LMFDB - Embedded newform 2601.2.a.bk.1.4

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:	$6$
Coefficient field:	6.6.45769536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 15x^{4} + 72x^{2} - 109 $ x^6 - 15x^4 + 72x^2 - 109
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.76653$ 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)

$f(q)$	$=$	$q+1.76653 q^{2} +1.12061 q^{4} -3.31998 q^{5} +4.87939 q^{7} -1.55346 q^{8} +O(q^{10})$	$q+1.76653 q^{2} +1.12061 q^{4} -3.31998 q^{5} +4.87939 q^{7} -1.55346 q^{8} -5.86484 q^{10} +2.38003 q^{11} -1.83750 q^{13} +8.61956 q^{14} -4.98545 q^{16} +1.94356 q^{19} -3.72042 q^{20} +4.20439 q^{22} +7.17948 q^{23} +6.02229 q^{25} -3.24599 q^{26} +5.46791 q^{28} -6.02646 q^{29} +5.65270 q^{31} -5.70002 q^{32} -16.1995 q^{35} +1.57398 q^{37} +3.43335 q^{38} +5.15745 q^{40} +10.4995 q^{41} +5.75877 q^{43} +2.66710 q^{44} +12.6827 q^{46} -2.91954 q^{47} +16.8084 q^{49} +10.6385 q^{50} -2.05913 q^{52} -9.23307 q^{53} -7.90167 q^{55} -7.57992 q^{56} -10.6459 q^{58} -0.726880 q^{59} +7.22668 q^{61} +9.98565 q^{62} -0.0983261 q^{64} +6.10046 q^{65} +5.14796 q^{67} -28.6168 q^{70} +9.34644 q^{71} -1.34730 q^{73} +2.78047 q^{74} +2.17799 q^{76} +11.6131 q^{77} +10.7588 q^{79} +16.5516 q^{80} +18.5476 q^{82} -5.37358 q^{83} +10.1730 q^{86} -3.69728 q^{88} +11.1130 q^{89} -8.96585 q^{91} +8.04543 q^{92} -5.15745 q^{94} -6.45260 q^{95} +1.08378 q^{97} +29.6925 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 18 q^{4} + 18 q^{7}+O(q^{10}) $ 6 * q + 18 * q^4 + 18 * q^7	$ 6 q + 18 q^{4} + 18 q^{7} + 12 q^{10} - 6 q^{13} + 6 q^{16} - 18 q^{19} + 24 q^{22} + 24 q^{25} + 42 q^{28} + 36 q^{31} - 6 q^{37} + 66 q^{40} + 12 q^{43} + 18 q^{46} + 24 q^{49} - 24 q^{52} - 24 q^{55} + 18 q^{58} + 30 q^{61} - 24 q^{64} - 18 q^{70} - 6 q^{73} - 78 q^{76} + 42 q^{79} + 6 q^{82} - 12 q^{91} - 66 q^{94} - 6 q^{97}+O(q^{100}) $ 6 * q + 18 * q^4 + 18 * q^7 + 12 * q^10 - 6 * q^13 + 6 * q^16 - 18 * q^19 + 24 * q^22 + 24 * q^25 + 42 * q^28 + 36 * q^31 - 6 * q^37 + 66 * q^40 + 12 * q^43 + 18 * q^46 + 24 * q^49 - 24 * q^52 - 24 * q^55 + 18 * q^58 + 30 * q^61 - 24 * q^64 - 18 * q^70 - 6 * q^73 - 78 * q^76 + 42 * q^79 + 6 * q^82 - 12 * q^91 - 66 * q^94 - 6 * q^97

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 $

$2$	1.76653	1.24912	0.624561	−	0.780976i	$-0.285278\pi$
$2$	1.76653	1.24912	0.624561	+	0.780976i	$0.285278\pi$
$3$	0	0
$3$	0	0
$4$	1.12061	0.560307
$5$	−3.31998	−1.48474	−0.742371	−	0.669989i	$-0.766299\pi$
$5$	−3.31998	−1.48474	−0.742371	+	0.669989i	$0.766299\pi$
$6$	0	0
$7$	4.87939	1.84423	0.922117	−	0.386911i	$-0.126458\pi$
$7$	4.87939	1.84423	0.922117	+	0.386911i	$0.126458\pi$
$8$	−1.55346	−0.549230
$9$	0	0
$10$	−5.86484	−1.85462
$11$	2.38003	0.717607	0.358804	−	0.933413i	$-0.383185\pi$
$11$	2.38003	0.717607	0.358804	+	0.933413i	$0.383185\pi$
$12$	0	0
$13$	−1.83750	−0.509630	−0.254815	−	0.966990i	$-0.582015\pi$
$13$	−1.83750	−0.509630	−0.254815	+	0.966990i	$0.582015\pi$
$14$	8.61956	2.30367
$15$	0	0
$16$	−4.98545	−1.24636
$17$	0	0
$17$	0	0
$18$	0	0
$19$	1.94356	0.445884	0.222942	−	0.974832i	$-0.428434\pi$
$19$	1.94356	0.445884	0.222942	+	0.974832i	$0.428434\pi$
$20$	−3.72042	−0.831912
$21$	0	0
$22$	4.20439	0.896380
$23$	7.17948	1.49702	0.748512	−	0.663121i	$-0.230769\pi$
$23$	7.17948	1.49702	0.748512	+	0.663121i	$0.230769\pi$
$24$	0	0
$25$	6.02229	1.20446
$26$	−3.24599	−0.636590
$27$	0	0
$28$	5.46791	1.03334
$29$	−6.02646	−1.11909	−0.559543	−	0.828802i	$-0.689023\pi$
$29$	−6.02646	−1.11909	−0.559543	+	0.828802i	$0.689023\pi$
$30$	0	0
$31$	5.65270	1.01526	0.507628	−	0.861576i	$-0.330523\pi$
$31$	5.65270	1.01526	0.507628	+	0.861576i	$0.330523\pi$
$32$	−5.70002	−1.00763
$33$	0	0
$34$	0	0
$35$	−16.1995	−2.73821
$36$	0	0
$37$	1.57398	0.258760	0.129380	−	0.991595i	$-0.458701\pi$
$37$	1.57398	0.258760	0.129380	+	0.991595i	$0.458701\pi$
$38$	3.43335	0.556964
$39$	0	0
$40$	5.15745	0.815465
$41$	10.4995	1.63974	0.819870	−	0.572550i	$-0.194046\pi$
$41$	10.4995	1.63974	0.819870	+	0.572550i	$0.194046\pi$
$42$	0	0
$43$	5.75877	0.878204	0.439102	−	0.898437i	$-0.355297\pi$
$43$	5.75877	0.878204	0.439102	+	0.898437i	$0.355297\pi$
$44$	2.66710	0.402081
$45$	0	0
$46$	12.6827	1.86997
$47$	−2.91954	−0.425859	−0.212930	−	0.977068i	$-0.568301\pi$
$47$	−2.91954	−0.425859	−0.212930	+	0.977068i	$0.568301\pi$
$48$	0	0
$49$	16.8084	2.40120
$50$	10.6385	1.50452
$51$	0	0
$52$	−2.05913	−0.285549
$53$	−9.23307	−1.26826	−0.634130	−	0.773227i	$-0.718642\pi$
$53$	−9.23307	−1.26826	−0.634130	+	0.773227i	$0.718642\pi$
$54$	0	0
$55$	−7.90167	−1.06546
$56$	−7.57992	−1.01291
$57$	0	0
$58$	−10.6459	−1.39787
$59$	−0.726880	−0.0946317	−0.0473158	−	0.998880i	$-0.515067\pi$
$59$	−0.726880	−0.0946317	−0.0473158	+	0.998880i	$0.515067\pi$
$60$	0	0
$61$	7.22668	0.925282	0.462641	−	0.886546i	$-0.346902\pi$
$61$	7.22668	0.925282	0.462641	+	0.886546i	$0.346902\pi$
$62$	9.98565	1.26818
$63$	0	0
$64$	−0.0983261	−0.0122908
$65$	6.10046	0.756669
$66$	0	0
$67$	5.14796	0.628923	0.314461	−	0.949270i	$-0.398176\pi$
$67$	5.14796	0.628923	0.314461	+	0.949270i	$0.398176\pi$
$68$	0	0
$69$	0	0
$70$	−28.6168	−3.42036
$71$	9.34644	1.10922	0.554609	−	0.832111i	$-0.312868\pi$
$71$	9.34644	1.10922	0.554609	+	0.832111i	$0.312868\pi$
$72$	0	0
$73$	−1.34730	−0.157689	−0.0788446	−	0.996887i	$-0.525123\pi$
$73$	−1.34730	−0.157689	−0.0788446	+	0.996887i	$0.525123\pi$
$74$	2.78047	0.323223
$75$	0	0
$76$	2.17799	0.249832
$77$	11.6131	1.32344
$78$	0	0
$79$	10.7588	1.21046	0.605228	−	0.796052i	$-0.293082\pi$
$79$	10.7588	1.21046	0.605228	+	0.796052i	$0.293082\pi$
$80$	16.5516	1.85053
$81$	0	0
$82$	18.5476	2.04824
$83$	−5.37358	−0.589827	−0.294913	−	0.955524i	$-0.595291\pi$
$83$	−5.37358	−0.589827	−0.294913	+	0.955524i	$0.595291\pi$
$84$	0	0
$85$	0	0
$86$	10.1730	1.09698
$87$	0	0
$88$	−3.69728	−0.394131
$89$	11.1130	1.17797	0.588986	−	0.808143i	$-0.299527\pi$
$89$	11.1130	1.17797	0.588986	+	0.808143i	$0.299527\pi$
$90$	0	0
$91$	−8.96585	−0.939877
$92$	8.04543	0.838794
$93$	0	0
$94$	−5.15745	−0.531950
$95$	−6.45260	−0.662022
$96$	0	0
$97$	1.08378	0.110041	0.0550205	−	0.998485i	$-0.482478\pi$
$97$	1.08378	0.110041	0.0550205	+	0.998485i	$0.482478\pi$
$98$	29.6925	2.99939
$99$	0	0
$100$	6.74867	0.674867
$101$	13.3796	1.33132	0.665661	−	0.746254i	$-0.268149\pi$
$101$	13.3796	1.33132	0.665661	+	0.746254i	$0.268149\pi$
$102$	0	0
$103$	11.6604	1.14894	0.574469	−	0.818526i	$-0.305209\pi$
$103$	11.6604	1.14894	0.574469	+	0.818526i	$0.305209\pi$
$104$	2.85447	0.279904
$105$	0	0
$106$	−16.3105	−1.58421
$107$	−5.08651	−0.491731	−0.245866	−	0.969304i	$-0.579072\pi$
$107$	−5.08651	−0.491731	−0.245866	+	0.969304i	$0.579072\pi$
$108$	0	0
$109$	−5.55169	−0.531755	−0.265878	−	0.964007i	$-0.585662\pi$
$109$	−5.55169	−0.531755	−0.265878	+	0.964007i	$0.585662\pi$
$110$	−13.9585	−1.33089
$111$	0	0
$112$	−24.3259	−2.29859
$113$	−17.2785	−1.62542	−0.812712	−	0.582665i	$-0.802010\pi$
$113$	−17.2785	−1.62542	−0.812712	+	0.582665i	$0.802010\pi$
$114$	0	0
$115$	−23.8357	−2.22269
$116$	−6.75334	−0.627032
$117$	0	0
$118$	−1.28405	−0.118207
$119$	0	0
$120$	0	0
$121$	−5.33544	−0.485040
$122$	12.7661	1.15579
$123$	0	0
$124$	6.33450	0.568855
$125$	−3.39398	−0.303567
$126$	0	0
$127$	−10.8598	−0.963650	−0.481825	−	0.876267i	$-0.660026\pi$
$127$	−10.8598	−0.963650	−0.481825	+	0.876267i	$0.660026\pi$
$128$	11.2263	0.992278
$129$	0	0
$130$	10.7766	0.945172
$131$	−7.06610	−0.617368	−0.308684	−	0.951165i	$-0.599889\pi$
$131$	−7.06610	−0.617368	−0.308684	+	0.951165i	$0.599889\pi$
$132$	0	0
$133$	9.48339	0.822314
$134$	9.09400	0.785602
$135$	0	0
$136$	0	0
$137$	0.539510	0.0460934	0.0230467	−	0.999734i	$-0.492663\pi$
$137$	0.539510	0.0460934	0.0230467	+	0.999734i	$0.492663\pi$
$138$	0	0
$139$	8.40373	0.712795	0.356398	−	0.934334i	$-0.384005\pi$
$139$	8.40373	0.712795	0.356398	+	0.934334i	$0.384005\pi$
$140$	−18.1534	−1.53424
$141$	0	0
$142$	16.5107	1.38555
$143$	−4.37330	−0.365714
$144$	0	0
$145$	20.0077	1.66155
$146$	−2.38003	−0.196973
$147$	0	0
$148$	1.76382	0.144985
$149$	−16.1255	−1.32105	−0.660525	−	0.750804i	$-0.729666\pi$
$149$	−16.1255	−1.32105	−0.660525	+	0.750804i	$0.729666\pi$
$150$	0	0
$151$	−10.2121	−0.831052	−0.415526	−	0.909581i	$-0.636402\pi$
$151$	−10.2121	−0.831052	−0.415526	+	0.909581i	$0.636402\pi$
$152$	−3.01924	−0.244893
$153$	0	0
$154$	20.5149	1.65313
$155$	−18.7669	−1.50739
$156$	0	0
$157$	15.1702	1.21072	0.605359	−	0.795953i	$-0.293030\pi$
$157$	15.1702	1.21072	0.605359	+	0.795953i	$0.293030\pi$
$158$	19.0056	1.51201
$159$	0	0
$160$	18.9240	1.49607
$161$	35.0314	2.76086
$162$	0	0
$163$	−17.1506	−1.34334	−0.671671	−	0.740849i	$-0.734423\pi$
$163$	−17.1506	−1.34334	−0.671671	+	0.740849i	$0.734423\pi$
$164$	11.7658	0.918758
$165$	0	0
$166$	−9.49256	−0.736766
$167$	−9.84658	−0.761951	−0.380976	−	0.924585i	$-0.624412\pi$
$167$	−9.84658	−0.761951	−0.380976	+	0.924585i	$0.624412\pi$
$168$	0	0
$169$	−9.62361	−0.740278
$170$	0	0
$171$	0	0
$172$	6.45336	0.492064
$173$	−9.15907	−0.696351	−0.348176	−	0.937429i	$-0.613199\pi$
$173$	−9.15907	−0.696351	−0.348176	+	0.937429i	$0.613199\pi$
$174$	0	0
$175$	29.3851	2.22130
$176$	−11.8655	−0.894399
$177$	0	0
$178$	19.6313	1.47143
$179$	−1.01395	−0.0757860	−0.0378930	−	0.999282i	$-0.512065\pi$
$179$	−1.01395	−0.0757860	−0.0378930	+	0.999282i	$0.512065\pi$
$180$	0	0
$181$	21.9736	1.63328	0.816642	−	0.577144i	$-0.195833\pi$
$181$	21.9736	1.63328	0.816642	+	0.577144i	$0.195833\pi$
$182$	−15.8384	−1.17402
$183$	0	0
$184$	−11.1530	−0.822211
$185$	−5.22558	−0.384192
$186$	0	0
$187$	0	0
$188$	−3.27168	−0.238612
$189$	0	0
$190$	−11.3987	−0.826947
$191$	1.47946	0.107050	0.0535249	−	0.998567i	$-0.482954\pi$
$191$	1.47946	0.107050	0.0535249	+	0.998567i	$0.482954\pi$
$192$	0	0
$193$	−4.69965	−0.338288	−0.169144	−	0.985591i	$-0.554100\pi$
$193$	−4.69965	−0.338288	−0.169144	+	0.985591i	$0.554100\pi$
$194$	1.91452	0.137455
$195$	0	0
$196$	18.8357	1.34541
$197$	13.1319	0.935612	0.467806	−	0.883831i	$-0.345045\pi$
$197$	13.1319	0.935612	0.467806	+	0.883831i	$0.345045\pi$
$198$	0	0
$199$	1.77332	0.125707	0.0628536	−	0.998023i	$-0.479980\pi$
$199$	1.77332	0.125707	0.0628536	+	0.998023i	$0.479980\pi$
$200$	−9.35537	−0.661524
$201$	0	0
$202$	23.6355	1.66299
$203$	−29.4054	−2.06386
$204$	0	0
$205$	−34.8580	−2.43459
$206$	20.5985	1.43516
$207$	0	0
$208$	9.16075	0.635184
$209$	4.62575	0.319970
$210$	0	0
$211$	−0.887126	−0.0610723	−0.0305361	−	0.999534i	$-0.509721\pi$
$211$	−0.887126	−0.0610723	−0.0305361	+	0.999534i	$0.509721\pi$
$212$	−10.3467	−0.710615
$213$	0	0
$214$	−8.98545	−0.614233
$215$	−19.1190	−1.30391
$216$	0	0
$217$	27.5817	1.87237
$218$	−9.80720	−0.664228
$219$	0	0
$220$	−8.85473	−0.596986
$221$	0	0
$222$	0	0
$223$	−0.288171	−0.0192974	−0.00964868	−	0.999953i	$-0.503071\pi$
$223$	−0.288171	−0.0192974	−0.00964868	+	0.999953i	$0.503071\pi$
$224$	−27.8126	−1.85831
$225$	0	0
$226$	−30.5229	−2.03035
$227$	−8.40649	−0.557959	−0.278979	−	0.960297i	$-0.589996\pi$
$227$	−8.40649	−0.557959	−0.278979	+	0.960297i	$0.589996\pi$
$228$	0	0
$229$	−3.87939	−0.256357	−0.128178	−	0.991751i	$-0.540913\pi$
$229$	−3.87939	−0.256357	−0.128178	+	0.991751i	$0.540913\pi$
$230$	−42.1065	−2.77642
$231$	0	0
$232$	9.36184	0.614635
$233$	12.7798	0.837232	0.418616	−	0.908163i	$-0.362515\pi$
$233$	12.7798	0.837232	0.418616	+	0.908163i	$0.362515\pi$
$234$	0	0
$235$	9.69284	0.632291
$236$	−0.814552	−0.0530228
$237$	0	0
$238$	0	0
$239$	8.87200	0.573882	0.286941	−	0.957948i	$-0.407362\pi$
$239$	8.87200	0.573882	0.286941	+	0.957948i	$0.407362\pi$
$240$	0	0
$241$	−14.7374	−0.949320	−0.474660	−	0.880169i	$-0.657429\pi$
$241$	−14.7374	−0.949320	−0.474660	+	0.880169i	$0.657429\pi$
$242$	−9.42519	−0.605874
$243$	0	0
$244$	8.09833	0.518442
$245$	−55.8036	−3.56516
$246$	0	0
$247$	−3.57129	−0.227236
$248$	−8.78123	−0.557609
$249$	0	0
$250$	−5.99556	−0.379192
$251$	−16.3385	−1.03128	−0.515640	−	0.856805i	$-0.672446\pi$
$251$	−16.3385	−1.03128	−0.515640	+	0.856805i	$0.672446\pi$
$252$	0	0
$253$	17.0874	1.07428
$254$	−19.1841	−1.20372
$255$	0	0
$256$	20.0283	1.25177
$257$	27.0254	1.68580	0.842898	−	0.538073i	$-0.180847\pi$
$257$	27.0254	1.68580	0.842898	+	0.538073i	$0.180847\pi$
$258$	0	0
$259$	7.68004	0.477215
$260$	6.83626	0.423967
$261$	0	0
$262$	−12.4825	−0.771169
$263$	−29.1047	−1.79467	−0.897335	−	0.441349i	$-0.854500\pi$
$263$	−29.1047	−1.79467	−0.897335	+	0.441349i	$0.854500\pi$
$264$	0	0
$265$	30.6536	1.88304
$266$	16.7527	1.02717
$267$	0	0
$268$	5.76888	0.352390
$269$	7.65391	0.466667	0.233334	−	0.972397i	$-0.425037\pi$
$269$	7.65391	0.466667	0.233334	+	0.972397i	$0.425037\pi$
$270$	0	0
$271$	−17.7219	−1.07653	−0.538265	−	0.842775i	$-0.680920\pi$
$271$	−17.7219	−1.07653	−0.538265	+	0.842775i	$0.680920\pi$
$272$	0	0
$273$	0	0
$274$	0.953058	0.0575764
$275$	14.3333	0.864328
$276$	0	0
$277$	8.68004	0.521533	0.260767	−	0.965402i	$-0.416025\pi$
$277$	8.68004	0.521533	0.260767	+	0.965402i	$0.416025\pi$
$278$	14.8454	0.890369
$279$	0	0
$280$	25.1652	1.50391
$281$	−5.83909	−0.348331	−0.174165	−	0.984716i	$-0.555723\pi$
$281$	−5.83909	−0.348331	−0.174165	+	0.984716i	$0.555723\pi$
$282$	0	0
$283$	−25.7033	−1.52790	−0.763950	−	0.645275i	$-0.776743\pi$
$283$	−25.7033	−1.52790	−0.763950	+	0.645275i	$0.776743\pi$
$284$	10.4738	0.621503
$285$	0	0
$286$	−7.72556	−0.456822
$287$	51.2309	3.02406
$288$	0	0
$289$	0	0
$290$	35.3442	2.07548
$291$	0	0
$292$	−1.50980	−0.0883544
$293$	17.3871	1.01577	0.507883	−	0.861426i	$-0.330428\pi$
$293$	17.3871	1.01577	0.507883	+	0.861426i	$0.330428\pi$
$294$	0	0
$295$	2.41323	0.140504
$296$	−2.44511	−0.142119
$297$	0	0
$298$	−28.4861	−1.65015
$299$	−13.1923	−0.762928
$300$	0	0
$301$	28.0993	1.61961
$302$	−18.0400	−1.03809
$303$	0	0
$304$	−9.68954	−0.555733
$305$	−23.9925	−1.37380
$306$	0	0
$307$	−14.4192	−0.822948	−0.411474	−	0.911421i	$-0.634986\pi$
$307$	−14.4192	−0.822948	−0.411474	+	0.911421i	$0.634986\pi$
$308$	13.0138	0.741531
$309$	0	0
$310$	−33.1522	−1.88292
$311$	−17.3782	−0.985427	−0.492714	−	0.870192i	$-0.663995\pi$
$311$	−17.3782	−0.985427	−0.492714	+	0.870192i	$0.663995\pi$
$312$	0	0
$313$	1.46522	0.0828193	0.0414096	−	0.999142i	$-0.486815\pi$
$313$	1.46522	0.0828193	0.0414096	+	0.999142i	$0.486815\pi$
$314$	26.7986	1.51233
$315$	0	0
$316$	12.0564	0.678228
$317$	0.113372	0.00636759	0.00318379	−	0.999995i	$-0.498987\pi$
$317$	0.113372	0.00636759	0.00318379	+	0.999995i	$0.498987\pi$
$318$	0	0
$319$	−14.3432	−0.803064
$320$	0.326441	0.0182486
$321$	0	0
$322$	61.8839	3.44866
$323$	0	0
$324$	0	0
$325$	−11.0659	−0.613827
$326$	−30.2971	−1.67800
$327$	0	0
$328$	−16.3105	−0.900594
$329$	−14.2456	−0.785384
$330$	0	0
$331$	−32.7229	−1.79861	−0.899306	−	0.437320i	$-0.855928\pi$
$331$	−32.7229	−1.79861	−0.899306	+	0.437320i	$0.855928\pi$
$332$	−6.02171	−0.330484
$333$	0	0
$334$	−17.3942	−0.951770
$335$	−17.0911	−0.933788
$336$	0	0
$337$	−1.13341	−0.0617407	−0.0308703	−	0.999523i	$-0.509828\pi$
$337$	−1.13341	−0.0617407	−0.0308703	+	0.999523i	$0.509828\pi$
$338$	−17.0004	−0.924697
$339$	0	0
$340$	0	0
$341$	13.4536	0.728555
$342$	0	0
$343$	47.8590	2.58414
$344$	−8.94600	−0.482336
$345$	0	0
$346$	−16.1797	−0.869828
$347$	2.59310	0.139205	0.0696025	−	0.997575i	$-0.477827\pi$
$347$	2.59310	0.139205	0.0696025	+	0.997575i	$0.477827\pi$
$348$	0	0
$349$	−16.6391	−0.890670	−0.445335	−	0.895364i	$-0.646915\pi$
$349$	−16.6391	−0.890670	−0.445335	+	0.895364i	$0.646915\pi$
$350$	51.9095	2.77468
$351$	0	0
$352$	−13.5662	−0.723083
$353$	−19.4801	−1.03682	−0.518410	−	0.855132i	$-0.673476\pi$
$353$	−19.4801	−1.03682	−0.518410	+	0.855132i	$0.673476\pi$
$354$	0	0
$355$	−31.0300	−1.64690
$356$	12.4534	0.660027
$357$	0	0
$358$	−1.79116	−0.0946660
$359$	16.7784	0.885528	0.442764	−	0.896638i	$-0.353998\pi$
$359$	16.7784	0.885528	0.442764	+	0.896638i	$0.353998\pi$
$360$	0	0
$361$	−15.2226	−0.801188
$362$	38.8169	2.04017
$363$	0	0
$364$	−10.0473	−0.526620
$365$	4.47300	0.234128
$366$	0	0
$367$	3.12742	0.163250	0.0816250	−	0.996663i	$-0.473989\pi$
$367$	3.12742	0.163250	0.0816250	+	0.996663i	$0.473989\pi$
$368$	−35.7929	−1.86584
$369$	0	0
$370$	−9.23112	−0.479903
$371$	−45.0517	−2.33897
$372$	0	0
$373$	−27.3678	−1.41705	−0.708526	−	0.705684i	$-0.750640\pi$
$373$	−27.3678	−1.41705	−0.708526	+	0.705684i	$0.750640\pi$
$374$	0	0
$375$	0	0
$376$	4.53539	0.233895
$377$	11.0736	0.570319
$378$	0	0
$379$	17.7520	0.911857	0.455929	−	0.890016i	$-0.349307\pi$
$379$	17.7520	0.911857	0.455929	+	0.890016i	$0.349307\pi$
$380$	−7.23088	−0.370936
$381$	0	0
$382$	2.61350	0.133718
$383$	1.55346	0.0793779	0.0396890	−	0.999212i	$-0.487363\pi$
$383$	1.55346	0.0793779	0.0396890	+	0.999212i	$0.487363\pi$
$384$	0	0
$385$	−38.5553	−1.96496
$386$	−8.30205	−0.422563
$387$	0	0
$388$	1.21450	0.0616568
$389$	−14.0582	−0.712780	−0.356390	−	0.934337i	$-0.615992\pi$
$389$	−14.0582	−0.712780	−0.356390	+	0.934337i	$0.615992\pi$
$390$	0	0
$391$	0	0
$392$	−26.1111	−1.31881
$393$	0	0
$394$	23.1979	1.16869
$395$	−35.7189	−1.79721
$396$	0	0
$397$	−23.7547	−1.19221	−0.596106	−	0.802906i	$-0.703286\pi$
$397$	−23.7547	−1.19221	−0.596106	+	0.802906i	$0.703286\pi$
$398$	3.13261	0.157024
$399$	0	0
$400$	−30.0238	−1.50119
$401$	−25.1801	−1.25743	−0.628717	−	0.777634i	$-0.716420\pi$
$401$	−25.1801	−1.25743	−0.628717	+	0.777634i	$0.716420\pi$
$402$	0	0
$403$	−10.3868	−0.517404
$404$	14.9934	0.745950
$405$	0	0
$406$	−51.9454	−2.57801
$407$	3.74612	0.185688
$408$	0	0
$409$	19.2499	0.951846	0.475923	−	0.879487i	$-0.342114\pi$
$409$	19.2499	0.951846	0.475923	+	0.879487i	$0.342114\pi$
$410$	−61.5776	−3.04110
$411$	0	0
$412$	13.0669	0.643758
$413$	−3.54673	−0.174523
$414$	0	0
$415$	17.8402	0.875740
$416$	10.4738	0.513518
$417$	0	0
$418$	8.17150	0.399681
$419$	−0.800878	−0.0391254	−0.0195627	−	0.999809i	$-0.506227\pi$
$419$	−0.800878	−0.0391254	−0.0195627	+	0.999809i	$0.506227\pi$
$420$	0	0
$421$	22.4611	1.09469	0.547344	−	0.836908i	$-0.315639\pi$
$421$	22.4611	1.09469	0.547344	+	0.836908i	$0.315639\pi$
$422$	−1.56713	−0.0762868
$423$	0	0
$424$	14.3432	0.696566
$425$	0	0
$426$	0	0
$427$	35.2618	1.70644
$428$	−5.70002	−0.275521
$429$	0	0
$430$	−33.7743	−1.62874
$431$	16.0515	0.773173	0.386586	−	0.922253i	$-0.373654\pi$
$431$	16.0515	0.773173	0.386586	+	0.922253i	$0.373654\pi$
$432$	0	0
$433$	−29.9445	−1.43904	−0.719520	−	0.694471i	$-0.755638\pi$
$433$	−29.9445	−1.43904	−0.719520	+	0.694471i	$0.755638\pi$
$434$	48.7238	2.33882
$435$	0	0
$436$	−6.22130	−0.297946
$437$	13.9538	0.667499
$438$	0	0
$439$	32.2344	1.53847	0.769233	−	0.638969i	$-0.220639\pi$
$439$	32.2344	1.53847	0.769233	+	0.638969i	$0.220639\pi$
$440$	12.2749	0.585183
$441$	0	0
$442$	0	0
$443$	0.800878	0.0380509	0.0190254	−	0.999819i	$-0.493944\pi$
$443$	0.800878	0.0380509	0.0190254	+	0.999819i	$0.493944\pi$
$444$	0	0
$445$	−36.8949	−1.74898
$446$	−0.509062	−0.0241048
$447$	0	0
$448$	−0.479771	−0.0226670
$449$	22.7261	1.07251	0.536255	−	0.844056i	$-0.319839\pi$
$449$	22.7261	1.07251	0.536255	+	0.844056i	$0.319839\pi$
$450$	0	0
$451$	24.9891	1.17669
$452$	−19.3625	−0.910737
$453$	0	0
$454$	−14.8503	−0.696959
$455$	29.7665	1.39547
$456$	0	0
$457$	32.1438	1.50363	0.751813	−	0.659377i	$-0.229180\pi$
$457$	32.1438	1.50363	0.751813	+	0.659377i	$0.229180\pi$
$458$	−6.85304	−0.320221
$459$	0	0
$460$	−26.7107	−1.24539
$461$	−18.0054	−0.838594	−0.419297	−	0.907849i	$-0.637723\pi$
$461$	−18.0054	−0.838594	−0.419297	+	0.907849i	$0.637723\pi$
$462$	0	0
$463$	8.01279	0.372386	0.186193	−	0.982513i	$-0.440385\pi$
$463$	8.01279	0.372386	0.186193	+	0.982513i	$0.440385\pi$
$464$	30.0446	1.39479
$465$	0	0
$466$	22.5758	1.04581
$467$	−9.23307	−0.427256	−0.213628	−	0.976915i	$-0.568528\pi$
$467$	−9.23307	−0.427256	−0.213628	+	0.976915i	$0.568528\pi$
$468$	0	0
$469$	25.1189	1.15988
$470$	17.1227	0.789809
$471$	0	0
$472$	1.12918	0.0519746
$473$	13.7061	0.630206
$474$	0	0
$475$	11.7047	0.537048
$476$	0	0
$477$	0	0
$478$	15.6726	0.716849
$479$	38.2637	1.74832	0.874158	−	0.485643i	$-0.161414\pi$
$479$	38.2637	1.74832	0.874158	+	0.485643i	$0.161414\pi$
$480$	0	0
$481$	−2.89218	−0.131872
$482$	−26.0340	−1.18582
$483$	0	0
$484$	−5.97897	−0.271771
$485$	−3.59813	−0.163382
$486$	0	0
$487$	−17.4097	−0.788910	−0.394455	−	0.918915i	$-0.629067\pi$
$487$	−17.4097	−0.788910	−0.394455	+	0.918915i	$0.629067\pi$
$488$	−11.2263	−0.508193
$489$	0	0
$490$	−98.5785	−4.45332
$491$	−2.13234	−0.0962312	−0.0481156	−	0.998842i	$-0.515322\pi$
$491$	−2.13234	−0.0962312	−0.0481156	+	0.998842i	$0.515322\pi$
$492$	0	0
$493$	0	0
$494$	−6.30878	−0.283845
$495$	0	0
$496$	−28.1813	−1.26538
$497$	45.6049	2.04566
$498$	0	0
$499$	4.04694	0.181166	0.0905830	−	0.995889i	$-0.471127\pi$
$499$	4.04694	0.181166	0.0905830	+	0.995889i	$0.471127\pi$
$500$	−3.80335	−0.170091
$501$	0	0
$502$	−28.8625	−1.28820
$503$	−20.7119	−0.923496	−0.461748	−	0.887011i	$-0.652778\pi$
$503$	−20.7119	−0.923496	−0.461748	+	0.887011i	$0.652778\pi$
$504$	0	0
$505$	−44.4201	−1.97667
$506$	30.1853	1.34190
$507$	0	0
$508$	−12.1696	−0.539940
$509$	18.1534	0.804634	0.402317	−	0.915500i	$-0.368205\pi$
$509$	18.1534	0.804634	0.402317	+	0.915500i	$0.368205\pi$
$510$	0	0
$511$	−6.57398	−0.290816
$512$	12.9278	0.571333
$513$	0	0
$514$	47.7410	2.10577
$515$	−38.7125	−1.70588
$516$	0	0
$517$	−6.94862	−0.305600
$518$	13.5670	0.596100
$519$	0	0
$520$	−9.47680	−0.415585
$521$	2.15329	0.0943374	0.0471687	−	0.998887i	$-0.484980\pi$
$521$	2.15329	0.0943374	0.0471687	+	0.998887i	$0.484980\pi$
$522$	0	0
$523$	5.83069	0.254958	0.127479	−	0.991841i	$-0.459311\pi$
$523$	5.83069	0.254958	0.127479	+	0.991841i	$0.459311\pi$
$524$	−7.91838	−0.345916
$525$	0	0
$526$	−51.4142	−2.24176
$527$	0	0
$528$	0	0
$529$	28.5449	1.24108
$530$	54.1505	2.35215
$531$	0	0
$532$	10.6272	0.460749
$533$	−19.2927	−0.835660
$534$	0	0
$535$	16.8871	0.730094
$536$	−7.99713	−0.345423
$537$	0	0
$538$	13.5208	0.582925
$539$	40.0046	1.72312
$540$	0	0
$541$	−39.4567	−1.69637	−0.848187	−	0.529697i	$-0.822306\pi$
$541$	−39.4567	−1.69637	−0.848187	+	0.529697i	$0.822306\pi$
$542$	−31.3063	−1.34472
$543$	0	0
$544$	0	0
$545$	18.4315	0.789519
$546$	0	0
$547$	5.13516	0.219564	0.109782	−	0.993956i	$-0.464985\pi$
$547$	5.13516	0.219564	0.109782	+	0.993956i	$0.464985\pi$
$548$	0.604583	0.0258265
$549$	0	0
$550$	25.3201	1.07965
$551$	−11.7128	−0.498982
$552$	0	0
$553$	52.4962	2.23236
$554$	15.3335	0.651459
$555$	0	0
$556$	9.41735	0.399384
$557$	39.0252	1.65355	0.826776	−	0.562531i	$-0.190172\pi$
$557$	39.0252	1.65355	0.826776	+	0.562531i	$0.190172\pi$
$558$	0	0
$559$	−10.5817	−0.447559
$560$	80.7617	3.41281
$561$	0	0
$562$	−10.3149	−0.435108
$563$	28.6045	1.20554	0.602769	−	0.797916i	$-0.294064\pi$
$563$	28.6045	1.20554	0.602769	+	0.797916i	$0.294064\pi$
$564$	0	0
$565$	57.3643	2.41334
$566$	−45.4055	−1.90854
$567$	0	0
$568$	−14.5193	−0.609216
$569$	−26.6643	−1.11783	−0.558913	−	0.829226i	$-0.688781\pi$
$569$	−26.6643	−1.11783	−0.558913	+	0.829226i	$0.688781\pi$
$570$	0	0
$571$	33.9760	1.42185	0.710925	−	0.703268i	$-0.248277\pi$
$571$	33.9760	1.42185	0.710925	+	0.703268i	$0.248277\pi$
$572$	−4.90079	−0.204912
$573$	0	0
$574$	90.5007	3.77743
$575$	43.2369	1.80310
$576$	0	0
$577$	28.5408	1.18817	0.594084	−	0.804403i	$-0.297515\pi$
$577$	28.5408	1.18817	0.594084	+	0.804403i	$0.297515\pi$
$578$	0	0
$579$	0	0
$580$	22.4210	0.930980
$581$	−26.2198	−1.08778
$582$	0	0
$583$	−21.9750	−0.910112
$584$	2.09297	0.0866076
$585$	0	0
$586$	30.7148	1.26882
$587$	44.1768	1.82337	0.911686	−	0.410887i	$-0.134781\pi$
$587$	44.1768	1.82337	0.911686	+	0.410887i	$0.134781\pi$
$588$	0	0
$589$	10.9864	0.452686
$590$	4.26303	0.175506
$591$	0	0
$592$	−7.84699	−0.322509
$593$	−21.5731	−0.885899	−0.442950	−	0.896547i	$-0.646068\pi$
$593$	−21.5731	−0.885899	−0.442950	+	0.896547i	$0.646068\pi$
$594$	0	0
$595$	0	0
$596$	−18.0704	−0.740194
$597$	0	0
$598$	−23.3045	−0.952991
$599$	15.9864	0.653187	0.326593	−	0.945165i	$-0.394099\pi$
$599$	15.9864	0.653187	0.326593	+	0.945165i	$0.394099\pi$
$600$	0	0
$601$	8.17799	0.333587	0.166793	−	0.985992i	$-0.446659\pi$
$601$	8.17799	0.333587	0.166793	+	0.985992i	$0.446659\pi$
$602$	49.6381	2.02310
$603$	0	0
$604$	−11.4439	−0.465644
$605$	17.7136	0.720159
$606$	0	0
$607$	−5.26445	−0.213678	−0.106839	−	0.994276i	$-0.534073\pi$
$607$	−5.26445	−0.213678	−0.106839	+	0.994276i	$0.534073\pi$
$608$	−11.0783	−0.449286
$609$	0	0
$610$	−42.3833	−1.71605
$611$	5.36465	0.217031
$612$	0	0
$613$	−22.1334	−0.893960	−0.446980	−	0.894544i	$-0.647500\pi$
$613$	−22.1334	−0.893960	−0.446980	+	0.894544i	$0.647500\pi$
$614$	−25.4719	−1.02796
$615$	0	0
$616$	−18.0405	−0.726871
$617$	−5.48695	−0.220896	−0.110448	−	0.993882i	$-0.535229\pi$
$617$	−5.48695	−0.220896	−0.110448	+	0.993882i	$0.535229\pi$
$618$	0	0
$619$	−12.5003	−0.502430	−0.251215	−	0.967931i	$-0.580830\pi$
$619$	−12.5003	−0.502430	−0.251215	+	0.967931i	$0.580830\pi$
$620$	−21.0304	−0.844603
$621$	0	0
$622$	−30.6990	−1.23092
$623$	54.2245	2.17246
$624$	0	0
$625$	−18.8435	−0.753739
$626$	2.58835	0.103451
$627$	0	0
$628$	17.0000	0.678374
$629$	0	0
$630$	0	0
$631$	33.2891	1.32522	0.662609	−	0.748965i	$-0.269449\pi$
$631$	33.2891	1.32522	0.662609	+	0.748965i	$0.269449\pi$
$632$	−16.7133	−0.664819
$633$	0	0
$634$	0.200274	0.00795390
$635$	36.0543	1.43077
$636$	0	0
$637$	−30.8854	−1.22372
$638$	−25.3376	−1.00313
$639$	0	0
$640$	−37.2713	−1.47328
$641$	−8.98063	−0.354713	−0.177357	−	0.984147i	$-0.556755\pi$
$641$	−8.98063	−0.354713	−0.177357	+	0.984147i	$0.556755\pi$
$642$	0	0
$643$	−32.3105	−1.27420	−0.637100	−	0.770781i	$-0.719866\pi$
$643$	−32.3105	−1.27420	−0.637100	+	0.770781i	$0.719866\pi$
$644$	39.2567	1.54693
$645$	0	0
$646$	0	0
$647$	−41.0579	−1.61415	−0.807076	−	0.590447i	$-0.798951\pi$
$647$	−41.0579	−1.61415	−0.807076	+	0.590447i	$0.798951\pi$
$648$	0	0
$649$	−1.73000	−0.0679084
$650$	−19.5483	−0.766746
$651$	0	0
$652$	−19.2193	−0.752685
$653$	36.2584	1.41890	0.709451	−	0.704754i	$-0.248943\pi$
$653$	36.2584	1.41890	0.709451	+	0.704754i	$0.248943\pi$
$654$	0	0
$655$	23.4593	0.916633
$656$	−52.3446	−2.04371
$657$	0	0
$658$	−25.1652	−0.981041
$659$	−11.7265	−0.456799	−0.228399	−	0.973568i	$-0.573349\pi$
$659$	−11.7265	−0.456799	−0.228399	+	0.973568i	$0.573349\pi$
$660$	0	0
$661$	−9.32594	−0.362737	−0.181369	−	0.983415i	$-0.558053\pi$
$661$	−9.32594	−0.362737	−0.181369	+	0.983415i	$0.558053\pi$
$662$	−57.8058	−2.24669
$663$	0	0
$664$	8.34762	0.323951
$665$	−31.4847	−1.22092
$666$	0	0
$667$	−43.2668	−1.67530
$668$	−11.0342	−0.426927
$669$	0	0
$670$	−30.1919	−1.16642
$671$	17.1997	0.663989
$672$	0	0
$673$	16.1566	0.622792	0.311396	−	0.950280i	$-0.399203\pi$
$673$	16.1566	0.622792	0.311396	+	0.950280i	$0.399203\pi$
$674$	−2.00219	−0.0771217
$675$	0	0
$676$	−10.7844	−0.414783
$677$	43.8384	1.68485	0.842423	−	0.538817i	$-0.181129\pi$
$677$	43.8384	1.68485	0.842423	+	0.538817i	$0.181129\pi$
$678$	0	0
$679$	5.28817	0.202941
$680$	0	0
$681$	0	0
$682$	23.7662	0.910054
$683$	−13.3587	−0.511156	−0.255578	−	0.966788i	$-0.582266\pi$
$683$	−13.3587	−0.511156	−0.255578	+	0.966788i	$0.582266\pi$
$684$	0	0
$685$	−1.79116	−0.0684369
$686$	84.5441	3.22791
$687$	0	0
$688$	−28.7101	−1.09456
$689$	16.9657	0.646343
$690$	0	0
$691$	30.6486	1.16593	0.582964	−	0.812498i	$-0.301893\pi$
$691$	30.6486	1.16593	0.582964	+	0.812498i	$0.301893\pi$
$692$	−10.2638	−0.390171
$693$	0	0
$694$	4.58079	0.173884
$695$	−27.9003	−1.05832
$696$	0	0
$697$	0	0
$698$	−29.3934	−1.11256
$699$	0	0
$700$	32.9293	1.24461
$701$	5.45232	0.205931	0.102966	−	0.994685i	$-0.467167\pi$
$701$	5.45232	0.205931	0.102966	+	0.994685i	$0.467167\pi$
$702$	0	0
$703$	3.05913	0.115377
$704$	−0.234019	−0.00881994
$705$	0	0
$706$	−34.4121	−1.29512
$707$	65.2844	2.45527
$708$	0	0
$709$	−37.8598	−1.42185	−0.710927	−	0.703266i	$-0.751724\pi$
$709$	−37.8598	−1.42185	−0.710927	+	0.703266i	$0.751724\pi$
$710$	−54.8154	−2.05718
$711$	0	0
$712$	−17.2635	−0.646978
$713$	40.5835	1.51986
$714$	0	0
$715$	14.5193	0.542991
$716$	−1.13624	−0.0424634
$717$	0	0
$718$	29.6394	1.10613
$719$	36.0800	1.34556	0.672779	−	0.739844i	$-0.265101\pi$
$719$	36.0800	1.34556	0.672779	+	0.739844i	$0.265101\pi$
$720$	0	0
$721$	56.8958	2.11891
$722$	−26.8911	−1.00078
$723$	0	0
$724$	24.6239	0.915141
$725$	−36.2931	−1.34789
$726$	0	0
$727$	−29.4347	−1.09167	−0.545836	−	0.837892i	$-0.683788\pi$
$727$	−29.4347	−1.09167	−0.545836	+	0.837892i	$0.683788\pi$
$728$	13.9281	0.516208
$729$	0	0
$730$	7.90167	0.292454
$731$	0	0
$732$	0	0
$733$	−27.4361	−1.01338	−0.506688	−	0.862129i	$-0.669130\pi$
$733$	−27.4361	−1.01338	−0.506688	+	0.862129i	$0.669130\pi$
$734$	5.52467	0.203919
$735$	0	0
$736$	−40.9231	−1.50845
$737$	12.2523	0.451320
$738$	0	0
$739$	−4.12742	−0.151830	−0.0759148	−	0.997114i	$-0.524188\pi$
$739$	−4.12742	−0.151830	−0.0759148	+	0.997114i	$0.524188\pi$
$740$	−5.85586	−0.215266
$741$	0	0
$742$	−79.5850	−2.92166
$743$	−43.6856	−1.60267	−0.801335	−	0.598216i	$-0.795876\pi$
$743$	−43.6856	−1.60267	−0.801335	+	0.598216i	$0.795876\pi$
$744$	0	0
$745$	53.5363	1.96142
$746$	−48.3460	−1.77007
$747$	0	0
$748$	0	0
$749$	−24.8190	−0.906868
$750$	0	0
$751$	23.8033	0.868597	0.434298	−	0.900769i	$-0.356996\pi$
$751$	23.8033	0.868597	0.434298	+	0.900769i	$0.356996\pi$
$752$	14.5552	0.530775
$753$	0	0
$754$	19.5618	0.712399
$755$	33.9041	1.23390
$756$	0	0
$757$	−45.8813	−1.66758	−0.833791	−	0.552080i	$-0.813834\pi$
$757$	−45.8813	−1.66758	−0.833791	+	0.552080i	$0.813834\pi$
$758$	31.3593	1.13902
$759$	0	0
$760$	10.0238	0.363603
$761$	27.1387	0.983779	0.491889	−	0.870658i	$-0.336306\pi$
$761$	27.1387	0.983779	0.491889	+	0.870658i	$0.336306\pi$
$762$	0	0
$763$	−27.0888	−0.980682
$764$	1.65790	0.0599808
$765$	0	0
$766$	2.74422	0.0991528
$767$	1.33564	0.0482271
$768$	0	0
$769$	−8.24216	−0.297220	−0.148610	−	0.988896i	$-0.547480\pi$
$769$	−8.24216	−0.297220	−0.148610	+	0.988896i	$0.547480\pi$
$770$	−68.1090	−2.45448
$771$	0	0
$772$	−5.26649	−0.189545
$773$	31.0979	1.11852	0.559258	−	0.828994i	$-0.311086\pi$
$773$	31.0979	1.11852	0.559258	+	0.828994i	$0.311086\pi$
$774$	0	0
$775$	34.0422	1.22283
$776$	−1.68360	−0.0604378
$777$	0	0
$778$	−24.8342	−0.890349
$779$	20.4064	0.731134
$780$	0	0
$781$	22.2449	0.795983
$782$	0	0
$783$	0	0
$784$	−83.7975	−2.99277
$785$	−50.3650	−1.79760
$786$	0	0
$787$	50.1644	1.78817	0.894083	−	0.447901i	$-0.147828\pi$
$787$	50.1644	1.78817	0.894083	+	0.447901i	$0.147828\pi$
$788$	14.7158	0.524230
$789$	0	0
$790$	−63.0984	−2.24494
$791$	−84.3084	−2.99766
$792$	0	0
$793$	−13.2790	−0.471551
$794$	−41.9632	−1.48922
$795$	0	0
$796$	1.98721	0.0704347
$797$	18.1187	0.641799	0.320900	−	0.947113i	$-0.396015\pi$
$797$	18.1187	0.641799	0.320900	+	0.947113i	$0.396015\pi$
$798$	0	0
$799$	0	0
$800$	−34.3272	−1.21365
$801$	0	0
$802$	−44.4813	−1.57069
$803$	−3.20661	−0.113159
$804$	0	0
$805$	−116.304	−4.09917
$806$	−18.3486	−0.646302
$807$	0	0
$808$	−20.7847	−0.731203
$809$	−13.0065	−0.457286	−0.228643	−	0.973510i	$-0.573429\pi$
$809$	−13.0065	−0.457286	−0.228643	+	0.973510i	$0.573429\pi$
$810$	0	0
$811$	27.6450	0.970746	0.485373	−	0.874307i	$-0.338684\pi$
$811$	27.6450	0.970746	0.485373	+	0.874307i	$0.338684\pi$
$812$	−32.9521	−1.15639
$813$	0	0
$814$	6.61762	0.231948
$815$	56.9399	1.99452
$816$	0	0
$817$	11.1925	0.391577
$818$	34.0055	1.18897
$819$	0	0
$820$	−39.0624	−1.36412
$821$	2.95417	0.103101	0.0515506	−	0.998670i	$-0.483584\pi$
$821$	2.95417	0.103101	0.0515506	+	0.998670i	$0.483584\pi$
$822$	0	0
$823$	−42.3492	−1.47620	−0.738100	−	0.674692i	$-0.764277\pi$
$823$	−42.3492	−1.47620	−0.738100	+	0.674692i	$0.764277\pi$
$824$	−18.1140	−0.631031
$825$	0	0
$826$	−6.26539	−0.218001
$827$	34.9921	1.21679	0.608397	−	0.793633i	$-0.291813\pi$
$827$	34.9921	1.21679	0.608397	+	0.793633i	$0.291813\pi$
$828$	0	0
$829$	−18.1634	−0.630842	−0.315421	−	0.948952i	$-0.602146\pi$
$829$	−18.1634	−0.630842	−0.315421	+	0.948952i	$0.602146\pi$
$830$	31.5152	1.09391
$831$	0	0
$832$	0.180674	0.00626374
$833$	0	0
$834$	0	0
$835$	32.6905	1.13130
$836$	5.18368	0.179281
$837$	0	0
$838$	−1.41477	−0.0488725
$839$	−18.5932	−0.641908	−0.320954	−	0.947095i	$-0.604004\pi$
$839$	−18.5932	−0.641908	−0.320954	+	0.947095i	$0.604004\pi$
$840$	0	0
$841$	7.31820	0.252352
$842$	39.6781	1.36740
$843$	0	0
$844$	−0.994127	−0.0342192
$845$	31.9502	1.09912
$846$	0	0
$847$	−26.0337	−0.894527
$848$	46.0310	1.58071
$849$	0	0
$850$	0	0
$851$	11.3003	0.387371
$852$	0	0
$853$	47.1052	1.61285	0.806426	−	0.591334i	$-0.201399\pi$
$853$	47.1052	1.61285	0.806426	+	0.591334i	$0.201399\pi$
$854$	62.2908	2.13155
$855$	0	0
$856$	7.90167	0.270074
$857$	−20.8598	−0.712559	−0.356279	−	0.934379i	$-0.615955\pi$
$857$	−20.8598	−0.712559	−0.356279	+	0.934379i	$0.615955\pi$
$858$	0	0
$859$	−28.2823	−0.964980	−0.482490	−	0.875902i	$-0.660267\pi$
$859$	−28.2823	−0.964980	−0.482490	+	0.875902i	$0.660267\pi$
$860$	−21.4251	−0.730588
$861$	0	0
$862$	28.3554	0.965787
$863$	−20.4200	−0.695106	−0.347553	−	0.937660i	$-0.612987\pi$
$863$	−20.4200	−0.695106	−0.347553	+	0.937660i	$0.612987\pi$
$864$	0	0
$865$	30.4080	1.03390
$866$	−52.8977	−1.79754
$867$	0	0
$868$	30.9085	1.04910
$869$	25.6062	0.868632
$870$	0	0
$871$	−9.45935	−0.320518
$872$	8.62431	0.292056
$873$	0	0
$874$	24.6497	0.833788
$875$	−16.5605	−0.559849
$876$	0	0
$877$	−35.3536	−1.19381	−0.596903	−	0.802313i	$-0.703602\pi$
$877$	−35.3536	−1.19381	−0.596903	+	0.802313i	$0.703602\pi$
$878$	56.9430	1.92173
$879$	0	0
$880$	39.3934	1.32795
$881$	2.01897	0.0680208	0.0340104	−	0.999421i	$-0.489172\pi$
$881$	2.01897	0.0680208	0.0340104	+	0.999421i	$0.489172\pi$
$882$	0	0
$883$	11.6277	0.391304	0.195652	−	0.980673i	$-0.437318\pi$
$883$	11.6277	0.391304	0.195652	+	0.980673i	$0.437318\pi$
$884$	0	0
$885$	0	0
$886$	1.41477	0.0475302
$887$	−8.78123	−0.294845	−0.147422	−	0.989074i	$-0.547098\pi$
$887$	−8.78123	−0.294845	−0.147422	+	0.989074i	$0.547098\pi$
$888$	0	0
$889$	−52.9891	−1.77720
$890$	−65.1757	−2.18470
$891$	0	0
$892$	−0.322929	−0.0108125
$893$	−5.67432	−0.189884
$894$	0	0
$895$	3.36629	0.112523
$896$	54.7776	1.82999
$897$	0	0
$898$	40.1462	1.33970
$899$	−34.0658	−1.13616
$900$	0	0
$901$	0	0
$902$	44.1439	1.46983
$903$	0	0
$904$	26.8414	0.892732
$905$	−72.9520	−2.42501
$906$	0	0
$907$	14.6108	0.485144	0.242572	−	0.970133i	$-0.422009\pi$
$907$	14.6108	0.485144	0.242572	+	0.970133i	$0.422009\pi$
$908$	−9.42044	−0.312628
$909$	0	0
$910$	52.5833	1.74312
$911$	−41.0048	−1.35855	−0.679276	−	0.733883i	$-0.737706\pi$
$911$	−41.0048	−1.35855	−0.679276	+	0.733883i	$0.737706\pi$
$912$	0	0
$913$	−12.7893	−0.423264
$914$	56.7829	1.87821
$915$	0	0
$916$	−4.34730	−0.143639
$917$	−34.4782	−1.13857
$918$	0	0
$919$	−54.7401	−1.80571	−0.902855	−	0.429946i	$-0.858533\pi$
$919$	−54.7401	−1.80571	−0.902855	+	0.429946i	$0.858533\pi$
$920$	37.0278	1.22077
$921$	0	0
$922$	−31.8070	−1.04751
$923$	−17.1741	−0.565291
$924$	0	0
$925$	9.47895	0.311666
$926$	14.1548	0.465156
$927$	0	0
$928$	34.3509	1.12762
$929$	22.1126	0.725490	0.362745	−	0.931889i	$-0.381840\pi$
$929$	22.1126	0.725490	0.362745	+	0.931889i	$0.381840\pi$
$930$	0	0
$931$	32.6682	1.07066
$932$	14.3212	0.469107
$933$	0	0
$934$	−16.3105	−0.533695
$935$	0	0
$936$	0	0
$937$	−12.1584	−0.397197	−0.198599	−	0.980081i	$-0.563639\pi$
$937$	−12.1584	−0.397197	−0.198599	+	0.980081i	$0.563639\pi$
$938$	44.3731	1.44883
$939$	0	0
$940$	10.8619	0.354277
$941$	51.8397	1.68992	0.844962	−	0.534826i	$-0.179623\pi$
$941$	51.8397	1.68992	0.844962	+	0.534826i	$0.179623\pi$
$942$	0	0
$943$	75.3806	2.45473
$944$	3.62382	0.117945
$945$	0	0
$946$	24.2121	0.787204
$947$	7.99238	0.259717	0.129859	−	0.991533i	$-0.458548\pi$
$947$	7.99238	0.259717	0.129859	+	0.991533i	$0.458548\pi$
$948$	0	0
$949$	2.47565	0.0803631
$950$	20.6767	0.670839
$951$	0	0
$952$	0	0
$953$	24.6936	0.799906	0.399953	−	0.916536i	$-0.369027\pi$
$953$	24.6936	0.799906	0.399953	+	0.916536i	$0.369027\pi$
$954$	0	0
$955$	−4.91178	−0.158941
$956$	9.94210	0.321550
$957$	0	0
$958$	67.5939	2.18386
$959$	2.63248	0.0850071
$960$	0	0
$961$	0.953058	0.0307438
$962$	−5.10911	−0.164724
$963$	0	0
$964$	−16.5150	−0.531911
$965$	15.6027	0.502270
$966$	0	0
$967$	44.4124	1.42821	0.714103	−	0.700040i	$-0.246835\pi$
$967$	44.4124	1.42821	0.714103	+	0.700040i	$0.246835\pi$
$968$	8.28837	0.266398
$969$	0	0
$970$	−6.35618	−0.204085
$971$	−49.0382	−1.57371	−0.786856	−	0.617137i	$-0.788293\pi$
$971$	−49.0382	−1.57371	−0.786856	+	0.617137i	$0.788293\pi$
$972$	0	0
$973$	41.0051	1.31456
$974$	−30.7547	−0.985445
$975$	0	0
$976$	−36.0283	−1.15324
$977$	−43.0464	−1.37718	−0.688588	−	0.725152i	$-0.741769\pi$
$977$	−43.0464	−1.37718	−0.688588	+	0.725152i	$0.741769\pi$
$978$	0	0
$979$	26.4492	0.845322
$980$	−62.5343	−1.99759
$981$	0	0
$982$	−3.76684	−0.120205
$983$	−55.4168	−1.76752	−0.883761	−	0.467938i	$-0.844997\pi$
$983$	−55.4168	−1.76752	−0.883761	+	0.467938i	$0.844997\pi$
$984$	0	0
$985$	−43.5978	−1.38914
$986$	0	0
$987$	0	0
$988$	−4.00204	−0.127322
$989$	41.3450	1.31469
$990$	0	0
$991$	38.2222	1.21417	0.607085	−	0.794637i	$-0.292339\pi$
$991$	38.2222	1.21417	0.607085	+	0.794637i	$0.292339\pi$
$992$	−32.2205	−1.02300
$993$	0	0
$994$	80.5622	2.55528
$995$	−5.88739	−0.186643
$996$	0	0
$997$	18.0169	0.570601	0.285301	−	0.958438i	$-0.407907\pi$
$997$	18.0169	0.570601	0.285301	+	0.958438i	$0.407907\pi$
$998$	7.14903	0.226299
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.bk.1.4	yes	6
3.2	odd	2	inner	2601.2.a.bk.1.3	yes	6
17.16	even	2		2601.2.a.bj.1.4	yes	6
51.50	odd	2		2601.2.a.bj.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2601.2.a.bj.1.3	✓	6	51.50	odd	2
2601.2.a.bj.1.4	yes	6	17.16	even	2
2601.2.a.bk.1.3	yes	6	3.2	odd	2	inner
2601.2.a.bk.1.4	yes	6	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)

\(f(q)\)	\(=\)	\(q+1.76653 q^{2} +1.12061 q^{4} -3.31998 q^{5} +4.87939 q^{7} -1.55346 q^{8} +O(q^{10})\)	\(q+1.76653 q^{2} +1.12061 q^{4} -3.31998 q^{5} +4.87939 q^{7} -1.55346 q^{8} -5.86484 q^{10} +2.38003 q^{11} -1.83750 q^{13} +8.61956 q^{14} -4.98545 q^{16} +1.94356 q^{19} -3.72042 q^{20} +4.20439 q^{22} +7.17948 q^{23} +6.02229 q^{25} -3.24599 q^{26} +5.46791 q^{28} -6.02646 q^{29} +5.65270 q^{31} -5.70002 q^{32} -16.1995 q^{35} +1.57398 q^{37} +3.43335 q^{38} +5.15745 q^{40} +10.4995 q^{41} +5.75877 q^{43} +2.66710 q^{44} +12.6827 q^{46} -2.91954 q^{47} +16.8084 q^{49} +10.6385 q^{50} -2.05913 q^{52} -9.23307 q^{53} -7.90167 q^{55} -7.57992 q^{56} -10.6459 q^{58} -0.726880 q^{59} +7.22668 q^{61} +9.98565 q^{62} -0.0983261 q^{64} +6.10046 q^{65} +5.14796 q^{67} -28.6168 q^{70} +9.34644 q^{71} -1.34730 q^{73} +2.78047 q^{74} +2.17799 q^{76} +11.6131 q^{77} +10.7588 q^{79} +16.5516 q^{80} +18.5476 q^{82} -5.37358 q^{83} +10.1730 q^{86} -3.69728 q^{88} +11.1130 q^{89} -8.96585 q^{91} +8.04543 q^{92} -5.15745 q^{94} -6.45260 q^{95} +1.08378 q^{97} +29.6925 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 18 q^{4} + 18 q^{7}+O(q^{10}) \) 6 * q + 18 * q^4 + 18 * q^7	\( 6 q + 18 q^{4} + 18 q^{7} + 12 q^{10} - 6 q^{13} + 6 q^{16} - 18 q^{19} + 24 q^{22} + 24 q^{25} + 42 q^{28} + 36 q^{31} - 6 q^{37} + 66 q^{40} + 12 q^{43} + 18 q^{46} + 24 q^{49} - 24 q^{52} - 24 q^{55} + 18 q^{58} + 30 q^{61} - 24 q^{64} - 18 q^{70} - 6 q^{73} - 78 q^{76} + 42 q^{79} + 6 q^{82} - 12 q^{91} - 66 q^{94} - 6 q^{97}+O(q^{100}) \) 6 * q + 18 * q^4 + 18 * q^7 + 12 * q^10 - 6 * q^13 + 6 * q^16 - 18 * q^19 + 24 * q^22 + 24 * q^25 + 42 * q^28 + 36 * q^31 - 6 * q^37 + 66 * q^40 + 12 * q^43 + 18 * q^46 + 24 * q^49 - 24 * q^52 - 24 * q^55 + 18 * q^58 + 30 * q^61 - 24 * q^64 - 18 * q^70 - 6 * q^73 - 78 * q^76 + 42 * q^79 + 6 * q^82 - 12 * q^91 - 66 * q^94 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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