LMFDB - Embedded newform 9016.2.a.br.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9016,2,Mod(1,9016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("9016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9016 = 2^{3} \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9016.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:	$71.9931224624$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 4 x^{10} - 14 x^{9} + 61 x^{8} + 71 x^{7} - 343 x^{6} - 152 x^{5} + 867 x^{4} + 102 x^{3} + \cdots + 243 $ x^11 - 4x^10 - 14x^9 + 61x^8 + 71x^7 - 343x^6 - 152x^5 + 867x^4 + 102x^3 - 918x^2 + 35x + 243
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1288)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$1.99086$ of defining polynomial
Character	$\chi$	$=$	9016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.99086 q^{3} +1.57733 q^{5} +0.963536 q^{9} +O(q^{10})$	$q+1.99086 q^{3} +1.57733 q^{5} +0.963536 q^{9} -5.45206 q^{11} +1.27655 q^{13} +3.14026 q^{15} -3.10267 q^{17} +1.80039 q^{19} -1.00000 q^{23} -2.51202 q^{25} -4.05432 q^{27} +8.86478 q^{29} +5.60436 q^{31} -10.8543 q^{33} +0.200405 q^{37} +2.54144 q^{39} -0.263965 q^{41} +11.2203 q^{43} +1.51982 q^{45} +8.45367 q^{47} -6.17699 q^{51} +13.4123 q^{53} -8.59972 q^{55} +3.58434 q^{57} -5.28941 q^{59} +1.84750 q^{61} +2.01355 q^{65} -3.22727 q^{67} -1.99086 q^{69} +6.24792 q^{71} -2.95569 q^{73} -5.00109 q^{75} +9.32631 q^{79} -10.9622 q^{81} +12.2378 q^{83} -4.89394 q^{85} +17.6486 q^{87} -13.1025 q^{89} +11.1575 q^{93} +2.83982 q^{95} +6.66077 q^{97} -5.25326 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 4 q^{3} + 3 q^{5} + 11 q^{9}+O(q^{10}) $ 11 * q + 4 * q^3 + 3 * q^5 + 11 * q^9	$ 11 q + 4 q^{3} + 3 q^{5} + 11 q^{9} - 13 q^{13} + 7 q^{17} + 8 q^{19} - 11 q^{23} + 6 q^{25} + 25 q^{27} - 3 q^{29} + 12 q^{31} - 2 q^{33} - q^{37} - 21 q^{39} + 12 q^{41} + 9 q^{43} + 19 q^{45} + 17 q^{47} + 19 q^{51} - 5 q^{53} + 21 q^{55} + 11 q^{57} + 33 q^{59} - 15 q^{61} - 9 q^{65} - 5 q^{67} - 4 q^{69} - 9 q^{71} + 5 q^{73} + 44 q^{75} + 11 q^{79} - 13 q^{81} + 51 q^{83} + 33 q^{85} + 4 q^{87} + 26 q^{89} + 6 q^{93} - 19 q^{95} + 21 q^{97} + 15 q^{99}+O(q^{100}) $ 11 * q + 4 * q^3 + 3 * q^5 + 11 * q^9 - 13 * q^13 + 7 * q^17 + 8 * q^19 - 11 * q^23 + 6 * q^25 + 25 * q^27 - 3 * q^29 + 12 * q^31 - 2 * q^33 - q^37 - 21 * q^39 + 12 * q^41 + 9 * q^43 + 19 * q^45 + 17 * q^47 + 19 * q^51 - 5 * q^53 + 21 * q^55 + 11 * q^57 + 33 * q^59 - 15 * q^61 - 9 * q^65 - 5 * q^67 - 4 * q^69 - 9 * q^71 + 5 * q^73 + 44 * q^75 + 11 * q^79 - 13 * q^81 + 51 * q^83 + 33 * q^85 + 4 * q^87 + 26 * q^89 + 6 * q^93 - 19 * q^95 + 21 * q^97 + 15 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.99086	1.14943	0.574713	−	0.818355i	$-0.305114\pi$
$3$	1.99086	1.14943	0.574713	+	0.818355i	$0.305114\pi$
$4$	0	0
$5$	1.57733	0.705405	0.352703	−	0.935736i	$-0.385263\pi$
$5$	1.57733	0.705405	0.352703	+	0.935736i	$0.385263\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.963536	0.321179
$10$	0	0
$11$	−5.45206	−1.64386	−0.821929	−	0.569589i	$-0.807102\pi$
$11$	−5.45206	−1.64386	−0.821929	+	0.569589i	$0.807102\pi$
$12$	0	0
$13$	1.27655	0.354052	0.177026	−	0.984206i	$-0.443352\pi$
$13$	1.27655	0.354052	0.177026	+	0.984206i	$0.443352\pi$
$14$	0	0
$15$	3.14026	0.810811
$16$	0	0
$17$	−3.10267	−0.752508	−0.376254	−	0.926517i	$-0.622788\pi$
$17$	−3.10267	−0.752508	−0.376254	+	0.926517i	$0.622788\pi$
$18$	0	0
$19$	1.80039	0.413038	0.206519	−	0.978443i	$-0.433786\pi$
$19$	1.80039	0.413038	0.206519	+	0.978443i	$0.433786\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−2.51202	−0.502404
$26$	0	0
$27$	−4.05432	−0.780254
$28$	0	0
$29$	8.86478	1.64615	0.823074	−	0.567934i	$-0.192257\pi$
$29$	8.86478	1.64615	0.823074	+	0.567934i	$0.192257\pi$
$30$	0	0
$31$	5.60436	1.00657	0.503286	−	0.864120i	$-0.332124\pi$
$31$	5.60436	1.00657	0.503286	+	0.864120i	$0.332124\pi$
$32$	0	0
$33$	−10.8543	−1.88949
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.200405	0.0329464	0.0164732	−	0.999864i	$-0.494756\pi$
$37$	0.200405	0.0329464	0.0164732	+	0.999864i	$0.494756\pi$
$38$	0	0
$39$	2.54144	0.406956
$40$	0	0
$41$	−0.263965	−0.0412244	−0.0206122	−	0.999788i	$-0.506562\pi$
$41$	−0.263965	−0.0412244	−0.0206122	+	0.999788i	$0.506562\pi$
$42$	0	0
$43$	11.2203	1.71109	0.855543	−	0.517732i	$-0.173224\pi$
$43$	11.2203	1.71109	0.855543	+	0.517732i	$0.173224\pi$
$44$	0	0
$45$	1.51982	0.226561
$46$	0	0
$47$	8.45367	1.23309	0.616547	−	0.787318i	$-0.288531\pi$
$47$	8.45367	1.23309	0.616547	+	0.787318i	$0.288531\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−6.17699	−0.864951
$52$	0	0
$53$	13.4123	1.84232	0.921159	−	0.389186i	$-0.127244\pi$
$53$	13.4123	1.84232	0.921159	+	0.389186i	$0.127244\pi$
$54$	0	0
$55$	−8.59972	−1.15959
$56$	0	0
$57$	3.58434	0.474757
$58$	0	0
$59$	−5.28941	−0.688623	−0.344311	−	0.938856i	$-0.611888\pi$
$59$	−5.28941	−0.688623	−0.344311	+	0.938856i	$0.611888\pi$
$60$	0	0
$61$	1.84750	0.236548	0.118274	−	0.992981i	$-0.462264\pi$
$61$	1.84750	0.236548	0.118274	+	0.992981i	$0.462264\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.01355	0.249750
$66$	0	0
$67$	−3.22727	−0.394273	−0.197137	−	0.980376i	$-0.563164\pi$
$67$	−3.22727	−0.394273	−0.197137	+	0.980376i	$0.563164\pi$
$68$	0	0
$69$	−1.99086	−0.239672
$70$	0	0
$71$	6.24792	0.741492	0.370746	−	0.928734i	$-0.379102\pi$
$71$	6.24792	0.741492	0.370746	+	0.928734i	$0.379102\pi$
$72$	0	0
$73$	−2.95569	−0.345938	−0.172969	−	0.984927i	$-0.555336\pi$
$73$	−2.95569	−0.345938	−0.172969	+	0.984927i	$0.555336\pi$
$74$	0	0
$75$	−5.00109	−0.577476
$76$	0	0
$77$	0	0
$78$	0	0
$79$	9.32631	1.04929	0.524646	−	0.851320i	$-0.324198\pi$
$79$	9.32631	1.04929	0.524646	+	0.851320i	$0.324198\pi$
$80$	0	0
$81$	−10.9622	−1.21802
$82$	0	0
$83$	12.2378	1.34327	0.671637	−	0.740881i	$-0.265592\pi$
$83$	12.2378	1.34327	0.671637	+	0.740881i	$0.265592\pi$
$84$	0	0
$85$	−4.89394	−0.530823
$86$	0	0
$87$	17.6486	1.89212
$88$	0	0
$89$	−13.1025	−1.38887	−0.694434	−	0.719557i	$-0.744345\pi$
$89$	−13.1025	−1.38887	−0.694434	+	0.719557i	$0.744345\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	11.1575	1.15698
$94$	0	0
$95$	2.83982	0.291359
$96$	0	0
$97$	6.66077	0.676299	0.338150	−	0.941092i	$-0.390199\pi$
$97$	6.66077	0.676299	0.338150	+	0.941092i	$0.390199\pi$
$98$	0	0
$99$	−5.25326	−0.527973
$100$	0	0
$101$	16.9028	1.68189	0.840944	−	0.541122i	$-0.182000\pi$
$101$	16.9028	1.68189	0.840944	+	0.541122i	$0.182000\pi$
$102$	0	0
$103$	11.5797	1.14098	0.570489	−	0.821305i	$-0.306754\pi$
$103$	11.5797	1.14098	0.570489	+	0.821305i	$0.306754\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.4068	1.68278	0.841389	−	0.540430i	$-0.181738\pi$
$107$	17.4068	1.68278	0.841389	+	0.540430i	$0.181738\pi$
$108$	0	0
$109$	10.4530	1.00122	0.500608	−	0.865674i	$-0.333110\pi$
$109$	10.4530	1.00122	0.500608	+	0.865674i	$0.333110\pi$
$110$	0	0
$111$	0.398980	0.0378695
$112$	0	0
$113$	−17.3127	−1.62865	−0.814323	−	0.580412i	$-0.802892\pi$
$113$	−17.3127	−1.62865	−0.814323	+	0.580412i	$0.802892\pi$
$114$	0	0
$115$	−1.57733	−0.147087
$116$	0	0
$117$	1.23000	0.113714
$118$	0	0
$119$	0	0
$120$	0	0
$121$	18.7250	1.70227
$122$	0	0
$123$	−0.525518	−0.0473843
$124$	0	0
$125$	−11.8490	−1.05980
$126$	0	0
$127$	1.77902	0.157862	0.0789310	−	0.996880i	$-0.474849\pi$
$127$	1.77902	0.157862	0.0789310	+	0.996880i	$0.474849\pi$
$128$	0	0
$129$	22.3382	1.96677
$130$	0	0
$131$	9.11487	0.796370	0.398185	−	0.917305i	$-0.369640\pi$
$131$	9.11487	0.796370	0.398185	+	0.917305i	$0.369640\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−6.39502	−0.550395
$136$	0	0
$137$	−0.423265	−0.0361620	−0.0180810	−	0.999837i	$-0.505756\pi$
$137$	−0.423265	−0.0361620	−0.0180810	+	0.999837i	$0.505756\pi$
$138$	0	0
$139$	−13.6537	−1.15809	−0.579044	−	0.815296i	$-0.696574\pi$
$139$	−13.6537	−1.15809	−0.579044	+	0.815296i	$0.696574\pi$
$140$	0	0
$141$	16.8301	1.41735
$142$	0	0
$143$	−6.95984	−0.582011
$144$	0	0
$145$	13.9827	1.16120
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.2179	1.16477	0.582386	−	0.812912i	$-0.302119\pi$
$149$	14.2179	1.16477	0.582386	+	0.812912i	$0.302119\pi$
$150$	0	0
$151$	−2.79445	−0.227409	−0.113705	−	0.993515i	$-0.536272\pi$
$151$	−2.79445	−0.227409	−0.113705	+	0.993515i	$0.536272\pi$
$152$	0	0
$153$	−2.98953	−0.241689
$154$	0	0
$155$	8.83994	0.710041
$156$	0	0
$157$	−13.5411	−1.08070	−0.540349	−	0.841441i	$-0.681708\pi$
$157$	−13.5411	−1.08070	−0.540349	+	0.841441i	$0.681708\pi$
$158$	0	0
$159$	26.7020	2.11761
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0.120993	0.00947687	0.00473844	−	0.999989i	$-0.498492\pi$
$163$	0.120993	0.00947687	0.00473844	+	0.999989i	$0.498492\pi$
$164$	0	0
$165$	−17.1209	−1.33286
$166$	0	0
$167$	13.1831	1.02014	0.510068	−	0.860134i	$-0.329620\pi$
$167$	13.1831	1.02014	0.510068	+	0.860134i	$0.329620\pi$
$168$	0	0
$169$	−11.3704	−0.874647
$170$	0	0
$171$	1.73474	0.132659
$172$	0	0
$173$	−12.6980	−0.965409	−0.482704	−	0.875783i	$-0.660345\pi$
$173$	−12.6980	−0.965409	−0.482704	+	0.875783i	$0.660345\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−10.5305	−0.791520
$178$	0	0
$179$	16.5116	1.23413	0.617067	−	0.786910i	$-0.288321\pi$
$179$	16.5116	1.23413	0.617067	+	0.786910i	$0.288321\pi$
$180$	0	0
$181$	6.28327	0.467032	0.233516	−	0.972353i	$-0.424977\pi$
$181$	6.28327	0.467032	0.233516	+	0.972353i	$0.424977\pi$
$182$	0	0
$183$	3.67812	0.271895
$184$	0	0
$185$	0.316106	0.0232406
$186$	0	0
$187$	16.9159	1.23702
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.76942	−0.345103	−0.172552	−	0.985000i	$-0.555201\pi$
$191$	−4.76942	−0.345103	−0.172552	+	0.985000i	$0.555201\pi$
$192$	0	0
$193$	−4.56772	−0.328791	−0.164396	−	0.986394i	$-0.552567\pi$
$193$	−4.56772	−0.328791	−0.164396	+	0.986394i	$0.552567\pi$
$194$	0	0
$195$	4.00870	0.287069
$196$	0	0
$197$	−5.45536	−0.388678	−0.194339	−	0.980934i	$-0.562256\pi$
$197$	−5.45536	−0.388678	−0.194339	+	0.980934i	$0.562256\pi$
$198$	0	0
$199$	−20.6182	−1.46158	−0.730792	−	0.682600i	$-0.760849\pi$
$199$	−20.6182	−1.46158	−0.730792	+	0.682600i	$0.760849\pi$
$200$	0	0
$201$	−6.42504	−0.453188
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.416360	−0.0290799
$206$	0	0
$207$	−0.963536	−0.0669704
$208$	0	0
$209$	−9.81585	−0.678977
$210$	0	0
$211$	−6.76242	−0.465544	−0.232772	−	0.972531i	$-0.574780\pi$
$211$	−6.76242	−0.465544	−0.232772	+	0.972531i	$0.574780\pi$
$212$	0	0
$213$	12.4388	0.852290
$214$	0	0
$215$	17.6982	1.20701
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−5.88438	−0.397630
$220$	0	0
$221$	−3.96072	−0.266427
$222$	0	0
$223$	5.28283	0.353764	0.176882	−	0.984232i	$-0.443399\pi$
$223$	5.28283	0.353764	0.176882	+	0.984232i	$0.443399\pi$
$224$	0	0
$225$	−2.42042	−0.161361
$226$	0	0
$227$	29.2634	1.94228	0.971140	−	0.238509i	$-0.0766587\pi$
$227$	29.2634	1.94228	0.971140	+	0.238509i	$0.0766587\pi$
$228$	0	0
$229$	−3.89618	−0.257467	−0.128733	−	0.991679i	$-0.541091\pi$
$229$	−3.89618	−0.257467	−0.128733	+	0.991679i	$0.541091\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.32280	0.545245	0.272622	−	0.962121i	$-0.412109\pi$
$233$	8.32280	0.545245	0.272622	+	0.962121i	$0.412109\pi$
$234$	0	0
$235$	13.3343	0.869831
$236$	0	0
$237$	18.5674	1.20608
$238$	0	0
$239$	−18.2789	−1.18237	−0.591183	−	0.806537i	$-0.701339\pi$
$239$	−18.2789	−1.18237	−0.591183	+	0.806537i	$0.701339\pi$
$240$	0	0
$241$	−10.4970	−0.676170	−0.338085	−	0.941116i	$-0.609779\pi$
$241$	−10.4970	−0.676170	−0.338085	+	0.941116i	$0.609779\pi$
$242$	0	0
$243$	−9.66129	−0.619772
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.29830	0.146237
$248$	0	0
$249$	24.3638	1.54399
$250$	0	0
$251$	15.8932	1.00317	0.501585	−	0.865108i	$-0.332751\pi$
$251$	15.8932	1.00317	0.501585	+	0.865108i	$0.332751\pi$
$252$	0	0
$253$	5.45206	0.342768
$254$	0	0
$255$	−9.74317	−0.610141
$256$	0	0
$257$	−0.308565	−0.0192477	−0.00962387	−	0.999954i	$-0.503063\pi$
$257$	−0.308565	−0.0192477	−0.00962387	+	0.999954i	$0.503063\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8.54154	0.528708
$262$	0	0
$263$	−15.4165	−0.950621	−0.475310	−	0.879818i	$-0.657664\pi$
$263$	−15.4165	−0.950621	−0.475310	+	0.879818i	$0.657664\pi$
$264$	0	0
$265$	21.1556	1.29958
$266$	0	0
$267$	−26.0854	−1.59640
$268$	0	0
$269$	−17.7644	−1.08311	−0.541557	−	0.840664i	$-0.682165\pi$
$269$	−17.7644	−1.08311	−0.541557	+	0.840664i	$0.682165\pi$
$270$	0	0
$271$	5.29206	0.321470	0.160735	−	0.986998i	$-0.448614\pi$
$271$	5.29206	0.321470	0.160735	+	0.986998i	$0.448614\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	13.6957	0.825881
$276$	0	0
$277$	3.03657	0.182450	0.0912248	−	0.995830i	$-0.470922\pi$
$277$	3.03657	0.182450	0.0912248	+	0.995830i	$0.470922\pi$
$278$	0	0
$279$	5.40000	0.323290
$280$	0	0
$281$	29.4879	1.75910	0.879551	−	0.475805i	$-0.157843\pi$
$281$	29.4879	1.75910	0.879551	+	0.475805i	$0.157843\pi$
$282$	0	0
$283$	−29.3775	−1.74631	−0.873157	−	0.487439i	$-0.837931\pi$
$283$	−29.3775	−1.74631	−0.873157	+	0.487439i	$0.837931\pi$
$284$	0	0
$285$	5.65369	0.334896
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−7.37345	−0.433732
$290$	0	0
$291$	13.2607	0.777355
$292$	0	0
$293$	27.6513	1.61541	0.807704	−	0.589588i	$-0.200710\pi$
$293$	27.6513	1.61541	0.807704	+	0.589588i	$0.200710\pi$
$294$	0	0
$295$	−8.34317	−0.485758
$296$	0	0
$297$	22.1044	1.28263
$298$	0	0
$299$	−1.27655	−0.0738249
$300$	0	0
$301$	0	0
$302$	0	0
$303$	33.6511	1.93320
$304$	0	0
$305$	2.91413	0.166862
$306$	0	0
$307$	−26.8408	−1.53189	−0.765943	−	0.642909i	$-0.777727\pi$
$307$	−26.8408	−1.53189	−0.765943	+	0.642909i	$0.777727\pi$
$308$	0	0
$309$	23.0535	1.31147
$310$	0	0
$311$	29.7633	1.68772	0.843861	−	0.536562i	$-0.180277\pi$
$311$	29.7633	1.68772	0.843861	+	0.536562i	$0.180277\pi$
$312$	0	0
$313$	7.10498	0.401597	0.200799	−	0.979633i	$-0.435646\pi$
$313$	7.10498	0.401597	0.200799	+	0.979633i	$0.435646\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.39859	−0.190884	−0.0954420	−	0.995435i	$-0.530426\pi$
$317$	−3.39859	−0.190884	−0.0954420	+	0.995435i	$0.530426\pi$
$318$	0	0
$319$	−48.3313	−2.70604
$320$	0	0
$321$	34.6545	1.93423
$322$	0	0
$323$	−5.58602	−0.310815
$324$	0	0
$325$	−3.20672	−0.177877
$326$	0	0
$327$	20.8105	1.15082
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−6.49659	−0.357085	−0.178543	−	0.983932i	$-0.557138\pi$
$331$	−6.49659	−0.357085	−0.178543	+	0.983932i	$0.557138\pi$
$332$	0	0
$333$	0.193098	0.0105817
$334$	0	0
$335$	−5.09047	−0.278122
$336$	0	0
$337$	19.8062	1.07891	0.539457	−	0.842013i	$-0.318629\pi$
$337$	19.8062	1.07891	0.539457	+	0.842013i	$0.318629\pi$
$338$	0	0
$339$	−34.4673	−1.87201
$340$	0	0
$341$	−30.5553	−1.65466
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3.14026	−0.169066
$346$	0	0
$347$	−31.7637	−1.70516	−0.852581	−	0.522594i	$-0.824964\pi$
$347$	−31.7637	−1.70516	−0.852581	+	0.522594i	$0.824964\pi$
$348$	0	0
$349$	−7.00623	−0.375035	−0.187517	−	0.982261i	$-0.560044\pi$
$349$	−7.00623	−0.375035	−0.187517	+	0.982261i	$0.560044\pi$
$350$	0	0
$351$	−5.17555	−0.276250
$352$	0	0
$353$	−27.3059	−1.45334	−0.726672	−	0.686984i	$-0.758934\pi$
$353$	−27.3059	−1.45334	−0.726672	+	0.686984i	$0.758934\pi$
$354$	0	0
$355$	9.85506	0.523052
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.8119	0.570631	0.285316	−	0.958434i	$-0.407902\pi$
$359$	10.8119	0.570631	0.285316	+	0.958434i	$0.407902\pi$
$360$	0	0
$361$	−15.7586	−0.829399
$362$	0	0
$363$	37.2789	1.95663
$364$	0	0
$365$	−4.66212	−0.244026
$366$	0	0
$367$	−4.18029	−0.218209	−0.109105	−	0.994030i	$-0.534798\pi$
$367$	−4.18029	−0.218209	−0.109105	+	0.994030i	$0.534798\pi$
$368$	0	0
$369$	−0.254340	−0.0132404
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−26.5671	−1.37559	−0.687796	−	0.725904i	$-0.741422\pi$
$373$	−26.5671	−1.37559	−0.687796	+	0.725904i	$0.741422\pi$
$374$	0	0
$375$	−23.5897	−1.21816
$376$	0	0
$377$	11.3164	0.582822
$378$	0	0
$379$	−22.8995	−1.17627	−0.588135	−	0.808762i	$-0.700138\pi$
$379$	−22.8995	−1.17627	−0.588135	+	0.808762i	$0.700138\pi$
$380$	0	0
$381$	3.54178	0.181451
$382$	0	0
$383$	23.9892	1.22579	0.612897	−	0.790163i	$-0.290004\pi$
$383$	23.9892	1.22579	0.612897	+	0.790163i	$0.290004\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.8112	0.549564
$388$	0	0
$389$	13.3399	0.676362	0.338181	−	0.941081i	$-0.390188\pi$
$389$	13.3399	0.676362	0.338181	+	0.941081i	$0.390188\pi$
$390$	0	0
$391$	3.10267	0.156909
$392$	0	0
$393$	18.1465	0.915367
$394$	0	0
$395$	14.7107	0.740176
$396$	0	0
$397$	0.770174	0.0386539	0.0193270	−	0.999813i	$-0.493848\pi$
$397$	0.770174	0.0386539	0.0193270	+	0.999813i	$0.493848\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−20.4491	−1.02118	−0.510589	−	0.859825i	$-0.670573\pi$
$401$	−20.4491	−1.02118	−0.510589	+	0.859825i	$0.670573\pi$
$402$	0	0
$403$	7.15426	0.356379
$404$	0	0
$405$	−17.2911	−0.859200
$406$	0	0
$407$	−1.09262	−0.0541593
$408$	0	0
$409$	−12.7722	−0.631546	−0.315773	−	0.948835i	$-0.602264\pi$
$409$	−12.7722	−0.631546	−0.315773	+	0.948835i	$0.602264\pi$
$410$	0	0
$411$	−0.842663	−0.0415655
$412$	0	0
$413$	0	0
$414$	0	0
$415$	19.3031	0.947552
$416$	0	0
$417$	−27.1826	−1.33114
$418$	0	0
$419$	−0.796047	−0.0388894	−0.0194447	−	0.999811i	$-0.506190\pi$
$419$	−0.796047	−0.0388894	−0.0194447	+	0.999811i	$0.506190\pi$
$420$	0	0
$421$	1.80814	0.0881234	0.0440617	−	0.999029i	$-0.485970\pi$
$421$	1.80814	0.0881234	0.0440617	+	0.999029i	$0.485970\pi$
$422$	0	0
$423$	8.14542	0.396044
$424$	0	0
$425$	7.79396	0.378063
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−13.8561	−0.668978
$430$	0	0
$431$	−10.0003	−0.481696	−0.240848	−	0.970563i	$-0.577426\pi$
$431$	−10.0003	−0.481696	−0.240848	+	0.970563i	$0.577426\pi$
$432$	0	0
$433$	0.963187	0.0462878	0.0231439	−	0.999732i	$-0.492632\pi$
$433$	0.963187	0.0462878	0.0231439	+	0.999732i	$0.492632\pi$
$434$	0	0
$435$	27.8377	1.33471
$436$	0	0
$437$	−1.80039	−0.0861245
$438$	0	0
$439$	−26.5236	−1.26590	−0.632952	−	0.774191i	$-0.718157\pi$
$439$	−26.5236	−1.26590	−0.632952	+	0.774191i	$0.718157\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.9737	0.616397	0.308198	−	0.951322i	$-0.400274\pi$
$443$	12.9737	0.616397	0.308198	+	0.951322i	$0.400274\pi$
$444$	0	0
$445$	−20.6671	−0.979714
$446$	0	0
$447$	28.3058	1.33882
$448$	0	0
$449$	−31.4221	−1.48290	−0.741451	−	0.671008i	$-0.765862\pi$
$449$	−31.4221	−1.48290	−0.741451	+	0.671008i	$0.765862\pi$
$450$	0	0
$451$	1.43915	0.0677670
$452$	0	0
$453$	−5.56337	−0.261390
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.66627	−0.0779448	−0.0389724	−	0.999240i	$-0.512408\pi$
$457$	−1.66627	−0.0779448	−0.0389724	+	0.999240i	$0.512408\pi$
$458$	0	0
$459$	12.5792	0.587147
$460$	0	0
$461$	13.3335	0.621003	0.310501	−	0.950573i	$-0.399503\pi$
$461$	13.3335	0.621003	0.310501	+	0.950573i	$0.399503\pi$
$462$	0	0
$463$	−7.93411	−0.368729	−0.184365	−	0.982858i	$-0.559023\pi$
$463$	−7.93411	−0.368729	−0.184365	+	0.982858i	$0.559023\pi$
$464$	0	0
$465$	17.5991	0.816140
$466$	0	0
$467$	1.40728	0.0651212	0.0325606	−	0.999470i	$-0.489634\pi$
$467$	1.40728	0.0651212	0.0325606	+	0.999470i	$0.489634\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−26.9585	−1.24218
$472$	0	0
$473$	−61.1740	−2.81278
$474$	0	0
$475$	−4.52262	−0.207512
$476$	0	0
$477$	12.9232	0.591714
$478$	0	0
$479$	29.4268	1.34455	0.672273	−	0.740303i	$-0.265318\pi$
$479$	29.4268	1.34455	0.672273	+	0.740303i	$0.265318\pi$
$480$	0	0
$481$	0.255828	0.0116647
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.5063	0.477065
$486$	0	0
$487$	−32.1746	−1.45797	−0.728986	−	0.684529i	$-0.760008\pi$
$487$	−32.1746	−1.45797	−0.728986	+	0.684529i	$0.760008\pi$
$488$	0	0
$489$	0.240880	0.0108930
$490$	0	0
$491$	−30.6562	−1.38350	−0.691748	−	0.722139i	$-0.743159\pi$
$491$	−30.6562	−1.38350	−0.691748	+	0.722139i	$0.743159\pi$
$492$	0	0
$493$	−27.5045	−1.23874
$494$	0	0
$495$	−8.28615	−0.372435
$496$	0	0
$497$	0	0
$498$	0	0
$499$	40.7239	1.82305	0.911525	−	0.411244i	$-0.134906\pi$
$499$	40.7239	1.82305	0.911525	+	0.411244i	$0.134906\pi$
$500$	0	0
$501$	26.2457	1.17257
$502$	0	0
$503$	8.26818	0.368660	0.184330	−	0.982864i	$-0.440988\pi$
$503$	8.26818	0.368660	0.184330	+	0.982864i	$0.440988\pi$
$504$	0	0
$505$	26.6613	1.18641
$506$	0	0
$507$	−22.6369	−1.00534
$508$	0	0
$509$	13.7582	0.609821	0.304910	−	0.952381i	$-0.401373\pi$
$509$	13.7582	0.609821	0.304910	+	0.952381i	$0.401373\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−7.29937	−0.322275
$514$	0	0
$515$	18.2650	0.804851
$516$	0	0
$517$	−46.0899	−2.02703
$518$	0	0
$519$	−25.2799	−1.10967
$520$	0	0
$521$	2.36866	0.103773	0.0518863	−	0.998653i	$-0.483477\pi$
$521$	2.36866	0.103773	0.0518863	+	0.998653i	$0.483477\pi$
$522$	0	0
$523$	−3.04284	−0.133054	−0.0665271	−	0.997785i	$-0.521192\pi$
$523$	−3.04284	−0.133054	−0.0665271	+	0.997785i	$0.521192\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−17.3885	−0.757453
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.09654	−0.221171
$532$	0	0
$533$	−0.336965	−0.0145956
$534$	0	0
$535$	27.4563	1.18704
$536$	0	0
$537$	32.8723	1.41855
$538$	0	0
$539$	0	0
$540$	0	0
$541$	22.7295	0.977216	0.488608	−	0.872503i	$-0.337505\pi$
$541$	22.7295	0.977216	0.488608	+	0.872503i	$0.337505\pi$
$542$	0	0
$543$	12.5091	0.536818
$544$	0	0
$545$	16.4879	0.706262
$546$	0	0
$547$	22.4608	0.960355	0.480178	−	0.877171i	$-0.340572\pi$
$547$	22.4608	0.960355	0.480178	+	0.877171i	$0.340572\pi$
$548$	0	0
$549$	1.78013	0.0759743
$550$	0	0
$551$	15.9601	0.679923
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.629324	0.0267133
$556$	0	0
$557$	8.59982	0.364386	0.182193	−	0.983263i	$-0.441680\pi$
$557$	8.59982	0.364386	0.182193	+	0.983263i	$0.441680\pi$
$558$	0	0
$559$	14.3233	0.605813
$560$	0	0
$561$	33.6773	1.42186
$562$	0	0
$563$	−32.4372	−1.36706	−0.683532	−	0.729921i	$-0.739557\pi$
$563$	−32.4372	−1.36706	−0.683532	+	0.729921i	$0.739557\pi$
$564$	0	0
$565$	−27.3080	−1.14886
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18.3889	−0.770905	−0.385452	−	0.922728i	$-0.625955\pi$
$569$	−18.3889	−0.770905	−0.385452	+	0.922728i	$0.625955\pi$
$570$	0	0
$571$	−16.4359	−0.687820	−0.343910	−	0.939003i	$-0.611751\pi$
$571$	−16.4359	−0.687820	−0.343910	+	0.939003i	$0.611751\pi$
$572$	0	0
$573$	−9.49527	−0.396671
$574$	0	0
$575$	2.51202	0.104758
$576$	0	0
$577$	−6.85543	−0.285395	−0.142698	−	0.989766i	$-0.545578\pi$
$577$	−6.85543	−0.285395	−0.142698	+	0.989766i	$0.545578\pi$
$578$	0	0
$579$	−9.09370	−0.377921
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−73.1246	−3.02851
$584$	0	0
$585$	1.94013	0.0802144
$586$	0	0
$587$	3.02849	0.124999	0.0624996	−	0.998045i	$-0.480093\pi$
$587$	3.02849	0.124999	0.0624996	+	0.998045i	$0.480093\pi$
$588$	0	0
$589$	10.0900	0.415753
$590$	0	0
$591$	−10.8609	−0.446757
$592$	0	0
$593$	−28.2962	−1.16199	−0.580993	−	0.813908i	$-0.697336\pi$
$593$	−28.2962	−1.16199	−0.580993	+	0.813908i	$0.697336\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−41.0480	−1.67998
$598$	0	0
$599$	−18.3945	−0.751579	−0.375790	−	0.926705i	$-0.622628\pi$
$599$	−18.3945	−0.751579	−0.375790	+	0.926705i	$0.622628\pi$
$600$	0	0
$601$	32.0486	1.30729	0.653645	−	0.756801i	$-0.273239\pi$
$601$	32.0486	1.30729	0.653645	+	0.756801i	$0.273239\pi$
$602$	0	0
$603$	−3.10959	−0.126632
$604$	0	0
$605$	29.5355	1.20079
$606$	0	0
$607$	−11.6560	−0.473104	−0.236552	−	0.971619i	$-0.576017\pi$
$607$	−11.6560	−0.473104	−0.236552	+	0.971619i	$0.576017\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.7915	0.436579
$612$	0	0
$613$	−10.0064	−0.404156	−0.202078	−	0.979369i	$-0.564769\pi$
$613$	−10.0064	−0.404156	−0.202078	+	0.979369i	$0.564769\pi$
$614$	0	0
$615$	−0.828916	−0.0334251
$616$	0	0
$617$	25.3705	1.02138	0.510688	−	0.859766i	$-0.329391\pi$
$617$	25.3705	1.02138	0.510688	+	0.859766i	$0.329391\pi$
$618$	0	0
$619$	42.5887	1.71178	0.855891	−	0.517156i	$-0.173009\pi$
$619$	42.5887	1.71178	0.855891	+	0.517156i	$0.173009\pi$
$620$	0	0
$621$	4.05432	0.162694
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−6.12967	−0.245187
$626$	0	0
$627$	−19.5420	−0.780433
$628$	0	0
$629$	−0.621791	−0.0247924
$630$	0	0
$631$	47.3317	1.88424	0.942122	−	0.335271i	$-0.108828\pi$
$631$	47.3317	1.88424	0.942122	+	0.335271i	$0.108828\pi$
$632$	0	0
$633$	−13.4631	−0.535109
$634$	0	0
$635$	2.80610	0.111357
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.02010	0.238152
$640$	0	0
$641$	4.81767	0.190287	0.0951433	−	0.995464i	$-0.469669\pi$
$641$	4.81767	0.190287	0.0951433	+	0.995464i	$0.469669\pi$
$642$	0	0
$643$	6.67119	0.263086	0.131543	−	0.991310i	$-0.458007\pi$
$643$	6.67119	0.263086	0.131543	+	0.991310i	$0.458007\pi$
$644$	0	0
$645$	35.2347	1.38737
$646$	0	0
$647$	18.1524	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$647$	18.1524	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$648$	0	0
$649$	28.8382	1.13200
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−31.0688	−1.21582	−0.607908	−	0.794008i	$-0.707991\pi$
$653$	−31.0688	−1.21582	−0.607908	+	0.794008i	$0.707991\pi$
$654$	0	0
$655$	14.3772	0.561763
$656$	0	0
$657$	−2.84792	−0.111108
$658$	0	0
$659$	−9.69081	−0.377500	−0.188750	−	0.982025i	$-0.560444\pi$
$659$	−9.69081	−0.377500	−0.188750	+	0.982025i	$0.560444\pi$
$660$	0	0
$661$	−9.12336	−0.354857	−0.177429	−	0.984134i	$-0.556778\pi$
$661$	−9.12336	−0.354857	−0.177429	+	0.984134i	$0.556778\pi$
$662$	0	0
$663$	−7.88525	−0.306238
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.86478	−0.343246
$668$	0	0
$669$	10.5174	0.406626
$670$	0	0
$671$	−10.0727	−0.388852
$672$	0	0
$673$	−16.6361	−0.641276	−0.320638	−	0.947202i	$-0.603897\pi$
$673$	−16.6361	−0.641276	−0.320638	+	0.947202i	$0.603897\pi$
$674$	0	0
$675$	10.1845	0.392003
$676$	0	0
$677$	21.8170	0.838494	0.419247	−	0.907872i	$-0.362294\pi$
$677$	21.8170	0.838494	0.419247	+	0.907872i	$0.362294\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	58.2594	2.23251
$682$	0	0
$683$	−19.1748	−0.733705	−0.366853	−	0.930279i	$-0.619565\pi$
$683$	−19.1748	−0.733705	−0.366853	+	0.930279i	$0.619565\pi$
$684$	0	0
$685$	−0.667630	−0.0255088
$686$	0	0
$687$	−7.75676	−0.295939
$688$	0	0
$689$	17.1215	0.652276
$690$	0	0
$691$	26.6183	1.01261	0.506303	−	0.862356i	$-0.331012\pi$
$691$	26.6183	1.01261	0.506303	+	0.862356i	$0.331012\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−21.5364	−0.816922
$696$	0	0
$697$	0.818995	0.0310216
$698$	0	0
$699$	16.5696	0.626718
$700$	0	0
$701$	−47.1534	−1.78096	−0.890481	−	0.455021i	$-0.849632\pi$
$701$	−47.1534	−1.78096	−0.890481	+	0.455021i	$0.849632\pi$
$702$	0	0
$703$	0.360808	0.0136081
$704$	0	0
$705$	26.5467	0.999806
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−21.7819	−0.818038	−0.409019	−	0.912526i	$-0.634129\pi$
$709$	−21.7819	−0.818038	−0.409019	+	0.912526i	$0.634129\pi$
$710$	0	0
$711$	8.98624	0.337010
$712$	0	0
$713$	−5.60436	−0.209885
$714$	0	0
$715$	−10.9780	−0.410554
$716$	0	0
$717$	−36.3909	−1.35904
$718$	0	0
$719$	−15.5941	−0.581561	−0.290781	−	0.956790i	$-0.593915\pi$
$719$	−15.5941	−0.581561	−0.290781	+	0.956790i	$0.593915\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−20.8981	−0.777207
$724$	0	0
$725$	−22.2685	−0.827031
$726$	0	0
$727$	13.0959	0.485702	0.242851	−	0.970064i	$-0.421917\pi$
$727$	13.0959	0.485702	0.242851	+	0.970064i	$0.421917\pi$
$728$	0	0
$729$	13.6523	0.505641
$730$	0	0
$731$	−34.8130	−1.28760
$732$	0	0
$733$	5.15728	0.190489	0.0952443	−	0.995454i	$-0.469637\pi$
$733$	5.15728	0.190489	0.0952443	+	0.995454i	$0.469637\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	17.5953	0.648129
$738$	0	0
$739$	−13.4955	−0.496441	−0.248220	−	0.968704i	$-0.579846\pi$
$739$	−13.4955	−0.496441	−0.248220	+	0.968704i	$0.579846\pi$
$740$	0	0
$741$	4.57559	0.168089
$742$	0	0
$743$	−1.00770	−0.0369689	−0.0184844	−	0.999829i	$-0.505884\pi$
$743$	−1.00770	−0.0369689	−0.0184844	+	0.999829i	$0.505884\pi$
$744$	0	0
$745$	22.4263	0.821636
$746$	0	0
$747$	11.7916	0.431431
$748$	0	0
$749$	0	0
$750$	0	0
$751$	16.3929	0.598184	0.299092	−	0.954224i	$-0.403316\pi$
$751$	16.3929	0.598184	0.299092	+	0.954224i	$0.403316\pi$
$752$	0	0
$753$	31.6412	1.15307
$754$	0	0
$755$	−4.40778	−0.160416
$756$	0	0
$757$	15.7508	0.572471	0.286236	−	0.958159i	$-0.407596\pi$
$757$	15.7508	0.572471	0.286236	+	0.958159i	$0.407596\pi$
$758$	0	0
$759$	10.8543	0.393987
$760$	0	0
$761$	−0.655661	−0.0237677	−0.0118838	−	0.999929i	$-0.503783\pi$
$761$	−0.655661	−0.0237677	−0.0118838	+	0.999929i	$0.503783\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.71549	−0.170489
$766$	0	0
$767$	−6.75221	−0.243808
$768$	0	0
$769$	−47.1576	−1.70055	−0.850274	−	0.526340i	$-0.823564\pi$
$769$	−47.1576	−1.70055	−0.850274	+	0.526340i	$0.823564\pi$
$770$	0	0
$771$	−0.614310	−0.0221238
$772$	0	0
$773$	5.90163	0.212267	0.106133	−	0.994352i	$-0.466153\pi$
$773$	5.90163	0.212267	0.106133	+	0.994352i	$0.466153\pi$
$774$	0	0
$775$	−14.0783	−0.505706
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.475240	−0.0170272
$780$	0	0
$781$	−34.0641	−1.21891
$782$	0	0
$783$	−35.9407	−1.28441
$784$	0	0
$785$	−21.3589	−0.762330
$786$	0	0
$787$	44.0582	1.57050	0.785252	−	0.619177i	$-0.212534\pi$
$787$	44.0582	1.57050	0.785252	+	0.619177i	$0.212534\pi$
$788$	0	0
$789$	−30.6921	−1.09267
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.35843	0.0837504
$794$	0	0
$795$	42.1180	1.49377
$796$	0	0
$797$	47.8542	1.69508	0.847542	−	0.530729i	$-0.178082\pi$
$797$	47.8542	1.69508	0.847542	+	0.530729i	$0.178082\pi$
$798$	0	0
$799$	−26.2289	−0.927912
$800$	0	0
$801$	−12.6248	−0.446075
$802$	0	0
$803$	16.1146	0.568673
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−35.3665	−1.24496
$808$	0	0
$809$	−40.7189	−1.43160	−0.715800	−	0.698306i	$-0.753938\pi$
$809$	−40.7189	−1.43160	−0.715800	+	0.698306i	$0.753938\pi$
$810$	0	0
$811$	24.9654	0.876654	0.438327	−	0.898815i	$-0.355571\pi$
$811$	24.9654	0.876654	0.438327	+	0.898815i	$0.355571\pi$
$812$	0	0
$813$	10.5358	0.369505
$814$	0	0
$815$	0.190846	0.00668503
$816$	0	0
$817$	20.2010	0.706744
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.4842	0.575304	0.287652	−	0.957735i	$-0.407125\pi$
$821$	16.4842	0.575304	0.287652	+	0.957735i	$0.407125\pi$
$822$	0	0
$823$	−26.5869	−0.926763	−0.463381	−	0.886159i	$-0.653364\pi$
$823$	−26.5869	−0.926763	−0.463381	+	0.886159i	$0.653364\pi$
$824$	0	0
$825$	27.2662	0.949288
$826$	0	0
$827$	−51.5586	−1.79287	−0.896434	−	0.443176i	$-0.853852\pi$
$827$	−51.5586	−1.79287	−0.896434	+	0.443176i	$0.853852\pi$
$828$	0	0
$829$	−20.1134	−0.698566	−0.349283	−	0.937017i	$-0.613575\pi$
$829$	−20.1134	−0.698566	−0.349283	+	0.937017i	$0.613575\pi$
$830$	0	0
$831$	6.04539	0.209712
$832$	0	0
$833$	0	0
$834$	0	0
$835$	20.7941	0.719609
$836$	0	0
$837$	−22.7219	−0.785383
$838$	0	0
$839$	44.0617	1.52118	0.760589	−	0.649234i	$-0.224910\pi$
$839$	44.0617	1.52118	0.760589	+	0.649234i	$0.224910\pi$
$840$	0	0
$841$	49.5843	1.70980
$842$	0	0
$843$	58.7064	2.02196
$844$	0	0
$845$	−17.9349	−0.616981
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−58.4867	−2.00726
$850$	0	0
$851$	−0.200405	−0.00686981
$852$	0	0
$853$	−11.0200	−0.377319	−0.188659	−	0.982043i	$-0.560414\pi$
$853$	−11.0200	−0.377319	−0.188659	+	0.982043i	$0.560414\pi$
$854$	0	0
$855$	2.73627	0.0935785
$856$	0	0
$857$	2.49851	0.0853474	0.0426737	−	0.999089i	$-0.486412\pi$
$857$	2.49851	0.0853474	0.0426737	+	0.999089i	$0.486412\pi$
$858$	0	0
$859$	45.7866	1.56222	0.781109	−	0.624395i	$-0.214654\pi$
$859$	45.7866	1.56222	0.781109	+	0.624395i	$0.214654\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.97756	0.0673170	0.0336585	−	0.999433i	$-0.489284\pi$
$863$	1.97756	0.0673170	0.0336585	+	0.999433i	$0.489284\pi$
$864$	0	0
$865$	−20.0289	−0.681004
$866$	0	0
$867$	−14.6795	−0.498543
$868$	0	0
$869$	−50.8476	−1.72489
$870$	0	0
$871$	−4.11977	−0.139593
$872$	0	0
$873$	6.41790	0.217213
$874$	0	0
$875$	0	0
$876$	0	0
$877$	42.2078	1.42526	0.712628	−	0.701542i	$-0.247505\pi$
$877$	42.2078	1.42526	0.712628	+	0.701542i	$0.247505\pi$
$878$	0	0
$879$	55.0500	1.85679
$880$	0	0
$881$	−17.7954	−0.599544	−0.299772	−	0.954011i	$-0.596911\pi$
$881$	−17.7954	−0.599544	−0.299772	+	0.954011i	$0.596911\pi$
$882$	0	0
$883$	39.0564	1.31435	0.657176	−	0.753737i	$-0.271751\pi$
$883$	39.0564	1.31435	0.657176	+	0.753737i	$0.271751\pi$
$884$	0	0
$885$	−16.6101	−0.558342
$886$	0	0
$887$	21.5543	0.723724	0.361862	−	0.932232i	$-0.382141\pi$
$887$	21.5543	0.723724	0.361862	+	0.932232i	$0.382141\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	59.7666	2.00226
$892$	0	0
$893$	15.2199	0.509315
$894$	0	0
$895$	26.0443	0.870565
$896$	0	0
$897$	−2.54144	−0.0848562
$898$	0	0
$899$	49.6814	1.65697
$900$	0	0
$901$	−41.6139	−1.38636
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.91081	0.329446
$906$	0	0
$907$	16.1356	0.535774	0.267887	−	0.963450i	$-0.413675\pi$
$907$	16.1356	0.535774	0.267887	+	0.963450i	$0.413675\pi$
$908$	0	0
$909$	16.2864	0.540187
$910$	0	0
$911$	−37.7250	−1.24988	−0.624942	−	0.780671i	$-0.714877\pi$
$911$	−37.7250	−1.24988	−0.624942	+	0.780671i	$0.714877\pi$
$912$	0	0
$913$	−66.7213	−2.20815
$914$	0	0
$915$	5.80163	0.191796
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−34.4543	−1.13654	−0.568270	−	0.822842i	$-0.692387\pi$
$919$	−34.4543	−1.13654	−0.568270	+	0.822842i	$0.692387\pi$
$920$	0	0
$921$	−53.4364	−1.76079
$922$	0	0
$923$	7.97580	0.262527
$924$	0	0
$925$	−0.503422	−0.0165524
$926$	0	0
$927$	11.1574	0.366458
$928$	0	0
$929$	−0.000727706 0	−2.38753e−5 0	−1.19376e−5	−	1.00000i	$-0.500004\pi$
$929$	−0.000727706 0	−2.38753e−5 0	−1.19376e−5		1.00000i	$0.500004\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	59.2547	1.93991
$934$	0	0
$935$	26.6821	0.872597
$936$	0	0
$937$	−8.03992	−0.262653	−0.131326	−	0.991339i	$-0.541924\pi$
$937$	−8.03992	−0.262653	−0.131326	+	0.991339i	$0.541924\pi$
$938$	0	0
$939$	14.1450	0.461606
$940$	0	0
$941$	18.4735	0.602217	0.301109	−	0.953590i	$-0.402643\pi$
$941$	18.4735	0.602217	0.301109	+	0.953590i	$0.402643\pi$
$942$	0	0
$943$	0.263965	0.00859587
$944$	0	0
$945$	0	0
$946$	0	0
$947$	35.0162	1.13787	0.568937	−	0.822381i	$-0.307355\pi$
$947$	35.0162	1.13787	0.568937	+	0.822381i	$0.307355\pi$
$948$	0	0
$949$	−3.77310	−0.122480
$950$	0	0
$951$	−6.76613	−0.219407
$952$	0	0
$953$	26.7745	0.867311	0.433656	−	0.901079i	$-0.357223\pi$
$953$	26.7745	0.867311	0.433656	+	0.901079i	$0.357223\pi$
$954$	0	0
$955$	−7.52297	−0.243438
$956$	0	0
$957$	−96.2211	−3.11039
$958$	0	0
$959$	0	0
$960$	0	0
$961$	0.408832	0.0131881
$962$	0	0
$963$	16.7721	0.540473
$964$	0	0
$965$	−7.20481	−0.231931
$966$	0	0
$967$	38.6501	1.24290	0.621451	−	0.783453i	$-0.286543\pi$
$967$	38.6501	1.24290	0.621451	+	0.783453i	$0.286543\pi$
$968$	0	0
$969$	−11.1210	−0.357258
$970$	0	0
$971$	41.0472	1.31727	0.658634	−	0.752463i	$-0.271135\pi$
$971$	41.0472	1.31727	0.658634	+	0.752463i	$0.271135\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−6.38415	−0.204456
$976$	0	0
$977$	−40.9412	−1.30983	−0.654913	−	0.755704i	$-0.727295\pi$
$977$	−40.9412	−1.30983	−0.654913	+	0.755704i	$0.727295\pi$
$978$	0	0
$979$	71.4359	2.28310
$980$	0	0
$981$	10.0718	0.321569
$982$	0	0
$983$	36.4716	1.16326	0.581631	−	0.813452i	$-0.302415\pi$
$983$	36.4716	1.16326	0.581631	+	0.813452i	$0.302415\pi$
$984$	0	0
$985$	−8.60492	−0.274176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.2203	−0.356786
$990$	0	0
$991$	42.5796	1.35259	0.676293	−	0.736633i	$-0.263585\pi$
$991$	42.5796	1.35259	0.676293	+	0.736633i	$0.263585\pi$
$992$	0	0
$993$	−12.9338	−0.410443
$994$	0	0
$995$	−32.5218	−1.03101
$996$	0	0
$997$	−53.2816	−1.68744	−0.843722	−	0.536780i	$-0.819641\pi$
$997$	−53.2816	−1.68744	−0.843722	+	0.536780i	$0.819641\pi$
$998$	0	0
$999$	−0.812508	−0.0257066

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9016.2.a.br.1.9		11
7.3	odd	6		1288.2.q.d.737.9	✓	22
7.5	odd	6		1288.2.q.d.921.9	yes	22
7.6	odd	2		9016.2.a.bk.1.3		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.q.d.737.9	✓	22	7.3	odd	6
1288.2.q.d.921.9	yes	22	7.5	odd	6
9016.2.a.bk.1.3		11	7.6	odd	2
9016.2.a.br.1.9		11	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:	\( N \)	\(=\)	\( 9016 = 2^{3} \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.99086 q^{3} +1.57733 q^{5} +0.963536 q^{9} +O(q^{10})\)	\(q+1.99086 q^{3} +1.57733 q^{5} +0.963536 q^{9} -5.45206 q^{11} +1.27655 q^{13} +3.14026 q^{15} -3.10267 q^{17} +1.80039 q^{19} -1.00000 q^{23} -2.51202 q^{25} -4.05432 q^{27} +8.86478 q^{29} +5.60436 q^{31} -10.8543 q^{33} +0.200405 q^{37} +2.54144 q^{39} -0.263965 q^{41} +11.2203 q^{43} +1.51982 q^{45} +8.45367 q^{47} -6.17699 q^{51} +13.4123 q^{53} -8.59972 q^{55} +3.58434 q^{57} -5.28941 q^{59} +1.84750 q^{61} +2.01355 q^{65} -3.22727 q^{67} -1.99086 q^{69} +6.24792 q^{71} -2.95569 q^{73} -5.00109 q^{75} +9.32631 q^{79} -10.9622 q^{81} +12.2378 q^{83} -4.89394 q^{85} +17.6486 q^{87} -13.1025 q^{89} +11.1575 q^{93} +2.83982 q^{95} +6.66077 q^{97} -5.25326 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 4 q^{3} + 3 q^{5} + 11 q^{9}+O(q^{10}) \) 11 * q + 4 * q^3 + 3 * q^5 + 11 * q^9	\( 11 q + 4 q^{3} + 3 q^{5} + 11 q^{9} - 13 q^{13} + 7 q^{17} + 8 q^{19} - 11 q^{23} + 6 q^{25} + 25 q^{27} - 3 q^{29} + 12 q^{31} - 2 q^{33} - q^{37} - 21 q^{39} + 12 q^{41} + 9 q^{43} + 19 q^{45} + 17 q^{47} + 19 q^{51} - 5 q^{53} + 21 q^{55} + 11 q^{57} + 33 q^{59} - 15 q^{61} - 9 q^{65} - 5 q^{67} - 4 q^{69} - 9 q^{71} + 5 q^{73} + 44 q^{75} + 11 q^{79} - 13 q^{81} + 51 q^{83} + 33 q^{85} + 4 q^{87} + 26 q^{89} + 6 q^{93} - 19 q^{95} + 21 q^{97} + 15 q^{99}+O(q^{100}) \) 11 * q + 4 * q^3 + 3 * q^5 + 11 * q^9 - 13 * q^13 + 7 * q^17 + 8 * q^19 - 11 * q^23 + 6 * q^25 + 25 * q^27 - 3 * q^29 + 12 * q^31 - 2 * q^33 - q^37 - 21 * q^39 + 12 * q^41 + 9 * q^43 + 19 * q^45 + 17 * q^47 + 19 * q^51 - 5 * q^53 + 21 * q^55 + 11 * q^57 + 33 * q^59 - 15 * q^61 - 9 * q^65 - 5 * q^67 - 4 * q^69 - 9 * q^71 + 5 * q^73 + 44 * q^75 + 11 * q^79 - 13 * q^81 + 51 * q^83 + 33 * q^85 + 4 * q^87 + 26 * q^89 + 6 * q^93 - 19 * q^95 + 21 * q^97 + 15 * q^99

Introduction

L-functions

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Database

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