LMFDB - Embedded newform 3024.2.a.bm.1.1

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			1.1
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)

$f(q)$	$=$	$q+0.267949 q^{5} -1.00000 q^{7} -5.19615 q^{11} +3.46410 q^{13} +4.00000 q^{17} +5.92820 q^{19} -0.267949 q^{23} -4.92820 q^{25} -4.92820 q^{29} +1.53590 q^{31} -0.267949 q^{35} -0.464102 q^{37} +9.19615 q^{41} -5.46410 q^{43} +1.46410 q^{47} +1.00000 q^{49} +8.00000 q^{53} -1.39230 q^{55} -9.46410 q^{59} +10.9282 q^{61} +0.928203 q^{65} +8.53590 q^{67} -14.6603 q^{71} +8.39230 q^{73} +5.19615 q^{77} +12.3923 q^{79} +12.9282 q^{83} +1.07180 q^{85} +14.2679 q^{89} -3.46410 q^{91} +1.58846 q^{95} -1.07180 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.267949	0.119831	0.0599153	−	0.998203i	$-0.480917\pi$
$5$	0.267949	0.119831	0.0599153	+	0.998203i	$0.480917\pi$
$6$	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	$-0.786490\pi$
$11$	−5.19615	−1.56670	−0.783349	+	0.621582i	$0.786490\pi$
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	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
$19$	5.92820	1.36002	0.680012	−	0.733201i	$-0.261975\pi$
$19$	5.92820	1.36002	0.680012	+	0.733201i	$0.261975\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.267949	−0.0558713	−0.0279356	−	0.999610i	$-0.508893\pi$
$23$	−0.267949	−0.0558713	−0.0279356	+	0.999610i	$0.508893\pi$
$24$	0	0
$25$	−4.92820	−0.985641
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.92820	−0.915144	−0.457572	−	0.889172i	$-0.651281\pi$
$29$	−4.92820	−0.915144	−0.457572	+	0.889172i	$0.651281\pi$
$30$	0	0
$31$	1.53590	0.275855	0.137928	−	0.990442i	$-0.455956\pi$
$31$	1.53590	0.275855	0.137928	+	0.990442i	$0.455956\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.267949	−0.0452917
$36$	0	0
$37$	−0.464102	−0.0762978	−0.0381489	−	0.999272i	$-0.512146\pi$
$37$	−0.464102	−0.0762978	−0.0381489	+	0.999272i	$0.512146\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.19615	1.43620	0.718099	−	0.695941i	$-0.245013\pi$
$41$	9.19615	1.43620	0.718099	+	0.695941i	$0.245013\pi$
$42$	0	0
$43$	−5.46410	−0.833268	−0.416634	−	0.909074i	$-0.636790\pi$
$43$	−5.46410	−0.833268	−0.416634	+	0.909074i	$0.636790\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.46410	0.213561	0.106781	−	0.994283i	$-0.465946\pi$
$47$	1.46410	0.213561	0.106781	+	0.994283i	$0.465946\pi$
$48$	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
$55$	−1.39230	−0.187738
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.46410	−1.23212	−0.616061	−	0.787699i	$-0.711272\pi$
$59$	−9.46410	−1.23212	−0.616061	+	0.787699i	$0.711272\pi$
$60$	0	0
$61$	10.9282	1.39921	0.699607	−	0.714528i	$-0.253359\pi$
$61$	10.9282	1.39921	0.699607	+	0.714528i	$0.253359\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.928203	0.115129
$66$	0	0
$67$	8.53590	1.04283	0.521413	−	0.853304i	$-0.325405\pi$
$67$	8.53590	1.04283	0.521413	+	0.853304i	$0.325405\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.6603	−1.73985	−0.869926	−	0.493182i	$-0.835834\pi$
$71$	−14.6603	−1.73985	−0.869926	+	0.493182i	$0.835834\pi$
$72$	0	0
$73$	8.39230	0.982245	0.491122	−	0.871091i	$-0.336587\pi$
$73$	8.39230	0.982245	0.491122	+	0.871091i	$0.336587\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.19615	0.592157
$78$	0	0
$79$	12.3923	1.39424	0.697122	−	0.716953i	$-0.254464\pi$
$79$	12.3923	1.39424	0.697122	+	0.716953i	$0.254464\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.9282	1.41905	0.709527	−	0.704678i	$-0.248908\pi$
$83$	12.9282	1.41905	0.709527	+	0.704678i	$0.248908\pi$
$84$	0	0
$85$	1.07180	0.116253
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.2679	1.51240	0.756200	−	0.654341i	$-0.227054\pi$
$89$	14.2679	1.51240	0.756200	+	0.654341i	$0.227054\pi$
$90$	0	0
$91$	−3.46410	−0.363137
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.58846	0.162972
$96$	0	0
$97$	−1.07180	−0.108824	−0.0544122	−	0.998519i	$-0.517329\pi$
$97$	−1.07180	−0.108824	−0.0544122	+	0.998519i	$0.517329\pi$
$98$	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
$103$	12.4641	1.22812	0.614062	−	0.789258i	$-0.289534\pi$
$103$	12.4641	1.22812	0.614062	+	0.789258i	$0.289534\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.46410	0.334887	0.167444	−	0.985882i	$-0.446449\pi$
$107$	3.46410	0.334887	0.167444	+	0.985882i	$0.446449\pi$
$108$	0	0
$109$	10.4641	1.00228	0.501140	−	0.865366i	$-0.332914\pi$
$109$	10.4641	1.00228	0.501140	+	0.865366i	$0.332914\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.46410	−0.325875	−0.162938	−	0.986636i	$-0.552097\pi$
$113$	−3.46410	−0.325875	−0.162938	+	0.986636i	$0.552097\pi$
$114$	0	0
$115$	−0.0717968	−0.00669508
$116$	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
$125$	−2.66025	−0.237940
$126$	0	0
$127$	14.3923	1.27711	0.638555	−	0.769576i	$-0.279532\pi$
$127$	14.3923	1.27711	0.638555	+	0.769576i	$0.279532\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.3923	−1.08272	−0.541360	−	0.840791i	$-0.682091\pi$
$131$	−12.3923	−1.08272	−0.541360	+	0.840791i	$0.682091\pi$
$132$	0	0
$133$	−5.92820	−0.514040
$134$	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	$-0.405073\pi$
$139$	6.92820	0.587643	0.293821	+	0.955860i	$0.405073\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−18.0000	−1.50524
$144$	0	0
$145$	−1.32051	−0.109662
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.5359	−1.35467	−0.677337	−	0.735673i	$-0.736866\pi$
$149$	−16.5359	−1.35467	−0.677337	+	0.735673i	$0.736866\pi$
$150$	0	0
$151$	7.07180	0.575495	0.287747	−	0.957706i	$-0.407094\pi$
$151$	7.07180	0.575495	0.287747	+	0.957706i	$0.407094\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.411543	0.0330559
$156$	0	0
$157$	1.60770	0.128308	0.0641540	−	0.997940i	$-0.479565\pi$
$157$	1.60770	0.128308	0.0641540	+	0.997940i	$0.479565\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.267949	0.0211174
$162$	0	0
$163$	−14.3923	−1.12729	−0.563646	−	0.826016i	$-0.690602\pi$
$163$	−14.3923	−1.12729	−0.563646	+	0.826016i	$0.690602\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.46410	−0.268060	−0.134030	−	0.990977i	$-0.542792\pi$
$167$	−3.46410	−0.268060	−0.134030	+	0.990977i	$0.542792\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.1244	1.07385	0.536927	−	0.843628i	$-0.319585\pi$
$173$	14.1244	1.07385	0.536927	+	0.843628i	$0.319585\pi$
$174$	0	0
$175$	4.92820	0.372537
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.4641	−0.856867	−0.428434	−	0.903573i	$-0.640934\pi$
$179$	−11.4641	−0.856867	−0.428434	+	0.903573i	$0.640934\pi$
$180$	0	0
$181$	−16.9282	−1.25826	−0.629132	−	0.777299i	$-0.716589\pi$
$181$	−16.9282	−1.25826	−0.629132	+	0.777299i	$0.716589\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.124356	−0.00914281
$186$	0	0
$187$	−20.7846	−1.51992
$188$	0	0
$189$	0	0
$190$	0	0
$191$	26.1244	1.89029	0.945146	−	0.326648i	$-0.105919\pi$
$191$	26.1244	1.89029	0.945146	+	0.326648i	$0.105919\pi$
$192$	0	0
$193$	26.7846	1.92800	0.963999	−	0.265905i	$-0.0856708\pi$
$193$	26.7846	1.92800	0.963999	+	0.265905i	$0.0856708\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.46410	0.246807	0.123404	−	0.992357i	$-0.460619\pi$
$197$	3.46410	0.246807	0.123404	+	0.992357i	$0.460619\pi$
$198$	0	0
$199$	10.4641	0.741780	0.370890	−	0.928677i	$-0.379053\pi$
$199$	10.4641	0.741780	0.370890	+	0.928677i	$0.379053\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.92820	0.345892
$204$	0	0
$205$	2.46410	0.172100
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−30.8038	−2.13075
$210$	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
$215$	−1.46410	−0.0998509
$216$	0	0
$217$	−1.53590	−0.104264
$218$	0	0
$219$	0	0
$220$	0	0
$221$	13.8564	0.932083
$222$	0	0
$223$	−6.46410	−0.432868	−0.216434	−	0.976297i	$-0.569443\pi$
$223$	−6.46410	−0.432868	−0.216434	+	0.976297i	$0.569443\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.3205	1.41509	0.707546	−	0.706667i	$-0.249802\pi$
$227$	21.3205	1.41509	0.707546	+	0.706667i	$0.249802\pi$
$228$	0	0
$229$	14.9282	0.986483	0.493242	−	0.869892i	$-0.335812\pi$
$229$	14.9282	0.986483	0.493242	+	0.869892i	$0.335812\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.46410	−0.488990	−0.244495	−	0.969651i	$-0.578622\pi$
$233$	−7.46410	−0.488990	−0.244495	+	0.969651i	$0.578622\pi$
$234$	0	0
$235$	0.392305	0.0255911
$236$	0	0
$237$	0	0
$238$	0	0
$239$	26.3923	1.70718	0.853588	−	0.520948i	$-0.174422\pi$
$239$	26.3923	1.70718	0.853588	+	0.520948i	$0.174422\pi$
$240$	0	0
$241$	−17.4641	−1.12496	−0.562481	−	0.826810i	$-0.690153\pi$
$241$	−17.4641	−1.12496	−0.562481	+	0.826810i	$0.690153\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.267949	0.0171186
$246$	0	0
$247$	20.5359	1.30667
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.07180	0.0676512	0.0338256	−	0.999428i	$-0.489231\pi$
$251$	1.07180	0.0676512	0.0338256	+	0.999428i	$0.489231\pi$
$252$	0	0
$253$	1.39230	0.0875335
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−0.124356	−0.00775709	−0.00387855	−	0.999992i	$-0.501235\pi$
$257$	−0.124356	−0.00775709	−0.00387855	+	0.999992i	$0.501235\pi$
$258$	0	0
$259$	0.464102	0.0288379
$260$	0	0
$261$	0	0
$262$	0	0
$263$	14.6603	0.903990	0.451995	−	0.892020i	$-0.350712\pi$
$263$	14.6603	0.903990	0.451995	+	0.892020i	$0.350712\pi$
$264$	0	0
$265$	2.14359	0.131680
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.58846	0.584619	0.292309	−	0.956324i	$-0.405576\pi$
$269$	9.58846	0.584619	0.292309	+	0.956324i	$0.405576\pi$
$270$	0	0
$271$	9.07180	0.551072	0.275536	−	0.961291i	$-0.411145\pi$
$271$	9.07180	0.551072	0.275536	+	0.961291i	$0.411145\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	25.6077	1.54420
$276$	0	0
$277$	17.3923	1.04500	0.522501	−	0.852639i	$-0.324999\pi$
$277$	17.3923	1.04500	0.522501	+	0.852639i	$0.324999\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.3205	−0.675325	−0.337662	−	0.941267i	$-0.609636\pi$
$281$	−11.3205	−0.675325	−0.337662	+	0.941267i	$0.609636\pi$
$282$	0	0
$283$	−1.07180	−0.0637117	−0.0318559	−	0.999492i	$-0.510142\pi$
$283$	−1.07180	−0.0637117	−0.0318559	+	0.999492i	$0.510142\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.19615	−0.542832
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.07180	0.296298	0.148149	−	0.988965i	$-0.452669\pi$
$293$	5.07180	0.296298	0.148149	+	0.988965i	$0.452669\pi$
$294$	0	0
$295$	−2.53590	−0.147646
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.928203	−0.0536794
$300$	0	0
$301$	5.46410	0.314946
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.92820	0.167668
$306$	0	0
$307$	−6.07180	−0.346536	−0.173268	−	0.984875i	$-0.555433\pi$
$307$	−6.07180	−0.346536	−0.173268	+	0.984875i	$0.555433\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.8564	−1.12595	−0.562977	−	0.826473i	$-0.690344\pi$
$311$	−19.8564	−1.12595	−0.562977	+	0.826473i	$0.690344\pi$
$312$	0	0
$313$	−26.3923	−1.49178	−0.745891	−	0.666068i	$-0.767976\pi$
$313$	−26.3923	−1.49178	−0.745891	+	0.666068i	$0.767976\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−33.4641	−1.87953	−0.939766	−	0.341820i	$-0.888957\pi$
$317$	−33.4641	−1.87953	−0.939766	+	0.341820i	$0.888957\pi$
$318$	0	0
$319$	25.6077	1.43376
$320$	0	0
$321$	0	0
$322$	0	0
$323$	23.7128	1.31942
$324$	0	0
$325$	−17.0718	−0.946973
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.46410	−0.0807185
$330$	0	0
$331$	−1.07180	−0.0589113	−0.0294556	−	0.999566i	$-0.509377\pi$
$331$	−1.07180	−0.0589113	−0.0294556	+	0.999566i	$0.509377\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.28719	0.124962
$336$	0	0
$337$	−7.92820	−0.431877	−0.215938	−	0.976407i	$-0.569281\pi$
$337$	−7.92820	−0.431877	−0.215938	+	0.976407i	$0.569281\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−7.98076	−0.432182
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−34.5167	−1.85295	−0.926476	−	0.376355i	$-0.877177\pi$
$347$	−34.5167	−1.85295	−0.926476	+	0.376355i	$0.877177\pi$
$348$	0	0
$349$	−4.92820	−0.263801	−0.131900	−	0.991263i	$-0.542108\pi$
$349$	−4.92820	−0.263801	−0.131900	+	0.991263i	$0.542108\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	29.4449	1.56719	0.783596	−	0.621271i	$-0.213383\pi$
$353$	29.4449	1.56719	0.783596	+	0.621271i	$0.213383\pi$
$354$	0	0
$355$	−3.92820	−0.208487
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.5359	1.08384	0.541922	−	0.840429i	$-0.317697\pi$
$359$	20.5359	1.08384	0.541922	+	0.840429i	$0.317697\pi$
$360$	0	0
$361$	16.1436	0.849663
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.24871	0.117703
$366$	0	0
$367$	−22.3205	−1.16512	−0.582561	−	0.812787i	$-0.697949\pi$
$367$	−22.3205	−1.16512	−0.582561	+	0.812787i	$0.697949\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.00000	−0.415339
$372$	0	0
$373$	−9.53590	−0.493750	−0.246875	−	0.969047i	$-0.579404\pi$
$373$	−9.53590	−0.493750	−0.246875	+	0.969047i	$0.579404\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−17.0718	−0.879242
$378$	0	0
$379$	−23.3205	−1.19789	−0.598947	−	0.800789i	$-0.704414\pi$
$379$	−23.3205	−1.19789	−0.598947	+	0.800789i	$0.704414\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8.53590	0.436164	0.218082	−	0.975930i	$-0.430020\pi$
$383$	8.53590	0.436164	0.218082	+	0.975930i	$0.430020\pi$
$384$	0	0
$385$	1.39230	0.0709584
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−23.7128	−1.20229	−0.601144	−	0.799141i	$-0.705288\pi$
$389$	−23.7128	−1.20229	−0.601144	+	0.799141i	$0.705288\pi$
$390$	0	0
$391$	−1.07180	−0.0542031
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.32051	0.167073
$396$	0	0
$397$	23.8564	1.19732	0.598659	−	0.801004i	$-0.295700\pi$
$397$	23.8564	1.19732	0.598659	+	0.801004i	$0.295700\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−38.7846	−1.93681	−0.968405	−	0.249381i	$-0.919773\pi$
$401$	−38.7846	−1.93681	−0.968405	+	0.249381i	$0.919773\pi$
$402$	0	0
$403$	5.32051	0.265033
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.41154	0.119536
$408$	0	0
$409$	2.53590	0.125392	0.0626961	−	0.998033i	$-0.480030\pi$
$409$	2.53590	0.125392	0.0626961	+	0.998033i	$0.480030\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.46410	0.465698
$414$	0	0
$415$	3.46410	0.170046
$416$	0	0
$417$	0	0
$418$	0	0
$419$	16.2487	0.793801	0.396901	−	0.917862i	$-0.370086\pi$
$419$	16.2487	0.793801	0.396901	+	0.917862i	$0.370086\pi$
$420$	0	0
$421$	−13.5359	−0.659699	−0.329850	−	0.944033i	$-0.606998\pi$
$421$	−13.5359	−0.659699	−0.329850	+	0.944033i	$0.606998\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−19.7128	−0.956212
$426$	0	0
$427$	−10.9282	−0.528853
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.0526	0.628720	0.314360	−	0.949304i	$-0.398210\pi$
$431$	13.0526	0.628720	0.314360	+	0.949304i	$0.398210\pi$
$432$	0	0
$433$	9.32051	0.447915	0.223958	−	0.974599i	$-0.428102\pi$
$433$	9.32051	0.447915	0.223958	+	0.974599i	$0.428102\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.58846	−0.0759862
$438$	0	0
$439$	27.7128	1.32266	0.661330	−	0.750095i	$-0.269992\pi$
$439$	27.7128	1.32266	0.661330	+	0.750095i	$0.269992\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.73205	−0.462384	−0.231192	−	0.972908i	$-0.574263\pi$
$443$	−9.73205	−0.462384	−0.231192	+	0.972908i	$0.574263\pi$
$444$	0	0
$445$	3.82309	0.181232
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.67949	−0.126453	−0.0632265	−	0.997999i	$-0.520139\pi$
$449$	−2.67949	−0.126453	−0.0632265	+	0.997999i	$0.520139\pi$
$450$	0	0
$451$	−47.7846	−2.25009
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.928203	−0.0435148
$456$	0	0
$457$	14.8564	0.694953	0.347477	−	0.937689i	$-0.387039\pi$
$457$	14.8564	0.694953	0.347477	+	0.937689i	$0.387039\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−33.5885	−1.56437	−0.782185	−	0.623046i	$-0.785895\pi$
$461$	−33.5885	−1.56437	−0.782185	+	0.623046i	$0.785895\pi$
$462$	0	0
$463$	3.07180	0.142759	0.0713793	−	0.997449i	$-0.477260\pi$
$463$	3.07180	0.142759	0.0713793	+	0.997449i	$0.477260\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−35.3205	−1.63444	−0.817219	−	0.576327i	$-0.804485\pi$
$467$	−35.3205	−1.63444	−0.817219	+	0.576327i	$0.804485\pi$
$468$	0	0
$469$	−8.53590	−0.394151
$470$	0	0
$471$	0	0
$472$	0	0
$473$	28.3923	1.30548
$474$	0	0
$475$	−29.2154	−1.34049
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5.85641	−0.267586	−0.133793	−	0.991009i	$-0.542716\pi$
$479$	−5.85641	−0.267586	−0.133793	+	0.991009i	$0.542716\pi$
$480$	0	0
$481$	−1.60770	−0.0733046
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.287187	−0.0130405
$486$	0	0
$487$	33.1769	1.50339	0.751695	−	0.659511i	$-0.229237\pi$
$487$	33.1769	1.50339	0.751695	+	0.659511i	$0.229237\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	31.0526	1.40138	0.700691	−	0.713465i	$-0.252875\pi$
$491$	31.0526	1.40138	0.700691	+	0.713465i	$0.252875\pi$
$492$	0	0
$493$	−19.7128	−0.887820
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.6603	0.657602
$498$	0	0
$499$	3.60770	0.161503	0.0807513	−	0.996734i	$-0.474268\pi$
$499$	3.60770	0.161503	0.0807513	+	0.996734i	$0.474268\pi$
$500$	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
$509$	−5.07180	−0.224803	−0.112402	−	0.993663i	$-0.535854\pi$
$509$	−5.07180	−0.224803	−0.112402	+	0.993663i	$0.535854\pi$
$510$	0	0
$511$	−8.39230	−0.371254
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.33975	0.147167
$516$	0	0
$517$	−7.60770	−0.334586
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.2679	0.449847	0.224924	−	0.974376i	$-0.427787\pi$
$521$	10.2679	0.449847	0.224924	+	0.974376i	$0.427787\pi$
$522$	0	0
$523$	21.7846	0.952574	0.476287	−	0.879290i	$-0.341982\pi$
$523$	21.7846	0.952574	0.476287	+	0.879290i	$0.341982\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.14359	0.267619
$528$	0	0
$529$	−22.9282	−0.996878
$530$	0	0
$531$	0	0
$532$	0	0
$533$	31.8564	1.37985
$534$	0	0
$535$	0.928203	0.0401297
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.19615	−0.223814
$540$	0	0
$541$	−41.2487	−1.77342	−0.886710	−	0.462325i	$-0.847015\pi$
$541$	−41.2487	−1.77342	−0.886710	+	0.462325i	$0.847015\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.80385	0.120104
$546$	0	0
$547$	−21.0718	−0.900965	−0.450482	−	0.892785i	$-0.648748\pi$
$547$	−21.0718	−0.900965	−0.450482	+	0.892785i	$0.648748\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−29.2154	−1.24462
$552$	0	0
$553$	−12.3923	−0.526974
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.46410	0.316264	0.158132	−	0.987418i	$-0.449453\pi$
$557$	7.46410	0.316264	0.158132	+	0.987418i	$0.449453\pi$
$558$	0	0
$559$	−18.9282	−0.800578
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.0718	−1.14094	−0.570470	−	0.821318i	$-0.693239\pi$
$563$	−27.0718	−1.14094	−0.570470	+	0.821318i	$0.693239\pi$
$564$	0	0
$565$	−0.928203	−0.0390498
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17.3205	0.726113	0.363057	−	0.931767i	$-0.381733\pi$
$569$	17.3205	0.726113	0.363057	+	0.931767i	$0.381733\pi$
$570$	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
$575$	1.32051	0.0550690
$576$	0	0
$577$	−38.7846	−1.61462	−0.807312	−	0.590125i	$-0.799079\pi$
$577$	−38.7846	−1.61462	−0.807312	+	0.590125i	$0.799079\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−12.9282	−0.536352
$582$	0	0
$583$	−41.5692	−1.72162
$584$	0	0
$585$	0	0
$586$	0	0
$587$	39.1769	1.61700	0.808502	−	0.588493i	$-0.200279\pi$
$587$	39.1769	1.61700	0.808502	+	0.588493i	$0.200279\pi$
$588$	0	0
$589$	9.10512	0.375170
$590$	0	0
$591$	0	0
$592$	0	0
$593$	32.9090	1.35141	0.675705	−	0.737172i	$-0.263839\pi$
$593$	32.9090	1.35141	0.675705	+	0.737172i	$0.263839\pi$
$594$	0	0
$595$	−1.07180	−0.0439394
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14.1244	0.577106	0.288553	−	0.957464i	$-0.406826\pi$
$599$	14.1244	0.577106	0.288553	+	0.957464i	$0.406826\pi$
$600$	0	0
$601$	−30.3923	−1.23973	−0.619864	−	0.784709i	$-0.712812\pi$
$601$	−30.3923	−1.23973	−0.619864	+	0.784709i	$0.712812\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.28719	0.174299
$606$	0	0
$607$	1.07180	0.0435029	0.0217514	−	0.999763i	$-0.493076\pi$
$607$	1.07180	0.0435029	0.0217514	+	0.999763i	$0.493076\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.07180	0.205183
$612$	0	0
$613$	−42.1769	−1.70351	−0.851755	−	0.523941i	$-0.824461\pi$
$613$	−42.1769	−1.70351	−0.851755	+	0.523941i	$0.824461\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.39230	0.176828	0.0884138	−	0.996084i	$-0.471820\pi$
$617$	4.39230	0.176828	0.0884138	+	0.996084i	$0.471820\pi$
$618$	0	0
$619$	−41.7846	−1.67947	−0.839733	−	0.543000i	$-0.817288\pi$
$619$	−41.7846	−1.67947	−0.839733	+	0.543000i	$0.817288\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−14.2679	−0.571633
$624$	0	0
$625$	23.9282	0.957128
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.85641	−0.0740198
$630$	0	0
$631$	−2.24871	−0.0895198	−0.0447599	−	0.998998i	$-0.514252\pi$
$631$	−2.24871	−0.0895198	−0.0447599	+	0.998998i	$0.514252\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.85641	0.153037
$636$	0	0
$637$	3.46410	0.137253
$638$	0	0
$639$	0	0
$640$	0	0
$641$	42.1051	1.66305	0.831526	−	0.555486i	$-0.187468\pi$
$641$	42.1051	1.66305	0.831526	+	0.555486i	$0.187468\pi$
$642$	0	0
$643$	−45.7846	−1.80557	−0.902784	−	0.430093i	$-0.858481\pi$
$643$	−45.7846	−1.80557	−0.902784	+	0.430093i	$0.858481\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.3205	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$647$	11.3205	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$648$	0	0
$649$	49.1769	1.93036
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−37.4641	−1.46608	−0.733042	−	0.680184i	$-0.761900\pi$
$653$	−37.4641	−1.46608	−0.733042	+	0.680184i	$0.761900\pi$
$654$	0	0
$655$	−3.32051	−0.129743
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−23.3397	−0.909187	−0.454594	−	0.890699i	$-0.650216\pi$
$659$	−23.3397	−0.909187	−0.454594	+	0.890699i	$0.650216\pi$
$660$	0	0
$661$	41.0333	1.59601	0.798006	−	0.602650i	$-0.205888\pi$
$661$	41.0333	1.59601	0.798006	+	0.602650i	$0.205888\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.58846	−0.0615977
$666$	0	0
$667$	1.32051	0.0511303
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−56.7846	−2.19215
$672$	0	0
$673$	−31.0718	−1.19773	−0.598865	−	0.800850i	$-0.704381\pi$
$673$	−31.0718	−1.19773	−0.598865	+	0.800850i	$0.704381\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.3731	1.62853	0.814265	−	0.580494i	$-0.197140\pi$
$677$	42.3731	1.62853	0.814265	+	0.580494i	$0.197140\pi$
$678$	0	0
$679$	1.07180	0.0411318
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−18.8038	−0.719509	−0.359755	−	0.933047i	$-0.617140\pi$
$683$	−18.8038	−0.719509	−0.359755	+	0.933047i	$0.617140\pi$
$684$	0	0
$685$	−2.67949	−0.102378
$686$	0	0
$687$	0	0
$688$	0	0
$689$	27.7128	1.05577
$690$	0	0
$691$	5.85641	0.222788	0.111394	−	0.993776i	$-0.464468\pi$
$691$	5.85641	0.222788	0.111394	+	0.993776i	$0.464468\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.85641	0.0704175
$696$	0	0
$697$	36.7846	1.39332
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−39.8564	−1.50536	−0.752678	−	0.658389i	$-0.771238\pi$
$701$	−39.8564	−1.50536	−0.752678	+	0.658389i	$0.771238\pi$
$702$	0	0
$703$	−2.75129	−0.103767
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−20.4641	−0.768545	−0.384273	−	0.923220i	$-0.625548\pi$
$709$	−20.4641	−0.768545	−0.384273	+	0.923220i	$0.625548\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.411543	−0.0154124
$714$	0	0
$715$	−4.82309	−0.180373
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4.39230	−0.163805	−0.0819027	−	0.996640i	$-0.526100\pi$
$719$	−4.39230	−0.163805	−0.0819027	+	0.996640i	$0.526100\pi$
$720$	0	0
$721$	−12.4641	−0.464187
$722$	0	0
$723$	0	0
$724$	0	0
$725$	24.2872	0.902003
$726$	0	0
$727$	24.7846	0.919210	0.459605	−	0.888123i	$-0.347991\pi$
$727$	24.7846	0.919210	0.459605	+	0.888123i	$0.347991\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−21.8564	−0.808388
$732$	0	0
$733$	−26.2487	−0.969518	−0.484759	−	0.874648i	$-0.661093\pi$
$733$	−26.2487	−0.969518	−0.484759	+	0.874648i	$0.661093\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−44.3538	−1.63379
$738$	0	0
$739$	−39.5692	−1.45558	−0.727789	−	0.685802i	$-0.759452\pi$
$739$	−39.5692	−1.45558	−0.727789	+	0.685802i	$0.759452\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−38.3731	−1.40777	−0.703886	−	0.710313i	$-0.748553\pi$
$743$	−38.3731	−1.40777	−0.703886	+	0.710313i	$0.748553\pi$
$744$	0	0
$745$	−4.43078	−0.162331
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.46410	−0.126576
$750$	0	0
$751$	41.1769	1.50257	0.751283	−	0.659980i	$-0.229435\pi$
$751$	41.1769	1.50257	0.751283	+	0.659980i	$0.229435\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.89488	0.0689618
$756$	0	0
$757$	−34.7846	−1.26427	−0.632134	−	0.774859i	$-0.717821\pi$
$757$	−34.7846	−1.26427	−0.632134	+	0.774859i	$0.717821\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	13.8564	0.502294	0.251147	−	0.967949i	$-0.419192\pi$
$761$	13.8564	0.502294	0.251147	+	0.967949i	$0.419192\pi$
$762$	0	0
$763$	−10.4641	−0.378826
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−32.7846	−1.18378
$768$	0	0
$769$	−46.1051	−1.66259	−0.831297	−	0.555829i	$-0.812401\pi$
$769$	−46.1051	−1.66259	−0.831297	+	0.555829i	$0.812401\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	9.87564	0.355202	0.177601	−	0.984103i	$-0.443166\pi$
$773$	9.87564	0.355202	0.177601	+	0.984103i	$0.443166\pi$
$774$	0	0
$775$	−7.56922	−0.271894
$776$	0	0
$777$	0	0
$778$	0	0
$779$	54.5167	1.95326
$780$	0	0
$781$	76.1769	2.72582
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.430781	0.0153752
$786$	0	0
$787$	23.7128	0.845270	0.422635	−	0.906300i	$-0.361105\pi$
$787$	23.7128	0.845270	0.422635	+	0.906300i	$0.361105\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.46410	0.123169
$792$	0	0
$793$	37.8564	1.34432
$794$	0	0
$795$	0	0
$796$	0	0
$797$	50.6603	1.79448	0.897239	−	0.441545i	$-0.145569\pi$
$797$	50.6603	1.79448	0.897239	+	0.441545i	$0.145569\pi$
$798$	0	0
$799$	5.85641	0.207185
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−43.6077	−1.53888
$804$	0	0
$805$	0.0717968	0.00253050
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−45.7128	−1.60718	−0.803588	−	0.595185i	$-0.797079\pi$
$809$	−45.7128	−1.60718	−0.803588	+	0.595185i	$0.797079\pi$
$810$	0	0
$811$	−2.07180	−0.0727506	−0.0363753	−	0.999338i	$-0.511581\pi$
$811$	−2.07180	−0.0727506	−0.0363753	+	0.999338i	$0.511581\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.85641	−0.135084
$816$	0	0
$817$	−32.3923	−1.13326
$818$	0	0
$819$	0	0
$820$	0	0
$821$	52.3923	1.82850	0.914252	−	0.405146i	$-0.132779\pi$
$821$	52.3923	1.82850	0.914252	+	0.405146i	$0.132779\pi$
$822$	0	0
$823$	23.8564	0.831582	0.415791	−	0.909460i	$-0.363505\pi$
$823$	23.8564	0.831582	0.415791	+	0.909460i	$0.363505\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25.9808	−0.903440	−0.451720	−	0.892160i	$-0.649189\pi$
$827$	−25.9808	−0.903440	−0.451720	+	0.892160i	$0.649189\pi$
$828$	0	0
$829$	−20.3923	−0.708254	−0.354127	−	0.935197i	$-0.615222\pi$
$829$	−20.3923	−0.708254	−0.354127	+	0.935197i	$0.615222\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	−0.928203	−0.0321218
$836$	0	0
$837$	0	0
$838$	0	0
$839$	38.7846	1.33899	0.669497	−	0.742815i	$-0.266510\pi$
$839$	38.7846	1.33899	0.669497	+	0.742815i	$0.266510\pi$
$840$	0	0
$841$	−4.71281	−0.162511
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.267949	−0.00921773
$846$	0	0
$847$	−16.0000	−0.549767
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.124356	0.00426286
$852$	0	0
$853$	−38.9282	−1.33288	−0.666438	−	0.745560i	$-0.732182\pi$
$853$	−38.9282	−1.33288	−0.666438	+	0.745560i	$0.732182\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.4449	−0.595905	−0.297953	−	0.954581i	$-0.596304\pi$
$857$	−17.4449	−0.595905	−0.297953	+	0.954581i	$0.596304\pi$
$858$	0	0
$859$	−43.9282	−1.49881	−0.749405	−	0.662111i	$-0.769661\pi$
$859$	−43.9282	−1.49881	−0.749405	+	0.662111i	$0.769661\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.32051	−0.0449506	−0.0224753	−	0.999747i	$-0.507155\pi$
$863$	−1.32051	−0.0449506	−0.0224753	+	0.999747i	$0.507155\pi$
$864$	0	0
$865$	3.78461	0.128681
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−64.3923	−2.18436
$870$	0	0
$871$	29.5692	1.00191
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.66025	0.0899330
$876$	0	0
$877$	12.1436	0.410060	0.205030	−	0.978756i	$-0.434271\pi$
$877$	12.1436	0.410060	0.205030	+	0.978756i	$0.434271\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	21.4449	0.722496	0.361248	−	0.932470i	$-0.382351\pi$
$881$	21.4449	0.722496	0.361248	+	0.932470i	$0.382351\pi$
$882$	0	0
$883$	−21.0718	−0.709122	−0.354561	−	0.935033i	$-0.615370\pi$
$883$	−21.0718	−0.709122	−0.354561	+	0.935033i	$0.615370\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.71281	0.326124	0.163062	−	0.986616i	$-0.447863\pi$
$887$	9.71281	0.326124	0.163062	+	0.986616i	$0.447863\pi$
$888$	0	0
$889$	−14.3923	−0.482702
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.67949	0.290448
$894$	0	0
$895$	−3.07180	−0.102679
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.56922	−0.252448
$900$	0	0
$901$	32.0000	1.06607
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.53590	−0.150778
$906$	0	0
$907$	1.21539	0.0403564	0.0201782	−	0.999796i	$-0.493577\pi$
$907$	1.21539	0.0403564	0.0201782	+	0.999796i	$0.493577\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−19.4641	−0.644874	−0.322437	−	0.946591i	$-0.604502\pi$
$911$	−19.4641	−0.644874	−0.322437	+	0.946591i	$0.604502\pi$
$912$	0	0
$913$	−67.1769	−2.22323
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.3923	0.409230
$918$	0	0
$919$	2.92820	0.0965925	0.0482963	−	0.998833i	$-0.484621\pi$
$919$	2.92820	0.0965925	0.0482963	+	0.998833i	$0.484621\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−50.7846	−1.67160
$924$	0	0
$925$	2.28719	0.0752022
$926$	0	0
$927$	0	0
$928$	0	0
$929$	22.9282	0.752250	0.376125	−	0.926569i	$-0.377256\pi$
$929$	22.9282	0.752250	0.376125	+	0.926569i	$0.377256\pi$
$930$	0	0
$931$	5.92820	0.194289
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.56922	−0.182133
$936$	0	0
$937$	3.21539	0.105042	0.0525211	−	0.998620i	$-0.483274\pi$
$937$	3.21539	0.105042	0.0525211	+	0.998620i	$0.483274\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	55.9808	1.82492	0.912460	−	0.409165i	$-0.134180\pi$
$941$	55.9808	1.82492	0.912460	+	0.409165i	$0.134180\pi$
$942$	0	0
$943$	−2.46410	−0.0802422
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.1244	−0.913919	−0.456959	−	0.889488i	$-0.651062\pi$
$947$	−28.1244	−0.913919	−0.456959	+	0.889488i	$0.651062\pi$
$948$	0	0
$949$	29.0718	0.943710
$950$	0	0
$951$	0	0
$952$	0	0
$953$	40.9282	1.32579	0.662897	−	0.748710i	$-0.269327\pi$
$953$	40.9282	1.32579	0.662897	+	0.748710i	$0.269327\pi$
$954$	0	0
$955$	7.00000	0.226515
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10.0000	0.322917
$960$	0	0
$961$	−28.6410	−0.923904
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7.17691	0.231033
$966$	0	0
$967$	−45.7128	−1.47002	−0.735012	−	0.678054i	$-0.762824\pi$
$967$	−45.7128	−1.47002	−0.735012	+	0.678054i	$0.762824\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.6410	1.17587	0.587933	−	0.808910i	$-0.299942\pi$
$971$	36.6410	1.17587	0.587933	+	0.808910i	$0.299942\pi$
$972$	0	0
$973$	−6.92820	−0.222108
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−22.6410	−0.724350	−0.362175	−	0.932110i	$-0.617966\pi$
$977$	−22.6410	−0.724350	−0.362175	+	0.932110i	$0.617966\pi$
$978$	0	0
$979$	−74.1384	−2.36947
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−45.4641	−1.45008	−0.725040	−	0.688707i	$-0.758179\pi$
$983$	−45.4641	−1.45008	−0.725040	+	0.688707i	$0.758179\pi$
$984$	0	0
$985$	0.928203	0.0295750
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.46410	0.0465557
$990$	0	0
$991$	6.67949	0.212181	0.106091	−	0.994356i	$-0.466167\pi$
$991$	6.67949	0.212181	0.106091	+	0.994356i	$0.466167\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.80385	0.0888879
$996$	0	0
$997$	37.7128	1.19438	0.597188	−	0.802101i	$-0.296284\pi$
$997$	37.7128	1.19438	0.597188	+	0.802101i	$0.296284\pi$
$998$	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.1.1		2
3.2	odd	2		3024.2.a.be.1.2		2
4.3	odd	2		1512.2.a.r.1.1	yes	2
12.11	even	2		1512.2.a.m.1.2	✓	2

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

\(f(q)\)	\(=\)	\(q+0.267949 q^{5} -1.00000 q^{7} -5.19615 q^{11} +3.46410 q^{13} +4.00000 q^{17} +5.92820 q^{19} -0.267949 q^{23} -4.92820 q^{25} -4.92820 q^{29} +1.53590 q^{31} -0.267949 q^{35} -0.464102 q^{37} +9.19615 q^{41} -5.46410 q^{43} +1.46410 q^{47} +1.00000 q^{49} +8.00000 q^{53} -1.39230 q^{55} -9.46410 q^{59} +10.9282 q^{61} +0.928203 q^{65} +8.53590 q^{67} -14.6603 q^{71} +8.39230 q^{73} +5.19615 q^{77} +12.3923 q^{79} +12.9282 q^{83} +1.07180 q^{85} +14.2679 q^{89} -3.46410 q^{91} +1.58846 q^{95} -1.07180 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 2 q^{7} + 8 q^{17} - 2 q^{19} - 4 q^{23} + 4 q^{25} + 4 q^{29} + 10 q^{31} - 4 q^{35} + 6 q^{37} + 8 q^{41} - 4 q^{43} - 4 q^{47} + 2 q^{49} + 16 q^{53} + 18 q^{55} - 12 q^{59} + 8 q^{61}+ \cdots - 16 q^{97}+O(q^{100}) \) 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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