LMFDB - Embedded newform 8450.2.a.cs.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8450,2,Mod(1,8450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8450.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,8,0,8,0,0,10,8,16,0,0,0,0,10,0,8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.4735897080$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 20x^{6} + 132x^{4} - 332x^{2} + 256 $ x^8 - 20x^6 + 132x^4 - 332*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.40987$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.40987 q^{3} +1.00000 q^{4} +2.40987 q^{6} -1.40561 q^{7} +1.00000 q^{8} +2.80745 q^{9} +5.30846 q^{11} +2.40987 q^{12} -1.40561 q^{14} +1.00000 q^{16} +7.04051 q^{17} +2.80745 q^{18} +2.64187 q^{19} -3.38732 q^{21} +5.30846 q^{22} -2.66659 q^{23} +2.40987 q^{24} -0.464013 q^{27} -1.40561 q^{28} -4.21306 q^{29} +5.07645 q^{31} +1.00000 q^{32} +12.7927 q^{33} +7.04051 q^{34} +2.80745 q^{36} +0.825990 q^{37} +2.64187 q^{38} +2.81100 q^{41} -3.38732 q^{42} +5.28374 q^{43} +5.30846 q^{44} -2.66659 q^{46} -1.79070 q^{47} +2.40987 q^{48} -5.02427 q^{49} +16.9667 q^{51} -7.93952 q^{53} -0.464013 q^{54} -1.40561 q^{56} +6.36656 q^{57} -4.21306 q^{58} -6.46419 q^{59} -14.7351 q^{61} +5.07645 q^{62} -3.94617 q^{63} +1.00000 q^{64} +12.7927 q^{66} +4.00000 q^{67} +7.04051 q^{68} -6.42612 q^{69} +9.63946 q^{71} +2.80745 q^{72} +7.04281 q^{73} +0.825990 q^{74} +2.64187 q^{76} -7.46160 q^{77} -4.05956 q^{79} -9.54057 q^{81} +2.81100 q^{82} +8.55910 q^{83} -3.38732 q^{84} +5.28374 q^{86} -10.1529 q^{87} +5.30846 q^{88} -15.1600 q^{89} -2.66659 q^{92} +12.2336 q^{93} -1.79070 q^{94} +2.40987 q^{96} +4.36656 q^{97} -5.02427 q^{98} +14.9033 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} + 10 q^{7} + 8 q^{8} + 16 q^{9} + 10 q^{14} + 8 q^{16} + 16 q^{18} + 10 q^{28} - 6 q^{29} + 8 q^{32} + 20 q^{33} + 16 q^{36} + 40 q^{37} - 6 q^{47} + 30 q^{49} + 20 q^{51} + 10 q^{56}+ \cdots + 30 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 + 10 * q^7 + 8 * q^8 + 16 * q^9 + 10 * q^14 + 8 * q^16 + 16 * q^18 + 10 * q^28 - 6 * q^29 + 8 * q^32 + 20 * q^33 + 16 * q^36 + 40 * q^37 - 6 * q^47 + 30 * q^49 + 20 * q^51 + 10 * q^56 + 24 * q^57 - 6 * q^58 + 10 * q^61 + 50 * q^63 + 8 * q^64 + 20 * q^66 + 32 * q^67 + 4 * q^69 + 16 * q^72 + 26 * q^73 + 40 * q^74 - 4 * q^79 - 16 * q^81 + 48 * q^83 + 36 * q^93 - 6 * q^94 + 8 * q^97 + 30 * 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
$2$	1.00000	0.707107
$3$	2.40987	1.39134	0.695668	−	0.718363i	$-0.255108\pi$
$3$	2.40987	1.39134	0.695668	+	0.718363i	$0.255108\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.40987	0.983824
$7$	−1.40561	−0.531269	−0.265635	−	0.964074i	$-0.585581\pi$
$7$	−1.40561	−0.531269	−0.265635	+	0.964074i	$0.585581\pi$
$8$	1.00000	0.353553
$9$	2.80745	0.935818
$10$	0	0
$11$	5.30846	1.60056	0.800280	−	0.599626i	$-0.204684\pi$
$11$	5.30846	1.60056	0.800280	+	0.599626i	$0.204684\pi$
$12$	2.40987	0.695668
$13$	0	0
$13$	0	0
$14$	−1.40561	−0.375664
$15$	0	0
$16$	1.00000	0.250000
$17$	7.04051	1.70757	0.853787	−	0.520622i	$-0.174300\pi$
$17$	7.04051	1.70757	0.853787	+	0.520622i	$0.174300\pi$
$18$	2.80745	0.661723
$19$	2.64187	0.606087	0.303044	−	0.952977i	$-0.401997\pi$
$19$	2.64187	0.606087	0.303044	+	0.952977i	$0.401997\pi$
$20$	0	0
$21$	−3.38732	−0.739174
$22$	5.30846	1.13177
$23$	−2.66659	−0.556022	−0.278011	−	0.960578i	$-0.589675\pi$
$23$	−2.66659	−0.556022	−0.278011	+	0.960578i	$0.589675\pi$
$24$	2.40987	0.491912
$25$	0	0
$26$	0	0
$27$	−0.464013	−0.0892993
$28$	−1.40561	−0.265635
$29$	−4.21306	−0.782345	−0.391173	−	0.920317i	$-0.627930\pi$
$29$	−4.21306	−0.782345	−0.391173	+	0.920317i	$0.627930\pi$
$30$	0	0
$31$	5.07645	0.911758	0.455879	−	0.890042i	$-0.349325\pi$
$31$	5.07645	0.911758	0.455879	+	0.890042i	$0.349325\pi$
$32$	1.00000	0.176777
$33$	12.7927	2.22692
$34$	7.04051	1.20744
$35$	0	0
$36$	2.80745	0.467909
$37$	0.825990	0.135792	0.0678960	−	0.997692i	$-0.478371\pi$
$37$	0.825990	0.135792	0.0678960	+	0.997692i	$0.478371\pi$
$38$	2.64187	0.428568
$39$	0	0
$40$	0	0
$41$	2.81100	0.439005	0.219502	−	0.975612i	$-0.429557\pi$
$41$	2.81100	0.439005	0.219502	+	0.975612i	$0.429557\pi$
$42$	−3.38732	−0.522675
$43$	5.28374	0.805763	0.402882	−	0.915252i	$-0.368009\pi$
$43$	5.28374	0.805763	0.402882	+	0.915252i	$0.368009\pi$
$44$	5.30846	0.800280
$45$	0	0
$46$	−2.66659	−0.393167
$47$	−1.79070	−0.261200	−0.130600	−	0.991435i	$-0.541690\pi$
$47$	−1.79070	−0.261200	−0.130600	+	0.991435i	$0.541690\pi$
$48$	2.40987	0.347834
$49$	−5.02427	−0.717753
$50$	0	0
$51$	16.9667	2.37581
$52$	0	0
$53$	−7.93952	−1.09058	−0.545288	−	0.838248i	$-0.683580\pi$
$53$	−7.93952	−1.09058	−0.545288	+	0.838248i	$0.683580\pi$
$54$	−0.464013	−0.0631441
$55$	0	0
$56$	−1.40561	−0.187832
$57$	6.36656	0.843271
$58$	−4.21306	−0.553202
$59$	−6.46419	−0.841566	−0.420783	−	0.907161i	$-0.638245\pi$
$59$	−6.46419	−0.841566	−0.420783	+	0.907161i	$0.638245\pi$
$60$	0	0
$61$	−14.7351	−1.88663	−0.943317	−	0.331892i	$-0.892313\pi$
$61$	−14.7351	−1.88663	−0.943317	+	0.331892i	$0.892313\pi$
$62$	5.07645	0.644710
$63$	−3.94617	−0.497171
$64$	1.00000	0.125000
$65$	0	0
$66$	12.7927	1.57467
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	7.04051	0.853787
$69$	−6.42612	−0.773614
$70$	0	0
$71$	9.63946	1.14399	0.571997	−	0.820256i	$-0.306169\pi$
$71$	9.63946	1.14399	0.571997	+	0.820256i	$0.306169\pi$
$72$	2.80745	0.330861
$73$	7.04281	0.824298	0.412149	−	0.911116i	$-0.364778\pi$
$73$	7.04281	0.824298	0.412149	+	0.911116i	$0.364778\pi$
$74$	0.825990	0.0960194
$75$	0	0
$76$	2.64187	0.303044
$77$	−7.46160	−0.850329
$78$	0	0
$79$	−4.05956	−0.456736	−0.228368	−	0.973575i	$-0.573339\pi$
$79$	−4.05956	−0.456736	−0.228368	+	0.973575i	$0.573339\pi$
$80$	0	0
$81$	−9.54057	−1.06006
$82$	2.81100	0.310423
$83$	8.55910	0.939484	0.469742	−	0.882804i	$-0.344347\pi$
$83$	8.55910	0.939484	0.469742	+	0.882804i	$0.344347\pi$
$84$	−3.38732	−0.369587
$85$	0	0
$86$	5.28374	0.569761
$87$	−10.1529	−1.08851
$88$	5.30846	0.565884
$89$	−15.1600	−1.60695	−0.803477	−	0.595336i	$-0.797019\pi$
$89$	−15.1600	−1.60695	−0.803477	+	0.595336i	$0.797019\pi$
$90$	0	0
$91$	0	0
$92$	−2.66659	−0.278011
$93$	12.2336	1.26856
$94$	−1.79070	−0.184697
$95$	0	0
$96$	2.40987	0.245956
$97$	4.36656	0.443357	0.221678	−	0.975120i	$-0.428847\pi$
$97$	4.36656	0.443357	0.221678	+	0.975120i	$0.428847\pi$
$98$	−5.02427	−0.507528
$99$	14.9033	1.49783
$100$	0	0
$101$	8.38707	0.834545	0.417272	−	0.908782i	$-0.362986\pi$
$101$	8.38707	0.834545	0.417272	+	0.908782i	$0.362986\pi$
$102$	16.9667	1.67995
$103$	5.56518	0.548354	0.274177	−	0.961679i	$-0.411595\pi$
$103$	5.56518	0.548354	0.274177	+	0.961679i	$0.411595\pi$
$104$	0	0
$105$	0	0
$106$	−7.93952	−0.771154
$107$	15.1847	1.46796	0.733980	−	0.679172i	$-0.237661\pi$
$107$	15.1847	1.46796	0.733980	+	0.679172i	$0.237661\pi$
$108$	−0.464013	−0.0446496
$109$	0.667003	0.0638873	0.0319436	−	0.999490i	$-0.489830\pi$
$109$	0.667003	0.0638873	0.0319436	+	0.999490i	$0.489830\pi$
$110$	0	0
$111$	1.99053	0.188932
$112$	−1.40561	−0.132817
$113$	3.36912	0.316940	0.158470	−	0.987364i	$-0.449344\pi$
$113$	3.36912	0.316940	0.158470	+	0.987364i	$0.449344\pi$
$114$	6.36656	0.596283
$115$	0	0
$116$	−4.21306	−0.391173
$117$	0	0
$118$	−6.46419	−0.595077
$119$	−9.89618	−0.907182
$120$	0	0
$121$	17.1797	1.56179
$122$	−14.7351	−1.33405
$123$	6.77414	0.610803
$124$	5.07645	0.455879
$125$	0	0
$126$	−3.94617	−0.351553
$127$	−10.8489	−0.962686	−0.481343	−	0.876532i	$-0.659851\pi$
$127$	−10.8489	−0.962686	−0.481343	+	0.876532i	$0.659851\pi$
$128$	1.00000	0.0883883
$129$	12.7331	1.12109
$130$	0	0
$131$	15.3500	1.34114	0.670568	−	0.741848i	$-0.266051\pi$
$131$	15.3500	1.34114	0.670568	+	0.741848i	$0.266051\pi$
$132$	12.7927	1.11346
$133$	−3.71343	−0.321995
$134$	4.00000	0.345547
$135$	0	0
$136$	7.04051	0.603719
$137$	11.3908	0.973184	0.486592	−	0.873629i	$-0.338240\pi$
$137$	11.3908	0.973184	0.486592	+	0.873629i	$0.338240\pi$
$138$	−6.42612	−0.547028
$139$	−11.3500	−0.962694	−0.481347	−	0.876530i	$-0.659852\pi$
$139$	−11.3500	−0.962694	−0.481347	+	0.876530i	$0.659852\pi$
$140$	0	0
$141$	−4.31535	−0.363418
$142$	9.63946	0.808926
$143$	0	0
$144$	2.80745	0.233954
$145$	0	0
$146$	7.04281	0.582867
$147$	−12.1078	−0.998636
$148$	0.825990	0.0678960
$149$	1.96406	0.160902	0.0804509	−	0.996759i	$-0.474364\pi$
$149$	1.96406	0.160902	0.0804509	+	0.996759i	$0.474364\pi$
$150$	0	0
$151$	10.3602	0.843101	0.421550	−	0.906805i	$-0.361486\pi$
$151$	10.3602	0.843101	0.421550	+	0.906805i	$0.361486\pi$
$152$	2.64187	0.214284
$153$	19.7659	1.59798
$154$	−7.46160	−0.601273
$155$	0	0
$156$	0	0
$157$	−5.83105	−0.465368	−0.232684	−	0.972552i	$-0.574751\pi$
$157$	−5.83105	−0.465368	−0.232684	+	0.972552i	$0.574751\pi$
$158$	−4.05956	−0.322961
$159$	−19.1332	−1.51736
$160$	0	0
$161$	3.74817	0.295397
$162$	−9.54057	−0.749578
$163$	−11.7518	−0.920475	−0.460238	−	0.887796i	$-0.652236\pi$
$163$	−11.7518	−0.920475	−0.460238	+	0.887796i	$0.652236\pi$
$164$	2.81100	0.219502
$165$	0	0
$166$	8.55910	0.664315
$167$	−11.1760	−0.864824	−0.432412	−	0.901676i	$-0.642337\pi$
$167$	−11.1760	−0.864824	−0.432412	+	0.901676i	$0.642337\pi$
$168$	−3.38732	−0.261338
$169$	0	0
$170$	0	0
$171$	7.41693	0.567187
$172$	5.28374	0.402882
$173$	−0.533400	−0.0405537	−0.0202768	−	0.999794i	$-0.506455\pi$
$173$	−0.533400	−0.0405537	−0.0202768	+	0.999794i	$0.506455\pi$
$174$	−10.1529	−0.769690
$175$	0	0
$176$	5.30846	0.400140
$177$	−15.5778	−1.17090
$178$	−15.1600	−1.13629
$179$	9.23733	0.690430	0.345215	−	0.938524i	$-0.387806\pi$
$179$	9.23733	0.690430	0.345215	+	0.938524i	$0.387806\pi$
$180$	0	0
$181$	4.76464	0.354153	0.177077	−	0.984197i	$-0.443336\pi$
$181$	4.76464	0.354153	0.177077	+	0.984197i	$0.443336\pi$
$182$	0	0
$183$	−35.5096	−2.62494
$184$	−2.66659	−0.196583
$185$	0	0
$186$	12.2336	0.897009
$187$	37.3743	2.73308
$188$	−1.79070	−0.130600
$189$	0.652219	0.0474420
$190$	0	0
$191$	−17.9444	−1.29841	−0.649205	−	0.760613i	$-0.724898\pi$
$191$	−17.9444	−1.29841	−0.649205	+	0.760613i	$0.724898\pi$
$192$	2.40987	0.173917
$193$	20.9462	1.50774	0.753869	−	0.657024i	$-0.228185\pi$
$193$	20.9462	1.50774	0.753869	+	0.657024i	$0.228185\pi$
$194$	4.36656	0.313501
$195$	0	0
$196$	−5.02427	−0.358877
$197$	−26.7724	−1.90745	−0.953726	−	0.300678i	$-0.902787\pi$
$197$	−26.7724	−1.90745	−0.953726	+	0.300678i	$0.902787\pi$
$198$	14.9033	1.05913
$199$	14.3890	1.02001	0.510006	−	0.860171i	$-0.329643\pi$
$199$	14.3890	1.02001	0.510006	+	0.860171i	$0.329643\pi$
$200$	0	0
$201$	9.63946	0.679915
$202$	8.38707	0.590112
$203$	5.92190	0.415636
$204$	16.9667	1.18791
$205$	0	0
$206$	5.56518	0.387745
$207$	−7.48632	−0.520335
$208$	0	0
$209$	14.0243	0.970079
$210$	0	0
$211$	−24.3908	−1.67913	−0.839567	−	0.543256i	$-0.817191\pi$
$211$	−24.3908	−1.67913	−0.839567	+	0.543256i	$0.817191\pi$
$212$	−7.93952	−0.545288
$213$	23.2298	1.59168
$214$	15.1847	1.03800
$215$	0	0
$216$	−0.464013	−0.0315721
$217$	−7.13549	−0.484389
$218$	0.667003	0.0451751
$219$	16.9722	1.14688
$220$	0	0
$221$	0	0
$222$	1.99053	0.133595
$223$	2.97949	0.199521	0.0997606	−	0.995011i	$-0.468192\pi$
$223$	2.97949	0.199521	0.0997606	+	0.995011i	$0.468192\pi$
$224$	−1.40561	−0.0939160
$225$	0	0
$226$	3.36912	0.224110
$227$	3.96293	0.263029	0.131514	−	0.991314i	$-0.458016\pi$
$227$	3.96293	0.263029	0.131514	+	0.991314i	$0.458016\pi$
$228$	6.36656	0.421636
$229$	−9.85151	−0.651006	−0.325503	−	0.945541i	$-0.605534\pi$
$229$	−9.85151	−0.651006	−0.325503	+	0.945541i	$0.605534\pi$
$230$	0	0
$231$	−17.9815	−1.18309
$232$	−4.21306	−0.276601
$233$	−19.3821	−1.26976	−0.634881	−	0.772610i	$-0.718951\pi$
$233$	−19.3821	−1.26976	−0.634881	+	0.772610i	$0.718951\pi$
$234$	0	0
$235$	0	0
$236$	−6.46419	−0.420783
$237$	−9.78300	−0.635474
$238$	−9.89618	−0.641474
$239$	30.0138	1.94143	0.970715	−	0.240232i	$-0.0772237\pi$
$239$	30.0138	1.94143	0.970715	+	0.240232i	$0.0772237\pi$
$240$	0	0
$241$	−19.1467	−1.23334	−0.616672	−	0.787220i	$-0.711520\pi$
$241$	−19.1467	−1.23334	−0.616672	+	0.787220i	$0.711520\pi$
$242$	17.1797	1.10436
$243$	−21.5994	−1.38561
$244$	−14.7351	−0.943317
$245$	0	0
$246$	6.77414	0.431903
$247$	0	0
$248$	5.07645	0.322355
$249$	20.6263	1.30714
$250$	0	0
$251$	−17.0430	−1.07574	−0.537872	−	0.843026i	$-0.680772\pi$
$251$	−17.0430	−1.07574	−0.537872	+	0.843026i	$0.680772\pi$
$252$	−3.94617	−0.248586
$253$	−14.1555	−0.889947
$254$	−10.8489	−0.680722
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.22526	0.513077	0.256539	−	0.966534i	$-0.417418\pi$
$257$	8.22526	0.513077	0.256539	+	0.966534i	$0.417418\pi$
$258$	12.7331	0.792729
$259$	−1.16102	−0.0721421
$260$	0	0
$261$	−11.8280	−0.732133
$262$	15.3500	0.948326
$263$	−10.4296	−0.643116	−0.321558	−	0.946890i	$-0.604206\pi$
$263$	−10.4296	−0.643116	−0.321558	+	0.946890i	$0.604206\pi$
$264$	12.7927	0.787335
$265$	0	0
$266$	−3.71343	−0.227685
$267$	−36.5335	−2.23581
$268$	4.00000	0.244339
$269$	22.6412	1.38046	0.690228	−	0.723592i	$-0.257510\pi$
$269$	22.6412	1.38046	0.690228	+	0.723592i	$0.257510\pi$
$270$	0	0
$271$	−0.252421	−0.0153335	−0.00766675	−	0.999971i	$-0.502440\pi$
$271$	−0.252421	−0.0153335	−0.00766675	+	0.999971i	$0.502440\pi$
$272$	7.04051	0.426894
$273$	0	0
$274$	11.3908	0.688145
$275$	0	0
$276$	−6.42612	−0.386807
$277$	4.65536	0.279713	0.139857	−	0.990172i	$-0.455336\pi$
$277$	4.65536	0.279713	0.139857	+	0.990172i	$0.455336\pi$
$278$	−11.3500	−0.680727
$279$	14.2519	0.853239
$280$	0	0
$281$	3.70262	0.220880	0.110440	−	0.993883i	$-0.464774\pi$
$281$	3.70262	0.220880	0.110440	+	0.993883i	$0.464774\pi$
$282$	−4.31535	−0.256975
$283$	5.34796	0.317903	0.158952	−	0.987286i	$-0.449189\pi$
$283$	5.34796	0.317903	0.158952	+	0.987286i	$0.449189\pi$
$284$	9.63946	0.571997
$285$	0	0
$286$	0	0
$287$	−3.95116	−0.233230
$288$	2.80745	0.165431
$289$	32.5688	1.91581
$290$	0	0
$291$	10.5228	0.616858
$292$	7.04281	0.412149
$293$	24.1742	1.41227	0.706136	−	0.708076i	$-0.250437\pi$
$293$	24.1742	1.41227	0.706136	+	0.708076i	$0.250437\pi$
$294$	−12.1078	−0.706142
$295$	0	0
$296$	0.825990	0.0480097
$297$	−2.46319	−0.142929
$298$	1.96406	0.113775
$299$	0	0
$300$	0	0
$301$	−7.42686	−0.428077
$302$	10.3602	0.596162
$303$	20.2117	1.16113
$304$	2.64187	0.151522
$305$	0	0
$306$	19.7659	1.12994
$307$	27.5443	1.57204	0.786019	−	0.618203i	$-0.212139\pi$
$307$	27.5443	1.57204	0.786019	+	0.618203i	$0.212139\pi$
$308$	−7.46160	−0.425164
$309$	13.4113	0.762944
$310$	0	0
$311$	−12.8262	−0.727306	−0.363653	−	0.931534i	$-0.618471\pi$
$311$	−12.8262	−0.727306	−0.363653	+	0.931534i	$0.618471\pi$
$312$	0	0
$313$	−2.72604	−0.154085	−0.0770425	−	0.997028i	$-0.524548\pi$
$313$	−2.72604	−0.154085	−0.0770425	+	0.997028i	$0.524548\pi$
$314$	−5.83105	−0.329065
$315$	0	0
$316$	−4.05956	−0.228368
$317$	1.89961	0.106692	0.0533462	−	0.998576i	$-0.483011\pi$
$317$	1.89961	0.106692	0.0533462	+	0.998576i	$0.483011\pi$
$318$	−19.1332	−1.07294
$319$	−22.3649	−1.25219
$320$	0	0
$321$	36.5931	2.04243
$322$	3.74817	0.208877
$323$	18.6001	1.03494
$324$	−9.54057	−0.530031
$325$	0	0
$326$	−11.7518	−0.650874
$327$	1.60739	0.0888887
$328$	2.81100	0.155212
$329$	2.51702	0.138768
$330$	0	0
$331$	−8.98724	−0.493984	−0.246992	−	0.969018i	$-0.579442\pi$
$331$	−8.98724	−0.493984	−0.246992	+	0.969018i	$0.579442\pi$
$332$	8.55910	0.469742
$333$	2.31893	0.127076
$334$	−11.1760	−0.611523
$335$	0	0
$336$	−3.38732	−0.184794
$337$	1.37859	0.0750967	0.0375484	−	0.999295i	$-0.488045\pi$
$337$	1.37859	0.0750967	0.0375484	+	0.999295i	$0.488045\pi$
$338$	0	0
$339$	8.11912	0.440970
$340$	0	0
$341$	26.9481	1.45932
$342$	7.41693	0.401062
$343$	16.9014	0.912589
$344$	5.28374	0.284880
$345$	0	0
$346$	−0.533400	−0.0286758
$347$	11.8958	0.638598	0.319299	−	0.947654i	$-0.396553\pi$
$347$	11.8958	0.638598	0.319299	+	0.947654i	$0.396553\pi$
$348$	−10.1529	−0.544253
$349$	4.56777	0.244507	0.122254	−	0.992499i	$-0.460988\pi$
$349$	4.56777	0.244507	0.122254	+	0.992499i	$0.460988\pi$
$350$	0	0
$351$	0	0
$352$	5.30846	0.282942
$353$	−0.842019	−0.0448161	−0.0224081	−	0.999749i	$-0.507133\pi$
$353$	−0.842019	−0.0448161	−0.0224081	+	0.999749i	$0.507133\pi$
$354$	−15.5778	−0.827952
$355$	0	0
$356$	−15.1600	−0.803477
$357$	−23.8485	−1.26220
$358$	9.23733	0.488208
$359$	8.05922	0.425349	0.212675	−	0.977123i	$-0.431783\pi$
$359$	8.05922	0.425349	0.212675	+	0.977123i	$0.431783\pi$
$360$	0	0
$361$	−12.0205	−0.632659
$362$	4.76464	0.250424
$363$	41.4009	2.17298
$364$	0	0
$365$	0	0
$366$	−35.5096	−1.85612
$367$	−16.5182	−0.862245	−0.431122	−	0.902293i	$-0.641882\pi$
$367$	−16.5182	−0.862245	−0.431122	+	0.902293i	$0.641882\pi$
$368$	−2.66659	−0.139005
$369$	7.89176	0.410828
$370$	0	0
$371$	11.1598	0.579390
$372$	12.2336	0.634281
$373$	−26.5545	−1.37494	−0.687471	−	0.726212i	$-0.741279\pi$
$373$	−26.5545	−1.37494	−0.687471	+	0.726212i	$0.741279\pi$
$374$	37.3743	1.93258
$375$	0	0
$376$	−1.79070	−0.0923483
$377$	0	0
$378$	0.652219	0.0335465
$379$	1.15573	0.0593659	0.0296830	−	0.999559i	$-0.490550\pi$
$379$	1.15573	0.0593659	0.0296830	+	0.999559i	$0.490550\pi$
$380$	0	0
$381$	−26.1445	−1.33942
$382$	−17.9444	−0.918115
$383$	−35.0077	−1.78881	−0.894405	−	0.447258i	$-0.852401\pi$
$383$	−35.0077	−1.78881	−0.894405	+	0.447258i	$0.852401\pi$
$384$	2.40987	0.122978
$385$	0	0
$386$	20.9462	1.06613
$387$	14.8339	0.754048
$388$	4.36656	0.221678
$389$	11.3135	0.573615	0.286807	−	0.957988i	$-0.407406\pi$
$389$	11.3135	0.573615	0.286807	+	0.957988i	$0.407406\pi$
$390$	0	0
$391$	−18.7741	−0.949449
$392$	−5.02427	−0.253764
$393$	36.9914	1.86597
$394$	−26.7724	−1.34877
$395$	0	0
$396$	14.9033	0.748917
$397$	18.0057	0.903682	0.451841	−	0.892099i	$-0.350767\pi$
$397$	18.0057	0.903682	0.451841	+	0.892099i	$0.350767\pi$
$398$	14.3890	0.721258
$399$	−8.94887	−0.448004
$400$	0	0
$401$	−4.58861	−0.229144	−0.114572	−	0.993415i	$-0.536550\pi$
$401$	−4.58861	−0.229144	−0.114572	+	0.993415i	$0.536550\pi$
$402$	9.63946	0.480773
$403$	0	0
$404$	8.38707	0.417272
$405$	0	0
$406$	5.92190	0.293899
$407$	4.38474	0.217343
$408$	16.9667	0.839976
$409$	−26.2500	−1.29798	−0.648991	−	0.760796i	$-0.724809\pi$
$409$	−26.2500	−1.29798	−0.648991	+	0.760796i	$0.724809\pi$
$410$	0	0
$411$	27.4504	1.35403
$412$	5.56518	0.274177
$413$	9.08611	0.447098
$414$	−7.48632	−0.367933
$415$	0	0
$416$	0	0
$417$	−27.3520	−1.33943
$418$	14.0243	0.685950
$419$	12.0774	0.590018	0.295009	−	0.955494i	$-0.404677\pi$
$419$	12.0774	0.590018	0.295009	+	0.955494i	$0.404677\pi$
$420$	0	0
$421$	3.11716	0.151921	0.0759604	−	0.997111i	$-0.475798\pi$
$421$	3.11716	0.151921	0.0759604	+	0.997111i	$0.475798\pi$
$422$	−24.3908	−1.18733
$423$	−5.02731	−0.244436
$424$	−7.93952	−0.385577
$425$	0	0
$426$	23.2298	1.12549
$427$	20.7117	1.00231
$428$	15.1847	0.733980
$429$	0	0
$430$	0	0
$431$	5.23001	0.251921	0.125960	−	0.992035i	$-0.459799\pi$
$431$	5.23001	0.251921	0.125960	+	0.992035i	$0.459799\pi$
$432$	−0.464013	−0.0223248
$433$	−22.3063	−1.07197	−0.535986	−	0.844227i	$-0.680060\pi$
$433$	−22.3063	−1.07197	−0.535986	+	0.844227i	$0.680060\pi$
$434$	−7.13549	−0.342515
$435$	0	0
$436$	0.667003	0.0319436
$437$	−7.04478	−0.336998
$438$	16.9722	0.810964
$439$	−5.24835	−0.250490	−0.125245	−	0.992126i	$-0.539972\pi$
$439$	−5.24835	−0.250490	−0.125245	+	0.992126i	$0.539972\pi$
$440$	0	0
$441$	−14.1054	−0.671686
$442$	0	0
$443$	1.11623	0.0530338	0.0265169	−	0.999648i	$-0.491558\pi$
$443$	1.11623	0.0530338	0.0265169	+	0.999648i	$0.491558\pi$
$444$	1.99053	0.0944661
$445$	0	0
$446$	2.97949	0.141083
$447$	4.73311	0.223869
$448$	−1.40561	−0.0664086
$449$	−34.4883	−1.62760	−0.813802	−	0.581142i	$-0.802606\pi$
$449$	−34.4883	−1.62760	−0.813802	+	0.581142i	$0.802606\pi$
$450$	0	0
$451$	14.9221	0.702654
$452$	3.36912	0.158470
$453$	24.9667	1.17304
$454$	3.96293	0.185989
$455$	0	0
$456$	6.36656	0.298141
$457$	−4.07540	−0.190639	−0.0953196	−	0.995447i	$-0.530387\pi$
$457$	−4.07540	−0.190639	−0.0953196	+	0.995447i	$0.530387\pi$
$458$	−9.85151	−0.460331
$459$	−3.26689	−0.152485
$460$	0	0
$461$	−30.3417	−1.41315	−0.706576	−	0.707637i	$-0.749761\pi$
$461$	−30.3417	−1.41315	−0.706576	+	0.707637i	$0.749761\pi$
$462$	−17.9815	−0.836573
$463$	25.2291	1.17250	0.586248	−	0.810132i	$-0.300605\pi$
$463$	25.2291	1.17250	0.586248	+	0.810132i	$0.300605\pi$
$464$	−4.21306	−0.195586
$465$	0	0
$466$	−19.3821	−0.897858
$467$	−21.9997	−1.01802	−0.509012	−	0.860759i	$-0.669989\pi$
$467$	−21.9997	−1.01802	−0.509012	+	0.860759i	$0.669989\pi$
$468$	0	0
$469$	−5.62242	−0.259619
$470$	0	0
$471$	−14.0520	−0.647484
$472$	−6.46419	−0.297538
$473$	28.0485	1.28967
$474$	−9.78300	−0.449348
$475$	0	0
$476$	−9.89618	−0.453591
$477$	−22.2898	−1.02058
$478$	30.0138	1.37280
$479$	−22.1081	−1.01015	−0.505073	−	0.863077i	$-0.668534\pi$
$479$	−22.1081	−1.01015	−0.505073	+	0.863077i	$0.668534\pi$
$480$	0	0
$481$	0	0
$482$	−19.1467	−0.872107
$483$	9.03259	0.410997
$484$	17.1797	0.780897
$485$	0	0
$486$	−21.5994	−0.979771
$487$	3.66873	0.166246	0.0831231	−	0.996539i	$-0.473511\pi$
$487$	3.66873	0.166246	0.0831231	+	0.996539i	$0.473511\pi$
$488$	−14.7351	−0.667026
$489$	−28.3204	−1.28069
$490$	0	0
$491$	−26.4689	−1.19453	−0.597263	−	0.802045i	$-0.703745\pi$
$491$	−26.4689	−1.19453	−0.597263	+	0.802045i	$0.703745\pi$
$492$	6.77414	0.305402
$493$	−29.6621	−1.33591
$494$	0	0
$495$	0	0
$496$	5.07645	0.227939
$497$	−13.5493	−0.607769
$498$	20.6263	0.924286
$499$	−37.0404	−1.65816	−0.829078	−	0.559133i	$-0.811134\pi$
$499$	−37.0404	−1.65816	−0.829078	+	0.559133i	$0.811134\pi$
$500$	0	0
$501$	−26.9326	−1.20326
$502$	−17.0430	−0.760666
$503$	34.0196	1.51686	0.758429	−	0.651756i	$-0.225967\pi$
$503$	34.0196	1.51686	0.758429	+	0.651756i	$0.225967\pi$
$504$	−3.94617	−0.175777
$505$	0	0
$506$	−14.1555	−0.629288
$507$	0	0
$508$	−10.8489	−0.481343
$509$	16.1439	0.715568	0.357784	−	0.933804i	$-0.383532\pi$
$509$	16.1439	0.715568	0.357784	+	0.933804i	$0.383532\pi$
$510$	0	0
$511$	−9.89942	−0.437924
$512$	1.00000	0.0441942
$513$	−1.22586	−0.0541231
$514$	8.22526	0.362801
$515$	0	0
$516$	12.7331	0.560544
$517$	−9.50586	−0.418067
$518$	−1.16102	−0.0510121
$519$	−1.28542	−0.0564238
$520$	0	0
$521$	−36.5483	−1.60121	−0.800605	−	0.599193i	$-0.795488\pi$
$521$	−36.5483	−1.60121	−0.800605	+	0.599193i	$0.795488\pi$
$522$	−11.8280	−0.517696
$523$	0.667003	0.0291660	0.0145830	−	0.999894i	$-0.495358\pi$
$523$	0.667003	0.0291660	0.0145830	+	0.999894i	$0.495358\pi$
$524$	15.3500	0.670568
$525$	0	0
$526$	−10.4296	−0.454752
$527$	35.7408	1.55689
$528$	12.7927	0.556730
$529$	−15.8893	−0.690840
$530$	0	0
$531$	−18.1479	−0.787552
$532$	−3.71343	−0.160998
$533$	0	0
$534$	−36.5335	−1.58096
$535$	0	0
$536$	4.00000	0.172774
$537$	22.2607	0.960621
$538$	22.6412	0.976129
$539$	−26.6711	−1.14881
$540$	0	0
$541$	−7.70275	−0.331167	−0.165584	−	0.986196i	$-0.552951\pi$
$541$	−7.70275	−0.331167	−0.165584	+	0.986196i	$0.552951\pi$
$542$	−0.252421	−0.0108424
$543$	11.4822	0.492746
$544$	7.04051	0.301859
$545$	0	0
$546$	0	0
$547$	−27.5445	−1.17772	−0.588858	−	0.808236i	$-0.700422\pi$
$547$	−27.5445	−1.17772	−0.588858	+	0.808236i	$0.700422\pi$
$548$	11.3908	0.486592
$549$	−41.3681	−1.76555
$550$	0	0
$551$	−11.1304	−0.474169
$552$	−6.42612	−0.273514
$553$	5.70614	0.242650
$554$	4.65536	0.197787
$555$	0	0
$556$	−11.3500	−0.481347
$557$	−4.16273	−0.176381	−0.0881903	−	0.996104i	$-0.528108\pi$
$557$	−4.16273	−0.176381	−0.0881903	+	0.996104i	$0.528108\pi$
$558$	14.2519	0.603331
$559$	0	0
$560$	0	0
$561$	90.0670	3.80263
$562$	3.70262	0.156186
$563$	31.5897	1.33135	0.665674	−	0.746242i	$-0.268144\pi$
$563$	31.5897	1.33135	0.665674	+	0.746242i	$0.268144\pi$
$564$	−4.31535	−0.181709
$565$	0	0
$566$	5.34796	0.224791
$567$	13.4103	0.563179
$568$	9.63946	0.404463
$569$	−23.9166	−1.00264	−0.501318	−	0.865263i	$-0.667151\pi$
$569$	−23.9166	−1.00264	−0.501318	+	0.865263i	$0.667151\pi$
$570$	0	0
$571$	−0.653231	−0.0273369	−0.0136684	−	0.999907i	$-0.504351\pi$
$571$	−0.653231	−0.0273369	−0.0136684	+	0.999907i	$0.504351\pi$
$572$	0	0
$573$	−43.2436	−1.80653
$574$	−3.95116	−0.164918
$575$	0	0
$576$	2.80745	0.116977
$577$	−21.3167	−0.887425	−0.443712	−	0.896169i	$-0.646339\pi$
$577$	−21.3167	−0.887425	−0.443712	+	0.896169i	$0.646339\pi$
$578$	32.5688	1.35468
$579$	50.4775	2.09777
$580$	0	0
$581$	−12.0307	−0.499119
$582$	10.5228	0.436185
$583$	−42.1466	−1.74553
$584$	7.04281	0.291433
$585$	0	0
$586$	24.1742	0.998627
$587$	15.2298	0.628602	0.314301	−	0.949323i	$-0.398230\pi$
$587$	15.2298	0.628602	0.314301	+	0.949323i	$0.398230\pi$
$588$	−12.1078	−0.499318
$589$	13.4113	0.552605
$590$	0	0
$591$	−64.5178	−2.65391
$592$	0.825990	0.0339480
$593$	−9.70527	−0.398548	−0.199274	−	0.979944i	$-0.563858\pi$
$593$	−9.70527	−0.398548	−0.199274	+	0.979944i	$0.563858\pi$
$594$	−2.46319	−0.101066
$595$	0	0
$596$	1.96406	0.0804509
$597$	34.6757	1.41918
$598$	0	0
$599$	1.20026	0.0490411	0.0245206	−	0.999699i	$-0.492194\pi$
$599$	1.20026	0.0490411	0.0245206	+	0.999699i	$0.492194\pi$
$600$	0	0
$601$	14.0736	0.574075	0.287037	−	0.957919i	$-0.407330\pi$
$601$	14.0736	0.574075	0.287037	+	0.957919i	$0.407330\pi$
$602$	−7.42686	−0.302696
$603$	11.2298	0.457313
$604$	10.3602	0.421550
$605$	0	0
$606$	20.2117	0.821045
$607$	−31.5294	−1.27974	−0.639870	−	0.768484i	$-0.721012\pi$
$607$	−31.5294	−1.27974	−0.639870	+	0.768484i	$0.721012\pi$
$608$	2.64187	0.107142
$609$	14.2710	0.578290
$610$	0	0
$611$	0	0
$612$	19.7659	0.798989
$613$	−2.83351	−0.114444	−0.0572222	−	0.998361i	$-0.518224\pi$
$613$	−2.83351	−0.114444	−0.0572222	+	0.998361i	$0.518224\pi$
$614$	27.5443	1.11160
$615$	0	0
$616$	−7.46160	−0.300637
$617$	2.25408	0.0907460	0.0453730	−	0.998970i	$-0.485552\pi$
$617$	2.25408	0.0907460	0.0453730	+	0.998970i	$0.485552\pi$
$618$	13.4113	0.539483
$619$	−6.37526	−0.256243	−0.128122	−	0.991758i	$-0.540895\pi$
$619$	−6.37526	−0.256243	−0.128122	+	0.991758i	$0.540895\pi$
$620$	0	0
$621$	1.23733	0.0496524
$622$	−12.8262	−0.514283
$623$	21.3090	0.853725
$624$	0	0
$625$	0	0
$626$	−2.72604	−0.108955
$627$	33.7966	1.34971
$628$	−5.83105	−0.232684
$629$	5.81539	0.231875
$630$	0	0
$631$	−30.8334	−1.22746	−0.613729	−	0.789517i	$-0.710331\pi$
$631$	−30.8334	−1.22746	−0.613729	+	0.789517i	$0.710331\pi$
$632$	−4.05956	−0.161481
$633$	−58.7786	−2.33624
$634$	1.89961	0.0754430
$635$	0	0
$636$	−19.1332	−0.758680
$637$	0	0
$638$	−22.3649	−0.885433
$639$	27.0623	1.07057
$640$	0	0
$641$	19.1890	0.757919	0.378960	−	0.925413i	$-0.376282\pi$
$641$	19.1890	0.757919	0.378960	+	0.925413i	$0.376282\pi$
$642$	36.5931	1.44421
$643$	24.5555	0.968376	0.484188	−	0.874964i	$-0.339115\pi$
$643$	24.5555	0.968376	0.484188	+	0.874964i	$0.339115\pi$
$644$	3.74817	0.147699
$645$	0	0
$646$	18.6001	0.731812
$647$	29.3667	1.15453	0.577263	−	0.816558i	$-0.304121\pi$
$647$	29.3667	1.15453	0.577263	+	0.816558i	$0.304121\pi$
$648$	−9.54057	−0.374789
$649$	−34.3149	−1.34698
$650$	0	0
$651$	−17.1956	−0.673948
$652$	−11.7518	−0.460238
$653$	25.0514	0.980337	0.490168	−	0.871628i	$-0.336935\pi$
$653$	25.0514	0.980337	0.490168	+	0.871628i	$0.336935\pi$
$654$	1.60739	0.0628538
$655$	0	0
$656$	2.81100	0.109751
$657$	19.7724	0.771393
$658$	2.51702	0.0981236
$659$	35.0933	1.36704	0.683521	−	0.729931i	$-0.260448\pi$
$659$	35.0933	1.36704	0.683521	+	0.729931i	$0.260448\pi$
$660$	0	0
$661$	−31.1291	−1.21078	−0.605392	−	0.795928i	$-0.706984\pi$
$661$	−31.1291	−1.21078	−0.605392	+	0.795928i	$0.706984\pi$
$662$	−8.98724	−0.349299
$663$	0	0
$664$	8.55910	0.332158
$665$	0	0
$666$	2.31893	0.0898566
$667$	11.2345	0.435001
$668$	−11.1760	−0.432412
$669$	7.18017	0.277601
$670$	0	0
$671$	−78.2206	−3.01967
$672$	−3.38732	−0.130669
$673$	0.193846	0.00747220	0.00373610	−	0.999993i	$-0.498811\pi$
$673$	0.193846	0.00747220	0.00373610	+	0.999993i	$0.498811\pi$
$674$	1.37859	0.0531014
$675$	0	0
$676$	0	0
$677$	−21.9008	−0.841718	−0.420859	−	0.907126i	$-0.638271\pi$
$677$	−21.9008	−0.841718	−0.420859	+	0.907126i	$0.638271\pi$
$678$	8.11912	0.311813
$679$	−6.13766	−0.235542
$680$	0	0
$681$	9.55012	0.365961
$682$	26.9481	1.03190
$683$	−5.11821	−0.195843	−0.0979214	−	0.995194i	$-0.531219\pi$
$683$	−5.11821	−0.195843	−0.0979214	+	0.995194i	$0.531219\pi$
$684$	7.41693	0.283593
$685$	0	0
$686$	16.9014	0.645298
$687$	−23.7408	−0.905769
$688$	5.28374	0.201441
$689$	0	0
$690$	0	0
$691$	−1.91684	−0.0729199	−0.0364600	−	0.999335i	$-0.511608\pi$
$691$	−1.91684	−0.0729199	−0.0364600	+	0.999335i	$0.511608\pi$
$692$	−0.533400	−0.0202768
$693$	−20.9481	−0.795753
$694$	11.8958	0.451557
$695$	0	0
$696$	−10.1529	−0.384845
$697$	19.7909	0.749633
$698$	4.56777	0.172893
$699$	−46.7082	−1.76667
$700$	0	0
$701$	23.4448	0.885500	0.442750	−	0.896645i	$-0.354003\pi$
$701$	23.4448	0.885500	0.442750	+	0.896645i	$0.354003\pi$
$702$	0	0
$703$	2.18216	0.0823017
$704$	5.30846	0.200070
$705$	0	0
$706$	−0.842019	−0.0316898
$707$	−11.7889	−0.443368
$708$	−15.5778	−0.585451
$709$	5.00101	0.187817	0.0939085	−	0.995581i	$-0.470064\pi$
$709$	5.00101	0.187817	0.0939085	+	0.995581i	$0.470064\pi$
$710$	0	0
$711$	−11.3970	−0.427422
$712$	−15.1600	−0.568144
$713$	−13.5368	−0.506957
$714$	−23.8485	−0.892507
$715$	0	0
$716$	9.23733	0.345215
$717$	72.3292	2.70118
$718$	8.05922	0.300767
$719$	−45.4777	−1.69603	−0.848016	−	0.529971i	$-0.822203\pi$
$719$	−45.4777	−1.69603	−0.848016	+	0.529971i	$0.822203\pi$
$720$	0	0
$721$	−7.82245	−0.291323
$722$	−12.0205	−0.447357
$723$	−46.1409	−1.71600
$724$	4.76464	0.177077
$725$	0	0
$726$	41.4009	1.53653
$727$	−6.38432	−0.236781	−0.118391	−	0.992967i	$-0.537774\pi$
$727$	−6.38432	−0.236781	−0.118391	+	0.992967i	$0.537774\pi$
$728$	0	0
$729$	−23.4301	−0.867780
$730$	0	0
$731$	37.2003	1.37590
$732$	−35.5096	−1.31247
$733$	−46.0006	−1.69907	−0.849536	−	0.527530i	$-0.823118\pi$
$733$	−46.0006	−1.69907	−0.849536	+	0.527530i	$0.823118\pi$
$734$	−16.5182	−0.609699
$735$	0	0
$736$	−2.66659	−0.0982917
$737$	21.2338	0.782159
$738$	7.89176	0.290500
$739$	−4.78499	−0.176019	−0.0880094	−	0.996120i	$-0.528051\pi$
$739$	−4.78499	−0.176019	−0.0880094	+	0.996120i	$0.528051\pi$
$740$	0	0
$741$	0	0
$742$	11.1598	0.409691
$743$	−15.8150	−0.580195	−0.290098	−	0.956997i	$-0.593688\pi$
$743$	−15.8150	−0.580195	−0.290098	+	0.956997i	$0.593688\pi$
$744$	12.2336	0.448504
$745$	0	0
$746$	−26.5545	−0.972231
$747$	24.0293	0.879185
$748$	37.3743	1.36654
$749$	−21.3437	−0.779881
$750$	0	0
$751$	−9.88840	−0.360833	−0.180416	−	0.983590i	$-0.557745\pi$
$751$	−9.88840	−0.360833	−0.180416	+	0.983590i	$0.557745\pi$
$752$	−1.79070	−0.0653001
$753$	−41.0713	−1.49672
$754$	0	0
$755$	0	0
$756$	0.652219	0.0237210
$757$	−8.74091	−0.317694	−0.158847	−	0.987303i	$-0.550778\pi$
$757$	−8.74091	−0.317694	−0.158847	+	0.987303i	$0.550778\pi$
$758$	1.15573	0.0419780
$759$	−34.1128	−1.23822
$760$	0	0
$761$	49.4743	1.79344	0.896721	−	0.442595i	$-0.145942\pi$
$761$	49.4743	1.79344	0.896721	+	0.442595i	$0.145942\pi$
$762$	−26.1445	−0.947114
$763$	−0.937543	−0.0339413
$764$	−17.9444	−0.649205
$765$	0	0
$766$	−35.0077	−1.26488
$767$	0	0
$768$	2.40987	0.0869585
$769$	−23.3722	−0.842823	−0.421412	−	0.906870i	$-0.638465\pi$
$769$	−23.3722	−0.842823	−0.421412	+	0.906870i	$0.638465\pi$
$770$	0	0
$771$	19.8218	0.713863
$772$	20.9462	0.753869
$773$	1.65771	0.0596238	0.0298119	−	0.999556i	$-0.490509\pi$
$773$	1.65771	0.0596238	0.0298119	+	0.999556i	$0.490509\pi$
$774$	14.8339	0.533192
$775$	0	0
$776$	4.36656	0.156750
$777$	−2.79789	−0.100374
$778$	11.3135	0.405607
$779$	7.42631	0.266075
$780$	0	0
$781$	51.1707	1.83103
$782$	−18.7741	−0.671362
$783$	1.95491	0.0698629
$784$	−5.02427	−0.179438
$785$	0	0
$786$	36.9914	1.31944
$787$	−43.3447	−1.54507	−0.772536	−	0.634971i	$-0.781012\pi$
$787$	−43.3447	−1.54507	−0.772536	+	0.634971i	$0.781012\pi$
$788$	−26.7724	−0.953726
$789$	−25.1339	−0.894791
$790$	0	0
$791$	−4.73565	−0.168380
$792$	14.9033	0.529564
$793$	0	0
$794$	18.0057	0.639000
$795$	0	0
$796$	14.3890	0.510006
$797$	−20.4233	−0.723430	−0.361715	−	0.932289i	$-0.617809\pi$
$797$	−20.4233	−0.723430	−0.361715	+	0.932289i	$0.617809\pi$
$798$	−8.94887	−0.316787
$799$	−12.6074	−0.446019
$800$	0	0
$801$	−42.5609	−1.50382
$802$	−4.58861	−0.162029
$803$	37.3865	1.31934
$804$	9.63946	0.339958
$805$	0	0
$806$	0	0
$807$	54.5621	1.92068
$808$	8.38707	0.295056
$809$	23.0235	0.809465	0.404732	−	0.914435i	$-0.367365\pi$
$809$	23.0235	0.809465	0.404732	+	0.914435i	$0.367365\pi$
$810$	0	0
$811$	−5.67837	−0.199395	−0.0996973	−	0.995018i	$-0.531787\pi$
$811$	−5.67837	−0.199395	−0.0996973	+	0.995018i	$0.531787\pi$
$812$	5.92190	0.207818
$813$	−0.608301	−0.0213340
$814$	4.38474	0.153685
$815$	0	0
$816$	16.9667	0.593953
$817$	13.9590	0.488363
$818$	−26.2500	−0.917811
$819$	0	0
$820$	0	0
$821$	−12.9626	−0.452397	−0.226198	−	0.974081i	$-0.572630\pi$
$821$	−12.9626	−0.452397	−0.226198	+	0.974081i	$0.572630\pi$
$822$	27.4504	0.957441
$823$	−8.18710	−0.285384	−0.142692	−	0.989767i	$-0.545576\pi$
$823$	−8.18710	−0.285384	−0.142692	+	0.989767i	$0.545576\pi$
$824$	5.56518	0.193872
$825$	0	0
$826$	9.08611	0.316146
$827$	−9.13747	−0.317741	−0.158870	−	0.987299i	$-0.550785\pi$
$827$	−9.13747	−0.317741	−0.158870	+	0.987299i	$0.550785\pi$
$828$	−7.48632	−0.260168
$829$	−4.90515	−0.170363	−0.0851814	−	0.996365i	$-0.527147\pi$
$829$	−4.90515	−0.170363	−0.0851814	+	0.996365i	$0.527147\pi$
$830$	0	0
$831$	11.2188	0.389176
$832$	0	0
$833$	−35.3734	−1.22562
$834$	−27.3520	−0.947121
$835$	0	0
$836$	14.0243	0.485040
$837$	−2.35554	−0.0814193
$838$	12.0774	0.417206
$839$	2.25773	0.0779455	0.0389727	−	0.999240i	$-0.487591\pi$
$839$	2.25773	0.0779455	0.0389727	+	0.999240i	$0.487591\pi$
$840$	0	0
$841$	−11.2501	−0.387936
$842$	3.11716	0.107424
$843$	8.92282	0.307318
$844$	−24.3908	−0.839567
$845$	0	0
$846$	−5.02731	−0.172842
$847$	−24.1480	−0.829734
$848$	−7.93952	−0.272644
$849$	12.8879	0.442310
$850$	0	0
$851$	−2.20257	−0.0755033
$852$	23.2298	0.795840
$853$	14.7832	0.506166	0.253083	−	0.967445i	$-0.418555\pi$
$853$	14.7832	0.506166	0.253083	+	0.967445i	$0.418555\pi$
$854$	20.7117	0.708741
$855$	0	0
$856$	15.1847	0.519002
$857$	19.3235	0.660079	0.330039	−	0.943967i	$-0.392938\pi$
$857$	19.3235	0.660079	0.330039	+	0.943967i	$0.392938\pi$
$858$	0	0
$859$	−48.8245	−1.66587	−0.832935	−	0.553371i	$-0.813341\pi$
$859$	−48.8245	−1.66587	−0.832935	+	0.553371i	$0.813341\pi$
$860$	0	0
$861$	−9.52177	−0.324501
$862$	5.23001	0.178135
$863$	15.2366	0.518660	0.259330	−	0.965789i	$-0.416498\pi$
$863$	15.2366	0.518660	0.259330	+	0.965789i	$0.416498\pi$
$864$	−0.464013	−0.0157860
$865$	0	0
$866$	−22.3063	−0.757998
$867$	78.4864	2.66554
$868$	−7.13549	−0.242194
$869$	−21.5500	−0.731034
$870$	0	0
$871$	0	0
$872$	0.667003	0.0225876
$873$	12.2589	0.414901
$874$	−7.04478	−0.238293
$875$	0	0
$876$	16.9722	0.573438
$877$	−0.445906	−0.0150572	−0.00752859	−	0.999972i	$-0.502396\pi$
$877$	−0.445906	−0.0150572	−0.00752859	+	0.999972i	$0.502396\pi$
$878$	−5.24835	−0.177123
$879$	58.2566	1.96495
$880$	0	0
$881$	−38.8107	−1.30756	−0.653782	−	0.756683i	$-0.726819\pi$
$881$	−38.8107	−1.30756	−0.653782	+	0.756683i	$0.726819\pi$
$882$	−14.1054	−0.474954
$883$	25.2239	0.848853	0.424427	−	0.905462i	$-0.360476\pi$
$883$	25.2239	0.848853	0.424427	+	0.905462i	$0.360476\pi$
$884$	0	0
$885$	0	0
$886$	1.11623	0.0375005
$887$	−0.261895	−0.00879357	−0.00439678	−	0.999990i	$-0.501400\pi$
$887$	−0.261895	−0.00879357	−0.00439678	+	0.999990i	$0.501400\pi$
$888$	1.99053	0.0667977
$889$	15.2493	0.511446
$890$	0	0
$891$	−50.6457	−1.69670
$892$	2.97949	0.0997606
$893$	−4.73080	−0.158310
$894$	4.73311	0.158299
$895$	0	0
$896$	−1.40561	−0.0469580
$897$	0	0
$898$	−34.4883	−1.15089
$899$	−21.3874	−0.713310
$900$	0	0
$901$	−55.8983	−1.86224
$902$	14.9221	0.496851
$903$	−17.8977	−0.595600
$904$	3.36912	0.112055
$905$	0	0
$906$	24.9667	0.829463
$907$	38.0630	1.26386	0.631930	−	0.775025i	$-0.282263\pi$
$907$	38.0630	1.26386	0.631930	+	0.775025i	$0.282263\pi$
$908$	3.96293	0.131514
$909$	23.5463	0.780982
$910$	0	0
$911$	−50.4261	−1.67069	−0.835346	−	0.549725i	$-0.814733\pi$
$911$	−50.4261	−1.67069	−0.835346	+	0.549725i	$0.814733\pi$
$912$	6.36656	0.210818
$913$	45.4357	1.50370
$914$	−4.07540	−0.134802
$915$	0	0
$916$	−9.85151	−0.325503
$917$	−21.5760	−0.712504
$918$	−3.26689	−0.107823
$919$	−6.72916	−0.221975	−0.110987	−	0.993822i	$-0.535401\pi$
$919$	−6.72916	−0.221975	−0.110987	+	0.993822i	$0.535401\pi$
$920$	0	0
$921$	66.3781	2.18723
$922$	−30.3417	−0.999249
$923$	0	0
$924$	−17.9815	−0.591547
$925$	0	0
$926$	25.2291	0.829079
$927$	15.6240	0.513159
$928$	−4.21306	−0.138300
$929$	15.9956	0.524800	0.262400	−	0.964959i	$-0.415486\pi$
$929$	15.9956	0.524800	0.262400	+	0.964959i	$0.415486\pi$
$930$	0	0
$931$	−13.2735	−0.435021
$932$	−19.3821	−0.634881
$933$	−30.9094	−1.01193
$934$	−21.9997	−0.719852
$935$	0	0
$936$	0	0
$937$	32.2129	1.05235	0.526176	−	0.850376i	$-0.323625\pi$
$937$	32.2129	1.05235	0.526176	+	0.850376i	$0.323625\pi$
$938$	−5.62242	−0.183579
$939$	−6.56940	−0.214384
$940$	0	0
$941$	14.9874	0.488576	0.244288	−	0.969703i	$-0.421446\pi$
$941$	14.9874	0.488576	0.244288	+	0.969703i	$0.421446\pi$
$942$	−14.0520	−0.457840
$943$	−7.49578	−0.244096
$944$	−6.46419	−0.210391
$945$	0	0
$946$	28.0485	0.911937
$947$	14.8776	0.483456	0.241728	−	0.970344i	$-0.422286\pi$
$947$	14.8776	0.483456	0.241728	+	0.970344i	$0.422286\pi$
$948$	−9.78300	−0.317737
$949$	0	0
$950$	0	0
$951$	4.57779	0.148445
$952$	−9.89618	−0.320737
$953$	6.62205	0.214509	0.107255	−	0.994232i	$-0.465794\pi$
$953$	6.62205	0.214509	0.107255	+	0.994232i	$0.465794\pi$
$954$	−22.2898	−0.721660
$955$	0	0
$956$	30.0138	0.970715
$957$	−53.8963	−1.74222
$958$	−22.1081	−0.714281
$959$	−16.0110	−0.517023
$960$	0	0
$961$	−5.22962	−0.168697
$962$	0	0
$963$	42.6303	1.37374
$964$	−19.1467	−0.616672
$965$	0	0
$966$	9.03259	0.290619
$967$	24.8355	0.798655	0.399328	−	0.916808i	$-0.369244\pi$
$967$	24.8355	0.798655	0.399328	+	0.916808i	$0.369244\pi$
$968$	17.1797	0.552178
$969$	44.8238	1.43995
$970$	0	0
$971$	−23.0829	−0.740766	−0.370383	−	0.928879i	$-0.620774\pi$
$971$	−23.0829	−0.740766	−0.370383	+	0.928879i	$0.620774\pi$
$972$	−21.5994	−0.692803
$973$	15.9536	0.511450
$974$	3.66873	0.117554
$975$	0	0
$976$	−14.7351	−0.471659
$977$	−8.01979	−0.256576	−0.128288	−	0.991737i	$-0.540948\pi$
$977$	−8.01979	−0.256576	−0.128288	+	0.991737i	$0.540948\pi$
$978$	−28.3204	−0.905585
$979$	−80.4761	−2.57203
$980$	0	0
$981$	1.87258	0.0597868
$982$	−26.4689	−0.844657
$983$	16.0606	0.512254	0.256127	−	0.966643i	$-0.417553\pi$
$983$	16.0606	0.512254	0.256127	+	0.966643i	$0.417553\pi$
$984$	6.77414	0.215952
$985$	0	0
$986$	−29.6621	−0.944633
$987$	6.06568	0.193073
$988$	0	0
$989$	−14.0896	−0.448022
$990$	0	0
$991$	−34.4747	−1.09512	−0.547562	−	0.836765i	$-0.684444\pi$
$991$	−34.4747	−1.09512	−0.547562	+	0.836765i	$0.684444\pi$
$992$	5.07645	0.161178
$993$	−21.6581	−0.687297
$994$	−13.5493	−0.429757
$995$	0	0
$996$	20.6263	0.653569
$997$	−44.2466	−1.40130	−0.700652	−	0.713503i	$-0.747108\pi$
$997$	−44.2466	−1.40130	−0.700652	+	0.713503i	$0.747108\pi$
$998$	−37.0404	−1.17249
$999$	−0.383270	−0.0121261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.cs.1.7		8
5.2	odd	4		1690.2.b.e.339.10		16
5.3	odd	4		1690.2.b.e.339.7		16
5.4	even	2		8450.2.a.cr.1.2		8
13.2	odd	12		650.2.m.e.251.5		16
13.7	odd	12		650.2.m.e.101.5		16
13.12	even	2		8450.2.a.cr.1.7		8
65.2	even	12		130.2.m.a.69.4	yes	8
65.7	even	12		130.2.m.b.49.1	yes	8
65.8	even	4		1690.2.c.f.1689.7		8
65.12	odd	4		1690.2.b.e.339.2		16
65.18	even	4		1690.2.c.e.1689.7		8
65.28	even	12		130.2.m.b.69.1	yes	8
65.33	even	12		130.2.m.a.49.4	✓	8
65.38	odd	4		1690.2.b.e.339.15		16
65.47	even	4		1690.2.c.e.1689.2		8
65.54	odd	12		650.2.m.e.251.4		16
65.57	even	4		1690.2.c.f.1689.2		8
65.59	odd	12		650.2.m.e.101.4		16
65.64	even	2	inner	8450.2.a.cs.1.2		8
195.2	odd	12		1170.2.bj.b.199.1		8
195.98	odd	12		1170.2.bj.b.829.1		8
195.137	odd	12		1170.2.bj.a.829.4		8
195.158	odd	12		1170.2.bj.a.199.4		8
260.7	odd	12		1040.2.df.a.49.4		8
260.67	odd	12		1040.2.df.c.849.1		8
260.163	odd	12		1040.2.df.c.49.1		8
260.223	odd	12		1040.2.df.a.849.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.m.a.49.4	✓	8	65.33	even	12
130.2.m.a.69.4	yes	8	65.2	even	12
130.2.m.b.49.1	yes	8	65.7	even	12
130.2.m.b.69.1	yes	8	65.28	even	12
650.2.m.e.101.4		16	65.59	odd	12
650.2.m.e.101.5		16	13.7	odd	12
650.2.m.e.251.4		16	65.54	odd	12
650.2.m.e.251.5		16	13.2	odd	12
1040.2.df.a.49.4		8	260.7	odd	12
1040.2.df.a.849.4		8	260.223	odd	12
1040.2.df.c.49.1		8	260.163	odd	12
1040.2.df.c.849.1		8	260.67	odd	12
1170.2.bj.a.199.4		8	195.158	odd	12
1170.2.bj.a.829.4		8	195.137	odd	12
1170.2.bj.b.199.1		8	195.2	odd	12
1170.2.bj.b.829.1		8	195.98	odd	12
1690.2.b.e.339.2		16	65.12	odd	4
1690.2.b.e.339.7		16	5.3	odd	4
1690.2.b.e.339.10		16	5.2	odd	4
1690.2.b.e.339.15		16	65.38	odd	4
1690.2.c.e.1689.2		8	65.47	even	4
1690.2.c.e.1689.7		8	65.18	even	4
1690.2.c.f.1689.2		8	65.57	even	4
1690.2.c.f.1689.7		8	65.8	even	4
8450.2.a.cr.1.2		8	5.4	even	2
8450.2.a.cr.1.7		8	13.12	even	2
8450.2.a.cs.1.2		8	65.64	even	2	inner
8450.2.a.cs.1.7		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.40987 q^{3} +1.00000 q^{4} +2.40987 q^{6} -1.40561 q^{7} +1.00000 q^{8} +2.80745 q^{9} +5.30846 q^{11} +2.40987 q^{12} -1.40561 q^{14} +1.00000 q^{16} +7.04051 q^{17} +2.80745 q^{18} +2.64187 q^{19} -3.38732 q^{21} +5.30846 q^{22} -2.66659 q^{23} +2.40987 q^{24} -0.464013 q^{27} -1.40561 q^{28} -4.21306 q^{29} +5.07645 q^{31} +1.00000 q^{32} +12.7927 q^{33} +7.04051 q^{34} +2.80745 q^{36} +0.825990 q^{37} +2.64187 q^{38} +2.81100 q^{41} -3.38732 q^{42} +5.28374 q^{43} +5.30846 q^{44} -2.66659 q^{46} -1.79070 q^{47} +2.40987 q^{48} -5.02427 q^{49} +16.9667 q^{51} -7.93952 q^{53} -0.464013 q^{54} -1.40561 q^{56} +6.36656 q^{57} -4.21306 q^{58} -6.46419 q^{59} -14.7351 q^{61} +5.07645 q^{62} -3.94617 q^{63} +1.00000 q^{64} +12.7927 q^{66} +4.00000 q^{67} +7.04051 q^{68} -6.42612 q^{69} +9.63946 q^{71} +2.80745 q^{72} +7.04281 q^{73} +0.825990 q^{74} +2.64187 q^{76} -7.46160 q^{77} -4.05956 q^{79} -9.54057 q^{81} +2.81100 q^{82} +8.55910 q^{83} -3.38732 q^{84} +5.28374 q^{86} -10.1529 q^{87} +5.30846 q^{88} -15.1600 q^{89} -2.66659 q^{92} +12.2336 q^{93} -1.79070 q^{94} +2.40987 q^{96} +4.36656 q^{97} -5.02427 q^{98} +14.9033 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} + 10 q^{7} + 8 q^{8} + 16 q^{9} + 10 q^{14} + 8 q^{16} + 16 q^{18} + 10 q^{28} - 6 q^{29} + 8 q^{32} + 20 q^{33} + 16 q^{36} + 40 q^{37} - 6 q^{47} + 30 q^{49} + 20 q^{51} + 10 q^{56}+ \cdots + 30 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 + 10 * q^7 + 8 * q^8 + 16 * q^9 + 10 * q^14 + 8 * q^16 + 16 * q^18 + 10 * q^28 - 6 * q^29 + 8 * q^32 + 20 * q^33 + 16 * q^36 + 40 * q^37 - 6 * q^47 + 30 * q^49 + 20 * q^51 + 10 * q^56 + 24 * q^57 - 6 * q^58 + 10 * q^61 + 50 * q^63 + 8 * q^64 + 20 * q^66 + 32 * q^67 + 4 * q^69 + 16 * q^72 + 26 * q^73 + 40 * q^74 - 4 * q^79 - 16 * q^81 + 48 * q^83 + 36 * q^93 - 6 * q^94 + 8 * q^97 + 30 * q^98

Introduction

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