LMFDB - Embedded newform 6025.2.a.m.1.3

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			1.3
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)

$f(q)$	$=$	$q-2.67715 q^{2} +1.88460 q^{3} +5.16713 q^{4} -5.04535 q^{6} -1.59154 q^{7} -8.47889 q^{8} +0.551710 q^{9} +O(q^{10})$	\(q-2.67715 q^{2} +1.88460 q^{3} +5.16713 q^{4} -5.04535 q^{6} -1.59154 q^{7} -8.47889 q^{8} +0.551710 q^{9} +2.39627 q^{11} +9.73797 q^{12} +0.0275426 q^{13} +4.26080 q^{14} +12.3650 q^{16} -3.49494 q^{17} -1.47701 q^{18} +1.43682 q^{19} -2.99942 q^{21} -6.41518 q^{22} +7.45333 q^{23} -15.9793 q^{24} -0.0737356 q^{26} -4.61404 q^{27} -8.22372 q^{28} +2.60700 q^{29} +2.43312 q^{31} -16.1452 q^{32} +4.51601 q^{33} +9.35647 q^{34} +2.85076 q^{36} -8.17337 q^{37} -3.84659 q^{38} +0.0519067 q^{39} -7.97879 q^{41} +8.02990 q^{42} +3.35161 q^{43} +12.3818 q^{44} -19.9537 q^{46} -4.93313 q^{47} +23.3030 q^{48} -4.46699 q^{49} -6.58655 q^{51} +0.142316 q^{52} -12.5545 q^{53} +12.3525 q^{54} +13.4945 q^{56} +2.70783 q^{57} -6.97933 q^{58} -1.43698 q^{59} +3.26250 q^{61} -6.51383 q^{62} -0.878072 q^{63} +18.4930 q^{64} -12.0900 q^{66} -11.2230 q^{67} -18.0588 q^{68} +14.0465 q^{69} -6.36666 q^{71} -4.67789 q^{72} +3.44783 q^{73} +21.8813 q^{74} +7.42424 q^{76} -3.81377 q^{77} -0.138962 q^{78} +8.60027 q^{79} -10.3507 q^{81} +21.3604 q^{82} +0.554585 q^{83} -15.4984 q^{84} -8.97277 q^{86} +4.91315 q^{87} -20.3177 q^{88} -7.65069 q^{89} -0.0438353 q^{91} +38.5124 q^{92} +4.58546 q^{93} +13.2067 q^{94} -30.4271 q^{96} +14.2551 q^{97} +11.9588 q^{98} +1.32205 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.67715	−1.89303	−0.946515	−	0.322658i	$-0.895423\pi$
$2$	−2.67715	−1.89303	−0.946515	+	0.322658i	$0.895423\pi$
$3$	1.88460	1.08807	0.544037	−	0.839061i	$-0.316895\pi$
$3$	1.88460	1.08807	0.544037	+	0.839061i	$0.316895\pi$
$4$	5.16713	2.58357
$5$	0	0
$5$	0	0
$6$	−5.04535	−2.05976
$7$	−1.59154	−0.601547	−0.300774	−	0.953696i	$-0.597245\pi$
$7$	−1.59154	−0.601547	−0.300774	+	0.953696i	$0.597245\pi$
$8$	−8.47889	−2.99774
$9$	0.551710	0.183903
$10$	0	0
$11$	2.39627	0.722503	0.361251	−	0.932468i	$-0.382350\pi$
$11$	2.39627	0.722503	0.361251	+	0.932468i	$0.382350\pi$
$12$	9.73797	2.81111
$13$	0.0275426	0.00763894	0.00381947	−	0.999993i	$-0.498784\pi$
$13$	0.0275426	0.00763894	0.00381947	+	0.999993i	$0.498784\pi$
$14$	4.26080	1.13875
$15$	0	0
$16$	12.3650	3.09125
$17$	−3.49494	−0.847646	−0.423823	−	0.905745i	$-0.639312\pi$
$17$	−3.49494	−0.847646	−0.423823	+	0.905745i	$0.639312\pi$
$18$	−1.47701	−0.348135
$19$	1.43682	0.329629	0.164815	−	0.986325i	$-0.447297\pi$
$19$	1.43682	0.329629	0.164815	+	0.986325i	$0.447297\pi$
$20$	0	0
$21$	−2.99942	−0.654528
$22$	−6.41518	−1.36772
$23$	7.45333	1.55413	0.777064	−	0.629422i	$-0.216708\pi$
$23$	7.45333	1.55413	0.777064	+	0.629422i	$0.216708\pi$
$24$	−15.9793	−3.26176
$25$	0	0
$26$	−0.0737356	−0.0144608
$27$	−4.61404	−0.887973
$28$	−8.22372	−1.55414
$29$	2.60700	0.484108	0.242054	−	0.970263i	$-0.422179\pi$
$29$	2.60700	0.484108	0.242054	+	0.970263i	$0.422179\pi$
$30$	0	0
$31$	2.43312	0.437001	0.218501	−	0.975837i	$-0.429883\pi$
$31$	2.43312	0.437001	0.218501	+	0.975837i	$0.429883\pi$
$32$	−16.1452	−2.85409
$33$	4.51601	0.786136
$34$	9.35647	1.60462
$35$	0	0
$36$	2.85076	0.475127
$37$	−8.17337	−1.34369	−0.671846	−	0.740690i	$-0.734499\pi$
$37$	−8.17337	−1.34369	−0.671846	+	0.740690i	$0.734499\pi$
$38$	−3.84659	−0.623999
$39$	0.0519067	0.00831173
$40$	0	0
$41$	−7.97879	−1.24608	−0.623039	−	0.782191i	$-0.714102\pi$
$41$	−7.97879	−1.24608	−0.623039	+	0.782191i	$0.714102\pi$
$42$	8.02990	1.23904
$43$	3.35161	0.511116	0.255558	−	0.966794i	$-0.417741\pi$
$43$	3.35161	0.511116	0.255558	+	0.966794i	$0.417741\pi$
$44$	12.3818	1.86663
$45$	0	0
$46$	−19.9537	−2.94201
$47$	−4.93313	−0.719571	−0.359785	−	0.933035i	$-0.617150\pi$
$47$	−4.93313	−0.719571	−0.359785	+	0.933035i	$0.617150\pi$
$48$	23.3030	3.36350
$49$	−4.46699	−0.638141
$50$	0	0
$51$	−6.58655	−0.922301
$52$	0.142316	0.0197357
$53$	−12.5545	−1.72449	−0.862245	−	0.506492i	$-0.830942\pi$
$53$	−12.5545	−1.72449	−0.862245	+	0.506492i	$0.830942\pi$
$54$	12.3525	1.68096
$55$	0	0
$56$	13.4945	1.80328
$57$	2.70783	0.358661
$58$	−6.97933	−0.916431
$59$	−1.43698	−0.187079	−0.0935395	−	0.995616i	$-0.529818\pi$
$59$	−1.43698	−0.187079	−0.0935395	+	0.995616i	$0.529818\pi$
$60$	0	0
$61$	3.26250	0.417720	0.208860	−	0.977946i	$-0.433025\pi$
$61$	3.26250	0.417720	0.208860	+	0.977946i	$0.433025\pi$
$62$	−6.51383	−0.827257
$63$	−0.878072	−0.110627
$64$	18.4930	2.31163
$65$	0	0
$66$	−12.0900	−1.48818
$67$	−11.2230	−1.37111	−0.685555	−	0.728021i	$-0.740440\pi$
$67$	−11.2230	−1.37111	−0.685555	+	0.728021i	$0.740440\pi$
$68$	−18.0588	−2.18995
$69$	14.0465	1.69100
$70$	0	0
$71$	−6.36666	−0.755584	−0.377792	−	0.925890i	$-0.623317\pi$
$71$	−6.36666	−0.755584	−0.377792	+	0.925890i	$0.623317\pi$
$72$	−4.67789	−0.551295
$73$	3.44783	0.403538	0.201769	−	0.979433i	$-0.435331\pi$
$73$	3.44783	0.403538	0.201769	+	0.979433i	$0.435331\pi$
$74$	21.8813	2.54365
$75$	0	0
$76$	7.42424	0.851619
$77$	−3.81377	−0.434620
$78$	−0.138962	−0.0157344
$79$	8.60027	0.967606	0.483803	−	0.875177i	$-0.339255\pi$
$79$	8.60027	0.967606	0.483803	+	0.875177i	$0.339255\pi$
$80$	0	0
$81$	−10.3507	−1.15008
$82$	21.3604	2.35886
$83$	0.554585	0.0608737	0.0304368	−	0.999537i	$-0.490310\pi$
$83$	0.554585	0.0608737	0.0304368	+	0.999537i	$0.490310\pi$
$84$	−15.4984	−1.69102
$85$	0	0
$86$	−8.97277	−0.967559
$87$	4.91315	0.526745
$88$	−20.3177	−2.16588
$89$	−7.65069	−0.810971	−0.405486	−	0.914101i	$-0.632898\pi$
$89$	−7.65069	−0.810971	−0.405486	+	0.914101i	$0.632898\pi$
$90$	0	0
$91$	−0.0438353	−0.00459518
$92$	38.5124	4.01519
$93$	4.58546	0.475490
$94$	13.2067	1.36217
$95$	0	0
$96$	−30.4271	−3.10546
$97$	14.2551	1.44739	0.723694	−	0.690121i	$-0.242443\pi$
$97$	14.2551	1.44739	0.723694	+	0.690121i	$0.242443\pi$
$98$	11.9588	1.20802
$99$	1.32205	0.132871
$100$	0	0
$101$	12.4952	1.24332	0.621659	−	0.783288i	$-0.286459\pi$
$101$	12.4952	1.24332	0.621659	+	0.783288i	$0.286459\pi$
$102$	17.6332	1.74595
$103$	4.79357	0.472325	0.236162	−	0.971714i	$-0.424110\pi$
$103$	4.79357	0.472325	0.236162	+	0.971714i	$0.424110\pi$
$104$	−0.233531	−0.0228996
$105$	0	0
$106$	33.6102	3.26451
$107$	−1.02545	−0.0991339	−0.0495670	−	0.998771i	$-0.515784\pi$
$107$	−1.02545	−0.0991339	−0.0495670	+	0.998771i	$0.515784\pi$
$108$	−23.8414	−2.29414
$109$	1.83691	0.175944	0.0879720	−	0.996123i	$-0.471961\pi$
$109$	1.83691	0.175944	0.0879720	+	0.996123i	$0.471961\pi$
$110$	0	0
$111$	−15.4035	−1.46204
$112$	−19.6794	−1.85953
$113$	−11.0964	−1.04386	−0.521931	−	0.852988i	$-0.674788\pi$
$113$	−11.0964	−1.04386	−0.521931	+	0.852988i	$0.674788\pi$
$114$	−7.24927	−0.678956
$115$	0	0
$116$	13.4707	1.25072
$117$	0.0151955	0.00140483
$118$	3.84701	0.354146
$119$	5.56235	0.509899
$120$	0	0
$121$	−5.25789	−0.477990
$122$	−8.73420	−0.790757
$123$	−15.0368	−1.35582
$124$	12.5723	1.12902
$125$	0	0
$126$	2.35073	0.209420
$127$	3.29225	0.292140	0.146070	−	0.989274i	$-0.453338\pi$
$127$	3.29225	0.292140	0.146070	+	0.989274i	$0.453338\pi$
$128$	−17.2183	−1.52190
$129$	6.31645	0.556132
$130$	0	0
$131$	3.50073	0.305860	0.152930	−	0.988237i	$-0.451129\pi$
$131$	3.50073	0.305860	0.152930	+	0.988237i	$0.451129\pi$
$132$	23.3348	2.03103
$133$	−2.28676	−0.198288
$134$	30.0457	2.59555
$135$	0	0
$136$	29.6332	2.54102
$137$	−2.94398	−0.251521	−0.125760	−	0.992061i	$-0.540137\pi$
$137$	−2.94398	−0.251521	−0.125760	+	0.992061i	$0.540137\pi$
$138$	−37.6047	−3.20112
$139$	11.1797	0.948247	0.474124	−	0.880458i	$-0.342765\pi$
$139$	11.1797	0.948247	0.474124	+	0.880458i	$0.342765\pi$
$140$	0	0
$141$	−9.29696	−0.782945
$142$	17.0445	1.43034
$143$	0.0659995	0.00551916
$144$	6.82190	0.568491
$145$	0	0
$146$	−9.23035	−0.763909
$147$	−8.41847	−0.694344
$148$	−42.2329	−3.47152
$149$	−10.8371	−0.887811	−0.443905	−	0.896074i	$-0.646407\pi$
$149$	−10.8371	−0.887811	−0.443905	+	0.896074i	$0.646407\pi$
$150$	0	0
$151$	21.0824	1.71566	0.857832	−	0.513931i	$-0.171811\pi$
$151$	21.0824	1.71566	0.857832	+	0.513931i	$0.171811\pi$
$152$	−12.1826	−0.988143
$153$	−1.92819	−0.155885
$154$	10.2100	0.822748
$155$	0	0
$156$	0.268209	0.0214739
$157$	8.99973	0.718257	0.359128	−	0.933288i	$-0.383074\pi$
$157$	8.99973	0.718257	0.359128	+	0.933288i	$0.383074\pi$
$158$	−23.0242	−1.83171
$159$	−23.6601	−1.87637
$160$	0	0
$161$	−11.8623	−0.934881
$162$	27.7105	2.17714
$163$	−8.95348	−0.701290	−0.350645	−	0.936508i	$-0.614038\pi$
$163$	−8.95348	−0.701290	−0.350645	+	0.936508i	$0.614038\pi$
$164$	−41.2275	−3.21932
$165$	0	0
$166$	−1.48471	−0.115236
$167$	4.08733	0.316287	0.158144	−	0.987416i	$-0.449449\pi$
$167$	4.08733	0.316287	0.158144	+	0.987416i	$0.449449\pi$
$168$	25.4318	1.96210
$169$	−12.9992	−0.999942
$170$	0	0
$171$	0.792709	0.0606200
$172$	17.3182	1.32050
$173$	−13.0826	−0.994650	−0.497325	−	0.867564i	$-0.665684\pi$
$173$	−13.0826	−0.994650	−0.497325	+	0.867564i	$0.665684\pi$
$174$	−13.1532	−0.997144
$175$	0	0
$176$	29.6299	2.23344
$177$	−2.70813	−0.203556
$178$	20.4820	1.53519
$179$	21.0512	1.57344	0.786721	−	0.617309i	$-0.211777\pi$
$179$	21.0512	1.57344	0.786721	+	0.617309i	$0.211777\pi$
$180$	0	0
$181$	−12.7934	−0.950928	−0.475464	−	0.879735i	$-0.657720\pi$
$181$	−12.7934	−0.950928	−0.475464	+	0.879735i	$0.657720\pi$
$182$	0.117354	0.00869883
$183$	6.14850	0.454510
$184$	−63.1960	−4.65887
$185$	0	0
$186$	−12.2760	−0.900116
$187$	−8.37481	−0.612427
$188$	−25.4901	−1.85906
$189$	7.34345	0.534158
$190$	0	0
$191$	−12.0543	−0.872216	−0.436108	−	0.899894i	$-0.643644\pi$
$191$	−12.0543	−0.872216	−0.436108	+	0.899894i	$0.643644\pi$
$192$	34.8519	2.51522
$193$	−11.8654	−0.854091	−0.427045	−	0.904230i	$-0.640446\pi$
$193$	−11.8654	−0.854091	−0.427045	+	0.904230i	$0.640446\pi$
$194$	−38.1631	−2.73995
$195$	0	0
$196$	−23.0815	−1.64868
$197$	−17.3571	−1.23664	−0.618321	−	0.785925i	$-0.712187\pi$
$197$	−17.3571	−1.23664	−0.618321	+	0.785925i	$0.712187\pi$
$198$	−3.53932	−0.251528
$199$	4.64364	0.329179	0.164589	−	0.986362i	$-0.447370\pi$
$199$	4.64364	0.329179	0.164589	+	0.986362i	$0.447370\pi$
$200$	0	0
$201$	−21.1509	−1.49187
$202$	−33.4515	−2.35364
$203$	−4.14916	−0.291214
$204$	−34.0336	−2.38283
$205$	0	0
$206$	−12.8331	−0.894125
$207$	4.11208	0.285809
$208$	0.340564	0.0236139
$209$	3.44301	0.238158
$210$	0	0
$211$	−7.05588	−0.485747	−0.242873	−	0.970058i	$-0.578090\pi$
$211$	−7.05588	−0.485747	−0.242873	+	0.970058i	$0.578090\pi$
$212$	−64.8706	−4.45533
$213$	−11.9986	−0.822131
$214$	2.74528	0.187664
$215$	0	0
$216$	39.1219	2.66191
$217$	−3.87242	−0.262877
$218$	−4.91768	−0.333068
$219$	6.49777	0.439079
$220$	0	0
$221$	−0.0962596	−0.00647512
$222$	41.2375	2.76768
$223$	7.73879	0.518228	0.259114	−	0.965847i	$-0.416570\pi$
$223$	7.73879	0.518228	0.259114	+	0.965847i	$0.416570\pi$
$224$	25.6957	1.71687
$225$	0	0
$226$	29.7068	1.97606
$227$	−17.6652	−1.17248	−0.586239	−	0.810138i	$-0.699392\pi$
$227$	−17.6652	−1.17248	−0.586239	+	0.810138i	$0.699392\pi$
$228$	13.9917	0.926624
$229$	−11.5229	−0.761455	−0.380728	−	0.924687i	$-0.624326\pi$
$229$	−11.5229	−0.761455	−0.380728	+	0.924687i	$0.624326\pi$
$230$	0	0
$231$	−7.18743	−0.472898
$232$	−22.1045	−1.45123
$233$	−13.2245	−0.866367	−0.433184	−	0.901306i	$-0.642610\pi$
$233$	−13.2245	−0.866367	−0.433184	+	0.901306i	$0.642610\pi$
$234$	−0.0406807	−0.00265938
$235$	0	0
$236$	−7.42507	−0.483331
$237$	16.2081	1.05283
$238$	−14.8912	−0.965255
$239$	2.54491	0.164616	0.0823082	−	0.996607i	$-0.473771\pi$
$239$	2.54491	0.164616	0.0823082	+	0.996607i	$0.473771\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	14.0762	0.904850
$243$	−5.66487	−0.363402
$244$	16.8578	1.07921
$245$	0	0
$246$	40.2558	2.56662
$247$	0.0395738	0.00251802
$248$	−20.6302	−1.31002
$249$	1.04517	0.0662350
$250$	0	0
$251$	−14.0948	−0.889656	−0.444828	−	0.895616i	$-0.646735\pi$
$251$	−14.0948	−0.889656	−0.444828	+	0.895616i	$0.646735\pi$
$252$	−4.53711	−0.285811
$253$	17.8602	1.12286
$254$	−8.81385	−0.553030
$255$	0	0
$256$	9.10992	0.569370
$257$	−21.5811	−1.34619	−0.673097	−	0.739554i	$-0.735036\pi$
$257$	−21.5811	−1.34619	−0.673097	+	0.739554i	$0.735036\pi$
$258$	−16.9101	−1.05278
$259$	13.0083	0.808295
$260$	0	0
$261$	1.43831	0.0890291
$262$	−9.37198	−0.579003
$263$	−2.37822	−0.146648	−0.0733238	−	0.997308i	$-0.523361\pi$
$263$	−2.37822	−0.146648	−0.0733238	+	0.997308i	$0.523361\pi$
$264$	−38.2907	−2.35663
$265$	0	0
$266$	6.12201	0.375365
$267$	−14.4185	−0.882396
$268$	−57.9908	−3.54235
$269$	−11.0411	−0.673189	−0.336595	−	0.941650i	$-0.609275\pi$
$269$	−11.0411	−0.673189	−0.336595	+	0.941650i	$0.609275\pi$
$270$	0	0
$271$	−16.4996	−1.00228	−0.501140	−	0.865366i	$-0.667086\pi$
$271$	−16.4996	−1.00228	−0.501140	+	0.865366i	$0.667086\pi$
$272$	−43.2149	−2.62029
$273$	−0.0826119	−0.00499990
$274$	7.88147	0.476137
$275$	0	0
$276$	72.5803	4.36882
$277$	2.78767	0.167495	0.0837475	−	0.996487i	$-0.473311\pi$
$277$	2.78767	0.167495	0.0837475	+	0.996487i	$0.473311\pi$
$278$	−29.9297	−1.79506
$279$	1.34238	0.0803661
$280$	0	0
$281$	9.29721	0.554625	0.277312	−	0.960780i	$-0.410556\pi$
$281$	9.29721	0.554625	0.277312	+	0.960780i	$0.410556\pi$
$282$	24.8894	1.48214
$283$	23.0593	1.37073	0.685367	−	0.728198i	$-0.259642\pi$
$283$	23.0593	1.37073	0.685367	+	0.728198i	$0.259642\pi$
$284$	−32.8974	−1.95210
$285$	0	0
$286$	−0.176691	−0.0104479
$287$	12.6986	0.749575
$288$	−8.90745	−0.524877
$289$	−4.78542	−0.281496
$290$	0	0
$291$	26.8652	1.57486
$292$	17.8154	1.04257
$293$	8.78431	0.513185	0.256592	−	0.966520i	$-0.417400\pi$
$293$	8.78431	0.513185	0.256592	+	0.966520i	$0.417400\pi$
$294$	22.5375	1.31441
$295$	0	0
$296$	69.3011	4.02804
$297$	−11.0565	−0.641563
$298$	29.0126	1.68065
$299$	0.205284	0.0118719
$300$	0	0
$301$	−5.33424	−0.307461
$302$	−56.4408	−3.24780
$303$	23.5484	1.35282
$304$	17.7663	1.01897
$305$	0	0
$306$	5.16206	0.295095
$307$	21.8404	1.24650	0.623249	−	0.782023i	$-0.285812\pi$
$307$	21.8404	1.24650	0.623249	+	0.782023i	$0.285812\pi$
$308$	−19.7063	−1.12287
$309$	9.03395	0.513924
$310$	0	0
$311$	−25.4118	−1.44097	−0.720484	−	0.693471i	$-0.756080\pi$
$311$	−25.4118	−1.44097	−0.720484	+	0.693471i	$0.756080\pi$
$312$	−0.440111	−0.0249164
$313$	−16.1164	−0.910954	−0.455477	−	0.890247i	$-0.650531\pi$
$313$	−16.1164	−0.910954	−0.455477	+	0.890247i	$0.650531\pi$
$314$	−24.0936	−1.35968
$315$	0	0
$316$	44.4387	2.49987
$317$	−5.80010	−0.325766	−0.162883	−	0.986645i	$-0.552079\pi$
$317$	−5.80010	−0.325766	−0.162883	+	0.986645i	$0.552079\pi$
$318$	63.3417	3.55203
$319$	6.24708	0.349769
$320$	0	0
$321$	−1.93256	−0.107865
$322$	31.7572	1.76976
$323$	−5.02160	−0.279409
$324$	−53.4837	−2.97132
$325$	0	0
$326$	23.9698	1.32756
$327$	3.46184	0.191440
$328$	67.6513	3.73542
$329$	7.85129	0.432856
$330$	0	0
$331$	11.4808	0.631041	0.315521	−	0.948919i	$-0.397821\pi$
$331$	11.4808	0.631041	0.315521	+	0.948919i	$0.397821\pi$
$332$	2.86562	0.157271
$333$	−4.50933	−0.247110
$334$	−10.9424	−0.598741
$335$	0	0
$336$	−37.0878	−2.02331
$337$	15.8128	0.861379	0.430690	−	0.902500i	$-0.358270\pi$
$337$	15.8128	0.861379	0.430690	+	0.902500i	$0.358270\pi$
$338$	34.8009	1.89292
$339$	−20.9123	−1.13580
$340$	0	0
$341$	5.83042	0.315735
$342$	−2.12220	−0.114756
$343$	18.2502	0.985419
$344$	−28.4180	−1.53219
$345$	0	0
$346$	35.0240	1.88290
$347$	22.4772	1.20664	0.603320	−	0.797499i	$-0.293844\pi$
$347$	22.4772	1.20664	0.603320	+	0.797499i	$0.293844\pi$
$348$	25.3869	1.36088
$349$	−7.48633	−0.400734	−0.200367	−	0.979721i	$-0.564213\pi$
$349$	−7.48633	−0.400734	−0.200367	+	0.979721i	$0.564213\pi$
$350$	0	0
$351$	−0.127083	−0.00678317
$352$	−38.6882	−2.06209
$353$	−3.84764	−0.204789	−0.102395	−	0.994744i	$-0.532650\pi$
$353$	−3.84764	−0.204789	−0.102395	+	0.994744i	$0.532650\pi$
$354$	7.25007	0.385337
$355$	0	0
$356$	−39.5321	−2.09520
$357$	10.4828	0.554808
$358$	−56.3573	−2.97857
$359$	−24.6548	−1.30123	−0.650614	−	0.759408i	$-0.725489\pi$
$359$	−24.6548	−1.30123	−0.650614	+	0.759408i	$0.725489\pi$
$360$	0	0
$361$	−16.9355	−0.891344
$362$	34.2499	1.80014
$363$	−9.90901	−0.520088
$364$	−0.226503	−0.0118720
$365$	0	0
$366$	−16.4605	−0.860402
$367$	1.51800	0.0792387	0.0396194	−	0.999215i	$-0.487385\pi$
$367$	1.51800	0.0792387	0.0396194	+	0.999215i	$0.487385\pi$
$368$	92.1604	4.80419
$369$	−4.40198	−0.229158
$370$	0	0
$371$	19.9810	1.03736
$372$	23.6937	1.22846
$373$	−19.4909	−1.00920	−0.504600	−	0.863353i	$-0.668360\pi$
$373$	−19.4909	−1.00920	−0.504600	+	0.863353i	$0.668360\pi$
$374$	22.4206	1.15934
$375$	0	0
$376$	41.8274	2.15709
$377$	0.0718035	0.00369807
$378$	−19.6595	−1.01118
$379$	16.3391	0.839281	0.419640	−	0.907690i	$-0.362156\pi$
$379$	16.3391	0.839281	0.419640	+	0.907690i	$0.362156\pi$
$380$	0	0
$381$	6.20457	0.317870
$382$	32.2711	1.65113
$383$	−6.49369	−0.331812	−0.165906	−	0.986142i	$-0.553055\pi$
$383$	−6.49369	−0.331812	−0.165906	+	0.986142i	$0.553055\pi$
$384$	−32.4496	−1.65594
$385$	0	0
$386$	31.7655	1.61682
$387$	1.84912	0.0939961
$388$	73.6581	3.73942
$389$	−2.98582	−0.151387	−0.0756936	−	0.997131i	$-0.524117\pi$
$389$	−2.98582	−0.151387	−0.0756936	+	0.997131i	$0.524117\pi$
$390$	0	0
$391$	−26.0489	−1.31735
$392$	37.8751	1.91298
$393$	6.59747	0.332798
$394$	46.4676	2.34100
$395$	0	0
$396$	6.83119	0.343280
$397$	−5.46972	−0.274517	−0.137259	−	0.990535i	$-0.543829\pi$
$397$	−5.46972	−0.274517	−0.137259	+	0.990535i	$0.543829\pi$
$398$	−12.4317	−0.623146
$399$	−4.30963	−0.215752
$400$	0	0
$401$	27.0246	1.34955	0.674773	−	0.738025i	$-0.264242\pi$
$401$	27.0246	1.34955	0.674773	+	0.738025i	$0.264242\pi$
$402$	56.6241	2.82415
$403$	0.0670145	0.00333823
$404$	64.5643	3.21219
$405$	0	0
$406$	11.1079	0.551276
$407$	−19.5856	−0.970822
$408$	55.8466	2.76482
$409$	−21.2203	−1.04928	−0.524639	−	0.851325i	$-0.675800\pi$
$409$	−21.2203	−1.04928	−0.524639	+	0.851325i	$0.675800\pi$
$410$	0	0
$411$	−5.54822	−0.273673
$412$	24.7690	1.22028
$413$	2.28702	0.112537
$414$	−11.0087	−0.541046
$415$	0	0
$416$	−0.444680	−0.0218022
$417$	21.0692	1.03176
$418$	−9.21746	−0.450841
$419$	−31.4097	−1.53446	−0.767232	−	0.641369i	$-0.778367\pi$
$419$	−31.4097	−1.53446	−0.767232	+	0.641369i	$0.778367\pi$
$420$	0	0
$421$	−25.7953	−1.25719	−0.628593	−	0.777734i	$-0.716369\pi$
$421$	−25.7953	−1.25719	−0.628593	+	0.777734i	$0.716369\pi$
$422$	18.8897	0.919534
$423$	−2.72166	−0.132332
$424$	106.448	5.16957
$425$	0	0
$426$	32.1221	1.55632
$427$	−5.19241	−0.251278
$428$	−5.29863	−0.256119
$429$	0.124383	0.00600525
$430$	0	0
$431$	4.29683	0.206971	0.103485	−	0.994631i	$-0.467000\pi$
$431$	4.29683	0.206971	0.103485	+	0.994631i	$0.467000\pi$
$432$	−57.0526	−2.74494
$433$	27.3144	1.31265	0.656324	−	0.754479i	$-0.272110\pi$
$433$	27.3144	1.31265	0.656324	+	0.754479i	$0.272110\pi$
$434$	10.3670	0.497634
$435$	0	0
$436$	9.49156	0.454563
$437$	10.7091	0.512286
$438$	−17.3955	−0.831189
$439$	34.3159	1.63781	0.818904	−	0.573931i	$-0.194582\pi$
$439$	34.3159	1.63781	0.818904	+	0.573931i	$0.194582\pi$
$440$	0	0
$441$	−2.46448	−0.117356
$442$	0.257701	0.0122576
$443$	17.6022	0.836307	0.418154	−	0.908376i	$-0.362677\pi$
$443$	17.6022	0.836307	0.418154	+	0.908376i	$0.362677\pi$
$444$	−79.5920	−3.77727
$445$	0	0
$446$	−20.7179	−0.981021
$447$	−20.4236	−0.966003
$448$	−29.4325	−1.39055
$449$	−13.8318	−0.652761	−0.326380	−	0.945239i	$-0.605829\pi$
$449$	−13.8318	−0.652761	−0.326380	+	0.945239i	$0.605829\pi$
$450$	0	0
$451$	−19.1193	−0.900295
$452$	−57.3366	−2.69689
$453$	39.7319	1.86677
$454$	47.2923	2.21954
$455$	0	0
$456$	−22.9594	−1.07517
$457$	−35.9522	−1.68177	−0.840886	−	0.541213i	$-0.817965\pi$
$457$	−35.9522	−1.68177	−0.840886	+	0.541213i	$0.817965\pi$
$458$	30.8486	1.44146
$459$	16.1258	0.752687
$460$	0	0
$461$	−38.7381	−1.80421	−0.902106	−	0.431514i	$-0.857980\pi$
$461$	−38.7381	−1.80421	−0.902106	+	0.431514i	$0.857980\pi$
$462$	19.2418	0.895210
$463$	−3.68719	−0.171358	−0.0856792	−	0.996323i	$-0.527306\pi$
$463$	−3.68719	−0.171358	−0.0856792	+	0.996323i	$0.527306\pi$
$464$	32.2355	1.49650
$465$	0	0
$466$	35.4040	1.64006
$467$	−19.4573	−0.900375	−0.450188	−	0.892934i	$-0.648643\pi$
$467$	−19.4573	−0.900375	−0.450188	+	0.892934i	$0.648643\pi$
$468$	0.0785173	0.00362947
$469$	17.8619	0.824787
$470$	0	0
$471$	16.9609	0.781516
$472$	12.1840	0.560814
$473$	8.03137	0.369283
$474$	−43.3914	−1.99303
$475$	0	0
$476$	28.7414	1.31736
$477$	−6.92643	−0.317140
$478$	−6.81310	−0.311624
$479$	−34.2074	−1.56298	−0.781489	−	0.623919i	$-0.785539\pi$
$479$	−34.2074	−1.56298	−0.781489	+	0.623919i	$0.785539\pi$
$480$	0	0
$481$	−0.225116	−0.0102644
$482$	−2.67715	−0.121941
$483$	−22.3557	−1.01722
$484$	−27.1682	−1.23492
$485$	0	0
$486$	15.1657	0.687931
$487$	18.9925	0.860634	0.430317	−	0.902678i	$-0.358402\pi$
$487$	18.9925	0.860634	0.430317	+	0.902678i	$0.358402\pi$
$488$	−27.6624	−1.25222
$489$	−16.8737	−0.763055
$490$	0	0
$491$	−8.59531	−0.387901	−0.193951	−	0.981011i	$-0.562130\pi$
$491$	−8.59531	−0.387901	−0.193951	+	0.981011i	$0.562130\pi$
$492$	−77.6972	−3.50286
$493$	−9.11130	−0.410352
$494$	−0.105945	−0.00476669
$495$	0	0
$496$	30.0855	1.35088
$497$	10.1328	0.454520
$498$	−2.79808	−0.125385
$499$	2.75601	0.123376	0.0616880	−	0.998095i	$-0.480352\pi$
$499$	2.75601	0.123376	0.0616880	+	0.998095i	$0.480352\pi$
$500$	0	0
$501$	7.70297	0.344144
$502$	37.7339	1.68415
$503$	18.5524	0.827211	0.413606	−	0.910456i	$-0.364269\pi$
$503$	18.5524	0.827211	0.413606	+	0.910456i	$0.364269\pi$
$504$	7.44507	0.331630
$505$	0	0
$506$	−47.8144	−2.12561
$507$	−24.4983	−1.08801
$508$	17.0115	0.754763
$509$	−29.4409	−1.30494	−0.652471	−	0.757813i	$-0.726268\pi$
$509$	−29.4409	−1.30494	−0.652471	+	0.757813i	$0.726268\pi$
$510$	0	0
$511$	−5.48737	−0.242747
$512$	10.0480	0.444062
$513$	−6.62955	−0.292702
$514$	57.7759	2.54839
$515$	0	0
$516$	32.6379	1.43680
$517$	−11.8211	−0.519892
$518$	−34.8251	−1.53013
$519$	−24.6554	−1.08225
$520$	0	0
$521$	2.36830	0.103757	0.0518786	−	0.998653i	$-0.483479\pi$
$521$	2.36830	0.103757	0.0518786	+	0.998653i	$0.483479\pi$
$522$	−3.85057	−0.168535
$523$	−18.3423	−0.802055	−0.401027	−	0.916066i	$-0.631347\pi$
$523$	−18.3423	−0.802055	−0.401027	+	0.916066i	$0.631347\pi$
$524$	18.0887	0.790210
$525$	0	0
$526$	6.36686	0.277608
$527$	−8.50360	−0.370423
$528$	55.8404	2.43014
$529$	32.5522	1.41531
$530$	0	0
$531$	−0.792797	−0.0344045
$532$	−11.8160	−0.512289
$533$	−0.219757	−0.00951871
$534$	38.6004	1.67040
$535$	0	0
$536$	95.1587	4.11023
$537$	39.6731	1.71202
$538$	29.5588	1.27437
$539$	−10.7041	−0.461058
$540$	0	0
$541$	−33.3120	−1.43219	−0.716097	−	0.698001i	$-0.754073\pi$
$541$	−33.3120	−1.43219	−0.716097	+	0.698001i	$0.754073\pi$
$542$	44.1720	1.89735
$543$	−24.1105	−1.03468
$544$	56.4263	2.41926
$545$	0	0
$546$	0.221164	0.00946496
$547$	15.4782	0.661802	0.330901	−	0.943666i	$-0.392647\pi$
$547$	15.4782	0.661802	0.330901	+	0.943666i	$0.392647\pi$
$548$	−15.2119	−0.649821
$549$	1.79996	0.0768202
$550$	0	0
$551$	3.74579	0.159576
$552$	−119.099	−5.06919
$553$	−13.6877	−0.582061
$554$	−7.46302	−0.317073
$555$	0	0
$556$	57.7668	2.44986
$557$	14.5442	0.616258	0.308129	−	0.951345i	$-0.400297\pi$
$557$	14.5442	0.616258	0.308129	+	0.951345i	$0.400297\pi$
$558$	−3.59375	−0.152135
$559$	0.0923121	0.00390439
$560$	0	0
$561$	−15.7832	−0.666365
$562$	−24.8900	−1.04992
$563$	−39.9984	−1.68573	−0.842867	−	0.538122i	$-0.819134\pi$
$563$	−39.9984	−1.68573	−0.842867	+	0.538122i	$0.819134\pi$
$564$	−48.0386	−2.02279
$565$	0	0
$566$	−61.7332	−2.59484
$567$	16.4737	0.691829
$568$	53.9822	2.26504
$569$	6.86796	0.287920	0.143960	−	0.989584i	$-0.454016\pi$
$569$	6.86796	0.287920	0.143960	+	0.989584i	$0.454016\pi$
$570$	0	0
$571$	39.3835	1.64815	0.824074	−	0.566483i	$-0.191696\pi$
$571$	39.3835	1.64815	0.824074	+	0.566483i	$0.191696\pi$
$572$	0.341028	0.0142591
$573$	−22.7175	−0.949035
$574$	−33.9961	−1.41897
$575$	0	0
$576$	10.2028	0.425117
$577$	−2.65581	−0.110563	−0.0552814	−	0.998471i	$-0.517606\pi$
$577$	−2.65581	−0.110563	−0.0552814	+	0.998471i	$0.517606\pi$
$578$	12.8113	0.532880
$579$	−22.3615	−0.929313
$580$	0	0
$581$	−0.882647	−0.0366184
$582$	−71.9221	−2.98127
$583$	−30.0839	−1.24595
$584$	−29.2337	−1.20970
$585$	0	0
$586$	−23.5169	−0.971475
$587$	36.8821	1.52229	0.761143	−	0.648584i	$-0.224638\pi$
$587$	36.8821	1.52229	0.761143	+	0.648584i	$0.224638\pi$
$588$	−43.4994	−1.79388
$589$	3.49596	0.144048
$590$	0	0
$591$	−32.7112	−1.34556
$592$	−101.064	−4.15369
$593$	−37.8794	−1.55552	−0.777760	−	0.628562i	$-0.783644\pi$
$593$	−37.8794	−1.55552	−0.777760	+	0.628562i	$0.783644\pi$
$594$	29.5999	1.21450
$595$	0	0
$596$	−55.9968	−2.29372
$597$	8.75139	0.358171
$598$	−0.549576	−0.0224738
$599$	−31.8571	−1.30165	−0.650823	−	0.759230i	$-0.725576\pi$
$599$	−31.8571	−1.30165	−0.650823	+	0.759230i	$0.725576\pi$
$600$	0	0
$601$	−21.5297	−0.878214	−0.439107	−	0.898435i	$-0.644705\pi$
$601$	−21.5297	−0.878214	−0.439107	+	0.898435i	$0.644705\pi$
$602$	14.2806	0.582033
$603$	−6.19185	−0.252152
$604$	108.936	4.43253
$605$	0	0
$606$	−63.0426	−2.56093
$607$	−31.3114	−1.27089	−0.635446	−	0.772146i	$-0.719184\pi$
$607$	−31.3114	−1.27089	−0.635446	+	0.772146i	$0.719184\pi$
$608$	−23.1977	−0.940791
$609$	−7.81949	−0.316862
$610$	0	0
$611$	−0.135871	−0.00549676
$612$	−9.96323	−0.402740
$613$	−2.52691	−0.102061	−0.0510304	−	0.998697i	$-0.516251\pi$
$613$	−2.52691	−0.102061	−0.0510304	+	0.998697i	$0.516251\pi$
$614$	−58.4701	−2.35966
$615$	0	0
$616$	32.3365	1.30288
$617$	−9.74436	−0.392293	−0.196147	−	0.980575i	$-0.562843\pi$
$617$	−9.74436	−0.392293	−0.196147	+	0.980575i	$0.562843\pi$
$618$	−24.1852	−0.972873
$619$	−40.5750	−1.63085	−0.815423	−	0.578866i	$-0.803495\pi$
$619$	−40.5750	−1.63085	−0.815423	+	0.578866i	$0.803495\pi$
$620$	0	0
$621$	−34.3900	−1.38002
$622$	68.0311	2.72780
$623$	12.1764	0.487838
$624$	0.641826	0.0256936
$625$	0	0
$626$	43.1461	1.72446
$627$	6.48869	0.259133
$628$	46.5028	1.85566
$629$	28.5654	1.13898
$630$	0	0
$631$	−48.0502	−1.91285	−0.956423	−	0.291984i	$-0.905685\pi$
$631$	−48.0502	−1.91285	−0.956423	+	0.291984i	$0.905685\pi$
$632$	−72.9207	−2.90063
$633$	−13.2975	−0.528528
$634$	15.5277	0.616686
$635$	0	0
$636$	−122.255	−4.84773
$637$	−0.123032	−0.00487472
$638$	−16.7244	−0.662124
$639$	−3.51256	−0.138955
$640$	0	0
$641$	23.4644	0.926787	0.463393	−	0.886153i	$-0.346632\pi$
$641$	23.4644	0.926787	0.463393	+	0.886153i	$0.346632\pi$
$642$	5.17375	0.204192
$643$	−18.1291	−0.714941	−0.357470	−	0.933925i	$-0.616361\pi$
$643$	−18.1291	−0.714941	−0.357470	+	0.933925i	$0.616361\pi$
$644$	−61.2941	−2.41533
$645$	0	0
$646$	13.4436	0.528930
$647$	−2.00410	−0.0787892	−0.0393946	−	0.999224i	$-0.512543\pi$
$647$	−2.00410	−0.0787892	−0.0393946	+	0.999224i	$0.512543\pi$
$648$	87.7628	3.44765
$649$	−3.44339	−0.135165
$650$	0	0
$651$	−7.29796	−0.286029
$652$	−46.2638	−1.81183
$653$	10.1560	0.397434	0.198717	−	0.980057i	$-0.436323\pi$
$653$	10.1560	0.397434	0.198717	+	0.980057i	$0.436323\pi$
$654$	−9.26786	−0.362402
$655$	0	0
$656$	−98.6577	−3.85194
$657$	1.90220	0.0742120
$658$	−21.0191	−0.819409
$659$	39.6153	1.54319	0.771596	−	0.636113i	$-0.219459\pi$
$659$	39.6153	1.54319	0.771596	+	0.636113i	$0.219459\pi$
$660$	0	0
$661$	−7.96149	−0.309666	−0.154833	−	0.987941i	$-0.549484\pi$
$661$	−7.96149	−0.309666	−0.154833	+	0.987941i	$0.549484\pi$
$662$	−30.7358	−1.19458
$663$	−0.181411	−0.00704541
$664$	−4.70227	−0.182483
$665$	0	0
$666$	12.0722	0.467787
$667$	19.4308	0.752365
$668$	21.1198	0.817149
$669$	14.5845	0.563870
$670$	0	0
$671$	7.81783	0.301804
$672$	48.4262	1.86808
$673$	37.9765	1.46389	0.731944	−	0.681365i	$-0.238613\pi$
$673$	37.9765	1.46389	0.731944	+	0.681365i	$0.238613\pi$
$674$	−42.3333	−1.63062
$675$	0	0
$676$	−67.1688	−2.58342
$677$	34.5977	1.32970	0.664849	−	0.746978i	$-0.268496\pi$
$677$	34.5977	1.32970	0.664849	+	0.746978i	$0.268496\pi$
$678$	55.9853	2.15010
$679$	−22.6877	−0.870672
$680$	0	0
$681$	−33.2918	−1.27574
$682$	−15.6089	−0.597696
$683$	32.5089	1.24392	0.621959	−	0.783049i	$-0.286337\pi$
$683$	32.5089	1.24392	0.621959	+	0.783049i	$0.286337\pi$
$684$	4.09603	0.156616
$685$	0	0
$686$	−48.8586	−1.86543
$687$	−21.7161	−0.828519
$688$	41.4427	1.57999
$689$	−0.345783	−0.0131733
$690$	0	0
$691$	−17.6525	−0.671532	−0.335766	−	0.941945i	$-0.608995\pi$
$691$	−17.6525	−0.671532	−0.335766	+	0.941945i	$0.608995\pi$
$692$	−67.5994	−2.56975
$693$	−2.10410	−0.0799281
$694$	−60.1749	−2.28421
$695$	0	0
$696$	−41.6580	−1.57904
$697$	27.8854	1.05623
$698$	20.0420	0.758602
$699$	−24.9229	−0.942671
$700$	0	0
$701$	−3.77846	−0.142710	−0.0713552	−	0.997451i	$-0.522732\pi$
$701$	−3.77846	−0.142710	−0.0713552	+	0.997451i	$0.522732\pi$
$702$	0.340219	0.0128408
$703$	−11.7437	−0.442921
$704$	44.3143	1.67016
$705$	0	0
$706$	10.3007	0.387673
$707$	−19.8866	−0.747914
$708$	−13.9933	−0.525899
$709$	−26.0264	−0.977440	−0.488720	−	0.872441i	$-0.662536\pi$
$709$	−26.0264	−0.977440	−0.488720	+	0.872441i	$0.662536\pi$
$710$	0	0
$711$	4.74486	0.177946
$712$	64.8693	2.43108
$713$	18.1349	0.679156
$714$	−28.0640	−1.05027
$715$	0	0
$716$	108.774	4.06509
$717$	4.79613	0.179115
$718$	66.0045	2.46327
$719$	−16.5753	−0.618155	−0.309078	−	0.951037i	$-0.600020\pi$
$719$	−16.5753	−0.618155	−0.309078	+	0.951037i	$0.600020\pi$
$720$	0	0
$721$	−7.62918	−0.284126
$722$	45.3390	1.68734
$723$	1.88460	0.0700890
$724$	−66.1053	−2.45678
$725$	0	0
$726$	26.5279	0.984543
$727$	−10.2796	−0.381250	−0.190625	−	0.981663i	$-0.561051\pi$
$727$	−10.2796	−0.381250	−0.190625	+	0.981663i	$0.561051\pi$
$728$	0.371674	0.0137752
$729$	20.3762	0.754675
$730$	0	0
$731$	−11.7137	−0.433246
$732$	31.7701	1.17426
$733$	5.66934	0.209402	0.104701	−	0.994504i	$-0.466611\pi$
$733$	5.66934	0.209402	0.104701	+	0.994504i	$0.466611\pi$
$734$	−4.06390	−0.150001
$735$	0	0
$736$	−120.335	−4.43562
$737$	−26.8934	−0.990630
$738$	11.7848	0.433803
$739$	−2.14371	−0.0788577	−0.0394289	−	0.999222i	$-0.512554\pi$
$739$	−2.14371	−0.0788577	−0.0394289	+	0.999222i	$0.512554\pi$
$740$	0	0
$741$	0.0745807	0.00273979
$742$	−53.4921	−1.96376
$743$	13.6302	0.500045	0.250022	−	0.968240i	$-0.419562\pi$
$743$	13.6302	0.500045	0.250022	+	0.968240i	$0.419562\pi$
$744$	−38.8796	−1.42539
$745$	0	0
$746$	52.1801	1.91045
$747$	0.305971	0.0111949
$748$	−43.2738	−1.58225
$749$	1.63205	0.0596337
$750$	0	0
$751$	43.7821	1.59763	0.798816	−	0.601575i	$-0.205460\pi$
$751$	43.7821	1.59763	0.798816	+	0.601575i	$0.205460\pi$
$752$	−60.9981	−2.22437
$753$	−26.5630	−0.968011
$754$	−0.192229	−0.00700056
$755$	0	0
$756$	37.9446	1.38003
$757$	−11.6837	−0.424652	−0.212326	−	0.977199i	$-0.568104\pi$
$757$	−11.6837	−0.424652	−0.212326	+	0.977199i	$0.568104\pi$
$758$	−43.7421	−1.58878
$759$	33.6593	1.22176
$760$	0	0
$761$	2.17359	0.0787926	0.0393963	−	0.999224i	$-0.487457\pi$
$761$	2.17359	0.0787926	0.0393963	+	0.999224i	$0.487457\pi$
$762$	−16.6106	−0.601737
$763$	−2.92352	−0.105839
$764$	−62.2860	−2.25343
$765$	0	0
$766$	17.3846	0.628130
$767$	−0.0395782	−0.00142909
$768$	17.1685	0.619517
$769$	27.1479	0.978978	0.489489	−	0.872009i	$-0.337183\pi$
$769$	27.1479	0.978978	0.489489	+	0.872009i	$0.337183\pi$
$770$	0	0
$771$	−40.6718	−1.46476
$772$	−61.3101	−2.20660
$773$	21.8735	0.786734	0.393367	−	0.919382i	$-0.371310\pi$
$773$	21.8735	0.786734	0.393367	+	0.919382i	$0.371310\pi$
$774$	−4.95037	−0.177938
$775$	0	0
$776$	−120.868	−4.33889
$777$	24.5154	0.879484
$778$	7.99349	0.286581
$779$	−11.4641	−0.410744
$780$	0	0
$781$	−15.2562	−0.545912
$782$	69.7369	2.49379
$783$	−12.0288	−0.429874
$784$	−55.2342	−1.97265
$785$	0	0
$786$	−17.6624	−0.629997
$787$	23.2483	0.828714	0.414357	−	0.910114i	$-0.364007\pi$
$787$	23.2483	0.828714	0.414357	+	0.910114i	$0.364007\pi$
$788$	−89.6865	−3.19495
$789$	−4.48199	−0.159563
$790$	0	0
$791$	17.6604	0.627933
$792$	−11.2095	−0.398312
$793$	0.0898577	0.00319094
$794$	14.6433	0.519670
$795$	0	0
$796$	23.9943	0.850455
$797$	43.7810	1.55080	0.775402	−	0.631468i	$-0.217547\pi$
$797$	43.7810	1.55080	0.775402	+	0.631468i	$0.217547\pi$
$798$	11.5375	0.408424
$799$	17.2410	0.609941
$800$	0	0
$801$	−4.22097	−0.149140
$802$	−72.3490	−2.55473
$803$	8.26193	0.291557
$804$	−109.289	−3.85434
$805$	0	0
$806$	−0.179408	−0.00631937
$807$	−20.8081	−0.732479
$808$	−105.945	−3.72714
$809$	−36.5184	−1.28392	−0.641959	−	0.766739i	$-0.721878\pi$
$809$	−36.5184	−1.28392	−0.641959	+	0.766739i	$0.721878\pi$
$810$	0	0
$811$	28.9480	1.01650	0.508250	−	0.861209i	$-0.330292\pi$
$811$	28.9480	1.01650	0.508250	+	0.861209i	$0.330292\pi$
$812$	−21.4392	−0.752370
$813$	−31.0952	−1.09055
$814$	52.4336	1.83780
$815$	0	0
$816$	−81.4426	−2.85106
$817$	4.81567	0.168479
$818$	56.8101	1.98632
$819$	−0.0241844	−0.000845070 0
$820$	0	0
$821$	−37.3534	−1.30364	−0.651821	−	0.758373i	$-0.725994\pi$
$821$	−37.3534	−1.30364	−0.651821	+	0.758373i	$0.725994\pi$
$822$	14.8534	0.518072
$823$	35.2373	1.22829	0.614147	−	0.789192i	$-0.289500\pi$
$823$	35.2373	1.22829	0.614147	+	0.789192i	$0.289500\pi$
$824$	−40.6441	−1.41591
$825$	0	0
$826$	−6.12269	−0.213036
$827$	−51.8325	−1.80239	−0.901196	−	0.433412i	$-0.857310\pi$
$827$	−51.8325	−1.80239	−0.901196	+	0.433412i	$0.857310\pi$
$828$	21.2477	0.738408
$829$	−55.9515	−1.94328	−0.971638	−	0.236473i	$-0.924009\pi$
$829$	−55.9515	−1.94328	−0.971638	+	0.236473i	$0.924009\pi$
$830$	0	0
$831$	5.25365	0.182247
$832$	0.509346	0.0176584
$833$	15.6118	0.540918
$834$	−56.4054	−1.95316
$835$	0	0
$836$	17.7905	0.615297
$837$	−11.2265	−0.388045
$838$	84.0885	2.90479
$839$	32.2425	1.11314	0.556568	−	0.830802i	$-0.312118\pi$
$839$	32.2425	1.11314	0.556568	+	0.830802i	$0.312118\pi$
$840$	0	0
$841$	−22.2036	−0.765640
$842$	69.0579	2.37989
$843$	17.5215	0.603473
$844$	−36.4587	−1.25496
$845$	0	0
$846$	7.28629	0.250508
$847$	8.36816	0.287533
$848$	−155.236	−5.33082
$849$	43.4575	1.49146
$850$	0	0
$851$	−60.9188	−2.08827
$852$	−61.9984	−2.12403
$853$	19.8494	0.679632	0.339816	−	0.940492i	$-0.389635\pi$
$853$	19.8494	0.679632	0.339816	+	0.940492i	$0.389635\pi$
$854$	13.9009	0.475678
$855$	0	0
$856$	8.69467	0.297178
$857$	23.2116	0.792892	0.396446	−	0.918058i	$-0.370243\pi$
$857$	23.2116	0.792892	0.396446	+	0.918058i	$0.370243\pi$
$858$	−0.332991	−0.0113681
$859$	23.2438	0.793068	0.396534	−	0.918020i	$-0.370213\pi$
$859$	23.2438	0.793068	0.396534	+	0.918020i	$0.370213\pi$
$860$	0	0
$861$	23.9318	0.815592
$862$	−11.5033	−0.391802
$863$	10.4157	0.354554	0.177277	−	0.984161i	$-0.443271\pi$
$863$	10.4157	0.354554	0.177277	+	0.984161i	$0.443271\pi$
$864$	74.4945	2.53435
$865$	0	0
$866$	−73.1248	−2.48488
$867$	−9.01860	−0.306288
$868$	−20.0093	−0.679160
$869$	20.6086	0.699098
$870$	0	0
$871$	−0.309111	−0.0104738
$872$	−15.5750	−0.527434
$873$	7.86470	0.266180
$874$	−28.6699	−0.969773
$875$	0	0
$876$	33.5748	1.13439
$877$	−40.4928	−1.36734	−0.683672	−	0.729790i	$-0.739618\pi$
$877$	−40.4928	−1.36734	−0.683672	+	0.729790i	$0.739618\pi$
$878$	−91.8688	−3.10042
$879$	16.5549	0.558383
$880$	0	0
$881$	−36.9663	−1.24542	−0.622712	−	0.782451i	$-0.713969\pi$
$881$	−36.9663	−1.24542	−0.622712	+	0.782451i	$0.713969\pi$
$882$	6.59779	0.222159
$883$	39.2198	1.31985	0.659925	−	0.751332i	$-0.270588\pi$
$883$	39.2198	1.31985	0.659925	+	0.751332i	$0.270588\pi$
$884$	−0.497386	−0.0167289
$885$	0	0
$886$	−47.1238	−1.58316
$887$	−40.5461	−1.36140	−0.680702	−	0.732561i	$-0.738325\pi$
$887$	−40.5461	−1.36140	−0.680702	+	0.732561i	$0.738325\pi$
$888$	130.605	4.38280
$889$	−5.23976	−0.175736
$890$	0	0
$891$	−24.8032	−0.830938
$892$	39.9873	1.33888
$893$	−7.08802	−0.237192
$894$	54.6770	1.82867
$895$	0	0
$896$	27.4037	0.915494
$897$	0.386878	0.0129175
$898$	37.0297	1.23570
$899$	6.34314	0.211556
$900$	0	0
$901$	43.8771	1.46176
$902$	51.1853	1.70429
$903$	−10.0529	−0.334540
$904$	94.0852	3.12923
$905$	0	0
$906$	−106.368	−3.53385
$907$	−2.04922	−0.0680432	−0.0340216	−	0.999421i	$-0.510831\pi$
$907$	−2.04922	−0.0680432	−0.0340216	+	0.999421i	$0.510831\pi$
$908$	−91.2783	−3.02918
$909$	6.89372	0.228650
$910$	0	0
$911$	46.4301	1.53830	0.769150	−	0.639069i	$-0.220680\pi$
$911$	46.4301	1.53830	0.769150	+	0.639069i	$0.220680\pi$
$912$	33.4823	1.10871
$913$	1.32894	0.0439814
$914$	96.2493	3.18364
$915$	0	0
$916$	−59.5404	−1.96727
$917$	−5.57157	−0.183989
$918$	−43.1711	−1.42486
$919$	11.0824	0.365575	0.182787	−	0.983152i	$-0.441488\pi$
$919$	11.0824	0.365575	0.182787	+	0.983152i	$0.441488\pi$
$920$	0	0
$921$	41.1604	1.35628
$922$	103.708	3.41543
$923$	−0.175354	−0.00577186
$924$	−37.1384	−1.22176
$925$	0	0
$926$	9.87116	0.324387
$927$	2.64466	0.0868621
$928$	−42.0904	−1.38169
$929$	−8.65542	−0.283975	−0.141988	−	0.989868i	$-0.545349\pi$
$929$	−8.65542	−0.283975	−0.141988	+	0.989868i	$0.545349\pi$
$930$	0	0
$931$	−6.41826	−0.210350
$932$	−68.3329	−2.23832
$933$	−47.8910	−1.56788
$934$	52.0900	1.70444
$935$	0	0
$936$	−0.128841	−0.00421131
$937$	−31.9032	−1.04223	−0.521116	−	0.853486i	$-0.674484\pi$
$937$	−31.9032	−1.04223	−0.521116	+	0.853486i	$0.674484\pi$
$938$	−47.8191	−1.56135
$939$	−30.3730	−0.991185
$940$	0	0
$941$	−16.9163	−0.551456	−0.275728	−	0.961236i	$-0.588919\pi$
$941$	−16.9163	−0.551456	−0.275728	+	0.961236i	$0.588919\pi$
$942$	−45.4068	−1.47943
$943$	−59.4686	−1.93656
$944$	−17.7683	−0.578308
$945$	0	0
$946$	−21.5012	−0.699064
$947$	53.3429	1.73341	0.866706	−	0.498820i	$-0.166233\pi$
$947$	53.3429	1.73341	0.866706	+	0.498820i	$0.166233\pi$
$948$	83.7492	2.72005
$949$	0.0949621	0.00308260
$950$	0	0
$951$	−10.9309	−0.354458
$952$	−47.1625	−1.52855
$953$	−25.4882	−0.825643	−0.412822	−	0.910812i	$-0.635457\pi$
$953$	−25.4882	−0.825643	−0.412822	+	0.910812i	$0.635457\pi$
$954$	18.5431	0.600355
$955$	0	0
$956$	13.1499	0.425297
$957$	11.7732	0.380574
$958$	91.5785	2.95877
$959$	4.68547	0.151302
$960$	0	0
$961$	−25.0799	−0.809030
$962$	0.602668	0.0194308
$963$	−0.565751	−0.0182311
$964$	5.16713	0.166422
$965$	0	0
$966$	59.8495	1.92563
$967$	−42.0466	−1.35213	−0.676064	−	0.736843i	$-0.736316\pi$
$967$	−42.0466	−1.35213	−0.676064	+	0.736843i	$0.736316\pi$
$968$	44.5810	1.43289
$969$	−9.46369	−0.304018
$970$	0	0
$971$	33.5536	1.07679	0.538393	−	0.842694i	$-0.319032\pi$
$971$	33.5536	1.07679	0.538393	+	0.842694i	$0.319032\pi$
$972$	−29.2711	−0.938873
$973$	−17.7929	−0.570416
$974$	−50.8459	−1.62921
$975$	0	0
$976$	40.3408	1.29128
$977$	47.4264	1.51731	0.758653	−	0.651495i	$-0.225858\pi$
$977$	47.4264	1.51731	0.758653	+	0.651495i	$0.225858\pi$
$978$	45.1734	1.44449
$979$	−18.3331	−0.585929
$980$	0	0
$981$	1.01344	0.0323567
$982$	23.0109	0.734309
$983$	−47.4785	−1.51433	−0.757165	−	0.653224i	$-0.773416\pi$
$983$	−47.4785	−1.51433	−0.757165	+	0.653224i	$0.773416\pi$
$984$	127.495	4.06441
$985$	0	0
$986$	24.3923	0.776809
$987$	14.7965	0.470979
$988$	0.204483	0.00650547
$989$	24.9807	0.794340
$990$	0	0
$991$	−1.81124	−0.0575361	−0.0287680	−	0.999586i	$-0.509158\pi$
$991$	−1.81124	−0.0575361	−0.0287680	+	0.999586i	$0.509158\pi$
$992$	−39.2831	−1.24724
$993$	21.6367	0.686619
$994$	−27.1271	−0.860420
$995$	0	0
$996$	5.40054	0.171123
$997$	9.89751	0.313457	0.156729	−	0.987642i	$-0.449905\pi$
$997$	9.89751	0.313457	0.156729	+	0.987642i	$0.449905\pi$
$998$	−7.37825	−0.233555
$999$	37.7123	1.19316

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.m.1.3	✓	40
5.4	even	2		6025.2.a.n.1.38	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.m.1.3	✓	40	1.1	even	1	trivial
6025.2.a.n.1.38	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)

\(f(q)\)	\(=\)	\(q-2.67715 q^{2} +1.88460 q^{3} +5.16713 q^{4} -5.04535 q^{6} -1.59154 q^{7} -8.47889 q^{8} +0.551710 q^{9} +O(q^{10})\)	\(q-2.67715 q^{2} +1.88460 q^{3} +5.16713 q^{4} -5.04535 q^{6} -1.59154 q^{7} -8.47889 q^{8} +0.551710 q^{9} +2.39627 q^{11} +9.73797 q^{12} +0.0275426 q^{13} +4.26080 q^{14} +12.3650 q^{16} -3.49494 q^{17} -1.47701 q^{18} +1.43682 q^{19} -2.99942 q^{21} -6.41518 q^{22} +7.45333 q^{23} -15.9793 q^{24} -0.0737356 q^{26} -4.61404 q^{27} -8.22372 q^{28} +2.60700 q^{29} +2.43312 q^{31} -16.1452 q^{32} +4.51601 q^{33} +9.35647 q^{34} +2.85076 q^{36} -8.17337 q^{37} -3.84659 q^{38} +0.0519067 q^{39} -7.97879 q^{41} +8.02990 q^{42} +3.35161 q^{43} +12.3818 q^{44} -19.9537 q^{46} -4.93313 q^{47} +23.3030 q^{48} -4.46699 q^{49} -6.58655 q^{51} +0.142316 q^{52} -12.5545 q^{53} +12.3525 q^{54} +13.4945 q^{56} +2.70783 q^{57} -6.97933 q^{58} -1.43698 q^{59} +3.26250 q^{61} -6.51383 q^{62} -0.878072 q^{63} +18.4930 q^{64} -12.0900 q^{66} -11.2230 q^{67} -18.0588 q^{68} +14.0465 q^{69} -6.36666 q^{71} -4.67789 q^{72} +3.44783 q^{73} +21.8813 q^{74} +7.42424 q^{76} -3.81377 q^{77} -0.138962 q^{78} +8.60027 q^{79} -10.3507 q^{81} +21.3604 q^{82} +0.554585 q^{83} -15.4984 q^{84} -8.97277 q^{86} +4.91315 q^{87} -20.3177 q^{88} -7.65069 q^{89} -0.0438353 q^{91} +38.5124 q^{92} +4.58546 q^{93} +13.2067 q^{94} -30.4271 q^{96} +14.2551 q^{97} +11.9588 q^{98} +1.32205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 9 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 20 q^{7} - 27 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q - 9 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 20 * q^7 - 27 * q^8 + 38 * q^9	\( 40 q - 9 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 20 q^{7} - 27 q^{8} + 38 q^{9} + q^{11} - 26 q^{12} - 11 q^{13} - q^{14} + 43 q^{16} - 20 q^{17} - 18 q^{18} + 2 q^{21} - 23 q^{22} - 79 q^{23} - 2 q^{24} + 2 q^{26} - 26 q^{27} - 30 q^{28} + 2 q^{29} + q^{31} - 68 q^{32} - 20 q^{33} + 5 q^{34} + 32 q^{36} - 16 q^{37} - 45 q^{38} - 2 q^{39} - 2 q^{41} - 19 q^{42} - 25 q^{43} + 3 q^{44} + 14 q^{46} - 88 q^{47} - 75 q^{48} + 40 q^{49} - 10 q^{51} - 18 q^{52} - 34 q^{53} + 4 q^{54} - 15 q^{56} - 51 q^{57} - 53 q^{58} + q^{59} + 9 q^{61} - 39 q^{62} - 110 q^{63} + 17 q^{64} + 26 q^{66} - 30 q^{67} - 44 q^{68} - 7 q^{69} + 5 q^{71} - 18 q^{72} - 23 q^{73} - 18 q^{74} + 43 q^{76} - 30 q^{77} - 46 q^{78} + 5 q^{79} + 44 q^{81} - 5 q^{82} - 65 q^{83} - 65 q^{84} + 40 q^{86} - 33 q^{87} - 71 q^{88} - 9 q^{89} + q^{91} - 117 q^{92} - 68 q^{93} - 72 q^{94} + 83 q^{96} + 8 q^{97} - 76 q^{98} - 40 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 20 * q^7 - 27 * q^8 + 38 * q^9 + q^11 - 26 * q^12 - 11 * q^13 - q^14 + 43 * q^16 - 20 * q^17 - 18 * q^18 + 2 * q^21 - 23 * q^22 - 79 * q^23 - 2 * q^24 + 2 * q^26 - 26 * q^27 - 30 * q^28 + 2 * q^29 + q^31 - 68 * q^32 - 20 * q^33 + 5 * q^34 + 32 * q^36 - 16 * q^37 - 45 * q^38 - 2 * q^39 - 2 * q^41 - 19 * q^42 - 25 * q^43 + 3 * q^44 + 14 * q^46 - 88 * q^47 - 75 * q^48 + 40 * q^49 - 10 * q^51 - 18 * q^52 - 34 * q^53 + 4 * q^54 - 15 * q^56 - 51 * q^57 - 53 * q^58 + q^59 + 9 * q^61 - 39 * q^62 - 110 * q^63 + 17 * q^64 + 26 * q^66 - 30 * q^67 - 44 * q^68 - 7 * q^69 + 5 * q^71 - 18 * q^72 - 23 * q^73 - 18 * q^74 + 43 * q^76 - 30 * q^77 - 46 * q^78 + 5 * q^79 + 44 * q^81 - 5 * q^82 - 65 * q^83 - 65 * q^84 + 40 * q^86 - 33 * q^87 - 71 * q^88 - 9 * q^89 + q^91 - 117 * q^92 - 68 * q^93 - 72 * q^94 + 83 * q^96 + 8 * q^97 - 76 * q^98 - 40 * q^99

Introduction

L-functions

Modular forms

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