LMFDB - Embedded newform 6025.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, 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("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.61803 q^{3} -1.00000 q^{4} -1.61803 q^{6} +1.23607 q^{7} +3.00000 q^{8} -0.381966 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.61803 q^{3} -1.00000 q^{4} -1.61803 q^{6} +1.23607 q^{7} +3.00000 q^{8} -0.381966 q^{9} +3.61803 q^{11} -1.61803 q^{12} +0.381966 q^{13} -1.23607 q^{14} -1.00000 q^{16} +3.85410 q^{17} +0.381966 q^{18} +0.618034 q^{19} +2.00000 q^{21} -3.61803 q^{22} -8.47214 q^{23} +4.85410 q^{24} -0.381966 q^{26} -5.47214 q^{27} -1.23607 q^{28} -7.09017 q^{29} -1.52786 q^{31} -5.00000 q^{32} +5.85410 q^{33} -3.85410 q^{34} +0.381966 q^{36} -4.47214 q^{37} -0.618034 q^{38} +0.618034 q^{39} -4.09017 q^{41} -2.00000 q^{42} -6.47214 q^{43} -3.61803 q^{44} +8.47214 q^{46} -7.85410 q^{47} -1.61803 q^{48} -5.47214 q^{49} +6.23607 q^{51} -0.381966 q^{52} +4.00000 q^{53} +5.47214 q^{54} +3.70820 q^{56} +1.00000 q^{57} +7.09017 q^{58} -2.76393 q^{59} +4.85410 q^{61} +1.52786 q^{62} -0.472136 q^{63} +7.00000 q^{64} -5.85410 q^{66} -12.7984 q^{67} -3.85410 q^{68} -13.7082 q^{69} -14.7984 q^{71} -1.14590 q^{72} +1.09017 q^{73} +4.47214 q^{74} -0.618034 q^{76} +4.47214 q^{77} -0.618034 q^{78} +14.6525 q^{79} -7.70820 q^{81} +4.09017 q^{82} +13.3820 q^{83} -2.00000 q^{84} +6.47214 q^{86} -11.4721 q^{87} +10.8541 q^{88} +16.4721 q^{89} +0.472136 q^{91} +8.47214 q^{92} -2.47214 q^{93} +7.85410 q^{94} -8.09017 q^{96} +12.7639 q^{97} +5.47214 q^{98} -1.38197 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + q^{3} - 2 q^{4} - q^{6} - 2 q^{7} + 6 q^{8} - 3 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + q^3 - 2 * q^4 - q^6 - 2 * q^7 + 6 * q^8 - 3 * q^9	$ 2 q - 2 q^{2} + q^{3} - 2 q^{4} - q^{6} - 2 q^{7} + 6 q^{8} - 3 q^{9} + 5 q^{11} - q^{12} + 3 q^{13} + 2 q^{14} - 2 q^{16} + q^{17} + 3 q^{18} - q^{19} + 4 q^{21} - 5 q^{22} - 8 q^{23} + 3 q^{24} - 3 q^{26} - 2 q^{27} + 2 q^{28} - 3 q^{29} - 12 q^{31} - 10 q^{32} + 5 q^{33} - q^{34} + 3 q^{36} + q^{38} - q^{39} + 3 q^{41} - 4 q^{42} - 4 q^{43} - 5 q^{44} + 8 q^{46} - 9 q^{47} - q^{48} - 2 q^{49} + 8 q^{51} - 3 q^{52} + 8 q^{53} + 2 q^{54} - 6 q^{56} + 2 q^{57} + 3 q^{58} - 10 q^{59} + 3 q^{61} + 12 q^{62} + 8 q^{63} + 14 q^{64} - 5 q^{66} - q^{67} - q^{68} - 14 q^{69} - 5 q^{71} - 9 q^{72} - 9 q^{73} + q^{76} + q^{78} - 2 q^{79} - 2 q^{81} - 3 q^{82} + 29 q^{83} - 4 q^{84} + 4 q^{86} - 14 q^{87} + 15 q^{88} + 24 q^{89} - 8 q^{91} + 8 q^{92} + 4 q^{93} + 9 q^{94} - 5 q^{96} + 30 q^{97} + 2 q^{98} - 5 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + q^3 - 2 * q^4 - q^6 - 2 * q^7 + 6 * q^8 - 3 * q^9 + 5 * q^11 - q^12 + 3 * q^13 + 2 * q^14 - 2 * q^16 + q^17 + 3 * q^18 - q^19 + 4 * q^21 - 5 * q^22 - 8 * q^23 + 3 * q^24 - 3 * q^26 - 2 * q^27 + 2 * q^28 - 3 * q^29 - 12 * q^31 - 10 * q^32 + 5 * q^33 - q^34 + 3 * q^36 + q^38 - q^39 + 3 * q^41 - 4 * q^42 - 4 * q^43 - 5 * q^44 + 8 * q^46 - 9 * q^47 - q^48 - 2 * q^49 + 8 * q^51 - 3 * q^52 + 8 * q^53 + 2 * q^54 - 6 * q^56 + 2 * q^57 + 3 * q^58 - 10 * q^59 + 3 * q^61 + 12 * q^62 + 8 * q^63 + 14 * q^64 - 5 * q^66 - q^67 - q^68 - 14 * q^69 - 5 * q^71 - 9 * q^72 - 9 * q^73 + q^76 + q^78 - 2 * q^79 - 2 * q^81 - 3 * q^82 + 29 * q^83 - 4 * q^84 + 4 * q^86 - 14 * q^87 + 15 * q^88 + 24 * q^89 - 8 * q^91 + 8 * q^92 + 4 * q^93 + 9 * q^94 - 5 * q^96 + 30 * q^97 + 2 * q^98 - 5 * q^99

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 $

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	1.61803	0.934172	0.467086	−	0.884212i	$-0.345304\pi$
$3$	1.61803	0.934172	0.467086	+	0.884212i	$0.345304\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.61803	−0.660560
$7$	1.23607	0.467190	0.233595	−	0.972334i	$-0.424951\pi$
$7$	1.23607	0.467190	0.233595	+	0.972334i	$0.424951\pi$
$8$	3.00000	1.06066
$9$	−0.381966	−0.127322
$10$	0	0
$11$	3.61803	1.09088	0.545439	−	0.838150i	$-0.316363\pi$
$11$	3.61803	1.09088	0.545439	+	0.838150i	$0.316363\pi$
$12$	−1.61803	−0.467086
$13$	0.381966	0.105938	0.0529692	−	0.998596i	$-0.483131\pi$
$13$	0.381966	0.105938	0.0529692	+	0.998596i	$0.483131\pi$
$14$	−1.23607	−0.330353
$15$	0	0
$16$	−1.00000	−0.250000
$17$	3.85410	0.934757	0.467379	−	0.884057i	$-0.345199\pi$
$17$	3.85410	0.934757	0.467379	+	0.884057i	$0.345199\pi$
$18$	0.381966	0.0900303
$19$	0.618034	0.141787	0.0708934	−	0.997484i	$-0.477415\pi$
$19$	0.618034	0.141787	0.0708934	+	0.997484i	$0.477415\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	−3.61803	−0.771367
$23$	−8.47214	−1.76656	−0.883281	−	0.468844i	$-0.844671\pi$
$23$	−8.47214	−1.76656	−0.883281	+	0.468844i	$0.844671\pi$
$24$	4.85410	0.990839
$25$	0	0
$26$	−0.381966	−0.0749097
$27$	−5.47214	−1.05311
$28$	−1.23607	−0.233595
$29$	−7.09017	−1.31661	−0.658306	−	0.752751i	$-0.728727\pi$
$29$	−7.09017	−1.31661	−0.658306	+	0.752751i	$0.728727\pi$
$30$	0	0
$31$	−1.52786	−0.274412	−0.137206	−	0.990543i	$-0.543812\pi$
$31$	−1.52786	−0.274412	−0.137206	+	0.990543i	$0.543812\pi$
$32$	−5.00000	−0.883883
$33$	5.85410	1.01907
$34$	−3.85410	−0.660973
$35$	0	0
$36$	0.381966	0.0636610
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	−0.618034	−0.100258
$39$	0.618034	0.0989646
$40$	0	0
$41$	−4.09017	−0.638777	−0.319389	−	0.947624i	$-0.603478\pi$
$41$	−4.09017	−0.638777	−0.319389	+	0.947624i	$0.603478\pi$
$42$	−2.00000	−0.308607
$43$	−6.47214	−0.986991	−0.493496	−	0.869748i	$-0.664281\pi$
$43$	−6.47214	−0.986991	−0.493496	+	0.869748i	$0.664281\pi$
$44$	−3.61803	−0.545439
$45$	0	0
$46$	8.47214	1.24915
$47$	−7.85410	−1.14564	−0.572819	−	0.819682i	$-0.694150\pi$
$47$	−7.85410	−1.14564	−0.572819	+	0.819682i	$0.694150\pi$
$48$	−1.61803	−0.233543
$49$	−5.47214	−0.781734
$50$	0	0
$51$	6.23607	0.873224
$52$	−0.381966	−0.0529692
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	5.47214	0.744663
$55$	0	0
$56$	3.70820	0.495530
$57$	1.00000	0.132453
$58$	7.09017	0.930985
$59$	−2.76393	−0.359833	−0.179917	−	0.983682i	$-0.557583\pi$
$59$	−2.76393	−0.359833	−0.179917	+	0.983682i	$0.557583\pi$
$60$	0	0
$61$	4.85410	0.621504	0.310752	−	0.950491i	$-0.399419\pi$
$61$	4.85410	0.621504	0.310752	+	0.950491i	$0.399419\pi$
$62$	1.52786	0.194039
$63$	−0.472136	−0.0594835
$64$	7.00000	0.875000
$65$	0	0
$66$	−5.85410	−0.720590
$67$	−12.7984	−1.56357	−0.781785	−	0.623548i	$-0.785691\pi$
$67$	−12.7984	−1.56357	−0.781785	+	0.623548i	$0.785691\pi$
$68$	−3.85410	−0.467379
$69$	−13.7082	−1.65027
$70$	0	0
$71$	−14.7984	−1.75624	−0.878122	−	0.478437i	$-0.841204\pi$
$71$	−14.7984	−1.75624	−0.878122	+	0.478437i	$0.841204\pi$
$72$	−1.14590	−0.135045
$73$	1.09017	0.127595	0.0637974	−	0.997963i	$-0.479679\pi$
$73$	1.09017	0.127595	0.0637974	+	0.997963i	$0.479679\pi$
$74$	4.47214	0.519875
$75$	0	0
$76$	−0.618034	−0.0708934
$77$	4.47214	0.509647
$78$	−0.618034	−0.0699786
$79$	14.6525	1.64853	0.824266	−	0.566203i	$-0.191588\pi$
$79$	14.6525	1.64853	0.824266	+	0.566203i	$0.191588\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	4.09017	0.451684
$83$	13.3820	1.46886	0.734431	−	0.678684i	$-0.237449\pi$
$83$	13.3820	1.46886	0.734431	+	0.678684i	$0.237449\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	6.47214	0.697908
$87$	−11.4721	−1.22994
$88$	10.8541	1.15705
$89$	16.4721	1.74604	0.873021	−	0.487682i	$-0.162157\pi$
$89$	16.4721	1.74604	0.873021	+	0.487682i	$0.162157\pi$
$90$	0	0
$91$	0.472136	0.0494933
$92$	8.47214	0.883281
$93$	−2.47214	−0.256349
$94$	7.85410	0.810089
$95$	0	0
$96$	−8.09017	−0.825700
$97$	12.7639	1.29598	0.647990	−	0.761648i	$-0.275610\pi$
$97$	12.7639	1.29598	0.647990	+	0.761648i	$0.275610\pi$
$98$	5.47214	0.552769
$99$	−1.38197	−0.138893
$100$	0	0
$101$	2.76393	0.275022	0.137511	−	0.990500i	$-0.456090\pi$
$101$	2.76393	0.275022	0.137511	+	0.990500i	$0.456090\pi$
$102$	−6.23607	−0.617463
$103$	−18.6525	−1.83788	−0.918942	−	0.394394i	$-0.870955\pi$
$103$	−18.6525	−1.83788	−0.918942	+	0.394394i	$0.870955\pi$
$104$	1.14590	0.112365
$105$	0	0
$106$	−4.00000	−0.388514
$107$	−8.09017	−0.782106	−0.391053	−	0.920368i	$-0.627889\pi$
$107$	−8.09017	−0.782106	−0.391053	+	0.920368i	$0.627889\pi$
$108$	5.47214	0.526557
$109$	−14.4721	−1.38618	−0.693090	−	0.720851i	$-0.743751\pi$
$109$	−14.4721	−1.38618	−0.693090	+	0.720851i	$0.743751\pi$
$110$	0	0
$111$	−7.23607	−0.686817
$112$	−1.23607	−0.116797
$113$	21.1246	1.98724	0.993618	−	0.112796i	$-0.0359807\pi$
$113$	21.1246	1.98724	0.993618	+	0.112796i	$0.0359807\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	7.09017	0.658306
$117$	−0.145898	−0.0134883
$118$	2.76393	0.254441
$119$	4.76393	0.436709
$120$	0	0
$121$	2.09017	0.190015
$122$	−4.85410	−0.439470
$123$	−6.61803	−0.596728
$124$	1.52786	0.137206
$125$	0	0
$126$	0.472136	0.0420612
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	3.00000	0.265165
$129$	−10.4721	−0.922020
$130$	0	0
$131$	−15.3820	−1.34393	−0.671964	−	0.740584i	$-0.734549\pi$
$131$	−15.3820	−1.34393	−0.671964	+	0.740584i	$0.734549\pi$
$132$	−5.85410	−0.509534
$133$	0.763932	0.0662413
$134$	12.7984	1.10561
$135$	0	0
$136$	11.5623	0.991460
$137$	1.90983	0.163168	0.0815839	−	0.996666i	$-0.474002\pi$
$137$	1.90983	0.163168	0.0815839	+	0.996666i	$0.474002\pi$
$138$	13.7082	1.16692
$139$	1.90983	0.161990	0.0809948	−	0.996715i	$-0.474190\pi$
$139$	1.90983	0.161990	0.0809948	+	0.996715i	$0.474190\pi$
$140$	0	0
$141$	−12.7082	−1.07022
$142$	14.7984	1.24185
$143$	1.38197	0.115566
$144$	0.381966	0.0318305
$145$	0	0
$146$	−1.09017	−0.0902231
$147$	−8.85410	−0.730274
$148$	4.47214	0.367607
$149$	−4.94427	−0.405051	−0.202525	−	0.979277i	$-0.564915\pi$
$149$	−4.94427	−0.405051	−0.202525	+	0.979277i	$0.564915\pi$
$150$	0	0
$151$	−10.9443	−0.890632	−0.445316	−	0.895373i	$-0.646909\pi$
$151$	−10.9443	−0.890632	−0.445316	+	0.895373i	$0.646909\pi$
$152$	1.85410	0.150388
$153$	−1.47214	−0.119015
$154$	−4.47214	−0.360375
$155$	0	0
$156$	−0.618034	−0.0494823
$157$	10.3262	0.824124	0.412062	−	0.911156i	$-0.364809\pi$
$157$	10.3262	0.824124	0.412062	+	0.911156i	$0.364809\pi$
$158$	−14.6525	−1.16569
$159$	6.47214	0.513274
$160$	0	0
$161$	−10.4721	−0.825320
$162$	7.70820	0.605614
$163$	21.8885	1.71444	0.857222	−	0.514948i	$-0.172189\pi$
$163$	21.8885	1.71444	0.857222	+	0.514948i	$0.172189\pi$
$164$	4.09017	0.319389
$165$	0	0
$166$	−13.3820	−1.03864
$167$	−17.7082	−1.37030	−0.685151	−	0.728401i	$-0.740264\pi$
$167$	−17.7082	−1.37030	−0.685151	+	0.728401i	$0.740264\pi$
$168$	6.00000	0.462910
$169$	−12.8541	−0.988777
$170$	0	0
$171$	−0.236068	−0.0180526
$172$	6.47214	0.493496
$173$	−9.32624	−0.709061	−0.354530	−	0.935045i	$-0.615359\pi$
$173$	−9.32624	−0.709061	−0.354530	+	0.935045i	$0.615359\pi$
$174$	11.4721	0.869700
$175$	0	0
$176$	−3.61803	−0.272720
$177$	−4.47214	−0.336146
$178$	−16.4721	−1.23464
$179$	−12.9443	−0.967500	−0.483750	−	0.875206i	$-0.660726\pi$
$179$	−12.9443	−0.967500	−0.483750	+	0.875206i	$0.660726\pi$
$180$	0	0
$181$	−5.61803	−0.417585	−0.208793	−	0.977960i	$-0.566953\pi$
$181$	−5.61803	−0.417585	−0.208793	+	0.977960i	$0.566953\pi$
$182$	−0.472136	−0.0349970
$183$	7.85410	0.580592
$184$	−25.4164	−1.87372
$185$	0	0
$186$	2.47214	0.181266
$187$	13.9443	1.01971
$188$	7.85410	0.572819
$189$	−6.76393	−0.492004
$190$	0	0
$191$	10.6525	0.770786	0.385393	−	0.922753i	$-0.374066\pi$
$191$	10.6525	0.770786	0.385393	+	0.922753i	$0.374066\pi$
$192$	11.3262	0.817401
$193$	−8.47214	−0.609838	−0.304919	−	0.952378i	$-0.598629\pi$
$193$	−8.47214	−0.609838	−0.304919	+	0.952378i	$0.598629\pi$
$194$	−12.7639	−0.916397
$195$	0	0
$196$	5.47214	0.390867
$197$	−2.94427	−0.209771	−0.104885	−	0.994484i	$-0.533448\pi$
$197$	−2.94427	−0.209771	−0.104885	+	0.994484i	$0.533448\pi$
$198$	1.38197	0.0982120
$199$	−17.1459	−1.21544	−0.607720	−	0.794151i	$-0.707916\pi$
$199$	−17.1459	−1.21544	−0.607720	+	0.794151i	$0.707916\pi$
$200$	0	0
$201$	−20.7082	−1.46064
$202$	−2.76393	−0.194470
$203$	−8.76393	−0.615107
$204$	−6.23607	−0.436612
$205$	0	0
$206$	18.6525	1.29958
$207$	3.23607	0.224922
$208$	−0.381966	−0.0264846
$209$	2.23607	0.154672
$210$	0	0
$211$	−10.7639	−0.741020	−0.370510	−	0.928829i	$-0.620817\pi$
$211$	−10.7639	−0.741020	−0.370510	+	0.928829i	$0.620817\pi$
$212$	−4.00000	−0.274721
$213$	−23.9443	−1.64063
$214$	8.09017	0.553033
$215$	0	0
$216$	−16.4164	−1.11700
$217$	−1.88854	−0.128203
$218$	14.4721	0.980177
$219$	1.76393	0.119195
$220$	0	0
$221$	1.47214	0.0990266
$222$	7.23607	0.485653
$223$	−6.61803	−0.443176	−0.221588	−	0.975140i	$-0.571124\pi$
$223$	−6.61803	−0.443176	−0.221588	+	0.975140i	$0.571124\pi$
$224$	−6.18034	−0.412941
$225$	0	0
$226$	−21.1246	−1.40519
$227$	27.5967	1.83166	0.915830	−	0.401566i	$-0.131534\pi$
$227$	27.5967	1.83166	0.915830	+	0.401566i	$0.131534\pi$
$228$	−1.00000	−0.0662266
$229$	25.0344	1.65432	0.827161	−	0.561965i	$-0.189954\pi$
$229$	25.0344	1.65432	0.827161	+	0.561965i	$0.189954\pi$
$230$	0	0
$231$	7.23607	0.476098
$232$	−21.2705	−1.39648
$233$	−14.9443	−0.979032	−0.489516	−	0.871994i	$-0.662826\pi$
$233$	−14.9443	−0.979032	−0.489516	+	0.871994i	$0.662826\pi$
$234$	0.145898	0.00953765
$235$	0	0
$236$	2.76393	0.179917
$237$	23.7082	1.54001
$238$	−4.76393	−0.308800
$239$	−10.4721	−0.677386	−0.338693	−	0.940897i	$-0.609985\pi$
$239$	−10.4721	−0.677386	−0.338693	+	0.940897i	$0.609985\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−2.09017	−0.134361
$243$	3.94427	0.253025
$244$	−4.85410	−0.310752
$245$	0	0
$246$	6.61803	0.421950
$247$	0.236068	0.0150206
$248$	−4.58359	−0.291058
$249$	21.6525	1.37217
$250$	0	0
$251$	7.70820	0.486538	0.243269	−	0.969959i	$-0.421780\pi$
$251$	7.70820	0.486538	0.243269	+	0.969959i	$0.421780\pi$
$252$	0.472136	0.0297418
$253$	−30.6525	−1.92710
$254$	−10.0000	−0.627456
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−9.12461	−0.569178	−0.284589	−	0.958650i	$-0.591857\pi$
$257$	−9.12461	−0.569178	−0.284589	+	0.958650i	$0.591857\pi$
$258$	10.4721	0.651967
$259$	−5.52786	−0.343485
$260$	0	0
$261$	2.70820	0.167634
$262$	15.3820	0.950301
$263$	14.4721	0.892390	0.446195	−	0.894936i	$-0.352779\pi$
$263$	14.4721	0.892390	0.446195	+	0.894936i	$0.352779\pi$
$264$	17.5623	1.08089
$265$	0	0
$266$	−0.763932	−0.0468397
$267$	26.6525	1.63111
$268$	12.7984	0.781785
$269$	1.70820	0.104151	0.0520755	−	0.998643i	$-0.483416\pi$
$269$	1.70820	0.104151	0.0520755	+	0.998643i	$0.483416\pi$
$270$	0	0
$271$	−5.81966	−0.353519	−0.176760	−	0.984254i	$-0.556562\pi$
$271$	−5.81966	−0.353519	−0.176760	+	0.984254i	$0.556562\pi$
$272$	−3.85410	−0.233689
$273$	0.763932	0.0462353
$274$	−1.90983	−0.115377
$275$	0	0
$276$	13.7082	0.825137
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−1.90983	−0.114544
$279$	0.583592	0.0349387
$280$	0	0
$281$	−9.03444	−0.538950	−0.269475	−	0.963007i	$-0.586850\pi$
$281$	−9.03444	−0.538950	−0.269475	+	0.963007i	$0.586850\pi$
$282$	12.7082	0.756763
$283$	−6.76393	−0.402074	−0.201037	−	0.979584i	$-0.564431\pi$
$283$	−6.76393	−0.402074	−0.201037	+	0.979584i	$0.564431\pi$
$284$	14.7984	0.878122
$285$	0	0
$286$	−1.38197	−0.0817174
$287$	−5.05573	−0.298430
$288$	1.90983	0.112538
$289$	−2.14590	−0.126229
$290$	0	0
$291$	20.6525	1.21067
$292$	−1.09017	−0.0637974
$293$	−5.67376	−0.331465	−0.165732	−	0.986171i	$-0.552999\pi$
$293$	−5.67376	−0.331465	−0.165732	+	0.986171i	$0.552999\pi$
$294$	8.85410	0.516382
$295$	0	0
$296$	−13.4164	−0.779813
$297$	−19.7984	−1.14882
$298$	4.94427	0.286414
$299$	−3.23607	−0.187147
$300$	0	0
$301$	−8.00000	−0.461112
$302$	10.9443	0.629772
$303$	4.47214	0.256917
$304$	−0.618034	−0.0354467
$305$	0	0
$306$	1.47214	0.0841564
$307$	−29.7082	−1.69554	−0.847768	−	0.530367i	$-0.822054\pi$
$307$	−29.7082	−1.69554	−0.847768	+	0.530367i	$0.822054\pi$
$308$	−4.47214	−0.254824
$309$	−30.1803	−1.71690
$310$	0	0
$311$	−30.2148	−1.71332	−0.856662	−	0.515879i	$-0.827465\pi$
$311$	−30.2148	−1.71332	−0.856662	+	0.515879i	$0.827465\pi$
$312$	1.85410	0.104968
$313$	−12.2918	−0.694773	−0.347387	−	0.937722i	$-0.612931\pi$
$313$	−12.2918	−0.694773	−0.347387	+	0.937722i	$0.612931\pi$
$314$	−10.3262	−0.582743
$315$	0	0
$316$	−14.6525	−0.824266
$317$	15.0344	0.844418	0.422209	−	0.906498i	$-0.361255\pi$
$317$	15.0344	0.844418	0.422209	+	0.906498i	$0.361255\pi$
$318$	−6.47214	−0.362939
$319$	−25.6525	−1.43626
$320$	0	0
$321$	−13.0902	−0.730622
$322$	10.4721	0.583589
$323$	2.38197	0.132536
$324$	7.70820	0.428234
$325$	0	0
$326$	−21.8885	−1.21229
$327$	−23.4164	−1.29493
$328$	−12.2705	−0.677526
$329$	−9.70820	−0.535231
$330$	0	0
$331$	−14.1803	−0.779422	−0.389711	−	0.920937i	$-0.627425\pi$
$331$	−14.1803	−0.779422	−0.389711	+	0.920937i	$0.627425\pi$
$332$	−13.3820	−0.734431
$333$	1.70820	0.0936090
$334$	17.7082	0.968950
$335$	0	0
$336$	−2.00000	−0.109109
$337$	26.6525	1.45185	0.725926	−	0.687772i	$-0.241411\pi$
$337$	26.6525	1.45185	0.725926	+	0.687772i	$0.241411\pi$
$338$	12.8541	0.699171
$339$	34.1803	1.85642
$340$	0	0
$341$	−5.52786	−0.299351
$342$	0.236068	0.0127651
$343$	−15.4164	−0.832408
$344$	−19.4164	−1.04686
$345$	0	0
$346$	9.32624	0.501382
$347$	12.7984	0.687053	0.343526	−	0.939143i	$-0.388379\pi$
$347$	12.7984	0.687053	0.343526	+	0.939143i	$0.388379\pi$
$348$	11.4721	0.614971
$349$	−14.5066	−0.776519	−0.388260	−	0.921550i	$-0.626924\pi$
$349$	−14.5066	−0.776519	−0.388260	+	0.921550i	$0.626924\pi$
$350$	0	0
$351$	−2.09017	−0.111565
$352$	−18.0902	−0.964209
$353$	14.6738	0.781006	0.390503	−	0.920602i	$-0.372301\pi$
$353$	14.6738	0.781006	0.390503	+	0.920602i	$0.372301\pi$
$354$	4.47214	0.237691
$355$	0	0
$356$	−16.4721	−0.873021
$357$	7.70820	0.407961
$358$	12.9443	0.684126
$359$	17.7082	0.934603	0.467302	−	0.884098i	$-0.345226\pi$
$359$	17.7082	0.934603	0.467302	+	0.884098i	$0.345226\pi$
$360$	0	0
$361$	−18.6180	−0.979897
$362$	5.61803	0.295277
$363$	3.38197	0.177507
$364$	−0.472136	−0.0247466
$365$	0	0
$366$	−7.85410	−0.410540
$367$	−6.18034	−0.322611	−0.161306	−	0.986905i	$-0.551570\pi$
$367$	−6.18034	−0.322611	−0.161306	+	0.986905i	$0.551570\pi$
$368$	8.47214	0.441641
$369$	1.56231	0.0813304
$370$	0	0
$371$	4.94427	0.256694
$372$	2.47214	0.128174
$373$	−8.14590	−0.421779	−0.210889	−	0.977510i	$-0.567636\pi$
$373$	−8.14590	−0.421779	−0.210889	+	0.977510i	$0.567636\pi$
$374$	−13.9443	−0.721041
$375$	0	0
$376$	−23.5623	−1.21513
$377$	−2.70820	−0.139480
$378$	6.76393	0.347899
$379$	−36.7426	−1.88734	−0.943671	−	0.330884i	$-0.892653\pi$
$379$	−36.7426	−1.88734	−0.943671	+	0.330884i	$0.892653\pi$
$380$	0	0
$381$	16.1803	0.828944
$382$	−10.6525	−0.545028
$383$	15.5279	0.793437	0.396718	−	0.917940i	$-0.370149\pi$
$383$	15.5279	0.793437	0.396718	+	0.917940i	$0.370149\pi$
$384$	4.85410	0.247710
$385$	0	0
$386$	8.47214	0.431220
$387$	2.47214	0.125666
$388$	−12.7639	−0.647990
$389$	−23.7082	−1.20205	−0.601027	−	0.799229i	$-0.705242\pi$
$389$	−23.7082	−1.20205	−0.601027	+	0.799229i	$0.705242\pi$
$390$	0	0
$391$	−32.6525	−1.65131
$392$	−16.4164	−0.829154
$393$	−24.8885	−1.25546
$394$	2.94427	0.148330
$395$	0	0
$396$	1.38197	0.0694464
$397$	31.3050	1.57115	0.785575	−	0.618766i	$-0.212367\pi$
$397$	31.3050	1.57115	0.785575	+	0.618766i	$0.212367\pi$
$398$	17.1459	0.859446
$399$	1.23607	0.0618808
$400$	0	0
$401$	6.79837	0.339495	0.169747	−	0.985488i	$-0.445705\pi$
$401$	6.79837	0.339495	0.169747	+	0.985488i	$0.445705\pi$
$402$	20.7082	1.03283
$403$	−0.583592	−0.0290708
$404$	−2.76393	−0.137511
$405$	0	0
$406$	8.76393	0.434947
$407$	−16.1803	−0.802030
$408$	18.7082	0.926194
$409$	15.0557	0.744458	0.372229	−	0.928141i	$-0.378594\pi$
$409$	15.0557	0.744458	0.372229	+	0.928141i	$0.378594\pi$
$410$	0	0
$411$	3.09017	0.152427
$412$	18.6525	0.918942
$413$	−3.41641	−0.168110
$414$	−3.23607	−0.159044
$415$	0	0
$416$	−1.90983	−0.0936371
$417$	3.09017	0.151326
$418$	−2.23607	−0.109370
$419$	27.7984	1.35804	0.679020	−	0.734120i	$-0.262405\pi$
$419$	27.7984	1.35804	0.679020	+	0.734120i	$0.262405\pi$
$420$	0	0
$421$	34.3607	1.67464	0.837319	−	0.546715i	$-0.184122\pi$
$421$	34.3607	1.67464	0.837319	+	0.546715i	$0.184122\pi$
$422$	10.7639	0.523980
$423$	3.00000	0.145865
$424$	12.0000	0.582772
$425$	0	0
$426$	23.9443	1.16010
$427$	6.00000	0.290360
$428$	8.09017	0.391053
$429$	2.23607	0.107958
$430$	0	0
$431$	22.4508	1.08142	0.540710	−	0.841209i	$-0.318156\pi$
$431$	22.4508	1.08142	0.540710	+	0.841209i	$0.318156\pi$
$432$	5.47214	0.263278
$433$	18.6525	0.896381	0.448190	−	0.893938i	$-0.352069\pi$
$433$	18.6525	0.896381	0.448190	+	0.893938i	$0.352069\pi$
$434$	1.88854	0.0906530
$435$	0	0
$436$	14.4721	0.693090
$437$	−5.23607	−0.250475
$438$	−1.76393	−0.0842839
$439$	10.8541	0.518038	0.259019	−	0.965872i	$-0.416601\pi$
$439$	10.8541	0.518038	0.259019	+	0.965872i	$0.416601\pi$
$440$	0	0
$441$	2.09017	0.0995319
$442$	−1.47214	−0.0700224
$443$	14.9443	0.710024	0.355012	−	0.934862i	$-0.384477\pi$
$443$	14.9443	0.710024	0.355012	+	0.934862i	$0.384477\pi$
$444$	7.23607	0.343409
$445$	0	0
$446$	6.61803	0.313373
$447$	−8.00000	−0.378387
$448$	8.65248	0.408791
$449$	−13.2361	−0.624649	−0.312324	−	0.949976i	$-0.601108\pi$
$449$	−13.2361	−0.624649	−0.312324	+	0.949976i	$0.601108\pi$
$450$	0	0
$451$	−14.7984	−0.696828
$452$	−21.1246	−0.993618
$453$	−17.7082	−0.832004
$454$	−27.5967	−1.29518
$455$	0	0
$456$	3.00000	0.140488
$457$	24.6525	1.15319	0.576597	−	0.817029i	$-0.304380\pi$
$457$	24.6525	1.15319	0.576597	+	0.817029i	$0.304380\pi$
$458$	−25.0344	−1.16978
$459$	−21.0902	−0.984405
$460$	0	0
$461$	−36.0000	−1.67669	−0.838344	−	0.545142i	$-0.816476\pi$
$461$	−36.0000	−1.67669	−0.838344	+	0.545142i	$0.816476\pi$
$462$	−7.23607	−0.336652
$463$	−4.29180	−0.199457	−0.0997283	−	0.995015i	$-0.531797\pi$
$463$	−4.29180	−0.199457	−0.0997283	+	0.995015i	$0.531797\pi$
$464$	7.09017	0.329153
$465$	0	0
$466$	14.9443	0.692280
$467$	−8.67376	−0.401374	−0.200687	−	0.979655i	$-0.564317\pi$
$467$	−8.67376	−0.401374	−0.200687	+	0.979655i	$0.564317\pi$
$468$	0.145898	0.00674414
$469$	−15.8197	−0.730484
$470$	0	0
$471$	16.7082	0.769873
$472$	−8.29180	−0.381661
$473$	−23.4164	−1.07669
$474$	−23.7082	−1.08895
$475$	0	0
$476$	−4.76393	−0.218354
$477$	−1.52786	−0.0699561
$478$	10.4721	0.478984
$479$	−38.9443	−1.77941	−0.889705	−	0.456537i	$-0.849090\pi$
$479$	−38.9443	−1.77941	−0.889705	+	0.456537i	$0.849090\pi$
$480$	0	0
$481$	−1.70820	−0.0778874
$482$	−1.00000	−0.0455488
$483$	−16.9443	−0.770991
$484$	−2.09017	−0.0950077
$485$	0	0
$486$	−3.94427	−0.178916
$487$	7.32624	0.331984	0.165992	−	0.986127i	$-0.446917\pi$
$487$	7.32624	0.331984	0.165992	+	0.986127i	$0.446917\pi$
$488$	14.5623	0.659205
$489$	35.4164	1.60159
$490$	0	0
$491$	−3.41641	−0.154180	−0.0770902	−	0.997024i	$-0.524563\pi$
$491$	−3.41641	−0.154180	−0.0770902	+	0.997024i	$0.524563\pi$
$492$	6.61803	0.298364
$493$	−27.3262	−1.23071
$494$	−0.236068	−0.0106212
$495$	0	0
$496$	1.52786	0.0686031
$497$	−18.2918	−0.820499
$498$	−21.6525	−0.970271
$499$	41.3050	1.84906	0.924532	−	0.381105i	$-0.124456\pi$
$499$	41.3050	1.84906	0.924532	+	0.381105i	$0.124456\pi$
$500$	0	0
$501$	−28.6525	−1.28010
$502$	−7.70820	−0.344034
$503$	−26.6525	−1.18838	−0.594188	−	0.804327i	$-0.702526\pi$
$503$	−26.6525	−1.18838	−0.594188	+	0.804327i	$0.702526\pi$
$504$	−1.41641	−0.0630918
$505$	0	0
$506$	30.6525	1.36267
$507$	−20.7984	−0.923688
$508$	−10.0000	−0.443678
$509$	2.96556	0.131446	0.0657230	−	0.997838i	$-0.479065\pi$
$509$	2.96556	0.131446	0.0657230	+	0.997838i	$0.479065\pi$
$510$	0	0
$511$	1.34752	0.0596110
$512$	11.0000	0.486136
$513$	−3.38197	−0.149317
$514$	9.12461	0.402469
$515$	0	0
$516$	10.4721	0.461010
$517$	−28.4164	−1.24975
$518$	5.52786	0.242880
$519$	−15.0902	−0.662385
$520$	0	0
$521$	3.23607	0.141775	0.0708874	−	0.997484i	$-0.477417\pi$
$521$	3.23607	0.141775	0.0708874	+	0.997484i	$0.477417\pi$
$522$	−2.70820	−0.118535
$523$	6.56231	0.286950	0.143475	−	0.989654i	$-0.454172\pi$
$523$	6.56231	0.286950	0.143475	+	0.989654i	$0.454172\pi$
$524$	15.3820	0.671964
$525$	0	0
$526$	−14.4721	−0.631015
$527$	−5.88854	−0.256509
$528$	−5.85410	−0.254767
$529$	48.7771	2.12074
$530$	0	0
$531$	1.05573	0.0458147
$532$	−0.763932	−0.0331207
$533$	−1.56231	−0.0676710
$534$	−26.6525	−1.15337
$535$	0	0
$536$	−38.3951	−1.65842
$537$	−20.9443	−0.903812
$538$	−1.70820	−0.0736459
$539$	−19.7984	−0.852776
$540$	0	0
$541$	34.7984	1.49610	0.748049	−	0.663643i	$-0.230991\pi$
$541$	34.7984	1.49610	0.748049	+	0.663643i	$0.230991\pi$
$542$	5.81966	0.249976
$543$	−9.09017	−0.390097
$544$	−19.2705	−0.826216
$545$	0	0
$546$	−0.763932	−0.0326933
$547$	9.70820	0.415093	0.207546	−	0.978225i	$-0.433452\pi$
$547$	9.70820	0.415093	0.207546	+	0.978225i	$0.433452\pi$
$548$	−1.90983	−0.0815839
$549$	−1.85410	−0.0791311
$550$	0	0
$551$	−4.38197	−0.186678
$552$	−41.1246	−1.75038
$553$	18.1115	0.770177
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−1.90983	−0.0809948
$557$	−45.8885	−1.94436	−0.972180	−	0.234235i	$-0.924741\pi$
$557$	−45.8885	−1.94436	−0.972180	+	0.234235i	$0.924741\pi$
$558$	−0.583592	−0.0247054
$559$	−2.47214	−0.104560
$560$	0	0
$561$	22.5623	0.952581
$562$	9.03444	0.381095
$563$	7.72949	0.325759	0.162880	−	0.986646i	$-0.447922\pi$
$563$	7.72949	0.325759	0.162880	+	0.986646i	$0.447922\pi$
$564$	12.7082	0.535112
$565$	0	0
$566$	6.76393	0.284309
$567$	−9.52786	−0.400133
$568$	−44.3951	−1.86278
$569$	23.5066	0.985447	0.492724	−	0.870186i	$-0.336001\pi$
$569$	23.5066	0.985447	0.492724	+	0.870186i	$0.336001\pi$
$570$	0	0
$571$	0.326238	0.0136526	0.00682632	−	0.999977i	$-0.497827\pi$
$571$	0.326238	0.0136526	0.00682632	+	0.999977i	$0.497827\pi$
$572$	−1.38197	−0.0577829
$573$	17.2361	0.720047
$574$	5.05573	0.211022
$575$	0	0
$576$	−2.67376	−0.111407
$577$	22.8541	0.951429	0.475714	−	0.879600i	$-0.342190\pi$
$577$	22.8541	0.951429	0.475714	+	0.879600i	$0.342190\pi$
$578$	2.14590	0.0892576
$579$	−13.7082	−0.569694
$580$	0	0
$581$	16.5410	0.686237
$582$	−20.6525	−0.856073
$583$	14.4721	0.599375
$584$	3.27051	0.135335
$585$	0	0
$586$	5.67376	0.234381
$587$	−20.4721	−0.844975	−0.422488	−	0.906369i	$-0.638843\pi$
$587$	−20.4721	−0.844975	−0.422488	+	0.906369i	$0.638843\pi$
$588$	8.85410	0.365137
$589$	−0.944272	−0.0389080
$590$	0	0
$591$	−4.76393	−0.195962
$592$	4.47214	0.183804
$593$	−40.8328	−1.67680	−0.838401	−	0.545053i	$-0.816509\pi$
$593$	−40.8328	−1.67680	−0.838401	+	0.545053i	$0.816509\pi$
$594$	19.7984	0.812337
$595$	0	0
$596$	4.94427	0.202525
$597$	−27.7426	−1.13543
$598$	3.23607	0.132333
$599$	19.4508	0.794740	0.397370	−	0.917658i	$-0.369923\pi$
$599$	19.4508	0.794740	0.397370	+	0.917658i	$0.369923\pi$
$600$	0	0
$601$	38.7426	1.58035	0.790173	−	0.612884i	$-0.209991\pi$
$601$	38.7426	1.58035	0.790173	+	0.612884i	$0.209991\pi$
$602$	8.00000	0.326056
$603$	4.88854	0.199077
$604$	10.9443	0.445316
$605$	0	0
$606$	−4.47214	−0.181668
$607$	44.3607	1.80054	0.900272	−	0.435327i	$-0.143367\pi$
$607$	44.3607	1.80054	0.900272	+	0.435327i	$0.143367\pi$
$608$	−3.09017	−0.125323
$609$	−14.1803	−0.574616
$610$	0	0
$611$	−3.00000	−0.121367
$612$	1.47214	0.0595076
$613$	26.6869	1.07787	0.538937	−	0.842346i	$-0.318826\pi$
$613$	26.6869	1.07787	0.538937	+	0.842346i	$0.318826\pi$
$614$	29.7082	1.19893
$615$	0	0
$616$	13.4164	0.540562
$617$	−19.0557	−0.767155	−0.383577	−	0.923509i	$-0.625308\pi$
$617$	−19.0557	−0.767155	−0.383577	+	0.923509i	$0.625308\pi$
$618$	30.1803	1.21403
$619$	11.9656	0.480936	0.240468	−	0.970657i	$-0.422699\pi$
$619$	11.9656	0.480936	0.240468	+	0.970657i	$0.422699\pi$
$620$	0	0
$621$	46.3607	1.86039
$622$	30.2148	1.21150
$623$	20.3607	0.815733
$624$	−0.618034	−0.0247412
$625$	0	0
$626$	12.2918	0.491279
$627$	3.61803	0.144490
$628$	−10.3262	−0.412062
$629$	−17.2361	−0.687247
$630$	0	0
$631$	16.9098	0.673170	0.336585	−	0.941653i	$-0.390728\pi$
$631$	16.9098	0.673170	0.336585	+	0.941653i	$0.390728\pi$
$632$	43.9574	1.74853
$633$	−17.4164	−0.692240
$634$	−15.0344	−0.597094
$635$	0	0
$636$	−6.47214	−0.256637
$637$	−2.09017	−0.0828155
$638$	25.6525	1.01559
$639$	5.65248	0.223608
$640$	0	0
$641$	−14.1459	−0.558729	−0.279365	−	0.960185i	$-0.590124\pi$
$641$	−14.1459	−0.558729	−0.279365	+	0.960185i	$0.590124\pi$
$642$	13.0902	0.516628
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	10.4721	0.412660
$645$	0	0
$646$	−2.38197	−0.0937172
$647$	−3.81966	−0.150166	−0.0750832	−	0.997177i	$-0.523922\pi$
$647$	−3.81966	−0.150166	−0.0750832	+	0.997177i	$0.523922\pi$
$648$	−23.1246	−0.908421
$649$	−10.0000	−0.392534
$650$	0	0
$651$	−3.05573	−0.119763
$652$	−21.8885	−0.857222
$653$	18.3262	0.717161	0.358581	−	0.933499i	$-0.383261\pi$
$653$	18.3262	0.717161	0.358581	+	0.933499i	$0.383261\pi$
$654$	23.4164	0.915654
$655$	0	0
$656$	4.09017	0.159694
$657$	−0.416408	−0.0162456
$658$	9.70820	0.378465
$659$	−14.7639	−0.575121	−0.287561	−	0.957762i	$-0.592844\pi$
$659$	−14.7639	−0.575121	−0.287561	+	0.957762i	$0.592844\pi$
$660$	0	0
$661$	−40.0689	−1.55850	−0.779249	−	0.626714i	$-0.784399\pi$
$661$	−40.0689	−1.55850	−0.779249	+	0.626714i	$0.784399\pi$
$662$	14.1803	0.551135
$663$	2.38197	0.0925079
$664$	40.1459	1.55796
$665$	0	0
$666$	−1.70820	−0.0661916
$667$	60.0689	2.32588
$668$	17.7082	0.685151
$669$	−10.7082	−0.414003
$670$	0	0
$671$	17.5623	0.677985
$672$	−10.0000	−0.385758
$673$	−5.34752	−0.206132	−0.103066	−	0.994675i	$-0.532865\pi$
$673$	−5.34752	−0.206132	−0.103066	+	0.994675i	$0.532865\pi$
$674$	−26.6525	−1.02662
$675$	0	0
$676$	12.8541	0.494389
$677$	−12.3820	−0.475878	−0.237939	−	0.971280i	$-0.576472\pi$
$677$	−12.3820	−0.475878	−0.237939	+	0.971280i	$0.576472\pi$
$678$	−34.1803	−1.31269
$679$	15.7771	0.605469
$680$	0	0
$681$	44.6525	1.71109
$682$	5.52786	0.211673
$683$	16.3607	0.626024	0.313012	−	0.949749i	$-0.398662\pi$
$683$	16.3607	0.626024	0.313012	+	0.949749i	$0.398662\pi$
$684$	0.236068	0.00902628
$685$	0	0
$686$	15.4164	0.588601
$687$	40.5066	1.54542
$688$	6.47214	0.246748
$689$	1.52786	0.0582070
$690$	0	0
$691$	−28.1803	−1.07203	−0.536015	−	0.844208i	$-0.680071\pi$
$691$	−28.1803	−1.07203	−0.536015	+	0.844208i	$0.680071\pi$
$692$	9.32624	0.354530
$693$	−1.70820	−0.0648893
$694$	−12.7984	−0.485820
$695$	0	0
$696$	−34.4164	−1.30455
$697$	−15.7639	−0.597102
$698$	14.5066	0.549082
$699$	−24.1803	−0.914584
$700$	0	0
$701$	−34.1803	−1.29097	−0.645487	−	0.763771i	$-0.723345\pi$
$701$	−34.1803	−1.29097	−0.645487	+	0.763771i	$0.723345\pi$
$702$	2.09017	0.0788884
$703$	−2.76393	−0.104244
$704$	25.3262	0.954519
$705$	0	0
$706$	−14.6738	−0.552254
$707$	3.41641	0.128487
$708$	4.47214	0.168073
$709$	−18.4721	−0.693736	−0.346868	−	0.937914i	$-0.612755\pi$
$709$	−18.4721	−0.693736	−0.346868	+	0.937914i	$0.612755\pi$
$710$	0	0
$711$	−5.59675	−0.209894
$712$	49.4164	1.85196
$713$	12.9443	0.484767
$714$	−7.70820	−0.288472
$715$	0	0
$716$	12.9443	0.483750
$717$	−16.9443	−0.632795
$718$	−17.7082	−0.660864
$719$	15.5967	0.581661	0.290830	−	0.956775i	$-0.406069\pi$
$719$	15.5967	0.581661	0.290830	+	0.956775i	$0.406069\pi$
$720$	0	0
$721$	−23.0557	−0.858640
$722$	18.6180	0.692891
$723$	1.61803	0.0601753
$724$	5.61803	0.208793
$725$	0	0
$726$	−3.38197	−0.125517
$727$	−44.7984	−1.66148	−0.830740	−	0.556661i	$-0.812082\pi$
$727$	−44.7984	−1.66148	−0.830740	+	0.556661i	$0.812082\pi$
$728$	1.41641	0.0524956
$729$	29.5066	1.09284
$730$	0	0
$731$	−24.9443	−0.922597
$732$	−7.85410	−0.290296
$733$	−18.1803	−0.671506	−0.335753	−	0.941950i	$-0.608991\pi$
$733$	−18.1803	−0.671506	−0.335753	+	0.941950i	$0.608991\pi$
$734$	6.18034	0.228121
$735$	0	0
$736$	42.3607	1.56144
$737$	−46.3050	−1.70566
$738$	−1.56231	−0.0575093
$739$	−7.23607	−0.266183	−0.133092	−	0.991104i	$-0.542490\pi$
$739$	−7.23607	−0.266183	−0.133092	+	0.991104i	$0.542490\pi$
$740$	0	0
$741$	0.381966	0.0140319
$742$	−4.94427	−0.181510
$743$	18.9230	0.694217	0.347109	−	0.937825i	$-0.387164\pi$
$743$	18.9230	0.694217	0.347109	+	0.937825i	$0.387164\pi$
$744$	−7.41641	−0.271899
$745$	0	0
$746$	8.14590	0.298243
$747$	−5.11146	−0.187018
$748$	−13.9443	−0.509853
$749$	−10.0000	−0.365392
$750$	0	0
$751$	9.14590	0.333739	0.166869	−	0.985979i	$-0.446634\pi$
$751$	9.14590	0.333739	0.166869	+	0.985979i	$0.446634\pi$
$752$	7.85410	0.286410
$753$	12.4721	0.454510
$754$	2.70820	0.0986270
$755$	0	0
$756$	6.76393	0.246002
$757$	−7.85410	−0.285462	−0.142731	−	0.989762i	$-0.545588\pi$
$757$	−7.85410	−0.285462	−0.142731	+	0.989762i	$0.545588\pi$
$758$	36.7426	1.33455
$759$	−49.5967	−1.80025
$760$	0	0
$761$	−24.8328	−0.900189	−0.450094	−	0.892981i	$-0.648610\pi$
$761$	−24.8328	−0.900189	−0.450094	+	0.892981i	$0.648610\pi$
$762$	−16.1803	−0.586152
$763$	−17.8885	−0.647609
$764$	−10.6525	−0.385393
$765$	0	0
$766$	−15.5279	−0.561045
$767$	−1.05573	−0.0381201
$768$	−27.5066	−0.992558
$769$	−3.41641	−0.123199	−0.0615994	−	0.998101i	$-0.519620\pi$
$769$	−3.41641	−0.123199	−0.0615994	+	0.998101i	$0.519620\pi$
$770$	0	0
$771$	−14.7639	−0.531710
$772$	8.47214	0.304919
$773$	37.0132	1.33127	0.665635	−	0.746277i	$-0.268161\pi$
$773$	37.0132	1.33127	0.665635	+	0.746277i	$0.268161\pi$
$774$	−2.47214	−0.0888591
$775$	0	0
$776$	38.2918	1.37460
$777$	−8.94427	−0.320874
$778$	23.7082	0.849980
$779$	−2.52786	−0.0905701
$780$	0	0
$781$	−53.5410	−1.91585
$782$	32.6525	1.16765
$783$	38.7984	1.38654
$784$	5.47214	0.195433
$785$	0	0
$786$	24.8885	0.887745
$787$	6.96556	0.248295	0.124148	−	0.992264i	$-0.460380\pi$
$787$	6.96556	0.248295	0.124148	+	0.992264i	$0.460380\pi$
$788$	2.94427	0.104885
$789$	23.4164	0.833646
$790$	0	0
$791$	26.1115	0.928417
$792$	−4.14590	−0.147318
$793$	1.85410	0.0658411
$794$	−31.3050	−1.11097
$795$	0	0
$796$	17.1459	0.607720
$797$	−34.0344	−1.20556	−0.602781	−	0.797907i	$-0.705941\pi$
$797$	−34.0344	−1.20556	−0.602781	+	0.797907i	$0.705941\pi$
$798$	−1.23607	−0.0437563
$799$	−30.2705	−1.07089
$800$	0	0
$801$	−6.29180	−0.222310
$802$	−6.79837	−0.240059
$803$	3.94427	0.139190
$804$	20.7082	0.730322
$805$	0	0
$806$	0.583592	0.0205562
$807$	2.76393	0.0972950
$808$	8.29180	0.291704
$809$	−33.8885	−1.19146	−0.595729	−	0.803186i	$-0.703137\pi$
$809$	−33.8885	−1.19146	−0.595729	+	0.803186i	$0.703137\pi$
$810$	0	0
$811$	−25.5066	−0.895657	−0.447829	−	0.894119i	$-0.647803\pi$
$811$	−25.5066	−0.895657	−0.447829	+	0.894119i	$0.647803\pi$
$812$	8.76393	0.307554
$813$	−9.41641	−0.330248
$814$	16.1803	0.567121
$815$	0	0
$816$	−6.23607	−0.218306
$817$	−4.00000	−0.139942
$818$	−15.0557	−0.526411
$819$	−0.180340	−0.00630159
$820$	0	0
$821$	8.45085	0.294937	0.147468	−	0.989067i	$-0.452888\pi$
$821$	8.45085	0.294937	0.147468	+	0.989067i	$0.452888\pi$
$822$	−3.09017	−0.107782
$823$	39.4164	1.37397	0.686985	−	0.726672i	$-0.258934\pi$
$823$	39.4164	1.37397	0.686985	+	0.726672i	$0.258934\pi$
$824$	−55.9574	−1.94937
$825$	0	0
$826$	3.41641	0.118872
$827$	10.9443	0.380570	0.190285	−	0.981729i	$-0.439059\pi$
$827$	10.9443	0.380570	0.190285	+	0.981729i	$0.439059\pi$
$828$	−3.23607	−0.112461
$829$	−10.7984	−0.375043	−0.187522	−	0.982260i	$-0.560045\pi$
$829$	−10.7984	−0.375043	−0.187522	+	0.982260i	$0.560045\pi$
$830$	0	0
$831$	12.9443	0.449032
$832$	2.67376	0.0926960
$833$	−21.0902	−0.730731
$834$	−3.09017	−0.107004
$835$	0	0
$836$	−2.23607	−0.0773360
$837$	8.36068	0.288987
$838$	−27.7984	−0.960279
$839$	−23.8885	−0.824724	−0.412362	−	0.911020i	$-0.635296\pi$
$839$	−23.8885	−0.824724	−0.412362	+	0.911020i	$0.635296\pi$
$840$	0	0
$841$	21.2705	0.733466
$842$	−34.3607	−1.18415
$843$	−14.6180	−0.503472
$844$	10.7639	0.370510
$845$	0	0
$846$	−3.00000	−0.103142
$847$	2.58359	0.0887733
$848$	−4.00000	−0.137361
$849$	−10.9443	−0.375606
$850$	0	0
$851$	37.8885	1.29880
$852$	23.9443	0.820317
$853$	−19.9787	−0.684058	−0.342029	−	0.939689i	$-0.611114\pi$
$853$	−19.9787	−0.684058	−0.342029	+	0.939689i	$0.611114\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	−24.2705	−0.829549
$857$	20.2918	0.693155	0.346577	−	0.938021i	$-0.387344\pi$
$857$	20.2918	0.693155	0.346577	+	0.938021i	$0.387344\pi$
$858$	−2.23607	−0.0763381
$859$	47.7426	1.62896	0.814479	−	0.580193i	$-0.197023\pi$
$859$	47.7426	1.62896	0.814479	+	0.580193i	$0.197023\pi$
$860$	0	0
$861$	−8.18034	−0.278785
$862$	−22.4508	−0.764679
$863$	23.2361	0.790965	0.395482	−	0.918474i	$-0.370577\pi$
$863$	23.2361	0.790965	0.395482	+	0.918474i	$0.370577\pi$
$864$	27.3607	0.930829
$865$	0	0
$866$	−18.6525	−0.633837
$867$	−3.47214	−0.117920
$868$	1.88854	0.0641014
$869$	53.0132	1.79835
$870$	0	0
$871$	−4.88854	−0.165642
$872$	−43.4164	−1.47027
$873$	−4.87539	−0.165007
$874$	5.23607	0.177113
$875$	0	0
$876$	−1.76393	−0.0595977
$877$	32.5410	1.09883	0.549416	−	0.835549i	$-0.314850\pi$
$877$	32.5410	1.09883	0.549416	+	0.835549i	$0.314850\pi$
$878$	−10.8541	−0.366308
$879$	−9.18034	−0.309645
$880$	0	0
$881$	5.20163	0.175247	0.0876236	−	0.996154i	$-0.472073\pi$
$881$	5.20163	0.175247	0.0876236	+	0.996154i	$0.472073\pi$
$882$	−2.09017	−0.0703797
$883$	−17.4377	−0.586825	−0.293413	−	0.955986i	$-0.594791\pi$
$883$	−17.4377	−0.586825	−0.293413	+	0.955986i	$0.594791\pi$
$884$	−1.47214	−0.0495133
$885$	0	0
$886$	−14.9443	−0.502063
$887$	−41.9787	−1.40951	−0.704754	−	0.709452i	$-0.748942\pi$
$887$	−41.9787	−1.40951	−0.704754	+	0.709452i	$0.748942\pi$
$888$	−21.7082	−0.728480
$889$	12.3607	0.414564
$890$	0	0
$891$	−27.8885	−0.934301
$892$	6.61803	0.221588
$893$	−4.85410	−0.162436
$894$	8.00000	0.267560
$895$	0	0
$896$	3.70820	0.123882
$897$	−5.23607	−0.174827
$898$	13.2361	0.441693
$899$	10.8328	0.361295
$900$	0	0
$901$	15.4164	0.513595
$902$	14.7984	0.492732
$903$	−12.9443	−0.430758
$904$	63.3738	2.10778
$905$	0	0
$906$	17.7082	0.588316
$907$	−28.1803	−0.935713	−0.467856	−	0.883804i	$-0.654974\pi$
$907$	−28.1803	−0.935713	−0.467856	+	0.883804i	$0.654974\pi$
$908$	−27.5967	−0.915830
$909$	−1.05573	−0.0350163
$910$	0	0
$911$	32.6525	1.08182	0.540912	−	0.841079i	$-0.318079\pi$
$911$	32.6525	1.08182	0.540912	+	0.841079i	$0.318079\pi$
$912$	−1.00000	−0.0331133
$913$	48.4164	1.60235
$914$	−24.6525	−0.815431
$915$	0	0
$916$	−25.0344	−0.827161
$917$	−19.0132	−0.627870
$918$	21.0902	0.696079
$919$	−14.9443	−0.492966	−0.246483	−	0.969147i	$-0.579275\pi$
$919$	−14.9443	−0.492966	−0.246483	+	0.969147i	$0.579275\pi$
$920$	0	0
$921$	−48.0689	−1.58392
$922$	36.0000	1.18560
$923$	−5.65248	−0.186054
$924$	−7.23607	−0.238049
$925$	0	0
$926$	4.29180	0.141037
$927$	7.12461	0.234003
$928$	35.4508	1.16373
$929$	11.2361	0.368643	0.184322	−	0.982866i	$-0.440991\pi$
$929$	11.2361	0.368643	0.184322	+	0.982866i	$0.440991\pi$
$930$	0	0
$931$	−3.38197	−0.110839
$932$	14.9443	0.489516
$933$	−48.8885	−1.60054
$934$	8.67376	0.283814
$935$	0	0
$936$	−0.437694	−0.0143065
$937$	−24.3607	−0.795829	−0.397914	−	0.917423i	$-0.630266\pi$
$937$	−24.3607	−0.795829	−0.397914	+	0.917423i	$0.630266\pi$
$938$	15.8197	0.516530
$939$	−19.8885	−0.649038
$940$	0	0
$941$	−24.6525	−0.803648	−0.401824	−	0.915717i	$-0.631624\pi$
$941$	−24.6525	−0.803648	−0.401824	+	0.915717i	$0.631624\pi$
$942$	−16.7082	−0.544383
$943$	34.6525	1.12844
$944$	2.76393	0.0899583
$945$	0	0
$946$	23.4164	0.761333
$947$	3.12461	0.101536	0.0507681	−	0.998710i	$-0.483833\pi$
$947$	3.12461	0.101536	0.0507681	+	0.998710i	$0.483833\pi$
$948$	−23.7082	−0.770007
$949$	0.416408	0.0135172
$950$	0	0
$951$	24.3262	0.788832
$952$	14.2918	0.463200
$953$	−27.6180	−0.894636	−0.447318	−	0.894375i	$-0.647621\pi$
$953$	−27.6180	−0.894636	−0.447318	+	0.894375i	$0.647621\pi$
$954$	1.52786	0.0494664
$955$	0	0
$956$	10.4721	0.338693
$957$	−41.5066	−1.34172
$958$	38.9443	1.25823
$959$	2.36068	0.0762303
$960$	0	0
$961$	−28.6656	−0.924698
$962$	1.70820	0.0550747
$963$	3.09017	0.0995793
$964$	−1.00000	−0.0322078
$965$	0	0
$966$	16.9443	0.545173
$967$	24.9443	0.802154	0.401077	−	0.916044i	$-0.368636\pi$
$967$	24.9443	0.802154	0.401077	+	0.916044i	$0.368636\pi$
$968$	6.27051	0.201542
$969$	3.85410	0.123812
$970$	0	0
$971$	−36.7214	−1.17844	−0.589222	−	0.807971i	$-0.700566\pi$
$971$	−36.7214	−1.17844	−0.589222	+	0.807971i	$0.700566\pi$
$972$	−3.94427	−0.126513
$973$	2.36068	0.0756799
$974$	−7.32624	−0.234748
$975$	0	0
$976$	−4.85410	−0.155376
$977$	−22.9443	−0.734052	−0.367026	−	0.930211i	$-0.619624\pi$
$977$	−22.9443	−0.734052	−0.367026	+	0.930211i	$0.619624\pi$
$978$	−35.4164	−1.13249
$979$	59.5967	1.90472
$980$	0	0
$981$	5.52786	0.176491
$982$	3.41641	0.109022
$983$	−49.5967	−1.58189	−0.790945	−	0.611887i	$-0.790411\pi$
$983$	−49.5967	−1.58189	−0.790945	+	0.611887i	$0.790411\pi$
$984$	−19.8541	−0.632926
$985$	0	0
$986$	27.3262	0.870245
$987$	−15.7082	−0.499998
$988$	−0.236068	−0.00751032
$989$	54.8328	1.74358
$990$	0	0
$991$	24.5836	0.780924	0.390462	−	0.920619i	$-0.372315\pi$
$991$	24.5836	0.780924	0.390462	+	0.920619i	$0.372315\pi$
$992$	7.63932	0.242549
$993$	−22.9443	−0.728114
$994$	18.2918	0.580181
$995$	0	0
$996$	−21.6525	−0.686085
$997$	−21.1459	−0.669697	−0.334849	−	0.942272i	$-0.608685\pi$
$997$	−21.1459	−0.669697	−0.334849	+	0.942272i	$0.608685\pi$
$998$	−41.3050	−1.30749
$999$	24.4721	0.774264

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.a.1.2		2
5.2	odd	4		1205.2.b.b.724.1	✓	4
5.3	odd	4		1205.2.b.b.724.4	yes	4
5.4	even	2		6025.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.b.724.1	✓	4	5.2	odd	4
1205.2.b.b.724.4	yes	4	5.3	odd	4
6025.2.a.a.1.2		2	1.1	even	1	trivial
6025.2.a.d.1.1		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:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.61803 q^{3} -1.00000 q^{4} -1.61803 q^{6} +1.23607 q^{7} +3.00000 q^{8} -0.381966 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.61803 q^{3} -1.00000 q^{4} -1.61803 q^{6} +1.23607 q^{7} +3.00000 q^{8} -0.381966 q^{9} +3.61803 q^{11} -1.61803 q^{12} +0.381966 q^{13} -1.23607 q^{14} -1.00000 q^{16} +3.85410 q^{17} +0.381966 q^{18} +0.618034 q^{19} +2.00000 q^{21} -3.61803 q^{22} -8.47214 q^{23} +4.85410 q^{24} -0.381966 q^{26} -5.47214 q^{27} -1.23607 q^{28} -7.09017 q^{29} -1.52786 q^{31} -5.00000 q^{32} +5.85410 q^{33} -3.85410 q^{34} +0.381966 q^{36} -4.47214 q^{37} -0.618034 q^{38} +0.618034 q^{39} -4.09017 q^{41} -2.00000 q^{42} -6.47214 q^{43} -3.61803 q^{44} +8.47214 q^{46} -7.85410 q^{47} -1.61803 q^{48} -5.47214 q^{49} +6.23607 q^{51} -0.381966 q^{52} +4.00000 q^{53} +5.47214 q^{54} +3.70820 q^{56} +1.00000 q^{57} +7.09017 q^{58} -2.76393 q^{59} +4.85410 q^{61} +1.52786 q^{62} -0.472136 q^{63} +7.00000 q^{64} -5.85410 q^{66} -12.7984 q^{67} -3.85410 q^{68} -13.7082 q^{69} -14.7984 q^{71} -1.14590 q^{72} +1.09017 q^{73} +4.47214 q^{74} -0.618034 q^{76} +4.47214 q^{77} -0.618034 q^{78} +14.6525 q^{79} -7.70820 q^{81} +4.09017 q^{82} +13.3820 q^{83} -2.00000 q^{84} +6.47214 q^{86} -11.4721 q^{87} +10.8541 q^{88} +16.4721 q^{89} +0.472136 q^{91} +8.47214 q^{92} -2.47214 q^{93} +7.85410 q^{94} -8.09017 q^{96} +12.7639 q^{97} +5.47214 q^{98} -1.38197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + q^{3} - 2 q^{4} - q^{6} - 2 q^{7} + 6 q^{8} - 3 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + q^3 - 2 * q^4 - q^6 - 2 * q^7 + 6 * q^8 - 3 * q^9	\( 2 q - 2 q^{2} + q^{3} - 2 q^{4} - q^{6} - 2 q^{7} + 6 q^{8} - 3 q^{9} + 5 q^{11} - q^{12} + 3 q^{13} + 2 q^{14} - 2 q^{16} + q^{17} + 3 q^{18} - q^{19} + 4 q^{21} - 5 q^{22} - 8 q^{23} + 3 q^{24} - 3 q^{26} - 2 q^{27} + 2 q^{28} - 3 q^{29} - 12 q^{31} - 10 q^{32} + 5 q^{33} - q^{34} + 3 q^{36} + q^{38} - q^{39} + 3 q^{41} - 4 q^{42} - 4 q^{43} - 5 q^{44} + 8 q^{46} - 9 q^{47} - q^{48} - 2 q^{49} + 8 q^{51} - 3 q^{52} + 8 q^{53} + 2 q^{54} - 6 q^{56} + 2 q^{57} + 3 q^{58} - 10 q^{59} + 3 q^{61} + 12 q^{62} + 8 q^{63} + 14 q^{64} - 5 q^{66} - q^{67} - q^{68} - 14 q^{69} - 5 q^{71} - 9 q^{72} - 9 q^{73} + q^{76} + q^{78} - 2 q^{79} - 2 q^{81} - 3 q^{82} + 29 q^{83} - 4 q^{84} + 4 q^{86} - 14 q^{87} + 15 q^{88} + 24 q^{89} - 8 q^{91} + 8 q^{92} + 4 q^{93} + 9 q^{94} - 5 q^{96} + 30 q^{97} + 2 q^{98} - 5 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + q^3 - 2 * q^4 - q^6 - 2 * q^7 + 6 * q^8 - 3 * q^9 + 5 * q^11 - q^12 + 3 * q^13 + 2 * q^14 - 2 * q^16 + q^17 + 3 * q^18 - q^19 + 4 * q^21 - 5 * q^22 - 8 * q^23 + 3 * q^24 - 3 * q^26 - 2 * q^27 + 2 * q^28 - 3 * q^29 - 12 * q^31 - 10 * q^32 + 5 * q^33 - q^34 + 3 * q^36 + q^38 - q^39 + 3 * q^41 - 4 * q^42 - 4 * q^43 - 5 * q^44 + 8 * q^46 - 9 * q^47 - q^48 - 2 * q^49 + 8 * q^51 - 3 * q^52 + 8 * q^53 + 2 * q^54 - 6 * q^56 + 2 * q^57 + 3 * q^58 - 10 * q^59 + 3 * q^61 + 12 * q^62 + 8 * q^63 + 14 * q^64 - 5 * q^66 - q^67 - q^68 - 14 * q^69 - 5 * q^71 - 9 * q^72 - 9 * q^73 + q^76 + q^78 - 2 * q^79 - 2 * q^81 - 3 * q^82 + 29 * q^83 - 4 * q^84 + 4 * q^86 - 14 * q^87 + 15 * q^88 + 24 * q^89 - 8 * q^91 + 8 * q^92 + 4 * q^93 + 9 * q^94 - 5 * q^96 + 30 * q^97 + 2 * q^98 - 5 * q^99

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