LMFDB - Embedded newform 7225.2.a.p.1.1

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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.41421 q^{3} -1.00000 q^{4} -1.41421 q^{6} +4.24264 q^{7} -3.00000 q^{8} -1.00000 q^{9} +4.24264 q^{11} +1.41421 q^{12} +4.24264 q^{14} -1.00000 q^{16} -1.00000 q^{18} -6.00000 q^{19} -6.00000 q^{21} +4.24264 q^{22} -1.41421 q^{23} +4.24264 q^{24} +5.65685 q^{27} -4.24264 q^{28} +4.24264 q^{29} +1.41421 q^{31} +5.00000 q^{32} -6.00000 q^{33} +1.00000 q^{36} -4.24264 q^{37} -6.00000 q^{38} -4.24264 q^{41} -6.00000 q^{42} +12.0000 q^{43} -4.24264 q^{44} -1.41421 q^{46} +2.00000 q^{47} +1.41421 q^{48} +11.0000 q^{49} +2.00000 q^{53} +5.65685 q^{54} -12.7279 q^{56} +8.48528 q^{57} +4.24264 q^{58} +6.00000 q^{59} +1.41421 q^{61} +1.41421 q^{62} -4.24264 q^{63} +7.00000 q^{64} -6.00000 q^{66} -6.00000 q^{67} +2.00000 q^{69} -4.24264 q^{71} +3.00000 q^{72} -4.24264 q^{73} -4.24264 q^{74} +6.00000 q^{76} +18.0000 q^{77} -9.89949 q^{79} -5.00000 q^{81} -4.24264 q^{82} -4.00000 q^{83} +6.00000 q^{84} +12.0000 q^{86} -6.00000 q^{87} -12.7279 q^{88} +6.00000 q^{89} +1.41421 q^{92} -2.00000 q^{93} +2.00000 q^{94} -7.07107 q^{96} +4.24264 q^{97} +11.0000 q^{98} -4.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} - 2 q^{9} - 2 q^{16} - 2 q^{18} - 12 q^{19} - 12 q^{21} + 10 q^{32} - 12 q^{33} + 2 q^{36} - 12 q^{38} - 12 q^{42} + 24 q^{43} + 4 q^{47} + 22 q^{49} + 4 q^{53} + 12 q^{59}+ \cdots + 22 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 - 2 * q^9 - 2 * q^16 - 2 * q^18 - 12 * q^19 - 12 * q^21 + 10 * q^32 - 12 * q^33 + 2 * q^36 - 12 * q^38 - 12 * q^42 + 24 * q^43 + 4 * q^47 + 22 * q^49 + 4 * q^53 + 12 * q^59 + 14 * q^64 - 12 * q^66 - 12 * q^67 + 4 * q^69 + 6 * q^72 + 12 * q^76 + 36 * q^77 - 10 * q^81 - 8 * q^83 + 12 * q^84 + 24 * q^86 - 12 * q^87 + 12 * q^89 - 4 * q^93 + 4 * q^94 + 22 * q^98

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.41421	−0.577350
$7$	4.24264	1.60357	0.801784	−	0.597614i	$-0.203885\pi$
$7$	4.24264	1.60357	0.801784	+	0.597614i	$0.203885\pi$
$8$	−3.00000	−1.06066
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	1.41421	0.408248
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	4.24264	1.13389
$15$	0	0
$16$	−1.00000	−0.250000
$17$	0	0
$17$	0	0
$18$	−1.00000	−0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−6.00000	−1.30931
$22$	4.24264	0.904534
$23$	−1.41421	−0.294884	−0.147442	−	0.989071i	$-0.547104\pi$
$23$	−1.41421	−0.294884	−0.147442	+	0.989071i	$0.547104\pi$
$24$	4.24264	0.866025
$25$	0	0
$26$	0	0
$27$	5.65685	1.08866
$28$	−4.24264	−0.801784
$29$	4.24264	0.787839	0.393919	−	0.919145i	$-0.371119\pi$
$29$	4.24264	0.787839	0.393919	+	0.919145i	$0.371119\pi$
$30$	0	0
$31$	1.41421	0.254000	0.127000	−	0.991903i	$-0.459465\pi$
$31$	1.41421	0.254000	0.127000	+	0.991903i	$0.459465\pi$
$32$	5.00000	0.883883
$33$	−6.00000	−1.04447
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.24264	−0.697486	−0.348743	−	0.937218i	$-0.613391\pi$
$37$	−4.24264	−0.697486	−0.348743	+	0.937218i	$0.613391\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	0	0
$41$	−4.24264	−0.662589	−0.331295	−	0.943527i	$-0.607485\pi$
$41$	−4.24264	−0.662589	−0.331295	+	0.943527i	$0.607485\pi$
$42$	−6.00000	−0.925820
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	−4.24264	−0.639602
$45$	0	0
$46$	−1.41421	−0.208514
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	1.41421	0.204124
$49$	11.0000	1.57143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	5.65685	0.769800
$55$	0	0
$56$	−12.7279	−1.70084
$57$	8.48528	1.12390
$58$	4.24264	0.557086
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	1.41421	0.181071	0.0905357	−	0.995893i	$-0.471142\pi$
$61$	1.41421	0.181071	0.0905357	+	0.995893i	$0.471142\pi$
$62$	1.41421	0.179605
$63$	−4.24264	−0.534522
$64$	7.00000	0.875000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	−4.24264	−0.503509	−0.251754	−	0.967791i	$-0.581008\pi$
$71$	−4.24264	−0.503509	−0.251754	+	0.967791i	$0.581008\pi$
$72$	3.00000	0.353553
$73$	−4.24264	−0.496564	−0.248282	−	0.968688i	$-0.579866\pi$
$73$	−4.24264	−0.496564	−0.248282	+	0.968688i	$0.579866\pi$
$74$	−4.24264	−0.493197
$75$	0	0
$76$	6.00000	0.688247
$77$	18.0000	2.05129
$78$	0	0
$79$	−9.89949	−1.11378	−0.556890	−	0.830586i	$-0.688006\pi$
$79$	−9.89949	−1.11378	−0.556890	+	0.830586i	$0.688006\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	−4.24264	−0.468521
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	6.00000	0.654654
$85$	0	0
$86$	12.0000	1.29399
$87$	−6.00000	−0.643268
$88$	−12.7279	−1.35680
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	1.41421	0.147442
$93$	−2.00000	−0.207390
$94$	2.00000	0.206284
$95$	0	0
$96$	−7.07107	−0.721688
$97$	4.24264	0.430775	0.215387	−	0.976529i	$-0.430899\pi$
$97$	4.24264	0.430775	0.215387	+	0.976529i	$0.430899\pi$
$98$	11.0000	1.11117
$99$	−4.24264	−0.426401
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	0	0
$106$	2.00000	0.194257
$107$	−12.7279	−1.23045	−0.615227	−	0.788350i	$-0.710936\pi$
$107$	−12.7279	−1.23045	−0.615227	+	0.788350i	$0.710936\pi$
$108$	−5.65685	−0.544331
$109$	9.89949	0.948200	0.474100	−	0.880471i	$-0.342774\pi$
$109$	9.89949	0.948200	0.474100	+	0.880471i	$0.342774\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	−4.24264	−0.400892
$113$	12.7279	1.19734	0.598671	−	0.800995i	$-0.295696\pi$
$113$	12.7279	1.19734	0.598671	+	0.800995i	$0.295696\pi$
$114$	8.48528	0.794719
$115$	0	0
$116$	−4.24264	−0.393919
$117$	0	0
$118$	6.00000	0.552345
$119$	0	0
$120$	0	0
$121$	7.00000	0.636364
$122$	1.41421	0.128037
$123$	6.00000	0.541002
$124$	−1.41421	−0.127000
$125$	0	0
$126$	−4.24264	−0.377964
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−3.00000	−0.265165
$129$	−16.9706	−1.49417
$130$	0	0
$131$	4.24264	0.370681	0.185341	−	0.982674i	$-0.440661\pi$
$131$	4.24264	0.370681	0.185341	+	0.982674i	$0.440661\pi$
$132$	6.00000	0.522233
$133$	−25.4558	−2.20730
$134$	−6.00000	−0.518321
$135$	0	0
$136$	0	0
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	2.00000	0.170251
$139$	9.89949	0.839664	0.419832	−	0.907602i	$-0.362089\pi$
$139$	9.89949	0.839664	0.419832	+	0.907602i	$0.362089\pi$
$140$	0	0
$141$	−2.82843	−0.238197
$142$	−4.24264	−0.356034
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−4.24264	−0.351123
$147$	−15.5563	−1.28307
$148$	4.24264	0.348743
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	18.0000	1.45999
$153$	0	0
$154$	18.0000	1.45048
$155$	0	0
$156$	0	0
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	−9.89949	−0.787562
$159$	−2.82843	−0.224309
$160$	0	0
$161$	−6.00000	−0.472866
$162$	−5.00000	−0.392837
$163$	12.7279	0.996928	0.498464	−	0.866910i	$-0.333898\pi$
$163$	12.7279	0.996928	0.498464	+	0.866910i	$0.333898\pi$
$164$	4.24264	0.331295
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−4.24264	−0.328305	−0.164153	−	0.986435i	$-0.552489\pi$
$167$	−4.24264	−0.328305	−0.164153	+	0.986435i	$0.552489\pi$
$168$	18.0000	1.38873
$169$	−13.0000	−1.00000
$170$	0	0
$171$	6.00000	0.458831
$172$	−12.0000	−0.914991
$173$	21.2132	1.61281	0.806405	−	0.591364i	$-0.201410\pi$
$173$	21.2132	1.61281	0.806405	+	0.591364i	$0.201410\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	−4.24264	−0.319801
$177$	−8.48528	−0.637793
$178$	6.00000	0.449719
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	15.5563	1.15629	0.578147	−	0.815933i	$-0.303776\pi$
$181$	15.5563	1.15629	0.578147	+	0.815933i	$0.303776\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	4.24264	0.312772
$185$	0	0
$186$	−2.00000	−0.146647
$187$	0	0
$188$	−2.00000	−0.145865
$189$	24.0000	1.74574
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−9.89949	−0.714435
$193$	−4.24264	−0.305392	−0.152696	−	0.988273i	$-0.548796\pi$
$193$	−4.24264	−0.305392	−0.152696	+	0.988273i	$0.548796\pi$
$194$	4.24264	0.304604
$195$	0	0
$196$	−11.0000	−0.785714
$197$	18.3848	1.30986	0.654931	−	0.755689i	$-0.272698\pi$
$197$	18.3848	1.30986	0.654931	+	0.755689i	$0.272698\pi$
$198$	−4.24264	−0.301511
$199$	1.41421	0.100251	0.0501255	−	0.998743i	$-0.484038\pi$
$199$	1.41421	0.100251	0.0501255	+	0.998743i	$0.484038\pi$
$200$	0	0
$201$	8.48528	0.598506
$202$	−6.00000	−0.422159
$203$	18.0000	1.26335
$204$	0	0
$205$	0	0
$206$	−6.00000	−0.418040
$207$	1.41421	0.0982946
$208$	0	0
$209$	−25.4558	−1.76082
$210$	0	0
$211$	24.0416	1.65509	0.827547	−	0.561396i	$-0.189736\pi$
$211$	24.0416	1.65509	0.827547	+	0.561396i	$0.189736\pi$
$212$	−2.00000	−0.137361
$213$	6.00000	0.411113
$214$	−12.7279	−0.870063
$215$	0	0
$216$	−16.9706	−1.15470
$217$	6.00000	0.407307
$218$	9.89949	0.670478
$219$	6.00000	0.405442
$220$	0	0
$221$	0	0
$222$	6.00000	0.402694
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	21.2132	1.41737
$225$	0	0
$226$	12.7279	0.846649
$227$	−21.2132	−1.40797	−0.703985	−	0.710215i	$-0.748598\pi$
$227$	−21.2132	−1.40797	−0.703985	+	0.710215i	$0.748598\pi$
$228$	−8.48528	−0.561951
$229$	24.0000	1.58596	0.792982	−	0.609245i	$-0.208527\pi$
$229$	24.0000	1.58596	0.792982	+	0.609245i	$0.208527\pi$
$230$	0	0
$231$	−25.4558	−1.67487
$232$	−12.7279	−0.835629
$233$	7.07107	0.463241	0.231621	−	0.972806i	$-0.425597\pi$
$233$	7.07107	0.463241	0.231621	+	0.972806i	$0.425597\pi$
$234$	0	0
$235$	0	0
$236$	−6.00000	−0.390567
$237$	14.0000	0.909398
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	26.8701	1.73085	0.865426	−	0.501036i	$-0.167048\pi$
$241$	26.8701	1.73085	0.865426	+	0.501036i	$0.167048\pi$
$242$	7.00000	0.449977
$243$	−9.89949	−0.635053
$244$	−1.41421	−0.0905357
$245$	0	0
$246$	6.00000	0.382546
$247$	0	0
$248$	−4.24264	−0.269408
$249$	5.65685	0.358489
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	4.24264	0.267261
$253$	−6.00000	−0.377217
$254$	0	0
$255$	0	0
$256$	−17.0000	−1.06250
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	−16.9706	−1.05654
$259$	−18.0000	−1.11847
$260$	0	0
$261$	−4.24264	−0.262613
$262$	4.24264	0.262111
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	18.0000	1.10782
$265$	0	0
$266$	−25.4558	−1.56080
$267$	−8.48528	−0.519291
$268$	6.00000	0.366508
$269$	−4.24264	−0.258678	−0.129339	−	0.991600i	$-0.541286\pi$
$269$	−4.24264	−0.258678	−0.129339	+	0.991600i	$0.541286\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	0	0
$274$	−4.00000	−0.241649
$275$	0	0
$276$	−2.00000	−0.120386
$277$	−29.6985	−1.78441	−0.892205	−	0.451632i	$-0.850842\pi$
$277$	−29.6985	−1.78441	−0.892205	+	0.451632i	$0.850842\pi$
$278$	9.89949	0.593732
$279$	−1.41421	−0.0846668
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	−2.82843	−0.168430
$283$	−12.7279	−0.756596	−0.378298	−	0.925684i	$-0.623491\pi$
$283$	−12.7279	−0.756596	−0.378298	+	0.925684i	$0.623491\pi$
$284$	4.24264	0.251754
$285$	0	0
$286$	0	0
$287$	−18.0000	−1.06251
$288$	−5.00000	−0.294628
$289$	0	0
$290$	0	0
$291$	−6.00000	−0.351726
$292$	4.24264	0.248282
$293$	4.00000	0.233682	0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000	0.233682	0.116841	+	0.993151i	$0.462723\pi$
$294$	−15.5563	−0.907265
$295$	0	0
$296$	12.7279	0.739795
$297$	24.0000	1.39262
$298$	−18.0000	−1.04271
$299$	0	0
$300$	0	0
$301$	50.9117	2.93450
$302$	10.0000	0.575435
$303$	8.48528	0.487467
$304$	6.00000	0.344124
$305$	0	0
$306$	0	0
$307$	18.0000	1.02731	0.513657	−	0.857996i	$-0.328290\pi$
$307$	18.0000	1.02731	0.513657	+	0.857996i	$0.328290\pi$
$308$	−18.0000	−1.02565
$309$	8.48528	0.482711
$310$	0	0
$311$	4.24264	0.240578	0.120289	−	0.992739i	$-0.461618\pi$
$311$	4.24264	0.240578	0.120289	+	0.992739i	$0.461618\pi$
$312$	0	0
$313$	4.24264	0.239808	0.119904	−	0.992785i	$-0.461741\pi$
$313$	4.24264	0.239808	0.119904	+	0.992785i	$0.461741\pi$
$314$	12.0000	0.677199
$315$	0	0
$316$	9.89949	0.556890
$317$	21.2132	1.19145	0.595726	−	0.803188i	$-0.296864\pi$
$317$	21.2132	1.19145	0.595726	+	0.803188i	$0.296864\pi$
$318$	−2.82843	−0.158610
$319$	18.0000	1.00781
$320$	0	0
$321$	18.0000	1.00466
$322$	−6.00000	−0.334367
$323$	0	0
$324$	5.00000	0.277778
$325$	0	0
$326$	12.7279	0.704934
$327$	−14.0000	−0.774202
$328$	12.7279	0.702782
$329$	8.48528	0.467809
$330$	0	0
$331$	18.0000	0.989369	0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000	0.989369	0.494685	+	0.869072i	$0.335284\pi$
$332$	4.00000	0.219529
$333$	4.24264	0.232495
$334$	−4.24264	−0.232147
$335$	0	0
$336$	6.00000	0.327327
$337$	−21.2132	−1.15556	−0.577778	−	0.816194i	$-0.696080\pi$
$337$	−21.2132	−1.15556	−0.577778	+	0.816194i	$0.696080\pi$
$338$	−13.0000	−0.707107
$339$	−18.0000	−0.977626
$340$	0	0
$341$	6.00000	0.324918
$342$	6.00000	0.324443
$343$	16.9706	0.916324
$344$	−36.0000	−1.94099
$345$	0	0
$346$	21.2132	1.14043
$347$	−12.7279	−0.683271	−0.341635	−	0.939833i	$-0.610981\pi$
$347$	−12.7279	−0.683271	−0.341635	+	0.939833i	$0.610981\pi$
$348$	6.00000	0.321634
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	0	0
$351$	0	0
$352$	21.2132	1.13067
$353$	−16.0000	−0.851594	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	−16.0000	−0.851594	−0.425797	+	0.904819i	$0.640006\pi$
$354$	−8.48528	−0.450988
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	6.00000	0.317110
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	15.5563	0.817624
$363$	−9.89949	−0.519589
$364$	0	0
$365$	0	0
$366$	−2.00000	−0.104542
$367$	4.24264	0.221464	0.110732	−	0.993850i	$-0.464680\pi$
$367$	4.24264	0.221464	0.110732	+	0.993850i	$0.464680\pi$
$368$	1.41421	0.0737210
$369$	4.24264	0.220863
$370$	0	0
$371$	8.48528	0.440534
$372$	2.00000	0.103695
$373$	−24.0000	−1.24267	−0.621336	−	0.783544i	$-0.713410\pi$
$373$	−24.0000	−1.24267	−0.621336	+	0.783544i	$0.713410\pi$
$374$	0	0
$375$	0	0
$376$	−6.00000	−0.309426
$377$	0	0
$378$	24.0000	1.23443
$379$	−15.5563	−0.799076	−0.399538	−	0.916717i	$-0.630829\pi$
$379$	−15.5563	−0.799076	−0.399538	+	0.916717i	$0.630829\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	4.24264	0.216506
$385$	0	0
$386$	−4.24264	−0.215945
$387$	−12.0000	−0.609994
$388$	−4.24264	−0.215387
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	−33.0000	−1.66675
$393$	−6.00000	−0.302660
$394$	18.3848	0.926212
$395$	0	0
$396$	4.24264	0.213201
$397$	38.1838	1.91639	0.958194	−	0.286119i	$-0.0923652\pi$
$397$	38.1838	1.91639	0.958194	+	0.286119i	$0.0923652\pi$
$398$	1.41421	0.0708881
$399$	36.0000	1.80225
$400$	0	0
$401$	12.7279	0.635602	0.317801	−	0.948157i	$-0.397056\pi$
$401$	12.7279	0.635602	0.317801	+	0.948157i	$0.397056\pi$
$402$	8.48528	0.423207
$403$	0	0
$404$	6.00000	0.298511
$405$	0	0
$406$	18.0000	0.893325
$407$	−18.0000	−0.892227
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	5.65685	0.279032
$412$	6.00000	0.295599
$413$	25.4558	1.25260
$414$	1.41421	0.0695048
$415$	0	0
$416$	0	0
$417$	−14.0000	−0.685583
$418$	−25.4558	−1.24509
$419$	−21.2132	−1.03633	−0.518166	−	0.855280i	$-0.673385\pi$
$419$	−21.2132	−1.03633	−0.518166	+	0.855280i	$0.673385\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	24.0416	1.17033
$423$	−2.00000	−0.0972433
$424$	−6.00000	−0.291386
$425$	0	0
$426$	6.00000	0.290701
$427$	6.00000	0.290360
$428$	12.7279	0.615227
$429$	0	0
$430$	0	0
$431$	−12.7279	−0.613082	−0.306541	−	0.951857i	$-0.599172\pi$
$431$	−12.7279	−0.613082	−0.306541	+	0.951857i	$0.599172\pi$
$432$	−5.65685	−0.272166
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	6.00000	0.288009
$435$	0	0
$436$	−9.89949	−0.474100
$437$	8.48528	0.405906
$438$	6.00000	0.286691
$439$	26.8701	1.28244	0.641219	−	0.767358i	$-0.278429\pi$
$439$	26.8701	1.28244	0.641219	+	0.767358i	$0.278429\pi$
$440$	0	0
$441$	−11.0000	−0.523810
$442$	0	0
$443$	22.0000	1.04525	0.522626	−	0.852562i	$-0.324953\pi$
$443$	22.0000	1.04525	0.522626	+	0.852562i	$0.324953\pi$
$444$	−6.00000	−0.284747
$445$	0	0
$446$	0	0
$447$	25.4558	1.20402
$448$	29.6985	1.40312
$449$	21.2132	1.00111	0.500556	−	0.865704i	$-0.333129\pi$
$449$	21.2132	1.00111	0.500556	+	0.865704i	$0.333129\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	−12.7279	−0.598671
$453$	−14.1421	−0.664455
$454$	−21.2132	−0.995585
$455$	0	0
$456$	−25.4558	−1.19208
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	24.0000	1.12145
$459$	0	0
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	−25.4558	−1.18431
$463$	−18.0000	−0.836531	−0.418265	−	0.908325i	$-0.637362\pi$
$463$	−18.0000	−0.836531	−0.418265	+	0.908325i	$0.637362\pi$
$464$	−4.24264	−0.196960
$465$	0	0
$466$	7.07107	0.327561
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	0	0
$469$	−25.4558	−1.17544
$470$	0	0
$471$	−16.9706	−0.781962
$472$	−18.0000	−0.828517
$473$	50.9117	2.34092
$474$	14.0000	0.643041
$475$	0	0
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	24.0000	1.09773
$479$	−12.7279	−0.581554	−0.290777	−	0.956791i	$-0.593914\pi$
$479$	−12.7279	−0.581554	−0.290777	+	0.956791i	$0.593914\pi$
$480$	0	0
$481$	0	0
$482$	26.8701	1.22390
$483$	8.48528	0.386094
$484$	−7.00000	−0.318182
$485$	0	0
$486$	−9.89949	−0.449050
$487$	−4.24264	−0.192252	−0.0961262	−	0.995369i	$-0.530645\pi$
$487$	−4.24264	−0.192252	−0.0961262	+	0.995369i	$0.530645\pi$
$488$	−4.24264	−0.192055
$489$	−18.0000	−0.813988
$490$	0	0
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	−6.00000	−0.270501
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−1.41421	−0.0635001
$497$	−18.0000	−0.807410
$498$	5.65685	0.253490
$499$	−32.5269	−1.45610	−0.728052	−	0.685522i	$-0.759574\pi$
$499$	−32.5269	−1.45610	−0.728052	+	0.685522i	$0.759574\pi$
$500$	0	0
$501$	6.00000	0.268060
$502$	12.0000	0.535586
$503$	−4.24264	−0.189170	−0.0945850	−	0.995517i	$-0.530152\pi$
$503$	−4.24264	−0.189170	−0.0945850	+	0.995517i	$0.530152\pi$
$504$	12.7279	0.566947
$505$	0	0
$506$	−6.00000	−0.266733
$507$	18.3848	0.816497
$508$	0	0
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−18.0000	−0.796273
$512$	−11.0000	−0.486136
$513$	−33.9411	−1.49854
$514$	14.0000	0.617514
$515$	0	0
$516$	16.9706	0.747087
$517$	8.48528	0.373182
$518$	−18.0000	−0.790875
$519$	−30.0000	−1.31685
$520$	0	0
$521$	−12.7279	−0.557620	−0.278810	−	0.960346i	$-0.589940\pi$
$521$	−12.7279	−0.557620	−0.278810	+	0.960346i	$0.589940\pi$
$522$	−4.24264	−0.185695
$523$	−6.00000	−0.262362	−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000	−0.262362	−0.131181	+	0.991358i	$0.541877\pi$
$524$	−4.24264	−0.185341
$525$	0	0
$526$	−16.0000	−0.697633
$527$	0	0
$528$	6.00000	0.261116
$529$	−21.0000	−0.913043
$530$	0	0
$531$	−6.00000	−0.260378
$532$	25.4558	1.10365
$533$	0	0
$534$	−8.48528	−0.367194
$535$	0	0
$536$	18.0000	0.777482
$537$	−8.48528	−0.366167
$538$	−4.24264	−0.182913
$539$	46.6690	2.01018
$540$	0	0
$541$	−1.41421	−0.0608018	−0.0304009	−	0.999538i	$-0.509678\pi$
$541$	−1.41421	−0.0608018	−0.0304009	+	0.999538i	$0.509678\pi$
$542$	24.0000	1.03089
$543$	−22.0000	−0.944110
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12.7279	0.544207	0.272103	−	0.962268i	$-0.412281\pi$
$547$	12.7279	0.544207	0.272103	+	0.962268i	$0.412281\pi$
$548$	4.00000	0.170872
$549$	−1.41421	−0.0603572
$550$	0	0
$551$	−25.4558	−1.08446
$552$	−6.00000	−0.255377
$553$	−42.0000	−1.78602
$554$	−29.6985	−1.26177
$555$	0	0
$556$	−9.89949	−0.419832
$557$	−32.0000	−1.35588	−0.677942	−	0.735116i	$-0.737128\pi$
$557$	−32.0000	−1.35588	−0.677942	+	0.735116i	$0.737128\pi$
$558$	−1.41421	−0.0598684
$559$	0	0
$560$	0	0
$561$	0	0
$562$	24.0000	1.01238
$563$	−28.0000	−1.18006	−0.590030	−	0.807382i	$-0.700884\pi$
$563$	−28.0000	−1.18006	−0.590030	+	0.807382i	$0.700884\pi$
$564$	2.82843	0.119098
$565$	0	0
$566$	−12.7279	−0.534994
$567$	−21.2132	−0.890871
$568$	12.7279	0.534052
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−32.5269	−1.36121	−0.680604	−	0.732651i	$-0.738283\pi$
$571$	−32.5269	−1.36121	−0.680604	+	0.732651i	$0.738283\pi$
$572$	0	0
$573$	0	0
$574$	−18.0000	−0.751305
$575$	0	0
$576$	−7.00000	−0.291667
$577$	−36.0000	−1.49870	−0.749350	−	0.662174i	$-0.769634\pi$
$577$	−36.0000	−1.49870	−0.749350	+	0.662174i	$0.769634\pi$
$578$	0	0
$579$	6.00000	0.249351
$580$	0	0
$581$	−16.9706	−0.704058
$582$	−6.00000	−0.248708
$583$	8.48528	0.351424
$584$	12.7279	0.526685
$585$	0	0
$586$	4.00000	0.165238
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	15.5563	0.641533
$589$	−8.48528	−0.349630
$590$	0	0
$591$	−26.0000	−1.06950
$592$	4.24264	0.174371
$593$	−2.00000	−0.0821302	−0.0410651	−	0.999156i	$-0.513075\pi$
$593$	−2.00000	−0.0821302	−0.0410651	+	0.999156i	$0.513075\pi$
$594$	24.0000	0.984732
$595$	0	0
$596$	18.0000	0.737309
$597$	−2.00000	−0.0818546
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−41.0122	−1.67292	−0.836461	−	0.548026i	$-0.815379\pi$
$601$	−41.0122	−1.67292	−0.836461	+	0.548026i	$0.815379\pi$
$602$	50.9117	2.07501
$603$	6.00000	0.244339
$604$	−10.0000	−0.406894
$605$	0	0
$606$	8.48528	0.344691
$607$	−46.6690	−1.89424	−0.947119	−	0.320882i	$-0.896021\pi$
$607$	−46.6690	−1.89424	−0.947119	+	0.320882i	$0.896021\pi$
$608$	−30.0000	−1.21666
$609$	−25.4558	−1.03152
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−36.0000	−1.45403	−0.727013	−	0.686624i	$-0.759092\pi$
$613$	−36.0000	−1.45403	−0.727013	+	0.686624i	$0.759092\pi$
$614$	18.0000	0.726421
$615$	0	0
$616$	−54.0000	−2.17572
$617$	21.2132	0.854011	0.427006	−	0.904249i	$-0.359568\pi$
$617$	21.2132	0.854011	0.427006	+	0.904249i	$0.359568\pi$
$618$	8.48528	0.341328
$619$	−18.3848	−0.738947	−0.369473	−	0.929241i	$-0.620462\pi$
$619$	−18.3848	−0.738947	−0.369473	+	0.929241i	$0.620462\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	4.24264	0.170114
$623$	25.4558	1.01987
$624$	0	0
$625$	0	0
$626$	4.24264	0.169570
$627$	36.0000	1.43770
$628$	−12.0000	−0.478852
$629$	0	0
$630$	0	0
$631$	34.0000	1.35352	0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000	1.35352	0.676759	+	0.736204i	$0.263384\pi$
$632$	29.6985	1.18134
$633$	−34.0000	−1.35138
$634$	21.2132	0.842484
$635$	0	0
$636$	2.82843	0.112154
$637$	0	0
$638$	18.0000	0.712627
$639$	4.24264	0.167836
$640$	0	0
$641$	12.7279	0.502723	0.251361	−	0.967893i	$-0.419122\pi$
$641$	12.7279	0.502723	0.251361	+	0.967893i	$0.419122\pi$
$642$	18.0000	0.710403
$643$	−12.7279	−0.501940	−0.250970	−	0.967995i	$-0.580750\pi$
$643$	−12.7279	−0.501940	−0.250970	+	0.967995i	$0.580750\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	0	0
$647$	46.0000	1.80845	0.904223	−	0.427060i	$-0.140451\pi$
$647$	46.0000	1.80845	0.904223	+	0.427060i	$0.140451\pi$
$648$	15.0000	0.589256
$649$	25.4558	0.999229
$650$	0	0
$651$	−8.48528	−0.332564
$652$	−12.7279	−0.498464
$653$	−29.6985	−1.16219	−0.581096	−	0.813835i	$-0.697376\pi$
$653$	−29.6985	−1.16219	−0.581096	+	0.813835i	$0.697376\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	4.24264	0.165647
$657$	4.24264	0.165521
$658$	8.48528	0.330791
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−36.0000	−1.40024	−0.700119	−	0.714026i	$-0.746870\pi$
$661$	−36.0000	−1.40024	−0.700119	+	0.714026i	$0.746870\pi$
$662$	18.0000	0.699590
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	4.24264	0.164399
$667$	−6.00000	−0.232321
$668$	4.24264	0.164153
$669$	0	0
$670$	0	0
$671$	6.00000	0.231627
$672$	−30.0000	−1.15728
$673$	21.2132	0.817709	0.408854	−	0.912600i	$-0.365928\pi$
$673$	21.2132	0.817709	0.408854	+	0.912600i	$0.365928\pi$
$674$	−21.2132	−0.817102
$675$	0	0
$676$	13.0000	0.500000
$677$	−15.5563	−0.597879	−0.298940	−	0.954272i	$-0.596633\pi$
$677$	−15.5563	−0.597879	−0.298940	+	0.954272i	$0.596633\pi$
$678$	−18.0000	−0.691286
$679$	18.0000	0.690777
$680$	0	0
$681$	30.0000	1.14960
$682$	6.00000	0.229752
$683$	21.2132	0.811701	0.405850	−	0.913940i	$-0.366975\pi$
$683$	21.2132	0.811701	0.405850	+	0.913940i	$0.366975\pi$
$684$	−6.00000	−0.229416
$685$	0	0
$686$	16.9706	0.647939
$687$	−33.9411	−1.29493
$688$	−12.0000	−0.457496
$689$	0	0
$690$	0	0
$691$	26.8701	1.02219	0.511093	−	0.859526i	$-0.329241\pi$
$691$	26.8701	1.02219	0.511093	+	0.859526i	$0.329241\pi$
$692$	−21.2132	−0.806405
$693$	−18.0000	−0.683763
$694$	−12.7279	−0.483145
$695$	0	0
$696$	18.0000	0.682288
$697$	0	0
$698$	28.0000	1.05982
$699$	−10.0000	−0.378235
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	25.4558	0.960085
$704$	29.6985	1.11930
$705$	0	0
$706$	−16.0000	−0.602168
$707$	−25.4558	−0.957366
$708$	8.48528	0.318896
$709$	−7.07107	−0.265560	−0.132780	−	0.991146i	$-0.542390\pi$
$709$	−7.07107	−0.265560	−0.132780	+	0.991146i	$0.542390\pi$
$710$	0	0
$711$	9.89949	0.371260
$712$	−18.0000	−0.674579
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	0	0
$716$	−6.00000	−0.224231
$717$	−33.9411	−1.26755
$718$	30.0000	1.11959
$719$	29.6985	1.10757	0.553783	−	0.832661i	$-0.313184\pi$
$719$	29.6985	1.10757	0.553783	+	0.832661i	$0.313184\pi$
$720$	0	0
$721$	−25.4558	−0.948025
$722$	17.0000	0.632674
$723$	−38.0000	−1.41324
$724$	−15.5563	−0.578147
$725$	0	0
$726$	−9.89949	−0.367405
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	0	0
$732$	2.00000	0.0739221
$733$	42.0000	1.55131	0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000	1.55131	0.775653	+	0.631160i	$0.217421\pi$
$734$	4.24264	0.156599
$735$	0	0
$736$	−7.07107	−0.260643
$737$	−25.4558	−0.937678
$738$	4.24264	0.156174
$739$	−34.0000	−1.25071	−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000	−1.25071	−0.625355	+	0.780340i	$0.715046\pi$
$740$	0	0
$741$	0	0
$742$	8.48528	0.311504
$743$	12.7279	0.466942	0.233471	−	0.972364i	$-0.424992\pi$
$743$	12.7279	0.466942	0.233471	+	0.972364i	$0.424992\pi$
$744$	6.00000	0.219971
$745$	0	0
$746$	−24.0000	−0.878702
$747$	4.00000	0.146352
$748$	0	0
$749$	−54.0000	−1.97312
$750$	0	0
$751$	−26.8701	−0.980502	−0.490251	−	0.871581i	$-0.663095\pi$
$751$	−26.8701	−0.980502	−0.490251	+	0.871581i	$0.663095\pi$
$752$	−2.00000	−0.0729325
$753$	−16.9706	−0.618442
$754$	0	0
$755$	0	0
$756$	−24.0000	−0.872872
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	−15.5563	−0.565032
$759$	8.48528	0.307996
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	42.0000	1.52050
$764$	0	0
$765$	0	0
$766$	16.0000	0.578103
$767$	0	0
$768$	24.0416	0.867528
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	−19.7990	−0.713043
$772$	4.24264	0.152696
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	−12.7279	−0.456906
$777$	25.4558	0.913223
$778$	0	0
$779$	25.4558	0.912050
$780$	0	0
$781$	−18.0000	−0.644091
$782$	0	0
$783$	24.0000	0.857690
$784$	−11.0000	−0.392857
$785$	0	0
$786$	−6.00000	−0.214013
$787$	38.1838	1.36110	0.680552	−	0.732700i	$-0.261740\pi$
$787$	38.1838	1.36110	0.680552	+	0.732700i	$0.261740\pi$
$788$	−18.3848	−0.654931
$789$	22.6274	0.805557
$790$	0	0
$791$	54.0000	1.92002
$792$	12.7279	0.452267
$793$	0	0
$794$	38.1838	1.35509
$795$	0	0
$796$	−1.41421	−0.0501255
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	36.0000	1.27439
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	12.7279	0.449439
$803$	−18.0000	−0.635206
$804$	−8.48528	−0.299253
$805$	0	0
$806$	0	0
$807$	6.00000	0.211210
$808$	18.0000	0.633238
$809$	4.24264	0.149163	0.0745817	−	0.997215i	$-0.476238\pi$
$809$	4.24264	0.149163	0.0745817	+	0.997215i	$0.476238\pi$
$810$	0	0
$811$	−1.41421	−0.0496598	−0.0248299	−	0.999692i	$-0.507904\pi$
$811$	−1.41421	−0.0496598	−0.0248299	+	0.999692i	$0.507904\pi$
$812$	−18.0000	−0.631676
$813$	−33.9411	−1.19037
$814$	−18.0000	−0.630900
$815$	0	0
$816$	0	0
$817$	−72.0000	−2.51896
$818$	30.0000	1.04893
$819$	0	0
$820$	0	0
$821$	−21.2132	−0.740346	−0.370173	−	0.928963i	$-0.620702\pi$
$821$	−21.2132	−0.740346	−0.370173	+	0.928963i	$0.620702\pi$
$822$	5.65685	0.197305
$823$	−55.1543	−1.92256	−0.961280	−	0.275575i	$-0.911132\pi$
$823$	−55.1543	−1.92256	−0.961280	+	0.275575i	$0.911132\pi$
$824$	18.0000	0.627060
$825$	0	0
$826$	25.4558	0.885722
$827$	18.3848	0.639301	0.319651	−	0.947535i	$-0.396434\pi$
$827$	18.3848	0.639301	0.319651	+	0.947535i	$0.396434\pi$
$828$	−1.41421	−0.0491473
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	42.0000	1.45696
$832$	0	0
$833$	0	0
$834$	−14.0000	−0.484780
$835$	0	0
$836$	25.4558	0.880409
$837$	8.00000	0.276520
$838$	−21.2132	−0.732798
$839$	−4.24264	−0.146472	−0.0732361	−	0.997315i	$-0.523333\pi$
$839$	−4.24264	−0.146472	−0.0732361	+	0.997315i	$0.523333\pi$
$840$	0	0
$841$	−11.0000	−0.379310
$842$	6.00000	0.206774
$843$	−33.9411	−1.16899
$844$	−24.0416	−0.827547
$845$	0	0
$846$	−2.00000	−0.0687614
$847$	29.6985	1.02045
$848$	−2.00000	−0.0686803
$849$	18.0000	0.617758
$850$	0	0
$851$	6.00000	0.205677
$852$	−6.00000	−0.205557
$853$	4.24264	0.145265	0.0726326	−	0.997359i	$-0.476860\pi$
$853$	4.24264	0.145265	0.0726326	+	0.997359i	$0.476860\pi$
$854$	6.00000	0.205316
$855$	0	0
$856$	38.1838	1.30509
$857$	−15.5563	−0.531395	−0.265697	−	0.964056i	$-0.585602\pi$
$857$	−15.5563	−0.531395	−0.265697	+	0.964056i	$0.585602\pi$
$858$	0	0
$859$	30.0000	1.02359	0.511793	−	0.859109i	$-0.328981\pi$
$859$	30.0000	1.02359	0.511793	+	0.859109i	$0.328981\pi$
$860$	0	0
$861$	25.4558	0.867533
$862$	−12.7279	−0.433515
$863$	38.0000	1.29354	0.646768	−	0.762687i	$-0.276120\pi$
$863$	38.0000	1.29354	0.646768	+	0.762687i	$0.276120\pi$
$864$	28.2843	0.962250
$865$	0	0
$866$	30.0000	1.01944
$867$	0	0
$868$	−6.00000	−0.203653
$869$	−42.0000	−1.42475
$870$	0	0
$871$	0	0
$872$	−29.6985	−1.00572
$873$	−4.24264	−0.143592
$874$	8.48528	0.287019
$875$	0	0
$876$	−6.00000	−0.202721
$877$	−38.1838	−1.28937	−0.644687	−	0.764447i	$-0.723012\pi$
$877$	−38.1838	−1.28937	−0.644687	+	0.764447i	$0.723012\pi$
$878$	26.8701	0.906821
$879$	−5.65685	−0.190801
$880$	0	0
$881$	4.24264	0.142938	0.0714691	−	0.997443i	$-0.477231\pi$
$881$	4.24264	0.142938	0.0714691	+	0.997443i	$0.477231\pi$
$882$	−11.0000	−0.370389
$883$	6.00000	0.201916	0.100958	−	0.994891i	$-0.467809\pi$
$883$	6.00000	0.201916	0.100958	+	0.994891i	$0.467809\pi$
$884$	0	0
$885$	0	0
$886$	22.0000	0.739104
$887$	24.0416	0.807239	0.403619	−	0.914927i	$-0.367752\pi$
$887$	24.0416	0.807239	0.403619	+	0.914927i	$0.367752\pi$
$888$	−18.0000	−0.604040
$889$	0	0
$890$	0	0
$891$	−21.2132	−0.710669
$892$	0	0
$893$	−12.0000	−0.401565
$894$	25.4558	0.851371
$895$	0	0
$896$	−12.7279	−0.425210
$897$	0	0
$898$	21.2132	0.707894
$899$	6.00000	0.200111
$900$	0	0
$901$	0	0
$902$	−18.0000	−0.599334
$903$	−72.0000	−2.39601
$904$	−38.1838	−1.26997
$905$	0	0
$906$	−14.1421	−0.469841
$907$	−21.2132	−0.704373	−0.352186	−	0.935930i	$-0.614562\pi$
$907$	−21.2132	−0.704373	−0.352186	+	0.935930i	$0.614562\pi$
$908$	21.2132	0.703985
$909$	6.00000	0.199007
$910$	0	0
$911$	29.6985	0.983955	0.491977	−	0.870608i	$-0.336274\pi$
$911$	29.6985	0.983955	0.491977	+	0.870608i	$0.336274\pi$
$912$	−8.48528	−0.280976
$913$	−16.9706	−0.561644
$914$	6.00000	0.198462
$915$	0	0
$916$	−24.0000	−0.792982
$917$	18.0000	0.594412
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	−25.4558	−0.838799
$922$	12.0000	0.395199
$923$	0	0
$924$	25.4558	0.837436
$925$	0	0
$926$	−18.0000	−0.591517
$927$	6.00000	0.197066
$928$	21.2132	0.696358
$929$	29.6985	0.974376	0.487188	−	0.873297i	$-0.338023\pi$
$929$	29.6985	0.974376	0.487188	+	0.873297i	$0.338023\pi$
$930$	0	0
$931$	−66.0000	−2.16306
$932$	−7.07107	−0.231621
$933$	−6.00000	−0.196431
$934$	−28.0000	−0.916188
$935$	0	0
$936$	0	0
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	−25.4558	−0.831163
$939$	−6.00000	−0.195803
$940$	0	0
$941$	21.2132	0.691531	0.345765	−	0.938321i	$-0.387619\pi$
$941$	21.2132	0.691531	0.345765	+	0.938321i	$0.387619\pi$
$942$	−16.9706	−0.552931
$943$	6.00000	0.195387
$944$	−6.00000	−0.195283
$945$	0	0
$946$	50.9117	1.65528
$947$	15.5563	0.505513	0.252757	−	0.967530i	$-0.418663\pi$
$947$	15.5563	0.505513	0.252757	+	0.967530i	$0.418663\pi$
$948$	−14.0000	−0.454699
$949$	0	0
$950$	0	0
$951$	−30.0000	−0.972817
$952$	0	0
$953$	4.00000	0.129573	0.0647864	−	0.997899i	$-0.479363\pi$
$953$	4.00000	0.129573	0.0647864	+	0.997899i	$0.479363\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	−24.0000	−0.776215
$957$	−25.4558	−0.822871
$958$	−12.7279	−0.411220
$959$	−16.9706	−0.548008
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	12.7279	0.410152
$964$	−26.8701	−0.865426
$965$	0	0
$966$	8.48528	0.273009
$967$	−24.0000	−0.771788	−0.385894	−	0.922543i	$-0.626107\pi$
$967$	−24.0000	−0.771788	−0.385894	+	0.922543i	$0.626107\pi$
$968$	−21.0000	−0.674966
$969$	0	0
$970$	0	0
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	9.89949	0.317526
$973$	42.0000	1.34646
$974$	−4.24264	−0.135943
$975$	0	0
$976$	−1.41421	−0.0452679
$977$	46.0000	1.47167	0.735835	−	0.677161i	$-0.236790\pi$
$977$	46.0000	1.47167	0.735835	+	0.677161i	$0.236790\pi$
$978$	−18.0000	−0.575577
$979$	25.4558	0.813572
$980$	0	0
$981$	−9.89949	−0.316067
$982$	6.00000	0.191468
$983$	−29.6985	−0.947235	−0.473617	−	0.880731i	$-0.657052\pi$
$983$	−29.6985	−0.947235	−0.473617	+	0.880731i	$0.657052\pi$
$984$	−18.0000	−0.573819
$985$	0	0
$986$	0	0
$987$	−12.0000	−0.381964
$988$	0	0
$989$	−16.9706	−0.539633
$990$	0	0
$991$	−18.3848	−0.584012	−0.292006	−	0.956417i	$-0.594323\pi$
$991$	−18.3848	−0.584012	−0.292006	+	0.956417i	$0.594323\pi$
$992$	7.07107	0.224507
$993$	−25.4558	−0.807817
$994$	−18.0000	−0.570925
$995$	0	0
$996$	−5.65685	−0.179244
$997$	4.24264	0.134366	0.0671829	−	0.997741i	$-0.478599\pi$
$997$	4.24264	0.134366	0.0671829	+	0.997741i	$0.478599\pi$
$998$	−32.5269	−1.02962
$999$	−24.0000	−0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.p.1.1		2
5.2	odd	4		1445.2.b.a.579.4		4
5.3	odd	4		1445.2.b.a.579.1		4
5.4	even	2		7225.2.a.i.1.2		2
17.8	even	8		425.2.e.b.251.1		2
17.15	even	8		425.2.e.b.276.1		2
17.16	even	2	inner	7225.2.a.p.1.2		2
85.8	odd	8		85.2.j.a.64.1	yes	2
85.32	odd	8		85.2.j.a.4.1	✓	2
85.33	odd	4		1445.2.b.a.579.2		4
85.42	odd	8		85.2.j.b.64.1	yes	2
85.49	even	8		425.2.e.a.276.1		2
85.59	even	8		425.2.e.a.251.1		2
85.67	odd	4		1445.2.b.a.579.3		4
85.83	odd	8		85.2.j.b.4.1	yes	2
85.84	even	2		7225.2.a.i.1.1		2
255.8	even	8		765.2.t.b.64.1		2
255.32	even	8		765.2.t.b.514.1		2
255.83	even	8		765.2.t.a.514.1		2
255.212	even	8		765.2.t.a.64.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.j.a.4.1	✓	2	85.32	odd	8
85.2.j.a.64.1	yes	2	85.8	odd	8
85.2.j.b.4.1	yes	2	85.83	odd	8
85.2.j.b.64.1	yes	2	85.42	odd	8
425.2.e.a.251.1		2	85.59	even	8
425.2.e.a.276.1		2	85.49	even	8
425.2.e.b.251.1		2	17.8	even	8
425.2.e.b.276.1		2	17.15	even	8
765.2.t.a.64.1		2	255.212	even	8
765.2.t.a.514.1		2	255.83	even	8
765.2.t.b.64.1		2	255.8	even	8
765.2.t.b.514.1		2	255.32	even	8
1445.2.b.a.579.1		4	5.3	odd	4
1445.2.b.a.579.2		4	85.33	odd	4
1445.2.b.a.579.3		4	85.67	odd	4
1445.2.b.a.579.4		4	5.2	odd	4
7225.2.a.i.1.1		2	85.84	even	2
7225.2.a.i.1.2		2	5.4	even	2
7225.2.a.p.1.1		2	1.1	even	1	trivial
7225.2.a.p.1.2		2	17.16	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.41421 q^{3} -1.00000 q^{4} -1.41421 q^{6} +4.24264 q^{7} -3.00000 q^{8} -1.00000 q^{9} +4.24264 q^{11} +1.41421 q^{12} +4.24264 q^{14} -1.00000 q^{16} -1.00000 q^{18} -6.00000 q^{19} -6.00000 q^{21} +4.24264 q^{22} -1.41421 q^{23} +4.24264 q^{24} +5.65685 q^{27} -4.24264 q^{28} +4.24264 q^{29} +1.41421 q^{31} +5.00000 q^{32} -6.00000 q^{33} +1.00000 q^{36} -4.24264 q^{37} -6.00000 q^{38} -4.24264 q^{41} -6.00000 q^{42} +12.0000 q^{43} -4.24264 q^{44} -1.41421 q^{46} +2.00000 q^{47} +1.41421 q^{48} +11.0000 q^{49} +2.00000 q^{53} +5.65685 q^{54} -12.7279 q^{56} +8.48528 q^{57} +4.24264 q^{58} +6.00000 q^{59} +1.41421 q^{61} +1.41421 q^{62} -4.24264 q^{63} +7.00000 q^{64} -6.00000 q^{66} -6.00000 q^{67} +2.00000 q^{69} -4.24264 q^{71} +3.00000 q^{72} -4.24264 q^{73} -4.24264 q^{74} +6.00000 q^{76} +18.0000 q^{77} -9.89949 q^{79} -5.00000 q^{81} -4.24264 q^{82} -4.00000 q^{83} +6.00000 q^{84} +12.0000 q^{86} -6.00000 q^{87} -12.7279 q^{88} +6.00000 q^{89} +1.41421 q^{92} -2.00000 q^{93} +2.00000 q^{94} -7.07107 q^{96} +4.24264 q^{97} +11.0000 q^{98} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} - 2 q^{9} - 2 q^{16} - 2 q^{18} - 12 q^{19} - 12 q^{21} + 10 q^{32} - 12 q^{33} + 2 q^{36} - 12 q^{38} - 12 q^{42} + 24 q^{43} + 4 q^{47} + 22 q^{49} + 4 q^{53} + 12 q^{59}+ \cdots + 22 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 - 2 * q^9 - 2 * q^16 - 2 * q^18 - 12 * q^19 - 12 * q^21 + 10 * q^32 - 12 * q^33 + 2 * q^36 - 12 * q^38 - 12 * q^42 + 24 * q^43 + 4 * q^47 + 22 * q^49 + 4 * q^53 + 12 * q^59 + 14 * q^64 - 12 * q^66 - 12 * q^67 + 4 * q^69 + 6 * q^72 + 12 * q^76 + 36 * q^77 - 10 * q^81 - 8 * q^83 + 12 * q^84 + 24 * q^86 - 12 * q^87 + 12 * q^89 - 4 * q^93 + 4 * q^94 + 22 * q^98

Introduction

L-functions

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