LMFDB - Embedded newform 9016.2.a.bl.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:	$1$
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} - 4x^{10} - 14x^{9} + 63x^{8} + 51x^{7} - 305x^{6} + 16x^{5} + 429x^{4} - 234x^{3} - 42x^{2} + 39x - 3 $ x^11 - 4x^10 - 14x^9 + 63x^8 + 51x^7 - 305x^6 + 16x^5 + 429x^4 - 234x^3 - 42x^2 + 39x - 3
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			$-1.59775$ 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)

$f(q)$	$=$	$q+1.59775 q^{3} +3.07107 q^{5} -0.447181 q^{9} +O(q^{10})$	$q+1.59775 q^{3} +3.07107 q^{5} -0.447181 q^{9} -4.02111 q^{11} -0.691911 q^{13} +4.90681 q^{15} -1.37169 q^{17} -0.475349 q^{19} +1.00000 q^{23} +4.43145 q^{25} -5.50775 q^{27} -4.32012 q^{29} +2.13467 q^{31} -6.42474 q^{33} -7.96971 q^{37} -1.10550 q^{39} -6.24839 q^{41} -8.22237 q^{43} -1.37332 q^{45} +2.96445 q^{47} -2.19163 q^{51} -11.0872 q^{53} -12.3491 q^{55} -0.759491 q^{57} +1.89317 q^{59} +11.5427 q^{61} -2.12490 q^{65} -1.64390 q^{67} +1.59775 q^{69} -9.61707 q^{71} +2.60236 q^{73} +7.08038 q^{75} -14.9241 q^{79} -7.45849 q^{81} +5.38588 q^{83} -4.21256 q^{85} -6.90249 q^{87} -1.41428 q^{89} +3.41068 q^{93} -1.45983 q^{95} +13.9016 q^{97} +1.79816 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 4 q^{3} - q^{5} + 11 q^{9}+O(q^{10}) $ 11 * q - 4 * q^3 - q^5 + 11 * q^9	$ 11 q - 4 q^{3} - q^{5} + 11 q^{9} + 3 q^{13} - 8 q^{15} - 5 q^{17} - 12 q^{19} + 11 q^{23} + 22 q^{25} - 19 q^{27} - 15 q^{29} - 16 q^{31} - 4 q^{33} + 3 q^{37} - q^{39} - 28 q^{41} - 9 q^{43} + 19 q^{45} - 31 q^{47} - 15 q^{51} + 13 q^{53} - 35 q^{55} - 21 q^{57} - 11 q^{59} + 19 q^{61} - 7 q^{65} + 19 q^{67} - 4 q^{69} - 5 q^{71} + 5 q^{73} - 28 q^{75} - 13 q^{79} + 35 q^{81} - 17 q^{83} - 39 q^{85} - 4 q^{87} - 10 q^{89} - 6 q^{93} + 33 q^{95} - 35 q^{97} + 13 q^{99}+O(q^{100}) $ 11 * q - 4 * q^3 - q^5 + 11 * q^9 + 3 * q^13 - 8 * q^15 - 5 * q^17 - 12 * q^19 + 11 * q^23 + 22 * q^25 - 19 * q^27 - 15 * q^29 - 16 * q^31 - 4 * q^33 + 3 * q^37 - q^39 - 28 * q^41 - 9 * q^43 + 19 * q^45 - 31 * q^47 - 15 * q^51 + 13 * q^53 - 35 * q^55 - 21 * q^57 - 11 * q^59 + 19 * q^61 - 7 * q^65 + 19 * q^67 - 4 * q^69 - 5 * q^71 + 5 * q^73 - 28 * q^75 - 13 * q^79 + 35 * q^81 - 17 * q^83 - 39 * q^85 - 4 * q^87 - 10 * q^89 - 6 * q^93 + 33 * q^95 - 35 * q^97 + 13 * 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
$3$	1.59775	0.922464	0.461232	−	0.887280i	$-0.347408\pi$
$3$	1.59775	0.922464	0.461232	+	0.887280i	$0.347408\pi$
$4$	0	0
$5$	3.07107	1.37342	0.686712	−	0.726930i	$-0.259054\pi$
$5$	3.07107	1.37342	0.686712	+	0.726930i	$0.259054\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.447181	−0.149060
$10$	0	0
$11$	−4.02111	−1.21241	−0.606205	−	0.795309i	$-0.707309\pi$
$11$	−4.02111	−1.21241	−0.606205	+	0.795309i	$0.707309\pi$
$12$	0	0
$13$	−0.691911	−0.191901	−0.0959507	−	0.995386i	$-0.530589\pi$
$13$	−0.691911	−0.191901	−0.0959507	+	0.995386i	$0.530589\pi$
$14$	0	0
$15$	4.90681	1.26693
$16$	0	0
$17$	−1.37169	−0.332684	−0.166342	−	0.986068i	$-0.553196\pi$
$17$	−1.37169	−0.332684	−0.166342	+	0.986068i	$0.553196\pi$
$18$	0	0
$19$	−0.475349	−0.109052	−0.0545262	−	0.998512i	$-0.517365\pi$
$19$	−0.475349	−0.109052	−0.0545262	+	0.998512i	$0.517365\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	4.43145	0.886291
$26$	0	0
$27$	−5.50775	−1.05997
$28$	0	0
$29$	−4.32012	−0.802226	−0.401113	−	0.916029i	$-0.631377\pi$
$29$	−4.32012	−0.802226	−0.401113	+	0.916029i	$0.631377\pi$
$30$	0	0
$31$	2.13467	0.383398	0.191699	−	0.981454i	$-0.438600\pi$
$31$	2.13467	0.383398	0.191699	+	0.981454i	$0.438600\pi$
$32$	0	0
$33$	−6.42474	−1.11840
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.96971	−1.31021	−0.655106	−	0.755537i	$-0.727376\pi$
$37$	−7.96971	−1.31021	−0.655106	+	0.755537i	$0.727376\pi$
$38$	0	0
$39$	−1.10550	−0.177022
$40$	0	0
$41$	−6.24839	−0.975834	−0.487917	−	0.872890i	$-0.662243\pi$
$41$	−6.24839	−0.975834	−0.487917	+	0.872890i	$0.662243\pi$
$42$	0	0
$43$	−8.22237	−1.25390	−0.626950	−	0.779060i	$-0.715697\pi$
$43$	−8.22237	−1.25390	−0.626950	+	0.779060i	$0.715697\pi$
$44$	0	0
$45$	−1.37332	−0.204723
$46$	0	0
$47$	2.96445	0.432409	0.216205	−	0.976348i	$-0.430632\pi$
$47$	2.96445	0.432409	0.216205	+	0.976348i	$0.430632\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.19163	−0.306889
$52$	0	0
$53$	−11.0872	−1.52294	−0.761471	−	0.648199i	$-0.775523\pi$
$53$	−11.0872	−1.52294	−0.761471	+	0.648199i	$0.775523\pi$
$54$	0	0
$55$	−12.3491	−1.66515
$56$	0	0
$57$	−0.759491	−0.100597
$58$	0	0
$59$	1.89317	0.246470	0.123235	−	0.992378i	$-0.460673\pi$
$59$	1.89317	0.246470	0.123235	+	0.992378i	$0.460673\pi$
$60$	0	0
$61$	11.5427	1.47789	0.738947	−	0.673764i	$-0.235323\pi$
$61$	11.5427	1.47789	0.738947	+	0.673764i	$0.235323\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.12490	−0.263562
$66$	0	0
$67$	−1.64390	−0.200835	−0.100417	−	0.994945i	$-0.532018\pi$
$67$	−1.64390	−0.200835	−0.100417	+	0.994945i	$0.532018\pi$
$68$	0	0
$69$	1.59775	0.192347
$70$	0	0
$71$	−9.61707	−1.14134	−0.570668	−	0.821181i	$-0.693316\pi$
$71$	−9.61707	−1.14134	−0.570668	+	0.821181i	$0.693316\pi$
$72$	0	0
$73$	2.60236	0.304583	0.152291	−	0.988336i	$-0.451335\pi$
$73$	2.60236	0.304583	0.152291	+	0.988336i	$0.451335\pi$
$74$	0	0
$75$	7.08038	0.817571
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.9241	−1.67910	−0.839548	−	0.543285i	$-0.817180\pi$
$79$	−14.9241	−1.67910	−0.839548	+	0.543285i	$0.817180\pi$
$80$	0	0
$81$	−7.45849	−0.828721
$82$	0	0
$83$	5.38588	0.591177	0.295588	−	0.955315i	$-0.404484\pi$
$83$	5.38588	0.591177	0.295588	+	0.955315i	$0.404484\pi$
$84$	0	0
$85$	−4.21256	−0.456916
$86$	0	0
$87$	−6.90249	−0.740025
$88$	0	0
$89$	−1.41428	−0.149913	−0.0749565	−	0.997187i	$-0.523882\pi$
$89$	−1.41428	−0.149913	−0.0749565	+	0.997187i	$0.523882\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.41068	0.353671
$94$	0	0
$95$	−1.45983	−0.149775
$96$	0	0
$97$	13.9016	1.41150	0.705749	−	0.708462i	$-0.250611\pi$
$97$	13.9016	1.41150	0.705749	+	0.708462i	$0.250611\pi$
$98$	0	0
$99$	1.79816	0.180722
$100$	0	0
$101$	3.52841	0.351090	0.175545	−	0.984471i	$-0.443831\pi$
$101$	3.52841	0.351090	0.175545	+	0.984471i	$0.443831\pi$
$102$	0	0
$103$	−4.86071	−0.478940	−0.239470	−	0.970904i	$-0.576974\pi$
$103$	−4.86071	−0.478940	−0.239470	+	0.970904i	$0.576974\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.31748	0.900755	0.450377	−	0.892838i	$-0.351289\pi$
$107$	9.31748	0.900755	0.450377	+	0.892838i	$0.351289\pi$
$108$	0	0
$109$	−0.884809	−0.0847493	−0.0423747	−	0.999102i	$-0.513492\pi$
$109$	−0.884809	−0.0847493	−0.0423747	+	0.999102i	$0.513492\pi$
$110$	0	0
$111$	−12.7336	−1.20862
$112$	0	0
$113$	8.03855	0.756204	0.378102	−	0.925764i	$-0.376577\pi$
$113$	8.03855	0.756204	0.378102	+	0.925764i	$0.376577\pi$
$114$	0	0
$115$	3.07107	0.286379
$116$	0	0
$117$	0.309409	0.0286049
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.16931	0.469937
$122$	0	0
$123$	−9.98339	−0.900172
$124$	0	0
$125$	−1.74604	−0.156171
$126$	0	0
$127$	−3.92308	−0.348117	−0.174059	−	0.984735i	$-0.555688\pi$
$127$	−3.92308	−0.348117	−0.174059	+	0.984735i	$0.555688\pi$
$128$	0	0
$129$	−13.1373	−1.15668
$130$	0	0
$131$	−17.3444	−1.51539	−0.757694	−	0.652610i	$-0.773674\pi$
$131$	−17.3444	−1.51539	−0.757694	+	0.652610i	$0.773674\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−16.9147	−1.45578
$136$	0	0
$137$	−5.95677	−0.508921	−0.254461	−	0.967083i	$-0.581898\pi$
$137$	−5.95677	−0.508921	−0.254461	+	0.967083i	$0.581898\pi$
$138$	0	0
$139$	17.2978	1.46718	0.733591	−	0.679592i	$-0.237843\pi$
$139$	17.2978	1.46718	0.733591	+	0.679592i	$0.237843\pi$
$140$	0	0
$141$	4.73646	0.398882
$142$	0	0
$143$	2.78225	0.232663
$144$	0	0
$145$	−13.2674	−1.10180
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.58570	0.703368	0.351684	−	0.936119i	$-0.385609\pi$
$149$	8.58570	0.703368	0.351684	+	0.936119i	$0.385609\pi$
$150$	0	0
$151$	3.23223	0.263035	0.131517	−	0.991314i	$-0.458015\pi$
$151$	3.23223	0.263035	0.131517	+	0.991314i	$0.458015\pi$
$152$	0	0
$153$	0.613394	0.0495899
$154$	0	0
$155$	6.55572	0.526568
$156$	0	0
$157$	8.08788	0.645483	0.322741	−	0.946487i	$-0.395396\pi$
$157$	8.08788	0.645483	0.322741	+	0.946487i	$0.395396\pi$
$158$	0	0
$159$	−17.7146	−1.40486
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−25.2383	−1.97682	−0.988410	−	0.151811i	$-0.951490\pi$
$163$	−25.2383	−1.97682	−0.988410	+	0.151811i	$0.951490\pi$
$164$	0	0
$165$	−19.7308	−1.53604
$166$	0	0
$167$	−13.3163	−1.03044	−0.515221	−	0.857057i	$-0.672290\pi$
$167$	−13.3163	−1.03044	−0.515221	+	0.857057i	$0.672290\pi$
$168$	0	0
$169$	−12.5213	−0.963174
$170$	0	0
$171$	0.212567	0.0162554
$172$	0	0
$173$	9.64041	0.732947	0.366473	−	0.930429i	$-0.380565\pi$
$173$	9.64041	0.732947	0.366473	+	0.930429i	$0.380565\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.02482	0.227359
$178$	0	0
$179$	13.0611	0.976229	0.488115	−	0.872780i	$-0.337685\pi$
$179$	13.0611	0.976229	0.488115	+	0.872780i	$0.337685\pi$
$180$	0	0
$181$	11.6259	0.864147	0.432074	−	0.901838i	$-0.357782\pi$
$181$	11.6259	0.864147	0.432074	+	0.901838i	$0.357782\pi$
$182$	0	0
$183$	18.4424	1.36330
$184$	0	0
$185$	−24.4755	−1.79948
$186$	0	0
$187$	5.51572	0.403349
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.6102	−0.840081	−0.420041	−	0.907505i	$-0.637984\pi$
$191$	−11.6102	−0.840081	−0.420041	+	0.907505i	$0.637984\pi$
$192$	0	0
$193$	26.9118	1.93716	0.968578	−	0.248711i	$-0.0800070\pi$
$193$	26.9118	1.93716	0.968578	+	0.248711i	$0.0800070\pi$
$194$	0	0
$195$	−3.39507	−0.243126
$196$	0	0
$197$	−20.6388	−1.47046	−0.735228	−	0.677820i	$-0.762925\pi$
$197$	−20.6388	−1.47046	−0.735228	+	0.677820i	$0.762925\pi$
$198$	0	0
$199$	−8.59158	−0.609041	−0.304521	−	0.952506i	$-0.598496\pi$
$199$	−8.59158	−0.609041	−0.304521	+	0.952506i	$0.598496\pi$
$200$	0	0
$201$	−2.62656	−0.185263
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−19.1892	−1.34023
$206$	0	0
$207$	−0.447181	−0.0310812
$208$	0	0
$209$	1.91143	0.132216
$210$	0	0
$211$	3.29359	0.226740	0.113370	−	0.993553i	$-0.463835\pi$
$211$	3.29359	0.226740	0.113370	+	0.993553i	$0.463835\pi$
$212$	0	0
$213$	−15.3657	−1.05284
$214$	0	0
$215$	−25.2514	−1.72213
$216$	0	0
$217$	0	0
$218$	0	0
$219$	4.15793	0.280967
$220$	0	0
$221$	0.949088	0.0638425
$222$	0	0
$223$	−21.1250	−1.41463	−0.707316	−	0.706897i	$-0.750094\pi$
$223$	−21.1250	−1.41463	−0.707316	+	0.706897i	$0.750094\pi$
$224$	0	0
$225$	−1.98166	−0.132111
$226$	0	0
$227$	−25.7312	−1.70784	−0.853920	−	0.520404i	$-0.825781\pi$
$227$	−25.7312	−1.70784	−0.853920	+	0.520404i	$0.825781\pi$
$228$	0	0
$229$	10.2017	0.674150	0.337075	−	0.941478i	$-0.390562\pi$
$229$	10.2017	0.674150	0.337075	+	0.941478i	$0.390562\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.65894	0.632778	0.316389	−	0.948630i	$-0.397530\pi$
$233$	9.65894	0.632778	0.316389	+	0.948630i	$0.397530\pi$
$234$	0	0
$235$	9.10402	0.593881
$236$	0	0
$237$	−23.8451	−1.54891
$238$	0	0
$239$	1.25565	0.0812215	0.0406107	−	0.999175i	$-0.487070\pi$
$239$	1.25565	0.0812215	0.0406107	+	0.999175i	$0.487070\pi$
$240$	0	0
$241$	11.8102	0.760765	0.380382	−	0.924829i	$-0.375792\pi$
$241$	11.8102	0.760765	0.380382	+	0.924829i	$0.375792\pi$
$242$	0	0
$243$	4.60641	0.295501
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.328899	0.0209273
$248$	0	0
$249$	8.60531	0.545339
$250$	0	0
$251$	17.6474	1.11389	0.556947	−	0.830548i	$-0.311973\pi$
$251$	17.6474	1.11389	0.556947	+	0.830548i	$0.311973\pi$
$252$	0	0
$253$	−4.02111	−0.252805
$254$	0	0
$255$	−6.73063	−0.421488
$256$	0	0
$257$	0.552035	0.0344350	0.0172175	−	0.999852i	$-0.494519\pi$
$257$	0.552035	0.0344350	0.0172175	+	0.999852i	$0.494519\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.93187	0.119580
$262$	0	0
$263$	−1.94549	−0.119964	−0.0599821	−	0.998199i	$-0.519104\pi$
$263$	−1.94549	−0.119964	−0.0599821	+	0.998199i	$0.519104\pi$
$264$	0	0
$265$	−34.0495	−2.09164
$266$	0	0
$267$	−2.25967	−0.138289
$268$	0	0
$269$	−27.0606	−1.64992	−0.824958	−	0.565194i	$-0.808801\pi$
$269$	−27.0606	−1.64992	−0.824958	+	0.565194i	$0.808801\pi$
$270$	0	0
$271$	−13.6177	−0.827216	−0.413608	−	0.910455i	$-0.635732\pi$
$271$	−13.6177	−0.827216	−0.413608	+	0.910455i	$0.635732\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−17.8194	−1.07455
$276$	0	0
$277$	28.7941	1.73007	0.865034	−	0.501713i	$-0.167297\pi$
$277$	28.7941	1.73007	0.865034	+	0.501713i	$0.167297\pi$
$278$	0	0
$279$	−0.954583	−0.0571494
$280$	0	0
$281$	−26.0709	−1.55526	−0.777629	−	0.628724i	$-0.783578\pi$
$281$	−26.0709	−1.55526	−0.777629	+	0.628724i	$0.783578\pi$
$282$	0	0
$283$	−2.65544	−0.157849	−0.0789247	−	0.996881i	$-0.525149\pi$
$283$	−2.65544	−0.157849	−0.0789247	+	0.996881i	$0.525149\pi$
$284$	0	0
$285$	−2.33245	−0.138162
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.1185	−0.889321
$290$	0	0
$291$	22.2114	1.30206
$292$	0	0
$293$	8.92203	0.521230	0.260615	−	0.965443i	$-0.416075\pi$
$293$	8.92203	0.521230	0.260615	+	0.965443i	$0.416075\pi$
$294$	0	0
$295$	5.81405	0.338507
$296$	0	0
$297$	22.1473	1.28511
$298$	0	0
$299$	−0.691911	−0.0400142
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.63753	0.323868
$304$	0	0
$305$	35.4485	2.02977
$306$	0	0
$307$	−13.1992	−0.753319	−0.376660	−	0.926352i	$-0.622927\pi$
$307$	−13.1992	−0.753319	−0.376660	+	0.926352i	$0.622927\pi$
$308$	0	0
$309$	−7.76623	−0.441805
$310$	0	0
$311$	13.8154	0.783401	0.391700	−	0.920093i	$-0.371887\pi$
$311$	13.8154	0.783401	0.391700	+	0.920093i	$0.371887\pi$
$312$	0	0
$313$	1.17649	0.0664991	0.0332496	−	0.999447i	$-0.489414\pi$
$313$	1.17649	0.0664991	0.0332496	+	0.999447i	$0.489414\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.77408	−0.268139	−0.134070	−	0.990972i	$-0.542805\pi$
$317$	−4.77408	−0.268139	−0.134070	+	0.990972i	$0.542805\pi$
$318$	0	0
$319$	17.3717	0.972627
$320$	0	0
$321$	14.8870	0.830914
$322$	0	0
$323$	0.652032	0.0362800
$324$	0	0
$325$	−3.06617	−0.170081
$326$	0	0
$327$	−1.41371	−0.0781782
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−0.0884339	−0.00486077	−0.00243038	−	0.999997i	$-0.500774\pi$
$331$	−0.0884339	−0.00486077	−0.00243038	+	0.999997i	$0.500774\pi$
$332$	0	0
$333$	3.56390	0.195300
$334$	0	0
$335$	−5.04854	−0.275831
$336$	0	0
$337$	−1.60561	−0.0874634	−0.0437317	−	0.999043i	$-0.513925\pi$
$337$	−1.60561	−0.0874634	−0.0437317	+	0.999043i	$0.513925\pi$
$338$	0	0
$339$	12.8436	0.697571
$340$	0	0
$341$	−8.58374	−0.464836
$342$	0	0
$343$	0	0
$344$	0	0
$345$	4.90681	0.264174
$346$	0	0
$347$	35.7201	1.91756	0.958778	−	0.284158i	$-0.0917139\pi$
$347$	35.7201	1.91756	0.958778	+	0.284158i	$0.0917139\pi$
$348$	0	0
$349$	8.58333	0.459455	0.229727	−	0.973255i	$-0.426217\pi$
$349$	8.58333	0.459455	0.229727	+	0.973255i	$0.426217\pi$
$350$	0	0
$351$	3.81087	0.203409
$352$	0	0
$353$	29.3036	1.55967	0.779837	−	0.625982i	$-0.215302\pi$
$353$	29.3036	1.55967	0.779837	+	0.625982i	$0.215302\pi$
$354$	0	0
$355$	−29.5347	−1.56754
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.66959	0.140895	0.0704477	−	0.997515i	$-0.477557\pi$
$359$	2.66959	0.140895	0.0704477	+	0.997515i	$0.477557\pi$
$360$	0	0
$361$	−18.7740	−0.988108
$362$	0	0
$363$	8.25929	0.433500
$364$	0	0
$365$	7.99202	0.418321
$366$	0	0
$367$	33.8670	1.76784	0.883921	−	0.467636i	$-0.154894\pi$
$367$	33.8670	1.76784	0.883921	+	0.467636i	$0.154894\pi$
$368$	0	0
$369$	2.79416	0.145458
$370$	0	0
$371$	0	0
$372$	0	0
$373$	12.0277	0.622769	0.311385	−	0.950284i	$-0.399207\pi$
$373$	12.0277	0.622769	0.311385	+	0.950284i	$0.399207\pi$
$374$	0	0
$375$	−2.78974	−0.144062
$376$	0	0
$377$	2.98914	0.153948
$378$	0	0
$379$	−35.3658	−1.81662	−0.908311	−	0.418296i	$-0.862627\pi$
$379$	−35.3658	−1.81662	−0.908311	+	0.418296i	$0.862627\pi$
$380$	0	0
$381$	−6.26812	−0.321126
$382$	0	0
$383$	−32.9182	−1.68204	−0.841020	−	0.541003i	$-0.818045\pi$
$383$	−32.9182	−1.68204	−0.841020	+	0.541003i	$0.818045\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.67688	0.186906
$388$	0	0
$389$	7.68289	0.389538	0.194769	−	0.980849i	$-0.437604\pi$
$389$	7.68289	0.389538	0.194769	+	0.980849i	$0.437604\pi$
$390$	0	0
$391$	−1.37169	−0.0693694
$392$	0	0
$393$	−27.7121	−1.39789
$394$	0	0
$395$	−45.8330	−2.30611
$396$	0	0
$397$	15.4049	0.773148	0.386574	−	0.922258i	$-0.373658\pi$
$397$	15.4049	0.773148	0.386574	+	0.922258i	$0.373658\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.3950	1.06841	0.534207	−	0.845353i	$-0.320610\pi$
$401$	21.3950	1.06841	0.534207	+	0.845353i	$0.320610\pi$
$402$	0	0
$403$	−1.47700	−0.0735746
$404$	0	0
$405$	−22.9055	−1.13818
$406$	0	0
$407$	32.0471	1.58851
$408$	0	0
$409$	1.44888	0.0716426	0.0358213	−	0.999358i	$-0.488595\pi$
$409$	1.44888	0.0716426	0.0358213	+	0.999358i	$0.488595\pi$
$410$	0	0
$411$	−9.51746	−0.469462
$412$	0	0
$413$	0	0
$414$	0	0
$415$	16.5404	0.811936
$416$	0	0
$417$	27.6377	1.35342
$418$	0	0
$419$	27.2624	1.33186	0.665929	−	0.746015i	$-0.268036\pi$
$419$	27.2624	1.33186	0.665929	+	0.746015i	$0.268036\pi$
$420$	0	0
$421$	12.7079	0.619345	0.309672	−	0.950843i	$-0.399781\pi$
$421$	12.7079	0.619345	0.309672	+	0.950843i	$0.399781\pi$
$422$	0	0
$423$	−1.32564	−0.0644550
$424$	0	0
$425$	−6.07859	−0.294855
$426$	0	0
$427$	0	0
$428$	0	0
$429$	4.44535	0.214623
$430$	0	0
$431$	6.90525	0.332614	0.166307	−	0.986074i	$-0.446816\pi$
$431$	6.90525	0.332614	0.166307	+	0.986074i	$0.446816\pi$
$432$	0	0
$433$	10.9197	0.524768	0.262384	−	0.964964i	$-0.415491\pi$
$433$	10.9197	0.524768	0.262384	+	0.964964i	$0.415491\pi$
$434$	0	0
$435$	−21.1980	−1.01637
$436$	0	0
$437$	−0.475349	−0.0227390
$438$	0	0
$439$	−21.2003	−1.01183	−0.505917	−	0.862582i	$-0.668846\pi$
$439$	−21.2003	−1.01183	−0.505917	+	0.862582i	$0.668846\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10.1155	−0.480602	−0.240301	−	0.970698i	$-0.577246\pi$
$443$	−10.1155	−0.480602	−0.240301	+	0.970698i	$0.577246\pi$
$444$	0	0
$445$	−4.34334	−0.205894
$446$	0	0
$447$	13.7178	0.648831
$448$	0	0
$449$	8.05346	0.380066	0.190033	−	0.981778i	$-0.439140\pi$
$449$	8.05346	0.380066	0.190033	+	0.981778i	$0.439140\pi$
$450$	0	0
$451$	25.1254	1.18311
$452$	0	0
$453$	5.16430	0.242640
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−36.7771	−1.72036	−0.860179	−	0.509992i	$-0.829648\pi$
$457$	−36.7771	−1.72036	−0.860179	+	0.509992i	$0.829648\pi$
$458$	0	0
$459$	7.55493	0.352634
$460$	0	0
$461$	−23.3835	−1.08908	−0.544540	−	0.838735i	$-0.683296\pi$
$461$	−23.3835	−1.08908	−0.544540	+	0.838735i	$0.683296\pi$
$462$	0	0
$463$	5.79098	0.269130	0.134565	−	0.990905i	$-0.457036\pi$
$463$	5.79098	0.269130	0.134565	+	0.990905i	$0.457036\pi$
$464$	0	0
$465$	10.4744	0.485740
$466$	0	0
$467$	−37.1150	−1.71748	−0.858740	−	0.512412i	$-0.828752\pi$
$467$	−37.1150	−1.71748	−0.858740	+	0.512412i	$0.828752\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	12.9224	0.595435
$472$	0	0
$473$	33.0630	1.52024
$474$	0	0
$475$	−2.10649	−0.0966522
$476$	0	0
$477$	4.95798	0.227010
$478$	0	0
$479$	3.09026	0.141197	0.0705987	−	0.997505i	$-0.477509\pi$
$479$	3.09026	0.141197	0.0705987	+	0.997505i	$0.477509\pi$
$480$	0	0
$481$	5.51433	0.251432
$482$	0	0
$483$	0	0
$484$	0	0
$485$	42.6929	1.93858
$486$	0	0
$487$	12.5413	0.568299	0.284149	−	0.958780i	$-0.408289\pi$
$487$	12.5413	0.568299	0.284149	+	0.958780i	$0.408289\pi$
$488$	0	0
$489$	−40.3247	−1.82354
$490$	0	0
$491$	−28.1331	−1.26963	−0.634814	−	0.772665i	$-0.718923\pi$
$491$	−28.1331	−1.26963	−0.634814	+	0.772665i	$0.718923\pi$
$492$	0	0
$493$	5.92587	0.266888
$494$	0	0
$495$	5.52227	0.248208
$496$	0	0
$497$	0	0
$498$	0	0
$499$	13.6164	0.609554	0.304777	−	0.952424i	$-0.401418\pi$
$499$	13.6164	0.609554	0.304777	+	0.952424i	$0.401418\pi$
$500$	0	0
$501$	−21.2761	−0.950546
$502$	0	0
$503$	−7.60295	−0.338999	−0.169499	−	0.985530i	$-0.554215\pi$
$503$	−7.60295	−0.338999	−0.169499	+	0.985530i	$0.554215\pi$
$504$	0	0
$505$	10.8360	0.482195
$506$	0	0
$507$	−20.0059	−0.888493
$508$	0	0
$509$	34.5006	1.52921	0.764606	−	0.644498i	$-0.222934\pi$
$509$	34.5006	1.52921	0.764606	+	0.644498i	$0.222934\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2.61810	0.115592
$514$	0	0
$515$	−14.9276	−0.657788
$516$	0	0
$517$	−11.9204	−0.524257
$518$	0	0
$519$	15.4030	0.676117
$520$	0	0
$521$	−13.8248	−0.605675	−0.302837	−	0.953042i	$-0.597934\pi$
$521$	−13.8248	−0.605675	−0.302837	+	0.953042i	$0.597934\pi$
$522$	0	0
$523$	9.72600	0.425288	0.212644	−	0.977130i	$-0.431792\pi$
$523$	9.72600	0.425288	0.212644	+	0.977130i	$0.431792\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.92811	−0.127550
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−0.846588	−0.0367388
$532$	0	0
$533$	4.32333	0.187264
$534$	0	0
$535$	28.6146	1.23712
$536$	0	0
$537$	20.8684	0.900536
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−28.1469	−1.21013	−0.605065	−	0.796176i	$-0.706853\pi$
$541$	−28.1469	−1.21013	−0.605065	+	0.796176i	$0.706853\pi$
$542$	0	0
$543$	18.5754	0.797145
$544$	0	0
$545$	−2.71731	−0.116397
$546$	0	0
$547$	35.3250	1.51039	0.755193	−	0.655503i	$-0.227543\pi$
$547$	35.3250	1.51039	0.755193	+	0.655503i	$0.227543\pi$
$548$	0	0
$549$	−5.16168	−0.220295
$550$	0	0
$551$	2.05356	0.0874847
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−39.1059	−1.65995
$556$	0	0
$557$	12.2217	0.517849	0.258924	−	0.965898i	$-0.416632\pi$
$557$	12.2217	0.517849	0.258924	+	0.965898i	$0.416632\pi$
$558$	0	0
$559$	5.68914	0.240625
$560$	0	0
$561$	8.81277	0.372075
$562$	0	0
$563$	−6.04651	−0.254830	−0.127415	−	0.991850i	$-0.540668\pi$
$563$	−6.04651	−0.254830	−0.127415	+	0.991850i	$0.540668\pi$
$564$	0	0
$565$	24.6869	1.03859
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.7241	0.784955	0.392477	−	0.919762i	$-0.371618\pi$
$569$	18.7241	0.784955	0.392477	+	0.919762i	$0.371618\pi$
$570$	0	0
$571$	6.18730	0.258930	0.129465	−	0.991584i	$-0.458674\pi$
$571$	6.18730	0.258930	0.129465	+	0.991584i	$0.458674\pi$
$572$	0	0
$573$	−18.5502	−0.774945
$574$	0	0
$575$	4.43145	0.184804
$576$	0	0
$577$	10.8994	0.453748	0.226874	−	0.973924i	$-0.427149\pi$
$577$	10.8994	0.453748	0.226874	+	0.973924i	$0.427149\pi$
$578$	0	0
$579$	42.9985	1.78696
$580$	0	0
$581$	0	0
$582$	0	0
$583$	44.5828	1.84643
$584$	0	0
$585$	0.950216	0.0392866
$586$	0	0
$587$	−1.11608	−0.0460654	−0.0230327	−	0.999735i	$-0.507332\pi$
$587$	−1.11608	−0.0460654	−0.0230327	+	0.999735i	$0.507332\pi$
$588$	0	0
$589$	−1.01471	−0.0418105
$590$	0	0
$591$	−32.9758	−1.35644
$592$	0	0
$593$	−10.8131	−0.444042	−0.222021	−	0.975042i	$-0.571265\pi$
$593$	−10.8131	−0.444042	−0.222021	+	0.975042i	$0.571265\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−13.7272	−0.561818
$598$	0	0
$599$	−6.20397	−0.253487	−0.126744	−	0.991936i	$-0.540453\pi$
$599$	−6.20397	−0.253487	−0.126744	+	0.991936i	$0.540453\pi$
$600$	0	0
$601$	−15.5717	−0.635181	−0.317591	−	0.948228i	$-0.602874\pi$
$601$	−15.5717	−0.635181	−0.317591	+	0.948228i	$0.602874\pi$
$602$	0	0
$603$	0.735122	0.0299365
$604$	0	0
$605$	15.8753	0.645423
$606$	0	0
$607$	−7.72027	−0.313356	−0.156678	−	0.987650i	$-0.550079\pi$
$607$	−7.72027	−0.313356	−0.156678	+	0.987650i	$0.550079\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.05113	−0.0829800
$612$	0	0
$613$	−25.4094	−1.02627	−0.513137	−	0.858306i	$-0.671517\pi$
$613$	−25.4094	−1.02627	−0.513137	+	0.858306i	$0.671517\pi$
$614$	0	0
$615$	−30.6597	−1.23632
$616$	0	0
$617$	46.2975	1.86387	0.931933	−	0.362630i	$-0.118121\pi$
$617$	46.2975	1.86387	0.931933	+	0.362630i	$0.118121\pi$
$618$	0	0
$619$	−4.89485	−0.196741	−0.0983703	−	0.995150i	$-0.531363\pi$
$619$	−4.89485	−0.196741	−0.0983703	+	0.995150i	$0.531363\pi$
$620$	0	0
$621$	−5.50775	−0.221018
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−27.5195	−1.10078
$626$	0	0
$627$	3.05399	0.121965
$628$	0	0
$629$	10.9320	0.435887
$630$	0	0
$631$	45.4349	1.80873	0.904367	−	0.426756i	$-0.140344\pi$
$631$	45.4349	1.80873	0.904367	+	0.426756i	$0.140344\pi$
$632$	0	0
$633$	5.26234	0.209159
$634$	0	0
$635$	−12.0480	−0.478112
$636$	0	0
$637$	0	0
$638$	0	0
$639$	4.30057	0.170128
$640$	0	0
$641$	−8.68276	−0.342948	−0.171474	−	0.985189i	$-0.554853\pi$
$641$	−8.68276	−0.342948	−0.171474	+	0.985189i	$0.554853\pi$
$642$	0	0
$643$	−45.5312	−1.79558	−0.897788	−	0.440427i	$-0.854827\pi$
$643$	−45.5312	−1.79558	−0.897788	+	0.440427i	$0.854827\pi$
$644$	0	0
$645$	−40.3456	−1.58861
$646$	0	0
$647$	27.6835	1.08835	0.544175	−	0.838972i	$-0.316843\pi$
$647$	27.6835	1.08835	0.544175	+	0.838972i	$0.316843\pi$
$648$	0	0
$649$	−7.61264	−0.298822
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20.6090	−0.806494	−0.403247	−	0.915091i	$-0.632118\pi$
$653$	−20.6090	−0.806494	−0.403247	+	0.915091i	$0.632118\pi$
$654$	0	0
$655$	−53.2658	−2.08127
$656$	0	0
$657$	−1.16372	−0.0454012
$658$	0	0
$659$	7.35098	0.286353	0.143177	−	0.989697i	$-0.454268\pi$
$659$	7.35098	0.286353	0.143177	+	0.989697i	$0.454268\pi$
$660$	0	0
$661$	31.2848	1.21684	0.608419	−	0.793616i	$-0.291804\pi$
$661$	31.2848	1.21684	0.608419	+	0.793616i	$0.291804\pi$
$662$	0	0
$663$	1.51641	0.0588924
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.32012	−0.167276
$668$	0	0
$669$	−33.7525	−1.30495
$670$	0	0
$671$	−46.4145	−1.79181
$672$	0	0
$673$	−32.2705	−1.24394	−0.621968	−	0.783043i	$-0.713667\pi$
$673$	−32.2705	−1.24394	−0.621968	+	0.783043i	$0.713667\pi$
$674$	0	0
$675$	−24.4073	−0.939439
$676$	0	0
$677$	−32.8675	−1.26320	−0.631601	−	0.775294i	$-0.717602\pi$
$677$	−32.8675	−1.26320	−0.631601	+	0.775294i	$0.717602\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−41.1121	−1.57542
$682$	0	0
$683$	22.1865	0.848945	0.424472	−	0.905441i	$-0.360460\pi$
$683$	22.1865	0.848945	0.424472	+	0.905441i	$0.360460\pi$
$684$	0	0
$685$	−18.2936	−0.698964
$686$	0	0
$687$	16.2999	0.621879
$688$	0	0
$689$	7.67134	0.292255
$690$	0	0
$691$	−39.9626	−1.52025	−0.760124	−	0.649778i	$-0.774862\pi$
$691$	−39.9626	−1.52025	−0.760124	+	0.649778i	$0.774862\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	53.1228	2.01506
$696$	0	0
$697$	8.57086	0.324644
$698$	0	0
$699$	15.4326	0.583715
$700$	0	0
$701$	−42.5053	−1.60540	−0.802702	−	0.596380i	$-0.796605\pi$
$701$	−42.5053	−1.60540	−0.802702	+	0.596380i	$0.796605\pi$
$702$	0	0
$703$	3.78839	0.142882
$704$	0	0
$705$	14.5460	0.547834
$706$	0	0
$707$	0	0
$708$	0	0
$709$	7.28902	0.273745	0.136872	−	0.990589i	$-0.456295\pi$
$709$	7.28902	0.273745	0.136872	+	0.990589i	$0.456295\pi$
$710$	0	0
$711$	6.67378	0.250286
$712$	0	0
$713$	2.13467	0.0799440
$714$	0	0
$715$	8.54447	0.319545
$716$	0	0
$717$	2.00623	0.0749239
$718$	0	0
$719$	−4.68172	−0.174599	−0.0872994	−	0.996182i	$-0.527824\pi$
$719$	−4.68172	−0.174599	−0.0872994	+	0.996182i	$0.527824\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	18.8699	0.701778
$724$	0	0
$725$	−19.1444	−0.711006
$726$	0	0
$727$	−37.0061	−1.37248	−0.686240	−	0.727375i	$-0.740740\pi$
$727$	−37.0061	−1.37248	−0.686240	+	0.727375i	$0.740740\pi$
$728$	0	0
$729$	29.7354	1.10131
$730$	0	0
$731$	11.2785	0.417152
$732$	0	0
$733$	18.2970	0.675816	0.337908	−	0.941179i	$-0.390281\pi$
$733$	18.2970	0.675816	0.337908	+	0.941179i	$0.390281\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.61032	0.243494
$738$	0	0
$739$	22.0023	0.809367	0.404684	−	0.914457i	$-0.367382\pi$
$739$	22.0023	0.809367	0.404684	+	0.914457i	$0.367382\pi$
$740$	0	0
$741$	0.525500	0.0193047
$742$	0	0
$743$	8.12518	0.298084	0.149042	−	0.988831i	$-0.452381\pi$
$743$	8.12518	0.298084	0.149042	+	0.988831i	$0.452381\pi$
$744$	0	0
$745$	26.3673	0.966021
$746$	0	0
$747$	−2.40846	−0.0881209
$748$	0	0
$749$	0	0
$750$	0	0
$751$	46.3174	1.69015	0.845073	−	0.534650i	$-0.179557\pi$
$751$	46.3174	1.69015	0.845073	+	0.534650i	$0.179557\pi$
$752$	0	0
$753$	28.1962	1.02753
$754$	0	0
$755$	9.92638	0.361258
$756$	0	0
$757$	−49.7354	−1.80766	−0.903832	−	0.427888i	$-0.859258\pi$
$757$	−49.7354	−1.80766	−0.903832	+	0.427888i	$0.859258\pi$
$758$	0	0
$759$	−6.42474	−0.233203
$760$	0	0
$761$	−21.8067	−0.790493	−0.395247	−	0.918575i	$-0.629341\pi$
$761$	−21.8067	−0.790493	−0.395247	+	0.918575i	$0.629341\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.88377	0.0681080
$766$	0	0
$767$	−1.30990	−0.0472979
$768$	0	0
$769$	−5.15879	−0.186031	−0.0930153	−	0.995665i	$-0.529651\pi$
$769$	−5.15879	−0.186031	−0.0930153	+	0.995665i	$0.529651\pi$
$770$	0	0
$771$	0.882016	0.0317650
$772$	0	0
$773$	45.8430	1.64886	0.824430	−	0.565964i	$-0.191496\pi$
$773$	45.8430	1.64886	0.824430	+	0.565964i	$0.191496\pi$
$774$	0	0
$775$	9.45969	0.339802
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.97016	0.106417
$780$	0	0
$781$	38.6713	1.38377
$782$	0	0
$783$	23.7941	0.850333
$784$	0	0
$785$	24.8384	0.886521
$786$	0	0
$787$	41.8558	1.49200	0.745998	−	0.665948i	$-0.231973\pi$
$787$	41.8558	1.49200	0.745998	+	0.665948i	$0.231973\pi$
$788$	0	0
$789$	−3.10842	−0.110663
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−7.98653	−0.283610
$794$	0	0
$795$	−54.4028	−1.92947
$796$	0	0
$797$	5.90017	0.208995	0.104497	−	0.994525i	$-0.466677\pi$
$797$	5.90017	0.208995	0.104497	+	0.994525i	$0.466677\pi$
$798$	0	0
$799$	−4.06631	−0.143856
$800$	0	0
$801$	0.632437	0.0223460
$802$	0	0
$803$	−10.4644	−0.369279
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−43.2362	−1.52199
$808$	0	0
$809$	10.2565	0.360599	0.180299	−	0.983612i	$-0.442293\pi$
$809$	10.2565	0.360599	0.180299	+	0.983612i	$0.442293\pi$
$810$	0	0
$811$	5.55142	0.194937	0.0974684	−	0.995239i	$-0.468926\pi$
$811$	5.55142	0.194937	0.0974684	+	0.995239i	$0.468926\pi$
$812$	0	0
$813$	−21.7577	−0.763077
$814$	0	0
$815$	−77.5086	−2.71501
$816$	0	0
$817$	3.90849	0.136741
$818$	0	0
$819$	0	0
$820$	0	0
$821$	44.9464	1.56864	0.784320	−	0.620356i	$-0.213012\pi$
$821$	44.9464	1.56864	0.784320	+	0.620356i	$0.213012\pi$
$822$	0	0
$823$	−22.3398	−0.778715	−0.389357	−	0.921087i	$-0.627303\pi$
$823$	−22.3398	−0.778715	−0.389357	+	0.921087i	$0.627303\pi$
$824$	0	0
$825$	−28.4710	−0.991232
$826$	0	0
$827$	−1.36558	−0.0474858	−0.0237429	−	0.999718i	$-0.507558\pi$
$827$	−1.36558	−0.0474858	−0.0237429	+	0.999718i	$0.507558\pi$
$828$	0	0
$829$	−18.7358	−0.650720	−0.325360	−	0.945590i	$-0.605486\pi$
$829$	−18.7358	−0.650720	−0.325360	+	0.945590i	$0.605486\pi$
$830$	0	0
$831$	46.0059	1.59593
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−40.8951	−1.41523
$836$	0	0
$837$	−11.7572	−0.406389
$838$	0	0
$839$	−46.9151	−1.61969	−0.809844	−	0.586645i	$-0.800448\pi$
$839$	−46.9151	−1.61969	−0.809844	+	0.586645i	$0.800448\pi$
$840$	0	0
$841$	−10.3366	−0.356434
$842$	0	0
$843$	−41.6548	−1.43467
$844$	0	0
$845$	−38.4536	−1.32285
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4.24274	−0.145610
$850$	0	0
$851$	−7.96971	−0.273198
$852$	0	0
$853$	−19.9761	−0.683967	−0.341984	−	0.939706i	$-0.611099\pi$
$853$	−19.9761	−0.683967	−0.341984	+	0.939706i	$0.611099\pi$
$854$	0	0
$855$	0.652807	0.0223255
$856$	0	0
$857$	31.0555	1.06083	0.530417	−	0.847737i	$-0.322035\pi$
$857$	31.0555	1.06083	0.530417	+	0.847737i	$0.322035\pi$
$858$	0	0
$859$	−28.5565	−0.974336	−0.487168	−	0.873308i	$-0.661970\pi$
$859$	−28.5565	−0.974336	−0.487168	+	0.873308i	$0.661970\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−41.5072	−1.41292	−0.706461	−	0.707752i	$-0.749710\pi$
$863$	−41.5072	−1.41292	−0.706461	+	0.707752i	$0.749710\pi$
$864$	0	0
$865$	29.6063	1.00665
$866$	0	0
$867$	−24.1556	−0.820367
$868$	0	0
$869$	60.0116	2.03575
$870$	0	0
$871$	1.13743	0.0385405
$872$	0	0
$873$	−6.21655	−0.210398
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−20.2679	−0.684397	−0.342199	−	0.939628i	$-0.611172\pi$
$877$	−20.2679	−0.684397	−0.342199	+	0.939628i	$0.611172\pi$
$878$	0	0
$879$	14.2552	0.480816
$880$	0	0
$881$	−35.0903	−1.18222	−0.591111	−	0.806590i	$-0.701311\pi$
$881$	−35.0903	−1.18222	−0.591111	+	0.806590i	$0.701311\pi$
$882$	0	0
$883$	−7.61249	−0.256181	−0.128090	−	0.991762i	$-0.540885\pi$
$883$	−7.61249	−0.256181	−0.128090	+	0.991762i	$0.540885\pi$
$884$	0	0
$885$	9.28942	0.312261
$886$	0	0
$887$	16.1435	0.542044	0.271022	−	0.962573i	$-0.412638\pi$
$887$	16.1435	0.542044	0.271022	+	0.962573i	$0.412638\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	29.9914	1.00475
$892$	0	0
$893$	−1.40915	−0.0471553
$894$	0	0
$895$	40.1114	1.34078
$896$	0	0
$897$	−1.10550	−0.0369117
$898$	0	0
$899$	−9.22203	−0.307572
$900$	0	0
$901$	15.2082	0.506659
$902$	0	0
$903$	0	0
$904$	0	0
$905$	35.7040	1.18684
$906$	0	0
$907$	−19.0546	−0.632697	−0.316348	−	0.948643i	$-0.602457\pi$
$907$	−19.0546	−0.632697	−0.316348	+	0.948643i	$0.602457\pi$
$908$	0	0
$909$	−1.57784	−0.0523335
$910$	0	0
$911$	1.88286	0.0623818	0.0311909	−	0.999513i	$-0.490070\pi$
$911$	1.88286	0.0623818	0.0311909	+	0.999513i	$0.490070\pi$
$912$	0	0
$913$	−21.6572	−0.716748
$914$	0	0
$915$	56.6380	1.87239
$916$	0	0
$917$	0	0
$918$	0	0
$919$	10.3158	0.340287	0.170143	−	0.985419i	$-0.445577\pi$
$919$	10.3158	0.340287	0.170143	+	0.985419i	$0.445577\pi$
$920$	0	0
$921$	−21.0891	−0.694910
$922$	0	0
$923$	6.65415	0.219024
$924$	0	0
$925$	−35.3174	−1.16123
$926$	0	0
$927$	2.17362	0.0713909
$928$	0	0
$929$	22.0544	0.723581	0.361790	−	0.932259i	$-0.382166\pi$
$929$	22.0544	0.723581	0.361790	+	0.932259i	$0.382166\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	22.0737	0.722659
$934$	0	0
$935$	16.9391	0.553969
$936$	0	0
$937$	−23.2760	−0.760395	−0.380197	−	0.924905i	$-0.624144\pi$
$937$	−23.2760	−0.760395	−0.380197	+	0.924905i	$0.624144\pi$
$938$	0	0
$939$	1.87974	0.0613431
$940$	0	0
$941$	49.8799	1.62604	0.813018	−	0.582238i	$-0.197823\pi$
$941$	49.8799	1.62604	0.813018	+	0.582238i	$0.197823\pi$
$942$	0	0
$943$	−6.24839	−0.203476
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9.91284	−0.322124	−0.161062	−	0.986944i	$-0.551492\pi$
$947$	−9.91284	−0.322124	−0.161062	+	0.986944i	$0.551492\pi$
$948$	0	0
$949$	−1.80060	−0.0584499
$950$	0	0
$951$	−7.62781	−0.247349
$952$	0	0
$953$	−55.0233	−1.78238	−0.891189	−	0.453632i	$-0.850128\pi$
$953$	−55.0233	−1.78238	−0.891189	+	0.453632i	$0.850128\pi$
$954$	0	0
$955$	−35.6556	−1.15379
$956$	0	0
$957$	27.7557	0.897213
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−26.4432	−0.853006
$962$	0	0
$963$	−4.16660	−0.134267
$964$	0	0
$965$	82.6480	2.66053
$966$	0	0
$967$	−9.76414	−0.313994	−0.156997	−	0.987599i	$-0.550181\pi$
$967$	−9.76414	−0.313994	−0.156997	+	0.987599i	$0.550181\pi$
$968$	0	0
$969$	1.04179	0.0334670
$970$	0	0
$971$	−55.6371	−1.78548	−0.892740	−	0.450573i	$-0.851220\pi$
$971$	−55.6371	−1.78548	−0.892740	+	0.450573i	$0.851220\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−4.89899	−0.156893
$976$	0	0
$977$	−11.5429	−0.369291	−0.184646	−	0.982805i	$-0.559114\pi$
$977$	−11.5429	−0.369291	−0.184646	+	0.982805i	$0.559114\pi$
$978$	0	0
$979$	5.68696	0.181756
$980$	0	0
$981$	0.395669	0.0126327
$982$	0	0
$983$	−38.4656	−1.22686	−0.613431	−	0.789748i	$-0.710211\pi$
$983$	−38.4656	−1.22686	−0.613431	+	0.789748i	$0.710211\pi$
$984$	0	0
$985$	−63.3832	−2.01956
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.22237	−0.261456
$990$	0	0
$991$	−20.8646	−0.662785	−0.331392	−	0.943493i	$-0.607518\pi$
$991$	−20.8646	−0.662785	−0.331392	+	0.943493i	$0.607518\pi$
$992$	0	0
$993$	−0.141296	−0.00448388
$994$	0	0
$995$	−26.3853	−0.836471
$996$	0	0
$997$	−40.4801	−1.28202	−0.641010	−	0.767533i	$-0.721484\pi$
$997$	−40.4801	−1.28202	−0.641010	+	0.767533i	$0.721484\pi$
$998$	0	0
$999$	43.8952	1.38878

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.q.a.737.9	✓	22	7.3	odd	6
1288.2.q.a.921.9	yes	22	7.5	odd	6
9016.2.a.bl.1.9		11	1.1	even	1	trivial
9016.2.a.bq.1.3		11	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9016 = 2^{3} \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.59775 q^{3} +3.07107 q^{5} -0.447181 q^{9} +O(q^{10})\)	\(q+1.59775 q^{3} +3.07107 q^{5} -0.447181 q^{9} -4.02111 q^{11} -0.691911 q^{13} +4.90681 q^{15} -1.37169 q^{17} -0.475349 q^{19} +1.00000 q^{23} +4.43145 q^{25} -5.50775 q^{27} -4.32012 q^{29} +2.13467 q^{31} -6.42474 q^{33} -7.96971 q^{37} -1.10550 q^{39} -6.24839 q^{41} -8.22237 q^{43} -1.37332 q^{45} +2.96445 q^{47} -2.19163 q^{51} -11.0872 q^{53} -12.3491 q^{55} -0.759491 q^{57} +1.89317 q^{59} +11.5427 q^{61} -2.12490 q^{65} -1.64390 q^{67} +1.59775 q^{69} -9.61707 q^{71} +2.60236 q^{73} +7.08038 q^{75} -14.9241 q^{79} -7.45849 q^{81} +5.38588 q^{83} -4.21256 q^{85} -6.90249 q^{87} -1.41428 q^{89} +3.41068 q^{93} -1.45983 q^{95} +13.9016 q^{97} +1.79816 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 4 q^{3} - q^{5} + 11 q^{9}+O(q^{10}) \) 11 * q - 4 * q^3 - q^5 + 11 * q^9	\( 11 q - 4 q^{3} - q^{5} + 11 q^{9} + 3 q^{13} - 8 q^{15} - 5 q^{17} - 12 q^{19} + 11 q^{23} + 22 q^{25} - 19 q^{27} - 15 q^{29} - 16 q^{31} - 4 q^{33} + 3 q^{37} - q^{39} - 28 q^{41} - 9 q^{43} + 19 q^{45} - 31 q^{47} - 15 q^{51} + 13 q^{53} - 35 q^{55} - 21 q^{57} - 11 q^{59} + 19 q^{61} - 7 q^{65} + 19 q^{67} - 4 q^{69} - 5 q^{71} + 5 q^{73} - 28 q^{75} - 13 q^{79} + 35 q^{81} - 17 q^{83} - 39 q^{85} - 4 q^{87} - 10 q^{89} - 6 q^{93} + 33 q^{95} - 35 q^{97} + 13 q^{99}+O(q^{100}) \) 11 * q - 4 * q^3 - q^5 + 11 * q^9 + 3 * q^13 - 8 * q^15 - 5 * q^17 - 12 * q^19 + 11 * q^23 + 22 * q^25 - 19 * q^27 - 15 * q^29 - 16 * q^31 - 4 * q^33 + 3 * q^37 - q^39 - 28 * q^41 - 9 * q^43 + 19 * q^45 - 31 * q^47 - 15 * q^51 + 13 * q^53 - 35 * q^55 - 21 * q^57 - 11 * q^59 + 19 * q^61 - 7 * q^65 + 19 * q^67 - 4 * q^69 - 5 * q^71 + 5 * q^73 - 28 * q^75 - 13 * q^79 + 35 * q^81 - 17 * q^83 - 39 * q^85 - 4 * q^87 - 10 * q^89 - 6 * q^93 + 33 * q^95 - 35 * q^97 + 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more