LMFDB - Embedded newform 4896.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4896,2,Mod(1,4896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4896, 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("4896.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4896 = 2^{5} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4896.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:	$39.0947568296$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	4896.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.622216 q^{5} -1.52543 q^{7} -1.09679 q^{11} +2.42864 q^{13} -1.00000 q^{17} +5.80642 q^{19} -8.57628 q^{23} -4.61285 q^{25} +3.37778 q^{29} -3.33185 q^{31} -0.949145 q^{35} -3.37778 q^{37} +6.85728 q^{41} -7.05086 q^{43} +1.24443 q^{47} -4.67307 q^{49} +10.8573 q^{53} -0.682439 q^{55} +4.56199 q^{59} -14.9906 q^{61} +1.51114 q^{65} -11.6128 q^{67} -12.2810 q^{71} +13.6128 q^{73} +1.67307 q^{77} +12.1891 q^{79} -7.05086 q^{83} -0.622216 q^{85} -7.67307 q^{89} -3.70471 q^{91} +3.61285 q^{95} +5.61285 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{5} + 2 q^{7} - 10 q^{11} - 6 q^{13} - 3 q^{17} + 4 q^{19} - 6 q^{23} + 13 q^{25} + 10 q^{29} + 10 q^{31} - 16 q^{35} - 10 q^{37} - 6 q^{41} - 8 q^{43} + 4 q^{47} - q^{49} + 6 q^{53} - 16 q^{55}+ \cdots - 10 q^{97}+O(q^{100}) $ 3 * q + 2 * q^5 + 2 * q^7 - 10 * q^11 - 6 * q^13 - 3 * q^17 + 4 * q^19 - 6 * q^23 + 13 * q^25 + 10 * q^29 + 10 * q^31 - 16 * q^35 - 10 * q^37 - 6 * q^41 - 8 * q^43 + 4 * q^47 - q^49 + 6 * q^53 - 16 * q^55 - 18 * q^61 + 4 * q^65 - 8 * q^67 - 30 * q^71 + 14 * q^73 - 8 * q^77 - 10 * q^79 - 8 * q^83 - 2 * q^85 - 10 * q^89 - 24 * q^91 - 16 * q^95 - 10 * q^97

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
$5$	0.622216	0.278263	0.139132	−	0.990274i	$-0.455569\pi$
$5$	0.622216	0.278263	0.139132	+	0.990274i	$0.455569\pi$
$6$	0	0
$7$	−1.52543	−0.576557	−0.288279	−	0.957547i	$-0.593083\pi$
$7$	−1.52543	−0.576557	−0.288279	+	0.957547i	$0.593083\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.09679	−0.330694	−0.165347	−	0.986235i	$-0.552874\pi$
$11$	−1.09679	−0.330694	−0.165347	+	0.986235i	$0.552874\pi$
$12$	0	0
$13$	2.42864	0.673583	0.336792	−	0.941579i	$-0.390658\pi$
$13$	2.42864	0.673583	0.336792	+	0.941579i	$0.390658\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	5.80642	1.33208	0.666042	−	0.745914i	$-0.267987\pi$
$19$	5.80642	1.33208	0.666042	+	0.745914i	$0.267987\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.57628	−1.78828	−0.894139	−	0.447789i	$-0.852212\pi$
$23$	−8.57628	−1.78828	−0.894139	+	0.447789i	$0.852212\pi$
$24$	0	0
$25$	−4.61285	−0.922570
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.37778	0.627239	0.313619	−	0.949549i	$-0.398458\pi$
$29$	3.37778	0.627239	0.313619	+	0.949549i	$0.398458\pi$
$30$	0	0
$31$	−3.33185	−0.598418	−0.299209	−	0.954188i	$-0.596723\pi$
$31$	−3.33185	−0.598418	−0.299209	+	0.954188i	$0.596723\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.949145	−0.160435
$36$	0	0
$37$	−3.37778	−0.555304	−0.277652	−	0.960682i	$-0.589556\pi$
$37$	−3.37778	−0.555304	−0.277652	+	0.960682i	$0.589556\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.85728	1.07093	0.535464	−	0.844558i	$-0.320137\pi$
$41$	6.85728	1.07093	0.535464	+	0.844558i	$0.320137\pi$
$42$	0	0
$43$	−7.05086	−1.07525	−0.537623	−	0.843186i	$-0.680677\pi$
$43$	−7.05086	−1.07525	−0.537623	+	0.843186i	$0.680677\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.24443	0.181519	0.0907595	−	0.995873i	$-0.471071\pi$
$47$	1.24443	0.181519	0.0907595	+	0.995873i	$0.471071\pi$
$48$	0	0
$49$	−4.67307	−0.667582
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.8573	1.49136	0.745681	−	0.666303i	$-0.232124\pi$
$53$	10.8573	1.49136	0.745681	+	0.666303i	$0.232124\pi$
$54$	0	0
$55$	−0.682439	−0.0920200
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.56199	0.593921	0.296960	−	0.954890i	$-0.404027\pi$
$59$	4.56199	0.593921	0.296960	+	0.954890i	$0.404027\pi$
$60$	0	0
$61$	−14.9906	−1.91935	−0.959677	−	0.281105i	$-0.909299\pi$
$61$	−14.9906	−1.91935	−0.959677	+	0.281105i	$0.909299\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.51114	0.187434
$66$	0	0
$67$	−11.6128	−1.41874	−0.709368	−	0.704839i	$-0.751019\pi$
$67$	−11.6128	−1.41874	−0.709368	+	0.704839i	$0.751019\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.2810	−1.45749	−0.728743	−	0.684787i	$-0.759895\pi$
$71$	−12.2810	−1.45749	−0.728743	+	0.684787i	$0.759895\pi$
$72$	0	0
$73$	13.6128	1.59326	0.796632	−	0.604465i	$-0.206613\pi$
$73$	13.6128	1.59326	0.796632	+	0.604465i	$0.206613\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.67307	0.190664
$78$	0	0
$79$	12.1891	1.37138	0.685692	−	0.727892i	$-0.259500\pi$
$79$	12.1891	1.37138	0.685692	+	0.727892i	$0.259500\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7.05086	−0.773932	−0.386966	−	0.922094i	$-0.626477\pi$
$83$	−7.05086	−0.773932	−0.386966	+	0.922094i	$0.626477\pi$
$84$	0	0
$85$	−0.622216	−0.0674888
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.67307	−0.813344	−0.406672	−	0.913574i	$-0.633311\pi$
$89$	−7.67307	−0.813344	−0.406672	+	0.913574i	$0.633311\pi$
$90$	0	0
$91$	−3.70471	−0.388360
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.61285	0.370670
$96$	0	0
$97$	5.61285	0.569898	0.284949	−	0.958543i	$-0.408023\pi$
$97$	5.61285	0.569898	0.284949	+	0.958543i	$0.408023\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.4286	−1.03769	−0.518844	−	0.854869i	$-0.673638\pi$
$101$	−10.4286	−1.03769	−0.518844	+	0.854869i	$0.673638\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.46520	0.334994	0.167497	−	0.985873i	$-0.446432\pi$
$107$	3.46520	0.334994	0.167497	+	0.985873i	$0.446432\pi$
$108$	0	0
$109$	5.47949	0.524840	0.262420	−	0.964954i	$-0.415479\pi$
$109$	5.47949	0.524840	0.262420	+	0.964954i	$0.415479\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.4701	1.73752	0.868762	−	0.495230i	$-0.164916\pi$
$113$	18.4701	1.73752	0.868762	+	0.495230i	$0.164916\pi$
$114$	0	0
$115$	−5.33630	−0.497612
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.52543	0.139836
$120$	0	0
$121$	−9.79706	−0.890641
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.98126	−0.534981
$126$	0	0
$127$	−4.94914	−0.439166	−0.219583	−	0.975594i	$-0.570470\pi$
$127$	−4.94914	−0.439166	−0.219583	+	0.975594i	$0.570470\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.65878	−0.494410	−0.247205	−	0.968963i	$-0.579512\pi$
$131$	−5.65878	−0.494410	−0.247205	+	0.968963i	$0.579512\pi$
$132$	0	0
$133$	−8.85728	−0.768023
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.7971	−1.43507	−0.717535	−	0.696523i	$-0.754730\pi$
$137$	−16.7971	−1.43507	−0.717535	+	0.696523i	$0.754730\pi$
$138$	0	0
$139$	−10.8113	−0.917006	−0.458503	−	0.888693i	$-0.651614\pi$
$139$	−10.8113	−0.917006	−0.458503	+	0.888693i	$0.651614\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.66370	−0.222750
$144$	0	0
$145$	2.10171	0.174538
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.48886	−0.695435	−0.347717	−	0.937599i	$-0.613043\pi$
$149$	−8.48886	−0.695435	−0.347717	+	0.937599i	$0.613043\pi$
$150$	0	0
$151$	−0.561993	−0.0457343	−0.0228672	−	0.999739i	$-0.507279\pi$
$151$	−0.561993	−0.0457343	−0.0228672	+	0.999739i	$0.507279\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.07313	−0.166518
$156$	0	0
$157$	−16.1017	−1.28506	−0.642528	−	0.766262i	$-0.722114\pi$
$157$	−16.1017	−1.28506	−0.642528	+	0.766262i	$0.722114\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	13.0825	1.03105
$162$	0	0
$163$	9.95407	0.779663	0.389831	−	0.920886i	$-0.372533\pi$
$163$	9.95407	0.779663	0.389831	+	0.920886i	$0.372533\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.1383	−1.63573	−0.817864	−	0.575411i	$-0.804842\pi$
$167$	−21.1383	−1.63573	−0.817864	+	0.575411i	$0.804842\pi$
$168$	0	0
$169$	−7.10171	−0.546285
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.99063	0.531488	0.265744	−	0.964044i	$-0.414382\pi$
$173$	6.99063	0.531488	0.265744	+	0.964044i	$0.414382\pi$
$174$	0	0
$175$	7.03657	0.531914
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−17.4193	−1.30198	−0.650989	−	0.759087i	$-0.725646\pi$
$179$	−17.4193	−1.30198	−0.650989	+	0.759087i	$0.725646\pi$
$180$	0	0
$181$	−6.99063	−0.519610	−0.259805	−	0.965661i	$-0.583658\pi$
$181$	−6.99063	−0.519610	−0.259805	+	0.965661i	$0.583658\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.10171	−0.154521
$186$	0	0
$187$	1.09679	0.0802051
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.24443	−0.668904	−0.334452	−	0.942413i	$-0.608551\pi$
$191$	−9.24443	−0.668904	−0.334452	+	0.942413i	$0.608551\pi$
$192$	0	0
$193$	18.8573	1.35738	0.678688	−	0.734426i	$-0.262549\pi$
$193$	18.8573	1.35738	0.678688	+	0.734426i	$0.262549\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.86665	−0.132993	−0.0664965	−	0.997787i	$-0.521182\pi$
$197$	−1.86665	−0.132993	−0.0664965	+	0.997787i	$0.521182\pi$
$198$	0	0
$199$	−17.7003	−1.25474	−0.627369	−	0.778722i	$-0.715868\pi$
$199$	−17.7003	−1.25474	−0.627369	+	0.778722i	$0.715868\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.15257	−0.361639
$204$	0	0
$205$	4.26671	0.298000
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.36842	−0.440513
$210$	0	0
$211$	−5.65878	−0.389567	−0.194783	−	0.980846i	$-0.562400\pi$
$211$	−5.65878	−0.389567	−0.194783	+	0.980846i	$0.562400\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.38715	−0.299201
$216$	0	0
$217$	5.08250	0.345022
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.42864	−0.163368
$222$	0	0
$223$	−26.9304	−1.80339	−0.901697	−	0.432369i	$-0.857678\pi$
$223$	−26.9304	−1.80339	−0.901697	+	0.432369i	$0.857678\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.8622	1.45105	0.725523	−	0.688198i	$-0.241598\pi$
$227$	21.8622	1.45105	0.725523	+	0.688198i	$0.241598\pi$
$228$	0	0
$229$	−3.67307	−0.242723	−0.121362	−	0.992608i	$-0.538726\pi$
$229$	−3.67307	−0.242723	−0.121362	+	0.992608i	$0.538726\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.3461	1.13638	0.568192	−	0.822896i	$-0.307643\pi$
$233$	17.3461	1.13638	0.568192	+	0.822896i	$0.307643\pi$
$234$	0	0
$235$	0.774305	0.0505101
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.1427	−0.720763	−0.360381	−	0.932805i	$-0.617353\pi$
$239$	−11.1427	−0.720763	−0.360381	+	0.932805i	$0.617353\pi$
$240$	0	0
$241$	−0.755569	−0.0486705	−0.0243352	−	0.999704i	$-0.507747\pi$
$241$	−0.755569	−0.0486705	−0.0243352	+	0.999704i	$0.507747\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.90766	−0.185763
$246$	0	0
$247$	14.1017	0.897270
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	9.40636	0.591373
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.4286	−0.650521	−0.325260	−	0.945625i	$-0.605452\pi$
$257$	−10.4286	−0.650521	−0.325260	+	0.945625i	$0.605452\pi$
$258$	0	0
$259$	5.15257	0.320165
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.56199	−0.527955	−0.263978	−	0.964529i	$-0.585034\pi$
$263$	−8.56199	−0.527955	−0.263978	+	0.964529i	$0.585034\pi$
$264$	0	0
$265$	6.75557	0.414991
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−16.5018	−1.00613	−0.503065	−	0.864248i	$-0.667794\pi$
$269$	−16.5018	−1.00613	−0.503065	+	0.864248i	$0.667794\pi$
$270$	0	0
$271$	18.3684	1.11580	0.557901	−	0.829908i	$-0.311607\pi$
$271$	18.3684	1.11580	0.557901	+	0.829908i	$0.311607\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.05932	0.305088
$276$	0	0
$277$	29.5625	1.77624	0.888118	−	0.459615i	$-0.152013\pi$
$277$	29.5625	1.77624	0.888118	+	0.459615i	$0.152013\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	20.1017	1.19917	0.599584	−	0.800312i	$-0.295333\pi$
$281$	20.1017	1.19917	0.599584	+	0.800312i	$0.295333\pi$
$282$	0	0
$283$	−9.39207	−0.558301	−0.279150	−	0.960247i	$-0.590053\pi$
$283$	−9.39207	−0.558301	−0.279150	+	0.960247i	$0.590053\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.4603	−0.617451
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−31.7146	−1.85278	−0.926392	−	0.376560i	$-0.877107\pi$
$293$	−31.7146	−1.85278	−0.926392	+	0.376560i	$0.877107\pi$
$294$	0	0
$295$	2.83854	0.165266
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−20.8287	−1.20455
$300$	0	0
$301$	10.7556	0.619941
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.32741	−0.534086
$306$	0	0
$307$	11.1427	0.635949	0.317974	−	0.948099i	$-0.396997\pi$
$307$	11.1427	0.635949	0.317974	+	0.948099i	$0.396997\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−17.5254	−0.993776	−0.496888	−	0.867815i	$-0.665524\pi$
$311$	−17.5254	−0.993776	−0.496888	+	0.867815i	$0.665524\pi$
$312$	0	0
$313$	19.4479	1.09926	0.549629	−	0.835409i	$-0.314769\pi$
$313$	19.4479	1.09926	0.549629	+	0.835409i	$0.314769\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.6222	0.933597	0.466798	−	0.884364i	$-0.345407\pi$
$317$	16.6222	0.933597	0.466798	+	0.884364i	$0.345407\pi$
$318$	0	0
$319$	−3.70471	−0.207424
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.80642	−0.323078
$324$	0	0
$325$	−11.2029	−0.621428
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.89829	−0.104656
$330$	0	0
$331$	−32.7654	−1.80095	−0.900475	−	0.434908i	$-0.856781\pi$
$331$	−32.7654	−1.80095	−0.900475	+	0.434908i	$0.856781\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.22570	−0.394782
$336$	0	0
$337$	16.1017	0.877116	0.438558	−	0.898703i	$-0.355489\pi$
$337$	16.1017	0.877116	0.438558	+	0.898703i	$0.355489\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.65433	0.197893
$342$	0	0
$343$	17.8064	0.961457
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−0.830082	−0.0445611	−0.0222806	−	0.999752i	$-0.507093\pi$
$347$	−0.830082	−0.0445611	−0.0222806	+	0.999752i	$0.507093\pi$
$348$	0	0
$349$	−2.85728	−0.152947	−0.0764733	−	0.997072i	$-0.524366\pi$
$349$	−2.85728	−0.152947	−0.0764733	+	0.997072i	$0.524366\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.61285	0.0858432	0.0429216	−	0.999078i	$-0.486333\pi$
$353$	1.61285	0.0858432	0.0429216	+	0.999078i	$0.486333\pi$
$354$	0	0
$355$	−7.64143	−0.405565
$356$	0	0
$357$	0	0
$358$	0	0
$359$	25.6227	1.35231	0.676157	−	0.736758i	$-0.263644\pi$
$359$	25.6227	1.35231	0.676157	+	0.736758i	$0.263644\pi$
$360$	0	0
$361$	14.7146	0.774450
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.47013	0.443347
$366$	0	0
$367$	11.7190	0.611727	0.305864	−	0.952075i	$-0.401055\pi$
$367$	11.7190	0.611727	0.305864	+	0.952075i	$0.401055\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−16.5620	−0.859856
$372$	0	0
$373$	−33.0005	−1.70870	−0.854350	−	0.519698i	$-0.826044\pi$
$373$	−33.0005	−1.70870	−0.854350	+	0.519698i	$0.826044\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.20342	0.422498
$378$	0	0
$379$	2.90321	0.149128	0.0745640	−	0.997216i	$-0.476243\pi$
$379$	2.90321	0.149128	0.0745640	+	0.997216i	$0.476243\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.6637	1.56684	0.783421	−	0.621491i	$-0.213473\pi$
$383$	30.6637	1.56684	0.783421	+	0.621491i	$0.213473\pi$
$384$	0	0
$385$	1.04101	0.0530548
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.16193	0.312422	0.156211	−	0.987724i	$-0.450072\pi$
$389$	6.16193	0.312422	0.156211	+	0.987724i	$0.450072\pi$
$390$	0	0
$391$	8.57628	0.433721
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7.58427	0.381606
$396$	0	0
$397$	−16.8889	−0.847631	−0.423815	−	0.905749i	$-0.639309\pi$
$397$	−16.8889	−0.847631	−0.423815	+	0.905749i	$0.639309\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.47013	−0.323103	−0.161551	−	0.986864i	$-0.551650\pi$
$401$	−6.47013	−0.323103	−0.161551	+	0.986864i	$0.551650\pi$
$402$	0	0
$403$	−8.09187	−0.403085
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.70471	0.183636
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6.95899	−0.342429
$414$	0	0
$415$	−4.38715	−0.215357
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−17.3002	−0.845170	−0.422585	−	0.906323i	$-0.638877\pi$
$419$	−17.3002	−0.845170	−0.422585	+	0.906323i	$0.638877\pi$
$420$	0	0
$421$	−29.5714	−1.44122	−0.720610	−	0.693341i	$-0.756138\pi$
$421$	−29.5714	−1.44122	−0.720610	+	0.693341i	$0.756138\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.61285	0.223756
$426$	0	0
$427$	22.8671	1.10662
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−9.70027	−0.467245	−0.233623	−	0.972327i	$-0.575058\pi$
$431$	−9.70027	−0.467245	−0.233623	+	0.972327i	$0.575058\pi$
$432$	0	0
$433$	1.83807	0.0883318	0.0441659	−	0.999024i	$-0.485937\pi$
$433$	1.83807	0.0883318	0.0441659	+	0.999024i	$0.485937\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−49.7975	−2.38214
$438$	0	0
$439$	−15.9224	−0.759936	−0.379968	−	0.925000i	$-0.624065\pi$
$439$	−15.9224	−0.759936	−0.379968	+	0.925000i	$0.624065\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.10171	0.289901	0.144951	−	0.989439i	$-0.453698\pi$
$443$	6.10171	0.289901	0.144951	+	0.989439i	$0.453698\pi$
$444$	0	0
$445$	−4.77430	−0.226324
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−9.22570	−0.435387	−0.217694	−	0.976017i	$-0.569853\pi$
$449$	−9.22570	−0.435387	−0.217694	+	0.976017i	$0.569853\pi$
$450$	0	0
$451$	−7.52098	−0.354149
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.30513	−0.108066
$456$	0	0
$457$	11.0192	0.515457	0.257729	−	0.966217i	$-0.417026\pi$
$457$	11.0192	0.515457	0.257729	+	0.966217i	$0.417026\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−25.3461	−1.18049	−0.590244	−	0.807225i	$-0.700968\pi$
$461$	−25.3461	−1.18049	−0.590244	+	0.807225i	$0.700968\pi$
$462$	0	0
$463$	−25.1240	−1.16761	−0.583805	−	0.811894i	$-0.698437\pi$
$463$	−25.1240	−1.16761	−0.583805	+	0.811894i	$0.698437\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.1338	−0.885408	−0.442704	−	0.896668i	$-0.645981\pi$
$467$	−19.1338	−0.885408	−0.442704	+	0.896668i	$0.645981\pi$
$468$	0	0
$469$	17.7146	0.817982
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.73329	0.355577
$474$	0	0
$475$	−26.7841	−1.22894
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−24.0143	−1.09724	−0.548620	−	0.836072i	$-0.684847\pi$
$479$	−24.0143	−1.09724	−0.548620	+	0.836072i	$0.684847\pi$
$480$	0	0
$481$	−8.20342	−0.374044
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.49240	0.158582
$486$	0	0
$487$	−0.484417	−0.0219510	−0.0109755	−	0.999940i	$-0.503494\pi$
$487$	−0.484417	−0.0219510	−0.0109755	+	0.999940i	$0.503494\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.60300	−0.0723425	−0.0361713	−	0.999346i	$-0.511516\pi$
$491$	−1.60300	−0.0723425	−0.0361713	+	0.999346i	$0.511516\pi$
$492$	0	0
$493$	−3.37778	−0.152128
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.7338	0.840324
$498$	0	0
$499$	13.1798	0.590007	0.295004	−	0.955496i	$-0.404679\pi$
$499$	13.1798	0.590007	0.295004	+	0.955496i	$0.404679\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	36.8716	1.64402	0.822011	−	0.569472i	$-0.192852\pi$
$503$	36.8716	1.64402	0.822011	+	0.569472i	$0.192852\pi$
$504$	0	0
$505$	−6.48886	−0.288751
$506$	0	0
$507$	0	0
$508$	0	0
$509$	17.2257	0.763516	0.381758	−	0.924262i	$-0.375319\pi$
$509$	17.2257	0.763516	0.381758	+	0.924262i	$0.375319\pi$
$510$	0	0
$511$	−20.7654	−0.918608
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.36488	−0.0600272
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−39.0607	−1.71128	−0.855640	−	0.517571i	$-0.826836\pi$
$521$	−39.0607	−1.71128	−0.855640	+	0.517571i	$0.826836\pi$
$522$	0	0
$523$	1.71456	0.0749724	0.0374862	−	0.999297i	$-0.488065\pi$
$523$	1.71456	0.0749724	0.0374862	+	0.999297i	$0.488065\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.33185	0.145138
$528$	0	0
$529$	50.5526	2.19794
$530$	0	0
$531$	0	0
$532$	0	0
$533$	16.6539	0.721359
$534$	0	0
$535$	2.15610	0.0932165
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.12537	0.220765
$540$	0	0
$541$	41.9496	1.80356	0.901778	−	0.432200i	$-0.142263\pi$
$541$	41.9496	1.80356	0.901778	+	0.432200i	$0.142263\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.40943	0.146044
$546$	0	0
$547$	1.27163	0.0543709	0.0271855	−	0.999630i	$-0.491346\pi$
$547$	1.27163	0.0543709	0.0271855	+	0.999630i	$0.491346\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	19.6128	0.835535
$552$	0	0
$553$	−18.5936	−0.790682
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.0005	1.05930	0.529652	−	0.848215i	$-0.322322\pi$
$557$	25.0005	1.05930	0.529652	+	0.848215i	$0.322322\pi$
$558$	0	0
$559$	−17.1240	−0.724267
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−29.5022	−1.24337	−0.621686	−	0.783267i	$-0.713552\pi$
$563$	−29.5022	−1.24337	−0.621686	+	0.783267i	$0.713552\pi$
$564$	0	0
$565$	11.4924	0.483489
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−26.3497	−1.10464	−0.552318	−	0.833633i	$-0.686257\pi$
$569$	−26.3497	−1.10464	−0.552318	+	0.833633i	$0.686257\pi$
$570$	0	0
$571$	26.6079	1.11351	0.556754	−	0.830678i	$-0.312047\pi$
$571$	26.6079	1.11351	0.556754	+	0.830678i	$0.312047\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	39.5611	1.64981
$576$	0	0
$577$	−8.14320	−0.339006	−0.169503	−	0.985530i	$-0.554216\pi$
$577$	−8.14320	−0.339006	−0.169503	+	0.985530i	$0.554216\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.7556	0.446216
$582$	0	0
$583$	−11.9081	−0.493185
$584$	0	0
$585$	0	0
$586$	0	0
$587$	29.8064	1.23024	0.615121	−	0.788432i	$-0.289107\pi$
$587$	29.8064	1.23024	0.615121	+	0.788432i	$0.289107\pi$
$588$	0	0
$589$	−19.3461	−0.797144
$590$	0	0
$591$	0	0
$592$	0	0
$593$	29.8163	1.22441	0.612204	−	0.790700i	$-0.290283\pi$
$593$	29.8163	1.22441	0.612204	+	0.790700i	$0.290283\pi$
$594$	0	0
$595$	0.949145	0.0389111
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−31.8163	−1.29998	−0.649989	−	0.759944i	$-0.725226\pi$
$599$	−31.8163	−1.29998	−0.649989	+	0.759944i	$0.725226\pi$
$600$	0	0
$601$	−2.20342	−0.0898794	−0.0449397	−	0.998990i	$-0.514310\pi$
$601$	−2.20342	−0.0898794	−0.0449397	+	0.998990i	$0.514310\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.09588	−0.247833
$606$	0	0
$607$	−9.43356	−0.382896	−0.191448	−	0.981503i	$-0.561318\pi$
$607$	−9.43356	−0.382896	−0.191448	+	0.981503i	$0.561318\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.02227	0.122268
$612$	0	0
$613$	−13.0223	−0.525965	−0.262982	−	0.964801i	$-0.584706\pi$
$613$	−13.0223	−0.525965	−0.262982	+	0.964801i	$0.584706\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.85728	0.276064	0.138032	−	0.990428i	$-0.455922\pi$
$617$	6.85728	0.276064	0.138032	+	0.990428i	$0.455922\pi$
$618$	0	0
$619$	5.56691	0.223753	0.111877	−	0.993722i	$-0.464314\pi$
$619$	5.56691	0.223753	0.111877	+	0.993722i	$0.464314\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	11.7047	0.468939
$624$	0	0
$625$	19.3426	0.773704
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.37778	0.134681
$630$	0	0
$631$	30.0098	1.19467	0.597337	−	0.801991i	$-0.296226\pi$
$631$	30.0098	1.19467	0.597337	+	0.801991i	$0.296226\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.07944	−0.122204
$636$	0	0
$637$	−11.3492	−0.449672
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−5.07944	−0.200626	−0.100313	−	0.994956i	$-0.531984\pi$
$641$	−5.07944	−0.200626	−0.100313	+	0.994956i	$0.531984\pi$
$642$	0	0
$643$	41.3546	1.63087	0.815433	−	0.578851i	$-0.196499\pi$
$643$	41.3546	1.63087	0.815433	+	0.578851i	$0.196499\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−35.3087	−1.38813	−0.694064	−	0.719914i	$-0.744181\pi$
$647$	−35.3087	−1.38813	−0.694064	+	0.719914i	$0.744181\pi$
$648$	0	0
$649$	−5.00354	−0.196406
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−16.2351	−0.635327	−0.317664	−	0.948203i	$-0.602898\pi$
$653$	−16.2351	−0.635327	−0.317664	+	0.948203i	$0.602898\pi$
$654$	0	0
$655$	−3.52098	−0.137576
$656$	0	0
$657$	0	0
$658$	0	0
$659$	29.4479	1.14713	0.573563	−	0.819162i	$-0.305561\pi$
$659$	29.4479	1.14713	0.573563	+	0.819162i	$0.305561\pi$
$660$	0	0
$661$	−34.6735	−1.34864	−0.674322	−	0.738437i	$-0.735564\pi$
$661$	−34.6735	−1.34864	−0.674322	+	0.738437i	$0.735564\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.51114	−0.213713
$666$	0	0
$667$	−28.9688	−1.12168
$668$	0	0
$669$	0	0
$670$	0	0
$671$	16.4415	0.634719
$672$	0	0
$673$	23.5111	0.906288	0.453144	−	0.891437i	$-0.350302\pi$
$673$	23.5111	0.906288	0.453144	+	0.891437i	$0.350302\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.25734	0.278922	0.139461	−	0.990228i	$-0.455463\pi$
$677$	7.25734	0.278922	0.139461	+	0.990228i	$0.455463\pi$
$678$	0	0
$679$	−8.56199	−0.328579
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−33.0049	−1.26290	−0.631449	−	0.775417i	$-0.717540\pi$
$683$	−33.0049	−1.26290	−0.631449	+	0.775417i	$0.717540\pi$
$684$	0	0
$685$	−10.4514	−0.399327
$686$	0	0
$687$	0	0
$688$	0	0
$689$	26.3684	1.00456
$690$	0	0
$691$	23.1985	0.882512	0.441256	−	0.897381i	$-0.354533\pi$
$691$	23.1985	0.882512	0.441256	+	0.897381i	$0.354533\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.72699	−0.255169
$696$	0	0
$697$	−6.85728	−0.259738
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−12.9175	−0.487887	−0.243944	−	0.969789i	$-0.578441\pi$
$701$	−12.9175	−0.487887	−0.243944	+	0.969789i	$0.578441\pi$
$702$	0	0
$703$	−19.6128	−0.739713
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.9081	0.598287
$708$	0	0
$709$	−34.4197	−1.29266	−0.646330	−	0.763058i	$-0.723697\pi$
$709$	−34.4197	−1.29266	−0.646330	+	0.763058i	$0.723697\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	28.5749	1.07014
$714$	0	0
$715$	−1.65740	−0.0619832
$716$	0	0
$717$	0	0
$718$	0	0
$719$	18.1704	0.677641	0.338821	−	0.940851i	$-0.389972\pi$
$719$	18.1704	0.677641	0.338821	+	0.940851i	$0.389972\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−15.5812	−0.578671
$726$	0	0
$727$	−6.75557	−0.250550	−0.125275	−	0.992122i	$-0.539981\pi$
$727$	−6.75557	−0.250550	−0.125275	+	0.992122i	$0.539981\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7.05086	0.260785
$732$	0	0
$733$	−1.73329	−0.0640207	−0.0320103	−	0.999488i	$-0.510191\pi$
$733$	−1.73329	−0.0640207	−0.0320103	+	0.999488i	$0.510191\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.7368	0.469167
$738$	0	0
$739$	43.9081	1.61519	0.807593	−	0.589740i	$-0.200770\pi$
$739$	43.9081	1.61519	0.807593	+	0.589740i	$0.200770\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13.8207	0.507033	0.253516	−	0.967331i	$-0.418413\pi$
$743$	13.8207	0.507033	0.253516	+	0.967331i	$0.418413\pi$
$744$	0	0
$745$	−5.28190	−0.193514
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.28592	−0.193143
$750$	0	0
$751$	3.69042	0.134665	0.0673327	−	0.997731i	$-0.478551\pi$
$751$	3.69042	0.134665	0.0673327	+	0.997731i	$0.478551\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−0.349681	−0.0127262
$756$	0	0
$757$	−40.4099	−1.46872	−0.734361	−	0.678759i	$-0.762518\pi$
$757$	−40.4099	−1.46872	−0.734361	+	0.678759i	$0.762518\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	32.9175	1.19326	0.596629	−	0.802517i	$-0.296506\pi$
$761$	32.9175	1.19326	0.596629	+	0.802517i	$0.296506\pi$
$762$	0	0
$763$	−8.35857	−0.302601
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.0794	0.400055
$768$	0	0
$769$	37.0005	1.33427	0.667136	−	0.744936i	$-0.267520\pi$
$769$	37.0005	1.33427	0.667136	+	0.744936i	$0.267520\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3.67307	0.132111	0.0660556	−	0.997816i	$-0.478959\pi$
$773$	3.67307	0.132111	0.0660556	+	0.997816i	$0.478959\pi$
$774$	0	0
$775$	15.3693	0.552082
$776$	0	0
$777$	0	0
$778$	0	0
$779$	39.8163	1.42657
$780$	0	0
$781$	13.4697	0.481982
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−10.0187	−0.357584
$786$	0	0
$787$	2.16638	0.0772231	0.0386115	−	0.999254i	$-0.487707\pi$
$787$	2.16638	0.0772231	0.0386115	+	0.999254i	$0.487707\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−28.1748	−1.00178
$792$	0	0
$793$	−36.4068	−1.29284
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.1847	0.573291	0.286645	−	0.958037i	$-0.407460\pi$
$797$	16.1847	0.573291	0.286645	+	0.958037i	$0.407460\pi$
$798$	0	0
$799$	−1.24443	−0.0440248
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−14.9304	−0.526883
$804$	0	0
$805$	8.14013	0.286902
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−4.67259	−0.164280	−0.0821398	−	0.996621i	$-0.526175\pi$
$809$	−4.67259	−0.164280	−0.0821398	+	0.996621i	$0.526175\pi$
$810$	0	0
$811$	37.0879	1.30233	0.651166	−	0.758935i	$-0.274280\pi$
$811$	37.0879	1.30233	0.651166	+	0.758935i	$0.274280\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.19358	0.216952
$816$	0	0
$817$	−40.9403	−1.43232
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−23.7275	−0.828094	−0.414047	−	0.910255i	$-0.635885\pi$
$821$	−23.7275	−0.828094	−0.414047	+	0.910255i	$0.635885\pi$
$822$	0	0
$823$	33.7003	1.17472	0.587359	−	0.809327i	$-0.300168\pi$
$823$	33.7003	1.17472	0.587359	+	0.809327i	$0.300168\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.86220	0.342942	0.171471	−	0.985189i	$-0.445148\pi$
$827$	9.86220	0.342942	0.171471	+	0.985189i	$0.445148\pi$
$828$	0	0
$829$	−16.1017	−0.559236	−0.279618	−	0.960111i	$-0.590208\pi$
$829$	−16.1017	−0.559236	−0.279618	+	0.960111i	$0.590208\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.67307	0.161912
$834$	0	0
$835$	−13.1526	−0.455163
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10.6494	0.367659	0.183829	−	0.982958i	$-0.441151\pi$
$839$	10.6494	0.367659	0.183829	+	0.982958i	$0.441151\pi$
$840$	0	0
$841$	−17.5906	−0.606571
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.41880	−0.152011
$846$	0	0
$847$	14.9447	0.513506
$848$	0	0
$849$	0	0
$850$	0	0
$851$	28.9688	0.993039
$852$	0	0
$853$	−36.0513	−1.23437	−0.617187	−	0.786816i	$-0.711728\pi$
$853$	−36.0513	−1.23437	−0.617187	+	0.786816i	$0.711728\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.24443	−0.247465	−0.123733	−	0.992316i	$-0.539486\pi$
$857$	−7.24443	−0.247465	−0.123733	+	0.992316i	$0.539486\pi$
$858$	0	0
$859$	39.6414	1.35255	0.676274	−	0.736650i	$-0.263594\pi$
$859$	39.6414	1.35255	0.676274	+	0.736650i	$0.263594\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16.5906	0.564750	0.282375	−	0.959304i	$-0.408878\pi$
$863$	16.5906	0.564750	0.282375	+	0.959304i	$0.408878\pi$
$864$	0	0
$865$	4.34968	0.147894
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−13.3689	−0.453509
$870$	0	0
$871$	−28.2034	−0.955636
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.12399	0.308447
$876$	0	0
$877$	39.1941	1.32349	0.661745	−	0.749729i	$-0.269816\pi$
$877$	39.1941	1.32349	0.661745	+	0.749729i	$0.269816\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−30.5531	−1.02936	−0.514680	−	0.857382i	$-0.672089\pi$
$881$	−30.5531	−1.02936	−0.514680	+	0.857382i	$0.672089\pi$
$882$	0	0
$883$	45.3274	1.52539	0.762694	−	0.646759i	$-0.223876\pi$
$883$	45.3274	1.52539	0.762694	+	0.646759i	$0.223876\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.8243	0.900670	0.450335	−	0.892860i	$-0.351305\pi$
$887$	26.8243	0.900670	0.450335	+	0.892860i	$0.351305\pi$
$888$	0	0
$889$	7.54956	0.253204
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.22570	0.241799
$894$	0	0
$895$	−10.8385	−0.362293
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−11.2543	−0.375351
$900$	0	0
$901$	−10.8573	−0.361708
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.34968	−0.144588
$906$	0	0
$907$	−49.5580	−1.64555	−0.822774	−	0.568369i	$-0.807575\pi$
$907$	−49.5580	−1.64555	−0.822774	+	0.568369i	$0.807575\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4.19802	0.139087	0.0695433	−	0.997579i	$-0.477846\pi$
$911$	4.19802	0.139087	0.0695433	+	0.997579i	$0.477846\pi$
$912$	0	0
$913$	7.73329	0.255935
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.63206	0.285056
$918$	0	0
$919$	49.5941	1.63596	0.817979	−	0.575248i	$-0.195094\pi$
$919$	49.5941	1.63596	0.817979	+	0.575248i	$0.195094\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−29.8261	−0.981738
$924$	0	0
$925$	15.5812	0.512307
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0.755569	0.0247894	0.0123947	−	0.999923i	$-0.496055\pi$
$929$	0.755569	0.0247894	0.0123947	+	0.999923i	$0.496055\pi$
$930$	0	0
$931$	−27.1338	−0.889275
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.682439	0.0223181
$936$	0	0
$937$	−4.28544	−0.139999	−0.0699996	−	0.997547i	$-0.522300\pi$
$937$	−4.28544	−0.139999	−0.0699996	+	0.997547i	$0.522300\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	16.4385	0.535879	0.267940	−	0.963436i	$-0.413657\pi$
$941$	16.4385	0.535879	0.267940	+	0.963436i	$0.413657\pi$
$942$	0	0
$943$	−58.8100	−1.91512
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.0370	0.846090	0.423045	−	0.906109i	$-0.360961\pi$
$947$	26.0370	0.846090	0.423045	+	0.906109i	$0.360961\pi$
$948$	0	0
$949$	33.0607	1.07320
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−17.3876	−0.563241	−0.281620	−	0.959526i	$-0.590872\pi$
$953$	−17.3876	−0.563241	−0.281620	+	0.959526i	$0.590872\pi$
$954$	0	0
$955$	−5.75203	−0.186131
$956$	0	0
$957$	0	0
$958$	0	0
$959$	25.6227	0.827400
$960$	0	0
$961$	−19.8988	−0.641896
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11.7333	0.377708
$966$	0	0
$967$	−8.44155	−0.271462	−0.135731	−	0.990746i	$-0.543338\pi$
$967$	−8.44155	−0.271462	−0.135731	+	0.990746i	$0.543338\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	30.8671	0.990573	0.495287	−	0.868730i	$-0.335063\pi$
$971$	30.8671	0.990573	0.495287	+	0.868730i	$0.335063\pi$
$972$	0	0
$973$	16.4919	0.528707
$974$	0	0
$975$	0	0
$976$	0	0
$977$	58.5531	1.87328	0.936640	−	0.350294i	$-0.113918\pi$
$977$	58.5531	1.87328	0.936640	+	0.350294i	$0.113918\pi$
$978$	0	0
$979$	8.41573	0.268968
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−26.1990	−0.835618	−0.417809	−	0.908535i	$-0.637202\pi$
$983$	−26.1990	−0.835618	−0.417809	+	0.908535i	$0.637202\pi$
$984$	0	0
$985$	−1.16146	−0.0370071
$986$	0	0
$987$	0	0
$988$	0	0
$989$	60.4701	1.92284
$990$	0	0
$991$	56.4844	1.79429	0.897143	−	0.441740i	$-0.145638\pi$
$991$	56.4844	1.79429	0.897143	+	0.441740i	$0.145638\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11.0134	−0.349148
$996$	0	0
$997$	−22.9077	−0.725493	−0.362746	−	0.931888i	$-0.618161\pi$
$997$	−22.9077	−0.725493	−0.362746	+	0.931888i	$0.618161\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4896.2.a.bf.1.2		3
3.2	odd	2		544.2.a.j.1.3	yes	3
4.3	odd	2		4896.2.a.be.1.2		3
8.3	odd	2		9792.2.a.dc.1.2		3
8.5	even	2		9792.2.a.dd.1.2		3
12.11	even	2		544.2.a.i.1.1	✓	3
24.5	odd	2		1088.2.a.u.1.1		3
24.11	even	2		1088.2.a.v.1.3		3
51.50	odd	2		9248.2.a.t.1.1		3
204.203	even	2		9248.2.a.u.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.2.a.i.1.1	✓	3	12.11	even	2
544.2.a.j.1.3	yes	3	3.2	odd	2
1088.2.a.u.1.1		3	24.5	odd	2
1088.2.a.v.1.3		3	24.11	even	2
4896.2.a.be.1.2		3	4.3	odd	2
4896.2.a.bf.1.2		3	1.1	even	1	trivial
9248.2.a.t.1.1		3	51.50	odd	2
9248.2.a.u.1.3		3	204.203	even	2
9792.2.a.dc.1.2		3	8.3	odd	2
9792.2.a.dd.1.2		3	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4896 = 2^{5} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4896.a (trivial)

\(f(q)\)	\(=\)	\(q+0.622216 q^{5} -1.52543 q^{7} -1.09679 q^{11} +2.42864 q^{13} -1.00000 q^{17} +5.80642 q^{19} -8.57628 q^{23} -4.61285 q^{25} +3.37778 q^{29} -3.33185 q^{31} -0.949145 q^{35} -3.37778 q^{37} +6.85728 q^{41} -7.05086 q^{43} +1.24443 q^{47} -4.67307 q^{49} +10.8573 q^{53} -0.682439 q^{55} +4.56199 q^{59} -14.9906 q^{61} +1.51114 q^{65} -11.6128 q^{67} -12.2810 q^{71} +13.6128 q^{73} +1.67307 q^{77} +12.1891 q^{79} -7.05086 q^{83} -0.622216 q^{85} -7.67307 q^{89} -3.70471 q^{91} +3.61285 q^{95} +5.61285 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{5} + 2 q^{7} - 10 q^{11} - 6 q^{13} - 3 q^{17} + 4 q^{19} - 6 q^{23} + 13 q^{25} + 10 q^{29} + 10 q^{31} - 16 q^{35} - 10 q^{37} - 6 q^{41} - 8 q^{43} + 4 q^{47} - q^{49} + 6 q^{53} - 16 q^{55}+ \cdots - 10 q^{97}+O(q^{100}) \) 3 * q + 2 * q^5 + 2 * q^7 - 10 * q^11 - 6 * q^13 - 3 * q^17 + 4 * q^19 - 6 * q^23 + 13 * q^25 + 10 * q^29 + 10 * q^31 - 16 * q^35 - 10 * q^37 - 6 * q^41 - 8 * q^43 + 4 * q^47 - q^49 + 6 * q^53 - 16 * q^55 - 18 * q^61 + 4 * q^65 - 8 * q^67 - 30 * q^71 + 14 * q^73 - 8 * q^77 - 10 * q^79 - 8 * q^83 - 2 * q^85 - 10 * q^89 - 24 * q^91 - 16 * q^95 - 10 * q^97

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