LMFDB - Embedded newform 6625.2.a.b.1.2

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			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+0.618034 q^{2} +2.00000 q^{3} -1.61803 q^{4} +1.23607 q^{6} +3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.618034 q^{2} +2.00000 q^{3} -1.61803 q^{4} +1.23607 q^{6} +3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -3.23607 q^{12} +2.38197 q^{13} +1.85410 q^{14} +1.85410 q^{16} -1.47214 q^{17} +0.618034 q^{18} -6.23607 q^{19} +6.00000 q^{21} +3.38197 q^{23} -4.47214 q^{24} +1.47214 q^{26} -4.00000 q^{27} -4.85410 q^{28} -1.52786 q^{29} +4.23607 q^{31} +5.61803 q^{32} -0.909830 q^{34} -1.61803 q^{36} +10.7082 q^{37} -3.85410 q^{38} +4.76393 q^{39} -1.09017 q^{41} +3.70820 q^{42} -0.236068 q^{43} +2.09017 q^{46} +11.2361 q^{47} +3.70820 q^{48} +2.00000 q^{49} -2.94427 q^{51} -3.85410 q^{52} +1.00000 q^{53} -2.47214 q^{54} -6.70820 q^{56} -12.4721 q^{57} -0.944272 q^{58} +10.0902 q^{59} +6.14590 q^{61} +2.61803 q^{62} +3.00000 q^{63} -0.236068 q^{64} +0.145898 q^{67} +2.38197 q^{68} +6.76393 q^{69} -4.38197 q^{71} -2.23607 q^{72} +15.2361 q^{73} +6.61803 q^{74} +10.0902 q^{76} +2.94427 q^{78} -11.4721 q^{79} -11.0000 q^{81} -0.673762 q^{82} +9.94427 q^{83} -9.70820 q^{84} -0.145898 q^{86} -3.05573 q^{87} +0.618034 q^{89} +7.14590 q^{91} -5.47214 q^{92} +8.47214 q^{93} +6.94427 q^{94} +11.2361 q^{96} +15.5623 q^{97} +1.23607 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	1.23607	0.504623
$7$	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	$0.5\pi$
$12$	−3.23607	−0.934172
$13$	2.38197	0.660639	0.330319	−	0.943869i	$-0.392844\pi$
$13$	2.38197	0.660639	0.330319	+	0.943869i	$0.392844\pi$
$14$	1.85410	0.495530
$15$	0	0
$16$	1.85410	0.463525
$17$	−1.47214	−0.357045	−0.178523	−	0.983936i	$-0.557132\pi$
$17$	−1.47214	−0.357045	−0.178523	+	0.983936i	$0.557132\pi$
$18$	0.618034	0.145672
$19$	−6.23607	−1.43065	−0.715326	−	0.698791i	$-0.753722\pi$
$19$	−6.23607	−1.43065	−0.715326	+	0.698791i	$0.753722\pi$
$20$	0	0
$21$	6.00000	1.30931
$22$	0	0
$23$	3.38197	0.705189	0.352594	−	0.935776i	$-0.385300\pi$
$23$	3.38197	0.705189	0.352594	+	0.935776i	$0.385300\pi$
$24$	−4.47214	−0.912871
$25$	0	0
$26$	1.47214	0.288710
$27$	−4.00000	−0.769800
$28$	−4.85410	−0.917339
$29$	−1.52786	−0.283717	−0.141859	−	0.989887i	$-0.545308\pi$
$29$	−1.52786	−0.283717	−0.141859	+	0.989887i	$0.545308\pi$
$30$	0	0
$31$	4.23607	0.760820	0.380410	−	0.924818i	$-0.375783\pi$
$31$	4.23607	0.760820	0.380410	+	0.924818i	$0.375783\pi$
$32$	5.61803	0.993137
$33$	0	0
$34$	−0.909830	−0.156035
$35$	0	0
$36$	−1.61803	−0.269672
$37$	10.7082	1.76042	0.880209	−	0.474586i	$-0.157402\pi$
$37$	10.7082	1.76042	0.880209	+	0.474586i	$0.157402\pi$
$38$	−3.85410	−0.625218
$39$	4.76393	0.762840
$40$	0	0
$41$	−1.09017	−0.170256	−0.0851280	−	0.996370i	$-0.527130\pi$
$41$	−1.09017	−0.170256	−0.0851280	+	0.996370i	$0.527130\pi$
$42$	3.70820	0.572188
$43$	−0.236068	−0.0360000	−0.0180000	−	0.999838i	$-0.505730\pi$
$43$	−0.236068	−0.0360000	−0.0180000	+	0.999838i	$0.505730\pi$
$44$	0	0
$45$	0	0
$46$	2.09017	0.308179
$47$	11.2361	1.63895	0.819474	−	0.573116i	$-0.194265\pi$
$47$	11.2361	1.63895	0.819474	+	0.573116i	$0.194265\pi$
$48$	3.70820	0.535233
$49$	2.00000	0.285714
$50$	0	0
$51$	−2.94427	−0.412281
$52$	−3.85410	−0.534468
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	−2.47214	−0.336415
$55$	0	0
$56$	−6.70820	−0.896421
$57$	−12.4721	−1.65197
$58$	−0.944272	−0.123989
$59$	10.0902	1.31363	0.656814	−	0.754053i	$-0.271904\pi$
$59$	10.0902	1.31363	0.656814	+	0.754053i	$0.271904\pi$
$60$	0	0
$61$	6.14590	0.786902	0.393451	−	0.919346i	$-0.371281\pi$
$61$	6.14590	0.786902	0.393451	+	0.919346i	$0.371281\pi$
$62$	2.61803	0.332491
$63$	3.00000	0.377964
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	0	0
$67$	0.145898	0.0178243	0.00891214	−	0.999960i	$-0.497163\pi$
$67$	0.145898	0.0178243	0.00891214	+	0.999960i	$0.497163\pi$
$68$	2.38197	0.288856
$69$	6.76393	0.814282
$70$	0	0
$71$	−4.38197	−0.520044	−0.260022	−	0.965603i	$-0.583730\pi$
$71$	−4.38197	−0.520044	−0.260022	+	0.965603i	$0.583730\pi$
$72$	−2.23607	−0.263523
$73$	15.2361	1.78325	0.891623	−	0.452778i	$-0.149567\pi$
$73$	15.2361	1.78325	0.891623	+	0.452778i	$0.149567\pi$
$74$	6.61803	0.769331
$75$	0	0
$76$	10.0902	1.15742
$77$	0	0
$78$	2.94427	0.333373
$79$	−11.4721	−1.29072	−0.645358	−	0.763880i	$-0.723292\pi$
$79$	−11.4721	−1.29072	−0.645358	+	0.763880i	$0.723292\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	−0.673762	−0.0744046
$83$	9.94427	1.09153	0.545763	−	0.837940i	$-0.316240\pi$
$83$	9.94427	1.09153	0.545763	+	0.837940i	$0.316240\pi$
$84$	−9.70820	−1.05925
$85$	0	0
$86$	−0.145898	−0.0157326
$87$	−3.05573	−0.327608
$88$	0	0
$89$	0.618034	0.0655115	0.0327557	−	0.999463i	$-0.489572\pi$
$89$	0.618034	0.0655115	0.0327557	+	0.999463i	$0.489572\pi$
$90$	0	0
$91$	7.14590	0.749094
$92$	−5.47214	−0.570510
$93$	8.47214	0.878520
$94$	6.94427	0.716247
$95$	0	0
$96$	11.2361	1.14678
$97$	15.5623	1.58011	0.790056	−	0.613034i	$-0.210051\pi$
$97$	15.5623	1.58011	0.790056	+	0.613034i	$0.210051\pi$
$98$	1.23607	0.124862
$99$	0	0
$100$	0	0
$101$	−1.61803	−0.161000	−0.0805002	−	0.996755i	$-0.525652\pi$
$101$	−1.61803	−0.161000	−0.0805002	+	0.996755i	$0.525652\pi$
$102$	−1.81966	−0.180173
$103$	1.61803	0.159430	0.0797148	−	0.996818i	$-0.474599\pi$
$103$	1.61803	0.159430	0.0797148	+	0.996818i	$0.474599\pi$
$104$	−5.32624	−0.522281
$105$	0	0
$106$	0.618034	0.0600288
$107$	8.94427	0.864675	0.432338	−	0.901712i	$-0.357689\pi$
$107$	8.94427	0.864675	0.432338	+	0.901712i	$0.357689\pi$
$108$	6.47214	0.622782
$109$	−18.2361	−1.74670	−0.873349	−	0.487095i	$-0.838057\pi$
$109$	−18.2361	−1.74670	−0.873349	+	0.487095i	$0.838057\pi$
$110$	0	0
$111$	21.4164	2.03276
$112$	5.56231	0.525589
$113$	12.3820	1.16480	0.582399	−	0.812903i	$-0.302114\pi$
$113$	12.3820	1.16480	0.582399	+	0.812903i	$0.302114\pi$
$114$	−7.70820	−0.721939
$115$	0	0
$116$	2.47214	0.229532
$117$	2.38197	0.220213
$118$	6.23607	0.574077
$119$	−4.41641	−0.404851
$120$	0	0
$121$	−11.0000	−1.00000
$122$	3.79837	0.343889
$123$	−2.18034	−0.196595
$124$	−6.85410	−0.615517
$125$	0	0
$126$	1.85410	0.165177
$127$	−12.2705	−1.08883	−0.544416	−	0.838815i	$-0.683249\pi$
$127$	−12.2705	−1.08883	−0.544416	+	0.838815i	$0.683249\pi$
$128$	−11.3820	−1.00603
$129$	−0.472136	−0.0415693
$130$	0	0
$131$	−7.14590	−0.624340	−0.312170	−	0.950026i	$-0.601056\pi$
$131$	−7.14590	−0.624340	−0.312170	+	0.950026i	$0.601056\pi$
$132$	0	0
$133$	−18.7082	−1.62221
$134$	0.0901699	0.00778950
$135$	0	0
$136$	3.29180	0.282269
$137$	3.61803	0.309110	0.154555	−	0.987984i	$-0.450606\pi$
$137$	3.61803	0.309110	0.154555	+	0.987984i	$0.450606\pi$
$138$	4.18034	0.355854
$139$	19.4164	1.64688	0.823439	−	0.567405i	$-0.192052\pi$
$139$	19.4164	1.64688	0.823439	+	0.567405i	$0.192052\pi$
$140$	0	0
$141$	22.4721	1.89250
$142$	−2.70820	−0.227267
$143$	0	0
$144$	1.85410	0.154508
$145$	0	0
$146$	9.41641	0.779307
$147$	4.00000	0.329914
$148$	−17.3262	−1.42421
$149$	−12.1803	−0.997852	−0.498926	−	0.866644i	$-0.666272\pi$
$149$	−12.1803	−0.997852	−0.498926	+	0.866644i	$0.666272\pi$
$150$	0	0
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	13.9443	1.13103
$153$	−1.47214	−0.119015
$154$	0	0
$155$	0	0
$156$	−7.70820	−0.617150
$157$	7.18034	0.573054	0.286527	−	0.958072i	$-0.407499\pi$
$157$	7.18034	0.573054	0.286527	+	0.958072i	$0.407499\pi$
$158$	−7.09017	−0.564064
$159$	2.00000	0.158610
$160$	0	0
$161$	10.1459	0.799609
$162$	−6.79837	−0.534131
$163$	−15.7082	−1.23036	−0.615181	−	0.788386i	$-0.710917\pi$
$163$	−15.7082	−1.23036	−0.615181	+	0.788386i	$0.710917\pi$
$164$	1.76393	0.137740
$165$	0	0
$166$	6.14590	0.477014
$167$	14.1803	1.09731	0.548654	−	0.836050i	$-0.315141\pi$
$167$	14.1803	1.09731	0.548654	+	0.836050i	$0.315141\pi$
$168$	−13.4164	−1.03510
$169$	−7.32624	−0.563557
$170$	0	0
$171$	−6.23607	−0.476884
$172$	0.381966	0.0291246
$173$	17.3820	1.32153	0.660763	−	0.750594i	$-0.270233\pi$
$173$	17.3820	1.32153	0.660763	+	0.750594i	$0.270233\pi$
$174$	−1.88854	−0.143170
$175$	0	0
$176$	0	0
$177$	20.1803	1.51685
$178$	0.381966	0.0286296
$179$	−24.8885	−1.86026	−0.930129	−	0.367234i	$-0.880305\pi$
$179$	−24.8885	−1.86026	−0.930129	+	0.367234i	$0.880305\pi$
$180$	0	0
$181$	−11.5279	−0.856859	−0.428430	−	0.903575i	$-0.640933\pi$
$181$	−11.5279	−0.856859	−0.428430	+	0.903575i	$0.640933\pi$
$182$	4.41641	0.327366
$183$	12.2918	0.908636
$184$	−7.56231	−0.557501
$185$	0	0
$186$	5.23607	0.383927
$187$	0	0
$188$	−18.1803	−1.32594
$189$	−12.0000	−0.872872
$190$	0	0
$191$	15.1803	1.09841	0.549205	−	0.835687i	$-0.314930\pi$
$191$	15.1803	1.09841	0.549205	+	0.835687i	$0.314930\pi$
$192$	−0.472136	−0.0340735
$193$	4.56231	0.328402	0.164201	−	0.986427i	$-0.447495\pi$
$193$	4.56231	0.328402	0.164201	+	0.986427i	$0.447495\pi$
$194$	9.61803	0.690535
$195$	0	0
$196$	−3.23607	−0.231148
$197$	−14.5279	−1.03507	−0.517534	−	0.855663i	$-0.673150\pi$
$197$	−14.5279	−1.03507	−0.517534	+	0.855663i	$0.673150\pi$
$198$	0	0
$199$	4.61803	0.327364	0.163682	−	0.986513i	$-0.447663\pi$
$199$	4.61803	0.327364	0.163682	+	0.986513i	$0.447663\pi$
$200$	0	0
$201$	0.291796	0.0205817
$202$	−1.00000	−0.0703598
$203$	−4.58359	−0.321705
$204$	4.76393	0.333542
$205$	0	0
$206$	1.00000	0.0696733
$207$	3.38197	0.235063
$208$	4.41641	0.306223
$209$	0	0
$210$	0	0
$211$	18.4164	1.26784	0.633919	−	0.773400i	$-0.281445\pi$
$211$	18.4164	1.26784	0.633919	+	0.773400i	$0.281445\pi$
$212$	−1.61803	−0.111127
$213$	−8.76393	−0.600495
$214$	5.52786	0.377877
$215$	0	0
$216$	8.94427	0.608581
$217$	12.7082	0.862689
$218$	−11.2705	−0.763335
$219$	30.4721	2.05912
$220$	0	0
$221$	−3.50658	−0.235878
$222$	13.2361	0.888347
$223$	19.9443	1.33557	0.667784	−	0.744355i	$-0.267243\pi$
$223$	19.9443	1.33557	0.667784	+	0.744355i	$0.267243\pi$
$224$	16.8541	1.12611
$225$	0	0
$226$	7.65248	0.509035
$227$	9.94427	0.660025	0.330012	−	0.943977i	$-0.392947\pi$
$227$	9.94427	0.660025	0.330012	+	0.943977i	$0.392947\pi$
$228$	20.1803	1.33648
$229$	5.14590	0.340051	0.170025	−	0.985440i	$-0.445615\pi$
$229$	5.14590	0.340051	0.170025	+	0.985440i	$0.445615\pi$
$230$	0	0
$231$	0	0
$232$	3.41641	0.224298
$233$	0.236068	0.0154653	0.00773266	−	0.999970i	$-0.497539\pi$
$233$	0.236068	0.0154653	0.00773266	+	0.999970i	$0.497539\pi$
$234$	1.47214	0.0962365
$235$	0	0
$236$	−16.3262	−1.06275
$237$	−22.9443	−1.49039
$238$	−2.72949	−0.176927
$239$	−30.0689	−1.94499	−0.972497	−	0.232915i	$-0.925174\pi$
$239$	−30.0689	−1.94499	−0.972497	+	0.232915i	$0.925174\pi$
$240$	0	0
$241$	7.27051	0.468335	0.234167	−	0.972196i	$-0.424764\pi$
$241$	7.27051	0.468335	0.234167	+	0.972196i	$0.424764\pi$
$242$	−6.79837	−0.437016
$243$	−10.0000	−0.641500
$244$	−9.94427	−0.636617
$245$	0	0
$246$	−1.34752	−0.0859150
$247$	−14.8541	−0.945144
$248$	−9.47214	−0.601481
$249$	19.8885	1.26039
$250$	0	0
$251$	3.52786	0.222677	0.111338	−	0.993783i	$-0.464486\pi$
$251$	3.52786	0.222677	0.111338	+	0.993783i	$0.464486\pi$
$252$	−4.85410	−0.305780
$253$	0	0
$254$	−7.58359	−0.475837
$255$	0	0
$256$	−6.56231	−0.410144
$257$	20.6180	1.28612	0.643059	−	0.765817i	$-0.277665\pi$
$257$	20.6180	1.28612	0.643059	+	0.765817i	$0.277665\pi$
$258$	−0.291796	−0.0181664
$259$	32.1246	1.99613
$260$	0	0
$261$	−1.52786	−0.0945724
$262$	−4.41641	−0.272847
$263$	−11.5623	−0.712962	−0.356481	−	0.934303i	$-0.616024\pi$
$263$	−11.5623	−0.712962	−0.356481	+	0.934303i	$0.616024\pi$
$264$	0	0
$265$	0	0
$266$	−11.5623	−0.708930
$267$	1.23607	0.0756461
$268$	−0.236068	−0.0144201
$269$	9.27051	0.565233	0.282616	−	0.959233i	$-0.408798\pi$
$269$	9.27051	0.565233	0.282616	+	0.959233i	$0.408798\pi$
$270$	0	0
$271$	4.70820	0.286003	0.143002	−	0.989722i	$-0.454325\pi$
$271$	4.70820	0.286003	0.143002	+	0.989722i	$0.454325\pi$
$272$	−2.72949	−0.165500
$273$	14.2918	0.864979
$274$	2.23607	0.135086
$275$	0	0
$276$	−10.9443	−0.658768
$277$	12.4164	0.746030	0.373015	−	0.927825i	$-0.378324\pi$
$277$	12.4164	0.746030	0.373015	+	0.927825i	$0.378324\pi$
$278$	12.0000	0.719712
$279$	4.23607	0.253607
$280$	0	0
$281$	6.76393	0.403502	0.201751	−	0.979437i	$-0.435337\pi$
$281$	6.76393	0.403502	0.201751	+	0.979437i	$0.435337\pi$
$282$	13.8885	0.827051
$283$	−3.18034	−0.189052	−0.0945258	−	0.995522i	$-0.530133\pi$
$283$	−3.18034	−0.189052	−0.0945258	+	0.995522i	$0.530133\pi$
$284$	7.09017	0.420724
$285$	0	0
$286$	0	0
$287$	−3.27051	−0.193052
$288$	5.61803	0.331046
$289$	−14.8328	−0.872519
$290$	0	0
$291$	31.1246	1.82456
$292$	−24.6525	−1.44268
$293$	14.2918	0.834936	0.417468	−	0.908692i	$-0.362918\pi$
$293$	14.2918	0.834936	0.417468	+	0.908692i	$0.362918\pi$
$294$	2.47214	0.144178
$295$	0	0
$296$	−23.9443	−1.39173
$297$	0	0
$298$	−7.52786	−0.436077
$299$	8.05573	0.465875
$300$	0	0
$301$	−0.708204	−0.0408202
$302$	5.56231	0.320075
$303$	−3.23607	−0.185907
$304$	−11.5623	−0.663144
$305$	0	0
$306$	−0.909830	−0.0520115
$307$	−8.85410	−0.505330	−0.252665	−	0.967554i	$-0.581307\pi$
$307$	−8.85410	−0.505330	−0.252665	+	0.967554i	$0.581307\pi$
$308$	0	0
$309$	3.23607	0.184093
$310$	0	0
$311$	10.4721	0.593820	0.296910	−	0.954905i	$-0.404044\pi$
$311$	10.4721	0.593820	0.296910	+	0.954905i	$0.404044\pi$
$312$	−10.6525	−0.603078
$313$	−20.1246	−1.13751	−0.568755	−	0.822507i	$-0.692575\pi$
$313$	−20.1246	−1.13751	−0.568755	+	0.822507i	$0.692575\pi$
$314$	4.43769	0.250434
$315$	0	0
$316$	18.5623	1.04421
$317$	0.819660	0.0460367	0.0230183	−	0.999735i	$-0.492672\pi$
$317$	0.819660	0.0460367	0.0230183	+	0.999735i	$0.492672\pi$
$318$	1.23607	0.0693153
$319$	0	0
$320$	0	0
$321$	17.8885	0.998441
$322$	6.27051	0.349442
$323$	9.18034	0.510808
$324$	17.7984	0.988799
$325$	0	0
$326$	−9.70820	−0.537688
$327$	−36.4721	−2.01691
$328$	2.43769	0.134599
$329$	33.7082	1.85839
$330$	0	0
$331$	10.1459	0.557669	0.278834	−	0.960339i	$-0.410052\pi$
$331$	10.1459	0.557669	0.278834	+	0.960339i	$0.410052\pi$
$332$	−16.0902	−0.883063
$333$	10.7082	0.586806
$334$	8.76393	0.479541
$335$	0	0
$336$	11.1246	0.606897
$337$	−12.4164	−0.676365	−0.338182	−	0.941081i	$-0.609812\pi$
$337$	−12.4164	−0.676365	−0.338182	+	0.941081i	$0.609812\pi$
$338$	−4.52786	−0.246283
$339$	24.7639	1.34499
$340$	0	0
$341$	0	0
$342$	−3.85410	−0.208406
$343$	−15.0000	−0.809924
$344$	0.527864	0.0284605
$345$	0	0
$346$	10.7426	0.577528
$347$	24.2148	1.29992	0.649959	−	0.759969i	$-0.274786\pi$
$347$	24.2148	1.29992	0.649959	+	0.759969i	$0.274786\pi$
$348$	4.94427	0.265041
$349$	4.41641	0.236405	0.118202	−	0.992990i	$-0.462287\pi$
$349$	4.41641	0.236405	0.118202	+	0.992990i	$0.462287\pi$
$350$	0	0
$351$	−9.52786	−0.508560
$352$	0	0
$353$	−15.2705	−0.812767	−0.406384	−	0.913703i	$-0.633210\pi$
$353$	−15.2705	−0.812767	−0.406384	+	0.913703i	$0.633210\pi$
$354$	12.4721	0.662887
$355$	0	0
$356$	−1.00000	−0.0529999
$357$	−8.83282	−0.467482
$358$	−15.3820	−0.812962
$359$	15.7639	0.831989	0.415994	−	0.909367i	$-0.363434\pi$
$359$	15.7639	0.831989	0.415994	+	0.909367i	$0.363434\pi$
$360$	0	0
$361$	19.8885	1.04677
$362$	−7.12461	−0.374461
$363$	−22.0000	−1.15470
$364$	−11.5623	−0.606029
$365$	0	0
$366$	7.59675	0.397088
$367$	−34.3262	−1.79182	−0.895908	−	0.444241i	$-0.853474\pi$
$367$	−34.3262	−1.79182	−0.895908	+	0.444241i	$0.853474\pi$
$368$	6.27051	0.326873
$369$	−1.09017	−0.0567520
$370$	0	0
$371$	3.00000	0.155752
$372$	−13.7082	−0.710737
$373$	−15.8541	−0.820894	−0.410447	−	0.911884i	$-0.634627\pi$
$373$	−15.8541	−0.820894	−0.410447	+	0.911884i	$0.634627\pi$
$374$	0	0
$375$	0	0
$376$	−25.1246	−1.29570
$377$	−3.63932	−0.187435
$378$	−7.41641	−0.381459
$379$	−35.4721	−1.82208	−0.911041	−	0.412317i	$-0.864720\pi$
$379$	−35.4721	−1.82208	−0.911041	+	0.412317i	$0.864720\pi$
$380$	0	0
$381$	−24.5410	−1.25727
$382$	9.38197	0.480023
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	−22.7639	−1.16167
$385$	0	0
$386$	2.81966	0.143517
$387$	−0.236068	−0.0120000
$388$	−25.1803	−1.27834
$389$	−21.0902	−1.06931	−0.534657	−	0.845069i	$-0.679559\pi$
$389$	−21.0902	−1.06931	−0.534657	+	0.845069i	$0.679559\pi$
$390$	0	0
$391$	−4.97871	−0.251784
$392$	−4.47214	−0.225877
$393$	−14.2918	−0.720926
$394$	−8.97871	−0.452341
$395$	0	0
$396$	0	0
$397$	−22.1246	−1.11040	−0.555201	−	0.831716i	$-0.687359\pi$
$397$	−22.1246	−1.11040	−0.555201	+	0.831716i	$0.687359\pi$
$398$	2.85410	0.143063
$399$	−37.4164	−1.87316
$400$	0	0
$401$	1.32624	0.0662292	0.0331146	−	0.999452i	$-0.489457\pi$
$401$	1.32624	0.0662292	0.0331146	+	0.999452i	$0.489457\pi$
$402$	0.180340	0.00899454
$403$	10.0902	0.502627
$404$	2.61803	0.130252
$405$	0	0
$406$	−2.83282	−0.140590
$407$	0	0
$408$	6.58359	0.325936
$409$	14.9787	0.740650	0.370325	−	0.928902i	$-0.379246\pi$
$409$	14.9787	0.740650	0.370325	+	0.928902i	$0.379246\pi$
$410$	0	0
$411$	7.23607	0.356929
$412$	−2.61803	−0.128981
$413$	30.2705	1.48951
$414$	2.09017	0.102726
$415$	0	0
$416$	13.3820	0.656105
$417$	38.8328	1.90165
$418$	0	0
$419$	36.0344	1.76040	0.880199	−	0.474605i	$-0.157409\pi$
$419$	36.0344	1.76040	0.880199	+	0.474605i	$0.157409\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	11.3820	0.554065
$423$	11.2361	0.546316
$424$	−2.23607	−0.108593
$425$	0	0
$426$	−5.41641	−0.262426
$427$	18.4377	0.892263
$428$	−14.4721	−0.699537
$429$	0	0
$430$	0	0
$431$	8.52786	0.410773	0.205386	−	0.978681i	$-0.434155\pi$
$431$	8.52786	0.410773	0.205386	+	0.978681i	$0.434155\pi$
$432$	−7.41641	−0.356822
$433$	26.9787	1.29651	0.648257	−	0.761422i	$-0.275498\pi$
$433$	26.9787	1.29651	0.648257	+	0.761422i	$0.275498\pi$
$434$	7.85410	0.377009
$435$	0	0
$436$	29.5066	1.41311
$437$	−21.0902	−1.00888
$438$	18.8328	0.899867
$439$	18.7082	0.892894	0.446447	−	0.894810i	$-0.352689\pi$
$439$	18.7082	0.892894	0.446447	+	0.894810i	$0.352689\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	−2.16718	−0.103082
$443$	37.4164	1.77771	0.888854	−	0.458191i	$-0.151503\pi$
$443$	37.4164	1.77771	0.888854	+	0.458191i	$0.151503\pi$
$444$	−34.6525	−1.64453
$445$	0	0
$446$	12.3262	0.583664
$447$	−24.3607	−1.15222
$448$	−0.708204	−0.0334595
$449$	−2.09017	−0.0986412	−0.0493206	−	0.998783i	$-0.515706\pi$
$449$	−2.09017	−0.0986412	−0.0493206	+	0.998783i	$0.515706\pi$
$450$	0	0
$451$	0	0
$452$	−20.0344	−0.942341
$453$	18.0000	0.845714
$454$	6.14590	0.288441
$455$	0	0
$456$	27.8885	1.30600
$457$	3.79837	0.177680	0.0888402	−	0.996046i	$-0.471684\pi$
$457$	3.79837	0.177680	0.0888402	+	0.996046i	$0.471684\pi$
$458$	3.18034	0.148608
$459$	5.88854	0.274854
$460$	0	0
$461$	−9.03444	−0.420776	−0.210388	−	0.977618i	$-0.567473\pi$
$461$	−9.03444	−0.420776	−0.210388	+	0.977618i	$0.567473\pi$
$462$	0	0
$463$	−18.4377	−0.856872	−0.428436	−	0.903572i	$-0.640935\pi$
$463$	−18.4377	−0.856872	−0.428436	+	0.903572i	$0.640935\pi$
$464$	−2.83282	−0.131510
$465$	0	0
$466$	0.145898	0.00675860
$467$	10.3475	0.478826	0.239413	−	0.970918i	$-0.423045\pi$
$467$	10.3475	0.478826	0.239413	+	0.970918i	$0.423045\pi$
$468$	−3.85410	−0.178156
$469$	0.437694	0.0202108
$470$	0	0
$471$	14.3607	0.661705
$472$	−22.5623	−1.03851
$473$	0	0
$474$	−14.1803	−0.651325
$475$	0	0
$476$	7.14590	0.327532
$477$	1.00000	0.0457869
$478$	−18.5836	−0.849994
$479$	−7.11146	−0.324931	−0.162465	−	0.986714i	$-0.551945\pi$
$479$	−7.11146	−0.324931	−0.162465	+	0.986714i	$0.551945\pi$
$480$	0	0
$481$	25.5066	1.16300
$482$	4.49342	0.204670
$483$	20.2918	0.923309
$484$	17.7984	0.809017
$485$	0	0
$486$	−6.18034	−0.280346
$487$	−32.5410	−1.47457	−0.737287	−	0.675579i	$-0.763894\pi$
$487$	−32.5410	−1.47457	−0.737287	+	0.675579i	$0.763894\pi$
$488$	−13.7426	−0.622100
$489$	−31.4164	−1.42070
$490$	0	0
$491$	−29.3607	−1.32503	−0.662514	−	0.749049i	$-0.730511\pi$
$491$	−29.3607	−1.32503	−0.662514	+	0.749049i	$0.730511\pi$
$492$	3.52786	0.159048
$493$	2.24922	0.101300
$494$	−9.18034	−0.413043
$495$	0	0
$496$	7.85410	0.352660
$497$	−13.1459	−0.589674
$498$	12.2918	0.550809
$499$	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
$501$	28.3607	1.26706
$502$	2.18034	0.0973133
$503$	10.4164	0.464445	0.232222	−	0.972663i	$-0.425400\pi$
$503$	10.4164	0.464445	0.232222	+	0.972663i	$0.425400\pi$
$504$	−6.70820	−0.298807
$505$	0	0
$506$	0	0
$507$	−14.6525	−0.650739
$508$	19.8541	0.880883
$509$	35.1246	1.55687	0.778436	−	0.627725i	$-0.216014\pi$
$509$	35.1246	1.55687	0.778436	+	0.627725i	$0.216014\pi$
$510$	0	0
$511$	45.7082	2.02201
$512$	18.7082	0.826794
$513$	24.9443	1.10132
$514$	12.7426	0.562054
$515$	0	0
$516$	0.763932	0.0336302
$517$	0	0
$518$	19.8541	0.872339
$519$	34.7639	1.52597
$520$	0	0
$521$	−36.5967	−1.60333	−0.801666	−	0.597772i	$-0.796053\pi$
$521$	−36.5967	−1.60333	−0.801666	+	0.597772i	$0.796053\pi$
$522$	−0.944272	−0.0413297
$523$	17.2705	0.755187	0.377593	−	0.925972i	$-0.376752\pi$
$523$	17.2705	0.755187	0.377593	+	0.925972i	$0.376752\pi$
$524$	11.5623	0.505102
$525$	0	0
$526$	−7.14590	−0.311576
$527$	−6.23607	−0.271647
$528$	0	0
$529$	−11.5623	−0.502709
$530$	0	0
$531$	10.0902	0.437876
$532$	30.2705	1.31239
$533$	−2.59675	−0.112478
$534$	0.763932	0.0330586
$535$	0	0
$536$	−0.326238	−0.0140913
$537$	−49.7771	−2.14804
$538$	5.72949	0.247016
$539$	0	0
$540$	0	0
$541$	−14.7082	−0.632355	−0.316178	−	0.948700i	$-0.602400\pi$
$541$	−14.7082	−0.632355	−0.316178	+	0.948700i	$0.602400\pi$
$542$	2.90983	0.124988
$543$	−23.0557	−0.989416
$544$	−8.27051	−0.354595
$545$	0	0
$546$	8.83282	0.378010
$547$	7.43769	0.318013	0.159006	−	0.987278i	$-0.449171\pi$
$547$	7.43769	0.318013	0.159006	+	0.987278i	$0.449171\pi$
$548$	−5.85410	−0.250075
$549$	6.14590	0.262301
$550$	0	0
$551$	9.52786	0.405901
$552$	−15.1246	−0.643746
$553$	−34.4164	−1.46353
$554$	7.67376	0.326027
$555$	0	0
$556$	−31.4164	−1.33235
$557$	−42.4508	−1.79870	−0.899350	−	0.437229i	$-0.855960\pi$
$557$	−42.4508	−1.79870	−0.899350	+	0.437229i	$0.855960\pi$
$558$	2.61803	0.110830
$559$	−0.562306	−0.0237830
$560$	0	0
$561$	0	0
$562$	4.18034	0.176337
$563$	−5.50658	−0.232075	−0.116037	−	0.993245i	$-0.537019\pi$
$563$	−5.50658	−0.232075	−0.116037	+	0.993245i	$0.537019\pi$
$564$	−36.3607	−1.53106
$565$	0	0
$566$	−1.96556	−0.0826186
$567$	−33.0000	−1.38587
$568$	9.79837	0.411131
$569$	23.5623	0.987783	0.493892	−	0.869523i	$-0.335574\pi$
$569$	23.5623	0.987783	0.493892	+	0.869523i	$0.335574\pi$
$570$	0	0
$571$	−15.7984	−0.661141	−0.330571	−	0.943781i	$-0.607241\pi$
$571$	−15.7984	−0.661141	−0.330571	+	0.943781i	$0.607241\pi$
$572$	0	0
$573$	30.3607	1.26834
$574$	−2.02129	−0.0843669
$575$	0	0
$576$	−0.236068	−0.00983617
$577$	5.81966	0.242276	0.121138	−	0.992636i	$-0.461346\pi$
$577$	5.81966	0.242276	0.121138	+	0.992636i	$0.461346\pi$
$578$	−9.16718	−0.381305
$579$	9.12461	0.379206
$580$	0	0
$581$	29.8328	1.23767
$582$	19.2361	0.797361
$583$	0	0
$584$	−34.0689	−1.40978
$585$	0	0
$586$	8.83282	0.364880
$587$	−33.3607	−1.37694	−0.688471	−	0.725264i	$-0.741718\pi$
$587$	−33.3607	−1.37694	−0.688471	+	0.725264i	$0.741718\pi$
$588$	−6.47214	−0.266906
$589$	−26.4164	−1.08847
$590$	0	0
$591$	−29.0557	−1.19519
$592$	19.8541	0.815999
$593$	−36.9787	−1.51853	−0.759267	−	0.650779i	$-0.774442\pi$
$593$	−36.9787	−1.51853	−0.759267	+	0.650779i	$0.774442\pi$
$594$	0	0
$595$	0	0
$596$	19.7082	0.807279
$597$	9.23607	0.378007
$598$	4.97871	0.203595
$599$	13.7426	0.561509	0.280755	−	0.959780i	$-0.409415\pi$
$599$	13.7426	0.561509	0.280755	+	0.959780i	$0.409415\pi$
$600$	0	0
$601$	21.7082	0.885496	0.442748	−	0.896646i	$-0.354004\pi$
$601$	21.7082	0.885496	0.442748	+	0.896646i	$0.354004\pi$
$602$	−0.437694	−0.0178391
$603$	0.145898	0.00594143
$604$	−14.5623	−0.592532
$605$	0	0
$606$	−2.00000	−0.0812444
$607$	9.43769	0.383064	0.191532	−	0.981486i	$-0.438654\pi$
$607$	9.43769	0.383064	0.191532	+	0.981486i	$0.438654\pi$
$608$	−35.0344	−1.42083
$609$	−9.16718	−0.371473
$610$	0	0
$611$	26.7639	1.08275
$612$	2.38197	0.0962853
$613$	−5.87539	−0.237305	−0.118652	−	0.992936i	$-0.537857\pi$
$613$	−5.87539	−0.237305	−0.118652	+	0.992936i	$0.537857\pi$
$614$	−5.47214	−0.220837
$615$	0	0
$616$	0	0
$617$	−23.3607	−0.940466	−0.470233	−	0.882542i	$-0.655830\pi$
$617$	−23.3607	−0.940466	−0.470233	+	0.882542i	$0.655830\pi$
$618$	2.00000	0.0804518
$619$	−43.5066	−1.74868	−0.874339	−	0.485317i	$-0.838704\pi$
$619$	−43.5066	−1.74868	−0.874339	+	0.485317i	$0.838704\pi$
$620$	0	0
$621$	−13.5279	−0.542854
$622$	6.47214	0.259509
$623$	1.85410	0.0742830
$624$	8.83282	0.353596
$625$	0	0
$626$	−12.4377	−0.497110
$627$	0	0
$628$	−11.6180	−0.463610
$629$	−15.7639	−0.628549
$630$	0	0
$631$	−46.9443	−1.86882	−0.934411	−	0.356197i	$-0.884073\pi$
$631$	−46.9443	−1.86882	−0.934411	+	0.356197i	$0.884073\pi$
$632$	25.6525	1.02040
$633$	36.8328	1.46397
$634$	0.506578	0.0201188
$635$	0	0
$636$	−3.23607	−0.128318
$637$	4.76393	0.188754
$638$	0	0
$639$	−4.38197	−0.173348
$640$	0	0
$641$	29.1803	1.15255	0.576277	−	0.817254i	$-0.304505\pi$
$641$	29.1803	1.15255	0.576277	+	0.817254i	$0.304505\pi$
$642$	11.0557	0.436335
$643$	−30.7771	−1.21373	−0.606865	−	0.794805i	$-0.707573\pi$
$643$	−30.7771	−1.21373	−0.606865	+	0.794805i	$0.707573\pi$
$644$	−16.4164	−0.646897
$645$	0	0
$646$	5.67376	0.223231
$647$	−20.5066	−0.806197	−0.403098	−	0.915157i	$-0.632067\pi$
$647$	−20.5066	−0.806197	−0.403098	+	0.915157i	$0.632067\pi$
$648$	24.5967	0.966252
$649$	0	0
$650$	0	0
$651$	25.4164	0.996148
$652$	25.4164	0.995383
$653$	3.05573	0.119580	0.0597899	−	0.998211i	$-0.480957\pi$
$653$	3.05573	0.119580	0.0597899	+	0.998211i	$0.480957\pi$
$654$	−22.5410	−0.881424
$655$	0	0
$656$	−2.02129	−0.0789180
$657$	15.2361	0.594416
$658$	20.8328	0.812148
$659$	41.6312	1.62172	0.810860	−	0.585240i	$-0.199000\pi$
$659$	41.6312	1.62172	0.810860	+	0.585240i	$0.199000\pi$
$660$	0	0
$661$	−5.85410	−0.227698	−0.113849	−	0.993498i	$-0.536318\pi$
$661$	−5.85410	−0.227698	−0.113849	+	0.993498i	$0.536318\pi$
$662$	6.27051	0.243710
$663$	−7.01316	−0.272368
$664$	−22.2361	−0.862927
$665$	0	0
$666$	6.61803	0.256444
$667$	−5.16718	−0.200074
$668$	−22.9443	−0.887741
$669$	39.8885	1.54218
$670$	0	0
$671$	0	0
$672$	33.7082	1.30032
$673$	11.0000	0.424019	0.212009	−	0.977268i	$-0.431999\pi$
$673$	11.0000	0.424019	0.212009	+	0.977268i	$0.431999\pi$
$674$	−7.67376	−0.295582
$675$	0	0
$676$	11.8541	0.455927
$677$	20.1803	0.775593	0.387797	−	0.921745i	$-0.373236\pi$
$677$	20.1803	0.775593	0.387797	+	0.921745i	$0.373236\pi$
$678$	15.3050	0.587783
$679$	46.6869	1.79168
$680$	0	0
$681$	19.8885	0.762131
$682$	0	0
$683$	4.88854	0.187055	0.0935275	−	0.995617i	$-0.470186\pi$
$683$	4.88854	0.187055	0.0935275	+	0.995617i	$0.470186\pi$
$684$	10.0902	0.385807
$685$	0	0
$686$	−9.27051	−0.353950
$687$	10.2918	0.392657
$688$	−0.437694	−0.0166869
$689$	2.38197	0.0907457
$690$	0	0
$691$	−40.1246	−1.52641	−0.763206	−	0.646155i	$-0.776376\pi$
$691$	−40.1246	−1.52641	−0.763206	+	0.646155i	$0.776376\pi$
$692$	−28.1246	−1.06914
$693$	0	0
$694$	14.9656	0.568085
$695$	0	0
$696$	6.83282	0.258997
$697$	1.60488	0.0607891
$698$	2.72949	0.103313
$699$	0.472136	0.0178578
$700$	0	0
$701$	−21.3262	−0.805481	−0.402740	−	0.915314i	$-0.631942\pi$
$701$	−21.3262	−0.805481	−0.402740	+	0.915314i	$0.631942\pi$
$702$	−5.88854	−0.222249
$703$	−66.7771	−2.51855
$704$	0	0
$705$	0	0
$706$	−9.43769	−0.355192
$707$	−4.85410	−0.182557
$708$	−32.6525	−1.22716
$709$	−6.14590	−0.230814	−0.115407	−	0.993318i	$-0.536817\pi$
$709$	−6.14590	−0.230814	−0.115407	+	0.993318i	$0.536817\pi$
$710$	0	0
$711$	−11.4721	−0.430239
$712$	−1.38197	−0.0517914
$713$	14.3262	0.536522
$714$	−5.45898	−0.204297
$715$	0	0
$716$	40.2705	1.50498
$717$	−60.1378	−2.24589
$718$	9.74265	0.363592
$719$	1.09017	0.0406565	0.0203282	−	0.999793i	$-0.493529\pi$
$719$	1.09017	0.0406565	0.0203282	+	0.999793i	$0.493529\pi$
$720$	0	0
$721$	4.85410	0.180776
$722$	12.2918	0.457453
$723$	14.5410	0.540786
$724$	18.6525	0.693214
$725$	0	0
$726$	−13.5967	−0.504623
$727$	−24.8541	−0.921788	−0.460894	−	0.887455i	$-0.652471\pi$
$727$	−24.8541	−0.921788	−0.460894	+	0.887455i	$0.652471\pi$
$728$	−15.9787	−0.592211
$729$	13.0000	0.481481
$730$	0	0
$731$	0.347524	0.0128536
$732$	−19.8885	−0.735102
$733$	2.52786	0.0933688	0.0466844	−	0.998910i	$-0.485134\pi$
$733$	2.52786	0.0933688	0.0466844	+	0.998910i	$0.485134\pi$
$734$	−21.2148	−0.783052
$735$	0	0
$736$	19.0000	0.700349
$737$	0	0
$738$	−0.673762	−0.0248015
$739$	26.6312	0.979644	0.489822	−	0.871822i	$-0.337062\pi$
$739$	26.6312	0.979644	0.489822	+	0.871822i	$0.337062\pi$
$740$	0	0
$741$	−29.7082	−1.09136
$742$	1.85410	0.0680662
$743$	−23.3607	−0.857020	−0.428510	−	0.903537i	$-0.640961\pi$
$743$	−23.3607	−0.857020	−0.428510	+	0.903537i	$0.640961\pi$
$744$	−18.9443	−0.694531
$745$	0	0
$746$	−9.79837	−0.358744
$747$	9.94427	0.363842
$748$	0	0
$749$	26.8328	0.980450
$750$	0	0
$751$	−21.3951	−0.780719	−0.390360	−	0.920662i	$-0.627649\pi$
$751$	−21.3951	−0.780719	−0.390360	+	0.920662i	$0.627649\pi$
$752$	20.8328	0.759695
$753$	7.05573	0.257125
$754$	−2.24922	−0.0819119
$755$	0	0
$756$	19.4164	0.706168
$757$	20.1246	0.731441	0.365721	−	0.930725i	$-0.380823\pi$
$757$	20.1246	0.731441	0.365721	+	0.930725i	$0.380823\pi$
$758$	−21.9230	−0.796279
$759$	0	0
$760$	0	0
$761$	−31.1459	−1.12904	−0.564519	−	0.825420i	$-0.690938\pi$
$761$	−31.1459	−1.12904	−0.564519	+	0.825420i	$0.690938\pi$
$762$	−15.1672	−0.549449
$763$	−54.7082	−1.98057
$764$	−24.5623	−0.888633
$765$	0	0
$766$	12.9787	0.468940
$767$	24.0344	0.867833
$768$	−13.1246	−0.473594
$769$	−25.7082	−0.927062	−0.463531	−	0.886081i	$-0.653418\pi$
$769$	−25.7082	−0.927062	−0.463531	+	0.886081i	$0.653418\pi$
$770$	0	0
$771$	41.2361	1.48508
$772$	−7.38197	−0.265683
$773$	−24.4508	−0.879436	−0.439718	−	0.898136i	$-0.644922\pi$
$773$	−24.4508	−0.879436	−0.439718	+	0.898136i	$0.644922\pi$
$774$	−0.145898	−0.00524420
$775$	0	0
$776$	−34.7984	−1.24919
$777$	64.2492	2.30493
$778$	−13.0344	−0.467307
$779$	6.79837	0.243577
$780$	0	0
$781$	0	0
$782$	−3.07701	−0.110034
$783$	6.11146	0.218406
$784$	3.70820	0.132436
$785$	0	0
$786$	−8.83282	−0.315056
$787$	49.5066	1.76472	0.882359	−	0.470576i	$-0.155954\pi$
$787$	49.5066	1.76472	0.882359	+	0.470576i	$0.155954\pi$
$788$	23.5066	0.837387
$789$	−23.1246	−0.823258
$790$	0	0
$791$	37.1459	1.32076
$792$	0	0
$793$	14.6393	0.519858
$794$	−13.6738	−0.485264
$795$	0	0
$796$	−7.47214	−0.264843
$797$	−4.96556	−0.175889	−0.0879445	−	0.996125i	$-0.528030\pi$
$797$	−4.96556	−0.175889	−0.0879445	+	0.996125i	$0.528030\pi$
$798$	−23.1246	−0.818602
$799$	−16.5410	−0.585179
$800$	0	0
$801$	0.618034	0.0218372
$802$	0.819660	0.0289432
$803$	0	0
$804$	−0.472136	−0.0166510
$805$	0	0
$806$	6.23607	0.219656
$807$	18.5410	0.652675
$808$	3.61803	0.127282
$809$	−15.7639	−0.554230	−0.277115	−	0.960837i	$-0.589378\pi$
$809$	−15.7639	−0.554230	−0.277115	+	0.960837i	$0.589378\pi$
$810$	0	0
$811$	−48.0132	−1.68597	−0.842985	−	0.537937i	$-0.819204\pi$
$811$	−48.0132	−1.68597	−0.842985	+	0.537937i	$0.819204\pi$
$812$	7.41641	0.260265
$813$	9.41641	0.330248
$814$	0	0
$815$	0	0
$816$	−5.45898	−0.191103
$817$	1.47214	0.0515035
$818$	9.25735	0.323676
$819$	7.14590	0.249698
$820$	0	0
$821$	−30.4508	−1.06274	−0.531371	−	0.847139i	$-0.678323\pi$
$821$	−30.4508	−1.06274	−0.531371	+	0.847139i	$0.678323\pi$
$822$	4.47214	0.155984
$823$	−42.8328	−1.49306	−0.746529	−	0.665353i	$-0.768281\pi$
$823$	−42.8328	−1.49306	−0.746529	+	0.665353i	$0.768281\pi$
$824$	−3.61803	−0.126040
$825$	0	0
$826$	18.7082	0.650942
$827$	−8.50658	−0.295803	−0.147901	−	0.989002i	$-0.547252\pi$
$827$	−8.50658	−0.295803	−0.147901	+	0.989002i	$0.547252\pi$
$828$	−5.47214	−0.190170
$829$	21.2705	0.738755	0.369378	−	0.929279i	$-0.379571\pi$
$829$	21.2705	0.738755	0.369378	+	0.929279i	$0.379571\pi$
$830$	0	0
$831$	24.8328	0.861441
$832$	−0.562306	−0.0194944
$833$	−2.94427	−0.102013
$834$	24.0000	0.831052
$835$	0	0
$836$	0	0
$837$	−16.9443	−0.585680
$838$	22.2705	0.769322
$839$	18.5279	0.639653	0.319826	−	0.947476i	$-0.396375\pi$
$839$	18.5279	0.639653	0.319826	+	0.947476i	$0.396375\pi$
$840$	0	0
$841$	−26.6656	−0.919505
$842$	1.23607	0.0425977
$843$	13.5279	0.465924
$844$	−29.7984	−1.02570
$845$	0	0
$846$	6.94427	0.238749
$847$	−33.0000	−1.13389
$848$	1.85410	0.0636701
$849$	−6.36068	−0.218298
$850$	0	0
$851$	36.2148	1.24143
$852$	14.1803	0.485810
$853$	13.5623	0.464365	0.232182	−	0.972672i	$-0.425413\pi$
$853$	13.5623	0.464365	0.232182	+	0.972672i	$0.425413\pi$
$854$	11.3951	0.389933
$855$	0	0
$856$	−20.0000	−0.683586
$857$	41.7984	1.42781	0.713903	−	0.700245i	$-0.246926\pi$
$857$	41.7984	1.42781	0.713903	+	0.700245i	$0.246926\pi$
$858$	0	0
$859$	−35.8328	−1.22260	−0.611300	−	0.791399i	$-0.709353\pi$
$859$	−35.8328	−1.22260	−0.611300	+	0.791399i	$0.709353\pi$
$860$	0	0
$861$	−6.54102	−0.222917
$862$	5.27051	0.179514
$863$	−25.0902	−0.854079	−0.427040	−	0.904233i	$-0.640444\pi$
$863$	−25.0902	−0.854079	−0.427040	+	0.904233i	$0.640444\pi$
$864$	−22.4721	−0.764518
$865$	0	0
$866$	16.6738	0.566598
$867$	−29.6656	−1.00750
$868$	−20.5623	−0.697930
$869$	0	0
$870$	0	0
$871$	0.347524	0.0117754
$872$	40.7771	1.38089
$873$	15.5623	0.526704
$874$	−13.0344	−0.440897
$875$	0	0
$876$	−49.3050	−1.66586
$877$	19.0689	0.643910	0.321955	−	0.946755i	$-0.395660\pi$
$877$	19.0689	0.643910	0.321955	+	0.946755i	$0.395660\pi$
$878$	11.5623	0.390209
$879$	28.5836	0.964101
$880$	0	0
$881$	−36.6180	−1.23369	−0.616846	−	0.787084i	$-0.711590\pi$
$881$	−36.6180	−1.23369	−0.616846	+	0.787084i	$0.711590\pi$
$882$	1.23607	0.0416206
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	5.67376	0.190829
$885$	0	0
$886$	23.1246	0.776887
$887$	30.4853	1.02360	0.511798	−	0.859106i	$-0.328980\pi$
$887$	30.4853	1.02360	0.511798	+	0.859106i	$0.328980\pi$
$888$	−47.8885	−1.60703
$889$	−36.8115	−1.23462
$890$	0	0
$891$	0	0
$892$	−32.2705	−1.08050
$893$	−70.0689	−2.34477
$894$	−15.0557	−0.503539
$895$	0	0
$896$	−34.1459	−1.14073
$897$	16.1115	0.537946
$898$	−1.29180	−0.0431078
$899$	−6.47214	−0.215858
$900$	0	0
$901$	−1.47214	−0.0490440
$902$	0	0
$903$	−1.41641	−0.0471351
$904$	−27.6869	−0.920853
$905$	0	0
$906$	11.1246	0.369590
$907$	10.7082	0.355560	0.177780	−	0.984070i	$-0.443108\pi$
$907$	10.7082	0.355560	0.177780	+	0.984070i	$0.443108\pi$
$908$	−16.0902	−0.533971
$909$	−1.61803	−0.0536668
$910$	0	0
$911$	−31.3050	−1.03718	−0.518590	−	0.855023i	$-0.673543\pi$
$911$	−31.3050	−1.03718	−0.518590	+	0.855023i	$0.673543\pi$
$912$	−23.1246	−0.765732
$913$	0	0
$914$	2.34752	0.0776492
$915$	0	0
$916$	−8.32624	−0.275107
$917$	−21.4377	−0.707935
$918$	3.63932	0.120115
$919$	14.5279	0.479230	0.239615	−	0.970868i	$-0.422979\pi$
$919$	14.5279	0.479230	0.239615	+	0.970868i	$0.422979\pi$
$920$	0	0
$921$	−17.7082	−0.583505
$922$	−5.58359	−0.183886
$923$	−10.4377	−0.343561
$924$	0	0
$925$	0	0
$926$	−11.3951	−0.374467
$927$	1.61803	0.0531432
$928$	−8.58359	−0.281770
$929$	14.2361	0.467070	0.233535	−	0.972348i	$-0.424971\pi$
$929$	14.2361	0.467070	0.233535	+	0.972348i	$0.424971\pi$
$930$	0	0
$931$	−12.4721	−0.408758
$932$	−0.381966	−0.0125117
$933$	20.9443	0.685685
$934$	6.39512	0.209255
$935$	0	0
$936$	−5.32624	−0.174094
$937$	55.2361	1.80448	0.902242	−	0.431230i	$-0.141920\pi$
$937$	55.2361	1.80448	0.902242	+	0.431230i	$0.141920\pi$
$938$	0.270510	0.00883246
$939$	−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	$0.578622\pi$
$942$	8.87539	0.289176
$943$	−3.68692	−0.120063
$944$	18.7082	0.608900
$945$	0	0
$946$	0	0
$947$	−45.2361	−1.46997	−0.734987	−	0.678081i	$-0.762812\pi$
$947$	−45.2361	−1.46997	−0.734987	+	0.678081i	$0.762812\pi$
$948$	37.1246	1.20575
$949$	36.2918	1.17808
$950$	0	0
$951$	1.63932	0.0531586
$952$	9.87539	0.320063
$953$	9.05573	0.293344	0.146672	−	0.989185i	$-0.453144\pi$
$953$	9.05573	0.293344	0.146672	+	0.989185i	$0.453144\pi$
$954$	0.618034	0.0200096
$955$	0	0
$956$	48.6525	1.57353
$957$	0	0
$958$	−4.39512	−0.142000
$959$	10.8541	0.350497
$960$	0	0
$961$	−13.0557	−0.421153
$962$	15.7639	0.508250
$963$	8.94427	0.288225
$964$	−11.7639	−0.378891
$965$	0	0
$966$	12.5410	0.403501
$967$	9.70820	0.312195	0.156097	−	0.987742i	$-0.450109\pi$
$967$	9.70820	0.312195	0.156097	+	0.987742i	$0.450109\pi$
$968$	24.5967	0.790569
$969$	18.3607	0.589830
$970$	0	0
$971$	−49.9443	−1.60279	−0.801394	−	0.598137i	$-0.795908\pi$
$971$	−49.9443	−1.60279	−0.801394	+	0.598137i	$0.795908\pi$
$972$	16.1803	0.518985
$973$	58.2492	1.86738
$974$	−20.1115	−0.644413
$975$	0	0
$976$	11.3951	0.364749
$977$	5.25735	0.168198	0.0840988	−	0.996457i	$-0.473199\pi$
$977$	5.25735	0.168198	0.0840988	+	0.996457i	$0.473199\pi$
$978$	−19.4164	−0.620868
$979$	0	0
$980$	0	0
$981$	−18.2361	−0.582233
$982$	−18.1459	−0.579059
$983$	−45.8115	−1.46116	−0.730580	−	0.682827i	$-0.760750\pi$
$983$	−45.8115	−1.46116	−0.730580	+	0.682827i	$0.760750\pi$
$984$	4.87539	0.155422
$985$	0	0
$986$	1.39010	0.0442697
$987$	67.4164	2.14589
$988$	24.0344	0.764637
$989$	−0.798374	−0.0253868
$990$	0	0
$991$	38.8328	1.23357	0.616783	−	0.787134i	$-0.288436\pi$
$991$	38.8328	1.23357	0.616783	+	0.787134i	$0.288436\pi$
$992$	23.7984	0.755599
$993$	20.2918	0.643941
$994$	−8.12461	−0.257697
$995$	0	0
$996$	−32.1803	−1.01967
$997$	−25.2492	−0.799651	−0.399825	−	0.916591i	$-0.630929\pi$
$997$	−25.2492	−0.799651	−0.399825	+	0.916591i	$0.630929\pi$
$998$	3.70820	0.117381
$999$	−42.8328	−1.35517

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6625.2.a.b.1.2	✓	2	1.1	even	1	trivial
6625.2.a.c.1.1	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)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} +2.00000 q^{3} -1.61803 q^{4} +1.23607 q^{6} +3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.618034 q^{2} +2.00000 q^{3} -1.61803 q^{4} +1.23607 q^{6} +3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -3.23607 q^{12} +2.38197 q^{13} +1.85410 q^{14} +1.85410 q^{16} -1.47214 q^{17} +0.618034 q^{18} -6.23607 q^{19} +6.00000 q^{21} +3.38197 q^{23} -4.47214 q^{24} +1.47214 q^{26} -4.00000 q^{27} -4.85410 q^{28} -1.52786 q^{29} +4.23607 q^{31} +5.61803 q^{32} -0.909830 q^{34} -1.61803 q^{36} +10.7082 q^{37} -3.85410 q^{38} +4.76393 q^{39} -1.09017 q^{41} +3.70820 q^{42} -0.236068 q^{43} +2.09017 q^{46} +11.2361 q^{47} +3.70820 q^{48} +2.00000 q^{49} -2.94427 q^{51} -3.85410 q^{52} +1.00000 q^{53} -2.47214 q^{54} -6.70820 q^{56} -12.4721 q^{57} -0.944272 q^{58} +10.0902 q^{59} +6.14590 q^{61} +2.61803 q^{62} +3.00000 q^{63} -0.236068 q^{64} +0.145898 q^{67} +2.38197 q^{68} +6.76393 q^{69} -4.38197 q^{71} -2.23607 q^{72} +15.2361 q^{73} +6.61803 q^{74} +10.0902 q^{76} +2.94427 q^{78} -11.4721 q^{79} -11.0000 q^{81} -0.673762 q^{82} +9.94427 q^{83} -9.70820 q^{84} -0.145898 q^{86} -3.05573 q^{87} +0.618034 q^{89} +7.14590 q^{91} -5.47214 q^{92} +8.47214 q^{93} +6.94427 q^{94} +11.2361 q^{96} +15.5623 q^{97} +1.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 4 q^{3} - q^{4} - 2 q^{6} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) 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

Groups

Database

Properties

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