LMFDB - Embedded newform 2601.2.a.bf.1.2

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:	4.4.7232.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 4x + 4 $ x^4 - 2x^3 - 5x^2 + 4*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.63640$ 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)

$f(q)$	$=$	$q-0.222191 q^{2} -1.95063 q^{4} -0.636405 q^{5} +1.72844 q^{7} +0.877796 q^{8} +O(q^{10})$	$q-0.222191 q^{2} -1.95063 q^{4} -0.636405 q^{5} +1.72844 q^{7} +0.877796 q^{8} +0.141404 q^{10} +4.95063 q^{11} +2.50625 q^{13} -0.384044 q^{14} +3.70622 q^{16} -0.950631 q^{19} +1.24139 q^{20} -1.09999 q^{22} -2.12220 q^{23} -4.59499 q^{25} -0.556867 q^{26} -3.37155 q^{28} +9.58704 q^{29} -5.27281 q^{31} -2.57908 q^{32} -1.09999 q^{35} -8.48705 q^{37} +0.211222 q^{38} -0.558634 q^{40} +6.92171 q^{41} +7.15061 q^{43} -9.65685 q^{44} +0.471535 q^{46} -8.10124 q^{47} -4.01250 q^{49} +1.02097 q^{50} -4.88877 q^{52} +6.44438 q^{53} -3.15061 q^{55} +1.51722 q^{56} -2.13016 q^{58} +10.1125 q^{59} -2.96983 q^{61} +1.17157 q^{62} -6.83940 q^{64} -1.59499 q^{65} +7.70129 q^{67} +0.244408 q^{70} -0.384044 q^{71} -6.51420 q^{73} +1.88575 q^{74} +1.85433 q^{76} +8.55687 q^{77} +2.37280 q^{79} -2.35866 q^{80} -1.53794 q^{82} +13.3581 q^{83} -1.58880 q^{86} +4.34564 q^{88} +1.98875 q^{89} +4.33190 q^{91} +4.13964 q^{92} +1.80002 q^{94} +0.604986 q^{95} -3.04142 q^{97} +0.891542 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 6 q^{4} + 6 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 6 * q^4 + 6 * q^5 - 4 * q^7 + 6 * q^8	$ 4 q + 2 q^{2} + 6 q^{4} + 6 q^{5} - 4 q^{7} + 6 q^{8} + 12 q^{10} + 6 q^{11} + 2 q^{13} + 4 q^{14} + 6 q^{16} + 10 q^{19} + 16 q^{20} - 4 q^{22} - 6 q^{23} + 2 q^{25} + 20 q^{26} - 24 q^{28} + 16 q^{29} - 4 q^{31} + 14 q^{32} - 4 q^{35} - 12 q^{37} + 12 q^{38} + 8 q^{40} - 14 q^{41} + 14 q^{43} - 16 q^{44} + 12 q^{46} - 4 q^{47} + 30 q^{50} - 8 q^{52} + 20 q^{53} + 2 q^{55} - 16 q^{56} - 8 q^{58} + 24 q^{59} - 12 q^{61} + 16 q^{62} - 2 q^{64} + 14 q^{65} + 4 q^{67} - 4 q^{70} + 4 q^{71} - 20 q^{73} + 12 q^{74} + 40 q^{76} + 12 q^{77} - 8 q^{79} + 4 q^{82} + 4 q^{83} + 4 q^{86} - 16 q^{88} - 4 q^{89} + 28 q^{91} + 16 q^{92} + 8 q^{94} + 22 q^{95} - 24 q^{97} - 38 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 6 * q^4 + 6 * q^5 - 4 * q^7 + 6 * q^8 + 12 * q^10 + 6 * q^11 + 2 * q^13 + 4 * q^14 + 6 * q^16 + 10 * q^19 + 16 * q^20 - 4 * q^22 - 6 * q^23 + 2 * q^25 + 20 * q^26 - 24 * q^28 + 16 * q^29 - 4 * q^31 + 14 * q^32 - 4 * q^35 - 12 * q^37 + 12 * q^38 + 8 * q^40 - 14 * q^41 + 14 * q^43 - 16 * q^44 + 12 * q^46 - 4 * q^47 + 30 * q^50 - 8 * q^52 + 20 * q^53 + 2 * q^55 - 16 * q^56 - 8 * q^58 + 24 * q^59 - 12 * q^61 + 16 * q^62 - 2 * q^64 + 14 * q^65 + 4 * q^67 - 4 * q^70 + 4 * q^71 - 20 * q^73 + 12 * q^74 + 40 * q^76 + 12 * q^77 - 8 * q^79 + 4 * q^82 + 4 * q^83 + 4 * q^86 - 16 * q^88 - 4 * q^89 + 28 * q^91 + 16 * q^92 + 8 * q^94 + 22 * q^95 - 24 * q^97 - 38 * 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 $

$2$	−0.222191	−0.157113	−0.0785565	−	0.996910i	$-0.525031\pi$
$2$	−0.222191	−0.157113	−0.0785565	+	0.996910i	$0.525031\pi$
$3$	0	0
$3$	0	0
$4$	−1.95063	−0.975315
$5$	−0.636405	−0.284609	−0.142304	−	0.989823i	$-0.545451\pi$
$5$	−0.636405	−0.284609	−0.142304	+	0.989823i	$0.545451\pi$
$6$	0	0
$7$	1.72844	0.653289	0.326644	−	0.945147i	$-0.394082\pi$
$7$	1.72844	0.653289	0.326644	+	0.945147i	$0.394082\pi$
$8$	0.877796	0.310348
$9$	0	0
$10$	0.141404	0.0447158
$11$	4.95063	1.49267	0.746336	−	0.665570i	$-0.231811\pi$
$11$	4.95063	1.49267	0.746336	+	0.665570i	$0.231811\pi$
$12$	0	0
$13$	2.50625	0.695108	0.347554	−	0.937660i	$-0.387012\pi$
$13$	2.50625	0.695108	0.347554	+	0.937660i	$0.387012\pi$
$14$	−0.384044	−0.102640
$15$	0	0
$16$	3.70622	0.926556
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−0.950631	−0.218090	−0.109045	−	0.994037i	$-0.534779\pi$
$19$	−0.950631	−0.218090	−0.109045	+	0.994037i	$0.534779\pi$
$20$	1.24139	0.277584
$21$	0	0
$22$	−1.09999	−0.234518
$23$	−2.12220	−0.442510	−0.221255	−	0.975216i	$-0.571015\pi$
$23$	−2.12220	−0.442510	−0.221255	+	0.975216i	$0.571015\pi$
$24$	0	0
$25$	−4.59499	−0.918998
$26$	−0.556867	−0.109211
$27$	0	0
$28$	−3.37155	−0.637163
$29$	9.58704	1.78027	0.890134	−	0.455699i	$-0.150611\pi$
$29$	9.58704	1.78027	0.890134	+	0.455699i	$0.150611\pi$
$30$	0	0
$31$	−5.27281	−0.947025	−0.473512	−	0.880787i	$-0.657014\pi$
$31$	−5.27281	−0.947025	−0.473512	+	0.880787i	$0.657014\pi$
$32$	−2.57908	−0.455922
$33$	0	0
$34$	0	0
$35$	−1.09999	−0.185932
$36$	0	0
$37$	−8.48705	−1.39526	−0.697631	−	0.716457i	$-0.745763\pi$
$37$	−8.48705	−1.39526	−0.697631	+	0.716457i	$0.745763\pi$
$38$	0.211222	0.0342647
$39$	0	0
$40$	−0.558634	−0.0883278
$41$	6.92171	1.08099	0.540495	−	0.841347i	$-0.318237\pi$
$41$	6.92171	1.08099	0.540495	+	0.841347i	$0.318237\pi$
$42$	0	0
$43$	7.15061	1.09046	0.545229	−	0.838287i	$-0.316443\pi$
$43$	7.15061	1.09046	0.545229	+	0.838287i	$0.316443\pi$
$44$	−9.65685	−1.45583
$45$	0	0
$46$	0.471535	0.0695241
$47$	−8.10124	−1.18169	−0.590843	−	0.806786i	$-0.701205\pi$
$47$	−8.10124	−1.18169	−0.590843	+	0.806786i	$0.701205\pi$
$48$	0	0
$49$	−4.01250	−0.573214
$50$	1.02097	0.144387
$51$	0	0
$52$	−4.88877	−0.677950
$53$	6.44438	0.885204	0.442602	−	0.896718i	$-0.354055\pi$
$53$	6.44438	0.885204	0.442602	+	0.896718i	$0.354055\pi$
$54$	0	0
$55$	−3.15061	−0.424828
$56$	1.51722	0.202747
$57$	0	0
$58$	−2.13016	−0.279703
$59$	10.1125	1.31653	0.658267	−	0.752785i	$-0.271290\pi$
$59$	10.1125	1.31653	0.658267	+	0.752785i	$0.271290\pi$
$60$	0	0
$61$	−2.96983	−0.380248	−0.190124	−	0.981760i	$-0.560889\pi$
$61$	−2.96983	−0.380248	−0.190124	+	0.981760i	$0.560889\pi$
$62$	1.17157	0.148790
$63$	0	0
$64$	−6.83940	−0.854925
$65$	−1.59499	−0.197834
$66$	0	0
$67$	7.70129	0.940862	0.470431	−	0.882437i	$-0.344098\pi$
$67$	7.70129	0.940862	0.470431	+	0.882437i	$0.344098\pi$
$68$	0	0
$69$	0	0
$70$	0.244408	0.0292123
$71$	−0.384044	−0.0455777	−0.0227888	−	0.999740i	$-0.507255\pi$
$71$	−0.384044	−0.0455777	−0.0227888	+	0.999740i	$0.507255\pi$
$72$	0	0
$73$	−6.51420	−0.762430	−0.381215	−	0.924487i	$-0.624494\pi$
$73$	−6.51420	−0.762430	−0.381215	+	0.924487i	$0.624494\pi$
$74$	1.88575	0.219214
$75$	0	0
$76$	1.85433	0.212706
$77$	8.55687	0.975145
$78$	0	0
$79$	2.37280	0.266961	0.133480	−	0.991051i	$-0.457385\pi$
$79$	2.37280	0.266961	0.133480	+	0.991051i	$0.457385\pi$
$80$	−2.35866	−0.263706
$81$	0	0
$82$	−1.53794	−0.169838
$83$	13.3581	1.46625	0.733123	−	0.680096i	$-0.238062\pi$
$83$	13.3581	1.46625	0.733123	+	0.680096i	$0.238062\pi$
$84$	0	0
$85$	0	0
$86$	−1.58880	−0.171325
$87$	0	0
$88$	4.34564	0.463247
$89$	1.98875	0.210807	0.105404	−	0.994430i	$-0.466387\pi$
$89$	1.98875	0.210807	0.105404	+	0.994430i	$0.466387\pi$
$90$	0	0
$91$	4.33190	0.454106
$92$	4.13964	0.431587
$93$	0	0
$94$	1.80002	0.185658
$95$	0.604986	0.0620703
$96$	0	0
$97$	−3.04142	−0.308809	−0.154405	−	0.988008i	$-0.549346\pi$
$97$	−3.04142	−0.308809	−0.154405	+	0.988008i	$0.549346\pi$
$98$	0.891542	0.0900594
$99$	0	0
$100$	8.96313	0.896313
$101$	18.5456	1.84536	0.922679	−	0.385569i	$-0.125995\pi$
$101$	18.5456	1.84536	0.922679	+	0.385569i	$0.125995\pi$
$102$	0	0
$103$	3.39501	0.334521	0.167260	−	0.985913i	$-0.446508\pi$
$103$	3.39501	0.334521	0.167260	+	0.985913i	$0.446508\pi$
$104$	2.19998	0.215725
$105$	0	0
$106$	−1.43189	−0.139077
$107$	−13.1531	−1.27156	−0.635779	−	0.771871i	$-0.719321\pi$
$107$	−13.1531	−1.27156	−0.635779	+	0.771871i	$0.719321\pi$
$108$	0	0
$109$	17.3896	1.66562	0.832809	−	0.553561i	$-0.186731\pi$
$109$	17.3896	1.66562	0.832809	+	0.553561i	$0.186731\pi$
$110$	0.700037	0.0667460
$111$	0	0
$112$	6.40598	0.605309
$113$	0.776042	0.0730039	0.0365019	−	0.999334i	$-0.488378\pi$
$113$	0.776042	0.0730039	0.0365019	+	0.999334i	$0.488378\pi$
$114$	0	0
$115$	1.35058	0.125942
$116$	−18.7008	−1.73632
$117$	0	0
$118$	−2.24691	−0.206845
$119$	0	0
$120$	0	0
$121$	13.5087	1.22807
$122$	0.659871	0.0597419
$123$	0	0
$124$	10.2853	0.923648
$125$	6.10630	0.546164
$126$	0	0
$127$	10.8519	0.962950	0.481475	−	0.876460i	$-0.340101\pi$
$127$	10.8519	0.962950	0.481475	+	0.876460i	$0.340101\pi$
$128$	6.67782	0.590242
$129$	0	0
$130$	0.354393	0.0310823
$131$	0.706223	0.0617030	0.0308515	−	0.999524i	$-0.490178\pi$
$131$	0.706223	0.0617030	0.0308515	+	0.999524i	$0.490178\pi$
$132$	0	0
$133$	−1.64311	−0.142476
$134$	−1.71116	−0.147822
$135$	0	0
$136$	0	0
$137$	−12.3125	−1.05192	−0.525962	−	0.850508i	$-0.676295\pi$
$137$	−12.3125	−1.05192	−0.525962	+	0.850508i	$0.676295\pi$
$138$	0	0
$139$	13.3444	1.13186	0.565928	−	0.824454i	$-0.308518\pi$
$139$	13.3444	1.13186	0.565928	+	0.824454i	$0.308518\pi$
$140$	2.14567	0.181342
$141$	0	0
$142$	0.0853313	0.00716085
$143$	12.4075	1.03757
$144$	0	0
$145$	−6.10124	−0.506680
$146$	1.44740	0.119788
$147$	0	0
$148$	16.5551	1.36082
$149$	−0.356892	−0.0292377	−0.0146189	−	0.999893i	$-0.504653\pi$
$149$	−0.356892	−0.0292377	−0.0146189	+	0.999893i	$0.504653\pi$
$150$	0	0
$151$	9.13317	0.743247	0.371624	−	0.928384i	$-0.378801\pi$
$151$	9.13317	0.743247	0.371624	+	0.928384i	$0.378801\pi$
$152$	−0.834460	−0.0676837
$153$	0	0
$154$	−1.90126	−0.153208
$155$	3.35564	0.269532
$156$	0	0
$157$	17.5975	1.40443	0.702216	−	0.711964i	$-0.252194\pi$
$157$	17.5975	1.40443	0.702216	+	0.711964i	$0.252194\pi$
$158$	−0.527215	−0.0419430
$159$	0	0
$160$	1.64134	0.129759
$161$	−3.66810	−0.289087
$162$	0	0
$163$	−16.5901	−1.29943	−0.649717	−	0.760177i	$-0.725112\pi$
$163$	−16.5901	−1.29943	−0.649717	+	0.760177i	$0.725112\pi$
$164$	−13.5017	−1.05431
$165$	0	0
$166$	−2.96806	−0.230366
$167$	−14.2234	−1.10064	−0.550321	−	0.834953i	$-0.685495\pi$
$167$	−14.2234	−1.10064	−0.550321	+	0.834953i	$0.685495\pi$
$168$	0	0
$169$	−6.71872	−0.516825
$170$	0	0
$171$	0	0
$172$	−13.9482	−1.06354
$173$	−1.89331	−0.143946	−0.0719728	−	0.997407i	$-0.522929\pi$
$173$	−1.89331	−0.143946	−0.0719728	+	0.997407i	$0.522929\pi$
$174$	0	0
$175$	−7.94216	−0.600371
$176$	18.3481	1.38304
$177$	0	0
$178$	−0.441884	−0.0331206
$179$	22.2469	1.66281	0.831406	−	0.555666i	$-0.187536\pi$
$179$	22.2469	1.66281	0.831406	+	0.555666i	$0.187536\pi$
$180$	0	0
$181$	11.6142	0.863276	0.431638	−	0.902047i	$-0.357936\pi$
$181$	11.6142	0.863276	0.431638	+	0.902047i	$0.357936\pi$
$182$	−0.962511	−0.0713460
$183$	0	0
$184$	−1.86286	−0.137332
$185$	5.40120	0.397104
$186$	0	0
$187$	0	0
$188$	15.8025	1.15252
$189$	0	0
$190$	−0.134423	−0.00975205
$191$	−6.63311	−0.479955	−0.239978	−	0.970778i	$-0.577140\pi$
$191$	−6.63311	−0.479955	−0.239978	+	0.970778i	$0.577140\pi$
$192$	0	0
$193$	−2.91769	−0.210020	−0.105010	−	0.994471i	$-0.533487\pi$
$193$	−2.91769	−0.210020	−0.105010	+	0.994471i	$0.533487\pi$
$194$	0.675776	0.0485179
$195$	0	0
$196$	7.82690	0.559064
$197$	4.53517	0.323117	0.161559	−	0.986863i	$-0.448348\pi$
$197$	4.53517	0.323117	0.161559	+	0.986863i	$0.448348\pi$
$198$	0	0
$199$	−2.46029	−0.174405	−0.0872026	−	0.996191i	$-0.527793\pi$
$199$	−2.46029	−0.174405	−0.0872026	+	0.996191i	$0.527793\pi$
$200$	−4.03346	−0.285209
$201$	0	0
$202$	−4.12068	−0.289930
$203$	16.5706	1.16303
$204$	0	0
$205$	−4.40501	−0.307659
$206$	−0.754343	−0.0525576
$207$	0	0
$208$	9.28872	0.644057
$209$	−4.70622	−0.325536
$210$	0	0
$211$	−19.9038	−1.37023	−0.685116	−	0.728434i	$-0.740248\pi$
$211$	−19.9038	−1.37023	−0.685116	+	0.728434i	$0.740248\pi$
$212$	−12.5706	−0.863353
$213$	0	0
$214$	2.92251	0.199778
$215$	−4.55068	−0.310354
$216$	0	0
$217$	−9.11373	−0.618681
$218$	−3.86381	−0.261690
$219$	0	0
$220$	6.14567	0.414341
$221$	0	0
$222$	0	0
$223$	14.3631	0.961823	0.480911	−	0.876769i	$-0.340306\pi$
$223$	14.3631	0.961823	0.480911	+	0.876769i	$0.340306\pi$
$224$	−4.45779	−0.297849
$225$	0	0
$226$	−0.172430	−0.0114699
$227$	−10.8744	−0.721758	−0.360879	−	0.932613i	$-0.617523\pi$
$227$	−10.8744	−0.721758	−0.360879	+	0.932613i	$0.617523\pi$
$228$	0	0
$229$	18.9900	1.25489	0.627447	−	0.778659i	$-0.284100\pi$
$229$	18.9900	1.25489	0.627447	+	0.778659i	$0.284100\pi$
$230$	−0.300087	−0.0197872
$231$	0	0
$232$	8.41546	0.552502
$233$	7.08079	0.463878	0.231939	−	0.972730i	$-0.425493\pi$
$233$	7.08079	0.463878	0.231939	+	0.972730i	$0.425493\pi$
$234$	0	0
$235$	5.15567	0.336319
$236$	−19.7257	−1.28404
$237$	0	0
$238$	0	0
$239$	5.28872	0.342099	0.171049	−	0.985262i	$-0.445284\pi$
$239$	5.28872	0.342099	0.171049	+	0.985262i	$0.445284\pi$
$240$	0	0
$241$	−5.79826	−0.373499	−0.186749	−	0.982408i	$-0.559795\pi$
$241$	−5.79826	−0.373499	−0.186749	+	0.982408i	$0.559795\pi$
$242$	−3.00153	−0.192945
$243$	0	0
$244$	5.79304	0.370862
$245$	2.55357	0.163142
$246$	0	0
$247$	−2.38252	−0.151596
$248$	−4.62845	−0.293907
$249$	0	0
$250$	−1.35677	−0.0858095
$251$	8.33190	0.525905	0.262952	−	0.964809i	$-0.415304\pi$
$251$	8.33190	0.525905	0.262952	+	0.964809i	$0.415304\pi$
$252$	0	0
$253$	−10.5062	−0.660522
$254$	−2.41120	−0.151292
$255$	0	0
$256$	12.1950	0.762190
$257$	4.25565	0.265460	0.132730	−	0.991152i	$-0.457626\pi$
$257$	4.25565	0.265460	0.132730	+	0.991152i	$0.457626\pi$
$258$	0	0
$259$	−14.6694	−0.911509
$260$	3.11123	0.192951
$261$	0	0
$262$	−0.156917	−0.00969435
$263$	9.09999	0.561129	0.280565	−	0.959835i	$-0.409478\pi$
$263$	9.09999	0.561129	0.280565	+	0.959835i	$0.409478\pi$
$264$	0	0
$265$	−4.10124	−0.251937
$266$	0.365084	0.0223848
$267$	0	0
$268$	−15.0224	−0.917637
$269$	21.3820	1.30368	0.651842	−	0.758355i	$-0.273997\pi$
$269$	21.3820	1.30368	0.651842	+	0.758355i	$0.273997\pi$
$270$	0	0
$271$	−19.3213	−1.17368	−0.586842	−	0.809702i	$-0.699629\pi$
$271$	−19.3213	−1.17368	−0.586842	+	0.809702i	$0.699629\pi$
$272$	0	0
$273$	0	0
$274$	2.73572	0.165271
$275$	−22.7481	−1.37176
$276$	0	0
$277$	−3.34013	−0.200689	−0.100344	−	0.994953i	$-0.531994\pi$
$277$	−3.34013	−0.200689	−0.100344	+	0.994953i	$0.531994\pi$
$278$	−2.96501	−0.177829
$279$	0	0
$280$	−0.965565	−0.0577035
$281$	−4.67754	−0.279039	−0.139519	−	0.990219i	$-0.544556\pi$
$281$	−4.67754	−0.279039	−0.139519	+	0.990219i	$0.544556\pi$
$282$	0	0
$283$	−10.0582	−0.597897	−0.298948	−	0.954269i	$-0.596636\pi$
$283$	−10.0582	−0.597897	−0.298948	+	0.954269i	$0.596636\pi$
$284$	0.749129	0.0444526
$285$	0	0
$286$	−2.75684	−0.163015
$287$	11.9638	0.706198
$288$	0	0
$289$	0	0
$290$	1.35564	0.0796061
$291$	0	0
$292$	12.7068	0.743609
$293$	−5.76934	−0.337048	−0.168524	−	0.985698i	$-0.553900\pi$
$293$	−5.76934	−0.337048	−0.168524	+	0.985698i	$0.553900\pi$
$294$	0	0
$295$	−6.43563	−0.374697
$296$	−7.44990	−0.433017
$297$	0	0
$298$	0.0792983	0.00459363
$299$	−5.31877	−0.307592
$300$	0	0
$301$	12.3594	0.712383
$302$	−2.02931	−0.116774
$303$	0	0
$304$	−3.52325	−0.202072
$305$	1.89001	0.108222
$306$	0	0
$307$	−13.2375	−0.755502	−0.377751	−	0.925907i	$-0.623302\pi$
$307$	−13.2375	−0.755502	−0.377751	+	0.925907i	$0.623302\pi$
$308$	−16.6913	−0.951074
$309$	0	0
$310$	−0.745595	−0.0423469
$311$	−1.29530	−0.0734499	−0.0367250	−	0.999325i	$-0.511693\pi$
$311$	−1.29530	−0.0734499	−0.0367250	+	0.999325i	$0.511693\pi$
$312$	0	0
$313$	19.1007	1.07964	0.539818	−	0.841782i	$-0.318493\pi$
$313$	19.1007	1.07964	0.539818	+	0.841782i	$0.318493\pi$
$314$	−3.91001	−0.220655
$315$	0	0
$316$	−4.62845	−0.260371
$317$	15.1043	0.848339	0.424170	−	0.905583i	$-0.360566\pi$
$317$	15.1043	0.848339	0.424170	+	0.905583i	$0.360566\pi$
$318$	0	0
$319$	47.4619	2.65735
$320$	4.35263	0.243319
$321$	0	0
$322$	0.815020	0.0454193
$323$	0	0
$324$	0	0
$325$	−11.5162	−0.638803
$326$	3.68617	0.204158
$327$	0	0
$328$	6.07585	0.335483
$329$	−14.0025	−0.771983
$330$	0	0
$331$	−22.6769	−1.24644	−0.623218	−	0.782048i	$-0.714175\pi$
$331$	−22.6769	−1.24644	−0.623218	+	0.782048i	$0.714175\pi$
$332$	−26.0568	−1.43005
$333$	0	0
$334$	3.16033	0.172925
$335$	−4.90114	−0.267778
$336$	0	0
$337$	1.61669	0.0880666	0.0440333	−	0.999030i	$-0.485979\pi$
$337$	1.61669	0.0880666	0.0440333	+	0.999030i	$0.485979\pi$
$338$	1.49284	0.0811999
$339$	0	0
$340$	0	0
$341$	−26.1037	−1.41360
$342$	0	0
$343$	−19.0344	−1.02776
$344$	6.27677	0.338421
$345$	0	0
$346$	0.420677	0.0226157
$347$	−15.3606	−0.824602	−0.412301	−	0.911048i	$-0.635275\pi$
$347$	−15.3606	−0.824602	−0.412301	+	0.911048i	$0.635275\pi$
$348$	0	0
$349$	−15.1087	−0.808749	−0.404374	−	0.914594i	$-0.632511\pi$
$349$	−15.1087	−0.808749	−0.404374	+	0.914594i	$0.632511\pi$
$350$	1.76468	0.0943261
$351$	0	0
$352$	−12.7681	−0.680541
$353$	31.2482	1.66317	0.831586	−	0.555396i	$-0.187433\pi$
$353$	31.2482	1.66317	0.831586	+	0.555396i	$0.187433\pi$
$354$	0	0
$355$	0.244408	0.0129718
$356$	−3.87932	−0.205604
$357$	0	0
$358$	−4.94307	−0.261249
$359$	−2.72503	−0.143822	−0.0719108	−	0.997411i	$-0.522910\pi$
$359$	−2.72503	−0.143822	−0.0719108	+	0.997411i	$0.522910\pi$
$360$	0	0
$361$	−18.0963	−0.952437
$362$	−2.58057	−0.135632
$363$	0	0
$364$	−8.44994	−0.442897
$365$	4.14567	0.216994
$366$	0	0
$367$	18.4515	0.963163	0.481581	−	0.876401i	$-0.340063\pi$
$367$	18.4515	0.963163	0.481581	+	0.876401i	$0.340063\pi$
$368$	−7.86536	−0.410010
$369$	0	0
$370$	−1.20010	−0.0623902
$371$	11.1387	0.578294
$372$	0	0
$373$	24.4049	1.26364	0.631820	−	0.775115i	$-0.282308\pi$
$373$	24.4049	1.26364	0.631820	+	0.775115i	$0.282308\pi$
$374$	0	0
$375$	0	0
$376$	−7.11123	−0.366734
$377$	24.0275	1.23748
$378$	0	0
$379$	5.89057	0.302578	0.151289	−	0.988490i	$-0.451658\pi$
$379$	5.89057	0.302578	0.151289	+	0.988490i	$0.451658\pi$
$380$	−1.18010	−0.0605381
$381$	0	0
$382$	1.47382	0.0754072
$383$	−30.7756	−1.57256	−0.786279	−	0.617871i	$-0.787995\pi$
$383$	−30.7756	−1.57256	−0.786279	+	0.617871i	$0.787995\pi$
$384$	0	0
$385$	−5.44563	−0.277535
$386$	0.648284	0.0329968
$387$	0	0
$388$	5.93268	0.301186
$389$	−19.5244	−0.989925	−0.494963	−	0.868914i	$-0.664818\pi$
$389$	−19.5244	−0.989925	−0.494963	+	0.868914i	$0.664818\pi$
$390$	0	0
$391$	0	0
$392$	−3.52215	−0.177896
$393$	0	0
$394$	−1.00768	−0.0507659
$395$	−1.51006	−0.0759794
$396$	0	0
$397$	0.904279	0.0453844	0.0226922	−	0.999742i	$-0.492776\pi$
$397$	0.904279	0.0453844	0.0226922	+	0.999742i	$0.492776\pi$
$398$	0.546655	0.0274013
$399$	0	0
$400$	−17.0301	−0.851503
$401$	1.26486	0.0631639	0.0315820	−	0.999501i	$-0.489945\pi$
$401$	1.26486	0.0631639	0.0315820	+	0.999501i	$0.489945\pi$
$402$	0	0
$403$	−13.2150	−0.658285
$404$	−36.1757	−1.79981
$405$	0	0
$406$	−3.68185	−0.182727
$407$	−42.0162	−2.08267
$408$	0	0
$409$	28.6350	1.41591	0.707954	−	0.706258i	$-0.249618\pi$
$409$	28.6350	1.41591	0.707954	+	0.706258i	$0.249618\pi$
$410$	0.978756	0.0483373
$411$	0	0
$412$	−6.62242	−0.326263
$413$	17.4788	0.860076
$414$	0	0
$415$	−8.50119	−0.417307
$416$	−6.46382	−0.316915
$417$	0	0
$418$	1.04568	0.0511460
$419$	23.9581	1.17043	0.585214	−	0.810879i	$-0.301010\pi$
$419$	23.9581	1.17043	0.585214	+	0.810879i	$0.301010\pi$
$420$	0	0
$421$	−12.5312	−0.610735	−0.305368	−	0.952235i	$-0.598779\pi$
$421$	−12.5312	−0.610735	−0.305368	+	0.952235i	$0.598779\pi$
$422$	4.42244	0.215281
$423$	0	0
$424$	5.65685	0.274721
$425$	0	0
$426$	0	0
$427$	−5.13317	−0.248412
$428$	25.6569	1.24017
$429$	0	0
$430$	1.01112	0.0487606
$431$	−17.6728	−0.851267	−0.425633	−	0.904896i	$-0.639949\pi$
$431$	−17.6728	−0.851267	−0.425633	+	0.904896i	$0.639949\pi$
$432$	0	0
$433$	−14.0076	−0.673160	−0.336580	−	0.941655i	$-0.609270\pi$
$433$	−14.0076	−0.673160	−0.336580	+	0.941655i	$0.609270\pi$
$434$	2.02499	0.0972028
$435$	0	0
$436$	−33.9206	−1.62450
$437$	2.01743	0.0965069
$438$	0	0
$439$	11.5740	0.552398	0.276199	−	0.961100i	$-0.410925\pi$
$439$	11.5740	0.552398	0.276199	+	0.961100i	$0.410925\pi$
$440$	−2.76559	−0.131844
$441$	0	0
$442$	0	0
$443$	35.6174	1.69223	0.846117	−	0.532997i	$-0.178934\pi$
$443$	35.6174	1.69223	0.846117	+	0.532997i	$0.178934\pi$
$444$	0	0
$445$	−1.26565	−0.0599977
$446$	−3.19135	−0.151115
$447$	0	0
$448$	−11.8215	−0.558513
$449$	−22.1461	−1.04514	−0.522569	−	0.852597i	$-0.675026\pi$
$449$	−22.1461	−1.04514	−0.522569	+	0.852597i	$0.675026\pi$
$450$	0	0
$451$	34.2668	1.61356
$452$	−1.51377	−0.0712018
$453$	0	0
$454$	2.41619	0.113398
$455$	−2.75684	−0.129243
$456$	0	0
$457$	9.90632	0.463398	0.231699	−	0.972787i	$-0.425572\pi$
$457$	9.90632	0.463398	0.231699	+	0.972787i	$0.425572\pi$
$458$	−4.21941	−0.197160
$459$	0	0
$460$	−2.63449	−0.122834
$461$	9.62367	0.448219	0.224109	−	0.974564i	$-0.428053\pi$
$461$	9.62367	0.448219	0.224109	+	0.974564i	$0.428053\pi$
$462$	0	0
$463$	15.3013	0.711113	0.355557	−	0.934655i	$-0.384291\pi$
$463$	15.3013	0.711113	0.355557	+	0.934655i	$0.384291\pi$
$464$	35.5317	1.64952
$465$	0	0
$466$	−1.57329	−0.0728812
$467$	22.3150	1.03261	0.516307	−	0.856404i	$-0.327307\pi$
$467$	22.3150	1.03261	0.516307	+	0.856404i	$0.327307\pi$
$468$	0	0
$469$	13.3112	0.614655
$470$	−1.14554	−0.0528400
$471$	0	0
$472$	8.87670	0.408583
$473$	35.4000	1.62769
$474$	0	0
$475$	4.36814	0.200424
$476$	0	0
$477$	0	0
$478$	−1.17511	−0.0537481
$479$	−15.5571	−0.710824	−0.355412	−	0.934710i	$-0.615659\pi$
$479$	−15.5571	−0.710824	−0.355412	+	0.934710i	$0.615659\pi$
$480$	0	0
$481$	−21.2707	−0.969858
$482$	1.28832	0.0586815
$483$	0	0
$484$	−26.3506	−1.19775
$485$	1.93557	0.0878898
$486$	0	0
$487$	0.0296516	0.00134364	0.000671822	−	1.00000i	$-0.499786\pi$
$487$	0.0296516	0.00134364	0.000671822		1.00000i	$0.499786\pi$
$488$	−2.60691	−0.118009
$489$	0	0
$490$	−0.567382	−0.0256317
$491$	−6.66935	−0.300984	−0.150492	−	0.988611i	$-0.548086\pi$
$491$	−6.66935	−0.300984	−0.150492	+	0.988611i	$0.548086\pi$
$492$	0	0
$493$	0	0
$494$	0.529375	0.0238177
$495$	0	0
$496$	−19.5422	−0.877471
$497$	−0.663798	−0.0297754
$498$	0	0
$499$	18.6469	0.834748	0.417374	−	0.908735i	$-0.362951\pi$
$499$	18.6469	0.834748	0.417374	+	0.908735i	$0.362951\pi$
$500$	−11.9111	−0.532682
$501$	0	0
$502$	−1.85128	−0.0826265
$503$	25.7396	1.14767	0.573837	−	0.818970i	$-0.305454\pi$
$503$	25.7396	1.14767	0.573837	+	0.818970i	$0.305454\pi$
$504$	0	0
$505$	−11.8025	−0.525205
$506$	2.33440	0.103777
$507$	0	0
$508$	−21.1680	−0.939180
$509$	−41.1950	−1.82594	−0.912968	−	0.408032i	$-0.866215\pi$
$509$	−41.1950	−1.82594	−0.912968	+	0.408032i	$0.866215\pi$
$510$	0	0
$511$	−11.2594	−0.498087
$512$	−16.0653	−0.709992
$513$	0	0
$514$	−0.945570	−0.0417073
$515$	−2.16060	−0.0952076
$516$	0	0
$517$	−40.1062	−1.76387
$518$	3.25940	0.143210
$519$	0	0
$520$	−1.40007	−0.0613973
$521$	8.98205	0.393511	0.196755	−	0.980453i	$-0.436960\pi$
$521$	8.98205	0.393511	0.196755	+	0.980453i	$0.436960\pi$
$522$	0	0
$523$	−5.38995	−0.235686	−0.117843	−	0.993032i	$-0.537598\pi$
$523$	−5.38995	−0.235686	−0.117843	+	0.993032i	$0.537598\pi$
$524$	−1.37758	−0.0601799
$525$	0	0
$526$	−2.02194	−0.0881607
$527$	0	0
$528$	0	0
$529$	−18.4963	−0.804185
$530$	0.911259	0.0395826
$531$	0	0
$532$	3.20510	0.138959
$533$	17.3475	0.751405
$534$	0	0
$535$	8.37070	0.361897
$536$	6.76016	0.291994
$537$	0	0
$538$	−4.75090	−0.204826
$539$	−19.8644	−0.855620
$540$	0	0
$541$	−1.22765	−0.0527806	−0.0263903	−	0.999652i	$-0.508401\pi$
$541$	−1.22765	−0.0527806	−0.0263903	+	0.999652i	$0.508401\pi$
$542$	4.29302	0.184401
$543$	0	0
$544$	0	0
$545$	−11.0668	−0.474050
$546$	0	0
$547$	−13.5274	−0.578391	−0.289196	−	0.957270i	$-0.593388\pi$
$547$	−13.5274	−0.578391	−0.289196	+	0.957270i	$0.593388\pi$
$548$	24.0171	1.02596
$549$	0	0
$550$	5.05443	0.215522
$551$	−9.11373	−0.388258
$552$	0	0
$553$	4.10124	0.174402
$554$	0.742148	0.0315308
$555$	0	0
$556$	−26.0300	−1.10392
$557$	18.3599	0.777936	0.388968	−	0.921251i	$-0.372832\pi$
$557$	18.3599	0.777936	0.388968	+	0.921251i	$0.372832\pi$
$558$	0	0
$559$	17.9212	0.757986
$560$	−4.07680	−0.172276
$561$	0	0
$562$	1.03931	0.0438406
$563$	−3.33315	−0.140475	−0.0702377	−	0.997530i	$-0.522376\pi$
$563$	−3.33315	−0.140475	−0.0702377	+	0.997530i	$0.522376\pi$
$564$	0	0
$565$	−0.493877	−0.0207775
$566$	2.23484	0.0939374
$567$	0	0
$568$	−0.337113	−0.0141449
$569$	−29.4369	−1.23406	−0.617029	−	0.786940i	$-0.711664\pi$
$569$	−29.4369	−1.23406	−0.617029	+	0.786940i	$0.711664\pi$
$570$	0	0
$571$	25.8275	1.08085	0.540424	−	0.841393i	$-0.318264\pi$
$571$	25.8275	1.08085	0.540424	+	0.841393i	$0.318264\pi$
$572$	−24.2025	−1.01196
$573$	0	0
$574$	−2.65824	−0.110953
$575$	9.75150	0.406666
$576$	0	0
$577$	−22.1850	−0.923575	−0.461788	−	0.886990i	$-0.652792\pi$
$577$	−22.1850	−0.923575	−0.461788	+	0.886990i	$0.652792\pi$
$578$	0	0
$579$	0	0
$580$	11.9013	0.494173
$581$	23.0887	0.957882
$582$	0	0
$583$	31.9038	1.32132
$584$	−5.71814	−0.236618
$585$	0	0
$586$	1.28190	0.0529547
$587$	−32.1482	−1.32690	−0.663448	−	0.748222i	$-0.730908\pi$
$587$	−32.1482	−1.32690	−0.663448	+	0.748222i	$0.730908\pi$
$588$	0	0
$589$	5.01250	0.206536
$590$	1.42994	0.0588698
$591$	0	0
$592$	−31.4549	−1.29279
$593$	29.3406	1.20488	0.602438	−	0.798166i	$-0.294196\pi$
$593$	29.3406	1.20488	0.602438	+	0.798166i	$0.294196\pi$
$594$	0	0
$595$	0	0
$596$	0.696164	0.0285160
$597$	0	0
$598$	1.18178	0.0483268
$599$	36.2632	1.48167	0.740836	−	0.671686i	$-0.234430\pi$
$599$	36.2632	1.48167	0.740836	+	0.671686i	$0.234430\pi$
$600$	0	0
$601$	−45.7383	−1.86570	−0.932851	−	0.360262i	$-0.882687\pi$
$601$	−45.7383	−1.86570	−0.932851	+	0.360262i	$0.882687\pi$
$602$	−2.74615	−0.111925
$603$	0	0
$604$	−17.8155	−0.724900
$605$	−8.59703	−0.349519
$606$	0	0
$607$	33.6640	1.36638	0.683190	−	0.730241i	$-0.260592\pi$
$607$	33.6640	1.36638	0.683190	+	0.730241i	$0.260592\pi$
$608$	2.45176	0.0994318
$609$	0	0
$610$	−0.419945	−0.0170031
$611$	−20.3037	−0.821400
$612$	0	0
$613$	−5.89120	−0.237943	−0.118972	−	0.992898i	$-0.537960\pi$
$613$	−5.89120	−0.237943	−0.118972	+	0.992898i	$0.537960\pi$
$614$	2.94125	0.118699
$615$	0	0
$616$	7.51118	0.302634
$617$	−35.2179	−1.41782	−0.708909	−	0.705300i	$-0.750812\pi$
$617$	−35.2179	−1.41782	−0.708909	+	0.705300i	$0.750812\pi$
$618$	0	0
$619$	32.8288	1.31950	0.659750	−	0.751485i	$-0.270662\pi$
$619$	32.8288	1.31950	0.659750	+	0.751485i	$0.270662\pi$
$620$	−6.54562	−0.262878
$621$	0	0
$622$	0.287805	0.0115399
$623$	3.43744	0.137718
$624$	0	0
$625$	19.0889	0.763555
$626$	−4.24402	−0.169625
$627$	0	0
$628$	−34.3262	−1.36976
$629$	0	0
$630$	0	0
$631$	10.7057	0.426186	0.213093	−	0.977032i	$-0.431646\pi$
$631$	10.7057	0.426186	0.213093	+	0.977032i	$0.431646\pi$
$632$	2.08283	0.0828506
$633$	0	0
$634$	−3.35603	−0.133285
$635$	−6.90620	−0.274064
$636$	0	0
$637$	−10.0563	−0.398446
$638$	−10.5456	−0.417505
$639$	0	0
$640$	−4.24980	−0.167988
$641$	−36.9437	−1.45919	−0.729593	−	0.683881i	$-0.760291\pi$
$641$	−36.9437	−1.45919	−0.729593	+	0.683881i	$0.760291\pi$
$642$	0	0
$643$	39.3137	1.55038	0.775191	−	0.631727i	$-0.217654\pi$
$643$	39.3137	1.55038	0.775191	+	0.631727i	$0.217654\pi$
$644$	7.15511	0.281951
$645$	0	0
$646$	0	0
$647$	14.8806	0.585016	0.292508	−	0.956263i	$-0.405510\pi$
$647$	14.8806	0.585016	0.292508	+	0.956263i	$0.405510\pi$
$648$	0	0
$649$	50.0632	1.96515
$650$	2.55880	0.100364
$651$	0	0
$652$	32.3611	1.26736
$653$	24.2908	0.950571	0.475285	−	0.879832i	$-0.342345\pi$
$653$	24.2908	0.950571	0.475285	+	0.879832i	$0.342345\pi$
$654$	0	0
$655$	−0.449444	−0.0175612
$656$	25.6534	1.00160
$657$	0	0
$658$	3.11123	0.121289
$659$	9.14192	0.356119	0.178059	−	0.984020i	$-0.443018\pi$
$659$	9.14192	0.356119	0.178059	+	0.984020i	$0.443018\pi$
$660$	0	0
$661$	−16.7951	−0.653253	−0.326627	−	0.945153i	$-0.605912\pi$
$661$	−16.7951	−0.653253	−0.326627	+	0.945153i	$0.605912\pi$
$662$	5.03861	0.195831
$663$	0	0
$664$	11.7257	0.455046
$665$	1.04568	0.0405498
$666$	0	0
$667$	−20.3456	−0.787787
$668$	27.7447	1.07347
$669$	0	0
$670$	1.08899	0.0420714
$671$	−14.7025	−0.567585
$672$	0	0
$673$	−18.6313	−0.718186	−0.359093	−	0.933302i	$-0.616914\pi$
$673$	−18.6313	−0.718186	−0.359093	+	0.933302i	$0.616914\pi$
$674$	−0.359214	−0.0138364
$675$	0	0
$676$	13.1057	0.504067
$677$	10.8010	0.415117	0.207559	−	0.978223i	$-0.433448\pi$
$677$	10.8010	0.415117	0.207559	+	0.978223i	$0.433448\pi$
$678$	0	0
$679$	−5.25690	−0.201741
$680$	0	0
$681$	0	0
$682$	5.80002	0.222094
$683$	−22.2199	−0.850221	−0.425111	−	0.905141i	$-0.639765\pi$
$683$	−22.2199	−0.850221	−0.425111	+	0.905141i	$0.639765\pi$
$684$	0	0
$685$	7.83571	0.299387
$686$	4.22929	0.161475
$687$	0	0
$688$	26.5017	1.01037
$689$	16.1512	0.615313
$690$	0	0
$691$	38.5413	1.46618	0.733090	−	0.680132i	$-0.238077\pi$
$691$	38.5413	1.46618	0.733090	+	0.680132i	$0.238077\pi$
$692$	3.69315	0.140392
$693$	0	0
$694$	3.41300	0.129556
$695$	−8.49244	−0.322137
$696$	0	0
$697$	0	0
$698$	3.35702	0.127065
$699$	0	0
$700$	15.4922	0.585551
$701$	−18.0843	−0.683034	−0.341517	−	0.939876i	$-0.610941\pi$
$701$	−18.0843	−0.683034	−0.341517	+	0.939876i	$0.610941\pi$
$702$	0	0
$703$	8.06805	0.304292
$704$	−33.8593	−1.27612
$705$	0	0
$706$	−6.94307	−0.261306
$707$	32.0550	1.20555
$708$	0	0
$709$	−52.2614	−1.96272	−0.981359	−	0.192185i	$-0.938443\pi$
$709$	−52.2614	−1.96272	−0.981359	+	0.192185i	$0.938443\pi$
$710$	−0.0543053	−0.00203804
$711$	0	0
$712$	1.74572	0.0654236
$713$	11.1900	0.419068
$714$	0	0
$715$	−7.89620	−0.295301
$716$	−43.3955	−1.62177
$717$	0	0
$718$	0.605478	0.0225962
$719$	2.25831	0.0842206	0.0421103	−	0.999113i	$-0.486592\pi$
$719$	2.25831	0.0842206	0.0421103	+	0.999113i	$0.486592\pi$
$720$	0	0
$721$	5.86808	0.218539
$722$	4.02084	0.149640
$723$	0	0
$724$	−22.6550	−0.841966
$725$	−44.0523	−1.63606
$726$	0	0
$727$	−39.5601	−1.46720	−0.733601	−	0.679581i	$-0.762162\pi$
$727$	−39.5601	−1.46720	−0.733601	+	0.679581i	$0.762162\pi$
$728$	3.80252	0.140931
$729$	0	0
$730$	−0.921132	−0.0340926
$731$	0	0
$732$	0	0
$733$	−13.5137	−0.499139	−0.249570	−	0.968357i	$-0.580289\pi$
$733$	−13.5137	−0.499139	−0.249570	+	0.968357i	$0.580289\pi$
$734$	−4.09977	−0.151325
$735$	0	0
$736$	5.47334	0.201750
$737$	38.1262	1.40440
$738$	0	0
$739$	−10.7282	−0.394642	−0.197321	−	0.980339i	$-0.563224\pi$
$739$	−10.7282	−0.394642	−0.197321	+	0.980339i	$0.563224\pi$
$740$	−10.5357	−0.387302
$741$	0	0
$742$	−2.47493	−0.0908575
$743$	−43.9921	−1.61392	−0.806958	−	0.590609i	$-0.798887\pi$
$743$	−43.9921	−1.61392	−0.806958	+	0.590609i	$0.798887\pi$
$744$	0	0
$745$	0.227128	0.00832131
$746$	−5.42257	−0.198534
$747$	0	0
$748$	0	0
$749$	−22.7343	−0.830695
$750$	0	0
$751$	12.5047	0.456304	0.228152	−	0.973626i	$-0.426732\pi$
$751$	12.5047	0.456304	0.228152	+	0.973626i	$0.426732\pi$
$752$	−30.0250	−1.09490
$753$	0	0
$754$	−5.33870	−0.194424
$755$	−5.81240	−0.211535
$756$	0	0
$757$	−35.8474	−1.30290	−0.651449	−	0.758693i	$-0.725838\pi$
$757$	−35.8474	−1.30290	−0.651449	+	0.758693i	$0.725838\pi$
$758$	−1.30883	−0.0475390
$759$	0	0
$760$	0.531055	0.0192634
$761$	23.4468	0.849944	0.424972	−	0.905206i	$-0.360284\pi$
$761$	23.4468	0.849944	0.424972	+	0.905206i	$0.360284\pi$
$762$	0	0
$763$	30.0568	1.08813
$764$	12.9388	0.468108
$765$	0	0
$766$	6.83807	0.247070
$767$	25.3444	0.915133
$768$	0	0
$769$	22.8938	0.825573	0.412786	−	0.910828i	$-0.364556\pi$
$769$	22.8938	0.825573	0.412786	+	0.910828i	$0.364556\pi$
$770$	1.20997	0.0436044
$771$	0	0
$772$	5.69133	0.204835
$773$	−34.1287	−1.22753	−0.613763	−	0.789491i	$-0.710345\pi$
$773$	−34.1287	−1.22753	−0.613763	+	0.789491i	$0.710345\pi$
$774$	0	0
$775$	24.2285	0.870313
$776$	−2.66974	−0.0958382
$777$	0	0
$778$	4.33815	0.155530
$779$	−6.57999	−0.235753
$780$	0	0
$781$	−1.90126	−0.0680325
$782$	0	0
$783$	0	0
$784$	−14.8712	−0.531115
$785$	−11.1991	−0.399714
$786$	0	0
$787$	−13.1730	−0.469568	−0.234784	−	0.972048i	$-0.575438\pi$
$787$	−13.1730	−0.469568	−0.234784	+	0.972048i	$0.575438\pi$
$788$	−8.84644	−0.315141
$789$	0	0
$790$	0.335522	0.0119373
$791$	1.34134	0.0476926
$792$	0	0
$793$	−7.44313	−0.264313
$794$	−0.200923	−0.00713049
$795$	0	0
$796$	4.79911	0.170100
$797$	−36.3506	−1.28761	−0.643803	−	0.765191i	$-0.722644\pi$
$797$	−36.3506	−1.28761	−0.643803	+	0.765191i	$0.722644\pi$
$798$	0	0
$799$	0	0
$800$	11.8509	0.418991
$801$	0	0
$802$	−0.281040	−0.00992388
$803$	−32.2494	−1.13806
$804$	0	0
$805$	2.33440	0.0822767
$806$	2.93625	0.103425
$807$	0	0
$808$	16.2793	0.572703
$809$	−45.1267	−1.58657	−0.793285	−	0.608851i	$-0.791631\pi$
$809$	−45.1267	−1.58657	−0.793285	+	0.608851i	$0.791631\pi$
$810$	0	0
$811$	3.32815	0.116867	0.0584336	−	0.998291i	$-0.481389\pi$
$811$	3.32815	0.116867	0.0584336	+	0.998291i	$0.481389\pi$
$812$	−32.3232	−1.13432
$813$	0	0
$814$	9.33565	0.327214
$815$	10.5580	0.369830
$816$	0	0
$817$	−6.79759	−0.237817
$818$	−6.36244	−0.222458
$819$	0	0
$820$	8.59255	0.300065
$821$	−7.46983	−0.260699	−0.130349	−	0.991468i	$-0.541610\pi$
$821$	−7.46983	−0.260699	−0.130349	+	0.991468i	$0.541610\pi$
$822$	0	0
$823$	−54.1834	−1.88871	−0.944357	−	0.328923i	$-0.893314\pi$
$823$	−54.1834	−1.88871	−0.944357	+	0.328923i	$0.893314\pi$
$824$	2.98013	0.103818
$825$	0	0
$826$	−3.88364	−0.135129
$827$	−4.95063	−0.172150	−0.0860752	−	0.996289i	$-0.527433\pi$
$827$	−4.95063	−0.172150	−0.0860752	+	0.996289i	$0.527433\pi$
$828$	0	0
$829$	−0.0762440	−0.00264806	−0.00132403	−	0.999999i	$-0.500421\pi$
$829$	−0.0762440	−0.00264806	−0.00132403	+	0.999999i	$0.500421\pi$
$830$	1.88889	0.0655643
$831$	0	0
$832$	−17.1412	−0.594265
$833$	0	0
$834$	0	0
$835$	9.05187	0.313253
$836$	9.18010	0.317501
$837$	0	0
$838$	−5.32328	−0.183890
$839$	−36.4479	−1.25832	−0.629160	−	0.777276i	$-0.716601\pi$
$839$	−36.4479	−1.25832	−0.629160	+	0.777276i	$0.716601\pi$
$840$	0	0
$841$	62.9113	2.16935
$842$	2.78433	0.0959545
$843$	0	0
$844$	38.8249	1.33641
$845$	4.27583	0.147093
$846$	0	0
$847$	23.3491	0.802283
$848$	23.8843	0.820191
$849$	0	0
$850$	0	0
$851$	18.0112	0.617418
$852$	0	0
$853$	−39.9684	−1.36849	−0.684245	−	0.729252i	$-0.739868\pi$
$853$	−39.9684	−1.36849	−0.684245	+	0.729252i	$0.739868\pi$
$854$	1.14055	0.0390287
$855$	0	0
$856$	−11.5457	−0.394625
$857$	−2.95962	−0.101099	−0.0505493	−	0.998722i	$-0.516097\pi$
$857$	−2.95962	−0.101099	−0.0505493	+	0.998722i	$0.516097\pi$
$858$	0	0
$859$	−22.4494	−0.765963	−0.382981	−	0.923756i	$-0.625103\pi$
$859$	−22.4494	−0.765963	−0.382981	+	0.923756i	$0.625103\pi$
$860$	8.87670	0.302693
$861$	0	0
$862$	3.92673	0.133745
$863$	27.8981	0.949661	0.474831	−	0.880077i	$-0.342509\pi$
$863$	27.8981	0.949661	0.474831	+	0.880077i	$0.342509\pi$
$864$	0	0
$865$	1.20491	0.0409682
$866$	3.11236	0.105762
$867$	0	0
$868$	17.7775	0.603409
$869$	11.7468	0.398484
$870$	0	0
$871$	19.3013	0.654001
$872$	15.2645	0.516921
$873$	0	0
$874$	−0.448256	−0.0151625
$875$	10.5544	0.356803
$876$	0	0
$877$	−20.6961	−0.698858	−0.349429	−	0.936963i	$-0.613624\pi$
$877$	−20.6961	−0.698858	−0.349429	+	0.936963i	$0.613624\pi$
$878$	−2.57165	−0.0867889
$879$	0	0
$880$	−11.6768	−0.393627
$881$	−22.2399	−0.749282	−0.374641	−	0.927170i	$-0.622234\pi$
$881$	−22.2399	−0.749282	−0.374641	+	0.927170i	$0.622234\pi$
$882$	0	0
$883$	4.23253	0.142436	0.0712180	−	0.997461i	$-0.477311\pi$
$883$	4.23253	0.142436	0.0712180	+	0.997461i	$0.477311\pi$
$884$	0	0
$885$	0	0
$886$	−7.91388	−0.265872
$887$	−36.3047	−1.21899	−0.609496	−	0.792789i	$-0.708628\pi$
$887$	−36.3047	−1.21899	−0.609496	+	0.792789i	$0.708628\pi$
$888$	0	0
$889$	18.7568	0.629084
$890$	0.281217	0.00942642
$891$	0	0
$892$	−28.0171	−0.938081
$893$	7.70129	0.257714
$894$	0	0
$895$	−14.1580	−0.473251
$896$	11.5422	0.385598
$897$	0	0
$898$	4.92066	0.164205
$899$	−50.5506	−1.68596
$900$	0	0
$901$	0	0
$902$	−7.61380	−0.253512
$903$	0	0
$904$	0.681206	0.0226566
$905$	−7.39133	−0.245696
$906$	0	0
$907$	−24.3718	−0.809251	−0.404626	−	0.914482i	$-0.632598\pi$
$907$	−24.3718	−0.809251	−0.404626	+	0.914482i	$0.632598\pi$
$908$	21.2119	0.703942
$909$	0	0
$910$	0.612546	0.0203057
$911$	26.9178	0.891826	0.445913	−	0.895076i	$-0.352879\pi$
$911$	26.9178	0.891826	0.445913	+	0.895076i	$0.352879\pi$
$912$	0	0
$913$	66.1312	2.18862
$914$	−2.20110	−0.0728059
$915$	0	0
$916$	−37.0425	−1.22392
$917$	1.22066	0.0403099
$918$	0	0
$919$	−23.8000	−0.785088	−0.392544	−	0.919733i	$-0.628405\pi$
$919$	−23.8000	−0.785088	−0.392544	+	0.919733i	$0.628405\pi$
$920$	1.18553	0.0390859
$921$	0	0
$922$	−2.13830	−0.0704210
$923$	−0.962511	−0.0316814
$924$	0	0
$925$	38.9979	1.28224
$926$	−3.39983	−0.111725
$927$	0	0
$928$	−24.7258	−0.811663
$929$	−5.82742	−0.191191	−0.0955957	−	0.995420i	$-0.530476\pi$
$929$	−5.82742	−0.191191	−0.0955957	+	0.995420i	$0.530476\pi$
$930$	0	0
$931$	3.81440	0.125012
$932$	−13.8120	−0.452427
$933$	0	0
$934$	−4.95819	−0.162237
$935$	0	0
$936$	0	0
$937$	55.4424	1.81123	0.905613	−	0.424106i	$-0.139411\pi$
$937$	55.4424	1.81123	0.905613	+	0.424106i	$0.139411\pi$
$938$	−2.95764	−0.0965702
$939$	0	0
$940$	−10.0568	−0.328017
$941$	−21.1696	−0.690109	−0.345054	−	0.938583i	$-0.612140\pi$
$941$	−21.1696	−0.690109	−0.345054	+	0.938583i	$0.612140\pi$
$942$	0	0
$943$	−14.6893	−0.478349
$944$	37.4791	1.21984
$945$	0	0
$946$	−7.86558	−0.255732
$947$	37.5831	1.22129	0.610643	−	0.791906i	$-0.290911\pi$
$947$	37.5831	1.22129	0.610643	+	0.791906i	$0.290911\pi$
$948$	0	0
$949$	−16.3262	−0.529971
$950$	−0.970563	−0.0314892
$951$	0	0
$952$	0	0
$953$	−21.2794	−0.689307	−0.344654	−	0.938730i	$-0.612004\pi$
$953$	−21.2794	−0.689307	−0.344654	+	0.938730i	$0.612004\pi$
$954$	0	0
$955$	4.22134	0.136599
$956$	−10.3163	−0.333654
$957$	0	0
$958$	3.45666	0.111680
$959$	−21.2813	−0.687210
$960$	0	0
$961$	−3.19748	−0.103144
$962$	4.72616	0.152377
$963$	0	0
$964$	11.3103	0.364279
$965$	1.85683	0.0597734
$966$	0	0
$967$	−26.2320	−0.843563	−0.421782	−	0.906697i	$-0.638595\pi$
$967$	−26.2320	−0.843563	−0.421782	+	0.906697i	$0.638595\pi$
$968$	11.8579	0.381128
$969$	0	0
$970$	−0.430067	−0.0138086
$971$	20.8243	0.668285	0.334142	−	0.942523i	$-0.391553\pi$
$971$	20.8243	0.668285	0.334142	+	0.942523i	$0.391553\pi$
$972$	0	0
$973$	23.0650	0.739429
$974$	−0.00658834	−0.000211104 0
$975$	0	0
$976$	−11.0069	−0.352321
$977$	−16.2900	−0.521162	−0.260581	−	0.965452i	$-0.583914\pi$
$977$	−16.2900	−0.521162	−0.260581	+	0.965452i	$0.583914\pi$
$978$	0	0
$979$	9.84558	0.314666
$980$	−4.98108	−0.159115
$981$	0	0
$982$	1.48187	0.0472884
$983$	0.931671	0.0297157	0.0148578	−	0.999890i	$-0.495270\pi$
$983$	0.931671	0.0297157	0.0148578	+	0.999890i	$0.495270\pi$
$984$	0	0
$985$	−2.88620	−0.0919621
$986$	0	0
$987$	0	0
$988$	4.64741	0.147854
$989$	−15.1750	−0.482538
$990$	0	0
$991$	21.6972	0.689234	0.344617	−	0.938743i	$-0.388009\pi$
$991$	21.6972	0.689234	0.344617	+	0.938743i	$0.388009\pi$
$992$	13.5990	0.431769
$993$	0	0
$994$	0.147490	0.00467810
$995$	1.56574	0.0496373
$996$	0	0
$997$	3.22709	0.102203	0.0511015	−	0.998693i	$-0.483727\pi$
$997$	3.22709	0.102203	0.0511015	+	0.998693i	$0.483727\pi$
$998$	−4.14317	−0.131150
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.bf.1.2		4
3.2	odd	2		867.2.a.k.1.3		4
17.2	even	8		153.2.f.b.55.2		8
17.9	even	8		153.2.f.b.64.3		8
17.16	even	2		2601.2.a.be.1.2		4
51.2	odd	8		51.2.e.a.4.3	✓	8
51.5	even	16		867.2.h.k.688.2		16
51.8	odd	8		867.2.e.g.829.2		8
51.11	even	16		867.2.h.i.733.4		16
51.14	even	16		867.2.h.i.757.4		16
51.20	even	16		867.2.h.i.757.3		16
51.23	even	16		867.2.h.i.733.3		16
51.26	odd	8		51.2.e.a.13.2	yes	8
51.29	even	16		867.2.h.k.688.1		16
51.32	odd	8		867.2.e.g.616.3		8
51.38	odd	4		867.2.d.f.577.4		8
51.41	even	16		867.2.h.k.712.2		16
51.44	even	16		867.2.h.k.712.1		16
51.47	odd	4		867.2.d.f.577.3		8
51.50	odd	2		867.2.a.l.1.3		4
68.19	odd	8		2448.2.be.x.1585.3		8
68.43	odd	8		2448.2.be.x.1441.3		8
204.155	even	8		816.2.bd.e.769.3		8
204.179	even	8		816.2.bd.e.625.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.e.a.4.3	✓	8	51.2	odd	8
51.2.e.a.13.2	yes	8	51.26	odd	8
153.2.f.b.55.2		8	17.2	even	8
153.2.f.b.64.3		8	17.9	even	8
816.2.bd.e.625.3		8	204.179	even	8
816.2.bd.e.769.3		8	204.155	even	8
867.2.a.k.1.3		4	3.2	odd	2
867.2.a.l.1.3		4	51.50	odd	2
867.2.d.f.577.3		8	51.47	odd	4
867.2.d.f.577.4		8	51.38	odd	4
867.2.e.g.616.3		8	51.32	odd	8
867.2.e.g.829.2		8	51.8	odd	8
867.2.h.i.733.3		16	51.23	even	16
867.2.h.i.733.4		16	51.11	even	16
867.2.h.i.757.3		16	51.20	even	16
867.2.h.i.757.4		16	51.14	even	16
867.2.h.k.688.1		16	51.29	even	16
867.2.h.k.688.2		16	51.5	even	16
867.2.h.k.712.1		16	51.44	even	16
867.2.h.k.712.2		16	51.41	even	16
2448.2.be.x.1441.3		8	68.43	odd	8
2448.2.be.x.1585.3		8	68.19	odd	8
2601.2.a.be.1.2		4	17.16	even	2
2601.2.a.bf.1.2		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)

\(f(q)\)	\(=\)	\(q-0.222191 q^{2} -1.95063 q^{4} -0.636405 q^{5} +1.72844 q^{7} +0.877796 q^{8} +O(q^{10})\)	\(q-0.222191 q^{2} -1.95063 q^{4} -0.636405 q^{5} +1.72844 q^{7} +0.877796 q^{8} +0.141404 q^{10} +4.95063 q^{11} +2.50625 q^{13} -0.384044 q^{14} +3.70622 q^{16} -0.950631 q^{19} +1.24139 q^{20} -1.09999 q^{22} -2.12220 q^{23} -4.59499 q^{25} -0.556867 q^{26} -3.37155 q^{28} +9.58704 q^{29} -5.27281 q^{31} -2.57908 q^{32} -1.09999 q^{35} -8.48705 q^{37} +0.211222 q^{38} -0.558634 q^{40} +6.92171 q^{41} +7.15061 q^{43} -9.65685 q^{44} +0.471535 q^{46} -8.10124 q^{47} -4.01250 q^{49} +1.02097 q^{50} -4.88877 q^{52} +6.44438 q^{53} -3.15061 q^{55} +1.51722 q^{56} -2.13016 q^{58} +10.1125 q^{59} -2.96983 q^{61} +1.17157 q^{62} -6.83940 q^{64} -1.59499 q^{65} +7.70129 q^{67} +0.244408 q^{70} -0.384044 q^{71} -6.51420 q^{73} +1.88575 q^{74} +1.85433 q^{76} +8.55687 q^{77} +2.37280 q^{79} -2.35866 q^{80} -1.53794 q^{82} +13.3581 q^{83} -1.58880 q^{86} +4.34564 q^{88} +1.98875 q^{89} +4.33190 q^{91} +4.13964 q^{92} +1.80002 q^{94} +0.604986 q^{95} -3.04142 q^{97} +0.891542 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 6 q^{4} + 6 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 6 * q^4 + 6 * q^5 - 4 * q^7 + 6 * q^8	\( 4 q + 2 q^{2} + 6 q^{4} + 6 q^{5} - 4 q^{7} + 6 q^{8} + 12 q^{10} + 6 q^{11} + 2 q^{13} + 4 q^{14} + 6 q^{16} + 10 q^{19} + 16 q^{20} - 4 q^{22} - 6 q^{23} + 2 q^{25} + 20 q^{26} - 24 q^{28} + 16 q^{29} - 4 q^{31} + 14 q^{32} - 4 q^{35} - 12 q^{37} + 12 q^{38} + 8 q^{40} - 14 q^{41} + 14 q^{43} - 16 q^{44} + 12 q^{46} - 4 q^{47} + 30 q^{50} - 8 q^{52} + 20 q^{53} + 2 q^{55} - 16 q^{56} - 8 q^{58} + 24 q^{59} - 12 q^{61} + 16 q^{62} - 2 q^{64} + 14 q^{65} + 4 q^{67} - 4 q^{70} + 4 q^{71} - 20 q^{73} + 12 q^{74} + 40 q^{76} + 12 q^{77} - 8 q^{79} + 4 q^{82} + 4 q^{83} + 4 q^{86} - 16 q^{88} - 4 q^{89} + 28 q^{91} + 16 q^{92} + 8 q^{94} + 22 q^{95} - 24 q^{97} - 38 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 6 * q^4 + 6 * q^5 - 4 * q^7 + 6 * q^8 + 12 * q^10 + 6 * q^11 + 2 * q^13 + 4 * q^14 + 6 * q^16 + 10 * q^19 + 16 * q^20 - 4 * q^22 - 6 * q^23 + 2 * q^25 + 20 * q^26 - 24 * q^28 + 16 * q^29 - 4 * q^31 + 14 * q^32 - 4 * q^35 - 12 * q^37 + 12 * q^38 + 8 * q^40 - 14 * q^41 + 14 * q^43 - 16 * q^44 + 12 * q^46 - 4 * q^47 + 30 * q^50 - 8 * q^52 + 20 * q^53 + 2 * q^55 - 16 * q^56 - 8 * q^58 + 24 * q^59 - 12 * q^61 + 16 * q^62 - 2 * q^64 + 14 * q^65 + 4 * q^67 - 4 * q^70 + 4 * q^71 - 20 * q^73 + 12 * q^74 + 40 * q^76 + 12 * q^77 - 8 * q^79 + 4 * q^82 + 4 * q^83 + 4 * q^86 - 16 * q^88 - 4 * q^89 + 28 * q^91 + 16 * q^92 + 8 * q^94 + 22 * q^95 - 24 * q^97 - 38 * q^98

Introduction

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