LMFDB - Embedded newform 6664.2.a.t.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6664,2,Mod(1,6664)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6664 = 2^{3} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6664.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$53.2123079070$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.15751800.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 6x^{3} + 16x^{2} - 7x - 1 $ x^6 - x^5 - 8x^4 + 6x^3 + 16x^2 - 7x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.66208$ of defining polynomial
Character	$\chi$	$=$	6664.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.29428 q^{3} +1.75649 q^{5} -1.32483 q^{9} +0.221093 q^{11} -4.11360 q^{13} -2.27339 q^{15} +1.00000 q^{17} +0.347250 q^{19} -5.77360 q^{23} -1.91475 q^{25} +5.59756 q^{27} +8.59467 q^{29} +6.49819 q^{31} -0.286157 q^{33} +9.04411 q^{37} +5.32416 q^{39} -7.17538 q^{41} +6.05010 q^{43} -2.32705 q^{45} +7.97382 q^{47} -1.29428 q^{51} -11.5617 q^{53} +0.388347 q^{55} -0.449440 q^{57} -14.3481 q^{59} -6.78173 q^{61} -7.22549 q^{65} -3.18814 q^{67} +7.47268 q^{69} +11.8751 q^{71} +6.96039 q^{73} +2.47823 q^{75} +11.0481 q^{79} -3.27033 q^{81} -16.2749 q^{83} +1.75649 q^{85} -11.1239 q^{87} -4.88198 q^{89} -8.41050 q^{93} +0.609940 q^{95} -1.92023 q^{97} -0.292910 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 4 q^{5} + 4 q^{9} + 2 q^{11} - 6 q^{13} - 16 q^{15} + 6 q^{17} + 4 q^{19} - 4 q^{23} + 8 q^{25} + 14 q^{27} - 6 q^{29} - 16 q^{31} - 8 q^{33} - 12 q^{37} + 10 q^{39} - 22 q^{41} + 2 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 4 * q^5 + 4 * q^9 + 2 * q^11 - 6 * q^13 - 16 * q^15 + 6 * q^17 + 4 * q^19 - 4 * q^23 + 8 * q^25 + 14 * q^27 - 6 * q^29 - 16 * q^31 - 8 * q^33 - 12 * q^37 + 10 * q^39 - 22 * q^41 + 2 * q^43 - 30 * q^45 + 2 * q^47 + 2 * q^51 + 6 * q^53 - 2 * q^55 + 6 * q^57 - 20 * q^59 + 6 * q^61 + 12 * q^65 - 2 * q^67 - 46 * q^69 - 8 * q^71 - 18 * q^73 + 50 * q^75 + 10 * q^79 + 30 * q^81 - 6 * q^83 - 4 * q^85 - 60 * q^87 - 6 * q^89 + 8 * q^93 - 10 * q^95 - 36 * q^97 - 8 * q^99

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
$3$	−1.29428	−0.747255	−0.373627	−	0.927579i	$-0.621886\pi$
$3$	−1.29428	−0.747255	−0.373627	+	0.927579i	$0.621886\pi$
$4$	0	0
$5$	1.75649	0.785525	0.392763	−	0.919640i	$-0.371519\pi$
$5$	1.75649	0.785525	0.392763	+	0.919640i	$0.371519\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.32483	−0.441610
$10$	0	0
$11$	0.221093	0.0666620	0.0333310	−	0.999444i	$-0.489388\pi$
$11$	0.221093	0.0666620	0.0333310	+	0.999444i	$0.489388\pi$
$12$	0	0
$13$	−4.11360	−1.14091	−0.570454	−	0.821330i	$-0.693233\pi$
$13$	−4.11360	−1.14091	−0.570454	+	0.821330i	$0.693233\pi$
$14$	0	0
$15$	−2.27339	−0.586987
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0.347250	0.0796646	0.0398323	−	0.999206i	$-0.487318\pi$
$19$	0.347250	0.0796646	0.0398323	+	0.999206i	$0.487318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.77360	−1.20388	−0.601940	−	0.798541i	$-0.705605\pi$
$23$	−5.77360	−1.20388	−0.601940	+	0.798541i	$0.705605\pi$
$24$	0	0
$25$	−1.91475	−0.382950
$26$	0	0
$27$	5.59756	1.07725
$28$	0	0
$29$	8.59467	1.59599	0.797995	−	0.602664i	$-0.205894\pi$
$29$	8.59467	1.59599	0.797995	+	0.602664i	$0.205894\pi$
$30$	0	0
$31$	6.49819	1.16711	0.583555	−	0.812074i	$-0.301661\pi$
$31$	6.49819	1.16711	0.583555	+	0.812074i	$0.301661\pi$
$32$	0	0
$33$	−0.286157	−0.0498135
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.04411	1.48684	0.743421	−	0.668823i	$-0.233202\pi$
$37$	9.04411	1.48684	0.743421	+	0.668823i	$0.233202\pi$
$38$	0	0
$39$	5.32416	0.852549
$40$	0	0
$41$	−7.17538	−1.12061	−0.560303	−	0.828288i	$-0.689315\pi$
$41$	−7.17538	−1.12061	−0.560303	+	0.828288i	$0.689315\pi$
$42$	0	0
$43$	6.05010	0.922632	0.461316	−	0.887236i	$-0.347377\pi$
$43$	6.05010	0.922632	0.461316	+	0.887236i	$0.347377\pi$
$44$	0	0
$45$	−2.32705	−0.346896
$46$	0	0
$47$	7.97382	1.16310	0.581551	−	0.813510i	$-0.302446\pi$
$47$	7.97382	1.16310	0.581551	+	0.813510i	$0.302446\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.29428	−0.181236
$52$	0	0
$53$	−11.5617	−1.58813	−0.794063	−	0.607836i	$-0.792038\pi$
$53$	−11.5617	−1.58813	−0.794063	+	0.607836i	$0.792038\pi$
$54$	0	0
$55$	0.388347	0.0523647
$56$	0	0
$57$	−0.449440	−0.0595298
$58$	0	0
$59$	−14.3481	−1.86796	−0.933979	−	0.357329i	$-0.883688\pi$
$59$	−14.3481	−1.86796	−0.933979	+	0.357329i	$0.883688\pi$
$60$	0	0
$61$	−6.78173	−0.868312	−0.434156	−	0.900838i	$-0.642953\pi$
$61$	−6.78173	−0.868312	−0.434156	+	0.900838i	$0.642953\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7.22549	−0.896211
$66$	0	0
$67$	−3.18814	−0.389494	−0.194747	−	0.980854i	$-0.562389\pi$
$67$	−3.18814	−0.389494	−0.194747	+	0.980854i	$0.562389\pi$
$68$	0	0
$69$	7.47268	0.899605
$70$	0	0
$71$	11.8751	1.40932	0.704660	−	0.709545i	$-0.251099\pi$
$71$	11.8751	1.40932	0.704660	+	0.709545i	$0.251099\pi$
$72$	0	0
$73$	6.96039	0.814652	0.407326	−	0.913283i	$-0.366461\pi$
$73$	6.96039	0.814652	0.407326	+	0.913283i	$0.366461\pi$
$74$	0	0
$75$	2.47823	0.286162
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.0481	1.24301	0.621503	−	0.783412i	$-0.286522\pi$
$79$	11.0481	1.24301	0.621503	+	0.783412i	$0.286522\pi$
$80$	0	0
$81$	−3.27033	−0.363371
$82$	0	0
$83$	−16.2749	−1.78640	−0.893201	−	0.449657i	$-0.851546\pi$
$83$	−16.2749	−1.78640	−0.893201	+	0.449657i	$0.851546\pi$
$84$	0	0
$85$	1.75649	0.190518
$86$	0	0
$87$	−11.1239	−1.19261
$88$	0	0
$89$	−4.88198	−0.517489	−0.258744	−	0.965946i	$-0.583309\pi$
$89$	−4.88198	−0.517489	−0.258744	+	0.965946i	$0.583309\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−8.41050	−0.872128
$94$	0	0
$95$	0.609940	0.0625785
$96$	0	0
$97$	−1.92023	−0.194970	−0.0974849	−	0.995237i	$-0.531080\pi$
$97$	−1.92023	−0.194970	−0.0974849	+	0.995237i	$0.531080\pi$
$98$	0	0
$99$	−0.292910	−0.0294386
$100$	0	0
$101$	−17.2228	−1.71373	−0.856865	−	0.515541i	$-0.827591\pi$
$101$	−17.2228	−1.71373	−0.856865	+	0.515541i	$0.827591\pi$
$102$	0	0
$103$	−0.324544	−0.0319783	−0.0159891	−	0.999872i	$-0.505090\pi$
$103$	−0.324544	−0.0319783	−0.0159891	+	0.999872i	$0.505090\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.36785	0.905625	0.452812	−	0.891606i	$-0.350421\pi$
$107$	9.36785	0.905625	0.452812	+	0.891606i	$0.350421\pi$
$108$	0	0
$109$	−13.2046	−1.26477	−0.632386	−	0.774653i	$-0.717924\pi$
$109$	−13.2046	−1.26477	−0.632386	+	0.774653i	$0.717924\pi$
$110$	0	0
$111$	−11.7056	−1.11105
$112$	0	0
$113$	−1.43010	−0.134532	−0.0672661	−	0.997735i	$-0.521428\pi$
$113$	−1.43010	−0.134532	−0.0672661	+	0.997735i	$0.521428\pi$
$114$	0	0
$115$	−10.1413	−0.945678
$116$	0	0
$117$	5.44982	0.503836
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.9511	−0.995556
$122$	0	0
$123$	9.28698	0.837379
$124$	0	0
$125$	−12.1457	−1.08634
$126$	0	0
$127$	−20.6878	−1.83575	−0.917873	−	0.396875i	$-0.870095\pi$
$127$	−20.6878	−1.83575	−0.917873	+	0.396875i	$0.870095\pi$
$128$	0	0
$129$	−7.83055	−0.689442
$130$	0	0
$131$	1.71540	0.149876	0.0749378	−	0.997188i	$-0.476124\pi$
$131$	1.71540	0.149876	0.0749378	+	0.997188i	$0.476124\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	9.83204	0.846207
$136$	0	0
$137$	−7.45883	−0.637251	−0.318625	−	0.947881i	$-0.603221\pi$
$137$	−7.45883	−0.637251	−0.318625	+	0.947881i	$0.603221\pi$
$138$	0	0
$139$	2.25977	0.191671	0.0958354	−	0.995397i	$-0.469448\pi$
$139$	2.25977	0.191671	0.0958354	+	0.995397i	$0.469448\pi$
$140$	0	0
$141$	−10.3204	−0.869134
$142$	0	0
$143$	−0.909487	−0.0760551
$144$	0	0
$145$	15.0964	1.25369
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.0395	1.64170	0.820850	−	0.571144i	$-0.193500\pi$
$149$	20.0395	1.64170	0.820850	+	0.571144i	$0.193500\pi$
$150$	0	0
$151$	−16.5493	−1.34677	−0.673383	−	0.739294i	$-0.735160\pi$
$151$	−16.5493	−1.34677	−0.673383	+	0.739294i	$0.735160\pi$
$152$	0	0
$153$	−1.32483	−0.107106
$154$	0	0
$155$	11.4140	0.916794
$156$	0	0
$157$	3.10918	0.248139	0.124070	−	0.992274i	$-0.460405\pi$
$157$	3.10918	0.248139	0.124070	+	0.992274i	$0.460405\pi$
$158$	0	0
$159$	14.9641	1.18673
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−14.7490	−1.15523	−0.577614	−	0.816310i	$-0.696016\pi$
$163$	−14.7490	−1.15523	−0.577614	+	0.816310i	$0.696016\pi$
$164$	0	0
$165$	−0.502631	−0.0391297
$166$	0	0
$167$	9.97009	0.771509	0.385754	−	0.922602i	$-0.373941\pi$
$167$	9.97009	0.771509	0.385754	+	0.922602i	$0.373941\pi$
$168$	0	0
$169$	3.92170	0.301669
$170$	0	0
$171$	−0.460047	−0.0351807
$172$	0	0
$173$	1.71541	0.130420	0.0652102	−	0.997872i	$-0.479228\pi$
$173$	1.71541	0.130420	0.0652102	+	0.997872i	$0.479228\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	18.5704	1.39584
$178$	0	0
$179$	5.04041	0.376738	0.188369	−	0.982098i	$-0.439680\pi$
$179$	5.04041	0.376738	0.188369	+	0.982098i	$0.439680\pi$
$180$	0	0
$181$	−19.2854	−1.43347	−0.716737	−	0.697343i	$-0.754365\pi$
$181$	−19.2854	−1.43347	−0.716737	+	0.697343i	$0.754365\pi$
$182$	0	0
$183$	8.77748	0.648850
$184$	0	0
$185$	15.8859	1.16795
$186$	0	0
$187$	0.221093	0.0161679
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.7050	1.13637	0.568187	−	0.822899i	$-0.307645\pi$
$191$	15.7050	1.13637	0.568187	+	0.822899i	$0.307645\pi$
$192$	0	0
$193$	9.53724	0.686506	0.343253	−	0.939243i	$-0.388471\pi$
$193$	9.53724	0.686506	0.343253	+	0.939243i	$0.388471\pi$
$194$	0	0
$195$	9.35183	0.669698
$196$	0	0
$197$	7.83077	0.557919	0.278959	−	0.960303i	$-0.410010\pi$
$197$	7.83077	0.557919	0.278959	+	0.960303i	$0.410010\pi$
$198$	0	0
$199$	−21.2158	−1.50395	−0.751975	−	0.659191i	$-0.770899\pi$
$199$	−21.2158	−1.50395	−0.751975	+	0.659191i	$0.770899\pi$
$200$	0	0
$201$	4.12636	0.291051
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−12.6035	−0.880264
$206$	0	0
$207$	7.64904	0.531645
$208$	0	0
$209$	0.0767745	0.00531060
$210$	0	0
$211$	15.8080	1.08827	0.544134	−	0.838998i	$-0.316858\pi$
$211$	15.8080	1.08827	0.544134	+	0.838998i	$0.316858\pi$
$212$	0	0
$213$	−15.3698	−1.05312
$214$	0	0
$215$	10.6269	0.724751
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−9.00872	−0.608753
$220$	0	0
$221$	−4.11360	−0.276711
$222$	0	0
$223$	6.80165	0.455472	0.227736	−	0.973723i	$-0.426868\pi$
$223$	6.80165	0.455472	0.227736	+	0.973723i	$0.426868\pi$
$224$	0	0
$225$	2.53672	0.169115
$226$	0	0
$227$	17.4926	1.16103	0.580513	−	0.814251i	$-0.302852\pi$
$227$	17.4926	1.16103	0.580513	+	0.814251i	$0.302852\pi$
$228$	0	0
$229$	−23.1693	−1.53107	−0.765537	−	0.643392i	$-0.777527\pi$
$229$	−23.1693	−1.53107	−0.765537	+	0.643392i	$0.777527\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.49404	−0.556463	−0.278232	−	0.960514i	$-0.589748\pi$
$233$	−8.49404	−0.556463	−0.278232	+	0.960514i	$0.589748\pi$
$234$	0	0
$235$	14.0059	0.913646
$236$	0	0
$237$	−14.2994	−0.928843
$238$	0	0
$239$	1.57059	0.101593	0.0507966	−	0.998709i	$-0.483824\pi$
$239$	1.57059	0.101593	0.0507966	+	0.998709i	$0.483824\pi$
$240$	0	0
$241$	−25.6050	−1.64936	−0.824680	−	0.565599i	$-0.808645\pi$
$241$	−25.6050	−1.64936	−0.824680	+	0.565599i	$0.808645\pi$
$242$	0	0
$243$	−12.5599	−0.805720
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.42845	−0.0908899
$248$	0	0
$249$	21.0643	1.33490
$250$	0	0
$251$	−1.31691	−0.0831226	−0.0415613	−	0.999136i	$-0.513233\pi$
$251$	−1.31691	−0.0831226	−0.0415613	+	0.999136i	$0.513233\pi$
$252$	0	0
$253$	−1.27650	−0.0802530
$254$	0	0
$255$	−2.27339	−0.142365
$256$	0	0
$257$	−25.9322	−1.61761	−0.808803	−	0.588080i	$-0.799884\pi$
$257$	−25.9322	−1.61761	−0.808803	+	0.588080i	$0.799884\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−11.3865	−0.704806
$262$	0	0
$263$	19.4338	1.19834	0.599169	−	0.800623i	$-0.295498\pi$
$263$	19.4338	1.19834	0.599169	+	0.800623i	$0.295498\pi$
$264$	0	0
$265$	−20.3080	−1.24751
$266$	0	0
$267$	6.31866	0.386696
$268$	0	0
$269$	−8.42441	−0.513645	−0.256823	−	0.966459i	$-0.582676\pi$
$269$	−8.42441	−0.513645	−0.256823	+	0.966459i	$0.582676\pi$
$270$	0	0
$271$	−4.25002	−0.258170	−0.129085	−	0.991634i	$-0.541204\pi$
$271$	−4.25002	−0.258170	−0.129085	+	0.991634i	$0.541204\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.423338	−0.0255282
$276$	0	0
$277$	18.0331	1.08350	0.541751	−	0.840539i	$-0.317761\pi$
$277$	18.0331	1.08350	0.541751	+	0.840539i	$0.317761\pi$
$278$	0	0
$279$	−8.60900	−0.515407
$280$	0	0
$281$	−22.2919	−1.32982	−0.664911	−	0.746923i	$-0.731530\pi$
$281$	−22.2919	−1.32982	−0.664911	+	0.746923i	$0.731530\pi$
$282$	0	0
$283$	−17.2009	−1.02248	−0.511242	−	0.859437i	$-0.670815\pi$
$283$	−17.2009	−1.02248	−0.511242	+	0.859437i	$0.670815\pi$
$284$	0	0
$285$	−0.789435	−0.0467621
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	2.48532	0.145692
$292$	0	0
$293$	−15.7587	−0.920632	−0.460316	−	0.887755i	$-0.652264\pi$
$293$	−15.7587	−0.920632	−0.460316	+	0.887755i	$0.652264\pi$
$294$	0	0
$295$	−25.2022	−1.46733
$296$	0	0
$297$	1.23758	0.0718116
$298$	0	0
$299$	23.7503	1.37351
$300$	0	0
$301$	0	0
$302$	0	0
$303$	22.2912	1.28059
$304$	0	0
$305$	−11.9120	−0.682081
$306$	0	0
$307$	8.60510	0.491119	0.245559	−	0.969382i	$-0.421028\pi$
$307$	8.60510	0.491119	0.245559	+	0.969382i	$0.421028\pi$
$308$	0	0
$309$	0.420052	0.0238959
$310$	0	0
$311$	−32.1821	−1.82488	−0.912439	−	0.409213i	$-0.865803\pi$
$311$	−32.1821	−1.82488	−0.912439	+	0.409213i	$0.865803\pi$
$312$	0	0
$313$	13.5676	0.766886	0.383443	−	0.923564i	$-0.374738\pi$
$313$	13.5676	0.766886	0.383443	+	0.923564i	$0.374738\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.4960	−0.589514	−0.294757	−	0.955572i	$-0.595239\pi$
$317$	−10.4960	−0.589514	−0.294757	+	0.955572i	$0.595239\pi$
$318$	0	0
$319$	1.90022	0.106392
$320$	0	0
$321$	−12.1247	−0.676733
$322$	0	0
$323$	0.347250	0.0193215
$324$	0	0
$325$	7.87652	0.436911
$326$	0	0
$327$	17.0905	0.945108
$328$	0	0
$329$	0	0
$330$	0	0
$331$	15.6571	0.860589	0.430295	−	0.902688i	$-0.358410\pi$
$331$	15.6571	0.860589	0.430295	+	0.902688i	$0.358410\pi$
$332$	0	0
$333$	−11.9819	−0.656605
$334$	0	0
$335$	−5.59994	−0.305957
$336$	0	0
$337$	−1.58505	−0.0863432	−0.0431716	−	0.999068i	$-0.513746\pi$
$337$	−1.58505	−0.0863432	−0.0431716	+	0.999068i	$0.513746\pi$
$338$	0	0
$339$	1.85095	0.100530
$340$	0	0
$341$	1.43670	0.0778018
$342$	0	0
$343$	0	0
$344$	0	0
$345$	13.1257	0.706662
$346$	0	0
$347$	0.982978	0.0527690	0.0263845	−	0.999652i	$-0.491601\pi$
$347$	0.982978	0.0527690	0.0263845	+	0.999652i	$0.491601\pi$
$348$	0	0
$349$	5.13822	0.275042	0.137521	−	0.990499i	$-0.456086\pi$
$349$	5.13822	0.275042	0.137521	+	0.990499i	$0.456086\pi$
$350$	0	0
$351$	−23.0261	−1.22904
$352$	0	0
$353$	−16.5818	−0.882559	−0.441280	−	0.897370i	$-0.645475\pi$
$353$	−16.5818	−0.882559	−0.441280	+	0.897370i	$0.645475\pi$
$354$	0	0
$355$	20.8585	1.10706
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.9442	0.894279	0.447139	−	0.894464i	$-0.352443\pi$
$359$	16.9442	0.894279	0.447139	+	0.894464i	$0.352443\pi$
$360$	0	0
$361$	−18.8794	−0.993654
$362$	0	0
$363$	14.1739	0.743934
$364$	0	0
$365$	12.2258	0.639930
$366$	0	0
$367$	−14.8844	−0.776958	−0.388479	−	0.921458i	$-0.626999\pi$
$367$	−14.8844	−0.776958	−0.388479	+	0.921458i	$0.626999\pi$
$368$	0	0
$369$	9.50616	0.494871
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−3.67248	−0.190154	−0.0950770	−	0.995470i	$-0.530310\pi$
$373$	−3.67248	−0.190154	−0.0950770	+	0.995470i	$0.530310\pi$
$374$	0	0
$375$	15.7199	0.811775
$376$	0	0
$377$	−35.3550	−1.82088
$378$	0	0
$379$	−12.9081	−0.663044	−0.331522	−	0.943448i	$-0.607562\pi$
$379$	−12.9081	−0.663044	−0.331522	+	0.943448i	$0.607562\pi$
$380$	0	0
$381$	26.7759	1.37177
$382$	0	0
$383$	−5.06434	−0.258776	−0.129388	−	0.991594i	$-0.541301\pi$
$383$	−5.06434	−0.258776	−0.129388	+	0.991594i	$0.541301\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.01536	−0.407444
$388$	0	0
$389$	37.9647	1.92489	0.962444	−	0.271480i	$-0.0875129\pi$
$389$	37.9647	1.92489	0.962444	+	0.271480i	$0.0875129\pi$
$390$	0	0
$391$	−5.77360	−0.291984
$392$	0	0
$393$	−2.22022	−0.111995
$394$	0	0
$395$	19.4058	0.976413
$396$	0	0
$397$	25.7985	1.29479	0.647396	−	0.762154i	$-0.275858\pi$
$397$	25.7985	1.29479	0.647396	+	0.762154i	$0.275858\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.8993	−0.793973	−0.396986	−	0.917824i	$-0.629944\pi$
$401$	−15.8993	−0.793973	−0.396986	+	0.917824i	$0.629944\pi$
$402$	0	0
$403$	−26.7309	−1.33156
$404$	0	0
$405$	−5.74430	−0.285437
$406$	0	0
$407$	1.99959	0.0991159
$408$	0	0
$409$	−30.5484	−1.51052	−0.755260	−	0.655425i	$-0.772490\pi$
$409$	−30.5484	−1.51052	−0.755260	+	0.655425i	$0.772490\pi$
$410$	0	0
$411$	9.65384	0.476189
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−28.5867	−1.40326
$416$	0	0
$417$	−2.92478	−0.143227
$418$	0	0
$419$	12.4866	0.610012	0.305006	−	0.952350i	$-0.401342\pi$
$419$	12.4866	0.610012	0.305006	+	0.952350i	$0.401342\pi$
$420$	0	0
$421$	−3.20785	−0.156341	−0.0781704	−	0.996940i	$-0.524908\pi$
$421$	−3.20785	−0.156341	−0.0781704	+	0.996940i	$0.524908\pi$
$422$	0	0
$423$	−10.5640	−0.513637
$424$	0	0
$425$	−1.91475	−0.0928791
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.17713	0.0568326
$430$	0	0
$431$	−26.2084	−1.26241	−0.631207	−	0.775615i	$-0.717440\pi$
$431$	−26.2084	−1.26241	−0.631207	+	0.775615i	$0.717440\pi$
$432$	0	0
$433$	−4.47940	−0.215266	−0.107633	−	0.994191i	$-0.534327\pi$
$433$	−4.47940	−0.215266	−0.107633	+	0.994191i	$0.534327\pi$
$434$	0	0
$435$	−19.5391	−0.936827
$436$	0	0
$437$	−2.00488	−0.0959066
$438$	0	0
$439$	33.9573	1.62070	0.810348	−	0.585949i	$-0.199278\pi$
$439$	33.9573	1.62070	0.810348	+	0.585949i	$0.199278\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.65893	−0.126330	−0.0631649	−	0.998003i	$-0.520119\pi$
$443$	−2.65893	−0.126330	−0.0631649	+	0.998003i	$0.520119\pi$
$444$	0	0
$445$	−8.57513	−0.406500
$446$	0	0
$447$	−25.9368	−1.22677
$448$	0	0
$449$	−3.71667	−0.175400	−0.0877001	−	0.996147i	$-0.527952\pi$
$449$	−3.71667	−0.175400	−0.0877001	+	0.996147i	$0.527952\pi$
$450$	0	0
$451$	−1.58642	−0.0747018
$452$	0	0
$453$	21.4195	1.00638
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11.2976	−0.528481	−0.264240	−	0.964457i	$-0.585121\pi$
$457$	−11.2976	−0.528481	−0.264240	+	0.964457i	$0.585121\pi$
$458$	0	0
$459$	5.59756	0.261272
$460$	0	0
$461$	−31.4569	−1.46509	−0.732546	−	0.680717i	$-0.761668\pi$
$461$	−31.4569	−1.46509	−0.732546	+	0.680717i	$0.761668\pi$
$462$	0	0
$463$	−1.04755	−0.0486836	−0.0243418	−	0.999704i	$-0.507749\pi$
$463$	−1.04755	−0.0486836	−0.0243418	+	0.999704i	$0.507749\pi$
$464$	0	0
$465$	−14.7729	−0.685079
$466$	0	0
$467$	−18.8507	−0.872306	−0.436153	−	0.899873i	$-0.643659\pi$
$467$	−18.8507	−0.872306	−0.436153	+	0.899873i	$0.643659\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−4.02416	−0.185423
$472$	0	0
$473$	1.33763	0.0615045
$474$	0	0
$475$	−0.664898	−0.0305076
$476$	0	0
$477$	15.3173	0.701332
$478$	0	0
$479$	−22.5428	−1.03001	−0.515003	−	0.857189i	$-0.672209\pi$
$479$	−22.5428	−1.03001	−0.515003	+	0.857189i	$0.672209\pi$
$480$	0	0
$481$	−37.2039	−1.69635
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.37286	−0.153154
$486$	0	0
$487$	1.39837	0.0633664	0.0316832	−	0.999498i	$-0.489913\pi$
$487$	1.39837	0.0633664	0.0316832	+	0.999498i	$0.489913\pi$
$488$	0	0
$489$	19.0893	0.863249
$490$	0	0
$491$	−12.6812	−0.572293	−0.286146	−	0.958186i	$-0.592374\pi$
$491$	−12.6812	−0.572293	−0.286146	+	0.958186i	$0.592374\pi$
$492$	0	0
$493$	8.59467	0.387085
$494$	0	0
$495$	−0.514493	−0.0231248
$496$	0	0
$497$	0	0
$498$	0	0
$499$	8.88953	0.397950	0.198975	−	0.980005i	$-0.436239\pi$
$499$	8.88953	0.397950	0.198975	+	0.980005i	$0.436239\pi$
$500$	0	0
$501$	−12.9041	−0.576514
$502$	0	0
$503$	0.263810	0.0117627	0.00588136	−	0.999983i	$-0.498128\pi$
$503$	0.263810	0.0117627	0.00588136	+	0.999983i	$0.498128\pi$
$504$	0	0
$505$	−30.2516	−1.34618
$506$	0	0
$507$	−5.07579	−0.225424
$508$	0	0
$509$	−25.1765	−1.11593	−0.557965	−	0.829864i	$-0.688418\pi$
$509$	−25.1765	−1.11593	−0.557965	+	0.829864i	$0.688418\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.94375	0.0858187
$514$	0	0
$515$	−0.570057	−0.0251197
$516$	0	0
$517$	1.76295	0.0775347
$518$	0	0
$519$	−2.22023	−0.0974573
$520$	0	0
$521$	41.7031	1.82705	0.913524	−	0.406786i	$-0.133350\pi$
$521$	41.7031	1.82705	0.913524	+	0.406786i	$0.133350\pi$
$522$	0	0
$523$	9.76281	0.426898	0.213449	−	0.976954i	$-0.431530\pi$
$523$	9.76281	0.426898	0.213449	+	0.976954i	$0.431530\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.49819	0.283066
$528$	0	0
$529$	10.3345	0.449326
$530$	0	0
$531$	19.0087	0.824909
$532$	0	0
$533$	29.5166	1.27851
$534$	0	0
$535$	16.4545	0.711391
$536$	0	0
$537$	−6.52372	−0.281519
$538$	0	0
$539$	0	0
$540$	0	0
$541$	41.7460	1.79480	0.897401	−	0.441216i	$-0.145453\pi$
$541$	41.7460	1.79480	0.897401	+	0.441216i	$0.145453\pi$
$542$	0	0
$543$	24.9608	1.07117
$544$	0	0
$545$	−23.1937	−0.993511
$546$	0	0
$547$	−9.94333	−0.425146	−0.212573	−	0.977145i	$-0.568184\pi$
$547$	−9.94333	−0.425146	−0.212573	+	0.977145i	$0.568184\pi$
$548$	0	0
$549$	8.98464	0.383455
$550$	0	0
$551$	2.98450	0.127144
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−20.5608	−0.872758
$556$	0	0
$557$	16.2207	0.687291	0.343646	−	0.939099i	$-0.388338\pi$
$557$	16.2207	0.687291	0.343646	+	0.939099i	$0.388338\pi$
$558$	0	0
$559$	−24.8877	−1.05264
$560$	0	0
$561$	−0.286157	−0.0120815
$562$	0	0
$563$	−2.37117	−0.0999329	−0.0499664	−	0.998751i	$-0.515911\pi$
$563$	−2.37117	−0.0999329	−0.0499664	+	0.998751i	$0.515911\pi$
$564$	0	0
$565$	−2.51195	−0.105678
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−42.3068	−1.77359	−0.886796	−	0.462161i	$-0.847074\pi$
$569$	−42.3068	−1.77359	−0.886796	+	0.462161i	$0.847074\pi$
$570$	0	0
$571$	−21.8460	−0.914226	−0.457113	−	0.889409i	$-0.651117\pi$
$571$	−21.8460	−0.914226	−0.457113	+	0.889409i	$0.651117\pi$
$572$	0	0
$573$	−20.3267	−0.849162
$574$	0	0
$575$	11.0550	0.461026
$576$	0	0
$577$	−2.81255	−0.117088	−0.0585441	−	0.998285i	$-0.518646\pi$
$577$	−2.81255	−0.117088	−0.0585441	+	0.998285i	$0.518646\pi$
$578$	0	0
$579$	−12.3439	−0.512995
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.55621	−0.105868
$584$	0	0
$585$	9.57254	0.395776
$586$	0	0
$587$	45.5253	1.87903	0.939516	−	0.342505i	$-0.111275\pi$
$587$	45.5253	1.87903	0.939516	+	0.342505i	$0.111275\pi$
$588$	0	0
$589$	2.25650	0.0929773
$590$	0	0
$591$	−10.1352	−0.416908
$592$	0	0
$593$	−17.2590	−0.708740	−0.354370	−	0.935105i	$-0.615305\pi$
$593$	−17.2590	−0.708740	−0.354370	+	0.935105i	$0.615305\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	27.4593	1.12383
$598$	0	0
$599$	−15.9024	−0.649754	−0.324877	−	0.945756i	$-0.605323\pi$
$599$	−15.9024	−0.649754	−0.324877	+	0.945756i	$0.605323\pi$
$600$	0	0
$601$	35.0052	1.42789	0.713946	−	0.700201i	$-0.246906\pi$
$601$	35.0052	1.42789	0.713946	+	0.700201i	$0.246906\pi$
$602$	0	0
$603$	4.22375	0.172004
$604$	0	0
$605$	−19.2355	−0.782034
$606$	0	0
$607$	38.8739	1.57784	0.788922	−	0.614493i	$-0.210639\pi$
$607$	38.8739	1.57784	0.788922	+	0.614493i	$0.210639\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−32.8011	−1.32699
$612$	0	0
$613$	−7.47434	−0.301886	−0.150943	−	0.988542i	$-0.548231\pi$
$613$	−7.47434	−0.301886	−0.150943	+	0.988542i	$0.548231\pi$
$614$	0	0
$615$	16.3125	0.657782
$616$	0	0
$617$	2.89649	0.116608	0.0583041	−	0.998299i	$-0.481431\pi$
$617$	2.89649	0.116608	0.0583041	+	0.998299i	$0.481431\pi$
$618$	0	0
$619$	−23.0211	−0.925298	−0.462649	−	0.886542i	$-0.653101\pi$
$619$	−23.0211	−0.925298	−0.462649	+	0.886542i	$0.653101\pi$
$620$	0	0
$621$	−32.3181	−1.29688
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−11.7600	−0.470399
$626$	0	0
$627$	−0.0993679	−0.00396837
$628$	0	0
$629$	9.04411	0.360612
$630$	0	0
$631$	−8.30072	−0.330446	−0.165223	−	0.986256i	$-0.552834\pi$
$631$	−8.30072	−0.330446	−0.165223	+	0.986256i	$0.552834\pi$
$632$	0	0
$633$	−20.4601	−0.813214
$634$	0	0
$635$	−36.3379	−1.44202
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−15.7326	−0.622370
$640$	0	0
$641$	24.0704	0.950722	0.475361	−	0.879791i	$-0.342317\pi$
$641$	24.0704	0.950722	0.475361	+	0.879791i	$0.342317\pi$
$642$	0	0
$643$	29.7528	1.17334	0.586668	−	0.809828i	$-0.300439\pi$
$643$	29.7528	1.17334	0.586668	+	0.809828i	$0.300439\pi$
$644$	0	0
$645$	−13.7543	−0.541574
$646$	0	0
$647$	−28.7782	−1.13139	−0.565694	−	0.824615i	$-0.691392\pi$
$647$	−28.7782	−1.13139	−0.565694	+	0.824615i	$0.691392\pi$
$648$	0	0
$649$	−3.17225	−0.124522
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−28.1677	−1.10229	−0.551144	−	0.834410i	$-0.685808\pi$
$653$	−28.1677	−1.10229	−0.551144	+	0.834410i	$0.685808\pi$
$654$	0	0
$655$	3.01309	0.117731
$656$	0	0
$657$	−9.22134	−0.359759
$658$	0	0
$659$	−46.2520	−1.80172	−0.900862	−	0.434107i	$-0.857064\pi$
$659$	−46.2520	−1.80172	−0.900862	+	0.434107i	$0.857064\pi$
$660$	0	0
$661$	−24.1010	−0.937421	−0.468710	−	0.883352i	$-0.655281\pi$
$661$	−24.1010	−0.937421	−0.468710	+	0.883352i	$0.655281\pi$
$662$	0	0
$663$	5.32416	0.206773
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−49.6222	−1.92138
$668$	0	0
$669$	−8.80326	−0.340354
$670$	0	0
$671$	−1.49939	−0.0578834
$672$	0	0
$673$	−21.7745	−0.839346	−0.419673	−	0.907675i	$-0.637855\pi$
$673$	−21.7745	−0.839346	−0.419673	+	0.907675i	$0.637855\pi$
$674$	0	0
$675$	−10.7179	−0.412533
$676$	0	0
$677$	7.59712	0.291981	0.145991	−	0.989286i	$-0.453363\pi$
$677$	7.59712	0.291981	0.145991	+	0.989286i	$0.453363\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−22.6404	−0.867582
$682$	0	0
$683$	−24.1319	−0.923381	−0.461690	−	0.887041i	$-0.652757\pi$
$683$	−24.1319	−0.923381	−0.461690	+	0.887041i	$0.652757\pi$
$684$	0	0
$685$	−13.1013	−0.500576
$686$	0	0
$687$	29.9877	1.14410
$688$	0	0
$689$	47.5603	1.81190
$690$	0	0
$691$	47.5473	1.80878	0.904392	−	0.426703i	$-0.140325\pi$
$691$	47.5473	1.80878	0.904392	+	0.426703i	$0.140325\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.96925	0.150562
$696$	0	0
$697$	−7.17538	−0.271787
$698$	0	0
$699$	10.9937	0.415820
$700$	0	0
$701$	−11.2887	−0.426369	−0.213184	−	0.977012i	$-0.568384\pi$
$701$	−11.2887	−0.426369	−0.213184	+	0.977012i	$0.568384\pi$
$702$	0	0
$703$	3.14057	0.118449
$704$	0	0
$705$	−18.1276	−0.682726
$706$	0	0
$707$	0	0
$708$	0	0
$709$	31.4948	1.18281	0.591406	−	0.806374i	$-0.298573\pi$
$709$	31.4948	1.18281	0.591406	+	0.806374i	$0.298573\pi$
$710$	0	0
$711$	−14.6368	−0.548924
$712$	0	0
$713$	−37.5180	−1.40506
$714$	0	0
$715$	−1.59750	−0.0597432
$716$	0	0
$717$	−2.03279	−0.0759160
$718$	0	0
$719$	17.9567	0.669673	0.334836	−	0.942276i	$-0.391319\pi$
$719$	17.9567	0.669673	0.334836	+	0.942276i	$0.391319\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	33.1401	1.23249
$724$	0	0
$725$	−16.4567	−0.611185
$726$	0	0
$727$	−24.1635	−0.896176	−0.448088	−	0.893989i	$-0.647895\pi$
$727$	−24.1635	−0.896176	−0.448088	+	0.893989i	$0.647895\pi$
$728$	0	0
$729$	26.0671	0.965449
$730$	0	0
$731$	6.05010	0.223771
$732$	0	0
$733$	9.90212	0.365743	0.182872	−	0.983137i	$-0.441461\pi$
$733$	9.90212	0.365743	0.182872	+	0.983137i	$0.441461\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.704876	−0.0259644
$738$	0	0
$739$	−48.7915	−1.79483	−0.897413	−	0.441192i	$-0.854556\pi$
$739$	−48.7915	−1.79483	−0.897413	+	0.441192i	$0.854556\pi$
$740$	0	0
$741$	1.84882	0.0679179
$742$	0	0
$743$	−18.7369	−0.687390	−0.343695	−	0.939081i	$-0.611679\pi$
$743$	−18.7369	−0.687390	−0.343695	+	0.939081i	$0.611679\pi$
$744$	0	0
$745$	35.1991	1.28960
$746$	0	0
$747$	21.5615	0.788893
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−33.7387	−1.23114	−0.615571	−	0.788081i	$-0.711075\pi$
$751$	−33.7387	−1.23114	−0.615571	+	0.788081i	$0.711075\pi$
$752$	0	0
$753$	1.70445	0.0621138
$754$	0	0
$755$	−29.0687	−1.05792
$756$	0	0
$757$	31.7560	1.15419	0.577095	−	0.816677i	$-0.304186\pi$
$757$	31.7560	1.15419	0.577095	+	0.816677i	$0.304186\pi$
$758$	0	0
$759$	1.65216	0.0599694
$760$	0	0
$761$	2.97707	0.107919	0.0539593	−	0.998543i	$-0.482816\pi$
$761$	2.97707	0.107919	0.0539593	+	0.998543i	$0.482816\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.32705	−0.0841346
$766$	0	0
$767$	59.0221	2.13117
$768$	0	0
$769$	−26.0967	−0.941071	−0.470535	−	0.882381i	$-0.655939\pi$
$769$	−26.0967	−0.941071	−0.470535	+	0.882381i	$0.655939\pi$
$770$	0	0
$771$	33.5636	1.20876
$772$	0	0
$773$	50.2181	1.80622	0.903110	−	0.429409i	$-0.141278\pi$
$773$	50.2181	1.80622	0.903110	+	0.429409i	$0.141278\pi$
$774$	0	0
$775$	−12.4424	−0.446945
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.49165	−0.0892726
$780$	0	0
$781$	2.62551	0.0939481
$782$	0	0
$783$	48.1092	1.71928
$784$	0	0
$785$	5.46123	0.194920
$786$	0	0
$787$	−12.7574	−0.454752	−0.227376	−	0.973807i	$-0.573015\pi$
$787$	−12.7574	−0.454752	−0.227376	+	0.973807i	$0.573015\pi$
$788$	0	0
$789$	−25.1528	−0.895463
$790$	0	0
$791$	0	0
$792$	0	0
$793$	27.8973	0.990663
$794$	0	0
$795$	26.2843	0.932209
$796$	0	0
$797$	−28.1558	−0.997330	−0.498665	−	0.866795i	$-0.666176\pi$
$797$	−28.1558	−0.997330	−0.498665	+	0.866795i	$0.666176\pi$
$798$	0	0
$799$	7.97382	0.282094
$800$	0	0
$801$	6.46779	0.228528
$802$	0	0
$803$	1.53889	0.0543063
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.9036	0.383824
$808$	0	0
$809$	39.5645	1.39101	0.695506	−	0.718520i	$-0.255180\pi$
$809$	39.5645	1.39101	0.695506	+	0.718520i	$0.255180\pi$
$810$	0	0
$811$	33.2336	1.16699	0.583495	−	0.812117i	$-0.301685\pi$
$811$	33.2336	1.16699	0.583495	+	0.812117i	$0.301685\pi$
$812$	0	0
$813$	5.50073	0.192919
$814$	0	0
$815$	−25.9064	−0.907460
$816$	0	0
$817$	2.10090	0.0735011
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−21.2159	−0.740439	−0.370219	−	0.928944i	$-0.620717\pi$
$821$	−21.2159	−0.740439	−0.370219	+	0.928944i	$0.620717\pi$
$822$	0	0
$823$	−28.9374	−1.00869	−0.504347	−	0.863501i	$-0.668267\pi$
$823$	−28.9374	−1.00869	−0.504347	+	0.863501i	$0.668267\pi$
$824$	0	0
$825$	0.547919	0.0190761
$826$	0	0
$827$	46.3822	1.61287	0.806433	−	0.591325i	$-0.201395\pi$
$827$	46.3822	1.61287	0.806433	+	0.591325i	$0.201395\pi$
$828$	0	0
$829$	−17.5216	−0.608552	−0.304276	−	0.952584i	$-0.598414\pi$
$829$	−17.5216	−0.608552	−0.304276	+	0.952584i	$0.598414\pi$
$830$	0	0
$831$	−23.3399	−0.809653
$832$	0	0
$833$	0	0
$834$	0	0
$835$	17.5123	0.606039
$836$	0	0
$837$	36.3740	1.25727
$838$	0	0
$839$	11.5741	0.399584	0.199792	−	0.979838i	$-0.435973\pi$
$839$	11.5741	0.399584	0.199792	+	0.979838i	$0.435973\pi$
$840$	0	0
$841$	44.8684	1.54719
$842$	0	0
$843$	28.8520	0.993716
$844$	0	0
$845$	6.88841	0.236969
$846$	0	0
$847$	0	0
$848$	0	0
$849$	22.2628	0.764057
$850$	0	0
$851$	−52.2171	−1.78998
$852$	0	0
$853$	51.3617	1.75859	0.879295	−	0.476277i	$-0.158014\pi$
$853$	51.3617	1.75859	0.879295	+	0.476277i	$0.158014\pi$
$854$	0	0
$855$	−0.808067	−0.0276353
$856$	0	0
$857$	−36.2390	−1.23790	−0.618950	−	0.785430i	$-0.712442\pi$
$857$	−36.2390	−1.23790	−0.618950	+	0.785430i	$0.712442\pi$
$858$	0	0
$859$	−24.4763	−0.835119	−0.417560	−	0.908650i	$-0.637115\pi$
$859$	−24.4763	−0.835119	−0.417560	+	0.908650i	$0.637115\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	25.5929	0.871192	0.435596	−	0.900142i	$-0.356538\pi$
$863$	25.5929	0.871192	0.435596	+	0.900142i	$0.356538\pi$
$864$	0	0
$865$	3.01310	0.102448
$866$	0	0
$867$	−1.29428	−0.0439562
$868$	0	0
$869$	2.44265	0.0828613
$870$	0	0
$871$	13.1147	0.444376
$872$	0	0
$873$	2.54398	0.0861006
$874$	0	0
$875$	0	0
$876$	0	0
$877$	9.71423	0.328026	0.164013	−	0.986458i	$-0.447556\pi$
$877$	9.71423	0.328026	0.164013	+	0.986458i	$0.447556\pi$
$878$	0	0
$879$	20.3962	0.687947
$880$	0	0
$881$	39.0076	1.31420	0.657100	−	0.753803i	$-0.271783\pi$
$881$	39.0076	1.31420	0.657100	+	0.753803i	$0.271783\pi$
$882$	0	0
$883$	34.0396	1.14552	0.572761	−	0.819722i	$-0.305872\pi$
$883$	34.0396	1.14552	0.572761	+	0.819722i	$0.305872\pi$
$884$	0	0
$885$	32.6188	1.09647
$886$	0	0
$887$	−55.3660	−1.85901	−0.929505	−	0.368810i	$-0.879765\pi$
$887$	−55.3660	−1.85901	−0.929505	+	0.368810i	$0.879765\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−0.723047	−0.0242230
$892$	0	0
$893$	2.76891	0.0926580
$894$	0	0
$895$	8.85341	0.295937
$896$	0	0
$897$	−30.7396	−1.02637
$898$	0	0
$899$	55.8498	1.86270
$900$	0	0
$901$	−11.5617	−0.385177
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−33.8746	−1.12603
$906$	0	0
$907$	4.88850	0.162320	0.0811599	−	0.996701i	$-0.474138\pi$
$907$	4.88850	0.162320	0.0811599	+	0.996701i	$0.474138\pi$
$908$	0	0
$909$	22.8173	0.756800
$910$	0	0
$911$	38.8077	1.28576	0.642879	−	0.765968i	$-0.277740\pi$
$911$	38.8077	1.28576	0.642879	+	0.765968i	$0.277740\pi$
$912$	0	0
$913$	−3.59826	−0.119085
$914$	0	0
$915$	15.4175	0.509688
$916$	0	0
$917$	0	0
$918$	0	0
$919$	3.96699	0.130859	0.0654295	−	0.997857i	$-0.479158\pi$
$919$	3.96699	0.130859	0.0654295	+	0.997857i	$0.479158\pi$
$920$	0	0
$921$	−11.1374	−0.366991
$922$	0	0
$923$	−48.8496	−1.60790
$924$	0	0
$925$	−17.3172	−0.569387
$926$	0	0
$927$	0.429966	0.0141219
$928$	0	0
$929$	0.787446	0.0258353	0.0129176	−	0.999917i	$-0.495888\pi$
$929$	0.787446	0.0258353	0.0129176	+	0.999917i	$0.495888\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	41.6527	1.36365
$934$	0	0
$935$	0.388347	0.0127003
$936$	0	0
$937$	1.89922	0.0620448	0.0310224	−	0.999519i	$-0.490124\pi$
$937$	1.89922	0.0620448	0.0310224	+	0.999519i	$0.490124\pi$
$938$	0	0
$939$	−17.5603	−0.573060
$940$	0	0
$941$	−2.45806	−0.0801306	−0.0400653	−	0.999197i	$-0.512757\pi$
$941$	−2.45806	−0.0801306	−0.0400653	+	0.999197i	$0.512757\pi$
$942$	0	0
$943$	41.4278	1.34907
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−12.6039	−0.409573	−0.204787	−	0.978807i	$-0.565650\pi$
$947$	−12.6039	−0.409573	−0.204787	+	0.978807i	$0.565650\pi$
$948$	0	0
$949$	−28.6323	−0.929443
$950$	0	0
$951$	13.5848	0.440517
$952$	0	0
$953$	−45.4002	−1.47066	−0.735329	−	0.677710i	$-0.762972\pi$
$953$	−45.4002	−1.47066	−0.735329	+	0.677710i	$0.762972\pi$
$954$	0	0
$955$	27.5857	0.892651
$956$	0	0
$957$	−2.45942	−0.0795019
$958$	0	0
$959$	0	0
$960$	0	0
$961$	11.2265	0.362144
$962$	0	0
$963$	−12.4108	−0.399933
$964$	0	0
$965$	16.7520	0.539268
$966$	0	0
$967$	−10.6379	−0.342092	−0.171046	−	0.985263i	$-0.554715\pi$
$967$	−10.6379	−0.342092	−0.171046	+	0.985263i	$0.554715\pi$
$968$	0	0
$969$	−0.449440	−0.0144381
$970$	0	0
$971$	−37.2922	−1.19676	−0.598381	−	0.801212i	$-0.704189\pi$
$971$	−37.2922	−1.19676	−0.598381	+	0.801212i	$0.704189\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−10.1945	−0.326484
$976$	0	0
$977$	−12.8992	−0.412683	−0.206342	−	0.978480i	$-0.566156\pi$
$977$	−12.8992	−0.412683	−0.206342	+	0.978480i	$0.566156\pi$
$978$	0	0
$979$	−1.07937	−0.0344968
$980$	0	0
$981$	17.4939	0.558536
$982$	0	0
$983$	−1.86470	−0.0594748	−0.0297374	−	0.999558i	$-0.509467\pi$
$983$	−1.86470	−0.0594748	−0.0297374	+	0.999558i	$0.509467\pi$
$984$	0	0
$985$	13.7546	0.438259
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−34.9309	−1.11074
$990$	0	0
$991$	16.2988	0.517747	0.258874	−	0.965911i	$-0.416649\pi$
$991$	16.2988	0.517747	0.258874	+	0.965911i	$0.416649\pi$
$992$	0	0
$993$	−20.2647	−0.643080
$994$	0	0
$995$	−37.2653	−1.18139
$996$	0	0
$997$	3.00252	0.0950908	0.0475454	−	0.998869i	$-0.484860\pi$
$997$	3.00252	0.0950908	0.0475454	+	0.998869i	$0.484860\pi$
$998$	0	0
$999$	50.6249	1.60170

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6664.2.a.t.1.2	yes	6
7.6	odd	2		6664.2.a.q.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6664.2.a.q.1.5	✓	6	7.6	odd	2
6664.2.a.t.1.2	yes	6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6664.a (trivial)

\(f(q)\)	\(=\)	\(q-1.29428 q^{3} +1.75649 q^{5} -1.32483 q^{9} +0.221093 q^{11} -4.11360 q^{13} -2.27339 q^{15} +1.00000 q^{17} +0.347250 q^{19} -5.77360 q^{23} -1.91475 q^{25} +5.59756 q^{27} +8.59467 q^{29} +6.49819 q^{31} -0.286157 q^{33} +9.04411 q^{37} +5.32416 q^{39} -7.17538 q^{41} +6.05010 q^{43} -2.32705 q^{45} +7.97382 q^{47} -1.29428 q^{51} -11.5617 q^{53} +0.388347 q^{55} -0.449440 q^{57} -14.3481 q^{59} -6.78173 q^{61} -7.22549 q^{65} -3.18814 q^{67} +7.47268 q^{69} +11.8751 q^{71} +6.96039 q^{73} +2.47823 q^{75} +11.0481 q^{79} -3.27033 q^{81} -16.2749 q^{83} +1.75649 q^{85} -11.1239 q^{87} -4.88198 q^{89} -8.41050 q^{93} +0.609940 q^{95} -1.92023 q^{97} -0.292910 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 4 q^{5} + 4 q^{9} + 2 q^{11} - 6 q^{13} - 16 q^{15} + 6 q^{17} + 4 q^{19} - 4 q^{23} + 8 q^{25} + 14 q^{27} - 6 q^{29} - 16 q^{31} - 8 q^{33} - 12 q^{37} + 10 q^{39} - 22 q^{41} + 2 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 4 * q^5 + 4 * q^9 + 2 * q^11 - 6 * q^13 - 16 * q^15 + 6 * q^17 + 4 * q^19 - 4 * q^23 + 8 * q^25 + 14 * q^27 - 6 * q^29 - 16 * q^31 - 8 * q^33 - 12 * q^37 + 10 * q^39 - 22 * q^41 + 2 * q^43 - 30 * q^45 + 2 * q^47 + 2 * q^51 + 6 * q^53 - 2 * q^55 + 6 * q^57 - 20 * q^59 + 6 * q^61 + 12 * q^65 - 2 * q^67 - 46 * q^69 - 8 * q^71 - 18 * q^73 + 50 * q^75 + 10 * q^79 + 30 * q^81 - 6 * q^83 - 4 * q^85 - 60 * q^87 - 6 * q^89 + 8 * q^93 - 10 * q^95 - 36 * q^97 - 8 * q^99

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