LMFDB - Embedded newform 7225.2.a.bq.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 21 x^{8} - 232 x^{7} + 44 x^{6} + 424 x^{5} - 137 x^{4} + \cdots + 17 $ x^12 - 4x^11 - 10x^10 + 52x^9 + 21x^8 - 232x^7 + 44x^6 + 424x^5 - 137x^4 - 316x^3 + 46x^2 + 92*x + 17
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.301687$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.301687 q^{2} +1.06101 q^{3} -1.90899 q^{4} -0.320094 q^{6} +2.50984 q^{7} +1.17929 q^{8} -1.87425 q^{9} +2.44557 q^{11} -2.02546 q^{12} +5.61335 q^{13} -0.757185 q^{14} +3.46219 q^{16} +0.565436 q^{18} -7.13107 q^{19} +2.66297 q^{21} -0.737797 q^{22} +0.860805 q^{23} +1.25124 q^{24} -1.69348 q^{26} -5.17165 q^{27} -4.79124 q^{28} +3.75143 q^{29} -2.24733 q^{31} -3.40308 q^{32} +2.59479 q^{33} +3.57791 q^{36} +5.10557 q^{37} +2.15135 q^{38} +5.95585 q^{39} +12.3496 q^{41} -0.803384 q^{42} +2.62014 q^{43} -4.66856 q^{44} -0.259694 q^{46} +2.30114 q^{47} +3.67344 q^{48} -0.700712 q^{49} -10.7158 q^{52} -2.77475 q^{53} +1.56022 q^{54} +2.95983 q^{56} -7.56616 q^{57} -1.13176 q^{58} +7.44167 q^{59} +0.906291 q^{61} +0.677989 q^{62} -4.70406 q^{63} -5.89773 q^{64} -0.782813 q^{66} -6.69889 q^{67} +0.913326 q^{69} -0.240766 q^{71} -2.21028 q^{72} +6.45746 q^{73} -1.54028 q^{74} +13.6131 q^{76} +6.13799 q^{77} -1.79680 q^{78} +14.7096 q^{79} +0.135560 q^{81} -3.72571 q^{82} -13.8984 q^{83} -5.08358 q^{84} -0.790460 q^{86} +3.98032 q^{87} +2.88404 q^{88} -0.395163 q^{89} +14.0886 q^{91} -1.64326 q^{92} -2.38445 q^{93} -0.694222 q^{94} -3.61071 q^{96} +8.95923 q^{97} +0.211396 q^{98} -4.58361 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} - 8 q^{3} + 12 q^{4} + 8 q^{6} - 16 q^{7} + 12 q^{8} + 12 q^{9} + 16 q^{11} - 16 q^{12} + 8 q^{13} - 16 q^{14} + 12 q^{16} - 4 q^{18} + 16 q^{21} - 16 q^{22} - 16 q^{23} + 16 q^{26} - 32 q^{27}+ \cdots + 56 q^{99}+O(q^{100}) $ 12 * q + 4 * q^2 - 8 * q^3 + 12 * q^4 + 8 * q^6 - 16 * q^7 + 12 * q^8 + 12 * q^9 + 16 * q^11 - 16 * q^12 + 8 * q^13 - 16 * q^14 + 12 * q^16 - 4 * q^18 + 16 * q^21 - 16 * q^22 - 16 * q^23 + 16 * q^26 - 32 * q^27 - 40 * q^28 + 16 * q^29 + 24 * q^31 + 28 * q^32 + 12 * q^36 - 24 * q^37 + 24 * q^38 + 8 * q^39 + 8 * q^41 + 16 * q^43 + 8 * q^44 + 40 * q^46 + 32 * q^47 + 24 * q^48 + 20 * q^49 + 24 * q^52 + 8 * q^54 - 24 * q^56 - 32 * q^57 + 16 * q^58 + 8 * q^59 + 24 * q^61 - 8 * q^62 - 48 * q^63 + 36 * q^64 + 40 * q^66 + 8 * q^67 + 48 * q^69 - 16 * q^71 - 12 * q^72 - 16 * q^73 + 16 * q^76 - 24 * q^77 + 24 * q^78 + 40 * q^79 + 36 * q^81 + 16 * q^82 + 40 * q^83 + 32 * q^84 - 8 * q^86 + 32 * q^87 - 48 * q^88 + 8 * q^89 + 72 * q^91 + 8 * q^92 - 24 * q^93 - 16 * q^94 + 8 * q^96 - 32 * q^97 + 60 * q^98 + 56 * 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 $

$2$	−0.301687	−0.213325	−0.106662	−	0.994295i	$-0.534016\pi$
$2$	−0.301687	−0.213325	−0.106662	+	0.994295i	$0.534016\pi$
$3$	1.06101	0.612577	0.306288	−	0.951939i	$-0.400913\pi$
$3$	1.06101	0.612577	0.306288	+	0.951939i	$0.400913\pi$
$4$	−1.90899	−0.954493
$5$	0	0
$5$	0	0
$6$	−0.320094	−0.130678
$7$	2.50984	0.948630	0.474315	−	0.880355i	$-0.342696\pi$
$7$	2.50984	0.948630	0.474315	+	0.880355i	$0.342696\pi$
$8$	1.17929	0.416942
$9$	−1.87425	−0.624750
$10$	0	0
$11$	2.44557	0.737368	0.368684	−	0.929555i	$-0.379808\pi$
$11$	2.44557	0.737368	0.368684	+	0.929555i	$0.379808\pi$
$12$	−2.02546	−0.584700
$13$	5.61335	1.55686	0.778432	−	0.627729i	$-0.216015\pi$
$13$	5.61335	1.55686	0.778432	+	0.627729i	$0.216015\pi$
$14$	−0.757185	−0.202366
$15$	0	0
$16$	3.46219	0.865549
$17$	0	0
$17$	0	0
$18$	0.565436	0.133275
$19$	−7.13107	−1.63598	−0.817990	−	0.575233i	$-0.804911\pi$
$19$	−7.13107	−1.63598	−0.817990	+	0.575233i	$0.804911\pi$
$20$	0	0
$21$	2.66297	0.581108
$22$	−0.737797	−0.157299
$23$	0.860805	0.179490	0.0897451	−	0.995965i	$-0.471395\pi$
$23$	0.860805	0.179490	0.0897451	+	0.995965i	$0.471395\pi$
$24$	1.25124	0.255409
$25$	0	0
$26$	−1.69348	−0.332118
$27$	−5.17165	−0.995284
$28$	−4.79124	−0.905460
$29$	3.75143	0.696624	0.348312	−	0.937379i	$-0.386755\pi$
$29$	3.75143	0.696624	0.348312	+	0.937379i	$0.386755\pi$
$30$	0	0
$31$	−2.24733	−0.403632	−0.201816	−	0.979423i	$-0.564684\pi$
$31$	−2.24733	−0.403632	−0.201816	+	0.979423i	$0.564684\pi$
$32$	−3.40308	−0.601585
$33$	2.59479	0.451694
$34$	0	0
$35$	0	0
$36$	3.57791	0.596319
$37$	5.10557	0.839351	0.419675	−	0.907674i	$-0.362144\pi$
$37$	5.10557	0.839351	0.419675	+	0.907674i	$0.362144\pi$
$38$	2.15135	0.348995
$39$	5.95585	0.953699
$40$	0	0
$41$	12.3496	1.92868	0.964340	−	0.264666i	$-0.0852617\pi$
$41$	12.3496	1.92868	0.964340	+	0.264666i	$0.0852617\pi$
$42$	−0.803384	−0.123965
$43$	2.62014	0.399567	0.199783	−	0.979840i	$-0.435976\pi$
$43$	2.62014	0.399567	0.199783	+	0.979840i	$0.435976\pi$
$44$	−4.66856	−0.703812
$45$	0	0
$46$	−0.259694	−0.0382897
$47$	2.30114	0.335655	0.167828	−	0.985816i	$-0.446325\pi$
$47$	2.30114	0.335655	0.167828	+	0.985816i	$0.446325\pi$
$48$	3.67344	0.530215
$49$	−0.700712	−0.100102
$50$	0	0
$51$	0	0
$52$	−10.7158	−1.48602
$53$	−2.77475	−0.381141	−0.190571	−	0.981674i	$-0.561034\pi$
$53$	−2.77475	−0.381141	−0.190571	+	0.981674i	$0.561034\pi$
$54$	1.56022	0.212319
$55$	0	0
$56$	2.95983	0.395523
$57$	−7.56616	−1.00216
$58$	−1.13176	−0.148607
$59$	7.44167	0.968823	0.484412	−	0.874840i	$-0.339034\pi$
$59$	7.44167	0.968823	0.484412	+	0.874840i	$0.339034\pi$
$60$	0	0
$61$	0.906291	0.116039	0.0580193	−	0.998315i	$-0.481522\pi$
$61$	0.906291	0.116039	0.0580193	+	0.998315i	$0.481522\pi$
$62$	0.677989	0.0861047
$63$	−4.70406	−0.592656
$64$	−5.89773	−0.737216
$65$	0	0
$66$	−0.782813	−0.0963576
$67$	−6.69889	−0.818399	−0.409200	−	0.912445i	$-0.634192\pi$
$67$	−6.69889	−0.818399	−0.409200	+	0.912445i	$0.634192\pi$
$68$	0	0
$69$	0.913326	0.109952
$70$	0	0
$71$	−0.240766	−0.0285737	−0.0142868	−	0.999898i	$-0.504548\pi$
$71$	−0.240766	−0.0285737	−0.0142868	+	0.999898i	$0.504548\pi$
$72$	−2.21028	−0.260484
$73$	6.45746	0.755788	0.377894	−	0.925849i	$-0.376648\pi$
$73$	6.45746	0.755788	0.377894	+	0.925849i	$0.376648\pi$
$74$	−1.54028	−0.179054
$75$	0	0
$76$	13.6131	1.56153
$77$	6.13799	0.699489
$78$	−1.79680	−0.203448
$79$	14.7096	1.65496	0.827479	−	0.561497i	$-0.189774\pi$
$79$	14.7096	1.65496	0.827479	+	0.561497i	$0.189774\pi$
$80$	0	0
$81$	0.135560	0.0150623
$82$	−3.72571	−0.411435
$83$	−13.8984	−1.52555	−0.762775	−	0.646664i	$-0.776164\pi$
$83$	−13.8984	−1.52555	−0.762775	+	0.646664i	$0.776164\pi$
$84$	−5.08358	−0.554664
$85$	0	0
$86$	−0.790460	−0.0852375
$87$	3.98032	0.426736
$88$	2.88404	0.307439
$89$	−0.395163	−0.0418872	−0.0209436	−	0.999781i	$-0.506667\pi$
$89$	−0.395163	−0.0418872	−0.0209436	+	0.999781i	$0.506667\pi$
$90$	0	0
$91$	14.0886	1.47689
$92$	−1.64326	−0.171322
$93$	−2.38445	−0.247255
$94$	−0.694222	−0.0716036
$95$	0	0
$96$	−3.61071	−0.368517
$97$	8.95923	0.909672	0.454836	−	0.890575i	$-0.349698\pi$
$97$	8.95923	0.909672	0.454836	+	0.890575i	$0.349698\pi$
$98$	0.211396	0.0213542
$99$	−4.58361	−0.460671
$100$	0	0
$101$	−15.2882	−1.52124	−0.760619	−	0.649199i	$-0.775104\pi$
$101$	−15.2882	−1.52124	−0.760619	+	0.649199i	$0.775104\pi$
$102$	0	0
$103$	14.7746	1.45579	0.727894	−	0.685690i	$-0.240499\pi$
$103$	14.7746	1.45579	0.727894	+	0.685690i	$0.240499\pi$
$104$	6.61977	0.649122
$105$	0	0
$106$	0.837105	0.0813068
$107$	−1.47078	−0.142185	−0.0710926	−	0.997470i	$-0.522649\pi$
$107$	−1.47078	−0.142185	−0.0710926	+	0.997470i	$0.522649\pi$
$108$	9.87260	0.949991
$109$	9.43771	0.903969	0.451984	−	0.892026i	$-0.350716\pi$
$109$	9.43771	0.903969	0.451984	+	0.892026i	$0.350716\pi$
$110$	0	0
$111$	5.41708	0.514167
$112$	8.68955	0.821085
$113$	−14.9610	−1.40742	−0.703708	−	0.710489i	$-0.748474\pi$
$113$	−14.9610	−1.40742	−0.703708	+	0.710489i	$0.748474\pi$
$114$	2.28261	0.213786
$115$	0	0
$116$	−7.16143	−0.664922
$117$	−10.5208	−0.972651
$118$	−2.24505	−0.206674
$119$	0	0
$120$	0	0
$121$	−5.01917	−0.456288
$122$	−0.273416	−0.0247539
$123$	13.1031	1.18146
$124$	4.29011	0.385264
$125$	0	0
$126$	1.41915	0.126428
$127$	5.95484	0.528406	0.264203	−	0.964467i	$-0.414891\pi$
$127$	5.95484	0.528406	0.264203	+	0.964467i	$0.414891\pi$
$128$	8.58542	0.758851
$129$	2.78000	0.244765
$130$	0	0
$131$	4.18730	0.365846	0.182923	−	0.983127i	$-0.441444\pi$
$131$	4.18730	0.365846	0.182923	+	0.983127i	$0.441444\pi$
$132$	−4.95341	−0.431139
$133$	−17.8978	−1.55194
$134$	2.02097	0.174585
$135$	0	0
$136$	0	0
$137$	−7.25998	−0.620262	−0.310131	−	0.950694i	$-0.600373\pi$
$137$	−7.25998	−0.620262	−0.310131	+	0.950694i	$0.600373\pi$
$138$	−0.275538	−0.0234554
$139$	−10.3932	−0.881540	−0.440770	−	0.897620i	$-0.645295\pi$
$139$	−10.3932	−0.881540	−0.440770	+	0.897620i	$0.645295\pi$
$140$	0	0
$141$	2.44154	0.205615
$142$	0.0726360	0.00609548
$143$	13.7279	1.14798
$144$	−6.48902	−0.540751
$145$	0	0
$146$	−1.94813	−0.161228
$147$	−0.743466	−0.0613200
$148$	−9.74646	−0.801154
$149$	−6.01765	−0.492985	−0.246492	−	0.969145i	$-0.579278\pi$
$149$	−6.01765	−0.492985	−0.246492	+	0.969145i	$0.579278\pi$
$150$	0	0
$151$	−10.4183	−0.847828	−0.423914	−	0.905702i	$-0.639344\pi$
$151$	−10.4183	−0.847828	−0.423914	+	0.905702i	$0.639344\pi$
$152$	−8.40959	−0.682108
$153$	0	0
$154$	−1.85175	−0.149218
$155$	0	0
$156$	−11.3696	−0.910298
$157$	−13.4073	−1.07002	−0.535008	−	0.844847i	$-0.679691\pi$
$157$	−13.4073	−1.07002	−0.535008	+	0.844847i	$0.679691\pi$
$158$	−4.43769	−0.353044
$159$	−2.94405	−0.233478
$160$	0	0
$161$	2.16048	0.170270
$162$	−0.0408968	−0.00321315
$163$	1.57513	0.123374	0.0616870	−	0.998096i	$-0.480352\pi$
$163$	1.57513	0.123374	0.0616870	+	0.998096i	$0.480352\pi$
$164$	−23.5752	−1.84091
$165$	0	0
$166$	4.19297	0.325438
$167$	17.6049	1.36231	0.681154	−	0.732140i	$-0.261478\pi$
$167$	17.6049	1.36231	0.681154	+	0.732140i	$0.261478\pi$
$168$	3.14042	0.242288
$169$	18.5098	1.42383
$170$	0	0
$171$	13.3654	1.02208
$172$	−5.00180	−0.381384
$173$	−18.8429	−1.43260	−0.716300	−	0.697793i	$-0.754166\pi$
$173$	−18.8429	−1.43260	−0.716300	+	0.697793i	$0.754166\pi$
$174$	−1.20081	−0.0910333
$175$	0	0
$176$	8.46705	0.638228
$177$	7.89572	0.593479
$178$	0.119215	0.00893557
$179$	3.31384	0.247688	0.123844	−	0.992302i	$-0.460478\pi$
$179$	3.31384	0.247688	0.123844	+	0.992302i	$0.460478\pi$
$180$	0	0
$181$	−15.8652	−1.17925	−0.589624	−	0.807678i	$-0.700724\pi$
$181$	−15.8652	−1.17925	−0.589624	+	0.807678i	$0.700724\pi$
$182$	−4.25035	−0.315057
$183$	0.961587	0.0710826
$184$	1.01514	0.0748370
$185$	0	0
$186$	0.719356	0.0527457
$187$	0	0
$188$	−4.39284	−0.320380
$189$	−12.9800	−0.944156
$190$	0	0
$191$	27.4943	1.98941	0.994707	−	0.102748i	$-0.0327636\pi$
$191$	27.4943	1.98941	0.994707	+	0.102748i	$0.0327636\pi$
$192$	−6.25757	−0.451601
$193$	14.7613	1.06254	0.531270	−	0.847203i	$-0.321715\pi$
$193$	14.7613	1.06254	0.531270	+	0.847203i	$0.321715\pi$
$194$	−2.70288	−0.194056
$195$	0	0
$196$	1.33765	0.0955464
$197$	−0.376406	−0.0268178	−0.0134089	−	0.999910i	$-0.504268\pi$
$197$	−0.376406	−0.0268178	−0.0134089	+	0.999910i	$0.504268\pi$
$198$	1.38282	0.0982724
$199$	−7.53004	−0.533791	−0.266895	−	0.963726i	$-0.585998\pi$
$199$	−7.53004	−0.533791	−0.266895	+	0.963726i	$0.585998\pi$
$200$	0	0
$201$	−7.10761	−0.501332
$202$	4.61226	0.324518
$203$	9.41549	0.660838
$204$	0	0
$205$	0	0
$206$	−4.45731	−0.310556
$207$	−1.61336	−0.112137
$208$	19.4345	1.34754
$209$	−17.4395	−1.20632
$210$	0	0
$211$	8.69248	0.598415	0.299207	−	0.954188i	$-0.403278\pi$
$211$	8.69248	0.598415	0.299207	+	0.954188i	$0.403278\pi$
$212$	5.29695	0.363796
$213$	−0.255456	−0.0175036
$214$	0.443714	0.0303316
$215$	0	0
$216$	−6.09887	−0.414975
$217$	−5.64043	−0.382897
$218$	−2.84723	−0.192839
$219$	6.85145	0.462978
$220$	0	0
$221$	0	0
$222$	−1.63426	−0.109684
$223$	−6.46310	−0.432801	−0.216401	−	0.976305i	$-0.569432\pi$
$223$	−6.46310	−0.432801	−0.216401	+	0.976305i	$0.569432\pi$
$224$	−8.54117	−0.570681
$225$	0	0
$226$	4.51355	0.300237
$227$	3.06060	0.203139	0.101570	−	0.994828i	$-0.467614\pi$
$227$	3.06060	0.203139	0.101570	+	0.994828i	$0.467614\pi$
$228$	14.4437	0.956557
$229$	−7.55923	−0.499528	−0.249764	−	0.968307i	$-0.580353\pi$
$229$	−7.55923	−0.499528	−0.249764	+	0.968307i	$0.580353\pi$
$230$	0	0
$231$	6.51250	0.428491
$232$	4.42403	0.290452
$233$	22.3478	1.46406	0.732028	−	0.681275i	$-0.238574\pi$
$233$	22.3478	1.46406	0.732028	+	0.681275i	$0.238574\pi$
$234$	3.17400	0.207491
$235$	0	0
$236$	−14.2060	−0.924735
$237$	15.6071	1.01379
$238$	0	0
$239$	−3.45981	−0.223797	−0.111898	−	0.993720i	$-0.535693\pi$
$239$	−3.45981	−0.223797	−0.111898	+	0.993720i	$0.535693\pi$
$240$	0	0
$241$	18.0262	1.16117	0.580585	−	0.814199i	$-0.302824\pi$
$241$	18.0262	1.16117	0.580585	+	0.814199i	$0.302824\pi$
$242$	1.51422	0.0973376
$243$	15.6588	1.00451
$244$	−1.73010	−0.110758
$245$	0	0
$246$	−3.95302	−0.252036
$247$	−40.0292	−2.54700
$248$	−2.65025	−0.168291
$249$	−14.7464	−0.934516
$250$	0	0
$251$	24.0478	1.51788	0.758941	−	0.651159i	$-0.225717\pi$
$251$	24.0478	1.51788	0.758941	+	0.651159i	$0.225717\pi$
$252$	8.97999	0.565686
$253$	2.10516	0.132350
$254$	−1.79650	−0.112722
$255$	0	0
$256$	9.20534	0.575334
$257$	14.8017	0.923302	0.461651	−	0.887062i	$-0.347257\pi$
$257$	14.8017	0.923302	0.461651	+	0.887062i	$0.347257\pi$
$258$	−0.838689	−0.0522145
$259$	12.8142	0.796233
$260$	0	0
$261$	−7.03113	−0.435216
$262$	−1.26325	−0.0780440
$263$	24.5163	1.51174	0.755871	−	0.654720i	$-0.227214\pi$
$263$	24.5163	1.51174	0.755871	+	0.654720i	$0.227214\pi$
$264$	3.06000	0.188330
$265$	0	0
$266$	5.39954	0.331067
$267$	−0.419273	−0.0256591
$268$	12.7881	0.781156
$269$	6.59468	0.402085	0.201042	−	0.979583i	$-0.435567\pi$
$269$	6.59468	0.402085	0.201042	+	0.979583i	$0.435567\pi$
$270$	0	0
$271$	26.1956	1.59127	0.795634	−	0.605778i	$-0.207138\pi$
$271$	26.1956	1.59127	0.795634	+	0.605778i	$0.207138\pi$
$272$	0	0
$273$	14.9482	0.904707
$274$	2.19024	0.132317
$275$	0	0
$276$	−1.74353	−0.104948
$277$	−9.56504	−0.574708	−0.287354	−	0.957825i	$-0.592776\pi$
$277$	−9.56504	−0.574708	−0.287354	+	0.957825i	$0.592776\pi$
$278$	3.13549	0.188054
$279$	4.21205	0.252169
$280$	0	0
$281$	12.4173	0.740755	0.370377	−	0.928881i	$-0.379228\pi$
$281$	12.4173	0.740755	0.370377	+	0.928881i	$0.379228\pi$
$282$	−0.736580	−0.0438627
$283$	−16.8433	−1.00123	−0.500616	−	0.865670i	$-0.666893\pi$
$283$	−16.8433	−1.00123	−0.500616	+	0.865670i	$0.666893\pi$
$284$	0.459619	0.0272734
$285$	0	0
$286$	−4.14152	−0.244893
$287$	30.9954	1.82960
$288$	6.37822	0.375840
$289$	0	0
$290$	0	0
$291$	9.50586	0.557244
$292$	−12.3272	−0.721394
$293$	20.8806	1.21986	0.609930	−	0.792456i	$-0.291198\pi$
$293$	20.8806	1.21986	0.609930	+	0.792456i	$0.291198\pi$
$294$	0.224294	0.0130811
$295$	0	0
$296$	6.02095	0.349960
$297$	−12.6476	−0.733890
$298$	1.81544	0.105166
$299$	4.83200	0.279442
$300$	0	0
$301$	6.57612	0.379041
$302$	3.14306	0.180863
$303$	−16.2210	−0.931875
$304$	−24.6891	−1.41602
$305$	0	0
$306$	0	0
$307$	29.9529	1.70950	0.854751	−	0.519038i	$-0.173710\pi$
$307$	29.9529	1.70950	0.854751	+	0.519038i	$0.173710\pi$
$308$	−11.7173	−0.667657
$309$	15.6761	0.891782
$310$	0	0
$311$	0.210323	0.0119264	0.00596318	−	0.999982i	$-0.498102\pi$
$311$	0.210323	0.0119264	0.00596318	+	0.999982i	$0.498102\pi$
$312$	7.02367	0.397637
$313$	32.4242	1.83273	0.916363	−	0.400349i	$-0.131111\pi$
$313$	32.4242	1.83273	0.916363	+	0.400349i	$0.131111\pi$
$314$	4.04479	0.228261
$315$	0	0
$316$	−28.0804	−1.57965
$317$	33.4391	1.87813	0.939063	−	0.343746i	$-0.111696\pi$
$317$	33.4391	1.87813	0.939063	+	0.343746i	$0.111696\pi$
$318$	0.888180	0.0498067
$319$	9.17441	0.513668
$320$	0	0
$321$	−1.56051	−0.0870994
$322$	−0.651789	−0.0363228
$323$	0	0
$324$	−0.258783	−0.0143768
$325$	0	0
$326$	−0.475197	−0.0263187
$327$	10.0135	0.553750
$328$	14.5637	0.804147
$329$	5.77548	0.318413
$330$	0	0
$331$	−0.885340	−0.0486627	−0.0243313	−	0.999704i	$-0.507746\pi$
$331$	−0.885340	−0.0486627	−0.0243313	+	0.999704i	$0.507746\pi$
$332$	26.5319	1.45613
$333$	−9.56912	−0.524384
$334$	−5.31117	−0.290614
$335$	0	0
$336$	9.21973	0.502977
$337$	−17.1565	−0.934575	−0.467287	−	0.884105i	$-0.654769\pi$
$337$	−17.1565	−0.934575	−0.467287	+	0.884105i	$0.654769\pi$
$338$	−5.58415	−0.303738
$339$	−15.8739	−0.862150
$340$	0	0
$341$	−5.49600	−0.297625
$342$	−4.03216	−0.218034
$343$	−19.3275	−1.04359
$344$	3.08990	0.166596
$345$	0	0
$346$	5.68466	0.305609
$347$	−30.9101	−1.65934	−0.829671	−	0.558253i	$-0.811472\pi$
$347$	−30.9101	−1.65934	−0.829671	+	0.558253i	$0.811472\pi$
$348$	−7.59838	−0.407316
$349$	3.37170	0.180483	0.0902414	−	0.995920i	$-0.471236\pi$
$349$	3.37170	0.180483	0.0902414	+	0.995920i	$0.471236\pi$
$350$	0	0
$351$	−29.0303	−1.54952
$352$	−8.32247	−0.443589
$353$	−23.7918	−1.26631	−0.633155	−	0.774025i	$-0.718240\pi$
$353$	−23.7918	−1.26631	−0.633155	+	0.774025i	$0.718240\pi$
$354$	−2.38203	−0.126604
$355$	0	0
$356$	0.754360	0.0399810
$357$	0	0
$358$	−0.999743	−0.0528381
$359$	31.7129	1.67374	0.836871	−	0.547399i	$-0.184382\pi$
$359$	31.7129	1.67374	0.836871	+	0.547399i	$0.184382\pi$
$360$	0	0
$361$	31.8521	1.67643
$362$	4.78631	0.251563
$363$	−5.32541	−0.279512
$364$	−26.8950	−1.40968
$365$	0	0
$366$	−0.290098	−0.0151637
$367$	−4.51687	−0.235779	−0.117889	−	0.993027i	$-0.537613\pi$
$367$	−4.51687	−0.235779	−0.117889	+	0.993027i	$0.537613\pi$
$368$	2.98027	0.155358
$369$	−23.1462	−1.20494
$370$	0	0
$371$	−6.96417	−0.361562
$372$	4.55187	0.236003
$373$	−5.12748	−0.265491	−0.132745	−	0.991150i	$-0.542379\pi$
$373$	−5.12748	−0.265491	−0.132745	+	0.991150i	$0.542379\pi$
$374$	0	0
$375$	0	0
$376$	2.71371	0.139949
$377$	21.0581	1.08455
$378$	3.91589	0.201412
$379$	5.53013	0.284064	0.142032	−	0.989862i	$-0.454636\pi$
$379$	5.53013	0.284064	0.142032	+	0.989862i	$0.454636\pi$
$380$	0	0
$381$	6.31816	0.323689
$382$	−8.29466	−0.424391
$383$	−20.4344	−1.04415	−0.522075	−	0.852899i	$-0.674842\pi$
$383$	−20.4344	−1.04415	−0.522075	+	0.852899i	$0.674842\pi$
$384$	9.10925	0.464854
$385$	0	0
$386$	−4.45328	−0.226666
$387$	−4.91079	−0.249629
$388$	−17.1030	−0.868275
$389$	5.18629	0.262955	0.131478	−	0.991319i	$-0.458028\pi$
$389$	5.18629	0.262955	0.131478	+	0.991319i	$0.458028\pi$
$390$	0	0
$391$	0	0
$392$	−0.826343	−0.0417366
$393$	4.44278	0.224109
$394$	0.113557	0.00572091
$395$	0	0
$396$	8.75005	0.439707
$397$	−6.17353	−0.309841	−0.154920	−	0.987927i	$-0.549512\pi$
$397$	−6.17353	−0.309841	−0.154920	+	0.987927i	$0.549512\pi$
$398$	2.27171	0.113871
$399$	−18.9898	−0.950681
$400$	0	0
$401$	9.49644	0.474229	0.237115	−	0.971482i	$-0.423798\pi$
$401$	9.49644	0.474229	0.237115	+	0.971482i	$0.423798\pi$
$402$	2.14427	0.106947
$403$	−12.6150	−0.628400
$404$	29.1850	1.45201
$405$	0	0
$406$	−2.84053	−0.140973
$407$	12.4860	0.618910
$408$	0	0
$409$	14.3886	0.711469	0.355734	−	0.934587i	$-0.384231\pi$
$409$	14.3886	0.711469	0.355734	+	0.934587i	$0.384231\pi$
$410$	0	0
$411$	−7.70294	−0.379958
$412$	−28.2046	−1.38954
$413$	18.6774	0.919055
$414$	0.486730	0.0239215
$415$	0	0
$416$	−19.1027	−0.936586
$417$	−11.0273	−0.540011
$418$	5.26128	0.257338
$419$	21.2233	1.03683	0.518413	−	0.855130i	$-0.326523\pi$
$419$	21.2233	1.03683	0.518413	+	0.855130i	$0.326523\pi$
$420$	0	0
$421$	−20.1672	−0.982887	−0.491444	−	0.870909i	$-0.663531\pi$
$421$	−20.1672	−0.982887	−0.491444	+	0.870909i	$0.663531\pi$
$422$	−2.62241	−0.127657
$423$	−4.31290	−0.209701
$424$	−3.27223	−0.158914
$425$	0	0
$426$	0.0770678	0.00373395
$427$	2.27464	0.110078
$428$	2.80769	0.135715
$429$	14.5655	0.703227
$430$	0	0
$431$	19.9686	0.961851	0.480926	−	0.876761i	$-0.340301\pi$
$431$	19.9686	0.961851	0.480926	+	0.876761i	$0.340301\pi$
$432$	−17.9052	−0.861466
$433$	18.2236	0.875770	0.437885	−	0.899031i	$-0.355728\pi$
$433$	18.2236	0.875770	0.437885	+	0.899031i	$0.355728\pi$
$434$	1.70164	0.0816815
$435$	0	0
$436$	−18.0165	−0.862832
$437$	−6.13846	−0.293642
$438$	−2.06699	−0.0987648
$439$	14.7332	0.703178	0.351589	−	0.936155i	$-0.385642\pi$
$439$	14.7332	0.703178	0.351589	+	0.936155i	$0.385642\pi$
$440$	0	0
$441$	1.31331	0.0625386
$442$	0	0
$443$	23.3335	1.10861	0.554305	−	0.832314i	$-0.312984\pi$
$443$	23.3335	1.10861	0.554305	+	0.832314i	$0.312984\pi$
$444$	−10.3411	−0.490768
$445$	0	0
$446$	1.94983	0.0923272
$447$	−6.38481	−0.301991
$448$	−14.8023	−0.699345
$449$	−1.71162	−0.0807764	−0.0403882	−	0.999184i	$-0.512859\pi$
$449$	−1.71162	−0.0807764	−0.0403882	+	0.999184i	$0.512859\pi$
$450$	0	0
$451$	30.2018	1.42215
$452$	28.5604	1.34337
$453$	−11.0539	−0.519359
$454$	−0.923344	−0.0433347
$455$	0	0
$456$	−8.92269	−0.417843
$457$	13.2168	0.618258	0.309129	−	0.951020i	$-0.399963\pi$
$457$	13.2168	0.618258	0.309129	+	0.951020i	$0.399963\pi$
$458$	2.28052	0.106562
$459$	0	0
$460$	0	0
$461$	−29.2906	−1.36420	−0.682099	−	0.731260i	$-0.738933\pi$
$461$	−29.2906	−1.36420	−0.682099	+	0.731260i	$0.738933\pi$
$462$	−1.96473	−0.0914077
$463$	−9.80371	−0.455617	−0.227808	−	0.973706i	$-0.573156\pi$
$463$	−9.80371	−0.455617	−0.227808	+	0.973706i	$0.573156\pi$
$464$	12.9882	0.602962
$465$	0	0
$466$	−6.74205	−0.312319
$467$	−13.6819	−0.633125	−0.316562	−	0.948572i	$-0.602529\pi$
$467$	−13.6819	−0.633125	−0.316562	+	0.948572i	$0.602529\pi$
$468$	20.0841	0.928388
$469$	−16.8131	−0.776358
$470$	0	0
$471$	−14.2253	−0.655466
$472$	8.77589	0.403943
$473$	6.40773	0.294628
$474$	−4.70845	−0.216266
$475$	0	0
$476$	0	0
$477$	5.20057	0.238118
$478$	1.04378	0.0477414
$479$	11.0100	0.503060	0.251530	−	0.967850i	$-0.419066\pi$
$479$	11.0100	0.503060	0.251530	+	0.967850i	$0.419066\pi$
$480$	0	0
$481$	28.6594	1.30676
$482$	−5.43827	−0.247707
$483$	2.29230	0.104303
$484$	9.58153	0.435524
$485$	0	0
$486$	−4.72404	−0.214287
$487$	−10.0942	−0.457413	−0.228707	−	0.973495i	$-0.573450\pi$
$487$	−10.0942	−0.457413	−0.228707	+	0.973495i	$0.573450\pi$
$488$	1.06878	0.0483813
$489$	1.67124	0.0755760
$490$	0	0
$491$	17.0827	0.770933	0.385466	−	0.922722i	$-0.374041\pi$
$491$	17.0827	0.770933	0.385466	+	0.922722i	$0.374041\pi$
$492$	−25.0136	−1.12770
$493$	0	0
$494$	12.0763	0.543338
$495$	0	0
$496$	−7.78068	−0.349363
$497$	−0.604284	−0.0271059
$498$	4.44880	0.199356
$499$	31.3595	1.40384	0.701921	−	0.712254i	$-0.252326\pi$
$499$	31.3595	1.40384	0.701921	+	0.712254i	$0.252326\pi$
$500$	0	0
$501$	18.6791	0.834519
$502$	−7.25490	−0.323802
$503$	−8.67535	−0.386815	−0.193407	−	0.981119i	$-0.561954\pi$
$503$	−8.67535	−0.386815	−0.193407	+	0.981119i	$0.561954\pi$
$504$	−5.54745	−0.247103
$505$	0	0
$506$	−0.635099	−0.0282336
$507$	19.6391	0.872203
$508$	−11.3677	−0.504360
$509$	14.1196	0.625840	0.312920	−	0.949780i	$-0.398693\pi$
$509$	14.1196	0.625840	0.312920	+	0.949780i	$0.398693\pi$
$510$	0	0
$511$	16.2072	0.716963
$512$	−19.9480	−0.881584
$513$	36.8794	1.62826
$514$	−4.46546	−0.196963
$515$	0	0
$516$	−5.30698	−0.233627
$517$	5.62760	0.247501
$518$	−3.86586	−0.169856
$519$	−19.9926	−0.877577
$520$	0	0
$521$	−17.6222	−0.772044	−0.386022	−	0.922490i	$-0.626151\pi$
$521$	−17.6222	−0.772044	−0.386022	+	0.922490i	$0.626151\pi$
$522$	2.12120	0.0928423
$523$	26.3853	1.15375	0.576875	−	0.816833i	$-0.304272\pi$
$523$	26.3853	1.15375	0.576875	+	0.816833i	$0.304272\pi$
$524$	−7.99349	−0.349197
$525$	0	0
$526$	−7.39626	−0.322492
$527$	0	0
$528$	8.98366	0.390963
$529$	−22.2590	−0.967783
$530$	0	0
$531$	−13.9476	−0.605272
$532$	34.1667	1.48131
$533$	69.3226	3.00269
$534$	0.126489	0.00547372
$535$	0	0
$536$	−7.89992	−0.341225
$537$	3.51603	0.151728
$538$	−1.98953	−0.0857746
$539$	−1.71364	−0.0738118
$540$	0	0
$541$	−4.40821	−0.189524	−0.0947620	−	0.995500i	$-0.530209\pi$
$541$	−4.40821	−0.189524	−0.0947620	+	0.995500i	$0.530209\pi$
$542$	−7.90286	−0.339457
$543$	−16.8331	−0.722379
$544$	0	0
$545$	0	0
$546$	−4.50968	−0.192996
$547$	−22.6638	−0.969034	−0.484517	−	0.874782i	$-0.661005\pi$
$547$	−22.6638	−0.969034	−0.484517	+	0.874782i	$0.661005\pi$
$548$	13.8592	0.592035
$549$	−1.69861	−0.0724951
$550$	0	0
$551$	−26.7517	−1.13966
$552$	1.07708	0.0458434
$553$	36.9187	1.56994
$554$	2.88565	0.122599
$555$	0	0
$556$	19.8405	0.841423
$557$	−11.4954	−0.487076	−0.243538	−	0.969891i	$-0.578308\pi$
$557$	−11.4954	−0.487076	−0.243538	+	0.969891i	$0.578308\pi$
$558$	−1.27072	−0.0537939
$559$	14.7078	0.622072
$560$	0	0
$561$	0	0
$562$	−3.74614	−0.158021
$563$	−13.5441	−0.570817	−0.285409	−	0.958406i	$-0.592129\pi$
$563$	−13.5441	−0.570817	−0.285409	+	0.958406i	$0.592129\pi$
$564$	−4.66086	−0.196258
$565$	0	0
$566$	5.08141	0.213587
$567$	0.340234	0.0142885
$568$	−0.283933	−0.0119136
$569$	15.8764	0.665574	0.332787	−	0.943002i	$-0.392011\pi$
$569$	15.8764	0.665574	0.332787	+	0.943002i	$0.392011\pi$
$570$	0	0
$571$	−3.54043	−0.148162	−0.0740812	−	0.997252i	$-0.523602\pi$
$571$	−3.54043	−0.148162	−0.0740812	+	0.997252i	$0.523602\pi$
$572$	−26.2063	−1.09574
$573$	29.1718	1.21867
$574$	−9.35092	−0.390300
$575$	0	0
$576$	11.0538	0.460575
$577$	18.5078	0.770492	0.385246	−	0.922814i	$-0.374117\pi$
$577$	18.5078	0.770492	0.385246	+	0.922814i	$0.374117\pi$
$578$	0	0
$579$	15.6619	0.650887
$580$	0	0
$581$	−34.8828	−1.44718
$582$	−2.86779	−0.118874
$583$	−6.78585	−0.281041
$584$	7.61521	0.315120
$585$	0	0
$586$	−6.29941	−0.260226
$587$	13.7261	0.566535	0.283268	−	0.959041i	$-0.408582\pi$
$587$	13.7261	0.566535	0.283268	+	0.959041i	$0.408582\pi$
$588$	1.41926	0.0585295
$589$	16.0258	0.660333
$590$	0	0
$591$	−0.399372	−0.0164280
$592$	17.6765	0.726499
$593$	−20.3238	−0.834598	−0.417299	−	0.908769i	$-0.637023\pi$
$593$	−20.3238	−0.834598	−0.417299	+	0.908769i	$0.637023\pi$
$594$	3.81563	0.156557
$595$	0	0
$596$	11.4876	0.470550
$597$	−7.98948	−0.326988
$598$	−1.45775	−0.0596119
$599$	−26.7277	−1.09206	−0.546032	−	0.837764i	$-0.683862\pi$
$599$	−26.7277	−1.09206	−0.546032	+	0.837764i	$0.683862\pi$
$600$	0	0
$601$	−27.6591	−1.12824	−0.564119	−	0.825693i	$-0.690784\pi$
$601$	−27.6591	−1.12824	−0.564119	+	0.825693i	$0.690784\pi$
$602$	−1.98393	−0.0808588
$603$	12.5554	0.511295
$604$	19.8883	0.809245
$605$	0	0
$606$	4.89367	0.198792
$607$	−23.6186	−0.958649	−0.479325	−	0.877638i	$-0.659118\pi$
$607$	−23.6186	−0.958649	−0.479325	+	0.877638i	$0.659118\pi$
$608$	24.2676	0.984180
$609$	9.98997	0.404814
$610$	0	0
$611$	12.9171	0.522570
$612$	0	0
$613$	−12.0396	−0.486275	−0.243137	−	0.969992i	$-0.578177\pi$
$613$	−12.0396	−0.486275	−0.243137	+	0.969992i	$0.578177\pi$
$614$	−9.03639	−0.364679
$615$	0	0
$616$	7.23847	0.291646
$617$	−15.5478	−0.625931	−0.312966	−	0.949764i	$-0.601322\pi$
$617$	−15.5478	−0.625931	−0.312966	+	0.949764i	$0.601322\pi$
$618$	−4.72927	−0.190239
$619$	33.4929	1.34619	0.673097	−	0.739554i	$-0.264964\pi$
$619$	33.4929	1.34619	0.673097	+	0.739554i	$0.264964\pi$
$620$	0	0
$621$	−4.45178	−0.178644
$622$	−0.0634518	−0.00254419
$623$	−0.991795	−0.0397354
$624$	20.6203	0.825473
$625$	0	0
$626$	−9.78196	−0.390966
$627$	−18.5036	−0.738963
$628$	25.5942	1.02132
$629$	0	0
$630$	0	0
$631$	−0.544614	−0.0216807	−0.0108404	−	0.999941i	$-0.503451\pi$
$631$	−0.544614	−0.0216807	−0.0108404	+	0.999941i	$0.503451\pi$
$632$	17.3469	0.690021
$633$	9.22284	0.366575
$634$	−10.0881	−0.400651
$635$	0	0
$636$	5.62014	0.222853
$637$	−3.93335	−0.155845
$638$	−2.76780	−0.109578
$639$	0.451256	0.0178514
$640$	0	0
$641$	−12.4241	−0.490723	−0.245361	−	0.969432i	$-0.578907\pi$
$641$	−12.4241	−0.490723	−0.245361	+	0.969432i	$0.578907\pi$
$642$	0.470786	0.0185805
$643$	−38.3641	−1.51293	−0.756467	−	0.654032i	$-0.773076\pi$
$643$	−38.3641	−1.51293	−0.756467	+	0.654032i	$0.773076\pi$
$644$	−4.12433	−0.162521
$645$	0	0
$646$	0	0
$647$	−19.6602	−0.772923	−0.386462	−	0.922305i	$-0.626303\pi$
$647$	−19.6602	−0.772923	−0.386462	+	0.922305i	$0.626303\pi$
$648$	0.159865	0.00628008
$649$	18.1992	0.714379
$650$	0	0
$651$	−5.98457	−0.234554
$652$	−3.00691	−0.117760
$653$	−31.1272	−1.21810	−0.609051	−	0.793131i	$-0.708449\pi$
$653$	−31.1272	−1.21810	−0.609051	+	0.793131i	$0.708449\pi$
$654$	−3.02095	−0.118129
$655$	0	0
$656$	42.7566	1.66937
$657$	−12.1029	−0.472179
$658$	−1.74239	−0.0679253
$659$	−4.15956	−0.162033	−0.0810167	−	0.996713i	$-0.525817\pi$
$659$	−4.15956	−0.162033	−0.0810167	+	0.996713i	$0.525817\pi$
$660$	0	0
$661$	−6.86044	−0.266840	−0.133420	−	0.991060i	$-0.542596\pi$
$661$	−6.86044	−0.266840	−0.133420	+	0.991060i	$0.542596\pi$
$662$	0.267095	0.0103810
$663$	0	0
$664$	−16.3903	−0.636066
$665$	0	0
$666$	2.88688	0.111864
$667$	3.22925	0.125037
$668$	−33.6075	−1.30031
$669$	−6.85744	−0.265124
$670$	0	0
$671$	2.21640	0.0855632
$672$	−9.06230	−0.349586
$673$	−17.2806	−0.666116	−0.333058	−	0.942906i	$-0.608081\pi$
$673$	−17.2806	−0.666116	−0.333058	+	0.942906i	$0.608081\pi$
$674$	5.17589	0.199368
$675$	0	0
$676$	−35.3348	−1.35903
$677$	−8.13384	−0.312609	−0.156304	−	0.987709i	$-0.549958\pi$
$677$	−8.13384	−0.312609	−0.156304	+	0.987709i	$0.549958\pi$
$678$	4.78894	0.183918
$679$	22.4862	0.862942
$680$	0	0
$681$	3.24734	0.124438
$682$	1.65807	0.0634908
$683$	−29.1370	−1.11489	−0.557447	−	0.830212i	$-0.688219\pi$
$683$	−29.1370	−1.11489	−0.557447	+	0.830212i	$0.688219\pi$
$684$	−25.5143	−0.975566
$685$	0	0
$686$	5.83086	0.222623
$687$	−8.02044	−0.305999
$688$	9.07142	0.345845
$689$	−15.5756	−0.593385
$690$	0	0
$691$	37.3851	1.42219	0.711097	−	0.703093i	$-0.248198\pi$
$691$	37.3851	1.42219	0.711097	+	0.703093i	$0.248198\pi$
$692$	35.9708	1.36741
$693$	−11.5041	−0.437006
$694$	9.32517	0.353979
$695$	0	0
$696$	4.69395	0.177924
$697$	0	0
$698$	−1.01720	−0.0385015
$699$	23.7114	0.896846
$700$	0	0
$701$	14.6423	0.553031	0.276515	−	0.961009i	$-0.410820\pi$
$701$	14.6423	0.553031	0.276515	+	0.961009i	$0.410820\pi$
$702$	8.75805	0.330551
$703$	−36.4082	−1.37316
$704$	−14.4233	−0.543599
$705$	0	0
$706$	7.17767	0.270135
$707$	−38.3710	−1.44309
$708$	−15.0728	−0.566471
$709$	28.3593	1.06506	0.532528	−	0.846413i	$-0.321242\pi$
$709$	28.3593	1.06506	0.532528	+	0.846413i	$0.321242\pi$
$710$	0	0
$711$	−27.5694	−1.03393
$712$	−0.466011	−0.0174645
$713$	−1.93451	−0.0724480
$714$	0	0
$715$	0	0
$716$	−6.32608	−0.236417
$717$	−3.67091	−0.137093
$718$	−9.56736	−0.357051
$719$	−12.6354	−0.471222	−0.235611	−	0.971848i	$-0.575709\pi$
$719$	−12.6354	−0.471222	−0.235611	+	0.971848i	$0.575709\pi$
$720$	0	0
$721$	37.0819	1.38100
$722$	−9.60936	−0.357624
$723$	19.1261	0.711306
$724$	30.2863	1.12558
$725$	0	0
$726$	1.60661	0.0596268
$727$	15.0242	0.557218	0.278609	−	0.960405i	$-0.410127\pi$
$727$	15.0242	0.557218	0.278609	+	0.960405i	$0.410127\pi$
$728$	16.6145	0.615776
$729$	16.2075	0.600277
$730$	0	0
$731$	0	0
$732$	−1.83566	−0.0678478
$733$	1.76679	0.0652578	0.0326289	−	0.999468i	$-0.489612\pi$
$733$	1.76679	0.0652578	0.0326289	+	0.999468i	$0.489612\pi$
$734$	1.36268	0.0502974
$735$	0	0
$736$	−2.92939	−0.107979
$737$	−16.3826	−0.603461
$738$	6.98290	0.257044
$739$	19.7149	0.725225	0.362613	−	0.931940i	$-0.381885\pi$
$739$	19.7149	0.725225	0.362613	+	0.931940i	$0.381885\pi$
$740$	0	0
$741$	−42.4715	−1.56023
$742$	2.10100	0.0771301
$743$	−28.3615	−1.04048	−0.520242	−	0.854019i	$-0.674158\pi$
$743$	−28.3615	−1.04048	−0.520242	+	0.854019i	$0.674158\pi$
$744$	−2.81195	−0.103091
$745$	0	0
$746$	1.54689	0.0566357
$747$	26.0491	0.953087
$748$	0	0
$749$	−3.69141	−0.134881
$750$	0	0
$751$	−17.9518	−0.655072	−0.327536	−	0.944839i	$-0.606218\pi$
$751$	−17.9518	−0.655072	−0.327536	+	0.944839i	$0.606218\pi$
$752$	7.96698	0.290526
$753$	25.5150	0.929819
$754$	−6.35296	−0.231361
$755$	0	0
$756$	24.7786	0.901190
$757$	43.9323	1.59675	0.798374	−	0.602162i	$-0.205694\pi$
$757$	43.9323	1.59675	0.798374	+	0.602162i	$0.205694\pi$
$758$	−1.66837	−0.0605978
$759$	2.23361	0.0810748
$760$	0	0
$761$	13.2781	0.481331	0.240666	−	0.970608i	$-0.422634\pi$
$761$	13.2781	0.481331	0.240666	+	0.970608i	$0.422634\pi$
$762$	−1.90611	−0.0690510
$763$	23.6871	0.857532
$764$	−52.4861	−1.89888
$765$	0	0
$766$	6.16480	0.222743
$767$	41.7728	1.50833
$768$	9.76700	0.352436
$769$	24.8906	0.897578	0.448789	−	0.893638i	$-0.351856\pi$
$769$	24.8906	0.897578	0.448789	+	0.893638i	$0.351856\pi$
$770$	0	0
$771$	15.7048	0.565593
$772$	−28.1790	−1.01419
$773$	−53.7048	−1.93163	−0.965813	−	0.259238i	$-0.916529\pi$
$773$	−53.7048	−1.93163	−0.965813	+	0.259238i	$0.916529\pi$
$774$	1.48152	0.0532521
$775$	0	0
$776$	10.5655	0.379280
$777$	13.5960	0.487754
$778$	−1.56463	−0.0560949
$779$	−88.0657	−3.15528
$780$	0	0
$781$	−0.588811	−0.0210693
$782$	0	0
$783$	−19.4011	−0.693339
$784$	−2.42600	−0.0866429
$785$	0	0
$786$	−1.34033	−0.0478079
$787$	−51.4777	−1.83498	−0.917491	−	0.397757i	$-0.869789\pi$
$787$	−51.4777	−1.83498	−0.917491	+	0.397757i	$0.869789\pi$
$788$	0.718554	0.0255974
$789$	26.0122	0.926058
$790$	0	0
$791$	−37.5498	−1.33512
$792$	−5.40541	−0.192073
$793$	5.08733	0.180656
$794$	1.86247	0.0660967
$795$	0	0
$796$	14.3747	0.509499
$797$	18.9183	0.670119	0.335059	−	0.942197i	$-0.391244\pi$
$797$	18.9183	0.670119	0.335059	+	0.942197i	$0.391244\pi$
$798$	5.72898	0.202804
$799$	0	0
$800$	0	0
$801$	0.740634	0.0261690
$802$	−2.86495	−0.101165
$803$	15.7922	0.557294
$804$	13.5683	0.478518
$805$	0	0
$806$	3.80579	0.134053
$807$	6.99705	0.246308
$808$	−18.0293	−0.634267
$809$	−30.2347	−1.06300	−0.531498	−	0.847059i	$-0.678371\pi$
$809$	−30.2347	−1.06300	−0.531498	+	0.847059i	$0.678371\pi$
$810$	0	0
$811$	17.1714	0.602969	0.301484	−	0.953471i	$-0.402518\pi$
$811$	17.1714	0.602969	0.301484	+	0.953471i	$0.402518\pi$
$812$	−17.9740	−0.630765
$813$	27.7939	0.974773
$814$	−3.76688	−0.132029
$815$	0	0
$816$	0	0
$817$	−18.6844	−0.653683
$818$	−4.34084	−0.151774
$819$	−26.4056	−0.922686
$820$	0	0
$821$	−43.7843	−1.52808	−0.764041	−	0.645168i	$-0.776787\pi$
$821$	−43.7843	−1.52808	−0.764041	+	0.645168i	$0.776787\pi$
$822$	2.32388	0.0810545
$823$	−25.2696	−0.880842	−0.440421	−	0.897791i	$-0.645171\pi$
$823$	−25.2696	−0.880842	−0.440421	+	0.897791i	$0.645171\pi$
$824$	17.4236	0.606979
$825$	0	0
$826$	−5.63472	−0.196057
$827$	22.0678	0.767371	0.383686	−	0.923464i	$-0.374655\pi$
$827$	22.0678	0.767371	0.383686	+	0.923464i	$0.374655\pi$
$828$	3.07989	0.107033
$829$	−20.9555	−0.727816	−0.363908	−	0.931435i	$-0.618558\pi$
$829$	−20.9555	−0.727816	−0.363908	+	0.931435i	$0.618558\pi$
$830$	0	0
$831$	−10.1486	−0.352052
$832$	−33.1060	−1.14774
$833$	0	0
$834$	3.32680	0.115198
$835$	0	0
$836$	33.2918	1.15142
$837$	11.6224	0.401728
$838$	−6.40279	−0.221181
$839$	−20.2251	−0.698247	−0.349123	−	0.937077i	$-0.613521\pi$
$839$	−20.2251	−0.698247	−0.349123	+	0.937077i	$0.613521\pi$
$840$	0	0
$841$	−14.9267	−0.514715
$842$	6.08417	0.209674
$843$	13.1749	0.453769
$844$	−16.5938	−0.571182
$845$	0	0
$846$	1.30115	0.0447343
$847$	−12.5973	−0.432849
$848$	−9.60672	−0.329896
$849$	−17.8710	−0.613331
$850$	0	0
$851$	4.39490	0.150655
$852$	0.487662	0.0167070
$853$	11.4942	0.393553	0.196776	−	0.980448i	$-0.436953\pi$
$853$	11.4942	0.393553	0.196776	+	0.980448i	$0.436953\pi$
$854$	−0.686230	−0.0234823
$855$	0	0
$856$	−1.73447	−0.0592830
$857$	28.0863	0.959411	0.479705	−	0.877430i	$-0.340744\pi$
$857$	28.0863	0.959411	0.479705	+	0.877430i	$0.340744\pi$
$858$	−4.39421	−0.150016
$859$	−3.15015	−0.107482	−0.0537409	−	0.998555i	$-0.517115\pi$
$859$	−3.15015	−0.107482	−0.0537409	+	0.998555i	$0.517115\pi$
$860$	0	0
$861$	32.8866	1.12077
$862$	−6.02425	−0.205187
$863$	34.5368	1.17565	0.587823	−	0.808989i	$-0.299985\pi$
$863$	34.5368	1.17565	0.587823	+	0.808989i	$0.299985\pi$
$864$	17.5995	0.598747
$865$	0	0
$866$	−5.49781	−0.186823
$867$	0	0
$868$	10.7675	0.365473
$869$	35.9734	1.22031
$870$	0	0
$871$	−37.6032	−1.27414
$872$	11.1298	0.376902
$873$	−16.7918	−0.568317
$874$	1.85189	0.0626412
$875$	0	0
$876$	−13.0793	−0.441909
$877$	−15.8248	−0.534367	−0.267184	−	0.963646i	$-0.586093\pi$
$877$	−15.8248	−0.534367	−0.267184	+	0.963646i	$0.586093\pi$
$878$	−4.44481	−0.150005
$879$	22.1546	0.747257
$880$	0	0
$881$	31.5651	1.06346	0.531728	−	0.846915i	$-0.321543\pi$
$881$	31.5651	1.06346	0.531728	+	0.846915i	$0.321543\pi$
$882$	−0.396208	−0.0133410
$883$	−20.2779	−0.682406	−0.341203	−	0.939990i	$-0.610834\pi$
$883$	−20.2779	−0.682406	−0.341203	+	0.939990i	$0.610834\pi$
$884$	0	0
$885$	0	0
$886$	−7.03942	−0.236494
$887$	48.8392	1.63986	0.819930	−	0.572465i	$-0.194013\pi$
$887$	48.8392	1.63986	0.819930	+	0.572465i	$0.194013\pi$
$888$	6.38831	0.214378
$889$	14.9457	0.501262
$890$	0	0
$891$	0.331523	0.0111064
$892$	12.3380	0.413106
$893$	−16.4096	−0.549125
$894$	1.92621	0.0644222
$895$	0	0
$896$	21.5480	0.719869
$897$	5.12682	0.171180
$898$	0.516374	0.0172316
$899$	−8.43070	−0.281180
$900$	0	0
$901$	0	0
$902$	−9.11148	−0.303379
$903$	6.97735	0.232192
$904$	−17.6434	−0.586810
$905$	0	0
$906$	3.33483	0.110792
$907$	−41.7064	−1.38484	−0.692419	−	0.721495i	$-0.743455\pi$
$907$	−41.7064	−1.38484	−0.692419	+	0.721495i	$0.743455\pi$
$908$	−5.84265	−0.193895
$909$	28.6540	0.950393
$910$	0	0
$911$	−6.40146	−0.212090	−0.106045	−	0.994361i	$-0.533819\pi$
$911$	−6.40146	−0.212090	−0.106045	+	0.994361i	$0.533819\pi$
$912$	−26.1955	−0.867420
$913$	−33.9896	−1.12489
$914$	−3.98735	−0.131890
$915$	0	0
$916$	14.4305	0.476796
$917$	10.5094	0.347052
$918$	0	0
$919$	−23.6812	−0.781170	−0.390585	−	0.920567i	$-0.627727\pi$
$919$	−23.6812	−0.781170	−0.390585	+	0.920567i	$0.627727\pi$
$920$	0	0
$921$	31.7804	1.04720
$922$	8.83658	0.291017
$923$	−1.35151	−0.0444854
$924$	−12.4323	−0.408991
$925$	0	0
$926$	2.95765	0.0971944
$927$	−27.6914	−0.909503
$928$	−12.7664	−0.419078
$929$	54.8951	1.80105	0.900524	−	0.434805i	$-0.143183\pi$
$929$	54.8951	1.80105	0.900524	+	0.434805i	$0.143183\pi$
$930$	0	0
$931$	4.99683	0.163764
$932$	−42.6617	−1.39743
$933$	0.223156	0.00730580
$934$	4.12766	0.135061
$935$	0	0
$936$	−12.4071	−0.405539
$937$	−6.77360	−0.221284	−0.110642	−	0.993860i	$-0.535291\pi$
$937$	−6.77360	−0.221284	−0.110642	+	0.993860i	$0.535291\pi$
$938$	5.07230	0.165616
$939$	34.4025	1.12268
$940$	0	0
$941$	25.2189	0.822111	0.411056	−	0.911610i	$-0.365160\pi$
$941$	25.2189	0.822111	0.411056	+	0.911610i	$0.365160\pi$
$942$	4.29158	0.139827
$943$	10.6306	0.346179
$944$	25.7645	0.838564
$945$	0	0
$946$	−1.93313	−0.0628514
$947$	24.5012	0.796183	0.398091	−	0.917346i	$-0.369673\pi$
$947$	24.5012	0.796183	0.398091	+	0.917346i	$0.369673\pi$
$948$	−29.7937	−0.967654
$949$	36.2480	1.17666
$950$	0	0
$951$	35.4793	1.15050
$952$	0	0
$953$	−30.3936	−0.984545	−0.492272	−	0.870441i	$-0.663834\pi$
$953$	−30.3936	−0.984545	−0.492272	+	0.870441i	$0.663834\pi$
$954$	−1.56894	−0.0507964
$955$	0	0
$956$	6.60473	0.213612
$957$	9.73417	0.314661
$958$	−3.32157	−0.107315
$959$	−18.2214	−0.588399
$960$	0	0
$961$	−25.9495	−0.837081
$962$	−8.64616	−0.278763
$963$	2.75660	0.0888302
$964$	−34.4118	−1.10833
$965$	0	0
$966$	−0.691557	−0.0222505
$967$	−43.9516	−1.41339	−0.706694	−	0.707520i	$-0.749814\pi$
$967$	−43.9516	−1.41339	−0.706694	+	0.707520i	$0.749814\pi$
$968$	−5.91906	−0.190246
$969$	0	0
$970$	0	0
$971$	42.3938	1.36048	0.680241	−	0.732989i	$-0.261875\pi$
$971$	42.3938	1.36048	0.680241	+	0.732989i	$0.261875\pi$
$972$	−29.8924	−0.958798
$973$	−26.0853	−0.836255
$974$	3.04530	0.0975776
$975$	0	0
$976$	3.13775	0.100437
$977$	23.7293	0.759168	0.379584	−	0.925157i	$-0.376067\pi$
$977$	23.7293	0.759168	0.379584	+	0.925157i	$0.376067\pi$
$978$	−0.504191	−0.0161222
$979$	−0.966400	−0.0308863
$980$	0	0
$981$	−17.6886	−0.564754
$982$	−5.15363	−0.164459
$983$	28.0626	0.895057	0.447529	−	0.894270i	$-0.352304\pi$
$983$	28.0626	0.895057	0.447529	+	0.894270i	$0.352304\pi$
$984$	15.4523	0.492602
$985$	0	0
$986$	0	0
$987$	6.12786	0.195052
$988$	76.4152	2.43109
$989$	2.25543	0.0717184
$990$	0	0
$991$	44.5466	1.41507	0.707535	−	0.706679i	$-0.249807\pi$
$991$	44.5466	1.41507	0.707535	+	0.706679i	$0.249807\pi$
$992$	7.64783	0.242819
$993$	−0.939358	−0.0298096
$994$	0.182305	0.00578235
$995$	0	0
$996$	28.1507	0.891989
$997$	20.5479	0.650757	0.325379	−	0.945584i	$-0.394508\pi$
$997$	20.5479	0.650757	0.325379	+	0.945584i	$0.394508\pi$
$998$	−9.46074	−0.299474
$999$	−26.4042	−0.835392

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bq.1.6		12
5.4	even	2		1445.2.a.q.1.7		12
17.11	odd	16		425.2.m.b.376.3		24
17.14	odd	16		425.2.m.b.26.3		24
17.16	even	2		7225.2.a.bs.1.6		12
85.4	even	4		1445.2.d.j.866.12		24
85.14	odd	16		85.2.l.a.26.4	✓	24
85.28	even	16		425.2.n.c.274.3		24
85.48	even	16		425.2.n.f.349.4		24
85.62	even	16		425.2.n.f.274.4		24
85.64	even	4		1445.2.d.j.866.11		24
85.79	odd	16		85.2.l.a.36.4	yes	24
85.82	even	16		425.2.n.c.349.3		24
85.84	even	2		1445.2.a.p.1.7		12
255.14	even	16		765.2.be.b.451.3		24
255.164	even	16		765.2.be.b.631.3		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.l.a.26.4	✓	24	85.14	odd	16
85.2.l.a.36.4	yes	24	85.79	odd	16
425.2.m.b.26.3		24	17.14	odd	16
425.2.m.b.376.3		24	17.11	odd	16
425.2.n.c.274.3		24	85.28	even	16
425.2.n.c.349.3		24	85.82	even	16
425.2.n.f.274.4		24	85.62	even	16
425.2.n.f.349.4		24	85.48	even	16
765.2.be.b.451.3		24	255.14	even	16
765.2.be.b.631.3		24	255.164	even	16
1445.2.a.p.1.7		12	85.84	even	2
1445.2.a.q.1.7		12	5.4	even	2
1445.2.d.j.866.11		24	85.64	even	4
1445.2.d.j.866.12		24	85.4	even	4
7225.2.a.bq.1.6		12	1.1	even	1	trivial
7225.2.a.bs.1.6		12	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q-0.301687 q^{2} +1.06101 q^{3} -1.90899 q^{4} -0.320094 q^{6} +2.50984 q^{7} +1.17929 q^{8} -1.87425 q^{9} +2.44557 q^{11} -2.02546 q^{12} +5.61335 q^{13} -0.757185 q^{14} +3.46219 q^{16} +0.565436 q^{18} -7.13107 q^{19} +2.66297 q^{21} -0.737797 q^{22} +0.860805 q^{23} +1.25124 q^{24} -1.69348 q^{26} -5.17165 q^{27} -4.79124 q^{28} +3.75143 q^{29} -2.24733 q^{31} -3.40308 q^{32} +2.59479 q^{33} +3.57791 q^{36} +5.10557 q^{37} +2.15135 q^{38} +5.95585 q^{39} +12.3496 q^{41} -0.803384 q^{42} +2.62014 q^{43} -4.66856 q^{44} -0.259694 q^{46} +2.30114 q^{47} +3.67344 q^{48} -0.700712 q^{49} -10.7158 q^{52} -2.77475 q^{53} +1.56022 q^{54} +2.95983 q^{56} -7.56616 q^{57} -1.13176 q^{58} +7.44167 q^{59} +0.906291 q^{61} +0.677989 q^{62} -4.70406 q^{63} -5.89773 q^{64} -0.782813 q^{66} -6.69889 q^{67} +0.913326 q^{69} -0.240766 q^{71} -2.21028 q^{72} +6.45746 q^{73} -1.54028 q^{74} +13.6131 q^{76} +6.13799 q^{77} -1.79680 q^{78} +14.7096 q^{79} +0.135560 q^{81} -3.72571 q^{82} -13.8984 q^{83} -5.08358 q^{84} -0.790460 q^{86} +3.98032 q^{87} +2.88404 q^{88} -0.395163 q^{89} +14.0886 q^{91} -1.64326 q^{92} -2.38445 q^{93} -0.694222 q^{94} -3.61071 q^{96} +8.95923 q^{97} +0.211396 q^{98} -4.58361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} - 8 q^{3} + 12 q^{4} + 8 q^{6} - 16 q^{7} + 12 q^{8} + 12 q^{9} + 16 q^{11} - 16 q^{12} + 8 q^{13} - 16 q^{14} + 12 q^{16} - 4 q^{18} + 16 q^{21} - 16 q^{22} - 16 q^{23} + 16 q^{26} - 32 q^{27}+ \cdots + 56 q^{99}+O(q^{100}) \) 12 * q + 4 * q^2 - 8 * q^3 + 12 * q^4 + 8 * q^6 - 16 * q^7 + 12 * q^8 + 12 * q^9 + 16 * q^11 - 16 * q^12 + 8 * q^13 - 16 * q^14 + 12 * q^16 - 4 * q^18 + 16 * q^21 - 16 * q^22 - 16 * q^23 + 16 * q^26 - 32 * q^27 - 40 * q^28 + 16 * q^29 + 24 * q^31 + 28 * q^32 + 12 * q^36 - 24 * q^37 + 24 * q^38 + 8 * q^39 + 8 * q^41 + 16 * q^43 + 8 * q^44 + 40 * q^46 + 32 * q^47 + 24 * q^48 + 20 * q^49 + 24 * q^52 + 8 * q^54 - 24 * q^56 - 32 * q^57 + 16 * q^58 + 8 * q^59 + 24 * q^61 - 8 * q^62 - 48 * q^63 + 36 * q^64 + 40 * q^66 + 8 * q^67 + 48 * q^69 - 16 * q^71 - 12 * q^72 - 16 * q^73 + 16 * q^76 - 24 * q^77 + 24 * q^78 + 40 * q^79 + 36 * q^81 + 16 * q^82 + 40 * q^83 + 32 * q^84 - 8 * q^86 + 32 * q^87 - 48 * q^88 + 8 * q^89 + 72 * q^91 + 8 * q^92 - 24 * q^93 - 16 * q^94 + 8 * q^96 - 32 * q^97 + 60 * q^98 + 56 * q^99

Introduction

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