LMFDB - Embedded newform 6048.2.a.bi.1.1

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:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-1.82843 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-1.82843 q^{5} +1.00000 q^{7} -4.82843 q^{11} +2.82843 q^{13} -0.171573 q^{17} +6.82843 q^{19} +4.00000 q^{23} -1.65685 q^{25} -2.82843 q^{29} -6.82843 q^{31} -1.82843 q^{35} +2.65685 q^{37} -3.82843 q^{41} +7.82843 q^{43} -8.65685 q^{47} +1.00000 q^{49} +2.00000 q^{53} +8.82843 q^{55} -0.656854 q^{59} +3.17157 q^{61} -5.17157 q^{65} +4.00000 q^{67} -1.65685 q^{71} -5.65685 q^{73} -4.82843 q^{77} -1.82843 q^{79} +5.34315 q^{83} +0.313708 q^{85} +6.00000 q^{89} +2.82843 q^{91} -12.4853 q^{95} -18.1421 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 4 q^{11} - 6 q^{17} + 8 q^{19} + 8 q^{23} + 8 q^{25} - 8 q^{31} + 2 q^{35} - 6 q^{37} - 2 q^{41} + 10 q^{43} - 6 q^{47} + 2 q^{49} + 4 q^{53} + 12 q^{55} + 10 q^{59} + 12 q^{61} - 16 q^{65} + 8 q^{67} + 8 q^{71} - 4 q^{77} + 2 q^{79} + 22 q^{83} - 22 q^{85} + 12 q^{89} - 8 q^{95} - 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^7 - 4 * q^11 - 6 * q^17 + 8 * q^19 + 8 * q^23 + 8 * q^25 - 8 * q^31 + 2 * q^35 - 6 * q^37 - 2 * q^41 + 10 * q^43 - 6 * q^47 + 2 * q^49 + 4 * q^53 + 12 * q^55 + 10 * q^59 + 12 * q^61 - 16 * q^65 + 8 * q^67 + 8 * q^71 - 4 * q^77 + 2 * q^79 + 22 * q^83 - 22 * q^85 + 12 * q^89 - 8 * 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
$5$	−1.82843	−0.817697	−0.408849	−	0.912602i	$-0.634070\pi$
$5$	−1.82843	−0.817697	−0.408849	+	0.912602i	$0.634070\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.82843	−1.45583	−0.727913	−	0.685670i	$-0.759509\pi$
$11$	−4.82843	−1.45583	−0.727913	+	0.685670i	$0.759509\pi$
$12$	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
$17$	−0.171573	−0.0416125	−0.0208063	−	0.999784i	$-0.506623\pi$
$17$	−0.171573	−0.0416125	−0.0208063	+	0.999784i	$0.506623\pi$
$18$	0	0
$19$	6.82843	1.56655	0.783274	−	0.621676i	$-0.213548\pi$
$19$	6.82843	1.56655	0.783274	+	0.621676i	$0.213548\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−1.65685	−0.331371
$26$	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
$31$	−6.82843	−1.22642	−0.613211	−	0.789919i	$-0.710122\pi$
$31$	−6.82843	−1.22642	−0.613211	+	0.789919i	$0.710122\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.82843	−0.309061
$36$	0	0
$37$	2.65685	0.436784	0.218392	−	0.975861i	$-0.429919\pi$
$37$	2.65685	0.436784	0.218392	+	0.975861i	$0.429919\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.82843	−0.597900	−0.298950	−	0.954269i	$-0.596636\pi$
$41$	−3.82843	−0.597900	−0.298950	+	0.954269i	$0.596636\pi$
$42$	0	0
$43$	7.82843	1.19382	0.596912	−	0.802307i	$-0.296394\pi$
$43$	7.82843	1.19382	0.596912	+	0.802307i	$0.296394\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.65685	−1.26273	−0.631366	−	0.775485i	$-0.717505\pi$
$47$	−8.65685	−1.26273	−0.631366	+	0.775485i	$0.717505\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	8.82843	1.19042
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.656854	−0.0855151	−0.0427576	−	0.999085i	$-0.513614\pi$
$59$	−0.656854	−0.0855151	−0.0427576	+	0.999085i	$0.513614\pi$
$60$	0	0
$61$	3.17157	0.406078	0.203039	−	0.979171i	$-0.434918\pi$
$61$	3.17157	0.406078	0.203039	+	0.979171i	$0.434918\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.17157	−0.641455
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.65685	−0.196632	−0.0983162	−	0.995155i	$-0.531346\pi$
$71$	−1.65685	−0.196632	−0.0983162	+	0.995155i	$0.531346\pi$
$72$	0	0
$73$	−5.65685	−0.662085	−0.331042	−	0.943616i	$-0.607400\pi$
$73$	−5.65685	−0.662085	−0.331042	+	0.943616i	$0.607400\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.82843	−0.550250
$78$	0	0
$79$	−1.82843	−0.205714	−0.102857	−	0.994696i	$-0.532798\pi$
$79$	−1.82843	−0.205714	−0.102857	+	0.994696i	$0.532798\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.34315	0.586486	0.293243	−	0.956038i	$-0.405265\pi$
$83$	5.34315	0.586486	0.293243	+	0.956038i	$0.405265\pi$
$84$	0	0
$85$	0.313708	0.0340265
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−12.4853	−1.28096
$96$	0	0
$97$	−18.1421	−1.84205	−0.921027	−	0.389498i	$-0.872649\pi$
$97$	−18.1421	−1.84205	−0.921027	+	0.389498i	$0.872649\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.9706	1.88764	0.943821	−	0.330458i	$-0.107203\pi$
$101$	18.9706	1.88764	0.943821	+	0.330458i	$0.107203\pi$
$102$	0	0
$103$	2.34315	0.230877	0.115439	−	0.993315i	$-0.463173\pi$
$103$	2.34315	0.230877	0.115439	+	0.993315i	$0.463173\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.65685	0.546869	0.273434	−	0.961891i	$-0.411840\pi$
$107$	5.65685	0.546869	0.273434	+	0.961891i	$0.411840\pi$
$108$	0	0
$109$	−10.6569	−1.02074	−0.510371	−	0.859954i	$-0.670492\pi$
$109$	−10.6569	−1.02074	−0.510371	+	0.859954i	$0.670492\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.17157	0.674645	0.337322	−	0.941389i	$-0.390479\pi$
$113$	7.17157	0.674645	0.337322	+	0.941389i	$0.390479\pi$
$114$	0	0
$115$	−7.31371	−0.682007
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.171573	−0.0157281
$120$	0	0
$121$	12.3137	1.11943
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.1716	1.08866
$126$	0	0
$127$	12.1716	1.08005	0.540026	−	0.841648i	$-0.318414\pi$
$127$	12.1716	1.08005	0.540026	+	0.841648i	$0.318414\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	6.82843	0.592100
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.8284	−1.26688	−0.633439	−	0.773793i	$-0.718357\pi$
$137$	−14.8284	−1.26688	−0.633439	+	0.773793i	$0.718357\pi$
$138$	0	0
$139$	11.1716	0.947560	0.473780	−	0.880643i	$-0.342889\pi$
$139$	11.1716	0.947560	0.473780	+	0.880643i	$0.342889\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−13.6569	−1.14204
$144$	0	0
$145$	5.17157	0.429476
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.1421	1.65011	0.825054	−	0.565054i	$-0.191145\pi$
$149$	20.1421	1.65011	0.825054	+	0.565054i	$0.191145\pi$
$150$	0	0
$151$	17.8284	1.45086	0.725428	−	0.688298i	$-0.241642\pi$
$151$	17.8284	1.45086	0.725428	+	0.688298i	$0.241642\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	12.4853	1.00284
$156$	0	0
$157$	2.82843	0.225733	0.112867	−	0.993610i	$-0.463997\pi$
$157$	2.82843	0.225733	0.112867	+	0.993610i	$0.463997\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	−1.48528	−0.116336	−0.0581681	−	0.998307i	$-0.518526\pi$
$163$	−1.48528	−0.116336	−0.0581681	+	0.998307i	$0.518526\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.9706	1.39060	0.695302	−	0.718718i	$-0.255271\pi$
$167$	17.9706	1.39060	0.695302	+	0.718718i	$0.255271\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	−1.65685	−0.125246
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.14214	0.309598	0.154799	−	0.987946i	$-0.450527\pi$
$179$	4.14214	0.309598	0.154799	+	0.987946i	$0.450527\pi$
$180$	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
$185$	−4.85786	−0.357157
$186$	0	0
$187$	0.828427	0.0605806
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.6274	1.78198	0.890989	−	0.454026i	$-0.150013\pi$
$191$	24.6274	1.78198	0.890989	+	0.454026i	$0.150013\pi$
$192$	0	0
$193$	10.3137	0.742397	0.371198	−	0.928554i	$-0.378947\pi$
$193$	10.3137	0.742397	0.371198	+	0.928554i	$0.378947\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.7990	−0.840643	−0.420322	−	0.907375i	$-0.638083\pi$
$197$	−11.7990	−0.840643	−0.420322	+	0.907375i	$0.638083\pi$
$198$	0	0
$199$	18.9706	1.34479	0.672394	−	0.740194i	$-0.265266\pi$
$199$	18.9706	1.34479	0.672394	+	0.740194i	$0.265266\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.82843	−0.198517
$204$	0	0
$205$	7.00000	0.488901
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−32.9706	−2.28062
$210$	0	0
$211$	16.9706	1.16830	0.584151	−	0.811645i	$-0.301428\pi$
$211$	16.9706	1.16830	0.584151	+	0.811645i	$0.301428\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−14.3137	−0.976187
$216$	0	0
$217$	−6.82843	−0.463544
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.485281	−0.0326436
$222$	0	0
$223$	18.1421	1.21489	0.607444	−	0.794363i	$-0.292195\pi$
$223$	18.1421	1.21489	0.607444	+	0.794363i	$0.292195\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.6569	−1.17193	−0.585963	−	0.810338i	$-0.699284\pi$
$227$	−17.6569	−1.17193	−0.585963	+	0.810338i	$0.699284\pi$
$228$	0	0
$229$	1.31371	0.0868123	0.0434062	−	0.999058i	$-0.486179\pi$
$229$	1.31371	0.0868123	0.0434062	+	0.999058i	$0.486179\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.1421	0.664433	0.332217	−	0.943203i	$-0.392203\pi$
$233$	10.1421	0.664433	0.332217	+	0.943203i	$0.392203\pi$
$234$	0	0
$235$	15.8284	1.03253
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.1421	0.785409	0.392705	−	0.919665i	$-0.371539\pi$
$239$	12.1421	0.785409	0.392705	+	0.919665i	$0.371539\pi$
$240$	0	0
$241$	−18.1421	−1.16864	−0.584319	−	0.811524i	$-0.698638\pi$
$241$	−18.1421	−1.16864	−0.584319	+	0.811524i	$0.698638\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.82843	−0.116814
$246$	0	0
$247$	19.3137	1.22890
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.97056	0.250620	0.125310	−	0.992118i	$-0.460008\pi$
$251$	3.97056	0.250620	0.125310	+	0.992118i	$0.460008\pi$
$252$	0	0
$253$	−19.3137	−1.21424
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	2.65685	0.165089
$260$	0	0
$261$	0	0
$262$	0	0
$263$	10.8284	0.667709	0.333855	−	0.942625i	$-0.391651\pi$
$263$	10.8284	0.667709	0.333855	+	0.942625i	$0.391651\pi$
$264$	0	0
$265$	−3.65685	−0.224639
$266$	0	0
$267$	0	0
$268$	0	0
$269$	15.1421	0.923232	0.461616	−	0.887080i	$-0.347270\pi$
$269$	15.1421	0.923232	0.461616	+	0.887080i	$0.347270\pi$
$270$	0	0
$271$	−6.00000	−0.364474	−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000	−0.364474	−0.182237	+	0.983255i	$0.558334\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.00000	0.482418
$276$	0	0
$277$	−24.6569	−1.48149	−0.740743	−	0.671788i	$-0.765527\pi$
$277$	−24.6569	−1.48149	−0.740743	+	0.671788i	$0.765527\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−32.9706	−1.96686	−0.983429	−	0.181291i	$-0.941972\pi$
$281$	−32.9706	−1.96686	−0.983429	+	0.181291i	$0.941972\pi$
$282$	0	0
$283$	23.1716	1.37741	0.688704	−	0.725043i	$-0.258180\pi$
$283$	23.1716	1.37741	0.688704	+	0.725043i	$0.258180\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.82843	−0.225985
$288$	0	0
$289$	−16.9706	−0.998268
$290$	0	0
$291$	0	0
$292$	0	0
$293$	12.1716	0.711071	0.355535	−	0.934663i	$-0.384299\pi$
$293$	12.1716	0.711071	0.355535	+	0.934663i	$0.384299\pi$
$294$	0	0
$295$	1.20101	0.0699255
$296$	0	0
$297$	0	0
$298$	0	0
$299$	11.3137	0.654289
$300$	0	0
$301$	7.82843	0.451223
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5.79899	−0.332049
$306$	0	0
$307$	−24.2843	−1.38598	−0.692988	−	0.720949i	$-0.743706\pi$
$307$	−24.2843	−1.38598	−0.692988	+	0.720949i	$0.743706\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.97056	0.565379	0.282689	−	0.959212i	$-0.408773\pi$
$311$	9.97056	0.565379	0.282689	+	0.959212i	$0.408773\pi$
$312$	0	0
$313$	24.6274	1.39202	0.696012	−	0.718030i	$-0.254956\pi$
$313$	24.6274	1.39202	0.696012	+	0.718030i	$0.254956\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−29.6569	−1.66569	−0.832847	−	0.553503i	$-0.813291\pi$
$317$	−29.6569	−1.66569	−0.832847	+	0.553503i	$0.813291\pi$
$318$	0	0
$319$	13.6569	0.764637
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.17157	−0.0651881
$324$	0	0
$325$	−4.68629	−0.259949
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8.65685	−0.477268
$330$	0	0
$331$	28.7990	1.58294	0.791468	−	0.611211i	$-0.209317\pi$
$331$	28.7990	1.58294	0.791468	+	0.611211i	$0.209317\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.31371	−0.399591
$336$	0	0
$337$	−20.3137	−1.10656	−0.553279	−	0.832996i	$-0.686624\pi$
$337$	−20.3137	−1.10656	−0.553279	+	0.832996i	$0.686624\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	32.9706	1.78546
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.14214	−0.329727	−0.164864	−	0.986316i	$-0.552718\pi$
$347$	−6.14214	−0.329727	−0.164864	+	0.986316i	$0.552718\pi$
$348$	0	0
$349$	−20.2843	−1.08579	−0.542896	−	0.839800i	$-0.682672\pi$
$349$	−20.2843	−1.08579	−0.542896	+	0.839800i	$0.682672\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.85786	−0.258558	−0.129279	−	0.991608i	$-0.541266\pi$
$353$	−4.85786	−0.258558	−0.129279	+	0.991608i	$0.541266\pi$
$354$	0	0
$355$	3.02944	0.160786
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.3431	0.862558	0.431279	−	0.902219i	$-0.358062\pi$
$359$	16.3431	0.862558	0.431279	+	0.902219i	$0.358062\pi$
$360$	0	0
$361$	27.6274	1.45407
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.3431	0.541385
$366$	0	0
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.00000	0.103835
$372$	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
$379$	−17.4853	−0.898159	−0.449079	−	0.893492i	$-0.648248\pi$
$379$	−17.4853	−0.898159	−0.449079	+	0.893492i	$0.648248\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.00000	0.0510976	0.0255488	−	0.999674i	$-0.491867\pi$
$383$	1.00000	0.0510976	0.0255488	+	0.999674i	$0.491867\pi$
$384$	0	0
$385$	8.82843	0.449938
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.79899	−0.0912124	−0.0456062	−	0.998959i	$-0.514522\pi$
$389$	−1.79899	−0.0912124	−0.0456062	+	0.998959i	$0.514522\pi$
$390$	0	0
$391$	−0.686292	−0.0347073
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.34315	0.168212
$396$	0	0
$397$	14.3431	0.719862	0.359931	−	0.932979i	$-0.382800\pi$
$397$	14.3431	0.719862	0.359931	+	0.932979i	$0.382800\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	25.3137	1.26411	0.632053	−	0.774925i	$-0.282212\pi$
$401$	25.3137	1.26411	0.632053	+	0.774925i	$0.282212\pi$
$402$	0	0
$403$	−19.3137	−0.962084
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.8284	−0.635882
$408$	0	0
$409$	−3.02944	−0.149796	−0.0748980	−	0.997191i	$-0.523863\pi$
$409$	−3.02944	−0.149796	−0.0748980	+	0.997191i	$0.523863\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.656854	−0.0323217
$414$	0	0
$415$	−9.76955	−0.479568
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.6569	−0.911447	−0.455723	−	0.890121i	$-0.650619\pi$
$419$	−18.6569	−0.911447	−0.455723	+	0.890121i	$0.650619\pi$
$420$	0	0
$421$	14.9706	0.729621	0.364810	−	0.931082i	$-0.381134\pi$
$421$	14.9706	0.729621	0.364810	+	0.931082i	$0.381134\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.284271	0.0137892
$426$	0	0
$427$	3.17157	0.153483
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.9706	1.20279	0.601395	−	0.798952i	$-0.294612\pi$
$431$	24.9706	1.20279	0.601395	+	0.798952i	$0.294612\pi$
$432$	0	0
$433$	34.4853	1.65726	0.828628	−	0.559799i	$-0.189122\pi$
$433$	34.4853	1.65726	0.828628	+	0.559799i	$0.189122\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	27.3137	1.30659
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−34.8284	−1.65475	−0.827374	−	0.561651i	$-0.810166\pi$
$443$	−34.8284	−1.65475	−0.827374	+	0.561651i	$0.810166\pi$
$444$	0	0
$445$	−10.9706	−0.520055
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.17157	0.0552899	0.0276450	−	0.999618i	$-0.491199\pi$
$449$	1.17157	0.0552899	0.0276450	+	0.999618i	$0.491199\pi$
$450$	0	0
$451$	18.4853	0.870438
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.17157	−0.242447
$456$	0	0
$457$	41.3137	1.93257	0.966287	−	0.257468i	$-0.0828881\pi$
$457$	41.3137	1.93257	0.966287	+	0.257468i	$0.0828881\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−0.514719	−0.0239728	−0.0119864	−	0.999928i	$-0.503815\pi$
$461$	−0.514719	−0.0239728	−0.0119864	+	0.999928i	$0.503815\pi$
$462$	0	0
$463$	−18.1716	−0.844505	−0.422252	−	0.906478i	$-0.638760\pi$
$463$	−18.1716	−0.844505	−0.422252	+	0.906478i	$0.638760\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	40.2843	1.86413	0.932067	−	0.362286i	$-0.118004\pi$
$467$	40.2843	1.86413	0.932067	+	0.362286i	$0.118004\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−37.7990	−1.73800
$474$	0	0
$475$	−11.3137	−0.519109
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.00000	−0.0456912	−0.0228456	−	0.999739i	$-0.507273\pi$
$479$	−1.00000	−0.0456912	−0.0228456	+	0.999739i	$0.507273\pi$
$480$	0	0
$481$	7.51472	0.342642
$482$	0	0
$483$	0	0
$484$	0	0
$485$	33.1716	1.50624
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−30.1421	−1.36030	−0.680148	−	0.733075i	$-0.738084\pi$
$491$	−30.1421	−1.36030	−0.680148	+	0.733075i	$0.738084\pi$
$492$	0	0
$493$	0.485281	0.0218560
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.65685	−0.0743201
$498$	0	0
$499$	43.7696	1.95939	0.979697	−	0.200483i	$-0.0642512\pi$
$499$	43.7696	1.95939	0.979697	+	0.200483i	$0.0642512\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.62742	−0.429265	−0.214633	−	0.976695i	$-0.568855\pi$
$503$	−9.62742	−0.429265	−0.214633	+	0.976695i	$0.568855\pi$
$504$	0	0
$505$	−34.6863	−1.54352
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.48528	0.243131	0.121565	−	0.992583i	$-0.461209\pi$
$509$	5.48528	0.243131	0.121565	+	0.992583i	$0.461209\pi$
$510$	0	0
$511$	−5.65685	−0.250244
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.28427	−0.188788
$516$	0	0
$517$	41.7990	1.83832
$518$	0	0
$519$	0	0
$520$	0	0
$521$	38.4558	1.68478	0.842391	−	0.538867i	$-0.181148\pi$
$521$	38.4558	1.68478	0.842391	+	0.538867i	$0.181148\pi$
$522$	0	0
$523$	40.4264	1.76772	0.883862	−	0.467748i	$-0.154935\pi$
$523$	40.4264	1.76772	0.883862	+	0.467748i	$0.154935\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.17157	0.0510345
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−10.8284	−0.469031
$534$	0	0
$535$	−10.3431	−0.447173
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.82843	−0.207975
$540$	0	0
$541$	−27.2843	−1.17304	−0.586521	−	0.809934i	$-0.699503\pi$
$541$	−27.2843	−1.17304	−0.586521	+	0.809934i	$0.699503\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	19.4853	0.834658
$546$	0	0
$547$	5.48528	0.234534	0.117267	−	0.993100i	$-0.462587\pi$
$547$	5.48528	0.234534	0.117267	+	0.993100i	$0.462587\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−19.3137	−0.822792
$552$	0	0
$553$	−1.82843	−0.0777526
$554$	0	0
$555$	0	0
$556$	0	0
$557$	35.3137	1.49629	0.748145	−	0.663535i	$-0.230945\pi$
$557$	35.3137	1.49629	0.748145	+	0.663535i	$0.230945\pi$
$558$	0	0
$559$	22.1421	0.936513
$560$	0	0
$561$	0	0
$562$	0	0
$563$	39.3137	1.65688	0.828438	−	0.560081i	$-0.189230\pi$
$563$	39.3137	1.65688	0.828438	+	0.560081i	$0.189230\pi$
$564$	0	0
$565$	−13.1127	−0.551655
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−26.3431	−1.10436	−0.552181	−	0.833724i	$-0.686204\pi$
$569$	−26.3431	−1.10436	−0.552181	+	0.833724i	$0.686204\pi$
$570$	0	0
$571$	−31.7696	−1.32951	−0.664757	−	0.747059i	$-0.731465\pi$
$571$	−31.7696	−1.32951	−0.664757	+	0.747059i	$0.731465\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.62742	−0.276382
$576$	0	0
$577$	10.9706	0.456711	0.228355	−	0.973578i	$-0.426665\pi$
$577$	10.9706	0.456711	0.228355	+	0.973578i	$0.426665\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.34315	0.221671
$582$	0	0
$583$	−9.65685	−0.399946
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.97056	0.205157	0.102579	−	0.994725i	$-0.467291\pi$
$587$	4.97056	0.205157	0.102579	+	0.994725i	$0.467291\pi$
$588$	0	0
$589$	−46.6274	−1.92125
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−11.4853	−0.471644	−0.235822	−	0.971796i	$-0.575778\pi$
$593$	−11.4853	−0.471644	−0.235822	+	0.971796i	$0.575778\pi$
$594$	0	0
$595$	0.313708	0.0128608
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−27.1716	−1.11020	−0.555100	−	0.831783i	$-0.687320\pi$
$599$	−27.1716	−1.11020	−0.555100	+	0.831783i	$0.687320\pi$
$600$	0	0
$601$	19.1127	0.779623	0.389812	−	0.920895i	$-0.372540\pi$
$601$	19.1127	0.779623	0.389812	+	0.920895i	$0.372540\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−22.5147	−0.915353
$606$	0	0
$607$	−4.14214	−0.168124	−0.0840620	−	0.996461i	$-0.526789\pi$
$607$	−4.14214	−0.168124	−0.0840620	+	0.996461i	$0.526789\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−24.4853	−0.990568
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.3137	−1.42168	−0.710838	−	0.703356i	$-0.751684\pi$
$617$	−35.3137	−1.42168	−0.710838	+	0.703356i	$0.751684\pi$
$618$	0	0
$619$	−1.17157	−0.0470895	−0.0235447	−	0.999723i	$-0.507495\pi$
$619$	−1.17157	−0.0470895	−0.0235447	+	0.999723i	$0.507495\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.455844	−0.0181757
$630$	0	0
$631$	−28.1127	−1.11915	−0.559574	−	0.828780i	$-0.689035\pi$
$631$	−28.1127	−1.11915	−0.559574	+	0.828780i	$0.689035\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−22.2548	−0.883156
$636$	0	0
$637$	2.82843	0.112066
$638$	0	0
$639$	0	0
$640$	0	0
$641$	11.4558	0.452479	0.226239	−	0.974072i	$-0.427357\pi$
$641$	11.4558	0.452479	0.226239	+	0.974072i	$0.427357\pi$
$642$	0	0
$643$	26.0000	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21.6569	−0.851419	−0.425709	−	0.904860i	$-0.639975\pi$
$647$	−21.6569	−0.851419	−0.425709	+	0.904860i	$0.639975\pi$
$648$	0	0
$649$	3.17157	0.124495
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.48528	0.253789	0.126894	−	0.991916i	$-0.459499\pi$
$653$	6.48528	0.253789	0.126894	+	0.991916i	$0.459499\pi$
$654$	0	0
$655$	7.31371	0.285770
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.3431	0.480821	0.240410	−	0.970671i	$-0.422718\pi$
$659$	12.3431	0.480821	0.240410	+	0.970671i	$0.422718\pi$
$660$	0	0
$661$	−17.1127	−0.665607	−0.332803	−	0.942996i	$-0.607995\pi$
$661$	−17.1127	−0.665607	−0.332803	+	0.942996i	$0.607995\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12.4853	−0.484158
$666$	0	0
$667$	−11.3137	−0.438069
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−15.3137	−0.591179
$672$	0	0
$673$	−35.6569	−1.37447	−0.687235	−	0.726435i	$-0.741176\pi$
$673$	−35.6569	−1.37447	−0.687235	+	0.726435i	$0.741176\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−25.3137	−0.972885	−0.486442	−	0.873713i	$-0.661706\pi$
$677$	−25.3137	−0.972885	−0.486442	+	0.873713i	$0.661706\pi$
$678$	0	0
$679$	−18.1421	−0.696231
$680$	0	0
$681$	0	0
$682$	0	0
$683$	21.1716	0.810108	0.405054	−	0.914293i	$-0.367253\pi$
$683$	21.1716	0.810108	0.405054	+	0.914293i	$0.367253\pi$
$684$	0	0
$685$	27.1127	1.03592
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.65685	0.215509
$690$	0	0
$691$	−20.8284	−0.792351	−0.396175	−	0.918175i	$-0.629663\pi$
$691$	−20.8284	−0.792351	−0.396175	+	0.918175i	$0.629663\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−20.4264	−0.774818
$696$	0	0
$697$	0.656854	0.0248801
$698$	0	0
$699$	0	0
$700$	0	0
$701$	46.4853	1.75572	0.877862	−	0.478913i	$-0.158969\pi$
$701$	46.4853	1.75572	0.877862	+	0.478913i	$0.158969\pi$
$702$	0	0
$703$	18.1421	0.684244
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.9706	0.713461
$708$	0	0
$709$	31.2843	1.17491	0.587453	−	0.809258i	$-0.300131\pi$
$709$	31.2843	1.17491	0.587453	+	0.809258i	$0.300131\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−27.3137	−1.02291
$714$	0	0
$715$	24.9706	0.933846
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.34315	−0.0500909	−0.0250454	−	0.999686i	$-0.507973\pi$
$719$	−1.34315	−0.0500909	−0.0250454	+	0.999686i	$0.507973\pi$
$720$	0	0
$721$	2.34315	0.0872633
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.68629	0.174044
$726$	0	0
$727$	10.1421	0.376151	0.188075	−	0.982155i	$-0.439775\pi$
$727$	10.1421	0.376151	0.188075	+	0.982155i	$0.439775\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.34315	−0.0496780
$732$	0	0
$733$	2.68629	0.0992204	0.0496102	−	0.998769i	$-0.484202\pi$
$733$	2.68629	0.0992204	0.0496102	+	0.998769i	$0.484202\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19.3137	−0.711430
$738$	0	0
$739$	37.9411	1.39569	0.697843	−	0.716250i	$-0.254143\pi$
$739$	37.9411	1.39569	0.697843	+	0.716250i	$0.254143\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.17157	0.0429808	0.0214904	−	0.999769i	$-0.493159\pi$
$743$	1.17157	0.0429808	0.0214904	+	0.999769i	$0.493159\pi$
$744$	0	0
$745$	−36.8284	−1.34929
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.65685	0.206697
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−32.5980	−1.18636
$756$	0	0
$757$	−35.6274	−1.29490	−0.647450	−	0.762108i	$-0.724165\pi$
$757$	−35.6274	−1.29490	−0.647450	+	0.762108i	$0.724165\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.17157	−0.0787195	−0.0393597	−	0.999225i	$-0.512532\pi$
$761$	−2.17157	−0.0787195	−0.0393597	+	0.999225i	$0.512532\pi$
$762$	0	0
$763$	−10.6569	−0.385804
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.85786	−0.0670836
$768$	0	0
$769$	4.34315	0.156618	0.0783089	−	0.996929i	$-0.475048\pi$
$769$	4.34315	0.156618	0.0783089	+	0.996929i	$0.475048\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	37.8284	1.36059	0.680297	−	0.732937i	$-0.261851\pi$
$773$	37.8284	1.36059	0.680297	+	0.732937i	$0.261851\pi$
$774$	0	0
$775$	11.3137	0.406400
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−26.1421	−0.936639
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.17157	−0.184581
$786$	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
$791$	7.17157	0.254992
$792$	0	0
$793$	8.97056	0.318554
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13.3137	−0.471596	−0.235798	−	0.971802i	$-0.575770\pi$
$797$	−13.3137	−0.471596	−0.235798	+	0.971802i	$0.575770\pi$
$798$	0	0
$799$	1.48528	0.0525455
$800$	0	0
$801$	0	0
$802$	0	0
$803$	27.3137	0.963880
$804$	0	0
$805$	−7.31371	−0.257774
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−31.9411	−1.12299	−0.561495	−	0.827480i	$-0.689774\pi$
$809$	−31.9411	−1.12299	−0.561495	+	0.827480i	$0.689774\pi$
$810$	0	0
$811$	−42.7696	−1.50184	−0.750921	−	0.660392i	$-0.770390\pi$
$811$	−42.7696	−1.50184	−0.750921	+	0.660392i	$0.770390\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.71573	0.0951278
$816$	0	0
$817$	53.4558	1.87018
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.6569	0.755829	0.377915	−	0.925840i	$-0.376641\pi$
$821$	21.6569	0.755829	0.377915	+	0.925840i	$0.376641\pi$
$822$	0	0
$823$	−26.1716	−0.912284	−0.456142	−	0.889907i	$-0.650769\pi$
$823$	−26.1716	−0.912284	−0.456142	+	0.889907i	$0.650769\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.17157	−0.110286	−0.0551432	−	0.998478i	$-0.517562\pi$
$827$	−3.17157	−0.110286	−0.0551432	+	0.998478i	$0.517562\pi$
$828$	0	0
$829$	9.17157	0.318542	0.159271	−	0.987235i	$-0.449086\pi$
$829$	9.17157	0.318542	0.159271	+	0.987235i	$0.449086\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−0.171573	−0.00594465
$834$	0	0
$835$	−32.8579	−1.13709
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24.3137	0.839402	0.419701	−	0.907662i	$-0.362135\pi$
$839$	24.3137	0.839402	0.419701	+	0.907662i	$0.362135\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9.14214	0.314499
$846$	0	0
$847$	12.3137	0.423104
$848$	0	0
$849$	0	0
$850$	0	0
$851$	10.6274	0.364303
$852$	0	0
$853$	14.9706	0.512582	0.256291	−	0.966600i	$-0.417499\pi$
$853$	14.9706	0.512582	0.256291	+	0.966600i	$0.417499\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−43.1421	−1.47371	−0.736854	−	0.676052i	$-0.763689\pi$
$857$	−43.1421	−1.47371	−0.736854	+	0.676052i	$0.763689\pi$
$858$	0	0
$859$	−11.6569	−0.397727	−0.198863	−	0.980027i	$-0.563725\pi$
$859$	−11.6569	−0.397727	−0.198863	+	0.980027i	$0.563725\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18.6863	−0.636089	−0.318044	−	0.948076i	$-0.603026\pi$
$863$	−18.6863	−0.636089	−0.318044	+	0.948076i	$0.603026\pi$
$864$	0	0
$865$	−3.65685	−0.124337
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8.82843	0.299484
$870$	0	0
$871$	11.3137	0.383350
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.1716	0.411474
$876$	0	0
$877$	−31.9706	−1.07957	−0.539785	−	0.841803i	$-0.681494\pi$
$877$	−31.9706	−1.07957	−0.539785	+	0.841803i	$0.681494\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	26.9706	0.908661	0.454331	−	0.890833i	$-0.349878\pi$
$881$	26.9706	0.908661	0.454331	+	0.890833i	$0.349878\pi$
$882$	0	0
$883$	30.7990	1.03647	0.518234	−	0.855239i	$-0.326590\pi$
$883$	30.7990	1.03647	0.518234	+	0.855239i	$0.326590\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	52.5980	1.76607	0.883034	−	0.469310i	$-0.155497\pi$
$887$	52.5980	1.76607	0.883034	+	0.469310i	$0.155497\pi$
$888$	0	0
$889$	12.1716	0.408221
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−59.1127	−1.97813
$894$	0	0
$895$	−7.57359	−0.253157
$896$	0	0
$897$	0	0
$898$	0	0
$899$	19.3137	0.644148
$900$	0	0
$901$	−0.343146	−0.0114318
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−35.9411	−1.19472
$906$	0	0
$907$	−29.7696	−0.988482	−0.494241	−	0.869325i	$-0.664554\pi$
$907$	−29.7696	−0.988482	−0.494241	+	0.869325i	$0.664554\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	30.4853	1.01002	0.505011	−	0.863113i	$-0.331488\pi$
$911$	30.4853	1.01002	0.505011	+	0.863113i	$0.331488\pi$
$912$	0	0
$913$	−25.7990	−0.853822
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.00000	−0.132092
$918$	0	0
$919$	−45.0833	−1.48716	−0.743580	−	0.668647i	$-0.766874\pi$
$919$	−45.0833	−1.48716	−0.743580	+	0.668647i	$0.766874\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−4.68629	−0.154251
$924$	0	0
$925$	−4.40202	−0.144738
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−15.8284	−0.519314	−0.259657	−	0.965701i	$-0.583609\pi$
$929$	−15.8284	−0.519314	−0.259657	+	0.965701i	$0.583609\pi$
$930$	0	0
$931$	6.82843	0.223793
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.51472	−0.0495366
$936$	0	0
$937$	33.1716	1.08367	0.541834	−	0.840486i	$-0.317730\pi$
$937$	33.1716	1.08367	0.541834	+	0.840486i	$0.317730\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−42.7990	−1.39521	−0.697604	−	0.716484i	$-0.745750\pi$
$941$	−42.7990	−1.39521	−0.697604	+	0.716484i	$0.745750\pi$
$942$	0	0
$943$	−15.3137	−0.498683
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−57.9411	−1.88283	−0.941417	−	0.337245i	$-0.890505\pi$
$947$	−57.9411	−1.88283	−0.941417	+	0.337245i	$0.890505\pi$
$948$	0	0
$949$	−16.0000	−0.519382
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.31371	0.107342	0.0536708	−	0.998559i	$-0.482908\pi$
$953$	3.31371	0.107342	0.0536708	+	0.998559i	$0.482908\pi$
$954$	0	0
$955$	−45.0294	−1.45712
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−14.8284	−0.478835
$960$	0	0
$961$	15.6274	0.504110
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−18.8579	−0.607056
$966$	0	0
$967$	−5.65685	−0.181912	−0.0909561	−	0.995855i	$-0.528992\pi$
$967$	−5.65685	−0.181912	−0.0909561	+	0.995855i	$0.528992\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−10.6569	−0.341995	−0.170997	−	0.985271i	$-0.554699\pi$
$971$	−10.6569	−0.341995	−0.170997	+	0.985271i	$0.554699\pi$
$972$	0	0
$973$	11.1716	0.358144
$974$	0	0
$975$	0	0
$976$	0	0
$977$	42.9706	1.37475	0.687375	−	0.726303i	$-0.258763\pi$
$977$	42.9706	1.37475	0.687375	+	0.726303i	$0.258763\pi$
$978$	0	0
$979$	−28.9706	−0.925903
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.71573	−0.0866183	−0.0433091	−	0.999062i	$-0.513790\pi$
$983$	−2.71573	−0.0866183	−0.0433091	+	0.999062i	$0.513790\pi$
$984$	0	0
$985$	21.5736	0.687392
$986$	0	0
$987$	0	0
$988$	0	0
$989$	31.3137	0.995718
$990$	0	0
$991$	27.4264	0.871229	0.435614	−	0.900133i	$-0.356531\pi$
$991$	27.4264	0.871229	0.435614	+	0.900133i	$0.356531\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−34.6863	−1.09963
$996$	0	0
$997$	−17.8579	−0.565564	−0.282782	−	0.959184i	$-0.591257\pi$
$997$	−17.8579	−0.565564	−0.282782	+	0.959184i	$0.591257\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.a.bi.1.1	yes	2
3.2	odd	2		6048.2.a.bb.1.2	yes	2
4.3	odd	2		6048.2.a.bh.1.1	yes	2
12.11	even	2		6048.2.a.y.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.a.y.1.2	✓	2	12.11	even	2
6048.2.a.bb.1.2	yes	2	3.2	odd	2
6048.2.a.bh.1.1	yes	2	4.3	odd	2
6048.2.a.bi.1.1	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)

\(f(q)\)	\(=\)	\(q-1.82843 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-1.82843 q^{5} +1.00000 q^{7} -4.82843 q^{11} +2.82843 q^{13} -0.171573 q^{17} +6.82843 q^{19} +4.00000 q^{23} -1.65685 q^{25} -2.82843 q^{29} -6.82843 q^{31} -1.82843 q^{35} +2.65685 q^{37} -3.82843 q^{41} +7.82843 q^{43} -8.65685 q^{47} +1.00000 q^{49} +2.00000 q^{53} +8.82843 q^{55} -0.656854 q^{59} +3.17157 q^{61} -5.17157 q^{65} +4.00000 q^{67} -1.65685 q^{71} -5.65685 q^{73} -4.82843 q^{77} -1.82843 q^{79} +5.34315 q^{83} +0.313708 q^{85} +6.00000 q^{89} +2.82843 q^{91} -12.4853 q^{95} -18.1421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + 2 * q^7	\( 2 q + 2 q^{5} + 2 q^{7} - 4 q^{11} - 6 q^{17} + 8 q^{19} + 8 q^{23} + 8 q^{25} - 8 q^{31} + 2 q^{35} - 6 q^{37} - 2 q^{41} + 10 q^{43} - 6 q^{47} + 2 q^{49} + 4 q^{53} + 12 q^{55} + 10 q^{59} + 12 q^{61} - 16 q^{65} + 8 q^{67} + 8 q^{71} - 4 q^{77} + 2 q^{79} + 22 q^{83} - 22 q^{85} + 12 q^{89} - 8 q^{95} - 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^7 - 4 * q^11 - 6 * q^17 + 8 * q^19 + 8 * q^23 + 8 * q^25 - 8 * q^31 + 2 * q^35 - 6 * q^37 - 2 * q^41 + 10 * q^43 - 6 * q^47 + 2 * q^49 + 4 * q^53 + 12 * q^55 + 10 * q^59 + 12 * q^61 - 16 * q^65 + 8 * q^67 + 8 * q^71 - 4 * q^77 + 2 * q^79 + 22 * q^83 - 22 * q^85 + 12 * q^89 - 8 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more