LMFDB - Embedded newform 6025.2.a.a.1.1

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			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -0.618034 q^{3} -1.00000 q^{4} +0.618034 q^{6} -3.23607 q^{7} +3.00000 q^{8} -2.61803 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.618034 q^{3} -1.00000 q^{4} +0.618034 q^{6} -3.23607 q^{7} +3.00000 q^{8} -2.61803 q^{9} +1.38197 q^{11} +0.618034 q^{12} +2.61803 q^{13} +3.23607 q^{14} -1.00000 q^{16} -2.85410 q^{17} +2.61803 q^{18} -1.61803 q^{19} +2.00000 q^{21} -1.38197 q^{22} +0.472136 q^{23} -1.85410 q^{24} -2.61803 q^{26} +3.47214 q^{27} +3.23607 q^{28} +4.09017 q^{29} -10.4721 q^{31} -5.00000 q^{32} -0.854102 q^{33} +2.85410 q^{34} +2.61803 q^{36} +4.47214 q^{37} +1.61803 q^{38} -1.61803 q^{39} +7.09017 q^{41} -2.00000 q^{42} +2.47214 q^{43} -1.38197 q^{44} -0.472136 q^{46} -1.14590 q^{47} +0.618034 q^{48} +3.47214 q^{49} +1.76393 q^{51} -2.61803 q^{52} +4.00000 q^{53} -3.47214 q^{54} -9.70820 q^{56} +1.00000 q^{57} -4.09017 q^{58} -7.23607 q^{59} -1.85410 q^{61} +10.4721 q^{62} +8.47214 q^{63} +7.00000 q^{64} +0.854102 q^{66} +11.7984 q^{67} +2.85410 q^{68} -0.291796 q^{69} +9.79837 q^{71} -7.85410 q^{72} -10.0902 q^{73} -4.47214 q^{74} +1.61803 q^{76} -4.47214 q^{77} +1.61803 q^{78} -16.6525 q^{79} +5.70820 q^{81} -7.09017 q^{82} +15.6180 q^{83} -2.00000 q^{84} -2.47214 q^{86} -2.52786 q^{87} +4.14590 q^{88} +7.52786 q^{89} -8.47214 q^{91} -0.472136 q^{92} +6.47214 q^{93} +1.14590 q^{94} +3.09017 q^{96} +17.2361 q^{97} -3.47214 q^{98} -3.61803 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.615027\pi$
$3$	−0.618034	−0.356822	−0.178411	−	0.983956i	$-0.557096\pi$
$3$	−0.618034	−0.356822	−0.178411	+	0.983956i	$0.557096\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0.618034	0.252311
$7$	−3.23607	−1.22312	−0.611559	−	0.791199i	$-0.709457\pi$
$7$	−3.23607	−1.22312	−0.611559	+	0.791199i	$0.709457\pi$
$8$	3.00000	1.06066
$9$	−2.61803	−0.872678
$10$	0	0
$11$	1.38197	0.416678	0.208339	−	0.978057i	$-0.433194\pi$
$11$	1.38197	0.416678	0.208339	+	0.978057i	$0.433194\pi$
$12$	0.618034	0.178411
$13$	2.61803	0.726112	0.363056	−	0.931767i	$-0.381733\pi$
$13$	2.61803	0.726112	0.363056	+	0.931767i	$0.381733\pi$
$14$	3.23607	0.864876
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−2.85410	−0.692221	−0.346111	−	0.938194i	$-0.612498\pi$
$17$	−2.85410	−0.692221	−0.346111	+	0.938194i	$0.612498\pi$
$18$	2.61803	0.617077
$19$	−1.61803	−0.371202	−0.185601	−	0.982625i	$-0.559423\pi$
$19$	−1.61803	−0.371202	−0.185601	+	0.982625i	$0.559423\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	−1.38197	−0.294636
$23$	0.472136	0.0984472	0.0492236	−	0.998788i	$-0.484325\pi$
$23$	0.472136	0.0984472	0.0492236	+	0.998788i	$0.484325\pi$
$24$	−1.85410	−0.378467
$25$	0	0
$26$	−2.61803	−0.513439
$27$	3.47214	0.668213
$28$	3.23607	0.611559
$29$	4.09017	0.759525	0.379763	−	0.925084i	$-0.376006\pi$
$29$	4.09017	0.759525	0.379763	+	0.925084i	$0.376006\pi$
$30$	0	0
$31$	−10.4721	−1.88085	−0.940426	−	0.340000i	$-0.889573\pi$
$31$	−10.4721	−1.88085	−0.940426	+	0.340000i	$0.889573\pi$
$32$	−5.00000	−0.883883
$33$	−0.854102	−0.148680
$34$	2.85410	0.489474
$35$	0	0
$36$	2.61803	0.436339
$37$	4.47214	0.735215	0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214	0.735215	0.367607	+	0.929981i	$0.380177\pi$
$38$	1.61803	0.262480
$39$	−1.61803	−0.259093
$40$	0	0
$41$	7.09017	1.10730	0.553649	−	0.832750i	$-0.313235\pi$
$41$	7.09017	1.10730	0.553649	+	0.832750i	$0.313235\pi$
$42$	−2.00000	−0.308607
$43$	2.47214	0.376997	0.188499	−	0.982073i	$-0.439638\pi$
$43$	2.47214	0.376997	0.188499	+	0.982073i	$0.439638\pi$
$44$	−1.38197	−0.208339
$45$	0	0
$46$	−0.472136	−0.0696126
$47$	−1.14590	−0.167146	−0.0835732	−	0.996502i	$-0.526633\pi$
$47$	−1.14590	−0.167146	−0.0835732	+	0.996502i	$0.526633\pi$
$48$	0.618034	0.0892055
$49$	3.47214	0.496019
$50$	0	0
$51$	1.76393	0.247000
$52$	−2.61803	−0.363056
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	−3.47214	−0.472498
$55$	0	0
$56$	−9.70820	−1.29731
$57$	1.00000	0.132453
$58$	−4.09017	−0.537066
$59$	−7.23607	−0.942056	−0.471028	−	0.882118i	$-0.656117\pi$
$59$	−7.23607	−0.942056	−0.471028	+	0.882118i	$0.656117\pi$
$60$	0	0
$61$	−1.85410	−0.237393	−0.118697	−	0.992931i	$-0.537872\pi$
$61$	−1.85410	−0.237393	−0.118697	+	0.992931i	$0.537872\pi$
$62$	10.4721	1.32996
$63$	8.47214	1.06739
$64$	7.00000	0.875000
$65$	0	0
$66$	0.854102	0.105133
$67$	11.7984	1.44140	0.720700	−	0.693247i	$-0.243820\pi$
$67$	11.7984	1.44140	0.720700	+	0.693247i	$0.243820\pi$
$68$	2.85410	0.346111
$69$	−0.291796	−0.0351281
$70$	0	0
$71$	9.79837	1.16285	0.581427	−	0.813599i	$-0.302495\pi$
$71$	9.79837	1.16285	0.581427	+	0.813599i	$0.302495\pi$
$72$	−7.85410	−0.925615
$73$	−10.0902	−1.18097	−0.590483	−	0.807050i	$-0.701063\pi$
$73$	−10.0902	−1.18097	−0.590483	+	0.807050i	$0.701063\pi$
$74$	−4.47214	−0.519875
$75$	0	0
$76$	1.61803	0.185601
$77$	−4.47214	−0.509647
$78$	1.61803	0.183206
$79$	−16.6525	−1.87355	−0.936775	−	0.349932i	$-0.886205\pi$
$79$	−16.6525	−1.87355	−0.936775	+	0.349932i	$0.886205\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	−7.09017	−0.782978
$83$	15.6180	1.71430	0.857151	−	0.515065i	$-0.172232\pi$
$83$	15.6180	1.71430	0.857151	+	0.515065i	$0.172232\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	−2.47214	−0.266577
$87$	−2.52786	−0.271015
$88$	4.14590	0.441954
$89$	7.52786	0.797952	0.398976	−	0.916961i	$-0.369366\pi$
$89$	7.52786	0.797952	0.398976	+	0.916961i	$0.369366\pi$
$90$	0	0
$91$	−8.47214	−0.888121
$92$	−0.472136	−0.0492236
$93$	6.47214	0.671129
$94$	1.14590	0.118190
$95$	0	0
$96$	3.09017	0.315389
$97$	17.2361	1.75006	0.875029	−	0.484071i	$-0.160842\pi$
$97$	17.2361	1.75006	0.875029	+	0.484071i	$0.160842\pi$
$98$	−3.47214	−0.350739
$99$	−3.61803	−0.363626
$100$	0	0
$101$	7.23607	0.720016	0.360008	−	0.932949i	$-0.382774\pi$
$101$	7.23607	0.720016	0.360008	+	0.932949i	$0.382774\pi$
$102$	−1.76393	−0.174655
$103$	12.6525	1.24669	0.623343	−	0.781949i	$-0.285774\pi$
$103$	12.6525	1.24669	0.623343	+	0.781949i	$0.285774\pi$
$104$	7.85410	0.770158
$105$	0	0
$106$	−4.00000	−0.388514
$107$	3.09017	0.298738	0.149369	−	0.988782i	$-0.452276\pi$
$107$	3.09017	0.298738	0.149369	+	0.988782i	$0.452276\pi$
$108$	−3.47214	−0.334106
$109$	−5.52786	−0.529473	−0.264737	−	0.964321i	$-0.585285\pi$
$109$	−5.52786	−0.529473	−0.264737	+	0.964321i	$0.585285\pi$
$110$	0	0
$111$	−2.76393	−0.262341
$112$	3.23607	0.305780
$113$	−19.1246	−1.79909	−0.899546	−	0.436826i	$-0.856103\pi$
$113$	−19.1246	−1.79909	−0.899546	+	0.436826i	$0.856103\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	−4.09017	−0.379763
$117$	−6.85410	−0.633662
$118$	7.23607	0.666134
$119$	9.23607	0.846669
$120$	0	0
$121$	−9.09017	−0.826379
$122$	1.85410	0.167863
$123$	−4.38197	−0.395109
$124$	10.4721	0.940426
$125$	0	0
$126$	−8.47214	−0.754758
$127$	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
$129$	−1.52786	−0.134521
$130$	0	0
$131$	−17.6180	−1.53929	−0.769647	−	0.638469i	$-0.779568\pi$
$131$	−17.6180	−1.53929	−0.769647	+	0.638469i	$0.779568\pi$
$132$	0.854102	0.0743400
$133$	5.23607	0.454025
$134$	−11.7984	−1.01922
$135$	0	0
$136$	−8.56231	−0.734212
$137$	13.0902	1.11837	0.559184	−	0.829043i	$-0.311114\pi$
$137$	13.0902	1.11837	0.559184	+	0.829043i	$0.311114\pi$
$138$	0.291796	0.0248393
$139$	13.0902	1.11029	0.555147	−	0.831752i	$-0.312662\pi$
$139$	13.0902	1.11029	0.555147	+	0.831752i	$0.312662\pi$
$140$	0	0
$141$	0.708204	0.0596415
$142$	−9.79837	−0.822261
$143$	3.61803	0.302555
$144$	2.61803	0.218169
$145$	0	0
$146$	10.0902	0.835068
$147$	−2.14590	−0.176991
$148$	−4.47214	−0.367607
$149$	12.9443	1.06044	0.530218	−	0.847861i	$-0.322110\pi$
$149$	12.9443	1.06044	0.530218	+	0.847861i	$0.322110\pi$
$150$	0	0
$151$	6.94427	0.565117	0.282558	−	0.959250i	$-0.408817\pi$
$151$	6.94427	0.565117	0.282558	+	0.959250i	$0.408817\pi$
$152$	−4.85410	−0.393720
$153$	7.47214	0.604086
$154$	4.47214	0.360375
$155$	0	0
$156$	1.61803	0.129546
$157$	−5.32624	−0.425080	−0.212540	−	0.977152i	$-0.568174\pi$
$157$	−5.32624	−0.425080	−0.212540	+	0.977152i	$0.568174\pi$
$158$	16.6525	1.32480
$159$	−2.47214	−0.196053
$160$	0	0
$161$	−1.52786	−0.120413
$162$	−5.70820	−0.448479
$163$	−13.8885	−1.08783	−0.543917	−	0.839139i	$-0.683060\pi$
$163$	−13.8885	−1.08783	−0.543917	+	0.839139i	$0.683060\pi$
$164$	−7.09017	−0.553649
$165$	0	0
$166$	−15.6180	−1.21219
$167$	−4.29180	−0.332109	−0.166055	−	0.986117i	$-0.553103\pi$
$167$	−4.29180	−0.332109	−0.166055	+	0.986117i	$0.553103\pi$
$168$	6.00000	0.462910
$169$	−6.14590	−0.472761
$170$	0	0
$171$	4.23607	0.323940
$172$	−2.47214	−0.188499
$173$	6.32624	0.480975	0.240487	−	0.970652i	$-0.422693\pi$
$173$	6.32624	0.480975	0.240487	+	0.970652i	$0.422693\pi$
$174$	2.52786	0.191637
$175$	0	0
$176$	−1.38197	−0.104170
$177$	4.47214	0.336146
$178$	−7.52786	−0.564237
$179$	4.94427	0.369552	0.184776	−	0.982781i	$-0.440844\pi$
$179$	4.94427	0.369552	0.184776	+	0.982781i	$0.440844\pi$
$180$	0	0
$181$	−3.38197	−0.251380	−0.125690	−	0.992070i	$-0.540114\pi$
$181$	−3.38197	−0.251380	−0.125690	+	0.992070i	$0.540114\pi$
$182$	8.47214	0.627996
$183$	1.14590	0.0847072
$184$	1.41641	0.104419
$185$	0	0
$186$	−6.47214	−0.474560
$187$	−3.94427	−0.288434
$188$	1.14590	0.0835732
$189$	−11.2361	−0.817304
$190$	0	0
$191$	−20.6525	−1.49436	−0.747180	−	0.664621i	$-0.768593\pi$
$191$	−20.6525	−1.49436	−0.747180	+	0.664621i	$0.768593\pi$
$192$	−4.32624	−0.312219
$193$	0.472136	0.0339851	0.0169925	−	0.999856i	$-0.494591\pi$
$193$	0.472136	0.0339851	0.0169925	+	0.999856i	$0.494591\pi$
$194$	−17.2361	−1.23748
$195$	0	0
$196$	−3.47214	−0.248010
$197$	14.9443	1.06474	0.532368	−	0.846513i	$-0.321302\pi$
$197$	14.9443	1.06474	0.532368	+	0.846513i	$0.321302\pi$
$198$	3.61803	0.257122
$199$	−23.8541	−1.69097	−0.845486	−	0.533997i	$-0.820689\pi$
$199$	−23.8541	−1.69097	−0.845486	+	0.533997i	$0.820689\pi$
$200$	0	0
$201$	−7.29180	−0.514324
$202$	−7.23607	−0.509128
$203$	−13.2361	−0.928990
$204$	−1.76393	−0.123500
$205$	0	0
$206$	−12.6525	−0.881540
$207$	−1.23607	−0.0859127
$208$	−2.61803	−0.181528
$209$	−2.23607	−0.154672
$210$	0	0
$211$	−15.2361	−1.04889	−0.524447	−	0.851443i	$-0.675728\pi$
$211$	−15.2361	−1.04889	−0.524447	+	0.851443i	$0.675728\pi$
$212$	−4.00000	−0.274721
$213$	−6.05573	−0.414932
$214$	−3.09017	−0.211240
$215$	0	0
$216$	10.4164	0.708747
$217$	33.8885	2.30050
$218$	5.52786	0.374394
$219$	6.23607	0.421394
$220$	0	0
$221$	−7.47214	−0.502630
$222$	2.76393	0.185503
$223$	−4.38197	−0.293438	−0.146719	−	0.989178i	$-0.546871\pi$
$223$	−4.38197	−0.293438	−0.146719	+	0.989178i	$0.546871\pi$
$224$	16.1803	1.08109
$225$	0	0
$226$	19.1246	1.27215
$227$	−21.5967	−1.43343	−0.716713	−	0.697368i	$-0.754354\pi$
$227$	−21.5967	−1.43343	−0.716713	+	0.697368i	$0.754354\pi$
$228$	−1.00000	−0.0662266
$229$	−4.03444	−0.266603	−0.133302	−	0.991076i	$-0.542558\pi$
$229$	−4.03444	−0.266603	−0.133302	+	0.991076i	$0.542558\pi$
$230$	0	0
$231$	2.76393	0.181853
$232$	12.2705	0.805598
$233$	2.94427	0.192886	0.0964428	−	0.995339i	$-0.469254\pi$
$233$	2.94427	0.192886	0.0964428	+	0.995339i	$0.469254\pi$
$234$	6.85410	0.448067
$235$	0	0
$236$	7.23607	0.471028
$237$	10.2918	0.668524
$238$	−9.23607	−0.598685
$239$	−1.52786	−0.0988293	−0.0494147	−	0.998778i	$-0.515736\pi$
$239$	−1.52786	−0.0988293	−0.0494147	+	0.998778i	$0.515736\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	9.09017	0.584338
$243$	−13.9443	−0.894525
$244$	1.85410	0.118697
$245$	0	0
$246$	4.38197	0.279384
$247$	−4.23607	−0.269535
$248$	−31.4164	−1.99494
$249$	−9.65248	−0.611701
$250$	0	0
$251$	−5.70820	−0.360299	−0.180149	−	0.983639i	$-0.557658\pi$
$251$	−5.70820	−0.360299	−0.180149	+	0.983639i	$0.557658\pi$
$252$	−8.47214	−0.533694
$253$	0.652476	0.0410208
$254$	−10.0000	−0.627456
$255$	0	0
$256$	−17.0000	−1.06250
$257$	31.1246	1.94150	0.970750	−	0.240093i	$-0.0771778\pi$
$257$	31.1246	1.94150	0.970750	+	0.240093i	$0.0771778\pi$
$258$	1.52786	0.0951207
$259$	−14.4721	−0.899255
$260$	0	0
$261$	−10.7082	−0.662821
$262$	17.6180	1.08845
$263$	5.52786	0.340863	0.170431	−	0.985370i	$-0.445484\pi$
$263$	5.52786	0.340863	0.170431	+	0.985370i	$0.445484\pi$
$264$	−2.56231	−0.157699
$265$	0	0
$266$	−5.23607	−0.321044
$267$	−4.65248	−0.284727
$268$	−11.7984	−0.720700
$269$	−11.7082	−0.713862	−0.356931	−	0.934131i	$-0.616177\pi$
$269$	−11.7082	−0.713862	−0.356931	+	0.934131i	$0.616177\pi$
$270$	0	0
$271$	−28.1803	−1.71183	−0.855917	−	0.517113i	$-0.827007\pi$
$271$	−28.1803	−1.71183	−0.855917	+	0.517113i	$0.827007\pi$
$272$	2.85410	0.173055
$273$	5.23607	0.316901
$274$	−13.0902	−0.790806
$275$	0	0
$276$	0.291796	0.0175641
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−13.0902	−0.785096
$279$	27.4164	1.64138
$280$	0	0
$281$	20.0344	1.19515	0.597577	−	0.801811i	$-0.296130\pi$
$281$	20.0344	1.19515	0.597577	+	0.801811i	$0.296130\pi$
$282$	−0.708204	−0.0421729
$283$	−11.2361	−0.667915	−0.333957	−	0.942588i	$-0.608384\pi$
$283$	−11.2361	−0.667915	−0.333957	+	0.942588i	$0.608384\pi$
$284$	−9.79837	−0.581427
$285$	0	0
$286$	−3.61803	−0.213939
$287$	−22.9443	−1.35436
$288$	13.0902	0.771346
$289$	−8.85410	−0.520830
$290$	0	0
$291$	−10.6525	−0.624459
$292$	10.0902	0.590483
$293$	−21.3262	−1.24589	−0.622946	−	0.782265i	$-0.714064\pi$
$293$	−21.3262	−1.24589	−0.622946	+	0.782265i	$0.714064\pi$
$294$	2.14590	0.125151
$295$	0	0
$296$	13.4164	0.779813
$297$	4.79837	0.278430
$298$	−12.9443	−0.749842
$299$	1.23607	0.0714837
$300$	0	0
$301$	−8.00000	−0.461112
$302$	−6.94427	−0.399598
$303$	−4.47214	−0.256917
$304$	1.61803	0.0928006
$305$	0	0
$306$	−7.47214	−0.427154
$307$	−16.2918	−0.929822	−0.464911	−	0.885357i	$-0.653914\pi$
$307$	−16.2918	−0.929822	−0.464911	+	0.885357i	$0.653914\pi$
$308$	4.47214	0.254824
$309$	−7.81966	−0.444845
$310$	0	0
$311$	21.2148	1.20298	0.601490	−	0.798880i	$-0.294574\pi$
$311$	21.2148	1.20298	0.601490	+	0.798880i	$0.294574\pi$
$312$	−4.85410	−0.274809
$313$	−25.7082	−1.45311	−0.726557	−	0.687106i	$-0.758881\pi$
$313$	−25.7082	−1.45311	−0.726557	+	0.687106i	$0.758881\pi$
$314$	5.32624	0.300577
$315$	0	0
$316$	16.6525	0.936775
$317$	−14.0344	−0.788253	−0.394126	−	0.919056i	$-0.628953\pi$
$317$	−14.0344	−0.788253	−0.394126	+	0.919056i	$0.628953\pi$
$318$	2.47214	0.138631
$319$	5.65248	0.316478
$320$	0	0
$321$	−1.90983	−0.106596
$322$	1.52786	0.0851445
$323$	4.61803	0.256954
$324$	−5.70820	−0.317122
$325$	0	0
$326$	13.8885	0.769215
$327$	3.41641	0.188928
$328$	21.2705	1.17447
$329$	3.70820	0.204440
$330$	0	0
$331$	8.18034	0.449632	0.224816	−	0.974401i	$-0.427822\pi$
$331$	8.18034	0.449632	0.224816	+	0.974401i	$0.427822\pi$
$332$	−15.6180	−0.857151
$333$	−11.7082	−0.641606
$334$	4.29180	0.234837
$335$	0	0
$336$	−2.00000	−0.109109
$337$	−4.65248	−0.253437	−0.126718	−	0.991939i	$-0.540444\pi$
$337$	−4.65248	−0.253437	−0.126718	+	0.991939i	$0.540444\pi$
$338$	6.14590	0.334293
$339$	11.8197	0.641956
$340$	0	0
$341$	−14.4721	−0.783710
$342$	−4.23607	−0.229060
$343$	11.4164	0.616428
$344$	7.41641	0.399866
$345$	0	0
$346$	−6.32624	−0.340101
$347$	−11.7984	−0.633370	−0.316685	−	0.948531i	$-0.602570\pi$
$347$	−11.7984	−0.633370	−0.316685	+	0.948531i	$0.602570\pi$
$348$	2.52786	0.135508
$349$	23.5066	1.25828	0.629139	−	0.777293i	$-0.283408\pi$
$349$	23.5066	1.25828	0.629139	+	0.777293i	$0.283408\pi$
$350$	0	0
$351$	9.09017	0.485197
$352$	−6.90983	−0.368295
$353$	30.3262	1.61410	0.807051	−	0.590481i	$-0.201062\pi$
$353$	30.3262	1.61410	0.807051	+	0.590481i	$0.201062\pi$
$354$	−4.47214	−0.237691
$355$	0	0
$356$	−7.52786	−0.398976
$357$	−5.70820	−0.302110
$358$	−4.94427	−0.261313
$359$	4.29180	0.226512	0.113256	−	0.993566i	$-0.463872\pi$
$359$	4.29180	0.226512	0.113256	+	0.993566i	$0.463872\pi$
$360$	0	0
$361$	−16.3820	−0.862209
$362$	3.38197	0.177752
$363$	5.61803	0.294870
$364$	8.47214	0.444061
$365$	0	0
$366$	−1.14590	−0.0598970
$367$	16.1803	0.844607	0.422303	−	0.906455i	$-0.361222\pi$
$367$	16.1803	0.844607	0.422303	+	0.906455i	$0.361222\pi$
$368$	−0.472136	−0.0246118
$369$	−18.5623	−0.966315
$370$	0	0
$371$	−12.9443	−0.672033
$372$	−6.47214	−0.335565
$373$	−14.8541	−0.769116	−0.384558	−	0.923101i	$-0.625646\pi$
$373$	−14.8541	−0.769116	−0.384558	+	0.923101i	$0.625646\pi$
$374$	3.94427	0.203953
$375$	0	0
$376$	−3.43769	−0.177286
$377$	10.7082	0.551501
$378$	11.2361	0.577921
$379$	5.74265	0.294980	0.147490	−	0.989064i	$-0.452881\pi$
$379$	5.74265	0.294980	0.147490	+	0.989064i	$0.452881\pi$
$380$	0	0
$381$	−6.18034	−0.316628
$382$	20.6525	1.05667
$383$	24.4721	1.25047	0.625234	−	0.780437i	$-0.285004\pi$
$383$	24.4721	1.25047	0.625234	+	0.780437i	$0.285004\pi$
$384$	−1.85410	−0.0946167
$385$	0	0
$386$	−0.472136	−0.0240311
$387$	−6.47214	−0.328997
$388$	−17.2361	−0.875029
$389$	−10.2918	−0.521815	−0.260907	−	0.965364i	$-0.584022\pi$
$389$	−10.2918	−0.521815	−0.260907	+	0.965364i	$0.584022\pi$
$390$	0	0
$391$	−1.34752	−0.0681472
$392$	10.4164	0.526108
$393$	10.8885	0.549254
$394$	−14.9443	−0.752882
$395$	0	0
$396$	3.61803	0.181813
$397$	−31.3050	−1.57115	−0.785575	−	0.618766i	$-0.787633\pi$
$397$	−31.3050	−1.57115	−0.785575	+	0.618766i	$0.787633\pi$
$398$	23.8541	1.19570
$399$	−3.23607	−0.162006
$400$	0	0
$401$	−17.7984	−0.888808	−0.444404	−	0.895826i	$-0.646585\pi$
$401$	−17.7984	−0.888808	−0.444404	+	0.895826i	$0.646585\pi$
$402$	7.29180	0.363682
$403$	−27.4164	−1.36571
$404$	−7.23607	−0.360008
$405$	0	0
$406$	13.2361	0.656895
$407$	6.18034	0.306348
$408$	5.29180	0.261983
$409$	32.9443	1.62899	0.814495	−	0.580171i	$-0.197014\pi$
$409$	32.9443	1.62899	0.814495	+	0.580171i	$0.197014\pi$
$410$	0	0
$411$	−8.09017	−0.399059
$412$	−12.6525	−0.623343
$413$	23.4164	1.15225
$414$	1.23607	0.0607494
$415$	0	0
$416$	−13.0902	−0.641798
$417$	−8.09017	−0.396177
$418$	2.23607	0.109370
$419$	3.20163	0.156410	0.0782048	−	0.996937i	$-0.475081\pi$
$419$	3.20163	0.156410	0.0782048	+	0.996937i	$0.475081\pi$
$420$	0	0
$421$	−10.3607	−0.504949	−0.252474	−	0.967604i	$-0.581244\pi$
$421$	−10.3607	−0.504949	−0.252474	+	0.967604i	$0.581244\pi$
$422$	15.2361	0.741680
$423$	3.00000	0.145865
$424$	12.0000	0.582772
$425$	0	0
$426$	6.05573	0.293401
$427$	6.00000	0.290360
$428$	−3.09017	−0.149369
$429$	−2.23607	−0.107958
$430$	0	0
$431$	−33.4508	−1.61127	−0.805635	−	0.592412i	$-0.798176\pi$
$431$	−33.4508	−1.61127	−0.805635	+	0.592412i	$0.798176\pi$
$432$	−3.47214	−0.167053
$433$	−12.6525	−0.608039	−0.304020	−	0.952666i	$-0.598329\pi$
$433$	−12.6525	−0.608039	−0.304020	+	0.952666i	$0.598329\pi$
$434$	−33.8885	−1.62670
$435$	0	0
$436$	5.52786	0.264737
$437$	−0.763932	−0.0365438
$438$	−6.23607	−0.297971
$439$	4.14590	0.197873	0.0989365	−	0.995094i	$-0.468456\pi$
$439$	4.14590	0.197873	0.0989365	+	0.995094i	$0.468456\pi$
$440$	0	0
$441$	−9.09017	−0.432865
$442$	7.47214	0.355413
$443$	−2.94427	−0.139887	−0.0699433	−	0.997551i	$-0.522282\pi$
$443$	−2.94427	−0.139887	−0.0699433	+	0.997551i	$0.522282\pi$
$444$	2.76393	0.131170
$445$	0	0
$446$	4.38197	0.207492
$447$	−8.00000	−0.378387
$448$	−22.6525	−1.07023
$449$	−8.76393	−0.413596	−0.206798	−	0.978384i	$-0.566304\pi$
$449$	−8.76393	−0.413596	−0.206798	+	0.978384i	$0.566304\pi$
$450$	0	0
$451$	9.79837	0.461387
$452$	19.1246	0.899546
$453$	−4.29180	−0.201646
$454$	21.5967	1.01359
$455$	0	0
$456$	3.00000	0.140488
$457$	−6.65248	−0.311190	−0.155595	−	0.987821i	$-0.549729\pi$
$457$	−6.65248	−0.311190	−0.155595	+	0.987821i	$0.549729\pi$
$458$	4.03444	0.188517
$459$	−9.90983	−0.462551
$460$	0	0
$461$	−36.0000	−1.67669	−0.838344	−	0.545142i	$-0.816476\pi$
$461$	−36.0000	−1.67669	−0.838344	+	0.545142i	$0.816476\pi$
$462$	−2.76393	−0.128590
$463$	−17.7082	−0.822970	−0.411485	−	0.911417i	$-0.634990\pi$
$463$	−17.7082	−0.822970	−0.411485	+	0.911417i	$0.634990\pi$
$464$	−4.09017	−0.189881
$465$	0	0
$466$	−2.94427	−0.136391
$467$	−24.3262	−1.12568	−0.562842	−	0.826564i	$-0.690292\pi$
$467$	−24.3262	−1.12568	−0.562842	+	0.826564i	$0.690292\pi$
$468$	6.85410	0.316831
$469$	−38.1803	−1.76300
$470$	0	0
$471$	3.29180	0.151678
$472$	−21.7082	−0.999201
$473$	3.41641	0.157087
$474$	−10.2918	−0.472718
$475$	0	0
$476$	−9.23607	−0.423334
$477$	−10.4721	−0.479486
$478$	1.52786	0.0698829
$479$	−21.0557	−0.962061	−0.481030	−	0.876704i	$-0.659737\pi$
$479$	−21.0557	−0.962061	−0.481030	+	0.876704i	$0.659737\pi$
$480$	0	0
$481$	11.7082	0.533848
$482$	−1.00000	−0.0455488
$483$	0.944272	0.0429659
$484$	9.09017	0.413190
$485$	0	0
$486$	13.9443	0.632525
$487$	−8.32624	−0.377298	−0.188649	−	0.982045i	$-0.560411\pi$
$487$	−8.32624	−0.377298	−0.188649	+	0.982045i	$0.560411\pi$
$488$	−5.56231	−0.251794
$489$	8.58359	0.388163
$490$	0	0
$491$	23.4164	1.05677	0.528384	−	0.849006i	$-0.322798\pi$
$491$	23.4164	1.05677	0.528384	+	0.849006i	$0.322798\pi$
$492$	4.38197	0.197554
$493$	−11.6738	−0.525760
$494$	4.23607	0.190590
$495$	0	0
$496$	10.4721	0.470213
$497$	−31.7082	−1.42231
$498$	9.65248	0.432538
$499$	−21.3050	−0.953741	−0.476870	−	0.878974i	$-0.658229\pi$
$499$	−21.3050	−0.953741	−0.476870	+	0.878974i	$0.658229\pi$
$500$	0	0
$501$	2.65248	0.118504
$502$	5.70820	0.254770
$503$	4.65248	0.207444	0.103722	−	0.994606i	$-0.466925\pi$
$503$	4.65248	0.207444	0.103722	+	0.994606i	$0.466925\pi$
$504$	25.4164	1.13214
$505$	0	0
$506$	−0.652476	−0.0290061
$507$	3.79837	0.168692
$508$	−10.0000	−0.443678
$509$	32.0344	1.41990	0.709951	−	0.704251i	$-0.248717\pi$
$509$	32.0344	1.41990	0.709951	+	0.704251i	$0.248717\pi$
$510$	0	0
$511$	32.6525	1.44446
$512$	11.0000	0.486136
$513$	−5.61803	−0.248042
$514$	−31.1246	−1.37285
$515$	0	0
$516$	1.52786	0.0672605
$517$	−1.58359	−0.0696463
$518$	14.4721	0.635869
$519$	−3.90983	−0.171622
$520$	0	0
$521$	−1.23607	−0.0541531	−0.0270766	−	0.999633i	$-0.508620\pi$
$521$	−1.23607	−0.0541531	−0.0270766	+	0.999633i	$0.508620\pi$
$522$	10.7082	0.468685
$523$	−13.5623	−0.593038	−0.296519	−	0.955027i	$-0.595826\pi$
$523$	−13.5623	−0.593038	−0.296519	+	0.955027i	$0.595826\pi$
$524$	17.6180	0.769647
$525$	0	0
$526$	−5.52786	−0.241026
$527$	29.8885	1.30197
$528$	0.854102	0.0371700
$529$	−22.7771	−0.990308
$530$	0	0
$531$	18.9443	0.822111
$532$	−5.23607	−0.227012
$533$	18.5623	0.804023
$534$	4.65248	0.201332
$535$	0	0
$536$	35.3951	1.52884
$537$	−3.05573	−0.131864
$538$	11.7082	0.504777
$539$	4.79837	0.206681
$540$	0	0
$541$	10.2016	0.438602	0.219301	−	0.975657i	$-0.429622\pi$
$541$	10.2016	0.438602	0.219301	+	0.975657i	$0.429622\pi$
$542$	28.1803	1.21045
$543$	2.09017	0.0896978
$544$	14.2705	0.611843
$545$	0	0
$546$	−5.23607	−0.224083
$547$	−3.70820	−0.158551	−0.0792757	−	0.996853i	$-0.525261\pi$
$547$	−3.70820	−0.158551	−0.0792757	+	0.996853i	$0.525261\pi$
$548$	−13.0902	−0.559184
$549$	4.85410	0.207168
$550$	0	0
$551$	−6.61803	−0.281938
$552$	−0.875388	−0.0372590
$553$	53.8885	2.29157
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−13.0902	−0.555147
$557$	−10.1115	−0.428436	−0.214218	−	0.976786i	$-0.568720\pi$
$557$	−10.1115	−0.428436	−0.214218	+	0.976786i	$0.568720\pi$
$558$	−27.4164	−1.16063
$559$	6.47214	0.273742
$560$	0	0
$561$	2.43769	0.102920
$562$	−20.0344	−0.845102
$563$	41.2705	1.73934	0.869672	−	0.493629i	$-0.164330\pi$
$563$	41.2705	1.73934	0.869672	+	0.493629i	$0.164330\pi$
$564$	−0.708204	−0.0298208
$565$	0	0
$566$	11.2361	0.472287
$567$	−18.4721	−0.775757
$568$	29.3951	1.23339
$569$	−14.5066	−0.608147	−0.304074	−	0.952649i	$-0.598347\pi$
$569$	−14.5066	−0.608147	−0.304074	+	0.952649i	$0.598347\pi$
$570$	0	0
$571$	−15.3262	−0.641383	−0.320692	−	0.947184i	$-0.603915\pi$
$571$	−15.3262	−0.641383	−0.320692	+	0.947184i	$0.603915\pi$
$572$	−3.61803	−0.151278
$573$	12.7639	0.533221
$574$	22.9443	0.957675
$575$	0	0
$576$	−18.3262	−0.763593
$577$	16.1459	0.672163	0.336081	−	0.941833i	$-0.390898\pi$
$577$	16.1459	0.672163	0.336081	+	0.941833i	$0.390898\pi$
$578$	8.85410	0.368282
$579$	−0.291796	−0.0121266
$580$	0	0
$581$	−50.5410	−2.09679
$582$	10.6525	0.441559
$583$	5.52786	0.228941
$584$	−30.2705	−1.25260
$585$	0	0
$586$	21.3262	0.880979
$587$	−11.5279	−0.475806	−0.237903	−	0.971289i	$-0.576460\pi$
$587$	−11.5279	−0.475806	−0.237903	+	0.971289i	$0.576460\pi$
$588$	2.14590	0.0884953
$589$	16.9443	0.698177
$590$	0	0
$591$	−9.23607	−0.379921
$592$	−4.47214	−0.183804
$593$	12.8328	0.526981	0.263490	−	0.964662i	$-0.415126\pi$
$593$	12.8328	0.526981	0.263490	+	0.964662i	$0.415126\pi$
$594$	−4.79837	−0.196880
$595$	0	0
$596$	−12.9443	−0.530218
$597$	14.7426	0.603376
$598$	−1.23607	−0.0505466
$599$	−36.4508	−1.48934	−0.744671	−	0.667432i	$-0.767393\pi$
$599$	−36.4508	−1.48934	−0.744671	+	0.667432i	$0.767393\pi$
$600$	0	0
$601$	−3.74265	−0.152666	−0.0763329	−	0.997082i	$-0.524321\pi$
$601$	−3.74265	−0.152666	−0.0763329	+	0.997082i	$0.524321\pi$
$602$	8.00000	0.326056
$603$	−30.8885	−1.25788
$604$	−6.94427	−0.282558
$605$	0	0
$606$	4.47214	0.181668
$607$	−0.360680	−0.0146395	−0.00731977	−	0.999973i	$-0.502330\pi$
$607$	−0.360680	−0.0146395	−0.00731977	+	0.999973i	$0.502330\pi$
$608$	8.09017	0.328100
$609$	8.18034	0.331484
$610$	0	0
$611$	−3.00000	−0.121367
$612$	−7.47214	−0.302043
$613$	−33.6869	−1.36060	−0.680301	−	0.732933i	$-0.738151\pi$
$613$	−33.6869	−1.36060	−0.680301	+	0.732933i	$0.738151\pi$
$614$	16.2918	0.657483
$615$	0	0
$616$	−13.4164	−0.540562
$617$	−36.9443	−1.48732	−0.743660	−	0.668558i	$-0.766912\pi$
$617$	−36.9443	−1.48732	−0.743660	+	0.668558i	$0.766912\pi$
$618$	7.81966	0.314553
$619$	41.0344	1.64931	0.824657	−	0.565634i	$-0.191368\pi$
$619$	41.0344	1.64931	0.824657	+	0.565634i	$0.191368\pi$
$620$	0	0
$621$	1.63932	0.0657837
$622$	−21.2148	−0.850635
$623$	−24.3607	−0.975990
$624$	1.61803	0.0647732
$625$	0	0
$626$	25.7082	1.02751
$627$	1.38197	0.0551904
$628$	5.32624	0.212540
$629$	−12.7639	−0.508931
$630$	0	0
$631$	28.0902	1.11825	0.559126	−	0.829083i	$-0.311137\pi$
$631$	28.0902	1.11825	0.559126	+	0.829083i	$0.311137\pi$
$632$	−49.9574	−1.98720
$633$	9.41641	0.374269
$634$	14.0344	0.557379
$635$	0	0
$636$	2.47214	0.0980266
$637$	9.09017	0.360166
$638$	−5.65248	−0.223784
$639$	−25.6525	−1.01480
$640$	0	0
$641$	−20.8541	−0.823688	−0.411844	−	0.911254i	$-0.635115\pi$
$641$	−20.8541	−0.823688	−0.411844	+	0.911254i	$0.635115\pi$
$642$	1.90983	0.0753750
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	1.52786	0.0602063
$645$	0	0
$646$	−4.61803	−0.181694
$647$	−26.1803	−1.02926	−0.514628	−	0.857414i	$-0.672070\pi$
$647$	−26.1803	−1.02926	−0.514628	+	0.857414i	$0.672070\pi$
$648$	17.1246	0.672718
$649$	−10.0000	−0.392534
$650$	0	0
$651$	−20.9443	−0.820871
$652$	13.8885	0.543917
$653$	2.67376	0.104632	0.0523162	−	0.998631i	$-0.483340\pi$
$653$	2.67376	0.104632	0.0523162	+	0.998631i	$0.483340\pi$
$654$	−3.41641	−0.133592
$655$	0	0
$656$	−7.09017	−0.276825
$657$	26.4164	1.03060
$658$	−3.70820	−0.144561
$659$	−19.2361	−0.749331	−0.374665	−	0.927160i	$-0.622242\pi$
$659$	−19.2361	−0.749331	−0.374665	+	0.927160i	$0.622242\pi$
$660$	0	0
$661$	18.0689	0.702798	0.351399	−	0.936226i	$-0.385706\pi$
$661$	18.0689	0.702798	0.351399	+	0.936226i	$0.385706\pi$
$662$	−8.18034	−0.317938
$663$	4.61803	0.179350
$664$	46.8541	1.81829
$665$	0	0
$666$	11.7082	0.453684
$667$	1.93112	0.0747731
$668$	4.29180	0.166055
$669$	2.70820	0.104705
$670$	0	0
$671$	−2.56231	−0.0989167
$672$	−10.0000	−0.385758
$673$	−36.6525	−1.41285	−0.706425	−	0.707788i	$-0.749693\pi$
$673$	−36.6525	−1.41285	−0.706425	+	0.707788i	$0.749693\pi$
$674$	4.65248	0.179207
$675$	0	0
$676$	6.14590	0.236381
$677$	−14.6180	−0.561817	−0.280908	−	0.959735i	$-0.590636\pi$
$677$	−14.6180	−0.561817	−0.280908	+	0.959735i	$0.590636\pi$
$678$	−11.8197	−0.453931
$679$	−55.7771	−2.14053
$680$	0	0
$681$	13.3475	0.511478
$682$	14.4721	0.554167
$683$	−28.3607	−1.08519	−0.542596	−	0.839994i	$-0.682558\pi$
$683$	−28.3607	−1.08519	−0.542596	+	0.839994i	$0.682558\pi$
$684$	−4.23607	−0.161970
$685$	0	0
$686$	−11.4164	−0.435880
$687$	2.49342	0.0951300
$688$	−2.47214	−0.0942493
$689$	10.4721	0.398957
$690$	0	0
$691$	−5.81966	−0.221390	−0.110695	−	0.993854i	$-0.535308\pi$
$691$	−5.81966	−0.221390	−0.110695	+	0.993854i	$0.535308\pi$
$692$	−6.32624	−0.240487
$693$	11.7082	0.444758
$694$	11.7984	0.447860
$695$	0	0
$696$	−7.58359	−0.287455
$697$	−20.2361	−0.766496
$698$	−23.5066	−0.889737
$699$	−1.81966	−0.0688259
$700$	0	0
$701$	−11.8197	−0.446422	−0.223211	−	0.974770i	$-0.571654\pi$
$701$	−11.8197	−0.446422	−0.223211	+	0.974770i	$0.571654\pi$
$702$	−9.09017	−0.343086
$703$	−7.23607	−0.272913
$704$	9.67376	0.364594
$705$	0	0
$706$	−30.3262	−1.14134
$707$	−23.4164	−0.880665
$708$	−4.47214	−0.168073
$709$	−9.52786	−0.357826	−0.178913	−	0.983865i	$-0.557258\pi$
$709$	−9.52786	−0.357826	−0.178913	+	0.983865i	$0.557258\pi$
$710$	0	0
$711$	43.5967	1.63501
$712$	22.5836	0.846356
$713$	−4.94427	−0.185164
$714$	5.70820	0.213624
$715$	0	0
$716$	−4.94427	−0.184776
$717$	0.944272	0.0352645
$718$	−4.29180	−0.160168
$719$	−33.5967	−1.25295	−0.626474	−	0.779443i	$-0.715502\pi$
$719$	−33.5967	−1.25295	−0.626474	+	0.779443i	$0.715502\pi$
$720$	0	0
$721$	−40.9443	−1.52484
$722$	16.3820	0.609674
$723$	−0.618034	−0.0229849
$724$	3.38197	0.125690
$725$	0	0
$726$	−5.61803	−0.208505
$727$	−20.2016	−0.749237	−0.374618	−	0.927179i	$-0.622226\pi$
$727$	−20.2016	−0.749237	−0.374618	+	0.927179i	$0.622226\pi$
$728$	−25.4164	−0.941995
$729$	−8.50658	−0.315058
$730$	0	0
$731$	−7.05573	−0.260966
$732$	−1.14590	−0.0423536
$733$	4.18034	0.154404	0.0772022	−	0.997015i	$-0.475401\pi$
$733$	4.18034	0.154404	0.0772022	+	0.997015i	$0.475401\pi$
$734$	−16.1803	−0.597227
$735$	0	0
$736$	−2.36068	−0.0870158
$737$	16.3050	0.600601
$738$	18.5623	0.683288
$739$	−2.76393	−0.101673	−0.0508364	−	0.998707i	$-0.516189\pi$
$739$	−2.76393	−0.101673	−0.0508364	+	0.998707i	$0.516189\pi$
$740$	0	0
$741$	2.61803	0.0961759
$742$	12.9443	0.475199
$743$	−45.9230	−1.68475	−0.842375	−	0.538891i	$-0.818843\pi$
$743$	−45.9230	−1.68475	−0.842375	+	0.538891i	$0.818843\pi$
$744$	19.4164	0.711840
$745$	0	0
$746$	14.8541	0.543847
$747$	−40.8885	−1.49603
$748$	3.94427	0.144217
$749$	−10.0000	−0.365392
$750$	0	0
$751$	15.8541	0.578524	0.289262	−	0.957250i	$-0.406590\pi$
$751$	15.8541	0.578524	0.289262	+	0.957250i	$0.406590\pi$
$752$	1.14590	0.0417866
$753$	3.52786	0.128563
$754$	−10.7082	−0.389970
$755$	0	0
$756$	11.2361	0.408652
$757$	−1.14590	−0.0416484	−0.0208242	−	0.999783i	$-0.506629\pi$
$757$	−1.14590	−0.0416484	−0.0208242	+	0.999783i	$0.506629\pi$
$758$	−5.74265	−0.208582
$759$	−0.403252	−0.0146371
$760$	0	0
$761$	28.8328	1.04519	0.522594	−	0.852581i	$-0.324964\pi$
$761$	28.8328	1.04519	0.522594	+	0.852581i	$0.324964\pi$
$762$	6.18034	0.223890
$763$	17.8885	0.647609
$764$	20.6525	0.747180
$765$	0	0
$766$	−24.4721	−0.884214
$767$	−18.9443	−0.684038
$768$	10.5066	0.379123
$769$	23.4164	0.844417	0.422209	−	0.906499i	$-0.361255\pi$
$769$	23.4164	0.844417	0.422209	+	0.906499i	$0.361255\pi$
$770$	0	0
$771$	−19.2361	−0.692770
$772$	−0.472136	−0.0169925
$773$	−39.0132	−1.40321	−0.701603	−	0.712568i	$-0.747532\pi$
$773$	−39.0132	−1.40321	−0.701603	+	0.712568i	$0.747532\pi$
$774$	6.47214	0.232636
$775$	0	0
$776$	51.7082	1.85622
$777$	8.94427	0.320874
$778$	10.2918	0.368979
$779$	−11.4721	−0.411032
$780$	0	0
$781$	13.5410	0.484536
$782$	1.34752	0.0481874
$783$	14.2016	0.507525
$784$	−3.47214	−0.124005
$785$	0	0
$786$	−10.8885	−0.388381
$787$	36.0344	1.28449	0.642245	−	0.766500i	$-0.278003\pi$
$787$	36.0344	1.28449	0.642245	+	0.766500i	$0.278003\pi$
$788$	−14.9443	−0.532368
$789$	−3.41641	−0.121627
$790$	0	0
$791$	61.8885	2.20050
$792$	−10.8541	−0.385684
$793$	−4.85410	−0.172374
$794$	31.3050	1.11097
$795$	0	0
$796$	23.8541	0.845486
$797$	−4.96556	−0.175889	−0.0879445	−	0.996125i	$-0.528030\pi$
$797$	−4.96556	−0.175889	−0.0879445	+	0.996125i	$0.528030\pi$
$798$	3.23607	0.114556
$799$	3.27051	0.115702
$800$	0	0
$801$	−19.7082	−0.696355
$802$	17.7984	0.628482
$803$	−13.9443	−0.492083
$804$	7.29180	0.257162
$805$	0	0
$806$	27.4164	0.965702
$807$	7.23607	0.254722
$808$	21.7082	0.763692
$809$	1.88854	0.0663977	0.0331988	−	0.999449i	$-0.489431\pi$
$809$	1.88854	0.0663977	0.0331988	+	0.999449i	$0.489431\pi$
$810$	0	0
$811$	12.5066	0.439165	0.219583	−	0.975594i	$-0.429530\pi$
$811$	12.5066	0.439165	0.219583	+	0.975594i	$0.429530\pi$
$812$	13.2361	0.464495
$813$	17.4164	0.610820
$814$	−6.18034	−0.216621
$815$	0	0
$816$	−1.76393	−0.0617500
$817$	−4.00000	−0.139942
$818$	−32.9443	−1.15187
$819$	22.1803	0.775044
$820$	0	0
$821$	−47.4508	−1.65605	−0.828023	−	0.560694i	$-0.810534\pi$
$821$	−47.4508	−1.65605	−0.828023	+	0.560694i	$0.810534\pi$
$822$	8.09017	0.282177
$823$	12.5836	0.438636	0.219318	−	0.975653i	$-0.429617\pi$
$823$	12.5836	0.438636	0.219318	+	0.975653i	$0.429617\pi$
$824$	37.9574	1.32231
$825$	0	0
$826$	−23.4164	−0.814761
$827$	−6.94427	−0.241476	−0.120738	−	0.992684i	$-0.538526\pi$
$827$	−6.94427	−0.241476	−0.120738	+	0.992684i	$0.538526\pi$
$828$	1.23607	0.0429563
$829$	13.7984	0.479237	0.239619	−	0.970867i	$-0.422978\pi$
$829$	13.7984	0.479237	0.239619	+	0.970867i	$0.422978\pi$
$830$	0	0
$831$	−4.94427	−0.171515
$832$	18.3262	0.635348
$833$	−9.90983	−0.343355
$834$	8.09017	0.280140
$835$	0	0
$836$	2.23607	0.0773360
$837$	−36.3607	−1.25681
$838$	−3.20163	−0.110598
$839$	11.8885	0.410438	0.205219	−	0.978716i	$-0.434209\pi$
$839$	11.8885	0.410438	0.205219	+	0.978716i	$0.434209\pi$
$840$	0	0
$841$	−12.2705	−0.423121
$842$	10.3607	0.357053
$843$	−12.3820	−0.426458
$844$	15.2361	0.524447
$845$	0	0
$846$	−3.00000	−0.103142
$847$	29.4164	1.01076
$848$	−4.00000	−0.137361
$849$	6.94427	0.238327
$850$	0	0
$851$	2.11146	0.0723798
$852$	6.05573	0.207466
$853$	26.9787	0.923734	0.461867	−	0.886949i	$-0.347180\pi$
$853$	26.9787	0.923734	0.461867	+	0.886949i	$0.347180\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	9.27051	0.316860
$857$	33.7082	1.15145	0.575725	−	0.817643i	$-0.304720\pi$
$857$	33.7082	1.15145	0.575725	+	0.817643i	$0.304720\pi$
$858$	2.23607	0.0763381
$859$	5.25735	0.179379	0.0896893	−	0.995970i	$-0.471413\pi$
$859$	5.25735	0.179379	0.0896893	+	0.995970i	$0.471413\pi$
$860$	0	0
$861$	14.1803	0.483265
$862$	33.4508	1.13934
$863$	18.7639	0.638732	0.319366	−	0.947632i	$-0.396530\pi$
$863$	18.7639	0.638732	0.319366	+	0.947632i	$0.396530\pi$
$864$	−17.3607	−0.590622
$865$	0	0
$866$	12.6525	0.429949
$867$	5.47214	0.185843
$868$	−33.8885	−1.15025
$869$	−23.0132	−0.780668
$870$	0	0
$871$	30.8885	1.04662
$872$	−16.5836	−0.561591
$873$	−45.1246	−1.52724
$874$	0.763932	0.0258404
$875$	0	0
$876$	−6.23607	−0.210697
$877$	−34.5410	−1.16637	−0.583184	−	0.812340i	$-0.698193\pi$
$877$	−34.5410	−1.16637	−0.583184	+	0.812340i	$0.698193\pi$
$878$	−4.14590	−0.139917
$879$	13.1803	0.444562
$880$	0	0
$881$	29.7984	1.00393	0.501966	−	0.864887i	$-0.332610\pi$
$881$	29.7984	1.00393	0.501966	+	0.864887i	$0.332610\pi$
$882$	9.09017	0.306082
$883$	−37.5623	−1.26407	−0.632036	−	0.774939i	$-0.717781\pi$
$883$	−37.5623	−1.26407	−0.632036	+	0.774939i	$0.717781\pi$
$884$	7.47214	0.251315
$885$	0	0
$886$	2.94427	0.0989147
$887$	4.97871	0.167169	0.0835844	−	0.996501i	$-0.473363\pi$
$887$	4.97871	0.167169	0.0835844	+	0.996501i	$0.473363\pi$
$888$	−8.29180	−0.278254
$889$	−32.3607	−1.08534
$890$	0	0
$891$	7.88854	0.264276
$892$	4.38197	0.146719
$893$	1.85410	0.0620452
$894$	8.00000	0.267560
$895$	0	0
$896$	−9.70820	−0.324328
$897$	−0.763932	−0.0255069
$898$	8.76393	0.292456
$899$	−42.8328	−1.42855
$900$	0	0
$901$	−11.4164	−0.380336
$902$	−9.79837	−0.326250
$903$	4.94427	0.164535
$904$	−57.3738	−1.90823
$905$	0	0
$906$	4.29180	0.142585
$907$	−5.81966	−0.193239	−0.0966193	−	0.995321i	$-0.530803\pi$
$907$	−5.81966	−0.193239	−0.0966193	+	0.995321i	$0.530803\pi$
$908$	21.5967	0.716713
$909$	−18.9443	−0.628342
$910$	0	0
$911$	1.34752	0.0446455	0.0223227	−	0.999751i	$-0.492894\pi$
$911$	1.34752	0.0446455	0.0223227	+	0.999751i	$0.492894\pi$
$912$	−1.00000	−0.0331133
$913$	21.5836	0.714313
$914$	6.65248	0.220044
$915$	0	0
$916$	4.03444	0.133302
$917$	57.0132	1.88274
$918$	9.90983	0.327073
$919$	2.94427	0.0971226	0.0485613	−	0.998820i	$-0.484536\pi$
$919$	2.94427	0.0971226	0.0485613	+	0.998820i	$0.484536\pi$
$920$	0	0
$921$	10.0689	0.331781
$922$	36.0000	1.18560
$923$	25.6525	0.844362
$924$	−2.76393	−0.0909267
$925$	0	0
$926$	17.7082	0.581928
$927$	−33.1246	−1.08795
$928$	−20.4508	−0.671332
$929$	6.76393	0.221917	0.110959	−	0.993825i	$-0.464608\pi$
$929$	6.76393	0.221917	0.110959	+	0.993825i	$0.464608\pi$
$930$	0	0
$931$	−5.61803	−0.184124
$932$	−2.94427	−0.0964428
$933$	−13.1115	−0.429250
$934$	24.3262	0.795979
$935$	0	0
$936$	−20.5623	−0.672100
$937$	20.3607	0.665154	0.332577	−	0.943076i	$-0.392082\pi$
$937$	20.3607	0.665154	0.332577	+	0.943076i	$0.392082\pi$
$938$	38.1803	1.24663
$939$	15.8885	0.518503
$940$	0	0
$941$	6.65248	0.216865	0.108432	−	0.994104i	$-0.465417\pi$
$941$	6.65248	0.216865	0.108432	+	0.994104i	$0.465417\pi$
$942$	−3.29180	−0.107253
$943$	3.34752	0.109010
$944$	7.23607	0.235514
$945$	0	0
$946$	−3.41641	−0.111077
$947$	−37.1246	−1.20639	−0.603194	−	0.797595i	$-0.706105\pi$
$947$	−37.1246	−1.20639	−0.603194	+	0.797595i	$0.706105\pi$
$948$	−10.2918	−0.334262
$949$	−26.4164	−0.857513
$950$	0	0
$951$	8.67376	0.281266
$952$	27.7082	0.898028
$953$	−25.3820	−0.822203	−0.411101	−	0.911590i	$-0.634856\pi$
$953$	−25.3820	−0.822203	−0.411101	+	0.911590i	$0.634856\pi$
$954$	10.4721	0.339048
$955$	0	0
$956$	1.52786	0.0494147
$957$	−3.49342	−0.112926
$958$	21.0557	0.680280
$959$	−42.3607	−1.36790
$960$	0	0
$961$	78.6656	2.53760
$962$	−11.7082	−0.377488
$963$	−8.09017	−0.260702
$964$	−1.00000	−0.0322078
$965$	0	0
$966$	−0.944272	−0.0303815
$967$	7.05573	0.226897	0.113448	−	0.993544i	$-0.463810\pi$
$967$	7.05573	0.226897	0.113448	+	0.993544i	$0.463810\pi$
$968$	−27.2705	−0.876507
$969$	−2.85410	−0.0916870
$970$	0	0
$971$	52.7214	1.69191	0.845954	−	0.533255i	$-0.179031\pi$
$971$	52.7214	1.69191	0.845954	+	0.533255i	$0.179031\pi$
$972$	13.9443	0.447263
$973$	−42.3607	−1.35802
$974$	8.32624	0.266790
$975$	0	0
$976$	1.85410	0.0593484
$977$	−5.05573	−0.161747	−0.0808735	−	0.996724i	$-0.525771\pi$
$977$	−5.05573	−0.161747	−0.0808735	+	0.996724i	$0.525771\pi$
$978$	−8.58359	−0.274473
$979$	10.4033	0.332489
$980$	0	0
$981$	14.4721	0.462060
$982$	−23.4164	−0.747248
$983$	−0.403252	−0.0128617	−0.00643087	−	0.999979i	$-0.502047\pi$
$983$	−0.403252	−0.0128617	−0.00643087	+	0.999979i	$0.502047\pi$
$984$	−13.1459	−0.419076
$985$	0	0
$986$	11.6738	0.371768
$987$	−2.29180	−0.0729487
$988$	4.23607	0.134767
$989$	1.16718	0.0371143
$990$	0	0
$991$	51.4164	1.63330	0.816648	−	0.577136i	$-0.195830\pi$
$991$	51.4164	1.63330	0.816648	+	0.577136i	$0.195830\pi$
$992$	52.3607	1.66245
$993$	−5.05573	−0.160439
$994$	31.7082	1.00572
$995$	0	0
$996$	9.65248	0.305850
$997$	−27.8541	−0.882148	−0.441074	−	0.897471i	$-0.645402\pi$
$997$	−27.8541	−0.882148	−0.441074	+	0.897471i	$0.645402\pi$
$998$	21.3050	0.674396
$999$	15.5279	0.491280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.a.1.1		2
5.2	odd	4		1205.2.b.b.724.2	✓	4
5.3	odd	4		1205.2.b.b.724.3	yes	4
5.4	even	2		6025.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.b.724.2	✓	4	5.2	odd	4
1205.2.b.b.724.3	yes	4	5.3	odd	4
6025.2.a.a.1.1		2	1.1	even	1	trivial
6025.2.a.d.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.618034 q^{3} -1.00000 q^{4} +0.618034 q^{6} -3.23607 q^{7} +3.00000 q^{8} -2.61803 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.618034 q^{3} -1.00000 q^{4} +0.618034 q^{6} -3.23607 q^{7} +3.00000 q^{8} -2.61803 q^{9} +1.38197 q^{11} +0.618034 q^{12} +2.61803 q^{13} +3.23607 q^{14} -1.00000 q^{16} -2.85410 q^{17} +2.61803 q^{18} -1.61803 q^{19} +2.00000 q^{21} -1.38197 q^{22} +0.472136 q^{23} -1.85410 q^{24} -2.61803 q^{26} +3.47214 q^{27} +3.23607 q^{28} +4.09017 q^{29} -10.4721 q^{31} -5.00000 q^{32} -0.854102 q^{33} +2.85410 q^{34} +2.61803 q^{36} +4.47214 q^{37} +1.61803 q^{38} -1.61803 q^{39} +7.09017 q^{41} -2.00000 q^{42} +2.47214 q^{43} -1.38197 q^{44} -0.472136 q^{46} -1.14590 q^{47} +0.618034 q^{48} +3.47214 q^{49} +1.76393 q^{51} -2.61803 q^{52} +4.00000 q^{53} -3.47214 q^{54} -9.70820 q^{56} +1.00000 q^{57} -4.09017 q^{58} -7.23607 q^{59} -1.85410 q^{61} +10.4721 q^{62} +8.47214 q^{63} +7.00000 q^{64} +0.854102 q^{66} +11.7984 q^{67} +2.85410 q^{68} -0.291796 q^{69} +9.79837 q^{71} -7.85410 q^{72} -10.0902 q^{73} -4.47214 q^{74} +1.61803 q^{76} -4.47214 q^{77} +1.61803 q^{78} -16.6525 q^{79} +5.70820 q^{81} -7.09017 q^{82} +15.6180 q^{83} -2.00000 q^{84} -2.47214 q^{86} -2.52786 q^{87} +4.14590 q^{88} +7.52786 q^{89} -8.47214 q^{91} -0.472136 q^{92} +6.47214 q^{93} +1.14590 q^{94} +3.09017 q^{96} +17.2361 q^{97} -3.47214 q^{98} -3.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + q^{3} - 2 q^{4} - q^{6} - 2 q^{7} + 6 q^{8} - 3 q^{9}+O(q^{10}) \) 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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