LMFDB - Embedded newform 7225.2.a.bk.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7225,2,Mod(1,7225)] 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("7225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [8,0,0,8,0,6,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$57.6919154604$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.4217732978944.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} + 40x^{4} - 47x^{2} + 16 $ x^8 - 12x^6 + 40x^4 - 47*x^2 + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1445)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.22010$ 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)

$f(q)$	$=$	\(q-1.22010 q^{2} +2.38602 q^{3} -0.511354 q^{4} -2.91119 q^{6} -1.22010 q^{7} +3.06410 q^{8} +2.69310 q^{9} +5.09294 q^{11} -1.22010 q^{12} -2.60778 q^{13} +1.48865 q^{14} -2.71581 q^{16} -3.28586 q^{18} +8.51548 q^{19} -2.91119 q^{21} -6.21390 q^{22} +2.89653 q^{23} +7.31102 q^{24} +3.18175 q^{26} -0.732262 q^{27} +0.623903 q^{28} +5.49277 q^{29} +3.18175 q^{31} -2.81465 q^{32} +12.1519 q^{33} -1.37713 q^{36} -3.09561 q^{37} -10.3897 q^{38} -6.22221 q^{39} -0.181750 q^{41} +3.55194 q^{42} +6.70174 q^{43} -2.60429 q^{44} -3.53406 q^{46} +8.71332 q^{47} -6.47999 q^{48} -5.51135 q^{49} +1.33350 q^{52} -0.167574 q^{53} +0.893434 q^{54} -3.73852 q^{56} +20.3181 q^{57} -6.70174 q^{58} -13.7336 q^{59} +0.293269 q^{61} -3.88206 q^{62} -3.28586 q^{63} +8.86577 q^{64} -14.8265 q^{66} +1.56304 q^{67} +6.91119 q^{69} +6.93390 q^{71} +8.25195 q^{72} -8.04302 q^{73} +3.77696 q^{74} -4.35442 q^{76} -6.21390 q^{77} +7.59173 q^{78} -0.0888115 q^{79} -9.82650 q^{81} +0.221753 q^{82} -14.8138 q^{83} +1.48865 q^{84} -8.17680 q^{86} +13.1059 q^{87} +15.6053 q^{88} -2.79554 q^{89} +3.18175 q^{91} -1.48115 q^{92} +7.59173 q^{93} -10.6311 q^{94} -6.71581 q^{96} -3.54691 q^{97} +6.72441 q^{98} +13.7158 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4} + 6 q^{6} + 8 q^{9} + 10 q^{11} + 24 q^{14} + 16 q^{16} - 4 q^{19} + 6 q^{21} + 12 q^{24} + 24 q^{26} - 4 q^{29} + 24 q^{31} - 18 q^{36} + 26 q^{39} + 22 q^{44} + 8 q^{46} - 32 q^{49} + 28 q^{54}+ \cdots + 72 q^{99}+O(q^{100}) $ 8 * q + 8 * q^4 + 6 * q^6 + 8 * q^9 + 10 * q^11 + 24 * q^14 + 16 * q^16 - 4 * q^19 + 6 * q^21 + 12 * q^24 + 24 * q^26 - 4 * q^29 + 24 * q^31 - 18 * q^36 + 26 * q^39 + 22 * q^44 + 8 * q^46 - 32 * q^49 + 28 * q^54 + 32 * q^56 - 22 * q^59 + 6 * q^61 + 90 * q^64 + 26 * q^69 + 2 * q^71 + 20 * q^74 - 66 * q^76 - 30 * q^79 + 40 * q^81 + 24 * q^84 - 38 * q^86 - 48 * q^89 + 24 * q^91 + 42 * q^94 - 16 * q^96 + 72 * 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 $

$2$	−1.22010	−0.862742	−0.431371	−	0.902175i	$-0.641970\pi$
$2$	−1.22010	−0.862742	−0.431371	+	0.902175i	$0.641970\pi$
$3$	2.38602	1.37757	0.688785	−	0.724965i	$-0.258144\pi$
$3$	2.38602	1.37757	0.688785	+	0.724965i	$0.258144\pi$
$4$	−0.511354	−0.255677
$5$	0	0
$5$	0	0
$6$	−2.91119	−1.18849
$7$	−1.22010	−0.461155	−0.230577	−	0.973054i	$-0.574061\pi$
$7$	−1.22010	−0.461155	−0.230577	+	0.973054i	$0.574061\pi$
$8$	3.06410	1.08332
$9$	2.69310	0.897701
$10$	0	0
$11$	5.09294	1.53558	0.767789	−	0.640702i	$-0.221357\pi$
$11$	5.09294	1.53558	0.767789	+	0.640702i	$0.221357\pi$
$12$	−1.22010	−0.352213
$13$	−2.60778	−0.723267	−0.361633	−	0.932320i	$-0.617781\pi$
$13$	−2.60778	−0.723267	−0.361633	+	0.932320i	$0.617781\pi$
$14$	1.48865	0.397858
$15$	0	0
$16$	−2.71581	−0.678953
$17$	0	0
$17$	0	0
$18$	−3.28586	−0.774484
$19$	8.51548	1.95359	0.976793	−	0.214187i	$-0.0687103\pi$
$19$	8.51548	1.95359	0.976793	+	0.214187i	$0.0687103\pi$
$20$	0	0
$21$	−2.91119	−0.635273
$22$	−6.21390	−1.32481
$23$	2.89653	0.603968	0.301984	−	0.953313i	$-0.402351\pi$
$23$	2.89653	0.603968	0.301984	+	0.953313i	$0.402351\pi$
$24$	7.31102	1.49236
$25$	0	0
$26$	3.18175	0.623993
$27$	−0.732262	−0.140924
$28$	0.623903	0.117907
$29$	5.49277	1.01998	0.509991	−	0.860180i	$-0.329649\pi$
$29$	5.49277	1.01998	0.509991	+	0.860180i	$0.329649\pi$
$30$	0	0
$31$	3.18175	0.571459	0.285730	−	0.958310i	$-0.407764\pi$
$31$	3.18175	0.571459	0.285730	+	0.958310i	$0.407764\pi$
$32$	−2.81465	−0.497564
$33$	12.1519	2.11537
$34$	0	0
$35$	0	0
$36$	−1.37713	−0.229521
$37$	−3.09561	−0.508916	−0.254458	−	0.967084i	$-0.581897\pi$
$37$	−3.09561	−0.508916	−0.254458	+	0.967084i	$0.581897\pi$
$38$	−10.3897	−1.68544
$39$	−6.22221	−0.996351
$40$	0	0
$41$	−0.181750	−0.0283846	−0.0141923	−	0.999899i	$-0.504518\pi$
$41$	−0.181750	−0.0283846	−0.0141923	+	0.999899i	$0.504518\pi$
$42$	3.55194	0.548077
$43$	6.70174	1.02201	0.511003	−	0.859579i	$-0.329274\pi$
$43$	6.70174	1.02201	0.511003	+	0.859579i	$0.329274\pi$
$44$	−2.60429	−0.392612
$45$	0	0
$46$	−3.53406	−0.521069
$47$	8.71332	1.27097	0.635484	−	0.772114i	$-0.280801\pi$
$47$	8.71332	1.27097	0.635484	+	0.772114i	$0.280801\pi$
$48$	−6.47999	−0.935305
$49$	−5.51135	−0.787336
$50$	0	0
$51$	0	0
$52$	1.33350	0.184923
$53$	−0.167574	−0.0230180	−0.0115090	−	0.999934i	$-0.503664\pi$
$53$	−0.167574	−0.0230180	−0.0115090	+	0.999934i	$0.503664\pi$
$54$	0.893434	0.121581
$55$	0	0
$56$	−3.73852	−0.499580
$57$	20.3181	2.69120
$58$	−6.70174	−0.879981
$59$	−13.7336	−1.78796	−0.893979	−	0.448109i	$-0.852098\pi$
$59$	−13.7336	−1.78796	−0.893979	+	0.448109i	$0.852098\pi$
$60$	0	0
$61$	0.293269	0.0375492	0.0187746	−	0.999824i	$-0.494024\pi$
$61$	0.293269	0.0375492	0.0187746	+	0.999824i	$0.494024\pi$
$62$	−3.88206	−0.493022
$63$	−3.28586	−0.413979
$64$	8.86577	1.10822
$65$	0	0
$66$	−14.8265	−1.82502
$67$	1.56304	0.190955	0.0954776	−	0.995432i	$-0.469562\pi$
$67$	1.56304	0.190955	0.0954776	+	0.995432i	$0.469562\pi$
$68$	0	0
$69$	6.91119	0.832009
$70$	0	0
$71$	6.93390	0.822902	0.411451	−	0.911432i	$-0.365022\pi$
$71$	6.93390	0.822902	0.411451	+	0.911432i	$0.365022\pi$
$72$	8.25195	0.972502
$73$	−8.04302	−0.941364	−0.470682	−	0.882303i	$-0.655992\pi$
$73$	−8.04302	−0.941364	−0.470682	+	0.882303i	$0.655992\pi$
$74$	3.77696	0.439063
$75$	0	0
$76$	−4.35442	−0.499486
$77$	−6.21390	−0.708140
$78$	7.59173	0.859594
$79$	−0.0888115	−0.00999207	−0.00499604	−	0.999988i	$-0.501590\pi$
$79$	−0.0888115	−0.00999207	−0.00499604	+	0.999988i	$0.501590\pi$
$80$	0	0
$81$	−9.82650	−1.09183
$82$	0.221753	0.0244886
$83$	−14.8138	−1.62603	−0.813014	−	0.582244i	$-0.802175\pi$
$83$	−14.8138	−1.62603	−0.813014	+	0.582244i	$0.802175\pi$
$84$	1.48865	0.162425
$85$	0	0
$86$	−8.17680	−0.881727
$87$	13.1059	1.40510
$88$	15.6053	1.66353
$89$	−2.79554	−0.296327	−0.148163	−	0.988963i	$-0.547336\pi$
$89$	−2.79554	−0.296327	−0.148163	+	0.988963i	$0.547336\pi$
$90$	0	0
$91$	3.18175	0.333538
$92$	−1.48115	−0.154421
$93$	7.59173	0.787225
$94$	−10.6311	−1.09652
$95$	0	0
$96$	−6.71581	−0.685430
$97$	−3.54691	−0.360134	−0.180067	−	0.983654i	$-0.557631\pi$
$97$	−3.54691	−0.360134	−0.180067	+	0.983654i	$0.557631\pi$
$98$	6.72441	0.679268
$99$	13.7158	1.37849
$100$	0	0
$101$	11.9855	1.19261	0.596303	−	0.802759i	$-0.296636\pi$
$101$	11.9855	1.19261	0.596303	+	0.802759i	$0.296636\pi$
$102$	0	0
$103$	16.6578	1.64134	0.820672	−	0.571399i	$-0.193599\pi$
$103$	16.6578	1.64134	0.820672	+	0.571399i	$0.193599\pi$
$104$	−7.99050	−0.783533
$105$	0	0
$106$	0.204457	0.0198586
$107$	−15.5904	−1.50718	−0.753591	−	0.657343i	$-0.771680\pi$
$107$	−15.5904	−1.50718	−0.753591	+	0.657343i	$0.771680\pi$
$108$	0.374445	0.0360310
$109$	5.13423	0.491770	0.245885	−	0.969299i	$-0.420922\pi$
$109$	5.13423	0.491770	0.245885	+	0.969299i	$0.420922\pi$
$110$	0	0
$111$	−7.38621	−0.701068
$112$	3.31356	0.313102
$113$	11.7497	1.10532	0.552660	−	0.833407i	$-0.313613\pi$
$113$	11.7497	1.10532	0.552660	+	0.833407i	$0.313613\pi$
$114$	−24.7902	−2.32181
$115$	0	0
$116$	−2.80875	−0.260786
$117$	−7.02301	−0.649278
$118$	16.7563	1.54255
$119$	0	0
$120$	0	0
$121$	14.9380	1.35800
$122$	−0.357817	−0.0323952
$123$	−0.433660	−0.0391018
$124$	−1.62700	−0.146109
$125$	0	0
$126$	4.00908	0.357157
$127$	−4.14814	−0.368088	−0.184044	−	0.982918i	$-0.558919\pi$
$127$	−4.14814	−0.368088	−0.184044	+	0.982918i	$0.558919\pi$
$128$	−5.18785	−0.458545
$129$	15.9905	1.40788
$130$	0	0
$131$	−2.93390	−0.256336	−0.128168	−	0.991752i	$-0.540910\pi$
$131$	−2.93390	−0.256336	−0.128168	+	0.991752i	$0.540910\pi$
$132$	−6.21390	−0.540851
$133$	−10.3897	−0.900905
$134$	−1.90706	−0.164745
$135$	0	0
$136$	0	0
$137$	2.65211	0.226585	0.113292	−	0.993562i	$-0.463860\pi$
$137$	2.65211	0.226585	0.113292	+	0.993562i	$0.463860\pi$
$138$	−8.43235	−0.717809
$139$	7.14785	0.606273	0.303137	−	0.952947i	$-0.401966\pi$
$139$	7.14785	0.606273	0.303137	+	0.952947i	$0.401966\pi$
$140$	0	0
$141$	20.7902	1.75085
$142$	−8.46005	−0.709952
$143$	−13.2812	−1.11063
$144$	−7.31396	−0.609497
$145$	0	0
$146$	9.81330	0.812154
$147$	−13.1502	−1.08461
$148$	1.58295	0.130118
$149$	14.7521	1.20854	0.604271	−	0.796779i	$-0.293464\pi$
$149$	14.7521	1.20854	0.604271	+	0.796779i	$0.293464\pi$
$150$	0	0
$151$	16.9236	1.37722	0.688610	−	0.725132i	$-0.258221\pi$
$151$	16.9236	1.37722	0.688610	+	0.725132i	$0.258221\pi$
$152$	26.0923	2.11637
$153$	0	0
$154$	7.58158	0.610942
$155$	0	0
$156$	3.18175	0.254744
$157$	11.3515	0.905952	0.452976	−	0.891523i	$-0.350362\pi$
$157$	11.3515	0.905952	0.452976	+	0.891523i	$0.350362\pi$
$158$	0.108359	0.00862058
$159$	−0.399835	−0.0317090
$160$	0	0
$161$	−3.53406	−0.278523
$162$	11.9893	0.941971
$163$	9.83285	0.770168	0.385084	−	0.922882i	$-0.374172\pi$
$163$	9.83285	0.770168	0.385084	+	0.922882i	$0.374172\pi$
$164$	0.0929385	0.00725728
$165$	0	0
$166$	18.0744	1.40284
$167$	−5.59503	−0.432957	−0.216478	−	0.976287i	$-0.569457\pi$
$167$	−5.59503	−0.432957	−0.216478	+	0.976287i	$0.569457\pi$
$168$	−8.92019	−0.688207
$169$	−6.19950	−0.476885
$170$	0	0
$171$	22.9331	1.75374
$172$	−3.42696	−0.261303
$173$	−20.3773	−1.54926	−0.774630	−	0.632414i	$-0.782064\pi$
$173$	−20.3773	−1.54926	−0.774630	+	0.632414i	$0.782064\pi$
$174$	−15.9905	−1.21224
$175$	0	0
$176$	−13.8315	−1.04259
$177$	−32.7686	−2.46304
$178$	3.41084	0.255654
$179$	−1.82944	−0.136739	−0.0683694	−	0.997660i	$-0.521780\pi$
$179$	−1.82944	−0.136739	−0.0683694	+	0.997660i	$0.521780\pi$
$180$	0	0
$181$	−17.5060	−1.30121	−0.650605	−	0.759417i	$-0.725484\pi$
$181$	−17.5060	−1.30121	−0.650605	+	0.759417i	$0.725484\pi$
$182$	−3.88206	−0.287757
$183$	0.699745	0.0517267
$184$	8.87528	0.654294
$185$	0	0
$186$	−9.26267	−0.679172
$187$	0	0
$188$	−4.45558	−0.324957
$189$	0.893434	0.0649877
$190$	0	0
$191$	−10.0793	−0.729313	−0.364657	−	0.931142i	$-0.618814\pi$
$191$	−10.0793	−0.729313	−0.364657	+	0.931142i	$0.618814\pi$
$192$	21.1539	1.52665
$193$	5.02531	0.361730	0.180865	−	0.983508i	$-0.442110\pi$
$193$	5.02531	0.361730	0.180865	+	0.983508i	$0.442110\pi$
$194$	4.32759	0.310703
$195$	0	0
$196$	2.81825	0.201304
$197$	−5.23822	−0.373208	−0.186604	−	0.982435i	$-0.559748\pi$
$197$	−5.23822	−0.373208	−0.186604	+	0.982435i	$0.559748\pi$
$198$	−16.7347	−1.18928
$199$	−9.75215	−0.691311	−0.345656	−	0.938361i	$-0.612343\pi$
$199$	−9.75215	−0.691311	−0.345656	+	0.938361i	$0.612343\pi$
$200$	0	0
$201$	3.72944	0.263054
$202$	−14.6236	−1.02891
$203$	−6.70174	−0.470370
$204$	0	0
$205$	0	0
$206$	−20.3242	−1.41606
$207$	7.80066	0.542183
$208$	7.08223	0.491064
$209$	43.3688	2.99988
$210$	0	0
$211$	16.5857	1.14181	0.570904	−	0.821017i	$-0.306593\pi$
$211$	16.5857	1.14181	0.570904	+	0.821017i	$0.306593\pi$
$212$	0.0856895	0.00588518
$213$	16.5444	1.13361
$214$	19.0219	1.30031
$215$	0	0
$216$	−2.24373	−0.152666
$217$	−3.88206	−0.263531
$218$	−6.26427	−0.424270
$219$	−19.1908	−1.29680
$220$	0	0
$221$	0	0
$222$	9.01192	0.604840
$223$	20.9685	1.40416	0.702078	−	0.712101i	$-0.252256\pi$
$223$	20.9685	1.40416	0.702078	+	0.712101i	$0.252256\pi$
$224$	3.43415	0.229454
$225$	0	0
$226$	−14.3358	−0.953606
$227$	3.36949	0.223641	0.111820	−	0.993728i	$-0.464332\pi$
$227$	3.36949	0.223641	0.111820	+	0.993728i	$0.464332\pi$
$228$	−10.3897	−0.688078
$229$	10.9897	0.726218	0.363109	−	0.931747i	$-0.381715\pi$
$229$	10.9897	0.726218	0.363109	+	0.931747i	$0.381715\pi$
$230$	0	0
$231$	−14.8265	−0.975512
$232$	16.8304	1.10497
$233$	4.22499	0.276788	0.138394	−	0.990377i	$-0.455806\pi$
$233$	4.22499	0.276788	0.138394	+	0.990377i	$0.455806\pi$
$234$	8.56878	0.560159
$235$	0	0
$236$	7.02271	0.457139
$237$	−0.211906	−0.0137648
$238$	0	0
$239$	−7.05578	−0.456400	−0.228200	−	0.973614i	$-0.573284\pi$
$239$	−7.05578	−0.456400	−0.228200	+	0.973614i	$0.573284\pi$
$240$	0	0
$241$	1.85459	0.119464	0.0597322	−	0.998214i	$-0.480975\pi$
$241$	1.85459	0.119464	0.0597322	+	0.998214i	$0.480975\pi$
$242$	−18.2259	−1.17160
$243$	−21.2495	−1.36315
$244$	−0.149964	−0.00960045
$245$	0	0
$246$	0.529108	0.0337347
$247$	−22.2065	−1.41296
$248$	9.74922	0.619076
$249$	−35.3461	−2.23997
$250$	0	0
$251$	22.3833	1.41282	0.706410	−	0.707803i	$-0.250314\pi$
$251$	22.3833	1.41282	0.706410	+	0.707803i	$0.250314\pi$
$252$	1.68024	0.105845
$253$	14.7519	0.927441
$254$	5.06115	0.317565
$255$	0	0
$256$	−11.4019	−0.712616
$257$	9.88702	0.616736	0.308368	−	0.951267i	$-0.400217\pi$
$257$	9.88702	0.616736	0.308368	+	0.951267i	$0.400217\pi$
$258$	−19.5100	−1.21464
$259$	3.77696	0.234689
$260$	0	0
$261$	14.7926	0.915639
$262$	3.57965	0.221151
$263$	−13.3798	−0.825031	−0.412515	−	0.910951i	$-0.635350\pi$
$263$	−13.3798	−0.825031	−0.412515	+	0.910951i	$0.635350\pi$
$264$	37.2346	2.29163
$265$	0	0
$266$	12.6765	0.777249
$267$	−6.67023	−0.408211
$268$	−0.799264	−0.0488228
$269$	13.5341	0.825186	0.412593	−	0.910915i	$-0.364623\pi$
$269$	13.5341	0.825186	0.412593	+	0.910915i	$0.364623\pi$
$270$	0	0
$271$	0.170561	0.0103608	0.00518041	−	0.999987i	$-0.498351\pi$
$271$	0.170561	0.0103608	0.00518041	+	0.999987i	$0.498351\pi$
$272$	0	0
$273$	7.59173	0.459472
$274$	−3.23584	−0.195484
$275$	0	0
$276$	−3.53406	−0.212725
$277$	−26.7893	−1.60961	−0.804806	−	0.593538i	$-0.797731\pi$
$277$	−26.7893	−1.60961	−0.804806	+	0.593538i	$0.797731\pi$
$278$	−8.72110	−0.523057
$279$	8.56878	0.513000
$280$	0	0
$281$	11.4267	0.681658	0.340829	−	0.940125i	$-0.389292\pi$
$281$	11.4267	0.681658	0.340829	+	0.940125i	$0.389292\pi$
$282$	−25.3661	−1.51053
$283$	−15.2082	−0.904033	−0.452016	−	0.892010i	$-0.649295\pi$
$283$	−15.2082	−0.904033	−0.452016	+	0.892010i	$0.649295\pi$
$284$	−3.54567	−0.210397
$285$	0	0
$286$	16.2045	0.958190
$287$	0.221753	0.0130897
$288$	−7.58014	−0.446664
$289$	0	0
$290$	0	0
$291$	−8.46300	−0.496110
$292$	4.11283	0.240685
$293$	22.5189	1.31557	0.657785	−	0.753205i	$-0.271493\pi$
$293$	22.5189	1.31557	0.657785	+	0.753205i	$0.271493\pi$
$294$	16.0446	0.935739
$295$	0	0
$296$	−9.48529	−0.551321
$297$	−3.72937	−0.216400
$298$	−17.9991	−1.04266
$299$	−7.55350	−0.436830
$300$	0	0
$301$	−8.17680	−0.471303
$302$	−20.6485	−1.18819
$303$	28.5978	1.64290
$304$	−23.1264	−1.32639
$305$	0	0
$306$	0	0
$307$	−17.0837	−0.975018	−0.487509	−	0.873118i	$-0.662094\pi$
$307$	−17.0837	−0.975018	−0.487509	+	0.873118i	$0.662094\pi$
$308$	3.17750	0.181055
$309$	39.7459	2.26107
$310$	0	0
$311$	4.23254	0.240005	0.120003	−	0.992774i	$-0.461710\pi$
$311$	4.23254	0.240005	0.120003	+	0.992774i	$0.461710\pi$
$312$	−19.0655	−1.07937
$313$	30.7671	1.73906	0.869529	−	0.493881i	$-0.164422\pi$
$313$	30.7671	1.73906	0.869529	+	0.493881i	$0.164422\pi$
$314$	−13.8500	−0.781603
$315$	0	0
$316$	0.0454141	0.00255474
$317$	−8.07279	−0.453413	−0.226706	−	0.973963i	$-0.572796\pi$
$317$	−8.07279	−0.453413	−0.226706	+	0.973963i	$0.572796\pi$
$318$	0.487839	0.0273567
$319$	27.9744	1.56626
$320$	0	0
$321$	−37.1991	−2.07625
$322$	4.31191	0.240293
$323$	0	0
$324$	5.02482	0.279157
$325$	0	0
$326$	−11.9971	−0.664456
$327$	12.2504	0.677447
$328$	−0.556901	−0.0307497
$329$	−10.6311	−0.586113
$330$	0	0
$331$	−6.97023	−0.383119	−0.191559	−	0.981481i	$-0.561354\pi$
$331$	−6.97023	−0.383119	−0.191559	+	0.981481i	$0.561354\pi$
$332$	7.57510	0.415738
$333$	−8.33681	−0.456854
$334$	6.82650	0.373530
$335$	0	0
$336$	7.90624	0.431321
$337$	33.1481	1.80569	0.902845	−	0.429965i	$-0.141474\pi$
$337$	33.1481	1.80569	0.902845	+	0.429965i	$0.141474\pi$
$338$	7.56402	0.411429
$339$	28.0351	1.52266
$340$	0	0
$341$	16.2045	0.877521
$342$	−27.9807	−1.51302
$343$	15.2651	0.824239
$344$	20.5348	1.10716
$345$	0	0
$346$	24.8624	1.33661
$347$	−2.60778	−0.139993	−0.0699964	−	0.997547i	$-0.522299\pi$
$347$	−2.60778	−0.139993	−0.0699964	+	0.997547i	$0.522299\pi$
$348$	−6.70174	−0.359251
$349$	19.5494	1.04645	0.523227	−	0.852193i	$-0.324728\pi$
$349$	19.5494	1.04645	0.523227	+	0.852193i	$0.324728\pi$
$350$	0	0
$351$	1.90958	0.101926
$352$	−14.3348	−0.764049
$353$	7.75252	0.412625	0.206313	−	0.978486i	$-0.433854\pi$
$353$	7.75252	0.412625	0.206313	+	0.978486i	$0.433854\pi$
$354$	39.9810	2.12497
$355$	0	0
$356$	1.42951	0.0757639
$357$	0	0
$358$	2.23210	0.117970
$359$	8.62911	0.455427	0.227713	−	0.973728i	$-0.426875\pi$
$359$	8.62911	0.455427	0.227713	+	0.973728i	$0.426875\pi$
$360$	0	0
$361$	53.5134	2.81650
$362$	21.3591	1.12261
$363$	35.6425	1.87074
$364$	−1.62700	−0.0852779
$365$	0	0
$366$	−0.853760	−0.0446267
$367$	23.4513	1.22415	0.612074	−	0.790801i	$-0.290336\pi$
$367$	23.4513	1.22415	0.612074	+	0.790801i	$0.290336\pi$
$368$	−7.86643	−0.410066
$369$	−0.489472	−0.0254809
$370$	0	0
$371$	0.204457	0.0106149
$372$	−3.88206	−0.201275
$373$	−17.1484	−0.887911	−0.443956	−	0.896049i	$-0.646425\pi$
$373$	−17.1484	−0.887911	−0.443956	+	0.896049i	$0.646425\pi$
$374$	0	0
$375$	0	0
$376$	26.6985	1.37687
$377$	−14.3239	−0.737720
$378$	−1.09008	−0.0560676
$379$	−5.81994	−0.298950	−0.149475	−	0.988765i	$-0.547758\pi$
$379$	−5.81994	−0.298950	−0.149475	+	0.988765i	$0.547758\pi$
$380$	0	0
$381$	−9.89756	−0.507067
$382$	12.2978	0.629209
$383$	−18.7137	−0.956227	−0.478114	−	0.878298i	$-0.658679\pi$
$383$	−18.7137	−0.956227	−0.478114	+	0.878298i	$0.658679\pi$
$384$	−12.3783	−0.631678
$385$	0	0
$386$	−6.13138	−0.312079
$387$	18.0485	0.917456
$388$	1.81372	0.0920779
$389$	4.25611	0.215793	0.107897	−	0.994162i	$-0.465588\pi$
$389$	4.25611	0.215793	0.107897	+	0.994162i	$0.465588\pi$
$390$	0	0
$391$	0	0
$392$	−16.8874	−0.852941
$393$	−7.00034	−0.353120
$394$	6.39116	0.321982
$395$	0	0
$396$	−7.01363	−0.352448
$397$	−2.37320	−0.119108	−0.0595538	−	0.998225i	$-0.518968\pi$
$397$	−2.37320	−0.119108	−0.0595538	+	0.998225i	$0.518968\pi$
$398$	11.8986	0.596423
$399$	−24.7902	−1.24106
$400$	0	0
$401$	32.0215	1.59908	0.799538	−	0.600616i	$-0.205078\pi$
$401$	32.0215	1.59908	0.799538	+	0.600616i	$0.205078\pi$
$402$	−4.55029	−0.226948
$403$	−8.29729	−0.413317
$404$	−6.12885	−0.304922
$405$	0	0
$406$	8.17680	0.405808
$407$	−15.7658	−0.781481
$408$	0	0
$409$	5.58654	0.276236	0.138118	−	0.990416i	$-0.455895\pi$
$409$	5.58654	0.276236	0.138118	+	0.990416i	$0.455895\pi$
$410$	0	0
$411$	6.32799	0.312137
$412$	−8.51804	−0.419654
$413$	16.7563	0.824525
$414$	−9.51759	−0.467764
$415$	0	0
$416$	7.33997	0.359872
$417$	17.0549	0.835184
$418$	−52.9143	−2.58812
$419$	17.3222	0.846245	0.423123	−	0.906072i	$-0.360934\pi$
$419$	17.3222	0.846245	0.423123	+	0.906072i	$0.360934\pi$
$420$	0	0
$421$	4.03634	0.196719	0.0983595	−	0.995151i	$-0.468641\pi$
$421$	4.03634	0.196719	0.0983595	+	0.995151i	$0.468641\pi$
$422$	−20.2362	−0.985085
$423$	23.4659	1.14095
$424$	−0.513464	−0.0249360
$425$	0	0
$426$	−20.1859	−0.978009
$427$	−0.357817	−0.0173160
$428$	7.97221	0.385352
$429$	−31.6893	−1.52998
$430$	0	0
$431$	−7.85871	−0.378541	−0.189270	−	0.981925i	$-0.560612\pi$
$431$	−7.85871	−0.378541	−0.189270	+	0.981925i	$0.560612\pi$
$432$	1.98868	0.0956806
$433$	−1.37385	−0.0660228	−0.0330114	−	0.999455i	$-0.510510\pi$
$433$	−1.37385	−0.0660228	−0.0330114	+	0.999455i	$0.510510\pi$
$434$	4.73650	0.227359
$435$	0	0
$436$	−2.62540	−0.125734
$437$	24.6654	1.17990
$438$	23.4147	1.11880
$439$	3.99376	0.190612	0.0953059	−	0.995448i	$-0.469617\pi$
$439$	3.99376	0.190612	0.0953059	+	0.995448i	$0.469617\pi$
$440$	0	0
$441$	−14.8426	−0.706793
$442$	0	0
$443$	−27.6102	−1.31180	−0.655900	−	0.754848i	$-0.727711\pi$
$443$	−27.6102	−1.31180	−0.655900	+	0.754848i	$0.727711\pi$
$444$	3.77696	0.179247
$445$	0	0
$446$	−25.5837	−1.21142
$447$	35.1990	1.66485
$448$	−10.8171	−0.511062
$449$	7.78848	0.367561	0.183781	−	0.982967i	$-0.441166\pi$
$449$	7.78848	0.367561	0.183781	+	0.982967i	$0.441166\pi$
$450$	0	0
$451$	−0.925642	−0.0435868
$452$	−6.00826	−0.282605
$453$	40.3800	1.89722
$454$	−4.11112	−0.192944
$455$	0	0
$456$	62.2569	2.91545
$457$	−15.9443	−0.745841	−0.372920	−	0.927863i	$-0.621644\pi$
$457$	−15.9443	−0.745841	−0.372920	+	0.927863i	$0.621644\pi$
$458$	−13.4085	−0.626539
$459$	0	0
$460$	0	0
$461$	15.3288	0.713932	0.356966	−	0.934117i	$-0.383811\pi$
$461$	15.3288	0.713932	0.356966	+	0.934117i	$0.383811\pi$
$462$	18.0898	0.841615
$463$	−6.34996	−0.295108	−0.147554	−	0.989054i	$-0.547140\pi$
$463$	−6.34996	−0.295108	−0.147554	+	0.989054i	$0.547140\pi$
$464$	−14.9173	−0.692520
$465$	0	0
$466$	−5.15492	−0.238797
$467$	−26.8800	−1.24386	−0.621929	−	0.783074i	$-0.713651\pi$
$467$	−26.8800	−1.24386	−0.621929	+	0.783074i	$0.713651\pi$
$468$	3.59124	0.166005
$469$	−1.90706	−0.0880599
$470$	0	0
$471$	27.0851	1.24801
$472$	−42.0811	−1.93694
$473$	34.1315	1.56937
$474$	0.258547	0.0118755
$475$	0	0
$476$	0	0
$477$	−0.451294	−0.0206633
$478$	8.60876	0.393756
$479$	29.2788	1.33778	0.668892	−	0.743360i	$-0.266769\pi$
$479$	29.2788	1.33778	0.668892	+	0.743360i	$0.266769\pi$
$480$	0	0
$481$	8.07267	0.368082
$482$	−2.26278	−0.103067
$483$	−8.43235	−0.383685
$484$	−7.63861	−0.347210
$485$	0	0
$486$	25.9265	1.17605
$487$	23.0166	1.04298	0.521492	−	0.853256i	$-0.325376\pi$
$487$	23.0166	1.04298	0.521492	+	0.853256i	$0.325376\pi$
$488$	0.898606	0.0406780
$489$	23.4614	1.06096
$490$	0	0
$491$	−9.99917	−0.451256	−0.225628	−	0.974213i	$-0.572443\pi$
$491$	−9.99917	−0.451256	−0.225628	+	0.974213i	$0.572443\pi$
$492$	0.221753	0.00999741
$493$	0	0
$494$	27.0941	1.21902
$495$	0	0
$496$	−8.64103	−0.387994
$497$	−8.46005	−0.379485
$498$	43.1258	1.93251
$499$	19.1289	0.856325	0.428163	−	0.903702i	$-0.359161\pi$
$499$	19.1289	0.856325	0.428163	+	0.903702i	$0.359161\pi$
$500$	0	0
$501$	−13.3499	−0.596428
$502$	−27.3099	−1.21890
$503$	−31.4994	−1.40449	−0.702243	−	0.711937i	$-0.747818\pi$
$503$	−31.4994	−1.40449	−0.702243	+	0.711937i	$0.747818\pi$
$504$	−10.0682	−0.448474
$505$	0	0
$506$	−17.9988	−0.800142
$507$	−14.7922	−0.656943
$508$	2.12117	0.0941116
$509$	−21.5494	−0.955159	−0.477580	−	0.878588i	$-0.658486\pi$
$509$	−21.5494	−0.955159	−0.477580	+	0.878588i	$0.658486\pi$
$510$	0	0
$511$	9.81330	0.434115
$512$	24.2871	1.07335
$513$	−6.23556	−0.275307
$514$	−12.0632	−0.532084
$515$	0	0
$516$	−8.17680	−0.359963
$517$	44.3764	1.95167
$518$	−4.60828	−0.202476
$519$	−48.6208	−2.13422
$520$	0	0
$521$	20.8311	0.912625	0.456312	−	0.889820i	$-0.349170\pi$
$521$	20.8311	0.912625	0.456312	+	0.889820i	$0.349170\pi$
$522$	−18.0485	−0.789960
$523$	−41.1113	−1.79767	−0.898835	−	0.438287i	$-0.855585\pi$
$523$	−41.1113	−1.79767	−0.898835	+	0.438287i	$0.855585\pi$
$524$	1.50026	0.0655391
$525$	0	0
$526$	16.3247	0.711789
$527$	0	0
$528$	−33.0022	−1.43623
$529$	−14.6101	−0.635222
$530$	0	0
$531$	−36.9859	−1.60505
$532$	5.31283	0.230341
$533$	0.473963	0.0205296
$534$	8.13835	0.352181
$535$	0	0
$536$	4.78931	0.206866
$537$	−4.36508	−0.188367
$538$	−16.5129	−0.711923
$539$	−28.0690	−1.20902
$540$	0	0
$541$	−40.3482	−1.73471	−0.867353	−	0.497694i	$-0.834180\pi$
$541$	−40.3482	−1.73471	−0.867353	+	0.497694i	$0.834180\pi$
$542$	−0.208101	−0.00893871
$543$	−41.7697	−1.79251
$544$	0	0
$545$	0	0
$546$	−9.26267	−0.396406
$547$	1.63485	0.0699011	0.0349506	−	0.999389i	$-0.488873\pi$
$547$	1.63485	0.0699011	0.0349506	+	0.999389i	$0.488873\pi$
$548$	−1.35616	−0.0579325
$549$	0.789802	0.0337079
$550$	0	0
$551$	46.7736	1.99262
$552$	21.1766	0.901336
$553$	0.108359	0.00460789
$554$	32.6856	1.38868
$555$	0	0
$556$	−3.65508	−0.155010
$557$	13.0683	0.553723	0.276861	−	0.960910i	$-0.410706\pi$
$557$	13.0683	0.553723	0.276861	+	0.960910i	$0.410706\pi$
$558$	−10.4548	−0.442586
$559$	−17.4766	−0.739183
$560$	0	0
$561$	0	0
$562$	−13.9417	−0.588095
$563$	−16.1788	−0.681856	−0.340928	−	0.940089i	$-0.610741\pi$
$563$	−16.1788	−0.681856	−0.340928	+	0.940089i	$0.610741\pi$
$564$	−10.6311	−0.447651
$565$	0	0
$566$	18.5555	0.779947
$567$	11.9893	0.503504
$568$	21.2462	0.891470
$569$	−30.5580	−1.28106	−0.640528	−	0.767935i	$-0.721285\pi$
$569$	−30.5580	−1.28106	−0.640528	+	0.767935i	$0.721285\pi$
$570$	0	0
$571$	−28.4965	−1.19254	−0.596270	−	0.802784i	$-0.703351\pi$
$571$	−28.4965	−1.19254	−0.596270	+	0.802784i	$0.703351\pi$
$572$	6.79141	0.283963
$573$	−24.0495	−1.00468
$574$	−0.270561	−0.0112930
$575$	0	0
$576$	23.8764	0.994852
$577$	−47.3380	−1.97071	−0.985353	−	0.170525i	$-0.945453\pi$
$577$	−47.3380	−1.97071	−0.985353	+	0.170525i	$0.945453\pi$
$578$	0	0
$579$	11.9905	0.498308
$580$	0	0
$581$	18.0744	0.749851
$582$	10.3257	0.428015
$583$	−0.853443	−0.0353460
$584$	−24.6447	−1.01980
$585$	0	0
$586$	−27.4754	−1.13500
$587$	16.6921	0.688956	0.344478	−	0.938794i	$-0.388056\pi$
$587$	16.6921	0.688956	0.344478	+	0.938794i	$0.388056\pi$
$588$	6.72441	0.277310
$589$	27.0941	1.11639
$590$	0	0
$591$	−12.4985	−0.514120
$592$	8.40710	0.345530
$593$	8.64023	0.354812	0.177406	−	0.984138i	$-0.443230\pi$
$593$	8.64023	0.354812	0.177406	+	0.984138i	$0.443230\pi$
$594$	4.55020	0.186697
$595$	0	0
$596$	−7.54356	−0.308996
$597$	−23.2688	−0.952330
$598$	9.21604	0.376872
$599$	−16.6378	−0.679801	−0.339900	−	0.940461i	$-0.610393\pi$
$599$	−16.6378	−0.679801	−0.339900	+	0.940461i	$0.610393\pi$
$600$	0	0
$601$	−2.12885	−0.0868377	−0.0434188	−	0.999057i	$-0.513825\pi$
$601$	−2.12885	−0.0868377	−0.0434188	+	0.999057i	$0.513825\pi$
$602$	9.97652	0.406613
$603$	4.20942	0.171421
$604$	−8.65393	−0.352123
$605$	0	0
$606$	−34.8922	−1.41740
$607$	27.8320	1.12966	0.564832	−	0.825206i	$-0.308941\pi$
$607$	27.8320	1.12966	0.564832	+	0.825206i	$0.308941\pi$
$608$	−23.9681	−0.972034
$609$	−15.9905	−0.647968
$610$	0	0
$611$	−22.7224	−0.919249
$612$	0	0
$613$	18.2060	0.735333	0.367666	−	0.929958i	$-0.380157\pi$
$613$	18.2060	0.735333	0.367666	+	0.929958i	$0.380157\pi$
$614$	20.8438	0.841189
$615$	0	0
$616$	−19.0400	−0.767145
$617$	−23.8562	−0.960416	−0.480208	−	0.877155i	$-0.659439\pi$
$617$	−23.8562	−0.960416	−0.480208	+	0.877155i	$0.659439\pi$
$618$	−48.4941	−1.95072
$619$	25.9281	1.04214	0.521070	−	0.853514i	$-0.325533\pi$
$619$	25.9281	1.04214	0.521070	+	0.853514i	$0.325533\pi$
$620$	0	0
$621$	−2.12102	−0.0851136
$622$	−5.16412	−0.207063
$623$	3.41084	0.136653
$624$	16.8983	0.676475
$625$	0	0
$626$	−37.5390	−1.50036
$627$	103.479	4.13255
$628$	−5.80465	−0.231631
$629$	0	0
$630$	0	0
$631$	−29.8191	−1.18708	−0.593540	−	0.804804i	$-0.702270\pi$
$631$	−29.8191	−1.18708	−0.593540	+	0.804804i	$0.702270\pi$
$632$	−0.272128	−0.0108247
$633$	39.5739	1.57292
$634$	9.84961	0.391178
$635$	0	0
$636$	0.204457	0.00810725
$637$	14.3724	0.569454
$638$	−34.1315	−1.35128
$639$	18.6737	0.738720
$640$	0	0
$641$	−2.91036	−0.114952	−0.0574762	−	0.998347i	$-0.518305\pi$
$641$	−2.91036	−0.114952	−0.0574762	+	0.998347i	$0.518305\pi$
$642$	45.3866	1.79127
$643$	−6.02970	−0.237788	−0.118894	−	0.992907i	$-0.537935\pi$
$643$	−6.02970	−0.237788	−0.118894	+	0.992907i	$0.537935\pi$
$644$	1.80715	0.0712119
$645$	0	0
$646$	0	0
$647$	23.9114	0.940055	0.470028	−	0.882652i	$-0.344244\pi$
$647$	23.9114	0.940055	0.470028	+	0.882652i	$0.344244\pi$
$648$	−30.1094	−1.18281
$649$	−69.9442	−2.74555
$650$	0	0
$651$	−9.26267	−0.363033
$652$	−5.02806	−0.196914
$653$	13.9545	0.546082	0.273041	−	0.962002i	$-0.411970\pi$
$653$	13.9545	0.546082	0.273041	+	0.962002i	$0.411970\pi$
$654$	−14.9467	−0.584462
$655$	0	0
$656$	0.493599	0.0192718
$657$	−21.6607	−0.845064
$658$	12.9710	0.505664
$659$	−26.5601	−1.03463	−0.517317	−	0.855794i	$-0.673069\pi$
$659$	−26.5601	−1.03463	−0.517317	+	0.855794i	$0.673069\pi$
$660$	0	0
$661$	−5.90789	−0.229790	−0.114895	−	0.993378i	$-0.536653\pi$
$661$	−5.90789	−0.229790	−0.114895	+	0.993378i	$0.536653\pi$
$662$	8.50439	0.330532
$663$	0	0
$664$	−45.3911	−1.76152
$665$	0	0
$666$	10.1718	0.394147
$667$	15.9100	0.616037
$668$	2.86104	0.110697
$669$	50.0313	1.93432
$670$	0	0
$671$	1.49360	0.0576597
$672$	8.19397	0.316089
$673$	21.5332	0.830044	0.415022	−	0.909811i	$-0.363774\pi$
$673$	21.5332	0.830044	0.415022	+	0.909811i	$0.363774\pi$
$674$	−40.4440	−1.55784
$675$	0	0
$676$	3.17014	0.121928
$677$	−1.33748	−0.0514034	−0.0257017	−	0.999670i	$-0.508182\pi$
$677$	−1.33748	−0.0514034	−0.0257017	+	0.999670i	$0.508182\pi$
$678$	−34.2056	−1.31366
$679$	4.32759	0.166078
$680$	0	0
$681$	8.03967	0.308081
$682$	−19.7711	−0.757074
$683$	−2.97842	−0.113966	−0.0569830	−	0.998375i	$-0.518148\pi$
$683$	−2.97842	−0.113966	−0.0569830	+	0.998375i	$0.518148\pi$
$684$	−11.7269	−0.448389
$685$	0	0
$686$	−18.6250	−0.711105
$687$	26.2216	1.00042
$688$	−18.2007	−0.693893
$689$	0.436995	0.0166482
$690$	0	0
$691$	27.7356	1.05511	0.527556	−	0.849520i	$-0.323109\pi$
$691$	27.7356	1.05511	0.527556	+	0.849520i	$0.323109\pi$
$692$	10.4200	0.396110
$693$	−16.7347	−0.635698
$694$	3.18175	0.120778
$695$	0	0
$696$	40.1578	1.52218
$697$	0	0
$698$	−23.8522	−0.902820
$699$	10.0809	0.381296
$700$	0	0
$701$	18.9711	0.716528	0.358264	−	0.933620i	$-0.383369\pi$
$701$	18.9711	0.716528	0.358264	+	0.933620i	$0.383369\pi$
$702$	−2.32987	−0.0879354
$703$	−26.3606	−0.994211
$704$	45.1528	1.70176
$705$	0	0
$706$	−9.45886	−0.355989
$707$	−14.6236	−0.549976
$708$	16.7563	0.629742
$709$	−21.6039	−0.811350	−0.405675	−	0.914017i	$-0.632964\pi$
$709$	−21.6039	−0.811350	−0.405675	+	0.914017i	$0.632964\pi$
$710$	0	0
$711$	−0.239179	−0.00896990
$712$	−8.56584	−0.321018
$713$	9.21604	0.345143
$714$	0	0
$715$	0	0
$716$	0.935490	0.0349609
$717$	−16.8352	−0.628724
$718$	−10.5284	−0.392916
$719$	−32.9236	−1.22784	−0.613921	−	0.789367i	$-0.710409\pi$
$719$	−32.9236	−1.22784	−0.613921	+	0.789367i	$0.710409\pi$
$720$	0	0
$721$	−20.3242	−0.756914
$722$	−65.2918	−2.42991
$723$	4.42508	0.164571
$724$	8.95174	0.332689
$725$	0	0
$726$	−43.4874	−1.61397
$727$	26.9451	0.999338	0.499669	−	0.866217i	$-0.333455\pi$
$727$	26.9451	0.999338	0.499669	+	0.866217i	$0.333455\pi$
$728$	9.74922	0.361330
$729$	−21.2222	−0.786008
$730$	0	0
$731$	0	0
$732$	−0.357817	−0.0132253
$733$	23.7123	0.875835	0.437918	−	0.899015i	$-0.355716\pi$
$733$	23.7123	0.875835	0.437918	+	0.899015i	$0.355716\pi$
$734$	−28.6129	−1.05612
$735$	0	0
$736$	−8.15271	−0.300513
$737$	7.96044	0.293227
$738$	0.597205	0.0219834
$739$	−12.1673	−0.447581	−0.223791	−	0.974637i	$-0.571843\pi$
$739$	−12.1673	−0.447581	−0.223791	+	0.974637i	$0.571843\pi$
$740$	0	0
$741$	−52.9851	−1.94646
$742$	−0.249458	−0.00915790
$743$	48.1778	1.76747	0.883736	−	0.467986i	$-0.155020\pi$
$743$	48.1778	1.76747	0.883736	+	0.467986i	$0.155020\pi$
$744$	23.2618	0.852821
$745$	0	0
$746$	20.9228	0.766038
$747$	−39.8952	−1.45969
$748$	0	0
$749$	19.0219	0.695044
$750$	0	0
$751$	27.0124	0.985696	0.492848	−	0.870115i	$-0.335956\pi$
$751$	27.0124	0.985696	0.492848	+	0.870115i	$0.335956\pi$
$752$	−23.6637	−0.862927
$753$	53.4070	1.94626
$754$	17.4766	0.636461
$755$	0	0
$756$	−0.456860	−0.0166159
$757$	−9.53731	−0.346639	−0.173320	−	0.984866i	$-0.555449\pi$
$757$	−9.53731	−0.346639	−0.173320	+	0.984866i	$0.555449\pi$
$758$	7.10091	0.257917
$759$	35.1983	1.27762
$760$	0	0
$761$	35.0640	1.27107	0.635535	−	0.772072i	$-0.280780\pi$
$761$	35.0640	1.27107	0.635535	+	0.772072i	$0.280780\pi$
$762$	12.0760	0.437468
$763$	−6.26427	−0.226782
$764$	5.15409	0.186468
$765$	0	0
$766$	22.8326	0.824977
$767$	35.8141	1.29317
$768$	−27.2051	−0.981679
$769$	23.1144	0.833527	0.416763	−	0.909015i	$-0.363164\pi$
$769$	23.1144	0.833527	0.416763	+	0.909015i	$0.363164\pi$
$770$	0	0
$771$	23.5907	0.849597
$772$	−2.56971	−0.0924859
$773$	−32.1951	−1.15798	−0.578988	−	0.815336i	$-0.696552\pi$
$773$	−32.1951	−1.15798	−0.578988	+	0.815336i	$0.696552\pi$
$774$	−22.0210	−0.791527
$775$	0	0
$776$	−10.8681	−0.390142
$777$	9.01192	0.323301
$778$	−5.19288	−0.186174
$779$	−1.54769	−0.0554517
$780$	0	0
$781$	35.3139	1.26363
$782$	0	0
$783$	−4.02215	−0.143740
$784$	14.9678	0.534564
$785$	0	0
$786$	8.54112	0.304652
$787$	−31.8413	−1.13502	−0.567510	−	0.823367i	$-0.692093\pi$
$787$	−31.8413	−1.13502	−0.567510	+	0.823367i	$0.692093\pi$
$788$	2.67858	0.0954206
$789$	−31.9244	−1.13654
$790$	0	0
$791$	−14.3358	−0.509724
$792$	42.0267	1.49335
$793$	−0.764779	−0.0271581
$794$	2.89554	0.102759
$795$	0	0
$796$	4.98679	0.176752
$797$	28.0182	0.992456	0.496228	−	0.868192i	$-0.334718\pi$
$797$	28.0182	0.992456	0.496228	+	0.868192i	$0.334718\pi$
$798$	30.2465	1.07071
$799$	0	0
$800$	0	0
$801$	−7.52869	−0.266013
$802$	−39.0694	−1.37959
$803$	−40.9626	−1.44554
$804$	−1.90706	−0.0672569
$805$	0	0
$806$	10.1235	0.356586
$807$	32.2926	1.13675
$808$	36.7250	1.29198
$809$	−5.11363	−0.179786	−0.0898928	−	0.995951i	$-0.528652\pi$
$809$	−5.11363	−0.179786	−0.0898928	+	0.995951i	$0.528652\pi$
$810$	0	0
$811$	7.64274	0.268373	0.134186	−	0.990956i	$-0.457158\pi$
$811$	7.64274	0.268373	0.134186	+	0.990956i	$0.457158\pi$
$812$	3.42696	0.120263
$813$	0.406961	0.0142728
$814$	19.2358	0.674216
$815$	0	0
$816$	0	0
$817$	57.0685	1.99657
$818$	−6.81614	−0.238321
$819$	8.56878	0.299417
$820$	0	0
$821$	26.0661	0.909713	0.454857	−	0.890565i	$-0.349690\pi$
$821$	26.0661	0.909713	0.454857	+	0.890565i	$0.349690\pi$
$822$	−7.72079	−0.269293
$823$	−28.2352	−0.984216	−0.492108	−	0.870534i	$-0.663773\pi$
$823$	−28.2352	−0.984216	−0.492108	+	0.870534i	$0.663773\pi$
$824$	51.0413	1.77811
$825$	0	0
$826$	−20.4444	−0.711352
$827$	26.4070	0.918263	0.459131	−	0.888368i	$-0.348161\pi$
$827$	26.4070	0.918263	0.459131	+	0.888368i	$0.348161\pi$
$828$	−3.98889	−0.138624
$829$	−19.6406	−0.682147	−0.341074	−	0.940037i	$-0.610791\pi$
$829$	−19.6406	−0.682147	−0.341074	+	0.940037i	$0.610791\pi$
$830$	0	0
$831$	−63.9198	−2.21735
$832$	−23.1200	−0.801540
$833$	0	0
$834$	−20.8087	−0.720548
$835$	0	0
$836$	−22.1768	−0.767001
$837$	−2.32987	−0.0805322
$838$	−21.1348	−0.730091
$839$	−26.1879	−0.904106	−0.452053	−	0.891991i	$-0.649308\pi$
$839$	−26.1879	−0.904106	−0.452053	+	0.891991i	$0.649308\pi$
$840$	0	0
$841$	1.17056	0.0403642
$842$	−4.92474	−0.169718
$843$	27.2643	0.939032
$844$	−8.48116	−0.291934
$845$	0	0
$846$	−28.6307	−0.984345
$847$	−18.2259	−0.626249
$848$	0.455099	0.0156282
$849$	−36.2871	−1.24537
$850$	0	0
$851$	−8.96655	−0.307369
$852$	−8.46005	−0.289837
$853$	14.4807	0.495811	0.247905	−	0.968784i	$-0.420258\pi$
$853$	14.4807	0.495811	0.247905	+	0.968784i	$0.420258\pi$
$854$	0.436573	0.0149392
$855$	0	0
$856$	−47.7707	−1.63277
$857$	38.1094	1.30179	0.650895	−	0.759167i	$-0.274394\pi$
$857$	38.1094	1.30179	0.650895	+	0.759167i	$0.274394\pi$
$858$	38.6642	1.31997
$859$	−5.28290	−0.180250	−0.0901252	−	0.995930i	$-0.528727\pi$
$859$	−5.28290	−0.180250	−0.0901252	+	0.995930i	$0.528727\pi$
$860$	0	0
$861$	0.529108	0.0180320
$862$	9.58842	0.326583
$863$	13.9555	0.475051	0.237526	−	0.971381i	$-0.423664\pi$
$863$	13.9555	0.475051	0.237526	+	0.971381i	$0.423664\pi$
$864$	2.06106	0.0701186
$865$	0	0
$866$	1.67623	0.0569606
$867$	0	0
$868$	1.98510	0.0673788
$869$	−0.452311	−0.0153436
$870$	0	0
$871$	−4.07605	−0.138112
$872$	15.7318	0.532746
$873$	−9.55219	−0.323293
$874$	−30.0942	−1.01795
$875$	0	0
$876$	9.81330	0.331561
$877$	27.2887	0.921474	0.460737	−	0.887537i	$-0.347585\pi$
$877$	27.2887	0.921474	0.460737	+	0.887537i	$0.347585\pi$
$878$	−4.87279	−0.164449
$879$	53.7307	1.81229
$880$	0	0
$881$	1.38294	0.0465925	0.0232963	−	0.999729i	$-0.492584\pi$
$881$	1.38294	0.0465925	0.0232963	+	0.999729i	$0.492584\pi$
$882$	18.1095	0.609779
$883$	−38.9311	−1.31013	−0.655067	−	0.755570i	$-0.727360\pi$
$883$	−38.9311	−1.31013	−0.655067	+	0.755570i	$0.727360\pi$
$884$	0	0
$885$	0	0
$886$	33.6872	1.13174
$887$	11.5882	0.389093	0.194547	−	0.980893i	$-0.437676\pi$
$887$	11.5882	0.389093	0.194547	+	0.980893i	$0.437676\pi$
$888$	−22.6321	−0.759484
$889$	5.06115	0.169746
$890$	0	0
$891$	−50.0458	−1.67660
$892$	−10.7223	−0.359010
$893$	74.1981	2.48294
$894$	−42.9463	−1.43634
$895$	0	0
$896$	6.32970	0.211460
$897$	−18.0228	−0.601765
$898$	−9.50273	−0.317110
$899$	17.4766	0.582878
$900$	0	0
$901$	0	0
$902$	1.12938	0.0376041
$903$	−19.5100	−0.649253
$904$	36.0024	1.19742
$905$	0	0
$906$	−49.2677	−1.63681
$907$	−40.1287	−1.33245	−0.666226	−	0.745750i	$-0.732091\pi$
$907$	−40.1287	−1.33245	−0.666226	+	0.745750i	$0.732091\pi$
$908$	−1.72300	−0.0571797
$909$	32.2783	1.07060
$910$	0	0
$911$	38.1808	1.26499	0.632494	−	0.774566i	$-0.282031\pi$
$911$	38.1808	1.26499	0.632494	+	0.774566i	$0.282031\pi$
$912$	−55.1802	−1.82720
$913$	−75.4459	−2.49689
$914$	19.4536	0.643468
$915$	0	0
$916$	−5.61961	−0.185677
$917$	3.57965	0.118210
$918$	0	0
$919$	18.9678	0.625690	0.312845	−	0.949804i	$-0.398718\pi$
$919$	18.9678	0.625690	0.312845	+	0.949804i	$0.398718\pi$
$920$	0	0
$921$	−40.7621	−1.34316
$922$	−18.7027	−0.615939
$923$	−18.0820	−0.595178
$924$	7.58158	0.249416
$925$	0	0
$926$	7.74760	0.254602
$927$	44.8612	1.47344
$928$	−15.4602	−0.507506
$929$	−25.1479	−0.825076	−0.412538	−	0.910940i	$-0.635358\pi$
$929$	−25.1479	−0.825076	−0.412538	+	0.910940i	$0.635358\pi$
$930$	0	0
$931$	−46.9318	−1.53813
$932$	−2.16046	−0.0707684
$933$	10.0989	0.330624
$934$	32.7963	1.07313
$935$	0	0
$936$	−21.5192	−0.703378
$937$	−3.62000	−0.118260	−0.0591301	−	0.998250i	$-0.518833\pi$
$937$	−3.62000	−0.118260	−0.0591301	+	0.998250i	$0.518833\pi$
$938$	2.32681	0.0759730
$939$	73.4110	2.39568
$940$	0	0
$941$	−25.9083	−0.844585	−0.422292	−	0.906460i	$-0.638774\pi$
$941$	−25.9083	−0.844585	−0.422292	+	0.906460i	$0.638774\pi$
$942$	−33.0465	−1.07671
$943$	−0.526445	−0.0171434
$944$	37.2978	1.21394
$945$	0	0
$946$	−41.6439	−1.35396
$947$	3.84826	0.125052	0.0625259	−	0.998043i	$-0.480084\pi$
$947$	3.84826	0.125052	0.0625259	+	0.998043i	$0.480084\pi$
$948$	0.108359	0.00351934
$949$	20.9744	0.680858
$950$	0	0
$951$	−19.2618	−0.624608
$952$	0	0
$953$	24.1253	0.781494	0.390747	−	0.920498i	$-0.372217\pi$
$953$	24.1253	0.781494	0.390747	+	0.920498i	$0.372217\pi$
$954$	0.550624	0.0178271
$955$	0	0
$956$	3.60800	0.116691
$957$	66.7474	2.15764
$958$	−35.7231	−1.15416
$959$	−3.23584	−0.104491
$960$	0	0
$961$	−20.8765	−0.673434
$962$	−9.84947	−0.317560
$963$	−41.9866	−1.35300
$964$	−0.948349	−0.0305443
$965$	0	0
$966$	10.2883	0.331021
$967$	−47.4022	−1.52435	−0.762177	−	0.647369i	$-0.775869\pi$
$967$	−47.4022	−1.52435	−0.762177	+	0.647369i	$0.775869\pi$
$968$	45.7717	1.47116
$969$	0	0
$970$	0	0
$971$	21.6700	0.695423	0.347711	−	0.937602i	$-0.386959\pi$
$971$	21.6700	0.695423	0.347711	+	0.937602i	$0.386959\pi$
$972$	10.8660	0.348527
$973$	−8.72110	−0.279586
$974$	−28.0826	−0.899825
$975$	0	0
$976$	−0.796462	−0.0254941
$977$	24.7334	0.791290	0.395645	−	0.918403i	$-0.370521\pi$
$977$	24.7334	0.791290	0.395645	+	0.918403i	$0.370521\pi$
$978$	−28.6253	−0.915335
$979$	−14.2375	−0.455033
$980$	0	0
$981$	13.8270	0.441462
$982$	12.2000	0.389318
$983$	−29.5399	−0.942176	−0.471088	−	0.882086i	$-0.656139\pi$
$983$	−29.5399	−0.942176	−0.471088	+	0.882086i	$0.656139\pi$
$984$	−1.32878	−0.0423599
$985$	0	0
$986$	0	0
$987$	−25.3661	−0.807412
$988$	11.3554	0.361262
$989$	19.4118	0.617259
$990$	0	0
$991$	39.3751	1.25079	0.625395	−	0.780308i	$-0.284938\pi$
$991$	39.3751	1.25079	0.625395	+	0.780308i	$0.284938\pi$
$992$	−8.95550	−0.284337
$993$	−16.6311	−0.527773
$994$	10.3221	0.327398
$995$	0	0
$996$	18.0744	0.572708
$997$	27.2169	0.861968	0.430984	−	0.902360i	$-0.358167\pi$
$997$	27.2169	0.861968	0.430984	+	0.902360i	$0.358167\pi$
$998$	−23.3391	−0.738787
$999$	2.26680	0.0717184

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bk.1.3		8
5.2	odd	4		1445.2.b.c.579.3	✓	8
5.3	odd	4		1445.2.b.c.579.6	yes	8
5.4	even	2	inner	7225.2.a.bk.1.6		8
17.16	even	2		7225.2.a.bj.1.3		8
85.33	odd	4		1445.2.b.d.579.6	yes	8
85.67	odd	4		1445.2.b.d.579.3	yes	8
85.84	even	2		7225.2.a.bj.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1445.2.b.c.579.3	✓	8	5.2	odd	4
1445.2.b.c.579.6	yes	8	5.3	odd	4
1445.2.b.d.579.3	yes	8	85.67	odd	4
1445.2.b.d.579.6	yes	8	85.33	odd	4
7225.2.a.bj.1.3		8	17.16	even	2
7225.2.a.bj.1.6		8	85.84	even	2
7225.2.a.bk.1.3		8	1.1	even	1	trivial
7225.2.a.bk.1.6		8	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.22010 q^{2} +2.38602 q^{3} -0.511354 q^{4} -2.91119 q^{6} -1.22010 q^{7} +3.06410 q^{8} +2.69310 q^{9} +5.09294 q^{11} -1.22010 q^{12} -2.60778 q^{13} +1.48865 q^{14} -2.71581 q^{16} -3.28586 q^{18} +8.51548 q^{19} -2.91119 q^{21} -6.21390 q^{22} +2.89653 q^{23} +7.31102 q^{24} +3.18175 q^{26} -0.732262 q^{27} +0.623903 q^{28} +5.49277 q^{29} +3.18175 q^{31} -2.81465 q^{32} +12.1519 q^{33} -1.37713 q^{36} -3.09561 q^{37} -10.3897 q^{38} -6.22221 q^{39} -0.181750 q^{41} +3.55194 q^{42} +6.70174 q^{43} -2.60429 q^{44} -3.53406 q^{46} +8.71332 q^{47} -6.47999 q^{48} -5.51135 q^{49} +1.33350 q^{52} -0.167574 q^{53} +0.893434 q^{54} -3.73852 q^{56} +20.3181 q^{57} -6.70174 q^{58} -13.7336 q^{59} +0.293269 q^{61} -3.88206 q^{62} -3.28586 q^{63} +8.86577 q^{64} -14.8265 q^{66} +1.56304 q^{67} +6.91119 q^{69} +6.93390 q^{71} +8.25195 q^{72} -8.04302 q^{73} +3.77696 q^{74} -4.35442 q^{76} -6.21390 q^{77} +7.59173 q^{78} -0.0888115 q^{79} -9.82650 q^{81} +0.221753 q^{82} -14.8138 q^{83} +1.48865 q^{84} -8.17680 q^{86} +13.1059 q^{87} +15.6053 q^{88} -2.79554 q^{89} +3.18175 q^{91} -1.48115 q^{92} +7.59173 q^{93} -10.6311 q^{94} -6.71581 q^{96} -3.54691 q^{97} +6.72441 q^{98} +13.7158 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} + 6 q^{6} + 8 q^{9} + 10 q^{11} + 24 q^{14} + 16 q^{16} - 4 q^{19} + 6 q^{21} + 12 q^{24} + 24 q^{26} - 4 q^{29} + 24 q^{31} - 18 q^{36} + 26 q^{39} + 22 q^{44} + 8 q^{46} - 32 q^{49} + 28 q^{54}+ \cdots + 72 q^{99}+O(q^{100}) \) 8 * q + 8 * q^4 + 6 * q^6 + 8 * q^9 + 10 * q^11 + 24 * q^14 + 16 * q^16 - 4 * q^19 + 6 * q^21 + 12 * q^24 + 24 * q^26 - 4 * q^29 + 24 * q^31 - 18 * q^36 + 26 * q^39 + 22 * q^44 + 8 * q^46 - 32 * q^49 + 28 * q^54 + 32 * q^56 - 22 * q^59 + 6 * q^61 + 90 * q^64 + 26 * q^69 + 2 * q^71 + 20 * q^74 - 66 * q^76 - 30 * q^79 + 40 * q^81 + 24 * q^84 - 38 * q^86 - 48 * q^89 + 24 * q^91 + 42 * q^94 - 16 * q^96 + 72 * q^99

Introduction

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