LMFDB - Embedded newform 8001.2.a.o.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} + \cdots - 8 $ x^13 - 4x^12 - 10x^11 + 53x^10 + 19x^9 - 242x^8 + 61x^7 + 467x^6 - 211x^5 - 372x^4 + 146x^3 + 116x^2 - 12x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-1.58856$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.58856 q^{2} +0.523519 q^{4} -3.57671 q^{5} +1.00000 q^{7} -2.34548 q^{8} +O(q^{10})$	\(q+1.58856 q^{2} +0.523519 q^{4} -3.57671 q^{5} +1.00000 q^{7} -2.34548 q^{8} -5.68181 q^{10} +2.48473 q^{11} +0.477380 q^{13} +1.58856 q^{14} -4.77297 q^{16} +2.92733 q^{17} -1.96364 q^{19} -1.87248 q^{20} +3.94714 q^{22} +1.07941 q^{23} +7.79283 q^{25} +0.758347 q^{26} +0.523519 q^{28} -5.99492 q^{29} +6.77344 q^{31} -2.89119 q^{32} +4.65024 q^{34} -3.57671 q^{35} +4.95829 q^{37} -3.11935 q^{38} +8.38908 q^{40} +2.60754 q^{41} -2.46629 q^{43} +1.30080 q^{44} +1.71470 q^{46} -12.0814 q^{47} +1.00000 q^{49} +12.3794 q^{50} +0.249918 q^{52} -0.254045 q^{53} -8.88714 q^{55} -2.34548 q^{56} -9.52329 q^{58} +9.98644 q^{59} +8.89920 q^{61} +10.7600 q^{62} +4.95311 q^{64} -1.70745 q^{65} -11.0484 q^{67} +1.53251 q^{68} -5.68181 q^{70} -4.92950 q^{71} +1.24866 q^{73} +7.87653 q^{74} -1.02800 q^{76} +2.48473 q^{77} -7.06010 q^{79} +17.0715 q^{80} +4.14223 q^{82} -6.48027 q^{83} -10.4702 q^{85} -3.91784 q^{86} -5.82787 q^{88} -8.47932 q^{89} +0.477380 q^{91} +0.565091 q^{92} -19.1920 q^{94} +7.02335 q^{95} -8.21793 q^{97} +1.58856 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * q^97 - 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 $

$2$	1.58856	1.12328	0.561640	−	0.827381i	$-0.310170\pi$
$2$	1.58856	1.12328	0.561640	+	0.827381i	$0.310170\pi$
$3$	0	0
$3$	0	0
$4$	0.523519	0.261760
$5$	−3.57671	−1.59955	−0.799776	−	0.600299i	$-0.795048\pi$
$5$	−3.57671	−1.59955	−0.799776	+	0.600299i	$0.795048\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.34548	−0.829251
$9$	0	0
$10$	−5.68181	−1.79675
$11$	2.48473	0.749173	0.374587	−	0.927192i	$-0.377785\pi$
$11$	2.48473	0.749173	0.374587	+	0.927192i	$0.377785\pi$
$12$	0	0
$13$	0.477380	0.132401	0.0662007	−	0.997806i	$-0.478912\pi$
$13$	0.477380	0.132401	0.0662007	+	0.997806i	$0.478912\pi$
$14$	1.58856	0.424560
$15$	0	0
$16$	−4.77297	−1.19324
$17$	2.92733	0.709982	0.354991	−	0.934870i	$-0.384484\pi$
$17$	2.92733	0.709982	0.354991	+	0.934870i	$0.384484\pi$
$18$	0	0
$19$	−1.96364	−0.450489	−0.225245	−	0.974302i	$-0.572318\pi$
$19$	−1.96364	−0.450489	−0.225245	+	0.974302i	$0.572318\pi$
$20$	−1.87248	−0.418698
$21$	0	0
$22$	3.94714	0.841532
$23$	1.07941	0.225072	0.112536	−	0.993648i	$-0.464103\pi$
$23$	1.07941	0.225072	0.112536	+	0.993648i	$0.464103\pi$
$24$	0	0
$25$	7.79283	1.55857
$26$	0.758347	0.148724
$27$	0	0
$28$	0.523519	0.0989359
$29$	−5.99492	−1.11323	−0.556615	−	0.830771i	$-0.687900\pi$
$29$	−5.99492	−1.11323	−0.556615	+	0.830771i	$0.687900\pi$
$30$	0	0
$31$	6.77344	1.21655	0.608273	−	0.793728i	$-0.291862\pi$
$31$	6.77344	1.21655	0.608273	+	0.793728i	$0.291862\pi$
$32$	−2.89119	−0.511094
$33$	0	0
$34$	4.65024	0.797509
$35$	−3.57671	−0.604574
$36$	0	0
$37$	4.95829	0.815137	0.407569	−	0.913175i	$-0.366377\pi$
$37$	4.95829	0.815137	0.407569	+	0.913175i	$0.366377\pi$
$38$	−3.11935	−0.506026
$39$	0	0
$40$	8.38908	1.32643
$41$	2.60754	0.407229	0.203614	−	0.979051i	$-0.434731\pi$
$41$	2.60754	0.407229	0.203614	+	0.979051i	$0.434731\pi$
$42$	0	0
$43$	−2.46629	−0.376106	−0.188053	−	0.982159i	$-0.560218\pi$
$43$	−2.46629	−0.376106	−0.188053	+	0.982159i	$0.560218\pi$
$44$	1.30080	0.196103
$45$	0	0
$46$	1.71470	0.252819
$47$	−12.0814	−1.76225	−0.881126	−	0.472882i	$-0.843214\pi$
$47$	−12.0814	−1.76225	−0.881126	+	0.472882i	$0.843214\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	12.3794	1.75071
$51$	0	0
$52$	0.249918	0.0346574
$53$	−0.254045	−0.0348958	−0.0174479	−	0.999848i	$-0.505554\pi$
$53$	−0.254045	−0.0348958	−0.0174479	+	0.999848i	$0.505554\pi$
$54$	0	0
$55$	−8.88714	−1.19834
$56$	−2.34548	−0.313427
$57$	0	0
$58$	−9.52329	−1.25047
$59$	9.98644	1.30012	0.650062	−	0.759881i	$-0.274743\pi$
$59$	9.98644	1.30012	0.650062	+	0.759881i	$0.274743\pi$
$60$	0	0
$61$	8.89920	1.13943	0.569713	−	0.821844i	$-0.307054\pi$
$61$	8.89920	1.13943	0.569713	+	0.821844i	$0.307054\pi$
$62$	10.7600	1.36652
$63$	0	0
$64$	4.95311	0.619139
$65$	−1.70745	−0.211783
$66$	0	0
$67$	−11.0484	−1.34977	−0.674887	−	0.737921i	$-0.735808\pi$
$67$	−11.0484	−1.34977	−0.674887	+	0.737921i	$0.735808\pi$
$68$	1.53251	0.185845
$69$	0	0
$70$	−5.68181	−0.679106
$71$	−4.92950	−0.585025	−0.292512	−	0.956262i	$-0.594491\pi$
$71$	−4.92950	−0.585025	−0.292512	+	0.956262i	$0.594491\pi$
$72$	0	0
$73$	1.24866	0.146145	0.0730724	−	0.997327i	$-0.476720\pi$
$73$	1.24866	0.146145	0.0730724	+	0.997327i	$0.476720\pi$
$74$	7.87653	0.915628
$75$	0	0
$76$	−1.02800	−0.117920
$77$	2.48473	0.283161
$78$	0	0
$79$	−7.06010	−0.794323	−0.397162	−	0.917749i	$-0.630005\pi$
$79$	−7.06010	−0.794323	−0.397162	+	0.917749i	$0.630005\pi$
$80$	17.0715	1.90865
$81$	0	0
$82$	4.14223	0.457432
$83$	−6.48027	−0.711302	−0.355651	−	0.934619i	$-0.615741\pi$
$83$	−6.48027	−0.711302	−0.355651	+	0.934619i	$0.615741\pi$
$84$	0	0
$85$	−10.4702	−1.13565
$86$	−3.91784	−0.422472
$87$	0	0
$88$	−5.82787	−0.621253
$89$	−8.47932	−0.898806	−0.449403	−	0.893329i	$-0.648363\pi$
$89$	−8.47932	−0.898806	−0.449403	+	0.893329i	$0.648363\pi$
$90$	0	0
$91$	0.477380	0.0500431
$92$	0.565091	0.0589148
$93$	0	0
$94$	−19.1920	−1.97950
$95$	7.02335	0.720581
$96$	0	0
$97$	−8.21793	−0.834405	−0.417202	−	0.908814i	$-0.636989\pi$
$97$	−8.21793	−0.834405	−0.417202	+	0.908814i	$0.636989\pi$
$98$	1.58856	0.160469
$99$	0	0
$100$	4.07970	0.407970
$101$	−1.20095	−0.119499	−0.0597493	−	0.998213i	$-0.519030\pi$
$101$	−1.20095	−0.119499	−0.0597493	+	0.998213i	$0.519030\pi$
$102$	0	0
$103$	3.15831	0.311197	0.155599	−	0.987820i	$-0.450269\pi$
$103$	3.15831	0.311197	0.155599	+	0.987820i	$0.450269\pi$
$104$	−1.11968	−0.109794
$105$	0	0
$106$	−0.403566	−0.0391978
$107$	−10.4900	−1.01411	−0.507053	−	0.861915i	$-0.669265\pi$
$107$	−10.4900	−1.01411	−0.507053	+	0.861915i	$0.669265\pi$
$108$	0	0
$109$	13.4906	1.29217	0.646084	−	0.763267i	$-0.276406\pi$
$109$	13.4906	1.29217	0.646084	+	0.763267i	$0.276406\pi$
$110$	−14.1177	−1.34607
$111$	0	0
$112$	−4.77297	−0.451003
$113$	−10.9519	−1.03027	−0.515133	−	0.857111i	$-0.672257\pi$
$113$	−10.9519	−1.03027	−0.515133	+	0.857111i	$0.672257\pi$
$114$	0	0
$115$	−3.86072	−0.360014
$116$	−3.13846	−0.291399
$117$	0	0
$118$	15.8640	1.46040
$119$	2.92733	0.268348
$120$	0	0
$121$	−4.82613	−0.438739
$122$	14.1369	1.27990
$123$	0	0
$124$	3.54603	0.318443
$125$	−9.98912	−0.893454
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	13.6507	1.20656
$129$	0	0
$130$	−2.71238	−0.237892
$131$	−6.78368	−0.592693	−0.296347	−	0.955080i	$-0.595768\pi$
$131$	−6.78368	−0.592693	−0.296347	+	0.955080i	$0.595768\pi$
$132$	0	0
$133$	−1.96364	−0.170269
$134$	−17.5510	−1.51617
$135$	0	0
$136$	−6.86598	−0.588753
$137$	−1.14240	−0.0976021	−0.0488010	−	0.998809i	$-0.515540\pi$
$137$	−1.14240	−0.0976021	−0.0488010	+	0.998809i	$0.515540\pi$
$138$	0	0
$139$	3.86056	0.327449	0.163724	−	0.986506i	$-0.447649\pi$
$139$	3.86056	0.327449	0.163724	+	0.986506i	$0.447649\pi$
$140$	−1.87248	−0.158253
$141$	0	0
$142$	−7.83081	−0.657147
$143$	1.18616	0.0991917
$144$	0	0
$145$	21.4421	1.78067
$146$	1.98357	0.164162
$147$	0	0
$148$	2.59576	0.213370
$149$	−22.7264	−1.86182	−0.930911	−	0.365245i	$-0.880985\pi$
$149$	−22.7264	−1.86182	−0.930911	+	0.365245i	$0.880985\pi$
$150$	0	0
$151$	−13.4219	−1.09226	−0.546130	−	0.837700i	$-0.683900\pi$
$151$	−13.4219	−1.09226	−0.546130	+	0.837700i	$0.683900\pi$
$152$	4.60566	0.373569
$153$	0	0
$154$	3.94714	0.318069
$155$	−24.2266	−1.94593
$156$	0	0
$157$	−12.2685	−0.979131	−0.489565	−	0.871967i	$-0.662845\pi$
$157$	−12.2685	−0.979131	−0.489565	+	0.871967i	$0.662845\pi$
$158$	−11.2154	−0.892248
$159$	0	0
$160$	10.3409	0.817521
$161$	1.07941	0.0850692
$162$	0	0
$163$	0.962043	0.0753530	0.0376765	−	0.999290i	$-0.488004\pi$
$163$	0.962043	0.0753530	0.0376765	+	0.999290i	$0.488004\pi$
$164$	1.36510	0.106596
$165$	0	0
$166$	−10.2943	−0.798992
$167$	−1.61913	−0.125292	−0.0626461	−	0.998036i	$-0.519954\pi$
$167$	−1.61913	−0.125292	−0.0626461	+	0.998036i	$0.519954\pi$
$168$	0	0
$169$	−12.7721	−0.982470
$170$	−16.6325	−1.27566
$171$	0	0
$172$	−1.29115	−0.0984493
$173$	16.2581	1.23608	0.618039	−	0.786148i	$-0.287927\pi$
$173$	16.2581	1.23608	0.618039	+	0.786148i	$0.287927\pi$
$174$	0	0
$175$	7.79283	0.589082
$176$	−11.8595	−0.893945
$177$	0	0
$178$	−13.4699	−1.00961
$179$	−15.8589	−1.18535	−0.592675	−	0.805442i	$-0.701928\pi$
$179$	−15.8589	−1.18535	−0.592675	+	0.805442i	$0.701928\pi$
$180$	0	0
$181$	10.0700	0.748501	0.374250	−	0.927328i	$-0.377900\pi$
$181$	10.0700	0.748501	0.374250	+	0.927328i	$0.377900\pi$
$182$	0.758347	0.0562124
$183$	0	0
$184$	−2.53173	−0.186641
$185$	−17.7343	−1.30385
$186$	0	0
$187$	7.27362	0.531899
$188$	−6.32484	−0.461286
$189$	0	0
$190$	11.1570	0.809414
$191$	3.66146	0.264934	0.132467	−	0.991187i	$-0.457710\pi$
$191$	3.66146	0.264934	0.132467	+	0.991187i	$0.457710\pi$
$192$	0	0
$193$	24.6937	1.77749	0.888747	−	0.458399i	$-0.151577\pi$
$193$	24.6937	1.77749	0.888747	+	0.458399i	$0.151577\pi$
$194$	−13.0547	−0.937271
$195$	0	0
$196$	0.523519	0.0373942
$197$	−17.7356	−1.26361	−0.631804	−	0.775129i	$-0.717685\pi$
$197$	−17.7356	−1.26361	−0.631804	+	0.775129i	$0.717685\pi$
$198$	0	0
$199$	−18.3092	−1.29790	−0.648952	−	0.760829i	$-0.724792\pi$
$199$	−18.3092	−1.29790	−0.648952	+	0.760829i	$0.724792\pi$
$200$	−18.2779	−1.29244
$201$	0	0
$202$	−1.90777	−0.134230
$203$	−5.99492	−0.420761
$204$	0	0
$205$	−9.32639	−0.651383
$206$	5.01716	0.349562
$207$	0	0
$208$	−2.27852	−0.157987
$209$	−4.87910	−0.337494
$210$	0	0
$211$	−7.38421	−0.508350	−0.254175	−	0.967158i	$-0.581804\pi$
$211$	−7.38421	−0.508350	−0.254175	+	0.967158i	$0.581804\pi$
$212$	−0.132998	−0.00913431
$213$	0	0
$214$	−16.6640	−1.13913
$215$	8.82119	0.601600
$216$	0	0
$217$	6.77344	0.459811
$218$	21.4306	1.45147
$219$	0	0
$220$	−4.65259	−0.313678
$221$	1.39745	0.0940026
$222$	0	0
$223$	−13.4908	−0.903412	−0.451706	−	0.892167i	$-0.649184\pi$
$223$	−13.4908	−0.903412	−0.451706	+	0.892167i	$0.649184\pi$
$224$	−2.89119	−0.193175
$225$	0	0
$226$	−17.3977	−1.15728
$227$	−9.66789	−0.641681	−0.320840	−	0.947133i	$-0.603965\pi$
$227$	−9.66789	−0.641681	−0.320840	+	0.947133i	$0.603965\pi$
$228$	0	0
$229$	−6.40726	−0.423404	−0.211702	−	0.977334i	$-0.567901\pi$
$229$	−6.40726	−0.423404	−0.211702	+	0.977334i	$0.567901\pi$
$230$	−6.13299	−0.404397
$231$	0	0
$232$	14.0609	0.923147
$233$	−7.44888	−0.487992	−0.243996	−	0.969776i	$-0.578458\pi$
$233$	−7.44888	−0.487992	−0.243996	+	0.969776i	$0.578458\pi$
$234$	0	0
$235$	43.2116	2.81881
$236$	5.22810	0.340320
$237$	0	0
$238$	4.65024	0.301430
$239$	−22.1746	−1.43436	−0.717179	−	0.696890i	$-0.754567\pi$
$239$	−22.1746	−1.43436	−0.717179	+	0.696890i	$0.754567\pi$
$240$	0	0
$241$	6.59447	0.424787	0.212394	−	0.977184i	$-0.431874\pi$
$241$	6.59447	0.424787	0.212394	+	0.977184i	$0.431874\pi$
$242$	−7.66659	−0.492827
$243$	0	0
$244$	4.65890	0.298256
$245$	−3.57671	−0.228507
$246$	0	0
$247$	−0.937401	−0.0596454
$248$	−15.8870	−1.00882
$249$	0	0
$250$	−15.8683	−1.00360
$251$	2.14270	0.135246	0.0676229	−	0.997711i	$-0.478459\pi$
$251$	2.14270	0.135246	0.0676229	+	0.997711i	$0.478459\pi$
$252$	0	0
$253$	2.68203	0.168618
$254$	−1.58856	−0.0996751
$255$	0	0
$256$	11.7787	0.736168
$257$	−28.5552	−1.78122	−0.890612	−	0.454764i	$-0.849724\pi$
$257$	−28.5552	−1.78122	−0.890612	+	0.454764i	$0.849724\pi$
$258$	0	0
$259$	4.95829	0.308093
$260$	−0.893883	−0.0554363
$261$	0	0
$262$	−10.7763	−0.665761
$263$	11.2798	0.695542	0.347771	−	0.937579i	$-0.386939\pi$
$263$	11.2798	0.695542	0.347771	+	0.937579i	$0.386939\pi$
$264$	0	0
$265$	0.908645	0.0558176
$266$	−3.11935	−0.191260
$267$	0	0
$268$	−5.78404	−0.353316
$269$	8.17992	0.498739	0.249369	−	0.968408i	$-0.419777\pi$
$269$	8.17992	0.498739	0.249369	+	0.968408i	$0.419777\pi$
$270$	0	0
$271$	−0.247885	−0.0150579	−0.00752896	−	0.999972i	$-0.502397\pi$
$271$	−0.247885	−0.0150579	−0.00752896	+	0.999972i	$0.502397\pi$
$272$	−13.9720	−0.847180
$273$	0	0
$274$	−1.81477	−0.109635
$275$	19.3630	1.16764
$276$	0	0
$277$	21.5369	1.29403	0.647013	−	0.762479i	$-0.276018\pi$
$277$	21.5369	1.29403	0.647013	+	0.762479i	$0.276018\pi$
$278$	6.13273	0.367817
$279$	0	0
$280$	8.38908	0.501343
$281$	12.4596	0.743278	0.371639	−	0.928377i	$-0.378796\pi$
$281$	12.4596	0.743278	0.371639	+	0.928377i	$0.378796\pi$
$282$	0	0
$283$	8.60593	0.511570	0.255785	−	0.966734i	$-0.417666\pi$
$283$	8.60593	0.511570	0.255785	+	0.966734i	$0.417666\pi$
$284$	−2.58069	−0.153136
$285$	0	0
$286$	1.88429	0.111420
$287$	2.60754	0.153918
$288$	0	0
$289$	−8.43074	−0.495926
$290$	34.0620	2.00019
$291$	0	0
$292$	0.653699	0.0382548
$293$	16.8805	0.986167	0.493084	−	0.869982i	$-0.335870\pi$
$293$	16.8805	0.986167	0.493084	+	0.869982i	$0.335870\pi$
$294$	0	0
$295$	−35.7186	−2.07961
$296$	−11.6295	−0.675953
$297$	0	0
$298$	−36.1023	−2.09135
$299$	0.515288	0.0297999
$300$	0	0
$301$	−2.46629	−0.142155
$302$	−21.3215	−1.22692
$303$	0	0
$304$	9.37237	0.537542
$305$	−31.8298	−1.82257
$306$	0	0
$307$	16.3200	0.931434	0.465717	−	0.884934i	$-0.345796\pi$
$307$	16.3200	0.931434	0.465717	+	0.884934i	$0.345796\pi$
$308$	1.30080	0.0741201
$309$	0	0
$310$	−38.4854	−2.18582
$311$	−29.0604	−1.64786	−0.823931	−	0.566690i	$-0.808224\pi$
$311$	−29.0604	−1.64786	−0.823931	+	0.566690i	$0.808224\pi$
$312$	0	0
$313$	25.9671	1.46775	0.733875	−	0.679285i	$-0.237710\pi$
$313$	25.9671	1.46775	0.733875	+	0.679285i	$0.237710\pi$
$314$	−19.4892	−1.09984
$315$	0	0
$316$	−3.69610	−0.207922
$317$	−17.5779	−0.987272	−0.493636	−	0.869669i	$-0.664333\pi$
$317$	−17.5779	−0.987272	−0.493636	+	0.869669i	$0.664333\pi$
$318$	0	0
$319$	−14.8957	−0.834002
$320$	−17.7158	−0.990345
$321$	0	0
$322$	1.71470	0.0955566
$323$	−5.74821	−0.319839
$324$	0	0
$325$	3.72014	0.206356
$326$	1.52826	0.0846426
$327$	0	0
$328$	−6.11592	−0.337695
$329$	−12.0814	−0.666069
$330$	0	0
$331$	−15.3981	−0.846356	−0.423178	−	0.906047i	$-0.639085\pi$
$331$	−15.3981	−0.846356	−0.423178	+	0.906047i	$0.639085\pi$
$332$	−3.39255	−0.186190
$333$	0	0
$334$	−2.57209	−0.140738
$335$	39.5168	2.15903
$336$	0	0
$337$	−8.13632	−0.443214	−0.221607	−	0.975136i	$-0.571130\pi$
$337$	−8.13632	−0.443214	−0.221607	+	0.975136i	$0.571130\pi$
$338$	−20.2892	−1.10359
$339$	0	0
$340$	−5.48135	−0.297268
$341$	16.8302	0.911404
$342$	0	0
$343$	1.00000	0.0539949
$344$	5.78462	0.311886
$345$	0	0
$346$	25.8269	1.38846
$347$	21.0011	1.12740	0.563699	−	0.825980i	$-0.309378\pi$
$347$	21.0011	1.12740	0.563699	+	0.825980i	$0.309378\pi$
$348$	0	0
$349$	5.77386	0.309068	0.154534	−	0.987988i	$-0.450612\pi$
$349$	5.77386	0.309068	0.154534	+	0.987988i	$0.450612\pi$
$350$	12.3794	0.661705
$351$	0	0
$352$	−7.18381	−0.382898
$353$	13.6757	0.727886	0.363943	−	0.931421i	$-0.381430\pi$
$353$	13.6757	0.727886	0.363943	+	0.931421i	$0.381430\pi$
$354$	0	0
$355$	17.6314	0.935777
$356$	−4.43909	−0.235271
$357$	0	0
$358$	−25.1928	−1.33148
$359$	−12.2071	−0.644265	−0.322132	−	0.946695i	$-0.604400\pi$
$359$	−12.2071	−0.644265	−0.322132	+	0.946695i	$0.604400\pi$
$360$	0	0
$361$	−15.1441	−0.797060
$362$	15.9969	0.840776
$363$	0	0
$364$	0.249918	0.0130993
$365$	−4.46610	−0.233766
$366$	0	0
$367$	4.41002	0.230201	0.115101	−	0.993354i	$-0.463281\pi$
$367$	4.41002	0.230201	0.115101	+	0.993354i	$0.463281\pi$
$368$	−5.15198	−0.268565
$369$	0	0
$370$	−28.1720	−1.46459
$371$	−0.254045	−0.0131894
$372$	0	0
$373$	26.9854	1.39725	0.698625	−	0.715488i	$-0.253796\pi$
$373$	26.9854	1.39725	0.698625	+	0.715488i	$0.253796\pi$
$374$	11.5546	0.597472
$375$	0	0
$376$	28.3366	1.46135
$377$	−2.86186	−0.147393
$378$	0	0
$379$	−22.1339	−1.13694	−0.568470	−	0.822704i	$-0.692465\pi$
$379$	−22.1339	−1.13694	−0.568470	+	0.822704i	$0.692465\pi$
$380$	3.67686	0.188619
$381$	0	0
$382$	5.81645	0.297595
$383$	−23.5449	−1.20309	−0.601545	−	0.798839i	$-0.705448\pi$
$383$	−23.5449	−1.20309	−0.601545	+	0.798839i	$0.705448\pi$
$384$	0	0
$385$	−8.88714	−0.452931
$386$	39.2274	1.99662
$387$	0	0
$388$	−4.30225	−0.218414
$389$	−23.9241	−1.21300	−0.606501	−	0.795083i	$-0.707427\pi$
$389$	−23.9241	−1.21300	−0.606501	+	0.795083i	$0.707427\pi$
$390$	0	0
$391$	3.15978	0.159797
$392$	−2.34548	−0.118464
$393$	0	0
$394$	−28.1740	−1.41939
$395$	25.2519	1.27056
$396$	0	0
$397$	−6.46262	−0.324349	−0.162175	−	0.986762i	$-0.551851\pi$
$397$	−6.46262	−0.324349	−0.162175	+	0.986762i	$0.551851\pi$
$398$	−29.0852	−1.45791
$399$	0	0
$400$	−37.1949	−1.85974
$401$	−13.4295	−0.670636	−0.335318	−	0.942105i	$-0.608844\pi$
$401$	−13.4295	−0.670636	−0.335318	+	0.942105i	$0.608844\pi$
$402$	0	0
$403$	3.23351	0.161073
$404$	−0.628718	−0.0312799
$405$	0	0
$406$	−9.52329	−0.472633
$407$	12.3200	0.610679
$408$	0	0
$409$	−30.6625	−1.51616	−0.758081	−	0.652161i	$-0.773863\pi$
$409$	−30.6625	−1.51616	−0.758081	+	0.652161i	$0.773863\pi$
$410$	−14.8155	−0.731687
$411$	0	0
$412$	1.65344	0.0814589
$413$	9.98644	0.491401
$414$	0	0
$415$	23.1780	1.13776
$416$	−1.38020	−0.0676696
$417$	0	0
$418$	−7.75074	−0.379101
$419$	26.5173	1.29545	0.647727	−	0.761872i	$-0.275720\pi$
$419$	26.5173	1.29545	0.647727	+	0.761872i	$0.275720\pi$
$420$	0	0
$421$	−4.12021	−0.200807	−0.100403	−	0.994947i	$-0.532013\pi$
$421$	−4.12021	−0.200807	−0.100403	+	0.994947i	$0.532013\pi$
$422$	−11.7303	−0.571020
$423$	0	0
$424$	0.595857	0.0289374
$425$	22.8122	1.10655
$426$	0	0
$427$	8.89920	0.430662
$428$	−5.49172	−0.265452
$429$	0	0
$430$	14.0130	0.675766
$431$	−10.7589	−0.518236	−0.259118	−	0.965846i	$-0.583432\pi$
$431$	−10.7589	−0.518236	−0.259118	+	0.965846i	$0.583432\pi$
$432$	0	0
$433$	−4.39734	−0.211323	−0.105661	−	0.994402i	$-0.533696\pi$
$433$	−4.39734	−0.211323	−0.105661	+	0.994402i	$0.533696\pi$
$434$	10.7600	0.516497
$435$	0	0
$436$	7.06260	0.338237
$437$	−2.11956	−0.101393
$438$	0	0
$439$	18.9542	0.904632	0.452316	−	0.891858i	$-0.350598\pi$
$439$	18.9542	0.904632	0.452316	+	0.891858i	$0.350598\pi$
$440$	20.8446	0.993726
$441$	0	0
$442$	2.21993	0.105591
$443$	36.2425	1.72193	0.860967	−	0.508660i	$-0.169859\pi$
$443$	36.2425	1.72193	0.860967	+	0.508660i	$0.169859\pi$
$444$	0	0
$445$	30.3280	1.43769
$446$	−21.4310	−1.01479
$447$	0	0
$448$	4.95311	0.234013
$449$	−17.9491	−0.847072	−0.423536	−	0.905879i	$-0.639211\pi$
$449$	−17.9491	−0.847072	−0.423536	+	0.905879i	$0.639211\pi$
$450$	0	0
$451$	6.47902	0.305085
$452$	−5.73352	−0.269682
$453$	0	0
$454$	−15.3580	−0.720787
$455$	−1.70745	−0.0800465
$456$	0	0
$457$	18.1951	0.851132	0.425566	−	0.904927i	$-0.360075\pi$
$457$	18.1951	0.851132	0.425566	+	0.904927i	$0.360075\pi$
$458$	−10.1783	−0.475601
$459$	0	0
$460$	−2.02116	−0.0942372
$461$	−8.88699	−0.413908	−0.206954	−	0.978351i	$-0.566355\pi$
$461$	−8.88699	−0.413908	−0.206954	+	0.978351i	$0.566355\pi$
$462$	0	0
$463$	−34.6096	−1.60845	−0.804223	−	0.594328i	$-0.797418\pi$
$463$	−34.6096	−1.60845	−0.804223	+	0.594328i	$0.797418\pi$
$464$	28.6136	1.32835
$465$	0	0
$466$	−11.8330	−0.548152
$467$	−11.2894	−0.522410	−0.261205	−	0.965283i	$-0.584120\pi$
$467$	−11.2894	−0.522410	−0.261205	+	0.965283i	$0.584120\pi$
$468$	0	0
$469$	−11.0484	−0.510167
$470$	68.6441	3.16632
$471$	0	0
$472$	−23.4230	−1.07813
$473$	−6.12805	−0.281768
$474$	0	0
$475$	−15.3023	−0.702117
$476$	1.53251	0.0702427
$477$	0	0
$478$	−35.2257	−1.61119
$479$	23.5409	1.07561	0.537805	−	0.843069i	$-0.319253\pi$
$479$	23.5409	1.07561	0.537805	+	0.843069i	$0.319253\pi$
$480$	0	0
$481$	2.36699	0.107925
$482$	10.4757	0.477155
$483$	0	0
$484$	−2.52657	−0.114844
$485$	29.3931	1.33467
$486$	0	0
$487$	3.81924	0.173066	0.0865330	−	0.996249i	$-0.472421\pi$
$487$	3.81924	0.173066	0.0865330	+	0.996249i	$0.472421\pi$
$488$	−20.8729	−0.944870
$489$	0	0
$490$	−5.68181	−0.256678
$491$	3.24874	0.146614	0.0733068	−	0.997309i	$-0.476645\pi$
$491$	3.24874	0.146614	0.0733068	+	0.997309i	$0.476645\pi$
$492$	0	0
$493$	−17.5491	−0.790372
$494$	−1.48912	−0.0669986
$495$	0	0
$496$	−32.3294	−1.45163
$497$	−4.92950	−0.221119
$498$	0	0
$499$	4.24713	0.190128	0.0950639	−	0.995471i	$-0.469694\pi$
$499$	4.24713	0.190128	0.0950639	+	0.995471i	$0.469694\pi$
$500$	−5.22950	−0.233870
$501$	0	0
$502$	3.40380	0.151919
$503$	−7.96728	−0.355244	−0.177622	−	0.984099i	$-0.556840\pi$
$503$	−7.96728	−0.355244	−0.177622	+	0.984099i	$0.556840\pi$
$504$	0	0
$505$	4.29543	0.191144
$506$	4.26057	0.189405
$507$	0	0
$508$	−0.523519	−0.0232274
$509$	20.0689	0.889537	0.444768	−	0.895646i	$-0.353286\pi$
$509$	20.0689	0.889537	0.444768	+	0.895646i	$0.353286\pi$
$510$	0	0
$511$	1.24866	0.0552376
$512$	−8.59023	−0.379638
$513$	0	0
$514$	−45.3616	−2.00081
$515$	−11.2963	−0.497776
$516$	0	0
$517$	−30.0190	−1.32023
$518$	7.87653	0.346075
$519$	0	0
$520$	4.00478	0.175621
$521$	−15.9440	−0.698521	−0.349260	−	0.937026i	$-0.613567\pi$
$521$	−15.9440	−0.698521	−0.349260	+	0.937026i	$0.613567\pi$
$522$	0	0
$523$	16.4975	0.721387	0.360693	−	0.932684i	$-0.382540\pi$
$523$	16.4975	0.721387	0.360693	+	0.932684i	$0.382540\pi$
$524$	−3.55139	−0.155143
$525$	0	0
$526$	17.9186	0.781289
$527$	19.8281	0.863726
$528$	0	0
$529$	−21.8349	−0.949343
$530$	1.44344	0.0626988
$531$	0	0
$532$	−1.02800	−0.0445695
$533$	1.24479	0.0539177
$534$	0	0
$535$	37.5196	1.62212
$536$	25.9137	1.11930
$537$	0	0
$538$	12.9943	0.560223
$539$	2.48473	0.107025
$540$	0	0
$541$	−25.0911	−1.07875	−0.539375	−	0.842065i	$-0.681340\pi$
$541$	−25.0911	−1.07875	−0.539375	+	0.842065i	$0.681340\pi$
$542$	−0.393779	−0.0169143
$543$	0	0
$544$	−8.46345	−0.362868
$545$	−48.2520	−2.06689
$546$	0	0
$547$	4.58420	0.196006	0.0980030	−	0.995186i	$-0.468755\pi$
$547$	4.58420	0.196006	0.0980030	+	0.995186i	$0.468755\pi$
$548$	−0.598070	−0.0255483
$549$	0	0
$550$	30.7593	1.31158
$551$	11.7718	0.501498
$552$	0	0
$553$	−7.06010	−0.300226
$554$	34.2126	1.45355
$555$	0	0
$556$	2.02108	0.0857129
$557$	−33.5821	−1.42292	−0.711459	−	0.702728i	$-0.751965\pi$
$557$	−33.5821	−1.42292	−0.711459	+	0.702728i	$0.751965\pi$
$558$	0	0
$559$	−1.17736	−0.0497969
$560$	17.0715	0.721402
$561$	0	0
$562$	19.7928	0.834910
$563$	−24.8052	−1.04541	−0.522706	−	0.852513i	$-0.675078\pi$
$563$	−24.8052	−1.04541	−0.522706	+	0.852513i	$0.675078\pi$
$564$	0	0
$565$	39.1716	1.64796
$566$	13.6710	0.574636
$567$	0	0
$568$	11.5620	0.485132
$569$	7.31381	0.306611	0.153305	−	0.988179i	$-0.451008\pi$
$569$	7.31381	0.306611	0.153305	+	0.988179i	$0.451008\pi$
$570$	0	0
$571$	23.2477	0.972884	0.486442	−	0.873713i	$-0.338294\pi$
$571$	23.2477	0.972884	0.486442	+	0.873713i	$0.338294\pi$
$572$	0.620978	0.0259644
$573$	0	0
$574$	4.14223	0.172893
$575$	8.41164	0.350789
$576$	0	0
$577$	43.7722	1.82226	0.911130	−	0.412119i	$-0.135211\pi$
$577$	43.7722	1.82226	0.911130	+	0.412119i	$0.135211\pi$
$578$	−13.3927	−0.557064
$579$	0	0
$580$	11.2253	0.466107
$581$	−6.48027	−0.268847
$582$	0	0
$583$	−0.631233	−0.0261430
$584$	−2.92871	−0.121191
$585$	0	0
$586$	26.8156	1.10774
$587$	−2.57469	−0.106269	−0.0531345	−	0.998587i	$-0.516921\pi$
$587$	−2.57469	−0.106269	−0.0531345	+	0.998587i	$0.516921\pi$
$588$	0	0
$589$	−13.3006	−0.548041
$590$	−56.7410	−2.33599
$591$	0	0
$592$	−23.6657	−0.972655
$593$	36.2317	1.48786	0.743929	−	0.668259i	$-0.232960\pi$
$593$	36.2317	1.48786	0.743929	+	0.668259i	$0.232960\pi$
$594$	0	0
$595$	−10.4702	−0.429236
$596$	−11.8977	−0.487350
$597$	0	0
$598$	0.818565	0.0334736
$599$	1.34487	0.0549500	0.0274750	−	0.999622i	$-0.491253\pi$
$599$	1.34487	0.0549500	0.0274750	+	0.999622i	$0.491253\pi$
$600$	0	0
$601$	4.65233	0.189772	0.0948862	−	0.995488i	$-0.469751\pi$
$601$	4.65233	0.189772	0.0948862	+	0.995488i	$0.469751\pi$
$602$	−3.91784	−0.159679
$603$	0	0
$604$	−7.02664	−0.285910
$605$	17.2617	0.701786
$606$	0	0
$607$	−14.0853	−0.571703	−0.285852	−	0.958274i	$-0.592276\pi$
$607$	−14.0853	−0.571703	−0.285852	+	0.958274i	$0.592276\pi$
$608$	5.67724	0.230242
$609$	0	0
$610$	−50.5636	−2.04726
$611$	−5.76742	−0.233325
$612$	0	0
$613$	25.6928	1.03772	0.518861	−	0.854858i	$-0.326356\pi$
$613$	25.6928	1.03772	0.518861	+	0.854858i	$0.326356\pi$
$614$	25.9254	1.04626
$615$	0	0
$616$	−5.82787	−0.234812
$617$	−47.0479	−1.89408	−0.947038	−	0.321122i	$-0.895940\pi$
$617$	−47.0479	−1.89408	−0.947038	+	0.321122i	$0.895940\pi$
$618$	0	0
$619$	15.8138	0.635611	0.317805	−	0.948156i	$-0.397054\pi$
$619$	15.8138	0.635611	0.317805	+	0.948156i	$0.397054\pi$
$620$	−12.6831	−0.509366
$621$	0	0
$622$	−46.1641	−1.85101
$623$	−8.47932	−0.339717
$624$	0	0
$625$	−3.23599	−0.129440
$626$	41.2503	1.64870
$627$	0	0
$628$	−6.42278	−0.256297
$629$	14.5145	0.578732
$630$	0	0
$631$	45.6190	1.81606	0.908032	−	0.418900i	$-0.137584\pi$
$631$	45.6190	1.81606	0.908032	+	0.418900i	$0.137584\pi$
$632$	16.5593	0.658694
$633$	0	0
$634$	−27.9235	−1.10898
$635$	3.57671	0.141937
$636$	0	0
$637$	0.477380	0.0189145
$638$	−23.6628	−0.936818
$639$	0	0
$640$	−48.8245	−1.92996
$641$	33.9261	1.34000	0.670000	−	0.742361i	$-0.266294\pi$
$641$	33.9261	1.34000	0.670000	+	0.742361i	$0.266294\pi$
$642$	0	0
$643$	8.05267	0.317566	0.158783	−	0.987313i	$-0.449243\pi$
$643$	8.05267	0.317566	0.158783	+	0.987313i	$0.449243\pi$
$644$	0.565091	0.0222677
$645$	0	0
$646$	−9.13137	−0.359269
$647$	0.994279	0.0390891	0.0195446	−	0.999809i	$-0.493778\pi$
$647$	0.994279	0.0390891	0.0195446	+	0.999809i	$0.493778\pi$
$648$	0	0
$649$	24.8136	0.974018
$650$	5.90967	0.231796
$651$	0	0
$652$	0.503648	0.0197244
$653$	−23.2886	−0.911354	−0.455677	−	0.890145i	$-0.650603\pi$
$653$	−23.2886	−0.911354	−0.455677	+	0.890145i	$0.650603\pi$
$654$	0	0
$655$	24.2632	0.948043
$656$	−12.4457	−0.485922
$657$	0	0
$658$	−19.1920	−0.748182
$659$	18.1773	0.708088	0.354044	−	0.935229i	$-0.384806\pi$
$659$	18.1773	0.708088	0.354044	+	0.935229i	$0.384806\pi$
$660$	0	0
$661$	−37.8810	−1.47340	−0.736701	−	0.676219i	$-0.763617\pi$
$661$	−37.8810	−1.47340	−0.736701	+	0.676219i	$0.763617\pi$
$662$	−24.4608	−0.950695
$663$	0	0
$664$	15.1993	0.589848
$665$	7.02335	0.272354
$666$	0	0
$667$	−6.47097	−0.250557
$668$	−0.847648	−0.0327965
$669$	0	0
$670$	62.7747	2.42520
$671$	22.1121	0.853628
$672$	0	0
$673$	17.3738	0.669711	0.334855	−	0.942270i	$-0.391313\pi$
$673$	17.3738	0.669711	0.334855	+	0.942270i	$0.391313\pi$
$674$	−12.9250	−0.497853
$675$	0	0
$676$	−6.68645	−0.257171
$677$	21.8592	0.840117	0.420059	−	0.907497i	$-0.362010\pi$
$677$	21.8592	0.840117	0.420059	+	0.907497i	$0.362010\pi$
$678$	0	0
$679$	−8.21793	−0.315375
$680$	24.5576	0.941741
$681$	0	0
$682$	26.7357	1.02376
$683$	15.8806	0.607653	0.303827	−	0.952727i	$-0.401736\pi$
$683$	15.8806	0.607653	0.303827	+	0.952727i	$0.401736\pi$
$684$	0	0
$685$	4.08604	0.156120
$686$	1.58856	0.0606515
$687$	0	0
$688$	11.7715	0.448785
$689$	−0.121276	−0.00462025
$690$	0	0
$691$	5.91148	0.224883	0.112442	−	0.993658i	$-0.464133\pi$
$691$	5.91148	0.224883	0.112442	+	0.993658i	$0.464133\pi$
$692$	8.51141	0.323555
$693$	0	0
$694$	33.3615	1.26639
$695$	−13.8081	−0.523771
$696$	0	0
$697$	7.63312	0.289125
$698$	9.17211	0.347170
$699$	0	0
$700$	4.07970	0.154198
$701$	29.8056	1.12574	0.562872	−	0.826544i	$-0.309697\pi$
$701$	29.8056	1.12574	0.562872	+	0.826544i	$0.309697\pi$
$702$	0	0
$703$	−9.73627	−0.367210
$704$	12.3071	0.463843
$705$	0	0
$706$	21.7247	0.817620
$707$	−1.20095	−0.0451662
$708$	0	0
$709$	−17.6729	−0.663719	−0.331859	−	0.943329i	$-0.607676\pi$
$709$	−17.6729	−0.663719	−0.331859	+	0.943329i	$0.607676\pi$
$710$	28.0085	1.05114
$711$	0	0
$712$	19.8880	0.745336
$713$	7.31131	0.273811
$714$	0	0
$715$	−4.24255	−0.158662
$716$	−8.30245	−0.310277
$717$	0	0
$718$	−19.3917	−0.723690
$719$	23.2946	0.868740	0.434370	−	0.900734i	$-0.356971\pi$
$719$	23.2946	0.868740	0.434370	+	0.900734i	$0.356971\pi$
$720$	0	0
$721$	3.15831	0.117622
$722$	−24.0573	−0.895322
$723$	0	0
$724$	5.27186	0.195927
$725$	−46.7174	−1.73504
$726$	0	0
$727$	−32.3572	−1.20006	−0.600032	−	0.799976i	$-0.704845\pi$
$727$	−32.3572	−1.20006	−0.600032	+	0.799976i	$0.704845\pi$
$728$	−1.11968	−0.0414983
$729$	0	0
$730$	−7.09466	−0.262585
$731$	−7.21964	−0.267028
$732$	0	0
$733$	−15.8526	−0.585530	−0.292765	−	0.956184i	$-0.594575\pi$
$733$	−15.8526	−0.585530	−0.292765	+	0.956184i	$0.594575\pi$
$734$	7.00558	0.258581
$735$	0	0
$736$	−3.12077	−0.115033
$737$	−27.4522	−1.01121
$738$	0	0
$739$	−20.5904	−0.757432	−0.378716	−	0.925513i	$-0.623634\pi$
$739$	−20.5904	−0.757432	−0.378716	+	0.925513i	$0.623634\pi$
$740$	−9.28427	−0.341296
$741$	0	0
$742$	−0.403566	−0.0148154
$743$	21.0394	0.771861	0.385931	−	0.922528i	$-0.373880\pi$
$743$	21.0394	0.771861	0.385931	+	0.922528i	$0.373880\pi$
$744$	0	0
$745$	81.2858	2.97808
$746$	42.8679	1.56950
$747$	0	0
$748$	3.80788	0.139230
$749$	−10.4900	−0.383296
$750$	0	0
$751$	8.79716	0.321013	0.160506	−	0.987035i	$-0.448687\pi$
$751$	8.79716	0.321013	0.160506	+	0.987035i	$0.448687\pi$
$752$	57.6641	2.10279
$753$	0	0
$754$	−4.54623	−0.165564
$755$	48.0063	1.74713
$756$	0	0
$757$	−43.2173	−1.57076	−0.785380	−	0.619013i	$-0.787533\pi$
$757$	−43.2173	−1.57076	−0.785380	+	0.619013i	$0.787533\pi$
$758$	−35.1610	−1.27710
$759$	0	0
$760$	−16.4731	−0.597542
$761$	−0.234033	−0.00848368	−0.00424184	−	0.999991i	$-0.501350\pi$
$761$	−0.234033	−0.00848368	−0.00424184	+	0.999991i	$0.501350\pi$
$762$	0	0
$763$	13.4906	0.488393
$764$	1.91685	0.0693491
$765$	0	0
$766$	−37.4025	−1.35141
$767$	4.76733	0.172138
$768$	0	0
$769$	−31.8474	−1.14845	−0.574224	−	0.818698i	$-0.694696\pi$
$769$	−31.8474	−1.14845	−0.574224	+	0.818698i	$0.694696\pi$
$770$	−14.1177	−0.508768
$771$	0	0
$772$	12.9276	0.465276
$773$	−19.5786	−0.704192	−0.352096	−	0.935964i	$-0.614531\pi$
$773$	−19.5786	−0.704192	−0.352096	+	0.935964i	$0.614531\pi$
$774$	0	0
$775$	52.7843	1.89607
$776$	19.2750	0.691931
$777$	0	0
$778$	−38.0049	−1.36254
$779$	−5.12025	−0.183452
$780$	0	0
$781$	−12.2485	−0.438285
$782$	5.01950	0.179497
$783$	0	0
$784$	−4.77297	−0.170463
$785$	43.8807	1.56617
$786$	0	0
$787$	33.9347	1.20964	0.604821	−	0.796361i	$-0.293245\pi$
$787$	33.9347	1.20964	0.604821	+	0.796361i	$0.293245\pi$
$788$	−9.28492	−0.330761
$789$	0	0
$790$	40.1141	1.42720
$791$	−10.9519	−0.389404
$792$	0	0
$793$	4.24830	0.150862
$794$	−10.2662	−0.364335
$795$	0	0
$796$	−9.58522	−0.339739
$797$	−42.0680	−1.49013	−0.745063	−	0.666995i	$-0.767580\pi$
$797$	−42.0680	−1.49013	−0.745063	+	0.666995i	$0.767580\pi$
$798$	0	0
$799$	−35.3662	−1.25117
$800$	−22.5305	−0.796574
$801$	0	0
$802$	−21.3335	−0.753313
$803$	3.10258	0.109488
$804$	0	0
$805$	−3.86072	−0.136073
$806$	5.13662	0.180930
$807$	0	0
$808$	2.81679	0.0990943
$809$	15.3134	0.538392	0.269196	−	0.963085i	$-0.413242\pi$
$809$	15.3134	0.538392	0.269196	+	0.963085i	$0.413242\pi$
$810$	0	0
$811$	48.5802	1.70588	0.852941	−	0.522008i	$-0.174817\pi$
$811$	48.5802	1.70588	0.852941	+	0.522008i	$0.174817\pi$
$812$	−3.13846	−0.110138
$813$	0	0
$814$	19.5710	0.685964
$815$	−3.44094	−0.120531
$816$	0	0
$817$	4.84289	0.169431
$818$	−48.7091	−1.70308
$819$	0	0
$820$	−4.88255	−0.170506
$821$	33.4998	1.16915	0.584576	−	0.811339i	$-0.301261\pi$
$821$	33.4998	1.16915	0.584576	+	0.811339i	$0.301261\pi$
$822$	0	0
$823$	−19.0305	−0.663360	−0.331680	−	0.943392i	$-0.607615\pi$
$823$	−19.0305	−0.663360	−0.331680	+	0.943392i	$0.607615\pi$
$824$	−7.40774	−0.258061
$825$	0	0
$826$	15.8640	0.551981
$827$	37.1919	1.29329	0.646645	−	0.762791i	$-0.276172\pi$
$827$	37.1919	1.29329	0.646645	+	0.762791i	$0.276172\pi$
$828$	0	0
$829$	−20.7791	−0.721688	−0.360844	−	0.932626i	$-0.617511\pi$
$829$	−20.7791	−0.721688	−0.360844	+	0.932626i	$0.617511\pi$
$830$	36.8196	1.27803
$831$	0	0
$832$	2.36452	0.0819750
$833$	2.92733	0.101426
$834$	0	0
$835$	5.79116	0.200411
$836$	−2.55430	−0.0883425
$837$	0	0
$838$	42.1243	1.45516
$839$	30.6926	1.05963	0.529814	−	0.848114i	$-0.322262\pi$
$839$	30.6926	1.05963	0.529814	+	0.848114i	$0.322262\pi$
$840$	0	0
$841$	6.93910	0.239279
$842$	−6.54520	−0.225563
$843$	0	0
$844$	−3.86578	−0.133066
$845$	45.6821	1.57151
$846$	0	0
$847$	−4.82613	−0.165828
$848$	1.21255	0.0416391
$849$	0	0
$850$	36.2385	1.24297
$851$	5.35201	0.183465
$852$	0	0
$853$	−14.3744	−0.492171	−0.246086	−	0.969248i	$-0.579144\pi$
$853$	−14.3744	−0.492171	−0.246086	+	0.969248i	$0.579144\pi$
$854$	14.1369	0.483755
$855$	0	0
$856$	24.6040	0.840949
$857$	−28.6832	−0.979800	−0.489900	−	0.871779i	$-0.662967\pi$
$857$	−28.6832	−0.979800	−0.489900	+	0.871779i	$0.662967\pi$
$858$	0	0
$859$	2.16240	0.0737801	0.0368900	−	0.999319i	$-0.488255\pi$
$859$	2.16240	0.0737801	0.0368900	+	0.999319i	$0.488255\pi$
$860$	4.61806	0.157475
$861$	0	0
$862$	−17.0911	−0.582125
$863$	30.3176	1.03202	0.516012	−	0.856581i	$-0.327416\pi$
$863$	30.3176	1.03202	0.516012	+	0.856581i	$0.327416\pi$
$864$	0	0
$865$	−58.1503	−1.97717
$866$	−6.98544	−0.237375
$867$	0	0
$868$	3.54603	0.120360
$869$	−17.5424	−0.595086
$870$	0	0
$871$	−5.27428	−0.178712
$872$	−31.6419	−1.07153
$873$	0	0
$874$	−3.36705	−0.113892
$875$	−9.98912	−0.337694
$876$	0	0
$877$	2.24322	0.0757482	0.0378741	−	0.999283i	$-0.487941\pi$
$877$	2.24322	0.0757482	0.0378741	+	0.999283i	$0.487941\pi$
$878$	30.1098	1.01616
$879$	0	0
$880$	42.4180	1.42991
$881$	−44.4609	−1.49793	−0.748963	−	0.662612i	$-0.769448\pi$
$881$	−44.4609	−1.49793	−0.748963	+	0.662612i	$0.769448\pi$
$882$	0	0
$883$	−5.61418	−0.188932	−0.0944660	−	0.995528i	$-0.530114\pi$
$883$	−5.61418	−0.188932	−0.0944660	+	0.995528i	$0.530114\pi$
$884$	0.731592	0.0246061
$885$	0	0
$886$	57.5734	1.93422
$887$	−3.60496	−0.121043	−0.0605213	−	0.998167i	$-0.519276\pi$
$887$	−3.60496	−0.121043	−0.0605213	+	0.998167i	$0.519276\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	48.1779	1.61493
$891$	0	0
$892$	−7.06271	−0.236477
$893$	23.7235	0.793875
$894$	0	0
$895$	56.7227	1.89603
$896$	13.6507	0.456037
$897$	0	0
$898$	−28.5133	−0.951500
$899$	−40.6063	−1.35430
$900$	0	0
$901$	−0.743674	−0.0247754
$902$	10.2923	0.342696
$903$	0	0
$904$	25.6873	0.854348
$905$	−36.0176	−1.19727
$906$	0	0
$907$	−40.1524	−1.33324	−0.666620	−	0.745398i	$-0.732260\pi$
$907$	−40.1524	−1.33324	−0.666620	+	0.745398i	$0.732260\pi$
$908$	−5.06133	−0.167966
$909$	0	0
$910$	−2.71238	−0.0899146
$911$	−31.1955	−1.03355	−0.516777	−	0.856120i	$-0.672868\pi$
$911$	−31.1955	−1.03355	−0.516777	+	0.856120i	$0.672868\pi$
$912$	0	0
$913$	−16.1017	−0.532888
$914$	28.9040	0.956060
$915$	0	0
$916$	−3.35433	−0.110830
$917$	−6.78368	−0.224017
$918$	0	0
$919$	−34.1283	−1.12579	−0.562894	−	0.826529i	$-0.690312\pi$
$919$	−34.1283	−1.12579	−0.562894	+	0.826529i	$0.690312\pi$
$920$	9.05524	0.298542
$921$	0	0
$922$	−14.1175	−0.464935
$923$	−2.35325	−0.0774581
$924$	0	0
$925$	38.6391	1.27044
$926$	−54.9794	−1.80674
$927$	0	0
$928$	17.3324	0.568965
$929$	39.3611	1.29140	0.645699	−	0.763592i	$-0.276566\pi$
$929$	39.3611	1.29140	0.645699	+	0.763592i	$0.276566\pi$
$930$	0	0
$931$	−1.96364	−0.0643556
$932$	−3.89963	−0.127737
$933$	0	0
$934$	−17.9338	−0.586813
$935$	−26.0156	−0.850801
$936$	0	0
$937$	45.8260	1.49707	0.748535	−	0.663096i	$-0.230758\pi$
$937$	45.8260	1.49707	0.748535	+	0.663096i	$0.230758\pi$
$938$	−17.5510	−0.573060
$939$	0	0
$940$	22.6221	0.737851
$941$	−4.51164	−0.147075	−0.0735376	−	0.997292i	$-0.523429\pi$
$941$	−4.51164	−0.147075	−0.0735376	+	0.997292i	$0.523429\pi$
$942$	0	0
$943$	2.81460	0.0916558
$944$	−47.6649	−1.55136
$945$	0	0
$946$	−9.73478	−0.316505
$947$	−7.22710	−0.234849	−0.117425	−	0.993082i	$-0.537464\pi$
$947$	−7.22710	−0.234849	−0.117425	+	0.993082i	$0.537464\pi$
$948$	0	0
$949$	0.596087	0.0193498
$950$	−24.3086	−0.788674
$951$	0	0
$952$	−6.86598	−0.222528
$953$	27.5338	0.891906	0.445953	−	0.895056i	$-0.352865\pi$
$953$	27.5338	0.891906	0.445953	+	0.895056i	$0.352865\pi$
$954$	0	0
$955$	−13.0960	−0.423776
$956$	−11.6088	−0.375457
$957$	0	0
$958$	37.3961	1.20821
$959$	−1.14240	−0.0368901
$960$	0	0
$961$	14.8795	0.479985
$962$	3.76010	0.121230
$963$	0	0
$964$	3.45233	0.111192
$965$	−88.3222	−2.84319
$966$	0	0
$967$	−27.1488	−0.873047	−0.436523	−	0.899693i	$-0.643790\pi$
$967$	−27.1488	−0.873047	−0.436523	+	0.899693i	$0.643790\pi$
$968$	11.3196	0.363825
$969$	0	0
$970$	46.6927	1.49921
$971$	14.8963	0.478044	0.239022	−	0.971014i	$-0.423173\pi$
$971$	14.8963	0.478044	0.239022	+	0.971014i	$0.423173\pi$
$972$	0	0
$973$	3.86056	0.123764
$974$	6.06708	0.194402
$975$	0	0
$976$	−42.4756	−1.35961
$977$	−26.3876	−0.844213	−0.422107	−	0.906546i	$-0.638709\pi$
$977$	−26.3876	−0.844213	−0.422107	+	0.906546i	$0.638709\pi$
$978$	0	0
$979$	−21.0688	−0.673362
$980$	−1.87248	−0.0598140
$981$	0	0
$982$	5.16082	0.164688
$983$	−53.6989	−1.71273	−0.856364	−	0.516372i	$-0.827282\pi$
$983$	−53.6989	−1.71273	−0.856364	+	0.516372i	$0.827282\pi$
$984$	0	0
$985$	63.4349	2.02120
$986$	−27.8778	−0.887810
$987$	0	0
$988$	−0.490748	−0.0156128
$989$	−2.66213	−0.0846508
$990$	0	0
$991$	−52.4107	−1.66488	−0.832440	−	0.554115i	$-0.813057\pi$
$991$	−52.4107	−1.66488	−0.832440	+	0.554115i	$0.813057\pi$
$992$	−19.5833	−0.621770
$993$	0	0
$994$	−7.83081	−0.248378
$995$	65.4866	2.07606
$996$	0	0
$997$	47.8562	1.51562	0.757810	−	0.652476i	$-0.226269\pi$
$997$	47.8562	1.51562	0.757810	+	0.652476i	$0.226269\pi$
$998$	6.74682	0.213567
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.o.1.11		13
3.2	odd	2		2667.2.a.l.1.3	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.l.1.3	✓	13	3.2	odd	2
8001.2.a.o.1.11		13	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+1.58856 q^{2} +0.523519 q^{4} -3.57671 q^{5} +1.00000 q^{7} -2.34548 q^{8} +O(q^{10})\)	\(q+1.58856 q^{2} +0.523519 q^{4} -3.57671 q^{5} +1.00000 q^{7} -2.34548 q^{8} -5.68181 q^{10} +2.48473 q^{11} +0.477380 q^{13} +1.58856 q^{14} -4.77297 q^{16} +2.92733 q^{17} -1.96364 q^{19} -1.87248 q^{20} +3.94714 q^{22} +1.07941 q^{23} +7.79283 q^{25} +0.758347 q^{26} +0.523519 q^{28} -5.99492 q^{29} +6.77344 q^{31} -2.89119 q^{32} +4.65024 q^{34} -3.57671 q^{35} +4.95829 q^{37} -3.11935 q^{38} +8.38908 q^{40} +2.60754 q^{41} -2.46629 q^{43} +1.30080 q^{44} +1.71470 q^{46} -12.0814 q^{47} +1.00000 q^{49} +12.3794 q^{50} +0.249918 q^{52} -0.254045 q^{53} -8.88714 q^{55} -2.34548 q^{56} -9.52329 q^{58} +9.98644 q^{59} +8.89920 q^{61} +10.7600 q^{62} +4.95311 q^{64} -1.70745 q^{65} -11.0484 q^{67} +1.53251 q^{68} -5.68181 q^{70} -4.92950 q^{71} +1.24866 q^{73} +7.87653 q^{74} -1.02800 q^{76} +2.48473 q^{77} -7.06010 q^{79} +17.0715 q^{80} +4.14223 q^{82} -6.48027 q^{83} -10.4702 q^{85} -3.91784 q^{86} -5.82787 q^{88} -8.47932 q^{89} +0.477380 q^{91} +0.565091 q^{92} -19.1920 q^{94} +7.02335 q^{95} -8.21793 q^{97} +1.58856 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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