−1.15091	−0.575454	+	0.817834i	$0.695175\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.7984	1.55908	0.779542	−	0.626350i	$-0.215452\pi$
$233$	23.7984	1.55908	0.779542	+	0.626350i	$0.215452\pi$
$234$	0	0
$235$	−15.4721	−1.00929
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−19.7426	−1.27705	−0.638523	−	0.769603i	$-0.720454\pi$
$239$	−19.7426	−1.27705	−0.638523	+	0.769603i	$0.720454\pi$
$240$	0	0
$241$	0.763932	0.0492092	0.0246046	−	0.999697i	$-0.492167\pi$
$241$	0.763932	0.0492092	0.0246046	+	0.999697i	$0.492167\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−11.2361	−0.717846
$246$	0	0
$247$	29.0344	1.84742
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.32624	0.399309	0.199654	−	0.979866i	$-0.436018\pi$
$251$	6.32624	0.399309	0.199654	+	0.979866i	$0.436018\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.8541	−0.677060	−0.338530	−	0.940956i	$-0.609930\pi$
$257$	−10.8541	−0.677060	−0.338530	+	0.940956i	$0.609930\pi$
$258$	0	0
$259$	−0.527864	−0.0327999
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−22.7426	−1.40237	−0.701186	−	0.712979i	$-0.747346\pi$
$263$	−22.7426	−1.40237	−0.701186	+	0.712979i	$0.747346\pi$
$264$	0	0
$265$	−10.7082	−0.657800
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.41641	−0.0863599	−0.0431800	−	0.999067i	$-0.513749\pi$
$269$	−1.41641	−0.0863599	−0.0431800	+	0.999067i	$0.513749\pi$
$270$	0	0
$271$	13.6180	0.827237	0.413618	−	0.910450i	$-0.364265\pi$
$271$	13.6180	0.827237	0.413618	+	0.910450i	$0.364265\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6.43769	−0.386804	−0.193402	−	0.981120i	$-0.561952\pi$
$277$	−6.43769	−0.386804	−0.193402	+	0.981120i	$0.561952\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.1803	1.08455	0.542274	−	0.840202i	$-0.317563\pi$
$281$	18.1803	1.08455	0.542274	+	0.840202i	$0.317563\pi$
$282$	0	0
$283$	−24.6525	−1.46544	−0.732719	−	0.680532i	$-0.761749\pi$
$283$	−24.6525	−1.46544	−0.732719	+	0.680532i	$0.761749\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.05573	0.121346
$288$	0	0
$289$	−16.9787	−0.998748
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.7639	1.03778	0.518890	−	0.854841i	$-0.326345\pi$
$293$	17.7639	1.03778	0.518890	+	0.854841i	$0.326345\pi$
$294$	0	0
$295$	−19.0902	−1.11147
$296$	0	0
$297$	0	0
$298$	0	0
$299$	21.1803	1.22489
$300$	0	0
$301$	−0.596748	−0.0343960
$302$	0	0
$303$	0	0
$304$	0	0
$305$	17.5623	1.00561
$306$	0	0
$307$	31.7426	1.81165	0.905824	−	0.423654i	$-0.139253\pi$
$307$	31.7426	1.81165	0.905824	+	0.423654i	$0.139253\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.52786	−0.256752	−0.128376	−	0.991726i	$-0.540976\pi$
$311$	−4.52786	−0.256752	−0.128376	+	0.991726i	$0.540976\pi$
$312$	0	0
$313$	10.4164	0.588770	0.294385	−	0.955687i	$-0.404885\pi$
$313$	10.4164	0.588770	0.294385	+	0.955687i	$0.404885\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.2361	1.47356	0.736782	−	0.676130i	$-0.236344\pi$
$317$	26.2361	1.47356	0.736782	+	0.676130i	$0.236344\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.00000	−0.0556415
$324$	0	0
$325$	−10.0902	−0.559702
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.25735	0.124452
$330$	0	0
$331$	11.9443	0.656517	0.328258	−	0.944588i	$-0.393538\pi$
$331$	11.9443	0.656517	0.328258	+	0.944588i	$0.393538\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.09017	−0.387377
$336$	0	0
$337$	12.1803	0.663505	0.331753	−	0.943366i	$-0.392360\pi$
$337$	12.1803	0.663505	0.331753	+	0.943366i	$0.392360\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	3.29180	0.177740
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.52786	−0.296751	−0.148376	−	0.988931i	$-0.547404\pi$
$347$	−5.52786	−0.296751	−0.148376	+	0.988931i	$0.547404\pi$
$348$	0	0
$349$	−19.2918	−1.03267	−0.516333	−	0.856388i	$-0.672703\pi$
$349$	−19.2918	−1.03267	−0.516333	+	0.856388i	$0.672703\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−32.9443	−1.75345	−0.876723	−	0.480995i	$-0.840276\pi$
$353$	−32.9443	−1.75345	−0.876723	+	0.480995i	$0.840276\pi$
$354$	0	0
$355$	23.5623	1.25056
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.1803	0.959522	0.479761	−	0.877399i	$-0.340723\pi$
$359$	18.1803	0.959522	0.479761	+	0.877399i	$0.340723\pi$
$360$	0	0
$361$	27.9787	1.47256
$362$	0	0
$363$	0	0
$364$	0	0
$365$	25.4164	1.33036
$366$	0	0
$367$	−22.0902	−1.15310	−0.576549	−	0.817063i	$-0.695601\pi$
$367$	−22.0902	−1.15310	−0.576549	+	0.817063i	$0.695601\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.56231	0.0811109
$372$	0	0
$373$	−22.4164	−1.16068	−0.580339	−	0.814375i	$-0.697080\pi$
$373$	−22.4164	−1.16068	−0.580339	+	0.814375i	$0.697080\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.47214	0.436337
$378$	0	0
$379$	−21.7639	−1.11794	−0.558969	−	0.829189i	$-0.688803\pi$
$379$	−21.7639	−1.11794	−0.558969	+	0.829189i	$0.688803\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.00000	0.255488	0.127744	−	0.991807i	$-0.459226\pi$
$383$	5.00000	0.255488	0.127744	+	0.991807i	$0.459226\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.38197	0.272877	0.136438	−	0.990649i	$-0.456434\pi$
$389$	5.38197	0.272877	0.136438	+	0.990649i	$0.456434\pi$
$390$	0	0
$391$	−0.729490	−0.0368919
$392$	0	0
$393$	0	0
$394$	0	0
$395$	14.0902	0.708953
$396$	0	0
$397$	5.76393	0.289283	0.144642	−	0.989484i	$-0.453797\pi$
$397$	5.76393	0.289283	0.144642	+	0.989484i	$0.453797\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.14590	−0.306912	−0.153456	−	0.988156i	$-0.549040\pi$
$401$	−6.14590	−0.306912	−0.153456	+	0.988156i	$0.549040\pi$
$402$	0	0
$403$	14.3262	0.713641
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−23.4164	−1.15787	−0.578933	−	0.815375i	$-0.696531\pi$
$409$	−23.4164	−1.15787	−0.578933	+	0.815375i	$0.696531\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.78522	0.137052
$414$	0	0
$415$	−2.85410	−0.140102
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11.7984	0.576388	0.288194	−	0.957572i	$-0.406945\pi$
$419$	11.7984	0.576388	0.288194	+	0.957572i	$0.406945\pi$
$420$	0	0
$421$	−8.67376	−0.422733	−0.211367	−	0.977407i	$-0.567791\pi$
$421$	−8.67376	−0.422733	−0.211367	+	0.977407i	$0.567791\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.347524	0.0168574
$426$	0	0
$427$	−2.56231	−0.123999
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−30.5066	−1.46945	−0.734725	−	0.678365i	$-0.762689\pi$
$431$	−30.5066	−1.46945	−0.734725	+	0.678365i	$0.762689\pi$
$432$	0	0
$433$	−3.52786	−0.169538	−0.0847692	−	0.996401i	$-0.527015\pi$
$433$	−3.52786	−0.169538	−0.0847692	+	0.996401i	$0.527015\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	34.2705	1.63938
$438$	0	0
$439$	−3.47214	−0.165716	−0.0828580	−	0.996561i	$-0.526405\pi$
$439$	−3.47214	−0.165716	−0.0828580	+	0.996561i	$0.526405\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.7082	0.936365	0.468183	−	0.883632i	$-0.344909\pi$
$443$	19.7082	0.936365	0.468183	+	0.883632i	$0.344909\pi$
$444$	0	0
$445$	8.38197	0.397343
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.9443	−0.516492	−0.258246	−	0.966079i	$-0.583145\pi$
$449$	−10.9443	−0.516492	−0.258246	+	0.966079i	$0.583145\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.61803	−0.0758546
$456$	0	0
$457$	−16.7984	−0.785795	−0.392897	−	0.919582i	$-0.628527\pi$
$457$	−16.7984	−0.785795	−0.392897	+	0.919582i	$0.628527\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	16.2705	0.757793	0.378897	−	0.925439i	$-0.376304\pi$
$461$	16.2705	0.757793	0.378897	+	0.925439i	$0.376304\pi$
$462$	0	0
$463$	16.0344	0.745184	0.372592	−	0.927995i	$-0.378469\pi$
$463$	16.0344	0.745184	0.372592	+	0.927995i	$0.378469\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.2361	0.566218	0.283109	−	0.959088i	$-0.408634\pi$
$467$	12.2361	0.566218	0.283109	+	0.959088i	$0.408634\pi$
$468$	0	0
$469$	1.03444	0.0477661
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−16.3262	−0.749099
$476$	0	0
$477$	0	0
$478$	0	0
$479$	26.2148	1.19778	0.598892	−	0.800830i	$-0.295608\pi$
$479$	26.2148	1.19778	0.598892	+	0.800830i	$0.295608\pi$
$480$	0	0
$481$	9.47214	0.431892
$482$	0	0
$483$	0	0
$484$	0	0
$485$	24.3262	1.10460
$486$	0	0
$487$	−35.3607	−1.60235	−0.801173	−	0.598433i	$-0.795790\pi$
$487$	−35.3607	−1.60235	−0.801173	+	0.598433i	$0.795790\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−23.0902	−1.04204	−0.521022	−	0.853543i	$-0.674449\pi$
$491$	−23.0902	−1.04204	−0.521022	+	0.853543i	$0.674449\pi$
$492$	0	0
$493$	−0.291796	−0.0131418
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.43769	−0.154202
$498$	0	0
$499$	22.0902	0.988892	0.494446	−	0.869208i	$-0.335371\pi$
$499$	22.0902	0.988892	0.494446	+	0.869208i	$0.335371\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21.7082	−0.967921	−0.483960	−	0.875090i	$-0.660802\pi$
$503$	−21.7082	−0.967921	−0.483960	+	0.875090i	$0.660802\pi$
$504$	0	0
$505$	26.2705	1.16902
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−17.2705	−0.765502	−0.382751	−	0.923852i	$-0.625023\pi$
$509$	−17.2705	−0.765502	−0.382751	+	0.923852i	$0.625023\pi$
$510$	0	0
$511$	−3.70820	−0.164041
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−15.2361	−0.671381
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.00000	0.175243	0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000	0.175243	0.0876216	+	0.996154i	$0.472073\pi$
$522$	0	0
$523$	3.67376	0.160642	0.0803212	−	0.996769i	$-0.474405\pi$
$523$	3.67376	0.160642	0.0803212	+	0.996769i	$0.474405\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.493422	−0.0214938
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−36.8885	−1.59782
$534$	0	0
$535$	−25.3262	−1.09495
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	29.5066	1.26859	0.634293	−	0.773092i	$-0.281291\pi$
$541$	29.5066	1.26859	0.634293	+	0.773092i	$0.281291\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−14.4721	−0.619918
$546$	0	0
$547$	27.8541	1.19096	0.595478	−	0.803372i	$-0.296963\pi$
$547$	27.8541	1.19096	0.595478	+	0.803372i	$0.296963\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13.7082	0.583989
$552$	0	0
$553$	−2.05573	−0.0874185
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.67376	−0.198034	−0.0990168	−	0.995086i	$-0.531570\pi$
$557$	−4.67376	−0.198034	−0.0990168	+	0.995086i	$0.531570\pi$
$558$	0	0
$559$	10.7082	0.452909
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.8328	−1.17301	−0.586507	−	0.809944i	$-0.699497\pi$
$563$	−27.8328	−1.17301	−0.586507	+	0.809944i	$0.699497\pi$
$564$	0	0
$565$	12.8541	0.540776
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.81966	0.160128	0.0800642	−	0.996790i	$-0.474487\pi$
$569$	3.81966	0.160128	0.0800642	+	0.996790i	$0.474487\pi$
$570$	0	0
$571$	−13.6180	−0.569897	−0.284948	−	0.958543i	$-0.591976\pi$
$571$	−13.6180	−0.569897	−0.284948	+	0.958543i	$0.591976\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−11.9098	−0.496674
$576$	0	0
$577$	−30.0689	−1.25178	−0.625892	−	0.779910i	$-0.715265\pi$
$577$	−30.0689	−1.25178	−0.625892	+	0.779910i	$0.715265\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.416408	0.0172755
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.5279	−0.599629	−0.299815	−	0.953997i	$-0.596925\pi$
$587$	−14.5279	−0.599629	−0.299815	+	0.953997i	$0.596925\pi$
$588$	0	0
$589$	23.1803	0.955129
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−19.6180	−0.805616	−0.402808	−	0.915284i	$-0.631966\pi$
$593$	−19.6180	−0.805616	−0.402808	+	0.915284i	$0.631966\pi$
$594$	0	0
$595$	0.0557281	0.00228463
$596$	0	0
$597$	0	0
$598$	0	0
$599$	35.5967	1.45444	0.727222	−	0.686403i	$-0.240811\pi$
$599$	35.5967	1.45444	0.727222	+	0.686403i	$0.240811\pi$
$600$	0	0
$601$	7.11146	0.290082	0.145041	−	0.989426i	$-0.453669\pi$
$601$	7.11146	0.290082	0.145041	+	0.989426i	$0.453669\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.50658	0.101739	0.0508694	−	0.998705i	$-0.483801\pi$
$607$	2.50658	0.101739	0.0508694	+	0.998705i	$0.483801\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−40.5066	−1.63872
$612$	0	0
$613$	25.2361	1.01928	0.509638	−	0.860389i	$-0.329779\pi$
$613$	25.2361	1.01928	0.509638	+	0.860389i	$0.329779\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13.2918	−0.535108	−0.267554	−	0.963543i	$-0.586215\pi$
$617$	−13.2918	−0.535108	−0.267554	+	0.963543i	$0.586215\pi$
$618$	0	0
$619$	26.4164	1.06177	0.530883	−	0.847445i	$-0.321861\pi$
$619$	26.4164	1.06177	0.530883	+	0.847445i	$0.321861\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.22291	−0.0489949
$624$	0	0
$625$	−7.41641	−0.296656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.326238	−0.0130080
$630$	0	0
$631$	27.1591	1.08118	0.540592	−	0.841285i	$-0.318200\pi$
$631$	27.1591	1.08118	0.540592	+	0.841285i	$0.318200\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.76393	−0.109683
$636$	0	0
$637$	−29.4164	−1.16552
$638$	0	0
$639$	0	0
$640$	0	0
$641$	26.4508	1.04475	0.522373	−	0.852717i	$-0.325047\pi$
$641$	26.4508	1.04475	0.522373	+	0.852717i	$0.325047\pi$
$642$	0	0
$643$	19.8541	0.782969	0.391485	−	0.920185i	$-0.371962\pi$
$643$	19.8541	0.782969	0.391485	+	0.920185i	$0.371962\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−14.3262	−0.563223	−0.281611	−	0.959529i	$-0.590869\pi$
$647$	−14.3262	−0.563223	−0.281611	+	0.959529i	$0.590869\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	48.3951	1.89385	0.946924	−	0.321458i	$-0.104173\pi$
$653$	48.3951	1.89385	0.946924	+	0.321458i	$0.104173\pi$
$654$	0	0
$655$	−3.47214	−0.135668
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0.652476	0.0254169	0.0127084	−	0.999919i	$-0.495955\pi$
$659$	0.652476	0.0254169	0.0127084	+	0.999919i	$0.495955\pi$
$660$	0	0
$661$	−42.0344	−1.63495	−0.817475	−	0.575964i	$-0.804627\pi$
$661$	−42.0344	−1.63495	−0.817475	+	0.575964i	$0.804627\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.61803	−0.101523
$666$	0	0
$667$	10.0000	0.387202
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	35.9443	1.38555	0.692775	−	0.721154i	$-0.256388\pi$
$673$	35.9443	1.38555	0.692775	+	0.721154i	$0.256388\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−42.2492	−1.62377	−0.811885	−	0.583818i	$-0.801558\pi$
$677$	−42.2492	−1.62377	−0.811885	+	0.583818i	$0.801558\pi$
$678$	0	0
$679$	−3.54915	−0.136204
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−17.9443	−0.686618	−0.343309	−	0.939222i	$-0.611548\pi$
$683$	−17.9443	−0.686618	−0.343309	+	0.939222i	$0.611548\pi$
$684$	0	0
$685$	23.0344	0.880101
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−28.0344	−1.06803
$690$	0	0
$691$	−23.1246	−0.879702	−0.439851	−	0.898071i	$-0.644969\pi$
$691$	−23.1246	−0.879702	−0.439851	+	0.898071i	$0.644969\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−18.7082	−0.709643
$696$	0	0
$697$	1.27051	0.0481240
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−27.9098	−1.05414	−0.527070	−	0.849822i	$-0.676709\pi$
$701$	−27.9098	−1.05414	−0.527070	+	0.849822i	$0.676709\pi$
$702$	0	0
$703$	15.3262	0.578040
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.83282	−0.144148
$708$	0	0
$709$	−35.2705	−1.32461	−0.662306	−	0.749234i	$-0.730422\pi$
$709$	−35.2705	−1.32461	−0.662306	+	0.749234i	$0.730422\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	16.9098	0.633278
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0.527864	0.0196860	0.00984300	−	0.999952i	$-0.496867\pi$
$719$	0.527864	0.0196860	0.00984300	+	0.999952i	$0.496867\pi$
$720$	0	0
$721$	2.22291	0.0827856
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.76393	−0.176928
$726$	0	0
$727$	−48.9787	−1.81652	−0.908260	−	0.418406i	$-0.862589\pi$
$727$	−48.9787	−1.81652	−0.908260	+	0.418406i	$0.862589\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.368810	−0.0136409
$732$	0	0
$733$	−51.4853	−1.90165	−0.950826	−	0.309725i	$-0.899763\pi$
$733$	−51.4853	−1.90165	−0.950826	+	0.309725i	$0.899763\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−38.4853	−1.41570	−0.707852	−	0.706361i	$-0.750336\pi$
$739$	−38.4853	−1.41570	−0.707852	+	0.706361i	$0.750336\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−37.7639	−1.38542	−0.692712	−	0.721214i	$-0.743584\pi$
$743$	−37.7639	−1.38542	−0.692712	+	0.721214i	$0.743584\pi$
$744$	0	0
$745$	35.9787	1.31816
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.69505	0.135014
$750$	0	0
$751$	15.6869	0.572424	0.286212	−	0.958166i	$-0.407604\pi$
$751$	15.6869	0.572424	0.286212	+	0.958166i	$0.407604\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−7.23607	−0.263347
$756$	0	0
$757$	43.4721	1.58002	0.790011	−	0.613093i	$-0.210075\pi$
$757$	43.4721	1.58002	0.790011	+	0.613093i	$0.210075\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−35.6525	−1.29240	−0.646201	−	0.763168i	$-0.723643\pi$
$761$	−35.6525	−1.29240	−0.646201	+	0.763168i	$0.723643\pi$
$762$	0	0
$763$	2.11146	0.0764398
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−49.9787	−1.80463
$768$	0	0
$769$	−9.97871	−0.359842	−0.179921	−	0.983681i	$-0.557584\pi$
$769$	−9.97871	−0.359842	−0.179921	+	0.983681i	$0.557584\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	24.2361	0.871711	0.435855	−	0.900017i	$-0.356446\pi$
$773$	24.2361	0.871711	0.435855	+	0.900017i	$0.356446\pi$
$774$	0	0
$775$	−8.05573	−0.289370
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−59.6869	−2.13851
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.70820	−0.132351
$786$	0	0
$787$	−15.7082	−0.559937	−0.279968	−	0.960009i	$-0.590324\pi$
$787$	−15.7082	−0.559937	−0.279968	+	0.960009i	$0.590324\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.87539	−0.0666811
$792$	0	0
$793$	45.9787	1.63275
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26.0689	−0.923407	−0.461704	−	0.887034i	$-0.652762\pi$
$797$	−26.0689	−0.923407	−0.461704	+	0.887034i	$0.652762\pi$
$798$	0	0
$799$	1.39512	0.0493559
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−1.90983	−0.0673127
$806$	0	0
$807$	0	0
$808$	0	0
$809$	38.3394	1.34794	0.673971	−	0.738758i	$-0.264587\pi$
$809$	38.3394	1.34794	0.673971	+	0.738758i	$0.264587\pi$
$810$	0	0
$811$	12.3262	0.432833	0.216416	−	0.976301i	$-0.430563\pi$
$811$	12.3262	0.432833	0.216416	+	0.976301i	$0.430563\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−20.4164	−0.715156
$816$	0	0
$817$	17.3262	0.606168
$818$	0	0
$819$	0	0
$820$	0	0
$821$	26.1246	0.911755	0.455878	−	0.890042i	$-0.349325\pi$
$821$	26.1246	0.911755	0.455878	+	0.890042i	$0.349325\pi$
$822$	0	0
$823$	18.1246	0.631784	0.315892	−	0.948795i	$-0.397696\pi$
$823$	18.1246	0.631784	0.315892	+	0.948795i	$0.397696\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−48.6525	−1.69181	−0.845906	−	0.533332i	$-0.820940\pi$
$827$	−48.6525	−1.69181	−0.845906	+	0.533332i	$0.820940\pi$
$828$	0	0
$829$	13.4508	0.467167	0.233584	−	0.972337i	$-0.424955\pi$
$829$	13.4508	0.467167	0.233584	+	0.972337i	$0.424955\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.01316	0.0351038
$834$	0	0
$835$	7.38197	0.255463
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4.59675	−0.158697	−0.0793487	−	0.996847i	$-0.525284\pi$
$839$	−4.59675	−0.158697	−0.0793487	+	0.996847i	$0.525284\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.00000	0.275208
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.1803	0.383257
$852$	0	0
$853$	13.3607	0.457461	0.228730	−	0.973490i	$-0.426543\pi$
$853$	13.3607	0.457461	0.228730	+	0.973490i	$0.426543\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−29.1803	−0.996781	−0.498391	−	0.866953i	$-0.666075\pi$
$857$	−29.1803	−0.996781	−0.498391	+	0.866953i	$0.666075\pi$
$858$	0	0
$859$	−58.1246	−1.98319	−0.991593	−	0.129395i	$-0.958696\pi$
$859$	−58.1246	−1.98319	−0.991593	+	0.129395i	$0.958696\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18.5836	−0.632593	−0.316296	−	0.948660i	$-0.602439\pi$
$863$	−18.5836	−0.632593	−0.316296	+	0.948660i	$0.602439\pi$
$864$	0	0
$865$	40.9787	1.39332
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−18.5623	−0.628960
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.81966	0.0953219
$876$	0	0
$877$	−23.5836	−0.796361	−0.398181	−	0.917307i	$-0.630358\pi$
$877$	−23.5836	−0.796361	−0.398181	+	0.917307i	$0.630358\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−22.5066	−0.758266	−0.379133	−	0.925342i	$-0.623778\pi$
$881$	−22.5066	−0.758266	−0.379133	+	0.925342i	$0.623778\pi$
$882$	0	0
$883$	−36.2492	−1.21988	−0.609942	−	0.792446i	$-0.708807\pi$
$883$	−36.2492	−1.21988	−0.609942	+	0.792446i	$0.708807\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−28.5279	−0.957872	−0.478936	−	0.877850i	$-0.658977\pi$
$887$	−28.5279	−0.957872	−0.478936	+	0.877850i	$0.658977\pi$
$888$	0	0
$889$	0.403252	0.0135246
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−65.5410	−2.19325
$894$	0	0
$895$	−23.7984	−0.795492
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.76393	0.225590
$900$	0	0
$901$	0.965558	0.0321674
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.09017	0.268926
$906$	0	0
$907$	−4.27051	−0.141800	−0.0709000	−	0.997483i	$-0.522587\pi$
$907$	−4.27051	−0.141800	−0.0709000	+	0.997483i	$0.522587\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−19.8328	−0.657090	−0.328545	−	0.944488i	$-0.606558\pi$
$911$	−19.8328	−0.657090	−0.328545	+	0.944488i	$0.606558\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.506578	0.0167287
$918$	0	0
$919$	1.58359	0.0522379	0.0261189	−	0.999659i	$-0.491685\pi$
$919$	1.58359	0.0522379	0.0261189	+	0.999659i	$0.491685\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	61.6869	2.03045
$924$	0	0
$925$	−5.32624	−0.175126
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−8.23607	−0.270217	−0.135108	−	0.990831i	$-0.543138\pi$
$929$	−8.23607	−0.270217	−0.135108	+	0.990831i	$0.543138\pi$
$930$	0	0
$931$	−47.5967	−1.55992
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−7.65248	−0.249995	−0.124998	−	0.992157i	$-0.539892\pi$
$937$	−7.65248	−0.249995	−0.124998	+	0.992157i	$0.539892\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−17.0902	−0.557124	−0.278562	−	0.960418i	$-0.589858\pi$
$941$	−17.0902	−0.557124	−0.278562	+	0.960418i	$0.589858\pi$
$942$	0	0
$943$	−43.5410	−1.41789
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.9230	1.06985	0.534927	−	0.844899i	$-0.320339\pi$
$947$	32.9230	1.06985	0.534927	+	0.844899i	$0.320339\pi$
$948$	0	0
$949$	66.5410	2.16001
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−60.1803	−1.94943	−0.974716	−	0.223446i	$-0.928269\pi$
$953$	−60.1803	−1.94943	−0.974716	+	0.223446i	$0.928269\pi$
$954$	0	0
$955$	7.32624	0.237071
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.36068	−0.108522
$960$	0	0
$961$	−19.5623	−0.631042
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11.0000	0.354103
$966$	0	0
$967$	−46.1591	−1.48438	−0.742188	−	0.670192i	$-0.766212\pi$
$967$	−46.1591	−1.48438	−0.742188	+	0.670192i	$0.766212\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	44.4721	1.42718	0.713589	−	0.700564i	$-0.247068\pi$
$971$	44.4721	1.42718	0.713589	+	0.700564i	$0.247068\pi$
$972$	0	0
$973$	2.72949	0.0875034
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.8197	0.666080	0.333040	−	0.942913i	$-0.391926\pi$
$977$	20.8197	0.666080	0.333040	+	0.942913i	$0.391926\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	14.3607	0.458035	0.229017	−	0.973422i	$-0.426449\pi$
$983$	14.3607	0.458035	0.229017	+	0.973422i	$0.426449\pi$
$984$	0	0
$985$	5.94427	0.189400
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.6393	0.401907
$990$	0	0
$991$	33.1459	1.05291	0.526457	−	0.850202i	$-0.323520\pi$
$991$	33.1459	1.05291	0.526457	+	0.850202i	$0.323520\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	29.0344	0.920454
$996$	0	0
$997$	−32.4508	−1.02773	−0.513864	−	0.857871i	$-0.671787\pi$
$997$	−32.4508	−1.02773	−0.513864	+	0.857871i	$0.671787\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4356.2.a.w.1.2		2
3.2	odd	2		1452.2.a.m.1.1		2
11.2	odd	10		396.2.j.b.37.1		4
11.6	odd	10		396.2.j.b.289.1		4
11.10	odd	2		4356.2.a.r.1.2		2
12.11	even	2		5808.2.a.bn.1.1		2
33.2	even	10		132.2.i.a.37.1	yes	4
33.5	odd	10		1452.2.i.g.1213.1		4
33.8	even	10		1452.2.i.c.493.1		4
33.14	odd	10		1452.2.i.f.493.1		4
33.17	even	10		132.2.i.a.25.1	✓	4
33.20	odd	10		1452.2.i.g.565.1		4
33.26	odd	10		1452.2.i.f.1237.1		4
33.29	even	10		1452.2.i.c.1237.1		4
33.32	even	2		1452.2.a.l.1.1		2
132.35	odd	10		528.2.y.i.433.1		4
132.83	odd	10		528.2.y.i.289.1		4
132.131	odd	2		5808.2.a.bq.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.2.i.a.25.1	✓	4	33.17	even	10
132.2.i.a.37.1	yes	4	33.2	even	10
396.2.j.b.37.1		4	11.2	odd	10
396.2.j.b.289.1		4	11.6	odd	10
528.2.y.i.289.1		4	132.83	odd	10
528.2.y.i.433.1		4	132.35	odd	10
1452.2.a.l.1.1		2	33.32	even	2
1452.2.a.m.1.1		2	3.2	odd	2
1452.2.i.c.493.1		4	33.8	even	10
1452.2.i.c.1237.1		4	33.29	even	10
1452.2.i.f.493.1		4	33.14	odd	10
1452.2.i.f.1237.1		4	33.26	odd	10
1452.2.i.g.565.1		4	33.20	odd	10
1452.2.i.g.1213.1		4	33.5	odd	10
4356.2.a.r.1.2		2	11.10	odd	2
4356.2.a.w.1.2		2	1.1	even	1	trivial
5808.2.a.bn.1.1		2	12.11	even	2
5808.2.a.bq.1.1		2	132.131	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:	\( N \)	\(=\)	\( 4356 = 2^{2} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4356.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{5} -0.236068 q^{7} +O(q^{10})\)	\(q+1.61803 q^{5} -0.236068 q^{7} +4.23607 q^{13} -0.145898 q^{17} +6.85410 q^{19} +5.00000 q^{23} -2.38197 q^{25} +2.00000 q^{29} +3.38197 q^{31} -0.381966 q^{35} +2.23607 q^{37} -8.70820 q^{41} +2.52786 q^{43} -9.56231 q^{47} -6.94427 q^{49} -6.61803 q^{53} -11.7984 q^{59} +10.8541 q^{61} +6.85410 q^{65} -4.38197 q^{67} +14.5623 q^{71} +15.7082 q^{73} +8.70820 q^{79} -1.76393 q^{83} -0.236068 q^{85} +5.18034 q^{89} -1.00000 q^{91} +11.0902 q^{95} +15.0344 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q + q^5 + 4 * q^7	\( 2 q + q^{5} + 4 q^{7} + 4 q^{13} - 7 q^{17} + 7 q^{19} + 10 q^{23} - 7 q^{25} + 4 q^{29} + 9 q^{31} - 3 q^{35} - 4 q^{41} + 14 q^{43} + q^{47} + 4 q^{49} - 11 q^{53} + q^{59} + 15 q^{61} + 7 q^{65} - 11 q^{67} + 9 q^{71} + 18 q^{73} + 4 q^{79} - 8 q^{83} + 4 q^{85} - 12 q^{89} - 2 q^{91} + 11 q^{95} + q^{97}+O(q^{100}) \) 2 * q + q^5 + 4 * q^7 + 4 * q^13 - 7 * q^17 + 7 * q^19 + 10 * q^23 - 7 * q^25 + 4 * q^29 + 9 * q^31 - 3 * q^35 - 4 * q^41 + 14 * q^43 + q^47 + 4 * q^49 - 11 * q^53 + q^59 + 15 * q^61 + 7 * q^65 - 11 * q^67 + 9 * q^71 + 18 * q^73 + 4 * q^79 - 8 * q^83 + 4 * q^85 - 12 * q^89 - 2 * q^91 + 11 * q^95 + q^97

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