LMFDB - Embedded newform 7248.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7248,2,Mod(1,7248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7248, 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("7248.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7248 = 2^{4} \cdot 3 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7248.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:	$57.8755713850$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 453)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	7248.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -0.267949 q^{11} -3.46410 q^{13} +2.00000 q^{15} -1.00000 q^{21} -9.46410 q^{23} -1.00000 q^{25} +1.00000 q^{27} +3.46410 q^{29} -2.53590 q^{31} -0.267949 q^{33} -2.00000 q^{35} +7.92820 q^{37} -3.46410 q^{39} -0.803848 q^{41} -3.46410 q^{43} +2.00000 q^{45} -3.73205 q^{47} -6.00000 q^{49} +7.73205 q^{53} -0.535898 q^{55} -10.6603 q^{59} -0.535898 q^{61} -1.00000 q^{63} -6.92820 q^{65} -5.00000 q^{67} -9.46410 q^{69} +0.928203 q^{71} -1.07180 q^{73} -1.00000 q^{75} +0.267949 q^{77} -1.92820 q^{79} +1.00000 q^{81} -1.46410 q^{83} +3.46410 q^{87} -6.92820 q^{89} +3.46410 q^{91} -2.53590 q^{93} -7.92820 q^{97} -0.267949 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{11} + 4 q^{15} - 2 q^{21} - 12 q^{23} - 2 q^{25} + 2 q^{27} - 12 q^{31} - 4 q^{33} - 4 q^{35} + 2 q^{37} - 12 q^{41} + 4 q^{45} - 4 q^{47} - 12 q^{49} + 12 q^{53} - 8 q^{55} - 4 q^{59} - 8 q^{61} - 2 q^{63} - 10 q^{67} - 12 q^{69} - 12 q^{71} - 16 q^{73} - 2 q^{75} + 4 q^{77} + 10 q^{79} + 2 q^{81} + 4 q^{83} - 12 q^{93} - 2 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^11 + 4 * q^15 - 2 * q^21 - 12 * q^23 - 2 * q^25 + 2 * q^27 - 12 * q^31 - 4 * q^33 - 4 * q^35 + 2 * q^37 - 12 * q^41 + 4 * q^45 - 4 * q^47 - 12 * q^49 + 12 * q^53 - 8 * q^55 - 4 * q^59 - 8 * q^61 - 2 * q^63 - 10 * q^67 - 12 * q^69 - 12 * q^71 - 16 * q^73 - 2 * q^75 + 4 * q^77 + 10 * q^79 + 2 * q^81 + 4 * q^83 - 12 * q^93 - 2 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.267949	−0.0807897	−0.0403949	−	0.999184i	$-0.512862\pi$
$11$	−0.267949	−0.0807897	−0.0403949	+	0.999184i	$0.512862\pi$
$12$	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
$15$	2.00000	0.516398
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−9.46410	−1.97340	−0.986701	−	0.162547i	$-0.948029\pi$
$23$	−9.46410	−1.97340	−0.986701	+	0.162547i	$0.948029\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	−2.53590	−0.455461	−0.227730	−	0.973724i	$-0.573130\pi$
$31$	−2.53590	−0.455461	−0.227730	+	0.973724i	$0.573130\pi$
$32$	0	0
$33$	−0.267949	−0.0466440
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	7.92820	1.30339	0.651694	−	0.758482i	$-0.274059\pi$
$37$	7.92820	1.30339	0.651694	+	0.758482i	$0.274059\pi$
$38$	0	0
$39$	−3.46410	−0.554700
$40$	0	0
$41$	−0.803848	−0.125540	−0.0627700	−	0.998028i	$-0.519993\pi$
$41$	−0.803848	−0.125540	−0.0627700	+	0.998028i	$0.519993\pi$
$42$	0	0
$43$	−3.46410	−0.528271	−0.264135	−	0.964486i	$-0.585087\pi$
$43$	−3.46410	−0.528271	−0.264135	+	0.964486i	$0.585087\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	−3.73205	−0.544376	−0.272188	−	0.962244i	$-0.587747\pi$
$47$	−3.73205	−0.544376	−0.272188	+	0.962244i	$0.587747\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.73205	1.06208	0.531039	−	0.847347i	$-0.321802\pi$
$53$	7.73205	1.06208	0.531039	+	0.847347i	$0.321802\pi$
$54$	0	0
$55$	−0.535898	−0.0722605
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.6603	−1.38785	−0.693923	−	0.720049i	$-0.744119\pi$
$59$	−10.6603	−1.38785	−0.693923	+	0.720049i	$0.744119\pi$
$60$	0	0
$61$	−0.535898	−0.0686148	−0.0343074	−	0.999411i	$-0.510923\pi$
$61$	−0.535898	−0.0686148	−0.0343074	+	0.999411i	$0.510923\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−6.92820	−0.859338
$66$	0	0
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	0	0
$69$	−9.46410	−1.13934
$70$	0	0
$71$	0.928203	0.110157	0.0550787	−	0.998482i	$-0.482459\pi$
$71$	0.928203	0.110157	0.0550787	+	0.998482i	$0.482459\pi$
$72$	0	0
$73$	−1.07180	−0.125444	−0.0627222	−	0.998031i	$-0.519978\pi$
$73$	−1.07180	−0.125444	−0.0627222	+	0.998031i	$0.519978\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0.267949	0.0305356
$78$	0	0
$79$	−1.92820	−0.216940	−0.108470	−	0.994100i	$-0.534595\pi$
$79$	−1.92820	−0.216940	−0.108470	+	0.994100i	$0.534595\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.46410	−0.160706	−0.0803530	−	0.996766i	$-0.525605\pi$
$83$	−1.46410	−0.160706	−0.0803530	+	0.996766i	$0.525605\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.46410	0.371391
$88$	0	0
$89$	−6.92820	−0.734388	−0.367194	−	0.930144i	$-0.619682\pi$
$89$	−6.92820	−0.734388	−0.367194	+	0.930144i	$0.619682\pi$
$90$	0	0
$91$	3.46410	0.363137
$92$	0	0
$93$	−2.53590	−0.262960
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.92820	−0.804987	−0.402494	−	0.915423i	$-0.631856\pi$
$97$	−7.92820	−0.804987	−0.402494	+	0.915423i	$0.631856\pi$
$98$	0	0
$99$	−0.267949	−0.0269299
$100$	0	0
$101$	−14.6603	−1.45875	−0.729375	−	0.684114i	$-0.760189\pi$
$101$	−14.6603	−1.45875	−0.729375	+	0.684114i	$0.760189\pi$
$102$	0	0
$103$	−1.46410	−0.144262	−0.0721311	−	0.997395i	$-0.522980\pi$
$103$	−1.46410	−0.144262	−0.0721311	+	0.997395i	$0.522980\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	0	0
$109$	12.3923	1.18697	0.593484	−	0.804846i	$-0.297752\pi$
$109$	12.3923	1.18697	0.593484	+	0.804846i	$0.297752\pi$
$110$	0	0
$111$	7.92820	0.752512
$112$	0	0
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	0	0
$115$	−18.9282	−1.76506
$116$	0	0
$117$	−3.46410	−0.320256
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.9282	−0.993473
$122$	0	0
$123$	−0.803848	−0.0724805
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−12.3923	−1.09964	−0.549820	−	0.835283i	$-0.685304\pi$
$127$	−12.3923	−1.09964	−0.549820	+	0.835283i	$0.685304\pi$
$128$	0	0
$129$	−3.46410	−0.304997
$130$	0	0
$131$	−11.4641	−1.00162	−0.500812	−	0.865556i	$-0.666965\pi$
$131$	−11.4641	−1.00162	−0.500812	+	0.865556i	$0.666965\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.00000	0.172133
$136$	0	0
$137$	−19.3205	−1.65066	−0.825331	−	0.564649i	$-0.809012\pi$
$137$	−19.3205	−1.65066	−0.825331	+	0.564649i	$0.809012\pi$
$138$	0	0
$139$	17.4641	1.48129	0.740643	−	0.671899i	$-0.234521\pi$
$139$	17.4641	1.48129	0.740643	+	0.671899i	$0.234521\pi$
$140$	0	0
$141$	−3.73205	−0.314295
$142$	0	0
$143$	0.928203	0.0776203
$144$	0	0
$145$	6.92820	0.575356
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.07180	−0.407377
$156$	0	0
$157$	23.4641	1.87264	0.936320	−	0.351149i	$-0.114209\pi$
$157$	23.4641	1.87264	0.936320	+	0.351149i	$0.114209\pi$
$158$	0	0
$159$	7.73205	0.613192
$160$	0	0
$161$	9.46410	0.745876
$162$	0	0
$163$	−1.07180	−0.0839496	−0.0419748	−	0.999119i	$-0.513365\pi$
$163$	−1.07180	−0.0839496	−0.0419748	+	0.999119i	$0.513365\pi$
$164$	0	0
$165$	−0.535898	−0.0417196
$166$	0	0
$167$	10.1244	0.783446	0.391723	−	0.920083i	$-0.371879\pi$
$167$	10.1244	0.783446	0.391723	+	0.920083i	$0.371879\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−10.6603	−0.801274
$178$	0	0
$179$	−7.46410	−0.557893	−0.278947	−	0.960307i	$-0.589985\pi$
$179$	−7.46410	−0.557893	−0.278947	+	0.960307i	$0.589985\pi$
$180$	0	0
$181$	−12.3923	−0.921113	−0.460556	−	0.887630i	$-0.652350\pi$
$181$	−12.3923	−0.921113	−0.460556	+	0.887630i	$0.652350\pi$
$182$	0	0
$183$	−0.535898	−0.0396147
$184$	0	0
$185$	15.8564	1.16579
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	21.0526	1.52331	0.761655	−	0.647983i	$-0.224387\pi$
$191$	21.0526	1.52331	0.761655	+	0.647983i	$0.224387\pi$
$192$	0	0
$193$	10.8564	0.781461	0.390731	−	0.920505i	$-0.372222\pi$
$193$	10.8564	0.781461	0.390731	+	0.920505i	$0.372222\pi$
$194$	0	0
$195$	−6.92820	−0.496139
$196$	0	0
$197$	−21.8564	−1.55720	−0.778602	−	0.627518i	$-0.784071\pi$
$197$	−21.8564	−1.55720	−0.778602	+	0.627518i	$0.784071\pi$
$198$	0	0
$199$	−2.92820	−0.207575	−0.103787	−	0.994600i	$-0.533096\pi$
$199$	−2.92820	−0.207575	−0.103787	+	0.994600i	$0.533096\pi$
$200$	0	0
$201$	−5.00000	−0.352673
$202$	0	0
$203$	−3.46410	−0.243132
$204$	0	0
$205$	−1.60770	−0.112286
$206$	0	0
$207$	−9.46410	−0.657801
$208$	0	0
$209$	0	0
$210$	0	0
$211$	3.92820	0.270429	0.135214	−	0.990816i	$-0.456828\pi$
$211$	3.92820	0.270429	0.135214	+	0.990816i	$0.456828\pi$
$212$	0	0
$213$	0.928203	0.0635994
$214$	0	0
$215$	−6.92820	−0.472500
$216$	0	0
$217$	2.53590	0.172148
$218$	0	0
$219$	−1.07180	−0.0724253
$220$	0	0
$221$	0	0
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	−14.3923	−0.955251	−0.477625	−	0.878564i	$-0.658502\pi$
$227$	−14.3923	−0.955251	−0.477625	+	0.878564i	$0.658502\pi$
$228$	0	0
$229$	−0.0717968	−0.00474446	−0.00237223	−	0.999997i	$-0.500755\pi$
$229$	−0.0717968	−0.00474446	−0.00237223	+	0.999997i	$0.500755\pi$
$230$	0	0
$231$	0.267949	0.0176298
$232$	0	0
$233$	6.66025	0.436328	0.218164	−	0.975912i	$-0.429993\pi$
$233$	6.66025	0.436328	0.218164	+	0.975912i	$0.429993\pi$
$234$	0	0
$235$	−7.46410	−0.486904
$236$	0	0
$237$	−1.92820	−0.125250
$238$	0	0
$239$	−20.5359	−1.32836	−0.664178	−	0.747574i	$-0.731218\pi$
$239$	−20.5359	−1.32836	−0.664178	+	0.747574i	$0.731218\pi$
$240$	0	0
$241$	−25.0000	−1.61039	−0.805196	−	0.593009i	$-0.797940\pi$
$241$	−25.0000	−1.61039	−0.805196	+	0.593009i	$0.797940\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−12.0000	−0.766652
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1.46410	−0.0927837
$250$	0	0
$251$	19.4641	1.22856	0.614282	−	0.789087i	$-0.289446\pi$
$251$	19.4641	1.22856	0.614282	+	0.789087i	$0.289446\pi$
$252$	0	0
$253$	2.53590	0.159431
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.8564	1.11385	0.556926	−	0.830562i	$-0.311981\pi$
$257$	17.8564	1.11385	0.556926	+	0.830562i	$0.311981\pi$
$258$	0	0
$259$	−7.92820	−0.492635
$260$	0	0
$261$	3.46410	0.214423
$262$	0	0
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	15.4641	0.949952
$266$	0	0
$267$	−6.92820	−0.423999
$268$	0	0
$269$	14.9282	0.910189	0.455094	−	0.890443i	$-0.349606\pi$
$269$	14.9282	0.910189	0.455094	+	0.890443i	$0.349606\pi$
$270$	0	0
$271$	26.7128	1.62269	0.811344	−	0.584569i	$-0.198736\pi$
$271$	26.7128	1.62269	0.811344	+	0.584569i	$0.198736\pi$
$272$	0	0
$273$	3.46410	0.209657
$274$	0	0
$275$	0.267949	0.0161579
$276$	0	0
$277$	−22.7846	−1.36899	−0.684497	−	0.729015i	$-0.739978\pi$
$277$	−22.7846	−1.36899	−0.684497	+	0.729015i	$0.739978\pi$
$278$	0	0
$279$	−2.53590	−0.151820
$280$	0	0
$281$	−21.5885	−1.28786	−0.643930	−	0.765085i	$-0.722697\pi$
$281$	−21.5885	−1.28786	−0.643930	+	0.765085i	$0.722697\pi$
$282$	0	0
$283$	5.00000	0.297219	0.148610	−	0.988896i	$-0.452520\pi$
$283$	5.00000	0.297219	0.148610	+	0.988896i	$0.452520\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.803848	0.0474496
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−7.92820	−0.464760
$292$	0	0
$293$	10.6603	0.622779	0.311389	−	0.950282i	$-0.399206\pi$
$293$	10.6603	0.622779	0.311389	+	0.950282i	$0.399206\pi$
$294$	0	0
$295$	−21.3205	−1.24133
$296$	0	0
$297$	−0.267949	−0.0155480
$298$	0	0
$299$	32.7846	1.89598
$300$	0	0
$301$	3.46410	0.199667
$302$	0	0
$303$	−14.6603	−0.842210
$304$	0	0
$305$	−1.07180	−0.0613709
$306$	0	0
$307$	14.9282	0.851998	0.425999	−	0.904724i	$-0.359923\pi$
$307$	14.9282	0.851998	0.425999	+	0.904724i	$0.359923\pi$
$308$	0	0
$309$	−1.46410	−0.0832898
$310$	0	0
$311$	−13.3205	−0.755337	−0.377668	−	0.925941i	$-0.623274\pi$
$311$	−13.3205	−0.755337	−0.377668	+	0.925941i	$0.623274\pi$
$312$	0	0
$313$	1.14359	0.0646397	0.0323199	−	0.999478i	$-0.489710\pi$
$313$	1.14359	0.0646397	0.0323199	+	0.999478i	$0.489710\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−21.5885	−1.21253	−0.606264	−	0.795263i	$-0.707333\pi$
$317$	−21.5885	−1.21253	−0.606264	+	0.795263i	$0.707333\pi$
$318$	0	0
$319$	−0.928203	−0.0519694
$320$	0	0
$321$	−2.00000	−0.111629
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3.46410	0.192154
$326$	0	0
$327$	12.3923	0.685296
$328$	0	0
$329$	3.73205	0.205755
$330$	0	0
$331$	0.928203	0.0510187	0.0255093	−	0.999675i	$-0.491879\pi$
$331$	0.928203	0.0510187	0.0255093	+	0.999675i	$0.491879\pi$
$332$	0	0
$333$	7.92820	0.434463
$334$	0	0
$335$	−10.0000	−0.546358
$336$	0	0
$337$	−15.3205	−0.834561	−0.417281	−	0.908778i	$-0.637017\pi$
$337$	−15.3205	−0.834561	−0.417281	+	0.908778i	$0.637017\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	0.679492	0.0367966
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	−18.9282	−1.01906
$346$	0	0
$347$	20.2679	1.08804	0.544020	−	0.839072i	$-0.316901\pi$
$347$	20.2679	1.08804	0.544020	+	0.839072i	$0.316901\pi$
$348$	0	0
$349$	−20.8564	−1.11642	−0.558209	−	0.829700i	$-0.688511\pi$
$349$	−20.8564	−1.11642	−0.558209	+	0.829700i	$0.688511\pi$
$350$	0	0
$351$	−3.46410	−0.184900
$352$	0	0
$353$	17.0526	0.907616	0.453808	−	0.891099i	$-0.350065\pi$
$353$	17.0526	0.907616	0.453808	+	0.891099i	$0.350065\pi$
$354$	0	0
$355$	1.85641	0.0985278
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.32051	−0.386362	−0.193181	−	0.981163i	$-0.561880\pi$
$359$	−7.32051	−0.386362	−0.193181	+	0.981163i	$0.561880\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−10.9282	−0.573582
$364$	0	0
$365$	−2.14359	−0.112201
$366$	0	0
$367$	9.92820	0.518248	0.259124	−	0.965844i	$-0.416566\pi$
$367$	9.92820	0.518248	0.259124	+	0.965844i	$0.416566\pi$
$368$	0	0
$369$	−0.803848	−0.0418466
$370$	0	0
$371$	−7.73205	−0.401428
$372$	0	0
$373$	6.14359	0.318103	0.159052	−	0.987270i	$-0.449156\pi$
$373$	6.14359	0.318103	0.159052	+	0.987270i	$0.449156\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	11.9282	0.612711	0.306355	−	0.951917i	$-0.400890\pi$
$379$	11.9282	0.612711	0.306355	+	0.951917i	$0.400890\pi$
$380$	0	0
$381$	−12.3923	−0.634877
$382$	0	0
$383$	31.7321	1.62143	0.810716	−	0.585440i	$-0.199078\pi$
$383$	31.7321	1.62143	0.810716	+	0.585440i	$0.199078\pi$
$384$	0	0
$385$	0.535898	0.0273119
$386$	0	0
$387$	−3.46410	−0.176090
$388$	0	0
$389$	13.0526	0.661791	0.330896	−	0.943667i	$-0.392649\pi$
$389$	13.0526	0.661791	0.330896	+	0.943667i	$0.392649\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−11.4641	−0.578287
$394$	0	0
$395$	−3.85641	−0.194037
$396$	0	0
$397$	−15.0000	−0.752828	−0.376414	−	0.926451i	$-0.622843\pi$
$397$	−15.0000	−0.752828	−0.376414	+	0.926451i	$0.622843\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12.3923	0.618842	0.309421	−	0.950925i	$-0.399865\pi$
$401$	12.3923	0.618842	0.309421	+	0.950925i	$0.399865\pi$
$402$	0	0
$403$	8.78461	0.437593
$404$	0	0
$405$	2.00000	0.0993808
$406$	0	0
$407$	−2.12436	−0.105300
$408$	0	0
$409$	−5.07180	−0.250784	−0.125392	−	0.992107i	$-0.540019\pi$
$409$	−5.07180	−0.250784	−0.125392	+	0.992107i	$0.540019\pi$
$410$	0	0
$411$	−19.3205	−0.953010
$412$	0	0
$413$	10.6603	0.524557
$414$	0	0
$415$	−2.92820	−0.143740
$416$	0	0
$417$	17.4641	0.855221
$418$	0	0
$419$	22.3923	1.09394	0.546968	−	0.837154i	$-0.315782\pi$
$419$	22.3923	1.09394	0.546968	+	0.837154i	$0.315782\pi$
$420$	0	0
$421$	25.1769	1.22705	0.613524	−	0.789676i	$-0.289751\pi$
$421$	25.1769	1.22705	0.613524	+	0.789676i	$0.289751\pi$
$422$	0	0
$423$	−3.73205	−0.181459
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.535898	0.0259339
$428$	0	0
$429$	0.928203	0.0448141
$430$	0	0
$431$	33.7128	1.62389	0.811945	−	0.583735i	$-0.198409\pi$
$431$	33.7128	1.62389	0.811945	+	0.583735i	$0.198409\pi$
$432$	0	0
$433$	−1.85641	−0.0892132	−0.0446066	−	0.999005i	$-0.514203\pi$
$433$	−1.85641	−0.0892132	−0.0446066	+	0.999005i	$0.514203\pi$
$434$	0	0
$435$	6.92820	0.332182
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−9.46410	−0.451697	−0.225848	−	0.974162i	$-0.572515\pi$
$439$	−9.46410	−0.451697	−0.225848	+	0.974162i	$0.572515\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	5.32051	0.252785	0.126392	−	0.991980i	$-0.459660\pi$
$443$	5.32051	0.252785	0.126392	+	0.991980i	$0.459660\pi$
$444$	0	0
$445$	−13.8564	−0.656857
$446$	0	0
$447$	4.00000	0.189194
$448$	0	0
$449$	9.33975	0.440770	0.220385	−	0.975413i	$-0.429269\pi$
$449$	9.33975	0.440770	0.220385	+	0.975413i	$0.429269\pi$
$450$	0	0
$451$	0.215390	0.0101423
$452$	0	0
$453$	−1.00000	−0.0469841
$454$	0	0
$455$	6.92820	0.324799
$456$	0	0
$457$	−32.9282	−1.54032	−0.770158	−	0.637853i	$-0.779823\pi$
$457$	−32.9282	−1.54032	−0.770158	+	0.637853i	$0.779823\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.5359	1.04960	0.524801	−	0.851225i	$-0.324140\pi$
$461$	22.5359	1.04960	0.524801	+	0.851225i	$0.324140\pi$
$462$	0	0
$463$	−34.2487	−1.59167	−0.795836	−	0.605512i	$-0.792968\pi$
$463$	−34.2487	−1.59167	−0.795836	+	0.605512i	$0.792968\pi$
$464$	0	0
$465$	−5.07180	−0.235199
$466$	0	0
$467$	−32.9282	−1.52374	−0.761868	−	0.647733i	$-0.775717\pi$
$467$	−32.9282	−1.52374	−0.761868	+	0.647733i	$0.775717\pi$
$468$	0	0
$469$	5.00000	0.230879
$470$	0	0
$471$	23.4641	1.08117
$472$	0	0
$473$	0.928203	0.0426788
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.73205	0.354026
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−27.4641	−1.25226
$482$	0	0
$483$	9.46410	0.430632
$484$	0	0
$485$	−15.8564	−0.720002
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	−1.07180	−0.0484683
$490$	0	0
$491$	10.3923	0.468998	0.234499	−	0.972116i	$-0.424655\pi$
$491$	10.3923	0.468998	0.234499	+	0.972116i	$0.424655\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−0.535898	−0.0240868
$496$	0	0
$497$	−0.928203	−0.0416356
$498$	0	0
$499$	12.8564	0.575532	0.287766	−	0.957701i	$-0.407087\pi$
$499$	12.8564	0.575532	0.287766	+	0.957701i	$0.407087\pi$
$500$	0	0
$501$	10.1244	0.452323
$502$	0	0
$503$	−23.4449	−1.04535	−0.522677	−	0.852531i	$-0.675067\pi$
$503$	−23.4449	−1.04535	−0.522677	+	0.852531i	$0.675067\pi$
$504$	0	0
$505$	−29.3205	−1.30475
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	1.07180	0.0474135
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.92820	−0.129032
$516$	0	0
$517$	1.00000	0.0439799
$518$	0	0
$519$	4.00000	0.175581
$520$	0	0
$521$	21.1769	0.927777	0.463889	−	0.885893i	$-0.346454\pi$
$521$	21.1769	0.927777	0.463889	+	0.885893i	$0.346454\pi$
$522$	0	0
$523$	−27.6410	−1.20866	−0.604329	−	0.796735i	$-0.706559\pi$
$523$	−27.6410	−1.20866	−0.604329	+	0.796735i	$0.706559\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	0	0
$528$	0	0
$529$	66.5692	2.89431
$530$	0	0
$531$	−10.6603	−0.462616
$532$	0	0
$533$	2.78461	0.120615
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	−7.46410	−0.322100
$538$	0	0
$539$	1.60770	0.0692483
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	0	0
$543$	−12.3923	−0.531805
$544$	0	0
$545$	24.7846	1.06166
$546$	0	0
$547$	−33.4641	−1.43082	−0.715411	−	0.698704i	$-0.753760\pi$
$547$	−33.4641	−1.43082	−0.715411	+	0.698704i	$0.753760\pi$
$548$	0	0
$549$	−0.535898	−0.0228716
$550$	0	0
$551$	0	0
$552$	0	0
$553$	1.92820	0.0819955
$554$	0	0
$555$	15.8564	0.673067
$556$	0	0
$557$	13.8756	0.587930	0.293965	−	0.955816i	$-0.405025\pi$
$557$	13.8756	0.587930	0.293965	+	0.955816i	$0.405025\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−24.2679	−1.02277	−0.511386	−	0.859351i	$-0.670868\pi$
$563$	−24.2679	−1.02277	−0.511386	+	0.859351i	$0.670868\pi$
$564$	0	0
$565$	24.0000	1.00969
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	23.6077	0.989686	0.494843	−	0.868982i	$-0.335226\pi$
$569$	23.6077	0.989686	0.494843	+	0.868982i	$0.335226\pi$
$570$	0	0
$571$	3.07180	0.128551	0.0642753	−	0.997932i	$-0.479526\pi$
$571$	3.07180	0.128551	0.0642753	+	0.997932i	$0.479526\pi$
$572$	0	0
$573$	21.0526	0.879483
$574$	0	0
$575$	9.46410	0.394680
$576$	0	0
$577$	−6.07180	−0.252772	−0.126386	−	0.991981i	$-0.540338\pi$
$577$	−6.07180	−0.252772	−0.126386	+	0.991981i	$0.540338\pi$
$578$	0	0
$579$	10.8564	0.451177
$580$	0	0
$581$	1.46410	0.0607412
$582$	0	0
$583$	−2.07180	−0.0858051
$584$	0	0
$585$	−6.92820	−0.286446
$586$	0	0
$587$	2.78461	0.114933	0.0574666	−	0.998347i	$-0.481698\pi$
$587$	2.78461	0.114933	0.0574666	+	0.998347i	$0.481698\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−21.8564	−0.899052
$592$	0	0
$593$	−18.1244	−0.744278	−0.372139	−	0.928177i	$-0.621376\pi$
$593$	−18.1244	−0.744278	−0.372139	+	0.928177i	$0.621376\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.92820	−0.119843
$598$	0	0
$599$	−28.9282	−1.18197	−0.590987	−	0.806681i	$-0.701262\pi$
$599$	−28.9282	−1.18197	−0.590987	+	0.806681i	$0.701262\pi$
$600$	0	0
$601$	−37.7846	−1.54127	−0.770633	−	0.637279i	$-0.780060\pi$
$601$	−37.7846	−1.54127	−0.770633	+	0.637279i	$0.780060\pi$
$602$	0	0
$603$	−5.00000	−0.203616
$604$	0	0
$605$	−21.8564	−0.888589
$606$	0	0
$607$	27.7128	1.12483	0.562414	−	0.826856i	$-0.309873\pi$
$607$	27.7128	1.12483	0.562414	+	0.826856i	$0.309873\pi$
$608$	0	0
$609$	−3.46410	−0.140372
$610$	0	0
$611$	12.9282	0.523019
$612$	0	0
$613$	13.0000	0.525065	0.262533	−	0.964923i	$-0.415442\pi$
$613$	13.0000	0.525065	0.262533	+	0.964923i	$0.415442\pi$
$614$	0	0
$615$	−1.60770	−0.0648285
$616$	0	0
$617$	−6.92820	−0.278919	−0.139459	−	0.990228i	$-0.544536\pi$
$617$	−6.92820	−0.278919	−0.139459	+	0.990228i	$0.544536\pi$
$618$	0	0
$619$	−26.8564	−1.07945	−0.539725	−	0.841841i	$-0.681472\pi$
$619$	−26.8564	−1.07945	−0.539725	+	0.841841i	$0.681472\pi$
$620$	0	0
$621$	−9.46410	−0.379781
$622$	0	0
$623$	6.92820	0.277573
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	2.14359	0.0853351	0.0426676	−	0.999089i	$-0.486414\pi$
$631$	2.14359	0.0853351	0.0426676	+	0.999089i	$0.486414\pi$
$632$	0	0
$633$	3.92820	0.156132
$634$	0	0
$635$	−24.7846	−0.983547
$636$	0	0
$637$	20.7846	0.823516
$638$	0	0
$639$	0.928203	0.0367192
$640$	0	0
$641$	−5.85641	−0.231314	−0.115657	−	0.993289i	$-0.536897\pi$
$641$	−5.85641	−0.231314	−0.115657	+	0.993289i	$0.536897\pi$
$642$	0	0
$643$	−12.1436	−0.478897	−0.239448	−	0.970909i	$-0.576967\pi$
$643$	−12.1436	−0.478897	−0.239448	+	0.970909i	$0.576967\pi$
$644$	0	0
$645$	−6.92820	−0.272798
$646$	0	0
$647$	−8.24871	−0.324290	−0.162145	−	0.986767i	$-0.551841\pi$
$647$	−8.24871	−0.324290	−0.162145	+	0.986767i	$0.551841\pi$
$648$	0	0
$649$	2.85641	0.112124
$650$	0	0
$651$	2.53590	0.0993897
$652$	0	0
$653$	−38.1051	−1.49117	−0.745584	−	0.666411i	$-0.767829\pi$
$653$	−38.1051	−1.49117	−0.745584	+	0.666411i	$0.767829\pi$
$654$	0	0
$655$	−22.9282	−0.895879
$656$	0	0
$657$	−1.07180	−0.0418148
$658$	0	0
$659$	37.8372	1.47393	0.736963	−	0.675933i	$-0.236259\pi$
$659$	37.8372	1.47393	0.736963	+	0.675933i	$0.236259\pi$
$660$	0	0
$661$	−10.5359	−0.409799	−0.204899	−	0.978783i	$-0.565687\pi$
$661$	−10.5359	−0.409799	−0.204899	+	0.978783i	$0.565687\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−32.7846	−1.26943
$668$	0	0
$669$	4.00000	0.154649
$670$	0	0
$671$	0.143594	0.00554337
$672$	0	0
$673$	−46.7128	−1.80065	−0.900323	−	0.435222i	$-0.856670\pi$
$673$	−46.7128	−1.80065	−0.900323	+	0.435222i	$0.856670\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	8.80385	0.338359	0.169180	−	0.985585i	$-0.445888\pi$
$677$	8.80385	0.338359	0.169180	+	0.985585i	$0.445888\pi$
$678$	0	0
$679$	7.92820	0.304257
$680$	0	0
$681$	−14.3923	−0.551514
$682$	0	0
$683$	−14.5359	−0.556201	−0.278100	−	0.960552i	$-0.589705\pi$
$683$	−14.5359	−0.556201	−0.278100	+	0.960552i	$0.589705\pi$
$684$	0	0
$685$	−38.6410	−1.47640
$686$	0	0
$687$	−0.0717968	−0.00273922
$688$	0	0
$689$	−26.7846	−1.02041
$690$	0	0
$691$	−26.0718	−0.991818	−0.495909	−	0.868375i	$-0.665165\pi$
$691$	−26.0718	−0.991818	−0.495909	+	0.868375i	$0.665165\pi$
$692$	0	0
$693$	0.267949	0.0101785
$694$	0	0
$695$	34.9282	1.32490
$696$	0	0
$697$	0	0
$698$	0	0
$699$	6.66025	0.251914
$700$	0	0
$701$	0.784610	0.0296343	0.0148171	−	0.999890i	$-0.495283\pi$
$701$	0.784610	0.0296343	0.0148171	+	0.999890i	$0.495283\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−7.46410	−0.281114
$706$	0	0
$707$	14.6603	0.551356
$708$	0	0
$709$	46.7846	1.75703	0.878516	−	0.477712i	$-0.158534\pi$
$709$	46.7846	1.75703	0.878516	+	0.477712i	$0.158534\pi$
$710$	0	0
$711$	−1.92820	−0.0723133
$712$	0	0
$713$	24.0000	0.898807
$714$	0	0
$715$	1.85641	0.0694257
$716$	0	0
$717$	−20.5359	−0.766927
$718$	0	0
$719$	−2.24871	−0.0838628	−0.0419314	−	0.999120i	$-0.513351\pi$
$719$	−2.24871	−0.0838628	−0.0419314	+	0.999120i	$0.513351\pi$
$720$	0	0
$721$	1.46410	0.0545260
$722$	0	0
$723$	−25.0000	−0.929760
$724$	0	0
$725$	−3.46410	−0.128654
$726$	0	0
$727$	21.7128	0.805284	0.402642	−	0.915358i	$-0.368092\pi$
$727$	21.7128	0.805284	0.402642	+	0.915358i	$0.368092\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−5.07180	−0.187331	−0.0936655	−	0.995604i	$-0.529858\pi$
$733$	−5.07180	−0.187331	−0.0936655	+	0.995604i	$0.529858\pi$
$734$	0	0
$735$	−12.0000	−0.442627
$736$	0	0
$737$	1.33975	0.0493502
$738$	0	0
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	41.5885	1.52573	0.762866	−	0.646556i	$-0.223791\pi$
$743$	41.5885	1.52573	0.762866	+	0.646556i	$0.223791\pi$
$744$	0	0
$745$	8.00000	0.293097
$746$	0	0
$747$	−1.46410	−0.0535687
$748$	0	0
$749$	2.00000	0.0730784
$750$	0	0
$751$	16.7846	0.612479	0.306240	−	0.951954i	$-0.400929\pi$
$751$	16.7846	0.612479	0.306240	+	0.951954i	$0.400929\pi$
$752$	0	0
$753$	19.4641	0.709311
$754$	0	0
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	−43.6410	−1.58616	−0.793080	−	0.609118i	$-0.791524\pi$
$757$	−43.6410	−1.58616	−0.793080	+	0.609118i	$0.791524\pi$
$758$	0	0
$759$	2.53590	0.0920473
$760$	0	0
$761$	11.4449	0.414876	0.207438	−	0.978248i	$-0.433487\pi$
$761$	11.4449	0.414876	0.207438	+	0.978248i	$0.433487\pi$
$762$	0	0
$763$	−12.3923	−0.448632
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36.9282	1.33340
$768$	0	0
$769$	−13.0718	−0.471381	−0.235691	−	0.971828i	$-0.575735\pi$
$769$	−13.0718	−0.471381	−0.235691	+	0.971828i	$0.575735\pi$
$770$	0	0
$771$	17.8564	0.643083
$772$	0	0
$773$	−46.6410	−1.67756	−0.838780	−	0.544470i	$-0.816731\pi$
$773$	−46.6410	−1.67756	−0.838780	+	0.544470i	$0.816731\pi$
$774$	0	0
$775$	2.53590	0.0910922
$776$	0	0
$777$	−7.92820	−0.284423
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−0.248711	−0.00889959
$782$	0	0
$783$	3.46410	0.123797
$784$	0	0
$785$	46.9282	1.67494
$786$	0	0
$787$	−29.8564	−1.06427	−0.532133	−	0.846661i	$-0.678609\pi$
$787$	−29.8564	−1.06427	−0.532133	+	0.846661i	$0.678609\pi$
$788$	0	0
$789$	−8.00000	−0.284808
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	1.85641	0.0659229
$794$	0	0
$795$	15.4641	0.548455
$796$	0	0
$797$	31.3205	1.10943	0.554715	−	0.832041i	$-0.312827\pi$
$797$	31.3205	1.10943	0.554715	+	0.832041i	$0.312827\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.92820	−0.244796
$802$	0	0
$803$	0.287187	0.0101346
$804$	0	0
$805$	18.9282	0.667132
$806$	0	0
$807$	14.9282	0.525498
$808$	0	0
$809$	46.3731	1.63039	0.815195	−	0.579186i	$-0.196630\pi$
$809$	46.3731	1.63039	0.815195	+	0.579186i	$0.196630\pi$
$810$	0	0
$811$	38.7128	1.35939	0.679695	−	0.733495i	$-0.262112\pi$
$811$	38.7128	1.35939	0.679695	+	0.733495i	$0.262112\pi$
$812$	0	0
$813$	26.7128	0.936859
$814$	0	0
$815$	−2.14359	−0.0750868
$816$	0	0
$817$	0	0
$818$	0	0
$819$	3.46410	0.121046
$820$	0	0
$821$	−13.0526	−0.455537	−0.227769	−	0.973715i	$-0.573143\pi$
$821$	−13.0526	−0.455537	−0.227769	+	0.973715i	$0.573143\pi$
$822$	0	0
$823$	21.1769	0.738181	0.369090	−	0.929393i	$-0.379669\pi$
$823$	21.1769	0.738181	0.369090	+	0.929393i	$0.379669\pi$
$824$	0	0
$825$	0.267949	0.00932879
$826$	0	0
$827$	51.1962	1.78026	0.890132	−	0.455702i	$-0.150612\pi$
$827$	51.1962	1.78026	0.890132	+	0.455702i	$0.150612\pi$
$828$	0	0
$829$	40.7128	1.41401	0.707007	−	0.707206i	$-0.250045\pi$
$829$	40.7128	1.41401	0.707007	+	0.707206i	$0.250045\pi$
$830$	0	0
$831$	−22.7846	−0.790389
$832$	0	0
$833$	0	0
$834$	0	0
$835$	20.2487	0.700736
$836$	0	0
$837$	−2.53590	−0.0876535
$838$	0	0
$839$	19.4641	0.671975	0.335988	−	0.941866i	$-0.390930\pi$
$839$	19.4641	0.671975	0.335988	+	0.941866i	$0.390930\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	−21.5885	−0.743546
$844$	0	0
$845$	−2.00000	−0.0688021
$846$	0	0
$847$	10.9282	0.375498
$848$	0	0
$849$	5.00000	0.171600
$850$	0	0
$851$	−75.0333	−2.57211
$852$	0	0
$853$	−23.8564	−0.816828	−0.408414	−	0.912797i	$-0.633918\pi$
$853$	−23.8564	−0.816828	−0.408414	+	0.912797i	$0.633918\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.803848	0.0274589	0.0137295	−	0.999906i	$-0.495630\pi$
$857$	0.803848	0.0274589	0.0137295	+	0.999906i	$0.495630\pi$
$858$	0	0
$859$	−50.7128	−1.73030	−0.865149	−	0.501514i	$-0.832777\pi$
$859$	−50.7128	−1.73030	−0.865149	+	0.501514i	$0.832777\pi$
$860$	0	0
$861$	0.803848	0.0273951
$862$	0	0
$863$	−56.1051	−1.90984	−0.954920	−	0.296863i	$-0.904060\pi$
$863$	−56.1051	−1.90984	−0.954920	+	0.296863i	$0.904060\pi$
$864$	0	0
$865$	8.00000	0.272008
$866$	0	0
$867$	−17.0000	−0.577350
$868$	0	0
$869$	0.516660	0.0175265
$870$	0	0
$871$	17.3205	0.586883
$872$	0	0
$873$	−7.92820	−0.268329
$874$	0	0
$875$	12.0000	0.405674
$876$	0	0
$877$	25.1769	0.850164	0.425082	−	0.905155i	$-0.360245\pi$
$877$	25.1769	0.850164	0.425082	+	0.905155i	$0.360245\pi$
$878$	0	0
$879$	10.6603	0.359561
$880$	0	0
$881$	50.1244	1.68873	0.844366	−	0.535766i	$-0.179977\pi$
$881$	50.1244	1.68873	0.844366	+	0.535766i	$0.179977\pi$
$882$	0	0
$883$	26.1051	0.878507	0.439254	−	0.898363i	$-0.355243\pi$
$883$	26.1051	0.878507	0.439254	+	0.898363i	$0.355243\pi$
$884$	0	0
$885$	−21.3205	−0.716681
$886$	0	0
$887$	3.71281	0.124664	0.0623320	−	0.998055i	$-0.480146\pi$
$887$	3.71281	0.124664	0.0623320	+	0.998055i	$0.480146\pi$
$888$	0	0
$889$	12.3923	0.415625
$890$	0	0
$891$	−0.267949	−0.00897664
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−14.9282	−0.498995
$896$	0	0
$897$	32.7846	1.09465
$898$	0	0
$899$	−8.78461	−0.292983
$900$	0	0
$901$	0	0
$902$	0	0
$903$	3.46410	0.115278
$904$	0	0
$905$	−24.7846	−0.823868
$906$	0	0
$907$	−22.9282	−0.761318	−0.380659	−	0.924715i	$-0.624303\pi$
$907$	−22.9282	−0.761318	−0.380659	+	0.924715i	$0.624303\pi$
$908$	0	0
$909$	−14.6603	−0.486250
$910$	0	0
$911$	−39.1769	−1.29799	−0.648995	−	0.760793i	$-0.724810\pi$
$911$	−39.1769	−1.29799	−0.648995	+	0.760793i	$0.724810\pi$
$912$	0	0
$913$	0.392305	0.0129834
$914$	0	0
$915$	−1.07180	−0.0354325
$916$	0	0
$917$	11.4641	0.378578
$918$	0	0
$919$	49.8564	1.64461	0.822306	−	0.569046i	$-0.192687\pi$
$919$	49.8564	1.64461	0.822306	+	0.569046i	$0.192687\pi$
$920$	0	0
$921$	14.9282	0.491901
$922$	0	0
$923$	−3.21539	−0.105836
$924$	0	0
$925$	−7.92820	−0.260678
$926$	0	0
$927$	−1.46410	−0.0480874
$928$	0	0
$929$	−37.8372	−1.24140	−0.620699	−	0.784049i	$-0.713151\pi$
$929$	−37.8372	−1.24140	−0.620699	+	0.784049i	$0.713151\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−13.3205	−0.436094
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−24.1436	−0.788737	−0.394368	−	0.918952i	$-0.629037\pi$
$937$	−24.1436	−0.788737	−0.394368	+	0.918952i	$0.629037\pi$
$938$	0	0
$939$	1.14359	0.0373198
$940$	0	0
$941$	7.98076	0.260165	0.130083	−	0.991503i	$-0.458476\pi$
$941$	7.98076	0.260165	0.130083	+	0.991503i	$0.458476\pi$
$942$	0	0
$943$	7.60770	0.247741
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	−6.53590	−0.212388	−0.106194	−	0.994345i	$-0.533866\pi$
$947$	−6.53590	−0.212388	−0.106194	+	0.994345i	$0.533866\pi$
$948$	0	0
$949$	3.71281	0.120523
$950$	0	0
$951$	−21.5885	−0.700054
$952$	0	0
$953$	0.928203	0.0300675	0.0150337	−	0.999887i	$-0.495214\pi$
$953$	0.928203	0.0300675	0.0150337	+	0.999887i	$0.495214\pi$
$954$	0	0
$955$	42.1051	1.36249
$956$	0	0
$957$	−0.928203	−0.0300045
$958$	0	0
$959$	19.3205	0.623892
$960$	0	0
$961$	−24.5692	−0.792555
$962$	0	0
$963$	−2.00000	−0.0644491
$964$	0	0
$965$	21.7128	0.698960
$966$	0	0
$967$	−19.7128	−0.633921	−0.316961	−	0.948439i	$-0.602662\pi$
$967$	−19.7128	−0.633921	−0.316961	+	0.948439i	$0.602662\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	57.3205	1.83950	0.919751	−	0.392502i	$-0.128390\pi$
$971$	57.3205	1.83950	0.919751	+	0.392502i	$0.128390\pi$
$972$	0	0
$973$	−17.4641	−0.559873
$974$	0	0
$975$	3.46410	0.110940
$976$	0	0
$977$	6.41154	0.205123	0.102562	−	0.994727i	$-0.467296\pi$
$977$	6.41154	0.205123	0.102562	+	0.994727i	$0.467296\pi$
$978$	0	0
$979$	1.85641	0.0593310
$980$	0	0
$981$	12.3923	0.395656
$982$	0	0
$983$	55.1769	1.75987	0.879935	−	0.475094i	$-0.157586\pi$
$983$	55.1769	1.75987	0.879935	+	0.475094i	$0.157586\pi$
$984$	0	0
$985$	−43.7128	−1.39281
$986$	0	0
$987$	3.73205	0.118792
$988$	0	0
$989$	32.7846	1.04249
$990$	0	0
$991$	43.8564	1.39314	0.696572	−	0.717487i	$-0.254708\pi$
$991$	43.8564	1.39314	0.696572	+	0.717487i	$0.254708\pi$
$992$	0	0
$993$	0.928203	0.0294556
$994$	0	0
$995$	−5.85641	−0.185661
$996$	0	0
$997$	9.78461	0.309882	0.154941	−	0.987924i	$-0.450481\pi$
$997$	9.78461	0.309882	0.154941	+	0.987924i	$0.450481\pi$
$998$	0	0
$999$	7.92820	0.250837

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7248.2.a.t.1.2		2
4.3	odd	2		453.2.a.c.1.1	✓	2
12.11	even	2		1359.2.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
453.2.a.c.1.1	✓	2	4.3	odd	2
1359.2.a.b.1.2		2	12.11	even	2
7248.2.a.t.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:	\( N \)	\(=\)	\( 7248 = 2^{4} \cdot 3 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7248.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -0.267949 q^{11} -3.46410 q^{13} +2.00000 q^{15} -1.00000 q^{21} -9.46410 q^{23} -1.00000 q^{25} +1.00000 q^{27} +3.46410 q^{29} -2.53590 q^{31} -0.267949 q^{33} -2.00000 q^{35} +7.92820 q^{37} -3.46410 q^{39} -0.803848 q^{41} -3.46410 q^{43} +2.00000 q^{45} -3.73205 q^{47} -6.00000 q^{49} +7.73205 q^{53} -0.535898 q^{55} -10.6603 q^{59} -0.535898 q^{61} -1.00000 q^{63} -6.92820 q^{65} -5.00000 q^{67} -9.46410 q^{69} +0.928203 q^{71} -1.07180 q^{73} -1.00000 q^{75} +0.267949 q^{77} -1.92820 q^{79} +1.00000 q^{81} -1.46410 q^{83} +3.46410 q^{87} -6.92820 q^{89} +3.46410 q^{91} -2.53590 q^{93} -7.92820 q^{97} -0.267949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{11} + 4 q^{15} - 2 q^{21} - 12 q^{23} - 2 q^{25} + 2 q^{27} - 12 q^{31} - 4 q^{33} - 4 q^{35} + 2 q^{37} - 12 q^{41} + 4 q^{45} - 4 q^{47} - 12 q^{49} + 12 q^{53} - 8 q^{55} - 4 q^{59} - 8 q^{61} - 2 q^{63} - 10 q^{67} - 12 q^{69} - 12 q^{71} - 16 q^{73} - 2 q^{75} + 4 q^{77} + 10 q^{79} + 2 q^{81} + 4 q^{83} - 12 q^{93} - 2 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^11 + 4 * q^15 - 2 * q^21 - 12 * q^23 - 2 * q^25 + 2 * q^27 - 12 * q^31 - 4 * q^33 - 4 * q^35 + 2 * q^37 - 12 * q^41 + 4 * q^45 - 4 * q^47 - 12 * q^49 + 12 * q^53 - 8 * q^55 - 4 * q^59 - 8 * q^61 - 2 * q^63 - 10 * q^67 - 12 * q^69 - 12 * q^71 - 16 * q^73 - 2 * q^75 + 4 * q^77 + 10 * q^79 + 2 * q^81 + 4 * q^83 - 12 * q^93 - 2 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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