LMFDB - Embedded newform 6025.2.a.n.1.26

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

Embedding invariants

Embedding label			1.26
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)

$f(q)$	$=$	$q+1.00174 q^{2} -2.95041 q^{3} -0.996521 q^{4} -2.95554 q^{6} +1.49397 q^{7} -3.00173 q^{8} +5.70493 q^{9} +O(q^{10})$	\(q+1.00174 q^{2} -2.95041 q^{3} -0.996521 q^{4} -2.95554 q^{6} +1.49397 q^{7} -3.00173 q^{8} +5.70493 q^{9} +0.117623 q^{11} +2.94015 q^{12} +5.56865 q^{13} +1.49657 q^{14} -1.01391 q^{16} +3.01555 q^{17} +5.71484 q^{18} +4.81620 q^{19} -4.40783 q^{21} +0.117828 q^{22} +8.22107 q^{23} +8.85634 q^{24} +5.57833 q^{26} -7.98065 q^{27} -1.48877 q^{28} +6.36189 q^{29} -0.882496 q^{31} +4.98779 q^{32} -0.347038 q^{33} +3.02079 q^{34} -5.68508 q^{36} -6.53654 q^{37} +4.82457 q^{38} -16.4298 q^{39} -7.03844 q^{41} -4.41549 q^{42} -12.9858 q^{43} -0.117214 q^{44} +8.23536 q^{46} -6.49192 q^{47} +2.99144 q^{48} -4.76805 q^{49} -8.89710 q^{51} -5.54928 q^{52} +9.80086 q^{53} -7.99452 q^{54} -4.48450 q^{56} -14.2098 q^{57} +6.37295 q^{58} +10.1833 q^{59} -2.66070 q^{61} -0.884030 q^{62} +8.52300 q^{63} +7.02427 q^{64} -0.347641 q^{66} +12.4990 q^{67} -3.00505 q^{68} -24.2555 q^{69} -0.718468 q^{71} -17.1246 q^{72} +15.3038 q^{73} -6.54790 q^{74} -4.79944 q^{76} +0.175726 q^{77} -16.4584 q^{78} +9.25785 q^{79} +6.43141 q^{81} -7.05067 q^{82} +2.92059 q^{83} +4.39249 q^{84} -13.0084 q^{86} -18.7702 q^{87} -0.353074 q^{88} -16.1812 q^{89} +8.31941 q^{91} -8.19247 q^{92} +2.60373 q^{93} -6.50320 q^{94} -14.7160 q^{96} -0.213462 q^{97} -4.77634 q^{98} +0.671033 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9	$ 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9} + q^{11} + 26 q^{12} + 11 q^{13} - q^{14} + 43 q^{16} + 20 q^{17} + 18 q^{18} + 2 q^{21} + 23 q^{22} + 79 q^{23} - 2 q^{24} + 2 q^{26} + 26 q^{27} + 30 q^{28} + 2 q^{29} + q^{31} + 68 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 16 q^{37} + 45 q^{38} - 2 q^{39} - 2 q^{41} + 19 q^{42} + 25 q^{43} + 3 q^{44} + 14 q^{46} + 88 q^{47} + 75 q^{48} + 40 q^{49} - 10 q^{51} + 18 q^{52} + 34 q^{53} + 4 q^{54} - 15 q^{56} + 51 q^{57} + 53 q^{58} + q^{59} + 9 q^{61} + 39 q^{62} + 110 q^{63} + 17 q^{64} + 26 q^{66} + 30 q^{67} + 44 q^{68} - 7 q^{69} + 5 q^{71} + 18 q^{72} + 23 q^{73} - 18 q^{74} + 43 q^{76} + 30 q^{77} + 46 q^{78} + 5 q^{79} + 44 q^{81} + 5 q^{82} + 65 q^{83} - 65 q^{84} + 40 q^{86} + 33 q^{87} + 71 q^{88} - 9 q^{89} + q^{91} + 117 q^{92} + 68 q^{93} - 72 q^{94} + 83 q^{96} - 8 q^{97} + 76 q^{98} - 40 q^{99}+O(q^{100}) $ 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9 + q^11 + 26 * q^12 + 11 * q^13 - q^14 + 43 * q^16 + 20 * q^17 + 18 * q^18 + 2 * q^21 + 23 * q^22 + 79 * q^23 - 2 * q^24 + 2 * q^26 + 26 * q^27 + 30 * q^28 + 2 * q^29 + q^31 + 68 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 16 * q^37 + 45 * q^38 - 2 * q^39 - 2 * q^41 + 19 * q^42 + 25 * q^43 + 3 * q^44 + 14 * q^46 + 88 * q^47 + 75 * q^48 + 40 * q^49 - 10 * q^51 + 18 * q^52 + 34 * q^53 + 4 * q^54 - 15 * q^56 + 51 * q^57 + 53 * q^58 + q^59 + 9 * q^61 + 39 * q^62 + 110 * q^63 + 17 * q^64 + 26 * q^66 + 30 * q^67 + 44 * q^68 - 7 * q^69 + 5 * q^71 + 18 * q^72 + 23 * q^73 - 18 * q^74 + 43 * q^76 + 30 * q^77 + 46 * q^78 + 5 * q^79 + 44 * q^81 + 5 * q^82 + 65 * q^83 - 65 * q^84 + 40 * q^86 + 33 * q^87 + 71 * q^88 - 9 * q^89 + q^91 + 117 * q^92 + 68 * q^93 - 72 * q^94 + 83 * q^96 - 8 * q^97 + 76 * q^98 - 40 * 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 $

$2$	1.00174	0.708336	0.354168	−	0.935182i	$-0.384764\pi$
$2$	1.00174	0.708336	0.354168	+	0.935182i	$0.384764\pi$
$3$	−2.95041	−1.70342	−0.851710	−	0.524013i	$-0.824434\pi$
$3$	−2.95041	−1.70342	−0.851710	+	0.524013i	$0.824434\pi$
$4$	−0.996521	−0.498260
$5$	0	0
$5$	0	0
$6$	−2.95554	−1.20659
$7$	1.49397	0.564668	0.282334	−	0.959316i	$-0.408891\pi$
$7$	1.49397	0.564668	0.282334	+	0.959316i	$0.408891\pi$
$8$	−3.00173	−1.06127
$9$	5.70493	1.90164
$10$	0	0
$11$	0.117623	0.0354648	0.0177324	−	0.999843i	$-0.494355\pi$
$11$	0.117623	0.0354648	0.0177324	+	0.999843i	$0.494355\pi$
$12$	2.94015	0.848747
$13$	5.56865	1.54447	0.772233	−	0.635339i	$-0.219140\pi$
$13$	5.56865	1.54447	0.772233	+	0.635339i	$0.219140\pi$
$14$	1.49657	0.399975
$15$	0	0
$16$	−1.01391	−0.253476
$17$	3.01555	0.731377	0.365689	−	0.930737i	$-0.380834\pi$
$17$	3.01555	0.731377	0.365689	+	0.930737i	$0.380834\pi$
$18$	5.71484	1.34700
$19$	4.81620	1.10491	0.552456	−	0.833542i	$-0.313691\pi$
$19$	4.81620	1.10491	0.552456	+	0.833542i	$0.313691\pi$
$20$	0	0
$21$	−4.40783	−0.961867
$22$	0.117828	0.0251210
$23$	8.22107	1.71421	0.857106	−	0.515140i	$-0.172260\pi$
$23$	8.22107	1.71421	0.857106	+	0.515140i	$0.172260\pi$
$24$	8.85634	1.80779
$25$	0	0
$26$	5.57833	1.09400
$27$	−7.98065	−1.53588
$28$	−1.48877	−0.281352
$29$	6.36189	1.18137	0.590687	−	0.806901i	$-0.298857\pi$
$29$	6.36189	1.18137	0.590687	+	0.806901i	$0.298857\pi$
$30$	0	0
$31$	−0.882496	−0.158501	−0.0792505	−	0.996855i	$-0.525253\pi$
$31$	−0.882496	−0.158501	−0.0792505	+	0.996855i	$0.525253\pi$
$32$	4.98779	0.881725
$33$	−0.347038	−0.0604115
$34$	3.02079	0.518061
$35$	0	0
$36$	−5.68508	−0.947513
$37$	−6.53654	−1.07460	−0.537300	−	0.843391i	$-0.680556\pi$
$37$	−6.53654	−1.07460	−0.537300	+	0.843391i	$0.680556\pi$
$38$	4.82457	0.782649
$39$	−16.4298	−2.63088
$40$	0	0
$41$	−7.03844	−1.09922	−0.549610	−	0.835422i	$-0.685224\pi$
$41$	−7.03844	−1.09922	−0.549610	+	0.835422i	$0.685224\pi$
$42$	−4.41549	−0.681325
$43$	−12.9858	−1.98031	−0.990157	−	0.139961i	$-0.955302\pi$
$43$	−12.9858	−1.98031	−0.990157	+	0.139961i	$0.955302\pi$
$44$	−0.117214	−0.0176707
$45$	0	0
$46$	8.23536	1.21424
$47$	−6.49192	−0.946944	−0.473472	−	0.880809i	$-0.656999\pi$
$47$	−6.49192	−0.946944	−0.473472	+	0.880809i	$0.656999\pi$
$48$	2.99144	0.431777
$49$	−4.76805	−0.681150
$50$	0	0
$51$	−8.89710	−1.24584
$52$	−5.54928	−0.769546
$53$	9.80086	1.34625	0.673126	−	0.739528i	$-0.264951\pi$
$53$	9.80086	1.34625	0.673126	+	0.739528i	$0.264951\pi$
$54$	−7.99452	−1.08792
$55$	0	0
$56$	−4.48450	−0.599266
$57$	−14.2098	−1.88213
$58$	6.37295	0.836809
$59$	10.1833	1.32575	0.662874	−	0.748731i	$-0.269337\pi$
$59$	10.1833	1.32575	0.662874	+	0.748731i	$0.269337\pi$
$60$	0	0
$61$	−2.66070	−0.340668	−0.170334	−	0.985386i	$-0.554485\pi$
$61$	−2.66070	−0.340668	−0.170334	+	0.985386i	$0.554485\pi$
$62$	−0.884030	−0.112272
$63$	8.52300	1.07380
$64$	7.02427	0.878034
$65$	0	0
$66$	−0.347641	−0.0427916
$67$	12.4990	1.52699	0.763495	−	0.645813i	$-0.223482\pi$
$67$	12.4990	1.52699	0.763495	+	0.645813i	$0.223482\pi$
$68$	−3.00505	−0.364416
$69$	−24.2555	−2.92002
$70$	0	0
$71$	−0.718468	−0.0852665	−0.0426332	−	0.999091i	$-0.513575\pi$
$71$	−0.718468	−0.0852665	−0.0426332	+	0.999091i	$0.513575\pi$
$72$	−17.1246	−2.01816
$73$	15.3038	1.79118	0.895589	−	0.444881i	$-0.146754\pi$
$73$	15.3038	1.79118	0.895589	+	0.444881i	$0.146754\pi$
$74$	−6.54790	−0.761178
$75$	0	0
$76$	−4.79944	−0.550534
$77$	0.175726	0.0200258
$78$	−16.4584	−1.86354
$79$	9.25785	1.04159	0.520795	−	0.853682i	$-0.325636\pi$
$79$	9.25785	1.04159	0.520795	+	0.853682i	$0.325636\pi$
$80$	0	0
$81$	6.43141	0.714602
$82$	−7.05067	−0.778616
$83$	2.92059	0.320577	0.160288	−	0.987070i	$-0.448758\pi$
$83$	2.92059	0.320577	0.160288	+	0.987070i	$0.448758\pi$
$84$	4.39249	0.479260
$85$	0	0
$86$	−13.0084	−1.40273
$87$	−18.7702	−2.01238
$88$	−0.353074	−0.0376378
$89$	−16.1812	−1.71520	−0.857600	−	0.514318i	$-0.828045\pi$
$89$	−16.1812	−1.71520	−0.857600	+	0.514318i	$0.828045\pi$
$90$	0	0
$91$	8.31941	0.872111
$92$	−8.19247	−0.854124
$93$	2.60373	0.269994
$94$	−6.50320	−0.670754
$95$	0	0
$96$	−14.7160	−1.50195
$97$	−0.213462	−0.0216737	−0.0108369	−	0.999941i	$-0.503450\pi$
$97$	−0.213462	−0.0216737	−0.0108369	+	0.999941i	$0.503450\pi$
$98$	−4.77634	−0.482483
$99$	0.671033	0.0674414
$100$	0	0
$101$	−15.4473	−1.53707	−0.768534	−	0.639809i	$-0.779014\pi$
$101$	−15.4473	−1.53707	−0.768534	+	0.639809i	$0.779014\pi$
$102$	−8.91256	−0.882475
$103$	15.0505	1.48297	0.741486	−	0.670968i	$-0.234121\pi$
$103$	15.0505	1.48297	0.741486	+	0.670968i	$0.234121\pi$
$104$	−16.7156	−1.63910
$105$	0	0
$106$	9.81790	0.953598
$107$	6.78921	0.656338	0.328169	−	0.944619i	$-0.393568\pi$
$107$	6.78921	0.656338	0.328169	+	0.944619i	$0.393568\pi$
$108$	7.95288	0.765266
$109$	−16.0852	−1.54068	−0.770342	−	0.637631i	$-0.779914\pi$
$109$	−16.0852	−1.54068	−0.770342	+	0.637631i	$0.779914\pi$
$110$	0	0
$111$	19.2855	1.83050
$112$	−1.51475	−0.143130
$113$	0.120165	0.0113042	0.00565208	−	0.999984i	$-0.498201\pi$
$113$	0.120165	0.0113042	0.00565208	+	0.999984i	$0.498201\pi$
$114$	−14.2345	−1.33318
$115$	0	0
$116$	−6.33976	−0.588632
$117$	31.7688	2.93702
$118$	10.2010	0.939075
$119$	4.50514	0.412985
$120$	0	0
$121$	−10.9862	−0.998742
$122$	−2.66533	−0.241308
$123$	20.7663	1.87243
$124$	0.879426	0.0789748
$125$	0	0
$126$	8.53781	0.760609
$127$	13.8723	1.23097	0.615486	−	0.788148i	$-0.288960\pi$
$127$	13.8723	1.23097	0.615486	+	0.788148i	$0.288960\pi$
$128$	−2.93910	−0.259782
$129$	38.3134	3.37331
$130$	0	0
$131$	−0.573929	−0.0501444	−0.0250722	−	0.999686i	$-0.507982\pi$
$131$	−0.573929	−0.0501444	−0.0250722	+	0.999686i	$0.507982\pi$
$132$	0.345830	0.0301006
$133$	7.19526	0.623909
$134$	12.5207	1.08162
$135$	0	0
$136$	−9.05185	−0.776190
$137$	−2.98381	−0.254924	−0.127462	−	0.991843i	$-0.540683\pi$
$137$	−2.98381	−0.254924	−0.127462	+	0.991843i	$0.540683\pi$
$138$	−24.2977	−2.06836
$139$	19.6360	1.66550	0.832751	−	0.553648i	$-0.186765\pi$
$139$	19.6360	1.66550	0.832751	+	0.553648i	$0.186765\pi$
$140$	0	0
$141$	19.1538	1.61304
$142$	−0.719717	−0.0603973
$143$	0.655004	0.0547742
$144$	−5.78426	−0.482021
$145$	0	0
$146$	15.3304	1.26876
$147$	14.0677	1.16029
$148$	6.51380	0.535431
$149$	−6.59757	−0.540494	−0.270247	−	0.962791i	$-0.587105\pi$
$149$	−6.59757	−0.540494	−0.270247	+	0.962791i	$0.587105\pi$
$150$	0	0
$151$	−8.13493	−0.662011	−0.331005	−	0.943629i	$-0.607388\pi$
$151$	−8.13493	−0.662011	−0.331005	+	0.943629i	$0.607388\pi$
$152$	−14.4569	−1.17261
$153$	17.2035	1.39082
$154$	0.176031	0.0141850
$155$	0	0
$156$	16.3727	1.31086
$157$	17.3907	1.38793	0.693965	−	0.720008i	$-0.255862\pi$
$157$	17.3907	1.38793	0.693965	+	0.720008i	$0.255862\pi$
$158$	9.27394	0.737795
$159$	−28.9166	−2.29323
$160$	0	0
$161$	12.2820	0.967960
$162$	6.44259	0.506178
$163$	−21.7370	−1.70257	−0.851286	−	0.524702i	$-0.824177\pi$
$163$	−21.7370	−1.70257	−0.851286	+	0.524702i	$0.824177\pi$
$164$	7.01395	0.547697
$165$	0	0
$166$	2.92567	0.227076
$167$	−0.517692	−0.0400602	−0.0200301	−	0.999799i	$-0.506376\pi$
$167$	−0.517692	−0.0400602	−0.0200301	+	0.999799i	$0.506376\pi$
$168$	13.2311	1.02080
$169$	18.0099	1.38538
$170$	0	0
$171$	27.4761	2.10115
$172$	12.9406	0.986712
$173$	−10.9319	−0.831139	−0.415569	−	0.909562i	$-0.636418\pi$
$173$	−10.9319	−0.831139	−0.415569	+	0.909562i	$0.636418\pi$
$174$	−18.8028	−1.42544
$175$	0	0
$176$	−0.119259	−0.00898949
$177$	−30.0448	−2.25831
$178$	−16.2093	−1.21494
$179$	11.5026	0.859745	0.429873	−	0.902890i	$-0.358558\pi$
$179$	11.5026	0.859745	0.429873	+	0.902890i	$0.358558\pi$
$180$	0	0
$181$	22.9491	1.70580	0.852898	−	0.522078i	$-0.174843\pi$
$181$	22.9491	1.70580	0.852898	+	0.522078i	$0.174843\pi$
$182$	8.33387	0.617747
$183$	7.85017	0.580302
$184$	−24.6774	−1.81924
$185$	0	0
$186$	2.60825	0.191246
$187$	0.354699	0.0259381
$188$	6.46933	0.471824
$189$	−11.9229	−0.867260
$190$	0	0
$191$	−1.30038	−0.0940922	−0.0470461	−	0.998893i	$-0.514981\pi$
$191$	−1.30038	−0.0940922	−0.0470461	+	0.998893i	$0.514981\pi$
$192$	−20.7245	−1.49566
$193$	1.26277	0.0908959	0.0454479	−	0.998967i	$-0.485528\pi$
$193$	1.26277	0.0908959	0.0454479	+	0.998967i	$0.485528\pi$
$194$	−0.213833	−0.0153523
$195$	0	0
$196$	4.75146	0.339390
$197$	−6.53924	−0.465901	−0.232951	−	0.972489i	$-0.574838\pi$
$197$	−6.53924	−0.465901	−0.232951	+	0.972489i	$0.574838\pi$
$198$	0.672200	0.0477711
$199$	2.48860	0.176412	0.0882062	−	0.996102i	$-0.471887\pi$
$199$	2.48860	0.176412	0.0882062	+	0.996102i	$0.471887\pi$
$200$	0	0
$201$	−36.8771	−2.60111
$202$	−15.4742	−1.08876
$203$	9.50448	0.667084
$204$	8.86614	0.620754
$205$	0	0
$206$	15.0767	1.05044
$207$	46.9006	3.25982
$208$	−5.64609	−0.391486
$209$	0.566498	0.0391855
$210$	0	0
$211$	−27.4526	−1.88992	−0.944959	−	0.327189i	$-0.893899\pi$
$211$	−27.4526	−1.88992	−0.944959	+	0.327189i	$0.893899\pi$
$212$	−9.76676	−0.670784
$213$	2.11978	0.145245
$214$	6.80101	0.464908
$215$	0	0
$216$	23.9557	1.62998
$217$	−1.31842	−0.0895004
$218$	−16.1132	−1.09132
$219$	−45.1526	−3.05113
$220$	0	0
$221$	16.7925	1.12959
$222$	19.3190	1.29661
$223$	11.3598	0.760708	0.380354	−	0.924841i	$-0.375802\pi$
$223$	11.3598	0.760708	0.380354	+	0.924841i	$0.375802\pi$
$224$	7.45161	0.497882
$225$	0	0
$226$	0.120374	0.00800715
$227$	−0.641141	−0.0425540	−0.0212770	−	0.999774i	$-0.506773\pi$
$227$	−0.641141	−0.0425540	−0.0212770	+	0.999774i	$0.506773\pi$
$228$	14.1603	0.937791
$229$	−19.6284	−1.29708	−0.648542	−	0.761179i	$-0.724621\pi$
$229$	−19.6284	−1.29708	−0.648542	+	0.761179i	$0.724621\pi$
$230$	0	0
$231$	−0.518464	−0.0341124
$232$	−19.0967	−1.25376
$233$	22.7056	1.48750	0.743748	−	0.668461i	$-0.233046\pi$
$233$	22.7056	1.48750	0.743748	+	0.668461i	$0.233046\pi$
$234$	31.8240	2.08040
$235$	0	0
$236$	−10.1478	−0.660567
$237$	−27.3145	−1.77426
$238$	4.51297	0.292532
$239$	27.5965	1.78507	0.892533	−	0.450982i	$-0.148926\pi$
$239$	27.5965	1.78507	0.892533	+	0.450982i	$0.148926\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−11.0053	−0.707445
$243$	4.96663	0.318609
$244$	2.65145	0.169741
$245$	0	0
$246$	20.8024	1.32631
$247$	26.8198	1.70650
$248$	2.64901	0.168213
$249$	−8.61695	−0.546077
$250$	0	0
$251$	−1.99620	−0.125999	−0.0629996	−	0.998014i	$-0.520067\pi$
$251$	−1.99620	−0.125999	−0.0629996	+	0.998014i	$0.520067\pi$
$252$	−8.49334	−0.535030
$253$	0.966991	0.0607942
$254$	13.8965	0.871941
$255$	0	0
$256$	−16.9928	−1.06205
$257$	−9.12665	−0.569305	−0.284652	−	0.958631i	$-0.591878\pi$
$257$	−9.12665	−0.569305	−0.284652	+	0.958631i	$0.591878\pi$
$258$	38.3800	2.38944
$259$	−9.76540	−0.606793
$260$	0	0
$261$	36.2941	2.24655
$262$	−0.574927	−0.0355191
$263$	4.14621	0.255667	0.127833	−	0.991796i	$-0.459198\pi$
$263$	4.14621	0.255667	0.127833	+	0.991796i	$0.459198\pi$
$264$	1.04171	0.0641130
$265$	0	0
$266$	7.20777	0.441937
$267$	47.7411	2.92171
$268$	−12.4555	−0.760839
$269$	−6.29978	−0.384105	−0.192052	−	0.981385i	$-0.561514\pi$
$269$	−6.29978	−0.384105	−0.192052	+	0.981385i	$0.561514\pi$
$270$	0	0
$271$	23.4880	1.42680	0.713398	−	0.700759i	$-0.247155\pi$
$271$	23.4880	1.42680	0.713398	+	0.700759i	$0.247155\pi$
$272$	−3.05748	−0.185387
$273$	−24.5457	−1.48557
$274$	−2.98899	−0.180572
$275$	0	0
$276$	24.1711	1.45493
$277$	−12.3279	−0.740709	−0.370355	−	0.928890i	$-0.620764\pi$
$277$	−12.3279	−0.740709	−0.370355	+	0.928890i	$0.620764\pi$
$278$	19.6701	1.17973
$279$	−5.03458	−0.301412
$280$	0	0
$281$	−2.70463	−0.161345	−0.0806723	−	0.996741i	$-0.525707\pi$
$281$	−2.70463	−0.161345	−0.0806723	+	0.996741i	$0.525707\pi$
$282$	19.1871	1.14258
$283$	−27.4623	−1.63247	−0.816233	−	0.577723i	$-0.803941\pi$
$283$	−27.4623	−1.63247	−0.816233	+	0.577723i	$0.803941\pi$
$284$	0.715968	0.0424849
$285$	0	0
$286$	0.656143	0.0387985
$287$	−10.5152	−0.620694
$288$	28.4550	1.67673
$289$	−7.90649	−0.465087
$290$	0	0
$291$	0.629800	0.0369195
$292$	−15.2506	−0.892473
$293$	8.31232	0.485611	0.242805	−	0.970075i	$-0.421932\pi$
$293$	8.31232	0.485611	0.242805	+	0.970075i	$0.421932\pi$
$294$	14.0922	0.821872
$295$	0	0
$296$	19.6209	1.14044
$297$	−0.938711	−0.0544696
$298$	−6.60904	−0.382852
$299$	45.7803	2.64754
$300$	0	0
$301$	−19.4004	−1.11822
$302$	−8.14907	−0.468926
$303$	45.5760	2.61827
$304$	−4.88317	−0.280069
$305$	0	0
$306$	17.2334	0.985166
$307$	6.78168	0.387051	0.193525	−	0.981095i	$-0.438008\pi$
$307$	6.78168	0.387051	0.193525	+	0.981095i	$0.438008\pi$
$308$	−0.175115	−0.00997808
$309$	−44.4053	−2.52613
$310$	0	0
$311$	−26.4595	−1.50038	−0.750191	−	0.661221i	$-0.770039\pi$
$311$	−26.4595	−1.50038	−0.750191	+	0.661221i	$0.770039\pi$
$312$	49.3179	2.79207
$313$	−10.2693	−0.580453	−0.290226	−	0.956958i	$-0.593731\pi$
$313$	−10.2693	−0.580453	−0.290226	+	0.956958i	$0.593731\pi$
$314$	17.4210	0.983121
$315$	0	0
$316$	−9.22563	−0.518982
$317$	19.0759	1.07141	0.535704	−	0.844406i	$-0.320046\pi$
$317$	19.0759	1.07141	0.535704	+	0.844406i	$0.320046\pi$
$318$	−28.9668	−1.62438
$319$	0.748308	0.0418972
$320$	0	0
$321$	−20.0310	−1.11802
$322$	12.3034	0.685641
$323$	14.5235	0.808108
$324$	−6.40904	−0.356058
$325$	0	0
$326$	−21.7748	−1.20599
$327$	47.4580	2.62443
$328$	21.1275	1.16657
$329$	−9.69874	−0.534709
$330$	0	0
$331$	−10.1361	−0.557131	−0.278565	−	0.960417i	$-0.589859\pi$
$331$	−10.1361	−0.557131	−0.278565	+	0.960417i	$0.589859\pi$
$332$	−2.91043	−0.159731
$333$	−37.2905	−2.04351
$334$	−0.518592	−0.0283761
$335$	0	0
$336$	4.46912	0.243811
$337$	24.5351	1.33651	0.668257	−	0.743930i	$-0.267041\pi$
$337$	24.5351	1.33651	0.668257	+	0.743930i	$0.267041\pi$
$338$	18.0412	0.981313
$339$	−0.354536	−0.0192558
$340$	0	0
$341$	−0.103802	−0.00562121
$342$	27.5238	1.48832
$343$	−17.5811	−0.949292
$344$	38.9798	2.10165
$345$	0	0
$346$	−10.9509	−0.588725
$347$	15.0724	0.809127	0.404564	−	0.914510i	$-0.367423\pi$
$347$	15.0724	0.809127	0.404564	+	0.914510i	$0.367423\pi$
$348$	18.7049	1.00269
$349$	−0.641561	−0.0343420	−0.0171710	−	0.999853i	$-0.505466\pi$
$349$	−0.641561	−0.0343420	−0.0171710	+	0.999853i	$0.505466\pi$
$350$	0	0
$351$	−44.4415	−2.37211
$352$	0.586681	0.0312702
$353$	4.72402	0.251434	0.125717	−	0.992066i	$-0.459877\pi$
$353$	4.72402	0.251434	0.125717	+	0.992066i	$0.459877\pi$
$354$	−30.0970	−1.59964
$355$	0	0
$356$	16.1249	0.854616
$357$	−13.2920	−0.703488
$358$	11.5226	0.608988
$359$	21.5749	1.13868	0.569340	−	0.822102i	$-0.307199\pi$
$359$	21.5749	1.13868	0.569340	+	0.822102i	$0.307199\pi$
$360$	0	0
$361$	4.19579	0.220831
$362$	22.9890	1.20828
$363$	32.4137	1.70128
$364$	−8.29046	−0.434538
$365$	0	0
$366$	7.86382	0.411048
$367$	16.6949	0.871465	0.435733	−	0.900076i	$-0.356489\pi$
$367$	16.6949	0.871465	0.435733	+	0.900076i	$0.356489\pi$
$368$	−8.33539	−0.434512
$369$	−40.1538	−2.09032
$370$	0	0
$371$	14.6422	0.760185
$372$	−2.59467	−0.134527
$373$	18.5092	0.958369	0.479185	−	0.877714i	$-0.340932\pi$
$373$	18.5092	0.958369	0.479185	+	0.877714i	$0.340932\pi$
$374$	0.355315	0.0183729
$375$	0	0
$376$	19.4870	1.00496
$377$	35.4272	1.82459
$378$	−11.9436	−0.614312
$379$	21.0734	1.08247	0.541233	−	0.840872i	$-0.317958\pi$
$379$	21.0734	1.08247	0.541233	+	0.840872i	$0.317958\pi$
$380$	0	0
$381$	−40.9291	−2.09686
$382$	−1.30264	−0.0666489
$383$	−5.20993	−0.266215	−0.133108	−	0.991102i	$-0.542496\pi$
$383$	−5.20993	−0.266215	−0.133108	+	0.991102i	$0.542496\pi$
$384$	8.67155	0.442518
$385$	0	0
$386$	1.26496	0.0643848
$387$	−74.0830	−3.76585
$388$	0.212719	0.0107992
$389$	−22.4883	−1.14020	−0.570101	−	0.821574i	$-0.693096\pi$
$389$	−22.4883	−1.14020	−0.570101	+	0.821574i	$0.693096\pi$
$390$	0	0
$391$	24.7910	1.25374
$392$	14.3124	0.722885
$393$	1.69333	0.0854171
$394$	−6.55060	−0.330015
$395$	0	0
$396$	−0.668698	−0.0336034
$397$	37.6626	1.89023	0.945115	−	0.326738i	$-0.105949\pi$
$397$	37.6626	1.89023	0.945115	+	0.326738i	$0.105949\pi$
$398$	2.49293	0.124959
$399$	−21.2290	−1.06278
$400$	0	0
$401$	31.1613	1.55612	0.778060	−	0.628190i	$-0.216204\pi$
$401$	31.1613	1.55612	0.778060	+	0.628190i	$0.216204\pi$
$402$	−36.9412	−1.84246
$403$	−4.91432	−0.244800
$404$	15.3936	0.765860
$405$	0	0
$406$	9.52100	0.472519
$407$	−0.768851	−0.0381105
$408$	26.7067	1.32218
$409$	−1.39415	−0.0689364	−0.0344682	−	0.999406i	$-0.510974\pi$
$409$	−1.39415	−0.0689364	−0.0344682	+	0.999406i	$0.510974\pi$
$410$	0	0
$411$	8.80346	0.434242
$412$	−14.9982	−0.738906
$413$	15.2135	0.748607
$414$	46.9821	2.30905
$415$	0	0
$416$	27.7753	1.36180
$417$	−57.9342	−2.83705
$418$	0.567483	0.0277565
$419$	6.38600	0.311977	0.155988	−	0.987759i	$-0.450144\pi$
$419$	6.38600	0.311977	0.155988	+	0.987759i	$0.450144\pi$
$420$	0	0
$421$	−15.8431	−0.772147	−0.386073	−	0.922468i	$-0.626169\pi$
$421$	−15.8431	−0.772147	−0.386073	+	0.922468i	$0.626169\pi$
$422$	−27.5004	−1.33870
$423$	−37.0359	−1.80075
$424$	−29.4195	−1.42874
$425$	0	0
$426$	2.12346	0.102882
$427$	−3.97502	−0.192364
$428$	−6.76559	−0.327027
$429$	−1.93253	−0.0933035
$430$	0	0
$431$	−36.3307	−1.74999	−0.874994	−	0.484133i	$-0.839135\pi$
$431$	−36.3307	−1.74999	−0.874994	+	0.484133i	$0.839135\pi$
$432$	8.09162	0.389308
$433$	−36.9821	−1.77725	−0.888624	−	0.458636i	$-0.848338\pi$
$433$	−36.9821	−1.77725	−0.888624	+	0.458636i	$0.848338\pi$
$434$	−1.32072	−0.0633964
$435$	0	0
$436$	16.0292	0.767661
$437$	39.5943	1.89405
$438$	−45.2311	−2.16123
$439$	−1.39790	−0.0667181	−0.0333590	−	0.999443i	$-0.510620\pi$
$439$	−1.39790	−0.0667181	−0.0333590	+	0.999443i	$0.510620\pi$
$440$	0	0
$441$	−27.2014	−1.29530
$442$	16.8217	0.800127
$443$	11.3005	0.536901	0.268450	−	0.963293i	$-0.413488\pi$
$443$	11.3005	0.536901	0.268450	+	0.963293i	$0.413488\pi$
$444$	−19.2184	−0.912064
$445$	0	0
$446$	11.3795	0.538837
$447$	19.4656	0.920689
$448$	10.4941	0.495798
$449$	25.9125	1.22289	0.611444	−	0.791288i	$-0.290589\pi$
$449$	25.9125	1.22289	0.611444	+	0.791288i	$0.290589\pi$
$450$	0	0
$451$	−0.827885	−0.0389836
$452$	−0.119747	−0.00563242
$453$	24.0014	1.12768
$454$	−0.642255	−0.0301425
$455$	0	0
$456$	42.6539	1.99745
$457$	2.29759	0.107477	0.0537383	−	0.998555i	$-0.482886\pi$
$457$	2.29759	0.107477	0.0537383	+	0.998555i	$0.482886\pi$
$458$	−19.6626	−0.918771
$459$	−24.0660	−1.12330
$460$	0	0
$461$	−20.1050	−0.936385	−0.468192	−	0.883627i	$-0.655095\pi$
$461$	−20.1050	−0.936385	−0.468192	+	0.883627i	$0.655095\pi$
$462$	−0.519365	−0.0241631
$463$	−23.2993	−1.08281	−0.541404	−	0.840762i	$-0.682107\pi$
$463$	−23.2993	−1.08281	−0.541404	+	0.840762i	$0.682107\pi$
$464$	−6.45036	−0.299450
$465$	0	0
$466$	22.7451	1.05365
$467$	−9.85964	−0.456250	−0.228125	−	0.973632i	$-0.573259\pi$
$467$	−9.85964	−0.456250	−0.228125	+	0.973632i	$0.573259\pi$
$468$	−31.6582	−1.46340
$469$	18.6731	0.862243
$470$	0	0
$471$	−51.3098	−2.36423
$472$	−30.5674	−1.40698
$473$	−1.52743	−0.0702314
$474$	−27.3619	−1.25678
$475$	0	0
$476$	−4.48946	−0.205774
$477$	55.9132	2.56009
$478$	27.6444	1.26443
$479$	17.4252	0.796179	0.398089	−	0.917347i	$-0.369673\pi$
$479$	17.4252	0.796179	0.398089	+	0.917347i	$0.369673\pi$
$480$	0	0
$481$	−36.3997	−1.65969
$482$	1.00174	0.0456279
$483$	−36.2371	−1.64884
$484$	10.9479	0.497634
$485$	0	0
$486$	4.97526	0.225682
$487$	14.9615	0.677970	0.338985	−	0.940792i	$-0.389916\pi$
$487$	14.9615	0.677970	0.338985	+	0.940792i	$0.389916\pi$
$488$	7.98671	0.361542
$489$	64.1330	2.90020
$490$	0	0
$491$	15.0287	0.678234	0.339117	−	0.940744i	$-0.389872\pi$
$491$	15.0287	0.678234	0.339117	+	0.940744i	$0.389872\pi$
$492$	−20.6940	−0.932959
$493$	19.1846	0.864030
$494$	26.8664	1.20878
$495$	0	0
$496$	0.894768	0.0401763
$497$	−1.07337	−0.0481473
$498$	−8.63193	−0.386806
$499$	0.947417	0.0424122	0.0212061	−	0.999775i	$-0.493249\pi$
$499$	0.947417	0.0424122	0.0212061	+	0.999775i	$0.493249\pi$
$500$	0	0
$501$	1.52740	0.0682394
$502$	−1.99967	−0.0892497
$503$	21.8928	0.976151	0.488075	−	0.872802i	$-0.337699\pi$
$503$	21.8928	0.976151	0.488075	+	0.872802i	$0.337699\pi$
$504$	−25.5837	−1.13959
$505$	0	0
$506$	0.968671	0.0430627
$507$	−53.1366	−2.35988
$508$	−13.8241	−0.613344
$509$	−25.2513	−1.11925	−0.559623	−	0.828747i	$-0.689054\pi$
$509$	−25.2513	−1.11925	−0.559623	+	0.828747i	$0.689054\pi$
$510$	0	0
$511$	22.8635	1.01142
$512$	−11.1441	−0.492504
$513$	−38.4364	−1.69701
$514$	−9.14251	−0.403259
$515$	0	0
$516$	−38.1801	−1.68079
$517$	−0.763602	−0.0335832
$518$	−9.78238	−0.429813
$519$	32.2537	1.41578
$520$	0	0
$521$	42.3792	1.85667	0.928333	−	0.371749i	$-0.121242\pi$
$521$	42.3792	1.85667	0.928333	+	0.371749i	$0.121242\pi$
$522$	36.3572	1.59131
$523$	37.5455	1.64175	0.820876	−	0.571107i	$-0.193486\pi$
$523$	37.5455	1.64175	0.820876	+	0.571107i	$0.193486\pi$
$524$	0.571932	0.0249850
$525$	0	0
$526$	4.15342	0.181098
$527$	−2.66121	−0.115924
$528$	0.351863	0.0153129
$529$	44.5860	1.93852
$530$	0	0
$531$	58.0948	2.52110
$532$	−7.17023	−0.310869
$533$	−39.1946	−1.69771
$534$	47.8241	2.06955
$535$	0	0
$536$	−37.5185	−1.62055
$537$	−33.9374	−1.46451
$538$	−6.31074	−0.272075
$539$	−0.560835	−0.0241569
$540$	0	0
$541$	44.6646	1.92028	0.960141	−	0.279515i	$-0.0901736\pi$
$541$	44.6646	1.92028	0.960141	+	0.279515i	$0.0901736\pi$
$542$	23.5289	1.01065
$543$	−67.7094	−2.90569
$544$	15.0409	0.644874
$545$	0	0
$546$	−24.5883	−1.05228
$547$	−17.7546	−0.759131	−0.379565	−	0.925165i	$-0.623926\pi$
$547$	−17.7546	−0.759131	−0.379565	+	0.925165i	$0.623926\pi$
$548$	2.97342	0.127018
$549$	−15.1791	−0.647829
$550$	0	0
$551$	30.6401	1.30531
$552$	72.8086	3.09894
$553$	13.8310	0.588152
$554$	−12.3493	−0.524671
$555$	0	0
$556$	−19.5677	−0.829853
$557$	−2.15600	−0.0913525	−0.0456763	−	0.998956i	$-0.514544\pi$
$557$	−2.15600	−0.0913525	−0.0456763	+	0.998956i	$0.514544\pi$
$558$	−5.04333	−0.213501
$559$	−72.3134	−3.05853
$560$	0	0
$561$	−1.04651	−0.0441836
$562$	−2.70933	−0.114286
$563$	−44.6127	−1.88020	−0.940101	−	0.340896i	$-0.889270\pi$
$563$	−44.6127	−1.88020	−0.940101	+	0.340896i	$0.889270\pi$
$564$	−19.0872	−0.803716
$565$	0	0
$566$	−27.5101	−1.15633
$567$	9.60835	0.403513
$568$	2.15665	0.0904909
$569$	4.47247	0.187496	0.0937479	−	0.995596i	$-0.470115\pi$
$569$	4.47247	0.187496	0.0937479	+	0.995596i	$0.470115\pi$
$570$	0	0
$571$	−3.27924	−0.137232	−0.0686159	−	0.997643i	$-0.521858\pi$
$571$	−3.27924	−0.137232	−0.0686159	+	0.997643i	$0.521858\pi$
$572$	−0.652725	−0.0272918
$573$	3.83666	0.160279
$574$	−10.5335	−0.439660
$575$	0	0
$576$	40.0730	1.66971
$577$	−28.9044	−1.20330	−0.601652	−	0.798758i	$-0.705491\pi$
$577$	−28.9044	−1.20330	−0.601652	+	0.798758i	$0.705491\pi$
$578$	−7.92023	−0.329438
$579$	−3.72568	−0.154834
$580$	0	0
$581$	4.36328	0.181019
$582$	0.630894	0.0261514
$583$	1.15281	0.0477446
$584$	−45.9380	−1.90093
$585$	0	0
$586$	8.32677	0.343976
$587$	37.3590	1.54197	0.770985	−	0.636853i	$-0.219764\pi$
$587$	37.3590	1.54197	0.770985	+	0.636853i	$0.219764\pi$
$588$	−14.0188	−0.578124
$589$	−4.25028	−0.175130
$590$	0	0
$591$	19.2934	0.793626
$592$	6.62744	0.272386
$593$	−11.0546	−0.453958	−0.226979	−	0.973900i	$-0.572885\pi$
$593$	−11.0546	−0.453958	−0.226979	+	0.973900i	$0.572885\pi$
$594$	−0.940343	−0.0385827
$595$	0	0
$596$	6.57462	0.269307
$597$	−7.34240	−0.300505
$598$	45.8599	1.87535
$599$	13.6158	0.556327	0.278163	−	0.960534i	$-0.410274\pi$
$599$	13.6158	0.556327	0.278163	+	0.960534i	$0.410274\pi$
$600$	0	0
$601$	1.09654	0.0447287	0.0223643	−	0.999750i	$-0.492881\pi$
$601$	1.09654	0.0447287	0.0223643	+	0.999750i	$0.492881\pi$
$602$	−19.4341	−0.792075
$603$	71.3057	2.90379
$604$	8.10662	0.329854
$605$	0	0
$606$	45.6552	1.85462
$607$	−39.0159	−1.58360	−0.791802	−	0.610777i	$-0.790857\pi$
$607$	−39.0159	−1.58360	−0.791802	+	0.610777i	$0.790857\pi$
$608$	24.0222	0.974229
$609$	−28.0421	−1.13632
$610$	0	0
$611$	−36.1513	−1.46252
$612$	−17.1436	−0.692989
$613$	1.72568	0.0696997	0.0348498	−	0.999393i	$-0.488905\pi$
$613$	1.72568	0.0696997	0.0348498	+	0.999393i	$0.488905\pi$
$614$	6.79346	0.274162
$615$	0	0
$616$	−0.527482	−0.0212528
$617$	5.33029	0.214589	0.107295	−	0.994227i	$-0.465781\pi$
$617$	5.33029	0.214589	0.107295	+	0.994227i	$0.465781\pi$
$618$	−44.4824	−1.78935
$619$	−31.3838	−1.26142	−0.630711	−	0.776018i	$-0.717237\pi$
$619$	−31.3838	−1.26142	−0.630711	+	0.776018i	$0.717237\pi$
$620$	0	0
$621$	−65.6095	−2.63282
$622$	−26.5055	−1.06277
$623$	−24.1742	−0.968518
$624$	16.6583	0.666865
$625$	0	0
$626$	−10.2871	−0.411155
$627$	−1.67140	−0.0667494
$628$	−17.3302	−0.691551
$629$	−19.7112	−0.785939
$630$	0	0
$631$	30.2224	1.20313	0.601567	−	0.798822i	$-0.294543\pi$
$631$	30.2224	1.20313	0.601567	+	0.798822i	$0.294543\pi$
$632$	−27.7895	−1.10541
$633$	80.9966	3.21933
$634$	19.1090	0.758916
$635$	0	0
$636$	28.8160	1.14263
$637$	−26.5516	−1.05201
$638$	0.749608	0.0296773
$639$	−4.09881	−0.162146
$640$	0	0
$641$	6.01295	0.237497	0.118749	−	0.992924i	$-0.462112\pi$
$641$	6.01295	0.237497	0.118749	+	0.992924i	$0.462112\pi$
$642$	−20.0658	−0.791934
$643$	−9.39170	−0.370372	−0.185186	−	0.982703i	$-0.559289\pi$
$643$	−9.39170	−0.370372	−0.185186	+	0.982703i	$0.559289\pi$
$644$	−12.2393	−0.482296
$645$	0	0
$646$	14.5487	0.572412
$647$	19.3567	0.760992	0.380496	−	0.924783i	$-0.375753\pi$
$647$	19.3567	0.760992	0.380496	+	0.924783i	$0.375753\pi$
$648$	−19.3054	−0.758386
$649$	1.19779	0.0470174
$650$	0	0
$651$	3.88989	0.152457
$652$	21.6613	0.848324
$653$	1.48107	0.0579586	0.0289793	−	0.999580i	$-0.490774\pi$
$653$	1.48107	0.0579586	0.0289793	+	0.999580i	$0.490774\pi$
$654$	47.5405	1.85898
$655$	0	0
$656$	7.13631	0.278626
$657$	87.3073	3.40618
$658$	−9.71560	−0.378753
$659$	−8.38318	−0.326562	−0.163281	−	0.986580i	$-0.552208\pi$
$659$	−8.38318	−0.326562	−0.163281	+	0.986580i	$0.552208\pi$
$660$	0	0
$661$	−7.44801	−0.289694	−0.144847	−	0.989454i	$-0.546269\pi$
$661$	−7.44801	−0.289694	−0.144847	+	0.989454i	$0.546269\pi$
$662$	−10.1537	−0.394636
$663$	−49.5449	−1.92416
$664$	−8.76683	−0.340219
$665$	0	0
$666$	−37.3553	−1.44749
$667$	52.3016	2.02512
$668$	0.515891	0.0199604
$669$	−33.5161	−1.29581
$670$	0	0
$671$	−0.312961	−0.0120817
$672$	−21.9853	−0.848102
$673$	−23.9244	−0.922217	−0.461108	−	0.887344i	$-0.652548\pi$
$673$	−23.9244	−0.922217	−0.461108	+	0.887344i	$0.652548\pi$
$674$	24.5778	0.946701
$675$	0	0
$676$	−17.9472	−0.690279
$677$	−24.6748	−0.948331	−0.474165	−	0.880436i	$-0.657250\pi$
$677$	−24.6748	−0.948331	−0.474165	+	0.880436i	$0.657250\pi$
$678$	−0.355152	−0.0136395
$679$	−0.318905	−0.0122385
$680$	0	0
$681$	1.89163	0.0724874
$682$	−0.103983	−0.00398170
$683$	−5.12928	−0.196266	−0.0981332	−	0.995173i	$-0.531287\pi$
$683$	−5.12928	−0.196266	−0.0981332	+	0.995173i	$0.531287\pi$
$684$	−27.3805	−1.04692
$685$	0	0
$686$	−17.6117	−0.672417
$687$	57.9120	2.20948
$688$	13.1664	0.501963
$689$	54.5776	2.07924
$690$	0	0
$691$	37.7587	1.43641	0.718204	−	0.695833i	$-0.244965\pi$
$691$	37.7587	1.43641	0.718204	+	0.695833i	$0.244965\pi$
$692$	10.8939	0.414123
$693$	1.00250	0.0380820
$694$	15.0986	0.573134
$695$	0	0
$696$	56.3430	2.13568
$697$	−21.2247	−0.803944
$698$	−0.642676	−0.0243257
$699$	−66.9909	−2.53383
$700$	0	0
$701$	−35.2086	−1.32981	−0.664905	−	0.746928i	$-0.731528\pi$
$701$	−35.2086	−1.32981	−0.664905	+	0.746928i	$0.731528\pi$
$702$	−44.5187	−1.68025
$703$	−31.4813	−1.18734
$704$	0.826219	0.0311393
$705$	0	0
$706$	4.73224	0.178100
$707$	−23.0779	−0.867933
$708$	29.9403	1.12522
$709$	−15.3688	−0.577187	−0.288594	−	0.957452i	$-0.593188\pi$
$709$	−15.3688	−0.577187	−0.288594	+	0.957452i	$0.593188\pi$
$710$	0	0
$711$	52.8153	1.98073
$712$	48.5715	1.82029
$713$	−7.25506	−0.271704
$714$	−13.3151	−0.498305
$715$	0	0
$716$	−11.4626	−0.428377
$717$	−81.4209	−3.04072
$718$	21.6124	0.806568
$719$	−42.7904	−1.59581	−0.797907	−	0.602780i	$-0.794060\pi$
$719$	−42.7904	−1.59581	−0.797907	+	0.602780i	$0.794060\pi$
$720$	0	0
$721$	22.4851	0.837387
$722$	4.20309	0.156423
$723$	−2.95041	−0.109727
$724$	−22.8693	−0.849930
$725$	0	0
$726$	32.4700	1.20508
$727$	−25.1777	−0.933789	−0.466895	−	0.884313i	$-0.654627\pi$
$727$	−25.1777	−0.933789	−0.466895	+	0.884313i	$0.654627\pi$
$728$	−24.9726	−0.925546
$729$	−33.9478	−1.25733
$730$	0	0
$731$	−39.1592	−1.44836
$732$	−7.82286	−0.289141
$733$	8.36399	0.308931	0.154466	−	0.987998i	$-0.450634\pi$
$733$	8.36399	0.308931	0.154466	+	0.987998i	$0.450634\pi$
$734$	16.7239	0.617290
$735$	0	0
$736$	41.0050	1.51146
$737$	1.47017	0.0541544
$738$	−40.2236	−1.48065
$739$	4.45075	0.163723	0.0818617	−	0.996644i	$-0.473913\pi$
$739$	4.45075	0.163723	0.0818617	+	0.996644i	$0.473913\pi$
$740$	0	0
$741$	−79.1293	−2.90689
$742$	14.6677	0.538466
$743$	7.31475	0.268352	0.134176	−	0.990958i	$-0.457161\pi$
$743$	7.31475	0.268352	0.134176	+	0.990958i	$0.457161\pi$
$744$	−7.81568	−0.286537
$745$	0	0
$746$	18.5414	0.678847
$747$	16.6618	0.609622
$748$	−0.353465	−0.0129239
$749$	10.1429	0.370613
$750$	0	0
$751$	16.5040	0.602240	0.301120	−	0.953586i	$-0.402639\pi$
$751$	16.5040	0.602240	0.301120	+	0.953586i	$0.402639\pi$
$752$	6.58219	0.240028
$753$	5.88962	0.214630
$754$	35.4888	1.29242
$755$	0	0
$756$	11.8814	0.432121
$757$	−4.51718	−0.164180	−0.0820898	−	0.996625i	$-0.526159\pi$
$757$	−4.51718	−0.164180	−0.0820898	+	0.996625i	$0.526159\pi$
$758$	21.1100	0.766750
$759$	−2.85302	−0.103558
$760$	0	0
$761$	−28.4449	−1.03113	−0.515563	−	0.856852i	$-0.672417\pi$
$761$	−28.4449	−1.03113	−0.515563	+	0.856852i	$0.672417\pi$
$762$	−41.0003	−1.48528
$763$	−24.0308	−0.869974
$764$	1.29586	0.0468824
$765$	0	0
$766$	−5.21899	−0.188570
$767$	56.7071	2.04757
$768$	50.1356	1.80911
$769$	11.0400	0.398114	0.199057	−	0.979988i	$-0.436212\pi$
$769$	11.0400	0.398114	0.199057	+	0.979988i	$0.436212\pi$
$770$	0	0
$771$	26.9274	0.969766
$772$	−1.25837	−0.0452898
$773$	41.7511	1.50168	0.750841	−	0.660483i	$-0.229648\pi$
$773$	41.7511	1.50168	0.750841	+	0.660483i	$0.229648\pi$
$774$	−74.2117	−2.66749
$775$	0	0
$776$	0.640754	0.0230017
$777$	28.8120	1.03362
$778$	−22.5274	−0.807646
$779$	−33.8985	−1.21454
$780$	0	0
$781$	−0.0845087	−0.00302396
$782$	24.8341	0.888066
$783$	−50.7720	−1.81444
$784$	4.83435	0.172655
$785$	0	0
$786$	1.69627	0.0605040
$787$	−3.16487	−0.112816	−0.0564078	−	0.998408i	$-0.517965\pi$
$787$	−3.16487	−0.112816	−0.0564078	+	0.998408i	$0.517965\pi$
$788$	6.51648	0.232140
$789$	−12.2330	−0.435508
$790$	0	0
$791$	0.179523	0.00638310
$792$	−2.01426	−0.0715736
$793$	−14.8165	−0.526151
$794$	37.7280	1.33892
$795$	0	0
$796$	−2.47994	−0.0878993
$797$	48.9804	1.73498	0.867488	−	0.497458i	$-0.165733\pi$
$797$	48.9804	1.73498	0.867488	+	0.497458i	$0.165733\pi$
$798$	−21.2659	−0.752804
$799$	−19.5767	−0.692573
$800$	0	0
$801$	−92.3123	−3.26170
$802$	31.2154	1.10226
$803$	1.80009	0.0635238
$804$	36.7488	1.29603
$805$	0	0
$806$	−4.92286	−0.173400
$807$	18.5870	0.654292
$808$	46.3687	1.63125
$809$	48.9970	1.72264	0.861322	−	0.508060i	$-0.169637\pi$
$809$	48.9970	1.72264	0.861322	+	0.508060i	$0.169637\pi$
$810$	0	0
$811$	−32.4582	−1.13976	−0.569881	−	0.821728i	$-0.693011\pi$
$811$	−32.4582	−1.13976	−0.569881	+	0.821728i	$0.693011\pi$
$812$	−9.47141	−0.332381
$813$	−69.2993	−2.43043
$814$	−0.770187	−0.0269950
$815$	0	0
$816$	9.02082	0.315792
$817$	−62.5422	−2.18807
$818$	−1.39658	−0.0488301
$819$	47.4616	1.65844
$820$	0	0
$821$	−43.4241	−1.51551	−0.757756	−	0.652538i	$-0.773704\pi$
$821$	−43.4241	−1.51551	−0.757756	+	0.652538i	$0.773704\pi$
$822$	8.81876	0.307589
$823$	−20.9838	−0.731449	−0.365725	−	0.930723i	$-0.619179\pi$
$823$	−20.9838	−0.731449	−0.365725	+	0.930723i	$0.619179\pi$
$824$	−45.1776	−1.57384
$825$	0	0
$826$	15.2399	0.530265
$827$	−50.9820	−1.77282	−0.886408	−	0.462904i	$-0.846807\pi$
$827$	−50.9820	−1.77282	−0.886408	+	0.462904i	$0.846807\pi$
$828$	−46.7374	−1.62424
$829$	25.3871	0.881730	0.440865	−	0.897573i	$-0.354672\pi$
$829$	25.3871	0.881730	0.440865	+	0.897573i	$0.354672\pi$
$830$	0	0
$831$	36.3723	1.26174
$832$	39.1157	1.35609
$833$	−14.3783	−0.498178
$834$	−58.0349	−2.00958
$835$	0	0
$836$	−0.564527	−0.0195246
$837$	7.04289	0.243438
$838$	6.39710	0.220984
$839$	11.6977	0.403851	0.201926	−	0.979401i	$-0.435280\pi$
$839$	11.6977	0.403851	0.201926	+	0.979401i	$0.435280\pi$
$840$	0	0
$841$	11.4737	0.395644
$842$	−15.8707	−0.546939
$843$	7.97976	0.274838
$844$	27.3571	0.941671
$845$	0	0
$846$	−37.1003	−1.27553
$847$	−16.4130	−0.563958
$848$	−9.93715	−0.341243
$849$	81.0251	2.78078
$850$	0	0
$851$	−53.7374	−1.84209
$852$	−2.11240	−0.0723697
$853$	−41.8267	−1.43212	−0.716060	−	0.698039i	$-0.754056\pi$
$853$	−41.8267	−1.43212	−0.716060	+	0.698039i	$0.754056\pi$
$854$	−3.98192	−0.136259
$855$	0	0
$856$	−20.3794	−0.696553
$857$	22.7190	0.776067	0.388033	−	0.921645i	$-0.373155\pi$
$857$	22.7190	0.776067	0.388033	+	0.921645i	$0.373155\pi$
$858$	−1.93589	−0.0660902
$859$	−17.9255	−0.611610	−0.305805	−	0.952094i	$-0.598926\pi$
$859$	−17.9255	−0.611610	−0.305805	+	0.952094i	$0.598926\pi$
$860$	0	0
$861$	31.0242	1.05730
$862$	−36.3939	−1.23958
$863$	1.68482	0.0573518	0.0286759	−	0.999589i	$-0.490871\pi$
$863$	1.68482	0.0573518	0.0286759	+	0.999589i	$0.490871\pi$
$864$	−39.8058	−1.35422
$865$	0	0
$866$	−37.0464	−1.25889
$867$	23.3274	0.792240
$868$	1.31384	0.0445945
$869$	1.08894	0.0369397
$870$	0	0
$871$	69.6024	2.35839
$872$	48.2834	1.63508
$873$	−1.21778	−0.0412157
$874$	39.6632	1.34163
$875$	0	0
$876$	44.9955	1.52026
$877$	−32.0371	−1.08182	−0.540908	−	0.841082i	$-0.681919\pi$
$877$	−32.0371	−1.08182	−0.540908	+	0.841082i	$0.681919\pi$
$878$	−1.40033	−0.0472588
$879$	−24.5248	−0.827200
$880$	0	0
$881$	43.2384	1.45674	0.728370	−	0.685184i	$-0.240278\pi$
$881$	43.2384	1.45674	0.728370	+	0.685184i	$0.240278\pi$
$882$	−27.2487	−0.917510
$883$	41.9013	1.41009	0.705046	−	0.709161i	$-0.250926\pi$
$883$	41.9013	1.41009	0.705046	+	0.709161i	$0.250926\pi$
$884$	−16.7341	−0.562829
$885$	0	0
$886$	11.3201	0.380306
$887$	−3.79584	−0.127452	−0.0637259	−	0.997967i	$-0.520298\pi$
$887$	−3.79584	−0.127452	−0.0637259	+	0.997967i	$0.520298\pi$
$888$	−57.8898	−1.94265
$889$	20.7249	0.695090
$890$	0	0
$891$	0.756485	0.0253432
$892$	−11.3203	−0.379031
$893$	−31.2664	−1.04629
$894$	19.4994	0.652157
$895$	0	0
$896$	−4.39093	−0.146691
$897$	−135.071	−4.50988
$898$	25.9576	0.866215
$899$	−5.61435	−0.187249
$900$	0	0
$901$	29.5549	0.984618
$902$	−0.829324	−0.0276135
$903$	57.2391	1.90480
$904$	−0.360703	−0.0119968
$905$	0	0
$906$	24.0431	0.798778
$907$	5.90799	0.196172	0.0980858	−	0.995178i	$-0.468728\pi$
$907$	5.90799	0.196172	0.0980858	+	0.995178i	$0.468728\pi$
$908$	0.638910	0.0212030
$909$	−88.1260	−2.92295
$910$	0	0
$911$	9.89943	0.327983	0.163991	−	0.986462i	$-0.447563\pi$
$911$	9.89943	0.327983	0.163991	+	0.986462i	$0.447563\pi$
$912$	14.4074	0.477076
$913$	0.343530	0.0113692
$914$	2.30158	0.0761296
$915$	0	0
$916$	19.5602	0.646286
$917$	−0.857434	−0.0283150
$918$	−24.1078	−0.795677
$919$	2.08772	0.0688675	0.0344338	−	0.999407i	$-0.489037\pi$
$919$	2.08772	0.0688675	0.0344338	+	0.999407i	$0.489037\pi$
$920$	0	0
$921$	−20.0087	−0.659310
$922$	−20.1400	−0.663275
$923$	−4.00090	−0.131691
$924$	0.516660	0.0169969
$925$	0	0
$926$	−23.3398	−0.766992
$927$	85.8622	2.82008
$928$	31.7318	1.04165
$929$	6.19462	0.203239	0.101619	−	0.994823i	$-0.467598\pi$
$929$	6.19462	0.203239	0.101619	+	0.994823i	$0.467598\pi$
$930$	0	0
$931$	−22.9639	−0.752611
$932$	−22.6266	−0.741160
$933$	78.0665	2.55578
$934$	−9.87678	−0.323178
$935$	0	0
$936$	−95.3612	−3.11698
$937$	−23.6766	−0.773481	−0.386741	−	0.922188i	$-0.626399\pi$
$937$	−23.6766	−0.773481	−0.386741	+	0.922188i	$0.626399\pi$
$938$	18.7055	0.610757
$939$	30.2985	0.988755
$940$	0	0
$941$	−27.7804	−0.905615	−0.452808	−	0.891608i	$-0.649578\pi$
$941$	−27.7804	−0.905615	−0.452808	+	0.891608i	$0.649578\pi$
$942$	−51.3990	−1.67467
$943$	−57.8635	−1.88429
$944$	−10.3249	−0.336046
$945$	0	0
$946$	−1.53009	−0.0497474
$947$	32.2510	1.04802	0.524008	−	0.851713i	$-0.324436\pi$
$947$	32.2510	1.04802	0.524008	+	0.851713i	$0.324436\pi$
$948$	27.2194	0.884046
$949$	85.2218	2.76642
$950$	0	0
$951$	−56.2816	−1.82506
$952$	−13.5232	−0.438289
$953$	51.3936	1.66480	0.832401	−	0.554174i	$-0.186966\pi$
$953$	51.3936	1.66480	0.832401	+	0.554174i	$0.186966\pi$
$954$	56.0104	1.81340
$955$	0	0
$956$	−27.5004	−0.889428
$957$	−2.20782	−0.0713685
$958$	17.4555	0.563962
$959$	−4.45772	−0.143947
$960$	0	0
$961$	−30.2212	−0.974877
$962$	−36.4630	−1.17561
$963$	38.7320	1.24812
$964$	−0.996521	−0.0320958
$965$	0	0
$966$	−36.3001	−1.16794
$967$	6.01501	0.193430	0.0967149	−	0.995312i	$-0.469167\pi$
$967$	6.01501	0.193430	0.0967149	+	0.995312i	$0.469167\pi$
$968$	32.9775	1.05994
$969$	−42.8502	−1.37655
$970$	0	0
$971$	26.1797	0.840148	0.420074	−	0.907490i	$-0.362004\pi$
$971$	26.1797	0.840148	0.420074	+	0.907490i	$0.362004\pi$
$972$	−4.94935	−0.158750
$973$	29.3356	0.940455
$974$	14.9875	0.480230
$975$	0	0
$976$	2.69770	0.0863514
$977$	33.2956	1.06522	0.532610	−	0.846361i	$-0.321211\pi$
$977$	33.2956	1.06522	0.532610	+	0.846361i	$0.321211\pi$
$978$	64.2445	2.05431
$979$	−1.90328	−0.0608292
$980$	0	0
$981$	−91.7649	−2.92983
$982$	15.0548	0.480418
$983$	19.6537	0.626857	0.313428	−	0.949612i	$-0.398522\pi$
$983$	19.6537	0.626857	0.313428	+	0.949612i	$0.398522\pi$
$984$	−62.3348	−1.98716
$985$	0	0
$986$	19.2179	0.612023
$987$	28.6153	0.910834
$988$	−26.7264	−0.850281
$989$	−106.757	−3.39468
$990$	0	0
$991$	−0.177474	−0.00563763	−0.00281882	−	0.999996i	$-0.500897\pi$
$991$	−0.177474	−0.00563763	−0.00281882	+	0.999996i	$0.500897\pi$
$992$	−4.40171	−0.139754
$993$	29.9057	0.949029
$994$	−1.07524	−0.0341044
$995$	0	0
$996$	8.58697	0.272089
$997$	−3.89349	−0.123308	−0.0616540	−	0.998098i	$-0.519638\pi$
$997$	−3.89349	−0.123308	−0.0616540	+	0.998098i	$0.519638\pi$
$998$	0.949064	0.0300421
$999$	52.1658	1.65045

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.n.1.26	yes	40
5.4	even	2		6025.2.a.m.1.15	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.m.1.15	✓	40	5.4	even	2
6025.2.a.n.1.26	yes	40	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)

\(f(q)\)	\(=\)	\(q+1.00174 q^{2} -2.95041 q^{3} -0.996521 q^{4} -2.95554 q^{6} +1.49397 q^{7} -3.00173 q^{8} +5.70493 q^{9} +O(q^{10})\)	\(q+1.00174 q^{2} -2.95041 q^{3} -0.996521 q^{4} -2.95554 q^{6} +1.49397 q^{7} -3.00173 q^{8} +5.70493 q^{9} +0.117623 q^{11} +2.94015 q^{12} +5.56865 q^{13} +1.49657 q^{14} -1.01391 q^{16} +3.01555 q^{17} +5.71484 q^{18} +4.81620 q^{19} -4.40783 q^{21} +0.117828 q^{22} +8.22107 q^{23} +8.85634 q^{24} +5.57833 q^{26} -7.98065 q^{27} -1.48877 q^{28} +6.36189 q^{29} -0.882496 q^{31} +4.98779 q^{32} -0.347038 q^{33} +3.02079 q^{34} -5.68508 q^{36} -6.53654 q^{37} +4.82457 q^{38} -16.4298 q^{39} -7.03844 q^{41} -4.41549 q^{42} -12.9858 q^{43} -0.117214 q^{44} +8.23536 q^{46} -6.49192 q^{47} +2.99144 q^{48} -4.76805 q^{49} -8.89710 q^{51} -5.54928 q^{52} +9.80086 q^{53} -7.99452 q^{54} -4.48450 q^{56} -14.2098 q^{57} +6.37295 q^{58} +10.1833 q^{59} -2.66070 q^{61} -0.884030 q^{62} +8.52300 q^{63} +7.02427 q^{64} -0.347641 q^{66} +12.4990 q^{67} -3.00505 q^{68} -24.2555 q^{69} -0.718468 q^{71} -17.1246 q^{72} +15.3038 q^{73} -6.54790 q^{74} -4.79944 q^{76} +0.175726 q^{77} -16.4584 q^{78} +9.25785 q^{79} +6.43141 q^{81} -7.05067 q^{82} +2.92059 q^{83} +4.39249 q^{84} -13.0084 q^{86} -18.7702 q^{87} -0.353074 q^{88} -16.1812 q^{89} +8.31941 q^{91} -8.19247 q^{92} +2.60373 q^{93} -6.50320 q^{94} -14.7160 q^{96} -0.213462 q^{97} -4.77634 q^{98} +0.671033 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9	\( 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9} + q^{11} + 26 q^{12} + 11 q^{13} - q^{14} + 43 q^{16} + 20 q^{17} + 18 q^{18} + 2 q^{21} + 23 q^{22} + 79 q^{23} - 2 q^{24} + 2 q^{26} + 26 q^{27} + 30 q^{28} + 2 q^{29} + q^{31} + 68 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 16 q^{37} + 45 q^{38} - 2 q^{39} - 2 q^{41} + 19 q^{42} + 25 q^{43} + 3 q^{44} + 14 q^{46} + 88 q^{47} + 75 q^{48} + 40 q^{49} - 10 q^{51} + 18 q^{52} + 34 q^{53} + 4 q^{54} - 15 q^{56} + 51 q^{57} + 53 q^{58} + q^{59} + 9 q^{61} + 39 q^{62} + 110 q^{63} + 17 q^{64} + 26 q^{66} + 30 q^{67} + 44 q^{68} - 7 q^{69} + 5 q^{71} + 18 q^{72} + 23 q^{73} - 18 q^{74} + 43 q^{76} + 30 q^{77} + 46 q^{78} + 5 q^{79} + 44 q^{81} + 5 q^{82} + 65 q^{83} - 65 q^{84} + 40 q^{86} + 33 q^{87} + 71 q^{88} - 9 q^{89} + q^{91} + 117 q^{92} + 68 q^{93} - 72 q^{94} + 83 q^{96} - 8 q^{97} + 76 q^{98} - 40 q^{99}+O(q^{100}) \) 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9 + q^11 + 26 * q^12 + 11 * q^13 - q^14 + 43 * q^16 + 20 * q^17 + 18 * q^18 + 2 * q^21 + 23 * q^22 + 79 * q^23 - 2 * q^24 + 2 * q^26 + 26 * q^27 + 30 * q^28 + 2 * q^29 + q^31 + 68 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 16 * q^37 + 45 * q^38 - 2 * q^39 - 2 * q^41 + 19 * q^42 + 25 * q^43 + 3 * q^44 + 14 * q^46 + 88 * q^47 + 75 * q^48 + 40 * q^49 - 10 * q^51 + 18 * q^52 + 34 * q^53 + 4 * q^54 - 15 * q^56 + 51 * q^57 + 53 * q^58 + q^59 + 9 * q^61 + 39 * q^62 + 110 * q^63 + 17 * q^64 + 26 * q^66 + 30 * q^67 + 44 * q^68 - 7 * q^69 + 5 * q^71 + 18 * q^72 + 23 * q^73 - 18 * q^74 + 43 * q^76 + 30 * q^77 + 46 * q^78 + 5 * q^79 + 44 * q^81 + 5 * q^82 + 65 * q^83 - 65 * q^84 + 40 * q^86 + 33 * q^87 + 71 * q^88 - 9 * q^89 + q^91 + 117 * q^92 + 68 * q^93 - 72 * q^94 + 83 * q^96 - 8 * q^97 + 76 * q^98 - 40 * q^99

Introduction

L-functions

Modular forms

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Database

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