LMFDB - Embedded newform 8001.2.a.v.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:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + \cdots + 210 $ x^19 - 4x^18 - 22x^17 + 101x^16 + 178x^15 - 1035x^14 - 583x^13 + 5572x^12 + 21x^11 - 17032x^10 + 4985x^9 + 29792x^8 - 13249x^7 - 28600x^6 + 14000x^5 + 13725x^4 - 5723x^3 - 2913x^2 + 608x + 210
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			$2.26573$ 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)

$f(q)$	$=$	$q-2.26573 q^{2} +3.13354 q^{4} +0.263451 q^{5} +1.00000 q^{7} -2.56830 q^{8} +O(q^{10})$	\(q-2.26573 q^{2} +3.13354 q^{4} +0.263451 q^{5} +1.00000 q^{7} -2.56830 q^{8} -0.596909 q^{10} -2.11462 q^{11} +5.02935 q^{13} -2.26573 q^{14} -0.448003 q^{16} -0.600740 q^{17} -0.662362 q^{19} +0.825534 q^{20} +4.79117 q^{22} +0.862531 q^{23} -4.93059 q^{25} -11.3952 q^{26} +3.13354 q^{28} -7.49718 q^{29} -6.27258 q^{31} +6.15165 q^{32} +1.36112 q^{34} +0.263451 q^{35} +1.88794 q^{37} +1.50074 q^{38} -0.676621 q^{40} -9.92148 q^{41} -0.837435 q^{43} -6.62626 q^{44} -1.95426 q^{46} +6.32880 q^{47} +1.00000 q^{49} +11.1714 q^{50} +15.7597 q^{52} -2.22979 q^{53} -0.557099 q^{55} -2.56830 q^{56} +16.9866 q^{58} +8.92253 q^{59} +12.6716 q^{61} +14.2120 q^{62} -13.0420 q^{64} +1.32499 q^{65} -6.63659 q^{67} -1.88244 q^{68} -0.596909 q^{70} -3.04160 q^{71} +3.58423 q^{73} -4.27757 q^{74} -2.07554 q^{76} -2.11462 q^{77} -6.24082 q^{79} -0.118027 q^{80} +22.4794 q^{82} +15.5940 q^{83} -0.158265 q^{85} +1.89740 q^{86} +5.43098 q^{88} -10.2988 q^{89} +5.02935 q^{91} +2.70278 q^{92} -14.3394 q^{94} -0.174500 q^{95} +10.8058 q^{97} -2.26573 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * 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 $

$2$	−2.26573	−1.60211	−0.801057	−	0.598588i	$-0.795729\pi$
$2$	−2.26573	−1.60211	−0.801057	+	0.598588i	$0.795729\pi$
$3$	0	0
$3$	0	0
$4$	3.13354	1.56677
$5$	0.263451	0.117819	0.0589094	−	0.998263i	$-0.481238\pi$
$5$	0.263451	0.117819	0.0589094	+	0.998263i	$0.481238\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.56830	−0.908031
$9$	0	0
$10$	−0.596909	−0.188759
$11$	−2.11462	−0.637583	−0.318791	−	0.947825i	$-0.603277\pi$
$11$	−2.11462	−0.637583	−0.318791	+	0.947825i	$0.603277\pi$
$12$	0	0
$13$	5.02935	1.39489	0.697445	−	0.716638i	$-0.254320\pi$
$13$	5.02935	1.39489	0.697445	+	0.716638i	$0.254320\pi$
$14$	−2.26573	−0.605542
$15$	0	0
$16$	−0.448003	−0.112001
$17$	−0.600740	−0.145701	−0.0728504	−	0.997343i	$-0.523210\pi$
$17$	−0.600740	−0.145701	−0.0728504	+	0.997343i	$0.523210\pi$
$18$	0	0
$19$	−0.662362	−0.151956	−0.0759782	−	0.997109i	$-0.524208\pi$
$19$	−0.662362	−0.151956	−0.0759782	+	0.997109i	$0.524208\pi$
$20$	0.825534	0.184595
$21$	0	0
$22$	4.79117	1.02148
$23$	0.862531	0.179850	0.0899251	−	0.995949i	$-0.471337\pi$
$23$	0.862531	0.179850	0.0899251	+	0.995949i	$0.471337\pi$
$24$	0	0
$25$	−4.93059	−0.986119
$26$	−11.3952	−2.23477
$27$	0	0
$28$	3.13354	0.592184
$29$	−7.49718	−1.39219	−0.696095	−	0.717949i	$-0.745081\pi$
$29$	−7.49718	−1.39219	−0.696095	+	0.717949i	$0.745081\pi$
$30$	0	0
$31$	−6.27258	−1.12659	−0.563294	−	0.826256i	$-0.690466\pi$
$31$	−6.27258	−1.12659	−0.563294	+	0.826256i	$0.690466\pi$
$32$	6.15165	1.08747
$33$	0	0
$34$	1.36112	0.233429
$35$	0.263451	0.0445313
$36$	0	0
$37$	1.88794	0.310376	0.155188	−	0.987885i	$-0.450402\pi$
$37$	1.88794	0.310376	0.155188	+	0.987885i	$0.450402\pi$
$38$	1.50074	0.243451
$39$	0	0
$40$	−0.676621	−0.106983
$41$	−9.92148	−1.54948	−0.774738	−	0.632283i	$-0.782118\pi$
$41$	−9.92148	−1.54948	−0.774738	+	0.632283i	$0.782118\pi$
$42$	0	0
$43$	−0.837435	−0.127708	−0.0638538	−	0.997959i	$-0.520339\pi$
$43$	−0.837435	−0.127708	−0.0638538	+	0.997959i	$0.520339\pi$
$44$	−6.62626	−0.998946
$45$	0	0
$46$	−1.95426	−0.288140
$47$	6.32880	0.923150	0.461575	−	0.887101i	$-0.347284\pi$
$47$	6.32880	0.923150	0.461575	+	0.887101i	$0.347284\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	11.1714	1.57988
$51$	0	0
$52$	15.7597	2.18547
$53$	−2.22979	−0.306285	−0.153143	−	0.988204i	$-0.548939\pi$
$53$	−2.22979	−0.306285	−0.153143	+	0.988204i	$0.548939\pi$
$54$	0	0
$55$	−0.557099	−0.0751192
$56$	−2.56830	−0.343204
$57$	0	0
$58$	16.9866	2.23045
$59$	8.92253	1.16161	0.580807	−	0.814041i	$-0.302737\pi$
$59$	8.92253	1.16161	0.580807	+	0.814041i	$0.302737\pi$
$60$	0	0
$61$	12.6716	1.62243	0.811215	−	0.584748i	$-0.198806\pi$
$61$	12.6716	1.62243	0.811215	+	0.584748i	$0.198806\pi$
$62$	14.2120	1.80492
$63$	0	0
$64$	−13.0420	−1.63025
$65$	1.32499	0.164344
$66$	0	0
$67$	−6.63659	−0.810788	−0.405394	−	0.914142i	$-0.632866\pi$
$67$	−6.63659	−0.810788	−0.405394	+	0.914142i	$0.632866\pi$
$68$	−1.88244	−0.228280
$69$	0	0
$70$	−0.596909	−0.0713443
$71$	−3.04160	−0.360971	−0.180486	−	0.983578i	$-0.557767\pi$
$71$	−3.04160	−0.360971	−0.180486	+	0.983578i	$0.557767\pi$
$72$	0	0
$73$	3.58423	0.419502	0.209751	−	0.977755i	$-0.432735\pi$
$73$	3.58423	0.419502	0.209751	+	0.977755i	$0.432735\pi$
$74$	−4.27757	−0.497257
$75$	0	0
$76$	−2.07554	−0.238081
$77$	−2.11462	−0.240984
$78$	0	0
$79$	−6.24082	−0.702147	−0.351073	−	0.936348i	$-0.614183\pi$
$79$	−6.24082	−0.702147	−0.351073	+	0.936348i	$0.614183\pi$
$80$	−0.118027	−0.0131958
$81$	0	0
$82$	22.4794	2.48244
$83$	15.5940	1.71166	0.855832	−	0.517254i	$-0.173046\pi$
$83$	15.5940	1.71166	0.855832	+	0.517254i	$0.173046\pi$
$84$	0	0
$85$	−0.158265	−0.0171663
$86$	1.89740	0.204602
$87$	0	0
$88$	5.43098	0.578945
$89$	−10.2988	−1.09167	−0.545837	−	0.837891i	$-0.683788\pi$
$89$	−10.2988	−1.09167	−0.545837	+	0.837891i	$0.683788\pi$
$90$	0	0
$91$	5.02935	0.527219
$92$	2.70278	0.281784
$93$	0	0
$94$	−14.3394	−1.47899
$95$	−0.174500	−0.0179033
$96$	0	0
$97$	10.8058	1.09716	0.548580	−	0.836098i	$-0.315169\pi$
$97$	10.8058	1.09716	0.548580	+	0.836098i	$0.315169\pi$
$98$	−2.26573	−0.228873
$99$	0	0
$100$	−15.4502	−1.54502
$101$	11.1783	1.11229	0.556143	−	0.831086i	$-0.312281\pi$
$101$	11.1783	1.11229	0.556143	+	0.831086i	$0.312281\pi$
$102$	0	0
$103$	8.91300	0.878224	0.439112	−	0.898432i	$-0.355293\pi$
$103$	8.91300	0.878224	0.439112	+	0.898432i	$0.355293\pi$
$104$	−12.9169	−1.26660
$105$	0	0
$106$	5.05210	0.490704
$107$	−1.40233	−0.135568	−0.0677840	−	0.997700i	$-0.521593\pi$
$107$	−1.40233	−0.135568	−0.0677840	+	0.997700i	$0.521593\pi$
$108$	0	0
$109$	−7.84506	−0.751421	−0.375710	−	0.926737i	$-0.622601\pi$
$109$	−7.84506	−0.751421	−0.375710	+	0.926737i	$0.622601\pi$
$110$	1.26224	0.120350
$111$	0	0
$112$	−0.448003	−0.0423323
$113$	5.51409	0.518722	0.259361	−	0.965780i	$-0.416488\pi$
$113$	5.51409	0.518722	0.259361	+	0.965780i	$0.416488\pi$
$114$	0	0
$115$	0.227234	0.0211897
$116$	−23.4927	−2.18124
$117$	0	0
$118$	−20.2161	−1.86104
$119$	−0.600740	−0.0550697
$120$	0	0
$121$	−6.52837	−0.593488
$122$	−28.7104	−2.59932
$123$	0	0
$124$	−19.6554	−1.76511
$125$	−2.61622	−0.234002
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	17.2464	1.52438
$129$	0	0
$130$	−3.00206	−0.263298
$131$	−4.56918	−0.399211	−0.199606	−	0.979876i	$-0.563966\pi$
$131$	−4.56918	−0.399211	−0.199606	+	0.979876i	$0.563966\pi$
$132$	0	0
$133$	−0.662362	−0.0574341
$134$	15.0367	1.29898
$135$	0	0
$136$	1.54288	0.132301
$137$	11.8489	1.01232	0.506158	−	0.862441i	$-0.331065\pi$
$137$	11.8489	1.01232	0.506158	+	0.862441i	$0.331065\pi$
$138$	0	0
$139$	15.4594	1.31125	0.655623	−	0.755088i	$-0.272406\pi$
$139$	15.4594	1.31125	0.655623	+	0.755088i	$0.272406\pi$
$140$	0.825534	0.0697703
$141$	0	0
$142$	6.89144	0.578317
$143$	−10.6352	−0.889358
$144$	0	0
$145$	−1.97514	−0.164026
$146$	−8.12090	−0.672090
$147$	0	0
$148$	5.91594	0.486287
$149$	19.2171	1.57433	0.787163	−	0.616745i	$-0.211549\pi$
$149$	19.2171	1.57433	0.787163	+	0.616745i	$0.211549\pi$
$150$	0	0
$151$	−19.4578	−1.58345	−0.791725	−	0.610878i	$-0.790817\pi$
$151$	−19.4578	−1.58345	−0.791725	+	0.610878i	$0.790817\pi$
$152$	1.70115	0.137981
$153$	0	0
$154$	4.79117	0.386083
$155$	−1.65252	−0.132733
$156$	0	0
$157$	20.3132	1.62117	0.810583	−	0.585623i	$-0.199150\pi$
$157$	20.3132	1.62117	0.810583	+	0.585623i	$0.199150\pi$
$158$	14.1400	1.12492
$159$	0	0
$160$	1.62066	0.128124
$161$	0.862531	0.0679770
$162$	0	0
$163$	18.2949	1.43297	0.716483	−	0.697605i	$-0.245751\pi$
$163$	18.2949	1.43297	0.716483	+	0.697605i	$0.245751\pi$
$164$	−31.0894	−2.42767
$165$	0	0
$166$	−35.3318	−2.74228
$167$	−0.426687	−0.0330181	−0.0165090	−	0.999864i	$-0.505255\pi$
$167$	−0.426687	−0.0330181	−0.0165090	+	0.999864i	$0.505255\pi$
$168$	0	0
$169$	12.2944	0.945720
$170$	0.358587	0.0275024
$171$	0	0
$172$	−2.62414	−0.200089
$173$	18.0358	1.37124	0.685620	−	0.727960i	$-0.259531\pi$
$173$	18.0358	1.37124	0.685620	+	0.727960i	$0.259531\pi$
$174$	0	0
$175$	−4.93059	−0.372718
$176$	0.947357	0.0714097
$177$	0	0
$178$	23.3344	1.74899
$179$	3.90602	0.291950	0.145975	−	0.989288i	$-0.453368\pi$
$179$	3.90602	0.291950	0.145975	+	0.989288i	$0.453368\pi$
$180$	0	0
$181$	−19.6214	−1.45845	−0.729223	−	0.684276i	$-0.760118\pi$
$181$	−19.6214	−1.45845	−0.729223	+	0.684276i	$0.760118\pi$
$182$	−11.3952	−0.844665
$183$	0	0
$184$	−2.21524	−0.163310
$185$	0.497380	0.0365681
$186$	0	0
$187$	1.27034	0.0928963
$188$	19.8316	1.44636
$189$	0	0
$190$	0.395370	0.0286832
$191$	18.2968	1.32391	0.661957	−	0.749542i	$-0.269726\pi$
$191$	18.2968	1.32391	0.661957	+	0.749542i	$0.269726\pi$
$192$	0	0
$193$	−8.29105	−0.596803	−0.298401	−	0.954440i	$-0.596453\pi$
$193$	−8.29105	−0.596803	−0.298401	+	0.954440i	$0.596453\pi$
$194$	−24.4830	−1.75778
$195$	0	0
$196$	3.13354	0.223824
$197$	22.0383	1.57016	0.785082	−	0.619391i	$-0.212621\pi$
$197$	22.0383	1.57016	0.785082	+	0.619391i	$0.212621\pi$
$198$	0	0
$199$	−7.76159	−0.550204	−0.275102	−	0.961415i	$-0.588712\pi$
$199$	−7.76159	−0.550204	−0.275102	+	0.961415i	$0.588712\pi$
$200$	12.6632	0.895427
$201$	0	0
$202$	−25.3271	−1.78201
$203$	−7.49718	−0.526199
$204$	0	0
$205$	−2.61382	−0.182557
$206$	−20.1945	−1.40702
$207$	0	0
$208$	−2.25316	−0.156229
$209$	1.40065	0.0968847
$210$	0	0
$211$	−6.35929	−0.437792	−0.218896	−	0.975748i	$-0.570246\pi$
$211$	−6.35929	−0.437792	−0.218896	+	0.975748i	$0.570246\pi$
$212$	−6.98714	−0.479878
$213$	0	0
$214$	3.17729	0.217195
$215$	−0.220623	−0.0150464
$216$	0	0
$217$	−6.27258	−0.425811
$218$	17.7748	1.20386
$219$	0	0
$220$	−1.74569	−0.117695
$221$	−3.02133	−0.203237
$222$	0	0
$223$	−10.9206	−0.731298	−0.365649	−	0.930753i	$-0.619153\pi$
$223$	−10.9206	−0.731298	−0.365649	+	0.930753i	$0.619153\pi$
$224$	6.15165	0.411025
$225$	0	0
$226$	−12.4934	−0.831051
$227$	−1.39196	−0.0923874	−0.0461937	−	0.998933i	$-0.514709\pi$
$227$	−1.39196	−0.0923874	−0.0461937	+	0.998933i	$0.514709\pi$
$228$	0	0
$229$	5.71394	0.377588	0.188794	−	0.982017i	$-0.439542\pi$
$229$	5.71394	0.377588	0.188794	+	0.982017i	$0.439542\pi$
$230$	−0.514852	−0.0339484
$231$	0	0
$232$	19.2550	1.26415
$233$	−14.1302	−0.925697	−0.462849	−	0.886437i	$-0.653173\pi$
$233$	−14.1302	−0.925697	−0.462849	+	0.886437i	$0.653173\pi$
$234$	0	0
$235$	1.66733	0.108764
$236$	27.9591	1.81998
$237$	0	0
$238$	1.36112	0.0882280
$239$	23.6713	1.53117	0.765585	−	0.643335i	$-0.222450\pi$
$239$	23.6713	1.53117	0.765585	+	0.643335i	$0.222450\pi$
$240$	0	0
$241$	18.5876	1.19734	0.598668	−	0.800997i	$-0.295697\pi$
$241$	18.5876	1.19734	0.598668	+	0.800997i	$0.295697\pi$
$242$	14.7915	0.950836
$243$	0	0
$244$	39.7069	2.54198
$245$	0.263451	0.0168313
$246$	0	0
$247$	−3.33125	−0.211963
$248$	16.1099	1.02298
$249$	0	0
$250$	5.92766	0.374898
$251$	23.4333	1.47910	0.739548	−	0.673104i	$-0.235039\pi$
$251$	23.4333	1.47910	0.739548	+	0.673104i	$0.235039\pi$
$252$	0	0
$253$	−1.82393	−0.114669
$254$	−2.26573	−0.142165
$255$	0	0
$256$	−12.9916	−0.811976
$257$	7.24896	0.452178	0.226089	−	0.974107i	$-0.427406\pi$
$257$	7.24896	0.452178	0.226089	+	0.974107i	$0.427406\pi$
$258$	0	0
$259$	1.88794	0.117311
$260$	4.15190	0.257490
$261$	0	0
$262$	10.3525	0.639582
$263$	3.53120	0.217743	0.108871	−	0.994056i	$-0.465276\pi$
$263$	3.53120	0.217743	0.108871	+	0.994056i	$0.465276\pi$
$264$	0	0
$265$	−0.587440	−0.0360861
$266$	1.50074	0.0920160
$267$	0	0
$268$	−20.7960	−1.27032
$269$	17.7944	1.08494	0.542472	−	0.840074i	$-0.317488\pi$
$269$	17.7944	1.08494	0.542472	+	0.840074i	$0.317488\pi$
$270$	0	0
$271$	12.7007	0.771511	0.385756	−	0.922601i	$-0.373941\pi$
$271$	12.7007	0.771511	0.385756	+	0.922601i	$0.373941\pi$
$272$	0.269133	0.0163186
$273$	0	0
$274$	−26.8463	−1.62185
$275$	10.4263	0.628732
$276$	0	0
$277$	−5.23820	−0.314733	−0.157366	−	0.987540i	$-0.550300\pi$
$277$	−5.23820	−0.314733	−0.157366	+	0.987540i	$0.550300\pi$
$278$	−35.0268	−2.10077
$279$	0	0
$280$	−0.676621	−0.0404358
$281$	−17.9732	−1.07219	−0.536096	−	0.844157i	$-0.680101\pi$
$281$	−17.9732	−1.07219	−0.536096	+	0.844157i	$0.680101\pi$
$282$	0	0
$283$	−14.1901	−0.843514	−0.421757	−	0.906709i	$-0.638586\pi$
$283$	−14.1901	−0.843514	−0.421757	+	0.906709i	$0.638586\pi$
$284$	−9.53097	−0.565559
$285$	0	0
$286$	24.0965	1.42485
$287$	−9.92148	−0.585647
$288$	0	0
$289$	−16.6391	−0.978771
$290$	4.47513	0.262789
$291$	0	0
$292$	11.2313	0.657263
$293$	−20.3093	−1.18648	−0.593241	−	0.805025i	$-0.702152\pi$
$293$	−20.3093	−1.18648	−0.593241	+	0.805025i	$0.702152\pi$
$294$	0	0
$295$	2.35065	0.136860
$296$	−4.84880	−0.281831
$297$	0	0
$298$	−43.5408	−2.52225
$299$	4.33797	0.250871
$300$	0	0
$301$	−0.837435	−0.0482690
$302$	44.0861	2.53687
$303$	0	0
$304$	0.296740	0.0170192
$305$	3.33834	0.191153
$306$	0	0
$307$	−17.4945	−0.998463	−0.499231	−	0.866469i	$-0.666384\pi$
$307$	−17.4945	−0.998463	−0.499231	+	0.866469i	$0.666384\pi$
$308$	−6.62626	−0.377566
$309$	0	0
$310$	3.74416	0.212654
$311$	−9.84459	−0.558235	−0.279118	−	0.960257i	$-0.590042\pi$
$311$	−9.84459	−0.558235	−0.279118	+	0.960257i	$0.590042\pi$
$312$	0	0
$313$	22.4434	1.26857	0.634287	−	0.773097i	$-0.281294\pi$
$313$	22.4434	1.26857	0.634287	+	0.773097i	$0.281294\pi$
$314$	−46.0242	−2.59729
$315$	0	0
$316$	−19.5559	−1.10010
$317$	−14.8472	−0.833905	−0.416952	−	0.908928i	$-0.636902\pi$
$317$	−14.8472	−0.833905	−0.416952	+	0.908928i	$0.636902\pi$
$318$	0	0
$319$	15.8537	0.887637
$320$	−3.43592	−0.192074
$321$	0	0
$322$	−1.95426	−0.108907
$323$	0.397908	0.0221402
$324$	0	0
$325$	−24.7977	−1.37553
$326$	−41.4513	−2.29577
$327$	0	0
$328$	25.4813	1.40697
$329$	6.32880	0.348918
$330$	0	0
$331$	27.0357	1.48601	0.743007	−	0.669284i	$-0.233399\pi$
$331$	27.0357	1.48601	0.743007	+	0.669284i	$0.233399\pi$
$332$	48.8644	2.68178
$333$	0	0
$334$	0.966759	0.0528987
$335$	−1.74841	−0.0955261
$336$	0	0
$337$	12.7721	0.695742	0.347871	−	0.937542i	$-0.386905\pi$
$337$	12.7721	0.695742	0.347871	+	0.937542i	$0.386905\pi$
$338$	−27.8557	−1.51515
$339$	0	0
$340$	−0.495931	−0.0268956
$341$	13.2641	0.718294
$342$	0	0
$343$	1.00000	0.0539949
$344$	2.15079	0.115963
$345$	0	0
$346$	−40.8644	−2.19688
$347$	1.00797	0.0541108	0.0270554	−	0.999634i	$-0.491387\pi$
$347$	1.00797	0.0541108	0.0270554	+	0.999634i	$0.491387\pi$
$348$	0	0
$349$	21.1658	1.13298	0.566490	−	0.824068i	$-0.308301\pi$
$349$	21.1658	1.13298	0.566490	+	0.824068i	$0.308301\pi$
$350$	11.1714	0.597137
$351$	0	0
$352$	−13.0084	−0.693351
$353$	−25.1557	−1.33890	−0.669450	−	0.742857i	$-0.733470\pi$
$353$	−25.1557	−1.33890	−0.669450	+	0.742857i	$0.733470\pi$
$354$	0	0
$355$	−0.801311	−0.0425292
$356$	−32.2718	−1.71040
$357$	0	0
$358$	−8.84999	−0.467737
$359$	−24.8593	−1.31202	−0.656012	−	0.754750i	$-0.727758\pi$
$359$	−24.8593	−1.31202	−0.656012	+	0.754750i	$0.727758\pi$
$360$	0	0
$361$	−18.5613	−0.976909
$362$	44.4568	2.33660
$363$	0	0
$364$	15.7597	0.826031
$365$	0.944267	0.0494252
$366$	0	0
$367$	17.4263	0.909648	0.454824	−	0.890581i	$-0.349702\pi$
$367$	17.4263	0.909648	0.454824	+	0.890581i	$0.349702\pi$
$368$	−0.386416	−0.0201433
$369$	0	0
$370$	−1.12693	−0.0585862
$371$	−2.22979	−0.115765
$372$	0	0
$373$	−3.72752	−0.193004	−0.0965018	−	0.995333i	$-0.530765\pi$
$373$	−3.72752	−0.193004	−0.0965018	+	0.995333i	$0.530765\pi$
$374$	−2.87825	−0.148831
$375$	0	0
$376$	−16.2543	−0.838249
$377$	−37.7059	−1.94195
$378$	0	0
$379$	−24.7896	−1.27336	−0.636678	−	0.771130i	$-0.719692\pi$
$379$	−24.7896	−1.27336	−0.636678	+	0.771130i	$0.719692\pi$
$380$	−0.546803	−0.0280504
$381$	0	0
$382$	−41.4558	−2.12106
$383$	12.0043	0.613393	0.306696	−	0.951807i	$-0.400776\pi$
$383$	12.0043	0.613393	0.306696	+	0.951807i	$0.400776\pi$
$384$	0	0
$385$	−0.557099	−0.0283924
$386$	18.7853	0.956146
$387$	0	0
$388$	33.8603	1.71900
$389$	3.92849	0.199182	0.0995911	−	0.995028i	$-0.468247\pi$
$389$	3.92849	0.199182	0.0995911	+	0.995028i	$0.468247\pi$
$390$	0	0
$391$	−0.518157	−0.0262043
$392$	−2.56830	−0.129719
$393$	0	0
$394$	−49.9329	−2.51558
$395$	−1.64415	−0.0827261
$396$	0	0
$397$	−8.49822	−0.426514	−0.213257	−	0.976996i	$-0.568407\pi$
$397$	−8.49822	−0.426514	−0.213257	+	0.976996i	$0.568407\pi$
$398$	17.5857	0.881490
$399$	0	0
$400$	2.20892	0.110446
$401$	22.7863	1.13790	0.568948	−	0.822374i	$-0.307351\pi$
$401$	22.7863	1.13790	0.568948	+	0.822374i	$0.307351\pi$
$402$	0	0
$403$	−31.5470	−1.57147
$404$	35.0278	1.74270
$405$	0	0
$406$	16.9866	0.843031
$407$	−3.99228	−0.197890
$408$	0	0
$409$	−0.459834	−0.0227373	−0.0113687	−	0.999935i	$-0.503619\pi$
$409$	−0.459834	−0.0227373	−0.0113687	+	0.999935i	$0.503619\pi$
$410$	5.92222	0.292478
$411$	0	0
$412$	27.9293	1.37598
$413$	8.92253	0.439049
$414$	0	0
$415$	4.10825	0.201666
$416$	30.9388	1.51690
$417$	0	0
$418$	−3.17349	−0.155220
$419$	7.58777	0.370687	0.185343	−	0.982674i	$-0.440660\pi$
$419$	7.58777	0.370687	0.185343	+	0.982674i	$0.440660\pi$
$420$	0	0
$421$	−7.86449	−0.383292	−0.191646	−	0.981464i	$-0.561383\pi$
$421$	−7.86449	−0.383292	−0.191646	+	0.981464i	$0.561383\pi$
$422$	14.4085	0.701393
$423$	0	0
$424$	5.72677	0.278116
$425$	2.96200	0.143678
$426$	0	0
$427$	12.6716	0.613221
$428$	−4.39425	−0.212404
$429$	0	0
$430$	0.499873	0.0241060
$431$	−28.4816	−1.37191	−0.685954	−	0.727645i	$-0.740615\pi$
$431$	−28.4816	−1.37191	−0.685954	+	0.727645i	$0.740615\pi$
$432$	0	0
$433$	3.03914	0.146052	0.0730260	−	0.997330i	$-0.476734\pi$
$433$	3.03914	0.146052	0.0730260	+	0.997330i	$0.476734\pi$
$434$	14.2120	0.682197
$435$	0	0
$436$	−24.5828	−1.17730
$437$	−0.571308	−0.0273294
$438$	0	0
$439$	34.0162	1.62350	0.811752	−	0.584002i	$-0.198514\pi$
$439$	34.0162	1.62350	0.811752	+	0.584002i	$0.198514\pi$
$440$	1.43080	0.0682106
$441$	0	0
$442$	6.84553	0.325609
$443$	20.9252	0.994186	0.497093	−	0.867697i	$-0.334401\pi$
$443$	20.9252	0.994186	0.497093	+	0.867697i	$0.334401\pi$
$444$	0	0
$445$	−2.71324	−0.128620
$446$	24.7432	1.17162
$447$	0	0
$448$	−13.0420	−0.616176
$449$	−2.80824	−0.132529	−0.0662646	−	0.997802i	$-0.521108\pi$
$449$	−2.80824	−0.132529	−0.0662646	+	0.997802i	$0.521108\pi$
$450$	0	0
$451$	20.9802	0.987919
$452$	17.2786	0.812718
$453$	0	0
$454$	3.15380	0.148015
$455$	1.32499	0.0621163
$456$	0	0
$457$	−7.78721	−0.364271	−0.182135	−	0.983273i	$-0.558301\pi$
$457$	−7.78721	−0.364271	−0.182135	+	0.983273i	$0.558301\pi$
$458$	−12.9463	−0.604939
$459$	0	0
$460$	0.712048	0.0331994
$461$	2.68061	0.124849	0.0624243	−	0.998050i	$-0.480117\pi$
$461$	2.68061	0.124849	0.0624243	+	0.998050i	$0.480117\pi$
$462$	0	0
$463$	4.95916	0.230472	0.115236	−	0.993338i	$-0.463238\pi$
$463$	4.95916	0.230472	0.115236	+	0.993338i	$0.463238\pi$
$464$	3.35876	0.155926
$465$	0	0
$466$	32.0151	1.48307
$467$	39.4769	1.82677	0.913386	−	0.407095i	$-0.133458\pi$
$467$	39.4769	1.82677	0.913386	+	0.407095i	$0.133458\pi$
$468$	0	0
$469$	−6.63659	−0.306449
$470$	−3.77772	−0.174253
$471$	0	0
$472$	−22.9157	−1.05478
$473$	1.77086	0.0814242
$474$	0	0
$475$	3.26584	0.149847
$476$	−1.88244	−0.0862816
$477$	0	0
$478$	−53.6328	−2.45311
$479$	−31.0109	−1.41692	−0.708462	−	0.705749i	$-0.750610\pi$
$479$	−31.0109	−1.41692	−0.708462	+	0.705749i	$0.750610\pi$
$480$	0	0
$481$	9.49512	0.432940
$482$	−42.1146	−1.91827
$483$	0	0
$484$	−20.4569	−0.929860
$485$	2.84679	0.129266
$486$	0	0
$487$	39.6366	1.79611	0.898053	−	0.439887i	$-0.144981\pi$
$487$	39.6366	1.79611	0.898053	+	0.439887i	$0.144981\pi$
$488$	−32.5444	−1.47322
$489$	0	0
$490$	−0.596909	−0.0269656
$491$	−16.5621	−0.747438	−0.373719	−	0.927542i	$-0.621918\pi$
$491$	−16.5621	−0.747438	−0.373719	+	0.927542i	$0.621918\pi$
$492$	0	0
$493$	4.50385	0.202843
$494$	7.54773	0.339588
$495$	0	0
$496$	2.81013	0.126179
$497$	−3.04160	−0.136434
$498$	0	0
$499$	39.7879	1.78115	0.890576	−	0.454834i	$-0.150301\pi$
$499$	39.7879	1.78115	0.890576	+	0.454834i	$0.150301\pi$
$500$	−8.19804	−0.366628
$501$	0	0
$502$	−53.0935	−2.36968
$503$	−18.0707	−0.805734	−0.402867	−	0.915259i	$-0.631986\pi$
$503$	−18.0707	−0.805734	−0.402867	+	0.915259i	$0.631986\pi$
$504$	0	0
$505$	2.94494	0.131048
$506$	4.13253	0.183713
$507$	0	0
$508$	3.13354	0.139028
$509$	−23.0487	−1.02162	−0.510808	−	0.859695i	$-0.670654\pi$
$509$	−23.0487	−1.02162	−0.510808	+	0.859695i	$0.670654\pi$
$510$	0	0
$511$	3.58423	0.158557
$512$	−5.05717	−0.223497
$513$	0	0
$514$	−16.4242	−0.724441
$515$	2.34814	0.103471
$516$	0	0
$517$	−13.3830	−0.588585
$518$	−4.27757	−0.187946
$519$	0	0
$520$	−3.40296	−0.149230
$521$	−23.3186	−1.02161	−0.510803	−	0.859698i	$-0.670652\pi$
$521$	−23.3186	−1.02161	−0.510803	+	0.859698i	$0.670652\pi$
$522$	0	0
$523$	5.13133	0.224377	0.112189	−	0.993687i	$-0.464214\pi$
$523$	5.13133	0.224377	0.112189	+	0.993687i	$0.464214\pi$
$524$	−14.3177	−0.625473
$525$	0	0
$526$	−8.00074	−0.348849
$527$	3.76819	0.164145
$528$	0	0
$529$	−22.2560	−0.967654
$530$	1.33098	0.0578141
$531$	0	0
$532$	−2.07554	−0.0899861
$533$	−49.8986	−2.16135
$534$	0	0
$535$	−0.369444	−0.0159725
$536$	17.0447	0.736221
$537$	0	0
$538$	−40.3173	−1.73820
$539$	−2.11462	−0.0910832
$540$	0	0
$541$	31.1702	1.34011	0.670055	−	0.742312i	$-0.266271\pi$
$541$	31.1702	1.34011	0.670055	+	0.742312i	$0.266271\pi$
$542$	−28.7763	−1.23605
$543$	0	0
$544$	−3.69554	−0.158445
$545$	−2.06679	−0.0885315
$546$	0	0
$547$	−7.59702	−0.324825	−0.162413	−	0.986723i	$-0.551928\pi$
$547$	−7.59702	−0.324825	−0.162413	+	0.986723i	$0.551928\pi$
$548$	37.1289	1.58607
$549$	0	0
$550$	−23.6233	−1.00730
$551$	4.96585	0.211552
$552$	0	0
$553$	−6.24082	−0.265387
$554$	11.8684	0.504238
$555$	0	0
$556$	48.4425	2.05442
$557$	11.1855	0.473947	0.236973	−	0.971516i	$-0.423845\pi$
$557$	11.1855	0.473947	0.236973	+	0.971516i	$0.423845\pi$
$558$	0	0
$559$	−4.21176	−0.178138
$560$	−0.118027	−0.00498754
$561$	0	0
$562$	40.7225	1.71777
$563$	28.2073	1.18879	0.594397	−	0.804172i	$-0.297391\pi$
$563$	28.2073	1.18879	0.594397	+	0.804172i	$0.297391\pi$
$564$	0	0
$565$	1.45269	0.0611152
$566$	32.1510	1.35141
$567$	0	0
$568$	7.81173	0.327773
$569$	−24.8095	−1.04007	−0.520035	−	0.854145i	$-0.674081\pi$
$569$	−24.8095	−1.04007	−0.520035	+	0.854145i	$0.674081\pi$
$570$	0	0
$571$	−21.0354	−0.880304	−0.440152	−	0.897923i	$-0.645076\pi$
$571$	−21.0354	−0.880304	−0.440152	+	0.897923i	$0.645076\pi$
$572$	−33.3258	−1.39342
$573$	0	0
$574$	22.4794	0.938273
$575$	−4.25279	−0.177354
$576$	0	0
$577$	−13.9122	−0.579171	−0.289585	−	0.957152i	$-0.593517\pi$
$577$	−13.9122	−0.579171	−0.289585	+	0.957152i	$0.593517\pi$
$578$	37.6998	1.56810
$579$	0	0
$580$	−6.18917	−0.256991
$581$	15.5940	0.646948
$582$	0	0
$583$	4.71516	0.195282
$584$	−9.20537	−0.380921
$585$	0	0
$586$	46.0154	1.90088
$587$	−27.0344	−1.11583	−0.557914	−	0.829899i	$-0.688398\pi$
$587$	−27.0344	−1.11583	−0.557914	+	0.829899i	$0.688398\pi$
$588$	0	0
$589$	4.15472	0.171192
$590$	−5.32594	−0.219265
$591$	0	0
$592$	−0.845803	−0.0347623
$593$	29.7989	1.22369	0.611847	−	0.790976i	$-0.290427\pi$
$593$	29.7989	1.22369	0.611847	+	0.790976i	$0.290427\pi$
$594$	0	0
$595$	−0.158265	−0.00648825
$596$	60.2176	2.46661
$597$	0	0
$598$	−9.82868	−0.401924
$599$	−29.2831	−1.19648	−0.598238	−	0.801319i	$-0.704132\pi$
$599$	−29.2831	−1.19648	−0.598238	+	0.801319i	$0.704132\pi$
$600$	0	0
$601$	−6.01295	−0.245274	−0.122637	−	0.992452i	$-0.539135\pi$
$601$	−6.01295	−0.245274	−0.122637	+	0.992452i	$0.539135\pi$
$602$	1.89740	0.0773324
$603$	0	0
$604$	−60.9717	−2.48090
$605$	−1.71990	−0.0699241
$606$	0	0
$607$	8.18284	0.332131	0.166066	−	0.986115i	$-0.446894\pi$
$607$	8.18284	0.332131	0.166066	+	0.986115i	$0.446894\pi$
$608$	−4.07462	−0.165248
$609$	0	0
$610$	−7.56378	−0.306249
$611$	31.8298	1.28769
$612$	0	0
$613$	−33.3302	−1.34619	−0.673097	−	0.739555i	$-0.735036\pi$
$613$	−33.3302	−1.34619	−0.673097	+	0.739555i	$0.735036\pi$
$614$	39.6378	1.59965
$615$	0	0
$616$	5.43098	0.218821
$617$	15.3915	0.619638	0.309819	−	0.950796i	$-0.399732\pi$
$617$	15.3915	0.619638	0.309819	+	0.950796i	$0.399732\pi$
$618$	0	0
$619$	16.0862	0.646558	0.323279	−	0.946304i	$-0.395215\pi$
$619$	16.0862	0.646558	0.323279	+	0.946304i	$0.395215\pi$
$620$	−5.17823	−0.207963
$621$	0	0
$622$	22.3052	0.894357
$623$	−10.2988	−0.412614
$624$	0	0
$625$	23.9637	0.958549
$626$	−50.8507	−2.03240
$627$	0	0
$628$	63.6521	2.54000
$629$	−1.13416	−0.0452220
$630$	0	0
$631$	2.67650	0.106550	0.0532749	−	0.998580i	$-0.483034\pi$
$631$	2.67650	0.106550	0.0532749	+	0.998580i	$0.483034\pi$
$632$	16.0283	0.637571
$633$	0	0
$634$	33.6399	1.33601
$635$	0.263451	0.0104547
$636$	0	0
$637$	5.02935	0.199270
$638$	−35.9202	−1.42210
$639$	0	0
$640$	4.54357	0.179600
$641$	47.2540	1.86642	0.933210	−	0.359333i	$-0.116996\pi$
$641$	47.2540	1.86642	0.933210	+	0.359333i	$0.116996\pi$
$642$	0	0
$643$	14.0330	0.553409	0.276704	−	0.960955i	$-0.410758\pi$
$643$	14.0330	0.553409	0.276704	+	0.960955i	$0.410758\pi$
$644$	2.70278	0.106504
$645$	0	0
$646$	−0.901552	−0.0354711
$647$	18.8486	0.741017	0.370508	−	0.928829i	$-0.379183\pi$
$647$	18.8486	0.741017	0.370508	+	0.928829i	$0.379183\pi$
$648$	0	0
$649$	−18.8678	−0.740625
$650$	56.1849	2.20375
$651$	0	0
$652$	57.3277	2.24513
$653$	−42.9357	−1.68020	−0.840101	−	0.542429i	$-0.817505\pi$
$653$	−42.9357	−1.68020	−0.840101	+	0.542429i	$0.817505\pi$
$654$	0	0
$655$	−1.20376	−0.0470346
$656$	4.44485	0.173542
$657$	0	0
$658$	−14.3394	−0.559007
$659$	6.16411	0.240120	0.120060	−	0.992767i	$-0.461691\pi$
$659$	6.16411	0.240120	0.120060	+	0.992767i	$0.461691\pi$
$660$	0	0
$661$	17.9305	0.697416	0.348708	−	0.937231i	$-0.386620\pi$
$661$	17.9305	0.697416	0.348708	+	0.937231i	$0.386620\pi$
$662$	−61.2555	−2.38076
$663$	0	0
$664$	−40.0501	−1.55424
$665$	−0.174500	−0.00676682
$666$	0	0
$667$	−6.46655	−0.250386
$668$	−1.33704	−0.0517317
$669$	0	0
$670$	3.96144	0.153044
$671$	−26.7956	−1.03443
$672$	0	0
$673$	27.6482	1.06576	0.532879	−	0.846191i	$-0.321110\pi$
$673$	27.6482	1.06576	0.532879	+	0.846191i	$0.321110\pi$
$674$	−28.9382	−1.11466
$675$	0	0
$676$	38.5249	1.48173
$677$	44.0086	1.69139	0.845694	−	0.533668i	$-0.179187\pi$
$677$	44.0086	1.69139	0.845694	+	0.533668i	$0.179187\pi$
$678$	0	0
$679$	10.8058	0.414688
$680$	0.406473	0.0155875
$681$	0	0
$682$	−30.0530	−1.15079
$683$	−12.3742	−0.473484	−0.236742	−	0.971573i	$-0.576080\pi$
$683$	−12.3742	−0.473484	−0.236742	+	0.971573i	$0.576080\pi$
$684$	0	0
$685$	3.12159	0.119270
$686$	−2.26573	−0.0865060
$687$	0	0
$688$	0.375173	0.0143034
$689$	−11.2144	−0.427234
$690$	0	0
$691$	29.0744	1.10604	0.553021	−	0.833168i	$-0.313475\pi$
$691$	29.0744	1.10604	0.553021	+	0.833168i	$0.313475\pi$
$692$	56.5160	2.14842
$693$	0	0
$694$	−2.28380	−0.0866918
$695$	4.07278	0.154489
$696$	0	0
$697$	5.96023	0.225760
$698$	−47.9561	−1.81516
$699$	0	0
$700$	−15.4502	−0.583963
$701$	−9.79244	−0.369855	−0.184928	−	0.982752i	$-0.559205\pi$
$701$	−9.79244	−0.369855	−0.184928	+	0.982752i	$0.559205\pi$
$702$	0	0
$703$	−1.25050	−0.0471635
$704$	27.5789	1.03942
$705$	0	0
$706$	56.9960	2.14507
$707$	11.1783	0.420405
$708$	0	0
$709$	0.289511	0.0108728	0.00543641	−	0.999985i	$-0.498270\pi$
$709$	0.289511	0.0108728	0.00543641	+	0.999985i	$0.498270\pi$
$710$	1.81556	0.0681366
$711$	0	0
$712$	26.4505	0.991275
$713$	−5.41030	−0.202617
$714$	0	0
$715$	−2.80185	−0.104783
$716$	12.2397	0.457418
$717$	0	0
$718$	56.3246	2.10201
$719$	35.0204	1.30604	0.653021	−	0.757340i	$-0.273501\pi$
$719$	35.0204	1.30604	0.653021	+	0.757340i	$0.273501\pi$
$720$	0	0
$721$	8.91300	0.331938
$722$	42.0549	1.56512
$723$	0	0
$724$	−61.4844	−2.28505
$725$	36.9655	1.37287
$726$	0	0
$727$	17.1376	0.635597	0.317798	−	0.948158i	$-0.397057\pi$
$727$	17.1376	0.635597	0.317798	+	0.948158i	$0.397057\pi$
$728$	−12.9169	−0.478731
$729$	0	0
$730$	−2.13946	−0.0791848
$731$	0.503081	0.0186071
$732$	0	0
$733$	15.9224	0.588106	0.294053	−	0.955789i	$-0.404996\pi$
$733$	15.9224	0.588106	0.294053	+	0.955789i	$0.404996\pi$
$734$	−39.4834	−1.45736
$735$	0	0
$736$	5.30599	0.195581
$737$	14.0339	0.516944
$738$	0	0
$739$	−26.5239	−0.975697	−0.487848	−	0.872928i	$-0.662218\pi$
$739$	−26.5239	−0.975697	−0.487848	+	0.872928i	$0.662218\pi$
$740$	1.55856	0.0572938
$741$	0	0
$742$	5.05210	0.185469
$743$	23.2387	0.852544	0.426272	−	0.904595i	$-0.359827\pi$
$743$	23.2387	0.852544	0.426272	+	0.904595i	$0.359827\pi$
$744$	0	0
$745$	5.06276	0.185485
$746$	8.44556	0.309214
$747$	0	0
$748$	3.98066	0.145547
$749$	−1.40233	−0.0512399
$750$	0	0
$751$	−3.26661	−0.119200	−0.0596001	−	0.998222i	$-0.518983\pi$
$751$	−3.26661	−0.119200	−0.0596001	+	0.998222i	$0.518983\pi$
$752$	−2.83532	−0.103393
$753$	0	0
$754$	85.4315	3.11123
$755$	−5.12616	−0.186560
$756$	0	0
$757$	5.68679	0.206690	0.103345	−	0.994646i	$-0.467045\pi$
$757$	5.68679	0.206690	0.103345	+	0.994646i	$0.467045\pi$
$758$	56.1666	2.04006
$759$	0	0
$760$	0.448168	0.0162568
$761$	41.2816	1.49646	0.748229	−	0.663440i	$-0.230904\pi$
$761$	41.2816	1.49646	0.748229	+	0.663440i	$0.230904\pi$
$762$	0	0
$763$	−7.84506	−0.284010
$764$	57.3339	2.07427
$765$	0	0
$766$	−27.1986	−0.982726
$767$	44.8745	1.62033
$768$	0	0
$769$	−15.3798	−0.554611	−0.277306	−	0.960782i	$-0.589441\pi$
$769$	−15.3798	−0.554611	−0.277306	+	0.960782i	$0.589441\pi$
$770$	1.26224	0.0454879
$771$	0	0
$772$	−25.9803	−0.935053
$773$	−43.4059	−1.56120	−0.780602	−	0.625029i	$-0.785087\pi$
$773$	−43.4059	−1.56120	−0.780602	+	0.625029i	$0.785087\pi$
$774$	0	0
$775$	30.9276	1.11095
$776$	−27.7525	−0.996256
$777$	0	0
$778$	−8.90090	−0.319113
$779$	6.57162	0.235453
$780$	0	0
$781$	6.43183	0.230149
$782$	1.17400	0.0419823
$783$	0	0
$784$	−0.448003	−0.0160001
$785$	5.35152	0.191004
$786$	0	0
$787$	3.18555	0.113553	0.0567763	−	0.998387i	$-0.481918\pi$
$787$	3.18555	0.113553	0.0567763	+	0.998387i	$0.481918\pi$
$788$	69.0580	2.46009
$789$	0	0
$790$	3.72520	0.132537
$791$	5.51409	0.196058
$792$	0	0
$793$	63.7299	2.26311
$794$	19.2547	0.683324
$795$	0	0
$796$	−24.3212	−0.862044
$797$	−44.1424	−1.56361	−0.781803	−	0.623526i	$-0.785700\pi$
$797$	−44.1424	−1.56361	−0.781803	+	0.623526i	$0.785700\pi$
$798$	0	0
$799$	−3.80196	−0.134504
$800$	−30.3313	−1.07237
$801$	0	0
$802$	−51.6278	−1.82304
$803$	−7.57929	−0.267467
$804$	0	0
$805$	0.227234	0.00800896
$806$	71.4771	2.51767
$807$	0	0
$808$	−28.7093	−1.00999
$809$	−31.3325	−1.10159	−0.550797	−	0.834639i	$-0.685676\pi$
$809$	−31.3325	−1.10159	−0.550797	+	0.834639i	$0.685676\pi$
$810$	0	0
$811$	−6.35201	−0.223049	−0.111525	−	0.993762i	$-0.535573\pi$
$811$	−6.35201	−0.223049	−0.111525	+	0.993762i	$0.535573\pi$
$812$	−23.4927	−0.824433
$813$	0	0
$814$	9.04544	0.317043
$815$	4.81980	0.168830
$816$	0	0
$817$	0.554686	0.0194060
$818$	1.04186	0.0364278
$819$	0	0
$820$	−8.19052	−0.286025
$821$	−19.9589	−0.696571	−0.348285	−	0.937389i	$-0.613236\pi$
$821$	−19.9589	−0.696571	−0.348285	+	0.937389i	$0.613236\pi$
$822$	0	0
$823$	−3.54944	−0.123726	−0.0618629	−	0.998085i	$-0.519704\pi$
$823$	−3.54944	−0.123726	−0.0618629	+	0.998085i	$0.519704\pi$
$824$	−22.8913	−0.797455
$825$	0	0
$826$	−20.2161	−0.703407
$827$	33.1724	1.15352	0.576758	−	0.816915i	$-0.304317\pi$
$827$	33.1724	1.15352	0.576758	+	0.816915i	$0.304317\pi$
$828$	0	0
$829$	17.7237	0.615570	0.307785	−	0.951456i	$-0.400412\pi$
$829$	17.7237	0.615570	0.307785	+	0.951456i	$0.400412\pi$
$830$	−9.30819	−0.323092
$831$	0	0
$832$	−65.5928	−2.27402
$833$	−0.600740	−0.0208144
$834$	0	0
$835$	−0.112411	−0.00389015
$836$	4.38898	0.151796
$837$	0	0
$838$	−17.1919	−0.593883
$839$	−38.2507	−1.32056	−0.660280	−	0.751019i	$-0.729562\pi$
$839$	−38.2507	−1.32056	−0.660280	+	0.751019i	$0.729562\pi$
$840$	0	0
$841$	27.2077	0.938196
$842$	17.8188	0.614078
$843$	0	0
$844$	−19.9271	−0.685919
$845$	3.23896	0.111424
$846$	0	0
$847$	−6.52837	−0.224318
$848$	0.998952	0.0343041
$849$	0	0
$850$	−6.71111	−0.230189
$851$	1.62841	0.0558211
$852$	0	0
$853$	29.5909	1.01317	0.506587	−	0.862189i	$-0.330907\pi$
$853$	29.5909	1.01317	0.506587	+	0.862189i	$0.330907\pi$
$854$	−28.7104	−0.982450
$855$	0	0
$856$	3.60159	0.123100
$857$	15.8769	0.542344	0.271172	−	0.962531i	$-0.412589\pi$
$857$	15.8769	0.542344	0.271172	+	0.962531i	$0.412589\pi$
$858$	0	0
$859$	42.5121	1.45049	0.725247	−	0.688489i	$-0.241726\pi$
$859$	42.5121	1.45049	0.725247	+	0.688489i	$0.241726\pi$
$860$	−0.691331	−0.0235742
$861$	0	0
$862$	64.5316	2.19795
$863$	−44.1539	−1.50302	−0.751508	−	0.659724i	$-0.770673\pi$
$863$	−44.1539	−1.50302	−0.751508	+	0.659724i	$0.770673\pi$
$864$	0	0
$865$	4.75156	0.161558
$866$	−6.88589	−0.233992
$867$	0	0
$868$	−19.6554	−0.667147
$869$	13.1970	0.447677
$870$	0	0
$871$	−33.3777	−1.13096
$872$	20.1485	0.682313
$873$	0	0
$874$	1.29443	0.0437848
$875$	−2.61622	−0.0884445
$876$	0	0
$877$	4.55932	0.153957	0.0769786	−	0.997033i	$-0.475473\pi$
$877$	4.55932	0.153957	0.0769786	+	0.997033i	$0.475473\pi$
$878$	−77.0716	−2.60104
$879$	0	0
$880$	0.249582	0.00841340
$881$	−30.1946	−1.01728	−0.508642	−	0.860978i	$-0.669852\pi$
$881$	−30.1946	−1.01728	−0.508642	+	0.860978i	$0.669852\pi$
$882$	0	0
$883$	50.7580	1.70814	0.854071	−	0.520156i	$-0.174126\pi$
$883$	50.7580	1.70814	0.854071	+	0.520156i	$0.174126\pi$
$884$	−9.46747	−0.318425
$885$	0	0
$886$	−47.4109	−1.59280
$887$	41.3459	1.38826	0.694129	−	0.719851i	$-0.255790\pi$
$887$	41.3459	1.38826	0.694129	+	0.719851i	$0.255790\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	6.14747	0.206064
$891$	0	0
$892$	−34.2202	−1.14578
$893$	−4.19196	−0.140279
$894$	0	0
$895$	1.02904	0.0343971
$896$	17.2464	0.576160
$897$	0	0
$898$	6.36273	0.212327
$899$	47.0267	1.56843
$900$	0	0
$901$	1.33952	0.0446260
$902$	−47.5355	−1.58276
$903$	0	0
$904$	−14.1618	−0.471015
$905$	−5.16927	−0.171832
$906$	0	0
$907$	6.47378	0.214958	0.107479	−	0.994207i	$-0.465722\pi$
$907$	6.47378	0.214958	0.107479	+	0.994207i	$0.465722\pi$
$908$	−4.36175	−0.144750
$909$	0	0
$910$	−3.00206	−0.0995175
$911$	17.7715	0.588797	0.294399	−	0.955683i	$-0.404881\pi$
$911$	17.7715	0.588797	0.294399	+	0.955683i	$0.404881\pi$
$912$	0	0
$913$	−32.9754	−1.09133
$914$	17.6437	0.583603
$915$	0	0
$916$	17.9049	0.591593
$917$	−4.56918	−0.150888
$918$	0	0
$919$	−1.18053	−0.0389421	−0.0194710	−	0.999810i	$-0.506198\pi$
$919$	−1.18053	−0.0389421	−0.0194710	+	0.999810i	$0.506198\pi$
$920$	−0.583606	−0.0192409
$921$	0	0
$922$	−6.07355	−0.200022
$923$	−15.2973	−0.503515
$924$	0	0
$925$	−9.30867	−0.306067
$926$	−11.2361	−0.369242
$927$	0	0
$928$	−46.1200	−1.51396
$929$	29.3261	0.962159	0.481080	−	0.876677i	$-0.340245\pi$
$929$	29.3261	0.962159	0.481080	+	0.876677i	$0.340245\pi$
$930$	0	0
$931$	−0.662362	−0.0217081
$932$	−44.2774	−1.45036
$933$	0	0
$934$	−89.4440	−2.92670
$935$	0.334672	0.0109449
$936$	0	0
$937$	24.0660	0.786202	0.393101	−	0.919495i	$-0.371402\pi$
$937$	24.0660	0.786202	0.393101	+	0.919495i	$0.371402\pi$
$938$	15.0367	0.490966
$939$	0	0
$940$	5.22464	0.170409
$941$	33.5049	1.09223	0.546115	−	0.837710i	$-0.316106\pi$
$941$	33.5049	1.09223	0.546115	+	0.837710i	$0.316106\pi$
$942$	0	0
$943$	−8.55758	−0.278673
$944$	−3.99732	−0.130102
$945$	0	0
$946$	−4.01229	−0.130451
$947$	−0.188355	−0.00612073	−0.00306036	−	0.999995i	$-0.500974\pi$
$947$	−0.188355	−0.00612073	−0.00306036	+	0.999995i	$0.500974\pi$
$948$	0	0
$949$	18.0263	0.585159
$950$	−7.39952	−0.240072
$951$	0	0
$952$	1.54288	0.0500050
$953$	38.4600	1.24584	0.622921	−	0.782285i	$-0.285946\pi$
$953$	38.4600	1.24584	0.622921	+	0.782285i	$0.285946\pi$
$954$	0	0
$955$	4.82032	0.155982
$956$	74.1750	2.39899
$957$	0	0
$958$	70.2623	2.27007
$959$	11.8489	0.382620
$960$	0	0
$961$	8.34528	0.269203
$962$	−21.5134	−0.693619
$963$	0	0
$964$	58.2452	1.87595
$965$	−2.18428	−0.0703146
$966$	0	0
$967$	31.6878	1.01901	0.509505	−	0.860468i	$-0.329829\pi$
$967$	31.6878	1.01901	0.509505	+	0.860468i	$0.329829\pi$
$968$	16.7668	0.538906
$969$	0	0
$970$	−6.45006	−0.207099
$971$	8.21103	0.263504	0.131752	−	0.991283i	$-0.457940\pi$
$971$	8.21103	0.263504	0.131752	+	0.991283i	$0.457940\pi$
$972$	0	0
$973$	15.4594	0.495604
$974$	−89.8059	−2.87757
$975$	0	0
$976$	−5.67691	−0.181713
$977$	−37.8453	−1.21078	−0.605389	−	0.795929i	$-0.706983\pi$
$977$	−37.8453	−1.21078	−0.605389	+	0.795929i	$0.706983\pi$
$978$	0	0
$979$	21.7782	0.696033
$980$	0.825534	0.0263707
$981$	0	0
$982$	37.5253	1.19748
$983$	54.3534	1.73360	0.866802	−	0.498652i	$-0.166171\pi$
$983$	54.3534	1.73360	0.866802	+	0.498652i	$0.166171\pi$
$984$	0	0
$985$	5.80601	0.184995
$986$	−10.2045	−0.324978
$987$	0	0
$988$	−10.4386	−0.332097
$989$	−0.722314	−0.0229682
$990$	0	0
$991$	11.9772	0.380469	0.190235	−	0.981739i	$-0.439075\pi$
$991$	11.9772	0.380469	0.190235	+	0.981739i	$0.439075\pi$
$992$	−38.5868	−1.22513
$993$	0	0
$994$	6.89144	0.218583
$995$	−2.04480	−0.0648244
$996$	0	0
$997$	−41.7518	−1.32229	−0.661146	−	0.750258i	$-0.729929\pi$
$997$	−41.7518	−1.32229	−0.661146	+	0.750258i	$0.729929\pi$
$998$	−90.1488	−2.85361
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.v.1.4		19
3.2	odd	2		2667.2.a.q.1.16	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.q.1.16	✓	19	3.2	odd	2
8001.2.a.v.1.4		19	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)

\(f(q)\)	\(=\)	\(q-2.26573 q^{2} +3.13354 q^{4} +0.263451 q^{5} +1.00000 q^{7} -2.56830 q^{8} +O(q^{10})\)	\(q-2.26573 q^{2} +3.13354 q^{4} +0.263451 q^{5} +1.00000 q^{7} -2.56830 q^{8} -0.596909 q^{10} -2.11462 q^{11} +5.02935 q^{13} -2.26573 q^{14} -0.448003 q^{16} -0.600740 q^{17} -0.662362 q^{19} +0.825534 q^{20} +4.79117 q^{22} +0.862531 q^{23} -4.93059 q^{25} -11.3952 q^{26} +3.13354 q^{28} -7.49718 q^{29} -6.27258 q^{31} +6.15165 q^{32} +1.36112 q^{34} +0.263451 q^{35} +1.88794 q^{37} +1.50074 q^{38} -0.676621 q^{40} -9.92148 q^{41} -0.837435 q^{43} -6.62626 q^{44} -1.95426 q^{46} +6.32880 q^{47} +1.00000 q^{49} +11.1714 q^{50} +15.7597 q^{52} -2.22979 q^{53} -0.557099 q^{55} -2.56830 q^{56} +16.9866 q^{58} +8.92253 q^{59} +12.6716 q^{61} +14.2120 q^{62} -13.0420 q^{64} +1.32499 q^{65} -6.63659 q^{67} -1.88244 q^{68} -0.596909 q^{70} -3.04160 q^{71} +3.58423 q^{73} -4.27757 q^{74} -2.07554 q^{76} -2.11462 q^{77} -6.24082 q^{79} -0.118027 q^{80} +22.4794 q^{82} +15.5940 q^{83} -0.158265 q^{85} +1.89740 q^{86} +5.43098 q^{88} -10.2988 q^{89} +5.02935 q^{91} +2.70278 q^{92} -14.3394 q^{94} -0.174500 q^{95} +10.8058 q^{97} -2.26573 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more