LMFDB - Embedded newform 6025.2.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.1098672178$
Analytic rank:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-2.60048 q^{2} -3.03019 q^{3} +4.76251 q^{4} +7.87996 q^{6} -1.49919 q^{7} -7.18385 q^{8} +6.18207 q^{9} +O(q^{10})$	\(q-2.60048 q^{2} -3.03019 q^{3} +4.76251 q^{4} +7.87996 q^{6} -1.49919 q^{7} -7.18385 q^{8} +6.18207 q^{9} -4.55264 q^{11} -14.4313 q^{12} -6.26213 q^{13} +3.89862 q^{14} +9.15646 q^{16} +5.97793 q^{17} -16.0764 q^{18} -3.57983 q^{19} +4.54284 q^{21} +11.8391 q^{22} -6.76317 q^{23} +21.7684 q^{24} +16.2846 q^{26} -9.64228 q^{27} -7.13990 q^{28} -3.27850 q^{29} -4.86195 q^{31} -9.44350 q^{32} +13.7954 q^{33} -15.5455 q^{34} +29.4421 q^{36} +9.31160 q^{37} +9.30929 q^{38} +18.9755 q^{39} +1.97370 q^{41} -11.8136 q^{42} +5.98429 q^{43} -21.6820 q^{44} +17.5875 q^{46} -8.42175 q^{47} -27.7458 q^{48} -4.75243 q^{49} -18.1143 q^{51} -29.8234 q^{52} +11.0696 q^{53} +25.0746 q^{54} +10.7700 q^{56} +10.8476 q^{57} +8.52567 q^{58} +8.77550 q^{59} +9.19955 q^{61} +12.6434 q^{62} -9.26810 q^{63} +6.24475 q^{64} -35.8747 q^{66} +15.1272 q^{67} +28.4700 q^{68} +20.4937 q^{69} -8.51912 q^{71} -44.4110 q^{72} -1.35081 q^{73} -24.2147 q^{74} -17.0490 q^{76} +6.82528 q^{77} -49.3454 q^{78} +0.518054 q^{79} +10.6718 q^{81} -5.13256 q^{82} -12.6728 q^{83} +21.6353 q^{84} -15.5620 q^{86} +9.93448 q^{87} +32.7055 q^{88} +6.25837 q^{89} +9.38812 q^{91} -32.2096 q^{92} +14.7327 q^{93} +21.9006 q^{94} +28.6156 q^{96} -7.07145 q^{97} +12.3586 q^{98} -28.1448 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q - 11 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 16 * q^7 - 33 * q^8 + 38 * q^9	$ 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9} + q^{11} - 14 q^{12} - 9 q^{13} - q^{14} + 43 q^{16} - 12 q^{17} - 42 q^{18} + 2 q^{21} - 5 q^{22} - 77 q^{23} - 2 q^{24} + 2 q^{26} - 38 q^{27} - 42 q^{28} + 2 q^{29} + q^{31} - 72 q^{32} - 20 q^{33} + 5 q^{34} + 32 q^{36} - 28 q^{37} - 23 q^{38} - 2 q^{39} - 2 q^{41} - 37 q^{42} - 31 q^{43} + 3 q^{44} + 14 q^{46} - 96 q^{47} - 13 q^{48} + 40 q^{49} - 10 q^{51} - 42 q^{52} - 54 q^{53} + 4 q^{54} - 15 q^{56} - 37 q^{57} - 27 q^{58} + q^{59} + 5 q^{61} - 39 q^{62} - 70 q^{63} + 65 q^{64} - 52 q^{66} - 34 q^{67} - 52 q^{68} + 21 q^{69} - 9 q^{71} - 70 q^{72} - 25 q^{73} + 22 q^{74} - 47 q^{76} - 54 q^{77} - 58 q^{78} + 13 q^{79} + 12 q^{81} + 5 q^{82} - 63 q^{83} + 95 q^{84} - 18 q^{86} - 47 q^{87} - 13 q^{88} + 19 q^{89} - 31 q^{91} - 137 q^{92} - 52 q^{93} + 120 q^{94} - 49 q^{96} - 36 q^{97} - 64 q^{98} + 16 q^{99}+O(q^{100}) $ 40 * q - 11 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 16 * q^7 - 33 * q^8 + 38 * q^9 + q^11 - 14 * q^12 - 9 * q^13 - q^14 + 43 * q^16 - 12 * q^17 - 42 * q^18 + 2 * q^21 - 5 * q^22 - 77 * q^23 - 2 * q^24 + 2 * q^26 - 38 * q^27 - 42 * q^28 + 2 * q^29 + q^31 - 72 * q^32 - 20 * q^33 + 5 * q^34 + 32 * q^36 - 28 * q^37 - 23 * q^38 - 2 * q^39 - 2 * q^41 - 37 * q^42 - 31 * q^43 + 3 * q^44 + 14 * q^46 - 96 * q^47 - 13 * q^48 + 40 * q^49 - 10 * q^51 - 42 * q^52 - 54 * q^53 + 4 * q^54 - 15 * q^56 - 37 * q^57 - 27 * q^58 + q^59 + 5 * q^61 - 39 * q^62 - 70 * q^63 + 65 * q^64 - 52 * q^66 - 34 * q^67 - 52 * q^68 + 21 * q^69 - 9 * q^71 - 70 * q^72 - 25 * q^73 + 22 * q^74 - 47 * q^76 - 54 * q^77 - 58 * q^78 + 13 * q^79 + 12 * q^81 + 5 * q^82 - 63 * q^83 + 95 * q^84 - 18 * q^86 - 47 * q^87 - 13 * q^88 + 19 * q^89 - 31 * q^91 - 137 * q^92 - 52 * q^93 + 120 * q^94 - 49 * q^96 - 36 * q^97 - 64 * q^98 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.60048	−1.83882	−0.919409	−	0.393302i	$-0.871333\pi$
$2$	−2.60048	−1.83882	−0.919409	+	0.393302i	$0.871333\pi$
$3$	−3.03019	−1.74948	−0.874741	−	0.484590i	$-0.838969\pi$
$3$	−3.03019	−1.74948	−0.874741	+	0.484590i	$0.838969\pi$
$4$	4.76251	2.38125
$5$	0	0
$5$	0	0
$6$	7.87996	3.21698
$7$	−1.49919	−0.566641	−0.283320	−	0.959025i	$-0.591436\pi$
$7$	−1.49919	−0.566641	−0.283320	+	0.959025i	$0.591436\pi$
$8$	−7.18385	−2.53987
$9$	6.18207	2.06069
$10$	0	0
$11$	−4.55264	−1.37267	−0.686337	−	0.727284i	$-0.740782\pi$
$11$	−4.55264	−1.37267	−0.686337	+	0.727284i	$0.740782\pi$
$12$	−14.4313	−4.16596
$13$	−6.26213	−1.73680	−0.868401	−	0.495862i	$-0.834852\pi$
$13$	−6.26213	−1.73680	−0.868401	+	0.495862i	$0.834852\pi$
$14$	3.89862	1.04195
$15$	0	0
$16$	9.15646	2.28911
$17$	5.97793	1.44986	0.724931	−	0.688821i	$-0.241872\pi$
$17$	5.97793	1.44986	0.724931	+	0.688821i	$0.241872\pi$
$18$	−16.0764	−3.78923
$19$	−3.57983	−0.821270	−0.410635	−	0.911800i	$-0.634693\pi$
$19$	−3.57983	−0.821270	−0.410635	+	0.911800i	$0.634693\pi$
$20$	0	0
$21$	4.54284	0.991328
$22$	11.8391	2.52410
$23$	−6.76317	−1.41022	−0.705109	−	0.709099i	$-0.749102\pi$
$23$	−6.76317	−1.41022	−0.705109	+	0.709099i	$0.749102\pi$
$24$	21.7684	4.44347
$25$	0	0
$26$	16.2846	3.19366
$27$	−9.64228	−1.85566
$28$	−7.13990	−1.34931
$29$	−3.27850	−0.608802	−0.304401	−	0.952544i	$-0.598456\pi$
$29$	−3.27850	−0.608802	−0.304401	+	0.952544i	$0.598456\pi$
$30$	0	0
$31$	−4.86195	−0.873232	−0.436616	−	0.899648i	$-0.643823\pi$
$31$	−4.86195	−0.873232	−0.436616	+	0.899648i	$0.643823\pi$
$32$	−9.44350	−1.66939
$33$	13.7954	2.40147
$34$	−15.5455	−2.66603
$35$	0	0
$36$	29.4421	4.90702
$37$	9.31160	1.53082	0.765409	−	0.643544i	$-0.222537\pi$
$37$	9.31160	1.53082	0.765409	+	0.643544i	$0.222537\pi$
$38$	9.30929	1.51017
$39$	18.9755	3.03851
$40$	0	0
$41$	1.97370	0.308240	0.154120	−	0.988052i	$-0.450746\pi$
$41$	1.97370	0.308240	0.154120	+	0.988052i	$0.450746\pi$
$42$	−11.8136	−1.82287
$43$	5.98429	0.912595	0.456298	−	0.889827i	$-0.349175\pi$
$43$	5.98429	0.912595	0.456298	+	0.889827i	$0.349175\pi$
$44$	−21.6820	−3.26868
$45$	0	0
$46$	17.5875	2.59313
$47$	−8.42175	−1.22844	−0.614219	−	0.789135i	$-0.710529\pi$
$47$	−8.42175	−1.22844	−0.614219	+	0.789135i	$0.710529\pi$
$48$	−27.7458	−4.00477
$49$	−4.75243	−0.678918
$50$	0	0
$51$	−18.1143	−2.53651
$52$	−29.8234	−4.13577
$53$	11.0696	1.52052	0.760261	−	0.649618i	$-0.225071\pi$
$53$	11.0696	1.52052	0.760261	+	0.649618i	$0.225071\pi$
$54$	25.0746	3.41222
$55$	0	0
$56$	10.7700	1.43920
$57$	10.8476	1.43680
$58$	8.52567	1.11948
$59$	8.77550	1.14247	0.571237	−	0.820785i	$-0.306464\pi$
$59$	8.77550	1.14247	0.571237	+	0.820785i	$0.306464\pi$
$60$	0	0
$61$	9.19955	1.17788	0.588941	−	0.808176i	$-0.299545\pi$
$61$	9.19955	1.17788	0.588941	+	0.808176i	$0.299545\pi$
$62$	12.6434	1.60572
$63$	−9.26810	−1.16767
$64$	6.24475	0.780593
$65$	0	0
$66$	−35.8747	−4.41587
$67$	15.1272	1.84808	0.924038	−	0.382301i	$-0.124868\pi$
$67$	15.1272	1.84808	0.924038	+	0.382301i	$0.124868\pi$
$68$	28.4700	3.45249
$69$	20.4937	2.46715
$70$	0	0
$71$	−8.51912	−1.01103	−0.505517	−	0.862817i	$-0.668698\pi$
$71$	−8.51912	−1.01103	−0.505517	+	0.862817i	$0.668698\pi$
$72$	−44.4110	−5.23389
$73$	−1.35081	−0.158101	−0.0790505	−	0.996871i	$-0.525189\pi$
$73$	−1.35081	−0.158101	−0.0790505	+	0.996871i	$0.525189\pi$
$74$	−24.2147	−2.81490
$75$	0	0
$76$	−17.0490	−1.95565
$77$	6.82528	0.777813
$78$	−49.3454	−5.58726
$79$	0.518054	0.0582856	0.0291428	−	0.999575i	$-0.490722\pi$
$79$	0.518054	0.0582856	0.0291428	+	0.999575i	$0.490722\pi$
$80$	0	0
$81$	10.6718	1.18575
$82$	−5.13256	−0.566797
$83$	−12.6728	−1.39102	−0.695510	−	0.718516i	$-0.744822\pi$
$83$	−12.6728	−1.39102	−0.695510	+	0.718516i	$0.744822\pi$
$84$	21.6353	2.36060
$85$	0	0
$86$	−15.5620	−1.67810
$87$	9.93448	1.06509
$88$	32.7055	3.48642
$89$	6.25837	0.663386	0.331693	−	0.943387i	$-0.392380\pi$
$89$	6.25837	0.663386	0.331693	+	0.943387i	$0.392380\pi$
$90$	0	0
$91$	9.38812	0.984143
$92$	−32.2096	−3.35809
$93$	14.7327	1.52771
$94$	21.9006	2.25888
$95$	0	0
$96$	28.6156	2.92057
$97$	−7.07145	−0.717997	−0.358998	−	0.933338i	$-0.616882\pi$
$97$	−7.07145	−0.717997	−0.358998	+	0.933338i	$0.616882\pi$
$98$	12.3586	1.24841
$99$	−28.1448	−2.82865
$100$	0	0
$101$	−6.03034	−0.600042	−0.300021	−	0.953933i	$-0.596994\pi$
$101$	−6.03034	−0.600042	−0.300021	+	0.953933i	$0.596994\pi$
$102$	47.1059	4.66418
$103$	4.88888	0.481715	0.240858	−	0.970560i	$-0.422571\pi$
$103$	4.88888	0.481715	0.240858	+	0.970560i	$0.422571\pi$
$104$	44.9862	4.41126
$105$	0	0
$106$	−28.7862	−2.79596
$107$	−3.88545	−0.375620	−0.187810	−	0.982205i	$-0.560139\pi$
$107$	−3.88545	−0.375620	−0.187810	+	0.982205i	$0.560139\pi$
$108$	−45.9214	−4.41879
$109$	−7.54032	−0.722232	−0.361116	−	0.932521i	$-0.617604\pi$
$109$	−7.54032	−0.722232	−0.361116	+	0.932521i	$0.617604\pi$
$110$	0	0
$111$	−28.2160	−2.67814
$112$	−13.7273	−1.29711
$113$	−11.9513	−1.12429	−0.562143	−	0.827040i	$-0.690023\pi$
$113$	−11.9513	−1.12429	−0.562143	+	0.827040i	$0.690023\pi$
$114$	−28.2090	−2.64201
$115$	0	0
$116$	−15.6139	−1.44971
$117$	−38.7129	−3.57901
$118$	−22.8205	−2.10080
$119$	−8.96206	−0.821551
$120$	0	0
$121$	9.72657	0.884233
$122$	−23.9233	−2.16591
$123$	−5.98068	−0.539260
$124$	−23.1551	−2.07939
$125$	0	0
$126$	24.1015	2.14713
$127$	−2.79240	−0.247785	−0.123893	−	0.992296i	$-0.539538\pi$
$127$	−2.79240	−0.247785	−0.123893	+	0.992296i	$0.539538\pi$
$128$	2.64765	0.234022
$129$	−18.1335	−1.59657
$130$	0	0
$131$	7.74233	0.676451	0.338225	−	0.941065i	$-0.390173\pi$
$131$	7.74233	0.676451	0.338225	+	0.941065i	$0.390173\pi$
$132$	65.7006	5.71851
$133$	5.36685	0.465365
$134$	−39.3379	−3.39828
$135$	0	0
$136$	−42.9446	−3.68247
$137$	−17.4242	−1.48865	−0.744326	−	0.667817i	$-0.767229\pi$
$137$	−17.4242	−1.48865	−0.744326	+	0.667817i	$0.767229\pi$
$138$	−53.2935	−4.53664
$139$	23.3060	1.97679	0.988394	−	0.151909i	$-0.0485420\pi$
$139$	23.3060	1.97679	0.988394	+	0.151909i	$0.0485420\pi$
$140$	0	0
$141$	25.5195	2.14913
$142$	22.1538	1.85911
$143$	28.5093	2.38406
$144$	56.6058	4.71715
$145$	0	0
$146$	3.51277	0.290719
$147$	14.4008	1.18776
$148$	44.3466	3.64527
$149$	9.16928	0.751177	0.375588	−	0.926787i	$-0.377441\pi$
$149$	9.16928	0.751177	0.375588	+	0.926787i	$0.377441\pi$
$150$	0	0
$151$	−2.92734	−0.238224	−0.119112	−	0.992881i	$-0.538005\pi$
$151$	−2.92734	−0.238224	−0.119112	+	0.992881i	$0.538005\pi$
$152$	25.7170	2.08592
$153$	36.9560	2.98772
$154$	−17.7490	−1.43026
$155$	0	0
$156$	90.3708	7.23545
$157$	10.8882	0.868973	0.434487	−	0.900678i	$-0.356930\pi$
$157$	10.8882	0.868973	0.434487	+	0.900678i	$0.356930\pi$
$158$	−1.34719	−0.107177
$159$	−33.5429	−2.66013
$160$	0	0
$161$	10.1393	0.799087
$162$	−27.7517	−2.18038
$163$	−8.98983	−0.704137	−0.352069	−	0.935974i	$-0.614522\pi$
$163$	−8.98983	−0.704137	−0.352069	+	0.935974i	$0.614522\pi$
$164$	9.39975	0.733997
$165$	0	0
$166$	32.9554	2.55783
$167$	21.4312	1.65840	0.829198	−	0.558955i	$-0.188797\pi$
$167$	21.4312	1.65840	0.829198	+	0.558955i	$0.188797\pi$
$168$	−32.6350	−2.51785
$169$	26.2143	2.01648
$170$	0	0
$171$	−22.1308	−1.69238
$172$	28.5002	2.17312
$173$	−1.39282	−0.105894	−0.0529472	−	0.998597i	$-0.516861\pi$
$173$	−1.39282	−0.105894	−0.0529472	+	0.998597i	$0.516861\pi$
$174$	−25.8344	−1.95850
$175$	0	0
$176$	−41.6861	−3.14221
$177$	−26.5915	−1.99874
$178$	−16.2748	−1.21985
$179$	−2.97054	−0.222028	−0.111014	−	0.993819i	$-0.535410\pi$
$179$	−2.97054	−0.222028	−0.111014	+	0.993819i	$0.535410\pi$
$180$	0	0
$181$	25.3686	1.88564	0.942818	−	0.333307i	$-0.108165\pi$
$181$	25.3686	1.88564	0.942818	+	0.333307i	$0.108165\pi$
$182$	−24.4137	−1.80966
$183$	−27.8764	−2.06068
$184$	48.5856	3.58178
$185$	0	0
$186$	−38.3120	−2.80917
$187$	−27.2154	−1.99019
$188$	−40.1087	−2.92522
$189$	14.4556	1.05149
$190$	0	0
$191$	4.80399	0.347605	0.173802	−	0.984781i	$-0.444395\pi$
$191$	4.80399	0.347605	0.173802	+	0.984781i	$0.444395\pi$
$192$	−18.9228	−1.36563
$193$	18.0251	1.29748	0.648739	−	0.761011i	$-0.275297\pi$
$193$	18.0251	1.29748	0.648739	+	0.761011i	$0.275297\pi$
$194$	18.3892	1.32027
$195$	0	0
$196$	−22.6335	−1.61668
$197$	−4.09578	−0.291812	−0.145906	−	0.989298i	$-0.546610\pi$
$197$	−4.09578	−0.291812	−0.145906	+	0.989298i	$0.546610\pi$
$198$	73.1899	5.20138
$199$	13.3877	0.949032	0.474516	−	0.880247i	$-0.342623\pi$
$199$	13.3877	0.949032	0.474516	+	0.880247i	$0.342623\pi$
$200$	0	0
$201$	−45.8382	−3.23318
$202$	15.6818	1.10337
$203$	4.91509	0.344972
$204$	−86.2694	−6.04007
$205$	0	0
$206$	−12.7134	−0.885787
$207$	−41.8104	−2.90602
$208$	−57.3389	−3.97574
$209$	16.2977	1.12734
$210$	0	0
$211$	−7.13653	−0.491299	−0.245649	−	0.969359i	$-0.579001\pi$
$211$	−7.13653	−0.491299	−0.245649	+	0.969359i	$0.579001\pi$
$212$	52.7189	3.62075
$213$	25.8146	1.76879
$214$	10.1040	0.690698
$215$	0	0
$216$	69.2687	4.71314
$217$	7.28899	0.494809
$218$	19.6085	1.32805
$219$	4.09323	0.276595
$220$	0	0
$221$	−37.4346	−2.51812
$222$	73.3751	4.92461
$223$	−8.24028	−0.551810	−0.275905	−	0.961185i	$-0.588977\pi$
$223$	−8.24028	−0.551810	−0.275905	+	0.961185i	$0.588977\pi$
$224$	14.1576	0.945945
$225$	0	0
$226$	31.0792	2.06736
$227$	−0.914947	−0.0607272	−0.0303636	−	0.999539i	$-0.509667\pi$
$227$	−0.914947	−0.0607272	−0.0303636	+	0.999539i	$0.509667\pi$
$228$	51.6617	3.42138
$229$	23.1533	1.53001	0.765006	−	0.644023i	$-0.222736\pi$
$229$	23.1533	1.53001	0.765006	+	0.644023i	$0.222736\pi$
$230$	0	0
$231$	−20.6819	−1.36077
$232$	23.5522	1.54628
$233$	9.03132	0.591662	0.295831	−	0.955240i	$-0.404404\pi$
$233$	9.03132	0.591662	0.295831	+	0.955240i	$0.404404\pi$
$234$	100.672	6.58115
$235$	0	0
$236$	41.7934	2.72052
$237$	−1.56980	−0.101970
$238$	23.3057	1.51068
$239$	−13.2115	−0.854579	−0.427290	−	0.904115i	$-0.640532\pi$
$239$	−13.2115	−0.854579	−0.427290	+	0.904115i	$0.640532\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	−25.2938	−1.62594
$243$	−3.41066	−0.218794
$244$	43.8129	2.80484
$245$	0	0
$246$	15.5527	0.991601
$247$	22.4174	1.42638
$248$	34.9275	2.21790
$249$	38.4010	2.43357
$250$	0	0
$251$	−25.7618	−1.62607	−0.813035	−	0.582215i	$-0.802186\pi$
$251$	−25.7618	−1.62607	−0.813035	+	0.582215i	$0.802186\pi$
$252$	−44.1394	−2.78052
$253$	30.7903	1.93577
$254$	7.26158	0.455632
$255$	0	0
$256$	−19.3747	−1.21092
$257$	11.5847	0.722636	0.361318	−	0.932443i	$-0.382327\pi$
$257$	11.5847	0.722636	0.361318	+	0.932443i	$0.382327\pi$
$258$	47.1559	2.93580
$259$	−13.9599	−0.867424
$260$	0	0
$261$	−20.2679	−1.25455
$262$	−20.1338	−1.24387
$263$	21.9605	1.35414	0.677071	−	0.735918i	$-0.263249\pi$
$263$	21.9605	1.35414	0.677071	+	0.735918i	$0.263249\pi$
$264$	−99.1040	−6.09943
$265$	0	0
$266$	−13.9564	−0.855722
$267$	−18.9641	−1.16058
$268$	72.0432	4.40074
$269$	5.58960	0.340804	0.170402	−	0.985375i	$-0.445493\pi$
$269$	5.58960	0.340804	0.170402	+	0.985375i	$0.445493\pi$
$270$	0	0
$271$	4.54437	0.276051	0.138025	−	0.990429i	$-0.455924\pi$
$271$	4.54437	0.276051	0.138025	+	0.990429i	$0.455924\pi$
$272$	54.7367	3.31890
$273$	−28.4478	−1.72174
$274$	45.3114	2.73736
$275$	0	0
$276$	97.6014	5.87491
$277$	6.30539	0.378854	0.189427	−	0.981895i	$-0.439337\pi$
$277$	6.30539	0.378854	0.189427	+	0.981895i	$0.439337\pi$
$278$	−60.6068	−3.63496
$279$	−30.0569	−1.79946
$280$	0	0
$281$	19.4809	1.16214	0.581068	−	0.813855i	$-0.302635\pi$
$281$	19.4809	1.16214	0.581068	+	0.813855i	$0.302635\pi$
$282$	−66.3631	−3.95186
$283$	28.9071	1.71835	0.859174	−	0.511684i	$-0.170978\pi$
$283$	28.9071	1.71835	0.859174	+	0.511684i	$0.170978\pi$
$284$	−40.5724	−2.40753
$285$	0	0
$286$	−74.1378	−4.38386
$287$	−2.95895	−0.174661
$288$	−58.3804	−3.44010
$289$	18.7357	1.10210
$290$	0	0
$291$	21.4278	1.25612
$292$	−6.43326	−0.376478
$293$	6.04800	0.353328	0.176664	−	0.984271i	$-0.443469\pi$
$293$	6.04800	0.353328	0.176664	+	0.984271i	$0.443469\pi$
$294$	−37.4490	−2.18407
$295$	0	0
$296$	−66.8932	−3.88809
$297$	43.8979	2.54721
$298$	−23.8445	−1.38128
$299$	42.3518	2.44927
$300$	0	0
$301$	−8.97158	−0.517113
$302$	7.61251	0.438050
$303$	18.2731	1.04976
$304$	−32.7786	−1.87998
$305$	0	0
$306$	−96.1034	−5.49387
$307$	−7.56756	−0.431904	−0.215952	−	0.976404i	$-0.569285\pi$
$307$	−7.56756	−0.431904	−0.215952	+	0.976404i	$0.569285\pi$
$308$	32.5054	1.85217
$309$	−14.8142	−0.842753
$310$	0	0
$311$	−8.19176	−0.464512	−0.232256	−	0.972655i	$-0.574611\pi$
$311$	−8.19176	−0.464512	−0.232256	+	0.972655i	$0.574611\pi$
$312$	−136.317	−7.71742
$313$	−11.2293	−0.634719	−0.317360	−	0.948305i	$-0.602796\pi$
$313$	−11.2293	−0.634719	−0.317360	+	0.948305i	$0.602796\pi$
$314$	−28.3146	−1.59788
$315$	0	0
$316$	2.46724	0.138793
$317$	−3.49350	−0.196214	−0.0981071	−	0.995176i	$-0.531279\pi$
$317$	−3.49350	−0.196214	−0.0981071	+	0.995176i	$0.531279\pi$
$318$	87.2277	4.89149
$319$	14.9258	0.835686
$320$	0	0
$321$	11.7737	0.657141
$322$	−26.3670	−1.46938
$323$	−21.4000	−1.19073
$324$	50.8244	2.82358
$325$	0	0
$326$	23.3779	1.29478
$327$	22.8486	1.26353
$328$	−14.1787	−0.782890
$329$	12.6258	0.696083
$330$	0	0
$331$	−3.11295	−0.171103	−0.0855515	−	0.996334i	$-0.527265\pi$
$331$	−3.11295	−0.171103	−0.0855515	+	0.996334i	$0.527265\pi$
$332$	−60.3543	−3.31237
$333$	57.5650	3.15454
$334$	−55.7315	−3.04949
$335$	0	0
$336$	41.5963	2.26926
$337$	−11.3728	−0.619517	−0.309758	−	0.950815i	$-0.600248\pi$
$337$	−11.3728	−0.619517	−0.309758	+	0.950815i	$0.600248\pi$
$338$	−68.1698	−3.70795
$339$	36.2148	1.96692
$340$	0	0
$341$	22.1347	1.19866
$342$	57.5507	3.11198
$343$	17.6191	0.951343
$344$	−42.9902	−2.31788
$345$	0	0
$346$	3.62201	0.194720
$347$	−34.6263	−1.85884	−0.929418	−	0.369029i	$-0.879690\pi$
$347$	−34.6263	−1.85884	−0.929418	+	0.369029i	$0.879690\pi$
$348$	47.3130	2.53624
$349$	−22.4992	−1.20435	−0.602177	−	0.798363i	$-0.705700\pi$
$349$	−22.4992	−1.20435	−0.602177	+	0.798363i	$0.705700\pi$
$350$	0	0
$351$	60.3812	3.22291
$352$	42.9929	2.29153
$353$	−27.5462	−1.46614	−0.733068	−	0.680155i	$-0.761913\pi$
$353$	−27.5462	−1.46614	−0.733068	+	0.680155i	$0.761913\pi$
$354$	69.1506	3.67531
$355$	0	0
$356$	29.8055	1.57969
$357$	27.1568	1.43729
$358$	7.72482	0.408270
$359$	12.4969	0.659562	0.329781	−	0.944057i	$-0.393025\pi$
$359$	12.4969	0.659562	0.329781	+	0.944057i	$0.393025\pi$
$360$	0	0
$361$	−6.18479	−0.325515
$362$	−65.9707	−3.46734
$363$	−29.4734	−1.54695
$364$	44.7110	2.34349
$365$	0	0
$366$	72.4921	3.78922
$367$	19.0123	0.992432	0.496216	−	0.868199i	$-0.334722\pi$
$367$	19.0123	0.992432	0.496216	+	0.868199i	$0.334722\pi$
$368$	−61.9266	−3.22815
$369$	12.2015	0.635186
$370$	0	0
$371$	−16.5954	−0.861589
$372$	70.1644	3.63785
$373$	−3.09225	−0.160111	−0.0800554	−	0.996790i	$-0.525510\pi$
$373$	−3.09225	−0.160111	−0.0800554	+	0.996790i	$0.525510\pi$
$374$	70.7732	3.65959
$375$	0	0
$376$	60.5006	3.12008
$377$	20.5304	1.05737
$378$	−37.5916	−1.93350
$379$	23.2733	1.19547	0.597736	−	0.801693i	$-0.296067\pi$
$379$	23.2733	1.19547	0.597736	+	0.801693i	$0.296067\pi$
$380$	0	0
$381$	8.46150	0.433496
$382$	−12.4927	−0.639182
$383$	−26.2747	−1.34258	−0.671288	−	0.741196i	$-0.734259\pi$
$383$	−26.2747	−1.34258	−0.671288	+	0.741196i	$0.734259\pi$
$384$	−8.02290	−0.409417
$385$	0	0
$386$	−46.8740	−2.38583
$387$	36.9953	1.88058
$388$	−33.6778	−1.70973
$389$	−14.6530	−0.742934	−0.371467	−	0.928446i	$-0.621145\pi$
$389$	−14.6530	−0.742934	−0.371467	+	0.928446i	$0.621145\pi$
$390$	0	0
$391$	−40.4298	−2.04462
$392$	34.1407	1.72437
$393$	−23.4608	−1.18344
$394$	10.6510	0.536589
$395$	0	0
$396$	−134.040	−6.73574
$397$	−6.01879	−0.302075	−0.151037	−	0.988528i	$-0.548261\pi$
$397$	−6.01879	−0.302075	−0.151037	+	0.988528i	$0.548261\pi$
$398$	−34.8146	−1.74510
$399$	−16.2626	−0.814148
$400$	0	0
$401$	−14.7053	−0.734349	−0.367175	−	0.930152i	$-0.619675\pi$
$401$	−14.7053	−0.734349	−0.367175	+	0.930152i	$0.619675\pi$
$402$	119.201	5.94522
$403$	30.4462	1.51663
$404$	−28.7195	−1.42885
$405$	0	0
$406$	−12.7816	−0.634340
$407$	−42.3924	−2.10131
$408$	130.130	6.44241
$409$	−11.7163	−0.579331	−0.289666	−	0.957128i	$-0.593544\pi$
$409$	−11.7163	−0.579331	−0.289666	+	0.957128i	$0.593544\pi$
$410$	0	0
$411$	52.7987	2.60437
$412$	23.2833	1.14709
$413$	−13.1561	−0.647372
$414$	108.727	5.34364
$415$	0	0
$416$	59.1365	2.89940
$417$	−70.6217	−3.45836
$418$	−42.3819	−2.07297
$419$	−1.49375	−0.0729744	−0.0364872	−	0.999334i	$-0.511617\pi$
$419$	−1.49375	−0.0729744	−0.0364872	+	0.999334i	$0.511617\pi$
$420$	0	0
$421$	31.7207	1.54597	0.772987	−	0.634422i	$-0.218762\pi$
$421$	31.7207	1.54597	0.772987	+	0.634422i	$0.218762\pi$
$422$	18.5584	0.903409
$423$	−52.0638	−2.53143
$424$	−79.5221	−3.86193
$425$	0	0
$426$	−67.1304	−3.25248
$427$	−13.7919	−0.667436
$428$	−18.5045	−0.894447
$429$	−86.3885	−4.17088
$430$	0	0
$431$	24.4771	1.17902	0.589510	−	0.807761i	$-0.299321\pi$
$431$	24.4771	1.17902	0.589510	+	0.807761i	$0.299321\pi$
$432$	−88.2891	−4.24781
$433$	38.0111	1.82670	0.913348	−	0.407179i	$-0.133488\pi$
$433$	38.0111	1.82670	0.913348	+	0.407179i	$0.133488\pi$
$434$	−18.9549	−0.909864
$435$	0	0
$436$	−35.9108	−1.71982
$437$	24.2110	1.15817
$438$	−10.6444	−0.508608
$439$	5.68949	0.271544	0.135772	−	0.990740i	$-0.456648\pi$
$439$	5.68949	0.271544	0.135772	+	0.990740i	$0.456648\pi$
$440$	0	0
$441$	−29.3798	−1.39904
$442$	97.3480	4.63037
$443$	16.7617	0.796373	0.398186	−	0.917305i	$-0.369640\pi$
$443$	16.7617	0.796373	0.398186	+	0.917305i	$0.369640\pi$
$444$	−134.379	−6.37733
$445$	0	0
$446$	21.4287	1.01468
$447$	−27.7847	−1.31417
$448$	−9.36206	−0.442316
$449$	−14.3859	−0.678913	−0.339456	−	0.940622i	$-0.610243\pi$
$449$	−14.3859	−0.678913	−0.339456	+	0.940622i	$0.610243\pi$
$450$	0	0
$451$	−8.98554	−0.423113
$452$	−56.9183	−2.67721
$453$	8.87042	0.416769
$454$	2.37930	0.111666
$455$	0	0
$456$	−77.9274	−3.64929
$457$	−11.6826	−0.546489	−0.273245	−	0.961945i	$-0.588097\pi$
$457$	−11.6826	−0.546489	−0.273245	+	0.961945i	$0.588097\pi$
$458$	−60.2097	−2.81341
$459$	−57.6409	−2.69045
$460$	0	0
$461$	−39.8411	−1.85558	−0.927792	−	0.373097i	$-0.878296\pi$
$461$	−39.8411	−1.85558	−0.927792	+	0.373097i	$0.878296\pi$
$462$	53.7829	2.50221
$463$	−16.5972	−0.771335	−0.385668	−	0.922638i	$-0.626029\pi$
$463$	−16.5972	−0.771335	−0.385668	+	0.922638i	$0.626029\pi$
$464$	−30.0194	−1.39362
$465$	0	0
$466$	−23.4858	−1.08796
$467$	−3.59284	−0.166257	−0.0831284	−	0.996539i	$-0.526491\pi$
$467$	−3.59284	−0.166257	−0.0831284	+	0.996539i	$0.526491\pi$
$468$	−184.371	−8.52253
$469$	−22.6785	−1.04719
$470$	0	0
$471$	−32.9934	−1.52025
$472$	−63.0419	−2.90174
$473$	−27.2443	−1.25270
$474$	4.08225	0.187504
$475$	0	0
$476$	−42.6819	−1.95632
$477$	68.4328	3.13332
$478$	34.3562	1.57142
$479$	21.6089	0.987336	0.493668	−	0.869650i	$-0.335656\pi$
$479$	21.6089	0.987336	0.493668	+	0.869650i	$0.335656\pi$
$480$	0	0
$481$	−58.3105	−2.65873
$482$	2.60048	0.118449
$483$	−30.7240	−1.39799
$484$	46.3228	2.10558
$485$	0	0
$486$	8.86937	0.402323
$487$	6.73402	0.305148	0.152574	−	0.988292i	$-0.451244\pi$
$487$	6.73402	0.305148	0.152574	+	0.988292i	$0.451244\pi$
$488$	−66.0882	−2.99167
$489$	27.2409	1.23188
$490$	0	0
$491$	25.1718	1.13599	0.567993	−	0.823033i	$-0.307720\pi$
$491$	25.1718	1.13599	0.567993	+	0.823033i	$0.307720\pi$
$492$	−28.4830	−1.28411
$493$	−19.5986	−0.882678
$494$	−58.2960	−2.62286
$495$	0	0
$496$	−44.5183	−1.99893
$497$	12.7718	0.572893
$498$	−99.8612	−4.47489
$499$	−20.5744	−0.921035	−0.460518	−	0.887651i	$-0.652336\pi$
$499$	−20.5744	−0.921035	−0.460518	+	0.887651i	$0.652336\pi$
$500$	0	0
$501$	−64.9407	−2.90134
$502$	66.9931	2.99005
$503$	22.6590	1.01032	0.505158	−	0.863027i	$-0.331434\pi$
$503$	22.6590	1.01032	0.505158	+	0.863027i	$0.331434\pi$
$504$	66.5806	2.96574
$505$	0	0
$506$	−80.0696	−3.55953
$507$	−79.4343	−3.52780
$508$	−13.2988	−0.590039
$509$	−17.1862	−0.761767	−0.380884	−	0.924623i	$-0.624380\pi$
$509$	−17.1862	−0.761767	−0.380884	+	0.924623i	$0.624380\pi$
$510$	0	0
$511$	2.02513	0.0895864
$512$	45.0882	1.99263
$513$	34.5178	1.52400
$514$	−30.1259	−1.32880
$515$	0	0
$516$	−86.3611	−3.80184
$517$	38.3412	1.68625
$518$	36.3024	1.59503
$519$	4.22052	0.185260
$520$	0	0
$521$	−16.6666	−0.730179	−0.365089	−	0.930973i	$-0.618962\pi$
$521$	−16.6666	−0.730179	−0.365089	+	0.930973i	$0.618962\pi$
$522$	52.7063	2.30689
$523$	21.5198	0.940995	0.470497	−	0.882401i	$-0.344075\pi$
$523$	21.5198	0.940995	0.470497	+	0.882401i	$0.344075\pi$
$524$	36.8729	1.61080
$525$	0	0
$526$	−57.1078	−2.49002
$527$	−29.0644	−1.26607
$528$	126.317	5.49724
$529$	22.7404	0.988714
$530$	0	0
$531$	54.2508	2.35428
$532$	25.5597	1.10815
$533$	−12.3596	−0.535352
$534$	49.3157	2.13410
$535$	0	0
$536$	−108.671	−4.69388
$537$	9.00129	0.388435
$538$	−14.5357	−0.626676
$539$	21.6361	0.931934
$540$	0	0
$541$	−5.37353	−0.231026	−0.115513	−	0.993306i	$-0.536851\pi$
$541$	−5.37353	−0.231026	−0.115513	+	0.993306i	$0.536851\pi$
$542$	−11.8176	−0.507608
$543$	−76.8719	−3.29889
$544$	−56.4526	−2.42039
$545$	0	0
$546$	73.9781	3.16597
$547$	14.6263	0.625377	0.312688	−	0.949856i	$-0.398770\pi$
$547$	14.6263	0.625377	0.312688	+	0.949856i	$0.398770\pi$
$548$	−82.9830	−3.54486
$549$	56.8723	2.42725
$550$	0	0
$551$	11.7365	0.499991
$552$	−147.224	−6.26625
$553$	−0.776662	−0.0330270
$554$	−16.3970	−0.696644
$555$	0	0
$556$	110.995	4.70724
$557$	−15.2637	−0.646744	−0.323372	−	0.946272i	$-0.604816\pi$
$557$	−15.2637	−0.646744	−0.323372	+	0.946272i	$0.604816\pi$
$558$	78.1625	3.30888
$559$	−37.4744	−1.58500
$560$	0	0
$561$	82.4679	3.48180
$562$	−50.6599	−2.13696
$563$	−33.9873	−1.43239	−0.716197	−	0.697898i	$-0.754119\pi$
$563$	−33.9873	−1.43239	−0.716197	+	0.697898i	$0.754119\pi$
$564$	121.537	5.11763
$565$	0	0
$566$	−75.1723	−3.15973
$567$	−15.9990	−0.671895
$568$	61.2001	2.56790
$569$	7.00290	0.293577	0.146788	−	0.989168i	$-0.453106\pi$
$569$	7.00290	0.293577	0.146788	+	0.989168i	$0.453106\pi$
$570$	0	0
$571$	33.1462	1.38712	0.693562	−	0.720396i	$-0.256040\pi$
$571$	33.1462	1.38712	0.693562	+	0.720396i	$0.256040\pi$
$572$	135.776	5.67706
$573$	−14.5570	−0.608128
$574$	7.69469	0.321170
$575$	0	0
$576$	38.6055	1.60856
$577$	9.55039	0.397588	0.198794	−	0.980041i	$-0.436298\pi$
$577$	9.55039	0.397588	0.198794	+	0.980041i	$0.436298\pi$
$578$	−48.7218	−2.02656
$579$	−54.6196	−2.26991
$580$	0	0
$581$	18.9989	0.788209
$582$	−55.7227	−2.30978
$583$	−50.3958	−2.08718
$584$	9.70405	0.401556
$585$	0	0
$586$	−15.7277	−0.649706
$587$	24.6231	1.01630	0.508151	−	0.861268i	$-0.330329\pi$
$587$	24.6231	1.01630	0.508151	+	0.861268i	$0.330329\pi$
$588$	68.5838	2.82835
$589$	17.4050	0.717160
$590$	0	0
$591$	12.4110	0.510520
$592$	85.2613	3.50422
$593$	−19.7588	−0.811398	−0.405699	−	0.914007i	$-0.632972\pi$
$593$	−19.7588	−0.811398	−0.405699	+	0.914007i	$0.632972\pi$
$594$	−114.156	−4.68386
$595$	0	0
$596$	43.6688	1.78874
$597$	−40.5674	−1.66031
$598$	−110.135	−4.50376
$599$	20.0514	0.819277	0.409639	−	0.912248i	$-0.365655\pi$
$599$	20.0514	0.819277	0.409639	+	0.912248i	$0.365655\pi$
$600$	0	0
$601$	3.59401	0.146603	0.0733013	−	0.997310i	$-0.476647\pi$
$601$	3.59401	0.146603	0.0733013	+	0.997310i	$0.476647\pi$
$602$	23.3304	0.950878
$603$	93.5171	3.80831
$604$	−13.9415	−0.567271
$605$	0	0
$606$	−47.5189	−1.93032
$607$	−36.4529	−1.47958	−0.739790	−	0.672838i	$-0.765075\pi$
$607$	−36.4529	−1.47958	−0.739790	+	0.672838i	$0.765075\pi$
$608$	33.8062	1.37102
$609$	−14.8937	−0.603522
$610$	0	0
$611$	52.7381	2.13356
$612$	176.003	7.11451
$613$	20.5652	0.830619	0.415309	−	0.909680i	$-0.363673\pi$
$613$	20.5652	0.830619	0.415309	+	0.909680i	$0.363673\pi$
$614$	19.6793	0.794192
$615$	0	0
$616$	−49.0318	−1.97555
$617$	−35.9462	−1.44714	−0.723570	−	0.690251i	$-0.757500\pi$
$617$	−35.9462	−1.44714	−0.723570	+	0.690251i	$0.757500\pi$
$618$	38.5242	1.54967
$619$	−44.5867	−1.79209	−0.896046	−	0.443962i	$-0.853572\pi$
$619$	−44.5867	−1.79209	−0.896046	+	0.443962i	$0.853572\pi$
$620$	0	0
$621$	65.2124	2.61688
$622$	21.3025	0.854154
$623$	−9.38248	−0.375901
$624$	173.748	6.95549
$625$	0	0
$626$	29.2017	1.16713
$627$	−49.3852	−1.97225
$628$	51.8551	2.06925
$629$	55.6642	2.21948
$630$	0	0
$631$	24.0156	0.956045	0.478023	−	0.878348i	$-0.341354\pi$
$631$	24.0156	0.956045	0.478023	+	0.878348i	$0.341354\pi$
$632$	−3.72162	−0.148038
$633$	21.6251	0.859519
$634$	9.08477	0.360802
$635$	0	0
$636$	−159.748	−6.33443
$637$	29.7603	1.17915
$638$	−38.8144	−1.53667
$639$	−52.6658	−2.08343
$640$	0	0
$641$	−5.55232	−0.219303	−0.109652	−	0.993970i	$-0.534974\pi$
$641$	−5.55232	−0.219303	−0.109652	+	0.993970i	$0.534974\pi$
$642$	−30.6172	−1.20836
$643$	30.5971	1.20663	0.603316	−	0.797502i	$-0.293846\pi$
$643$	30.5971	1.20663	0.603316	+	0.797502i	$0.293846\pi$
$644$	48.2884	1.90283
$645$	0	0
$646$	55.6503	2.18953
$647$	−17.9287	−0.704849	−0.352424	−	0.935840i	$-0.614643\pi$
$647$	−17.9287	−0.704849	−0.352424	+	0.935840i	$0.614643\pi$
$648$	−76.6644	−3.01166
$649$	−39.9517	−1.56824
$650$	0	0
$651$	−22.0870	−0.865660
$652$	−42.8141	−1.67673
$653$	11.1915	0.437956	0.218978	−	0.975730i	$-0.429728\pi$
$653$	11.1915	0.437956	0.218978	+	0.975730i	$0.429728\pi$
$654$	−59.4174	−2.32341
$655$	0	0
$656$	18.0721	0.705596
$657$	−8.35083	−0.325797
$658$	−32.8332	−1.27997
$659$	6.49274	0.252921	0.126461	−	0.991972i	$-0.459638\pi$
$659$	6.49274	0.252921	0.126461	+	0.991972i	$0.459638\pi$
$660$	0	0
$661$	−16.8310	−0.654651	−0.327326	−	0.944912i	$-0.606147\pi$
$661$	−16.8310	−0.654651	−0.327326	+	0.944912i	$0.606147\pi$
$662$	8.09516	0.314627
$663$	113.434	4.40541
$664$	91.0395	3.53302
$665$	0	0
$666$	−149.697	−5.80063
$667$	22.1730	0.858543
$668$	102.066	3.94906
$669$	24.9696	0.965382
$670$	0	0
$671$	−41.8823	−1.61685
$672$	−42.9003	−1.65491
$673$	−31.6078	−1.21839	−0.609196	−	0.793020i	$-0.708508\pi$
$673$	−31.6078	−1.21839	−0.609196	+	0.793020i	$0.708508\pi$
$674$	29.5748	1.13918
$675$	0	0
$676$	124.846	4.80176
$677$	4.17745	0.160552	0.0802762	−	0.996773i	$-0.474420\pi$
$677$	4.17745	0.160552	0.0802762	+	0.996773i	$0.474420\pi$
$678$	−94.1760	−3.61681
$679$	10.6014	0.406846
$680$	0	0
$681$	2.77247	0.106241
$682$	−57.5610	−2.20412
$683$	14.0000	0.535695	0.267847	−	0.963461i	$-0.413688\pi$
$683$	14.0000	0.535695	0.267847	+	0.963461i	$0.413688\pi$
$684$	−105.398	−4.02999
$685$	0	0
$686$	−45.8182	−1.74935
$687$	−70.1589	−2.67673
$688$	54.7949	2.08903
$689$	−69.3190	−2.64085
$690$	0	0
$691$	−15.3813	−0.585132	−0.292566	−	0.956245i	$-0.594509\pi$
$691$	−15.3813	−0.585132	−0.292566	+	0.956245i	$0.594509\pi$
$692$	−6.63333	−0.252161
$693$	42.1943	1.60283
$694$	90.0450	3.41806
$695$	0	0
$696$	−71.3678	−2.70519
$697$	11.7986	0.446905
$698$	58.5087	2.21459
$699$	−27.3666	−1.03510
$700$	0	0
$701$	4.05598	0.153192	0.0765961	−	0.997062i	$-0.475595\pi$
$701$	4.05598	0.153192	0.0765961	+	0.997062i	$0.475595\pi$
$702$	−157.020	−5.92635
$703$	−33.3340	−1.25722
$704$	−28.4301	−1.07150
$705$	0	0
$706$	71.6334	2.69596
$707$	9.04063	0.340008
$708$	−126.642	−4.75950
$709$	−49.4736	−1.85802	−0.929010	−	0.370054i	$-0.879339\pi$
$709$	−49.4736	−1.85802	−0.929010	+	0.370054i	$0.879339\pi$
$710$	0	0
$711$	3.20265	0.120109
$712$	−44.9592	−1.68492
$713$	32.8822	1.23145
$714$	−70.6207	−2.64291
$715$	0	0
$716$	−14.1472	−0.528705
$717$	40.0333	1.49507
$718$	−32.4980	−1.21282
$719$	21.0548	0.785211	0.392605	−	0.919707i	$-0.371574\pi$
$719$	21.0548	0.785211	0.392605	+	0.919707i	$0.371574\pi$
$720$	0	0
$721$	−7.32936	−0.272960
$722$	16.0834	0.598564
$723$	3.03019	0.112694
$724$	120.818	4.49018
$725$	0	0
$726$	76.6450	2.84456
$727$	7.42139	0.275244	0.137622	−	0.990485i	$-0.456054\pi$
$727$	7.42139	0.275244	0.137622	+	0.990485i	$0.456054\pi$
$728$	−67.4429	−2.49960
$729$	−21.6803	−0.802975
$730$	0	0
$731$	35.7737	1.32314
$732$	−132.762	−4.90701
$733$	28.9906	1.07079	0.535396	−	0.844601i	$-0.320162\pi$
$733$	28.9906	1.07079	0.535396	+	0.844601i	$0.320162\pi$
$734$	−49.4410	−1.82490
$735$	0	0
$736$	63.8680	2.35421
$737$	−68.8685	−2.53681
$738$	−31.7299	−1.16799
$739$	−34.4204	−1.26617	−0.633087	−	0.774080i	$-0.718213\pi$
$739$	−34.4204	−1.26617	−0.633087	+	0.774080i	$0.718213\pi$
$740$	0	0
$741$	−67.9290	−2.49543
$742$	43.1560	1.58431
$743$	27.4714	1.00783	0.503915	−	0.863754i	$-0.331893\pi$
$743$	27.4714	1.00783	0.503915	+	0.863754i	$0.331893\pi$
$744$	−105.837	−3.88018
$745$	0	0
$746$	8.04135	0.294415
$747$	−78.3441	−2.86646
$748$	−129.614	−4.73914
$749$	5.82502	0.212842
$750$	0	0
$751$	−26.9123	−0.982044	−0.491022	−	0.871147i	$-0.663376\pi$
$751$	−26.9123	−0.982044	−0.491022	+	0.871147i	$0.663376\pi$
$752$	−77.1134	−2.81204
$753$	78.0632	2.84478
$754$	−53.3889	−1.94431
$755$	0	0
$756$	68.8450	2.50387
$757$	18.5116	0.672815	0.336408	−	0.941716i	$-0.390788\pi$
$757$	18.5116	0.672815	0.336408	+	0.941716i	$0.390788\pi$
$758$	−60.5219	−2.19825
$759$	−93.3005	−3.38659
$760$	0	0
$761$	49.6546	1.79998	0.899989	−	0.435913i	$-0.143575\pi$
$761$	49.6546	1.79998	0.899989	+	0.435913i	$0.143575\pi$
$762$	−22.0040	−0.797120
$763$	11.3044	0.409246
$764$	22.8790	0.827735
$765$	0	0
$766$	68.3270	2.46876
$767$	−54.9533	−1.98425
$768$	58.7090	2.11848
$769$	−8.64855	−0.311875	−0.155937	−	0.987767i	$-0.549840\pi$
$769$	−8.64855	−0.311875	−0.155937	+	0.987767i	$0.549840\pi$
$770$	0	0
$771$	−35.1040	−1.26424
$772$	85.8448	3.08962
$773$	13.2600	0.476927	0.238464	−	0.971151i	$-0.423356\pi$
$773$	13.2600	0.476927	0.238464	+	0.971151i	$0.423356\pi$
$774$	−96.2055	−3.45804
$775$	0	0
$776$	50.8002	1.82362
$777$	42.3011	1.51754
$778$	38.1047	1.36612
$779$	−7.06551	−0.253148
$780$	0	0
$781$	38.7845	1.38782
$782$	105.137	3.75969
$783$	31.6122	1.12973
$784$	−43.5154	−1.55412
$785$	0	0
$786$	61.0093	2.17613
$787$	34.9043	1.24420	0.622101	−	0.782937i	$-0.286279\pi$
$787$	34.9043	1.24420	0.622101	+	0.782937i	$0.286279\pi$
$788$	−19.5062	−0.694878
$789$	−66.5445	−2.36905
$790$	0	0
$791$	17.9173	0.637066
$792$	202.188	7.18443
$793$	−57.6088	−2.04575
$794$	15.6518	0.555460
$795$	0	0
$796$	63.7592	2.25989
$797$	−2.48784	−0.0881240	−0.0440620	−	0.999029i	$-0.514030\pi$
$797$	−2.48784	−0.0881240	−0.0440620	+	0.999029i	$0.514030\pi$
$798$	42.2906	1.49707
$799$	−50.3447	−1.78107
$800$	0	0
$801$	38.6897	1.36703
$802$	38.2410	1.35034
$803$	6.14978	0.217021
$804$	−218.305	−7.69901
$805$	0	0
$806$	−79.1748	−2.78881
$807$	−16.9376	−0.596230
$808$	43.3211	1.52403
$809$	−29.9445	−1.05279	−0.526397	−	0.850239i	$-0.676457\pi$
$809$	−29.9445	−1.05279	−0.526397	+	0.850239i	$0.676457\pi$
$810$	0	0
$811$	30.6585	1.07656	0.538282	−	0.842765i	$-0.319074\pi$
$811$	30.6585	1.07656	0.538282	+	0.842765i	$0.319074\pi$
$812$	23.4082	0.821465
$813$	−13.7703	−0.482946
$814$	110.241	3.86393
$815$	0	0
$816$	−165.863	−5.80636
$817$	−21.4227	−0.749487
$818$	30.4679	1.06529
$819$	58.0380	2.02801
$820$	0	0
$821$	11.4371	0.399158	0.199579	−	0.979882i	$-0.436043\pi$
$821$	11.4371	0.399158	0.199579	+	0.979882i	$0.436043\pi$
$822$	−137.302	−4.78896
$823$	−23.3278	−0.813155	−0.406577	−	0.913616i	$-0.633278\pi$
$823$	−23.3278	−0.813155	−0.406577	+	0.913616i	$0.633278\pi$
$824$	−35.1210	−1.22350
$825$	0	0
$826$	34.2123	1.19040
$827$	−50.7680	−1.76538	−0.882688	−	0.469959i	$-0.844269\pi$
$827$	−50.7680	−1.76538	−0.882688	+	0.469959i	$0.844269\pi$
$828$	−199.122	−6.91997
$829$	44.3059	1.53881	0.769404	−	0.638763i	$-0.220554\pi$
$829$	44.3059	1.53881	0.769404	+	0.638763i	$0.220554\pi$
$830$	0	0
$831$	−19.1065	−0.662799
$832$	−39.1054	−1.35574
$833$	−28.4097	−0.984338
$834$	183.650	6.35929
$835$	0	0
$836$	77.6179	2.68447
$837$	46.8803	1.62042
$838$	3.88447	0.134187
$839$	49.0647	1.69390	0.846951	−	0.531671i	$-0.178436\pi$
$839$	49.0647	1.69390	0.846951	+	0.531671i	$0.178436\pi$
$840$	0	0
$841$	−18.2515	−0.629361
$842$	−82.4892	−2.84277
$843$	−59.0310	−2.03314
$844$	−33.9878	−1.16991
$845$	0	0
$846$	135.391	4.65484
$847$	−14.5820	−0.501043
$848$	101.358	3.48065
$849$	−87.5940	−3.00622
$850$	0	0
$851$	−62.9759	−2.15879
$852$	122.942	4.21193
$853$	−23.1165	−0.791495	−0.395747	−	0.918359i	$-0.629514\pi$
$853$	−23.1165	−0.791495	−0.395747	+	0.918359i	$0.629514\pi$
$854$	35.8655	1.22729
$855$	0	0
$856$	27.9125	0.954028
$857$	−14.1682	−0.483977	−0.241989	−	0.970279i	$-0.577800\pi$
$857$	−14.1682	−0.483977	−0.241989	+	0.970279i	$0.577800\pi$
$858$	224.652	7.66949
$859$	0.410976	0.0140223	0.00701117	−	0.999975i	$-0.497768\pi$
$859$	0.410976	0.0140223	0.00701117	+	0.999975i	$0.497768\pi$
$860$	0	0
$861$	8.96618	0.305567
$862$	−63.6522	−2.16800
$863$	−18.5166	−0.630314	−0.315157	−	0.949040i	$-0.602057\pi$
$863$	−18.5166	−0.630314	−0.315157	+	0.949040i	$0.602057\pi$
$864$	91.0569	3.09782
$865$	0	0
$866$	−98.8472	−3.35896
$867$	−56.7728	−1.92810
$868$	34.7139	1.17827
$869$	−2.35852	−0.0800072
$870$	0	0
$871$	−94.7282	−3.20974
$872$	54.1685	1.83438
$873$	−43.7162	−1.47957
$874$	−62.9603	−2.12966
$875$	0	0
$876$	19.4940	0.658642
$877$	−47.7399	−1.61206	−0.806031	−	0.591873i	$-0.798389\pi$
$877$	−47.7399	−1.61206	−0.806031	+	0.591873i	$0.798389\pi$
$878$	−14.7954	−0.499321
$879$	−18.3266	−0.618141
$880$	0	0
$881$	−37.0152	−1.24707	−0.623536	−	0.781794i	$-0.714305\pi$
$881$	−37.0152	−1.24707	−0.623536	+	0.781794i	$0.714305\pi$
$882$	76.4017	2.57258
$883$	−47.6233	−1.60265	−0.801325	−	0.598229i	$-0.795871\pi$
$883$	−47.6233	−1.60265	−0.801325	+	0.598229i	$0.795871\pi$
$884$	−178.283	−5.99629
$885$	0	0
$886$	−43.5885	−1.46438
$887$	14.1873	0.476361	0.238181	−	0.971221i	$-0.423449\pi$
$887$	14.1873	0.476361	0.238181	+	0.971221i	$0.423449\pi$
$888$	202.699	6.80214
$889$	4.18633	0.140405
$890$	0	0
$891$	−48.5847	−1.62765
$892$	−39.2444	−1.31400
$893$	30.1485	1.00888
$894$	72.2536	2.41652
$895$	0	0
$896$	−3.96934	−0.132606
$897$	−128.334	−4.28496
$898$	37.4103	1.24840
$899$	15.9399	0.531625
$900$	0	0
$901$	66.1731	2.20455
$902$	23.3667	0.778027
$903$	27.1856	0.904681
$904$	85.8565	2.85555
$905$	0	0
$906$	−23.0674	−0.766362
$907$	28.4593	0.944976	0.472488	−	0.881337i	$-0.343356\pi$
$907$	28.4593	0.944976	0.472488	+	0.881337i	$0.343356\pi$
$908$	−4.35744	−0.144607
$909$	−37.2800	−1.23650
$910$	0	0
$911$	−32.9339	−1.09115	−0.545574	−	0.838063i	$-0.683688\pi$
$911$	−32.9339	−1.09115	−0.545574	+	0.838063i	$0.683688\pi$
$912$	99.3255	3.28899
$913$	57.6947	1.90942
$914$	30.3804	1.00489
$915$	0	0
$916$	110.268	3.64335
$917$	−11.6072	−0.383304
$918$	149.894	4.94725
$919$	−42.6489	−1.40686	−0.703429	−	0.710766i	$-0.748349\pi$
$919$	−42.6489	−1.40686	−0.703429	+	0.710766i	$0.748349\pi$
$920$	0	0
$921$	22.9312	0.755608
$922$	103.606	3.41208
$923$	53.3479	1.75597
$924$	−98.4977	−3.24034
$925$	0	0
$926$	43.1606	1.41835
$927$	30.2234	0.992666
$928$	30.9605	1.01633
$929$	31.2382	1.02489	0.512446	−	0.858720i	$-0.328740\pi$
$929$	31.2382	1.02489	0.512446	+	0.858720i	$0.328740\pi$
$930$	0	0
$931$	17.0129	0.557575
$932$	43.0117	1.40890
$933$	24.8226	0.812656
$934$	9.34312	0.305716
$935$	0	0
$936$	278.108	9.09024
$937$	13.7164	0.448094	0.224047	−	0.974578i	$-0.428073\pi$
$937$	13.7164	0.448094	0.224047	+	0.974578i	$0.428073\pi$
$938$	58.9750	1.92560
$939$	34.0270	1.11043
$940$	0	0
$941$	−37.5334	−1.22355	−0.611777	−	0.791030i	$-0.709545\pi$
$941$	−37.5334	−1.22355	−0.611777	+	0.791030i	$0.709545\pi$
$942$	85.7986	2.79547
$943$	−13.3484	−0.434685
$944$	80.3525	2.61525
$945$	0	0
$946$	70.8484	2.30348
$947$	−26.1999	−0.851381	−0.425690	−	0.904869i	$-0.639969\pi$
$947$	−26.1999	−0.851381	−0.425690	+	0.904869i	$0.639969\pi$
$948$	−7.47620	−0.242816
$949$	8.45898	0.274590
$950$	0	0
$951$	10.5860	0.343273
$952$	64.3821	2.08664
$953$	−14.6924	−0.475932	−0.237966	−	0.971273i	$-0.576481\pi$
$953$	−14.6924	−0.475932	−0.237966	+	0.971273i	$0.576481\pi$
$954$	−177.958	−5.76161
$955$	0	0
$956$	−62.9197	−2.03497
$957$	−45.2281	−1.46202
$958$	−56.1936	−1.81553
$959$	26.1222	0.843530
$960$	0	0
$961$	−7.36142	−0.237465
$962$	151.635	4.88892
$963$	−24.0201	−0.774037
$964$	−4.76251	−0.153390
$965$	0	0
$966$	79.8971	2.57065
$967$	34.7139	1.11633	0.558163	−	0.829732i	$-0.311507\pi$
$967$	34.7139	1.11633	0.558163	+	0.829732i	$0.311507\pi$
$968$	−69.8742	−2.24584
$969$	64.8462	2.08316
$970$	0	0
$971$	24.1393	0.774667	0.387334	−	0.921940i	$-0.373396\pi$
$971$	24.1393	0.774667	0.387334	+	0.921940i	$0.373396\pi$
$972$	−16.2433	−0.521004
$973$	−34.9401	−1.12013
$974$	−17.5117	−0.561111
$975$	0	0
$976$	84.2353	2.69631
$977$	46.2874	1.48087	0.740433	−	0.672130i	$-0.234621\pi$
$977$	46.2874	1.48087	0.740433	+	0.672130i	$0.234621\pi$
$978$	−70.8395	−2.26520
$979$	−28.4921	−0.910612
$980$	0	0
$981$	−46.6148	−1.48830
$982$	−65.4588	−2.08887
$983$	−55.9477	−1.78446	−0.892228	−	0.451585i	$-0.850859\pi$
$983$	−55.9477	−1.78446	−0.892228	+	0.451585i	$0.850859\pi$
$984$	42.9643	1.36965
$985$	0	0
$986$	50.9659	1.62309
$987$	−38.2586	−1.21779
$988$	106.763	3.39658
$989$	−40.4727	−1.28696
$990$	0	0
$991$	−32.9227	−1.04582	−0.522912	−	0.852386i	$-0.675155\pi$
$991$	−32.9227	−1.04582	−0.522912	+	0.852386i	$0.675155\pi$
$992$	45.9139	1.45777
$993$	9.43283	0.299342
$994$	−33.2128	−1.05345
$995$	0	0
$996$	182.885	5.79494
$997$	0.524168	0.0166006	0.00830028	−	0.999966i	$-0.497358\pi$
$997$	0.524168	0.0166006	0.00830028	+	0.999966i	$0.497358\pi$
$998$	53.5033	1.69362
$999$	−89.7851	−2.84067

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.l.1.4	✓	40
5.4	even	2		6025.2.a.o.1.37	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.l.1.4	✓	40	1.1	even	1	trivial
6025.2.a.o.1.37	yes	40	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.60048 q^{2} -3.03019 q^{3} +4.76251 q^{4} +7.87996 q^{6} -1.49919 q^{7} -7.18385 q^{8} +6.18207 q^{9} +O(q^{10})\)	\(q-2.60048 q^{2} -3.03019 q^{3} +4.76251 q^{4} +7.87996 q^{6} -1.49919 q^{7} -7.18385 q^{8} +6.18207 q^{9} -4.55264 q^{11} -14.4313 q^{12} -6.26213 q^{13} +3.89862 q^{14} +9.15646 q^{16} +5.97793 q^{17} -16.0764 q^{18} -3.57983 q^{19} +4.54284 q^{21} +11.8391 q^{22} -6.76317 q^{23} +21.7684 q^{24} +16.2846 q^{26} -9.64228 q^{27} -7.13990 q^{28} -3.27850 q^{29} -4.86195 q^{31} -9.44350 q^{32} +13.7954 q^{33} -15.5455 q^{34} +29.4421 q^{36} +9.31160 q^{37} +9.30929 q^{38} +18.9755 q^{39} +1.97370 q^{41} -11.8136 q^{42} +5.98429 q^{43} -21.6820 q^{44} +17.5875 q^{46} -8.42175 q^{47} -27.7458 q^{48} -4.75243 q^{49} -18.1143 q^{51} -29.8234 q^{52} +11.0696 q^{53} +25.0746 q^{54} +10.7700 q^{56} +10.8476 q^{57} +8.52567 q^{58} +8.77550 q^{59} +9.19955 q^{61} +12.6434 q^{62} -9.26810 q^{63} +6.24475 q^{64} -35.8747 q^{66} +15.1272 q^{67} +28.4700 q^{68} +20.4937 q^{69} -8.51912 q^{71} -44.4110 q^{72} -1.35081 q^{73} -24.2147 q^{74} -17.0490 q^{76} +6.82528 q^{77} -49.3454 q^{78} +0.518054 q^{79} +10.6718 q^{81} -5.13256 q^{82} -12.6728 q^{83} +21.6353 q^{84} -15.5620 q^{86} +9.93448 q^{87} +32.7055 q^{88} +6.25837 q^{89} +9.38812 q^{91} -32.2096 q^{92} +14.7327 q^{93} +21.9006 q^{94} +28.6156 q^{96} -7.07145 q^{97} +12.3586 q^{98} -28.1448 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q - 11 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 16 * q^7 - 33 * q^8 + 38 * q^9	\( 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9} + q^{11} - 14 q^{12} - 9 q^{13} - q^{14} + 43 q^{16} - 12 q^{17} - 42 q^{18} + 2 q^{21} - 5 q^{22} - 77 q^{23} - 2 q^{24} + 2 q^{26} - 38 q^{27} - 42 q^{28} + 2 q^{29} + q^{31} - 72 q^{32} - 20 q^{33} + 5 q^{34} + 32 q^{36} - 28 q^{37} - 23 q^{38} - 2 q^{39} - 2 q^{41} - 37 q^{42} - 31 q^{43} + 3 q^{44} + 14 q^{46} - 96 q^{47} - 13 q^{48} + 40 q^{49} - 10 q^{51} - 42 q^{52} - 54 q^{53} + 4 q^{54} - 15 q^{56} - 37 q^{57} - 27 q^{58} + q^{59} + 5 q^{61} - 39 q^{62} - 70 q^{63} + 65 q^{64} - 52 q^{66} - 34 q^{67} - 52 q^{68} + 21 q^{69} - 9 q^{71} - 70 q^{72} - 25 q^{73} + 22 q^{74} - 47 q^{76} - 54 q^{77} - 58 q^{78} + 13 q^{79} + 12 q^{81} + 5 q^{82} - 63 q^{83} + 95 q^{84} - 18 q^{86} - 47 q^{87} - 13 q^{88} + 19 q^{89} - 31 q^{91} - 137 q^{92} - 52 q^{93} + 120 q^{94} - 49 q^{96} - 36 q^{97} - 64 q^{98} + 16 q^{99}+O(q^{100}) \) 40 * q - 11 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 16 * q^7 - 33 * q^8 + 38 * q^9 + q^11 - 14 * q^12 - 9 * q^13 - q^14 + 43 * q^16 - 12 * q^17 - 42 * q^18 + 2 * q^21 - 5 * q^22 - 77 * q^23 - 2 * q^24 + 2 * q^26 - 38 * q^27 - 42 * q^28 + 2 * q^29 + q^31 - 72 * q^32 - 20 * q^33 + 5 * q^34 + 32 * q^36 - 28 * q^37 - 23 * q^38 - 2 * q^39 - 2 * q^41 - 37 * q^42 - 31 * q^43 + 3 * q^44 + 14 * q^46 - 96 * q^47 - 13 * q^48 + 40 * q^49 - 10 * q^51 - 42 * q^52 - 54 * q^53 + 4 * q^54 - 15 * q^56 - 37 * q^57 - 27 * q^58 + q^59 + 5 * q^61 - 39 * q^62 - 70 * q^63 + 65 * q^64 - 52 * q^66 - 34 * q^67 - 52 * q^68 + 21 * q^69 - 9 * q^71 - 70 * q^72 - 25 * q^73 + 22 * q^74 - 47 * q^76 - 54 * q^77 - 58 * q^78 + 13 * q^79 + 12 * q^81 + 5 * q^82 - 63 * q^83 + 95 * q^84 - 18 * q^86 - 47 * q^87 - 13 * q^88 + 19 * q^89 - 31 * q^91 - 137 * q^92 - 52 * q^93 + 120 * q^94 - 49 * q^96 - 36 * q^97 - 64 * q^98 + 16 * q^99

Introduction

L-functions

Modular forms

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Database

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