LMFDB - Embedded newform 6012.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.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:	$48.0060616952$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.161121.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 3x^{2} + 5x + 1 $ x^5 - x^4 - 6x^3 + 3x^2 + 5*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2004)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.31991$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.28459 q^{5} -1.96468 q^{7} +O(q^{10})$	$q-1.28459 q^{5} -1.96468 q^{7} +2.40916 q^{11} -3.92621 q^{13} +3.63982 q^{17} +2.30861 q^{19} +4.55057 q^{23} -3.34982 q^{25} +5.81156 q^{29} -4.65708 q^{31} +2.52381 q^{35} +4.09300 q^{37} -7.34227 q^{41} -3.88636 q^{43} -7.51721 q^{47} -3.14003 q^{49} +10.6471 q^{53} -3.09479 q^{55} -10.8022 q^{59} -14.0025 q^{61} +5.04358 q^{65} +0.286799 q^{67} +10.1569 q^{71} +3.44823 q^{73} -4.73323 q^{77} +5.99698 q^{79} -10.0713 q^{83} -4.67569 q^{85} +9.05989 q^{89} +7.71375 q^{91} -2.96563 q^{95} +4.02676 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 7 q^{5} - 2 q^{7}+O(q^{10}) $ 5 * q + 7 * q^5 - 2 * q^7	$ 5 q + 7 q^{5} - 2 q^{7} + 5 q^{11} - 8 q^{13} + 7 q^{17} + 2 q^{19} + 13 q^{23} + 2 q^{25} + 11 q^{29} - 12 q^{31} + 12 q^{35} - 7 q^{37} + 12 q^{41} + 19 q^{47} - 9 q^{49} + 21 q^{53} - q^{55} + 7 q^{59} - 6 q^{61} - 14 q^{65} + 10 q^{67} + 35 q^{71} - 8 q^{73} + 6 q^{77} + 11 q^{83} + 5 q^{85} + 32 q^{89} + 5 q^{91} + 19 q^{95} + 11 q^{97}+O(q^{100}) $ 5 * q + 7 * q^5 - 2 * q^7 + 5 * q^11 - 8 * q^13 + 7 * q^17 + 2 * q^19 + 13 * q^23 + 2 * q^25 + 11 * q^29 - 12 * q^31 + 12 * q^35 - 7 * q^37 + 12 * q^41 + 19 * q^47 - 9 * q^49 + 21 * q^53 - q^55 + 7 * q^59 - 6 * q^61 - 14 * q^65 + 10 * q^67 + 35 * q^71 - 8 * q^73 + 6 * q^77 + 11 * q^83 + 5 * q^85 + 32 * q^89 + 5 * q^91 + 19 * q^95 + 11 * 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
$5$	−1.28459	−0.574487	−0.287244	−	0.957858i	$-0.592739\pi$
$5$	−1.28459	−0.574487	−0.287244	+	0.957858i	$0.592739\pi$
$6$	0	0
$7$	−1.96468	−0.742579	−0.371290	−	0.928517i	$-0.621084\pi$
$7$	−1.96468	−0.742579	−0.371290	+	0.928517i	$0.621084\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.40916	0.726390	0.363195	−	0.931713i	$-0.381686\pi$
$11$	2.40916	0.726390	0.363195	+	0.931713i	$0.381686\pi$
$12$	0	0
$13$	−3.92621	−1.08893	−0.544467	−	0.838782i	$-0.683268\pi$
$13$	−3.92621	−1.08893	−0.544467	+	0.838782i	$0.683268\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.63982	0.882787	0.441393	−	0.897314i	$-0.354484\pi$
$17$	3.63982	0.882787	0.441393	+	0.897314i	$0.354484\pi$
$18$	0	0
$19$	2.30861	0.529632	0.264816	−	0.964299i	$-0.414689\pi$
$19$	2.30861	0.529632	0.264816	+	0.964299i	$0.414689\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.55057	0.948860	0.474430	−	0.880293i	$-0.342654\pi$
$23$	4.55057	0.948860	0.474430	+	0.880293i	$0.342654\pi$
$24$	0	0
$25$	−3.34982	−0.669965
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.81156	1.07918	0.539590	−	0.841928i	$-0.318579\pi$
$29$	5.81156	1.07918	0.539590	+	0.841928i	$0.318579\pi$
$30$	0	0
$31$	−4.65708	−0.836436	−0.418218	−	0.908347i	$-0.637345\pi$
$31$	−4.65708	−0.836436	−0.418218	+	0.908347i	$0.637345\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.52381	0.426602
$36$	0	0
$37$	4.09300	0.672885	0.336442	−	0.941704i	$-0.390776\pi$
$37$	4.09300	0.672885	0.336442	+	0.941704i	$0.390776\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.34227	−1.14667	−0.573335	−	0.819321i	$-0.694351\pi$
$41$	−7.34227	−1.14667	−0.573335	+	0.819321i	$0.694351\pi$
$42$	0	0
$43$	−3.88636	−0.592664	−0.296332	−	0.955085i	$-0.595763\pi$
$43$	−3.88636	−0.592664	−0.296332	+	0.955085i	$0.595763\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.51721	−1.09650	−0.548249	−	0.836315i	$-0.684705\pi$
$47$	−7.51721	−1.09650	−0.548249	+	0.836315i	$0.684705\pi$
$48$	0	0
$49$	−3.14003	−0.448576
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.6471	1.46250	0.731248	−	0.682111i	$-0.238938\pi$
$53$	10.6471	1.46250	0.731248	+	0.682111i	$0.238938\pi$
$54$	0	0
$55$	−3.09479	−0.417302
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.8022	−1.40633	−0.703164	−	0.711028i	$-0.748230\pi$
$59$	−10.8022	−1.40633	−0.703164	+	0.711028i	$0.748230\pi$
$60$	0	0
$61$	−14.0025	−1.79284	−0.896418	−	0.443209i	$-0.853840\pi$
$61$	−14.0025	−1.79284	−0.896418	+	0.443209i	$0.853840\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.04358	0.625579
$66$	0	0
$67$	0.286799	0.0350381	0.0175191	−	0.999847i	$-0.494423\pi$
$67$	0.286799	0.0350381	0.0175191	+	0.999847i	$0.494423\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.1569	1.20540	0.602699	−	0.797968i	$-0.294092\pi$
$71$	10.1569	1.20540	0.602699	+	0.797968i	$0.294092\pi$
$72$	0	0
$73$	3.44823	0.403585	0.201792	−	0.979428i	$-0.435323\pi$
$73$	3.44823	0.403585	0.201792	+	0.979428i	$0.435323\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.73323	−0.539402
$78$	0	0
$79$	5.99698	0.674713	0.337356	−	0.941377i	$-0.390467\pi$
$79$	5.99698	0.674713	0.337356	+	0.941377i	$0.390467\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.0713	−1.10547	−0.552737	−	0.833356i	$-0.686417\pi$
$83$	−10.0713	−1.10547	−0.552737	+	0.833356i	$0.686417\pi$
$84$	0	0
$85$	−4.67569	−0.507150
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.05989	0.960346	0.480173	−	0.877174i	$-0.340574\pi$
$89$	9.05989	0.960346	0.480173	+	0.877174i	$0.340574\pi$
$90$	0	0
$91$	7.71375	0.808620
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.96563	−0.304267
$96$	0	0
$97$	4.02676	0.408855	0.204428	−	0.978882i	$-0.434467\pi$
$97$	4.02676	0.408855	0.204428	+	0.978882i	$0.434467\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.06980	−0.106450	−0.0532248	−	0.998583i	$-0.516950\pi$
$101$	−1.06980	−0.106450	−0.0532248	+	0.998583i	$0.516950\pi$
$102$	0	0
$103$	15.5422	1.53142	0.765709	−	0.643188i	$-0.222388\pi$
$103$	15.5422	1.53142	0.765709	+	0.643188i	$0.222388\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.54641	0.246171	0.123085	−	0.992396i	$-0.460721\pi$
$107$	2.54641	0.246171	0.123085	+	0.992396i	$0.460721\pi$
$108$	0	0
$109$	15.5695	1.49129	0.745646	−	0.666343i	$-0.232141\pi$
$109$	15.5695	1.49129	0.745646	+	0.666343i	$0.232141\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.22887	0.303746	0.151873	−	0.988400i	$-0.451469\pi$
$113$	3.22887	0.303746	0.151873	+	0.988400i	$0.451469\pi$
$114$	0	0
$115$	−5.84563	−0.545108
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.15109	−0.655539
$120$	0	0
$121$	−5.19594	−0.472358
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.7261	0.959373
$126$	0	0
$127$	0.940660	0.0834701	0.0417351	−	0.999129i	$-0.486711\pi$
$127$	0.940660	0.0834701	0.0417351	+	0.999129i	$0.486711\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.7284	1.19946	0.599729	−	0.800203i	$-0.295275\pi$
$131$	13.7284	1.19946	0.599729	+	0.800203i	$0.295275\pi$
$132$	0	0
$133$	−4.53569	−0.393294
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.2672	1.47524	0.737620	−	0.675216i	$-0.235950\pi$
$137$	17.2672	1.47524	0.737620	+	0.675216i	$0.235950\pi$
$138$	0	0
$139$	10.0634	0.853570	0.426785	−	0.904353i	$-0.359646\pi$
$139$	10.0634	0.853570	0.426785	+	0.904353i	$0.359646\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−9.45888	−0.790991
$144$	0	0
$145$	−7.46548	−0.619975
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.8300	1.54261	0.771306	−	0.636464i	$-0.219604\pi$
$149$	18.8300	1.54261	0.771306	+	0.636464i	$0.219604\pi$
$150$	0	0
$151$	10.6354	0.865498	0.432749	−	0.901514i	$-0.357544\pi$
$151$	10.6354	0.865498	0.432749	+	0.901514i	$0.357544\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.98244	0.480521
$156$	0	0
$157$	22.2179	1.77318	0.886592	−	0.462553i	$-0.153066\pi$
$157$	22.2179	1.77318	0.886592	+	0.462553i	$0.153066\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.94042	−0.704604
$162$	0	0
$163$	−21.9987	−1.72307	−0.861535	−	0.507699i	$-0.830496\pi$
$163$	−21.9987	−1.72307	−0.861535	+	0.507699i	$0.830496\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	2.41512	0.185778
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−15.8426	−1.20449	−0.602246	−	0.798311i	$-0.705727\pi$
$173$	−15.8426	−1.20449	−0.602246	+	0.798311i	$0.705727\pi$
$174$	0	0
$175$	6.58133	0.497502
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.43373	0.256649	0.128324	−	0.991732i	$-0.459040\pi$
$179$	3.43373	0.256649	0.128324	+	0.991732i	$0.459040\pi$
$180$	0	0
$181$	−11.3243	−0.841725	−0.420863	−	0.907124i	$-0.638273\pi$
$181$	−11.3243	−0.841725	−0.420863	+	0.907124i	$0.638273\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.25783	−0.386564
$186$	0	0
$187$	8.76893	0.641247
$188$	0	0
$189$	0	0
$190$	0	0
$191$	26.0824	1.88726	0.943628	−	0.331007i	$-0.107388\pi$
$191$	26.0824	1.88726	0.943628	+	0.331007i	$0.107388\pi$
$192$	0	0
$193$	25.5074	1.83606	0.918031	−	0.396508i	$-0.129778\pi$
$193$	25.5074	1.83606	0.918031	+	0.396508i	$0.129778\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.73730	0.337519	0.168759	−	0.985657i	$-0.446024\pi$
$197$	4.73730	0.337519	0.168759	+	0.985657i	$0.446024\pi$
$198$	0	0
$199$	−13.8698	−0.983206	−0.491603	−	0.870819i	$-0.663589\pi$
$199$	−13.8698	−0.983206	−0.491603	+	0.870819i	$0.663589\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−11.4179	−0.801376
$204$	0	0
$205$	9.43182	0.658747
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.56182	0.384719
$210$	0	0
$211$	17.5267	1.20659	0.603294	−	0.797519i	$-0.293854\pi$
$211$	17.5267	1.20659	0.603294	+	0.797519i	$0.293854\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.99238	0.340478
$216$	0	0
$217$	9.14967	0.621120
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−14.2907	−0.961297
$222$	0	0
$223$	17.9825	1.20420	0.602099	−	0.798422i	$-0.294331\pi$
$223$	17.9825	1.20420	0.602099	+	0.798422i	$0.294331\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.80164	−0.318696	−0.159348	−	0.987222i	$-0.550939\pi$
$227$	−4.80164	−0.318696	−0.159348	+	0.987222i	$0.550939\pi$
$228$	0	0
$229$	−27.8758	−1.84208	−0.921041	−	0.389465i	$-0.872660\pi$
$229$	−27.8758	−1.84208	−0.921041	+	0.389465i	$0.872660\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.52381	−0.361877	−0.180939	−	0.983494i	$-0.557914\pi$
$233$	−5.52381	−0.361877	−0.180939	+	0.983494i	$0.557914\pi$
$234$	0	0
$235$	9.65655	0.629924
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.2120	1.50146	0.750729	−	0.660611i	$-0.229703\pi$
$239$	23.2120	1.50146	0.750729	+	0.660611i	$0.229703\pi$
$240$	0	0
$241$	3.27780	0.211142	0.105571	−	0.994412i	$-0.466333\pi$
$241$	3.27780	0.211142	0.105571	+	0.994412i	$0.466333\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.03366	0.257701
$246$	0	0
$247$	−9.06410	−0.576735
$248$	0	0
$249$	0	0
$250$	0	0
$251$	11.2250	0.708518	0.354259	−	0.935147i	$-0.384733\pi$
$251$	11.2250	0.708518	0.354259	+	0.935147i	$0.384733\pi$
$252$	0	0
$253$	10.9631	0.689242
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.9517	−0.807907	−0.403954	−	0.914779i	$-0.632364\pi$
$257$	−12.9517	−0.807907	−0.403954	+	0.914779i	$0.632364\pi$
$258$	0	0
$259$	−8.04143	−0.499670
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.16998	0.257132	0.128566	−	0.991701i	$-0.458963\pi$
$263$	4.16998	0.257132	0.128566	+	0.991701i	$0.458963\pi$
$264$	0	0
$265$	−13.6772	−0.840185
$266$	0	0
$267$	0	0
$268$	0	0
$269$	31.2329	1.90430	0.952152	−	0.305625i	$-0.0988654\pi$
$269$	31.2329	1.90430	0.952152	+	0.305625i	$0.0988654\pi$
$270$	0	0
$271$	19.9256	1.21039	0.605197	−	0.796076i	$-0.293095\pi$
$271$	19.9256	1.21039	0.605197	+	0.796076i	$0.293095\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.07027	−0.486655
$276$	0	0
$277$	2.20031	0.132204	0.0661021	−	0.997813i	$-0.478944\pi$
$277$	2.20031	0.132204	0.0661021	+	0.997813i	$0.478944\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−26.4717	−1.57917	−0.789585	−	0.613641i	$-0.789704\pi$
$281$	−26.4717	−1.57917	−0.789585	+	0.613641i	$0.789704\pi$
$282$	0	0
$283$	14.2466	0.846870	0.423435	−	0.905926i	$-0.360824\pi$
$283$	14.2466	0.846870	0.423435	+	0.905926i	$0.360824\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14.4252	0.851494
$288$	0	0
$289$	−3.75169	−0.220687
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−12.2255	−0.714221	−0.357111	−	0.934062i	$-0.616238\pi$
$293$	−12.2255	−0.714221	−0.357111	+	0.934062i	$0.616238\pi$
$294$	0	0
$295$	13.8764	0.807917
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−17.8665	−1.03325
$300$	0	0
$301$	7.63545	0.440100
$302$	0	0
$303$	0	0
$304$	0	0
$305$	17.9875	1.02996
$306$	0	0
$307$	1.04146	0.0594395	0.0297197	−	0.999558i	$-0.490539\pi$
$307$	1.04146	0.0594395	0.0297197	+	0.999558i	$0.490539\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.61814	0.0917564	0.0458782	−	0.998947i	$-0.485391\pi$
$311$	1.61814	0.0917564	0.0458782	+	0.998947i	$0.485391\pi$
$312$	0	0
$313$	−25.1205	−1.41990	−0.709948	−	0.704254i	$-0.751281\pi$
$313$	−25.1205	−1.41990	−0.709948	+	0.704254i	$0.751281\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.6518	1.44075	0.720374	−	0.693586i	$-0.243970\pi$
$317$	25.6518	1.44075	0.720374	+	0.693586i	$0.243970\pi$
$318$	0	0
$319$	14.0010	0.783905
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.40294	0.467552
$324$	0	0
$325$	13.1521	0.729547
$326$	0	0
$327$	0	0
$328$	0	0
$329$	14.7689	0.814236
$330$	0	0
$331$	17.7667	0.976544	0.488272	−	0.872691i	$-0.337627\pi$
$331$	17.7667	0.976544	0.488272	+	0.872691i	$0.337627\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.368420	−0.0201290
$336$	0	0
$337$	−19.2213	−1.04705	−0.523526	−	0.852010i	$-0.675384\pi$
$337$	−19.2213	−1.04705	−0.523526	+	0.852010i	$0.675384\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11.2197	−0.607578
$342$	0	0
$343$	19.9219	1.07568
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.1326	−0.758677	−0.379339	−	0.925258i	$-0.623848\pi$
$347$	−14.1326	−0.758677	−0.379339	+	0.925258i	$0.623848\pi$
$348$	0	0
$349$	−28.1862	−1.50877	−0.754387	−	0.656429i	$-0.772066\pi$
$349$	−28.1862	−1.50877	−0.754387	+	0.656429i	$0.772066\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	16.0991	0.856870	0.428435	−	0.903573i	$-0.359065\pi$
$353$	16.0991	0.856870	0.428435	+	0.903573i	$0.359065\pi$
$354$	0	0
$355$	−13.0474	−0.692486
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.09816	0.163515	0.0817574	−	0.996652i	$-0.473947\pi$
$359$	3.09816	0.163515	0.0817574	+	0.996652i	$0.473947\pi$
$360$	0	0
$361$	−13.6703	−0.719490
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.42957	−0.231854
$366$	0	0
$367$	−15.5041	−0.809307	−0.404654	−	0.914470i	$-0.632608\pi$
$367$	−15.5041	−0.809307	−0.404654	+	0.914470i	$0.632608\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−20.9182	−1.08602
$372$	0	0
$373$	−8.04456	−0.416532	−0.208266	−	0.978072i	$-0.566782\pi$
$373$	−8.04456	−0.416532	−0.208266	+	0.978072i	$0.566782\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−22.8174	−1.17516
$378$	0	0
$379$	18.3818	0.944210	0.472105	−	0.881542i	$-0.343494\pi$
$379$	18.3818	0.944210	0.472105	+	0.881542i	$0.343494\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.38490	−0.0707649	−0.0353825	−	0.999374i	$-0.511265\pi$
$383$	−1.38490	−0.0707649	−0.0353825	+	0.999374i	$0.511265\pi$
$384$	0	0
$385$	6.08028	0.309880
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−18.6320	−0.944680	−0.472340	−	0.881416i	$-0.656591\pi$
$389$	−18.6320	−0.944680	−0.472340	+	0.881416i	$0.656591\pi$
$390$	0	0
$391$	16.5633	0.837641
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.70367	−0.387614
$396$	0	0
$397$	36.8598	1.84994	0.924969	−	0.380042i	$-0.124090\pi$
$397$	36.8598	1.84994	0.924969	+	0.380042i	$0.124090\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.06368	0.252868	0.126434	−	0.991975i	$-0.459647\pi$
$401$	5.06368	0.252868	0.126434	+	0.991975i	$0.459647\pi$
$402$	0	0
$403$	18.2847	0.910824
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.86070	0.488777
$408$	0	0
$409$	24.0361	1.18851	0.594254	−	0.804278i	$-0.297447\pi$
$409$	24.0361	1.18851	0.594254	+	0.804278i	$0.297447\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	21.2229	1.04431
$414$	0	0
$415$	12.9376	0.635080
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−10.8318	−0.529170	−0.264585	−	0.964362i	$-0.585235\pi$
$419$	−10.8318	−0.529170	−0.264585	+	0.964362i	$0.585235\pi$
$420$	0	0
$421$	2.18637	0.106557	0.0532785	−	0.998580i	$-0.483033\pi$
$421$	2.18637	0.106557	0.0532785	+	0.998580i	$0.483033\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−12.1928	−0.591436
$426$	0	0
$427$	27.5104	1.33132
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.3902	−0.933994	−0.466997	−	0.884259i	$-0.654664\pi$
$431$	−19.3902	−0.933994	−0.466997	+	0.884259i	$0.654664\pi$
$432$	0	0
$433$	18.7996	0.903449	0.451725	−	0.892157i	$-0.350809\pi$
$433$	18.7996	0.903449	0.451725	+	0.892157i	$0.350809\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.5055	0.502547
$438$	0	0
$439$	−8.05917	−0.384643	−0.192322	−	0.981332i	$-0.561602\pi$
$439$	−8.05917	−0.384643	−0.192322	+	0.981332i	$0.561602\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	28.1425	1.33709	0.668546	−	0.743671i	$-0.266917\pi$
$443$	28.1425	1.33709	0.668546	+	0.743671i	$0.266917\pi$
$444$	0	0
$445$	−11.6383	−0.551706
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−28.9364	−1.36559	−0.682797	−	0.730608i	$-0.739237\pi$
$449$	−28.9364	−1.36559	−0.682797	+	0.730608i	$0.739237\pi$
$450$	0	0
$451$	−17.6887	−0.832929
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−9.90902	−0.464542
$456$	0	0
$457$	−7.80085	−0.364909	−0.182454	−	0.983214i	$-0.558404\pi$
$457$	−7.80085	−0.364909	−0.182454	+	0.983214i	$0.558404\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13.6981	0.637984	0.318992	−	0.947757i	$-0.396656\pi$
$461$	13.6981	0.637984	0.318992	+	0.947757i	$0.396656\pi$
$462$	0	0
$463$	39.1886	1.82125	0.910625	−	0.413235i	$-0.135601\pi$
$463$	39.1886	1.82125	0.910625	+	0.413235i	$0.135601\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.2581	−0.798610	−0.399305	−	0.916818i	$-0.630748\pi$
$467$	−17.2581	−0.798610	−0.399305	+	0.916818i	$0.630748\pi$
$468$	0	0
$469$	−0.563469	−0.0260186
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−9.36287	−0.430505
$474$	0	0
$475$	−7.73344	−0.354835
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.46912	0.112817	0.0564084	−	0.998408i	$-0.482035\pi$
$479$	2.46912	0.112817	0.0564084	+	0.998408i	$0.482035\pi$
$480$	0	0
$481$	−16.0700	−0.732727
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.17274	−0.234882
$486$	0	0
$487$	17.6955	0.801861	0.400930	−	0.916108i	$-0.368687\pi$
$487$	17.6955	0.801861	0.400930	+	0.916108i	$0.368687\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−14.9043	−0.672623	−0.336312	−	0.941751i	$-0.609180\pi$
$491$	−14.9043	−0.672623	−0.336312	+	0.941751i	$0.609180\pi$
$492$	0	0
$493$	21.1530	0.952685
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−19.9550	−0.895104
$498$	0	0
$499$	14.7483	0.660226	0.330113	−	0.943941i	$-0.392913\pi$
$499$	14.7483	0.660226	0.330113	+	0.943941i	$0.392913\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21.6170	−0.963856	−0.481928	−	0.876211i	$-0.660063\pi$
$503$	−21.6170	−0.963856	−0.481928	+	0.876211i	$0.660063\pi$
$504$	0	0
$505$	1.37426	0.0611539
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11.3422	0.502735	0.251368	−	0.967892i	$-0.419120\pi$
$509$	11.3422	0.502735	0.251368	+	0.967892i	$0.419120\pi$
$510$	0	0
$511$	−6.77467	−0.299694
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−19.9654	−0.879779
$516$	0	0
$517$	−18.1102	−0.796484
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.90072	0.258515	0.129258	−	0.991611i	$-0.458741\pi$
$521$	5.90072	0.258515	0.129258	+	0.991611i	$0.458741\pi$
$522$	0	0
$523$	7.21958	0.315690	0.157845	−	0.987464i	$-0.449545\pi$
$523$	7.21958	0.315690	0.157845	+	0.987464i	$0.449545\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.9509	−0.738394
$528$	0	0
$529$	−2.29229	−0.0996648
$530$	0	0
$531$	0	0
$532$	0	0
$533$	28.8273	1.24865
$534$	0	0
$535$	−3.27110	−0.141422
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−7.56484	−0.325841
$540$	0	0
$541$	−4.15094	−0.178463	−0.0892314	−	0.996011i	$-0.528441\pi$
$541$	−4.15094	−0.178463	−0.0892314	+	0.996011i	$0.528441\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−20.0005	−0.856728
$546$	0	0
$547$	−5.37704	−0.229906	−0.114953	−	0.993371i	$-0.536672\pi$
$547$	−5.37704	−0.229906	−0.114953	+	0.993371i	$0.536672\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13.4166	0.571568
$552$	0	0
$553$	−11.7822	−0.501028
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.1551	0.557399	0.278700	−	0.960378i	$-0.410097\pi$
$557$	13.1551	0.557399	0.278700	+	0.960378i	$0.410097\pi$
$558$	0	0
$559$	15.2587	0.645372
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.08117	−0.214146	−0.107073	−	0.994251i	$-0.534148\pi$
$563$	−5.08117	−0.214146	−0.107073	+	0.994251i	$0.534148\pi$
$564$	0	0
$565$	−4.14778	−0.174498
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16.5777	0.694972	0.347486	−	0.937685i	$-0.387035\pi$
$569$	16.5777	0.694972	0.347486	+	0.937685i	$0.387035\pi$
$570$	0	0
$571$	30.5518	1.27855	0.639277	−	0.768977i	$-0.279234\pi$
$571$	30.5518	1.27855	0.639277	+	0.768977i	$0.279234\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−15.2436	−0.635703
$576$	0	0
$577$	−23.2441	−0.967663	−0.483832	−	0.875161i	$-0.660755\pi$
$577$	−23.2441	−0.967663	−0.483832	+	0.875161i	$0.660755\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	19.7870	0.820902
$582$	0	0
$583$	25.6507	1.06234
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.3846	0.800088	0.400044	−	0.916496i	$-0.368995\pi$
$587$	19.3846	0.800088	0.400044	+	0.916496i	$0.368995\pi$
$588$	0	0
$589$	−10.7514	−0.443003
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.34754	−0.383857	−0.191929	−	0.981409i	$-0.561474\pi$
$593$	−9.34754	−0.383857	−0.191929	+	0.981409i	$0.561474\pi$
$594$	0	0
$595$	9.18623	0.376599
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.66606	−0.0680732	−0.0340366	−	0.999421i	$-0.510836\pi$
$599$	−1.66606	−0.0680732	−0.0340366	+	0.999421i	$0.510836\pi$
$600$	0	0
$601$	−26.5815	−1.08428	−0.542141	−	0.840287i	$-0.682386\pi$
$601$	−26.5815	−1.08428	−0.542141	+	0.840287i	$0.682386\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6.67466	0.271363
$606$	0	0
$607$	−30.8790	−1.25334	−0.626670	−	0.779285i	$-0.715583\pi$
$607$	−30.8790	−1.25334	−0.626670	+	0.779285i	$0.715583\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	29.5141	1.19401
$612$	0	0
$613$	14.9530	0.603947	0.301973	−	0.953316i	$-0.402355\pi$
$613$	14.9530	0.603947	0.301973	+	0.953316i	$0.402355\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20.3705	−0.820087	−0.410043	−	0.912066i	$-0.634486\pi$
$617$	−20.3705	−0.820087	−0.410043	+	0.912066i	$0.634486\pi$
$618$	0	0
$619$	0.534153	0.0214694	0.0107347	−	0.999942i	$-0.496583\pi$
$619$	0.534153	0.0214694	0.0107347	+	0.999942i	$0.496583\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−17.7998	−0.713133
$624$	0	0
$625$	2.97043	0.118817
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.8978	0.594014
$630$	0	0
$631$	−16.3425	−0.650583	−0.325292	−	0.945614i	$-0.605462\pi$
$631$	−16.3425	−0.650583	−0.325292	+	0.945614i	$0.605462\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.20836	−0.0479525
$636$	0	0
$637$	12.3284	0.488470
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−45.9146	−1.81352	−0.906759	−	0.421649i	$-0.861452\pi$
$641$	−45.9146	−1.81352	−0.906759	+	0.421649i	$0.861452\pi$
$642$	0	0
$643$	−2.16517	−0.0853860	−0.0426930	−	0.999088i	$-0.513594\pi$
$643$	−2.16517	−0.0853860	−0.0426930	+	0.999088i	$0.513594\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	29.7954	1.17138	0.585688	−	0.810536i	$-0.300824\pi$
$647$	29.7954	1.17138	0.585688	+	0.810536i	$0.300824\pi$
$648$	0	0
$649$	−26.0243	−1.02154
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.26237	−0.362464	−0.181232	−	0.983440i	$-0.558009\pi$
$653$	−9.26237	−0.362464	−0.181232	+	0.983440i	$0.558009\pi$
$654$	0	0
$655$	−17.6354	−0.689073
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−0.492474	−0.0191841	−0.00959203	−	0.999954i	$-0.503053\pi$
$659$	−0.492474	−0.0191841	−0.00959203	+	0.999954i	$0.503053\pi$
$660$	0	0
$661$	−8.06786	−0.313804	−0.156902	−	0.987614i	$-0.550151\pi$
$661$	−8.06786	−0.313804	−0.156902	+	0.987614i	$0.550151\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.82651	0.225942
$666$	0	0
$667$	26.4459	1.02399
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−33.7343	−1.30230
$672$	0	0
$673$	6.59943	0.254389	0.127195	−	0.991878i	$-0.459403\pi$
$673$	6.59943	0.254389	0.127195	+	0.991878i	$0.459403\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.6380	0.562584	0.281292	−	0.959622i	$-0.409237\pi$
$677$	14.6380	0.562584	0.281292	+	0.959622i	$0.409237\pi$
$678$	0	0
$679$	−7.91130	−0.303608
$680$	0	0
$681$	0	0
$682$	0	0
$683$	10.3419	0.395724	0.197862	−	0.980230i	$-0.436600\pi$
$683$	10.3419	0.395724	0.197862	+	0.980230i	$0.436600\pi$
$684$	0	0
$685$	−22.1814	−0.847506
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−41.8029	−1.59256
$690$	0	0
$691$	−34.5829	−1.31559	−0.657797	−	0.753195i	$-0.728512\pi$
$691$	−34.5829	−1.31559	−0.657797	+	0.753195i	$0.728512\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.9274	−0.490365
$696$	0	0
$697$	−26.7246	−1.01227
$698$	0	0
$699$	0	0
$700$	0	0
$701$	38.5480	1.45594	0.727969	−	0.685610i	$-0.240465\pi$
$701$	38.5480	1.45594	0.727969	+	0.685610i	$0.240465\pi$
$702$	0	0
$703$	9.44915	0.356381
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.10183	0.0790473
$708$	0	0
$709$	16.9239	0.635592	0.317796	−	0.948159i	$-0.397057\pi$
$709$	16.9239	0.635592	0.317796	+	0.948159i	$0.397057\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−21.1924	−0.793660
$714$	0	0
$715$	12.1508	0.454414
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−16.1218	−0.601241	−0.300621	−	0.953744i	$-0.597194\pi$
$719$	−16.1218	−0.601241	−0.300621	+	0.953744i	$0.597194\pi$
$720$	0	0
$721$	−30.5354	−1.13720
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−19.4677	−0.723012
$726$	0	0
$727$	10.8925	0.403980	0.201990	−	0.979388i	$-0.435259\pi$
$727$	10.8925	0.403980	0.201990	+	0.979388i	$0.435259\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−14.1457	−0.523196
$732$	0	0
$733$	39.2054	1.44809	0.724043	−	0.689755i	$-0.242282\pi$
$733$	39.2054	1.44809	0.724043	+	0.689755i	$0.242282\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.690946	0.0254513
$738$	0	0
$739$	−46.8075	−1.72184	−0.860921	−	0.508738i	$-0.830112\pi$
$739$	−46.8075	−1.72184	−0.860921	+	0.508738i	$0.830112\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	33.4808	1.22829	0.614145	−	0.789193i	$-0.289501\pi$
$743$	33.4808	1.22829	0.614145	+	0.789193i	$0.289501\pi$
$744$	0	0
$745$	−24.1888	−0.886211
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.00289	−0.182801
$750$	0	0
$751$	−24.7166	−0.901922	−0.450961	−	0.892544i	$-0.648919\pi$
$751$	−24.7166	−0.901922	−0.450961	+	0.892544i	$0.648919\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−13.6622	−0.497218
$756$	0	0
$757$	−26.5140	−0.963667	−0.481834	−	0.876263i	$-0.660029\pi$
$757$	−26.5140	−0.963667	−0.481834	+	0.876263i	$0.660029\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−25.2602	−0.915681	−0.457840	−	0.889034i	$-0.651377\pi$
$761$	−25.2602	−0.915681	−0.457840	+	0.889034i	$0.651377\pi$
$762$	0	0
$763$	−30.5892	−1.10740
$764$	0	0
$765$	0	0
$766$	0	0
$767$	42.4117	1.53140
$768$	0	0
$769$	−19.6768	−0.709564	−0.354782	−	0.934949i	$-0.615445\pi$
$769$	−19.6768	−0.709564	−0.354782	+	0.934949i	$0.615445\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	36.8440	1.32519	0.662594	−	0.748979i	$-0.269456\pi$
$773$	36.8440	1.32519	0.662594	+	0.748979i	$0.269456\pi$
$774$	0	0
$775$	15.6004	0.560382
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−16.9505	−0.607313
$780$	0	0
$781$	24.4695	0.875589
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−28.5410	−1.01867
$786$	0	0
$787$	5.36433	0.191218	0.0956088	−	0.995419i	$-0.469520\pi$
$787$	5.36433	0.191218	0.0956088	+	0.995419i	$0.469520\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.34369	−0.225556
$792$	0	0
$793$	54.9767	1.95228
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.20815	0.255326	0.127663	−	0.991818i	$-0.459252\pi$
$797$	7.20815	0.255326	0.127663	+	0.991818i	$0.459252\pi$
$798$	0	0
$799$	−27.3613	−0.967973
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.30734	0.293160
$804$	0	0
$805$	11.4848	0.404786
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23.3611	−0.821334	−0.410667	−	0.911785i	$-0.634704\pi$
$809$	−23.3611	−0.821334	−0.410667	+	0.911785i	$0.634704\pi$
$810$	0	0
$811$	47.5681	1.67034	0.835170	−	0.549991i	$-0.185369\pi$
$811$	47.5681	1.67034	0.835170	+	0.549991i	$0.185369\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	28.2593	0.989881
$816$	0	0
$817$	−8.97209	−0.313894
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.28529	−0.0448568	−0.0224284	−	0.999748i	$-0.507140\pi$
$821$	−1.28529	−0.0448568	−0.0224284	+	0.999748i	$0.507140\pi$
$822$	0	0
$823$	15.1103	0.526712	0.263356	−	0.964699i	$-0.415171\pi$
$823$	15.1103	0.526712	0.263356	+	0.964699i	$0.415171\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.1009	1.11626	0.558129	−	0.829754i	$-0.311519\pi$
$827$	32.1009	1.11626	0.558129	+	0.829754i	$0.311519\pi$
$828$	0	0
$829$	−39.5385	−1.37323	−0.686614	−	0.727022i	$-0.740904\pi$
$829$	−39.5385	−1.37323	−0.686614	+	0.727022i	$0.740904\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−11.4292	−0.395997
$834$	0	0
$835$	−1.28459	−0.0444551
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.69426	0.0930161	0.0465081	−	0.998918i	$-0.485191\pi$
$839$	2.69426	0.0930161	0.0465081	+	0.998918i	$0.485191\pi$
$840$	0	0
$841$	4.77421	0.164628
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.10244	−0.106727
$846$	0	0
$847$	10.2084	0.350763
$848$	0	0
$849$	0	0
$850$	0	0
$851$	18.6255	0.638473
$852$	0	0
$853$	48.3525	1.65556	0.827779	−	0.561054i	$-0.189604\pi$
$853$	48.3525	1.65556	0.827779	+	0.561054i	$0.189604\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	36.9684	1.26282	0.631408	−	0.775450i	$-0.282477\pi$
$857$	36.9684	1.26282	0.631408	+	0.775450i	$0.282477\pi$
$858$	0	0
$859$	57.7504	1.97042	0.985210	−	0.171352i	$-0.0548136\pi$
$859$	57.7504	1.97042	0.985210	+	0.171352i	$0.0548136\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.667087	−0.0227079	−0.0113540	−	0.999936i	$-0.503614\pi$
$863$	−0.667087	−0.0227079	−0.0113540	+	0.999936i	$0.503614\pi$
$864$	0	0
$865$	20.3513	0.691965
$866$	0	0
$867$	0	0
$868$	0	0
$869$	14.4477	0.490105
$870$	0	0
$871$	−1.12603	−0.0381542
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−21.0734	−0.712411
$876$	0	0
$877$	−17.1291	−0.578407	−0.289204	−	0.957268i	$-0.593390\pi$
$877$	−17.1291	−0.578407	−0.289204	+	0.957268i	$0.593390\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18.5491	0.624935	0.312468	−	0.949928i	$-0.398844\pi$
$881$	18.5491	0.624935	0.312468	+	0.949928i	$0.398844\pi$
$882$	0	0
$883$	−18.0598	−0.607761	−0.303880	−	0.952710i	$-0.598282\pi$
$883$	−18.0598	−0.607761	−0.303880	+	0.952710i	$0.598282\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14.4311	0.484549	0.242274	−	0.970208i	$-0.422107\pi$
$887$	14.4311	0.484549	0.242274	+	0.970208i	$0.422107\pi$
$888$	0	0
$889$	−1.84810	−0.0619832
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−17.3543	−0.580740
$894$	0	0
$895$	−4.41094	−0.147442
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−27.0649	−0.902664
$900$	0	0
$901$	38.7537	1.29107
$902$	0	0
$903$	0	0
$904$	0	0
$905$	14.5470	0.483560
$906$	0	0
$907$	53.6448	1.78125	0.890624	−	0.454741i	$-0.150268\pi$
$907$	53.6448	1.78125	0.890624	+	0.454741i	$0.150268\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	39.7916	1.31835	0.659177	−	0.751988i	$-0.270905\pi$
$911$	39.7916	1.31835	0.659177	+	0.751988i	$0.270905\pi$
$912$	0	0
$913$	−24.2635	−0.803005
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−26.9720	−0.890693
$918$	0	0
$919$	−25.7644	−0.849889	−0.424945	−	0.905219i	$-0.639706\pi$
$919$	−25.7644	−0.849889	−0.424945	+	0.905219i	$0.639706\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−39.8780	−1.31260
$924$	0	0
$925$	−13.7108	−0.450809
$926$	0	0
$927$	0	0
$928$	0	0
$929$	24.8575	0.815549	0.407775	−	0.913083i	$-0.366305\pi$
$929$	24.8575	0.815549	0.407775	+	0.913083i	$0.366305\pi$
$930$	0	0
$931$	−7.24911	−0.237580
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−11.2645	−0.368388
$936$	0	0
$937$	12.4739	0.407505	0.203752	−	0.979022i	$-0.434686\pi$
$937$	12.4739	0.407505	0.203752	+	0.979022i	$0.434686\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	8.43127	0.274852	0.137426	−	0.990512i	$-0.456117\pi$
$941$	8.43127	0.274852	0.137426	+	0.990512i	$0.456117\pi$
$942$	0	0
$943$	−33.4115	−1.08803
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.1816	1.50070	0.750350	−	0.661041i	$-0.229885\pi$
$947$	46.1816	1.50070	0.750350	+	0.661041i	$0.229885\pi$
$948$	0	0
$949$	−13.5385	−0.439477
$950$	0	0
$951$	0	0
$952$	0	0
$953$	27.1218	0.878563	0.439281	−	0.898350i	$-0.355233\pi$
$953$	27.1218	0.878563	0.439281	+	0.898350i	$0.355233\pi$
$954$	0	0
$955$	−33.5053	−1.08420
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−33.9246	−1.09548
$960$	0	0
$961$	−9.31164	−0.300375
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−32.7666	−1.05479
$966$	0	0
$967$	−51.9680	−1.67118	−0.835590	−	0.549354i	$-0.814874\pi$
$967$	−51.9680	−1.67118	−0.835590	+	0.549354i	$0.814874\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.3326	1.16597	0.582985	−	0.812483i	$-0.301885\pi$
$971$	36.3326	1.16597	0.582985	+	0.812483i	$0.301885\pi$
$972$	0	0
$973$	−19.7714	−0.633843
$974$	0	0
$975$	0	0
$976$	0	0
$977$	56.2379	1.79921	0.899605	−	0.436704i	$-0.143854\pi$
$977$	56.2379	1.79921	0.899605	+	0.436704i	$0.143854\pi$
$978$	0	0
$979$	21.8267	0.697586
$980$	0	0
$981$	0	0
$982$	0	0
$983$	45.9094	1.46428	0.732141	−	0.681153i	$-0.238521\pi$
$983$	45.9094	1.46428	0.732141	+	0.681153i	$0.238521\pi$
$984$	0	0
$985$	−6.08550	−0.193900
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−17.6852	−0.562355
$990$	0	0
$991$	−23.0349	−0.731726	−0.365863	−	0.930669i	$-0.619226\pi$
$991$	−23.0349	−0.731726	−0.365863	+	0.930669i	$0.619226\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	17.8171	0.564839
$996$	0	0
$997$	−49.2868	−1.56093	−0.780464	−	0.625201i	$-0.785017\pi$
$997$	−49.2868	−1.56093	−0.780464	+	0.625201i	$0.785017\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.f.1.1		5
3.2	odd	2		2004.2.a.b.1.5	✓	5
12.11	even	2		8016.2.a.q.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.b.1.5	✓	5	3.2	odd	2
6012.2.a.f.1.1		5	1.1	even	1	trivial
8016.2.a.q.1.5		5	12.11	even	2

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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q-1.28459 q^{5} -1.96468 q^{7} +O(q^{10})\)	\(q-1.28459 q^{5} -1.96468 q^{7} +2.40916 q^{11} -3.92621 q^{13} +3.63982 q^{17} +2.30861 q^{19} +4.55057 q^{23} -3.34982 q^{25} +5.81156 q^{29} -4.65708 q^{31} +2.52381 q^{35} +4.09300 q^{37} -7.34227 q^{41} -3.88636 q^{43} -7.51721 q^{47} -3.14003 q^{49} +10.6471 q^{53} -3.09479 q^{55} -10.8022 q^{59} -14.0025 q^{61} +5.04358 q^{65} +0.286799 q^{67} +10.1569 q^{71} +3.44823 q^{73} -4.73323 q^{77} +5.99698 q^{79} -10.0713 q^{83} -4.67569 q^{85} +9.05989 q^{89} +7.71375 q^{91} -2.96563 q^{95} +4.02676 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 7 q^{5} - 2 q^{7}+O(q^{10}) \) 5 * q + 7 * q^5 - 2 * q^7	\( 5 q + 7 q^{5} - 2 q^{7} + 5 q^{11} - 8 q^{13} + 7 q^{17} + 2 q^{19} + 13 q^{23} + 2 q^{25} + 11 q^{29} - 12 q^{31} + 12 q^{35} - 7 q^{37} + 12 q^{41} + 19 q^{47} - 9 q^{49} + 21 q^{53} - q^{55} + 7 q^{59} - 6 q^{61} - 14 q^{65} + 10 q^{67} + 35 q^{71} - 8 q^{73} + 6 q^{77} + 11 q^{83} + 5 q^{85} + 32 q^{89} + 5 q^{91} + 19 q^{95} + 11 q^{97}+O(q^{100}) \) 5 * q + 7 * q^5 - 2 * q^7 + 5 * q^11 - 8 * q^13 + 7 * q^17 + 2 * q^19 + 13 * q^23 + 2 * q^25 + 11 * q^29 - 12 * q^31 + 12 * q^35 - 7 * q^37 + 12 * q^41 + 19 * q^47 - 9 * q^49 + 21 * q^53 - q^55 + 7 * q^59 - 6 * q^61 - 14 * q^65 + 10 * q^67 + 35 * q^71 - 8 * q^73 + 6 * q^77 + 11 * q^83 + 5 * q^85 + 32 * q^89 + 5 * q^91 + 19 * q^95 + 11 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more