LMFDB - Embedded newform 8001.2.a.s.1.10

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} + \cdots - 20 $ x^16 - 4x^15 - 18x^14 + 83x^13 + 112x^12 - 668x^11 - 235x^10 + 2648x^9 - 298x^8 - 5422x^7 + 2075x^6 + 5385x^5 - 3163x^4 - 1882x^3 + 1614x^2 - 200*x - 20
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			1.10
Root			$-0.0643619$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0643619 q^{2} -1.99586 q^{4} -3.34910 q^{5} -1.00000 q^{7} -0.257181 q^{8} +O(q^{10})$	\(q+0.0643619 q^{2} -1.99586 q^{4} -3.34910 q^{5} -1.00000 q^{7} -0.257181 q^{8} -0.215554 q^{10} +1.12754 q^{11} +1.58117 q^{13} -0.0643619 q^{14} +3.97516 q^{16} -4.13963 q^{17} +1.41071 q^{19} +6.68432 q^{20} +0.0725704 q^{22} +4.54286 q^{23} +6.21645 q^{25} +0.101767 q^{26} +1.99586 q^{28} -8.98971 q^{29} -5.91120 q^{31} +0.770210 q^{32} -0.266434 q^{34} +3.34910 q^{35} -1.75634 q^{37} +0.0907962 q^{38} +0.861323 q^{40} -1.45003 q^{41} -4.28589 q^{43} -2.25040 q^{44} +0.292387 q^{46} -1.99378 q^{47} +1.00000 q^{49} +0.400102 q^{50} -3.15579 q^{52} -8.37372 q^{53} -3.77623 q^{55} +0.257181 q^{56} -0.578594 q^{58} -8.12664 q^{59} +3.22482 q^{61} -0.380456 q^{62} -7.90075 q^{64} -5.29550 q^{65} +2.70132 q^{67} +8.26210 q^{68} +0.215554 q^{70} -2.64923 q^{71} +11.1956 q^{73} -0.113041 q^{74} -2.81559 q^{76} -1.12754 q^{77} -11.1257 q^{79} -13.3132 q^{80} -0.0933266 q^{82} +5.61080 q^{83} +13.8640 q^{85} -0.275848 q^{86} -0.289981 q^{88} -7.01954 q^{89} -1.58117 q^{91} -9.06690 q^{92} -0.128324 q^{94} -4.72462 q^{95} -8.30013 q^{97} +0.0643619 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.0643619	0.0455107	0.0227554	−	0.999741i	$-0.492756\pi$
$2$	0.0643619	0.0455107	0.0227554	+	0.999741i	$0.492756\pi$
$3$	0	0
$3$	0	0
$4$	−1.99586	−0.997929
$5$	−3.34910	−1.49776	−0.748881	−	0.662705i	$-0.769408\pi$
$5$	−3.34910	−1.49776	−0.748881	+	0.662705i	$0.769408\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−0.257181	−0.0909271
$9$	0	0
$10$	−0.215554	−0.0681642
$11$	1.12754	0.339965	0.169983	−	0.985447i	$-0.445629\pi$
$11$	1.12754	0.339965	0.169983	+	0.985447i	$0.445629\pi$
$12$	0	0
$13$	1.58117	0.438538	0.219269	−	0.975664i	$-0.429633\pi$
$13$	1.58117	0.438538	0.219269	+	0.975664i	$0.429633\pi$
$14$	−0.0643619	−0.0172014
$15$	0	0
$16$	3.97516	0.993791
$17$	−4.13963	−1.00401	−0.502003	−	0.864866i	$-0.667404\pi$
$17$	−4.13963	−1.00401	−0.502003	+	0.864866i	$0.667404\pi$
$18$	0	0
$19$	1.41071	0.323640	0.161820	−	0.986820i	$-0.448264\pi$
$19$	1.41071	0.323640	0.161820	+	0.986820i	$0.448264\pi$
$20$	6.68432	1.49466
$21$	0	0
$22$	0.0725704	0.0154721
$23$	4.54286	0.947251	0.473626	−	0.880726i	$-0.342945\pi$
$23$	4.54286	0.947251	0.473626	+	0.880726i	$0.342945\pi$
$24$	0	0
$25$	6.21645	1.24329
$26$	0.101767	0.0199582
$27$	0	0
$28$	1.99586	0.377182
$29$	−8.98971	−1.66935	−0.834674	−	0.550745i	$-0.814344\pi$
$29$	−8.98971	−1.66935	−0.834674	+	0.550745i	$0.814344\pi$
$30$	0	0
$31$	−5.91120	−1.06168	−0.530841	−	0.847471i	$-0.678124\pi$
$31$	−5.91120	−1.06168	−0.530841	+	0.847471i	$0.678124\pi$
$32$	0.770210	0.136155
$33$	0	0
$34$	−0.266434	−0.0456931
$35$	3.34910	0.566101
$36$	0	0
$37$	−1.75634	−0.288740	−0.144370	−	0.989524i	$-0.546116\pi$
$37$	−1.75634	−0.288740	−0.144370	+	0.989524i	$0.546116\pi$
$38$	0.0907962	0.0147291
$39$	0	0
$40$	0.861323	0.136187
$41$	−1.45003	−0.226457	−0.113228	−	0.993569i	$-0.536119\pi$
$41$	−1.45003	−0.226457	−0.113228	+	0.993569i	$0.536119\pi$
$42$	0	0
$43$	−4.28589	−0.653592	−0.326796	−	0.945095i	$-0.605969\pi$
$43$	−4.28589	−0.653592	−0.326796	+	0.945095i	$0.605969\pi$
$44$	−2.25040	−0.339261
$45$	0	0
$46$	0.292387	0.0431101
$47$	−1.99378	−0.290823	−0.145412	−	0.989371i	$-0.546451\pi$
$47$	−1.99378	−0.290823	−0.145412	+	0.989371i	$0.546451\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.400102	0.0565830
$51$	0	0
$52$	−3.15579	−0.437630
$53$	−8.37372	−1.15022	−0.575110	−	0.818076i	$-0.695041\pi$
$53$	−8.37372	−1.15022	−0.575110	+	0.818076i	$0.695041\pi$
$54$	0	0
$55$	−3.77623	−0.509187
$56$	0.257181	0.0343672
$57$	0	0
$58$	−0.578594	−0.0759732
$59$	−8.12664	−1.05800	−0.528999	−	0.848622i	$-0.677433\pi$
$59$	−8.12664	−1.05800	−0.528999	+	0.848622i	$0.677433\pi$
$60$	0	0
$61$	3.22482	0.412896	0.206448	−	0.978458i	$-0.433810\pi$
$61$	3.22482	0.412896	0.206448	+	0.978458i	$0.433810\pi$
$62$	−0.380456	−0.0483179
$63$	0	0
$64$	−7.90075	−0.987594
$65$	−5.29550	−0.656826
$66$	0	0
$67$	2.70132	0.330019	0.165010	−	0.986292i	$-0.447234\pi$
$67$	2.70132	0.330019	0.165010	+	0.986292i	$0.447234\pi$
$68$	8.26210	1.00193
$69$	0	0
$70$	0.215554	0.0257636
$71$	−2.64923	−0.314406	−0.157203	−	0.987566i	$-0.550248\pi$
$71$	−2.64923	−0.314406	−0.157203	+	0.987566i	$0.550248\pi$
$72$	0	0
$73$	11.1956	1.31034	0.655171	−	0.755480i	$-0.272596\pi$
$73$	11.1956	1.31034	0.655171	+	0.755480i	$0.272596\pi$
$74$	−0.113041	−0.0131408
$75$	0	0
$76$	−2.81559	−0.322970
$77$	−1.12754	−0.128495
$78$	0	0
$79$	−11.1257	−1.25174	−0.625869	−	0.779928i	$-0.715256\pi$
$79$	−11.1257	−1.25174	−0.625869	+	0.779928i	$0.715256\pi$
$80$	−13.3132	−1.48846
$81$	0	0
$82$	−0.0933266	−0.0103062
$83$	5.61080	0.615865	0.307932	−	0.951408i	$-0.400363\pi$
$83$	5.61080	0.615865	0.307932	+	0.951408i	$0.400363\pi$
$84$	0	0
$85$	13.8640	1.50376
$86$	−0.275848	−0.0297454
$87$	0	0
$88$	−0.289981	−0.0309121
$89$	−7.01954	−0.744070	−0.372035	−	0.928219i	$-0.621340\pi$
$89$	−7.01954	−0.744070	−0.372035	+	0.928219i	$0.621340\pi$
$90$	0	0
$91$	−1.58117	−0.165752
$92$	−9.06690	−0.945289
$93$	0	0
$94$	−0.128324	−0.0132356
$95$	−4.72462	−0.484736
$96$	0	0
$97$	−8.30013	−0.842751	−0.421375	−	0.906886i	$-0.638453\pi$
$97$	−8.30013	−0.842751	−0.421375	+	0.906886i	$0.638453\pi$
$98$	0.0643619	0.00650153
$99$	0	0
$100$	−12.4072	−1.24072
$101$	−12.8152	−1.27516	−0.637581	−	0.770383i	$-0.720065\pi$
$101$	−12.8152	−1.27516	−0.637581	+	0.770383i	$0.720065\pi$
$102$	0	0
$103$	12.8018	1.26140	0.630699	−	0.776028i	$-0.282768\pi$
$103$	12.8018	1.26140	0.630699	+	0.776028i	$0.282768\pi$
$104$	−0.406647	−0.0398750
$105$	0	0
$106$	−0.538948	−0.0523473
$107$	−7.45768	−0.720961	−0.360481	−	0.932767i	$-0.617387\pi$
$107$	−7.45768	−0.720961	−0.360481	+	0.932767i	$0.617387\pi$
$108$	0	0
$109$	−1.81533	−0.173877	−0.0869387	−	0.996214i	$-0.527708\pi$
$109$	−1.81533	−0.173877	−0.0869387	+	0.996214i	$0.527708\pi$
$110$	−0.243045	−0.0231735
$111$	0	0
$112$	−3.97516	−0.375618
$113$	−5.05222	−0.475273	−0.237637	−	0.971354i	$-0.576373\pi$
$113$	−5.05222	−0.475273	−0.237637	+	0.971354i	$0.576373\pi$
$114$	0	0
$115$	−15.2145	−1.41876
$116$	17.9422	1.66589
$117$	0	0
$118$	−0.523046	−0.0481503
$119$	4.13963	0.379479
$120$	0	0
$121$	−9.72866	−0.884424
$122$	0.207556	0.0187912
$123$	0	0
$124$	11.7979	1.05948
$125$	−4.07401	−0.364391
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−2.04893	−0.181101
$129$	0	0
$130$	−0.340828	−0.0298926
$131$	4.59478	0.401448	0.200724	−	0.979648i	$-0.435671\pi$
$131$	4.59478	0.401448	0.200724	+	0.979648i	$0.435671\pi$
$132$	0	0
$133$	−1.41071	−0.122324
$134$	0.173862	0.0150194
$135$	0	0
$136$	1.06463	0.0912915
$137$	−7.31232	−0.624733	−0.312367	−	0.949962i	$-0.601122\pi$
$137$	−7.31232	−0.624733	−0.312367	+	0.949962i	$0.601122\pi$
$138$	0	0
$139$	−1.75924	−0.149217	−0.0746084	−	0.997213i	$-0.523771\pi$
$139$	−1.75924	−0.149217	−0.0746084	+	0.997213i	$0.523771\pi$
$140$	−6.68432	−0.564928
$141$	0	0
$142$	−0.170509	−0.0143088
$143$	1.78283	0.149088
$144$	0	0
$145$	30.1074	2.50028
$146$	0.720568	0.0596346
$147$	0	0
$148$	3.50540	0.288142
$149$	12.6683	1.03783	0.518914	−	0.854827i	$-0.326337\pi$
$149$	12.6683	1.03783	0.518914	+	0.854827i	$0.326337\pi$
$150$	0	0
$151$	−1.26503	−0.102947	−0.0514733	−	0.998674i	$-0.516392\pi$
$151$	−1.26503	−0.102947	−0.0514733	+	0.998674i	$0.516392\pi$
$152$	−0.362809	−0.0294277
$153$	0	0
$154$	−0.0725704	−0.00584789
$155$	19.7972	1.59015
$156$	0	0
$157$	19.4880	1.55531	0.777655	−	0.628692i	$-0.216409\pi$
$157$	19.4880	1.55531	0.777655	+	0.628692i	$0.216409\pi$
$158$	−0.716070	−0.0569675
$159$	0	0
$160$	−2.57951	−0.203928
$161$	−4.54286	−0.358027
$162$	0	0
$163$	−5.18214	−0.405897	−0.202948	−	0.979189i	$-0.565052\pi$
$163$	−5.18214	−0.405897	−0.202948	+	0.979189i	$0.565052\pi$
$164$	2.89405	0.225988
$165$	0	0
$166$	0.361121	0.0280284
$167$	5.11077	0.395483	0.197741	−	0.980254i	$-0.436639\pi$
$167$	5.11077	0.395483	0.197741	+	0.980254i	$0.436639\pi$
$168$	0	0
$169$	−10.4999	−0.807684
$170$	0.892313	0.0684373
$171$	0	0
$172$	8.55403	0.652238
$173$	3.64191	0.276890	0.138445	−	0.990370i	$-0.455790\pi$
$173$	3.64191	0.276890	0.138445	+	0.990370i	$0.455790\pi$
$174$	0	0
$175$	−6.21645	−0.469920
$176$	4.48214	0.337854
$177$	0	0
$178$	−0.451790	−0.0338631
$179$	4.50889	0.337010	0.168505	−	0.985701i	$-0.446106\pi$
$179$	4.50889	0.337010	0.168505	+	0.985701i	$0.446106\pi$
$180$	0	0
$181$	9.87908	0.734306	0.367153	−	0.930160i	$-0.380332\pi$
$181$	9.87908	0.734306	0.367153	+	0.930160i	$0.380332\pi$
$182$	−0.101767	−0.00754348
$183$	0	0
$184$	−1.16834	−0.0861309
$185$	5.88214	0.432464
$186$	0	0
$187$	−4.66758	−0.341328
$188$	3.97931	0.290221
$189$	0	0
$190$	−0.304085	−0.0220607
$191$	−11.0971	−0.802955	−0.401477	−	0.915869i	$-0.631503\pi$
$191$	−11.0971	−0.802955	−0.401477	+	0.915869i	$0.631503\pi$
$192$	0	0
$193$	13.4560	0.968582	0.484291	−	0.874907i	$-0.339078\pi$
$193$	13.4560	0.968582	0.484291	+	0.874907i	$0.339078\pi$
$194$	−0.534212	−0.0383542
$195$	0	0
$196$	−1.99586	−0.142561
$197$	17.7892	1.26743	0.633714	−	0.773567i	$-0.281530\pi$
$197$	17.7892	1.26743	0.633714	+	0.773567i	$0.281530\pi$
$198$	0	0
$199$	24.2628	1.71994	0.859970	−	0.510344i	$-0.170482\pi$
$199$	24.2628	1.71994	0.859970	+	0.510344i	$0.170482\pi$
$200$	−1.59875	−0.113049
$201$	0	0
$202$	−0.824812	−0.0580336
$203$	8.98971	0.630954
$204$	0	0
$205$	4.85629	0.339178
$206$	0.823946	0.0574071
$207$	0	0
$208$	6.28541	0.435815
$209$	1.59063	0.110026
$210$	0	0
$211$	−22.9327	−1.57875	−0.789375	−	0.613911i	$-0.789595\pi$
$211$	−22.9327	−1.57875	−0.789375	+	0.613911i	$0.789595\pi$
$212$	16.7128	1.14784
$213$	0	0
$214$	−0.479990	−0.0328114
$215$	14.3539	0.978925
$216$	0	0
$217$	5.91120	0.401278
$218$	−0.116838	−0.00791328
$219$	0	0
$220$	7.53682	0.508132
$221$	−6.54546	−0.440295
$222$	0	0
$223$	−5.95734	−0.398933	−0.199467	−	0.979905i	$-0.563921\pi$
$223$	−5.95734	−0.398933	−0.199467	+	0.979905i	$0.563921\pi$
$224$	−0.770210	−0.0514618
$225$	0	0
$226$	−0.325170	−0.0216300
$227$	−27.1134	−1.79958	−0.899788	−	0.436327i	$-0.856279\pi$
$227$	−27.1134	−1.79958	−0.899788	+	0.436327i	$0.856279\pi$
$228$	0	0
$229$	−2.25571	−0.149062	−0.0745308	−	0.997219i	$-0.523746\pi$
$229$	−2.25571	−0.149062	−0.0745308	+	0.997219i	$0.523746\pi$
$230$	−0.979232	−0.0645686
$231$	0	0
$232$	2.31198	0.151789
$233$	24.8581	1.62851	0.814254	−	0.580509i	$-0.197146\pi$
$233$	24.8581	1.62851	0.814254	+	0.580509i	$0.197146\pi$
$234$	0	0
$235$	6.67737	0.435584
$236$	16.2196	1.05581
$237$	0	0
$238$	0.266434	0.0172704
$239$	−24.9794	−1.61579	−0.807893	−	0.589329i	$-0.799392\pi$
$239$	−24.9794	−1.61579	−0.807893	+	0.589329i	$0.799392\pi$
$240$	0	0
$241$	10.3175	0.664610	0.332305	−	0.943172i	$-0.392174\pi$
$241$	10.3175	0.664610	0.332305	+	0.943172i	$0.392174\pi$
$242$	−0.626155	−0.0402507
$243$	0	0
$244$	−6.43629	−0.412041
$245$	−3.34910	−0.213966
$246$	0	0
$247$	2.23058	0.141929
$248$	1.52025	0.0965358
$249$	0	0
$250$	−0.262211	−0.0165837
$251$	−12.0809	−0.762540	−0.381270	−	0.924464i	$-0.624513\pi$
$251$	−12.0809	−0.762540	−0.381270	+	0.924464i	$0.624513\pi$
$252$	0	0
$253$	5.12224	0.322033
$254$	−0.0643619	−0.00403842
$255$	0	0
$256$	15.6696	0.979352
$257$	0.876282	0.0546610	0.0273305	−	0.999626i	$-0.491299\pi$
$257$	0.876282	0.0546610	0.0273305	+	0.999626i	$0.491299\pi$
$258$	0	0
$259$	1.75634	0.109133
$260$	10.5691	0.655465
$261$	0	0
$262$	0.295729	0.0182702
$263$	3.67810	0.226802	0.113401	−	0.993549i	$-0.463826\pi$
$263$	3.67810	0.226802	0.113401	+	0.993549i	$0.463826\pi$
$264$	0	0
$265$	28.0444	1.72275
$266$	−0.0907962	−0.00556707
$267$	0	0
$268$	−5.39146	−0.329336
$269$	2.62233	0.159886	0.0799432	−	0.996799i	$-0.474526\pi$
$269$	2.62233	0.159886	0.0799432	+	0.996799i	$0.474526\pi$
$270$	0	0
$271$	26.2552	1.59489	0.797446	−	0.603390i	$-0.206184\pi$
$271$	26.2552	1.59489	0.797446	+	0.603390i	$0.206184\pi$
$272$	−16.4557	−0.997773
$273$	0	0
$274$	−0.470634	−0.0284321
$275$	7.00928	0.422676
$276$	0	0
$277$	0.694219	0.0417116	0.0208558	−	0.999782i	$-0.493361\pi$
$277$	0.694219	0.0417116	0.0208558	+	0.999782i	$0.493361\pi$
$278$	−0.113228	−0.00679096
$279$	0	0
$280$	−0.861323	−0.0514739
$281$	0.817269	0.0487542	0.0243771	−	0.999703i	$-0.492240\pi$
$281$	0.817269	0.0487542	0.0243771	+	0.999703i	$0.492240\pi$
$282$	0	0
$283$	28.4832	1.69315	0.846576	−	0.532267i	$-0.178660\pi$
$283$	28.4832	1.69315	0.846576	+	0.532267i	$0.178660\pi$
$284$	5.28748	0.313754
$285$	0	0
$286$	0.114746	0.00678509
$287$	1.45003	0.0855925
$288$	0	0
$289$	0.136503	0.00802961
$290$	1.93777	0.113790
$291$	0	0
$292$	−22.3448	−1.30763
$293$	19.5729	1.14346	0.571730	−	0.820442i	$-0.306272\pi$
$293$	19.5729	1.14346	0.571730	+	0.820442i	$0.306272\pi$
$294$	0	0
$295$	27.2169	1.58463
$296$	0.451696	0.0262543
$297$	0	0
$298$	0.815355	0.0472323
$299$	7.18304	0.415406
$300$	0	0
$301$	4.28589	0.247035
$302$	−0.0814196	−0.00468517
$303$	0	0
$304$	5.60782	0.321631
$305$	−10.8002	−0.618420
$306$	0	0
$307$	−19.9142	−1.13656	−0.568281	−	0.822835i	$-0.692391\pi$
$307$	−19.9142	−1.13656	−0.568281	+	0.822835i	$0.692391\pi$
$308$	2.25040	0.128229
$309$	0	0
$310$	1.27418	0.0723688
$311$	31.0691	1.76177	0.880884	−	0.473333i	$-0.156949\pi$
$311$	31.0691	1.76177	0.880884	+	0.473333i	$0.156949\pi$
$312$	0	0
$313$	6.81585	0.385255	0.192627	−	0.981272i	$-0.438299\pi$
$313$	6.81585	0.385255	0.192627	+	0.981272i	$0.438299\pi$
$314$	1.25428	0.0707832
$315$	0	0
$316$	22.2053	1.24915
$317$	−0.448737	−0.0252036	−0.0126018	−	0.999921i	$-0.504011\pi$
$317$	−0.448737	−0.0252036	−0.0126018	+	0.999921i	$0.504011\pi$
$318$	0	0
$319$	−10.1362	−0.567520
$320$	26.4604	1.47918
$321$	0	0
$322$	−0.292387	−0.0162941
$323$	−5.83983	−0.324937
$324$	0	0
$325$	9.82928	0.545230
$326$	−0.333532	−0.0184726
$327$	0	0
$328$	0.372920	0.0205910
$329$	1.99378	0.109921
$330$	0	0
$331$	10.3376	0.568204	0.284102	−	0.958794i	$-0.408305\pi$
$331$	10.3376	0.568204	0.284102	+	0.958794i	$0.408305\pi$
$332$	−11.1983	−0.614589
$333$	0	0
$334$	0.328938	0.0179987
$335$	−9.04699	−0.494290
$336$	0	0
$337$	−23.9653	−1.30547	−0.652735	−	0.757586i	$-0.726379\pi$
$337$	−23.9653	−1.30547	−0.652735	+	0.757586i	$0.726379\pi$
$338$	−0.675793	−0.0367583
$339$	0	0
$340$	−27.6706	−1.50065
$341$	−6.66510	−0.360935
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	1.10225	0.0594293
$345$	0	0
$346$	0.234400	0.0126014
$347$	34.5404	1.85423	0.927113	−	0.374782i	$-0.122282\pi$
$347$	34.5404	1.85423	0.927113	+	0.374782i	$0.122282\pi$
$348$	0	0
$349$	−8.99216	−0.481339	−0.240670	−	0.970607i	$-0.577367\pi$
$349$	−8.99216	−0.481339	−0.240670	+	0.970607i	$0.577367\pi$
$350$	−0.400102	−0.0213864
$351$	0	0
$352$	0.868441	0.0462881
$353$	6.13449	0.326506	0.163253	−	0.986584i	$-0.447801\pi$
$353$	6.13449	0.326506	0.163253	+	0.986584i	$0.447801\pi$
$354$	0	0
$355$	8.87252	0.470905
$356$	14.0100	0.742528
$357$	0	0
$358$	0.290200	0.0153376
$359$	31.1838	1.64582	0.822909	−	0.568173i	$-0.192349\pi$
$359$	31.1838	1.64582	0.822909	+	0.568173i	$0.192349\pi$
$360$	0	0
$361$	−17.0099	−0.895257
$362$	0.635836	0.0334188
$363$	0	0
$364$	3.15579	0.165409
$365$	−37.4951	−1.96258
$366$	0	0
$367$	6.41580	0.334902	0.167451	−	0.985880i	$-0.446446\pi$
$367$	6.41580	0.334902	0.167451	+	0.985880i	$0.446446\pi$
$368$	18.0586	0.941369
$369$	0	0
$370$	0.378586	0.0196817
$371$	8.37372	0.434742
$372$	0	0
$373$	−14.7244	−0.762401	−0.381201	−	0.924492i	$-0.624489\pi$
$373$	−14.7244	−0.762401	−0.381201	+	0.924492i	$0.624489\pi$
$374$	−0.300414	−0.0155341
$375$	0	0
$376$	0.512763	0.0264437
$377$	−14.2143	−0.732072
$378$	0	0
$379$	−17.9184	−0.920408	−0.460204	−	0.887813i	$-0.652224\pi$
$379$	−17.9184	−0.920408	−0.460204	+	0.887813i	$0.652224\pi$
$380$	9.42967	0.483732
$381$	0	0
$382$	−0.714227	−0.0365430
$383$	14.8889	0.760788	0.380394	−	0.924825i	$-0.375788\pi$
$383$	14.8889	0.760788	0.380394	+	0.924825i	$0.375788\pi$
$384$	0	0
$385$	3.77623	0.192455
$386$	0.866051	0.0440808
$387$	0	0
$388$	16.5659	0.841005
$389$	26.9900	1.36845	0.684225	−	0.729271i	$-0.260141\pi$
$389$	26.9900	1.36845	0.684225	+	0.729271i	$0.260141\pi$
$390$	0	0
$391$	−18.8057	−0.951047
$392$	−0.257181	−0.0129896
$393$	0	0
$394$	1.14495	0.0576815
$395$	37.2610	1.87481
$396$	0	0
$397$	−11.3273	−0.568501	−0.284251	−	0.958750i	$-0.591745\pi$
$397$	−11.3273	−0.568501	−0.284251	+	0.958750i	$0.591745\pi$
$398$	1.56160	0.0782757
$399$	0	0
$400$	24.7114	1.23557
$401$	−1.40249	−0.0700371	−0.0350185	−	0.999387i	$-0.511149\pi$
$401$	−1.40249	−0.0700371	−0.0350185	+	0.999387i	$0.511149\pi$
$402$	0	0
$403$	−9.34662	−0.465588
$404$	25.5774	1.27252
$405$	0	0
$406$	0.578594	0.0287152
$407$	−1.98034	−0.0981616
$408$	0	0
$409$	−12.0991	−0.598263	−0.299132	−	0.954212i	$-0.596697\pi$
$409$	−12.0991	−0.598263	−0.299132	+	0.954212i	$0.596697\pi$
$410$	0.312560	0.0154362
$411$	0	0
$412$	−25.5505	−1.25878
$413$	8.12664	0.399886
$414$	0	0
$415$	−18.7911	−0.922419
$416$	1.21783	0.0597093
$417$	0	0
$418$	0.102376	0.00500738
$419$	10.2416	0.500333	0.250166	−	0.968203i	$-0.419515\pi$
$419$	10.2416	0.500333	0.250166	+	0.968203i	$0.419515\pi$
$420$	0	0
$421$	19.8420	0.967038	0.483519	−	0.875334i	$-0.339358\pi$
$421$	19.8420	0.967038	0.483519	+	0.875334i	$0.339358\pi$
$422$	−1.47599	−0.0718500
$423$	0	0
$424$	2.15356	0.104586
$425$	−25.7338	−1.24827
$426$	0	0
$427$	−3.22482	−0.156060
$428$	14.8845	0.719468
$429$	0	0
$430$	0.923841	0.0445516
$431$	−33.4368	−1.61059	−0.805296	−	0.592873i	$-0.797994\pi$
$431$	−33.4368	−1.61059	−0.805296	+	0.592873i	$0.797994\pi$
$432$	0	0
$433$	−9.75610	−0.468848	−0.234424	−	0.972134i	$-0.575320\pi$
$433$	−9.75610	−0.468848	−0.234424	+	0.972134i	$0.575320\pi$
$434$	0.380456	0.0182625
$435$	0	0
$436$	3.62315	0.173517
$437$	6.40868	0.306569
$438$	0	0
$439$	−17.3795	−0.829480	−0.414740	−	0.909940i	$-0.636128\pi$
$439$	−17.3795	−0.829480	−0.414740	+	0.909940i	$0.636128\pi$
$440$	0.971175	0.0462989
$441$	0	0
$442$	−0.421278	−0.0200381
$443$	−12.0778	−0.573835	−0.286918	−	0.957955i	$-0.592631\pi$
$443$	−12.0778	−0.573835	−0.286918	+	0.957955i	$0.592631\pi$
$444$	0	0
$445$	23.5091	1.11444
$446$	−0.383426	−0.0181557
$447$	0	0
$448$	7.90075	0.373275
$449$	37.3969	1.76487	0.882435	−	0.470434i	$-0.155903\pi$
$449$	37.3969	1.76487	0.882435	+	0.470434i	$0.155903\pi$
$450$	0	0
$451$	−1.63496	−0.0769874
$452$	10.0835	0.474289
$453$	0	0
$454$	−1.74507	−0.0819000
$455$	5.29550	0.248257
$456$	0	0
$457$	19.9365	0.932589	0.466295	−	0.884630i	$-0.345589\pi$
$457$	19.9365	0.932589	0.466295	+	0.884630i	$0.345589\pi$
$458$	−0.145182	−0.00678390
$459$	0	0
$460$	30.3659	1.41582
$461$	−18.2364	−0.849355	−0.424678	−	0.905345i	$-0.639613\pi$
$461$	−18.2364	−0.849355	−0.424678	+	0.905345i	$0.639613\pi$
$462$	0	0
$463$	−11.2836	−0.524393	−0.262197	−	0.965014i	$-0.584447\pi$
$463$	−11.2836	−0.524393	−0.262197	+	0.965014i	$0.584447\pi$
$464$	−35.7356	−1.65898
$465$	0	0
$466$	1.59991	0.0741145
$467$	−1.07958	−0.0499568	−0.0249784	−	0.999688i	$-0.507952\pi$
$467$	−1.07958	−0.0499568	−0.0249784	+	0.999688i	$0.507952\pi$
$468$	0	0
$469$	−2.70132	−0.124736
$470$	0.429768	0.0198237
$471$	0	0
$472$	2.09002	0.0962008
$473$	−4.83250	−0.222199
$474$	0	0
$475$	8.76964	0.402379
$476$	−8.26210	−0.378693
$477$	0	0
$478$	−1.60772	−0.0735356
$479$	7.79439	0.356135	0.178067	−	0.984018i	$-0.443016\pi$
$479$	7.79439	0.356135	0.178067	+	0.984018i	$0.443016\pi$
$480$	0	0
$481$	−2.77707	−0.126624
$482$	0.664055	0.0302469
$483$	0	0
$484$	19.4170	0.882592
$485$	27.7980	1.26224
$486$	0	0
$487$	3.44378	0.156053	0.0780263	−	0.996951i	$-0.475138\pi$
$487$	3.44378	0.156053	0.0780263	+	0.996951i	$0.475138\pi$
$488$	−0.829363	−0.0375435
$489$	0	0
$490$	−0.215554	−0.00973774
$491$	−8.23053	−0.371438	−0.185719	−	0.982603i	$-0.559461\pi$
$491$	−8.23053	−0.371438	−0.185719	+	0.982603i	$0.559461\pi$
$492$	0	0
$493$	37.2140	1.67604
$494$	0.143564	0.00645927
$495$	0	0
$496$	−23.4980	−1.05509
$497$	2.64923	0.118834
$498$	0	0
$499$	−25.0343	−1.12069	−0.560346	−	0.828259i	$-0.689332\pi$
$499$	−25.0343	−1.12069	−0.560346	+	0.828259i	$0.689332\pi$
$500$	8.13115	0.363636
$501$	0	0
$502$	−0.777550	−0.0347037
$503$	−22.7540	−1.01455	−0.507274	−	0.861785i	$-0.669347\pi$
$503$	−22.7540	−1.01455	−0.507274	+	0.861785i	$0.669347\pi$
$504$	0	0
$505$	42.9194	1.90989
$506$	0.329677	0.0146559
$507$	0	0
$508$	1.99586	0.0885519
$509$	30.5098	1.35232	0.676162	−	0.736753i	$-0.263642\pi$
$509$	30.5098	1.35232	0.676162	+	0.736753i	$0.263642\pi$
$510$	0	0
$511$	−11.1956	−0.495263
$512$	5.10638	0.225672
$513$	0	0
$514$	0.0563991	0.00248766
$515$	−42.8744	−1.88927
$516$	0	0
$517$	−2.24807	−0.0988698
$518$	0.113041	0.00496674
$519$	0	0
$520$	1.36190	0.0597233
$521$	19.7590	0.865656	0.432828	−	0.901477i	$-0.357516\pi$
$521$	19.7590	0.865656	0.432828	+	0.901477i	$0.357516\pi$
$522$	0	0
$523$	4.51884	0.197595	0.0987975	−	0.995108i	$-0.468500\pi$
$523$	4.51884	0.197595	0.0987975	+	0.995108i	$0.468500\pi$
$524$	−9.17054	−0.400617
$525$	0	0
$526$	0.236729	0.0103219
$527$	24.4702	1.06594
$528$	0	0
$529$	−2.36245	−0.102715
$530$	1.80499	0.0784038
$531$	0	0
$532$	2.81559	0.122071
$533$	−2.29275	−0.0993098
$534$	0	0
$535$	24.9765	1.07983
$536$	−0.694728	−0.0300077
$537$	0	0
$538$	0.168778	0.00727655
$539$	1.12754	0.0485665
$540$	0	0
$541$	36.4935	1.56898	0.784489	−	0.620142i	$-0.212925\pi$
$541$	36.4935	1.56898	0.784489	+	0.620142i	$0.212925\pi$
$542$	1.68984	0.0725847
$543$	0	0
$544$	−3.18838	−0.136701
$545$	6.07973	0.260427
$546$	0	0
$547$	7.65446	0.327281	0.163641	−	0.986520i	$-0.447676\pi$
$547$	7.65446	0.327281	0.163641	+	0.986520i	$0.447676\pi$
$548$	14.5943	0.623440
$549$	0	0
$550$	0.451130	0.0192363
$551$	−12.6819	−0.540268
$552$	0	0
$553$	11.1257	0.473113
$554$	0.0446812	0.00189832
$555$	0	0
$556$	3.51119	0.148908
$557$	26.4021	1.11869	0.559346	−	0.828934i	$-0.311052\pi$
$557$	26.4021	1.11869	0.559346	+	0.828934i	$0.311052\pi$
$558$	0	0
$559$	−6.77673	−0.286625
$560$	13.3132	0.562586
$561$	0	0
$562$	0.0526009	0.00221884
$563$	27.9838	1.17938	0.589688	−	0.807631i	$-0.299251\pi$
$563$	27.9838	1.17938	0.589688	+	0.807631i	$0.299251\pi$
$564$	0	0
$565$	16.9204	0.711846
$566$	1.83323	0.0770566
$567$	0	0
$568$	0.681331	0.0285880
$569$	32.5017	1.36254	0.681271	−	0.732032i	$-0.261428\pi$
$569$	32.5017	1.36254	0.681271	+	0.732032i	$0.261428\pi$
$570$	0	0
$571$	−11.6819	−0.488873	−0.244437	−	0.969665i	$-0.578603\pi$
$571$	−11.6819	−0.488873	−0.244437	+	0.969665i	$0.578603\pi$
$572$	−3.55828	−0.148779
$573$	0	0
$574$	0.0933266	0.00389538
$575$	28.2405	1.17771
$576$	0	0
$577$	20.2574	0.843328	0.421664	−	0.906752i	$-0.361446\pi$
$577$	20.2574	0.843328	0.421664	+	0.906752i	$0.361446\pi$
$578$	0.00878561	0.000365433 0
$579$	0	0
$580$	−60.0901	−2.49511
$581$	−5.61080	−0.232775
$582$	0	0
$583$	−9.44169	−0.391035
$584$	−2.87929	−0.119146
$585$	0	0
$586$	1.25975	0.0520397
$587$	−0.691503	−0.0285414	−0.0142707	−	0.999898i	$-0.504543\pi$
$587$	−0.691503	−0.0285414	−0.0142707	+	0.999898i	$0.504543\pi$
$588$	0	0
$589$	−8.33902	−0.343603
$590$	1.75173	0.0721176
$591$	0	0
$592$	−6.98172	−0.286947
$593$	3.16511	0.129976	0.0649878	−	0.997886i	$-0.479299\pi$
$593$	3.16511	0.129976	0.0649878	+	0.997886i	$0.479299\pi$
$594$	0	0
$595$	−13.8640	−0.568369
$596$	−25.2841	−1.03568
$597$	0	0
$598$	0.462314	0.0189054
$599$	39.0051	1.59371	0.796853	−	0.604174i	$-0.206497\pi$
$599$	39.0051	1.59371	0.796853	+	0.604174i	$0.206497\pi$
$600$	0	0
$601$	16.9396	0.690980	0.345490	−	0.938422i	$-0.387713\pi$
$601$	16.9396	0.690980	0.345490	+	0.938422i	$0.387713\pi$
$602$	0.275848	0.0112427
$603$	0	0
$604$	2.52482	0.102733
$605$	32.5822	1.32466
$606$	0	0
$607$	−23.4278	−0.950906	−0.475453	−	0.879741i	$-0.657716\pi$
$607$	−23.4278	−0.950906	−0.475453	+	0.879741i	$0.657716\pi$
$608$	1.08655	0.0440653
$609$	0	0
$610$	−0.695124	−0.0281447
$611$	−3.15251	−0.127537
$612$	0	0
$613$	−46.4858	−1.87755	−0.938773	−	0.344537i	$-0.888036\pi$
$613$	−46.4858	−1.87755	−0.938773	+	0.344537i	$0.888036\pi$
$614$	−1.28171	−0.0517257
$615$	0	0
$616$	0.289981	0.0116837
$617$	−32.2732	−1.29927	−0.649635	−	0.760246i	$-0.725078\pi$
$617$	−32.2732	−1.29927	−0.649635	+	0.760246i	$0.725078\pi$
$618$	0	0
$619$	−37.6206	−1.51210	−0.756049	−	0.654515i	$-0.772873\pi$
$619$	−37.6206	−1.51210	−0.756049	+	0.654515i	$0.772873\pi$
$620$	−39.5124	−1.58685
$621$	0	0
$622$	1.99967	0.0801793
$623$	7.01954	0.281232
$624$	0	0
$625$	−17.4380	−0.697520
$626$	0.438681	0.0175332
$627$	0	0
$628$	−38.8952	−1.55209
$629$	7.27058	0.289897
$630$	0	0
$631$	48.1604	1.91724	0.958618	−	0.284696i	$-0.0918926\pi$
$631$	48.1604	1.91724	0.958618	+	0.284696i	$0.0918926\pi$
$632$	2.86131	0.113817
$633$	0	0
$634$	−0.0288816	−0.00114703
$635$	3.34910	0.132905
$636$	0	0
$637$	1.58117	0.0626483
$638$	−0.652387	−0.0258282
$639$	0	0
$640$	6.86206	0.271247
$641$	8.46967	0.334532	0.167266	−	0.985912i	$-0.446506\pi$
$641$	8.46967	0.334532	0.167266	+	0.985912i	$0.446506\pi$
$642$	0	0
$643$	−15.9873	−0.630479	−0.315240	−	0.949012i	$-0.602085\pi$
$643$	−15.9873	−0.630479	−0.315240	+	0.949012i	$0.602085\pi$
$644$	9.06690	0.357286
$645$	0	0
$646$	−0.375862	−0.0147881
$647$	13.1042	0.515180	0.257590	−	0.966254i	$-0.417072\pi$
$647$	13.1042	0.515180	0.257590	+	0.966254i	$0.417072\pi$
$648$	0	0
$649$	−9.16309	−0.359683
$650$	0.632630	0.0248138
$651$	0	0
$652$	10.3428	0.405056
$653$	35.4675	1.38795	0.693975	−	0.719999i	$-0.255858\pi$
$653$	35.4675	1.38795	0.693975	+	0.719999i	$0.255858\pi$
$654$	0	0
$655$	−15.3884	−0.601274
$656$	−5.76410	−0.225050
$657$	0	0
$658$	0.128324	0.00500257
$659$	−30.8047	−1.19998	−0.599990	−	0.800007i	$-0.704829\pi$
$659$	−30.8047	−1.19998	−0.599990	+	0.800007i	$0.704829\pi$
$660$	0	0
$661$	5.56012	0.216264	0.108132	−	0.994137i	$-0.465513\pi$
$661$	5.56012	0.216264	0.108132	+	0.994137i	$0.465513\pi$
$662$	0.665344	0.0258593
$663$	0	0
$664$	−1.44299	−0.0559988
$665$	4.72462	0.183213
$666$	0	0
$667$	−40.8390	−1.58129
$668$	−10.2004	−0.394664
$669$	0	0
$670$	−0.582281	−0.0224955
$671$	3.63611	0.140370
$672$	0	0
$673$	−30.2783	−1.16714	−0.583571	−	0.812062i	$-0.698345\pi$
$673$	−30.2783	−1.16714	−0.583571	+	0.812062i	$0.698345\pi$
$674$	−1.54245	−0.0594129
$675$	0	0
$676$	20.9563	0.806011
$677$	15.9811	0.614205	0.307102	−	0.951676i	$-0.400641\pi$
$677$	15.9811	0.614205	0.307102	+	0.951676i	$0.400641\pi$
$678$	0	0
$679$	8.30013	0.318530
$680$	−3.56556	−0.136733
$681$	0	0
$682$	−0.428978	−0.0164264
$683$	−29.1314	−1.11468	−0.557340	−	0.830284i	$-0.688178\pi$
$683$	−29.1314	−1.11468	−0.557340	+	0.830284i	$0.688178\pi$
$684$	0	0
$685$	24.4897	0.935702
$686$	−0.0643619	−0.00245735
$687$	0	0
$688$	−17.0371	−0.649534
$689$	−13.2403	−0.504415
$690$	0	0
$691$	13.1079	0.498649	0.249325	−	0.968420i	$-0.419791\pi$
$691$	13.1079	0.498649	0.249325	+	0.968420i	$0.419791\pi$
$692$	−7.26874	−0.276316
$693$	0	0
$694$	2.22308	0.0843871
$695$	5.89187	0.223491
$696$	0	0
$697$	6.00258	0.227364
$698$	−0.578752	−0.0219061
$699$	0	0
$700$	12.4072	0.468946
$701$	−16.9057	−0.638519	−0.319260	−	0.947667i	$-0.603434\pi$
$701$	−16.9057	−0.638519	−0.319260	+	0.947667i	$0.603434\pi$
$702$	0	0
$703$	−2.47769	−0.0934479
$704$	−8.90840	−0.335748
$705$	0	0
$706$	0.394827	0.0148595
$707$	12.8152	0.481966
$708$	0	0
$709$	−6.78349	−0.254759	−0.127380	−	0.991854i	$-0.540657\pi$
$709$	−6.78349	−0.254759	−0.127380	+	0.991854i	$0.540657\pi$
$710$	0.571052	0.0214312
$711$	0	0
$712$	1.80529	0.0676561
$713$	−26.8537	−1.00568
$714$	0	0
$715$	−5.97087	−0.223298
$716$	−8.99910	−0.336312
$717$	0	0
$718$	2.00705	0.0749024
$719$	21.2093	0.790974	0.395487	−	0.918471i	$-0.370576\pi$
$719$	21.2093	0.790974	0.395487	+	0.918471i	$0.370576\pi$
$720$	0	0
$721$	−12.8018	−0.476763
$722$	−1.09479	−0.0407438
$723$	0	0
$724$	−19.7172	−0.732786
$725$	−55.8841	−2.07548
$726$	0	0
$727$	16.9250	0.627715	0.313857	−	0.949470i	$-0.398379\pi$
$727$	16.9250	0.627715	0.313857	+	0.949470i	$0.398379\pi$
$728$	0.406647	0.0150713
$729$	0	0
$730$	−2.41325	−0.0893185
$731$	17.7420	0.656211
$732$	0	0
$733$	3.35622	0.123965	0.0619825	−	0.998077i	$-0.480258\pi$
$733$	3.35622	0.123965	0.0619825	+	0.998077i	$0.480258\pi$
$734$	0.412933	0.0152416
$735$	0	0
$736$	3.49896	0.128973
$737$	3.04584	0.112195
$738$	0	0
$739$	35.0282	1.28853	0.644266	−	0.764801i	$-0.277163\pi$
$739$	35.0282	1.28853	0.644266	+	0.764801i	$0.277163\pi$
$740$	−11.7399	−0.431568
$741$	0	0
$742$	0.538948	0.0197854
$743$	23.8534	0.875097	0.437549	−	0.899195i	$-0.355847\pi$
$743$	23.8534	0.875097	0.437549	+	0.899195i	$0.355847\pi$
$744$	0	0
$745$	−42.4274	−1.55442
$746$	−0.947690	−0.0346974
$747$	0	0
$748$	9.31583	0.340621
$749$	7.45768	0.272498
$750$	0	0
$751$	−41.0931	−1.49951	−0.749754	−	0.661717i	$-0.769828\pi$
$751$	−41.0931	−1.49951	−0.749754	+	0.661717i	$0.769828\pi$
$752$	−7.92561	−0.289017
$753$	0	0
$754$	−0.914857	−0.0333171
$755$	4.23671	0.154190
$756$	0	0
$757$	3.61470	0.131379	0.0656893	−	0.997840i	$-0.479075\pi$
$757$	3.61470	0.131379	0.0656893	+	0.997840i	$0.479075\pi$
$758$	−1.15326	−0.0418884
$759$	0	0
$760$	1.21508	0.0440756
$761$	−6.62933	−0.240313	−0.120157	−	0.992755i	$-0.538340\pi$
$761$	−6.62933	−0.240313	−0.120157	+	0.992755i	$0.538340\pi$
$762$	0	0
$763$	1.81533	0.0657195
$764$	22.1481	0.801292
$765$	0	0
$766$	0.958278	0.0346240
$767$	−12.8496	−0.463973
$768$	0	0
$769$	−15.1383	−0.545901	−0.272951	−	0.962028i	$-0.588000\pi$
$769$	−15.1383	−0.545901	−0.272951	+	0.962028i	$0.588000\pi$
$770$	0.243045	0.00875875
$771$	0	0
$772$	−26.8562	−0.966576
$773$	−12.4529	−0.447899	−0.223950	−	0.974601i	$-0.571895\pi$
$773$	−12.4529	−0.447899	−0.223950	+	0.974601i	$0.571895\pi$
$774$	0	0
$775$	−36.7467	−1.31998
$776$	2.13464	0.0766289
$777$	0	0
$778$	1.73713	0.0622791
$779$	−2.04558	−0.0732904
$780$	0	0
$781$	−2.98710	−0.106887
$782$	−1.21037	−0.0432828
$783$	0	0
$784$	3.97516	0.141970
$785$	−65.2671	−2.32948
$786$	0	0
$787$	14.0610	0.501219	0.250609	−	0.968088i	$-0.419369\pi$
$787$	14.0610	0.501219	0.250609	+	0.968088i	$0.419369\pi$
$788$	−35.5047	−1.26480
$789$	0	0
$790$	2.39819	0.0853237
$791$	5.05222	0.179636
$792$	0	0
$793$	5.09900	0.181071
$794$	−0.729047	−0.0258729
$795$	0	0
$796$	−48.4250	−1.71638
$797$	−5.90795	−0.209270	−0.104635	−	0.994511i	$-0.533367\pi$
$797$	−5.90795	−0.209270	−0.104635	+	0.994511i	$0.533367\pi$
$798$	0	0
$799$	8.25352	0.291988
$800$	4.78798	0.169280
$801$	0	0
$802$	−0.0902670	−0.00318744
$803$	12.6234	0.445471
$804$	0	0
$805$	15.2145	0.536240
$806$	−0.601566	−0.0211893
$807$	0	0
$808$	3.29583	0.115947
$809$	−24.6974	−0.868313	−0.434156	−	0.900838i	$-0.642953\pi$
$809$	−24.6974	−0.868313	−0.434156	+	0.900838i	$0.642953\pi$
$810$	0	0
$811$	7.64665	0.268510	0.134255	−	0.990947i	$-0.457136\pi$
$811$	7.64665	0.268510	0.134255	+	0.990947i	$0.457136\pi$
$812$	−17.9422	−0.629647
$813$	0	0
$814$	−0.127458	−0.00446740
$815$	17.3555	0.607936
$816$	0	0
$817$	−6.04617	−0.211529
$818$	−0.778722	−0.0272274
$819$	0	0
$820$	−9.69246	−0.338475
$821$	−21.5493	−0.752075	−0.376038	−	0.926604i	$-0.622714\pi$
$821$	−21.5493	−0.752075	−0.376038	+	0.926604i	$0.622714\pi$
$822$	0	0
$823$	26.3722	0.919277	0.459639	−	0.888106i	$-0.347979\pi$
$823$	26.3722	0.919277	0.459639	+	0.888106i	$0.347979\pi$
$824$	−3.29237	−0.114695
$825$	0	0
$826$	0.523046	0.0181991
$827$	−24.2793	−0.844273	−0.422136	−	0.906532i	$-0.638720\pi$
$827$	−24.2793	−0.844273	−0.422136	+	0.906532i	$0.638720\pi$
$828$	0	0
$829$	−34.1781	−1.18706	−0.593528	−	0.804813i	$-0.702265\pi$
$829$	−34.1781	−1.18706	−0.593528	+	0.804813i	$0.702265\pi$
$830$	−1.20943	−0.0419799
$831$	0	0
$832$	−12.4924	−0.433098
$833$	−4.13963	−0.143430
$834$	0	0
$835$	−17.1165	−0.592339
$836$	−3.17468	−0.109799
$837$	0	0
$838$	0.659165	0.0227705
$839$	−26.9370	−0.929970	−0.464985	−	0.885319i	$-0.653940\pi$
$839$	−26.9370	−0.929970	−0.464985	+	0.885319i	$0.653940\pi$
$840$	0	0
$841$	51.8149	1.78672
$842$	1.27707	0.0440106
$843$	0	0
$844$	45.7703	1.57548
$845$	35.1652	1.20972
$846$	0	0
$847$	9.72866	0.334281
$848$	−33.2869	−1.14308
$849$	0	0
$850$	−1.65627	−0.0568097
$851$	−7.97879	−0.273509
$852$	0	0
$853$	−41.2185	−1.41129	−0.705647	−	0.708563i	$-0.749344\pi$
$853$	−41.2185	−1.41129	−0.705647	+	0.708563i	$0.749344\pi$
$854$	−0.207556	−0.00710241
$855$	0	0
$856$	1.91797	0.0655549
$857$	0.0250159	0.000854527 0	0.000427263	−	1.00000i	$-0.499864\pi$
$857$	0.0250159	0.000854527 0	0.000427263		1.00000i	$0.499864\pi$
$858$	0	0
$859$	−23.8364	−0.813288	−0.406644	−	0.913587i	$-0.633301\pi$
$859$	−23.8364	−0.813288	−0.406644	+	0.913587i	$0.633301\pi$
$860$	−28.6483	−0.976898
$861$	0	0
$862$	−2.15205	−0.0732992
$863$	46.7979	1.59302	0.796509	−	0.604627i	$-0.206678\pi$
$863$	46.7979	1.59302	0.796509	+	0.604627i	$0.206678\pi$
$864$	0	0
$865$	−12.1971	−0.414715
$866$	−0.627921	−0.0213376
$867$	0	0
$868$	−11.7979	−0.400447
$869$	−12.5446	−0.425548
$870$	0	0
$871$	4.27126	0.144726
$872$	0.466869	0.0158102
$873$	0	0
$874$	0.412474	0.0139522
$875$	4.07401	0.137727
$876$	0	0
$877$	−35.6455	−1.20366	−0.601832	−	0.798623i	$-0.705562\pi$
$877$	−35.6455	−1.20366	−0.601832	+	0.798623i	$0.705562\pi$
$878$	−1.11858	−0.0377502
$879$	0	0
$880$	−15.0111	−0.506025
$881$	28.1228	0.947482	0.473741	−	0.880664i	$-0.342903\pi$
$881$	28.1228	0.947482	0.473741	+	0.880664i	$0.342903\pi$
$882$	0	0
$883$	−43.5319	−1.46497	−0.732483	−	0.680785i	$-0.761639\pi$
$883$	−43.5319	−1.46497	−0.732483	+	0.680785i	$0.761639\pi$
$884$	13.0638	0.439383
$885$	0	0
$886$	−0.777352	−0.0261157
$887$	−21.4396	−0.719872	−0.359936	−	0.932977i	$-0.617201\pi$
$887$	−21.4396	−0.719872	−0.359936	+	0.932977i	$0.617201\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	1.51309	0.0507189
$891$	0	0
$892$	11.8900	0.398107
$893$	−2.81266	−0.0941221
$894$	0	0
$895$	−15.1007	−0.504761
$896$	2.04893	0.0684499
$897$	0	0
$898$	2.40694	0.0803205
$899$	53.1400	1.77232
$900$	0	0
$901$	34.6641	1.15483
$902$	−0.105229	−0.00350375
$903$	0	0
$904$	1.29933	0.0432152
$905$	−33.0860	−1.09982
$906$	0	0
$907$	−17.6827	−0.587145	−0.293573	−	0.955937i	$-0.594844\pi$
$907$	−17.6827	−0.587145	−0.293573	+	0.955937i	$0.594844\pi$
$908$	54.1144	1.79585
$909$	0	0
$910$	0.340828	0.0112983
$911$	11.2583	0.373005	0.186503	−	0.982454i	$-0.440285\pi$
$911$	11.2583	0.373005	0.186503	+	0.982454i	$0.440285\pi$
$912$	0	0
$913$	6.32638	0.209373
$914$	1.28315	0.0424428
$915$	0	0
$916$	4.50208	0.148753
$917$	−4.59478	−0.151733
$918$	0	0
$919$	40.4072	1.33291	0.666455	−	0.745545i	$-0.267811\pi$
$919$	40.4072	1.33291	0.666455	+	0.745545i	$0.267811\pi$
$920$	3.91287	0.129003
$921$	0	0
$922$	−1.17373	−0.0386547
$923$	−4.18888	−0.137879
$924$	0	0
$925$	−10.9182	−0.358988
$926$	−0.726233	−0.0238655
$927$	0	0
$928$	−6.92397	−0.227290
$929$	−22.0644	−0.723908	−0.361954	−	0.932196i	$-0.617890\pi$
$929$	−22.0644	−0.723908	−0.361954	+	0.932196i	$0.617890\pi$
$930$	0	0
$931$	1.41071	0.0462343
$932$	−49.6132	−1.62513
$933$	0	0
$934$	−0.0694835	−0.00227357
$935$	15.6322	0.511227
$936$	0	0
$937$	−18.3151	−0.598329	−0.299164	−	0.954202i	$-0.596708\pi$
$937$	−18.3151	−0.598329	−0.299164	+	0.954202i	$0.596708\pi$
$938$	−0.173862	−0.00567680
$939$	0	0
$940$	−13.3271	−0.434682
$941$	45.4360	1.48117	0.740585	−	0.671963i	$-0.234548\pi$
$941$	45.4360	1.48117	0.740585	+	0.671963i	$0.234548\pi$
$942$	0	0
$943$	−6.58728	−0.214511
$944$	−32.3047	−1.05143
$945$	0	0
$946$	−0.311029	−0.0101124
$947$	−14.0468	−0.456461	−0.228230	−	0.973607i	$-0.573294\pi$
$947$	−14.0468	−0.456461	−0.228230	+	0.973607i	$0.573294\pi$
$948$	0	0
$949$	17.7021	0.574635
$950$	0.564430	0.0183125
$951$	0	0
$952$	−1.06463	−0.0345049
$953$	49.8415	1.61453	0.807263	−	0.590193i	$-0.200948\pi$
$953$	49.8415	1.61453	0.807263	+	0.590193i	$0.200948\pi$
$954$	0	0
$955$	37.1651	1.20263
$956$	49.8554	1.61244
$957$	0	0
$958$	0.501661	0.0162079
$959$	7.31232	0.236127
$960$	0	0
$961$	3.94229	0.127171
$962$	−0.178737	−0.00576272
$963$	0	0
$964$	−20.5923	−0.663234
$965$	−45.0653	−1.45070
$966$	0	0
$967$	−1.72049	−0.0553271	−0.0276635	−	0.999617i	$-0.508807\pi$
$967$	−1.72049	−0.0553271	−0.0276635	+	0.999617i	$0.508807\pi$
$968$	2.50202	0.0804181
$969$	0	0
$970$	1.78913	0.0574454
$971$	46.2773	1.48511	0.742554	−	0.669786i	$-0.233614\pi$
$971$	46.2773	1.48511	0.742554	+	0.669786i	$0.233614\pi$
$972$	0	0
$973$	1.75924	0.0563987
$974$	0.221648	0.00710207
$975$	0	0
$976$	12.8192	0.410332
$977$	−55.5513	−1.77724	−0.888622	−	0.458640i	$-0.848337\pi$
$977$	−55.5513	−1.77724	−0.888622	+	0.458640i	$0.848337\pi$
$978$	0	0
$979$	−7.91479	−0.252958
$980$	6.68432	0.213523
$981$	0	0
$982$	−0.529732	−0.0169044
$983$	12.0011	0.382776	0.191388	−	0.981514i	$-0.438701\pi$
$983$	12.0011	0.382776	0.191388	+	0.981514i	$0.438701\pi$
$984$	0	0
$985$	−59.5778	−1.89831
$986$	2.39516	0.0762776
$987$	0	0
$988$	−4.45192	−0.141635
$989$	−19.4702	−0.619116
$990$	0	0
$991$	53.7528	1.70752	0.853758	−	0.520670i	$-0.174318\pi$
$991$	53.7528	1.70752	0.853758	+	0.520670i	$0.174318\pi$
$992$	−4.55287	−0.144554
$993$	0	0
$994$	0.170509	0.00540822
$995$	−81.2583	−2.57606
$996$	0	0
$997$	31.3231	0.992011	0.496006	−	0.868319i	$-0.334800\pi$
$997$	31.3231	0.992011	0.496006	+	0.868319i	$0.334800\pi$
$998$	−1.61126	−0.0510034
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.s.1.10		16
3.2	odd	2		2667.2.a.n.1.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.n.1.7	✓	16	3.2	odd	2
8001.2.a.s.1.10		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0643619 q^{2} -1.99586 q^{4} -3.34910 q^{5} -1.00000 q^{7} -0.257181 q^{8} +O(q^{10})\)	\(q+0.0643619 q^{2} -1.99586 q^{4} -3.34910 q^{5} -1.00000 q^{7} -0.257181 q^{8} -0.215554 q^{10} +1.12754 q^{11} +1.58117 q^{13} -0.0643619 q^{14} +3.97516 q^{16} -4.13963 q^{17} +1.41071 q^{19} +6.68432 q^{20} +0.0725704 q^{22} +4.54286 q^{23} +6.21645 q^{25} +0.101767 q^{26} +1.99586 q^{28} -8.98971 q^{29} -5.91120 q^{31} +0.770210 q^{32} -0.266434 q^{34} +3.34910 q^{35} -1.75634 q^{37} +0.0907962 q^{38} +0.861323 q^{40} -1.45003 q^{41} -4.28589 q^{43} -2.25040 q^{44} +0.292387 q^{46} -1.99378 q^{47} +1.00000 q^{49} +0.400102 q^{50} -3.15579 q^{52} -8.37372 q^{53} -3.77623 q^{55} +0.257181 q^{56} -0.578594 q^{58} -8.12664 q^{59} +3.22482 q^{61} -0.380456 q^{62} -7.90075 q^{64} -5.29550 q^{65} +2.70132 q^{67} +8.26210 q^{68} +0.215554 q^{70} -2.64923 q^{71} +11.1956 q^{73} -0.113041 q^{74} -2.81559 q^{76} -1.12754 q^{77} -11.1257 q^{79} -13.3132 q^{80} -0.0933266 q^{82} +5.61080 q^{83} +13.8640 q^{85} -0.275848 q^{86} -0.289981 q^{88} -7.01954 q^{89} -1.58117 q^{91} -9.06690 q^{92} -0.128324 q^{94} -4.72462 q^{95} -8.30013 q^{97} +0.0643619 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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