LMFDB - Embedded newform 2601.2.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2601,2,Mod(1,2601)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2601, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2601.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2601 = 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2601.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:	$20.7690895657$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 153)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.765367$ of defining polynomial
Character	$\chi$	$=$	2601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{4} +2.93015 q^{5} -1.53073 q^{7} -3.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} -1.00000 q^{4} +2.93015 q^{5} -1.53073 q^{7} -3.00000 q^{8} +2.93015 q^{10} +2.16478 q^{11} +4.24264 q^{13} -1.53073 q^{14} -1.00000 q^{16} -2.82843 q^{19} -2.93015 q^{20} +2.16478 q^{22} +8.92177 q^{23} +3.58579 q^{25} +4.24264 q^{26} +1.53073 q^{28} +0.317025 q^{29} -3.69552 q^{31} +5.00000 q^{32} -4.48528 q^{35} -9.68714 q^{37} -2.82843 q^{38} -8.79045 q^{40} +0.317025 q^{41} +5.65685 q^{43} -2.16478 q^{44} +8.92177 q^{46} +2.82843 q^{47} -4.65685 q^{49} +3.58579 q^{50} -4.24264 q^{52} +8.24264 q^{53} +6.34315 q^{55} +4.59220 q^{56} +0.317025 q^{58} +12.4853 q^{59} +11.6662 q^{61} -3.69552 q^{62} +7.00000 q^{64} +12.4316 q^{65} +12.4853 q^{67} -4.48528 q^{70} -8.92177 q^{71} -3.82683 q^{73} -9.68714 q^{74} +2.82843 q^{76} -3.31371 q^{77} +11.9832 q^{79} -2.93015 q^{80} +0.317025 q^{82} +4.48528 q^{83} +5.65685 q^{86} -6.49435 q^{88} +13.4142 q^{89} -6.49435 q^{91} -8.92177 q^{92} +2.82843 q^{94} -8.28772 q^{95} +2.48181 q^{97} -4.65685 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 - 4 * q^4 - 12 * q^8	$ 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8} - 4 q^{16} + 20 q^{25} + 20 q^{32} + 16 q^{35} + 4 q^{49} + 20 q^{50} + 16 q^{53} + 48 q^{55} + 16 q^{59} + 28 q^{64} + 16 q^{67} + 16 q^{70} + 32 q^{77} - 16 q^{83} + 48 q^{89} + 4 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 - 4 * q^4 - 12 * q^8 - 4 * q^16 + 20 * q^25 + 20 * q^32 + 16 * q^35 + 4 * q^49 + 20 * q^50 + 16 * q^53 + 48 * q^55 + 16 * q^59 + 28 * q^64 + 16 * q^67 + 16 * q^70 + 32 * q^77 - 16 * q^83 + 48 * q^89 + 4 * 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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	2.93015	1.31040	0.655202	−	0.755454i	$-0.272584\pi$
$5$	2.93015	1.31040	0.655202	+	0.755454i	$0.272584\pi$
$6$	0	0
$7$	−1.53073	−0.578563	−0.289281	−	0.957244i	$-0.593416\pi$
$7$	−1.53073	−0.578563	−0.289281	+	0.957244i	$0.593416\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	2.93015	0.926595
$11$	2.16478	0.652707	0.326354	−	0.945248i	$-0.394180\pi$
$11$	2.16478	0.652707	0.326354	+	0.945248i	$0.394180\pi$
$12$	0	0
$13$	4.24264	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$14$	−1.53073	−0.409106
$15$	0	0
$16$	−1.00000	−0.250000
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	−2.93015	−0.655202
$21$	0	0
$22$	2.16478	0.461534
$23$	8.92177	1.86032	0.930159	−	0.367157i	$-0.119669\pi$
$23$	8.92177	1.86032	0.930159	+	0.367157i	$0.119669\pi$
$24$	0	0
$25$	3.58579	0.717157
$26$	4.24264	0.832050
$27$	0	0
$28$	1.53073	0.289281
$29$	0.317025	0.0588701	0.0294351	−	0.999567i	$-0.490629\pi$
$29$	0.317025	0.0588701	0.0294351	+	0.999567i	$0.490629\pi$
$30$	0	0
$31$	−3.69552	−0.663735	−0.331867	−	0.943326i	$-0.607679\pi$
$31$	−3.69552	−0.663735	−0.331867	+	0.943326i	$0.607679\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	0	0
$35$	−4.48528	−0.758151
$36$	0	0
$37$	−9.68714	−1.59256	−0.796278	−	0.604931i	$-0.793201\pi$
$37$	−9.68714	−1.59256	−0.796278	+	0.604931i	$0.793201\pi$
$38$	−2.82843	−0.458831
$39$	0	0
$40$	−8.79045	−1.38989
$41$	0.317025	0.0495110	0.0247555	−	0.999694i	$-0.492119\pi$
$41$	0.317025	0.0495110	0.0247555	+	0.999694i	$0.492119\pi$
$42$	0	0
$43$	5.65685	0.862662	0.431331	−	0.902194i	$-0.358044\pi$
$43$	5.65685	0.862662	0.431331	+	0.902194i	$0.358044\pi$
$44$	−2.16478	−0.326354
$45$	0	0
$46$	8.92177	1.31544
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	−4.65685	−0.665265
$50$	3.58579	0.507107
$51$	0	0
$52$	−4.24264	−0.588348
$53$	8.24264	1.13221	0.566107	−	0.824332i	$-0.308449\pi$
$53$	8.24264	1.13221	0.566107	+	0.824332i	$0.308449\pi$
$54$	0	0
$55$	6.34315	0.855310
$56$	4.59220	0.613659
$57$	0	0
$58$	0.317025	0.0416275
$59$	12.4853	1.62545	0.812723	−	0.582651i	$-0.197984\pi$
$59$	12.4853	1.62545	0.812723	+	0.582651i	$0.197984\pi$
$60$	0	0
$61$	11.6662	1.49371	0.746853	−	0.664990i	$-0.231564\pi$
$61$	11.6662	1.49371	0.746853	+	0.664990i	$0.231564\pi$
$62$	−3.69552	−0.469331
$63$	0	0
$64$	7.00000	0.875000
$65$	12.4316	1.54195
$66$	0	0
$67$	12.4853	1.52532	0.762660	−	0.646800i	$-0.223893\pi$
$67$	12.4853	1.52532	0.762660	+	0.646800i	$0.223893\pi$
$68$	0	0
$69$	0	0
$70$	−4.48528	−0.536094
$71$	−8.92177	−1.05882	−0.529410	−	0.848366i	$-0.677587\pi$
$71$	−8.92177	−1.05882	−0.529410	+	0.848366i	$0.677587\pi$
$72$	0	0
$73$	−3.82683	−0.447897	−0.223949	−	0.974601i	$-0.571895\pi$
$73$	−3.82683	−0.447897	−0.223949	+	0.974601i	$0.571895\pi$
$74$	−9.68714	−1.12611
$75$	0	0
$76$	2.82843	0.324443
$77$	−3.31371	−0.377632
$78$	0	0
$79$	11.9832	1.34822	0.674110	−	0.738631i	$-0.264527\pi$
$79$	11.9832	1.34822	0.674110	+	0.738631i	$0.264527\pi$
$80$	−2.93015	−0.327601
$81$	0	0
$82$	0.317025	0.0350096
$83$	4.48528	0.492324	0.246162	−	0.969229i	$-0.420831\pi$
$83$	4.48528	0.492324	0.246162	+	0.969229i	$0.420831\pi$
$84$	0	0
$85$	0	0
$86$	5.65685	0.609994
$87$	0	0
$88$	−6.49435	−0.692300
$89$	13.4142	1.42190	0.710952	−	0.703241i	$-0.248264\pi$
$89$	13.4142	1.42190	0.710952	+	0.703241i	$0.248264\pi$
$90$	0	0
$91$	−6.49435	−0.680793
$92$	−8.92177	−0.930159
$93$	0	0
$94$	2.82843	0.291730
$95$	−8.28772	−0.850302
$96$	0	0
$97$	2.48181	0.251990	0.125995	−	0.992031i	$-0.459788\pi$
$97$	2.48181	0.251990	0.125995	+	0.992031i	$0.459788\pi$
$98$	−4.65685	−0.470413
$99$	0	0
$100$	−3.58579	−0.358579
$101$	−16.2426	−1.61620	−0.808102	−	0.589043i	$-0.799505\pi$
$101$	−16.2426	−1.61620	−0.808102	+	0.589043i	$0.799505\pi$
$102$	0	0
$103$	16.4853	1.62434	0.812172	−	0.583419i	$-0.198285\pi$
$103$	16.4853	1.62434	0.812172	+	0.583419i	$0.198285\pi$
$104$	−12.7279	−1.24808
$105$	0	0
$106$	8.24264	0.800596
$107$	−10.4525	−1.01048	−0.505241	−	0.862978i	$-0.668596\pi$
$107$	−10.4525	−1.01048	−0.505241	+	0.862978i	$0.668596\pi$
$108$	0	0
$109$	−11.6662	−1.11742	−0.558710	−	0.829363i	$-0.688704\pi$
$109$	−11.6662	−1.11742	−0.558710	+	0.829363i	$0.688704\pi$
$110$	6.34315	0.604795
$111$	0	0
$112$	1.53073	0.144641
$113$	−3.37849	−0.317822	−0.158911	−	0.987293i	$-0.550798\pi$
$113$	−3.37849	−0.317822	−0.158911	+	0.987293i	$0.550798\pi$
$114$	0	0
$115$	26.1421	2.43777
$116$	−0.317025	−0.0294351
$117$	0	0
$118$	12.4853	1.14936
$119$	0	0
$120$	0	0
$121$	−6.31371	−0.573973
$122$	11.6662	1.05621
$123$	0	0
$124$	3.69552	0.331867
$125$	−4.14386	−0.370638
$126$	0	0
$127$	6.34315	0.562863	0.281432	−	0.959581i	$-0.409191\pi$
$127$	6.34315	0.562863	0.281432	+	0.959581i	$0.409191\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	12.4316	1.09032
$131$	1.26810	0.110795	0.0553973	−	0.998464i	$-0.482357\pi$
$131$	1.26810	0.110795	0.0553973	+	0.998464i	$0.482357\pi$
$132$	0	0
$133$	4.32957	0.375421
$134$	12.4853	1.07856
$135$	0	0
$136$	0	0
$137$	−7.75736	−0.662756	−0.331378	−	0.943498i	$-0.607514\pi$
$137$	−7.75736	−0.662756	−0.331378	+	0.943498i	$0.607514\pi$
$138$	0	0
$139$	−0.896683	−0.0760557	−0.0380278	−	0.999277i	$-0.512108\pi$
$139$	−0.896683	−0.0760557	−0.0380278	+	0.999277i	$0.512108\pi$
$140$	4.48528	0.379075
$141$	0	0
$142$	−8.92177	−0.748698
$143$	9.18440	0.768038
$144$	0	0
$145$	0.928932	0.0771436
$146$	−3.82683	−0.316711
$147$	0	0
$148$	9.68714	0.796278
$149$	17.3137	1.41839	0.709197	−	0.705010i	$-0.249058\pi$
$149$	17.3137	1.41839	0.709197	+	0.705010i	$0.249058\pi$
$150$	0	0
$151$	6.34315	0.516198	0.258099	−	0.966118i	$-0.416904\pi$
$151$	6.34315	0.516198	0.258099	+	0.966118i	$0.416904\pi$
$152$	8.48528	0.688247
$153$	0	0
$154$	−3.31371	−0.267026
$155$	−10.8284	−0.869760
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	11.9832	0.953335
$159$	0	0
$160$	14.6508	1.15824
$161$	−13.6569	−1.07631
$162$	0	0
$163$	12.6173	0.988262	0.494131	−	0.869387i	$-0.335486\pi$
$163$	12.6173	0.988262	0.494131	+	0.869387i	$0.335486\pi$
$164$	−0.317025	−0.0247555
$165$	0	0
$166$	4.48528	0.348125
$167$	−6.75699	−0.522871	−0.261436	−	0.965221i	$-0.584196\pi$
$167$	−6.75699	−0.522871	−0.261436	+	0.965221i	$0.584196\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	0	0
$172$	−5.65685	−0.431331
$173$	4.46088	0.339155	0.169577	−	0.985517i	$-0.445760\pi$
$173$	4.46088	0.339155	0.169577	+	0.985517i	$0.445760\pi$
$174$	0	0
$175$	−5.48888	−0.414921
$176$	−2.16478	−0.163177
$177$	0	0
$178$	13.4142	1.00544
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	5.09494	0.378704	0.189352	−	0.981909i	$-0.439361\pi$
$181$	5.09494	0.378704	0.189352	+	0.981909i	$0.439361\pi$
$182$	−6.49435	−0.481393
$183$	0	0
$184$	−26.7653	−1.97316
$185$	−28.3848	−2.08689
$186$	0	0
$187$	0	0
$188$	−2.82843	−0.206284
$189$	0	0
$190$	−8.28772	−0.601254
$191$	−14.8284	−1.07295	−0.536474	−	0.843917i	$-0.680244\pi$
$191$	−14.8284	−1.07295	−0.536474	+	0.843917i	$0.680244\pi$
$192$	0	0
$193$	−12.9343	−0.931032	−0.465516	−	0.885039i	$-0.654131\pi$
$193$	−12.9343	−0.931032	−0.465516	+	0.885039i	$0.654131\pi$
$194$	2.48181	0.178184
$195$	0	0
$196$	4.65685	0.332632
$197$	8.86738	0.631774	0.315887	−	0.948797i	$-0.397698\pi$
$197$	8.86738	0.631774	0.315887	+	0.948797i	$0.397698\pi$
$198$	0	0
$199$	1.53073	0.108511	0.0542554	−	0.998527i	$-0.482721\pi$
$199$	1.53073	0.108511	0.0542554	+	0.998527i	$0.482721\pi$
$200$	−10.7574	−0.760660
$201$	0	0
$202$	−16.2426	−1.14283
$203$	−0.485281	−0.0340601
$204$	0	0
$205$	0.928932	0.0648794
$206$	16.4853	1.14858
$207$	0	0
$208$	−4.24264	−0.294174
$209$	−6.12293	−0.423532
$210$	0	0
$211$	−3.43289	−0.236330	−0.118165	−	0.992994i	$-0.537701\pi$
$211$	−3.43289	−0.236330	−0.118165	+	0.992994i	$0.537701\pi$
$212$	−8.24264	−0.566107
$213$	0	0
$214$	−10.4525	−0.714518
$215$	16.5754	1.13044
$216$	0	0
$217$	5.65685	0.384012
$218$	−11.6662	−0.790136
$219$	0	0
$220$	−6.34315	−0.427655
$221$	0	0
$222$	0	0
$223$	9.17157	0.614174	0.307087	−	0.951681i	$-0.400646\pi$
$223$	9.17157	0.614174	0.307087	+	0.951681i	$0.400646\pi$
$224$	−7.65367	−0.511382
$225$	0	0
$226$	−3.37849	−0.224734
$227$	13.5140	0.896954	0.448477	−	0.893794i	$-0.351967\pi$
$227$	13.5140	0.896954	0.448477	+	0.893794i	$0.351967\pi$
$228$	0	0
$229$	−22.6274	−1.49526	−0.747631	−	0.664114i	$-0.768809\pi$
$229$	−22.6274	−1.49526	−0.747631	+	0.664114i	$0.768809\pi$
$230$	26.1421	1.72376
$231$	0	0
$232$	−0.951076	−0.0624412
$233$	1.66205	0.108885	0.0544423	−	0.998517i	$-0.482662\pi$
$233$	1.66205	0.108885	0.0544423	+	0.998517i	$0.482662\pi$
$234$	0	0
$235$	8.28772	0.540631
$236$	−12.4853	−0.812723
$237$	0	0
$238$	0	0
$239$	20.4853	1.32508	0.662541	−	0.749025i	$-0.269478\pi$
$239$	20.4853	1.32508	0.662541	+	0.749025i	$0.269478\pi$
$240$	0	0
$241$	−12.9343	−0.833172	−0.416586	−	0.909096i	$-0.636774\pi$
$241$	−12.9343	−0.833172	−0.416586	+	0.909096i	$0.636774\pi$
$242$	−6.31371	−0.405861
$243$	0	0
$244$	−11.6662	−0.746853
$245$	−13.6453	−0.871765
$246$	0	0
$247$	−12.0000	−0.763542
$248$	11.0866	0.703997
$249$	0	0
$250$	−4.14386	−0.262081
$251$	−22.6274	−1.42823	−0.714115	−	0.700028i	$-0.753171\pi$
$251$	−22.6274	−1.42823	−0.714115	+	0.700028i	$0.753171\pi$
$252$	0	0
$253$	19.3137	1.21424
$254$	6.34315	0.398004
$255$	0	0
$256$	−17.0000	−1.06250
$257$	2.97056	0.185299	0.0926493	−	0.995699i	$-0.470466\pi$
$257$	2.97056	0.185299	0.0926493	+	0.995699i	$0.470466\pi$
$258$	0	0
$259$	14.8284	0.921394
$260$	−12.4316	−0.770974
$261$	0	0
$262$	1.26810	0.0783436
$263$	−11.5147	−0.710028	−0.355014	−	0.934861i	$-0.615524\pi$
$263$	−11.5147	−0.710028	−0.355014	+	0.934861i	$0.615524\pi$
$264$	0	0
$265$	24.1522	1.48366
$266$	4.32957	0.265463
$267$	0	0
$268$	−12.4853	−0.762660
$269$	−16.8925	−1.02995	−0.514976	−	0.857205i	$-0.672199\pi$
$269$	−16.8925	−1.02995	−0.514976	+	0.857205i	$0.672199\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	0	0
$273$	0	0
$274$	−7.75736	−0.468639
$275$	7.76245	0.468094
$276$	0	0
$277$	5.35757	0.321905	0.160953	−	0.986962i	$-0.448543\pi$
$277$	5.35757	0.321905	0.160953	+	0.986962i	$0.448543\pi$
$278$	−0.896683	−0.0537795
$279$	0	0
$280$	13.4558	0.804140
$281$	8.72792	0.520664	0.260332	−	0.965519i	$-0.416168\pi$
$281$	8.72792	0.520664	0.260332	+	0.965519i	$0.416168\pi$
$282$	0	0
$283$	−14.7821	−0.878703	−0.439352	−	0.898315i	$-0.644792\pi$
$283$	−14.7821	−0.878703	−0.439352	+	0.898315i	$0.644792\pi$
$284$	8.92177	0.529410
$285$	0	0
$286$	9.18440	0.543085
$287$	−0.485281	−0.0286453
$288$	0	0
$289$	0	0
$290$	0.928932	0.0545488
$291$	0	0
$292$	3.82683	0.223949
$293$	−18.3431	−1.07162	−0.535809	−	0.844339i	$-0.679993\pi$
$293$	−18.3431	−1.07162	−0.535809	+	0.844339i	$0.679993\pi$
$294$	0	0
$295$	36.5838	2.12999
$296$	29.0614	1.68916
$297$	0	0
$298$	17.3137	1.00296
$299$	37.8519	2.18903
$300$	0	0
$301$	−8.65914	−0.499104
$302$	6.34315	0.365007
$303$	0	0
$304$	2.82843	0.162221
$305$	34.1838	1.95736
$306$	0	0
$307$	24.9706	1.42515	0.712573	−	0.701598i	$-0.247530\pi$
$307$	24.9706	1.42515	0.712573	+	0.701598i	$0.247530\pi$
$308$	3.31371	0.188816
$309$	0	0
$310$	−10.8284	−0.615013
$311$	−5.86030	−0.332307	−0.166154	−	0.986100i	$-0.553135\pi$
$311$	−5.86030	−0.332307	−0.166154	+	0.986100i	$0.553135\pi$
$312$	0	0
$313$	−21.4077	−1.21004	−0.605019	−	0.796211i	$-0.706834\pi$
$313$	−21.4077	−1.21004	−0.605019	+	0.796211i	$0.706834\pi$
$314$	0	0
$315$	0	0
$316$	−11.9832	−0.674110
$317$	−26.8966	−1.51067	−0.755333	−	0.655342i	$-0.772525\pi$
$317$	−26.8966	−1.51067	−0.755333	+	0.655342i	$0.772525\pi$
$318$	0	0
$319$	0.686292	0.0384249
$320$	20.5111	1.14660
$321$	0	0
$322$	−13.6569	−0.761067
$323$	0	0
$324$	0	0
$325$	15.2132	0.843877
$326$	12.6173	0.698807
$327$	0	0
$328$	−0.951076	−0.0525144
$329$	−4.32957	−0.238697
$330$	0	0
$331$	−0.686292	−0.0377220	−0.0188610	−	0.999822i	$-0.506004\pi$
$331$	−0.686292	−0.0377220	−0.0188610	+	0.999822i	$0.506004\pi$
$332$	−4.48528	−0.246162
$333$	0	0
$334$	−6.75699	−0.369726
$335$	36.5838	1.99878
$336$	0	0
$337$	14.8365	0.808194	0.404097	−	0.914716i	$-0.367586\pi$
$337$	14.8365	0.808194	0.404097	+	0.914716i	$0.367586\pi$
$338$	5.00000	0.271964
$339$	0	0
$340$	0	0
$341$	−8.00000	−0.433224
$342$	0	0
$343$	17.8435	0.963461
$344$	−16.9706	−0.914991
$345$	0	0
$346$	4.46088	0.239819
$347$	−5.22625	−0.280560	−0.140280	−	0.990112i	$-0.544800\pi$
$347$	−5.22625	−0.280560	−0.140280	+	0.990112i	$0.544800\pi$
$348$	0	0
$349$	−7.07107	−0.378506	−0.189253	−	0.981928i	$-0.560607\pi$
$349$	−7.07107	−0.378506	−0.189253	+	0.981928i	$0.560607\pi$
$350$	−5.48888	−0.293393
$351$	0	0
$352$	10.8239	0.576917
$353$	−5.65685	−0.301084	−0.150542	−	0.988604i	$-0.548102\pi$
$353$	−5.65685	−0.301084	−0.150542	+	0.988604i	$0.548102\pi$
$354$	0	0
$355$	−26.1421	−1.38748
$356$	−13.4142	−0.710952
$357$	0	0
$358$	−16.0000	−0.845626
$359$	−0.970563	−0.0512243	−0.0256122	−	0.999672i	$-0.508153\pi$
$359$	−0.970563	−0.0512243	−0.0256122	+	0.999672i	$0.508153\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	5.09494	0.267784
$363$	0	0
$364$	6.49435	0.340397
$365$	−11.2132	−0.586926
$366$	0	0
$367$	−20.6424	−1.07752	−0.538762	−	0.842458i	$-0.681108\pi$
$367$	−20.6424	−1.07752	−0.538762	+	0.842458i	$0.681108\pi$
$368$	−8.92177	−0.465079
$369$	0	0
$370$	−28.3848	−1.47565
$371$	−12.6173	−0.655057
$372$	0	0
$373$	−5.21320	−0.269929	−0.134965	−	0.990850i	$-0.543092\pi$
$373$	−5.21320	−0.269929	−0.134965	+	0.990850i	$0.543092\pi$
$374$	0	0
$375$	0	0
$376$	−8.48528	−0.437595
$377$	1.34502	0.0692723
$378$	0	0
$379$	18.7402	0.962621	0.481310	−	0.876550i	$-0.340161\pi$
$379$	18.7402	0.962621	0.481310	+	0.876550i	$0.340161\pi$
$380$	8.28772	0.425151
$381$	0	0
$382$	−14.8284	−0.758688
$383$	−24.4853	−1.25114	−0.625570	−	0.780168i	$-0.715133\pi$
$383$	−24.4853	−1.25114	−0.625570	+	0.780168i	$0.715133\pi$
$384$	0	0
$385$	−9.70967	−0.494850
$386$	−12.9343	−0.658339
$387$	0	0
$388$	−2.48181	−0.125995
$389$	−8.97056	−0.454826	−0.227413	−	0.973798i	$-0.573027\pi$
$389$	−8.97056	−0.454826	−0.227413	+	0.973798i	$0.573027\pi$
$390$	0	0
$391$	0	0
$392$	13.9706	0.705620
$393$	0	0
$394$	8.86738	0.446732
$395$	35.1127	1.76671
$396$	0	0
$397$	17.0782	0.857129	0.428564	−	0.903511i	$-0.359019\pi$
$397$	17.0782	0.857129	0.428564	+	0.903511i	$0.359019\pi$
$398$	1.53073	0.0767287
$399$	0	0
$400$	−3.58579	−0.179289
$401$	−22.4901	−1.12310	−0.561552	−	0.827442i	$-0.689795\pi$
$401$	−22.4901	−1.12310	−0.561552	+	0.827442i	$0.689795\pi$
$402$	0	0
$403$	−15.6788	−0.781014
$404$	16.2426	0.808102
$405$	0	0
$406$	−0.485281	−0.0240841
$407$	−20.9706	−1.03947
$408$	0	0
$409$	0.970563	0.0479912	0.0239956	−	0.999712i	$-0.492361\pi$
$409$	0.970563	0.0479912	0.0239956	+	0.999712i	$0.492361\pi$
$410$	0.928932	0.0458767
$411$	0	0
$412$	−16.4853	−0.812172
$413$	−19.1116	−0.940422
$414$	0	0
$415$	13.1426	0.645143
$416$	21.2132	1.04006
$417$	0	0
$418$	−6.12293	−0.299483
$419$	27.0279	1.32040	0.660201	−	0.751089i	$-0.270471\pi$
$419$	27.0279	1.32040	0.660201	+	0.751089i	$0.270471\pi$
$420$	0	0
$421$	−11.2721	−0.549367	−0.274684	−	0.961535i	$-0.588573\pi$
$421$	−11.2721	−0.549367	−0.274684	+	0.961535i	$0.588573\pi$
$422$	−3.43289	−0.167110
$423$	0	0
$424$	−24.7279	−1.20089
$425$	0	0
$426$	0	0
$427$	−17.8579	−0.864203
$428$	10.4525	0.505241
$429$	0	0
$430$	16.5754	0.799339
$431$	−31.6201	−1.52309	−0.761544	−	0.648113i	$-0.775559\pi$
$431$	−31.6201	−1.52309	−0.761544	+	0.648113i	$0.775559\pi$
$432$	0	0
$433$	0.343146	0.0164905	0.00824527	−	0.999966i	$-0.497375\pi$
$433$	0.343146	0.0164905	0.00824527	+	0.999966i	$0.497375\pi$
$434$	5.65685	0.271538
$435$	0	0
$436$	11.6662	0.558710
$437$	−25.2346	−1.20713
$438$	0	0
$439$	5.48888	0.261970	0.130985	−	0.991384i	$-0.458186\pi$
$439$	5.48888	0.261970	0.130985	+	0.991384i	$0.458186\pi$
$440$	−19.0294	−0.907193
$441$	0	0
$442$	0	0
$443$	4.97056	0.236159	0.118079	−	0.993004i	$-0.462326\pi$
$443$	4.97056	0.236159	0.118079	+	0.993004i	$0.462326\pi$
$444$	0	0
$445$	39.3057	1.86327
$446$	9.17157	0.434287
$447$	0	0
$448$	−10.7151	−0.506243
$449$	−20.5111	−0.967977	−0.483988	−	0.875074i	$-0.660812\pi$
$449$	−20.5111	−0.967977	−0.483988	+	0.875074i	$0.660812\pi$
$450$	0	0
$451$	0.686292	0.0323162
$452$	3.37849	0.158911
$453$	0	0
$454$	13.5140	0.634242
$455$	−19.0294	−0.892114
$456$	0	0
$457$	28.2843	1.32308	0.661541	−	0.749909i	$-0.269903\pi$
$457$	28.2843	1.32308	0.661541	+	0.749909i	$0.269903\pi$
$458$	−22.6274	−1.05731
$459$	0	0
$460$	−26.1421	−1.21888
$461$	−32.7279	−1.52429	−0.762146	−	0.647406i	$-0.775854\pi$
$461$	−32.7279	−1.52429	−0.762146	+	0.647406i	$0.775854\pi$
$462$	0	0
$463$	−3.51472	−0.163343	−0.0816714	−	0.996659i	$-0.526026\pi$
$463$	−3.51472	−0.163343	−0.0816714	+	0.996659i	$0.526026\pi$
$464$	−0.317025	−0.0147175
$465$	0	0
$466$	1.66205	0.0769930
$467$	−0.970563	−0.0449123	−0.0224561	−	0.999748i	$-0.507149\pi$
$467$	−0.970563	−0.0449123	−0.0224561	+	0.999748i	$0.507149\pi$
$468$	0	0
$469$	−19.1116	−0.882494
$470$	8.28772	0.382284
$471$	0	0
$472$	−37.4558	−1.72404
$473$	12.2459	0.563066
$474$	0	0
$475$	−10.1421	−0.465353
$476$	0	0
$477$	0	0
$478$	20.4853	0.936975
$479$	−31.0949	−1.42076	−0.710381	−	0.703818i	$-0.751477\pi$
$479$	−31.0949	−1.42076	−0.710381	+	0.703818i	$0.751477\pi$
$480$	0	0
$481$	−41.0990	−1.87395
$482$	−12.9343	−0.589142
$483$	0	0
$484$	6.31371	0.286987
$485$	7.27208	0.330208
$486$	0	0
$487$	−41.5474	−1.88269	−0.941346	−	0.337443i	$-0.890438\pi$
$487$	−41.5474	−1.88269	−0.941346	+	0.337443i	$0.890438\pi$
$488$	−34.9986	−1.58431
$489$	0	0
$490$	−13.6453	−0.616431
$491$	29.9411	1.35122	0.675612	−	0.737257i	$-0.263879\pi$
$491$	29.9411	1.35122	0.675612	+	0.737257i	$0.263879\pi$
$492$	0	0
$493$	0	0
$494$	−12.0000	−0.539906
$495$	0	0
$496$	3.69552	0.165934
$497$	13.6569	0.612594
$498$	0	0
$499$	27.7708	1.24319	0.621595	−	0.783338i	$-0.286485\pi$
$499$	27.7708	1.24319	0.621595	+	0.783338i	$0.286485\pi$
$500$	4.14386	0.185319
$501$	0	0
$502$	−22.6274	−1.00991
$503$	−43.7122	−1.94903	−0.974515	−	0.224324i	$-0.927983\pi$
$503$	−43.7122	−1.94903	−0.974515	+	0.224324i	$0.927983\pi$
$504$	0	0
$505$	−47.5934	−2.11788
$506$	19.3137	0.858599
$507$	0	0
$508$	−6.34315	−0.281432
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	5.85786	0.259137
$512$	−11.0000	−0.486136
$513$	0	0
$514$	2.97056	0.131026
$515$	48.3044	2.12854
$516$	0	0
$517$	6.12293	0.269286
$518$	14.8284	0.651524
$519$	0	0
$520$	−37.2947	−1.63548
$521$	−8.52782	−0.373611	−0.186805	−	0.982397i	$-0.559813\pi$
$521$	−8.52782	−0.373611	−0.186805	+	0.982397i	$0.559813\pi$
$522$	0	0
$523$	−8.48528	−0.371035	−0.185518	−	0.982641i	$-0.559396\pi$
$523$	−8.48528	−0.371035	−0.185518	+	0.982641i	$0.559396\pi$
$524$	−1.26810	−0.0553973
$525$	0	0
$526$	−11.5147	−0.502066
$527$	0	0
$528$	0	0
$529$	56.5980	2.46078
$530$	24.1522	1.04910
$531$	0	0
$532$	−4.32957	−0.187711
$533$	1.34502	0.0582595
$534$	0	0
$535$	−30.6274	−1.32414
$536$	−37.4558	−1.61785
$537$	0	0
$538$	−16.8925	−0.728286
$539$	−10.0811	−0.434223
$540$	0	0
$541$	18.8715	0.811351	0.405675	−	0.914017i	$-0.367036\pi$
$541$	18.8715	0.811351	0.405675	+	0.914017i	$0.367036\pi$
$542$	4.00000	0.171815
$543$	0	0
$544$	0	0
$545$	−34.1838	−1.46427
$546$	0	0
$547$	−32.2542	−1.37909	−0.689545	−	0.724243i	$-0.742189\pi$
$547$	−32.2542	−1.37909	−0.689545	+	0.724243i	$0.742189\pi$
$548$	7.75736	0.331378
$549$	0	0
$550$	7.76245	0.330992
$551$	−0.896683	−0.0382000
$552$	0	0
$553$	−18.3431	−0.780030
$554$	5.35757	0.227621
$555$	0	0
$556$	0.896683	0.0380278
$557$	36.0416	1.52713	0.763566	−	0.645729i	$-0.223447\pi$
$557$	36.0416	1.52713	0.763566	+	0.645729i	$0.223447\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	4.48528	0.189538
$561$	0	0
$562$	8.72792	0.368165
$563$	−4.48528	−0.189032	−0.0945160	−	0.995523i	$-0.530130\pi$
$563$	−4.48528	−0.189032	−0.0945160	+	0.995523i	$0.530130\pi$
$564$	0	0
$565$	−9.89949	−0.416475
$566$	−14.7821	−0.621337
$567$	0	0
$568$	26.7653	1.12305
$569$	−0.242641	−0.0101720	−0.00508601	−	0.999987i	$-0.501619\pi$
$569$	−0.242641	−0.0101720	−0.00508601	+	0.999987i	$0.501619\pi$
$570$	0	0
$571$	19.1116	0.799797	0.399899	−	0.916559i	$-0.369045\pi$
$571$	19.1116	0.799797	0.399899	+	0.916559i	$0.369045\pi$
$572$	−9.18440	−0.384019
$573$	0	0
$574$	−0.485281	−0.0202553
$575$	31.9916	1.33414
$576$	0	0
$577$	4.72792	0.196826	0.0984130	−	0.995146i	$-0.468623\pi$
$577$	4.72792	0.196826	0.0984130	+	0.995146i	$0.468623\pi$
$578$	0	0
$579$	0	0
$580$	−0.928932	−0.0385718
$581$	−6.86577	−0.284840
$582$	0	0
$583$	17.8435	0.739004
$584$	11.4805	0.475067
$585$	0	0
$586$	−18.3431	−0.757748
$587$	12.9706	0.535352	0.267676	−	0.963509i	$-0.413744\pi$
$587$	12.9706	0.535352	0.267676	+	0.963509i	$0.413744\pi$
$588$	0	0
$589$	10.4525	0.430688
$590$	36.5838	1.50613
$591$	0	0
$592$	9.68714	0.398139
$593$	32.7279	1.34397	0.671987	−	0.740563i	$-0.265441\pi$
$593$	32.7279	1.34397	0.671987	+	0.740563i	$0.265441\pi$
$594$	0	0
$595$	0	0
$596$	−17.3137	−0.709197
$597$	0	0
$598$	37.8519	1.54788
$599$	−16.2843	−0.665357	−0.332679	−	0.943040i	$-0.607952\pi$
$599$	−16.2843	−0.665357	−0.332679	+	0.943040i	$0.607952\pi$
$600$	0	0
$601$	33.3141	1.35891	0.679454	−	0.733718i	$-0.262217\pi$
$601$	33.3141	1.35891	0.679454	+	0.733718i	$0.262217\pi$
$602$	−8.65914	−0.352920
$603$	0	0
$604$	−6.34315	−0.258099
$605$	−18.5001	−0.752137
$606$	0	0
$607$	−45.5055	−1.84701	−0.923506	−	0.383583i	$-0.874690\pi$
$607$	−45.5055	−1.84701	−0.923506	+	0.383583i	$0.874690\pi$
$608$	−14.1421	−0.573539
$609$	0	0
$610$	34.1838	1.38406
$611$	12.0000	0.485468
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	24.9706	1.00773
$615$	0	0
$616$	9.94113	0.400539
$617$	−14.6508	−0.589817	−0.294909	−	0.955525i	$-0.595289\pi$
$617$	−14.6508	−0.589817	−0.294909	+	0.955525i	$0.595289\pi$
$618$	0	0
$619$	−25.6060	−1.02919	−0.514596	−	0.857433i	$-0.672058\pi$
$619$	−25.6060	−1.02919	−0.514596	+	0.857433i	$0.672058\pi$
$620$	10.8284	0.434880
$621$	0	0
$622$	−5.86030	−0.234977
$623$	−20.5336	−0.822661
$624$	0	0
$625$	−30.0711	−1.20284
$626$	−21.4077	−0.855625
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	9.17157	0.365115	0.182557	−	0.983195i	$-0.441563\pi$
$631$	9.17157	0.365115	0.182557	+	0.983195i	$0.441563\pi$
$632$	−35.9497	−1.43000
$633$	0	0
$634$	−26.8966	−1.06820
$635$	18.5864	0.737578
$636$	0	0
$637$	−19.7574	−0.782815
$638$	0.686292	0.0271705
$639$	0	0
$640$	−8.79045	−0.347473
$641$	23.8352	0.941432	0.470716	−	0.882285i	$-0.343996\pi$
$641$	23.8352	0.941432	0.470716	+	0.882285i	$0.343996\pi$
$642$	0	0
$643$	−23.9665	−0.945146	−0.472573	−	0.881292i	$-0.656675\pi$
$643$	−23.9665	−0.945146	−0.472573	+	0.881292i	$0.656675\pi$
$644$	13.6569	0.538155
$645$	0	0
$646$	0	0
$647$	44.4853	1.74890	0.874448	−	0.485118i	$-0.161224\pi$
$647$	44.4853	1.74890	0.874448	+	0.485118i	$0.161224\pi$
$648$	0	0
$649$	27.0279	1.06094
$650$	15.2132	0.596711
$651$	0	0
$652$	−12.6173	−0.494131
$653$	−10.7695	−0.421444	−0.210722	−	0.977546i	$-0.567582\pi$
$653$	−10.7695	−0.421444	−0.210722	+	0.977546i	$0.567582\pi$
$654$	0	0
$655$	3.71573	0.145186
$656$	−0.317025	−0.0123778
$657$	0	0
$658$	−4.32957	−0.168784
$659$	−2.14214	−0.0834458	−0.0417229	−	0.999129i	$-0.513285\pi$
$659$	−2.14214	−0.0834458	−0.0417229	+	0.999129i	$0.513285\pi$
$660$	0	0
$661$	−9.89949	−0.385046	−0.192523	−	0.981292i	$-0.561667\pi$
$661$	−9.89949	−0.385046	−0.192523	+	0.981292i	$0.561667\pi$
$662$	−0.686292	−0.0266735
$663$	0	0
$664$	−13.4558	−0.522188
$665$	12.6863	0.491953
$666$	0	0
$667$	2.82843	0.109517
$668$	6.75699	0.261436
$669$	0	0
$670$	36.5838	1.41335
$671$	25.2548	0.974952
$672$	0	0
$673$	−41.8644	−1.61375	−0.806877	−	0.590719i	$-0.798844\pi$
$673$	−41.8644	−1.61375	−0.806877	+	0.590719i	$0.798844\pi$
$674$	14.8365	0.571479
$675$	0	0
$676$	−5.00000	−0.192308
$677$	17.7122	0.680736	0.340368	−	0.940292i	$-0.389448\pi$
$677$	17.7122	0.680736	0.340368	+	0.940292i	$0.389448\pi$
$678$	0	0
$679$	−3.79899	−0.145792
$680$	0	0
$681$	0	0
$682$	−8.00000	−0.306336
$683$	2.16478	0.0828332	0.0414166	−	0.999142i	$-0.486813\pi$
$683$	2.16478	0.0828332	0.0414166	+	0.999142i	$0.486813\pi$
$684$	0	0
$685$	−22.7302	−0.868478
$686$	17.8435	0.681270
$687$	0	0
$688$	−5.65685	−0.215666
$689$	34.9706	1.33227
$690$	0	0
$691$	−17.4721	−0.664671	−0.332335	−	0.943161i	$-0.607837\pi$
$691$	−17.4721	−0.664671	−0.332335	+	0.943161i	$0.607837\pi$
$692$	−4.46088	−0.169577
$693$	0	0
$694$	−5.22625	−0.198386
$695$	−2.62742	−0.0996636
$696$	0	0
$697$	0	0
$698$	−7.07107	−0.267644
$699$	0	0
$700$	5.48888	0.207460
$701$	−10.5858	−0.399820	−0.199910	−	0.979814i	$-0.564065\pi$
$701$	−10.5858	−0.399820	−0.199910	+	0.979814i	$0.564065\pi$
$702$	0	0
$703$	27.3994	1.03339
$704$	15.1535	0.571119
$705$	0	0
$706$	−5.65685	−0.212899
$707$	24.8632	0.935075
$708$	0	0
$709$	12.2233	0.459057	0.229529	−	0.973302i	$-0.426282\pi$
$709$	12.2233	0.459057	0.229529	+	0.973302i	$0.426282\pi$
$710$	−26.1421	−0.981097
$711$	0	0
$712$	−40.2426	−1.50816
$713$	−32.9706	−1.23476
$714$	0	0
$715$	26.9117	1.00644
$716$	16.0000	0.597948
$717$	0	0
$718$	−0.970563	−0.0362211
$719$	31.9916	1.19308	0.596542	−	0.802582i	$-0.296541\pi$
$719$	31.9916	1.19308	0.596542	+	0.802582i	$0.296541\pi$
$720$	0	0
$721$	−25.2346	−0.939785
$722$	−11.0000	−0.409378
$723$	0	0
$724$	−5.09494	−0.189352
$725$	1.13679	0.0422191
$726$	0	0
$727$	−16.9706	−0.629403	−0.314702	−	0.949191i	$-0.601904\pi$
$727$	−16.9706	−0.629403	−0.314702	+	0.949191i	$0.601904\pi$
$728$	19.4831	0.722090
$729$	0	0
$730$	−11.2132	−0.415019
$731$	0	0
$732$	0	0
$733$	−5.61522	−0.207403	−0.103702	−	0.994608i	$-0.533069\pi$
$733$	−5.61522	−0.207403	−0.103702	+	0.994608i	$0.533069\pi$
$734$	−20.6424	−0.761924
$735$	0	0
$736$	44.6088	1.64430
$737$	27.0279	0.995587
$738$	0	0
$739$	26.8284	0.986900	0.493450	−	0.869774i	$-0.335736\pi$
$739$	26.8284	0.986900	0.493450	+	0.869774i	$0.335736\pi$
$740$	28.3848	1.04345
$741$	0	0
$742$	−12.6173	−0.463195
$743$	−22.8072	−0.836714	−0.418357	−	0.908283i	$-0.637394\pi$
$743$	−22.8072	−0.836714	−0.418357	+	0.908283i	$0.637394\pi$
$744$	0	0
$745$	50.7318	1.85867
$746$	−5.21320	−0.190869
$747$	0	0
$748$	0	0
$749$	16.0000	0.584627
$750$	0	0
$751$	38.8573	1.41792	0.708962	−	0.705247i	$-0.249164\pi$
$751$	38.8573	1.41792	0.708962	+	0.705247i	$0.249164\pi$
$752$	−2.82843	−0.103142
$753$	0	0
$754$	1.34502	0.0489829
$755$	18.5864	0.676428
$756$	0	0
$757$	−35.3137	−1.28350	−0.641749	−	0.766915i	$-0.721791\pi$
$757$	−35.3137	−1.28350	−0.641749	+	0.766915i	$0.721791\pi$
$758$	18.7402	0.680676
$759$	0	0
$760$	24.8632	0.901882
$761$	−14.8701	−0.539039	−0.269520	−	0.962995i	$-0.586865\pi$
$761$	−14.8701	−0.539039	−0.269520	+	0.962995i	$0.586865\pi$
$762$	0	0
$763$	17.8579	0.646498
$764$	14.8284	0.536474
$765$	0	0
$766$	−24.4853	−0.884689
$767$	52.9706	1.91266
$768$	0	0
$769$	−45.2132	−1.63043	−0.815215	−	0.579159i	$-0.803381\pi$
$769$	−45.2132	−1.63043	−0.815215	+	0.579159i	$0.803381\pi$
$770$	−9.70967	−0.349912
$771$	0	0
$772$	12.9343	0.465516
$773$	8.97056	0.322649	0.161324	−	0.986901i	$-0.448423\pi$
$773$	8.97056	0.322649	0.161324	+	0.986901i	$0.448423\pi$
$774$	0	0
$775$	−13.2513	−0.476002
$776$	−7.44543	−0.267275
$777$	0	0
$778$	−8.97056	−0.321610
$779$	−0.896683	−0.0321270
$780$	0	0
$781$	−19.3137	−0.691099
$782$	0	0
$783$	0	0
$784$	4.65685	0.166316
$785$	0	0
$786$	0	0
$787$	−3.43289	−0.122369	−0.0611846	−	0.998126i	$-0.519488\pi$
$787$	−3.43289	−0.122369	−0.0611846	+	0.998126i	$0.519488\pi$
$788$	−8.86738	−0.315887
$789$	0	0
$790$	35.1127	1.24925
$791$	5.17157	0.183880
$792$	0	0
$793$	49.4955	1.75764
$794$	17.0782	0.606082
$795$	0	0
$796$	−1.53073	−0.0542554
$797$	17.2132	0.609723	0.304861	−	0.952397i	$-0.401390\pi$
$797$	17.2132	0.609723	0.304861	+	0.952397i	$0.401390\pi$
$798$	0	0
$799$	0	0
$800$	17.9289	0.633883
$801$	0	0
$802$	−22.4901	−0.794154
$803$	−8.28427	−0.292346
$804$	0	0
$805$	−40.0166	−1.41040
$806$	−15.6788	−0.552261
$807$	0	0
$808$	48.7279	1.71424
$809$	16.7068	0.587378	0.293689	−	0.955901i	$-0.405117\pi$
$809$	16.7068	0.587378	0.293689	+	0.955901i	$0.405117\pi$
$810$	0	0
$811$	19.1116	0.671100	0.335550	−	0.942022i	$-0.391078\pi$
$811$	19.1116	0.671100	0.335550	+	0.942022i	$0.391078\pi$
$812$	0.485281	0.0170300
$813$	0	0
$814$	−20.9706	−0.735018
$815$	36.9706	1.29502
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0.970563	0.0339349
$819$	0	0
$820$	−0.928932	−0.0324397
$821$	32.9426	1.14971	0.574853	−	0.818257i	$-0.305059\pi$
$821$	32.9426	1.14971	0.574853	+	0.818257i	$0.305059\pi$
$822$	0	0
$823$	14.1480	0.493169	0.246585	−	0.969121i	$-0.420692\pi$
$823$	14.1480	0.493169	0.246585	+	0.969121i	$0.420692\pi$
$824$	−49.4558	−1.72288
$825$	0	0
$826$	−19.1116	−0.664979
$827$	37.3266	1.29797	0.648987	−	0.760800i	$-0.275193\pi$
$827$	37.3266	1.29797	0.648987	+	0.760800i	$0.275193\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	13.1426	0.456185
$831$	0	0
$832$	29.6985	1.02961
$833$	0	0
$834$	0	0
$835$	−19.7990	−0.685172
$836$	6.12293	0.211766
$837$	0	0
$838$	27.0279	0.933665
$839$	11.6118	0.400885	0.200442	−	0.979706i	$-0.435762\pi$
$839$	11.6118	0.400885	0.200442	+	0.979706i	$0.435762\pi$
$840$	0	0
$841$	−28.8995	−0.996534
$842$	−11.2721	−0.388461
$843$	0	0
$844$	3.43289	0.118165
$845$	14.6508	0.504001
$846$	0	0
$847$	9.66461	0.332080
$848$	−8.24264	−0.283053
$849$	0	0
$850$	0	0
$851$	−86.4264	−2.96266
$852$	0	0
$853$	−5.09494	−0.174447	−0.0872236	−	0.996189i	$-0.527799\pi$
$853$	−5.09494	−0.174447	−0.0872236	+	0.996189i	$0.527799\pi$
$854$	−17.8579	−0.611084
$855$	0	0
$856$	31.3575	1.07178
$857$	7.59928	0.259586	0.129793	−	0.991541i	$-0.458569\pi$
$857$	7.59928	0.259586	0.129793	+	0.991541i	$0.458569\pi$
$858$	0	0
$859$	34.6274	1.18147	0.590736	−	0.806865i	$-0.298837\pi$
$859$	34.6274	1.18147	0.590736	+	0.806865i	$0.298837\pi$
$860$	−16.5754	−0.565218
$861$	0	0
$862$	−31.6201	−1.07699
$863$	40.2843	1.37129	0.685646	−	0.727935i	$-0.259520\pi$
$863$	40.2843	1.37129	0.685646	+	0.727935i	$0.259520\pi$
$864$	0	0
$865$	13.0711	0.444430
$866$	0.343146	0.0116606
$867$	0	0
$868$	−5.65685	−0.192006
$869$	25.9411	0.879992
$870$	0	0
$871$	52.9706	1.79484
$872$	34.9986	1.18520
$873$	0	0
$874$	−25.2346	−0.853572
$875$	6.34315	0.214437
$876$	0	0
$877$	43.0237	1.45281	0.726404	−	0.687268i	$-0.241190\pi$
$877$	43.0237	1.45281	0.726404	+	0.687268i	$0.241190\pi$
$878$	5.48888	0.185241
$879$	0	0
$880$	−6.34315	−0.213827
$881$	−45.0028	−1.51618	−0.758091	−	0.652148i	$-0.773868\pi$
$881$	−45.0028	−1.51618	−0.758091	+	0.652148i	$0.773868\pi$
$882$	0	0
$883$	−21.4558	−0.722047	−0.361023	−	0.932557i	$-0.617573\pi$
$883$	−21.4558	−0.722047	−0.361023	+	0.932557i	$0.617573\pi$
$884$	0	0
$885$	0	0
$886$	4.97056	0.166989
$887$	−11.6118	−0.389887	−0.194943	−	0.980815i	$-0.562452\pi$
$887$	−11.6118	−0.389887	−0.194943	+	0.980815i	$0.562452\pi$
$888$	0	0
$889$	−9.70967	−0.325652
$890$	39.3057	1.31753
$891$	0	0
$892$	−9.17157	−0.307087
$893$	−8.00000	−0.267710
$894$	0	0
$895$	−46.8824	−1.56711
$896$	4.59220	0.153415
$897$	0	0
$898$	−20.5111	−0.684463
$899$	−1.17157	−0.0390741
$900$	0	0
$901$	0	0
$902$	0.686292	0.0228510
$903$	0	0
$904$	10.1355	0.337101
$905$	14.9289	0.496254
$906$	0	0
$907$	33.1509	1.10076	0.550378	−	0.834915i	$-0.314483\pi$
$907$	33.1509	1.10076	0.550378	+	0.834915i	$0.314483\pi$
$908$	−13.5140	−0.448477
$909$	0	0
$910$	−19.0294	−0.630820
$911$	−8.55035	−0.283286	−0.141643	−	0.989918i	$-0.545238\pi$
$911$	−8.55035	−0.283286	−0.141643	+	0.989918i	$0.545238\pi$
$912$	0	0
$913$	9.70967	0.321343
$914$	28.2843	0.935561
$915$	0	0
$916$	22.6274	0.747631
$917$	−1.94113	−0.0641016
$918$	0	0
$919$	12.9706	0.427859	0.213930	−	0.976849i	$-0.431374\pi$
$919$	12.9706	0.427859	0.213930	+	0.976849i	$0.431374\pi$
$920$	−78.4264	−2.58564
$921$	0	0
$922$	−32.7279	−1.07784
$923$	−37.8519	−1.24591
$924$	0	0
$925$	−34.7360	−1.14211
$926$	−3.51472	−0.115501
$927$	0	0
$928$	1.58513	0.0520343
$929$	−22.3044	−0.731784	−0.365892	−	0.930657i	$-0.619236\pi$
$929$	−22.3044	−0.731784	−0.365892	+	0.930657i	$0.619236\pi$
$930$	0	0
$931$	13.1716	0.431681
$932$	−1.66205	−0.0544423
$933$	0	0
$934$	−0.970563	−0.0317578
$935$	0	0
$936$	0	0
$937$	−24.0416	−0.785406	−0.392703	−	0.919665i	$-0.628460\pi$
$937$	−24.0416	−0.785406	−0.392703	+	0.919665i	$0.628460\pi$
$938$	−19.1116	−0.624017
$939$	0	0
$940$	−8.28772	−0.270316
$941$	−12.8574	−0.419139	−0.209569	−	0.977794i	$-0.567206\pi$
$941$	−12.8574	−0.419139	−0.209569	+	0.977794i	$0.567206\pi$
$942$	0	0
$943$	2.82843	0.0921063
$944$	−12.4853	−0.406361
$945$	0	0
$946$	12.2459	0.398148
$947$	−45.7682	−1.48727	−0.743633	−	0.668588i	$-0.766899\pi$
$947$	−45.7682	−1.48727	−0.743633	+	0.668588i	$0.766899\pi$
$948$	0	0
$949$	−16.2359	−0.527039
$950$	−10.1421	−0.329054
$951$	0	0
$952$	0	0
$953$	33.2132	1.07588	0.537941	−	0.842983i	$-0.319202\pi$
$953$	33.2132	1.07588	0.537941	+	0.842983i	$0.319202\pi$
$954$	0	0
$955$	−43.4495	−1.40599
$956$	−20.4853	−0.662541
$957$	0	0
$958$	−31.0949	−1.00463
$959$	11.8745	0.383446
$960$	0	0
$961$	−17.3431	−0.559456
$962$	−41.0990	−1.32509
$963$	0	0
$964$	12.9343	0.416586
$965$	−37.8995	−1.22003
$966$	0	0
$967$	−55.1127	−1.77230	−0.886152	−	0.463394i	$-0.846632\pi$
$967$	−55.1127	−1.77230	−0.886152	+	0.463394i	$0.846632\pi$
$968$	18.9411	0.608791
$969$	0	0
$970$	7.27208	0.233492
$971$	−40.4853	−1.29923	−0.649617	−	0.760261i	$-0.725071\pi$
$971$	−40.4853	−1.29923	−0.649617	+	0.760261i	$0.725071\pi$
$972$	0	0
$973$	1.37258	0.0440030
$974$	−41.5474	−1.33126
$975$	0	0
$976$	−11.6662	−0.373426
$977$	−8.72792	−0.279231	−0.139615	−	0.990206i	$-0.544587\pi$
$977$	−8.72792	−0.279231	−0.139615	+	0.990206i	$0.544587\pi$
$978$	0	0
$979$	29.0389	0.928087
$980$	13.6453	0.435883
$981$	0	0
$982$	29.9411	0.955460
$983$	−41.9188	−1.33700	−0.668501	−	0.743711i	$-0.733064\pi$
$983$	−41.9188	−1.33700	−0.668501	+	0.743711i	$0.733064\pi$
$984$	0	0
$985$	25.9828	0.827879
$986$	0	0
$987$	0	0
$988$	12.0000	0.381771
$989$	50.4692	1.60483
$990$	0	0
$991$	4.96362	0.157675	0.0788373	−	0.996887i	$-0.474879\pi$
$991$	4.96362	0.157675	0.0788373	+	0.996887i	$0.474879\pi$
$992$	−18.4776	−0.586664
$993$	0	0
$994$	13.6569	0.433169
$995$	4.48528	0.142193
$996$	0	0
$997$	1.84776	0.0585191	0.0292596	−	0.999572i	$-0.490685\pi$
$997$	1.84776	0.0585191	0.0292596	+	0.999572i	$0.490685\pi$
$998$	27.7708	0.879069
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.bg.1.3		4
3.2	odd	2		2601.2.a.ba.1.2		4
17.11	odd	16		153.2.l.a.19.1	✓	4
17.14	odd	16		153.2.l.a.145.1	yes	4
17.16	even	2	inner	2601.2.a.bg.1.2		4
51.11	even	16		153.2.l.b.19.1	yes	4
51.14	even	16		153.2.l.b.145.1	yes	4
51.50	odd	2		2601.2.a.ba.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.2.l.a.19.1	✓	4	17.11	odd	16
153.2.l.a.145.1	yes	4	17.14	odd	16
153.2.l.b.19.1	yes	4	51.11	even	16
153.2.l.b.145.1	yes	4	51.14	even	16
2601.2.a.ba.1.2		4	3.2	odd	2
2601.2.a.ba.1.3		4	51.50	odd	2
2601.2.a.bg.1.2		4	17.16	even	2	inner
2601.2.a.bg.1.3		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2601.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{4} +2.93015 q^{5} -1.53073 q^{7} -3.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{4} +2.93015 q^{5} -1.53073 q^{7} -3.00000 q^{8} +2.93015 q^{10} +2.16478 q^{11} +4.24264 q^{13} -1.53073 q^{14} -1.00000 q^{16} -2.82843 q^{19} -2.93015 q^{20} +2.16478 q^{22} +8.92177 q^{23} +3.58579 q^{25} +4.24264 q^{26} +1.53073 q^{28} +0.317025 q^{29} -3.69552 q^{31} +5.00000 q^{32} -4.48528 q^{35} -9.68714 q^{37} -2.82843 q^{38} -8.79045 q^{40} +0.317025 q^{41} +5.65685 q^{43} -2.16478 q^{44} +8.92177 q^{46} +2.82843 q^{47} -4.65685 q^{49} +3.58579 q^{50} -4.24264 q^{52} +8.24264 q^{53} +6.34315 q^{55} +4.59220 q^{56} +0.317025 q^{58} +12.4853 q^{59} +11.6662 q^{61} -3.69552 q^{62} +7.00000 q^{64} +12.4316 q^{65} +12.4853 q^{67} -4.48528 q^{70} -8.92177 q^{71} -3.82683 q^{73} -9.68714 q^{74} +2.82843 q^{76} -3.31371 q^{77} +11.9832 q^{79} -2.93015 q^{80} +0.317025 q^{82} +4.48528 q^{83} +5.65685 q^{86} -6.49435 q^{88} +13.4142 q^{89} -6.49435 q^{91} -8.92177 q^{92} +2.82843 q^{94} -8.28772 q^{95} +2.48181 q^{97} -4.65685 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 - 4 * q^4 - 12 * q^8	\( 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8} - 4 q^{16} + 20 q^{25} + 20 q^{32} + 16 q^{35} + 4 q^{49} + 20 q^{50} + 16 q^{53} + 48 q^{55} + 16 q^{59} + 28 q^{64} + 16 q^{67} + 16 q^{70} + 32 q^{77} - 16 q^{83} + 48 q^{89} + 4 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^4 - 12 * q^8 - 4 * q^16 + 20 * q^25 + 20 * q^32 + 16 * q^35 + 4 * q^49 + 20 * q^50 + 16 * q^53 + 48 * q^55 + 16 * q^59 + 28 * q^64 + 16 * q^67 + 16 * q^70 + 32 * q^77 - 16 * q^83 + 48 * q^89 + 4 * q^98

Introduction

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