LMFDB - Embedded newform 3024.2.a.bm.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
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			$1.73205$ 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)

$f(q)$	$=$	$q+3.73205 q^{5} -1.00000 q^{7} +5.19615 q^{11} -3.46410 q^{13} +4.00000 q^{17} -7.92820 q^{19} -3.73205 q^{23} +8.92820 q^{25} +8.92820 q^{29} +8.46410 q^{31} -3.73205 q^{35} +6.46410 q^{37} -1.19615 q^{41} +1.46410 q^{43} -5.46410 q^{47} +1.00000 q^{49} +8.00000 q^{53} +19.3923 q^{55} -2.53590 q^{59} -2.92820 q^{61} -12.9282 q^{65} +15.4641 q^{67} +2.66025 q^{71} -12.3923 q^{73} -5.19615 q^{77} -8.39230 q^{79} -0.928203 q^{83} +14.9282 q^{85} +17.7321 q^{89} +3.46410 q^{91} -29.5885 q^{95} -14.9282 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 2 q^{7} + 8 q^{17} - 2 q^{19} - 4 q^{23} + 4 q^{25} + 4 q^{29} + 10 q^{31} - 4 q^{35} + 6 q^{37} + 8 q^{41} - 4 q^{43} - 4 q^{47} + 2 q^{49} + 16 q^{53} + 18 q^{55} - 12 q^{59} + 8 q^{61}+ \cdots - 16 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 - 2 * q^7 + 8 * q^17 - 2 * q^19 - 4 * q^23 + 4 * q^25 + 4 * q^29 + 10 * q^31 - 4 * q^35 + 6 * q^37 + 8 * q^41 - 4 * q^43 - 4 * q^47 + 2 * q^49 + 16 * q^53 + 18 * q^55 - 12 * q^59 + 8 * q^61 - 12 * q^65 + 24 * q^67 - 12 * q^71 - 4 * q^73 + 4 * q^79 + 12 * q^83 + 16 * q^85 + 32 * q^89 - 28 * q^95 - 16 * 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
$5$	3.73205	1.66902	0.834512	−	0.550990i	$-0.185750\pi$
$5$	3.73205	1.66902	0.834512	+	0.550990i	$0.185750\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.19615	1.56670	0.783349	−	0.621582i	$-0.213510\pi$
$11$	5.19615	1.56670	0.783349	+	0.621582i	$0.213510\pi$
$12$	0	0
$13$	−3.46410	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−7.92820	−1.81885	−0.909427	−	0.415863i	$-0.863480\pi$
$19$	−7.92820	−1.81885	−0.909427	+	0.415863i	$0.863480\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.73205	−0.778186	−0.389093	−	0.921198i	$-0.627212\pi$
$23$	−3.73205	−0.778186	−0.389093	+	0.921198i	$0.627212\pi$
$24$	0	0
$25$	8.92820	1.78564
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.92820	1.65793	0.828963	−	0.559304i	$-0.188931\pi$
$29$	8.92820	1.65793	0.828963	+	0.559304i	$0.188931\pi$
$30$	0	0
$31$	8.46410	1.52020	0.760099	−	0.649808i	$-0.225151\pi$
$31$	8.46410	1.52020	0.760099	+	0.649808i	$0.225151\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.73205	−0.630832
$36$	0	0
$37$	6.46410	1.06269	0.531346	−	0.847155i	$-0.321686\pi$
$37$	6.46410	1.06269	0.531346	+	0.847155i	$0.321686\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.19615	−0.186808	−0.0934038	−	0.995628i	$-0.529775\pi$
$41$	−1.19615	−0.186808	−0.0934038	+	0.995628i	$0.529775\pi$
$42$	0	0
$43$	1.46410	0.223273	0.111637	−	0.993749i	$-0.464391\pi$
$43$	1.46410	0.223273	0.111637	+	0.993749i	$0.464391\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.46410	−0.797021	−0.398511	−	0.917164i	$-0.630473\pi$
$47$	−5.46410	−0.797021	−0.398511	+	0.917164i	$0.630473\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	0	0
$55$	19.3923	2.61486
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.53590	−0.330146	−0.165073	−	0.986281i	$-0.552786\pi$
$59$	−2.53590	−0.330146	−0.165073	+	0.986281i	$0.552786\pi$
$60$	0	0
$61$	−2.92820	−0.374918	−0.187459	−	0.982272i	$-0.560025\pi$
$61$	−2.92820	−0.374918	−0.187459	+	0.982272i	$0.560025\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.9282	−1.60355
$66$	0	0
$67$	15.4641	1.88924	0.944620	−	0.328165i	$-0.106430\pi$
$67$	15.4641	1.88924	0.944620	+	0.328165i	$0.106430\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.66025	0.315714	0.157857	−	0.987462i	$-0.449541\pi$
$71$	2.66025	0.315714	0.157857	+	0.987462i	$0.449541\pi$
$72$	0	0
$73$	−12.3923	−1.45041	−0.725205	−	0.688533i	$-0.758255\pi$
$73$	−12.3923	−1.45041	−0.725205	+	0.688533i	$0.758255\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.19615	−0.592157
$78$	0	0
$79$	−8.39230	−0.944208	−0.472104	−	0.881543i	$-0.656505\pi$
$79$	−8.39230	−0.944208	−0.472104	+	0.881543i	$0.656505\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.928203	−0.101884	−0.0509418	−	0.998702i	$-0.516222\pi$
$83$	−0.928203	−0.101884	−0.0509418	+	0.998702i	$0.516222\pi$
$84$	0	0
$85$	14.9282	1.61919
$86$	0	0
$87$	0	0
$88$	0	0
$89$	17.7321	1.87959	0.939797	−	0.341734i	$-0.111014\pi$
$89$	17.7321	1.87959	0.939797	+	0.341734i	$0.111014\pi$
$90$	0	0
$91$	3.46410	0.363137
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−29.5885	−3.03571
$96$	0	0
$97$	−14.9282	−1.51573	−0.757865	−	0.652412i	$-0.773757\pi$
$97$	−14.9282	−1.51573	−0.757865	+	0.652412i	$0.773757\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	5.53590	0.545468	0.272734	−	0.962089i	$-0.412072\pi$
$103$	5.53590	0.545468	0.272734	+	0.962089i	$0.412072\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.46410	−0.334887	−0.167444	−	0.985882i	$-0.553551\pi$
$107$	−3.46410	−0.334887	−0.167444	+	0.985882i	$0.553551\pi$
$108$	0	0
$109$	3.53590	0.338678	0.169339	−	0.985558i	$-0.445837\pi$
$109$	3.53590	0.338678	0.169339	+	0.985558i	$0.445837\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.46410	0.325875	0.162938	−	0.986636i	$-0.447903\pi$
$113$	3.46410	0.325875	0.162938	+	0.986636i	$0.447903\pi$
$114$	0	0
$115$	−13.9282	−1.29881
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	16.0000	1.45455
$122$	0	0
$123$	0	0
$124$	0	0
$125$	14.6603	1.31125
$126$	0	0
$127$	−6.39230	−0.567225	−0.283613	−	0.958939i	$-0.591533\pi$
$127$	−6.39230	−0.567225	−0.283613	+	0.958939i	$0.591533\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.39230	0.733239	0.366620	−	0.930371i	$-0.380515\pi$
$131$	8.39230	0.733239	0.366620	+	0.930371i	$0.380515\pi$
$132$	0	0
$133$	7.92820	0.687462
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	−6.92820	−0.587643	−0.293821	−	0.955860i	$-0.594927\pi$
$139$	−6.92820	−0.587643	−0.293821	+	0.955860i	$0.594927\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−18.0000	−1.50524
$144$	0	0
$145$	33.3205	2.76712
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−23.4641	−1.92225	−0.961127	−	0.276106i	$-0.910956\pi$
$149$	−23.4641	−1.92225	−0.961127	+	0.276106i	$0.910956\pi$
$150$	0	0
$151$	20.9282	1.70311	0.851557	−	0.524263i	$-0.175659\pi$
$151$	20.9282	1.70311	0.851557	+	0.524263i	$0.175659\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	31.5885	2.53725
$156$	0	0
$157$	22.3923	1.78710	0.893550	−	0.448963i	$-0.148207\pi$
$157$	22.3923	1.78710	0.893550	+	0.448963i	$0.148207\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.73205	0.294127
$162$	0	0
$163$	6.39230	0.500684	0.250342	−	0.968157i	$-0.419457\pi$
$163$	6.39230	0.500684	0.250342	+	0.968157i	$0.419457\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.46410	0.268060	0.134030	−	0.990977i	$-0.457208\pi$
$167$	3.46410	0.268060	0.134030	+	0.990977i	$0.457208\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−10.1244	−0.769741	−0.384870	−	0.922971i	$-0.625754\pi$
$173$	−10.1244	−0.769741	−0.384870	+	0.922971i	$0.625754\pi$
$174$	0	0
$175$	−8.92820	−0.674909
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.53590	−0.339029	−0.169514	−	0.985528i	$-0.554220\pi$
$179$	−4.53590	−0.339029	−0.169514	+	0.985528i	$0.554220\pi$
$180$	0	0
$181$	−3.07180	−0.228325	−0.114162	−	0.993462i	$-0.536418\pi$
$181$	−3.07180	−0.228325	−0.114162	+	0.993462i	$0.536418\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	24.1244	1.77366
$186$	0	0
$187$	20.7846	1.51992
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.87564	0.135717	0.0678584	−	0.997695i	$-0.478383\pi$
$191$	1.87564	0.135717	0.0678584	+	0.997695i	$0.478383\pi$
$192$	0	0
$193$	−14.7846	−1.06422	−0.532110	−	0.846675i	$-0.678601\pi$
$193$	−14.7846	−1.06422	−0.532110	+	0.846675i	$0.678601\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.46410	−0.246807	−0.123404	−	0.992357i	$-0.539381\pi$
$197$	−3.46410	−0.246807	−0.123404	+	0.992357i	$0.539381\pi$
$198$	0	0
$199$	3.53590	0.250653	0.125327	−	0.992116i	$-0.460002\pi$
$199$	3.53590	0.250653	0.125327	+	0.992116i	$0.460002\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−8.92820	−0.626637
$204$	0	0
$205$	−4.46410	−0.311786
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−41.1962	−2.84960
$210$	0	0
$211$	−2.00000	−0.137686	−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000	−0.137686	−0.0688428	+	0.997628i	$0.521931\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.46410	0.372649
$216$	0	0
$217$	−8.46410	−0.574581
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−13.8564	−0.932083
$222$	0	0
$223$	0.464102	0.0310785	0.0155393	−	0.999879i	$-0.495053\pi$
$223$	0.464102	0.0310785	0.0155393	+	0.999879i	$0.495053\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.3205	−0.884113	−0.442057	−	0.896987i	$-0.645751\pi$
$227$	−13.3205	−0.884113	−0.442057	+	0.896987i	$0.645751\pi$
$228$	0	0
$229$	1.07180	0.0708263	0.0354132	−	0.999373i	$-0.488725\pi$
$229$	1.07180	0.0708263	0.0354132	+	0.999373i	$0.488725\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.535898	−0.0351079	−0.0175539	−	0.999846i	$-0.505588\pi$
$233$	−0.535898	−0.0351079	−0.0175539	+	0.999846i	$0.505588\pi$
$234$	0	0
$235$	−20.3923	−1.33025
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.60770	0.362732	0.181366	−	0.983416i	$-0.441948\pi$
$239$	5.60770	0.362732	0.181366	+	0.983416i	$0.441948\pi$
$240$	0	0
$241$	−10.5359	−0.678677	−0.339338	−	0.940664i	$-0.610203\pi$
$241$	−10.5359	−0.678677	−0.339338	+	0.940664i	$0.610203\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.73205	0.238432
$246$	0	0
$247$	27.4641	1.74750
$248$	0	0
$249$	0	0
$250$	0	0
$251$	14.9282	0.942260	0.471130	−	0.882064i	$-0.343846\pi$
$251$	14.9282	0.942260	0.471130	+	0.882064i	$0.343846\pi$
$252$	0	0
$253$	−19.3923	−1.21918
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.1244	1.50484	0.752418	−	0.658686i	$-0.228887\pi$
$257$	24.1244	1.50484	0.752418	+	0.658686i	$0.228887\pi$
$258$	0	0
$259$	−6.46410	−0.401660
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.66025	−0.164038	−0.0820191	−	0.996631i	$-0.526137\pi$
$263$	−2.66025	−0.164038	−0.0820191	+	0.996631i	$0.526137\pi$
$264$	0	0
$265$	29.8564	1.83406
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−21.5885	−1.31627	−0.658136	−	0.752899i	$-0.728655\pi$
$269$	−21.5885	−1.31627	−0.658136	+	0.752899i	$0.728655\pi$
$270$	0	0
$271$	22.9282	1.39279	0.696395	−	0.717659i	$-0.254786\pi$
$271$	22.9282	1.39279	0.696395	+	0.717659i	$0.254786\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	46.3923	2.79756
$276$	0	0
$277$	−3.39230	−0.203824	−0.101912	−	0.994793i	$-0.532496\pi$
$277$	−3.39230	−0.203824	−0.101912	+	0.994793i	$0.532496\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	23.3205	1.39118	0.695592	−	0.718437i	$-0.255142\pi$
$281$	23.3205	1.39118	0.695592	+	0.718437i	$0.255142\pi$
$282$	0	0
$283$	−14.9282	−0.887390	−0.443695	−	0.896178i	$-0.646333\pi$
$283$	−14.9282	−0.887390	−0.443695	+	0.896178i	$0.646333\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.19615	0.0706066
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.9282	1.10580	0.552899	−	0.833248i	$-0.313522\pi$
$293$	18.9282	1.10580	0.552899	+	0.833248i	$0.313522\pi$
$294$	0	0
$295$	−9.46410	−0.551021
$296$	0	0
$297$	0	0
$298$	0	0
$299$	12.9282	0.747657
$300$	0	0
$301$	−1.46410	−0.0843894
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−10.9282	−0.625747
$306$	0	0
$307$	−19.9282	−1.13736	−0.568681	−	0.822558i	$-0.692546\pi$
$307$	−19.9282	−1.13736	−0.568681	+	0.822558i	$0.692546\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.85641	0.445496	0.222748	−	0.974876i	$-0.428497\pi$
$311$	7.85641	0.445496	0.222748	+	0.974876i	$0.428497\pi$
$312$	0	0
$313$	−5.60770	−0.316966	−0.158483	−	0.987362i	$-0.550660\pi$
$313$	−5.60770	−0.316966	−0.158483	+	0.987362i	$0.550660\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.5359	−1.49040	−0.745202	−	0.666838i	$-0.767647\pi$
$317$	−26.5359	−1.49040	−0.745202	+	0.666838i	$0.767647\pi$
$318$	0	0
$319$	46.3923	2.59747
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−31.7128	−1.76455
$324$	0	0
$325$	−30.9282	−1.71559
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.46410	0.301246
$330$	0	0
$331$	−14.9282	−0.820528	−0.410264	−	0.911967i	$-0.634563\pi$
$331$	−14.9282	−0.820528	−0.410264	+	0.911967i	$0.634563\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	57.7128	3.15319
$336$	0	0
$337$	5.92820	0.322930	0.161465	−	0.986878i	$-0.448378\pi$
$337$	5.92820	0.322930	0.161465	+	0.986878i	$0.448378\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	43.9808	2.38169
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.5167	0.564564	0.282282	−	0.959331i	$-0.408909\pi$
$347$	10.5167	0.564564	0.282282	+	0.959331i	$0.408909\pi$
$348$	0	0
$349$	8.92820	0.477916	0.238958	−	0.971030i	$-0.423194\pi$
$349$	8.92820	0.477916	0.238958	+	0.971030i	$0.423194\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−29.4449	−1.56719	−0.783596	−	0.621271i	$-0.786617\pi$
$353$	−29.4449	−1.56719	−0.783596	+	0.621271i	$0.786617\pi$
$354$	0	0
$355$	9.92820	0.526934
$356$	0	0
$357$	0	0
$358$	0	0
$359$	27.4641	1.44950	0.724750	−	0.689012i	$-0.241955\pi$
$359$	27.4641	1.44950	0.724750	+	0.689012i	$0.241955\pi$
$360$	0	0
$361$	43.8564	2.30823
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−46.2487	−2.42077
$366$	0	0
$367$	12.3205	0.643125	0.321563	−	0.946888i	$-0.395792\pi$
$367$	12.3205	0.643125	0.321563	+	0.946888i	$0.395792\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.00000	−0.415339
$372$	0	0
$373$	−16.4641	−0.852479	−0.426239	−	0.904610i	$-0.640162\pi$
$373$	−16.4641	−0.852479	−0.426239	+	0.904610i	$0.640162\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−30.9282	−1.59288
$378$	0	0
$379$	11.3205	0.581495	0.290748	−	0.956800i	$-0.406096\pi$
$379$	11.3205	0.581495	0.290748	+	0.956800i	$0.406096\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	15.4641	0.790179	0.395089	−	0.918643i	$-0.370714\pi$
$383$	15.4641	0.790179	0.395089	+	0.918643i	$0.370714\pi$
$384$	0	0
$385$	−19.3923	−0.988323
$386$	0	0
$387$	0	0
$388$	0	0
$389$	31.7128	1.60790	0.803952	−	0.594695i	$-0.202727\pi$
$389$	31.7128	1.60790	0.803952	+	0.594695i	$0.202727\pi$
$390$	0	0
$391$	−14.9282	−0.754952
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−31.3205	−1.57591
$396$	0	0
$397$	−3.85641	−0.193547	−0.0967737	−	0.995306i	$-0.530852\pi$
$397$	−3.85641	−0.193547	−0.0967737	+	0.995306i	$0.530852\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.78461	0.139057	0.0695284	−	0.997580i	$-0.477851\pi$
$401$	2.78461	0.139057	0.0695284	+	0.997580i	$0.477851\pi$
$402$	0	0
$403$	−29.3205	−1.46056
$404$	0	0
$405$	0	0
$406$	0	0
$407$	33.5885	1.66492
$408$	0	0
$409$	9.46410	0.467970	0.233985	−	0.972240i	$-0.424823\pi$
$409$	9.46410	0.467970	0.233985	+	0.972240i	$0.424823\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.53590	0.124783
$414$	0	0
$415$	−3.46410	−0.170046
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−32.2487	−1.57545	−0.787726	−	0.616025i	$-0.788742\pi$
$419$	−32.2487	−1.57545	−0.787726	+	0.616025i	$0.788742\pi$
$420$	0	0
$421$	−20.4641	−0.997359	−0.498680	−	0.866786i	$-0.666182\pi$
$421$	−20.4641	−0.997359	−0.498680	+	0.866786i	$0.666182\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	35.7128	1.73233
$426$	0	0
$427$	2.92820	0.141706
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−25.0526	−1.20674	−0.603370	−	0.797462i	$-0.706176\pi$
$431$	−25.0526	−1.20674	−0.603370	+	0.797462i	$0.706176\pi$
$432$	0	0
$433$	−25.3205	−1.21683	−0.608413	−	0.793621i	$-0.708194\pi$
$433$	−25.3205	−1.21683	−0.608413	+	0.793621i	$0.708194\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	29.5885	1.41541
$438$	0	0
$439$	−27.7128	−1.32266	−0.661330	−	0.750095i	$-0.730008\pi$
$439$	−27.7128	−1.32266	−0.661330	+	0.750095i	$0.730008\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.26795	−0.297799	−0.148900	−	0.988852i	$-0.547573\pi$
$443$	−6.26795	−0.297799	−0.148900	+	0.988852i	$0.547573\pi$
$444$	0	0
$445$	66.1769	3.13709
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−37.3205	−1.76126	−0.880632	−	0.473801i	$-0.842882\pi$
$449$	−37.3205	−1.76126	−0.880632	+	0.473801i	$0.842882\pi$
$450$	0	0
$451$	−6.21539	−0.292671
$452$	0	0
$453$	0	0
$454$	0	0
$455$	12.9282	0.606084
$456$	0	0
$457$	−12.8564	−0.601397	−0.300699	−	0.953719i	$-0.597220\pi$
$457$	−12.8564	−0.601397	−0.300699	+	0.953719i	$0.597220\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.41154	−0.112317	−0.0561584	−	0.998422i	$-0.517885\pi$
$461$	−2.41154	−0.112317	−0.0561584	+	0.998422i	$0.517885\pi$
$462$	0	0
$463$	16.9282	0.786720	0.393360	−	0.919384i	$-0.371313\pi$
$463$	16.9282	0.786720	0.393360	+	0.919384i	$0.371313\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−0.679492	−0.0314431	−0.0157216	−	0.999876i	$-0.505005\pi$
$467$	−0.679492	−0.0314431	−0.0157216	+	0.999876i	$0.505005\pi$
$468$	0	0
$469$	−15.4641	−0.714066
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.60770	0.349802
$474$	0	0
$475$	−70.7846	−3.24782
$476$	0	0
$477$	0	0
$478$	0	0
$479$	21.8564	0.998645	0.499322	−	0.866416i	$-0.333582\pi$
$479$	21.8564	0.998645	0.499322	+	0.866416i	$0.333582\pi$
$480$	0	0
$481$	−22.3923	−1.02100
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−55.7128	−2.52979
$486$	0	0
$487$	−29.1769	−1.32213	−0.661066	−	0.750328i	$-0.729896\pi$
$487$	−29.1769	−1.32213	−0.661066	+	0.750328i	$0.729896\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7.05256	−0.318278	−0.159139	−	0.987256i	$-0.550872\pi$
$491$	−7.05256	−0.318278	−0.159139	+	0.987256i	$0.550872\pi$
$492$	0	0
$493$	35.7128	1.60842
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.66025	−0.119329
$498$	0	0
$499$	24.3923	1.09195	0.545975	−	0.837802i	$-0.316159\pi$
$499$	24.3923	1.09195	0.545975	+	0.837802i	$0.316159\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14.0000	0.624229	0.312115	−	0.950044i	$-0.398963\pi$
$503$	14.0000	0.624229	0.312115	+	0.950044i	$0.398963\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−18.9282	−0.838978	−0.419489	−	0.907760i	$-0.637791\pi$
$509$	−18.9282	−0.838978	−0.419489	+	0.907760i	$0.637791\pi$
$510$	0	0
$511$	12.3923	0.548203
$512$	0	0
$513$	0	0
$514$	0	0
$515$	20.6603	0.910400
$516$	0	0
$517$	−28.3923	−1.24869
$518$	0	0
$519$	0	0
$520$	0	0
$521$	13.7321	0.601612	0.300806	−	0.953685i	$-0.402744\pi$
$521$	13.7321	0.601612	0.300806	+	0.953685i	$0.402744\pi$
$522$	0	0
$523$	−19.7846	−0.865121	−0.432560	−	0.901605i	$-0.642390\pi$
$523$	−19.7846	−0.865121	−0.432560	+	0.901605i	$0.642390\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	33.8564	1.47481
$528$	0	0
$529$	−9.07180	−0.394426
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.14359	0.179479
$534$	0	0
$535$	−12.9282	−0.558935
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.19615	0.223814
$540$	0	0
$541$	7.24871	0.311646	0.155823	−	0.987785i	$-0.450197\pi$
$541$	7.24871	0.311646	0.155823	+	0.987785i	$0.450197\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13.1962	0.565261
$546$	0	0
$547$	−34.9282	−1.49342	−0.746711	−	0.665149i	$-0.768368\pi$
$547$	−34.9282	−1.49342	−0.746711	+	0.665149i	$0.768368\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−70.7846	−3.01553
$552$	0	0
$553$	8.39230	0.356877
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.535898	0.0227067	0.0113534	−	0.999936i	$-0.496386\pi$
$557$	0.535898	0.0227067	0.0113534	+	0.999936i	$0.496386\pi$
$558$	0	0
$559$	−5.07180	−0.214514
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−40.9282	−1.72492	−0.862459	−	0.506127i	$-0.831077\pi$
$563$	−40.9282	−1.72492	−0.862459	+	0.506127i	$0.831077\pi$
$564$	0	0
$565$	12.9282	0.543894
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17.3205	−0.726113	−0.363057	−	0.931767i	$-0.618267\pi$
$569$	−17.3205	−0.726113	−0.363057	+	0.931767i	$0.618267\pi$
$570$	0	0
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−33.3205	−1.38956
$576$	0	0
$577$	2.78461	0.115925	0.0579624	−	0.998319i	$-0.481540\pi$
$577$	2.78461	0.115925	0.0579624	+	0.998319i	$0.481540\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.928203	0.0385084
$582$	0	0
$583$	41.5692	1.72162
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.1769	−0.956614	−0.478307	−	0.878193i	$-0.658749\pi$
$587$	−23.1769	−0.956614	−0.478307	+	0.878193i	$0.658749\pi$
$588$	0	0
$589$	−67.1051	−2.76502
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−32.9090	−1.35141	−0.675705	−	0.737172i	$-0.736161\pi$
$593$	−32.9090	−1.35141	−0.675705	+	0.737172i	$0.736161\pi$
$594$	0	0
$595$	−14.9282	−0.611997
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10.1244	−0.413670	−0.206835	−	0.978376i	$-0.566316\pi$
$599$	−10.1244	−0.413670	−0.206835	+	0.978376i	$0.566316\pi$
$600$	0	0
$601$	−9.60770	−0.391906	−0.195953	−	0.980613i	$-0.562780\pi$
$601$	−9.60770	−0.391906	−0.195953	+	0.980613i	$0.562780\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	59.7128	2.42767
$606$	0	0
$607$	14.9282	0.605917	0.302959	−	0.953004i	$-0.402026\pi$
$607$	14.9282	0.605917	0.302959	+	0.953004i	$0.402026\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	18.9282	0.765753
$612$	0	0
$613$	20.1769	0.814938	0.407469	−	0.913219i	$-0.366411\pi$
$613$	20.1769	0.814938	0.407469	+	0.913219i	$0.366411\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16.3923	−0.659929	−0.329965	−	0.943993i	$-0.607037\pi$
$617$	−16.3923	−0.659929	−0.329965	+	0.943993i	$0.607037\pi$
$618$	0	0
$619$	−0.215390	−0.00865727	−0.00432863	−	0.999991i	$-0.501378\pi$
$619$	−0.215390	−0.00865727	−0.00432863	+	0.999991i	$0.501378\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−17.7321	−0.710420
$624$	0	0
$625$	10.0718	0.402872
$626$	0	0
$627$	0	0
$628$	0	0
$629$	25.8564	1.03096
$630$	0	0
$631$	46.2487	1.84113	0.920566	−	0.390587i	$-0.127728\pi$
$631$	46.2487	1.84113	0.920566	+	0.390587i	$0.127728\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−23.8564	−0.946713
$636$	0	0
$637$	−3.46410	−0.137253
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−34.1051	−1.34707	−0.673536	−	0.739155i	$-0.735225\pi$
$641$	−34.1051	−1.34707	−0.673536	+	0.739155i	$0.735225\pi$
$642$	0	0
$643$	−4.21539	−0.166239	−0.0831194	−	0.996540i	$-0.526488\pi$
$643$	−4.21539	−0.166239	−0.0831194	+	0.996540i	$0.526488\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23.3205	−0.916824	−0.458412	−	0.888740i	$-0.651582\pi$
$647$	−23.3205	−0.916824	−0.458412	+	0.888740i	$0.651582\pi$
$648$	0	0
$649$	−13.1769	−0.517239
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.5359	−1.19496	−0.597481	−	0.801883i	$-0.703832\pi$
$653$	−30.5359	−1.19496	−0.597481	+	0.801883i	$0.703832\pi$
$654$	0	0
$655$	31.3205	1.22379
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−40.6603	−1.58390	−0.791949	−	0.610587i	$-0.790934\pi$
$659$	−40.6603	−1.58390	−0.791949	+	0.610587i	$0.790934\pi$
$660$	0	0
$661$	−49.0333	−1.90718	−0.953588	−	0.301116i	$-0.902641\pi$
$661$	−49.0333	−1.90718	−0.953588	+	0.301116i	$0.902641\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	29.5885	1.14739
$666$	0	0
$667$	−33.3205	−1.29018
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−15.2154	−0.587384
$672$	0	0
$673$	−44.9282	−1.73185	−0.865927	−	0.500170i	$-0.833271\pi$
$673$	−44.9282	−1.73185	−0.865927	+	0.500170i	$0.833271\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−30.3731	−1.16733	−0.583666	−	0.811994i	$-0.698382\pi$
$677$	−30.3731	−1.16733	−0.583666	+	0.811994i	$0.698382\pi$
$678$	0	0
$679$	14.9282	0.572892
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−29.1962	−1.11716	−0.558580	−	0.829451i	$-0.688654\pi$
$683$	−29.1962	−1.11716	−0.558580	+	0.829451i	$0.688654\pi$
$684$	0	0
$685$	−37.3205	−1.42594
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−27.7128	−1.05577
$690$	0	0
$691$	−21.8564	−0.831457	−0.415728	−	0.909489i	$-0.636473\pi$
$691$	−21.8564	−0.831457	−0.415728	+	0.909489i	$0.636473\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−25.8564	−0.980789
$696$	0	0
$697$	−4.78461	−0.181230
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−12.1436	−0.458657	−0.229329	−	0.973349i	$-0.573653\pi$
$701$	−12.1436	−0.458657	−0.229329	+	0.973349i	$0.573653\pi$
$702$	0	0
$703$	−51.2487	−1.93288
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−13.5359	−0.508351	−0.254176	−	0.967158i	$-0.581804\pi$
$709$	−13.5359	−0.508351	−0.254176	+	0.967158i	$0.581804\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−31.5885	−1.18300
$714$	0	0
$715$	−67.1769	−2.51227
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16.3923	0.611330	0.305665	−	0.952139i	$-0.401121\pi$
$719$	16.3923	0.611330	0.305665	+	0.952139i	$0.401121\pi$
$720$	0	0
$721$	−5.53590	−0.206168
$722$	0	0
$723$	0	0
$724$	0	0
$725$	79.7128	2.96046
$726$	0	0
$727$	−16.7846	−0.622507	−0.311253	−	0.950327i	$-0.600749\pi$
$727$	−16.7846	−0.622507	−0.311253	+	0.950327i	$0.600749\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.85641	0.216607
$732$	0	0
$733$	22.2487	0.821775	0.410887	−	0.911686i	$-0.365219\pi$
$733$	22.2487	0.821775	0.410887	+	0.911686i	$0.365219\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	80.3538	2.95987
$738$	0	0
$739$	43.5692	1.60272	0.801360	−	0.598183i	$-0.204110\pi$
$739$	43.5692	1.60272	0.801360	+	0.598183i	$0.204110\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	34.3731	1.26103	0.630513	−	0.776179i	$-0.282845\pi$
$743$	34.3731	1.26103	0.630513	+	0.776179i	$0.282845\pi$
$744$	0	0
$745$	−87.5692	−3.20829
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.46410	0.126576
$750$	0	0
$751$	−21.1769	−0.772757	−0.386378	−	0.922340i	$-0.626274\pi$
$751$	−21.1769	−0.772757	−0.386378	+	0.922340i	$0.626274\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	78.1051	2.84254
$756$	0	0
$757$	6.78461	0.246591	0.123295	−	0.992370i	$-0.460654\pi$
$757$	6.78461	0.246591	0.123295	+	0.992370i	$0.460654\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−13.8564	−0.502294	−0.251147	−	0.967949i	$-0.580808\pi$
$761$	−13.8564	−0.502294	−0.251147	+	0.967949i	$0.580808\pi$
$762$	0	0
$763$	−3.53590	−0.128008
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.78461	0.317194
$768$	0	0
$769$	30.1051	1.08562	0.542809	−	0.839856i	$-0.317361\pi$
$769$	30.1051	1.08562	0.542809	+	0.839856i	$0.317361\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	34.1244	1.22737	0.613684	−	0.789552i	$-0.289687\pi$
$773$	34.1244	1.22737	0.613684	+	0.789552i	$0.289687\pi$
$774$	0	0
$775$	75.5692	2.71453
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.48334	0.339776
$780$	0	0
$781$	13.8231	0.494629
$782$	0	0
$783$	0	0
$784$	0	0
$785$	83.5692	2.98271
$786$	0	0
$787$	−31.7128	−1.13044	−0.565220	−	0.824940i	$-0.691209\pi$
$787$	−31.7128	−1.13044	−0.565220	+	0.824940i	$0.691209\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.46410	−0.123169
$792$	0	0
$793$	10.1436	0.360210
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.3397	1.18095	0.590477	−	0.807054i	$-0.298940\pi$
$797$	33.3397	1.18095	0.590477	+	0.807054i	$0.298940\pi$
$798$	0	0
$799$	−21.8564	−0.773224
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−64.3923	−2.27236
$804$	0	0
$805$	13.9282	0.490905
$806$	0	0
$807$	0	0
$808$	0	0
$809$	9.71281	0.341484	0.170742	−	0.985316i	$-0.445383\pi$
$809$	9.71281	0.341484	0.170742	+	0.985316i	$0.445383\pi$
$810$	0	0
$811$	−15.9282	−0.559315	−0.279657	−	0.960100i	$-0.590221\pi$
$811$	−15.9282	−0.559315	−0.279657	+	0.960100i	$0.590221\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	23.8564	0.835653
$816$	0	0
$817$	−11.6077	−0.406102
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.6077	1.10312	0.551558	−	0.834136i	$-0.314034\pi$
$821$	31.6077	1.10312	0.551558	+	0.834136i	$0.314034\pi$
$822$	0	0
$823$	−3.85641	−0.134426	−0.0672129	−	0.997739i	$-0.521411\pi$
$823$	−3.85641	−0.134426	−0.0672129	+	0.997739i	$0.521411\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.9808	0.903440	0.451720	−	0.892160i	$-0.350811\pi$
$827$	25.9808	0.903440	0.451720	+	0.892160i	$0.350811\pi$
$828$	0	0
$829$	0.392305	0.0136253	0.00681266	−	0.999977i	$-0.497831\pi$
$829$	0.392305	0.0136253	0.00681266	+	0.999977i	$0.497831\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	12.9282	0.447399
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.78461	−0.0961354	−0.0480677	−	0.998844i	$-0.515306\pi$
$839$	−2.78461	−0.0961354	−0.0480677	+	0.998844i	$0.515306\pi$
$840$	0	0
$841$	50.7128	1.74872
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.73205	−0.128386
$846$	0	0
$847$	−16.0000	−0.549767
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−24.1244	−0.826972
$852$	0	0
$853$	−25.0718	−0.858442	−0.429221	−	0.903200i	$-0.641212\pi$
$853$	−25.0718	−0.858442	−0.429221	+	0.903200i	$0.641212\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41.4449	1.41573	0.707865	−	0.706348i	$-0.249659\pi$
$857$	41.4449	1.41573	0.707865	+	0.706348i	$0.249659\pi$
$858$	0	0
$859$	−30.0718	−1.02604	−0.513018	−	0.858378i	$-0.671473\pi$
$859$	−30.0718	−1.02604	−0.513018	+	0.858378i	$0.671473\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	33.3205	1.13424	0.567122	−	0.823634i	$-0.308057\pi$
$863$	33.3205	1.13424	0.567122	+	0.823634i	$0.308057\pi$
$864$	0	0
$865$	−37.7846	−1.28472
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−43.6077	−1.47929
$870$	0	0
$871$	−53.5692	−1.81512
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−14.6603	−0.495607
$876$	0	0
$877$	39.8564	1.34586	0.672928	−	0.739708i	$-0.265037\pi$
$877$	39.8564	1.34586	0.672928	+	0.739708i	$0.265037\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−37.4449	−1.26155	−0.630775	−	0.775966i	$-0.717263\pi$
$881$	−37.4449	−1.26155	−0.630775	+	0.775966i	$0.717263\pi$
$882$	0	0
$883$	−34.9282	−1.17543	−0.587714	−	0.809069i	$-0.699972\pi$
$883$	−34.9282	−1.17543	−0.587714	+	0.809069i	$0.699972\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−45.7128	−1.53489	−0.767443	−	0.641117i	$-0.778471\pi$
$887$	−45.7128	−1.53489	−0.767443	+	0.641117i	$0.778471\pi$
$888$	0	0
$889$	6.39230	0.214391
$890$	0	0
$891$	0	0
$892$	0	0
$893$	43.3205	1.44967
$894$	0	0
$895$	−16.9282	−0.565847
$896$	0	0
$897$	0	0
$898$	0	0
$899$	75.5692	2.52037
$900$	0	0
$901$	32.0000	1.06607
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11.4641	−0.381080
$906$	0	0
$907$	42.7846	1.42064	0.710320	−	0.703879i	$-0.248550\pi$
$907$	42.7846	1.42064	0.710320	+	0.703879i	$0.248550\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−12.5359	−0.415333	−0.207666	−	0.978200i	$-0.566587\pi$
$911$	−12.5359	−0.415333	−0.207666	+	0.978200i	$0.566587\pi$
$912$	0	0
$913$	−4.82309	−0.159621
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.39230	−0.277138
$918$	0	0
$919$	−10.9282	−0.360488	−0.180244	−	0.983622i	$-0.557689\pi$
$919$	−10.9282	−0.360488	−0.180244	+	0.983622i	$0.557689\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9.21539	−0.303328
$924$	0	0
$925$	57.7128	1.89759
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9.07180	0.297636	0.148818	−	0.988865i	$-0.452453\pi$
$929$	9.07180	0.297636	0.148818	+	0.988865i	$0.452453\pi$
$930$	0	0
$931$	−7.92820	−0.259836
$932$	0	0
$933$	0	0
$934$	0	0
$935$	77.5692	2.53678
$936$	0	0
$937$	44.7846	1.46305	0.731525	−	0.681815i	$-0.238809\pi$
$937$	44.7846	1.46305	0.731525	+	0.681815i	$0.238809\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.01924	0.131023	0.0655117	−	0.997852i	$-0.479132\pi$
$941$	4.01924	0.131023	0.0655117	+	0.997852i	$0.479132\pi$
$942$	0	0
$943$	4.46410	0.145371
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.87564	−0.125942	−0.0629708	−	0.998015i	$-0.520057\pi$
$947$	−3.87564	−0.125942	−0.0629708	+	0.998015i	$0.520057\pi$
$948$	0	0
$949$	42.9282	1.39351
$950$	0	0
$951$	0	0
$952$	0	0
$953$	27.0718	0.876942	0.438471	−	0.898745i	$-0.355520\pi$
$953$	27.0718	0.876942	0.438471	+	0.898745i	$0.355520\pi$
$954$	0	0
$955$	7.00000	0.226515
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10.0000	0.322917
$960$	0	0
$961$	40.6410	1.31100
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−55.1769	−1.77621
$966$	0	0
$967$	9.71281	0.312343	0.156172	−	0.987730i	$-0.450085\pi$
$967$	9.71281	0.312343	0.156172	+	0.987730i	$0.450085\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−32.6410	−1.04750	−0.523750	−	0.851872i	$-0.675467\pi$
$971$	−32.6410	−1.04750	−0.523750	+	0.851872i	$0.675467\pi$
$972$	0	0
$973$	6.92820	0.222108
$974$	0	0
$975$	0	0
$976$	0	0
$977$	46.6410	1.49218	0.746089	−	0.665846i	$-0.231929\pi$
$977$	46.6410	1.49218	0.746089	+	0.665846i	$0.231929\pi$
$978$	0	0
$979$	92.1384	2.94476
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−38.5359	−1.22910	−0.614552	−	0.788876i	$-0.710663\pi$
$983$	−38.5359	−1.22910	−0.614552	+	0.788876i	$0.710663\pi$
$984$	0	0
$985$	−12.9282	−0.411927
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.46410	−0.173748
$990$	0	0
$991$	41.3205	1.31259	0.656295	−	0.754505i	$-0.272123\pi$
$991$	41.3205	1.31259	0.656295	+	0.754505i	$0.272123\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	13.1962	0.418346
$996$	0	0
$997$	−17.7128	−0.560970	−0.280485	−	0.959858i	$-0.590495\pi$
$997$	−17.7128	−0.560970	−0.280485	+	0.959858i	$0.590495\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.a.bm.1.2		2
3.2	odd	2		3024.2.a.be.1.1		2
4.3	odd	2		1512.2.a.r.1.2	yes	2
12.11	even	2		1512.2.a.m.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.a.m.1.1	✓	2	12.11	even	2
1512.2.a.r.1.2	yes	2	4.3	odd	2
3024.2.a.be.1.1		2	3.2	odd	2
3024.2.a.bm.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.73205 q^{5} -1.00000 q^{7} +5.19615 q^{11} -3.46410 q^{13} +4.00000 q^{17} -7.92820 q^{19} -3.73205 q^{23} +8.92820 q^{25} +8.92820 q^{29} +8.46410 q^{31} -3.73205 q^{35} +6.46410 q^{37} -1.19615 q^{41} +1.46410 q^{43} -5.46410 q^{47} +1.00000 q^{49} +8.00000 q^{53} +19.3923 q^{55} -2.53590 q^{59} -2.92820 q^{61} -12.9282 q^{65} +15.4641 q^{67} +2.66025 q^{71} -12.3923 q^{73} -5.19615 q^{77} -8.39230 q^{79} -0.928203 q^{83} +14.9282 q^{85} +17.7321 q^{89} +3.46410 q^{91} -29.5885 q^{95} -14.9282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 2 q^{7} + 8 q^{17} - 2 q^{19} - 4 q^{23} + 4 q^{25} + 4 q^{29} + 10 q^{31} - 4 q^{35} + 6 q^{37} + 8 q^{41} - 4 q^{43} - 4 q^{47} + 2 q^{49} + 16 q^{53} + 18 q^{55} - 12 q^{59} + 8 q^{61}+ \cdots - 16 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 - 2 * q^7 + 8 * q^17 - 2 * q^19 - 4 * q^23 + 4 * q^25 + 4 * q^29 + 10 * q^31 - 4 * q^35 + 6 * q^37 + 8 * q^41 - 4 * q^43 - 4 * q^47 + 2 * q^49 + 16 * q^53 + 18 * q^55 - 12 * q^59 + 8 * q^61 - 12 * q^65 + 24 * q^67 - 12 * q^71 - 4 * q^73 + 4 * q^79 + 12 * q^83 + 16 * q^85 + 32 * q^89 - 28 * q^95 - 16 * q^97

Introduction

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