LMFDB - Embedded newform 8001.2.a.o.1.9

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			1.9
Root			$-0.423652$ 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)

$f(q)$	$=$	$q+0.423652 q^{2} -1.82052 q^{4} -2.66196 q^{5} +1.00000 q^{7} -1.61857 q^{8} +O(q^{10})$	\(q+0.423652 q^{2} -1.82052 q^{4} -2.66196 q^{5} +1.00000 q^{7} -1.61857 q^{8} -1.12774 q^{10} -3.97552 q^{11} +4.95058 q^{13} +0.423652 q^{14} +2.95533 q^{16} -5.63178 q^{17} +3.72073 q^{19} +4.84615 q^{20} -1.68424 q^{22} +4.13626 q^{23} +2.08603 q^{25} +2.09732 q^{26} -1.82052 q^{28} -7.04468 q^{29} -2.57265 q^{31} +4.48917 q^{32} -2.38592 q^{34} -2.66196 q^{35} +2.65212 q^{37} +1.57630 q^{38} +4.30857 q^{40} -2.05217 q^{41} +4.03075 q^{43} +7.23751 q^{44} +1.75233 q^{46} +7.52083 q^{47} +1.00000 q^{49} +0.883750 q^{50} -9.01263 q^{52} -0.214398 q^{53} +10.5827 q^{55} -1.61857 q^{56} -2.98449 q^{58} +12.5754 q^{59} -2.37431 q^{61} -1.08991 q^{62} -4.00881 q^{64} -13.1783 q^{65} +4.26602 q^{67} +10.2528 q^{68} -1.12774 q^{70} +13.7119 q^{71} -10.8892 q^{73} +1.12358 q^{74} -6.77367 q^{76} -3.97552 q^{77} +2.69299 q^{79} -7.86696 q^{80} -0.869404 q^{82} -4.22143 q^{83} +14.9916 q^{85} +1.70763 q^{86} +6.43466 q^{88} -4.08362 q^{89} +4.95058 q^{91} -7.53014 q^{92} +3.18621 q^{94} -9.90444 q^{95} +0.135110 q^{97} +0.423652 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.423652	0.299567	0.149784	−	0.988719i	$-0.452142\pi$
$2$	0.423652	0.299567	0.149784	+	0.988719i	$0.452142\pi$
$3$	0	0
$3$	0	0
$4$	−1.82052	−0.910259
$5$	−2.66196	−1.19046	−0.595232	−	0.803554i	$-0.702940\pi$
$5$	−2.66196	−1.19046	−0.595232	+	0.803554i	$0.702940\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.61857	−0.572251
$9$	0	0
$10$	−1.12774	−0.356624
$11$	−3.97552	−1.19866	−0.599332	−	0.800500i	$-0.704567\pi$
$11$	−3.97552	−1.19866	−0.599332	+	0.800500i	$0.704567\pi$
$12$	0	0
$13$	4.95058	1.37304	0.686522	−	0.727109i	$-0.259136\pi$
$13$	4.95058	1.37304	0.686522	+	0.727109i	$0.259136\pi$
$14$	0.423652	0.113226
$15$	0	0
$16$	2.95533	0.738832
$17$	−5.63178	−1.36591	−0.682954	−	0.730462i	$-0.739305\pi$
$17$	−5.63178	−1.36591	−0.682954	+	0.730462i	$0.739305\pi$
$18$	0	0
$19$	3.72073	0.853595	0.426797	−	0.904347i	$-0.359642\pi$
$19$	3.72073	0.853595	0.426797	+	0.904347i	$0.359642\pi$
$20$	4.84615	1.08363
$21$	0	0
$22$	−1.68424	−0.359081
$23$	4.13626	0.862469	0.431235	−	0.902240i	$-0.358078\pi$
$23$	4.13626	0.862469	0.431235	+	0.902240i	$0.358078\pi$
$24$	0	0
$25$	2.08603	0.417205
$26$	2.09732	0.411319
$27$	0	0
$28$	−1.82052	−0.344046
$29$	−7.04468	−1.30816	−0.654082	−	0.756423i	$-0.726945\pi$
$29$	−7.04468	−1.30816	−0.654082	+	0.756423i	$0.726945\pi$
$30$	0	0
$31$	−2.57265	−0.462062	−0.231031	−	0.972946i	$-0.574210\pi$
$31$	−2.57265	−0.462062	−0.231031	+	0.972946i	$0.574210\pi$
$32$	4.48917	0.793581
$33$	0	0
$34$	−2.38592	−0.409181
$35$	−2.66196	−0.449953
$36$	0	0
$37$	2.65212	0.436006	0.218003	−	0.975948i	$-0.430046\pi$
$37$	2.65212	0.436006	0.218003	+	0.975948i	$0.430046\pi$
$38$	1.57630	0.255709
$39$	0	0
$40$	4.30857	0.681245
$41$	−2.05217	−0.320494	−0.160247	−	0.987077i	$-0.551229\pi$
$41$	−2.05217	−0.320494	−0.160247	+	0.987077i	$0.551229\pi$
$42$	0	0
$43$	4.03075	0.614683	0.307342	−	0.951599i	$-0.400561\pi$
$43$	4.03075	0.614683	0.307342	+	0.951599i	$0.400561\pi$
$44$	7.23751	1.09110
$45$	0	0
$46$	1.75233	0.258368
$47$	7.52083	1.09703	0.548513	−	0.836142i	$-0.315194\pi$
$47$	7.52083	1.09703	0.548513	+	0.836142i	$0.315194\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.883750	0.124981
$51$	0	0
$52$	−9.01263	−1.24983
$53$	−0.214398	−0.0294499	−0.0147249	−	0.999892i	$-0.504687\pi$
$53$	−0.214398	−0.0294499	−0.0147249	+	0.999892i	$0.504687\pi$
$54$	0	0
$55$	10.5827	1.42697
$56$	−1.61857	−0.216291
$57$	0	0
$58$	−2.98449	−0.391883
$59$	12.5754	1.63718	0.818591	−	0.574376i	$-0.194755\pi$
$59$	12.5754	1.63718	0.818591	+	0.574376i	$0.194755\pi$
$60$	0	0
$61$	−2.37431	−0.303999	−0.151999	−	0.988381i	$-0.548571\pi$
$61$	−2.37431	−0.303999	−0.151999	+	0.988381i	$0.548571\pi$
$62$	−1.08991	−0.138419
$63$	0	0
$64$	−4.00881	−0.501101
$65$	−13.1783	−1.63456
$66$	0	0
$67$	4.26602	0.521177	0.260588	−	0.965450i	$-0.416083\pi$
$67$	4.26602	0.521177	0.260588	+	0.965450i	$0.416083\pi$
$68$	10.2528	1.24333
$69$	0	0
$70$	−1.12774	−0.134791
$71$	13.7119	1.62731	0.813653	−	0.581351i	$-0.197476\pi$
$71$	13.7119	1.62731	0.813653	+	0.581351i	$0.197476\pi$
$72$	0	0
$73$	−10.8892	−1.27449	−0.637244	−	0.770662i	$-0.719926\pi$
$73$	−10.8892	−1.27449	−0.637244	+	0.770662i	$0.719926\pi$
$74$	1.12358	0.130613
$75$	0	0
$76$	−6.77367	−0.776993
$77$	−3.97552	−0.453053
$78$	0	0
$79$	2.69299	0.302985	0.151492	−	0.988458i	$-0.451592\pi$
$79$	2.69299	0.302985	0.151492	+	0.988458i	$0.451592\pi$
$80$	−7.86696	−0.879553
$81$	0	0
$82$	−0.869404	−0.0960096
$83$	−4.22143	−0.463362	−0.231681	−	0.972792i	$-0.574423\pi$
$83$	−4.22143	−0.463362	−0.231681	+	0.972792i	$0.574423\pi$
$84$	0	0
$85$	14.9916	1.62606
$86$	1.70763	0.184139
$87$	0	0
$88$	6.43466	0.685937
$89$	−4.08362	−0.432863	−0.216431	−	0.976298i	$-0.569442\pi$
$89$	−4.08362	−0.432863	−0.216431	+	0.976298i	$0.569442\pi$
$90$	0	0
$91$	4.95058	0.518962
$92$	−7.53014	−0.785071
$93$	0	0
$94$	3.18621	0.328633
$95$	−9.90444	−1.01617
$96$	0	0
$97$	0.135110	0.0137183	0.00685917	−	0.999976i	$-0.497817\pi$
$97$	0.135110	0.0137183	0.00685917	+	0.999976i	$0.497817\pi$
$98$	0.423652	0.0427953
$99$	0	0
$100$	−3.79765	−0.379765
$101$	−0.725428	−0.0721828	−0.0360914	−	0.999348i	$-0.511491\pi$
$101$	−0.725428	−0.0721828	−0.0360914	+	0.999348i	$0.511491\pi$
$102$	0	0
$103$	14.2476	1.40386	0.701930	−	0.712245i	$-0.252322\pi$
$103$	14.2476	1.40386	0.701930	+	0.712245i	$0.252322\pi$
$104$	−8.01287	−0.785726
$105$	0	0
$106$	−0.0908303	−0.00882222
$107$	18.0628	1.74620	0.873098	−	0.487544i	$-0.162107\pi$
$107$	18.0628	1.74620	0.873098	+	0.487544i	$0.162107\pi$
$108$	0	0
$109$	−7.40505	−0.709276	−0.354638	−	0.935004i	$-0.615396\pi$
$109$	−7.40505	−0.709276	−0.354638	+	0.935004i	$0.615396\pi$
$110$	4.48337	0.427473
$111$	0	0
$112$	2.95533	0.279252
$113$	−18.6046	−1.75017	−0.875087	−	0.483966i	$-0.839196\pi$
$113$	−18.6046	−1.75017	−0.875087	+	0.483966i	$0.839196\pi$
$114$	0	0
$115$	−11.0105	−1.02674
$116$	12.8250	1.19077
$117$	0	0
$118$	5.32761	0.490446
$119$	−5.63178	−0.516264
$120$	0	0
$121$	4.80477	0.436797
$122$	−1.00588	−0.0910681
$123$	0	0
$124$	4.68357	0.420597
$125$	7.75688	0.693796
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−10.6767	−0.943694
$129$	0	0
$130$	−5.58299	−0.489661
$131$	−11.7101	−1.02312	−0.511559	−	0.859248i	$-0.670932\pi$
$131$	−11.7101	−1.02312	−0.511559	+	0.859248i	$0.670932\pi$
$132$	0	0
$133$	3.72073	0.322629
$134$	1.80731	0.156127
$135$	0	0
$136$	9.11543	0.781642
$137$	−2.39147	−0.204317	−0.102159	−	0.994768i	$-0.532575\pi$
$137$	−2.39147	−0.204317	−0.102159	+	0.994768i	$0.532575\pi$
$138$	0	0
$139$	−2.14800	−0.182191	−0.0910953	−	0.995842i	$-0.529037\pi$
$139$	−2.14800	−0.182191	−0.0910953	+	0.995842i	$0.529037\pi$
$140$	4.84615	0.409574
$141$	0	0
$142$	5.80909	0.487488
$143$	−19.6811	−1.64582
$144$	0	0
$145$	18.7527	1.55732
$146$	−4.61324	−0.381795
$147$	0	0
$148$	−4.82823	−0.396878
$149$	2.05953	0.168723	0.0843615	−	0.996435i	$-0.473115\pi$
$149$	2.05953	0.168723	0.0843615	+	0.996435i	$0.473115\pi$
$150$	0	0
$151$	12.3000	1.00096	0.500478	−	0.865749i	$-0.333157\pi$
$151$	12.3000	1.00096	0.500478	+	0.865749i	$0.333157\pi$
$152$	−6.02227	−0.488471
$153$	0	0
$154$	−1.68424	−0.135720
$155$	6.84830	0.550069
$156$	0	0
$157$	−19.3425	−1.54370	−0.771850	−	0.635804i	$-0.780669\pi$
$157$	−19.3425	−1.54370	−0.771850	+	0.635804i	$0.780669\pi$
$158$	1.14089	0.0907643
$159$	0	0
$160$	−11.9500	−0.944730
$161$	4.13626	0.325983
$162$	0	0
$163$	−12.4805	−0.977551	−0.488775	−	0.872410i	$-0.662556\pi$
$163$	−12.4805	−0.977551	−0.488775	+	0.872410i	$0.662556\pi$
$164$	3.73601	0.291733
$165$	0	0
$166$	−1.78842	−0.138808
$167$	−2.16315	−0.167390	−0.0836949	−	0.996491i	$-0.526672\pi$
$167$	−2.16315	−0.167390	−0.0836949	+	0.996491i	$0.526672\pi$
$168$	0	0
$169$	11.5083	0.885252
$170$	6.35121	0.487115
$171$	0	0
$172$	−7.33805	−0.559521
$173$	15.5879	1.18513	0.592563	−	0.805524i	$-0.298116\pi$
$173$	15.5879	1.18513	0.592563	+	0.805524i	$0.298116\pi$
$174$	0	0
$175$	2.08603	0.157689
$176$	−11.7490	−0.885612
$177$	0	0
$178$	−1.73003	−0.129671
$179$	11.1995	0.837090	0.418545	−	0.908196i	$-0.362540\pi$
$179$	11.1995	0.837090	0.418545	+	0.908196i	$0.362540\pi$
$180$	0	0
$181$	−6.72881	−0.500148	−0.250074	−	0.968227i	$-0.580455\pi$
$181$	−6.72881	−0.500148	−0.250074	+	0.968227i	$0.580455\pi$
$182$	2.09732	0.155464
$183$	0	0
$184$	−6.69483	−0.493549
$185$	−7.05983	−0.519049
$186$	0	0
$187$	22.3893	1.63726
$188$	−13.6918	−0.998578
$189$	0	0
$190$	−4.19604	−0.304413
$191$	−23.3444	−1.68914	−0.844570	−	0.535446i	$-0.820144\pi$
$191$	−23.3444	−1.68914	−0.844570	+	0.535446i	$0.820144\pi$
$192$	0	0
$193$	2.13924	0.153986	0.0769928	−	0.997032i	$-0.475468\pi$
$193$	2.13924	0.153986	0.0769928	+	0.997032i	$0.475468\pi$
$194$	0.0572397	0.00410957
$195$	0	0
$196$	−1.82052	−0.130037
$197$	−7.01767	−0.499988	−0.249994	−	0.968247i	$-0.580429\pi$
$197$	−7.01767	−0.499988	−0.249994	+	0.968247i	$0.580429\pi$
$198$	0	0
$199$	−18.7133	−1.32655	−0.663277	−	0.748374i	$-0.730835\pi$
$199$	−18.7133	−1.32655	−0.663277	+	0.748374i	$0.730835\pi$
$200$	−3.37638	−0.238746
$201$	0	0
$202$	−0.307329	−0.0216236
$203$	−7.04468	−0.494440
$204$	0	0
$205$	5.46278	0.381537
$206$	6.03604	0.420551
$207$	0	0
$208$	14.6306	1.01445
$209$	−14.7919	−1.02317
$210$	0	0
$211$	−11.4113	−0.785585	−0.392792	−	0.919627i	$-0.628491\pi$
$211$	−11.4113	−0.785585	−0.392792	+	0.919627i	$0.628491\pi$
$212$	0.390316	0.0268070
$213$	0	0
$214$	7.65234	0.523103
$215$	−10.7297	−0.731759
$216$	0	0
$217$	−2.57265	−0.174643
$218$	−3.13717	−0.212476
$219$	0	0
$220$	−19.2660	−1.29891
$221$	−27.8806	−1.87545
$222$	0	0
$223$	12.7903	0.856505	0.428252	−	0.903659i	$-0.359129\pi$
$223$	12.7903	0.856505	0.428252	+	0.903659i	$0.359129\pi$
$224$	4.48917	0.299945
$225$	0	0
$226$	−7.88188	−0.524295
$227$	23.6675	1.57087	0.785434	−	0.618945i	$-0.212440\pi$
$227$	23.6675	1.57087	0.785434	+	0.618945i	$0.212440\pi$
$228$	0	0
$229$	−16.7152	−1.10457	−0.552287	−	0.833654i	$-0.686245\pi$
$229$	−16.7152	−1.10457	−0.552287	+	0.833654i	$0.686245\pi$
$230$	−4.66464	−0.307577
$231$	0	0
$232$	11.4023	0.748599
$233$	−27.5835	−1.80706	−0.903528	−	0.428529i	$-0.859032\pi$
$233$	−27.5835	−1.80706	−0.903528	+	0.428529i	$0.859032\pi$
$234$	0	0
$235$	−20.0201	−1.30597
$236$	−22.8938	−1.49026
$237$	0	0
$238$	−2.38592	−0.154656
$239$	6.10076	0.394625	0.197313	−	0.980341i	$-0.436779\pi$
$239$	6.10076	0.394625	0.197313	+	0.980341i	$0.436779\pi$
$240$	0	0
$241$	−16.4442	−1.05926	−0.529632	−	0.848228i	$-0.677670\pi$
$241$	−16.4442	−1.05926	−0.529632	+	0.848228i	$0.677670\pi$
$242$	2.03555	0.130850
$243$	0	0
$244$	4.32247	0.276718
$245$	−2.66196	−0.170066
$246$	0	0
$247$	18.4198	1.17202
$248$	4.16402	0.264416
$249$	0	0
$250$	3.28622	0.207839
$251$	−31.2771	−1.97420	−0.987098	−	0.160118i	$-0.948813\pi$
$251$	−31.2771	−1.97420	−0.987098	+	0.160118i	$0.948813\pi$
$252$	0	0
$253$	−16.4438	−1.03381
$254$	−0.423652	−0.0265823
$255$	0	0
$256$	3.49442	0.218401
$257$	−15.4673	−0.964824	−0.482412	−	0.875944i	$-0.660239\pi$
$257$	−15.4673	−0.964824	−0.482412	+	0.875944i	$0.660239\pi$
$258$	0	0
$259$	2.65212	0.164795
$260$	23.9913	1.48787
$261$	0	0
$262$	−4.96102	−0.306493
$263$	5.45766	0.336534	0.168267	−	0.985741i	$-0.446183\pi$
$263$	5.45766	0.336534	0.168267	+	0.985741i	$0.446183\pi$
$264$	0	0
$265$	0.570720	0.0350590
$266$	1.57630	0.0966489
$267$	0	0
$268$	−7.76636	−0.474406
$269$	−24.3057	−1.48194	−0.740971	−	0.671536i	$-0.765635\pi$
$269$	−24.3057	−1.48194	−0.740971	+	0.671536i	$0.765635\pi$
$270$	0	0
$271$	26.4187	1.60482	0.802410	−	0.596774i	$-0.203551\pi$
$271$	26.4187	1.60482	0.802410	+	0.596774i	$0.203551\pi$
$272$	−16.6438	−1.00918
$273$	0	0
$274$	−1.01315	−0.0612068
$275$	−8.29304	−0.500089
$276$	0	0
$277$	−13.7390	−0.825494	−0.412747	−	0.910846i	$-0.635431\pi$
$277$	−13.7390	−0.825494	−0.412747	+	0.910846i	$0.635431\pi$
$278$	−0.910003	−0.0545784
$279$	0	0
$280$	4.30857	0.257486
$281$	5.06765	0.302311	0.151155	−	0.988510i	$-0.451701\pi$
$281$	5.06765	0.302311	0.151155	+	0.988510i	$0.451701\pi$
$282$	0	0
$283$	−27.6362	−1.64280	−0.821400	−	0.570353i	$-0.806806\pi$
$283$	−27.6362	−1.64280	−0.821400	+	0.570353i	$0.806806\pi$
$284$	−24.9628	−1.48127
$285$	0	0
$286$	−8.33796	−0.493034
$287$	−2.05217	−0.121135
$288$	0	0
$289$	14.7169	0.865702
$290$	7.94460	0.466523
$291$	0	0
$292$	19.8240	1.16011
$293$	−26.9116	−1.57219	−0.786097	−	0.618104i	$-0.787901\pi$
$293$	−26.9116	−1.57219	−0.786097	+	0.618104i	$0.787901\pi$
$294$	0	0
$295$	−33.4753	−1.94901
$296$	−4.29264	−0.249505
$297$	0	0
$298$	0.872522	0.0505439
$299$	20.4769	1.18421
$300$	0	0
$301$	4.03075	0.232328
$302$	5.21091	0.299854
$303$	0	0
$304$	10.9960	0.630663
$305$	6.32031	0.361900
$306$	0	0
$307$	−10.2148	−0.582989	−0.291494	−	0.956573i	$-0.594152\pi$
$307$	−10.2148	−0.582989	−0.291494	+	0.956573i	$0.594152\pi$
$308$	7.23751	0.412395
$309$	0	0
$310$	2.90130	0.164783
$311$	−9.69180	−0.549572	−0.274786	−	0.961505i	$-0.588607\pi$
$311$	−9.69180	−0.549572	−0.274786	+	0.961505i	$0.588607\pi$
$312$	0	0
$313$	13.6848	0.773509	0.386755	−	0.922183i	$-0.373596\pi$
$313$	13.6848	0.773509	0.386755	+	0.922183i	$0.373596\pi$
$314$	−8.19450	−0.462442
$315$	0	0
$316$	−4.90263	−0.275795
$317$	−14.1092	−0.792451	−0.396226	−	0.918153i	$-0.629680\pi$
$317$	−14.1092	−0.792451	−0.396226	+	0.918153i	$0.629680\pi$
$318$	0	0
$319$	28.0063	1.56805
$320$	10.6713	0.596543
$321$	0	0
$322$	1.75233	0.0976538
$323$	−20.9544	−1.16593
$324$	0	0
$325$	10.3271	0.572842
$326$	−5.28740	−0.292842
$327$	0	0
$328$	3.32157	0.183403
$329$	7.52083	0.414637
$330$	0	0
$331$	1.11891	0.0615011	0.0307505	−	0.999527i	$-0.490210\pi$
$331$	1.11891	0.0615011	0.0307505	+	0.999527i	$0.490210\pi$
$332$	7.68520	0.421780
$333$	0	0
$334$	−0.916424	−0.0501445
$335$	−11.3560	−0.620442
$336$	0	0
$337$	−23.4974	−1.27999	−0.639993	−	0.768381i	$-0.721063\pi$
$337$	−23.4974	−1.27999	−0.639993	+	0.768381i	$0.721063\pi$
$338$	4.87550	0.265192
$339$	0	0
$340$	−27.2924	−1.48014
$341$	10.2276	0.553858
$342$	0	0
$343$	1.00000	0.0539949
$344$	−6.52405	−0.351753
$345$	0	0
$346$	6.60384	0.355025
$347$	−6.87719	−0.369187	−0.184593	−	0.982815i	$-0.559097\pi$
$347$	−6.87719	−0.369187	−0.184593	+	0.982815i	$0.559097\pi$
$348$	0	0
$349$	−3.13512	−0.167819	−0.0839094	−	0.996473i	$-0.526741\pi$
$349$	−3.13512	−0.167819	−0.0839094	+	0.996473i	$0.526741\pi$
$350$	0.883750	0.0472384
$351$	0	0
$352$	−17.8468	−0.951238
$353$	29.3356	1.56138	0.780689	−	0.624920i	$-0.214869\pi$
$353$	29.3356	1.56138	0.780689	+	0.624920i	$0.214869\pi$
$354$	0	0
$355$	−36.5006	−1.93725
$356$	7.43430	0.394017
$357$	0	0
$358$	4.74469	0.250765
$359$	6.92912	0.365705	0.182852	−	0.983140i	$-0.441467\pi$
$359$	6.92912	0.365705	0.182852	+	0.983140i	$0.441467\pi$
$360$	0	0
$361$	−5.15614	−0.271376
$362$	−2.85067	−0.149828
$363$	0	0
$364$	−9.01263	−0.472390
$365$	28.9867	1.51723
$366$	0	0
$367$	20.4454	1.06724	0.533621	−	0.845724i	$-0.320831\pi$
$367$	20.4454	1.06724	0.533621	+	0.845724i	$0.320831\pi$
$368$	12.2240	0.637220
$369$	0	0
$370$	−2.99091	−0.155490
$371$	−0.214398	−0.0111310
$372$	0	0
$373$	−1.24610	−0.0645209	−0.0322604	−	0.999479i	$-0.510271\pi$
$373$	−1.24610	−0.0645209	−0.0322604	+	0.999479i	$0.510271\pi$
$374$	9.48526	0.490471
$375$	0	0
$376$	−12.1730	−0.627774
$377$	−34.8753	−1.79617
$378$	0	0
$379$	−11.8870	−0.610592	−0.305296	−	0.952258i	$-0.598755\pi$
$379$	−11.8870	−0.610592	−0.305296	+	0.952258i	$0.598755\pi$
$380$	18.0312	0.924982
$381$	0	0
$382$	−9.88989	−0.506011
$383$	14.6285	0.747481	0.373741	−	0.927533i	$-0.378075\pi$
$383$	14.6285	0.747481	0.373741	+	0.927533i	$0.378075\pi$
$384$	0	0
$385$	10.5827	0.539343
$386$	0.906292	0.0461290
$387$	0	0
$388$	−0.245970	−0.0124873
$389$	21.5646	1.09337	0.546685	−	0.837338i	$-0.315890\pi$
$389$	21.5646	1.09337	0.546685	+	0.837338i	$0.315890\pi$
$390$	0	0
$391$	−23.2945	−1.17805
$392$	−1.61857	−0.0817502
$393$	0	0
$394$	−2.97305	−0.149780
$395$	−7.16862	−0.360693
$396$	0	0
$397$	8.65271	0.434267	0.217133	−	0.976142i	$-0.430329\pi$
$397$	8.65271	0.434267	0.217133	+	0.976142i	$0.430329\pi$
$398$	−7.92794	−0.397392
$399$	0	0
$400$	6.16489	0.308245
$401$	25.4873	1.27277	0.636387	−	0.771370i	$-0.280428\pi$
$401$	25.4873	1.27277	0.636387	+	0.771370i	$0.280428\pi$
$402$	0	0
$403$	−12.7361	−0.634432
$404$	1.32065	0.0657050
$405$	0	0
$406$	−2.98449	−0.148118
$407$	−10.5436	−0.522625
$408$	0	0
$409$	38.0003	1.87900	0.939498	−	0.342555i	$-0.111292\pi$
$409$	38.0003	1.87900	0.939498	+	0.342555i	$0.111292\pi$
$410$	2.31432	0.114296
$411$	0	0
$412$	−25.9381	−1.27788
$413$	12.5754	0.618797
$414$	0	0
$415$	11.2373	0.551616
$416$	22.2240	1.08962
$417$	0	0
$418$	−6.26660	−0.306509
$419$	4.34686	0.212358	0.106179	−	0.994347i	$-0.466138\pi$
$419$	4.34686	0.212358	0.106179	+	0.994347i	$0.466138\pi$
$420$	0	0
$421$	29.4508	1.43535	0.717673	−	0.696380i	$-0.245207\pi$
$421$	29.4508	1.43535	0.717673	+	0.696380i	$0.245207\pi$
$422$	−4.83441	−0.235335
$423$	0	0
$424$	0.347019	0.0168527
$425$	−11.7480	−0.569864
$426$	0	0
$427$	−2.37431	−0.114901
$428$	−32.8837	−1.58949
$429$	0	0
$430$	−4.54565	−0.219211
$431$	−29.6562	−1.42849	−0.714245	−	0.699895i	$-0.753230\pi$
$431$	−29.6562	−1.42849	−0.714245	+	0.699895i	$0.753230\pi$
$432$	0	0
$433$	−9.59705	−0.461205	−0.230602	−	0.973048i	$-0.574070\pi$
$433$	−9.59705	−0.461205	−0.230602	+	0.973048i	$0.574070\pi$
$434$	−1.08991	−0.0523174
$435$	0	0
$436$	13.4810	0.645625
$437$	15.3899	0.736199
$438$	0	0
$439$	−29.8729	−1.42576	−0.712878	−	0.701288i	$-0.752609\pi$
$439$	−29.8729	−1.42576	−0.712878	+	0.701288i	$0.752609\pi$
$440$	−17.1288	−0.816584
$441$	0	0
$442$	−11.8117	−0.561824
$443$	−1.94573	−0.0924444	−0.0462222	−	0.998931i	$-0.514718\pi$
$443$	−1.94573	−0.0924444	−0.0462222	+	0.998931i	$0.514718\pi$
$444$	0	0
$445$	10.8704	0.515308
$446$	5.41865	0.256581
$447$	0	0
$448$	−4.00881	−0.189398
$449$	17.6859	0.834650	0.417325	−	0.908757i	$-0.362968\pi$
$449$	17.6859	0.834650	0.417325	+	0.908757i	$0.362968\pi$
$450$	0	0
$451$	8.15843	0.384165
$452$	33.8700	1.59311
$453$	0	0
$454$	10.0268	0.470581
$455$	−13.1783	−0.617806
$456$	0	0
$457$	−38.7242	−1.81144	−0.905721	−	0.423874i	$-0.860670\pi$
$457$	−38.7242	−1.81144	−0.905721	+	0.423874i	$0.860670\pi$
$458$	−7.08144	−0.330894
$459$	0	0
$460$	20.0449	0.934599
$461$	−2.11009	−0.0982766	−0.0491383	−	0.998792i	$-0.515648\pi$
$461$	−2.11009	−0.0982766	−0.0491383	+	0.998792i	$0.515648\pi$
$462$	0	0
$463$	−16.3478	−0.759747	−0.379873	−	0.925038i	$-0.624032\pi$
$463$	−16.3478	−0.759747	−0.379873	+	0.925038i	$0.624032\pi$
$464$	−20.8193	−0.966514
$465$	0	0
$466$	−11.6858	−0.541335
$467$	−15.8333	−0.732679	−0.366340	−	0.930481i	$-0.619389\pi$
$467$	−15.8333	−0.732679	−0.366340	+	0.930481i	$0.619389\pi$
$468$	0	0
$469$	4.26602	0.196986
$470$	−8.48157	−0.391226
$471$	0	0
$472$	−20.3542	−0.936880
$473$	−16.0243	−0.736799
$474$	0	0
$475$	7.76155	0.356124
$476$	10.2528	0.469935
$477$	0	0
$478$	2.58460	0.118217
$479$	22.1361	1.01142	0.505712	−	0.862702i	$-0.331230\pi$
$479$	22.1361	1.01142	0.505712	+	0.862702i	$0.331230\pi$
$480$	0	0
$481$	13.1295	0.598655
$482$	−6.96661	−0.317321
$483$	0	0
$484$	−8.74717	−0.397599
$485$	−0.359657	−0.0163312
$486$	0	0
$487$	−0.348140	−0.0157757	−0.00788787	−	0.999969i	$-0.502511\pi$
$487$	−0.348140	−0.0157757	−0.00788787	+	0.999969i	$0.502511\pi$
$488$	3.84298	0.173964
$489$	0	0
$490$	−1.12774	−0.0509463
$491$	7.28753	0.328882	0.164441	−	0.986387i	$-0.447418\pi$
$491$	7.28753	0.328882	0.164441	+	0.986387i	$0.447418\pi$
$492$	0	0
$493$	39.6741	1.78683
$494$	7.80359	0.351100
$495$	0	0
$496$	−7.60304	−0.341386
$497$	13.7119	0.615064
$498$	0	0
$499$	−16.5620	−0.741416	−0.370708	−	0.928749i	$-0.620885\pi$
$499$	−16.5620	−0.741416	−0.370708	+	0.928749i	$0.620885\pi$
$500$	−14.1215	−0.631535
$501$	0	0
$502$	−13.2506	−0.591404
$503$	19.6862	0.877766	0.438883	−	0.898544i	$-0.355374\pi$
$503$	19.6862	0.877766	0.438883	+	0.898544i	$0.355374\pi$
$504$	0	0
$505$	1.93106	0.0859310
$506$	−6.96644	−0.309696
$507$	0	0
$508$	1.82052	0.0807725
$509$	−0.855626	−0.0379249	−0.0189625	−	0.999820i	$-0.506036\pi$
$509$	−0.855626	−0.0379249	−0.0189625	+	0.999820i	$0.506036\pi$
$510$	0	0
$511$	−10.8892	−0.481711
$512$	22.8338	1.00912
$513$	0	0
$514$	−6.55276	−0.289030
$515$	−37.9266	−1.67125
$516$	0	0
$517$	−29.8992	−1.31497
$518$	1.12358	0.0493671
$519$	0	0
$520$	21.3299	0.935379
$521$	19.7053	0.863303	0.431652	−	0.902040i	$-0.357931\pi$
$521$	19.7053	0.863303	0.431652	+	0.902040i	$0.357931\pi$
$522$	0	0
$523$	−45.5771	−1.99295	−0.996473	−	0.0839120i	$-0.973259\pi$
$523$	−45.5771	−1.99295	−0.996473	+	0.0839120i	$0.973259\pi$
$524$	21.3185	0.931303
$525$	0	0
$526$	2.31215	0.100814
$527$	14.4886	0.631134
$528$	0	0
$529$	−5.89137	−0.256147
$530$	0.241787	0.0105025
$531$	0	0
$532$	−6.77367	−0.293676
$533$	−10.1594	−0.440053
$534$	0	0
$535$	−48.0824	−2.07878
$536$	−6.90485	−0.298244
$537$	0	0
$538$	−10.2971	−0.443942
$539$	−3.97552	−0.171238
$540$	0	0
$541$	26.4890	1.13885	0.569426	−	0.822043i	$-0.307166\pi$
$541$	26.4890	1.13885	0.569426	+	0.822043i	$0.307166\pi$
$542$	11.1923	0.480751
$543$	0	0
$544$	−25.2820	−1.08396
$545$	19.7120	0.844367
$546$	0	0
$547$	−31.6604	−1.35370	−0.676850	−	0.736121i	$-0.736655\pi$
$547$	−31.6604	−1.35370	−0.676850	+	0.736121i	$0.736655\pi$
$548$	4.35372	0.185982
$549$	0	0
$550$	−3.51337	−0.149810
$551$	−26.2114	−1.11664
$552$	0	0
$553$	2.69299	0.114517
$554$	−5.82054	−0.247291
$555$	0	0
$556$	3.91047	0.165841
$557$	−9.95634	−0.421864	−0.210932	−	0.977501i	$-0.567650\pi$
$557$	−9.95634	−0.421864	−0.210932	+	0.977501i	$0.567650\pi$
$558$	0	0
$559$	19.9546	0.843988
$560$	−7.86696	−0.332440
$561$	0	0
$562$	2.14692	0.0905624
$563$	−13.7616	−0.579983	−0.289992	−	0.957029i	$-0.593653\pi$
$563$	−13.7616	−0.579983	−0.289992	+	0.957029i	$0.593653\pi$
$564$	0	0
$565$	49.5247	2.08352
$566$	−11.7081	−0.492129
$567$	0	0
$568$	−22.1937	−0.931228
$569$	7.58374	0.317927	0.158963	−	0.987284i	$-0.449185\pi$
$569$	7.58374	0.317927	0.158963	+	0.987284i	$0.449185\pi$
$570$	0	0
$571$	0.400150	0.0167458	0.00837288	−	0.999965i	$-0.497335\pi$
$571$	0.400150	0.0167458	0.00837288	+	0.999965i	$0.497335\pi$
$572$	35.8299	1.49812
$573$	0	0
$574$	−0.869404	−0.0362882
$575$	8.62835	0.359827
$576$	0	0
$577$	−18.6822	−0.777748	−0.388874	−	0.921291i	$-0.627136\pi$
$577$	−18.6822	−0.777748	−0.388874	+	0.921291i	$0.627136\pi$
$578$	6.23486	0.259336
$579$	0	0
$580$	−34.1396	−1.41757
$581$	−4.22143	−0.175135
$582$	0	0
$583$	0.852345	0.0353005
$584$	17.6250	0.729327
$585$	0	0
$586$	−11.4012	−0.470978
$587$	−35.7593	−1.47595	−0.737973	−	0.674830i	$-0.764217\pi$
$587$	−35.7593	−1.47595	−0.737973	+	0.674830i	$0.764217\pi$
$588$	0	0
$589$	−9.57216	−0.394414
$590$	−14.1819	−0.583859
$591$	0	0
$592$	7.83788	0.322135
$593$	−27.4101	−1.12560	−0.562799	−	0.826594i	$-0.690276\pi$
$593$	−27.4101	−1.12560	−0.562799	+	0.826594i	$0.690276\pi$
$594$	0	0
$595$	14.9916	0.614594
$596$	−3.74941	−0.153582
$597$	0	0
$598$	8.67508	0.354750
$599$	13.9434	0.569711	0.284856	−	0.958570i	$-0.408054\pi$
$599$	13.9434	0.569711	0.284856	+	0.958570i	$0.408054\pi$
$600$	0	0
$601$	−26.8951	−1.09708	−0.548538	−	0.836126i	$-0.684815\pi$
$601$	−26.8951	−1.09708	−0.548538	+	0.836126i	$0.684815\pi$
$602$	1.70763	0.0695980
$603$	0	0
$604$	−22.3923	−0.911131
$605$	−12.7901	−0.519991
$606$	0	0
$607$	18.7939	0.762819	0.381410	−	0.924406i	$-0.375439\pi$
$607$	18.7939	0.762819	0.381410	+	0.924406i	$0.375439\pi$
$608$	16.7030	0.677397
$609$	0	0
$610$	2.67761	0.108413
$611$	37.2325	1.50627
$612$	0	0
$613$	6.13877	0.247943	0.123971	−	0.992286i	$-0.460437\pi$
$613$	6.13877	0.247943	0.123971	+	0.992286i	$0.460437\pi$
$614$	−4.32752	−0.174644
$615$	0	0
$616$	6.43466	0.259260
$617$	16.3314	0.657477	0.328739	−	0.944421i	$-0.393376\pi$
$617$	16.3314	0.657477	0.328739	+	0.944421i	$0.393376\pi$
$618$	0	0
$619$	18.1738	0.730465	0.365233	−	0.930916i	$-0.380989\pi$
$619$	18.1738	0.730465	0.365233	+	0.930916i	$0.380989\pi$
$620$	−12.4675	−0.500705
$621$	0	0
$622$	−4.10595	−0.164634
$623$	−4.08362	−0.163607
$624$	0	0
$625$	−31.0786	−1.24315
$626$	5.79759	0.231718
$627$	0	0
$628$	35.2134	1.40517
$629$	−14.9362	−0.595543
$630$	0	0
$631$	−22.8576	−0.909945	−0.454972	−	0.890506i	$-0.650351\pi$
$631$	−22.8576	−0.909945	−0.454972	+	0.890506i	$0.650351\pi$
$632$	−4.35879	−0.173383
$633$	0	0
$634$	−5.97739	−0.237393
$635$	2.66196	0.105637
$636$	0	0
$637$	4.95058	0.196149
$638$	11.8649	0.469737
$639$	0	0
$640$	28.4209	1.12343
$641$	−34.5503	−1.36465	−0.682327	−	0.731047i	$-0.739032\pi$
$641$	−34.5503	−1.36465	−0.682327	+	0.731047i	$0.739032\pi$
$642$	0	0
$643$	4.29860	0.169520	0.0847601	−	0.996401i	$-0.472988\pi$
$643$	4.29860	0.169520	0.0847601	+	0.996401i	$0.472988\pi$
$644$	−7.53014	−0.296729
$645$	0	0
$646$	−8.87735	−0.349275
$647$	17.3258	0.681147	0.340573	−	0.940218i	$-0.389379\pi$
$647$	17.3258	0.681147	0.340573	+	0.940218i	$0.389379\pi$
$648$	0	0
$649$	−49.9939	−1.96243
$650$	4.37508	0.171605
$651$	0	0
$652$	22.7210	0.889825
$653$	−29.1488	−1.14068	−0.570341	−	0.821408i	$-0.693189\pi$
$653$	−29.1488	−1.14068	−0.570341	+	0.821408i	$0.693189\pi$
$654$	0	0
$655$	31.1719	1.21799
$656$	−6.06482	−0.236791
$657$	0	0
$658$	3.18621	0.124212
$659$	35.0106	1.36382	0.681910	−	0.731436i	$-0.261150\pi$
$659$	35.0106	1.36382	0.681910	+	0.731436i	$0.261150\pi$
$660$	0	0
$661$	41.6401	1.61961	0.809806	−	0.586697i	$-0.199572\pi$
$661$	41.6401	1.61961	0.809806	+	0.586697i	$0.199572\pi$
$662$	0.474030	0.0184237
$663$	0	0
$664$	6.83269	0.265160
$665$	−9.90444	−0.384078
$666$	0	0
$667$	−29.1386	−1.12825
$668$	3.93806	0.152368
$669$	0	0
$670$	−4.81098	−0.185864
$671$	9.43910	0.364393
$672$	0	0
$673$	6.26678	0.241567	0.120783	−	0.992679i	$-0.461459\pi$
$673$	6.26678	0.241567	0.120783	+	0.992679i	$0.461459\pi$
$674$	−9.95473	−0.383442
$675$	0	0
$676$	−20.9510	−0.805809
$677$	35.5065	1.36463	0.682313	−	0.731060i	$-0.260974\pi$
$677$	35.5065	1.36463	0.682313	+	0.731060i	$0.260974\pi$
$678$	0	0
$679$	0.135110	0.00518505
$680$	−24.2649	−0.930517
$681$	0	0
$682$	4.33296	0.165918
$683$	−30.5511	−1.16900	−0.584502	−	0.811392i	$-0.698710\pi$
$683$	−30.5511	−1.16900	−0.584502	+	0.811392i	$0.698710\pi$
$684$	0	0
$685$	6.36601	0.243233
$686$	0.423652	0.0161751
$687$	0	0
$688$	11.9122	0.454148
$689$	−1.06140	−0.0404360
$690$	0	0
$691$	21.9818	0.836229	0.418114	−	0.908394i	$-0.362691\pi$
$691$	21.9818	0.836229	0.418114	+	0.908394i	$0.362691\pi$
$692$	−28.3781	−1.07877
$693$	0	0
$694$	−2.91354	−0.110596
$695$	5.71788	0.216892
$696$	0	0
$697$	11.5573	0.437766
$698$	−1.32820	−0.0502730
$699$	0	0
$700$	−3.79765	−0.143538
$701$	5.79388	0.218832	0.109416	−	0.993996i	$-0.465102\pi$
$701$	5.79388	0.218832	0.109416	+	0.993996i	$0.465102\pi$
$702$	0	0
$703$	9.86783	0.372172
$704$	15.9371	0.600652
$705$	0	0
$706$	12.4281	0.467737
$707$	−0.725428	−0.0272825
$708$	0	0
$709$	−12.2837	−0.461324	−0.230662	−	0.973034i	$-0.574089\pi$
$709$	−12.2837	−0.461324	−0.230662	+	0.973034i	$0.574089\pi$
$710$	−15.4636	−0.580337
$711$	0	0
$712$	6.60962	0.247706
$713$	−10.6412	−0.398515
$714$	0	0
$715$	52.3904	1.95929
$716$	−20.3889	−0.761969
$717$	0	0
$718$	2.93554	0.109553
$719$	25.6425	0.956302	0.478151	−	0.878278i	$-0.341307\pi$
$719$	25.6425	0.956302	0.478151	+	0.878278i	$0.341307\pi$
$720$	0	0
$721$	14.2476	0.530610
$722$	−2.18441	−0.0812953
$723$	0	0
$724$	12.2499	0.455265
$725$	−14.6954	−0.545774
$726$	0	0
$727$	41.3287	1.53280	0.766399	−	0.642365i	$-0.222047\pi$
$727$	41.3287	1.53280	0.766399	+	0.642365i	$0.222047\pi$
$728$	−8.01287	−0.296977
$729$	0	0
$730$	12.2803	0.454513
$731$	−22.7003	−0.839600
$732$	0	0
$733$	27.6156	1.02001	0.510004	−	0.860172i	$-0.329644\pi$
$733$	27.6156	1.02001	0.510004	+	0.860172i	$0.329644\pi$
$734$	8.66174	0.319711
$735$	0	0
$736$	18.5684	0.684439
$737$	−16.9596	−0.624716
$738$	0	0
$739$	−30.8185	−1.13368	−0.566839	−	0.823828i	$-0.691834\pi$
$739$	−30.8185	−1.13368	−0.566839	+	0.823828i	$0.691834\pi$
$740$	12.8526	0.472470
$741$	0	0
$742$	−0.0908303	−0.00333449
$743$	−24.3137	−0.891983	−0.445991	−	0.895037i	$-0.647149\pi$
$743$	−24.3137	−0.891983	−0.445991	+	0.895037i	$0.647149\pi$
$744$	0	0
$745$	−5.48237	−0.200859
$746$	−0.527915	−0.0193283
$747$	0	0
$748$	−40.7601	−1.49034
$749$	18.0628	0.660000
$750$	0	0
$751$	13.6195	0.496984	0.248492	−	0.968634i	$-0.420065\pi$
$751$	13.6195	0.496984	0.248492	+	0.968634i	$0.420065\pi$
$752$	22.2265	0.810517
$753$	0	0
$754$	−14.7750	−0.538073
$755$	−32.7420	−1.19160
$756$	0	0
$757$	33.4512	1.21580	0.607902	−	0.794012i	$-0.292011\pi$
$757$	33.4512	1.21580	0.607902	+	0.794012i	$0.292011\pi$
$758$	−5.03593	−0.182913
$759$	0	0
$760$	16.0310	0.581507
$761$	−18.4399	−0.668446	−0.334223	−	0.942494i	$-0.608474\pi$
$761$	−18.4399	−0.668446	−0.334223	+	0.942494i	$0.608474\pi$
$762$	0	0
$763$	−7.40505	−0.268081
$764$	42.4989	1.53755
$765$	0	0
$766$	6.19739	0.223921
$767$	62.2558	2.24793
$768$	0	0
$769$	42.6719	1.53879	0.769394	−	0.638774i	$-0.220558\pi$
$769$	42.6719	1.53879	0.769394	+	0.638774i	$0.220558\pi$
$770$	4.48337	0.161570
$771$	0	0
$772$	−3.89452	−0.140167
$773$	−17.6637	−0.635319	−0.317659	−	0.948205i	$-0.602897\pi$
$773$	−17.6637	−0.635319	−0.317659	+	0.948205i	$0.602897\pi$
$774$	0	0
$775$	−5.36663	−0.192775
$776$	−0.218685	−0.00785034
$777$	0	0
$778$	9.13590	0.327538
$779$	−7.63556	−0.273572
$780$	0	0
$781$	−54.5121	−1.95059
$782$	−9.86876	−0.352906
$783$	0	0
$784$	2.95533	0.105547
$785$	51.4890	1.83772
$786$	0	0
$787$	34.4904	1.22945	0.614724	−	0.788742i	$-0.289267\pi$
$787$	34.4904	1.22945	0.614724	+	0.788742i	$0.289267\pi$
$788$	12.7758	0.455119
$789$	0	0
$790$	−3.03700	−0.108052
$791$	−18.6046	−0.661503
$792$	0	0
$793$	−11.7542	−0.417404
$794$	3.66574	0.130092
$795$	0	0
$796$	34.0680	1.20751
$797$	11.8100	0.418333	0.209166	−	0.977880i	$-0.432925\pi$
$797$	11.8100	0.418333	0.209166	+	0.977880i	$0.432925\pi$
$798$	0	0
$799$	−42.3556	−1.49844
$800$	9.36453	0.331086
$801$	0	0
$802$	10.7977	0.381282
$803$	43.2904	1.52768
$804$	0	0
$805$	−11.0105	−0.388071
$806$	−5.39569	−0.190055
$807$	0	0
$808$	1.17416	0.0413067
$809$	−6.66413	−0.234298	−0.117149	−	0.993114i	$-0.537376\pi$
$809$	−6.66413	−0.234298	−0.117149	+	0.993114i	$0.537376\pi$
$810$	0	0
$811$	−1.22927	−0.0431654	−0.0215827	−	0.999767i	$-0.506871\pi$
$811$	−1.22927	−0.0431654	−0.0215827	+	0.999767i	$0.506871\pi$
$812$	12.8250	0.450069
$813$	0	0
$814$	−4.46680	−0.156561
$815$	33.2227	1.16374
$816$	0	0
$817$	14.9973	0.524691
$818$	16.0989	0.562885
$819$	0	0
$820$	−9.94510	−0.347298
$821$	−28.1827	−0.983582	−0.491791	−	0.870713i	$-0.663658\pi$
$821$	−28.1827	−0.983582	−0.491791	+	0.870713i	$0.663658\pi$
$822$	0	0
$823$	−24.6122	−0.857926	−0.428963	−	0.903322i	$-0.641121\pi$
$823$	−24.6122	−0.857926	−0.428963	+	0.903322i	$0.641121\pi$
$824$	−23.0608	−0.803361
$825$	0	0
$826$	5.32761	0.185371
$827$	1.75039	0.0608670	0.0304335	−	0.999537i	$-0.490311\pi$
$827$	1.75039	0.0608670	0.0304335	+	0.999537i	$0.490311\pi$
$828$	0	0
$829$	−34.2634	−1.19002	−0.595009	−	0.803719i	$-0.702851\pi$
$829$	−34.2634	−1.19002	−0.595009	+	0.803719i	$0.702851\pi$
$830$	4.76070	0.165246
$831$	0	0
$832$	−19.8459	−0.688034
$833$	−5.63178	−0.195130
$834$	0	0
$835$	5.75822	0.199271
$836$	26.9289	0.931354
$837$	0	0
$838$	1.84156	0.0636155
$839$	−35.2195	−1.21591	−0.607955	−	0.793971i	$-0.708010\pi$
$839$	−35.2195	−1.21591	−0.607955	+	0.793971i	$0.708010\pi$
$840$	0	0
$841$	20.6276	0.711296
$842$	12.4769	0.429983
$843$	0	0
$844$	20.7744	0.715086
$845$	−30.6346	−1.05386
$846$	0	0
$847$	4.80477	0.165094
$848$	−0.633617	−0.0217585
$849$	0	0
$850$	−4.97708	−0.170713
$851$	10.9699	0.376042
$852$	0	0
$853$	−0.785201	−0.0268848	−0.0134424	−	0.999910i	$-0.504279\pi$
$853$	−0.785201	−0.0268848	−0.0134424	+	0.999910i	$0.504279\pi$
$854$	−1.00588	−0.0344205
$855$	0	0
$856$	−29.2359	−0.999263
$857$	−33.4829	−1.14375	−0.571876	−	0.820340i	$-0.693784\pi$
$857$	−33.4829	−1.14375	−0.571876	+	0.820340i	$0.693784\pi$
$858$	0	0
$859$	−8.16211	−0.278488	−0.139244	−	0.990258i	$-0.544467\pi$
$859$	−8.16211	−0.278488	−0.139244	+	0.990258i	$0.544467\pi$
$860$	19.5336	0.666090
$861$	0	0
$862$	−12.5639	−0.427929
$863$	5.62343	0.191424	0.0957119	−	0.995409i	$-0.469487\pi$
$863$	5.62343	0.191424	0.0957119	+	0.995409i	$0.469487\pi$
$864$	0	0
$865$	−41.4943	−1.41085
$866$	−4.06581	−0.138162
$867$	0	0
$868$	4.68357	0.158971
$869$	−10.7060	−0.363177
$870$	0	0
$871$	21.1193	0.715599
$872$	11.9856	0.405884
$873$	0	0
$874$	6.51997	0.220541
$875$	7.75688	0.262230
$876$	0	0
$877$	2.77495	0.0937033	0.0468516	−	0.998902i	$-0.485081\pi$
$877$	2.77495	0.0937033	0.0468516	+	0.998902i	$0.485081\pi$
$878$	−12.6557	−0.427110
$879$	0	0
$880$	31.2753	1.05429
$881$	36.0262	1.21375	0.606876	−	0.794797i	$-0.292422\pi$
$881$	36.0262	1.21375	0.606876	+	0.794797i	$0.292422\pi$
$882$	0	0
$883$	9.10055	0.306258	0.153129	−	0.988206i	$-0.451065\pi$
$883$	9.10055	0.306258	0.153129	+	0.988206i	$0.451065\pi$
$884$	50.7571	1.70715
$885$	0	0
$886$	−0.824312	−0.0276933
$887$	−46.7522	−1.56979	−0.784893	−	0.619632i	$-0.787282\pi$
$887$	−46.7522	−1.56979	−0.784893	+	0.619632i	$0.787282\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	4.60528	0.154369
$891$	0	0
$892$	−23.2851	−0.779641
$893$	27.9830	0.936415
$894$	0	0
$895$	−29.8126	−0.996526
$896$	−10.6767	−0.356683
$897$	0	0
$898$	7.49268	0.250034
$899$	18.1235	0.604454
$900$	0	0
$901$	1.20744	0.0402258
$902$	3.45633	0.115083
$903$	0	0
$904$	30.1129	1.00154
$905$	17.9118	0.595409
$906$	0	0
$907$	50.9022	1.69018	0.845090	−	0.534623i	$-0.179547\pi$
$907$	50.9022	1.69018	0.845090	+	0.534623i	$0.179547\pi$
$908$	−43.0872	−1.42990
$909$	0	0
$910$	−5.58299	−0.185074
$911$	29.0201	0.961480	0.480740	−	0.876863i	$-0.340368\pi$
$911$	29.0201	0.961480	0.480740	+	0.876863i	$0.340368\pi$
$912$	0	0
$913$	16.7824	0.555416
$914$	−16.4056	−0.542649
$915$	0	0
$916$	30.4304	1.00545
$917$	−11.7101	−0.386702
$918$	0	0
$919$	−34.4150	−1.13525	−0.567623	−	0.823288i	$-0.692137\pi$
$919$	−34.4150	−1.13525	−0.567623	+	0.823288i	$0.692137\pi$
$920$	17.8214	0.587553
$921$	0	0
$922$	−0.893943	−0.0294405
$923$	67.8820	2.23436
$924$	0	0
$925$	5.53239	0.181904
$926$	−6.92578	−0.227595
$927$	0	0
$928$	−31.6248	−1.03813
$929$	−41.0030	−1.34526	−0.672632	−	0.739977i	$-0.734836\pi$
$929$	−41.0030	−1.34526	−0.672632	+	0.739977i	$0.734836\pi$
$930$	0	0
$931$	3.72073	0.121942
$932$	50.2163	1.64489
$933$	0	0
$934$	−6.70782	−0.219487
$935$	−59.5993	−1.94911
$936$	0	0
$937$	45.4052	1.48332	0.741662	−	0.670774i	$-0.234038\pi$
$937$	45.4052	1.48332	0.741662	+	0.670774i	$0.234038\pi$
$938$	1.80731	0.0590106
$939$	0	0
$940$	36.4470	1.18877
$941$	−6.54913	−0.213496	−0.106748	−	0.994286i	$-0.534044\pi$
$941$	−6.54913	−0.213496	−0.106748	+	0.994286i	$0.534044\pi$
$942$	0	0
$943$	−8.48828	−0.276417
$944$	37.1645	1.20960
$945$	0	0
$946$	−6.78874	−0.220721
$947$	30.9829	1.00681	0.503405	−	0.864051i	$-0.332081\pi$
$947$	30.9829	1.00681	0.503405	+	0.864051i	$0.332081\pi$
$948$	0	0
$949$	−53.9080	−1.74993
$950$	3.28820	0.106683
$951$	0	0
$952$	9.11543	0.295433
$953$	−51.1779	−1.65782	−0.828908	−	0.559385i	$-0.811037\pi$
$953$	−51.1779	−1.65782	−0.828908	+	0.559385i	$0.811037\pi$
$954$	0	0
$955$	62.1418	2.01086
$956$	−11.1065	−0.359211
$957$	0	0
$958$	9.37801	0.302990
$959$	−2.39147	−0.0772247
$960$	0	0
$961$	−24.3814	−0.786498
$962$	5.56236	0.179338
$963$	0	0
$964$	29.9370	0.964204
$965$	−5.69456	−0.183314
$966$	0	0
$967$	−38.9551	−1.25271	−0.626356	−	0.779538i	$-0.715454\pi$
$967$	−38.9551	−1.25271	−0.626356	+	0.779538i	$0.715454\pi$
$968$	−7.77686	−0.249958
$969$	0	0
$970$	−0.152370	−0.00489229
$971$	−42.6229	−1.36783	−0.683917	−	0.729559i	$-0.739725\pi$
$971$	−42.6229	−1.36783	−0.683917	+	0.729559i	$0.739725\pi$
$972$	0	0
$973$	−2.14800	−0.0688616
$974$	−0.147490	−0.00472589
$975$	0	0
$976$	−7.01685	−0.224604
$977$	−49.7039	−1.59017	−0.795084	−	0.606500i	$-0.792573\pi$
$977$	−49.7039	−1.59017	−0.795084	+	0.606500i	$0.792573\pi$
$978$	0	0
$979$	16.2345	0.518857
$980$	4.84615	0.154804
$981$	0	0
$982$	3.08738	0.0985222
$983$	20.5978	0.656968	0.328484	−	0.944510i	$-0.393462\pi$
$983$	20.5978	0.656968	0.328484	+	0.944510i	$0.393462\pi$
$984$	0	0
$985$	18.6808	0.595218
$986$	16.8080	0.535276
$987$	0	0
$988$	−33.5336	−1.06685
$989$	16.6722	0.530146
$990$	0	0
$991$	−40.6235	−1.29045	−0.645224	−	0.763994i	$-0.723236\pi$
$991$	−40.6235	−1.29045	−0.645224	+	0.763994i	$0.723236\pi$
$992$	−11.5491	−0.366684
$993$	0	0
$994$	5.80909	0.184253
$995$	49.8141	1.57921
$996$	0	0
$997$	−34.6749	−1.09817	−0.549083	−	0.835768i	$-0.685023\pi$
$997$	−34.6749	−1.09817	−0.549083	+	0.835768i	$0.685023\pi$
$998$	−7.01652	−0.222104
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.o.1.9		13
3.2	odd	2		2667.2.a.l.1.5	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.l.1.5	✓	13	3.2	odd	2
8001.2.a.o.1.9		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)

\(f(q)\)	\(=\)	\(q+0.423652 q^{2} -1.82052 q^{4} -2.66196 q^{5} +1.00000 q^{7} -1.61857 q^{8} +O(q^{10})\)	\(q+0.423652 q^{2} -1.82052 q^{4} -2.66196 q^{5} +1.00000 q^{7} -1.61857 q^{8} -1.12774 q^{10} -3.97552 q^{11} +4.95058 q^{13} +0.423652 q^{14} +2.95533 q^{16} -5.63178 q^{17} +3.72073 q^{19} +4.84615 q^{20} -1.68424 q^{22} +4.13626 q^{23} +2.08603 q^{25} +2.09732 q^{26} -1.82052 q^{28} -7.04468 q^{29} -2.57265 q^{31} +4.48917 q^{32} -2.38592 q^{34} -2.66196 q^{35} +2.65212 q^{37} +1.57630 q^{38} +4.30857 q^{40} -2.05217 q^{41} +4.03075 q^{43} +7.23751 q^{44} +1.75233 q^{46} +7.52083 q^{47} +1.00000 q^{49} +0.883750 q^{50} -9.01263 q^{52} -0.214398 q^{53} +10.5827 q^{55} -1.61857 q^{56} -2.98449 q^{58} +12.5754 q^{59} -2.37431 q^{61} -1.08991 q^{62} -4.00881 q^{64} -13.1783 q^{65} +4.26602 q^{67} +10.2528 q^{68} -1.12774 q^{70} +13.7119 q^{71} -10.8892 q^{73} +1.12358 q^{74} -6.77367 q^{76} -3.97552 q^{77} +2.69299 q^{79} -7.86696 q^{80} -0.869404 q^{82} -4.22143 q^{83} +14.9916 q^{85} +1.70763 q^{86} +6.43466 q^{88} -4.08362 q^{89} +4.95058 q^{91} -7.53014 q^{92} +3.18621 q^{94} -9.90444 q^{95} +0.135110 q^{97} +0.423652 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) \) 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

Properties

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