LMFDB - Embedded newform 9016.2.a.br.1.6

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			1.6
Root			$0.710583$ 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)

$f(q)$	$=$	$q+0.710583 q^{3} -1.82431 q^{5} -2.49507 q^{9} +O(q^{10})$	$q+0.710583 q^{3} -1.82431 q^{5} -2.49507 q^{9} -5.77063 q^{11} -5.36591 q^{13} -1.29632 q^{15} -4.21028 q^{17} -3.23858 q^{19} -1.00000 q^{23} -1.67189 q^{25} -3.90470 q^{27} -5.40844 q^{29} +1.66789 q^{31} -4.10051 q^{33} +9.55963 q^{37} -3.81293 q^{39} +9.89333 q^{41} -7.54130 q^{43} +4.55179 q^{45} -1.83849 q^{47} -2.99175 q^{51} -10.8389 q^{53} +10.5274 q^{55} -2.30128 q^{57} +6.38372 q^{59} -9.11785 q^{61} +9.78909 q^{65} -10.9619 q^{67} -0.710583 q^{69} +0.315288 q^{71} -14.7227 q^{73} -1.18802 q^{75} +10.8950 q^{79} +4.71060 q^{81} -8.54941 q^{83} +7.68085 q^{85} -3.84314 q^{87} +17.3245 q^{89} +1.18517 q^{93} +5.90818 q^{95} -8.01165 q^{97} +14.3981 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.710583	0.410255	0.205128	−	0.978735i	$-0.434239\pi$
$3$	0.710583	0.410255	0.205128	+	0.978735i	$0.434239\pi$
$4$	0	0
$5$	−1.82431	−0.815856	−0.407928	−	0.913014i	$-0.633749\pi$
$5$	−1.82431	−0.815856	−0.407928	+	0.913014i	$0.633749\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.49507	−0.831691
$10$	0	0
$11$	−5.77063	−1.73991	−0.869955	−	0.493132i	$-0.835852\pi$
$11$	−5.77063	−1.73991	−0.869955	+	0.493132i	$0.835852\pi$
$12$	0	0
$13$	−5.36591	−1.48824	−0.744119	−	0.668048i	$-0.767130\pi$
$13$	−5.36591	−1.48824	−0.744119	+	0.668048i	$0.767130\pi$
$14$	0	0
$15$	−1.29632	−0.334709
$16$	0	0
$17$	−4.21028	−1.02114	−0.510571	−	0.859835i	$-0.670566\pi$
$17$	−4.21028	−1.02114	−0.510571	+	0.859835i	$0.670566\pi$
$18$	0	0
$19$	−3.23858	−0.742982	−0.371491	−	0.928437i	$-0.621153\pi$
$19$	−3.23858	−0.742982	−0.371491	+	0.928437i	$0.621153\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.67189	−0.334379
$26$	0	0
$27$	−3.90470	−0.751460
$28$	0	0
$29$	−5.40844	−1.00432	−0.502161	−	0.864774i	$-0.667461\pi$
$29$	−5.40844	−1.00432	−0.502161	+	0.864774i	$0.667461\pi$
$30$	0	0
$31$	1.66789	0.299562	0.149781	−	0.988719i	$-0.452143\pi$
$31$	1.66789	0.299562	0.149781	+	0.988719i	$0.452143\pi$
$32$	0	0
$33$	−4.10051	−0.713807
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.55963	1.57159	0.785797	−	0.618484i	$-0.212253\pi$
$37$	9.55963	1.57159	0.785797	+	0.618484i	$0.212253\pi$
$38$	0	0
$39$	−3.81293	−0.610557
$40$	0	0
$41$	9.89333	1.54508	0.772539	−	0.634967i	$-0.218986\pi$
$41$	9.89333	1.54508	0.772539	+	0.634967i	$0.218986\pi$
$42$	0	0
$43$	−7.54130	−1.15004	−0.575019	−	0.818140i	$-0.695005\pi$
$43$	−7.54130	−1.15004	−0.575019	+	0.818140i	$0.695005\pi$
$44$	0	0
$45$	4.55179	0.678540
$46$	0	0
$47$	−1.83849	−0.268172	−0.134086	−	0.990970i	$-0.542810\pi$
$47$	−1.83849	−0.268172	−0.134086	+	0.990970i	$0.542810\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.99175	−0.418929
$52$	0	0
$53$	−10.8389	−1.48883	−0.744417	−	0.667715i	$-0.767272\pi$
$53$	−10.8389	−1.48883	−0.744417	+	0.667715i	$0.767272\pi$
$54$	0	0
$55$	10.5274	1.41952
$56$	0	0
$57$	−2.30128	−0.304812
$58$	0	0
$59$	6.38372	0.831090	0.415545	−	0.909573i	$-0.363591\pi$
$59$	6.38372	0.831090	0.415545	+	0.909573i	$0.363591\pi$
$60$	0	0
$61$	−9.11785	−1.16742	−0.583711	−	0.811962i	$-0.698400\pi$
$61$	−9.11785	−1.16742	−0.583711	+	0.811962i	$0.698400\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.78909	1.21419
$66$	0	0
$67$	−10.9619	−1.33921	−0.669606	−	0.742717i	$-0.733537\pi$
$67$	−10.9619	−1.33921	−0.669606	+	0.742717i	$0.733537\pi$
$68$	0	0
$69$	−0.710583	−0.0855441
$70$	0	0
$71$	0.315288	0.0374179	0.0187089	−	0.999825i	$-0.494044\pi$
$71$	0.315288	0.0374179	0.0187089	+	0.999825i	$0.494044\pi$
$72$	0	0
$73$	−14.7227	−1.72316	−0.861580	−	0.507622i	$-0.830525\pi$
$73$	−14.7227	−1.72316	−0.861580	+	0.507622i	$0.830525\pi$
$74$	0	0
$75$	−1.18802	−0.137181
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.8950	1.22578	0.612892	−	0.790166i	$-0.290006\pi$
$79$	10.8950	1.22578	0.612892	+	0.790166i	$0.290006\pi$
$80$	0	0
$81$	4.71060	0.523400
$82$	0	0
$83$	−8.54941	−0.938420	−0.469210	−	0.883087i	$-0.655461\pi$
$83$	−8.54941	−0.938420	−0.469210	+	0.883087i	$0.655461\pi$
$84$	0	0
$85$	7.68085	0.833105
$86$	0	0
$87$	−3.84314	−0.412028
$88$	0	0
$89$	17.3245	1.83639	0.918194	−	0.396130i	$-0.129647\pi$
$89$	17.3245	1.83639	0.918194	+	0.396130i	$0.129647\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.18517	0.122897
$94$	0	0
$95$	5.90818	0.606166
$96$	0	0
$97$	−8.01165	−0.813460	−0.406730	−	0.913548i	$-0.633331\pi$
$97$	−8.01165	−0.813460	−0.406730	+	0.913548i	$0.633331\pi$
$98$	0	0
$99$	14.3981	1.44707
$100$	0	0
$101$	−1.41746	−0.141042	−0.0705212	−	0.997510i	$-0.522466\pi$
$101$	−1.41746	−0.141042	−0.0705212	+	0.997510i	$0.522466\pi$
$102$	0	0
$103$	9.60107	0.946021	0.473011	−	0.881057i	$-0.343167\pi$
$103$	9.60107	0.946021	0.473011	+	0.881057i	$0.343167\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.36185	−0.711697	−0.355849	−	0.934544i	$-0.615808\pi$
$107$	−7.36185	−0.711697	−0.355849	+	0.934544i	$0.615808\pi$
$108$	0	0
$109$	11.9452	1.14414	0.572070	−	0.820205i	$-0.306140\pi$
$109$	11.9452	1.14414	0.572070	+	0.820205i	$0.306140\pi$
$110$	0	0
$111$	6.79291	0.644754
$112$	0	0
$113$	13.8151	1.29961	0.649806	−	0.760100i	$-0.274850\pi$
$113$	13.8151	1.29961	0.649806	+	0.760100i	$0.274850\pi$
$114$	0	0
$115$	1.82431	0.170118
$116$	0	0
$117$	13.3883	1.23775
$118$	0	0
$119$	0	0
$120$	0	0
$121$	22.3001	2.02728
$122$	0	0
$123$	7.03003	0.633876
$124$	0	0
$125$	12.1716	1.08866
$126$	0	0
$127$	−17.1252	−1.51962	−0.759810	−	0.650145i	$-0.774708\pi$
$127$	−17.1252	−1.51962	−0.759810	+	0.650145i	$0.774708\pi$
$128$	0	0
$129$	−5.35872	−0.471809
$130$	0	0
$131$	6.50378	0.568238	0.284119	−	0.958789i	$-0.408299\pi$
$131$	6.50378	0.568238	0.284119	+	0.958789i	$0.408299\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	7.12339	0.613084
$136$	0	0
$137$	−3.89953	−0.333160	−0.166580	−	0.986028i	$-0.553272\pi$
$137$	−3.89953	−0.333160	−0.166580	+	0.986028i	$0.553272\pi$
$138$	0	0
$139$	−4.85448	−0.411752	−0.205876	−	0.978578i	$-0.566004\pi$
$139$	−4.85448	−0.411752	−0.205876	+	0.978578i	$0.566004\pi$
$140$	0	0
$141$	−1.30640	−0.110019
$142$	0	0
$143$	30.9647	2.58940
$144$	0	0
$145$	9.86666	0.819382
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.59188	−0.621951	−0.310976	−	0.950418i	$-0.600656\pi$
$149$	−7.59188	−0.621951	−0.310976	+	0.950418i	$0.600656\pi$
$150$	0	0
$151$	−22.1085	−1.79916	−0.899581	−	0.436754i	$-0.856128\pi$
$151$	−22.1085	−1.79916	−0.899581	+	0.436754i	$0.856128\pi$
$152$	0	0
$153$	10.5049	0.849275
$154$	0	0
$155$	−3.04275	−0.244399
$156$	0	0
$157$	15.4263	1.23115	0.615577	−	0.788077i	$-0.288923\pi$
$157$	15.4263	1.23115	0.615577	+	0.788077i	$0.288923\pi$
$158$	0	0
$159$	−7.70191	−0.610801
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.63706	−0.206551	−0.103275	−	0.994653i	$-0.532932\pi$
$163$	−2.63706	−0.206551	−0.103275	+	0.994653i	$0.532932\pi$
$164$	0	0
$165$	7.48059	0.582363
$166$	0	0
$167$	9.72802	0.752777	0.376388	−	0.926462i	$-0.377166\pi$
$167$	9.72802	0.752777	0.376388	+	0.926462i	$0.377166\pi$
$168$	0	0
$169$	15.7930	1.21485
$170$	0	0
$171$	8.08050	0.617931
$172$	0	0
$173$	1.95315	0.148496	0.0742478	−	0.997240i	$-0.476344\pi$
$173$	1.95315	0.148496	0.0742478	+	0.997240i	$0.476344\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.53616	0.340959
$178$	0	0
$179$	1.32603	0.0991121	0.0495560	−	0.998771i	$-0.484219\pi$
$179$	1.32603	0.0991121	0.0495560	+	0.998771i	$0.484219\pi$
$180$	0	0
$181$	−8.65740	−0.643500	−0.321750	−	0.946825i	$-0.604271\pi$
$181$	−8.65740	−0.643500	−0.321750	+	0.946825i	$0.604271\pi$
$182$	0	0
$183$	−6.47899	−0.478941
$184$	0	0
$185$	−17.4397	−1.28219
$186$	0	0
$187$	24.2959	1.77670
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.8065	0.854291	0.427146	−	0.904183i	$-0.359519\pi$
$191$	11.8065	0.854291	0.427146	+	0.904183i	$0.359519\pi$
$192$	0	0
$193$	−23.0407	−1.65850	−0.829251	−	0.558876i	$-0.811233\pi$
$193$	−23.0407	−1.65850	−0.829251	+	0.558876i	$0.811233\pi$
$194$	0	0
$195$	6.95596	0.498127
$196$	0	0
$197$	−12.9587	−0.923271	−0.461636	−	0.887070i	$-0.652737\pi$
$197$	−12.9587	−0.923271	−0.461636	+	0.887070i	$0.652737\pi$
$198$	0	0
$199$	12.5176	0.887352	0.443676	−	0.896187i	$-0.353674\pi$
$199$	12.5176	0.887352	0.443676	+	0.896187i	$0.353674\pi$
$200$	0	0
$201$	−7.78935	−0.549418
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−18.0485	−1.26056
$206$	0	0
$207$	2.49507	0.173420
$208$	0	0
$209$	18.6887	1.29272
$210$	0	0
$211$	−0.280508	−0.0193109	−0.00965547	−	0.999953i	$-0.503073\pi$
$211$	−0.280508	−0.0193109	−0.00965547	+	0.999953i	$0.503073\pi$
$212$	0	0
$213$	0.224039	0.0153509
$214$	0	0
$215$	13.7577	0.938265
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−10.4617	−0.706935
$220$	0	0
$221$	22.5920	1.51970
$222$	0	0
$223$	−24.7990	−1.66067	−0.830334	−	0.557267i	$-0.811850\pi$
$223$	−24.7990	−1.66067	−0.830334	+	0.557267i	$0.811850\pi$
$224$	0	0
$225$	4.17149	0.278100
$226$	0	0
$227$	−13.5716	−0.900780	−0.450390	−	0.892832i	$-0.648715\pi$
$227$	−13.5716	−0.900780	−0.450390	+	0.892832i	$0.648715\pi$
$228$	0	0
$229$	8.57743	0.566812	0.283406	−	0.959000i	$-0.408536\pi$
$229$	8.57743	0.566812	0.283406	+	0.959000i	$0.408536\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.3594	−1.13725	−0.568625	−	0.822597i	$-0.692524\pi$
$233$	−17.3594	−1.13725	−0.568625	+	0.822597i	$0.692524\pi$
$234$	0	0
$235$	3.35398	0.218790
$236$	0	0
$237$	7.74181	0.502884
$238$	0	0
$239$	7.81259	0.505355	0.252677	−	0.967551i	$-0.418689\pi$
$239$	7.81259	0.505355	0.252677	+	0.967551i	$0.418689\pi$
$240$	0	0
$241$	−20.8604	−1.34374	−0.671868	−	0.740671i	$-0.734508\pi$
$241$	−20.8604	−1.34374	−0.671868	+	0.740671i	$0.734508\pi$
$242$	0	0
$243$	15.0614	0.966188
$244$	0	0
$245$	0	0
$246$	0	0
$247$	17.3780	1.10573
$248$	0	0
$249$	−6.07507	−0.384992
$250$	0	0
$251$	27.9679	1.76532	0.882659	−	0.470015i	$-0.155751\pi$
$251$	27.9679	1.76532	0.882659	+	0.470015i	$0.155751\pi$
$252$	0	0
$253$	5.77063	0.362796
$254$	0	0
$255$	5.45788	0.341786
$256$	0	0
$257$	8.36998	0.522105	0.261053	−	0.965325i	$-0.415930\pi$
$257$	8.36998	0.522105	0.261053	+	0.965325i	$0.415930\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	13.4944	0.835285
$262$	0	0
$263$	−0.521371	−0.0321491	−0.0160746	−	0.999871i	$-0.505117\pi$
$263$	−0.521371	−0.0321491	−0.0160746	+	0.999871i	$0.505117\pi$
$264$	0	0
$265$	19.7735	1.21467
$266$	0	0
$267$	12.3105	0.753388
$268$	0	0
$269$	−8.74556	−0.533226	−0.266613	−	0.963804i	$-0.585905\pi$
$269$	−8.74556	−0.533226	−0.266613	+	0.963804i	$0.585905\pi$
$270$	0	0
$271$	−0.415839	−0.0252604	−0.0126302	−	0.999920i	$-0.504020\pi$
$271$	−0.415839	−0.0252604	−0.0126302	+	0.999920i	$0.504020\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.64787	0.581789
$276$	0	0
$277$	−24.5877	−1.47733	−0.738667	−	0.674071i	$-0.764544\pi$
$277$	−24.5877	−1.47733	−0.738667	+	0.674071i	$0.764544\pi$
$278$	0	0
$279$	−4.16150	−0.249143
$280$	0	0
$281$	2.50209	0.149262	0.0746310	−	0.997211i	$-0.476222\pi$
$281$	2.50209	0.149262	0.0746310	+	0.997211i	$0.476222\pi$
$282$	0	0
$283$	4.19688	0.249479	0.124739	−	0.992190i	$-0.460191\pi$
$283$	4.19688	0.249479	0.124739	+	0.992190i	$0.460191\pi$
$284$	0	0
$285$	4.19825	0.248683
$286$	0	0
$287$	0	0
$288$	0	0
$289$	0.726441	0.0427318
$290$	0	0
$291$	−5.69294	−0.333726
$292$	0	0
$293$	0.289936	0.0169382	0.00846912	−	0.999964i	$-0.497304\pi$
$293$	0.289936	0.0169382	0.00846912	+	0.999964i	$0.497304\pi$
$294$	0	0
$295$	−11.6459	−0.678050
$296$	0	0
$297$	22.5326	1.30747
$298$	0	0
$299$	5.36591	0.310319
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.00722	−0.0578633
$304$	0	0
$305$	16.6338	0.952448
$306$	0	0
$307$	−31.4328	−1.79397	−0.896983	−	0.442064i	$-0.854246\pi$
$307$	−31.4328	−1.79397	−0.896983	+	0.442064i	$0.854246\pi$
$308$	0	0
$309$	6.82235	0.388110
$310$	0	0
$311$	−15.0694	−0.854510	−0.427255	−	0.904131i	$-0.640519\pi$
$311$	−15.0694	−0.854510	−0.427255	+	0.904131i	$0.640519\pi$
$312$	0	0
$313$	−18.9122	−1.06898	−0.534491	−	0.845174i	$-0.679497\pi$
$313$	−18.9122	−1.06898	−0.534491	+	0.845174i	$0.679497\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−33.6795	−1.89163	−0.945814	−	0.324708i	$-0.894734\pi$
$317$	−33.6795	−1.89163	−0.945814	+	0.324708i	$0.894734\pi$
$318$	0	0
$319$	31.2101	1.74743
$320$	0	0
$321$	−5.23120	−0.291977
$322$	0	0
$323$	13.6353	0.758690
$324$	0	0
$325$	8.97124	0.497635
$326$	0	0
$327$	8.48804	0.469390
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−10.1369	−0.557172	−0.278586	−	0.960411i	$-0.589866\pi$
$331$	−10.1369	−0.557172	−0.278586	+	0.960411i	$0.589866\pi$
$332$	0	0
$333$	−23.8520	−1.30708
$334$	0	0
$335$	19.9979	1.09260
$336$	0	0
$337$	13.1847	0.718215	0.359108	−	0.933296i	$-0.383081\pi$
$337$	13.1847	0.718215	0.359108	+	0.933296i	$0.383081\pi$
$338$	0	0
$339$	9.81674	0.533172
$340$	0	0
$341$	−9.62476	−0.521210
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.29632	0.0697917
$346$	0	0
$347$	−21.4893	−1.15360	−0.576802	−	0.816884i	$-0.695700\pi$
$347$	−21.4893	−1.15360	−0.576802	+	0.816884i	$0.695700\pi$
$348$	0	0
$349$	11.6538	0.623812	0.311906	−	0.950113i	$-0.399033\pi$
$349$	11.6538	0.623812	0.311906	+	0.950113i	$0.399033\pi$
$350$	0	0
$351$	20.9523	1.11835
$352$	0	0
$353$	25.5247	1.35854	0.679271	−	0.733888i	$-0.262296\pi$
$353$	25.5247	1.35854	0.679271	+	0.733888i	$0.262296\pi$
$354$	0	0
$355$	−0.575184	−0.0305276
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.7750	−1.62424	−0.812121	−	0.583489i	$-0.801687\pi$
$359$	−30.7750	−1.62424	−0.812121	+	0.583489i	$0.801687\pi$
$360$	0	0
$361$	−8.51158	−0.447978
$362$	0	0
$363$	15.8461	0.831704
$364$	0	0
$365$	26.8587	1.40585
$366$	0	0
$367$	6.78311	0.354076	0.177038	−	0.984204i	$-0.443349\pi$
$367$	6.78311	0.354076	0.177038	+	0.984204i	$0.443349\pi$
$368$	0	0
$369$	−24.6846	−1.28503
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−21.3020	−1.10298	−0.551488	−	0.834183i	$-0.685940\pi$
$373$	−21.3020	−1.10298	−0.551488	+	0.834183i	$0.685940\pi$
$374$	0	0
$375$	8.64893	0.446629
$376$	0	0
$377$	29.0212	1.49467
$378$	0	0
$379$	−11.2499	−0.577867	−0.288934	−	0.957349i	$-0.593301\pi$
$379$	−11.2499	−0.577867	−0.288934	+	0.957349i	$0.593301\pi$
$380$	0	0
$381$	−12.1689	−0.623432
$382$	0	0
$383$	16.8472	0.860851	0.430426	−	0.902626i	$-0.358364\pi$
$383$	16.8472	0.860851	0.430426	+	0.902626i	$0.358364\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	18.8161	0.956475
$388$	0	0
$389$	16.0705	0.814804	0.407402	−	0.913249i	$-0.366435\pi$
$389$	16.0705	0.814804	0.407402	+	0.913249i	$0.366435\pi$
$390$	0	0
$391$	4.21028	0.212923
$392$	0	0
$393$	4.62148	0.233123
$394$	0	0
$395$	−19.8759	−1.00006
$396$	0	0
$397$	9.17055	0.460257	0.230128	−	0.973160i	$-0.426085\pi$
$397$	9.17055	0.460257	0.230128	+	0.973160i	$0.426085\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.4385	−1.12052	−0.560262	−	0.828315i	$-0.689300\pi$
$401$	−22.4385	−1.12052	−0.560262	+	0.828315i	$0.689300\pi$
$402$	0	0
$403$	−8.94975	−0.445819
$404$	0	0
$405$	−8.59360	−0.427019
$406$	0	0
$407$	−55.1651	−2.73443
$408$	0	0
$409$	9.26421	0.458086	0.229043	−	0.973416i	$-0.426440\pi$
$409$	9.26421	0.458086	0.229043	+	0.973416i	$0.426440\pi$
$410$	0	0
$411$	−2.77094	−0.136680
$412$	0	0
$413$	0	0
$414$	0	0
$415$	15.5968	0.765616
$416$	0	0
$417$	−3.44951	−0.168923
$418$	0	0
$419$	15.3144	0.748157	0.374079	−	0.927397i	$-0.377959\pi$
$419$	15.3144	0.748157	0.374079	+	0.927397i	$0.377959\pi$
$420$	0	0
$421$	6.91095	0.336819	0.168410	−	0.985717i	$-0.446137\pi$
$421$	6.91095	0.336819	0.168410	+	0.985717i	$0.446137\pi$
$422$	0	0
$423$	4.58718	0.223036
$424$	0	0
$425$	7.03914	0.341448
$426$	0	0
$427$	0	0
$428$	0	0
$429$	22.0030	1.06231
$430$	0	0
$431$	−11.5959	−0.558553	−0.279276	−	0.960211i	$-0.590095\pi$
$431$	−11.5959	−0.558553	−0.279276	+	0.960211i	$0.590095\pi$
$432$	0	0
$433$	−14.8912	−0.715625	−0.357812	−	0.933793i	$-0.616477\pi$
$433$	−14.8912	−0.715625	−0.357812	+	0.933793i	$0.616477\pi$
$434$	0	0
$435$	7.01108	0.336156
$436$	0	0
$437$	3.23858	0.154922
$438$	0	0
$439$	3.18734	0.152123	0.0760616	−	0.997103i	$-0.475765\pi$
$439$	3.18734	0.152123	0.0760616	+	0.997103i	$0.475765\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	22.7155	1.07925	0.539624	−	0.841906i	$-0.318566\pi$
$443$	22.7155	1.07925	0.539624	+	0.841906i	$0.318566\pi$
$444$	0	0
$445$	−31.6052	−1.49823
$446$	0	0
$447$	−5.39466	−0.255159
$448$	0	0
$449$	22.9245	1.08188	0.540938	−	0.841062i	$-0.318069\pi$
$449$	22.9245	1.08188	0.540938	+	0.841062i	$0.318069\pi$
$450$	0	0
$451$	−57.0907	−2.68830
$452$	0	0
$453$	−15.7099	−0.738115
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.78898	0.177241	0.0886204	−	0.996065i	$-0.471754\pi$
$457$	3.78898	0.177241	0.0886204	+	0.996065i	$0.471754\pi$
$458$	0	0
$459$	16.4399	0.767348
$460$	0	0
$461$	−23.2945	−1.08493	−0.542466	−	0.840078i	$-0.682509\pi$
$461$	−23.2945	−1.08493	−0.542466	+	0.840078i	$0.682509\pi$
$462$	0	0
$463$	−8.37283	−0.389118	−0.194559	−	0.980891i	$-0.562328\pi$
$463$	−8.37283	−0.389118	−0.194559	+	0.980891i	$0.562328\pi$
$464$	0	0
$465$	−2.16212	−0.100266
$466$	0	0
$467$	−7.03109	−0.325360	−0.162680	−	0.986679i	$-0.552014\pi$
$467$	−7.03109	−0.325360	−0.162680	+	0.986679i	$0.552014\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	10.9617	0.505087
$472$	0	0
$473$	43.5180	2.00096
$474$	0	0
$475$	5.41456	0.248437
$476$	0	0
$477$	27.0438	1.23825
$478$	0	0
$479$	6.36519	0.290833	0.145417	−	0.989371i	$-0.453548\pi$
$479$	6.36519	0.290833	0.145417	+	0.989371i	$0.453548\pi$
$480$	0	0
$481$	−51.2962	−2.33890
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.6157	0.663666
$486$	0	0
$487$	−32.4724	−1.47146	−0.735732	−	0.677273i	$-0.763162\pi$
$487$	−32.4724	−1.47146	−0.735732	+	0.677273i	$0.763162\pi$
$488$	0	0
$489$	−1.87385	−0.0847385
$490$	0	0
$491$	−11.5365	−0.520636	−0.260318	−	0.965523i	$-0.583827\pi$
$491$	−11.5365	−0.520636	−0.260318	+	0.965523i	$0.583827\pi$
$492$	0	0
$493$	22.7710	1.02556
$494$	0	0
$495$	−26.2667	−1.18060
$496$	0	0
$497$	0	0
$498$	0	0
$499$	8.41561	0.376734	0.188367	−	0.982099i	$-0.439680\pi$
$499$	8.41561	0.376734	0.188367	+	0.982099i	$0.439680\pi$
$500$	0	0
$501$	6.91256	0.308830
$502$	0	0
$503$	33.9583	1.51413	0.757063	−	0.653342i	$-0.226634\pi$
$503$	33.9583	1.51413	0.757063	+	0.653342i	$0.226634\pi$
$504$	0	0
$505$	2.58588	0.115070
$506$	0	0
$507$	11.2223	0.498398
$508$	0	0
$509$	−0.988230	−0.0438025	−0.0219013	−	0.999760i	$-0.506972\pi$
$509$	−0.988230	−0.0438025	−0.0219013	+	0.999760i	$0.506972\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	12.6457	0.558321
$514$	0	0
$515$	−17.5153	−0.771817
$516$	0	0
$517$	10.6093	0.466595
$518$	0	0
$519$	1.38788	0.0609210
$520$	0	0
$521$	−18.9980	−0.832316	−0.416158	−	0.909292i	$-0.636624\pi$
$521$	−18.9980	−0.832316	−0.416158	+	0.909292i	$0.636624\pi$
$522$	0	0
$523$	26.5585	1.16132	0.580660	−	0.814146i	$-0.302795\pi$
$523$	26.5585	1.16132	0.580660	+	0.814146i	$0.302795\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.02228	−0.305895
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−15.9279	−0.691210
$532$	0	0
$533$	−53.0868	−2.29944
$534$	0	0
$535$	13.4303	0.580642
$536$	0	0
$537$	0.942253	0.0406612
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3.27656	−0.140870	−0.0704351	−	0.997516i	$-0.522439\pi$
$541$	−3.27656	−0.140870	−0.0704351	+	0.997516i	$0.522439\pi$
$542$	0	0
$543$	−6.15180	−0.263999
$544$	0	0
$545$	−21.7917	−0.933454
$546$	0	0
$547$	30.4657	1.30262	0.651311	−	0.758811i	$-0.274220\pi$
$547$	30.4657	1.30262	0.651311	+	0.758811i	$0.274220\pi$
$548$	0	0
$549$	22.7497	0.970934
$550$	0	0
$551$	17.5157	0.746193
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−12.3924	−0.526027
$556$	0	0
$557$	−28.5520	−1.20979	−0.604894	−	0.796306i	$-0.706784\pi$
$557$	−28.5520	−1.20979	−0.604894	+	0.796306i	$0.706784\pi$
$558$	0	0
$559$	40.4660	1.71153
$560$	0	0
$561$	17.2643	0.728898
$562$	0	0
$563$	−24.0084	−1.01183	−0.505916	−	0.862583i	$-0.668845\pi$
$563$	−24.0084	−1.01183	−0.505916	+	0.862583i	$0.668845\pi$
$564$	0	0
$565$	−25.2029	−1.06030
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17.8141	0.746808	0.373404	−	0.927669i	$-0.378191\pi$
$569$	17.8141	0.746808	0.373404	+	0.927669i	$0.378191\pi$
$570$	0	0
$571$	24.8853	1.04142	0.520709	−	0.853734i	$-0.325668\pi$
$571$	24.8853	1.04142	0.520709	+	0.853734i	$0.325668\pi$
$572$	0	0
$573$	8.38952	0.350477
$574$	0	0
$575$	1.67189	0.0697228
$576$	0	0
$577$	−0.925816	−0.0385422	−0.0192711	−	0.999814i	$-0.506135\pi$
$577$	−0.925816	−0.0385422	−0.0192711	+	0.999814i	$0.506135\pi$
$578$	0	0
$579$	−16.3723	−0.680409
$580$	0	0
$581$	0	0
$582$	0	0
$583$	62.5471	2.59044
$584$	0	0
$585$	−24.4245	−1.00983
$586$	0	0
$587$	−3.67887	−0.151843	−0.0759216	−	0.997114i	$-0.524190\pi$
$587$	−3.67887	−0.151843	−0.0759216	+	0.997114i	$0.524190\pi$
$588$	0	0
$589$	−5.40160	−0.222569
$590$	0	0
$591$	−9.20825	−0.378777
$592$	0	0
$593$	29.3857	1.20673	0.603363	−	0.797466i	$-0.293827\pi$
$593$	29.3857	1.20673	0.603363	+	0.797466i	$0.293827\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.89482	0.364041
$598$	0	0
$599$	1.09237	0.0446330	0.0223165	−	0.999751i	$-0.492896\pi$
$599$	1.09237	0.0446330	0.0223165	+	0.999751i	$0.492896\pi$
$600$	0	0
$601$	37.2297	1.51863	0.759316	−	0.650722i	$-0.225534\pi$
$601$	37.2297	1.51863	0.759316	+	0.650722i	$0.225534\pi$
$602$	0	0
$603$	27.3508	1.11381
$604$	0	0
$605$	−40.6823	−1.65397
$606$	0	0
$607$	23.1342	0.938990	0.469495	−	0.882935i	$-0.344436\pi$
$607$	23.1342	0.938990	0.469495	+	0.882935i	$0.344436\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.86521	0.399104
$612$	0	0
$613$	−45.6081	−1.84209	−0.921046	−	0.389453i	$-0.872664\pi$
$613$	−45.6081	−1.84209	−0.921046	+	0.389453i	$0.872664\pi$
$614$	0	0
$615$	−12.8250	−0.517152
$616$	0	0
$617$	18.5245	0.745767	0.372884	−	0.927878i	$-0.378369\pi$
$617$	18.5245	0.745767	0.372884	+	0.927878i	$0.378369\pi$
$618$	0	0
$619$	14.4628	0.581309	0.290654	−	0.956828i	$-0.406127\pi$
$619$	14.4628	0.581309	0.290654	+	0.956828i	$0.406127\pi$
$620$	0	0
$621$	3.90470	0.156690
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−13.8453	−0.553812
$626$	0	0
$627$	13.2798	0.530345
$628$	0	0
$629$	−40.2487	−1.60482
$630$	0	0
$631$	−21.4915	−0.855563	−0.427782	−	0.903882i	$-0.640705\pi$
$631$	−21.4915	−0.855563	−0.427782	+	0.903882i	$0.640705\pi$
$632$	0	0
$633$	−0.199324	−0.00792241
$634$	0	0
$635$	31.2418	1.23979
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−0.786668	−0.0311201
$640$	0	0
$641$	−7.33364	−0.289661	−0.144831	−	0.989456i	$-0.546264\pi$
$641$	−7.33364	−0.289661	−0.144831	+	0.989456i	$0.546264\pi$
$642$	0	0
$643$	20.5107	0.808864	0.404432	−	0.914568i	$-0.367469\pi$
$643$	20.5107	0.808864	0.404432	+	0.914568i	$0.367469\pi$
$644$	0	0
$645$	9.77596	0.384928
$646$	0	0
$647$	15.0921	0.593331	0.296666	−	0.954981i	$-0.404125\pi$
$647$	15.0921	0.593331	0.296666	+	0.954981i	$0.404125\pi$
$648$	0	0
$649$	−36.8381	−1.44602
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15.1018	0.590979	0.295490	−	0.955346i	$-0.404517\pi$
$653$	15.1018	0.590979	0.295490	+	0.955346i	$0.404517\pi$
$654$	0	0
$655$	−11.8649	−0.463601
$656$	0	0
$657$	36.7342	1.43314
$658$	0	0
$659$	−20.4769	−0.797665	−0.398832	−	0.917024i	$-0.630585\pi$
$659$	−20.4769	−0.797665	−0.398832	+	0.917024i	$0.630585\pi$
$660$	0	0
$661$	−31.1152	−1.21024	−0.605121	−	0.796134i	$-0.706875\pi$
$661$	−31.1152	−1.21024	−0.605121	+	0.796134i	$0.706875\pi$
$662$	0	0
$663$	16.0535	0.623465
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.40844	0.209416
$668$	0	0
$669$	−17.6218	−0.681297
$670$	0	0
$671$	52.6157	2.03121
$672$	0	0
$673$	−3.35334	−0.129262	−0.0646309	−	0.997909i	$-0.520587\pi$
$673$	−3.35334	−0.129262	−0.0646309	+	0.997909i	$0.520587\pi$
$674$	0	0
$675$	6.52825	0.251272
$676$	0	0
$677$	−47.8577	−1.83932	−0.919660	−	0.392715i	$-0.871536\pi$
$677$	−47.8577	−1.83932	−0.919660	+	0.392715i	$0.871536\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−9.64375	−0.369549
$682$	0	0
$683$	46.6991	1.78689	0.893446	−	0.449171i	$-0.148281\pi$
$683$	46.6991	1.78689	0.893446	+	0.449171i	$0.148281\pi$
$684$	0	0
$685$	7.11396	0.271810
$686$	0	0
$687$	6.09497	0.232538
$688$	0	0
$689$	58.1605	2.21574
$690$	0	0
$691$	10.1481	0.386052	0.193026	−	0.981194i	$-0.438170\pi$
$691$	10.1481	0.386052	0.193026	+	0.981194i	$0.438170\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.85608	0.335930
$696$	0	0
$697$	−41.6537	−1.57775
$698$	0	0
$699$	−12.3353	−0.466562
$700$	0	0
$701$	16.2419	0.613450	0.306725	−	0.951798i	$-0.400767\pi$
$701$	16.2419	0.613450	0.306725	+	0.951798i	$0.400767\pi$
$702$	0	0
$703$	−30.9597	−1.16767
$704$	0	0
$705$	2.38328	0.0897596
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.63185	0.0612855	0.0306427	−	0.999530i	$-0.490245\pi$
$709$	1.63185	0.0612855	0.0306427	+	0.999530i	$0.490245\pi$
$710$	0	0
$711$	−27.1839	−1.01947
$712$	0	0
$713$	−1.66789	−0.0624629
$714$	0	0
$715$	−56.4892	−2.11258
$716$	0	0
$717$	5.55149	0.207324
$718$	0	0
$719$	38.8491	1.44883	0.724413	−	0.689367i	$-0.242111\pi$
$719$	38.8491	1.44883	0.724413	+	0.689367i	$0.242111\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−14.8230	−0.551274
$724$	0	0
$725$	9.04233	0.335824
$726$	0	0
$727$	47.5946	1.76519	0.882593	−	0.470137i	$-0.155796\pi$
$727$	47.5946	1.76519	0.882593	+	0.470137i	$0.155796\pi$
$728$	0	0
$729$	−3.42946	−0.127017
$730$	0	0
$731$	31.7510	1.17435
$732$	0	0
$733$	11.3808	0.420359	0.210180	−	0.977663i	$-0.432595\pi$
$733$	11.3808	0.420359	0.210180	+	0.977663i	$0.432595\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	63.2571	2.33011
$738$	0	0
$739$	0.652395	0.0239987	0.0119994	−	0.999928i	$-0.496180\pi$
$739$	0.652395	0.0239987	0.0119994	+	0.999928i	$0.496180\pi$
$740$	0	0
$741$	12.3485	0.453633
$742$	0	0
$743$	−8.33443	−0.305761	−0.152880	−	0.988245i	$-0.548855\pi$
$743$	−8.33443	−0.305761	−0.152880	+	0.988245i	$0.548855\pi$
$744$	0	0
$745$	13.8499	0.507423
$746$	0	0
$747$	21.3314	0.780475
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−18.2478	−0.665871	−0.332936	−	0.942950i	$-0.608039\pi$
$751$	−18.2478	−0.665871	−0.332936	+	0.942950i	$0.608039\pi$
$752$	0	0
$753$	19.8735	0.724230
$754$	0	0
$755$	40.3327	1.46786
$756$	0	0
$757$	11.1850	0.406527	0.203264	−	0.979124i	$-0.434845\pi$
$757$	11.1850	0.406527	0.203264	+	0.979124i	$0.434845\pi$
$758$	0	0
$759$	4.10051	0.148839
$760$	0	0
$761$	1.29827	0.0470623	0.0235311	−	0.999723i	$-0.492509\pi$
$761$	1.29827	0.0470623	0.0235311	+	0.999723i	$0.492509\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−19.1643	−0.692886
$766$	0	0
$767$	−34.2545	−1.23686
$768$	0	0
$769$	−46.1638	−1.66471	−0.832354	−	0.554244i	$-0.813007\pi$
$769$	−46.1638	−1.66471	−0.832354	+	0.554244i	$0.813007\pi$
$770$	0	0
$771$	5.94756	0.214196
$772$	0	0
$773$	−27.3831	−0.984902	−0.492451	−	0.870340i	$-0.663899\pi$
$773$	−27.3831	−0.984902	−0.492451	+	0.870340i	$0.663899\pi$
$774$	0	0
$775$	−2.78853	−0.100167
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−32.0404	−1.14797
$780$	0	0
$781$	−1.81941	−0.0651037
$782$	0	0
$783$	21.1183	0.754708
$784$	0	0
$785$	−28.1424	−1.00444
$786$	0	0
$787$	44.3866	1.58221	0.791106	−	0.611679i	$-0.209506\pi$
$787$	44.3866	1.58221	0.791106	+	0.611679i	$0.209506\pi$
$788$	0	0
$789$	−0.370477	−0.0131893
$790$	0	0
$791$	0	0
$792$	0	0
$793$	48.9256	1.73740
$794$	0	0
$795$	14.0507	0.498326
$796$	0	0
$797$	4.59683	0.162828	0.0814141	−	0.996680i	$-0.474056\pi$
$797$	4.59683	0.162828	0.0814141	+	0.996680i	$0.474056\pi$
$798$	0	0
$799$	7.74057	0.273842
$800$	0	0
$801$	−43.2258	−1.52731
$802$	0	0
$803$	84.9591	2.99814
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.21444	−0.218759
$808$	0	0
$809$	−38.7456	−1.36222	−0.681112	−	0.732180i	$-0.738503\pi$
$809$	−38.7456	−1.36222	−0.681112	+	0.732180i	$0.738503\pi$
$810$	0	0
$811$	19.2047	0.674368	0.337184	−	0.941439i	$-0.390526\pi$
$811$	19.2047	0.674368	0.337184	+	0.941439i	$0.390526\pi$
$812$	0	0
$813$	−0.295488	−0.0103632
$814$	0	0
$815$	4.81082	0.168516
$816$	0	0
$817$	24.4231	0.854457
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.84661	0.0993475	0.0496738	−	0.998765i	$-0.484182\pi$
$821$	2.84661	0.0993475	0.0496738	+	0.998765i	$0.484182\pi$
$822$	0	0
$823$	38.0885	1.32768	0.663841	−	0.747874i	$-0.268925\pi$
$823$	38.0885	1.32768	0.663841	+	0.747874i	$0.268925\pi$
$824$	0	0
$825$	6.85561	0.238682
$826$	0	0
$827$	36.0886	1.25492	0.627462	−	0.778647i	$-0.284094\pi$
$827$	36.0886	1.25492	0.627462	+	0.778647i	$0.284094\pi$
$828$	0	0
$829$	−0.376284	−0.0130689	−0.00653443	−	0.999979i	$-0.502080\pi$
$829$	−0.376284	−0.0130689	−0.00653443	+	0.999979i	$0.502080\pi$
$830$	0	0
$831$	−17.4716	−0.606084
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−17.7469	−0.614157
$836$	0	0
$837$	−6.51261	−0.225109
$838$	0	0
$839$	−55.6489	−1.92121	−0.960606	−	0.277914i	$-0.910357\pi$
$839$	−55.6489	−1.92121	−0.960606	+	0.277914i	$0.910357\pi$
$840$	0	0
$841$	0.251188	0.00866167
$842$	0	0
$843$	1.77794	0.0612355
$844$	0	0
$845$	−28.8114	−0.991142
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.98223	0.102350
$850$	0	0
$851$	−9.55963	−0.327700
$852$	0	0
$853$	−21.0743	−0.721572	−0.360786	−	0.932649i	$-0.617491\pi$
$853$	−21.0743	−0.721572	−0.360786	+	0.932649i	$0.617491\pi$
$854$	0	0
$855$	−14.7413	−0.504143
$856$	0	0
$857$	−23.2322	−0.793598	−0.396799	−	0.917906i	$-0.629879\pi$
$857$	−23.2322	−0.793598	−0.396799	+	0.917906i	$0.629879\pi$
$858$	0	0
$859$	−40.0979	−1.36812	−0.684061	−	0.729425i	$-0.739788\pi$
$859$	−40.0979	−1.36812	−0.684061	+	0.729425i	$0.739788\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31.7021	−1.07915	−0.539576	−	0.841937i	$-0.681416\pi$
$863$	−31.7021	−1.07915	−0.539576	+	0.841937i	$0.681416\pi$
$864$	0	0
$865$	−3.56316	−0.121151
$866$	0	0
$867$	0.516196	0.0175309
$868$	0	0
$869$	−62.8711	−2.13275
$870$	0	0
$871$	58.8207	1.99306
$872$	0	0
$873$	19.9896	0.676547
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−29.3670	−0.991653	−0.495827	−	0.868421i	$-0.665135\pi$
$877$	−29.3670	−0.991653	−0.495827	+	0.868421i	$0.665135\pi$
$878$	0	0
$879$	0.206023	0.00694900
$880$	0	0
$881$	−21.1272	−0.711793	−0.355897	−	0.934525i	$-0.615824\pi$
$881$	−21.1272	−0.711793	−0.355897	+	0.934525i	$0.615824\pi$
$882$	0	0
$883$	−0.691836	−0.0232821	−0.0116411	−	0.999932i	$-0.503706\pi$
$883$	−0.691836	−0.0232821	−0.0116411	+	0.999932i	$0.503706\pi$
$884$	0	0
$885$	−8.27537	−0.278173
$886$	0	0
$887$	−37.1873	−1.24863	−0.624313	−	0.781174i	$-0.714621\pi$
$887$	−37.1873	−1.24863	−0.624313	+	0.781174i	$0.714621\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−27.1831	−0.910669
$892$	0	0
$893$	5.95412	0.199247
$894$	0	0
$895$	−2.41909	−0.0808612
$896$	0	0
$897$	3.81293	0.127310
$898$	0	0
$899$	−9.02067	−0.300856
$900$	0	0
$901$	45.6347	1.52031
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15.7938	0.525003
$906$	0	0
$907$	−47.6899	−1.58352	−0.791759	−	0.610834i	$-0.790834\pi$
$907$	−47.6899	−1.58352	−0.791759	+	0.610834i	$0.790834\pi$
$908$	0	0
$909$	3.53666	0.117304
$910$	0	0
$911$	−6.58857	−0.218289	−0.109145	−	0.994026i	$-0.534811\pi$
$911$	−6.58857	−0.218289	−0.109145	+	0.994026i	$0.534811\pi$
$912$	0	0
$913$	49.3355	1.63277
$914$	0	0
$915$	11.8197	0.390747
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−53.6369	−1.76932	−0.884659	−	0.466238i	$-0.845609\pi$
$919$	−53.6369	−1.76932	−0.884659	+	0.466238i	$0.845609\pi$
$920$	0	0
$921$	−22.3356	−0.735984
$922$	0	0
$923$	−1.69181	−0.0556866
$924$	0	0
$925$	−15.9827	−0.525508
$926$	0	0
$927$	−23.9554	−0.786797
$928$	0	0
$929$	−5.74670	−0.188543	−0.0942716	−	0.995547i	$-0.530052\pi$
$929$	−5.74670	−0.188543	−0.0942716	+	0.995547i	$0.530052\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−10.7081	−0.350567
$934$	0	0
$935$	−44.3233	−1.44953
$936$	0	0
$937$	−19.0373	−0.621921	−0.310961	−	0.950423i	$-0.600651\pi$
$937$	−19.0373	−0.621921	−0.310961	+	0.950423i	$0.600651\pi$
$938$	0	0
$939$	−13.4387	−0.438556
$940$	0	0
$941$	38.5821	1.25774	0.628870	−	0.777510i	$-0.283518\pi$
$941$	38.5821	1.25774	0.628870	+	0.777510i	$0.283518\pi$
$942$	0	0
$943$	−9.89333	−0.322171
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.5863	1.22139	0.610695	−	0.791866i	$-0.290890\pi$
$947$	37.5863	1.22139	0.610695	+	0.791866i	$0.290890\pi$
$948$	0	0
$949$	79.0007	2.56447
$950$	0	0
$951$	−23.9321	−0.776050
$952$	0	0
$953$	15.6442	0.506764	0.253382	−	0.967366i	$-0.418457\pi$
$953$	15.6442	0.506764	0.253382	+	0.967366i	$0.418457\pi$
$954$	0	0
$955$	−21.5388	−0.696979
$956$	0	0
$957$	22.1773	0.716891
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28.2181	−0.910263
$962$	0	0
$963$	18.3684	0.591912
$964$	0	0
$965$	42.0333	1.35310
$966$	0	0
$967$	−13.3728	−0.430042	−0.215021	−	0.976609i	$-0.568982\pi$
$967$	−13.3728	−0.430042	−0.215021	+	0.976609i	$0.568982\pi$
$968$	0	0
$969$	9.68903	0.311257
$970$	0	0
$971$	4.99499	0.160297	0.0801484	−	0.996783i	$-0.474461\pi$
$971$	4.99499	0.160297	0.0801484	+	0.996783i	$0.474461\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	6.37480	0.204157
$976$	0	0
$977$	−22.7928	−0.729208	−0.364604	−	0.931163i	$-0.618796\pi$
$977$	−22.7928	−0.729208	−0.364604	+	0.931163i	$0.618796\pi$
$978$	0	0
$979$	−99.9730	−3.19515
$980$	0	0
$981$	−29.8041	−0.951571
$982$	0	0
$983$	−15.3695	−0.490210	−0.245105	−	0.969497i	$-0.578822\pi$
$983$	−15.3695	−0.490210	−0.245105	+	0.969497i	$0.578822\pi$
$984$	0	0
$985$	23.6407	0.753257
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.54130	0.239799
$990$	0	0
$991$	32.1275	1.02056	0.510281	−	0.860008i	$-0.329541\pi$
$991$	32.1275	1.02056	0.510281	+	0.860008i	$0.329541\pi$
$992$	0	0
$993$	−7.20308	−0.228583
$994$	0	0
$995$	−22.8361	−0.723952
$996$	0	0
$997$	−47.3044	−1.49815	−0.749073	−	0.662488i	$-0.769501\pi$
$997$	−47.3044	−1.49815	−0.749073	+	0.662488i	$0.769501\pi$
$998$	0	0
$999$	−37.3275	−1.18099

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9016.2.a.br.1.6		11
7.3	odd	6		1288.2.q.d.737.6	✓	22
7.5	odd	6		1288.2.q.d.921.6	yes	22
7.6	odd	2		9016.2.a.bk.1.6		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.q.d.737.6	✓	22	7.3	odd	6
1288.2.q.d.921.6	yes	22	7.5	odd	6
9016.2.a.bk.1.6		11	7.6	odd	2
9016.2.a.br.1.6		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)

\(f(q)\)	\(=\)	\(q+0.710583 q^{3} -1.82431 q^{5} -2.49507 q^{9} +O(q^{10})\)	\(q+0.710583 q^{3} -1.82431 q^{5} -2.49507 q^{9} -5.77063 q^{11} -5.36591 q^{13} -1.29632 q^{15} -4.21028 q^{17} -3.23858 q^{19} -1.00000 q^{23} -1.67189 q^{25} -3.90470 q^{27} -5.40844 q^{29} +1.66789 q^{31} -4.10051 q^{33} +9.55963 q^{37} -3.81293 q^{39} +9.89333 q^{41} -7.54130 q^{43} +4.55179 q^{45} -1.83849 q^{47} -2.99175 q^{51} -10.8389 q^{53} +10.5274 q^{55} -2.30128 q^{57} +6.38372 q^{59} -9.11785 q^{61} +9.78909 q^{65} -10.9619 q^{67} -0.710583 q^{69} +0.315288 q^{71} -14.7227 q^{73} -1.18802 q^{75} +10.8950 q^{79} +4.71060 q^{81} -8.54941 q^{83} +7.68085 q^{85} -3.84314 q^{87} +17.3245 q^{89} +1.18517 q^{93} +5.90818 q^{95} -8.01165 q^{97} +14.3981 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 4 q^{3} + 3 q^{5} + 11 q^{9}+O(q^{10}) \) 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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