LMFDB - Embedded newform 525.2.a.j.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [525,2,Mod(1,525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("525.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-1,3,5,0,-1,3,-9,3,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$4.19214610612$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	525.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.90321 q^{2} +1.00000 q^{3} +1.62222 q^{4} +1.90321 q^{6} +1.00000 q^{7} -0.719004 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.62222 q^{12} +6.42864 q^{13} +1.90321 q^{14} -4.61285 q^{16} -4.42864 q^{17} +1.90321 q^{18} -2.42864 q^{19} +1.00000 q^{21} +3.80642 q^{22} -1.37778 q^{23} -0.719004 q^{24} +12.2351 q^{26} +1.00000 q^{27} +1.62222 q^{28} +0.755569 q^{29} +5.18421 q^{31} -7.34122 q^{32} +2.00000 q^{33} -8.42864 q^{34} +1.62222 q^{36} -7.61285 q^{37} -4.62222 q^{38} +6.42864 q^{39} -8.23506 q^{41} +1.90321 q^{42} -10.1017 q^{43} +3.24443 q^{44} -2.62222 q^{46} -2.75557 q^{47} -4.61285 q^{48} +1.00000 q^{49} -4.42864 q^{51} +10.4286 q^{52} -9.18421 q^{53} +1.90321 q^{54} -0.719004 q^{56} -2.42864 q^{57} +1.43801 q^{58} +14.1017 q^{59} +6.85728 q^{61} +9.86665 q^{62} +1.00000 q^{63} -4.74620 q^{64} +3.80642 q^{66} -2.75557 q^{67} -7.18421 q^{68} -1.37778 q^{69} +2.00000 q^{71} -0.719004 q^{72} +1.57136 q^{73} -14.4889 q^{74} -3.93978 q^{76} +2.00000 q^{77} +12.2351 q^{78} -4.85728 q^{79} +1.00000 q^{81} -15.6731 q^{82} -11.6128 q^{83} +1.62222 q^{84} -19.2257 q^{86} +0.755569 q^{87} -1.43801 q^{88} +4.62222 q^{89} +6.42864 q^{91} -2.23506 q^{92} +5.18421 q^{93} -5.24443 q^{94} -7.34122 q^{96} +11.9398 q^{97} +1.90321 q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{3} + 5 q^{4} - q^{6} + 3 q^{7} - 9 q^{8} + 3 q^{9} + 6 q^{11} + 5 q^{12} + 6 q^{13} - q^{14} + 13 q^{16} - q^{18} + 6 q^{19} + 3 q^{21} - 2 q^{22} - 4 q^{23} - 9 q^{24} + 10 q^{26}+ \cdots + 6 q^{99}+O(q^{100}) $ 3 * q - q^2 + 3 * q^3 + 5 * q^4 - q^6 + 3 * q^7 - 9 * q^8 + 3 * q^9 + 6 * q^11 + 5 * q^12 + 6 * q^13 - q^14 + 13 * q^16 - q^18 + 6 * q^19 + 3 * q^21 - 2 * q^22 - 4 * q^23 - 9 * q^24 + 10 * q^26 + 3 * q^27 + 5 * q^28 + 2 * q^29 + 2 * q^31 - 29 * q^32 + 6 * q^33 - 12 * q^34 + 5 * q^36 + 4 * q^37 - 14 * q^38 + 6 * q^39 + 2 * q^41 - q^42 - 4 * q^43 + 10 * q^44 - 8 * q^46 - 8 * q^47 + 13 * q^48 + 3 * q^49 + 18 * q^52 - 14 * q^53 - q^54 - 9 * q^56 + 6 * q^57 + 18 * q^58 + 16 * q^59 - 6 * q^61 + 30 * q^62 + 3 * q^63 + 13 * q^64 - 2 * q^66 - 8 * q^67 - 8 * q^68 - 4 * q^69 + 6 * q^71 - 9 * q^72 + 18 * q^73 - 44 * q^74 + 2 * q^76 + 6 * q^77 + 10 * q^78 + 12 * q^79 + 3 * q^81 - 34 * q^82 - 8 * q^83 + 5 * q^84 - 4 * q^86 + 2 * q^87 - 18 * q^88 + 14 * q^89 + 6 * q^91 + 20 * q^92 + 2 * q^93 - 16 * q^94 - 29 * q^96 + 22 * q^97 - q^98 + 6 * q^99

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 $

$2$	1.90321	1.34577	0.672887	−	0.739745i	$-0.265054\pi$
$2$	1.90321	1.34577	0.672887	+	0.739745i	$0.265054\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.62222	0.811108
$5$	0	0
$5$	0	0
$6$	1.90321	0.776983
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.719004	−0.254206
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	1.62222	0.468293
$13$	6.42864	1.78298	0.891492	−	0.453037i	$-0.149659\pi$
$13$	6.42864	1.78298	0.891492	+	0.453037i	$0.149659\pi$
$14$	1.90321	0.508655
$15$	0	0
$16$	−4.61285	−1.15321
$17$	−4.42864	−1.07410	−0.537051	−	0.843550i	$-0.680462\pi$
$17$	−4.42864	−1.07410	−0.537051	+	0.843550i	$0.680462\pi$
$18$	1.90321	0.448591
$19$	−2.42864	−0.557168	−0.278584	−	0.960412i	$-0.589865\pi$
$19$	−2.42864	−0.557168	−0.278584	+	0.960412i	$0.589865\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	3.80642	0.811532
$23$	−1.37778	−0.287288	−0.143644	−	0.989629i	$-0.545882\pi$
$23$	−1.37778	−0.287288	−0.143644	+	0.989629i	$0.545882\pi$
$24$	−0.719004	−0.146766
$25$	0	0
$26$	12.2351	2.39949
$27$	1.00000	0.192450
$28$	1.62222	0.306570
$29$	0.755569	0.140306	0.0701528	−	0.997536i	$-0.477651\pi$
$29$	0.755569	0.140306	0.0701528	+	0.997536i	$0.477651\pi$
$30$	0	0
$31$	5.18421	0.931111	0.465556	−	0.885019i	$-0.345855\pi$
$31$	5.18421	0.931111	0.465556	+	0.885019i	$0.345855\pi$
$32$	−7.34122	−1.29776
$33$	2.00000	0.348155
$34$	−8.42864	−1.44550
$35$	0	0
$36$	1.62222	0.270369
$37$	−7.61285	−1.25154	−0.625772	−	0.780006i	$-0.715216\pi$
$37$	−7.61285	−1.25154	−0.625772	+	0.780006i	$0.715216\pi$
$38$	−4.62222	−0.749822
$39$	6.42864	1.02941
$40$	0	0
$41$	−8.23506	−1.28610	−0.643050	−	0.765824i	$-0.722331\pi$
$41$	−8.23506	−1.28610	−0.643050	+	0.765824i	$0.722331\pi$
$42$	1.90321	0.293672
$43$	−10.1017	−1.54050	−0.770248	−	0.637744i	$-0.779868\pi$
$43$	−10.1017	−1.54050	−0.770248	+	0.637744i	$0.779868\pi$
$44$	3.24443	0.489116
$45$	0	0
$46$	−2.62222	−0.386625
$47$	−2.75557	−0.401941	−0.200971	−	0.979597i	$-0.564410\pi$
$47$	−2.75557	−0.401941	−0.200971	+	0.979597i	$0.564410\pi$
$48$	−4.61285	−0.665807
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.42864	−0.620134
$52$	10.4286	1.44619
$53$	−9.18421	−1.26155	−0.630774	−	0.775967i	$-0.717263\pi$
$53$	−9.18421	−1.26155	−0.630774	+	0.775967i	$0.717263\pi$
$54$	1.90321	0.258994
$55$	0	0
$56$	−0.719004	−0.0960809
$57$	−2.42864	−0.321681
$58$	1.43801	0.188820
$59$	14.1017	1.83589	0.917943	−	0.396712i	$-0.129849\pi$
$59$	14.1017	1.83589	0.917943	+	0.396712i	$0.129849\pi$
$60$	0	0
$61$	6.85728	0.877985	0.438992	−	0.898491i	$-0.355336\pi$
$61$	6.85728	0.877985	0.438992	+	0.898491i	$0.355336\pi$
$62$	9.86665	1.25307
$63$	1.00000	0.125988
$64$	−4.74620	−0.593275
$65$	0	0
$66$	3.80642	0.468538
$67$	−2.75557	−0.336646	−0.168323	−	0.985732i	$-0.553835\pi$
$67$	−2.75557	−0.336646	−0.168323	+	0.985732i	$0.553835\pi$
$68$	−7.18421	−0.871213
$69$	−1.37778	−0.165866
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	−0.719004	−0.0847354
$73$	1.57136	0.183914	0.0919569	−	0.995763i	$-0.470688\pi$
$73$	1.57136	0.183914	0.0919569	+	0.995763i	$0.470688\pi$
$74$	−14.4889	−1.68430
$75$	0	0
$76$	−3.93978	−0.451923
$77$	2.00000	0.227921
$78$	12.2351	1.38535
$79$	−4.85728	−0.546487	−0.273243	−	0.961945i	$-0.588096\pi$
$79$	−4.85728	−0.546487	−0.273243	+	0.961945i	$0.588096\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−15.6731	−1.73080
$83$	−11.6128	−1.27468	−0.637338	−	0.770585i	$-0.719964\pi$
$83$	−11.6128	−1.27468	−0.637338	+	0.770585i	$0.719964\pi$
$84$	1.62222	0.176998
$85$	0	0
$86$	−19.2257	−2.07316
$87$	0.755569	0.0810055
$88$	−1.43801	−0.153292
$89$	4.62222	0.489954	0.244977	−	0.969529i	$-0.421220\pi$
$89$	4.62222	0.489954	0.244977	+	0.969529i	$0.421220\pi$
$90$	0	0
$91$	6.42864	0.673905
$92$	−2.23506	−0.233021
$93$	5.18421	0.537577
$94$	−5.24443	−0.540922
$95$	0	0
$96$	−7.34122	−0.749260
$97$	11.9398	1.21230	0.606150	−	0.795350i	$-0.292713\pi$
$97$	11.9398	1.21230	0.606150	+	0.795350i	$0.292713\pi$
$98$	1.90321	0.192253
$99$	2.00000	0.201008
$100$	0	0
$101$	1.47949	0.147215	0.0736076	−	0.997287i	$-0.476549\pi$
$101$	1.47949	0.147215	0.0736076	+	0.997287i	$0.476549\pi$
$102$	−8.42864	−0.834560
$103$	8.85728	0.872734	0.436367	−	0.899769i	$-0.356265\pi$
$103$	8.85728	0.872734	0.436367	+	0.899769i	$0.356265\pi$
$104$	−4.62222	−0.453246
$105$	0	0
$106$	−17.4795	−1.69776
$107$	−1.76494	−0.170623	−0.0853114	−	0.996354i	$-0.527189\pi$
$107$	−1.76494	−0.170623	−0.0853114	+	0.996354i	$0.527189\pi$
$108$	1.62222	0.156098
$109$	5.61285	0.537613	0.268807	−	0.963194i	$-0.413371\pi$
$109$	5.61285	0.537613	0.268807	+	0.963194i	$0.413371\pi$
$110$	0	0
$111$	−7.61285	−0.722580
$112$	−4.61285	−0.435873
$113$	11.2859	1.06169	0.530845	−	0.847469i	$-0.321875\pi$
$113$	11.2859	1.06169	0.530845	+	0.847469i	$0.321875\pi$
$114$	−4.62222	−0.432910
$115$	0	0
$116$	1.22570	0.113803
$117$	6.42864	0.594328
$118$	26.8385	2.47069
$119$	−4.42864	−0.405973
$120$	0	0
$121$	−7.00000	−0.636364
$122$	13.0509	1.18157
$123$	−8.23506	−0.742531
$124$	8.40990	0.755232
$125$	0	0
$126$	1.90321	0.169552
$127$	12.8573	1.14090	0.570450	−	0.821333i	$-0.306769\pi$
$127$	12.8573	1.14090	0.570450	+	0.821333i	$0.306769\pi$
$128$	5.64941	0.499342
$129$	−10.1017	−0.889406
$130$	0	0
$131$	−2.10171	−0.183627	−0.0918136	−	0.995776i	$-0.529266\pi$
$131$	−2.10171	−0.183627	−0.0918136	+	0.995776i	$0.529266\pi$
$132$	3.24443	0.282391
$133$	−2.42864	−0.210590
$134$	−5.24443	−0.453050
$135$	0	0
$136$	3.18421	0.273044
$137$	−15.9398	−1.36183	−0.680914	−	0.732364i	$-0.738417\pi$
$137$	−15.9398	−1.36183	−0.680914	+	0.732364i	$0.738417\pi$
$138$	−2.62222	−0.223218
$139$	11.6731	0.990097	0.495048	−	0.868865i	$-0.335150\pi$
$139$	11.6731	0.990097	0.495048	+	0.868865i	$0.335150\pi$
$140$	0	0
$141$	−2.75557	−0.232061
$142$	3.80642	0.319428
$143$	12.8573	1.07518
$144$	−4.61285	−0.384404
$145$	0	0
$146$	2.99063	0.247506
$147$	1.00000	0.0824786
$148$	−12.3497	−1.01514
$149$	−21.2257	−1.73888	−0.869438	−	0.494041i	$-0.835519\pi$
$149$	−21.2257	−1.73888	−0.869438	+	0.494041i	$0.835519\pi$
$150$	0	0
$151$	16.8573	1.37183	0.685913	−	0.727684i	$-0.259403\pi$
$151$	16.8573	1.37183	0.685913	+	0.727684i	$0.259403\pi$
$152$	1.74620	0.141636
$153$	−4.42864	−0.358034
$154$	3.80642	0.306730
$155$	0	0
$156$	10.4286	0.834959
$157$	10.4286	0.832296	0.416148	−	0.909297i	$-0.363380\pi$
$157$	10.4286	0.832296	0.416148	+	0.909297i	$0.363380\pi$
$158$	−9.24443	−0.735447
$159$	−9.18421	−0.728355
$160$	0	0
$161$	−1.37778	−0.108585
$162$	1.90321	0.149530
$163$	−20.8573	−1.63367	−0.816834	−	0.576873i	$-0.804273\pi$
$163$	−20.8573	−1.63367	−0.816834	+	0.576873i	$0.804273\pi$
$164$	−13.3590	−1.04317
$165$	0	0
$166$	−22.1017	−1.71543
$167$	−15.3461	−1.18752	−0.593760	−	0.804642i	$-0.702357\pi$
$167$	−15.3461	−1.18752	−0.593760	+	0.804642i	$0.702357\pi$
$168$	−0.719004	−0.0554723
$169$	28.3274	2.17903
$170$	0	0
$171$	−2.42864	−0.185723
$172$	−16.3872	−1.24951
$173$	−2.06022	−0.156636	−0.0783179	−	0.996928i	$-0.524955\pi$
$173$	−2.06022	−0.156636	−0.0783179	+	0.996928i	$0.524955\pi$
$174$	1.43801	0.109015
$175$	0	0
$176$	−9.22570	−0.695413
$177$	14.1017	1.05995
$178$	8.79706	0.659367
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	−12.1017	−0.899513	−0.449757	−	0.893151i	$-0.648489\pi$
$181$	−12.1017	−0.899513	−0.449757	+	0.893151i	$0.648489\pi$
$182$	12.2351	0.906923
$183$	6.85728	0.506905
$184$	0.990632	0.0730304
$185$	0	0
$186$	9.86665	0.723458
$187$	−8.85728	−0.647708
$188$	−4.47013	−0.326017
$189$	1.00000	0.0727393
$190$	0	0
$191$	0.488863	0.0353729	0.0176864	−	0.999844i	$-0.494370\pi$
$191$	0.488863	0.0353729	0.0176864	+	0.999844i	$0.494370\pi$
$192$	−4.74620	−0.342528
$193$	22.9590	1.65262	0.826312	−	0.563212i	$-0.190435\pi$
$193$	22.9590	1.65262	0.826312	+	0.563212i	$0.190435\pi$
$194$	22.7239	1.63148
$195$	0	0
$196$	1.62222	0.115873
$197$	1.18421	0.0843713	0.0421857	−	0.999110i	$-0.486568\pi$
$197$	1.18421	0.0843713	0.0421857	+	0.999110i	$0.486568\pi$
$198$	3.80642	0.270511
$199$	8.79706	0.623607	0.311803	−	0.950147i	$-0.399067\pi$
$199$	8.79706	0.623607	0.311803	+	0.950147i	$0.399067\pi$
$200$	0	0
$201$	−2.75557	−0.194363
$202$	2.81579	0.198118
$203$	0.755569	0.0530305
$204$	−7.18421	−0.502995
$205$	0	0
$206$	16.8573	1.17450
$207$	−1.37778	−0.0957626
$208$	−29.6543	−2.05616
$209$	−4.85728	−0.335985
$210$	0	0
$211$	23.2257	1.59892	0.799461	−	0.600717i	$-0.205118\pi$
$211$	23.2257	1.59892	0.799461	+	0.600717i	$0.205118\pi$
$212$	−14.8988	−1.02325
$213$	2.00000	0.137038
$214$	−3.35905	−0.229620
$215$	0	0
$216$	−0.719004	−0.0489220
$217$	5.18421	0.351927
$218$	10.6824	0.723506
$219$	1.57136	0.106183
$220$	0	0
$221$	−28.4701	−1.91511
$222$	−14.4889	−0.972429
$223$	15.2257	1.01959	0.509794	−	0.860297i	$-0.329722\pi$
$223$	15.2257	1.01959	0.509794	+	0.860297i	$0.329722\pi$
$224$	−7.34122	−0.490506
$225$	0	0
$226$	21.4795	1.42879
$227$	14.3684	0.953665	0.476833	−	0.878994i	$-0.341785\pi$
$227$	14.3684	0.953665	0.476833	+	0.878994i	$0.341785\pi$
$228$	−3.93978	−0.260918
$229$	−5.61285	−0.370907	−0.185454	−	0.982653i	$-0.559375\pi$
$229$	−5.61285	−0.370907	−0.185454	+	0.982653i	$0.559375\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	−0.543257	−0.0356666
$233$	−23.2859	−1.52551	−0.762756	−	0.646687i	$-0.776154\pi$
$233$	−23.2859	−1.52551	−0.762756	+	0.646687i	$0.776154\pi$
$234$	12.2351	0.799831
$235$	0	0
$236$	22.8760	1.48910
$237$	−4.85728	−0.315514
$238$	−8.42864	−0.546348
$239$	−8.48886	−0.549099	−0.274549	−	0.961573i	$-0.588529\pi$
$239$	−8.48886	−0.549099	−0.274549	+	0.961573i	$0.588529\pi$
$240$	0	0
$241$	−7.24443	−0.466655	−0.233327	−	0.972398i	$-0.574961\pi$
$241$	−7.24443	−0.466655	−0.233327	+	0.972398i	$0.574961\pi$
$242$	−13.3225	−0.856402
$243$	1.00000	0.0641500
$244$	11.1240	0.712140
$245$	0	0
$246$	−15.6731	−0.999278
$247$	−15.6128	−0.993422
$248$	−3.72746	−0.236694
$249$	−11.6128	−0.735934
$250$	0	0
$251$	27.6128	1.74291	0.871454	−	0.490478i	$-0.163178\pi$
$251$	27.6128	1.74291	0.871454	+	0.490478i	$0.163178\pi$
$252$	1.62222	0.102190
$253$	−2.75557	−0.173241
$254$	24.4701	1.53539
$255$	0	0
$256$	20.2444	1.26528
$257$	0.428639	0.0267378	0.0133689	−	0.999911i	$-0.495744\pi$
$257$	0.428639	0.0267378	0.0133689	+	0.999911i	$0.495744\pi$
$258$	−19.2257	−1.19694
$259$	−7.61285	−0.473039
$260$	0	0
$261$	0.755569	0.0467685
$262$	−4.00000	−0.247121
$263$	9.37778	0.578259	0.289129	−	0.957290i	$-0.406634\pi$
$263$	9.37778	0.578259	0.289129	+	0.957290i	$0.406634\pi$
$264$	−1.43801	−0.0885032
$265$	0	0
$266$	−4.62222	−0.283406
$267$	4.62222	0.282875
$268$	−4.47013	−0.273056
$269$	1.74620	0.106468	0.0532339	−	0.998582i	$-0.483047\pi$
$269$	1.74620	0.106468	0.0532339	+	0.998582i	$0.483047\pi$
$270$	0	0
$271$	−2.69535	−0.163731	−0.0818653	−	0.996643i	$-0.526088\pi$
$271$	−2.69535	−0.163731	−0.0818653	+	0.996643i	$0.526088\pi$
$272$	20.4286	1.23867
$273$	6.42864	0.389079
$274$	−30.3368	−1.83271
$275$	0	0
$276$	−2.23506	−0.134535
$277$	5.12399	0.307870	0.153935	−	0.988081i	$-0.450805\pi$
$277$	5.12399	0.307870	0.153935	+	0.988081i	$0.450805\pi$
$278$	22.2163	1.33245
$279$	5.18421	0.310370
$280$	0	0
$281$	23.9813	1.43060	0.715301	−	0.698816i	$-0.246290\pi$
$281$	23.9813	1.43060	0.715301	+	0.698816i	$0.246290\pi$
$282$	−5.24443	−0.312301
$283$	−2.36842	−0.140788	−0.0703939	−	0.997519i	$-0.522426\pi$
$283$	−2.36842	−0.140788	−0.0703939	+	0.997519i	$0.522426\pi$
$284$	3.24443	0.192522
$285$	0	0
$286$	24.4701	1.44695
$287$	−8.23506	−0.486100
$288$	−7.34122	−0.432585
$289$	2.61285	0.153697
$290$	0	0
$291$	11.9398	0.699922
$292$	2.54909	0.149174
$293$	−8.42864	−0.492406	−0.246203	−	0.969218i	$-0.579183\pi$
$293$	−8.42864	−0.492406	−0.246203	+	0.969218i	$0.579183\pi$
$294$	1.90321	0.110998
$295$	0	0
$296$	5.47367	0.318150
$297$	2.00000	0.116052
$298$	−40.3970	−2.34014
$299$	−8.85728	−0.512230
$300$	0	0
$301$	−10.1017	−0.582253
$302$	32.0830	1.84617
$303$	1.47949	0.0849947
$304$	11.2029	0.642533
$305$	0	0
$306$	−8.42864	−0.481833
$307$	22.5718	1.28824	0.644121	−	0.764923i	$-0.277223\pi$
$307$	22.5718	1.28824	0.644121	+	0.764923i	$0.277223\pi$
$308$	3.24443	0.184869
$309$	8.85728	0.503873
$310$	0	0
$311$	−24.0830	−1.36562	−0.682810	−	0.730596i	$-0.739242\pi$
$311$	−24.0830	−1.36562	−0.682810	+	0.730596i	$0.739242\pi$
$312$	−4.62222	−0.261681
$313$	−9.65433	−0.545695	−0.272848	−	0.962057i	$-0.587965\pi$
$313$	−9.65433	−0.545695	−0.272848	+	0.962057i	$0.587965\pi$
$314$	19.8479	1.12008
$315$	0	0
$316$	−7.87955	−0.443260
$317$	6.04149	0.339324	0.169662	−	0.985502i	$-0.445732\pi$
$317$	6.04149	0.339324	0.169662	+	0.985502i	$0.445732\pi$
$318$	−17.4795	−0.980201
$319$	1.51114	0.0846075
$320$	0	0
$321$	−1.76494	−0.0985092
$322$	−2.62222	−0.146130
$323$	10.7556	0.598456
$324$	1.62222	0.0901231
$325$	0	0
$326$	−39.6958	−2.19855
$327$	5.61285	0.310391
$328$	5.92104	0.326935
$329$	−2.75557	−0.151919
$330$	0	0
$331$	13.5111	0.742639	0.371320	−	0.928505i	$-0.378905\pi$
$331$	13.5111	0.742639	0.371320	+	0.928505i	$0.378905\pi$
$332$	−18.8385	−1.03390
$333$	−7.61285	−0.417181
$334$	−29.2070	−1.59813
$335$	0	0
$336$	−4.61285	−0.251651
$337$	−10.4889	−0.571365	−0.285682	−	0.958324i	$-0.592220\pi$
$337$	−10.4889	−0.571365	−0.285682	+	0.958324i	$0.592220\pi$
$338$	53.9131	2.93248
$339$	11.2859	0.612967
$340$	0	0
$341$	10.3684	0.561481
$342$	−4.62222	−0.249941
$343$	1.00000	0.0539949
$344$	7.26317	0.391604
$345$	0	0
$346$	−3.92104	−0.210796
$347$	16.7239	0.897787	0.448894	−	0.893585i	$-0.351818\pi$
$347$	16.7239	0.897787	0.448894	+	0.893585i	$0.351818\pi$
$348$	1.22570	0.0657042
$349$	−16.3684	−0.876181	−0.438091	−	0.898931i	$-0.644345\pi$
$349$	−16.3684	−0.876181	−0.438091	+	0.898931i	$0.644345\pi$
$350$	0	0
$351$	6.42864	0.343135
$352$	−14.6824	−0.782577
$353$	−0.549086	−0.0292249	−0.0146124	−	0.999893i	$-0.504651\pi$
$353$	−0.549086	−0.0292249	−0.0146124	+	0.999893i	$0.504651\pi$
$354$	26.8385	1.42645
$355$	0	0
$356$	7.49823	0.397405
$357$	−4.42864	−0.234388
$358$	19.0321	1.00588
$359$	−0.285442	−0.0150651	−0.00753253	−	0.999972i	$-0.502398\pi$
$359$	−0.285442	−0.0150651	−0.00753253	+	0.999972i	$0.502398\pi$
$360$	0	0
$361$	−13.1017	−0.689564
$362$	−23.0321	−1.21054
$363$	−7.00000	−0.367405
$364$	10.4286	0.546609
$365$	0	0
$366$	13.0509	0.682179
$367$	−1.71456	−0.0894992	−0.0447496	−	0.998998i	$-0.514249\pi$
$367$	−1.71456	−0.0894992	−0.0447496	+	0.998998i	$0.514249\pi$
$368$	6.35551	0.331304
$369$	−8.23506	−0.428700
$370$	0	0
$371$	−9.18421	−0.476820
$372$	8.40990	0.436033
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	−16.8573	−0.871669
$375$	0	0
$376$	1.98126	0.102176
$377$	4.85728	0.250163
$378$	1.90321	0.0978907
$379$	−4.85728	−0.249502	−0.124751	−	0.992188i	$-0.539813\pi$
$379$	−4.85728	−0.249502	−0.124751	+	0.992188i	$0.539813\pi$
$380$	0	0
$381$	12.8573	0.658698
$382$	0.930409	0.0476039
$383$	8.38715	0.428563	0.214282	−	0.976772i	$-0.431259\pi$
$383$	8.38715	0.428563	0.214282	+	0.976772i	$0.431259\pi$
$384$	5.64941	0.288295
$385$	0	0
$386$	43.6958	2.22406
$387$	−10.1017	−0.513499
$388$	19.3689	0.983307
$389$	8.95899	0.454239	0.227119	−	0.973867i	$-0.427069\pi$
$389$	8.95899	0.454239	0.227119	+	0.973867i	$0.427069\pi$
$390$	0	0
$391$	6.10171	0.308577
$392$	−0.719004	−0.0363152
$393$	−2.10171	−0.106017
$394$	2.25380	0.113545
$395$	0	0
$396$	3.24443	0.163039
$397$	2.54909	0.127935	0.0639675	−	0.997952i	$-0.479625\pi$
$397$	2.54909	0.127935	0.0639675	+	0.997952i	$0.479625\pi$
$398$	16.7427	0.839234
$399$	−2.42864	−0.121584
$400$	0	0
$401$	0.958989	0.0478896	0.0239448	−	0.999713i	$-0.492377\pi$
$401$	0.958989	0.0478896	0.0239448	+	0.999713i	$0.492377\pi$
$402$	−5.24443	−0.261568
$403$	33.3274	1.66016
$404$	2.40006	0.119407
$405$	0	0
$406$	1.43801	0.0713671
$407$	−15.2257	−0.754710
$408$	3.18421	0.157642
$409$	−31.9813	−1.58137	−0.790686	−	0.612222i	$-0.790276\pi$
$409$	−31.9813	−1.58137	−0.790686	+	0.612222i	$0.790276\pi$
$410$	0	0
$411$	−15.9398	−0.786251
$412$	14.3684	0.707881
$413$	14.1017	0.693900
$414$	−2.62222	−0.128875
$415$	0	0
$416$	−47.1941	−2.31388
$417$	11.6731	0.571633
$418$	−9.24443	−0.452160
$419$	−0.470127	−0.0229672	−0.0114836	−	0.999934i	$-0.503655\pi$
$419$	−0.470127	−0.0229672	−0.0114836	+	0.999934i	$0.503655\pi$
$420$	0	0
$421$	−33.6128	−1.63819	−0.819095	−	0.573658i	$-0.805524\pi$
$421$	−33.6128	−1.63819	−0.819095	+	0.573658i	$0.805524\pi$
$422$	44.2034	2.15179
$423$	−2.75557	−0.133980
$424$	6.60348	0.320693
$425$	0	0
$426$	3.80642	0.184422
$427$	6.85728	0.331847
$428$	−2.86311	−0.138394
$429$	12.8573	0.620755
$430$	0	0
$431$	11.7146	0.564270	0.282135	−	0.959375i	$-0.408957\pi$
$431$	11.7146	0.564270	0.282135	+	0.959375i	$0.408957\pi$
$432$	−4.61285	−0.221936
$433$	0.0602231	0.00289414	0.00144707	−	0.999999i	$-0.499539\pi$
$433$	0.0602231	0.00289414	0.00144707	+	0.999999i	$0.499539\pi$
$434$	9.86665	0.473614
$435$	0	0
$436$	9.10525	0.436062
$437$	3.34614	0.160068
$438$	2.99063	0.142898
$439$	−22.4286	−1.07046	−0.535230	−	0.844706i	$-0.679775\pi$
$439$	−22.4286	−1.07046	−0.535230	+	0.844706i	$0.679775\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−54.1847	−2.57730
$443$	23.9496	1.13788	0.568940	−	0.822379i	$-0.307353\pi$
$443$	23.9496	1.13788	0.568940	+	0.822379i	$0.307353\pi$
$444$	−12.3497	−0.586090
$445$	0	0
$446$	28.9777	1.37214
$447$	−21.2257	−1.00394
$448$	−4.74620	−0.224237
$449$	29.4291	1.38885	0.694423	−	0.719567i	$-0.255660\pi$
$449$	29.4291	1.38885	0.694423	+	0.719567i	$0.255660\pi$
$450$	0	0
$451$	−16.4701	−0.775548
$452$	18.3082	0.861145
$453$	16.8573	0.792024
$454$	27.3461	1.28342
$455$	0	0
$456$	1.74620	0.0817733
$457$	3.14272	0.147010	0.0735051	−	0.997295i	$-0.476581\pi$
$457$	3.14272	0.147010	0.0735051	+	0.997295i	$0.476581\pi$
$458$	−10.6824	−0.499158
$459$	−4.42864	−0.206711
$460$	0	0
$461$	−3.37778	−0.157319	−0.0786596	−	0.996902i	$-0.525064\pi$
$461$	−3.37778	−0.157319	−0.0786596	+	0.996902i	$0.525064\pi$
$462$	3.80642	0.177091
$463$	−20.8573	−0.969320	−0.484660	−	0.874703i	$-0.661057\pi$
$463$	−20.8573	−0.969320	−0.484660	+	0.874703i	$0.661057\pi$
$464$	−3.48532	−0.161802
$465$	0	0
$466$	−44.3180	−2.05299
$467$	14.3684	0.664891	0.332446	−	0.943122i	$-0.392126\pi$
$467$	14.3684	0.664891	0.332446	+	0.943122i	$0.392126\pi$
$468$	10.4286	0.482064
$469$	−2.75557	−0.127240
$470$	0	0
$471$	10.4286	0.480526
$472$	−10.1392	−0.466694
$473$	−20.2034	−0.928954
$474$	−9.24443	−0.424611
$475$	0	0
$476$	−7.18421	−0.329288
$477$	−9.18421	−0.420516
$478$	−16.1561	−0.738963
$479$	6.36842	0.290980	0.145490	−	0.989360i	$-0.453524\pi$
$479$	6.36842	0.290980	0.145490	+	0.989360i	$0.453524\pi$
$480$	0	0
$481$	−48.9403	−2.23148
$482$	−13.7877	−0.628012
$483$	−1.37778	−0.0626914
$484$	−11.3555	−0.516160
$485$	0	0
$486$	1.90321	0.0863314
$487$	−17.3274	−0.785180	−0.392590	−	0.919714i	$-0.628421\pi$
$487$	−17.3274	−0.785180	−0.392590	+	0.919714i	$0.628421\pi$
$488$	−4.93041	−0.223189
$489$	−20.8573	−0.943199
$490$	0	0
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	−13.3590	−0.602272
$493$	−3.34614	−0.150703
$494$	−29.7146	−1.33692
$495$	0	0
$496$	−23.9140	−1.07377
$497$	2.00000	0.0897123
$498$	−22.1017	−0.990401
$499$	23.3461	1.04512	0.522558	−	0.852603i	$-0.324978\pi$
$499$	23.3461	1.04512	0.522558	+	0.852603i	$0.324978\pi$
$500$	0	0
$501$	−15.3461	−0.685615
$502$	52.5531	2.34556
$503$	−0.387152	−0.0172623	−0.00863113	−	0.999963i	$-0.502747\pi$
$503$	−0.387152	−0.0172623	−0.00863113	+	0.999963i	$0.502747\pi$
$504$	−0.719004	−0.0320270
$505$	0	0
$506$	−5.24443	−0.233143
$507$	28.3274	1.25806
$508$	20.8573	0.925392
$509$	−29.9496	−1.32749	−0.663747	−	0.747957i	$-0.731035\pi$
$509$	−29.9496	−1.32749	−0.663747	+	0.747957i	$0.731035\pi$
$510$	0	0
$511$	1.57136	0.0695129
$512$	27.2306	1.20343
$513$	−2.42864	−0.107227
$514$	0.815792	0.0359830
$515$	0	0
$516$	−16.3872	−0.721404
$517$	−5.51114	−0.242380
$518$	−14.4889	−0.636604
$519$	−2.06022	−0.0904338
$520$	0	0
$521$	−18.5205	−0.811398	−0.405699	−	0.914007i	$-0.632972\pi$
$521$	−18.5205	−0.811398	−0.405699	+	0.914007i	$0.632972\pi$
$522$	1.43801	0.0629399
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−3.40943	−0.148942
$525$	0	0
$526$	17.8479	0.778206
$527$	−22.9590	−1.00011
$528$	−9.22570	−0.401497
$529$	−21.1017	−0.917466
$530$	0	0
$531$	14.1017	0.611962
$532$	−3.93978	−0.170811
$533$	−52.9403	−2.29310
$534$	8.79706	0.380686
$535$	0	0
$536$	1.98126	0.0855776
$537$	10.0000	0.431532
$538$	3.32339	0.143282
$539$	2.00000	0.0861461
$540$	0	0
$541$	14.5906	0.627298	0.313649	−	0.949539i	$-0.398449\pi$
$541$	14.5906	0.627298	0.313649	+	0.949539i	$0.398449\pi$
$542$	−5.12981	−0.220344
$543$	−12.1017	−0.519334
$544$	32.5116	1.39392
$545$	0	0
$546$	12.2351	0.523612
$547$	18.7556	0.801930	0.400965	−	0.916093i	$-0.368675\pi$
$547$	18.7556	0.801930	0.400965	+	0.916093i	$0.368675\pi$
$548$	−25.8578	−1.10459
$549$	6.85728	0.292662
$550$	0	0
$551$	−1.83500	−0.0781738
$552$	0.990632	0.0421641
$553$	−4.85728	−0.206553
$554$	9.75203	0.414324
$555$	0	0
$556$	18.9362	0.803075
$557$	−31.8765	−1.35065	−0.675325	−	0.737520i	$-0.735997\pi$
$557$	−31.8765	−1.35065	−0.675325	+	0.737520i	$0.735997\pi$
$558$	9.86665	0.417688
$559$	−64.9403	−2.74668
$560$	0	0
$561$	−8.85728	−0.373955
$562$	45.6414	1.92527
$563$	2.01874	0.0850796	0.0425398	−	0.999095i	$-0.486455\pi$
$563$	2.01874	0.0850796	0.0425398	+	0.999095i	$0.486455\pi$
$564$	−4.47013	−0.188226
$565$	0	0
$566$	−4.50760	−0.189468
$567$	1.00000	0.0419961
$568$	−1.43801	−0.0603375
$569$	−28.9590	−1.21402	−0.607012	−	0.794693i	$-0.707632\pi$
$569$	−28.9590	−1.21402	−0.607012	+	0.794693i	$0.707632\pi$
$570$	0	0
$571$	8.97773	0.375706	0.187853	−	0.982197i	$-0.439847\pi$
$571$	8.97773	0.375706	0.187853	+	0.982197i	$0.439847\pi$
$572$	20.8573	0.872087
$573$	0.488863	0.0204225
$574$	−15.6731	−0.654181
$575$	0	0
$576$	−4.74620	−0.197758
$577$	−28.6766	−1.19382	−0.596911	−	0.802307i	$-0.703606\pi$
$577$	−28.6766	−1.19382	−0.596911	+	0.802307i	$0.703606\pi$
$578$	4.97280	0.206841
$579$	22.9590	0.954143
$580$	0	0
$581$	−11.6128	−0.481782
$582$	22.7239	0.941937
$583$	−18.3684	−0.760742
$584$	−1.12981	−0.0467520
$585$	0	0
$586$	−16.0415	−0.662668
$587$	−45.2070	−1.86589	−0.932945	−	0.360018i	$-0.882771\pi$
$587$	−45.2070	−1.86589	−0.932945	+	0.360018i	$0.882771\pi$
$588$	1.62222	0.0668990
$589$	−12.5906	−0.518786
$590$	0	0
$591$	1.18421	0.0487118
$592$	35.1169	1.44330
$593$	18.2636	0.749998	0.374999	−	0.927025i	$-0.377643\pi$
$593$	18.2636	0.749998	0.374999	+	0.927025i	$0.377643\pi$
$594$	3.80642	0.156179
$595$	0	0
$596$	−34.4327	−1.41042
$597$	8.79706	0.360040
$598$	−16.8573	−0.689345
$599$	−22.7368	−0.929002	−0.464501	−	0.885573i	$-0.653766\pi$
$599$	−22.7368	−0.929002	−0.464501	+	0.885573i	$0.653766\pi$
$600$	0	0
$601$	0.488863	0.0199411	0.00997056	−	0.999950i	$-0.496826\pi$
$601$	0.488863	0.0199411	0.00997056	+	0.999950i	$0.496826\pi$
$602$	−19.2257	−0.783581
$603$	−2.75557	−0.112215
$604$	27.3461	1.11270
$605$	0	0
$606$	2.81579	0.114384
$607$	−20.2034	−0.820032	−0.410016	−	0.912078i	$-0.634477\pi$
$607$	−20.2034	−0.820032	−0.410016	+	0.912078i	$0.634477\pi$
$608$	17.8292	0.723069
$609$	0.755569	0.0306172
$610$	0	0
$611$	−17.7146	−0.716654
$612$	−7.18421	−0.290404
$613$	10.3684	0.418776	0.209388	−	0.977833i	$-0.432853\pi$
$613$	10.3684	0.418776	0.209388	+	0.977833i	$0.432853\pi$
$614$	42.9590	1.73368
$615$	0	0
$616$	−1.43801	−0.0579390
$617$	−39.2859	−1.58159	−0.790796	−	0.612080i	$-0.790333\pi$
$617$	−39.2859	−1.58159	−0.790796	+	0.612080i	$0.790333\pi$
$618$	16.8573	0.678099
$619$	42.8988	1.72425	0.862123	−	0.506698i	$-0.169134\pi$
$619$	42.8988	1.72425	0.862123	+	0.506698i	$0.169134\pi$
$620$	0	0
$621$	−1.37778	−0.0552886
$622$	−45.8350	−1.83782
$623$	4.62222	0.185185
$624$	−29.6543	−1.18712
$625$	0	0
$626$	−18.3742	−0.734383
$627$	−4.85728	−0.193981
$628$	16.9175	0.675082
$629$	33.7146	1.34429
$630$	0	0
$631$	15.3461	0.610920	0.305460	−	0.952205i	$-0.401190\pi$
$631$	15.3461	0.610920	0.305460	+	0.952205i	$0.401190\pi$
$632$	3.49240	0.138920
$633$	23.2257	0.923139
$634$	11.4982	0.456653
$635$	0	0
$636$	−14.8988	−0.590775
$637$	6.42864	0.254712
$638$	2.87601	0.113863
$639$	2.00000	0.0791188
$640$	0	0
$641$	30.6735	1.21153	0.605766	−	0.795643i	$-0.292867\pi$
$641$	30.6735	1.21153	0.605766	+	0.795643i	$0.292867\pi$
$642$	−3.35905	−0.132571
$643$	−49.0607	−1.93477	−0.967383	−	0.253320i	$-0.918477\pi$
$643$	−49.0607	−1.93477	−0.967383	+	0.253320i	$0.918477\pi$
$644$	−2.23506	−0.0880738
$645$	0	0
$646$	20.4701	0.805386
$647$	−15.3461	−0.603319	−0.301660	−	0.953416i	$-0.597541\pi$
$647$	−15.3461	−0.603319	−0.301660	+	0.953416i	$0.597541\pi$
$648$	−0.719004	−0.0282451
$649$	28.2034	1.10708
$650$	0	0
$651$	5.18421	0.203185
$652$	−33.8350	−1.32508
$653$	−19.4697	−0.761906	−0.380953	−	0.924594i	$-0.624404\pi$
$653$	−19.4697	−0.761906	−0.380953	+	0.924594i	$0.624404\pi$
$654$	10.6824	0.417716
$655$	0	0
$656$	37.9871	1.48315
$657$	1.57136	0.0613046
$658$	−5.24443	−0.204449
$659$	30.9403	1.20526	0.602631	−	0.798020i	$-0.294119\pi$
$659$	30.9403	1.20526	0.602631	+	0.798020i	$0.294119\pi$
$660$	0	0
$661$	47.7975	1.85911	0.929554	−	0.368685i	$-0.120192\pi$
$661$	47.7975	1.85911	0.929554	+	0.368685i	$0.120192\pi$
$662$	25.7146	0.999425
$663$	−28.4701	−1.10569
$664$	8.34968	0.324030
$665$	0	0
$666$	−14.4889	−0.561432
$667$	−1.04101	−0.0403081
$668$	−24.8948	−0.963207
$669$	15.2257	0.588659
$670$	0	0
$671$	13.7146	0.529445
$672$	−7.34122	−0.283194
$673$	27.8163	1.07224	0.536119	−	0.844142i	$-0.319890\pi$
$673$	27.8163	1.07224	0.536119	+	0.844142i	$0.319890\pi$
$674$	−19.9625	−0.768928
$675$	0	0
$676$	45.9532	1.76743
$677$	−19.0005	−0.730248	−0.365124	−	0.930959i	$-0.618973\pi$
$677$	−19.0005	−0.730248	−0.365124	+	0.930959i	$0.618973\pi$
$678$	21.4795	0.824915
$679$	11.9398	0.458207
$680$	0	0
$681$	14.3684	0.550599
$682$	19.7333	0.755627
$683$	4.52051	0.172972	0.0864862	−	0.996253i	$-0.472436\pi$
$683$	4.52051	0.172972	0.0864862	+	0.996253i	$0.472436\pi$
$684$	−3.93978	−0.150641
$685$	0	0
$686$	1.90321	0.0726650
$687$	−5.61285	−0.214143
$688$	46.5977	1.77652
$689$	−59.0420	−2.24932
$690$	0	0
$691$	−1.18421	−0.0450494	−0.0225247	−	0.999746i	$-0.507170\pi$
$691$	−1.18421	−0.0450494	−0.0225247	+	0.999746i	$0.507170\pi$
$692$	−3.34213	−0.127049
$693$	2.00000	0.0759737
$694$	31.8292	1.20822
$695$	0	0
$696$	−0.543257	−0.0205921
$697$	36.4701	1.38140
$698$	−31.1526	−1.17914
$699$	−23.2859	−0.880754
$700$	0	0
$701$	−26.6735	−1.00745	−0.503723	−	0.863865i	$-0.668037\pi$
$701$	−26.6735	−1.00745	−0.503723	+	0.863865i	$0.668037\pi$
$702$	12.2351	0.461783
$703$	18.4889	0.697321
$704$	−9.49240	−0.357758
$705$	0	0
$706$	−1.04503	−0.0393301
$707$	1.47949	0.0556421
$708$	22.8760	0.859733
$709$	−18.2034	−0.683644	−0.341822	−	0.939765i	$-0.611044\pi$
$709$	−18.2034	−0.683644	−0.341822	+	0.939765i	$0.611044\pi$
$710$	0	0
$711$	−4.85728	−0.182162
$712$	−3.32339	−0.124549
$713$	−7.14272	−0.267497
$714$	−8.42864	−0.315434
$715$	0	0
$716$	16.2222	0.606250
$717$	−8.48886	−0.317022
$718$	−0.543257	−0.0202742
$719$	4.85728	0.181146	0.0905730	−	0.995890i	$-0.471130\pi$
$719$	4.85728	0.181146	0.0905730	+	0.995890i	$0.471130\pi$
$720$	0	0
$721$	8.85728	0.329862
$722$	−24.9353	−0.927997
$723$	−7.24443	−0.269423
$724$	−19.6316	−0.729602
$725$	0	0
$726$	−13.3225	−0.494444
$727$	21.0607	0.781098	0.390549	−	0.920582i	$-0.372285\pi$
$727$	21.0607	0.781098	0.390549	+	0.920582i	$0.372285\pi$
$728$	−4.62222	−0.171311
$729$	1.00000	0.0370370
$730$	0	0
$731$	44.7368	1.65465
$732$	11.1240	0.411154
$733$	9.45091	0.349077	0.174539	−	0.984650i	$-0.444157\pi$
$733$	9.45091	0.349077	0.174539	+	0.984650i	$0.444157\pi$
$734$	−3.26317	−0.120446
$735$	0	0
$736$	10.1146	0.372830
$737$	−5.51114	−0.203005
$738$	−15.6731	−0.576934
$739$	−8.20342	−0.301768	−0.150884	−	0.988551i	$-0.548212\pi$
$739$	−8.20342	−0.301768	−0.150884	+	0.988551i	$0.548212\pi$
$740$	0	0
$741$	−15.6128	−0.573552
$742$	−17.4795	−0.641692
$743$	8.33677	0.305847	0.152923	−	0.988238i	$-0.451131\pi$
$743$	8.33677	0.305847	0.152923	+	0.988238i	$0.451131\pi$
$744$	−3.72746	−0.136655
$745$	0	0
$746$	−30.4514	−1.11490
$747$	−11.6128	−0.424892
$748$	−14.3684	−0.525361
$749$	−1.76494	−0.0644894
$750$	0	0
$751$	−25.9180	−0.945760	−0.472880	−	0.881127i	$-0.656786\pi$
$751$	−25.9180	−0.945760	−0.472880	+	0.881127i	$0.656786\pi$
$752$	12.7110	0.463523
$753$	27.6128	1.00627
$754$	9.24443	0.336662
$755$	0	0
$756$	1.62222	0.0589994
$757$	8.94025	0.324939	0.162470	−	0.986714i	$-0.448054\pi$
$757$	8.94025	0.324939	0.162470	+	0.986714i	$0.448054\pi$
$758$	−9.24443	−0.335773
$759$	−2.75557	−0.100021
$760$	0	0
$761$	−0.825636	−0.0299293	−0.0149646	−	0.999888i	$-0.504764\pi$
$761$	−0.825636	−0.0299293	−0.0149646	+	0.999888i	$0.504764\pi$
$762$	24.4701	0.886459
$763$	5.61285	0.203199
$764$	0.793040	0.0286912
$765$	0	0
$766$	15.9625	0.576750
$767$	90.6548	3.27336
$768$	20.2444	0.730508
$769$	21.2257	0.765418	0.382709	−	0.923869i	$-0.374991\pi$
$769$	21.2257	0.765418	0.382709	+	0.923869i	$0.374991\pi$
$770$	0	0
$771$	0.428639	0.0154371
$772$	37.2444	1.34046
$773$	29.4893	1.06066	0.530329	−	0.847792i	$-0.322068\pi$
$773$	29.4893	1.06066	0.530329	+	0.847792i	$0.322068\pi$
$774$	−19.2257	−0.691053
$775$	0	0
$776$	−8.58474	−0.308174
$777$	−7.61285	−0.273109
$778$	17.0509	0.611303
$779$	20.0000	0.716574
$780$	0	0
$781$	4.00000	0.143131
$782$	11.6128	0.415275
$783$	0.755569	0.0270018
$784$	−4.61285	−0.164745
$785$	0	0
$786$	−4.00000	−0.142675
$787$	−34.4514	−1.22806	−0.614030	−	0.789283i	$-0.710453\pi$
$787$	−34.4514	−1.22806	−0.614030	+	0.789283i	$0.710453\pi$
$788$	1.92104	0.0684343
$789$	9.37778	0.333858
$790$	0	0
$791$	11.2859	0.401281
$792$	−1.43801	−0.0510974
$793$	44.0830	1.56543
$794$	4.85145	0.172172
$795$	0	0
$796$	14.2707	0.505812
$797$	18.9175	0.670092	0.335046	−	0.942202i	$-0.391248\pi$
$797$	18.9175	0.670092	0.335046	+	0.942202i	$0.391248\pi$
$798$	−4.62222	−0.163625
$799$	12.2034	0.431726
$800$	0	0
$801$	4.62222	0.163318
$802$	1.82516	0.0644486
$803$	3.14272	0.110904
$804$	−4.47013	−0.157649
$805$	0	0
$806$	63.4291	2.23420
$807$	1.74620	0.0614692
$808$	−1.06376	−0.0374230
$809$	21.2257	0.746256	0.373128	−	0.927780i	$-0.378285\pi$
$809$	21.2257	0.746256	0.373128	+	0.927780i	$0.378285\pi$
$810$	0	0
$811$	−21.5081	−0.755251	−0.377625	−	0.925958i	$-0.623259\pi$
$811$	−21.5081	−0.755251	−0.377625	+	0.925958i	$0.623259\pi$
$812$	1.22570	0.0430135
$813$	−2.69535	−0.0945299
$814$	−28.9777	−1.01567
$815$	0	0
$816$	20.4286	0.715145
$817$	24.5334	0.858315
$818$	−60.8671	−2.12817
$819$	6.42864	0.224635
$820$	0	0
$821$	−46.2034	−1.61251	−0.806255	−	0.591568i	$-0.798509\pi$
$821$	−46.2034	−1.61251	−0.806255	+	0.591568i	$0.798509\pi$
$822$	−30.3368	−1.05812
$823$	−17.8350	−0.621689	−0.310845	−	0.950461i	$-0.600612\pi$
$823$	−17.8350	−0.621689	−0.310845	+	0.950461i	$0.600612\pi$
$824$	−6.36842	−0.221854
$825$	0	0
$826$	26.8385	0.933832
$827$	35.2128	1.22447	0.612234	−	0.790676i	$-0.290271\pi$
$827$	35.2128	1.22447	0.612234	+	0.790676i	$0.290271\pi$
$828$	−2.23506	−0.0776738
$829$	14.3872	0.499686	0.249843	−	0.968286i	$-0.419621\pi$
$829$	14.3872	0.499686	0.249843	+	0.968286i	$0.419621\pi$
$830$	0	0
$831$	5.12399	0.177749
$832$	−30.5116	−1.05780
$833$	−4.42864	−0.153443
$834$	22.2163	0.769289
$835$	0	0
$836$	−7.87955	−0.272520
$837$	5.18421	0.179192
$838$	−0.894751	−0.0309087
$839$	1.51114	0.0521703	0.0260851	−	0.999660i	$-0.491696\pi$
$839$	1.51114	0.0521703	0.0260851	+	0.999660i	$0.491696\pi$
$840$	0	0
$841$	−28.4291	−0.980314
$842$	−63.9724	−2.20463
$843$	23.9813	0.825959
$844$	37.6771	1.29690
$845$	0	0
$846$	−5.24443	−0.180307
$847$	−7.00000	−0.240523
$848$	42.3654	1.45483
$849$	−2.36842	−0.0812838
$850$	0	0
$851$	10.4889	0.359554
$852$	3.24443	0.111152
$853$	−15.4064	−0.527504	−0.263752	−	0.964591i	$-0.584960\pi$
$853$	−15.4064	−0.527504	−0.263752	+	0.964591i	$0.584960\pi$
$854$	13.0509	0.446591
$855$	0	0
$856$	1.26900	0.0433734
$857$	19.8578	0.678328	0.339164	−	0.940727i	$-0.389856\pi$
$857$	19.8578	0.678328	0.339164	+	0.940727i	$0.389856\pi$
$858$	24.4701	0.835396
$859$	2.42864	0.0828641	0.0414321	−	0.999141i	$-0.486808\pi$
$859$	2.42864	0.0828641	0.0414321	+	0.999141i	$0.486808\pi$
$860$	0	0
$861$	−8.23506	−0.280650
$862$	22.2953	0.759380
$863$	−39.2958	−1.33764	−0.668822	−	0.743423i	$-0.733201\pi$
$863$	−39.2958	−1.33764	−0.668822	+	0.743423i	$0.733201\pi$
$864$	−7.34122	−0.249753
$865$	0	0
$866$	0.114617	0.00389485
$867$	2.61285	0.0887370
$868$	8.40990	0.285451
$869$	−9.71456	−0.329544
$870$	0	0
$871$	−17.7146	−0.600235
$872$	−4.03566	−0.136665
$873$	11.9398	0.404100
$874$	6.36842	0.215415
$875$	0	0
$876$	2.54909	0.0861256
$877$	−56.2864	−1.90066	−0.950328	−	0.311249i	$-0.899253\pi$
$877$	−56.2864	−1.90066	−0.950328	+	0.311249i	$0.899253\pi$
$878$	−42.6865	−1.44060
$879$	−8.42864	−0.284291
$880$	0	0
$881$	−2.33677	−0.0787279	−0.0393640	−	0.999225i	$-0.512533\pi$
$881$	−2.33677	−0.0787279	−0.0393640	+	0.999225i	$0.512533\pi$
$882$	1.90321	0.0640845
$883$	33.7146	1.13459	0.567293	−	0.823516i	$-0.307991\pi$
$883$	33.7146	1.13459	0.567293	+	0.823516i	$0.307991\pi$
$884$	−46.1847	−1.55336
$885$	0	0
$886$	45.5812	1.53133
$887$	47.8992	1.60830	0.804150	−	0.594427i	$-0.202621\pi$
$887$	47.8992	1.60830	0.804150	+	0.594427i	$0.202621\pi$
$888$	5.47367	0.183684
$889$	12.8573	0.431219
$890$	0	0
$891$	2.00000	0.0670025
$892$	24.6994	0.826996
$893$	6.69228	0.223949
$894$	−40.3970	−1.35108
$895$	0	0
$896$	5.64941	0.188734
$897$	−8.85728	−0.295736
$898$	56.0098	1.86907
$899$	3.91703	0.130640
$900$	0	0
$901$	40.6735	1.35503
$902$	−31.3461	−1.04371
$903$	−10.1017	−0.336164
$904$	−8.11462	−0.269888
$905$	0	0
$906$	32.0830	1.06589
$907$	23.7591	0.788908	0.394454	−	0.918916i	$-0.370934\pi$
$907$	23.7591	0.788908	0.394454	+	0.918916i	$0.370934\pi$
$908$	23.3087	0.773525
$909$	1.47949	0.0490717
$910$	0	0
$911$	22.9403	0.760045	0.380022	−	0.924977i	$-0.375916\pi$
$911$	22.9403	0.760045	0.380022	+	0.924977i	$0.375916\pi$
$912$	11.2029	0.370967
$913$	−23.2257	−0.768658
$914$	5.98126	0.197843
$915$	0	0
$916$	−9.10525	−0.300846
$917$	−2.10171	−0.0694046
$918$	−8.42864	−0.278187
$919$	16.9777	0.560043	0.280022	−	0.959994i	$-0.409658\pi$
$919$	16.9777	0.560043	0.280022	+	0.959994i	$0.409658\pi$
$920$	0	0
$921$	22.5718	0.743767
$922$	−6.42864	−0.211716
$923$	12.8573	0.423202
$924$	3.24443	0.106734
$925$	0	0
$926$	−39.6958	−1.30449
$927$	8.85728	0.290911
$928$	−5.54680	−0.182082
$929$	39.3403	1.29071	0.645357	−	0.763881i	$-0.276709\pi$
$929$	39.3403	1.29071	0.645357	+	0.763881i	$0.276709\pi$
$930$	0	0
$931$	−2.42864	−0.0795954
$932$	−37.7748	−1.23735
$933$	−24.0830	−0.788441
$934$	27.3461	0.894793
$935$	0	0
$936$	−4.62222	−0.151082
$937$	−17.7748	−0.580677	−0.290338	−	0.956924i	$-0.593768\pi$
$937$	−17.7748	−0.580677	−0.290338	+	0.956924i	$0.593768\pi$
$938$	−5.24443	−0.171237
$939$	−9.65433	−0.315057
$940$	0	0
$941$	35.5812	1.15991	0.579957	−	0.814647i	$-0.303069\pi$
$941$	35.5812	1.15991	0.579957	+	0.814647i	$0.303069\pi$
$942$	19.8479	0.646680
$943$	11.3461	0.369481
$944$	−65.0490	−2.11717
$945$	0	0
$946$	−38.4514	−1.25016
$947$	30.5018	0.991174	0.495587	−	0.868558i	$-0.334953\pi$
$947$	30.5018	0.991174	0.495587	+	0.868558i	$0.334953\pi$
$948$	−7.87955	−0.255916
$949$	10.1017	0.327915
$950$	0	0
$951$	6.04149	0.195909
$952$	3.18421	0.103201
$953$	−51.1655	−1.65741	−0.828706	−	0.559684i	$-0.810923\pi$
$953$	−51.1655	−1.65741	−0.828706	+	0.559684i	$0.810923\pi$
$954$	−17.4795	−0.565920
$955$	0	0
$956$	−13.7708	−0.445378
$957$	1.51114	0.0488481
$958$	12.1204	0.391594
$959$	−15.9398	−0.514722
$960$	0	0
$961$	−4.12399	−0.133032
$962$	−93.1437	−3.00307
$963$	−1.76494	−0.0568743
$964$	−11.7520	−0.378507
$965$	0	0
$966$	−2.62222	−0.0843684
$967$	47.8992	1.54034	0.770168	−	0.637841i	$-0.220172\pi$
$967$	47.8992	1.54034	0.770168	+	0.637841i	$0.220172\pi$
$968$	5.03303	0.161768
$969$	10.7556	0.345519
$970$	0	0
$971$	40.6735	1.30528	0.652638	−	0.757670i	$-0.273662\pi$
$971$	40.6735	1.30528	0.652638	+	0.757670i	$0.273662\pi$
$972$	1.62222	0.0520326
$973$	11.6731	0.374221
$974$	−32.9777	−1.05667
$975$	0	0
$976$	−31.6316	−1.01250
$977$	−27.4893	−0.879462	−0.439731	−	0.898130i	$-0.644926\pi$
$977$	−27.4893	−0.879462	−0.439731	+	0.898130i	$0.644926\pi$
$978$	−39.6958	−1.26933
$979$	9.24443	0.295453
$980$	0	0
$981$	5.61285	0.179204
$982$	3.80642	0.121468
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	5.92104	0.188756
$985$	0	0
$986$	−6.36842	−0.202812
$987$	−2.75557	−0.0877107
$988$	−25.3274	−0.805772
$989$	13.9180	0.442566
$990$	0	0
$991$	−34.6923	−1.10204	−0.551018	−	0.834493i	$-0.685761\pi$
$991$	−34.6923	−1.10204	−0.551018	+	0.834493i	$0.685761\pi$
$992$	−38.0584	−1.20836
$993$	13.5111	0.428763
$994$	3.80642	0.120732
$995$	0	0
$996$	−18.8385	−0.596922
$997$	−28.6766	−0.908197	−0.454099	−	0.890951i	$-0.650039\pi$
$997$	−28.6766	−0.908197	−0.454099	+	0.890951i	$0.650039\pi$
$998$	44.4327	1.40649
$999$	−7.61285	−0.240860

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.a.j.1.3		3
3.2	odd	2		1575.2.a.x.1.1		3
4.3	odd	2		8400.2.a.dg.1.3		3
5.2	odd	4		105.2.d.b.64.5	yes	6
5.3	odd	4		105.2.d.b.64.2	✓	6
5.4	even	2		525.2.a.k.1.1		3
7.6	odd	2		3675.2.a.bi.1.3		3
15.2	even	4		315.2.d.e.64.2		6
15.8	even	4		315.2.d.e.64.5		6
15.14	odd	2		1575.2.a.w.1.3		3
20.3	even	4		1680.2.t.k.1009.2		6
20.7	even	4		1680.2.t.k.1009.5		6
20.19	odd	2		8400.2.a.dj.1.1		3
35.2	odd	12		735.2.q.e.214.5		12
35.3	even	12		735.2.q.f.79.5		12
35.12	even	12		735.2.q.f.214.5		12
35.13	even	4		735.2.d.b.589.2		6
35.17	even	12		735.2.q.f.79.2		12
35.18	odd	12		735.2.q.e.79.5		12
35.23	odd	12		735.2.q.e.214.2		12
35.27	even	4		735.2.d.b.589.5		6
35.32	odd	12		735.2.q.e.79.2		12
35.33	even	12		735.2.q.f.214.2		12
35.34	odd	2		3675.2.a.bj.1.1		3
60.23	odd	4		5040.2.t.v.1009.4		6
60.47	odd	4		5040.2.t.v.1009.3		6
105.62	odd	4		2205.2.d.l.1324.2		6
105.83	odd	4		2205.2.d.l.1324.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.d.b.64.2	✓	6	5.3	odd	4
105.2.d.b.64.5	yes	6	5.2	odd	4
315.2.d.e.64.2		6	15.2	even	4
315.2.d.e.64.5		6	15.8	even	4
525.2.a.j.1.3		3	1.1	even	1	trivial
525.2.a.k.1.1		3	5.4	even	2
735.2.d.b.589.2		6	35.13	even	4
735.2.d.b.589.5		6	35.27	even	4
735.2.q.e.79.2		12	35.32	odd	12
735.2.q.e.79.5		12	35.18	odd	12
735.2.q.e.214.2		12	35.23	odd	12
735.2.q.e.214.5		12	35.2	odd	12
735.2.q.f.79.2		12	35.17	even	12
735.2.q.f.79.5		12	35.3	even	12
735.2.q.f.214.2		12	35.33	even	12
735.2.q.f.214.5		12	35.12	even	12
1575.2.a.w.1.3		3	15.14	odd	2
1575.2.a.x.1.1		3	3.2	odd	2
1680.2.t.k.1009.2		6	20.3	even	4
1680.2.t.k.1009.5		6	20.7	even	4
2205.2.d.l.1324.2		6	105.62	odd	4
2205.2.d.l.1324.5		6	105.83	odd	4
3675.2.a.bi.1.3		3	7.6	odd	2
3675.2.a.bj.1.1		3	35.34	odd	2
5040.2.t.v.1009.3		6	60.47	odd	4
5040.2.t.v.1009.4		6	60.23	odd	4
8400.2.a.dg.1.3		3	4.3	odd	2
8400.2.a.dj.1.1		3	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)

\(f(q)\)	\(=\)	\(q+1.90321 q^{2} +1.00000 q^{3} +1.62222 q^{4} +1.90321 q^{6} +1.00000 q^{7} -0.719004 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.62222 q^{12} +6.42864 q^{13} +1.90321 q^{14} -4.61285 q^{16} -4.42864 q^{17} +1.90321 q^{18} -2.42864 q^{19} +1.00000 q^{21} +3.80642 q^{22} -1.37778 q^{23} -0.719004 q^{24} +12.2351 q^{26} +1.00000 q^{27} +1.62222 q^{28} +0.755569 q^{29} +5.18421 q^{31} -7.34122 q^{32} +2.00000 q^{33} -8.42864 q^{34} +1.62222 q^{36} -7.61285 q^{37} -4.62222 q^{38} +6.42864 q^{39} -8.23506 q^{41} +1.90321 q^{42} -10.1017 q^{43} +3.24443 q^{44} -2.62222 q^{46} -2.75557 q^{47} -4.61285 q^{48} +1.00000 q^{49} -4.42864 q^{51} +10.4286 q^{52} -9.18421 q^{53} +1.90321 q^{54} -0.719004 q^{56} -2.42864 q^{57} +1.43801 q^{58} +14.1017 q^{59} +6.85728 q^{61} +9.86665 q^{62} +1.00000 q^{63} -4.74620 q^{64} +3.80642 q^{66} -2.75557 q^{67} -7.18421 q^{68} -1.37778 q^{69} +2.00000 q^{71} -0.719004 q^{72} +1.57136 q^{73} -14.4889 q^{74} -3.93978 q^{76} +2.00000 q^{77} +12.2351 q^{78} -4.85728 q^{79} +1.00000 q^{81} -15.6731 q^{82} -11.6128 q^{83} +1.62222 q^{84} -19.2257 q^{86} +0.755569 q^{87} -1.43801 q^{88} +4.62222 q^{89} +6.42864 q^{91} -2.23506 q^{92} +5.18421 q^{93} -5.24443 q^{94} -7.34122 q^{96} +11.9398 q^{97} +1.90321 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{3} + 5 q^{4} - q^{6} + 3 q^{7} - 9 q^{8} + 3 q^{9} + 6 q^{11} + 5 q^{12} + 6 q^{13} - q^{14} + 13 q^{16} - q^{18} + 6 q^{19} + 3 q^{21} - 2 q^{22} - 4 q^{23} - 9 q^{24} + 10 q^{26}+ \cdots + 6 q^{99}+O(q^{100}) \) 3 * q - q^2 + 3 * q^3 + 5 * q^4 - q^6 + 3 * q^7 - 9 * q^8 + 3 * q^9 + 6 * q^11 + 5 * q^12 + 6 * q^13 - q^14 + 13 * q^16 - q^18 + 6 * q^19 + 3 * q^21 - 2 * q^22 - 4 * q^23 - 9 * q^24 + 10 * q^26 + 3 * q^27 + 5 * q^28 + 2 * q^29 + 2 * q^31 - 29 * q^32 + 6 * q^33 - 12 * q^34 + 5 * q^36 + 4 * q^37 - 14 * q^38 + 6 * q^39 + 2 * q^41 - q^42 - 4 * q^43 + 10 * q^44 - 8 * q^46 - 8 * q^47 + 13 * q^48 + 3 * q^49 + 18 * q^52 - 14 * q^53 - q^54 - 9 * q^56 + 6 * q^57 + 18 * q^58 + 16 * q^59 - 6 * q^61 + 30 * q^62 + 3 * q^63 + 13 * q^64 - 2 * q^66 - 8 * q^67 - 8 * q^68 - 4 * q^69 + 6 * q^71 - 9 * q^72 + 18 * q^73 - 44 * q^74 + 2 * q^76 + 6 * q^77 + 10 * q^78 + 12 * q^79 + 3 * q^81 - 34 * q^82 - 8 * q^83 + 5 * q^84 - 4 * q^86 + 2 * q^87 - 18 * q^88 + 14 * q^89 + 6 * q^91 + 20 * q^92 + 2 * q^93 - 16 * q^94 - 29 * q^96 + 22 * q^97 - q^98 + 6 * q^99

Introduction

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