LMFDB - Embedded newform 4225.2.a.ca.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, 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("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 26x^{16} + 281x^{14} - 1632x^{12} + 5482x^{10} - 10620x^{8} + 11052x^{6} - 5165x^{4} + 760x^{2} - 29 $ x^18 - 26x^16 + 281x^14 - 1632x^12 + 5482x^10 - 10620x^8 + 11052x^6 - 5165x^4 + 760x^2 - 29
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 845)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$1.57695$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.57695 q^{2} -2.97563 q^{3} +0.486782 q^{4} -4.69244 q^{6} +2.50257 q^{7} -2.38627 q^{8} +5.85440 q^{9} +O(q^{10})$	\(q+1.57695 q^{2} -2.97563 q^{3} +0.486782 q^{4} -4.69244 q^{6} +2.50257 q^{7} -2.38627 q^{8} +5.85440 q^{9} +1.10010 q^{11} -1.44849 q^{12} +3.94644 q^{14} -4.73661 q^{16} -0.402080 q^{17} +9.23212 q^{18} -7.81790 q^{19} -7.44674 q^{21} +1.73480 q^{22} +2.14151 q^{23} +7.10068 q^{24} -8.49366 q^{27} +1.21821 q^{28} +5.91169 q^{29} -6.37466 q^{31} -2.69686 q^{32} -3.27348 q^{33} -0.634062 q^{34} +2.84982 q^{36} +5.92600 q^{37} -12.3285 q^{38} +3.16045 q^{41} -11.7432 q^{42} +10.1368 q^{43} +0.535507 q^{44} +3.37706 q^{46} -5.57769 q^{47} +14.0944 q^{48} -0.737134 q^{49} +1.19644 q^{51} +4.04902 q^{53} -13.3941 q^{54} -5.97182 q^{56} +23.2632 q^{57} +9.32246 q^{58} -11.4048 q^{59} -3.48524 q^{61} -10.0525 q^{62} +14.6511 q^{63} +5.22039 q^{64} -5.16213 q^{66} -1.62128 q^{67} -0.195725 q^{68} -6.37235 q^{69} +2.20326 q^{71} -13.9702 q^{72} -3.32283 q^{73} +9.34503 q^{74} -3.80562 q^{76} +2.75307 q^{77} -5.69016 q^{79} +7.71083 q^{81} +4.98389 q^{82} +6.03310 q^{83} -3.62494 q^{84} +15.9853 q^{86} -17.5910 q^{87} -2.62513 q^{88} -2.71010 q^{89} +1.04245 q^{92} +18.9687 q^{93} -8.79576 q^{94} +8.02488 q^{96} -4.06074 q^{97} -1.16243 q^{98} +6.44041 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9}+O(q^{10}) $ 18 * q + 16 * q^4 - 16 * q^6 + 18 * q^9	$ 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9} - 22 q^{11} + 4 q^{14} - 12 q^{16} - 28 q^{19} - 26 q^{21} - 34 q^{24} - 20 q^{29} - 32 q^{31} - 18 q^{34} + 32 q^{36} - 52 q^{41} - 50 q^{44} - 30 q^{46} + 44 q^{49} - 40 q^{51} - 90 q^{54} - 20 q^{56} - 76 q^{59} + 8 q^{61} - 68 q^{64} + 8 q^{66} + 30 q^{69} - 72 q^{71} + 30 q^{74} - 4 q^{76} + 16 q^{79} - 30 q^{81} - 78 q^{84} + 30 q^{86} - 94 q^{89} - 128 q^{94} + 18 q^{96} - 76 q^{99}+O(q^{100}) $ 18 * q + 16 * q^4 - 16 * q^6 + 18 * q^9 - 22 * q^11 + 4 * q^14 - 12 * q^16 - 28 * q^19 - 26 * q^21 - 34 * q^24 - 20 * q^29 - 32 * q^31 - 18 * q^34 + 32 * q^36 - 52 * q^41 - 50 * q^44 - 30 * q^46 + 44 * q^49 - 40 * q^51 - 90 * q^54 - 20 * q^56 - 76 * q^59 + 8 * q^61 - 68 * q^64 + 8 * q^66 + 30 * q^69 - 72 * q^71 + 30 * q^74 - 4 * q^76 + 16 * q^79 - 30 * q^81 - 78 * q^84 + 30 * q^86 - 94 * q^89 - 128 * q^94 + 18 * q^96 - 76 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	1.57695	1.11507	0.557537	−	0.830152i	$-0.311746\pi$
$2$	1.57695	1.11507	0.557537	+	0.830152i	$0.311746\pi$
$3$	−2.97563	−1.71798	−0.858992	−	0.511989i	$-0.828909\pi$
$3$	−2.97563	−1.71798	−0.858992	+	0.511989i	$0.828909\pi$
$4$	0.486782	0.243391
$5$	0	0
$5$	0	0
$6$	−4.69244	−1.91568
$7$	2.50257	0.945883	0.472942	−	0.881094i	$-0.343192\pi$
$7$	2.50257	0.945883	0.472942	+	0.881094i	$0.343192\pi$
$8$	−2.38627	−0.843675
$9$	5.85440	1.95147
$10$	0	0
$11$	1.10010	0.331691	0.165846	−	0.986152i	$-0.446965\pi$
$11$	1.10010	0.331691	0.165846	+	0.986152i	$0.446965\pi$
$12$	−1.44849	−0.418142
$13$	0	0
$13$	0	0
$14$	3.94644	1.05473
$15$	0	0
$16$	−4.73661	−1.18415
$17$	−0.402080	−0.0975187	−0.0487594	−	0.998811i	$-0.515527\pi$
$17$	−0.402080	−0.0975187	−0.0487594	+	0.998811i	$0.515527\pi$
$18$	9.23212	2.17603
$19$	−7.81790	−1.79355	−0.896775	−	0.442487i	$-0.854096\pi$
$19$	−7.81790	−1.79355	−0.896775	+	0.442487i	$0.854096\pi$
$20$	0	0
$21$	−7.44674	−1.62501
$22$	1.73480	0.369861
$23$	2.14151	0.446535	0.223268	−	0.974757i	$-0.428328\pi$
$23$	2.14151	0.446535	0.223268	+	0.974757i	$0.428328\pi$
$24$	7.10068	1.44942
$25$	0	0
$26$	0	0
$27$	−8.49366	−1.63461
$28$	1.21821	0.230220
$29$	5.91169	1.09777	0.548887	−	0.835897i	$-0.315052\pi$
$29$	5.91169	1.09777	0.548887	+	0.835897i	$0.315052\pi$
$30$	0	0
$31$	−6.37466	−1.14492	−0.572461	−	0.819932i	$-0.694011\pi$
$31$	−6.37466	−1.14492	−0.572461	+	0.819932i	$0.694011\pi$
$32$	−2.69686	−0.476742
$33$	−3.27348	−0.569841
$34$	−0.634062	−0.108741
$35$	0	0
$36$	2.84982	0.474970
$37$	5.92600	0.974228	0.487114	−	0.873338i	$-0.338050\pi$
$37$	5.92600	0.974228	0.487114	+	0.873338i	$0.338050\pi$
$38$	−12.3285	−1.99994
$39$	0	0
$40$	0	0
$41$	3.16045	0.493580	0.246790	−	0.969069i	$-0.420624\pi$
$41$	3.16045	0.493580	0.246790	+	0.969069i	$0.420624\pi$
$42$	−11.7432	−1.81201
$43$	10.1368	1.54585	0.772924	−	0.634499i	$-0.218794\pi$
$43$	10.1368	1.54585	0.772924	+	0.634499i	$0.218794\pi$
$44$	0.535507	0.0807308
$45$	0	0
$46$	3.37706	0.497920
$47$	−5.57769	−0.813590	−0.406795	−	0.913519i	$-0.633354\pi$
$47$	−5.57769	−0.813590	−0.406795	+	0.913519i	$0.633354\pi$
$48$	14.0944	2.03435
$49$	−0.737134	−0.105305
$50$	0	0
$51$	1.19644	0.167536
$52$	0	0
$53$	4.04902	0.556175	0.278088	−	0.960556i	$-0.410299\pi$
$53$	4.04902	0.556175	0.278088	+	0.960556i	$0.410299\pi$
$54$	−13.3941	−1.82271
$55$	0	0
$56$	−5.97182	−0.798018
$57$	23.2632	3.08129
$58$	9.32246	1.22410
$59$	−11.4048	−1.48477	−0.742387	−	0.669971i	$-0.766306\pi$
$59$	−11.4048	−1.48477	−0.742387	+	0.669971i	$0.766306\pi$
$60$	0	0
$61$	−3.48524	−0.446239	−0.223119	−	0.974791i	$-0.571624\pi$
$61$	−3.48524	−0.446239	−0.223119	+	0.974791i	$0.571624\pi$
$62$	−10.0525	−1.27667
$63$	14.6511	1.84586
$64$	5.22039	0.652549
$65$	0	0
$66$	−5.16213	−0.635415
$67$	−1.62128	−0.198071	−0.0990356	−	0.995084i	$-0.531576\pi$
$67$	−1.62128	−0.198071	−0.0990356	+	0.995084i	$0.531576\pi$
$68$	−0.195725	−0.0237352
$69$	−6.37235	−0.767141
$70$	0	0
$71$	2.20326	0.261479	0.130740	−	0.991417i	$-0.458265\pi$
$71$	2.20326	0.261479	0.130740	+	0.991417i	$0.458265\pi$
$72$	−13.9702	−1.64641
$73$	−3.32283	−0.388908	−0.194454	−	0.980912i	$-0.562293\pi$
$73$	−3.32283	−0.388908	−0.194454	+	0.980912i	$0.562293\pi$
$74$	9.34503	1.08634
$75$	0	0
$76$	−3.80562	−0.436534
$77$	2.75307	0.313741
$78$	0	0
$79$	−5.69016	−0.640193	−0.320097	−	0.947385i	$-0.603715\pi$
$79$	−5.69016	−0.640193	−0.320097	+	0.947385i	$0.603715\pi$
$80$	0	0
$81$	7.71083	0.856759
$82$	4.98389	0.550379
$83$	6.03310	0.662219	0.331109	−	0.943592i	$-0.392577\pi$
$83$	6.03310	0.662219	0.331109	+	0.943592i	$0.392577\pi$
$84$	−3.62494	−0.395513
$85$	0	0
$86$	15.9853	1.72373
$87$	−17.5910	−1.88596
$88$	−2.62513	−0.279840
$89$	−2.71010	−0.287270	−0.143635	−	0.989631i	$-0.545879\pi$
$89$	−2.71010	−0.287270	−0.143635	+	0.989631i	$0.545879\pi$
$90$	0	0
$91$	0	0
$92$	1.04245	0.108683
$93$	18.9687	1.96696
$94$	−8.79576	−0.907213
$95$	0	0
$96$	8.02488	0.819035
$97$	−4.06074	−0.412306	−0.206153	−	0.978520i	$-0.566094\pi$
$97$	−4.06074	−0.412306	−0.206153	+	0.978520i	$0.566094\pi$
$98$	−1.16243	−0.117423
$99$	6.44041	0.647285
$100$	0	0
$101$	6.02905	0.599913	0.299956	−	0.953953i	$-0.403028\pi$
$101$	6.02905	0.599913	0.299956	+	0.953953i	$0.403028\pi$
$102$	1.88674	0.186815
$103$	−5.20878	−0.513236	−0.256618	−	0.966513i	$-0.582608\pi$
$103$	−5.20878	−0.513236	−0.256618	+	0.966513i	$0.582608\pi$
$104$	0	0
$105$	0	0
$106$	6.38511	0.620177
$107$	−8.88002	−0.858464	−0.429232	−	0.903194i	$-0.641216\pi$
$107$	−8.88002	−0.858464	−0.429232	+	0.903194i	$0.641216\pi$
$108$	−4.13456	−0.397849
$109$	−3.08862	−0.295836	−0.147918	−	0.989000i	$-0.547257\pi$
$109$	−3.08862	−0.295836	−0.147918	+	0.989000i	$0.547257\pi$
$110$	0	0
$111$	−17.6336	−1.67371
$112$	−11.8537	−1.12007
$113$	−18.6573	−1.75513	−0.877563	−	0.479461i	$-0.840832\pi$
$113$	−18.6573	−1.75513	−0.877563	+	0.479461i	$0.840832\pi$
$114$	36.6850	3.43587
$115$	0	0
$116$	2.87771	0.267188
$117$	0	0
$118$	−17.9848	−1.65563
$119$	−1.00623	−0.0922413
$120$	0	0
$121$	−9.78979	−0.889981
$122$	−5.49606	−0.497590
$123$	−9.40436	−0.847962
$124$	−3.10307	−0.278664
$125$	0	0
$126$	23.1040	2.05827
$127$	8.89792	0.789563	0.394781	−	0.918775i	$-0.370820\pi$
$127$	8.89792	0.789563	0.394781	+	0.918775i	$0.370820\pi$
$128$	13.6260	1.20438
$129$	−30.1634	−2.65574
$130$	0	0
$131$	−9.45417	−0.826014	−0.413007	−	0.910728i	$-0.635522\pi$
$131$	−9.45417	−0.826014	−0.413007	+	0.910728i	$0.635522\pi$
$132$	−1.59347	−0.138694
$133$	−19.5649	−1.69649
$134$	−2.55669	−0.220864
$135$	0	0
$136$	0.959473	0.0822741
$137$	−8.60480	−0.735157	−0.367579	−	0.929992i	$-0.619813\pi$
$137$	−8.60480	−0.735157	−0.367579	+	0.929992i	$0.619813\pi$
$138$	−10.0489	−0.855419
$139$	−18.6163	−1.57901	−0.789507	−	0.613741i	$-0.789664\pi$
$139$	−18.6163	−1.57901	−0.789507	+	0.613741i	$0.789664\pi$
$140$	0	0
$141$	16.5972	1.39773
$142$	3.47444	0.291569
$143$	0	0
$144$	−27.7300	−2.31083
$145$	0	0
$146$	−5.23995	−0.433661
$147$	2.19344	0.180912
$148$	2.88467	0.237119
$149$	4.05456	0.332162	0.166081	−	0.986112i	$-0.446889\pi$
$149$	4.05456	0.332162	0.166081	+	0.986112i	$0.446889\pi$
$150$	0	0
$151$	−6.51555	−0.530228	−0.265114	−	0.964217i	$-0.585410\pi$
$151$	−6.51555	−0.530228	−0.265114	+	0.964217i	$0.585410\pi$
$152$	18.6557	1.51317
$153$	−2.35394	−0.190305
$154$	4.34146	0.349845
$155$	0	0
$156$	0	0
$157$	9.75491	0.778526	0.389263	−	0.921127i	$-0.372730\pi$
$157$	9.75491	0.778526	0.389263	+	0.921127i	$0.372730\pi$
$158$	−8.97312	−0.713863
$159$	−12.0484	−0.955500
$160$	0	0
$161$	5.35928	0.422370
$162$	12.1596	0.955350
$163$	−10.5746	−0.828269	−0.414135	−	0.910216i	$-0.635916\pi$
$163$	−10.5746	−0.828269	−0.414135	+	0.910216i	$0.635916\pi$
$164$	1.53845	0.120133
$165$	0	0
$166$	9.51392	0.738423
$167$	15.7085	1.21556	0.607780	−	0.794105i	$-0.292060\pi$
$167$	15.7085	1.21556	0.607780	+	0.794105i	$0.292060\pi$
$168$	17.7700	1.37098
$169$	0	0
$170$	0	0
$171$	−45.7691	−3.50005
$172$	4.93441	0.376246
$173$	−5.80716	−0.441510	−0.220755	−	0.975329i	$-0.570852\pi$
$173$	−5.80716	−0.441510	−0.220755	+	0.975329i	$0.570852\pi$
$174$	−27.7402	−2.10298
$175$	0	0
$176$	−5.21072	−0.392773
$177$	33.9364	2.55082
$178$	−4.27370	−0.320327
$179$	10.1488	0.758554	0.379277	−	0.925283i	$-0.376173\pi$
$179$	10.1488	0.758554	0.379277	+	0.925283i	$0.376173\pi$
$180$	0	0
$181$	−21.4176	−1.59196	−0.795979	−	0.605325i	$-0.793043\pi$
$181$	−21.4176	−1.59196	−0.795979	+	0.605325i	$0.793043\pi$
$182$	0	0
$183$	10.3708	0.766631
$184$	−5.11023	−0.376731
$185$	0	0
$186$	29.9127	2.19330
$187$	−0.442327	−0.0323461
$188$	−2.71512	−0.198021
$189$	−21.2560	−1.54615
$190$	0	0
$191$	−26.1789	−1.89424	−0.947119	−	0.320882i	$-0.896021\pi$
$191$	−26.1789	−1.89424	−0.947119	+	0.320882i	$0.896021\pi$
$192$	−15.5340	−1.12107
$193$	−19.2469	−1.38542	−0.692710	−	0.721216i	$-0.743583\pi$
$193$	−19.2469	−1.38542	−0.692710	+	0.721216i	$0.743583\pi$
$194$	−6.40360	−0.459752
$195$	0	0
$196$	−0.358824	−0.0256303
$197$	−5.47869	−0.390341	−0.195170	−	0.980769i	$-0.562526\pi$
$197$	−5.47869	−0.390341	−0.195170	+	0.980769i	$0.562526\pi$
$198$	10.1562	0.721771
$199$	−10.7974	−0.765407	−0.382704	−	0.923871i	$-0.625007\pi$
$199$	−10.7974	−0.765407	−0.382704	+	0.923871i	$0.625007\pi$
$200$	0	0
$201$	4.82434	0.340283
$202$	9.50753	0.668948
$203$	14.7944	1.03837
$204$	0.582407	0.0407767
$205$	0	0
$206$	−8.21400	−0.572296
$207$	12.5373	0.871400
$208$	0	0
$209$	−8.60044	−0.594905
$210$	0	0
$211$	4.85559	0.334272	0.167136	−	0.985934i	$-0.446548\pi$
$211$	4.85559	0.334272	0.167136	+	0.985934i	$0.446548\pi$
$212$	1.97099	0.135368
$213$	−6.55611	−0.449217
$214$	−14.0034	−0.957251
$215$	0	0
$216$	20.2682	1.37908
$217$	−15.9530	−1.08296
$218$	−4.87061	−0.329879
$219$	9.88752	0.668137
$220$	0	0
$221$	0	0
$222$	−27.8074	−1.86631
$223$	−24.0504	−1.61053	−0.805267	−	0.592913i	$-0.797978\pi$
$223$	−24.0504	−1.61053	−0.805267	+	0.592913i	$0.797978\pi$
$224$	−6.74909	−0.450943
$225$	0	0
$226$	−29.4216	−1.95710
$227$	−16.9275	−1.12352	−0.561759	−	0.827301i	$-0.689875\pi$
$227$	−16.9275	−1.12352	−0.561759	+	0.827301i	$0.689875\pi$
$228$	11.3241	0.749958
$229$	2.87668	0.190096	0.0950481	−	0.995473i	$-0.469700\pi$
$229$	2.87668	0.190096	0.0950481	+	0.995473i	$0.469700\pi$
$230$	0	0
$231$	−8.19213	−0.539003
$232$	−14.1069	−0.926164
$233$	14.5163	0.950995	0.475497	−	0.879717i	$-0.342268\pi$
$233$	14.5163	0.950995	0.475497	+	0.879717i	$0.342268\pi$
$234$	0	0
$235$	0	0
$236$	−5.55164	−0.361381
$237$	16.9318	1.09984
$238$	−1.58678	−0.102856
$239$	−19.9074	−1.28771	−0.643853	−	0.765149i	$-0.722665\pi$
$239$	−19.9074	−1.28771	−0.643853	+	0.765149i	$0.722665\pi$
$240$	0	0
$241$	13.3567	0.860382	0.430191	−	0.902738i	$-0.358446\pi$
$241$	13.3567	0.860382	0.430191	+	0.902738i	$0.358446\pi$
$242$	−15.4380	−0.992395
$243$	2.53637	0.162708
$244$	−1.69655	−0.108611
$245$	0	0
$246$	−14.8302	−0.945541
$247$	0	0
$248$	15.2117	0.965943
$249$	−17.9523	−1.13768
$250$	0	0
$251$	0.581179	0.0366837	0.0183419	−	0.999832i	$-0.494161\pi$
$251$	0.581179	0.0366837	0.0183419	+	0.999832i	$0.494161\pi$
$252$	7.13188	0.449266
$253$	2.35587	0.148112
$254$	14.0316	0.880421
$255$	0	0
$256$	11.0468	0.690428
$257$	−13.9879	−0.872543	−0.436271	−	0.899815i	$-0.643701\pi$
$257$	−13.9879	−0.872543	−0.436271	+	0.899815i	$0.643701\pi$
$258$	−47.5663	−2.96135
$259$	14.8302	0.921506
$260$	0	0
$261$	34.6094	2.14227
$262$	−14.9088	−0.921068
$263$	22.3185	1.37622	0.688109	−	0.725607i	$-0.258441\pi$
$263$	22.3185	1.37622	0.688109	+	0.725607i	$0.258441\pi$
$264$	7.81143	0.480760
$265$	0	0
$266$	−30.8529	−1.89171
$267$	8.06426	0.493525
$268$	−0.789211	−0.0482088
$269$	1.02418	0.0624454	0.0312227	−	0.999512i	$-0.490060\pi$
$269$	1.02418	0.0624454	0.0312227	+	0.999512i	$0.490060\pi$
$270$	0	0
$271$	−11.8065	−0.717191	−0.358596	−	0.933493i	$-0.616744\pi$
$271$	−11.8065	−0.717191	−0.358596	+	0.933493i	$0.616744\pi$
$272$	1.90450	0.115477
$273$	0	0
$274$	−13.5694	−0.819755
$275$	0	0
$276$	−3.10195	−0.186715
$277$	22.8154	1.37084	0.685422	−	0.728146i	$-0.259618\pi$
$277$	22.8154	1.37084	0.685422	+	0.728146i	$0.259618\pi$
$278$	−29.3571	−1.76072
$279$	−37.3198	−2.23428
$280$	0	0
$281$	−27.9854	−1.66947	−0.834733	−	0.550655i	$-0.814378\pi$
$281$	−27.9854	−1.66947	−0.834733	+	0.550655i	$0.814378\pi$
$282$	26.1730	1.55858
$283$	17.3539	1.03158	0.515791	−	0.856715i	$-0.327498\pi$
$283$	17.3539	1.03158	0.515791	+	0.856715i	$0.327498\pi$
$284$	1.07251	0.0636417
$285$	0	0
$286$	0	0
$287$	7.90926	0.466869
$288$	−15.7885	−0.930347
$289$	−16.8383	−0.990490
$290$	0	0
$291$	12.0833	0.708334
$292$	−1.61749	−0.0946567
$293$	19.2013	1.12175	0.560876	−	0.827900i	$-0.310464\pi$
$293$	19.2013	1.12175	0.560876	+	0.827900i	$0.310464\pi$
$294$	3.45895	0.201730
$295$	0	0
$296$	−14.1411	−0.821932
$297$	−9.34385	−0.542185
$298$	6.39385	0.370386
$299$	0	0
$300$	0	0
$301$	25.3681	1.46219
$302$	−10.2747	−0.591244
$303$	−17.9403	−1.03064
$304$	37.0303	2.12384
$305$	0	0
$306$	−3.71205	−0.212204
$307$	0.848426	0.0484222	0.0242111	−	0.999707i	$-0.492293\pi$
$307$	0.848426	0.0484222	0.0242111	+	0.999707i	$0.492293\pi$
$308$	1.34015	0.0763619
$309$	15.4994	0.881731
$310$	0	0
$311$	28.7388	1.62963	0.814815	−	0.579721i	$-0.196839\pi$
$311$	28.7388	1.62963	0.814815	+	0.579721i	$0.196839\pi$
$312$	0	0
$313$	−12.4034	−0.701082	−0.350541	−	0.936547i	$-0.614002\pi$
$313$	−12.4034	−0.701082	−0.350541	+	0.936547i	$0.614002\pi$
$314$	15.3830	0.868115
$315$	0	0
$316$	−2.76987	−0.155817
$317$	−19.2685	−1.08223	−0.541114	−	0.840949i	$-0.681997\pi$
$317$	−19.2685	−1.08223	−0.541114	+	0.840949i	$0.681997\pi$
$318$	−18.9998	−1.06545
$319$	6.50343	0.364122
$320$	0	0
$321$	26.4237	1.47483
$322$	8.45134	0.470974
$323$	3.14342	0.174905
$324$	3.75350	0.208528
$325$	0	0
$326$	−16.6757	−0.923582
$327$	9.19060	0.508242
$328$	−7.54171	−0.416421
$329$	−13.9586	−0.769561
$330$	0	0
$331$	−1.59138	−0.0874700	−0.0437350	−	0.999043i	$-0.513926\pi$
$331$	−1.59138	−0.0874700	−0.0437350	+	0.999043i	$0.513926\pi$
$332$	2.93681	0.161178
$333$	34.6932	1.90118
$334$	24.7716	1.35544
$335$	0	0
$336$	35.2723	1.92426
$337$	−21.5690	−1.17494	−0.587470	−	0.809246i	$-0.699876\pi$
$337$	−21.5690	−1.17494	−0.587470	+	0.809246i	$0.699876\pi$
$338$	0	0
$339$	55.5172	3.01528
$340$	0	0
$341$	−7.01274	−0.379761
$342$	−72.1758	−3.90282
$343$	−19.3627	−1.04549
$344$	−24.1892	−1.30419
$345$	0	0
$346$	−9.15762	−0.492317
$347$	−20.2120	−1.08504	−0.542519	−	0.840043i	$-0.682529\pi$
$347$	−20.2120	−1.08504	−0.542519	+	0.840043i	$0.682529\pi$
$348$	−8.56300	−0.459025
$349$	17.4541	0.934294	0.467147	−	0.884180i	$-0.345282\pi$
$349$	17.4541	0.934294	0.467147	+	0.884180i	$0.345282\pi$
$350$	0	0
$351$	0	0
$352$	−2.96681	−0.158131
$353$	15.3833	0.818768	0.409384	−	0.912362i	$-0.365744\pi$
$353$	15.3833	0.818768	0.409384	+	0.912362i	$0.365744\pi$
$354$	53.5162	2.84435
$355$	0	0
$356$	−1.31923	−0.0699189
$357$	2.99419	0.158469
$358$	16.0041	0.845844
$359$	−22.9477	−1.21114	−0.605568	−	0.795794i	$-0.707054\pi$
$359$	−22.9477	−1.21114	−0.605568	+	0.795794i	$0.707054\pi$
$360$	0	0
$361$	42.1196	2.21682
$362$	−33.7745	−1.77515
$363$	29.1308	1.52897
$364$	0	0
$365$	0	0
$366$	16.3543	0.854851
$367$	15.8623	0.828006	0.414003	−	0.910275i	$-0.364130\pi$
$367$	15.8623	0.828006	0.414003	+	0.910275i	$0.364130\pi$
$368$	−10.1435	−0.528766
$369$	18.5026	0.963206
$370$	0	0
$371$	10.1330	0.526077
$372$	9.23361	0.478740
$373$	−26.1131	−1.35208	−0.676041	−	0.736864i	$-0.736306\pi$
$373$	−26.1131	−1.35208	−0.676041	+	0.736864i	$0.736306\pi$
$374$	−0.697529	−0.0360684
$375$	0	0
$376$	13.3099	0.686406
$377$	0	0
$378$	−33.5197	−1.72407
$379$	−14.5391	−0.746822	−0.373411	−	0.927666i	$-0.621812\pi$
$379$	−14.5391	−0.746822	−0.373411	+	0.927666i	$0.621812\pi$
$380$	0	0
$381$	−26.4770	−1.35646
$382$	−41.2829	−2.11222
$383$	−20.2240	−1.03340	−0.516700	−	0.856167i	$-0.672840\pi$
$383$	−20.2240	−1.03340	−0.516700	+	0.856167i	$0.672840\pi$
$384$	−40.5461	−2.06911
$385$	0	0
$386$	−30.3514	−1.54485
$387$	59.3449	3.01667
$388$	−1.97670	−0.100352
$389$	14.2959	0.724833	0.362417	−	0.932016i	$-0.381952\pi$
$389$	14.2959	0.724833	0.362417	+	0.932016i	$0.381952\pi$
$390$	0	0
$391$	−0.861058	−0.0435456
$392$	1.75900	0.0888431
$393$	28.1321	1.41908
$394$	−8.63964	−0.435259
$395$	0	0
$396$	3.13508	0.157543
$397$	39.4218	1.97853	0.989263	−	0.146148i	$-0.0466875\pi$
$397$	39.4218	1.97853	0.989263	+	0.146148i	$0.0466875\pi$
$398$	−17.0270	−0.853486
$399$	58.2179	2.91454
$400$	0	0
$401$	−9.82719	−0.490747	−0.245373	−	0.969429i	$-0.578911\pi$
$401$	−9.82719	−0.490747	−0.245373	+	0.969429i	$0.578911\pi$
$402$	7.60777	0.379441
$403$	0	0
$404$	2.93483	0.146013
$405$	0	0
$406$	23.3301	1.15785
$407$	6.51917	0.323143
$408$	−2.85504	−0.141346
$409$	26.4785	1.30928	0.654639	−	0.755942i	$-0.272821\pi$
$409$	26.4785	1.30928	0.654639	+	0.755942i	$0.272821\pi$
$410$	0	0
$411$	25.6047	1.26299
$412$	−2.53554	−0.124917
$413$	−28.5413	−1.40442
$414$	19.7707	0.971675
$415$	0	0
$416$	0	0
$417$	55.3953	2.71272
$418$	−13.5625	−0.663364
$419$	11.9915	0.585821	0.292911	−	0.956140i	$-0.405376\pi$
$419$	11.9915	0.585821	0.292911	+	0.956140i	$0.405376\pi$
$420$	0	0
$421$	24.0331	1.17130	0.585650	−	0.810564i	$-0.300839\pi$
$421$	24.0331	1.17130	0.585650	+	0.810564i	$0.300839\pi$
$422$	7.65703	0.372738
$423$	−32.6541	−1.58769
$424$	−9.66206	−0.469231
$425$	0	0
$426$	−10.3387	−0.500911
$427$	−8.72206	−0.422090
$428$	−4.32264	−0.208943
$429$	0	0
$430$	0	0
$431$	−2.14102	−0.103129	−0.0515646	−	0.998670i	$-0.516421\pi$
$431$	−2.14102	−0.103129	−0.0515646	+	0.998670i	$0.516421\pi$
$432$	40.2311	1.93562
$433$	40.6751	1.95472	0.977360	−	0.211582i	$-0.0678616\pi$
$433$	40.6751	1.95472	0.977360	+	0.211582i	$0.0678616\pi$
$434$	−25.1572	−1.20758
$435$	0	0
$436$	−1.50349	−0.0720039
$437$	−16.7421	−0.800883
$438$	15.5922	0.745022
$439$	−1.22863	−0.0586395	−0.0293198	−	0.999570i	$-0.509334\pi$
$439$	−1.22863	−0.0586395	−0.0293198	+	0.999570i	$0.509334\pi$
$440$	0	0
$441$	−4.31548	−0.205499
$442$	0	0
$443$	16.6168	0.789487	0.394743	−	0.918791i	$-0.370833\pi$
$443$	16.6168	0.789487	0.394743	+	0.918791i	$0.370833\pi$
$444$	−8.58373	−0.407366
$445$	0	0
$446$	−37.9264	−1.79586
$447$	−12.0649	−0.570649
$448$	13.0644	0.617235
$449$	14.4632	0.682558	0.341279	−	0.939962i	$-0.389140\pi$
$449$	14.4632	0.682558	0.341279	+	0.939962i	$0.389140\pi$
$450$	0	0
$451$	3.47680	0.163716
$452$	−9.08202	−0.427182
$453$	19.3879	0.910923
$454$	−26.6939	−1.25281
$455$	0	0
$456$	−55.5124	−2.59961
$457$	3.41353	0.159678	0.0798391	−	0.996808i	$-0.474559\pi$
$457$	3.41353	0.159678	0.0798391	+	0.996808i	$0.474559\pi$
$458$	4.53639	0.211971
$459$	3.41513	0.159405
$460$	0	0
$461$	−11.2284	−0.522959	−0.261479	−	0.965209i	$-0.584210\pi$
$461$	−11.2284	−0.522959	−0.261479	+	0.965209i	$0.584210\pi$
$462$	−12.9186	−0.601028
$463$	16.1382	0.750007	0.375003	−	0.927023i	$-0.377642\pi$
$463$	16.1382	0.750007	0.375003	+	0.927023i	$0.377642\pi$
$464$	−28.0014	−1.29993
$465$	0	0
$466$	22.8915	1.06043
$467$	−21.8329	−1.01031	−0.505153	−	0.863030i	$-0.668564\pi$
$467$	−21.8329	−1.01031	−0.505153	+	0.863030i	$0.668564\pi$
$468$	0	0
$469$	−4.05738	−0.187352
$470$	0	0
$471$	−29.0270	−1.33750
$472$	27.2149	1.25267
$473$	11.1515	0.512744
$474$	26.7007	1.22640
$475$	0	0
$476$	−0.489817	−0.0224507
$477$	23.7046	1.08536
$478$	−31.3931	−1.43589
$479$	−23.3266	−1.06582	−0.532910	−	0.846172i	$-0.678901\pi$
$479$	−23.3266	−1.06582	−0.532910	+	0.846172i	$0.678901\pi$
$480$	0	0
$481$	0	0
$482$	21.0629	0.959390
$483$	−15.9473	−0.725626
$484$	−4.76550	−0.216613
$485$	0	0
$486$	3.99974	0.181432
$487$	25.0950	1.13716	0.568582	−	0.822627i	$-0.307492\pi$
$487$	25.0950	1.13716	0.568582	+	0.822627i	$0.307492\pi$
$488$	8.31673	0.376481
$489$	31.4663	1.42295
$490$	0	0
$491$	13.0772	0.590167	0.295084	−	0.955471i	$-0.404652\pi$
$491$	13.0772	0.590167	0.295084	+	0.955471i	$0.404652\pi$
$492$	−4.57788	−0.206387
$493$	−2.37697	−0.107053
$494$	0	0
$495$	0	0
$496$	30.1943	1.35576
$497$	5.51383	0.247329
$498$	−28.3099	−1.26860
$499$	18.6179	0.833450	0.416725	−	0.909033i	$-0.363178\pi$
$499$	18.6179	0.833450	0.416725	+	0.909033i	$0.363178\pi$
$500$	0	0
$501$	−46.7428	−2.08831
$502$	0.916493	0.0409051
$503$	31.8049	1.41811	0.709055	−	0.705153i	$-0.249122\pi$
$503$	31.8049	1.41811	0.709055	+	0.705153i	$0.249122\pi$
$504$	−34.9615	−1.55731
$505$	0	0
$506$	3.71509	0.165156
$507$	0	0
$508$	4.33135	0.192173
$509$	−6.25215	−0.277122	−0.138561	−	0.990354i	$-0.544248\pi$
$509$	−6.25215	−0.277122	−0.138561	+	0.990354i	$0.544248\pi$
$510$	0	0
$511$	−8.31562	−0.367861
$512$	−9.83171	−0.434504
$513$	66.4026	2.93175
$514$	−22.0583	−0.972950
$515$	0	0
$516$	−14.6830	−0.646384
$517$	−6.13600	−0.269861
$518$	23.3866	1.02755
$519$	17.2800	0.758507
$520$	0	0
$521$	3.57664	0.156695	0.0783476	−	0.996926i	$-0.475036\pi$
$521$	3.57664	0.156695	0.0783476	+	0.996926i	$0.475036\pi$
$522$	54.5775	2.38879
$523$	4.77249	0.208686	0.104343	−	0.994541i	$-0.466726\pi$
$523$	4.77249	0.208686	0.104343	+	0.994541i	$0.466726\pi$
$524$	−4.60212	−0.201045
$525$	0	0
$526$	35.1953	1.53459
$527$	2.56312	0.111651
$528$	15.5052	0.674778
$529$	−18.4139	−0.800606
$530$	0	0
$531$	−66.7681	−2.89749
$532$	−9.52383	−0.412910
$533$	0	0
$534$	12.7170	0.550317
$535$	0	0
$536$	3.86882	0.167108
$537$	−30.1990	−1.30318
$538$	1.61508	0.0696312
$539$	−0.810918	−0.0349287
$540$	0	0
$541$	2.58531	0.111151	0.0555756	−	0.998454i	$-0.482301\pi$
$541$	2.58531	0.111151	0.0555756	+	0.998454i	$0.482301\pi$
$542$	−18.6182	−0.799722
$543$	63.7309	2.73496
$544$	1.08435	0.0464913
$545$	0	0
$546$	0	0
$547$	−23.1852	−0.991328	−0.495664	−	0.868514i	$-0.665075\pi$
$547$	−23.1852	−0.991328	−0.495664	+	0.868514i	$0.665075\pi$
$548$	−4.18866	−0.178931
$549$	−20.4040	−0.870821
$550$	0	0
$551$	−46.2170	−1.96891
$552$	15.2062	0.647218
$553$	−14.2400	−0.605548
$554$	35.9788	1.52859
$555$	0	0
$556$	−9.06209	−0.384318
$557$	26.2187	1.11092	0.555462	−	0.831542i	$-0.312542\pi$
$557$	26.2187	1.11092	0.555462	+	0.831542i	$0.312542\pi$
$558$	−58.8516	−2.49139
$559$	0	0
$560$	0	0
$561$	1.31620	0.0555701
$562$	−44.1316	−1.86158
$563$	9.80593	0.413271	0.206635	−	0.978418i	$-0.433749\pi$
$563$	9.80593	0.413271	0.206635	+	0.978418i	$0.433749\pi$
$564$	8.07921	0.340196
$565$	0	0
$566$	27.3663	1.15029
$567$	19.2969	0.810394
$568$	−5.25759	−0.220604
$569$	11.3131	0.474271	0.237135	−	0.971477i	$-0.423792\pi$
$569$	11.3131	0.474271	0.237135	+	0.971477i	$0.423792\pi$
$570$	0	0
$571$	−19.2986	−0.807622	−0.403811	−	0.914842i	$-0.632315\pi$
$571$	−19.2986	−0.807622	−0.403811	+	0.914842i	$0.632315\pi$
$572$	0	0
$573$	77.8988	3.25427
$574$	12.4725	0.520594
$575$	0	0
$576$	30.5623	1.27343
$577$	−31.8045	−1.32404	−0.662019	−	0.749487i	$-0.730300\pi$
$577$	−31.8045	−1.32404	−0.662019	+	0.749487i	$0.730300\pi$
$578$	−26.5533	−1.10447
$579$	57.2717	2.38013
$580$	0	0
$581$	15.0983	0.626382
$582$	19.0548	0.789846
$583$	4.45431	0.184479
$584$	7.92918	0.328112
$585$	0	0
$586$	30.2796	1.25084
$587$	−28.9273	−1.19396	−0.596980	−	0.802256i	$-0.703633\pi$
$587$	−28.9273	−1.19396	−0.596980	+	0.802256i	$0.703633\pi$
$588$	1.06773	0.0440324
$589$	49.8365	2.05348
$590$	0	0
$591$	16.3026	0.670599
$592$	−28.0691	−1.15363
$593$	−43.9018	−1.80283	−0.901416	−	0.432954i	$-0.857471\pi$
$593$	−43.9018	−1.80283	−0.901416	+	0.432954i	$0.857471\pi$
$594$	−14.7348	−0.604577
$595$	0	0
$596$	1.97369	0.0808454
$597$	32.1291	1.31496
$598$	0	0
$599$	−38.7588	−1.58364	−0.791821	−	0.610753i	$-0.790867\pi$
$599$	−38.7588	−1.58364	−0.791821	+	0.610753i	$0.790867\pi$
$600$	0	0
$601$	−36.4774	−1.48794	−0.743972	−	0.668211i	$-0.767060\pi$
$601$	−36.4774	−1.48794	−0.743972	+	0.668211i	$0.767060\pi$
$602$	40.0043	1.63045
$603$	−9.49164	−0.386529
$604$	−3.17165	−0.129053
$605$	0	0
$606$	−28.2909	−1.14924
$607$	13.0299	0.528866	0.264433	−	0.964404i	$-0.414815\pi$
$607$	13.0299	0.528866	0.264433	+	0.964404i	$0.414815\pi$
$608$	21.0838	0.855061
$609$	−44.0228	−1.78390
$610$	0	0
$611$	0	0
$612$	−1.14586	−0.0463185
$613$	23.3590	0.943460	0.471730	−	0.881743i	$-0.343630\pi$
$613$	23.3590	0.943460	0.471730	+	0.881743i	$0.343630\pi$
$614$	1.33793	0.0539944
$615$	0	0
$616$	−6.56958	−0.264696
$617$	−29.1925	−1.17524	−0.587622	−	0.809136i	$-0.699936\pi$
$617$	−29.1925	−1.17524	−0.587622	+	0.809136i	$0.699936\pi$
$618$	24.4419	0.983196
$619$	−19.4985	−0.783711	−0.391856	−	0.920027i	$-0.628167\pi$
$619$	−19.4985	−0.783711	−0.391856	+	0.920027i	$0.628167\pi$
$620$	0	0
$621$	−18.1893	−0.729910
$622$	45.3198	1.81716
$623$	−6.78221	−0.271724
$624$	0	0
$625$	0	0
$626$	−19.5596	−0.781759
$627$	25.5918	1.02204
$628$	4.74852	0.189486
$629$	−2.38273	−0.0950055
$630$	0	0
$631$	23.6296	0.940679	0.470340	−	0.882485i	$-0.344131\pi$
$631$	23.6296	0.940679	0.470340	+	0.882485i	$0.344131\pi$
$632$	13.5783	0.540115
$633$	−14.4484	−0.574274
$634$	−30.3856	−1.20677
$635$	0	0
$636$	−5.86495	−0.232560
$637$	0	0
$638$	10.2556	0.406023
$639$	12.8988	0.510268
$640$	0	0
$641$	33.0492	1.30537	0.652683	−	0.757631i	$-0.273644\pi$
$641$	33.0492	1.30537	0.652683	+	0.757631i	$0.273644\pi$
$642$	41.6689	1.64454
$643$	1.56780	0.0618280	0.0309140	−	0.999522i	$-0.490158\pi$
$643$	1.56780	0.0618280	0.0309140	+	0.999522i	$0.490158\pi$
$644$	2.60880	0.102801
$645$	0	0
$646$	4.95703	0.195032
$647$	2.35134	0.0924405	0.0462203	−	0.998931i	$-0.485282\pi$
$647$	2.35134	0.0924405	0.0462203	+	0.998931i	$0.485282\pi$
$648$	−18.4002	−0.722826
$649$	−12.5463	−0.492487
$650$	0	0
$651$	47.4704	1.86051
$652$	−5.14754	−0.201593
$653$	−11.3315	−0.443436	−0.221718	−	0.975111i	$-0.571166\pi$
$653$	−11.3315	−0.443436	−0.221718	+	0.975111i	$0.571166\pi$
$654$	14.4932	0.566727
$655$	0	0
$656$	−14.9698	−0.584474
$657$	−19.4532	−0.758941
$658$	−22.0120	−0.858118
$659$	−26.0107	−1.01323	−0.506616	−	0.862172i	$-0.669104\pi$
$659$	−26.0107	−1.01323	−0.506616	+	0.862172i	$0.669104\pi$
$660$	0	0
$661$	−2.13040	−0.0828630	−0.0414315	−	0.999141i	$-0.513192\pi$
$661$	−2.13040	−0.0828630	−0.0414315	+	0.999141i	$0.513192\pi$
$662$	−2.50953	−0.0975355
$663$	0	0
$664$	−14.3966	−0.558698
$665$	0	0
$666$	54.7095	2.11995
$667$	12.6599	0.490195
$668$	7.64662	0.295857
$669$	71.5652	2.76687
$670$	0	0
$671$	−3.83410	−0.148014
$672$	20.0828	0.774712
$673$	10.4944	0.404530	0.202265	−	0.979331i	$-0.435170\pi$
$673$	10.4944	0.404530	0.202265	+	0.979331i	$0.435170\pi$
$674$	−34.0133	−1.31015
$675$	0	0
$676$	0	0
$677$	−18.0822	−0.694955	−0.347477	−	0.937688i	$-0.612962\pi$
$677$	−18.0822	−0.694955	−0.347477	+	0.937688i	$0.612962\pi$
$678$	87.5480	3.36226
$679$	−10.1623	−0.389993
$680$	0	0
$681$	50.3701	1.93018
$682$	−11.0588	−0.423462
$683$	−35.1572	−1.34525	−0.672626	−	0.739982i	$-0.734834\pi$
$683$	−35.1572	−1.34525	−0.672626	+	0.739982i	$0.734834\pi$
$684$	−22.2796	−0.851882
$685$	0	0
$686$	−30.5341	−1.16580
$687$	−8.55994	−0.326582
$688$	−48.0140	−1.83052
$689$	0	0
$690$	0	0
$691$	−21.3411	−0.811853	−0.405927	−	0.913906i	$-0.633051\pi$
$691$	−21.3411	−0.811853	−0.405927	+	0.913906i	$0.633051\pi$
$692$	−2.82682	−0.107460
$693$	16.1176	0.612256
$694$	−31.8734	−1.20990
$695$	0	0
$696$	41.9770	1.59114
$697$	−1.27076	−0.0481333
$698$	27.5242	1.04181
$699$	−43.1952	−1.63379
$700$	0	0
$701$	49.7048	1.87733	0.938663	−	0.344836i	$-0.112065\pi$
$701$	49.7048	1.87733	0.938663	+	0.344836i	$0.112065\pi$
$702$	0	0
$703$	−46.3289	−1.74733
$704$	5.74293	0.216445
$705$	0	0
$706$	24.2587	0.912987
$707$	15.0881	0.567448
$708$	16.5197	0.620847
$709$	−31.5404	−1.18452	−0.592262	−	0.805745i	$-0.701765\pi$
$709$	−31.5404	−1.18452	−0.592262	+	0.805745i	$0.701765\pi$
$710$	0	0
$711$	−33.3125	−1.24932
$712$	6.46703	0.242362
$713$	−13.6514	−0.511249
$714$	4.72169	0.176705
$715$	0	0
$716$	4.94024	0.184625
$717$	59.2373	2.21226
$718$	−36.1875	−1.35051
$719$	18.0924	0.674734	0.337367	−	0.941373i	$-0.390464\pi$
$719$	18.0924	0.674734	0.337367	+	0.941373i	$0.390464\pi$
$720$	0	0
$721$	−13.0353	−0.485461
$722$	66.4206	2.47192
$723$	−39.7447	−1.47812
$724$	−10.4257	−0.387468
$725$	0	0
$726$	45.9380	1.70492
$727$	−38.6078	−1.43189	−0.715943	−	0.698159i	$-0.754003\pi$
$727$	−38.6078	−1.43189	−0.715943	+	0.698159i	$0.754003\pi$
$728$	0	0
$729$	−30.6798	−1.13629
$730$	0	0
$731$	−4.07580	−0.150749
$732$	5.04832	0.186591
$733$	20.6452	0.762548	0.381274	−	0.924462i	$-0.375486\pi$
$733$	20.6452	0.762548	0.381274	+	0.924462i	$0.375486\pi$
$734$	25.0141	0.923289
$735$	0	0
$736$	−5.77535	−0.212882
$737$	−1.78357	−0.0656985
$738$	29.1777	1.07405
$739$	25.2014	0.927049	0.463525	−	0.886084i	$-0.346585\pi$
$739$	25.2014	0.927049	0.463525	+	0.886084i	$0.346585\pi$
$740$	0	0
$741$	0	0
$742$	15.9792	0.586615
$743$	32.2167	1.18192	0.590958	−	0.806702i	$-0.298750\pi$
$743$	32.2167	1.18192	0.590958	+	0.806702i	$0.298750\pi$
$744$	−45.2644	−1.65947
$745$	0	0
$746$	−41.1791	−1.50767
$747$	35.3202	1.29230
$748$	−0.215317	−0.00787276
$749$	−22.2229	−0.812007
$750$	0	0
$751$	−10.5385	−0.384555	−0.192278	−	0.981341i	$-0.561587\pi$
$751$	−10.5385	−0.384555	−0.192278	+	0.981341i	$0.561587\pi$
$752$	26.4193	0.963414
$753$	−1.72938	−0.0630220
$754$	0	0
$755$	0	0
$756$	−10.3470	−0.376318
$757$	−21.4833	−0.780824	−0.390412	−	0.920640i	$-0.627667\pi$
$757$	−21.4833	−0.780824	−0.390412	+	0.920640i	$0.627667\pi$
$758$	−22.9274	−0.832762
$759$	−7.01020	−0.254454
$760$	0	0
$761$	−18.0017	−0.652561	−0.326280	−	0.945273i	$-0.605795\pi$
$761$	−18.0017	−0.652561	−0.326280	+	0.945273i	$0.605795\pi$
$762$	−41.7529	−1.51255
$763$	−7.72949	−0.279826
$764$	−12.7434	−0.461041
$765$	0	0
$766$	−31.8923	−1.15232
$767$	0	0
$768$	−32.8714	−1.18614
$769$	37.4867	1.35181	0.675903	−	0.736991i	$-0.263754\pi$
$769$	37.4867	1.35181	0.675903	+	0.736991i	$0.263754\pi$
$770$	0	0
$771$	41.6230	1.49901
$772$	−9.36904	−0.337199
$773$	−2.65353	−0.0954409	−0.0477205	−	0.998861i	$-0.515196\pi$
$773$	−2.65353	−0.0954409	−0.0477205	+	0.998861i	$0.515196\pi$
$774$	93.5842	3.36381
$775$	0	0
$776$	9.69004	0.347852
$777$	−44.1294	−1.58313
$778$	22.5440	0.808243
$779$	−24.7081	−0.885260
$780$	0	0
$781$	2.42380	0.0867304
$782$	−1.35785	−0.0485566
$783$	−50.2119	−1.79443
$784$	3.49151	0.124697
$785$	0	0
$786$	44.3631	1.58238
$787$	−34.2486	−1.22083	−0.610415	−	0.792081i	$-0.708998\pi$
$787$	−34.2486	−1.22083	−0.610415	+	0.792081i	$0.708998\pi$
$788$	−2.66693	−0.0950055
$789$	−66.4118	−2.36432
$790$	0	0
$791$	−46.6911	−1.66014
$792$	−15.3686	−0.546099
$793$	0	0
$794$	62.1664	2.20620
$795$	0	0
$796$	−5.25598	−0.186293
$797$	4.92056	0.174295	0.0871476	−	0.996195i	$-0.472225\pi$
$797$	4.92056	0.174295	0.0871476	+	0.996195i	$0.472225\pi$
$798$	91.8069	3.24993
$799$	2.24268	0.0793403
$800$	0	0
$801$	−15.8660	−0.560598
$802$	−15.4970	−0.547219
$803$	−3.65543	−0.128997
$804$	2.34840	0.0828219
$805$	0	0
$806$	0	0
$807$	−3.04759	−0.107280
$808$	−14.3870	−0.506132
$809$	39.3217	1.38248	0.691239	−	0.722626i	$-0.257065\pi$
$809$	39.3217	1.38248	0.691239	+	0.722626i	$0.257065\pi$
$810$	0	0
$811$	14.2001	0.498632	0.249316	−	0.968422i	$-0.419794\pi$
$811$	14.2001	0.498632	0.249316	+	0.968422i	$0.419794\pi$
$812$	7.20167	0.252729
$813$	35.1317	1.23212
$814$	10.2804	0.360329
$815$	0	0
$816$	−5.66708	−0.198388
$817$	−79.2485	−2.77255
$818$	41.7554	1.45994
$819$	0	0
$820$	0	0
$821$	−38.1226	−1.33049	−0.665243	−	0.746627i	$-0.731672\pi$
$821$	−38.1226	−1.33049	−0.665243	+	0.746627i	$0.731672\pi$
$822$	40.3775	1.40833
$823$	18.3621	0.640063	0.320032	−	0.947407i	$-0.396306\pi$
$823$	18.3621	0.640063	0.320032	+	0.947407i	$0.396306\pi$
$824$	12.4296	0.433004
$825$	0	0
$826$	−45.0082	−1.56604
$827$	20.5433	0.714359	0.357180	−	0.934036i	$-0.383738\pi$
$827$	20.5433	0.714359	0.357180	+	0.934036i	$0.383738\pi$
$828$	6.10291	0.212091
$829$	46.0983	1.60106	0.800531	−	0.599291i	$-0.204551\pi$
$829$	46.0983	1.60106	0.800531	+	0.599291i	$0.204551\pi$
$830$	0	0
$831$	−67.8903	−2.35509
$832$	0	0
$833$	0.296387	0.0102692
$834$	87.3559	3.02489
$835$	0	0
$836$	−4.18654	−0.144795
$837$	54.1442	1.87150
$838$	18.9100	0.653235
$839$	−15.8711	−0.547932	−0.273966	−	0.961739i	$-0.588336\pi$
$839$	−15.8711	−0.547932	−0.273966	+	0.961739i	$0.588336\pi$
$840$	0	0
$841$	5.94809	0.205107
$842$	37.8991	1.30609
$843$	83.2742	2.86812
$844$	2.36361	0.0813589
$845$	0	0
$846$	−51.4939	−1.77040
$847$	−24.4996	−0.841818
$848$	−19.1786	−0.658596
$849$	−51.6388	−1.77224
$850$	0	0
$851$	12.6906	0.435027
$852$	−3.19140	−0.109335
$853$	49.0753	1.68031	0.840154	−	0.542348i	$-0.182465\pi$
$853$	49.0753	1.68031	0.840154	+	0.542348i	$0.182465\pi$
$854$	−13.7543	−0.470662
$855$	0	0
$856$	21.1902	0.724265
$857$	−10.6489	−0.363758	−0.181879	−	0.983321i	$-0.558218\pi$
$857$	−10.6489	−0.363758	−0.181879	+	0.983321i	$0.558218\pi$
$858$	0	0
$859$	35.5693	1.21361	0.606804	−	0.794852i	$-0.292451\pi$
$859$	35.5693	1.21361	0.606804	+	0.794852i	$0.292451\pi$
$860$	0	0
$861$	−23.5351	−0.802074
$862$	−3.37628	−0.114997
$863$	0.0297148	0.00101150	0.000505752	−	1.00000i	$-0.499839\pi$
$863$	0.0297148	0.00101150	0.000505752		1.00000i	$0.499839\pi$
$864$	22.9062	0.779286
$865$	0	0
$866$	64.1427	2.17966
$867$	50.1047	1.70165
$868$	−7.76566	−0.263584
$869$	−6.25972	−0.212347
$870$	0	0
$871$	0	0
$872$	7.37029	0.249590
$873$	−23.7732	−0.804601
$874$	−26.4015	−0.893045
$875$	0	0
$876$	4.81307	0.162619
$877$	10.4120	0.351587	0.175794	−	0.984427i	$-0.443751\pi$
$877$	10.4120	0.351587	0.175794	+	0.984427i	$0.443751\pi$
$878$	−1.93750	−0.0653875
$879$	−57.1361	−1.92715
$880$	0	0
$881$	−52.2077	−1.75892	−0.879461	−	0.475970i	$-0.842097\pi$
$881$	−52.2077	−1.75892	−0.879461	+	0.475970i	$0.842097\pi$
$882$	−6.80531	−0.229147
$883$	−57.3602	−1.93033	−0.965163	−	0.261650i	$-0.915733\pi$
$883$	−57.3602	−1.93033	−0.965163	+	0.261650i	$0.915733\pi$
$884$	0	0
$885$	0	0
$886$	26.2039	0.880337
$887$	−15.4404	−0.518438	−0.259219	−	0.965819i	$-0.583465\pi$
$887$	−15.4404	−0.518438	−0.259219	+	0.965819i	$0.583465\pi$
$888$	42.0786	1.41207
$889$	22.2677	0.746834
$890$	0	0
$891$	8.48266	0.284180
$892$	−11.7073	−0.391990
$893$	43.6058	1.45921
$894$	−19.0258	−0.636317
$895$	0	0
$896$	34.1001	1.13921
$897$	0	0
$898$	22.8077	0.761103
$899$	−37.6850	−1.25687
$900$	0	0
$901$	−1.62803	−0.0542375
$902$	5.48276	0.182556
$903$	−75.4861	−2.51202
$904$	44.5213	1.48076
$905$	0	0
$906$	30.5738	1.01575
$907$	−36.7881	−1.22153	−0.610765	−	0.791812i	$-0.709138\pi$
$907$	−36.7881	−1.22153	−0.610765	+	0.791812i	$0.709138\pi$
$908$	−8.24000	−0.273454
$909$	35.2965	1.17071
$910$	0	0
$911$	−4.72354	−0.156498	−0.0782490	−	0.996934i	$-0.524933\pi$
$911$	−4.72354	−0.156498	−0.0782490	+	0.996934i	$0.524933\pi$
$912$	−110.189	−3.64871
$913$	6.63699	0.219652
$914$	5.38298	0.178053
$915$	0	0
$916$	1.40032	0.0462677
$917$	−23.6597	−0.781313
$918$	5.38551	0.177748
$919$	14.5599	0.480285	0.240143	−	0.970738i	$-0.422806\pi$
$919$	14.5599	0.480285	0.240143	+	0.970738i	$0.422806\pi$
$920$	0	0
$921$	−2.52461	−0.0831886
$922$	−17.7067	−0.583138
$923$	0	0
$924$	−3.98778	−0.131188
$925$	0	0
$926$	25.4492	0.836313
$927$	−30.4943	−1.00156
$928$	−15.9430	−0.523355
$929$	−23.6896	−0.777230	−0.388615	−	0.921400i	$-0.627046\pi$
$929$	−23.6896	−0.777230	−0.388615	+	0.921400i	$0.627046\pi$
$930$	0	0
$931$	5.76284	0.188869
$932$	7.06628	0.231464
$933$	−85.5163	−2.79968
$934$	−34.4294	−1.12657
$935$	0	0
$936$	0	0
$937$	17.4179	0.569019	0.284510	−	0.958673i	$-0.408169\pi$
$937$	17.4179	0.569019	0.284510	+	0.958673i	$0.408169\pi$
$938$	−6.39829	−0.208912
$939$	36.9080	1.20445
$940$	0	0
$941$	34.6953	1.13103	0.565517	−	0.824736i	$-0.308677\pi$
$941$	34.6953	1.13103	0.565517	+	0.824736i	$0.308677\pi$
$942$	−45.7743	−1.49141
$943$	6.76814	0.220401
$944$	54.0199	1.75820
$945$	0	0
$946$	17.5853	0.571748
$947$	44.4101	1.44314	0.721568	−	0.692344i	$-0.243422\pi$
$947$	44.4101	1.44314	0.721568	+	0.692344i	$0.243422\pi$
$948$	8.24212	0.267692
$949$	0	0
$950$	0	0
$951$	57.3361	1.85925
$952$	2.40115	0.0778217
$953$	37.9862	1.23049	0.615247	−	0.788335i	$-0.289056\pi$
$953$	37.9862	1.23049	0.615247	+	0.788335i	$0.289056\pi$
$954$	37.3810	1.21026
$955$	0	0
$956$	−9.69059	−0.313416
$957$	−19.3518	−0.625556
$958$	−36.7850	−1.18847
$959$	−21.5341	−0.695373
$960$	0	0
$961$	9.63628	0.310848
$962$	0	0
$963$	−51.9872	−1.67526
$964$	6.50181	0.209409
$965$	0	0
$966$	−25.1481	−0.809126
$967$	10.9763	0.352974	0.176487	−	0.984303i	$-0.443527\pi$
$967$	10.9763	0.352974	0.176487	+	0.984303i	$0.443527\pi$
$968$	23.3611	0.750855
$969$	−9.35368	−0.300483
$970$	0	0
$971$	−0.742671	−0.0238334	−0.0119167	−	0.999929i	$-0.503793\pi$
$971$	−0.742671	−0.0238334	−0.0119167	+	0.999929i	$0.503793\pi$
$972$	1.23466	0.0396018
$973$	−46.5887	−1.49356
$974$	39.5737	1.26802
$975$	0	0
$976$	16.5082	0.528415
$977$	28.4157	0.909100	0.454550	−	0.890721i	$-0.349800\pi$
$977$	28.4157	0.909100	0.454550	+	0.890721i	$0.349800\pi$
$978$	49.6208	1.58670
$979$	−2.98137	−0.0952849
$980$	0	0
$981$	−18.0820	−0.577315
$982$	20.6222	0.658081
$983$	−38.7173	−1.23489	−0.617445	−	0.786614i	$-0.711832\pi$
$983$	−38.7173	−1.23489	−0.617445	+	0.786614i	$0.711832\pi$
$984$	22.4414	0.715405
$985$	0	0
$986$	−3.74838	−0.119373
$987$	41.5356	1.32209
$988$	0	0
$989$	21.7080	0.690276
$990$	0	0
$991$	40.7506	1.29449	0.647243	−	0.762284i	$-0.275922\pi$
$991$	40.7506	1.29449	0.647243	+	0.762284i	$0.275922\pi$
$992$	17.1916	0.545833
$993$	4.73536	0.150272
$994$	8.69505	0.275790
$995$	0	0
$996$	−8.73886	−0.276901
$997$	−4.68838	−0.148482	−0.0742412	−	0.997240i	$-0.523653\pi$
$997$	−4.68838	−0.148482	−0.0742412	+	0.997240i	$0.523653\pi$
$998$	29.3595	0.929359
$999$	−50.3334	−1.59248

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.ca.1.13		18
5.2	odd	4		845.2.b.g.339.13	yes	18
5.3	odd	4		845.2.b.g.339.6	✓	18
5.4	even	2	inner	4225.2.a.ca.1.6		18
13.12	even	2		4225.2.a.cb.1.6		18
65.2	even	12		845.2.l.g.654.11		72
65.3	odd	12		845.2.n.i.529.13		36
65.7	even	12		845.2.l.g.699.26		72
65.8	even	4		845.2.d.e.844.25		36
65.12	odd	4		845.2.b.h.339.6	yes	18
65.17	odd	12		845.2.n.h.484.6		36
65.18	even	4		845.2.d.e.844.11		36
65.22	odd	12		845.2.n.i.484.13		36
65.23	odd	12		845.2.n.h.529.6		36
65.28	even	12		845.2.l.g.654.26		72
65.32	even	12		845.2.l.g.699.12		72
65.33	even	12		845.2.l.g.699.11		72
65.37	even	12		845.2.l.g.654.25		72
65.38	odd	4		845.2.b.h.339.13	yes	18
65.42	odd	12		845.2.n.i.529.6		36
65.43	odd	12		845.2.n.h.484.13		36
65.47	even	4		845.2.d.e.844.12		36
65.48	odd	12		845.2.n.i.484.6		36
65.57	even	4		845.2.d.e.844.26		36
65.58	even	12		845.2.l.g.699.25		72
65.62	odd	12		845.2.n.h.529.13		36
65.63	even	12		845.2.l.g.654.12		72
65.64	even	2		4225.2.a.cb.1.13		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.b.g.339.6	✓	18	5.3	odd	4
845.2.b.g.339.13	yes	18	5.2	odd	4
845.2.b.h.339.6	yes	18	65.12	odd	4
845.2.b.h.339.13	yes	18	65.38	odd	4
845.2.d.e.844.11		36	65.18	even	4
845.2.d.e.844.12		36	65.47	even	4
845.2.d.e.844.25		36	65.8	even	4
845.2.d.e.844.26		36	65.57	even	4
845.2.l.g.654.11		72	65.2	even	12
845.2.l.g.654.12		72	65.63	even	12
845.2.l.g.654.25		72	65.37	even	12
845.2.l.g.654.26		72	65.28	even	12
845.2.l.g.699.11		72	65.33	even	12
845.2.l.g.699.12		72	65.32	even	12
845.2.l.g.699.25		72	65.58	even	12
845.2.l.g.699.26		72	65.7	even	12
845.2.n.h.484.6		36	65.17	odd	12
845.2.n.h.484.13		36	65.43	odd	12
845.2.n.h.529.6		36	65.23	odd	12
845.2.n.h.529.13		36	65.62	odd	12
845.2.n.i.484.6		36	65.48	odd	12
845.2.n.i.484.13		36	65.22	odd	12
845.2.n.i.529.6		36	65.42	odd	12
845.2.n.i.529.13		36	65.3	odd	12
4225.2.a.ca.1.6		18	5.4	even	2	inner
4225.2.a.ca.1.13		18	1.1	even	1	trivial
4225.2.a.cb.1.6		18	13.12	even	2
4225.2.a.cb.1.13		18	65.64	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q+1.57695 q^{2} -2.97563 q^{3} +0.486782 q^{4} -4.69244 q^{6} +2.50257 q^{7} -2.38627 q^{8} +5.85440 q^{9} +O(q^{10})\)	\(q+1.57695 q^{2} -2.97563 q^{3} +0.486782 q^{4} -4.69244 q^{6} +2.50257 q^{7} -2.38627 q^{8} +5.85440 q^{9} +1.10010 q^{11} -1.44849 q^{12} +3.94644 q^{14} -4.73661 q^{16} -0.402080 q^{17} +9.23212 q^{18} -7.81790 q^{19} -7.44674 q^{21} +1.73480 q^{22} +2.14151 q^{23} +7.10068 q^{24} -8.49366 q^{27} +1.21821 q^{28} +5.91169 q^{29} -6.37466 q^{31} -2.69686 q^{32} -3.27348 q^{33} -0.634062 q^{34} +2.84982 q^{36} +5.92600 q^{37} -12.3285 q^{38} +3.16045 q^{41} -11.7432 q^{42} +10.1368 q^{43} +0.535507 q^{44} +3.37706 q^{46} -5.57769 q^{47} +14.0944 q^{48} -0.737134 q^{49} +1.19644 q^{51} +4.04902 q^{53} -13.3941 q^{54} -5.97182 q^{56} +23.2632 q^{57} +9.32246 q^{58} -11.4048 q^{59} -3.48524 q^{61} -10.0525 q^{62} +14.6511 q^{63} +5.22039 q^{64} -5.16213 q^{66} -1.62128 q^{67} -0.195725 q^{68} -6.37235 q^{69} +2.20326 q^{71} -13.9702 q^{72} -3.32283 q^{73} +9.34503 q^{74} -3.80562 q^{76} +2.75307 q^{77} -5.69016 q^{79} +7.71083 q^{81} +4.98389 q^{82} +6.03310 q^{83} -3.62494 q^{84} +15.9853 q^{86} -17.5910 q^{87} -2.62513 q^{88} -2.71010 q^{89} +1.04245 q^{92} +18.9687 q^{93} -8.79576 q^{94} +8.02488 q^{96} -4.06074 q^{97} -1.16243 q^{98} +6.44041 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9}+O(q^{10}) \) 18 * q + 16 * q^4 - 16 * q^6 + 18 * q^9	\( 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9} - 22 q^{11} + 4 q^{14} - 12 q^{16} - 28 q^{19} - 26 q^{21} - 34 q^{24} - 20 q^{29} - 32 q^{31} - 18 q^{34} + 32 q^{36} - 52 q^{41} - 50 q^{44} - 30 q^{46} + 44 q^{49} - 40 q^{51} - 90 q^{54} - 20 q^{56} - 76 q^{59} + 8 q^{61} - 68 q^{64} + 8 q^{66} + 30 q^{69} - 72 q^{71} + 30 q^{74} - 4 q^{76} + 16 q^{79} - 30 q^{81} - 78 q^{84} + 30 q^{86} - 94 q^{89} - 128 q^{94} + 18 q^{96} - 76 q^{99}+O(q^{100}) \) 18 * q + 16 * q^4 - 16 * q^6 + 18 * q^9 - 22 * q^11 + 4 * q^14 - 12 * q^16 - 28 * q^19 - 26 * q^21 - 34 * q^24 - 20 * q^29 - 32 * q^31 - 18 * q^34 + 32 * q^36 - 52 * q^41 - 50 * q^44 - 30 * q^46 + 44 * q^49 - 40 * q^51 - 90 * q^54 - 20 * q^56 - 76 * q^59 + 8 * q^61 - 68 * q^64 + 8 * q^66 + 30 * q^69 - 72 * q^71 + 30 * q^74 - 4 * q^76 + 16 * q^79 - 30 * q^81 - 78 * q^84 + 30 * q^86 - 94 * q^89 - 128 * q^94 + 18 * q^96 - 76 * q^99

Introduction

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