LMFDB - Embedded newform 945.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [945,2,Mod(1,945)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(945, 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("945.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 945 = 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	945.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:	$7.54586299101$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	945.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})$	\(q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +0.618034 q^{10} -0.236068 q^{11} +3.61803 q^{13} -0.618034 q^{14} +1.85410 q^{16} -0.618034 q^{17} +4.09017 q^{19} -1.61803 q^{20} -0.145898 q^{22} +7.61803 q^{23} +1.00000 q^{25} +2.23607 q^{26} +1.61803 q^{28} +10.5623 q^{29} -6.70820 q^{31} +5.61803 q^{32} -0.381966 q^{34} -1.00000 q^{35} -6.70820 q^{37} +2.52786 q^{38} -2.23607 q^{40} -3.09017 q^{41} +7.94427 q^{43} +0.381966 q^{44} +4.70820 q^{46} +11.0000 q^{47} +1.00000 q^{49} +0.618034 q^{50} -5.85410 q^{52} -6.61803 q^{53} -0.236068 q^{55} +2.23607 q^{56} +6.52786 q^{58} +14.7082 q^{59} -7.14590 q^{61} -4.14590 q^{62} -0.236068 q^{64} +3.61803 q^{65} -4.32624 q^{67} +1.00000 q^{68} -0.618034 q^{70} -2.85410 q^{71} -3.94427 q^{73} -4.14590 q^{74} -6.61803 q^{76} +0.236068 q^{77} +3.38197 q^{79} +1.85410 q^{80} -1.90983 q^{82} -2.70820 q^{83} -0.618034 q^{85} +4.90983 q^{86} +0.527864 q^{88} -7.47214 q^{89} -3.61803 q^{91} -12.3262 q^{92} +6.79837 q^{94} +4.09017 q^{95} +12.3262 q^{97} +0.618034 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7	$ 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7} - q^{10} + 4 q^{11} + 5 q^{13} + q^{14} - 3 q^{16} + q^{17} - 3 q^{19} - q^{20} - 7 q^{22} + 13 q^{23} + 2 q^{25} + q^{28} + q^{29} + 9 q^{32} - 3 q^{34} - 2 q^{35} + 14 q^{38} + 5 q^{41} - 2 q^{43} + 3 q^{44} - 4 q^{46} + 22 q^{47} + 2 q^{49} - q^{50} - 5 q^{52} - 11 q^{53} + 4 q^{55} + 22 q^{58} + 16 q^{59} - 21 q^{61} - 15 q^{62} + 4 q^{64} + 5 q^{65} + 7 q^{67} + 2 q^{68} + q^{70} + q^{71} + 10 q^{73} - 15 q^{74} - 11 q^{76} - 4 q^{77} + 9 q^{79} - 3 q^{80} - 15 q^{82} + 8 q^{83} + q^{85} + 21 q^{86} + 10 q^{88} - 6 q^{89} - 5 q^{91} - 9 q^{92} - 11 q^{94} - 3 q^{95} + 9 q^{97} - q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7 - q^10 + 4 * q^11 + 5 * q^13 + q^14 - 3 * q^16 + q^17 - 3 * q^19 - q^20 - 7 * q^22 + 13 * q^23 + 2 * q^25 + q^28 + q^29 + 9 * q^32 - 3 * q^34 - 2 * q^35 + 14 * q^38 + 5 * q^41 - 2 * q^43 + 3 * q^44 - 4 * q^46 + 22 * q^47 + 2 * q^49 - q^50 - 5 * q^52 - 11 * q^53 + 4 * q^55 + 22 * q^58 + 16 * q^59 - 21 * q^61 - 15 * q^62 + 4 * q^64 + 5 * q^65 + 7 * q^67 + 2 * q^68 + q^70 + q^71 + 10 * q^73 - 15 * q^74 - 11 * q^76 - 4 * q^77 + 9 * q^79 - 3 * q^80 - 15 * q^82 + 8 * q^83 + q^85 + 21 * q^86 + 10 * q^88 - 6 * q^89 - 5 * q^91 - 9 * q^92 - 11 * q^94 - 3 * q^95 + 9 * q^97 - q^98

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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.23607	−0.790569
$9$	0	0
$10$	0.618034	0.195440
$11$	−0.236068	−0.0711772	−0.0355886	−	0.999367i	$-0.511331\pi$
$11$	−0.236068	−0.0711772	−0.0355886	+	0.999367i	$0.511331\pi$
$12$	0	0
$13$	3.61803	1.00346	0.501731	−	0.865024i	$-0.332697\pi$
$13$	3.61803	1.00346	0.501731	+	0.865024i	$0.332697\pi$
$14$	−0.618034	−0.165177
$15$	0	0
$16$	1.85410	0.463525
$17$	−0.618034	−0.149895	−0.0749476	−	0.997187i	$-0.523879\pi$
$17$	−0.618034	−0.149895	−0.0749476	+	0.997187i	$0.523879\pi$
$18$	0	0
$19$	4.09017	0.938349	0.469175	−	0.883105i	$-0.344551\pi$
$19$	4.09017	0.938349	0.469175	+	0.883105i	$0.344551\pi$
$20$	−1.61803	−0.361803
$21$	0	0
$22$	−0.145898	−0.0311056
$23$	7.61803	1.58847	0.794235	−	0.607611i	$-0.207872\pi$
$23$	7.61803	1.58847	0.794235	+	0.607611i	$0.207872\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.23607	0.438529
$27$	0	0
$28$	1.61803	0.305780
$29$	10.5623	1.96137	0.980685	−	0.195591i	$-0.0626626\pi$
$29$	10.5623	1.96137	0.980685	+	0.195591i	$0.0626626\pi$
$30$	0	0
$31$	−6.70820	−1.20483	−0.602414	−	0.798183i	$-0.705795\pi$
$31$	−6.70820	−1.20483	−0.602414	+	0.798183i	$0.705795\pi$
$32$	5.61803	0.993137
$33$	0	0
$34$	−0.381966	−0.0655066
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−6.70820	−1.10282	−0.551411	−	0.834234i	$-0.685910\pi$
$37$	−6.70820	−1.10282	−0.551411	+	0.834234i	$0.685910\pi$
$38$	2.52786	0.410074
$39$	0	0
$40$	−2.23607	−0.353553
$41$	−3.09017	−0.482603	−0.241302	−	0.970450i	$-0.577574\pi$
$41$	−3.09017	−0.482603	−0.241302	+	0.970450i	$0.577574\pi$
$42$	0	0
$43$	7.94427	1.21149	0.605745	−	0.795659i	$-0.292875\pi$
$43$	7.94427	1.21149	0.605745	+	0.795659i	$0.292875\pi$
$44$	0.381966	0.0575835
$45$	0	0
$46$	4.70820	0.694187
$47$	11.0000	1.60451	0.802257	−	0.596978i	$-0.203632\pi$
$47$	11.0000	1.60451	0.802257	+	0.596978i	$0.203632\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.618034	0.0874032
$51$	0	0
$52$	−5.85410	−0.811818
$53$	−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
$55$	−0.236068	−0.0318314
$56$	2.23607	0.298807
$57$	0	0
$58$	6.52786	0.857151
$59$	14.7082	1.91485	0.957423	−	0.288690i	$-0.0932198\pi$
$59$	14.7082	1.91485	0.957423	+	0.288690i	$0.0932198\pi$
$60$	0	0
$61$	−7.14590	−0.914938	−0.457469	−	0.889225i	$-0.651244\pi$
$61$	−7.14590	−0.914938	−0.457469	+	0.889225i	$0.651244\pi$
$62$	−4.14590	−0.526530
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	3.61803	0.448762
$66$	0	0
$67$	−4.32624	−0.528534	−0.264267	−	0.964450i	$-0.585130\pi$
$67$	−4.32624	−0.528534	−0.264267	+	0.964450i	$0.585130\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	−0.618034	−0.0738692
$71$	−2.85410	−0.338720	−0.169360	−	0.985554i	$-0.554170\pi$
$71$	−2.85410	−0.338720	−0.169360	+	0.985554i	$0.554170\pi$
$72$	0	0
$73$	−3.94427	−0.461642	−0.230821	−	0.972996i	$-0.574141\pi$
$73$	−3.94427	−0.461642	−0.230821	+	0.972996i	$0.574141\pi$
$74$	−4.14590	−0.481951
$75$	0	0
$76$	−6.61803	−0.759141
$77$	0.236068	0.0269024
$78$	0	0
$79$	3.38197	0.380501	0.190250	−	0.981736i	$-0.439070\pi$
$79$	3.38197	0.380501	0.190250	+	0.981736i	$0.439070\pi$
$80$	1.85410	0.207295
$81$	0	0
$82$	−1.90983	−0.210905
$83$	−2.70820	−0.297264	−0.148632	−	0.988893i	$-0.547487\pi$
$83$	−2.70820	−0.297264	−0.148632	+	0.988893i	$0.547487\pi$
$84$	0	0
$85$	−0.618034	−0.0670352
$86$	4.90983	0.529441
$87$	0	0
$88$	0.527864	0.0562705
$89$	−7.47214	−0.792045	−0.396022	−	0.918241i	$-0.629610\pi$
$89$	−7.47214	−0.792045	−0.396022	+	0.918241i	$0.629610\pi$
$90$	0	0
$91$	−3.61803	−0.379273
$92$	−12.3262	−1.28510
$93$	0	0
$94$	6.79837	0.701199
$95$	4.09017	0.419643
$96$	0	0
$97$	12.3262	1.25154	0.625770	−	0.780008i	$-0.284785\pi$
$97$	12.3262	1.25154	0.625770	+	0.780008i	$0.284785\pi$
$98$	0.618034	0.0624309
$99$	0	0
$100$	−1.61803	−0.161803
$101$	14.4721	1.44003	0.720016	−	0.693958i	$-0.244135\pi$
$101$	14.4721	1.44003	0.720016	+	0.693958i	$0.244135\pi$
$102$	0	0
$103$	8.38197	0.825900	0.412950	−	0.910754i	$-0.364498\pi$
$103$	8.38197	0.825900	0.412950	+	0.910754i	$0.364498\pi$
$104$	−8.09017	−0.793306
$105$	0	0
$106$	−4.09017	−0.397272
$107$	10.7082	1.03520	0.517601	−	0.855622i	$-0.326825\pi$
$107$	10.7082	1.03520	0.517601	+	0.855622i	$0.326825\pi$
$108$	0	0
$109$	−7.56231	−0.724338	−0.362169	−	0.932113i	$-0.617964\pi$
$109$	−7.56231	−0.724338	−0.362169	+	0.932113i	$0.617964\pi$
$110$	−0.145898	−0.0139108
$111$	0	0
$112$	−1.85410	−0.175196
$113$	−0.854102	−0.0803472	−0.0401736	−	0.999193i	$-0.512791\pi$
$113$	−0.854102	−0.0803472	−0.0401736	+	0.999193i	$0.512791\pi$
$114$	0	0
$115$	7.61803	0.710385
$116$	−17.0902	−1.58678
$117$	0	0
$118$	9.09017	0.836818
$119$	0.618034	0.0566551
$120$	0	0
$121$	−10.9443	−0.994934
$122$	−4.41641	−0.399843
$123$	0	0
$124$	10.8541	0.974727
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.0902	−0.984093	−0.492047	−	0.870569i	$-0.663751\pi$
$127$	−11.0902	−0.984093	−0.492047	+	0.870569i	$0.663751\pi$
$128$	−11.3820	−1.00603
$129$	0	0
$130$	2.23607	0.196116
$131$	6.61803	0.578220	0.289110	−	0.957296i	$-0.406641\pi$
$131$	6.61803	0.578220	0.289110	+	0.957296i	$0.406641\pi$
$132$	0	0
$133$	−4.09017	−0.354663
$134$	−2.67376	−0.230978
$135$	0	0
$136$	1.38197	0.118503
$137$	−21.8885	−1.87006	−0.935032	−	0.354563i	$-0.884630\pi$
$137$	−21.8885	−1.87006	−0.935032	+	0.354563i	$0.884630\pi$
$138$	0	0
$139$	−19.1803	−1.62686	−0.813428	−	0.581666i	$-0.802401\pi$
$139$	−19.1803	−1.62686	−0.813428	+	0.581666i	$0.802401\pi$
$140$	1.61803	0.136749
$141$	0	0
$142$	−1.76393	−0.148026
$143$	−0.854102	−0.0714236
$144$	0	0
$145$	10.5623	0.877152
$146$	−2.43769	−0.201745
$147$	0	0
$148$	10.8541	0.892202
$149$	8.79837	0.720791	0.360395	−	0.932800i	$-0.382642\pi$
$149$	8.79837	0.720791	0.360395	+	0.932800i	$0.382642\pi$
$150$	0	0
$151$	−17.9443	−1.46028	−0.730142	−	0.683295i	$-0.760546\pi$
$151$	−17.9443	−1.46028	−0.730142	+	0.683295i	$0.760546\pi$
$152$	−9.14590	−0.741830
$153$	0	0
$154$	0.145898	0.0117568
$155$	−6.70820	−0.538816
$156$	0	0
$157$	−0.763932	−0.0609684	−0.0304842	−	0.999535i	$-0.509705\pi$
$157$	−0.763932	−0.0609684	−0.0304842	+	0.999535i	$0.509705\pi$
$158$	2.09017	0.166285
$159$	0	0
$160$	5.61803	0.444145
$161$	−7.61803	−0.600385
$162$	0	0
$163$	14.9443	1.17053	0.585263	−	0.810844i	$-0.300991\pi$
$163$	14.9443	1.17053	0.585263	+	0.810844i	$0.300991\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	−1.67376	−0.129909
$167$	1.47214	0.113917	0.0569587	−	0.998377i	$-0.481860\pi$
$167$	1.47214	0.113917	0.0569587	+	0.998377i	$0.481860\pi$
$168$	0	0
$169$	0.0901699	0.00693615
$170$	−0.381966	−0.0292955
$171$	0	0
$172$	−12.8541	−0.980116
$173$	2.52786	0.192190	0.0960950	−	0.995372i	$-0.469365\pi$
$173$	2.52786	0.192190	0.0960950	+	0.995372i	$0.469365\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−0.437694	−0.0329924
$177$	0	0
$178$	−4.61803	−0.346136
$179$	−7.00000	−0.523205	−0.261602	−	0.965176i	$-0.584251\pi$
$179$	−7.00000	−0.523205	−0.261602	+	0.965176i	$0.584251\pi$
$180$	0	0
$181$	−10.5279	−0.782530	−0.391265	−	0.920278i	$-0.627962\pi$
$181$	−10.5279	−0.782530	−0.391265	+	0.920278i	$0.627962\pi$
$182$	−2.23607	−0.165748
$183$	0	0
$184$	−17.0344	−1.25580
$185$	−6.70820	−0.493197
$186$	0	0
$187$	0.145898	0.0106691
$188$	−17.7984	−1.29808
$189$	0	0
$190$	2.52786	0.183391
$191$	−5.79837	−0.419556	−0.209778	−	0.977749i	$-0.567274\pi$
$191$	−5.79837	−0.419556	−0.209778	+	0.977749i	$0.567274\pi$
$192$	0	0
$193$	5.85410	0.421387	0.210694	−	0.977552i	$-0.432428\pi$
$193$	5.85410	0.421387	0.210694	+	0.977552i	$0.432428\pi$
$194$	7.61803	0.546943
$195$	0	0
$196$	−1.61803	−0.115574
$197$	−10.5279	−0.750079	−0.375040	−	0.927009i	$-0.622371\pi$
$197$	−10.5279	−0.750079	−0.375040	+	0.927009i	$0.622371\pi$
$198$	0	0
$199$	−0.618034	−0.0438113	−0.0219056	−	0.999760i	$-0.506973\pi$
$199$	−0.618034	−0.0438113	−0.0219056	+	0.999760i	$0.506973\pi$
$200$	−2.23607	−0.158114
$201$	0	0
$202$	8.94427	0.629317
$203$	−10.5623	−0.741329
$204$	0	0
$205$	−3.09017	−0.215827
$206$	5.18034	0.360931
$207$	0	0
$208$	6.70820	0.465130
$209$	−0.965558	−0.0667891
$210$	0	0
$211$	23.1246	1.59196	0.795982	−	0.605320i	$-0.206955\pi$
$211$	23.1246	1.59196	0.795982	+	0.605320i	$0.206955\pi$
$212$	10.7082	0.735442
$213$	0	0
$214$	6.61803	0.452399
$215$	7.94427	0.541795
$216$	0	0
$217$	6.70820	0.455383
$218$	−4.67376	−0.316547
$219$	0	0
$220$	0.381966	0.0257521
$221$	−2.23607	−0.150414
$222$	0	0
$223$	−1.47214	−0.0985815	−0.0492908	−	0.998784i	$-0.515696\pi$
$223$	−1.47214	−0.0985815	−0.0492908	+	0.998784i	$0.515696\pi$
$224$	−5.61803	−0.375371
$225$	0	0
$226$	−0.527864	−0.0351130
$227$	27.5623	1.82937	0.914687	−	0.404162i	$-0.132437\pi$
$227$	27.5623	1.82937	0.914687	+	0.404162i	$0.132437\pi$
$228$	0	0
$229$	−0.0557281	−0.00368262	−0.00184131	−	0.999998i	$-0.500586\pi$
$229$	−0.0557281	−0.00368262	−0.00184131	+	0.999998i	$0.500586\pi$
$230$	4.70820	0.310450
$231$	0	0
$232$	−23.6180	−1.55060
$233$	5.90983	0.387166	0.193583	−	0.981084i	$-0.437989\pi$
$233$	5.90983	0.387166	0.193583	+	0.981084i	$0.437989\pi$
$234$	0	0
$235$	11.0000	0.717561
$236$	−23.7984	−1.54914
$237$	0	0
$238$	0.381966	0.0247592
$239$	−19.6525	−1.27121	−0.635606	−	0.772013i	$-0.719250\pi$
$239$	−19.6525	−1.27121	−0.635606	+	0.772013i	$0.719250\pi$
$240$	0	0
$241$	−5.38197	−0.346683	−0.173341	−	0.984862i	$-0.555456\pi$
$241$	−5.38197	−0.346683	−0.173341	+	0.984862i	$0.555456\pi$
$242$	−6.76393	−0.434802
$243$	0	0
$244$	11.5623	0.740201
$245$	1.00000	0.0638877
$246$	0	0
$247$	14.7984	0.941598
$248$	15.0000	0.952501
$249$	0	0
$250$	0.618034	0.0390879
$251$	−3.23607	−0.204259	−0.102129	−	0.994771i	$-0.532566\pi$
$251$	−3.23607	−0.204259	−0.102129	+	0.994771i	$0.532566\pi$
$252$	0	0
$253$	−1.79837	−0.113063
$254$	−6.85410	−0.430065
$255$	0	0
$256$	−6.56231	−0.410144
$257$	8.94427	0.557928	0.278964	−	0.960302i	$-0.410009\pi$
$257$	8.94427	0.557928	0.278964	+	0.960302i	$0.410009\pi$
$258$	0	0
$259$	6.70820	0.416828
$260$	−5.85410	−0.363056
$261$	0	0
$262$	4.09017	0.252692
$263$	11.7426	0.724083	0.362041	−	0.932162i	$-0.382080\pi$
$263$	11.7426	0.724083	0.362041	+	0.932162i	$0.382080\pi$
$264$	0	0
$265$	−6.61803	−0.406543
$266$	−2.52786	−0.154993
$267$	0	0
$268$	7.00000	0.427593
$269$	−2.70820	−0.165122	−0.0825611	−	0.996586i	$-0.526310\pi$
$269$	−2.70820	−0.165122	−0.0825611	+	0.996586i	$0.526310\pi$
$270$	0	0
$271$	−4.43769	−0.269571	−0.134785	−	0.990875i	$-0.543034\pi$
$271$	−4.43769	−0.269571	−0.134785	+	0.990875i	$0.543034\pi$
$272$	−1.14590	−0.0694803
$273$	0	0
$274$	−13.5279	−0.817248
$275$	−0.236068	−0.0142354
$276$	0	0
$277$	20.3262	1.22129	0.610643	−	0.791906i	$-0.290911\pi$
$277$	20.3262	1.22129	0.610643	+	0.791906i	$0.290911\pi$
$278$	−11.8541	−0.710962
$279$	0	0
$280$	2.23607	0.133631
$281$	−11.3820	−0.678991	−0.339496	−	0.940608i	$-0.610256\pi$
$281$	−11.3820	−0.678991	−0.339496	+	0.940608i	$0.610256\pi$
$282$	0	0
$283$	27.8541	1.65575	0.827877	−	0.560909i	$-0.189548\pi$
$283$	27.8541	1.65575	0.827877	+	0.560909i	$0.189548\pi$
$284$	4.61803	0.274030
$285$	0	0
$286$	−0.527864	−0.0312133
$287$	3.09017	0.182407
$288$	0	0
$289$	−16.6180	−0.977531
$290$	6.52786	0.383329
$291$	0	0
$292$	6.38197	0.373476
$293$	−19.7082	−1.15137	−0.575683	−	0.817673i	$-0.695264\pi$
$293$	−19.7082	−1.15137	−0.575683	+	0.817673i	$0.695264\pi$
$294$	0	0
$295$	14.7082	0.856345
$296$	15.0000	0.871857
$297$	0	0
$298$	5.43769	0.314997
$299$	27.5623	1.59397
$300$	0	0
$301$	−7.94427	−0.457900
$302$	−11.0902	−0.638168
$303$	0	0
$304$	7.58359	0.434949
$305$	−7.14590	−0.409173
$306$	0	0
$307$	−7.41641	−0.423277	−0.211638	−	0.977348i	$-0.567880\pi$
$307$	−7.41641	−0.423277	−0.211638	+	0.977348i	$0.567880\pi$
$308$	−0.381966	−0.0217645
$309$	0	0
$310$	−4.14590	−0.235471
$311$	−1.32624	−0.0752041	−0.0376020	−	0.999293i	$-0.511972\pi$
$311$	−1.32624	−0.0752041	−0.0376020	+	0.999293i	$0.511972\pi$
$312$	0	0
$313$	−0.819660	−0.0463299	−0.0231650	−	0.999732i	$-0.507374\pi$
$313$	−0.819660	−0.0463299	−0.0231650	+	0.999732i	$0.507374\pi$
$314$	−0.472136	−0.0266442
$315$	0	0
$316$	−5.47214	−0.307832
$317$	−23.9443	−1.34484	−0.672422	−	0.740168i	$-0.734746\pi$
$317$	−23.9443	−1.34484	−0.672422	+	0.740168i	$0.734746\pi$
$318$	0	0
$319$	−2.49342	−0.139605
$320$	−0.236068	−0.0131966
$321$	0	0
$322$	−4.70820	−0.262378
$323$	−2.52786	−0.140654
$324$	0	0
$325$	3.61803	0.200692
$326$	9.23607	0.511538
$327$	0	0
$328$	6.90983	0.381532
$329$	−11.0000	−0.606450
$330$	0	0
$331$	−25.1459	−1.38214	−0.691072	−	0.722786i	$-0.742861\pi$
$331$	−25.1459	−1.38214	−0.691072	+	0.722786i	$0.742861\pi$
$332$	4.38197	0.240492
$333$	0	0
$334$	0.909830	0.0497837
$335$	−4.32624	−0.236368
$336$	0	0
$337$	10.7426	0.585189	0.292595	−	0.956237i	$-0.405481\pi$
$337$	10.7426	0.585189	0.292595	+	0.956237i	$0.405481\pi$
$338$	0.0557281	0.00303121
$339$	0	0
$340$	1.00000	0.0542326
$341$	1.58359	0.0857563
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−17.7639	−0.957767
$345$	0	0
$346$	1.56231	0.0839901
$347$	−17.2361	−0.925281	−0.462640	−	0.886546i	$-0.653098\pi$
$347$	−17.2361	−0.925281	−0.462640	+	0.886546i	$0.653098\pi$
$348$	0	0
$349$	−13.3607	−0.715181	−0.357590	−	0.933879i	$-0.616402\pi$
$349$	−13.3607	−0.715181	−0.357590	+	0.933879i	$0.616402\pi$
$350$	−0.618034	−0.0330353
$351$	0	0
$352$	−1.32624	−0.0706887
$353$	27.3262	1.45443	0.727214	−	0.686410i	$-0.240815\pi$
$353$	27.3262	1.45443	0.727214	+	0.686410i	$0.240815\pi$
$354$	0	0
$355$	−2.85410	−0.151480
$356$	12.0902	0.640778
$357$	0	0
$358$	−4.32624	−0.228649
$359$	9.58359	0.505803	0.252901	−	0.967492i	$-0.418615\pi$
$359$	9.58359	0.505803	0.252901	+	0.967492i	$0.418615\pi$
$360$	0	0
$361$	−2.27051	−0.119501
$362$	−6.50658	−0.341978
$363$	0	0
$364$	5.85410	0.306838
$365$	−3.94427	−0.206453
$366$	0	0
$367$	−8.14590	−0.425212	−0.212606	−	0.977138i	$-0.568195\pi$
$367$	−8.14590	−0.425212	−0.212606	+	0.977138i	$0.568195\pi$
$368$	14.1246	0.736296
$369$	0	0
$370$	−4.14590	−0.215535
$371$	6.61803	0.343591
$372$	0	0
$373$	29.9787	1.55224	0.776119	−	0.630586i	$-0.217185\pi$
$373$	29.9787	1.55224	0.776119	+	0.630586i	$0.217185\pi$
$374$	0.0901699	0.00466258
$375$	0	0
$376$	−24.5967	−1.26848
$377$	38.2148	1.96816
$378$	0	0
$379$	−23.8885	−1.22707	−0.613536	−	0.789667i	$-0.710253\pi$
$379$	−23.8885	−1.22707	−0.613536	+	0.789667i	$0.710253\pi$
$380$	−6.61803	−0.339498
$381$	0	0
$382$	−3.58359	−0.183353
$383$	19.6525	1.00419	0.502097	−	0.864811i	$-0.332562\pi$
$383$	19.6525	1.00419	0.502097	+	0.864811i	$0.332562\pi$
$384$	0	0
$385$	0.236068	0.0120311
$386$	3.61803	0.184153
$387$	0	0
$388$	−19.9443	−1.01252
$389$	1.20163	0.0609249	0.0304624	−	0.999536i	$-0.490302\pi$
$389$	1.20163	0.0609249	0.0304624	+	0.999536i	$0.490302\pi$
$390$	0	0
$391$	−4.70820	−0.238104
$392$	−2.23607	−0.112938
$393$	0	0
$394$	−6.50658	−0.327797
$395$	3.38197	0.170165
$396$	0	0
$397$	26.9443	1.35229	0.676147	−	0.736767i	$-0.263648\pi$
$397$	26.9443	1.35229	0.676147	+	0.736767i	$0.263648\pi$
$398$	−0.381966	−0.0191462
$399$	0	0
$400$	1.85410	0.0927051
$401$	−19.6180	−0.979678	−0.489839	−	0.871813i	$-0.662944\pi$
$401$	−19.6180	−0.979678	−0.489839	+	0.871813i	$0.662944\pi$
$402$	0	0
$403$	−24.2705	−1.20900
$404$	−23.4164	−1.16501
$405$	0	0
$406$	−6.52786	−0.323972
$407$	1.58359	0.0784957
$408$	0	0
$409$	−30.9443	−1.53010	−0.765048	−	0.643973i	$-0.777285\pi$
$409$	−30.9443	−1.53010	−0.765048	+	0.643973i	$0.777285\pi$
$410$	−1.90983	−0.0943198
$411$	0	0
$412$	−13.5623	−0.668167
$413$	−14.7082	−0.723743
$414$	0	0
$415$	−2.70820	−0.132941
$416$	20.3262	0.996576
$417$	0	0
$418$	−0.596748	−0.0291879
$419$	−1.47214	−0.0719185	−0.0359593	−	0.999353i	$-0.511449\pi$
$419$	−1.47214	−0.0719185	−0.0359593	+	0.999353i	$0.511449\pi$
$420$	0	0
$421$	28.5623	1.39204	0.696021	−	0.718022i	$-0.254952\pi$
$421$	28.5623	1.39204	0.696021	+	0.718022i	$0.254952\pi$
$422$	14.2918	0.695714
$423$	0	0
$424$	14.7984	0.718673
$425$	−0.618034	−0.0299791
$426$	0	0
$427$	7.14590	0.345814
$428$	−17.3262	−0.837495
$429$	0	0
$430$	4.90983	0.236773
$431$	−25.4508	−1.22592	−0.612962	−	0.790112i	$-0.710022\pi$
$431$	−25.4508	−1.22592	−0.612962	+	0.790112i	$0.710022\pi$
$432$	0	0
$433$	30.2148	1.45203	0.726015	−	0.687679i	$-0.241370\pi$
$433$	30.2148	1.45203	0.726015	+	0.687679i	$0.241370\pi$
$434$	4.14590	0.199009
$435$	0	0
$436$	12.2361	0.586001
$437$	31.1591	1.49054
$438$	0	0
$439$	24.7082	1.17926	0.589629	−	0.807674i	$-0.299274\pi$
$439$	24.7082	1.17926	0.589629	+	0.807674i	$0.299274\pi$
$440$	0.527864	0.0251649
$441$	0	0
$442$	−1.38197	−0.0657334
$443$	19.6869	0.935354	0.467677	−	0.883900i	$-0.345091\pi$
$443$	19.6869	0.935354	0.467677	+	0.883900i	$0.345091\pi$
$444$	0	0
$445$	−7.47214	−0.354213
$446$	−0.909830	−0.0430817
$447$	0	0
$448$	0.236068	0.0111532
$449$	11.5279	0.544034	0.272017	−	0.962293i	$-0.412309\pi$
$449$	11.5279	0.544034	0.272017	+	0.962293i	$0.412309\pi$
$450$	0	0
$451$	0.729490	0.0343504
$452$	1.38197	0.0650022
$453$	0	0
$454$	17.0344	0.799466
$455$	−3.61803	−0.169616
$456$	0	0
$457$	1.67376	0.0782953	0.0391476	−	0.999233i	$-0.487536\pi$
$457$	1.67376	0.0782953	0.0391476	+	0.999233i	$0.487536\pi$
$458$	−0.0344419	−0.00160936
$459$	0	0
$460$	−12.3262	−0.574714
$461$	1.79837	0.0837586	0.0418793	−	0.999123i	$-0.486666\pi$
$461$	1.79837	0.0837586	0.0418793	+	0.999123i	$0.486666\pi$
$462$	0	0
$463$	1.61803	0.0751964	0.0375982	−	0.999293i	$-0.488029\pi$
$463$	1.61803	0.0751964	0.0375982	+	0.999293i	$0.488029\pi$
$464$	19.5836	0.909145
$465$	0	0
$466$	3.65248	0.169198
$467$	−0.944272	−0.0436957	−0.0218478	−	0.999761i	$-0.506955\pi$
$467$	−0.944272	−0.0436957	−0.0218478	+	0.999761i	$0.506955\pi$
$468$	0	0
$469$	4.32624	0.199767
$470$	6.79837	0.313586
$471$	0	0
$472$	−32.8885	−1.51382
$473$	−1.87539	−0.0862304
$474$	0	0
$475$	4.09017	0.187670
$476$	−1.00000	−0.0458349
$477$	0	0
$478$	−12.1459	−0.555540
$479$	37.0902	1.69469	0.847347	−	0.531040i	$-0.178199\pi$
$479$	37.0902	1.69469	0.847347	+	0.531040i	$0.178199\pi$
$480$	0	0
$481$	−24.2705	−1.10664
$482$	−3.32624	−0.151506
$483$	0	0
$484$	17.7082	0.804918
$485$	12.3262	0.559706
$486$	0	0
$487$	−24.4721	−1.10894	−0.554469	−	0.832204i	$-0.687079\pi$
$487$	−24.4721	−1.10894	−0.554469	+	0.832204i	$0.687079\pi$
$488$	15.9787	0.723322
$489$	0	0
$490$	0.618034	0.0279199
$491$	18.5066	0.835190	0.417595	−	0.908633i	$-0.362873\pi$
$491$	18.5066	0.835190	0.417595	+	0.908633i	$0.362873\pi$
$492$	0	0
$493$	−6.52786	−0.294000
$494$	9.14590	0.411493
$495$	0	0
$496$	−12.4377	−0.558469
$497$	2.85410	0.128024
$498$	0	0
$499$	−41.3607	−1.85156	−0.925779	−	0.378065i	$-0.876590\pi$
$499$	−41.3607	−1.85156	−0.925779	+	0.378065i	$0.876590\pi$
$500$	−1.61803	−0.0723607
$501$	0	0
$502$	−2.00000	−0.0892644
$503$	42.0902	1.87671	0.938354	−	0.345676i	$-0.112350\pi$
$503$	42.0902	1.87671	0.938354	+	0.345676i	$0.112350\pi$
$504$	0	0
$505$	14.4721	0.644002
$506$	−1.11146	−0.0494103
$507$	0	0
$508$	17.9443	0.796148
$509$	−30.3050	−1.34324	−0.671622	−	0.740894i	$-0.734402\pi$
$509$	−30.3050	−1.34324	−0.671622	+	0.740894i	$0.734402\pi$
$510$	0	0
$511$	3.94427	0.174484
$512$	18.7082	0.826794
$513$	0	0
$514$	5.52786	0.243824
$515$	8.38197	0.369354
$516$	0	0
$517$	−2.59675	−0.114205
$518$	4.14590	0.182160
$519$	0	0
$520$	−8.09017	−0.354777
$521$	33.8328	1.48224	0.741121	−	0.671371i	$-0.234294\pi$
$521$	33.8328	1.48224	0.741121	+	0.671371i	$0.234294\pi$
$522$	0	0
$523$	−25.7984	−1.12808	−0.564042	−	0.825746i	$-0.690754\pi$
$523$	−25.7984	−1.12808	−0.564042	+	0.825746i	$0.690754\pi$
$524$	−10.7082	−0.467790
$525$	0	0
$526$	7.25735	0.316436
$527$	4.14590	0.180598
$528$	0	0
$529$	35.0344	1.52324
$530$	−4.09017	−0.177666
$531$	0	0
$532$	6.61803	0.286928
$533$	−11.1803	−0.484274
$534$	0	0
$535$	10.7082	0.462956
$536$	9.67376	0.417843
$537$	0	0
$538$	−1.67376	−0.0721610
$539$	−0.236068	−0.0101682
$540$	0	0
$541$	24.5066	1.05362	0.526810	−	0.849983i	$-0.323388\pi$
$541$	24.5066	1.05362	0.526810	+	0.849983i	$0.323388\pi$
$542$	−2.74265	−0.117807
$543$	0	0
$544$	−3.47214	−0.148867
$545$	−7.56231	−0.323934
$546$	0	0
$547$	−25.3607	−1.08434	−0.542172	−	0.840267i	$-0.682398\pi$
$547$	−25.3607	−1.08434	−0.542172	+	0.840267i	$0.682398\pi$
$548$	35.4164	1.51291
$549$	0	0
$550$	−0.145898	−0.00622111
$551$	43.2016	1.84045
$552$	0	0
$553$	−3.38197	−0.143816
$554$	12.5623	0.533721
$555$	0	0
$556$	31.0344	1.31615
$557$	7.79837	0.330428	0.165214	−	0.986258i	$-0.447169\pi$
$557$	7.79837	0.330428	0.165214	+	0.986258i	$0.447169\pi$
$558$	0	0
$559$	28.7426	1.21568
$560$	−1.85410	−0.0783501
$561$	0	0
$562$	−7.03444	−0.296730
$563$	−38.6869	−1.63046	−0.815230	−	0.579138i	$-0.803389\pi$
$563$	−38.6869	−1.63046	−0.815230	+	0.579138i	$0.803389\pi$
$564$	0	0
$565$	−0.854102	−0.0359323
$566$	17.2148	0.723591
$567$	0	0
$568$	6.38197	0.267781
$569$	−42.2361	−1.77063	−0.885314	−	0.464994i	$-0.846057\pi$
$569$	−42.2361	−1.77063	−0.885314	+	0.464994i	$0.846057\pi$
$570$	0	0
$571$	43.7426	1.83057	0.915286	−	0.402804i	$-0.131964\pi$
$571$	43.7426	1.83057	0.915286	+	0.402804i	$0.131964\pi$
$572$	1.38197	0.0577829
$573$	0	0
$574$	1.90983	0.0797148
$575$	7.61803	0.317694
$576$	0	0
$577$	−18.1246	−0.754537	−0.377269	−	0.926104i	$-0.623137\pi$
$577$	−18.1246	−0.754537	−0.377269	+	0.926104i	$0.623137\pi$
$578$	−10.2705	−0.427197
$579$	0	0
$580$	−17.0902	−0.709631
$581$	2.70820	0.112355
$582$	0	0
$583$	1.56231	0.0647041
$584$	8.81966	0.364960
$585$	0	0
$586$	−12.1803	−0.503165
$587$	−31.7984	−1.31246	−0.656230	−	0.754561i	$-0.727850\pi$
$587$	−31.7984	−1.31246	−0.656230	+	0.754561i	$0.727850\pi$
$588$	0	0
$589$	−27.4377	−1.13055
$590$	9.09017	0.374236
$591$	0	0
$592$	−12.4377	−0.511186
$593$	28.3050	1.16235	0.581173	−	0.813780i	$-0.302594\pi$
$593$	28.3050	1.16235	0.581173	+	0.813780i	$0.302594\pi$
$594$	0	0
$595$	0.618034	0.0253369
$596$	−14.2361	−0.583132
$597$	0	0
$598$	17.0344	0.696590
$599$	−21.0557	−0.860314	−0.430157	−	0.902754i	$-0.641542\pi$
$599$	−21.0557	−0.860314	−0.430157	+	0.902754i	$0.641542\pi$
$600$	0	0
$601$	−23.7984	−0.970756	−0.485378	−	0.874304i	$-0.661318\pi$
$601$	−23.7984	−0.970756	−0.485378	+	0.874304i	$0.661318\pi$
$602$	−4.90983	−0.200110
$603$	0	0
$604$	29.0344	1.18139
$605$	−10.9443	−0.444948
$606$	0	0
$607$	−2.41641	−0.0980790	−0.0490395	−	0.998797i	$-0.515616\pi$
$607$	−2.41641	−0.0980790	−0.0490395	+	0.998797i	$0.515616\pi$
$608$	22.9787	0.931910
$609$	0	0
$610$	−4.41641	−0.178815
$611$	39.7984	1.61007
$612$	0	0
$613$	−20.7082	−0.836396	−0.418198	−	0.908356i	$-0.637338\pi$
$613$	−20.7082	−0.836396	−0.418198	+	0.908356i	$0.637338\pi$
$614$	−4.58359	−0.184979
$615$	0	0
$616$	−0.527864	−0.0212682
$617$	−13.2016	−0.531477	−0.265739	−	0.964045i	$-0.585616\pi$
$617$	−13.2016	−0.531477	−0.265739	+	0.964045i	$0.585616\pi$
$618$	0	0
$619$	19.5410	0.785420	0.392710	−	0.919662i	$-0.371538\pi$
$619$	19.5410	0.785420	0.392710	+	0.919662i	$0.371538\pi$
$620$	10.8541	0.435911
$621$	0	0
$622$	−0.819660	−0.0328654
$623$	7.47214	0.299365
$624$	0	0
$625$	1.00000	0.0400000
$626$	−0.506578	−0.0202469
$627$	0	0
$628$	1.23607	0.0493245
$629$	4.14590	0.165308
$630$	0	0
$631$	6.11146	0.243293	0.121647	−	0.992573i	$-0.461183\pi$
$631$	6.11146	0.243293	0.121647	+	0.992573i	$0.461183\pi$
$632$	−7.56231	−0.300812
$633$	0	0
$634$	−14.7984	−0.587719
$635$	−11.0902	−0.440100
$636$	0	0
$637$	3.61803	0.143352
$638$	−1.54102	−0.0610096
$639$	0	0
$640$	−11.3820	−0.449912
$641$	−34.2705	−1.35360	−0.676802	−	0.736165i	$-0.736635\pi$
$641$	−34.2705	−1.35360	−0.676802	+	0.736165i	$0.736635\pi$
$642$	0	0
$643$	−0.742646	−0.0292871	−0.0146435	−	0.999893i	$-0.504661\pi$
$643$	−0.742646	−0.0292871	−0.0146435	+	0.999893i	$0.504661\pi$
$644$	12.3262	0.485722
$645$	0	0
$646$	−1.56231	−0.0614681
$647$	−30.9443	−1.21654	−0.608272	−	0.793728i	$-0.708137\pi$
$647$	−30.9443	−1.21654	−0.608272	+	0.793728i	$0.708137\pi$
$648$	0	0
$649$	−3.47214	−0.136293
$650$	2.23607	0.0877058
$651$	0	0
$652$	−24.1803	−0.946975
$653$	−23.0344	−0.901407	−0.450704	−	0.892674i	$-0.648827\pi$
$653$	−23.0344	−0.901407	−0.450704	+	0.892674i	$0.648827\pi$
$654$	0	0
$655$	6.61803	0.258588
$656$	−5.72949	−0.223699
$657$	0	0
$658$	−6.79837	−0.265028
$659$	−5.23607	−0.203968	−0.101984	−	0.994786i	$-0.532519\pi$
$659$	−5.23607	−0.203968	−0.101984	+	0.994786i	$0.532519\pi$
$660$	0	0
$661$	35.2148	1.36970	0.684848	−	0.728686i	$-0.259869\pi$
$661$	35.2148	1.36970	0.684848	+	0.728686i	$0.259869\pi$
$662$	−15.5410	−0.604019
$663$	0	0
$664$	6.05573	0.235008
$665$	−4.09017	−0.158610
$666$	0	0
$667$	80.4640	3.11558
$668$	−2.38197	−0.0921610
$669$	0	0
$670$	−2.67376	−0.103296
$671$	1.68692	0.0651227
$672$	0	0
$673$	−37.5279	−1.44659	−0.723296	−	0.690538i	$-0.757374\pi$
$673$	−37.5279	−1.44659	−0.723296	+	0.690538i	$0.757374\pi$
$674$	6.63932	0.255737
$675$	0	0
$676$	−0.145898	−0.00561146
$677$	13.6525	0.524707	0.262354	−	0.964972i	$-0.415501\pi$
$677$	13.6525	0.524707	0.262354	+	0.964972i	$0.415501\pi$
$678$	0	0
$679$	−12.3262	−0.473038
$680$	1.38197	0.0529960
$681$	0	0
$682$	0.978714	0.0374769
$683$	3.97871	0.152241	0.0761206	−	0.997099i	$-0.475747\pi$
$683$	3.97871	0.152241	0.0761206	+	0.997099i	$0.475747\pi$
$684$	0	0
$685$	−21.8885	−0.836318
$686$	−0.618034	−0.0235966
$687$	0	0
$688$	14.7295	0.561557
$689$	−23.9443	−0.912204
$690$	0	0
$691$	−39.4721	−1.50159	−0.750795	−	0.660535i	$-0.770330\pi$
$691$	−39.4721	−1.50159	−0.750795	+	0.660535i	$0.770330\pi$
$692$	−4.09017	−0.155485
$693$	0	0
$694$	−10.6525	−0.404362
$695$	−19.1803	−0.727552
$696$	0	0
$697$	1.90983	0.0723400
$698$	−8.25735	−0.312545
$699$	0	0
$700$	1.61803	0.0611559
$701$	−28.8885	−1.09111	−0.545553	−	0.838077i	$-0.683680\pi$
$701$	−28.8885	−1.09111	−0.545553	+	0.838077i	$0.683680\pi$
$702$	0	0
$703$	−27.4377	−1.03483
$704$	0.0557281	0.00210033
$705$	0	0
$706$	16.8885	0.635609
$707$	−14.4721	−0.544281
$708$	0	0
$709$	39.5967	1.48709	0.743544	−	0.668688i	$-0.233144\pi$
$709$	39.5967	1.48709	0.743544	+	0.668688i	$0.233144\pi$
$710$	−1.76393	−0.0661992
$711$	0	0
$712$	16.7082	0.626166
$713$	−51.1033	−1.91383
$714$	0	0
$715$	−0.854102	−0.0319416
$716$	11.3262	0.423281
$717$	0	0
$718$	5.92299	0.221044
$719$	42.0344	1.56762	0.783810	−	0.621001i	$-0.213274\pi$
$719$	42.0344	1.56762	0.783810	+	0.621001i	$0.213274\pi$
$720$	0	0
$721$	−8.38197	−0.312161
$722$	−1.40325	−0.0522236
$723$	0	0
$724$	17.0344	0.633080
$725$	10.5623	0.392274
$726$	0	0
$727$	44.4508	1.64859	0.824295	−	0.566160i	$-0.191572\pi$
$727$	44.4508	1.64859	0.824295	+	0.566160i	$0.191572\pi$
$728$	8.09017	0.299842
$729$	0	0
$730$	−2.43769	−0.0902231
$731$	−4.90983	−0.181597
$732$	0	0
$733$	−9.56231	−0.353192	−0.176596	−	0.984283i	$-0.556509\pi$
$733$	−9.56231	−0.353192	−0.176596	+	0.984283i	$0.556509\pi$
$734$	−5.03444	−0.185825
$735$	0	0
$736$	42.7984	1.57757
$737$	1.02129	0.0376196
$738$	0	0
$739$	10.6525	0.391858	0.195929	−	0.980618i	$-0.437228\pi$
$739$	10.6525	0.391858	0.195929	+	0.980618i	$0.437228\pi$
$740$	10.8541	0.399005
$741$	0	0
$742$	4.09017	0.150155
$743$	12.3262	0.452206	0.226103	−	0.974103i	$-0.427401\pi$
$743$	12.3262	0.452206	0.226103	+	0.974103i	$0.427401\pi$
$744$	0	0
$745$	8.79837	0.322347
$746$	18.5279	0.678353
$747$	0	0
$748$	−0.236068	−0.00863150
$749$	−10.7082	−0.391269
$750$	0	0
$751$	−12.1115	−0.441953	−0.220977	−	0.975279i	$-0.570924\pi$
$751$	−12.1115	−0.441953	−0.220977	+	0.975279i	$0.570924\pi$
$752$	20.3951	0.743734
$753$	0	0
$754$	23.6180	0.860118
$755$	−17.9443	−0.653059
$756$	0	0
$757$	−27.2705	−0.991164	−0.495582	−	0.868561i	$-0.665045\pi$
$757$	−27.2705	−0.991164	−0.495582	+	0.868561i	$0.665045\pi$
$758$	−14.7639	−0.536250
$759$	0	0
$760$	−9.14590	−0.331757
$761$	−39.0344	−1.41500	−0.707499	−	0.706715i	$-0.750176\pi$
$761$	−39.0344	−1.41500	−0.707499	+	0.706715i	$0.750176\pi$
$762$	0	0
$763$	7.56231	0.273774
$764$	9.38197	0.339428
$765$	0	0
$766$	12.1459	0.438849
$767$	53.2148	1.92147
$768$	0	0
$769$	−26.5967	−0.959103	−0.479552	−	0.877514i	$-0.659201\pi$
$769$	−26.5967	−0.959103	−0.479552	+	0.877514i	$0.659201\pi$
$770$	0.145898	0.00525780
$771$	0	0
$772$	−9.47214	−0.340910
$773$	−33.3262	−1.19866	−0.599331	−	0.800502i	$-0.704567\pi$
$773$	−33.3262	−1.19866	−0.599331	+	0.800502i	$0.704567\pi$
$774$	0	0
$775$	−6.70820	−0.240966
$776$	−27.5623	−0.989429
$777$	0	0
$778$	0.742646	0.0266251
$779$	−12.6393	−0.452851
$780$	0	0
$781$	0.673762	0.0241091
$782$	−2.90983	−0.104055
$783$	0	0
$784$	1.85410	0.0662179
$785$	−0.763932	−0.0272659
$786$	0	0
$787$	1.23607	0.0440611	0.0220305	−	0.999757i	$-0.492987\pi$
$787$	1.23607	0.0440611	0.0220305	+	0.999757i	$0.492987\pi$
$788$	17.0344	0.606827
$789$	0	0
$790$	2.09017	0.0743649
$791$	0.854102	0.0303684
$792$	0	0
$793$	−25.8541	−0.918106
$794$	16.6525	0.590974
$795$	0	0
$796$	1.00000	0.0354441
$797$	−36.6869	−1.29952	−0.649759	−	0.760141i	$-0.725130\pi$
$797$	−36.6869	−1.29952	−0.649759	+	0.760141i	$0.725130\pi$
$798$	0	0
$799$	−6.79837	−0.240509
$800$	5.61803	0.198627
$801$	0	0
$802$	−12.1246	−0.428135
$803$	0.931116	0.0328584
$804$	0	0
$805$	−7.61803	−0.268500
$806$	−15.0000	−0.528352
$807$	0	0
$808$	−32.3607	−1.13844
$809$	−51.3607	−1.80575	−0.902873	−	0.429908i	$-0.858546\pi$
$809$	−51.3607	−1.80575	−0.902873	+	0.429908i	$0.858546\pi$
$810$	0	0
$811$	−7.32624	−0.257259	−0.128630	−	0.991693i	$-0.541058\pi$
$811$	−7.32624	−0.257259	−0.128630	+	0.991693i	$0.541058\pi$
$812$	17.0902	0.599747
$813$	0	0
$814$	0.978714	0.0343039
$815$	14.9443	0.523475
$816$	0	0
$817$	32.4934	1.13680
$818$	−19.1246	−0.668676
$819$	0	0
$820$	5.00000	0.174608
$821$	−56.4164	−1.96895	−0.984473	−	0.175535i	$-0.943834\pi$
$821$	−56.4164	−1.96895	−0.984473	+	0.175535i	$0.943834\pi$
$822$	0	0
$823$	−20.0557	−0.699099	−0.349549	−	0.936918i	$-0.613665\pi$
$823$	−20.0557	−0.699099	−0.349549	+	0.936918i	$0.613665\pi$
$824$	−18.7426	−0.652931
$825$	0	0
$826$	−9.09017	−0.316287
$827$	55.5410	1.93135	0.965675	−	0.259752i	$-0.0836410\pi$
$827$	55.5410	1.93135	0.965675	+	0.259752i	$0.0836410\pi$
$828$	0	0
$829$	31.3607	1.08920	0.544601	−	0.838695i	$-0.316681\pi$
$829$	31.3607	1.08920	0.544601	+	0.838695i	$0.316681\pi$
$830$	−1.67376	−0.0580971
$831$	0	0
$832$	−0.854102	−0.0296107
$833$	−0.618034	−0.0214136
$834$	0	0
$835$	1.47214	0.0509454
$836$	1.56231	0.0540335
$837$	0	0
$838$	−0.909830	−0.0314296
$839$	2.61803	0.0903846	0.0451923	−	0.998978i	$-0.485610\pi$
$839$	2.61803	0.0903846	0.0451923	+	0.998978i	$0.485610\pi$
$840$	0	0
$841$	82.5623	2.84698
$842$	17.6525	0.608344
$843$	0	0
$844$	−37.4164	−1.28793
$845$	0.0901699	0.00310194
$846$	0	0
$847$	10.9443	0.376050
$848$	−12.2705	−0.421371
$849$	0	0
$850$	−0.381966	−0.0131013
$851$	−51.1033	−1.75180
$852$	0	0
$853$	−55.8328	−1.91168	−0.955840	−	0.293889i	$-0.905050\pi$
$853$	−55.8328	−1.91168	−0.955840	+	0.293889i	$0.905050\pi$
$854$	4.41641	0.151126
$855$	0	0
$856$	−23.9443	−0.818398
$857$	−21.7639	−0.743442	−0.371721	−	0.928345i	$-0.621232\pi$
$857$	−21.7639	−0.743442	−0.371721	+	0.928345i	$0.621232\pi$
$858$	0	0
$859$	−27.5967	−0.941589	−0.470794	−	0.882243i	$-0.656033\pi$
$859$	−27.5967	−0.941589	−0.470794	+	0.882243i	$0.656033\pi$
$860$	−12.8541	−0.438321
$861$	0	0
$862$	−15.7295	−0.535749
$863$	19.7771	0.673220	0.336610	−	0.941644i	$-0.390720\pi$
$863$	19.7771	0.673220	0.336610	+	0.941644i	$0.390720\pi$
$864$	0	0
$865$	2.52786	0.0859500
$866$	18.6738	0.634560
$867$	0	0
$868$	−10.8541	−0.368412
$869$	−0.798374	−0.0270830
$870$	0	0
$871$	−15.6525	−0.530364
$872$	16.9098	0.572639
$873$	0	0
$874$	19.2574	0.651390
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	10.0557	0.339558	0.169779	−	0.985482i	$-0.445695\pi$
$877$	10.0557	0.339558	0.169779	+	0.985482i	$0.445695\pi$
$878$	15.2705	0.515355
$879$	0	0
$880$	−0.437694	−0.0147547
$881$	20.1115	0.677572	0.338786	−	0.940863i	$-0.389984\pi$
$881$	20.1115	0.677572	0.338786	+	0.940863i	$0.389984\pi$
$882$	0	0
$883$	11.6525	0.392137	0.196069	−	0.980590i	$-0.437182\pi$
$883$	11.6525	0.392137	0.196069	+	0.980590i	$0.437182\pi$
$884$	3.61803	0.121688
$885$	0	0
$886$	12.1672	0.408765
$887$	−19.2361	−0.645884	−0.322942	−	0.946419i	$-0.604672\pi$
$887$	−19.2361	−0.645884	−0.322942	+	0.946419i	$0.604672\pi$
$888$	0	0
$889$	11.0902	0.371952
$890$	−4.61803	−0.154797
$891$	0	0
$892$	2.38197	0.0797541
$893$	44.9919	1.50560
$894$	0	0
$895$	−7.00000	−0.233984
$896$	11.3820	0.380245
$897$	0	0
$898$	7.12461	0.237751
$899$	−70.8541	−2.36312
$900$	0	0
$901$	4.09017	0.136263
$902$	0.450850	0.0150117
$903$	0	0
$904$	1.90983	0.0635200
$905$	−10.5279	−0.349958
$906$	0	0
$907$	14.8541	0.493222	0.246611	−	0.969115i	$-0.420683\pi$
$907$	14.8541	0.493222	0.246611	+	0.969115i	$0.420683\pi$
$908$	−44.5967	−1.48000
$909$	0	0
$910$	−2.23607	−0.0741249
$911$	52.3607	1.73479	0.867393	−	0.497623i	$-0.165794\pi$
$911$	52.3607	1.73479	0.867393	+	0.497623i	$0.165794\pi$
$912$	0	0
$913$	0.639320	0.0211584
$914$	1.03444	0.0342163
$915$	0	0
$916$	0.0901699	0.00297930
$917$	−6.61803	−0.218547
$918$	0	0
$919$	−27.1591	−0.895895	−0.447947	−	0.894060i	$-0.647845\pi$
$919$	−27.1591	−0.895895	−0.447947	+	0.894060i	$0.647845\pi$
$920$	−17.0344	−0.561609
$921$	0	0
$922$	1.11146	0.0366039
$923$	−10.3262	−0.339892
$924$	0	0
$925$	−6.70820	−0.220564
$926$	1.00000	0.0328620
$927$	0	0
$928$	59.3394	1.94791
$929$	30.1033	0.987658	0.493829	−	0.869559i	$-0.335597\pi$
$929$	30.1033	0.987658	0.493829	+	0.869559i	$0.335597\pi$
$930$	0	0
$931$	4.09017	0.134050
$932$	−9.56231	−0.313224
$933$	0	0
$934$	−0.583592	−0.0190957
$935$	0.145898	0.00477138
$936$	0	0
$937$	9.05573	0.295838	0.147919	−	0.988999i	$-0.452743\pi$
$937$	9.05573	0.295838	0.147919	+	0.988999i	$0.452743\pi$
$938$	2.67376	0.0873014
$939$	0	0
$940$	−17.7984	−0.580519
$941$	−50.2361	−1.63765	−0.818825	−	0.574044i	$-0.805374\pi$
$941$	−50.2361	−1.63765	−0.818825	+	0.574044i	$0.805374\pi$
$942$	0	0
$943$	−23.5410	−0.766601
$944$	27.2705	0.887579
$945$	0	0
$946$	−1.15905	−0.0376841
$947$	−25.5623	−0.830663	−0.415332	−	0.909670i	$-0.636334\pi$
$947$	−25.5623	−0.830663	−0.415332	+	0.909670i	$0.636334\pi$
$948$	0	0
$949$	−14.2705	−0.463240
$950$	2.52786	0.0820147
$951$	0	0
$952$	−1.38197	−0.0447898
$953$	−4.34752	−0.140830	−0.0704151	−	0.997518i	$-0.522432\pi$
$953$	−4.34752	−0.140830	−0.0704151	+	0.997518i	$0.522432\pi$
$954$	0	0
$955$	−5.79837	−0.187631
$956$	31.7984	1.02843
$957$	0	0
$958$	22.9230	0.740608
$959$	21.8885	0.706818
$960$	0	0
$961$	14.0000	0.451613
$962$	−15.0000	−0.483619
$963$	0	0
$964$	8.70820	0.280472
$965$	5.85410	0.188450
$966$	0	0
$967$	−17.4721	−0.561866	−0.280933	−	0.959727i	$-0.590644\pi$
$967$	−17.4721	−0.561866	−0.280933	+	0.959727i	$0.590644\pi$
$968$	24.4721	0.786564
$969$	0	0
$970$	7.61803	0.244600
$971$	−58.2492	−1.86931	−0.934653	−	0.355560i	$-0.884290\pi$
$971$	−58.2492	−1.86931	−0.934653	+	0.355560i	$0.884290\pi$
$972$	0	0
$973$	19.1803	0.614893
$974$	−15.1246	−0.484624
$975$	0	0
$976$	−13.2492	−0.424097
$977$	−30.3607	−0.971324	−0.485662	−	0.874147i	$-0.661421\pi$
$977$	−30.3607	−0.971324	−0.485662	+	0.874147i	$0.661421\pi$
$978$	0	0
$979$	1.76393	0.0563755
$980$	−1.61803	−0.0516862
$981$	0	0
$982$	11.4377	0.364991
$983$	0.0212862	0.000678925 0	0.000339463	−	1.00000i	$-0.499892\pi$
$983$	0.0212862	0.000678925 0	0.000339463		1.00000i	$0.499892\pi$
$984$	0	0
$985$	−10.5279	−0.335446
$986$	−4.03444	−0.128483
$987$	0	0
$988$	−23.9443	−0.761769
$989$	60.5197	1.92442
$990$	0	0
$991$	−7.11146	−0.225903	−0.112951	−	0.993601i	$-0.536030\pi$
$991$	−7.11146	−0.225903	−0.112951	+	0.993601i	$0.536030\pi$
$992$	−37.6869	−1.19656
$993$	0	0
$994$	1.76393	0.0559485
$995$	−0.618034	−0.0195930
$996$	0	0
$997$	1.29180	0.0409116	0.0204558	−	0.999791i	$-0.493488\pi$
$997$	1.29180	0.0409116	0.0204558	+	0.999791i	$0.493488\pi$
$998$	−25.5623	−0.809161
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.a.e.1.2	✓	2
3.2	odd	2		945.2.a.g.1.1	yes	2
5.4	even	2		4725.2.a.be.1.1		2
7.6	odd	2		6615.2.a.o.1.2		2
15.14	odd	2		4725.2.a.w.1.2		2
21.20	even	2		6615.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.e.1.2	✓	2	1.1	even	1	trivial
945.2.a.g.1.1	yes	2	3.2	odd	2
4725.2.a.w.1.2		2	15.14	odd	2
4725.2.a.be.1.1		2	5.4	even	2
6615.2.a.o.1.2		2	7.6	odd	2
6615.2.a.r.1.1		2	21.20	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:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +0.618034 q^{10} -0.236068 q^{11} +3.61803 q^{13} -0.618034 q^{14} +1.85410 q^{16} -0.618034 q^{17} +4.09017 q^{19} -1.61803 q^{20} -0.145898 q^{22} +7.61803 q^{23} +1.00000 q^{25} +2.23607 q^{26} +1.61803 q^{28} +10.5623 q^{29} -6.70820 q^{31} +5.61803 q^{32} -0.381966 q^{34} -1.00000 q^{35} -6.70820 q^{37} +2.52786 q^{38} -2.23607 q^{40} -3.09017 q^{41} +7.94427 q^{43} +0.381966 q^{44} +4.70820 q^{46} +11.0000 q^{47} +1.00000 q^{49} +0.618034 q^{50} -5.85410 q^{52} -6.61803 q^{53} -0.236068 q^{55} +2.23607 q^{56} +6.52786 q^{58} +14.7082 q^{59} -7.14590 q^{61} -4.14590 q^{62} -0.236068 q^{64} +3.61803 q^{65} -4.32624 q^{67} +1.00000 q^{68} -0.618034 q^{70} -2.85410 q^{71} -3.94427 q^{73} -4.14590 q^{74} -6.61803 q^{76} +0.236068 q^{77} +3.38197 q^{79} +1.85410 q^{80} -1.90983 q^{82} -2.70820 q^{83} -0.618034 q^{85} +4.90983 q^{86} +0.527864 q^{88} -7.47214 q^{89} -3.61803 q^{91} -12.3262 q^{92} +6.79837 q^{94} +4.09017 q^{95} +12.3262 q^{97} +0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7	\( 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7} - q^{10} + 4 q^{11} + 5 q^{13} + q^{14} - 3 q^{16} + q^{17} - 3 q^{19} - q^{20} - 7 q^{22} + 13 q^{23} + 2 q^{25} + q^{28} + q^{29} + 9 q^{32} - 3 q^{34} - 2 q^{35} + 14 q^{38} + 5 q^{41} - 2 q^{43} + 3 q^{44} - 4 q^{46} + 22 q^{47} + 2 q^{49} - q^{50} - 5 q^{52} - 11 q^{53} + 4 q^{55} + 22 q^{58} + 16 q^{59} - 21 q^{61} - 15 q^{62} + 4 q^{64} + 5 q^{65} + 7 q^{67} + 2 q^{68} + q^{70} + q^{71} + 10 q^{73} - 15 q^{74} - 11 q^{76} - 4 q^{77} + 9 q^{79} - 3 q^{80} - 15 q^{82} + 8 q^{83} + q^{85} + 21 q^{86} + 10 q^{88} - 6 q^{89} - 5 q^{91} - 9 q^{92} - 11 q^{94} - 3 q^{95} + 9 q^{97} - q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7 - q^10 + 4 * q^11 + 5 * q^13 + q^14 - 3 * q^16 + q^17 - 3 * q^19 - q^20 - 7 * q^22 + 13 * q^23 + 2 * q^25 + q^28 + q^29 + 9 * q^32 - 3 * q^34 - 2 * q^35 + 14 * q^38 + 5 * q^41 - 2 * q^43 + 3 * q^44 - 4 * q^46 + 22 * q^47 + 2 * q^49 - q^50 - 5 * q^52 - 11 * q^53 + 4 * q^55 + 22 * q^58 + 16 * q^59 - 21 * q^61 - 15 * q^62 + 4 * q^64 + 5 * q^65 + 7 * q^67 + 2 * q^68 + q^70 + q^71 + 10 * q^73 - 15 * q^74 - 11 * q^76 - 4 * q^77 + 9 * q^79 - 3 * q^80 - 15 * q^82 + 8 * q^83 + q^85 + 21 * q^86 + 10 * q^88 - 6 * q^89 - 5 * q^91 - 9 * q^92 - 11 * q^94 - 3 * q^95 + 9 * q^97 - q^98

Introduction

L-functions

Modular forms

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Database

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