LMFDB - Embedded newform 1251.2.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1251,2,Mod(1,1251)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1251, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1251.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1251 = 3^{2} \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1251.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:	$9.98928529286$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 11x^{5} + 8x^{4} + 35x^{3} - 10x^{2} - 32x - 8 $ x^7 - x^6 - 11x^5 + 8x^4 + 35x^3 - 10x^2 - 32*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 139)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.308806$ of defining polynomial
Character	$\chi$	$=$	1251.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.308806 q^{2} -1.90464 q^{4} -2.83261 q^{5} +4.16776 q^{7} -1.20578 q^{8} +O(q^{10})$	\(q+0.308806 q^{2} -1.90464 q^{4} -2.83261 q^{5} +4.16776 q^{7} -1.20578 q^{8} -0.874728 q^{10} -1.93164 q^{11} +4.12937 q^{13} +1.28703 q^{14} +3.43693 q^{16} -1.03363 q^{17} -7.34157 q^{19} +5.39511 q^{20} -0.596503 q^{22} +4.09697 q^{23} +3.02371 q^{25} +1.27517 q^{26} -7.93807 q^{28} -4.49067 q^{29} -1.28386 q^{31} +3.47289 q^{32} -0.319192 q^{34} -11.8056 q^{35} -7.77849 q^{37} -2.26712 q^{38} +3.41550 q^{40} -11.5603 q^{41} -7.87867 q^{43} +3.67908 q^{44} +1.26517 q^{46} +6.84669 q^{47} +10.3702 q^{49} +0.933738 q^{50} -7.86495 q^{52} -3.68473 q^{53} +5.47160 q^{55} -5.02538 q^{56} -1.38675 q^{58} -1.24534 q^{59} +0.281096 q^{61} -0.396462 q^{62} -5.80140 q^{64} -11.6969 q^{65} -8.51902 q^{67} +1.96870 q^{68} -3.64565 q^{70} +6.40133 q^{71} -5.92608 q^{73} -2.40204 q^{74} +13.9830 q^{76} -8.05062 q^{77} -8.65375 q^{79} -9.73549 q^{80} -3.56988 q^{82} -15.8366 q^{83} +2.92789 q^{85} -2.43298 q^{86} +2.32913 q^{88} +3.23844 q^{89} +17.2102 q^{91} -7.80326 q^{92} +2.11430 q^{94} +20.7958 q^{95} -7.27325 q^{97} +3.20237 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 9 q^{4} - 11 q^{5} - 5 q^{7} - 6 q^{8}+O(q^{10}) $ 7 * q - q^2 + 9 * q^4 - 11 * q^5 - 5 * q^7 - 6 * q^8	$ 7 q - q^{2} + 9 q^{4} - 11 q^{5} - 5 q^{7} - 6 q^{8} - 4 q^{10} - 2 q^{11} + 6 q^{13} - 7 q^{14} + 5 q^{16} - 5 q^{17} - 10 q^{19} - 12 q^{20} - 18 q^{22} + q^{23} + 14 q^{25} + 8 q^{26} - 28 q^{28} - 30 q^{29} - 20 q^{31} + 12 q^{32} - 17 q^{34} + 7 q^{35} + 6 q^{37} - 6 q^{38} - 22 q^{40} - 19 q^{41} - 12 q^{43} - 25 q^{44} + 22 q^{46} + 3 q^{47} - 8 q^{49} - 12 q^{50} - 8 q^{52} - 38 q^{53} + 7 q^{55} - 21 q^{56} - 21 q^{58} + 14 q^{59} + 4 q^{61} + q^{62} - 16 q^{64} - 10 q^{65} + 9 q^{67} + 25 q^{68} + 20 q^{70} - 24 q^{71} - 5 q^{73} - 9 q^{74} + 3 q^{76} + 13 q^{77} + 8 q^{79} - 11 q^{80} + 56 q^{82} + 9 q^{83} - 22 q^{85} - 39 q^{86} - 29 q^{88} - 10 q^{89} + 7 q^{91} - 29 q^{92} - 36 q^{94} + 21 q^{95} - 5 q^{97} + 49 q^{98}+O(q^{100}) $ 7 * q - q^2 + 9 * q^4 - 11 * q^5 - 5 * q^7 - 6 * q^8 - 4 * q^10 - 2 * q^11 + 6 * q^13 - 7 * q^14 + 5 * q^16 - 5 * q^17 - 10 * q^19 - 12 * q^20 - 18 * q^22 + q^23 + 14 * q^25 + 8 * q^26 - 28 * q^28 - 30 * q^29 - 20 * q^31 + 12 * q^32 - 17 * q^34 + 7 * q^35 + 6 * q^37 - 6 * q^38 - 22 * q^40 - 19 * q^41 - 12 * q^43 - 25 * q^44 + 22 * q^46 + 3 * q^47 - 8 * q^49 - 12 * q^50 - 8 * q^52 - 38 * q^53 + 7 * q^55 - 21 * q^56 - 21 * q^58 + 14 * q^59 + 4 * q^61 + q^62 - 16 * q^64 - 10 * q^65 + 9 * q^67 + 25 * q^68 + 20 * q^70 - 24 * q^71 - 5 * q^73 - 9 * q^74 + 3 * q^76 + 13 * q^77 + 8 * q^79 - 11 * q^80 + 56 * q^82 + 9 * q^83 - 22 * q^85 - 39 * q^86 - 29 * q^88 - 10 * q^89 + 7 * q^91 - 29 * q^92 - 36 * q^94 + 21 * q^95 - 5 * q^97 + 49 * 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 $

$2$	0.308806	0.218359	0.109179	−	0.994022i	$-0.465178\pi$
$2$	0.308806	0.218359	0.109179	+	0.994022i	$0.465178\pi$
$3$	0	0
$3$	0	0
$4$	−1.90464	−0.952319
$5$	−2.83261	−1.26678	−0.633392	−	0.773831i	$-0.718338\pi$
$5$	−2.83261	−1.26678	−0.633392	+	0.773831i	$0.718338\pi$
$6$	0	0
$7$	4.16776	1.57526	0.787632	−	0.616146i	$-0.211307\pi$
$7$	4.16776	1.57526	0.787632	+	0.616146i	$0.211307\pi$
$8$	−1.20578	−0.426306
$9$	0	0
$10$	−0.874728	−0.276613
$11$	−1.93164	−0.582412	−0.291206	−	0.956660i	$-0.594057\pi$
$11$	−1.93164	−0.582412	−0.291206	+	0.956660i	$0.594057\pi$
$12$	0	0
$13$	4.12937	1.14528	0.572640	−	0.819807i	$-0.305919\pi$
$13$	4.12937	1.14528	0.572640	+	0.819807i	$0.305919\pi$
$14$	1.28703	0.343972
$15$	0	0
$16$	3.43693	0.859232
$17$	−1.03363	−0.250693	−0.125347	−	0.992113i	$-0.540004\pi$
$17$	−1.03363	−0.250693	−0.125347	+	0.992113i	$0.540004\pi$
$18$	0	0
$19$	−7.34157	−1.68427	−0.842135	−	0.539266i	$-0.818702\pi$
$19$	−7.34157	−1.68427	−0.842135	+	0.539266i	$0.818702\pi$
$20$	5.39511	1.20638
$21$	0	0
$22$	−0.596503	−0.127175
$23$	4.09697	0.854278	0.427139	−	0.904186i	$-0.359521\pi$
$23$	4.09697	0.854278	0.427139	+	0.904186i	$0.359521\pi$
$24$	0	0
$25$	3.02371	0.604741
$26$	1.27517	0.250082
$27$	0	0
$28$	−7.93807	−1.50015
$29$	−4.49067	−0.833897	−0.416948	−	0.908930i	$-0.636900\pi$
$29$	−4.49067	−0.833897	−0.416948	+	0.908930i	$0.636900\pi$
$30$	0	0
$31$	−1.28386	−0.230587	−0.115294	−	0.993331i	$-0.536781\pi$
$31$	−1.28386	−0.230587	−0.115294	+	0.993331i	$0.536781\pi$
$32$	3.47289	0.613927
$33$	0	0
$34$	−0.319192	−0.0547410
$35$	−11.8056	−1.99552
$36$	0	0
$37$	−7.77849	−1.27878	−0.639388	−	0.768884i	$-0.720812\pi$
$37$	−7.77849	−1.27878	−0.639388	+	0.768884i	$0.720812\pi$
$38$	−2.26712	−0.367775
$39$	0	0
$40$	3.41550	0.540038
$41$	−11.5603	−1.80541	−0.902706	−	0.430258i	$-0.858423\pi$
$41$	−11.5603	−1.80541	−0.902706	+	0.430258i	$0.858423\pi$
$42$	0	0
$43$	−7.87867	−1.20149	−0.600743	−	0.799442i	$-0.705129\pi$
$43$	−7.87867	−1.20149	−0.600743	+	0.799442i	$0.705129\pi$
$44$	3.67908	0.554643
$45$	0	0
$46$	1.26517	0.186539
$47$	6.84669	0.998692	0.499346	−	0.866403i	$-0.333574\pi$
$47$	6.84669	0.998692	0.499346	+	0.866403i	$0.333574\pi$
$48$	0	0
$49$	10.3702	1.48145
$50$	0.933738	0.132051
$51$	0	0
$52$	−7.86495	−1.09067
$53$	−3.68473	−0.506136	−0.253068	−	0.967449i	$-0.581440\pi$
$53$	−3.68473	−0.506136	−0.253068	+	0.967449i	$0.581440\pi$
$54$	0	0
$55$	5.47160	0.737791
$56$	−5.02538	−0.671544
$57$	0	0
$58$	−1.38675	−0.182089
$59$	−1.24534	−0.162129	−0.0810647	−	0.996709i	$-0.525832\pi$
$59$	−1.24534	−0.162129	−0.0810647	+	0.996709i	$0.525832\pi$
$60$	0	0
$61$	0.281096	0.0359907	0.0179953	−	0.999838i	$-0.494272\pi$
$61$	0.281096	0.0359907	0.0179953	+	0.999838i	$0.494272\pi$
$62$	−0.396462	−0.0503507
$63$	0	0
$64$	−5.80140	−0.725176
$65$	−11.6969	−1.45082
$66$	0	0
$67$	−8.51902	−1.04076	−0.520382	−	0.853934i	$-0.674210\pi$
$67$	−8.51902	−1.04076	−0.520382	+	0.853934i	$0.674210\pi$
$68$	1.96870	0.238740
$69$	0	0
$70$	−3.64565	−0.435739
$71$	6.40133	0.759698	0.379849	−	0.925048i	$-0.375976\pi$
$71$	6.40133	0.759698	0.379849	+	0.925048i	$0.375976\pi$
$72$	0	0
$73$	−5.92608	−0.693595	−0.346798	−	0.937940i	$-0.612731\pi$
$73$	−5.92608	−0.693595	−0.346798	+	0.937940i	$0.612731\pi$
$74$	−2.40204	−0.279232
$75$	0	0
$76$	13.9830	1.60396
$77$	−8.05062	−0.917453
$78$	0	0
$79$	−8.65375	−0.973623	−0.486811	−	0.873507i	$-0.661840\pi$
$79$	−8.65375	−0.973623	−0.486811	+	0.873507i	$0.661840\pi$
$80$	−9.73549	−1.08846
$81$	0	0
$82$	−3.56988	−0.394228
$83$	−15.8366	−1.73829	−0.869146	−	0.494555i	$-0.835331\pi$
$83$	−15.8366	−1.73829	−0.869146	+	0.494555i	$0.835331\pi$
$84$	0	0
$85$	2.92789	0.317574
$86$	−2.43298	−0.262355
$87$	0	0
$88$	2.32913	0.248286
$89$	3.23844	0.343274	0.171637	−	0.985160i	$-0.445094\pi$
$89$	3.23844	0.343274	0.171637	+	0.985160i	$0.445094\pi$
$90$	0	0
$91$	17.2102	1.80412
$92$	−7.80326	−0.813546
$93$	0	0
$94$	2.11430	0.218073
$95$	20.7958	2.13361
$96$	0	0
$97$	−7.27325	−0.738487	−0.369243	−	0.929333i	$-0.620383\pi$
$97$	−7.27325	−0.738487	−0.369243	+	0.929333i	$0.620383\pi$
$98$	3.20237	0.323489
$99$	0	0
$100$	−5.75907	−0.575907
$101$	13.0456	1.29809	0.649044	−	0.760751i	$-0.275169\pi$
$101$	13.0456	1.29809	0.649044	+	0.760751i	$0.275169\pi$
$102$	0	0
$103$	−11.9048	−1.17302	−0.586510	−	0.809942i	$-0.699498\pi$
$103$	−11.9048	−1.17302	−0.586510	+	0.809942i	$0.699498\pi$
$104$	−4.97909	−0.488240
$105$	0	0
$106$	−1.13786	−0.110519
$107$	−16.1058	−1.55700	−0.778502	−	0.627642i	$-0.784020\pi$
$107$	−16.1058	−1.55700	−0.778502	+	0.627642i	$0.784020\pi$
$108$	0	0
$109$	13.1464	1.25919	0.629597	−	0.776922i	$-0.283220\pi$
$109$	13.1464	1.25919	0.629597	+	0.776922i	$0.283220\pi$
$110$	1.68966	0.161103
$111$	0	0
$112$	14.3243	1.35352
$113$	14.7595	1.38846	0.694228	−	0.719755i	$-0.255746\pi$
$113$	14.7595	1.38846	0.694228	+	0.719755i	$0.255746\pi$
$114$	0	0
$115$	−11.6052	−1.08219
$116$	8.55311	0.794136
$117$	0	0
$118$	−0.384568	−0.0354024
$119$	−4.30793	−0.394908
$120$	0	0
$121$	−7.26876	−0.660796
$122$	0.0868042	0.00785888
$123$	0	0
$124$	2.44528	0.219593
$125$	5.59808	0.500707
$126$	0	0
$127$	11.0142	0.977350	0.488675	−	0.872466i	$-0.337480\pi$
$127$	11.0142	0.977350	0.488675	+	0.872466i	$0.337480\pi$
$128$	−8.73730	−0.772275
$129$	0	0
$130$	−3.61207	−0.316800
$131$	5.25402	0.459046	0.229523	−	0.973303i	$-0.426283\pi$
$131$	5.25402	0.459046	0.229523	+	0.973303i	$0.426283\pi$
$132$	0	0
$133$	−30.5979	−2.65317
$134$	−2.63072	−0.227260
$135$	0	0
$136$	1.24633	0.106872
$137$	1.68115	0.143630	0.0718151	−	0.997418i	$-0.477121\pi$
$137$	1.68115	0.143630	0.0718151	+	0.997418i	$0.477121\pi$
$138$	0	0
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	22.4855	1.90037
$141$	0	0
$142$	1.97677	0.165887
$143$	−7.97646	−0.667025
$144$	0	0
$145$	12.7203	1.05637
$146$	−1.83001	−0.151453
$147$	0	0
$148$	14.8152	1.21780
$149$	−13.9257	−1.14084	−0.570421	−	0.821353i	$-0.693220\pi$
$149$	−13.9257	−1.14084	−0.570421	+	0.821353i	$0.693220\pi$
$150$	0	0
$151$	8.63888	0.703022	0.351511	−	0.936184i	$-0.385668\pi$
$151$	8.63888	0.703022	0.351511	+	0.936184i	$0.385668\pi$
$152$	8.85228	0.718015
$153$	0	0
$154$	−2.48608	−0.200334
$155$	3.63667	0.292104
$156$	0	0
$157$	10.4362	0.832900	0.416450	−	0.909159i	$-0.363274\pi$
$157$	10.4362	0.832900	0.416450	+	0.909159i	$0.363274\pi$
$158$	−2.67233	−0.212599
$159$	0	0
$160$	−9.83737	−0.777712
$161$	17.0752	1.34571
$162$	0	0
$163$	−3.73406	−0.292474	−0.146237	−	0.989250i	$-0.546716\pi$
$163$	−3.73406	−0.292474	−0.146237	+	0.989250i	$0.546716\pi$
$164$	22.0182	1.71933
$165$	0	0
$166$	−4.89043	−0.379571
$167$	19.7503	1.52832	0.764161	−	0.645025i	$-0.223153\pi$
$167$	19.7503	1.52832	0.764161	+	0.645025i	$0.223153\pi$
$168$	0	0
$169$	4.05166	0.311666
$170$	0.904148	0.0693450
$171$	0	0
$172$	15.0060	1.14420
$173$	7.51428	0.571300	0.285650	−	0.958334i	$-0.407790\pi$
$173$	7.51428	0.571300	0.285650	+	0.958334i	$0.407790\pi$
$174$	0	0
$175$	12.6021	0.952627
$176$	−6.63892	−0.500427
$177$	0	0
$178$	1.00005	0.0749568
$179$	−24.2625	−1.81346	−0.906732	−	0.421708i	$-0.861431\pi$
$179$	−24.2625	−1.81346	−0.906732	+	0.421708i	$0.861431\pi$
$180$	0	0
$181$	−3.06020	−0.227463	−0.113732	−	0.993512i	$-0.536280\pi$
$181$	−3.06020	−0.227463	−0.113732	+	0.993512i	$0.536280\pi$
$182$	5.31461	0.393945
$183$	0	0
$184$	−4.94003	−0.364184
$185$	22.0335	1.61993
$186$	0	0
$187$	1.99661	0.146007
$188$	−13.0405	−0.951074
$189$	0	0
$190$	6.42187	0.465892
$191$	−8.83544	−0.639310	−0.319655	−	0.947534i	$-0.603567\pi$
$191$	−8.83544	−0.639310	−0.319655	+	0.947534i	$0.603567\pi$
$192$	0	0
$193$	14.9309	1.07475	0.537374	−	0.843344i	$-0.319416\pi$
$193$	14.9309	1.07475	0.537374	+	0.843344i	$0.319416\pi$
$194$	−2.24602	−0.161255
$195$	0	0
$196$	−19.7515	−1.41082
$197$	11.8981	0.847702	0.423851	−	0.905732i	$-0.360678\pi$
$197$	11.8981	0.847702	0.423851	+	0.905732i	$0.360678\pi$
$198$	0	0
$199$	−4.40723	−0.312420	−0.156210	−	0.987724i	$-0.549928\pi$
$199$	−4.40723	−0.312420	−0.156210	+	0.987724i	$0.549928\pi$
$200$	−3.64591	−0.257805
$201$	0	0
$202$	4.02857	0.283449
$203$	−18.7160	−1.31361
$204$	0	0
$205$	32.7458	2.28707
$206$	−3.67629	−0.256139
$207$	0	0
$208$	14.1923	0.984061
$209$	14.1813	0.980940
$210$	0	0
$211$	21.4218	1.47474	0.737368	−	0.675492i	$-0.236069\pi$
$211$	21.4218	1.47474	0.737368	+	0.675492i	$0.236069\pi$
$212$	7.01807	0.482003
$213$	0	0
$214$	−4.97356	−0.339985
$215$	22.3173	1.52202
$216$	0	0
$217$	−5.35079	−0.363236
$218$	4.05968	0.274956
$219$	0	0
$220$	−10.4214	−0.702612
$221$	−4.26825	−0.287114
$222$	0	0
$223$	−7.93581	−0.531421	−0.265711	−	0.964053i	$-0.585607\pi$
$223$	−7.93581	−0.531421	−0.265711	+	0.964053i	$0.585607\pi$
$224$	14.4742	0.967096
$225$	0	0
$226$	4.55782	0.303181
$227$	−15.9354	−1.05767	−0.528836	−	0.848724i	$-0.677371\pi$
$227$	−15.9354	−1.05767	−0.528836	+	0.848724i	$0.677371\pi$
$228$	0	0
$229$	13.4712	0.890203	0.445102	−	0.895480i	$-0.353168\pi$
$229$	13.4712	0.890203	0.445102	+	0.895480i	$0.353168\pi$
$230$	−3.58374	−0.236305
$231$	0	0
$232$	5.41474	0.355495
$233$	−18.4315	−1.20749	−0.603745	−	0.797177i	$-0.706326\pi$
$233$	−18.4315	−1.20749	−0.603745	+	0.797177i	$0.706326\pi$
$234$	0	0
$235$	−19.3940	−1.26513
$236$	2.37192	0.154399
$237$	0	0
$238$	−1.33031	−0.0862315
$239$	−0.889441	−0.0575331	−0.0287666	−	0.999586i	$-0.509158\pi$
$239$	−0.889441	−0.0575331	−0.0287666	+	0.999586i	$0.509158\pi$
$240$	0	0
$241$	−11.7554	−0.757233	−0.378616	−	0.925554i	$-0.623600\pi$
$241$	−11.7554	−0.757233	−0.378616	+	0.925554i	$0.623600\pi$
$242$	−2.24463	−0.144291
$243$	0	0
$244$	−0.535387	−0.0342746
$245$	−29.3747	−1.87668
$246$	0	0
$247$	−30.3160	−1.92896
$248$	1.54804	0.0983007
$249$	0	0
$250$	1.72872	0.109334
$251$	14.4551	0.912397	0.456199	−	0.889878i	$-0.349211\pi$
$251$	14.4551	0.912397	0.456199	+	0.889878i	$0.349211\pi$
$252$	0	0
$253$	−7.91389	−0.497542
$254$	3.40124	0.213413
$255$	0	0
$256$	8.90468	0.556543
$257$	−17.2989	−1.07907	−0.539537	−	0.841962i	$-0.681401\pi$
$257$	−17.2989	−1.07907	−0.539537	+	0.841962i	$0.681401\pi$
$258$	0	0
$259$	−32.4189	−2.01441
$260$	22.2784	1.38165
$261$	0	0
$262$	1.62247	0.100237
$263$	−4.00876	−0.247191	−0.123595	−	0.992333i	$-0.539442\pi$
$263$	−4.00876	−0.247191	−0.123595	+	0.992333i	$0.539442\pi$
$264$	0	0
$265$	10.4374	0.641165
$266$	−9.44880	−0.579343
$267$	0	0
$268$	16.2257	0.991140
$269$	−28.0301	−1.70902	−0.854511	−	0.519433i	$-0.826143\pi$
$269$	−28.0301	−1.70902	−0.854511	+	0.519433i	$0.826143\pi$
$270$	0	0
$271$	16.3058	0.990507	0.495253	−	0.868749i	$-0.335075\pi$
$271$	16.3058	0.990507	0.495253	+	0.868749i	$0.335075\pi$
$272$	−3.55252	−0.215403
$273$	0	0
$274$	0.519149	0.0313629
$275$	−5.84072	−0.352209
$276$	0	0
$277$	−26.3241	−1.58166	−0.790831	−	0.612035i	$-0.790351\pi$
$277$	−26.3241	−1.58166	−0.790831	+	0.612035i	$0.790351\pi$
$278$	0.308806	0.0185209
$279$	0	0
$280$	14.2350	0.850701
$281$	7.58167	0.452285	0.226142	−	0.974094i	$-0.427389\pi$
$281$	7.58167	0.452285	0.226142	+	0.974094i	$0.427389\pi$
$282$	0	0
$283$	−14.2221	−0.845417	−0.422709	−	0.906266i	$-0.638921\pi$
$283$	−14.2221	−0.845417	−0.422709	+	0.906266i	$0.638921\pi$
$284$	−12.1922	−0.723475
$285$	0	0
$286$	−2.46318	−0.145651
$287$	−48.1804	−2.84400
$288$	0	0
$289$	−15.9316	−0.937153
$290$	3.92812	0.230667
$291$	0	0
$292$	11.2870	0.660524
$293$	4.68983	0.273983	0.136991	−	0.990572i	$-0.456257\pi$
$293$	4.68983	0.273983	0.136991	+	0.990572i	$0.456257\pi$
$294$	0	0
$295$	3.52757	0.205383
$296$	9.37912	0.545150
$297$	0	0
$298$	−4.30035	−0.249113
$299$	16.9179	0.978388
$300$	0	0
$301$	−32.8364	−1.89266
$302$	2.66774	0.153511
$303$	0	0
$304$	−25.2324	−1.44718
$305$	−0.796238	−0.0455924
$306$	0	0
$307$	10.6916	0.610199	0.305100	−	0.952320i	$-0.401310\pi$
$307$	10.6916	0.610199	0.305100	+	0.952320i	$0.401310\pi$
$308$	15.3335	0.873708
$309$	0	0
$310$	1.12302	0.0637835
$311$	21.0455	1.19338	0.596690	−	0.802472i	$-0.296482\pi$
$311$	21.0455	1.19338	0.596690	+	0.802472i	$0.296482\pi$
$312$	0	0
$313$	−14.2033	−0.802819	−0.401409	−	0.915899i	$-0.631480\pi$
$313$	−14.2033	−0.802819	−0.401409	+	0.915899i	$0.631480\pi$
$314$	3.22276	0.181871
$315$	0	0
$316$	16.4823	0.927200
$317$	−2.85738	−0.160487	−0.0802433	−	0.996775i	$-0.525570\pi$
$317$	−2.85738	−0.160487	−0.0802433	+	0.996775i	$0.525570\pi$
$318$	0	0
$319$	8.67437	0.485672
$320$	16.4331	0.918641
$321$	0	0
$322$	5.27292	0.293848
$323$	7.58849	0.422235
$324$	0	0
$325$	12.4860	0.692598
$326$	−1.15310	−0.0638642
$327$	0	0
$328$	13.9391	0.769658
$329$	28.5353	1.57320
$330$	0	0
$331$	12.1172	0.666019	0.333010	−	0.942923i	$-0.391936\pi$
$331$	12.1172	0.666019	0.333010	+	0.942923i	$0.391936\pi$
$332$	30.1630	1.65541
$333$	0	0
$334$	6.09900	0.333722
$335$	24.1311	1.31842
$336$	0	0
$337$	−12.4929	−0.680532	−0.340266	−	0.940329i	$-0.610517\pi$
$337$	−12.4929	−0.680532	−0.340266	+	0.940329i	$0.610517\pi$
$338$	1.25117	0.0680549
$339$	0	0
$340$	−5.57657	−0.302432
$341$	2.47995	0.134297
$342$	0	0
$343$	14.0461	0.758418
$344$	9.49991	0.512201
$345$	0	0
$346$	2.32045	0.124748
$347$	11.1030	0.596039	0.298020	−	0.954560i	$-0.403674\pi$
$347$	11.1030	0.596039	0.298020	+	0.954560i	$0.403674\pi$
$348$	0	0
$349$	16.3372	0.874509	0.437254	−	0.899338i	$-0.355951\pi$
$349$	16.3372	0.874509	0.437254	+	0.899338i	$0.355951\pi$
$350$	3.89159	0.208014
$351$	0	0
$352$	−6.70839	−0.357558
$353$	26.2175	1.39542	0.697709	−	0.716382i	$-0.254203\pi$
$353$	26.2175	1.39542	0.697709	+	0.716382i	$0.254203\pi$
$354$	0	0
$355$	−18.1325	−0.962373
$356$	−6.16806	−0.326906
$357$	0	0
$358$	−7.49240	−0.395986
$359$	−18.6480	−0.984204	−0.492102	−	0.870537i	$-0.663771\pi$
$359$	−18.6480	−0.984204	−0.492102	+	0.870537i	$0.663771\pi$
$360$	0	0
$361$	34.8986	1.83677
$362$	−0.945009	−0.0496686
$363$	0	0
$364$	−32.7792	−1.71810
$365$	16.7863	0.878636
$366$	0	0
$367$	−3.92562	−0.204916	−0.102458	−	0.994737i	$-0.532671\pi$
$367$	−3.92562	−0.204916	−0.102458	+	0.994737i	$0.532671\pi$
$368$	14.0810	0.734023
$369$	0	0
$370$	6.80407	0.353727
$371$	−15.3570	−0.797297
$372$	0	0
$373$	14.3279	0.741870	0.370935	−	0.928659i	$-0.379037\pi$
$373$	14.3279	0.741870	0.370935	+	0.928659i	$0.379037\pi$
$374$	0.616565	0.0318818
$375$	0	0
$376$	−8.25557	−0.425749
$377$	−18.5436	−0.955045
$378$	0	0
$379$	5.65835	0.290650	0.145325	−	0.989384i	$-0.453577\pi$
$379$	5.65835	0.290650	0.145325	+	0.989384i	$0.453577\pi$
$380$	−39.6085	−2.03188
$381$	0	0
$382$	−2.72844	−0.139599
$383$	−4.23323	−0.216308	−0.108154	−	0.994134i	$-0.534494\pi$
$383$	−4.23323	−0.216308	−0.108154	+	0.994134i	$0.534494\pi$
$384$	0	0
$385$	22.8043	1.16221
$386$	4.61074	0.234681
$387$	0	0
$388$	13.8529	0.703275
$389$	5.21409	0.264365	0.132182	−	0.991225i	$-0.457802\pi$
$389$	5.21409	0.264365	0.132182	+	0.991225i	$0.457802\pi$
$390$	0	0
$391$	−4.23477	−0.214162
$392$	−12.5041	−0.631553
$393$	0	0
$394$	3.67419	0.185103
$395$	24.5127	1.23337
$396$	0	0
$397$	9.63263	0.483448	0.241724	−	0.970345i	$-0.422287\pi$
$397$	9.63263	0.483448	0.241724	+	0.970345i	$0.422287\pi$
$398$	−1.36098	−0.0682196
$399$	0	0
$400$	10.3923	0.519613
$401$	−24.8722	−1.24206	−0.621030	−	0.783787i	$-0.713286\pi$
$401$	−24.8722	−1.24206	−0.621030	+	0.783787i	$0.713286\pi$
$402$	0	0
$403$	−5.30151	−0.264087
$404$	−24.8472	−1.23619
$405$	0	0
$406$	−5.77962	−0.286837
$407$	15.0253	0.744775
$408$	0	0
$409$	−28.4449	−1.40651	−0.703255	−	0.710938i	$-0.748271\pi$
$409$	−28.4449	−1.40651	−0.703255	+	0.710938i	$0.748271\pi$
$410$	10.1121	0.499401
$411$	0	0
$412$	22.6744	1.11709
$413$	−5.19027	−0.255397
$414$	0	0
$415$	44.8590	2.20204
$416$	14.3408	0.703118
$417$	0	0
$418$	4.37926	0.214197
$419$	4.59075	0.224273	0.112136	−	0.993693i	$-0.464231\pi$
$419$	4.59075	0.224273	0.112136	+	0.993693i	$0.464231\pi$
$420$	0	0
$421$	20.9586	1.02146	0.510729	−	0.859742i	$-0.329375\pi$
$421$	20.9586	1.02146	0.510729	+	0.859742i	$0.329375\pi$
$422$	6.61517	0.322021
$423$	0	0
$424$	4.44295	0.215769
$425$	−3.12541	−0.151604
$426$	0	0
$427$	1.17154	0.0566948
$428$	30.6757	1.48276
$429$	0	0
$430$	6.89170	0.332347
$431$	−14.1727	−0.682676	−0.341338	−	0.939941i	$-0.610880\pi$
$431$	−14.1727	−0.682676	−0.341338	+	0.939941i	$0.610880\pi$
$432$	0	0
$433$	7.32714	0.352120	0.176060	−	0.984379i	$-0.443665\pi$
$433$	7.32714	0.352120	0.176060	+	0.984379i	$0.443665\pi$
$434$	−1.65236	−0.0793157
$435$	0	0
$436$	−25.0391	−1.19916
$437$	−30.0782	−1.43884
$438$	0	0
$439$	3.07088	0.146565	0.0732825	−	0.997311i	$-0.476653\pi$
$439$	3.07088	0.146565	0.0732825	+	0.997311i	$0.476653\pi$
$440$	−6.59752	−0.314525
$441$	0	0
$442$	−1.31806	−0.0626938
$443$	−14.9982	−0.712586	−0.356293	−	0.934374i	$-0.615960\pi$
$443$	−14.9982	−0.712586	−0.356293	+	0.934374i	$0.615960\pi$
$444$	0	0
$445$	−9.17325	−0.434854
$446$	−2.45063	−0.116041
$447$	0	0
$448$	−24.1788	−1.14234
$449$	22.4064	1.05742	0.528711	−	0.848802i	$-0.322676\pi$
$449$	22.4064	1.05742	0.528711	+	0.848802i	$0.322676\pi$
$450$	0	0
$451$	22.3303	1.05149
$452$	−28.1115	−1.32225
$453$	0	0
$454$	−4.92096	−0.230952
$455$	−48.7498	−2.28543
$456$	0	0
$457$	−12.9317	−0.604919	−0.302460	−	0.953162i	$-0.597808\pi$
$457$	−12.9317	−0.604919	−0.302460	+	0.953162i	$0.597808\pi$
$458$	4.15999	0.194384
$459$	0	0
$460$	22.1036	1.03059
$461$	39.7115	1.84955	0.924774	−	0.380517i	$-0.124254\pi$
$461$	39.7115	1.84955	0.924774	+	0.380517i	$0.124254\pi$
$462$	0	0
$463$	21.4987	0.999130	0.499565	−	0.866276i	$-0.333493\pi$
$463$	21.4987	0.999130	0.499565	+	0.866276i	$0.333493\pi$
$464$	−15.4341	−0.716510
$465$	0	0
$466$	−5.69177	−0.263666
$467$	19.2688	0.891655	0.445827	−	0.895119i	$-0.352910\pi$
$467$	19.2688	0.891655	0.445827	+	0.895119i	$0.352910\pi$
$468$	0	0
$469$	−35.5052	−1.63948
$470$	−5.98899	−0.276252
$471$	0	0
$472$	1.50160	0.0691168
$473$	15.2188	0.699761
$474$	0	0
$475$	−22.1987	−1.01855
$476$	8.20506	0.376078
$477$	0	0
$478$	−0.274665	−0.0125629
$479$	23.7465	1.08500	0.542502	−	0.840055i	$-0.317477\pi$
$479$	23.7465	1.08500	0.542502	+	0.840055i	$0.317477\pi$
$480$	0	0
$481$	−32.1202	−1.46456
$482$	−3.63014	−0.165348
$483$	0	0
$484$	13.8444	0.629289
$485$	20.6023	0.935503
$486$	0	0
$487$	−27.3504	−1.23937	−0.619683	−	0.784852i	$-0.712739\pi$
$487$	−27.3504	−1.23937	−0.619683	+	0.784852i	$0.712739\pi$
$488$	−0.338939	−0.0153430
$489$	0	0
$490$	−9.07109	−0.409790
$491$	29.0333	1.31025	0.655126	−	0.755519i	$-0.272615\pi$
$491$	29.0333	1.31025	0.655126	+	0.755519i	$0.272615\pi$
$492$	0	0
$493$	4.64171	0.209052
$494$	−9.36176	−0.421206
$495$	0	0
$496$	−4.41252	−0.198128
$497$	26.6792	1.19672
$498$	0	0
$499$	26.6880	1.19472	0.597359	−	0.801974i	$-0.296217\pi$
$499$	26.6880	1.19472	0.597359	+	0.801974i	$0.296217\pi$
$500$	−10.6623	−0.476833
$501$	0	0
$502$	4.46382	0.199230
$503$	−7.41588	−0.330658	−0.165329	−	0.986238i	$-0.552869\pi$
$503$	−7.41588	−0.330658	−0.165329	+	0.986238i	$0.552869\pi$
$504$	0	0
$505$	−36.9532	−1.64440
$506$	−2.44386	−0.108643
$507$	0	0
$508$	−20.9780	−0.930749
$509$	−36.1455	−1.60212	−0.801060	−	0.598583i	$-0.795730\pi$
$509$	−36.1455	−1.60212	−0.801060	+	0.598583i	$0.795730\pi$
$510$	0	0
$511$	−24.6985	−1.09260
$512$	20.2244	0.893801
$513$	0	0
$514$	−5.34199	−0.235625
$515$	33.7219	1.48596
$516$	0	0
$517$	−13.2254	−0.581651
$518$	−10.0111	−0.439864
$519$	0	0
$520$	14.1038	0.618494
$521$	−33.9636	−1.48797	−0.743986	−	0.668196i	$-0.767067\pi$
$521$	−33.9636	−1.48797	−0.743986	+	0.668196i	$0.767067\pi$
$522$	0	0
$523$	−11.7578	−0.514134	−0.257067	−	0.966394i	$-0.582756\pi$
$523$	−11.7578	−0.514134	−0.257067	+	0.966394i	$0.582756\pi$
$524$	−10.0070	−0.437158
$525$	0	0
$526$	−1.23793	−0.0539762
$527$	1.32704	0.0578066
$528$	0	0
$529$	−6.21480	−0.270209
$530$	3.22313	0.140004
$531$	0	0
$532$	58.2779	2.52667
$533$	−47.7366	−2.06770
$534$	0	0
$535$	45.6214	1.97239
$536$	10.2720	0.443684
$537$	0	0
$538$	−8.65585	−0.373180
$539$	−20.0315	−0.862818
$540$	0	0
$541$	0.804094	0.0345707	0.0172854	−	0.999851i	$-0.494498\pi$
$541$	0.804094	0.0345707	0.0172854	+	0.999851i	$0.494498\pi$
$542$	5.03533	0.216286
$543$	0	0
$544$	−3.58970	−0.153907
$545$	−37.2386	−1.59513
$546$	0	0
$547$	−4.34094	−0.185605	−0.0928027	−	0.995685i	$-0.529583\pi$
$547$	−4.34094	−0.185605	−0.0928027	+	0.995685i	$0.529583\pi$
$548$	−3.20198	−0.136782
$549$	0	0
$550$	−1.80365	−0.0769079
$551$	32.9686	1.40451
$552$	0	0
$553$	−36.0667	−1.53371
$554$	−8.12904	−0.345370
$555$	0	0
$556$	−1.90464	−0.0807747
$557$	29.2321	1.23861	0.619303	−	0.785152i	$-0.287415\pi$
$557$	29.2321	1.23861	0.619303	+	0.785152i	$0.287415\pi$
$558$	0	0
$559$	−32.5339	−1.37604
$560$	−40.5751	−1.71461
$561$	0	0
$562$	2.34126	0.0987603
$563$	−27.9325	−1.17721	−0.588606	−	0.808420i	$-0.700323\pi$
$563$	−27.9325	−1.17721	−0.588606	+	0.808420i	$0.700323\pi$
$564$	0	0
$565$	−41.8079	−1.75887
$566$	−4.39187	−0.184604
$567$	0	0
$568$	−7.71857	−0.323864
$569$	−21.5023	−0.901424	−0.450712	−	0.892669i	$-0.648830\pi$
$569$	−21.5023	−0.901424	−0.450712	+	0.892669i	$0.648830\pi$
$570$	0	0
$571$	19.6384	0.821843	0.410922	−	0.911671i	$-0.365207\pi$
$571$	19.6384	0.821843	0.410922	+	0.911671i	$0.365207\pi$
$572$	15.1923	0.635221
$573$	0	0
$574$	−14.8784	−0.621012
$575$	12.3881	0.516617
$576$	0	0
$577$	14.7632	0.614599	0.307299	−	0.951613i	$-0.400575\pi$
$577$	14.7632	0.614599	0.307299	+	0.951613i	$0.400575\pi$
$578$	−4.91977	−0.204636
$579$	0	0
$580$	−24.2277	−1.00600
$581$	−66.0031	−2.73827
$582$	0	0
$583$	7.11757	0.294780
$584$	7.14552	0.295684
$585$	0	0
$586$	1.44825	0.0598265
$587$	6.27328	0.258926	0.129463	−	0.991584i	$-0.458675\pi$
$587$	6.27328	0.258926	0.129463	+	0.991584i	$0.458675\pi$
$588$	0	0
$589$	9.42551	0.388371
$590$	1.08933	0.0448472
$591$	0	0
$592$	−26.7341	−1.09877
$593$	−40.8082	−1.67579	−0.837895	−	0.545831i	$-0.816214\pi$
$593$	−40.8082	−1.67579	−0.837895	+	0.545831i	$0.816214\pi$
$594$	0	0
$595$	12.2027	0.500263
$596$	26.5235	1.08645
$597$	0	0
$598$	5.22435	0.213639
$599$	10.3330	0.422193	0.211096	−	0.977465i	$-0.432297\pi$
$599$	10.3330	0.422193	0.211096	+	0.977465i	$0.432297\pi$
$600$	0	0
$601$	15.7883	0.644019	0.322009	−	0.946737i	$-0.395642\pi$
$601$	15.7883	0.644019	0.322009	+	0.946737i	$0.395642\pi$
$602$	−10.1401	−0.413278
$603$	0	0
$604$	−16.4539	−0.669502
$605$	20.5896	0.837086
$606$	0	0
$607$	12.8843	0.522958	0.261479	−	0.965209i	$-0.415790\pi$
$607$	12.8843	0.522958	0.261479	+	0.965209i	$0.415790\pi$
$608$	−25.4965	−1.03402
$609$	0	0
$610$	−0.245883	−0.00995551
$611$	28.2725	1.14378
$612$	0	0
$613$	6.99917	0.282694	0.141347	−	0.989960i	$-0.454857\pi$
$613$	6.99917	0.282694	0.141347	+	0.989960i	$0.454857\pi$
$614$	3.30162	0.133242
$615$	0	0
$616$	9.70723	0.391116
$617$	8.16208	0.328593	0.164297	−	0.986411i	$-0.447465\pi$
$617$	8.16208	0.328593	0.164297	+	0.986411i	$0.447465\pi$
$618$	0	0
$619$	−12.6031	−0.506559	−0.253280	−	0.967393i	$-0.581509\pi$
$619$	−12.6031	−0.506559	−0.253280	+	0.967393i	$0.581509\pi$
$620$	−6.92654	−0.278176
$621$	0	0
$622$	6.49897	0.260585
$623$	13.4970	0.540747
$624$	0	0
$625$	−30.9757	−1.23903
$626$	−4.38607	−0.175303
$627$	0	0
$628$	−19.8772	−0.793187
$629$	8.04011	0.320580
$630$	0	0
$631$	45.4769	1.81041	0.905203	−	0.424980i	$-0.139719\pi$
$631$	45.4769	1.81041	0.905203	+	0.424980i	$0.139719\pi$
$632$	10.4345	0.415061
$633$	0	0
$634$	−0.882377	−0.0350437
$635$	−31.1989	−1.23809
$636$	0	0
$637$	42.8223	1.69668
$638$	2.67870	0.106051
$639$	0	0
$640$	24.7494	0.978306
$641$	−12.1865	−0.481339	−0.240669	−	0.970607i	$-0.577367\pi$
$641$	−12.1865	−0.481339	−0.240669	+	0.970607i	$0.577367\pi$
$642$	0	0
$643$	29.7340	1.17259	0.586296	−	0.810097i	$-0.300585\pi$
$643$	29.7340	1.17259	0.586296	+	0.810097i	$0.300585\pi$
$644$	−32.5221	−1.28155
$645$	0	0
$646$	2.34337	0.0921987
$647$	32.8228	1.29040	0.645199	−	0.764015i	$-0.276774\pi$
$647$	32.8228	1.29040	0.645199	+	0.764015i	$0.276774\pi$
$648$	0	0
$649$	2.40555	0.0944262
$650$	3.85575	0.151235
$651$	0	0
$652$	7.11203	0.278529
$653$	−10.3258	−0.404078	−0.202039	−	0.979377i	$-0.564757\pi$
$653$	−10.3258	−0.404078	−0.202039	+	0.979377i	$0.564757\pi$
$654$	0	0
$655$	−14.8826	−0.581512
$656$	−39.7318	−1.55127
$657$	0	0
$658$	8.81188	0.343523
$659$	−25.8076	−1.00532	−0.502661	−	0.864484i	$-0.667646\pi$
$659$	−25.8076	−1.00532	−0.502661	+	0.864484i	$0.667646\pi$
$660$	0	0
$661$	−13.3955	−0.521025	−0.260513	−	0.965470i	$-0.583892\pi$
$661$	−13.3955	−0.521025	−0.260513	+	0.965470i	$0.583892\pi$
$662$	3.74185	0.145431
$663$	0	0
$664$	19.0954	0.741044
$665$	86.6719	3.36099
$666$	0	0
$667$	−18.3982	−0.712380
$668$	−37.6171	−1.45545
$669$	0	0
$670$	7.45182	0.287889
$671$	−0.542978	−0.0209614
$672$	0	0
$673$	−6.82079	−0.262922	−0.131461	−	0.991321i	$-0.541967\pi$
$673$	−6.82079	−0.262922	−0.131461	+	0.991321i	$0.541967\pi$
$674$	−3.85788	−0.148600
$675$	0	0
$676$	−7.71694	−0.296805
$677$	−10.5869	−0.406888	−0.203444	−	0.979087i	$-0.565213\pi$
$677$	−10.5869	−0.406888	−0.203444	+	0.979087i	$0.565213\pi$
$678$	0	0
$679$	−30.3131	−1.16331
$680$	−3.53037	−0.135384
$681$	0	0
$682$	0.765823	0.0293249
$683$	−43.3107	−1.65724	−0.828620	−	0.559812i	$-0.810873\pi$
$683$	−43.3107	−1.65724	−0.828620	+	0.559812i	$0.810873\pi$
$684$	0	0
$685$	−4.76205	−0.181948
$686$	4.33752	0.165607
$687$	0	0
$688$	−27.0784	−1.03236
$689$	−15.2156	−0.579667
$690$	0	0
$691$	−27.5238	−1.04706	−0.523528	−	0.852009i	$-0.675384\pi$
$691$	−27.5238	−1.04706	−0.523528	+	0.852009i	$0.675384\pi$
$692$	−14.3120	−0.544060
$693$	0	0
$694$	3.42867	0.130150
$695$	−2.83261	−0.107447
$696$	0	0
$697$	11.9491	0.452604
$698$	5.04501	0.190957
$699$	0	0
$700$	−24.0024	−0.907205
$701$	−51.4490	−1.94320	−0.971601	−	0.236625i	$-0.923959\pi$
$701$	−51.4490	−1.94320	−0.971601	+	0.236625i	$0.923959\pi$
$702$	0	0
$703$	57.1063	2.15381
$704$	11.2062	0.422351
$705$	0	0
$706$	8.09612	0.304702
$707$	54.3710	2.04483
$708$	0	0
$709$	16.8255	0.631894	0.315947	−	0.948777i	$-0.397678\pi$
$709$	16.8255	0.631894	0.315947	+	0.948777i	$0.397678\pi$
$710$	−5.59942	−0.210143
$711$	0	0
$712$	−3.90483	−0.146340
$713$	−5.25992	−0.196986
$714$	0	0
$715$	22.5942	0.844977
$716$	46.2113	1.72700
$717$	0	0
$718$	−5.75862	−0.214910
$719$	−26.2933	−0.980574	−0.490287	−	0.871561i	$-0.663108\pi$
$719$	−26.2933	−0.980574	−0.490287	+	0.871561i	$0.663108\pi$
$720$	0	0
$721$	−49.6165	−1.84781
$722$	10.7769	0.401074
$723$	0	0
$724$	5.82858	0.216618
$725$	−13.5785	−0.504292
$726$	0	0
$727$	23.5378	0.872967	0.436484	−	0.899712i	$-0.356224\pi$
$727$	23.5378	0.872967	0.436484	+	0.899712i	$0.356224\pi$
$728$	−20.7516	−0.769106
$729$	0	0
$730$	5.18371	0.191858
$731$	8.14366	0.301204
$732$	0	0
$733$	−10.0422	−0.370919	−0.185459	−	0.982652i	$-0.559377\pi$
$733$	−10.0422	−0.370919	−0.185459	+	0.982652i	$0.559377\pi$
$734$	−1.21225	−0.0447451
$735$	0	0
$736$	14.2284	0.524464
$737$	16.4557	0.606154
$738$	0	0
$739$	8.15975	0.300161	0.150081	−	0.988674i	$-0.452047\pi$
$739$	8.15975	0.300161	0.150081	+	0.988674i	$0.452047\pi$
$740$	−41.9658	−1.54269
$741$	0	0
$742$	−4.74234	−0.174097
$743$	−7.78206	−0.285496	−0.142748	−	0.989759i	$-0.545594\pi$
$743$	−7.78206	−0.285496	−0.142748	+	0.989759i	$0.545594\pi$
$744$	0	0
$745$	39.4463	1.44520
$746$	4.42454	0.161994
$747$	0	0
$748$	−3.80282	−0.139045
$749$	−67.1249	−2.45269
$750$	0	0
$751$	−8.30887	−0.303195	−0.151598	−	0.988442i	$-0.548442\pi$
$751$	−8.30887	−0.303195	−0.151598	+	0.988442i	$0.548442\pi$
$752$	23.5316	0.858108
$753$	0	0
$754$	−5.72638	−0.208542
$755$	−24.4706	−0.890577
$756$	0	0
$757$	−38.8376	−1.41158	−0.705788	−	0.708424i	$-0.749407\pi$
$757$	−38.8376	−1.41158	−0.705788	+	0.708424i	$0.749407\pi$
$758$	1.74733	0.0634659
$759$	0	0
$760$	−25.0751	−0.909569
$761$	25.3906	0.920408	0.460204	−	0.887813i	$-0.347776\pi$
$761$	25.3906	0.920408	0.460204	+	0.887813i	$0.347776\pi$
$762$	0	0
$763$	54.7909	1.98356
$764$	16.8283	0.608827
$765$	0	0
$766$	−1.30725	−0.0472327
$767$	−5.14246	−0.185684
$768$	0	0
$769$	39.8098	1.43558	0.717788	−	0.696261i	$-0.245155\pi$
$769$	39.8098	1.43558	0.717788	+	0.696261i	$0.245155\pi$
$770$	7.04210	0.253780
$771$	0	0
$772$	−28.4379	−1.02350
$773$	−9.95735	−0.358141	−0.179070	−	0.983836i	$-0.557309\pi$
$773$	−9.95735	−0.358141	−0.179070	+	0.983836i	$0.557309\pi$
$774$	0	0
$775$	−3.88200	−0.139446
$776$	8.76991	0.314821
$777$	0	0
$778$	1.61014	0.0577263
$779$	84.8706	3.04080
$780$	0	0
$781$	−12.3651	−0.442458
$782$	−1.30772	−0.0467641
$783$	0	0
$784$	35.6416	1.27291
$785$	−29.5618	−1.05510
$786$	0	0
$787$	30.2211	1.07727	0.538633	−	0.842541i	$-0.318941\pi$
$787$	30.2211	1.07727	0.538633	+	0.842541i	$0.318941\pi$
$788$	−22.6615	−0.807283
$789$	0	0
$790$	7.56968	0.269317
$791$	61.5139	2.18718
$792$	0	0
$793$	1.16075	0.0412194
$794$	2.97461	0.105565
$795$	0	0
$796$	8.39417	0.297524
$797$	10.6318	0.376597	0.188298	−	0.982112i	$-0.439703\pi$
$797$	10.6318	0.376597	0.188298	+	0.982112i	$0.439703\pi$
$798$	0	0
$799$	−7.07697	−0.250365
$800$	10.5010	0.371267
$801$	0	0
$802$	−7.68069	−0.271215
$803$	11.4471	0.403959
$804$	0	0
$805$	−48.3674	−1.70473
$806$	−1.63714	−0.0576657
$807$	0	0
$808$	−15.7301	−0.553383
$809$	15.9047	0.559178	0.279589	−	0.960120i	$-0.409802\pi$
$809$	15.9047	0.559178	0.279589	+	0.960120i	$0.409802\pi$
$810$	0	0
$811$	53.3099	1.87196	0.935981	−	0.352049i	$-0.114515\pi$
$811$	53.3099	1.87196	0.935981	+	0.352049i	$0.114515\pi$
$812$	35.6473	1.25097
$813$	0	0
$814$	4.63989	0.162628
$815$	10.5771	0.370501
$816$	0	0
$817$	57.8418	2.02363
$818$	−8.78395	−0.307124
$819$	0	0
$820$	−62.3690	−2.17802
$821$	16.7614	0.584975	0.292488	−	0.956269i	$-0.405517\pi$
$821$	16.7614	0.584975	0.292488	+	0.956269i	$0.405517\pi$
$822$	0	0
$823$	38.0271	1.32554	0.662770	−	0.748823i	$-0.269381\pi$
$823$	38.0271	1.32554	0.662770	+	0.748823i	$0.269381\pi$
$824$	14.3546	0.500065
$825$	0	0
$826$	−1.60279	−0.0557681
$827$	−40.6301	−1.41285	−0.706424	−	0.707789i	$-0.749693\pi$
$827$	−40.6301	−1.41285	−0.706424	+	0.707789i	$0.749693\pi$
$828$	0	0
$829$	−48.8098	−1.69523	−0.847617	−	0.530608i	$-0.821964\pi$
$829$	−48.8098	−1.69523	−0.847617	+	0.530608i	$0.821964\pi$
$830$	13.8527	0.480835
$831$	0	0
$832$	−23.9561	−0.830529
$833$	−10.7190	−0.371390
$834$	0	0
$835$	−55.9449	−1.93605
$836$	−27.0102	−0.934168
$837$	0	0
$838$	1.41765	0.0489719
$839$	13.9510	0.481641	0.240821	−	0.970570i	$-0.422583\pi$
$839$	13.9510	0.481641	0.240821	+	0.970570i	$0.422583\pi$
$840$	0	0
$841$	−8.83388	−0.304617
$842$	6.47213	0.223044
$843$	0	0
$844$	−40.8007	−1.40442
$845$	−11.4768	−0.394813
$846$	0	0
$847$	−30.2944	−1.04093
$848$	−12.6641	−0.434888
$849$	0	0
$850$	−0.965144	−0.0331042
$851$	−31.8683	−1.09243
$852$	0	0
$853$	−48.3304	−1.65480	−0.827401	−	0.561612i	$-0.810181\pi$
$853$	−48.3304	−1.65480	−0.827401	+	0.561612i	$0.810181\pi$
$854$	0.361779	0.0123798
$855$	0	0
$856$	19.4199	0.663760
$857$	47.8068	1.63305	0.816525	−	0.577310i	$-0.195897\pi$
$857$	47.8068	1.63305	0.816525	+	0.577310i	$0.195897\pi$
$858$	0	0
$859$	57.7110	1.96907	0.984537	−	0.175179i	$-0.0560503\pi$
$859$	57.7110	1.96907	0.984537	+	0.175179i	$0.0560503\pi$
$860$	−42.5063	−1.44945
$861$	0	0
$862$	−4.37662	−0.149068
$863$	−22.5913	−0.769018	−0.384509	−	0.923121i	$-0.625629\pi$
$863$	−22.5913	−0.769018	−0.384509	+	0.923121i	$0.625629\pi$
$864$	0	0
$865$	−21.2851	−0.723714
$866$	2.26266	0.0768884
$867$	0	0
$868$	10.1913	0.345916
$869$	16.7160	0.567050
$870$	0	0
$871$	−35.1781	−1.19197
$872$	−15.8516	−0.536802
$873$	0	0
$874$	−9.28833	−0.314182
$875$	23.3314	0.788746
$876$	0	0
$877$	8.20763	0.277152	0.138576	−	0.990352i	$-0.455748\pi$
$877$	8.20763	0.277152	0.138576	+	0.990352i	$0.455748\pi$
$878$	0.948305	0.0320037
$879$	0	0
$880$	18.8055	0.633933
$881$	−54.9341	−1.85078	−0.925388	−	0.379020i	$-0.876261\pi$
$881$	−54.9341	−1.85078	−0.925388	+	0.379020i	$0.876261\pi$
$882$	0	0
$883$	−42.9747	−1.44621	−0.723107	−	0.690736i	$-0.757286\pi$
$883$	−42.9747	−1.44621	−0.723107	+	0.690736i	$0.757286\pi$
$884$	8.12948	0.273424
$885$	0	0
$886$	−4.63153	−0.155599
$887$	16.0125	0.537647	0.268824	−	0.963189i	$-0.413365\pi$
$887$	16.0125	0.537647	0.268824	+	0.963189i	$0.413365\pi$
$888$	0	0
$889$	45.9044	1.53958
$890$	−2.83275	−0.0949541
$891$	0	0
$892$	15.1149	0.506083
$893$	−50.2654	−1.68207
$894$	0	0
$895$	68.7263	2.29727
$896$	−36.4149	−1.21654
$897$	0	0
$898$	6.91921	0.230897
$899$	5.76537	0.192286
$900$	0	0
$901$	3.80866	0.126885
$902$	6.89574	0.229603
$903$	0	0
$904$	−17.7966	−0.591907
$905$	8.66838	0.288147
$906$	0	0
$907$	17.4683	0.580026	0.290013	−	0.957023i	$-0.406340\pi$
$907$	17.4683	0.580026	0.290013	+	0.957023i	$0.406340\pi$
$908$	30.3513	1.00724
$909$	0	0
$910$	−15.0542	−0.499043
$911$	−16.6656	−0.552157	−0.276078	−	0.961135i	$-0.589035\pi$
$911$	−16.6656	−0.552157	−0.276078	+	0.961135i	$0.589035\pi$
$912$	0	0
$913$	30.5907	1.01240
$914$	−3.99338	−0.132089
$915$	0	0
$916$	−25.6578	−0.847758
$917$	21.8975	0.723118
$918$	0	0
$919$	18.5901	0.613230	0.306615	−	0.951834i	$-0.400804\pi$
$919$	18.5901	0.613230	0.306615	+	0.951834i	$0.400804\pi$
$920$	13.9932	0.461342
$921$	0	0
$922$	12.2631	0.403865
$923$	26.4334	0.870067
$924$	0	0
$925$	−23.5199	−0.773329
$926$	6.63893	0.218169
$927$	0	0
$928$	−15.5956	−0.511951
$929$	−6.08296	−0.199576	−0.0997878	−	0.995009i	$-0.531816\pi$
$929$	−6.08296	−0.199576	−0.0997878	+	0.995009i	$0.531816\pi$
$930$	0	0
$931$	−76.1334	−2.49517
$932$	35.1054	1.14992
$933$	0	0
$934$	5.95033	0.194701
$935$	−5.65563	−0.184959
$936$	0	0
$937$	−50.4851	−1.64928	−0.824639	−	0.565660i	$-0.808622\pi$
$937$	−50.4851	−1.64928	−0.824639	+	0.565660i	$0.808622\pi$
$938$	−10.9642	−0.357994
$939$	0	0
$940$	36.9386	1.20481
$941$	26.6303	0.868123	0.434061	−	0.900883i	$-0.357080\pi$
$941$	26.6303	0.868123	0.434061	+	0.900883i	$0.357080\pi$
$942$	0	0
$943$	−47.3622	−1.54232
$944$	−4.28014	−0.139307
$945$	0	0
$946$	4.69965	0.152799
$947$	41.7464	1.35658	0.678289	−	0.734796i	$-0.262722\pi$
$947$	41.7464	1.35658	0.678289	+	0.734796i	$0.262722\pi$
$948$	0	0
$949$	−24.4710	−0.794361
$950$	−6.85510	−0.222409
$951$	0	0
$952$	5.19440	0.168351
$953$	45.7416	1.48172	0.740858	−	0.671662i	$-0.234419\pi$
$953$	45.7416	1.48172	0.740858	+	0.671662i	$0.234419\pi$
$954$	0	0
$955$	25.0274	0.809867
$956$	1.69406	0.0547899
$957$	0	0
$958$	7.33305	0.236920
$959$	7.00662	0.226255
$960$	0	0
$961$	−29.3517	−0.946830
$962$	−9.91892	−0.319799
$963$	0	0
$964$	22.3898	0.721128
$965$	−42.2934	−1.36147
$966$	0	0
$967$	29.9836	0.964207	0.482104	−	0.876114i	$-0.339873\pi$
$967$	29.9836	0.964207	0.482104	+	0.876114i	$0.339873\pi$
$968$	8.76449	0.281701
$969$	0	0
$970$	6.36212	0.204275
$971$	32.2248	1.03414	0.517071	−	0.855942i	$-0.327022\pi$
$971$	32.2248	1.03414	0.517071	+	0.855942i	$0.327022\pi$
$972$	0	0
$973$	4.16776	0.133612
$974$	−8.44597	−0.270626
$975$	0	0
$976$	0.966108	0.0309244
$977$	1.45628	0.0465907	0.0232953	−	0.999729i	$-0.492584\pi$
$977$	1.45628	0.0465907	0.0232953	+	0.999729i	$0.492584\pi$
$978$	0	0
$979$	−6.25551	−0.199927
$980$	55.9483	1.78720
$981$	0	0
$982$	8.96564	0.286105
$983$	35.1255	1.12033	0.560164	−	0.828382i	$-0.310738\pi$
$983$	35.1255	1.12033	0.560164	+	0.828382i	$0.310738\pi$
$984$	0	0
$985$	−33.7026	−1.07385
$986$	1.43339	0.0456483
$987$	0	0
$988$	57.7410	1.83699
$989$	−32.2787	−1.02640
$990$	0	0
$991$	−9.58149	−0.304366	−0.152183	−	0.988352i	$-0.548630\pi$
$991$	−9.58149	−0.304366	−0.152183	+	0.988352i	$0.548630\pi$
$992$	−4.45869	−0.141564
$993$	0	0
$994$	8.23869	0.261315
$995$	12.4840	0.395769
$996$	0	0
$997$	−16.5332	−0.523613	−0.261807	−	0.965120i	$-0.584318\pi$
$997$	−16.5332	−0.523613	−0.261807	+	0.965120i	$0.584318\pi$
$998$	8.24140	0.260877
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1251.2.a.k.1.4		7
3.2	odd	2		139.2.a.c.1.4	✓	7
12.11	even	2		2224.2.a.o.1.5		7
15.14	odd	2		3475.2.a.e.1.4		7
21.20	even	2		6811.2.a.p.1.4		7
24.5	odd	2		8896.2.a.be.1.5		7
24.11	even	2		8896.2.a.bd.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
139.2.a.c.1.4	✓	7	3.2	odd	2
1251.2.a.k.1.4		7	1.1	even	1	trivial
2224.2.a.o.1.5		7	12.11	even	2
3475.2.a.e.1.4		7	15.14	odd	2
6811.2.a.p.1.4		7	21.20	even	2
8896.2.a.bd.1.3		7	24.11	even	2
8896.2.a.be.1.5		7	24.5	odd	2

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:	\( N \)	\(=\)	\( 1251 = 3^{2} \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1251.a (trivial)

\(f(q)\)	\(=\)	\(q+0.308806 q^{2} -1.90464 q^{4} -2.83261 q^{5} +4.16776 q^{7} -1.20578 q^{8} +O(q^{10})\)	\(q+0.308806 q^{2} -1.90464 q^{4} -2.83261 q^{5} +4.16776 q^{7} -1.20578 q^{8} -0.874728 q^{10} -1.93164 q^{11} +4.12937 q^{13} +1.28703 q^{14} +3.43693 q^{16} -1.03363 q^{17} -7.34157 q^{19} +5.39511 q^{20} -0.596503 q^{22} +4.09697 q^{23} +3.02371 q^{25} +1.27517 q^{26} -7.93807 q^{28} -4.49067 q^{29} -1.28386 q^{31} +3.47289 q^{32} -0.319192 q^{34} -11.8056 q^{35} -7.77849 q^{37} -2.26712 q^{38} +3.41550 q^{40} -11.5603 q^{41} -7.87867 q^{43} +3.67908 q^{44} +1.26517 q^{46} +6.84669 q^{47} +10.3702 q^{49} +0.933738 q^{50} -7.86495 q^{52} -3.68473 q^{53} +5.47160 q^{55} -5.02538 q^{56} -1.38675 q^{58} -1.24534 q^{59} +0.281096 q^{61} -0.396462 q^{62} -5.80140 q^{64} -11.6969 q^{65} -8.51902 q^{67} +1.96870 q^{68} -3.64565 q^{70} +6.40133 q^{71} -5.92608 q^{73} -2.40204 q^{74} +13.9830 q^{76} -8.05062 q^{77} -8.65375 q^{79} -9.73549 q^{80} -3.56988 q^{82} -15.8366 q^{83} +2.92789 q^{85} -2.43298 q^{86} +2.32913 q^{88} +3.23844 q^{89} +17.2102 q^{91} -7.80326 q^{92} +2.11430 q^{94} +20.7958 q^{95} -7.27325 q^{97} +3.20237 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 9 q^{4} - 11 q^{5} - 5 q^{7} - 6 q^{8}+O(q^{10}) \) 7 * q - q^2 + 9 * q^4 - 11 * q^5 - 5 * q^7 - 6 * q^8	\( 7 q - q^{2} + 9 q^{4} - 11 q^{5} - 5 q^{7} - 6 q^{8} - 4 q^{10} - 2 q^{11} + 6 q^{13} - 7 q^{14} + 5 q^{16} - 5 q^{17} - 10 q^{19} - 12 q^{20} - 18 q^{22} + q^{23} + 14 q^{25} + 8 q^{26} - 28 q^{28} - 30 q^{29} - 20 q^{31} + 12 q^{32} - 17 q^{34} + 7 q^{35} + 6 q^{37} - 6 q^{38} - 22 q^{40} - 19 q^{41} - 12 q^{43} - 25 q^{44} + 22 q^{46} + 3 q^{47} - 8 q^{49} - 12 q^{50} - 8 q^{52} - 38 q^{53} + 7 q^{55} - 21 q^{56} - 21 q^{58} + 14 q^{59} + 4 q^{61} + q^{62} - 16 q^{64} - 10 q^{65} + 9 q^{67} + 25 q^{68} + 20 q^{70} - 24 q^{71} - 5 q^{73} - 9 q^{74} + 3 q^{76} + 13 q^{77} + 8 q^{79} - 11 q^{80} + 56 q^{82} + 9 q^{83} - 22 q^{85} - 39 q^{86} - 29 q^{88} - 10 q^{89} + 7 q^{91} - 29 q^{92} - 36 q^{94} + 21 q^{95} - 5 q^{97} + 49 q^{98}+O(q^{100}) \) 7 * q - q^2 + 9 * q^4 - 11 * q^5 - 5 * q^7 - 6 * q^8 - 4 * q^10 - 2 * q^11 + 6 * q^13 - 7 * q^14 + 5 * q^16 - 5 * q^17 - 10 * q^19 - 12 * q^20 - 18 * q^22 + q^23 + 14 * q^25 + 8 * q^26 - 28 * q^28 - 30 * q^29 - 20 * q^31 + 12 * q^32 - 17 * q^34 + 7 * q^35 + 6 * q^37 - 6 * q^38 - 22 * q^40 - 19 * q^41 - 12 * q^43 - 25 * q^44 + 22 * q^46 + 3 * q^47 - 8 * q^49 - 12 * q^50 - 8 * q^52 - 38 * q^53 + 7 * q^55 - 21 * q^56 - 21 * q^58 + 14 * q^59 + 4 * q^61 + q^62 - 16 * q^64 - 10 * q^65 + 9 * q^67 + 25 * q^68 + 20 * q^70 - 24 * q^71 - 5 * q^73 - 9 * q^74 + 3 * q^76 + 13 * q^77 + 8 * q^79 - 11 * q^80 + 56 * q^82 + 9 * q^83 - 22 * q^85 - 39 * q^86 - 29 * q^88 - 10 * q^89 + 7 * q^91 - 29 * q^92 - 36 * q^94 + 21 * q^95 - 5 * q^97 + 49 * q^98

Introduction

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