LMFDB - Embedded newform 1925.2.a.bc.1.3

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			1.3
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	$-0.622525\pi$
$2$	−1.06204	−0.750974	−0.375487	+	0.926828i	$0.622525\pi$
$3$	−1.17047	−0.675772	−0.337886	−	0.941187i	$-0.609712\pi$
$3$	−1.17047	−0.675772	−0.337886	+	0.941187i	$0.609712\pi$
$4$	−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	$-0.896047\pi$
$13$	−6.82997	−1.89429	−0.947146	+	0.320803i	$0.896047\pi$
$14$	−1.06204	−0.283841
$15$	0	0
$16$	−1.49533	−0.373833
$17$	2.26325	0.548919	0.274460	−	0.961599i	$-0.411501\pi$
$17$	2.26325	0.548919	0.274460	+	0.961599i	$0.411501\pi$
$18$	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	$-0.929484\pi$
$23$	−9.35726	−1.95112	−0.975562	+	0.219724i	$0.929484\pi$
$24$	−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	$-0.682976\pi$
$37$	−6.61436	−1.08739	−0.543697	+	0.839281i	$0.682976\pi$
$38$	−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	$-0.623389\pi$
$43$	−4.95746	−0.756005	−0.378003	+	0.925804i	$0.623389\pi$
$44$	0.872076	0.131470
$45$	0	0
$46$	9.93777	1.46524
$47$	−1.24309	−0.181323	−0.0906613	−	0.995882i	$-0.528898\pi$
$47$	−1.24309	−0.181323	−0.0906613	+	0.995882i	$0.528898\pi$
$48$	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	$-0.411923\pi$
$53$	3.97763	0.546369	0.273185	+	0.961962i	$0.411923\pi$
$54$	−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	$-0.605413\pi$
$67$	−5.32284	−0.650288	−0.325144	+	0.945665i	$0.605413\pi$
$68$	−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	$-0.519940\pi$
$73$	−1.06974	−0.125204	−0.0626019	+	0.998039i	$0.519940\pi$
$74$	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	$-0.391832\pi$
$83$	6.07331	0.666633	0.333316	+	0.942815i	$0.391832\pi$
$84$	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	$-0.171960\pi$
$97$	16.8926	1.71518	0.857592	+	0.514331i	$0.171960\pi$
$98$	−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	$-0.820255\pi$
$103$	−17.1467	−1.68951	−0.844757	+	0.535150i	$0.820255\pi$
$104$	−20.8331	−2.04286
$105$	0	0
$106$	−4.22439	−0.410309
$107$	17.4535	1.68729	0.843645	−	0.536902i	$-0.180406\pi$
$107$	17.4535	1.68729	0.843645	+	0.536902i	$0.180406\pi$
$108$	−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	$-0.214265\pi$
$113$	16.6228	1.56374	0.781872	+	0.623439i	$0.214265\pi$
$114$	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	$-0.245675\pi$
$127$	16.1524	1.43330	0.716648	+	0.697435i	$0.245675\pi$
$128$	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	$-0.555542\pi$
$137$	−4.06402	−0.347213	−0.173606	+	0.984815i	$0.555542\pi$
$138$	−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	$-0.623786\pi$
$157$	−9.50165	−0.758314	−0.379157	+	0.925332i	$0.623786\pi$
$158$	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	$-0.133560\pi$
$163$	23.3193	1.82651	0.913256	+	0.407386i	$0.133560\pi$
$164$	−0.459842	−0.0359077
$165$	0	0
$166$	−6.45009	−0.500624
$167$	12.6057	0.975458	0.487729	−	0.872995i	$-0.337825\pi$
$167$	12.6057	0.975458	0.487729	+	0.872995i	$0.337825\pi$
$168$	−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	$-0.614591\pi$
$173$	−9.26682	−0.704543	−0.352272	+	0.935898i	$0.614591\pi$
$174$	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	$-0.334685\pi$
$193$	13.7901	0.992636	0.496318	+	0.868141i	$0.334685\pi$
$194$	−17.9406	−1.28806
$195$	0	0
$196$	−0.872076	−0.0622911
$197$	−11.4424	−0.815234	−0.407617	−	0.913153i	$-0.633640\pi$
$197$	−11.4424	−0.815234	−0.407617	+	0.913153i	$0.633640\pi$
$198$	−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	$-0.525181\pi$
$223$	−2.36026	−0.158055	−0.0790274	+	0.996872i	$0.525181\pi$
$224$	−4.51241	−0.301498
$225$	0	0
$226$	−17.6541	−1.17433
$227$	−28.4393	−1.88758	−0.943790	−	0.330545i	$-0.892768\pi$
$227$	−28.4393	−1.88758	−0.943790	+	0.330545i	$0.892768\pi$
$228$	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	$-0.698305\pi$
$233$	−17.8126	−1.16694	−0.583470	+	0.812135i	$0.698305\pi$
$234$	−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	$-0.571715\pi$
$257$	−7.16272	−0.446798	−0.223399	+	0.974727i	$0.571715\pi$
$258$	−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	$-0.569408\pi$
$263$	−7.01651	−0.432657	−0.216328	+	0.976321i	$0.569408\pi$
$264$	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	$-0.393924\pi$
$277$	10.8885	0.654227	0.327114	+	0.944985i	$0.393924\pi$
$278$	−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	$-0.0576087\pi$
$283$	33.0957	1.96733	0.983667	+	0.179997i	$0.0576087\pi$
$284$	−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	$-0.330689\pi$
$293$	17.3630	1.01436	0.507178	+	0.861841i	$0.330689\pi$
$294$	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	$-0.391396\pi$
$307$	11.7256	0.669217	0.334608	+	0.942357i	$0.391396\pi$
$308$	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	$-0.286027\pi$
$313$	22.0342	1.24545	0.622723	+	0.782443i	$0.286027\pi$
$314$	10.0911	0.569474
$315$	0	0
$316$	0.843146	0.0474307
$317$	−14.8993	−0.836827	−0.418414	−	0.908257i	$-0.637414\pi$
$317$	−14.8993	−0.836827	−0.418414	+	0.908257i	$0.637414\pi$
$318$	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	$-0.532024\pi$
$337$	−3.68755	−0.200874	−0.100437	+	0.994943i	$0.532024\pi$
$338$	−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	$-0.329606\pi$
$347$	19.0044	1.02021	0.510106	+	0.860112i	$0.329606\pi$
$348$	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	$-0.524161\pi$
$353$	−2.84951	−0.151664	−0.0758320	+	0.997121i	$0.524161\pi$
$354$	−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	$-0.530771\pi$
$367$	−3.69812	−0.193040	−0.0965201	+	0.995331i	$0.530771\pi$
$368$	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	$-0.206844\pi$
$373$	30.7541	1.59238	0.796192	+	0.605043i	$0.206844\pi$
$374$	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	$-0.626219\pi$
$383$	−15.1169	−0.772440	−0.386220	+	0.922407i	$0.626219\pi$
$384$	−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	$-0.491819\pi$
$397$	1.02403	0.0513946	0.0256973	+	0.999670i	$0.491819\pi$
$398$	−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	$-0.614501\pi$
$433$	−14.6496	−0.704016	−0.352008	+	0.935997i	$0.614501\pi$
$434$	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	$-0.356433\pi$
$443$	18.3489	0.871784	0.435892	+	0.899999i	$0.356433\pi$
$444$	−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	$-0.590995\pi$
$457$	−12.0566	−0.563984	−0.281992	+	0.959417i	$0.590995\pi$
$458$	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	$-0.780047\pi$
$463$	−33.1629	−1.54121	−0.770606	+	0.637311i	$0.780047\pi$
$464$	−7.42312	−0.344610
$465$	0	0
$466$	18.9176	0.876341
$467$	20.8688	0.965695	0.482847	−	0.875705i	$-0.339603\pi$
$467$	20.8688	0.965695	0.482847	+	0.875705i	$0.339603\pi$
$468$	−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	$-0.458039\pi$
$487$	5.80132	0.262883	0.131441	+	0.991324i	$0.458039\pi$
$488$	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	$-0.379294\pi$
$503$	16.6048	0.740370	0.370185	+	0.928958i	$0.379294\pi$
$504$	−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	$-0.542272\pi$
$523$	−6.05630	−0.264823	−0.132412	+	0.991195i	$0.542272\pi$
$524$	−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	$-0.402412\pi$
$547$	14.1171	0.603603	0.301802	+	0.953371i	$0.402412\pi$
$548$	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	$-0.513200\pi$
$557$	−1.95691	−0.0829168	−0.0414584	+	0.999140i	$0.513200\pi$
$558$	−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	$-0.606027\pi$
$563$	−15.5164	−0.653939	−0.326970	+	0.945035i	$0.606027\pi$
$564$	−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	$-0.551355\pi$
$577$	−7.71735	−0.321278	−0.160639	+	0.987013i	$0.551355\pi$
$578$	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	$-0.281971\pi$
$587$	30.6554	1.26528	0.632641	+	0.774445i	$0.281971\pi$
$588$	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	$-0.836636\pi$
$593$	−42.4286	−1.74233	−0.871167	+	0.490986i	$0.836636\pi$
$594$	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	$-0.518564\pi$
$607$	−2.87209	−0.116575	−0.0582873	+	0.998300i	$0.518564\pi$
$608$	−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	$-0.260757\pi$
$613$	33.8112	1.36562	0.682811	+	0.730595i	$0.260757\pi$
$614$	−12.4531	−0.502564
$615$	0	0
$616$	−3.05025	−0.122898
$617$	26.0998	1.05074	0.525369	−	0.850875i	$-0.323927\pi$
$617$	26.0998	1.05074	0.525369	+	0.850875i	$0.323927\pi$
$618$	−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	$-0.264474\pi$
$643$	34.1937	1.34847	0.674235	+	0.738517i	$0.264474\pi$
$644$	8.16024	0.321559
$645$	0	0
$646$	−15.4229	−0.606804
$647$	25.8239	1.01524	0.507620	−	0.861581i	$-0.330525\pi$
$647$	25.8239	1.01524	0.507620	+	0.861581i	$0.330525\pi$
$648$	−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	$-0.364218\pi$
$653$	21.1459	0.827502	0.413751	+	0.910390i	$0.364218\pi$
$654$	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	$-0.376828\pi$
$673$	19.5798	0.754745	0.377372	+	0.926062i	$0.376828\pi$
$674$	3.91632	0.150851
$675$	0	0
$676$	−29.3440	−1.12862
$677$	−40.2314	−1.54622	−0.773109	−	0.634273i	$-0.781299\pi$
$677$	−40.2314	−1.54622	−0.773109	+	0.634273i	$0.781299\pi$
$678$	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	$-0.244296\pi$
$683$	37.6157	1.43933	0.719663	+	0.694323i	$0.244296\pi$
$684$	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	$-0.555870\pi$
$727$	−9.41666	−0.349245	−0.174622	+	0.984636i	$0.555870\pi$
$728$	−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	$-0.458180\pi$
$733$	7.09353	0.262006	0.131003	+	0.991382i	$0.458180\pi$
$734$	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	$-0.507030\pi$
$743$	−1.20393	−0.0441681	−0.0220840	+	0.999756i	$0.507030\pi$
$744$	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	$-0.326200\pi$
$757$	28.5746	1.03856	0.519282	+	0.854603i	$0.326200\pi$
$758$	−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	$-0.480292\pi$
$773$	3.44069	0.123753	0.0618765	+	0.998084i	$0.480292\pi$
$774$	−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	$-0.406824\pi$
$787$	16.1901	0.577116	0.288558	+	0.957462i	$0.406824\pi$
$788$	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	$-0.616030\pi$
$797$	−20.1288	−0.712999	−0.356499	+	0.934296i	$0.616030\pi$
$798$	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	$-0.908267\pi$
$823$	−55.0098	−1.91752	−0.958761	+	0.284215i	$0.908267\pi$
$824$	−52.3017	−1.82202
$825$	0	0
$826$	12.3755	0.430598
$827$	44.3005	1.54048	0.770239	−	0.637755i	$-0.220137\pi$
$827$	44.3005	1.54048	0.770239	+	0.637755i	$0.220137\pi$
$828$	−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	$-0.648769\pi$
$853$	−26.3172	−0.901083	−0.450541	+	0.892755i	$0.648769\pi$
$854$	−3.61772	−0.123796
$855$	0	0
$856$	53.2375	1.81962
$857$	−39.8259	−1.36043	−0.680213	−	0.733014i	$-0.738113\pi$
$857$	−39.8259	−1.36043	−0.680213	+	0.733014i	$0.738113\pi$
$858$	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	$-0.370532\pi$
$863$	23.2437	0.791225	0.395613	+	0.918418i	$0.370532\pi$
$864$	−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	$-0.101580\pi$
$877$	56.2380	1.89902	0.949511	+	0.313735i	$0.101580\pi$
$878$	−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	$-0.444601\pi$
$883$	10.2912	0.346328	0.173164	+	0.984893i	$0.444601\pi$
$884$	13.4805	0.453398
$885$	0	0
$886$	−19.4872	−0.654687
$887$	−40.8093	−1.37024	−0.685120	−	0.728430i	$-0.740250\pi$
$887$	−40.8093	−1.37024	−0.685120	+	0.728430i	$0.740250\pi$
$888$	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	$-0.615420\pi$
$907$	−21.3652	−0.709420	−0.354710	+	0.934976i	$0.615420\pi$
$908$	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	$-0.893746\pi$
$937$	−57.8417	−1.88960	−0.944802	+	0.327641i	$0.893746\pi$
$938$	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	$-0.655885\pi$
$947$	−28.9507	−0.940773	−0.470386	+	0.882461i	$0.655885\pi$
$948$	−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	$-0.465117\pi$
$953$	6.75259	0.218738	0.109369	+	0.994001i	$0.465117\pi$
$954$	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	$-0.511131\pi$
$967$	−2.17445	−0.0699257	−0.0349628	+	0.999389i	$0.511131\pi$
$968$	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	$-0.239913\pi$
$977$	45.5825	1.45831	0.729156	+	0.684347i	$0.239913\pi$
$978$	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	$-0.506628\pi$
$983$	−1.30557	−0.0416413	−0.0208206	+	0.999783i	$0.506628\pi$
$984$	−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	$-0.278602\pi$
$997$	40.4670	1.28160	0.640801	+	0.767707i	$0.278602\pi$
$998$	−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.bc.1.3	yes	7
5.2	odd	4		1925.2.b.q.1849.5		14
5.3	odd	4		1925.2.b.q.1849.10		14
5.4	even	2		1925.2.a.ba.1.5	✓	7

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

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