LMFDB - Embedded newform 6025.2.a.k.1.25

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:	$0$
Dimension:	$25$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.25
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)

$f(q)$	$=$	$q+2.82262 q^{2} +2.39134 q^{3} +5.96719 q^{4} +6.74985 q^{6} -0.875274 q^{7} +11.1979 q^{8} +2.71851 q^{9} +O(q^{10})$	\(q+2.82262 q^{2} +2.39134 q^{3} +5.96719 q^{4} +6.74985 q^{6} -0.875274 q^{7} +11.1979 q^{8} +2.71851 q^{9} -3.86157 q^{11} +14.2696 q^{12} +0.582652 q^{13} -2.47057 q^{14} +19.6730 q^{16} +2.93934 q^{17} +7.67332 q^{18} +2.08453 q^{19} -2.09308 q^{21} -10.8998 q^{22} -0.349389 q^{23} +26.7779 q^{24} +1.64460 q^{26} -0.673145 q^{27} -5.22293 q^{28} +3.11646 q^{29} -4.04618 q^{31} +33.1336 q^{32} -9.23433 q^{33} +8.29663 q^{34} +16.2218 q^{36} +5.70743 q^{37} +5.88384 q^{38} +1.39332 q^{39} -11.6663 q^{41} -5.90797 q^{42} +7.84129 q^{43} -23.0427 q^{44} -0.986192 q^{46} -5.82783 q^{47} +47.0448 q^{48} -6.23389 q^{49} +7.02895 q^{51} +3.47679 q^{52} -10.7860 q^{53} -1.90003 q^{54} -9.80121 q^{56} +4.98482 q^{57} +8.79657 q^{58} +1.51778 q^{59} -4.55591 q^{61} -11.4208 q^{62} -2.37944 q^{63} +54.1777 q^{64} -26.0650 q^{66} +3.14491 q^{67} +17.5396 q^{68} -0.835508 q^{69} +5.72049 q^{71} +30.4415 q^{72} +11.4155 q^{73} +16.1099 q^{74} +12.4388 q^{76} +3.37994 q^{77} +3.93281 q^{78} -10.7741 q^{79} -9.76524 q^{81} -32.9295 q^{82} -11.0219 q^{83} -12.4898 q^{84} +22.1330 q^{86} +7.45250 q^{87} -43.2414 q^{88} -7.49366 q^{89} -0.509980 q^{91} -2.08487 q^{92} -9.67580 q^{93} -16.4498 q^{94} +79.2337 q^{96} -3.87107 q^{97} -17.5959 q^{98} -10.4977 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9}+O(q^{10}) $ 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9	$ 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9} + 10 q^{11} - 22 q^{12} - 10 q^{13} + 13 q^{14} + 54 q^{16} - q^{17} + 13 q^{18} + 50 q^{19} + 9 q^{21} - 11 q^{22} + 31 q^{23} + 22 q^{24} + 8 q^{26} - 42 q^{27} - 14 q^{28} + 4 q^{29} + 34 q^{31} + 44 q^{32} - 28 q^{33} + 33 q^{34} + 83 q^{36} - 14 q^{37} + 10 q^{38} + 23 q^{39} + 11 q^{41} - 23 q^{42} - 49 q^{43} + 20 q^{44} + 27 q^{46} + 28 q^{47} - 30 q^{48} + 66 q^{49} + 49 q^{51} - 39 q^{52} + 16 q^{53} + 5 q^{54} + 51 q^{56} - 10 q^{57} + 8 q^{58} + 30 q^{59} + 35 q^{61} + 18 q^{62} + 73 q^{64} - 13 q^{66} - 37 q^{67} - 11 q^{68} - 4 q^{69} + 12 q^{71} + 90 q^{72} - 36 q^{73} - 12 q^{74} + 57 q^{76} + 31 q^{77} + 9 q^{78} + 16 q^{79} + 65 q^{81} + 11 q^{82} - 43 q^{83} - 62 q^{84} - 9 q^{86} + 22 q^{87} - 20 q^{88} + 38 q^{89} + 86 q^{91} + 119 q^{92} - 10 q^{93} - 18 q^{94} - 34 q^{96} - 17 q^{97} + 32 q^{98} + 26 q^{99}+O(q^{100}) $ 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9 + 10 * q^11 - 22 * q^12 - 10 * q^13 + 13 * q^14 + 54 * q^16 - q^17 + 13 * q^18 + 50 * q^19 + 9 * q^21 - 11 * q^22 + 31 * q^23 + 22 * q^24 + 8 * q^26 - 42 * q^27 - 14 * q^28 + 4 * q^29 + 34 * q^31 + 44 * q^32 - 28 * q^33 + 33 * q^34 + 83 * q^36 - 14 * q^37 + 10 * q^38 + 23 * q^39 + 11 * q^41 - 23 * q^42 - 49 * q^43 + 20 * q^44 + 27 * q^46 + 28 * q^47 - 30 * q^48 + 66 * q^49 + 49 * q^51 - 39 * q^52 + 16 * q^53 + 5 * q^54 + 51 * q^56 - 10 * q^57 + 8 * q^58 + 30 * q^59 + 35 * q^61 + 18 * q^62 + 73 * q^64 - 13 * q^66 - 37 * q^67 - 11 * q^68 - 4 * q^69 + 12 * q^71 + 90 * q^72 - 36 * q^73 - 12 * q^74 + 57 * q^76 + 31 * q^77 + 9 * q^78 + 16 * q^79 + 65 * q^81 + 11 * q^82 - 43 * q^83 - 62 * q^84 - 9 * q^86 + 22 * q^87 - 20 * q^88 + 38 * q^89 + 86 * q^91 + 119 * q^92 - 10 * q^93 - 18 * q^94 - 34 * q^96 - 17 * q^97 + 32 * q^98 + 26 * 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 $

$2$	2.82262	1.99589	0.997947	−	0.0640413i	$-0.0203989\pi$
$2$	2.82262	1.99589	0.997947	+	0.0640413i	$0.0203989\pi$
$3$	2.39134	1.38064	0.690320	−	0.723504i	$-0.257470\pi$
$3$	2.39134	1.38064	0.690320	+	0.723504i	$0.257470\pi$
$4$	5.96719	2.98359
$5$	0	0
$5$	0	0
$6$	6.74985	2.75561
$7$	−0.875274	−0.330823	−0.165411	−	0.986225i	$-0.552895\pi$
$7$	−0.875274	−0.330823	−0.165411	+	0.986225i	$0.552895\pi$
$8$	11.1979	3.95905
$9$	2.71851	0.906169
$10$	0	0
$11$	−3.86157	−1.16431	−0.582154	−	0.813079i	$-0.697790\pi$
$11$	−3.86157	−1.16431	−0.582154	+	0.813079i	$0.697790\pi$
$12$	14.2696	4.11927
$13$	0.582652	0.161598	0.0807992	−	0.996730i	$-0.474253\pi$
$13$	0.582652	0.161598	0.0807992	+	0.996730i	$0.474253\pi$
$14$	−2.47057	−0.660287
$15$	0	0
$16$	19.6730	4.91824
$17$	2.93934	0.712894	0.356447	−	0.934316i	$-0.383988\pi$
$17$	2.93934	0.712894	0.356447	+	0.934316i	$0.383988\pi$
$18$	7.67332	1.80862
$19$	2.08453	0.478224	0.239112	−	0.970992i	$-0.423144\pi$
$19$	2.08453	0.478224	0.239112	+	0.970992i	$0.423144\pi$
$20$	0	0
$21$	−2.09308	−0.456747
$22$	−10.8998	−2.32384
$23$	−0.349389	−0.0728526	−0.0364263	−	0.999336i	$-0.511597\pi$
$23$	−0.349389	−0.0728526	−0.0364263	+	0.999336i	$0.511597\pi$
$24$	26.7779	5.46602
$25$	0	0
$26$	1.64460	0.322534
$27$	−0.673145	−0.129547
$28$	−5.22293	−0.987041
$29$	3.11646	0.578711	0.289356	−	0.957222i	$-0.406559\pi$
$29$	3.11646	0.578711	0.289356	+	0.957222i	$0.406559\pi$
$30$	0	0
$31$	−4.04618	−0.726716	−0.363358	−	0.931650i	$-0.618370\pi$
$31$	−4.04618	−0.726716	−0.363358	+	0.931650i	$0.618370\pi$
$32$	33.1336	5.85725
$33$	−9.23433	−1.60749
$34$	8.29663	1.42286
$35$	0	0
$36$	16.2218	2.70364
$37$	5.70743	0.938296	0.469148	−	0.883120i	$-0.344561\pi$
$37$	5.70743	0.938296	0.469148	+	0.883120i	$0.344561\pi$
$38$	5.88384	0.954485
$39$	1.39332	0.223109
$40$	0	0
$41$	−11.6663	−1.82197	−0.910984	−	0.412441i	$-0.864676\pi$
$41$	−11.6663	−1.82197	−0.910984	+	0.412441i	$0.864676\pi$
$42$	−5.90797	−0.911619
$43$	7.84129	1.19579	0.597893	−	0.801576i	$-0.296005\pi$
$43$	7.84129	1.19579	0.597893	+	0.801576i	$0.296005\pi$
$44$	−23.0427	−3.47382
$45$	0	0
$46$	−0.986192	−0.145406
$47$	−5.82783	−0.850076	−0.425038	−	0.905175i	$-0.639739\pi$
$47$	−5.82783	−0.850076	−0.425038	+	0.905175i	$0.639739\pi$
$48$	47.0448	6.79033
$49$	−6.23389	−0.890556
$50$	0	0
$51$	7.02895	0.984250
$52$	3.47679	0.482144
$53$	−10.7860	−1.48157	−0.740784	−	0.671743i	$-0.765546\pi$
$53$	−10.7860	−1.48157	−0.740784	+	0.671743i	$0.765546\pi$
$54$	−1.90003	−0.258562
$55$	0	0
$56$	−9.80121	−1.30974
$57$	4.98482	0.660256
$58$	8.79657	1.15505
$59$	1.51778	0.197599	0.0987993	−	0.995107i	$-0.468500\pi$
$59$	1.51778	0.197599	0.0987993	+	0.995107i	$0.468500\pi$
$60$	0	0
$61$	−4.55591	−0.583325	−0.291662	−	0.956521i	$-0.594208\pi$
$61$	−4.55591	−0.583325	−0.291662	+	0.956521i	$0.594208\pi$
$62$	−11.4208	−1.45045
$63$	−2.37944	−0.299781
$64$	54.1777	6.77221
$65$	0	0
$66$	−26.0650	−3.20838
$67$	3.14491	0.384212	0.192106	−	0.981374i	$-0.438468\pi$
$67$	3.14491	0.384212	0.192106	+	0.981374i	$0.438468\pi$
$68$	17.5396	2.12699
$69$	−0.835508	−0.100583
$70$	0	0
$71$	5.72049	0.678897	0.339449	−	0.940625i	$-0.389760\pi$
$71$	5.72049	0.678897	0.339449	+	0.940625i	$0.389760\pi$
$72$	30.4415	3.58756
$73$	11.4155	1.33608	0.668039	−	0.744126i	$-0.267134\pi$
$73$	11.4155	1.33608	0.668039	+	0.744126i	$0.267134\pi$
$74$	16.1099	1.87274
$75$	0	0
$76$	12.4388	1.42683
$77$	3.37994	0.385179
$78$	3.93281	0.445303
$79$	−10.7741	−1.21218	−0.606091	−	0.795396i	$-0.707263\pi$
$79$	−10.7741	−1.21218	−0.606091	+	0.795396i	$0.707263\pi$
$80$	0	0
$81$	−9.76524	−1.08503
$82$	−32.9295	−3.63646
$83$	−11.0219	−1.20981	−0.604906	−	0.796297i	$-0.706789\pi$
$83$	−11.0219	−1.20981	−0.604906	+	0.796297i	$0.706789\pi$
$84$	−12.4898	−1.36275
$85$	0	0
$86$	22.1330	2.38666
$87$	7.45250	0.798992
$88$	−43.2414	−4.60955
$89$	−7.49366	−0.794326	−0.397163	−	0.917748i	$-0.630005\pi$
$89$	−7.49366	−0.794326	−0.397163	+	0.917748i	$0.630005\pi$
$90$	0	0
$91$	−0.509980	−0.0534604
$92$	−2.08487	−0.217363
$93$	−9.67580	−1.00333
$94$	−16.4498	−1.69666
$95$	0	0
$96$	79.2337	8.08676
$97$	−3.87107	−0.393047	−0.196524	−	0.980499i	$-0.562965\pi$
$97$	−3.87107	−0.393047	−0.196524	+	0.980499i	$0.562965\pi$
$98$	−17.5959	−1.77746
$99$	−10.4977	−1.05506
$100$	0	0
$101$	14.9796	1.49053	0.745263	−	0.666771i	$-0.232324\pi$
$101$	14.9796	1.49053	0.745263	+	0.666771i	$0.232324\pi$
$102$	19.8401	1.96446
$103$	−19.8287	−1.95378	−0.976891	−	0.213738i	$-0.931436\pi$
$103$	−19.8287	−1.95378	−0.976891	+	0.213738i	$0.931436\pi$
$104$	6.52446	0.639776
$105$	0	0
$106$	−30.4447	−2.95706
$107$	2.47924	0.239678	0.119839	−	0.992793i	$-0.461762\pi$
$107$	2.47924	0.239678	0.119839	+	0.992793i	$0.461762\pi$
$108$	−4.01679	−0.386515
$109$	2.46586	0.236187	0.118093	−	0.993003i	$-0.462322\pi$
$109$	2.46586	0.236187	0.118093	+	0.993003i	$0.462322\pi$
$110$	0	0
$111$	13.6484	1.29545
$112$	−17.2193	−1.62707
$113$	17.5877	1.65451	0.827255	−	0.561827i	$-0.189901\pi$
$113$	17.5877	1.65451	0.827255	+	0.561827i	$0.189901\pi$
$114$	14.0703	1.31780
$115$	0	0
$116$	18.5965	1.72664
$117$	1.58394	0.146436
$118$	4.28413	0.394386
$119$	−2.57272	−0.235841
$120$	0	0
$121$	3.91174	0.355613
$122$	−12.8596	−1.16425
$123$	−27.8981	−2.51548
$124$	−24.1443	−2.16823
$125$	0	0
$126$	−6.71626	−0.598332
$127$	−5.83290	−0.517586	−0.258793	−	0.965933i	$-0.583325\pi$
$127$	−5.83290	−0.517586	−0.258793	+	0.965933i	$0.583325\pi$
$128$	86.6558	7.65936
$129$	18.7512	1.65095
$130$	0	0
$131$	17.7946	1.55472	0.777361	−	0.629054i	$-0.216558\pi$
$131$	17.7946	1.55472	0.777361	+	0.629054i	$0.216558\pi$
$132$	−55.1030	−4.79610
$133$	−1.82454	−0.158207
$134$	8.87690	0.766847
$135$	0	0
$136$	32.9143	2.82238
$137$	−18.5668	−1.58627	−0.793135	−	0.609046i	$-0.791553\pi$
$137$	−18.5668	−1.58627	−0.793135	+	0.609046i	$0.791553\pi$
$138$	−2.35832	−0.200754
$139$	6.46549	0.548396	0.274198	−	0.961673i	$-0.411588\pi$
$139$	6.46549	0.548396	0.274198	+	0.961673i	$0.411588\pi$
$140$	0	0
$141$	−13.9363	−1.17365
$142$	16.1468	1.35501
$143$	−2.24995	−0.188150
$144$	53.4811	4.45676
$145$	0	0
$146$	32.2215	2.66667
$147$	−14.9074	−1.22954
$148$	34.0573	2.79949
$149$	20.0818	1.64517	0.822584	−	0.568643i	$-0.192531\pi$
$149$	20.0818	1.64517	0.822584	+	0.568643i	$0.192531\pi$
$150$	0	0
$151$	3.95379	0.321755	0.160877	−	0.986974i	$-0.448568\pi$
$151$	3.95379	0.321755	0.160877	+	0.986974i	$0.448568\pi$
$152$	23.3423	1.89331
$153$	7.99060	0.646002
$154$	9.54028	0.768777
$155$	0	0
$156$	8.31419	0.665668
$157$	−20.0077	−1.59679	−0.798395	−	0.602135i	$-0.794317\pi$
$157$	−20.0077	−1.59679	−0.798395	+	0.602135i	$0.794317\pi$
$158$	−30.4112	−2.41939
$159$	−25.7930	−2.04551
$160$	0	0
$161$	0.305811	0.0241013
$162$	−27.5636	−2.16560
$163$	−10.4933	−0.821895	−0.410947	−	0.911659i	$-0.634802\pi$
$163$	−10.4933	−0.821895	−0.410947	+	0.911659i	$0.634802\pi$
$164$	−69.6150	−5.43602
$165$	0	0
$166$	−31.1107	−2.41466
$167$	18.8619	1.45958	0.729789	−	0.683672i	$-0.239618\pi$
$167$	18.8619	1.45958	0.729789	+	0.683672i	$0.239618\pi$
$168$	−23.4380	−1.80828
$169$	−12.6605	−0.973886
$170$	0	0
$171$	5.66681	0.433352
$172$	46.7905	3.56774
$173$	0.345627	0.0262775	0.0131388	−	0.999914i	$-0.495818\pi$
$173$	0.345627	0.0262775	0.0131388	+	0.999914i	$0.495818\pi$
$174$	21.0356	1.59470
$175$	0	0
$176$	−75.9686	−5.72635
$177$	3.62954	0.272813
$178$	−21.1518	−1.58539
$179$	−7.98056	−0.596495	−0.298248	−	0.954489i	$-0.596402\pi$
$179$	−7.98056	−0.596495	−0.298248	+	0.954489i	$0.596402\pi$
$180$	0	0
$181$	6.75810	0.502325	0.251163	−	0.967945i	$-0.419187\pi$
$181$	6.75810	0.502325	0.251163	+	0.967945i	$0.419187\pi$
$182$	−1.43948	−0.106701
$183$	−10.8947	−0.805362
$184$	−3.91241	−0.288427
$185$	0	0
$186$	−27.3111	−2.00255
$187$	−11.3505	−0.830027
$188$	−34.7758	−2.53628
$189$	0.589187	0.0428570
$190$	0	0
$191$	12.9405	0.936345	0.468172	−	0.883637i	$-0.344913\pi$
$191$	12.9405	0.936345	0.468172	+	0.883637i	$0.344913\pi$
$192$	129.557	9.34999
$193$	−8.90975	−0.641338	−0.320669	−	0.947191i	$-0.603908\pi$
$193$	−8.90975	−0.641338	−0.320669	+	0.947191i	$0.603908\pi$
$194$	−10.9265	−0.784481
$195$	0	0
$196$	−37.1988	−2.65706
$197$	16.0917	1.14649	0.573245	−	0.819384i	$-0.305684\pi$
$197$	16.0917	1.14649	0.573245	+	0.819384i	$0.305684\pi$
$198$	−29.6311	−2.10579
$199$	−12.9171	−0.915668	−0.457834	−	0.889038i	$-0.651375\pi$
$199$	−12.9171	−0.915668	−0.457834	+	0.889038i	$0.651375\pi$
$200$	0	0
$201$	7.52056	0.530459
$202$	42.2817	2.97493
$203$	−2.72775	−0.191451
$204$	41.9431	2.93660
$205$	0	0
$206$	−55.9690	−3.89954
$207$	−0.949816	−0.0660168
$208$	11.4625	0.794781
$209$	−8.04956	−0.556800
$210$	0	0
$211$	16.8567	1.16046	0.580231	−	0.814452i	$-0.302962\pi$
$211$	16.8567	1.16046	0.580231	+	0.814452i	$0.302962\pi$
$212$	−64.3620	−4.42040
$213$	13.6796	0.937313
$214$	6.99797	0.478371
$215$	0	0
$216$	−7.53780	−0.512882
$217$	3.54152	0.240414
$218$	6.96019	0.471404
$219$	27.2982	1.84464
$220$	0	0
$221$	1.71261	0.115203
$222$	38.5243	2.58558
$223$	−19.1445	−1.28201	−0.641005	−	0.767537i	$-0.721482\pi$
$223$	−19.1445	−1.28201	−0.641005	+	0.767537i	$0.721482\pi$
$224$	−29.0010	−1.93771
$225$	0	0
$226$	49.6434	3.30223
$227$	22.0068	1.46064	0.730321	−	0.683104i	$-0.239371\pi$
$227$	22.0068	1.46064	0.730321	+	0.683104i	$0.239371\pi$
$228$	29.7454	1.96994
$229$	14.2044	0.938653	0.469326	−	0.883025i	$-0.344497\pi$
$229$	14.2044	0.938653	0.469326	+	0.883025i	$0.344497\pi$
$230$	0	0
$231$	8.08257	0.531794
$232$	34.8977	2.29114
$233$	−23.0830	−1.51222	−0.756108	−	0.654447i	$-0.772902\pi$
$233$	−23.0830	−1.51222	−0.756108	+	0.654447i	$0.772902\pi$
$234$	4.47087	0.292270
$235$	0	0
$236$	9.05690	0.589554
$237$	−25.7645	−1.67359
$238$	−7.26183	−0.470714
$239$	2.58179	0.167002	0.0835011	−	0.996508i	$-0.473390\pi$
$239$	2.58179	0.167002	0.0835011	+	0.996508i	$0.473390\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	11.0414	0.709766
$243$	−21.3326	−1.36849
$244$	−27.1860	−1.74041
$245$	0	0
$246$	−78.7457	−5.02064
$247$	1.21455	0.0772803
$248$	−45.3086	−2.87710
$249$	−26.3571	−1.67031
$250$	0	0
$251$	10.6808	0.674165	0.337083	−	0.941475i	$-0.390560\pi$
$251$	10.6808	0.674165	0.337083	+	0.941475i	$0.390560\pi$
$252$	−14.1986	−0.894426
$253$	1.34919	0.0848229
$254$	−16.4641	−1.03305
$255$	0	0
$256$	136.241	8.51507
$257$	−4.68459	−0.292217	−0.146108	−	0.989269i	$-0.546675\pi$
$257$	−4.68459	−0.292217	−0.146108	+	0.989269i	$0.546675\pi$
$258$	52.9275	3.29512
$259$	−4.99557	−0.310409
$260$	0	0
$261$	8.47211	0.524410
$262$	50.2275	3.10306
$263$	−13.5038	−0.832682	−0.416341	−	0.909208i	$-0.636688\pi$
$263$	−13.5038	−0.832682	−0.416341	+	0.909208i	$0.636688\pi$
$264$	−103.405	−6.36413
$265$	0	0
$266$	−5.14997	−0.315765
$267$	−17.9199	−1.09668
$268$	18.7663	1.14633
$269$	−11.5617	−0.704931	−0.352466	−	0.935825i	$-0.614657\pi$
$269$	−11.5617	−0.704931	−0.352466	+	0.935825i	$0.614657\pi$
$270$	0	0
$271$	−0.360776	−0.0219156	−0.0109578	−	0.999940i	$-0.503488\pi$
$271$	−0.360776	−0.0219156	−0.0109578	+	0.999940i	$0.503488\pi$
$272$	57.8255	3.50618
$273$	−1.21954	−0.0738097
$274$	−52.4071	−3.16603
$275$	0	0
$276$	−4.98563	−0.300100
$277$	14.0418	0.843688	0.421844	−	0.906669i	$-0.361383\pi$
$277$	14.0418	0.843688	0.421844	+	0.906669i	$0.361383\pi$
$278$	18.2496	1.09454
$279$	−10.9996	−0.658527
$280$	0	0
$281$	−29.4531	−1.75702	−0.878511	−	0.477722i	$-0.841463\pi$
$281$	−29.4531	−1.75702	−0.878511	+	0.477722i	$0.841463\pi$
$282$	−39.3370	−2.34248
$283$	−18.0379	−1.07224	−0.536122	−	0.844141i	$-0.680111\pi$
$283$	−18.0379	−1.07224	−0.536122	+	0.844141i	$0.680111\pi$
$284$	34.1353	2.02555
$285$	0	0
$286$	−6.35076	−0.375528
$287$	10.2112	0.602749
$288$	90.0739	5.30766
$289$	−8.36031	−0.491783
$290$	0	0
$291$	−9.25703	−0.542657
$292$	68.1182	3.98632
$293$	−13.5687	−0.792692	−0.396346	−	0.918101i	$-0.629722\pi$
$293$	−13.5687	−0.792692	−0.396346	+	0.918101i	$0.629722\pi$
$294$	−42.0778	−2.45403
$295$	0	0
$296$	63.9111	3.71476
$297$	2.59940	0.150832
$298$	56.6834	3.28358
$299$	−0.203572	−0.0117729
$300$	0	0
$301$	−6.86328	−0.395593
$302$	11.1601	0.642189
$303$	35.8213	2.05788
$304$	41.0089	2.35202
$305$	0	0
$306$	22.5544	1.28935
$307$	−7.21586	−0.411831	−0.205916	−	0.978570i	$-0.566017\pi$
$307$	−7.21586	−0.411831	−0.205916	+	0.978570i	$0.566017\pi$
$308$	20.1687	1.14922
$309$	−47.4172	−2.69747
$310$	0	0
$311$	27.9272	1.58361	0.791803	−	0.610777i	$-0.209143\pi$
$311$	27.9272	1.58361	0.791803	+	0.610777i	$0.209143\pi$
$312$	15.6022	0.883301
$313$	−1.75711	−0.0993176	−0.0496588	−	0.998766i	$-0.515813\pi$
$313$	−1.75711	−0.0993176	−0.0496588	+	0.998766i	$0.515813\pi$
$314$	−56.4742	−3.18702
$315$	0	0
$316$	−64.2911	−3.61666
$317$	21.5589	1.21087	0.605433	−	0.795896i	$-0.293000\pi$
$317$	21.5589	1.21087	0.605433	+	0.795896i	$0.293000\pi$
$318$	−72.8037	−4.08263
$319$	−12.0344	−0.673798
$320$	0	0
$321$	5.92872	0.330909
$322$	0.863189	0.0481036
$323$	6.12713	0.340923
$324$	−58.2710	−3.23728
$325$	0	0
$326$	−29.6185	−1.64042
$327$	5.89671	0.326089
$328$	−130.638	−7.21326
$329$	5.10095	0.281224
$330$	0	0
$331$	31.6052	1.73718	0.868589	−	0.495533i	$-0.165027\pi$
$331$	31.6052	1.73718	0.868589	+	0.495533i	$0.165027\pi$
$332$	−65.7698	−3.60959
$333$	15.5157	0.850254
$334$	53.2400	2.91316
$335$	0	0
$336$	−41.1771	−2.24639
$337$	4.42993	0.241314	0.120657	−	0.992694i	$-0.461500\pi$
$337$	4.42993	0.241314	0.120657	+	0.992694i	$0.461500\pi$
$338$	−35.7358	−1.94377
$339$	42.0581	2.28428
$340$	0	0
$341$	15.6246	0.846121
$342$	15.9953	0.864924
$343$	11.5833	0.625439
$344$	87.8058	4.73417
$345$	0	0
$346$	0.975574	0.0524472
$347$	−32.8108	−1.76138	−0.880689	−	0.473695i	$-0.842920\pi$
$347$	−32.8108	−1.76138	−0.880689	+	0.473695i	$0.842920\pi$
$348$	44.4705	2.38387
$349$	−26.7722	−1.43308	−0.716542	−	0.697544i	$-0.754276\pi$
$349$	−26.7722	−1.43308	−0.716542	+	0.697544i	$0.754276\pi$
$350$	0	0
$351$	−0.392209	−0.0209346
$352$	−127.948	−6.81964
$353$	−11.7696	−0.626434	−0.313217	−	0.949682i	$-0.601407\pi$
$353$	−11.7696	−0.626434	−0.313217	+	0.949682i	$0.601407\pi$
$354$	10.2448	0.544505
$355$	0	0
$356$	−44.7161	−2.36995
$357$	−6.15226	−0.325612
$358$	−22.5261	−1.19054
$359$	27.2430	1.43783	0.718916	−	0.695097i	$-0.244639\pi$
$359$	27.2430	1.43783	0.718916	+	0.695097i	$0.244639\pi$
$360$	0	0
$361$	−14.6547	−0.771302
$362$	19.0755	1.00259
$363$	9.35430	0.490973
$364$	−3.04315	−0.159504
$365$	0	0
$366$	−30.7517	−1.60742
$367$	−7.35808	−0.384089	−0.192044	−	0.981386i	$-0.561512\pi$
$367$	−7.35808	−0.384089	−0.192044	+	0.981386i	$0.561512\pi$
$368$	−6.87352	−0.358307
$369$	−31.7149	−1.65101
$370$	0	0
$371$	9.44070	0.490137
$372$	−57.7373	−2.99354
$373$	−16.8192	−0.870865	−0.435433	−	0.900221i	$-0.643405\pi$
$373$	−16.8192	−0.870865	−0.435433	+	0.900221i	$0.643405\pi$
$374$	−32.0380	−1.65665
$375$	0	0
$376$	−65.2593	−3.36549
$377$	1.81581	0.0935189
$378$	1.66305	0.0855381
$379$	−17.7702	−0.912792	−0.456396	−	0.889777i	$-0.650860\pi$
$379$	−17.7702	−0.912792	−0.456396	+	0.889777i	$0.650860\pi$
$380$	0	0
$381$	−13.9484	−0.714600
$382$	36.5262	1.86885
$383$	5.54121	0.283142	0.141571	−	0.989928i	$-0.454785\pi$
$383$	5.54121	0.283142	0.141571	+	0.989928i	$0.454785\pi$
$384$	207.223	10.5748
$385$	0	0
$386$	−25.1489	−1.28004
$387$	21.3166	1.08358
$388$	−23.0994	−1.17269
$389$	0.784106	0.0397557	0.0198779	−	0.999802i	$-0.493672\pi$
$389$	0.784106	0.0397557	0.0198779	+	0.999802i	$0.493672\pi$
$390$	0	0
$391$	−1.02697	−0.0519362
$392$	−69.8064	−3.52575
$393$	42.5530	2.14651
$394$	45.4209	2.28827
$395$	0	0
$396$	−62.6418	−3.14787
$397$	−30.0330	−1.50732	−0.753658	−	0.657267i	$-0.771712\pi$
$397$	−30.0330	−1.50732	−0.753658	+	0.657267i	$0.771712\pi$
$398$	−36.4600	−1.82758
$399$	−4.36309	−0.218427
$400$	0	0
$401$	−5.90379	−0.294821	−0.147410	−	0.989075i	$-0.547094\pi$
$401$	−5.90379	−0.294821	−0.147410	+	0.989075i	$0.547094\pi$
$402$	21.2277	1.05874
$403$	−2.35751	−0.117436
$404$	89.3861	4.44713
$405$	0	0
$406$	−7.69941	−0.382116
$407$	−22.0397	−1.09246
$408$	78.7093	3.89669
$409$	8.25561	0.408214	0.204107	−	0.978949i	$-0.434571\pi$
$409$	8.25561	0.408214	0.204107	+	0.978949i	$0.434571\pi$
$410$	0	0
$411$	−44.3996	−2.19007
$412$	−118.322	−5.82929
$413$	−1.32848	−0.0653701
$414$	−2.68097	−0.131763
$415$	0	0
$416$	19.3053	0.946523
$417$	15.4612	0.757138
$418$	−22.7209	−1.11131
$419$	−10.2035	−0.498472	−0.249236	−	0.968443i	$-0.580179\pi$
$419$	−10.2035	−0.498472	−0.249236	+	0.968443i	$0.580179\pi$
$420$	0	0
$421$	19.0614	0.928994	0.464497	−	0.885575i	$-0.346235\pi$
$421$	19.0614	0.928994	0.464497	+	0.885575i	$0.346235\pi$
$422$	47.5800	2.31616
$423$	−15.8430	−0.770313
$424$	−120.780	−5.86560
$425$	0	0
$426$	38.6124	1.87078
$427$	3.98767	0.192977
$428$	14.7941	0.715101
$429$	−5.38040	−0.259768
$430$	0	0
$431$	4.82231	0.232283	0.116141	−	0.993233i	$-0.462947\pi$
$431$	4.82231	0.232283	0.116141	+	0.993233i	$0.462947\pi$
$432$	−13.2428	−0.637143
$433$	0.0847613	0.00407337	0.00203669	−	0.999998i	$-0.499352\pi$
$433$	0.0847613	0.00407337	0.00203669	+	0.999998i	$0.499352\pi$
$434$	9.99637	0.479841
$435$	0	0
$436$	14.7143	0.704685
$437$	−0.728312	−0.0348399
$438$	77.0526	3.68172
$439$	20.8527	0.995247	0.497624	−	0.867393i	$-0.334206\pi$
$439$	20.8527	0.995247	0.497624	+	0.867393i	$0.334206\pi$
$440$	0	0
$441$	−16.9469	−0.806995
$442$	4.83404	0.229932
$443$	37.6155	1.78717	0.893583	−	0.448897i	$-0.148183\pi$
$443$	37.6155	1.78717	0.893583	+	0.448897i	$0.148183\pi$
$444$	81.4426	3.86510
$445$	0	0
$446$	−54.0377	−2.55876
$447$	48.0225	2.27139
$448$	−47.4203	−2.24040
$449$	−2.90799	−0.137237	−0.0686183	−	0.997643i	$-0.521859\pi$
$449$	−2.90799	−0.137237	−0.0686183	+	0.997643i	$0.521859\pi$
$450$	0	0
$451$	45.0502	2.12133
$452$	104.949	4.93639
$453$	9.45486	0.444228
$454$	62.1168	2.91529
$455$	0	0
$456$	55.8194	2.61398
$457$	−20.1272	−0.941510	−0.470755	−	0.882264i	$-0.656018\pi$
$457$	−20.1272	−0.941510	−0.470755	+	0.882264i	$0.656018\pi$
$458$	40.0936	1.87345
$459$	−1.97860	−0.0923531
$460$	0	0
$461$	−24.1119	−1.12300	−0.561502	−	0.827476i	$-0.689776\pi$
$461$	−24.1119	−1.12300	−0.561502	+	0.827476i	$0.689776\pi$
$462$	22.8140	1.06141
$463$	6.04174	0.280784	0.140392	−	0.990096i	$-0.455164\pi$
$463$	6.04174	0.280784	0.140392	+	0.990096i	$0.455164\pi$
$464$	61.3099	2.84624
$465$	0	0
$466$	−65.1545	−3.01822
$467$	3.37791	0.156311	0.0781555	−	0.996941i	$-0.475097\pi$
$467$	3.37791	0.156311	0.0781555	+	0.996941i	$0.475097\pi$
$468$	9.45169	0.436904
$469$	−2.75266	−0.127106
$470$	0	0
$471$	−47.8452	−2.20459
$472$	16.9959	0.782302
$473$	−30.2797	−1.39226
$474$	−72.7235	−3.34030
$475$	0	0
$476$	−15.3519	−0.703655
$477$	−29.3218	−1.34255
$478$	7.28741	0.333319
$479$	−1.50496	−0.0687634	−0.0343817	−	0.999409i	$-0.510946\pi$
$479$	−1.50496	−0.0687634	−0.0343817	+	0.999409i	$0.510946\pi$
$480$	0	0
$481$	3.32544	0.151627
$482$	2.82262	0.128567
$483$	0.731298	0.0332752
$484$	23.3421	1.06100
$485$	0	0
$486$	−60.2138	−2.73135
$487$	5.05466	0.229049	0.114524	−	0.993420i	$-0.463466\pi$
$487$	5.05466	0.229049	0.114524	+	0.993420i	$0.463466\pi$
$488$	−51.0165	−2.30941
$489$	−25.0929	−1.13474
$490$	0	0
$491$	−2.67521	−0.120731	−0.0603653	−	0.998176i	$-0.519227\pi$
$491$	−2.67521	−0.120731	−0.0603653	+	0.998176i	$0.519227\pi$
$492$	−166.473	−7.50519
$493$	9.16031	0.412559
$494$	3.42823	0.154243
$495$	0	0
$496$	−79.6004	−3.57417
$497$	−5.00700	−0.224595
$498$	−74.3962	−3.33377
$499$	−0.844066	−0.0377856	−0.0188928	−	0.999822i	$-0.506014\pi$
$499$	−0.844066	−0.0377856	−0.0188928	+	0.999822i	$0.506014\pi$
$500$	0	0
$501$	45.1052	2.01515
$502$	30.1478	1.34556
$503$	−31.8346	−1.41944	−0.709718	−	0.704486i	$-0.751178\pi$
$503$	−31.8346	−1.41944	−0.709718	+	0.704486i	$0.751178\pi$
$504$	−26.6447	−1.18685
$505$	0	0
$506$	3.80825	0.169298
$507$	−30.2756	−1.34459
$508$	−34.8060	−1.54427
$509$	39.1345	1.73461	0.867304	−	0.497779i	$-0.165851\pi$
$509$	39.1345	1.73461	0.867304	+	0.497779i	$0.165851\pi$
$510$	0	0
$511$	−9.99166	−0.442005
$512$	211.246	9.33582
$513$	−1.40319	−0.0619524
$514$	−13.2228	−0.583233
$515$	0	0
$516$	111.892	4.92577
$517$	22.5046	0.989750
$518$	−14.1006	−0.619544
$519$	0.826512	0.0362798
$520$	0	0
$521$	−8.26390	−0.362048	−0.181024	−	0.983479i	$-0.557941\pi$
$521$	−8.26390	−0.362048	−0.181024	+	0.983479i	$0.557941\pi$
$522$	23.9135	1.04667
$523$	24.9962	1.09301	0.546503	−	0.837457i	$-0.315959\pi$
$523$	24.9962	1.09301	0.546503	+	0.837457i	$0.315959\pi$
$524$	106.184	4.63866
$525$	0	0
$526$	−38.1162	−1.66195
$527$	−11.8931	−0.518071
$528$	−181.667	−7.90603
$529$	−22.8779	−0.994692
$530$	0	0
$531$	4.12611	0.179058
$532$	−10.8874	−0.472027
$533$	−6.79738	−0.294427
$534$	−50.5810	−2.18886
$535$	0	0
$536$	35.2163	1.52111
$537$	−19.0842	−0.823545
$538$	−32.6344	−1.40697
$539$	24.0726	1.03688
$540$	0	0
$541$	37.4455	1.60991	0.804955	−	0.593336i	$-0.202190\pi$
$541$	37.4455	1.60991	0.804955	+	0.593336i	$0.202190\pi$
$542$	−1.01834	−0.0437412
$543$	16.1609	0.693531
$544$	97.3908	4.17559
$545$	0	0
$546$	−3.44229	−0.147316
$547$	40.9984	1.75297	0.876483	−	0.481433i	$-0.159884\pi$
$547$	40.9984	1.75297	0.876483	+	0.481433i	$0.159884\pi$
$548$	−110.792	−4.73279
$549$	−12.3853	−0.528591
$550$	0	0
$551$	6.49635	0.276754
$552$	−9.35591	−0.398214
$553$	9.43029	0.401017
$554$	39.6346	1.68391
$555$	0	0
$556$	38.5808	1.63619
$557$	34.6422	1.46784	0.733918	−	0.679238i	$-0.237690\pi$
$557$	34.6422	1.46784	0.733918	+	0.679238i	$0.237690\pi$
$558$	−31.0476	−1.31435
$559$	4.56874	0.193237
$560$	0	0
$561$	−27.1428	−1.14597
$562$	−83.1348	−3.50683
$563$	14.9416	0.629712	0.314856	−	0.949139i	$-0.398044\pi$
$563$	14.9416	0.629712	0.314856	+	0.949139i	$0.398044\pi$
$564$	−83.1607	−3.50170
$565$	0	0
$566$	−50.9142	−2.14008
$567$	8.54727	0.358951
$568$	64.0573	2.68779
$569$	−33.8353	−1.41845	−0.709224	−	0.704983i	$-0.750954\pi$
$569$	−33.8353	−1.41845	−0.709224	+	0.704983i	$0.750954\pi$
$570$	0	0
$571$	12.5673	0.525925	0.262962	−	0.964806i	$-0.415300\pi$
$571$	12.5673	0.525925	0.262962	+	0.964806i	$0.415300\pi$
$572$	−13.4259	−0.561364
$573$	30.9452	1.29276
$574$	28.8224	1.20302
$575$	0	0
$576$	147.282	6.13676
$577$	21.5147	0.895668	0.447834	−	0.894117i	$-0.352196\pi$
$577$	21.5147	0.895668	0.447834	+	0.894117i	$0.352196\pi$
$578$	−23.5980	−0.981547
$579$	−21.3062	−0.885457
$580$	0	0
$581$	9.64719	0.400233
$582$	−26.1291	−1.08309
$583$	41.6509	1.72500
$584$	127.829	5.28960
$585$	0	0
$586$	−38.2993	−1.58213
$587$	24.6605	1.01785	0.508924	−	0.860812i	$-0.330044\pi$
$587$	24.6605	1.01785	0.508924	+	0.860812i	$0.330044\pi$
$588$	−88.9551	−3.66844
$589$	−8.43439	−0.347533
$590$	0	0
$591$	38.4808	1.58289
$592$	112.282	4.61477
$593$	25.4243	1.04405	0.522026	−	0.852930i	$-0.325177\pi$
$593$	25.4243	1.04405	0.522026	+	0.852930i	$0.325177\pi$
$594$	7.33712	0.301046
$595$	0	0
$596$	119.832	4.90852
$597$	−30.8891	−1.26421
$598$	−0.574607	−0.0234974
$599$	13.4986	0.551537	0.275768	−	0.961224i	$-0.411068\pi$
$599$	13.4986	0.551537	0.275768	+	0.961224i	$0.411068\pi$
$600$	0	0
$601$	−16.8884	−0.688893	−0.344447	−	0.938806i	$-0.611933\pi$
$601$	−16.8884	−0.688893	−0.344447	+	0.938806i	$0.611933\pi$
$602$	−19.3724	−0.789562
$603$	8.54947	0.348161
$604$	23.5930	0.959986
$605$	0	0
$606$	101.110	4.10731
$607$	−26.6758	−1.08274	−0.541369	−	0.840785i	$-0.682094\pi$
$607$	−26.6758	−1.08274	−0.541369	+	0.840785i	$0.682094\pi$
$608$	69.0680	2.80108
$609$	−6.52299	−0.264325
$610$	0	0
$611$	−3.39559	−0.137371
$612$	47.6814	1.92741
$613$	6.89142	0.278342	0.139171	−	0.990268i	$-0.455556\pi$
$613$	6.89142	0.278342	0.139171	+	0.990268i	$0.455556\pi$
$614$	−20.3676	−0.821971
$615$	0	0
$616$	37.8481	1.52494
$617$	−4.43494	−0.178544	−0.0892719	−	0.996007i	$-0.528454\pi$
$617$	−4.43494	−0.178544	−0.0892719	+	0.996007i	$0.528454\pi$
$618$	−133.841	−5.38387
$619$	7.38111	0.296672	0.148336	−	0.988937i	$-0.452608\pi$
$619$	7.38111	0.296672	0.148336	+	0.988937i	$0.452608\pi$
$620$	0	0
$621$	0.235190	0.00943783
$622$	78.8279	3.16071
$623$	6.55901	0.262781
$624$	27.4107	1.09731
$625$	0	0
$626$	−4.95965	−0.198227
$627$	−19.2492	−0.768741
$628$	−119.390	−4.76417
$629$	16.7760	0.668905
$630$	0	0
$631$	−39.4029	−1.56860	−0.784301	−	0.620380i	$-0.786978\pi$
$631$	−39.4029	−1.56860	−0.784301	+	0.620380i	$0.786978\pi$
$632$	−120.647	−4.79908
$633$	40.3101	1.60218
$634$	60.8525	2.41676
$635$	0	0
$636$	−153.911	−6.10299
$637$	−3.63219	−0.143913
$638$	−33.9686	−1.34483
$639$	15.5512	0.615196
$640$	0	0
$641$	33.5054	1.32339	0.661693	−	0.749775i	$-0.269838\pi$
$641$	33.5054	1.32339	0.661693	+	0.749775i	$0.269838\pi$
$642$	16.7345	0.660459
$643$	10.6479	0.419914	0.209957	−	0.977711i	$-0.432668\pi$
$643$	10.6479	0.419914	0.209957	+	0.977711i	$0.432668\pi$
$644$	1.82483	0.0719085
$645$	0	0
$646$	17.2946	0.680446
$647$	11.2025	0.440417	0.220209	−	0.975453i	$-0.429326\pi$
$647$	11.2025	0.440417	0.220209	+	0.975453i	$0.429326\pi$
$648$	−109.350	−4.29567
$649$	−5.86103	−0.230066
$650$	0	0
$651$	8.46898	0.331925
$652$	−62.6152	−2.45220
$653$	6.39411	0.250221	0.125110	−	0.992143i	$-0.460071\pi$
$653$	6.39411	0.250221	0.125110	+	0.992143i	$0.460071\pi$
$654$	16.6442	0.650839
$655$	0	0
$656$	−229.511	−8.96089
$657$	31.0330	1.21071
$658$	14.3980	0.561294
$659$	4.95824	0.193145	0.0965727	−	0.995326i	$-0.469212\pi$
$659$	4.95824	0.193145	0.0965727	+	0.995326i	$0.469212\pi$
$660$	0	0
$661$	−17.7462	−0.690246	−0.345123	−	0.938557i	$-0.612163\pi$
$661$	−17.7462	−0.690246	−0.345123	+	0.938557i	$0.612163\pi$
$662$	89.2095	3.46722
$663$	4.09543	0.159053
$664$	−123.422	−4.78970
$665$	0	0
$666$	43.7949	1.69702
$667$	−1.08885	−0.0421606
$668$	112.553	4.35479
$669$	−45.7810	−1.77000
$670$	0	0
$671$	17.5930	0.679170
$672$	−69.3512	−2.67528
$673$	30.4347	1.17317	0.586586	−	0.809887i	$-0.300472\pi$
$673$	30.4347	1.17317	0.586586	+	0.809887i	$0.300472\pi$
$674$	12.5040	0.481637
$675$	0	0
$676$	−75.5477	−2.90568
$677$	−8.31341	−0.319510	−0.159755	−	0.987157i	$-0.551070\pi$
$677$	−8.31341	−0.319510	−0.159755	+	0.987157i	$0.551070\pi$
$678$	118.714	4.55919
$679$	3.38824	0.130029
$680$	0	0
$681$	52.6257	2.01662
$682$	44.1024	1.68877
$683$	−19.7934	−0.757373	−0.378687	−	0.925525i	$-0.623624\pi$
$683$	−19.7934	−0.757373	−0.378687	+	0.925525i	$0.623624\pi$
$684$	33.8149	1.29295
$685$	0	0
$686$	32.6952	1.24831
$687$	33.9675	1.29594
$688$	154.262	5.88117
$689$	−6.28447	−0.239419
$690$	0	0
$691$	50.0679	1.90467	0.952337	−	0.305048i	$-0.0986725\pi$
$691$	50.0679	1.90467	0.952337	+	0.305048i	$0.0986725\pi$
$692$	2.06242	0.0784015
$693$	9.18838	0.349038
$694$	−92.6125	−3.51552
$695$	0	0
$696$	83.4522	3.16325
$697$	−34.2911	−1.29887
$698$	−75.5679	−2.86029
$699$	−55.1993	−2.08783
$700$	0	0
$701$	−2.90213	−0.109612	−0.0548059	−	0.998497i	$-0.517454\pi$
$701$	−2.90213	−0.109612	−0.0548059	+	0.998497i	$0.517454\pi$
$702$	−1.10706	−0.0417832
$703$	11.8973	0.448716
$704$	−209.211	−7.88493
$705$	0	0
$706$	−33.2212	−1.25030
$707$	−13.1113	−0.493100
$708$	21.6581	0.813962
$709$	8.06909	0.303041	0.151520	−	0.988454i	$-0.451583\pi$
$709$	8.06909	0.303041	0.151520	+	0.988454i	$0.451583\pi$
$710$	0	0
$711$	−29.2895	−1.09844
$712$	−83.9130	−3.14477
$713$	1.41369	0.0529431
$714$	−17.3655	−0.649887
$715$	0	0
$716$	−47.6215	−1.77970
$717$	6.17394	0.230570
$718$	76.8967	2.86976
$719$	10.4158	0.388443	0.194221	−	0.980958i	$-0.437782\pi$
$719$	10.4158	0.388443	0.194221	+	0.980958i	$0.437782\pi$
$720$	0	0
$721$	17.3556	0.646355
$722$	−41.3648	−1.53944
$723$	2.39134	0.0889349
$724$	40.3269	1.49874
$725$	0	0
$726$	26.4036	0.979931
$727$	−21.2166	−0.786879	−0.393439	−	0.919351i	$-0.628715\pi$
$727$	−21.2166	−0.786879	−0.393439	+	0.919351i	$0.628715\pi$
$728$	−5.71069	−0.211652
$729$	−21.7177	−0.804360
$730$	0	0
$731$	23.0482	0.852468
$732$	−65.0110	−2.40287
$733$	−16.0821	−0.594007	−0.297003	−	0.954876i	$-0.595987\pi$
$733$	−16.0821	−0.594007	−0.297003	+	0.954876i	$0.595987\pi$
$734$	−20.7691	−0.766600
$735$	0	0
$736$	−11.5765	−0.426716
$737$	−12.1443	−0.447341
$738$	−89.5191	−3.29525
$739$	−12.6559	−0.465555	−0.232777	−	0.972530i	$-0.574781\pi$
$739$	−12.6559	−0.465555	−0.232777	+	0.972530i	$0.574781\pi$
$740$	0	0
$741$	2.90441	0.106696
$742$	26.6475	0.978261
$743$	−0.448745	−0.0164629	−0.00823143	−	0.999966i	$-0.502620\pi$
$743$	−0.448745	−0.0164629	−0.00823143	+	0.999966i	$0.502620\pi$
$744$	−108.348	−3.97224
$745$	0	0
$746$	−47.4742	−1.73815
$747$	−29.9631	−1.09629
$748$	−67.7303	−2.47647
$749$	−2.17002	−0.0792908
$750$	0	0
$751$	9.51089	0.347057	0.173529	−	0.984829i	$-0.444483\pi$
$751$	9.51089	0.347057	0.173529	+	0.984829i	$0.444483\pi$
$752$	−114.651	−4.18088
$753$	25.5414	0.930780
$754$	5.12534	0.186654
$755$	0	0
$756$	3.51579	0.127868
$757$	−28.1606	−1.02351	−0.511757	−	0.859130i	$-0.671005\pi$
$757$	−28.1606	−1.02351	−0.511757	+	0.859130i	$0.671005\pi$
$758$	−50.1584	−1.82184
$759$	3.22637	0.117110
$760$	0	0
$761$	8.38239	0.303861	0.151931	−	0.988391i	$-0.451451\pi$
$761$	8.38239	0.303861	0.151931	+	0.988391i	$0.451451\pi$
$762$	−39.3711	−1.42627
$763$	−2.15830	−0.0781359
$764$	77.2187	2.79367
$765$	0	0
$766$	15.6407	0.565123
$767$	0.884339	0.0319316
$768$	325.799	11.7563
$769$	53.2813	1.92137	0.960686	−	0.277637i	$-0.0895512\pi$
$769$	53.2813	1.92137	0.960686	+	0.277637i	$0.0895512\pi$
$770$	0	0
$771$	−11.2024	−0.403446
$772$	−53.1662	−1.91349
$773$	−22.0092	−0.791618	−0.395809	−	0.918333i	$-0.629536\pi$
$773$	−22.0092	−0.791618	−0.395809	+	0.918333i	$0.629536\pi$
$774$	60.1687	2.16272
$775$	0	0
$776$	−43.3477	−1.55609
$777$	−11.9461	−0.428564
$778$	2.21323	0.0793483
$779$	−24.3187	−0.871309
$780$	0	0
$781$	−22.0901	−0.790445
$782$	−2.89875	−0.103659
$783$	−2.09783	−0.0749702
$784$	−122.639	−4.37997
$785$	0	0
$786$	120.111	4.28422
$787$	42.1989	1.50423	0.752115	−	0.659032i	$-0.229034\pi$
$787$	42.1989	1.50423	0.752115	+	0.659032i	$0.229034\pi$
$788$	96.0225	3.42066
$789$	−32.2923	−1.14964
$790$	0	0
$791$	−15.3940	−0.547349
$792$	−117.552	−4.17703
$793$	−2.65451	−0.0942644
$794$	−84.7719	−3.00844
$795$	0	0
$796$	−77.0787	−2.73198
$797$	40.1568	1.42243	0.711214	−	0.702976i	$-0.248146\pi$
$797$	40.1568	1.42243	0.711214	+	0.702976i	$0.248146\pi$
$798$	−12.3153	−0.435958
$799$	−17.1299	−0.606014
$800$	0	0
$801$	−20.3716	−0.719794
$802$	−16.6641	−0.588432
$803$	−44.0816	−1.55561
$804$	44.8766	1.58267
$805$	0	0
$806$	−6.65437	−0.234390
$807$	−27.6480	−0.973257
$808$	167.740	5.90106
$809$	35.1051	1.23423	0.617114	−	0.786874i	$-0.288302\pi$
$809$	35.1051	1.23423	0.617114	+	0.786874i	$0.288302\pi$
$810$	0	0
$811$	28.9245	1.01568	0.507838	−	0.861453i	$-0.330445\pi$
$811$	28.9245	1.01568	0.507838	+	0.861453i	$0.330445\pi$
$812$	−16.2770	−0.571212
$813$	−0.862739	−0.0302576
$814$	−62.2096	−2.18044
$815$	0	0
$816$	138.280	4.84078
$817$	16.3454	0.571854
$818$	23.3025	0.814752
$819$	−1.38638	−0.0484442
$820$	0	0
$821$	−3.08954	−0.107826	−0.0539129	−	0.998546i	$-0.517169\pi$
$821$	−3.08954	−0.107826	−0.0539129	+	0.998546i	$0.517169\pi$
$822$	−125.323	−4.37115
$823$	30.6217	1.06741	0.533703	−	0.845672i	$-0.320800\pi$
$823$	30.6217	1.06741	0.533703	+	0.845672i	$0.320800\pi$
$824$	−222.040	−7.73511
$825$	0	0
$826$	−3.74979	−0.130472
$827$	20.3196	0.706581	0.353290	−	0.935514i	$-0.385063\pi$
$827$	20.3196	0.706581	0.353290	+	0.935514i	$0.385063\pi$
$828$	−5.66773	−0.196967
$829$	20.4430	0.710015	0.355007	−	0.934864i	$-0.384478\pi$
$829$	20.4430	0.710015	0.355007	+	0.934864i	$0.384478\pi$
$830$	0	0
$831$	33.5786	1.16483
$832$	31.5667	1.09438
$833$	−18.3235	−0.634872
$834$	43.6411	1.51117
$835$	0	0
$836$	−48.0333	−1.66127
$837$	2.72367	0.0941438
$838$	−28.8005	−0.994897
$839$	−20.6751	−0.713784	−0.356892	−	0.934146i	$-0.616164\pi$
$839$	−20.6751	−0.713784	−0.356892	+	0.934146i	$0.616164\pi$
$840$	0	0
$841$	−19.2877	−0.665093
$842$	53.8030	1.85417
$843$	−70.4323	−2.42582
$844$	100.587	3.46235
$845$	0	0
$846$	−44.7188	−1.53746
$847$	−3.42385	−0.117645
$848$	−212.192	−7.28672
$849$	−43.1348	−1.48038
$850$	0	0
$851$	−1.99411	−0.0683573
$852$	81.6290	2.79656
$853$	44.7345	1.53168	0.765839	−	0.643032i	$-0.222324\pi$
$853$	44.7345	1.53168	0.765839	+	0.643032i	$0.222324\pi$
$854$	11.2557	0.385162
$855$	0	0
$856$	27.7623	0.948895
$857$	25.2320	0.861908	0.430954	−	0.902374i	$-0.358177\pi$
$857$	25.2320	0.861908	0.430954	+	0.902374i	$0.358177\pi$
$858$	−15.1868	−0.518470
$859$	29.3453	1.00125	0.500624	−	0.865665i	$-0.333104\pi$
$859$	29.3453	1.00125	0.500624	+	0.865665i	$0.333104\pi$
$860$	0	0
$861$	24.4185	0.832179
$862$	13.6116	0.463611
$863$	−53.4748	−1.82030	−0.910152	−	0.414273i	$-0.864036\pi$
$863$	−53.4748	−1.82030	−0.910152	+	0.414273i	$0.864036\pi$
$864$	−22.3037	−0.758788
$865$	0	0
$866$	0.239249	0.00813002
$867$	−19.9923	−0.678975
$868$	21.1329	0.717298
$869$	41.6050	1.41135
$870$	0	0
$871$	1.83239	0.0620881
$872$	27.6124	0.935074
$873$	−10.5235	−0.356167
$874$	−2.05575	−0.0695367
$875$	0	0
$876$	162.894	5.50367
$877$	55.8081	1.88451	0.942253	−	0.334901i	$-0.108703\pi$
$877$	55.8081	1.88451	0.942253	+	0.334901i	$0.108703\pi$
$878$	58.8594	1.98641
$879$	−32.4474	−1.09442
$880$	0	0
$881$	−51.5903	−1.73812	−0.869061	−	0.494705i	$-0.835276\pi$
$881$	−51.5903	−1.73812	−0.869061	+	0.494705i	$0.835276\pi$
$882$	−47.8346	−1.61068
$883$	−15.8447	−0.533216	−0.266608	−	0.963805i	$-0.585903\pi$
$883$	−15.8447	−0.533216	−0.266608	+	0.963805i	$0.585903\pi$
$884$	10.2195	0.343718
$885$	0	0
$886$	106.174	3.56700
$887$	12.1704	0.408641	0.204321	−	0.978904i	$-0.434501\pi$
$887$	12.1704	0.408641	0.204321	+	0.978904i	$0.434501\pi$
$888$	152.833	5.12874
$889$	5.10538	0.171229
$890$	0	0
$891$	37.7092	1.26331
$892$	−114.239	−3.82500
$893$	−12.1483	−0.406527
$894$	135.549	4.53345
$895$	0	0
$896$	−75.8476	−2.53389
$897$	−0.486810	−0.0162541
$898$	−8.20815	−0.273910
$899$	−12.6097	−0.420559
$900$	0	0
$901$	−31.7036	−1.05620
$902$	127.160	4.23396
$903$	−16.4124	−0.546172
$904$	196.945	6.55028
$905$	0	0
$906$	26.6875	0.886632
$907$	−2.07612	−0.0689366	−0.0344683	−	0.999406i	$-0.510974\pi$
$907$	−2.07612	−0.0689366	−0.0344683	+	0.999406i	$0.510974\pi$
$908$	131.319	4.35796
$909$	40.7221	1.35067
$910$	0	0
$911$	36.6227	1.21336	0.606681	−	0.794945i	$-0.292500\pi$
$911$	36.6227	1.21336	0.606681	+	0.794945i	$0.292500\pi$
$912$	98.0662	3.24730
$913$	42.5619	1.40859
$914$	−56.8114	−1.87916
$915$	0	0
$916$	84.7603	2.80056
$917$	−15.5752	−0.514338
$918$	−5.58484	−0.184327
$919$	−3.00350	−0.0990763	−0.0495382	−	0.998772i	$-0.515775\pi$
$919$	−3.00350	−0.0990763	−0.0495382	+	0.998772i	$0.515775\pi$
$920$	0	0
$921$	−17.2556	−0.568591
$922$	−68.0588	−2.24140
$923$	3.33305	0.109709
$924$	48.2303	1.58666
$925$	0	0
$926$	17.0536	0.560414
$927$	−53.9045	−1.77046
$928$	103.259	3.38966
$929$	−16.4969	−0.541245	−0.270623	−	0.962686i	$-0.587230\pi$
$929$	−16.4969	−0.541245	−0.270623	+	0.962686i	$0.587230\pi$
$930$	0	0
$931$	−12.9947	−0.425885
$932$	−137.741	−4.51184
$933$	66.7834	2.18639
$934$	9.53456	0.311980
$935$	0	0
$936$	17.7368	0.579745
$937$	−42.2935	−1.38167	−0.690835	−	0.723013i	$-0.742757\pi$
$937$	−42.2935	−1.38167	−0.690835	+	0.723013i	$0.742757\pi$
$938$	−7.76972	−0.253690
$939$	−4.20184	−0.137122
$940$	0	0
$941$	17.0813	0.556834	0.278417	−	0.960460i	$-0.410190\pi$
$941$	17.0813	0.556834	0.278417	+	0.960460i	$0.410190\pi$
$942$	−135.049	−4.40013
$943$	4.07607	0.132735
$944$	29.8593	0.971838
$945$	0	0
$946$	−85.4682	−2.77881
$947$	−14.3511	−0.466347	−0.233174	−	0.972435i	$-0.574911\pi$
$947$	−14.3511	−0.466347	−0.233174	+	0.972435i	$0.574911\pi$
$948$	−153.742	−4.99330
$949$	6.65124	0.215908
$950$	0	0
$951$	51.5545	1.67177
$952$	−28.8090	−0.933707
$953$	15.9982	0.518233	0.259117	−	0.965846i	$-0.416569\pi$
$953$	15.9982	0.518233	0.259117	+	0.965846i	$0.416569\pi$
$954$	−82.7643	−2.67959
$955$	0	0
$956$	15.4060	0.498267
$957$	−28.7784	−0.930273
$958$	−4.24793	−0.137244
$959$	16.2511	0.524774
$960$	0	0
$961$	−14.6284	−0.471884
$962$	9.38646	0.302632
$963$	6.73984	0.217188
$964$	5.96719	0.192190
$965$	0	0
$966$	2.06418	0.0664138
$967$	−9.15411	−0.294376	−0.147188	−	0.989109i	$-0.547022\pi$
$967$	−9.15411	−0.294376	−0.147188	+	0.989109i	$0.547022\pi$
$968$	43.8032	1.40789
$969$	14.6521	0.470692
$970$	0	0
$971$	46.2181	1.48321	0.741605	−	0.670837i	$-0.234065\pi$
$971$	46.2181	1.48321	0.741605	+	0.670837i	$0.234065\pi$
$972$	−127.296	−4.08301
$973$	−5.65908	−0.181422
$974$	14.2674	0.457157
$975$	0	0
$976$	−89.6284	−2.86893
$977$	10.6150	0.339604	0.169802	−	0.985478i	$-0.445687\pi$
$977$	10.6150	0.339604	0.169802	+	0.985478i	$0.445687\pi$
$978$	−70.8278	−2.26482
$979$	28.9373	0.924840
$980$	0	0
$981$	6.70346	0.214025
$982$	−7.55111	−0.240966
$983$	−7.16851	−0.228640	−0.114320	−	0.993444i	$-0.536469\pi$
$983$	−7.16851	−0.228640	−0.114320	+	0.993444i	$0.536469\pi$
$984$	−312.399	−9.95892
$985$	0	0
$986$	25.8561	0.823425
$987$	12.1981	0.388270
$988$	7.24748	0.230573
$989$	−2.73966	−0.0871161
$990$	0	0
$991$	−53.7983	−1.70896	−0.854480	−	0.519484i	$-0.826124\pi$
$991$	−53.7983	−1.70896	−0.854480	+	0.519484i	$0.826124\pi$
$992$	−134.065	−4.25656
$993$	75.5788	2.39842
$994$	−14.1329	−0.448267
$995$	0	0
$996$	−157.278	−4.98354
$997$	−4.14619	−0.131311	−0.0656556	−	0.997842i	$-0.520914\pi$
$997$	−4.14619	−0.131311	−0.0656556	+	0.997842i	$0.520914\pi$
$998$	−2.38248	−0.0754160
$999$	−3.84193	−0.121553

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.k.1.25		25
5.4	even	2		1205.2.a.d.1.1	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.a.d.1.1	✓	25	5.4	even	2
6025.2.a.k.1.25		25	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.82262 q^{2} +2.39134 q^{3} +5.96719 q^{4} +6.74985 q^{6} -0.875274 q^{7} +11.1979 q^{8} +2.71851 q^{9} +O(q^{10})\)	\(q+2.82262 q^{2} +2.39134 q^{3} +5.96719 q^{4} +6.74985 q^{6} -0.875274 q^{7} +11.1979 q^{8} +2.71851 q^{9} -3.86157 q^{11} +14.2696 q^{12} +0.582652 q^{13} -2.47057 q^{14} +19.6730 q^{16} +2.93934 q^{17} +7.67332 q^{18} +2.08453 q^{19} -2.09308 q^{21} -10.8998 q^{22} -0.349389 q^{23} +26.7779 q^{24} +1.64460 q^{26} -0.673145 q^{27} -5.22293 q^{28} +3.11646 q^{29} -4.04618 q^{31} +33.1336 q^{32} -9.23433 q^{33} +8.29663 q^{34} +16.2218 q^{36} +5.70743 q^{37} +5.88384 q^{38} +1.39332 q^{39} -11.6663 q^{41} -5.90797 q^{42} +7.84129 q^{43} -23.0427 q^{44} -0.986192 q^{46} -5.82783 q^{47} +47.0448 q^{48} -6.23389 q^{49} +7.02895 q^{51} +3.47679 q^{52} -10.7860 q^{53} -1.90003 q^{54} -9.80121 q^{56} +4.98482 q^{57} +8.79657 q^{58} +1.51778 q^{59} -4.55591 q^{61} -11.4208 q^{62} -2.37944 q^{63} +54.1777 q^{64} -26.0650 q^{66} +3.14491 q^{67} +17.5396 q^{68} -0.835508 q^{69} +5.72049 q^{71} +30.4415 q^{72} +11.4155 q^{73} +16.1099 q^{74} +12.4388 q^{76} +3.37994 q^{77} +3.93281 q^{78} -10.7741 q^{79} -9.76524 q^{81} -32.9295 q^{82} -11.0219 q^{83} -12.4898 q^{84} +22.1330 q^{86} +7.45250 q^{87} -43.2414 q^{88} -7.49366 q^{89} -0.509980 q^{91} -2.08487 q^{92} -9.67580 q^{93} -16.4498 q^{94} +79.2337 q^{96} -3.87107 q^{97} -17.5959 q^{98} -10.4977 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9}+O(q^{10}) \) 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9	\( 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9} + 10 q^{11} - 22 q^{12} - 10 q^{13} + 13 q^{14} + 54 q^{16} - q^{17} + 13 q^{18} + 50 q^{19} + 9 q^{21} - 11 q^{22} + 31 q^{23} + 22 q^{24} + 8 q^{26} - 42 q^{27} - 14 q^{28} + 4 q^{29} + 34 q^{31} + 44 q^{32} - 28 q^{33} + 33 q^{34} + 83 q^{36} - 14 q^{37} + 10 q^{38} + 23 q^{39} + 11 q^{41} - 23 q^{42} - 49 q^{43} + 20 q^{44} + 27 q^{46} + 28 q^{47} - 30 q^{48} + 66 q^{49} + 49 q^{51} - 39 q^{52} + 16 q^{53} + 5 q^{54} + 51 q^{56} - 10 q^{57} + 8 q^{58} + 30 q^{59} + 35 q^{61} + 18 q^{62} + 73 q^{64} - 13 q^{66} - 37 q^{67} - 11 q^{68} - 4 q^{69} + 12 q^{71} + 90 q^{72} - 36 q^{73} - 12 q^{74} + 57 q^{76} + 31 q^{77} + 9 q^{78} + 16 q^{79} + 65 q^{81} + 11 q^{82} - 43 q^{83} - 62 q^{84} - 9 q^{86} + 22 q^{87} - 20 q^{88} + 38 q^{89} + 86 q^{91} + 119 q^{92} - 10 q^{93} - 18 q^{94} - 34 q^{96} - 17 q^{97} + 32 q^{98} + 26 q^{99}+O(q^{100}) \) 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9 + 10 * q^11 - 22 * q^12 - 10 * q^13 + 13 * q^14 + 54 * q^16 - q^17 + 13 * q^18 + 50 * q^19 + 9 * q^21 - 11 * q^22 + 31 * q^23 + 22 * q^24 + 8 * q^26 - 42 * q^27 - 14 * q^28 + 4 * q^29 + 34 * q^31 + 44 * q^32 - 28 * q^33 + 33 * q^34 + 83 * q^36 - 14 * q^37 + 10 * q^38 + 23 * q^39 + 11 * q^41 - 23 * q^42 - 49 * q^43 + 20 * q^44 + 27 * q^46 + 28 * q^47 - 30 * q^48 + 66 * q^49 + 49 * q^51 - 39 * q^52 + 16 * q^53 + 5 * q^54 + 51 * q^56 - 10 * q^57 + 8 * q^58 + 30 * q^59 + 35 * q^61 + 18 * q^62 + 73 * q^64 - 13 * q^66 - 37 * q^67 - 11 * q^68 - 4 * q^69 + 12 * q^71 + 90 * q^72 - 36 * q^73 - 12 * q^74 + 57 * q^76 + 31 * q^77 + 9 * q^78 + 16 * q^79 + 65 * q^81 + 11 * q^82 - 43 * q^83 - 62 * q^84 - 9 * q^86 + 22 * q^87 - 20 * q^88 + 38 * q^89 + 86 * q^91 + 119 * q^92 - 10 * q^93 - 18 * q^94 - 34 * q^96 - 17 * q^97 + 32 * q^98 + 26 * q^99

Introduction

L-functions

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