LMFDB - Embedded newform 8046.2.a.h.1.8

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 9x^{7} + 25x^{6} + 29x^{5} - 58x^{4} - 43x^{3} + 34x^{2} + 25x + 2 $ x^9 - 3x^8 - 9x^7 + 25x^6 + 29x^5 - 58x^4 - 43x^3 + 34x^2 + 25x + 2
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			1.8
Root			$-0.0929510$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.96687 q^{5} -4.72097 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.96687 q^{5} -4.72097 q^{7} +1.00000 q^{8} +1.96687 q^{10} +2.36478 q^{11} -0.742721 q^{13} -4.72097 q^{14} +1.00000 q^{16} -1.40102 q^{17} -2.69068 q^{19} +1.96687 q^{20} +2.36478 q^{22} -3.63674 q^{23} -1.13141 q^{25} -0.742721 q^{26} -4.72097 q^{28} +0.980272 q^{29} +3.58715 q^{31} +1.00000 q^{32} -1.40102 q^{34} -9.28555 q^{35} +10.6294 q^{37} -2.69068 q^{38} +1.96687 q^{40} -6.88728 q^{41} -1.80326 q^{43} +2.36478 q^{44} -3.63674 q^{46} -2.73029 q^{47} +15.2876 q^{49} -1.13141 q^{50} -0.742721 q^{52} +1.45122 q^{53} +4.65123 q^{55} -4.72097 q^{56} +0.980272 q^{58} -2.49561 q^{59} -7.04971 q^{61} +3.58715 q^{62} +1.00000 q^{64} -1.46084 q^{65} -4.95825 q^{67} -1.40102 q^{68} -9.28555 q^{70} -14.3379 q^{71} -7.71974 q^{73} +10.6294 q^{74} -2.69068 q^{76} -11.1641 q^{77} +2.59929 q^{79} +1.96687 q^{80} -6.88728 q^{82} -10.3231 q^{83} -2.75562 q^{85} -1.80326 q^{86} +2.36478 q^{88} +1.95400 q^{89} +3.50637 q^{91} -3.63674 q^{92} -2.73029 q^{94} -5.29222 q^{95} +0.529108 q^{97} +15.2876 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8}+O(q^{10}) $ 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^5 - 4 * q^7 + 9 * q^8	$ 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8} - 4 q^{10} - 4 q^{11} - 8 q^{13} - 4 q^{14} + 9 q^{16} - q^{17} - 10 q^{19} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 3 q^{25} - 8 q^{26} - 4 q^{28} - 4 q^{29} - 17 q^{31} + 9 q^{32} - q^{34} - 10 q^{35} - 11 q^{37} - 10 q^{38} - 4 q^{40} - 16 q^{43} - 4 q^{44} - 8 q^{46} - 7 q^{47} - 5 q^{49} - 3 q^{50} - 8 q^{52} - 12 q^{53} - 23 q^{55} - 4 q^{56} - 4 q^{58} - 6 q^{59} - 13 q^{61} - 17 q^{62} + 9 q^{64} + 24 q^{65} - 14 q^{67} - q^{68} - 10 q^{70} - 30 q^{71} - 12 q^{73} - 11 q^{74} - 10 q^{76} - 12 q^{77} - 35 q^{79} - 4 q^{80} - 5 q^{83} - 27 q^{85} - 16 q^{86} - 4 q^{88} - 23 q^{89} - 28 q^{91} - 8 q^{92} - 7 q^{94} - 32 q^{95} - 21 q^{97} - 5 q^{98}+O(q^{100}) $ 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^5 - 4 * q^7 + 9 * q^8 - 4 * q^10 - 4 * q^11 - 8 * q^13 - 4 * q^14 + 9 * q^16 - q^17 - 10 * q^19 - 4 * q^20 - 4 * q^22 - 8 * q^23 - 3 * q^25 - 8 * q^26 - 4 * q^28 - 4 * q^29 - 17 * q^31 + 9 * q^32 - q^34 - 10 * q^35 - 11 * q^37 - 10 * q^38 - 4 * q^40 - 16 * q^43 - 4 * q^44 - 8 * q^46 - 7 * q^47 - 5 * q^49 - 3 * q^50 - 8 * q^52 - 12 * q^53 - 23 * q^55 - 4 * q^56 - 4 * q^58 - 6 * q^59 - 13 * q^61 - 17 * q^62 + 9 * q^64 + 24 * q^65 - 14 * q^67 - q^68 - 10 * q^70 - 30 * q^71 - 12 * q^73 - 11 * q^74 - 10 * q^76 - 12 * q^77 - 35 * q^79 - 4 * q^80 - 5 * q^83 - 27 * q^85 - 16 * q^86 - 4 * q^88 - 23 * q^89 - 28 * q^91 - 8 * q^92 - 7 * q^94 - 32 * q^95 - 21 * q^97 - 5 * 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
$5$	1.96687	0.879612	0.439806	−	0.898093i	$-0.355047\pi$
$5$	1.96687	0.879612	0.439806	+	0.898093i	$0.355047\pi$
$6$	0	0
$7$	−4.72097	−1.78436	−0.892180	−	0.451681i	$-0.850825\pi$
$7$	−4.72097	−1.78436	−0.892180	+	0.451681i	$0.850825\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.96687	0.621980
$11$	2.36478	0.713009	0.356504	−	0.934294i	$-0.383968\pi$
$11$	2.36478	0.713009	0.356504	+	0.934294i	$0.383968\pi$
$12$	0	0
$13$	−0.742721	−0.205994	−0.102997	−	0.994682i	$-0.532843\pi$
$13$	−0.742721	−0.205994	−0.102997	+	0.994682i	$0.532843\pi$
$14$	−4.72097	−1.26173
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.40102	−0.339796	−0.169898	−	0.985462i	$-0.554344\pi$
$17$	−1.40102	−0.339796	−0.169898	+	0.985462i	$0.554344\pi$
$18$	0	0
$19$	−2.69068	−0.617284	−0.308642	−	0.951178i	$-0.599874\pi$
$19$	−2.69068	−0.617284	−0.308642	+	0.951178i	$0.599874\pi$
$20$	1.96687	0.439806
$21$	0	0
$22$	2.36478	0.504173
$23$	−3.63674	−0.758313	−0.379157	−	0.925333i	$-0.623786\pi$
$23$	−3.63674	−0.758313	−0.379157	+	0.925333i	$0.623786\pi$
$24$	0	0
$25$	−1.13141	−0.226283
$26$	−0.742721	−0.145660
$27$	0	0
$28$	−4.72097	−0.892180
$29$	0.980272	0.182032	0.0910159	−	0.995849i	$-0.470989\pi$
$29$	0.980272	0.182032	0.0910159	+	0.995849i	$0.470989\pi$
$30$	0	0
$31$	3.58715	0.644271	0.322135	−	0.946694i	$-0.395599\pi$
$31$	3.58715	0.644271	0.322135	+	0.946694i	$0.395599\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.40102	−0.240272
$35$	−9.28555	−1.56954
$36$	0	0
$37$	10.6294	1.74747	0.873735	−	0.486403i	$-0.161691\pi$
$37$	10.6294	1.74747	0.873735	+	0.486403i	$0.161691\pi$
$38$	−2.69068	−0.436486
$39$	0	0
$40$	1.96687	0.310990
$41$	−6.88728	−1.07561	−0.537806	−	0.843069i	$-0.680747\pi$
$41$	−6.88728	−1.07561	−0.537806	+	0.843069i	$0.680747\pi$
$42$	0	0
$43$	−1.80326	−0.274995	−0.137497	−	0.990502i	$-0.543906\pi$
$43$	−1.80326	−0.274995	−0.137497	+	0.990502i	$0.543906\pi$
$44$	2.36478	0.356504
$45$	0	0
$46$	−3.63674	−0.536208
$47$	−2.73029	−0.398254	−0.199127	−	0.979974i	$-0.563811\pi$
$47$	−2.73029	−0.398254	−0.199127	+	0.979974i	$0.563811\pi$
$48$	0	0
$49$	15.2876	2.18394
$50$	−1.13141	−0.160006
$51$	0	0
$52$	−0.742721	−0.102997
$53$	1.45122	0.199341	0.0996705	−	0.995020i	$-0.468221\pi$
$53$	1.45122	0.199341	0.0996705	+	0.995020i	$0.468221\pi$
$54$	0	0
$55$	4.65123	0.627171
$56$	−4.72097	−0.630866
$57$	0	0
$58$	0.980272	0.128716
$59$	−2.49561	−0.324900	−0.162450	−	0.986717i	$-0.551940\pi$
$59$	−2.49561	−0.324900	−0.162450	+	0.986717i	$0.551940\pi$
$60$	0	0
$61$	−7.04971	−0.902623	−0.451311	−	0.892367i	$-0.649044\pi$
$61$	−7.04971	−0.902623	−0.451311	+	0.892367i	$0.649044\pi$
$62$	3.58715	0.455568
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.46084	−0.181195
$66$	0	0
$67$	−4.95825	−0.605747	−0.302873	−	0.953031i	$-0.597946\pi$
$67$	−4.95825	−0.605747	−0.302873	+	0.953031i	$0.597946\pi$
$68$	−1.40102	−0.169898
$69$	0	0
$70$	−9.28555	−1.10983
$71$	−14.3379	−1.70160	−0.850798	−	0.525493i	$-0.823881\pi$
$71$	−14.3379	−1.70160	−0.850798	+	0.525493i	$0.823881\pi$
$72$	0	0
$73$	−7.71974	−0.903528	−0.451764	−	0.892138i	$-0.649205\pi$
$73$	−7.71974	−0.903528	−0.451764	+	0.892138i	$0.649205\pi$
$74$	10.6294	1.23565
$75$	0	0
$76$	−2.69068	−0.308642
$77$	−11.1641	−1.27226
$78$	0	0
$79$	2.59929	0.292443	0.146221	−	0.989252i	$-0.453289\pi$
$79$	2.59929	0.292443	0.146221	+	0.989252i	$0.453289\pi$
$80$	1.96687	0.219903
$81$	0	0
$82$	−6.88728	−0.760572
$83$	−10.3231	−1.13311	−0.566555	−	0.824024i	$-0.691724\pi$
$83$	−10.3231	−1.13311	−0.566555	+	0.824024i	$0.691724\pi$
$84$	0	0
$85$	−2.75562	−0.298889
$86$	−1.80326	−0.194451
$87$	0	0
$88$	2.36478	0.252087
$89$	1.95400	0.207123	0.103562	−	0.994623i	$-0.466976\pi$
$89$	1.95400	0.207123	0.103562	+	0.994623i	$0.466976\pi$
$90$	0	0
$91$	3.50637	0.367567
$92$	−3.63674	−0.379157
$93$	0	0
$94$	−2.73029	−0.281608
$95$	−5.29222	−0.542970
$96$	0	0
$97$	0.529108	0.0537228	0.0268614	−	0.999639i	$-0.491449\pi$
$97$	0.529108	0.0537228	0.0268614	+	0.999639i	$0.491449\pi$
$98$	15.2876	1.54428
$99$	0	0
$100$	−1.13141	−0.113141
$101$	4.78364	0.475989	0.237995	−	0.971266i	$-0.423510\pi$
$101$	4.78364	0.475989	0.237995	+	0.971266i	$0.423510\pi$
$102$	0	0
$103$	−1.60204	−0.157854	−0.0789268	−	0.996880i	$-0.525149\pi$
$103$	−1.60204	−0.157854	−0.0789268	+	0.996880i	$0.525149\pi$
$104$	−0.742721	−0.0728298
$105$	0	0
$106$	1.45122	0.140955
$107$	−14.1668	−1.36956	−0.684780	−	0.728750i	$-0.740102\pi$
$107$	−14.1668	−1.36956	−0.684780	+	0.728750i	$0.740102\pi$
$108$	0	0
$109$	−14.8290	−1.42036	−0.710179	−	0.704021i	$-0.751386\pi$
$109$	−14.8290	−1.42036	−0.710179	+	0.704021i	$0.751386\pi$
$110$	4.65123	0.443477
$111$	0	0
$112$	−4.72097	−0.446090
$113$	10.0435	0.944811	0.472405	−	0.881381i	$-0.343386\pi$
$113$	10.0435	0.944811	0.472405	+	0.881381i	$0.343386\pi$
$114$	0	0
$115$	−7.15301	−0.667021
$116$	0.980272	0.0910159
$117$	0	0
$118$	−2.49561	−0.229739
$119$	6.61415	0.606318
$120$	0	0
$121$	−5.40780	−0.491618
$122$	−7.04971	−0.638251
$123$	0	0
$124$	3.58715	0.322135
$125$	−12.0597	−1.07865
$126$	0	0
$127$	−10.1650	−0.901996	−0.450998	−	0.892525i	$-0.648932\pi$
$127$	−10.1650	−0.901996	−0.450998	+	0.892525i	$0.648932\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−1.46084	−0.128124
$131$	−14.2956	−1.24901	−0.624505	−	0.781021i	$-0.714699\pi$
$131$	−14.2956	−1.24901	−0.624505	+	0.781021i	$0.714699\pi$
$132$	0	0
$133$	12.7026	1.10146
$134$	−4.95825	−0.428328
$135$	0	0
$136$	−1.40102	−0.120136
$137$	−2.27871	−0.194683	−0.0973417	−	0.995251i	$-0.531034\pi$
$137$	−2.27871	−0.194683	−0.0973417	+	0.995251i	$0.531034\pi$
$138$	0	0
$139$	2.63621	0.223601	0.111800	−	0.993731i	$-0.464338\pi$
$139$	2.63621	0.223601	0.111800	+	0.993731i	$0.464338\pi$
$140$	−9.28555	−0.784772
$141$	0	0
$142$	−14.3379	−1.20321
$143$	−1.75638	−0.146875
$144$	0	0
$145$	1.92807	0.160117
$146$	−7.71974	−0.638890
$147$	0	0
$148$	10.6294	0.873735
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−8.80220	−0.716313	−0.358156	−	0.933662i	$-0.616595\pi$
$151$	−8.80220	−0.716313	−0.358156	+	0.933662i	$0.616595\pi$
$152$	−2.69068	−0.218243
$153$	0	0
$154$	−11.1641	−0.899627
$155$	7.05546	0.566708
$156$	0	0
$157$	23.7916	1.89878	0.949389	−	0.314102i	$-0.101703\pi$
$157$	23.7916	1.89878	0.949389	+	0.314102i	$0.101703\pi$
$158$	2.59929	0.206788
$159$	0	0
$160$	1.96687	0.155495
$161$	17.1690	1.35310
$162$	0	0
$163$	12.9127	1.01140	0.505699	−	0.862710i	$-0.331235\pi$
$163$	12.9127	1.01140	0.505699	+	0.862710i	$0.331235\pi$
$164$	−6.88728	−0.537806
$165$	0	0
$166$	−10.3231	−0.801230
$167$	12.1490	0.940119	0.470060	−	0.882635i	$-0.344232\pi$
$167$	12.1490	0.940119	0.470060	+	0.882635i	$0.344232\pi$
$168$	0	0
$169$	−12.4484	−0.957567
$170$	−2.75562	−0.211346
$171$	0	0
$172$	−1.80326	−0.137497
$173$	4.61956	0.351219	0.175609	−	0.984460i	$-0.443810\pi$
$173$	4.61956	0.351219	0.175609	+	0.984460i	$0.443810\pi$
$174$	0	0
$175$	5.34137	0.403770
$176$	2.36478	0.178252
$177$	0	0
$178$	1.95400	0.146458
$179$	15.0982	1.12850	0.564248	−	0.825605i	$-0.309166\pi$
$179$	15.0982	1.12850	0.564248	+	0.825605i	$0.309166\pi$
$180$	0	0
$181$	−18.5993	−1.38248	−0.691238	−	0.722627i	$-0.742934\pi$
$181$	−18.5993	−1.38248	−0.691238	+	0.722627i	$0.742934\pi$
$182$	3.50637	0.259909
$183$	0	0
$184$	−3.63674	−0.268104
$185$	20.9067	1.53709
$186$	0	0
$187$	−3.31310	−0.242278
$188$	−2.73029	−0.199127
$189$	0	0
$190$	−5.29222	−0.383938
$191$	−16.7618	−1.21284	−0.606420	−	0.795145i	$-0.707395\pi$
$191$	−16.7618	−1.21284	−0.606420	+	0.795145i	$0.707395\pi$
$192$	0	0
$193$	9.50979	0.684530	0.342265	−	0.939604i	$-0.388806\pi$
$193$	9.50979	0.684530	0.342265	+	0.939604i	$0.388806\pi$
$194$	0.529108	0.0379878
$195$	0	0
$196$	15.2876	1.09197
$197$	−3.67474	−0.261815	−0.130907	−	0.991395i	$-0.541789\pi$
$197$	−3.67474	−0.261815	−0.130907	+	0.991395i	$0.541789\pi$
$198$	0	0
$199$	−3.86258	−0.273811	−0.136906	−	0.990584i	$-0.543716\pi$
$199$	−3.86258	−0.273811	−0.136906	+	0.990584i	$0.543716\pi$
$200$	−1.13141	−0.0800031
$201$	0	0
$202$	4.78364	0.336575
$203$	−4.62783	−0.324810
$204$	0	0
$205$	−13.5464	−0.946121
$206$	−1.60204	−0.111619
$207$	0	0
$208$	−0.742721	−0.0514985
$209$	−6.36287	−0.440129
$210$	0	0
$211$	−2.33825	−0.160971	−0.0804857	−	0.996756i	$-0.525647\pi$
$211$	−2.33825	−0.160971	−0.0804857	+	0.996756i	$0.525647\pi$
$212$	1.45122	0.0996705
$213$	0	0
$214$	−14.1668	−0.968425
$215$	−3.54678	−0.241889
$216$	0	0
$217$	−16.9348	−1.14961
$218$	−14.8290	−1.00434
$219$	0	0
$220$	4.65123	0.313586
$221$	1.04056	0.0699959
$222$	0	0
$223$	−0.0107588	−0.000720463 0	−0.000360232	−	1.00000i	$-0.500115\pi$
$223$	−0.0107588	−0.000720463 0	−0.000360232		1.00000i	$0.500115\pi$
$224$	−4.72097	−0.315433
$225$	0	0
$226$	10.0435	0.668082
$227$	−13.1054	−0.869839	−0.434919	−	0.900469i	$-0.643223\pi$
$227$	−13.1054	−0.869839	−0.434919	+	0.900469i	$0.643223\pi$
$228$	0	0
$229$	−1.29209	−0.0853838	−0.0426919	−	0.999088i	$-0.513593\pi$
$229$	−1.29209	−0.0853838	−0.0426919	+	0.999088i	$0.513593\pi$
$230$	−7.15301	−0.471655
$231$	0	0
$232$	0.980272	0.0643580
$233$	−15.7092	−1.02915	−0.514573	−	0.857447i	$-0.672050\pi$
$233$	−15.7092	−1.02915	−0.514573	+	0.857447i	$0.672050\pi$
$234$	0	0
$235$	−5.37014	−0.350309
$236$	−2.49561	−0.162450
$237$	0	0
$238$	6.61415	0.428732
$239$	−17.2429	−1.11535	−0.557674	−	0.830060i	$-0.688306\pi$
$239$	−17.2429	−1.11535	−0.557674	+	0.830060i	$0.688306\pi$
$240$	0	0
$241$	−14.6956	−0.946626	−0.473313	−	0.880894i	$-0.656942\pi$
$241$	−14.6956	−0.946626	−0.473313	+	0.880894i	$0.656942\pi$
$242$	−5.40780	−0.347627
$243$	0	0
$244$	−7.04971	−0.451311
$245$	30.0687	1.92102
$246$	0	0
$247$	1.99842	0.127157
$248$	3.58715	0.227784
$249$	0	0
$250$	−12.0597	−0.762723
$251$	14.2238	0.897801	0.448901	−	0.893582i	$-0.351816\pi$
$251$	14.2238	0.897801	0.448901	+	0.893582i	$0.351816\pi$
$252$	0	0
$253$	−8.60011	−0.540684
$254$	−10.1650	−0.637807
$255$	0	0
$256$	1.00000	0.0625000
$257$	17.3481	1.08215	0.541073	−	0.840976i	$-0.318018\pi$
$257$	17.3481	1.08215	0.541073	+	0.840976i	$0.318018\pi$
$258$	0	0
$259$	−50.1813	−3.11811
$260$	−1.46084	−0.0905973
$261$	0	0
$262$	−14.2956	−0.883183
$263$	7.74953	0.477856	0.238928	−	0.971037i	$-0.423204\pi$
$263$	7.74953	0.477856	0.238928	+	0.971037i	$0.423204\pi$
$264$	0	0
$265$	2.85437	0.175343
$266$	12.7026	0.778847
$267$	0	0
$268$	−4.95825	−0.302873
$269$	20.9413	1.27682	0.638408	−	0.769698i	$-0.279593\pi$
$269$	20.9413	1.27682	0.638408	+	0.769698i	$0.279593\pi$
$270$	0	0
$271$	−12.6221	−0.766739	−0.383369	−	0.923595i	$-0.625236\pi$
$271$	−12.6221	−0.766739	−0.383369	+	0.923595i	$0.625236\pi$
$272$	−1.40102	−0.0849490
$273$	0	0
$274$	−2.27871	−0.137662
$275$	−2.67555	−0.161342
$276$	0	0
$277$	−23.0650	−1.38584	−0.692920	−	0.721014i	$-0.743676\pi$
$277$	−23.0650	−1.38584	−0.692920	+	0.721014i	$0.743676\pi$
$278$	2.63621	0.158109
$279$	0	0
$280$	−9.28555	−0.554917
$281$	15.8213	0.943821	0.471911	−	0.881646i	$-0.343565\pi$
$281$	15.8213	0.943821	0.471911	+	0.881646i	$0.343565\pi$
$282$	0	0
$283$	8.98703	0.534224	0.267112	−	0.963666i	$-0.413931\pi$
$283$	8.98703	0.534224	0.267112	+	0.963666i	$0.413931\pi$
$284$	−14.3379	−0.850798
$285$	0	0
$286$	−1.75638	−0.103857
$287$	32.5146	1.91928
$288$	0	0
$289$	−15.0372	−0.884539
$290$	1.92807	0.113220
$291$	0	0
$292$	−7.71974	−0.451764
$293$	14.8801	0.869305	0.434652	−	0.900598i	$-0.356871\pi$
$293$	14.8801	0.869305	0.434652	+	0.900598i	$0.356871\pi$
$294$	0	0
$295$	−4.90854	−0.285786
$296$	10.6294	0.617824
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	2.70109	0.156208
$300$	0	0
$301$	8.51314	0.490689
$302$	−8.80220	−0.506510
$303$	0	0
$304$	−2.69068	−0.154321
$305$	−13.8659	−0.793958
$306$	0	0
$307$	19.0452	1.08697	0.543483	−	0.839420i	$-0.317105\pi$
$307$	19.0452	1.08697	0.543483	+	0.839420i	$0.317105\pi$
$308$	−11.1641	−0.636132
$309$	0	0
$310$	7.05546	0.400723
$311$	4.29069	0.243303	0.121651	−	0.992573i	$-0.461181\pi$
$311$	4.29069	0.243303	0.121651	+	0.992573i	$0.461181\pi$
$312$	0	0
$313$	7.58104	0.428506	0.214253	−	0.976778i	$-0.431268\pi$
$313$	7.58104	0.428506	0.214253	+	0.976778i	$0.431268\pi$
$314$	23.7916	1.34264
$315$	0	0
$316$	2.59929	0.146221
$317$	−9.19583	−0.516489	−0.258245	−	0.966080i	$-0.583144\pi$
$317$	−9.19583	−0.516489	−0.258245	+	0.966080i	$0.583144\pi$
$318$	0	0
$319$	2.31813	0.129790
$320$	1.96687	0.109951
$321$	0	0
$322$	17.1690	0.956788
$323$	3.76968	0.209751
$324$	0	0
$325$	0.840326	0.0466129
$326$	12.9127	0.715167
$327$	0	0
$328$	−6.88728	−0.380286
$329$	12.8896	0.710629
$330$	0	0
$331$	17.4276	0.957907	0.478953	−	0.877840i	$-0.341016\pi$
$331$	17.4276	0.957907	0.478953	+	0.877840i	$0.341016\pi$
$332$	−10.3231	−0.566555
$333$	0	0
$334$	12.1490	0.664765
$335$	−9.75224	−0.532822
$336$	0	0
$337$	−19.1104	−1.04101	−0.520504	−	0.853859i	$-0.674256\pi$
$337$	−19.1104	−1.04101	−0.520504	+	0.853859i	$0.674256\pi$
$338$	−12.4484	−0.677102
$339$	0	0
$340$	−2.75562	−0.149444
$341$	8.48283	0.459371
$342$	0	0
$343$	−39.1254	−2.11257
$344$	−1.80326	−0.0972253
$345$	0	0
$346$	4.61956	0.248349
$347$	2.29033	0.122951	0.0614756	−	0.998109i	$-0.480419\pi$
$347$	2.29033	0.122951	0.0614756	+	0.998109i	$0.480419\pi$
$348$	0	0
$349$	1.71719	0.0919192	0.0459596	−	0.998943i	$-0.485365\pi$
$349$	1.71719	0.0919192	0.0459596	+	0.998943i	$0.485365\pi$
$350$	5.34137	0.285508
$351$	0	0
$352$	2.36478	0.126043
$353$	5.27408	0.280711	0.140356	−	0.990101i	$-0.455175\pi$
$353$	5.27408	0.280711	0.140356	+	0.990101i	$0.455175\pi$
$354$	0	0
$355$	−28.2008	−1.49674
$356$	1.95400	0.103562
$357$	0	0
$358$	15.0982	0.797967
$359$	−8.22530	−0.434115	−0.217057	−	0.976159i	$-0.569646\pi$
$359$	−8.22530	−0.434115	−0.217057	+	0.976159i	$0.569646\pi$
$360$	0	0
$361$	−11.7603	−0.618961
$362$	−18.5993	−0.977558
$363$	0	0
$364$	3.50637	0.183784
$365$	−15.1837	−0.794754
$366$	0	0
$367$	10.6889	0.557957	0.278979	−	0.960297i	$-0.410004\pi$
$367$	10.6889	0.557957	0.278979	+	0.960297i	$0.410004\pi$
$368$	−3.63674	−0.189578
$369$	0	0
$370$	20.9067	1.08689
$371$	−6.85119	−0.355696
$372$	0	0
$373$	17.4257	0.902266	0.451133	−	0.892457i	$-0.351020\pi$
$373$	17.4257	0.902266	0.451133	+	0.892457i	$0.351020\pi$
$374$	−3.31310	−0.171316
$375$	0	0
$376$	−2.73029	−0.140804
$377$	−0.728069	−0.0374974
$378$	0	0
$379$	−26.0915	−1.34023	−0.670116	−	0.742257i	$-0.733756\pi$
$379$	−26.0915	−1.34023	−0.670116	+	0.742257i	$0.733756\pi$
$380$	−5.29222	−0.271485
$381$	0	0
$382$	−16.7618	−0.857607
$383$	25.7062	1.31352	0.656762	−	0.754098i	$-0.271926\pi$
$383$	25.7062	1.31352	0.656762	+	0.754098i	$0.271926\pi$
$384$	0	0
$385$	−21.9583	−1.11910
$386$	9.50979	0.484036
$387$	0	0
$388$	0.529108	0.0268614
$389$	−26.0895	−1.32279	−0.661396	−	0.750037i	$-0.730036\pi$
$389$	−26.0895	−1.32279	−0.661396	+	0.750037i	$0.730036\pi$
$390$	0	0
$391$	5.09513	0.257672
$392$	15.2876	0.772139
$393$	0	0
$394$	−3.67474	−0.185131
$395$	5.11247	0.257236
$396$	0	0
$397$	33.2276	1.66765	0.833823	−	0.552033i	$-0.186148\pi$
$397$	33.2276	1.66765	0.833823	+	0.552033i	$0.186148\pi$
$398$	−3.86258	−0.193614
$399$	0	0
$400$	−1.13141	−0.0565707
$401$	−9.03686	−0.451279	−0.225640	−	0.974211i	$-0.572447\pi$
$401$	−9.03686	−0.451279	−0.225640	+	0.974211i	$0.572447\pi$
$402$	0	0
$403$	−2.66425	−0.132716
$404$	4.78364	0.237995
$405$	0	0
$406$	−4.62783	−0.229676
$407$	25.1363	1.24596
$408$	0	0
$409$	−0.663717	−0.0328187	−0.0164093	−	0.999865i	$-0.505223\pi$
$409$	−0.663717	−0.0328187	−0.0164093	+	0.999865i	$0.505223\pi$
$410$	−13.5464	−0.669008
$411$	0	0
$412$	−1.60204	−0.0789268
$413$	11.7817	0.579739
$414$	0	0
$415$	−20.3043	−0.996697
$416$	−0.742721	−0.0364149
$417$	0	0
$418$	−6.36287	−0.311218
$419$	−23.3834	−1.14235	−0.571176	−	0.820827i	$-0.693513\pi$
$419$	−23.3834	−1.14235	−0.571176	+	0.820827i	$0.693513\pi$
$420$	0	0
$421$	8.86133	0.431875	0.215937	−	0.976407i	$-0.430719\pi$
$421$	8.86133	0.431875	0.215937	+	0.976407i	$0.430719\pi$
$422$	−2.33825	−0.113824
$423$	0	0
$424$	1.45122	0.0704777
$425$	1.58513	0.0768900
$426$	0	0
$427$	33.2815	1.61060
$428$	−14.1668	−0.684780
$429$	0	0
$430$	−3.54678	−0.171041
$431$	−11.0794	−0.533675	−0.266838	−	0.963741i	$-0.585979\pi$
$431$	−11.0794	−0.533675	−0.266838	+	0.963741i	$0.585979\pi$
$432$	0	0
$433$	−4.17583	−0.200678	−0.100339	−	0.994953i	$-0.531993\pi$
$433$	−4.17583	−0.200678	−0.100339	+	0.994953i	$0.531993\pi$
$434$	−16.9348	−0.812898
$435$	0	0
$436$	−14.8290	−0.710179
$437$	9.78530	0.468095
$438$	0	0
$439$	−2.56687	−0.122510	−0.0612550	−	0.998122i	$-0.519510\pi$
$439$	−2.56687	−0.122510	−0.0612550	+	0.998122i	$0.519510\pi$
$440$	4.65123	0.221738
$441$	0	0
$442$	1.04056	0.0494946
$443$	0.520689	0.0247387	0.0123693	−	0.999923i	$-0.496063\pi$
$443$	0.520689	0.0247387	0.0123693	+	0.999923i	$0.496063\pi$
$444$	0	0
$445$	3.84326	0.182188
$446$	−0.0107588	−0.000509444 0
$447$	0	0
$448$	−4.72097	−0.223045
$449$	−11.8948	−0.561348	−0.280674	−	0.959803i	$-0.590558\pi$
$449$	−11.8948	−0.561348	−0.280674	+	0.959803i	$0.590558\pi$
$450$	0	0
$451$	−16.2869	−0.766921
$452$	10.0435	0.472405
$453$	0	0
$454$	−13.1054	−0.615069
$455$	6.89657	0.323316
$456$	0	0
$457$	−9.72256	−0.454802	−0.227401	−	0.973801i	$-0.573023\pi$
$457$	−9.72256	−0.454802	−0.227401	+	0.973801i	$0.573023\pi$
$458$	−1.29209	−0.0603755
$459$	0	0
$460$	−7.15301	−0.333511
$461$	21.1562	0.985341	0.492670	−	0.870216i	$-0.336021\pi$
$461$	21.1562	0.985341	0.492670	+	0.870216i	$0.336021\pi$
$462$	0	0
$463$	−10.3735	−0.482098	−0.241049	−	0.970513i	$-0.577491\pi$
$463$	−10.3735	−0.482098	−0.241049	+	0.970513i	$0.577491\pi$
$464$	0.980272	0.0455080
$465$	0	0
$466$	−15.7092	−0.727716
$467$	−24.4398	−1.13094	−0.565470	−	0.824769i	$-0.691305\pi$
$467$	−24.4398	−1.13094	−0.565470	+	0.824769i	$0.691305\pi$
$468$	0	0
$469$	23.4078	1.08087
$470$	−5.37014	−0.247706
$471$	0	0
$472$	−2.49561	−0.114870
$473$	−4.26432	−0.196074
$474$	0	0
$475$	3.04427	0.139681
$476$	6.61415	0.303159
$477$	0	0
$478$	−17.2429	−0.788670
$479$	−25.4413	−1.16244	−0.581222	−	0.813745i	$-0.697425\pi$
$479$	−25.4413	−1.16244	−0.581222	+	0.813745i	$0.697425\pi$
$480$	0	0
$481$	−7.89471	−0.359968
$482$	−14.6956	−0.669366
$483$	0	0
$484$	−5.40780	−0.245809
$485$	1.04069	0.0472552
$486$	0	0
$487$	14.1757	0.642360	0.321180	−	0.947018i	$-0.395921\pi$
$487$	14.1757	0.642360	0.321180	+	0.947018i	$0.395921\pi$
$488$	−7.04971	−0.319125
$489$	0	0
$490$	30.0687	1.35836
$491$	27.1441	1.22500	0.612499	−	0.790471i	$-0.290164\pi$
$491$	27.1441	1.22500	0.612499	+	0.790471i	$0.290164\pi$
$492$	0	0
$493$	−1.37338	−0.0618537
$494$	1.99842	0.0899134
$495$	0	0
$496$	3.58715	0.161068
$497$	67.6888	3.03626
$498$	0	0
$499$	24.7065	1.10601	0.553007	−	0.833176i	$-0.313480\pi$
$499$	24.7065	1.10601	0.553007	+	0.833176i	$0.313480\pi$
$500$	−12.0597	−0.539327
$501$	0	0
$502$	14.2238	0.634841
$503$	3.30721	0.147461	0.0737305	−	0.997278i	$-0.476510\pi$
$503$	3.30721	0.147461	0.0737305	+	0.997278i	$0.476510\pi$
$504$	0	0
$505$	9.40880	0.418686
$506$	−8.60011	−0.382321
$507$	0	0
$508$	−10.1650	−0.450998
$509$	−10.6388	−0.471556	−0.235778	−	0.971807i	$-0.575764\pi$
$509$	−10.6388	−0.471556	−0.235778	+	0.971807i	$0.575764\pi$
$510$	0	0
$511$	36.4447	1.61222
$512$	1.00000	0.0441942
$513$	0	0
$514$	17.3481	0.765193
$515$	−3.15101	−0.138850
$516$	0	0
$517$	−6.45656	−0.283959
$518$	−50.1813	−2.20484
$519$	0	0
$520$	−1.46084	−0.0640620
$521$	14.5721	0.638417	0.319209	−	0.947684i	$-0.396583\pi$
$521$	14.5721	0.638417	0.319209	+	0.947684i	$0.396583\pi$
$522$	0	0
$523$	−16.0901	−0.703570	−0.351785	−	0.936081i	$-0.614425\pi$
$523$	−16.0901	−0.703570	−0.351785	+	0.936081i	$0.614425\pi$
$524$	−14.2956	−0.624505
$525$	0	0
$526$	7.74953	0.337896
$527$	−5.02565	−0.218921
$528$	0	0
$529$	−9.77411	−0.424961
$530$	2.85437	0.123986
$531$	0	0
$532$	12.7026	0.550728
$533$	5.11533	0.221569
$534$	0	0
$535$	−27.8643	−1.20468
$536$	−4.95825	−0.214164
$537$	0	0
$538$	20.9413	0.902845
$539$	36.1518	1.55717
$540$	0	0
$541$	0.563076	0.0242085	0.0121043	−	0.999927i	$-0.496147\pi$
$541$	0.563076	0.0242085	0.0121043	+	0.999927i	$0.496147\pi$
$542$	−12.6221	−0.542166
$543$	0	0
$544$	−1.40102	−0.0600680
$545$	−29.1667	−1.24936
$546$	0	0
$547$	9.14176	0.390873	0.195437	−	0.980716i	$-0.437388\pi$
$547$	9.14176	0.390873	0.195437	+	0.980716i	$0.437388\pi$
$548$	−2.27871	−0.0973417
$549$	0	0
$550$	−2.67555	−0.114086
$551$	−2.63760	−0.112365
$552$	0	0
$553$	−12.2712	−0.521823
$554$	−23.0650	−0.979937
$555$	0	0
$556$	2.63621	0.111800
$557$	−14.7514	−0.625035	−0.312518	−	0.949912i	$-0.601172\pi$
$557$	−14.7514	−0.625035	−0.312518	+	0.949912i	$0.601172\pi$
$558$	0	0
$559$	1.33932	0.0566472
$560$	−9.28555	−0.392386
$561$	0	0
$562$	15.8213	0.667382
$563$	−2.58785	−0.109065	−0.0545324	−	0.998512i	$-0.517367\pi$
$563$	−2.58785	−0.109065	−0.0545324	+	0.998512i	$0.517367\pi$
$564$	0	0
$565$	19.7542	0.831067
$566$	8.98703	0.377753
$567$	0	0
$568$	−14.3379	−0.601605
$569$	36.0992	1.51336	0.756678	−	0.653788i	$-0.226821\pi$
$569$	36.0992	1.51336	0.756678	+	0.653788i	$0.226821\pi$
$570$	0	0
$571$	11.7259	0.490714	0.245357	−	0.969433i	$-0.421095\pi$
$571$	11.7259	0.490714	0.245357	+	0.969433i	$0.421095\pi$
$572$	−1.75638	−0.0734377
$573$	0	0
$574$	32.5146	1.35713
$575$	4.11466	0.171593
$576$	0	0
$577$	7.71463	0.321164	0.160582	−	0.987022i	$-0.448663\pi$
$577$	7.71463	0.321164	0.160582	+	0.987022i	$0.448663\pi$
$578$	−15.0372	−0.625463
$579$	0	0
$580$	1.92807	0.0800587
$581$	48.7352	2.02188
$582$	0	0
$583$	3.43183	0.142132
$584$	−7.71974	−0.319445
$585$	0	0
$586$	14.8801	0.614691
$587$	31.9581	1.31905	0.659526	−	0.751681i	$-0.270757\pi$
$587$	31.9581	1.31905	0.659526	+	0.751681i	$0.270757\pi$
$588$	0	0
$589$	−9.65186	−0.397698
$590$	−4.90854	−0.202081
$591$	0	0
$592$	10.6294	0.436867
$593$	−9.78725	−0.401914	−0.200957	−	0.979600i	$-0.564405\pi$
$593$	−9.78725	−0.401914	−0.200957	+	0.979600i	$0.564405\pi$
$594$	0	0
$595$	13.0092	0.533325
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	2.70109	0.110456
$599$	1.20677	0.0493071	0.0246535	−	0.999696i	$-0.492152\pi$
$599$	1.20677	0.0493071	0.0246535	+	0.999696i	$0.492152\pi$
$600$	0	0
$601$	−39.4430	−1.60891	−0.804456	−	0.594012i	$-0.797543\pi$
$601$	−39.4430	−1.60891	−0.804456	+	0.594012i	$0.797543\pi$
$602$	8.51314	0.346970
$603$	0	0
$604$	−8.80220	−0.358156
$605$	−10.6365	−0.432433
$606$	0	0
$607$	−34.7465	−1.41032	−0.705159	−	0.709049i	$-0.749124\pi$
$607$	−34.7465	−1.41032	−0.705159	+	0.709049i	$0.749124\pi$
$608$	−2.69068	−0.109121
$609$	0	0
$610$	−13.8659	−0.561413
$611$	2.02785	0.0820380
$612$	0	0
$613$	−7.46159	−0.301371	−0.150685	−	0.988582i	$-0.548148\pi$
$613$	−7.46159	−0.301371	−0.150685	+	0.988582i	$0.548148\pi$
$614$	19.0452	0.768600
$615$	0	0
$616$	−11.1641	−0.449813
$617$	15.1694	0.610698	0.305349	−	0.952240i	$-0.401227\pi$
$617$	15.1694	0.610698	0.305349	+	0.952240i	$0.401227\pi$
$618$	0	0
$619$	29.1682	1.17237	0.586184	−	0.810178i	$-0.300630\pi$
$619$	29.1682	1.17237	0.586184	+	0.810178i	$0.300630\pi$
$620$	7.05546	0.283354
$621$	0	0
$622$	4.29069	0.172041
$623$	−9.22476	−0.369582
$624$	0	0
$625$	−18.0628	−0.722513
$626$	7.58104	0.302999
$627$	0	0
$628$	23.7916	0.949389
$629$	−14.8920	−0.593783
$630$	0	0
$631$	23.2185	0.924314	0.462157	−	0.886798i	$-0.347076\pi$
$631$	23.2185	0.924314	0.462157	+	0.886798i	$0.347076\pi$
$632$	2.59929	0.103394
$633$	0	0
$634$	−9.19583	−0.365213
$635$	−19.9932	−0.793406
$636$	0	0
$637$	−11.3544	−0.449878
$638$	2.31813	0.0917756
$639$	0	0
$640$	1.96687	0.0777474
$641$	45.2775	1.78835	0.894177	−	0.447714i	$-0.147762\pi$
$641$	45.2775	1.78835	0.894177	+	0.447714i	$0.147762\pi$
$642$	0	0
$643$	1.17400	0.0462980	0.0231490	−	0.999732i	$-0.492631\pi$
$643$	1.17400	0.0462980	0.0231490	+	0.999732i	$0.492631\pi$
$644$	17.1690	0.676552
$645$	0	0
$646$	3.76968	0.148316
$647$	−43.5948	−1.71389	−0.856945	−	0.515409i	$-0.827640\pi$
$647$	−43.5948	−1.71389	−0.856945	+	0.515409i	$0.827640\pi$
$648$	0	0
$649$	−5.90157	−0.231657
$650$	0.840326	0.0329603
$651$	0	0
$652$	12.9127	0.505699
$653$	16.5167	0.646350	0.323175	−	0.946339i	$-0.395250\pi$
$653$	16.5167	0.646350	0.323175	+	0.946339i	$0.395250\pi$
$654$	0	0
$655$	−28.1175	−1.09864
$656$	−6.88728	−0.268903
$657$	0	0
$658$	12.8896	0.502491
$659$	−30.3444	−1.18205	−0.591024	−	0.806654i	$-0.701276\pi$
$659$	−30.3444	−1.18205	−0.591024	+	0.806654i	$0.701276\pi$
$660$	0	0
$661$	45.5678	1.77238	0.886192	−	0.463319i	$-0.153341\pi$
$661$	45.5678	1.77238	0.886192	+	0.463319i	$0.153341\pi$
$662$	17.4276	0.677342
$663$	0	0
$664$	−10.3231	−0.400615
$665$	24.9844	0.968854
$666$	0	0
$667$	−3.56499	−0.138037
$668$	12.1490	0.470060
$669$	0	0
$670$	−9.75224	−0.376762
$671$	−16.6710	−0.643578
$672$	0	0
$673$	23.6770	0.912680	0.456340	−	0.889805i	$-0.349160\pi$
$673$	23.6770	0.912680	0.456340	+	0.889805i	$0.349160\pi$
$674$	−19.1104	−0.736104
$675$	0	0
$676$	−12.4484	−0.478783
$677$	31.7520	1.22033	0.610164	−	0.792276i	$-0.291104\pi$
$677$	31.7520	1.22033	0.610164	+	0.792276i	$0.291104\pi$
$678$	0	0
$679$	−2.49790	−0.0958608
$680$	−2.75562	−0.105673
$681$	0	0
$682$	8.48283	0.324824
$683$	19.1101	0.731228	0.365614	−	0.930767i	$-0.380859\pi$
$683$	19.1101	0.731228	0.365614	+	0.930767i	$0.380859\pi$
$684$	0	0
$685$	−4.48193	−0.171246
$686$	−39.1254	−1.49381
$687$	0	0
$688$	−1.80326	−0.0687487
$689$	−1.07786	−0.0410630
$690$	0	0
$691$	−21.2510	−0.808426	−0.404213	−	0.914665i	$-0.632454\pi$
$691$	−21.2510	−0.808426	−0.404213	+	0.914665i	$0.632454\pi$
$692$	4.61956	0.175609
$693$	0	0
$694$	2.29033	0.0869396
$695$	5.18509	0.196682
$696$	0	0
$697$	9.64918	0.365489
$698$	1.71719	0.0649967
$699$	0	0
$700$	5.34137	0.201885
$701$	41.6697	1.57384	0.786922	−	0.617053i	$-0.211673\pi$
$701$	41.6697	1.57384	0.786922	+	0.617053i	$0.211673\pi$
$702$	0	0
$703$	−28.6004	−1.07868
$704$	2.36478	0.0891261
$705$	0	0
$706$	5.27408	0.198493
$707$	−22.5834	−0.849336
$708$	0	0
$709$	13.8554	0.520349	0.260174	−	0.965562i	$-0.416220\pi$
$709$	13.8554	0.520349	0.260174	+	0.965562i	$0.416220\pi$
$710$	−28.2008	−1.05836
$711$	0	0
$712$	1.95400	0.0732291
$713$	−13.0455	−0.488559
$714$	0	0
$715$	−3.45456	−0.129193
$716$	15.0982	0.564248
$717$	0	0
$718$	−8.22530	−0.306966
$719$	1.00407	0.0374455	0.0187228	−	0.999825i	$-0.494040\pi$
$719$	1.00407	0.0374455	0.0187228	+	0.999825i	$0.494040\pi$
$720$	0	0
$721$	7.56318	0.281668
$722$	−11.7603	−0.437671
$723$	0	0
$724$	−18.5993	−0.691238
$725$	−1.10909	−0.0411907
$726$	0	0
$727$	0.418696	0.0155286	0.00776429	−	0.999970i	$-0.497529\pi$
$727$	0.418696	0.0155286	0.00776429	+	0.999970i	$0.497529\pi$
$728$	3.50637	0.129955
$729$	0	0
$730$	−15.1837	−0.561976
$731$	2.52640	0.0934421
$732$	0	0
$733$	−22.0840	−0.815690	−0.407845	−	0.913051i	$-0.633720\pi$
$733$	−22.0840	−0.815690	−0.407845	+	0.913051i	$0.633720\pi$
$734$	10.6889	0.394535
$735$	0	0
$736$	−3.63674	−0.134052
$737$	−11.7252	−0.431903
$738$	0	0
$739$	−16.6984	−0.614259	−0.307130	−	0.951668i	$-0.599369\pi$
$739$	−16.6984	−0.614259	−0.307130	+	0.951668i	$0.599369\pi$
$740$	20.9067	0.768547
$741$	0	0
$742$	−6.85119	−0.251515
$743$	15.4316	0.566132	0.283066	−	0.959100i	$-0.408648\pi$
$743$	15.4316	0.566132	0.283066	+	0.959100i	$0.408648\pi$
$744$	0	0
$745$	−1.96687	−0.0720606
$746$	17.4257	0.637999
$747$	0	0
$748$	−3.31310	−0.121139
$749$	66.8812	2.44379
$750$	0	0
$751$	−24.8441	−0.906573	−0.453287	−	0.891365i	$-0.649749\pi$
$751$	−24.8441	−0.906573	−0.453287	+	0.891365i	$0.649749\pi$
$752$	−2.73029	−0.0995636
$753$	0	0
$754$	−0.728069	−0.0265147
$755$	−17.3128	−0.630077
$756$	0	0
$757$	−3.28099	−0.119250	−0.0596248	−	0.998221i	$-0.518990\pi$
$757$	−3.28099	−0.119250	−0.0596248	+	0.998221i	$0.518990\pi$
$758$	−26.0915	−0.947687
$759$	0	0
$760$	−5.29222	−0.191969
$761$	11.8041	0.427898	0.213949	−	0.976845i	$-0.431367\pi$
$761$	11.8041	0.427898	0.213949	+	0.976845i	$0.431367\pi$
$762$	0	0
$763$	70.0071	2.53443
$764$	−16.7618	−0.606420
$765$	0	0
$766$	25.7062	0.928801
$767$	1.85354	0.0669275
$768$	0	0
$769$	45.5064	1.64100	0.820502	−	0.571644i	$-0.193694\pi$
$769$	45.5064	1.64100	0.820502	+	0.571644i	$0.193694\pi$
$770$	−21.9583	−0.791322
$771$	0	0
$772$	9.50979	0.342265
$773$	−32.3417	−1.16325	−0.581625	−	0.813457i	$-0.697583\pi$
$773$	−32.3417	−1.16325	−0.581625	+	0.813457i	$0.697583\pi$
$774$	0	0
$775$	−4.05855	−0.145788
$776$	0.529108	0.0189939
$777$	0	0
$778$	−26.0895	−0.935355
$779$	18.5314	0.663958
$780$	0	0
$781$	−33.9060	−1.21325
$782$	5.09513	0.182201
$783$	0	0
$784$	15.2876	0.545985
$785$	46.7951	1.67019
$786$	0	0
$787$	26.6379	0.949538	0.474769	−	0.880110i	$-0.342532\pi$
$787$	26.6379	0.949538	0.474769	+	0.880110i	$0.342532\pi$
$788$	−3.67474	−0.130907
$789$	0	0
$790$	5.11247	0.181893
$791$	−47.4149	−1.68588
$792$	0	0
$793$	5.23597	0.185935
$794$	33.2276	1.17920
$795$	0	0
$796$	−3.86258	−0.136906
$797$	24.6220	0.872156	0.436078	−	0.899909i	$-0.356367\pi$
$797$	24.6220	0.872156	0.436078	+	0.899909i	$0.356367\pi$
$798$	0	0
$799$	3.82518	0.135325
$800$	−1.13141	−0.0400015
$801$	0	0
$802$	−9.03686	−0.319103
$803$	−18.2555	−0.644223
$804$	0	0
$805$	33.7691	1.19021
$806$	−2.66425	−0.0938443
$807$	0	0
$808$	4.78364	0.168288
$809$	−20.3222	−0.714492	−0.357246	−	0.934010i	$-0.616284\pi$
$809$	−20.3222	−0.714492	−0.357246	+	0.934010i	$0.616284\pi$
$810$	0	0
$811$	−25.0866	−0.880910	−0.440455	−	0.897775i	$-0.645183\pi$
$811$	−25.0866	−0.880910	−0.440455	+	0.897775i	$0.645183\pi$
$812$	−4.62783	−0.162405
$813$	0	0
$814$	25.1363	0.881028
$815$	25.3976	0.889638
$816$	0	0
$817$	4.85200	0.169750
$818$	−0.663717	−0.0232063
$819$	0	0
$820$	−13.5464	−0.473060
$821$	27.8602	0.972329	0.486165	−	0.873867i	$-0.338396\pi$
$821$	27.8602	0.972329	0.486165	+	0.873867i	$0.338396\pi$
$822$	0	0
$823$	2.46781	0.0860226	0.0430113	−	0.999075i	$-0.486305\pi$
$823$	2.46781	0.0860226	0.0430113	+	0.999075i	$0.486305\pi$
$824$	−1.60204	−0.0558097
$825$	0	0
$826$	11.7817	0.409937
$827$	29.2732	1.01793	0.508964	−	0.860788i	$-0.330028\pi$
$827$	29.2732	1.01793	0.508964	+	0.860788i	$0.330028\pi$
$828$	0	0
$829$	−4.23609	−0.147125	−0.0735627	−	0.997291i	$-0.523437\pi$
$829$	−4.23609	−0.147125	−0.0735627	+	0.997291i	$0.523437\pi$
$830$	−20.3043	−0.704771
$831$	0	0
$832$	−0.742721	−0.0257492
$833$	−21.4181	−0.742094
$834$	0	0
$835$	23.8956	0.826940
$836$	−6.36287	−0.220064
$837$	0	0
$838$	−23.3834	−0.807765
$839$	−28.5329	−0.985066	−0.492533	−	0.870294i	$-0.663929\pi$
$839$	−28.5329	−0.985066	−0.492533	+	0.870294i	$0.663929\pi$
$840$	0	0
$841$	−28.0391	−0.966864
$842$	8.86133	0.305381
$843$	0	0
$844$	−2.33825	−0.0804857
$845$	−24.4843	−0.842287
$846$	0	0
$847$	25.5301	0.877224
$848$	1.45122	0.0498352
$849$	0	0
$850$	1.58513	0.0543695
$851$	−38.6565	−1.32513
$852$	0	0
$853$	−11.3607	−0.388983	−0.194491	−	0.980904i	$-0.562306\pi$
$853$	−11.3607	−0.388983	−0.194491	+	0.980904i	$0.562306\pi$
$854$	33.2815	1.13887
$855$	0	0
$856$	−14.1668	−0.484212
$857$	54.4857	1.86120	0.930599	−	0.366041i	$-0.119287\pi$
$857$	54.4857	1.86120	0.930599	+	0.366041i	$0.119287\pi$
$858$	0	0
$859$	−44.3209	−1.51221	−0.756105	−	0.654450i	$-0.772900\pi$
$859$	−44.3209	−1.51221	−0.756105	+	0.654450i	$0.772900\pi$
$860$	−3.54678	−0.120944
$861$	0	0
$862$	−11.0794	−0.377366
$863$	−39.7757	−1.35398	−0.676990	−	0.735992i	$-0.736716\pi$
$863$	−39.7757	−1.35398	−0.676990	+	0.735992i	$0.736716\pi$
$864$	0	0
$865$	9.08609	0.308936
$866$	−4.17583	−0.141901
$867$	0	0
$868$	−16.9348	−0.574805
$869$	6.14675	0.208514
$870$	0	0
$871$	3.68260	0.124780
$872$	−14.8290	−0.502172
$873$	0	0
$874$	9.78530	0.330993
$875$	56.9335	1.92470
$876$	0	0
$877$	46.3619	1.56553	0.782765	−	0.622318i	$-0.213809\pi$
$877$	46.3619	1.56553	0.782765	+	0.622318i	$0.213809\pi$
$878$	−2.56687	−0.0866276
$879$	0	0
$880$	4.65123	0.156793
$881$	14.5453	0.490045	0.245023	−	0.969517i	$-0.421205\pi$
$881$	14.5453	0.490045	0.245023	+	0.969517i	$0.421205\pi$
$882$	0	0
$883$	−57.5300	−1.93604	−0.968019	−	0.250877i	$-0.919281\pi$
$883$	−57.5300	−1.93604	−0.968019	+	0.250877i	$0.919281\pi$
$884$	1.04056	0.0349980
$885$	0	0
$886$	0.520689	0.0174929
$887$	27.8500	0.935113	0.467556	−	0.883963i	$-0.345134\pi$
$887$	27.8500	0.935113	0.467556	+	0.883963i	$0.345134\pi$
$888$	0	0
$889$	47.9886	1.60948
$890$	3.84326	0.128826
$891$	0	0
$892$	−0.0107588	−0.000360232 0
$893$	7.34635	0.245836
$894$	0	0
$895$	29.6963	0.992638
$896$	−4.72097	−0.157717
$897$	0	0
$898$	−11.8948	−0.396933
$899$	3.51638	0.117278
$900$	0	0
$901$	−2.03319	−0.0677353
$902$	−16.2869	−0.542295
$903$	0	0
$904$	10.0435	0.334041
$905$	−36.5824	−1.21604
$906$	0	0
$907$	59.4138	1.97280	0.986401	−	0.164356i	$-0.0525547\pi$
$907$	59.4138	1.97280	0.986401	+	0.164356i	$0.0525547\pi$
$908$	−13.1054	−0.434919
$909$	0	0
$910$	6.89657	0.228619
$911$	−35.8757	−1.18861	−0.594307	−	0.804238i	$-0.702574\pi$
$911$	−35.8757	−1.18861	−0.594307	+	0.804238i	$0.702574\pi$
$912$	0	0
$913$	−24.4119	−0.807917
$914$	−9.72256	−0.321594
$915$	0	0
$916$	−1.29209	−0.0426919
$917$	67.4889	2.22868
$918$	0	0
$919$	−35.9219	−1.18495	−0.592477	−	0.805588i	$-0.701850\pi$
$919$	−35.9219	−1.18495	−0.592477	+	0.805588i	$0.701850\pi$
$920$	−7.15301	−0.235828
$921$	0	0
$922$	21.1562	0.696741
$923$	10.6491	0.350518
$924$	0	0
$925$	−12.0263	−0.395422
$926$	−10.3735	−0.340895
$927$	0	0
$928$	0.980272	0.0321790
$929$	12.3339	0.404662	0.202331	−	0.979317i	$-0.435148\pi$
$929$	12.3339	0.404662	0.202331	+	0.979317i	$0.435148\pi$
$930$	0	0
$931$	−41.1339	−1.34811
$932$	−15.7092	−0.514573
$933$	0	0
$934$	−24.4398	−0.799695
$935$	−6.51644	−0.213110
$936$	0	0
$937$	12.2201	0.399212	0.199606	−	0.979876i	$-0.436034\pi$
$937$	12.2201	0.399212	0.199606	+	0.979876i	$0.436034\pi$
$938$	23.4078	0.764290
$939$	0	0
$940$	−5.37014	−0.175155
$941$	−43.5255	−1.41889	−0.709446	−	0.704760i	$-0.751055\pi$
$941$	−43.5255	−1.41889	−0.709446	+	0.704760i	$0.751055\pi$
$942$	0	0
$943$	25.0472	0.815651
$944$	−2.49561	−0.0812251
$945$	0	0
$946$	−4.26432	−0.138645
$947$	20.9375	0.680377	0.340188	−	0.940357i	$-0.389509\pi$
$947$	20.9375	0.680377	0.340188	+	0.940357i	$0.389509\pi$
$948$	0	0
$949$	5.73362	0.186121
$950$	3.04427	0.0987692
$951$	0	0
$952$	6.61415	0.214366
$953$	−26.0035	−0.842336	−0.421168	−	0.906983i	$-0.638380\pi$
$953$	−26.0035	−0.842336	−0.421168	+	0.906983i	$0.638380\pi$
$954$	0	0
$955$	−32.9683	−1.06683
$956$	−17.2429	−0.557674
$957$	0	0
$958$	−25.4413	−0.821973
$959$	10.7577	0.347385
$960$	0	0
$961$	−18.1324	−0.584915
$962$	−7.89471	−0.254536
$963$	0	0
$964$	−14.6956	−0.473313
$965$	18.7045	0.602120
$966$	0	0
$967$	−32.4654	−1.04402	−0.522008	−	0.852941i	$-0.674817\pi$
$967$	−32.4654	−1.04402	−0.522008	+	0.852941i	$0.674817\pi$
$968$	−5.40780	−0.173813
$969$	0	0
$970$	1.04069	0.0334145
$971$	56.3554	1.80853	0.904265	−	0.426971i	$-0.140419\pi$
$971$	56.3554	1.80853	0.904265	+	0.426971i	$0.140419\pi$
$972$	0	0
$973$	−12.4455	−0.398984
$974$	14.1757	0.454217
$975$	0	0
$976$	−7.04971	−0.225656
$977$	−22.5573	−0.721673	−0.360837	−	0.932629i	$-0.617509\pi$
$977$	−22.5573	−0.721673	−0.360837	+	0.932629i	$0.617509\pi$
$978$	0	0
$979$	4.62078	0.147681
$980$	30.0687	0.960509
$981$	0	0
$982$	27.1441	0.866204
$983$	−2.66000	−0.0848407	−0.0424204	−	0.999100i	$-0.513507\pi$
$983$	−2.66000	−0.0848407	−0.0424204	+	0.999100i	$0.513507\pi$
$984$	0	0
$985$	−7.22775	−0.230295
$986$	−1.37338	−0.0437372
$987$	0	0
$988$	1.99842	0.0635784
$989$	6.55800	0.208532
$990$	0	0
$991$	−18.0611	−0.573730	−0.286865	−	0.957971i	$-0.592613\pi$
$991$	−18.0611	−0.573730	−0.286865	+	0.957971i	$0.592613\pi$
$992$	3.58715	0.113892
$993$	0	0
$994$	67.6888	2.14696
$995$	−7.59721	−0.240848
$996$	0	0
$997$	48.9423	1.55002	0.775009	−	0.631950i	$-0.217745\pi$
$997$	48.9423	1.55002	0.775009	+	0.631950i	$0.217745\pi$
$998$	24.7065	0.782070
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.h.1.8	yes	9
3.2	odd	2		8046.2.a.g.1.2	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.g.1.2	✓	9	3.2	odd	2
8046.2.a.h.1.8	yes	9	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.96687 q^{5} -4.72097 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.96687 q^{5} -4.72097 q^{7} +1.00000 q^{8} +1.96687 q^{10} +2.36478 q^{11} -0.742721 q^{13} -4.72097 q^{14} +1.00000 q^{16} -1.40102 q^{17} -2.69068 q^{19} +1.96687 q^{20} +2.36478 q^{22} -3.63674 q^{23} -1.13141 q^{25} -0.742721 q^{26} -4.72097 q^{28} +0.980272 q^{29} +3.58715 q^{31} +1.00000 q^{32} -1.40102 q^{34} -9.28555 q^{35} +10.6294 q^{37} -2.69068 q^{38} +1.96687 q^{40} -6.88728 q^{41} -1.80326 q^{43} +2.36478 q^{44} -3.63674 q^{46} -2.73029 q^{47} +15.2876 q^{49} -1.13141 q^{50} -0.742721 q^{52} +1.45122 q^{53} +4.65123 q^{55} -4.72097 q^{56} +0.980272 q^{58} -2.49561 q^{59} -7.04971 q^{61} +3.58715 q^{62} +1.00000 q^{64} -1.46084 q^{65} -4.95825 q^{67} -1.40102 q^{68} -9.28555 q^{70} -14.3379 q^{71} -7.71974 q^{73} +10.6294 q^{74} -2.69068 q^{76} -11.1641 q^{77} +2.59929 q^{79} +1.96687 q^{80} -6.88728 q^{82} -10.3231 q^{83} -2.75562 q^{85} -1.80326 q^{86} +2.36478 q^{88} +1.95400 q^{89} +3.50637 q^{91} -3.63674 q^{92} -2.73029 q^{94} -5.29222 q^{95} +0.529108 q^{97} +15.2876 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8}+O(q^{10}) \) 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^5 - 4 * q^7 + 9 * q^8	\( 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8} - 4 q^{10} - 4 q^{11} - 8 q^{13} - 4 q^{14} + 9 q^{16} - q^{17} - 10 q^{19} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 3 q^{25} - 8 q^{26} - 4 q^{28} - 4 q^{29} - 17 q^{31} + 9 q^{32} - q^{34} - 10 q^{35} - 11 q^{37} - 10 q^{38} - 4 q^{40} - 16 q^{43} - 4 q^{44} - 8 q^{46} - 7 q^{47} - 5 q^{49} - 3 q^{50} - 8 q^{52} - 12 q^{53} - 23 q^{55} - 4 q^{56} - 4 q^{58} - 6 q^{59} - 13 q^{61} - 17 q^{62} + 9 q^{64} + 24 q^{65} - 14 q^{67} - q^{68} - 10 q^{70} - 30 q^{71} - 12 q^{73} - 11 q^{74} - 10 q^{76} - 12 q^{77} - 35 q^{79} - 4 q^{80} - 5 q^{83} - 27 q^{85} - 16 q^{86} - 4 q^{88} - 23 q^{89} - 28 q^{91} - 8 q^{92} - 7 q^{94} - 32 q^{95} - 21 q^{97} - 5 q^{98}+O(q^{100}) \) 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^5 - 4 * q^7 + 9 * q^8 - 4 * q^10 - 4 * q^11 - 8 * q^13 - 4 * q^14 + 9 * q^16 - q^17 - 10 * q^19 - 4 * q^20 - 4 * q^22 - 8 * q^23 - 3 * q^25 - 8 * q^26 - 4 * q^28 - 4 * q^29 - 17 * q^31 + 9 * q^32 - q^34 - 10 * q^35 - 11 * q^37 - 10 * q^38 - 4 * q^40 - 16 * q^43 - 4 * q^44 - 8 * q^46 - 7 * q^47 - 5 * q^49 - 3 * q^50 - 8 * q^52 - 12 * q^53 - 23 * q^55 - 4 * q^56 - 4 * q^58 - 6 * q^59 - 13 * q^61 - 17 * q^62 + 9 * q^64 + 24 * q^65 - 14 * q^67 - q^68 - 10 * q^70 - 30 * q^71 - 12 * q^73 - 11 * q^74 - 10 * q^76 - 12 * q^77 - 35 * q^79 - 4 * q^80 - 5 * q^83 - 27 * q^85 - 16 * q^86 - 4 * q^88 - 23 * q^89 - 28 * q^91 - 8 * q^92 - 7 * q^94 - 32 * q^95 - 21 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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