LMFDB - Embedded newform 9016.2.a.bl.1.4

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			1.4
Root			$0.947093$ 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)

$f(q)$	$=$	$q-0.947093 q^{3} -3.46093 q^{5} -2.10301 q^{9} +O(q^{10})$	$q-0.947093 q^{3} -3.46093 q^{5} -2.10301 q^{9} +4.20604 q^{11} -1.85995 q^{13} +3.27783 q^{15} +0.291599 q^{17} -4.29796 q^{19} +1.00000 q^{23} +6.97805 q^{25} +4.83303 q^{27} +5.69091 q^{29} -2.11455 q^{31} -3.98352 q^{33} -10.2558 q^{37} +1.76155 q^{39} -9.06366 q^{41} +3.38944 q^{43} +7.27839 q^{45} +10.1023 q^{47} -0.276171 q^{51} +1.74108 q^{53} -14.5568 q^{55} +4.07057 q^{57} +2.45458 q^{59} -1.74995 q^{61} +6.43718 q^{65} +7.42413 q^{67} -0.947093 q^{69} +13.6649 q^{71} -8.50121 q^{73} -6.60886 q^{75} -6.85148 q^{79} +1.73171 q^{81} +11.6754 q^{83} -1.00920 q^{85} -5.38983 q^{87} +2.10584 q^{89} +2.00268 q^{93} +14.8749 q^{95} +2.71273 q^{97} -8.84537 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.947093	−0.546805	−0.273402	−	0.961900i	$-0.588149\pi$
$3$	−0.947093	−0.546805	−0.273402	+	0.961900i	$0.588149\pi$
$4$	0	0
$5$	−3.46093	−1.54778	−0.773888	−	0.633323i	$-0.781691\pi$
$5$	−3.46093	−1.54778	−0.773888	+	0.633323i	$0.781691\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.10301	−0.701005
$10$	0	0
$11$	4.20604	1.26817	0.634085	−	0.773263i	$-0.281377\pi$
$11$	4.20604	1.26817	0.634085	+	0.773263i	$0.281377\pi$
$12$	0	0
$13$	−1.85995	−0.515859	−0.257929	−	0.966164i	$-0.583040\pi$
$13$	−1.85995	−0.515859	−0.257929	+	0.966164i	$0.583040\pi$
$14$	0	0
$15$	3.27783	0.846331
$16$	0	0
$17$	0.291599	0.0707231	0.0353615	−	0.999375i	$-0.488742\pi$
$17$	0.291599	0.0707231	0.0353615	+	0.999375i	$0.488742\pi$
$18$	0	0
$19$	−4.29796	−0.986019	−0.493009	−	0.870024i	$-0.664103\pi$
$19$	−4.29796	−0.986019	−0.493009	+	0.870024i	$0.664103\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	6.97805	1.39561
$26$	0	0
$27$	4.83303	0.930117
$28$	0	0
$29$	5.69091	1.05678	0.528388	−	0.849003i	$-0.322797\pi$
$29$	5.69091	1.05678	0.528388	+	0.849003i	$0.322797\pi$
$30$	0	0
$31$	−2.11455	−0.379785	−0.189892	−	0.981805i	$-0.560814\pi$
$31$	−2.11455	−0.379785	−0.189892	+	0.981805i	$0.560814\pi$
$32$	0	0
$33$	−3.98352	−0.693441
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.2558	−1.68604	−0.843020	−	0.537882i	$-0.819225\pi$
$37$	−10.2558	−1.68604	−0.843020	+	0.537882i	$0.819225\pi$
$38$	0	0
$39$	1.76155	0.282074
$40$	0	0
$41$	−9.06366	−1.41551	−0.707753	−	0.706460i	$-0.750291\pi$
$41$	−9.06366	−1.41551	−0.707753	+	0.706460i	$0.750291\pi$
$42$	0	0
$43$	3.38944	0.516885	0.258442	−	0.966027i	$-0.416791\pi$
$43$	3.38944	0.516885	0.258442	+	0.966027i	$0.416791\pi$
$44$	0	0
$45$	7.27839	1.08500
$46$	0	0
$47$	10.1023	1.47358	0.736789	−	0.676122i	$-0.236341\pi$
$47$	10.1023	1.47358	0.736789	+	0.676122i	$0.236341\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.276171	−0.0386717
$52$	0	0
$53$	1.74108	0.239155	0.119578	−	0.992825i	$-0.461846\pi$
$53$	1.74108	0.239155	0.119578	+	0.992825i	$0.461846\pi$
$54$	0	0
$55$	−14.5568	−1.96284
$56$	0	0
$57$	4.07057	0.539160
$58$	0	0
$59$	2.45458	0.319559	0.159780	−	0.987153i	$-0.448922\pi$
$59$	2.45458	0.319559	0.159780	+	0.987153i	$0.448922\pi$
$60$	0	0
$61$	−1.74995	−0.224058	−0.112029	−	0.993705i	$-0.535735\pi$
$61$	−1.74995	−0.224058	−0.112029	+	0.993705i	$0.535735\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.43718	0.798433
$66$	0	0
$67$	7.42413	0.907002	0.453501	−	0.891256i	$-0.350175\pi$
$67$	7.42413	0.907002	0.453501	+	0.891256i	$0.350175\pi$
$68$	0	0
$69$	−0.947093	−0.114017
$70$	0	0
$71$	13.6649	1.62172	0.810860	−	0.585240i	$-0.199000\pi$
$71$	13.6649	1.62172	0.810860	+	0.585240i	$0.199000\pi$
$72$	0	0
$73$	−8.50121	−0.994991	−0.497496	−	0.867467i	$-0.665747\pi$
$73$	−8.50121	−0.994991	−0.497496	+	0.867467i	$0.665747\pi$
$74$	0	0
$75$	−6.60886	−0.763126
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.85148	−0.770852	−0.385426	−	0.922739i	$-0.625945\pi$
$79$	−6.85148	−0.770852	−0.385426	+	0.922739i	$0.625945\pi$
$80$	0	0
$81$	1.73171	0.192413
$82$	0	0
$83$	11.6754	1.28154	0.640770	−	0.767733i	$-0.278615\pi$
$83$	11.6754	1.28154	0.640770	+	0.767733i	$0.278615\pi$
$84$	0	0
$85$	−1.00920	−0.109463
$86$	0	0
$87$	−5.38983	−0.577850
$88$	0	0
$89$	2.10584	0.223219	0.111609	−	0.993752i	$-0.464399\pi$
$89$	2.10584	0.223219	0.111609	+	0.993752i	$0.464399\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.00268	0.207668
$94$	0	0
$95$	14.8749	1.52614
$96$	0	0
$97$	2.71273	0.275436	0.137718	−	0.990471i	$-0.456023\pi$
$97$	2.71273	0.275436	0.137718	+	0.990471i	$0.456023\pi$
$98$	0	0
$99$	−8.84537	−0.888993
$100$	0	0
$101$	8.46370	0.842170	0.421085	−	0.907021i	$-0.361649\pi$
$101$	8.46370	0.842170	0.421085	+	0.907021i	$0.361649\pi$
$102$	0	0
$103$	18.3614	1.80920	0.904600	−	0.426262i	$-0.140170\pi$
$103$	18.3614	1.80920	0.904600	+	0.426262i	$0.140170\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.26011	0.411841	0.205920	−	0.978569i	$-0.433981\pi$
$107$	4.26011	0.411841	0.205920	+	0.978569i	$0.433981\pi$
$108$	0	0
$109$	8.77154	0.840161	0.420080	−	0.907487i	$-0.362002\pi$
$109$	8.77154	0.840161	0.420080	+	0.907487i	$0.362002\pi$
$110$	0	0
$111$	9.71318	0.921934
$112$	0	0
$113$	−7.79442	−0.733237	−0.366619	−	0.930371i	$-0.619485\pi$
$113$	−7.79442	−0.733237	−0.366619	+	0.930371i	$0.619485\pi$
$114$	0	0
$115$	−3.46093	−0.322734
$116$	0	0
$117$	3.91151	0.361619
$118$	0	0
$119$	0	0
$120$	0	0
$121$	6.69080	0.608255
$122$	0	0
$123$	8.58413	0.774005
$124$	0	0
$125$	−6.84590	−0.612316
$126$	0	0
$127$	−4.53846	−0.402724	−0.201362	−	0.979517i	$-0.564537\pi$
$127$	−4.53846	−0.402724	−0.201362	+	0.979517i	$0.564537\pi$
$128$	0	0
$129$	−3.21012	−0.282635
$130$	0	0
$131$	−7.42777	−0.648967	−0.324484	−	0.945891i	$-0.605191\pi$
$131$	−7.42777	−0.648967	−0.324484	+	0.945891i	$0.605191\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−16.7268	−1.43961
$136$	0	0
$137$	−15.8076	−1.35053	−0.675267	−	0.737573i	$-0.735972\pi$
$137$	−15.8076	−1.35053	−0.675267	+	0.737573i	$0.735972\pi$
$138$	0	0
$139$	−3.93804	−0.334020	−0.167010	−	0.985955i	$-0.553411\pi$
$139$	−3.93804	−0.334020	−0.167010	+	0.985955i	$0.553411\pi$
$140$	0	0
$141$	−9.56786	−0.805759
$142$	0	0
$143$	−7.82305	−0.654196
$144$	0	0
$145$	−19.6959	−1.63565
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.62401	0.788429	0.394215	−	0.919018i	$-0.371017\pi$
$149$	9.62401	0.788429	0.394215	+	0.919018i	$0.371017\pi$
$150$	0	0
$151$	5.33251	0.433954	0.216977	−	0.976177i	$-0.430380\pi$
$151$	5.33251	0.433954	0.216977	+	0.976177i	$0.430380\pi$
$152$	0	0
$153$	−0.613236	−0.0495772
$154$	0	0
$155$	7.31832	0.587822
$156$	0	0
$157$	22.2446	1.77532	0.887658	−	0.460503i	$-0.152331\pi$
$157$	22.2446	1.77532	0.887658	+	0.460503i	$0.152331\pi$
$158$	0	0
$159$	−1.64896	−0.130771
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−8.74905	−0.685278	−0.342639	−	0.939467i	$-0.611321\pi$
$163$	−8.74905	−0.685278	−0.342639	+	0.939467i	$0.611321\pi$
$164$	0	0
$165$	13.7867	1.07329
$166$	0	0
$167$	−9.21828	−0.713332	−0.356666	−	0.934232i	$-0.616087\pi$
$167$	−9.21828	−0.713332	−0.356666	+	0.934232i	$0.616087\pi$
$168$	0	0
$169$	−9.54057	−0.733890
$170$	0	0
$171$	9.03867	0.691204
$172$	0	0
$173$	−19.0974	−1.45195	−0.725975	−	0.687721i	$-0.758611\pi$
$173$	−19.0974	−1.45195	−0.725975	+	0.687721i	$0.758611\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.32472	−0.174736
$178$	0	0
$179$	11.2520	0.841013	0.420506	−	0.907290i	$-0.361852\pi$
$179$	11.2520	0.841013	0.420506	+	0.907290i	$0.361852\pi$
$180$	0	0
$181$	−23.9799	−1.78241	−0.891204	−	0.453602i	$-0.850139\pi$
$181$	−23.9799	−1.78241	−0.891204	+	0.453602i	$0.850139\pi$
$182$	0	0
$183$	1.65736	0.122516
$184$	0	0
$185$	35.4946	2.60961
$186$	0	0
$187$	1.22648	0.0896889
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.8589	1.29222	0.646112	−	0.763243i	$-0.276394\pi$
$191$	17.8589	1.29222	0.646112	+	0.763243i	$0.276394\pi$
$192$	0	0
$193$	20.4425	1.47149	0.735743	−	0.677261i	$-0.236833\pi$
$193$	20.4425	1.47149	0.735743	+	0.677261i	$0.236833\pi$
$194$	0	0
$195$	−6.09661	−0.436587
$196$	0	0
$197$	−19.6772	−1.40195	−0.700973	−	0.713188i	$-0.747251\pi$
$197$	−19.6772	−1.40195	−0.700973	+	0.713188i	$0.747251\pi$
$198$	0	0
$199$	−7.23867	−0.513136	−0.256568	−	0.966526i	$-0.582592\pi$
$199$	−7.23867	−0.513136	−0.256568	+	0.966526i	$0.582592\pi$
$200$	0	0
$201$	−7.03135	−0.495953
$202$	0	0
$203$	0	0
$204$	0	0
$205$	31.3687	2.19089
$206$	0	0
$207$	−2.10301	−0.146170
$208$	0	0
$209$	−18.0774	−1.25044
$210$	0	0
$211$	19.9757	1.37518	0.687591	−	0.726098i	$-0.258668\pi$
$211$	19.9757	1.37518	0.687591	+	0.726098i	$0.258668\pi$
$212$	0	0
$213$	−12.9419	−0.886764
$214$	0	0
$215$	−11.7306	−0.800022
$216$	0	0
$217$	0	0
$218$	0	0
$219$	8.05144	0.544066
$220$	0	0
$221$	−0.542360	−0.0364831
$222$	0	0
$223$	−11.3214	−0.758135	−0.379068	−	0.925369i	$-0.623755\pi$
$223$	−11.3214	−0.758135	−0.379068	+	0.925369i	$0.623755\pi$
$224$	0	0
$225$	−14.6749	−0.978329
$226$	0	0
$227$	−29.2356	−1.94043	−0.970217	−	0.242237i	$-0.922119\pi$
$227$	−29.2356	−1.94043	−0.970217	+	0.242237i	$0.922119\pi$
$228$	0	0
$229$	9.73165	0.643085	0.321543	−	0.946895i	$-0.395799\pi$
$229$	9.73165	0.643085	0.321543	+	0.946895i	$0.395799\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−28.1549	−1.84449	−0.922246	−	0.386605i	$-0.873648\pi$
$233$	−28.1549	−1.84449	−0.922246	+	0.386605i	$0.873648\pi$
$234$	0	0
$235$	−34.9635	−2.28077
$236$	0	0
$237$	6.48899	0.421505
$238$	0	0
$239$	−4.32770	−0.279936	−0.139968	−	0.990156i	$-0.544700\pi$
$239$	−4.32770	−0.279936	−0.139968	+	0.990156i	$0.544700\pi$
$240$	0	0
$241$	−6.37150	−0.410424	−0.205212	−	0.978718i	$-0.565788\pi$
$241$	−6.37150	−0.410424	−0.205212	+	0.978718i	$0.565788\pi$
$242$	0	0
$243$	−16.1392	−1.03533
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.99401	0.508646
$248$	0	0
$249$	−11.0577	−0.700751
$250$	0	0
$251$	−12.9351	−0.816455	−0.408227	−	0.912880i	$-0.633853\pi$
$251$	−12.9351	−0.816455	−0.408227	+	0.912880i	$0.633853\pi$
$252$	0	0
$253$	4.20604	0.264432
$254$	0	0
$255$	0.955810	0.0598551
$256$	0	0
$257$	6.38551	0.398317	0.199159	−	0.979967i	$-0.436179\pi$
$257$	6.38551	0.398317	0.199159	+	0.979967i	$0.436179\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−11.9681	−0.740805
$262$	0	0
$263$	−26.4370	−1.63017	−0.815087	−	0.579339i	$-0.803311\pi$
$263$	−26.4370	−1.63017	−0.815087	+	0.579339i	$0.803311\pi$
$264$	0	0
$265$	−6.02575	−0.370159
$266$	0	0
$267$	−1.99443	−0.122057
$268$	0	0
$269$	1.98559	0.121064	0.0605319	−	0.998166i	$-0.480720\pi$
$269$	1.98559	0.121064	0.0605319	+	0.998166i	$0.480720\pi$
$270$	0	0
$271$	10.2061	0.619974	0.309987	−	0.950741i	$-0.399675\pi$
$271$	10.2061	0.619974	0.309987	+	0.950741i	$0.399675\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	29.3500	1.76987
$276$	0	0
$277$	12.2493	0.735987	0.367994	−	0.929828i	$-0.380045\pi$
$277$	12.2493	0.735987	0.367994	+	0.929828i	$0.380045\pi$
$278$	0	0
$279$	4.44693	0.266231
$280$	0	0
$281$	−9.64716	−0.575501	−0.287751	−	0.957705i	$-0.592907\pi$
$281$	−9.64716	−0.575501	−0.287751	+	0.957705i	$0.592907\pi$
$282$	0	0
$283$	−20.4180	−1.21372	−0.606861	−	0.794808i	$-0.707571\pi$
$283$	−20.4180	−1.21372	−0.606861	+	0.794808i	$0.707571\pi$
$284$	0	0
$285$	−14.0880	−0.834498
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.9150	−0.994998
$290$	0	0
$291$	−2.56921	−0.150609
$292$	0	0
$293$	15.3677	0.897791	0.448895	−	0.893584i	$-0.351818\pi$
$293$	15.3677	0.897791	0.448895	+	0.893584i	$0.351818\pi$
$294$	0	0
$295$	−8.49514	−0.494606
$296$	0	0
$297$	20.3279	1.17955
$298$	0	0
$299$	−1.85995	−0.107564
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−8.01592	−0.460502
$304$	0	0
$305$	6.05644	0.346791
$306$	0	0
$307$	5.39179	0.307726	0.153863	−	0.988092i	$-0.450829\pi$
$307$	5.39179	0.307726	0.153863	+	0.988092i	$0.450829\pi$
$308$	0	0
$309$	−17.3899	−0.989279
$310$	0	0
$311$	−4.08198	−0.231468	−0.115734	−	0.993280i	$-0.536922\pi$
$311$	−4.08198	−0.231468	−0.115734	+	0.993280i	$0.536922\pi$
$312$	0	0
$313$	31.4829	1.77952	0.889760	−	0.456429i	$-0.150872\pi$
$313$	31.4829	1.77952	0.889760	+	0.456429i	$0.150872\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.51776	−0.141411	−0.0707057	−	0.997497i	$-0.522525\pi$
$317$	−2.51776	−0.141411	−0.0707057	+	0.997497i	$0.522525\pi$
$318$	0	0
$319$	23.9362	1.34017
$320$	0	0
$321$	−4.03472	−0.225196
$322$	0	0
$323$	−1.25328	−0.0697343
$324$	0	0
$325$	−12.9789	−0.719937
$326$	0	0
$327$	−8.30746	−0.459404
$328$	0	0
$329$	0	0
$330$	0	0
$331$	24.0197	1.32024	0.660122	−	0.751159i	$-0.270505\pi$
$331$	24.0197	1.32024	0.660122	+	0.751159i	$0.270505\pi$
$332$	0	0
$333$	21.5681	1.18192
$334$	0	0
$335$	−25.6944	−1.40384
$336$	0	0
$337$	−16.2044	−0.882709	−0.441355	−	0.897333i	$-0.645502\pi$
$337$	−16.2044	−0.882709	−0.441355	+	0.897333i	$0.645502\pi$
$338$	0	0
$339$	7.38204	0.400937
$340$	0	0
$341$	−8.89390	−0.481632
$342$	0	0
$343$	0	0
$344$	0	0
$345$	3.27783	0.176472
$346$	0	0
$347$	−18.6349	−1.00037	−0.500186	−	0.865918i	$-0.666735\pi$
$347$	−18.6349	−1.00037	−0.500186	+	0.865918i	$0.666735\pi$
$348$	0	0
$349$	20.7929	1.11302	0.556510	−	0.830841i	$-0.312140\pi$
$349$	20.7929	1.11302	0.556510	+	0.830841i	$0.312140\pi$
$350$	0	0
$351$	−8.98922	−0.479809
$352$	0	0
$353$	16.6049	0.883792	0.441896	−	0.897066i	$-0.354306\pi$
$353$	16.6049	0.883792	0.441896	+	0.897066i	$0.354306\pi$
$354$	0	0
$355$	−47.2932	−2.51006
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−21.8462	−1.15300	−0.576500	−	0.817097i	$-0.695582\pi$
$359$	−21.8462	−1.15300	−0.576500	+	0.817097i	$0.695582\pi$
$360$	0	0
$361$	−0.527565	−0.0277666
$362$	0	0
$363$	−6.33681	−0.332596
$364$	0	0
$365$	29.4221	1.54002
$366$	0	0
$367$	−18.3640	−0.958593	−0.479297	−	0.877653i	$-0.659108\pi$
$367$	−18.3640	−0.958593	−0.479297	+	0.877653i	$0.659108\pi$
$368$	0	0
$369$	19.0610	0.992276
$370$	0	0
$371$	0	0
$372$	0	0
$373$	22.8934	1.18537	0.592687	−	0.805433i	$-0.298067\pi$
$373$	22.8934	1.18537	0.592687	+	0.805433i	$0.298067\pi$
$374$	0	0
$375$	6.48370	0.334817
$376$	0	0
$377$	−10.5848	−0.545147
$378$	0	0
$379$	−3.57092	−0.183426	−0.0917129	−	0.995785i	$-0.529234\pi$
$379$	−3.57092	−0.183426	−0.0917129	+	0.995785i	$0.529234\pi$
$380$	0	0
$381$	4.29835	0.220211
$382$	0	0
$383$	−26.6064	−1.35952	−0.679762	−	0.733433i	$-0.737917\pi$
$383$	−26.6064	−1.35952	−0.679762	+	0.733433i	$0.737917\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.12804	−0.362339
$388$	0	0
$389$	−9.96677	−0.505336	−0.252668	−	0.967553i	$-0.581308\pi$
$389$	−9.96677	−0.505336	−0.252668	+	0.967553i	$0.581308\pi$
$390$	0	0
$391$	0.291599	0.0147468
$392$	0	0
$393$	7.03479	0.354858
$394$	0	0
$395$	23.7125	1.19311
$396$	0	0
$397$	21.2350	1.06575	0.532877	−	0.846192i	$-0.321111\pi$
$397$	21.2350	1.06575	0.532877	+	0.846192i	$0.321111\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−35.9906	−1.79728	−0.898642	−	0.438682i	$-0.855446\pi$
$401$	−35.9906	−1.79728	−0.898642	+	0.438682i	$0.855446\pi$
$402$	0	0
$403$	3.93297	0.195915
$404$	0	0
$405$	−5.99334	−0.297812
$406$	0	0
$407$	−43.1363	−2.13819
$408$	0	0
$409$	−30.7955	−1.52274	−0.761370	−	0.648318i	$-0.775473\pi$
$409$	−30.7955	−1.52274	−0.761370	+	0.648318i	$0.775473\pi$
$410$	0	0
$411$	14.9713	0.738479
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−40.4077	−1.98354
$416$	0	0
$417$	3.72969	0.182644
$418$	0	0
$419$	−36.3512	−1.77587	−0.887935	−	0.459968i	$-0.847861\pi$
$419$	−36.3512	−1.77587	−0.887935	+	0.459968i	$0.847861\pi$
$420$	0	0
$421$	9.43869	0.460014	0.230007	−	0.973189i	$-0.426125\pi$
$421$	9.43869	0.460014	0.230007	+	0.973189i	$0.426125\pi$
$422$	0	0
$423$	−21.2454	−1.03299
$424$	0	0
$425$	2.03479	0.0987019
$426$	0	0
$427$	0	0
$428$	0	0
$429$	7.40916	0.357717
$430$	0	0
$431$	15.9121	0.766460	0.383230	−	0.923653i	$-0.374812\pi$
$431$	15.9121	0.766460	0.383230	+	0.923653i	$0.374812\pi$
$432$	0	0
$433$	12.5654	0.603855	0.301927	−	0.953331i	$-0.402370\pi$
$433$	12.5654	0.603855	0.301927	+	0.953331i	$0.402370\pi$
$434$	0	0
$435$	18.6538	0.894382
$436$	0	0
$437$	−4.29796	−0.205599
$438$	0	0
$439$	−23.5111	−1.12212	−0.561062	−	0.827774i	$-0.689607\pi$
$439$	−23.5111	−1.12212	−0.561062	+	0.827774i	$0.689607\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.15710	−0.387556	−0.193778	−	0.981045i	$-0.562074\pi$
$443$	−8.15710	−0.387556	−0.193778	+	0.981045i	$0.562074\pi$
$444$	0	0
$445$	−7.28818	−0.345493
$446$	0	0
$447$	−9.11483	−0.431117
$448$	0	0
$449$	−20.6140	−0.972837	−0.486419	−	0.873726i	$-0.661697\pi$
$449$	−20.6140	−0.972837	−0.486419	+	0.873726i	$0.661697\pi$
$450$	0	0
$451$	−38.1221	−1.79510
$452$	0	0
$453$	−5.05039	−0.237288
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16.4608	0.770006	0.385003	−	0.922915i	$-0.374200\pi$
$457$	16.4608	0.770006	0.385003	+	0.922915i	$0.374200\pi$
$458$	0	0
$459$	1.40931	0.0657808
$460$	0	0
$461$	−15.0007	−0.698653	−0.349327	−	0.937001i	$-0.613590\pi$
$461$	−15.0007	−0.698653	−0.349327	+	0.937001i	$0.613590\pi$
$462$	0	0
$463$	−3.04516	−0.141521	−0.0707603	−	0.997493i	$-0.522543\pi$
$463$	−3.04516	−0.141521	−0.0707603	+	0.997493i	$0.522543\pi$
$464$	0	0
$465$	−6.93113	−0.321424
$466$	0	0
$467$	−1.30212	−0.0602549	−0.0301275	−	0.999546i	$-0.509591\pi$
$467$	−1.30212	−0.0602549	−0.0301275	+	0.999546i	$0.509591\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−21.0678	−0.970751
$472$	0	0
$473$	14.2561	0.655498
$474$	0	0
$475$	−29.9914	−1.37610
$476$	0	0
$477$	−3.66151	−0.167649
$478$	0	0
$479$	−3.34560	−0.152865	−0.0764323	−	0.997075i	$-0.524353\pi$
$479$	−3.34560	−0.152865	−0.0764323	+	0.997075i	$0.524353\pi$
$480$	0	0
$481$	19.0753	0.869758
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.38856	−0.426313
$486$	0	0
$487$	−33.6980	−1.52700	−0.763501	−	0.645807i	$-0.776521\pi$
$487$	−33.6980	−1.52700	−0.763501	+	0.645807i	$0.776521\pi$
$488$	0	0
$489$	8.28616	0.374713
$490$	0	0
$491$	24.3715	1.09987	0.549935	−	0.835208i	$-0.314653\pi$
$491$	24.3715	1.09987	0.549935	+	0.835208i	$0.314653\pi$
$492$	0	0
$493$	1.65946	0.0747385
$494$	0	0
$495$	30.6132	1.37596
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−13.5692	−0.607441	−0.303721	−	0.952761i	$-0.598229\pi$
$499$	−13.5692	−0.607441	−0.303721	+	0.952761i	$0.598229\pi$
$500$	0	0
$501$	8.73057	0.390053
$502$	0	0
$503$	−1.36598	−0.0609062	−0.0304531	−	0.999536i	$-0.509695\pi$
$503$	−1.36598	−0.0609062	−0.0304531	+	0.999536i	$0.509695\pi$
$504$	0	0
$505$	−29.2923	−1.30349
$506$	0	0
$507$	9.03581	0.401294
$508$	0	0
$509$	−15.4251	−0.683705	−0.341853	−	0.939754i	$-0.611054\pi$
$509$	−15.4251	−0.683705	−0.341853	+	0.939754i	$0.611054\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−20.7722	−0.917113
$514$	0	0
$515$	−63.5475	−2.80024
$516$	0	0
$517$	42.4909	1.86875
$518$	0	0
$519$	18.0870	0.793933
$520$	0	0
$521$	23.9354	1.04863	0.524314	−	0.851525i	$-0.324322\pi$
$521$	23.9354	1.04863	0.524314	+	0.851525i	$0.324322\pi$
$522$	0	0
$523$	−11.5921	−0.506887	−0.253444	−	0.967350i	$-0.581563\pi$
$523$	−11.5921	−0.506887	−0.253444	+	0.967350i	$0.581563\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.616601	−0.0268596
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.16202	−0.224013
$532$	0	0
$533$	16.8580	0.730201
$534$	0	0
$535$	−14.7440	−0.637437
$536$	0	0
$537$	−10.6567	−0.459870
$538$	0	0
$539$	0	0
$540$	0	0
$541$	33.9706	1.46051	0.730256	−	0.683174i	$-0.239401\pi$
$541$	33.9706	1.46051	0.730256	+	0.683174i	$0.239401\pi$
$542$	0	0
$543$	22.7112	0.974629
$544$	0	0
$545$	−30.3577	−1.30038
$546$	0	0
$547$	−36.1828	−1.54706	−0.773531	−	0.633758i	$-0.781512\pi$
$547$	−36.1828	−1.54706	−0.773531	+	0.633758i	$0.781512\pi$
$548$	0	0
$549$	3.68016	0.157065
$550$	0	0
$551$	−24.4593	−1.04200
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−33.6167	−1.42695
$556$	0	0
$557$	23.0462	0.976497	0.488249	−	0.872705i	$-0.337636\pi$
$557$	23.0462	0.976497	0.488249	+	0.872705i	$0.337636\pi$
$558$	0	0
$559$	−6.30421	−0.266640
$560$	0	0
$561$	−1.16159	−0.0490423
$562$	0	0
$563$	38.0718	1.60453	0.802267	−	0.596966i	$-0.203627\pi$
$563$	38.0718	1.60453	0.802267	+	0.596966i	$0.203627\pi$
$564$	0	0
$565$	26.9760	1.13489
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−31.5147	−1.32116	−0.660582	−	0.750754i	$-0.729691\pi$
$569$	−31.5147	−1.32116	−0.660582	+	0.750754i	$0.729691\pi$
$570$	0	0
$571$	−6.70108	−0.280431	−0.140216	−	0.990121i	$-0.544780\pi$
$571$	−6.70108	−0.280431	−0.140216	+	0.990121i	$0.544780\pi$
$572$	0	0
$573$	−16.9140	−0.706594
$574$	0	0
$575$	6.97805	0.291005
$576$	0	0
$577$	−17.8233	−0.741992	−0.370996	−	0.928635i	$-0.620984\pi$
$577$	−17.8233	−0.741992	−0.370996	+	0.928635i	$0.620984\pi$
$578$	0	0
$579$	−19.3610	−0.804615
$580$	0	0
$581$	0	0
$582$	0	0
$583$	7.32305	0.303290
$584$	0	0
$585$	−13.5375	−0.559706
$586$	0	0
$587$	7.27955	0.300459	0.150230	−	0.988651i	$-0.451999\pi$
$587$	7.27955	0.300459	0.150230	+	0.988651i	$0.451999\pi$
$588$	0	0
$589$	9.08826	0.374475
$590$	0	0
$591$	18.6362	0.766590
$592$	0	0
$593$	−4.75316	−0.195189	−0.0975944	−	0.995226i	$-0.531115\pi$
$593$	−4.75316	−0.195189	−0.0975944	+	0.995226i	$0.531115\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.85570	0.280585
$598$	0	0
$599$	−31.0491	−1.26863	−0.634316	−	0.773074i	$-0.718718\pi$
$599$	−31.0491	−1.26863	−0.634316	+	0.773074i	$0.718718\pi$
$600$	0	0
$601$	−29.4043	−1.19943	−0.599713	−	0.800215i	$-0.704718\pi$
$601$	−29.4043	−1.19943	−0.599713	+	0.800215i	$0.704718\pi$
$602$	0	0
$603$	−15.6131	−0.635813
$604$	0	0
$605$	−23.1564	−0.941442
$606$	0	0
$607$	23.0974	0.937493	0.468746	−	0.883333i	$-0.344706\pi$
$607$	23.0974	0.937493	0.468746	+	0.883333i	$0.344706\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−18.7899	−0.760158
$612$	0	0
$613$	−27.8380	−1.12436	−0.562182	−	0.827013i	$-0.690038\pi$
$613$	−27.8380	−1.12436	−0.562182	+	0.827013i	$0.690038\pi$
$614$	0	0
$615$	−29.7091	−1.19799
$616$	0	0
$617$	19.7367	0.794569	0.397284	−	0.917696i	$-0.369953\pi$
$617$	19.7367	0.794569	0.397284	+	0.917696i	$0.369953\pi$
$618$	0	0
$619$	−25.3652	−1.01952	−0.509758	−	0.860318i	$-0.670265\pi$
$619$	−25.3652	−1.01952	−0.509758	+	0.860318i	$0.670265\pi$
$620$	0	0
$621$	4.83303	0.193943
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−11.1971	−0.447882
$626$	0	0
$627$	17.1210	0.683746
$628$	0	0
$629$	−2.99057	−0.119242
$630$	0	0
$631$	29.6304	1.17957	0.589784	−	0.807561i	$-0.299213\pi$
$631$	29.6304	1.17957	0.589784	+	0.807561i	$0.299213\pi$
$632$	0	0
$633$	−18.9188	−0.751956
$634$	0	0
$635$	15.7073	0.623326
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−28.7374	−1.13683
$640$	0	0
$641$	5.32204	0.210208	0.105104	−	0.994461i	$-0.466482\pi$
$641$	5.32204	0.210208	0.105104	+	0.994461i	$0.466482\pi$
$642$	0	0
$643$	11.8947	0.469080	0.234540	−	0.972106i	$-0.424642\pi$
$643$	11.8947	0.469080	0.234540	+	0.972106i	$0.424642\pi$
$644$	0	0
$645$	11.1100	0.437456
$646$	0	0
$647$	20.2361	0.795561	0.397781	−	0.917481i	$-0.369781\pi$
$647$	20.2361	0.795561	0.397781	+	0.917481i	$0.369781\pi$
$648$	0	0
$649$	10.3241	0.405255
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.4579	−1.19191	−0.595955	−	0.803018i	$-0.703226\pi$
$653$	−30.4579	−1.19191	−0.595955	+	0.803018i	$0.703226\pi$
$654$	0	0
$655$	25.7070	1.00446
$656$	0	0
$657$	17.8782	0.697494
$658$	0	0
$659$	−22.4648	−0.875106	−0.437553	−	0.899193i	$-0.644155\pi$
$659$	−22.4648	−0.875106	−0.437553	+	0.899193i	$0.644155\pi$
$660$	0	0
$661$	−21.6470	−0.841971	−0.420986	−	0.907067i	$-0.638316\pi$
$661$	−21.6470	−0.841971	−0.420986	+	0.907067i	$0.638316\pi$
$662$	0	0
$663$	0.513666	0.0199491
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.69091	0.220353
$668$	0	0
$669$	10.7224	0.414552
$670$	0	0
$671$	−7.36035	−0.284143
$672$	0	0
$673$	−18.9426	−0.730185	−0.365093	−	0.930971i	$-0.618963\pi$
$673$	−18.9426	−0.730185	−0.365093	+	0.930971i	$0.618963\pi$
$674$	0	0
$675$	33.7251	1.29808
$676$	0	0
$677$	−18.0507	−0.693743	−0.346872	−	0.937913i	$-0.612756\pi$
$677$	−18.0507	−0.693743	−0.346872	+	0.937913i	$0.612756\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	27.6888	1.06104
$682$	0	0
$683$	34.4517	1.31826	0.659129	−	0.752029i	$-0.270925\pi$
$683$	34.4517	1.31826	0.659129	+	0.752029i	$0.270925\pi$
$684$	0	0
$685$	54.7090	2.09033
$686$	0	0
$687$	−9.21678	−0.351642
$688$	0	0
$689$	−3.23833	−0.123370
$690$	0	0
$691$	−16.1464	−0.614239	−0.307120	−	0.951671i	$-0.599365\pi$
$691$	−16.1464	−0.614239	−0.307120	+	0.951671i	$0.599365\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.6293	0.516988
$696$	0	0
$697$	−2.64295	−0.100109
$698$	0	0
$699$	26.6653	1.00858
$700$	0	0
$701$	30.7261	1.16051	0.580254	−	0.814435i	$-0.302953\pi$
$701$	30.7261	1.16051	0.580254	+	0.814435i	$0.302953\pi$
$702$	0	0
$703$	44.0789	1.66247
$704$	0	0
$705$	33.1137	1.24713
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−45.4814	−1.70809	−0.854045	−	0.520200i	$-0.825858\pi$
$709$	−45.4814	−1.70809	−0.854045	+	0.520200i	$0.825858\pi$
$710$	0	0
$711$	14.4088	0.540371
$712$	0	0
$713$	−2.11455	−0.0791906
$714$	0	0
$715$	27.0750	1.01255
$716$	0	0
$717$	4.09874	0.153070
$718$	0	0
$719$	24.8338	0.926146	0.463073	−	0.886320i	$-0.346747\pi$
$719$	24.8338	0.926146	0.463073	+	0.886320i	$0.346747\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	6.03440	0.224422
$724$	0	0
$725$	39.7115	1.47485
$726$	0	0
$727$	32.9715	1.22284	0.611422	−	0.791305i	$-0.290598\pi$
$727$	32.9715	1.22284	0.611422	+	0.791305i	$0.290598\pi$
$728$	0	0
$729$	10.0902	0.373710
$730$	0	0
$731$	0.988357	0.0365557
$732$	0	0
$733$	−2.33041	−0.0860756	−0.0430378	−	0.999073i	$-0.513704\pi$
$733$	−2.33041	−0.0860756	−0.0430378	+	0.999073i	$0.513704\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	31.2262	1.15023
$738$	0	0
$739$	15.7195	0.578250	0.289125	−	0.957291i	$-0.406636\pi$
$739$	15.7195	0.578250	0.289125	+	0.957291i	$0.406636\pi$
$740$	0	0
$741$	−7.57107	−0.278130
$742$	0	0
$743$	30.6295	1.12369	0.561843	−	0.827244i	$-0.310092\pi$
$743$	30.6295	1.12369	0.561843	+	0.827244i	$0.310092\pi$
$744$	0	0
$745$	−33.3080	−1.22031
$746$	0	0
$747$	−24.5535	−0.898365
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−17.9907	−0.656491	−0.328245	−	0.944593i	$-0.606457\pi$
$751$	−17.9907	−0.656491	−0.328245	+	0.944593i	$0.606457\pi$
$752$	0	0
$753$	12.2507	0.446441
$754$	0	0
$755$	−18.4555	−0.671663
$756$	0	0
$757$	19.2892	0.701079	0.350539	−	0.936548i	$-0.385998\pi$
$757$	19.2892	0.701079	0.350539	+	0.936548i	$0.385998\pi$
$758$	0	0
$759$	−3.98352	−0.144592
$760$	0	0
$761$	21.1364	0.766194	0.383097	−	0.923708i	$-0.374857\pi$
$761$	21.1364	0.766194	0.383097	+	0.923708i	$0.374857\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.12237	0.0767344
$766$	0	0
$767$	−4.56541	−0.164847
$768$	0	0
$769$	−30.5788	−1.10270	−0.551349	−	0.834275i	$-0.685887\pi$
$769$	−30.5788	−1.10270	−0.551349	+	0.834275i	$0.685887\pi$
$770$	0	0
$771$	−6.04768	−0.217802
$772$	0	0
$773$	−43.1171	−1.55081	−0.775407	−	0.631461i	$-0.782455\pi$
$773$	−43.1171	−1.55081	−0.775407	+	0.631461i	$0.782455\pi$
$774$	0	0
$775$	−14.7555	−0.530032
$776$	0	0
$777$	0	0
$778$	0	0
$779$	38.9552	1.39572
$780$	0	0
$781$	57.4750	2.05662
$782$	0	0
$783$	27.5044	0.982926
$784$	0	0
$785$	−76.9872	−2.74779
$786$	0	0
$787$	38.9120	1.38706	0.693531	−	0.720427i	$-0.256054\pi$
$787$	38.9120	1.38706	0.693531	+	0.720427i	$0.256054\pi$
$788$	0	0
$789$	25.0383	0.891387
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3.25482	0.115582
$794$	0	0
$795$	5.70695	0.202405
$796$	0	0
$797$	40.6055	1.43832	0.719161	−	0.694843i	$-0.244526\pi$
$797$	40.6055	1.43832	0.719161	+	0.694843i	$0.244526\pi$
$798$	0	0
$799$	2.94583	0.104216
$800$	0	0
$801$	−4.42862	−0.156477
$802$	0	0
$803$	−35.7564	−1.26182
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.88054	−0.0661983
$808$	0	0
$809$	36.5718	1.28580	0.642898	−	0.765951i	$-0.277732\pi$
$809$	36.5718	1.28580	0.642898	+	0.765951i	$0.277732\pi$
$810$	0	0
$811$	−41.6094	−1.46111	−0.730553	−	0.682856i	$-0.760737\pi$
$811$	−41.6094	−1.46111	−0.730553	+	0.682856i	$0.760737\pi$
$812$	0	0
$813$	−9.66609	−0.339005
$814$	0	0
$815$	30.2799	1.06066
$816$	0	0
$817$	−14.5677	−0.509658
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−55.9242	−1.95177	−0.975885	−	0.218287i	$-0.929953\pi$
$821$	−55.9242	−1.95177	−0.975885	+	0.218287i	$0.929953\pi$
$822$	0	0
$823$	7.60832	0.265209	0.132605	−	0.991169i	$-0.457666\pi$
$823$	7.60832	0.265209	0.132605	+	0.991169i	$0.457666\pi$
$824$	0	0
$825$	−27.7972	−0.967773
$826$	0	0
$827$	10.5766	0.367784	0.183892	−	0.982946i	$-0.441130\pi$
$827$	10.5766	0.367784	0.183892	+	0.982946i	$0.441130\pi$
$828$	0	0
$829$	25.5560	0.887597	0.443799	−	0.896127i	$-0.353631\pi$
$829$	25.5560	0.887597	0.443799	+	0.896127i	$0.353631\pi$
$830$	0	0
$831$	−11.6012	−0.402441
$832$	0	0
$833$	0	0
$834$	0	0
$835$	31.9038	1.10408
$836$	0	0
$837$	−10.2197	−0.353244
$838$	0	0
$839$	56.3099	1.94403	0.972017	−	0.234912i	$-0.0754803\pi$
$839$	56.3099	1.94403	0.972017	+	0.234912i	$0.0754803\pi$
$840$	0	0
$841$	3.38650	0.116776
$842$	0	0
$843$	9.13676	0.314687
$844$	0	0
$845$	33.0193	1.13590
$846$	0	0
$847$	0	0
$848$	0	0
$849$	19.3377	0.663668
$850$	0	0
$851$	−10.2558	−0.351564
$852$	0	0
$853$	3.35036	0.114714	0.0573571	−	0.998354i	$-0.481733\pi$
$853$	3.35036	0.114714	0.0573571	+	0.998354i	$0.481733\pi$
$854$	0	0
$855$	−31.2822	−1.06983
$856$	0	0
$857$	3.12327	0.106689	0.0533444	−	0.998576i	$-0.483012\pi$
$857$	3.12327	0.106689	0.0533444	+	0.998576i	$0.483012\pi$
$858$	0	0
$859$	−10.6210	−0.362385	−0.181192	−	0.983448i	$-0.557996\pi$
$859$	−10.6210	−0.362385	−0.181192	+	0.983448i	$0.557996\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−47.5294	−1.61792	−0.808961	−	0.587863i	$-0.799969\pi$
$863$	−47.5294	−1.61792	−0.808961	+	0.587863i	$0.799969\pi$
$864$	0	0
$865$	66.0949	2.24729
$866$	0	0
$867$	16.0201	0.544070
$868$	0	0
$869$	−28.8176	−0.977571
$870$	0	0
$871$	−13.8086	−0.467885
$872$	0	0
$873$	−5.70490	−0.193082
$874$	0	0
$875$	0	0
$876$	0	0
$877$	39.5376	1.33509	0.667545	−	0.744570i	$-0.267345\pi$
$877$	39.5376	1.33509	0.667545	+	0.744570i	$0.267345\pi$
$878$	0	0
$879$	−14.5546	−0.490916
$880$	0	0
$881$	14.8998	0.501988	0.250994	−	0.967989i	$-0.419242\pi$
$881$	14.8998	0.501988	0.250994	+	0.967989i	$0.419242\pi$
$882$	0	0
$883$	15.6543	0.526809	0.263404	−	0.964686i	$-0.415155\pi$
$883$	15.6543	0.526809	0.263404	+	0.964686i	$0.415155\pi$
$884$	0	0
$885$	8.04569	0.270453
$886$	0	0
$887$	−12.8799	−0.432465	−0.216232	−	0.976342i	$-0.569377\pi$
$887$	−12.8799	−0.432465	−0.216232	+	0.976342i	$0.569377\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	7.28366	0.244012
$892$	0	0
$893$	−43.4194	−1.45298
$894$	0	0
$895$	−38.9424	−1.30170
$896$	0	0
$897$	1.76155	0.0588165
$898$	0	0
$899$	−12.0337	−0.401348
$900$	0	0
$901$	0.507696	0.0169138
$902$	0	0
$903$	0	0
$904$	0	0
$905$	82.9927	2.75877
$906$	0	0
$907$	41.3799	1.37400	0.686998	−	0.726659i	$-0.258928\pi$
$907$	41.3799	1.37400	0.686998	+	0.726659i	$0.258928\pi$
$908$	0	0
$909$	−17.7993	−0.590365
$910$	0	0
$911$	24.7825	0.821080	0.410540	−	0.911843i	$-0.365340\pi$
$911$	24.7825	0.821080	0.410540	+	0.911843i	$0.365340\pi$
$912$	0	0
$913$	49.1071	1.62521
$914$	0	0
$915$	−5.73601	−0.189627
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−18.9727	−0.625852	−0.312926	−	0.949777i	$-0.601309\pi$
$919$	−18.9727	−0.625852	−0.312926	+	0.949777i	$0.601309\pi$
$920$	0	0
$921$	−5.10653	−0.168266
$922$	0	0
$923$	−25.4160	−0.836579
$924$	0	0
$925$	−71.5654	−2.35305
$926$	0	0
$927$	−38.6142	−1.26826
$928$	0	0
$929$	−28.8534	−0.946651	−0.473325	−	0.880888i	$-0.656946\pi$
$929$	−28.8534	−0.946651	−0.473325	+	0.880888i	$0.656946\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.86601	0.126568
$934$	0	0
$935$	−4.24475	−0.138818
$936$	0	0
$937$	23.9024	0.780858	0.390429	−	0.920633i	$-0.372327\pi$
$937$	23.9024	0.780858	0.390429	+	0.920633i	$0.372327\pi$
$938$	0	0
$939$	−29.8173	−0.973049
$940$	0	0
$941$	−2.31508	−0.0754693	−0.0377347	−	0.999288i	$-0.512014\pi$
$941$	−2.31508	−0.0754693	−0.0377347	+	0.999288i	$0.512014\pi$
$942$	0	0
$943$	−9.06366	−0.295153
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−55.6174	−1.80732	−0.903661	−	0.428248i	$-0.859131\pi$
$947$	−55.6174	−1.80732	−0.903661	+	0.428248i	$0.859131\pi$
$948$	0	0
$949$	15.8119	0.513275
$950$	0	0
$951$	2.38455	0.0773244
$952$	0	0
$953$	38.2771	1.23992	0.619958	−	0.784635i	$-0.287149\pi$
$953$	38.2771	1.23992	0.619958	+	0.784635i	$0.287149\pi$
$954$	0	0
$955$	−61.8084	−2.00007
$956$	0	0
$957$	−22.6698	−0.732812
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−26.5287	−0.855763
$962$	0	0
$963$	−8.95908	−0.288702
$964$	0	0
$965$	−70.7502	−2.27753
$966$	0	0
$967$	55.3206	1.77899	0.889495	−	0.456946i	$-0.151057\pi$
$967$	55.3206	1.77899	0.889495	+	0.456946i	$0.151057\pi$
$968$	0	0
$969$	1.18697	0.0381310
$970$	0	0
$971$	−6.99759	−0.224563	−0.112282	−	0.993676i	$-0.535816\pi$
$971$	−6.99759	−0.224563	−0.112282	+	0.993676i	$0.535816\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	12.2922	0.393665
$976$	0	0
$977$	−28.3943	−0.908414	−0.454207	−	0.890896i	$-0.650077\pi$
$977$	−28.3943	−0.908414	−0.454207	+	0.890896i	$0.650077\pi$
$978$	0	0
$979$	8.85726	0.283079
$980$	0	0
$981$	−18.4467	−0.588957
$982$	0	0
$983$	−22.5719	−0.719932	−0.359966	−	0.932965i	$-0.617212\pi$
$983$	−22.5719	−0.719932	−0.359966	+	0.932965i	$0.617212\pi$
$984$	0	0
$985$	68.1016	2.16990
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.38944	0.107778
$990$	0	0
$991$	−38.1925	−1.21323	−0.606613	−	0.794997i	$-0.707472\pi$
$991$	−38.1925	−1.21323	−0.606613	+	0.794997i	$0.707472\pi$
$992$	0	0
$993$	−22.7489	−0.721915
$994$	0	0
$995$	25.0525	0.794219
$996$	0	0
$997$	35.2549	1.11653	0.558267	−	0.829661i	$-0.311467\pi$
$997$	35.2549	1.11653	0.558267	+	0.829661i	$0.311467\pi$
$998$	0	0
$999$	−49.5665	−1.56821

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9016.2.a.bl.1.4		11
7.3	odd	6		1288.2.q.a.737.4	✓	22
7.5	odd	6		1288.2.q.a.921.4	yes	22
7.6	odd	2		9016.2.a.bq.1.8		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.q.a.737.4	✓	22	7.3	odd	6
1288.2.q.a.921.4	yes	22	7.5	odd	6
9016.2.a.bl.1.4		11	1.1	even	1	trivial
9016.2.a.bq.1.8		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)

\(f(q)\)	\(=\)	\(q-0.947093 q^{3} -3.46093 q^{5} -2.10301 q^{9} +O(q^{10})\)	\(q-0.947093 q^{3} -3.46093 q^{5} -2.10301 q^{9} +4.20604 q^{11} -1.85995 q^{13} +3.27783 q^{15} +0.291599 q^{17} -4.29796 q^{19} +1.00000 q^{23} +6.97805 q^{25} +4.83303 q^{27} +5.69091 q^{29} -2.11455 q^{31} -3.98352 q^{33} -10.2558 q^{37} +1.76155 q^{39} -9.06366 q^{41} +3.38944 q^{43} +7.27839 q^{45} +10.1023 q^{47} -0.276171 q^{51} +1.74108 q^{53} -14.5568 q^{55} +4.07057 q^{57} +2.45458 q^{59} -1.74995 q^{61} +6.43718 q^{65} +7.42413 q^{67} -0.947093 q^{69} +13.6649 q^{71} -8.50121 q^{73} -6.60886 q^{75} -6.85148 q^{79} +1.73171 q^{81} +11.6754 q^{83} -1.00920 q^{85} -5.38983 q^{87} +2.10584 q^{89} +2.00268 q^{93} +14.8749 q^{95} +2.71273 q^{97} -8.84537 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 4 q^{3} - q^{5} + 11 q^{9}+O(q^{10}) \) 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

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