LMFDB - Embedded newform 7605.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.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:	$60.7262307372$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +0.828427 q^{7} -1.58579 q^{8} +O(q^{10})$	$q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +0.828427 q^{7} -1.58579 q^{8} +0.414214 q^{10} +0.585786 q^{11} +0.343146 q^{14} +3.00000 q^{16} +4.82843 q^{17} -3.41421 q^{19} -1.82843 q^{20} +0.242641 q^{22} +1.41421 q^{23} +1.00000 q^{25} -1.51472 q^{28} -5.65685 q^{29} -10.2426 q^{31} +4.41421 q^{32} +2.00000 q^{34} +0.828427 q^{35} -8.48528 q^{37} -1.41421 q^{38} -1.58579 q^{40} -8.82843 q^{41} +3.07107 q^{43} -1.07107 q^{44} +0.585786 q^{46} +0.828427 q^{47} -6.31371 q^{49} +0.414214 q^{50} +14.4853 q^{53} +0.585786 q^{55} -1.31371 q^{56} -2.34315 q^{58} +10.2426 q^{59} -8.00000 q^{61} -4.24264 q^{62} -4.17157 q^{64} +2.00000 q^{67} -8.82843 q^{68} +0.343146 q^{70} -7.89949 q^{71} +8.48528 q^{73} -3.51472 q^{74} +6.24264 q^{76} +0.485281 q^{77} +8.48528 q^{79} +3.00000 q^{80} -3.65685 q^{82} -8.82843 q^{83} +4.82843 q^{85} +1.27208 q^{86} -0.928932 q^{88} +6.00000 q^{89} -2.58579 q^{92} +0.343146 q^{94} -3.41421 q^{95} -3.65685 q^{97} -2.61522 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 4 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 4 * q^7 - 6 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 4 q^{7} - 6 q^{8} - 2 q^{10} + 4 q^{11} + 12 q^{14} + 6 q^{16} + 4 q^{17} - 4 q^{19} + 2 q^{20} - 8 q^{22} + 2 q^{25} - 20 q^{28} - 12 q^{31} + 6 q^{32} + 4 q^{34} - 4 q^{35} - 6 q^{40} - 12 q^{41} - 8 q^{43} + 12 q^{44} + 4 q^{46} - 4 q^{47} + 10 q^{49} - 2 q^{50} + 12 q^{53} + 4 q^{55} + 20 q^{56} - 16 q^{58} + 12 q^{59} - 16 q^{61} - 14 q^{64} + 4 q^{67} - 12 q^{68} + 12 q^{70} + 4 q^{71} - 24 q^{74} + 4 q^{76} - 16 q^{77} + 6 q^{80} + 4 q^{82} - 12 q^{83} + 4 q^{85} + 28 q^{86} - 16 q^{88} + 12 q^{89} - 8 q^{92} + 12 q^{94} - 4 q^{95} + 4 q^{97} - 42 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 4 * q^7 - 6 * q^8 - 2 * q^10 + 4 * q^11 + 12 * q^14 + 6 * q^16 + 4 * q^17 - 4 * q^19 + 2 * q^20 - 8 * q^22 + 2 * q^25 - 20 * q^28 - 12 * q^31 + 6 * q^32 + 4 * q^34 - 4 * q^35 - 6 * q^40 - 12 * q^41 - 8 * q^43 + 12 * q^44 + 4 * q^46 - 4 * q^47 + 10 * q^49 - 2 * q^50 + 12 * q^53 + 4 * q^55 + 20 * q^56 - 16 * q^58 + 12 * q^59 - 16 * q^61 - 14 * q^64 + 4 * q^67 - 12 * q^68 + 12 * q^70 + 4 * q^71 - 24 * q^74 + 4 * q^76 - 16 * q^77 + 6 * q^80 + 4 * q^82 - 12 * q^83 + 4 * q^85 + 28 * q^86 - 16 * q^88 + 12 * q^89 - 8 * q^92 + 12 * q^94 - 4 * q^95 + 4 * q^97 - 42 * q^98

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 $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.828427	0.313116	0.156558	−	0.987669i	$-0.449960\pi$
$7$	0.828427	0.313116	0.156558	+	0.987669i	$0.449960\pi$
$8$	−1.58579	−0.560660
$9$	0	0
$10$	0.414214	0.130986
$11$	0.585786	0.176621	0.0883106	−	0.996093i	$-0.471853\pi$
$11$	0.585786	0.176621	0.0883106	+	0.996093i	$0.471853\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0.343146	0.0917096
$15$	0	0
$16$	3.00000	0.750000
$17$	4.82843	1.17107	0.585533	−	0.810649i	$-0.300885\pi$
$17$	4.82843	1.17107	0.585533	+	0.810649i	$0.300885\pi$
$18$	0	0
$19$	−3.41421	−0.783274	−0.391637	−	0.920120i	$-0.628091\pi$
$19$	−3.41421	−0.783274	−0.391637	+	0.920120i	$0.628091\pi$
$20$	−1.82843	−0.408849
$21$	0	0
$22$	0.242641	0.0517312
$23$	1.41421	0.294884	0.147442	−	0.989071i	$-0.452896\pi$
$23$	1.41421	0.294884	0.147442	+	0.989071i	$0.452896\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−1.51472	−0.286255
$29$	−5.65685	−1.05045	−0.525226	−	0.850963i	$-0.676019\pi$
$29$	−5.65685	−1.05045	−0.525226	+	0.850963i	$0.676019\pi$
$30$	0	0
$31$	−10.2426	−1.83963	−0.919816	−	0.392349i	$-0.871662\pi$
$31$	−10.2426	−1.83963	−0.919816	+	0.392349i	$0.871662\pi$
$32$	4.41421	0.780330
$33$	0	0
$34$	2.00000	0.342997
$35$	0.828427	0.140030
$36$	0	0
$37$	−8.48528	−1.39497	−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528	−1.39497	−0.697486	+	0.716599i	$0.745698\pi$
$38$	−1.41421	−0.229416
$39$	0	0
$40$	−1.58579	−0.250735
$41$	−8.82843	−1.37877	−0.689384	−	0.724396i	$-0.742119\pi$
$41$	−8.82843	−1.37877	−0.689384	+	0.724396i	$0.742119\pi$
$42$	0	0
$43$	3.07107	0.468333	0.234167	−	0.972196i	$-0.424764\pi$
$43$	3.07107	0.468333	0.234167	+	0.972196i	$0.424764\pi$
$44$	−1.07107	−0.161470
$45$	0	0
$46$	0.585786	0.0863695
$47$	0.828427	0.120839	0.0604193	−	0.998173i	$-0.480756\pi$
$47$	0.828427	0.120839	0.0604193	+	0.998173i	$0.480756\pi$
$48$	0	0
$49$	−6.31371	−0.901958
$50$	0.414214	0.0585786
$51$	0	0
$52$	0	0
$53$	14.4853	1.98971	0.994853	−	0.101327i	$-0.0323087\pi$
$53$	14.4853	1.98971	0.994853	+	0.101327i	$0.0323087\pi$
$54$	0	0
$55$	0.585786	0.0789874
$56$	−1.31371	−0.175552
$57$	0	0
$58$	−2.34315	−0.307670
$59$	10.2426	1.33348	0.666739	−	0.745291i	$-0.267690\pi$
$59$	10.2426	1.33348	0.666739	+	0.745291i	$0.267690\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	−4.24264	−0.538816
$63$	0	0
$64$	−4.17157	−0.521447
$65$	0	0
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−8.82843	−1.07060
$69$	0	0
$70$	0.343146	0.0410138
$71$	−7.89949	−0.937498	−0.468749	−	0.883332i	$-0.655295\pi$
$71$	−7.89949	−0.937498	−0.468749	+	0.883332i	$0.655295\pi$
$72$	0	0
$73$	8.48528	0.993127	0.496564	−	0.868000i	$-0.334595\pi$
$73$	8.48528	0.993127	0.496564	+	0.868000i	$0.334595\pi$
$74$	−3.51472	−0.408578
$75$	0	0
$76$	6.24264	0.716080
$77$	0.485281	0.0553029
$78$	0	0
$79$	8.48528	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	−3.65685	−0.403832
$83$	−8.82843	−0.969046	−0.484523	−	0.874779i	$-0.661007\pi$
$83$	−8.82843	−0.969046	−0.484523	+	0.874779i	$0.661007\pi$
$84$	0	0
$85$	4.82843	0.523716
$86$	1.27208	0.137172
$87$	0	0
$88$	−0.928932	−0.0990245
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	−2.58579	−0.269587
$93$	0	0
$94$	0.343146	0.0353928
$95$	−3.41421	−0.350291
$96$	0	0
$97$	−3.65685	−0.371297	−0.185649	−	0.982616i	$-0.559439\pi$
$97$	−3.65685	−0.371297	−0.185649	+	0.982616i	$0.559439\pi$
$98$	−2.61522	−0.264177
$99$	0	0
$100$	−1.82843	−0.182843
$101$	−7.65685	−0.761885	−0.380943	−	0.924599i	$-0.624401\pi$
$101$	−7.65685	−0.761885	−0.380943	+	0.924599i	$0.624401\pi$
$102$	0	0
$103$	17.4142	1.71587	0.857937	−	0.513755i	$-0.171746\pi$
$103$	17.4142	1.71587	0.857937	+	0.513755i	$0.171746\pi$
$104$	0	0
$105$	0	0
$106$	6.00000	0.582772
$107$	6.58579	0.636672	0.318336	−	0.947978i	$-0.396876\pi$
$107$	6.58579	0.636672	0.318336	+	0.947978i	$0.396876\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0.242641	0.0231349
$111$	0	0
$112$	2.48528	0.234837
$113$	3.17157	0.298356	0.149178	−	0.988810i	$-0.452337\pi$
$113$	3.17157	0.298356	0.149178	+	0.988810i	$0.452337\pi$
$114$	0	0
$115$	1.41421	0.131876
$116$	10.3431	0.960337
$117$	0	0
$118$	4.24264	0.390567
$119$	4.00000	0.366679
$120$	0	0
$121$	−10.6569	−0.968805
$122$	−3.31371	−0.300009
$123$	0	0
$124$	18.7279	1.68182
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.41421	−0.835376	−0.417688	−	0.908590i	$-0.637160\pi$
$127$	−9.41421	−0.835376	−0.417688	+	0.908590i	$0.637160\pi$
$128$	−10.5563	−0.933058
$129$	0	0
$130$	0	0
$131$	−16.9706	−1.48272	−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706	−1.48272	−0.741362	+	0.671105i	$0.765820\pi$
$132$	0	0
$133$	−2.82843	−0.245256
$134$	0.828427	0.0715652
$135$	0	0
$136$	−7.65685	−0.656570
$137$	5.31371	0.453981	0.226990	−	0.973897i	$-0.427111\pi$
$137$	5.31371	0.453981	0.226990	+	0.973897i	$0.427111\pi$
$138$	0	0
$139$	−12.4853	−1.05899	−0.529494	−	0.848314i	$-0.677618\pi$
$139$	−12.4853	−1.05899	−0.529494	+	0.848314i	$0.677618\pi$
$140$	−1.51472	−0.128017
$141$	0	0
$142$	−3.27208	−0.274587
$143$	0	0
$144$	0	0
$145$	−5.65685	−0.469776
$146$	3.51472	0.290880
$147$	0	0
$148$	15.5147	1.27530
$149$	−0.343146	−0.0281116	−0.0140558	−	0.999901i	$-0.504474\pi$
$149$	−0.343146	−0.0281116	−0.0140558	+	0.999901i	$0.504474\pi$
$150$	0	0
$151$	−18.2426	−1.48457	−0.742283	−	0.670087i	$-0.766257\pi$
$151$	−18.2426	−1.48457	−0.742283	+	0.670087i	$0.766257\pi$
$152$	5.41421	0.439151
$153$	0	0
$154$	0.201010	0.0161979
$155$	−10.2426	−0.822709
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	3.51472	0.279616
$159$	0	0
$160$	4.41421	0.348974
$161$	1.17157	0.0923329
$162$	0	0
$163$	14.9706	1.17258	0.586292	−	0.810099i	$-0.300587\pi$
$163$	14.9706	1.17258	0.586292	+	0.810099i	$0.300587\pi$
$164$	16.1421	1.26049
$165$	0	0
$166$	−3.65685	−0.283827
$167$	−8.82843	−0.683164	−0.341582	−	0.939852i	$-0.610963\pi$
$167$	−8.82843	−0.683164	−0.341582	+	0.939852i	$0.610963\pi$
$168$	0	0
$169$	0	0
$170$	2.00000	0.153393
$171$	0	0
$172$	−5.61522	−0.428157
$173$	−11.1716	−0.849359	−0.424679	−	0.905344i	$-0.639613\pi$
$173$	−11.1716	−0.849359	−0.424679	+	0.905344i	$0.639613\pi$
$174$	0	0
$175$	0.828427	0.0626232
$176$	1.75736	0.132466
$177$	0	0
$178$	2.48528	0.186280
$179$	5.65685	0.422813	0.211407	−	0.977398i	$-0.432196\pi$
$179$	5.65685	0.422813	0.211407	+	0.977398i	$0.432196\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	−2.24264	−0.165330
$185$	−8.48528	−0.623850
$186$	0	0
$187$	2.82843	0.206835
$188$	−1.51472	−0.110472
$189$	0	0
$190$	−1.41421	−0.102598
$191$	−13.6569	−0.988175	−0.494088	−	0.869412i	$-0.664498\pi$
$191$	−13.6569	−0.988175	−0.494088	+	0.869412i	$0.664498\pi$
$192$	0	0
$193$	−15.6569	−1.12701	−0.563503	−	0.826114i	$-0.690546\pi$
$193$	−15.6569	−1.12701	−0.563503	+	0.826114i	$0.690546\pi$
$194$	−1.51472	−0.108750
$195$	0	0
$196$	11.5442	0.824583
$197$	−22.9706	−1.63658	−0.818292	−	0.574802i	$-0.805079\pi$
$197$	−22.9706	−1.63658	−0.818292	+	0.574802i	$0.805079\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	−1.58579	−0.112132
$201$	0	0
$202$	−3.17157	−0.223151
$203$	−4.68629	−0.328913
$204$	0	0
$205$	−8.82843	−0.616604
$206$	7.21320	0.502568
$207$	0	0
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−19.3137	−1.32961	−0.664805	−	0.747017i	$-0.731485\pi$
$211$	−19.3137	−1.32961	−0.664805	+	0.747017i	$0.731485\pi$
$212$	−26.4853	−1.81902
$213$	0	0
$214$	2.72792	0.186477
$215$	3.07107	0.209445
$216$	0	0
$217$	−8.48528	−0.576018
$218$	0.828427	0.0561082
$219$	0	0
$220$	−1.07107	−0.0722114
$221$	0	0
$222$	0	0
$223$	−26.4853	−1.77359	−0.886793	−	0.462167i	$-0.847072\pi$
$223$	−26.4853	−1.77359	−0.886793	+	0.462167i	$0.847072\pi$
$224$	3.65685	0.244334
$225$	0	0
$226$	1.31371	0.0873866
$227$	−27.6569	−1.83565	−0.917825	−	0.396985i	$-0.870056\pi$
$227$	−27.6569	−1.83565	−0.917825	+	0.396985i	$0.870056\pi$
$228$	0	0
$229$	−0.828427	−0.0547440	−0.0273720	−	0.999625i	$-0.508714\pi$
$229$	−0.828427	−0.0547440	−0.0273720	+	0.999625i	$0.508714\pi$
$230$	0.585786	0.0386256
$231$	0	0
$232$	8.97056	0.588946
$233$	−24.6274	−1.61340	−0.806698	−	0.590964i	$-0.798747\pi$
$233$	−24.6274	−1.61340	−0.806698	+	0.590964i	$0.798747\pi$
$234$	0	0
$235$	0.828427	0.0540406
$236$	−18.7279	−1.21908
$237$	0	0
$238$	1.65685	0.107398
$239$	−0.585786	−0.0378914	−0.0189457	−	0.999821i	$-0.506031\pi$
$239$	−0.585786	−0.0378914	−0.0189457	+	0.999821i	$0.506031\pi$
$240$	0	0
$241$	−2.48528	−0.160091	−0.0800455	−	0.996791i	$-0.525507\pi$
$241$	−2.48528	−0.160091	−0.0800455	+	0.996791i	$0.525507\pi$
$242$	−4.41421	−0.283756
$243$	0	0
$244$	14.6274	0.936424
$245$	−6.31371	−0.403368
$246$	0	0
$247$	0	0
$248$	16.2426	1.03141
$249$	0	0
$250$	0.414214	0.0261972
$251$	19.7990	1.24970	0.624851	−	0.780744i	$-0.285160\pi$
$251$	19.7990	1.24970	0.624851	+	0.780744i	$0.285160\pi$
$252$	0	0
$253$	0.828427	0.0520828
$254$	−3.89949	−0.244676
$255$	0	0
$256$	3.97056	0.248160
$257$	−16.3431	−1.01946	−0.509729	−	0.860335i	$-0.670254\pi$
$257$	−16.3431	−1.01946	−0.509729	+	0.860335i	$0.670254\pi$
$258$	0	0
$259$	−7.02944	−0.436788
$260$	0	0
$261$	0	0
$262$	−7.02944	−0.434280
$263$	13.4142	0.827156	0.413578	−	0.910469i	$-0.364279\pi$
$263$	13.4142	0.827156	0.413578	+	0.910469i	$0.364279\pi$
$264$	0	0
$265$	14.4853	0.889824
$266$	−1.17157	−0.0718337
$267$	0	0
$268$	−3.65685	−0.223378
$269$	2.68629	0.163786	0.0818930	−	0.996641i	$-0.473903\pi$
$269$	2.68629	0.163786	0.0818930	+	0.996641i	$0.473903\pi$
$270$	0	0
$271$	−1.27208	−0.0772732	−0.0386366	−	0.999253i	$-0.512301\pi$
$271$	−1.27208	−0.0772732	−0.0386366	+	0.999253i	$0.512301\pi$
$272$	14.4853	0.878299
$273$	0	0
$274$	2.20101	0.132968
$275$	0.585786	0.0353243
$276$	0	0
$277$	−7.17157	−0.430898	−0.215449	−	0.976515i	$-0.569122\pi$
$277$	−7.17157	−0.430898	−0.215449	+	0.976515i	$0.569122\pi$
$278$	−5.17157	−0.310170
$279$	0	0
$280$	−1.31371	−0.0785091
$281$	−17.7990	−1.06180	−0.530899	−	0.847435i	$-0.678146\pi$
$281$	−17.7990	−1.06180	−0.530899	+	0.847435i	$0.678146\pi$
$282$	0	0
$283$	8.72792	0.518821	0.259411	−	0.965767i	$-0.416472\pi$
$283$	8.72792	0.518821	0.259411	+	0.965767i	$0.416472\pi$
$284$	14.4437	0.857073
$285$	0	0
$286$	0	0
$287$	−7.31371	−0.431715
$288$	0	0
$289$	6.31371	0.371395
$290$	−2.34315	−0.137594
$291$	0	0
$292$	−15.5147	−0.907930
$293$	−2.14214	−0.125145	−0.0625724	−	0.998040i	$-0.519930\pi$
$293$	−2.14214	−0.125145	−0.0625724	+	0.998040i	$0.519930\pi$
$294$	0	0
$295$	10.2426	0.596350
$296$	13.4558	0.782105
$297$	0	0
$298$	−0.142136	−0.00823370
$299$	0	0
$300$	0	0
$301$	2.54416	0.146643
$302$	−7.55635	−0.434819
$303$	0	0
$304$	−10.2426	−0.587456
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−19.1716	−1.09418	−0.547090	−	0.837074i	$-0.684264\pi$
$307$	−19.1716	−1.09418	−0.547090	+	0.837074i	$0.684264\pi$
$308$	−0.887302	−0.0505587
$309$	0	0
$310$	−4.24264	−0.240966
$311$	8.48528	0.481156	0.240578	−	0.970630i	$-0.422663\pi$
$311$	8.48528	0.481156	0.240578	+	0.970630i	$0.422663\pi$
$312$	0	0
$313$	0.828427	0.0468255	0.0234127	−	0.999726i	$-0.492547\pi$
$313$	0.828427	0.0468255	0.0234127	+	0.999726i	$0.492547\pi$
$314$	7.45584	0.420758
$315$	0	0
$316$	−15.5147	−0.872771
$317$	26.1421	1.46829	0.734144	−	0.678993i	$-0.237584\pi$
$317$	26.1421	1.46829	0.734144	+	0.678993i	$0.237584\pi$
$318$	0	0
$319$	−3.31371	−0.185532
$320$	−4.17157	−0.233198
$321$	0	0
$322$	0.485281	0.0270437
$323$	−16.4853	−0.917266
$324$	0	0
$325$	0	0
$326$	6.20101	0.343442
$327$	0	0
$328$	14.0000	0.773021
$329$	0.686292	0.0378365
$330$	0	0
$331$	−22.0416	−1.21152	−0.605759	−	0.795648i	$-0.707130\pi$
$331$	−22.0416	−1.21152	−0.605759	+	0.795648i	$0.707130\pi$
$332$	16.1421	0.885915
$333$	0	0
$334$	−3.65685	−0.200094
$335$	2.00000	0.109272
$336$	0	0
$337$	7.17157	0.390660	0.195330	−	0.980738i	$-0.437422\pi$
$337$	7.17157	0.390660	0.195330	+	0.980738i	$0.437422\pi$
$338$	0	0
$339$	0	0
$340$	−8.82843	−0.478789
$341$	−6.00000	−0.324918
$342$	0	0
$343$	−11.0294	−0.595534
$344$	−4.87006	−0.262576
$345$	0	0
$346$	−4.62742	−0.248771
$347$	−4.24264	−0.227757	−0.113878	−	0.993495i	$-0.536327\pi$
$347$	−4.24264	−0.227757	−0.113878	+	0.993495i	$0.536327\pi$
$348$	0	0
$349$	−1.51472	−0.0810810	−0.0405405	−	0.999178i	$-0.512908\pi$
$349$	−1.51472	−0.0810810	−0.0405405	+	0.999178i	$0.512908\pi$
$350$	0.343146	0.0183419
$351$	0	0
$352$	2.58579	0.137823
$353$	9.17157	0.488154	0.244077	−	0.969756i	$-0.421515\pi$
$353$	9.17157	0.488154	0.244077	+	0.969756i	$0.421515\pi$
$354$	0	0
$355$	−7.89949	−0.419262
$356$	−10.9706	−0.581439
$357$	0	0
$358$	2.34315	0.123839
$359$	−27.8995	−1.47248	−0.736240	−	0.676721i	$-0.763400\pi$
$359$	−27.8995	−1.47248	−0.736240	+	0.676721i	$0.763400\pi$
$360$	0	0
$361$	−7.34315	−0.386481
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.48528	0.444140
$366$	0	0
$367$	4.44365	0.231957	0.115978	−	0.993252i	$-0.463000\pi$
$367$	4.44365	0.231957	0.115978	+	0.993252i	$0.463000\pi$
$368$	4.24264	0.221163
$369$	0	0
$370$	−3.51472	−0.182722
$371$	12.0000	0.623009
$372$	0	0
$373$	−25.3137	−1.31069	−0.655347	−	0.755328i	$-0.727478\pi$
$373$	−25.3137	−1.31069	−0.655347	+	0.755328i	$0.727478\pi$
$374$	1.17157	0.0605806
$375$	0	0
$376$	−1.31371	−0.0677493
$377$	0	0
$378$	0	0
$379$	−14.9289	−0.766848	−0.383424	−	0.923572i	$-0.625255\pi$
$379$	−14.9289	−0.766848	−0.383424	+	0.923572i	$0.625255\pi$
$380$	6.24264	0.320241
$381$	0	0
$382$	−5.65685	−0.289430
$383$	33.1127	1.69198	0.845990	−	0.533199i	$-0.179010\pi$
$383$	33.1127	1.69198	0.845990	+	0.533199i	$0.179010\pi$
$384$	0	0
$385$	0.485281	0.0247322
$386$	−6.48528	−0.330092
$387$	0	0
$388$	6.68629	0.339445
$389$	16.6274	0.843044	0.421522	−	0.906818i	$-0.361496\pi$
$389$	16.6274	0.843044	0.421522	+	0.906818i	$0.361496\pi$
$390$	0	0
$391$	6.82843	0.345328
$392$	10.0122	0.505692
$393$	0	0
$394$	−9.51472	−0.479345
$395$	8.48528	0.426941
$396$	0	0
$397$	27.7990	1.39519	0.697596	−	0.716492i	$-0.254253\pi$
$397$	27.7990	1.39519	0.697596	+	0.716492i	$0.254253\pi$
$398$	1.65685	0.0830506
$399$	0	0
$400$	3.00000	0.150000
$401$	17.3137	0.864605	0.432303	−	0.901729i	$-0.357701\pi$
$401$	17.3137	0.864605	0.432303	+	0.901729i	$0.357701\pi$
$402$	0	0
$403$	0	0
$404$	14.0000	0.696526
$405$	0	0
$406$	−1.94113	−0.0963364
$407$	−4.97056	−0.246382
$408$	0	0
$409$	−12.8284	−0.634325	−0.317162	−	0.948371i	$-0.602730\pi$
$409$	−12.8284	−0.634325	−0.317162	+	0.948371i	$0.602730\pi$
$410$	−3.65685	−0.180599
$411$	0	0
$412$	−31.8406	−1.56867
$413$	8.48528	0.417533
$414$	0	0
$415$	−8.82843	−0.433370
$416$	0	0
$417$	0	0
$418$	−0.828427	−0.0405197
$419$	−5.17157	−0.252648	−0.126324	−	0.991989i	$-0.540318\pi$
$419$	−5.17157	−0.252648	−0.126324	+	0.991989i	$0.540318\pi$
$420$	0	0
$421$	1.02944	0.0501717	0.0250859	−	0.999685i	$-0.492014\pi$
$421$	1.02944	0.0501717	0.0250859	+	0.999685i	$0.492014\pi$
$422$	−8.00000	−0.389434
$423$	0	0
$424$	−22.9706	−1.11555
$425$	4.82843	0.234213
$426$	0	0
$427$	−6.62742	−0.320723
$428$	−12.0416	−0.582054
$429$	0	0
$430$	1.27208	0.0613450
$431$	3.61522	0.174139	0.0870696	−	0.996202i	$-0.472250\pi$
$431$	3.61522	0.174139	0.0870696	+	0.996202i	$0.472250\pi$
$432$	0	0
$433$	−3.65685	−0.175737	−0.0878686	−	0.996132i	$-0.528006\pi$
$433$	−3.65685	−0.175737	−0.0878686	+	0.996132i	$0.528006\pi$
$434$	−3.51472	−0.168712
$435$	0	0
$436$	−3.65685	−0.175132
$437$	−4.82843	−0.230975
$438$	0	0
$439$	−32.9706	−1.57360	−0.786800	−	0.617209i	$-0.788263\pi$
$439$	−32.9706	−1.57360	−0.786800	+	0.617209i	$0.788263\pi$
$440$	−0.928932	−0.0442851
$441$	0	0
$442$	0	0
$443$	6.58579	0.312900	0.156450	−	0.987686i	$-0.449995\pi$
$443$	6.58579	0.312900	0.156450	+	0.987686i	$0.449995\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	−10.9706	−0.519471
$447$	0	0
$448$	−3.45584	−0.163273
$449$	29.1127	1.37391	0.686957	−	0.726698i	$-0.258946\pi$
$449$	29.1127	1.37391	0.686957	+	0.726698i	$0.258946\pi$
$450$	0	0
$451$	−5.17157	−0.243520
$452$	−5.79899	−0.272762
$453$	0	0
$454$	−11.4558	−0.537649
$455$	0	0
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−0.343146	−0.0160341
$459$	0	0
$460$	−2.58579	−0.120563
$461$	26.4853	1.23354	0.616771	−	0.787142i	$-0.288440\pi$
$461$	26.4853	1.23354	0.616771	+	0.787142i	$0.288440\pi$
$462$	0	0
$463$	15.6569	0.727636	0.363818	−	0.931470i	$-0.381473\pi$
$463$	15.6569	0.727636	0.363818	+	0.931470i	$0.381473\pi$
$464$	−16.9706	−0.787839
$465$	0	0
$466$	−10.2010	−0.472553
$467$	10.5858	0.489852	0.244926	−	0.969542i	$-0.421236\pi$
$467$	10.5858	0.489852	0.244926	+	0.969542i	$0.421236\pi$
$468$	0	0
$469$	1.65685	0.0765064
$470$	0.343146	0.0158281
$471$	0	0
$472$	−16.2426	−0.747628
$473$	1.79899	0.0827176
$474$	0	0
$475$	−3.41421	−0.156655
$476$	−7.31371	−0.335223
$477$	0	0
$478$	−0.242641	−0.0110981
$479$	−5.27208	−0.240887	−0.120444	−	0.992720i	$-0.538432\pi$
$479$	−5.27208	−0.240887	−0.120444	+	0.992720i	$0.538432\pi$
$480$	0	0
$481$	0	0
$482$	−1.02944	−0.0468896
$483$	0	0
$484$	19.4853	0.885695
$485$	−3.65685	−0.166049
$486$	0	0
$487$	−22.9706	−1.04090	−0.520448	−	0.853894i	$-0.674235\pi$
$487$	−22.9706	−1.04090	−0.520448	+	0.853894i	$0.674235\pi$
$488$	12.6863	0.574281
$489$	0	0
$490$	−2.61522	−0.118144
$491$	−10.8284	−0.488680	−0.244340	−	0.969690i	$-0.578571\pi$
$491$	−10.8284	−0.488680	−0.244340	+	0.969690i	$0.578571\pi$
$492$	0	0
$493$	−27.3137	−1.23015
$494$	0	0
$495$	0	0
$496$	−30.7279	−1.37972
$497$	−6.54416	−0.293546
$498$	0	0
$499$	−10.4437	−0.467522	−0.233761	−	0.972294i	$-0.575103\pi$
$499$	−10.4437	−0.467522	−0.233761	+	0.972294i	$0.575103\pi$
$500$	−1.82843	−0.0817697
$501$	0	0
$502$	8.20101	0.366029
$503$	−18.1005	−0.807062	−0.403531	−	0.914966i	$-0.632217\pi$
$503$	−18.1005	−0.807062	−0.403531	+	0.914966i	$0.632217\pi$
$504$	0	0
$505$	−7.65685	−0.340726
$506$	0.343146	0.0152547
$507$	0	0
$508$	17.2132	0.763712
$509$	−21.1127	−0.935804	−0.467902	−	0.883780i	$-0.654990\pi$
$509$	−21.1127	−0.935804	−0.467902	+	0.883780i	$0.654990\pi$
$510$	0	0
$511$	7.02944	0.310964
$512$	22.7574	1.00574
$513$	0	0
$514$	−6.76955	−0.298592
$515$	17.4142	0.767362
$516$	0	0
$517$	0.485281	0.0213427
$518$	−2.91169	−0.127932
$519$	0	0
$520$	0	0
$521$	6.34315	0.277898	0.138949	−	0.990300i	$-0.455628\pi$
$521$	6.34315	0.277898	0.138949	+	0.990300i	$0.455628\pi$
$522$	0	0
$523$	−28.2426	−1.23496	−0.617482	−	0.786585i	$-0.711847\pi$
$523$	−28.2426	−1.23496	−0.617482	+	0.786585i	$0.711847\pi$
$524$	31.0294	1.35553
$525$	0	0
$526$	5.55635	0.242268
$527$	−49.4558	−2.15433
$528$	0	0
$529$	−21.0000	−0.913043
$530$	6.00000	0.260623
$531$	0	0
$532$	5.17157	0.224216
$533$	0	0
$534$	0	0
$535$	6.58579	0.284728
$536$	−3.17157	−0.136991
$537$	0	0
$538$	1.11270	0.0479718
$539$	−3.69848	−0.159305
$540$	0	0
$541$	12.8284	0.551537	0.275769	−	0.961224i	$-0.411068\pi$
$541$	12.8284	0.551537	0.275769	+	0.961224i	$0.411068\pi$
$542$	−0.526912	−0.0226328
$543$	0	0
$544$	21.3137	0.913818
$545$	2.00000	0.0856706
$546$	0	0
$547$	−29.2132	−1.24907	−0.624533	−	0.780998i	$-0.714711\pi$
$547$	−29.2132	−1.24907	−0.624533	+	0.780998i	$0.714711\pi$
$548$	−9.71573	−0.415035
$549$	0	0
$550$	0.242641	0.0103462
$551$	19.3137	0.822792
$552$	0	0
$553$	7.02944	0.298922
$554$	−2.97056	−0.126207
$555$	0	0
$556$	22.8284	0.968141
$557$	3.79899	0.160968	0.0804842	−	0.996756i	$-0.474353\pi$
$557$	3.79899	0.160968	0.0804842	+	0.996756i	$0.474353\pi$
$558$	0	0
$559$	0	0
$560$	2.48528	0.105022
$561$	0	0
$562$	−7.37258	−0.310994
$563$	16.2426	0.684546	0.342273	−	0.939601i	$-0.388803\pi$
$563$	16.2426	0.684546	0.342273	+	0.939601i	$0.388803\pi$
$564$	0	0
$565$	3.17157	0.133429
$566$	3.61522	0.151959
$567$	0	0
$568$	12.5269	0.525618
$569$	21.6569	0.907903	0.453951	−	0.891027i	$-0.350014\pi$
$569$	21.6569	0.907903	0.453951	+	0.891027i	$0.350014\pi$
$570$	0	0
$571$	−28.4853	−1.19207	−0.596036	−	0.802958i	$-0.703258\pi$
$571$	−28.4853	−1.19207	−0.596036	+	0.802958i	$0.703258\pi$
$572$	0	0
$573$	0	0
$574$	−3.02944	−0.126446
$575$	1.41421	0.0589768
$576$	0	0
$577$	29.1716	1.21443	0.607214	−	0.794538i	$-0.292287\pi$
$577$	29.1716	1.21443	0.607214	+	0.794538i	$0.292287\pi$
$578$	2.61522	0.108779
$579$	0	0
$580$	10.3431	0.429476
$581$	−7.31371	−0.303424
$582$	0	0
$583$	8.48528	0.351424
$584$	−13.4558	−0.556807
$585$	0	0
$586$	−0.887302	−0.0366541
$587$	31.6569	1.30662	0.653309	−	0.757091i	$-0.273380\pi$
$587$	31.6569	1.30662	0.653309	+	0.757091i	$0.273380\pi$
$588$	0	0
$589$	34.9706	1.44094
$590$	4.24264	0.174667
$591$	0	0
$592$	−25.4558	−1.04623
$593$	20.6274	0.847066	0.423533	−	0.905881i	$-0.360790\pi$
$593$	20.6274	0.847066	0.423533	+	0.905881i	$0.360790\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	0.627417	0.0257000
$597$	0	0
$598$	0	0
$599$	25.4558	1.04010	0.520049	−	0.854137i	$-0.325914\pi$
$599$	25.4558	1.04010	0.520049	+	0.854137i	$0.325914\pi$
$600$	0	0
$601$	0.627417	0.0255929	0.0127964	−	0.999918i	$-0.495927\pi$
$601$	0.627417	0.0255929	0.0127964	+	0.999918i	$0.495927\pi$
$602$	1.05382	0.0429507
$603$	0	0
$604$	33.3553	1.35721
$605$	−10.6569	−0.433263
$606$	0	0
$607$	40.2426	1.63340	0.816699	−	0.577064i	$-0.195802\pi$
$607$	40.2426	1.63340	0.816699	+	0.577064i	$0.195802\pi$
$608$	−15.0711	−0.611213
$609$	0	0
$610$	−3.31371	−0.134168
$611$	0	0
$612$	0	0
$613$	37.3137	1.50709	0.753543	−	0.657398i	$-0.228343\pi$
$613$	37.3137	1.50709	0.753543	+	0.657398i	$0.228343\pi$
$614$	−7.94113	−0.320478
$615$	0	0
$616$	−0.769553	−0.0310062
$617$	−22.9706	−0.924760	−0.462380	−	0.886682i	$-0.653004\pi$
$617$	−22.9706	−0.924760	−0.462380	+	0.886682i	$0.653004\pi$
$618$	0	0
$619$	−10.2426	−0.411686	−0.205843	−	0.978585i	$-0.565994\pi$
$619$	−10.2426	−0.411686	−0.205843	+	0.978585i	$0.565994\pi$
$620$	18.7279	0.752131
$621$	0	0
$622$	3.51472	0.140927
$623$	4.97056	0.199141
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.343146	0.0137149
$627$	0	0
$628$	−32.9117	−1.31332
$629$	−40.9706	−1.63360
$630$	0	0
$631$	18.2426	0.726228	0.363114	−	0.931745i	$-0.381714\pi$
$631$	18.2426	0.726228	0.363114	+	0.931745i	$0.381714\pi$
$632$	−13.4558	−0.535245
$633$	0	0
$634$	10.8284	0.430052
$635$	−9.41421	−0.373592
$636$	0	0
$637$	0	0
$638$	−1.37258	−0.0543411
$639$	0	0
$640$	−10.5563	−0.417276
$641$	−36.3431	−1.43547	−0.717734	−	0.696317i	$-0.754821\pi$
$641$	−36.3431	−1.43547	−0.717734	+	0.696317i	$0.754821\pi$
$642$	0	0
$643$	−26.4853	−1.04448	−0.522239	−	0.852799i	$-0.674903\pi$
$643$	−26.4853	−1.04448	−0.522239	+	0.852799i	$0.674903\pi$
$644$	−2.14214	−0.0844120
$645$	0	0
$646$	−6.82843	−0.268661
$647$	−6.58579	−0.258914	−0.129457	−	0.991585i	$-0.541323\pi$
$647$	−6.58579	−0.258914	−0.129457	+	0.991585i	$0.541323\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	0	0
$651$	0	0
$652$	−27.3726	−1.07199
$653$	−13.0294	−0.509881	−0.254941	−	0.966957i	$-0.582056\pi$
$653$	−13.0294	−0.509881	−0.254941	+	0.966957i	$0.582056\pi$
$654$	0	0
$655$	−16.9706	−0.663095
$656$	−26.4853	−1.03408
$657$	0	0
$658$	0.284271	0.0110820
$659$	−46.1421	−1.79744	−0.898721	−	0.438520i	$-0.855503\pi$
$659$	−46.1421	−1.79744	−0.898721	+	0.438520i	$0.855503\pi$
$660$	0	0
$661$	49.5980	1.92914	0.964569	−	0.263831i	$-0.0849861\pi$
$661$	49.5980	1.92914	0.964569	+	0.263831i	$0.0849861\pi$
$662$	−9.12994	−0.354845
$663$	0	0
$664$	14.0000	0.543305
$665$	−2.82843	−0.109682
$666$	0	0
$667$	−8.00000	−0.309761
$668$	16.1421	0.624558
$669$	0	0
$670$	0.828427	0.0320049
$671$	−4.68629	−0.180912
$672$	0	0
$673$	−10.4853	−0.404178	−0.202089	−	0.979367i	$-0.564773\pi$
$673$	−10.4853	−0.404178	−0.202089	+	0.979367i	$0.564773\pi$
$674$	2.97056	0.114422
$675$	0	0
$676$	0	0
$677$	−8.14214	−0.312928	−0.156464	−	0.987684i	$-0.550009\pi$
$677$	−8.14214	−0.312928	−0.156464	+	0.987684i	$0.550009\pi$
$678$	0	0
$679$	−3.02944	−0.116259
$680$	−7.65685	−0.293627
$681$	0	0
$682$	−2.48528	−0.0951663
$683$	33.3137	1.27471	0.637357	−	0.770569i	$-0.280028\pi$
$683$	33.3137	1.27471	0.637357	+	0.770569i	$0.280028\pi$
$684$	0	0
$685$	5.31371	0.203026
$686$	−4.56854	−0.174428
$687$	0	0
$688$	9.21320	0.351250
$689$	0	0
$690$	0	0
$691$	−21.0711	−0.801581	−0.400791	−	0.916170i	$-0.631265\pi$
$691$	−21.0711	−0.801581	−0.400791	+	0.916170i	$0.631265\pi$
$692$	20.4264	0.776495
$693$	0	0
$694$	−1.75736	−0.0667084
$695$	−12.4853	−0.473594
$696$	0	0
$697$	−42.6274	−1.61463
$698$	−0.627417	−0.0237481
$699$	0	0
$700$	−1.51472	−0.0572510
$701$	−37.3137	−1.40932	−0.704660	−	0.709545i	$-0.748900\pi$
$701$	−37.3137	−1.40932	−0.704660	+	0.709545i	$0.748900\pi$
$702$	0	0
$703$	28.9706	1.09265
$704$	−2.44365	−0.0920986
$705$	0	0
$706$	3.79899	0.142977
$707$	−6.34315	−0.238559
$708$	0	0
$709$	17.1127	0.642681	0.321340	−	0.946964i	$-0.395867\pi$
$709$	17.1127	0.642681	0.321340	+	0.946964i	$0.395867\pi$
$710$	−3.27208	−0.122799
$711$	0	0
$712$	−9.51472	−0.356579
$713$	−14.4853	−0.542478
$714$	0	0
$715$	0	0
$716$	−10.3431	−0.386542
$717$	0	0
$718$	−11.5563	−0.431279
$719$	4.97056	0.185371	0.0926854	−	0.995695i	$-0.470455\pi$
$719$	4.97056	0.185371	0.0926854	+	0.995695i	$0.470455\pi$
$720$	0	0
$721$	14.4264	0.537267
$722$	−3.04163	−0.113198
$723$	0	0
$724$	0	0
$725$	−5.65685	−0.210090
$726$	0	0
$727$	19.3553	0.717850	0.358925	−	0.933366i	$-0.383143\pi$
$727$	19.3553	0.717850	0.358925	+	0.933366i	$0.383143\pi$
$728$	0	0
$729$	0	0
$730$	3.51472	0.130086
$731$	14.8284	0.548449
$732$	0	0
$733$	1.31371	0.0485229	0.0242615	−	0.999706i	$-0.492277\pi$
$733$	1.31371	0.0485229	0.0242615	+	0.999706i	$0.492277\pi$
$734$	1.84062	0.0679385
$735$	0	0
$736$	6.24264	0.230107
$737$	1.17157	0.0431554
$738$	0	0
$739$	−30.7279	−1.13034	−0.565172	−	0.824973i	$-0.691190\pi$
$739$	−30.7279	−1.13034	−0.565172	+	0.824973i	$0.691190\pi$
$740$	15.5147	0.570332
$741$	0	0
$742$	4.97056	0.182475
$743$	−38.4853	−1.41189	−0.705944	−	0.708268i	$-0.749477\pi$
$743$	−38.4853	−1.41189	−0.705944	+	0.708268i	$0.749477\pi$
$744$	0	0
$745$	−0.343146	−0.0125719
$746$	−10.4853	−0.383893
$747$	0	0
$748$	−5.17157	−0.189091
$749$	5.45584	0.199352
$750$	0	0
$751$	−44.4853	−1.62329	−0.811645	−	0.584150i	$-0.801428\pi$
$751$	−44.4853	−1.62329	−0.811645	+	0.584150i	$0.801428\pi$
$752$	2.48528	0.0906289
$753$	0	0
$754$	0	0
$755$	−18.2426	−0.663918
$756$	0	0
$757$	4.14214	0.150548	0.0752742	−	0.997163i	$-0.476017\pi$
$757$	4.14214	0.150548	0.0752742	+	0.997163i	$0.476017\pi$
$758$	−6.18377	−0.224605
$759$	0	0
$760$	5.41421	0.196394
$761$	−36.6274	−1.32774	−0.663871	−	0.747847i	$-0.731088\pi$
$761$	−36.6274	−1.32774	−0.663871	+	0.747847i	$0.731088\pi$
$762$	0	0
$763$	1.65685	0.0599822
$764$	24.9706	0.903403
$765$	0	0
$766$	13.7157	0.495569
$767$	0	0
$768$	0	0
$769$	−10.9706	−0.395609	−0.197804	−	0.980242i	$-0.563381\pi$
$769$	−10.9706	−0.395609	−0.197804	+	0.980242i	$0.563381\pi$
$770$	0.201010	0.00724390
$771$	0	0
$772$	28.6274	1.03032
$773$	6.14214	0.220917	0.110459	−	0.993881i	$-0.464768\pi$
$773$	6.14214	0.220917	0.110459	+	0.993881i	$0.464768\pi$
$774$	0	0
$775$	−10.2426	−0.367927
$776$	5.79899	0.208172
$777$	0	0
$778$	6.88730	0.246922
$779$	30.1421	1.07995
$780$	0	0
$781$	−4.62742	−0.165582
$782$	2.82843	0.101144
$783$	0	0
$784$	−18.9411	−0.676469
$785$	18.0000	0.642448
$786$	0	0
$787$	−5.51472	−0.196578	−0.0982892	−	0.995158i	$-0.531337\pi$
$787$	−5.51472	−0.196578	−0.0982892	+	0.995158i	$0.531337\pi$
$788$	42.0000	1.49619
$789$	0	0
$790$	3.51472	0.125048
$791$	2.62742	0.0934202
$792$	0	0
$793$	0	0
$794$	11.5147	0.408642
$795$	0	0
$796$	−7.31371	−0.259228
$797$	−10.9706	−0.388597	−0.194299	−	0.980942i	$-0.562243\pi$
$797$	−10.9706	−0.388597	−0.194299	+	0.980942i	$0.562243\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	4.41421	0.156066
$801$	0	0
$802$	7.17157	0.253237
$803$	4.97056	0.175407
$804$	0	0
$805$	1.17157	0.0412925
$806$	0	0
$807$	0	0
$808$	12.1421	0.427159
$809$	45.2548	1.59108	0.795538	−	0.605904i	$-0.207189\pi$
$809$	45.2548	1.59108	0.795538	+	0.605904i	$0.207189\pi$
$810$	0	0
$811$	−8.38478	−0.294429	−0.147215	−	0.989105i	$-0.547031\pi$
$811$	−8.38478	−0.294429	−0.147215	+	0.989105i	$0.547031\pi$
$812$	8.56854	0.300697
$813$	0	0
$814$	−2.05887	−0.0721635
$815$	14.9706	0.524396
$816$	0	0
$817$	−10.4853	−0.366834
$818$	−5.31371	−0.185789
$819$	0	0
$820$	16.1421	0.563708
$821$	39.2548	1.37000	0.685002	−	0.728542i	$-0.259801\pi$
$821$	39.2548	1.37000	0.685002	+	0.728542i	$0.259801\pi$
$822$	0	0
$823$	34.3848	1.19858	0.599289	−	0.800533i	$-0.295450\pi$
$823$	34.3848	1.19858	0.599289	+	0.800533i	$0.295450\pi$
$824$	−27.6152	−0.962022
$825$	0	0
$826$	3.51472	0.122293
$827$	27.8579	0.968713	0.484356	−	0.874871i	$-0.339054\pi$
$827$	27.8579	0.968713	0.484356	+	0.874871i	$0.339054\pi$
$828$	0	0
$829$	7.02944	0.244142	0.122071	−	0.992521i	$-0.461046\pi$
$829$	7.02944	0.244142	0.122071	+	0.992521i	$0.461046\pi$
$830$	−3.65685	−0.126931
$831$	0	0
$832$	0	0
$833$	−30.4853	−1.05625
$834$	0	0
$835$	−8.82843	−0.305520
$836$	3.65685	0.126475
$837$	0	0
$838$	−2.14214	−0.0739988
$839$	−18.7279	−0.646560	−0.323280	−	0.946303i	$-0.604786\pi$
$839$	−18.7279	−0.646560	−0.323280	+	0.946303i	$0.604786\pi$
$840$	0	0
$841$	3.00000	0.103448
$842$	0.426407	0.0146950
$843$	0	0
$844$	35.3137	1.21555
$845$	0	0
$846$	0	0
$847$	−8.82843	−0.303348
$848$	43.4558	1.49228
$849$	0	0
$850$	2.00000	0.0685994
$851$	−12.0000	−0.411355
$852$	0	0
$853$	−37.4558	−1.28246	−0.641232	−	0.767347i	$-0.721576\pi$
$853$	−37.4558	−1.28246	−0.641232	+	0.767347i	$0.721576\pi$
$854$	−2.74517	−0.0939376
$855$	0	0
$856$	−10.4437	−0.356957
$857$	−0.343146	−0.0117216	−0.00586082	−	0.999983i	$-0.501866\pi$
$857$	−0.343146	−0.0117216	−0.00586082	+	0.999983i	$0.501866\pi$
$858$	0	0
$859$	11.7990	0.402576	0.201288	−	0.979532i	$-0.435487\pi$
$859$	11.7990	0.402576	0.201288	+	0.979532i	$0.435487\pi$
$860$	−5.61522	−0.191478
$861$	0	0
$862$	1.49747	0.0510042
$863$	19.4558	0.662285	0.331142	−	0.943581i	$-0.392566\pi$
$863$	19.4558	0.662285	0.331142	+	0.943581i	$0.392566\pi$
$864$	0	0
$865$	−11.1716	−0.379845
$866$	−1.51472	−0.0514722
$867$	0	0
$868$	15.5147	0.526604
$869$	4.97056	0.168615
$870$	0	0
$871$	0	0
$872$	−3.17157	−0.107403
$873$	0	0
$874$	−2.00000	−0.0676510
$875$	0.828427	0.0280059
$876$	0	0
$877$	−2.68629	−0.0907096	−0.0453548	−	0.998971i	$-0.514442\pi$
$877$	−2.68629	−0.0907096	−0.0453548	+	0.998971i	$0.514442\pi$
$878$	−13.6569	−0.460897
$879$	0	0
$880$	1.75736	0.0592406
$881$	−52.9706	−1.78462	−0.892312	−	0.451420i	$-0.850918\pi$
$881$	−52.9706	−1.78462	−0.892312	+	0.451420i	$0.850918\pi$
$882$	0	0
$883$	−32.2426	−1.08505	−0.542526	−	0.840039i	$-0.682532\pi$
$883$	−32.2426	−1.08505	−0.542526	+	0.840039i	$0.682532\pi$
$884$	0	0
$885$	0	0
$886$	2.72792	0.0916463
$887$	−14.3848	−0.482994	−0.241497	−	0.970402i	$-0.577638\pi$
$887$	−14.3848	−0.482994	−0.241497	+	0.970402i	$0.577638\pi$
$888$	0	0
$889$	−7.79899	−0.261570
$890$	2.48528	0.0833068
$891$	0	0
$892$	48.4264	1.62144
$893$	−2.82843	−0.0946497
$894$	0	0
$895$	5.65685	0.189088
$896$	−8.74517	−0.292155
$897$	0	0
$898$	12.0589	0.402410
$899$	57.9411	1.93244
$900$	0	0
$901$	69.9411	2.33008
$902$	−2.14214	−0.0713253
$903$	0	0
$904$	−5.02944	−0.167277
$905$	0	0
$906$	0	0
$907$	−33.2132	−1.10283	−0.551413	−	0.834232i	$-0.685911\pi$
$907$	−33.2132	−1.10283	−0.551413	+	0.834232i	$0.685911\pi$
$908$	50.5685	1.67818
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	−5.17157	−0.171154
$914$	7.45584	0.246617
$915$	0	0
$916$	1.51472	0.0500477
$917$	−14.0589	−0.464265
$918$	0	0
$919$	−16.4853	−0.543799	−0.271900	−	0.962326i	$-0.587652\pi$
$919$	−16.4853	−0.543799	−0.271900	+	0.962326i	$0.587652\pi$
$920$	−2.24264	−0.0739377
$921$	0	0
$922$	10.9706	0.361296
$923$	0	0
$924$	0	0
$925$	−8.48528	−0.278994
$926$	6.48528	0.213120
$927$	0	0
$928$	−24.9706	−0.819699
$929$	11.1716	0.366527	0.183264	−	0.983064i	$-0.441334\pi$
$929$	11.1716	0.366527	0.183264	+	0.983064i	$0.441334\pi$
$930$	0	0
$931$	21.5563	0.706481
$932$	45.0294	1.47499
$933$	0	0
$934$	4.38478	0.143474
$935$	2.82843	0.0924995
$936$	0	0
$937$	10.9706	0.358393	0.179196	−	0.983813i	$-0.442650\pi$
$937$	10.9706	0.358393	0.179196	+	0.983813i	$0.442650\pi$
$938$	0.686292	0.0224082
$939$	0	0
$940$	−1.51472	−0.0494047
$941$	−54.7696	−1.78544	−0.892718	−	0.450615i	$-0.851205\pi$
$941$	−54.7696	−1.78544	−0.892718	+	0.450615i	$0.851205\pi$
$942$	0	0
$943$	−12.4853	−0.406577
$944$	30.7279	1.00011
$945$	0	0
$946$	0.745166	0.0242274
$947$	−45.1127	−1.46597	−0.732983	−	0.680247i	$-0.761872\pi$
$947$	−45.1127	−1.46597	−0.732983	+	0.680247i	$0.761872\pi$
$948$	0	0
$949$	0	0
$950$	−1.41421	−0.0458831
$951$	0	0
$952$	−6.34315	−0.205583
$953$	55.2548	1.78988	0.894940	−	0.446187i	$-0.147218\pi$
$953$	55.2548	1.78988	0.894940	+	0.446187i	$0.147218\pi$
$954$	0	0
$955$	−13.6569	−0.441925
$956$	1.07107	0.0346408
$957$	0	0
$958$	−2.18377	−0.0705543
$959$	4.40202	0.142149
$960$	0	0
$961$	73.9117	2.38425
$962$	0	0
$963$	0	0
$964$	4.54416	0.146357
$965$	−15.6569	−0.504012
$966$	0	0
$967$	19.9411	0.641263	0.320632	−	0.947204i	$-0.396105\pi$
$967$	19.9411	0.641263	0.320632	+	0.947204i	$0.396105\pi$
$968$	16.8995	0.543170
$969$	0	0
$970$	−1.51472	−0.0486347
$971$	12.2843	0.394221	0.197111	−	0.980381i	$-0.436844\pi$
$971$	12.2843	0.394221	0.197111	+	0.980381i	$0.436844\pi$
$972$	0	0
$973$	−10.3431	−0.331586
$974$	−9.51472	−0.304871
$975$	0	0
$976$	−24.0000	−0.768221
$977$	−56.4853	−1.80712	−0.903562	−	0.428457i	$-0.859057\pi$
$977$	−56.4853	−1.80712	−0.903562	+	0.428457i	$0.859057\pi$
$978$	0	0
$979$	3.51472	0.112331
$980$	11.5442	0.368765
$981$	0	0
$982$	−4.48528	−0.143131
$983$	−34.9706	−1.11539	−0.557694	−	0.830047i	$-0.688314\pi$
$983$	−34.9706	−1.11539	−0.557694	+	0.830047i	$0.688314\pi$
$984$	0	0
$985$	−22.9706	−0.731903
$986$	−11.3137	−0.360302
$987$	0	0
$988$	0	0
$989$	4.34315	0.138104
$990$	0	0
$991$	15.0294	0.477426	0.238713	−	0.971090i	$-0.423275\pi$
$991$	15.0294	0.477426	0.238713	+	0.971090i	$0.423275\pi$
$992$	−45.2132	−1.43552
$993$	0	0
$994$	−2.71068	−0.0859775
$995$	4.00000	0.126809
$996$	0	0
$997$	23.1716	0.733851	0.366926	−	0.930250i	$-0.380410\pi$
$997$	23.1716	0.733851	0.366926	+	0.930250i	$0.380410\pi$
$998$	−4.32590	−0.136934
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.x.1.2		2
3.2	odd	2		845.2.a.g.1.1		2
13.12	even	2		585.2.a.m.1.1		2
15.14	odd	2		4225.2.a.r.1.2		2
39.2	even	12		845.2.m.f.316.2		8
39.5	even	4		845.2.c.b.506.3		4
39.8	even	4		845.2.c.b.506.2		4
39.11	even	12		845.2.m.f.316.3		8
39.17	odd	6		845.2.e.h.146.1		4
39.20	even	12		845.2.m.f.361.2		8
39.23	odd	6		845.2.e.h.191.1		4
39.29	odd	6		845.2.e.c.191.2		4
39.32	even	12		845.2.m.f.361.3		8
39.35	odd	6		845.2.e.c.146.2		4
39.38	odd	2		65.2.a.b.1.2	✓	2
52.51	odd	2		9360.2.a.cd.1.2		2
65.12	odd	4		2925.2.c.r.2224.2		4
65.38	odd	4		2925.2.c.r.2224.3		4
65.64	even	2		2925.2.a.u.1.2		2
156.155	even	2		1040.2.a.j.1.1		2
195.38	even	4		325.2.b.f.274.2		4
195.77	even	4		325.2.b.f.274.3		4
195.194	odd	2		325.2.a.i.1.1		2
273.272	even	2		3185.2.a.j.1.2		2
312.77	odd	2		4160.2.a.bf.1.1		2
312.155	even	2		4160.2.a.z.1.2		2
429.428	even	2		7865.2.a.j.1.1		2
780.779	even	2		5200.2.a.bu.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.b.1.2	✓	2	39.38	odd	2
325.2.a.i.1.1		2	195.194	odd	2
325.2.b.f.274.2		4	195.38	even	4
325.2.b.f.274.3		4	195.77	even	4
585.2.a.m.1.1		2	13.12	even	2
845.2.a.g.1.1		2	3.2	odd	2
845.2.c.b.506.2		4	39.8	even	4
845.2.c.b.506.3		4	39.5	even	4
845.2.e.c.146.2		4	39.35	odd	6
845.2.e.c.191.2		4	39.29	odd	6
845.2.e.h.146.1		4	39.17	odd	6
845.2.e.h.191.1		4	39.23	odd	6
845.2.m.f.316.2		8	39.2	even	12
845.2.m.f.316.3		8	39.11	even	12
845.2.m.f.361.2		8	39.20	even	12
845.2.m.f.361.3		8	39.32	even	12
1040.2.a.j.1.1		2	156.155	even	2
2925.2.a.u.1.2		2	65.64	even	2
2925.2.c.r.2224.2		4	65.12	odd	4
2925.2.c.r.2224.3		4	65.38	odd	4
3185.2.a.j.1.2		2	273.272	even	2
4160.2.a.z.1.2		2	312.155	even	2
4160.2.a.bf.1.1		2	312.77	odd	2
4225.2.a.r.1.2		2	15.14	odd	2
5200.2.a.bu.1.2		2	780.779	even	2
7605.2.a.x.1.2		2	1.1	even	1	trivial
7865.2.a.j.1.1		2	429.428	even	2
9360.2.a.cd.1.2		2	52.51	odd	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:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +0.828427 q^{7} -1.58579 q^{8} +O(q^{10})\)	\(q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +0.828427 q^{7} -1.58579 q^{8} +0.414214 q^{10} +0.585786 q^{11} +0.343146 q^{14} +3.00000 q^{16} +4.82843 q^{17} -3.41421 q^{19} -1.82843 q^{20} +0.242641 q^{22} +1.41421 q^{23} +1.00000 q^{25} -1.51472 q^{28} -5.65685 q^{29} -10.2426 q^{31} +4.41421 q^{32} +2.00000 q^{34} +0.828427 q^{35} -8.48528 q^{37} -1.41421 q^{38} -1.58579 q^{40} -8.82843 q^{41} +3.07107 q^{43} -1.07107 q^{44} +0.585786 q^{46} +0.828427 q^{47} -6.31371 q^{49} +0.414214 q^{50} +14.4853 q^{53} +0.585786 q^{55} -1.31371 q^{56} -2.34315 q^{58} +10.2426 q^{59} -8.00000 q^{61} -4.24264 q^{62} -4.17157 q^{64} +2.00000 q^{67} -8.82843 q^{68} +0.343146 q^{70} -7.89949 q^{71} +8.48528 q^{73} -3.51472 q^{74} +6.24264 q^{76} +0.485281 q^{77} +8.48528 q^{79} +3.00000 q^{80} -3.65685 q^{82} -8.82843 q^{83} +4.82843 q^{85} +1.27208 q^{86} -0.928932 q^{88} +6.00000 q^{89} -2.58579 q^{92} +0.343146 q^{94} -3.41421 q^{95} -3.65685 q^{97} -2.61522 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 4 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 4 * q^7 - 6 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 4 q^{7} - 6 q^{8} - 2 q^{10} + 4 q^{11} + 12 q^{14} + 6 q^{16} + 4 q^{17} - 4 q^{19} + 2 q^{20} - 8 q^{22} + 2 q^{25} - 20 q^{28} - 12 q^{31} + 6 q^{32} + 4 q^{34} - 4 q^{35} - 6 q^{40} - 12 q^{41} - 8 q^{43} + 12 q^{44} + 4 q^{46} - 4 q^{47} + 10 q^{49} - 2 q^{50} + 12 q^{53} + 4 q^{55} + 20 q^{56} - 16 q^{58} + 12 q^{59} - 16 q^{61} - 14 q^{64} + 4 q^{67} - 12 q^{68} + 12 q^{70} + 4 q^{71} - 24 q^{74} + 4 q^{76} - 16 q^{77} + 6 q^{80} + 4 q^{82} - 12 q^{83} + 4 q^{85} + 28 q^{86} - 16 q^{88} + 12 q^{89} - 8 q^{92} + 12 q^{94} - 4 q^{95} + 4 q^{97} - 42 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 4 * q^7 - 6 * q^8 - 2 * q^10 + 4 * q^11 + 12 * q^14 + 6 * q^16 + 4 * q^17 - 4 * q^19 + 2 * q^20 - 8 * q^22 + 2 * q^25 - 20 * q^28 - 12 * q^31 + 6 * q^32 + 4 * q^34 - 4 * q^35 - 6 * q^40 - 12 * q^41 - 8 * q^43 + 12 * q^44 + 4 * q^46 - 4 * q^47 + 10 * q^49 - 2 * q^50 + 12 * q^53 + 4 * q^55 + 20 * q^56 - 16 * q^58 + 12 * q^59 - 16 * q^61 - 14 * q^64 + 4 * q^67 - 12 * q^68 + 12 * q^70 + 4 * q^71 - 24 * q^74 + 4 * q^76 - 16 * q^77 + 6 * q^80 + 4 * q^82 - 12 * q^83 + 4 * q^85 + 28 * q^86 - 16 * q^88 + 12 * q^89 - 8 * q^92 + 12 * q^94 - 4 * q^95 + 4 * q^97 - 42 * q^98

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