LMFDB - Embedded newform 6025.2.a.j.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.1098672178$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-2.03980 q^{2} +3.04377 q^{3} +2.16080 q^{4} -6.20869 q^{6} -3.73609 q^{7} -0.327997 q^{8} +6.26453 q^{9} +O(q^{10})$	\(q-2.03980 q^{2} +3.04377 q^{3} +2.16080 q^{4} -6.20869 q^{6} -3.73609 q^{7} -0.327997 q^{8} +6.26453 q^{9} -2.44956 q^{11} +6.57697 q^{12} -2.67600 q^{13} +7.62089 q^{14} -3.65255 q^{16} +4.31888 q^{17} -12.7784 q^{18} +4.90092 q^{19} -11.3718 q^{21} +4.99662 q^{22} -3.82036 q^{23} -0.998349 q^{24} +5.45852 q^{26} +9.93647 q^{27} -8.07294 q^{28} -1.38616 q^{29} +7.03224 q^{31} +8.10647 q^{32} -7.45590 q^{33} -8.80967 q^{34} +13.5364 q^{36} -9.99789 q^{37} -9.99691 q^{38} -8.14513 q^{39} +0.473620 q^{41} +23.1962 q^{42} -6.70759 q^{43} -5.29301 q^{44} +7.79279 q^{46} -4.83412 q^{47} -11.1175 q^{48} +6.95838 q^{49} +13.1457 q^{51} -5.78230 q^{52} +4.16301 q^{53} -20.2684 q^{54} +1.22543 q^{56} +14.9173 q^{57} +2.82750 q^{58} +9.97892 q^{59} -0.618642 q^{61} -14.3444 q^{62} -23.4049 q^{63} -9.23052 q^{64} +15.2086 q^{66} -12.8919 q^{67} +9.33224 q^{68} -11.6283 q^{69} -12.6851 q^{71} -2.05475 q^{72} +10.7512 q^{73} +20.3937 q^{74} +10.5899 q^{76} +9.15178 q^{77} +16.6145 q^{78} -6.32161 q^{79} +11.4507 q^{81} -0.966092 q^{82} -9.56865 q^{83} -24.5722 q^{84} +13.6822 q^{86} -4.21916 q^{87} +0.803450 q^{88} -11.7630 q^{89} +9.99779 q^{91} -8.25504 q^{92} +21.4045 q^{93} +9.86065 q^{94} +24.6742 q^{96} -6.00321 q^{97} -14.1937 q^{98} -15.3453 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9}+O(q^{10}) $ 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9	$ 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9} + 2 q^{11} - 20 q^{12} - 14 q^{13} - 5 q^{14} + 38 q^{16} - 7 q^{17} - 9 q^{18} + 30 q^{19} + q^{21} - q^{22} - 43 q^{23} - 6 q^{24} - 22 q^{26} - 42 q^{27} - 32 q^{28} - 4 q^{29} + 14 q^{31} - 26 q^{32} - 4 q^{33} + 7 q^{34} + 15 q^{36} - 16 q^{37} - 14 q^{38} - 21 q^{39} - q^{41} + 25 q^{42} - 35 q^{43} - 52 q^{44} - 27 q^{46} - 50 q^{47} - 26 q^{48} + 46 q^{49} - 7 q^{51} - 3 q^{52} - 4 q^{53} - 31 q^{54} - 51 q^{56} - 2 q^{58} + 6 q^{59} + 19 q^{61} - 28 q^{63} + 49 q^{64} - 27 q^{66} - 65 q^{67} + 25 q^{68} + 2 q^{69} - 34 q^{71} + 10 q^{72} - 8 q^{73} - 42 q^{74} + 71 q^{76} - q^{77} + 59 q^{78} - 12 q^{79} + 29 q^{81} - 11 q^{82} - 41 q^{83} - 10 q^{84} - 13 q^{86} - 40 q^{87} + 52 q^{88} - 24 q^{89} + 46 q^{91} - 85 q^{92} + 30 q^{93} + 14 q^{94} - 30 q^{96} - 9 q^{97} + 64 q^{98} + 2 q^{99}+O(q^{100}) $ 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9 + 2 * q^11 - 20 * q^12 - 14 * q^13 - 5 * q^14 + 38 * q^16 - 7 * q^17 - 9 * q^18 + 30 * q^19 + q^21 - q^22 - 43 * q^23 - 6 * q^24 - 22 * q^26 - 42 * q^27 - 32 * q^28 - 4 * q^29 + 14 * q^31 - 26 * q^32 - 4 * q^33 + 7 * q^34 + 15 * q^36 - 16 * q^37 - 14 * q^38 - 21 * q^39 - q^41 + 25 * q^42 - 35 * q^43 - 52 * q^44 - 27 * q^46 - 50 * q^47 - 26 * q^48 + 46 * q^49 - 7 * q^51 - 3 * q^52 - 4 * q^53 - 31 * q^54 - 51 * q^56 - 2 * q^58 + 6 * q^59 + 19 * q^61 - 28 * q^63 + 49 * q^64 - 27 * q^66 - 65 * q^67 + 25 * q^68 + 2 * q^69 - 34 * q^71 + 10 * q^72 - 8 * q^73 - 42 * q^74 + 71 * q^76 - q^77 + 59 * q^78 - 12 * q^79 + 29 * q^81 - 11 * q^82 - 41 * q^83 - 10 * q^84 - 13 * q^86 - 40 * q^87 + 52 * q^88 - 24 * q^89 + 46 * q^91 - 85 * q^92 + 30 * q^93 + 14 * q^94 - 30 * q^96 - 9 * q^97 + 64 * q^98 + 2 * 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 $

$2$	−2.03980	−1.44236	−0.721179	−	0.692748i	$-0.756400\pi$
$2$	−2.03980	−1.44236	−0.721179	+	0.692748i	$0.756400\pi$
$3$	3.04377	1.75732	0.878660	−	0.477447i	$-0.158438\pi$
$3$	3.04377	1.75732	0.878660	+	0.477447i	$0.158438\pi$
$4$	2.16080	1.08040
$5$	0	0
$5$	0	0
$6$	−6.20869	−2.53469
$7$	−3.73609	−1.41211	−0.706055	−	0.708157i	$-0.749527\pi$
$7$	−3.73609	−1.41211	−0.706055	+	0.708157i	$0.749527\pi$
$8$	−0.327997	−0.115965
$9$	6.26453	2.08818
$10$	0	0
$11$	−2.44956	−0.738570	−0.369285	−	0.929316i	$-0.620397\pi$
$11$	−2.44956	−0.738570	−0.369285	+	0.929316i	$0.620397\pi$
$12$	6.57697	1.89861
$13$	−2.67600	−0.742189	−0.371095	−	0.928595i	$-0.621017\pi$
$13$	−2.67600	−0.742189	−0.371095	+	0.928595i	$0.621017\pi$
$14$	7.62089	2.03677
$15$	0	0
$16$	−3.65255	−0.913137
$17$	4.31888	1.04748	0.523742	−	0.851877i	$-0.324536\pi$
$17$	4.31888	1.04748	0.523742	+	0.851877i	$0.324536\pi$
$18$	−12.7784	−3.01190
$19$	4.90092	1.12435	0.562174	−	0.827019i	$-0.309965\pi$
$19$	4.90092	1.12435	0.562174	+	0.827019i	$0.309965\pi$
$20$	0	0
$21$	−11.3718	−2.48153
$22$	4.99662	1.06528
$23$	−3.82036	−0.796601	−0.398300	−	0.917255i	$-0.630400\pi$
$23$	−3.82036	−0.796601	−0.398300	+	0.917255i	$0.630400\pi$
$24$	−0.998349	−0.203787
$25$	0	0
$26$	5.45852	1.07050
$27$	9.93647	1.91227
$28$	−8.07294	−1.52564
$29$	−1.38616	−0.257404	−0.128702	−	0.991683i	$-0.541081\pi$
$29$	−1.38616	−0.257404	−0.128702	+	0.991683i	$0.541081\pi$
$30$	0	0
$31$	7.03224	1.26303	0.631514	−	0.775365i	$-0.282434\pi$
$31$	7.03224	1.26303	0.631514	+	0.775365i	$0.282434\pi$
$32$	8.10647	1.43304
$33$	−7.45590	−1.29790
$34$	−8.80967	−1.51085
$35$	0	0
$36$	13.5364	2.25606
$37$	−9.99789	−1.64364	−0.821822	−	0.569745i	$-0.807042\pi$
$37$	−9.99789	−1.64364	−0.821822	+	0.569745i	$0.807042\pi$
$38$	−9.99691	−1.62171
$39$	−8.14513	−1.30426
$40$	0	0
$41$	0.473620	0.0739671	0.0369835	−	0.999316i	$-0.488225\pi$
$41$	0.473620	0.0739671	0.0369835	+	0.999316i	$0.488225\pi$
$42$	23.1962	3.57926
$43$	−6.70759	−1.02290	−0.511449	−	0.859314i	$-0.670891\pi$
$43$	−6.70759	−1.02290	−0.511449	+	0.859314i	$0.670891\pi$
$44$	−5.29301	−0.797951
$45$	0	0
$46$	7.79279	1.14898
$47$	−4.83412	−0.705128	−0.352564	−	0.935788i	$-0.614690\pi$
$47$	−4.83412	−0.705128	−0.352564	+	0.935788i	$0.614690\pi$
$48$	−11.1175	−1.60467
$49$	6.95838	0.994055
$50$	0	0
$51$	13.1457	1.84076
$52$	−5.78230	−0.801861
$53$	4.16301	0.571833	0.285917	−	0.958254i	$-0.407702\pi$
$53$	4.16301	0.571833	0.285917	+	0.958254i	$0.407702\pi$
$54$	−20.2684	−2.75819
$55$	0	0
$56$	1.22543	0.163755
$57$	14.9173	1.97584
$58$	2.82750	0.371269
$59$	9.97892	1.29914	0.649572	−	0.760300i	$-0.274948\pi$
$59$	9.97892	1.29914	0.649572	+	0.760300i	$0.274948\pi$
$60$	0	0
$61$	−0.618642	−0.0792089	−0.0396045	−	0.999215i	$-0.512610\pi$
$61$	−0.618642	−0.0792089	−0.0396045	+	0.999215i	$0.512610\pi$
$62$	−14.3444	−1.82174
$63$	−23.4049	−2.94873
$64$	−9.23052	−1.15381
$65$	0	0
$66$	15.2086	1.87204
$67$	−12.8919	−1.57499	−0.787496	−	0.616320i	$-0.788623\pi$
$67$	−12.8919	−1.57499	−0.787496	+	0.616320i	$0.788623\pi$
$68$	9.33224	1.13170
$69$	−11.6283	−1.39988
$70$	0	0
$71$	−12.6851	−1.50545	−0.752725	−	0.658335i	$-0.771261\pi$
$71$	−12.6851	−1.50545	−0.752725	+	0.658335i	$0.771261\pi$
$72$	−2.05475	−0.242155
$73$	10.7512	1.25833	0.629167	−	0.777270i	$-0.283396\pi$
$73$	10.7512	1.25833	0.629167	+	0.777270i	$0.283396\pi$
$74$	20.3937	2.37072
$75$	0	0
$76$	10.5899	1.21474
$77$	9.15178	1.04294
$78$	16.6145	1.88122
$79$	−6.32161	−0.711237	−0.355618	−	0.934631i	$-0.615730\pi$
$79$	−6.32161	−0.711237	−0.355618	+	0.934631i	$0.615730\pi$
$80$	0	0
$81$	11.4507	1.27230
$82$	−0.966092	−0.106687
$83$	−9.56865	−1.05030	−0.525148	−	0.851011i	$-0.675990\pi$
$83$	−9.56865	−1.05030	−0.525148	+	0.851011i	$0.675990\pi$
$84$	−24.5722	−2.68104
$85$	0	0
$86$	13.6822	1.47539
$87$	−4.21916	−0.452341
$88$	0.803450	0.0856480
$89$	−11.7630	−1.24687	−0.623436	−	0.781874i	$-0.714264\pi$
$89$	−11.7630	−1.24687	−0.623436	+	0.781874i	$0.714264\pi$
$90$	0	0
$91$	9.99779	1.04805
$92$	−8.25504	−0.860647
$93$	21.4045	2.21954
$94$	9.86065	1.01705
$95$	0	0
$96$	24.6742	2.51830
$97$	−6.00321	−0.609533	−0.304767	−	0.952427i	$-0.598578\pi$
$97$	−6.00321	−0.609533	−0.304767	+	0.952427i	$0.598578\pi$
$98$	−14.1937	−1.43378
$99$	−15.3453	−1.54226
$100$	0	0
$101$	6.17281	0.614218	0.307109	−	0.951674i	$-0.400638\pi$
$101$	6.17281	0.614218	0.307109	+	0.951674i	$0.400638\pi$
$102$	−26.8146	−2.65504
$103$	−5.38299	−0.530402	−0.265201	−	0.964193i	$-0.585438\pi$
$103$	−5.38299	−0.530402	−0.265201	+	0.964193i	$0.585438\pi$
$104$	0.877722	0.0860677
$105$	0	0
$106$	−8.49172	−0.824789
$107$	8.04322	0.777567	0.388784	−	0.921329i	$-0.372895\pi$
$107$	8.04322	0.777567	0.388784	+	0.921329i	$0.372895\pi$
$108$	21.4707	2.06602
$109$	0.227800	0.0218193	0.0109096	−	0.999940i	$-0.496527\pi$
$109$	0.227800	0.0218193	0.0109096	+	0.999940i	$0.496527\pi$
$110$	0	0
$111$	−30.4313	−2.88841
$112$	13.6462	1.28945
$113$	13.5662	1.27620	0.638099	−	0.769954i	$-0.279721\pi$
$113$	13.5662	1.27620	0.638099	+	0.769954i	$0.279721\pi$
$114$	−30.4283	−2.84987
$115$	0	0
$116$	−2.99522	−0.278099
$117$	−16.7639	−1.54982
$118$	−20.3550	−1.87383
$119$	−16.1357	−1.47916
$120$	0	0
$121$	−4.99965	−0.454514
$122$	1.26191	0.114248
$123$	1.44159	0.129984
$124$	15.1953	1.36457
$125$	0	0
$126$	47.7413	4.25313
$127$	8.98673	0.797443	0.398722	−	0.917072i	$-0.369454\pi$
$127$	8.98673	0.797443	0.398722	+	0.917072i	$0.369454\pi$
$128$	2.61550	0.231180
$129$	−20.4164	−1.79756
$130$	0	0
$131$	−10.7498	−0.939211	−0.469606	−	0.882876i	$-0.655604\pi$
$131$	−10.7498	−0.939211	−0.469606	+	0.882876i	$0.655604\pi$
$132$	−16.1107	−1.40226
$133$	−18.3103	−1.58770
$134$	26.2969	2.27170
$135$	0	0
$136$	−1.41658	−0.121471
$137$	15.0016	1.28167	0.640835	−	0.767679i	$-0.278588\pi$
$137$	15.0016	1.28167	0.640835	+	0.767679i	$0.278588\pi$
$138$	23.7195	2.01913
$139$	19.3429	1.64064	0.820321	−	0.571904i	$-0.193795\pi$
$139$	19.3429	1.64064	0.820321	+	0.571904i	$0.193795\pi$
$140$	0	0
$141$	−14.7139	−1.23914
$142$	25.8752	2.17140
$143$	6.55503	0.548159
$144$	−22.8815	−1.90679
$145$	0	0
$146$	−21.9304	−1.81497
$147$	21.1797	1.74687
$148$	−21.6034	−1.77579
$149$	−5.49710	−0.450340	−0.225170	−	0.974320i	$-0.572294\pi$
$149$	−5.49710	−0.450340	−0.225170	+	0.974320i	$0.572294\pi$
$150$	0	0
$151$	−15.1309	−1.23134	−0.615669	−	0.788005i	$-0.711114\pi$
$151$	−15.1309	−1.23134	−0.615669	+	0.788005i	$0.711114\pi$
$152$	−1.60749	−0.130385
$153$	27.0558	2.18733
$154$	−18.6678	−1.50430
$155$	0	0
$156$	−17.6000	−1.40913
$157$	0.105404	0.00841215	0.00420608	−	0.999991i	$-0.498661\pi$
$157$	0.105404	0.00841215	0.00420608	+	0.999991i	$0.498661\pi$
$158$	12.8948	1.02586
$159$	12.6712	1.00489
$160$	0	0
$161$	14.2732	1.12489
$162$	−23.3572	−1.83512
$163$	0.0587651	0.00460284	0.00230142	−	0.999997i	$-0.499267\pi$
$163$	0.0587651	0.00460284	0.00230142	+	0.999997i	$0.499267\pi$
$164$	1.02340	0.0799140
$165$	0	0
$166$	19.5182	1.51490
$167$	18.7953	1.45442	0.727210	−	0.686415i	$-0.240816\pi$
$167$	18.7953	1.45442	0.727210	+	0.686415i	$0.240816\pi$
$168$	3.72992	0.287770
$169$	−5.83902	−0.449155
$170$	0	0
$171$	30.7019	2.34784
$172$	−14.4938	−1.10514
$173$	−26.1815	−1.99054	−0.995270	−	0.0971429i	$-0.969030\pi$
$173$	−26.1815	−1.99054	−0.995270	+	0.0971429i	$0.969030\pi$
$174$	8.60625	0.652438
$175$	0	0
$176$	8.94713	0.674416
$177$	30.3735	2.28301
$178$	23.9941	1.79844
$179$	0.720874	0.0538807	0.0269403	−	0.999637i	$-0.491424\pi$
$179$	0.720874	0.0538807	0.0269403	+	0.999637i	$0.491424\pi$
$180$	0	0
$181$	−2.57112	−0.191110	−0.0955549	−	0.995424i	$-0.530463\pi$
$181$	−2.57112	−0.191110	−0.0955549	+	0.995424i	$0.530463\pi$
$182$	−20.3935	−1.51167
$183$	−1.88300	−0.139196
$184$	1.25307	0.0923775
$185$	0	0
$186$	−43.6610	−3.20138
$187$	−10.5794	−0.773640
$188$	−10.4456	−0.761820
$189$	−37.1236	−2.70034
$190$	0	0
$191$	7.81831	0.565713	0.282856	−	0.959162i	$-0.408718\pi$
$191$	7.81831	0.565713	0.282856	+	0.959162i	$0.408718\pi$
$192$	−28.0956	−2.02762
$193$	−4.14559	−0.298406	−0.149203	−	0.988807i	$-0.547671\pi$
$193$	−4.14559	−0.298406	−0.149203	+	0.988807i	$0.547671\pi$
$194$	12.2454	0.879166
$195$	0	0
$196$	15.0357	1.07398
$197$	−21.0383	−1.49891	−0.749457	−	0.662053i	$-0.769685\pi$
$197$	−21.0383	−1.49891	−0.749457	+	0.662053i	$0.769685\pi$
$198$	31.3015	2.22450
$199$	−2.70670	−0.191873	−0.0959363	−	0.995387i	$-0.530585\pi$
$199$	−2.70670	−0.191873	−0.0959363	+	0.995387i	$0.530585\pi$
$200$	0	0
$201$	−39.2398	−2.76776
$202$	−12.5913	−0.885923
$203$	5.17883	0.363482
$204$	28.4052	1.98876
$205$	0	0
$206$	10.9803	0.765030
$207$	−23.9328	−1.66344
$208$	9.77422	0.677720
$209$	−12.0051	−0.830410
$210$	0	0
$211$	−21.6056	−1.48739	−0.743695	−	0.668519i	$-0.766929\pi$
$211$	−21.6056	−1.48739	−0.743695	+	0.668519i	$0.766929\pi$
$212$	8.99542	0.617808
$213$	−38.6106	−2.64556
$214$	−16.4066	−1.12153
$215$	0	0
$216$	−3.25914	−0.221756
$217$	−26.2731	−1.78353
$218$	−0.464667	−0.0314712
$219$	32.7242	2.21130
$220$	0	0
$221$	−11.5573	−0.777431
$222$	62.0738	4.16612
$223$	−21.8054	−1.46019	−0.730097	−	0.683343i	$-0.760525\pi$
$223$	−21.8054	−1.46019	−0.730097	+	0.683343i	$0.760525\pi$
$224$	−30.2865	−2.02360
$225$	0	0
$226$	−27.6723	−1.84074
$227$	−10.9124	−0.724282	−0.362141	−	0.932123i	$-0.617954\pi$
$227$	−10.9124	−0.724282	−0.362141	+	0.932123i	$0.617954\pi$
$228$	32.2332	2.13470
$229$	−22.4483	−1.48343	−0.741714	−	0.670716i	$-0.765987\pi$
$229$	−22.4483	−1.48343	−0.741714	+	0.670716i	$0.765987\pi$
$230$	0	0
$231$	27.8559	1.83278
$232$	0.454658	0.0298497
$233$	1.99905	0.130962	0.0654812	−	0.997854i	$-0.479142\pi$
$233$	1.99905	0.130962	0.0654812	+	0.997854i	$0.479142\pi$
$234$	34.1950	2.23540
$235$	0	0
$236$	21.5624	1.40360
$237$	−19.2415	−1.24987
$238$	32.9138	2.13348
$239$	−26.6281	−1.72243	−0.861214	−	0.508242i	$-0.830296\pi$
$239$	−26.6281	−1.72243	−0.861214	+	0.508242i	$0.830296\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	10.1983	0.655572
$243$	5.04398	0.323571
$244$	−1.33676	−0.0855773
$245$	0	0
$246$	−2.94056	−0.187483
$247$	−13.1149	−0.834479
$248$	−2.30656	−0.146466
$249$	−29.1248	−1.84571
$250$	0	0
$251$	−18.8864	−1.19210	−0.596049	−	0.802948i	$-0.703264\pi$
$251$	−18.8864	−1.19210	−0.596049	+	0.802948i	$0.703264\pi$
$252$	−50.5732	−3.18581
$253$	9.35821	0.588346
$254$	−18.3312	−1.15020
$255$	0	0
$256$	13.1259	0.820371
$257$	−7.66605	−0.478195	−0.239097	−	0.970996i	$-0.576852\pi$
$257$	−7.66605	−0.478195	−0.239097	+	0.970996i	$0.576852\pi$
$258$	41.6454	2.59273
$259$	37.3530	2.32100
$260$	0	0
$261$	−8.68365	−0.537504
$262$	21.9274	1.35468
$263$	−25.8557	−1.59433	−0.797165	−	0.603762i	$-0.793668\pi$
$263$	−25.8557	−1.59433	−0.797165	+	0.603762i	$0.793668\pi$
$264$	2.44552	0.150511
$265$	0	0
$266$	37.3494	2.29004
$267$	−35.8038	−2.19115
$268$	−27.8567	−1.70162
$269$	−22.6904	−1.38346	−0.691730	−	0.722156i	$-0.743151\pi$
$269$	−22.6904	−1.38346	−0.691730	+	0.722156i	$0.743151\pi$
$270$	0	0
$271$	−10.7599	−0.653615	−0.326807	−	0.945091i	$-0.605973\pi$
$271$	−10.7599	−0.653615	−0.326807	+	0.945091i	$0.605973\pi$
$272$	−15.7749	−0.956495
$273$	30.4310	1.84177
$274$	−30.6002	−1.84863
$275$	0	0
$276$	−25.1264	−1.51243
$277$	26.7310	1.60611	0.803054	−	0.595907i	$-0.203207\pi$
$277$	26.7310	1.60611	0.803054	+	0.595907i	$0.203207\pi$
$278$	−39.4557	−2.36639
$279$	44.0537	2.63742
$280$	0	0
$281$	−13.6392	−0.813646	−0.406823	−	0.913507i	$-0.633363\pi$
$281$	−13.6392	−0.813646	−0.406823	+	0.913507i	$0.633363\pi$
$282$	30.0135	1.78728
$283$	0.969212	0.0576137	0.0288068	−	0.999585i	$-0.490829\pi$
$283$	0.969212	0.0576137	0.0288068	+	0.999585i	$0.490829\pi$
$284$	−27.4100	−1.62649
$285$	0	0
$286$	−13.3710	−0.790642
$287$	−1.76949	−0.104450
$288$	50.7832	2.99243
$289$	1.65276	0.0972211
$290$	0	0
$291$	−18.2724	−1.07115
$292$	23.2312	1.35950
$293$	19.8960	1.16234	0.581168	−	0.813783i	$-0.302596\pi$
$293$	19.8960	1.16234	0.581168	+	0.813783i	$0.302596\pi$
$294$	−43.2024	−2.51962
$295$	0	0
$296$	3.27928	0.190604
$297$	−24.3400	−1.41235
$298$	11.2130	0.649552
$299$	10.2233	0.591229
$300$	0	0
$301$	25.0602	1.44444
$302$	30.8641	1.77603
$303$	18.7886	1.07938
$304$	−17.9008	−1.02668
$305$	0	0
$306$	−55.1885	−3.15491
$307$	31.2993	1.78634	0.893172	−	0.449716i	$-0.148475\pi$
$307$	31.2993	1.78634	0.893172	+	0.449716i	$0.148475\pi$
$308$	19.7752	1.12679
$309$	−16.3846	−0.932087
$310$	0	0
$311$	25.6258	1.45311	0.726554	−	0.687110i	$-0.241121\pi$
$311$	25.6258	1.45311	0.726554	+	0.687110i	$0.241121\pi$
$312$	2.67158	0.151249
$313$	12.2655	0.693287	0.346644	−	0.937997i	$-0.387321\pi$
$313$	12.2655	0.693287	0.346644	+	0.937997i	$0.387321\pi$
$314$	−0.215003	−0.0121333
$315$	0	0
$316$	−13.6597	−0.768420
$317$	−30.4734	−1.71155	−0.855777	−	0.517344i	$-0.826921\pi$
$317$	−30.4734	−1.71155	−0.855777	+	0.517344i	$0.826921\pi$
$318$	−25.8468	−1.44942
$319$	3.39549	0.190111
$320$	0	0
$321$	24.4817	1.36644
$322$	−29.1146	−1.62249
$323$	21.1665	1.17774
$324$	24.7427	1.37460
$325$	0	0
$326$	−0.119869	−0.00663894
$327$	0.693370	0.0383435
$328$	−0.155346	−0.00857756
$329$	18.0607	0.995719
$330$	0	0
$331$	−4.18161	−0.229842	−0.114921	−	0.993375i	$-0.536661\pi$
$331$	−4.18161	−0.229842	−0.114921	+	0.993375i	$0.536661\pi$
$332$	−20.6759	−1.13474
$333$	−62.6321	−3.43222
$334$	−38.3386	−2.09780
$335$	0	0
$336$	41.5360	2.26598
$337$	22.0539	1.20135	0.600677	−	0.799492i	$-0.294898\pi$
$337$	22.0539	1.20135	0.600677	+	0.799492i	$0.294898\pi$
$338$	11.9104	0.647843
$339$	41.2923	2.24269
$340$	0	0
$341$	−17.2259	−0.932834
$342$	−62.6259	−3.38642
$343$	0.155490	0.00839569
$344$	2.20007	0.118620
$345$	0	0
$346$	53.4051	2.87107
$347$	−24.9969	−1.34190	−0.670952	−	0.741501i	$-0.734114\pi$
$347$	−24.9969	−1.34190	−0.670952	+	0.741501i	$0.734114\pi$
$348$	−9.11675	−0.488709
$349$	10.5881	0.566768	0.283384	−	0.959007i	$-0.408543\pi$
$349$	10.5881	0.566768	0.283384	+	0.959007i	$0.408543\pi$
$350$	0	0
$351$	−26.5900	−1.41927
$352$	−19.8573	−1.05840
$353$	−29.6990	−1.58072	−0.790360	−	0.612643i	$-0.790107\pi$
$353$	−29.6990	−1.58072	−0.790360	+	0.612643i	$0.790107\pi$
$354$	−61.9560	−3.29293
$355$	0	0
$356$	−25.4174	−1.34712
$357$	−49.1135	−2.59936
$358$	−1.47044	−0.0777153
$359$	11.2005	0.591140	0.295570	−	0.955321i	$-0.404490\pi$
$359$	11.2005	0.591140	0.295570	+	0.955321i	$0.404490\pi$
$360$	0	0
$361$	5.01900	0.264158
$362$	5.24458	0.275649
$363$	−15.2178	−0.798727
$364$	21.6032	1.13232
$365$	0	0
$366$	3.84095	0.200770
$367$	6.87644	0.358947	0.179474	−	0.983763i	$-0.442561\pi$
$367$	6.87644	0.358947	0.179474	+	0.983763i	$0.442561\pi$
$368$	13.9541	0.727406
$369$	2.96701	0.154456
$370$	0	0
$371$	−15.5534	−0.807491
$372$	46.2508	2.39799
$373$	27.0850	1.40241	0.701203	−	0.712962i	$-0.252647\pi$
$373$	27.0850	1.40241	0.701203	+	0.712962i	$0.252647\pi$
$374$	21.5798	1.11587
$375$	0	0
$376$	1.58558	0.0817699
$377$	3.70937	0.191042
$378$	75.7248	3.89486
$379$	−15.9631	−0.819969	−0.409985	−	0.912092i	$-0.634466\pi$
$379$	−15.9631	−0.819969	−0.409985	+	0.912092i	$0.634466\pi$
$380$	0	0
$381$	27.3535	1.40136
$382$	−15.9478	−0.815961
$383$	−14.1150	−0.721245	−0.360622	−	0.932712i	$-0.617436\pi$
$383$	−14.1150	−0.721245	−0.360622	+	0.932712i	$0.617436\pi$
$384$	7.96097	0.406257
$385$	0	0
$386$	8.45618	0.430408
$387$	−42.0199	−2.13599
$388$	−12.9717	−0.658539
$389$	27.5795	1.39833	0.699167	−	0.714959i	$-0.253554\pi$
$389$	27.5795	1.39833	0.699167	+	0.714959i	$0.253554\pi$
$390$	0	0
$391$	−16.4997	−0.834426
$392$	−2.28233	−0.115275
$393$	−32.7198	−1.65050
$394$	42.9139	2.16197
$395$	0	0
$396$	−33.1582	−1.66626
$397$	12.2577	0.615194	0.307597	−	0.951517i	$-0.400475\pi$
$397$	12.2577	0.615194	0.307597	+	0.951517i	$0.400475\pi$
$398$	5.52113	0.276749
$399$	−55.7323	−2.79010
$400$	0	0
$401$	−24.6308	−1.23000	−0.615001	−	0.788527i	$-0.710844\pi$
$401$	−24.6308	−1.23000	−0.615001	+	0.788527i	$0.710844\pi$
$402$	80.0416	3.99211
$403$	−18.8183	−0.937405
$404$	13.3382	0.663601
$405$	0	0
$406$	−10.5638	−0.524272
$407$	24.4904	1.21395
$408$	−4.31175	−0.213464
$409$	38.8172	1.91939	0.959694	−	0.281045i	$-0.0906812\pi$
$409$	38.8172	1.91939	0.959694	+	0.281045i	$0.0906812\pi$
$410$	0	0
$411$	45.6613	2.25230
$412$	−11.6316	−0.573046
$413$	−37.2822	−1.83454
$414$	48.8182	2.39928
$415$	0	0
$416$	−21.6929	−1.06358
$417$	58.8753	2.88313
$418$	24.4880	1.19775
$419$	25.8577	1.26323	0.631616	−	0.775282i	$-0.282392\pi$
$419$	25.8577	1.26323	0.631616	+	0.775282i	$0.282392\pi$
$420$	0	0
$421$	25.6578	1.25049	0.625243	−	0.780430i	$-0.285000\pi$
$421$	25.6578	1.25049	0.625243	+	0.780430i	$0.285000\pi$
$422$	44.0712	2.14535
$423$	−30.2835	−1.47243
$424$	−1.36546	−0.0663124
$425$	0	0
$426$	78.7581	3.81584
$427$	2.31130	0.111852
$428$	17.3798	0.840083
$429$	19.9520	0.963291
$430$	0	0
$431$	12.6987	0.611674	0.305837	−	0.952084i	$-0.401064\pi$
$431$	12.6987	0.611674	0.305837	+	0.952084i	$0.401064\pi$
$432$	−36.2934	−1.74617
$433$	−4.18248	−0.200997	−0.100498	−	0.994937i	$-0.532044\pi$
$433$	−4.18248	−0.200997	−0.100498	+	0.994937i	$0.532044\pi$
$434$	53.5919	2.57250
$435$	0	0
$436$	0.492230	0.0235735
$437$	−18.7233	−0.895657
$438$	−66.7510	−3.18949
$439$	10.3172	0.492414	0.246207	−	0.969217i	$-0.420816\pi$
$439$	10.3172	0.492414	0.246207	+	0.969217i	$0.420816\pi$
$440$	0	0
$441$	43.5910	2.07576
$442$	23.5747	1.12133
$443$	−16.6403	−0.790604	−0.395302	−	0.918551i	$-0.629360\pi$
$443$	−16.6403	−0.790604	−0.395302	+	0.918551i	$0.629360\pi$
$444$	−65.7558	−3.12063
$445$	0	0
$446$	44.4786	2.10612
$447$	−16.7319	−0.791391
$448$	34.4861	1.62931
$449$	0.552311	0.0260652	0.0130326	−	0.999915i	$-0.495851\pi$
$449$	0.552311	0.0260652	0.0130326	+	0.999915i	$0.495851\pi$
$450$	0	0
$451$	−1.16016	−0.0546299
$452$	29.3138	1.37880
$453$	−46.0551	−2.16386
$454$	22.2592	1.04467
$455$	0	0
$456$	−4.89283	−0.229128
$457$	−32.6774	−1.52859	−0.764293	−	0.644869i	$-0.776912\pi$
$457$	−32.6774	−1.52859	−0.764293	+	0.644869i	$0.776912\pi$
$458$	45.7902	2.13964
$459$	42.9145	2.00308
$460$	0	0
$461$	−29.0243	−1.35180	−0.675898	−	0.736995i	$-0.736244\pi$
$461$	−29.0243	−1.35180	−0.675898	+	0.736995i	$0.736244\pi$
$462$	−56.8206	−2.64353
$463$	−1.62873	−0.0756936	−0.0378468	−	0.999284i	$-0.512050\pi$
$463$	−1.62873	−0.0756936	−0.0378468	+	0.999284i	$0.512050\pi$
$464$	5.06302	0.235045
$465$	0	0
$466$	−4.07768	−0.188895
$467$	−2.54149	−0.117606	−0.0588030	−	0.998270i	$-0.518728\pi$
$467$	−2.54149	−0.117606	−0.0588030	+	0.998270i	$0.518728\pi$
$468$	−36.2234	−1.67443
$469$	48.1652	2.22406
$470$	0	0
$471$	0.320825	0.0147828
$472$	−3.27306	−0.150655
$473$	16.4306	0.755482
$474$	39.2489	1.80276
$475$	0	0
$476$	−34.8661	−1.59808
$477$	26.0793	1.19409
$478$	54.3161	2.48436
$479$	3.61801	0.165311	0.0826555	−	0.996578i	$-0.473660\pi$
$479$	3.61801	0.165311	0.0826555	+	0.996578i	$0.473660\pi$
$480$	0	0
$481$	26.7544	1.21989
$482$	2.03980	0.0929105
$483$	43.4444	1.97679
$484$	−10.8032	−0.491057
$485$	0	0
$486$	−10.2887	−0.466706
$487$	−8.19674	−0.371430	−0.185715	−	0.982604i	$-0.559460\pi$
$487$	−8.19674	−0.371430	−0.185715	+	0.982604i	$0.559460\pi$
$488$	0.202913	0.00918544
$489$	0.178867	0.00808866
$490$	0	0
$491$	−37.3799	−1.68693	−0.843466	−	0.537183i	$-0.819488\pi$
$491$	−37.3799	−1.68693	−0.843466	+	0.537183i	$0.819488\pi$
$492$	3.11499	0.140434
$493$	−5.98667	−0.269626
$494$	26.7517	1.20362
$495$	0	0
$496$	−25.6856	−1.15332
$497$	47.3929	2.12586
$498$	59.4088	2.66217
$499$	−39.8651	−1.78461	−0.892304	−	0.451436i	$-0.850912\pi$
$499$	−39.8651	−1.78461	−0.892304	+	0.451436i	$0.850912\pi$
$500$	0	0
$501$	57.2084	2.55588
$502$	38.5245	1.71943
$503$	20.7402	0.924760	0.462380	−	0.886682i	$-0.346996\pi$
$503$	20.7402	0.924760	0.462380	+	0.886682i	$0.346996\pi$
$504$	7.67673	0.341949
$505$	0	0
$506$	−19.0889	−0.848606
$507$	−17.7726	−0.789310
$508$	19.4185	0.861557
$509$	−17.4911	−0.775280	−0.387640	−	0.921811i	$-0.626710\pi$
$509$	−17.4911	−0.775280	−0.387640	+	0.921811i	$0.626710\pi$
$510$	0	0
$511$	−40.1675	−1.77691
$512$	−32.0053	−1.41445
$513$	48.6978	2.15006
$514$	15.6372	0.689729
$515$	0	0
$516$	−44.1156	−1.94208
$517$	11.8415	0.520787
$518$	−76.1929	−3.34772
$519$	−79.6904	−3.49802
$520$	0	0
$521$	30.7289	1.34626	0.673129	−	0.739525i	$-0.264950\pi$
$521$	30.7289	1.34626	0.673129	+	0.739525i	$0.264950\pi$
$522$	17.7129	0.775274
$523$	24.6264	1.07684	0.538420	−	0.842677i	$-0.319022\pi$
$523$	24.6264	1.07684	0.538420	+	0.842677i	$0.319022\pi$
$524$	−23.2281	−1.01472
$525$	0	0
$526$	52.7405	2.29960
$527$	30.3714	1.32300
$528$	27.2330	1.18516
$529$	−8.40482	−0.365427
$530$	0	0
$531$	62.5132	2.71284
$532$	−39.5648	−1.71535
$533$	−1.26741	−0.0548976
$534$	73.0326	3.16043
$535$	0	0
$536$	4.22850	0.182643
$537$	2.19418	0.0946856
$538$	46.2840	1.99545
$539$	−17.0450	−0.734179
$540$	0	0
$541$	−21.0892	−0.906694	−0.453347	−	0.891334i	$-0.649770\pi$
$541$	−21.0892	−0.906694	−0.453347	+	0.891334i	$0.649770\pi$
$542$	21.9480	0.942747
$543$	−7.82589	−0.335841
$544$	35.0109	1.50108
$545$	0	0
$546$	−62.0732	−2.65649
$547$	13.6988	0.585720	0.292860	−	0.956155i	$-0.405393\pi$
$547$	13.6988	0.585720	0.292860	+	0.956155i	$0.405393\pi$
$548$	32.4153	1.38471
$549$	−3.87550	−0.165402
$550$	0	0
$551$	−6.79347	−0.289411
$552$	3.81406	0.162337
$553$	23.6181	1.00434
$554$	−54.5259	−2.31658
$555$	0	0
$556$	41.7961	1.77255
$557$	28.8798	1.22367	0.611837	−	0.790984i	$-0.290431\pi$
$557$	28.8798	1.22367	0.611837	+	0.790984i	$0.290431\pi$
$558$	−89.8608	−3.80411
$559$	17.9495	0.759184
$560$	0	0
$561$	−32.2011	−1.35953
$562$	27.8213	1.17357
$563$	23.9667	1.01008	0.505038	−	0.863097i	$-0.331479\pi$
$563$	23.9667	1.01008	0.505038	+	0.863097i	$0.331479\pi$
$564$	−31.7938	−1.33876
$565$	0	0
$566$	−1.97700	−0.0830996
$567$	−42.7810	−1.79663
$568$	4.16070	0.174579
$569$	−3.75161	−0.157276	−0.0786378	−	0.996903i	$-0.525057\pi$
$569$	−3.75161	−0.157276	−0.0786378	+	0.996903i	$0.525057\pi$
$570$	0	0
$571$	−33.9368	−1.42021	−0.710106	−	0.704095i	$-0.751353\pi$
$571$	−33.9368	−1.42021	−0.710106	+	0.704095i	$0.751353\pi$
$572$	14.1641	0.592231
$573$	23.7971	0.994139
$574$	3.60941	0.150654
$575$	0	0
$576$	−57.8248	−2.40937
$577$	29.9776	1.24798	0.623991	−	0.781432i	$-0.285510\pi$
$577$	29.9776	1.24798	0.623991	+	0.781432i	$0.285510\pi$
$578$	−3.37130	−0.140228
$579$	−12.6182	−0.524395
$580$	0	0
$581$	35.7494	1.48313
$582$	37.2721	1.54498
$583$	−10.1975	−0.422339
$584$	−3.52637	−0.145922
$585$	0	0
$586$	−40.5839	−1.67651
$587$	−11.1712	−0.461085	−0.230542	−	0.973062i	$-0.574050\pi$
$587$	−11.1712	−0.461085	−0.230542	+	0.973062i	$0.574050\pi$
$588$	45.7651	1.88732
$589$	34.4644	1.42008
$590$	0	0
$591$	−64.0356	−2.63407
$592$	36.5178	1.50087
$593$	−47.4079	−1.94681	−0.973405	−	0.229091i	$-0.926425\pi$
$593$	−47.4079	−1.94681	−0.973405	+	0.229091i	$0.926425\pi$
$594$	49.6488	2.03711
$595$	0	0
$596$	−11.8781	−0.486547
$597$	−8.23856	−0.337182
$598$	−20.8535	−0.852764
$599$	14.7236	0.601592	0.300796	−	0.953689i	$-0.402748\pi$
$599$	14.7236	0.601592	0.300796	+	0.953689i	$0.402748\pi$
$600$	0	0
$601$	−10.3891	−0.423778	−0.211889	−	0.977294i	$-0.567962\pi$
$601$	−10.3891	−0.423778	−0.211889	+	0.977294i	$0.567962\pi$
$602$	−51.1178	−2.08341
$603$	−80.7614	−3.28886
$604$	−32.6949	−1.33034
$605$	0	0
$606$	−38.3251	−1.55685
$607$	−28.9124	−1.17352	−0.586760	−	0.809761i	$-0.699597\pi$
$607$	−28.9124	−1.17352	−0.586760	+	0.809761i	$0.699597\pi$
$608$	39.7292	1.61123
$609$	15.7632	0.638755
$610$	0	0
$611$	12.9361	0.523339
$612$	58.4621	2.36319
$613$	7.61596	0.307606	0.153803	−	0.988102i	$-0.450848\pi$
$613$	7.61596	0.307606	0.153803	+	0.988102i	$0.450848\pi$
$614$	−63.8444	−2.57655
$615$	0	0
$616$	−3.00176	−0.120944
$617$	0.589909	0.0237489	0.0118744	−	0.999929i	$-0.496220\pi$
$617$	0.589909	0.0237489	0.0118744	+	0.999929i	$0.496220\pi$
$618$	33.4213	1.34440
$619$	9.31049	0.374220	0.187110	−	0.982339i	$-0.440088\pi$
$619$	9.31049	0.374220	0.187110	+	0.982339i	$0.440088\pi$
$620$	0	0
$621$	−37.9609	−1.52332
$622$	−52.2717	−2.09590
$623$	43.9475	1.76072
$624$	29.7505	1.19097
$625$	0	0
$626$	−25.0192	−0.999969
$627$	−36.5407	−1.45930
$628$	0.227757	0.00908848
$629$	−43.1797	−1.72169
$630$	0	0
$631$	29.1514	1.16050	0.580249	−	0.814439i	$-0.302955\pi$
$631$	29.1514	1.16050	0.580249	+	0.814439i	$0.302955\pi$
$632$	2.07347	0.0824783
$633$	−65.7625	−2.61382
$634$	62.1597	2.46868
$635$	0	0
$636$	27.3800	1.08569
$637$	−18.6206	−0.737777
$638$	−6.92613	−0.274208
$639$	−79.4664	−3.14364
$640$	0	0
$641$	−32.8869	−1.29895	−0.649477	−	0.760381i	$-0.725012\pi$
$641$	−32.8869	−1.29895	−0.649477	+	0.760381i	$0.725012\pi$
$642$	−49.9379	−1.97089
$643$	−33.8492	−1.33488	−0.667441	−	0.744663i	$-0.732610\pi$
$643$	−33.8492	−1.33488	−0.667441	+	0.744663i	$0.732610\pi$
$644$	30.8416	1.21533
$645$	0	0
$646$	−43.1755	−1.69872
$647$	−19.4225	−0.763575	−0.381788	−	0.924250i	$-0.624691\pi$
$647$	−19.4225	−0.763575	−0.381788	+	0.924250i	$0.624691\pi$
$648$	−3.75581	−0.147542
$649$	−24.4440	−0.959510
$650$	0	0
$651$	−79.9692	−3.13424
$652$	0.126980	0.00497290
$653$	1.17925	0.0461475	0.0230737	−	0.999734i	$-0.492655\pi$
$653$	1.17925	0.0461475	0.0230737	+	0.999734i	$0.492655\pi$
$654$	−1.41434	−0.0553050
$655$	0	0
$656$	−1.72992	−0.0675420
$657$	67.3513	2.62763
$658$	−36.8403	−1.43618
$659$	15.0358	0.585711	0.292855	−	0.956157i	$-0.405395\pi$
$659$	15.0358	0.585711	0.292855	+	0.956157i	$0.405395\pi$
$660$	0	0
$661$	43.5945	1.69563	0.847814	−	0.530293i	$-0.177918\pi$
$661$	43.5945	1.69563	0.847814	+	0.530293i	$0.177918\pi$
$662$	8.52966	0.331515
$663$	−35.1779	−1.36620
$664$	3.13849	0.121797
$665$	0	0
$666$	127.757	4.95049
$667$	5.29564	0.205048
$668$	40.6128	1.57136
$669$	−66.3705	−2.56603
$670$	0	0
$671$	1.51540	0.0585014
$672$	−92.1852	−3.55612
$673$	37.2055	1.43416	0.717082	−	0.696988i	$-0.245477\pi$
$673$	37.2055	1.43416	0.717082	+	0.696988i	$0.245477\pi$
$674$	−44.9857	−1.73278
$675$	0	0
$676$	−12.6169	−0.485267
$677$	25.5395	0.981561	0.490781	−	0.871283i	$-0.336712\pi$
$677$	25.5395	0.981561	0.490781	+	0.871283i	$0.336712\pi$
$678$	−84.2282	−3.23476
$679$	22.4285	0.860728
$680$	0	0
$681$	−33.2148	−1.27280
$682$	35.1374	1.34548
$683$	19.8871	0.760957	0.380478	−	0.924790i	$-0.375759\pi$
$683$	19.8871	0.760957	0.380478	+	0.924790i	$0.375759\pi$
$684$	66.3407	2.53660
$685$	0	0
$686$	−0.317170	−0.0121096
$687$	−68.3276	−2.60686
$688$	24.4998	0.934046
$689$	−11.1402	−0.424409
$690$	0	0
$691$	−10.6290	−0.404346	−0.202173	−	0.979350i	$-0.564800\pi$
$691$	−10.6290	−0.404346	−0.202173	+	0.979350i	$0.564800\pi$
$692$	−56.5729	−2.15058
$693$	57.3316	2.17785
$694$	50.9888	1.93551
$695$	0	0
$696$	1.38387	0.0524556
$697$	2.04551	0.0774793
$698$	−21.5976	−0.817483
$699$	6.08466	0.230143
$700$	0	0
$701$	37.0593	1.39971	0.699854	−	0.714285i	$-0.253248\pi$
$701$	37.0593	1.39971	0.699854	+	0.714285i	$0.253248\pi$
$702$	54.2384	2.04710
$703$	−48.9988	−1.84803
$704$	22.6107	0.852173
$705$	0	0
$706$	60.5802	2.27997
$707$	−23.0622	−0.867343
$708$	65.6311	2.46657
$709$	17.5461	0.658957	0.329478	−	0.944163i	$-0.393127\pi$
$709$	17.5461	0.658957	0.329478	+	0.944163i	$0.393127\pi$
$710$	0	0
$711$	−39.6019	−1.48519
$712$	3.85822	0.144593
$713$	−26.8657	−1.00613
$714$	100.182	3.74921
$715$	0	0
$716$	1.55766	0.0582127
$717$	−81.0498	−3.02686
$718$	−22.8468	−0.852636
$719$	−11.0085	−0.410547	−0.205273	−	0.978705i	$-0.565808\pi$
$719$	−11.0085	−0.410547	−0.205273	+	0.978705i	$0.565808\pi$
$720$	0	0
$721$	20.1114	0.748986
$722$	−10.2378	−0.381011
$723$	−3.04377	−0.113199
$724$	−5.55567	−0.206475
$725$	0	0
$726$	31.0413	1.15205
$727$	6.22011	0.230691	0.115346	−	0.993325i	$-0.463202\pi$
$727$	6.22011	0.230691	0.115346	+	0.993325i	$0.463202\pi$
$728$	−3.27925	−0.121537
$729$	−18.9995	−0.703685
$730$	0	0
$731$	−28.9693	−1.07147
$732$	−4.06879	−0.150387
$733$	26.1126	0.964490	0.482245	−	0.876037i	$-0.339822\pi$
$733$	26.1126	0.964490	0.482245	+	0.876037i	$0.339822\pi$
$734$	−14.0266	−0.517731
$735$	0	0
$736$	−30.9697	−1.14156
$737$	31.5794	1.16324
$738$	−6.05211	−0.222781
$739$	46.0505	1.69399	0.846997	−	0.531598i	$-0.178408\pi$
$739$	46.0505	1.69399	0.846997	+	0.531598i	$0.178408\pi$
$740$	0	0
$741$	−39.9186	−1.46645
$742$	31.7258	1.16469
$743$	−25.8774	−0.949348	−0.474674	−	0.880162i	$-0.657434\pi$
$743$	−25.8774	−0.949348	−0.474674	+	0.880162i	$0.657434\pi$
$744$	−7.02062	−0.257389
$745$	0	0
$746$	−55.2480	−2.02277
$747$	−59.9431	−2.19320
$748$	−22.8599	−0.835840
$749$	−30.0502	−1.09801
$750$	0	0
$751$	−35.0661	−1.27958	−0.639790	−	0.768550i	$-0.720979\pi$
$751$	−35.0661	−1.27958	−0.639790	+	0.768550i	$0.720979\pi$
$752$	17.6568	0.643878
$753$	−57.4858	−2.09490
$754$	−7.56639	−0.275552
$755$	0	0
$756$	−80.2166	−2.91745
$757$	−40.6694	−1.47816	−0.739078	−	0.673620i	$-0.764738\pi$
$757$	−40.6694	−1.47816	−0.739078	+	0.673620i	$0.764738\pi$
$758$	32.5616	1.18269
$759$	28.4842	1.03391
$760$	0	0
$761$	9.35590	0.339151	0.169576	−	0.985517i	$-0.445760\pi$
$761$	9.35590	0.339151	0.169576	+	0.985517i	$0.445760\pi$
$762$	−55.7958	−2.02127
$763$	−0.851081	−0.0308112
$764$	16.8938	0.611196
$765$	0	0
$766$	28.7919	1.04029
$767$	−26.7036	−0.964211
$768$	39.9523	1.44165
$769$	35.4175	1.27719	0.638594	−	0.769544i	$-0.279516\pi$
$769$	35.4175	1.27719	0.638594	+	0.769544i	$0.279516\pi$
$770$	0	0
$771$	−23.3337	−0.840342
$772$	−8.95778	−0.322397
$773$	17.4611	0.628031	0.314015	−	0.949418i	$-0.398326\pi$
$773$	17.4611	0.628031	0.314015	+	0.949418i	$0.398326\pi$
$774$	85.7123	3.08087
$775$	0	0
$776$	1.96904	0.0706843
$777$	113.694	4.07875
$778$	−56.2567	−2.01690
$779$	2.32117	0.0831647
$780$	0	0
$781$	31.0730	1.11188
$782$	33.6562	1.20354
$783$	−13.7736	−0.492227
$784$	−25.4158	−0.907708
$785$	0	0
$786$	66.7419	2.38061
$787$	44.5610	1.58843	0.794213	−	0.607639i	$-0.207883\pi$
$787$	44.5610	1.58843	0.794213	+	0.607639i	$0.207883\pi$
$788$	−45.4595	−1.61943
$789$	−78.6987	−2.80175
$790$	0	0
$791$	−50.6845	−1.80213
$792$	5.03323	0.178848
$793$	1.65549	0.0587880
$794$	−25.0032	−0.887331
$795$	0	0
$796$	−5.84862	−0.207299
$797$	−31.8324	−1.12756	−0.563780	−	0.825925i	$-0.690654\pi$
$797$	−31.8324	−1.12756	−0.563780	+	0.825925i	$0.690654\pi$
$798$	113.683	4.02433
$799$	−20.8780	−0.738610
$800$	0	0
$801$	−73.6894	−2.60369
$802$	50.2419	1.77410
$803$	−26.3358	−0.929369
$804$	−84.7894	−2.99029
$805$	0	0
$806$	38.3856	1.35207
$807$	−69.0644	−2.43118
$808$	−2.02467	−0.0712275
$809$	−2.61203	−0.0918341	−0.0459171	−	0.998945i	$-0.514621\pi$
$809$	−2.61203	−0.0918341	−0.0459171	+	0.998945i	$0.514621\pi$
$810$	0	0
$811$	−49.5214	−1.73893	−0.869466	−	0.493993i	$-0.835537\pi$
$811$	−49.5214	−1.73893	−0.869466	+	0.493993i	$0.835537\pi$
$812$	11.1904	0.392706
$813$	−32.7505	−1.14861
$814$	−49.9557	−1.75095
$815$	0	0
$816$	−48.0152	−1.68087
$817$	−32.8734	−1.15009
$818$	−79.1795	−2.76845
$819$	62.6314	2.18852
$820$	0	0
$821$	0.473974	0.0165418	0.00827091	−	0.999966i	$-0.497367\pi$
$821$	0.473974	0.0165418	0.00827091	+	0.999966i	$0.497367\pi$
$822$	−93.1400	−3.24863
$823$	5.60333	0.195320	0.0976599	−	0.995220i	$-0.468864\pi$
$823$	5.60333	0.195320	0.0976599	+	0.995220i	$0.468864\pi$
$824$	1.76561	0.0615079
$825$	0	0
$826$	76.0483	2.64606
$827$	−9.70539	−0.337489	−0.168745	−	0.985660i	$-0.553971\pi$
$827$	−9.70539	−0.337489	−0.168745	+	0.985660i	$0.553971\pi$
$828$	−51.7139	−1.79718
$829$	−12.2162	−0.424286	−0.212143	−	0.977239i	$-0.568044\pi$
$829$	−12.2162	−0.424286	−0.212143	+	0.977239i	$0.568044\pi$
$830$	0	0
$831$	81.3628	2.82245
$832$	24.7009	0.856349
$833$	30.0524	1.04126
$834$	−120.094	−4.15851
$835$	0	0
$836$	−25.9406	−0.897174
$837$	69.8756	2.41526
$838$	−52.7446	−1.82203
$839$	−4.49074	−0.155037	−0.0775187	−	0.996991i	$-0.524700\pi$
$839$	−4.49074	−0.155037	−0.0775187	+	0.996991i	$0.524700\pi$
$840$	0	0
$841$	−27.0786	−0.933743
$842$	−52.3369	−1.80365
$843$	−41.5145	−1.42984
$844$	−46.6854	−1.60698
$845$	0	0
$846$	61.7723	2.12378
$847$	18.6792	0.641824
$848$	−15.2056	−0.522162
$849$	2.95006	0.101246
$850$	0	0
$851$	38.1956	1.30933
$852$	−83.4298	−2.85826
$853$	27.5711	0.944015	0.472008	−	0.881595i	$-0.343530\pi$
$853$	27.5711	0.944015	0.472008	+	0.881595i	$0.343530\pi$
$854$	−4.71460	−0.161330
$855$	0	0
$856$	−2.63816	−0.0901703
$857$	41.2192	1.40802	0.704011	−	0.710189i	$-0.251390\pi$
$857$	41.2192	1.40802	0.704011	+	0.710189i	$0.251390\pi$
$858$	−40.6981	−1.38941
$859$	42.7734	1.45941	0.729705	−	0.683763i	$-0.239658\pi$
$859$	42.7734	1.45941	0.729705	+	0.683763i	$0.239658\pi$
$860$	0	0
$861$	−5.38591	−0.183551
$862$	−25.9028	−0.882254
$863$	−20.3390	−0.692347	−0.346174	−	0.938170i	$-0.612519\pi$
$863$	−20.3390	−0.692347	−0.346174	+	0.938170i	$0.612519\pi$
$864$	80.5497	2.74036
$865$	0	0
$866$	8.53143	0.289910
$867$	5.03061	0.170849
$868$	−56.7709	−1.92693
$869$	15.4852	0.525298
$870$	0	0
$871$	34.4986	1.16894
$872$	−0.0747178	−0.00253026
$873$	−37.6073	−1.27281
$874$	38.1918	1.29186
$875$	0	0
$876$	70.7105	2.38908
$877$	−3.21300	−0.108495	−0.0542476	−	0.998528i	$-0.517276\pi$
$877$	−3.21300	−0.108495	−0.0542476	+	0.998528i	$0.517276\pi$
$878$	−21.0451	−0.710238
$879$	60.5588	2.04260
$880$	0	0
$881$	−46.0744	−1.55228	−0.776142	−	0.630558i	$-0.782826\pi$
$881$	−46.0744	−1.55228	−0.776142	+	0.630558i	$0.782826\pi$
$882$	−88.9170	−2.99399
$883$	21.9223	0.737744	0.368872	−	0.929480i	$-0.379744\pi$
$883$	21.9223	0.737744	0.368872	+	0.929480i	$0.379744\pi$
$884$	−24.9731	−0.839936
$885$	0	0
$886$	33.9429	1.14033
$887$	−19.8287	−0.665782	−0.332891	−	0.942965i	$-0.608024\pi$
$887$	−19.8287	−0.665782	−0.332891	+	0.942965i	$0.608024\pi$
$888$	9.98138	0.334953
$889$	−33.5752	−1.12608
$890$	0	0
$891$	−28.0493	−0.939686
$892$	−47.1170	−1.57759
$893$	−23.6916	−0.792809
$894$	34.1298	1.14147
$895$	0	0
$896$	−9.77174	−0.326451
$897$	31.1174	1.03898
$898$	−1.12661	−0.0375953
$899$	−9.74782	−0.325108
$900$	0	0
$901$	17.9796	0.598986
$902$	2.36650	0.0787959
$903$	76.2774	2.53835
$904$	−4.44967	−0.147994
$905$	0	0
$906$	93.9433	3.12106
$907$	40.4798	1.34411	0.672056	−	0.740501i	$-0.265412\pi$
$907$	40.4798	1.34411	0.672056	+	0.740501i	$0.265412\pi$
$908$	−23.5795	−0.782514
$909$	38.6698	1.28260
$910$	0	0
$911$	31.2481	1.03530	0.517649	−	0.855593i	$-0.326807\pi$
$911$	31.2481	1.03530	0.517649	+	0.855593i	$0.326807\pi$
$912$	−54.4860	−1.80421
$913$	23.4390	0.775717
$914$	66.6556	2.20477
$915$	0	0
$916$	−48.5063	−1.60269
$917$	40.1621	1.32627
$918$	−87.5371	−2.88915
$919$	6.52543	0.215254	0.107627	−	0.994191i	$-0.465675\pi$
$919$	6.52543	0.215254	0.107627	+	0.994191i	$0.465675\pi$
$920$	0	0
$921$	95.2677	3.13918
$922$	59.2039	1.94978
$923$	33.9455	1.11733
$924$	60.1910	1.98014
$925$	0	0
$926$	3.32229	0.109177
$927$	−33.7219	−1.10757
$928$	−11.2369	−0.368869
$929$	−40.7019	−1.33539	−0.667693	−	0.744437i	$-0.732718\pi$
$929$	−40.7019	−1.33539	−0.667693	+	0.744437i	$0.732718\pi$
$930$	0	0
$931$	34.1025	1.11766
$932$	4.31955	0.141492
$933$	77.9991	2.55358
$934$	5.18414	0.169630
$935$	0	0
$936$	5.49851	0.179725
$937$	−31.0396	−1.01402	−0.507010	−	0.861940i	$-0.669249\pi$
$937$	−31.0396	−1.01402	−0.507010	+	0.861940i	$0.669249\pi$
$938$	−98.2475	−3.20789
$939$	37.3333	1.21833
$940$	0	0
$941$	44.6429	1.45532	0.727658	−	0.685940i	$-0.240609\pi$
$941$	44.6429	1.45532	0.727658	+	0.685940i	$0.240609\pi$
$942$	−0.654420	−0.0213222
$943$	−1.80940	−0.0589222
$944$	−36.4485	−1.18630
$945$	0	0
$946$	−33.5153	−1.08968
$947$	−6.57190	−0.213558	−0.106779	−	0.994283i	$-0.534054\pi$
$947$	−6.57190	−0.213558	−0.106779	+	0.994283i	$0.534054\pi$
$948$	−41.5771	−1.35036
$949$	−28.7703	−0.933923
$950$	0	0
$951$	−92.7539	−3.00775
$952$	5.29248	0.171530
$953$	27.7405	0.898604	0.449302	−	0.893380i	$-0.351673\pi$
$953$	27.7405	0.898604	0.449302	+	0.893380i	$0.351673\pi$
$954$	−53.1966	−1.72230
$955$	0	0
$956$	−57.5380	−1.86091
$957$	10.3351	0.334086
$958$	−7.38002	−0.238438
$959$	−56.0472	−1.80986
$960$	0	0
$961$	18.4524	0.595238
$962$	−54.5737	−1.75953
$963$	50.3870	1.62370
$964$	−2.16080	−0.0695946
$965$	0	0
$966$	−88.6181	−2.85124
$967$	−31.5996	−1.01617	−0.508087	−	0.861306i	$-0.669647\pi$
$967$	−31.5996	−1.01617	−0.508087	+	0.861306i	$0.669647\pi$
$968$	1.63987	0.0527075
$969$	64.4259	2.06966
$970$	0	0
$971$	−41.1090	−1.31925	−0.659625	−	0.751595i	$-0.729285\pi$
$971$	−41.1090	−1.31925	−0.659625	+	0.751595i	$0.729285\pi$
$972$	10.8990	0.349586
$973$	−72.2668	−2.31677
$974$	16.7197	0.535735
$975$	0	0
$976$	2.25962	0.0723286
$977$	−29.3395	−0.938653	−0.469327	−	0.883025i	$-0.655503\pi$
$977$	−29.3395	−0.938653	−0.469327	+	0.883025i	$0.655503\pi$
$978$	−0.364854	−0.0116668
$979$	28.8141	0.920903
$980$	0	0
$981$	1.42706	0.0455625
$982$	76.2477	2.43316
$983$	−11.8109	−0.376709	−0.188354	−	0.982101i	$-0.560315\pi$
$983$	−11.8109	−0.376709	−0.188354	+	0.982101i	$0.560315\pi$
$984$	−0.472838	−0.0150735
$985$	0	0
$986$	12.2116	0.388898
$987$	54.9726	1.74980
$988$	−28.3386	−0.901570
$989$	25.6254	0.814842
$990$	0	0
$991$	4.64162	0.147446	0.0737229	−	0.997279i	$-0.476512\pi$
$991$	4.64162	0.147446	0.0737229	+	0.997279i	$0.476512\pi$
$992$	57.0066	1.80996
$993$	−12.7278	−0.403906
$994$	−96.6721	−3.06625
$995$	0	0
$996$	−62.9327	−1.99410
$997$	36.4753	1.15518	0.577592	−	0.816325i	$-0.303992\pi$
$997$	36.4753	1.15518	0.577592	+	0.816325i	$0.303992\pi$
$998$	81.3170	2.57404
$999$	−99.3438	−3.14310

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.j.1.6		25
5.4	even	2		1205.2.a.e.1.20	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.a.e.1.20	✓	25	5.4	even	2
6025.2.a.j.1.6		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)

\(f(q)\)	\(=\)	\(q-2.03980 q^{2} +3.04377 q^{3} +2.16080 q^{4} -6.20869 q^{6} -3.73609 q^{7} -0.327997 q^{8} +6.26453 q^{9} +O(q^{10})\)	\(q-2.03980 q^{2} +3.04377 q^{3} +2.16080 q^{4} -6.20869 q^{6} -3.73609 q^{7} -0.327997 q^{8} +6.26453 q^{9} -2.44956 q^{11} +6.57697 q^{12} -2.67600 q^{13} +7.62089 q^{14} -3.65255 q^{16} +4.31888 q^{17} -12.7784 q^{18} +4.90092 q^{19} -11.3718 q^{21} +4.99662 q^{22} -3.82036 q^{23} -0.998349 q^{24} +5.45852 q^{26} +9.93647 q^{27} -8.07294 q^{28} -1.38616 q^{29} +7.03224 q^{31} +8.10647 q^{32} -7.45590 q^{33} -8.80967 q^{34} +13.5364 q^{36} -9.99789 q^{37} -9.99691 q^{38} -8.14513 q^{39} +0.473620 q^{41} +23.1962 q^{42} -6.70759 q^{43} -5.29301 q^{44} +7.79279 q^{46} -4.83412 q^{47} -11.1175 q^{48} +6.95838 q^{49} +13.1457 q^{51} -5.78230 q^{52} +4.16301 q^{53} -20.2684 q^{54} +1.22543 q^{56} +14.9173 q^{57} +2.82750 q^{58} +9.97892 q^{59} -0.618642 q^{61} -14.3444 q^{62} -23.4049 q^{63} -9.23052 q^{64} +15.2086 q^{66} -12.8919 q^{67} +9.33224 q^{68} -11.6283 q^{69} -12.6851 q^{71} -2.05475 q^{72} +10.7512 q^{73} +20.3937 q^{74} +10.5899 q^{76} +9.15178 q^{77} +16.6145 q^{78} -6.32161 q^{79} +11.4507 q^{81} -0.966092 q^{82} -9.56865 q^{83} -24.5722 q^{84} +13.6822 q^{86} -4.21916 q^{87} +0.803450 q^{88} -11.7630 q^{89} +9.99779 q^{91} -8.25504 q^{92} +21.4045 q^{93} +9.86065 q^{94} +24.6742 q^{96} -6.00321 q^{97} -14.1937 q^{98} -15.3453 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9}+O(q^{10}) \) 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9	\( 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9} + 2 q^{11} - 20 q^{12} - 14 q^{13} - 5 q^{14} + 38 q^{16} - 7 q^{17} - 9 q^{18} + 30 q^{19} + q^{21} - q^{22} - 43 q^{23} - 6 q^{24} - 22 q^{26} - 42 q^{27} - 32 q^{28} - 4 q^{29} + 14 q^{31} - 26 q^{32} - 4 q^{33} + 7 q^{34} + 15 q^{36} - 16 q^{37} - 14 q^{38} - 21 q^{39} - q^{41} + 25 q^{42} - 35 q^{43} - 52 q^{44} - 27 q^{46} - 50 q^{47} - 26 q^{48} + 46 q^{49} - 7 q^{51} - 3 q^{52} - 4 q^{53} - 31 q^{54} - 51 q^{56} - 2 q^{58} + 6 q^{59} + 19 q^{61} - 28 q^{63} + 49 q^{64} - 27 q^{66} - 65 q^{67} + 25 q^{68} + 2 q^{69} - 34 q^{71} + 10 q^{72} - 8 q^{73} - 42 q^{74} + 71 q^{76} - q^{77} + 59 q^{78} - 12 q^{79} + 29 q^{81} - 11 q^{82} - 41 q^{83} - 10 q^{84} - 13 q^{86} - 40 q^{87} + 52 q^{88} - 24 q^{89} + 46 q^{91} - 85 q^{92} + 30 q^{93} + 14 q^{94} - 30 q^{96} - 9 q^{97} + 64 q^{98} + 2 q^{99}+O(q^{100}) \) 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9 + 2 * q^11 - 20 * q^12 - 14 * q^13 - 5 * q^14 + 38 * q^16 - 7 * q^17 - 9 * q^18 + 30 * q^19 + q^21 - q^22 - 43 * q^23 - 6 * q^24 - 22 * q^26 - 42 * q^27 - 32 * q^28 - 4 * q^29 + 14 * q^31 - 26 * q^32 - 4 * q^33 + 7 * q^34 + 15 * q^36 - 16 * q^37 - 14 * q^38 - 21 * q^39 - q^41 + 25 * q^42 - 35 * q^43 - 52 * q^44 - 27 * q^46 - 50 * q^47 - 26 * q^48 + 46 * q^49 - 7 * q^51 - 3 * q^52 - 4 * q^53 - 31 * q^54 - 51 * q^56 - 2 * q^58 + 6 * q^59 + 19 * q^61 - 28 * q^63 + 49 * q^64 - 27 * q^66 - 65 * q^67 + 25 * q^68 + 2 * q^69 - 34 * q^71 + 10 * q^72 - 8 * q^73 - 42 * q^74 + 71 * q^76 - q^77 + 59 * q^78 - 12 * q^79 + 29 * q^81 - 11 * q^82 - 41 * q^83 - 10 * q^84 - 13 * q^86 - 40 * q^87 + 52 * q^88 - 24 * q^89 + 46 * q^91 - 85 * q^92 + 30 * q^93 + 14 * q^94 - 30 * q^96 - 9 * q^97 + 64 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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