LMFDB - Embedded newform 8046.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, 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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 17x^{6} - 2x^{5} + 71x^{4} - 18x^{3} - 81x^{2} + 36x + 9 $ x^8 - 17x^6 - 2x^5 + 71x^4 - 18x^3 - 81x^2 + 36x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.29304$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.29304 q^{5} -0.779884 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.29304 q^{5} -0.779884 q^{7} -1.00000 q^{8} +1.29304 q^{10} +2.38592 q^{11} +0.594787 q^{13} +0.779884 q^{14} +1.00000 q^{16} +5.75484 q^{17} +0.870692 q^{19} -1.29304 q^{20} -2.38592 q^{22} -2.02570 q^{23} -3.32806 q^{25} -0.594787 q^{26} -0.779884 q^{28} -3.56389 q^{29} -4.32965 q^{31} -1.00000 q^{32} -5.75484 q^{34} +1.00842 q^{35} -5.50475 q^{37} -0.870692 q^{38} +1.29304 q^{40} +1.77482 q^{41} +2.79986 q^{43} +2.38592 q^{44} +2.02570 q^{46} +5.52119 q^{47} -6.39178 q^{49} +3.32806 q^{50} +0.594787 q^{52} -0.474793 q^{53} -3.08508 q^{55} +0.779884 q^{56} +3.56389 q^{58} -3.90074 q^{59} +6.01222 q^{61} +4.32965 q^{62} +1.00000 q^{64} -0.769082 q^{65} -10.3329 q^{67} +5.75484 q^{68} -1.00842 q^{70} -0.746834 q^{71} -9.67976 q^{73} +5.50475 q^{74} +0.870692 q^{76} -1.86074 q^{77} -4.04561 q^{79} -1.29304 q^{80} -1.77482 q^{82} -2.37305 q^{83} -7.44123 q^{85} -2.79986 q^{86} -2.38592 q^{88} +1.20491 q^{89} -0.463865 q^{91} -2.02570 q^{92} -5.52119 q^{94} -1.12584 q^{95} +0.348072 q^{97} +6.39178 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} - 5 q^{7} - 8 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^4 - 5 * q^7 - 8 * q^8	$ 8 q - 8 q^{2} + 8 q^{4} - 5 q^{7} - 8 q^{8} + 6 q^{11} - 8 q^{13} + 5 q^{14} + 8 q^{16} - 5 q^{17} - 14 q^{19} - 6 q^{22} + 21 q^{23} - 6 q^{25} + 8 q^{26} - 5 q^{28} + 3 q^{29} - 4 q^{31} - 8 q^{32} + 5 q^{34} - 2 q^{35} - 3 q^{37} + 14 q^{38} + 7 q^{41} - 12 q^{43} + 6 q^{44} - 21 q^{46} + 25 q^{47} - 7 q^{49} + 6 q^{50} - 8 q^{52} + 3 q^{53} - 9 q^{55} + 5 q^{56} - 3 q^{58} + 2 q^{59} - 17 q^{61} + 4 q^{62} + 8 q^{64} + 32 q^{65} - 14 q^{67} - 5 q^{68} + 2 q^{70} + 7 q^{71} - 10 q^{73} + 3 q^{74} - 14 q^{76} + 12 q^{77} - 33 q^{79} - 7 q^{82} + 13 q^{83} - 33 q^{85} + 12 q^{86} - 6 q^{88} - 22 q^{89} - 22 q^{91} + 21 q^{92} - 25 q^{94} - 14 q^{95} - 11 q^{97} + 7 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 - 5 * q^7 - 8 * q^8 + 6 * q^11 - 8 * q^13 + 5 * q^14 + 8 * q^16 - 5 * q^17 - 14 * q^19 - 6 * q^22 + 21 * q^23 - 6 * q^25 + 8 * q^26 - 5 * q^28 + 3 * q^29 - 4 * q^31 - 8 * q^32 + 5 * q^34 - 2 * q^35 - 3 * q^37 + 14 * q^38 + 7 * q^41 - 12 * q^43 + 6 * q^44 - 21 * q^46 + 25 * q^47 - 7 * q^49 + 6 * q^50 - 8 * q^52 + 3 * q^53 - 9 * q^55 + 5 * q^56 - 3 * q^58 + 2 * q^59 - 17 * q^61 + 4 * q^62 + 8 * q^64 + 32 * q^65 - 14 * q^67 - 5 * q^68 + 2 * q^70 + 7 * q^71 - 10 * q^73 + 3 * q^74 - 14 * q^76 + 12 * q^77 - 33 * q^79 - 7 * q^82 + 13 * q^83 - 33 * q^85 + 12 * q^86 - 6 * q^88 - 22 * q^89 - 22 * q^91 + 21 * q^92 - 25 * q^94 - 14 * q^95 - 11 * q^97 + 7 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.29304	−0.578264	−0.289132	−	0.957289i	$-0.593367\pi$
$5$	−1.29304	−0.578264	−0.289132	+	0.957289i	$0.593367\pi$
$6$	0	0
$7$	−0.779884	−0.294769	−0.147384	−	0.989079i	$-0.547085\pi$
$7$	−0.779884	−0.294769	−0.147384	+	0.989079i	$0.547085\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.29304	0.408894
$11$	2.38592	0.719382	0.359691	−	0.933071i	$-0.382882\pi$
$11$	2.38592	0.719382	0.359691	+	0.933071i	$0.382882\pi$
$12$	0	0
$13$	0.594787	0.164964	0.0824821	−	0.996593i	$-0.473715\pi$
$13$	0.594787	0.164964	0.0824821	+	0.996593i	$0.473715\pi$
$14$	0.779884	0.208433
$15$	0	0
$16$	1.00000	0.250000
$17$	5.75484	1.39575	0.697877	−	0.716217i	$-0.254128\pi$
$17$	5.75484	1.39575	0.697877	+	0.716217i	$0.254128\pi$
$18$	0	0
$19$	0.870692	0.199750	0.0998752	−	0.995000i	$-0.468156\pi$
$19$	0.870692	0.199750	0.0998752	+	0.995000i	$0.468156\pi$
$20$	−1.29304	−0.289132
$21$	0	0
$22$	−2.38592	−0.508680
$23$	−2.02570	−0.422389	−0.211194	−	0.977444i	$-0.567735\pi$
$23$	−2.02570	−0.422389	−0.211194	+	0.977444i	$0.567735\pi$
$24$	0	0
$25$	−3.32806	−0.665611
$26$	−0.594787	−0.116647
$27$	0	0
$28$	−0.779884	−0.147384
$29$	−3.56389	−0.661797	−0.330899	−	0.943666i	$-0.607352\pi$
$29$	−3.56389	−0.661797	−0.330899	+	0.943666i	$0.607352\pi$
$30$	0	0
$31$	−4.32965	−0.777628	−0.388814	−	0.921316i	$-0.627115\pi$
$31$	−4.32965	−0.777628	−0.388814	+	0.921316i	$0.627115\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−5.75484	−0.986948
$35$	1.00842	0.170454
$36$	0	0
$37$	−5.50475	−0.904976	−0.452488	−	0.891771i	$-0.649463\pi$
$37$	−5.50475	−0.904976	−0.452488	+	0.891771i	$0.649463\pi$
$38$	−0.870692	−0.141245
$39$	0	0
$40$	1.29304	0.204447
$41$	1.77482	0.277180	0.138590	−	0.990350i	$-0.455743\pi$
$41$	1.77482	0.277180	0.138590	+	0.990350i	$0.455743\pi$
$42$	0	0
$43$	2.79986	0.426975	0.213488	−	0.976946i	$-0.431518\pi$
$43$	2.79986	0.426975	0.213488	+	0.976946i	$0.431518\pi$
$44$	2.38592	0.359691
$45$	0	0
$46$	2.02570	0.298674
$47$	5.52119	0.805348	0.402674	−	0.915344i	$-0.368081\pi$
$47$	5.52119	0.805348	0.402674	+	0.915344i	$0.368081\pi$
$48$	0	0
$49$	−6.39178	−0.913112
$50$	3.32806	0.470658
$51$	0	0
$52$	0.594787	0.0824821
$53$	−0.474793	−0.0652178	−0.0326089	−	0.999468i	$-0.510382\pi$
$53$	−0.474793	−0.0652178	−0.0326089	+	0.999468i	$0.510382\pi$
$54$	0	0
$55$	−3.08508	−0.415993
$56$	0.779884	0.104216
$57$	0	0
$58$	3.56389	0.467961
$59$	−3.90074	−0.507833	−0.253917	−	0.967226i	$-0.581719\pi$
$59$	−3.90074	−0.507833	−0.253917	+	0.967226i	$0.581719\pi$
$60$	0	0
$61$	6.01222	0.769785	0.384893	−	0.922961i	$-0.374238\pi$
$61$	6.01222	0.769785	0.384893	+	0.922961i	$0.374238\pi$
$62$	4.32965	0.549866
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.769082	−0.0953928
$66$	0	0
$67$	−10.3329	−1.26236	−0.631182	−	0.775635i	$-0.717430\pi$
$67$	−10.3329	−1.26236	−0.631182	+	0.775635i	$0.717430\pi$
$68$	5.75484	0.697877
$69$	0	0
$70$	−1.00842	−0.120529
$71$	−0.746834	−0.0886329	−0.0443164	−	0.999018i	$-0.514111\pi$
$71$	−0.746834	−0.0886329	−0.0443164	+	0.999018i	$0.514111\pi$
$72$	0	0
$73$	−9.67976	−1.13293	−0.566465	−	0.824086i	$-0.691689\pi$
$73$	−9.67976	−1.13293	−0.566465	+	0.824086i	$0.691689\pi$
$74$	5.50475	0.639914
$75$	0	0
$76$	0.870692	0.0998752
$77$	−1.86074	−0.212051
$78$	0	0
$79$	−4.04561	−0.455167	−0.227583	−	0.973759i	$-0.573082\pi$
$79$	−4.04561	−0.455167	−0.227583	+	0.973759i	$0.573082\pi$
$80$	−1.29304	−0.144566
$81$	0	0
$82$	−1.77482	−0.195996
$83$	−2.37305	−0.260476	−0.130238	−	0.991483i	$-0.541574\pi$
$83$	−2.37305	−0.260476	−0.130238	+	0.991483i	$0.541574\pi$
$84$	0	0
$85$	−7.44123	−0.807114
$86$	−2.79986	−0.301917
$87$	0	0
$88$	−2.38592	−0.254340
$89$	1.20491	0.127721	0.0638603	−	0.997959i	$-0.479659\pi$
$89$	1.20491	0.127721	0.0638603	+	0.997959i	$0.479659\pi$
$90$	0	0
$91$	−0.463865	−0.0486263
$92$	−2.02570	−0.211194
$93$	0	0
$94$	−5.52119	−0.569467
$95$	−1.12584	−0.115508
$96$	0	0
$97$	0.348072	0.0353414	0.0176707	−	0.999844i	$-0.494375\pi$
$97$	0.348072	0.0353414	0.0176707	+	0.999844i	$0.494375\pi$
$98$	6.39178	0.645667
$99$	0	0
$100$	−3.32806	−0.332806
$101$	2.13166	0.212108	0.106054	−	0.994360i	$-0.466178\pi$
$101$	2.13166	0.212108	0.106054	+	0.994360i	$0.466178\pi$
$102$	0	0
$103$	−9.08487	−0.895159	−0.447580	−	0.894244i	$-0.647714\pi$
$103$	−9.08487	−0.895159	−0.447580	+	0.894244i	$0.647714\pi$
$104$	−0.594787	−0.0583237
$105$	0	0
$106$	0.474793	0.0461160
$107$	10.8250	1.04649	0.523245	−	0.852182i	$-0.324721\pi$
$107$	10.8250	1.04649	0.523245	+	0.852182i	$0.324721\pi$
$108$	0	0
$109$	19.1261	1.83194	0.915972	−	0.401243i	$-0.131422\pi$
$109$	19.1261	1.83194	0.915972	+	0.401243i	$0.131422\pi$
$110$	3.08508	0.294151
$111$	0	0
$112$	−0.779884	−0.0736921
$113$	11.5324	1.08488	0.542441	−	0.840094i	$-0.317500\pi$
$113$	11.5324	1.08488	0.542441	+	0.840094i	$0.317500\pi$
$114$	0	0
$115$	2.61931	0.244252
$116$	−3.56389	−0.330899
$117$	0	0
$118$	3.90074	0.359092
$119$	−4.48811	−0.411425
$120$	0	0
$121$	−5.30738	−0.482490
$122$	−6.01222	−0.544320
$123$	0	0
$124$	−4.32965	−0.388814
$125$	10.7685	0.963162
$126$	0	0
$127$	17.0207	1.51034	0.755170	−	0.655529i	$-0.227554\pi$
$127$	17.0207	1.51034	0.755170	+	0.655529i	$0.227554\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0.769082	0.0674529
$131$	18.2646	1.59579	0.797893	−	0.602799i	$-0.205948\pi$
$131$	18.2646	1.59579	0.797893	+	0.602799i	$0.205948\pi$
$132$	0	0
$133$	−0.679039	−0.0588801
$134$	10.3329	0.892626
$135$	0	0
$136$	−5.75484	−0.493474
$137$	9.70537	0.829186	0.414593	−	0.910007i	$-0.363924\pi$
$137$	9.70537	0.829186	0.414593	+	0.910007i	$0.363924\pi$
$138$	0	0
$139$	−2.83764	−0.240686	−0.120343	−	0.992732i	$-0.538399\pi$
$139$	−2.83764	−0.240686	−0.120343	+	0.992732i	$0.538399\pi$
$140$	1.00842	0.0852270
$141$	0	0
$142$	0.746834	0.0626729
$143$	1.41911	0.118672
$144$	0	0
$145$	4.60824	0.382693
$146$	9.67976	0.801103
$147$	0	0
$148$	−5.50475	−0.452488
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	6.40326	0.521090	0.260545	−	0.965462i	$-0.416098\pi$
$151$	6.40326	0.521090	0.260545	+	0.965462i	$0.416098\pi$
$152$	−0.870692	−0.0706224
$153$	0	0
$154$	1.86074	0.149943
$155$	5.59840	0.449674
$156$	0	0
$157$	20.8841	1.66673	0.833366	−	0.552722i	$-0.186411\pi$
$157$	20.8841	1.66673	0.833366	+	0.552722i	$0.186411\pi$
$158$	4.04561	0.321851
$159$	0	0
$160$	1.29304	0.102224
$161$	1.57981	0.124507
$162$	0	0
$163$	−17.1731	−1.34510	−0.672550	−	0.740051i	$-0.734801\pi$
$163$	−17.1731	−1.34510	−0.672550	+	0.740051i	$0.734801\pi$
$164$	1.77482	0.138590
$165$	0	0
$166$	2.37305	0.184185
$167$	−4.93234	−0.381676	−0.190838	−	0.981622i	$-0.561121\pi$
$167$	−4.93234	−0.381676	−0.190838	+	0.981622i	$0.561121\pi$
$168$	0	0
$169$	−12.6462	−0.972787
$170$	7.44123	0.570716
$171$	0	0
$172$	2.79986	0.213488
$173$	−13.3499	−1.01498	−0.507489	−	0.861659i	$-0.669426\pi$
$173$	−13.3499	−1.01498	−0.507489	+	0.861659i	$0.669426\pi$
$174$	0	0
$175$	2.59550	0.196201
$176$	2.38592	0.179846
$177$	0	0
$178$	−1.20491	−0.0903121
$179$	−21.4101	−1.60027	−0.800133	−	0.599823i	$-0.795238\pi$
$179$	−21.4101	−1.60027	−0.800133	+	0.599823i	$0.795238\pi$
$180$	0	0
$181$	−22.9119	−1.70303	−0.851513	−	0.524333i	$-0.824315\pi$
$181$	−22.9119	−1.70303	−0.851513	+	0.524333i	$0.824315\pi$
$182$	0.463865	0.0343840
$183$	0	0
$184$	2.02570	0.149337
$185$	7.11785	0.523315
$186$	0	0
$187$	13.7306	1.00408
$188$	5.52119	0.402674
$189$	0	0
$190$	1.12584	0.0816768
$191$	19.5028	1.41117	0.705587	−	0.708623i	$-0.250683\pi$
$191$	19.5028	1.41117	0.705587	+	0.708623i	$0.250683\pi$
$192$	0	0
$193$	−4.66227	−0.335597	−0.167799	−	0.985821i	$-0.553666\pi$
$193$	−4.66227	−0.335597	−0.167799	+	0.985821i	$0.553666\pi$
$194$	−0.348072	−0.0249901
$195$	0	0
$196$	−6.39178	−0.456556
$197$	−15.9087	−1.13345	−0.566724	−	0.823908i	$-0.691789\pi$
$197$	−15.9087	−1.13345	−0.566724	+	0.823908i	$0.691789\pi$
$198$	0	0
$199$	−11.8997	−0.843548	−0.421774	−	0.906701i	$-0.638592\pi$
$199$	−11.8997	−0.843548	−0.421774	+	0.906701i	$0.638592\pi$
$200$	3.32806	0.235329
$201$	0	0
$202$	−2.13166	−0.149983
$203$	2.77942	0.195077
$204$	0	0
$205$	−2.29490	−0.160283
$206$	9.08487	0.632973
$207$	0	0
$208$	0.594787	0.0412411
$209$	2.07740	0.143697
$210$	0	0
$211$	−27.8174	−1.91503	−0.957515	−	0.288385i	$-0.906882\pi$
$211$	−27.8174	−1.91503	−0.957515	+	0.288385i	$0.906882\pi$
$212$	−0.474793	−0.0326089
$213$	0	0
$214$	−10.8250	−0.739980
$215$	−3.62033	−0.246904
$216$	0	0
$217$	3.37663	0.229220
$218$	−19.1261	−1.29538
$219$	0	0
$220$	−3.08508	−0.207996
$221$	3.42291	0.230250
$222$	0	0
$223$	5.80380	0.388651	0.194325	−	0.980937i	$-0.437748\pi$
$223$	5.80380	0.388651	0.194325	+	0.980937i	$0.437748\pi$
$224$	0.779884	0.0521082
$225$	0	0
$226$	−11.5324	−0.767127
$227$	−25.9424	−1.72185	−0.860927	−	0.508728i	$-0.830116\pi$
$227$	−25.9424	−1.72185	−0.860927	+	0.508728i	$0.830116\pi$
$228$	0	0
$229$	−6.92543	−0.457645	−0.228823	−	0.973468i	$-0.573488\pi$
$229$	−6.92543	−0.457645	−0.228823	+	0.973468i	$0.573488\pi$
$230$	−2.61931	−0.172712
$231$	0	0
$232$	3.56389	0.233981
$233$	−16.6879	−1.09326	−0.546631	−	0.837374i	$-0.684090\pi$
$233$	−16.6879	−1.09326	−0.546631	+	0.837374i	$0.684090\pi$
$234$	0	0
$235$	−7.13910	−0.465703
$236$	−3.90074	−0.253917
$237$	0	0
$238$	4.48811	0.290921
$239$	28.0190	1.81240	0.906199	−	0.422851i	$-0.138971\pi$
$239$	28.0190	1.81240	0.906199	+	0.422851i	$0.138971\pi$
$240$	0	0
$241$	−14.3237	−0.922673	−0.461337	−	0.887225i	$-0.652630\pi$
$241$	−14.3237	−0.922673	−0.461337	+	0.887225i	$0.652630\pi$
$242$	5.30738	0.341172
$243$	0	0
$244$	6.01222	0.384893
$245$	8.26481	0.528019
$246$	0	0
$247$	0.517876	0.0329517
$248$	4.32965	0.274933
$249$	0	0
$250$	−10.7685	−0.681059
$251$	−13.5183	−0.853266	−0.426633	−	0.904425i	$-0.640300\pi$
$251$	−13.5183	−0.853266	−0.426633	+	0.904425i	$0.640300\pi$
$252$	0	0
$253$	−4.83317	−0.303859
$254$	−17.0207	−1.06797
$255$	0	0
$256$	1.00000	0.0625000
$257$	−26.6913	−1.66496	−0.832480	−	0.554055i	$-0.813080\pi$
$257$	−26.6913	−1.66496	−0.832480	+	0.554055i	$0.813080\pi$
$258$	0	0
$259$	4.29307	0.266758
$260$	−0.769082	−0.0476964
$261$	0	0
$262$	−18.2646	−1.12839
$263$	10.6622	0.657457	0.328729	−	0.944424i	$-0.393380\pi$
$263$	10.6622	0.657457	0.328729	+	0.944424i	$0.393380\pi$
$264$	0	0
$265$	0.613925	0.0377131
$266$	0.679039	0.0416345
$267$	0	0
$268$	−10.3329	−0.631182
$269$	13.1092	0.799283	0.399641	−	0.916672i	$-0.369135\pi$
$269$	13.1092	0.799283	0.399641	+	0.916672i	$0.369135\pi$
$270$	0	0
$271$	−0.550318	−0.0334295	−0.0167147	−	0.999860i	$-0.505321\pi$
$271$	−0.550318	−0.0334295	−0.0167147	+	0.999860i	$0.505321\pi$
$272$	5.75484	0.348939
$273$	0	0
$274$	−9.70537	−0.586323
$275$	−7.94047	−0.478829
$276$	0	0
$277$	18.8271	1.13121	0.565605	−	0.824677i	$-0.308643\pi$
$277$	18.8271	1.13121	0.565605	+	0.824677i	$0.308643\pi$
$278$	2.83764	0.170190
$279$	0	0
$280$	−1.00842	−0.0602646
$281$	13.3144	0.794272	0.397136	−	0.917760i	$-0.370004\pi$
$281$	13.3144	0.794272	0.397136	+	0.917760i	$0.370004\pi$
$282$	0	0
$283$	−17.5326	−1.04220	−0.521101	−	0.853495i	$-0.674479\pi$
$283$	−17.5326	−1.04220	−0.521101	+	0.853495i	$0.674479\pi$
$284$	−0.746834	−0.0443164
$285$	0	0
$286$	−1.41911	−0.0839140
$287$	−1.38415	−0.0817039
$288$	0	0
$289$	16.1182	0.948131
$290$	−4.60824	−0.270605
$291$	0	0
$292$	−9.67976	−0.566465
$293$	−31.2279	−1.82435	−0.912176	−	0.409799i	$-0.865599\pi$
$293$	−31.2279	−1.82435	−0.912176	+	0.409799i	$0.865599\pi$
$294$	0	0
$295$	5.04380	0.293661
$296$	5.50475	0.319957
$297$	0	0
$298$	1.00000	0.0579284
$299$	−1.20486	−0.0696790
$300$	0	0
$301$	−2.18357	−0.125859
$302$	−6.40326	−0.368466
$303$	0	0
$304$	0.870692	0.0499376
$305$	−7.77402	−0.445139
$306$	0	0
$307$	31.7057	1.80954	0.904769	−	0.425904i	$-0.140044\pi$
$307$	31.7057	1.80954	0.904769	+	0.425904i	$0.140044\pi$
$308$	−1.86074	−0.106026
$309$	0	0
$310$	−5.59840	−0.317968
$311$	−27.2312	−1.54414	−0.772069	−	0.635539i	$-0.780778\pi$
$311$	−27.2312	−1.54414	−0.772069	+	0.635539i	$0.780778\pi$
$312$	0	0
$313$	−11.4451	−0.646914	−0.323457	−	0.946243i	$-0.604845\pi$
$313$	−11.4451	−0.646914	−0.323457	+	0.946243i	$0.604845\pi$
$314$	−20.8841	−1.17856
$315$	0	0
$316$	−4.04561	−0.227583
$317$	−12.2603	−0.688607	−0.344304	−	0.938858i	$-0.611885\pi$
$317$	−12.2603	−0.688607	−0.344304	+	0.938858i	$0.611885\pi$
$318$	0	0
$319$	−8.50315	−0.476085
$320$	−1.29304	−0.0722830
$321$	0	0
$322$	−1.57981	−0.0880396
$323$	5.01069	0.278803
$324$	0	0
$325$	−1.97948	−0.109802
$326$	17.1731	0.951130
$327$	0	0
$328$	−1.77482	−0.0979979
$329$	−4.30589	−0.237391
$330$	0	0
$331$	−24.8970	−1.36846	−0.684232	−	0.729265i	$-0.739862\pi$
$331$	−24.8970	−1.36846	−0.684232	+	0.729265i	$0.739862\pi$
$332$	−2.37305	−0.130238
$333$	0	0
$334$	4.93234	0.269886
$335$	13.3608	0.729979
$336$	0	0
$337$	−17.4078	−0.948264	−0.474132	−	0.880454i	$-0.657238\pi$
$337$	−17.4078	−0.948264	−0.474132	+	0.880454i	$0.657238\pi$
$338$	12.6462	0.687864
$339$	0	0
$340$	−7.44123	−0.403557
$341$	−10.3302	−0.559412
$342$	0	0
$343$	10.4440	0.563925
$344$	−2.79986	−0.150959
$345$	0	0
$346$	13.3499	0.717697
$347$	15.9545	0.856483	0.428241	−	0.903664i	$-0.359133\pi$
$347$	15.9545	0.856483	0.428241	+	0.903664i	$0.359133\pi$
$348$	0	0
$349$	9.98835	0.534664	0.267332	−	0.963604i	$-0.413858\pi$
$349$	9.98835	0.534664	0.267332	+	0.963604i	$0.413858\pi$
$350$	−2.59550	−0.138735
$351$	0	0
$352$	−2.38592	−0.127170
$353$	8.12410	0.432402	0.216201	−	0.976349i	$-0.430633\pi$
$353$	8.12410	0.432402	0.216201	+	0.976349i	$0.430633\pi$
$354$	0	0
$355$	0.965684	0.0512532
$356$	1.20491	0.0638603
$357$	0	0
$358$	21.4101	1.13156
$359$	4.46644	0.235730	0.117865	−	0.993030i	$-0.462395\pi$
$359$	4.46644	0.235730	0.117865	+	0.993030i	$0.462395\pi$
$360$	0	0
$361$	−18.2419	−0.960100
$362$	22.9119	1.20422
$363$	0	0
$364$	−0.463865	−0.0243131
$365$	12.5163	0.655133
$366$	0	0
$367$	−10.0697	−0.525633	−0.262817	−	0.964846i	$-0.584651\pi$
$367$	−10.0697	−0.525633	−0.262817	+	0.964846i	$0.584651\pi$
$368$	−2.02570	−0.105597
$369$	0	0
$370$	−7.11785	−0.370039
$371$	0.370284	0.0192242
$372$	0	0
$373$	−0.716141	−0.0370804	−0.0185402	−	0.999828i	$-0.505902\pi$
$373$	−0.716141	−0.0370804	−0.0185402	+	0.999828i	$0.505902\pi$
$374$	−13.7306	−0.709992
$375$	0	0
$376$	−5.52119	−0.284733
$377$	−2.11975	−0.109173
$378$	0	0
$379$	14.8847	0.764577	0.382289	−	0.924043i	$-0.375136\pi$
$379$	14.8847	0.764577	0.382289	+	0.924043i	$0.375136\pi$
$380$	−1.12584	−0.0577542
$381$	0	0
$382$	−19.5028	−0.997850
$383$	12.9505	0.661742	0.330871	−	0.943676i	$-0.392658\pi$
$383$	12.9505	0.661742	0.330871	+	0.943676i	$0.392658\pi$
$384$	0	0
$385$	2.40601	0.122622
$386$	4.66227	0.237303
$387$	0	0
$388$	0.348072	0.0176707
$389$	−31.1953	−1.58166	−0.790831	−	0.612034i	$-0.790351\pi$
$389$	−31.1953	−1.58166	−0.790831	+	0.612034i	$0.790351\pi$
$390$	0	0
$391$	−11.6576	−0.589551
$392$	6.39178	0.322834
$393$	0	0
$394$	15.9087	0.801468
$395$	5.23112	0.263206
$396$	0	0
$397$	−3.25142	−0.163184	−0.0815920	−	0.996666i	$-0.526000\pi$
$397$	−3.25142	−0.163184	−0.0815920	+	0.996666i	$0.526000\pi$
$398$	11.8997	0.596478
$399$	0	0
$400$	−3.32806	−0.166403
$401$	−23.9551	−1.19626	−0.598129	−	0.801400i	$-0.704089\pi$
$401$	−23.9551	−1.19626	−0.598129	+	0.801400i	$0.704089\pi$
$402$	0	0
$403$	−2.57522	−0.128281
$404$	2.13166	0.106054
$405$	0	0
$406$	−2.77942	−0.137940
$407$	−13.1339	−0.651023
$408$	0	0
$409$	1.59986	0.0791082	0.0395541	−	0.999217i	$-0.487406\pi$
$409$	1.59986	0.0791082	0.0395541	+	0.999217i	$0.487406\pi$
$410$	2.29490	0.113337
$411$	0	0
$412$	−9.08487	−0.447580
$413$	3.04213	0.149693
$414$	0	0
$415$	3.06844	0.150624
$416$	−0.594787	−0.0291618
$417$	0	0
$418$	−2.07740	−0.101609
$419$	−34.0470	−1.66330	−0.831652	−	0.555297i	$-0.812605\pi$
$419$	−34.0470	−1.66330	−0.831652	+	0.555297i	$0.812605\pi$
$420$	0	0
$421$	6.51403	0.317474	0.158737	−	0.987321i	$-0.449258\pi$
$421$	6.51403	0.317474	0.158737	+	0.987321i	$0.449258\pi$
$422$	27.8174	1.35413
$423$	0	0
$424$	0.474793	0.0230580
$425$	−19.1524	−0.929030
$426$	0	0
$427$	−4.68883	−0.226909
$428$	10.8250	0.523245
$429$	0	0
$430$	3.62033	0.174588
$431$	21.3497	1.02838	0.514190	−	0.857677i	$-0.328093\pi$
$431$	21.3497	1.02838	0.514190	+	0.857677i	$0.328093\pi$
$432$	0	0
$433$	−4.14179	−0.199042	−0.0995208	−	0.995035i	$-0.531731\pi$
$433$	−4.14179	−0.199042	−0.0995208	+	0.995035i	$0.531731\pi$
$434$	−3.37663	−0.162083
$435$	0	0
$436$	19.1261	0.915972
$437$	−1.76376	−0.0843723
$438$	0	0
$439$	−5.93107	−0.283075	−0.141537	−	0.989933i	$-0.545205\pi$
$439$	−5.93107	−0.283075	−0.141537	+	0.989933i	$0.545205\pi$
$440$	3.08508	0.147076
$441$	0	0
$442$	−3.42291	−0.162811
$443$	14.5778	0.692614	0.346307	−	0.938121i	$-0.387436\pi$
$443$	14.5778	0.692614	0.346307	+	0.938121i	$0.387436\pi$
$444$	0	0
$445$	−1.55800	−0.0738562
$446$	−5.80380	−0.274818
$447$	0	0
$448$	−0.779884	−0.0368461
$449$	−4.78348	−0.225746	−0.112873	−	0.993609i	$-0.536005\pi$
$449$	−4.78348	−0.225746	−0.112873	+	0.993609i	$0.536005\pi$
$450$	0	0
$451$	4.23457	0.199398
$452$	11.5324	0.542441
$453$	0	0
$454$	25.9424	1.21754
$455$	0.599795	0.0281188
$456$	0	0
$457$	18.1331	0.848232	0.424116	−	0.905608i	$-0.360585\pi$
$457$	18.1331	0.848232	0.424116	+	0.905608i	$0.360585\pi$
$458$	6.92543	0.323604
$459$	0	0
$460$	2.61931	0.122126
$461$	8.51787	0.396717	0.198358	−	0.980130i	$-0.436439\pi$
$461$	8.51787	0.396717	0.198358	+	0.980130i	$0.436439\pi$
$462$	0	0
$463$	−25.9271	−1.20493	−0.602467	−	0.798144i	$-0.705815\pi$
$463$	−25.9271	−1.20493	−0.602467	+	0.798144i	$0.705815\pi$
$464$	−3.56389	−0.165449
$465$	0	0
$466$	16.6879	0.773053
$467$	7.05773	0.326593	0.163296	−	0.986577i	$-0.447787\pi$
$467$	7.05773	0.326593	0.163296	+	0.986577i	$0.447787\pi$
$468$	0	0
$469$	8.05846	0.372105
$470$	7.13910	0.329302
$471$	0	0
$472$	3.90074	0.179546
$473$	6.68025	0.307158
$474$	0	0
$475$	−2.89771	−0.132956
$476$	−4.48811	−0.205712
$477$	0	0
$478$	−28.0190	−1.28156
$479$	34.9661	1.59764	0.798822	−	0.601568i	$-0.205457\pi$
$479$	34.9661	1.59764	0.798822	+	0.601568i	$0.205457\pi$
$480$	0	0
$481$	−3.27415	−0.149289
$482$	14.3237	0.652429
$483$	0	0
$484$	−5.30738	−0.241245
$485$	−0.450070	−0.0204366
$486$	0	0
$487$	−18.4898	−0.837855	−0.418927	−	0.908020i	$-0.637594\pi$
$487$	−18.4898	−0.837855	−0.418927	+	0.908020i	$0.637594\pi$
$488$	−6.01222	−0.272160
$489$	0	0
$490$	−8.26481	−0.373366
$491$	−19.4078	−0.875863	−0.437932	−	0.899008i	$-0.644289\pi$
$491$	−19.4078	−0.875863	−0.437932	+	0.899008i	$0.644289\pi$
$492$	0	0
$493$	−20.5096	−0.923706
$494$	−0.517876	−0.0233003
$495$	0	0
$496$	−4.32965	−0.194407
$497$	0.582444	0.0261262
$498$	0	0
$499$	−30.5320	−1.36680	−0.683400	−	0.730044i	$-0.739499\pi$
$499$	−30.5320	−1.36680	−0.683400	+	0.730044i	$0.739499\pi$
$500$	10.7685	0.481581
$501$	0	0
$502$	13.5183	0.603350
$503$	−13.7400	−0.612635	−0.306317	−	0.951929i	$-0.599097\pi$
$503$	−13.7400	−0.612635	−0.306317	+	0.951929i	$0.599097\pi$
$504$	0	0
$505$	−2.75631	−0.122654
$506$	4.83317	0.214861
$507$	0	0
$508$	17.0207	0.755170
$509$	11.1971	0.496304	0.248152	−	0.968721i	$-0.420177\pi$
$509$	11.1971	0.496304	0.248152	+	0.968721i	$0.420177\pi$
$510$	0	0
$511$	7.54909	0.333952
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	26.6913	1.17730
$515$	11.7471	0.517638
$516$	0	0
$517$	13.1731	0.579353
$518$	−4.29307	−0.188627
$519$	0	0
$520$	0.769082	0.0337265
$521$	−20.5620	−0.900839	−0.450420	−	0.892817i	$-0.648726\pi$
$521$	−20.5620	−0.900839	−0.450420	+	0.892817i	$0.648726\pi$
$522$	0	0
$523$	5.63207	0.246273	0.123137	−	0.992390i	$-0.460705\pi$
$523$	5.63207	0.246273	0.123137	+	0.992390i	$0.460705\pi$
$524$	18.2646	0.797893
$525$	0	0
$526$	−10.6622	−0.464892
$527$	−24.9165	−1.08538
$528$	0	0
$529$	−18.8965	−0.821588
$530$	−0.613925	−0.0266672
$531$	0	0
$532$	−0.679039	−0.0294401
$533$	1.05564	0.0457248
$534$	0	0
$535$	−13.9971	−0.605147
$536$	10.3329	0.446313
$537$	0	0
$538$	−13.1092	−0.565178
$539$	−15.2503	−0.656876
$540$	0	0
$541$	42.8650	1.84291	0.921455	−	0.388484i	$-0.127001\pi$
$541$	42.8650	1.84291	0.921455	+	0.388484i	$0.127001\pi$
$542$	0.550318	0.0236382
$543$	0	0
$544$	−5.75484	−0.246737
$545$	−24.7307	−1.05935
$546$	0	0
$547$	9.38703	0.401361	0.200680	−	0.979657i	$-0.435685\pi$
$547$	9.38703	0.401361	0.200680	+	0.979657i	$0.435685\pi$
$548$	9.70537	0.414593
$549$	0	0
$550$	7.94047	0.338583
$551$	−3.10305	−0.132194
$552$	0	0
$553$	3.15511	0.134169
$554$	−18.8271	−0.799886
$555$	0	0
$556$	−2.83764	−0.120343
$557$	9.37235	0.397119	0.198560	−	0.980089i	$-0.436374\pi$
$557$	9.37235	0.397119	0.198560	+	0.980089i	$0.436374\pi$
$558$	0	0
$559$	1.66532	0.0704356
$560$	1.00842	0.0426135
$561$	0	0
$562$	−13.3144	−0.561635
$563$	−21.8213	−0.919659	−0.459830	−	0.888007i	$-0.652090\pi$
$563$	−21.8213	−0.919659	−0.459830	+	0.888007i	$0.652090\pi$
$564$	0	0
$565$	−14.9119	−0.627348
$566$	17.5326	0.736948
$567$	0	0
$568$	0.746834	0.0313365
$569$	−31.0932	−1.30349	−0.651747	−	0.758436i	$-0.725964\pi$
$569$	−31.0932	−1.30349	−0.651747	+	0.758436i	$0.725964\pi$
$570$	0	0
$571$	−8.05805	−0.337219	−0.168609	−	0.985683i	$-0.553928\pi$
$571$	−8.05805	−0.337219	−0.168609	+	0.985683i	$0.553928\pi$
$572$	1.41911	0.0593361
$573$	0	0
$574$	1.38415	0.0577734
$575$	6.74166	0.281146
$576$	0	0
$577$	−22.7923	−0.948855	−0.474427	−	0.880295i	$-0.657345\pi$
$577$	−22.7923	−0.948855	−0.474427	+	0.880295i	$0.657345\pi$
$578$	−16.1182	−0.670430
$579$	0	0
$580$	4.60824	0.191347
$581$	1.85071	0.0767802
$582$	0	0
$583$	−1.13282	−0.0469165
$584$	9.67976	0.400551
$585$	0	0
$586$	31.2279	1.29001
$587$	29.4324	1.21480	0.607402	−	0.794395i	$-0.292212\pi$
$587$	29.4324	1.21480	0.607402	+	0.794395i	$0.292212\pi$
$588$	0	0
$589$	−3.76979	−0.155332
$590$	−5.04380	−0.207650
$591$	0	0
$592$	−5.50475	−0.226244
$593$	10.4387	0.428667	0.214334	−	0.976761i	$-0.431242\pi$
$593$	10.4387	0.428667	0.214334	+	0.976761i	$0.431242\pi$
$594$	0	0
$595$	5.80330	0.237912
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	1.20486	0.0492705
$599$	−25.8287	−1.05533	−0.527666	−	0.849452i	$-0.676933\pi$
$599$	−25.8287	−1.05533	−0.527666	+	0.849452i	$0.676933\pi$
$600$	0	0
$601$	−33.4123	−1.36292	−0.681458	−	0.731857i	$-0.738654\pi$
$601$	−33.4123	−1.36292	−0.681458	+	0.731857i	$0.738654\pi$
$602$	2.18357	0.0889957
$603$	0	0
$604$	6.40326	0.260545
$605$	6.86264	0.279006
$606$	0	0
$607$	−39.3275	−1.59626	−0.798128	−	0.602488i	$-0.794176\pi$
$607$	−39.3275	−1.59626	−0.798128	+	0.602488i	$0.794176\pi$
$608$	−0.870692	−0.0353112
$609$	0	0
$610$	7.77402	0.314761
$611$	3.28393	0.132854
$612$	0	0
$613$	8.22742	0.332302	0.166151	−	0.986100i	$-0.446866\pi$
$613$	8.22742	0.332302	0.166151	+	0.986100i	$0.446866\pi$
$614$	−31.7057	−1.27954
$615$	0	0
$616$	1.86074	0.0749714
$617$	−11.0842	−0.446233	−0.223116	−	0.974792i	$-0.571623\pi$
$617$	−11.0842	−0.446233	−0.223116	+	0.974792i	$0.571623\pi$
$618$	0	0
$619$	−47.3737	−1.90411	−0.952055	−	0.305928i	$-0.901033\pi$
$619$	−47.3737	−1.90411	−0.952055	+	0.305928i	$0.901033\pi$
$620$	5.59840	0.224837
$621$	0	0
$622$	27.2312	1.09187
$623$	−0.939693	−0.0376480
$624$	0	0
$625$	2.71623	0.108649
$626$	11.4451	0.457437
$627$	0	0
$628$	20.8841	0.833366
$629$	−31.6790	−1.26312
$630$	0	0
$631$	22.9726	0.914525	0.457262	−	0.889332i	$-0.348830\pi$
$631$	22.9726	0.914525	0.457262	+	0.889332i	$0.348830\pi$
$632$	4.04561	0.160926
$633$	0	0
$634$	12.2603	0.486919
$635$	−22.0083	−0.873374
$636$	0	0
$637$	−3.80175	−0.150631
$638$	8.50315	0.336643
$639$	0	0
$640$	1.29304	0.0511118
$641$	13.6911	0.540767	0.270383	−	0.962753i	$-0.412850\pi$
$641$	13.6911	0.540767	0.270383	+	0.962753i	$0.412850\pi$
$642$	0	0
$643$	31.2826	1.23366	0.616832	−	0.787095i	$-0.288416\pi$
$643$	31.2826	1.23366	0.616832	+	0.787095i	$0.288416\pi$
$644$	1.57981	0.0622534
$645$	0	0
$646$	−5.01069	−0.197143
$647$	14.6018	0.574056	0.287028	−	0.957922i	$-0.407333\pi$
$647$	14.6018	0.574056	0.287028	+	0.957922i	$0.407333\pi$
$648$	0	0
$649$	−9.30685	−0.365326
$650$	1.97948	0.0776418
$651$	0	0
$652$	−17.1731	−0.672550
$653$	−11.3495	−0.444142	−0.222071	−	0.975031i	$-0.571282\pi$
$653$	−11.3495	−0.444142	−0.222071	+	0.975031i	$0.571282\pi$
$654$	0	0
$655$	−23.6168	−0.922785
$656$	1.77482	0.0692950
$657$	0	0
$658$	4.30589	0.167861
$659$	6.22965	0.242673	0.121336	−	0.992611i	$-0.461282\pi$
$659$	6.22965	0.242673	0.121336	+	0.992611i	$0.461282\pi$
$660$	0	0
$661$	30.0378	1.16834	0.584168	−	0.811632i	$-0.301421\pi$
$661$	30.0378	1.16834	0.584168	+	0.811632i	$0.301421\pi$
$662$	24.8970	0.967650
$663$	0	0
$664$	2.37305	0.0920923
$665$	0.878022	0.0340482
$666$	0	0
$667$	7.21938	0.279535
$668$	−4.93234	−0.190838
$669$	0	0
$670$	−13.3608	−0.516173
$671$	14.3447	0.553770
$672$	0	0
$673$	27.6940	1.06753	0.533763	−	0.845634i	$-0.320777\pi$
$673$	27.6940	1.06753	0.533763	+	0.845634i	$0.320777\pi$
$674$	17.4078	0.670524
$675$	0	0
$676$	−12.6462	−0.486393
$677$	9.60925	0.369313	0.184657	−	0.982803i	$-0.440883\pi$
$677$	9.60925	0.369313	0.184657	+	0.982803i	$0.440883\pi$
$678$	0	0
$679$	−0.271456	−0.0104175
$680$	7.44123	0.285358
$681$	0	0
$682$	10.3302	0.395564
$683$	30.7126	1.17519	0.587593	−	0.809156i	$-0.300076\pi$
$683$	30.7126	1.17519	0.587593	+	0.809156i	$0.300076\pi$
$684$	0	0
$685$	−12.5494	−0.479488
$686$	−10.4440	−0.398755
$687$	0	0
$688$	2.79986	0.106744
$689$	−0.282401	−0.0107586
$690$	0	0
$691$	−13.4405	−0.511303	−0.255651	−	0.966769i	$-0.582290\pi$
$691$	−13.4405	−0.511303	−0.255651	+	0.966769i	$0.582290\pi$
$692$	−13.3499	−0.507489
$693$	0	0
$694$	−15.9545	−0.605625
$695$	3.66918	0.139180
$696$	0	0
$697$	10.2138	0.386875
$698$	−9.98835	−0.378064
$699$	0	0
$700$	2.59550	0.0981006
$701$	−18.8840	−0.713239	−0.356619	−	0.934250i	$-0.616071\pi$
$701$	−18.8840	−0.713239	−0.356619	+	0.934250i	$0.616071\pi$
$702$	0	0
$703$	−4.79294	−0.180769
$704$	2.38592	0.0899228
$705$	0	0
$706$	−8.12410	−0.305754
$707$	−1.66245	−0.0625227
$708$	0	0
$709$	12.3538	0.463956	0.231978	−	0.972721i	$-0.425480\pi$
$709$	12.3538	0.463956	0.231978	+	0.972721i	$0.425480\pi$
$710$	−0.965684	−0.0362415
$711$	0	0
$712$	−1.20491	−0.0451560
$713$	8.77059	0.328461
$714$	0	0
$715$	−1.83497	−0.0686239
$716$	−21.4101	−0.800133
$717$	0	0
$718$	−4.46644	−0.166686
$719$	−5.97953	−0.222999	−0.111499	−	0.993764i	$-0.535565\pi$
$719$	−5.97953	−0.222999	−0.111499	+	0.993764i	$0.535565\pi$
$720$	0	0
$721$	7.08515	0.263865
$722$	18.2419	0.678893
$723$	0	0
$724$	−22.9119	−0.851513
$725$	11.8608	0.440499
$726$	0	0
$727$	−5.25473	−0.194887	−0.0974436	−	0.995241i	$-0.531067\pi$
$727$	−5.25473	−0.194887	−0.0974436	+	0.995241i	$0.531067\pi$
$728$	0.463865	0.0171920
$729$	0	0
$730$	−12.5163	−0.463249
$731$	16.1128	0.595953
$732$	0	0
$733$	−5.34221	−0.197319	−0.0986596	−	0.995121i	$-0.531455\pi$
$733$	−5.34221	−0.197319	−0.0986596	+	0.995121i	$0.531455\pi$
$734$	10.0697	0.371679
$735$	0	0
$736$	2.02570	0.0746684
$737$	−24.6535	−0.908122
$738$	0	0
$739$	2.42226	0.0891043	0.0445522	−	0.999007i	$-0.485814\pi$
$739$	2.42226	0.0891043	0.0445522	+	0.999007i	$0.485814\pi$
$740$	7.11785	0.261657
$741$	0	0
$742$	−0.370284	−0.0135935
$743$	−24.4929	−0.898558	−0.449279	−	0.893391i	$-0.648319\pi$
$743$	−24.4929	−0.898558	−0.449279	+	0.893391i	$0.648319\pi$
$744$	0	0
$745$	1.29304	0.0473732
$746$	0.716141	0.0262198
$747$	0	0
$748$	13.7306	0.502040
$749$	−8.44223	−0.308472
$750$	0	0
$751$	−6.85738	−0.250229	−0.125115	−	0.992142i	$-0.539930\pi$
$751$	−6.85738	−0.250229	−0.125115	+	0.992142i	$0.539930\pi$
$752$	5.52119	0.201337
$753$	0	0
$754$	2.11975	0.0771969
$755$	−8.27965	−0.301327
$756$	0	0
$757$	−2.09210	−0.0760386	−0.0380193	−	0.999277i	$-0.512105\pi$
$757$	−2.09210	−0.0760386	−0.0380193	+	0.999277i	$0.512105\pi$
$758$	−14.8847	−0.540638
$759$	0	0
$760$	1.12584	0.0408384
$761$	−24.4188	−0.885181	−0.442591	−	0.896724i	$-0.645940\pi$
$761$	−24.4188	−0.885181	−0.442591	+	0.896724i	$0.645940\pi$
$762$	0	0
$763$	−14.9161	−0.539999
$764$	19.5028	0.705587
$765$	0	0
$766$	−12.9505	−0.467922
$767$	−2.32011	−0.0837743
$768$	0	0
$769$	−7.74055	−0.279131	−0.139566	−	0.990213i	$-0.544571\pi$
$769$	−7.74055	−0.279131	−0.139566	+	0.990213i	$0.544571\pi$
$770$	−2.40601	−0.0867065
$771$	0	0
$772$	−4.66227	−0.167799
$773$	1.54253	0.0554811	0.0277406	−	0.999615i	$-0.491169\pi$
$773$	1.54253	0.0554811	0.0277406	+	0.999615i	$0.491169\pi$
$774$	0	0
$775$	14.4093	0.517598
$776$	−0.348072	−0.0124951
$777$	0	0
$778$	31.1953	1.11840
$779$	1.54532	0.0553668
$780$	0	0
$781$	−1.78189	−0.0637609
$782$	11.6576	0.416875
$783$	0	0
$784$	−6.39178	−0.228278
$785$	−27.0039	−0.963810
$786$	0	0
$787$	−18.1132	−0.645666	−0.322833	−	0.946456i	$-0.604635\pi$
$787$	−18.1132	−0.645666	−0.322833	+	0.946456i	$0.604635\pi$
$788$	−15.9087	−0.566724
$789$	0	0
$790$	−5.23112	−0.186115
$791$	−8.99398	−0.319789
$792$	0	0
$793$	3.57599	0.126987
$794$	3.25142	0.115388
$795$	0	0
$796$	−11.8997	−0.421774
$797$	−44.8283	−1.58790	−0.793950	−	0.607983i	$-0.791979\pi$
$797$	−44.8283	−1.58790	−0.793950	+	0.607983i	$0.791979\pi$
$798$	0	0
$799$	31.7736	1.12407
$800$	3.32806	0.117665
$801$	0	0
$802$	23.9551	0.845883
$803$	−23.0951	−0.815010
$804$	0	0
$805$	−2.04276	−0.0719978
$806$	2.57522	0.0907083
$807$	0	0
$808$	−2.13166	−0.0749915
$809$	28.9393	1.01745	0.508726	−	0.860928i	$-0.330117\pi$
$809$	28.9393	1.01745	0.508726	+	0.860928i	$0.330117\pi$
$810$	0	0
$811$	1.65931	0.0582663	0.0291331	−	0.999576i	$-0.490725\pi$
$811$	1.65931	0.0582663	0.0291331	+	0.999576i	$0.490725\pi$
$812$	2.77942	0.0975385
$813$	0	0
$814$	13.1339	0.460343
$815$	22.2054	0.777823
$816$	0	0
$817$	2.43782	0.0852885
$818$	−1.59986	−0.0559380
$819$	0	0
$820$	−2.29490	−0.0801415
$821$	29.0022	1.01218	0.506092	−	0.862479i	$-0.331090\pi$
$821$	29.0022	1.01218	0.506092	+	0.862479i	$0.331090\pi$
$822$	0	0
$823$	−5.54558	−0.193307	−0.0966533	−	0.995318i	$-0.530814\pi$
$823$	−5.54558	−0.193307	−0.0966533	+	0.995318i	$0.530814\pi$
$824$	9.08487	0.316487
$825$	0	0
$826$	−3.04213	−0.105849
$827$	−47.5180	−1.65236	−0.826182	−	0.563404i	$-0.809492\pi$
$827$	−47.5180	−1.65236	−0.826182	+	0.563404i	$0.809492\pi$
$828$	0	0
$829$	45.2374	1.57116	0.785580	−	0.618761i	$-0.212365\pi$
$829$	45.2374	1.57116	0.785580	+	0.618761i	$0.212365\pi$
$830$	−3.06844	−0.106507
$831$	0	0
$832$	0.594787	0.0206205
$833$	−36.7837	−1.27448
$834$	0	0
$835$	6.37770	0.220709
$836$	2.07740	0.0718484
$837$	0	0
$838$	34.0470	1.17613
$839$	−26.7379	−0.923095	−0.461547	−	0.887116i	$-0.652706\pi$
$839$	−26.7379	−0.923095	−0.461547	+	0.887116i	$0.652706\pi$
$840$	0	0
$841$	−16.2987	−0.562025
$842$	−6.51403	−0.224488
$843$	0	0
$844$	−27.8174	−0.957515
$845$	16.3520	0.562527
$846$	0	0
$847$	4.13915	0.142223
$848$	−0.474793	−0.0163045
$849$	0	0
$850$	19.1524	0.656923
$851$	11.1510	0.382251
$852$	0	0
$853$	10.5864	0.362473	0.181236	−	0.983440i	$-0.441990\pi$
$853$	10.5864	0.362473	0.181236	+	0.983440i	$0.441990\pi$
$854$	4.68883	0.160449
$855$	0	0
$856$	−10.8250	−0.369990
$857$	22.6214	0.772732	0.386366	−	0.922346i	$-0.373730\pi$
$857$	22.6214	0.772732	0.386366	+	0.922346i	$0.373730\pi$
$858$	0	0
$859$	−7.10914	−0.242561	−0.121280	−	0.992618i	$-0.538700\pi$
$859$	−7.10914	−0.242561	−0.121280	+	0.992618i	$0.538700\pi$
$860$	−3.62033	−0.123452
$861$	0	0
$862$	−21.3497	−0.727174
$863$	9.65027	0.328499	0.164249	−	0.986419i	$-0.447480\pi$
$863$	9.65027	0.328499	0.164249	+	0.986419i	$0.447480\pi$
$864$	0	0
$865$	17.2620	0.586924
$866$	4.14179	0.140744
$867$	0	0
$868$	3.37663	0.114610
$869$	−9.65250	−0.327439
$870$	0	0
$871$	−6.14587	−0.208245
$872$	−19.1261	−0.647690
$873$	0	0
$874$	1.76376	0.0596602
$875$	−8.39817	−0.283910
$876$	0	0
$877$	5.21578	0.176125	0.0880623	−	0.996115i	$-0.471933\pi$
$877$	5.21578	0.176125	0.0880623	+	0.996115i	$0.471933\pi$
$878$	5.93107	0.200164
$879$	0	0
$880$	−3.08508	−0.103998
$881$	−31.0749	−1.04694	−0.523470	−	0.852044i	$-0.675363\pi$
$881$	−31.0749	−1.04694	−0.523470	+	0.852044i	$0.675363\pi$
$882$	0	0
$883$	−0.543730	−0.0182980	−0.00914898	−	0.999958i	$-0.502912\pi$
$883$	−0.543730	−0.0182980	−0.00914898	+	0.999958i	$0.502912\pi$
$884$	3.42291	0.115125
$885$	0	0
$886$	−14.5778	−0.489752
$887$	25.5773	0.858801	0.429401	−	0.903114i	$-0.358725\pi$
$887$	25.5773	0.858801	0.429401	+	0.903114i	$0.358725\pi$
$888$	0	0
$889$	−13.2741	−0.445201
$890$	1.55800	0.0522242
$891$	0	0
$892$	5.80380	0.194325
$893$	4.80725	0.160869
$894$	0	0
$895$	27.6840	0.925375
$896$	0.779884	0.0260541
$897$	0	0
$898$	4.78348	0.159627
$899$	15.4304	0.514632
$900$	0	0
$901$	−2.73236	−0.0910281
$902$	−4.23457	−0.140996
$903$	0	0
$904$	−11.5324	−0.383564
$905$	29.6259	0.984798
$906$	0	0
$907$	−42.5914	−1.41422	−0.707112	−	0.707101i	$-0.750003\pi$
$907$	−42.5914	−1.41422	−0.707112	+	0.707101i	$0.750003\pi$
$908$	−25.9424	−0.860927
$909$	0	0
$910$	−0.599795	−0.0198830
$911$	−35.7002	−1.18280	−0.591401	−	0.806378i	$-0.701425\pi$
$911$	−35.7002	−1.18280	−0.591401	+	0.806378i	$0.701425\pi$
$912$	0	0
$913$	−5.66191	−0.187382
$914$	−18.1331	−0.599790
$915$	0	0
$916$	−6.92543	−0.228823
$917$	−14.2443	−0.470388
$918$	0	0
$919$	−31.1592	−1.02785	−0.513924	−	0.857836i	$-0.671808\pi$
$919$	−31.1592	−1.02785	−0.513924	+	0.857836i	$0.671808\pi$
$920$	−2.61931	−0.0863561
$921$	0	0
$922$	−8.51787	−0.280521
$923$	−0.444207	−0.0146213
$924$	0	0
$925$	18.3201	0.602362
$926$	25.9271	0.852016
$927$	0	0
$928$	3.56389	0.116990
$929$	−14.7832	−0.485021	−0.242510	−	0.970149i	$-0.577971\pi$
$929$	−14.7832	−0.485021	−0.242510	+	0.970149i	$0.577971\pi$
$930$	0	0
$931$	−5.56527	−0.182394
$932$	−16.6879	−0.546631
$933$	0	0
$934$	−7.05773	−0.230936
$935$	−17.7542	−0.580623
$936$	0	0
$937$	−17.0908	−0.558333	−0.279167	−	0.960243i	$-0.590058\pi$
$937$	−17.0908	−0.558333	−0.279167	+	0.960243i	$0.590058\pi$
$938$	−8.05846	−0.263118
$939$	0	0
$940$	−7.13910	−0.232852
$941$	16.9606	0.552899	0.276449	−	0.961028i	$-0.410842\pi$
$941$	16.9606	0.552899	0.276449	+	0.961028i	$0.410842\pi$
$942$	0	0
$943$	−3.59525	−0.117078
$944$	−3.90074	−0.126958
$945$	0	0
$946$	−6.68025	−0.217194
$947$	2.67669	0.0869807	0.0434903	−	0.999054i	$-0.486152\pi$
$947$	2.67669	0.0869807	0.0434903	+	0.999054i	$0.486152\pi$
$948$	0	0
$949$	−5.75740	−0.186893
$950$	2.89771	0.0940141
$951$	0	0
$952$	4.48811	0.145461
$953$	−25.7225	−0.833233	−0.416616	−	0.909082i	$-0.636784\pi$
$953$	−25.7225	−0.833233	−0.416616	+	0.909082i	$0.636784\pi$
$954$	0	0
$955$	−25.2179	−0.816031
$956$	28.0190	0.906199
$957$	0	0
$958$	−34.9661	−1.12970
$959$	−7.56907	−0.244418
$960$	0	0
$961$	−12.2541	−0.395294
$962$	3.27415	0.105563
$963$	0	0
$964$	−14.3237	−0.461337
$965$	6.02848	0.194064
$966$	0	0
$967$	16.0735	0.516890	0.258445	−	0.966026i	$-0.416790\pi$
$967$	16.0735	0.516890	0.258445	+	0.966026i	$0.416790\pi$
$968$	5.30738	0.170586
$969$	0	0
$970$	0.450070	0.0144509
$971$	5.25984	0.168796	0.0843981	−	0.996432i	$-0.473103\pi$
$971$	5.25984	0.168796	0.0843981	+	0.996432i	$0.473103\pi$
$972$	0	0
$973$	2.21303	0.0709466
$974$	18.4898	0.592453
$975$	0	0
$976$	6.01222	0.192446
$977$	16.1658	0.517189	0.258594	−	0.965986i	$-0.416741\pi$
$977$	16.1658	0.517189	0.258594	+	0.965986i	$0.416741\pi$
$978$	0	0
$979$	2.87483	0.0918799
$980$	8.26481	0.264010
$981$	0	0
$982$	19.4078	0.619329
$983$	35.6753	1.13786	0.568932	−	0.822384i	$-0.307357\pi$
$983$	35.6753	1.13786	0.568932	+	0.822384i	$0.307357\pi$
$984$	0	0
$985$	20.5705	0.655432
$986$	20.5096	0.653159
$987$	0	0
$988$	0.517876	0.0164758
$989$	−5.67170	−0.180349
$990$	0	0
$991$	47.3848	1.50523	0.752614	−	0.658462i	$-0.228793\pi$
$991$	47.3848	1.50523	0.752614	+	0.658462i	$0.228793\pi$
$992$	4.32965	0.137467
$993$	0	0
$994$	−0.582444	−0.0184740
$995$	15.3868	0.487793
$996$	0	0
$997$	42.2764	1.33891	0.669454	−	0.742854i	$-0.266528\pi$
$997$	42.2764	1.33891	0.669454	+	0.742854i	$0.266528\pi$
$998$	30.5320	0.966473
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.e.1.3	✓	8
3.2	odd	2		8046.2.a.f.1.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.e.1.3	✓	8	1.1	even	1	trivial
8046.2.a.f.1.6	yes	8	3.2	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.29304 q^{5} -0.779884 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.29304 q^{5} -0.779884 q^{7} -1.00000 q^{8} +1.29304 q^{10} +2.38592 q^{11} +0.594787 q^{13} +0.779884 q^{14} +1.00000 q^{16} +5.75484 q^{17} +0.870692 q^{19} -1.29304 q^{20} -2.38592 q^{22} -2.02570 q^{23} -3.32806 q^{25} -0.594787 q^{26} -0.779884 q^{28} -3.56389 q^{29} -4.32965 q^{31} -1.00000 q^{32} -5.75484 q^{34} +1.00842 q^{35} -5.50475 q^{37} -0.870692 q^{38} +1.29304 q^{40} +1.77482 q^{41} +2.79986 q^{43} +2.38592 q^{44} +2.02570 q^{46} +5.52119 q^{47} -6.39178 q^{49} +3.32806 q^{50} +0.594787 q^{52} -0.474793 q^{53} -3.08508 q^{55} +0.779884 q^{56} +3.56389 q^{58} -3.90074 q^{59} +6.01222 q^{61} +4.32965 q^{62} +1.00000 q^{64} -0.769082 q^{65} -10.3329 q^{67} +5.75484 q^{68} -1.00842 q^{70} -0.746834 q^{71} -9.67976 q^{73} +5.50475 q^{74} +0.870692 q^{76} -1.86074 q^{77} -4.04561 q^{79} -1.29304 q^{80} -1.77482 q^{82} -2.37305 q^{83} -7.44123 q^{85} -2.79986 q^{86} -2.38592 q^{88} +1.20491 q^{89} -0.463865 q^{91} -2.02570 q^{92} -5.52119 q^{94} -1.12584 q^{95} +0.348072 q^{97} +6.39178 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 5 q^{7} - 8 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^4 - 5 * q^7 - 8 * q^8	\( 8 q - 8 q^{2} + 8 q^{4} - 5 q^{7} - 8 q^{8} + 6 q^{11} - 8 q^{13} + 5 q^{14} + 8 q^{16} - 5 q^{17} - 14 q^{19} - 6 q^{22} + 21 q^{23} - 6 q^{25} + 8 q^{26} - 5 q^{28} + 3 q^{29} - 4 q^{31} - 8 q^{32} + 5 q^{34} - 2 q^{35} - 3 q^{37} + 14 q^{38} + 7 q^{41} - 12 q^{43} + 6 q^{44} - 21 q^{46} + 25 q^{47} - 7 q^{49} + 6 q^{50} - 8 q^{52} + 3 q^{53} - 9 q^{55} + 5 q^{56} - 3 q^{58} + 2 q^{59} - 17 q^{61} + 4 q^{62} + 8 q^{64} + 32 q^{65} - 14 q^{67} - 5 q^{68} + 2 q^{70} + 7 q^{71} - 10 q^{73} + 3 q^{74} - 14 q^{76} + 12 q^{77} - 33 q^{79} - 7 q^{82} + 13 q^{83} - 33 q^{85} + 12 q^{86} - 6 q^{88} - 22 q^{89} - 22 q^{91} + 21 q^{92} - 25 q^{94} - 14 q^{95} - 11 q^{97} + 7 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 - 5 * q^7 - 8 * q^8 + 6 * q^11 - 8 * q^13 + 5 * q^14 + 8 * q^16 - 5 * q^17 - 14 * q^19 - 6 * q^22 + 21 * q^23 - 6 * q^25 + 8 * q^26 - 5 * q^28 + 3 * q^29 - 4 * q^31 - 8 * q^32 + 5 * q^34 - 2 * q^35 - 3 * q^37 + 14 * q^38 + 7 * q^41 - 12 * q^43 + 6 * q^44 - 21 * q^46 + 25 * q^47 - 7 * q^49 + 6 * q^50 - 8 * q^52 + 3 * q^53 - 9 * q^55 + 5 * q^56 - 3 * q^58 + 2 * q^59 - 17 * q^61 + 4 * q^62 + 8 * q^64 + 32 * q^65 - 14 * q^67 - 5 * q^68 + 2 * q^70 + 7 * q^71 - 10 * q^73 + 3 * q^74 - 14 * q^76 + 12 * q^77 - 33 * q^79 - 7 * q^82 + 13 * q^83 - 33 * q^85 + 12 * q^86 - 6 * q^88 - 22 * q^89 - 22 * q^91 + 21 * q^92 - 25 * q^94 - 14 * q^95 - 11 * q^97 + 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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