LMFDB - Embedded newform 8001.2.a.o.1.4

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			1.4
Root			$1.45503$ 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)

$f(q)$	$=$	$q-1.45503 q^{2} +0.117115 q^{4} +0.420019 q^{5} +1.00000 q^{7} +2.73966 q^{8} +O(q^{10})$	\(q-1.45503 q^{2} +0.117115 q^{4} +0.420019 q^{5} +1.00000 q^{7} +2.73966 q^{8} -0.611141 q^{10} -4.09314 q^{11} +5.95755 q^{13} -1.45503 q^{14} -4.22051 q^{16} +7.99832 q^{17} +1.20264 q^{19} +0.0491906 q^{20} +5.95565 q^{22} +0.806334 q^{23} -4.82358 q^{25} -8.66843 q^{26} +0.117115 q^{28} -5.46630 q^{29} +3.58774 q^{31} +0.661667 q^{32} -11.6378 q^{34} +0.420019 q^{35} -3.79745 q^{37} -1.74988 q^{38} +1.15071 q^{40} -9.24830 q^{41} -4.78854 q^{43} -0.479369 q^{44} -1.17324 q^{46} -10.3421 q^{47} +1.00000 q^{49} +7.01846 q^{50} +0.697719 q^{52} -3.29407 q^{53} -1.71920 q^{55} +2.73966 q^{56} +7.95363 q^{58} -5.19779 q^{59} -4.46437 q^{61} -5.22027 q^{62} +7.47828 q^{64} +2.50229 q^{65} +10.1874 q^{67} +0.936724 q^{68} -0.611141 q^{70} -8.24577 q^{71} +8.96444 q^{73} +5.52541 q^{74} +0.140847 q^{76} -4.09314 q^{77} -11.1975 q^{79} -1.77270 q^{80} +13.4566 q^{82} -14.0246 q^{83} +3.35945 q^{85} +6.96747 q^{86} -11.2138 q^{88} -12.7389 q^{89} +5.95755 q^{91} +0.0944338 q^{92} +15.0480 q^{94} +0.505131 q^{95} -8.63018 q^{97} -1.45503 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.45503	−1.02886	−0.514431	−	0.857532i	$-0.671997\pi$
$2$	−1.45503	−1.02886	−0.514431	+	0.857532i	$0.671997\pi$
$3$	0	0
$3$	0	0
$4$	0.117115	0.0585575
$5$	0.420019	0.187838	0.0939191	−	0.995580i	$-0.470060\pi$
$5$	0.420019	0.187838	0.0939191	+	0.995580i	$0.470060\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.73966	0.968615
$9$	0	0
$10$	−0.611141	−0.193260
$11$	−4.09314	−1.23413	−0.617065	−	0.786912i	$-0.711678\pi$
$11$	−4.09314	−1.23413	−0.617065	+	0.786912i	$0.711678\pi$
$12$	0	0
$13$	5.95755	1.65233	0.826164	−	0.563430i	$-0.190518\pi$
$13$	5.95755	1.65233	0.826164	+	0.563430i	$0.190518\pi$
$14$	−1.45503	−0.388873
$15$	0	0
$16$	−4.22051	−1.05513
$17$	7.99832	1.93988	0.969939	−	0.243348i	$-0.0782457\pi$
$17$	7.99832	1.93988	0.969939	+	0.243348i	$0.0782457\pi$
$18$	0	0
$19$	1.20264	0.275904	0.137952	−	0.990439i	$-0.455948\pi$
$19$	1.20264	0.275904	0.137952	+	0.990439i	$0.455948\pi$
$20$	0.0491906	0.0109993
$21$	0	0
$22$	5.95565	1.26975
$23$	0.806334	0.168132	0.0840661	−	0.996460i	$-0.473209\pi$
$23$	0.806334	0.168132	0.0840661	+	0.996460i	$0.473209\pi$
$24$	0	0
$25$	−4.82358	−0.964717
$26$	−8.66843	−1.70002
$27$	0	0
$28$	0.117115	0.0221327
$29$	−5.46630	−1.01507	−0.507533	−	0.861632i	$-0.669442\pi$
$29$	−5.46630	−1.01507	−0.507533	+	0.861632i	$0.669442\pi$
$30$	0	0
$31$	3.58774	0.644376	0.322188	−	0.946676i	$-0.395582\pi$
$31$	3.58774	0.644376	0.322188	+	0.946676i	$0.395582\pi$
$32$	0.661667	0.116967
$33$	0	0
$34$	−11.6378	−1.99587
$35$	0.420019	0.0709962
$36$	0	0
$37$	−3.79745	−0.624297	−0.312149	−	0.950033i	$-0.601049\pi$
$37$	−3.79745	−0.624297	−0.312149	+	0.950033i	$0.601049\pi$
$38$	−1.74988	−0.283867
$39$	0	0
$40$	1.15071	0.181943
$41$	−9.24830	−1.44434	−0.722171	−	0.691715i	$-0.756855\pi$
$41$	−9.24830	−1.44434	−0.722171	+	0.691715i	$0.756855\pi$
$42$	0	0
$43$	−4.78854	−0.730245	−0.365123	−	0.930959i	$-0.618973\pi$
$43$	−4.78854	−0.730245	−0.365123	+	0.930959i	$0.618973\pi$
$44$	−0.479369	−0.0722676
$45$	0	0
$46$	−1.17324	−0.172985
$47$	−10.3421	−1.50855	−0.754273	−	0.656561i	$-0.772010\pi$
$47$	−10.3421	−1.50855	−0.754273	+	0.656561i	$0.772010\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	7.01846	0.992561
$51$	0	0
$52$	0.697719	0.0967563
$53$	−3.29407	−0.452475	−0.226238	−	0.974072i	$-0.572643\pi$
$53$	−3.29407	−0.452475	−0.226238	+	0.974072i	$0.572643\pi$
$54$	0	0
$55$	−1.71920	−0.231817
$56$	2.73966	0.366102
$57$	0	0
$58$	7.95363	1.04436
$59$	−5.19779	−0.676695	−0.338347	−	0.941021i	$-0.609868\pi$
$59$	−5.19779	−0.676695	−0.338347	+	0.941021i	$0.609868\pi$
$60$	0	0
$61$	−4.46437	−0.571605	−0.285802	−	0.958289i	$-0.592260\pi$
$61$	−4.46437	−0.571605	−0.285802	+	0.958289i	$0.592260\pi$
$62$	−5.22027	−0.662974
$63$	0	0
$64$	7.47828	0.934785
$65$	2.50229	0.310371
$66$	0	0
$67$	10.1874	1.24459	0.622297	−	0.782781i	$-0.286200\pi$
$67$	10.1874	1.24459	0.622297	+	0.782781i	$0.286200\pi$
$68$	0.936724	0.113594
$69$	0	0
$70$	−0.611141	−0.0730453
$71$	−8.24577	−0.978592	−0.489296	−	0.872118i	$-0.662746\pi$
$71$	−8.24577	−0.978592	−0.489296	+	0.872118i	$0.662746\pi$
$72$	0	0
$73$	8.96444	1.04921	0.524604	−	0.851346i	$-0.324213\pi$
$73$	8.96444	1.04921	0.524604	+	0.851346i	$0.324213\pi$
$74$	5.52541	0.642316
$75$	0	0
$76$	0.140847	0.0161563
$77$	−4.09314	−0.466457
$78$	0	0
$79$	−11.1975	−1.25982	−0.629911	−	0.776667i	$-0.716909\pi$
$79$	−11.1975	−1.25982	−0.629911	+	0.776667i	$0.716909\pi$
$80$	−1.77270	−0.198194
$81$	0	0
$82$	13.4566	1.48603
$83$	−14.0246	−1.53940	−0.769699	−	0.638407i	$-0.779594\pi$
$83$	−14.0246	−1.53940	−0.769699	+	0.638407i	$0.779594\pi$
$84$	0	0
$85$	3.35945	0.364383
$86$	6.96747	0.751322
$87$	0	0
$88$	−11.2138	−1.19540
$89$	−12.7389	−1.35032	−0.675162	−	0.737669i	$-0.735926\pi$
$89$	−12.7389	−1.35032	−0.675162	+	0.737669i	$0.735926\pi$
$90$	0	0
$91$	5.95755	0.624521
$92$	0.0944338	0.00984540
$93$	0	0
$94$	15.0480	1.55209
$95$	0.505131	0.0518254
$96$	0	0
$97$	−8.63018	−0.876262	−0.438131	−	0.898911i	$-0.644359\pi$
$97$	−8.63018	−0.876262	−0.438131	+	0.898911i	$0.644359\pi$
$98$	−1.45503	−0.146980
$99$	0	0
$100$	−0.564914	−0.0564914
$101$	13.9999	1.39305	0.696523	−	0.717535i	$-0.254730\pi$
$101$	13.9999	1.39305	0.696523	+	0.717535i	$0.254730\pi$
$102$	0	0
$103$	18.1046	1.78390	0.891950	−	0.452133i	$-0.149337\pi$
$103$	18.1046	1.78390	0.891950	+	0.452133i	$0.149337\pi$
$104$	16.3216	1.60047
$105$	0	0
$106$	4.79297	0.465534
$107$	8.11725	0.784725	0.392362	−	0.919811i	$-0.371658\pi$
$107$	8.11725	0.784725	0.392362	+	0.919811i	$0.371658\pi$
$108$	0	0
$109$	5.74750	0.550511	0.275256	−	0.961371i	$-0.411238\pi$
$109$	5.74750	0.550511	0.275256	+	0.961371i	$0.411238\pi$
$110$	2.50149	0.238507
$111$	0	0
$112$	−4.22051	−0.398801
$113$	−14.6228	−1.37559	−0.687797	−	0.725903i	$-0.741422\pi$
$113$	−14.6228	−1.37559	−0.687797	+	0.725903i	$0.741422\pi$
$114$	0	0
$115$	0.338676	0.0315817
$116$	−0.640186	−0.0594398
$117$	0	0
$118$	7.56294	0.696225
$119$	7.99832	0.733205
$120$	0	0
$121$	5.75383	0.523075
$122$	6.49580	0.588102
$123$	0	0
$124$	0.420178	0.0377331
$125$	−4.12609	−0.369049
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−12.2045	−1.07873
$129$	0	0
$130$	−3.64091	−0.319329
$131$	8.56457	0.748290	0.374145	−	0.927370i	$-0.377936\pi$
$131$	8.56457	0.748290	0.374145	+	0.927370i	$0.377936\pi$
$132$	0	0
$133$	1.20264	0.104282
$134$	−14.8230	−1.28052
$135$	0	0
$136$	21.9127	1.87899
$137$	−11.6339	−0.993952	−0.496976	−	0.867764i	$-0.665556\pi$
$137$	−11.6339	−0.993952	−0.496976	+	0.867764i	$0.665556\pi$
$138$	0	0
$139$	−5.36556	−0.455101	−0.227550	−	0.973766i	$-0.573072\pi$
$139$	−5.36556	−0.455101	−0.227550	+	0.973766i	$0.573072\pi$
$140$	0.0491906	0.00415736
$141$	0	0
$142$	11.9978	1.00684
$143$	−24.3851	−2.03919
$144$	0	0
$145$	−2.29595	−0.190668
$146$	−13.0435	−1.07949
$147$	0	0
$148$	−0.444739	−0.0365573
$149$	2.26030	0.185171	0.0925856	−	0.995705i	$-0.470487\pi$
$149$	2.26030	0.185171	0.0925856	+	0.995705i	$0.470487\pi$
$150$	0	0
$151$	5.84583	0.475727	0.237863	−	0.971299i	$-0.423553\pi$
$151$	5.84583	0.475727	0.237863	+	0.971299i	$0.423553\pi$
$152$	3.29482	0.267245
$153$	0	0
$154$	5.95565	0.479920
$155$	1.50692	0.121039
$156$	0	0
$157$	−2.77780	−0.221692	−0.110846	−	0.993838i	$-0.535356\pi$
$157$	−2.77780	−0.221692	−0.110846	+	0.993838i	$0.535356\pi$
$158$	16.2928	1.29618
$159$	0	0
$160$	0.277913	0.0219709
$161$	0.806334	0.0635480
$162$	0	0
$163$	−7.22816	−0.566153	−0.283076	−	0.959097i	$-0.591355\pi$
$163$	−7.22816	−0.566153	−0.283076	+	0.959097i	$0.591355\pi$
$164$	−1.08311	−0.0845770
$165$	0	0
$166$	20.4062	1.58383
$167$	−19.3352	−1.49620	−0.748102	−	0.663584i	$-0.769035\pi$
$167$	−19.3352	−1.49620	−0.748102	+	0.663584i	$0.769035\pi$
$168$	0	0
$169$	22.4925	1.73019
$170$	−4.88810	−0.374900
$171$	0	0
$172$	−0.560810	−0.0427614
$173$	−10.9047	−0.829067	−0.414533	−	0.910034i	$-0.636055\pi$
$173$	−10.9047	−0.829067	−0.414533	+	0.910034i	$0.636055\pi$
$174$	0	0
$175$	−4.82358	−0.364629
$176$	17.2752	1.30216
$177$	0	0
$178$	18.5355	1.38930
$179$	24.2428	1.81199	0.905995	−	0.423289i	$-0.139124\pi$
$179$	24.2428	1.81199	0.905995	+	0.423289i	$0.139124\pi$
$180$	0	0
$181$	−2.20670	−0.164023	−0.0820114	−	0.996631i	$-0.526134\pi$
$181$	−2.20670	−0.164023	−0.0820114	+	0.996631i	$0.526134\pi$
$182$	−8.66843	−0.642546
$183$	0	0
$184$	2.20908	0.162855
$185$	−1.59500	−0.117267
$186$	0	0
$187$	−32.7383	−2.39406
$188$	−1.21121	−0.0883367
$189$	0	0
$190$	−0.734982	−0.0533212
$191$	21.7204	1.57164	0.785818	−	0.618458i	$-0.212242\pi$
$191$	21.7204	1.57164	0.785818	+	0.618458i	$0.212242\pi$
$192$	0	0
$193$	−6.27244	−0.451500	−0.225750	−	0.974185i	$-0.572483\pi$
$193$	−6.27244	−0.451500	−0.225750	+	0.974185i	$0.572483\pi$
$194$	12.5572	0.901553
$195$	0	0
$196$	0.117115	0.00836536
$197$	−20.0593	−1.42917	−0.714584	−	0.699550i	$-0.753384\pi$
$197$	−20.0593	−1.42917	−0.714584	+	0.699550i	$0.753384\pi$
$198$	0	0
$199$	−0.345721	−0.0245075	−0.0122538	−	0.999925i	$-0.503901\pi$
$199$	−0.345721	−0.0245075	−0.0122538	+	0.999925i	$0.503901\pi$
$200$	−13.2150	−0.934439
$201$	0	0
$202$	−20.3703	−1.43325
$203$	−5.46630	−0.383659
$204$	0	0
$205$	−3.88446	−0.271303
$206$	−26.3428	−1.83539
$207$	0	0
$208$	−25.1439	−1.74342
$209$	−4.92257	−0.340501
$210$	0	0
$211$	−17.4577	−1.20184	−0.600919	−	0.799310i	$-0.705199\pi$
$211$	−17.4577	−1.20184	−0.600919	+	0.799310i	$0.705199\pi$
$212$	−0.385785	−0.0264958
$213$	0	0
$214$	−11.8109	−0.807374
$215$	−2.01128	−0.137168
$216$	0	0
$217$	3.58774	0.243551
$218$	−8.36280	−0.566400
$219$	0	0
$220$	−0.201344	−0.0135746
$221$	47.6504	3.20532
$222$	0	0
$223$	−8.39861	−0.562412	−0.281206	−	0.959647i	$-0.590734\pi$
$223$	−8.39861	−0.562412	−0.281206	+	0.959647i	$0.590734\pi$
$224$	0.661667	0.0442095
$225$	0	0
$226$	21.2766	1.41530
$227$	−7.77384	−0.515968	−0.257984	−	0.966149i	$-0.583058\pi$
$227$	−7.77384	−0.515968	−0.257984	+	0.966149i	$0.583058\pi$
$228$	0	0
$229$	−25.0564	−1.65577	−0.827885	−	0.560897i	$-0.810456\pi$
$229$	−25.0564	−1.65577	−0.827885	+	0.560897i	$0.810456\pi$
$230$	−0.492783	−0.0324932
$231$	0	0
$232$	−14.9758	−0.983208
$233$	30.1760	1.97689	0.988446	−	0.151571i	$-0.0484332\pi$
$233$	30.1760	1.97689	0.988446	+	0.151571i	$0.0484332\pi$
$234$	0	0
$235$	−4.34387	−0.283363
$236$	−0.608739	−0.0396256
$237$	0	0
$238$	−11.6378	−0.754367
$239$	5.76988	0.373222	0.186611	−	0.982434i	$-0.440250\pi$
$239$	5.76988	0.373222	0.186611	+	0.982434i	$0.440250\pi$
$240$	0	0
$241$	−11.9312	−0.768559	−0.384279	−	0.923217i	$-0.625550\pi$
$241$	−11.9312	−0.768559	−0.384279	+	0.923217i	$0.625550\pi$
$242$	−8.37199	−0.538172
$243$	0	0
$244$	−0.522845	−0.0334717
$245$	0.420019	0.0268340
$246$	0	0
$247$	7.16478	0.455884
$248$	9.82916	0.624152
$249$	0	0
$250$	6.00359	0.379701
$251$	−27.8947	−1.76070	−0.880350	−	0.474325i	$-0.842692\pi$
$251$	−27.8947	−1.76070	−0.880350	+	0.474325i	$0.842692\pi$
$252$	0	0
$253$	−3.30044	−0.207497
$254$	1.45503	0.0912968
$255$	0	0
$256$	2.80131	0.175082
$257$	1.37008	0.0854634	0.0427317	−	0.999087i	$-0.486394\pi$
$257$	1.37008	0.0854634	0.0427317	+	0.999087i	$0.486394\pi$
$258$	0	0
$259$	−3.79745	−0.235962
$260$	0.293055	0.0181745
$261$	0	0
$262$	−12.4617	−0.769888
$263$	−6.11351	−0.376975	−0.188488	−	0.982076i	$-0.560358\pi$
$263$	−6.11351	−0.376975	−0.188488	+	0.982076i	$0.560358\pi$
$264$	0	0
$265$	−1.38357	−0.0849921
$266$	−1.74988	−0.107292
$267$	0	0
$268$	1.19310	0.0728803
$269$	14.3108	0.872547	0.436274	−	0.899814i	$-0.356298\pi$
$269$	14.3108	0.872547	0.436274	+	0.899814i	$0.356298\pi$
$270$	0	0
$271$	−19.0355	−1.15632	−0.578162	−	0.815922i	$-0.696230\pi$
$271$	−19.0355	−1.15632	−0.578162	+	0.815922i	$0.696230\pi$
$272$	−33.7570	−2.04682
$273$	0	0
$274$	16.9277	1.02264
$275$	19.7436	1.19059
$276$	0	0
$277$	17.7110	1.06415	0.532076	−	0.846697i	$-0.321412\pi$
$277$	17.7110	1.06415	0.532076	+	0.846697i	$0.321412\pi$
$278$	7.80705	0.468236
$279$	0	0
$280$	1.15071	0.0687680
$281$	27.6078	1.64694	0.823472	−	0.567357i	$-0.192034\pi$
$281$	27.6078	1.64694	0.823472	+	0.567357i	$0.192034\pi$
$282$	0	0
$283$	11.8324	0.703360	0.351680	−	0.936120i	$-0.385610\pi$
$283$	11.8324	0.703360	0.351680	+	0.936120i	$0.385610\pi$
$284$	−0.965703	−0.0573039
$285$	0	0
$286$	35.4811	2.09804
$287$	−9.24830	−0.545910
$288$	0	0
$289$	46.9732	2.76313
$290$	3.34068	0.196171
$291$	0	0
$292$	1.04987	0.0614390
$293$	−17.9876	−1.05085	−0.525423	−	0.850841i	$-0.676093\pi$
$293$	−17.9876	−1.05085	−0.525423	+	0.850841i	$0.676093\pi$
$294$	0	0
$295$	−2.18317	−0.127109
$296$	−10.4037	−0.604703
$297$	0	0
$298$	−3.28881	−0.190516
$299$	4.80378	0.277810
$300$	0	0
$301$	−4.78854	−0.276007
$302$	−8.50586	−0.489457
$303$	0	0
$304$	−5.07575	−0.291114
$305$	−1.87512	−0.107369
$306$	0	0
$307$	13.1148	0.748500	0.374250	−	0.927328i	$-0.377900\pi$
$307$	13.1148	0.748500	0.374250	+	0.927328i	$0.377900\pi$
$308$	−0.479369	−0.0273146
$309$	0	0
$310$	−2.19261	−0.124532
$311$	9.74688	0.552695	0.276347	−	0.961058i	$-0.410876\pi$
$311$	9.74688	0.552695	0.276347	+	0.961058i	$0.410876\pi$
$312$	0	0
$313$	20.0884	1.13547	0.567733	−	0.823213i	$-0.307821\pi$
$313$	20.0884	1.13547	0.567733	+	0.823213i	$0.307821\pi$
$314$	4.04178	0.228091
$315$	0	0
$316$	−1.31140	−0.0737721
$317$	23.1449	1.29995	0.649974	−	0.759956i	$-0.274780\pi$
$317$	23.1449	1.29995	0.649974	+	0.759956i	$0.274780\pi$
$318$	0	0
$319$	22.3743	1.25272
$320$	3.14102	0.175588
$321$	0	0
$322$	−1.17324	−0.0653821
$323$	9.61909	0.535220
$324$	0	0
$325$	−28.7368	−1.59403
$326$	10.5172	0.582493
$327$	0	0
$328$	−25.3371	−1.39901
$329$	−10.3421	−0.570177
$330$	0	0
$331$	−12.8235	−0.704842	−0.352421	−	0.935841i	$-0.614642\pi$
$331$	−12.8235	−0.704842	−0.352421	+	0.935841i	$0.614642\pi$
$332$	−1.64249	−0.0901434
$333$	0	0
$334$	28.1333	1.53939
$335$	4.27892	0.233782
$336$	0	0
$337$	10.3028	0.561228	0.280614	−	0.959821i	$-0.409462\pi$
$337$	10.3028	0.561228	0.280614	+	0.959821i	$0.409462\pi$
$338$	−32.7272	−1.78013
$339$	0	0
$340$	0.393442	0.0213374
$341$	−14.6851	−0.795244
$342$	0	0
$343$	1.00000	0.0539949
$344$	−13.1189	−0.707326
$345$	0	0
$346$	15.8666	0.852996
$347$	−19.3807	−1.04041	−0.520206	−	0.854041i	$-0.674145\pi$
$347$	−19.3807	−1.04041	−0.520206	+	0.854041i	$0.674145\pi$
$348$	0	0
$349$	26.2455	1.40489	0.702446	−	0.711737i	$-0.252091\pi$
$349$	26.2455	1.40489	0.702446	+	0.711737i	$0.252091\pi$
$350$	7.01846	0.375153
$351$	0	0
$352$	−2.70830	−0.144353
$353$	−22.3694	−1.19060	−0.595300	−	0.803503i	$-0.702967\pi$
$353$	−22.3694	−1.19060	−0.595300	+	0.803503i	$0.702967\pi$
$354$	0	0
$355$	−3.46338	−0.183817
$356$	−1.49192	−0.0790716
$357$	0	0
$358$	−35.2740	−1.86429
$359$	−26.1003	−1.37752	−0.688759	−	0.724990i	$-0.741844\pi$
$359$	−26.1003	−1.37752	−0.688759	+	0.724990i	$0.741844\pi$
$360$	0	0
$361$	−17.5537	−0.923877
$362$	3.21082	0.168757
$363$	0	0
$364$	0.697719	0.0365704
$365$	3.76524	0.197081
$366$	0	0
$367$	−34.4353	−1.79751	−0.898754	−	0.438454i	$-0.855526\pi$
$367$	−34.4353	−1.79751	−0.898754	+	0.438454i	$0.855526\pi$
$368$	−3.40314	−0.177401
$369$	0	0
$370$	2.32078	0.120652
$371$	−3.29407	−0.171019
$372$	0	0
$373$	1.95485	0.101218	0.0506091	−	0.998719i	$-0.483884\pi$
$373$	1.95485	0.101218	0.0506091	+	0.998719i	$0.483884\pi$
$374$	47.6352	2.46316
$375$	0	0
$376$	−28.3337	−1.46120
$377$	−32.5658	−1.67722
$378$	0	0
$379$	−29.7272	−1.52698	−0.763492	−	0.645817i	$-0.776517\pi$
$379$	−29.7272	−1.52698	−0.763492	+	0.645817i	$0.776517\pi$
$380$	0.0591585	0.00303477
$381$	0	0
$382$	−31.6039	−1.61700
$383$	35.6298	1.82060	0.910300	−	0.413949i	$-0.135851\pi$
$383$	35.6298	1.82060	0.910300	+	0.413949i	$0.135851\pi$
$384$	0	0
$385$	−1.71920	−0.0876185
$386$	9.12660	0.464532
$387$	0	0
$388$	−1.01072	−0.0513118
$389$	28.9904	1.46987	0.734935	−	0.678138i	$-0.237213\pi$
$389$	28.9904	1.46987	0.734935	+	0.678138i	$0.237213\pi$
$390$	0	0
$391$	6.44932	0.326156
$392$	2.73966	0.138374
$393$	0	0
$394$	29.1869	1.47042
$395$	−4.70318	−0.236643
$396$	0	0
$397$	−3.59161	−0.180258	−0.0901289	−	0.995930i	$-0.528728\pi$
$397$	−3.59161	−0.180258	−0.0901289	+	0.995930i	$0.528728\pi$
$398$	0.503035	0.0252149
$399$	0	0
$400$	20.3580	1.01790
$401$	−9.61960	−0.480380	−0.240190	−	0.970726i	$-0.577210\pi$
$401$	−9.61960	−0.480380	−0.240190	+	0.970726i	$0.577210\pi$
$402$	0	0
$403$	21.3741	1.06472
$404$	1.63960	0.0815733
$405$	0	0
$406$	7.95363	0.394732
$407$	15.5435	0.770464
$408$	0	0
$409$	31.7875	1.57179	0.785894	−	0.618361i	$-0.212203\pi$
$409$	31.7875	1.57179	0.785894	+	0.618361i	$0.212203\pi$
$410$	5.65201	0.279133
$411$	0	0
$412$	2.12032	0.104461
$413$	−5.19779	−0.255766
$414$	0	0
$415$	−5.89060	−0.289158
$416$	3.94192	0.193268
$417$	0	0
$418$	7.16249	0.350329
$419$	−21.3043	−1.04078	−0.520392	−	0.853928i	$-0.674214\pi$
$419$	−21.3043	−1.04078	−0.520392	+	0.853928i	$0.674214\pi$
$420$	0	0
$421$	−25.4806	−1.24185	−0.620924	−	0.783871i	$-0.713242\pi$
$421$	−25.4806	−1.24185	−0.620924	+	0.783871i	$0.713242\pi$
$422$	25.4015	1.23653
$423$	0	0
$424$	−9.02461	−0.438274
$425$	−38.5806	−1.87143
$426$	0	0
$427$	−4.46437	−0.216046
$428$	0.950653	0.0459515
$429$	0	0
$430$	2.92647	0.141127
$431$	27.6736	1.33299	0.666495	−	0.745510i	$-0.267794\pi$
$431$	27.6736	1.33299	0.666495	+	0.745510i	$0.267794\pi$
$432$	0	0
$433$	28.1288	1.35178	0.675892	−	0.737000i	$-0.263758\pi$
$433$	28.1288	1.35178	0.675892	+	0.737000i	$0.263758\pi$
$434$	−5.22027	−0.250581
$435$	0	0
$436$	0.673119	0.0322366
$437$	0.969728	0.0463884
$438$	0	0
$439$	22.3769	1.06799	0.533995	−	0.845488i	$-0.320690\pi$
$439$	22.3769	1.06799	0.533995	+	0.845488i	$0.320690\pi$
$440$	−4.71001	−0.224541
$441$	0	0
$442$	−69.3329	−3.29783
$443$	−27.7545	−1.31866	−0.659328	−	0.751856i	$-0.729159\pi$
$443$	−27.7545	−1.31866	−0.659328	+	0.751856i	$0.729159\pi$
$444$	0	0
$445$	−5.35060	−0.253643
$446$	12.2202	0.578645
$447$	0	0
$448$	7.47828	0.353316
$449$	−0.741298	−0.0349840	−0.0174920	−	0.999847i	$-0.505568\pi$
$449$	−0.741298	−0.0349840	−0.0174920	+	0.999847i	$0.505568\pi$
$450$	0	0
$451$	37.8546	1.78250
$452$	−1.71255	−0.0805514
$453$	0	0
$454$	11.3112	0.530860
$455$	2.50229	0.117309
$456$	0	0
$457$	18.5086	0.865798	0.432899	−	0.901442i	$-0.357491\pi$
$457$	18.5086	0.865798	0.432899	+	0.901442i	$0.357491\pi$
$458$	36.4578	1.70356
$459$	0	0
$460$	0.0396640	0.00184934
$461$	40.2184	1.87316	0.936579	−	0.350456i	$-0.113973\pi$
$461$	40.2184	1.87316	0.936579	+	0.350456i	$0.113973\pi$
$462$	0	0
$463$	−33.6466	−1.56369	−0.781844	−	0.623474i	$-0.785721\pi$
$463$	−33.6466	−1.56369	−0.781844	+	0.623474i	$0.785721\pi$
$464$	23.0706	1.07103
$465$	0	0
$466$	−43.9070	−2.03395
$467$	−14.6853	−0.679554	−0.339777	−	0.940506i	$-0.610352\pi$
$467$	−14.6853	−0.679554	−0.339777	+	0.940506i	$0.610352\pi$
$468$	0	0
$469$	10.1874	0.470412
$470$	6.32046	0.291541
$471$	0	0
$472$	−14.2402	−0.655456
$473$	19.6002	0.901217
$474$	0	0
$475$	−5.80103	−0.266169
$476$	0.936724	0.0429347
$477$	0	0
$478$	−8.39535	−0.383994
$479$	−10.4227	−0.476227	−0.238113	−	0.971237i	$-0.576529\pi$
$479$	−10.4227	−0.476227	−0.238113	+	0.971237i	$0.576529\pi$
$480$	0	0
$481$	−22.6235	−1.03154
$482$	17.3603	0.790741
$483$	0	0
$484$	0.673859	0.0306300
$485$	−3.62484	−0.164596
$486$	0	0
$487$	−33.8424	−1.53355	−0.766773	−	0.641918i	$-0.778139\pi$
$487$	−33.8424	−1.53355	−0.766773	+	0.641918i	$0.778139\pi$
$488$	−12.2308	−0.553665
$489$	0	0
$490$	−0.611141	−0.0276085
$491$	−23.3096	−1.05195	−0.525975	−	0.850500i	$-0.676299\pi$
$491$	−23.3096	−1.05195	−0.525975	+	0.850500i	$0.676299\pi$
$492$	0	0
$493$	−43.7212	−1.96910
$494$	−10.4250	−0.469042
$495$	0	0
$496$	−15.1421	−0.679900
$497$	−8.24577	−0.369873
$498$	0	0
$499$	−0.990923	−0.0443598	−0.0221799	−	0.999754i	$-0.507061\pi$
$499$	−0.990923	−0.0443598	−0.0221799	+	0.999754i	$0.507061\pi$
$500$	−0.483228	−0.0216106
$501$	0	0
$502$	40.5877	1.81152
$503$	−12.6071	−0.562124	−0.281062	−	0.959690i	$-0.590687\pi$
$503$	−12.6071	−0.562124	−0.281062	+	0.959690i	$0.590687\pi$
$504$	0	0
$505$	5.88024	0.261667
$506$	4.80224	0.213486
$507$	0	0
$508$	−0.117115	−0.00519614
$509$	−32.4113	−1.43661	−0.718303	−	0.695730i	$-0.755081\pi$
$509$	−32.4113	−1.43661	−0.718303	+	0.695730i	$0.755081\pi$
$510$	0	0
$511$	8.96444	0.396563
$512$	20.3329	0.898597
$513$	0	0
$514$	−1.99351	−0.0879300
$515$	7.60429	0.335085
$516$	0	0
$517$	42.3316	1.86174
$518$	5.52541	0.242773
$519$	0	0
$520$	6.85541	0.300629
$521$	−7.88673	−0.345524	−0.172762	−	0.984964i	$-0.555269\pi$
$521$	−7.88673	−0.345524	−0.172762	+	0.984964i	$0.555269\pi$
$522$	0	0
$523$	−5.81467	−0.254258	−0.127129	−	0.991886i	$-0.540576\pi$
$523$	−5.81467	−0.254258	−0.127129	+	0.991886i	$0.540576\pi$
$524$	1.00304	0.0438180
$525$	0	0
$526$	8.89534	0.387855
$527$	28.6959	1.25001
$528$	0	0
$529$	−22.3498	−0.971732
$530$	2.01314	0.0874452
$531$	0	0
$532$	0.140847	0.00610649
$533$	−55.0972	−2.38653
$534$	0	0
$535$	3.40940	0.147401
$536$	27.9101	1.20553
$537$	0	0
$538$	−20.8227	−0.897731
$539$	−4.09314	−0.176304
$540$	0	0
$541$	11.4384	0.491777	0.245889	−	0.969298i	$-0.420920\pi$
$541$	11.4384	0.491777	0.245889	+	0.969298i	$0.420920\pi$
$542$	27.6972	1.18970
$543$	0	0
$544$	5.29223	0.226902
$545$	2.41406	0.103407
$546$	0	0
$547$	−11.5776	−0.495021	−0.247510	−	0.968885i	$-0.579612\pi$
$547$	−11.5776	−0.495021	−0.247510	+	0.968885i	$0.579612\pi$
$548$	−1.36251	−0.0582034
$549$	0	0
$550$	−28.7276	−1.22495
$551$	−6.57398	−0.280061
$552$	0	0
$553$	−11.1975	−0.476168
$554$	−25.7701	−1.09486
$555$	0	0
$556$	−0.628388	−0.0266496
$557$	4.77210	0.202201	0.101100	−	0.994876i	$-0.467764\pi$
$557$	4.77210	0.202201	0.101100	+	0.994876i	$0.467764\pi$
$558$	0	0
$559$	−28.5280	−1.20660
$560$	−1.77270	−0.0749101
$561$	0	0
$562$	−40.1702	−1.69448
$563$	24.0710	1.01447	0.507236	−	0.861807i	$-0.330667\pi$
$563$	24.0710	1.01447	0.507236	+	0.861807i	$0.330667\pi$
$564$	0	0
$565$	−6.14184	−0.258389
$566$	−17.2164	−0.723661
$567$	0	0
$568$	−22.5906	−0.947879
$569$	29.5305	1.23798	0.618991	−	0.785398i	$-0.287542\pi$
$569$	29.5305	1.23798	0.618991	+	0.785398i	$0.287542\pi$
$570$	0	0
$571$	−10.5476	−0.441402	−0.220701	−	0.975342i	$-0.570834\pi$
$571$	−10.5476	−0.441402	−0.220701	+	0.975342i	$0.570834\pi$
$572$	−2.85587	−0.119410
$573$	0	0
$574$	13.4566	0.561666
$575$	−3.88942	−0.162200
$576$	0	0
$577$	−23.4100	−0.974573	−0.487286	−	0.873242i	$-0.662013\pi$
$577$	−23.4100	−0.974573	−0.487286	+	0.873242i	$0.662013\pi$
$578$	−68.3474	−2.84288
$579$	0	0
$580$	−0.268890	−0.0111651
$581$	−14.0246	−0.581838
$582$	0	0
$583$	13.4831	0.558413
$584$	24.5595	1.01628
$585$	0	0
$586$	26.1725	1.08118
$587$	37.0926	1.53098	0.765488	−	0.643451i	$-0.222498\pi$
$587$	37.0926	1.53098	0.765488	+	0.643451i	$0.222498\pi$
$588$	0	0
$589$	4.31475	0.177786
$590$	3.17658	0.130778
$591$	0	0
$592$	16.0272	0.658714
$593$	13.9145	0.571400	0.285700	−	0.958319i	$-0.407774\pi$
$593$	13.9145	0.571400	0.285700	+	0.958319i	$0.407774\pi$
$594$	0	0
$595$	3.35945	0.137724
$596$	0.264715	0.0108432
$597$	0	0
$598$	−6.98964	−0.285828
$599$	−21.8080	−0.891051	−0.445526	−	0.895269i	$-0.646983\pi$
$599$	−21.8080	−0.891051	−0.445526	+	0.895269i	$0.646983\pi$
$600$	0	0
$601$	17.0329	0.694785	0.347393	−	0.937720i	$-0.387067\pi$
$601$	17.0329	0.694785	0.347393	+	0.937720i	$0.387067\pi$
$602$	6.96747	0.283973
$603$	0	0
$604$	0.684635	0.0278574
$605$	2.41672	0.0982535
$606$	0	0
$607$	−38.0454	−1.54422	−0.772108	−	0.635491i	$-0.780798\pi$
$607$	−38.0454	−1.54422	−0.772108	+	0.635491i	$0.780798\pi$
$608$	0.795746	0.0322718
$609$	0	0
$610$	2.72836	0.110468
$611$	−61.6134	−2.49261
$612$	0	0
$613$	16.0498	0.648245	0.324123	−	0.946015i	$-0.394931\pi$
$613$	16.0498	0.648245	0.324123	+	0.946015i	$0.394931\pi$
$614$	−19.0824	−0.770103
$615$	0	0
$616$	−11.2138	−0.451817
$617$	−28.5918	−1.15106	−0.575532	−	0.817779i	$-0.695205\pi$
$617$	−28.5918	−1.15106	−0.575532	+	0.817779i	$0.695205\pi$
$618$	0	0
$619$	−0.179381	−0.00720994	−0.00360497	−	0.999994i	$-0.501148\pi$
$619$	−0.179381	−0.00720994	−0.00360497	+	0.999994i	$0.501148\pi$
$620$	0.176483	0.00708772
$621$	0	0
$622$	−14.1820	−0.568647
$623$	−12.7389	−0.510375
$624$	0	0
$625$	22.3849	0.895395
$626$	−29.2293	−1.16824
$627$	0	0
$628$	−0.325322	−0.0129818
$629$	−30.3732	−1.21106
$630$	0	0
$631$	−29.5538	−1.17652	−0.588258	−	0.808673i	$-0.700186\pi$
$631$	−29.5538	−1.17652	−0.588258	+	0.808673i	$0.700186\pi$
$632$	−30.6774	−1.22028
$633$	0	0
$634$	−33.6766	−1.33747
$635$	−0.420019	−0.0166680
$636$	0	0
$637$	5.95755	0.236047
$638$	−32.5554	−1.28888
$639$	0	0
$640$	−5.12611	−0.202627
$641$	4.17299	0.164823	0.0824116	−	0.996598i	$-0.473738\pi$
$641$	4.17299	0.164823	0.0824116	+	0.996598i	$0.473738\pi$
$642$	0	0
$643$	−0.715991	−0.0282359	−0.0141180	−	0.999900i	$-0.504494\pi$
$643$	−0.715991	−0.0282359	−0.0141180	+	0.999900i	$0.504494\pi$
$644$	0.0944338	0.00372121
$645$	0	0
$646$	−13.9961	−0.550668
$647$	−18.4983	−0.727245	−0.363622	−	0.931546i	$-0.618460\pi$
$647$	−18.4983	−0.727245	−0.363622	+	0.931546i	$0.618460\pi$
$648$	0	0
$649$	21.2753	0.835129
$650$	41.8129	1.64004
$651$	0	0
$652$	−0.846526	−0.0331525
$653$	−24.6148	−0.963251	−0.481626	−	0.876377i	$-0.659954\pi$
$653$	−24.6148	−0.963251	−0.481626	+	0.876377i	$0.659954\pi$
$654$	0	0
$655$	3.59728	0.140558
$656$	39.0326	1.52397
$657$	0	0
$658$	15.0480	0.586633
$659$	29.6799	1.15616	0.578082	−	0.815979i	$-0.303801\pi$
$659$	29.6799	1.15616	0.578082	+	0.815979i	$0.303801\pi$
$660$	0	0
$661$	0.127223	0.00494841	0.00247420	−	0.999997i	$-0.499212\pi$
$661$	0.127223	0.00494841	0.00247420	+	0.999997i	$0.499212\pi$
$662$	18.6586	0.725186
$663$	0	0
$664$	−38.4225	−1.49108
$665$	0.505131	0.0195881
$666$	0	0
$667$	−4.40766	−0.170665
$668$	−2.26445	−0.0876140
$669$	0	0
$670$	−6.22596	−0.240530
$671$	18.2733	0.705434
$672$	0	0
$673$	18.7015	0.720891	0.360445	−	0.932780i	$-0.382625\pi$
$673$	18.7015	0.720891	0.360445	+	0.932780i	$0.382625\pi$
$674$	−14.9908	−0.577426
$675$	0	0
$676$	2.63420	0.101316
$677$	−37.6012	−1.44513	−0.722567	−	0.691301i	$-0.757038\pi$
$677$	−37.6012	−1.44513	−0.722567	+	0.691301i	$0.757038\pi$
$678$	0	0
$679$	−8.63018	−0.331196
$680$	9.20373	0.352947
$681$	0	0
$682$	21.3673	0.818196
$683$	−1.76839	−0.0676657	−0.0338328	−	0.999428i	$-0.510771\pi$
$683$	−1.76839	−0.0676657	−0.0338328	+	0.999428i	$0.510771\pi$
$684$	0	0
$685$	−4.88647	−0.186702
$686$	−1.45503	−0.0555533
$687$	0	0
$688$	20.2101	0.770503
$689$	−19.6246	−0.747637
$690$	0	0
$691$	−6.84776	−0.260501	−0.130250	−	0.991481i	$-0.541578\pi$
$691$	−6.84776	−0.260501	−0.130250	+	0.991481i	$0.541578\pi$
$692$	−1.27710	−0.0485481
$693$	0	0
$694$	28.1996	1.07044
$695$	−2.25364	−0.0854854
$696$	0	0
$697$	−73.9708	−2.80185
$698$	−38.1881	−1.44544
$699$	0	0
$700$	−0.564914	−0.0213518
$701$	27.6613	1.04475	0.522376	−	0.852715i	$-0.325046\pi$
$701$	27.6613	1.04475	0.522376	+	0.852715i	$0.325046\pi$
$702$	0	0
$703$	−4.56696	−0.172246
$704$	−30.6097	−1.15365
$705$	0	0
$706$	32.5481	1.22496
$707$	13.9999	0.526522
$708$	0	0
$709$	45.0037	1.69015	0.845075	−	0.534648i	$-0.179556\pi$
$709$	45.0037	1.69015	0.845075	+	0.534648i	$0.179556\pi$
$710$	5.03932	0.189122
$711$	0	0
$712$	−34.9003	−1.30794
$713$	2.89291	0.108340
$714$	0	0
$715$	−10.2422	−0.383037
$716$	2.83919	0.106106
$717$	0	0
$718$	37.9767	1.41728
$719$	−0.0406017	−0.00151419	−0.000757094	−	1.00000i	$-0.500241\pi$
$719$	−0.0406017	−0.00151419	−0.000757094		1.00000i	$0.500241\pi$
$720$	0	0
$721$	18.1046	0.674251
$722$	25.5411	0.950542
$723$	0	0
$724$	−0.258438	−0.00960477
$725$	26.3671	0.979251
$726$	0	0
$727$	−29.7498	−1.10336	−0.551679	−	0.834056i	$-0.686013\pi$
$727$	−29.7498	−1.10336	−0.551679	+	0.834056i	$0.686013\pi$
$728$	16.3216	0.604921
$729$	0	0
$730$	−5.47853	−0.202770
$731$	−38.3003	−1.41659
$732$	0	0
$733$	−3.84361	−0.141967	−0.0709835	−	0.997477i	$-0.522614\pi$
$733$	−3.84361	−0.141967	−0.0709835	+	0.997477i	$0.522614\pi$
$734$	50.1044	1.84939
$735$	0	0
$736$	0.533524	0.0196660
$737$	−41.6987	−1.53599
$738$	0	0
$739$	−13.5757	−0.499391	−0.249695	−	0.968324i	$-0.580331\pi$
$739$	−13.5757	−0.499391	−0.249695	+	0.968324i	$0.580331\pi$
$740$	−0.186799	−0.00686686
$741$	0	0
$742$	4.79297	0.175955
$743$	−7.62422	−0.279706	−0.139853	−	0.990172i	$-0.544663\pi$
$743$	−7.62422	−0.279706	−0.139853	+	0.990172i	$0.544663\pi$
$744$	0	0
$745$	0.949370	0.0347822
$746$	−2.84437	−0.104140
$747$	0	0
$748$	−3.83415	−0.140190
$749$	8.11725	0.296598
$750$	0	0
$751$	−38.7369	−1.41353	−0.706765	−	0.707449i	$-0.749846\pi$
$751$	−38.7369	−1.41353	−0.706765	+	0.707449i	$0.749846\pi$
$752$	43.6488	1.59171
$753$	0	0
$754$	47.3842	1.72563
$755$	2.45536	0.0893597
$756$	0	0
$757$	13.7525	0.499841	0.249921	−	0.968266i	$-0.419595\pi$
$757$	13.7525	0.499841	0.249921	+	0.968266i	$0.419595\pi$
$758$	43.2540	1.57106
$759$	0	0
$760$	1.38389	0.0501988
$761$	−29.3366	−1.06345	−0.531725	−	0.846917i	$-0.678456\pi$
$761$	−29.3366	−1.06345	−0.531725	+	0.846917i	$0.678456\pi$
$762$	0	0
$763$	5.74750	0.208074
$764$	2.54379	0.0920311
$765$	0	0
$766$	−51.8425	−1.87315
$767$	−30.9661	−1.11812
$768$	0	0
$769$	−29.4417	−1.06170	−0.530848	−	0.847467i	$-0.678127\pi$
$769$	−29.4417	−1.06170	−0.530848	+	0.847467i	$0.678127\pi$
$770$	2.50149	0.0901474
$771$	0	0
$772$	−0.734598	−0.0264387
$773$	14.9074	0.536180	0.268090	−	0.963394i	$-0.413608\pi$
$773$	14.9074	0.536180	0.268090	+	0.963394i	$0.413608\pi$
$774$	0	0
$775$	−17.3057	−0.621641
$776$	−23.6437	−0.848760
$777$	0	0
$778$	−42.1819	−1.51229
$779$	−11.1224	−0.398500
$780$	0	0
$781$	33.7511	1.20771
$782$	−9.38395	−0.335570
$783$	0	0
$784$	−4.22051	−0.150733
$785$	−1.16673	−0.0416423
$786$	0	0
$787$	−52.7973	−1.88202	−0.941010	−	0.338379i	$-0.890121\pi$
$787$	−52.7973	−1.88202	−0.941010	+	0.338379i	$0.890121\pi$
$788$	−2.34925	−0.0836885
$789$	0	0
$790$	6.84328	0.243473
$791$	−14.6228	−0.519926
$792$	0	0
$793$	−26.5968	−0.944478
$794$	5.22590	0.185460
$795$	0	0
$796$	−0.0404891	−0.00143510
$797$	19.3955	0.687023	0.343511	−	0.939148i	$-0.388384\pi$
$797$	19.3955	0.687023	0.343511	+	0.939148i	$0.388384\pi$
$798$	0	0
$799$	−82.7192	−2.92639
$800$	−3.19161	−0.112840
$801$	0	0
$802$	13.9968	0.494245
$803$	−36.6927	−1.29486
$804$	0	0
$805$	0.338676	0.0119367
$806$	−31.1000	−1.09545
$807$	0	0
$808$	38.3550	1.34932
$809$	−16.1411	−0.567490	−0.283745	−	0.958900i	$-0.591577\pi$
$809$	−16.1411	−0.567490	−0.283745	+	0.958900i	$0.591577\pi$
$810$	0	0
$811$	−29.7370	−1.04421	−0.522104	−	0.852882i	$-0.674853\pi$
$811$	−29.7370	−1.04421	−0.522104	+	0.852882i	$0.674853\pi$
$812$	−0.640186	−0.0224661
$813$	0	0
$814$	−22.6163	−0.792701
$815$	−3.03596	−0.106345
$816$	0	0
$817$	−5.75888	−0.201478
$818$	−46.2517	−1.61715
$819$	0	0
$820$	−0.454929	−0.0158868
$821$	−13.5406	−0.472570	−0.236285	−	0.971684i	$-0.575930\pi$
$821$	−13.5406	−0.472570	−0.236285	+	0.971684i	$0.575930\pi$
$822$	0	0
$823$	−20.0304	−0.698216	−0.349108	−	0.937083i	$-0.613515\pi$
$823$	−20.0304	−0.698216	−0.349108	+	0.937083i	$0.613515\pi$
$824$	49.6004	1.72791
$825$	0	0
$826$	7.56294	0.263148
$827$	40.3235	1.40219	0.701094	−	0.713069i	$-0.252696\pi$
$827$	40.3235	1.40219	0.701094	+	0.713069i	$0.252696\pi$
$828$	0	0
$829$	−49.5541	−1.72108	−0.860542	−	0.509379i	$-0.829875\pi$
$829$	−49.5541	−1.72108	−0.860542	+	0.509379i	$0.829875\pi$
$830$	8.57100	0.297504
$831$	0	0
$832$	44.5523	1.54457
$833$	7.99832	0.277125
$834$	0	0
$835$	−8.12116	−0.281044
$836$	−0.576507	−0.0199389
$837$	0	0
$838$	30.9984	1.07082
$839$	34.0265	1.17473	0.587363	−	0.809324i	$-0.300166\pi$
$839$	34.0265	1.17473	0.587363	+	0.809324i	$0.300166\pi$
$840$	0	0
$841$	0.880417	0.0303592
$842$	37.0750	1.27769
$843$	0	0
$844$	−2.04456	−0.0703767
$845$	9.44726	0.324996
$846$	0	0
$847$	5.75383	0.197704
$848$	13.9027	0.477419
$849$	0	0
$850$	56.1359	1.92545
$851$	−3.06201	−0.104964
$852$	0	0
$853$	27.8105	0.952213	0.476107	−	0.879388i	$-0.342048\pi$
$853$	27.8105	0.952213	0.476107	+	0.879388i	$0.342048\pi$
$854$	6.49580	0.222282
$855$	0	0
$856$	22.2385	0.760096
$857$	55.1687	1.88453	0.942263	−	0.334874i	$-0.108694\pi$
$857$	55.1687	1.88453	0.942263	+	0.334874i	$0.108694\pi$
$858$	0	0
$859$	14.0090	0.477980	0.238990	−	0.971022i	$-0.423184\pi$
$859$	14.0090	0.477980	0.238990	+	0.971022i	$0.423184\pi$
$860$	−0.235551	−0.00803222
$861$	0	0
$862$	−40.2659	−1.37146
$863$	7.64204	0.260138	0.130069	−	0.991505i	$-0.458480\pi$
$863$	7.64204	0.260138	0.130069	+	0.991505i	$0.458480\pi$
$864$	0	0
$865$	−4.58017	−0.155731
$866$	−40.9283	−1.39080
$867$	0	0
$868$	0.420178	0.0142618
$869$	45.8332	1.55478
$870$	0	0
$871$	60.6922	2.05648
$872$	15.7462	0.533233
$873$	0	0
$874$	−1.41098	−0.0477272
$875$	−4.12609	−0.139487
$876$	0	0
$877$	0.769283	0.0259768	0.0129884	−	0.999916i	$-0.495866\pi$
$877$	0.769283	0.0259768	0.0129884	+	0.999916i	$0.495866\pi$
$878$	−32.5590	−1.09881
$879$	0	0
$880$	7.25590	0.244596
$881$	3.94890	0.133042	0.0665209	−	0.997785i	$-0.478810\pi$
$881$	3.94890	0.133042	0.0665209	+	0.997785i	$0.478810\pi$
$882$	0	0
$883$	37.5238	1.26278	0.631388	−	0.775467i	$-0.282486\pi$
$883$	37.5238	1.26278	0.631388	+	0.775467i	$0.282486\pi$
$884$	5.58058	0.187695
$885$	0	0
$886$	40.3836	1.35671
$887$	−23.7485	−0.797397	−0.398698	−	0.917082i	$-0.630538\pi$
$887$	−23.7485	−0.797397	−0.398698	+	0.917082i	$0.630538\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	7.78528	0.260963
$891$	0	0
$892$	−0.983604	−0.0329335
$893$	−12.4378	−0.416214
$894$	0	0
$895$	10.1824	0.340361
$896$	−12.2045	−0.407723
$897$	0	0
$898$	1.07861	0.0359937
$899$	−19.6116	−0.654085
$900$	0	0
$901$	−26.3470	−0.877746
$902$	−55.0796	−1.83395
$903$	0	0
$904$	−40.0614	−1.33242
$905$	−0.926857	−0.0308098
$906$	0	0
$907$	−25.2818	−0.839468	−0.419734	−	0.907647i	$-0.637877\pi$
$907$	−25.2818	−0.839468	−0.419734	+	0.907647i	$0.637877\pi$
$908$	−0.910434	−0.0302138
$909$	0	0
$910$	−3.64091	−0.120695
$911$	−3.77037	−0.124918	−0.0624589	−	0.998048i	$-0.519894\pi$
$911$	−3.77037	−0.124918	−0.0624589	+	0.998048i	$0.519894\pi$
$912$	0	0
$913$	57.4047	1.89982
$914$	−26.9306	−0.890787
$915$	0	0
$916$	−2.93448	−0.0969578
$917$	8.56457	0.282827
$918$	0	0
$919$	31.0902	1.02557	0.512785	−	0.858517i	$-0.328614\pi$
$919$	31.0902	1.02557	0.512785	+	0.858517i	$0.328614\pi$
$920$	0.927855	0.0305905
$921$	0	0
$922$	−58.5190	−1.92722
$923$	−49.1246	−1.61696
$924$	0	0
$925$	18.3173	0.602270
$926$	48.9568	1.60882
$927$	0	0
$928$	−3.61687	−0.118730
$929$	−29.7655	−0.976574	−0.488287	−	0.872683i	$-0.662378\pi$
$929$	−29.7655	−0.976574	−0.488287	+	0.872683i	$0.662378\pi$
$930$	0	0
$931$	1.20264	0.0394149
$932$	3.53406	0.115762
$933$	0	0
$934$	21.3676	0.699168
$935$	−13.7507	−0.449696
$936$	0	0
$937$	31.5815	1.03172	0.515862	−	0.856672i	$-0.327472\pi$
$937$	31.5815	1.03172	0.515862	+	0.856672i	$0.327472\pi$
$938$	−14.8230	−0.483989
$939$	0	0
$940$	−0.508732	−0.0165930
$941$	7.00125	0.228234	0.114117	−	0.993467i	$-0.463596\pi$
$941$	7.00125	0.228234	0.114117	+	0.993467i	$0.463596\pi$
$942$	0	0
$943$	−7.45721	−0.242840
$944$	21.9373	0.714000
$945$	0	0
$946$	−28.5189	−0.927228
$947$	−34.3629	−1.11664	−0.558322	−	0.829624i	$-0.688555\pi$
$947$	−34.3629	−1.11664	−0.558322	+	0.829624i	$0.688555\pi$
$948$	0	0
$949$	53.4061	1.73364
$950$	8.44067	0.273852
$951$	0	0
$952$	21.9127	0.710193
$953$	−1.56016	−0.0505386	−0.0252693	−	0.999681i	$-0.508044\pi$
$953$	−1.56016	−0.0505386	−0.0252693	+	0.999681i	$0.508044\pi$
$954$	0	0
$955$	9.12300	0.295213
$956$	0.675739	0.0218550
$957$	0	0
$958$	15.1654	0.489972
$959$	−11.6339	−0.375679
$960$	0	0
$961$	−18.1282	−0.584779
$962$	32.9179	1.06132
$963$	0	0
$964$	−1.39733	−0.0450049
$965$	−2.63455	−0.0848091
$966$	0	0
$967$	30.5017	0.980868	0.490434	−	0.871478i	$-0.336838\pi$
$967$	30.5017	0.980868	0.490434	+	0.871478i	$0.336838\pi$
$968$	15.7635	0.506658
$969$	0	0
$970$	5.27426	0.169346
$971$	25.2755	0.811131	0.405565	−	0.914066i	$-0.367075\pi$
$971$	25.2755	0.811131	0.405565	+	0.914066i	$0.367075\pi$
$972$	0	0
$973$	−5.36556	−0.172012
$974$	49.2418	1.57781
$975$	0	0
$976$	18.8420	0.603116
$977$	−32.0198	−1.02440	−0.512202	−	0.858865i	$-0.671170\pi$
$977$	−32.0198	−1.02440	−0.512202	+	0.858865i	$0.671170\pi$
$978$	0	0
$979$	52.1423	1.66647
$980$	0.0491906	0.00157134
$981$	0	0
$982$	33.9162	1.08231
$983$	−44.6458	−1.42398	−0.711989	−	0.702190i	$-0.752206\pi$
$983$	−44.6458	−1.42398	−0.711989	+	0.702190i	$0.752206\pi$
$984$	0	0
$985$	−8.42530	−0.268452
$986$	63.6157	2.02594
$987$	0	0
$988$	0.839104	0.0266955
$989$	−3.86116	−0.122778
$990$	0	0
$991$	23.0161	0.731129	0.365565	−	0.930786i	$-0.380876\pi$
$991$	23.0161	0.731129	0.365565	+	0.930786i	$0.380876\pi$
$992$	2.37389	0.0753710
$993$	0	0
$994$	11.9978	0.380548
$995$	−0.145209	−0.00460345
$996$	0	0
$997$	31.6803	1.00333	0.501663	−	0.865063i	$-0.332722\pi$
$997$	31.6803	1.00333	0.501663	+	0.865063i	$0.332722\pi$
$998$	1.44182	0.0456401
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.o.1.4		13
3.2	odd	2		2667.2.a.l.1.10	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.l.1.10	✓	13	3.2	odd	2
8001.2.a.o.1.4		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)

\(f(q)\)	\(=\)	\(q-1.45503 q^{2} +0.117115 q^{4} +0.420019 q^{5} +1.00000 q^{7} +2.73966 q^{8} +O(q^{10})\)	\(q-1.45503 q^{2} +0.117115 q^{4} +0.420019 q^{5} +1.00000 q^{7} +2.73966 q^{8} -0.611141 q^{10} -4.09314 q^{11} +5.95755 q^{13} -1.45503 q^{14} -4.22051 q^{16} +7.99832 q^{17} +1.20264 q^{19} +0.0491906 q^{20} +5.95565 q^{22} +0.806334 q^{23} -4.82358 q^{25} -8.66843 q^{26} +0.117115 q^{28} -5.46630 q^{29} +3.58774 q^{31} +0.661667 q^{32} -11.6378 q^{34} +0.420019 q^{35} -3.79745 q^{37} -1.74988 q^{38} +1.15071 q^{40} -9.24830 q^{41} -4.78854 q^{43} -0.479369 q^{44} -1.17324 q^{46} -10.3421 q^{47} +1.00000 q^{49} +7.01846 q^{50} +0.697719 q^{52} -3.29407 q^{53} -1.71920 q^{55} +2.73966 q^{56} +7.95363 q^{58} -5.19779 q^{59} -4.46437 q^{61} -5.22027 q^{62} +7.47828 q^{64} +2.50229 q^{65} +10.1874 q^{67} +0.936724 q^{68} -0.611141 q^{70} -8.24577 q^{71} +8.96444 q^{73} +5.52541 q^{74} +0.140847 q^{76} -4.09314 q^{77} -11.1975 q^{79} -1.77270 q^{80} +13.4566 q^{82} -14.0246 q^{83} +3.35945 q^{85} +6.96747 q^{86} -11.2138 q^{88} -12.7389 q^{89} +5.95755 q^{91} +0.0944338 q^{92} +15.0480 q^{94} +0.505131 q^{95} -8.63018 q^{97} -1.45503 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) \) 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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