LMFDB - Embedded newform 8046.2.a.t.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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 46 x^{14} + 192 x^{13} + 752 x^{12} - 3378 x^{11} - 5277 x^{10} + 27132 x^{9} + \cdots - 4260 $ x^16 - 4x^15 - 46x^14 + 192x^13 + 752x^12 - 3378x^11 - 5277x^10 + 27132x^9 + 15173x^8 - 102466x^7 - 16612x^6 + 186120x^5 + 117x^4 - 156722x^3 + 10545x^2 + 48580*x - 4260
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.23178$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.23178 q^{5} +3.10807 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -3.23178 q^{5} +3.10807 q^{7} +1.00000 q^{8} -3.23178 q^{10} -0.565542 q^{11} -0.755940 q^{13} +3.10807 q^{14} +1.00000 q^{16} -0.292670 q^{17} +3.68097 q^{19} -3.23178 q^{20} -0.565542 q^{22} -5.90880 q^{23} +5.44437 q^{25} -0.755940 q^{26} +3.10807 q^{28} -4.02755 q^{29} +2.64010 q^{31} +1.00000 q^{32} -0.292670 q^{34} -10.0446 q^{35} +4.18154 q^{37} +3.68097 q^{38} -3.23178 q^{40} +6.81984 q^{41} +7.25335 q^{43} -0.565542 q^{44} -5.90880 q^{46} -4.40956 q^{47} +2.66008 q^{49} +5.44437 q^{50} -0.755940 q^{52} +7.88563 q^{53} +1.82770 q^{55} +3.10807 q^{56} -4.02755 q^{58} -1.95898 q^{59} -11.6458 q^{61} +2.64010 q^{62} +1.00000 q^{64} +2.44303 q^{65} +0.800947 q^{67} -0.292670 q^{68} -10.0446 q^{70} +14.4325 q^{71} -2.54480 q^{73} +4.18154 q^{74} +3.68097 q^{76} -1.75774 q^{77} +12.4796 q^{79} -3.23178 q^{80} +6.81984 q^{82} +17.0090 q^{83} +0.945843 q^{85} +7.25335 q^{86} -0.565542 q^{88} -16.9302 q^{89} -2.34951 q^{91} -5.90880 q^{92} -4.40956 q^{94} -11.8961 q^{95} +13.8685 q^{97} +2.66008 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) $ 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8	$ 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8} + 4 q^{10} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 16 q^{16} + q^{17} + 10 q^{19} + 4 q^{20} + 6 q^{22} + 10 q^{23} + 28 q^{25} + 6 q^{26} + 6 q^{28} + 6 q^{29} + 21 q^{31} + 16 q^{32} + q^{34} + 16 q^{35} + 17 q^{37} + 10 q^{38} + 4 q^{40} - 4 q^{41} + 16 q^{43} + 6 q^{44} + 10 q^{46} + 25 q^{47} + 36 q^{49} + 28 q^{50} + 6 q^{52} + 14 q^{53} + 19 q^{55} + 6 q^{56} + 6 q^{58} + 6 q^{59} + 23 q^{61} + 21 q^{62} + 16 q^{64} + 20 q^{65} + 22 q^{67} + q^{68} + 16 q^{70} + 10 q^{71} + 16 q^{73} + 17 q^{74} + 10 q^{76} - 2 q^{77} + 37 q^{79} + 4 q^{80} - 4 q^{82} + 33 q^{83} + 43 q^{85} + 16 q^{86} + 6 q^{88} - 3 q^{89} + 28 q^{91} + 10 q^{92} + 25 q^{94} + 14 q^{95} - 3 q^{97} + 36 q^{98}+O(q^{100}) $ 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8 + 4 * q^10 + 6 * q^11 + 6 * q^13 + 6 * q^14 + 16 * q^16 + q^17 + 10 * q^19 + 4 * q^20 + 6 * q^22 + 10 * q^23 + 28 * q^25 + 6 * q^26 + 6 * q^28 + 6 * q^29 + 21 * q^31 + 16 * q^32 + q^34 + 16 * q^35 + 17 * q^37 + 10 * q^38 + 4 * q^40 - 4 * q^41 + 16 * q^43 + 6 * q^44 + 10 * q^46 + 25 * q^47 + 36 * q^49 + 28 * q^50 + 6 * q^52 + 14 * q^53 + 19 * q^55 + 6 * q^56 + 6 * q^58 + 6 * q^59 + 23 * q^61 + 21 * q^62 + 16 * q^64 + 20 * q^65 + 22 * q^67 + q^68 + 16 * q^70 + 10 * q^71 + 16 * q^73 + 17 * q^74 + 10 * q^76 - 2 * q^77 + 37 * q^79 + 4 * q^80 - 4 * q^82 + 33 * q^83 + 43 * q^85 + 16 * q^86 + 6 * q^88 - 3 * q^89 + 28 * q^91 + 10 * q^92 + 25 * q^94 + 14 * q^95 - 3 * q^97 + 36 * 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
$5$	−3.23178	−1.44529	−0.722647	−	0.691217i	$-0.757075\pi$
$5$	−3.23178	−1.44529	−0.722647	+	0.691217i	$0.757075\pi$
$6$	0	0
$7$	3.10807	1.17474	0.587369	−	0.809319i	$-0.300164\pi$
$7$	3.10807	1.17474	0.587369	+	0.809319i	$0.300164\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.23178	−1.02198
$11$	−0.565542	−0.170517	−0.0852586	−	0.996359i	$-0.527172\pi$
$11$	−0.565542	−0.170517	−0.0852586	+	0.996359i	$0.527172\pi$
$12$	0	0
$13$	−0.755940	−0.209660	−0.104830	−	0.994490i	$-0.533430\pi$
$13$	−0.755940	−0.209660	−0.104830	+	0.994490i	$0.533430\pi$
$14$	3.10807	0.830666
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.292670	−0.0709828	−0.0354914	−	0.999370i	$-0.511300\pi$
$17$	−0.292670	−0.0709828	−0.0354914	+	0.999370i	$0.511300\pi$
$18$	0	0
$19$	3.68097	0.844472	0.422236	−	0.906486i	$-0.361245\pi$
$19$	3.68097	0.844472	0.422236	+	0.906486i	$0.361245\pi$
$20$	−3.23178	−0.722647
$21$	0	0
$22$	−0.565542	−0.120574
$23$	−5.90880	−1.23207	−0.616035	−	0.787719i	$-0.711262\pi$
$23$	−5.90880	−1.23207	−0.616035	+	0.787719i	$0.711262\pi$
$24$	0	0
$25$	5.44437	1.08887
$26$	−0.755940	−0.148252
$27$	0	0
$28$	3.10807	0.587369
$29$	−4.02755	−0.747897	−0.373949	−	0.927449i	$-0.621996\pi$
$29$	−4.02755	−0.747897	−0.373949	+	0.927449i	$0.621996\pi$
$30$	0	0
$31$	2.64010	0.474177	0.237088	−	0.971488i	$-0.423807\pi$
$31$	2.64010	0.474177	0.237088	+	0.971488i	$0.423807\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.292670	−0.0501924
$35$	−10.0446	−1.69784
$36$	0	0
$37$	4.18154	0.687440	0.343720	−	0.939072i	$-0.388313\pi$
$37$	4.18154	0.687440	0.343720	+	0.939072i	$0.388313\pi$
$38$	3.68097	0.597132
$39$	0	0
$40$	−3.23178	−0.510989
$41$	6.81984	1.06508	0.532540	−	0.846405i	$-0.321238\pi$
$41$	6.81984	1.06508	0.532540	+	0.846405i	$0.321238\pi$
$42$	0	0
$43$	7.25335	1.10613	0.553063	−	0.833139i	$-0.313459\pi$
$43$	7.25335	1.10613	0.553063	+	0.833139i	$0.313459\pi$
$44$	−0.565542	−0.0852586
$45$	0	0
$46$	−5.90880	−0.871206
$47$	−4.40956	−0.643201	−0.321600	−	0.946875i	$-0.604221\pi$
$47$	−4.40956	−0.643201	−0.321600	+	0.946875i	$0.604221\pi$
$48$	0	0
$49$	2.66008	0.380011
$50$	5.44437	0.769950
$51$	0	0
$52$	−0.755940	−0.104830
$53$	7.88563	1.08317	0.541587	−	0.840645i	$-0.317824\pi$
$53$	7.88563	1.08317	0.541587	+	0.840645i	$0.317824\pi$
$54$	0	0
$55$	1.82770	0.246447
$56$	3.10807	0.415333
$57$	0	0
$58$	−4.02755	−0.528843
$59$	−1.95898	−0.255038	−0.127519	−	0.991836i	$-0.540701\pi$
$59$	−1.95898	−0.255038	−0.127519	+	0.991836i	$0.540701\pi$
$60$	0	0
$61$	−11.6458	−1.49109	−0.745545	−	0.666455i	$-0.767811\pi$
$61$	−11.6458	−1.49109	−0.745545	+	0.666455i	$0.767811\pi$
$62$	2.64010	0.335294
$63$	0	0
$64$	1.00000	0.125000
$65$	2.44303	0.303020
$66$	0	0
$67$	0.800947	0.0978513	0.0489256	−	0.998802i	$-0.484420\pi$
$67$	0.800947	0.0978513	0.0489256	+	0.998802i	$0.484420\pi$
$68$	−0.292670	−0.0354914
$69$	0	0
$70$	−10.0446	−1.20056
$71$	14.4325	1.71282	0.856409	−	0.516298i	$-0.172690\pi$
$71$	14.4325	1.71282	0.856409	+	0.516298i	$0.172690\pi$
$72$	0	0
$73$	−2.54480	−0.297846	−0.148923	−	0.988849i	$-0.547581\pi$
$73$	−2.54480	−0.297846	−0.148923	+	0.988849i	$0.547581\pi$
$74$	4.18154	0.486094
$75$	0	0
$76$	3.68097	0.422236
$77$	−1.75774	−0.200313
$78$	0	0
$79$	12.4796	1.40407	0.702034	−	0.712144i	$-0.252276\pi$
$79$	12.4796	1.40407	0.702034	+	0.712144i	$0.252276\pi$
$80$	−3.23178	−0.361323
$81$	0	0
$82$	6.81984	0.753125
$83$	17.0090	1.86698	0.933489	−	0.358606i	$-0.116748\pi$
$83$	17.0090	1.86698	0.933489	+	0.358606i	$0.116748\pi$
$84$	0	0
$85$	0.945843	0.102591
$86$	7.25335	0.782149
$87$	0	0
$88$	−0.565542	−0.0602869
$89$	−16.9302	−1.79460	−0.897299	−	0.441424i	$-0.854473\pi$
$89$	−16.9302	−1.79460	−0.897299	+	0.441424i	$0.854473\pi$
$90$	0	0
$91$	−2.34951	−0.246296
$92$	−5.90880	−0.616035
$93$	0	0
$94$	−4.40956	−0.454812
$95$	−11.8961	−1.22051
$96$	0	0
$97$	13.8685	1.40813	0.704065	−	0.710135i	$-0.251366\pi$
$97$	13.8685	1.40813	0.704065	+	0.710135i	$0.251366\pi$
$98$	2.66008	0.268709
$99$	0	0
$100$	5.44437	0.544437
$101$	0.569545	0.0566718	0.0283359	−	0.999598i	$-0.490979\pi$
$101$	0.569545	0.0566718	0.0283359	+	0.999598i	$0.490979\pi$
$102$	0	0
$103$	−7.92510	−0.780883	−0.390442	−	0.920628i	$-0.627678\pi$
$103$	−7.92510	−0.780883	−0.390442	+	0.920628i	$0.627678\pi$
$104$	−0.755940	−0.0741261
$105$	0	0
$106$	7.88563	0.765920
$107$	−17.1379	−1.65678	−0.828392	−	0.560149i	$-0.810744\pi$
$107$	−17.1379	−1.65678	−0.828392	+	0.560149i	$0.810744\pi$
$108$	0	0
$109$	7.78332	0.745507	0.372754	−	0.927930i	$-0.378414\pi$
$109$	7.78332	0.745507	0.372754	+	0.927930i	$0.378414\pi$
$110$	1.82770	0.174265
$111$	0	0
$112$	3.10807	0.293685
$113$	9.42136	0.886287	0.443144	−	0.896451i	$-0.353863\pi$
$113$	9.42136	0.886287	0.443144	+	0.896451i	$0.353863\pi$
$114$	0	0
$115$	19.0959	1.78070
$116$	−4.02755	−0.373949
$117$	0	0
$118$	−1.95898	−0.180339
$119$	−0.909637	−0.0833863
$120$	0	0
$121$	−10.6802	−0.970924
$122$	−11.6458	−1.05436
$123$	0	0
$124$	2.64010	0.237088
$125$	−1.43611	−0.128449
$126$	0	0
$127$	6.06582	0.538255	0.269127	−	0.963105i	$-0.413265\pi$
$127$	6.06582	0.538255	0.269127	+	0.963105i	$0.413265\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	2.44303	0.214268
$131$	−3.03159	−0.264872	−0.132436	−	0.991192i	$-0.542280\pi$
$131$	−3.03159	−0.264872	−0.132436	+	0.991192i	$0.542280\pi$
$132$	0	0
$133$	11.4407	0.992035
$134$	0.800947	0.0691913
$135$	0	0
$136$	−0.292670	−0.0250962
$137$	6.42223	0.548688	0.274344	−	0.961632i	$-0.411539\pi$
$137$	6.42223	0.548688	0.274344	+	0.961632i	$0.411539\pi$
$138$	0	0
$139$	−4.35852	−0.369685	−0.184842	−	0.982768i	$-0.559177\pi$
$139$	−4.35852	−0.369685	−0.184842	+	0.982768i	$0.559177\pi$
$140$	−10.0446	−0.848921
$141$	0	0
$142$	14.4325	1.21115
$143$	0.427516	0.0357507
$144$	0	0
$145$	13.0161	1.08093
$146$	−2.54480	−0.210609
$147$	0	0
$148$	4.18154	0.343720
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−5.99666	−0.488001	−0.244001	−	0.969775i	$-0.578460\pi$
$151$	−5.99666	−0.488001	−0.244001	+	0.969775i	$0.578460\pi$
$152$	3.68097	0.298566
$153$	0	0
$154$	−1.75774	−0.141643
$155$	−8.53222	−0.685325
$156$	0	0
$157$	12.5575	1.00220	0.501098	−	0.865390i	$-0.332929\pi$
$157$	12.5575	1.00220	0.501098	+	0.865390i	$0.332929\pi$
$158$	12.4796	0.992825
$159$	0	0
$160$	−3.23178	−0.255494
$161$	−18.3650	−1.44736
$162$	0	0
$163$	6.12608	0.479832	0.239916	−	0.970794i	$-0.422880\pi$
$163$	6.12608	0.479832	0.239916	+	0.970794i	$0.422880\pi$
$164$	6.81984	0.532540
$165$	0	0
$166$	17.0090	1.32015
$167$	−11.9476	−0.924532	−0.462266	−	0.886741i	$-0.652963\pi$
$167$	−11.9476	−0.924532	−0.462266	+	0.886741i	$0.652963\pi$
$168$	0	0
$169$	−12.4286	−0.956043
$170$	0.945843	0.0725428
$171$	0	0
$172$	7.25335	0.553063
$173$	22.5164	1.71189	0.855944	−	0.517068i	$-0.172977\pi$
$173$	22.5164	1.71189	0.855944	+	0.517068i	$0.172977\pi$
$174$	0	0
$175$	16.9215	1.27914
$176$	−0.565542	−0.0426293
$177$	0	0
$178$	−16.9302	−1.26897
$179$	3.84611	0.287471	0.143736	−	0.989616i	$-0.454088\pi$
$179$	3.84611	0.287471	0.143736	+	0.989616i	$0.454088\pi$
$180$	0	0
$181$	6.25330	0.464804	0.232402	−	0.972620i	$-0.425341\pi$
$181$	6.25330	0.464804	0.232402	+	0.972620i	$0.425341\pi$
$182$	−2.34951	−0.174158
$183$	0	0
$184$	−5.90880	−0.435603
$185$	−13.5138	−0.993553
$186$	0	0
$187$	0.165517	0.0121038
$188$	−4.40956	−0.321600
$189$	0	0
$190$	−11.8961	−0.863031
$191$	17.5113	1.26707	0.633537	−	0.773713i	$-0.281603\pi$
$191$	17.5113	1.26707	0.633537	+	0.773713i	$0.281603\pi$
$192$	0	0
$193$	7.71204	0.555125	0.277562	−	0.960708i	$-0.410474\pi$
$193$	7.71204	0.555125	0.277562	+	0.960708i	$0.410474\pi$
$194$	13.8685	0.995699
$195$	0	0
$196$	2.66008	0.190006
$197$	13.9945	0.997067	0.498534	−	0.866870i	$-0.333872\pi$
$197$	13.9945	0.997067	0.498534	+	0.866870i	$0.333872\pi$
$198$	0	0
$199$	3.75774	0.266379	0.133190	−	0.991091i	$-0.457478\pi$
$199$	3.75774	0.266379	0.133190	+	0.991091i	$0.457478\pi$
$200$	5.44437	0.384975
$201$	0	0
$202$	0.569545	0.0400730
$203$	−12.5179	−0.878584
$204$	0	0
$205$	−22.0402	−1.53935
$206$	−7.92510	−0.552168
$207$	0	0
$208$	−0.755940	−0.0524150
$209$	−2.08174	−0.143997
$210$	0	0
$211$	0.452579	0.0311568	0.0155784	−	0.999879i	$-0.495041\pi$
$211$	0.452579	0.0311568	0.0155784	+	0.999879i	$0.495041\pi$
$212$	7.88563	0.541587
$213$	0	0
$214$	−17.1379	−1.17152
$215$	−23.4412	−1.59868
$216$	0	0
$217$	8.20562	0.557034
$218$	7.78332	0.527153
$219$	0	0
$220$	1.82770	0.123224
$221$	0.221241	0.0148823
$222$	0	0
$223$	−5.86934	−0.393040	−0.196520	−	0.980500i	$-0.562964\pi$
$223$	−5.86934	−0.393040	−0.196520	+	0.980500i	$0.562964\pi$
$224$	3.10807	0.207666
$225$	0	0
$226$	9.42136	0.626700
$227$	25.1098	1.66660	0.833299	−	0.552822i	$-0.186449\pi$
$227$	25.1098	1.66660	0.833299	+	0.552822i	$0.186449\pi$
$228$	0	0
$229$	15.6227	1.03238	0.516189	−	0.856475i	$-0.327350\pi$
$229$	15.6227	1.03238	0.516189	+	0.856475i	$0.327350\pi$
$230$	19.0959	1.25915
$231$	0	0
$232$	−4.02755	−0.264422
$233$	−14.5716	−0.954615	−0.477308	−	0.878736i	$-0.658387\pi$
$233$	−14.5716	−0.954615	−0.477308	+	0.878736i	$0.658387\pi$
$234$	0	0
$235$	14.2507	0.929614
$236$	−1.95898	−0.127519
$237$	0	0
$238$	−0.909637	−0.0589630
$239$	13.8869	0.898268	0.449134	−	0.893464i	$-0.351733\pi$
$239$	13.8869	0.898268	0.449134	+	0.893464i	$0.351733\pi$
$240$	0	0
$241$	21.9596	1.41454	0.707272	−	0.706941i	$-0.249925\pi$
$241$	21.9596	1.41454	0.707272	+	0.706941i	$0.249925\pi$
$242$	−10.6802	−0.686547
$243$	0	0
$244$	−11.6458	−0.745545
$245$	−8.59678	−0.549228
$246$	0	0
$247$	−2.78259	−0.177052
$248$	2.64010	0.167647
$249$	0	0
$250$	−1.43611	−0.0908273
$251$	17.5507	1.10779	0.553896	−	0.832586i	$-0.313141\pi$
$251$	17.5507	1.10779	0.553896	+	0.832586i	$0.313141\pi$
$252$	0	0
$253$	3.34167	0.210089
$254$	6.06582	0.380603
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.3184	−1.14267	−0.571336	−	0.820716i	$-0.693575\pi$
$257$	−18.3184	−1.14267	−0.571336	+	0.820716i	$0.693575\pi$
$258$	0	0
$259$	12.9965	0.807563
$260$	2.44303	0.151510
$261$	0	0
$262$	−3.03159	−0.187293
$263$	−1.06536	−0.0656932	−0.0328466	−	0.999460i	$-0.510457\pi$
$263$	−1.06536	−0.0656932	−0.0328466	+	0.999460i	$0.510457\pi$
$264$	0	0
$265$	−25.4846	−1.56550
$266$	11.4407	0.701474
$267$	0	0
$268$	0.800947	0.0489256
$269$	5.96084	0.363439	0.181719	−	0.983350i	$-0.441834\pi$
$269$	5.96084	0.363439	0.181719	+	0.983350i	$0.441834\pi$
$270$	0	0
$271$	5.48358	0.333104	0.166552	−	0.986033i	$-0.446737\pi$
$271$	5.48358	0.333104	0.166552	+	0.986033i	$0.446737\pi$
$272$	−0.292670	−0.0177457
$273$	0	0
$274$	6.42223	0.387981
$275$	−3.07902	−0.185672
$276$	0	0
$277$	32.4422	1.94926	0.974632	−	0.223813i	$-0.0718506\pi$
$277$	32.4422	1.94926	0.974632	+	0.223813i	$0.0718506\pi$
$278$	−4.35852	−0.261407
$279$	0	0
$280$	−10.0446	−0.600278
$281$	−0.115072	−0.00686464	−0.00343232	−	0.999994i	$-0.501093\pi$
$281$	−0.115072	−0.00686464	−0.00343232	+	0.999994i	$0.501093\pi$
$282$	0	0
$283$	−22.4356	−1.33366	−0.666828	−	0.745212i	$-0.732348\pi$
$283$	−22.4356	−1.33366	−0.666828	+	0.745212i	$0.732348\pi$
$284$	14.4325	0.856409
$285$	0	0
$286$	0.427516	0.0252795
$287$	21.1965	1.25119
$288$	0	0
$289$	−16.9143	−0.994961
$290$	13.0161	0.764334
$291$	0	0
$292$	−2.54480	−0.148923
$293$	13.2695	0.775215	0.387607	−	0.921825i	$-0.373302\pi$
$293$	13.2695	0.775215	0.387607	+	0.921825i	$0.373302\pi$
$294$	0	0
$295$	6.33099	0.368604
$296$	4.18154	0.243047
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	4.46670	0.258316
$300$	0	0
$301$	22.5439	1.29941
$302$	−5.99666	−0.345069
$303$	0	0
$304$	3.68097	0.211118
$305$	37.6366	2.15506
$306$	0	0
$307$	−4.66069	−0.266000	−0.133000	−	0.991116i	$-0.542461\pi$
$307$	−4.66069	−0.266000	−0.133000	+	0.991116i	$0.542461\pi$
$308$	−1.75774	−0.100157
$309$	0	0
$310$	−8.53222	−0.484598
$311$	−4.90285	−0.278015	−0.139007	−	0.990291i	$-0.544391\pi$
$311$	−4.90285	−0.278015	−0.139007	+	0.990291i	$0.544391\pi$
$312$	0	0
$313$	−4.14187	−0.234113	−0.117056	−	0.993125i	$-0.537346\pi$
$313$	−4.14187	−0.234113	−0.117056	+	0.993125i	$0.537346\pi$
$314$	12.5575	0.708660
$315$	0	0
$316$	12.4796	0.702034
$317$	12.1108	0.680208	0.340104	−	0.940388i	$-0.389538\pi$
$317$	12.1108	0.680208	0.340104	+	0.940388i	$0.389538\pi$
$318$	0	0
$319$	2.27775	0.127529
$320$	−3.23178	−0.180662
$321$	0	0
$322$	−18.3650	−1.02344
$323$	−1.07731	−0.0599430
$324$	0	0
$325$	−4.11562	−0.228293
$326$	6.12608	0.339292
$327$	0	0
$328$	6.81984	0.376563
$329$	−13.7052	−0.755593
$330$	0	0
$331$	−2.54691	−0.139991	−0.0699955	−	0.997547i	$-0.522299\pi$
$331$	−2.54691	−0.139991	−0.0699955	+	0.997547i	$0.522299\pi$
$332$	17.0090	0.933489
$333$	0	0
$334$	−11.9476	−0.653743
$335$	−2.58848	−0.141424
$336$	0	0
$337$	−9.45663	−0.515136	−0.257568	−	0.966260i	$-0.582921\pi$
$337$	−9.45663	−0.515136	−0.257568	+	0.966260i	$0.582921\pi$
$338$	−12.4286	−0.676024
$339$	0	0
$340$	0.945843	0.0512955
$341$	−1.49309	−0.0808553
$342$	0	0
$343$	−13.4888	−0.728325
$344$	7.25335	0.391075
$345$	0	0
$346$	22.5164	1.21049
$347$	−11.3451	−0.609036	−0.304518	−	0.952507i	$-0.598495\pi$
$347$	−11.3451	−0.609036	−0.304518	+	0.952507i	$0.598495\pi$
$348$	0	0
$349$	34.1773	1.82947	0.914733	−	0.404058i	$-0.132401\pi$
$349$	34.1773	1.82947	0.914733	+	0.404058i	$0.132401\pi$
$350$	16.9215	0.904491
$351$	0	0
$352$	−0.565542	−0.0301435
$353$	16.0944	0.856618	0.428309	−	0.903632i	$-0.359109\pi$
$353$	16.0944	0.856618	0.428309	+	0.903632i	$0.359109\pi$
$354$	0	0
$355$	−46.6425	−2.47552
$356$	−16.9302	−0.897299
$357$	0	0
$358$	3.84611	0.203273
$359$	10.6921	0.564308	0.282154	−	0.959369i	$-0.408951\pi$
$359$	10.6921	0.564308	0.282154	+	0.959369i	$0.408951\pi$
$360$	0	0
$361$	−5.45046	−0.286866
$362$	6.25330	0.328666
$363$	0	0
$364$	−2.34951	−0.123148
$365$	8.22421	0.430475
$366$	0	0
$367$	−34.8461	−1.81895	−0.909476	−	0.415757i	$-0.863517\pi$
$367$	−34.8461	−1.81895	−0.909476	+	0.415757i	$0.863517\pi$
$368$	−5.90880	−0.308018
$369$	0	0
$370$	−13.5138	−0.702548
$371$	24.5091	1.27245
$372$	0	0
$373$	35.2009	1.82263	0.911316	−	0.411709i	$-0.135068\pi$
$373$	35.2009	1.82263	0.911316	+	0.411709i	$0.135068\pi$
$374$	0.165517	0.00855868
$375$	0	0
$376$	−4.40956	−0.227406
$377$	3.04459	0.156804
$378$	0	0
$379$	19.8333	1.01877	0.509384	−	0.860539i	$-0.329873\pi$
$379$	19.8333	1.01877	0.509384	+	0.860539i	$0.329873\pi$
$380$	−11.8961	−0.610255
$381$	0	0
$382$	17.5113	0.895956
$383$	−0.838998	−0.0428708	−0.0214354	−	0.999770i	$-0.506824\pi$
$383$	−0.838998	−0.0428708	−0.0214354	+	0.999770i	$0.506824\pi$
$384$	0	0
$385$	5.68062	0.289511
$386$	7.71204	0.392532
$387$	0	0
$388$	13.8685	0.704065
$389$	18.1597	0.920735	0.460367	−	0.887728i	$-0.347718\pi$
$389$	18.1597	0.920735	0.460367	+	0.887728i	$0.347718\pi$
$390$	0	0
$391$	1.72933	0.0874559
$392$	2.66008	0.134354
$393$	0	0
$394$	13.9945	0.705033
$395$	−40.3313	−2.02929
$396$	0	0
$397$	−3.38229	−0.169752	−0.0848760	−	0.996392i	$-0.527049\pi$
$397$	−3.38229	−0.169752	−0.0848760	+	0.996392i	$0.527049\pi$
$398$	3.75774	0.188359
$399$	0	0
$400$	5.44437	0.272219
$401$	−3.08431	−0.154023	−0.0770115	−	0.997030i	$-0.524538\pi$
$401$	−3.08431	−0.154023	−0.0770115	+	0.997030i	$0.524538\pi$
$402$	0	0
$403$	−1.99576	−0.0994159
$404$	0.569545	0.0283359
$405$	0	0
$406$	−12.5179	−0.621253
$407$	−2.36483	−0.117220
$408$	0	0
$409$	0.462230	0.0228558	0.0114279	−	0.999935i	$-0.496362\pi$
$409$	0.462230	0.0228558	0.0114279	+	0.999935i	$0.496362\pi$
$410$	−22.0402	−1.08849
$411$	0	0
$412$	−7.92510	−0.390442
$413$	−6.08864	−0.299603
$414$	0	0
$415$	−54.9692	−2.69833
$416$	−0.755940	−0.0370630
$417$	0	0
$418$	−2.08174	−0.101821
$419$	20.5962	1.00619	0.503094	−	0.864231i	$-0.332195\pi$
$419$	20.5962	1.00619	0.503094	+	0.864231i	$0.332195\pi$
$420$	0	0
$421$	−10.0402	−0.489331	−0.244666	−	0.969608i	$-0.578678\pi$
$421$	−10.0402	−0.489331	−0.244666	+	0.969608i	$0.578678\pi$
$422$	0.452579	0.0220312
$423$	0	0
$424$	7.88563	0.382960
$425$	−1.59340	−0.0772914
$426$	0	0
$427$	−36.1959	−1.75164
$428$	−17.1379	−0.828392
$429$	0	0
$430$	−23.4412	−1.13044
$431$	−17.5848	−0.847031	−0.423515	−	0.905889i	$-0.639204\pi$
$431$	−17.5848	−0.847031	−0.423515	+	0.905889i	$0.639204\pi$
$432$	0	0
$433$	5.75874	0.276747	0.138374	−	0.990380i	$-0.455813\pi$
$433$	5.75874	0.276747	0.138374	+	0.990380i	$0.455813\pi$
$434$	8.20562	0.393882
$435$	0	0
$436$	7.78332	0.372754
$437$	−21.7501	−1.04045
$438$	0	0
$439$	−12.8706	−0.614282	−0.307141	−	0.951664i	$-0.599372\pi$
$439$	−12.8706	−0.614282	−0.307141	+	0.951664i	$0.599372\pi$
$440$	1.82770	0.0871323
$441$	0	0
$442$	0.221241	0.0105234
$443$	−22.4068	−1.06458	−0.532289	−	0.846563i	$-0.678668\pi$
$443$	−22.4068	−1.06458	−0.532289	+	0.846563i	$0.678668\pi$
$444$	0	0
$445$	54.7146	2.59372
$446$	−5.86934	−0.277921
$447$	0	0
$448$	3.10807	0.146842
$449$	30.8706	1.45687	0.728437	−	0.685113i	$-0.240247\pi$
$449$	30.8706	1.45687	0.728437	+	0.685113i	$0.240247\pi$
$450$	0	0
$451$	−3.85690	−0.181614
$452$	9.42136	0.443144
$453$	0	0
$454$	25.1098	1.17846
$455$	7.59310	0.355970
$456$	0	0
$457$	36.6093	1.71251	0.856255	−	0.516554i	$-0.172785\pi$
$457$	36.6093	1.71251	0.856255	+	0.516554i	$0.172785\pi$
$458$	15.6227	0.730001
$459$	0	0
$460$	19.0959	0.890352
$461$	7.02235	0.327063	0.163532	−	0.986538i	$-0.447711\pi$
$461$	7.02235	0.327063	0.163532	+	0.986538i	$0.447711\pi$
$462$	0	0
$463$	20.2488	0.941041	0.470521	−	0.882389i	$-0.344066\pi$
$463$	20.2488	0.941041	0.470521	+	0.882389i	$0.344066\pi$
$464$	−4.02755	−0.186974
$465$	0	0
$466$	−14.5716	−0.675015
$467$	15.1426	0.700714	0.350357	−	0.936616i	$-0.386060\pi$
$467$	15.1426	0.700714	0.350357	+	0.936616i	$0.386060\pi$
$468$	0	0
$469$	2.48940	0.114950
$470$	14.2507	0.657336
$471$	0	0
$472$	−1.95898	−0.0901694
$473$	−4.10207	−0.188614
$474$	0	0
$475$	20.0406	0.919524
$476$	−0.909637	−0.0416931
$477$	0	0
$478$	13.8869	0.635171
$479$	−7.50694	−0.343001	−0.171501	−	0.985184i	$-0.554862\pi$
$479$	−7.50694	−0.343001	−0.171501	+	0.985184i	$0.554862\pi$
$480$	0	0
$481$	−3.16099	−0.144129
$482$	21.9596	1.00023
$483$	0	0
$484$	−10.6802	−0.485462
$485$	−44.8198	−2.03516
$486$	0	0
$487$	−5.19502	−0.235409	−0.117705	−	0.993049i	$-0.537554\pi$
$487$	−5.19502	−0.235409	−0.117705	+	0.993049i	$0.537554\pi$
$488$	−11.6458	−0.527180
$489$	0	0
$490$	−8.59678	−0.388363
$491$	−27.6449	−1.24760	−0.623798	−	0.781586i	$-0.714411\pi$
$491$	−27.6449	−1.24760	−0.623798	+	0.781586i	$0.714411\pi$
$492$	0	0
$493$	1.17874	0.0530879
$494$	−2.78259	−0.125195
$495$	0	0
$496$	2.64010	0.118544
$497$	44.8571	2.01211
$498$	0	0
$499$	31.4700	1.40879	0.704395	−	0.709809i	$-0.251219\pi$
$499$	31.4700	1.40879	0.704395	+	0.709809i	$0.251219\pi$
$500$	−1.43611	−0.0642246
$501$	0	0
$502$	17.5507	0.783327
$503$	10.4697	0.466820	0.233410	−	0.972378i	$-0.425012\pi$
$503$	10.4697	0.466820	0.233410	+	0.972378i	$0.425012\pi$
$504$	0	0
$505$	−1.84064	−0.0819074
$506$	3.34167	0.148556
$507$	0	0
$508$	6.06582	0.269127
$509$	5.40639	0.239634	0.119817	−	0.992796i	$-0.461769\pi$
$509$	5.40639	0.239634	0.119817	+	0.992796i	$0.461769\pi$
$510$	0	0
$511$	−7.90940	−0.349891
$512$	1.00000	0.0441942
$513$	0	0
$514$	−18.3184	−0.807992
$515$	25.6121	1.12861
$516$	0	0
$517$	2.49379	0.109677
$518$	12.9965	0.571033
$519$	0	0
$520$	2.44303	0.107134
$521$	−25.4910	−1.11678	−0.558391	−	0.829578i	$-0.688581\pi$
$521$	−25.4910	−1.11678	−0.558391	+	0.829578i	$0.688581\pi$
$522$	0	0
$523$	−32.9082	−1.43897	−0.719487	−	0.694506i	$-0.755623\pi$
$523$	−32.9082	−1.43897	−0.719487	+	0.694506i	$0.755623\pi$
$524$	−3.03159	−0.132436
$525$	0	0
$526$	−1.06536	−0.0464521
$527$	−0.772678	−0.0336584
$528$	0	0
$529$	11.9140	0.517998
$530$	−25.4846	−1.10698
$531$	0	0
$532$	11.4407	0.496017
$533$	−5.15539	−0.223305
$534$	0	0
$535$	55.3859	2.39454
$536$	0.800947	0.0345957
$537$	0	0
$538$	5.96084	0.256990
$539$	−1.50439	−0.0647985
$540$	0	0
$541$	−30.2330	−1.29982	−0.649909	−	0.760012i	$-0.725193\pi$
$541$	−30.2330	−1.29982	−0.649909	+	0.760012i	$0.725193\pi$
$542$	5.48358	0.235540
$543$	0	0
$544$	−0.292670	−0.0125481
$545$	−25.1540	−1.07748
$546$	0	0
$547$	−8.55903	−0.365958	−0.182979	−	0.983117i	$-0.558574\pi$
$547$	−8.55903	−0.365958	−0.182979	+	0.983117i	$0.558574\pi$
$548$	6.42223	0.274344
$549$	0	0
$550$	−3.07902	−0.131290
$551$	−14.8253	−0.631579
$552$	0	0
$553$	38.7875	1.64941
$554$	32.4422	1.37834
$555$	0	0
$556$	−4.35852	−0.184842
$557$	−9.86540	−0.418010	−0.209005	−	0.977915i	$-0.567023\pi$
$557$	−9.86540	−0.418010	−0.209005	+	0.977915i	$0.567023\pi$
$558$	0	0
$559$	−5.48310	−0.231911
$560$	−10.0446	−0.424461
$561$	0	0
$562$	−0.115072	−0.00485403
$563$	−33.5154	−1.41250	−0.706252	−	0.707960i	$-0.749616\pi$
$563$	−33.5154	−1.41250	−0.706252	+	0.707960i	$0.749616\pi$
$564$	0	0
$565$	−30.4477	−1.28095
$566$	−22.4356	−0.943037
$567$	0	0
$568$	14.4325	0.605573
$569$	−18.0726	−0.757641	−0.378820	−	0.925470i	$-0.623670\pi$
$569$	−18.0726	−0.757641	−0.378820	+	0.925470i	$0.623670\pi$
$570$	0	0
$571$	28.9558	1.21176	0.605881	−	0.795556i	$-0.292821\pi$
$571$	28.9558	1.21176	0.605881	+	0.795556i	$0.292821\pi$
$572$	0.427516	0.0178753
$573$	0	0
$574$	21.1965	0.884725
$575$	−32.1697	−1.34157
$576$	0	0
$577$	−9.92418	−0.413149	−0.206574	−	0.978431i	$-0.566232\pi$
$577$	−9.92418	−0.413149	−0.206574	+	0.978431i	$0.566232\pi$
$578$	−16.9143	−0.703544
$579$	0	0
$580$	13.0161	0.540466
$581$	52.8650	2.19321
$582$	0	0
$583$	−4.45965	−0.184700
$584$	−2.54480	−0.105304
$585$	0	0
$586$	13.2695	0.548160
$587$	45.2180	1.86635	0.933173	−	0.359429i	$-0.117028\pi$
$587$	45.2180	1.86635	0.933173	+	0.359429i	$0.117028\pi$
$588$	0	0
$589$	9.71814	0.400429
$590$	6.33099	0.260643
$591$	0	0
$592$	4.18154	0.171860
$593$	−13.1229	−0.538892	−0.269446	−	0.963016i	$-0.586841\pi$
$593$	−13.1229	−0.538892	−0.269446	+	0.963016i	$0.586841\pi$
$594$	0	0
$595$	2.93974	0.120518
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	4.46670	0.182657
$599$	−18.2254	−0.744669	−0.372335	−	0.928099i	$-0.621443\pi$
$599$	−18.2254	−0.744669	−0.372335	+	0.928099i	$0.621443\pi$
$600$	0	0
$601$	2.32923	0.0950112	0.0475056	−	0.998871i	$-0.484873\pi$
$601$	2.32923	0.0950112	0.0475056	+	0.998871i	$0.484873\pi$
$602$	22.5439	0.918821
$603$	0	0
$604$	−5.99666	−0.244001
$605$	34.5159	1.40327
$606$	0	0
$607$	2.24602	0.0911633	0.0455816	−	0.998961i	$-0.485486\pi$
$607$	2.24602	0.0911633	0.0455816	+	0.998961i	$0.485486\pi$
$608$	3.68097	0.149283
$609$	0	0
$610$	37.6366	1.52386
$611$	3.33337	0.134854
$612$	0	0
$613$	−6.93101	−0.279941	−0.139971	−	0.990156i	$-0.544701\pi$
$613$	−6.93101	−0.279941	−0.139971	+	0.990156i	$0.544701\pi$
$614$	−4.66069	−0.188090
$615$	0	0
$616$	−1.75774	−0.0708214
$617$	8.91841	0.359042	0.179521	−	0.983754i	$-0.442545\pi$
$617$	8.91841	0.359042	0.179521	+	0.983754i	$0.442545\pi$
$618$	0	0
$619$	−49.3293	−1.98271	−0.991355	−	0.131206i	$-0.958115\pi$
$619$	−49.3293	−1.98271	−0.991355	+	0.131206i	$0.958115\pi$
$620$	−8.53222	−0.342662
$621$	0	0
$622$	−4.90285	−0.196586
$623$	−52.6202	−2.10818
$624$	0	0
$625$	−22.5807	−0.903227
$626$	−4.14187	−0.165543
$627$	0	0
$628$	12.5575	0.501098
$629$	−1.22381	−0.0487965
$630$	0	0
$631$	−12.1239	−0.482646	−0.241323	−	0.970445i	$-0.577581\pi$
$631$	−12.1239	−0.482646	−0.241323	+	0.970445i	$0.577581\pi$
$632$	12.4796	0.496413
$633$	0	0
$634$	12.1108	0.480980
$635$	−19.6034	−0.777936
$636$	0	0
$637$	−2.01086	−0.0796732
$638$	2.27775	0.0901769
$639$	0	0
$640$	−3.23178	−0.127747
$641$	−32.1081	−1.26819	−0.634096	−	0.773254i	$-0.718628\pi$
$641$	−32.1081	−1.26819	−0.634096	+	0.773254i	$0.718628\pi$
$642$	0	0
$643$	−33.6559	−1.32726	−0.663629	−	0.748062i	$-0.730984\pi$
$643$	−33.6559	−1.32726	−0.663629	+	0.748062i	$0.730984\pi$
$644$	−18.3650	−0.723681
$645$	0	0
$646$	−1.07731	−0.0423861
$647$	−4.35238	−0.171110	−0.0855549	−	0.996333i	$-0.527266\pi$
$647$	−4.35238	−0.171110	−0.0855549	+	0.996333i	$0.527266\pi$
$648$	0	0
$649$	1.10789	0.0434883
$650$	−4.11562	−0.161428
$651$	0	0
$652$	6.12608	0.239916
$653$	−33.5374	−1.31242	−0.656210	−	0.754578i	$-0.727842\pi$
$653$	−33.5374	−1.31242	−0.656210	+	0.754578i	$0.727842\pi$
$654$	0	0
$655$	9.79743	0.382817
$656$	6.81984	0.266270
$657$	0	0
$658$	−13.7052	−0.534285
$659$	−27.7008	−1.07907	−0.539536	−	0.841963i	$-0.681400\pi$
$659$	−27.7008	−1.07907	−0.539536	+	0.841963i	$0.681400\pi$
$660$	0	0
$661$	20.1619	0.784208	0.392104	−	0.919921i	$-0.371747\pi$
$661$	20.1619	0.784208	0.392104	+	0.919921i	$0.371747\pi$
$662$	−2.54691	−0.0989886
$663$	0	0
$664$	17.0090	0.660076
$665$	−36.9738	−1.43378
$666$	0	0
$667$	23.7980	0.921462
$668$	−11.9476	−0.462266
$669$	0	0
$670$	−2.58848	−0.100002
$671$	6.58618	0.254257
$672$	0	0
$673$	21.5456	0.830521	0.415261	−	0.909702i	$-0.363690\pi$
$673$	21.5456	0.830521	0.415261	+	0.909702i	$0.363690\pi$
$674$	−9.45663	−0.364256
$675$	0	0
$676$	−12.4286	−0.478021
$677$	15.3896	0.591470	0.295735	−	0.955270i	$-0.404436\pi$
$677$	15.3896	0.591470	0.295735	+	0.955270i	$0.404436\pi$
$678$	0	0
$679$	43.1042	1.65419
$680$	0.945843	0.0362714
$681$	0	0
$682$	−1.49309	−0.0571733
$683$	−6.30693	−0.241328	−0.120664	−	0.992693i	$-0.538502\pi$
$683$	−6.30693	−0.241328	−0.120664	+	0.992693i	$0.538502\pi$
$684$	0	0
$685$	−20.7552	−0.793015
$686$	−13.4888	−0.515003
$687$	0	0
$688$	7.25335	0.276531
$689$	−5.96106	−0.227098
$690$	0	0
$691$	−8.24301	−0.313579	−0.156789	−	0.987632i	$-0.550114\pi$
$691$	−8.24301	−0.313579	−0.156789	+	0.987632i	$0.550114\pi$
$692$	22.5164	0.855944
$693$	0	0
$694$	−11.3451	−0.430653
$695$	14.0858	0.534303
$696$	0	0
$697$	−1.99596	−0.0756024
$698$	34.1773	1.29363
$699$	0	0
$700$	16.9215	0.639571
$701$	38.9319	1.47044	0.735218	−	0.677831i	$-0.237080\pi$
$701$	38.9319	1.47044	0.735218	+	0.677831i	$0.237080\pi$
$702$	0	0
$703$	15.3921	0.580524
$704$	−0.565542	−0.0213147
$705$	0	0
$706$	16.0944	0.605721
$707$	1.77018	0.0665746
$708$	0	0
$709$	20.8384	0.782603	0.391302	−	0.920262i	$-0.372025\pi$
$709$	20.8384	0.782603	0.391302	+	0.920262i	$0.372025\pi$
$710$	−46.6425	−1.75046
$711$	0	0
$712$	−16.9302	−0.634486
$713$	−15.5999	−0.584219
$714$	0	0
$715$	−1.38163	−0.0516702
$716$	3.84611	0.143736
$717$	0	0
$718$	10.6921	0.399026
$719$	16.6753	0.621883	0.310941	−	0.950429i	$-0.399356\pi$
$719$	16.6753	0.621883	0.310941	+	0.950429i	$0.399356\pi$
$720$	0	0
$721$	−24.6317	−0.917334
$722$	−5.45046	−0.202845
$723$	0	0
$724$	6.25330	0.232402
$725$	−21.9275	−0.814366
$726$	0	0
$727$	−14.5111	−0.538185	−0.269093	−	0.963114i	$-0.586724\pi$
$727$	−14.5111	−0.538185	−0.269093	+	0.963114i	$0.586724\pi$
$728$	−2.34951	−0.0870788
$729$	0	0
$730$	8.22421	0.304392
$731$	−2.12284	−0.0785159
$732$	0	0
$733$	−5.24827	−0.193849	−0.0969247	−	0.995292i	$-0.530901\pi$
$733$	−5.24827	−0.193849	−0.0969247	+	0.995292i	$0.530901\pi$
$734$	−34.8461	−1.28619
$735$	0	0
$736$	−5.90880	−0.217801
$737$	−0.452969	−0.0166853
$738$	0	0
$739$	−2.25368	−0.0829028	−0.0414514	−	0.999141i	$-0.513198\pi$
$739$	−2.25368	−0.0829028	−0.0414514	+	0.999141i	$0.513198\pi$
$740$	−13.5138	−0.496777
$741$	0	0
$742$	24.5091	0.899756
$743$	22.8407	0.837943	0.418972	−	0.907999i	$-0.362391\pi$
$743$	22.8407	0.837943	0.418972	+	0.907999i	$0.362391\pi$
$744$	0	0
$745$	3.23178	0.118403
$746$	35.2009	1.28879
$747$	0	0
$748$	0.165517	0.00605190
$749$	−53.2658	−1.94629
$750$	0	0
$751$	32.6887	1.19283	0.596414	−	0.802677i	$-0.296592\pi$
$751$	32.6887	1.19283	0.596414	+	0.802677i	$0.296592\pi$
$752$	−4.40956	−0.160800
$753$	0	0
$754$	3.04459	0.110877
$755$	19.3799	0.705305
$756$	0	0
$757$	24.6766	0.896888	0.448444	−	0.893811i	$-0.351978\pi$
$757$	24.6766	0.896888	0.448444	+	0.893811i	$0.351978\pi$
$758$	19.8333	0.720378
$759$	0	0
$760$	−11.8961	−0.431516
$761$	−11.1859	−0.405488	−0.202744	−	0.979232i	$-0.564986\pi$
$761$	−11.1859	−0.405488	−0.202744	+	0.979232i	$0.564986\pi$
$762$	0	0
$763$	24.1911	0.875776
$764$	17.5113	0.633537
$765$	0	0
$766$	−0.838998	−0.0303142
$767$	1.48087	0.0534712
$768$	0	0
$769$	−41.3924	−1.49265	−0.746325	−	0.665582i	$-0.768183\pi$
$769$	−41.3924	−1.49265	−0.746325	+	0.665582i	$0.768183\pi$
$770$	5.68062	0.204716
$771$	0	0
$772$	7.71204	0.277562
$773$	−9.94554	−0.357716	−0.178858	−	0.983875i	$-0.557240\pi$
$773$	−9.94554	−0.357716	−0.178858	+	0.983875i	$0.557240\pi$
$774$	0	0
$775$	14.3737	0.516319
$776$	13.8685	0.497849
$777$	0	0
$778$	18.1597	0.651058
$779$	25.1036	0.899431
$780$	0	0
$781$	−8.16216	−0.292065
$782$	1.72933	0.0618406
$783$	0	0
$784$	2.66008	0.0950028
$785$	−40.5830	−1.44847
$786$	0	0
$787$	49.9316	1.77987	0.889935	−	0.456087i	$-0.150749\pi$
$787$	49.9316	1.77987	0.889935	+	0.456087i	$0.150749\pi$
$788$	13.9945	0.498534
$789$	0	0
$790$	−40.3313	−1.43492
$791$	29.2822	1.04116
$792$	0	0
$793$	8.80352	0.312622
$794$	−3.38229	−0.120033
$795$	0	0
$796$	3.75774	0.133190
$797$	−24.0151	−0.850657	−0.425329	−	0.905039i	$-0.639841\pi$
$797$	−24.0151	−0.850657	−0.425329	+	0.905039i	$0.639841\pi$
$798$	0	0
$799$	1.29055	0.0456562
$800$	5.44437	0.192488
$801$	0	0
$802$	−3.08431	−0.108911
$803$	1.43919	0.0507879
$804$	0	0
$805$	59.3514	2.09186
$806$	−1.99576	−0.0702977
$807$	0	0
$808$	0.569545	0.0200365
$809$	35.5688	1.25053	0.625266	−	0.780412i	$-0.284991\pi$
$809$	35.5688	1.25053	0.625266	+	0.780412i	$0.284991\pi$
$810$	0	0
$811$	33.7375	1.18468	0.592342	−	0.805687i	$-0.298203\pi$
$811$	33.7375	1.18468	0.592342	+	0.805687i	$0.298203\pi$
$812$	−12.5179	−0.439292
$813$	0	0
$814$	−2.36483	−0.0828874
$815$	−19.7981	−0.693498
$816$	0	0
$817$	26.6994	0.934093
$818$	0.462230	0.0161615
$819$	0	0
$820$	−22.0402	−0.769677
$821$	9.87137	0.344513	0.172257	−	0.985052i	$-0.444894\pi$
$821$	9.87137	0.344513	0.172257	+	0.985052i	$0.444894\pi$
$822$	0	0
$823$	−30.6298	−1.06769	−0.533845	−	0.845583i	$-0.679253\pi$
$823$	−30.6298	−1.06769	−0.533845	+	0.845583i	$0.679253\pi$
$824$	−7.92510	−0.276084
$825$	0	0
$826$	−6.08864	−0.211851
$827$	11.4752	0.399031	0.199516	−	0.979895i	$-0.436063\pi$
$827$	11.4752	0.399031	0.199516	+	0.979895i	$0.436063\pi$
$828$	0	0
$829$	17.5716	0.610286	0.305143	−	0.952306i	$-0.401296\pi$
$829$	17.5716	0.610286	0.305143	+	0.952306i	$0.401296\pi$
$830$	−54.9692	−1.90801
$831$	0	0
$832$	−0.755940	−0.0262075
$833$	−0.778525	−0.0269743
$834$	0	0
$835$	38.6119	1.33622
$836$	−2.08174	−0.0719986
$837$	0	0
$838$	20.5962	0.711483
$839$	−23.9857	−0.828079	−0.414040	−	0.910259i	$-0.635882\pi$
$839$	−23.9857	−0.828079	−0.414040	+	0.910259i	$0.635882\pi$
$840$	0	0
$841$	−12.7788	−0.440650
$842$	−10.0402	−0.346009
$843$	0	0
$844$	0.452579	0.0155784
$845$	40.1663	1.38176
$846$	0	0
$847$	−33.1947	−1.14058
$848$	7.88563	0.270794
$849$	0	0
$850$	−1.59340	−0.0546532
$851$	−24.7079	−0.846975
$852$	0	0
$853$	−33.4434	−1.14508	−0.572541	−	0.819876i	$-0.694042\pi$
$853$	−33.4434	−1.14508	−0.572541	+	0.819876i	$0.694042\pi$
$854$	−36.1959	−1.23860
$855$	0	0
$856$	−17.1379	−0.585762
$857$	−13.4927	−0.460902	−0.230451	−	0.973084i	$-0.574020\pi$
$857$	−13.4927	−0.460902	−0.230451	+	0.973084i	$0.574020\pi$
$858$	0	0
$859$	−8.06621	−0.275215	−0.137608	−	0.990487i	$-0.543941\pi$
$859$	−8.06621	−0.275215	−0.137608	+	0.990487i	$0.543941\pi$
$860$	−23.4412	−0.799338
$861$	0	0
$862$	−17.5848	−0.598941
$863$	−2.11525	−0.0720041	−0.0360021	−	0.999352i	$-0.511462\pi$
$863$	−2.11525	−0.0720041	−0.0360021	+	0.999352i	$0.511462\pi$
$864$	0	0
$865$	−72.7679	−2.47418
$866$	5.75874	0.195690
$867$	0	0
$868$	8.20562	0.278517
$869$	−7.05775	−0.239418
$870$	0	0
$871$	−0.605468	−0.0205155
$872$	7.78332	0.263577
$873$	0	0
$874$	−21.7501	−0.735709
$875$	−4.46351	−0.150894
$876$	0	0
$877$	−32.4357	−1.09528	−0.547638	−	0.836716i	$-0.684473\pi$
$877$	−32.4357	−1.09528	−0.547638	+	0.836716i	$0.684473\pi$
$878$	−12.8706	−0.434363
$879$	0	0
$880$	1.82770	0.0616119
$881$	−39.1640	−1.31947	−0.659735	−	0.751498i	$-0.729331\pi$
$881$	−39.1640	−1.31947	−0.659735	+	0.751498i	$0.729331\pi$
$882$	0	0
$883$	−36.3864	−1.22450	−0.612249	−	0.790665i	$-0.709735\pi$
$883$	−36.3864	−1.22450	−0.612249	+	0.790665i	$0.709735\pi$
$884$	0.221241	0.00744113
$885$	0	0
$886$	−22.4068	−0.752771
$887$	1.20854	0.0405787	0.0202894	−	0.999794i	$-0.493541\pi$
$887$	1.20854	0.0405787	0.0202894	+	0.999794i	$0.493541\pi$
$888$	0	0
$889$	18.8530	0.632309
$890$	54.7146	1.83404
$891$	0	0
$892$	−5.86934	−0.196520
$893$	−16.2315	−0.543165
$894$	0	0
$895$	−12.4298	−0.415481
$896$	3.10807	0.103833
$897$	0	0
$898$	30.8706	1.03017
$899$	−10.6331	−0.354635
$900$	0	0
$901$	−2.30788	−0.0768868
$902$	−3.85690	−0.128421
$903$	0	0
$904$	9.42136	0.313350
$905$	−20.2093	−0.671779
$906$	0	0
$907$	−16.0426	−0.532687	−0.266344	−	0.963878i	$-0.585816\pi$
$907$	−16.0426	−0.532687	−0.266344	+	0.963878i	$0.585816\pi$
$908$	25.1098	0.833299
$909$	0	0
$910$	7.59310	0.251709
$911$	−23.4751	−0.777765	−0.388883	−	0.921287i	$-0.627139\pi$
$911$	−23.4751	−0.777765	−0.388883	+	0.921287i	$0.627139\pi$
$912$	0	0
$913$	−9.61929	−0.318352
$914$	36.6093	1.21093
$915$	0	0
$916$	15.6227	0.516189
$917$	−9.42240	−0.311155
$918$	0	0
$919$	54.7165	1.80493	0.902465	−	0.430764i	$-0.141756\pi$
$919$	54.7165	1.80493	0.902465	+	0.430764i	$0.141756\pi$
$920$	19.0959	0.629574
$921$	0	0
$922$	7.02235	0.231269
$923$	−10.9101	−0.359110
$924$	0	0
$925$	22.7658	0.748536
$926$	20.2488	0.665417
$927$	0	0
$928$	−4.02755	−0.132211
$929$	−16.6444	−0.546084	−0.273042	−	0.962002i	$-0.588030\pi$
$929$	−16.6444	−0.546084	−0.273042	+	0.962002i	$0.588030\pi$
$930$	0	0
$931$	9.79167	0.320909
$932$	−14.5716	−0.477308
$933$	0	0
$934$	15.1426	0.495480
$935$	−0.534913	−0.0174935
$936$	0	0
$937$	32.5302	1.06272	0.531358	−	0.847147i	$-0.321682\pi$
$937$	32.5302	1.06272	0.531358	+	0.847147i	$0.321682\pi$
$938$	2.48940	0.0812817
$939$	0	0
$940$	14.2507	0.464807
$941$	−47.3270	−1.54282	−0.771408	−	0.636341i	$-0.780447\pi$
$941$	−47.3270	−1.54282	−0.771408	+	0.636341i	$0.780447\pi$
$942$	0	0
$943$	−40.2971	−1.31225
$944$	−1.95898	−0.0637594
$945$	0	0
$946$	−4.10207	−0.133370
$947$	−16.4092	−0.533226	−0.266613	−	0.963804i	$-0.585905\pi$
$947$	−16.4092	−0.533226	−0.266613	+	0.963804i	$0.585905\pi$
$948$	0	0
$949$	1.92372	0.0624464
$950$	20.0406	0.650202
$951$	0	0
$952$	−0.909637	−0.0294815
$953$	−32.6669	−1.05819	−0.529093	−	0.848564i	$-0.677468\pi$
$953$	−32.6669	−1.05819	−0.529093	+	0.848564i	$0.677468\pi$
$954$	0	0
$955$	−56.5926	−1.83129
$956$	13.8869	0.449134
$957$	0	0
$958$	−7.50694	−0.242538
$959$	19.9607	0.644565
$960$	0	0
$961$	−24.0299	−0.775156
$962$	−3.16099	−0.101914
$963$	0	0
$964$	21.9596	0.707272
$965$	−24.9236	−0.802318
$966$	0	0
$967$	−24.5048	−0.788020	−0.394010	−	0.919106i	$-0.628912\pi$
$967$	−24.5048	−0.788020	−0.394010	+	0.919106i	$0.628912\pi$
$968$	−10.6802	−0.343273
$969$	0	0
$970$	−44.8198	−1.43908
$971$	−4.93111	−0.158247	−0.0791235	−	0.996865i	$-0.525212\pi$
$971$	−4.93111	−0.158247	−0.0791235	+	0.996865i	$0.525212\pi$
$972$	0	0
$973$	−13.5466	−0.434283
$974$	−5.19502	−0.166459
$975$	0	0
$976$	−11.6458	−0.372772
$977$	7.47853	0.239259	0.119630	−	0.992819i	$-0.461829\pi$
$977$	7.47853	0.239259	0.119630	+	0.992819i	$0.461829\pi$
$978$	0	0
$979$	9.57473	0.306010
$980$	−8.59678	−0.274614
$981$	0	0
$982$	−27.6449	−0.882183
$983$	23.4457	0.747802	0.373901	−	0.927469i	$-0.378020\pi$
$983$	23.4457	0.747802	0.373901	+	0.927469i	$0.378020\pi$
$984$	0	0
$985$	−45.2271	−1.44105
$986$	1.17874	0.0375388
$987$	0	0
$988$	−2.78259	−0.0885261
$989$	−42.8586	−1.36283
$990$	0	0
$991$	−19.4356	−0.617393	−0.308697	−	0.951161i	$-0.599893\pi$
$991$	−19.4356	−0.617393	−0.308697	+	0.951161i	$0.599893\pi$
$992$	2.64010	0.0838234
$993$	0	0
$994$	44.8571	1.42278
$995$	−12.1442	−0.384996
$996$	0	0
$997$	6.15223	0.194843	0.0974215	−	0.995243i	$-0.468941\pi$
$997$	6.15223	0.194843	0.0974215	+	0.995243i	$0.468941\pi$
$998$	31.4700	0.996164
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.t.1.3	yes	16
3.2	odd	2		8046.2.a.s.1.14	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.s.1.14	✓	16	3.2	odd	2
8046.2.a.t.1.3	yes	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.23178 q^{5} +3.10807 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.23178 q^{5} +3.10807 q^{7} +1.00000 q^{8} -3.23178 q^{10} -0.565542 q^{11} -0.755940 q^{13} +3.10807 q^{14} +1.00000 q^{16} -0.292670 q^{17} +3.68097 q^{19} -3.23178 q^{20} -0.565542 q^{22} -5.90880 q^{23} +5.44437 q^{25} -0.755940 q^{26} +3.10807 q^{28} -4.02755 q^{29} +2.64010 q^{31} +1.00000 q^{32} -0.292670 q^{34} -10.0446 q^{35} +4.18154 q^{37} +3.68097 q^{38} -3.23178 q^{40} +6.81984 q^{41} +7.25335 q^{43} -0.565542 q^{44} -5.90880 q^{46} -4.40956 q^{47} +2.66008 q^{49} +5.44437 q^{50} -0.755940 q^{52} +7.88563 q^{53} +1.82770 q^{55} +3.10807 q^{56} -4.02755 q^{58} -1.95898 q^{59} -11.6458 q^{61} +2.64010 q^{62} +1.00000 q^{64} +2.44303 q^{65} +0.800947 q^{67} -0.292670 q^{68} -10.0446 q^{70} +14.4325 q^{71} -2.54480 q^{73} +4.18154 q^{74} +3.68097 q^{76} -1.75774 q^{77} +12.4796 q^{79} -3.23178 q^{80} +6.81984 q^{82} +17.0090 q^{83} +0.945843 q^{85} +7.25335 q^{86} -0.565542 q^{88} -16.9302 q^{89} -2.34951 q^{91} -5.90880 q^{92} -4.40956 q^{94} -11.8961 q^{95} +13.8685 q^{97} +2.66008 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) \) 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8	\( 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8} + 4 q^{10} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 16 q^{16} + q^{17} + 10 q^{19} + 4 q^{20} + 6 q^{22} + 10 q^{23} + 28 q^{25} + 6 q^{26} + 6 q^{28} + 6 q^{29} + 21 q^{31} + 16 q^{32} + q^{34} + 16 q^{35} + 17 q^{37} + 10 q^{38} + 4 q^{40} - 4 q^{41} + 16 q^{43} + 6 q^{44} + 10 q^{46} + 25 q^{47} + 36 q^{49} + 28 q^{50} + 6 q^{52} + 14 q^{53} + 19 q^{55} + 6 q^{56} + 6 q^{58} + 6 q^{59} + 23 q^{61} + 21 q^{62} + 16 q^{64} + 20 q^{65} + 22 q^{67} + q^{68} + 16 q^{70} + 10 q^{71} + 16 q^{73} + 17 q^{74} + 10 q^{76} - 2 q^{77} + 37 q^{79} + 4 q^{80} - 4 q^{82} + 33 q^{83} + 43 q^{85} + 16 q^{86} + 6 q^{88} - 3 q^{89} + 28 q^{91} + 10 q^{92} + 25 q^{94} + 14 q^{95} - 3 q^{97} + 36 q^{98}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8 + 4 * q^10 + 6 * q^11 + 6 * q^13 + 6 * q^14 + 16 * q^16 + q^17 + 10 * q^19 + 4 * q^20 + 6 * q^22 + 10 * q^23 + 28 * q^25 + 6 * q^26 + 6 * q^28 + 6 * q^29 + 21 * q^31 + 16 * q^32 + q^34 + 16 * q^35 + 17 * q^37 + 10 * q^38 + 4 * q^40 - 4 * q^41 + 16 * q^43 + 6 * q^44 + 10 * q^46 + 25 * q^47 + 36 * q^49 + 28 * q^50 + 6 * q^52 + 14 * q^53 + 19 * q^55 + 6 * q^56 + 6 * q^58 + 6 * q^59 + 23 * q^61 + 21 * q^62 + 16 * q^64 + 20 * q^65 + 22 * q^67 + q^68 + 16 * q^70 + 10 * q^71 + 16 * q^73 + 17 * q^74 + 10 * q^76 - 2 * q^77 + 37 * q^79 + 4 * q^80 - 4 * q^82 + 33 * q^83 + 43 * q^85 + 16 * q^86 + 6 * q^88 - 3 * q^89 + 28 * q^91 + 10 * q^92 + 25 * q^94 + 14 * q^95 - 3 * q^97 + 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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