LMFDB - Embedded newform 1925.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1925,2,Mod(1,1925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1925, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1925.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:	$15.3712023891$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.9921856.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 7x^{4} + 11x^{2} - 2x - 1 $ x^6 - 7x^4 + 11x^2 - 2*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 385)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.24274$ of defining polynomial
Character	$\chi$	$=$	1925.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.24274 q^{2} -3.30979 q^{3} +3.02989 q^{4} +7.42300 q^{6} +1.00000 q^{7} -2.30979 q^{8} +7.95470 q^{9} +O(q^{10})$	\(q-2.24274 q^{2} -3.30979 q^{3} +3.02989 q^{4} +7.42300 q^{6} +1.00000 q^{7} -2.30979 q^{8} +7.95470 q^{9} -1.00000 q^{11} -10.0283 q^{12} -2.34784 q^{13} -2.24274 q^{14} -0.879529 q^{16} +4.20155 q^{17} -17.8403 q^{18} -0.538260 q^{19} -3.30979 q^{21} +2.24274 q^{22} -0.251800 q^{23} +7.64491 q^{24} +5.26559 q^{26} -16.3990 q^{27} +3.02989 q^{28} -7.62047 q^{29} +0.441844 q^{31} +6.59213 q^{32} +3.30979 q^{33} -9.42300 q^{34} +24.1019 q^{36} -9.52973 q^{37} +1.20718 q^{38} +7.77084 q^{39} -0.122999 q^{41} +7.42300 q^{42} -4.54140 q^{43} -3.02989 q^{44} +0.564722 q^{46} +11.9033 q^{47} +2.91105 q^{48} +1.00000 q^{49} -13.9062 q^{51} -7.11369 q^{52} +10.3421 q^{53} +36.7787 q^{54} -2.30979 q^{56} +1.78153 q^{57} +17.0908 q^{58} -4.12232 q^{59} +15.5178 q^{61} -0.990943 q^{62} +7.95470 q^{63} -13.0254 q^{64} -7.42300 q^{66} -3.85982 q^{67} +12.7303 q^{68} +0.833404 q^{69} -0.782634 q^{71} -18.3737 q^{72} +15.1342 q^{73} +21.3727 q^{74} -1.63087 q^{76} -1.00000 q^{77} -17.4280 q^{78} -7.84140 q^{79} +30.4131 q^{81} +0.275855 q^{82} -2.41779 q^{83} -10.0283 q^{84} +10.1852 q^{86} +25.2221 q^{87} +2.30979 q^{88} +10.9371 q^{89} -2.34784 q^{91} -0.762927 q^{92} -1.46241 q^{93} -26.6961 q^{94} -21.8186 q^{96} -5.40327 q^{97} -2.24274 q^{98} -7.95470 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 2 * q^4 - 2 * q^6 + 6 * q^7 + 10 * q^9	$ 6 q - 6 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{7} + 10 q^{9} - 6 q^{11} - 12 q^{12} - 6 q^{13} - 6 q^{16} + 2 q^{17} - 14 q^{18} - 2 q^{19} - 6 q^{21} - 8 q^{23} + 22 q^{24} - 24 q^{27} + 2 q^{28} - 8 q^{29} + 4 q^{31} - 10 q^{32} + 6 q^{33} - 10 q^{34} + 26 q^{36} - 22 q^{37} + 10 q^{38} - 8 q^{39} + 4 q^{41} - 2 q^{42} - 30 q^{43} - 2 q^{44} - 8 q^{46} - 16 q^{47} + 8 q^{48} + 6 q^{49} - 4 q^{51} - 22 q^{52} - 6 q^{53} + 38 q^{54} - 18 q^{57} + 14 q^{58} + 14 q^{59} + 12 q^{61} + 14 q^{62} + 10 q^{63} - 22 q^{64} + 2 q^{66} - 46 q^{67} + 20 q^{68} + 12 q^{69} + 4 q^{71} - 32 q^{72} + 4 q^{73} - 8 q^{74} - 8 q^{76} - 6 q^{77} - 24 q^{78} - 8 q^{79} + 26 q^{81} + 10 q^{82} - 14 q^{83} - 12 q^{84} + 12 q^{86} + 2 q^{87} + 8 q^{89} - 6 q^{91} - 18 q^{92} - 12 q^{93} - 10 q^{94} - 16 q^{96} - 42 q^{97} - 10 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + 2 * q^4 - 2 * q^6 + 6 * q^7 + 10 * q^9 - 6 * q^11 - 12 * q^12 - 6 * q^13 - 6 * q^16 + 2 * q^17 - 14 * q^18 - 2 * q^19 - 6 * q^21 - 8 * q^23 + 22 * q^24 - 24 * q^27 + 2 * q^28 - 8 * q^29 + 4 * q^31 - 10 * q^32 + 6 * q^33 - 10 * q^34 + 26 * q^36 - 22 * q^37 + 10 * q^38 - 8 * q^39 + 4 * q^41 - 2 * q^42 - 30 * q^43 - 2 * q^44 - 8 * q^46 - 16 * q^47 + 8 * q^48 + 6 * q^49 - 4 * q^51 - 22 * q^52 - 6 * q^53 + 38 * q^54 - 18 * q^57 + 14 * q^58 + 14 * q^59 + 12 * q^61 + 14 * q^62 + 10 * q^63 - 22 * q^64 + 2 * q^66 - 46 * q^67 + 20 * q^68 + 12 * q^69 + 4 * q^71 - 32 * q^72 + 4 * q^73 - 8 * q^74 - 8 * q^76 - 6 * q^77 - 24 * q^78 - 8 * q^79 + 26 * q^81 + 10 * q^82 - 14 * q^83 - 12 * q^84 + 12 * q^86 + 2 * q^87 + 8 * q^89 - 6 * q^91 - 18 * q^92 - 12 * q^93 - 10 * q^94 - 16 * q^96 - 42 * q^97 - 10 * 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 $

$2$	−2.24274	−1.58586	−0.792929	−	0.609314i	$-0.791445\pi$
$2$	−2.24274	−1.58586	−0.792929	+	0.609314i	$0.791445\pi$
$3$	−3.30979	−1.91091	−0.955453	−	0.295142i	$-0.904633\pi$
$3$	−3.30979	−1.91091	−0.955453	+	0.295142i	$0.904633\pi$
$4$	3.02989	1.51495
$5$	0	0
$5$	0	0
$6$	7.42300	3.03043
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.30979	−0.816633
$9$	7.95470	2.65157
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−10.0283	−2.89492
$13$	−2.34784	−0.651172	−0.325586	−	0.945512i	$-0.605562\pi$
$13$	−2.34784	−0.651172	−0.325586	+	0.945512i	$0.605562\pi$
$14$	−2.24274	−0.599398
$15$	0	0
$16$	−0.879529	−0.219882
$17$	4.20155	1.01903	0.509513	−	0.860463i	$-0.329826\pi$
$17$	4.20155	1.01903	0.509513	+	0.860463i	$0.329826\pi$
$18$	−17.8403	−4.20501
$19$	−0.538260	−0.123485	−0.0617427	−	0.998092i	$-0.519666\pi$
$19$	−0.538260	−0.123485	−0.0617427	+	0.998092i	$0.519666\pi$
$20$	0	0
$21$	−3.30979	−0.722255
$22$	2.24274	0.478154
$23$	−0.251800	−0.0525039	−0.0262520	−	0.999655i	$-0.508357\pi$
$23$	−0.251800	−0.0525039	−0.0262520	+	0.999655i	$0.508357\pi$
$24$	7.64491	1.56051
$25$	0	0
$26$	5.26559	1.03267
$27$	−16.3990	−3.15599
$28$	3.02989	0.572596
$29$	−7.62047	−1.41509	−0.707543	−	0.706670i	$-0.750196\pi$
$29$	−7.62047	−1.41509	−0.707543	+	0.706670i	$0.750196\pi$
$30$	0	0
$31$	0.441844	0.0793576	0.0396788	−	0.999212i	$-0.487367\pi$
$31$	0.441844	0.0793576	0.0396788	+	0.999212i	$0.487367\pi$
$32$	6.59213	1.16534
$33$	3.30979	0.576160
$34$	−9.42300	−1.61603
$35$	0	0
$36$	24.1019	4.01698
$37$	−9.52973	−1.56668	−0.783339	−	0.621595i	$-0.786485\pi$
$37$	−9.52973	−1.56668	−0.783339	+	0.621595i	$0.786485\pi$
$38$	1.20718	0.195830
$39$	7.77084	1.24433
$40$	0	0
$41$	−0.122999	−0.0192092	−0.00960461	−	0.999954i	$-0.503057\pi$
$41$	−0.122999	−0.0192092	−0.00960461	+	0.999954i	$0.503057\pi$
$42$	7.42300	1.14539
$43$	−4.54140	−0.692557	−0.346279	−	0.938132i	$-0.612555\pi$
$43$	−4.54140	−0.692557	−0.346279	+	0.938132i	$0.612555\pi$
$44$	−3.02989	−0.456774
$45$	0	0
$46$	0.564722	0.0832638
$47$	11.9033	1.73628	0.868140	−	0.496319i	$-0.165315\pi$
$47$	11.9033	1.73628	0.868140	+	0.496319i	$0.165315\pi$
$48$	2.91105	0.420175
$49$	1.00000	0.142857
$50$	0	0
$51$	−13.9062	−1.94726
$52$	−7.11369	−0.986492
$53$	10.3421	1.42060	0.710301	−	0.703898i	$-0.248559\pi$
$53$	10.3421	1.42060	0.710301	+	0.703898i	$0.248559\pi$
$54$	36.7787	5.00495
$55$	0	0
$56$	−2.30979	−0.308658
$57$	1.78153	0.235969
$58$	17.0908	2.24413
$59$	−4.12232	−0.536680	−0.268340	−	0.963324i	$-0.586475\pi$
$59$	−4.12232	−0.536680	−0.268340	+	0.963324i	$0.586475\pi$
$60$	0	0
$61$	15.5178	1.98685	0.993427	−	0.114465i	$-0.0365152\pi$
$61$	15.5178	1.98685	0.993427	+	0.114465i	$0.0365152\pi$
$62$	−0.990943	−0.125850
$63$	7.95470	1.00220
$64$	−13.0254	−1.62817
$65$	0	0
$66$	−7.42300	−0.913708
$67$	−3.85982	−0.471552	−0.235776	−	0.971807i	$-0.575763\pi$
$67$	−3.85982	−0.471552	−0.235776	+	0.971807i	$0.575763\pi$
$68$	12.7303	1.54377
$69$	0.833404	0.100330
$70$	0	0
$71$	−0.782634	−0.0928815	−0.0464408	−	0.998921i	$-0.514788\pi$
$71$	−0.782634	−0.0928815	−0.0464408	+	0.998921i	$0.514788\pi$
$72$	−18.3737	−2.16536
$73$	15.1342	1.77133	0.885663	−	0.464328i	$-0.153704\pi$
$73$	15.1342	1.77133	0.885663	+	0.464328i	$0.153704\pi$
$74$	21.3727	2.48453
$75$	0	0
$76$	−1.63087	−0.187074
$77$	−1.00000	−0.113961
$78$	−17.4280	−1.97333
$79$	−7.84140	−0.882227	−0.441113	−	0.897451i	$-0.645416\pi$
$79$	−7.84140	−0.882227	−0.441113	+	0.897451i	$0.645416\pi$
$80$	0	0
$81$	30.4131	3.37923
$82$	0.275855	0.0304631
$83$	−2.41779	−0.265387	−0.132693	−	0.991157i	$-0.542363\pi$
$83$	−2.41779	−0.265387	−0.132693	+	0.991157i	$0.542363\pi$
$84$	−10.0283	−1.09418
$85$	0	0
$86$	10.1852	1.09830
$87$	25.2221	2.70410
$88$	2.30979	0.246224
$89$	10.9371	1.15933	0.579664	−	0.814855i	$-0.303184\pi$
$89$	10.9371	1.15933	0.579664	+	0.814855i	$0.303184\pi$
$90$	0	0
$91$	−2.34784	−0.246120
$92$	−0.762927	−0.0795407
$93$	−1.46241	−0.151645
$94$	−26.6961	−2.75350
$95$	0	0
$96$	−21.8186	−2.22685
$97$	−5.40327	−0.548619	−0.274310	−	0.961641i	$-0.588449\pi$
$97$	−5.40327	−0.548619	−0.274310	+	0.961641i	$0.588449\pi$
$98$	−2.24274	−0.226551
$99$	−7.95470	−0.799477
$100$	0	0
$101$	−2.27651	−0.226521	−0.113260	−	0.993565i	$-0.536129\pi$
$101$	−2.27651	−0.226521	−0.113260	+	0.993565i	$0.536129\pi$
$102$	31.1881	3.08809
$103$	−12.6197	−1.24346	−0.621728	−	0.783233i	$-0.713569\pi$
$103$	−12.6197	−1.24346	−0.621728	+	0.783233i	$0.713569\pi$
$104$	5.42300	0.531769
$105$	0	0
$106$	−23.1947	−2.25287
$107$	6.86412	0.663579	0.331790	−	0.943353i	$-0.392348\pi$
$107$	6.86412	0.663579	0.331790	+	0.943353i	$0.392348\pi$
$108$	−49.6872	−4.78115
$109$	−9.21061	−0.882216	−0.441108	−	0.897454i	$-0.645415\pi$
$109$	−9.21061	−0.882216	−0.441108	+	0.897454i	$0.645415\pi$
$110$	0	0
$111$	31.5414	2.99377
$112$	−0.879529	−0.0831077
$113$	2.53415	0.238392	0.119196	−	0.992871i	$-0.461968\pi$
$113$	2.53415	0.238392	0.119196	+	0.992871i	$0.461968\pi$
$114$	−3.99550	−0.374213
$115$	0	0
$116$	−23.0892	−2.14378
$117$	−18.6763	−1.72663
$118$	9.24529	0.851098
$119$	4.20155	0.385156
$120$	0	0
$121$	1.00000	0.0909091
$122$	−34.8025	−3.15087
$123$	0.407100	0.0367070
$124$	1.33874	0.120222
$125$	0	0
$126$	−17.8403	−1.58934
$127$	−18.4142	−1.63400	−0.817000	−	0.576638i	$-0.804364\pi$
$127$	−18.4142	−1.63400	−0.817000	+	0.576638i	$0.804364\pi$
$128$	16.0284	1.41672
$129$	15.0311	1.32341
$130$	0	0
$131$	−0.379317	−0.0331411	−0.0165705	−	0.999863i	$-0.505275\pi$
$131$	−0.379317	−0.0331411	−0.0165705	+	0.999863i	$0.505275\pi$
$132$	10.0283	0.872852
$133$	−0.538260	−0.0466731
$134$	8.65659	0.747815
$135$	0	0
$136$	−9.70470	−0.832171
$137$	−9.61797	−0.821718	−0.410859	−	0.911699i	$-0.634771\pi$
$137$	−9.61797	−0.821718	−0.410859	+	0.911699i	$0.634771\pi$
$138$	−1.86911	−0.159109
$139$	−2.06753	−0.175365	−0.0876827	−	0.996148i	$-0.527946\pi$
$139$	−2.06753	−0.175365	−0.0876827	+	0.996148i	$0.527946\pi$
$140$	0	0
$141$	−39.3975	−3.31787
$142$	1.75525	0.147297
$143$	2.34784	0.196336
$144$	−6.99639	−0.583032
$145$	0	0
$146$	−33.9422	−2.80907
$147$	−3.30979	−0.272987
$148$	−28.8741	−2.37343
$149$	−2.57341	−0.210822	−0.105411	−	0.994429i	$-0.533616\pi$
$149$	−2.57341	−0.210822	−0.105411	+	0.994429i	$0.533616\pi$
$150$	0	0
$151$	−1.49174	−0.121396	−0.0606980	−	0.998156i	$-0.519333\pi$
$151$	−1.49174	−0.121396	−0.0606980	+	0.998156i	$0.519333\pi$
$152$	1.24327	0.100842
$153$	33.4221	2.70201
$154$	2.24274	0.180725
$155$	0	0
$156$	23.5448	1.88509
$157$	1.69646	0.135392	0.0676960	−	0.997706i	$-0.478435\pi$
$157$	1.69646	0.135392	0.0676960	+	0.997706i	$0.478435\pi$
$158$	17.5863	1.39909
$159$	−34.2303	−2.71464
$160$	0	0
$161$	−0.251800	−0.0198446
$162$	−68.2087	−5.35899
$163$	−14.5006	−1.13578	−0.567888	−	0.823106i	$-0.692239\pi$
$163$	−14.5006	−1.13578	−0.567888	+	0.823106i	$0.692239\pi$
$164$	−0.372674	−0.0291009
$165$	0	0
$166$	5.42248	0.420866
$167$	2.85337	0.220800	0.110400	−	0.993887i	$-0.464787\pi$
$167$	2.85337	0.220800	0.110400	+	0.993887i	$0.464787\pi$
$168$	7.64491	0.589817
$169$	−7.48767	−0.575974
$170$	0	0
$171$	−4.28169	−0.327429
$172$	−13.7600	−1.04919
$173$	8.22125	0.625050	0.312525	−	0.949910i	$-0.398825\pi$
$173$	8.22125	0.625050	0.312525	+	0.949910i	$0.398825\pi$
$174$	−56.5668	−4.28832
$175$	0	0
$176$	0.879529	0.0662970
$177$	13.6440	1.02555
$178$	−24.5291	−1.83853
$179$	23.5710	1.76178	0.880889	−	0.473323i	$-0.156946\pi$
$179$	23.5710	1.76178	0.880889	+	0.473323i	$0.156946\pi$
$180$	0	0
$181$	0.991196	0.0736750	0.0368375	−	0.999321i	$-0.488272\pi$
$181$	0.991196	0.0736750	0.0368375	+	0.999321i	$0.488272\pi$
$182$	5.26559	0.390312
$183$	−51.3607	−3.79669
$184$	0.581604	0.0428764
$185$	0	0
$186$	3.27981	0.240487
$187$	−4.20155	−0.307248
$188$	36.0659	2.63037
$189$	−16.3990	−1.19285
$190$	0	0
$191$	−6.41346	−0.464062	−0.232031	−	0.972708i	$-0.574537\pi$
$191$	−6.41346	−0.464062	−0.232031	+	0.972708i	$0.574537\pi$
$192$	43.1113	3.11129
$193$	11.0625	0.796295	0.398148	−	0.917321i	$-0.369653\pi$
$193$	11.0625	0.796295	0.398148	+	0.917321i	$0.369653\pi$
$194$	12.1181	0.870032
$195$	0	0
$196$	3.02989	0.216421
$197$	12.5929	0.897207	0.448604	−	0.893731i	$-0.351922\pi$
$197$	12.5929	0.897207	0.448604	+	0.893731i	$0.351922\pi$
$198$	17.8403	1.26786
$199$	9.82997	0.696828	0.348414	−	0.937341i	$-0.386720\pi$
$199$	9.82997	0.696828	0.348414	+	0.937341i	$0.386720\pi$
$200$	0	0
$201$	12.7752	0.901093
$202$	5.10562	0.359230
$203$	−7.62047	−0.534852
$204$	−42.1345	−2.95000
$205$	0	0
$206$	28.3027	1.97195
$207$	−2.00299	−0.139218
$208$	2.06499	0.143181
$209$	0.538260	0.0372322
$210$	0	0
$211$	−10.6241	−0.731394	−0.365697	−	0.930734i	$-0.619169\pi$
$211$	−10.6241	−0.731394	−0.365697	+	0.930734i	$0.619169\pi$
$212$	31.3356	2.15214
$213$	2.59035	0.177488
$214$	−15.3945	−1.05234
$215$	0	0
$216$	37.8782	2.57728
$217$	0.441844	0.0299943
$218$	20.6570	1.39907
$219$	−50.0911	−3.38484
$220$	0	0
$221$	−9.86456	−0.663562
$222$	−70.7392	−4.74770
$223$	8.72692	0.584398	0.292199	−	0.956358i	$-0.405613\pi$
$223$	8.72692	0.584398	0.292199	+	0.956358i	$0.405613\pi$
$224$	6.59213	0.440455
$225$	0	0
$226$	−5.68343	−0.378056
$227$	−12.8005	−0.849597	−0.424799	−	0.905288i	$-0.639655\pi$
$227$	−12.8005	−0.849597	−0.424799	+	0.905288i	$0.639655\pi$
$228$	5.39784	0.357480
$229$	−19.8578	−1.31224	−0.656119	−	0.754657i	$-0.727803\pi$
$229$	−19.8578	−1.31224	−0.656119	+	0.754657i	$0.727803\pi$
$230$	0	0
$231$	3.30979	0.217768
$232$	17.6017	1.15561
$233$	−4.15433	−0.272159	−0.136080	−	0.990698i	$-0.543450\pi$
$233$	−4.15433	−0.272159	−0.136080	+	0.990698i	$0.543450\pi$
$234$	41.8862	2.73818
$235$	0	0
$236$	−12.4902	−0.813042
$237$	25.9534	1.68585
$238$	−9.42300	−0.610803
$239$	−13.0040	−0.841159	−0.420580	−	0.907256i	$-0.638173\pi$
$239$	−13.0040	−0.841159	−0.420580	+	0.907256i	$0.638173\pi$
$240$	0	0
$241$	20.4972	1.32034	0.660170	−	0.751116i	$-0.270484\pi$
$241$	20.4972	1.32034	0.660170	+	0.751116i	$0.270484\pi$
$242$	−2.24274	−0.144169
$243$	−51.4639	−3.30141
$244$	47.0174	3.00998
$245$	0	0
$246$	−0.913022	−0.0582121
$247$	1.26375	0.0804102
$248$	−1.02057	−0.0648060
$249$	8.00237	0.507130
$250$	0	0
$251$	−5.88412	−0.371402	−0.185701	−	0.982606i	$-0.559456\pi$
$251$	−5.88412	−0.371402	−0.185701	+	0.982606i	$0.559456\pi$
$252$	24.1019	1.51828
$253$	0.251800	0.0158305
$254$	41.2984	2.59129
$255$	0	0
$256$	−9.89667	−0.618542
$257$	−16.5283	−1.03101	−0.515504	−	0.856887i	$-0.672395\pi$
$257$	−16.5283	−1.03101	−0.515504	+	0.856887i	$0.672395\pi$
$258$	−33.7108	−2.09874
$259$	−9.52973	−0.592148
$260$	0	0
$261$	−60.6185	−3.75219
$262$	0.850710	0.0525570
$263$	−20.8553	−1.28600	−0.642998	−	0.765868i	$-0.722310\pi$
$263$	−20.8553	−1.28600	−0.642998	+	0.765868i	$0.722310\pi$
$264$	−7.64491	−0.470512
$265$	0	0
$266$	1.20718	0.0740169
$267$	−36.1994	−2.21537
$268$	−11.6949	−0.714377
$269$	13.9865	0.852771	0.426386	−	0.904542i	$-0.359787\pi$
$269$	13.9865	0.852771	0.426386	+	0.904542i	$0.359787\pi$
$270$	0	0
$271$	2.83550	0.172244	0.0861222	−	0.996285i	$-0.472552\pi$
$271$	2.83550	0.172244	0.0861222	+	0.996285i	$0.472552\pi$
$272$	−3.69539	−0.224066
$273$	7.77084	0.470312
$274$	21.5706	1.30313
$275$	0	0
$276$	2.52513	0.151995
$277$	21.3638	1.28362	0.641812	−	0.766862i	$-0.278183\pi$
$277$	21.3638	1.28362	0.641812	+	0.766862i	$0.278183\pi$
$278$	4.63693	0.278105
$279$	3.51474	0.210422
$280$	0	0
$281$	9.47504	0.565234	0.282617	−	0.959233i	$-0.408798\pi$
$281$	9.47504	0.565234	0.282617	+	0.959233i	$0.408798\pi$
$282$	88.3585	5.26167
$283$	−16.6002	−0.986777	−0.493389	−	0.869809i	$-0.664242\pi$
$283$	−16.6002	−0.986777	−0.493389	+	0.869809i	$0.664242\pi$
$284$	−2.37130	−0.140711
$285$	0	0
$286$	−5.26559	−0.311361
$287$	−0.122999	−0.00726040
$288$	52.4384	3.08996
$289$	0.653049	0.0384146
$290$	0	0
$291$	17.8837	1.04836
$292$	45.8551	2.68347
$293$	−19.9983	−1.16832	−0.584158	−	0.811640i	$-0.698575\pi$
$293$	−19.9983	−1.16832	−0.584158	+	0.811640i	$0.698575\pi$
$294$	7.42300	0.432918
$295$	0	0
$296$	22.0116	1.27940
$297$	16.3990	0.951566
$298$	5.77149	0.334333
$299$	0.591185	0.0341891
$300$	0	0
$301$	−4.54140	−0.261762
$302$	3.34559	0.192517
$303$	7.53475	0.432860
$304$	0.473415	0.0271522
$305$	0	0
$306$	−74.9571	−4.28501
$307$	−10.4700	−0.597556	−0.298778	−	0.954323i	$-0.596579\pi$
$307$	−10.4700	−0.597556	−0.298778	+	0.954323i	$0.596579\pi$
$308$	−3.02989	−0.172644
$309$	41.7685	2.37613
$310$	0	0
$311$	−22.2365	−1.26091	−0.630457	−	0.776224i	$-0.717132\pi$
$311$	−22.2365	−1.26091	−0.630457	+	0.776224i	$0.717132\pi$
$312$	−17.9490	−1.01616
$313$	−14.7407	−0.833193	−0.416596	−	0.909092i	$-0.636777\pi$
$313$	−14.7407	−0.833193	−0.416596	+	0.909092i	$0.636777\pi$
$314$	−3.80471	−0.214712
$315$	0	0
$316$	−23.7586	−1.33653
$317$	24.5109	1.37667	0.688333	−	0.725395i	$-0.258343\pi$
$317$	24.5109	1.37667	0.688333	+	0.725395i	$0.258343\pi$
$318$	76.7697	4.30503
$319$	7.62047	0.426665
$320$	0	0
$321$	−22.7188	−1.26804
$322$	0.564722	0.0314707
$323$	−2.26153	−0.125835
$324$	92.1485	5.11936
$325$	0	0
$326$	32.5211	1.80118
$327$	30.4852	1.68583
$328$	0.284102	0.0156869
$329$	11.9033	0.656252
$330$	0	0
$331$	−8.12733	−0.446719	−0.223359	−	0.974736i	$-0.571702\pi$
$331$	−8.12733	−0.446719	−0.223359	+	0.974736i	$0.571702\pi$
$332$	−7.32565	−0.402047
$333$	−75.8061	−4.15415
$334$	−6.39938	−0.350158
$335$	0	0
$336$	2.91105	0.158811
$337$	−18.7398	−1.02082	−0.510410	−	0.859931i	$-0.670506\pi$
$337$	−18.7398	−1.02082	−0.510410	+	0.859931i	$0.670506\pi$
$338$	16.7929	0.913414
$339$	−8.38748	−0.455546
$340$	0	0
$341$	−0.441844	−0.0239272
$342$	9.60274	0.519257
$343$	1.00000	0.0539949
$344$	10.4897	0.565565
$345$	0	0
$346$	−18.4381	−0.991240
$347$	9.58218	0.514399	0.257199	−	0.966358i	$-0.417200\pi$
$347$	9.58218	0.514399	0.257199	+	0.966358i	$0.417200\pi$
$348$	76.4204	4.09657
$349$	15.5178	0.830647	0.415324	−	0.909674i	$-0.363668\pi$
$349$	15.5178	0.830647	0.415324	+	0.909674i	$0.363668\pi$
$350$	0	0
$351$	38.5021	2.05509
$352$	−6.59213	−0.351362
$353$	−36.7877	−1.95801	−0.979005	−	0.203835i	$-0.934659\pi$
$353$	−36.7877	−1.95801	−0.979005	+	0.203835i	$0.934659\pi$
$354$	−30.6000	−1.62637
$355$	0	0
$356$	33.1382	1.75632
$357$	−13.9062	−0.735997
$358$	−52.8636	−2.79393
$359$	−26.6113	−1.40449	−0.702246	−	0.711935i	$-0.747819\pi$
$359$	−26.6113	−1.40449	−0.702246	+	0.711935i	$0.747819\pi$
$360$	0	0
$361$	−18.7103	−0.984751
$362$	−2.22300	−0.116838
$363$	−3.30979	−0.173719
$364$	−7.11369	−0.372859
$365$	0	0
$366$	115.189	6.02102
$367$	−24.1375	−1.25997	−0.629984	−	0.776608i	$-0.716938\pi$
$367$	−24.1375	−1.25997	−0.629984	+	0.776608i	$0.716938\pi$
$368$	0.221465	0.0115447
$369$	−0.978419	−0.0509345
$370$	0	0
$371$	10.3421	0.536937
$372$	−4.43095	−0.229734
$373$	38.5244	1.99472	0.997359	−	0.0726311i	$-0.0231396\pi$
$373$	38.5244	1.99472	0.997359	+	0.0726311i	$0.0231396\pi$
$374$	9.42300	0.487252
$375$	0	0
$376$	−27.4942	−1.41790
$377$	17.8916	0.921465
$378$	36.7787	1.89169
$379$	5.87740	0.301902	0.150951	−	0.988541i	$-0.451766\pi$
$379$	5.87740	0.301902	0.150951	+	0.988541i	$0.451766\pi$
$380$	0	0
$381$	60.9472	3.12242
$382$	14.3837	0.735937
$383$	4.32319	0.220905	0.110452	−	0.993881i	$-0.464770\pi$
$383$	4.32319	0.220905	0.110452	+	0.993881i	$0.464770\pi$
$384$	−53.0504	−2.70722
$385$	0	0
$386$	−24.8103	−1.26281
$387$	−36.1255	−1.83636
$388$	−16.3713	−0.831129
$389$	17.5755	0.891112	0.445556	−	0.895254i	$-0.353006\pi$
$389$	17.5755	0.891112	0.445556	+	0.895254i	$0.353006\pi$
$390$	0	0
$391$	−1.05795	−0.0535029
$392$	−2.30979	−0.116662
$393$	1.25546	0.0633295
$394$	−28.2426	−1.42284
$395$	0	0
$396$	−24.1019	−1.21117
$397$	−27.2878	−1.36953	−0.684767	−	0.728762i	$-0.740096\pi$
$397$	−27.2878	−1.36953	−0.684767	+	0.728762i	$0.740096\pi$
$398$	−22.0461	−1.10507
$399$	1.78153	0.0891879
$400$	0	0
$401$	16.1115	0.804570	0.402285	−	0.915515i	$-0.368216\pi$
$401$	16.1115	0.804570	0.402285	+	0.915515i	$0.368216\pi$
$402$	−28.6515	−1.42901
$403$	−1.03738	−0.0516754
$404$	−6.89757	−0.343167
$405$	0	0
$406$	17.0908	0.848200
$407$	9.52973	0.472371
$408$	32.1205	1.59020
$409$	−11.7215	−0.579592	−0.289796	−	0.957088i	$-0.593587\pi$
$409$	−11.7215	−0.579592	−0.289796	+	0.957088i	$0.593587\pi$
$410$	0	0
$411$	31.8334	1.57023
$412$	−38.2364	−1.88377
$413$	−4.12232	−0.202846
$414$	4.49219	0.220779
$415$	0	0
$416$	−15.4772	−0.758834
$417$	6.84307	0.335107
$418$	−1.20718	−0.0590450
$419$	10.4245	0.509272	0.254636	−	0.967037i	$-0.418044\pi$
$419$	10.4245	0.509272	0.254636	+	0.967037i	$0.418044\pi$
$420$	0	0
$421$	−38.8520	−1.89353	−0.946765	−	0.321927i	$-0.895669\pi$
$421$	−38.8520	−1.89353	−0.946765	+	0.321927i	$0.895669\pi$
$422$	23.8272	1.15989
$423$	94.6875	4.60386
$424$	−23.8881	−1.16011
$425$	0	0
$426$	−5.80949	−0.281471
$427$	15.5178	0.750960
$428$	20.7976	1.00529
$429$	−7.77084	−0.375180
$430$	0	0
$431$	4.44461	0.214090	0.107045	−	0.994254i	$-0.465861\pi$
$431$	4.44461	0.214090	0.107045	+	0.994254i	$0.465861\pi$
$432$	14.4234	0.693946
$433$	−12.4480	−0.598210	−0.299105	−	0.954220i	$-0.596688\pi$
$433$	−12.4480	−0.598210	−0.299105	+	0.954220i	$0.596688\pi$
$434$	−0.990943	−0.0475668
$435$	0	0
$436$	−27.9072	−1.33651
$437$	0.135534	0.00648346
$438$	112.341	5.36788
$439$	36.1563	1.72565	0.862824	−	0.505504i	$-0.168693\pi$
$439$	36.1563	1.72565	0.862824	+	0.505504i	$0.168693\pi$
$440$	0	0
$441$	7.95470	0.378795
$442$	22.1237	1.05232
$443$	−16.6004	−0.788710	−0.394355	−	0.918958i	$-0.629032\pi$
$443$	−16.6004	−0.788710	−0.394355	+	0.918958i	$0.629032\pi$
$444$	95.5670	4.53541
$445$	0	0
$446$	−19.5722	−0.926772
$447$	8.51743	0.402861
$448$	−13.0254	−0.615392
$449$	−13.0051	−0.613748	−0.306874	−	0.951750i	$-0.599283\pi$
$449$	−13.0051	−0.613748	−0.306874	+	0.951750i	$0.599283\pi$
$450$	0	0
$451$	0.122999	0.00579180
$452$	7.67819	0.361152
$453$	4.93734	0.231976
$454$	28.7082	1.34734
$455$	0	0
$456$	−4.11495	−0.192700
$457$	−20.3826	−0.953456	−0.476728	−	0.879051i	$-0.658177\pi$
$457$	−20.3826	−0.953456	−0.476728	+	0.879051i	$0.658177\pi$
$458$	44.5359	2.08102
$459$	−68.9012	−3.21603
$460$	0	0
$461$	−0.755909	−0.0352062	−0.0176031	−	0.999845i	$-0.505604\pi$
$461$	−0.755909	−0.0352062	−0.0176031	+	0.999845i	$0.505604\pi$
$462$	−7.42300	−0.345349
$463$	6.90413	0.320862	0.160431	−	0.987047i	$-0.448712\pi$
$463$	6.90413	0.320862	0.160431	+	0.987047i	$0.448712\pi$
$464$	6.70243	0.311152
$465$	0	0
$466$	9.31709	0.431606
$467$	−4.80911	−0.222539	−0.111269	−	0.993790i	$-0.535492\pi$
$467$	−4.80911	−0.222539	−0.111269	+	0.993790i	$0.535492\pi$
$468$	−56.5873	−2.61575
$469$	−3.85982	−0.178230
$470$	0	0
$471$	−5.61491	−0.258721
$472$	9.52168	0.438271
$473$	4.54140	0.208814
$474$	−58.2068	−2.67353
$475$	0	0
$476$	12.7303	0.583491
$477$	82.2685	3.76682
$478$	29.1646	1.33396
$479$	14.9237	0.681883	0.340941	−	0.940085i	$-0.389254\pi$
$479$	14.9237	0.681883	0.340941	+	0.940085i	$0.389254\pi$
$480$	0	0
$481$	22.3742	1.02018
$482$	−45.9699	−2.09387
$483$	0.833404	0.0379212
$484$	3.02989	0.137722
$485$	0	0
$486$	115.420	5.23557
$487$	−24.0009	−1.08758	−0.543792	−	0.839220i	$-0.683012\pi$
$487$	−24.0009	−1.08758	−0.543792	+	0.839220i	$0.683012\pi$
$488$	−35.8429	−1.62253
$489$	47.9939	2.17036
$490$	0	0
$491$	−4.31872	−0.194901	−0.0974507	−	0.995240i	$-0.531069\pi$
$491$	−4.31872	−0.194901	−0.0974507	+	0.995240i	$0.531069\pi$
$492$	1.23347	0.0556092
$493$	−32.0178	−1.44201
$494$	−2.83426	−0.127519
$495$	0	0
$496$	−0.388615	−0.0174493
$497$	−0.782634	−0.0351059
$498$	−17.9473	−0.804236
$499$	−20.8981	−0.935529	−0.467764	−	0.883853i	$-0.654940\pi$
$499$	−20.8981	−0.935529	−0.467764	+	0.883853i	$0.654940\pi$
$500$	0	0
$501$	−9.44405	−0.421929
$502$	13.1966	0.588991
$503$	30.7505	1.37110	0.685549	−	0.728026i	$-0.259562\pi$
$503$	30.7505	1.37110	0.685549	+	0.728026i	$0.259562\pi$
$504$	−18.3737	−0.818428
$505$	0	0
$506$	−0.564722	−0.0251050
$507$	24.7826	1.10063
$508$	−55.7932	−2.47542
$509$	9.38375	0.415927	0.207964	−	0.978137i	$-0.433316\pi$
$509$	9.38375	0.415927	0.207964	+	0.978137i	$0.433316\pi$
$510$	0	0
$511$	15.1342	0.669499
$512$	−9.86102	−0.435800
$513$	8.82692	0.389718
$514$	37.0688	1.63503
$515$	0	0
$516$	45.5426	2.00490
$517$	−11.9033	−0.523508
$518$	21.3727	0.939064
$519$	−27.2106	−1.19441
$520$	0	0
$521$	−10.6981	−0.468693	−0.234347	−	0.972153i	$-0.575295\pi$
$521$	−10.6981	−0.468693	−0.234347	+	0.972153i	$0.575295\pi$
$522$	135.952	5.95045
$523$	6.54553	0.286216	0.143108	−	0.989707i	$-0.454290\pi$
$523$	6.54553	0.286216	0.143108	+	0.989707i	$0.454290\pi$
$524$	−1.14929	−0.0502070
$525$	0	0
$526$	46.7732	2.03941
$527$	1.85643	0.0808674
$528$	−2.91105	−0.126687
$529$	−22.9366	−0.997243
$530$	0	0
$531$	−32.7918	−1.42304
$532$	−1.63087	−0.0707072
$533$	0.288781	0.0125085
$534$	81.1860	3.51326
$535$	0	0
$536$	8.91537	0.385085
$537$	−78.0149	−3.36659
$538$	−31.3681	−1.35237
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−3.55683	−0.152920	−0.0764600	−	0.997073i	$-0.524362\pi$
$541$	−3.55683	−0.152920	−0.0764600	+	0.997073i	$0.524362\pi$
$542$	−6.35930	−0.273155
$543$	−3.28065	−0.140786
$544$	27.6972	1.18751
$545$	0	0
$546$	−17.4280	−0.745849
$547$	−4.28956	−0.183408	−0.0917041	−	0.995786i	$-0.529231\pi$
$547$	−4.28956	−0.183408	−0.0917041	+	0.995786i	$0.529231\pi$
$548$	−29.1414	−1.24486
$549$	123.440	5.26827
$550$	0	0
$551$	4.10180	0.174742
$552$	−1.92499	−0.0819329
$553$	−7.84140	−0.333450
$554$	−47.9134	−2.03565
$555$	0	0
$556$	−6.26439	−0.265669
$557$	−3.55455	−0.150611	−0.0753056	−	0.997161i	$-0.523993\pi$
$557$	−3.55455	−0.150611	−0.0753056	+	0.997161i	$0.523993\pi$
$558$	−7.88265	−0.333699
$559$	10.6625	0.450974
$560$	0	0
$561$	13.9062	0.587122
$562$	−21.2501	−0.896381
$563$	−27.2285	−1.14754	−0.573772	−	0.819015i	$-0.694520\pi$
$563$	−27.2285	−1.14754	−0.573772	+	0.819015i	$0.694520\pi$
$564$	−119.370	−5.02640
$565$	0	0
$566$	37.2299	1.56489
$567$	30.4131	1.27723
$568$	1.80772	0.0758502
$569$	4.62626	0.193943	0.0969715	−	0.995287i	$-0.469084\pi$
$569$	4.62626	0.193943	0.0969715	+	0.995287i	$0.469084\pi$
$570$	0	0
$571$	7.69838	0.322167	0.161084	−	0.986941i	$-0.448501\pi$
$571$	7.69838	0.322167	0.161084	+	0.986941i	$0.448501\pi$
$572$	7.11369	0.297438
$573$	21.2272	0.886779
$574$	0.275855	0.0115140
$575$	0	0
$576$	−103.613	−4.31721
$577$	−4.33012	−0.180265	−0.0901327	−	0.995930i	$-0.528729\pi$
$577$	−4.33012	−0.180265	−0.0901327	+	0.995930i	$0.528729\pi$
$578$	−1.46462	−0.0609202
$579$	−36.6145	−1.52165
$580$	0	0
$581$	−2.41779	−0.100307
$582$	−40.1085	−1.66255
$583$	−10.3421	−0.428327
$584$	−34.9568	−1.44652
$585$	0	0
$586$	44.8511	1.85278
$587$	−33.3057	−1.37467	−0.687336	−	0.726340i	$-0.741220\pi$
$587$	−33.3057	−1.37467	−0.687336	+	0.726340i	$0.741220\pi$
$588$	−10.0283	−0.413560
$589$	−0.237827	−0.00979949
$590$	0	0
$591$	−41.6798	−1.71448
$592$	8.38167	0.344485
$593$	4.41566	0.181329	0.0906646	−	0.995881i	$-0.471101\pi$
$593$	4.41566	0.181329	0.0906646	+	0.995881i	$0.471101\pi$
$594$	−36.7787	−1.50905
$595$	0	0
$596$	−7.79715	−0.319384
$597$	−32.5351	−1.33157
$598$	−1.32588	−0.0542191
$599$	−46.7963	−1.91205	−0.956023	−	0.293291i	$-0.905249\pi$
$599$	−46.7963	−1.91205	−0.956023	+	0.293291i	$0.905249\pi$
$600$	0	0
$601$	−26.6654	−1.08771	−0.543853	−	0.839180i	$-0.683035\pi$
$601$	−26.6654	−1.08771	−0.543853	+	0.839180i	$0.683035\pi$
$602$	10.1852	0.415118
$603$	−30.7037	−1.25035
$604$	−4.51981	−0.183908
$605$	0	0
$606$	−16.8985	−0.686455
$607$	−31.1891	−1.26593	−0.632963	−	0.774182i	$-0.718161\pi$
$607$	−31.1891	−1.26593	−0.632963	+	0.774182i	$0.718161\pi$
$608$	−3.54828	−0.143902
$609$	25.2221	1.02205
$610$	0	0
$611$	−27.9471	−1.13062
$612$	101.265	4.09341
$613$	29.9411	1.20931	0.604655	−	0.796488i	$-0.293311\pi$
$613$	29.9411	1.20931	0.604655	+	0.796488i	$0.293311\pi$
$614$	23.4816	0.947640
$615$	0	0
$616$	2.30979	0.0930640
$617$	−39.2682	−1.58088	−0.790439	−	0.612541i	$-0.790147\pi$
$617$	−39.2682	−1.58088	−0.790439	+	0.612541i	$0.790147\pi$
$618$	−93.6761	−3.76820
$619$	−44.5424	−1.79031	−0.895156	−	0.445754i	$-0.852936\pi$
$619$	−44.5424	−1.79031	−0.895156	+	0.445754i	$0.852936\pi$
$620$	0	0
$621$	4.12926	0.165702
$622$	49.8706	1.99963
$623$	10.9371	0.438185
$624$	−6.83468	−0.273606
$625$	0	0
$626$	33.0596	1.32133
$627$	−1.78153	−0.0711473
$628$	5.14008	0.205112
$629$	−40.0397	−1.59649
$630$	0	0
$631$	−26.6772	−1.06200	−0.531000	−	0.847372i	$-0.678184\pi$
$631$	−26.6772	−1.06200	−0.531000	+	0.847372i	$0.678184\pi$
$632$	18.1120	0.720456
$633$	35.1636	1.39763
$634$	−54.9715	−2.18320
$635$	0	0
$636$	−103.714	−4.11253
$637$	−2.34784	−0.0930246
$638$	−17.0908	−0.676630
$639$	−6.22561	−0.246281
$640$	0	0
$641$	−10.1081	−0.399245	−0.199622	−	0.979873i	$-0.563972\pi$
$641$	−10.1081	−0.399245	−0.199622	+	0.979873i	$0.563972\pi$
$642$	50.9524	2.01093
$643$	17.8671	0.704610	0.352305	−	0.935885i	$-0.385398\pi$
$643$	17.8671	0.704610	0.352305	+	0.935885i	$0.385398\pi$
$644$	−0.762927	−0.0300635
$645$	0	0
$646$	5.07202	0.199556
$647$	−9.76951	−0.384079	−0.192039	−	0.981387i	$-0.561510\pi$
$647$	−9.76951	−0.384079	−0.192039	+	0.981387i	$0.561510\pi$
$648$	−70.2478	−2.75959
$649$	4.12232	0.161815
$650$	0	0
$651$	−1.46241	−0.0573164
$652$	−43.9353	−1.72064
$653$	−25.1830	−0.985487	−0.492744	−	0.870175i	$-0.664006\pi$
$653$	−25.1830	−0.985487	−0.492744	+	0.870175i	$0.664006\pi$
$654$	−68.3704	−2.67349
$655$	0	0
$656$	0.108181	0.00422376
$657$	120.388	4.69679
$658$	−26.6961	−1.04072
$659$	−27.0074	−1.05206	−0.526030	−	0.850466i	$-0.676320\pi$
$659$	−27.0074	−1.05206	−0.526030	+	0.850466i	$0.676320\pi$
$660$	0	0
$661$	−35.6463	−1.38648	−0.693240	−	0.720707i	$-0.743817\pi$
$661$	−35.6463	−1.38648	−0.693240	+	0.720707i	$0.743817\pi$
$662$	18.2275	0.708433
$663$	32.6496	1.26800
$664$	5.58458	0.216724
$665$	0	0
$666$	170.014	6.58789
$667$	1.91883	0.0742976
$668$	8.64541	0.334501
$669$	−28.8842	−1.11673
$670$	0	0
$671$	−15.5178	−0.599059
$672$	−21.8186	−0.841669
$673$	−0.686772	−0.0264731	−0.0132366	−	0.999912i	$-0.504213\pi$
$673$	−0.686772	−0.0264731	−0.0132366	+	0.999912i	$0.504213\pi$
$674$	42.0285	1.61888
$675$	0	0
$676$	−22.6868	−0.872571
$677$	−29.8583	−1.14755	−0.573775	−	0.819013i	$-0.694521\pi$
$677$	−29.8583	−1.14755	−0.573775	+	0.819013i	$0.694521\pi$
$678$	18.8110	0.722431
$679$	−5.40327	−0.207359
$680$	0	0
$681$	42.3669	1.62350
$682$	0.990943	0.0379452
$683$	28.9500	1.10774	0.553870	−	0.832603i	$-0.313151\pi$
$683$	28.9500	1.10774	0.553870	+	0.832603i	$0.313151\pi$
$684$	−12.9731	−0.496038
$685$	0	0
$686$	−2.24274	−0.0856283
$687$	65.7250	2.50757
$688$	3.99429	0.152281
$689$	−24.2816	−0.925056
$690$	0	0
$691$	44.6576	1.69886	0.849428	−	0.527705i	$-0.176947\pi$
$691$	44.6576	1.69886	0.849428	+	0.527705i	$0.176947\pi$
$692$	24.9095	0.946917
$693$	−7.95470	−0.302174
$694$	−21.4904	−0.815763
$695$	0	0
$696$	−58.2578	−2.20826
$697$	−0.516787	−0.0195747
$698$	−34.8024	−1.31729
$699$	13.7499	0.520071
$700$	0	0
$701$	−14.5419	−0.549238	−0.274619	−	0.961553i	$-0.588552\pi$
$701$	−14.5419	−0.549238	−0.274619	+	0.961553i	$0.588552\pi$
$702$	−86.3504	−3.25909
$703$	5.12947	0.193462
$704$	13.0254	0.490913
$705$	0	0
$706$	82.5053	3.10513
$707$	−2.27651	−0.0856168
$708$	41.3399	1.55365
$709$	23.4457	0.880523	0.440262	−	0.897869i	$-0.354886\pi$
$709$	23.4457	0.880523	0.440262	+	0.897869i	$0.354886\pi$
$710$	0	0
$711$	−62.3760	−2.33928
$712$	−25.2623	−0.946746
$713$	−0.111256	−0.00416658
$714$	31.1881	1.16719
$715$	0	0
$716$	71.4176	2.66900
$717$	43.0405	1.60738
$718$	59.6823	2.22732
$719$	−10.8117	−0.403210	−0.201605	−	0.979467i	$-0.564616\pi$
$719$	−10.8117	−0.403210	−0.201605	+	0.979467i	$0.564616\pi$
$720$	0	0
$721$	−12.6197	−0.469982
$722$	41.9623	1.56168
$723$	−67.8414	−2.52305
$724$	3.00322	0.111614
$725$	0	0
$726$	7.42300	0.275493
$727$	22.6040	0.838336	0.419168	−	0.907909i	$-0.362322\pi$
$727$	22.6040	0.838336	0.419168	+	0.907909i	$0.362322\pi$
$728$	5.42300	0.200990
$729$	79.0954	2.92946
$730$	0	0
$731$	−19.0809	−0.705734
$732$	−155.618	−5.75179
$733$	6.12939	0.226394	0.113197	−	0.993573i	$-0.463891\pi$
$733$	6.12939	0.226394	0.113197	+	0.993573i	$0.463891\pi$
$734$	54.1342	1.99813
$735$	0	0
$736$	−1.65990	−0.0611847
$737$	3.85982	0.142178
$738$	2.19434	0.0807749
$739$	46.0760	1.69493	0.847466	−	0.530850i	$-0.178127\pi$
$739$	46.0760	1.69493	0.847466	+	0.530850i	$0.178127\pi$
$740$	0	0
$741$	−4.18273	−0.153656
$742$	−23.1947	−0.851506
$743$	39.9599	1.46598	0.732992	−	0.680237i	$-0.238123\pi$
$743$	39.9599	1.46598	0.732992	+	0.680237i	$0.238123\pi$
$744$	3.37786	0.123838
$745$	0	0
$746$	−86.4003	−3.16334
$747$	−19.2328	−0.703690
$748$	−12.7303	−0.465464
$749$	6.86412	0.250809
$750$	0	0
$751$	−8.28182	−0.302208	−0.151104	−	0.988518i	$-0.548283\pi$
$751$	−8.28182	−0.302208	−0.151104	+	0.988518i	$0.548283\pi$
$752$	−10.4693	−0.381777
$753$	19.4752	0.709715
$754$	−40.1263	−1.46131
$755$	0	0
$756$	−49.6872	−1.80711
$757$	8.45772	0.307401	0.153700	−	0.988117i	$-0.450881\pi$
$757$	8.45772	0.307401	0.153700	+	0.988117i	$0.450881\pi$
$758$	−13.1815	−0.478774
$759$	−0.833404	−0.0302507
$760$	0	0
$761$	12.6362	0.458061	0.229031	−	0.973419i	$-0.426444\pi$
$761$	12.6362	0.458061	0.229031	+	0.973419i	$0.426444\pi$
$762$	−136.689	−4.95172
$763$	−9.21061	−0.333446
$764$	−19.4321	−0.703029
$765$	0	0
$766$	−9.69581	−0.350324
$767$	9.67852	0.349471
$768$	32.7559	1.18198
$769$	−47.2491	−1.70385	−0.851924	−	0.523666i	$-0.824564\pi$
$769$	−47.2491	−1.70385	−0.851924	+	0.523666i	$0.824564\pi$
$770$	0	0
$771$	54.7053	1.97016
$772$	33.5182	1.20635
$773$	−20.3641	−0.732447	−0.366224	−	0.930527i	$-0.619349\pi$
$773$	−20.3641	−0.732447	−0.366224	+	0.930527i	$0.619349\pi$
$774$	81.0201	2.91221
$775$	0	0
$776$	12.4804	0.448021
$777$	31.5414	1.13154
$778$	−39.4173	−1.41318
$779$	0.0662054	0.00237206
$780$	0	0
$781$	0.782634	0.0280048
$782$	2.37271	0.0848480
$783$	124.968	4.46599
$784$	−0.879529	−0.0314118
$785$	0	0
$786$	−2.81567	−0.100432
$787$	45.5014	1.62195	0.810974	−	0.585082i	$-0.198938\pi$
$787$	45.5014	1.62195	0.810974	+	0.585082i	$0.198938\pi$
$788$	38.1552	1.35922
$789$	69.0267	2.45742
$790$	0	0
$791$	2.53415	0.0901038
$792$	18.3737	0.652880
$793$	−36.4333	−1.29379
$794$	61.1995	2.17189
$795$	0	0
$796$	29.7838	1.05566
$797$	−22.4059	−0.793658	−0.396829	−	0.917893i	$-0.629889\pi$
$797$	−22.4059	−0.793658	−0.396829	+	0.917893i	$0.629889\pi$
$798$	−3.99550	−0.141439
$799$	50.0125	1.76932
$800$	0	0
$801$	87.0012	3.07403
$802$	−36.1339	−1.27593
$803$	−15.1342	−0.534075
$804$	38.7075	1.36511
$805$	0	0
$806$	2.32657	0.0819500
$807$	−46.2923	−1.62957
$808$	5.25825	0.184984
$809$	1.99982	0.0703099	0.0351550	−	0.999382i	$-0.488808\pi$
$809$	1.99982	0.0703099	0.0351550	+	0.999382i	$0.488808\pi$
$810$	0	0
$811$	17.0039	0.597088	0.298544	−	0.954396i	$-0.403499\pi$
$811$	17.0039	0.597088	0.298544	+	0.954396i	$0.403499\pi$
$812$	−23.0892	−0.810273
$813$	−9.38491	−0.329143
$814$	−21.3727	−0.749114
$815$	0	0
$816$	12.2310	0.428169
$817$	2.44445	0.0855206
$818$	26.2883	0.919151
$819$	−18.6763	−0.652603
$820$	0	0
$821$	−3.91259	−0.136550	−0.0682751	−	0.997667i	$-0.521750\pi$
$821$	−3.91259	−0.136550	−0.0682751	+	0.997667i	$0.521750\pi$
$822$	−71.3942	−2.49016
$823$	4.33750	0.151196	0.0755979	−	0.997138i	$-0.475913\pi$
$823$	4.33750	0.151196	0.0755979	+	0.997138i	$0.475913\pi$
$824$	29.1488	1.01545
$825$	0	0
$826$	9.24529	0.321685
$827$	36.3354	1.26350	0.631752	−	0.775170i	$-0.282336\pi$
$827$	36.3354	1.26350	0.631752	+	0.775170i	$0.282336\pi$
$828$	−6.06885	−0.210907
$829$	−33.3267	−1.15748	−0.578742	−	0.815511i	$-0.696456\pi$
$829$	−33.3267	−1.15748	−0.578742	+	0.815511i	$0.696456\pi$
$830$	0	0
$831$	−70.7095	−2.45289
$832$	30.5815	1.06022
$833$	4.20155	0.145575
$834$	−15.3473	−0.531432
$835$	0	0
$836$	1.63087	0.0564048
$837$	−7.24580	−0.250451
$838$	−23.3796	−0.807633
$839$	−22.7206	−0.784401	−0.392200	−	0.919880i	$-0.628286\pi$
$839$	−22.7206	−0.784401	−0.392200	+	0.919880i	$0.628286\pi$
$840$	0	0
$841$	29.0716	1.00247
$842$	87.1350	3.00287
$843$	−31.3604	−1.08011
$844$	−32.1900	−1.10802
$845$	0	0
$846$	−212.360	−7.30107
$847$	1.00000	0.0343604
$848$	−9.09621	−0.312365
$849$	54.9430	1.88564
$850$	0	0
$851$	2.39958	0.0822567
$852$	7.84849	0.268885
$853$	13.3339	0.456545	0.228273	−	0.973597i	$-0.426692\pi$
$853$	13.3339	0.456545	0.228273	+	0.973597i	$0.426692\pi$
$854$	−34.8025	−1.19092
$855$	0	0
$856$	−15.8547	−0.541901
$857$	−6.04170	−0.206380	−0.103190	−	0.994662i	$-0.532905\pi$
$857$	−6.04170	−0.206380	−0.103190	+	0.994662i	$0.532905\pi$
$858$	17.4280	0.594982
$859$	−28.1553	−0.960645	−0.480323	−	0.877092i	$-0.659480\pi$
$859$	−28.1553	−0.960645	−0.480323	+	0.877092i	$0.659480\pi$
$860$	0	0
$861$	0.407100	0.0138739
$862$	−9.96812	−0.339516
$863$	−38.0102	−1.29388	−0.646941	−	0.762540i	$-0.723952\pi$
$863$	−38.0102	−1.29388	−0.646941	+	0.762540i	$0.723952\pi$
$864$	−108.104	−3.67778
$865$	0	0
$866$	27.9176	0.948677
$867$	−2.16145	−0.0734068
$868$	1.33874	0.0454398
$869$	7.84140	0.266001
$870$	0	0
$871$	9.06223	0.307062
$872$	21.2746	0.720447
$873$	−42.9814	−1.45470
$874$	−0.303967	−0.0102819
$875$	0	0
$876$	−151.771	−5.12785
$877$	14.5034	0.489745	0.244873	−	0.969555i	$-0.421254\pi$
$877$	14.5034	0.489745	0.244873	+	0.969555i	$0.421254\pi$
$878$	−81.0894	−2.73663
$879$	66.1903	2.23254
$880$	0	0
$881$	30.0664	1.01296	0.506481	−	0.862251i	$-0.330946\pi$
$881$	30.0664	1.01296	0.506481	+	0.862251i	$0.330946\pi$
$882$	−17.8403	−0.600715
$883$	−3.93976	−0.132583	−0.0662917	−	0.997800i	$-0.521117\pi$
$883$	−3.93976	−0.132583	−0.0662917	+	0.997800i	$0.521117\pi$
$884$	−29.8886	−1.00526
$885$	0	0
$886$	37.2305	1.25078
$887$	21.6934	0.728393	0.364197	−	0.931322i	$-0.381344\pi$
$887$	21.6934	0.728393	0.364197	+	0.931322i	$0.381344\pi$
$888$	−72.8539	−2.44482
$889$	−18.4142	−0.617594
$890$	0	0
$891$	−30.4131	−1.01888
$892$	26.4416	0.885332
$893$	−6.40709	−0.214405
$894$	−19.1024	−0.638880
$895$	0	0
$896$	16.0284	0.535470
$897$	−1.95670	−0.0653322
$898$	29.1671	0.973317
$899$	−3.36706	−0.112298
$900$	0	0
$901$	43.4530	1.44763
$902$	−0.275855	−0.00918497
$903$	15.0311	0.500203
$904$	−5.85334	−0.194679
$905$	0	0
$906$	−11.0732	−0.367882
$907$	−27.9746	−0.928882	−0.464441	−	0.885604i	$-0.653745\pi$
$907$	−27.9746	−0.928882	−0.464441	+	0.885604i	$0.653745\pi$
$908$	−38.7841	−1.28709
$909$	−18.1089	−0.600635
$910$	0	0
$911$	12.6888	0.420397	0.210199	−	0.977659i	$-0.432589\pi$
$911$	12.6888	0.420397	0.210199	+	0.977659i	$0.432589\pi$
$912$	−1.56690	−0.0518854
$913$	2.41779	0.0800171
$914$	45.7128	1.51205
$915$	0	0
$916$	−60.1669	−1.98797
$917$	−0.379317	−0.0125261
$918$	154.528	5.10018
$919$	41.3180	1.36296	0.681478	−	0.731839i	$-0.261338\pi$
$919$	41.3180	1.36296	0.681478	+	0.731839i	$0.261338\pi$
$920$	0	0
$921$	34.6536	1.14187
$922$	1.69531	0.0558320
$923$	1.83750	0.0604819
$924$	10.0283	0.329907
$925$	0	0
$926$	−15.4842	−0.508842
$927$	−100.386	−3.29711
$928$	−50.2352	−1.64905
$929$	−13.5208	−0.443604	−0.221802	−	0.975092i	$-0.571194\pi$
$929$	−13.5208	−0.443604	−0.221802	+	0.975092i	$0.571194\pi$
$930$	0	0
$931$	−0.538260	−0.0176408
$932$	−12.5872	−0.412307
$933$	73.5980	2.40949
$934$	10.7856	0.352915
$935$	0	0
$936$	43.1383	1.41002
$937$	−45.0657	−1.47223	−0.736116	−	0.676855i	$-0.763342\pi$
$937$	−45.0657	−1.47223	−0.736116	+	0.676855i	$0.763342\pi$
$938$	8.65659	0.282648
$939$	48.7885	1.59215
$940$	0	0
$941$	−18.0399	−0.588085	−0.294043	−	0.955792i	$-0.595001\pi$
$941$	−18.0399	−0.588085	−0.294043	+	0.955792i	$0.595001\pi$
$942$	12.5928	0.410296
$943$	0.0309711	0.00100856
$944$	3.62570	0.118006
$945$	0	0
$946$	−10.1852	−0.331149
$947$	−28.8869	−0.938697	−0.469348	−	0.883013i	$-0.655511\pi$
$947$	−28.8869	−0.938697	−0.469348	+	0.883013i	$0.655511\pi$
$948$	78.6360	2.55398
$949$	−35.5327	−1.15344
$950$	0	0
$951$	−81.1257	−2.63068
$952$	−9.70470	−0.314531
$953$	5.35694	0.173528	0.0867642	−	0.996229i	$-0.472347\pi$
$953$	5.35694	0.173528	0.0867642	+	0.996229i	$0.472347\pi$
$954$	−184.507	−5.97364
$955$	0	0
$956$	−39.4008	−1.27431
$957$	−25.2221	−0.815316
$958$	−33.4701	−1.08137
$959$	−9.61797	−0.310580
$960$	0	0
$961$	−30.8048	−0.993702
$962$	−50.1796	−1.61786
$963$	54.6020	1.75952
$964$	62.1043	2.00025
$965$	0	0
$966$	−1.86911	−0.0601377
$967$	−38.6479	−1.24283	−0.621417	−	0.783480i	$-0.713443\pi$
$967$	−38.6479	−1.24283	−0.621417	+	0.783480i	$0.713443\pi$
$968$	−2.30979	−0.0742394
$969$	7.48518	0.240459
$970$	0	0
$971$	−3.53878	−0.113565	−0.0567824	−	0.998387i	$-0.518084\pi$
$971$	−3.53878	−0.113565	−0.0567824	+	0.998387i	$0.518084\pi$
$972$	−155.930	−5.00146
$973$	−2.06753	−0.0662819
$974$	53.8278	1.72475
$975$	0	0
$976$	−13.6484	−0.436874
$977$	−9.90538	−0.316901	−0.158451	−	0.987367i	$-0.550650\pi$
$977$	−9.90538	−0.316901	−0.158451	+	0.987367i	$0.550650\pi$
$978$	−107.638	−3.44189
$979$	−10.9371	−0.349551
$980$	0	0
$981$	−73.2676	−2.33925
$982$	9.68579	0.309086
$983$	13.2231	0.421753	0.210876	−	0.977513i	$-0.432368\pi$
$983$	13.2231	0.421753	0.210876	+	0.977513i	$0.432368\pi$
$984$	−0.940316	−0.0299762
$985$	0	0
$986$	71.8077	2.28682
$987$	−39.3975	−1.25404
$988$	3.82902	0.121817
$989$	1.14352	0.0363620
$990$	0	0
$991$	32.2644	1.02491	0.512456	−	0.858713i	$-0.328736\pi$
$991$	32.2644	1.02491	0.512456	+	0.858713i	$0.328736\pi$
$992$	2.91270	0.0924782
$993$	26.8998	0.853638
$994$	1.75525	0.0556730
$995$	0	0
$996$	24.2463	0.768274
$997$	24.4517	0.774393	0.387197	−	0.921997i	$-0.373443\pi$
$997$	24.4517	0.774393	0.387197	+	0.921997i	$0.373443\pi$
$998$	46.8691	1.48362
$999$	156.278	4.94441

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.a.y.1.1		6
5.2	odd	4		385.2.b.c.309.1	✓	12
5.3	odd	4		385.2.b.c.309.12	yes	12
5.4	even	2		1925.2.a.z.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.b.c.309.1	✓	12	5.2	odd	4
385.2.b.c.309.12	yes	12	5.3	odd	4
1925.2.a.y.1.1		6	1.1	even	1	trivial
1925.2.a.z.1.6		6	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:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1925.a (trivial)

\(f(q)\)	\(=\)	\(q-2.24274 q^{2} -3.30979 q^{3} +3.02989 q^{4} +7.42300 q^{6} +1.00000 q^{7} -2.30979 q^{8} +7.95470 q^{9} +O(q^{10})\)	\(q-2.24274 q^{2} -3.30979 q^{3} +3.02989 q^{4} +7.42300 q^{6} +1.00000 q^{7} -2.30979 q^{8} +7.95470 q^{9} -1.00000 q^{11} -10.0283 q^{12} -2.34784 q^{13} -2.24274 q^{14} -0.879529 q^{16} +4.20155 q^{17} -17.8403 q^{18} -0.538260 q^{19} -3.30979 q^{21} +2.24274 q^{22} -0.251800 q^{23} +7.64491 q^{24} +5.26559 q^{26} -16.3990 q^{27} +3.02989 q^{28} -7.62047 q^{29} +0.441844 q^{31} +6.59213 q^{32} +3.30979 q^{33} -9.42300 q^{34} +24.1019 q^{36} -9.52973 q^{37} +1.20718 q^{38} +7.77084 q^{39} -0.122999 q^{41} +7.42300 q^{42} -4.54140 q^{43} -3.02989 q^{44} +0.564722 q^{46} +11.9033 q^{47} +2.91105 q^{48} +1.00000 q^{49} -13.9062 q^{51} -7.11369 q^{52} +10.3421 q^{53} +36.7787 q^{54} -2.30979 q^{56} +1.78153 q^{57} +17.0908 q^{58} -4.12232 q^{59} +15.5178 q^{61} -0.990943 q^{62} +7.95470 q^{63} -13.0254 q^{64} -7.42300 q^{66} -3.85982 q^{67} +12.7303 q^{68} +0.833404 q^{69} -0.782634 q^{71} -18.3737 q^{72} +15.1342 q^{73} +21.3727 q^{74} -1.63087 q^{76} -1.00000 q^{77} -17.4280 q^{78} -7.84140 q^{79} +30.4131 q^{81} +0.275855 q^{82} -2.41779 q^{83} -10.0283 q^{84} +10.1852 q^{86} +25.2221 q^{87} +2.30979 q^{88} +10.9371 q^{89} -2.34784 q^{91} -0.762927 q^{92} -1.46241 q^{93} -26.6961 q^{94} -21.8186 q^{96} -5.40327 q^{97} -2.24274 q^{98} -7.95470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 2 * q^4 - 2 * q^6 + 6 * q^7 + 10 * q^9	\( 6 q - 6 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{7} + 10 q^{9} - 6 q^{11} - 12 q^{12} - 6 q^{13} - 6 q^{16} + 2 q^{17} - 14 q^{18} - 2 q^{19} - 6 q^{21} - 8 q^{23} + 22 q^{24} - 24 q^{27} + 2 q^{28} - 8 q^{29} + 4 q^{31} - 10 q^{32} + 6 q^{33} - 10 q^{34} + 26 q^{36} - 22 q^{37} + 10 q^{38} - 8 q^{39} + 4 q^{41} - 2 q^{42} - 30 q^{43} - 2 q^{44} - 8 q^{46} - 16 q^{47} + 8 q^{48} + 6 q^{49} - 4 q^{51} - 22 q^{52} - 6 q^{53} + 38 q^{54} - 18 q^{57} + 14 q^{58} + 14 q^{59} + 12 q^{61} + 14 q^{62} + 10 q^{63} - 22 q^{64} + 2 q^{66} - 46 q^{67} + 20 q^{68} + 12 q^{69} + 4 q^{71} - 32 q^{72} + 4 q^{73} - 8 q^{74} - 8 q^{76} - 6 q^{77} - 24 q^{78} - 8 q^{79} + 26 q^{81} + 10 q^{82} - 14 q^{83} - 12 q^{84} + 12 q^{86} + 2 q^{87} + 8 q^{89} - 6 q^{91} - 18 q^{92} - 12 q^{93} - 10 q^{94} - 16 q^{96} - 42 q^{97} - 10 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + 2 * q^4 - 2 * q^6 + 6 * q^7 + 10 * q^9 - 6 * q^11 - 12 * q^12 - 6 * q^13 - 6 * q^16 + 2 * q^17 - 14 * q^18 - 2 * q^19 - 6 * q^21 - 8 * q^23 + 22 * q^24 - 24 * q^27 + 2 * q^28 - 8 * q^29 + 4 * q^31 - 10 * q^32 + 6 * q^33 - 10 * q^34 + 26 * q^36 - 22 * q^37 + 10 * q^38 - 8 * q^39 + 4 * q^41 - 2 * q^42 - 30 * q^43 - 2 * q^44 - 8 * q^46 - 16 * q^47 + 8 * q^48 + 6 * q^49 - 4 * q^51 - 22 * q^52 - 6 * q^53 + 38 * q^54 - 18 * q^57 + 14 * q^58 + 14 * q^59 + 12 * q^61 + 14 * q^62 + 10 * q^63 - 22 * q^64 + 2 * q^66 - 46 * q^67 + 20 * q^68 + 12 * q^69 + 4 * q^71 - 32 * q^72 + 4 * q^73 - 8 * q^74 - 8 * q^76 - 6 * q^77 - 24 * q^78 - 8 * q^79 + 26 * q^81 + 10 * q^82 - 14 * q^83 - 12 * q^84 + 12 * q^86 + 2 * q^87 + 8 * q^89 - 6 * q^91 - 18 * q^92 - 12 * q^93 - 10 * q^94 - 16 * q^96 - 42 * q^97 - 10 * q^99

Introduction

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