LMFDB - Embedded newform 9576.2.a.ck.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9576.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:	$76.4647449756$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.18097.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + 6x + 4 $ x^4 - x^3 - 7x^2 + 6x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.37388$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.11245 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-1.11245 q^{5} -1.00000 q^{7} -5.56817 q^{11} -3.90225 q^{13} -2.20349 q^{17} -1.00000 q^{19} -8.48633 q^{23} -3.76246 q^{25} -9.96596 q^{29} -0.769163 q^{31} +1.11245 q^{35} -8.41248 q^{37} +3.91815 q^{41} +5.76246 q^{43} -8.37388 q^{47} +1.00000 q^{49} -0.170389 q^{53} +6.19429 q^{55} -0.887554 q^{59} +15.0898 q^{61} +4.34104 q^{65} +6.78757 q^{67} -8.76246 q^{71} +10.4943 q^{73} +5.56817 q^{77} +1.31268 q^{79} +17.7650 q^{83} +2.45127 q^{85} -9.57617 q^{89} +3.90225 q^{91} +1.11245 q^{95} -1.04204 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q + 3 * q^5 - 4 * q^7	$ 4 q + 3 q^{5} - 4 q^{7} - 6 q^{11} - 4 q^{13} + 9 q^{17} - 4 q^{19} - 22 q^{23} + 13 q^{25} + 6 q^{29} + 3 q^{31} - 3 q^{35} + 15 q^{37} + 20 q^{41} - 5 q^{43} - 29 q^{47} + 4 q^{49} - 14 q^{53} + 13 q^{55} - 11 q^{59} + 15 q^{61} + 2 q^{65} + 9 q^{67} - 7 q^{71} - 11 q^{73} + 6 q^{77} + 7 q^{79} + 15 q^{83} - 7 q^{85} + 19 q^{89} + 4 q^{91} - 3 q^{95} - 9 q^{97}+O(q^{100}) $ 4 * q + 3 * q^5 - 4 * q^7 - 6 * q^11 - 4 * q^13 + 9 * q^17 - 4 * q^19 - 22 * q^23 + 13 * q^25 + 6 * q^29 + 3 * q^31 - 3 * q^35 + 15 * q^37 + 20 * q^41 - 5 * q^43 - 29 * q^47 + 4 * q^49 - 14 * q^53 + 13 * q^55 - 11 * q^59 + 15 * q^61 + 2 * q^65 + 9 * q^67 - 7 * q^71 - 11 * q^73 + 6 * q^77 + 7 * q^79 + 15 * q^83 - 7 * q^85 + 19 * q^89 + 4 * q^91 - 3 * q^95 - 9 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	−1.11245	−0.497501	−0.248750	−	0.968568i	$-0.580020\pi$
$5$	−1.11245	−0.497501	−0.248750	+	0.968568i	$0.580020\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.56817	−1.67887	−0.839434	−	0.543462i	$-0.817113\pi$
$11$	−5.56817	−1.67887	−0.839434	+	0.543462i	$0.817113\pi$
$12$	0	0
$13$	−3.90225	−1.08229	−0.541145	−	0.840929i	$-0.682009\pi$
$13$	−3.90225	−1.08229	−0.541145	+	0.840929i	$0.682009\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.20349	−0.534426	−0.267213	−	0.963638i	$-0.586103\pi$
$17$	−2.20349	−0.534426	−0.267213	+	0.963638i	$0.586103\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.48633	−1.76952	−0.884761	−	0.466045i	$-0.845678\pi$
$23$	−8.48633	−1.76952	−0.884761	+	0.466045i	$0.845678\pi$
$24$	0	0
$25$	−3.76246	−0.752493
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.96596	−1.85063	−0.925316	−	0.379197i	$-0.876200\pi$
$29$	−9.96596	−1.85063	−0.925316	+	0.379197i	$0.876200\pi$
$30$	0	0
$31$	−0.769163	−0.138146	−0.0690729	−	0.997612i	$-0.522004\pi$
$31$	−0.769163	−0.138146	−0.0690729	+	0.997612i	$0.522004\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.11245	0.188038
$36$	0	0
$37$	−8.41248	−1.38300	−0.691502	−	0.722375i	$-0.743051\pi$
$37$	−8.41248	−1.38300	−0.691502	+	0.722375i	$0.743051\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.91815	0.611913	0.305956	−	0.952045i	$-0.401024\pi$
$41$	3.91815	0.611913	0.305956	+	0.952045i	$0.401024\pi$
$42$	0	0
$43$	5.76246	0.878768	0.439384	−	0.898299i	$-0.355197\pi$
$43$	5.76246	0.878768	0.439384	+	0.898299i	$0.355197\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.37388	−1.22146	−0.610728	−	0.791840i	$-0.709123\pi$
$47$	−8.37388	−1.22146	−0.610728	+	0.791840i	$0.709123\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.170389	−0.0234047	−0.0117024	−	0.999932i	$-0.503725\pi$
$53$	−0.170389	−0.0234047	−0.0117024	+	0.999932i	$0.503725\pi$
$54$	0	0
$55$	6.19429	0.835238
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.887554	−0.115550	−0.0577749	−	0.998330i	$-0.518401\pi$
$59$	−0.887554	−0.115550	−0.0577749	+	0.998330i	$0.518401\pi$
$60$	0	0
$61$	15.0898	1.93206	0.966028	−	0.258436i	$-0.0832071\pi$
$61$	15.0898	1.93206	0.966028	+	0.258436i	$0.0832071\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.34104	0.538440
$66$	0	0
$67$	6.78757	0.829234	0.414617	−	0.909996i	$-0.363916\pi$
$67$	6.78757	0.829234	0.414617	+	0.909996i	$0.363916\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.76246	−1.03991	−0.519957	−	0.854193i	$-0.674052\pi$
$71$	−8.76246	−1.03991	−0.519957	+	0.854193i	$0.674052\pi$
$72$	0	0
$73$	10.4943	1.22827	0.614134	−	0.789202i	$-0.289505\pi$
$73$	10.4943	1.22827	0.614134	+	0.789202i	$0.289505\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.56817	0.634552
$78$	0	0
$79$	1.31268	0.147688	0.0738441	−	0.997270i	$-0.476473\pi$
$79$	1.31268	0.147688	0.0738441	+	0.997270i	$0.476473\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	17.7650	1.94996	0.974979	−	0.222295i	$-0.0713550\pi$
$83$	17.7650	1.94996	0.974979	+	0.222295i	$0.0713550\pi$
$84$	0	0
$85$	2.45127	0.265877
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.57617	−1.01507	−0.507536	−	0.861630i	$-0.669444\pi$
$89$	−9.57617	−1.01507	−0.507536	+	0.861630i	$0.669444\pi$
$90$	0	0
$91$	3.90225	0.409067
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.11245	0.114135
$96$	0	0
$97$	−1.04204	−0.105803	−0.0529016	−	0.998600i	$-0.516847\pi$
$97$	−1.04204	−0.105803	−0.0529016	+	0.998600i	$0.516847\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.784312	0.0780419	0.0390210	−	0.999238i	$-0.487576\pi$
$101$	0.784312	0.0780419	0.0390210	+	0.999238i	$0.487576\pi$
$102$	0	0
$103$	−3.39778	−0.334794	−0.167397	−	0.985890i	$-0.553536\pi$
$103$	−3.39778	−0.334794	−0.167397	+	0.985890i	$0.553536\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.8933	−1.24644	−0.623222	−	0.782045i	$-0.714177\pi$
$107$	−12.8933	−1.24644	−0.623222	+	0.782045i	$0.714177\pi$
$108$	0	0
$109$	2.04650	0.196019	0.0980097	−	0.995185i	$-0.468752\pi$
$109$	2.04650	0.196019	0.0980097	+	0.995185i	$0.468752\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	17.8078	1.67521	0.837607	−	0.546274i	$-0.183954\pi$
$113$	17.8078	1.67521	0.837607	+	0.546274i	$0.183954\pi$
$114$	0	0
$115$	9.44058	0.880339
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.20349	0.201994
$120$	0	0
$121$	20.0046	1.81860
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.74777	0.871867
$126$	0	0
$127$	−6.60352	−0.585967	−0.292984	−	0.956117i	$-0.594648\pi$
$127$	−6.60352	−0.585967	−0.292984	+	0.956117i	$0.594648\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.93285	−0.430985	−0.215493	−	0.976505i	$-0.569136\pi$
$131$	−4.93285	−0.430985	−0.215493	+	0.976505i	$0.569136\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.0331	−1.28436	−0.642182	−	0.766552i	$-0.721971\pi$
$137$	−15.0331	−1.28436	−0.642182	+	0.766552i	$0.721971\pi$
$138$	0	0
$139$	−21.8342	−1.85195	−0.925975	−	0.377585i	$-0.876755\pi$
$139$	−21.8342	−1.85195	−0.925975	+	0.377585i	$0.876755\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	21.7284	1.81702
$144$	0	0
$145$	11.0866	0.920691
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.77242	−0.718665	−0.359332	−	0.933210i	$-0.616996\pi$
$149$	−8.77242	−0.718665	−0.359332	+	0.933210i	$0.616996\pi$
$150$	0	0
$151$	5.23084	0.425679	0.212840	−	0.977087i	$-0.431729\pi$
$151$	5.23084	0.425679	0.212840	+	0.977087i	$0.431729\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.855652	0.0687276
$156$	0	0
$157$	−20.3453	−1.62373	−0.811867	−	0.583842i	$-0.801549\pi$
$157$	−20.3453	−1.62373	−0.811867	+	0.583842i	$0.801549\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.48633	0.668816
$162$	0	0
$163$	3.22713	0.252768	0.126384	−	0.991981i	$-0.459663\pi$
$163$	3.22713	0.252768	0.126384	+	0.991981i	$0.459663\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.5341	−1.74374	−0.871872	−	0.489734i	$-0.837094\pi$
$167$	−22.5341	−1.74374	−0.871872	+	0.489734i	$0.837094\pi$
$168$	0	0
$169$	2.22758	0.171352
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.4364	1.70581	0.852903	−	0.522069i	$-0.174840\pi$
$173$	22.4364	1.70581	0.852903	+	0.522069i	$0.174840\pi$
$174$	0	0
$175$	3.76246	0.284416
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.19903	0.687568	0.343784	−	0.939049i	$-0.388291\pi$
$179$	9.19903	0.687568	0.343784	+	0.939049i	$0.388291\pi$
$180$	0	0
$181$	1.76470	0.131169	0.0655847	−	0.997847i	$-0.479109\pi$
$181$	1.76470	0.131169	0.0655847	+	0.997847i	$0.479109\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.35843	0.688046
$186$	0	0
$187$	12.2694	0.897230
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.4331	−1.18906	−0.594530	−	0.804074i	$-0.702662\pi$
$191$	−16.4331	−1.18906	−0.594530	+	0.804074i	$0.702662\pi$
$192$	0	0
$193$	−13.7284	−0.988194	−0.494097	−	0.869407i	$-0.664501\pi$
$193$	−13.7284	−0.988194	−0.494097	+	0.869407i	$0.664501\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.9147	1.63261	0.816303	−	0.577624i	$-0.196020\pi$
$197$	22.9147	1.63261	0.816303	+	0.577624i	$0.196020\pi$
$198$	0	0
$199$	11.1921	0.793384	0.396692	−	0.917952i	$-0.370158\pi$
$199$	11.1921	0.793384	0.396692	+	0.917952i	$0.370158\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.96596	0.699473
$204$	0	0
$205$	−4.35873	−0.304427
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.56817	0.385159
$210$	0	0
$211$	−6.85351	−0.471815	−0.235908	−	0.971775i	$-0.575806\pi$
$211$	−6.85351	−0.471815	−0.235908	+	0.971775i	$0.575806\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.41043	−0.437188
$216$	0	0
$217$	0.769163	0.0522142
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8.59859	0.578404
$222$	0	0
$223$	−25.5725	−1.71246	−0.856230	−	0.516596i	$-0.827199\pi$
$223$	−25.5725	−1.71246	−0.856230	+	0.516596i	$0.827199\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.0692	1.59753	0.798764	−	0.601644i	$-0.205487\pi$
$227$	24.0692	1.59753	0.798764	+	0.601644i	$0.205487\pi$
$228$	0	0
$229$	−18.3331	−1.21149	−0.605744	−	0.795660i	$-0.707124\pi$
$229$	−18.3331	−1.21149	−0.605744	+	0.795660i	$0.707124\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.92290	0.322510	0.161255	−	0.986913i	$-0.448446\pi$
$233$	4.92290	0.322510	0.161255	+	0.986913i	$0.448446\pi$
$234$	0	0
$235$	9.31549	0.607676
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.4679	0.806483	0.403241	−	0.915094i	$-0.367883\pi$
$239$	12.4679	0.806483	0.403241	+	0.915094i	$0.367883\pi$
$240$	0	0
$241$	2.64452	0.170349	0.0851744	−	0.996366i	$-0.472855\pi$
$241$	2.64452	0.170349	0.0851744	+	0.996366i	$0.472855\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.11245	−0.0710715
$246$	0	0
$247$	3.90225	0.248294
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.7374	0.803975	0.401988	−	0.915645i	$-0.368320\pi$
$251$	12.7374	0.803975	0.401988	+	0.915645i	$0.368320\pi$
$252$	0	0
$253$	47.2534	2.97079
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.55478	−0.408876	−0.204438	−	0.978880i	$-0.565537\pi$
$257$	−6.55478	−0.408876	−0.204438	+	0.978880i	$0.565537\pi$
$258$	0	0
$259$	8.41248	0.522726
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−22.4759	−1.38592	−0.692962	−	0.720974i	$-0.743695\pi$
$263$	−22.4759	−1.38592	−0.692962	+	0.720974i	$0.743695\pi$
$264$	0	0
$265$	0.189548	0.0116439
$266$	0	0
$267$	0	0
$268$	0	0
$269$	11.5535	0.704428	0.352214	−	0.935920i	$-0.385429\pi$
$269$	11.5535	0.704428	0.352214	+	0.935920i	$0.385429\pi$
$270$	0	0
$271$	23.6680	1.43773	0.718864	−	0.695151i	$-0.244663\pi$
$271$	23.6680	1.43773	0.718864	+	0.695151i	$0.244663\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	20.9501	1.26334
$276$	0	0
$277$	27.7364	1.66652	0.833260	−	0.552881i	$-0.186472\pi$
$277$	27.7364	1.66652	0.833260	+	0.552881i	$0.186472\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−25.3844	−1.51431	−0.757153	−	0.653238i	$-0.773410\pi$
$281$	−25.3844	−1.51431	−0.757153	+	0.653238i	$0.773410\pi$
$282$	0	0
$283$	−29.3217	−1.74299	−0.871497	−	0.490401i	$-0.836850\pi$
$283$	−29.3217	−1.74299	−0.871497	+	0.490401i	$0.836850\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.91815	−0.231281
$288$	0	0
$289$	−12.1446	−0.714389
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−13.0864	−0.764516	−0.382258	−	0.924056i	$-0.624853\pi$
$293$	−13.0864	−0.764516	−0.382258	+	0.924056i	$0.624853\pi$
$294$	0	0
$295$	0.987356	0.0574861
$296$	0	0
$297$	0	0
$298$	0	0
$299$	33.1158	1.91514
$300$	0	0
$301$	−5.76246	−0.332143
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−16.7866	−0.961200
$306$	0	0
$307$	−15.0958	−0.861562	−0.430781	−	0.902456i	$-0.641762\pi$
$307$	−15.0958	−0.861562	−0.430781	+	0.902456i	$0.641762\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−16.5702	−0.939611	−0.469806	−	0.882770i	$-0.655676\pi$
$311$	−16.5702	−0.939611	−0.469806	+	0.882770i	$0.655676\pi$
$312$	0	0
$313$	−6.51098	−0.368023	−0.184011	−	0.982924i	$-0.558908\pi$
$313$	−6.51098	−0.368023	−0.184011	+	0.982924i	$0.558908\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.38309	−0.302344	−0.151172	−	0.988507i	$-0.548305\pi$
$317$	−5.38309	−0.302344	−0.151172	+	0.988507i	$0.548305\pi$
$318$	0	0
$319$	55.4922	3.10697
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.20349	0.122606
$324$	0	0
$325$	14.6821	0.814416
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.37388	0.461667
$330$	0	0
$331$	−5.49227	−0.301883	−0.150941	−	0.988543i	$-0.548230\pi$
$331$	−5.49227	−0.301883	−0.150941	+	0.988543i	$0.548230\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.55080	−0.412544
$336$	0	0
$337$	−24.7742	−1.34954	−0.674768	−	0.738030i	$-0.735756\pi$
$337$	−24.7742	−1.34954	−0.674768	+	0.738030i	$0.735756\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.28283	0.231928
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.7804	0.578723	0.289362	−	0.957220i	$-0.406557\pi$
$347$	10.7804	0.578723	0.289362	+	0.957220i	$0.406557\pi$
$348$	0	0
$349$	−11.8446	−0.634026	−0.317013	−	0.948421i	$-0.602680\pi$
$349$	−11.8446	−0.634026	−0.317013	+	0.948421i	$0.602680\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.1423	0.646269	0.323135	−	0.946353i	$-0.395263\pi$
$353$	12.1423	0.646269	0.323135	+	0.946353i	$0.395263\pi$
$354$	0	0
$355$	9.74777	0.517358
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.23289	−0.223403	−0.111702	−	0.993742i	$-0.535630\pi$
$359$	−4.23289	−0.223403	−0.111702	+	0.993742i	$0.535630\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−11.6744	−0.611064
$366$	0	0
$367$	−8.48289	−0.442803	−0.221402	−	0.975183i	$-0.571063\pi$
$367$	−8.48289	−0.442803	−0.221402	+	0.975183i	$0.571063\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.170389	0.00884615
$372$	0	0
$373$	22.1727	1.14806	0.574030	−	0.818834i	$-0.305379\pi$
$373$	22.1727	1.14806	0.574030	+	0.818834i	$0.305379\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	38.8897	2.00292
$378$	0	0
$379$	33.0864	1.69953	0.849767	−	0.527158i	$-0.176742\pi$
$379$	33.0864	1.69953	0.849767	+	0.527158i	$0.176742\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−28.0525	−1.43342	−0.716709	−	0.697372i	$-0.754352\pi$
$383$	−28.0525	−1.43342	−0.716709	+	0.697372i	$0.754352\pi$
$384$	0	0
$385$	−6.19429	−0.315690
$386$	0	0
$387$	0	0
$388$	0	0
$389$	20.6533	1.04716	0.523581	−	0.851976i	$-0.324596\pi$
$389$	20.6533	1.04716	0.523581	+	0.851976i	$0.324596\pi$
$390$	0	0
$391$	18.6996	0.945678
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.46029	−0.0734750
$396$	0	0
$397$	8.98709	0.451049	0.225525	−	0.974237i	$-0.427590\pi$
$397$	8.98709	0.451049	0.225525	+	0.974237i	$0.427590\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.24629	−0.311925	−0.155962	−	0.987763i	$-0.549848\pi$
$401$	−6.24629	−0.311925	−0.155962	+	0.987763i	$0.549848\pi$
$402$	0	0
$403$	3.00147	0.149514
$404$	0	0
$405$	0	0
$406$	0	0
$407$	46.8422	2.32188
$408$	0	0
$409$	−0.661460	−0.0327071	−0.0163535	−	0.999866i	$-0.505206\pi$
$409$	−0.661460	−0.0327071	−0.0163535	+	0.999866i	$0.505206\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.887554	0.0436737
$414$	0	0
$415$	−19.7626	−0.970106
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7.13010	−0.348328	−0.174164	−	0.984717i	$-0.555722\pi$
$419$	−7.13010	−0.348328	−0.174164	+	0.984717i	$0.555722\pi$
$420$	0	0
$421$	26.6500	1.29884	0.649421	−	0.760429i	$-0.275011\pi$
$421$	26.6500	1.29884	0.649421	+	0.760429i	$0.275011\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.29057	0.402152
$426$	0	0
$427$	−15.0898	−0.730249
$428$	0	0
$429$	0	0
$430$	0	0
$431$	19.8807	0.957618	0.478809	−	0.877919i	$-0.341069\pi$
$431$	19.8807	0.957618	0.478809	+	0.877919i	$0.341069\pi$
$432$	0	0
$433$	−16.5468	−0.795187	−0.397594	−	0.917562i	$-0.630155\pi$
$433$	−16.5468	−0.795187	−0.397594	+	0.917562i	$0.630155\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.48633	0.405956
$438$	0	0
$439$	3.36364	0.160538	0.0802690	−	0.996773i	$-0.474422\pi$
$439$	3.36364	0.160538	0.0802690	+	0.996773i	$0.474422\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−33.4344	−1.58852	−0.794259	−	0.607580i	$-0.792141\pi$
$443$	−33.4344	−1.58852	−0.794259	+	0.607580i	$0.792141\pi$
$444$	0	0
$445$	10.6530	0.504999
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10.8033	0.509839	0.254920	−	0.966962i	$-0.417951\pi$
$449$	10.8033	0.509839	0.254920	+	0.966962i	$0.417951\pi$
$450$	0	0
$451$	−21.8170	−1.02732
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.34104	−0.203511
$456$	0	0
$457$	9.02064	0.421968	0.210984	−	0.977490i	$-0.432333\pi$
$457$	9.02064	0.421968	0.210984	+	0.977490i	$0.432333\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−36.6118	−1.70518	−0.852590	−	0.522580i	$-0.824970\pi$
$461$	−36.6118	−1.70518	−0.852590	+	0.522580i	$0.824970\pi$
$462$	0	0
$463$	−15.3794	−0.714740	−0.357370	−	0.933963i	$-0.616327\pi$
$463$	−15.3794	−0.714740	−0.357370	+	0.933963i	$0.616327\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−32.8989	−1.52238	−0.761190	−	0.648529i	$-0.775385\pi$
$467$	−32.8989	−1.52238	−0.761190	+	0.648529i	$0.775385\pi$
$468$	0	0
$469$	−6.78757	−0.313421
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−32.0864	−1.47533
$474$	0	0
$475$	3.76246	0.172634
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−25.9247	−1.18453	−0.592264	−	0.805744i	$-0.701766\pi$
$479$	−25.9247	−1.18453	−0.592264	+	0.805744i	$0.701766\pi$
$480$	0	0
$481$	32.8276	1.49681
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.15921	0.0526372
$486$	0	0
$487$	−2.13291	−0.0966512	−0.0483256	−	0.998832i	$-0.515389\pi$
$487$	−2.13291	−0.0966512	−0.0483256	+	0.998832i	$0.515389\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.76823	0.170058	0.0850288	−	0.996378i	$-0.472902\pi$
$491$	3.76823	0.170058	0.0850288	+	0.996378i	$0.472902\pi$
$492$	0	0
$493$	21.9599	0.989025
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.76246	0.393050
$498$	0	0
$499$	1.32934	0.0595093	0.0297546	−	0.999557i	$-0.490527\pi$
$499$	1.32934	0.0595093	0.0297546	+	0.999557i	$0.490527\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−22.1828	−0.989084	−0.494542	−	0.869154i	$-0.664664\pi$
$503$	−22.1828	−0.989084	−0.494542	+	0.869154i	$0.664664\pi$
$504$	0	0
$505$	−0.872504	−0.0388259
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8.72936	−0.386922	−0.193461	−	0.981108i	$-0.561971\pi$
$509$	−8.72936	−0.386922	−0.193461	+	0.981108i	$0.561971\pi$
$510$	0	0
$511$	−10.4943	−0.464242
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.77985	0.166560
$516$	0	0
$517$	46.6272	2.05066
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.7207	0.557304	0.278652	−	0.960392i	$-0.410112\pi$
$521$	12.7207	0.557304	0.278652	+	0.960392i	$0.410112\pi$
$522$	0	0
$523$	−21.7386	−0.950561	−0.475280	−	0.879834i	$-0.657653\pi$
$523$	−21.7386	−0.950561	−0.475280	+	0.879834i	$0.657653\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.69485	0.0738287
$528$	0	0
$529$	49.0178	2.13121
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−15.2896	−0.662267
$534$	0	0
$535$	14.3431	0.620107
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.56817	−0.239838
$540$	0	0
$541$	−16.3752	−0.704024	−0.352012	−	0.935995i	$-0.614502\pi$
$541$	−16.3752	−0.704024	−0.352012	+	0.935995i	$0.614502\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.27662	−0.0975198
$546$	0	0
$547$	−15.4242	−0.659491	−0.329745	−	0.944070i	$-0.606963\pi$
$547$	−15.4242	−0.659491	−0.329745	+	0.944070i	$0.606963\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.96596	0.424564
$552$	0	0
$553$	−1.31268	−0.0558209
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.17169	0.346246	0.173123	−	0.984900i	$-0.444614\pi$
$557$	8.17169	0.346246	0.173123	+	0.984900i	$0.444614\pi$
$558$	0	0
$559$	−22.4866	−0.951082
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13.4062	0.565005	0.282503	−	0.959266i	$-0.408835\pi$
$563$	13.4062	0.565005	0.282503	+	0.959266i	$0.408835\pi$
$564$	0	0
$565$	−19.8102	−0.833420
$566$	0	0
$567$	0	0
$568$	0	0
$569$	23.8571	1.00014	0.500070	−	0.865985i	$-0.333308\pi$
$569$	23.8571	1.00014	0.500070	+	0.865985i	$0.333308\pi$
$570$	0	0
$571$	−21.1561	−0.885353	−0.442677	−	0.896681i	$-0.645971\pi$
$571$	−21.1561	−0.885353	−0.442677	+	0.896681i	$0.645971\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	31.9295	1.33155
$576$	0	0
$577$	−1.65747	−0.0690014	−0.0345007	−	0.999405i	$-0.510984\pi$
$577$	−1.65747	−0.0690014	−0.0345007	+	0.999405i	$0.510984\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−17.7650	−0.737015
$582$	0	0
$583$	0.948755	0.0392934
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.9433	−0.451677	−0.225838	−	0.974165i	$-0.572512\pi$
$587$	−10.9433	−0.451677	−0.225838	+	0.974165i	$0.572512\pi$
$588$	0	0
$589$	0.769163	0.0316928
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21.5909	0.886631	0.443315	−	0.896366i	$-0.353802\pi$
$593$	21.5909	0.886631	0.443315	+	0.896366i	$0.353802\pi$
$594$	0	0
$595$	−2.45127	−0.100492
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9.29485	0.379777	0.189889	−	0.981806i	$-0.439187\pi$
$599$	9.29485	0.379777	0.189889	+	0.981806i	$0.439187\pi$
$600$	0	0
$601$	5.33957	0.217806	0.108903	−	0.994052i	$-0.465266\pi$
$601$	5.33957	0.217806	0.108903	+	0.994052i	$0.465266\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−22.2540	−0.904753
$606$	0	0
$607$	27.8292	1.12955	0.564775	−	0.825245i	$-0.308963\pi$
$607$	27.8292	1.12955	0.564775	+	0.825245i	$0.308963\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	32.6770	1.32197
$612$	0	0
$613$	−10.9854	−0.443696	−0.221848	−	0.975081i	$-0.571209\pi$
$613$	−10.9854	−0.443696	−0.221848	+	0.975081i	$0.571209\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−36.7312	−1.47874	−0.739372	−	0.673297i	$-0.764877\pi$
$617$	−36.7312	−1.47874	−0.739372	+	0.673297i	$0.764877\pi$
$618$	0	0
$619$	−14.9441	−0.600655	−0.300327	−	0.953836i	$-0.597096\pi$
$619$	−14.9441	−0.600655	−0.300327	+	0.953836i	$0.597096\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.57617	0.383661
$624$	0	0
$625$	7.96846	0.318739
$626$	0	0
$627$	0	0
$628$	0	0
$629$	18.5369	0.739113
$630$	0	0
$631$	−18.0231	−0.717490	−0.358745	−	0.933436i	$-0.616795\pi$
$631$	−18.0231	−0.717490	−0.358745	+	0.933436i	$0.616795\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.34605	0.291519
$636$	0	0
$637$	−3.90225	−0.154613
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−29.2150	−1.15392	−0.576962	−	0.816771i	$-0.695762\pi$
$641$	−29.2150	−1.15392	−0.576962	+	0.816771i	$0.695762\pi$
$642$	0	0
$643$	28.4913	1.12359	0.561794	−	0.827277i	$-0.310111\pi$
$643$	28.4913	1.12359	0.561794	+	0.827277i	$0.310111\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.34029	0.249263	0.124631	−	0.992203i	$-0.460225\pi$
$647$	6.34029	0.249263	0.124631	+	0.992203i	$0.460225\pi$
$648$	0	0
$649$	4.94206	0.193993
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3.78637	−0.148172	−0.0740860	−	0.997252i	$-0.523604\pi$
$653$	−3.78637	−0.148172	−0.0740860	+	0.997252i	$0.523604\pi$
$654$	0	0
$655$	5.48753	0.214416
$656$	0	0
$657$	0	0
$658$	0	0
$659$	30.9284	1.20480	0.602399	−	0.798195i	$-0.294212\pi$
$659$	30.9284	1.20480	0.602399	+	0.798195i	$0.294212\pi$
$660$	0	0
$661$	45.0867	1.75367	0.876834	−	0.480793i	$-0.159651\pi$
$661$	45.0867	1.75367	0.876834	+	0.480793i	$0.159651\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.11245	−0.0431388
$666$	0	0
$667$	84.5744	3.27473
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−84.0229	−3.24367
$672$	0	0
$673$	13.1396	0.506495	0.253247	−	0.967402i	$-0.418501\pi$
$673$	13.1396	0.506495	0.253247	+	0.967402i	$0.418501\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14.5968	−0.561001	−0.280501	−	0.959854i	$-0.590500\pi$
$677$	−14.5968	−0.561001	−0.280501	+	0.959854i	$0.590500\pi$
$678$	0	0
$679$	1.04204	0.0399899
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.57282	0.328030	0.164015	−	0.986458i	$-0.447555\pi$
$683$	8.57282	0.328030	0.164015	+	0.986458i	$0.447555\pi$
$684$	0	0
$685$	16.7235	0.638973
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.664901	0.0253307
$690$	0	0
$691$	−18.3045	−0.696336	−0.348168	−	0.937432i	$-0.613196\pi$
$691$	−18.3045	−0.696336	−0.348168	+	0.937432i	$0.613196\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	24.2893	0.921347
$696$	0	0
$697$	−8.63363	−0.327022
$698$	0	0
$699$	0	0
$700$	0	0
$701$	20.0115	0.755825	0.377912	−	0.925841i	$-0.376642\pi$
$701$	20.0115	0.755825	0.377912	+	0.925841i	$0.376642\pi$
$702$	0	0
$703$	8.41248	0.317283
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.784312	−0.0294971
$708$	0	0
$709$	−47.9313	−1.80010	−0.900049	−	0.435790i	$-0.856469\pi$
$709$	−47.9313	−1.80010	−0.900049	+	0.435790i	$0.856469\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.52737	0.244452
$714$	0	0
$715$	−24.1717	−0.903970
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22.9543	−0.856049	−0.428025	−	0.903767i	$-0.640790\pi$
$719$	−22.9543	−0.856049	−0.428025	+	0.903767i	$0.640790\pi$
$720$	0	0
$721$	3.39778	0.126540
$722$	0	0
$723$	0	0
$724$	0	0
$725$	37.4966	1.39259
$726$	0	0
$727$	−28.6053	−1.06091	−0.530456	−	0.847713i	$-0.677979\pi$
$727$	−28.6053	−1.06091	−0.530456	+	0.847713i	$0.677979\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−12.6976	−0.469636
$732$	0	0
$733$	6.69831	0.247408	0.123704	−	0.992319i	$-0.460523\pi$
$733$	6.69831	0.247408	0.123704	+	0.992319i	$0.460523\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−37.7944	−1.39217
$738$	0	0
$739$	29.7369	1.09389	0.546945	−	0.837169i	$-0.315791\pi$
$739$	29.7369	1.09389	0.546945	+	0.837169i	$0.315791\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21.0031	0.770528	0.385264	−	0.922806i	$-0.374110\pi$
$743$	21.0031	0.770528	0.385264	+	0.922806i	$0.374110\pi$
$744$	0	0
$745$	9.75884	0.357536
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.8933	0.471112
$750$	0	0
$751$	5.29306	0.193146	0.0965732	−	0.995326i	$-0.469212\pi$
$751$	5.29306	0.193146	0.0965732	+	0.995326i	$0.469212\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5.81902	−0.211776
$756$	0	0
$757$	29.7498	1.08128	0.540638	−	0.841255i	$-0.318183\pi$
$757$	29.7498	1.08128	0.540638	+	0.841255i	$0.318183\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	46.8412	1.69799	0.848997	−	0.528398i	$-0.177207\pi$
$761$	46.8412	1.69799	0.848997	+	0.528398i	$0.177207\pi$
$762$	0	0
$763$	−2.04650	−0.0740884
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.46346	0.125058
$768$	0	0
$769$	−17.2522	−0.622131	−0.311066	−	0.950388i	$-0.600686\pi$
$769$	−17.2522	−0.622131	−0.311066	+	0.950388i	$0.600686\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	20.7454	0.746159	0.373079	−	0.927799i	$-0.378302\pi$
$773$	20.7454	0.746159	0.373079	+	0.927799i	$0.378302\pi$
$774$	0	0
$775$	2.89395	0.103954
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.91815	−0.140382
$780$	0	0
$781$	48.7909	1.74588
$782$	0	0
$783$	0	0
$784$	0	0
$785$	22.6331	0.807809
$786$	0	0
$787$	−30.4834	−1.08662	−0.543309	−	0.839533i	$-0.682829\pi$
$787$	−30.4834	−1.08662	−0.543309	+	0.839533i	$0.682829\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−17.8078	−0.633171
$792$	0	0
$793$	−58.8844	−2.09105
$794$	0	0
$795$	0	0
$796$	0	0
$797$	43.3624	1.53597	0.767987	−	0.640466i	$-0.221259\pi$
$797$	43.3624	1.53597	0.767987	+	0.640466i	$0.221259\pi$
$798$	0	0
$799$	18.4518	0.652778
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−58.4342	−2.06210
$804$	0	0
$805$	−9.44058	−0.332737
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.2467	0.746996	0.373498	−	0.927631i	$-0.378158\pi$
$809$	21.2467	0.746996	0.373498	+	0.927631i	$0.378158\pi$
$810$	0	0
$811$	−8.19653	−0.287819	−0.143910	−	0.989591i	$-0.545967\pi$
$811$	−8.19653	−0.287819	−0.143910	+	0.989591i	$0.545967\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.59001	−0.125752
$816$	0	0
$817$	−5.76246	−0.201603
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.80451	−0.0629777	−0.0314888	−	0.999504i	$-0.510025\pi$
$821$	−1.80451	−0.0629777	−0.0314888	+	0.999504i	$0.510025\pi$
$822$	0	0
$823$	5.50742	0.191977	0.0959883	−	0.995382i	$-0.469399\pi$
$823$	5.50742	0.191977	0.0959883	+	0.995382i	$0.469399\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.5879	1.27229	0.636143	−	0.771571i	$-0.280529\pi$
$827$	36.5879	1.27229	0.636143	+	0.771571i	$0.280529\pi$
$828$	0	0
$829$	38.4907	1.33684	0.668419	−	0.743785i	$-0.266971\pi$
$829$	38.4907	1.33684	0.668419	+	0.743785i	$0.266971\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.20349	−0.0763465
$834$	0	0
$835$	25.0680	0.867514
$836$	0	0
$837$	0	0
$838$	0	0
$839$	28.5638	0.986132	0.493066	−	0.869992i	$-0.335876\pi$
$839$	28.5638	0.986132	0.493066	+	0.869992i	$0.335876\pi$
$840$	0	0
$841$	70.3203	2.42484
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.47806	−0.0852479
$846$	0	0
$847$	−20.0046	−0.687365
$848$	0	0
$849$	0	0
$850$	0	0
$851$	71.3911	2.44726
$852$	0	0
$853$	28.2140	0.966029	0.483014	−	0.875612i	$-0.339542\pi$
$853$	28.2140	0.966029	0.483014	+	0.875612i	$0.339542\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.2920	0.863960	0.431980	−	0.901883i	$-0.357815\pi$
$857$	25.2920	0.863960	0.431980	+	0.901883i	$0.357815\pi$
$858$	0	0
$859$	5.56921	0.190019	0.0950095	−	0.995476i	$-0.469712\pi$
$859$	5.56921	0.190019	0.0950095	+	0.995476i	$0.469712\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	14.5020	0.493653	0.246826	−	0.969060i	$-0.420612\pi$
$863$	14.5020	0.493653	0.246826	+	0.969060i	$0.420612\pi$
$864$	0	0
$865$	−24.9593	−0.848640
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7.30924	−0.247949
$870$	0	0
$871$	−26.4868	−0.897471
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.74777	−0.329535
$876$	0	0
$877$	−52.7538	−1.78137	−0.890685	−	0.454621i	$-0.849775\pi$
$877$	−52.7538	−1.78137	−0.890685	+	0.454621i	$0.849775\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	32.2929	1.08797	0.543987	−	0.839094i	$-0.316914\pi$
$881$	32.2929	1.08797	0.543987	+	0.839094i	$0.316914\pi$
$882$	0	0
$883$	25.6849	0.864366	0.432183	−	0.901786i	$-0.357743\pi$
$883$	25.6849	0.864366	0.432183	+	0.901786i	$0.357743\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−31.5163	−1.05821	−0.529106	−	0.848555i	$-0.677473\pi$
$887$	−31.5163	−1.05821	−0.529106	+	0.848555i	$0.677473\pi$
$888$	0	0
$889$	6.60352	0.221475
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.37388	0.280221
$894$	0	0
$895$	−10.2334	−0.342066
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.66545	0.255657
$900$	0	0
$901$	0.375451	0.0125081
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.96314	−0.0652568
$906$	0	0
$907$	−44.9865	−1.49375	−0.746876	−	0.664963i	$-0.768447\pi$
$907$	−44.9865	−1.49375	−0.746876	+	0.664963i	$0.768447\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−45.0556	−1.49276	−0.746380	−	0.665520i	$-0.768210\pi$
$911$	−45.0556	−1.49276	−0.746380	+	0.665520i	$0.768210\pi$
$912$	0	0
$913$	−98.9184	−3.27372
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.93285	0.162897
$918$	0	0
$919$	−10.0247	−0.330683	−0.165341	−	0.986236i	$-0.552873\pi$
$919$	−10.0247	−0.330683	−0.165341	+	0.986236i	$0.552873\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	34.1934	1.12549
$924$	0	0
$925$	31.6517	1.04070
$926$	0	0
$927$	0	0
$928$	0	0
$929$	26.4699	0.868451	0.434225	−	0.900804i	$-0.357022\pi$
$929$	26.4699	0.868451	0.434225	+	0.900804i	$0.357022\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−13.6491	−0.446373
$936$	0	0
$937$	5.58073	0.182315	0.0911573	−	0.995837i	$-0.470943\pi$
$937$	5.58073	0.182315	0.0911573	+	0.995837i	$0.470943\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−34.8612	−1.13644	−0.568222	−	0.822875i	$-0.692369\pi$
$941$	−34.8612	−1.13644	−0.568222	+	0.822875i	$0.692369\pi$
$942$	0	0
$943$	−33.2507	−1.08279
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7.02957	−0.228430	−0.114215	−	0.993456i	$-0.536435\pi$
$947$	−7.02957	−0.228430	−0.114215	+	0.993456i	$0.536435\pi$
$948$	0	0
$949$	−40.9515	−1.32934
$950$	0	0
$951$	0	0
$952$	0	0
$953$	60.3644	1.95540	0.977698	−	0.210018i	$-0.0673522\pi$
$953$	60.3644	1.95540	0.977698	+	0.210018i	$0.0673522\pi$
$954$	0	0
$955$	18.2810	0.591558
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.0331	0.485444
$960$	0	0
$961$	−30.4084	−0.980916
$962$	0	0
$963$	0	0
$964$	0	0
$965$	15.2721	0.491627
$966$	0	0
$967$	8.67767	0.279055	0.139527	−	0.990218i	$-0.455442\pi$
$967$	8.67767	0.279055	0.139527	+	0.990218i	$0.455442\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−48.8328	−1.56712	−0.783559	−	0.621317i	$-0.786598\pi$
$971$	−48.8328	−1.56712	−0.783559	+	0.621317i	$0.786598\pi$
$972$	0	0
$973$	21.8342	0.699971
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.95926	0.0946751	0.0473376	−	0.998879i	$-0.484926\pi$
$977$	2.95926	0.0946751	0.0473376	+	0.998879i	$0.484926\pi$
$978$	0	0
$979$	53.3218	1.70417
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−50.9910	−1.62636	−0.813180	−	0.582012i	$-0.802266\pi$
$983$	−50.9910	−1.62636	−0.813180	+	0.582012i	$0.802266\pi$
$984$	0	0
$985$	−25.4914	−0.812223
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−48.9022	−1.55500
$990$	0	0
$991$	10.6697	0.338935	0.169468	−	0.985536i	$-0.445795\pi$
$991$	10.6697	0.338935	0.169468	+	0.985536i	$0.445795\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−12.4506	−0.394709
$996$	0	0
$997$	33.5633	1.06296	0.531481	−	0.847070i	$-0.321636\pi$
$997$	33.5633	1.06296	0.531481	+	0.847070i	$0.321636\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.ck.1.2		4
3.2	odd	2		1064.2.a.g.1.2	✓	4
12.11	even	2		2128.2.a.u.1.3		4
21.20	even	2		7448.2.a.bk.1.3		4
24.5	odd	2		8512.2.a.bs.1.3		4
24.11	even	2		8512.2.a.bt.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.a.g.1.2	✓	4	3.2	odd	2
2128.2.a.u.1.3		4	12.11	even	2
7448.2.a.bk.1.3		4	21.20	even	2
8512.2.a.bs.1.3		4	24.5	odd	2
8512.2.a.bt.1.2		4	24.11	even	2
9576.2.a.ck.1.2		4	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:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q-1.11245 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-1.11245 q^{5} -1.00000 q^{7} -5.56817 q^{11} -3.90225 q^{13} -2.20349 q^{17} -1.00000 q^{19} -8.48633 q^{23} -3.76246 q^{25} -9.96596 q^{29} -0.769163 q^{31} +1.11245 q^{35} -8.41248 q^{37} +3.91815 q^{41} +5.76246 q^{43} -8.37388 q^{47} +1.00000 q^{49} -0.170389 q^{53} +6.19429 q^{55} -0.887554 q^{59} +15.0898 q^{61} +4.34104 q^{65} +6.78757 q^{67} -8.76246 q^{71} +10.4943 q^{73} +5.56817 q^{77} +1.31268 q^{79} +17.7650 q^{83} +2.45127 q^{85} -9.57617 q^{89} +3.90225 q^{91} +1.11245 q^{95} -1.04204 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q + 3 * q^5 - 4 * q^7	\( 4 q + 3 q^{5} - 4 q^{7} - 6 q^{11} - 4 q^{13} + 9 q^{17} - 4 q^{19} - 22 q^{23} + 13 q^{25} + 6 q^{29} + 3 q^{31} - 3 q^{35} + 15 q^{37} + 20 q^{41} - 5 q^{43} - 29 q^{47} + 4 q^{49} - 14 q^{53} + 13 q^{55} - 11 q^{59} + 15 q^{61} + 2 q^{65} + 9 q^{67} - 7 q^{71} - 11 q^{73} + 6 q^{77} + 7 q^{79} + 15 q^{83} - 7 q^{85} + 19 q^{89} + 4 q^{91} - 3 q^{95} - 9 q^{97}+O(q^{100}) \) 4 * q + 3 * q^5 - 4 * q^7 - 6 * q^11 - 4 * q^13 + 9 * q^17 - 4 * q^19 - 22 * q^23 + 13 * q^25 + 6 * q^29 + 3 * q^31 - 3 * q^35 + 15 * q^37 + 20 * q^41 - 5 * q^43 - 29 * q^47 + 4 * q^49 - 14 * q^53 + 13 * q^55 - 11 * q^59 + 15 * q^61 + 2 * q^65 + 9 * q^67 - 7 * q^71 - 11 * q^73 + 6 * q^77 + 7 * q^79 + 15 * q^83 - 7 * q^85 + 19 * q^89 + 4 * q^91 - 3 * q^95 - 9 * q^97

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