LMFDB - Embedded newform 1925.2.a.ba.1.5

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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 13x^{5} + 10x^{4} + 47x^{3} - 25x^{2} - 35x + 20 $ x^7 - x^6 - 13x^5 + 10x^4 + 47x^3 - 25x^2 - 35*x + 20
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.06204$ 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)

$f(q)$	$=$	$q+1.06204 q^{2} +1.17047 q^{3} -0.872076 q^{4} +1.24309 q^{6} -1.00000 q^{7} -3.05025 q^{8} -1.63000 q^{9} +O(q^{10})$	\(q+1.06204 q^{2} +1.17047 q^{3} -0.872076 q^{4} +1.24309 q^{6} -1.00000 q^{7} -3.05025 q^{8} -1.63000 q^{9} -1.00000 q^{11} -1.02074 q^{12} +6.82997 q^{13} -1.06204 q^{14} -1.49533 q^{16} -2.26325 q^{17} -1.73112 q^{18} +6.41641 q^{19} -1.17047 q^{21} -1.06204 q^{22} +9.35726 q^{23} -3.57024 q^{24} +7.25368 q^{26} -5.41928 q^{27} +0.872076 q^{28} +4.96420 q^{29} -0.856196 q^{31} +4.51241 q^{32} -1.17047 q^{33} -2.40366 q^{34} +1.42148 q^{36} +6.61436 q^{37} +6.81447 q^{38} +7.99428 q^{39} +0.527296 q^{41} -1.24309 q^{42} +4.95746 q^{43} +0.872076 q^{44} +9.93777 q^{46} +1.24309 q^{47} -1.75024 q^{48} +1.00000 q^{49} -2.64907 q^{51} -5.95625 q^{52} -3.97763 q^{53} -5.75548 q^{54} +3.05025 q^{56} +7.51023 q^{57} +5.27217 q^{58} -11.6526 q^{59} +3.40640 q^{61} -0.909312 q^{62} +1.63000 q^{63} +7.78301 q^{64} -1.24309 q^{66} +5.32284 q^{67} +1.97373 q^{68} +10.9524 q^{69} +4.43143 q^{71} +4.97190 q^{72} +1.06974 q^{73} +7.02470 q^{74} -5.59560 q^{76} +1.00000 q^{77} +8.49023 q^{78} -0.966826 q^{79} -1.45313 q^{81} +0.560008 q^{82} -6.07331 q^{83} +1.02074 q^{84} +5.26501 q^{86} +5.81046 q^{87} +3.05025 q^{88} -3.55617 q^{89} -6.82997 q^{91} -8.16024 q^{92} -1.00215 q^{93} +1.32020 q^{94} +5.28165 q^{96} -16.8926 q^{97} +1.06204 q^{98} +1.63000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 13 q^{4} - q^{6} - 7 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) $ 7 * q - q^2 + 13 * q^4 - q^6 - 7 * q^7 - 6 * q^8 + 13 * q^9	$ 7 q - q^{2} + 13 q^{4} - q^{6} - 7 q^{7} - 6 q^{8} + 13 q^{9} - 7 q^{11} + 21 q^{12} + 3 q^{13} + q^{14} + 29 q^{16} - 14 q^{18} + 18 q^{19} + q^{22} + 7 q^{23} - 22 q^{24} + 13 q^{26} - 6 q^{27} - 13 q^{28} - 2 q^{29} + 24 q^{31} - 33 q^{32} + 33 q^{34} + 44 q^{36} - 21 q^{37} - 9 q^{38} + 10 q^{39} - 10 q^{41} + q^{42} + 2 q^{43} - 13 q^{44} + 3 q^{46} - q^{47} + 77 q^{48} + 7 q^{49} + 29 q^{51} - 9 q^{52} + 11 q^{53} - 47 q^{54} + 6 q^{56} - 7 q^{57} + 33 q^{58} + q^{59} + 28 q^{61} - 16 q^{62} - 13 q^{63} + 48 q^{64} + q^{66} + 46 q^{68} + 33 q^{69} - 10 q^{71} - 36 q^{72} + 11 q^{73} - 6 q^{74} + 20 q^{76} + 7 q^{77} + 31 q^{78} + 19 q^{79} + 27 q^{81} - 8 q^{82} + 19 q^{83} - 21 q^{84} + 55 q^{86} + 12 q^{87} + 6 q^{88} - 2 q^{89} - 3 q^{91} - 40 q^{92} - 58 q^{93} + 21 q^{94} - 23 q^{96} - 8 q^{97} - q^{98} - 13 q^{99}+O(q^{100}) $ 7 * q - q^2 + 13 * q^4 - q^6 - 7 * q^7 - 6 * q^8 + 13 * q^9 - 7 * q^11 + 21 * q^12 + 3 * q^13 + q^14 + 29 * q^16 - 14 * q^18 + 18 * q^19 + q^22 + 7 * q^23 - 22 * q^24 + 13 * q^26 - 6 * q^27 - 13 * q^28 - 2 * q^29 + 24 * q^31 - 33 * q^32 + 33 * q^34 + 44 * q^36 - 21 * q^37 - 9 * q^38 + 10 * q^39 - 10 * q^41 + q^42 + 2 * q^43 - 13 * q^44 + 3 * q^46 - q^47 + 77 * q^48 + 7 * q^49 + 29 * q^51 - 9 * q^52 + 11 * q^53 - 47 * q^54 + 6 * q^56 - 7 * q^57 + 33 * q^58 + q^59 + 28 * q^61 - 16 * q^62 - 13 * q^63 + 48 * q^64 + q^66 + 46 * q^68 + 33 * q^69 - 10 * q^71 - 36 * q^72 + 11 * q^73 - 6 * q^74 + 20 * q^76 + 7 * q^77 + 31 * q^78 + 19 * q^79 + 27 * q^81 - 8 * q^82 + 19 * q^83 - 21 * q^84 + 55 * q^86 + 12 * q^87 + 6 * q^88 - 2 * q^89 - 3 * q^91 - 40 * q^92 - 58 * q^93 + 21 * q^94 - 23 * q^96 - 8 * q^97 - q^98 - 13 * 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 $

$2$	1.06204	0.750974	0.375487	−	0.926828i	$-0.377475\pi$
$2$	1.06204	0.750974	0.375487	+	0.926828i	$0.377475\pi$
$3$	1.17047	0.675772	0.337886	−	0.941187i	$-0.390288\pi$
$3$	1.17047	0.675772	0.337886	+	0.941187i	$0.390288\pi$
$4$	−0.872076	−0.436038
$5$	0	0
$5$	0	0
$6$	1.24309	0.507487
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.05025	−1.07843
$9$	−1.63000	−0.543332
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.02074	−0.294662
$13$	6.82997	1.89429	0.947146	−	0.320803i	$-0.103953\pi$
$13$	6.82997	1.89429	0.947146	+	0.320803i	$0.103953\pi$
$14$	−1.06204	−0.283841
$15$	0	0
$16$	−1.49533	−0.373833
$17$	−2.26325	−0.548919	−0.274460	−	0.961599i	$-0.588499\pi$
$17$	−2.26325	−0.548919	−0.274460	+	0.961599i	$0.588499\pi$
$18$	−1.73112	−0.408028
$19$	6.41641	1.47203	0.736013	−	0.676968i	$-0.236706\pi$
$19$	6.41641	1.47203	0.736013	+	0.676968i	$0.236706\pi$
$20$	0	0
$21$	−1.17047	−0.255418
$22$	−1.06204	−0.226427
$23$	9.35726	1.95112	0.975562	−	0.219724i	$-0.0705157\pi$
$23$	9.35726	1.95112	0.975562	+	0.219724i	$0.0705157\pi$
$24$	−3.57024	−0.728771
$25$	0	0
$26$	7.25368	1.42256
$27$	−5.41928	−1.04294
$28$	0.872076	0.164807
$29$	4.96420	0.921829	0.460914	−	0.887445i	$-0.347522\pi$
$29$	4.96420	0.921829	0.460914	+	0.887445i	$0.347522\pi$
$30$	0	0
$31$	−0.856196	−0.153777	−0.0768887	−	0.997040i	$-0.524499\pi$
$31$	−0.856196	−0.153777	−0.0768887	+	0.997040i	$0.524499\pi$
$32$	4.51241	0.797689
$33$	−1.17047	−0.203753
$34$	−2.40366	−0.412224
$35$	0	0
$36$	1.42148	0.236913
$37$	6.61436	1.08739	0.543697	−	0.839281i	$-0.317024\pi$
$37$	6.61436	1.08739	0.543697	+	0.839281i	$0.317024\pi$
$38$	6.81447	1.10545
$39$	7.99428	1.28011
$40$	0	0
$41$	0.527296	0.0823498	0.0411749	−	0.999152i	$-0.486890\pi$
$41$	0.527296	0.0823498	0.0411749	+	0.999152i	$0.486890\pi$
$42$	−1.24309	−0.191812
$43$	4.95746	0.756005	0.378003	−	0.925804i	$-0.376611\pi$
$43$	4.95746	0.756005	0.378003	+	0.925804i	$0.376611\pi$
$44$	0.872076	0.131470
$45$	0	0
$46$	9.93777	1.46524
$47$	1.24309	0.181323	0.0906613	−	0.995882i	$-0.471102\pi$
$47$	1.24309	0.181323	0.0906613	+	0.995882i	$0.471102\pi$
$48$	−1.75024	−0.252626
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.64907	−0.370945
$52$	−5.95625	−0.825983
$53$	−3.97763	−0.546369	−0.273185	−	0.961962i	$-0.588077\pi$
$53$	−3.97763	−0.546369	−0.273185	+	0.961962i	$0.588077\pi$
$54$	−5.75548	−0.783222
$55$	0	0
$56$	3.05025	0.407607
$57$	7.51023	0.994754
$58$	5.27217	0.692269
$59$	−11.6526	−1.51704	−0.758518	−	0.651652i	$-0.774076\pi$
$59$	−11.6526	−1.51704	−0.758518	+	0.651652i	$0.774076\pi$
$60$	0	0
$61$	3.40640	0.436145	0.218072	−	0.975933i	$-0.430023\pi$
$61$	3.40640	0.436145	0.218072	+	0.975933i	$0.430023\pi$
$62$	−0.909312	−0.115483
$63$	1.63000	0.205360
$64$	7.78301	0.972876
$65$	0	0
$66$	−1.24309	−0.153013
$67$	5.32284	0.650288	0.325144	−	0.945665i	$-0.394587\pi$
$67$	5.32284	0.650288	0.325144	+	0.945665i	$0.394587\pi$
$68$	1.97373	0.239350
$69$	10.9524	1.31852
$70$	0	0
$71$	4.43143	0.525914	0.262957	−	0.964808i	$-0.415302\pi$
$71$	4.43143	0.525914	0.262957	+	0.964808i	$0.415302\pi$
$72$	4.97190	0.585944
$73$	1.06974	0.125204	0.0626019	−	0.998039i	$-0.480060\pi$
$73$	1.06974	0.125204	0.0626019	+	0.998039i	$0.480060\pi$
$74$	7.02470	0.816605
$75$	0	0
$76$	−5.59560	−0.641859
$77$	1.00000	0.113961
$78$	8.49023	0.961329
$79$	−0.966826	−0.108776	−0.0543882	−	0.998520i	$-0.517321\pi$
$79$	−0.966826	−0.108776	−0.0543882	+	0.998520i	$0.517321\pi$
$80$	0	0
$81$	−1.45313	−0.161459
$82$	0.560008	0.0618426
$83$	−6.07331	−0.666633	−0.333316	−	0.942815i	$-0.608168\pi$
$83$	−6.07331	−0.666633	−0.333316	+	0.942815i	$0.608168\pi$
$84$	1.02074	0.111372
$85$	0	0
$86$	5.26501	0.567740
$87$	5.81046	0.622946
$88$	3.05025	0.325158
$89$	−3.55617	−0.376954	−0.188477	−	0.982078i	$-0.560355\pi$
$89$	−3.55617	−0.376954	−0.188477	+	0.982078i	$0.560355\pi$
$90$	0	0
$91$	−6.82997	−0.715975
$92$	−8.16024	−0.850764
$93$	−1.00215	−0.103918
$94$	1.32020	0.136169
$95$	0	0
$96$	5.28165	0.539056
$97$	−16.8926	−1.71518	−0.857592	−	0.514331i	$-0.828040\pi$
$97$	−16.8926	−1.71518	−0.857592	+	0.514331i	$0.828040\pi$
$98$	1.06204	0.107282
$99$	1.63000	0.163821
$100$	0	0
$101$	8.23058	0.818974	0.409487	−	0.912316i	$-0.365708\pi$
$101$	8.23058	0.818974	0.409487	+	0.912316i	$0.365708\pi$
$102$	−2.81342	−0.278570
$103$	17.1467	1.68951	0.844757	−	0.535150i	$-0.179745\pi$
$103$	17.1467	1.68951	0.844757	+	0.535150i	$0.179745\pi$
$104$	−20.8331	−2.04286
$105$	0	0
$106$	−4.22439	−0.410309
$107$	−17.4535	−1.68729	−0.843645	−	0.536902i	$-0.819594\pi$
$107$	−17.4535	−1.68729	−0.843645	+	0.536902i	$0.819594\pi$
$108$	4.72602	0.454762
$109$	9.79014	0.937725	0.468863	−	0.883271i	$-0.344664\pi$
$109$	9.79014	0.937725	0.468863	+	0.883271i	$0.344664\pi$
$110$	0	0
$111$	7.74193	0.734831
$112$	1.49533	0.141296
$113$	−16.6228	−1.56374	−0.781872	−	0.623439i	$-0.785735\pi$
$113$	−16.6228	−1.56374	−0.781872	+	0.623439i	$0.785735\pi$
$114$	7.97614	0.747034
$115$	0	0
$116$	−4.32916	−0.401952
$117$	−11.1328	−1.02923
$118$	−12.3755	−1.13925
$119$	2.26325	0.207472
$120$	0	0
$121$	1.00000	0.0909091
$122$	3.61772	0.327533
$123$	0.617186	0.0556497
$124$	0.746668	0.0670528
$125$	0	0
$126$	1.73112	0.154220
$127$	−16.1524	−1.43330	−0.716648	−	0.697435i	$-0.754325\pi$
$127$	−16.1524	−1.43330	−0.716648	+	0.697435i	$0.754325\pi$
$128$	−0.758967	−0.0670838
$129$	5.80257	0.510888
$130$	0	0
$131$	9.91338	0.866136	0.433068	−	0.901361i	$-0.357431\pi$
$131$	9.91338	0.866136	0.433068	+	0.901361i	$0.357431\pi$
$132$	1.02074	0.0888441
$133$	−6.41641	−0.556373
$134$	5.65305	0.488349
$135$	0	0
$136$	6.90349	0.591970
$137$	4.06402	0.347213	0.173606	−	0.984815i	$-0.444458\pi$
$137$	4.06402	0.347213	0.173606	+	0.984815i	$0.444458\pi$
$138$	11.6319	0.990171
$139$	14.0507	1.19176	0.595881	−	0.803073i	$-0.296803\pi$
$139$	14.0507	1.19176	0.595881	+	0.803073i	$0.296803\pi$
$140$	0	0
$141$	1.45500	0.122533
$142$	4.70634	0.394947
$143$	−6.82997	−0.571150
$144$	2.43738	0.203115
$145$	0	0
$146$	1.13611	0.0940249
$147$	1.17047	0.0965389
$148$	−5.76823	−0.474145
$149$	−4.68802	−0.384058	−0.192029	−	0.981389i	$-0.561507\pi$
$149$	−4.68802	−0.384058	−0.192029	+	0.981389i	$0.561507\pi$
$150$	0	0
$151$	4.15871	0.338431	0.169215	−	0.985579i	$-0.445877\pi$
$151$	4.15871	0.338431	0.169215	+	0.985579i	$0.445877\pi$
$152$	−19.5717	−1.58747
$153$	3.68909	0.298245
$154$	1.06204	0.0855814
$155$	0	0
$156$	−6.97162	−0.558177
$157$	9.50165	0.758314	0.379157	−	0.925332i	$-0.376214\pi$
$157$	9.50165	0.758314	0.379157	+	0.925332i	$0.376214\pi$
$158$	−1.02681	−0.0816883
$159$	−4.65570	−0.369221
$160$	0	0
$161$	−9.35726	−0.737456
$162$	−1.54328	−0.121251
$163$	−23.3193	−1.82651	−0.913256	−	0.407386i	$-0.866440\pi$
$163$	−23.3193	−1.82651	−0.913256	+	0.407386i	$0.866440\pi$
$164$	−0.459842	−0.0359077
$165$	0	0
$166$	−6.45009	−0.500624
$167$	−12.6057	−0.975458	−0.487729	−	0.872995i	$-0.662175\pi$
$167$	−12.6057	−0.975458	−0.487729	+	0.872995i	$0.662175\pi$
$168$	3.57024	0.275450
$169$	33.6484	2.58834
$170$	0	0
$171$	−10.4587	−0.799798
$172$	−4.32328	−0.329647
$173$	9.26682	0.704543	0.352272	−	0.935898i	$-0.385409\pi$
$173$	9.26682	0.704543	0.352272	+	0.935898i	$0.385409\pi$
$174$	6.17092	0.467817
$175$	0	0
$176$	1.49533	0.112715
$177$	−13.6390	−1.02517
$178$	−3.77679	−0.283082
$179$	−15.5397	−1.16149	−0.580747	−	0.814084i	$-0.697239\pi$
$179$	−15.5397	−1.16149	−0.580747	+	0.814084i	$0.697239\pi$
$180$	0	0
$181$	−3.33690	−0.248030	−0.124015	−	0.992280i	$-0.539577\pi$
$181$	−3.33690	−0.248030	−0.124015	+	0.992280i	$0.539577\pi$
$182$	−7.25368	−0.537679
$183$	3.98710	0.294735
$184$	−28.5420	−2.10415
$185$	0	0
$186$	−1.06432	−0.0780401
$187$	2.26325	0.165505
$188$	−1.08406	−0.0790636
$189$	5.41928	0.394195
$190$	0	0
$191$	20.9541	1.51618	0.758092	−	0.652147i	$-0.226132\pi$
$191$	20.9541	1.51618	0.758092	+	0.652147i	$0.226132\pi$
$192$	9.10979	0.657443
$193$	−13.7901	−0.992636	−0.496318	−	0.868141i	$-0.665315\pi$
$193$	−13.7901	−0.992636	−0.496318	+	0.868141i	$0.665315\pi$
$194$	−17.9406	−1.28806
$195$	0	0
$196$	−0.872076	−0.0622911
$197$	11.4424	0.815234	0.407617	−	0.913153i	$-0.366360\pi$
$197$	11.4424	0.815234	0.407617	+	0.913153i	$0.366360\pi$
$198$	1.73112	0.123025
$199$	21.8077	1.54591	0.772955	−	0.634461i	$-0.218778\pi$
$199$	21.8077	1.54591	0.772955	+	0.634461i	$0.218778\pi$
$200$	0	0
$201$	6.23023	0.439447
$202$	8.74119	0.615028
$203$	−4.96420	−0.348418
$204$	2.31019	0.161746
$205$	0	0
$206$	18.2104	1.26878
$207$	−15.2523	−1.06011
$208$	−10.2131	−0.708148
$209$	−6.41641	−0.443832
$210$	0	0
$211$	23.1975	1.59698	0.798490	−	0.602008i	$-0.205632\pi$
$211$	23.1975	1.59698	0.798490	+	0.602008i	$0.205632\pi$
$212$	3.46879	0.238238
$213$	5.18686	0.355398
$214$	−18.5362	−1.26711
$215$	0	0
$216$	16.5302	1.12474
$217$	0.856196	0.0581224
$218$	10.3975	0.704207
$219$	1.25210	0.0846093
$220$	0	0
$221$	−15.4579	−1.03981
$222$	8.22222	0.551839
$223$	2.36026	0.158055	0.0790274	−	0.996872i	$-0.474819\pi$
$223$	2.36026	0.158055	0.0790274	+	0.996872i	$0.474819\pi$
$224$	−4.51241	−0.301498
$225$	0	0
$226$	−17.6541	−1.17433
$227$	28.4393	1.88758	0.943790	−	0.330545i	$-0.107232\pi$
$227$	28.4393	1.88758	0.943790	+	0.330545i	$0.107232\pi$
$228$	−6.54949	−0.433751
$229$	−18.9280	−1.25080	−0.625400	−	0.780305i	$-0.715064\pi$
$229$	−18.9280	−1.25080	−0.625400	+	0.780305i	$0.715064\pi$
$230$	0	0
$231$	1.17047	0.0770114
$232$	−15.1421	−0.994125
$233$	17.8126	1.16694	0.583470	−	0.812135i	$-0.301695\pi$
$233$	17.8126	1.16694	0.583470	+	0.812135i	$0.301695\pi$
$234$	−11.8235	−0.772924
$235$	0	0
$236$	10.1619	0.661485
$237$	−1.13164	−0.0735081
$238$	2.40366	0.155806
$239$	−13.3135	−0.861179	−0.430589	−	0.902548i	$-0.641694\pi$
$239$	−13.3135	−0.861179	−0.430589	+	0.902548i	$0.641694\pi$
$240$	0	0
$241$	8.43397	0.543280	0.271640	−	0.962399i	$-0.412434\pi$
$241$	8.43397	0.543280	0.271640	+	0.962399i	$0.412434\pi$
$242$	1.06204	0.0682704
$243$	14.5570	0.933831
$244$	−2.97064	−0.190176
$245$	0	0
$246$	0.655474	0.0417915
$247$	43.8239	2.78845
$248$	2.61161	0.165838
$249$	−7.10864	−0.450492
$250$	0	0
$251$	13.9949	0.883350	0.441675	−	0.897175i	$-0.354384\pi$
$251$	13.9949	0.883350	0.441675	+	0.897175i	$0.354384\pi$
$252$	−1.42148	−0.0895448
$253$	−9.35726	−0.588286
$254$	−17.1545	−1.07637
$255$	0	0
$256$	−16.3721	−1.02325
$257$	7.16272	0.446798	0.223399	−	0.974727i	$-0.428285\pi$
$257$	7.16272	0.446798	0.223399	+	0.974727i	$0.428285\pi$
$258$	6.16255	0.383663
$259$	−6.61436	−0.410996
$260$	0	0
$261$	−8.09162	−0.500859
$262$	10.5284	0.650446
$263$	7.01651	0.432657	0.216328	−	0.976321i	$-0.430592\pi$
$263$	7.01651	0.432657	0.216328	+	0.976321i	$0.430592\pi$
$264$	3.57024	0.219733
$265$	0	0
$266$	−6.81447	−0.417822
$267$	−4.16240	−0.254735
$268$	−4.64192	−0.283550
$269$	−4.13429	−0.252072	−0.126036	−	0.992026i	$-0.540226\pi$
$269$	−4.13429	−0.252072	−0.126036	+	0.992026i	$0.540226\pi$
$270$	0	0
$271$	−12.8230	−0.778939	−0.389469	−	0.921039i	$-0.627342\pi$
$271$	−12.8230	−0.778939	−0.389469	+	0.921039i	$0.627342\pi$
$272$	3.38431	0.205204
$273$	−7.99428	−0.483836
$274$	4.31614	0.260748
$275$	0	0
$276$	−9.55134	−0.574923
$277$	−10.8885	−0.654227	−0.327114	−	0.944985i	$-0.606076\pi$
$277$	−10.8885	−0.654227	−0.327114	+	0.944985i	$0.606076\pi$
$278$	14.9223	0.894983
$279$	1.39560	0.0835521
$280$	0	0
$281$	−3.16841	−0.189011	−0.0945056	−	0.995524i	$-0.530127\pi$
$281$	−3.16841	−0.189011	−0.0945056	+	0.995524i	$0.530127\pi$
$282$	1.54526	0.0920190
$283$	−33.0957	−1.96733	−0.983667	−	0.179997i	$-0.942391\pi$
$283$	−33.0957	−1.96733	−0.983667	+	0.179997i	$0.942391\pi$
$284$	−3.86454	−0.229318
$285$	0	0
$286$	−7.25368	−0.428919
$287$	−0.527296	−0.0311253
$288$	−7.35520	−0.433409
$289$	−11.8777	−0.698687
$290$	0	0
$291$	−19.7723	−1.15907
$292$	−0.932897	−0.0545937
$293$	−17.3630	−1.01436	−0.507178	−	0.861841i	$-0.669311\pi$
$293$	−17.3630	−1.01436	−0.507178	+	0.861841i	$0.669311\pi$
$294$	1.24309	0.0724982
$295$	0	0
$296$	−20.1755	−1.17268
$297$	5.41928	0.314459
$298$	−4.97886	−0.288417
$299$	63.9098	3.69600
$300$	0	0
$301$	−4.95746	−0.285743
$302$	4.41670	0.254153
$303$	9.63367	0.553440
$304$	−9.59466	−0.550291
$305$	0	0
$306$	3.91795	0.223974
$307$	−11.7256	−0.669217	−0.334608	−	0.942357i	$-0.608604\pi$
$307$	−11.7256	−0.669217	−0.334608	+	0.942357i	$0.608604\pi$
$308$	−0.872076	−0.0496911
$309$	20.0697	1.14173
$310$	0	0
$311$	10.4874	0.594686	0.297343	−	0.954771i	$-0.403900\pi$
$311$	10.4874	0.594686	0.297343	+	0.954771i	$0.403900\pi$
$312$	−24.3846	−1.38051
$313$	−22.0342	−1.24545	−0.622723	−	0.782443i	$-0.713973\pi$
$313$	−22.0342	−1.24545	−0.622723	+	0.782443i	$0.713973\pi$
$314$	10.0911	0.569474
$315$	0	0
$316$	0.843146	0.0474307
$317$	14.8993	0.836827	0.418414	−	0.908257i	$-0.362586\pi$
$317$	14.8993	0.836827	0.418414	+	0.908257i	$0.362586\pi$
$318$	−4.94453	−0.277275
$319$	−4.96420	−0.277942
$320$	0	0
$321$	−20.4288	−1.14022
$322$	−9.93777	−0.553810
$323$	−14.5220	−0.808023
$324$	1.26724	0.0704022
$325$	0	0
$326$	−24.7660	−1.37166
$327$	11.4591	0.633689
$328$	−1.60839	−0.0888083
$329$	−1.24309	−0.0685335
$330$	0	0
$331$	13.5775	0.746290	0.373145	−	0.927773i	$-0.378279\pi$
$331$	13.5775	0.746290	0.373145	+	0.927773i	$0.378279\pi$
$332$	5.29639	0.290677
$333$	−10.7814	−0.590816
$334$	−13.3877	−0.732543
$335$	0	0
$336$	1.75024	0.0954836
$337$	3.68755	0.200874	0.100437	−	0.994943i	$-0.467976\pi$
$337$	3.68755	0.200874	0.100437	+	0.994943i	$0.467976\pi$
$338$	35.7359	1.94378
$339$	−19.4566	−1.05674
$340$	0	0
$341$	0.856196	0.0463656
$342$	−11.1075	−0.600628
$343$	−1.00000	−0.0539949
$344$	−15.1215	−0.815297
$345$	0	0
$346$	9.84171	0.529094
$347$	−19.0044	−1.02021	−0.510106	−	0.860112i	$-0.670394\pi$
$347$	−19.0044	−1.02021	−0.510106	+	0.860112i	$0.670394\pi$
$348$	−5.06716	−0.271628
$349$	−5.67289	−0.303663	−0.151831	−	0.988406i	$-0.548517\pi$
$349$	−5.67289	−0.303663	−0.151831	+	0.988406i	$0.548517\pi$
$350$	0	0
$351$	−37.0135	−1.97563
$352$	−4.51241	−0.240512
$353$	2.84951	0.151664	0.0758320	−	0.997121i	$-0.475839\pi$
$353$	2.84951	0.151664	0.0758320	+	0.997121i	$0.475839\pi$
$354$	−14.4851	−0.769876
$355$	0	0
$356$	3.10125	0.164366
$357$	2.64907	0.140204
$358$	−16.5038	−0.872251
$359$	20.9541	1.10592	0.552958	−	0.833209i	$-0.313499\pi$
$359$	20.9541	1.10592	0.552958	+	0.833209i	$0.313499\pi$
$360$	0	0
$361$	22.1703	1.16686
$362$	−3.54392	−0.186264
$363$	1.17047	0.0614339
$364$	5.95625	0.312192
$365$	0	0
$366$	4.23445	0.221338
$367$	3.69812	0.193040	0.0965201	−	0.995331i	$-0.469229\pi$
$367$	3.69812	0.193040	0.0965201	+	0.995331i	$0.469229\pi$
$368$	−13.9922	−0.729394
$369$	−0.859490	−0.0447433
$370$	0	0
$371$	3.97763	0.206508
$372$	0.873954	0.0453124
$373$	−30.7541	−1.59238	−0.796192	−	0.605043i	$-0.793156\pi$
$373$	−30.7541	−1.59238	−0.796192	+	0.605043i	$0.793156\pi$
$374$	2.40366	0.124290
$375$	0	0
$376$	−3.79172	−0.195543
$377$	33.9053	1.74621
$378$	5.75548	0.296030
$379$	18.6017	0.955506	0.477753	−	0.878494i	$-0.341451\pi$
$379$	18.6017	0.955506	0.477753	+	0.878494i	$0.341451\pi$
$380$	0	0
$381$	−18.9060	−0.968582
$382$	22.2540	1.13862
$383$	15.1169	0.772440	0.386220	−	0.922407i	$-0.373781\pi$
$383$	15.1169	0.772440	0.386220	+	0.922407i	$0.373781\pi$
$384$	−0.888349	−0.0453334
$385$	0	0
$386$	−14.6456	−0.745444
$387$	−8.08063	−0.410762
$388$	14.7316	0.747885
$389$	−13.9240	−0.705973	−0.352986	−	0.935628i	$-0.614834\pi$
$389$	−13.9240	−0.705973	−0.352986	+	0.935628i	$0.614834\pi$
$390$	0	0
$391$	−21.1778	−1.07101
$392$	−3.05025	−0.154061
$393$	11.6033	0.585311
$394$	12.1522	0.612220
$395$	0	0
$396$	−1.42148	−0.0714320
$397$	−1.02403	−0.0513946	−0.0256973	−	0.999670i	$-0.508181\pi$
$397$	−1.02403	−0.0513946	−0.0256973	+	0.999670i	$0.508181\pi$
$398$	23.1606	1.16094
$399$	−7.51023	−0.375982
$400$	0	0
$401$	3.00889	0.150257	0.0751284	−	0.997174i	$-0.476063\pi$
$401$	3.00889	0.150257	0.0751284	+	0.997174i	$0.476063\pi$
$402$	6.61674	0.330013
$403$	−5.84779	−0.291299
$404$	−7.17769	−0.357104
$405$	0	0
$406$	−5.27217	−0.261653
$407$	−6.61436	−0.327862
$408$	8.08035	0.400037
$409$	−31.1886	−1.54218	−0.771090	−	0.636727i	$-0.780288\pi$
$409$	−31.1886	−1.54218	−0.771090	+	0.636727i	$0.780288\pi$
$410$	0	0
$411$	4.75682	0.234637
$412$	−14.9532	−0.736692
$413$	11.6526	0.573385
$414$	−16.1985	−0.796113
$415$	0	0
$416$	30.8196	1.51105
$417$	16.4459	0.805360
$418$	−6.81447	−0.333307
$419$	−31.4805	−1.53792	−0.768962	−	0.639294i	$-0.779227\pi$
$419$	−31.4805	−1.53792	−0.768962	+	0.639294i	$0.779227\pi$
$420$	0	0
$421$	16.6545	0.811689	0.405845	−	0.913942i	$-0.366977\pi$
$421$	16.6545	0.811689	0.405845	+	0.913942i	$0.366977\pi$
$422$	24.6366	1.19929
$423$	−2.02622	−0.0985183
$424$	12.1328	0.589219
$425$	0	0
$426$	5.50864	0.266895
$427$	−3.40640	−0.164847
$428$	15.2207	0.735722
$429$	−7.99428	−0.385968
$430$	0	0
$431$	−31.0206	−1.49421	−0.747104	−	0.664707i	$-0.768556\pi$
$431$	−31.0206	−1.49421	−0.747104	+	0.664707i	$0.768556\pi$
$432$	8.10362	0.389886
$433$	14.6496	0.704016	0.352008	−	0.935997i	$-0.385499\pi$
$433$	14.6496	0.704016	0.352008	+	0.935997i	$0.385499\pi$
$434$	0.909312	0.0436484
$435$	0	0
$436$	−8.53775	−0.408884
$437$	60.0400	2.87210
$438$	1.32978	0.0635394
$439$	16.8201	0.802778	0.401389	−	0.915908i	$-0.368527\pi$
$439$	16.8201	0.802778	0.401389	+	0.915908i	$0.368527\pi$
$440$	0	0
$441$	−1.63000	−0.0776188
$442$	−16.4169	−0.780873
$443$	−18.3489	−0.871784	−0.435892	−	0.899999i	$-0.643567\pi$
$443$	−18.3489	−0.871784	−0.435892	+	0.899999i	$0.643567\pi$
$444$	−6.75155	−0.320414
$445$	0	0
$446$	2.50669	0.118695
$447$	−5.48720	−0.259536
$448$	−7.78301	−0.367713
$449$	11.1951	0.528329	0.264164	−	0.964478i	$-0.414904\pi$
$449$	11.1951	0.528329	0.264164	+	0.964478i	$0.414904\pi$
$450$	0	0
$451$	−0.527296	−0.0248294
$452$	14.4964	0.681852
$453$	4.86765	0.228702
$454$	30.2036	1.41752
$455$	0	0
$456$	−22.9081	−1.07277
$457$	12.0566	0.563984	0.281992	−	0.959417i	$-0.409005\pi$
$457$	12.0566	0.563984	0.281992	+	0.959417i	$0.409005\pi$
$458$	−20.1023	−0.939318
$459$	12.2652	0.572490
$460$	0	0
$461$	−41.1825	−1.91806	−0.959030	−	0.283303i	$-0.908570\pi$
$461$	−41.1825	−1.91806	−0.959030	+	0.283303i	$0.908570\pi$
$462$	1.24309	0.0578336
$463$	33.1629	1.54121	0.770606	−	0.637311i	$-0.219953\pi$
$463$	33.1629	1.54121	0.770606	+	0.637311i	$0.219953\pi$
$464$	−7.42312	−0.344610
$465$	0	0
$466$	18.9176	0.876341
$467$	−20.8688	−0.965695	−0.482847	−	0.875705i	$-0.660397\pi$
$467$	−20.8688	−0.965695	−0.482847	+	0.875705i	$0.660397\pi$
$468$	9.70866	0.448783
$469$	−5.32284	−0.245786
$470$	0	0
$471$	11.1214	0.512448
$472$	35.5433	1.63601
$473$	−4.95746	−0.227944
$474$	−1.20185	−0.0552027
$475$	0	0
$476$	−1.97373	−0.0904657
$477$	6.48351	0.296860
$478$	−14.1394	−0.646723
$479$	4.71764	0.215555	0.107777	−	0.994175i	$-0.465627\pi$
$479$	4.71764	0.215555	0.107777	+	0.994175i	$0.465627\pi$
$480$	0	0
$481$	45.1759	2.05984
$482$	8.95720	0.407989
$483$	−10.9524	−0.498352
$484$	−0.872076	−0.0396398
$485$	0	0
$486$	15.4601	0.701283
$487$	−5.80132	−0.262883	−0.131441	−	0.991324i	$-0.541961\pi$
$487$	−5.80132	−0.262883	−0.131441	+	0.991324i	$0.541961\pi$
$488$	−10.3904	−0.470350
$489$	−27.2946	−1.23431
$490$	0	0
$491$	−30.4064	−1.37222	−0.686111	−	0.727497i	$-0.740683\pi$
$491$	−30.4064	−1.37222	−0.686111	+	0.727497i	$0.740683\pi$
$492$	−0.538233	−0.0242654
$493$	−11.2352	−0.506010
$494$	46.5426	2.09405
$495$	0	0
$496$	1.28030	0.0574870
$497$	−4.43143	−0.198777
$498$	−7.54964	−0.338308
$499$	40.7188	1.82282	0.911411	−	0.411497i	$-0.134994\pi$
$499$	40.7188	1.82282	0.911411	+	0.411497i	$0.134994\pi$
$500$	0	0
$501$	−14.7546	−0.659187
$502$	14.8631	0.663373
$503$	−16.6048	−0.740370	−0.370185	−	0.928958i	$-0.620706\pi$
$503$	−16.6048	−0.740370	−0.370185	+	0.928958i	$0.620706\pi$
$504$	−4.97190	−0.221466
$505$	0	0
$506$	−9.93777	−0.441788
$507$	39.3846	1.74913
$508$	14.0861	0.624972
$509$	−8.91546	−0.395171	−0.197585	−	0.980286i	$-0.563310\pi$
$509$	−8.91546	−0.395171	−0.197585	+	0.980286i	$0.563310\pi$
$510$	0	0
$511$	−1.06974	−0.0473226
$512$	−15.8698	−0.701354
$513$	−34.7723	−1.53524
$514$	7.60708	0.335534
$515$	0	0
$516$	−5.06028	−0.222766
$517$	−1.24309	−0.0546708
$518$	−7.02470	−0.308648
$519$	10.8466	0.476111
$520$	0	0
$521$	−12.0474	−0.527806	−0.263903	−	0.964549i	$-0.585010\pi$
$521$	−12.0474	−0.527806	−0.263903	+	0.964549i	$0.585010\pi$
$522$	−8.59361	−0.376132
$523$	6.05630	0.264823	0.132412	−	0.991195i	$-0.457728\pi$
$523$	6.05630	0.264823	0.132412	+	0.991195i	$0.457728\pi$
$524$	−8.64522	−0.377668
$525$	0	0
$526$	7.45180	0.324914
$527$	1.93779	0.0844114
$528$	1.75024	0.0761696
$529$	64.5584	2.80689
$530$	0	0
$531$	18.9936	0.824253
$532$	5.59560	0.242600
$533$	3.60142	0.155995
$534$	−4.42063	−0.191299
$535$	0	0
$536$	−16.2360	−0.701288
$537$	−18.1888	−0.784905
$538$	−4.39077	−0.189300
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−2.81266	−0.120926	−0.0604629	−	0.998170i	$-0.519258\pi$
$541$	−2.81266	−0.120926	−0.0604629	+	0.998170i	$0.519258\pi$
$542$	−13.6185	−0.584963
$543$	−3.90575	−0.167612
$544$	−10.2127	−0.437867
$545$	0	0
$546$	−8.49023	−0.363348
$547$	−14.1171	−0.603603	−0.301802	−	0.953371i	$-0.597588\pi$
$547$	−14.1171	−0.603603	−0.301802	+	0.953371i	$0.597588\pi$
$548$	−3.54413	−0.151398
$549$	−5.55241	−0.236971
$550$	0	0
$551$	31.8523	1.35696
$552$	−33.4076	−1.42192
$553$	0.966826	0.0411136
$554$	−11.5640	−0.491308
$555$	0	0
$556$	−12.2533	−0.519654
$557$	1.95691	0.0829168	0.0414584	−	0.999140i	$-0.486800\pi$
$557$	1.95691	0.0829168	0.0414584	+	0.999140i	$0.486800\pi$
$558$	1.48217	0.0627455
$559$	33.8593	1.43209
$560$	0	0
$561$	2.64907	0.111844
$562$	−3.36497	−0.141943
$563$	15.5164	0.653939	0.326970	−	0.945035i	$-0.393973\pi$
$563$	15.5164	0.653939	0.326970	+	0.945035i	$0.393973\pi$
$564$	−1.26887	−0.0534290
$565$	0	0
$566$	−35.1489	−1.47742
$567$	1.45313	0.0610257
$568$	−13.5170	−0.567160
$569$	−37.7170	−1.58118	−0.790589	−	0.612347i	$-0.790226\pi$
$569$	−37.7170	−1.58118	−0.790589	+	0.612347i	$0.790226\pi$
$570$	0	0
$571$	−12.4679	−0.521767	−0.260883	−	0.965370i	$-0.584014\pi$
$571$	−12.4679	−0.521767	−0.260883	+	0.965370i	$0.584014\pi$
$572$	5.95625	0.249043
$573$	24.5262	1.02460
$574$	−0.560008	−0.0233743
$575$	0	0
$576$	−12.6863	−0.528594
$577$	7.71735	0.321278	0.160639	−	0.987013i	$-0.448645\pi$
$577$	7.71735	0.321278	0.160639	+	0.987013i	$0.448645\pi$
$578$	−12.6146	−0.524696
$579$	−16.1410	−0.670796
$580$	0	0
$581$	6.07331	0.251963
$582$	−20.9989	−0.870434
$583$	3.97763	0.164736
$584$	−3.26298	−0.135023
$585$	0	0
$586$	−18.4401	−0.761755
$587$	−30.6554	−1.26528	−0.632641	−	0.774445i	$-0.718029\pi$
$587$	−30.6554	−1.26528	−0.632641	+	0.774445i	$0.718029\pi$
$588$	−1.02074	−0.0420946
$589$	−5.49370	−0.226364
$590$	0	0
$591$	13.3930	0.550913
$592$	−9.89066	−0.406504
$593$	42.4286	1.74233	0.871167	−	0.490986i	$-0.163364\pi$
$593$	42.4286	1.74233	0.871167	+	0.490986i	$0.163364\pi$
$594$	5.75548	0.236150
$595$	0	0
$596$	4.08831	0.167464
$597$	25.5253	1.04468
$598$	67.8746	2.77560
$599$	8.49535	0.347111	0.173555	−	0.984824i	$-0.444474\pi$
$599$	8.49535	0.347111	0.173555	+	0.984824i	$0.444474\pi$
$600$	0	0
$601$	−23.1651	−0.944925	−0.472463	−	0.881351i	$-0.656635\pi$
$601$	−23.1651	−0.944925	−0.472463	+	0.881351i	$0.656635\pi$
$602$	−5.26501	−0.214586
$603$	−8.67620	−0.353322
$604$	−3.62671	−0.147569
$605$	0	0
$606$	10.2313	0.415619
$607$	2.87209	0.116575	0.0582873	−	0.998300i	$-0.481436\pi$
$607$	2.87209	0.116575	0.0582873	+	0.998300i	$0.481436\pi$
$608$	28.9535	1.17422
$609$	−5.81046	−0.235452
$610$	0	0
$611$	8.49023	0.343478
$612$	−3.21717	−0.130046
$613$	−33.8112	−1.36562	−0.682811	−	0.730595i	$-0.739243\pi$
$613$	−33.8112	−1.36562	−0.682811	+	0.730595i	$0.739243\pi$
$614$	−12.4531	−0.502564
$615$	0	0
$616$	−3.05025	−0.122898
$617$	−26.0998	−1.05074	−0.525369	−	0.850875i	$-0.676073\pi$
$617$	−26.0998	−1.05074	−0.525369	+	0.850875i	$0.676073\pi$
$618$	21.3148	0.857407
$619$	−27.1306	−1.09047	−0.545235	−	0.838283i	$-0.683560\pi$
$619$	−27.1306	−1.09047	−0.545235	+	0.838283i	$0.683560\pi$
$620$	0	0
$621$	−50.7096	−2.03491
$622$	11.1380	0.446594
$623$	3.55617	0.142475
$624$	−11.9541	−0.478547
$625$	0	0
$626$	−23.4011	−0.935297
$627$	−7.51023	−0.299930
$628$	−8.28616	−0.330654
$629$	−14.9700	−0.596892
$630$	0	0
$631$	−24.0346	−0.956802	−0.478401	−	0.878141i	$-0.658784\pi$
$631$	−24.0346	−0.956802	−0.478401	+	0.878141i	$0.658784\pi$
$632$	2.94906	0.117307
$633$	27.1520	1.07920
$634$	15.8236	0.628436
$635$	0	0
$636$	4.06013	0.160994
$637$	6.82997	0.270613
$638$	−5.27217	−0.208727
$639$	−7.22320	−0.285746
$640$	0	0
$641$	9.48821	0.374762	0.187381	−	0.982287i	$-0.440000\pi$
$641$	9.48821	0.374762	0.187381	+	0.982287i	$0.440000\pi$
$642$	−21.6961	−0.856278
$643$	−34.1937	−1.34847	−0.674235	−	0.738517i	$-0.735526\pi$
$643$	−34.1937	−1.34847	−0.674235	+	0.738517i	$0.735526\pi$
$644$	8.16024	0.321559
$645$	0	0
$646$	−15.4229	−0.606804
$647$	−25.8239	−1.01524	−0.507620	−	0.861581i	$-0.669475\pi$
$647$	−25.8239	−1.01524	−0.507620	+	0.861581i	$0.669475\pi$
$648$	4.43241	0.174122
$649$	11.6526	0.457403
$650$	0	0
$651$	1.00215	0.0392775
$652$	20.3362	0.796429
$653$	−21.1459	−0.827502	−0.413751	−	0.910390i	$-0.635782\pi$
$653$	−21.1459	−0.827502	−0.413751	+	0.910390i	$0.635782\pi$
$654$	12.1700	0.475884
$655$	0	0
$656$	−0.788483	−0.0307851
$657$	−1.74367	−0.0680272
$658$	−1.32020	−0.0514669
$659$	−12.9835	−0.505764	−0.252882	−	0.967497i	$-0.581379\pi$
$659$	−12.9835	−0.505764	−0.252882	+	0.967497i	$0.581379\pi$
$660$	0	0
$661$	32.6438	1.26970	0.634849	−	0.772636i	$-0.281062\pi$
$661$	32.6438	1.26970	0.634849	+	0.772636i	$0.281062\pi$
$662$	14.4199	0.560444
$663$	−18.0931	−0.702677
$664$	18.5251	0.718915
$665$	0	0
$666$	−11.4502	−0.443687
$667$	46.4513	1.79860
$668$	10.9931	0.425337
$669$	2.76262	0.106809
$670$	0	0
$671$	−3.40640	−0.131503
$672$	−5.28165	−0.203744
$673$	−19.5798	−0.754745	−0.377372	−	0.926062i	$-0.623172\pi$
$673$	−19.5798	−0.754745	−0.377372	+	0.926062i	$0.623172\pi$
$674$	3.91632	0.150851
$675$	0	0
$676$	−29.3440	−1.12862
$677$	40.2314	1.54622	0.773109	−	0.634273i	$-0.218701\pi$
$677$	40.2314	1.54622	0.773109	+	0.634273i	$0.218701\pi$
$678$	−20.6636	−0.793581
$679$	16.8926	0.648278
$680$	0	0
$681$	33.2874	1.27557
$682$	0.909312	0.0348194
$683$	−37.6157	−1.43933	−0.719663	−	0.694323i	$-0.755704\pi$
$683$	−37.6157	−1.43933	−0.719663	+	0.694323i	$0.755704\pi$
$684$	9.12080	0.348742
$685$	0	0
$686$	−1.06204	−0.0405488
$687$	−22.1547	−0.845256
$688$	−7.41304	−0.282620
$689$	−27.1671	−1.03498
$690$	0	0
$691$	14.7643	0.561659	0.280829	−	0.959758i	$-0.409391\pi$
$691$	14.7643	0.561659	0.280829	+	0.959758i	$0.409391\pi$
$692$	−8.08137	−0.307208
$693$	−1.63000	−0.0619184
$694$	−20.1834	−0.766152
$695$	0	0
$696$	−17.7234	−0.671802
$697$	−1.19340	−0.0452034
$698$	−6.02482	−0.228043
$699$	20.8491	0.788585
$700$	0	0
$701$	16.1271	0.609111	0.304555	−	0.952495i	$-0.401492\pi$
$701$	16.1271	0.609111	0.304555	+	0.952495i	$0.401492\pi$
$702$	−39.3097	−1.48365
$703$	42.4405	1.60067
$704$	−7.78301	−0.293333
$705$	0	0
$706$	3.02629	0.113896
$707$	−8.23058	−0.309543
$708$	11.8943	0.447013
$709$	−36.9892	−1.38916	−0.694579	−	0.719416i	$-0.744409\pi$
$709$	−36.9892	−1.38916	−0.694579	+	0.719416i	$0.744409\pi$
$710$	0	0
$711$	1.57592	0.0591017
$712$	10.8472	0.406517
$713$	−8.01165	−0.300039
$714$	2.81342	0.105289
$715$	0	0
$716$	13.5518	0.506455
$717$	−15.5831	−0.581961
$718$	22.2541	0.830515
$719$	6.71788	0.250535	0.125267	−	0.992123i	$-0.460021\pi$
$719$	6.71788	0.250535	0.125267	+	0.992123i	$0.460021\pi$
$720$	0	0
$721$	−17.1467	−0.638576
$722$	23.5457	0.876280
$723$	9.87173	0.367134
$724$	2.91003	0.108151
$725$	0	0
$726$	1.24309	0.0461352
$727$	9.41666	0.349245	0.174622	−	0.984636i	$-0.444130\pi$
$727$	9.41666	0.349245	0.174622	+	0.984636i	$0.444130\pi$
$728$	20.8331	0.772127
$729$	21.3979	0.792516
$730$	0	0
$731$	−11.2200	−0.414986
$732$	−3.47705	−0.128515
$733$	−7.09353	−0.262006	−0.131003	−	0.991382i	$-0.541820\pi$
$733$	−7.09353	−0.262006	−0.131003	+	0.991382i	$0.541820\pi$
$734$	3.92754	0.144968
$735$	0	0
$736$	42.2238	1.55639
$737$	−5.32284	−0.196069
$738$	−0.912811	−0.0336010
$739$	28.4731	1.04740	0.523701	−	0.851902i	$-0.324551\pi$
$739$	28.4731	1.04740	0.523701	+	0.851902i	$0.324551\pi$
$740$	0	0
$741$	51.2946	1.88435
$742$	4.22439	0.155082
$743$	1.20393	0.0441681	0.0220840	−	0.999756i	$-0.492970\pi$
$743$	1.20393	0.0441681	0.0220840	+	0.999756i	$0.492970\pi$
$744$	3.05682	0.112069
$745$	0	0
$746$	−32.6620	−1.19584
$747$	9.89947	0.362203
$748$	−1.97373	−0.0721667
$749$	17.4535	0.637735
$750$	0	0
$751$	13.1877	0.481227	0.240614	−	0.970621i	$-0.422651\pi$
$751$	13.1877	0.481227	0.240614	+	0.970621i	$0.422651\pi$
$752$	−1.85882	−0.0677844
$753$	16.3806	0.596943
$754$	36.0087	1.31136
$755$	0	0
$756$	−4.72602	−0.171884
$757$	−28.5746	−1.03856	−0.519282	−	0.854603i	$-0.673800\pi$
$757$	−28.5746	−1.03856	−0.519282	+	0.854603i	$0.673800\pi$
$758$	19.7557	0.717560
$759$	−10.9524	−0.397547
$760$	0	0
$761$	22.9475	0.831847	0.415924	−	0.909400i	$-0.363458\pi$
$761$	22.9475	0.831847	0.415924	+	0.909400i	$0.363458\pi$
$762$	−20.0788	−0.727380
$763$	−9.79014	−0.354427
$764$	−18.2736	−0.661114
$765$	0	0
$766$	16.0548	0.580082
$767$	−79.5867	−2.87371
$768$	−19.1630	−0.691487
$769$	−5.02937	−0.181364	−0.0906818	−	0.995880i	$-0.528905\pi$
$769$	−5.02937	−0.181364	−0.0906818	+	0.995880i	$0.528905\pi$
$770$	0	0
$771$	8.38376	0.301934
$772$	12.0260	0.432827
$773$	−3.44069	−0.123753	−0.0618765	−	0.998084i	$-0.519708\pi$
$773$	−3.44069	−0.123753	−0.0618765	+	0.998084i	$0.519708\pi$
$774$	−8.58194	−0.308471
$775$	0	0
$776$	51.5267	1.84970
$777$	−7.74193	−0.277740
$778$	−14.7878	−0.530167
$779$	3.38335	0.121221
$780$	0	0
$781$	−4.43143	−0.158569
$782$	−22.4917	−0.804301
$783$	−26.9024	−0.961413
$784$	−1.49533	−0.0534047
$785$	0	0
$786$	12.3232	0.439553
$787$	−16.1901	−0.577116	−0.288558	−	0.957462i	$-0.593176\pi$
$787$	−16.1901	−0.577116	−0.288558	+	0.957462i	$0.593176\pi$
$788$	−9.97861	−0.355473
$789$	8.21263	0.292377
$790$	0	0
$791$	16.6228	0.591040
$792$	−4.97190	−0.176669
$793$	23.2656	0.826185
$794$	−1.08756	−0.0385960
$795$	0	0
$796$	−19.0180	−0.674075
$797$	20.1288	0.712999	0.356499	−	0.934296i	$-0.383970\pi$
$797$	20.1288	0.712999	0.356499	+	0.934296i	$0.383970\pi$
$798$	−7.97614	−0.282352
$799$	−2.81342	−0.0995315
$800$	0	0
$801$	5.79654	0.204811
$802$	3.19555	0.112839
$803$	−1.06974	−0.0377504
$804$	−5.43324	−0.191615
$805$	0	0
$806$	−6.21057	−0.218758
$807$	−4.83907	−0.170343
$808$	−25.1054	−0.883203
$809$	−37.2899	−1.31104	−0.655522	−	0.755176i	$-0.727551\pi$
$809$	−37.2899	−1.31104	−0.655522	+	0.755176i	$0.727551\pi$
$810$	0	0
$811$	−14.5835	−0.512095	−0.256047	−	0.966664i	$-0.582420\pi$
$811$	−14.5835	−0.512095	−0.256047	+	0.966664i	$0.582420\pi$
$812$	4.32916	0.151924
$813$	−15.0089	−0.526385
$814$	−7.02470	−0.246216
$815$	0	0
$816$	3.96124	0.138671
$817$	31.8091	1.11286
$818$	−33.1235	−1.15814
$819$	11.1328	0.389012
$820$	0	0
$821$	−30.9951	−1.08174	−0.540869	−	0.841107i	$-0.681905\pi$
$821$	−30.9951	−1.08174	−0.540869	+	0.841107i	$0.681905\pi$
$822$	5.05192	0.176206
$823$	55.0098	1.91752	0.958761	−	0.284215i	$-0.0917330\pi$
$823$	55.0098	1.91752	0.958761	+	0.284215i	$0.0917330\pi$
$824$	−52.3017	−1.82202
$825$	0	0
$826$	12.3755	0.430598
$827$	−44.3005	−1.54048	−0.770239	−	0.637755i	$-0.779863\pi$
$827$	−44.3005	−1.54048	−0.770239	+	0.637755i	$0.779863\pi$
$828$	13.3012	0.462247
$829$	−9.84740	−0.342014	−0.171007	−	0.985270i	$-0.554702\pi$
$829$	−9.84740	−0.342014	−0.171007	+	0.985270i	$0.554702\pi$
$830$	0	0
$831$	−12.7447	−0.442109
$832$	53.1577	1.84291
$833$	−2.26325	−0.0784171
$834$	17.4662	0.604804
$835$	0	0
$836$	5.59560	0.193528
$837$	4.63997	0.160381
$838$	−33.4335	−1.15494
$839$	11.3177	0.390730	0.195365	−	0.980731i	$-0.437411\pi$
$839$	11.3177	0.390730	0.195365	+	0.980731i	$0.437411\pi$
$840$	0	0
$841$	−4.35673	−0.150232
$842$	17.6877	0.609558
$843$	−3.70853	−0.127729
$844$	−20.2300	−0.696344
$845$	0	0
$846$	−2.15193	−0.0739847
$847$	−1.00000	−0.0343604
$848$	5.94787	0.204251
$849$	−38.7376	−1.32947
$850$	0	0
$851$	61.8923	2.12164
$852$	−4.52334	−0.154967
$853$	26.3172	0.901083	0.450541	−	0.892755i	$-0.351231\pi$
$853$	26.3172	0.901083	0.450541	+	0.892755i	$0.351231\pi$
$854$	−3.61772	−0.123796
$855$	0	0
$856$	53.2375	1.81962
$857$	39.8259	1.36043	0.680213	−	0.733014i	$-0.261887\pi$
$857$	39.8259	1.36043	0.680213	+	0.733014i	$0.261887\pi$
$858$	−8.49023	−0.289852
$859$	16.5702	0.565367	0.282684	−	0.959213i	$-0.408775\pi$
$859$	16.5702	0.565367	0.282684	+	0.959213i	$0.408775\pi$
$860$	0	0
$861$	−0.617186	−0.0210336
$862$	−32.9450	−1.12211
$863$	−23.2437	−0.791225	−0.395613	−	0.918418i	$-0.629468\pi$
$863$	−23.2437	−0.791225	−0.395613	+	0.918418i	$0.629468\pi$
$864$	−24.4540	−0.831942
$865$	0	0
$866$	15.5585	0.528698
$867$	−13.9025	−0.472154
$868$	−0.746668	−0.0253436
$869$	0.966826	0.0327973
$870$	0	0
$871$	36.3548	1.23184
$872$	−29.8624	−1.01127
$873$	27.5349	0.931914
$874$	63.7648	2.15688
$875$	0	0
$876$	−1.09193	−0.0368929
$877$	−56.2380	−1.89902	−0.949511	−	0.313735i	$-0.898420\pi$
$877$	−56.2380	−1.89902	−0.949511	+	0.313735i	$0.898420\pi$
$878$	17.8636	0.602866
$879$	−20.3229	−0.685474
$880$	0	0
$881$	−50.4473	−1.69961	−0.849807	−	0.527094i	$-0.823282\pi$
$881$	−50.4473	−1.69961	−0.849807	+	0.527094i	$0.823282\pi$
$882$	−1.73112	−0.0582897
$883$	−10.2912	−0.346328	−0.173164	−	0.984893i	$-0.555399\pi$
$883$	−10.2912	−0.346328	−0.173164	+	0.984893i	$0.555399\pi$
$884$	13.4805	0.453398
$885$	0	0
$886$	−19.4872	−0.654687
$887$	40.8093	1.37024	0.685120	−	0.728430i	$-0.259750\pi$
$887$	40.8093	1.37024	0.685120	+	0.728430i	$0.259750\pi$
$888$	−23.6148	−0.792462
$889$	16.1524	0.541735
$890$	0	0
$891$	1.45313	0.0486817
$892$	−2.05833	−0.0689179
$893$	7.97614	0.266911
$894$	−5.82761	−0.194904
$895$	0	0
$896$	0.758967	0.0253553
$897$	74.8046	2.49765
$898$	11.8896	0.396761
$899$	−4.25033	−0.141756
$900$	0	0
$901$	9.00237	0.299913
$902$	−0.560008	−0.0186462
$903$	−5.80257	−0.193097
$904$	50.7038	1.68638
$905$	0	0
$906$	5.16963	0.171749
$907$	21.3652	0.709420	0.354710	−	0.934976i	$-0.384580\pi$
$907$	21.3652	0.709420	0.354710	+	0.934976i	$0.384580\pi$
$908$	−24.8012	−0.823057
$909$	−13.4158	−0.444974
$910$	0	0
$911$	17.4802	0.579144	0.289572	−	0.957156i	$-0.406487\pi$
$911$	17.4802	0.579144	0.289572	+	0.957156i	$0.406487\pi$
$912$	−11.2303	−0.371872
$913$	6.07331	0.200997
$914$	12.8046	0.423537
$915$	0	0
$916$	16.5067	0.545396
$917$	−9.91338	−0.327369
$918$	13.0261	0.429925
$919$	−22.1767	−0.731541	−0.365771	−	0.930705i	$-0.619195\pi$
$919$	−22.1767	−0.731541	−0.365771	+	0.930705i	$0.619195\pi$
$920$	0	0
$921$	−13.7245	−0.452238
$922$	−43.7374	−1.44041
$923$	30.2665	0.996234
$924$	−1.02074	−0.0335799
$925$	0	0
$926$	35.2203	1.15741
$927$	−27.9490	−0.917966
$928$	22.4005	0.735332
$929$	−8.89242	−0.291751	−0.145875	−	0.989303i	$-0.546600\pi$
$929$	−8.89242	−0.291751	−0.145875	+	0.989303i	$0.546600\pi$
$930$	0	0
$931$	6.41641	0.210289
$932$	−15.5339	−0.508830
$933$	12.2752	0.401872
$934$	−22.1635	−0.725212
$935$	0	0
$936$	33.9579	1.10995
$937$	57.8417	1.88960	0.944802	−	0.327641i	$-0.106254\pi$
$937$	57.8417	1.88960	0.944802	+	0.327641i	$0.106254\pi$
$938$	−5.65305	−0.184579
$939$	−25.7904	−0.841637
$940$	0	0
$941$	35.3237	1.15152	0.575760	−	0.817619i	$-0.304706\pi$
$941$	35.3237	1.15152	0.575760	+	0.817619i	$0.304706\pi$
$942$	11.8114	0.384835
$943$	4.93405	0.160675
$944$	17.4245	0.567118
$945$	0	0
$946$	−5.26501	−0.171180
$947$	28.9507	0.940773	0.470386	−	0.882461i	$-0.344115\pi$
$947$	28.9507	0.940773	0.470386	+	0.882461i	$0.344115\pi$
$948$	0.986879	0.0320523
$949$	7.30630	0.237173
$950$	0	0
$951$	17.4392	0.565505
$952$	−6.90349	−0.223743
$953$	−6.75259	−0.218738	−0.109369	−	0.994001i	$-0.534883\pi$
$953$	−6.75259	−0.218738	−0.109369	+	0.994001i	$0.534883\pi$
$954$	6.88573	0.222934
$955$	0	0
$956$	11.6104	0.375507
$957$	−5.81046	−0.187825
$958$	5.01031	0.161876
$959$	−4.06402	−0.131234
$960$	0	0
$961$	−30.2669	−0.976353
$962$	47.9785	1.54689
$963$	28.4490	0.916758
$964$	−7.35506	−0.236891
$965$	0	0
$966$	−11.6319	−0.374249
$967$	2.17445	0.0699257	0.0349628	−	0.999389i	$-0.488869\pi$
$967$	2.17445	0.0699257	0.0349628	+	0.999389i	$0.488869\pi$
$968$	−3.05025	−0.0980388
$969$	−16.9975	−0.546040
$970$	0	0
$971$	55.7879	1.79032	0.895159	−	0.445746i	$-0.147062\pi$
$971$	55.7879	1.79032	0.895159	+	0.445746i	$0.147062\pi$
$972$	−12.6948	−0.407186
$973$	−14.0507	−0.450444
$974$	−6.16122	−0.197418
$975$	0	0
$976$	−5.09370	−0.163045
$977$	−45.5825	−1.45831	−0.729156	−	0.684347i	$-0.760087\pi$
$977$	−45.5825	−1.45831	−0.729156	+	0.684347i	$0.760087\pi$
$978$	−28.9879	−0.926932
$979$	3.55617	0.113656
$980$	0	0
$981$	−15.9579	−0.509496
$982$	−32.2927	−1.03050
$983$	1.30557	0.0416413	0.0208206	−	0.999783i	$-0.493372\pi$
$983$	1.30557	0.0416413	0.0208206	+	0.999783i	$0.493372\pi$
$984$	−1.88257	−0.0600142
$985$	0	0
$986$	−11.9322	−0.380000
$987$	−1.45500	−0.0463131
$988$	−38.2177	−1.21587
$989$	46.3882	1.47506
$990$	0	0
$991$	34.4682	1.09492	0.547459	−	0.836833i	$-0.315595\pi$
$991$	34.4682	1.09492	0.547459	+	0.836833i	$0.315595\pi$
$992$	−3.86351	−0.122666
$993$	15.8921	0.504322
$994$	−4.70634	−0.149276
$995$	0	0
$996$	6.19928	0.196432
$997$	−40.4670	−1.28160	−0.640801	−	0.767707i	$-0.721398\pi$
$997$	−40.4670	−1.28160	−0.640801	+	0.767707i	$0.721398\pi$
$998$	43.2449	1.36889
$999$	−35.8451	−1.13409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.a.ba.1.5	✓	7
5.2	odd	4		1925.2.b.q.1849.10		14
5.3	odd	4		1925.2.b.q.1849.5		14
5.4	even	2		1925.2.a.bc.1.3	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1925.2.a.ba.1.5	✓	7	1.1	even	1	trivial
1925.2.a.bc.1.3	yes	7	5.4	even	2
1925.2.b.q.1849.5		14	5.3	odd	4
1925.2.b.q.1849.10		14	5.2	odd	4

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)

\(f(q)\)	\(=\)	\(q+1.06204 q^{2} +1.17047 q^{3} -0.872076 q^{4} +1.24309 q^{6} -1.00000 q^{7} -3.05025 q^{8} -1.63000 q^{9} +O(q^{10})\)	\(q+1.06204 q^{2} +1.17047 q^{3} -0.872076 q^{4} +1.24309 q^{6} -1.00000 q^{7} -3.05025 q^{8} -1.63000 q^{9} -1.00000 q^{11} -1.02074 q^{12} +6.82997 q^{13} -1.06204 q^{14} -1.49533 q^{16} -2.26325 q^{17} -1.73112 q^{18} +6.41641 q^{19} -1.17047 q^{21} -1.06204 q^{22} +9.35726 q^{23} -3.57024 q^{24} +7.25368 q^{26} -5.41928 q^{27} +0.872076 q^{28} +4.96420 q^{29} -0.856196 q^{31} +4.51241 q^{32} -1.17047 q^{33} -2.40366 q^{34} +1.42148 q^{36} +6.61436 q^{37} +6.81447 q^{38} +7.99428 q^{39} +0.527296 q^{41} -1.24309 q^{42} +4.95746 q^{43} +0.872076 q^{44} +9.93777 q^{46} +1.24309 q^{47} -1.75024 q^{48} +1.00000 q^{49} -2.64907 q^{51} -5.95625 q^{52} -3.97763 q^{53} -5.75548 q^{54} +3.05025 q^{56} +7.51023 q^{57} +5.27217 q^{58} -11.6526 q^{59} +3.40640 q^{61} -0.909312 q^{62} +1.63000 q^{63} +7.78301 q^{64} -1.24309 q^{66} +5.32284 q^{67} +1.97373 q^{68} +10.9524 q^{69} +4.43143 q^{71} +4.97190 q^{72} +1.06974 q^{73} +7.02470 q^{74} -5.59560 q^{76} +1.00000 q^{77} +8.49023 q^{78} -0.966826 q^{79} -1.45313 q^{81} +0.560008 q^{82} -6.07331 q^{83} +1.02074 q^{84} +5.26501 q^{86} +5.81046 q^{87} +3.05025 q^{88} -3.55617 q^{89} -6.82997 q^{91} -8.16024 q^{92} -1.00215 q^{93} +1.32020 q^{94} +5.28165 q^{96} -16.8926 q^{97} +1.06204 q^{98} +1.63000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 13 q^{4} - q^{6} - 7 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) \) 7 * q - q^2 + 13 * q^4 - q^6 - 7 * q^7 - 6 * q^8 + 13 * q^9	\( 7 q - q^{2} + 13 q^{4} - q^{6} - 7 q^{7} - 6 q^{8} + 13 q^{9} - 7 q^{11} + 21 q^{12} + 3 q^{13} + q^{14} + 29 q^{16} - 14 q^{18} + 18 q^{19} + q^{22} + 7 q^{23} - 22 q^{24} + 13 q^{26} - 6 q^{27} - 13 q^{28} - 2 q^{29} + 24 q^{31} - 33 q^{32} + 33 q^{34} + 44 q^{36} - 21 q^{37} - 9 q^{38} + 10 q^{39} - 10 q^{41} + q^{42} + 2 q^{43} - 13 q^{44} + 3 q^{46} - q^{47} + 77 q^{48} + 7 q^{49} + 29 q^{51} - 9 q^{52} + 11 q^{53} - 47 q^{54} + 6 q^{56} - 7 q^{57} + 33 q^{58} + q^{59} + 28 q^{61} - 16 q^{62} - 13 q^{63} + 48 q^{64} + q^{66} + 46 q^{68} + 33 q^{69} - 10 q^{71} - 36 q^{72} + 11 q^{73} - 6 q^{74} + 20 q^{76} + 7 q^{77} + 31 q^{78} + 19 q^{79} + 27 q^{81} - 8 q^{82} + 19 q^{83} - 21 q^{84} + 55 q^{86} + 12 q^{87} + 6 q^{88} - 2 q^{89} - 3 q^{91} - 40 q^{92} - 58 q^{93} + 21 q^{94} - 23 q^{96} - 8 q^{97} - q^{98} - 13 q^{99}+O(q^{100}) \) 7 * q - q^2 + 13 * q^4 - q^6 - 7 * q^7 - 6 * q^8 + 13 * q^9 - 7 * q^11 + 21 * q^12 + 3 * q^13 + q^14 + 29 * q^16 - 14 * q^18 + 18 * q^19 + q^22 + 7 * q^23 - 22 * q^24 + 13 * q^26 - 6 * q^27 - 13 * q^28 - 2 * q^29 + 24 * q^31 - 33 * q^32 + 33 * q^34 + 44 * q^36 - 21 * q^37 - 9 * q^38 + 10 * q^39 - 10 * q^41 + q^42 + 2 * q^43 - 13 * q^44 + 3 * q^46 - q^47 + 77 * q^48 + 7 * q^49 + 29 * q^51 - 9 * q^52 + 11 * q^53 - 47 * q^54 + 6 * q^56 - 7 * q^57 + 33 * q^58 + q^59 + 28 * q^61 - 16 * q^62 - 13 * q^63 + 48 * q^64 + q^66 + 46 * q^68 + 33 * q^69 - 10 * q^71 - 36 * q^72 + 11 * q^73 - 6 * q^74 + 20 * q^76 + 7 * q^77 + 31 * q^78 + 19 * q^79 + 27 * q^81 - 8 * q^82 + 19 * q^83 - 21 * q^84 + 55 * q^86 + 12 * q^87 + 6 * q^88 - 2 * q^89 - 3 * q^91 - 40 * q^92 - 58 * q^93 + 21 * q^94 - 23 * q^96 - 8 * q^97 - q^98 - 13 * q^99

Introduction

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