LMFDB - Embedded newform 6025.2.a.l.1.10

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:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-1.92684 q^{2} -1.97893 q^{3} +1.71272 q^{4} +3.81308 q^{6} -3.74943 q^{7} +0.553547 q^{8} +0.916158 q^{9} +O(q^{10})$	\(q-1.92684 q^{2} -1.97893 q^{3} +1.71272 q^{4} +3.81308 q^{6} -3.74943 q^{7} +0.553547 q^{8} +0.916158 q^{9} +0.698136 q^{11} -3.38935 q^{12} -4.67579 q^{13} +7.22455 q^{14} -4.49203 q^{16} -5.65460 q^{17} -1.76529 q^{18} +0.797009 q^{19} +7.41985 q^{21} -1.34520 q^{22} +2.87362 q^{23} -1.09543 q^{24} +9.00950 q^{26} +4.12377 q^{27} -6.42171 q^{28} +0.568476 q^{29} -5.19964 q^{31} +7.54834 q^{32} -1.38156 q^{33} +10.8955 q^{34} +1.56912 q^{36} -5.93333 q^{37} -1.53571 q^{38} +9.25304 q^{39} +4.54335 q^{41} -14.2969 q^{42} +7.00846 q^{43} +1.19571 q^{44} -5.53700 q^{46} +3.53934 q^{47} +8.88941 q^{48} +7.05822 q^{49} +11.1900 q^{51} -8.00830 q^{52} -3.89119 q^{53} -7.94586 q^{54} -2.07549 q^{56} -1.57722 q^{57} -1.09536 q^{58} +13.8320 q^{59} +0.157951 q^{61} +10.0189 q^{62} -3.43507 q^{63} -5.56039 q^{64} +2.66205 q^{66} -12.2343 q^{67} -9.68473 q^{68} -5.68668 q^{69} -6.17528 q^{71} +0.507137 q^{72} +4.72201 q^{73} +11.4326 q^{74} +1.36505 q^{76} -2.61761 q^{77} -17.8291 q^{78} -4.93666 q^{79} -10.9091 q^{81} -8.75432 q^{82} +1.99090 q^{83} +12.7081 q^{84} -13.5042 q^{86} -1.12497 q^{87} +0.386452 q^{88} +6.08236 q^{89} +17.5315 q^{91} +4.92170 q^{92} +10.2897 q^{93} -6.81975 q^{94} -14.9376 q^{96} +7.70151 q^{97} -13.6001 q^{98} +0.639603 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q - 11 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 16 * q^7 - 33 * q^8 + 38 * q^9	$ 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9} + q^{11} - 14 q^{12} - 9 q^{13} - q^{14} + 43 q^{16} - 12 q^{17} - 42 q^{18} + 2 q^{21} - 5 q^{22} - 77 q^{23} - 2 q^{24} + 2 q^{26} - 38 q^{27} - 42 q^{28} + 2 q^{29} + q^{31} - 72 q^{32} - 20 q^{33} + 5 q^{34} + 32 q^{36} - 28 q^{37} - 23 q^{38} - 2 q^{39} - 2 q^{41} - 37 q^{42} - 31 q^{43} + 3 q^{44} + 14 q^{46} - 96 q^{47} - 13 q^{48} + 40 q^{49} - 10 q^{51} - 42 q^{52} - 54 q^{53} + 4 q^{54} - 15 q^{56} - 37 q^{57} - 27 q^{58} + q^{59} + 5 q^{61} - 39 q^{62} - 70 q^{63} + 65 q^{64} - 52 q^{66} - 34 q^{67} - 52 q^{68} + 21 q^{69} - 9 q^{71} - 70 q^{72} - 25 q^{73} + 22 q^{74} - 47 q^{76} - 54 q^{77} - 58 q^{78} + 13 q^{79} + 12 q^{81} + 5 q^{82} - 63 q^{83} + 95 q^{84} - 18 q^{86} - 47 q^{87} - 13 q^{88} + 19 q^{89} - 31 q^{91} - 137 q^{92} - 52 q^{93} + 120 q^{94} - 49 q^{96} - 36 q^{97} - 64 q^{98} + 16 q^{99}+O(q^{100}) $ 40 * q - 11 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 16 * q^7 - 33 * q^8 + 38 * q^9 + q^11 - 14 * q^12 - 9 * q^13 - q^14 + 43 * q^16 - 12 * q^17 - 42 * q^18 + 2 * q^21 - 5 * q^22 - 77 * q^23 - 2 * q^24 + 2 * q^26 - 38 * q^27 - 42 * q^28 + 2 * q^29 + q^31 - 72 * q^32 - 20 * q^33 + 5 * q^34 + 32 * q^36 - 28 * q^37 - 23 * q^38 - 2 * q^39 - 2 * q^41 - 37 * q^42 - 31 * q^43 + 3 * q^44 + 14 * q^46 - 96 * q^47 - 13 * q^48 + 40 * q^49 - 10 * q^51 - 42 * q^52 - 54 * q^53 + 4 * q^54 - 15 * q^56 - 37 * q^57 - 27 * q^58 + q^59 + 5 * q^61 - 39 * q^62 - 70 * q^63 + 65 * q^64 - 52 * q^66 - 34 * q^67 - 52 * q^68 + 21 * q^69 - 9 * q^71 - 70 * q^72 - 25 * q^73 + 22 * q^74 - 47 * q^76 - 54 * q^77 - 58 * q^78 + 13 * q^79 + 12 * q^81 + 5 * q^82 - 63 * q^83 + 95 * q^84 - 18 * q^86 - 47 * q^87 - 13 * q^88 + 19 * q^89 - 31 * q^91 - 137 * q^92 - 52 * q^93 + 120 * q^94 - 49 * q^96 - 36 * q^97 - 64 * q^98 + 16 * 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 $

$2$	−1.92684	−1.36248	−0.681241	−	0.732059i	$-0.738560\pi$
$2$	−1.92684	−1.36248	−0.681241	+	0.732059i	$0.738560\pi$
$3$	−1.97893	−1.14253	−0.571267	−	0.820764i	$-0.693548\pi$
$3$	−1.97893	−1.14253	−0.571267	+	0.820764i	$0.693548\pi$
$4$	1.71272	0.856359
$5$	0	0
$5$	0	0
$6$	3.81308	1.55668
$7$	−3.74943	−1.41715	−0.708575	−	0.705635i	$-0.750662\pi$
$7$	−3.74943	−1.41715	−0.708575	+	0.705635i	$0.750662\pi$
$8$	0.553547	0.195709
$9$	0.916158	0.305386
$10$	0	0
$11$	0.698136	0.210496	0.105248	−	0.994446i	$-0.466436\pi$
$11$	0.698136	0.210496	0.105248	+	0.994446i	$0.466436\pi$
$12$	−3.38935	−0.978420
$13$	−4.67579	−1.29683	−0.648415	−	0.761287i	$-0.724568\pi$
$13$	−4.67579	−1.29683	−0.648415	+	0.761287i	$0.724568\pi$
$14$	7.22455	1.93084
$15$	0	0
$16$	−4.49203	−1.12301
$17$	−5.65460	−1.37144	−0.685721	−	0.727865i	$-0.740513\pi$
$17$	−5.65460	−1.37144	−0.685721	+	0.727865i	$0.740513\pi$
$18$	−1.76529	−0.416083
$19$	0.797009	0.182846	0.0914232	−	0.995812i	$-0.470858\pi$
$19$	0.797009	0.182846	0.0914232	+	0.995812i	$0.470858\pi$
$20$	0	0
$21$	7.41985	1.61914
$22$	−1.34520	−0.286797
$23$	2.87362	0.599191	0.299595	−	0.954066i	$-0.403148\pi$
$23$	2.87362	0.599191	0.299595	+	0.954066i	$0.403148\pi$
$24$	−1.09543	−0.223604
$25$	0	0
$26$	9.00950	1.76691
$27$	4.12377	0.793621
$28$	−6.42171	−1.21359
$29$	0.568476	0.105563	0.0527816	−	0.998606i	$-0.483191\pi$
$29$	0.568476	0.105563	0.0527816	+	0.998606i	$0.483191\pi$
$30$	0	0
$31$	−5.19964	−0.933884	−0.466942	−	0.884288i	$-0.654644\pi$
$31$	−5.19964	−0.933884	−0.466942	+	0.884288i	$0.654644\pi$
$32$	7.54834	1.33437
$33$	−1.38156	−0.240499
$34$	10.8955	1.86856
$35$	0	0
$36$	1.56912	0.261520
$37$	−5.93333	−0.975433	−0.487717	−	0.873002i	$-0.662170\pi$
$37$	−5.93333	−0.975433	−0.487717	+	0.873002i	$0.662170\pi$
$38$	−1.53571	−0.249125
$39$	9.25304	1.48167
$40$	0	0
$41$	4.54335	0.709553	0.354776	−	0.934951i	$-0.384557\pi$
$41$	4.54335	0.709553	0.354776	+	0.934951i	$0.384557\pi$
$42$	−14.2969	−2.20606
$43$	7.00846	1.06878	0.534390	−	0.845238i	$-0.320541\pi$
$43$	7.00846	1.06878	0.534390	+	0.845238i	$0.320541\pi$
$44$	1.19571	0.180260
$45$	0	0
$46$	−5.53700	−0.816387
$47$	3.53934	0.516266	0.258133	−	0.966109i	$-0.416893\pi$
$47$	3.53934	0.516266	0.258133	+	0.966109i	$0.416893\pi$
$48$	8.88941	1.28308
$49$	7.05822	1.00832
$50$	0	0
$51$	11.1900	1.56692
$52$	−8.00830	−1.11055
$53$	−3.89119	−0.534496	−0.267248	−	0.963628i	$-0.586114\pi$
$53$	−3.89119	−0.534496	−0.267248	+	0.963628i	$0.586114\pi$
$54$	−7.94586	−1.08129
$55$	0	0
$56$	−2.07549	−0.277349
$57$	−1.57722	−0.208908
$58$	−1.09536	−0.143828
$59$	13.8320	1.80078	0.900389	−	0.435086i	$-0.143282\pi$
$59$	13.8320	1.80078	0.900389	+	0.435086i	$0.143282\pi$
$60$	0	0
$61$	0.157951	0.0202236	0.0101118	−	0.999949i	$-0.496781\pi$
$61$	0.157951	0.0202236	0.0101118	+	0.999949i	$0.496781\pi$
$62$	10.0189	1.27240
$63$	−3.43507	−0.432778
$64$	−5.56039	−0.695049
$65$	0	0
$66$	2.66205	0.327676
$67$	−12.2343	−1.49466	−0.747328	−	0.664455i	$-0.768664\pi$
$67$	−12.2343	−1.49466	−0.747328	+	0.664455i	$0.768664\pi$
$68$	−9.68473	−1.17445
$69$	−5.68668	−0.684596
$70$	0	0
$71$	−6.17528	−0.732871	−0.366435	−	0.930444i	$-0.619422\pi$
$71$	−6.17528	−0.732871	−0.366435	+	0.930444i	$0.619422\pi$
$72$	0.507137	0.0597667
$73$	4.72201	0.552670	0.276335	−	0.961061i	$-0.410880\pi$
$73$	4.72201	0.552670	0.276335	+	0.961061i	$0.410880\pi$
$74$	11.4326	1.32901
$75$	0	0
$76$	1.36505	0.156582
$77$	−2.61761	−0.298305
$78$	−17.8291	−2.01875
$79$	−4.93666	−0.555417	−0.277709	−	0.960665i	$-0.589575\pi$
$79$	−4.93666	−0.555417	−0.277709	+	0.960665i	$0.589575\pi$
$80$	0	0
$81$	−10.9091	−1.21213
$82$	−8.75432	−0.966753
$83$	1.99090	0.218530	0.109265	−	0.994013i	$-0.465150\pi$
$83$	1.99090	0.218530	0.109265	+	0.994013i	$0.465150\pi$
$84$	12.7081	1.38657
$85$	0	0
$86$	−13.5042	−1.45619
$87$	−1.12497	−0.120610
$88$	0.386452	0.0411959
$89$	6.08236	0.644729	0.322365	−	0.946616i	$-0.395522\pi$
$89$	6.08236	0.644729	0.322365	+	0.946616i	$0.395522\pi$
$90$	0	0
$91$	17.5315	1.83780
$92$	4.92170	0.513122
$93$	10.2897	1.06699
$94$	−6.81975	−0.703403
$95$	0	0
$96$	−14.9376	−1.52457
$97$	7.70151	0.781970	0.390985	−	0.920397i	$-0.372135\pi$
$97$	7.70151	0.781970	0.390985	+	0.920397i	$0.372135\pi$
$98$	−13.6001	−1.37381
$99$	0.639603	0.0642825
$100$	0	0
$101$	5.53866	0.551118	0.275559	−	0.961284i	$-0.411137\pi$
$101$	5.53866	0.551118	0.275559	+	0.961284i	$0.411137\pi$
$102$	−21.5614	−2.13490
$103$	−1.88041	−0.185283	−0.0926414	−	0.995700i	$-0.529531\pi$
$103$	−1.88041	−0.185283	−0.0926414	+	0.995700i	$0.529531\pi$
$104$	−2.58827	−0.253801
$105$	0	0
$106$	7.49770	0.728241
$107$	7.94967	0.768524	0.384262	−	0.923224i	$-0.374456\pi$
$107$	7.94967	0.768524	0.384262	+	0.923224i	$0.374456\pi$
$108$	7.06286	0.679624
$109$	−5.51071	−0.527831	−0.263915	−	0.964546i	$-0.585014\pi$
$109$	−5.51071	−0.527831	−0.263915	+	0.964546i	$0.585014\pi$
$110$	0	0
$111$	11.7416	1.11447
$112$	16.8426	1.59147
$113$	6.23225	0.586280	0.293140	−	0.956069i	$-0.405300\pi$
$113$	6.23225	0.586280	0.293140	+	0.956069i	$0.405300\pi$
$114$	3.03906	0.284634
$115$	0	0
$116$	0.973638	0.0904001
$117$	−4.28376	−0.396034
$118$	−26.6521	−2.45353
$119$	21.2015	1.94354
$120$	0	0
$121$	−10.5126	−0.955691
$122$	−0.304347	−0.0275543
$123$	−8.99097	−0.810689
$124$	−8.90552	−0.799740
$125$	0	0
$126$	6.61883	0.589653
$127$	13.6028	1.20705	0.603527	−	0.797342i	$-0.293761\pi$
$127$	13.6028	1.20705	0.603527	+	0.797342i	$0.293761\pi$
$128$	−4.38269	−0.387379
$129$	−13.8692	−1.22112
$130$	0	0
$131$	−3.84906	−0.336294	−0.168147	−	0.985762i	$-0.553778\pi$
$131$	−3.84906	−0.336294	−0.168147	+	0.985762i	$0.553778\pi$
$132$	−2.36623	−0.205953
$133$	−2.98833	−0.259121
$134$	23.5735	2.03644
$135$	0	0
$136$	−3.13009	−0.268403
$137$	16.0210	1.36877	0.684383	−	0.729123i	$-0.260072\pi$
$137$	16.0210	1.36877	0.684383	+	0.729123i	$0.260072\pi$
$138$	10.9573	0.932750
$139$	−2.73566	−0.232035	−0.116018	−	0.993247i	$-0.537013\pi$
$139$	−2.73566	−0.232035	−0.116018	+	0.993247i	$0.537013\pi$
$140$	0	0
$141$	−7.00410	−0.589852
$142$	11.8988	0.998523
$143$	−3.26434	−0.272977
$144$	−4.11541	−0.342951
$145$	0	0
$146$	−9.09857	−0.753003
$147$	−13.9677	−1.15204
$148$	−10.1621	−0.835321
$149$	6.77335	0.554895	0.277447	−	0.960741i	$-0.410512\pi$
$149$	6.77335	0.554895	0.277447	+	0.960741i	$0.410512\pi$
$150$	0	0
$151$	4.34546	0.353628	0.176814	−	0.984244i	$-0.443421\pi$
$151$	4.34546	0.353628	0.176814	+	0.984244i	$0.443421\pi$
$152$	0.441182	0.0357846
$153$	−5.18050	−0.418819
$154$	5.04372	0.406435
$155$	0	0
$156$	15.8479	1.26884
$157$	−7.53813	−0.601608	−0.300804	−	0.953686i	$-0.597255\pi$
$157$	−7.53813	−0.601608	−0.300804	+	0.953686i	$0.597255\pi$
$158$	9.51215	0.756746
$159$	7.70039	0.610680
$160$	0	0
$161$	−10.7744	−0.849144
$162$	21.0202	1.65150
$163$	−8.72339	−0.683269	−0.341634	−	0.939833i	$-0.610980\pi$
$163$	−8.72339	−0.683269	−0.341634	+	0.939833i	$0.610980\pi$
$164$	7.78148	0.607632
$165$	0	0
$166$	−3.83615	−0.297743
$167$	−7.02667	−0.543740	−0.271870	−	0.962334i	$-0.587642\pi$
$167$	−7.02667	−0.543740	−0.271870	+	0.962334i	$0.587642\pi$
$168$	4.10724	0.316880
$169$	8.86297	0.681767
$170$	0	0
$171$	0.730186	0.0558387
$172$	12.0035	0.915259
$173$	17.0132	1.29349	0.646744	−	0.762707i	$-0.276130\pi$
$173$	17.0132	1.29349	0.646744	+	0.762707i	$0.276130\pi$
$174$	2.16764	0.164329
$175$	0	0
$176$	−3.13605	−0.236389
$177$	−27.3726	−2.05745
$178$	−11.7197	−0.878432
$179$	12.9003	0.964213	0.482107	−	0.876113i	$-0.339872\pi$
$179$	12.9003	0.964213	0.482107	+	0.876113i	$0.339872\pi$
$180$	0	0
$181$	14.2274	1.05752	0.528758	−	0.848772i	$-0.322658\pi$
$181$	14.2274	1.05752	0.528758	+	0.848772i	$0.322658\pi$
$182$	−33.7805	−2.50397
$183$	−0.312574	−0.0231061
$184$	1.59068	0.117267
$185$	0	0
$186$	−19.8267	−1.45376
$187$	−3.94768	−0.288683
$188$	6.06189	0.442109
$189$	−15.4618	−1.12468
$190$	0	0
$191$	−13.3875	−0.968688	−0.484344	−	0.874878i	$-0.660942\pi$
$191$	−13.3875	−0.968688	−0.484344	+	0.874878i	$0.660942\pi$
$192$	11.0036	0.794117
$193$	16.9893	1.22292	0.611460	−	0.791276i	$-0.290583\pi$
$193$	16.9893	1.22292	0.611460	+	0.791276i	$0.290583\pi$
$194$	−14.8396	−1.06542
$195$	0	0
$196$	12.0887	0.863481
$197$	−0.531396	−0.0378604	−0.0189302	−	0.999821i	$-0.506026\pi$
$197$	−0.531396	−0.0378604	−0.0189302	+	0.999821i	$0.506026\pi$
$198$	−1.23241	−0.0875838
$199$	16.3966	1.16232	0.581161	−	0.813789i	$-0.302599\pi$
$199$	16.3966	1.16232	0.581161	+	0.813789i	$0.302599\pi$
$200$	0	0
$201$	24.2108	1.70770
$202$	−10.6721	−0.750888
$203$	−2.13146	−0.149599
$204$	19.1654	1.34185
$205$	0	0
$206$	3.62326	0.252445
$207$	2.63269	0.182984
$208$	21.0038	1.45635
$209$	0.556421	0.0384884
$210$	0	0
$211$	2.17014	0.149399	0.0746993	−	0.997206i	$-0.476200\pi$
$211$	2.17014	0.149399	0.0746993	+	0.997206i	$0.476200\pi$
$212$	−6.66451	−0.457720
$213$	12.2204	0.837330
$214$	−15.3178	−1.04710
$215$	0	0
$216$	2.28270	0.155318
$217$	19.4957	1.32345
$218$	10.6183	0.719160
$219$	−9.34452	−0.631444
$220$	0	0
$221$	26.4397	1.77853
$222$	−22.6243	−1.51844
$223$	−2.53114	−0.169498	−0.0847488	−	0.996402i	$-0.527009\pi$
$223$	−2.53114	−0.169498	−0.0847488	+	0.996402i	$0.527009\pi$
$224$	−28.3020	−1.89100
$225$	0	0
$226$	−12.0086	−0.798797
$227$	−27.2508	−1.80870	−0.904351	−	0.426790i	$-0.859644\pi$
$227$	−27.2508	−1.80870	−0.904351	+	0.426790i	$0.859644\pi$
$228$	−2.70134	−0.178901
$229$	10.1576	0.671235	0.335618	−	0.941998i	$-0.391055\pi$
$229$	10.1576	0.671235	0.335618	+	0.941998i	$0.391055\pi$
$230$	0	0
$231$	5.18007	0.340823
$232$	0.314678	0.0206596
$233$	28.0714	1.83902	0.919508	−	0.393070i	$-0.128587\pi$
$233$	28.0714	1.83902	0.919508	+	0.393070i	$0.128587\pi$
$234$	8.25412	0.539589
$235$	0	0
$236$	23.6904	1.54211
$237$	9.76929	0.634584
$238$	−40.8519	−2.64804
$239$	22.7913	1.47425	0.737123	−	0.675758i	$-0.236184\pi$
$239$	22.7913	1.47425	0.737123	+	0.675758i	$0.236184\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	20.2561	1.30211
$243$	9.21706	0.591275
$244$	0.270526	0.0173186
$245$	0	0
$246$	17.3242	1.10455
$247$	−3.72664	−0.237121
$248$	−2.87825	−0.182769
$249$	−3.93986	−0.249678
$250$	0	0
$251$	1.86874	0.117954	0.0589770	−	0.998259i	$-0.481216\pi$
$251$	1.86874	0.117954	0.0589770	+	0.998259i	$0.481216\pi$
$252$	−5.88330	−0.370613
$253$	2.00618	0.126127
$254$	−26.2105	−1.64459
$255$	0	0
$256$	19.5655	1.22285
$257$	−0.966573	−0.0602932	−0.0301466	−	0.999545i	$-0.509597\pi$
$257$	−0.966573	−0.0602932	−0.0301466	+	0.999545i	$0.509597\pi$
$258$	26.7238	1.66375
$259$	22.2466	1.38234
$260$	0	0
$261$	0.520814	0.0322375
$262$	7.41653	0.458195
$263$	−4.57283	−0.281973	−0.140986	−	0.990012i	$-0.545027\pi$
$263$	−4.57283	−0.281973	−0.140986	+	0.990012i	$0.545027\pi$
$264$	−0.764760	−0.0470677
$265$	0	0
$266$	5.75803	0.353048
$267$	−12.0366	−0.736626
$268$	−20.9539	−1.27996
$269$	8.18085	0.498795	0.249398	−	0.968401i	$-0.419767\pi$
$269$	8.18085	0.498795	0.249398	+	0.968401i	$0.419767\pi$
$270$	0	0
$271$	−8.88291	−0.539598	−0.269799	−	0.962917i	$-0.586957\pi$
$271$	−8.88291	−0.539598	−0.269799	+	0.962917i	$0.586957\pi$
$272$	25.4006	1.54014
$273$	−34.6936	−2.09975
$274$	−30.8699	−1.86492
$275$	0	0
$276$	−9.73968	−0.586260
$277$	−4.57055	−0.274618	−0.137309	−	0.990528i	$-0.543845\pi$
$277$	−4.57055	−0.274618	−0.137309	+	0.990528i	$0.543845\pi$
$278$	5.27118	0.316144
$279$	−4.76370	−0.285195
$280$	0	0
$281$	−14.0161	−0.836128	−0.418064	−	0.908418i	$-0.637291\pi$
$281$	−14.0161	−0.836128	−0.418064	+	0.908418i	$0.637291\pi$
$282$	13.4958	0.803663
$283$	−18.3301	−1.08961	−0.544804	−	0.838563i	$-0.683396\pi$
$283$	−18.3301	−1.08961	−0.544804	+	0.838563i	$0.683396\pi$
$284$	−10.5765	−0.627600
$285$	0	0
$286$	6.28986	0.371927
$287$	−17.0350	−1.00554
$288$	6.91547	0.407498
$289$	14.9745	0.880851
$290$	0	0
$291$	−15.2407	−0.893427
$292$	8.08747	0.473284
$293$	7.61523	0.444886	0.222443	−	0.974946i	$-0.428597\pi$
$293$	7.61523	0.444886	0.222443	+	0.974946i	$0.428597\pi$
$294$	26.9135	1.56963
$295$	0	0
$296$	−3.28438	−0.190901
$297$	2.87896	0.167054
$298$	−13.0512	−0.756034
$299$	−13.4364	−0.777048
$300$	0	0
$301$	−26.2777	−1.51462
$302$	−8.37300	−0.481812
$303$	−10.9606	−0.629671
$304$	−3.58019	−0.205338
$305$	0	0
$306$	9.98201	0.570634
$307$	24.7729	1.41386	0.706932	−	0.707281i	$-0.250079\pi$
$307$	24.7729	1.41386	0.706932	+	0.707281i	$0.250079\pi$
$308$	−4.48323	−0.255456
$309$	3.72121	0.211692
$310$	0	0
$311$	−22.1846	−1.25797	−0.628986	−	0.777416i	$-0.716530\pi$
$311$	−22.1846	−1.25797	−0.628986	+	0.777416i	$0.716530\pi$
$312$	5.12200	0.289976
$313$	30.5525	1.72693	0.863465	−	0.504409i	$-0.168290\pi$
$313$	30.5525	1.72693	0.863465	+	0.504409i	$0.168290\pi$
$314$	14.5248	0.819681
$315$	0	0
$316$	−8.45510	−0.475637
$317$	−8.49126	−0.476916	−0.238458	−	0.971153i	$-0.576642\pi$
$317$	−8.49126	−0.476916	−0.238458	+	0.971153i	$0.576642\pi$
$318$	−14.8374	−0.832041
$319$	0.396873	0.0222206
$320$	0	0
$321$	−15.7318	−0.878065
$322$	20.7606	1.15694
$323$	−4.50676	−0.250763
$324$	−18.6843	−1.03801
$325$	0	0
$326$	16.8086	0.930942
$327$	10.9053	0.603065
$328$	2.51496	0.138866
$329$	−13.2705	−0.731626
$330$	0	0
$331$	−6.76018	−0.371573	−0.185787	−	0.982590i	$-0.559483\pi$
$331$	−6.76018	−0.371573	−0.185787	+	0.982590i	$0.559483\pi$
$332$	3.40986	0.187140
$333$	−5.43587	−0.297884
$334$	13.5393	0.740836
$335$	0	0
$336$	−33.3302	−1.81831
$337$	1.43313	0.0780674	0.0390337	−	0.999238i	$-0.487572\pi$
$337$	1.43313	0.0780674	0.0390337	+	0.999238i	$0.487572\pi$
$338$	−17.0775	−0.928895
$339$	−12.3332	−0.669846
$340$	0	0
$341$	−3.63006	−0.196579
$342$	−1.40695	−0.0760793
$343$	−0.218274	−0.0117857
$344$	3.87951	0.209169
$345$	0	0
$346$	−32.7817	−1.76236
$347$	16.3681	0.878686	0.439343	−	0.898319i	$-0.355211\pi$
$347$	16.3681	0.878686	0.439343	+	0.898319i	$0.355211\pi$
$348$	−1.92676	−0.103285
$349$	−27.5620	−1.47536	−0.737680	−	0.675151i	$-0.764079\pi$
$349$	−27.5620	−1.47536	−0.737680	+	0.675151i	$0.764079\pi$
$350$	0	0
$351$	−19.2819	−1.02919
$352$	5.26977	0.280880
$353$	8.42308	0.448315	0.224158	−	0.974553i	$-0.428037\pi$
$353$	8.42308	0.448315	0.224158	+	0.974553i	$0.428037\pi$
$354$	52.7427	2.80324
$355$	0	0
$356$	10.4174	0.552120
$357$	−41.9563	−2.22056
$358$	−24.8568	−1.31372
$359$	−28.2768	−1.49239	−0.746196	−	0.665726i	$-0.768122\pi$
$359$	−28.2768	−1.49239	−0.746196	+	0.665726i	$0.768122\pi$
$360$	0	0
$361$	−18.3648	−0.966567
$362$	−27.4140	−1.44085
$363$	20.8037	1.09191
$364$	30.0266	1.57382
$365$	0	0
$366$	0.602281	0.0314817
$367$	−31.5019	−1.64439	−0.822194	−	0.569208i	$-0.807250\pi$
$367$	−31.5019	−1.64439	−0.822194	+	0.569208i	$0.807250\pi$
$368$	−12.9084	−0.672896
$369$	4.16243	0.216687
$370$	0	0
$371$	14.5897	0.757461
$372$	17.6234	0.913730
$373$	11.5758	0.599374	0.299687	−	0.954038i	$-0.403118\pi$
$373$	11.5758	0.599374	0.299687	+	0.954038i	$0.403118\pi$
$374$	7.60655	0.393325
$375$	0	0
$376$	1.95919	0.101038
$377$	−2.65807	−0.136898
$378$	29.7924	1.53236
$379$	29.1806	1.49891	0.749454	−	0.662056i	$-0.230316\pi$
$379$	29.1806	1.49891	0.749454	+	0.662056i	$0.230316\pi$
$380$	0	0
$381$	−26.9190	−1.37910
$382$	25.7957	1.31982
$383$	−31.1678	−1.59260	−0.796301	−	0.604901i	$-0.793213\pi$
$383$	−31.1678	−1.59260	−0.796301	+	0.604901i	$0.793213\pi$
$384$	8.67304	0.442594
$385$	0	0
$386$	−32.7358	−1.66621
$387$	6.42085	0.326390
$388$	13.1905	0.669647
$389$	−16.3224	−0.827579	−0.413789	−	0.910373i	$-0.635795\pi$
$389$	−16.3224	−0.827579	−0.413789	+	0.910373i	$0.635795\pi$
$390$	0	0
$391$	−16.2491	−0.821755
$392$	3.90706	0.197336
$393$	7.61701	0.384227
$394$	1.02392	0.0515842
$395$	0	0
$396$	1.09546	0.0550489
$397$	21.0372	1.05583	0.527913	−	0.849298i	$-0.322975\pi$
$397$	21.0372	1.05583	0.527913	+	0.849298i	$0.322975\pi$
$398$	−31.5936	−1.58364
$399$	5.91369	0.296055
$400$	0	0
$401$	−15.8430	−0.791163	−0.395582	−	0.918431i	$-0.629457\pi$
$401$	−15.8430	−0.791163	−0.395582	+	0.918431i	$0.629457\pi$
$402$	−46.6503	−2.32671
$403$	24.3124	1.21109
$404$	9.48617	0.471954
$405$	0	0
$406$	4.10698	0.203826
$407$	−4.14227	−0.205325
$408$	6.19422	0.306660
$409$	29.4016	1.45381	0.726907	−	0.686735i	$-0.240957\pi$
$409$	29.4016	1.45381	0.726907	+	0.686735i	$0.240957\pi$
$410$	0	0
$411$	−31.7044	−1.56386
$412$	−3.22062	−0.158669
$413$	−51.8622	−2.55197
$414$	−5.07277	−0.249313
$415$	0	0
$416$	−35.2944	−1.73045
$417$	5.41367	0.265109
$418$	−1.07213	−0.0524398
$419$	29.5763	1.44490	0.722448	−	0.691425i	$-0.243017\pi$
$419$	29.5763	1.44490	0.722448	+	0.691425i	$0.243017\pi$
$420$	0	0
$421$	−8.95390	−0.436386	−0.218193	−	0.975906i	$-0.570016\pi$
$421$	−8.95390	−0.436386	−0.218193	+	0.975906i	$0.570016\pi$
$422$	−4.18151	−0.203553
$423$	3.24259	0.157660
$424$	−2.15396	−0.104605
$425$	0	0
$426$	−23.5468	−1.14085
$427$	−0.592227	−0.0286599
$428$	13.6155	0.658132
$429$	6.45989	0.311886
$430$	0	0
$431$	−12.0344	−0.579679	−0.289839	−	0.957075i	$-0.593602\pi$
$431$	−12.0344	−0.579679	−0.289839	+	0.957075i	$0.593602\pi$
$432$	−18.5241	−0.891243
$433$	−18.1591	−0.872670	−0.436335	−	0.899784i	$-0.643724\pi$
$433$	−18.1591	−0.872670	−0.436335	+	0.899784i	$0.643724\pi$
$434$	−37.5651	−1.80318
$435$	0	0
$436$	−9.43830	−0.452012
$437$	2.29030	0.109560
$438$	18.0054	0.860332
$439$	−8.59787	−0.410354	−0.205177	−	0.978725i	$-0.565777\pi$
$439$	−8.59787	−0.410354	−0.205177	+	0.978725i	$0.565777\pi$
$440$	0	0
$441$	6.46644	0.307926
$442$	−50.9451	−2.42321
$443$	26.4202	1.25526	0.627631	−	0.778511i	$-0.284025\pi$
$443$	26.4202	1.25526	0.627631	+	0.778511i	$0.284025\pi$
$444$	20.1101	0.954383
$445$	0	0
$446$	4.87711	0.230938
$447$	−13.4040	−0.633987
$448$	20.8483	0.984989
$449$	11.9923	0.565950	0.282975	−	0.959127i	$-0.408679\pi$
$449$	11.9923	0.565950	0.282975	+	0.959127i	$0.408679\pi$
$450$	0	0
$451$	3.17188	0.149358
$452$	10.6741	0.502066
$453$	−8.59935	−0.404033
$454$	52.5080	2.46432
$455$	0	0
$456$	−0.873068	−0.0408852
$457$	30.4011	1.42211	0.711053	−	0.703139i	$-0.248219\pi$
$457$	30.4011	1.42211	0.711053	+	0.703139i	$0.248219\pi$
$458$	−19.5722	−0.914547
$459$	−23.3183	−1.08840
$460$	0	0
$461$	−2.58678	−0.120478	−0.0602391	−	0.998184i	$-0.519186\pi$
$461$	−2.58678	−0.120478	−0.0602391	+	0.998184i	$0.519186\pi$
$462$	−9.98117	−0.464366
$463$	39.3392	1.82825	0.914124	−	0.405434i	$-0.132880\pi$
$463$	39.3392	1.82825	0.914124	+	0.405434i	$0.132880\pi$
$464$	−2.55361	−0.118548
$465$	0	0
$466$	−54.0891	−2.50563
$467$	−22.5514	−1.04355	−0.521777	−	0.853082i	$-0.674731\pi$
$467$	−22.5514	−1.04355	−0.521777	+	0.853082i	$0.674731\pi$
$468$	−7.33687	−0.339147
$469$	45.8716	2.11815
$470$	0	0
$471$	14.9174	0.687359
$472$	7.65669	0.352428
$473$	4.89286	0.224974
$474$	−18.8239	−0.864609
$475$	0	0
$476$	36.3122	1.66437
$477$	−3.56494	−0.163228
$478$	−43.9152	−2.00864
$479$	−36.0343	−1.64645	−0.823224	−	0.567716i	$-0.807827\pi$
$479$	−36.0343	−1.64645	−0.823224	+	0.567716i	$0.807827\pi$
$480$	0	0
$481$	27.7430	1.26497
$482$	1.92684	0.0877652
$483$	21.3218	0.970176
$484$	−18.0051	−0.818415
$485$	0	0
$486$	−17.7598	−0.805602
$487$	31.1634	1.41215	0.706075	−	0.708137i	$-0.250464\pi$
$487$	31.1634	1.41215	0.706075	+	0.708137i	$0.250464\pi$
$488$	0.0874335	0.00395793
$489$	17.2630	0.780659
$490$	0	0
$491$	−20.1073	−0.907431	−0.453715	−	0.891147i	$-0.649902\pi$
$491$	−20.1073	−0.907431	−0.453715	+	0.891147i	$0.649902\pi$
$492$	−15.3990	−0.694240
$493$	−3.21450	−0.144774
$494$	7.18065	0.323073
$495$	0	0
$496$	23.3570	1.04876
$497$	23.1538	1.03859
$498$	7.59148	0.340182
$499$	−32.3094	−1.44637	−0.723184	−	0.690656i	$-0.757322\pi$
$499$	−32.3094	−1.44637	−0.723184	+	0.690656i	$0.757322\pi$
$500$	0	0
$501$	13.9053	0.621242
$502$	−3.60077	−0.160710
$503$	−8.53483	−0.380549	−0.190275	−	0.981731i	$-0.560938\pi$
$503$	−8.53483	−0.380549	−0.190275	+	0.981731i	$0.560938\pi$
$504$	−1.90147	−0.0846984
$505$	0	0
$506$	−3.86558	−0.171846
$507$	−17.5392	−0.778942
$508$	23.2978	1.03367
$509$	−22.6149	−1.00239	−0.501193	−	0.865335i	$-0.667105\pi$
$509$	−22.6149	−1.00239	−0.501193	+	0.865335i	$0.667105\pi$
$510$	0	0
$511$	−17.7048	−0.783216
$512$	−28.9343	−1.27873
$513$	3.28668	0.145111
$514$	1.86243	0.0821484
$515$	0	0
$516$	−23.7541	−1.04572
$517$	2.47094	0.108672
$518$	−42.8657	−1.88341
$519$	−33.6679	−1.47786
$520$	0	0
$521$	10.6994	0.468747	0.234374	−	0.972147i	$-0.424696\pi$
$521$	10.6994	0.468747	0.234374	+	0.972147i	$0.424696\pi$
$522$	−1.00353	−0.0439231
$523$	−39.2992	−1.71843	−0.859217	−	0.511611i	$-0.829049\pi$
$523$	−39.2992	−1.71843	−0.859217	+	0.511611i	$0.829049\pi$
$524$	−6.59235	−0.287988
$525$	0	0
$526$	8.81111	0.384183
$527$	29.4019	1.28077
$528$	6.20602	0.270082
$529$	−14.7423	−0.640971
$530$	0	0
$531$	12.6723	0.549932
$532$	−5.11816	−0.221900
$533$	−21.2437	−0.920169
$534$	23.1925	1.00364
$535$	0	0
$536$	−6.77226	−0.292517
$537$	−25.5288	−1.10165
$538$	−15.7632	−0.679600
$539$	4.92760	0.212247
$540$	0	0
$541$	−25.2776	−1.08677	−0.543385	−	0.839483i	$-0.682858\pi$
$541$	−25.2776	−1.08677	−0.543385	+	0.839483i	$0.682858\pi$
$542$	17.1160	0.735193
$543$	−28.1551	−1.20825
$544$	−42.6828	−1.83001
$545$	0	0
$546$	66.8491	2.86088
$547$	−36.4817	−1.55984	−0.779922	−	0.625876i	$-0.784741\pi$
$547$	−36.4817	−1.55984	−0.779922	+	0.625876i	$0.784741\pi$
$548$	27.4394	1.17215
$549$	0.144708	0.00617600
$550$	0	0
$551$	0.453080	0.0193019
$552$	−3.14785	−0.133981
$553$	18.5096	0.787110
$554$	8.80673	0.374162
$555$	0	0
$556$	−4.68541	−0.198706
$557$	−40.8140	−1.72935	−0.864673	−	0.502336i	$-0.832474\pi$
$557$	−40.8140	−1.72935	−0.864673	+	0.502336i	$0.832474\pi$
$558$	9.17889	0.388573
$559$	−32.7700	−1.38602
$560$	0	0
$561$	7.81218	0.329830
$562$	27.0067	1.13921
$563$	2.45066	0.103283	0.0516415	−	0.998666i	$-0.483555\pi$
$563$	2.45066	0.103283	0.0516415	+	0.998666i	$0.483555\pi$
$564$	−11.9960	−0.505125
$565$	0	0
$566$	35.3191	1.48457
$567$	40.9030	1.71776
$568$	−3.41831	−0.143429
$569$	−4.85975	−0.203731	−0.101866	−	0.994798i	$-0.532481\pi$
$569$	−4.85975	−0.203731	−0.101866	+	0.994798i	$0.532481\pi$
$570$	0	0
$571$	15.2845	0.639638	0.319819	−	0.947479i	$-0.396378\pi$
$571$	15.2845	0.639638	0.319819	+	0.947479i	$0.396378\pi$
$572$	−5.59088	−0.233767
$573$	26.4930	1.10676
$574$	32.8237	1.37004
$575$	0	0
$576$	−5.09419	−0.212258
$577$	17.1153	0.712518	0.356259	−	0.934387i	$-0.384052\pi$
$577$	17.1153	0.712518	0.356259	+	0.934387i	$0.384052\pi$
$578$	−28.8534	−1.20014
$579$	−33.6207	−1.39723
$580$	0	0
$581$	−7.46475	−0.309690
$582$	29.3665	1.21728
$583$	−2.71658	−0.112509
$584$	2.61386	0.108162
$585$	0	0
$586$	−14.6733	−0.606150
$587$	−12.3850	−0.511184	−0.255592	−	0.966785i	$-0.582270\pi$
$587$	−12.3850	−0.511184	−0.255592	+	0.966785i	$0.582270\pi$
$588$	−23.9227	−0.986557
$589$	−4.14416	−0.170757
$590$	0	0
$591$	1.05160	0.0432568
$592$	26.6527	1.09542
$593$	−3.37993	−0.138797	−0.0693985	−	0.997589i	$-0.522108\pi$
$593$	−3.37993	−0.138797	−0.0693985	+	0.997589i	$0.522108\pi$
$594$	−5.54729	−0.227608
$595$	0	0
$596$	11.6008	0.475189
$597$	−32.4476	−1.32799
$598$	25.8898	1.05871
$599$	40.3761	1.64972	0.824861	−	0.565336i	$-0.191253\pi$
$599$	40.3761	1.64972	0.824861	+	0.565336i	$0.191253\pi$
$600$	0	0
$601$	−4.52198	−0.184455	−0.0922276	−	0.995738i	$-0.529399\pi$
$601$	−4.52198	−0.184455	−0.0922276	+	0.995738i	$0.529399\pi$
$602$	50.6330	2.06365
$603$	−11.2085	−0.456447
$604$	7.44254	0.302833
$605$	0	0
$606$	21.1194	0.857916
$607$	−16.9609	−0.688423	−0.344211	−	0.938892i	$-0.611854\pi$
$607$	−16.9609	−0.688423	−0.344211	+	0.938892i	$0.611854\pi$
$608$	6.01610	0.243985
$609$	4.21801	0.170922
$610$	0	0
$611$	−16.5492	−0.669509
$612$	−8.87274	−0.358659
$613$	33.3874	1.34850	0.674252	−	0.738502i	$-0.264466\pi$
$613$	33.3874	1.34850	0.674252	+	0.738502i	$0.264466\pi$
$614$	−47.7335	−1.92637
$615$	0	0
$616$	−1.44897	−0.0583808
$617$	11.7930	0.474768	0.237384	−	0.971416i	$-0.423710\pi$
$617$	11.7930	0.474768	0.237384	+	0.971416i	$0.423710\pi$
$618$	−7.17017	−0.288427
$619$	−37.6940	−1.51505	−0.757525	−	0.652806i	$-0.773592\pi$
$619$	−37.6940	−1.51505	−0.757525	+	0.652806i	$0.773592\pi$
$620$	0	0
$621$	11.8501	0.475530
$622$	42.7462	1.71397
$623$	−22.8054	−0.913679
$624$	−41.5650	−1.66393
$625$	0	0
$626$	−58.8698	−2.35291
$627$	−1.10112	−0.0439744
$628$	−12.9107	−0.515193
$629$	33.5506	1.33775
$630$	0	0
$631$	−8.35466	−0.332594	−0.166297	−	0.986076i	$-0.553181\pi$
$631$	−8.35466	−0.332594	−0.166297	+	0.986076i	$0.553181\pi$
$632$	−2.73267	−0.108700
$633$	−4.29455	−0.170693
$634$	16.3613	0.649790
$635$	0	0
$636$	13.1886	0.522961
$637$	−33.0027	−1.30761
$638$	−0.764712	−0.0302752
$639$	−5.65753	−0.223808
$640$	0	0
$641$	14.9207	0.589331	0.294665	−	0.955601i	$-0.404792\pi$
$641$	14.9207	0.589331	0.294665	+	0.955601i	$0.404792\pi$
$642$	30.3128	1.19635
$643$	21.2101	0.836445	0.418223	−	0.908345i	$-0.362653\pi$
$643$	21.2101	0.836445	0.418223	+	0.908345i	$0.362653\pi$
$644$	−18.4535	−0.727172
$645$	0	0
$646$	8.68382	0.341660
$647$	−48.0405	−1.88867	−0.944334	−	0.328988i	$-0.893292\pi$
$647$	−48.0405	−1.88867	−0.944334	+	0.328988i	$0.893292\pi$
$648$	−6.03872	−0.237223
$649$	9.65665	0.379057
$650$	0	0
$651$	−38.5806	−1.51209
$652$	−14.9407	−0.585123
$653$	−10.2639	−0.401656	−0.200828	−	0.979627i	$-0.564363\pi$
$653$	−10.2639	−0.401656	−0.200828	+	0.979627i	$0.564363\pi$
$654$	−21.0128	−0.821665
$655$	0	0
$656$	−20.4089	−0.796834
$657$	4.32611	0.168778
$658$	25.5701	0.996828
$659$	−7.09124	−0.276235	−0.138118	−	0.990416i	$-0.544105\pi$
$659$	−7.09124	−0.276235	−0.138118	+	0.990416i	$0.544105\pi$
$660$	0	0
$661$	−5.59563	−0.217645	−0.108822	−	0.994061i	$-0.534708\pi$
$661$	−5.59563	−0.217645	−0.108822	+	0.994061i	$0.534708\pi$
$662$	13.0258	0.506262
$663$	−52.3222	−2.03203
$664$	1.10206	0.0427682
$665$	0	0
$666$	10.4741	0.405861
$667$	1.63358	0.0632525
$668$	−12.0347	−0.465636
$669$	5.00895	0.193657
$670$	0	0
$671$	0.110271	0.00425698
$672$	56.0076	2.16054
$673$	−41.0740	−1.58329	−0.791644	−	0.610983i	$-0.790774\pi$
$673$	−41.0740	−1.58329	−0.791644	+	0.610983i	$0.790774\pi$
$674$	−2.76141	−0.106366
$675$	0	0
$676$	15.1798	0.583837
$677$	−6.53408	−0.251125	−0.125562	−	0.992086i	$-0.540074\pi$
$677$	−6.53408	−0.251125	−0.125562	+	0.992086i	$0.540074\pi$
$678$	23.7641	0.912653
$679$	−28.8762	−1.10817
$680$	0	0
$681$	53.9275	2.06650
$682$	6.99455	0.267835
$683$	−46.7203	−1.78770	−0.893851	−	0.448363i	$-0.852007\pi$
$683$	−46.7203	−1.78770	−0.893851	+	0.448363i	$0.852007\pi$
$684$	1.25060	0.0478180
$685$	0	0
$686$	0.420580	0.0160578
$687$	−20.1012	−0.766910
$688$	−31.4822	−1.20025
$689$	18.1944	0.693150
$690$	0	0
$691$	35.6615	1.35663	0.678314	−	0.734772i	$-0.262711\pi$
$691$	35.6615	1.35663	0.678314	+	0.734772i	$0.262711\pi$
$692$	29.1388	1.10769
$693$	−2.39815	−0.0910980
$694$	−31.5388	−1.19719
$695$	0	0
$696$	−0.622726	−0.0236044
$697$	−25.6908	−0.973110
$698$	53.1076	2.01015
$699$	−55.5512	−2.10114
$700$	0	0
$701$	−7.66032	−0.289326	−0.144663	−	0.989481i	$-0.546210\pi$
$701$	−7.66032	−0.289326	−0.144663	+	0.989481i	$0.546210\pi$
$702$	37.1531	1.40225
$703$	−4.72892	−0.178354
$704$	−3.88191	−0.146305
$705$	0	0
$706$	−16.2299	−0.610822
$707$	−20.7668	−0.781017
$708$	−46.8816	−1.76192
$709$	−43.2173	−1.62306	−0.811530	−	0.584310i	$-0.801365\pi$
$709$	−43.2173	−1.62306	−0.811530	+	0.584310i	$0.801365\pi$
$710$	0	0
$711$	−4.52276	−0.169617
$712$	3.36688	0.126179
$713$	−14.9418	−0.559574
$714$	80.8431	3.02548
$715$	0	0
$716$	22.0946	0.825712
$717$	−45.1024	−1.68438
$718$	54.4849	2.03336
$719$	−39.7493	−1.48240	−0.741199	−	0.671286i	$-0.765742\pi$
$719$	−39.7493	−1.48240	−0.741199	+	0.671286i	$0.765742\pi$
$720$	0	0
$721$	7.05048	0.262574
$722$	35.3860	1.31693
$723$	1.97893	0.0735971
$724$	24.3676	0.905614
$725$	0	0
$726$	−40.0854	−1.48771
$727$	27.2563	1.01088	0.505440	−	0.862862i	$-0.331330\pi$
$727$	27.2563	1.01088	0.505440	+	0.862862i	$0.331330\pi$
$728$	9.70453	0.359674
$729$	14.4875	0.536573
$730$	0	0
$731$	−39.6300	−1.46577
$732$	−0.535351	−0.0197871
$733$	5.66884	0.209383	0.104692	−	0.994505i	$-0.466614\pi$
$733$	5.66884	0.209383	0.104692	+	0.994505i	$0.466614\pi$
$734$	60.6992	2.24045
$735$	0	0
$736$	21.6910	0.799542
$737$	−8.54120	−0.314619
$738$	−8.02034	−0.295233
$739$	−2.42742	−0.0892940	−0.0446470	−	0.999003i	$-0.514216\pi$
$739$	−2.42742	−0.0892940	−0.0446470	+	0.999003i	$0.514216\pi$
$740$	0	0
$741$	7.37476	0.270919
$742$	−28.1121	−1.03203
$743$	−5.88156	−0.215774	−0.107887	−	0.994163i	$-0.534408\pi$
$743$	−5.88156	−0.215774	−0.107887	+	0.994163i	$0.534408\pi$
$744$	5.69585	0.208820
$745$	0	0
$746$	−22.3048	−0.816637
$747$	1.82398	0.0667360
$748$	−6.76126	−0.247216
$749$	−29.8067	−1.08911
$750$	0	0
$751$	19.7495	0.720670	0.360335	−	0.932823i	$-0.382662\pi$
$751$	19.7495	0.720670	0.360335	+	0.932823i	$0.382662\pi$
$752$	−15.8988	−0.579771
$753$	−3.69811	−0.134767
$754$	5.12168	0.186521
$755$	0	0
$756$	−26.4817	−0.963130
$757$	−13.2891	−0.482999	−0.241499	−	0.970401i	$-0.577639\pi$
$757$	−13.2891	−0.482999	−0.241499	+	0.970401i	$0.577639\pi$
$758$	−56.2264	−2.04224
$759$	−3.97008	−0.144105
$760$	0	0
$761$	40.5946	1.47155	0.735777	−	0.677224i	$-0.236817\pi$
$761$	40.5946	1.47155	0.735777	+	0.677224i	$0.236817\pi$
$762$	51.8686	1.87900
$763$	20.6620	0.748016
$764$	−22.9291	−0.829544
$765$	0	0
$766$	60.0555	2.16989
$767$	−64.6756	−2.33530
$768$	−38.7188	−1.39714
$769$	−2.05159	−0.0739823	−0.0369911	−	0.999316i	$-0.511777\pi$
$769$	−2.05159	−0.0739823	−0.0369911	+	0.999316i	$0.511777\pi$
$770$	0	0
$771$	1.91278	0.0688871
$772$	29.0979	1.04726
$773$	12.2422	0.440320	0.220160	−	0.975464i	$-0.429342\pi$
$773$	12.2422	0.440320	0.220160	+	0.975464i	$0.429342\pi$
$774$	−12.3720	−0.444701
$775$	0	0
$776$	4.26315	0.153038
$777$	−44.0244	−1.57937
$778$	31.4507	1.12756
$779$	3.62109	0.129739
$780$	0	0
$781$	−4.31118	−0.154266
$782$	31.3095	1.11963
$783$	2.34427	0.0837772
$784$	−31.7057	−1.13235
$785$	0	0
$786$	−14.6768	−0.523503
$787$	−37.5149	−1.33726	−0.668632	−	0.743594i	$-0.733120\pi$
$787$	−37.5149	−1.33726	−0.668632	+	0.743594i	$0.733120\pi$
$788$	−0.910132	−0.0324221
$789$	9.04930	0.322164
$790$	0	0
$791$	−23.3674	−0.830848
$792$	0.354051	0.0125806
$793$	−0.738546	−0.0262265
$794$	−40.5353	−1.43855
$795$	0	0
$796$	28.0827	0.995365
$797$	−13.6762	−0.484435	−0.242217	−	0.970222i	$-0.577875\pi$
$797$	−13.6762	−0.484435	−0.242217	+	0.970222i	$0.577875\pi$
$798$	−11.3947	−0.403369
$799$	−20.0135	−0.708028
$800$	0	0
$801$	5.57241	0.196891
$802$	30.5270	1.07795
$803$	3.29661	0.116335
$804$	41.4662	1.46240
$805$	0	0
$806$	−46.8462	−1.65009
$807$	−16.1893	−0.569891
$808$	3.06591	0.107858
$809$	−45.6843	−1.60617	−0.803087	−	0.595862i	$-0.796810\pi$
$809$	−45.6843	−1.60617	−0.803087	+	0.595862i	$0.796810\pi$
$810$	0	0
$811$	−24.9423	−0.875841	−0.437921	−	0.899014i	$-0.644285\pi$
$811$	−24.9423	−0.875841	−0.437921	+	0.899014i	$0.644285\pi$
$812$	−3.65059	−0.128111
$813$	17.5786	0.616510
$814$	7.98150	0.279752
$815$	0	0
$816$	−50.2661	−1.75966
$817$	5.58580	0.195422
$818$	−56.6522	−1.98080
$819$	16.0616	0.561239
$820$	0	0
$821$	5.06732	0.176851	0.0884254	−	0.996083i	$-0.471817\pi$
$821$	5.06732	0.176851	0.0884254	+	0.996083i	$0.471817\pi$
$822$	61.0893	2.13073
$823$	−0.229410	−0.00799674	−0.00399837	−	0.999992i	$-0.501273\pi$
$823$	−0.229410	−0.00799674	−0.00399837	+	0.999992i	$0.501273\pi$
$824$	−1.04090	−0.0362614
$825$	0	0
$826$	99.9303	3.47702
$827$	−9.37695	−0.326069	−0.163034	−	0.986620i	$-0.552128\pi$
$827$	−9.37695	−0.326069	−0.163034	+	0.986620i	$0.552128\pi$
$828$	4.50905	0.156700
$829$	−28.6245	−0.994169	−0.497085	−	0.867702i	$-0.665596\pi$
$829$	−28.6245	−0.994169	−0.497085	+	0.867702i	$0.665596\pi$
$830$	0	0
$831$	9.04479	0.313760
$832$	25.9992	0.901360
$833$	−39.9114	−1.38285
$834$	−10.4313	−0.361206
$835$	0	0
$836$	0.952992	0.0329599
$837$	−21.4422	−0.741150
$838$	−56.9888	−1.96865
$839$	−6.86107	−0.236870	−0.118435	−	0.992962i	$-0.537788\pi$
$839$	−6.86107	−0.236870	−0.118435	+	0.992962i	$0.537788\pi$
$840$	0	0
$841$	−28.6768	−0.988856
$842$	17.2527	0.594569
$843$	27.7368	0.955305
$844$	3.71684	0.127939
$845$	0	0
$846$	−6.24796	−0.214809
$847$	39.4163	1.35436
$848$	17.4794	0.600243
$849$	36.2739	1.24492
$850$	0	0
$851$	−17.0501	−0.584471
$852$	20.9301	0.717055
$853$	−10.5083	−0.359796	−0.179898	−	0.983685i	$-0.557577\pi$
$853$	−10.5083	−0.359796	−0.179898	+	0.983685i	$0.557577\pi$
$854$	1.14113	0.0390486
$855$	0	0
$856$	4.40052	0.150407
$857$	−11.2247	−0.383427	−0.191713	−	0.981451i	$-0.561404\pi$
$857$	−11.2247	−0.383427	−0.191713	+	0.981451i	$0.561404\pi$
$858$	−12.4472	−0.424940
$859$	−32.3311	−1.10312	−0.551561	−	0.834135i	$-0.685968\pi$
$859$	−32.3311	−1.10312	−0.551561	+	0.834135i	$0.685968\pi$
$860$	0	0
$861$	33.7110	1.14887
$862$	23.1885	0.789802
$863$	10.6333	0.361960	0.180980	−	0.983487i	$-0.442073\pi$
$863$	10.6333	0.361960	0.180980	+	0.983487i	$0.442073\pi$
$864$	31.1277	1.05898
$865$	0	0
$866$	34.9897	1.18900
$867$	−29.6334	−1.00640
$868$	33.3906	1.13335
$869$	−3.44646	−0.116913
$870$	0	0
$871$	57.2049	1.93831
$872$	−3.05044	−0.103301
$873$	7.05580	0.238803
$874$	−4.41304	−0.149273
$875$	0	0
$876$	−16.0045	−0.540743
$877$	45.2033	1.52641	0.763203	−	0.646158i	$-0.223625\pi$
$877$	45.2033	1.52641	0.763203	+	0.646158i	$0.223625\pi$
$878$	16.5667	0.559100
$879$	−15.0700	−0.508298
$880$	0	0
$881$	51.7627	1.74393	0.871965	−	0.489568i	$-0.162846\pi$
$881$	51.7627	1.74393	0.871965	+	0.489568i	$0.162846\pi$
$882$	−12.4598	−0.419543
$883$	42.7753	1.43950	0.719752	−	0.694231i	$-0.244255\pi$
$883$	42.7753	1.43950	0.719752	+	0.694231i	$0.244255\pi$
$884$	45.2837	1.52306
$885$	0	0
$886$	−50.9076	−1.71027
$887$	−22.6388	−0.760135	−0.380068	−	0.924959i	$-0.624099\pi$
$887$	−22.6388	−0.760135	−0.380068	+	0.924959i	$0.624099\pi$
$888$	6.49955	0.218111
$889$	−51.0028	−1.71058
$890$	0	0
$891$	−7.61606	−0.255148
$892$	−4.33513	−0.145151
$893$	2.82088	0.0943973
$894$	25.8273	0.863796
$895$	0	0
$896$	16.4326	0.548975
$897$	26.5897	0.887805
$898$	−23.1072	−0.771097
$899$	−2.95587	−0.0985838
$900$	0	0
$901$	22.0031	0.733030
$902$	−6.11171	−0.203498
$903$	52.0017	1.73051
$904$	3.44984	0.114740
$905$	0	0
$906$	16.5696	0.550487
$907$	45.7335	1.51856	0.759279	−	0.650766i	$-0.225552\pi$
$907$	45.7335	1.51856	0.759279	+	0.650766i	$0.225552\pi$
$908$	−46.6730	−1.54890
$909$	5.07429	0.168304
$910$	0	0
$911$	−35.4550	−1.17468	−0.587339	−	0.809341i	$-0.699824\pi$
$911$	−35.4550	−1.17468	−0.587339	+	0.809341i	$0.699824\pi$
$912$	7.08494	0.234606
$913$	1.38992	0.0459997
$914$	−58.5782	−1.93759
$915$	0	0
$916$	17.3972	0.574818
$917$	14.4318	0.476579
$918$	44.9306	1.48293
$919$	15.2959	0.504566	0.252283	−	0.967653i	$-0.418819\pi$
$919$	15.2959	0.504566	0.252283	+	0.967653i	$0.418819\pi$
$920$	0	0
$921$	−49.0238	−1.61539
$922$	4.98431	0.164149
$923$	28.8743	0.950408
$924$	8.87199	0.291867
$925$	0	0
$926$	−75.8004	−2.49096
$927$	−1.72276	−0.0565828
$928$	4.29105	0.140861
$929$	−16.9767	−0.556988	−0.278494	−	0.960438i	$-0.589835\pi$
$929$	−16.9767	−0.556988	−0.278494	+	0.960438i	$0.589835\pi$
$930$	0	0
$931$	5.62546	0.184367
$932$	48.0783	1.57486
$933$	43.9017	1.43728
$934$	43.4529	1.42182
$935$	0	0
$936$	−2.37126	−0.0775072
$937$	1.77519	0.0579928	0.0289964	−	0.999580i	$-0.490769\pi$
$937$	1.77519	0.0579928	0.0289964	+	0.999580i	$0.490769\pi$
$938$	−88.3873	−2.88595
$939$	−60.4612	−1.97308
$940$	0	0
$941$	53.5581	1.74594	0.872972	−	0.487771i	$-0.162190\pi$
$941$	53.5581	1.74594	0.872972	+	0.487771i	$0.162190\pi$
$942$	−28.7435	−0.936514
$943$	13.0559	0.425157
$944$	−62.1340	−2.02229
$945$	0	0
$946$	−9.42776	−0.306523
$947$	−34.8652	−1.13297	−0.566483	−	0.824074i	$-0.691696\pi$
$947$	−34.8652	−1.13297	−0.566483	+	0.824074i	$0.691696\pi$
$948$	16.7320	0.543431
$949$	−22.0791	−0.716718
$950$	0	0
$951$	16.8036	0.544894
$952$	11.7360	0.380367
$953$	−26.4851	−0.857935	−0.428968	−	0.903320i	$-0.641123\pi$
$953$	−26.4851	−0.857935	−0.428968	+	0.903320i	$0.641123\pi$
$954$	6.86908	0.222395
$955$	0	0
$956$	39.0351	1.26248
$957$	−0.785384	−0.0253879
$958$	69.4324	2.24326
$959$	−60.0696	−1.93975
$960$	0	0
$961$	−3.96370	−0.127861
$962$	−53.4563	−1.72350
$963$	7.28316	0.234696
$964$	−1.71272	−0.0551629
$965$	0	0
$966$	−41.0838	−1.32185
$967$	49.3662	1.58751	0.793755	−	0.608238i	$-0.208123\pi$
$967$	49.3662	1.58751	0.793755	+	0.608238i	$0.208123\pi$
$968$	−5.81923	−0.187037
$969$	8.91856	0.286506
$970$	0	0
$971$	38.1030	1.22278	0.611392	−	0.791328i	$-0.290610\pi$
$971$	38.1030	1.22278	0.611392	+	0.791328i	$0.290610\pi$
$972$	15.7862	0.506343
$973$	10.2571	0.328829
$974$	−60.0470	−1.92403
$975$	0	0
$976$	−0.709522	−0.0227112
$977$	24.5264	0.784669	0.392334	−	0.919823i	$-0.371668\pi$
$977$	24.5264	0.784669	0.392334	+	0.919823i	$0.371668\pi$
$978$	−33.2630	−1.06363
$979$	4.24632	0.135713
$980$	0	0
$981$	−5.04868	−0.161192
$982$	38.7436	1.23636
$983$	−5.40368	−0.172351	−0.0861753	−	0.996280i	$-0.527465\pi$
$983$	−5.40368	−0.172351	−0.0861753	+	0.996280i	$0.527465\pi$
$984$	−4.97693	−0.158659
$985$	0	0
$986$	6.19383	0.197252
$987$	26.2614	0.835909
$988$	−6.38269	−0.203060
$989$	20.1396	0.640403
$990$	0	0
$991$	29.7855	0.946167	0.473084	−	0.881018i	$-0.343141\pi$
$991$	29.7855	0.946167	0.473084	+	0.881018i	$0.343141\pi$
$992$	−39.2487	−1.24615
$993$	13.3779	0.424536
$994$	−44.6136	−1.41506
$995$	0	0
$996$	−6.74786	−0.213814
$997$	30.6856	0.971822	0.485911	−	0.874008i	$-0.338488\pi$
$997$	30.6856	0.971822	0.485911	+	0.874008i	$0.338488\pi$
$998$	62.2551	1.97065
$999$	−24.4677	−0.774124

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.l.1.10	✓	40
5.4	even	2		6025.2.a.o.1.31	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.l.1.10	✓	40	1.1	even	1	trivial
6025.2.a.o.1.31	yes	40	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)

\(f(q)\)	\(=\)	\(q-1.92684 q^{2} -1.97893 q^{3} +1.71272 q^{4} +3.81308 q^{6} -3.74943 q^{7} +0.553547 q^{8} +0.916158 q^{9} +O(q^{10})\)	\(q-1.92684 q^{2} -1.97893 q^{3} +1.71272 q^{4} +3.81308 q^{6} -3.74943 q^{7} +0.553547 q^{8} +0.916158 q^{9} +0.698136 q^{11} -3.38935 q^{12} -4.67579 q^{13} +7.22455 q^{14} -4.49203 q^{16} -5.65460 q^{17} -1.76529 q^{18} +0.797009 q^{19} +7.41985 q^{21} -1.34520 q^{22} +2.87362 q^{23} -1.09543 q^{24} +9.00950 q^{26} +4.12377 q^{27} -6.42171 q^{28} +0.568476 q^{29} -5.19964 q^{31} +7.54834 q^{32} -1.38156 q^{33} +10.8955 q^{34} +1.56912 q^{36} -5.93333 q^{37} -1.53571 q^{38} +9.25304 q^{39} +4.54335 q^{41} -14.2969 q^{42} +7.00846 q^{43} +1.19571 q^{44} -5.53700 q^{46} +3.53934 q^{47} +8.88941 q^{48} +7.05822 q^{49} +11.1900 q^{51} -8.00830 q^{52} -3.89119 q^{53} -7.94586 q^{54} -2.07549 q^{56} -1.57722 q^{57} -1.09536 q^{58} +13.8320 q^{59} +0.157951 q^{61} +10.0189 q^{62} -3.43507 q^{63} -5.56039 q^{64} +2.66205 q^{66} -12.2343 q^{67} -9.68473 q^{68} -5.68668 q^{69} -6.17528 q^{71} +0.507137 q^{72} +4.72201 q^{73} +11.4326 q^{74} +1.36505 q^{76} -2.61761 q^{77} -17.8291 q^{78} -4.93666 q^{79} -10.9091 q^{81} -8.75432 q^{82} +1.99090 q^{83} +12.7081 q^{84} -13.5042 q^{86} -1.12497 q^{87} +0.386452 q^{88} +6.08236 q^{89} +17.5315 q^{91} +4.92170 q^{92} +10.2897 q^{93} -6.81975 q^{94} -14.9376 q^{96} +7.70151 q^{97} -13.6001 q^{98} +0.639603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q - 11 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 16 * q^7 - 33 * q^8 + 38 * q^9	\( 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9} + q^{11} - 14 q^{12} - 9 q^{13} - q^{14} + 43 q^{16} - 12 q^{17} - 42 q^{18} + 2 q^{21} - 5 q^{22} - 77 q^{23} - 2 q^{24} + 2 q^{26} - 38 q^{27} - 42 q^{28} + 2 q^{29} + q^{31} - 72 q^{32} - 20 q^{33} + 5 q^{34} + 32 q^{36} - 28 q^{37} - 23 q^{38} - 2 q^{39} - 2 q^{41} - 37 q^{42} - 31 q^{43} + 3 q^{44} + 14 q^{46} - 96 q^{47} - 13 q^{48} + 40 q^{49} - 10 q^{51} - 42 q^{52} - 54 q^{53} + 4 q^{54} - 15 q^{56} - 37 q^{57} - 27 q^{58} + q^{59} + 5 q^{61} - 39 q^{62} - 70 q^{63} + 65 q^{64} - 52 q^{66} - 34 q^{67} - 52 q^{68} + 21 q^{69} - 9 q^{71} - 70 q^{72} - 25 q^{73} + 22 q^{74} - 47 q^{76} - 54 q^{77} - 58 q^{78} + 13 q^{79} + 12 q^{81} + 5 q^{82} - 63 q^{83} + 95 q^{84} - 18 q^{86} - 47 q^{87} - 13 q^{88} + 19 q^{89} - 31 q^{91} - 137 q^{92} - 52 q^{93} + 120 q^{94} - 49 q^{96} - 36 q^{97} - 64 q^{98} + 16 q^{99}+O(q^{100}) \) 40 * q - 11 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 16 * q^7 - 33 * q^8 + 38 * q^9 + q^11 - 14 * q^12 - 9 * q^13 - q^14 + 43 * q^16 - 12 * q^17 - 42 * q^18 + 2 * q^21 - 5 * q^22 - 77 * q^23 - 2 * q^24 + 2 * q^26 - 38 * q^27 - 42 * q^28 + 2 * q^29 + q^31 - 72 * q^32 - 20 * q^33 + 5 * q^34 + 32 * q^36 - 28 * q^37 - 23 * q^38 - 2 * q^39 - 2 * q^41 - 37 * q^42 - 31 * q^43 + 3 * q^44 + 14 * q^46 - 96 * q^47 - 13 * q^48 + 40 * q^49 - 10 * q^51 - 42 * q^52 - 54 * q^53 + 4 * q^54 - 15 * q^56 - 37 * q^57 - 27 * q^58 + q^59 + 5 * q^61 - 39 * q^62 - 70 * q^63 + 65 * q^64 - 52 * q^66 - 34 * q^67 - 52 * q^68 + 21 * q^69 - 9 * q^71 - 70 * q^72 - 25 * q^73 + 22 * q^74 - 47 * q^76 - 54 * q^77 - 58 * q^78 + 13 * q^79 + 12 * q^81 + 5 * q^82 - 63 * q^83 + 95 * q^84 - 18 * q^86 - 47 * q^87 - 13 * q^88 + 19 * q^89 - 31 * q^91 - 137 * q^92 - 52 * q^93 + 120 * q^94 - 49 * q^96 - 36 * q^97 - 64 * q^98 + 16 * q^99

Introduction

L-functions

Modular forms

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Database

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