LMFDB - Embedded newform 6625.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6625,2,Mod(1,6625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6625 = 5^{3} \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6625.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:	$52.9008913391$
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			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	6625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{2} +2.00000 q^{3} +0.618034 q^{4} -3.23607 q^{6} +3.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.61803 q^{2} +2.00000 q^{3} +0.618034 q^{4} -3.23607 q^{6} +3.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +1.23607 q^{12} +4.61803 q^{13} -4.85410 q^{14} -4.85410 q^{16} +7.47214 q^{17} -1.61803 q^{18} -1.76393 q^{19} +6.00000 q^{21} +5.61803 q^{23} +4.47214 q^{24} -7.47214 q^{26} -4.00000 q^{27} +1.85410 q^{28} -10.4721 q^{29} -0.236068 q^{31} +3.38197 q^{32} -12.0902 q^{34} +0.618034 q^{36} -2.70820 q^{37} +2.85410 q^{38} +9.23607 q^{39} +10.0902 q^{41} -9.70820 q^{42} +4.23607 q^{43} -9.09017 q^{46} +6.76393 q^{47} -9.70820 q^{48} +2.00000 q^{49} +14.9443 q^{51} +2.85410 q^{52} +1.00000 q^{53} +6.47214 q^{54} +6.70820 q^{56} -3.52786 q^{57} +16.9443 q^{58} -1.09017 q^{59} +12.8541 q^{61} +0.381966 q^{62} +3.00000 q^{63} +4.23607 q^{64} +6.85410 q^{67} +4.61803 q^{68} +11.2361 q^{69} -6.61803 q^{71} +2.23607 q^{72} +10.7639 q^{73} +4.38197 q^{74} -1.09017 q^{76} -14.9443 q^{78} -2.52786 q^{79} -11.0000 q^{81} -16.3262 q^{82} -7.94427 q^{83} +3.70820 q^{84} -6.85410 q^{86} -20.9443 q^{87} -1.61803 q^{89} +13.8541 q^{91} +3.47214 q^{92} -0.472136 q^{93} -10.9443 q^{94} +6.76393 q^{96} -4.56231 q^{97} -3.23607 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 4 q^{3} - q^{4} - 2 q^{6} + 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - q^2 + 4 * q^3 - q^4 - 2 * q^6 + 6 * q^7 + 2 * q^9	$ 2 q - q^{2} + 4 q^{3} - q^{4} - 2 q^{6} + 6 q^{7} + 2 q^{9} - 2 q^{12} + 7 q^{13} - 3 q^{14} - 3 q^{16} + 6 q^{17} - q^{18} - 8 q^{19} + 12 q^{21} + 9 q^{23} - 6 q^{26} - 8 q^{27} - 3 q^{28} - 12 q^{29} + 4 q^{31} + 9 q^{32} - 13 q^{34} - q^{36} + 8 q^{37} - q^{38} + 14 q^{39} + 9 q^{41} - 6 q^{42} + 4 q^{43} - 7 q^{46} + 18 q^{47} - 6 q^{48} + 4 q^{49} + 12 q^{51} - q^{52} + 2 q^{53} + 4 q^{54} - 16 q^{57} + 16 q^{58} + 9 q^{59} + 19 q^{61} + 3 q^{62} + 6 q^{63} + 4 q^{64} + 7 q^{67} + 7 q^{68} + 18 q^{69} - 11 q^{71} + 26 q^{73} + 11 q^{74} + 9 q^{76} - 12 q^{78} - 14 q^{79} - 22 q^{81} - 17 q^{82} + 2 q^{83} - 6 q^{84} - 7 q^{86} - 24 q^{87} - q^{89} + 21 q^{91} - 2 q^{92} + 8 q^{93} - 4 q^{94} + 18 q^{96} + 11 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - q^2 + 4 * q^3 - q^4 - 2 * q^6 + 6 * q^7 + 2 * q^9 - 2 * q^12 + 7 * q^13 - 3 * q^14 - 3 * q^16 + 6 * q^17 - q^18 - 8 * q^19 + 12 * q^21 + 9 * q^23 - 6 * q^26 - 8 * q^27 - 3 * q^28 - 12 * q^29 + 4 * q^31 + 9 * q^32 - 13 * q^34 - q^36 + 8 * q^37 - q^38 + 14 * q^39 + 9 * q^41 - 6 * q^42 + 4 * q^43 - 7 * q^46 + 18 * q^47 - 6 * q^48 + 4 * q^49 + 12 * q^51 - q^52 + 2 * q^53 + 4 * q^54 - 16 * q^57 + 16 * q^58 + 9 * q^59 + 19 * q^61 + 3 * q^62 + 6 * q^63 + 4 * q^64 + 7 * q^67 + 7 * q^68 + 18 * q^69 - 11 * q^71 + 26 * q^73 + 11 * q^74 + 9 * q^76 - 12 * q^78 - 14 * q^79 - 22 * q^81 - 17 * q^82 + 2 * q^83 - 6 * q^84 - 7 * q^86 - 24 * q^87 - q^89 + 21 * q^91 - 2 * q^92 + 8 * q^93 - 4 * q^94 + 18 * q^96 + 11 * q^97 - 2 * 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 $

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	−3.23607	−1.32112
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.23607	0.356822
$13$	4.61803	1.28081	0.640406	−	0.768036i	$-0.278766\pi$
$13$	4.61803	1.28081	0.640406	+	0.768036i	$0.278766\pi$
$14$	−4.85410	−1.29731
$15$	0	0
$16$	−4.85410	−1.21353
$17$	7.47214	1.81226	0.906130	−	0.423000i	$-0.139023\pi$
$17$	7.47214	1.81226	0.906130	+	0.423000i	$0.139023\pi$
$18$	−1.61803	−0.381374
$19$	−1.76393	−0.404674	−0.202337	−	0.979316i	$-0.564854\pi$
$19$	−1.76393	−0.404674	−0.202337	+	0.979316i	$0.564854\pi$
$20$	0	0
$21$	6.00000	1.30931
$22$	0	0
$23$	5.61803	1.17144	0.585721	−	0.810513i	$-0.300812\pi$
$23$	5.61803	1.17144	0.585721	+	0.810513i	$0.300812\pi$
$24$	4.47214	0.912871
$25$	0	0
$26$	−7.47214	−1.46541
$27$	−4.00000	−0.769800
$28$	1.85410	0.350392
$29$	−10.4721	−1.94463	−0.972313	−	0.233681i	$-0.924923\pi$
$29$	−10.4721	−1.94463	−0.972313	+	0.233681i	$0.924923\pi$
$30$	0	0
$31$	−0.236068	−0.0423991	−0.0211995	−	0.999775i	$-0.506749\pi$
$31$	−0.236068	−0.0423991	−0.0211995	+	0.999775i	$0.506749\pi$
$32$	3.38197	0.597853
$33$	0	0
$34$	−12.0902	−2.07345
$35$	0	0
$36$	0.618034	0.103006
$37$	−2.70820	−0.445226	−0.222613	−	0.974907i	$-0.571459\pi$
$37$	−2.70820	−0.445226	−0.222613	+	0.974907i	$0.571459\pi$
$38$	2.85410	0.462996
$39$	9.23607	1.47895
$40$	0	0
$41$	10.0902	1.57582	0.787910	−	0.615791i	$-0.211163\pi$
$41$	10.0902	1.57582	0.787910	+	0.615791i	$0.211163\pi$
$42$	−9.70820	−1.49801
$43$	4.23607	0.645994	0.322997	−	0.946400i	$-0.395310\pi$
$43$	4.23607	0.645994	0.322997	+	0.946400i	$0.395310\pi$
$44$	0	0
$45$	0	0
$46$	−9.09017	−1.34027
$47$	6.76393	0.986621	0.493310	−	0.869853i	$-0.335787\pi$
$47$	6.76393	0.986621	0.493310	+	0.869853i	$0.335787\pi$
$48$	−9.70820	−1.40126
$49$	2.00000	0.285714
$50$	0	0
$51$	14.9443	2.09262
$52$	2.85410	0.395793
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	6.47214	0.880746
$55$	0	0
$56$	6.70820	0.896421
$57$	−3.52786	−0.467277
$58$	16.9443	2.22489
$59$	−1.09017	−0.141928	−0.0709640	−	0.997479i	$-0.522608\pi$
$59$	−1.09017	−0.141928	−0.0709640	+	0.997479i	$0.522608\pi$
$60$	0	0
$61$	12.8541	1.64580	0.822900	−	0.568187i	$-0.192355\pi$
$61$	12.8541	1.64580	0.822900	+	0.568187i	$0.192355\pi$
$62$	0.381966	0.0485097
$63$	3.00000	0.377964
$64$	4.23607	0.529508
$65$	0	0
$66$	0	0
$67$	6.85410	0.837362	0.418681	−	0.908133i	$-0.362493\pi$
$67$	6.85410	0.837362	0.418681	+	0.908133i	$0.362493\pi$
$68$	4.61803	0.560019
$69$	11.2361	1.35266
$70$	0	0
$71$	−6.61803	−0.785416	−0.392708	−	0.919663i	$-0.628462\pi$
$71$	−6.61803	−0.785416	−0.392708	+	0.919663i	$0.628462\pi$
$72$	2.23607	0.263523
$73$	10.7639	1.25982	0.629911	−	0.776667i	$-0.283091\pi$
$73$	10.7639	1.25982	0.629911	+	0.776667i	$0.283091\pi$
$74$	4.38197	0.509393
$75$	0	0
$76$	−1.09017	−0.125051
$77$	0	0
$78$	−14.9443	−1.69211
$79$	−2.52786	−0.284407	−0.142203	−	0.989837i	$-0.545419\pi$
$79$	−2.52786	−0.284407	−0.142203	+	0.989837i	$0.545419\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	−16.3262	−1.80293
$83$	−7.94427	−0.871997	−0.435999	−	0.899947i	$-0.643605\pi$
$83$	−7.94427	−0.871997	−0.435999	+	0.899947i	$0.643605\pi$
$84$	3.70820	0.404598
$85$	0	0
$86$	−6.85410	−0.739097
$87$	−20.9443	−2.24546
$88$	0	0
$89$	−1.61803	−0.171511	−0.0857556	−	0.996316i	$-0.527330\pi$
$89$	−1.61803	−0.171511	−0.0857556	+	0.996316i	$0.527330\pi$
$90$	0	0
$91$	13.8541	1.45230
$92$	3.47214	0.361995
$93$	−0.472136	−0.0489582
$94$	−10.9443	−1.12882
$95$	0	0
$96$	6.76393	0.690341
$97$	−4.56231	−0.463232	−0.231616	−	0.972807i	$-0.574401\pi$
$97$	−4.56231	−0.463232	−0.231616	+	0.972807i	$0.574401\pi$
$98$	−3.23607	−0.326892
$99$	0	0
$100$	0	0
$101$	0.618034	0.0614967	0.0307483	−	0.999527i	$-0.490211\pi$
$101$	0.618034	0.0614967	0.0307483	+	0.999527i	$0.490211\pi$
$102$	−24.1803	−2.39421
$103$	−0.618034	−0.0608967	−0.0304483	−	0.999536i	$-0.509694\pi$
$103$	−0.618034	−0.0608967	−0.0304483	+	0.999536i	$0.509694\pi$
$104$	10.3262	1.01257
$105$	0	0
$106$	−1.61803	−0.157157
$107$	−8.94427	−0.864675	−0.432338	−	0.901712i	$-0.642311\pi$
$107$	−8.94427	−0.864675	−0.432338	+	0.901712i	$0.642311\pi$
$108$	−2.47214	−0.237881
$109$	−13.7639	−1.31835	−0.659173	−	0.751992i	$-0.729093\pi$
$109$	−13.7639	−1.31835	−0.659173	+	0.751992i	$0.729093\pi$
$110$	0	0
$111$	−5.41641	−0.514103
$112$	−14.5623	−1.37601
$113$	14.6180	1.37515	0.687574	−	0.726114i	$-0.258675\pi$
$113$	14.6180	1.37515	0.687574	+	0.726114i	$0.258675\pi$
$114$	5.70820	0.534622
$115$	0	0
$116$	−6.47214	−0.600923
$117$	4.61803	0.426937
$118$	1.76393	0.162383
$119$	22.4164	2.05491
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−20.7984	−1.88300
$123$	20.1803	1.81960
$124$	−0.145898	−0.0131020
$125$	0	0
$126$	−4.85410	−0.432438
$127$	21.2705	1.88745	0.943726	−	0.330728i	$-0.107294\pi$
$127$	21.2705	1.88745	0.943726	+	0.330728i	$0.107294\pi$
$128$	−13.6180	−1.20368
$129$	8.47214	0.745930
$130$	0	0
$131$	−13.8541	−1.21044	−0.605219	−	0.796059i	$-0.706915\pi$
$131$	−13.8541	−1.21044	−0.605219	+	0.796059i	$0.706915\pi$
$132$	0	0
$133$	−5.29180	−0.458857
$134$	−11.0902	−0.958045
$135$	0	0
$136$	16.7082	1.43272
$137$	1.38197	0.118069	0.0590347	−	0.998256i	$-0.481198\pi$
$137$	1.38197	0.118069	0.0590347	+	0.998256i	$0.481198\pi$
$138$	−18.1803	−1.54761
$139$	−7.41641	−0.629052	−0.314526	−	0.949249i	$-0.601845\pi$
$139$	−7.41641	−0.629052	−0.314526	+	0.949249i	$0.601845\pi$
$140$	0	0
$141$	13.5279	1.13925
$142$	10.7082	0.898613
$143$	0	0
$144$	−4.85410	−0.404508
$145$	0	0
$146$	−17.4164	−1.44139
$147$	4.00000	0.329914
$148$	−1.67376	−0.137582
$149$	10.1803	0.834006	0.417003	−	0.908905i	$-0.363080\pi$
$149$	10.1803	0.834006	0.417003	+	0.908905i	$0.363080\pi$
$150$	0	0
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	−3.94427	−0.319923
$153$	7.47214	0.604086
$154$	0	0
$155$	0	0
$156$	5.70820	0.457022
$157$	−15.1803	−1.21152	−0.605762	−	0.795646i	$-0.707131\pi$
$157$	−15.1803	−1.21152	−0.605762	+	0.795646i	$0.707131\pi$
$158$	4.09017	0.325396
$159$	2.00000	0.158610
$160$	0	0
$161$	16.8541	1.32829
$162$	17.7984	1.39837
$163$	−2.29180	−0.179507	−0.0897537	−	0.995964i	$-0.528608\pi$
$163$	−2.29180	−0.179507	−0.0897537	+	0.995964i	$0.528608\pi$
$164$	6.23607	0.486955
$165$	0	0
$166$	12.8541	0.997672
$167$	−8.18034	−0.633014	−0.316507	−	0.948590i	$-0.602510\pi$
$167$	−8.18034	−0.633014	−0.316507	+	0.948590i	$0.602510\pi$
$168$	13.4164	1.03510
$169$	8.32624	0.640480
$170$	0	0
$171$	−1.76393	−0.134891
$172$	2.61803	0.199623
$173$	19.6180	1.49153	0.745766	−	0.666208i	$-0.232084\pi$
$173$	19.6180	1.49153	0.745766	+	0.666208i	$0.232084\pi$
$174$	33.8885	2.56908
$175$	0	0
$176$	0	0
$177$	−2.18034	−0.163884
$178$	2.61803	0.196230
$179$	10.8885	0.813848	0.406924	−	0.913462i	$-0.366601\pi$
$179$	10.8885	0.813848	0.406924	+	0.913462i	$0.366601\pi$
$180$	0	0
$181$	−20.4721	−1.52168	−0.760841	−	0.648938i	$-0.775213\pi$
$181$	−20.4721	−1.52168	−0.760841	+	0.648938i	$0.775213\pi$
$182$	−22.4164	−1.66161
$183$	25.7082	1.90041
$184$	12.5623	0.926105
$185$	0	0
$186$	0.763932	0.0560142
$187$	0	0
$188$	4.18034	0.304883
$189$	−12.0000	−0.872872
$190$	0	0
$191$	−7.18034	−0.519551	−0.259776	−	0.965669i	$-0.583649\pi$
$191$	−7.18034	−0.519551	−0.259776	+	0.965669i	$0.583649\pi$
$192$	8.47214	0.611424
$193$	−15.5623	−1.12020	−0.560100	−	0.828425i	$-0.689237\pi$
$193$	−15.5623	−1.12020	−0.560100	+	0.828425i	$0.689237\pi$
$194$	7.38197	0.529994
$195$	0	0
$196$	1.23607	0.0882906
$197$	−23.4721	−1.67232	−0.836160	−	0.548485i	$-0.815205\pi$
$197$	−23.4721	−1.67232	−0.836160	+	0.548485i	$0.815205\pi$
$198$	0	0
$199$	2.38197	0.168853	0.0844265	−	0.996430i	$-0.473094\pi$
$199$	2.38197	0.168853	0.0844265	+	0.996430i	$0.473094\pi$
$200$	0	0
$201$	13.7082	0.966902
$202$	−1.00000	−0.0703598
$203$	−31.4164	−2.20500
$204$	9.23607	0.646654
$205$	0	0
$206$	1.00000	0.0696733
$207$	5.61803	0.390480
$208$	−22.4164	−1.55430
$209$	0	0
$210$	0	0
$211$	−8.41641	−0.579409	−0.289705	−	0.957116i	$-0.593557\pi$
$211$	−8.41641	−0.579409	−0.289705	+	0.957116i	$0.593557\pi$
$212$	0.618034	0.0424467
$213$	−13.2361	−0.906920
$214$	14.4721	0.989295
$215$	0	0
$216$	−8.94427	−0.608581
$217$	−0.708204	−0.0480760
$218$	22.2705	1.50835
$219$	21.5279	1.45472
$220$	0	0
$221$	34.5066	2.32116
$222$	8.76393	0.588197
$223$	2.05573	0.137662	0.0688309	−	0.997628i	$-0.478073\pi$
$223$	2.05573	0.137662	0.0688309	+	0.997628i	$0.478073\pi$
$224$	10.1459	0.677901
$225$	0	0
$226$	−23.6525	−1.57334
$227$	−7.94427	−0.527280	−0.263640	−	0.964621i	$-0.584923\pi$
$227$	−7.94427	−0.527280	−0.263640	+	0.964621i	$0.584923\pi$
$228$	−2.18034	−0.144397
$229$	11.8541	0.783341	0.391671	−	0.920106i	$-0.371897\pi$
$229$	11.8541	0.783341	0.391671	+	0.920106i	$0.371897\pi$
$230$	0	0
$231$	0	0
$232$	−23.4164	−1.53736
$233$	−4.23607	−0.277514	−0.138757	−	0.990326i	$-0.544311\pi$
$233$	−4.23607	−0.277514	−0.138757	+	0.990326i	$0.544311\pi$
$234$	−7.47214	−0.488469
$235$	0	0
$236$	−0.673762	−0.0438582
$237$	−5.05573	−0.328405
$238$	−36.2705	−2.35107
$239$	28.0689	1.81563	0.907813	−	0.419376i	$-0.137751\pi$
$239$	28.0689	1.81563	0.907813	+	0.419376i	$0.137751\pi$
$240$	0	0
$241$	−26.2705	−1.69223	−0.846116	−	0.532999i	$-0.821065\pi$
$241$	−26.2705	−1.69223	−0.846116	+	0.532999i	$0.821065\pi$
$242$	17.7984	1.14412
$243$	−10.0000	−0.641500
$244$	7.94427	0.508580
$245$	0	0
$246$	−32.6525	−2.08185
$247$	−8.14590	−0.518311
$248$	−0.527864	−0.0335194
$249$	−15.8885	−1.00690
$250$	0	0
$251$	12.4721	0.787234	0.393617	−	0.919274i	$-0.371224\pi$
$251$	12.4721	0.787234	0.393617	+	0.919274i	$0.371224\pi$
$252$	1.85410	0.116797
$253$	0	0
$254$	−34.4164	−2.15948
$255$	0	0
$256$	13.5623	0.847644
$257$	18.3820	1.14664	0.573318	−	0.819333i	$-0.305656\pi$
$257$	18.3820	1.14664	0.573318	+	0.819333i	$0.305656\pi$
$258$	−13.7082	−0.853435
$259$	−8.12461	−0.504839
$260$	0	0
$261$	−10.4721	−0.648209
$262$	22.4164	1.38489
$263$	8.56231	0.527974	0.263987	−	0.964526i	$-0.414962\pi$
$263$	8.56231	0.527974	0.263987	+	0.964526i	$0.414962\pi$
$264$	0	0
$265$	0	0
$266$	8.56231	0.524989
$267$	−3.23607	−0.198044
$268$	4.23607	0.258759
$269$	−24.2705	−1.47980	−0.739900	−	0.672717i	$-0.765127\pi$
$269$	−24.2705	−1.47980	−0.739900	+	0.672717i	$0.765127\pi$
$270$	0	0
$271$	−8.70820	−0.528986	−0.264493	−	0.964388i	$-0.585205\pi$
$271$	−8.70820	−0.528986	−0.264493	+	0.964388i	$0.585205\pi$
$272$	−36.2705	−2.19922
$273$	27.7082	1.67698
$274$	−2.23607	−0.135086
$275$	0	0
$276$	6.94427	0.417996
$277$	−14.4164	−0.866198	−0.433099	−	0.901346i	$-0.642580\pi$
$277$	−14.4164	−0.866198	−0.433099	+	0.901346i	$0.642580\pi$
$278$	12.0000	0.719712
$279$	−0.236068	−0.0141330
$280$	0	0
$281$	11.2361	0.670288	0.335144	−	0.942167i	$-0.391215\pi$
$281$	11.2361	0.670288	0.335144	+	0.942167i	$0.391215\pi$
$282$	−21.8885	−1.30344
$283$	19.1803	1.14015	0.570076	−	0.821592i	$-0.306914\pi$
$283$	19.1803	1.14015	0.570076	+	0.821592i	$0.306914\pi$
$284$	−4.09017	−0.242707
$285$	0	0
$286$	0	0
$287$	30.2705	1.78681
$288$	3.38197	0.199284
$289$	38.8328	2.28428
$290$	0	0
$291$	−9.12461	−0.534894
$292$	6.65248	0.389307
$293$	27.7082	1.61873	0.809365	−	0.587306i	$-0.199811\pi$
$293$	27.7082	1.61873	0.809365	+	0.587306i	$0.199811\pi$
$294$	−6.47214	−0.377463
$295$	0	0
$296$	−6.05573	−0.351982
$297$	0	0
$298$	−16.4721	−0.954205
$299$	25.9443	1.50040
$300$	0	0
$301$	12.7082	0.732489
$302$	−14.5623	−0.837967
$303$	1.23607	0.0710102
$304$	8.56231	0.491082
$305$	0	0
$306$	−12.0902	−0.691149
$307$	−2.14590	−0.122473	−0.0612364	−	0.998123i	$-0.519504\pi$
$307$	−2.14590	−0.122473	−0.0612364	+	0.998123i	$0.519504\pi$
$308$	0	0
$309$	−1.23607	−0.0703175
$310$	0	0
$311$	1.52786	0.0866372	0.0433186	−	0.999061i	$-0.486207\pi$
$311$	1.52786	0.0866372	0.0433186	+	0.999061i	$0.486207\pi$
$312$	20.6525	1.16922
$313$	20.1246	1.13751	0.568755	−	0.822507i	$-0.307425\pi$
$313$	20.1246	1.13751	0.568755	+	0.822507i	$0.307425\pi$
$314$	24.5623	1.38613
$315$	0	0
$316$	−1.56231	−0.0878866
$317$	23.1803	1.30194	0.650969	−	0.759104i	$-0.274363\pi$
$317$	23.1803	1.30194	0.650969	+	0.759104i	$0.274363\pi$
$318$	−3.23607	−0.181470
$319$	0	0
$320$	0	0
$321$	−17.8885	−0.998441
$322$	−27.2705	−1.51973
$323$	−13.1803	−0.733374
$324$	−6.79837	−0.377687
$325$	0	0
$326$	3.70820	0.205378
$327$	−27.5279	−1.52229
$328$	22.5623	1.24579
$329$	20.2918	1.11872
$330$	0	0
$331$	16.8541	0.926385	0.463193	−	0.886258i	$-0.346704\pi$
$331$	16.8541	0.926385	0.463193	+	0.886258i	$0.346704\pi$
$332$	−4.90983	−0.269462
$333$	−2.70820	−0.148409
$334$	13.2361	0.724245
$335$	0	0
$336$	−29.1246	−1.58888
$337$	14.4164	0.785312	0.392656	−	0.919685i	$-0.371556\pi$
$337$	14.4164	0.785312	0.392656	+	0.919685i	$0.371556\pi$
$338$	−13.4721	−0.732788
$339$	29.2361	1.58789
$340$	0	0
$341$	0	0
$342$	2.85410	0.154332
$343$	−15.0000	−0.809924
$344$	9.47214	0.510703
$345$	0	0
$346$	−31.7426	−1.70650
$347$	−27.2148	−1.46097	−0.730483	−	0.682931i	$-0.760705\pi$
$347$	−27.2148	−1.46097	−0.730483	+	0.682931i	$0.760705\pi$
$348$	−12.9443	−0.693886
$349$	−22.4164	−1.19992	−0.599961	−	0.800029i	$-0.704817\pi$
$349$	−22.4164	−1.19992	−0.599961	+	0.800029i	$0.704817\pi$
$350$	0	0
$351$	−18.4721	−0.985970
$352$	0	0
$353$	18.2705	0.972441	0.486221	−	0.873836i	$-0.338375\pi$
$353$	18.2705	0.972441	0.486221	+	0.873836i	$0.338375\pi$
$354$	3.52786	0.187504
$355$	0	0
$356$	−1.00000	−0.0529999
$357$	44.8328	2.37280
$358$	−17.6180	−0.931142
$359$	20.2361	1.06802	0.534009	−	0.845479i	$-0.320685\pi$
$359$	20.2361	1.06802	0.534009	+	0.845479i	$0.320685\pi$
$360$	0	0
$361$	−15.8885	−0.836239
$362$	33.1246	1.74099
$363$	−22.0000	−1.15470
$364$	8.56231	0.448787
$365$	0	0
$366$	−41.5967	−2.17430
$367$	−18.6738	−0.974762	−0.487381	−	0.873189i	$-0.662048\pi$
$367$	−18.6738	−0.974762	−0.487381	+	0.873189i	$0.662048\pi$
$368$	−27.2705	−1.42157
$369$	10.0902	0.525273
$370$	0	0
$371$	3.00000	0.155752
$372$	−0.291796	−0.0151289
$373$	−9.14590	−0.473557	−0.236778	−	0.971564i	$-0.576092\pi$
$373$	−9.14590	−0.473557	−0.236778	+	0.971564i	$0.576092\pi$
$374$	0	0
$375$	0	0
$376$	15.1246	0.779992
$377$	−48.3607	−2.49070
$378$	19.4164	0.998672
$379$	−26.5279	−1.36264	−0.681322	−	0.731983i	$-0.738595\pi$
$379$	−26.5279	−1.36264	−0.681322	+	0.731983i	$0.738595\pi$
$380$	0	0
$381$	42.5410	2.17944
$382$	11.6180	0.594430
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	−27.2361	−1.38988
$385$	0	0
$386$	25.1803	1.28165
$387$	4.23607	0.215331
$388$	−2.81966	−0.143147
$389$	−9.90983	−0.502448	−0.251224	−	0.967929i	$-0.580833\pi$
$389$	−9.90983	−0.502448	−0.251224	+	0.967929i	$0.580833\pi$
$390$	0	0
$391$	41.9787	2.12295
$392$	4.47214	0.225877
$393$	−27.7082	−1.39769
$394$	37.9787	1.91334
$395$	0	0
$396$	0	0
$397$	18.1246	0.909648	0.454824	−	0.890581i	$-0.349702\pi$
$397$	18.1246	0.909648	0.454824	+	0.890581i	$0.349702\pi$
$398$	−3.85410	−0.193189
$399$	−10.5836	−0.529842
$400$	0	0
$401$	−14.3262	−0.715418	−0.357709	−	0.933833i	$-0.616442\pi$
$401$	−14.3262	−0.715418	−0.357709	+	0.933833i	$0.616442\pi$
$402$	−22.1803	−1.10625
$403$	−1.09017	−0.0543052
$404$	0.381966	0.0190035
$405$	0	0
$406$	50.8328	2.52279
$407$	0	0
$408$	33.4164	1.65436
$409$	−31.9787	−1.58125	−0.790623	−	0.612303i	$-0.790243\pi$
$409$	−31.9787	−1.58125	−0.790623	+	0.612303i	$0.790243\pi$
$410$	0	0
$411$	2.76393	0.136335
$412$	−0.381966	−0.0188181
$413$	−3.27051	−0.160931
$414$	−9.09017	−0.446757
$415$	0	0
$416$	15.6180	0.765737
$417$	−14.8328	−0.726366
$418$	0	0
$419$	6.96556	0.340290	0.170145	−	0.985419i	$-0.445576\pi$
$419$	6.96556	0.340290	0.170145	+	0.985419i	$0.445576\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	13.6180	0.662916
$423$	6.76393	0.328874
$424$	2.23607	0.108593
$425$	0	0
$426$	21.4164	1.03763
$427$	38.5623	1.86616
$428$	−5.52786	−0.267199
$429$	0	0
$430$	0	0
$431$	17.4721	0.841603	0.420802	−	0.907153i	$-0.361749\pi$
$431$	17.4721	0.841603	0.420802	+	0.907153i	$0.361749\pi$
$432$	19.4164	0.934172
$433$	−19.9787	−0.960116	−0.480058	−	0.877237i	$-0.659384\pi$
$433$	−19.9787	−0.960116	−0.480058	+	0.877237i	$0.659384\pi$
$434$	1.14590	0.0550049
$435$	0	0
$436$	−8.50658	−0.407391
$437$	−9.90983	−0.474051
$438$	−34.8328	−1.66438
$439$	5.29180	0.252564	0.126282	−	0.991994i	$-0.459696\pi$
$439$	5.29180	0.252564	0.126282	+	0.991994i	$0.459696\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	−55.8328	−2.65570
$443$	10.5836	0.502842	0.251421	−	0.967878i	$-0.419102\pi$
$443$	10.5836	0.502842	0.251421	+	0.967878i	$0.419102\pi$
$444$	−3.34752	−0.158866
$445$	0	0
$446$	−3.32624	−0.157502
$447$	20.3607	0.963027
$448$	12.7082	0.600406
$449$	9.09017	0.428992	0.214496	−	0.976725i	$-0.431189\pi$
$449$	9.09017	0.428992	0.214496	+	0.976725i	$0.431189\pi$
$450$	0	0
$451$	0	0
$452$	9.03444	0.424944
$453$	18.0000	0.845714
$454$	12.8541	0.603273
$455$	0	0
$456$	−7.88854	−0.369415
$457$	−20.7984	−0.972907	−0.486453	−	0.873706i	$-0.661710\pi$
$457$	−20.7984	−0.972907	−0.486453	+	0.873706i	$0.661710\pi$
$458$	−19.1803	−0.896238
$459$	−29.8885	−1.39508
$460$	0	0
$461$	20.0344	0.933097	0.466548	−	0.884496i	$-0.345497\pi$
$461$	20.0344	0.933097	0.466548	+	0.884496i	$0.345497\pi$
$462$	0	0
$463$	−38.5623	−1.79214	−0.896071	−	0.443910i	$-0.853591\pi$
$463$	−38.5623	−1.79214	−0.896071	+	0.443910i	$0.853591\pi$
$464$	50.8328	2.35985
$465$	0	0
$466$	6.85410	0.317510
$467$	41.6525	1.92745	0.963723	−	0.266903i	$-0.0860004\pi$
$467$	41.6525	1.92745	0.963723	+	0.266903i	$0.0860004\pi$
$468$	2.85410	0.131931
$469$	20.5623	0.949479
$470$	0	0
$471$	−30.3607	−1.39895
$472$	−2.43769	−0.112204
$473$	0	0
$474$	8.18034	0.375735
$475$	0	0
$476$	13.8541	0.635002
$477$	1.00000	0.0457869
$478$	−45.4164	−2.07730
$479$	−42.8885	−1.95963	−0.979814	−	0.199912i	$-0.935934\pi$
$479$	−42.8885	−1.95963	−0.979814	+	0.199912i	$0.935934\pi$
$480$	0	0
$481$	−12.5066	−0.570251
$482$	42.5066	1.93612
$483$	33.7082	1.53378
$484$	−6.79837	−0.309017
$485$	0	0
$486$	16.1803	0.733955
$487$	34.5410	1.56520	0.782602	−	0.622523i	$-0.213892\pi$
$487$	34.5410	1.56520	0.782602	+	0.622523i	$0.213892\pi$
$488$	28.7426	1.30112
$489$	−4.58359	−0.207277
$490$	0	0
$491$	15.3607	0.693218	0.346609	−	0.938010i	$-0.387333\pi$
$491$	15.3607	0.693218	0.346609	+	0.938010i	$0.387333\pi$
$492$	12.4721	0.562287
$493$	−78.2492	−3.52417
$494$	13.1803	0.593012
$495$	0	0
$496$	1.14590	0.0514523
$497$	−19.8541	−0.890578
$498$	25.7082	1.15201
$499$	6.00000	0.268597	0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000	0.268597	0.134298	+	0.990941i	$0.457122\pi$
$500$	0	0
$501$	−16.3607	−0.730941
$502$	−20.1803	−0.900693
$503$	−16.4164	−0.731971	−0.365986	−	0.930620i	$-0.619268\pi$
$503$	−16.4164	−0.731971	−0.365986	+	0.930620i	$0.619268\pi$
$504$	6.70820	0.298807
$505$	0	0
$506$	0	0
$507$	16.6525	0.739562
$508$	13.1459	0.583255
$509$	−5.12461	−0.227144	−0.113572	−	0.993530i	$-0.536229\pi$
$509$	−5.12461	−0.227144	−0.113572	+	0.993530i	$0.536229\pi$
$510$	0	0
$511$	32.2918	1.42850
$512$	5.29180	0.233867
$513$	7.05573	0.311518
$514$	−29.7426	−1.31189
$515$	0	0
$516$	5.23607	0.230505
$517$	0	0
$518$	13.1459	0.577598
$519$	39.2361	1.72227
$520$	0	0
$521$	12.5967	0.551874	0.275937	−	0.961176i	$-0.411012\pi$
$521$	12.5967	0.551874	0.275937	+	0.961176i	$0.411012\pi$
$522$	16.9443	0.741631
$523$	−16.2705	−0.711460	−0.355730	−	0.934589i	$-0.615768\pi$
$523$	−16.2705	−0.711460	−0.355730	+	0.934589i	$0.615768\pi$
$524$	−8.56231	−0.374046
$525$	0	0
$526$	−13.8541	−0.604068
$527$	−1.76393	−0.0768381
$528$	0	0
$529$	8.56231	0.372274
$530$	0	0
$531$	−1.09017	−0.0473093
$532$	−3.27051	−0.141795
$533$	46.5967	2.01833
$534$	5.23607	0.226587
$535$	0	0
$536$	15.3262	0.661993
$537$	21.7771	0.939751
$538$	39.2705	1.69307
$539$	0	0
$540$	0	0
$541$	−1.29180	−0.0555387	−0.0277693	−	0.999614i	$-0.508840\pi$
$541$	−1.29180	−0.0555387	−0.0277693	+	0.999614i	$0.508840\pi$
$542$	14.0902	0.605225
$543$	−40.9443	−1.75709
$544$	25.2705	1.08346
$545$	0	0
$546$	−44.8328	−1.91867
$547$	27.5623	1.17848	0.589240	−	0.807958i	$-0.299427\pi$
$547$	27.5623	1.17848	0.589240	+	0.807958i	$0.299427\pi$
$548$	0.854102	0.0364854
$549$	12.8541	0.548600
$550$	0	0
$551$	18.4721	0.786939
$552$	25.1246	1.06937
$553$	−7.58359	−0.322487
$554$	23.3262	0.991037
$555$	0	0
$556$	−4.58359	−0.194388
$557$	13.4508	0.569931	0.284965	−	0.958538i	$-0.408018\pi$
$557$	13.4508	0.569931	0.284965	+	0.958538i	$0.408018\pi$
$558$	0.381966	0.0161699
$559$	19.5623	0.827397
$560$	0	0
$561$	0	0
$562$	−18.1803	−0.766891
$563$	32.5066	1.36999	0.684994	−	0.728548i	$-0.259805\pi$
$563$	32.5066	1.36999	0.684994	+	0.728548i	$0.259805\pi$
$564$	8.36068	0.352048
$565$	0	0
$566$	−31.0344	−1.30447
$567$	−33.0000	−1.38587
$568$	−14.7984	−0.620926
$569$	3.43769	0.144116	0.0720578	−	0.997400i	$-0.477043\pi$
$569$	3.43769	0.144116	0.0720578	+	0.997400i	$0.477043\pi$
$570$	0	0
$571$	8.79837	0.368200	0.184100	−	0.982907i	$-0.441063\pi$
$571$	8.79837	0.368200	0.184100	+	0.982907i	$0.441063\pi$
$572$	0	0
$573$	−14.3607	−0.599926
$574$	−48.9787	−2.04433
$575$	0	0
$576$	4.23607	0.176503
$577$	28.1803	1.17316	0.586581	−	0.809890i	$-0.300473\pi$
$577$	28.1803	1.17316	0.586581	+	0.809890i	$0.300473\pi$
$578$	−62.8328	−2.61350
$579$	−31.1246	−1.29349
$580$	0	0
$581$	−23.8328	−0.988752
$582$	14.7639	0.611985
$583$	0	0
$584$	24.0689	0.995977
$585$	0	0
$586$	−44.8328	−1.85203
$587$	11.3607	0.468905	0.234453	−	0.972128i	$-0.424670\pi$
$587$	11.3607	0.468905	0.234453	+	0.972128i	$0.424670\pi$
$588$	2.47214	0.101949
$589$	0.416408	0.0171578
$590$	0	0
$591$	−46.9443	−1.93103
$592$	13.1459	0.540293
$593$	9.97871	0.409777	0.204888	−	0.978785i	$-0.434317\pi$
$593$	9.97871	0.409777	0.204888	+	0.978785i	$0.434317\pi$
$594$	0	0
$595$	0	0
$596$	6.29180	0.257722
$597$	4.76393	0.194975
$598$	−41.9787	−1.71664
$599$	−28.7426	−1.17439	−0.587196	−	0.809445i	$-0.699768\pi$
$599$	−28.7426	−1.17439	−0.587196	+	0.809445i	$0.699768\pi$
$600$	0	0
$601$	8.29180	0.338229	0.169115	−	0.985596i	$-0.445909\pi$
$601$	8.29180	0.338229	0.169115	+	0.985596i	$0.445909\pi$
$602$	−20.5623	−0.838057
$603$	6.85410	0.279121
$604$	5.56231	0.226327
$605$	0	0
$606$	−2.00000	−0.0812444
$607$	29.5623	1.19990	0.599948	−	0.800039i	$-0.295188\pi$
$607$	29.5623	1.19990	0.599948	+	0.800039i	$0.295188\pi$
$608$	−5.96556	−0.241935
$609$	−62.8328	−2.54611
$610$	0	0
$611$	31.2361	1.26368
$612$	4.61803	0.186673
$613$	−46.1246	−1.86296	−0.931478	−	0.363798i	$-0.881480\pi$
$613$	−46.1246	−1.86296	−0.931478	+	0.363798i	$0.881480\pi$
$614$	3.47214	0.140124
$615$	0	0
$616$	0	0
$617$	21.3607	0.859949	0.429974	−	0.902841i	$-0.358523\pi$
$617$	21.3607	0.859949	0.429974	+	0.902841i	$0.358523\pi$
$618$	2.00000	0.0804518
$619$	−5.49342	−0.220799	−0.110400	−	0.993887i	$-0.535213\pi$
$619$	−5.49342	−0.220799	−0.110400	+	0.993887i	$0.535213\pi$
$620$	0	0
$621$	−22.4721	−0.901776
$622$	−2.47214	−0.0991236
$623$	−4.85410	−0.194475
$624$	−44.8328	−1.79475
$625$	0	0
$626$	−32.5623	−1.30145
$627$	0	0
$628$	−9.38197	−0.374381
$629$	−20.2361	−0.806865
$630$	0	0
$631$	−29.0557	−1.15669	−0.578345	−	0.815792i	$-0.696301\pi$
$631$	−29.0557	−1.15669	−0.578345	+	0.815792i	$0.696301\pi$
$632$	−5.65248	−0.224843
$633$	−16.8328	−0.669044
$634$	−37.5066	−1.48958
$635$	0	0
$636$	1.23607	0.0490133
$637$	9.23607	0.365946
$638$	0	0
$639$	−6.61803	−0.261805
$640$	0	0
$641$	6.81966	0.269360	0.134680	−	0.990889i	$-0.456999\pi$
$641$	6.81966	0.269360	0.134680	+	0.990889i	$0.456999\pi$
$642$	28.9443	1.14234
$643$	40.7771	1.60809	0.804046	−	0.594568i	$-0.202677\pi$
$643$	40.7771	1.60809	0.804046	+	0.594568i	$0.202677\pi$
$644$	10.4164	0.410464
$645$	0	0
$646$	21.3262	0.839070
$647$	17.5066	0.688255	0.344127	−	0.938923i	$-0.388175\pi$
$647$	17.5066	0.688255	0.344127	+	0.938923i	$0.388175\pi$
$648$	−24.5967	−0.966252
$649$	0	0
$650$	0	0
$651$	−1.41641	−0.0555134
$652$	−1.41641	−0.0554708
$653$	20.9443	0.819613	0.409806	−	0.912173i	$-0.365596\pi$
$653$	20.9443	0.819613	0.409806	+	0.912173i	$0.365596\pi$
$654$	44.5410	1.74169
$655$	0	0
$656$	−48.9787	−1.91230
$657$	10.7639	0.419941
$658$	−32.8328	−1.27996
$659$	−36.6312	−1.42695	−0.713474	−	0.700681i	$-0.752879\pi$
$659$	−36.6312	−1.42695	−0.713474	+	0.700681i	$0.752879\pi$
$660$	0	0
$661$	0.854102	0.0332207	0.0166104	−	0.999862i	$-0.494713\pi$
$661$	0.854102	0.0332207	0.0166104	+	0.999862i	$0.494713\pi$
$662$	−27.2705	−1.05990
$663$	69.0132	2.68025
$664$	−17.7639	−0.689374
$665$	0	0
$666$	4.38197	0.169798
$667$	−58.8328	−2.27802
$668$	−5.05573	−0.195612
$669$	4.11146	0.158958
$670$	0	0
$671$	0	0
$672$	20.2918	0.782773
$673$	11.0000	0.424019	0.212009	−	0.977268i	$-0.431999\pi$
$673$	11.0000	0.424019	0.212009	+	0.977268i	$0.431999\pi$
$674$	−23.3262	−0.898493
$675$	0	0
$676$	5.14590	0.197919
$677$	−2.18034	−0.0837973	−0.0418986	−	0.999122i	$-0.513341\pi$
$677$	−2.18034	−0.0837973	−0.0418986	+	0.999122i	$0.513341\pi$
$678$	−47.3050	−1.81674
$679$	−13.6869	−0.525256
$680$	0	0
$681$	−15.8885	−0.608850
$682$	0	0
$683$	−30.8885	−1.18192	−0.590959	−	0.806702i	$-0.701250\pi$
$683$	−30.8885	−1.18192	−0.590959	+	0.806702i	$0.701250\pi$
$684$	−1.09017	−0.0416837
$685$	0	0
$686$	24.2705	0.926652
$687$	23.7082	0.904524
$688$	−20.5623	−0.783931
$689$	4.61803	0.175933
$690$	0	0
$691$	0.124612	0.00474046	0.00237023	−	0.999997i	$-0.499246\pi$
$691$	0.124612	0.00474046	0.00237023	+	0.999997i	$0.499246\pi$
$692$	12.1246	0.460909
$693$	0	0
$694$	44.0344	1.67152
$695$	0	0
$696$	−46.8328	−1.77519
$697$	75.3951	2.85579
$698$	36.2705	1.37286
$699$	−8.47214	−0.320446
$700$	0	0
$701$	−5.67376	−0.214295	−0.107148	−	0.994243i	$-0.534172\pi$
$701$	−5.67376	−0.214295	−0.107148	+	0.994243i	$0.534172\pi$
$702$	29.8885	1.12807
$703$	4.77709	0.180171
$704$	0	0
$705$	0	0
$706$	−29.5623	−1.11259
$707$	1.85410	0.0697307
$708$	−1.34752	−0.0506431
$709$	−12.8541	−0.482746	−0.241373	−	0.970432i	$-0.577598\pi$
$709$	−12.8541	−0.482746	−0.241373	+	0.970432i	$0.577598\pi$
$710$	0	0
$711$	−2.52786	−0.0948023
$712$	−3.61803	−0.135592
$713$	−1.32624	−0.0496680
$714$	−72.5410	−2.71478
$715$	0	0
$716$	6.72949	0.251493
$717$	56.1378	2.09650
$718$	−32.7426	−1.22194
$719$	−10.0902	−0.376300	−0.188150	−	0.982140i	$-0.560249\pi$
$719$	−10.0902	−0.376300	−0.188150	+	0.982140i	$0.560249\pi$
$720$	0	0
$721$	−1.85410	−0.0690504
$722$	25.7082	0.956760
$723$	−52.5410	−1.95402
$724$	−12.6525	−0.470226
$725$	0	0
$726$	35.5967	1.32112
$727$	−18.1459	−0.672994	−0.336497	−	0.941685i	$-0.609242\pi$
$727$	−18.1459	−0.672994	−0.336497	+	0.941685i	$0.609242\pi$
$728$	30.9787	1.14815
$729$	13.0000	0.481481
$730$	0	0
$731$	31.6525	1.17071
$732$	15.8885	0.587257
$733$	11.4721	0.423733	0.211867	−	0.977299i	$-0.432046\pi$
$733$	11.4721	0.423733	0.211867	+	0.977299i	$0.432046\pi$
$734$	30.2148	1.11525
$735$	0	0
$736$	19.0000	0.700349
$737$	0	0
$738$	−16.3262	−0.600977
$739$	−51.6312	−1.89928	−0.949642	−	0.313337i	$-0.898553\pi$
$739$	−51.6312	−1.89928	−0.949642	+	0.313337i	$0.898553\pi$
$740$	0	0
$741$	−16.2918	−0.598494
$742$	−4.85410	−0.178200
$743$	21.3607	0.783647	0.391824	−	0.920040i	$-0.371844\pi$
$743$	21.3607	0.783647	0.391824	+	0.920040i	$0.371844\pi$
$744$	−1.05573	−0.0387049
$745$	0	0
$746$	14.7984	0.541807
$747$	−7.94427	−0.290666
$748$	0	0
$749$	−26.8328	−0.980450
$750$	0	0
$751$	52.3951	1.91193	0.955963	−	0.293489i	$-0.0948163\pi$
$751$	52.3951	1.91193	0.955963	+	0.293489i	$0.0948163\pi$
$752$	−32.8328	−1.19729
$753$	24.9443	0.909020
$754$	78.2492	2.84967
$755$	0	0
$756$	−7.41641	−0.269732
$757$	−20.1246	−0.731441	−0.365721	−	0.930725i	$-0.619177\pi$
$757$	−20.1246	−0.731441	−0.365721	+	0.930725i	$0.619177\pi$
$758$	42.9230	1.55903
$759$	0	0
$760$	0	0
$761$	−37.8541	−1.37221	−0.686105	−	0.727502i	$-0.740681\pi$
$761$	−37.8541	−1.37221	−0.686105	+	0.727502i	$0.740681\pi$
$762$	−68.8328	−2.49355
$763$	−41.2918	−1.49486
$764$	−4.43769	−0.160550
$765$	0	0
$766$	−33.9787	−1.22770
$767$	−5.03444	−0.181783
$768$	27.1246	0.978775
$769$	−12.2918	−0.443254	−0.221627	−	0.975132i	$-0.571137\pi$
$769$	−12.2918	−0.443254	−0.221627	+	0.975132i	$0.571137\pi$
$770$	0	0
$771$	36.7639	1.32402
$772$	−9.61803	−0.346161
$773$	31.4508	1.13121	0.565604	−	0.824677i	$-0.308643\pi$
$773$	31.4508	1.13121	0.565604	+	0.824677i	$0.308643\pi$
$774$	−6.85410	−0.246366
$775$	0	0
$776$	−10.2016	−0.366217
$777$	−16.2492	−0.582938
$778$	16.0344	0.574863
$779$	−17.7984	−0.637693
$780$	0	0
$781$	0	0
$782$	−67.9230	−2.42892
$783$	41.8885	1.49697
$784$	−9.70820	−0.346722
$785$	0	0
$786$	44.8328	1.59913
$787$	11.4934	0.409696	0.204848	−	0.978794i	$-0.434330\pi$
$787$	11.4934	0.409696	0.204848	+	0.978794i	$0.434330\pi$
$788$	−14.5066	−0.516775
$789$	17.1246	0.609652
$790$	0	0
$791$	43.8541	1.55927
$792$	0	0
$793$	59.3607	2.10796
$794$	−29.3262	−1.04075
$795$	0	0
$796$	1.47214	0.0521785
$797$	−34.0344	−1.20556	−0.602781	−	0.797907i	$-0.705941\pi$
$797$	−34.0344	−1.20556	−0.602781	+	0.797907i	$0.705941\pi$
$798$	17.1246	0.606205
$799$	50.5410	1.78801
$800$	0	0
$801$	−1.61803	−0.0571704
$802$	23.1803	0.818526
$803$	0	0
$804$	8.47214	0.298789
$805$	0	0
$806$	1.76393	0.0621319
$807$	−48.5410	−1.70872
$808$	1.38197	0.0486174
$809$	−20.2361	−0.711462	−0.355731	−	0.934588i	$-0.615768\pi$
$809$	−20.2361	−0.711462	−0.355731	+	0.934588i	$0.615768\pi$
$810$	0	0
$811$	28.0132	0.983675	0.491837	−	0.870687i	$-0.336325\pi$
$811$	28.0132	0.983675	0.491837	+	0.870687i	$0.336325\pi$
$812$	−19.4164	−0.681382
$813$	−17.4164	−0.610820
$814$	0	0
$815$	0	0
$816$	−72.5410	−2.53944
$817$	−7.47214	−0.261417
$818$	51.7426	1.80914
$819$	13.8541	0.484102
$820$	0	0
$821$	25.4508	0.888241	0.444120	−	0.895967i	$-0.353516\pi$
$821$	25.4508	0.888241	0.444120	+	0.895967i	$0.353516\pi$
$822$	−4.47214	−0.155984
$823$	10.8328	0.377608	0.188804	−	0.982015i	$-0.439539\pi$
$823$	10.8328	0.377608	0.188804	+	0.982015i	$0.439539\pi$
$824$	−1.38197	−0.0481431
$825$	0	0
$826$	5.29180	0.184125
$827$	29.5066	1.02604	0.513022	−	0.858375i	$-0.328526\pi$
$827$	29.5066	1.02604	0.513022	+	0.858375i	$0.328526\pi$
$828$	3.47214	0.120665
$829$	−12.2705	−0.426172	−0.213086	−	0.977033i	$-0.568352\pi$
$829$	−12.2705	−0.426172	−0.213086	+	0.977033i	$0.568352\pi$
$830$	0	0
$831$	−28.8328	−1.00020
$832$	19.5623	0.678201
$833$	14.9443	0.517788
$834$	24.0000	0.831052
$835$	0	0
$836$	0	0
$837$	0.944272	0.0326388
$838$	−11.2705	−0.389333
$839$	27.4721	0.948443	0.474222	−	0.880405i	$-0.342730\pi$
$839$	27.4721	0.948443	0.474222	+	0.880405i	$0.342730\pi$
$840$	0	0
$841$	80.6656	2.78157
$842$	−3.23607	−0.111522
$843$	22.4721	0.773981
$844$	−5.20163	−0.179047
$845$	0	0
$846$	−10.9443	−0.376272
$847$	−33.0000	−1.13389
$848$	−4.85410	−0.166691
$849$	38.3607	1.31654
$850$	0	0
$851$	−15.2148	−0.521556
$852$	−8.18034	−0.280254
$853$	−6.56231	−0.224689	−0.112345	−	0.993669i	$-0.535836\pi$
$853$	−6.56231	−0.224689	−0.112345	+	0.993669i	$0.535836\pi$
$854$	−62.3951	−2.13512
$855$	0	0
$856$	−20.0000	−0.683586
$857$	17.2016	0.587596	0.293798	−	0.955867i	$-0.405081\pi$
$857$	17.2016	0.587596	0.293798	+	0.955867i	$0.405081\pi$
$858$	0	0
$859$	17.8328	0.608448	0.304224	−	0.952601i	$-0.401603\pi$
$859$	17.8328	0.608448	0.304224	+	0.952601i	$0.401603\pi$
$860$	0	0
$861$	60.5410	2.06323
$862$	−28.2705	−0.962897
$863$	−13.9098	−0.473496	−0.236748	−	0.971571i	$-0.576082\pi$
$863$	−13.9098	−0.473496	−0.236748	+	0.971571i	$0.576082\pi$
$864$	−13.5279	−0.460227
$865$	0	0
$866$	32.3262	1.09849
$867$	77.6656	2.63766
$868$	−0.437694	−0.0148563
$869$	0	0
$870$	0	0
$871$	31.6525	1.07250
$872$	−30.7771	−1.04224
$873$	−4.56231	−0.154411
$874$	16.0344	0.542373
$875$	0	0
$876$	13.3050	0.449533
$877$	−39.0689	−1.31926	−0.659631	−	0.751589i	$-0.729288\pi$
$877$	−39.0689	−1.31926	−0.659631	+	0.751589i	$0.729288\pi$
$878$	−8.56231	−0.288964
$879$	55.4164	1.86915
$880$	0	0
$881$	−34.3820	−1.15836	−0.579179	−	0.815200i	$-0.696627\pi$
$881$	−34.3820	−1.15836	−0.579179	+	0.815200i	$0.696627\pi$
$882$	−3.23607	−0.108964
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	21.3262	0.717279
$885$	0	0
$886$	−17.1246	−0.575313
$887$	−54.4853	−1.82944	−0.914719	−	0.404092i	$-0.867588\pi$
$887$	−54.4853	−1.82944	−0.914719	+	0.404092i	$0.867588\pi$
$888$	−12.1115	−0.406434
$889$	63.8115	2.14017
$890$	0	0
$891$	0	0
$892$	1.27051	0.0425398
$893$	−11.9311	−0.399260
$894$	−32.9443	−1.10182
$895$	0	0
$896$	−40.8541	−1.36484
$897$	51.8885	1.73251
$898$	−14.7082	−0.490819
$899$	2.47214	0.0824504
$900$	0	0
$901$	7.47214	0.248933
$902$	0	0
$903$	25.4164	0.845805
$904$	32.6869	1.08715
$905$	0	0
$906$	−29.1246	−0.967600
$907$	−2.70820	−0.0899244	−0.0449622	−	0.998989i	$-0.514317\pi$
$907$	−2.70820	−0.0899244	−0.0449622	+	0.998989i	$0.514317\pi$
$908$	−4.90983	−0.162938
$909$	0.618034	0.0204989
$910$	0	0
$911$	31.3050	1.03718	0.518590	−	0.855023i	$-0.326457\pi$
$911$	31.3050	1.03718	0.518590	+	0.855023i	$0.326457\pi$
$912$	17.1246	0.567053
$913$	0	0
$914$	33.6525	1.11312
$915$	0	0
$916$	7.32624	0.242066
$917$	−41.5623	−1.37251
$918$	48.3607	1.59614
$919$	23.4721	0.774274	0.387137	−	0.922022i	$-0.373464\pi$
$919$	23.4721	0.774274	0.387137	+	0.922022i	$0.373464\pi$
$920$	0	0
$921$	−4.29180	−0.141419
$922$	−32.4164	−1.06758
$923$	−30.5623	−1.00597
$924$	0	0
$925$	0	0
$926$	62.3951	2.05043
$927$	−0.618034	−0.0202989
$928$	−35.4164	−1.16260
$929$	9.76393	0.320344	0.160172	−	0.987089i	$-0.448795\pi$
$929$	9.76393	0.320344	0.160172	+	0.987089i	$0.448795\pi$
$930$	0	0
$931$	−3.52786	−0.115621
$932$	−2.61803	−0.0857566
$933$	3.05573	0.100040
$934$	−67.3951	−2.20524
$935$	0	0
$936$	10.3262	0.337524
$937$	50.7639	1.65839	0.829193	−	0.558963i	$-0.188801\pi$
$937$	50.7639	1.65839	0.829193	+	0.558963i	$0.188801\pi$
$938$	−33.2705	−1.08632
$939$	40.2492	1.31348
$940$	0	0
$941$	−15.0000	−0.488986	−0.244493	−	0.969651i	$-0.578622\pi$
$941$	−15.0000	−0.488986	−0.244493	+	0.969651i	$0.578622\pi$
$942$	49.1246	1.60057
$943$	56.6869	1.84598
$944$	5.29180	0.172233
$945$	0	0
$946$	0	0
$947$	−40.7639	−1.32465	−0.662325	−	0.749217i	$-0.730430\pi$
$947$	−40.7639	−1.32465	−0.662325	+	0.749217i	$0.730430\pi$
$948$	−3.12461	−0.101483
$949$	49.7082	1.61360
$950$	0	0
$951$	46.3607	1.50335
$952$	50.1246	1.62455
$953$	26.9443	0.872811	0.436405	−	0.899750i	$-0.356251\pi$
$953$	26.9443	0.872811	0.436405	+	0.899750i	$0.356251\pi$
$954$	−1.61803	−0.0523858
$955$	0	0
$956$	17.3475	0.561059
$957$	0	0
$958$	69.3951	2.24205
$959$	4.14590	0.133878
$960$	0	0
$961$	−30.9443	−0.998202
$962$	20.2361	0.652437
$963$	−8.94427	−0.288225
$964$	−16.2361	−0.522929
$965$	0	0
$966$	−54.5410	−1.75483
$967$	−3.70820	−0.119248	−0.0596239	−	0.998221i	$-0.518990\pi$
$967$	−3.70820	−0.119248	−0.0596239	+	0.998221i	$0.518990\pi$
$968$	−24.5967	−0.790569
$969$	−26.3607	−0.846827
$970$	0	0
$971$	−32.0557	−1.02872	−0.514359	−	0.857575i	$-0.671970\pi$
$971$	−32.0557	−1.02872	−0.514359	+	0.857575i	$0.671970\pi$
$972$	−6.18034	−0.198234
$973$	−22.2492	−0.713277
$974$	−55.8885	−1.79078
$975$	0	0
$976$	−62.3951	−1.99722
$977$	47.7426	1.52742	0.763711	−	0.645558i	$-0.223375\pi$
$977$	47.7426	1.52742	0.763711	+	0.645558i	$0.223375\pi$
$978$	7.41641	0.237151
$979$	0	0
$980$	0	0
$981$	−13.7639	−0.439449
$982$	−24.8541	−0.793126
$983$	54.8115	1.74822	0.874108	−	0.485731i	$-0.161447\pi$
$983$	54.8115	1.74822	0.874108	+	0.485731i	$0.161447\pi$
$984$	45.1246	1.43852
$985$	0	0
$986$	126.610	4.03208
$987$	40.5836	1.29179
$988$	−5.03444	−0.160167
$989$	23.7984	0.756744
$990$	0	0
$991$	−14.8328	−0.471180	−0.235590	−	0.971853i	$-0.575702\pi$
$991$	−14.8328	−0.471180	−0.235590	+	0.971853i	$0.575702\pi$
$992$	−0.798374	−0.0253484
$993$	33.7082	1.06970
$994$	32.1246	1.01893
$995$	0	0
$996$	−9.81966	−0.311148
$997$	55.2492	1.74976	0.874880	−	0.484339i	$-0.160940\pi$
$997$	55.2492	1.74976	0.874880	+	0.484339i	$0.160940\pi$
$998$	−9.70820	−0.307308
$999$	10.8328	0.342735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6625.2.a.b.1.1	✓	2
5.4	even	2		6625.2.a.c.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6625.2.a.b.1.1	✓	2	1.1	even	1	trivial
6625.2.a.c.1.2	yes	2	5.4	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:	\( N \)	\(=\)	\( 6625 = 5^{3} \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6625.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} +2.00000 q^{3} +0.618034 q^{4} -3.23607 q^{6} +3.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.61803 q^{2} +2.00000 q^{3} +0.618034 q^{4} -3.23607 q^{6} +3.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +1.23607 q^{12} +4.61803 q^{13} -4.85410 q^{14} -4.85410 q^{16} +7.47214 q^{17} -1.61803 q^{18} -1.76393 q^{19} +6.00000 q^{21} +5.61803 q^{23} +4.47214 q^{24} -7.47214 q^{26} -4.00000 q^{27} +1.85410 q^{28} -10.4721 q^{29} -0.236068 q^{31} +3.38197 q^{32} -12.0902 q^{34} +0.618034 q^{36} -2.70820 q^{37} +2.85410 q^{38} +9.23607 q^{39} +10.0902 q^{41} -9.70820 q^{42} +4.23607 q^{43} -9.09017 q^{46} +6.76393 q^{47} -9.70820 q^{48} +2.00000 q^{49} +14.9443 q^{51} +2.85410 q^{52} +1.00000 q^{53} +6.47214 q^{54} +6.70820 q^{56} -3.52786 q^{57} +16.9443 q^{58} -1.09017 q^{59} +12.8541 q^{61} +0.381966 q^{62} +3.00000 q^{63} +4.23607 q^{64} +6.85410 q^{67} +4.61803 q^{68} +11.2361 q^{69} -6.61803 q^{71} +2.23607 q^{72} +10.7639 q^{73} +4.38197 q^{74} -1.09017 q^{76} -14.9443 q^{78} -2.52786 q^{79} -11.0000 q^{81} -16.3262 q^{82} -7.94427 q^{83} +3.70820 q^{84} -6.85410 q^{86} -20.9443 q^{87} -1.61803 q^{89} +13.8541 q^{91} +3.47214 q^{92} -0.472136 q^{93} -10.9443 q^{94} +6.76393 q^{96} -4.56231 q^{97} -3.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 4 q^{3} - q^{4} - 2 q^{6} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - q^2 + 4 * q^3 - q^4 - 2 * q^6 + 6 * q^7 + 2 * q^9	\( 2 q - q^{2} + 4 q^{3} - q^{4} - 2 q^{6} + 6 q^{7} + 2 q^{9} - 2 q^{12} + 7 q^{13} - 3 q^{14} - 3 q^{16} + 6 q^{17} - q^{18} - 8 q^{19} + 12 q^{21} + 9 q^{23} - 6 q^{26} - 8 q^{27} - 3 q^{28} - 12 q^{29} + 4 q^{31} + 9 q^{32} - 13 q^{34} - q^{36} + 8 q^{37} - q^{38} + 14 q^{39} + 9 q^{41} - 6 q^{42} + 4 q^{43} - 7 q^{46} + 18 q^{47} - 6 q^{48} + 4 q^{49} + 12 q^{51} - q^{52} + 2 q^{53} + 4 q^{54} - 16 q^{57} + 16 q^{58} + 9 q^{59} + 19 q^{61} + 3 q^{62} + 6 q^{63} + 4 q^{64} + 7 q^{67} + 7 q^{68} + 18 q^{69} - 11 q^{71} + 26 q^{73} + 11 q^{74} + 9 q^{76} - 12 q^{78} - 14 q^{79} - 22 q^{81} - 17 q^{82} + 2 q^{83} - 6 q^{84} - 7 q^{86} - 24 q^{87} - q^{89} + 21 q^{91} - 2 q^{92} + 8 q^{93} - 4 q^{94} + 18 q^{96} + 11 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - q^2 + 4 * q^3 - q^4 - 2 * q^6 + 6 * q^7 + 2 * q^9 - 2 * q^12 + 7 * q^13 - 3 * q^14 - 3 * q^16 + 6 * q^17 - q^18 - 8 * q^19 + 12 * q^21 + 9 * q^23 - 6 * q^26 - 8 * q^27 - 3 * q^28 - 12 * q^29 + 4 * q^31 + 9 * q^32 - 13 * q^34 - q^36 + 8 * q^37 - q^38 + 14 * q^39 + 9 * q^41 - 6 * q^42 + 4 * q^43 - 7 * q^46 + 18 * q^47 - 6 * q^48 + 4 * q^49 + 12 * q^51 - q^52 + 2 * q^53 + 4 * q^54 - 16 * q^57 + 16 * q^58 + 9 * q^59 + 19 * q^61 + 3 * q^62 + 6 * q^63 + 4 * q^64 + 7 * q^67 + 7 * q^68 + 18 * q^69 - 11 * q^71 + 26 * q^73 + 11 * q^74 + 9 * q^76 - 12 * q^78 - 14 * q^79 - 22 * q^81 - 17 * q^82 + 2 * q^83 - 6 * q^84 - 7 * q^86 - 24 * q^87 - q^89 + 21 * q^91 - 2 * q^92 + 8 * q^93 - 4 * q^94 + 18 * q^96 + 11 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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