LMFDB - Embedded newform 2448.4.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2448,4,Mod(1,2448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2448.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2448 = 2^{4} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2448.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:	$144.436675694$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19.9706 q^{5} +20.9706 q^{7} +16.0294 q^{11} -34.9411 q^{13} +17.0000 q^{17} +80.8823 q^{19} -115.971 q^{23} +273.823 q^{25} -154.118 q^{29} -299.941 q^{31} -418.794 q^{35} +315.529 q^{37} -132.265 q^{41} +23.1177 q^{43} +260.912 q^{47} +96.7645 q^{49} +676.087 q^{53} -320.117 q^{55} +629.294 q^{59} -461.852 q^{61} +697.794 q^{65} +789.470 q^{67} -686.412 q^{71} +484.912 q^{73} +336.146 q^{77} -254.000 q^{79} +548.912 q^{83} -339.500 q^{85} -925.145 q^{89} -732.735 q^{91} -1615.26 q^{95} +732.617 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5} + 8 q^{7} + 66 q^{11} - 2 q^{13} + 34 q^{17} + 26 q^{19} - 198 q^{23} + 344 q^{25} - 444 q^{29} - 532 q^{31} - 600 q^{35} + 88 q^{37} - 570 q^{41} + 182 q^{43} + 420 q^{47} - 78 q^{49} + 300 q^{53}+ \cdots + 1024 q^{97}+O(q^{100}) $ 2 * q - 6 * q^5 + 8 * q^7 + 66 * q^11 - 2 * q^13 + 34 * q^17 + 26 * q^19 - 198 * q^23 + 344 * q^25 - 444 * q^29 - 532 * q^31 - 600 * q^35 + 88 * q^37 - 570 * q^41 + 182 * q^43 + 420 * q^47 - 78 * q^49 + 300 * q^53 + 378 * q^55 + 444 * q^59 + 400 * q^61 + 1158 * q^65 + 968 * q^67 - 1848 * q^71 + 868 * q^73 - 312 * q^77 - 508 * q^79 + 996 * q^83 - 102 * q^85 + 288 * q^89 - 1160 * q^91 - 2382 * q^95 + 1024 * 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
$5$	−19.9706	−1.78622	−0.893111	−	0.449837i	$-0.851482\pi$
$5$	−19.9706	−1.78622	−0.893111	+	0.449837i	$0.851482\pi$
$6$	0	0
$7$	20.9706	1.13230	0.566152	−	0.824301i	$-0.308432\pi$
$7$	20.9706	1.13230	0.566152	+	0.824301i	$0.308432\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	16.0294	0.439369	0.219684	−	0.975571i	$-0.429497\pi$
$11$	16.0294	0.439369	0.219684	+	0.975571i	$0.429497\pi$
$12$	0	0
$13$	−34.9411	−0.745456	−0.372728	−	0.927941i	$-0.621578\pi$
$13$	−34.9411	−0.745456	−0.372728	+	0.927941i	$0.621578\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	80.8823	0.976614	0.488307	−	0.872672i	$-0.337615\pi$
$19$	80.8823	0.976614	0.488307	+	0.872672i	$0.337615\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−115.971	−1.05137	−0.525686	−	0.850679i	$-0.676191\pi$
$23$	−115.971	−1.05137	−0.525686	+	0.850679i	$0.676191\pi$
$24$	0	0
$25$	273.823	2.19059
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−154.118	−0.986860	−0.493430	−	0.869785i	$-0.664257\pi$
$29$	−154.118	−0.986860	−0.493430	+	0.869785i	$0.664257\pi$
$30$	0	0
$31$	−299.941	−1.73777	−0.868887	−	0.495010i	$-0.835164\pi$
$31$	−299.941	−1.73777	−0.868887	+	0.495010i	$0.835164\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−418.794	−2.02255
$36$	0	0
$37$	315.529	1.40196	0.700982	−	0.713179i	$-0.252745\pi$
$37$	315.529	1.40196	0.700982	+	0.713179i	$0.252745\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−132.265	−0.503813	−0.251906	−	0.967752i	$-0.581057\pi$
$41$	−132.265	−0.503813	−0.251906	+	0.967752i	$0.581057\pi$
$42$	0	0
$43$	23.1177	0.0819866	0.0409933	−	0.999159i	$-0.486948\pi$
$43$	23.1177	0.0819866	0.0409933	+	0.999159i	$0.486948\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	260.912	0.809742	0.404871	−	0.914374i	$-0.367316\pi$
$47$	260.912	0.809742	0.404871	+	0.914374i	$0.367316\pi$
$48$	0	0
$49$	96.7645	0.282112
$50$	0	0
$51$	0	0
$52$	0	0
$53$	676.087	1.75222	0.876111	−	0.482110i	$-0.160129\pi$
$53$	676.087	1.75222	0.876111	+	0.482110i	$0.160129\pi$
$54$	0	0
$55$	−320.117	−0.784810
$56$	0	0
$57$	0	0
$58$	0	0
$59$	629.294	1.38859	0.694297	−	0.719689i	$-0.255715\pi$
$59$	629.294	1.38859	0.694297	+	0.719689i	$0.255715\pi$
$60$	0	0
$61$	−461.852	−0.969411	−0.484706	−	0.874677i	$-0.661073\pi$
$61$	−461.852	−0.969411	−0.484706	+	0.874677i	$0.661073\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	697.794	1.33155
$66$	0	0
$67$	789.470	1.43954	0.719770	−	0.694213i	$-0.244247\pi$
$67$	789.470	1.43954	0.719770	+	0.694213i	$0.244247\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−686.412	−1.14735	−0.573677	−	0.819082i	$-0.694484\pi$
$71$	−686.412	−1.14735	−0.573677	+	0.819082i	$0.694484\pi$
$72$	0	0
$73$	484.912	0.777461	0.388730	−	0.921352i	$-0.372914\pi$
$73$	484.912	0.777461	0.388730	+	0.921352i	$0.372914\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	336.146	0.497499
$78$	0	0
$79$	−254.000	−0.361737	−0.180869	−	0.983507i	$-0.557891\pi$
$79$	−254.000	−0.361737	−0.180869	+	0.983507i	$0.557891\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	548.912	0.725914	0.362957	−	0.931806i	$-0.381767\pi$
$83$	548.912	0.725914	0.362957	+	0.931806i	$0.381767\pi$
$84$	0	0
$85$	−339.500	−0.433222
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−925.145	−1.10186	−0.550928	−	0.834553i	$-0.685726\pi$
$89$	−925.145	−1.10186	−0.550928	+	0.834553i	$0.685726\pi$
$90$	0	0
$91$	−732.735	−0.844082
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1615.26	−1.74445
$96$	0	0
$97$	732.617	0.766866	0.383433	−	0.923569i	$-0.374742\pi$
$97$	732.617	0.766866	0.383433	+	0.923569i	$0.374742\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	128.528	0.126624	0.0633120	−	0.997994i	$-0.479834\pi$
$101$	128.528	0.126624	0.0633120	+	0.997994i	$0.479834\pi$
$102$	0	0
$103$	−5.35325	−0.00512108	−0.00256054	−	0.999997i	$-0.500815\pi$
$103$	−5.35325	−0.00512108	−0.00256054	+	0.999997i	$0.500815\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	473.382	0.427697	0.213848	−	0.976867i	$-0.431400\pi$
$107$	473.382	0.427697	0.213848	+	0.976867i	$0.431400\pi$
$108$	0	0
$109$	−352.353	−0.309627	−0.154813	−	0.987944i	$-0.549478\pi$
$109$	−352.353	−0.309627	−0.154813	+	0.987944i	$0.549478\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	733.617	0.610734	0.305367	−	0.952235i	$-0.401221\pi$
$113$	733.617	0.610734	0.305367	+	0.952235i	$0.401221\pi$
$114$	0	0
$115$	2316.00	1.87798
$116$	0	0
$117$	0	0
$118$	0	0
$119$	356.500	0.274624
$120$	0	0
$121$	−1074.06	−0.806955
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2972.09	−2.12665
$126$	0	0
$127$	−340.410	−0.237846	−0.118923	−	0.992903i	$-0.537944\pi$
$127$	−340.410	−0.237846	−0.118923	+	0.992903i	$0.537944\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2133.85	−1.42317	−0.711586	−	0.702599i	$-0.752023\pi$
$131$	−2133.85	−1.42317	−0.711586	+	0.702599i	$0.752023\pi$
$132$	0	0
$133$	1696.15	1.10582
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−55.1455	−0.0343897	−0.0171949	−	0.999852i	$-0.505474\pi$
$137$	−55.1455	−0.0343897	−0.0171949	+	0.999852i	$0.505474\pi$
$138$	0	0
$139$	−256.266	−0.156375	−0.0781877	−	0.996939i	$-0.524913\pi$
$139$	−256.266	−0.156375	−0.0781877	+	0.996939i	$0.524913\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−560.087	−0.327530
$144$	0	0
$145$	3077.82	1.76275
$146$	0	0
$147$	0	0
$148$	0	0
$149$	84.3836	0.0463958	0.0231979	−	0.999731i	$-0.492615\pi$
$149$	84.3836	0.0463958	0.0231979	+	0.999731i	$0.492615\pi$
$150$	0	0
$151$	−1051.65	−0.566767	−0.283383	−	0.959007i	$-0.591457\pi$
$151$	−1051.65	−0.566767	−0.283383	+	0.959007i	$0.591457\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5989.99	3.10405
$156$	0	0
$157$	1587.47	0.806968	0.403484	−	0.914987i	$-0.367799\pi$
$157$	1587.47	0.806968	0.403484	+	0.914987i	$0.367799\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2431.97	−1.19047
$162$	0	0
$163$	−1749.59	−0.840727	−0.420363	−	0.907356i	$-0.638097\pi$
$163$	−1749.59	−0.840727	−0.420363	+	0.907356i	$0.638097\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1624.21	−0.752604	−0.376302	−	0.926497i	$-0.622804\pi$
$167$	−1624.21	−0.752604	−0.376302	+	0.926497i	$0.622804\pi$
$168$	0	0
$169$	−976.118	−0.444296
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3734.50	1.64120	0.820602	−	0.571500i	$-0.193638\pi$
$173$	3734.50	1.64120	0.820602	+	0.571500i	$0.193638\pi$
$174$	0	0
$175$	5742.23	2.48041
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1434.65	−0.599053	−0.299526	−	0.954088i	$-0.596829\pi$
$179$	−1434.65	−0.599053	−0.299526	+	0.954088i	$0.596829\pi$
$180$	0	0
$181$	−219.263	−0.0900426	−0.0450213	−	0.998986i	$-0.514336\pi$
$181$	−219.263	−0.0900426	−0.0450213	+	0.998986i	$0.514336\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6301.29	−2.50422
$186$	0	0
$187$	272.500	0.106563
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3116.44	−1.18062	−0.590308	−	0.807178i	$-0.700994\pi$
$191$	−3116.44	−1.18062	−0.590308	+	0.807178i	$0.700994\pi$
$192$	0	0
$193$	−3921.82	−1.46269	−0.731344	−	0.682009i	$-0.761106\pi$
$193$	−3921.82	−1.46269	−0.731344	+	0.682009i	$0.761106\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3141.68	−1.13622	−0.568110	−	0.822953i	$-0.692325\pi$
$197$	−3141.68	−1.13622	−0.568110	+	0.822953i	$0.692325\pi$
$198$	0	0
$199$	−1832.79	−0.652880	−0.326440	−	0.945218i	$-0.605849\pi$
$199$	−1832.79	−0.652880	−0.326440	+	0.945218i	$0.605849\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3231.94	−1.11743
$204$	0	0
$205$	2641.41	0.899921
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1296.50	0.429094
$210$	0	0
$211$	−4928.41	−1.60799	−0.803994	−	0.594637i	$-0.797296\pi$
$211$	−4928.41	−1.60799	−0.803994	+	0.594637i	$0.797296\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−461.674	−0.146446
$216$	0	0
$217$	−6289.93	−1.96769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−593.999	−0.180800
$222$	0	0
$223$	75.4727	0.0226638	0.0113319	−	0.999936i	$-0.496393\pi$
$223$	75.4727	0.0226638	0.0113319	+	0.999936i	$0.496393\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1629.97	−0.476587	−0.238293	−	0.971193i	$-0.576588\pi$
$227$	−1629.97	−0.476587	−0.238293	+	0.971193i	$0.576588\pi$
$228$	0	0
$229$	2000.35	0.577235	0.288617	−	0.957445i	$-0.406804\pi$
$229$	2000.35	0.577235	0.288617	+	0.957445i	$0.406804\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4225.26	1.18801	0.594004	−	0.804462i	$-0.297546\pi$
$233$	4225.26	1.18801	0.594004	+	0.804462i	$0.297546\pi$
$234$	0	0
$235$	−5210.55	−1.44638
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1614.97	−0.437086	−0.218543	−	0.975827i	$-0.570130\pi$
$239$	−1614.97	−0.437086	−0.218543	+	0.975827i	$0.570130\pi$
$240$	0	0
$241$	−6761.67	−1.80729	−0.903646	−	0.428280i	$-0.859120\pi$
$241$	−6761.67	−1.80729	−0.903646	+	0.428280i	$0.859120\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1932.44	−0.503915
$246$	0	0
$247$	−2826.12	−0.728022
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2880.29	0.724313	0.362156	−	0.932117i	$-0.382041\pi$
$251$	2880.29	0.724313	0.362156	+	0.932117i	$0.382041\pi$
$252$	0	0
$253$	−1858.94	−0.461940
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1460.44	0.354474	0.177237	−	0.984168i	$-0.443284\pi$
$257$	1460.44	0.354474	0.177237	+	0.984168i	$0.443284\pi$
$258$	0	0
$259$	6616.82	1.58745
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7601.29	−1.78219	−0.891094	−	0.453819i	$-0.850061\pi$
$263$	−7601.29	−1.78219	−0.891094	+	0.453819i	$0.850061\pi$
$264$	0	0
$265$	−13501.8	−3.12986
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6896.26	−1.56309	−0.781547	−	0.623846i	$-0.785569\pi$
$269$	−6896.26	−1.56309	−0.781547	+	0.623846i	$0.785569\pi$
$270$	0	0
$271$	−849.234	−0.190359	−0.0951795	−	0.995460i	$-0.530342\pi$
$271$	−849.234	−0.190359	−0.0951795	+	0.995460i	$0.530342\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4389.23	0.962476
$276$	0	0
$277$	4080.09	0.885013	0.442507	−	0.896765i	$-0.354089\pi$
$277$	4080.09	0.885013	0.442507	+	0.896765i	$0.354089\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4967.90	−1.05466	−0.527331	−	0.849660i	$-0.676807\pi$
$281$	−4967.90	−1.05466	−0.527331	+	0.849660i	$0.676807\pi$
$282$	0	0
$283$	1475.83	0.309996	0.154998	−	0.987915i	$-0.450463\pi$
$283$	1475.83	0.309996	0.154998	+	0.987915i	$0.450463\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2773.67	−0.570469
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−521.056	−0.103892	−0.0519461	−	0.998650i	$-0.516542\pi$
$293$	−521.056	−0.103892	−0.0519461	+	0.998650i	$0.516542\pi$
$294$	0	0
$295$	−12567.3	−2.48034
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4052.14	0.783751
$300$	0	0
$301$	484.792	0.0928337
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9223.44	1.73158
$306$	0	0
$307$	1718.23	0.319429	0.159715	−	0.987163i	$-0.448943\pi$
$307$	1718.23	0.319429	0.159715	+	0.987163i	$0.448943\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4916.05	0.896347	0.448173	−	0.893947i	$-0.352075\pi$
$311$	4916.05	0.896347	0.448173	+	0.893947i	$0.352075\pi$
$312$	0	0
$313$	7375.84	1.33197	0.665986	−	0.745964i	$-0.268011\pi$
$313$	7375.84	1.33197	0.665986	+	0.745964i	$0.268011\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−522.706	−0.0926124	−0.0463062	−	0.998927i	$-0.514745\pi$
$317$	−522.706	−0.0926124	−0.0463062	+	0.998927i	$0.514745\pi$
$318$	0	0
$319$	−2470.42	−0.433596
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1375.00	0.236864
$324$	0	0
$325$	−9567.70	−1.63299
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5471.46	0.916874
$330$	0	0
$331$	−5016.46	−0.833020	−0.416510	−	0.909131i	$-0.636747\pi$
$331$	−5016.46	−0.833020	−0.416510	+	0.909131i	$0.636747\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−15766.2	−2.57134
$336$	0	0
$337$	−10810.8	−1.74748	−0.873741	−	0.486392i	$-0.838313\pi$
$337$	−10810.8	−1.74748	−0.873741	+	0.486392i	$0.838313\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4807.89	−0.763524
$342$	0	0
$343$	−5163.70	−0.812867
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4137.35	−0.640070	−0.320035	−	0.947406i	$-0.603695\pi$
$347$	−4137.35	−0.640070	−0.320035	+	0.947406i	$0.603695\pi$
$348$	0	0
$349$	7531.11	1.15510	0.577552	−	0.816354i	$-0.304008\pi$
$349$	7531.11	1.15510	0.577552	+	0.816354i	$0.304008\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4872.29	0.734634	0.367317	−	0.930096i	$-0.380276\pi$
$353$	4872.29	0.734634	0.367317	+	0.930096i	$0.380276\pi$
$354$	0	0
$355$	13708.0	2.04943
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.9429	−0.00469604	−0.00234802	−	0.999997i	$-0.500747\pi$
$359$	−31.9429	−0.00469604	−0.00234802	+	0.999997i	$0.500747\pi$
$360$	0	0
$361$	−317.061	−0.0462256
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9683.96	−1.38872
$366$	0	0
$367$	−2262.42	−0.321791	−0.160895	−	0.986971i	$-0.551438\pi$
$367$	−2262.42	−0.321791	−0.160895	+	0.986971i	$0.551438\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	14177.9	1.98405
$372$	0	0
$373$	−5788.70	−0.803559	−0.401780	−	0.915736i	$-0.631608\pi$
$373$	−5788.70	−0.803559	−0.401780	+	0.915736i	$0.631608\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5385.05	0.735661
$378$	0	0
$379$	−10940.9	−1.48284	−0.741420	−	0.671041i	$-0.765847\pi$
$379$	−10940.9	−1.48284	−0.741420	+	0.671041i	$0.765847\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2414.03	0.322066	0.161033	−	0.986949i	$-0.448517\pi$
$383$	2414.03	0.322066	0.161033	+	0.986949i	$0.448517\pi$
$384$	0	0
$385$	−6713.03	−0.888644
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4479.41	0.583844	0.291922	−	0.956442i	$-0.405705\pi$
$389$	4479.41	0.583844	0.291922	+	0.956442i	$0.405705\pi$
$390$	0	0
$391$	−1971.50	−0.254995
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5072.52	0.646143
$396$	0	0
$397$	−1780.62	−0.225105	−0.112553	−	0.993646i	$-0.535903\pi$
$397$	−1780.62	−0.225105	−0.112553	+	0.993646i	$0.535903\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4067.97	0.506595	0.253297	−	0.967388i	$-0.418485\pi$
$401$	4067.97	0.506595	0.253297	+	0.967388i	$0.418485\pi$
$402$	0	0
$403$	10480.3	1.29543
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5057.75	0.615979
$408$	0	0
$409$	−632.302	−0.0764434	−0.0382217	−	0.999269i	$-0.512169\pi$
$409$	−632.302	−0.0764434	−0.0382217	+	0.999269i	$0.512169\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	13196.6	1.57231
$414$	0	0
$415$	−10962.1	−1.29664
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1107.06	−0.129077	−0.0645386	−	0.997915i	$-0.520558\pi$
$419$	−1107.06	−0.129077	−0.0645386	+	0.997915i	$0.520558\pi$
$420$	0	0
$421$	−15977.3	−1.84961	−0.924804	−	0.380444i	$-0.875771\pi$
$421$	−15977.3	−1.84961	−0.924804	+	0.380444i	$0.875771\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4655.00	0.531295
$426$	0	0
$427$	−9685.30	−1.09767
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10283.0	1.14922	0.574610	−	0.818427i	$-0.305154\pi$
$431$	10283.0	1.14922	0.574610	+	0.818427i	$0.305154\pi$
$432$	0	0
$433$	−7084.29	−0.786257	−0.393128	−	0.919484i	$-0.628607\pi$
$433$	−7084.29	−0.786257	−0.393128	+	0.919484i	$0.628607\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9379.96	−1.02678
$438$	0	0
$439$	5308.51	0.577133	0.288567	−	0.957460i	$-0.406821\pi$
$439$	5308.51	0.577133	0.288567	+	0.957460i	$0.406821\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3533.26	0.378939	0.189470	−	0.981887i	$-0.439323\pi$
$443$	3533.26	0.378939	0.189470	+	0.981887i	$0.439323\pi$
$444$	0	0
$445$	18475.7	1.96816
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4787.18	−0.503165	−0.251582	−	0.967836i	$-0.580951\pi$
$449$	−4787.18	−0.503165	−0.251582	+	0.967836i	$0.580951\pi$
$450$	0	0
$451$	−2120.13	−0.221360
$452$	0	0
$453$	0	0
$454$	0	0
$455$	14633.1	1.50772
$456$	0	0
$457$	13168.8	1.34795	0.673973	−	0.738756i	$-0.264586\pi$
$457$	13168.8	1.34795	0.673973	+	0.738756i	$0.264586\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7145.81	0.721938	0.360969	−	0.932578i	$-0.382446\pi$
$461$	7145.81	0.721938	0.360969	+	0.932578i	$0.382446\pi$
$462$	0	0
$463$	−9462.71	−0.949826	−0.474913	−	0.880033i	$-0.657520\pi$
$463$	−9462.71	−0.949826	−0.474913	+	0.880033i	$0.657520\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5306.09	−0.525774	−0.262887	−	0.964827i	$-0.584675\pi$
$467$	−5306.09	−0.525774	−0.262887	+	0.964827i	$0.584675\pi$
$468$	0	0
$469$	16555.6	1.63000
$470$	0	0
$471$	0	0
$472$	0	0
$473$	370.565	0.0360224
$474$	0	0
$475$	22147.5	2.13936
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11731.4	1.11904	0.559522	−	0.828815i	$-0.310985\pi$
$479$	11731.4	1.11904	0.559522	+	0.828815i	$0.310985\pi$
$480$	0	0
$481$	−11024.9	−1.04510
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14630.8	−1.36979
$486$	0	0
$487$	−11075.5	−1.03055	−0.515274	−	0.857026i	$-0.672310\pi$
$487$	−11075.5	−1.03055	−0.515274	+	0.857026i	$0.672310\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2009.14	0.184666	0.0923331	−	0.995728i	$-0.470568\pi$
$491$	2009.14	0.184666	0.0923331	+	0.995728i	$0.470568\pi$
$492$	0	0
$493$	−2620.00	−0.239349
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−14394.4	−1.29915
$498$	0	0
$499$	−9956.24	−0.893192	−0.446596	−	0.894736i	$-0.647364\pi$
$499$	−9956.24	−0.893192	−0.446596	+	0.894736i	$0.647364\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	11760.9	1.04253	0.521266	−	0.853394i	$-0.325460\pi$
$503$	11760.9	1.04253	0.521266	+	0.853394i	$0.325460\pi$
$504$	0	0
$505$	−2566.78	−0.226179
$506$	0	0
$507$	0	0
$508$	0	0
$509$	948.293	0.0825783	0.0412891	−	0.999147i	$-0.486854\pi$
$509$	948.293	0.0825783	0.0412891	+	0.999147i	$0.486854\pi$
$510$	0	0
$511$	10168.9	0.880322
$512$	0	0
$513$	0	0
$514$	0	0
$515$	106.907	0.00914738
$516$	0	0
$517$	4182.27	0.355775
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9796.08	0.823751	0.411875	−	0.911240i	$-0.364874\pi$
$521$	9796.08	0.823751	0.411875	+	0.911240i	$0.364874\pi$
$522$	0	0
$523$	4501.89	0.376394	0.188197	−	0.982131i	$-0.439736\pi$
$523$	4501.89	0.376394	0.188197	+	0.982131i	$0.439736\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5099.00	−0.421472
$528$	0	0
$529$	1282.17	0.105381
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4621.49	0.375570
$534$	0	0
$535$	−9453.70	−0.763961
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1551.08	0.123951
$540$	0	0
$541$	−8419.68	−0.669114	−0.334557	−	0.942376i	$-0.608587\pi$
$541$	−8419.68	−0.669114	−0.334557	+	0.942376i	$0.608587\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7036.69	0.553062
$546$	0	0
$547$	−4655.26	−0.363884	−0.181942	−	0.983309i	$-0.558238\pi$
$547$	−4655.26	−0.363884	−0.181942	+	0.983309i	$0.558238\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12465.4	−0.963781
$552$	0	0
$553$	−5326.52	−0.409596
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−17855.9	−1.35831	−0.679157	−	0.733993i	$-0.737654\pi$
$557$	−17855.9	−1.35831	−0.679157	+	0.733993i	$0.737654\pi$
$558$	0	0
$559$	−807.760	−0.0611174
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21434.5	−1.60454	−0.802269	−	0.596962i	$-0.796374\pi$
$563$	−21434.5	−1.60454	−0.802269	+	0.596962i	$0.796374\pi$
$564$	0	0
$565$	−14650.8	−1.09091
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14412.8	1.06189	0.530945	−	0.847406i	$-0.321837\pi$
$569$	14412.8	1.06189	0.530945	+	0.847406i	$0.321837\pi$
$570$	0	0
$571$	−4492.48	−0.329255	−0.164627	−	0.986356i	$-0.552642\pi$
$571$	−4492.48	−0.329255	−0.164627	+	0.986356i	$0.552642\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−31755.5	−2.30312
$576$	0	0
$577$	9544.29	0.688621	0.344310	−	0.938856i	$-0.388113\pi$
$577$	9544.29	0.688621	0.344310	+	0.938856i	$0.388113\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	11511.0	0.821956
$582$	0	0
$583$	10837.3	0.769872
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15671.9	−1.10195	−0.550977	−	0.834520i	$-0.685745\pi$
$587$	−15671.9	−1.10195	−0.550977	+	0.834520i	$0.685745\pi$
$588$	0	0
$589$	−24259.9	−1.69713
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6737.03	−0.466537	−0.233269	−	0.972412i	$-0.574942\pi$
$593$	−6737.03	−0.466537	−0.233269	+	0.972412i	$0.574942\pi$
$594$	0	0
$595$	−7119.50	−0.490539
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−25077.1	−1.71056	−0.855278	−	0.518169i	$-0.826614\pi$
$599$	−25077.1	−1.71056	−0.855278	+	0.518169i	$0.826614\pi$
$600$	0	0
$601$	18123.5	1.23007	0.615037	−	0.788499i	$-0.289141\pi$
$601$	18123.5	1.23007	0.615037	+	0.788499i	$0.289141\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	21449.5	1.44140
$606$	0	0
$607$	−18377.1	−1.22884	−0.614419	−	0.788980i	$-0.710610\pi$
$607$	−18377.1	−1.22884	−0.614419	+	0.788980i	$0.710610\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9116.55	−0.603627
$612$	0	0
$613$	19642.4	1.29421	0.647105	−	0.762401i	$-0.275979\pi$
$613$	19642.4	1.29421	0.647105	+	0.762401i	$0.275979\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8738.58	−0.570181	−0.285091	−	0.958501i	$-0.592024\pi$
$617$	−8738.58	−0.570181	−0.285091	+	0.958501i	$0.592024\pi$
$618$	0	0
$619$	−18996.2	−1.23348	−0.616739	−	0.787168i	$-0.711547\pi$
$619$	−18996.2	−1.23348	−0.616739	+	0.787168i	$0.711547\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−19400.8	−1.24764
$624$	0	0
$625$	25126.3	1.60808
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5363.99	0.340026
$630$	0	0
$631$	−18454.2	−1.16426	−0.582130	−	0.813096i	$-0.697781\pi$
$631$	−18454.2	−1.16426	−0.582130	+	0.813096i	$0.697781\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6798.17	0.424846
$636$	0	0
$637$	−3381.06	−0.210302
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2208.09	0.136060	0.0680300	−	0.997683i	$-0.478329\pi$
$641$	2208.09	0.136060	0.0680300	+	0.997683i	$0.478329\pi$
$642$	0	0
$643$	14048.7	0.861625	0.430813	−	0.902441i	$-0.358227\pi$
$643$	14048.7	0.861625	0.430813	+	0.902441i	$0.358227\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7959.60	0.483654	0.241827	−	0.970319i	$-0.422253\pi$
$647$	7959.60	0.483654	0.241827	+	0.970319i	$0.422253\pi$
$648$	0	0
$649$	10087.2	0.610105
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8533.37	−0.511388	−0.255694	−	0.966758i	$-0.582304\pi$
$653$	−8533.37	−0.511388	−0.255694	+	0.966758i	$0.582304\pi$
$654$	0	0
$655$	42614.2	2.54210
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8532.32	0.504358	0.252179	−	0.967681i	$-0.418853\pi$
$659$	8532.32	0.504358	0.252179	+	0.967681i	$0.418853\pi$
$660$	0	0
$661$	24194.0	1.42366	0.711828	−	0.702353i	$-0.247867\pi$
$661$	24194.0	1.42366	0.711828	+	0.702353i	$0.247867\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−33873.0	−1.97525
$666$	0	0
$667$	17873.1	1.03756
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7403.23	−0.425929
$672$	0	0
$673$	−10069.6	−0.576750	−0.288375	−	0.957518i	$-0.593115\pi$
$673$	−10069.6	−0.576750	−0.288375	+	0.957518i	$0.593115\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−13155.9	−0.746856	−0.373428	−	0.927659i	$-0.621818\pi$
$677$	−13155.9	−0.746856	−0.373428	+	0.927659i	$0.621818\pi$
$678$	0	0
$679$	15363.4	0.868326
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−21814.1	−1.22210	−0.611049	−	0.791593i	$-0.709252\pi$
$683$	−21814.1	−1.22210	−0.611049	+	0.791593i	$0.709252\pi$
$684$	0	0
$685$	1101.29	0.0614277
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−23623.3	−1.30620
$690$	0	0
$691$	8237.48	0.453500	0.226750	−	0.973953i	$-0.427190\pi$
$691$	8237.48	0.453500	0.226750	+	0.973953i	$0.427190\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5117.77	0.279321
$696$	0	0
$697$	−2248.50	−0.122192
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3684.13	−0.198499	−0.0992495	−	0.995063i	$-0.531644\pi$
$701$	−3684.13	−0.198499	−0.0992495	+	0.995063i	$0.531644\pi$
$702$	0	0
$703$	25520.7	1.36918
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2695.31	0.143377
$708$	0	0
$709$	26765.8	1.41779	0.708894	−	0.705315i	$-0.249194\pi$
$709$	26765.8	1.41779	0.708894	+	0.705315i	$0.249194\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	34784.3	1.82705
$714$	0	0
$715$	11185.2	0.585041
$716$	0	0
$717$	0	0
$718$	0	0
$719$	20163.3	1.04585	0.522924	−	0.852379i	$-0.324841\pi$
$719$	20163.3	1.04585	0.522924	+	0.852379i	$0.324841\pi$
$720$	0	0
$721$	−112.261	−0.00579862
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−42201.0	−2.16180
$726$	0	0
$727$	−23230.6	−1.18511	−0.592556	−	0.805529i	$-0.701881\pi$
$727$	−23230.6	−1.18511	−0.592556	+	0.805529i	$0.701881\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	393.002	0.0198847
$732$	0	0
$733$	−28594.4	−1.44087	−0.720436	−	0.693521i	$-0.756058\pi$
$733$	−28594.4	−1.44087	−0.720436	+	0.693521i	$0.756058\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12654.8	0.632489
$738$	0	0
$739$	−8294.73	−0.412891	−0.206446	−	0.978458i	$-0.566190\pi$
$739$	−8294.73	−0.412891	−0.206446	+	0.978458i	$0.566190\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5881.61	−0.290411	−0.145205	−	0.989402i	$-0.546384\pi$
$743$	−5881.61	−0.290411	−0.145205	+	0.989402i	$0.546384\pi$
$744$	0	0
$745$	−1685.19	−0.0828731
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9927.08	0.484283
$750$	0	0
$751$	−1255.39	−0.0609984	−0.0304992	−	0.999535i	$-0.509710\pi$
$751$	−1255.39	−0.0609984	−0.0304992	+	0.999535i	$0.509710\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	21002.0	1.01237
$756$	0	0
$757$	−38652.8	−1.85582	−0.927912	−	0.372799i	$-0.878398\pi$
$757$	−38652.8	−1.85582	−0.927912	+	0.372799i	$0.878398\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3286.48	0.156550	0.0782752	−	0.996932i	$-0.475059\pi$
$761$	3286.48	0.156550	0.0782752	+	0.996932i	$0.475059\pi$
$762$	0	0
$763$	−7389.05	−0.350592
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−21988.2	−1.03514
$768$	0	0
$769$	−29939.6	−1.40396	−0.701982	−	0.712195i	$-0.747701\pi$
$769$	−29939.6	−1.40396	−0.701982	+	0.712195i	$0.747701\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−40700.7	−1.89379	−0.946896	−	0.321541i	$-0.895799\pi$
$773$	−40700.7	−1.89379	−0.946896	+	0.321541i	$0.895799\pi$
$774$	0	0
$775$	−82130.9	−3.80675
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10697.9	−0.492030
$780$	0	0
$781$	−11002.8	−0.504112
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−31702.7	−1.44142
$786$	0	0
$787$	8126.31	0.368071	0.184035	−	0.982920i	$-0.441084\pi$
$787$	8126.31	0.368071	0.184035	+	0.982920i	$0.441084\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15384.4	0.691536
$792$	0	0
$793$	16137.6	0.722653
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11987.8	−0.532787	−0.266393	−	0.963864i	$-0.585832\pi$
$797$	−11987.8	−0.532787	−0.266393	+	0.963864i	$0.585832\pi$
$798$	0	0
$799$	4435.50	0.196391
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7772.86	0.341592
$804$	0	0
$805$	48567.8	2.12645
$806$	0	0
$807$	0	0
$808$	0	0
$809$	36146.3	1.57087	0.785436	−	0.618943i	$-0.212439\pi$
$809$	36146.3	1.57087	0.785436	+	0.618943i	$0.212439\pi$
$810$	0	0
$811$	6244.38	0.270370	0.135185	−	0.990820i	$-0.456837\pi$
$811$	6244.38	0.270370	0.135185	+	0.990820i	$0.456837\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	34940.3	1.50172
$816$	0	0
$817$	1869.82	0.0800692
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28413.7	−1.20785	−0.603924	−	0.797042i	$-0.706397\pi$
$821$	−28413.7	−1.20785	−0.603924	+	0.797042i	$0.706397\pi$
$822$	0	0
$823$	18803.1	0.796399	0.398199	−	0.917299i	$-0.369635\pi$
$823$	18803.1	0.796399	0.398199	+	0.917299i	$0.369635\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33860.3	1.42375	0.711873	−	0.702309i	$-0.247847\pi$
$827$	33860.3	1.42375	0.711873	+	0.702309i	$0.247847\pi$
$828$	0	0
$829$	−19746.0	−0.827270	−0.413635	−	0.910443i	$-0.635741\pi$
$829$	−19746.0	−0.827270	−0.413635	+	0.910443i	$0.635741\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1645.00	0.0684223
$834$	0	0
$835$	32436.3	1.34432
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31045.8	−1.27750	−0.638748	−	0.769416i	$-0.720547\pi$
$839$	−31045.8	−1.27750	−0.638748	+	0.769416i	$0.720547\pi$
$840$	0	0
$841$	−636.719	−0.0261068
$842$	0	0
$843$	0	0
$844$	0	0
$845$	19493.6	0.793611
$846$	0	0
$847$	−22523.6	−0.913718
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−36592.1	−1.47398
$852$	0	0
$853$	36020.9	1.44587	0.722937	−	0.690914i	$-0.242792\pi$
$853$	36020.9	1.44587	0.722937	+	0.690914i	$0.242792\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35927.0	1.43202	0.716010	−	0.698090i	$-0.245966\pi$
$857$	35927.0	1.43202	0.716010	+	0.698090i	$0.245966\pi$
$858$	0	0
$859$	20914.3	0.830718	0.415359	−	0.909658i	$-0.363656\pi$
$859$	20914.3	0.830718	0.415359	+	0.909658i	$0.363656\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−5395.57	−0.212824	−0.106412	−	0.994322i	$-0.533936\pi$
$863$	−5395.57	−0.212824	−0.106412	+	0.994322i	$0.533936\pi$
$864$	0	0
$865$	−74580.0	−2.93156
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4071.48	−0.158936
$870$	0	0
$871$	−27585.0	−1.07311
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−62326.3	−2.40802
$876$	0	0
$877$	−31305.2	−1.20536	−0.602681	−	0.797983i	$-0.705901\pi$
$877$	−31305.2	−1.20536	−0.602681	+	0.797983i	$0.705901\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−19975.4	−0.763891	−0.381945	−	0.924185i	$-0.624746\pi$
$881$	−19975.4	−0.763891	−0.381945	+	0.924185i	$0.624746\pi$
$882$	0	0
$883$	−32032.5	−1.22081	−0.610407	−	0.792088i	$-0.708994\pi$
$883$	−32032.5	−1.22081	−0.610407	+	0.792088i	$0.708994\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32668.4	1.23664	0.618319	−	0.785927i	$-0.287814\pi$
$887$	32668.4	1.23664	0.618319	+	0.785927i	$0.287814\pi$
$888$	0	0
$889$	−7138.58	−0.269314
$890$	0	0
$891$	0	0
$892$	0	0
$893$	21103.1	0.790805
$894$	0	0
$895$	28650.7	1.07004
$896$	0	0
$897$	0	0
$898$	0	0
$899$	46226.3	1.71494
$900$	0	0
$901$	11493.5	0.424976
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4378.81	0.160836
$906$	0	0
$907$	39521.1	1.44683	0.723417	−	0.690411i	$-0.242571\pi$
$907$	39521.1	1.44683	0.723417	+	0.690411i	$0.242571\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	47835.9	1.73971	0.869854	−	0.493310i	$-0.164213\pi$
$911$	47835.9	1.73971	0.869854	+	0.493310i	$0.164213\pi$
$912$	0	0
$913$	8798.75	0.318944
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−44748.1	−1.61146
$918$	0	0
$919$	39689.7	1.42464	0.712319	−	0.701856i	$-0.247645\pi$
$919$	39689.7	1.42464	0.712319	+	0.701856i	$0.247645\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	23984.0	0.855302
$924$	0	0
$925$	86399.2	3.07112
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4579.27	0.161723	0.0808617	−	0.996725i	$-0.474233\pi$
$929$	4579.27	0.161723	0.0808617	+	0.996725i	$0.474233\pi$
$930$	0	0
$931$	7826.53	0.275515
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5441.99	−0.190344
$936$	0	0
$937$	−38046.5	−1.32649	−0.663246	−	0.748401i	$-0.730822\pi$
$937$	−38046.5	−1.32649	−0.663246	+	0.748401i	$0.730822\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	9081.31	0.314604	0.157302	−	0.987551i	$-0.449720\pi$
$941$	9081.31	0.314604	0.157302	+	0.987551i	$0.449720\pi$
$942$	0	0
$943$	15338.8	0.529694
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39575.9	1.35802	0.679009	−	0.734130i	$-0.262410\pi$
$947$	39575.9	1.35802	0.679009	+	0.734130i	$0.262410\pi$
$948$	0	0
$949$	−16943.4	−0.579562
$950$	0	0
$951$	0	0
$952$	0	0
$953$	28601.6	0.972189	0.486095	−	0.873906i	$-0.338421\pi$
$953$	28601.6	0.972189	0.486095	+	0.873906i	$0.338421\pi$
$954$	0	0
$955$	62237.0	2.10884
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1156.43	−0.0389396
$960$	0	0
$961$	60173.7	2.01986
$962$	0	0
$963$	0	0
$964$	0	0
$965$	78320.9	2.61268
$966$	0	0
$967$	−2591.39	−0.0861773	−0.0430887	−	0.999071i	$-0.513720\pi$
$967$	−2591.39	−0.0861773	−0.0430887	+	0.999071i	$0.513720\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−49692.4	−1.64233	−0.821166	−	0.570689i	$-0.806676\pi$
$971$	−49692.4	−1.64233	−0.821166	+	0.570689i	$0.806676\pi$
$972$	0	0
$973$	−5374.04	−0.177064
$974$	0	0
$975$	0	0
$976$	0	0
$977$	44489.8	1.45686	0.728432	−	0.685119i	$-0.240250\pi$
$977$	44489.8	1.45686	0.728432	+	0.685119i	$0.240250\pi$
$978$	0	0
$979$	−14829.6	−0.484121
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−37606.9	−1.22022	−0.610109	−	0.792318i	$-0.708874\pi$
$983$	−37606.9	−1.22022	−0.610109	+	0.792318i	$0.708874\pi$
$984$	0	0
$985$	62741.0	2.02954
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2680.98	−0.0861983
$990$	0	0
$991$	55446.9	1.77732	0.888662	−	0.458563i	$-0.151636\pi$
$991$	55446.9	1.77732	0.888662	+	0.458563i	$0.151636\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	36601.9	1.16619
$996$	0	0
$997$	−39314.6	−1.24885	−0.624426	−	0.781084i	$-0.714667\pi$
$997$	−39314.6	−1.24885	−0.624426	+	0.781084i	$0.714667\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2448.4.a.v.1.1		2
3.2	odd	2		816.4.a.o.1.2		2
4.3	odd	2		153.4.a.e.1.1		2
12.11	even	2		51.4.a.d.1.2	✓	2
60.59	even	2		1275.4.a.m.1.1		2
84.83	odd	2		2499.4.a.l.1.2		2
204.203	even	2		867.4.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.a.d.1.2	✓	2	12.11	even	2
153.4.a.e.1.1		2	4.3	odd	2
816.4.a.o.1.2		2	3.2	odd	2
867.4.a.j.1.2		2	204.203	even	2
1275.4.a.m.1.1		2	60.59	even	2
2448.4.a.v.1.1		2	1.1	even	1	trivial
2499.4.a.l.1.2		2	84.83	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:	\( N \)	\(=\)	\( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2448.a (trivial)

\(f(q)\)	\(=\)	\(q-19.9706 q^{5} +20.9706 q^{7} +16.0294 q^{11} -34.9411 q^{13} +17.0000 q^{17} +80.8823 q^{19} -115.971 q^{23} +273.823 q^{25} -154.118 q^{29} -299.941 q^{31} -418.794 q^{35} +315.529 q^{37} -132.265 q^{41} +23.1177 q^{43} +260.912 q^{47} +96.7645 q^{49} +676.087 q^{53} -320.117 q^{55} +629.294 q^{59} -461.852 q^{61} +697.794 q^{65} +789.470 q^{67} -686.412 q^{71} +484.912 q^{73} +336.146 q^{77} -254.000 q^{79} +548.912 q^{83} -339.500 q^{85} -925.145 q^{89} -732.735 q^{91} -1615.26 q^{95} +732.617 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} + 8 q^{7} + 66 q^{11} - 2 q^{13} + 34 q^{17} + 26 q^{19} - 198 q^{23} + 344 q^{25} - 444 q^{29} - 532 q^{31} - 600 q^{35} + 88 q^{37} - 570 q^{41} + 182 q^{43} + 420 q^{47} - 78 q^{49} + 300 q^{53}+ \cdots + 1024 q^{97}+O(q^{100}) \) 2 * q - 6 * q^5 + 8 * q^7 + 66 * q^11 - 2 * q^13 + 34 * q^17 + 26 * q^19 - 198 * q^23 + 344 * q^25 - 444 * q^29 - 532 * q^31 - 600 * q^35 + 88 * q^37 - 570 * q^41 + 182 * q^43 + 420 * q^47 - 78 * q^49 + 300 * q^53 + 378 * q^55 + 444 * q^59 + 400 * q^61 + 1158 * q^65 + 968 * q^67 - 1848 * q^71 + 868 * q^73 - 312 * q^77 - 508 * q^79 + 996 * q^83 - 102 * q^85 + 288 * q^89 - 1160 * q^91 - 2382 * q^95 + 1024 * q^97

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