LMFDB - Embedded newform 6025.2.a.n.1.15

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			1.15
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)

$f(q)$	$=$	$q-0.460717 q^{2} +3.13691 q^{3} -1.78774 q^{4} -1.44523 q^{6} +1.56377 q^{7} +1.74508 q^{8} +6.84020 q^{9} +O(q^{10})$	\(q-0.460717 q^{2} +3.13691 q^{3} -1.78774 q^{4} -1.44523 q^{6} +1.56377 q^{7} +1.74508 q^{8} +6.84020 q^{9} -6.23889 q^{11} -5.60798 q^{12} -1.64617 q^{13} -0.720457 q^{14} +2.77149 q^{16} +5.60674 q^{17} -3.15139 q^{18} +3.84868 q^{19} +4.90541 q^{21} +2.87436 q^{22} +1.97859 q^{23} +5.47414 q^{24} +0.758419 q^{26} +12.0463 q^{27} -2.79562 q^{28} -3.28774 q^{29} +0.165861 q^{31} -4.76703 q^{32} -19.5708 q^{33} -2.58312 q^{34} -12.2285 q^{36} +0.397154 q^{37} -1.77315 q^{38} -5.16389 q^{39} +11.4703 q^{41} -2.26001 q^{42} +0.935446 q^{43} +11.1535 q^{44} -0.911571 q^{46} +8.85389 q^{47} +8.69392 q^{48} -4.55461 q^{49} +17.5878 q^{51} +2.94293 q^{52} -12.1915 q^{53} -5.54995 q^{54} +2.72890 q^{56} +12.0729 q^{57} +1.51472 q^{58} -0.836116 q^{59} +11.1724 q^{61} -0.0764151 q^{62} +10.6965 q^{63} -3.34674 q^{64} +9.01661 q^{66} -9.21225 q^{67} -10.0234 q^{68} +6.20666 q^{69} +0.539173 q^{71} +11.9367 q^{72} -0.619847 q^{73} -0.182975 q^{74} -6.88043 q^{76} -9.75621 q^{77} +2.37909 q^{78} +11.0121 q^{79} +17.2677 q^{81} -5.28456 q^{82} -2.18586 q^{83} -8.76961 q^{84} -0.430976 q^{86} -10.3134 q^{87} -10.8873 q^{88} +2.25777 q^{89} -2.57424 q^{91} -3.53721 q^{92} +0.520292 q^{93} -4.07913 q^{94} -14.9537 q^{96} -7.24352 q^{97} +2.09839 q^{98} -42.6752 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.460717	−0.325776	−0.162888	−	0.986645i	$-0.552081\pi$
$2$	−0.460717	−0.325776	−0.162888	+	0.986645i	$0.552081\pi$
$3$	3.13691	1.81110	0.905548	−	0.424245i	$-0.139460\pi$
$3$	3.13691	1.81110	0.905548	+	0.424245i	$0.139460\pi$
$4$	−1.78774	−0.893870
$5$	0	0
$5$	0	0
$6$	−1.44523	−0.590011
$7$	1.56377	0.591051	0.295525	−	0.955335i	$-0.404505\pi$
$7$	1.56377	0.591051	0.295525	+	0.955335i	$0.404505\pi$
$8$	1.74508	0.616977
$9$	6.84020	2.28007
$10$	0	0
$11$	−6.23889	−1.88110	−0.940548	−	0.339661i	$-0.889688\pi$
$11$	−6.23889	−1.88110	−0.940548	+	0.339661i	$0.889688\pi$
$12$	−5.60798	−1.61888
$13$	−1.64617	−0.456566	−0.228283	−	0.973595i	$-0.573311\pi$
$13$	−1.64617	−0.456566	−0.228283	+	0.973595i	$0.573311\pi$
$14$	−0.720457	−0.192550
$15$	0	0
$16$	2.77149	0.692874
$17$	5.60674	1.35983	0.679917	−	0.733289i	$-0.262016\pi$
$17$	5.60674	1.35983	0.679917	+	0.733289i	$0.262016\pi$
$18$	−3.15139	−0.742791
$19$	3.84868	0.882947	0.441474	−	0.897274i	$-0.354456\pi$
$19$	3.84868	0.882947	0.441474	+	0.897274i	$0.354456\pi$
$20$	0	0
$21$	4.90541	1.07045
$22$	2.87436	0.612816
$23$	1.97859	0.412565	0.206283	−	0.978492i	$-0.433863\pi$
$23$	1.97859	0.412565	0.206283	+	0.978492i	$0.433863\pi$
$24$	5.47414	1.11740
$25$	0	0
$26$	0.758419	0.148738
$27$	12.0463	2.31832
$28$	−2.79562	−0.528323
$29$	−3.28774	−0.610519	−0.305259	−	0.952269i	$-0.598743\pi$
$29$	−3.28774	−0.610519	−0.305259	+	0.952269i	$0.598743\pi$
$30$	0	0
$31$	0.165861	0.0297896	0.0148948	−	0.999889i	$-0.495259\pi$
$31$	0.165861	0.0297896	0.0148948	+	0.999889i	$0.495259\pi$
$32$	−4.76703	−0.842699
$33$	−19.5708	−3.40684
$34$	−2.58312	−0.443001
$35$	0	0
$36$	−12.2285	−2.03808
$37$	0.397154	0.0652917	0.0326458	−	0.999467i	$-0.489607\pi$
$37$	0.397154	0.0652917	0.0326458	+	0.999467i	$0.489607\pi$
$38$	−1.77315	−0.287643
$39$	−5.16389	−0.826884
$40$	0	0
$41$	11.4703	1.79136	0.895680	−	0.444699i	$-0.146689\pi$
$41$	11.4703	1.79136	0.895680	+	0.444699i	$0.146689\pi$
$42$	−2.26001	−0.348727
$43$	0.935446	0.142654	0.0713271	−	0.997453i	$-0.477277\pi$
$43$	0.935446	0.142654	0.0713271	+	0.997453i	$0.477277\pi$
$44$	11.1535	1.68145
$45$	0	0
$46$	−0.911571	−0.134404
$47$	8.85389	1.29147	0.645736	−	0.763561i	$-0.276551\pi$
$47$	8.85389	1.29147	0.645736	+	0.763561i	$0.276551\pi$
$48$	8.69392	1.25486
$49$	−4.55461	−0.650659
$50$	0	0
$51$	17.5878	2.46279
$52$	2.94293	0.408111
$53$	−12.1915	−1.67463	−0.837313	−	0.546723i	$-0.815875\pi$
$53$	−12.1915	−1.67463	−0.837313	+	0.546723i	$0.815875\pi$
$54$	−5.54995	−0.755253
$55$	0	0
$56$	2.72890	0.364665
$57$	12.0729	1.59910
$58$	1.51472	0.198892
$59$	−0.836116	−0.108853	−0.0544265	−	0.998518i	$-0.517333\pi$
$59$	−0.836116	−0.108853	−0.0544265	+	0.998518i	$0.517333\pi$
$60$	0	0
$61$	11.1724	1.43047	0.715237	−	0.698882i	$-0.246319\pi$
$61$	11.1724	1.43047	0.715237	+	0.698882i	$0.246319\pi$
$62$	−0.0764151	−0.00970473
$63$	10.6965	1.34763
$64$	−3.34674	−0.418342
$65$	0	0
$66$	9.01661	1.10987
$67$	−9.21225	−1.12546	−0.562728	−	0.826642i	$-0.690248\pi$
$67$	−9.21225	−1.12546	−0.562728	+	0.826642i	$0.690248\pi$
$68$	−10.0234	−1.21551
$69$	6.20666	0.747194
$70$	0	0
$71$	0.539173	0.0639880	0.0319940	−	0.999488i	$-0.489814\pi$
$71$	0.539173	0.0639880	0.0319940	+	0.999488i	$0.489814\pi$
$72$	11.9367	1.40675
$73$	−0.619847	−0.0725476	−0.0362738	−	0.999342i	$-0.511549\pi$
$73$	−0.619847	−0.0725476	−0.0362738	+	0.999342i	$0.511549\pi$
$74$	−0.182975	−0.0212705
$75$	0	0
$76$	−6.88043	−0.789240
$77$	−9.75621	−1.11182
$78$	2.37909	0.269379
$79$	11.0121	1.23895	0.619477	−	0.785015i	$-0.287345\pi$
$79$	11.0121	1.23895	0.619477	+	0.785015i	$0.287345\pi$
$80$	0	0
$81$	17.2677	1.91863
$82$	−5.28456	−0.583582
$83$	−2.18586	−0.239929	−0.119965	−	0.992778i	$-0.538278\pi$
$83$	−2.18586	−0.239929	−0.119965	+	0.992778i	$0.538278\pi$
$84$	−8.76961	−0.956842
$85$	0	0
$86$	−0.430976	−0.0464733
$87$	−10.3134	−1.10571
$88$	−10.8873	−1.16059
$89$	2.25777	0.239323	0.119662	−	0.992815i	$-0.461819\pi$
$89$	2.25777	0.239323	0.119662	+	0.992815i	$0.461819\pi$
$90$	0	0
$91$	−2.57424	−0.269854
$92$	−3.53721	−0.368779
$93$	0.520292	0.0539518
$94$	−4.07913	−0.420731
$95$	0	0
$96$	−14.9537	−1.52621
$97$	−7.24352	−0.735468	−0.367734	−	0.929931i	$-0.619866\pi$
$97$	−7.24352	−0.735468	−0.367734	+	0.929931i	$0.619866\pi$
$98$	2.09839	0.211969
$99$	−42.6752	−4.28902
$100$	0	0
$101$	−2.71147	−0.269802	−0.134901	−	0.990859i	$-0.543072\pi$
$101$	−2.71147	−0.269802	−0.134901	+	0.990859i	$0.543072\pi$
$102$	−8.10301	−0.802317
$103$	0.177602	0.0174996	0.00874980	−	0.999962i	$-0.497215\pi$
$103$	0.177602	0.0174996	0.00874980	+	0.999962i	$0.497215\pi$
$104$	−2.87269	−0.281691
$105$	0	0
$106$	5.61681	0.545553
$107$	16.3490	1.58052	0.790258	−	0.612774i	$-0.209947\pi$
$107$	16.3490	1.58052	0.790258	+	0.612774i	$0.209947\pi$
$108$	−21.5357	−2.07228
$109$	18.3591	1.75848	0.879242	−	0.476376i	$-0.158050\pi$
$109$	18.3591	1.75848	0.879242	+	0.476376i	$0.158050\pi$
$110$	0	0
$111$	1.24584	0.118249
$112$	4.33399	0.409523
$113$	7.57102	0.712221	0.356111	−	0.934444i	$-0.384103\pi$
$113$	7.57102	0.712221	0.356111	+	0.934444i	$0.384103\pi$
$114$	−5.56221	−0.520949
$115$	0	0
$116$	5.87763	0.545724
$117$	−11.2601	−1.04100
$118$	0.385213	0.0354617
$119$	8.76767	0.803731
$120$	0	0
$121$	27.9237	2.53852
$122$	−5.14729	−0.466014
$123$	35.9813	3.24432
$124$	−0.296517	−0.0266280
$125$	0	0
$126$	−4.92807	−0.439027
$127$	−7.81978	−0.693893	−0.346947	−	0.937885i	$-0.612782\pi$
$127$	−7.81978	−0.693893	−0.346947	+	0.937885i	$0.612782\pi$
$128$	11.0759	0.978985
$129$	2.93441	0.258360
$130$	0	0
$131$	6.96702	0.608711	0.304356	−	0.952558i	$-0.401559\pi$
$131$	6.96702	0.608711	0.304356	+	0.952558i	$0.401559\pi$
$132$	34.9875	3.04527
$133$	6.01846	0.521867
$134$	4.24424	0.366646
$135$	0	0
$136$	9.78418	0.838987
$137$	5.17568	0.442188	0.221094	−	0.975252i	$-0.429037\pi$
$137$	5.17568	0.442188	0.221094	+	0.975252i	$0.429037\pi$
$138$	−2.85951	−0.243418
$139$	−10.6546	−0.903714	−0.451857	−	0.892090i	$-0.649238\pi$
$139$	−10.6546	−0.903714	−0.451857	+	0.892090i	$0.649238\pi$
$140$	0	0
$141$	27.7738	2.33898
$142$	−0.248406	−0.0208458
$143$	10.2703	0.858844
$144$	18.9576	1.57980
$145$	0	0
$146$	0.285574	0.0236343
$147$	−14.2874	−1.17841
$148$	−0.710008	−0.0583623
$149$	8.67506	0.710689	0.355344	−	0.934735i	$-0.384364\pi$
$149$	8.67506	0.710689	0.355344	+	0.934735i	$0.384364\pi$
$150$	0	0
$151$	5.36349	0.436474	0.218237	−	0.975896i	$-0.429969\pi$
$151$	5.36349	0.436474	0.218237	+	0.975896i	$0.429969\pi$
$152$	6.71623	0.544758
$153$	38.3512	3.10051
$154$	4.49485	0.362205
$155$	0	0
$156$	9.23170	0.739127
$157$	19.1465	1.52806	0.764030	−	0.645180i	$-0.223218\pi$
$157$	19.1465	1.52806	0.764030	+	0.645180i	$0.223218\pi$
$158$	−5.07344	−0.403621
$159$	−38.2435	−3.03291
$160$	0	0
$161$	3.09407	0.243847
$162$	−7.95552	−0.625045
$163$	1.64100	0.128533	0.0642666	−	0.997933i	$-0.479529\pi$
$163$	1.64100	0.128533	0.0642666	+	0.997933i	$0.479529\pi$
$164$	−20.5059	−1.60124
$165$	0	0
$166$	1.00706	0.0781632
$167$	7.75478	0.600083	0.300041	−	0.953926i	$-0.403000\pi$
$167$	7.75478	0.600083	0.300041	+	0.953926i	$0.403000\pi$
$168$	8.56032	0.660443
$169$	−10.2901	−0.791547
$170$	0	0
$171$	26.3257	2.01318
$172$	−1.67233	−0.127514
$173$	−24.0284	−1.82685	−0.913423	−	0.407013i	$-0.866571\pi$
$173$	−24.0284	−1.82685	−0.913423	+	0.407013i	$0.866571\pi$
$174$	4.75153	0.360213
$175$	0	0
$176$	−17.2910	−1.30336
$177$	−2.62282	−0.197143
$178$	−1.04019	−0.0779658
$179$	−1.14342	−0.0854629	−0.0427314	−	0.999087i	$-0.513606\pi$
$179$	−1.14342	−0.0854629	−0.0427314	+	0.999087i	$0.513606\pi$
$180$	0	0
$181$	5.84022	0.434100	0.217050	−	0.976160i	$-0.430356\pi$
$181$	5.84022	0.434100	0.217050	+	0.976160i	$0.430356\pi$
$182$	1.18600	0.0879118
$183$	35.0466	2.59072
$184$	3.45279	0.254543
$185$	0	0
$186$	−0.239707	−0.0175762
$187$	−34.9798	−2.55798
$188$	−15.8284	−1.15441
$189$	18.8378	1.37024
$190$	0	0
$191$	24.6293	1.78211	0.891057	−	0.453891i	$-0.149964\pi$
$191$	24.6293	1.78211	0.891057	+	0.453891i	$0.149964\pi$
$192$	−10.4984	−0.757658
$193$	−8.15947	−0.587332	−0.293666	−	0.955908i	$-0.594875\pi$
$193$	−8.15947	−0.587332	−0.293666	+	0.955908i	$0.594875\pi$
$194$	3.33721	0.239598
$195$	0	0
$196$	8.14246	0.581605
$197$	20.3690	1.45123	0.725617	−	0.688098i	$-0.241554\pi$
$197$	20.3690	1.45123	0.725617	+	0.688098i	$0.241554\pi$
$198$	19.6612	1.39726
$199$	−4.91440	−0.348372	−0.174186	−	0.984713i	$-0.555729\pi$
$199$	−4.91440	−0.348372	−0.174186	+	0.984713i	$0.555729\pi$
$200$	0	0
$201$	−28.8980	−2.03831
$202$	1.24922	0.0878949
$203$	−5.14129	−0.360848
$204$	−31.4425	−2.20141
$205$	0	0
$206$	−0.0818241	−0.00570095
$207$	13.5340	0.940675
$208$	−4.56236	−0.316343
$209$	−24.0115	−1.66091
$210$	0	0
$211$	−24.5510	−1.69016	−0.845082	−	0.534637i	$-0.820448\pi$
$211$	−24.5510	−1.69016	−0.845082	+	0.534637i	$0.820448\pi$
$212$	21.7952	1.49690
$213$	1.69134	0.115888
$214$	−7.53225	−0.514894
$215$	0	0
$216$	21.0218	1.43035
$217$	0.259370	0.0176072
$218$	−8.45835	−0.572872
$219$	−1.94440	−0.131391
$220$	0	0
$221$	−9.22966	−0.620854
$222$	−0.573977	−0.0385228
$223$	−7.15892	−0.479397	−0.239699	−	0.970847i	$-0.577049\pi$
$223$	−7.15892	−0.479397	−0.239699	+	0.970847i	$0.577049\pi$
$224$	−7.45455	−0.498078
$225$	0	0
$226$	−3.48810	−0.232025
$227$	26.1878	1.73815	0.869073	−	0.494683i	$-0.164716\pi$
$227$	26.1878	1.73815	0.869073	+	0.494683i	$0.164716\pi$
$228$	−21.5833	−1.42939
$229$	4.52449	0.298987	0.149494	−	0.988763i	$-0.452236\pi$
$229$	4.52449	0.298987	0.149494	+	0.988763i	$0.452236\pi$
$230$	0	0
$231$	−30.6043	−2.01362
$232$	−5.73736	−0.376676
$233$	21.3515	1.39878	0.699391	−	0.714739i	$-0.253454\pi$
$233$	21.3515	1.39878	0.699391	+	0.714739i	$0.253454\pi$
$234$	5.18774	0.339133
$235$	0	0
$236$	1.49476	0.0973004
$237$	34.5438	2.24386
$238$	−4.03941	−0.261836
$239$	25.4300	1.64493	0.822466	−	0.568814i	$-0.192598\pi$
$239$	25.4300	1.64493	0.822466	+	0.568814i	$0.192598\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−12.8649	−0.826989
$243$	18.0282	1.15651
$244$	−19.9733	−1.27866
$245$	0	0
$246$	−16.5772	−1.05692
$247$	−6.33558	−0.403124
$248$	0.289441	0.0183795
$249$	−6.85684	−0.434535
$250$	0	0
$251$	−29.3573	−1.85302	−0.926508	−	0.376274i	$-0.877205\pi$
$251$	−29.3573	−1.85302	−0.926508	+	0.376274i	$0.877205\pi$
$252$	−19.1226	−1.20461
$253$	−12.3442	−0.776074
$254$	3.60271	0.226054
$255$	0	0
$256$	1.59060	0.0994127
$257$	22.1494	1.38164	0.690820	−	0.723027i	$-0.257250\pi$
$257$	22.1494	1.38164	0.690820	+	0.723027i	$0.257250\pi$
$258$	−1.35193	−0.0841676
$259$	0.621059	0.0385907
$260$	0	0
$261$	−22.4888	−1.39202
$262$	−3.20982	−0.198304
$263$	−17.3292	−1.06856	−0.534282	−	0.845306i	$-0.679418\pi$
$263$	−17.3292	−1.06856	−0.534282	+	0.845306i	$0.679418\pi$
$264$	−34.1526	−2.10195
$265$	0	0
$266$	−2.77281	−0.170012
$267$	7.08242	0.433437
$268$	16.4691	1.00601
$269$	−15.9261	−0.971032	−0.485516	−	0.874228i	$-0.661368\pi$
$269$	−15.9261	−0.971032	−0.485516	+	0.874228i	$0.661368\pi$
$270$	0	0
$271$	20.3504	1.23620	0.618099	−	0.786100i	$-0.287903\pi$
$271$	20.3504	1.23620	0.618099	+	0.786100i	$0.287903\pi$
$272$	15.5390	0.942193
$273$	−8.07516	−0.488731
$274$	−2.38452	−0.144054
$275$	0	0
$276$	−11.0959	−0.667895
$277$	−27.6708	−1.66258	−0.831289	−	0.555841i	$-0.812396\pi$
$277$	−27.6708	−1.66258	−0.831289	+	0.555841i	$0.812396\pi$
$278$	4.90877	0.294408
$279$	1.13452	0.0679222
$280$	0	0
$281$	−19.8732	−1.18554	−0.592769	−	0.805373i	$-0.701965\pi$
$281$	−19.8732	−1.18554	−0.592769	+	0.805373i	$0.701965\pi$
$282$	−12.7959	−0.761983
$283$	10.4690	0.622319	0.311160	−	0.950358i	$-0.399283\pi$
$283$	10.4690	0.622319	0.311160	+	0.950358i	$0.399283\pi$
$284$	−0.963900	−0.0571970
$285$	0	0
$286$	−4.73169	−0.279791
$287$	17.9369	1.05878
$288$	−32.6074	−1.92141
$289$	14.4355	0.849148
$290$	0	0
$291$	−22.7223	−1.33200
$292$	1.10813	0.0648481
$293$	10.2693	0.599940	0.299970	−	0.953949i	$-0.403023\pi$
$293$	10.2693	0.599940	0.299970	+	0.953949i	$0.403023\pi$
$294$	6.58245	0.383896
$295$	0	0
$296$	0.693063	0.0402835
$297$	−75.1558	−4.36098
$298$	−3.99675	−0.231525
$299$	−3.25710	−0.188363
$300$	0	0
$301$	1.46283	0.0843158
$302$	−2.47105	−0.142193
$303$	−8.50564	−0.488636
$304$	10.6666	0.611771
$305$	0	0
$306$	−17.6690	−1.01007
$307$	−18.7089	−1.06777	−0.533887	−	0.845556i	$-0.679269\pi$
$307$	−18.7089	−1.06777	−0.533887	+	0.845556i	$0.679269\pi$
$308$	17.4416	0.993825
$309$	0.557120	0.0316935
$310$	0	0
$311$	−8.38262	−0.475335	−0.237667	−	0.971347i	$-0.576383\pi$
$311$	−8.38262	−0.475335	−0.237667	+	0.971347i	$0.576383\pi$
$312$	−9.01138	−0.510169
$313$	−11.9859	−0.677485	−0.338742	−	0.940879i	$-0.610002\pi$
$313$	−11.9859	−0.677485	−0.338742	+	0.940879i	$0.610002\pi$
$314$	−8.82114	−0.497806
$315$	0	0
$316$	−19.6867	−1.10746
$317$	10.6885	0.600327	0.300163	−	0.953888i	$-0.402959\pi$
$317$	10.6885	0.600327	0.300163	+	0.953888i	$0.402959\pi$
$318$	17.6194	0.988049
$319$	20.5119	1.14844
$320$	0	0
$321$	51.2853	2.86246
$322$	−1.42549	−0.0794395
$323$	21.5785	1.20066
$324$	−30.8702	−1.71501
$325$	0	0
$326$	−0.756037	−0.0418730
$327$	57.5909	3.18478
$328$	20.0165	1.10523
$329$	13.8455	0.763325
$330$	0	0
$331$	−1.55229	−0.0853217	−0.0426608	−	0.999090i	$-0.513583\pi$
$331$	−1.55229	−0.0853217	−0.0426608	+	0.999090i	$0.513583\pi$
$332$	3.90775	0.214466
$333$	2.71661	0.148869
$334$	−3.57276	−0.195493
$335$	0	0
$336$	13.5953	0.741686
$337$	32.7770	1.78548	0.892738	−	0.450576i	$-0.148781\pi$
$337$	32.7770	1.78548	0.892738	+	0.450576i	$0.148781\pi$
$338$	4.74083	0.257867
$339$	23.7496	1.28990
$340$	0	0
$341$	−1.03479	−0.0560371
$342$	−12.1287	−0.655845
$343$	−18.0688	−0.975623
$344$	1.63242	0.0880144
$345$	0	0
$346$	11.0703	0.595142
$347$	−0.133187	−0.00714987	−0.00357494	−	0.999994i	$-0.501138\pi$
$347$	−0.133187	−0.00714987	−0.00357494	+	0.999994i	$0.501138\pi$
$348$	18.4376	0.988359
$349$	−17.7031	−0.947623	−0.473811	−	0.880626i	$-0.657122\pi$
$349$	−17.7031	−0.947623	−0.473811	+	0.880626i	$0.657122\pi$
$350$	0	0
$351$	−19.8304	−1.05847
$352$	29.7409	1.58520
$353$	−36.7532	−1.95618	−0.978089	−	0.208189i	$-0.933243\pi$
$353$	−36.7532	−1.95618	−0.978089	+	0.208189i	$0.933243\pi$
$354$	1.20838	0.0642245
$355$	0	0
$356$	−4.03631	−0.213924
$357$	27.5034	1.45563
$358$	0.526791	0.0278417
$359$	1.20900	0.0638087	0.0319043	−	0.999491i	$-0.489843\pi$
$359$	1.20900	0.0638087	0.0319043	+	0.999491i	$0.489843\pi$
$360$	0	0
$361$	−4.18769	−0.220405
$362$	−2.69069	−0.141420
$363$	87.5942	4.59750
$364$	4.60207	0.241214
$365$	0	0
$366$	−16.1466	−0.843995
$367$	26.4811	1.38230	0.691151	−	0.722710i	$-0.257104\pi$
$367$	26.4811	1.38230	0.691151	+	0.722710i	$0.257104\pi$
$368$	5.48366	0.285855
$369$	78.4591	4.08442
$370$	0	0
$371$	−19.0647	−0.989789
$372$	−0.930147	−0.0482259
$373$	1.46114	0.0756550	0.0378275	−	0.999284i	$-0.487956\pi$
$373$	1.46114	0.0756550	0.0378275	+	0.999284i	$0.487956\pi$
$374$	16.1158	0.833328
$375$	0	0
$376$	15.4507	0.796809
$377$	5.41219	0.278742
$378$	−8.67887	−0.446393
$379$	−22.8502	−1.17374	−0.586868	−	0.809683i	$-0.699639\pi$
$379$	−22.8502	−1.17374	−0.586868	+	0.809683i	$0.699639\pi$
$380$	0	0
$381$	−24.5299	−1.25671
$382$	−11.3471	−0.580570
$383$	1.62018	0.0827873	0.0413936	−	0.999143i	$-0.486820\pi$
$383$	1.62018	0.0827873	0.0413936	+	0.999143i	$0.486820\pi$
$384$	34.7442	1.77303
$385$	0	0
$386$	3.75921	0.191339
$387$	6.39863	0.325261
$388$	12.9495	0.657413
$389$	−26.5358	−1.34542	−0.672708	−	0.739908i	$-0.734869\pi$
$389$	−26.5358	−1.34542	−0.672708	+	0.739908i	$0.734869\pi$
$390$	0	0
$391$	11.0934	0.561020
$392$	−7.94814	−0.401442
$393$	21.8549	1.10243
$394$	−9.38436	−0.472777
$395$	0	0
$396$	76.2922	3.83383
$397$	12.4792	0.626315	0.313157	−	0.949701i	$-0.398613\pi$
$397$	12.4792	0.626315	0.313157	+	0.949701i	$0.398613\pi$
$398$	2.26414	0.113491
$399$	18.8794	0.945150
$400$	0	0
$401$	−16.3699	−0.817476	−0.408738	−	0.912652i	$-0.634031\pi$
$401$	−16.3699	−0.817476	−0.408738	+	0.912652i	$0.634031\pi$
$402$	13.3138	0.664032
$403$	−0.273036	−0.0136009
$404$	4.84741	0.241167
$405$	0	0
$406$	2.36868	0.117555
$407$	−2.47780	−0.122820
$408$	30.6921	1.51948
$409$	1.14611	0.0566715	0.0283357	−	0.999598i	$-0.490979\pi$
$409$	1.14611	0.0566715	0.0283357	+	0.999598i	$0.490979\pi$
$410$	0	0
$411$	16.2356	0.800845
$412$	−0.317506	−0.0156424
$413$	−1.30750	−0.0643377
$414$	−6.23532	−0.306449
$415$	0	0
$416$	7.84734	0.384748
$417$	−33.4226	−1.63671
$418$	11.0625	0.541084
$419$	34.6674	1.69361	0.846807	−	0.531901i	$-0.178522\pi$
$419$	34.6674	1.69361	0.846807	+	0.531901i	$0.178522\pi$
$420$	0	0
$421$	2.98107	0.145289	0.0726443	−	0.997358i	$-0.476856\pi$
$421$	2.98107	0.145289	0.0726443	+	0.997358i	$0.476856\pi$
$422$	11.3111	0.550615
$423$	60.5623	2.94464
$424$	−21.2750	−1.03321
$425$	0	0
$426$	−0.779227	−0.0377537
$427$	17.4710	0.845482
$428$	−29.2277	−1.41278
$429$	32.2169	1.55545
$430$	0	0
$431$	−10.7519	−0.517898	−0.258949	−	0.965891i	$-0.583376\pi$
$431$	−10.7519	−0.517898	−0.258949	+	0.965891i	$0.583376\pi$
$432$	33.3864	1.60630
$433$	−28.1073	−1.35075	−0.675376	−	0.737474i	$-0.736018\pi$
$433$	−28.1073	−1.35075	−0.675376	+	0.737474i	$0.736018\pi$
$434$	−0.119496	−0.00573599
$435$	0	0
$436$	−32.8213	−1.57186
$437$	7.61496	0.364273
$438$	0.895819	0.0428039
$439$	−19.4634	−0.928938	−0.464469	−	0.885589i	$-0.653755\pi$
$439$	−19.4634	−0.928938	−0.464469	+	0.885589i	$0.653755\pi$
$440$	0	0
$441$	−31.1544	−1.48355
$442$	4.25226	0.202259
$443$	−13.2766	−0.630792	−0.315396	−	0.948960i	$-0.602137\pi$
$443$	−13.2766	−0.630792	−0.315396	+	0.948960i	$0.602137\pi$
$444$	−2.22723	−0.105700
$445$	0	0
$446$	3.29824	0.156176
$447$	27.2129	1.28712
$448$	−5.23354	−0.247262
$449$	9.13708	0.431206	0.215603	−	0.976481i	$-0.430828\pi$
$449$	9.13708	0.431206	0.215603	+	0.976481i	$0.430828\pi$
$450$	0	0
$451$	−71.5619	−3.36972
$452$	−13.5350	−0.636633
$453$	16.8248	0.790496
$454$	−12.0652	−0.566246
$455$	0	0
$456$	21.0682	0.986609
$457$	−19.9016	−0.930959	−0.465479	−	0.885059i	$-0.654118\pi$
$457$	−19.9016	−0.930959	−0.465479	+	0.885059i	$0.654118\pi$
$458$	−2.08451	−0.0974028
$459$	67.5407	3.15253
$460$	0	0
$461$	12.8467	0.598330	0.299165	−	0.954201i	$-0.403292\pi$
$461$	12.8467	0.598330	0.299165	+	0.954201i	$0.403292\pi$
$462$	14.0999	0.655988
$463$	14.1128	0.655879	0.327940	−	0.944699i	$-0.393646\pi$
$463$	14.1128	0.655879	0.327940	+	0.944699i	$0.393646\pi$
$464$	−9.11196	−0.423012
$465$	0	0
$466$	−9.83699	−0.455690
$467$	12.3901	0.573345	0.286672	−	0.958029i	$-0.407451\pi$
$467$	12.3901	0.573345	0.286672	+	0.958029i	$0.407451\pi$
$468$	20.1302	0.930519
$469$	−14.4059	−0.665201
$470$	0	0
$471$	60.0610	2.76746
$472$	−1.45909	−0.0671598
$473$	−5.83614	−0.268346
$474$	−15.9149	−0.730997
$475$	0	0
$476$	−15.6743	−0.718431
$477$	−83.3920	−3.81826
$478$	−11.7160	−0.535879
$479$	−28.9441	−1.32249	−0.661245	−	0.750170i	$-0.729972\pi$
$479$	−28.9441	−1.32249	−0.661245	+	0.750170i	$0.729972\pi$
$480$	0	0
$481$	−0.653784	−0.0298100
$482$	−0.460717	−0.0209851
$483$	9.70581	0.441630
$484$	−49.9204	−2.26911
$485$	0	0
$486$	−8.30587	−0.376762
$487$	12.8942	0.584292	0.292146	−	0.956374i	$-0.405631\pi$
$487$	12.8942	0.584292	0.292146	+	0.956374i	$0.405631\pi$
$488$	19.4966	0.882570
$489$	5.14768	0.232786
$490$	0	0
$491$	15.9396	0.719344	0.359672	−	0.933079i	$-0.382889\pi$
$491$	15.9396	0.719344	0.359672	+	0.933079i	$0.382889\pi$
$492$	−64.3252	−2.90000
$493$	−18.4335	−0.830204
$494$	2.91891	0.131328
$495$	0	0
$496$	0.459684	0.0206404
$497$	0.843144	0.0378202
$498$	3.15906	0.141561
$499$	3.02622	0.135472	0.0677362	−	0.997703i	$-0.478422\pi$
$499$	3.02622	0.135472	0.0677362	+	0.997703i	$0.478422\pi$
$500$	0	0
$501$	24.3260	1.08681
$502$	13.5254	0.603668
$503$	26.4629	1.17992	0.589961	−	0.807432i	$-0.299143\pi$
$503$	26.4629	1.17992	0.589961	+	0.807432i	$0.299143\pi$
$504$	18.6662	0.831460
$505$	0	0
$506$	5.68719	0.252826
$507$	−32.2792	−1.43357
$508$	13.9797	0.620251
$509$	16.8536	0.747024	0.373512	−	0.927625i	$-0.378154\pi$
$509$	16.8536	0.747024	0.373512	+	0.927625i	$0.378154\pi$
$510$	0	0
$511$	−0.969300	−0.0428793
$512$	−22.8847	−1.01137
$513$	46.3625	2.04695
$514$	−10.2046	−0.450105
$515$	0	0
$516$	−5.24596	−0.230940
$517$	−55.2384	−2.42938
$518$	−0.286132	−0.0125719
$519$	−75.3749	−3.30859
$520$	0	0
$521$	−21.4237	−0.938591	−0.469295	−	0.883041i	$-0.655492\pi$
$521$	−21.4237	−0.938591	−0.469295	+	0.883041i	$0.655492\pi$
$522$	10.3610	0.453488
$523$	2.10170	0.0919007	0.0459504	−	0.998944i	$-0.485368\pi$
$523$	2.10170	0.0919007	0.0459504	+	0.998944i	$0.485368\pi$
$524$	−12.4552	−0.544109
$525$	0	0
$526$	7.98385	0.348112
$527$	0.929941	0.0405089
$528$	−54.2404	−2.36051
$529$	−19.0852	−0.829790
$530$	0	0
$531$	−5.71920	−0.248192
$532$	−10.7594	−0.466481
$533$	−18.8821	−0.817874
$534$	−3.26299	−0.141203
$535$	0	0
$536$	−16.0761	−0.694381
$537$	−3.58679	−0.154781
$538$	7.33742	0.316339
$539$	28.4157	1.22395
$540$	0	0
$541$	36.1902	1.55594	0.777968	−	0.628304i	$-0.216251\pi$
$541$	36.1902	1.55594	0.777968	+	0.628304i	$0.216251\pi$
$542$	−9.37577	−0.402724
$543$	18.3203	0.786197
$544$	−26.7275	−1.14593
$545$	0	0
$546$	3.72036	0.159217
$547$	4.54148	0.194180	0.0970898	−	0.995276i	$-0.469047\pi$
$547$	4.54148	0.194180	0.0970898	+	0.995276i	$0.469047\pi$
$548$	−9.25277	−0.395259
$549$	76.4211	3.26157
$550$	0	0
$551$	−12.6535	−0.539056
$552$	10.8311	0.461002
$553$	17.2204	0.732284
$554$	12.7484	0.541628
$555$	0	0
$556$	19.0477	0.807803
$557$	−24.9534	−1.05731	−0.528654	−	0.848837i	$-0.677303\pi$
$557$	−24.9534	−1.05731	−0.528654	+	0.848837i	$0.677303\pi$
$558$	−0.522695	−0.0221274
$559$	−1.53990	−0.0651310
$560$	0	0
$561$	−109.728	−4.63274
$562$	9.15593	0.386220
$563$	−32.2026	−1.35718	−0.678589	−	0.734518i	$-0.737408\pi$
$563$	−32.2026	−1.35718	−0.678589	+	0.734518i	$0.737408\pi$
$564$	−49.6524	−2.09074
$565$	0	0
$566$	−4.82326	−0.202737
$567$	27.0028	1.13401
$568$	0.940897	0.0394792
$569$	−36.2971	−1.52166	−0.760828	−	0.648954i	$-0.775207\pi$
$569$	−36.2971	−1.52166	−0.760828	+	0.648954i	$0.775207\pi$
$570$	0	0
$571$	17.6923	0.740401	0.370200	−	0.928952i	$-0.379289\pi$
$571$	17.6923	0.740401	0.370200	+	0.928952i	$0.379289\pi$
$572$	−18.3606	−0.767695
$573$	77.2599	3.22758
$574$	−8.26385	−0.344927
$575$	0	0
$576$	−22.8924	−0.953848
$577$	−2.04206	−0.0850121	−0.0425060	−	0.999096i	$-0.513534\pi$
$577$	−2.04206	−0.0850121	−0.0425060	+	0.999096i	$0.513534\pi$
$578$	−6.65068	−0.276632
$579$	−25.5955	−1.06371
$580$	0	0
$581$	−3.41819	−0.141810
$582$	10.4685	0.433934
$583$	76.0612	3.15013
$584$	−1.08168	−0.0447602
$585$	0	0
$586$	−4.73125	−0.195446
$587$	−27.7310	−1.14458	−0.572291	−	0.820050i	$-0.693945\pi$
$587$	−27.7310	−1.14458	−0.572291	+	0.820050i	$0.693945\pi$
$588$	25.5422	1.05334
$589$	0.638347	0.0263026
$590$	0	0
$591$	63.8958	2.62832
$592$	1.10071	0.0452389
$593$	29.7831	1.22305	0.611524	−	0.791226i	$-0.290557\pi$
$593$	29.7831	1.22305	0.611524	+	0.791226i	$0.290557\pi$
$594$	34.6255	1.42070
$595$	0	0
$596$	−15.5088	−0.635263
$597$	−15.4160	−0.630935
$598$	1.50060	0.0613642
$599$	20.2828	0.828733	0.414367	−	0.910110i	$-0.364003\pi$
$599$	20.2828	0.828733	0.414367	+	0.910110i	$0.364003\pi$
$600$	0	0
$601$	0.0237662	0.000969445 0	0.000484722	−	1.00000i	$-0.499846\pi$
$601$	0.0237662	0.000969445 0	0.000484722		1.00000i	$0.499846\pi$
$602$	−0.673948	−0.0274681
$603$	−63.0136	−2.56611
$604$	−9.58852	−0.390151
$605$	0	0
$606$	3.91869	0.159186
$607$	−26.3346	−1.06889	−0.534444	−	0.845204i	$-0.679479\pi$
$607$	−26.3346	−1.06889	−0.534444	+	0.845204i	$0.679479\pi$
$608$	−18.3467	−0.744059
$609$	−16.1277	−0.653529
$610$	0	0
$611$	−14.5750	−0.589642
$612$	−68.5620	−2.77145
$613$	28.6550	1.15737	0.578683	−	0.815553i	$-0.303567\pi$
$613$	28.6550	1.15737	0.578683	+	0.815553i	$0.303567\pi$
$614$	8.61951	0.347855
$615$	0	0
$616$	−17.0253	−0.685970
$617$	−39.0568	−1.57237	−0.786183	−	0.617994i	$-0.787946\pi$
$617$	−39.0568	−1.57237	−0.786183	+	0.617994i	$0.787946\pi$
$618$	−0.256675	−0.0103250
$619$	31.5865	1.26957	0.634785	−	0.772689i	$-0.281089\pi$
$619$	31.5865	1.26957	0.634785	+	0.772689i	$0.281089\pi$
$620$	0	0
$621$	23.8348	0.956458
$622$	3.86201	0.154853
$623$	3.53064	0.141452
$624$	−14.3117	−0.572926
$625$	0	0
$626$	5.52212	0.220708
$627$	−75.3218	−3.00806
$628$	−34.2290	−1.36589
$629$	2.22674	0.0887858
$630$	0	0
$631$	−31.0013	−1.23414	−0.617070	−	0.786908i	$-0.711681\pi$
$631$	−31.0013	−1.23414	−0.617070	+	0.786908i	$0.711681\pi$
$632$	19.2169	0.764406
$633$	−77.0144	−3.06105
$634$	−4.92438	−0.195572
$635$	0	0
$636$	68.3695	2.71103
$637$	7.49768	0.297069
$638$	−9.45016	−0.374135
$639$	3.68805	0.145897
$640$	0	0
$641$	40.0027	1.58001	0.790006	−	0.613099i	$-0.210077\pi$
$641$	40.0027	1.58001	0.790006	+	0.613099i	$0.210077\pi$
$642$	−23.6280	−0.932522
$643$	22.3354	0.880824	0.440412	−	0.897796i	$-0.354832\pi$
$643$	22.3354	0.880824	0.440412	+	0.897796i	$0.354832\pi$
$644$	−5.53139	−0.217967
$645$	0	0
$646$	−9.94159	−0.391147
$647$	18.8557	0.741295	0.370647	−	0.928774i	$-0.379136\pi$
$647$	18.8557	0.741295	0.370647	+	0.928774i	$0.379136\pi$
$648$	30.1334	1.18375
$649$	5.21643	0.204763
$650$	0	0
$651$	0.813619	0.0318882
$652$	−2.93369	−0.114892
$653$	−35.9274	−1.40595	−0.702974	−	0.711215i	$-0.748145\pi$
$653$	−35.9274	−1.40595	−0.702974	+	0.711215i	$0.748145\pi$
$654$	−26.5331	−1.03753
$655$	0	0
$656$	31.7899	1.24119
$657$	−4.23987	−0.165413
$658$	−6.37884	−0.248673
$659$	11.5076	0.448274	0.224137	−	0.974558i	$-0.428044\pi$
$659$	11.5076	0.448274	0.224137	+	0.974558i	$0.428044\pi$
$660$	0	0
$661$	2.26981	0.0882852	0.0441426	−	0.999025i	$-0.485944\pi$
$661$	2.26981	0.0882852	0.0441426	+	0.999025i	$0.485944\pi$
$662$	0.715167	0.0277957
$663$	−28.9526	−1.12443
$664$	−3.81449	−0.148031
$665$	0	0
$666$	−1.25159	−0.0484980
$667$	−6.50510	−0.251879
$668$	−13.8635	−0.536396
$669$	−22.4569	−0.868234
$670$	0	0
$671$	−69.7031	−2.69086
$672$	−23.3842	−0.902066
$673$	−24.3712	−0.939441	−0.469720	−	0.882815i	$-0.655645\pi$
$673$	−24.3712	−0.939441	−0.469720	+	0.882815i	$0.655645\pi$
$674$	−15.1009	−0.581665
$675$	0	0
$676$	18.3961	0.707541
$677$	2.60623	0.100166	0.0500829	−	0.998745i	$-0.484051\pi$
$677$	2.60623	0.100166	0.0500829	+	0.998745i	$0.484051\pi$
$678$	−10.9418	−0.420219
$679$	−11.3272	−0.434699
$680$	0	0
$681$	82.1488	3.14795
$682$	0.476746	0.0182555
$683$	−6.93762	−0.265461	−0.132730	−	0.991152i	$-0.542374\pi$
$683$	−6.93762	−0.265461	−0.132730	+	0.991152i	$0.542374\pi$
$684$	−47.0635	−1.79952
$685$	0	0
$686$	8.32460	0.317835
$687$	14.1929	0.541494
$688$	2.59258	0.0988413
$689$	20.0693	0.764578
$690$	0	0
$691$	−4.69414	−0.178574	−0.0892868	−	0.996006i	$-0.528459\pi$
$691$	−4.69414	−0.178574	−0.0892868	+	0.996006i	$0.528459\pi$
$692$	42.9565	1.63296
$693$	−66.7344	−2.53503
$694$	0.0613617	0.00232926
$695$	0	0
$696$	−17.9976	−0.682196
$697$	64.3110	2.43595
$698$	8.15610	0.308713
$699$	66.9777	2.53333
$700$	0	0
$701$	−1.76392	−0.0666223	−0.0333112	−	0.999445i	$-0.510605\pi$
$701$	−1.76392	−0.0666223	−0.0333112	+	0.999445i	$0.510605\pi$
$702$	9.13618	0.344823
$703$	1.52852	0.0576491
$704$	20.8799	0.786942
$705$	0	0
$706$	16.9328	0.637276
$707$	−4.24013	−0.159466
$708$	4.68892	0.176220
$709$	−26.5409	−0.996766	−0.498383	−	0.866957i	$-0.666073\pi$
$709$	−26.5409	−0.996766	−0.498383	+	0.866957i	$0.666073\pi$
$710$	0	0
$711$	75.3246	2.82489
$712$	3.93998	0.147657
$713$	0.328172	0.0122901
$714$	−12.6713	−0.474210
$715$	0	0
$716$	2.04413	0.0763927
$717$	79.7717	2.97913
$718$	−0.557007	−0.0207873
$719$	19.6415	0.732503	0.366252	−	0.930516i	$-0.380641\pi$
$719$	19.6415	0.732503	0.366252	+	0.930516i	$0.380641\pi$
$720$	0	0
$721$	0.277729	0.0103432
$722$	1.92934	0.0718025
$723$	3.13691	0.116663
$724$	−10.4408	−0.388029
$725$	0	0
$726$	−40.3561	−1.49776
$727$	22.6724	0.840874	0.420437	−	0.907322i	$-0.361877\pi$
$727$	22.6724	0.840874	0.420437	+	0.907322i	$0.361877\pi$
$728$	−4.49224	−0.166494
$729$	4.74958	0.175910
$730$	0	0
$731$	5.24480	0.193986
$732$	−62.6543	−2.31577
$733$	19.0027	0.701881	0.350941	−	0.936398i	$-0.385862\pi$
$733$	19.0027	0.701881	0.350941	+	0.936398i	$0.385862\pi$
$734$	−12.2003	−0.450321
$735$	0	0
$736$	−9.43200	−0.347668
$737$	57.4742	2.11709
$738$	−36.1474	−1.33061
$739$	−0.100236	−0.00368725	−0.00184362	−	0.999998i	$-0.500587\pi$
$739$	−0.100236	−0.00368725	−0.00184362	+	0.999998i	$0.500587\pi$
$740$	0	0
$741$	−19.8742	−0.730095
$742$	8.78342	0.322450
$743$	11.6994	0.429211	0.214606	−	0.976701i	$-0.431153\pi$
$743$	11.6994	0.429211	0.214606	+	0.976701i	$0.431153\pi$
$744$	0.907949	0.0332870
$745$	0	0
$746$	−0.673172	−0.0246466
$747$	−14.9517	−0.547055
$748$	62.5348	2.28650
$749$	25.5661	0.934165
$750$	0	0
$751$	−25.8880	−0.944667	−0.472334	−	0.881420i	$-0.656588\pi$
$751$	−25.8880	−0.944667	−0.472334	+	0.881420i	$0.656588\pi$
$752$	24.5385	0.894827
$753$	−92.0912	−3.35599
$754$	−2.49349	−0.0908075
$755$	0	0
$756$	−33.6770	−1.22482
$757$	−31.9067	−1.15967	−0.579834	−	0.814735i	$-0.696882\pi$
$757$	−31.9067	−1.15967	−0.579834	+	0.814735i	$0.696882\pi$
$758$	10.5275	0.382375
$759$	−38.7227	−1.40554
$760$	0	0
$761$	−45.1311	−1.63600	−0.818000	−	0.575218i	$-0.804917\pi$
$761$	−45.1311	−1.63600	−0.818000	+	0.575218i	$0.804917\pi$
$762$	11.3014	0.409405
$763$	28.7095	1.03935
$764$	−44.0308	−1.59298
$765$	0	0
$766$	−0.746444	−0.0269701
$767$	1.37639	0.0496986
$768$	4.98958	0.180046
$769$	−0.386985	−0.0139550	−0.00697751	−	0.999976i	$-0.502221\pi$
$769$	−0.386985	−0.0139550	−0.00697751	+	0.999976i	$0.502221\pi$
$770$	0	0
$771$	69.4805	2.50228
$772$	14.5870	0.524998
$773$	−33.7397	−1.21353	−0.606767	−	0.794880i	$-0.707534\pi$
$773$	−33.7397	−1.21353	−0.606767	+	0.794880i	$0.707534\pi$
$774$	−2.94796	−0.105962
$775$	0	0
$776$	−12.6405	−0.453767
$777$	1.94820	0.0698914
$778$	12.2255	0.438304
$779$	44.1455	1.58168
$780$	0	0
$781$	−3.36384	−0.120368
$782$	−5.11094	−0.182767
$783$	−39.6053	−1.41538
$784$	−12.6231	−0.450824
$785$	0	0
$786$	−10.0689	−0.359147
$787$	−8.95234	−0.319116	−0.159558	−	0.987189i	$-0.551007\pi$
$787$	−8.95234	−0.319116	−0.159558	+	0.987189i	$0.551007\pi$
$788$	−36.4146	−1.29722
$789$	−54.3601	−1.93527
$790$	0	0
$791$	11.8394	0.420959
$792$	−74.4715	−2.64623
$793$	−18.3916	−0.653105
$794$	−5.74939	−0.204038
$795$	0	0
$796$	8.78566	0.311400
$797$	−37.3700	−1.32371	−0.661856	−	0.749631i	$-0.730231\pi$
$797$	−37.3700	−1.32371	−0.661856	+	0.749631i	$0.730231\pi$
$798$	−8.69804	−0.307907
$799$	49.6414	1.75619
$800$	0	0
$801$	15.4436	0.545673
$802$	7.54191	0.266314
$803$	3.86716	0.136469
$804$	51.6621	1.82198
$805$	0	0
$806$	0.125792	0.00443085
$807$	−49.9587	−1.75863
$808$	−4.73172	−0.166461
$809$	−36.3719	−1.27877	−0.639385	−	0.768887i	$-0.720811\pi$
$809$	−36.3719	−1.27877	−0.639385	+	0.768887i	$0.720811\pi$
$810$	0	0
$811$	−53.9078	−1.89296	−0.946479	−	0.322766i	$-0.895387\pi$
$811$	−53.9078	−1.89296	−0.946479	+	0.322766i	$0.895387\pi$
$812$	9.19128	0.322551
$813$	63.8374	2.23887
$814$	1.14156	0.0400118
$815$	0	0
$816$	48.7446	1.70640
$817$	3.60023	0.125956
$818$	−0.528032	−0.0184622
$819$	−17.6083	−0.615284
$820$	0	0
$821$	−9.76390	−0.340762	−0.170381	−	0.985378i	$-0.554500\pi$
$821$	−9.76390	−0.340762	−0.170381	+	0.985378i	$0.554500\pi$
$822$	−7.48003	−0.260896
$823$	−27.2909	−0.951301	−0.475650	−	0.879634i	$-0.657787\pi$
$823$	−27.2909	−0.951301	−0.475650	+	0.879634i	$0.657787\pi$
$824$	0.309928	0.0107969
$825$	0	0
$826$	0.602385	0.0209597
$827$	−50.2553	−1.74755	−0.873773	−	0.486334i	$-0.838334\pi$
$827$	−50.2553	−1.74755	−0.873773	+	0.486334i	$0.838334\pi$
$828$	−24.1952	−0.840841
$829$	46.7978	1.62535	0.812677	−	0.582714i	$-0.198009\pi$
$829$	46.7978	1.62535	0.812677	+	0.582714i	$0.198009\pi$
$830$	0	0
$831$	−86.8008	−3.01109
$832$	5.50931	0.191001
$833$	−25.5365	−0.884788
$834$	15.3984	0.533202
$835$	0	0
$836$	42.9263	1.48464
$837$	1.99802	0.0690618
$838$	−15.9719	−0.551739
$839$	−31.0781	−1.07294	−0.536468	−	0.843921i	$-0.680242\pi$
$839$	−31.0781	−1.07294	−0.536468	+	0.843921i	$0.680242\pi$
$840$	0	0
$841$	−18.1907	−0.627267
$842$	−1.37343	−0.0473315
$843$	−62.3405	−2.14712
$844$	43.8909	1.51079
$845$	0	0
$846$	−27.9021	−0.959293
$847$	43.6664	1.50039
$848$	−33.7886	−1.16030
$849$	32.8404	1.12708
$850$	0	0
$851$	0.785805	0.0269371
$852$	−3.02367	−0.103589
$853$	−13.0908	−0.448220	−0.224110	−	0.974564i	$-0.571948\pi$
$853$	−13.0908	−0.448220	−0.224110	+	0.974564i	$0.571948\pi$
$854$	−8.04920	−0.275438
$855$	0	0
$856$	28.5302	0.975142
$857$	−10.7419	−0.366937	−0.183468	−	0.983026i	$-0.558733\pi$
$857$	−10.7419	−0.366937	−0.183468	+	0.983026i	$0.558733\pi$
$858$	−14.8429	−0.506728
$859$	32.5982	1.11224	0.556119	−	0.831103i	$-0.312290\pi$
$859$	32.5982	1.11224	0.556119	+	0.831103i	$0.312290\pi$
$860$	0	0
$861$	56.2666	1.91756
$862$	4.95356	0.168719
$863$	32.3524	1.10129	0.550644	−	0.834740i	$-0.314382\pi$
$863$	32.3524	1.10129	0.550644	+	0.834740i	$0.314382\pi$
$864$	−57.4252	−1.95365
$865$	0	0
$866$	12.9495	0.440042
$867$	45.2829	1.53789
$868$	−0.463685	−0.0157385
$869$	−68.7030	−2.33059
$870$	0	0
$871$	15.1650	0.513845
$872$	32.0380	1.08494
$873$	−49.5471	−1.67692
$874$	−3.50834	−0.118671
$875$	0	0
$876$	3.47609	0.117446
$877$	−21.9696	−0.741859	−0.370930	−	0.928661i	$-0.620961\pi$
$877$	−21.9696	−0.741859	−0.370930	+	0.928661i	$0.620961\pi$
$878$	8.96712	0.302626
$879$	32.2139	1.08655
$880$	0	0
$881$	51.0958	1.72146	0.860730	−	0.509062i	$-0.170008\pi$
$881$	51.0958	1.72146	0.860730	+	0.509062i	$0.170008\pi$
$882$	14.3534	0.483303
$883$	−40.1088	−1.34977	−0.674885	−	0.737923i	$-0.735807\pi$
$883$	−40.1088	−1.34977	−0.674885	+	0.737923i	$0.735807\pi$
$884$	16.5002	0.554963
$885$	0	0
$886$	6.11677	0.205497
$887$	−8.96502	−0.301016	−0.150508	−	0.988609i	$-0.548091\pi$
$887$	−8.96502	−0.301016	−0.150508	+	0.988609i	$0.548091\pi$
$888$	2.17408	0.0729572
$889$	−12.2284	−0.410126
$890$	0	0
$891$	−107.731	−3.60913
$892$	12.7983	0.428519
$893$	34.0757	1.14030
$894$	−12.5374	−0.419314
$895$	0	0
$896$	17.3203	0.578630
$897$	−10.2172	−0.341144
$898$	−4.20961	−0.140476
$899$	−0.545310	−0.0181871
$900$	0	0
$901$	−68.3544	−2.27721
$902$	32.9698	1.09777
$903$	4.58875	0.152704
$904$	13.2120	0.439425
$905$	0	0
$906$	−7.75145	−0.257525
$907$	36.3979	1.20857	0.604287	−	0.796767i	$-0.293458\pi$
$907$	36.3979	1.20857	0.604287	+	0.796767i	$0.293458\pi$
$908$	−46.8170	−1.55368
$909$	−18.5470	−0.615165
$910$	0	0
$911$	−17.2020	−0.569928	−0.284964	−	0.958538i	$-0.591982\pi$
$911$	−17.2020	−0.569928	−0.284964	+	0.958538i	$0.591982\pi$
$912$	33.4601	1.10797
$913$	13.6373	0.451330
$914$	9.16902	0.303284
$915$	0	0
$916$	−8.08862	−0.267256
$917$	10.8948	0.359779
$918$	−31.1171	−1.02702
$919$	−33.0226	−1.08931	−0.544657	−	0.838659i	$-0.683340\pi$
$919$	−33.0226	−1.08931	−0.544657	+	0.838659i	$0.683340\pi$
$920$	0	0
$921$	−58.6882	−1.93384
$922$	−5.91868	−0.194921
$923$	−0.887571	−0.0292148
$924$	54.7126	1.79991
$925$	0	0
$926$	−6.50202	−0.213670
$927$	1.21483	0.0399003
$928$	15.6728	0.514483
$929$	11.9266	0.391299	0.195650	−	0.980674i	$-0.437318\pi$
$929$	11.9266	0.391299	0.195650	+	0.980674i	$0.437318\pi$
$930$	0	0
$931$	−17.5292	−0.574497
$932$	−38.1709	−1.25033
$933$	−26.2955	−0.860876
$934$	−5.70832	−0.186782
$935$	0	0
$936$	−19.6498	−0.642274
$937$	3.61452	0.118081	0.0590406	−	0.998256i	$-0.481196\pi$
$937$	3.61452	0.118081	0.0590406	+	0.998256i	$0.481196\pi$
$938$	6.63703	0.216707
$939$	−37.5988	−1.22699
$940$	0	0
$941$	42.5409	1.38679	0.693397	−	0.720556i	$-0.256113\pi$
$941$	42.5409	1.38679	0.693397	+	0.720556i	$0.256113\pi$
$942$	−27.6711	−0.901573
$943$	22.6950	0.739052
$944$	−2.31729	−0.0754214
$945$	0	0
$946$	2.68881	0.0874207
$947$	27.2502	0.885511	0.442756	−	0.896642i	$-0.354001\pi$
$947$	27.2502	0.885511	0.442756	+	0.896642i	$0.354001\pi$
$948$	−61.7554	−2.00572
$949$	1.02037	0.0331228
$950$	0	0
$951$	33.5289	1.08725
$952$	15.3002	0.495884
$953$	51.7919	1.67770	0.838852	−	0.544359i	$-0.183227\pi$
$953$	51.7919	1.67770	0.838852	+	0.544359i	$0.183227\pi$
$954$	38.4201	1.24390
$955$	0	0
$956$	−45.4623	−1.47036
$957$	64.3439	2.07994
$958$	13.3350	0.430835
$959$	8.09359	0.261356
$960$	0	0
$961$	−30.9725	−0.999113
$962$	0.301209	0.00971137
$963$	111.830	3.60368
$964$	−1.78774	−0.0575792
$965$	0	0
$966$	−4.47163	−0.143872
$967$	19.5691	0.629299	0.314650	−	0.949208i	$-0.398113\pi$
$967$	19.5691	0.629299	0.314650	+	0.949208i	$0.398113\pi$
$968$	48.7290	1.56621
$969$	67.6899	2.17451
$970$	0	0
$971$	−5.00517	−0.160623	−0.0803117	−	0.996770i	$-0.525592\pi$
$971$	−5.00517	−0.160623	−0.0803117	+	0.996770i	$0.525592\pi$
$972$	−32.2296	−1.03377
$973$	−16.6614	−0.534141
$974$	−5.94058	−0.190348
$975$	0	0
$976$	30.9641	0.991137
$977$	42.5568	1.36151	0.680757	−	0.732510i	$-0.261651\pi$
$977$	42.5568	1.36151	0.680757	+	0.732510i	$0.261651\pi$
$978$	−2.37162	−0.0758361
$979$	−14.0860	−0.450190
$980$	0	0
$981$	125.580	4.00946
$982$	−7.34364	−0.234345
$983$	−29.2524	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$983$	−29.2524	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$984$	62.7901	2.00167
$985$	0	0
$986$	8.49263	0.270460
$987$	43.4320	1.38245
$988$	11.3264	0.360340
$989$	1.85087	0.0588541
$990$	0	0
$991$	−14.7307	−0.467935	−0.233967	−	0.972244i	$-0.575171\pi$
$991$	−14.7307	−0.467935	−0.233967	+	0.972244i	$0.575171\pi$
$992$	−0.790666	−0.0251037
$993$	−4.86940	−0.154526
$994$	−0.388451	−0.0123209
$995$	0	0
$996$	12.2583	0.388418
$997$	−57.7565	−1.82917	−0.914584	−	0.404396i	$-0.867482\pi$
$997$	−57.7565	−1.82917	−0.914584	+	0.404396i	$0.867482\pi$
$998$	−1.39423	−0.0441336
$999$	4.78425	0.151367

Twists

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

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

\(f(q)\)	\(=\)	\(q-0.460717 q^{2} +3.13691 q^{3} -1.78774 q^{4} -1.44523 q^{6} +1.56377 q^{7} +1.74508 q^{8} +6.84020 q^{9} +O(q^{10})\)	\(q-0.460717 q^{2} +3.13691 q^{3} -1.78774 q^{4} -1.44523 q^{6} +1.56377 q^{7} +1.74508 q^{8} +6.84020 q^{9} -6.23889 q^{11} -5.60798 q^{12} -1.64617 q^{13} -0.720457 q^{14} +2.77149 q^{16} +5.60674 q^{17} -3.15139 q^{18} +3.84868 q^{19} +4.90541 q^{21} +2.87436 q^{22} +1.97859 q^{23} +5.47414 q^{24} +0.758419 q^{26} +12.0463 q^{27} -2.79562 q^{28} -3.28774 q^{29} +0.165861 q^{31} -4.76703 q^{32} -19.5708 q^{33} -2.58312 q^{34} -12.2285 q^{36} +0.397154 q^{37} -1.77315 q^{38} -5.16389 q^{39} +11.4703 q^{41} -2.26001 q^{42} +0.935446 q^{43} +11.1535 q^{44} -0.911571 q^{46} +8.85389 q^{47} +8.69392 q^{48} -4.55461 q^{49} +17.5878 q^{51} +2.94293 q^{52} -12.1915 q^{53} -5.54995 q^{54} +2.72890 q^{56} +12.0729 q^{57} +1.51472 q^{58} -0.836116 q^{59} +11.1724 q^{61} -0.0764151 q^{62} +10.6965 q^{63} -3.34674 q^{64} +9.01661 q^{66} -9.21225 q^{67} -10.0234 q^{68} +6.20666 q^{69} +0.539173 q^{71} +11.9367 q^{72} -0.619847 q^{73} -0.182975 q^{74} -6.88043 q^{76} -9.75621 q^{77} +2.37909 q^{78} +11.0121 q^{79} +17.2677 q^{81} -5.28456 q^{82} -2.18586 q^{83} -8.76961 q^{84} -0.430976 q^{86} -10.3134 q^{87} -10.8873 q^{88} +2.25777 q^{89} -2.57424 q^{91} -3.53721 q^{92} +0.520292 q^{93} -4.07913 q^{94} -14.9537 q^{96} -7.24352 q^{97} +2.09839 q^{98} -42.6752 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9}+O(q^{10}) \) 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

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