LMFDB - Embedded newform 6048.2.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(1,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, 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("6048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.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:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	6048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+2.41421 q^{5} +1.00000 q^{7} +2.41421 q^{11} +2.82843 q^{13} +2.82843 q^{17} +3.82843 q^{19} +5.24264 q^{23} +0.828427 q^{25} -2.82843 q^{29} +4.65685 q^{31} +2.41421 q^{35} +0.171573 q^{37} -2.07107 q^{41} +4.82843 q^{43} +0.343146 q^{47} +1.00000 q^{49} -4.00000 q^{53} +5.82843 q^{55} -3.65685 q^{59} -11.3137 q^{61} +6.82843 q^{65} -4.48528 q^{67} +5.58579 q^{71} -3.17157 q^{73} +2.41421 q^{77} -4.82843 q^{79} +10.8284 q^{83} +6.82843 q^{85} +1.24264 q^{89} +2.82843 q^{91} +9.24264 q^{95} -9.65685 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 + 2 * q^7	$ 2 q + 2 q^{5} + 2 q^{7} + 2 q^{11} + 2 q^{19} + 2 q^{23} - 4 q^{25} - 2 q^{31} + 2 q^{35} + 6 q^{37} + 10 q^{41} + 4 q^{43} + 12 q^{47} + 2 q^{49} - 8 q^{53} + 6 q^{55} + 4 q^{59} + 8 q^{65} + 8 q^{67} + 14 q^{71} - 12 q^{73} + 2 q^{77} - 4 q^{79} + 16 q^{83} + 8 q^{85} - 6 q^{89} + 10 q^{95} - 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^7 + 2 * q^11 + 2 * q^19 + 2 * q^23 - 4 * q^25 - 2 * q^31 + 2 * q^35 + 6 * q^37 + 10 * q^41 + 4 * q^43 + 12 * q^47 + 2 * q^49 - 8 * q^53 + 6 * q^55 + 4 * q^59 + 8 * q^65 + 8 * q^67 + 14 * q^71 - 12 * q^73 + 2 * q^77 - 4 * q^79 + 16 * q^83 + 8 * q^85 - 6 * q^89 + 10 * q^95 - 8 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.41421	1.07967	0.539835	−	0.841771i	$-0.318487\pi$
$5$	2.41421	1.07967	0.539835	+	0.841771i	$0.318487\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.41421	0.727913	0.363956	−	0.931416i	$-0.381426\pi$
$11$	2.41421	0.727913	0.363956	+	0.931416i	$0.381426\pi$
$12$	0	0
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	3.82843	0.878301	0.439151	−	0.898413i	$-0.355279\pi$
$19$	3.82843	0.878301	0.439151	+	0.898413i	$0.355279\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.24264	1.09317	0.546583	−	0.837405i	$-0.315928\pi$
$23$	5.24264	1.09317	0.546583	+	0.837405i	$0.315928\pi$
$24$	0	0
$25$	0.828427	0.165685
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.82843	−0.525226	−0.262613	−	0.964901i	$-0.584584\pi$
$29$	−2.82843	−0.525226	−0.262613	+	0.964901i	$0.584584\pi$
$30$	0	0
$31$	4.65685	0.836396	0.418198	−	0.908356i	$-0.362662\pi$
$31$	4.65685	0.836396	0.418198	+	0.908356i	$0.362662\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.41421	0.408077
$36$	0	0
$37$	0.171573	0.0282064	0.0141032	−	0.999901i	$-0.495511\pi$
$37$	0.171573	0.0282064	0.0141032	+	0.999901i	$0.495511\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.07107	−0.323446	−0.161723	−	0.986836i	$-0.551705\pi$
$41$	−2.07107	−0.323446	−0.161723	+	0.986836i	$0.551705\pi$
$42$	0	0
$43$	4.82843	0.736328	0.368164	−	0.929761i	$-0.379986\pi$
$43$	4.82843	0.736328	0.368164	+	0.929761i	$0.379986\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.343146	0.0500530	0.0250265	−	0.999687i	$-0.492033\pi$
$47$	0.343146	0.0500530	0.0250265	+	0.999687i	$0.492033\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	5.82843	0.785905
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.65685	−0.476082	−0.238041	−	0.971255i	$-0.576505\pi$
$59$	−3.65685	−0.476082	−0.238041	+	0.971255i	$0.576505\pi$
$60$	0	0
$61$	−11.3137	−1.44857	−0.724286	−	0.689500i	$-0.757830\pi$
$61$	−11.3137	−1.44857	−0.724286	+	0.689500i	$0.757830\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.82843	0.846962
$66$	0	0
$67$	−4.48528	−0.547964	−0.273982	−	0.961735i	$-0.588341\pi$
$67$	−4.48528	−0.547964	−0.273982	+	0.961735i	$0.588341\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.58579	0.662911	0.331455	−	0.943471i	$-0.392460\pi$
$71$	5.58579	0.662911	0.331455	+	0.943471i	$0.392460\pi$
$72$	0	0
$73$	−3.17157	−0.371205	−0.185602	−	0.982625i	$-0.559424\pi$
$73$	−3.17157	−0.371205	−0.185602	+	0.982625i	$0.559424\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.41421	0.275125
$78$	0	0
$79$	−4.82843	−0.543240	−0.271620	−	0.962405i	$-0.587559\pi$
$79$	−4.82843	−0.543240	−0.271620	+	0.962405i	$0.587559\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.8284	1.18857	0.594287	−	0.804253i	$-0.297434\pi$
$83$	10.8284	1.18857	0.594287	+	0.804253i	$0.297434\pi$
$84$	0	0
$85$	6.82843	0.740647
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.24264	0.131720	0.0658598	−	0.997829i	$-0.479021\pi$
$89$	1.24264	0.131720	0.0658598	+	0.997829i	$0.479021\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	0	0
$93$	0	0
$94$	0	0
$95$	9.24264	0.948275
$96$	0	0
$97$	−9.65685	−0.980505	−0.490252	−	0.871580i	$-0.663095\pi$
$97$	−9.65685	−0.980505	−0.490252	+	0.871580i	$0.663095\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.4853	−1.24233	−0.621166	−	0.783679i	$-0.713341\pi$
$101$	−12.4853	−1.24233	−0.621166	+	0.783679i	$0.713341\pi$
$102$	0	0
$103$	−6.65685	−0.655919	−0.327960	−	0.944692i	$-0.606361\pi$
$103$	−6.65685	−0.655919	−0.327960	+	0.944692i	$0.606361\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.31371	−0.513696	−0.256848	−	0.966452i	$-0.582684\pi$
$107$	−5.31371	−0.513696	−0.256848	+	0.966452i	$0.582684\pi$
$108$	0	0
$109$	18.3137	1.75414	0.877068	−	0.480367i	$-0.159497\pi$
$109$	18.3137	1.75414	0.877068	+	0.480367i	$0.159497\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.1421	1.14224	0.571118	−	0.820868i	$-0.306510\pi$
$113$	12.1421	1.14224	0.571118	+	0.820868i	$0.306510\pi$
$114$	0	0
$115$	12.6569	1.18026
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.82843	0.259281
$120$	0	0
$121$	−5.17157	−0.470143
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.0711	−0.900784
$126$	0	0
$127$	−2.82843	−0.250982	−0.125491	−	0.992095i	$-0.540051\pi$
$127$	−2.82843	−0.250982	−0.125491	+	0.992095i	$0.540051\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	0	0
$133$	3.82843	0.331967
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.82843	−0.241649	−0.120824	−	0.992674i	$-0.538554\pi$
$137$	−2.82843	−0.241649	−0.120824	+	0.992674i	$0.538554\pi$
$138$	0	0
$139$	1.65685	0.140533	0.0702663	−	0.997528i	$-0.477615\pi$
$139$	1.65685	0.140533	0.0702663	+	0.997528i	$0.477615\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.82843	0.571022
$144$	0	0
$145$	−6.82843	−0.567070
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.17157	0.259825	0.129913	−	0.991525i	$-0.458530\pi$
$149$	3.17157	0.259825	0.129913	+	0.991525i	$0.458530\pi$
$150$	0	0
$151$	5.31371	0.432423	0.216212	−	0.976346i	$-0.430630\pi$
$151$	5.31371	0.432423	0.216212	+	0.976346i	$0.430630\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	11.2426	0.903031
$156$	0	0
$157$	−14.1421	−1.12867	−0.564333	−	0.825547i	$-0.690866\pi$
$157$	−14.1421	−1.12867	−0.564333	+	0.825547i	$0.690866\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.24264	0.413178
$162$	0	0
$163$	−16.4853	−1.29123	−0.645613	−	0.763664i	$-0.723398\pi$
$163$	−16.4853	−1.29123	−0.645613	+	0.763664i	$0.723398\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.51472	0.117212	0.0586062	−	0.998281i	$-0.481334\pi$
$167$	1.51472	0.117212	0.0586062	+	0.998281i	$0.481334\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.24264	0.702705	0.351352	−	0.936243i	$-0.385722\pi$
$173$	9.24264	0.702705	0.351352	+	0.936243i	$0.385722\pi$
$174$	0	0
$175$	0.828427	0.0626232
$176$	0	0
$177$	0	0
$178$	0	0
$179$	19.6569	1.46922	0.734611	−	0.678488i	$-0.237365\pi$
$179$	19.6569	1.46922	0.734611	+	0.678488i	$0.237365\pi$
$180$	0	0
$181$	19.6569	1.46108	0.730541	−	0.682869i	$-0.239268\pi$
$181$	19.6569	1.46108	0.730541	+	0.682869i	$0.239268\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.414214	0.0304536
$186$	0	0
$187$	6.82843	0.499344
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.4142	0.898261	0.449130	−	0.893466i	$-0.351734\pi$
$191$	12.4142	0.898261	0.449130	+	0.893466i	$0.351734\pi$
$192$	0	0
$193$	−3.65685	−0.263226	−0.131613	−	0.991301i	$-0.542016\pi$
$193$	−3.65685	−0.263226	−0.131613	+	0.991301i	$0.542016\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.8284	−1.76895	−0.884476	−	0.466586i	$-0.845484\pi$
$197$	−24.8284	−1.76895	−0.884476	+	0.466586i	$0.845484\pi$
$198$	0	0
$199$	−3.48528	−0.247065	−0.123533	−	0.992341i	$-0.539422\pi$
$199$	−3.48528	−0.247065	−0.123533	+	0.992341i	$0.539422\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.82843	−0.198517
$204$	0	0
$205$	−5.00000	−0.349215
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.24264	0.639327
$210$	0	0
$211$	−6.00000	−0.413057	−0.206529	−	0.978441i	$-0.566217\pi$
$211$	−6.00000	−0.413057	−0.206529	+	0.978441i	$0.566217\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.6569	0.794991
$216$	0	0
$217$	4.65685	0.316128
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	−24.7990	−1.66066	−0.830332	−	0.557270i	$-0.811849\pi$
$223$	−24.7990	−1.66066	−0.830332	+	0.557270i	$0.811849\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.8284	1.38243	0.691216	−	0.722649i	$-0.257075\pi$
$227$	20.8284	1.38243	0.691216	+	0.722649i	$0.257075\pi$
$228$	0	0
$229$	19.3137	1.27629	0.638143	−	0.769918i	$-0.279703\pi$
$229$	19.3137	1.27629	0.638143	+	0.769918i	$0.279703\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.8284	−0.840418	−0.420209	−	0.907427i	$-0.638043\pi$
$233$	−12.8284	−0.840418	−0.420209	+	0.907427i	$0.638043\pi$
$234$	0	0
$235$	0.828427	0.0540406
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.6569	1.01276	0.506379	−	0.862311i	$-0.330984\pi$
$239$	15.6569	1.01276	0.506379	+	0.862311i	$0.330984\pi$
$240$	0	0
$241$	−24.1421	−1.55513	−0.777566	−	0.628802i	$-0.783546\pi$
$241$	−24.1421	−1.55513	−0.777566	+	0.628802i	$0.783546\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.41421	0.154238
$246$	0	0
$247$	10.8284	0.688996
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.9706	1.57613	0.788064	−	0.615593i	$-0.211084\pi$
$251$	24.9706	1.57613	0.788064	+	0.615593i	$0.211084\pi$
$252$	0	0
$253$	12.6569	0.795730
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.27208	0.391241	0.195621	−	0.980680i	$-0.437328\pi$
$257$	6.27208	0.391241	0.195621	+	0.980680i	$0.437328\pi$
$258$	0	0
$259$	0.171573	0.0106610
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−13.3848	−0.825341	−0.412670	−	0.910880i	$-0.635404\pi$
$263$	−13.3848	−0.825341	−0.412670	+	0.910880i	$0.635404\pi$
$264$	0	0
$265$	−9.65685	−0.593216
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−15.5858	−0.950282	−0.475141	−	0.879910i	$-0.657603\pi$
$269$	−15.5858	−0.950282	−0.475141	+	0.879910i	$0.657603\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.00000	0.120605
$276$	0	0
$277$	13.8284	0.830870	0.415435	−	0.909623i	$-0.363629\pi$
$277$	13.8284	0.830870	0.415435	+	0.909623i	$0.363629\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−15.3137	−0.910305	−0.455153	−	0.890413i	$-0.650415\pi$
$283$	−15.3137	−0.910305	−0.455153	+	0.890413i	$0.650415\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.07107	−0.122251
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.17157	0.535809	0.267905	−	0.963445i	$-0.413669\pi$
$293$	9.17157	0.535809	0.267905	+	0.963445i	$0.413669\pi$
$294$	0	0
$295$	−8.82843	−0.514011
$296$	0	0
$297$	0	0
$298$	0	0
$299$	14.8284	0.857550
$300$	0	0
$301$	4.82843	0.278306
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−27.3137	−1.56398
$306$	0	0
$307$	−13.8284	−0.789230	−0.394615	−	0.918847i	$-0.629122\pi$
$307$	−13.8284	−0.789230	−0.394615	+	0.918847i	$0.629122\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−0.485281	−0.0275178	−0.0137589	−	0.999905i	$-0.504380\pi$
$311$	−0.485281	−0.0275178	−0.0137589	+	0.999905i	$0.504380\pi$
$312$	0	0
$313$	−18.8284	−1.06425	−0.532123	−	0.846667i	$-0.678606\pi$
$313$	−18.8284	−1.06425	−0.532123	+	0.846667i	$0.678606\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	34.2843	1.92560	0.962798	−	0.270221i	$-0.0870968\pi$
$317$	34.2843	1.92560	0.962798	+	0.270221i	$0.0870968\pi$
$318$	0	0
$319$	−6.82843	−0.382319
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10.8284	0.602510
$324$	0	0
$325$	2.34315	0.129974
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.343146	0.0189182
$330$	0	0
$331$	−34.6274	−1.90329	−0.951647	−	0.307192i	$-0.900611\pi$
$331$	−34.6274	−1.90329	−0.951647	+	0.307192i	$0.900611\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.8284	−0.591620
$336$	0	0
$337$	25.6274	1.39601	0.698007	−	0.716091i	$-0.254070\pi$
$337$	25.6274	1.39601	0.698007	+	0.716091i	$0.254070\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	11.2426	0.608823
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.0711	0.970106	0.485053	−	0.874485i	$-0.338800\pi$
$347$	18.0711	0.970106	0.485053	+	0.874485i	$0.338800\pi$
$348$	0	0
$349$	19.6569	1.05221	0.526104	−	0.850420i	$-0.323652\pi$
$349$	19.6569	1.05221	0.526104	+	0.850420i	$0.323652\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.414214	0.0220464	0.0110232	−	0.999939i	$-0.496491\pi$
$353$	0.414214	0.0220464	0.0110232	+	0.999939i	$0.496491\pi$
$354$	0	0
$355$	13.4853	0.715724
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.3137	1.12489	0.562447	−	0.826833i	$-0.309860\pi$
$359$	21.3137	1.12489	0.562447	+	0.826833i	$0.309860\pi$
$360$	0	0
$361$	−4.34315	−0.228587
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.65685	−0.400778
$366$	0	0
$367$	−2.51472	−0.131267	−0.0656336	−	0.997844i	$-0.520907\pi$
$367$	−2.51472	−0.131267	−0.0656336	+	0.997844i	$0.520907\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.00000	−0.207670
$372$	0	0
$373$	3.68629	0.190869	0.0954345	−	0.995436i	$-0.469576\pi$
$373$	3.68629	0.190869	0.0954345	+	0.995436i	$0.469576\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.00000	−0.412021
$378$	0	0
$379$	−14.4853	−0.744059	−0.372029	−	0.928221i	$-0.621338\pi$
$379$	−14.4853	−0.744059	−0.372029	+	0.928221i	$0.621338\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	35.4558	1.81171	0.905855	−	0.423589i	$-0.139230\pi$
$383$	35.4558	1.81171	0.905855	+	0.423589i	$0.139230\pi$
$384$	0	0
$385$	5.82843	0.297044
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−11.3137	−0.573628	−0.286814	−	0.957986i	$-0.592596\pi$
$389$	−11.3137	−0.573628	−0.286814	+	0.957986i	$0.592596\pi$
$390$	0	0
$391$	14.8284	0.749906
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11.6569	−0.586520
$396$	0	0
$397$	−3.65685	−0.183532	−0.0917661	−	0.995781i	$-0.529251\pi$
$397$	−3.65685	−0.183532	−0.0917661	+	0.995781i	$0.529251\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.8284	1.14000	0.569999	−	0.821646i	$-0.306944\pi$
$401$	22.8284	1.14000	0.569999	+	0.821646i	$0.306944\pi$
$402$	0	0
$403$	13.1716	0.656123
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.414214	0.0205318
$408$	0	0
$409$	−34.4853	−1.70519	−0.852594	−	0.522574i	$-0.824972\pi$
$409$	−34.4853	−1.70519	−0.852594	+	0.522574i	$0.824972\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.65685	−0.179942
$414$	0	0
$415$	26.1421	1.28327
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−0.142136	−0.00694378	−0.00347189	−	0.999994i	$-0.501105\pi$
$419$	−0.142136	−0.00694378	−0.00347189	+	0.999994i	$0.501105\pi$
$420$	0	0
$421$	10.5147	0.512456	0.256228	−	0.966616i	$-0.417520\pi$
$421$	10.5147	0.512456	0.256228	+	0.966616i	$0.417520\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.34315	0.113659
$426$	0	0
$427$	−11.3137	−0.547509
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.2132	−1.06997	−0.534986	−	0.844861i	$-0.679683\pi$
$431$	−22.2132	−1.06997	−0.534986	+	0.844861i	$0.679683\pi$
$432$	0	0
$433$	21.4558	1.03110	0.515551	−	0.856859i	$-0.327587\pi$
$433$	21.4558	1.03110	0.515551	+	0.856859i	$0.327587\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20.0711	0.960129
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.0710678	−0.00337653	−0.00168827	−	0.999999i	$-0.500537\pi$
$443$	−0.0710678	−0.00337653	−0.00168827	+	0.999999i	$0.500537\pi$
$444$	0	0
$445$	3.00000	0.142214
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−28.8284	−1.36050	−0.680249	−	0.732981i	$-0.738128\pi$
$449$	−28.8284	−1.36050	−0.680249	+	0.732981i	$0.738128\pi$
$450$	0	0
$451$	−5.00000	−0.235441
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.82843	0.320122
$456$	0	0
$457$	−8.65685	−0.404951	−0.202475	−	0.979287i	$-0.564899\pi$
$457$	−8.65685	−0.404951	−0.202475	+	0.979287i	$0.564899\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	25.2426	1.17567	0.587833	−	0.808982i	$-0.299981\pi$
$461$	25.2426	1.17567	0.587833	+	0.808982i	$0.299981\pi$
$462$	0	0
$463$	12.3431	0.573635	0.286817	−	0.957985i	$-0.407403\pi$
$463$	12.3431	0.573635	0.286817	+	0.957985i	$0.407403\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16.6274	−0.769425	−0.384713	−	0.923036i	$-0.625699\pi$
$467$	−16.6274	−0.769425	−0.384713	+	0.923036i	$0.625699\pi$
$468$	0	0
$469$	−4.48528	−0.207111
$470$	0	0
$471$	0	0
$472$	0	0
$473$	11.6569	0.535983
$474$	0	0
$475$	3.17157	0.145522
$476$	0	0
$477$	0	0
$478$	0	0
$479$	36.9706	1.68923	0.844614	−	0.535376i	$-0.179830\pi$
$479$	36.9706	1.68923	0.844614	+	0.535376i	$0.179830\pi$
$480$	0	0
$481$	0.485281	0.0221269
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−23.3137	−1.05862
$486$	0	0
$487$	22.4853	1.01891	0.509453	−	0.860499i	$-0.329848\pi$
$487$	22.4853	1.01891	0.509453	+	0.860499i	$0.329848\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−27.8701	−1.25776	−0.628879	−	0.777503i	$-0.716486\pi$
$491$	−27.8701	−1.25776	−0.628879	+	0.777503i	$0.716486\pi$
$492$	0	0
$493$	−8.00000	−0.360302
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.58579	0.250557
$498$	0	0
$499$	34.7696	1.55650	0.778249	−	0.627955i	$-0.216108\pi$
$499$	34.7696	1.55650	0.778249	+	0.627955i	$0.216108\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.1421	−0.452215	−0.226108	−	0.974102i	$-0.572600\pi$
$503$	−10.1421	−0.452215	−0.226108	+	0.974102i	$0.572600\pi$
$504$	0	0
$505$	−30.1421	−1.34131
$506$	0	0
$507$	0	0
$508$	0	0
$509$	32.4853	1.43988	0.719942	−	0.694034i	$-0.244168\pi$
$509$	32.4853	1.43988	0.719942	+	0.694034i	$0.244168\pi$
$510$	0	0
$511$	−3.17157	−0.140302
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.0711	−0.708176
$516$	0	0
$517$	0.828427	0.0364342
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−27.2426	−1.19352	−0.596761	−	0.802419i	$-0.703546\pi$
$521$	−27.2426	−1.19352	−0.596761	+	0.802419i	$0.703546\pi$
$522$	0	0
$523$	11.9706	0.523436	0.261718	−	0.965144i	$-0.415711\pi$
$523$	11.9706	0.523436	0.261718	+	0.965144i	$0.415711\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13.1716	0.573763
$528$	0	0
$529$	4.48528	0.195012
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.85786	−0.253732
$534$	0	0
$535$	−12.8284	−0.554621
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.41421	0.103988
$540$	0	0
$541$	−24.7990	−1.06619	−0.533096	−	0.846055i	$-0.678972\pi$
$541$	−24.7990	−1.06619	−0.533096	+	0.846055i	$0.678972\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	44.2132	1.89389
$546$	0	0
$547$	16.9706	0.725609	0.362804	−	0.931865i	$-0.381819\pi$
$547$	16.9706	0.725609	0.362804	+	0.931865i	$0.381819\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−10.8284	−0.461307
$552$	0	0
$553$	−4.82843	−0.205326
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.7696	1.30375	0.651874	−	0.758327i	$-0.273983\pi$
$557$	30.7696	1.30375	0.651874	+	0.758327i	$0.273983\pi$
$558$	0	0
$559$	13.6569	0.577623
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.7990	1.00301	0.501504	−	0.865155i	$-0.332780\pi$
$563$	23.7990	1.00301	0.501504	+	0.865155i	$0.332780\pi$
$564$	0	0
$565$	29.3137	1.23324
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9.79899	−0.410795	−0.205398	−	0.978679i	$-0.565849\pi$
$569$	−9.79899	−0.410795	−0.205398	+	0.978679i	$0.565849\pi$
$570$	0	0
$571$	12.6274	0.528441	0.264220	−	0.964462i	$-0.414885\pi$
$571$	12.6274	0.528441	0.264220	+	0.964462i	$0.414885\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.34315	0.181122
$576$	0	0
$577$	−39.9411	−1.66277	−0.831385	−	0.555696i	$-0.812452\pi$
$577$	−39.9411	−1.66277	−0.831385	+	0.555696i	$0.812452\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.8284	0.449239
$582$	0	0
$583$	−9.65685	−0.399946
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.4558	1.29832	0.649161	−	0.760651i	$-0.275120\pi$
$587$	31.4558	1.29832	0.649161	+	0.760651i	$0.275120\pi$
$588$	0	0
$589$	17.8284	0.734608
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−19.2426	−0.790201	−0.395100	−	0.918638i	$-0.629290\pi$
$593$	−19.2426	−0.790201	−0.395100	+	0.918638i	$0.629290\pi$
$594$	0	0
$595$	6.82843	0.279938
$596$	0	0
$597$	0	0
$598$	0	0
$599$	27.0416	1.10489	0.552446	−	0.833549i	$-0.313695\pi$
$599$	27.0416	1.10489	0.552446	+	0.833549i	$0.313695\pi$
$600$	0	0
$601$	−9.85786	−0.402111	−0.201055	−	0.979580i	$-0.564437\pi$
$601$	−9.85786	−0.402111	−0.201055	+	0.979580i	$0.564437\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−12.4853	−0.507599
$606$	0	0
$607$	−25.6569	−1.04138	−0.520690	−	0.853746i	$-0.674325\pi$
$607$	−25.6569	−1.04138	−0.520690	+	0.853746i	$0.674325\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.970563	0.0392648
$612$	0	0
$613$	8.02944	0.324306	0.162153	−	0.986766i	$-0.448156\pi$
$613$	8.02944	0.324306	0.162153	+	0.986766i	$0.448156\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−46.2843	−1.86333	−0.931667	−	0.363313i	$-0.881646\pi$
$617$	−46.2843	−1.86333	−0.931667	+	0.363313i	$0.881646\pi$
$618$	0	0
$619$	40.3137	1.62034	0.810172	−	0.586192i	$-0.199373\pi$
$619$	40.3137	1.62034	0.810172	+	0.586192i	$0.199373\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.24264	0.0497853
$624$	0	0
$625$	−28.4558	−1.13823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.485281	0.0193494
$630$	0	0
$631$	3.85786	0.153579	0.0767896	−	0.997047i	$-0.475533\pi$
$631$	3.85786	0.153579	0.0767896	+	0.997047i	$0.475533\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.82843	−0.270978
$636$	0	0
$637$	2.82843	0.112066
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−5.51472	−0.217818	−0.108909	−	0.994052i	$-0.534736\pi$
$641$	−5.51472	−0.217818	−0.108909	+	0.994052i	$0.534736\pi$
$642$	0	0
$643$	−3.48528	−0.137446	−0.0687230	−	0.997636i	$-0.521892\pi$
$643$	−3.48528	−0.137446	−0.0687230	+	0.997636i	$0.521892\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.6863	−0.420121	−0.210061	−	0.977688i	$-0.567366\pi$
$647$	−10.6863	−0.420121	−0.210061	+	0.977688i	$0.567366\pi$
$648$	0	0
$649$	−8.82843	−0.346546
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.02944	0.196817	0.0984085	−	0.995146i	$-0.468625\pi$
$653$	5.02944	0.196817	0.0984085	+	0.995146i	$0.468625\pi$
$654$	0	0
$655$	4.82843	0.188662
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−36.8995	−1.43740	−0.718700	−	0.695320i	$-0.755263\pi$
$659$	−36.8995	−1.43740	−0.718700	+	0.695320i	$0.755263\pi$
$660$	0	0
$661$	10.8284	0.421177	0.210589	−	0.977575i	$-0.432462\pi$
$661$	10.8284	0.421177	0.210589	+	0.977575i	$0.432462\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.24264	0.358414
$666$	0	0
$667$	−14.8284	−0.574159
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−27.3137	−1.05443
$672$	0	0
$673$	34.2843	1.32156	0.660781	−	0.750579i	$-0.270225\pi$
$673$	34.2843	1.32156	0.660781	+	0.750579i	$0.270225\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.04163	−0.193766	−0.0968828	−	0.995296i	$-0.530887\pi$
$677$	−5.04163	−0.193766	−0.0968828	+	0.995296i	$0.530887\pi$
$678$	0	0
$679$	−9.65685	−0.370596
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12.8995	0.493585	0.246793	−	0.969068i	$-0.420623\pi$
$683$	12.8995	0.493585	0.246793	+	0.969068i	$0.420623\pi$
$684$	0	0
$685$	−6.82843	−0.260901
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−11.3137	−0.431018
$690$	0	0
$691$	24.6863	0.939111	0.469555	−	0.882903i	$-0.344414\pi$
$691$	24.6863	0.939111	0.469555	+	0.882903i	$0.344414\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	−5.85786	−0.221882
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−31.5147	−1.19029	−0.595147	−	0.803617i	$-0.702906\pi$
$701$	−31.5147	−1.19029	−0.595147	+	0.803617i	$0.702906\pi$
$702$	0	0
$703$	0.656854	0.0247737
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12.4853	−0.469557
$708$	0	0
$709$	26.3137	0.988232	0.494116	−	0.869396i	$-0.335492\pi$
$709$	26.3137	0.988232	0.494116	+	0.869396i	$0.335492\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	24.4142	0.914319
$714$	0	0
$715$	16.4853	0.616515
$716$	0	0
$717$	0	0
$718$	0	0
$719$	7.65685	0.285552	0.142776	−	0.989755i	$-0.454397\pi$
$719$	7.65685	0.285552	0.142776	+	0.989755i	$0.454397\pi$
$720$	0	0
$721$	−6.65685	−0.247914
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.34315	−0.0870222
$726$	0	0
$727$	6.62742	0.245797	0.122899	−	0.992419i	$-0.460781\pi$
$727$	6.62742	0.245797	0.122899	+	0.992419i	$0.460781\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	13.6569	0.505117
$732$	0	0
$733$	−0.828427	−0.0305987	−0.0152993	−	0.999883i	$-0.504870\pi$
$733$	−0.828427	−0.0305987	−0.0152993	+	0.999883i	$0.504870\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.8284	−0.398870
$738$	0	0
$739$	−30.9706	−1.13927	−0.569635	−	0.821898i	$-0.692916\pi$
$739$	−30.9706	−1.13927	−0.569635	+	0.821898i	$0.692916\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.8995	0.619982	0.309991	−	0.950739i	$-0.399674\pi$
$743$	16.8995	0.619982	0.309991	+	0.950739i	$0.399674\pi$
$744$	0	0
$745$	7.65685	0.280525
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.31371	−0.194159
$750$	0	0
$751$	50.4853	1.84223	0.921117	−	0.389286i	$-0.127278\pi$
$751$	50.4853	1.84223	0.921117	+	0.389286i	$0.127278\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12.8284	0.466874
$756$	0	0
$757$	30.2843	1.10070	0.550350	−	0.834934i	$-0.314494\pi$
$757$	30.2843	1.10070	0.550350	+	0.834934i	$0.314494\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	47.7990	1.73271	0.866356	−	0.499427i	$-0.166456\pi$
$761$	47.7990	1.73271	0.866356	+	0.499427i	$0.166456\pi$
$762$	0	0
$763$	18.3137	0.663001
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.3431	−0.373469
$768$	0	0
$769$	1.85786	0.0669963	0.0334982	−	0.999439i	$-0.489335\pi$
$769$	1.85786	0.0669963	0.0334982	+	0.999439i	$0.489335\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−38.4142	−1.38166	−0.690832	−	0.723016i	$-0.742755\pi$
$773$	−38.4142	−1.38166	−0.690832	+	0.723016i	$0.742755\pi$
$774$	0	0
$775$	3.85786	0.138579
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.92893	−0.284083
$780$	0	0
$781$	13.4853	0.482541
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−34.1421	−1.21859
$786$	0	0
$787$	−25.9411	−0.924701	−0.462351	−	0.886697i	$-0.652994\pi$
$787$	−25.9411	−0.924701	−0.462351	+	0.886697i	$0.652994\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.1421	0.431725
$792$	0	0
$793$	−32.0000	−1.13635
$794$	0	0
$795$	0	0
$796$	0	0
$797$	40.8995	1.44873	0.724367	−	0.689414i	$-0.242132\pi$
$797$	40.8995	1.44873	0.724367	+	0.689414i	$0.242132\pi$
$798$	0	0
$799$	0.970563	0.0343360
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7.65685	−0.270205
$804$	0	0
$805$	12.6569	0.446095
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−48.4853	−1.70465	−0.852326	−	0.523011i	$-0.824809\pi$
$809$	−48.4853	−1.70465	−0.852326	+	0.523011i	$0.824809\pi$
$810$	0	0
$811$	−9.34315	−0.328082	−0.164041	−	0.986454i	$-0.552453\pi$
$811$	−9.34315	−0.328082	−0.164041	+	0.986454i	$0.552453\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−39.7990	−1.39410
$816$	0	0
$817$	18.4853	0.646718
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.28427	−0.219323	−0.109661	−	0.993969i	$-0.534977\pi$
$821$	−6.28427	−0.219323	−0.109661	+	0.993969i	$0.534977\pi$
$822$	0	0
$823$	−48.6274	−1.69505	−0.847523	−	0.530759i	$-0.821907\pi$
$823$	−48.6274	−1.69505	−0.847523	+	0.530759i	$0.821907\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.58579	0.263784	0.131892	−	0.991264i	$-0.457895\pi$
$827$	7.58579	0.263784	0.131892	+	0.991264i	$0.457895\pi$
$828$	0	0
$829$	8.14214	0.282788	0.141394	−	0.989953i	$-0.454842\pi$
$829$	8.14214	0.282788	0.141394	+	0.989953i	$0.454842\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.82843	0.0979992
$834$	0	0
$835$	3.65685	0.126551
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−34.1421	−1.17872	−0.589359	−	0.807871i	$-0.700620\pi$
$839$	−34.1421	−1.17872	−0.589359	+	0.807871i	$0.700620\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.0711	−0.415257
$846$	0	0
$847$	−5.17157	−0.177697
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.899495	0.0308343
$852$	0	0
$853$	11.0294	0.377641	0.188820	−	0.982012i	$-0.439534\pi$
$853$	11.0294	0.377641	0.188820	+	0.982012i	$0.439534\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.44365	−0.254270	−0.127135	−	0.991885i	$-0.540578\pi$
$857$	−7.44365	−0.254270	−0.127135	+	0.991885i	$0.540578\pi$
$858$	0	0
$859$	−46.1127	−1.57334	−0.786672	−	0.617371i	$-0.788198\pi$
$859$	−46.1127	−1.57334	−0.786672	+	0.617371i	$0.788198\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−35.6569	−1.21377	−0.606887	−	0.794788i	$-0.707582\pi$
$863$	−35.6569	−1.21377	−0.606887	+	0.794788i	$0.707582\pi$
$864$	0	0
$865$	22.3137	0.758689
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−11.6569	−0.395432
$870$	0	0
$871$	−12.6863	−0.429859
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−10.0711	−0.340464
$876$	0	0
$877$	39.9411	1.34872	0.674358	−	0.738405i	$-0.264420\pi$
$877$	39.9411	1.34872	0.674358	+	0.738405i	$0.264420\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	31.7279	1.06894	0.534470	−	0.845187i	$-0.320511\pi$
$881$	31.7279	1.06894	0.534470	+	0.845187i	$0.320511\pi$
$882$	0	0
$883$	−16.6863	−0.561538	−0.280769	−	0.959775i	$-0.590590\pi$
$883$	−16.6863	−0.561538	−0.280769	+	0.959775i	$0.590590\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32.7696	−1.10029	−0.550147	−	0.835068i	$-0.685428\pi$
$887$	−32.7696	−1.10029	−0.550147	+	0.835068i	$0.685428\pi$
$888$	0	0
$889$	−2.82843	−0.0948624
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.31371	0.0439616
$894$	0	0
$895$	47.4558	1.58627
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−13.1716	−0.439297
$900$	0	0
$901$	−11.3137	−0.376914
$902$	0	0
$903$	0	0
$904$	0	0
$905$	47.4558	1.57749
$906$	0	0
$907$	10.6863	0.354832	0.177416	−	0.984136i	$-0.443226\pi$
$907$	10.6863	0.354832	0.177416	+	0.984136i	$0.443226\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−9.02944	−0.299159	−0.149579	−	0.988750i	$-0.547792\pi$
$911$	−9.02944	−0.299159	−0.149579	+	0.988750i	$0.547792\pi$
$912$	0	0
$913$	26.1421	0.865178
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.00000	0.0660458
$918$	0	0
$919$	30.3431	1.00093	0.500464	−	0.865757i	$-0.333163\pi$
$919$	30.3431	1.00093	0.500464	+	0.865757i	$0.333163\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	15.7990	0.520030
$924$	0	0
$925$	0.142136	0.00467339
$926$	0	0
$927$	0	0
$928$	0	0
$929$	46.1421	1.51387	0.756937	−	0.653488i	$-0.226695\pi$
$929$	46.1421	1.51387	0.756937	+	0.653488i	$0.226695\pi$
$930$	0	0
$931$	3.82843	0.125472
$932$	0	0
$933$	0	0
$934$	0	0
$935$	16.4853	0.539126
$936$	0	0
$937$	−11.3137	−0.369603	−0.184801	−	0.982776i	$-0.559164\pi$
$937$	−11.3137	−0.369603	−0.184801	+	0.982776i	$0.559164\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	7.38478	0.240737	0.120368	−	0.992729i	$-0.461592\pi$
$941$	7.38478	0.240737	0.120368	+	0.992729i	$0.461592\pi$
$942$	0	0
$943$	−10.8579	−0.353581
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.72792	−0.121141	−0.0605706	−	0.998164i	$-0.519292\pi$
$947$	−3.72792	−0.121141	−0.0605706	+	0.998164i	$0.519292\pi$
$948$	0	0
$949$	−8.97056	−0.291197
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−53.1716	−1.72240	−0.861198	−	0.508269i	$-0.830285\pi$
$953$	−53.1716	−1.72240	−0.861198	+	0.508269i	$0.830285\pi$
$954$	0	0
$955$	29.9706	0.969825
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.82843	−0.0913347
$960$	0	0
$961$	−9.31371	−0.300442
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−8.82843	−0.284197
$966$	0	0
$967$	10.2843	0.330720	0.165360	−	0.986233i	$-0.447121\pi$
$967$	10.2843	0.330720	0.165360	+	0.986233i	$0.447121\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−41.1716	−1.32126	−0.660629	−	0.750712i	$-0.729710\pi$
$971$	−41.1716	−1.32126	−0.660629	+	0.750712i	$0.729710\pi$
$972$	0	0
$973$	1.65685	0.0531163
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.0294	0.480834	0.240417	−	0.970670i	$-0.422716\pi$
$977$	15.0294	0.480834	0.240417	+	0.970670i	$0.422716\pi$
$978$	0	0
$979$	3.00000	0.0958804
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.5980	−0.816449	−0.408224	−	0.912882i	$-0.633852\pi$
$983$	−25.5980	−0.816449	−0.408224	+	0.912882i	$0.633852\pi$
$984$	0	0
$985$	−59.9411	−1.90988
$986$	0	0
$987$	0	0
$988$	0	0
$989$	25.3137	0.804929
$990$	0	0
$991$	−37.4558	−1.18982	−0.594912	−	0.803791i	$-0.702813\pi$
$991$	−37.4558	−1.18982	−0.594912	+	0.803791i	$0.702813\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−8.41421	−0.266749
$996$	0	0
$997$	22.6863	0.718482	0.359241	−	0.933245i	$-0.383036\pi$
$997$	22.6863	0.718482	0.359241	+	0.933245i	$0.383036\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.a.bj.1.2	yes	2
3.2	odd	2		6048.2.a.ba.1.1	yes	2
4.3	odd	2		6048.2.a.bg.1.2	yes	2
12.11	even	2		6048.2.a.z.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.a.z.1.1	✓	2	12.11	even	2
6048.2.a.ba.1.1	yes	2	3.2	odd	2
6048.2.a.bg.1.2	yes	2	4.3	odd	2
6048.2.a.bj.1.2	yes	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:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+2.41421 q^{5} +1.00000 q^{7} +2.41421 q^{11} +2.82843 q^{13} +2.82843 q^{17} +3.82843 q^{19} +5.24264 q^{23} +0.828427 q^{25} -2.82843 q^{29} +4.65685 q^{31} +2.41421 q^{35} +0.171573 q^{37} -2.07107 q^{41} +4.82843 q^{43} +0.343146 q^{47} +1.00000 q^{49} -4.00000 q^{53} +5.82843 q^{55} -3.65685 q^{59} -11.3137 q^{61} +6.82843 q^{65} -4.48528 q^{67} +5.58579 q^{71} -3.17157 q^{73} +2.41421 q^{77} -4.82843 q^{79} +10.8284 q^{83} +6.82843 q^{85} +1.24264 q^{89} +2.82843 q^{91} +9.24264 q^{95} -9.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + 2 * q^7	\( 2 q + 2 q^{5} + 2 q^{7} + 2 q^{11} + 2 q^{19} + 2 q^{23} - 4 q^{25} - 2 q^{31} + 2 q^{35} + 6 q^{37} + 10 q^{41} + 4 q^{43} + 12 q^{47} + 2 q^{49} - 8 q^{53} + 6 q^{55} + 4 q^{59} + 8 q^{65} + 8 q^{67} + 14 q^{71} - 12 q^{73} + 2 q^{77} - 4 q^{79} + 16 q^{83} + 8 q^{85} - 6 q^{89} + 10 q^{95} - 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^7 + 2 * q^11 + 2 * q^19 + 2 * q^23 - 4 * q^25 - 2 * q^31 + 2 * q^35 + 6 * q^37 + 10 * q^41 + 4 * q^43 + 12 * q^47 + 2 * q^49 - 8 * q^53 + 6 * q^55 + 4 * q^59 + 8 * q^65 + 8 * q^67 + 14 * q^71 - 12 * q^73 + 2 * q^77 - 4 * q^79 + 16 * q^83 + 8 * q^85 - 6 * q^89 + 10 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more