LMFDB - Embedded newform 7225.2.a.r.1.2

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:	$1$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ 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)

$f(q)$	$=$	\(q-0.311108 q^{2} -2.21432 q^{3} -1.90321 q^{4} +0.688892 q^{6} +1.59210 q^{7} +1.21432 q^{8} +1.90321 q^{9} +1.31111 q^{11} +4.21432 q^{12} -3.52543 q^{13} -0.495316 q^{14} +3.42864 q^{16} -0.592104 q^{18} +4.42864 q^{19} -3.52543 q^{21} -0.407896 q^{22} -4.96989 q^{23} -2.68889 q^{24} +1.09679 q^{26} +2.42864 q^{27} -3.03011 q^{28} -8.42864 q^{29} +7.73975 q^{31} -3.49532 q^{32} -2.90321 q^{33} -3.62222 q^{36} -7.05086 q^{37} -1.37778 q^{38} +7.80642 q^{39} +3.67307 q^{41} +1.09679 q^{42} +2.47457 q^{43} -2.49532 q^{44} +1.54617 q^{46} -3.33185 q^{47} -7.59210 q^{48} -4.46520 q^{49} +6.70964 q^{52} -9.18421 q^{53} -0.755569 q^{54} +1.93332 q^{56} -9.80642 q^{57} +2.62222 q^{58} -1.37778 q^{59} +15.4193 q^{61} -2.40790 q^{62} +3.03011 q^{63} -5.76986 q^{64} +0.903212 q^{66} +9.13828 q^{67} +11.0049 q^{69} -10.5970 q^{71} +2.31111 q^{72} -5.57136 q^{73} +2.19358 q^{74} -8.42864 q^{76} +2.08742 q^{77} -2.42864 q^{78} +7.87310 q^{79} -11.0874 q^{81} -1.14272 q^{82} +7.19850 q^{83} +6.70964 q^{84} -0.769859 q^{86} +18.6637 q^{87} +1.59210 q^{88} +11.6271 q^{89} -5.61285 q^{91} +9.45875 q^{92} -17.1383 q^{93} +1.03657 q^{94} +7.73975 q^{96} +15.4795 q^{97} +1.38916 q^{98} +2.49532 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} + 2 q^{6} - 2 q^{7} - 3 q^{8} - q^{9} + 4 q^{11} + 6 q^{12} - 4 q^{13} + 12 q^{14} - 3 q^{16} + 5 q^{18} - 4 q^{21} - 8 q^{22} - 8 q^{23} - 8 q^{24} + 10 q^{26} - 6 q^{27} - 16 q^{28}+ \cdots - 6 q^{99}+O(q^{100}) $ 3 * q - q^2 + q^4 + 2 * q^6 - 2 * q^7 - 3 * q^8 - q^9 + 4 * q^11 + 6 * q^12 - 4 * q^13 + 12 * q^14 - 3 * q^16 + 5 * q^18 - 4 * q^21 - 8 * q^22 - 8 * q^23 - 8 * q^24 + 10 * q^26 - 6 * q^27 - 16 * q^28 - 12 * q^29 + 10 * q^31 + 3 * q^32 - 2 * q^33 - 11 * q^36 - 8 * q^37 - 4 * q^38 + 10 * q^39 - 2 * q^41 + 10 * q^42 + 14 * q^43 + 6 * q^44 - 22 * q^46 + 10 * q^47 - 16 * q^48 + 7 * q^49 - 14 * q^53 - 2 * q^54 + 6 * q^56 - 16 * q^57 + 8 * q^58 - 4 * q^59 + 6 * q^61 - 14 * q^62 + 16 * q^63 - 11 * q^64 - 4 * q^66 - 6 * q^67 + 8 * q^71 + 7 * q^72 - 30 * q^73 + 20 * q^74 - 12 * q^76 - 14 * q^77 + 6 * q^78 + 10 * q^79 - 13 * q^81 - 30 * q^82 + 2 * q^83 + 4 * q^86 + 16 * q^87 - 2 * q^88 + 2 * q^89 + 10 * q^91 + 22 * q^92 - 18 * q^93 - 4 * q^94 + 10 * q^96 + 20 * q^97 - 43 * 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 $

$2$	−0.311108	−0.219986	−0.109993	−	0.993932i	$-0.535083\pi$
$2$	−0.311108	−0.219986	−0.109993	+	0.993932i	$0.535083\pi$
$3$	−2.21432	−1.27844	−0.639219	−	0.769025i	$-0.720742\pi$
$3$	−2.21432	−1.27844	−0.639219	+	0.769025i	$0.720742\pi$
$4$	−1.90321	−0.951606
$5$	0	0
$5$	0	0
$6$	0.688892	0.281239
$7$	1.59210	0.601759	0.300879	−	0.953662i	$-0.402720\pi$
$7$	1.59210	0.601759	0.300879	+	0.953662i	$0.402720\pi$
$8$	1.21432	0.429327
$9$	1.90321	0.634404
$10$	0	0
$11$	1.31111	0.395314	0.197657	−	0.980271i	$-0.436667\pi$
$11$	1.31111	0.395314	0.197657	+	0.980271i	$0.436667\pi$
$12$	4.21432	1.21657
$13$	−3.52543	−0.977778	−0.488889	−	0.872346i	$-0.662598\pi$
$13$	−3.52543	−0.977778	−0.488889	+	0.872346i	$0.662598\pi$
$14$	−0.495316	−0.132379
$15$	0	0
$16$	3.42864	0.857160
$17$	0	0
$17$	0	0
$18$	−0.592104	−0.139560
$19$	4.42864	1.01600	0.508000	−	0.861357i	$-0.330385\pi$
$19$	4.42864	1.01600	0.508000	+	0.861357i	$0.330385\pi$
$20$	0	0
$21$	−3.52543	−0.769311
$22$	−0.407896	−0.0869637
$23$	−4.96989	−1.03629	−0.518147	−	0.855292i	$-0.673378\pi$
$23$	−4.96989	−1.03629	−0.518147	+	0.855292i	$0.673378\pi$
$24$	−2.68889	−0.548868
$25$	0	0
$26$	1.09679	0.215098
$27$	2.42864	0.467392
$28$	−3.03011	−0.572637
$29$	−8.42864	−1.56516	−0.782580	−	0.622551i	$-0.786096\pi$
$29$	−8.42864	−1.56516	−0.782580	+	0.622551i	$0.786096\pi$
$30$	0	0
$31$	7.73975	1.39010	0.695050	−	0.718962i	$-0.255382\pi$
$31$	7.73975	1.39010	0.695050	+	0.718962i	$0.255382\pi$
$32$	−3.49532	−0.617890
$33$	−2.90321	−0.505384
$34$	0	0
$35$	0	0
$36$	−3.62222	−0.603703
$37$	−7.05086	−1.15915	−0.579577	−	0.814918i	$-0.696782\pi$
$37$	−7.05086	−1.15915	−0.579577	+	0.814918i	$0.696782\pi$
$38$	−1.37778	−0.223506
$39$	7.80642	1.25003
$40$	0	0
$41$	3.67307	0.573637	0.286819	−	0.957985i	$-0.407402\pi$
$41$	3.67307	0.573637	0.286819	+	0.957985i	$0.407402\pi$
$42$	1.09679	0.169238
$43$	2.47457	0.377369	0.188684	−	0.982038i	$-0.439578\pi$
$43$	2.47457	0.377369	0.188684	+	0.982038i	$0.439578\pi$
$44$	−2.49532	−0.376183
$45$	0	0
$46$	1.54617	0.227970
$47$	−3.33185	−0.486000	−0.243000	−	0.970026i	$-0.578132\pi$
$47$	−3.33185	−0.486000	−0.243000	+	0.970026i	$0.578132\pi$
$48$	−7.59210	−1.09583
$49$	−4.46520	−0.637886
$50$	0	0
$51$	0	0
$52$	6.70964	0.930459
$53$	−9.18421	−1.26155	−0.630774	−	0.775967i	$-0.717263\pi$
$53$	−9.18421	−1.26155	−0.630774	+	0.775967i	$0.717263\pi$
$54$	−0.755569	−0.102820
$55$	0	0
$56$	1.93332	0.258351
$57$	−9.80642	−1.29889
$58$	2.62222	0.344314
$59$	−1.37778	−0.179372	−0.0896861	−	0.995970i	$-0.528586\pi$
$59$	−1.37778	−0.179372	−0.0896861	+	0.995970i	$0.528586\pi$
$60$	0	0
$61$	15.4193	1.97424	0.987118	−	0.159996i	$-0.0511480\pi$
$61$	15.4193	1.97424	0.987118	+	0.159996i	$0.0511480\pi$
$62$	−2.40790	−0.305803
$63$	3.03011	0.381758
$64$	−5.76986	−0.721232
$65$	0	0
$66$	0.903212	0.111178
$67$	9.13828	1.11642	0.558209	−	0.829700i	$-0.311489\pi$
$67$	9.13828	1.11642	0.558209	+	0.829700i	$0.311489\pi$
$68$	0	0
$69$	11.0049	1.32484
$70$	0	0
$71$	−10.5970	−1.25764	−0.628818	−	0.777553i	$-0.716461\pi$
$71$	−10.5970	−1.25764	−0.628818	+	0.777553i	$0.716461\pi$
$72$	2.31111	0.272367
$73$	−5.57136	−0.652078	−0.326039	−	0.945356i	$-0.605714\pi$
$73$	−5.57136	−0.652078	−0.326039	+	0.945356i	$0.605714\pi$
$74$	2.19358	0.254998
$75$	0	0
$76$	−8.42864	−0.966831
$77$	2.08742	0.237884
$78$	−2.42864	−0.274989
$79$	7.87310	0.885793	0.442897	−	0.896573i	$-0.353951\pi$
$79$	7.87310	0.885793	0.442897	+	0.896573i	$0.353951\pi$
$80$	0	0
$81$	−11.0874	−1.23194
$82$	−1.14272	−0.126192
$83$	7.19850	0.790138	0.395069	−	0.918651i	$-0.370721\pi$
$83$	7.19850	0.790138	0.395069	+	0.918651i	$0.370721\pi$
$84$	6.70964	0.732081
$85$	0	0
$86$	−0.769859	−0.0830160
$87$	18.6637	2.00096
$88$	1.59210	0.169719
$89$	11.6271	1.23247	0.616237	−	0.787561i	$-0.288656\pi$
$89$	11.6271	1.23247	0.616237	+	0.787561i	$0.288656\pi$
$90$	0	0
$91$	−5.61285	−0.588386
$92$	9.45875	0.986143
$93$	−17.1383	−1.77716
$94$	1.03657	0.106914
$95$	0	0
$96$	7.73975	0.789935
$97$	15.4795	1.57170	0.785852	−	0.618414i	$-0.212225\pi$
$97$	15.4795	1.57170	0.785852	+	0.618414i	$0.212225\pi$
$98$	1.38916	0.140326
$99$	2.49532	0.250789
$100$	0	0
$101$	−14.3827	−1.43113	−0.715566	−	0.698545i	$-0.753831\pi$
$101$	−14.3827	−1.43113	−0.715566	+	0.698545i	$0.753831\pi$
$102$	0	0
$103$	9.39207	0.925429	0.462714	−	0.886507i	$-0.346876\pi$
$103$	9.39207	0.925429	0.462714	+	0.886507i	$0.346876\pi$
$104$	−4.28100	−0.419786
$105$	0	0
$106$	2.85728	0.277523
$107$	−6.77631	−0.655091	−0.327545	−	0.944835i	$-0.606221\pi$
$107$	−6.77631	−0.655091	−0.327545	+	0.944835i	$0.606221\pi$
$108$	−4.62222	−0.444773
$109$	−2.85728	−0.273678	−0.136839	−	0.990593i	$-0.543694\pi$
$109$	−2.85728	−0.273678	−0.136839	+	0.990593i	$0.543694\pi$
$110$	0	0
$111$	15.6128	1.48191
$112$	5.45875	0.515803
$113$	5.86665	0.551888	0.275944	−	0.961174i	$-0.411010\pi$
$113$	5.86665	0.551888	0.275944	+	0.961174i	$0.411010\pi$
$114$	3.05086	0.285739
$115$	0	0
$116$	16.0415	1.48941
$117$	−6.70964	−0.620306
$118$	0.428639	0.0394595
$119$	0	0
$120$	0	0
$121$	−9.28100	−0.843727
$122$	−4.79706	−0.434305
$123$	−8.13335	−0.733360
$124$	−14.7304	−1.32283
$125$	0	0
$126$	−0.942691	−0.0839816
$127$	0.280996	0.0249344	0.0124672	−	0.999922i	$-0.496031\pi$
$127$	0.280996	0.0249344	0.0124672	+	0.999922i	$0.496031\pi$
$128$	8.78568	0.776552
$129$	−5.47949	−0.482443
$130$	0	0
$131$	13.4128	1.17188	0.585942	−	0.810353i	$-0.300725\pi$
$131$	13.4128	1.17188	0.585942	+	0.810353i	$0.300725\pi$
$132$	5.52543	0.480927
$133$	7.05086	0.611387
$134$	−2.84299	−0.245597
$135$	0	0
$136$	0	0
$137$	−10.2810	−0.878365	−0.439182	−	0.898398i	$-0.644732\pi$
$137$	−10.2810	−0.878365	−0.439182	+	0.898398i	$0.644732\pi$
$138$	−3.42372	−0.291446
$139$	18.3017	1.55233	0.776167	−	0.630528i	$-0.217162\pi$
$139$	18.3017	1.55233	0.776167	+	0.630528i	$0.217162\pi$
$140$	0	0
$141$	7.37778	0.621322
$142$	3.29682	0.276663
$143$	−4.62222	−0.386529
$144$	6.52543	0.543786
$145$	0	0
$146$	1.73329	0.143448
$147$	9.88739	0.815498
$148$	13.4193	1.10306
$149$	−4.91750	−0.402857	−0.201429	−	0.979503i	$-0.564558\pi$
$149$	−4.91750	−0.402857	−0.201429	+	0.979503i	$0.564558\pi$
$150$	0	0
$151$	−12.7239	−1.03546	−0.517729	−	0.855545i	$-0.673223\pi$
$151$	−12.7239	−1.03546	−0.517729	+	0.855545i	$0.673223\pi$
$152$	5.37778	0.436196
$153$	0	0
$154$	−0.649413	−0.0523312
$155$	0	0
$156$	−14.8573	−1.18953
$157$	9.18421	0.732980	0.366490	−	0.930422i	$-0.380559\pi$
$157$	9.18421	0.732980	0.366490	+	0.930422i	$0.380559\pi$
$158$	−2.44938	−0.194862
$159$	20.3368	1.61281
$160$	0	0
$161$	−7.91258	−0.623599
$162$	3.44938	0.271009
$163$	5.07160	0.397238	0.198619	−	0.980077i	$-0.436354\pi$
$163$	5.07160	0.397238	0.198619	+	0.980077i	$0.436354\pi$
$164$	−6.99063	−0.545877
$165$	0	0
$166$	−2.23951	−0.173820
$167$	19.8272	1.53427	0.767136	−	0.641484i	$-0.221681\pi$
$167$	19.8272	1.53427	0.767136	+	0.641484i	$0.221681\pi$
$168$	−4.28100	−0.330286
$169$	−0.571361	−0.0439508
$170$	0	0
$171$	8.42864	0.644554
$172$	−4.70964	−0.359106
$173$	2.48886	0.189225	0.0946124	−	0.995514i	$-0.469839\pi$
$173$	2.48886	0.189225	0.0946124	+	0.995514i	$0.469839\pi$
$174$	−5.80642	−0.440184
$175$	0	0
$176$	4.49532	0.338847
$177$	3.05086	0.229316
$178$	−3.61729	−0.271128
$179$	11.6128	0.867985	0.433992	−	0.900916i	$-0.357104\pi$
$179$	11.6128	0.867985	0.433992	+	0.900916i	$0.357104\pi$
$180$	0	0
$181$	−3.86665	−0.287406	−0.143703	−	0.989621i	$-0.545901\pi$
$181$	−3.86665	−0.287406	−0.143703	+	0.989621i	$0.545901\pi$
$182$	1.74620	0.129437
$183$	−34.1432	−2.52394
$184$	−6.03503	−0.444909
$185$	0	0
$186$	5.33185	0.390950
$187$	0	0
$188$	6.34122	0.462481
$189$	3.86665	0.281257
$190$	0	0
$191$	−14.9175	−1.07939	−0.539696	−	0.841860i	$-0.681461\pi$
$191$	−14.9175	−1.07939	−0.539696	+	0.841860i	$0.681461\pi$
$192$	12.7763	0.922051
$193$	17.7462	1.27740	0.638700	−	0.769456i	$-0.279473\pi$
$193$	17.7462	1.27740	0.638700	+	0.769456i	$0.279473\pi$
$194$	−4.81579	−0.345754
$195$	0	0
$196$	8.49823	0.607016
$197$	9.41927	0.671095	0.335548	−	0.942023i	$-0.391079\pi$
$197$	9.41927	0.671095	0.335548	+	0.942023i	$0.391079\pi$
$198$	−0.776312	−0.0551701
$199$	−7.21924	−0.511758	−0.255879	−	0.966709i	$-0.582365\pi$
$199$	−7.21924	−0.511758	−0.255879	+	0.966709i	$0.582365\pi$
$200$	0	0
$201$	−20.2351	−1.42727
$202$	4.47457	0.314830
$203$	−13.4193	−0.941848
$204$	0	0
$205$	0	0
$206$	−2.92195	−0.203582
$207$	−9.45875	−0.657429
$208$	−12.0874	−0.838112
$209$	5.80642	0.401639
$210$	0	0
$211$	14.8825	1.02455	0.512276	−	0.858821i	$-0.328803\pi$
$211$	14.8825	1.02455	0.512276	+	0.858821i	$0.328803\pi$
$212$	17.4795	1.20050
$213$	23.4652	1.60781
$214$	2.10816	0.144111
$215$	0	0
$216$	2.94914	0.200664
$217$	12.3225	0.836505
$218$	0.888922	0.0602054
$219$	12.3368	0.833642
$220$	0	0
$221$	0	0
$222$	−4.85728	−0.325999
$223$	−23.4652	−1.57135	−0.785673	−	0.618642i	$-0.787683\pi$
$223$	−23.4652	−1.57135	−0.785673	+	0.618642i	$0.787683\pi$
$224$	−5.56491	−0.371821
$225$	0	0
$226$	−1.82516	−0.121408
$227$	−4.34767	−0.288565	−0.144283	−	0.989537i	$-0.546087\pi$
$227$	−4.34767	−0.288565	−0.144283	+	0.989537i	$0.546087\pi$
$228$	18.6637	1.23603
$229$	−6.47457	−0.427852	−0.213926	−	0.976850i	$-0.568625\pi$
$229$	−6.47457	−0.427852	−0.213926	+	0.976850i	$0.568625\pi$
$230$	0	0
$231$	−4.62222	−0.304119
$232$	−10.2351	−0.671965
$233$	−18.5303	−1.21396	−0.606982	−	0.794716i	$-0.707620\pi$
$233$	−18.5303	−1.21396	−0.606982	+	0.794716i	$0.707620\pi$
$234$	2.08742	0.136459
$235$	0	0
$236$	2.62222	0.170692
$237$	−17.4336	−1.13243
$238$	0	0
$239$	−21.4193	−1.38550	−0.692749	−	0.721179i	$-0.743601\pi$
$239$	−21.4193	−1.38550	−0.692749	+	0.721179i	$0.743601\pi$
$240$	0	0
$241$	12.2351	0.788130	0.394065	−	0.919083i	$-0.371069\pi$
$241$	12.2351	0.788130	0.394065	+	0.919083i	$0.371069\pi$
$242$	2.88739	0.185608
$243$	17.2652	1.10756
$244$	−29.3461	−1.87869
$245$	0	0
$246$	2.53035	0.161329
$247$	−15.6128	−0.993422
$248$	9.39853	0.596807
$249$	−15.9398	−1.01014
$250$	0	0
$251$	−4.52051	−0.285332	−0.142666	−	0.989771i	$-0.545567\pi$
$251$	−4.52051	−0.285332	−0.142666	+	0.989771i	$0.545567\pi$
$252$	−5.76694	−0.363283
$253$	−6.51606	−0.409661
$254$	−0.0874201	−0.00548523
$255$	0	0
$256$	8.80642	0.550401
$257$	−6.18913	−0.386067	−0.193034	−	0.981192i	$-0.561833\pi$
$257$	−6.18913	−0.386067	−0.193034	+	0.981192i	$0.561833\pi$
$258$	1.70471	0.106131
$259$	−11.2257	−0.697531
$260$	0	0
$261$	−16.0415	−0.992943
$262$	−4.17283	−0.257798
$263$	8.18913	0.504963	0.252482	−	0.967602i	$-0.418753\pi$
$263$	8.18913	0.504963	0.252482	+	0.967602i	$0.418753\pi$
$264$	−3.52543	−0.216975
$265$	0	0
$266$	−2.19358	−0.134497
$267$	−25.7462	−1.57564
$268$	−17.3921	−1.06239
$269$	−3.89829	−0.237683	−0.118841	−	0.992913i	$-0.537918\pi$
$269$	−3.89829	−0.237683	−0.118841	+	0.992913i	$0.537918\pi$
$270$	0	0
$271$	8.85728	0.538041	0.269021	−	0.963134i	$-0.413300\pi$
$271$	8.85728	0.538041	0.269021	+	0.963134i	$0.413300\pi$
$272$	0	0
$273$	12.4286	0.752215
$274$	3.19850	0.193228
$275$	0	0
$276$	−20.9447	−1.26072
$277$	−1.14272	−0.0686595	−0.0343297	−	0.999411i	$-0.510930\pi$
$277$	−1.14272	−0.0686595	−0.0343297	+	0.999411i	$0.510930\pi$
$278$	−5.69381	−0.341492
$279$	14.7304	0.881885
$280$	0	0
$281$	14.6953	0.876651	0.438325	−	0.898816i	$-0.355572\pi$
$281$	14.6953	0.876651	0.438325	+	0.898816i	$0.355572\pi$
$282$	−2.29529	−0.136682
$283$	−1.97926	−0.117655	−0.0588273	−	0.998268i	$-0.518736\pi$
$283$	−1.97926	−0.117655	−0.0588273	+	0.998268i	$0.518736\pi$
$284$	20.1684	1.19677
$285$	0	0
$286$	1.43801	0.0850312
$287$	5.84791	0.345191
$288$	−6.65233	−0.391992
$289$	0	0
$290$	0	0
$291$	−34.2766	−2.00933
$292$	10.6035	0.620522
$293$	−2.94914	−0.172291	−0.0861454	−	0.996283i	$-0.527455\pi$
$293$	−2.94914	−0.172291	−0.0861454	+	0.996283i	$0.527455\pi$
$294$	−3.07604	−0.179399
$295$	0	0
$296$	−8.56199	−0.497656
$297$	3.18421	0.184767
$298$	1.52987	0.0886232
$299$	17.5210	1.01326
$300$	0	0
$301$	3.93978	0.227085
$302$	3.95851	0.227787
$303$	31.8479	1.82961
$304$	15.1842	0.870874
$305$	0	0
$306$	0	0
$307$	5.68736	0.324595	0.162297	−	0.986742i	$-0.448110\pi$
$307$	5.68736	0.324595	0.162297	+	0.986742i	$0.448110\pi$
$308$	−3.97280	−0.226371
$309$	−20.7971	−1.18310
$310$	0	0
$311$	−23.7210	−1.34510	−0.672548	−	0.740054i	$-0.734800\pi$
$311$	−23.7210	−1.34510	−0.672548	+	0.740054i	$0.734800\pi$
$312$	9.47949	0.536671
$313$	−30.8988	−1.74650	−0.873251	−	0.487271i	$-0.837992\pi$
$313$	−30.8988	−1.74650	−0.873251	+	0.487271i	$0.837992\pi$
$314$	−2.85728	−0.161246
$315$	0	0
$316$	−14.9842	−0.842926
$317$	13.7047	0.769733	0.384867	−	0.922972i	$-0.374247\pi$
$317$	13.7047	0.769733	0.384867	+	0.922972i	$0.374247\pi$
$318$	−6.32693	−0.354797
$319$	−11.0509	−0.618729
$320$	0	0
$321$	15.0049	0.837493
$322$	2.46167	0.137183
$323$	0	0
$324$	21.1017	1.17232
$325$	0	0
$326$	−1.57781	−0.0873870
$327$	6.32693	0.349880
$328$	4.46028	0.246278
$329$	−5.30465	−0.292455
$330$	0	0
$331$	−24.8988	−1.36856	−0.684280	−	0.729219i	$-0.739883\pi$
$331$	−24.8988	−1.36856	−0.684280	+	0.729219i	$0.739883\pi$
$332$	−13.7003	−0.751900
$333$	−13.4193	−0.735372
$334$	−6.16839	−0.337519
$335$	0	0
$336$	−12.0874	−0.659423
$337$	−5.28592	−0.287942	−0.143971	−	0.989582i	$-0.545987\pi$
$337$	−5.28592	−0.287942	−0.143971	+	0.989582i	$0.545987\pi$
$338$	0.177755	0.00966858
$339$	−12.9906	−0.705554
$340$	0	0
$341$	10.1476	0.549526
$342$	−2.62222	−0.141793
$343$	−18.2538	−0.985613
$344$	3.00492	0.162015
$345$	0	0
$346$	−0.774305	−0.0416269
$347$	−31.6019	−1.69648	−0.848241	−	0.529611i	$-0.822338\pi$
$347$	−31.6019	−1.69648	−0.848241	+	0.529611i	$0.822338\pi$
$348$	−35.5210	−1.90412
$349$	−26.5116	−1.41913	−0.709567	−	0.704638i	$-0.751109\pi$
$349$	−26.5116	−1.41913	−0.709567	+	0.704638i	$0.751109\pi$
$350$	0	0
$351$	−8.56199	−0.457005
$352$	−4.58274	−0.244261
$353$	3.71456	0.197706	0.0988530	−	0.995102i	$-0.468483\pi$
$353$	3.71456	0.197706	0.0988530	+	0.995102i	$0.468483\pi$
$354$	−0.949145	−0.0504465
$355$	0	0
$356$	−22.1289	−1.17283
$357$	0	0
$358$	−3.61285	−0.190945
$359$	−10.6637	−0.562809	−0.281404	−	0.959589i	$-0.590800\pi$
$359$	−10.6637	−0.562809	−0.281404	+	0.959589i	$0.590800\pi$
$360$	0	0
$361$	0.612848	0.0322551
$362$	1.20294	0.0632253
$363$	20.5511	1.07865
$364$	10.6824	0.559912
$365$	0	0
$366$	10.6222	0.555232
$367$	−10.8780	−0.567828	−0.283914	−	0.958850i	$-0.591633\pi$
$367$	−10.8780	−0.567828	−0.283914	+	0.958850i	$0.591633\pi$
$368$	−17.0400	−0.888269
$369$	6.99063	0.363918
$370$	0	0
$371$	−14.6222	−0.759148
$372$	32.6178	1.69115
$373$	16.5575	0.857317	0.428659	−	0.903467i	$-0.358986\pi$
$373$	16.5575	0.857317	0.428659	+	0.903467i	$0.358986\pi$
$374$	0	0
$375$	0	0
$376$	−4.04593	−0.208653
$377$	29.7146	1.53038
$378$	−1.20294	−0.0618728
$379$	8.16839	0.419582	0.209791	−	0.977746i	$-0.432722\pi$
$379$	8.16839	0.419582	0.209791	+	0.977746i	$0.432722\pi$
$380$	0	0
$381$	−0.622216	−0.0318771
$382$	4.64095	0.237452
$383$	36.7926	1.88001	0.940007	−	0.341154i	$-0.110818\pi$
$383$	36.7926	1.88001	0.940007	+	0.341154i	$0.110818\pi$
$384$	−19.4543	−0.992773
$385$	0	0
$386$	−5.52098	−0.281011
$387$	4.70964	0.239404
$388$	−29.4608	−1.49564
$389$	−12.5348	−0.635539	−0.317770	−	0.948168i	$-0.602934\pi$
$389$	−12.5348	−0.635539	−0.317770	+	0.948168i	$0.602934\pi$
$390$	0	0
$391$	0	0
$392$	−5.42219	−0.273862
$393$	−29.7003	−1.49818
$394$	−2.93041	−0.147632
$395$	0	0
$396$	−4.74912	−0.238652
$397$	32.1017	1.61114	0.805569	−	0.592502i	$-0.201860\pi$
$397$	32.1017	1.61114	0.805569	+	0.592502i	$0.201860\pi$
$398$	2.24596	0.112580
$399$	−15.6128	−0.781620
$400$	0	0
$401$	−13.0321	−0.650793	−0.325396	−	0.945578i	$-0.605498\pi$
$401$	−13.0321	−0.650793	−0.325396	+	0.945578i	$0.605498\pi$
$402$	6.29529	0.313980
$403$	−27.2859	−1.35921
$404$	27.3733	1.36187
$405$	0	0
$406$	4.17484	0.207194
$407$	−9.24443	−0.458229
$408$	0	0
$409$	3.12399	0.154471	0.0772356	−	0.997013i	$-0.475391\pi$
$409$	3.12399	0.154471	0.0772356	+	0.997013i	$0.475391\pi$
$410$	0	0
$411$	22.7654	1.12294
$412$	−17.8751	−0.880643
$413$	−2.19358	−0.107939
$414$	2.94269	0.144625
$415$	0	0
$416$	12.3225	0.604159
$417$	−40.5259	−1.98456
$418$	−1.80642	−0.0883551
$419$	13.1590	0.642860	0.321430	−	0.946933i	$-0.395836\pi$
$419$	13.1590	0.642860	0.321430	+	0.946933i	$0.395836\pi$
$420$	0	0
$421$	−27.6686	−1.34849	−0.674243	−	0.738509i	$-0.735530\pi$
$421$	−27.6686	−1.34849	−0.674243	+	0.738509i	$0.735530\pi$
$422$	−4.63005	−0.225387
$423$	−6.34122	−0.308321
$424$	−11.1526	−0.541616
$425$	0	0
$426$	−7.30021	−0.353696
$427$	24.5491	1.18801
$428$	12.8968	0.623388
$429$	10.2351	0.494154
$430$	0	0
$431$	17.6795	0.851593	0.425796	−	0.904819i	$-0.359994\pi$
$431$	17.6795	0.851593	0.425796	+	0.904819i	$0.359994\pi$
$432$	8.32693	0.400630
$433$	−24.9447	−1.19877	−0.599383	−	0.800462i	$-0.704587\pi$
$433$	−24.9447	−1.19877	−0.599383	+	0.800462i	$0.704587\pi$
$434$	−3.83362	−0.184020
$435$	0	0
$436$	5.43801	0.260433
$437$	−22.0098	−1.05287
$438$	−3.83807	−0.183390
$439$	11.9748	0.571527	0.285763	−	0.958300i	$-0.407753\pi$
$439$	11.9748	0.571527	0.285763	+	0.958300i	$0.407753\pi$
$440$	0	0
$441$	−8.49823	−0.404678
$442$	0	0
$443$	30.6909	1.45817	0.729084	−	0.684424i	$-0.239946\pi$
$443$	30.6909	1.45817	0.729084	+	0.684424i	$0.239946\pi$
$444$	−29.7146	−1.41019
$445$	0	0
$446$	7.30021	0.345675
$447$	10.8889	0.515028
$448$	−9.18622	−0.434008
$449$	−13.6543	−0.644388	−0.322194	−	0.946674i	$-0.604420\pi$
$449$	−13.6543	−0.644388	−0.322194	+	0.946674i	$0.604420\pi$
$450$	0	0
$451$	4.81579	0.226767
$452$	−11.1655	−0.525180
$453$	28.1748	1.32377
$454$	1.35260	0.0634804
$455$	0	0
$456$	−11.9081	−0.557649
$457$	−8.20787	−0.383948	−0.191974	−	0.981400i	$-0.561489\pi$
$457$	−8.20787	−0.383948	−0.191974	+	0.981400i	$0.561489\pi$
$458$	2.01429	0.0941216
$459$	0	0
$460$	0	0
$461$	−5.18421	−0.241453	−0.120726	−	0.992686i	$-0.538522\pi$
$461$	−5.18421	−0.241453	−0.120726	+	0.992686i	$0.538522\pi$
$462$	1.43801	0.0669022
$463$	−23.1985	−1.07813	−0.539063	−	0.842266i	$-0.681221\pi$
$463$	−23.1985	−1.07813	−0.539063	+	0.842266i	$0.681221\pi$
$464$	−28.8988	−1.34159
$465$	0	0
$466$	5.76494	0.267056
$467$	−35.2400	−1.63071	−0.815356	−	0.578960i	$-0.803459\pi$
$467$	−35.2400	−1.63071	−0.815356	+	0.578960i	$0.803459\pi$
$468$	12.7699	0.590287
$469$	14.5491	0.671814
$470$	0	0
$471$	−20.3368	−0.937069
$472$	−1.67307	−0.0770093
$473$	3.24443	0.149179
$474$	5.42372	0.249120
$475$	0	0
$476$	0	0
$477$	−17.4795	−0.800331
$478$	6.66370	0.304791
$479$	11.4445	0.522911	0.261455	−	0.965216i	$-0.415798\pi$
$479$	11.4445	0.522911	0.261455	+	0.965216i	$0.415798\pi$
$480$	0	0
$481$	24.8573	1.13339
$482$	−3.80642	−0.173378
$483$	17.5210	0.797232
$484$	17.6637	0.802896
$485$	0	0
$486$	−5.37133	−0.243649
$487$	−29.7540	−1.34828	−0.674142	−	0.738602i	$-0.735486\pi$
$487$	−29.7540	−1.34828	−0.674142	+	0.738602i	$0.735486\pi$
$488$	18.7239	0.847592
$489$	−11.2301	−0.507845
$490$	0	0
$491$	−25.4193	−1.14716	−0.573578	−	0.819151i	$-0.694445\pi$
$491$	−25.4193	−1.14716	−0.573578	+	0.819151i	$0.694445\pi$
$492$	15.4795	0.697870
$493$	0	0
$494$	4.85728	0.218539
$495$	0	0
$496$	26.5368	1.19154
$497$	−16.8716	−0.756793
$498$	4.95899	0.222218
$499$	−14.8035	−0.662696	−0.331348	−	0.943509i	$-0.607503\pi$
$499$	−14.8035	−0.662696	−0.331348	+	0.943509i	$0.607503\pi$
$500$	0	0
$501$	−43.9037	−1.96147
$502$	1.40636	0.0627691
$503$	4.34767	0.193853	0.0969266	−	0.995292i	$-0.469099\pi$
$503$	4.34767	0.193853	0.0969266	+	0.995292i	$0.469099\pi$
$504$	3.67952	0.163899
$505$	0	0
$506$	2.02720	0.0901199
$507$	1.26517	0.0561884
$508$	−0.534795	−0.0237277
$509$	−15.9813	−0.708357	−0.354179	−	0.935178i	$-0.615239\pi$
$509$	−15.9813	−0.708357	−0.354179	+	0.935178i	$0.615239\pi$
$510$	0	0
$511$	−8.87019	−0.392394
$512$	−20.3111	−0.897633
$513$	10.7556	0.474870
$514$	1.92549	0.0849296
$515$	0	0
$516$	10.4286	0.459095
$517$	−4.36842	−0.192123
$518$	3.49240	0.153447
$519$	−5.51114	−0.241912
$520$	0	0
$521$	−27.9081	−1.22268	−0.611339	−	0.791369i	$-0.709369\pi$
$521$	−27.9081	−1.22268	−0.611339	+	0.791369i	$0.709369\pi$
$522$	4.99063	0.218434
$523$	−15.8938	−0.694989	−0.347495	−	0.937682i	$-0.612968\pi$
$523$	−15.8938	−0.694989	−0.347495	+	0.937682i	$0.612968\pi$
$524$	−25.5274	−1.11517
$525$	0	0
$526$	−2.54770	−0.111085
$527$	0	0
$528$	−9.95407	−0.433195
$529$	1.69979	0.0739040
$530$	0	0
$531$	−2.62222	−0.113794
$532$	−13.4193	−0.581799
$533$	−12.9491	−0.560890
$534$	8.00984	0.346620
$535$	0	0
$536$	11.0968	0.479308
$537$	−25.7146	−1.10967
$538$	1.21279	0.0522870
$539$	−5.85436	−0.252165
$540$	0	0
$541$	−24.2449	−1.04237	−0.521185	−	0.853444i	$-0.674510\pi$
$541$	−24.2449	−1.04237	−0.521185	+	0.853444i	$0.674510\pi$
$542$	−2.75557	−0.118362
$543$	8.56199	0.367430
$544$	0	0
$545$	0	0
$546$	−3.86665	−0.165477
$547$	−11.7255	−0.501344	−0.250672	−	0.968072i	$-0.580652\pi$
$547$	−11.7255	−0.501344	−0.250672	+	0.968072i	$0.580652\pi$
$548$	19.5669	0.835857
$549$	29.3461	1.25246
$550$	0	0
$551$	−37.3274	−1.59020
$552$	13.3635	0.568788
$553$	12.5348	0.533034
$554$	0.355509	0.0151041
$555$	0	0
$556$	−34.8321	−1.47721
$557$	31.4336	1.33188	0.665941	−	0.746004i	$-0.268030\pi$
$557$	31.4336	1.33188	0.665941	+	0.746004i	$0.268030\pi$
$558$	−4.58274	−0.194003
$559$	−8.72393	−0.368983
$560$	0	0
$561$	0	0
$562$	−4.57184	−0.192851
$563$	11.9353	0.503014	0.251507	−	0.967855i	$-0.419074\pi$
$563$	11.9353	0.503014	0.251507	+	0.967855i	$0.419074\pi$
$564$	−14.0415	−0.591253
$565$	0	0
$566$	0.615762	0.0258824
$567$	−17.6523	−0.741328
$568$	−12.8682	−0.539937
$569$	−1.14272	−0.0479054	−0.0239527	−	0.999713i	$-0.507625\pi$
$569$	−1.14272	−0.0479054	−0.0239527	+	0.999713i	$0.507625\pi$
$570$	0	0
$571$	−26.7402	−1.11904	−0.559522	−	0.828816i	$-0.689015\pi$
$571$	−26.7402	−1.11904	−0.559522	+	0.828816i	$0.689015\pi$
$572$	8.79706	0.367823
$573$	33.0321	1.37994
$574$	−1.81933	−0.0759374
$575$	0	0
$576$	−10.9813	−0.457553
$577$	−27.9956	−1.16547	−0.582735	−	0.812662i	$-0.698017\pi$
$577$	−27.9956	−1.16547	−0.582735	+	0.812662i	$0.698017\pi$
$578$	0	0
$579$	−39.2958	−1.63308
$580$	0	0
$581$	11.4608	0.475472
$582$	10.6637	0.442025
$583$	−12.0415	−0.498707
$584$	−6.76541	−0.279955
$585$	0	0
$586$	0.917502	0.0379017
$587$	6.60793	0.272738	0.136369	−	0.990658i	$-0.456457\pi$
$587$	6.60793	0.272738	0.136369	+	0.990658i	$0.456457\pi$
$588$	−18.8178	−0.776033
$589$	34.2766	1.41234
$590$	0	0
$591$	−20.8573	−0.857954
$592$	−24.1748	−0.993580
$593$	−4.82564	−0.198165	−0.0990826	−	0.995079i	$-0.531591\pi$
$593$	−4.82564	−0.198165	−0.0990826	+	0.995079i	$0.531591\pi$
$594$	−0.990632	−0.0406461
$595$	0	0
$596$	9.35905	0.383362
$597$	15.9857	0.654252
$598$	−5.45091	−0.222904
$599$	−39.0005	−1.59352	−0.796758	−	0.604298i	$-0.793454\pi$
$599$	−39.0005	−1.59352	−0.796758	+	0.604298i	$0.793454\pi$
$600$	0	0
$601$	7.19405	0.293452	0.146726	−	0.989177i	$-0.453127\pi$
$601$	7.19405	0.293452	0.146726	+	0.989177i	$0.453127\pi$
$602$	−1.22570	−0.0499556
$603$	17.3921	0.708260
$604$	24.2163	0.985348
$605$	0	0
$606$	−9.90813	−0.402490
$607$	−7.23353	−0.293600	−0.146800	−	0.989166i	$-0.546897\pi$
$607$	−7.23353	−0.293600	−0.146800	+	0.989166i	$0.546897\pi$
$608$	−15.4795	−0.627776
$609$	29.7146	1.20409
$610$	0	0
$611$	11.7462	0.475200
$612$	0	0
$613$	−14.7654	−0.596369	−0.298185	−	0.954508i	$-0.596381\pi$
$613$	−14.7654	−0.596369	−0.298185	+	0.954508i	$0.596381\pi$
$614$	−1.76938	−0.0714065
$615$	0	0
$616$	2.53480	0.102130
$617$	−10.5205	−0.423540	−0.211770	−	0.977320i	$-0.567923\pi$
$617$	−10.5205	−0.423540	−0.211770	+	0.977320i	$0.567923\pi$
$618$	6.47013	0.260267
$619$	7.94623	0.319386	0.159693	−	0.987167i	$-0.448950\pi$
$619$	7.94623	0.319386	0.159693	+	0.987167i	$0.448950\pi$
$620$	0	0
$621$	−12.0701	−0.484355
$622$	7.37979	0.295903
$623$	18.5116	0.741652
$624$	26.7654	1.07147
$625$	0	0
$626$	9.61285	0.384207
$627$	−12.8573	−0.513470
$628$	−17.4795	−0.697508
$629$	0	0
$630$	0	0
$631$	14.2636	0.567827	0.283913	−	0.958850i	$-0.408367\pi$
$631$	14.2636	0.567827	0.283913	+	0.958850i	$0.408367\pi$
$632$	9.56046	0.380295
$633$	−32.9545	−1.30983
$634$	−4.26364	−0.169331
$635$	0	0
$636$	−38.7052	−1.53476
$637$	15.7418	0.623711
$638$	3.43801	0.136112
$639$	−20.1684	−0.797849
$640$	0	0
$641$	−25.4509	−1.00525	−0.502625	−	0.864504i	$-0.667632\pi$
$641$	−25.4509	−1.00525	−0.502625	+	0.864504i	$0.667632\pi$
$642$	−4.66815	−0.184237
$643$	−16.1126	−0.635419	−0.317710	−	0.948188i	$-0.602914\pi$
$643$	−16.1126	−0.635419	−0.317710	+	0.948188i	$0.602914\pi$
$644$	15.0593	0.593420
$645$	0	0
$646$	0	0
$647$	−40.4558	−1.59048	−0.795242	−	0.606293i	$-0.792656\pi$
$647$	−40.4558	−1.59048	−0.795242	+	0.606293i	$0.792656\pi$
$648$	−13.4637	−0.528903
$649$	−1.80642	−0.0709083
$650$	0	0
$651$	−27.2859	−1.06942
$652$	−9.65233	−0.378014
$653$	6.87601	0.269079	0.134540	−	0.990908i	$-0.457044\pi$
$653$	6.87601	0.269079	0.134540	+	0.990908i	$0.457044\pi$
$654$	−1.96836	−0.0769689
$655$	0	0
$656$	12.5936	0.491699
$657$	−10.6035	−0.413681
$658$	1.65032	0.0643361
$659$	16.0286	0.624385	0.312192	−	0.950019i	$-0.398937\pi$
$659$	16.0286	0.624385	0.312192	+	0.950019i	$0.398937\pi$
$660$	0	0
$661$	31.1655	1.21220	0.606098	−	0.795390i	$-0.292734\pi$
$661$	31.1655	1.21220	0.606098	+	0.795390i	$0.292734\pi$
$662$	7.74620	0.301065
$663$	0	0
$664$	8.74128	0.339227
$665$	0	0
$666$	4.17484	0.161772
$667$	41.8894	1.62196
$668$	−37.7353	−1.46002
$669$	51.9595	2.00887
$670$	0	0
$671$	20.2163	0.780443
$672$	12.3225	0.475350
$673$	19.8479	0.765081	0.382540	−	0.923939i	$-0.375049\pi$
$673$	19.8479	0.765081	0.382540	+	0.923939i	$0.375049\pi$
$674$	1.64449	0.0633434
$675$	0	0
$676$	1.08742	0.0418239
$677$	−28.8385	−1.10836	−0.554178	−	0.832398i	$-0.686967\pi$
$677$	−28.8385	−1.10836	−0.554178	+	0.832398i	$0.686967\pi$
$678$	4.04149	0.155212
$679$	24.6450	0.945787
$680$	0	0
$681$	9.62714	0.368913
$682$	−3.15701	−0.120888
$683$	19.2050	0.734857	0.367429	−	0.930052i	$-0.380238\pi$
$683$	19.2050	0.734857	0.367429	+	0.930052i	$0.380238\pi$
$684$	−16.0415	−0.613362
$685$	0	0
$686$	5.67890	0.216821
$687$	14.3368	0.546982
$688$	8.48442	0.323465
$689$	32.3783	1.23351
$690$	0	0
$691$	34.5555	1.31455	0.657277	−	0.753649i	$-0.271708\pi$
$691$	34.5555	1.31455	0.657277	+	0.753649i	$0.271708\pi$
$692$	−4.73683	−0.180067
$693$	3.97280	0.150914
$694$	9.83161	0.373203
$695$	0	0
$696$	22.6637	0.859065
$697$	0	0
$698$	8.24797	0.312190
$699$	41.0321	1.55198
$700$	0	0
$701$	−20.9131	−0.789875	−0.394938	−	0.918708i	$-0.629234\pi$
$701$	−20.9131	−0.789875	−0.394938	+	0.918708i	$0.629234\pi$
$702$	2.66370	0.100535
$703$	−31.2257	−1.17770
$704$	−7.56491	−0.285113
$705$	0	0
$706$	−1.15563	−0.0434926
$707$	−22.8988	−0.861197
$708$	−5.80642	−0.218219
$709$	−32.9906	−1.23899	−0.619495	−	0.785001i	$-0.712662\pi$
$709$	−32.9906	−1.23899	−0.619495	+	0.785001i	$0.712662\pi$
$710$	0	0
$711$	14.9842	0.561951
$712$	14.1191	0.529134
$713$	−38.4657	−1.44055
$714$	0	0
$715$	0	0
$716$	−22.1017	−0.825980
$717$	47.4291	1.77127
$718$	3.31756	0.123810
$719$	−10.5141	−0.392108	−0.196054	−	0.980593i	$-0.562813\pi$
$719$	−10.5141	−0.392108	−0.196054	+	0.980593i	$0.562813\pi$
$720$	0	0
$721$	14.9532	0.556885
$722$	−0.190662	−0.00709569
$723$	−27.0923	−1.00758
$724$	7.35905	0.273497
$725$	0	0
$726$	−6.39361	−0.237289
$727$	−43.2815	−1.60522	−0.802610	−	0.596503i	$-0.796556\pi$
$727$	−43.2815	−1.60522	−0.802610	+	0.596503i	$0.796556\pi$
$728$	−6.81579	−0.252610
$729$	−4.96836	−0.184013
$730$	0	0
$731$	0	0
$732$	64.9817	2.40179
$733$	−22.8859	−0.845308	−0.422654	−	0.906291i	$-0.638902\pi$
$733$	−22.8859	−0.845308	−0.422654	+	0.906291i	$0.638902\pi$
$734$	3.38424	0.124914
$735$	0	0
$736$	17.3713	0.640316
$737$	11.9813	0.441336
$738$	−2.17484	−0.0800570
$739$	−0.815792	−0.0300094	−0.0150047	−	0.999887i	$-0.504776\pi$
$739$	−0.815792	−0.0300094	−0.0150047	+	0.999887i	$0.504776\pi$
$740$	0	0
$741$	34.5718	1.27003
$742$	4.54909	0.167002
$743$	−30.1639	−1.10661	−0.553304	−	0.832980i	$-0.686633\pi$
$743$	−30.1639	−1.10661	−0.553304	+	0.832980i	$0.686633\pi$
$744$	−20.8113	−0.762981
$745$	0	0
$746$	−5.15118	−0.188598
$747$	13.7003	0.501267
$748$	0	0
$749$	−10.7886	−0.394207
$750$	0	0
$751$	2.17728	0.0794500	0.0397250	−	0.999211i	$-0.487352\pi$
$751$	2.17728	0.0794500	0.0397250	+	0.999211i	$0.487352\pi$
$752$	−11.4237	−0.416580
$753$	10.0098	0.364779
$754$	−9.24443	−0.336662
$755$	0	0
$756$	−7.35905	−0.267646
$757$	29.3230	1.06576	0.532881	−	0.846190i	$-0.321110\pi$
$757$	29.3230	1.06576	0.532881	+	0.846190i	$0.321110\pi$
$758$	−2.54125	−0.0923023
$759$	14.4286	0.523726
$760$	0	0
$761$	49.2212	1.78427	0.892134	−	0.451770i	$-0.149207\pi$
$761$	49.2212	1.78427	0.892134	+	0.451770i	$0.149207\pi$
$762$	0.193576	0.00701252
$763$	−4.54909	−0.164688
$764$	28.3912	1.02716
$765$	0	0
$766$	−11.4465	−0.413578
$767$	4.85728	0.175386
$768$	−19.5002	−0.703654
$769$	37.0178	1.33490	0.667449	−	0.744656i	$-0.267386\pi$
$769$	37.0178	1.33490	0.667449	+	0.744656i	$0.267386\pi$
$770$	0	0
$771$	13.7047	0.493563
$772$	−33.7748	−1.21558
$773$	−4.20787	−0.151346	−0.0756732	−	0.997133i	$-0.524111\pi$
$773$	−4.20787	−0.151346	−0.0756732	+	0.997133i	$0.524111\pi$
$774$	−1.46520	−0.0526657
$775$	0	0
$776$	18.7971	0.674775
$777$	24.8573	0.891750
$778$	3.89967	0.139810
$779$	16.2667	0.582815
$780$	0	0
$781$	−13.8938	−0.497161
$782$	0	0
$783$	−20.4701	−0.731543
$784$	−15.3096	−0.546771
$785$	0	0
$786$	9.23999	0.329579
$787$	−23.8557	−0.850366	−0.425183	−	0.905108i	$-0.639790\pi$
$787$	−23.8557	−0.850366	−0.425183	+	0.905108i	$0.639790\pi$
$788$	−17.9269	−0.638618
$789$	−18.1334	−0.645564
$790$	0	0
$791$	9.34031	0.332103
$792$	3.03011	0.107670
$793$	−54.3595	−1.93036
$794$	−9.98709	−0.354429
$795$	0	0
$796$	13.7397	0.486992
$797$	−34.4415	−1.21998	−0.609991	−	0.792408i	$-0.708827\pi$
$797$	−34.4415	−1.21998	−0.609991	+	0.792408i	$0.708827\pi$
$798$	4.85728	0.171946
$799$	0	0
$800$	0	0
$801$	22.1289	0.781886
$802$	4.05439	0.143166
$803$	−7.30465	−0.257776
$804$	38.5116	1.35820
$805$	0	0
$806$	8.48886	0.299007
$807$	8.63206	0.303863
$808$	−17.4652	−0.614424
$809$	40.2578	1.41539	0.707695	−	0.706518i	$-0.249735\pi$
$809$	40.2578	1.41539	0.707695	+	0.706518i	$0.249735\pi$
$810$	0	0
$811$	20.4953	0.719688	0.359844	−	0.933013i	$-0.382830\pi$
$811$	20.4953	0.719688	0.359844	+	0.933013i	$0.382830\pi$
$812$	25.5397	0.896268
$813$	−19.6128	−0.687853
$814$	2.87601	0.100804
$815$	0	0
$816$	0	0
$817$	10.9590	0.383407
$818$	−0.971896	−0.0339816
$819$	−10.6824	−0.373275
$820$	0	0
$821$	33.6958	1.17599	0.587996	−	0.808864i	$-0.299917\pi$
$821$	33.6958	1.17599	0.587996	+	0.808864i	$0.299917\pi$
$822$	−7.08250	−0.247030
$823$	−16.6113	−0.579034	−0.289517	−	0.957173i	$-0.593495\pi$
$823$	−16.6113	−0.579034	−0.289517	+	0.957173i	$0.593495\pi$
$824$	11.4050	0.397311
$825$	0	0
$826$	0.682439	0.0237451
$827$	0.316030	0.0109894	0.00549472	−	0.999985i	$-0.498251\pi$
$827$	0.316030	0.0109894	0.00549472	+	0.999985i	$0.498251\pi$
$828$	18.0020	0.625613
$829$	18.2034	0.632231	0.316115	−	0.948721i	$-0.397621\pi$
$829$	18.2034	0.632231	0.316115	+	0.948721i	$0.397621\pi$
$830$	0	0
$831$	2.53035	0.0877769
$832$	20.3412	0.705205
$833$	0	0
$834$	12.6079	0.436577
$835$	0	0
$836$	−11.0509	−0.382202
$837$	18.7971	0.649721
$838$	−4.09387	−0.141421
$839$	−2.63851	−0.0910916	−0.0455458	−	0.998962i	$-0.514503\pi$
$839$	−2.63851	−0.0910916	−0.0455458	+	0.998962i	$0.514503\pi$
$840$	0	0
$841$	42.0420	1.44972
$842$	8.60793	0.296649
$843$	−32.5402	−1.12074
$844$	−28.3245	−0.974969
$845$	0	0
$846$	1.97280	0.0678264
$847$	−14.7763	−0.507720
$848$	−31.4893	−1.08135
$849$	4.38271	0.150414
$850$	0	0
$851$	35.0420	1.20122
$852$	−44.6593	−1.53000
$853$	4.68244	0.160324	0.0801618	−	0.996782i	$-0.474456\pi$
$853$	4.68244	0.160324	0.0801618	+	0.996782i	$0.474456\pi$
$854$	−7.63741	−0.261347
$855$	0	0
$856$	−8.22861	−0.281248
$857$	47.3689	1.61809	0.809045	−	0.587746i	$-0.199985\pi$
$857$	47.3689	1.61809	0.809045	+	0.587746i	$0.199985\pi$
$858$	−3.18421	−0.108707
$859$	−16.1748	−0.551878	−0.275939	−	0.961175i	$-0.588989\pi$
$859$	−16.1748	−0.551878	−0.275939	+	0.961175i	$0.588989\pi$
$860$	0	0
$861$	−12.9491	−0.441306
$862$	−5.50024	−0.187339
$863$	32.5674	1.10861	0.554303	−	0.832315i	$-0.312985\pi$
$863$	32.5674	1.10861	0.554303	+	0.832315i	$0.312985\pi$
$864$	−8.48886	−0.288797
$865$	0	0
$866$	7.76049	0.263712
$867$	0	0
$868$	−23.4523	−0.796023
$869$	10.3225	0.350166
$870$	0	0
$871$	−32.2163	−1.09161
$872$	−3.46965	−0.117497
$873$	29.4608	0.997096
$874$	6.84743	0.231618
$875$	0	0
$876$	−23.4795	−0.793299
$877$	13.2543	0.447565	0.223783	−	0.974639i	$-0.428159\pi$
$877$	13.2543	0.447565	0.223783	+	0.974639i	$0.428159\pi$
$878$	−3.72546	−0.125728
$879$	6.53035	0.220263
$880$	0	0
$881$	2.78721	0.0939035	0.0469518	−	0.998897i	$-0.485049\pi$
$881$	2.78721	0.0939035	0.0469518	+	0.998897i	$0.485049\pi$
$882$	2.64387	0.0890236
$883$	27.1985	0.915302	0.457651	−	0.889132i	$-0.348691\pi$
$883$	27.1985	0.915302	0.457651	+	0.889132i	$0.348691\pi$
$884$	0	0
$885$	0	0
$886$	−9.54818	−0.320777
$887$	−48.2657	−1.62060	−0.810301	−	0.586014i	$-0.800696\pi$
$887$	−48.2657	−1.62060	−0.810301	+	0.586014i	$0.800696\pi$
$888$	18.9590	0.636222
$889$	0.447375	0.0150045
$890$	0	0
$891$	−14.5368	−0.487001
$892$	44.6593	1.49530
$893$	−14.7556	−0.493776
$894$	−3.38763	−0.113299
$895$	0	0
$896$	13.9877	0.467297
$897$	−38.7971	−1.29540
$898$	4.24797	0.141757
$899$	−65.2355	−2.17573
$900$	0	0
$901$	0	0
$902$	−1.49823	−0.0498856
$903$	−8.72393	−0.290314
$904$	7.12399	0.236940
$905$	0	0
$906$	−8.76541	−0.291211
$907$	−8.93825	−0.296790	−0.148395	−	0.988928i	$-0.547411\pi$
$907$	−8.93825	−0.296790	−0.148395	+	0.988928i	$0.547411\pi$
$908$	8.27454	0.274600
$909$	−27.3733	−0.907916
$910$	0	0
$911$	−51.6795	−1.71222	−0.856110	−	0.516794i	$-0.827125\pi$
$911$	−51.6795	−1.71222	−0.856110	+	0.516794i	$0.827125\pi$
$912$	−33.6227	−1.11336
$913$	9.43801	0.312352
$914$	2.55353	0.0844633
$915$	0	0
$916$	12.3225	0.407146
$917$	21.3546	0.705191
$918$	0	0
$919$	18.9719	0.625825	0.312913	−	0.949782i	$-0.398695\pi$
$919$	18.9719	0.625825	0.312913	+	0.949782i	$0.398695\pi$
$920$	0	0
$921$	−12.5936	−0.414974
$922$	1.61285	0.0531163
$923$	37.3590	1.22969
$924$	8.79706	0.289402
$925$	0	0
$926$	7.21723	0.237173
$927$	17.8751	0.587096
$928$	29.4608	0.967097
$929$	−50.3912	−1.65328	−0.826640	−	0.562731i	$-0.809751\pi$
$929$	−50.3912	−1.65328	−0.826640	+	0.562731i	$0.809751\pi$
$930$	0	0
$931$	−19.7748	−0.648092
$932$	35.2672	1.15521
$933$	52.5259	1.71962
$934$	10.9634	0.358735
$935$	0	0
$936$	−8.14764	−0.266314
$937$	58.1530	1.89978	0.949889	−	0.312589i	$-0.101196\pi$
$937$	58.1530	1.89978	0.949889	+	0.312589i	$0.101196\pi$
$938$	−4.52633	−0.147790
$939$	68.4197	2.23279
$940$	0	0
$941$	9.25734	0.301781	0.150890	−	0.988551i	$-0.451786\pi$
$941$	9.25734	0.301781	0.150890	+	0.988551i	$0.451786\pi$
$942$	6.32693	0.206142
$943$	−18.2548	−0.594457
$944$	−4.72393	−0.153751
$945$	0	0
$946$	−1.00937	−0.0328174
$947$	43.9516	1.42824	0.714118	−	0.700025i	$-0.246828\pi$
$947$	43.9516	1.42824	0.714118	+	0.700025i	$0.246828\pi$
$948$	33.1798	1.07763
$949$	19.6414	0.637588
$950$	0	0
$951$	−30.3466	−0.984057
$952$	0	0
$953$	38.1891	1.23707	0.618534	−	0.785758i	$-0.287727\pi$
$953$	38.1891	1.23707	0.618534	+	0.785758i	$0.287727\pi$
$954$	5.43801	0.176062
$955$	0	0
$956$	40.7654	1.31845
$957$	24.4701	0.791007
$958$	−3.56046	−0.115033
$959$	−16.3684	−0.528564
$960$	0	0
$961$	28.9037	0.932377
$962$	−7.73329	−0.249331
$963$	−12.8968	−0.415592
$964$	−23.2859	−0.749989
$965$	0	0
$966$	−5.45091	−0.175380
$967$	−7.36043	−0.236696	−0.118348	−	0.992972i	$-0.537760\pi$
$967$	−7.36043	−0.236696	−0.118348	+	0.992972i	$0.537760\pi$
$968$	−11.2701	−0.362235
$969$	0	0
$970$	0	0
$971$	35.2257	1.13045	0.565223	−	0.824938i	$-0.308790\pi$
$971$	35.2257	1.13045	0.565223	+	0.824938i	$0.308790\pi$
$972$	−32.8593	−1.05396
$973$	29.1383	0.934130
$974$	9.25671	0.296604
$975$	0	0
$976$	52.8671	1.69224
$977$	−46.9273	−1.50134	−0.750669	−	0.660678i	$-0.770269\pi$
$977$	−46.9273	−1.50134	−0.750669	+	0.660678i	$0.770269\pi$
$978$	3.49378	0.111719
$979$	15.2444	0.487214
$980$	0	0
$981$	−5.43801	−0.173622
$982$	7.90813	0.252359
$983$	8.21432	0.261996	0.130998	−	0.991383i	$-0.458182\pi$
$983$	8.21432	0.261996	0.130998	+	0.991383i	$0.458182\pi$
$984$	−9.87649	−0.314851
$985$	0	0
$986$	0	0
$987$	11.7462	0.373886
$988$	29.7146	0.945346
$989$	−12.2983	−0.391065
$990$	0	0
$991$	−17.7683	−0.564430	−0.282215	−	0.959351i	$-0.591069\pi$
$991$	−17.7683	−0.564430	−0.282215	+	0.959351i	$0.591069\pi$
$992$	−27.0529	−0.858929
$993$	55.1338	1.74962
$994$	5.24888	0.166484
$995$	0	0
$996$	30.3368	0.961257
$997$	11.6958	0.370410	0.185205	−	0.982700i	$-0.440705\pi$
$997$	11.6958	0.370410	0.185205	+	0.982700i	$0.440705\pi$
$998$	4.60549	0.145784
$999$	−17.1240	−0.541779

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.r.1.2		3
5.4	even	2		1445.2.a.k.1.2		3
17.4	even	4		425.2.d.c.101.4		6
17.13	even	4		425.2.d.c.101.3		6
17.16	even	2		7225.2.a.q.1.2		3
85.4	even	4		85.2.d.a.16.3	✓	6
85.13	odd	4		425.2.c.b.424.3		6
85.38	odd	4		425.2.c.a.424.3		6
85.47	odd	4		425.2.c.a.424.4		6
85.64	even	4		85.2.d.a.16.4	yes	6
85.72	odd	4		425.2.c.b.424.4		6
85.84	even	2		1445.2.a.j.1.2		3
255.89	odd	4		765.2.g.b.271.4		6
255.149	odd	4		765.2.g.b.271.3		6
340.259	odd	4		1360.2.c.f.1121.6		6
340.319	odd	4		1360.2.c.f.1121.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.d.a.16.3	✓	6	85.4	even	4
85.2.d.a.16.4	yes	6	85.64	even	4
425.2.c.a.424.3		6	85.38	odd	4
425.2.c.a.424.4		6	85.47	odd	4
425.2.c.b.424.3		6	85.13	odd	4
425.2.c.b.424.4		6	85.72	odd	4
425.2.d.c.101.3		6	17.13	even	4
425.2.d.c.101.4		6	17.4	even	4
765.2.g.b.271.3		6	255.149	odd	4
765.2.g.b.271.4		6	255.89	odd	4
1360.2.c.f.1121.1		6	340.319	odd	4
1360.2.c.f.1121.6		6	340.259	odd	4
1445.2.a.j.1.2		3	85.84	even	2
1445.2.a.k.1.2		3	5.4	even	2
7225.2.a.q.1.2		3	17.16	even	2
7225.2.a.r.1.2		3	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q-0.311108 q^{2} -2.21432 q^{3} -1.90321 q^{4} +0.688892 q^{6} +1.59210 q^{7} +1.21432 q^{8} +1.90321 q^{9} +1.31111 q^{11} +4.21432 q^{12} -3.52543 q^{13} -0.495316 q^{14} +3.42864 q^{16} -0.592104 q^{18} +4.42864 q^{19} -3.52543 q^{21} -0.407896 q^{22} -4.96989 q^{23} -2.68889 q^{24} +1.09679 q^{26} +2.42864 q^{27} -3.03011 q^{28} -8.42864 q^{29} +7.73975 q^{31} -3.49532 q^{32} -2.90321 q^{33} -3.62222 q^{36} -7.05086 q^{37} -1.37778 q^{38} +7.80642 q^{39} +3.67307 q^{41} +1.09679 q^{42} +2.47457 q^{43} -2.49532 q^{44} +1.54617 q^{46} -3.33185 q^{47} -7.59210 q^{48} -4.46520 q^{49} +6.70964 q^{52} -9.18421 q^{53} -0.755569 q^{54} +1.93332 q^{56} -9.80642 q^{57} +2.62222 q^{58} -1.37778 q^{59} +15.4193 q^{61} -2.40790 q^{62} +3.03011 q^{63} -5.76986 q^{64} +0.903212 q^{66} +9.13828 q^{67} +11.0049 q^{69} -10.5970 q^{71} +2.31111 q^{72} -5.57136 q^{73} +2.19358 q^{74} -8.42864 q^{76} +2.08742 q^{77} -2.42864 q^{78} +7.87310 q^{79} -11.0874 q^{81} -1.14272 q^{82} +7.19850 q^{83} +6.70964 q^{84} -0.769859 q^{86} +18.6637 q^{87} +1.59210 q^{88} +11.6271 q^{89} -5.61285 q^{91} +9.45875 q^{92} -17.1383 q^{93} +1.03657 q^{94} +7.73975 q^{96} +15.4795 q^{97} +1.38916 q^{98} +2.49532 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} + 2 q^{6} - 2 q^{7} - 3 q^{8} - q^{9} + 4 q^{11} + 6 q^{12} - 4 q^{13} + 12 q^{14} - 3 q^{16} + 5 q^{18} - 4 q^{21} - 8 q^{22} - 8 q^{23} - 8 q^{24} + 10 q^{26} - 6 q^{27} - 16 q^{28}+ \cdots - 6 q^{99}+O(q^{100}) \) 3 * q - q^2 + q^4 + 2 * q^6 - 2 * q^7 - 3 * q^8 - q^9 + 4 * q^11 + 6 * q^12 - 4 * q^13 + 12 * q^14 - 3 * q^16 + 5 * q^18 - 4 * q^21 - 8 * q^22 - 8 * q^23 - 8 * q^24 + 10 * q^26 - 6 * q^27 - 16 * q^28 - 12 * q^29 + 10 * q^31 + 3 * q^32 - 2 * q^33 - 11 * q^36 - 8 * q^37 - 4 * q^38 + 10 * q^39 - 2 * q^41 + 10 * q^42 + 14 * q^43 + 6 * q^44 - 22 * q^46 + 10 * q^47 - 16 * q^48 + 7 * q^49 - 14 * q^53 - 2 * q^54 + 6 * q^56 - 16 * q^57 + 8 * q^58 - 4 * q^59 + 6 * q^61 - 14 * q^62 + 16 * q^63 - 11 * q^64 - 4 * q^66 - 6 * q^67 + 8 * q^71 + 7 * q^72 - 30 * q^73 + 20 * q^74 - 12 * q^76 - 14 * q^77 + 6 * q^78 + 10 * q^79 - 13 * q^81 - 30 * q^82 + 2 * q^83 + 4 * q^86 + 16 * q^87 - 2 * q^88 + 2 * q^89 + 10 * q^91 + 22 * q^92 - 18 * q^93 - 4 * q^94 + 10 * q^96 + 20 * q^97 - 43 * q^98 - 6 * q^99

Introduction

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