LMFDB - Embedded newform 8001.2.a.u.1.12

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:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} + \cdots + 16 $ x^18 - 6x^17 - 11x^16 + 123x^15 - 35x^14 - 982x^13 + 988x^12 + 3872x^11 - 5421x^10 - 7882x^9 + 13376x^8 + 7948x^7 - 15795x^6 - 3858x^5 + 8199x^4 + 1453x^3 - 1610x^2 - 380*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$1.52505$ 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)

$f(q)$	$=$	$q+1.52505 q^{2} +0.325779 q^{4} +1.71268 q^{5} +1.00000 q^{7} -2.55327 q^{8} +O(q^{10})$	\(q+1.52505 q^{2} +0.325779 q^{4} +1.71268 q^{5} +1.00000 q^{7} -2.55327 q^{8} +2.61192 q^{10} +1.69211 q^{11} -1.16748 q^{13} +1.52505 q^{14} -4.54543 q^{16} -5.50471 q^{17} -0.828520 q^{19} +0.557955 q^{20} +2.58056 q^{22} +6.42297 q^{23} -2.06673 q^{25} -1.78047 q^{26} +0.325779 q^{28} +2.43481 q^{29} -0.583990 q^{31} -1.82546 q^{32} -8.39496 q^{34} +1.71268 q^{35} +11.0189 q^{37} -1.26353 q^{38} -4.37293 q^{40} +0.0442353 q^{41} +12.9529 q^{43} +0.551255 q^{44} +9.79535 q^{46} -3.58595 q^{47} +1.00000 q^{49} -3.15187 q^{50} -0.380341 q^{52} +12.7616 q^{53} +2.89805 q^{55} -2.55327 q^{56} +3.71321 q^{58} -1.61982 q^{59} -5.74621 q^{61} -0.890614 q^{62} +6.30693 q^{64} -1.99952 q^{65} -0.393750 q^{67} -1.79332 q^{68} +2.61192 q^{70} -10.0573 q^{71} +7.94732 q^{73} +16.8044 q^{74} -0.269914 q^{76} +1.69211 q^{77} +3.92230 q^{79} -7.78485 q^{80} +0.0674610 q^{82} -6.36257 q^{83} -9.42780 q^{85} +19.7538 q^{86} -4.32042 q^{88} +5.77339 q^{89} -1.16748 q^{91} +2.09247 q^{92} -5.46876 q^{94} -1.41899 q^{95} +15.1754 q^{97} +1.52505 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * 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 $

$2$	1.52505	1.07837	0.539187	−	0.842186i	$-0.318732\pi$
$2$	1.52505	1.07837	0.539187	+	0.842186i	$0.318732\pi$
$3$	0	0
$3$	0	0
$4$	0.325779	0.162890
$5$	1.71268	0.765933	0.382966	−	0.923762i	$-0.374902\pi$
$5$	1.71268	0.765933	0.382966	+	0.923762i	$0.374902\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.55327	−0.902718
$9$	0	0
$10$	2.61192	0.825962
$11$	1.69211	0.510191	0.255096	−	0.966916i	$-0.417893\pi$
$11$	1.69211	0.510191	0.255096	+	0.966916i	$0.417893\pi$
$12$	0	0
$13$	−1.16748	−0.323801	−0.161901	−	0.986807i	$-0.551762\pi$
$13$	−1.16748	−0.323801	−0.161901	+	0.986807i	$0.551762\pi$
$14$	1.52505	0.407587
$15$	0	0
$16$	−4.54543	−1.13636
$17$	−5.50471	−1.33509	−0.667544	−	0.744570i	$-0.732655\pi$
$17$	−5.50471	−1.33509	−0.667544	+	0.744570i	$0.732655\pi$
$18$	0	0
$19$	−0.828520	−0.190075	−0.0950377	−	0.995474i	$-0.530297\pi$
$19$	−0.828520	−0.190075	−0.0950377	+	0.995474i	$0.530297\pi$
$20$	0.557955	0.124762
$21$	0	0
$22$	2.58056	0.550177
$23$	6.42297	1.33928	0.669641	−	0.742685i	$-0.266448\pi$
$23$	6.42297	1.33928	0.669641	+	0.742685i	$0.266448\pi$
$24$	0	0
$25$	−2.06673	−0.413347
$26$	−1.78047	−0.349179
$27$	0	0
$28$	0.325779	0.0615664
$29$	2.43481	0.452133	0.226066	−	0.974112i	$-0.427413\pi$
$29$	2.43481	0.452133	0.226066	+	0.974112i	$0.427413\pi$
$30$	0	0
$31$	−0.583990	−0.104888	−0.0524438	−	0.998624i	$-0.516701\pi$
$31$	−0.583990	−0.104888	−0.0524438	+	0.998624i	$0.516701\pi$
$32$	−1.82546	−0.322699
$33$	0	0
$34$	−8.39496	−1.43972
$35$	1.71268	0.289495
$36$	0	0
$37$	11.0189	1.81150	0.905749	−	0.423814i	$-0.139309\pi$
$37$	11.0189	1.81150	0.905749	+	0.423814i	$0.139309\pi$
$38$	−1.26353	−0.204972
$39$	0	0
$40$	−4.37293	−0.691421
$41$	0.0442353	0.00690839	0.00345419	−	0.999994i	$-0.498900\pi$
$41$	0.0442353	0.00690839	0.00345419	+	0.999994i	$0.498900\pi$
$42$	0	0
$43$	12.9529	1.97529	0.987647	−	0.156695i	$-0.0500839\pi$
$43$	12.9529	1.97529	0.987647	+	0.156695i	$0.0500839\pi$
$44$	0.551255	0.0831048
$45$	0	0
$46$	9.79535	1.44425
$47$	−3.58595	−0.523065	−0.261533	−	0.965195i	$-0.584228\pi$
$47$	−3.58595	−0.523065	−0.261533	+	0.965195i	$0.584228\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.15187	−0.445742
$51$	0	0
$52$	−0.380341	−0.0527439
$53$	12.7616	1.75295	0.876473	−	0.481451i	$-0.159890\pi$
$53$	12.7616	1.75295	0.876473	+	0.481451i	$0.159890\pi$
$54$	0	0
$55$	2.89805	0.390772
$56$	−2.55327	−0.341195
$57$	0	0
$58$	3.71321	0.487568
$59$	−1.61982	−0.210883	−0.105442	−	0.994426i	$-0.533626\pi$
$59$	−1.61982	−0.210883	−0.105442	+	0.994426i	$0.533626\pi$
$60$	0	0
$61$	−5.74621	−0.735727	−0.367864	−	0.929880i	$-0.619911\pi$
$61$	−5.74621	−0.735727	−0.367864	+	0.929880i	$0.619911\pi$
$62$	−0.890614	−0.113108
$63$	0	0
$64$	6.30693	0.788366
$65$	−1.99952	−0.248010
$66$	0	0
$67$	−0.393750	−0.0481042	−0.0240521	−	0.999711i	$-0.507657\pi$
$67$	−0.393750	−0.0481042	−0.0240521	+	0.999711i	$0.507657\pi$
$68$	−1.79332	−0.217472
$69$	0	0
$70$	2.61192	0.312184
$71$	−10.0573	−1.19358	−0.596792	−	0.802396i	$-0.703558\pi$
$71$	−10.0573	−1.19358	−0.596792	+	0.802396i	$0.703558\pi$
$72$	0	0
$73$	7.94732	0.930163	0.465081	−	0.885268i	$-0.346025\pi$
$73$	7.94732	0.930163	0.465081	+	0.885268i	$0.346025\pi$
$74$	16.8044	1.95347
$75$	0	0
$76$	−0.269914	−0.0309613
$77$	1.69211	0.192834
$78$	0	0
$79$	3.92230	0.441293	0.220647	−	0.975354i	$-0.429183\pi$
$79$	3.92230	0.441293	0.220647	+	0.975354i	$0.429183\pi$
$80$	−7.78485	−0.870373
$81$	0	0
$82$	0.0674610	0.00744982
$83$	−6.36257	−0.698382	−0.349191	−	0.937051i	$-0.613544\pi$
$83$	−6.36257	−0.698382	−0.349191	+	0.937051i	$0.613544\pi$
$84$	0	0
$85$	−9.42780	−1.02259
$86$	19.7538	2.13010
$87$	0	0
$88$	−4.32042	−0.460559
$89$	5.77339	0.611978	0.305989	−	0.952035i	$-0.401013\pi$
$89$	5.77339	0.611978	0.305989	+	0.952035i	$0.401013\pi$
$90$	0	0
$91$	−1.16748	−0.122385
$92$	2.09247	0.218155
$93$	0	0
$94$	−5.46876	−0.564060
$95$	−1.41899	−0.145585
$96$	0	0
$97$	15.1754	1.54083	0.770415	−	0.637542i	$-0.220049\pi$
$97$	15.1754	1.54083	0.770415	+	0.637542i	$0.220049\pi$
$98$	1.52505	0.154053
$99$	0	0
$100$	−0.673298	−0.0673298
$101$	16.9988	1.69144	0.845722	−	0.533624i	$-0.179170\pi$
$101$	16.9988	1.69144	0.845722	+	0.533624i	$0.179170\pi$
$102$	0	0
$103$	0.475453	0.0468478	0.0234239	−	0.999726i	$-0.492543\pi$
$103$	0.475453	0.0468478	0.0234239	+	0.999726i	$0.492543\pi$
$104$	2.98090	0.292301
$105$	0	0
$106$	19.4621	1.89033
$107$	16.9210	1.63582	0.817908	−	0.575349i	$-0.195134\pi$
$107$	16.9210	1.63582	0.817908	+	0.575349i	$0.195134\pi$
$108$	0	0
$109$	3.98507	0.381700	0.190850	−	0.981619i	$-0.438876\pi$
$109$	3.98507	0.381700	0.190850	+	0.981619i	$0.438876\pi$
$110$	4.41967	0.421399
$111$	0	0
$112$	−4.54543	−0.429502
$113$	18.6198	1.75160	0.875802	−	0.482670i	$-0.160333\pi$
$113$	18.6198	1.75160	0.875802	+	0.482670i	$0.160333\pi$
$114$	0	0
$115$	11.0005	1.02580
$116$	0.793210	0.0736477
$117$	0	0
$118$	−2.47031	−0.227411
$119$	−5.50471	−0.504616
$120$	0	0
$121$	−8.13675	−0.739705
$122$	−8.76327	−0.793389
$123$	0	0
$124$	−0.190252	−0.0170851
$125$	−12.1030	−1.08253
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	13.2693	1.17285
$129$	0	0
$130$	−3.04937	−0.267448
$131$	10.2394	0.894622	0.447311	−	0.894379i	$-0.352382\pi$
$131$	10.2394	0.894622	0.447311	+	0.894379i	$0.352382\pi$
$132$	0	0
$133$	−0.828520	−0.0718418
$134$	−0.600488	−0.0518743
$135$	0	0
$136$	14.0550	1.20521
$137$	−14.4173	−1.23175	−0.615874	−	0.787844i	$-0.711197\pi$
$137$	−14.4173	−1.23175	−0.615874	+	0.787844i	$0.711197\pi$
$138$	0	0
$139$	2.41938	0.205209	0.102605	−	0.994722i	$-0.467282\pi$
$139$	2.41938	0.205209	0.102605	+	0.994722i	$0.467282\pi$
$140$	0.557955	0.0471558
$141$	0	0
$142$	−15.3379	−1.28713
$143$	−1.97551	−0.165201
$144$	0	0
$145$	4.17004	0.346303
$146$	12.1201	1.00306
$147$	0	0
$148$	3.58973	0.295074
$149$	−13.8882	−1.13776	−0.568882	−	0.822419i	$-0.692624\pi$
$149$	−13.8882	−1.13776	−0.568882	+	0.822419i	$0.692624\pi$
$150$	0	0
$151$	15.9548	1.29839	0.649193	−	0.760624i	$-0.275107\pi$
$151$	15.9548	1.29839	0.649193	+	0.760624i	$0.275107\pi$
$152$	2.11544	0.171584
$153$	0	0
$154$	2.58056	0.207947
$155$	−1.00019	−0.0803369
$156$	0	0
$157$	−8.84694	−0.706063	−0.353031	−	0.935611i	$-0.614849\pi$
$157$	−8.84694	−0.706063	−0.353031	+	0.935611i	$0.614849\pi$
$158$	5.98171	0.475879
$159$	0	0
$160$	−3.12643	−0.247166
$161$	6.42297	0.506201
$162$	0	0
$163$	−14.1224	−1.10615	−0.553077	−	0.833130i	$-0.686547\pi$
$163$	−14.1224	−1.10615	−0.553077	+	0.833130i	$0.686547\pi$
$164$	0.0144109	0.00112530
$165$	0	0
$166$	−9.70323	−0.753117
$167$	18.7284	1.44925	0.724624	−	0.689144i	$-0.242013\pi$
$167$	18.7284	1.44925	0.724624	+	0.689144i	$0.242013\pi$
$168$	0	0
$169$	−11.6370	−0.895153
$170$	−14.3779	−1.10273
$171$	0	0
$172$	4.21977	0.321755
$173$	22.7424	1.72907	0.864535	−	0.502573i	$-0.167613\pi$
$173$	22.7424	1.72907	0.864535	+	0.502573i	$0.167613\pi$
$174$	0	0
$175$	−2.06673	−0.156230
$176$	−7.69138	−0.579759
$177$	0	0
$178$	8.80471	0.659941
$179$	−4.09159	−0.305820	−0.152910	−	0.988240i	$-0.548864\pi$
$179$	−4.09159	−0.305820	−0.152910	+	0.988240i	$0.548864\pi$
$180$	0	0
$181$	−23.9770	−1.78220	−0.891099	−	0.453810i	$-0.850065\pi$
$181$	−23.9770	−1.78220	−0.891099	+	0.453810i	$0.850065\pi$
$182$	−1.78047	−0.131977
$183$	0	0
$184$	−16.3996	−1.20899
$185$	18.8719	1.38749
$186$	0	0
$187$	−9.31460	−0.681151
$188$	−1.16823	−0.0852018
$189$	0	0
$190$	−2.16403	−0.156995
$191$	5.39625	0.390459	0.195229	−	0.980758i	$-0.437455\pi$
$191$	5.39625	0.390459	0.195229	+	0.980758i	$0.437455\pi$
$192$	0	0
$193$	−9.73030	−0.700402	−0.350201	−	0.936675i	$-0.613887\pi$
$193$	−9.73030	−0.700402	−0.350201	+	0.936675i	$0.613887\pi$
$194$	23.1433	1.66159
$195$	0	0
$196$	0.325779	0.0232699
$197$	0.142596	0.0101595	0.00507976	−	0.999987i	$-0.498383\pi$
$197$	0.142596	0.0101595	0.00507976	+	0.999987i	$0.498383\pi$
$198$	0	0
$199$	−11.8974	−0.843386	−0.421693	−	0.906739i	$-0.638564\pi$
$199$	−11.8974	−0.843386	−0.421693	+	0.906739i	$0.638564\pi$
$200$	5.27693	0.373135
$201$	0	0
$202$	25.9240	1.82401
$203$	2.43481	0.170890
$204$	0	0
$205$	0.0757608	0.00529136
$206$	0.725090	0.0505194
$207$	0	0
$208$	5.30671	0.367954
$209$	−1.40195	−0.0969748
$210$	0	0
$211$	7.02643	0.483720	0.241860	−	0.970311i	$-0.422243\pi$
$211$	7.02643	0.483720	0.241860	+	0.970311i	$0.422243\pi$
$212$	4.15747	0.285537
$213$	0	0
$214$	25.8054	1.76402
$215$	22.1841	1.51294
$216$	0	0
$217$	−0.583990	−0.0396438
$218$	6.07743	0.411615
$219$	0	0
$220$	0.944122	0.0636527
$221$	6.42666	0.432304
$222$	0	0
$223$	−8.33132	−0.557907	−0.278953	−	0.960305i	$-0.589987\pi$
$223$	−8.33132	−0.557907	−0.278953	+	0.960305i	$0.589987\pi$
$224$	−1.82546	−0.121969
$225$	0	0
$226$	28.3961	1.88888
$227$	−7.54679	−0.500898	−0.250449	−	0.968130i	$-0.580578\pi$
$227$	−7.54679	−0.500898	−0.250449	+	0.968130i	$0.580578\pi$
$228$	0	0
$229$	−13.3110	−0.879614	−0.439807	−	0.898092i	$-0.644953\pi$
$229$	−13.3110	−0.879614	−0.439807	+	0.898092i	$0.644953\pi$
$230$	16.7763	1.10620
$231$	0	0
$232$	−6.21673	−0.408148
$233$	−26.4435	−1.73237	−0.866187	−	0.499720i	$-0.833436\pi$
$233$	−26.4435	−1.73237	−0.866187	+	0.499720i	$0.833436\pi$
$234$	0	0
$235$	−6.14159	−0.400633
$236$	−0.527705	−0.0343506
$237$	0	0
$238$	−8.39496	−0.544165
$239$	−13.5322	−0.875328	−0.437664	−	0.899139i	$-0.644194\pi$
$239$	−13.5322	−0.875328	−0.437664	+	0.899139i	$0.644194\pi$
$240$	0	0
$241$	4.26598	0.274796	0.137398	−	0.990516i	$-0.456126\pi$
$241$	4.26598	0.274796	0.137398	+	0.990516i	$0.456126\pi$
$242$	−12.4090	−0.797678
$243$	0	0
$244$	−1.87200	−0.119842
$245$	1.71268	0.109419
$246$	0	0
$247$	0.967283	0.0615467
$248$	1.49108	0.0946840
$249$	0	0
$250$	−18.4577	−1.16737
$251$	−11.6345	−0.734363	−0.367182	−	0.930149i	$-0.619677\pi$
$251$	−11.6345	−0.734363	−0.367182	+	0.930149i	$0.619677\pi$
$252$	0	0
$253$	10.8684	0.683290
$254$	1.52505	0.0956902
$255$	0	0
$256$	7.62251	0.476407
$257$	21.3143	1.32955	0.664774	−	0.747044i	$-0.268528\pi$
$257$	21.3143	1.32955	0.664774	+	0.747044i	$0.268528\pi$
$258$	0	0
$259$	11.0189	0.684682
$260$	−0.651402	−0.0403983
$261$	0	0
$262$	15.6156	0.964737
$263$	18.8019	1.15938	0.579688	−	0.814839i	$-0.303175\pi$
$263$	18.8019	1.15938	0.579688	+	0.814839i	$0.303175\pi$
$264$	0	0
$265$	21.8566	1.34264
$266$	−1.26353	−0.0774723
$267$	0	0
$268$	−0.128275	−0.00783566
$269$	19.1251	1.16608	0.583038	−	0.812445i	$-0.301864\pi$
$269$	19.1251	1.16608	0.583038	+	0.812445i	$0.301864\pi$
$270$	0	0
$271$	−24.9073	−1.51301	−0.756507	−	0.653986i	$-0.773095\pi$
$271$	−24.9073	−1.51301	−0.756507	+	0.653986i	$0.773095\pi$
$272$	25.0213	1.51714
$273$	0	0
$274$	−21.9870	−1.32829
$275$	−3.49715	−0.210886
$276$	0	0
$277$	16.2979	0.979245	0.489623	−	0.871934i	$-0.337135\pi$
$277$	16.2979	0.979245	0.489623	+	0.871934i	$0.337135\pi$
$278$	3.68968	0.221292
$279$	0	0
$280$	−4.37293	−0.261333
$281$	−18.9557	−1.13080	−0.565401	−	0.824816i	$-0.691279\pi$
$281$	−18.9557	−1.13080	−0.565401	+	0.824816i	$0.691279\pi$
$282$	0	0
$283$	−2.71943	−0.161654	−0.0808268	−	0.996728i	$-0.525756\pi$
$283$	−2.71943	−0.161654	−0.0808268	+	0.996728i	$0.525756\pi$
$284$	−3.27646	−0.194422
$285$	0	0
$286$	−3.01276	−0.178148
$287$	0.0442353	0.00261113
$288$	0	0
$289$	13.3019	0.782462
$290$	6.35953	0.373444
$291$	0	0
$292$	2.58907	0.151514
$293$	1.50441	0.0878883	0.0439442	−	0.999034i	$-0.486008\pi$
$293$	1.50441	0.0878883	0.0439442	+	0.999034i	$0.486008\pi$
$294$	0	0
$295$	−2.77424	−0.161522
$296$	−28.1343	−1.63527
$297$	0	0
$298$	−21.1802	−1.22693
$299$	−7.49870	−0.433661
$300$	0	0
$301$	12.9529	0.746591
$302$	24.3319	1.40015
$303$	0	0
$304$	3.76597	0.215993
$305$	−9.84142	−0.563518
$306$	0	0
$307$	−14.8091	−0.845197	−0.422599	−	0.906317i	$-0.638882\pi$
$307$	−14.8091	−0.845197	−0.422599	+	0.906317i	$0.638882\pi$
$308$	0.551255	0.0314107
$309$	0	0
$310$	−1.52534	−0.0866332
$311$	−18.8186	−1.06711	−0.533554	−	0.845766i	$-0.679144\pi$
$311$	−18.8186	−1.06711	−0.533554	+	0.845766i	$0.679144\pi$
$312$	0	0
$313$	−13.1089	−0.740961	−0.370480	−	0.928840i	$-0.620807\pi$
$313$	−13.1089	−0.740961	−0.370480	+	0.928840i	$0.620807\pi$
$314$	−13.4920	−0.761400
$315$	0	0
$316$	1.27780	0.0718820
$317$	26.8211	1.50642	0.753210	−	0.657780i	$-0.228504\pi$
$317$	26.8211	1.50642	0.753210	+	0.657780i	$0.228504\pi$
$318$	0	0
$319$	4.11997	0.230674
$320$	10.8017	0.603836
$321$	0	0
$322$	9.79535	0.545873
$323$	4.56076	0.253768
$324$	0	0
$325$	2.41288	0.133842
$326$	−21.5374	−1.19285
$327$	0	0
$328$	−0.112945	−0.00623633
$329$	−3.58595	−0.197700
$330$	0	0
$331$	1.70474	0.0937009	0.0468505	−	0.998902i	$-0.485082\pi$
$331$	1.70474	0.0937009	0.0468505	+	0.998902i	$0.485082\pi$
$332$	−2.07279	−0.113759
$333$	0	0
$334$	28.5618	1.56283
$335$	−0.674366	−0.0368446
$336$	0	0
$337$	4.03114	0.219590	0.109795	−	0.993954i	$-0.464981\pi$
$337$	4.03114	0.219590	0.109795	+	0.993954i	$0.464981\pi$
$338$	−17.7470	−0.965309
$339$	0	0
$340$	−3.07138	−0.166569
$341$	−0.988177	−0.0535128
$342$	0	0
$343$	1.00000	0.0539949
$344$	−33.0722	−1.78313
$345$	0	0
$346$	34.6833	1.86458
$347$	28.9241	1.55272	0.776362	−	0.630287i	$-0.217063\pi$
$347$	28.9241	1.55272	0.776362	+	0.630287i	$0.217063\pi$
$348$	0	0
$349$	−28.0120	−1.49945	−0.749723	−	0.661752i	$-0.769813\pi$
$349$	−28.0120	−1.49945	−0.749723	+	0.661752i	$0.769813\pi$
$350$	−3.15187	−0.168475
$351$	0	0
$352$	−3.08889	−0.164638
$353$	13.4125	0.713874	0.356937	−	0.934129i	$-0.383821\pi$
$353$	13.4125	0.713874	0.356937	+	0.934129i	$0.383821\pi$
$354$	0	0
$355$	−17.2249	−0.914205
$356$	1.88085	0.0996848
$357$	0	0
$358$	−6.23988	−0.329788
$359$	14.1328	0.745901	0.372951	−	0.927851i	$-0.378346\pi$
$359$	14.1328	0.745901	0.372951	+	0.927851i	$0.378346\pi$
$360$	0	0
$361$	−18.3136	−0.963871
$362$	−36.5662	−1.92187
$363$	0	0
$364$	−0.380341	−0.0199353
$365$	13.6112	0.712442
$366$	0	0
$367$	−0.149960	−0.00782787	−0.00391393	−	0.999992i	$-0.501246\pi$
$367$	−0.149960	−0.00782787	−0.00391393	+	0.999992i	$0.501246\pi$
$368$	−29.1951	−1.52190
$369$	0	0
$370$	28.7805	1.49623
$371$	12.7616	0.662551
$372$	0	0
$373$	32.4335	1.67934	0.839672	−	0.543093i	$-0.182747\pi$
$373$	32.4335	1.67934	0.839672	+	0.543093i	$0.182747\pi$
$374$	−14.2052	−0.734535
$375$	0	0
$376$	9.15592	0.472180
$377$	−2.84260	−0.146401
$378$	0	0
$379$	−3.44284	−0.176847	−0.0884234	−	0.996083i	$-0.528183\pi$
$379$	−3.44284	−0.176847	−0.0884234	+	0.996083i	$0.528183\pi$
$380$	−0.462276	−0.0237143
$381$	0	0
$382$	8.22955	0.421060
$383$	33.4195	1.70766	0.853828	−	0.520556i	$-0.174275\pi$
$383$	33.4195	1.70766	0.853828	+	0.520556i	$0.174275\pi$
$384$	0	0
$385$	2.89805	0.147698
$386$	−14.8392	−0.755295
$387$	0	0
$388$	4.94383	0.250985
$389$	29.8748	1.51471	0.757356	−	0.653002i	$-0.226491\pi$
$389$	29.8748	1.51471	0.757356	+	0.653002i	$0.226491\pi$
$390$	0	0
$391$	−35.3566	−1.78806
$392$	−2.55327	−0.128960
$393$	0	0
$394$	0.217466	0.0109558
$395$	6.71764	0.338001
$396$	0	0
$397$	20.8790	1.04788	0.523942	−	0.851754i	$-0.324461\pi$
$397$	20.8790	1.04788	0.523942	+	0.851754i	$0.324461\pi$
$398$	−18.1442	−0.909486
$399$	0	0
$400$	9.39418	0.469709
$401$	26.7500	1.33583	0.667915	−	0.744237i	$-0.267187\pi$
$401$	26.7500	1.33583	0.667915	+	0.744237i	$0.267187\pi$
$402$	0	0
$403$	0.681798	0.0339628
$404$	5.53785	0.275518
$405$	0	0
$406$	3.71321	0.184283
$407$	18.6453	0.924211
$408$	0	0
$409$	19.3447	0.956535	0.478267	−	0.878214i	$-0.341265\pi$
$409$	19.3447	0.956535	0.478267	+	0.878214i	$0.341265\pi$
$410$	0.115539	0.00570607
$411$	0	0
$412$	0.154893	0.00763101
$413$	−1.61982	−0.0797063
$414$	0	0
$415$	−10.8970	−0.534914
$416$	2.13119	0.104490
$417$	0	0
$418$	−2.13804	−0.104575
$419$	2.14664	0.104870	0.0524351	−	0.998624i	$-0.483302\pi$
$419$	2.14664	0.104870	0.0524351	+	0.998624i	$0.483302\pi$
$420$	0	0
$421$	−0.674689	−0.0328823	−0.0164412	−	0.999865i	$-0.505234\pi$
$421$	−0.674689	−0.0328823	−0.0164412	+	0.999865i	$0.505234\pi$
$422$	10.7157	0.521630
$423$	0	0
$424$	−32.5839	−1.58242
$425$	11.3768	0.551855
$426$	0	0
$427$	−5.74621	−0.278079
$428$	5.51251	0.266457
$429$	0	0
$430$	33.8319	1.63152
$431$	−7.06226	−0.340177	−0.170089	−	0.985429i	$-0.554405\pi$
$431$	−7.06226	−0.340177	−0.170089	+	0.985429i	$0.554405\pi$
$432$	0	0
$433$	6.73974	0.323891	0.161946	−	0.986800i	$-0.448223\pi$
$433$	6.73974	0.323891	0.161946	+	0.986800i	$0.448223\pi$
$434$	−0.890614	−0.0427508
$435$	0	0
$436$	1.29825	0.0621749
$437$	−5.32155	−0.254564
$438$	0	0
$439$	2.65835	0.126876	0.0634380	−	0.997986i	$-0.479793\pi$
$439$	2.65835	0.126876	0.0634380	+	0.997986i	$0.479793\pi$
$440$	−7.39950	−0.352757
$441$	0	0
$442$	9.80098	0.466185
$443$	−5.02361	−0.238679	−0.119339	−	0.992854i	$-0.538078\pi$
$443$	−5.02361	−0.238679	−0.119339	+	0.992854i	$0.538078\pi$
$444$	0	0
$445$	9.88795	0.468734
$446$	−12.7057	−0.601632
$447$	0	0
$448$	6.30693	0.297975
$449$	−16.7743	−0.791626	−0.395813	−	0.918331i	$-0.629537\pi$
$449$	−16.7743	−0.791626	−0.395813	+	0.918331i	$0.629537\pi$
$450$	0	0
$451$	0.0748511	0.00352460
$452$	6.06594	0.285318
$453$	0	0
$454$	−11.5092	−0.540155
$455$	−1.99952	−0.0937391
$456$	0	0
$457$	−17.6220	−0.824320	−0.412160	−	0.911111i	$-0.635226\pi$
$457$	−17.6220	−0.824320	−0.412160	+	0.911111i	$0.635226\pi$
$458$	−20.2999	−0.948552
$459$	0	0
$460$	3.58372	0.167092
$461$	0.380299	0.0177123	0.00885614	−	0.999961i	$-0.497181\pi$
$461$	0.380299	0.0177123	0.00885614	+	0.999961i	$0.497181\pi$
$462$	0	0
$463$	−8.78153	−0.408112	−0.204056	−	0.978959i	$-0.565413\pi$
$463$	−8.78153	−0.408112	−0.204056	+	0.978959i	$0.565413\pi$
$464$	−11.0672	−0.513784
$465$	0	0
$466$	−40.3277	−1.86815
$467$	19.8468	0.918400	0.459200	−	0.888333i	$-0.348136\pi$
$467$	19.8468	0.918400	0.459200	+	0.888333i	$0.348136\pi$
$468$	0	0
$469$	−0.393750	−0.0181817
$470$	−9.36623	−0.432032
$471$	0	0
$472$	4.13585	0.190368
$473$	21.9177	1.00778
$474$	0	0
$475$	1.71233	0.0785671
$476$	−1.79332	−0.0821967
$477$	0	0
$478$	−20.6374	−0.943930
$479$	−1.39194	−0.0635993	−0.0317997	−	0.999494i	$-0.510124\pi$
$479$	−1.39194	−0.0635993	−0.0317997	+	0.999494i	$0.510124\pi$
$480$	0	0
$481$	−12.8644	−0.586566
$482$	6.50584	0.296333
$483$	0	0
$484$	−2.65078	−0.120490
$485$	25.9906	1.18017
$486$	0	0
$487$	−5.98929	−0.271401	−0.135700	−	0.990750i	$-0.543328\pi$
$487$	−5.98929	−0.271401	−0.135700	+	0.990750i	$0.543328\pi$
$488$	14.6716	0.664154
$489$	0	0
$490$	2.61192	0.117995
$491$	1.88645	0.0851344	0.0425672	−	0.999094i	$-0.486446\pi$
$491$	1.88645	0.0851344	0.0425672	+	0.999094i	$0.486446\pi$
$492$	0	0
$493$	−13.4029	−0.603637
$494$	1.47515	0.0663703
$495$	0	0
$496$	2.65448	0.119190
$497$	−10.0573	−0.451132
$498$	0	0
$499$	18.0508	0.808066	0.404033	−	0.914744i	$-0.367608\pi$
$499$	18.0508	0.808066	0.404033	+	0.914744i	$0.367608\pi$
$500$	−3.94292	−0.176333
$501$	0	0
$502$	−17.7432	−0.791918
$503$	−40.4303	−1.80270	−0.901348	−	0.433095i	$-0.857421\pi$
$503$	−40.4303	−1.80270	−0.901348	+	0.433095i	$0.857421\pi$
$504$	0	0
$505$	29.1135	1.29553
$506$	16.5748	0.736841
$507$	0	0
$508$	0.325779	0.0144541
$509$	28.8945	1.28072	0.640362	−	0.768073i	$-0.278784\pi$
$509$	28.8945	1.28072	0.640362	+	0.768073i	$0.278784\pi$
$510$	0	0
$511$	7.94732	0.351569
$512$	−14.9139	−0.659108
$513$	0	0
$514$	32.5054	1.43375
$515$	0.814298	0.0358823
$516$	0	0
$517$	−6.06784	−0.266863
$518$	16.8044	0.738343
$519$	0	0
$520$	5.10532	0.223883
$521$	11.5348	0.505348	0.252674	−	0.967551i	$-0.418690\pi$
$521$	11.5348	0.505348	0.252674	+	0.967551i	$0.418690\pi$
$522$	0	0
$523$	8.32409	0.363987	0.181993	−	0.983300i	$-0.441745\pi$
$523$	8.32409	0.363987	0.181993	+	0.983300i	$0.441745\pi$
$524$	3.33579	0.145725
$525$	0	0
$526$	28.6739	1.25024
$527$	3.21470	0.140034
$528$	0	0
$529$	18.2545	0.793674
$530$	33.3324	1.44787
$531$	0	0
$532$	−0.269914	−0.0117023
$533$	−0.0516439	−0.00223695
$534$	0	0
$535$	28.9803	1.25293
$536$	1.00535	0.0434245
$537$	0	0
$538$	29.1667	1.25746
$539$	1.69211	0.0728845
$540$	0	0
$541$	28.0228	1.20479	0.602396	−	0.798197i	$-0.294213\pi$
$541$	28.0228	1.20479	0.602396	+	0.798197i	$0.294213\pi$
$542$	−37.9850	−1.63159
$543$	0	0
$544$	10.0486	0.430832
$545$	6.82514	0.292357
$546$	0	0
$547$	0.859530	0.0367509	0.0183754	−	0.999831i	$-0.494151\pi$
$547$	0.859530	0.0367509	0.0183754	+	0.999831i	$0.494151\pi$
$548$	−4.69684	−0.200639
$549$	0	0
$550$	−5.33333	−0.227414
$551$	−2.01729	−0.0859393
$552$	0	0
$553$	3.92230	0.166793
$554$	24.8551	1.05599
$555$	0	0
$556$	0.788183	0.0334264
$557$	34.8987	1.47871	0.739354	−	0.673317i	$-0.235131\pi$
$557$	34.8987	1.47871	0.739354	+	0.673317i	$0.235131\pi$
$558$	0	0
$559$	−15.1223	−0.639603
$560$	−7.78485	−0.328970
$561$	0	0
$562$	−28.9084	−1.21943
$563$	−13.3554	−0.562865	−0.281432	−	0.959581i	$-0.590810\pi$
$563$	−13.3554	−0.562865	−0.281432	+	0.959581i	$0.590810\pi$
$564$	0	0
$565$	31.8897	1.34161
$566$	−4.14727	−0.174323
$567$	0	0
$568$	25.6790	1.07747
$569$	3.71560	0.155766	0.0778831	−	0.996962i	$-0.475184\pi$
$569$	3.71560	0.155766	0.0778831	+	0.996962i	$0.475184\pi$
$570$	0	0
$571$	−5.10153	−0.213492	−0.106746	−	0.994286i	$-0.534043\pi$
$571$	−5.10153	−0.213492	−0.106746	+	0.994286i	$0.534043\pi$
$572$	−0.643581	−0.0269095
$573$	0	0
$574$	0.0674610	0.00281577
$575$	−13.2746	−0.553587
$576$	0	0
$577$	−13.8444	−0.576349	−0.288174	−	0.957578i	$-0.593048\pi$
$577$	−13.8444	−0.576349	−0.288174	+	0.957578i	$0.593048\pi$
$578$	20.2860	0.843786
$579$	0	0
$580$	1.35851	0.0564092
$581$	−6.36257	−0.263964
$582$	0	0
$583$	21.5941	0.894338
$584$	−20.2917	−0.839675
$585$	0	0
$586$	2.29429	0.0947764
$587$	42.5391	1.75578	0.877888	−	0.478866i	$-0.158952\pi$
$587$	42.5391	1.75578	0.877888	+	0.478866i	$0.158952\pi$
$588$	0	0
$589$	0.483847	0.0199366
$590$	−4.23085	−0.174181
$591$	0	0
$592$	−50.0857	−2.05851
$593$	−14.6580	−0.601933	−0.300966	−	0.953635i	$-0.597309\pi$
$593$	−14.6580	−0.601933	−0.300966	+	0.953635i	$0.597309\pi$
$594$	0	0
$595$	−9.42780	−0.386502
$596$	−4.52448	−0.185330
$597$	0	0
$598$	−11.4359	−0.467649
$599$	−40.4992	−1.65475	−0.827377	−	0.561647i	$-0.810168\pi$
$599$	−40.4992	−1.65475	−0.827377	+	0.561647i	$0.810168\pi$
$600$	0	0
$601$	13.8489	0.564908	0.282454	−	0.959281i	$-0.408851\pi$
$601$	13.8489	0.564908	0.282454	+	0.959281i	$0.408851\pi$
$602$	19.7538	0.805104
$603$	0	0
$604$	5.19775	0.211494
$605$	−13.9356	−0.566564
$606$	0	0
$607$	−43.2955	−1.75731	−0.878656	−	0.477456i	$-0.841559\pi$
$607$	−43.2955	−1.75731	−0.878656	+	0.477456i	$0.841559\pi$
$608$	1.51243	0.0613372
$609$	0	0
$610$	−15.0087	−0.607683
$611$	4.18654	0.169369
$612$	0	0
$613$	−10.6948	−0.431959	−0.215980	−	0.976398i	$-0.569295\pi$
$613$	−10.6948	−0.431959	−0.215980	+	0.976398i	$0.569295\pi$
$614$	−22.5846	−0.911439
$615$	0	0
$616$	−4.32042	−0.174075
$617$	−14.2139	−0.572230	−0.286115	−	0.958195i	$-0.592364\pi$
$617$	−14.2139	−0.572230	−0.286115	+	0.958195i	$0.592364\pi$
$618$	0	0
$619$	−4.95554	−0.199180	−0.0995899	−	0.995029i	$-0.531753\pi$
$619$	−4.95554	−0.199180	−0.0995899	+	0.995029i	$0.531753\pi$
$620$	−0.325840	−0.0130860
$621$	0	0
$622$	−28.6994	−1.15074
$623$	5.77339	0.231306
$624$	0	0
$625$	−10.3949	−0.415798
$626$	−19.9918	−0.799032
$627$	0	0
$628$	−2.88215	−0.115010
$629$	−60.6560	−2.41851
$630$	0	0
$631$	9.86390	0.392675	0.196338	−	0.980536i	$-0.437095\pi$
$631$	9.86390	0.392675	0.196338	+	0.980536i	$0.437095\pi$
$632$	−10.0147	−0.398363
$633$	0	0
$634$	40.9035	1.62448
$635$	1.71268	0.0679656
$636$	0	0
$637$	−1.16748	−0.0462574
$638$	6.28317	0.248753
$639$	0	0
$640$	22.7261	0.898326
$641$	−22.0104	−0.869358	−0.434679	−	0.900585i	$-0.643138\pi$
$641$	−22.0104	−0.869358	−0.434679	+	0.900585i	$0.643138\pi$
$642$	0	0
$643$	34.9113	1.37677	0.688383	−	0.725347i	$-0.258321\pi$
$643$	34.9113	1.37677	0.688383	+	0.725347i	$0.258321\pi$
$644$	2.09247	0.0824548
$645$	0	0
$646$	6.95539	0.273656
$647$	−5.34394	−0.210092	−0.105046	−	0.994467i	$-0.533499\pi$
$647$	−5.34394	−0.210092	−0.105046	+	0.994467i	$0.533499\pi$
$648$	0	0
$649$	−2.74093	−0.107591
$650$	3.67976	0.144332
$651$	0	0
$652$	−4.60079	−0.180181
$653$	−17.7216	−0.693500	−0.346750	−	0.937958i	$-0.612715\pi$
$653$	−17.7216	−0.693500	−0.346750	+	0.937958i	$0.612715\pi$
$654$	0	0
$655$	17.5368	0.685220
$656$	−0.201068	−0.00785039
$657$	0	0
$658$	−5.46876	−0.213195
$659$	−47.9307	−1.86711	−0.933557	−	0.358429i	$-0.883312\pi$
$659$	−47.9307	−1.86711	−0.933557	+	0.358429i	$0.883312\pi$
$660$	0	0
$661$	−33.4838	−1.30237	−0.651184	−	0.758920i	$-0.725727\pi$
$661$	−33.4838	−1.30237	−0.651184	+	0.758920i	$0.725727\pi$
$662$	2.59981	0.101045
$663$	0	0
$664$	16.2454	0.630442
$665$	−1.41899	−0.0550260
$666$	0	0
$667$	15.6387	0.605533
$668$	6.10133	0.236067
$669$	0	0
$670$	−1.02844	−0.0397322
$671$	−9.72325	−0.375362
$672$	0	0
$673$	−25.9171	−0.999031	−0.499515	−	0.866305i	$-0.666489\pi$
$673$	−25.9171	−0.999031	−0.499515	+	0.866305i	$0.666489\pi$
$674$	6.14769	0.236800
$675$	0	0
$676$	−3.79109	−0.145811
$677$	−42.5126	−1.63389	−0.816946	−	0.576714i	$-0.804335\pi$
$677$	−42.5126	−1.63389	−0.816946	+	0.576714i	$0.804335\pi$
$678$	0	0
$679$	15.1754	0.582379
$680$	24.0717	0.923109
$681$	0	0
$682$	−1.50702	−0.0577068
$683$	33.7802	1.29257	0.646283	−	0.763098i	$-0.276323\pi$
$683$	33.7802	1.29257	0.646283	+	0.763098i	$0.276323\pi$
$684$	0	0
$685$	−24.6921	−0.943437
$686$	1.52505	0.0582267
$687$	0	0
$688$	−58.8763	−2.24464
$689$	−14.8990	−0.567607
$690$	0	0
$691$	17.2866	0.657611	0.328806	−	0.944398i	$-0.393354\pi$
$691$	17.2866	0.657611	0.328806	+	0.944398i	$0.393354\pi$
$692$	7.40899	0.281647
$693$	0	0
$694$	44.1106	1.67442
$695$	4.14362	0.157176
$696$	0	0
$697$	−0.243502	−0.00922331
$698$	−42.7196	−1.61696
$699$	0	0
$700$	−0.673298	−0.0254483
$701$	−10.5109	−0.396991	−0.198495	−	0.980102i	$-0.563605\pi$
$701$	−10.5109	−0.396991	−0.198495	+	0.980102i	$0.563605\pi$
$702$	0	0
$703$	−9.12939	−0.344321
$704$	10.6720	0.402218
$705$	0	0
$706$	20.4547	0.769823
$707$	16.9988	0.639306
$708$	0	0
$709$	−0.557852	−0.0209506	−0.0104753	−	0.999945i	$-0.503334\pi$
$709$	−0.557852	−0.0209506	−0.0104753	+	0.999945i	$0.503334\pi$
$710$	−26.2689	−0.985854
$711$	0	0
$712$	−14.7410	−0.552443
$713$	−3.75095	−0.140474
$714$	0	0
$715$	−3.38342	−0.126533
$716$	−1.33295	−0.0498148
$717$	0	0
$718$	21.5532	0.804360
$719$	−31.7055	−1.18242	−0.591209	−	0.806519i	$-0.701349\pi$
$719$	−31.7055	−1.18242	−0.591209	+	0.806519i	$0.701349\pi$
$720$	0	0
$721$	0.475453	0.0177068
$722$	−27.9291	−1.03941
$723$	0	0
$724$	−7.81121	−0.290301
$725$	−5.03210	−0.186888
$726$	0	0
$727$	−23.0045	−0.853190	−0.426595	−	0.904443i	$-0.640287\pi$
$727$	−23.0045	−0.853190	−0.426595	+	0.904443i	$0.640287\pi$
$728$	2.98090	0.110480
$729$	0	0
$730$	20.7578	0.768279
$731$	−71.3018	−2.63719
$732$	0	0
$733$	−35.5374	−1.31260	−0.656302	−	0.754498i	$-0.727880\pi$
$733$	−35.5374	−1.31260	−0.656302	+	0.754498i	$0.727880\pi$
$734$	−0.228697	−0.00844136
$735$	0	0
$736$	−11.7249	−0.432185
$737$	−0.666269	−0.0245423
$738$	0	0
$739$	30.4169	1.11890	0.559451	−	0.828863i	$-0.311012\pi$
$739$	30.4169	1.11890	0.559451	+	0.828863i	$0.311012\pi$
$740$	6.14806	0.226007
$741$	0	0
$742$	19.4621	0.714478
$743$	−43.1393	−1.58263	−0.791313	−	0.611411i	$-0.790602\pi$
$743$	−43.1393	−1.58263	−0.791313	+	0.611411i	$0.790602\pi$
$744$	0	0
$745$	−23.7860	−0.871451
$746$	49.4628	1.81096
$747$	0	0
$748$	−3.03450	−0.110952
$749$	16.9210	0.618280
$750$	0	0
$751$	−36.1332	−1.31852	−0.659260	−	0.751915i	$-0.729130\pi$
$751$	−36.1332	−1.31852	−0.659260	+	0.751915i	$0.729130\pi$
$752$	16.2997	0.594389
$753$	0	0
$754$	−4.33511	−0.157875
$755$	27.3255	0.994477
$756$	0	0
$757$	−34.0480	−1.23750	−0.618748	−	0.785590i	$-0.712360\pi$
$757$	−34.0480	−1.23750	−0.618748	+	0.785590i	$0.712360\pi$
$758$	−5.25050	−0.190707
$759$	0	0
$760$	3.62306	0.131422
$761$	−49.9432	−1.81044	−0.905219	−	0.424945i	$-0.860293\pi$
$761$	−49.9432	−1.81044	−0.905219	+	0.424945i	$0.860293\pi$
$762$	0	0
$763$	3.98507	0.144269
$764$	1.75798	0.0636016
$765$	0	0
$766$	50.9664	1.84149
$767$	1.89112	0.0682843
$768$	0	0
$769$	−40.4520	−1.45874	−0.729368	−	0.684121i	$-0.760186\pi$
$769$	−40.4520	−1.45874	−0.729368	+	0.684121i	$0.760186\pi$
$770$	4.41967	0.159274
$771$	0	0
$772$	−3.16993	−0.114088
$773$	27.5266	0.990063	0.495032	−	0.868875i	$-0.335156\pi$
$773$	27.5266	0.990063	0.495032	+	0.868875i	$0.335156\pi$
$774$	0	0
$775$	1.20695	0.0433550
$776$	−38.7470	−1.39094
$777$	0	0
$778$	45.5606	1.63343
$779$	−0.0366498	−0.00131312
$780$	0	0
$781$	−17.0181	−0.608956
$782$	−53.9206	−1.92820
$783$	0	0
$784$	−4.54543	−0.162337
$785$	−15.1520	−0.540797
$786$	0	0
$787$	−40.4724	−1.44269	−0.721343	−	0.692578i	$-0.756475\pi$
$787$	−40.4724	−1.44269	−0.721343	+	0.692578i	$0.756475\pi$
$788$	0.0464547	0.00165488
$789$	0	0
$790$	10.2447	0.364491
$791$	18.6198	0.662044
$792$	0	0
$793$	6.70861	0.238230
$794$	31.8415	1.13001
$795$	0	0
$796$	−3.87593	−0.137379
$797$	42.8646	1.51834	0.759170	−	0.650892i	$-0.225605\pi$
$797$	42.8646	1.51834	0.759170	+	0.650892i	$0.225605\pi$
$798$	0	0
$799$	19.7396	0.698338
$800$	3.77274	0.133387
$801$	0	0
$802$	40.7951	1.44052
$803$	13.4478	0.474561
$804$	0	0
$805$	11.0005	0.387716
$806$	1.03978	0.0366246
$807$	0	0
$808$	−43.4025	−1.52690
$809$	24.1197	0.848003	0.424001	−	0.905662i	$-0.360625\pi$
$809$	24.1197	0.848003	0.424001	+	0.905662i	$0.360625\pi$
$810$	0	0
$811$	−4.65144	−0.163334	−0.0816670	−	0.996660i	$-0.526024\pi$
$811$	−4.65144	−0.163334	−0.0816670	+	0.996660i	$0.526024\pi$
$812$	0.793210	0.0278362
$813$	0	0
$814$	28.4350	0.996645
$815$	−24.1872	−0.847240
$816$	0	0
$817$	−10.7317	−0.375455
$818$	29.5017	1.03150
$819$	0	0
$820$	0.0246813	0.000861908 0
$821$	−11.1947	−0.390697	−0.195349	−	0.980734i	$-0.562584\pi$
$821$	−11.1947	−0.390697	−0.195349	+	0.980734i	$0.562584\pi$
$822$	0	0
$823$	47.6002	1.65924	0.829619	−	0.558330i	$-0.188558\pi$
$823$	47.6002	1.65924	0.829619	+	0.558330i	$0.188558\pi$
$824$	−1.21396	−0.0422903
$825$	0	0
$826$	−2.47031	−0.0859532
$827$	32.7249	1.13796	0.568979	−	0.822352i	$-0.307339\pi$
$827$	32.7249	1.13796	0.568979	+	0.822352i	$0.307339\pi$
$828$	0	0
$829$	31.8180	1.10508	0.552542	−	0.833485i	$-0.313658\pi$
$829$	31.8180	1.10508	0.552542	+	0.833485i	$0.313658\pi$
$830$	−16.6185	−0.576837
$831$	0	0
$832$	−7.36323	−0.255274
$833$	−5.50471	−0.190727
$834$	0	0
$835$	32.0758	1.11003
$836$	−0.456726	−0.0157962
$837$	0	0
$838$	3.27374	0.113089
$839$	4.50320	0.155468	0.0777338	−	0.996974i	$-0.475232\pi$
$839$	4.50320	0.155468	0.0777338	+	0.996974i	$0.475232\pi$
$840$	0	0
$841$	−23.0717	−0.795576
$842$	−1.02894	−0.0354594
$843$	0	0
$844$	2.28906	0.0787928
$845$	−19.9304	−0.685627
$846$	0	0
$847$	−8.13675	−0.279582
$848$	−58.0071	−1.99197
$849$	0	0
$850$	17.3502	0.595105
$851$	70.7741	2.42611
$852$	0	0
$853$	−43.9321	−1.50421	−0.752103	−	0.659046i	$-0.770960\pi$
$853$	−43.9321	−1.50421	−0.752103	+	0.659046i	$0.770960\pi$
$854$	−8.76327	−0.299873
$855$	0	0
$856$	−43.2039	−1.47668
$857$	8.78489	0.300086	0.150043	−	0.988679i	$-0.452059\pi$
$857$	8.78489	0.300086	0.150043	+	0.988679i	$0.452059\pi$
$858$	0	0
$859$	40.0693	1.36715	0.683573	−	0.729883i	$-0.260425\pi$
$859$	40.0693	1.36715	0.683573	+	0.729883i	$0.260425\pi$
$860$	7.22711	0.246443
$861$	0	0
$862$	−10.7703	−0.366838
$863$	−38.3279	−1.30470	−0.652348	−	0.757919i	$-0.726216\pi$
$863$	−38.3279	−1.30470	−0.652348	+	0.757919i	$0.726216\pi$
$864$	0	0
$865$	38.9504	1.32435
$866$	10.2784	0.349276
$867$	0	0
$868$	−0.190252	−0.00645756
$869$	6.63698	0.225144
$870$	0	0
$871$	0.459696	0.0155762
$872$	−10.1750	−0.344567
$873$	0	0
$874$	−8.11564	−0.274516
$875$	−12.1030	−0.409157
$876$	0	0
$877$	23.4617	0.792245	0.396122	−	0.918198i	$-0.370356\pi$
$877$	23.4617	0.792245	0.396122	+	0.918198i	$0.370356\pi$
$878$	4.05412	0.136820
$879$	0	0
$880$	−13.1729	−0.444057
$881$	38.6863	1.30338	0.651688	−	0.758487i	$-0.274061\pi$
$881$	38.6863	1.30338	0.651688	+	0.758487i	$0.274061\pi$
$882$	0	0
$883$	−8.55080	−0.287757	−0.143879	−	0.989595i	$-0.545958\pi$
$883$	−8.55080	−0.287757	−0.143879	+	0.989595i	$0.545958\pi$
$884$	2.09367	0.0704177
$885$	0	0
$886$	−7.66126	−0.257385
$887$	−53.2942	−1.78944	−0.894722	−	0.446623i	$-0.852627\pi$
$887$	−53.2942	−1.78944	−0.894722	+	0.446623i	$0.852627\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	15.0796	0.505470
$891$	0	0
$892$	−2.71417	−0.0908772
$893$	2.97103	0.0994219
$894$	0	0
$895$	−7.00758	−0.234237
$896$	13.2693	0.443297
$897$	0	0
$898$	−25.5816	−0.853669
$899$	−1.42190	−0.0474231
$900$	0	0
$901$	−70.2492	−2.34034
$902$	0.114152	0.00380084
$903$	0	0
$904$	−47.5414	−1.58120
$905$	−41.0649	−1.36504
$906$	0	0
$907$	24.0792	0.799536	0.399768	−	0.916616i	$-0.369091\pi$
$907$	24.0792	0.799536	0.399768	+	0.916616i	$0.369091\pi$
$908$	−2.45859	−0.0815911
$909$	0	0
$910$	−3.04937	−0.101086
$911$	43.2252	1.43212	0.716058	−	0.698041i	$-0.245945\pi$
$911$	43.2252	1.43212	0.716058	+	0.698041i	$0.245945\pi$
$912$	0	0
$913$	−10.7662	−0.356309
$914$	−26.8744	−0.888925
$915$	0	0
$916$	−4.33644	−0.143280
$917$	10.2394	0.338135
$918$	0	0
$919$	41.5792	1.37157	0.685786	−	0.727803i	$-0.259459\pi$
$919$	41.5792	1.37157	0.685786	+	0.727803i	$0.259459\pi$
$920$	−28.0872	−0.926007
$921$	0	0
$922$	0.579975	0.0191005
$923$	11.7417	0.386484
$924$	0	0
$925$	−22.7732	−0.748777
$926$	−13.3923	−0.440098
$927$	0	0
$928$	−4.44465	−0.145903
$929$	0.613766	0.0201370	0.0100685	−	0.999949i	$-0.496795\pi$
$929$	0.613766	0.0201370	0.0100685	+	0.999949i	$0.496795\pi$
$930$	0	0
$931$	−0.828520	−0.0271536
$932$	−8.61475	−0.282185
$933$	0	0
$934$	30.2674	0.990378
$935$	−15.9529	−0.521716
$936$	0	0
$937$	46.1622	1.50805	0.754026	−	0.656845i	$-0.228109\pi$
$937$	46.1622	1.50805	0.754026	+	0.656845i	$0.228109\pi$
$938$	−0.600488	−0.0196066
$939$	0	0
$940$	−2.00080	−0.0652589
$941$	−6.85694	−0.223530	−0.111765	−	0.993735i	$-0.535650\pi$
$941$	−6.85694	−0.223530	−0.111765	+	0.993735i	$0.535650\pi$
$942$	0	0
$943$	0.284122	0.00925227
$944$	7.36279	0.239638
$945$	0	0
$946$	33.4256	1.08676
$947$	−15.9566	−0.518519	−0.259259	−	0.965808i	$-0.583478\pi$
$947$	−15.9566	−0.518519	−0.259259	+	0.965808i	$0.583478\pi$
$948$	0	0
$949$	−9.27835	−0.301188
$950$	2.61139	0.0847246
$951$	0	0
$952$	14.0550	0.455526
$953$	−3.19350	−0.103447	−0.0517237	−	0.998661i	$-0.516472\pi$
$953$	−3.19350	−0.103447	−0.0517237	+	0.998661i	$0.516472\pi$
$954$	0	0
$955$	9.24203	0.299065
$956$	−4.40852	−0.142582
$957$	0	0
$958$	−2.12278	−0.0685838
$959$	−14.4173	−0.465557
$960$	0	0
$961$	−30.6590	−0.988999
$962$	−19.6189	−0.632537
$963$	0	0
$964$	1.38977	0.0447614
$965$	−16.6649	−0.536461
$966$	0	0
$967$	−32.8839	−1.05747	−0.528737	−	0.848786i	$-0.677334\pi$
$967$	−32.8839	−1.05747	−0.528737	+	0.848786i	$0.677334\pi$
$968$	20.7753	0.667745
$969$	0	0
$970$	39.6370	1.27267
$971$	49.2552	1.58068	0.790338	−	0.612671i	$-0.209905\pi$
$971$	49.2552	1.58068	0.790338	+	0.612671i	$0.209905\pi$
$972$	0	0
$973$	2.41938	0.0775618
$974$	−9.13397	−0.292671
$975$	0	0
$976$	26.1190	0.836049
$977$	16.7606	0.536220	0.268110	−	0.963388i	$-0.413601\pi$
$977$	16.7606	0.536220	0.268110	+	0.963388i	$0.413601\pi$
$978$	0	0
$979$	9.76923	0.312226
$980$	0.557955	0.0178232
$981$	0	0
$982$	2.87694	0.0918067
$983$	49.5398	1.58007	0.790037	−	0.613059i	$-0.210061\pi$
$983$	49.5398	1.58007	0.790037	+	0.613059i	$0.210061\pi$
$984$	0	0
$985$	0.244221	0.00778151
$986$	−20.4401	−0.650946
$987$	0	0
$988$	0.315120	0.0100253
$989$	83.1958	2.64547
$990$	0	0
$991$	−31.0914	−0.987652	−0.493826	−	0.869561i	$-0.664402\pi$
$991$	−31.0914	−0.987652	−0.493826	+	0.869561i	$0.664402\pi$
$992$	1.06605	0.0338471
$993$	0	0
$994$	−15.3379	−0.486489
$995$	−20.3765	−0.645977
$996$	0	0
$997$	−53.4798	−1.69372	−0.846861	−	0.531814i	$-0.821510\pi$
$997$	−53.4798	−1.69372	−0.846861	+	0.531814i	$0.821510\pi$
$998$	27.5284	0.871397
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.u.1.12		18
3.2	odd	2		2667.2.a.p.1.7	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.p.1.7	✓	18	3.2	odd	2
8001.2.a.u.1.12		18	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)

\(f(q)\)	\(=\)	\(q+1.52505 q^{2} +0.325779 q^{4} +1.71268 q^{5} +1.00000 q^{7} -2.55327 q^{8} +O(q^{10})\)	\(q+1.52505 q^{2} +0.325779 q^{4} +1.71268 q^{5} +1.00000 q^{7} -2.55327 q^{8} +2.61192 q^{10} +1.69211 q^{11} -1.16748 q^{13} +1.52505 q^{14} -4.54543 q^{16} -5.50471 q^{17} -0.828520 q^{19} +0.557955 q^{20} +2.58056 q^{22} +6.42297 q^{23} -2.06673 q^{25} -1.78047 q^{26} +0.325779 q^{28} +2.43481 q^{29} -0.583990 q^{31} -1.82546 q^{32} -8.39496 q^{34} +1.71268 q^{35} +11.0189 q^{37} -1.26353 q^{38} -4.37293 q^{40} +0.0442353 q^{41} +12.9529 q^{43} +0.551255 q^{44} +9.79535 q^{46} -3.58595 q^{47} +1.00000 q^{49} -3.15187 q^{50} -0.380341 q^{52} +12.7616 q^{53} +2.89805 q^{55} -2.55327 q^{56} +3.71321 q^{58} -1.61982 q^{59} -5.74621 q^{61} -0.890614 q^{62} +6.30693 q^{64} -1.99952 q^{65} -0.393750 q^{67} -1.79332 q^{68} +2.61192 q^{70} -10.0573 q^{71} +7.94732 q^{73} +16.8044 q^{74} -0.269914 q^{76} +1.69211 q^{77} +3.92230 q^{79} -7.78485 q^{80} +0.0674610 q^{82} -6.36257 q^{83} -9.42780 q^{85} +19.7538 q^{86} -4.32042 q^{88} +5.77339 q^{89} -1.16748 q^{91} +2.09247 q^{92} -5.46876 q^{94} -1.41899 q^{95} +15.1754 q^{97} +1.52505 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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