LMFDB - Embedded newform 3024.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(1,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, 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("3024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.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:	$24.1467615712$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1512)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	3024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.37228 q^{5} -1.00000 q^{7} -1.37228 q^{13} -1.00000 q^{17} -6.74456 q^{19} -5.37228 q^{23} +0.627719 q^{25} -0.627719 q^{29} -2.62772 q^{31} -2.37228 q^{35} +4.37228 q^{37} -5.11684 q^{41} -7.74456 q^{43} +0.372281 q^{47} +1.00000 q^{49} -12.1168 q^{53} -0.255437 q^{59} +1.25544 q^{61} -3.25544 q^{65} -4.62772 q^{67} -7.37228 q^{71} +10.0000 q^{73} +9.11684 q^{79} +15.8614 q^{83} -2.37228 q^{85} -2.62772 q^{89} +1.37228 q^{91} -16.0000 q^{95} +9.48913 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} - 2 q^{7} + 3 q^{13} - 2 q^{17} - 2 q^{19} - 5 q^{23} + 7 q^{25} - 7 q^{29} - 11 q^{31} + q^{35} + 3 q^{37} + 7 q^{41} - 4 q^{43} - 5 q^{47} + 2 q^{49} - 7 q^{53} - 12 q^{59} + 14 q^{61} - 18 q^{65}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q - q^5 - 2 * q^7 + 3 * q^13 - 2 * q^17 - 2 * q^19 - 5 * q^23 + 7 * q^25 - 7 * q^29 - 11 * q^31 + q^35 + 3 * q^37 + 7 * q^41 - 4 * q^43 - 5 * q^47 + 2 * q^49 - 7 * q^53 - 12 * q^59 + 14 * q^61 - 18 * q^65 - 15 * q^67 - 9 * q^71 + 20 * q^73 + q^79 + 3 * q^83 + q^85 - 11 * q^89 - 3 * q^91 - 32 * q^95 - 4 * 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
$5$	2.37228	1.06092	0.530458	−	0.847711i	$-0.322020\pi$
$5$	2.37228	1.06092	0.530458	+	0.847711i	$0.322020\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−1.37228	−0.380602	−0.190301	−	0.981726i	$-0.560946\pi$
$13$	−1.37228	−0.380602	−0.190301	+	0.981726i	$0.560946\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	−6.74456	−1.54731	−0.773654	−	0.633608i	$-0.781573\pi$
$19$	−6.74456	−1.54731	−0.773654	+	0.633608i	$0.781573\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.37228	−1.12020	−0.560099	−	0.828426i	$-0.689237\pi$
$23$	−5.37228	−1.12020	−0.560099	+	0.828426i	$0.689237\pi$
$24$	0	0
$25$	0.627719	0.125544
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.627719	−0.116564	−0.0582822	−	0.998300i	$-0.518562\pi$
$29$	−0.627719	−0.116564	−0.0582822	+	0.998300i	$0.518562\pi$
$30$	0	0
$31$	−2.62772	−0.471952	−0.235976	−	0.971759i	$-0.575829\pi$
$31$	−2.62772	−0.471952	−0.235976	+	0.971759i	$0.575829\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.37228	−0.400989
$36$	0	0
$37$	4.37228	0.718799	0.359399	−	0.933184i	$-0.382982\pi$
$37$	4.37228	0.718799	0.359399	+	0.933184i	$0.382982\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.11684	−0.799117	−0.399558	−	0.916708i	$-0.630837\pi$
$41$	−5.11684	−0.799117	−0.399558	+	0.916708i	$0.630837\pi$
$42$	0	0
$43$	−7.74456	−1.18103	−0.590517	−	0.807025i	$-0.701076\pi$
$43$	−7.74456	−1.18103	−0.590517	+	0.807025i	$0.701076\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.372281	0.0543028	0.0271514	−	0.999631i	$-0.491356\pi$
$47$	0.372281	0.0543028	0.0271514	+	0.999631i	$0.491356\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.1168	−1.66438	−0.832188	−	0.554493i	$-0.812912\pi$
$53$	−12.1168	−1.66438	−0.832188	+	0.554493i	$0.812912\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.255437	−0.0332551	−0.0166276	−	0.999862i	$-0.505293\pi$
$59$	−0.255437	−0.0332551	−0.0166276	+	0.999862i	$0.505293\pi$
$60$	0	0
$61$	1.25544	0.160742	0.0803711	−	0.996765i	$-0.474389\pi$
$61$	1.25544	0.160742	0.0803711	+	0.996765i	$0.474389\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.25544	−0.403787
$66$	0	0
$67$	−4.62772	−0.565366	−0.282683	−	0.959213i	$-0.591224\pi$
$67$	−4.62772	−0.565366	−0.282683	+	0.959213i	$0.591224\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.37228	−0.874929	−0.437464	−	0.899236i	$-0.644123\pi$
$71$	−7.37228	−0.874929	−0.437464	+	0.899236i	$0.644123\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	9.11684	1.02573	0.512863	−	0.858471i	$-0.328585\pi$
$79$	9.11684	1.02573	0.512863	+	0.858471i	$0.328585\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.8614	1.74102	0.870508	−	0.492155i	$-0.163791\pi$
$83$	15.8614	1.74102	0.870508	+	0.492155i	$0.163791\pi$
$84$	0	0
$85$	−2.37228	−0.257310
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.62772	−0.278538	−0.139269	−	0.990255i	$-0.544475\pi$
$89$	−2.62772	−0.278538	−0.139269	+	0.990255i	$0.544475\pi$
$90$	0	0
$91$	1.37228	0.143854
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−16.0000	−1.64157
$96$	0	0
$97$	9.48913	0.963475	0.481737	−	0.876316i	$-0.340006\pi$
$97$	9.48913	0.963475	0.481737	+	0.876316i	$0.340006\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−13.3723	−1.31761	−0.658805	−	0.752314i	$-0.728938\pi$
$103$	−13.3723	−1.31761	−0.658805	+	0.752314i	$0.728938\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.74456	−0.845369	−0.422684	−	0.906277i	$-0.638912\pi$
$107$	−8.74456	−0.845369	−0.422684	+	0.906277i	$0.638912\pi$
$108$	0	0
$109$	19.8614	1.90238	0.951189	−	0.308609i	$-0.0998636\pi$
$109$	19.8614	1.90238	0.951189	+	0.308609i	$0.0998636\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.74456	0.822619	0.411310	−	0.911496i	$-0.365071\pi$
$113$	8.74456	0.822619	0.411310	+	0.911496i	$0.365071\pi$
$114$	0	0
$115$	−12.7446	−1.18844
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.3723	−0.927725
$126$	0	0
$127$	11.1168	0.986460	0.493230	−	0.869899i	$-0.335816\pi$
$127$	11.1168	0.986460	0.493230	+	0.869899i	$0.335816\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−14.1168	−1.23339	−0.616697	−	0.787200i	$-0.711530\pi$
$131$	−14.1168	−1.23339	−0.616697	+	0.787200i	$0.711530\pi$
$132$	0	0
$133$	6.74456	0.584828
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	−2.74456	−0.232791	−0.116395	−	0.993203i	$-0.537134\pi$
$139$	−2.74456	−0.232791	−0.116395	+	0.993203i	$0.537134\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.48913	−0.123665
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.37228	0.276268	0.138134	−	0.990414i	$-0.455890\pi$
$149$	3.37228	0.276268	0.138134	+	0.990414i	$0.455890\pi$
$150$	0	0
$151$	5.86141	0.476995	0.238497	−	0.971143i	$-0.423345\pi$
$151$	5.86141	0.476995	0.238497	+	0.971143i	$0.423345\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.23369	−0.500702
$156$	0	0
$157$	10.1168	0.807412	0.403706	−	0.914889i	$-0.367722\pi$
$157$	10.1168	0.807412	0.403706	+	0.914889i	$0.367722\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.37228	0.423395
$162$	0	0
$163$	17.0000	1.33154	0.665771	−	0.746156i	$-0.268103\pi$
$163$	17.0000	1.33154	0.665771	+	0.746156i	$0.268103\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.3723	−1.73122	−0.865610	−	0.500720i	$-0.833069\pi$
$167$	−22.3723	−1.73122	−0.865610	+	0.500720i	$0.833069\pi$
$168$	0	0
$169$	−11.1168	−0.855142
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−11.2554	−0.855735	−0.427868	−	0.903841i	$-0.640735\pi$
$173$	−11.2554	−0.855735	−0.427868	+	0.903841i	$0.640735\pi$
$174$	0	0
$175$	−0.627719	−0.0474511
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.74456	0.354625	0.177313	−	0.984155i	$-0.443260\pi$
$179$	4.74456	0.354625	0.177313	+	0.984155i	$0.443260\pi$
$180$	0	0
$181$	−19.6060	−1.45730	−0.728650	−	0.684886i	$-0.759852\pi$
$181$	−19.6060	−1.45730	−0.728650	+	0.684886i	$0.759852\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.3723	0.762585
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.744563	0.0538747	0.0269373	−	0.999637i	$-0.491425\pi$
$191$	0.744563	0.0538747	0.0269373	+	0.999637i	$0.491425\pi$
$192$	0	0
$193$	11.2337	0.808619	0.404309	−	0.914622i	$-0.367512\pi$
$193$	11.2337	0.808619	0.404309	+	0.914622i	$0.367512\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.74456	−0.195542	−0.0977710	−	0.995209i	$-0.531171\pi$
$197$	−2.74456	−0.195542	−0.0977710	+	0.995209i	$0.531171\pi$
$198$	0	0
$199$	16.8614	1.19527	0.597637	−	0.801767i	$-0.296107\pi$
$199$	16.8614	1.19527	0.597637	+	0.801767i	$0.296107\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.627719	0.0440572
$204$	0	0
$205$	−12.1386	−0.847796
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−6.11684	−0.421101	−0.210550	−	0.977583i	$-0.567526\pi$
$211$	−6.11684	−0.421101	−0.210550	+	0.977583i	$0.567526\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−18.3723	−1.25298
$216$	0	0
$217$	2.62772	0.178381
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.37228	0.0923096
$222$	0	0
$223$	3.25544	0.218000	0.109000	−	0.994042i	$-0.465235\pi$
$223$	3.25544	0.218000	0.109000	+	0.994042i	$0.465235\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.6277	0.838131	0.419066	−	0.907956i	$-0.362358\pi$
$227$	12.6277	0.838131	0.419066	+	0.907956i	$0.362358\pi$
$228$	0	0
$229$	−6.74456	−0.445693	−0.222847	−	0.974854i	$-0.571535\pi$
$229$	−6.74456	−0.445693	−0.222847	+	0.974854i	$0.571535\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.744563	0.0487779	0.0243890	−	0.999703i	$-0.492236\pi$
$233$	0.744563	0.0487779	0.0243890	+	0.999703i	$0.492236\pi$
$234$	0	0
$235$	0.883156	0.0576107
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.2337	0.661962	0.330981	−	0.943637i	$-0.392620\pi$
$239$	10.2337	0.661962	0.330981	+	0.943637i	$0.392620\pi$
$240$	0	0
$241$	15.4891	0.997742	0.498871	−	0.866676i	$-0.333748\pi$
$241$	15.4891	0.997742	0.498871	+	0.866676i	$0.333748\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.37228	0.151559
$246$	0	0
$247$	9.25544	0.588909
$248$	0	0
$249$	0	0
$250$	0	0
$251$	26.3723	1.66460	0.832302	−	0.554323i	$-0.187023\pi$
$251$	26.3723	1.66460	0.832302	+	0.554323i	$0.187023\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.9783	−1.05907	−0.529537	−	0.848287i	$-0.677634\pi$
$257$	−16.9783	−1.05907	−0.529537	+	0.848287i	$0.677634\pi$
$258$	0	0
$259$	−4.37228	−0.271680
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.6277	−0.655333	−0.327667	−	0.944793i	$-0.606262\pi$
$263$	−10.6277	−0.655333	−0.327667	+	0.944793i	$0.606262\pi$
$264$	0	0
$265$	−28.7446	−1.76576
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.11684	−0.433922	−0.216961	−	0.976180i	$-0.569614\pi$
$269$	−7.11684	−0.433922	−0.216961	+	0.976180i	$0.569614\pi$
$270$	0	0
$271$	−17.6060	−1.06949	−0.534743	−	0.845015i	$-0.679592\pi$
$271$	−17.6060	−1.06949	−0.534743	+	0.845015i	$0.679592\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−29.3505	−1.76350	−0.881751	−	0.471715i	$-0.843635\pi$
$277$	−29.3505	−1.76350	−0.881751	+	0.471715i	$0.843635\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−28.9783	−1.72870	−0.864349	−	0.502893i	$-0.832269\pi$
$281$	−28.9783	−1.72870	−0.864349	+	0.502893i	$0.832269\pi$
$282$	0	0
$283$	−2.51087	−0.149256	−0.0746280	−	0.997211i	$-0.523777\pi$
$283$	−2.51087	−0.149256	−0.0746280	+	0.997211i	$0.523777\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.11684	0.302038
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0	0
$292$	0	0
$293$	28.3723	1.65753	0.828763	−	0.559599i	$-0.189045\pi$
$293$	28.3723	1.65753	0.828763	+	0.559599i	$0.189045\pi$
$294$	0	0
$295$	−0.605969	−0.0352809
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.37228	0.426350
$300$	0	0
$301$	7.74456	0.446389
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.97825	0.170534
$306$	0	0
$307$	31.4891	1.79718	0.898590	−	0.438790i	$-0.144593\pi$
$307$	31.4891	1.79718	0.898590	+	0.438790i	$0.144593\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−10.3723	−0.588158	−0.294079	−	0.955781i	$-0.595013\pi$
$311$	−10.3723	−0.588158	−0.294079	+	0.955781i	$0.595013\pi$
$312$	0	0
$313$	−0.233688	−0.0132088	−0.00660441	−	0.999978i	$-0.502102\pi$
$313$	−0.233688	−0.0132088	−0.00660441	+	0.999978i	$0.502102\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.4891	−0.645294	−0.322647	−	0.946519i	$-0.604573\pi$
$317$	−11.4891	−0.645294	−0.322647	+	0.946519i	$0.604573\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.74456	0.375278
$324$	0	0
$325$	−0.861407	−0.0477822
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.372281	−0.0205245
$330$	0	0
$331$	24.4891	1.34604	0.673022	−	0.739622i	$-0.264996\pi$
$331$	24.4891	1.34604	0.673022	+	0.739622i	$0.264996\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.9783	−0.599806
$336$	0	0
$337$	−30.4891	−1.66085	−0.830424	−	0.557132i	$-0.811902\pi$
$337$	−30.4891	−1.66085	−0.830424	+	0.557132i	$0.811902\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−0.510875	−0.0274252	−0.0137126	−	0.999906i	$-0.504365\pi$
$347$	−0.510875	−0.0274252	−0.0137126	+	0.999906i	$0.504365\pi$
$348$	0	0
$349$	−23.3723	−1.25109	−0.625545	−	0.780188i	$-0.715123\pi$
$349$	−23.3723	−1.25109	−0.625545	+	0.780188i	$0.715123\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.2554	0.652291	0.326146	−	0.945320i	$-0.394250\pi$
$353$	12.2554	0.652291	0.326146	+	0.945320i	$0.394250\pi$
$354$	0	0
$355$	−17.4891	−0.928226
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.13859	−0.271205	−0.135602	−	0.990763i	$-0.543297\pi$
$359$	−5.13859	−0.271205	−0.135602	+	0.990763i	$0.543297\pi$
$360$	0	0
$361$	26.4891	1.39416
$362$	0	0
$363$	0	0
$364$	0	0
$365$	23.7228	1.24171
$366$	0	0
$367$	−29.6060	−1.54542	−0.772710	−	0.634760i	$-0.781099\pi$
$367$	−29.6060	−1.54542	−0.772710	+	0.634760i	$0.781099\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	12.1168	0.629075
$372$	0	0
$373$	−20.3723	−1.05484	−0.527418	−	0.849606i	$-0.676840\pi$
$373$	−20.3723	−1.05484	−0.527418	+	0.849606i	$0.676840\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.861407	0.0443647
$378$	0	0
$379$	13.6277	0.700009	0.350004	−	0.936748i	$-0.386180\pi$
$379$	13.6277	0.700009	0.350004	+	0.936748i	$0.386180\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10.3723	−0.529999	−0.264999	−	0.964249i	$-0.585372\pi$
$383$	−10.3723	−0.529999	−0.264999	+	0.964249i	$0.585372\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−18.7446	−0.950387	−0.475194	−	0.879881i	$-0.657622\pi$
$389$	−18.7446	−0.950387	−0.475194	+	0.879881i	$0.657622\pi$
$390$	0	0
$391$	5.37228	0.271688
$392$	0	0
$393$	0	0
$394$	0	0
$395$	21.6277	1.08821
$396$	0	0
$397$	20.9783	1.05287	0.526434	−	0.850216i	$-0.323529\pi$
$397$	20.9783	1.05287	0.526434	+	0.850216i	$0.323529\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−32.2337	−1.60967	−0.804837	−	0.593496i	$-0.797747\pi$
$401$	−32.2337	−1.60967	−0.804837	+	0.593496i	$0.797747\pi$
$402$	0	0
$403$	3.60597	0.179626
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−14.7446	−0.729072	−0.364536	−	0.931189i	$-0.618772\pi$
$409$	−14.7446	−0.729072	−0.364536	+	0.931189i	$0.618772\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.255437	0.0125693
$414$	0	0
$415$	37.6277	1.84707
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−11.9783	−0.585176	−0.292588	−	0.956239i	$-0.594516\pi$
$419$	−11.9783	−0.585176	−0.292588	+	0.956239i	$0.594516\pi$
$420$	0	0
$421$	8.97825	0.437573	0.218787	−	0.975773i	$-0.429790\pi$
$421$	8.97825	0.437573	0.218787	+	0.975773i	$0.429790\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.627719	−0.0304488
$426$	0	0
$427$	−1.25544	−0.0607549
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−34.9783	−1.68484	−0.842422	−	0.538819i	$-0.818871\pi$
$431$	−34.9783	−1.68484	−0.842422	+	0.538819i	$0.818871\pi$
$432$	0	0
$433$	23.7228	1.14005	0.570023	−	0.821629i	$-0.306934\pi$
$433$	23.7228	1.14005	0.570023	+	0.821629i	$0.306934\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	36.2337	1.73329
$438$	0	0
$439$	7.37228	0.351860	0.175930	−	0.984403i	$-0.443707\pi$
$439$	7.37228	0.351860	0.175930	+	0.984403i	$0.443707\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	17.2554	0.819831	0.409915	−	0.912124i	$-0.365558\pi$
$443$	17.2554	0.819831	0.409915	+	0.912124i	$0.365558\pi$
$444$	0	0
$445$	−6.23369	−0.295505
$446$	0	0
$447$	0	0
$448$	0	0
$449$	28.7446	1.35654	0.678270	−	0.734813i	$-0.262730\pi$
$449$	28.7446	1.35654	0.678270	+	0.734813i	$0.262730\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.25544	0.152617
$456$	0	0
$457$	−29.6060	−1.38491	−0.692454	−	0.721462i	$-0.743471\pi$
$457$	−29.6060	−1.38491	−0.692454	+	0.721462i	$0.743471\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	39.3505	1.83274	0.916368	−	0.400336i	$-0.131107\pi$
$461$	39.3505	1.83274	0.916368	+	0.400336i	$0.131107\pi$
$462$	0	0
$463$	1.86141	0.0865069	0.0432535	−	0.999064i	$-0.486228\pi$
$463$	1.86141	0.0865069	0.0432535	+	0.999064i	$0.486228\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.4891	0.809300	0.404650	−	0.914472i	$-0.367393\pi$
$467$	17.4891	0.809300	0.404650	+	0.914472i	$0.367393\pi$
$468$	0	0
$469$	4.62772	0.213688
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.23369	−0.194255
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−20.6060	−0.941511	−0.470755	−	0.882264i	$-0.656019\pi$
$479$	−20.6060	−0.941511	−0.470755	+	0.882264i	$0.656019\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	22.5109	1.02217
$486$	0	0
$487$	−12.2337	−0.554361	−0.277181	−	0.960818i	$-0.589400\pi$
$487$	−12.2337	−0.554361	−0.277181	+	0.960818i	$0.589400\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	43.7228	1.97318	0.986591	−	0.163209i	$-0.0521846\pi$
$491$	43.7228	1.97318	0.986591	+	0.163209i	$0.0521846\pi$
$492$	0	0
$493$	0.627719	0.0282710
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.37228	0.330692
$498$	0	0
$499$	27.1168	1.21392	0.606958	−	0.794734i	$-0.292389\pi$
$499$	27.1168	1.21392	0.606958	+	0.794734i	$0.292389\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	35.1168	1.56578	0.782891	−	0.622158i	$-0.213744\pi$
$503$	35.1168	1.56578	0.782891	+	0.622158i	$0.213744\pi$
$504$	0	0
$505$	−14.2337	−0.633391
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−28.3723	−1.25758	−0.628790	−	0.777575i	$-0.716449\pi$
$509$	−28.3723	−1.25758	−0.628790	+	0.777575i	$0.716449\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−31.7228	−1.39787
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.48913	0.371915	0.185958	−	0.982558i	$-0.440461\pi$
$521$	8.48913	0.371915	0.185958	+	0.982558i	$0.440461\pi$
$522$	0	0
$523$	−5.76631	−0.252143	−0.126072	−	0.992021i	$-0.540237\pi$
$523$	−5.76631	−0.252143	−0.126072	+	0.992021i	$0.540237\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.62772	0.114465
$528$	0	0
$529$	5.86141	0.254844
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.02175	0.304146
$534$	0	0
$535$	−20.7446	−0.896866
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	16.6060	0.713946	0.356973	−	0.934115i	$-0.383809\pi$
$541$	16.6060	0.713946	0.356973	+	0.934115i	$0.383809\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	47.1168	2.01826
$546$	0	0
$547$	34.8397	1.48964	0.744818	−	0.667268i	$-0.232536\pi$
$547$	34.8397	1.48964	0.744818	+	0.667268i	$0.232536\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.23369	0.180361
$552$	0	0
$553$	−9.11684	−0.387688
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−32.6277	−1.38248	−0.691240	−	0.722625i	$-0.742935\pi$
$557$	−32.6277	−1.38248	−0.691240	+	0.722625i	$0.742935\pi$
$558$	0	0
$559$	10.6277	0.449505
$560$	0	0
$561$	0	0
$562$	0	0
$563$	32.6277	1.37509	0.687547	−	0.726140i	$-0.258687\pi$
$563$	32.6277	1.37509	0.687547	+	0.726140i	$0.258687\pi$
$564$	0	0
$565$	20.7446	0.872730
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.25544	−0.136475	−0.0682375	−	0.997669i	$-0.521738\pi$
$569$	−3.25544	−0.136475	−0.0682375	+	0.997669i	$0.521738\pi$
$570$	0	0
$571$	−17.0000	−0.711428	−0.355714	−	0.934595i	$-0.615762\pi$
$571$	−17.0000	−0.711428	−0.355714	+	0.934595i	$0.615762\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.37228	−0.140634
$576$	0	0
$577$	44.2337	1.84147	0.920736	−	0.390186i	$-0.127589\pi$
$577$	44.2337	1.84147	0.920736	+	0.390186i	$0.127589\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−15.8614	−0.658042
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.1168	0.582665	0.291332	−	0.956622i	$-0.405901\pi$
$587$	14.1168	0.582665	0.291332	+	0.956622i	$0.405901\pi$
$588$	0	0
$589$	17.7228	0.730256
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.6277	1.29879	0.649397	−	0.760449i	$-0.275021\pi$
$593$	31.6277	1.29879	0.649397	+	0.760449i	$0.275021\pi$
$594$	0	0
$595$	2.37228	0.0972541
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−27.3723	−1.11840	−0.559201	−	0.829032i	$-0.688892\pi$
$599$	−27.3723	−1.11840	−0.559201	+	0.829032i	$0.688892\pi$
$600$	0	0
$601$	30.2337	1.23326	0.616629	−	0.787254i	$-0.288498\pi$
$601$	30.2337	1.23326	0.616629	+	0.787254i	$0.288498\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−26.0951	−1.06092
$606$	0	0
$607$	−9.13859	−0.370924	−0.185462	−	0.982651i	$-0.559378\pi$
$607$	−9.13859	−0.370924	−0.185462	+	0.982651i	$0.559378\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.510875	−0.0206678
$612$	0	0
$613$	22.0000	0.888572	0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000	0.888572	0.444286	+	0.895885i	$0.353457\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	8.23369	0.330940	0.165470	−	0.986215i	$-0.447086\pi$
$619$	8.23369	0.330940	0.165470	+	0.986215i	$0.447086\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.62772	0.105277
$624$	0	0
$625$	−27.7446	−1.10978
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.37228	−0.174334
$630$	0	0
$631$	−11.8614	−0.472195	−0.236098	−	0.971729i	$-0.575868\pi$
$631$	−11.8614	−0.472195	−0.236098	+	0.971729i	$0.575868\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	26.3723	1.04655
$636$	0	0
$637$	−1.37228	−0.0543718
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.51087	0.0991736	0.0495868	−	0.998770i	$-0.484210\pi$
$641$	2.51087	0.0991736	0.0495868	+	0.998770i	$0.484210\pi$
$642$	0	0
$643$	−1.76631	−0.0696565	−0.0348283	−	0.999393i	$-0.511088\pi$
$643$	−1.76631	−0.0696565	−0.0348283	+	0.999393i	$0.511088\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.7446	0.579669	0.289834	−	0.957077i	$-0.406400\pi$
$647$	14.7446	0.579669	0.289834	+	0.957077i	$0.406400\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−17.6060	−0.688975	−0.344487	−	0.938791i	$-0.611947\pi$
$653$	−17.6060	−0.688975	−0.344487	+	0.938791i	$0.611947\pi$
$654$	0	0
$655$	−33.4891	−1.30853
$656$	0	0
$657$	0	0
$658$	0	0
$659$	37.4891	1.46037	0.730184	−	0.683250i	$-0.239434\pi$
$659$	37.4891	1.46037	0.730184	+	0.683250i	$0.239434\pi$
$660$	0	0
$661$	20.5109	0.797781	0.398890	−	0.916999i	$-0.369395\pi$
$661$	20.5109	0.797781	0.398890	+	0.916999i	$0.369395\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16.0000	0.620453
$666$	0	0
$667$	3.37228	0.130575
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	41.3723	1.59478	0.797392	−	0.603462i	$-0.206212\pi$
$673$	41.3723	1.59478	0.797392	+	0.603462i	$0.206212\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	26.2337	1.00824	0.504121	−	0.863633i	$-0.331816\pi$
$677$	26.2337	1.00824	0.504121	+	0.863633i	$0.331816\pi$
$678$	0	0
$679$	−9.48913	−0.364159
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26.2337	1.00380	0.501902	−	0.864924i	$-0.332634\pi$
$683$	26.2337	1.00380	0.501902	+	0.864924i	$0.332634\pi$
$684$	0	0
$685$	−33.2119	−1.26896
$686$	0	0
$687$	0	0
$688$	0	0
$689$	16.6277	0.633466
$690$	0	0
$691$	0.744563	0.0283245	0.0141622	−	0.999900i	$-0.495492\pi$
$691$	0.744563	0.0283245	0.0141622	+	0.999900i	$0.495492\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.51087	−0.246972
$696$	0	0
$697$	5.11684	0.193814
$698$	0	0
$699$	0	0
$700$	0	0
$701$	20.5109	0.774685	0.387343	−	0.921936i	$-0.373393\pi$
$701$	20.5109	0.774685	0.387343	+	0.921936i	$0.373393\pi$
$702$	0	0
$703$	−29.4891	−1.11220
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	−33.6277	−1.26292	−0.631458	−	0.775410i	$-0.717543\pi$
$709$	−33.6277	−1.26292	−0.631458	+	0.775410i	$0.717543\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	14.1168	0.528680
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−16.8832	−0.629635	−0.314818	−	0.949152i	$-0.601943\pi$
$719$	−16.8832	−0.629635	−0.314818	+	0.949152i	$0.601943\pi$
$720$	0	0
$721$	13.3723	0.498010
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.394031	−0.0146339
$726$	0	0
$727$	14.1168	0.523565	0.261782	−	0.965127i	$-0.415690\pi$
$727$	14.1168	0.523565	0.261782	+	0.965127i	$0.415690\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7.74456	0.286443
$732$	0	0
$733$	43.6060	1.61062	0.805312	−	0.592852i	$-0.201998\pi$
$733$	43.6060	1.61062	0.805312	+	0.592852i	$0.201998\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	33.8832	1.24305	0.621526	−	0.783393i	$-0.286513\pi$
$743$	33.8832	1.24305	0.621526	+	0.783393i	$0.286513\pi$
$744$	0	0
$745$	8.00000	0.293097
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8.74456	0.319519
$750$	0	0
$751$	−28.2337	−1.03026	−0.515131	−	0.857111i	$-0.672257\pi$
$751$	−28.2337	−1.03026	−0.515131	+	0.857111i	$0.672257\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	13.9049	0.506051
$756$	0	0
$757$	−15.1168	−0.549431	−0.274716	−	0.961526i	$-0.588584\pi$
$757$	−15.1168	−0.549431	−0.274716	+	0.961526i	$0.588584\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−9.51087	−0.344769	−0.172384	−	0.985030i	$-0.555147\pi$
$761$	−9.51087	−0.344769	−0.172384	+	0.985030i	$0.555147\pi$
$762$	0	0
$763$	−19.8614	−0.719031
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.350532	0.0126570
$768$	0	0
$769$	2.97825	0.107398	0.0536992	−	0.998557i	$-0.482899\pi$
$769$	2.97825	0.107398	0.0536992	+	0.998557i	$0.482899\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32.6060	1.17276	0.586378	−	0.810038i	$-0.300553\pi$
$773$	32.6060	1.17276	0.586378	+	0.810038i	$0.300553\pi$
$774$	0	0
$775$	−1.64947	−0.0592506
$776$	0	0
$777$	0	0
$778$	0	0
$779$	34.5109	1.23648
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	24.0000	0.856597
$786$	0	0
$787$	−32.9783	−1.17555	−0.587774	−	0.809025i	$-0.699996\pi$
$787$	−32.9783	−1.17555	−0.587774	+	0.809025i	$0.699996\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8.74456	−0.310921
$792$	0	0
$793$	−1.72281	−0.0611789
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.02175	0.248723	0.124362	−	0.992237i	$-0.460312\pi$
$797$	7.02175	0.248723	0.124362	+	0.992237i	$0.460312\pi$
$798$	0	0
$799$	−0.372281	−0.0131704
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	12.7446	0.449187
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−49.2119	−1.73020	−0.865100	−	0.501600i	$-0.832745\pi$
$809$	−49.2119	−1.73020	−0.865100	+	0.501600i	$0.832745\pi$
$810$	0	0
$811$	−0.510875	−0.0179392	−0.00896962	−	0.999960i	$-0.502855\pi$
$811$	−0.510875	−0.0179392	−0.00896962	+	0.999960i	$0.502855\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	40.3288	1.41266
$816$	0	0
$817$	52.2337	1.82743
$818$	0	0
$819$	0	0
$820$	0	0
$821$	50.5842	1.76540	0.882701	−	0.469936i	$-0.155723\pi$
$821$	50.5842	1.76540	0.882701	+	0.469936i	$0.155723\pi$
$822$	0	0
$823$	−26.6060	−0.927426	−0.463713	−	0.885986i	$-0.653483\pi$
$823$	−26.6060	−0.927426	−0.463713	+	0.885986i	$0.653483\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34.4674	−1.19855	−0.599274	−	0.800544i	$-0.704544\pi$
$827$	−34.4674	−1.19855	−0.599274	+	0.800544i	$0.704544\pi$
$828$	0	0
$829$	−12.2337	−0.424894	−0.212447	−	0.977173i	$-0.568143\pi$
$829$	−12.2337	−0.424894	−0.212447	+	0.977173i	$0.568143\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−53.0733	−1.83668
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.35053	0.115673	0.0578366	−	0.998326i	$-0.481580\pi$
$839$	3.35053	0.115673	0.0578366	+	0.998326i	$0.481580\pi$
$840$	0	0
$841$	−28.6060	−0.986413
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−26.3723	−0.907234
$846$	0	0
$847$	11.0000	0.377964
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−23.4891	−0.805197
$852$	0	0
$853$	−31.8397	−1.09017	−0.545085	−	0.838381i	$-0.683502\pi$
$853$	−31.8397	−1.09017	−0.545085	+	0.838381i	$0.683502\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	55.9783	1.91218	0.956090	−	0.293072i	$-0.0946776\pi$
$857$	55.9783	1.91218	0.956090	+	0.293072i	$0.0946776\pi$
$858$	0	0
$859$	12.9783	0.442812	0.221406	−	0.975182i	$-0.428935\pi$
$859$	12.9783	0.442812	0.221406	+	0.975182i	$0.428935\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−55.0951	−1.87546	−0.937729	−	0.347367i	$-0.887076\pi$
$863$	−55.0951	−1.87546	−0.937729	+	0.347367i	$0.887076\pi$
$864$	0	0
$865$	−26.7011	−0.907863
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	6.35053	0.215180
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10.3723	0.350647
$876$	0	0
$877$	32.6060	1.10103	0.550513	−	0.834827i	$-0.314432\pi$
$877$	32.6060	1.10103	0.550513	+	0.834827i	$0.314432\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−34.8614	−1.17451	−0.587255	−	0.809402i	$-0.699792\pi$
$881$	−34.8614	−1.17451	−0.587255	+	0.809402i	$0.699792\pi$
$882$	0	0
$883$	−33.7446	−1.13559	−0.567797	−	0.823168i	$-0.692204\pi$
$883$	−33.7446	−1.13559	−0.567797	+	0.823168i	$0.692204\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−26.8397	−0.901187	−0.450594	−	0.892729i	$-0.648788\pi$
$887$	−26.8397	−0.901187	−0.450594	+	0.892729i	$0.648788\pi$
$888$	0	0
$889$	−11.1168	−0.372847
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.51087	−0.0840232
$894$	0	0
$895$	11.2554	0.376228
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.64947	0.0550128
$900$	0	0
$901$	12.1168	0.403671
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−46.5109	−1.54607
$906$	0	0
$907$	−41.3505	−1.37302	−0.686511	−	0.727119i	$-0.740859\pi$
$907$	−41.3505	−1.37302	−0.686511	+	0.727119i	$0.740859\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−53.9565	−1.78766	−0.893829	−	0.448407i	$-0.851991\pi$
$911$	−53.9565	−1.78766	−0.893829	+	0.448407i	$0.851991\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.1168	0.466179
$918$	0	0
$919$	24.3723	0.803967	0.401984	−	0.915647i	$-0.368321\pi$
$919$	24.3723	0.803967	0.401984	+	0.915647i	$0.368321\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10.1168	0.333000
$924$	0	0
$925$	2.74456	0.0902407
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17.8614	0.586014	0.293007	−	0.956110i	$-0.405344\pi$
$929$	17.8614	0.586014	0.293007	+	0.956110i	$0.405344\pi$
$930$	0	0
$931$	−6.74456	−0.221044
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	28.4674	0.929989	0.464994	−	0.885314i	$-0.346056\pi$
$937$	28.4674	0.929989	0.464994	+	0.885314i	$0.346056\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−22.8832	−0.745970	−0.372985	−	0.927837i	$-0.621666\pi$
$941$	−22.8832	−0.745970	−0.372985	+	0.927837i	$0.621666\pi$
$942$	0	0
$943$	27.4891	0.895169
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.72281	0.185966	0.0929832	−	0.995668i	$-0.470360\pi$
$947$	5.72281	0.185966	0.0929832	+	0.995668i	$0.470360\pi$
$948$	0	0
$949$	−13.7228	−0.445461
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16.7446	0.542410	0.271205	−	0.962522i	$-0.412578\pi$
$953$	16.7446	0.542410	0.271205	+	0.962522i	$0.412578\pi$
$954$	0	0
$955$	1.76631	0.0571565
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.0000	0.452084
$960$	0	0
$961$	−24.0951	−0.777261
$962$	0	0
$963$	0	0
$964$	0	0
$965$	26.6495	0.857877
$966$	0	0
$967$	−29.4891	−0.948306	−0.474153	−	0.880442i	$-0.657246\pi$
$967$	−29.4891	−0.948306	−0.474153	+	0.880442i	$0.657246\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−43.7446	−1.40383	−0.701915	−	0.712261i	$-0.747671\pi$
$971$	−43.7446	−1.40383	−0.701915	+	0.712261i	$0.747671\pi$
$972$	0	0
$973$	2.74456	0.0879866
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.7446	0.471720	0.235860	−	0.971787i	$-0.424209\pi$
$977$	14.7446	0.471720	0.235860	+	0.971787i	$0.424209\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	13.6277	0.434657	0.217328	−	0.976099i	$-0.430266\pi$
$983$	13.6277	0.434657	0.217328	+	0.976099i	$0.430266\pi$
$984$	0	0
$985$	−6.51087	−0.207454
$986$	0	0
$987$	0	0
$988$	0	0
$989$	41.6060	1.32299
$990$	0	0
$991$	24.0951	0.765406	0.382703	−	0.923871i	$-0.374993\pi$
$991$	24.0951	0.765406	0.382703	+	0.923871i	$0.374993\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	40.0000	1.26809
$996$	0	0
$997$	−12.6277	−0.399924	−0.199962	−	0.979804i	$-0.564082\pi$
$997$	−12.6277	−0.399924	−0.199962	+	0.979804i	$0.564082\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.a.bf.1.2		2
3.2	odd	2		3024.2.a.bl.1.1		2
4.3	odd	2		1512.2.a.n.1.2	✓	2
12.11	even	2		1512.2.a.q.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.a.n.1.2	✓	2	4.3	odd	2
1512.2.a.q.1.1	yes	2	12.11	even	2
3024.2.a.bf.1.2		2	1.1	even	1	trivial
3024.2.a.bl.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.a (trivial)

\(f(q)\)	\(=\)	\(q+2.37228 q^{5} -1.00000 q^{7} -1.37228 q^{13} -1.00000 q^{17} -6.74456 q^{19} -5.37228 q^{23} +0.627719 q^{25} -0.627719 q^{29} -2.62772 q^{31} -2.37228 q^{35} +4.37228 q^{37} -5.11684 q^{41} -7.74456 q^{43} +0.372281 q^{47} +1.00000 q^{49} -12.1168 q^{53} -0.255437 q^{59} +1.25544 q^{61} -3.25544 q^{65} -4.62772 q^{67} -7.37228 q^{71} +10.0000 q^{73} +9.11684 q^{79} +15.8614 q^{83} -2.37228 q^{85} -2.62772 q^{89} +1.37228 q^{91} -16.0000 q^{95} +9.48913 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} - 2 q^{7} + 3 q^{13} - 2 q^{17} - 2 q^{19} - 5 q^{23} + 7 q^{25} - 7 q^{29} - 11 q^{31} + q^{35} + 3 q^{37} + 7 q^{41} - 4 q^{43} - 5 q^{47} + 2 q^{49} - 7 q^{53} - 12 q^{59} + 14 q^{61} - 18 q^{65}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q - q^5 - 2 * q^7 + 3 * q^13 - 2 * q^17 - 2 * q^19 - 5 * q^23 + 7 * q^25 - 7 * q^29 - 11 * q^31 + q^35 + 3 * q^37 + 7 * q^41 - 4 * q^43 - 5 * q^47 + 2 * q^49 - 7 * q^53 - 12 * q^59 + 14 * q^61 - 18 * q^65 - 15 * q^67 - 9 * q^71 + 20 * q^73 + q^79 + 3 * q^83 + q^85 - 11 * q^89 - 3 * q^91 - 32 * q^95 - 4 * q^97

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