LMFDB - Embedded newform 4225.2.a.ca.1.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4225,2,Mod(1,4225)] 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("4225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [18,0,0,16,0,-16,0] 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:	$33.7367948540$
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			1.18
Root			$2.51756$ 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)

$f(q)$	$=$	\(q+2.51756 q^{2} -2.61964 q^{3} +4.33809 q^{4} -6.59510 q^{6} -1.23067 q^{7} +5.88628 q^{8} +3.86253 q^{9} -2.94805 q^{11} -11.3643 q^{12} -3.09829 q^{14} +6.14287 q^{16} +5.86924 q^{17} +9.72414 q^{18} -4.76359 q^{19} +3.22393 q^{21} -7.42189 q^{22} -1.69519 q^{23} -15.4200 q^{24} -2.25952 q^{27} -5.33878 q^{28} +4.17106 q^{29} -2.50193 q^{31} +3.69246 q^{32} +7.72284 q^{33} +14.7762 q^{34} +16.7560 q^{36} -3.34159 q^{37} -11.9926 q^{38} -8.95956 q^{41} +8.11642 q^{42} +0.795573 q^{43} -12.7889 q^{44} -4.26775 q^{46} -10.0172 q^{47} -16.0921 q^{48} -5.48544 q^{49} -15.3753 q^{51} +7.14496 q^{53} -5.68847 q^{54} -7.24410 q^{56} +12.4789 q^{57} +10.5009 q^{58} -0.0348174 q^{59} +4.01406 q^{61} -6.29876 q^{62} -4.75352 q^{63} -2.98977 q^{64} +19.4427 q^{66} -7.22249 q^{67} +25.4613 q^{68} +4.44080 q^{69} -14.2161 q^{71} +22.7360 q^{72} -8.45275 q^{73} -8.41266 q^{74} -20.6649 q^{76} +3.62809 q^{77} +7.06674 q^{79} -5.66845 q^{81} -22.5562 q^{82} -3.50072 q^{83} +13.9857 q^{84} +2.00290 q^{86} -10.9267 q^{87} -17.3531 q^{88} -8.89032 q^{89} -7.35391 q^{92} +6.55417 q^{93} -25.2189 q^{94} -9.67292 q^{96} +10.6940 q^{97} -13.8099 q^{98} -11.3869 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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 $

$2$	2.51756	1.78018	0.890091	−	0.455783i	$-0.150641\pi$
$2$	2.51756	1.78018	0.890091	+	0.455783i	$0.150641\pi$
$3$	−2.61964	−1.51245	−0.756226	−	0.654311i	$-0.772959\pi$
$3$	−2.61964	−1.51245	−0.756226	+	0.654311i	$0.772959\pi$
$4$	4.33809	2.16905
$5$	0	0
$5$	0	0
$6$	−6.59510	−2.69244
$7$	−1.23067	−0.465151	−0.232576	−	0.972578i	$-0.574715\pi$
$7$	−1.23067	−0.465151	−0.232576	+	0.972578i	$0.574715\pi$
$8$	5.88628	2.08112
$9$	3.86253	1.28751
$10$	0	0
$11$	−2.94805	−0.888871	−0.444435	−	0.895811i	$-0.646596\pi$
$11$	−2.94805	−0.888871	−0.444435	+	0.895811i	$0.646596\pi$
$12$	−11.3643	−3.28058
$13$	0	0
$13$	0	0
$14$	−3.09829	−0.828054
$15$	0	0
$16$	6.14287	1.53572
$17$	5.86924	1.42350	0.711750	−	0.702433i	$-0.247903\pi$
$17$	5.86924	1.42350	0.711750	+	0.702433i	$0.247903\pi$
$18$	9.72414	2.29200
$19$	−4.76359	−1.09284	−0.546422	−	0.837510i	$-0.684010\pi$
$19$	−4.76359	−1.09284	−0.546422	+	0.837510i	$0.684010\pi$
$20$	0	0
$21$	3.22393	0.703519
$22$	−7.42189	−1.58235
$23$	−1.69519	−0.353472	−0.176736	−	0.984258i	$-0.556554\pi$
$23$	−1.69519	−0.353472	−0.176736	+	0.984258i	$0.556554\pi$
$24$	−15.4200	−3.14759
$25$	0	0
$26$	0	0
$27$	−2.25952	−0.434845
$28$	−5.33878	−1.00893
$29$	4.17106	0.774546	0.387273	−	0.921965i	$-0.373417\pi$
$29$	4.17106	0.774546	0.387273	+	0.921965i	$0.373417\pi$
$30$	0	0
$31$	−2.50193	−0.449360	−0.224680	−	0.974433i	$-0.572134\pi$
$31$	−2.50193	−0.449360	−0.224680	+	0.974433i	$0.572134\pi$
$32$	3.69246	0.652740
$33$	7.72284	1.34437
$34$	14.7762	2.53409
$35$	0	0
$36$	16.7560	2.79267
$37$	−3.34159	−0.549355	−0.274677	−	0.961536i	$-0.588571\pi$
$37$	−3.34159	−0.549355	−0.274677	+	0.961536i	$0.588571\pi$
$38$	−11.9926	−1.94546
$39$	0	0
$40$	0	0
$41$	−8.95956	−1.39925	−0.699624	−	0.714511i	$-0.746649\pi$
$41$	−8.95956	−1.39925	−0.699624	+	0.714511i	$0.746649\pi$
$42$	8.11642	1.25239
$43$	0.795573	0.121324	0.0606619	−	0.998158i	$-0.480679\pi$
$43$	0.795573	0.121324	0.0606619	+	0.998158i	$0.480679\pi$
$44$	−12.7889	−1.92800
$45$	0	0
$46$	−4.26775	−0.629245
$47$	−10.0172	−1.46116	−0.730580	−	0.682827i	$-0.760750\pi$
$47$	−10.0172	−1.46116	−0.730580	+	0.682827i	$0.760750\pi$
$48$	−16.0921	−2.32270
$49$	−5.48544	−0.783634
$50$	0	0
$51$	−15.3753	−2.15298
$52$	0	0
$53$	7.14496	0.981436	0.490718	−	0.871318i	$-0.336735\pi$
$53$	7.14496	0.981436	0.490718	+	0.871318i	$0.336735\pi$
$54$	−5.68847	−0.774103
$55$	0	0
$56$	−7.24410	−0.968034
$57$	12.4789	1.65287
$58$	10.5009	1.37883
$59$	−0.0348174	−0.00453284	−0.00226642	−	0.999997i	$-0.500721\pi$
$59$	−0.0348174	−0.00453284	−0.00226642	+	0.999997i	$0.500721\pi$
$60$	0	0
$61$	4.01406	0.513948	0.256974	−	0.966418i	$-0.417275\pi$
$61$	4.01406	0.513948	0.256974	+	0.966418i	$0.417275\pi$
$62$	−6.29876	−0.799943
$63$	−4.75352	−0.598887
$64$	−2.98977	−0.373721
$65$	0	0
$66$	19.4427	2.39323
$67$	−7.22249	−0.882367	−0.441184	−	0.897417i	$-0.645441\pi$
$67$	−7.22249	−0.882367	−0.441184	+	0.897417i	$0.645441\pi$
$68$	25.4613	3.08764
$69$	4.44080	0.534610
$70$	0	0
$71$	−14.2161	−1.68714	−0.843572	−	0.537015i	$-0.819552\pi$
$71$	−14.2161	−1.68714	−0.843572	+	0.537015i	$0.819552\pi$
$72$	22.7360	2.67946
$73$	−8.45275	−0.989320	−0.494660	−	0.869087i	$-0.664707\pi$
$73$	−8.45275	−0.989320	−0.494660	+	0.869087i	$0.664707\pi$
$74$	−8.41266	−0.977951
$75$	0	0
$76$	−20.6649	−2.37043
$77$	3.62809	0.413459
$78$	0	0
$79$	7.06674	0.795070	0.397535	−	0.917587i	$-0.369866\pi$
$79$	7.06674	0.795070	0.397535	+	0.917587i	$0.369866\pi$
$80$	0	0
$81$	−5.66845	−0.629828
$82$	−22.5562	−2.49092
$83$	−3.50072	−0.384254	−0.192127	−	0.981370i	$-0.561538\pi$
$83$	−3.50072	−0.384254	−0.192127	+	0.981370i	$0.561538\pi$
$84$	13.9857	1.52597
$85$	0	0
$86$	2.00290	0.215978
$87$	−10.9267	−1.17146
$88$	−17.3531	−1.84984
$89$	−8.89032	−0.942372	−0.471186	−	0.882034i	$-0.656174\pi$
$89$	−8.89032	−0.942372	−0.471186	+	0.882034i	$0.656174\pi$
$90$	0	0
$91$	0	0
$92$	−7.35391	−0.766698
$93$	6.55417	0.679636
$94$	−25.2189	−2.60113
$95$	0	0
$96$	−9.67292	−0.987238
$97$	10.6940	1.08581	0.542907	−	0.839793i	$-0.317324\pi$
$97$	10.6940	1.08581	0.542907	+	0.839793i	$0.317324\pi$
$98$	−13.8099	−1.39501
$99$	−11.3869	−1.14443
$100$	0	0
$101$	−12.9172	−1.28531	−0.642655	−	0.766156i	$-0.722167\pi$
$101$	−12.9172	−1.28531	−0.642655	+	0.766156i	$0.722167\pi$
$102$	−38.7082	−3.83269
$103$	−5.67077	−0.558757	−0.279379	−	0.960181i	$-0.590128\pi$
$103$	−5.67077	−0.558757	−0.279379	+	0.960181i	$0.590128\pi$
$104$	0	0
$105$	0	0
$106$	17.9878	1.74713
$107$	10.4504	1.01028	0.505138	−	0.863038i	$-0.331441\pi$
$107$	10.4504	1.01028	0.505138	+	0.863038i	$0.331441\pi$
$108$	−9.80201	−0.943199
$109$	5.38402	0.515696	0.257848	−	0.966186i	$-0.416987\pi$
$109$	5.38402	0.515696	0.257848	+	0.966186i	$0.416987\pi$
$110$	0	0
$111$	8.75379	0.830873
$112$	−7.55987	−0.714341
$113$	10.6715	1.00389	0.501945	−	0.864899i	$-0.332618\pi$
$113$	10.6715	1.00389	0.501945	+	0.864899i	$0.332618\pi$
$114$	31.4164	2.94241
$115$	0	0
$116$	18.0944	1.68003
$117$	0	0
$118$	−0.0876548	−0.00806928
$119$	−7.22313	−0.662143
$120$	0	0
$121$	−2.30900	−0.209909
$122$	10.1056	0.914920
$123$	23.4708	2.11630
$124$	−10.8536	−0.974684
$125$	0	0
$126$	−11.9673	−1.06613
$127$	7.79430	0.691632	0.345816	−	0.938302i	$-0.387602\pi$
$127$	7.79430	0.691632	0.345816	+	0.938302i	$0.387602\pi$
$128$	−14.9118	−1.31803
$129$	−2.08412	−0.183496
$130$	0	0
$131$	−14.4962	−1.26654	−0.633270	−	0.773931i	$-0.718288\pi$
$131$	−14.4962	−1.26654	−0.633270	+	0.773931i	$0.718288\pi$
$132$	33.5024	2.91601
$133$	5.86243	0.508338
$134$	−18.1830	−1.57077
$135$	0	0
$136$	34.5480	2.96247
$137$	2.26717	0.193697	0.0968486	−	0.995299i	$-0.469124\pi$
$137$	2.26717	0.193697	0.0968486	+	0.995299i	$0.469124\pi$
$138$	11.1800	0.951703
$139$	−11.3578	−0.963356	−0.481678	−	0.876348i	$-0.659973\pi$
$139$	−11.3578	−0.963356	−0.481678	+	0.876348i	$0.659973\pi$
$140$	0	0
$141$	26.2415	2.20993
$142$	−35.7899	−3.00342
$143$	0	0
$144$	23.7270	1.97725
$145$	0	0
$146$	−21.2803	−1.76117
$147$	14.3699	1.18521
$148$	−14.4962	−1.19158
$149$	−2.05699	−0.168515	−0.0842577	−	0.996444i	$-0.526852\pi$
$149$	−2.05699	−0.168515	−0.0842577	+	0.996444i	$0.526852\pi$
$150$	0	0
$151$	−5.25169	−0.427376	−0.213688	−	0.976902i	$-0.568548\pi$
$151$	−5.25169	−0.427376	−0.213688	+	0.976902i	$0.568548\pi$
$152$	−28.0399	−2.27433
$153$	22.6701	1.83277
$154$	9.13392	0.736033
$155$	0	0
$156$	0	0
$157$	−23.4521	−1.87168	−0.935842	−	0.352419i	$-0.885359\pi$
$157$	−23.4521	−1.87168	−0.935842	+	0.352419i	$0.885359\pi$
$158$	17.7909	1.41537
$159$	−18.7172	−1.48437
$160$	0	0
$161$	2.08623	0.164418
$162$	−14.2707	−1.12121
$163$	−7.10599	−0.556584	−0.278292	−	0.960496i	$-0.589768\pi$
$163$	−7.10599	−0.556584	−0.278292	+	0.960496i	$0.589768\pi$
$164$	−38.8674	−3.03503
$165$	0	0
$166$	−8.81326	−0.684041
$167$	20.4389	1.58161	0.790803	−	0.612070i	$-0.209663\pi$
$167$	20.4389	1.58161	0.790803	+	0.612070i	$0.209663\pi$
$168$	18.9770	1.46410
$169$	0	0
$170$	0	0
$171$	−18.3995	−1.40705
$172$	3.45127	0.263157
$173$	14.7412	1.12075	0.560376	−	0.828238i	$-0.310657\pi$
$173$	14.7412	1.12075	0.560376	+	0.828238i	$0.310657\pi$
$174$	−27.5085	−2.08542
$175$	0	0
$176$	−18.1095	−1.36505
$177$	0.0912092	0.00685570
$178$	−22.3819	−1.67759
$179$	−19.1009	−1.42767	−0.713835	−	0.700314i	$-0.753043\pi$
$179$	−19.1009	−1.42767	−0.713835	+	0.700314i	$0.753043\pi$
$180$	0	0
$181$	−12.2161	−0.908012	−0.454006	−	0.890999i	$-0.650006\pi$
$181$	−12.2161	−0.908012	−0.454006	+	0.890999i	$0.650006\pi$
$182$	0	0
$183$	−10.5154	−0.777321
$184$	−9.97840	−0.735617
$185$	0	0
$186$	16.5005	1.20988
$187$	−17.3028	−1.26531
$188$	−43.4556	−3.16933
$189$	2.78073	0.202269
$190$	0	0
$191$	4.01177	0.290282	0.145141	−	0.989411i	$-0.453636\pi$
$191$	4.01177	0.290282	0.145141	+	0.989411i	$0.453636\pi$
$192$	7.83213	0.565235
$193$	23.9928	1.72704	0.863520	−	0.504315i	$-0.168255\pi$
$193$	23.9928	1.72704	0.863520	+	0.504315i	$0.168255\pi$
$194$	26.9228	1.93295
$195$	0	0
$196$	−23.7964	−1.69974
$197$	6.93680	0.494227	0.247113	−	0.968987i	$-0.420518\pi$
$197$	6.93680	0.494227	0.247113	+	0.968987i	$0.420518\pi$
$198$	−28.6673	−2.03729
$199$	5.78686	0.410220	0.205110	−	0.978739i	$-0.434245\pi$
$199$	5.78686	0.410220	0.205110	+	0.978739i	$0.434245\pi$
$200$	0	0
$201$	18.9203	1.33454
$202$	−32.5198	−2.28809
$203$	−5.13321	−0.360281
$204$	−66.6996	−4.66990
$205$	0	0
$206$	−14.2765	−0.994690
$207$	−6.54774	−0.455099
$208$	0	0
$209$	14.0433	0.971397
$210$	0	0
$211$	−16.4258	−1.13080	−0.565398	−	0.824818i	$-0.691277\pi$
$211$	−16.4258	−1.13080	−0.565398	+	0.824818i	$0.691277\pi$
$212$	30.9955	2.12878
$213$	37.2412	2.55173
$214$	26.3094	1.79848
$215$	0	0
$216$	−13.3002	−0.904963
$217$	3.07906	0.209021
$218$	13.5546	0.918032
$219$	22.1432	1.49630
$220$	0	0
$221$	0	0
$222$	22.0382	1.47910
$223$	−21.7605	−1.45719	−0.728596	−	0.684944i	$-0.759827\pi$
$223$	−21.7605	−1.45719	−0.728596	+	0.684944i	$0.759827\pi$
$224$	−4.54421	−0.303623
$225$	0	0
$226$	26.8661	1.78711
$227$	16.1145	1.06955	0.534777	−	0.844993i	$-0.320396\pi$
$227$	16.1145	1.06955	0.534777	+	0.844993i	$0.320396\pi$
$228$	54.1347	3.58516
$229$	12.4251	0.821071	0.410536	−	0.911845i	$-0.365342\pi$
$229$	12.4251	0.821071	0.410536	+	0.911845i	$0.365342\pi$
$230$	0	0
$231$	−9.50430	−0.625337
$232$	24.5520	1.61192
$233$	−9.05571	−0.593259	−0.296630	−	0.954993i	$-0.595863\pi$
$233$	−9.05571	−0.593259	−0.296630	+	0.954993i	$0.595863\pi$
$234$	0	0
$235$	0	0
$236$	−0.151041	−0.00983194
$237$	−18.5123	−1.20251
$238$	−18.1846	−1.17873
$239$	21.4642	1.38841	0.694203	−	0.719779i	$-0.255757\pi$
$239$	21.4642	1.38841	0.694203	+	0.719779i	$0.255757\pi$
$240$	0	0
$241$	−9.06978	−0.584236	−0.292118	−	0.956382i	$-0.594360\pi$
$241$	−9.06978	−0.584236	−0.292118	+	0.956382i	$0.594360\pi$
$242$	−5.81304	−0.373676
$243$	21.6279	1.38743
$244$	17.4134	1.11478
$245$	0	0
$246$	59.0892	3.76739
$247$	0	0
$248$	−14.7271	−0.935171
$249$	9.17063	0.581165
$250$	0	0
$251$	25.4920	1.60904	0.804519	−	0.593926i	$-0.202423\pi$
$251$	25.4920	1.60904	0.804519	+	0.593926i	$0.202423\pi$
$252$	−20.6212	−1.29901
$253$	4.99752	0.314191
$254$	19.6226	1.23123
$255$	0	0
$256$	−31.5618	−1.97262
$257$	−24.8979	−1.55309	−0.776543	−	0.630064i	$-0.783028\pi$
$257$	−24.8979	−1.55309	−0.776543	+	0.630064i	$0.783028\pi$
$258$	−5.24688	−0.326657
$259$	4.11242	0.255533
$260$	0	0
$261$	16.1108	0.997235
$262$	−36.4951	−2.25467
$263$	−15.6411	−0.964469	−0.482235	−	0.876042i	$-0.660175\pi$
$263$	−15.6411	−0.964469	−0.482235	+	0.876042i	$0.660175\pi$
$264$	45.4588	2.79780
$265$	0	0
$266$	14.7590	0.904933
$267$	23.2895	1.42529
$268$	−31.3318	−1.91390
$269$	23.5391	1.43521	0.717603	−	0.696453i	$-0.245239\pi$
$269$	23.5391	1.43521	0.717603	+	0.696453i	$0.245239\pi$
$270$	0	0
$271$	2.80953	0.170667	0.0853333	−	0.996352i	$-0.472805\pi$
$271$	2.80953	0.170667	0.0853333	+	0.996352i	$0.472805\pi$
$272$	36.0540	2.18609
$273$	0	0
$274$	5.70772	0.344816
$275$	0	0
$276$	19.2646	1.15959
$277$	−10.9594	−0.658489	−0.329245	−	0.944245i	$-0.606794\pi$
$277$	−10.9594	−0.658489	−0.329245	+	0.944245i	$0.606794\pi$
$278$	−28.5939	−1.71495
$279$	−9.66379	−0.578556
$280$	0	0
$281$	−5.44514	−0.324830	−0.162415	−	0.986723i	$-0.551928\pi$
$281$	−5.44514	−0.324830	−0.162415	+	0.986723i	$0.551928\pi$
$282$	66.0645	3.93408
$283$	6.33040	0.376303	0.188152	−	0.982140i	$-0.439750\pi$
$283$	6.33040	0.376303	0.188152	+	0.982140i	$0.439750\pi$
$284$	−61.6709	−3.65950
$285$	0	0
$286$	0	0
$287$	11.0263	0.650862
$288$	14.2622	0.840410
$289$	17.4480	1.02635
$290$	0	0
$291$	−28.0145	−1.64224
$292$	−36.6688	−2.14588
$293$	−6.03480	−0.352557	−0.176278	−	0.984340i	$-0.556406\pi$
$293$	−6.03480	−0.352557	−0.176278	+	0.984340i	$0.556406\pi$
$294$	36.1770	2.10989
$295$	0	0
$296$	−19.6696	−1.14327
$297$	6.66118	0.386521
$298$	−5.17860	−0.299988
$299$	0	0
$300$	0	0
$301$	−0.979091	−0.0564339
$302$	−13.2214	−0.760808
$303$	33.8385	1.94397
$304$	−29.2621	−1.67830
$305$	0	0
$306$	57.0733	3.26266
$307$	−7.24198	−0.413321	−0.206661	−	0.978413i	$-0.566260\pi$
$307$	−7.24198	−0.413321	−0.206661	+	0.978413i	$0.566260\pi$
$308$	15.7390	0.896813
$309$	14.8554	0.845093
$310$	0	0
$311$	7.01177	0.397601	0.198800	−	0.980040i	$-0.436295\pi$
$311$	7.01177	0.397601	0.198800	+	0.980040i	$0.436295\pi$
$312$	0	0
$313$	−16.5018	−0.932739	−0.466369	−	0.884590i	$-0.654438\pi$
$313$	−16.5018	−0.932739	−0.466369	+	0.884590i	$0.654438\pi$
$314$	−59.0421	−3.33194
$315$	0	0
$316$	30.6562	1.72455
$317$	11.6000	0.651520	0.325760	−	0.945453i	$-0.394380\pi$
$317$	11.6000	0.651520	0.325760	+	0.945453i	$0.394380\pi$
$318$	−47.1217	−2.64246
$319$	−12.2965	−0.688471
$320$	0	0
$321$	−27.3763	−1.52799
$322$	5.25221	0.292694
$323$	−27.9587	−1.55566
$324$	−24.5903	−1.36613
$325$	0	0
$326$	−17.8897	−0.990821
$327$	−14.1042	−0.779965
$328$	−52.7385	−2.91200
$329$	12.3279	0.679661
$330$	0	0
$331$	−2.89281	−0.159003	−0.0795017	−	0.996835i	$-0.525333\pi$
$331$	−2.89281	−0.159003	−0.0795017	+	0.996835i	$0.525333\pi$
$332$	−15.1864	−0.833464
$333$	−12.9070	−0.707300
$334$	51.4560	2.81555
$335$	0	0
$336$	19.8042	1.08041
$337$	24.6017	1.34014	0.670071	−	0.742297i	$-0.266264\pi$
$337$	24.6017	1.34014	0.670071	+	0.742297i	$0.266264\pi$
$338$	0	0
$339$	−27.9555	−1.51834
$340$	0	0
$341$	7.37582	0.399423
$342$	−46.3219	−2.50480
$343$	15.3655	0.829660
$344$	4.68297	0.252489
$345$	0	0
$346$	37.1118	1.99514
$347$	22.2988	1.19706	0.598532	−	0.801099i	$-0.295751\pi$
$347$	22.2988	1.19706	0.598532	+	0.801099i	$0.295751\pi$
$348$	−47.4009	−2.54096
$349$	−0.844306	−0.0451947	−0.0225973	−	0.999745i	$-0.507194\pi$
$349$	−0.844306	−0.0451947	−0.0225973	+	0.999745i	$0.507194\pi$
$350$	0	0
$351$	0	0
$352$	−10.8855	−0.580202
$353$	−36.3560	−1.93503	−0.967517	−	0.252806i	$-0.918646\pi$
$353$	−36.3560	−1.93503	−0.967517	+	0.252806i	$0.918646\pi$
$354$	0.229624	0.0122044
$355$	0	0
$356$	−38.5670	−2.04405
$357$	18.9220	1.00146
$358$	−48.0877	−2.54151
$359$	−13.7611	−0.726281	−0.363140	−	0.931734i	$-0.618295\pi$
$359$	−13.7611	−0.726281	−0.363140	+	0.931734i	$0.618295\pi$
$360$	0	0
$361$	3.69184	0.194307
$362$	−30.7546	−1.61643
$363$	6.04875	0.317477
$364$	0	0
$365$	0	0
$366$	−26.4731	−1.38377
$367$	−12.2882	−0.641439	−0.320720	−	0.947174i	$-0.603925\pi$
$367$	−12.2882	−0.641439	−0.320720	+	0.947174i	$0.603925\pi$
$368$	−10.4134	−0.542834
$369$	−34.6066	−1.80155
$370$	0	0
$371$	−8.79312	−0.456516
$372$	28.4326	1.47416
$373$	18.3299	0.949088	0.474544	−	0.880232i	$-0.342613\pi$
$373$	18.3299	0.949088	0.474544	+	0.880232i	$0.342613\pi$
$374$	−43.5608	−2.25248
$375$	0	0
$376$	−58.9642	−3.04084
$377$	0	0
$378$	7.00066	0.360075
$379$	23.3019	1.19694	0.598469	−	0.801146i	$-0.295776\pi$
$379$	23.3019	1.19694	0.598469	+	0.801146i	$0.295776\pi$
$380$	0	0
$381$	−20.4183	−1.04606
$382$	10.0999	0.516754
$383$	17.2654	0.882223	0.441112	−	0.897452i	$-0.354584\pi$
$383$	17.2654	0.882223	0.441112	+	0.897452i	$0.354584\pi$
$384$	39.0637	1.99346
$385$	0	0
$386$	60.4032	3.07444
$387$	3.07292	0.156206
$388$	46.3917	2.35518
$389$	1.50480	0.0762963	0.0381482	−	0.999272i	$-0.487854\pi$
$389$	1.50480	0.0762963	0.0381482	+	0.999272i	$0.487854\pi$
$390$	0	0
$391$	−9.94951	−0.503168
$392$	−32.2889	−1.63083
$393$	37.9749	1.91558
$394$	17.4638	0.879813
$395$	0	0
$396$	−49.3976	−2.48232
$397$	5.69765	0.285957	0.142978	−	0.989726i	$-0.454332\pi$
$397$	5.69765	0.285957	0.142978	+	0.989726i	$0.454332\pi$
$398$	14.5688	0.730266
$399$	−15.3575	−0.768836
$400$	0	0
$401$	−11.8092	−0.589726	−0.294863	−	0.955540i	$-0.595274\pi$
$401$	−11.8092	−0.589726	−0.294863	+	0.955540i	$0.595274\pi$
$402$	47.6330	2.37572
$403$	0	0
$404$	−56.0361	−2.78790
$405$	0	0
$406$	−12.9232	−0.641365
$407$	9.85119	0.488305
$408$	−90.5035	−4.48059
$409$	7.54827	0.373238	0.186619	−	0.982432i	$-0.440247\pi$
$409$	7.54827	0.373238	0.186619	+	0.982432i	$0.440247\pi$
$410$	0	0
$411$	−5.93917	−0.292958
$412$	−24.6003	−1.21197
$413$	0.0428489	0.00210846
$414$	−16.4843	−0.810160
$415$	0	0
$416$	0	0
$417$	29.7534	1.45703
$418$	35.3549	1.72926
$419$	4.05446	0.198073	0.0990367	−	0.995084i	$-0.468424\pi$
$419$	4.05446	0.198073	0.0990367	+	0.995084i	$0.468424\pi$
$420$	0	0
$421$	2.77948	0.135464	0.0677318	−	0.997704i	$-0.478424\pi$
$421$	2.77948	0.135464	0.0677318	+	0.997704i	$0.478424\pi$
$422$	−41.3528	−2.01302
$423$	−38.6918	−1.88126
$424$	42.0573	2.04248
$425$	0	0
$426$	93.7569	4.54253
$427$	−4.94000	−0.239063
$428$	45.3347	2.19134
$429$	0	0
$430$	0	0
$431$	32.3535	1.55841	0.779206	−	0.626768i	$-0.215623\pi$
$431$	32.3535	1.55841	0.779206	+	0.626768i	$0.215623\pi$
$432$	−13.8799	−0.667799
$433$	−17.6445	−0.847941	−0.423971	−	0.905676i	$-0.639364\pi$
$433$	−17.6445	−0.847941	−0.423971	+	0.905676i	$0.639364\pi$
$434$	7.75172	0.372095
$435$	0	0
$436$	23.3564	1.11857
$437$	8.07522	0.386290
$438$	55.7468	2.66368
$439$	20.3325	0.970418	0.485209	−	0.874398i	$-0.338743\pi$
$439$	20.3325	0.970418	0.485209	+	0.874398i	$0.338743\pi$
$440$	0	0
$441$	−21.1877	−1.00894
$442$	0	0
$443$	20.4297	0.970647	0.485323	−	0.874335i	$-0.338702\pi$
$443$	20.4297	0.970647	0.485323	+	0.874335i	$0.338702\pi$
$444$	37.9747	1.80220
$445$	0	0
$446$	−54.7833	−2.59407
$447$	5.38859	0.254871
$448$	3.67943	0.173837
$449$	15.6858	0.740260	0.370130	−	0.928980i	$-0.379313\pi$
$449$	15.6858	0.740260	0.370130	+	0.928980i	$0.379313\pi$
$450$	0	0
$451$	26.4132	1.24375
$452$	46.2940	2.17749
$453$	13.7575	0.646386
$454$	40.5691	1.90400
$455$	0	0
$456$	73.4545	3.43982
$457$	26.3611	1.23312	0.616561	−	0.787307i	$-0.288525\pi$
$457$	26.3611	1.23312	0.616561	+	0.787307i	$0.288525\pi$
$458$	31.2808	1.46166
$459$	−13.2617	−0.619002
$460$	0	0
$461$	15.3348	0.714213	0.357107	−	0.934064i	$-0.383763\pi$
$461$	15.3348	0.714213	0.357107	+	0.934064i	$0.383763\pi$
$462$	−23.9276	−1.11321
$463$	−14.3752	−0.668074	−0.334037	−	0.942560i	$-0.608411\pi$
$463$	−14.3752	−0.668074	−0.334037	+	0.942560i	$0.608411\pi$
$464$	25.6223	1.18948
$465$	0	0
$466$	−22.7983	−1.05611
$467$	7.66475	0.354683	0.177341	−	0.984149i	$-0.443250\pi$
$467$	7.66475	0.354683	0.177341	+	0.984149i	$0.443250\pi$
$468$	0	0
$469$	8.88853	0.410434
$470$	0	0
$471$	61.4362	2.83083
$472$	−0.204945	−0.00943336
$473$	−2.34539	−0.107841
$474$	−46.6059	−2.14068
$475$	0	0
$476$	−31.3346	−1.43622
$477$	27.5976	1.26361
$478$	54.0374	2.47162
$479$	−24.3174	−1.11109	−0.555545	−	0.831486i	$-0.687490\pi$
$479$	−24.3174	−1.11109	−0.555545	+	0.831486i	$0.687490\pi$
$480$	0	0
$481$	0	0
$482$	−22.8337	−1.04005
$483$	−5.46518	−0.248675
$484$	−10.0167	−0.455302
$485$	0	0
$486$	54.4494	2.46988
$487$	28.3759	1.28584	0.642918	−	0.765935i	$-0.277724\pi$
$487$	28.3759	1.28584	0.642918	+	0.765935i	$0.277724\pi$
$488$	23.6279	1.06958
$489$	18.6152	0.841807
$490$	0	0
$491$	−1.43614	−0.0648120	−0.0324060	−	0.999475i	$-0.510317\pi$
$491$	−1.43614	−0.0648120	−0.0324060	+	0.999475i	$0.510317\pi$
$492$	101.819	4.59034
$493$	24.4809	1.10257
$494$	0	0
$495$	0	0
$496$	−15.3690	−0.690091
$497$	17.4954	0.784777
$498$	23.0876	1.03458
$499$	−16.7094	−0.748017	−0.374008	−	0.927425i	$-0.622017\pi$
$499$	−16.7094	−0.748017	−0.374008	+	0.927425i	$0.622017\pi$
$500$	0	0
$501$	−53.5425	−2.39210
$502$	64.1775	2.86438
$503$	−32.2369	−1.43737	−0.718686	−	0.695335i	$-0.755256\pi$
$503$	−32.2369	−1.43737	−0.718686	+	0.695335i	$0.755256\pi$
$504$	−27.9806	−1.24635
$505$	0	0
$506$	12.5815	0.559318
$507$	0	0
$508$	33.8124	1.50018
$509$	8.16399	0.361862	0.180931	−	0.983496i	$-0.442089\pi$
$509$	8.16399	0.361862	0.180931	+	0.983496i	$0.442089\pi$
$510$	0	0
$511$	10.4026	0.460183
$512$	−49.6351	−2.19358
$513$	10.7634	0.475218
$514$	−62.6818	−2.76477
$515$	0	0
$516$	−9.04110	−0.398012
$517$	29.5312	1.29878
$518$	10.3532	0.454895
$519$	−38.6167	−1.69508
$520$	0	0
$521$	35.3357	1.54808	0.774042	−	0.633135i	$-0.218232\pi$
$521$	35.3357	1.54808	0.774042	+	0.633135i	$0.218232\pi$
$522$	40.5599	1.77526
$523$	−30.1497	−1.31835	−0.659177	−	0.751988i	$-0.729095\pi$
$523$	−30.1497	−1.31835	−0.659177	+	0.751988i	$0.729095\pi$
$524$	−62.8860	−2.74719
$525$	0	0
$526$	−39.3773	−1.71693
$527$	−14.6844	−0.639665
$528$	47.4404	2.06458
$529$	−20.1263	−0.875057
$530$	0	0
$531$	−0.134483	−0.00583608
$532$	25.4318	1.10261
$533$	0	0
$534$	58.6325	2.53728
$535$	0	0
$536$	−42.5136	−1.83631
$537$	50.0376	2.15928
$538$	59.2611	2.55493
$539$	16.1714	0.696550
$540$	0	0
$541$	−17.5295	−0.753650	−0.376825	−	0.926284i	$-0.622984\pi$
$541$	−17.5295	−0.753650	−0.376825	+	0.926284i	$0.622984\pi$
$542$	7.07314	0.303817
$543$	32.0017	1.37332
$544$	21.6719	0.929176
$545$	0	0
$546$	0	0
$547$	19.5537	0.836055	0.418028	−	0.908434i	$-0.362722\pi$
$547$	19.5537	0.836055	0.418028	+	0.908434i	$0.362722\pi$
$548$	9.83518	0.420138
$549$	15.5044	0.661713
$550$	0	0
$551$	−19.8692	−0.846457
$552$	26.1398	1.11259
$553$	−8.69686	−0.369828
$554$	−27.5910	−1.17223
$555$	0	0
$556$	−49.2712	−2.08956
$557$	21.4861	0.910393	0.455197	−	0.890391i	$-0.349569\pi$
$557$	21.4861	0.910393	0.455197	+	0.890391i	$0.349569\pi$
$558$	−24.3291	−1.02993
$559$	0	0
$560$	0	0
$561$	45.3272	1.91372
$562$	−13.7084	−0.578256
$563$	23.2468	0.979737	0.489869	−	0.871796i	$-0.337045\pi$
$563$	23.2468	0.979737	0.489869	+	0.871796i	$0.337045\pi$
$564$	113.838	4.79345
$565$	0	0
$566$	15.9371	0.669888
$567$	6.97602	0.292965
$568$	−83.6802	−3.51114
$569$	−13.9322	−0.584067	−0.292033	−	0.956408i	$-0.594332\pi$
$569$	−13.9322	−0.584067	−0.292033	+	0.956408i	$0.594332\pi$
$570$	0	0
$571$	8.49820	0.355639	0.177819	−	0.984063i	$-0.443096\pi$
$571$	8.49820	0.355639	0.177819	+	0.984063i	$0.443096\pi$
$572$	0	0
$573$	−10.5094	−0.439037
$574$	27.7593	1.15865
$575$	0	0
$576$	−11.5481	−0.481170
$577$	2.02248	0.0841967	0.0420984	−	0.999113i	$-0.486596\pi$
$577$	2.02248	0.0841967	0.0420984	+	0.999113i	$0.486596\pi$
$578$	43.9263	1.82709
$579$	−62.8526	−2.61206
$580$	0	0
$581$	4.30824	0.178736
$582$	−70.5282	−2.92349
$583$	−21.0637	−0.872370
$584$	−49.7553	−2.05889
$585$	0	0
$586$	−15.1929	−0.627615
$587$	−6.16793	−0.254578	−0.127289	−	0.991866i	$-0.540628\pi$
$587$	−6.16793	−0.254578	−0.127289	+	0.991866i	$0.540628\pi$
$588$	62.3380	2.57077
$589$	11.9182	0.491081
$590$	0	0
$591$	−18.1719	−0.747494
$592$	−20.5270	−0.843654
$593$	16.6286	0.682857	0.341428	−	0.939908i	$-0.389089\pi$
$593$	16.6286	0.682857	0.341428	+	0.939908i	$0.389089\pi$
$594$	16.7699	0.688078
$595$	0	0
$596$	−8.92343	−0.365518
$597$	−15.1595	−0.620438
$598$	0	0
$599$	26.1127	1.06694	0.533468	−	0.845820i	$-0.320888\pi$
$599$	26.1127	1.06694	0.533468	+	0.845820i	$0.320888\pi$
$600$	0	0
$601$	25.0564	1.02207	0.511036	−	0.859559i	$-0.329262\pi$
$601$	25.0564	1.02207	0.511036	+	0.859559i	$0.329262\pi$
$602$	−2.46492	−0.100463
$603$	−27.8971	−1.13606
$604$	−22.7823	−0.926999
$605$	0	0
$606$	85.1903	3.46062
$607$	−2.29492	−0.0931479	−0.0465740	−	0.998915i	$-0.514830\pi$
$607$	−2.29492	−0.0931479	−0.0465740	+	0.998915i	$0.514830\pi$
$608$	−17.5894	−0.713343
$609$	13.4472	0.544907
$610$	0	0
$611$	0	0
$612$	98.3451	3.97537
$613$	46.0296	1.85912	0.929559	−	0.368674i	$-0.120188\pi$
$613$	46.0296	1.85912	0.929559	+	0.368674i	$0.120188\pi$
$614$	−18.2321	−0.735787
$615$	0	0
$616$	21.3560	0.860457
$617$	−27.1168	−1.09168	−0.545841	−	0.837889i	$-0.683790\pi$
$617$	−27.1168	−1.09168	−0.545841	+	0.837889i	$0.683790\pi$
$618$	37.3993	1.50442
$619$	−12.8016	−0.514539	−0.257270	−	0.966340i	$-0.582823\pi$
$619$	−12.8016	−0.514539	−0.257270	+	0.966340i	$0.582823\pi$
$620$	0	0
$621$	3.83033	0.153706
$622$	17.6525	0.707802
$623$	10.9411	0.438345
$624$	0	0
$625$	0	0
$626$	−41.5443	−1.66044
$627$	−36.7885	−1.46919
$628$	−101.738	−4.05977
$629$	−19.6126	−0.782007
$630$	0	0
$631$	−27.3951	−1.09058	−0.545290	−	0.838248i	$-0.683580\pi$
$631$	−27.3951	−1.09058	−0.545290	+	0.838248i	$0.683580\pi$
$632$	41.5968	1.65463
$633$	43.0296	1.71027
$634$	29.2036	1.15982
$635$	0	0
$636$	−81.1972	−3.21968
$637$	0	0
$638$	−30.9571	−1.22560
$639$	−54.9103	−2.17222
$640$	0	0
$641$	−28.2600	−1.11620	−0.558102	−	0.829772i	$-0.688470\pi$
$641$	−28.2600	−1.11620	−0.558102	+	0.829772i	$0.688470\pi$
$642$	−68.9213	−2.72011
$643$	3.87348	0.152755	0.0763776	−	0.997079i	$-0.475665\pi$
$643$	3.87348	0.152755	0.0763776	+	0.997079i	$0.475665\pi$
$644$	9.05027	0.356631
$645$	0	0
$646$	−70.3876	−2.76936
$647$	1.15655	0.0454688	0.0227344	−	0.999742i	$-0.492763\pi$
$647$	1.15655	0.0454688	0.0227344	+	0.999742i	$0.492763\pi$
$648$	−33.3661	−1.31075
$649$	0.102643	0.00402911
$650$	0	0
$651$	−8.06605	−0.316133
$652$	−30.8265	−1.20726
$653$	−14.4727	−0.566360	−0.283180	−	0.959067i	$-0.591389\pi$
$653$	−14.4727	−0.566360	−0.283180	+	0.959067i	$0.591389\pi$
$654$	−35.5082	−1.38848
$655$	0	0
$656$	−55.0374	−2.14885
$657$	−32.6490	−1.27376
$658$	31.0363	1.20992
$659$	9.18903	0.357954	0.178977	−	0.983853i	$-0.442721\pi$
$659$	9.18903	0.357954	0.178977	+	0.983853i	$0.442721\pi$
$660$	0	0
$661$	−20.6507	−0.803219	−0.401610	−	0.915811i	$-0.631549\pi$
$661$	−20.6507	−0.803219	−0.401610	+	0.915811i	$0.631549\pi$
$662$	−7.28282	−0.283055
$663$	0	0
$664$	−20.6062	−0.799676
$665$	0	0
$666$	−32.4941	−1.25912
$667$	−7.07075	−0.273781
$668$	88.6657	3.43058
$669$	57.0048	2.20393
$670$	0	0
$671$	−11.8336	−0.456833
$672$	11.9042	0.459215
$673$	−33.5772	−1.29431	−0.647153	−	0.762360i	$-0.724040\pi$
$673$	−33.5772	−1.29431	−0.647153	+	0.762360i	$0.724040\pi$
$674$	61.9363	2.38570
$675$	0	0
$676$	0	0
$677$	−26.9980	−1.03762	−0.518809	−	0.854890i	$-0.673625\pi$
$677$	−26.9980	−1.03762	−0.518809	+	0.854890i	$0.673625\pi$
$678$	−70.3797	−2.70291
$679$	−13.1609	−0.505068
$680$	0	0
$681$	−42.2141	−1.61765
$682$	18.5691	0.711046
$683$	47.5322	1.81877	0.909385	−	0.415956i	$-0.136553\pi$
$683$	47.5322	1.81877	0.909385	+	0.415956i	$0.136553\pi$
$684$	−79.8189	−3.05195
$685$	0	0
$686$	38.6836	1.47694
$687$	−32.5492	−1.24183
$688$	4.88710	0.186319
$689$	0	0
$690$	0	0
$691$	3.65882	0.139188	0.0695941	−	0.997575i	$-0.477830\pi$
$691$	3.65882	0.139188	0.0695941	+	0.997575i	$0.477830\pi$
$692$	63.9487	2.43096
$693$	14.0136	0.532333
$694$	56.1386	2.13099
$695$	0	0
$696$	−64.3175	−2.43795
$697$	−52.5858	−1.99183
$698$	−2.12559	−0.0804547
$699$	23.7227	0.897276
$700$	0	0
$701$	22.4645	0.848471	0.424235	−	0.905552i	$-0.360543\pi$
$701$	22.4645	0.848471	0.424235	+	0.905552i	$0.360543\pi$
$702$	0	0
$703$	15.9180	0.600359
$704$	8.81399	0.332190
$705$	0	0
$706$	−91.5283	−3.44471
$707$	15.8969	0.597864
$708$	0.395674	0.0148703
$709$	29.0023	1.08920	0.544602	−	0.838695i	$-0.316681\pi$
$709$	29.0023	1.08920	0.544602	+	0.838695i	$0.316681\pi$
$710$	0	0
$711$	27.2955	1.02366
$712$	−52.3309	−1.96118
$713$	4.24126	0.158837
$714$	47.6372	1.78278
$715$	0	0
$716$	−82.8616	−3.09668
$717$	−56.2286	−2.09990
$718$	−34.6443	−1.29291
$719$	−42.1565	−1.57217	−0.786086	−	0.618118i	$-0.787896\pi$
$719$	−42.1565	−1.57217	−0.786086	+	0.618118i	$0.787896\pi$
$720$	0	0
$721$	6.97887	0.259907
$722$	9.29441	0.345902
$723$	23.7596	0.883629
$724$	−52.9944	−1.96952
$725$	0	0
$726$	15.2281	0.565167
$727$	22.1957	0.823192	0.411596	−	0.911366i	$-0.364971\pi$
$727$	22.1957	0.823192	0.411596	+	0.911366i	$0.364971\pi$
$728$	0	0
$729$	−39.6520	−1.46859
$730$	0	0
$731$	4.66941	0.172704
$732$	−45.6168	−1.68605
$733$	46.0958	1.70259	0.851293	−	0.524690i	$-0.175819\pi$
$733$	46.0958	1.70259	0.851293	+	0.524690i	$0.175819\pi$
$734$	−30.9363	−1.14188
$735$	0	0
$736$	−6.25943	−0.230726
$737$	21.2923	0.784310
$738$	−87.1240	−3.20708
$739$	41.6553	1.53232	0.766158	−	0.642652i	$-0.222166\pi$
$739$	41.6553	1.53232	0.766158	+	0.642652i	$0.222166\pi$
$740$	0	0
$741$	0	0
$742$	−22.1372	−0.812682
$743$	−37.6731	−1.38209	−0.691046	−	0.722811i	$-0.742850\pi$
$743$	−37.6731	−1.38209	−0.691046	+	0.722811i	$0.742850\pi$
$744$	38.5797	1.41440
$745$	0	0
$746$	46.1467	1.68955
$747$	−13.5216	−0.494730
$748$	−75.0613	−2.74451
$749$	−12.8610	−0.469931
$750$	0	0
$751$	20.5336	0.749282	0.374641	−	0.927170i	$-0.377766\pi$
$751$	20.5336	0.749282	0.374641	+	0.927170i	$0.377766\pi$
$752$	−61.5344	−2.24393
$753$	−66.7799	−2.43359
$754$	0	0
$755$	0	0
$756$	12.0631	0.438730
$757$	31.6073	1.14879	0.574394	−	0.818579i	$-0.305238\pi$
$757$	31.6073	1.14879	0.574394	+	0.818579i	$0.305238\pi$
$758$	58.6638	2.13077
$759$	−13.0917	−0.475199
$760$	0	0
$761$	−5.08920	−0.184483	−0.0922417	−	0.995737i	$-0.529403\pi$
$761$	−5.08920	−0.184483	−0.0922417	+	0.995737i	$0.529403\pi$
$762$	−51.4042	−1.86218
$763$	−6.62598	−0.239876
$764$	17.4035	0.629635
$765$	0	0
$766$	43.4668	1.57052
$767$	0	0
$768$	82.6808	2.98348
$769$	27.5583	0.993776	0.496888	−	0.867815i	$-0.334476\pi$
$769$	27.5583	0.993776	0.496888	+	0.867815i	$0.334476\pi$
$770$	0	0
$771$	65.2235	2.34897
$772$	104.083	3.74603
$773$	−16.0622	−0.577716	−0.288858	−	0.957372i	$-0.593275\pi$
$773$	−16.0622	−0.577716	−0.288858	+	0.957372i	$0.593275\pi$
$774$	7.73626	0.278074
$775$	0	0
$776$	62.9481	2.25970
$777$	−10.7731	−0.386481
$778$	3.78842	0.135821
$779$	42.6797	1.52916
$780$	0	0
$781$	41.9099	1.49965
$782$	−25.0484	−0.895731
$783$	−9.42459	−0.336807
$784$	−33.6963	−1.20344
$785$	0	0
$786$	95.6040	3.41008
$787$	−22.6903	−0.808821	−0.404410	−	0.914578i	$-0.632523\pi$
$787$	−22.6903	−0.808821	−0.404410	+	0.914578i	$0.632523\pi$
$788$	30.0925	1.07200
$789$	40.9740	1.45871
$790$	0	0
$791$	−13.1331	−0.466961
$792$	−67.0267	−2.38169
$793$	0	0
$794$	14.3442	0.509055
$795$	0	0
$796$	25.1040	0.889786
$797$	−48.4419	−1.71590	−0.857950	−	0.513734i	$-0.828262\pi$
$797$	−48.4419	−1.71590	−0.857950	+	0.513734i	$0.828262\pi$
$798$	−38.6633	−1.36867
$799$	−58.7934	−2.07996
$800$	0	0
$801$	−34.3391	−1.21331
$802$	−29.7305	−1.04982
$803$	24.9191	0.879378
$804$	82.0782	2.89468
$805$	0	0
$806$	0	0
$807$	−61.6641	−2.17068
$808$	−76.0344	−2.67488
$809$	13.4880	0.474214	0.237107	−	0.971484i	$-0.423801\pi$
$809$	13.4880	0.474214	0.237107	+	0.971484i	$0.423801\pi$
$810$	0	0
$811$	−8.29712	−0.291351	−0.145676	−	0.989332i	$-0.546536\pi$
$811$	−8.29712	−0.291351	−0.145676	+	0.989332i	$0.546536\pi$
$812$	−22.2684	−0.781466
$813$	−7.35995	−0.258125
$814$	24.8009	0.869272
$815$	0	0
$816$	−94.4486	−3.30636
$817$	−3.78979	−0.132588
$818$	19.0032	0.664431
$819$	0	0
$820$	0	0
$821$	−5.55527	−0.193880	−0.0969401	−	0.995290i	$-0.530906\pi$
$821$	−5.55527	−0.193880	−0.0969401	+	0.995290i	$0.530906\pi$
$822$	−14.9522	−0.521518
$823$	−36.1618	−1.26052	−0.630261	−	0.776383i	$-0.717052\pi$
$823$	−36.1618	−1.26052	−0.630261	+	0.776383i	$0.717052\pi$
$824$	−33.3798	−1.16284
$825$	0	0
$826$	0.107875	0.00375343
$827$	44.8578	1.55986	0.779930	−	0.625867i	$-0.215255\pi$
$827$	44.8578	1.55986	0.779930	+	0.625867i	$0.215255\pi$
$828$	−28.4047	−0.987132
$829$	−43.9706	−1.52716	−0.763581	−	0.645712i	$-0.776561\pi$
$829$	−43.9706	−1.52716	−0.763581	+	0.645712i	$0.776561\pi$
$830$	0	0
$831$	28.7098	0.995933
$832$	0	0
$833$	−32.1954	−1.11550
$834$	74.9058	2.59378
$835$	0	0
$836$	60.9212	2.10700
$837$	5.65317	0.195402
$838$	10.2073	0.352607
$839$	−44.9857	−1.55308	−0.776539	−	0.630069i	$-0.783027\pi$
$839$	−44.9857	−1.55308	−0.776539	+	0.630069i	$0.783027\pi$
$840$	0	0
$841$	−11.6023	−0.400079
$842$	6.99751	0.241150
$843$	14.2643	0.491289
$844$	−71.2565	−2.45275
$845$	0	0
$846$	−97.4088	−3.34898
$847$	2.84163	0.0976394
$848$	43.8906	1.50721
$849$	−16.5834	−0.569141
$850$	0	0
$851$	5.66465	0.194182
$852$	161.556	5.53481
$853$	−22.7812	−0.780012	−0.390006	−	0.920812i	$-0.627527\pi$
$853$	−22.7812	−0.780012	−0.390006	+	0.920812i	$0.627527\pi$
$854$	−12.4367	−0.425576
$855$	0	0
$856$	61.5139	2.10250
$857$	−47.5063	−1.62278	−0.811391	−	0.584503i	$-0.801289\pi$
$857$	−47.5063	−1.62278	−0.811391	+	0.584503i	$0.801289\pi$
$858$	0	0
$859$	−1.32832	−0.0453215	−0.0226608	−	0.999743i	$-0.507214\pi$
$859$	−1.32832	−0.0453215	−0.0226608	+	0.999743i	$0.507214\pi$
$860$	0	0
$861$	−28.8850	−0.984397
$862$	81.4517	2.77426
$863$	−32.7971	−1.11643	−0.558213	−	0.829698i	$-0.688513\pi$
$863$	−32.7971	−1.11643	−0.558213	+	0.829698i	$0.688513\pi$
$864$	−8.34318	−0.283841
$865$	0	0
$866$	−44.4211	−1.50949
$867$	−45.7075	−1.55231
$868$	13.3573	0.453375
$869$	−20.8331	−0.706715
$870$	0	0
$871$	0	0
$872$	31.6919	1.07322
$873$	41.3060	1.39800
$874$	20.3298	0.687667
$875$	0	0
$876$	96.0593	3.24554
$877$	−11.6053	−0.391884	−0.195942	−	0.980615i	$-0.562776\pi$
$877$	−11.6053	−0.391884	−0.195942	+	0.980615i	$0.562776\pi$
$878$	51.1883	1.72752
$879$	15.8090	0.533225
$880$	0	0
$881$	−54.9898	−1.85265	−0.926326	−	0.376723i	$-0.877051\pi$
$881$	−54.9898	−1.85265	−0.926326	+	0.376723i	$0.877051\pi$
$882$	−53.3412	−1.79609
$883$	−37.4501	−1.26030	−0.630148	−	0.776475i	$-0.717006\pi$
$883$	−37.4501	−1.26030	−0.630148	+	0.776475i	$0.717006\pi$
$884$	0	0
$885$	0	0
$886$	51.4331	1.72793
$887$	9.48028	0.318317	0.159158	−	0.987253i	$-0.449122\pi$
$887$	9.48028	0.318317	0.159158	+	0.987253i	$0.449122\pi$
$888$	51.5273	1.72914
$889$	−9.59224	−0.321713
$890$	0	0
$891$	16.7109	0.559836
$892$	−94.3992	−3.16072
$893$	47.7179	1.59682
$894$	13.5661	0.453718
$895$	0	0
$896$	18.3516	0.613084
$897$	0	0
$898$	39.4900	1.31780
$899$	−10.4357	−0.348050
$900$	0	0
$901$	41.9355	1.39707
$902$	66.4968	2.21410
$903$	2.56487	0.0853535
$904$	62.8155	2.08921
$905$	0	0
$906$	34.6354	1.15068
$907$	19.8560	0.659308	0.329654	−	0.944102i	$-0.393068\pi$
$907$	19.8560	0.659308	0.329654	+	0.944102i	$0.393068\pi$
$908$	69.9060	2.31991
$909$	−49.8931	−1.65485
$910$	0	0
$911$	34.4106	1.14007	0.570037	−	0.821619i	$-0.306929\pi$
$911$	34.4106	1.14007	0.570037	+	0.821619i	$0.306929\pi$
$912$	76.6564	2.53835
$913$	10.3203	0.341552
$914$	66.3657	2.19518
$915$	0	0
$916$	53.9011	1.78094
$917$	17.8401	0.589133
$918$	−33.3870	−1.10194
$919$	−13.1333	−0.433228	−0.216614	−	0.976257i	$-0.569501\pi$
$919$	−13.1333	−0.433228	−0.216614	+	0.976257i	$0.569501\pi$
$920$	0	0
$921$	18.9714	0.625129
$922$	38.6063	1.27143
$923$	0	0
$924$	−41.2306	−1.35639
$925$	0	0
$926$	−36.1905	−1.18929
$927$	−21.9035	−0.719406
$928$	15.4014	0.505577
$929$	−32.7415	−1.07421	−0.537107	−	0.843514i	$-0.680483\pi$
$929$	−32.7415	−1.07421	−0.537107	+	0.843514i	$0.680483\pi$
$930$	0	0
$931$	26.1304	0.856390
$932$	−39.2845	−1.28681
$933$	−18.3683	−0.601352
$934$	19.2965	0.631399
$935$	0	0
$936$	0	0
$937$	38.4788	1.25705	0.628524	−	0.777790i	$-0.283659\pi$
$937$	38.4788	1.25705	0.628524	+	0.777790i	$0.283659\pi$
$938$	22.3774	0.730647
$939$	43.2289	1.41072
$940$	0	0
$941$	2.78860	0.0909057	0.0454528	−	0.998966i	$-0.485527\pi$
$941$	2.78860	0.0909057	0.0454528	+	0.998966i	$0.485527\pi$
$942$	154.669	5.03940
$943$	15.1882	0.494596
$944$	−0.213879	−0.00696116
$945$	0	0
$946$	−5.90465	−0.191977
$947$	−23.7581	−0.772034	−0.386017	−	0.922492i	$-0.626149\pi$
$947$	−23.7581	−0.772034	−0.386017	+	0.922492i	$0.626149\pi$
$948$	−80.3083	−2.60829
$949$	0	0
$950$	0	0
$951$	−30.3878	−0.985392
$952$	−42.5174	−1.37800
$953$	−21.8702	−0.708446	−0.354223	−	0.935161i	$-0.615255\pi$
$953$	−21.8702	−0.708446	−0.354223	+	0.935161i	$0.615255\pi$
$954$	69.4786	2.24945
$955$	0	0
$956$	93.1139	3.01152
$957$	32.2124	1.04128
$958$	−61.2205	−1.97794
$959$	−2.79014	−0.0900985
$960$	0	0
$961$	−24.7403	−0.798075
$962$	0	0
$963$	40.3649	1.30074
$964$	−39.3456	−1.26724
$965$	0	0
$966$	−13.7589	−0.442686
$967$	−19.4851	−0.626600	−0.313300	−	0.949654i	$-0.601434\pi$
$967$	−19.4851	−0.626600	−0.313300	+	0.949654i	$0.601434\pi$
$968$	−13.5914	−0.436845
$969$	73.2418	2.35287
$970$	0	0
$971$	−55.2070	−1.77168	−0.885839	−	0.463993i	$-0.846416\pi$
$971$	−55.2070	−1.77168	−0.885839	+	0.463993i	$0.846416\pi$
$972$	93.8238	3.00940
$973$	13.9778	0.448106
$974$	71.4380	2.28902
$975$	0	0
$976$	24.6578	0.789278
$977$	11.0757	0.354345	0.177172	−	0.984180i	$-0.443305\pi$
$977$	11.0757	0.354345	0.177172	+	0.984180i	$0.443305\pi$
$978$	46.8647	1.49857
$979$	26.2091	0.837646
$980$	0	0
$981$	20.7959	0.663963
$982$	−3.61556	−0.115377
$983$	−17.0548	−0.543965	−0.271982	−	0.962302i	$-0.587679\pi$
$983$	−17.0548	−0.543965	−0.271982	+	0.962302i	$0.587679\pi$
$984$	138.156	4.40426
$985$	0	0
$986$	61.6321	1.96277
$987$	−32.2948	−1.02795
$988$	0	0
$989$	−1.34865	−0.0428846
$990$	0	0
$991$	56.6005	1.79797	0.898987	−	0.437975i	$-0.144304\pi$
$991$	56.6005	1.79797	0.898987	+	0.437975i	$0.144304\pi$
$992$	−9.23828	−0.293316
$993$	7.57814	0.240485
$994$	44.0458	1.39705
$995$	0	0
$996$	39.7831	1.26057
$997$	−55.2477	−1.74971	−0.874855	−	0.484384i	$-0.839044\pi$
$997$	−55.2477	−1.74971	−0.874855	+	0.484384i	$0.839044\pi$
$998$	−42.0669	−1.33161
$999$	7.55040	0.238884

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.ca.1.18		18
5.2	odd	4		845.2.b.g.339.18	yes	18
5.3	odd	4		845.2.b.g.339.1	✓	18
5.4	even	2	inner	4225.2.a.ca.1.1		18
13.12	even	2		4225.2.a.cb.1.1		18
65.2	even	12		845.2.l.g.654.1		72
65.3	odd	12		845.2.n.i.529.18		36
65.7	even	12		845.2.l.g.699.36		72
65.8	even	4		845.2.d.e.844.35		36
65.12	odd	4		845.2.b.h.339.1	yes	18
65.17	odd	12		845.2.n.h.484.1		36
65.18	even	4		845.2.d.e.844.1		36
65.22	odd	12		845.2.n.i.484.18		36
65.23	odd	12		845.2.n.h.529.1		36
65.28	even	12		845.2.l.g.654.36		72
65.32	even	12		845.2.l.g.699.2		72
65.33	even	12		845.2.l.g.699.1		72
65.37	even	12		845.2.l.g.654.35		72
65.38	odd	4		845.2.b.h.339.18	yes	18
65.42	odd	12		845.2.n.i.529.1		36
65.43	odd	12		845.2.n.h.484.18		36
65.47	even	4		845.2.d.e.844.2		36
65.48	odd	12		845.2.n.i.484.1		36
65.57	even	4		845.2.d.e.844.36		36
65.58	even	12		845.2.l.g.699.35		72
65.62	odd	12		845.2.n.h.529.18		36
65.63	even	12		845.2.l.g.654.2		72
65.64	even	2		4225.2.a.cb.1.18		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.b.g.339.1	✓	18	5.3	odd	4
845.2.b.g.339.18	yes	18	5.2	odd	4
845.2.b.h.339.1	yes	18	65.12	odd	4
845.2.b.h.339.18	yes	18	65.38	odd	4
845.2.d.e.844.1		36	65.18	even	4
845.2.d.e.844.2		36	65.47	even	4
845.2.d.e.844.35		36	65.8	even	4
845.2.d.e.844.36		36	65.57	even	4
845.2.l.g.654.1		72	65.2	even	12
845.2.l.g.654.2		72	65.63	even	12
845.2.l.g.654.35		72	65.37	even	12
845.2.l.g.654.36		72	65.28	even	12
845.2.l.g.699.1		72	65.33	even	12
845.2.l.g.699.2		72	65.32	even	12
845.2.l.g.699.35		72	65.58	even	12
845.2.l.g.699.36		72	65.7	even	12
845.2.n.h.484.1		36	65.17	odd	12
845.2.n.h.484.18		36	65.43	odd	12
845.2.n.h.529.1		36	65.23	odd	12
845.2.n.h.529.18		36	65.62	odd	12
845.2.n.i.484.1		36	65.48	odd	12
845.2.n.i.484.18		36	65.22	odd	12
845.2.n.i.529.1		36	65.42	odd	12
845.2.n.i.529.18		36	65.3	odd	12
4225.2.a.ca.1.1		18	5.4	even	2	inner
4225.2.a.ca.1.18		18	1.1	even	1	trivial
4225.2.a.cb.1.1		18	13.12	even	2
4225.2.a.cb.1.18		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.51756 q^{2} -2.61964 q^{3} +4.33809 q^{4} -6.59510 q^{6} -1.23067 q^{7} +5.88628 q^{8} +3.86253 q^{9} -2.94805 q^{11} -11.3643 q^{12} -3.09829 q^{14} +6.14287 q^{16} +5.86924 q^{17} +9.72414 q^{18} -4.76359 q^{19} +3.22393 q^{21} -7.42189 q^{22} -1.69519 q^{23} -15.4200 q^{24} -2.25952 q^{27} -5.33878 q^{28} +4.17106 q^{29} -2.50193 q^{31} +3.69246 q^{32} +7.72284 q^{33} +14.7762 q^{34} +16.7560 q^{36} -3.34159 q^{37} -11.9926 q^{38} -8.95956 q^{41} +8.11642 q^{42} +0.795573 q^{43} -12.7889 q^{44} -4.26775 q^{46} -10.0172 q^{47} -16.0921 q^{48} -5.48544 q^{49} -15.3753 q^{51} +7.14496 q^{53} -5.68847 q^{54} -7.24410 q^{56} +12.4789 q^{57} +10.5009 q^{58} -0.0348174 q^{59} +4.01406 q^{61} -6.29876 q^{62} -4.75352 q^{63} -2.98977 q^{64} +19.4427 q^{66} -7.22249 q^{67} +25.4613 q^{68} +4.44080 q^{69} -14.2161 q^{71} +22.7360 q^{72} -8.45275 q^{73} -8.41266 q^{74} -20.6649 q^{76} +3.62809 q^{77} +7.06674 q^{79} -5.66845 q^{81} -22.5562 q^{82} -3.50072 q^{83} +13.9857 q^{84} +2.00290 q^{86} -10.9267 q^{87} -17.3531 q^{88} -8.89032 q^{89} -7.35391 q^{92} +6.55417 q^{93} -25.2189 q^{94} -9.67292 q^{96} +10.6940 q^{97} -13.8099 q^{98} -11.3869 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9} - 22 q^{11} + 4 q^{14} - 12 q^{16} - 28 q^{19} - 26 q^{21} - 34 q^{24} - 20 q^{29} - 32 q^{31} - 18 q^{34} + 32 q^{36} - 52 q^{41} - 50 q^{44} - 30 q^{46} + 44 q^{49}+ \cdots - 76 q^{99}+O(q^{100}) \) 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

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