LMFDB - Embedded newform 8281.2.a.co.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, 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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{10} + 88x^{8} - 197x^{6} + 172x^{4} - 36x^{2} + 1 $ x^12 - 16x^10 + 88x^8 - 197x^6 + 172x^4 - 36*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.34523$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.34523 q^{2} -2.05010 q^{3} -0.190366 q^{4} +3.56778 q^{5} +2.75785 q^{6} +2.94654 q^{8} +1.20292 q^{9} +O(q^{10})$	\(q-1.34523 q^{2} -2.05010 q^{3} -0.190366 q^{4} +3.56778 q^{5} +2.75785 q^{6} +2.94654 q^{8} +1.20292 q^{9} -4.79947 q^{10} -1.27867 q^{11} +0.390271 q^{12} -7.31431 q^{15} -3.58303 q^{16} -7.73920 q^{17} -1.61821 q^{18} -0.943878 q^{19} -0.679185 q^{20} +1.72010 q^{22} +1.64727 q^{23} -6.04071 q^{24} +7.72903 q^{25} +3.68419 q^{27} +4.04484 q^{29} +9.83940 q^{30} -5.15220 q^{31} -1.07309 q^{32} +2.62141 q^{33} +10.4110 q^{34} -0.228996 q^{36} +1.05608 q^{37} +1.26973 q^{38} +10.5126 q^{40} +4.19882 q^{41} +3.83065 q^{43} +0.243416 q^{44} +4.29176 q^{45} -2.21596 q^{46} -0.894217 q^{47} +7.34558 q^{48} -10.3973 q^{50} +15.8662 q^{51} -0.0799923 q^{53} -4.95607 q^{54} -4.56202 q^{55} +1.93505 q^{57} -5.44122 q^{58} +11.1847 q^{59} +1.39240 q^{60} +7.62392 q^{61} +6.93087 q^{62} +8.60961 q^{64} -3.52639 q^{66} -6.32103 q^{67} +1.47328 q^{68} -3.37708 q^{69} -11.4240 q^{71} +3.54446 q^{72} +0.760506 q^{73} -1.42067 q^{74} -15.8453 q^{75} +0.179683 q^{76} -2.85531 q^{79} -12.7834 q^{80} -11.1617 q^{81} -5.64837 q^{82} -2.32483 q^{83} -27.6117 q^{85} -5.15308 q^{86} -8.29233 q^{87} -3.76766 q^{88} -7.57626 q^{89} -5.77339 q^{90} -0.313586 q^{92} +10.5625 q^{93} +1.20292 q^{94} -3.36755 q^{95} +2.19995 q^{96} -0.478557 q^{97} -1.53815 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{3} + 8 q^{4} + 2 q^{9}+O(q^{10}) $ 12 * q - 6 * q^3 + 8 * q^4 + 2 * q^9	$ 12 q - 6 q^{3} + 8 q^{4} + 2 q^{9} - 24 q^{10} + 2 q^{12} + 16 q^{16} - 34 q^{17} + 30 q^{22} + 6 q^{23} - 10 q^{25} - 12 q^{27} + 2 q^{29} + 22 q^{30} - 26 q^{36} - 38 q^{38} - 2 q^{40} + 22 q^{43} + 38 q^{48} + 8 q^{51} + 16 q^{53} - 30 q^{55} + 10 q^{61} - 82 q^{62} - 2 q^{64} - 68 q^{66} - 22 q^{68} - 14 q^{69} + 66 q^{74} - 2 q^{75} + 70 q^{79} - 28 q^{81} - 10 q^{82} + 20 q^{87} - 28 q^{88} - 66 q^{92} + 2 q^{94} + 4 q^{95}+O(q^{100}) $ 12 * q - 6 * q^3 + 8 * q^4 + 2 * q^9 - 24 * q^10 + 2 * q^12 + 16 * q^16 - 34 * q^17 + 30 * q^22 + 6 * q^23 - 10 * q^25 - 12 * q^27 + 2 * q^29 + 22 * q^30 - 26 * q^36 - 38 * q^38 - 2 * q^40 + 22 * q^43 + 38 * q^48 + 8 * q^51 + 16 * q^53 - 30 * q^55 + 10 * q^61 - 82 * q^62 - 2 * q^64 - 68 * q^66 - 22 * q^68 - 14 * q^69 + 66 * q^74 - 2 * q^75 + 70 * q^79 - 28 * q^81 - 10 * q^82 + 20 * q^87 - 28 * q^88 - 66 * q^92 + 2 * q^94 + 4 * q^95

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.34523	−0.951219	−0.475609	−	0.879657i	$-0.657772\pi$
$2$	−1.34523	−0.951219	−0.475609	+	0.879657i	$0.657772\pi$
$3$	−2.05010	−1.18363	−0.591814	−	0.806075i	$-0.701588\pi$
$3$	−2.05010	−1.18363	−0.591814	+	0.806075i	$0.701588\pi$
$4$	−0.190366	−0.0951832
$5$	3.56778	1.59556	0.797779	−	0.602950i	$-0.206008\pi$
$5$	3.56778	1.59556	0.797779	+	0.602950i	$0.206008\pi$
$6$	2.75785	1.12589
$7$	0	0
$7$	0	0
$8$	2.94654	1.04176
$9$	1.20292	0.400975
$10$	−4.79947	−1.51772
$11$	−1.27867	−0.385534	−0.192767	−	0.981245i	$-0.561746\pi$
$11$	−1.27867	−0.385534	−0.192767	+	0.981245i	$0.561746\pi$
$12$	0.390271	0.112661
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−7.31431	−1.88855
$16$	−3.58303	−0.895757
$17$	−7.73920	−1.87703	−0.938515	−	0.345238i	$-0.887798\pi$
$17$	−7.73920	−1.87703	−0.938515	+	0.345238i	$0.887798\pi$
$18$	−1.61821	−0.381415
$19$	−0.943878	−0.216540	−0.108270	−	0.994121i	$-0.534531\pi$
$19$	−0.943878	−0.216540	−0.108270	+	0.994121i	$0.534531\pi$
$20$	−0.679185	−0.151870
$21$	0	0
$22$	1.72010	0.366727
$23$	1.64727	0.343481	0.171740	−	0.985142i	$-0.445061\pi$
$23$	1.64727	0.343481	0.171740	+	0.985142i	$0.445061\pi$
$24$	−6.04071	−1.23305
$25$	7.72903	1.54581
$26$	0	0
$27$	3.68419	0.709023
$28$	0	0
$29$	4.04484	0.751107	0.375554	−	0.926801i	$-0.377453\pi$
$29$	4.04484	0.751107	0.375554	+	0.926801i	$0.377453\pi$
$30$	9.83940	1.79642
$31$	−5.15220	−0.925362	−0.462681	−	0.886525i	$-0.653112\pi$
$31$	−5.15220	−0.925362	−0.462681	+	0.886525i	$0.653112\pi$
$32$	−1.07309	−0.189698
$33$	2.62141	0.456329
$34$	10.4110	1.78547
$35$	0	0
$36$	−0.228996	−0.0381661
$37$	1.05608	0.173619	0.0868094	−	0.996225i	$-0.472333\pi$
$37$	1.05608	0.173619	0.0868094	+	0.996225i	$0.472333\pi$
$38$	1.26973	0.205977
$39$	0	0
$40$	10.5126	1.66219
$41$	4.19882	0.655746	0.327873	−	0.944722i	$-0.393668\pi$
$41$	4.19882	0.655746	0.327873	+	0.944722i	$0.393668\pi$
$42$	0	0
$43$	3.83065	0.584168	0.292084	−	0.956393i	$-0.405651\pi$
$43$	3.83065	0.584168	0.292084	+	0.956393i	$0.405651\pi$
$44$	0.243416	0.0366964
$45$	4.29176	0.639778
$46$	−2.21596	−0.326725
$47$	−0.894217	−0.130435	−0.0652175	−	0.997871i	$-0.520774\pi$
$47$	−0.894217	−0.130435	−0.0652175	+	0.997871i	$0.520774\pi$
$48$	7.34558	1.06024
$49$	0	0
$50$	−10.3973	−1.47040
$51$	15.8662	2.22171
$52$	0	0
$53$	−0.0799923	−0.0109878	−0.00549389	−	0.999985i	$-0.501749\pi$
$53$	−0.0799923	−0.0109878	−0.00549389	+	0.999985i	$0.501749\pi$
$54$	−4.95607	−0.674436
$55$	−4.56202	−0.615142
$56$	0	0
$57$	1.93505	0.256303
$58$	−5.44122	−0.714467
$59$	11.1847	1.45613	0.728064	−	0.685509i	$-0.240420\pi$
$59$	11.1847	1.45613	0.728064	+	0.685509i	$0.240420\pi$
$60$	1.39240	0.179758
$61$	7.62392	0.976143	0.488072	−	0.872804i	$-0.337701\pi$
$61$	7.62392	0.976143	0.488072	+	0.872804i	$0.337701\pi$
$62$	6.93087	0.880221
$63$	0	0
$64$	8.60961	1.07620
$65$	0	0
$66$	−3.52639	−0.434069
$67$	−6.32103	−0.772237	−0.386119	−	0.922449i	$-0.626184\pi$
$67$	−6.32103	−0.772237	−0.386119	+	0.922449i	$0.626184\pi$
$68$	1.47328	0.178662
$69$	−3.37708	−0.406553
$70$	0	0
$71$	−11.4240	−1.35578	−0.677889	−	0.735165i	$-0.737105\pi$
$71$	−11.4240	−1.35578	−0.677889	+	0.735165i	$0.737105\pi$
$72$	3.54446	0.417719
$73$	0.760506	0.0890105	0.0445052	−	0.999009i	$-0.485829\pi$
$73$	0.760506	0.0890105	0.0445052	+	0.999009i	$0.485829\pi$
$74$	−1.42067	−0.165149
$75$	−15.8453	−1.82966
$76$	0.179683	0.0206110
$77$	0	0
$78$	0	0
$79$	−2.85531	−0.321247	−0.160624	−	0.987016i	$-0.551351\pi$
$79$	−2.85531	−0.321247	−0.160624	+	0.987016i	$0.551351\pi$
$80$	−12.7834	−1.42923
$81$	−11.1617	−1.24019
$82$	−5.64837	−0.623758
$83$	−2.32483	−0.255183	−0.127591	−	0.991827i	$-0.540725\pi$
$83$	−2.32483	−0.255183	−0.127591	+	0.991827i	$0.540725\pi$
$84$	0	0
$85$	−27.6117	−2.99491
$86$	−5.15308	−0.555671
$87$	−8.29233	−0.889032
$88$	−3.76766	−0.401634
$89$	−7.57626	−0.803082	−0.401541	−	0.915841i	$-0.631525\pi$
$89$	−7.57626	−0.803082	−0.401541	+	0.915841i	$0.631525\pi$
$90$	−5.77339	−0.608569
$91$	0	0
$92$	−0.313586	−0.0326936
$93$	10.5625	1.09528
$94$	1.20292	0.124072
$95$	−3.36755	−0.345503
$96$	2.19995	0.224532
$97$	−0.478557	−0.0485901	−0.0242951	−	0.999705i	$-0.507734\pi$
$97$	−0.478557	−0.0485901	−0.0242951	+	0.999705i	$0.507734\pi$
$98$	0	0
$99$	−1.53815	−0.154589
$100$	−1.47135	−0.147135
$101$	−2.87836	−0.286407	−0.143204	−	0.989693i	$-0.545740\pi$
$101$	−2.87836	−0.286407	−0.143204	+	0.989693i	$0.545740\pi$
$102$	−21.3436	−2.11333
$103$	11.3351	1.11688	0.558441	−	0.829544i	$-0.311400\pi$
$103$	11.3351	1.11688	0.558441	+	0.829544i	$0.311400\pi$
$104$	0	0
$105$	0	0
$106$	0.107608	0.0104518
$107$	−6.57206	−0.635345	−0.317673	−	0.948200i	$-0.602901\pi$
$107$	−6.57206	−0.635345	−0.317673	+	0.948200i	$0.602901\pi$
$108$	−0.701346	−0.0674871
$109$	−5.83914	−0.559288	−0.279644	−	0.960104i	$-0.590216\pi$
$109$	−5.83914	−0.559288	−0.279644	+	0.960104i	$0.590216\pi$
$110$	6.13694	0.585135
$111$	−2.16508	−0.205500
$112$	0	0
$113$	6.53233	0.614510	0.307255	−	0.951627i	$-0.400590\pi$
$113$	6.53233	0.614510	0.307255	+	0.951627i	$0.400590\pi$
$114$	−2.60308	−0.243800
$115$	5.87711	0.548043
$116$	−0.770001	−0.0714928
$117$	0	0
$118$	−15.0460	−1.38510
$119$	0	0
$120$	−21.5519	−1.96741
$121$	−9.36500	−0.851363
$122$	−10.2559	−0.928525
$123$	−8.60802	−0.776159
$124$	0.980805	0.0880789
$125$	9.73656	0.870865
$126$	0	0
$127$	14.7164	1.30586	0.652932	−	0.757416i	$-0.273539\pi$
$127$	14.7164	1.30586	0.652932	+	0.757416i	$0.273539\pi$
$128$	−9.43568	−0.834005
$129$	−7.85322	−0.691437
$130$	0	0
$131$	−11.1867	−0.977386	−0.488693	−	0.872456i	$-0.662526\pi$
$131$	−11.1867	−0.977386	−0.488693	+	0.872456i	$0.662526\pi$
$132$	−0.499028	−0.0434349
$133$	0	0
$134$	8.50322	0.734566
$135$	13.1444	1.13129
$136$	−22.8038	−1.95541
$137$	17.6308	1.50630	0.753151	−	0.657848i	$-0.228533\pi$
$137$	17.6308	1.50630	0.753151	+	0.657848i	$0.228533\pi$
$138$	4.54294	0.386721
$139$	5.85710	0.496793	0.248396	−	0.968658i	$-0.420096\pi$
$139$	5.85710	0.496793	0.248396	+	0.968658i	$0.420096\pi$
$140$	0	0
$141$	1.83324	0.154386
$142$	15.3678	1.28964
$143$	0	0
$144$	−4.31011	−0.359176
$145$	14.4311	1.19844
$146$	−1.02305	−0.0846684
$147$	0	0
$148$	−0.201043	−0.0165256
$149$	10.4790	0.858470	0.429235	−	0.903193i	$-0.358783\pi$
$149$	10.4790	0.858470	0.429235	+	0.903193i	$0.358783\pi$
$150$	21.3155	1.74041
$151$	4.71406	0.383625	0.191812	−	0.981432i	$-0.438564\pi$
$151$	4.71406	0.383625	0.191812	+	0.981432i	$0.438564\pi$
$152$	−2.78117	−0.225583
$153$	−9.30967	−0.752642
$154$	0	0
$155$	−18.3819	−1.47647
$156$	0	0
$157$	−9.00210	−0.718445	−0.359223	−	0.933252i	$-0.616958\pi$
$157$	−9.00210	−0.718445	−0.359223	+	0.933252i	$0.616958\pi$
$158$	3.84103	0.305576
$159$	0.163992	0.0130054
$160$	−3.82856	−0.302674
$161$	0	0
$162$	15.0151	1.17970
$163$	−12.0324	−0.942449	−0.471224	−	0.882013i	$-0.656188\pi$
$163$	−12.0324	−0.942449	−0.471224	+	0.882013i	$0.656188\pi$
$164$	−0.799315	−0.0624160
$165$	9.35261	0.728099
$166$	3.12742	0.242735
$167$	−19.4220	−1.50292	−0.751459	−	0.659780i	$-0.770650\pi$
$167$	−19.4220	−1.50292	−0.751459	+	0.659780i	$0.770650\pi$
$168$	0	0
$169$	0	0
$170$	37.1440	2.84882
$171$	−1.13541	−0.0868273
$172$	−0.729226	−0.0556030
$173$	−14.3795	−1.09325	−0.546627	−	0.837376i	$-0.684088\pi$
$173$	−14.3795	−1.09325	−0.546627	+	0.837376i	$0.684088\pi$
$174$	11.1551	0.845663
$175$	0	0
$176$	4.58152	0.345345
$177$	−22.9299	−1.72351
$178$	10.1918	0.763907
$179$	5.42606	0.405563	0.202781	−	0.979224i	$-0.435002\pi$
$179$	5.42606	0.405563	0.202781	+	0.979224i	$0.435002\pi$
$180$	−0.817008	−0.0608962
$181$	−15.4902	−1.15138	−0.575688	−	0.817669i	$-0.695266\pi$
$181$	−15.4902	−1.15138	−0.575688	+	0.817669i	$0.695266\pi$
$182$	0	0
$183$	−15.6298	−1.15539
$184$	4.85376	0.357824
$185$	3.76786	0.277019
$186$	−14.2090	−1.04185
$187$	9.89589	0.723659
$188$	0.170229	0.0124152
$189$	0	0
$190$	4.53011	0.328649
$191$	4.74622	0.343425	0.171712	−	0.985147i	$-0.445070\pi$
$191$	4.74622	0.343425	0.171712	+	0.985147i	$0.445070\pi$
$192$	−17.6506	−1.27382
$193$	21.0391	1.51443	0.757215	−	0.653166i	$-0.226559\pi$
$193$	21.0391	1.51443	0.757215	+	0.653166i	$0.226559\pi$
$194$	0.643768	0.0462198
$195$	0	0
$196$	0	0
$197$	5.81209	0.414094	0.207047	−	0.978331i	$-0.433615\pi$
$197$	5.81209	0.414094	0.207047	+	0.978331i	$0.433615\pi$
$198$	2.06915	0.147048
$199$	−10.6182	−0.752703	−0.376352	−	0.926477i	$-0.622821\pi$
$199$	−10.6182	−0.752703	−0.376352	+	0.926477i	$0.622821\pi$
$200$	22.7739	1.61036
$201$	12.9588	0.914041
$202$	3.87204	0.272436
$203$	0	0
$204$	−3.02038	−0.211469
$205$	14.9805	1.04628
$206$	−15.2483	−1.06240
$207$	1.98155	0.137727
$208$	0	0
$209$	1.20691	0.0834837
$210$	0	0
$211$	−4.66549	−0.321186	−0.160593	−	0.987021i	$-0.551341\pi$
$211$	−4.66549	−0.321186	−0.160593	+	0.987021i	$0.551341\pi$
$212$	0.0152278	0.00104585
$213$	23.4203	1.60474
$214$	8.84091	0.604352
$215$	13.6669	0.932074
$216$	10.8556	0.738631
$217$	0	0
$218$	7.85497	0.532005
$219$	−1.55912	−0.105355
$220$	0.868455	0.0585512
$221$	0	0
$222$	2.91252	0.195475
$223$	−24.2254	−1.62225	−0.811126	−	0.584871i	$-0.801145\pi$
$223$	−24.2254	−1.62225	−0.811126	+	0.584871i	$0.801145\pi$
$224$	0	0
$225$	9.29744	0.619829
$226$	−8.78747	−0.584534
$227$	15.3753	1.02049	0.510247	−	0.860028i	$-0.329554\pi$
$227$	15.3753	1.02049	0.510247	+	0.860028i	$0.329554\pi$
$228$	−0.368368	−0.0243958
$229$	16.3515	1.08054	0.540268	−	0.841493i	$-0.318323\pi$
$229$	16.3515	1.08054	0.540268	+	0.841493i	$0.318323\pi$
$230$	−7.90604	−0.521309
$231$	0	0
$232$	11.9183	0.782473
$233$	−29.1107	−1.90711	−0.953554	−	0.301223i	$-0.902605\pi$
$233$	−29.1107	−1.90711	−0.953554	+	0.301223i	$0.902605\pi$
$234$	0	0
$235$	−3.19037	−0.208117
$236$	−2.12920	−0.138599
$237$	5.85368	0.380237
$238$	0	0
$239$	−8.65409	−0.559787	−0.279893	−	0.960031i	$-0.590299\pi$
$239$	−8.65409	−0.559787	−0.279893	+	0.960031i	$0.590299\pi$
$240$	26.2074	1.69168
$241$	18.1982	1.17225	0.586124	−	0.810222i	$-0.300653\pi$
$241$	18.1982	1.17225	0.586124	+	0.810222i	$0.300653\pi$
$242$	12.5980	0.809833
$243$	11.8302	0.758905
$244$	−1.45134	−0.0929124
$245$	0	0
$246$	11.5797	0.738297
$247$	0	0
$248$	−15.1811	−0.964003
$249$	4.76614	0.302042
$250$	−13.0979	−0.828383
$251$	−15.8720	−1.00183	−0.500915	−	0.865497i	$-0.667003\pi$
$251$	−15.8720	−1.00183	−0.500915	+	0.865497i	$0.667003\pi$
$252$	0	0
$253$	−2.10632	−0.132423
$254$	−19.7968	−1.24216
$255$	56.6069	3.54486
$256$	−4.52609	−0.282880
$257$	24.3267	1.51746	0.758730	−	0.651406i	$-0.225820\pi$
$257$	24.3267	1.51746	0.758730	+	0.651406i	$0.225820\pi$
$258$	10.5644	0.657708
$259$	0	0
$260$	0	0
$261$	4.86563	0.301175
$262$	15.0486	0.929708
$263$	15.4345	0.951734	0.475867	−	0.879517i	$-0.342134\pi$
$263$	15.4345	0.951734	0.475867	+	0.879517i	$0.342134\pi$
$264$	7.72409	0.475385
$265$	−0.285395	−0.0175317
$266$	0	0
$267$	15.5321	0.950550
$268$	1.20331	0.0735040
$269$	13.0407	0.795106	0.397553	−	0.917579i	$-0.369859\pi$
$269$	13.0407	0.795106	0.397553	+	0.917579i	$0.369859\pi$
$270$	−17.6822	−1.07610
$271$	−26.9706	−1.63835	−0.819174	−	0.573544i	$-0.805568\pi$
$271$	−26.9706	−1.63835	−0.819174	+	0.573544i	$0.805568\pi$
$272$	27.7298	1.68136
$273$	0	0
$274$	−23.7174	−1.43282
$275$	−9.88289	−0.595961
$276$	0.642883	0.0386970
$277$	12.7015	0.763156	0.381578	−	0.924337i	$-0.375381\pi$
$277$	12.7015	0.763156	0.381578	+	0.924337i	$0.375381\pi$
$278$	−7.87912	−0.472558
$279$	−6.19770	−0.371047
$280$	0	0
$281$	26.7216	1.59408	0.797038	−	0.603930i	$-0.206399\pi$
$281$	26.7216	1.59408	0.797038	+	0.603930i	$0.206399\pi$
$282$	−2.46612	−0.146855
$283$	−14.7423	−0.876336	−0.438168	−	0.898893i	$-0.644373\pi$
$283$	−14.7423	−0.876336	−0.438168	+	0.898893i	$0.644373\pi$
$284$	2.17474	0.129047
$285$	6.90382	0.408947
$286$	0	0
$287$	0	0
$288$	−1.29085	−0.0760641
$289$	42.8952	2.52324
$290$	−19.4131	−1.13997
$291$	0.981092	0.0575126
$292$	−0.144775	−0.00847230
$293$	11.5831	0.676689	0.338345	−	0.941022i	$-0.390133\pi$
$293$	11.5831	0.676689	0.338345	+	0.941022i	$0.390133\pi$
$294$	0	0
$295$	39.9046	2.32334
$296$	3.11179	0.180869
$297$	−4.71087	−0.273353
$298$	−14.0966	−0.816593
$299$	0	0
$300$	3.01641	0.174153
$301$	0	0
$302$	−6.34147	−0.364911
$303$	5.90093	0.339000
$304$	3.38194	0.193968
$305$	27.2004	1.55749
$306$	12.5236	0.715927
$307$	−29.3335	−1.67415	−0.837076	−	0.547086i	$-0.815737\pi$
$307$	−29.3335	−1.67415	−0.837076	+	0.547086i	$0.815737\pi$
$308$	0	0
$309$	−23.2381	−1.32197
$310$	24.7278	1.40444
$311$	0.150654	0.00854282	0.00427141	−	0.999991i	$-0.498640\pi$
$311$	0.150654	0.00854282	0.00427141	+	0.999991i	$0.498640\pi$
$312$	0	0
$313$	10.5211	0.594690	0.297345	−	0.954770i	$-0.403899\pi$
$313$	10.5211	0.594690	0.297345	+	0.954770i	$0.403899\pi$
$314$	12.1099	0.683399
$315$	0	0
$316$	0.543555	0.0305773
$317$	−1.50676	−0.0846281	−0.0423140	−	0.999104i	$-0.513473\pi$
$317$	−1.50676	−0.0846281	−0.0423140	+	0.999104i	$0.513473\pi$
$318$	−0.220607	−0.0123710
$319$	−5.17202	−0.289578
$320$	30.7172	1.71714
$321$	13.4734	0.752012
$322$	0	0
$323$	7.30486	0.406453
$324$	2.12482	0.118046
$325$	0	0
$326$	16.1863	0.896475
$327$	11.9708	0.661989
$328$	12.3720	0.683129
$329$	0	0
$330$	−12.5814	−0.692582
$331$	−25.2509	−1.38791	−0.693957	−	0.720017i	$-0.744134\pi$
$331$	−25.2509	−1.38791	−0.693957	+	0.720017i	$0.744134\pi$
$332$	0.442569	0.0242891
$333$	1.27039	0.0696167
$334$	26.1270	1.42960
$335$	−22.5520	−1.23215
$336$	0	0
$337$	−32.1811	−1.75302	−0.876509	−	0.481386i	$-0.840134\pi$
$337$	−32.1811	−1.75302	−0.876509	+	0.481386i	$0.840134\pi$
$338$	0	0
$339$	−13.3920	−0.727351
$340$	5.25634	0.285065
$341$	6.58797	0.356759
$342$	1.52739	0.0825917
$343$	0	0
$344$	11.2871	0.608562
$345$	−12.0487	−0.648679
$346$	19.3437	1.03992
$347$	24.7638	1.32939	0.664695	−	0.747115i	$-0.268562\pi$
$347$	24.7638	1.32939	0.664695	+	0.747115i	$0.268562\pi$
$348$	1.57858	0.0846209
$349$	−11.5556	−0.618559	−0.309280	−	0.950971i	$-0.600088\pi$
$349$	−11.5556	−0.618559	−0.309280	+	0.950971i	$0.600088\pi$
$350$	0	0
$351$	0	0
$352$	1.37213	0.0731350
$353$	−20.0884	−1.06920	−0.534599	−	0.845106i	$-0.679537\pi$
$353$	−20.0884	−1.06920	−0.534599	+	0.845106i	$0.679537\pi$
$354$	30.8459	1.63944
$355$	−40.7582	−2.16322
$356$	1.44227	0.0764399
$357$	0	0
$358$	−7.29928	−0.385779
$359$	15.0510	0.794363	0.397181	−	0.917740i	$-0.369988\pi$
$359$	15.0510	0.794363	0.397181	+	0.917740i	$0.369988\pi$
$360$	12.6458	0.666495
$361$	−18.1091	−0.953110
$362$	20.8378	1.09521
$363$	19.1992	1.00770
$364$	0	0
$365$	2.71331	0.142021
$366$	21.0257	1.09903
$367$	−9.00355	−0.469982	−0.234991	−	0.971998i	$-0.575506\pi$
$367$	−9.00355	−0.469982	−0.234991	+	0.971998i	$0.575506\pi$
$368$	−5.90223	−0.307675
$369$	5.05087	0.262938
$370$	−5.06863	−0.263506
$371$	0	0
$372$	−2.01075	−0.104253
$373$	−16.1391	−0.835649	−0.417824	−	0.908528i	$-0.637207\pi$
$373$	−16.1391	−0.835649	−0.417824	+	0.908528i	$0.637207\pi$
$374$	−13.3122	−0.688358
$375$	−19.9610	−1.03078
$376$	−2.63484	−0.135882
$377$	0	0
$378$	0	0
$379$	−15.6655	−0.804685	−0.402342	−	0.915489i	$-0.631804\pi$
$379$	−15.6655	−0.804685	−0.402342	+	0.915489i	$0.631804\pi$
$380$	0.641068	0.0328861
$381$	−30.1700	−1.54566
$382$	−6.38474	−0.326672
$383$	−24.6328	−1.25868	−0.629339	−	0.777131i	$-0.716674\pi$
$383$	−24.6328	−1.25868	−0.629339	+	0.777131i	$0.716674\pi$
$384$	19.3441	0.987151
$385$	0	0
$386$	−28.3024	−1.44055
$387$	4.60798	0.234237
$388$	0.0911013	0.00462497
$389$	−18.8567	−0.956071	−0.478036	−	0.878340i	$-0.658651\pi$
$389$	−18.8567	−0.956071	−0.478036	+	0.878340i	$0.658651\pi$
$390$	0	0
$391$	−12.7486	−0.644724
$392$	0	0
$393$	22.9339	1.15686
$394$	−7.81857	−0.393894
$395$	−10.1871	−0.512569
$396$	0.292811	0.0147143
$397$	−14.5030	−0.727884	−0.363942	−	0.931422i	$-0.618569\pi$
$397$	−14.5030	−0.727884	−0.363942	+	0.931422i	$0.618569\pi$
$398$	14.2839	0.715985
$399$	0	0
$400$	−27.6933	−1.38467
$401$	20.9889	1.04814	0.524069	−	0.851676i	$-0.324413\pi$
$401$	20.9889	1.04814	0.524069	+	0.851676i	$0.324413\pi$
$402$	−17.4325	−0.869453
$403$	0	0
$404$	0.547943	0.0272612
$405$	−39.8226	−1.97880
$406$	0	0
$407$	−1.35038	−0.0669360
$408$	46.7502	2.31448
$409$	21.4276	1.05953	0.529763	−	0.848146i	$-0.322281\pi$
$409$	21.4276	1.05953	0.529763	+	0.848146i	$0.322281\pi$
$410$	−20.1521	−0.995242
$411$	−36.1450	−1.78290
$412$	−2.15782	−0.106308
$413$	0	0
$414$	−2.66563	−0.131009
$415$	−8.29446	−0.407159
$416$	0	0
$417$	−12.0077	−0.588018
$418$	−1.62357	−0.0794113
$419$	−7.96406	−0.389070	−0.194535	−	0.980896i	$-0.562320\pi$
$419$	−7.96406	−0.389070	−0.194535	+	0.980896i	$0.562320\pi$
$420$	0	0
$421$	2.81786	0.137334	0.0686670	−	0.997640i	$-0.478125\pi$
$421$	2.81786	0.137334	0.0686670	+	0.997640i	$0.478125\pi$
$422$	6.27614	0.305518
$423$	−1.07568	−0.0523011
$424$	−0.235700	−0.0114466
$425$	−59.8165	−2.90152
$426$	−31.5057	−1.52645
$427$	0	0
$428$	1.25110	0.0604742
$429$	0	0
$430$	−18.3851	−0.886606
$431$	−5.73626	−0.276306	−0.138153	−	0.990411i	$-0.544117\pi$
$431$	−5.73626	−0.276306	−0.138153	+	0.990411i	$0.544117\pi$
$432$	−13.2006	−0.635112
$433$	24.5257	1.17863	0.589314	−	0.807904i	$-0.299398\pi$
$433$	24.5257	1.17863	0.589314	+	0.807904i	$0.299398\pi$
$434$	0	0
$435$	−29.5852	−1.41850
$436$	1.11158	0.0532349
$437$	−1.55483	−0.0743774
$438$	2.09736	0.100216
$439$	−36.6423	−1.74884	−0.874420	−	0.485169i	$-0.838758\pi$
$439$	−36.6423	−1.74884	−0.874420	+	0.485169i	$0.838758\pi$
$440$	−13.4422	−0.640830
$441$	0	0
$442$	0	0
$443$	27.0933	1.28724	0.643622	−	0.765344i	$-0.277431\pi$
$443$	27.0933	1.28724	0.643622	+	0.765344i	$0.277431\pi$
$444$	0.412158	0.0195602
$445$	−27.0304	−1.28136
$446$	32.5886	1.54312
$447$	−21.4830	−1.01611
$448$	0	0
$449$	27.4324	1.29461	0.647307	−	0.762229i	$-0.275895\pi$
$449$	27.4324	1.29461	0.647307	+	0.762229i	$0.275895\pi$
$450$	−12.5072	−0.589593
$451$	−5.36892	−0.252812
$452$	−1.24354	−0.0584911
$453$	−9.66431	−0.454069
$454$	−20.6832	−0.970712
$455$	0	0
$456$	5.70169	0.267006
$457$	39.6639	1.85540	0.927700	−	0.373327i	$-0.121783\pi$
$457$	39.6639	1.85540	0.927700	+	0.373327i	$0.121783\pi$
$458$	−21.9964	−1.02783
$459$	−28.5127	−1.33086
$460$	−1.11880	−0.0521645
$461$	−4.89580	−0.228020	−0.114010	−	0.993480i	$-0.536370\pi$
$461$	−4.89580	−0.228020	−0.114010	+	0.993480i	$0.536370\pi$
$462$	0	0
$463$	4.71193	0.218982	0.109491	−	0.993988i	$-0.465078\pi$
$463$	4.71193	0.218982	0.109491	+	0.993988i	$0.465078\pi$
$464$	−14.4928	−0.672810
$465$	37.6848	1.74759
$466$	39.1605	1.81408
$467$	−32.0161	−1.48153	−0.740765	−	0.671764i	$-0.765537\pi$
$467$	−32.0161	−1.48153	−0.740765	+	0.671764i	$0.765537\pi$
$468$	0	0
$469$	0	0
$470$	4.29176	0.197964
$471$	18.4552	0.850372
$472$	32.9563	1.51693
$473$	−4.89814	−0.225217
$474$	−7.87452	−0.361689
$475$	−7.29526	−0.334730
$476$	0	0
$477$	−0.0962247	−0.00440582
$478$	11.6417	0.532480
$479$	−18.0245	−0.823560	−0.411780	−	0.911283i	$-0.635093\pi$
$479$	−18.0245	−0.823560	−0.411780	+	0.911283i	$0.635093\pi$
$480$	7.84894	0.358253
$481$	0	0
$482$	−24.4807	−1.11506
$483$	0	0
$484$	1.78278	0.0810355
$485$	−1.70739	−0.0775284
$486$	−15.9142	−0.721885
$487$	−17.6004	−0.797550	−0.398775	−	0.917049i	$-0.630565\pi$
$487$	−17.6004	−0.797550	−0.398775	+	0.917049i	$0.630565\pi$
$488$	22.4642	1.01691
$489$	24.6676	1.11551
$490$	0	0
$491$	3.86360	0.174362	0.0871810	−	0.996192i	$-0.472214\pi$
$491$	3.86360	0.174362	0.0871810	+	0.996192i	$0.472214\pi$
$492$	1.63868	0.0738773
$493$	−31.3038	−1.40985
$494$	0	0
$495$	−5.48776	−0.246656
$496$	18.4605	0.828899
$497$	0	0
$498$	−6.41153	−0.287308
$499$	12.6473	0.566169	0.283084	−	0.959095i	$-0.408642\pi$
$499$	12.6473	0.566169	0.283084	+	0.959095i	$0.408642\pi$
$500$	−1.85351	−0.0828917
$501$	39.8171	1.77890
$502$	21.3514	0.952959
$503$	−22.0360	−0.982537	−0.491268	−	0.871008i	$-0.663467\pi$
$503$	−22.0360	−0.982537	−0.491268	+	0.871008i	$0.663467\pi$
$504$	0	0
$505$	−10.2693	−0.456980
$506$	2.83348	0.125964
$507$	0	0
$508$	−2.80150	−0.124296
$509$	−15.6702	−0.694568	−0.347284	−	0.937760i	$-0.612896\pi$
$509$	−15.6702	−0.694568	−0.347284	+	0.937760i	$0.612896\pi$
$510$	−76.1491	−3.37194
$511$	0	0
$512$	24.9600	1.10309
$513$	−3.47743	−0.153532
$514$	−32.7249	−1.44344
$515$	40.4411	1.78205
$516$	1.49499	0.0658132
$517$	1.14341	0.0502871
$518$	0	0
$519$	29.4795	1.29401
$520$	0	0
$521$	25.2415	1.10585	0.552925	−	0.833231i	$-0.313512\pi$
$521$	25.2415	1.10585	0.552925	+	0.833231i	$0.313512\pi$
$522$	−6.54538	−0.286483
$523$	13.2477	0.579279	0.289640	−	0.957136i	$-0.406465\pi$
$523$	13.2477	0.579279	0.289640	+	0.957136i	$0.406465\pi$
$524$	2.12957	0.0930307
$525$	0	0
$526$	−20.7629	−0.905307
$527$	39.8738	1.73693
$528$	−9.39259	−0.408760
$529$	−20.2865	−0.882021
$530$	0.383920	0.0166764
$531$	13.4544	0.583871
$532$	0	0
$533$	0	0
$534$	−20.8942	−0.904181
$535$	−23.4477	−1.01373
$536$	−18.6252	−0.804485
$537$	−11.1240	−0.480036
$538$	−17.5427	−0.756320
$539$	0	0
$540$	−2.50225	−0.107680
$541$	−14.4034	−0.619250	−0.309625	−	0.950859i	$-0.600204\pi$
$541$	−14.4034	−0.619250	−0.309625	+	0.950859i	$0.600204\pi$
$542$	36.2816	1.55843
$543$	31.7565	1.36280
$544$	8.30488	0.356069
$545$	−20.8328	−0.892377
$546$	0	0
$547$	2.00679	0.0858042	0.0429021	−	0.999079i	$-0.486340\pi$
$547$	2.00679	0.0858042	0.0429021	+	0.999079i	$0.486340\pi$
$548$	−3.35631	−0.143375
$549$	9.17100	0.391409
$550$	13.2947	0.566889
$551$	−3.81783	−0.162645
$552$	−9.95071	−0.423530
$553$	0	0
$554$	−17.0863	−0.725928
$555$	−7.72451	−0.327887
$556$	−1.11499	−0.0472863
$557$	−8.57916	−0.363511	−0.181755	−	0.983344i	$-0.558178\pi$
$557$	−8.57916	−0.363511	−0.181755	+	0.983344i	$0.558178\pi$
$558$	8.33731	0.352946
$559$	0	0
$560$	0	0
$561$	−20.2876	−0.856543
$562$	−35.9466	−1.51631
$563$	12.7744	0.538375	0.269188	−	0.963088i	$-0.413245\pi$
$563$	12.7744	0.538375	0.269188	+	0.963088i	$0.413245\pi$
$564$	−0.348987	−0.0146950
$565$	23.3059	0.980487
$566$	19.8317	0.833587
$567$	0	0
$568$	−33.6612	−1.41239
$569$	−5.79116	−0.242778	−0.121389	−	0.992605i	$-0.538735\pi$
$569$	−5.79116	−0.242778	−0.121389	+	0.992605i	$0.538735\pi$
$570$	−9.28720	−0.388998
$571$	−44.1332	−1.84692	−0.923458	−	0.383700i	$-0.874650\pi$
$571$	−44.1332	−1.84692	−0.923458	+	0.383700i	$0.874650\pi$
$572$	0	0
$573$	−9.73025	−0.406487
$574$	0	0
$575$	12.7318	0.530954
$576$	10.3567	0.431529
$577$	−11.9330	−0.496776	−0.248388	−	0.968661i	$-0.579901\pi$
$577$	−11.9330	−0.496776	−0.248388	+	0.968661i	$0.579901\pi$
$578$	−57.7037	−2.40016
$579$	−43.1324	−1.79252
$580$	−2.74719	−0.114071
$581$	0	0
$582$	−1.31979	−0.0547071
$583$	0.102284	0.00423617
$584$	2.24086	0.0927274
$585$	0	0
$586$	−15.5818	−0.643679
$587$	−20.3516	−0.840000	−0.420000	−	0.907524i	$-0.637970\pi$
$587$	−20.3516	−0.840000	−0.420000	+	0.907524i	$0.637970\pi$
$588$	0	0
$589$	4.86304	0.200378
$590$	−53.6808	−2.21000
$591$	−11.9154	−0.490133
$592$	−3.78397	−0.155520
$593$	−18.1800	−0.746563	−0.373282	−	0.927718i	$-0.621768\pi$
$593$	−18.1800	−0.746563	−0.373282	+	0.927718i	$0.621768\pi$
$594$	6.33719	0.260018
$595$	0	0
$596$	−1.99484	−0.0817120
$597$	21.7684	0.890920
$598$	0	0
$599$	−38.2682	−1.56359	−0.781797	−	0.623532i	$-0.785697\pi$
$599$	−38.2682	−1.56359	−0.781797	+	0.623532i	$0.785697\pi$
$600$	−46.6888	−1.90606
$601$	26.8719	1.09613	0.548064	−	0.836436i	$-0.315365\pi$
$601$	26.8719	1.09613	0.548064	+	0.836436i	$0.315365\pi$
$602$	0	0
$603$	−7.60372	−0.309648
$604$	−0.897398	−0.0365146
$605$	−33.4122	−1.35840
$606$	−7.93809	−0.322463
$607$	9.40209	0.381619	0.190810	−	0.981627i	$-0.438889\pi$
$607$	9.40209	0.381619	0.190810	+	0.981627i	$0.438889\pi$
$608$	1.01287	0.0410773
$609$	0	0
$610$	−36.5908	−1.48152
$611$	0	0
$612$	1.77225	0.0716389
$613$	−13.2894	−0.536753	−0.268376	−	0.963314i	$-0.586487\pi$
$613$	−13.2894	−0.536753	−0.268376	+	0.963314i	$0.586487\pi$
$614$	39.4602	1.59248
$615$	−30.7115	−1.23841
$616$	0	0
$617$	11.2261	0.451947	0.225973	−	0.974133i	$-0.427444\pi$
$617$	11.2261	0.451947	0.225973	+	0.974133i	$0.427444\pi$
$618$	31.2606	1.25748
$619$	−9.28505	−0.373198	−0.186599	−	0.982436i	$-0.559746\pi$
$619$	−9.28505	−0.373198	−0.186599	+	0.982436i	$0.559746\pi$
$620$	3.49929	0.140535
$621$	6.06888	0.243536
$622$	−0.202664	−0.00812608
$623$	0	0
$624$	0	0
$625$	−3.90726	−0.156290
$626$	−14.1533	−0.565680
$627$	−2.47429	−0.0988137
$628$	1.71370	0.0683839
$629$	−8.17322	−0.325888
$630$	0	0
$631$	10.4026	0.414122	0.207061	−	0.978328i	$-0.433610\pi$
$631$	10.4026	0.414122	0.207061	+	0.978328i	$0.433610\pi$
$632$	−8.41327	−0.334662
$633$	9.56474	0.380164
$634$	2.02693	0.0804998
$635$	52.5046	2.08358
$636$	−0.0312187	−0.00123790
$637$	0	0
$638$	6.95754	0.275452
$639$	−13.7422	−0.543632
$640$	−33.6644	−1.33070
$641$	14.8591	0.586899	0.293449	−	0.955975i	$-0.405197\pi$
$641$	14.8591	0.586899	0.293449	+	0.955975i	$0.405197\pi$
$642$	−18.1248	−0.715328
$643$	2.29722	0.0905935	0.0452968	−	0.998974i	$-0.485577\pi$
$643$	2.29722	0.0905935	0.0452968	+	0.998974i	$0.485577\pi$
$644$	0	0
$645$	−28.0185	−1.10323
$646$	−9.82669	−0.386626
$647$	−7.99865	−0.314459	−0.157230	−	0.987562i	$-0.550256\pi$
$647$	−7.99865	−0.314459	−0.157230	+	0.987562i	$0.550256\pi$
$648$	−32.8885	−1.29198
$649$	−14.3016	−0.561387
$650$	0	0
$651$	0	0
$652$	2.29056	0.0897053
$653$	3.98444	0.155923	0.0779615	−	0.996956i	$-0.475159\pi$
$653$	3.98444	0.155923	0.0779615	+	0.996956i	$0.475159\pi$
$654$	−16.1035	−0.629696
$655$	−39.9116	−1.55948
$656$	−15.0445	−0.587389
$657$	0.914831	0.0356910
$658$	0	0
$659$	−27.5003	−1.07126	−0.535629	−	0.844453i	$-0.679925\pi$
$659$	−27.5003	−1.07126	−0.535629	+	0.844453i	$0.679925\pi$
$660$	−1.78042	−0.0693028
$661$	−6.98621	−0.271732	−0.135866	−	0.990727i	$-0.543382\pi$
$661$	−6.98621	−0.271732	−0.135866	+	0.990727i	$0.543382\pi$
$662$	33.9681	1.32021
$663$	0	0
$664$	−6.85019	−0.265839
$665$	0	0
$666$	−1.70896	−0.0662207
$667$	6.66296	0.257991
$668$	3.69729	0.143053
$669$	49.6646	1.92014
$670$	30.3376	1.17204
$671$	−9.74849	−0.376336
$672$	0	0
$673$	5.45566	0.210300	0.105150	−	0.994456i	$-0.466468\pi$
$673$	5.45566	0.210300	0.105150	+	0.994456i	$0.466468\pi$
$674$	43.2909	1.66750
$675$	28.4752	1.09601
$676$	0	0
$677$	−33.7922	−1.29874	−0.649371	−	0.760472i	$-0.724968\pi$
$677$	−33.7922	−1.29874	−0.649371	+	0.760472i	$0.724968\pi$
$678$	18.0152	0.691870
$679$	0	0
$680$	−81.3590	−3.11997
$681$	−31.5209	−1.20788
$682$	−8.86231	−0.339355
$683$	12.2988	0.470602	0.235301	−	0.971923i	$-0.424392\pi$
$683$	12.2988	0.470602	0.235301	+	0.971923i	$0.424392\pi$
$684$	0.216145	0.00826450
$685$	62.9028	2.40339
$686$	0	0
$687$	−33.5222	−1.27895
$688$	−13.7253	−0.523272
$689$	0	0
$690$	16.2082	0.617036
$691$	11.0897	0.421871	0.210935	−	0.977500i	$-0.432349\pi$
$691$	11.0897	0.421871	0.210935	+	0.977500i	$0.432349\pi$
$692$	2.73738	0.104059
$693$	0	0
$694$	−33.3129	−1.26454
$695$	20.8968	0.792662
$696$	−24.4337	−0.926156
$697$	−32.4955	−1.23086
$698$	15.5449	0.588385
$699$	59.6800	2.25731
$700$	0	0
$701$	−10.6470	−0.402133	−0.201066	−	0.979578i	$-0.564441\pi$
$701$	−10.6470	−0.402133	−0.201066	+	0.979578i	$0.564441\pi$
$702$	0	0
$703$	−0.996813	−0.0375955
$704$	−11.0089	−0.414912
$705$	6.54058	0.246333
$706$	27.0235	1.01704
$707$	0	0
$708$	4.36508	0.164050
$709$	−40.7069	−1.52878	−0.764391	−	0.644754i	$-0.776960\pi$
$709$	−40.7069	−1.52878	−0.764391	+	0.644754i	$0.776960\pi$
$710$	54.8290	2.05770
$711$	−3.43472	−0.128812
$712$	−22.3237	−0.836618
$713$	−8.48708	−0.317844
$714$	0	0
$715$	0	0
$716$	−1.03294	−0.0386028
$717$	17.7418	0.662579
$718$	−20.2470	−0.755612
$719$	9.77537	0.364560	0.182280	−	0.983247i	$-0.441652\pi$
$719$	9.77537	0.364560	0.182280	+	0.983247i	$0.441652\pi$
$720$	−15.3775	−0.573086
$721$	0	0
$722$	24.3608	0.906616
$723$	−37.3081	−1.38750
$724$	2.94881	0.109592
$725$	31.2627	1.16107
$726$	−25.8273	−0.958540
$727$	−12.2091	−0.452811	−0.226406	−	0.974033i	$-0.572697\pi$
$727$	−12.2091	−0.452811	−0.226406	+	0.974033i	$0.572697\pi$
$728$	0	0
$729$	9.23219	0.341933
$730$	−3.65002	−0.135093
$731$	−29.6461	−1.09650
$732$	2.97539	0.109974
$733$	−22.3153	−0.824236	−0.412118	−	0.911131i	$-0.635211\pi$
$733$	−22.3153	−0.824236	−0.412118	+	0.911131i	$0.635211\pi$
$734$	12.1118	0.447055
$735$	0	0
$736$	−1.76768	−0.0651576
$737$	8.08253	0.297724
$738$	−6.79456	−0.250111
$739$	−42.3729	−1.55871	−0.779357	−	0.626580i	$-0.784454\pi$
$739$	−42.3729	−1.55871	−0.779357	+	0.626580i	$0.784454\pi$
$740$	−0.717275	−0.0263675
$741$	0	0
$742$	0	0
$743$	−30.9801	−1.13655	−0.568276	−	0.822838i	$-0.692389\pi$
$743$	−30.9801	−1.13655	−0.568276	+	0.822838i	$0.692389\pi$
$744$	31.1229	1.14102
$745$	37.3866	1.36974
$746$	21.7107	0.794885
$747$	−2.79659	−0.102322
$748$	−1.88385	−0.0688802
$749$	0	0
$750$	26.8520	0.980497
$751$	22.5660	0.823444	0.411722	−	0.911309i	$-0.364927\pi$
$751$	22.5660	0.823444	0.411722	+	0.911309i	$0.364927\pi$
$752$	3.20400	0.116838
$753$	32.5392	1.18579
$754$	0	0
$755$	16.8187	0.612095
$756$	0	0
$757$	32.2808	1.17327	0.586633	−	0.809853i	$-0.300453\pi$
$757$	32.2808	1.17327	0.586633	+	0.809853i	$0.300453\pi$
$758$	21.0737	0.765431
$759$	4.31818	0.156740
$760$	−9.92260	−0.359931
$761$	−29.7517	−1.07850	−0.539249	−	0.842147i	$-0.681292\pi$
$761$	−29.7517	−1.07850	−0.539249	+	0.842147i	$0.681292\pi$
$762$	40.5855	1.47026
$763$	0	0
$764$	−0.903521	−0.0326883
$765$	−33.2148	−1.20088
$766$	33.1367	1.19728
$767$	0	0
$768$	9.27895	0.334825
$769$	41.8105	1.50773	0.753863	−	0.657032i	$-0.228188\pi$
$769$	41.8105	1.50773	0.753863	+	0.657032i	$0.228188\pi$
$770$	0	0
$771$	−49.8723	−1.79611
$772$	−4.00514	−0.144148
$773$	−41.4336	−1.49026	−0.745132	−	0.666917i	$-0.767613\pi$
$773$	−41.4336	−1.49026	−0.745132	+	0.666917i	$0.767613\pi$
$774$	−6.19877	−0.222810
$775$	−39.8215	−1.43043
$776$	−1.41009	−0.0506192
$777$	0	0
$778$	25.3665	0.909433
$779$	−3.96318	−0.141996
$780$	0	0
$781$	14.6075	0.522698
$782$	17.1497	0.613273
$783$	14.9020	0.532552
$784$	0	0
$785$	−32.1175	−1.14632
$786$	−30.8513	−1.10043
$787$	23.8627	0.850612	0.425306	−	0.905050i	$-0.360167\pi$
$787$	23.8627	0.850612	0.425306	+	0.905050i	$0.360167\pi$
$788$	−1.10643	−0.0394148
$789$	−31.6424	−1.12650
$790$	13.7040	0.487565
$791$	0	0
$792$	−4.53221	−0.161045
$793$	0	0
$794$	19.5098	0.692377
$795$	0.585088	0.0207509
$796$	2.02135	0.0716447
$797$	−50.8231	−1.80025	−0.900123	−	0.435636i	$-0.856524\pi$
$797$	−50.8231	−1.80025	−0.900123	+	0.435636i	$0.856524\pi$
$798$	0	0
$799$	6.92052	0.244830
$800$	−8.29397	−0.293236
$801$	−9.11367	−0.322016
$802$	−28.2349	−0.997008
$803$	−0.972438	−0.0343166
$804$	−2.46691	−0.0870014
$805$	0	0
$806$	0	0
$807$	−26.7348	−0.941110
$808$	−8.48119	−0.298367
$809$	4.41176	0.155109	0.0775547	−	0.996988i	$-0.475289\pi$
$809$	4.41176	0.155109	0.0775547	+	0.996988i	$0.475289\pi$
$810$	53.5704	1.88227
$811$	17.6493	0.619750	0.309875	−	0.950777i	$-0.399713\pi$
$811$	17.6493	0.619750	0.309875	+	0.950777i	$0.399713\pi$
$812$	0	0
$813$	55.2926	1.93920
$814$	1.81657	0.0636707
$815$	−42.9288	−1.50373
$816$	−56.8489	−1.99011
$817$	−3.61566	−0.126496
$818$	−28.8249	−1.00784
$819$	0	0
$820$	−2.85178	−0.0995884
$821$	−3.56043	−0.124260	−0.0621299	−	0.998068i	$-0.519789\pi$
$821$	−3.56043	−0.124260	−0.0621299	+	0.998068i	$0.519789\pi$
$822$	48.6232	1.69593
$823$	21.8665	0.762217	0.381109	−	0.924530i	$-0.375542\pi$
$823$	21.8665	0.762217	0.381109	+	0.924530i	$0.375542\pi$
$824$	33.3993	1.16352
$825$	20.2610	0.705396
$826$	0	0
$827$	18.1361	0.630653	0.315327	−	0.948983i	$-0.397886\pi$
$827$	18.1361	0.630653	0.315327	+	0.948983i	$0.397886\pi$
$828$	−0.377220	−0.0131093
$829$	−30.8994	−1.07318	−0.536590	−	0.843843i	$-0.680288\pi$
$829$	−30.8994	−1.07318	−0.536590	+	0.843843i	$0.680288\pi$
$830$	11.1579	0.387297
$831$	−26.0393	−0.903293
$832$	0	0
$833$	0	0
$834$	16.1530	0.559333
$835$	−69.2933	−2.39799
$836$	−0.229755	−0.00794625
$837$	−18.9817	−0.656103
$838$	10.7135	0.370090
$839$	15.3959	0.531526	0.265763	−	0.964038i	$-0.414376\pi$
$839$	15.3959	0.531526	0.265763	+	0.964038i	$0.414376\pi$
$840$	0	0
$841$	−12.6393	−0.435838
$842$	−3.79065	−0.130635
$843$	−54.7820	−1.88679
$844$	0.888153	0.0305715
$845$	0	0
$846$	1.44703	0.0497498
$847$	0	0
$848$	0.286615	0.00984239
$849$	30.2231	1.03726
$850$	80.4667	2.75998
$851$	1.73966	0.0596347
$852$	−4.45845	−0.152744
$853$	23.7772	0.814116	0.407058	−	0.913402i	$-0.366555\pi$
$853$	23.7772	0.814116	0.407058	+	0.913402i	$0.366555\pi$
$854$	0	0
$855$	−4.05090	−0.138538
$856$	−19.3648	−0.661877
$857$	30.1050	1.02837	0.514184	−	0.857680i	$-0.328095\pi$
$857$	30.1050	1.02837	0.514184	+	0.857680i	$0.328095\pi$
$858$	0	0
$859$	15.1343	0.516377	0.258188	−	0.966095i	$-0.416874\pi$
$859$	15.1343	0.516377	0.258188	+	0.966095i	$0.416874\pi$
$860$	−2.60172	−0.0887178
$861$	0	0
$862$	7.71657	0.262827
$863$	−18.2657	−0.621773	−0.310886	−	0.950447i	$-0.600626\pi$
$863$	−18.2657	−0.621773	−0.310886	+	0.950447i	$0.600626\pi$
$864$	−3.95348	−0.134500
$865$	−51.3029	−1.74435
$866$	−32.9925	−1.12113
$867$	−87.9395	−2.98658
$868$	0	0
$869$	3.65100	0.123852
$870$	39.7988	1.34931
$871$	0	0
$872$	−17.2053	−0.582643
$873$	−0.575668	−0.0194834
$874$	2.09159	0.0707492
$875$	0	0
$876$	0.296803	0.0100281
$877$	−6.99639	−0.236251	−0.118126	−	0.992999i	$-0.537689\pi$
$877$	−6.99639	−0.236251	−0.118126	+	0.992999i	$0.537689\pi$
$878$	49.2922	1.66353
$879$	−23.7465	−0.800948
$880$	16.3458	0.551018
$881$	−25.7746	−0.868368	−0.434184	−	0.900824i	$-0.642963\pi$
$881$	−25.7746	−0.868368	−0.434184	+	0.900824i	$0.642963\pi$
$882$	0	0
$883$	16.4526	0.553674	0.276837	−	0.960917i	$-0.410714\pi$
$883$	16.4526	0.553674	0.276837	+	0.960917i	$0.410714\pi$
$884$	0	0
$885$	−81.8086	−2.74997
$886$	−36.4467	−1.22445
$887$	−55.2455	−1.85496	−0.927481	−	0.373871i	$-0.878030\pi$
$887$	−55.2455	−1.85496	−0.927481	+	0.373871i	$0.878030\pi$
$888$	−6.37948	−0.214081
$889$	0	0
$890$	36.3620	1.21886
$891$	14.2722	0.478137
$892$	4.61170	0.154411
$893$	0.844032	0.0282444
$894$	28.8995	0.966542
$895$	19.3590	0.647099
$896$	0	0
$897$	0	0
$898$	−36.9028	−1.23146
$899$	−20.8398	−0.695046
$900$	−1.76992	−0.0589973
$901$	0.619076	0.0206244
$902$	7.22241	0.240480
$903$	0	0
$904$	19.2478	0.640171
$905$	−55.2655	−1.83709
$906$	13.0007	0.431919
$907$	47.8424	1.58858	0.794290	−	0.607538i	$-0.207843\pi$
$907$	47.8424	1.58858	0.794290	+	0.607538i	$0.207843\pi$
$908$	−2.92694	−0.0971338
$909$	−3.46245	−0.114842
$910$	0	0
$911$	−23.0711	−0.764380	−0.382190	−	0.924084i	$-0.624830\pi$
$911$	−23.0711	−0.764380	−0.382190	+	0.924084i	$0.624830\pi$
$912$	−6.93333	−0.229585
$913$	2.97269	0.0983817
$914$	−53.3569	−1.76489
$915$	−55.7637	−1.84349
$916$	−3.11277	−0.102849
$917$	0	0
$918$	38.3560	1.26594
$919$	−43.4368	−1.43285	−0.716424	−	0.697665i	$-0.754222\pi$
$919$	−43.4368	−1.43285	−0.716424	+	0.697665i	$0.754222\pi$
$920$	17.3171	0.570929
$921$	60.1367	1.98157
$922$	6.58595	0.216897
$923$	0	0
$924$	0	0
$925$	8.16249	0.268381
$926$	−6.33861	−0.208300
$927$	13.6353	0.447841
$928$	−4.34049	−0.142484
$929$	−12.7819	−0.419361	−0.209680	−	0.977770i	$-0.567242\pi$
$929$	−12.7819	−0.419361	−0.209680	+	0.977770i	$0.567242\pi$
$930$	−50.6945	−1.66234
$931$	0	0
$932$	5.54171	0.181525
$933$	−0.308857	−0.0101115
$934$	43.0689	1.40926
$935$	35.3063	1.15464
$936$	0	0
$937$	−16.2533	−0.530971	−0.265486	−	0.964115i	$-0.585532\pi$
$937$	−16.2533	−0.530971	−0.265486	+	0.964115i	$0.585532\pi$
$938$	0	0
$939$	−21.5694	−0.703891
$940$	0.607339	0.0198092
$941$	−45.1488	−1.47181	−0.735905	−	0.677085i	$-0.763243\pi$
$941$	−45.1488	−1.47181	−0.735905	+	0.677085i	$0.763243\pi$
$942$	−24.8265	−0.808890
$943$	6.91662	0.225236
$944$	−40.0752	−1.30434
$945$	0	0
$946$	6.58911	0.214230
$947$	19.8557	0.645225	0.322612	−	0.946531i	$-0.395439\pi$
$947$	19.8557	0.645225	0.322612	+	0.946531i	$0.395439\pi$
$948$	−1.11434	−0.0361922
$949$	0	0
$950$	9.81378	0.318401
$951$	3.08901	0.100168
$952$	0	0
$953$	15.7287	0.509501	0.254751	−	0.967007i	$-0.418007\pi$
$953$	15.7287	0.509501	0.254751	+	0.967007i	$0.418007\pi$
$954$	0.129444	0.00419090
$955$	16.9335	0.547954
$956$	1.64745	0.0532823
$957$	10.6032	0.342752
$958$	24.2470	0.783385
$959$	0	0
$960$	−62.9734	−2.03246
$961$	−4.45488	−0.143706
$962$	0	0
$963$	−7.90569	−0.254757
$964$	−3.46432	−0.111578
$965$	75.0629	2.41636
$966$	0	0
$967$	52.1912	1.67835	0.839177	−	0.543858i	$-0.183037\pi$
$967$	52.1912	1.67835	0.839177	+	0.543858i	$0.183037\pi$
$968$	−27.5943	−0.886915
$969$	−14.9757	−0.481089
$970$	2.29682	0.0737465
$971$	22.4584	0.720724	0.360362	−	0.932813i	$-0.382653\pi$
$971$	22.4584	0.720724	0.360362	+	0.932813i	$0.382653\pi$
$972$	−2.25207	−0.0722350
$973$	0	0
$974$	23.6765	0.758645
$975$	0	0
$976$	−27.3167	−0.874387
$977$	41.0345	1.31281	0.656405	−	0.754409i	$-0.272076\pi$
$977$	41.0345	1.31281	0.656405	+	0.754409i	$0.272076\pi$
$978$	−33.1835	−1.06109
$979$	9.68755	0.309616
$980$	0	0
$981$	−7.02404	−0.224260
$982$	−5.19742	−0.165856
$983$	−26.8328	−0.855832	−0.427916	−	0.903818i	$-0.640752\pi$
$983$	−26.8328	−0.855832	−0.427916	+	0.903818i	$0.640752\pi$
$984$	−25.3639	−0.808571
$985$	20.7362	0.660711
$986$	42.1107	1.34108
$987$	0	0
$988$	0	0
$989$	6.31013	0.200650
$990$	7.38228	0.234624
$991$	10.3751	0.329576	0.164788	−	0.986329i	$-0.447306\pi$
$991$	10.3751	0.329576	0.164788	+	0.986329i	$0.447306\pi$
$992$	5.52879	0.175539
$993$	51.7669	1.64277
$994$	0	0
$995$	−37.8833	−1.20098
$996$	−0.907312	−0.0287493
$997$	53.9097	1.70734	0.853669	−	0.520816i	$-0.174372\pi$
$997$	53.9097	1.70734	0.853669	+	0.520816i	$0.174372\pi$
$998$	−17.0134	−0.538550
$999$	3.89081	0.123100

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.co.1.4		12
7.3	odd	6		1183.2.e.j.170.9		24
7.5	odd	6		1183.2.e.j.508.9		24
7.6	odd	2		8281.2.a.cp.1.4		12
13.2	odd	12		637.2.q.i.589.2		12
13.7	odd	12		637.2.q.i.491.2		12
13.12	even	2	inner	8281.2.a.co.1.9		12
91.2	odd	12		637.2.k.i.459.5		12
91.12	odd	6		1183.2.e.j.508.4		24
91.20	even	12		637.2.q.g.491.2		12
91.33	even	12		91.2.u.b.88.5	yes	12
91.38	odd	6		1183.2.e.j.170.4		24
91.41	even	12		637.2.q.g.589.2		12
91.46	odd	12		637.2.k.i.569.2		12
91.54	even	12		91.2.k.b.4.5	✓	12
91.59	even	12		91.2.k.b.23.2	yes	12
91.67	odd	12		637.2.u.g.30.5		12
91.72	odd	12		637.2.u.g.361.5		12
91.80	even	12		91.2.u.b.30.5	yes	12
91.90	odd	2		8281.2.a.cp.1.9		12
273.59	odd	12		819.2.bm.f.478.5		12
273.80	odd	12		819.2.do.e.667.2		12
273.215	odd	12		819.2.do.e.361.2		12
273.236	odd	12		819.2.bm.f.550.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.k.b.4.5	✓	12	91.54	even	12
91.2.k.b.23.2	yes	12	91.59	even	12
91.2.u.b.30.5	yes	12	91.80	even	12
91.2.u.b.88.5	yes	12	91.33	even	12
637.2.k.i.459.5		12	91.2	odd	12
637.2.k.i.569.2		12	91.46	odd	12
637.2.q.g.491.2		12	91.20	even	12
637.2.q.g.589.2		12	91.41	even	12
637.2.q.i.491.2		12	13.7	odd	12
637.2.q.i.589.2		12	13.2	odd	12
637.2.u.g.30.5		12	91.67	odd	12
637.2.u.g.361.5		12	91.72	odd	12
819.2.bm.f.478.5		12	273.59	odd	12
819.2.bm.f.550.2		12	273.236	odd	12
819.2.do.e.361.2		12	273.215	odd	12
819.2.do.e.667.2		12	273.80	odd	12
1183.2.e.j.170.4		24	91.38	odd	6
1183.2.e.j.170.9		24	7.3	odd	6
1183.2.e.j.508.4		24	91.12	odd	6
1183.2.e.j.508.9		24	7.5	odd	6
8281.2.a.co.1.4		12	1.1	even	1	trivial
8281.2.a.co.1.9		12	13.12	even	2	inner
8281.2.a.cp.1.4		12	7.6	odd	2
8281.2.a.cp.1.9		12	91.90	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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q-1.34523 q^{2} -2.05010 q^{3} -0.190366 q^{4} +3.56778 q^{5} +2.75785 q^{6} +2.94654 q^{8} +1.20292 q^{9} +O(q^{10})\)	\(q-1.34523 q^{2} -2.05010 q^{3} -0.190366 q^{4} +3.56778 q^{5} +2.75785 q^{6} +2.94654 q^{8} +1.20292 q^{9} -4.79947 q^{10} -1.27867 q^{11} +0.390271 q^{12} -7.31431 q^{15} -3.58303 q^{16} -7.73920 q^{17} -1.61821 q^{18} -0.943878 q^{19} -0.679185 q^{20} +1.72010 q^{22} +1.64727 q^{23} -6.04071 q^{24} +7.72903 q^{25} +3.68419 q^{27} +4.04484 q^{29} +9.83940 q^{30} -5.15220 q^{31} -1.07309 q^{32} +2.62141 q^{33} +10.4110 q^{34} -0.228996 q^{36} +1.05608 q^{37} +1.26973 q^{38} +10.5126 q^{40} +4.19882 q^{41} +3.83065 q^{43} +0.243416 q^{44} +4.29176 q^{45} -2.21596 q^{46} -0.894217 q^{47} +7.34558 q^{48} -10.3973 q^{50} +15.8662 q^{51} -0.0799923 q^{53} -4.95607 q^{54} -4.56202 q^{55} +1.93505 q^{57} -5.44122 q^{58} +11.1847 q^{59} +1.39240 q^{60} +7.62392 q^{61} +6.93087 q^{62} +8.60961 q^{64} -3.52639 q^{66} -6.32103 q^{67} +1.47328 q^{68} -3.37708 q^{69} -11.4240 q^{71} +3.54446 q^{72} +0.760506 q^{73} -1.42067 q^{74} -15.8453 q^{75} +0.179683 q^{76} -2.85531 q^{79} -12.7834 q^{80} -11.1617 q^{81} -5.64837 q^{82} -2.32483 q^{83} -27.6117 q^{85} -5.15308 q^{86} -8.29233 q^{87} -3.76766 q^{88} -7.57626 q^{89} -5.77339 q^{90} -0.313586 q^{92} +10.5625 q^{93} +1.20292 q^{94} -3.36755 q^{95} +2.19995 q^{96} -0.478557 q^{97} -1.53815 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{3} + 8 q^{4} + 2 q^{9}+O(q^{10}) \) 12 * q - 6 * q^3 + 8 * q^4 + 2 * q^9	\( 12 q - 6 q^{3} + 8 q^{4} + 2 q^{9} - 24 q^{10} + 2 q^{12} + 16 q^{16} - 34 q^{17} + 30 q^{22} + 6 q^{23} - 10 q^{25} - 12 q^{27} + 2 q^{29} + 22 q^{30} - 26 q^{36} - 38 q^{38} - 2 q^{40} + 22 q^{43} + 38 q^{48} + 8 q^{51} + 16 q^{53} - 30 q^{55} + 10 q^{61} - 82 q^{62} - 2 q^{64} - 68 q^{66} - 22 q^{68} - 14 q^{69} + 66 q^{74} - 2 q^{75} + 70 q^{79} - 28 q^{81} - 10 q^{82} + 20 q^{87} - 28 q^{88} - 66 q^{92} + 2 q^{94} + 4 q^{95}+O(q^{100}) \) 12 * q - 6 * q^3 + 8 * q^4 + 2 * q^9 - 24 * q^10 + 2 * q^12 + 16 * q^16 - 34 * q^17 + 30 * q^22 + 6 * q^23 - 10 * q^25 - 12 * q^27 + 2 * q^29 + 22 * q^30 - 26 * q^36 - 38 * q^38 - 2 * q^40 + 22 * q^43 + 38 * q^48 + 8 * q^51 + 16 * q^53 - 30 * q^55 + 10 * q^61 - 82 * q^62 - 2 * q^64 - 68 * q^66 - 22 * q^68 - 14 * q^69 + 66 * q^74 - 2 * q^75 + 70 * q^79 - 28 * q^81 - 10 * q^82 + 20 * q^87 - 28 * q^88 - 66 * q^92 + 2 * q^94 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more