LMFDB - Embedded newform 6044.2.a.a.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6044,2,Mod(1,6044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6044 = 2^{2} \cdot 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6044.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:	$48.2615829817$
Analytic rank:	$1$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	6044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.95257 q^{3} -0.800433 q^{5} -2.68929 q^{7} +5.71766 q^{9} +O(q^{10})$	$q-2.95257 q^{3} -0.800433 q^{5} -2.68929 q^{7} +5.71766 q^{9} +0.219287 q^{11} +6.32566 q^{13} +2.36333 q^{15} -0.478791 q^{17} -4.74800 q^{19} +7.94032 q^{21} -5.64593 q^{23} -4.35931 q^{25} -8.02408 q^{27} +4.65439 q^{29} -2.65044 q^{31} -0.647461 q^{33} +2.15260 q^{35} -0.141964 q^{37} -18.6770 q^{39} -5.06259 q^{41} +5.18103 q^{43} -4.57660 q^{45} +0.0233786 q^{47} +0.232290 q^{49} +1.41366 q^{51} +9.61174 q^{53} -0.175525 q^{55} +14.0188 q^{57} -1.37065 q^{59} +11.7247 q^{61} -15.3765 q^{63} -5.06327 q^{65} -11.2848 q^{67} +16.6700 q^{69} +8.85701 q^{71} +5.49638 q^{73} +12.8712 q^{75} -0.589728 q^{77} +15.3579 q^{79} +6.53866 q^{81} +8.90129 q^{83} +0.383240 q^{85} -13.7424 q^{87} +3.19482 q^{89} -17.0116 q^{91} +7.82561 q^{93} +3.80046 q^{95} +16.1180 q^{97} +1.25381 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.95257	−1.70467	−0.852333	−	0.522999i	$-0.824813\pi$
$3$	−2.95257	−1.70467	−0.852333	+	0.522999i	$0.824813\pi$
$4$	0	0
$5$	−0.800433	−0.357964	−0.178982	−	0.983852i	$-0.557280\pi$
$5$	−0.800433	−0.357964	−0.178982	+	0.983852i	$0.557280\pi$
$6$	0	0
$7$	−2.68929	−1.01646	−0.508228	−	0.861222i	$-0.669699\pi$
$7$	−2.68929	−1.01646	−0.508228	+	0.861222i	$0.669699\pi$
$8$	0	0
$9$	5.71766	1.90589
$10$	0	0
$11$	0.219287	0.0661177	0.0330588	−	0.999453i	$-0.489475\pi$
$11$	0.219287	0.0661177	0.0330588	+	0.999453i	$0.489475\pi$
$12$	0	0
$13$	6.32566	1.75442	0.877212	−	0.480104i	$-0.159401\pi$
$13$	6.32566	1.75442	0.877212	+	0.480104i	$0.159401\pi$
$14$	0	0
$15$	2.36333	0.610210
$16$	0	0
$17$	−0.478791	−0.116124	−0.0580619	−	0.998313i	$-0.518492\pi$
$17$	−0.478791	−0.116124	−0.0580619	+	0.998313i	$0.518492\pi$
$18$	0	0
$19$	−4.74800	−1.08927	−0.544633	−	0.838675i	$-0.683331\pi$
$19$	−4.74800	−1.08927	−0.544633	+	0.838675i	$0.683331\pi$
$20$	0	0
$21$	7.94032	1.73272
$22$	0	0
$23$	−5.64593	−1.17726	−0.588628	−	0.808404i	$-0.700332\pi$
$23$	−5.64593	−1.17726	−0.588628	+	0.808404i	$0.700332\pi$
$24$	0	0
$25$	−4.35931	−0.871861
$26$	0	0
$27$	−8.02408	−1.54423
$28$	0	0
$29$	4.65439	0.864299	0.432149	−	0.901802i	$-0.357755\pi$
$29$	4.65439	0.864299	0.432149	+	0.901802i	$0.357755\pi$
$30$	0	0
$31$	−2.65044	−0.476033	−0.238017	−	0.971261i	$-0.576497\pi$
$31$	−2.65044	−0.476033	−0.238017	+	0.971261i	$0.576497\pi$
$32$	0	0
$33$	−0.647461	−0.112709
$34$	0	0
$35$	2.15260	0.363855
$36$	0	0
$37$	−0.141964	−0.0233387	−0.0116694	−	0.999932i	$-0.503715\pi$
$37$	−0.141964	−0.0233387	−0.0116694	+	0.999932i	$0.503715\pi$
$38$	0	0
$39$	−18.6770	−2.99071
$40$	0	0
$41$	−5.06259	−0.790643	−0.395322	−	0.918543i	$-0.629367\pi$
$41$	−5.06259	−0.790643	−0.395322	+	0.918543i	$0.629367\pi$
$42$	0	0
$43$	5.18103	0.790100	0.395050	−	0.918660i	$-0.370727\pi$
$43$	5.18103	0.790100	0.395050	+	0.918660i	$0.370727\pi$
$44$	0	0
$45$	−4.57660	−0.682240
$46$	0	0
$47$	0.0233786	0.00341012	0.00170506	−	0.999999i	$-0.499457\pi$
$47$	0.0233786	0.00341012	0.00170506	+	0.999999i	$0.499457\pi$
$48$	0	0
$49$	0.232290	0.0331843
$50$	0	0
$51$	1.41366	0.197952
$52$	0	0
$53$	9.61174	1.32027	0.660137	−	0.751145i	$-0.270498\pi$
$53$	9.61174	1.32027	0.660137	+	0.751145i	$0.270498\pi$
$54$	0	0
$55$	−0.175525	−0.0236678
$56$	0	0
$57$	14.0188	1.85683
$58$	0	0
$59$	−1.37065	−0.178443	−0.0892214	−	0.996012i	$-0.528438\pi$
$59$	−1.37065	−0.178443	−0.0892214	+	0.996012i	$0.528438\pi$
$60$	0	0
$61$	11.7247	1.50120	0.750600	−	0.660757i	$-0.229765\pi$
$61$	11.7247	1.50120	0.750600	+	0.660757i	$0.229765\pi$
$62$	0	0
$63$	−15.3765	−1.93725
$64$	0	0
$65$	−5.06327	−0.628021
$66$	0	0
$67$	−11.2848	−1.37866	−0.689332	−	0.724446i	$-0.742096\pi$
$67$	−11.2848	−1.37866	−0.689332	+	0.724446i	$0.742096\pi$
$68$	0	0
$69$	16.6700	2.00683
$70$	0	0
$71$	8.85701	1.05113	0.525567	−	0.850752i	$-0.323853\pi$
$71$	8.85701	1.05113	0.525567	+	0.850752i	$0.323853\pi$
$72$	0	0
$73$	5.49638	0.643303	0.321652	−	0.946858i	$-0.395762\pi$
$73$	5.49638	0.643303	0.321652	+	0.946858i	$0.395762\pi$
$74$	0	0
$75$	12.8712	1.48623
$76$	0	0
$77$	−0.589728	−0.0672057
$78$	0	0
$79$	15.3579	1.72790	0.863950	−	0.503578i	$-0.167983\pi$
$79$	15.3579	1.72790	0.863950	+	0.503578i	$0.167983\pi$
$80$	0	0
$81$	6.53866	0.726517
$82$	0	0
$83$	8.90129	0.977044	0.488522	−	0.872552i	$-0.337536\pi$
$83$	8.90129	0.977044	0.488522	+	0.872552i	$0.337536\pi$
$84$	0	0
$85$	0.383240	0.0415682
$86$	0	0
$87$	−13.7424	−1.47334
$88$	0	0
$89$	3.19482	0.338650	0.169325	−	0.985560i	$-0.445841\pi$
$89$	3.19482	0.338650	0.169325	+	0.985560i	$0.445841\pi$
$90$	0	0
$91$	−17.0116	−1.78330
$92$	0	0
$93$	7.82561	0.811477
$94$	0	0
$95$	3.80046	0.389919
$96$	0	0
$97$	16.1180	1.63654	0.818270	−	0.574834i	$-0.194933\pi$
$97$	16.1180	1.63654	0.818270	+	0.574834i	$0.194933\pi$
$98$	0	0
$99$	1.25381	0.126013
$100$	0	0
$101$	−6.90404	−0.686977	−0.343489	−	0.939157i	$-0.611609\pi$
$101$	−6.90404	−0.686977	−0.343489	+	0.939157i	$0.611609\pi$
$102$	0	0
$103$	8.22957	0.810884	0.405442	−	0.914121i	$-0.367118\pi$
$103$	8.22957	0.810884	0.405442	+	0.914121i	$0.367118\pi$
$104$	0	0
$105$	−6.35569	−0.620252
$106$	0	0
$107$	13.6879	1.32326	0.661629	−	0.749831i	$-0.269865\pi$
$107$	13.6879	1.32326	0.661629	+	0.749831i	$0.269865\pi$
$108$	0	0
$109$	2.88054	0.275906	0.137953	−	0.990439i	$-0.455948\pi$
$109$	2.88054	0.275906	0.137953	+	0.990439i	$0.455948\pi$
$110$	0	0
$111$	0.419158	0.0397848
$112$	0	0
$113$	−10.1991	−0.959446	−0.479723	−	0.877420i	$-0.659263\pi$
$113$	−10.1991	−0.959446	−0.479723	+	0.877420i	$0.659263\pi$
$114$	0	0
$115$	4.51918	0.421416
$116$	0	0
$117$	36.1680	3.34373
$118$	0	0
$119$	1.28761	0.118035
$120$	0	0
$121$	−10.9519	−0.995628
$122$	0	0
$123$	14.9476	1.34778
$124$	0	0
$125$	7.49150	0.670060
$126$	0	0
$127$	−5.18164	−0.459797	−0.229898	−	0.973215i	$-0.573839\pi$
$127$	−5.18164	−0.459797	−0.229898	+	0.973215i	$0.573839\pi$
$128$	0	0
$129$	−15.2973	−1.34686
$130$	0	0
$131$	6.54903	0.572192	0.286096	−	0.958201i	$-0.407642\pi$
$131$	6.54903	0.572192	0.286096	+	0.958201i	$0.407642\pi$
$132$	0	0
$133$	12.7688	1.10719
$134$	0	0
$135$	6.42274	0.552781
$136$	0	0
$137$	−13.0193	−1.11231	−0.556156	−	0.831078i	$-0.687724\pi$
$137$	−13.0193	−1.11231	−0.556156	+	0.831078i	$0.687724\pi$
$138$	0	0
$139$	−13.6273	−1.15585	−0.577925	−	0.816090i	$-0.696137\pi$
$139$	−13.6273	−1.15585	−0.577925	+	0.816090i	$0.696137\pi$
$140$	0	0
$141$	−0.0690268	−0.00581311
$142$	0	0
$143$	1.38714	0.115998
$144$	0	0
$145$	−3.72553	−0.309388
$146$	0	0
$147$	−0.685852	−0.0565681
$148$	0	0
$149$	−14.5056	−1.18835	−0.594173	−	0.804337i	$-0.702521\pi$
$149$	−14.5056	−1.18835	−0.594173	+	0.804337i	$0.702521\pi$
$150$	0	0
$151$	−18.9613	−1.54305	−0.771524	−	0.636201i	$-0.780505\pi$
$151$	−18.9613	−1.54305	−0.771524	+	0.636201i	$0.780505\pi$
$152$	0	0
$153$	−2.73756	−0.221319
$154$	0	0
$155$	2.12150	0.170403
$156$	0	0
$157$	−10.8425	−0.865328	−0.432664	−	0.901555i	$-0.642426\pi$
$157$	−10.8425	−0.865328	−0.432664	+	0.901555i	$0.642426\pi$
$158$	0	0
$159$	−28.3793	−2.25063
$160$	0	0
$161$	15.1835	1.19663
$162$	0	0
$163$	−12.7859	−1.00147	−0.500734	−	0.865601i	$-0.666936\pi$
$163$	−12.7859	−1.00147	−0.500734	+	0.865601i	$0.666936\pi$
$164$	0	0
$165$	0.518249	0.0403457
$166$	0	0
$167$	18.6479	1.44302	0.721508	−	0.692406i	$-0.243449\pi$
$167$	18.6479	1.44302	0.721508	+	0.692406i	$0.243449\pi$
$168$	0	0
$169$	27.0140	2.07800
$170$	0	0
$171$	−27.1475	−2.07602
$172$	0	0
$173$	−17.8672	−1.35842	−0.679210	−	0.733944i	$-0.737677\pi$
$173$	−17.8672	−1.35842	−0.679210	+	0.733944i	$0.737677\pi$
$174$	0	0
$175$	11.7234	0.886209
$176$	0	0
$177$	4.04692	0.304185
$178$	0	0
$179$	−12.9733	−0.969666	−0.484833	−	0.874607i	$-0.661120\pi$
$179$	−12.9733	−0.969666	−0.484833	+	0.874607i	$0.661120\pi$
$180$	0	0
$181$	−14.3323	−1.06531	−0.532654	−	0.846333i	$-0.678805\pi$
$181$	−14.3323	−1.06531	−0.532654	+	0.846333i	$0.678805\pi$
$182$	0	0
$183$	−34.6181	−2.55904
$184$	0	0
$185$	0.113633	0.00835444
$186$	0	0
$187$	−0.104993	−0.00767784
$188$	0	0
$189$	21.5791	1.56965
$190$	0	0
$191$	11.7454	0.849864	0.424932	−	0.905225i	$-0.360298\pi$
$191$	11.7454	0.849864	0.424932	+	0.905225i	$0.360298\pi$
$192$	0	0
$193$	16.2623	1.17059	0.585294	−	0.810821i	$-0.300979\pi$
$193$	16.2623	1.17059	0.585294	+	0.810821i	$0.300979\pi$
$194$	0	0
$195$	14.9496	1.07057
$196$	0	0
$197$	19.1167	1.36201	0.681004	−	0.732280i	$-0.261544\pi$
$197$	19.1167	1.36201	0.681004	+	0.732280i	$0.261544\pi$
$198$	0	0
$199$	7.81060	0.553678	0.276839	−	0.960916i	$-0.410713\pi$
$199$	7.81060	0.553678	0.276839	+	0.960916i	$0.410713\pi$
$200$	0	0
$201$	33.3193	2.35016
$202$	0	0
$203$	−12.5170	−0.878522
$204$	0	0
$205$	4.05226	0.283022
$206$	0	0
$207$	−32.2815	−2.24372
$208$	0	0
$209$	−1.04118	−0.0720197
$210$	0	0
$211$	−15.2502	−1.04987	−0.524935	−	0.851142i	$-0.675910\pi$
$211$	−15.2502	−1.04987	−0.524935	+	0.851142i	$0.675910\pi$
$212$	0	0
$213$	−26.1509	−1.79183
$214$	0	0
$215$	−4.14707	−0.282828
$216$	0	0
$217$	7.12781	0.483867
$218$	0	0
$219$	−16.2285	−1.09662
$220$	0	0
$221$	−3.02867	−0.203730
$222$	0	0
$223$	17.1853	1.15081	0.575407	−	0.817867i	$-0.304844\pi$
$223$	17.1853	1.15081	0.575407	+	0.817867i	$0.304844\pi$
$224$	0	0
$225$	−24.9250	−1.66167
$226$	0	0
$227$	−24.6551	−1.63641	−0.818207	−	0.574923i	$-0.805032\pi$
$227$	−24.6551	−1.63641	−0.818207	+	0.574923i	$0.805032\pi$
$228$	0	0
$229$	−1.62611	−0.107456	−0.0537281	−	0.998556i	$-0.517110\pi$
$229$	−1.62611	−0.107456	−0.0537281	+	0.998556i	$0.517110\pi$
$230$	0	0
$231$	1.74121	0.114563
$232$	0	0
$233$	−1.88833	−0.123709	−0.0618543	−	0.998085i	$-0.519701\pi$
$233$	−1.88833	−0.123709	−0.0618543	+	0.998085i	$0.519701\pi$
$234$	0	0
$235$	−0.0187130	−0.00122070
$236$	0	0
$237$	−45.3453	−2.94549
$238$	0	0
$239$	−17.1392	−1.10864	−0.554322	−	0.832302i	$-0.687022\pi$
$239$	−17.1392	−1.10864	−0.554322	+	0.832302i	$0.687022\pi$
$240$	0	0
$241$	−17.0954	−1.10121	−0.550607	−	0.834765i	$-0.685604\pi$
$241$	−17.0954	−1.10121	−0.550607	+	0.834765i	$0.685604\pi$
$242$	0	0
$243$	4.76640	0.305765
$244$	0	0
$245$	−0.185933	−0.0118788
$246$	0	0
$247$	−30.0342	−1.91103
$248$	0	0
$249$	−26.2817	−1.66553
$250$	0	0
$251$	12.4115	0.783406	0.391703	−	0.920092i	$-0.371886\pi$
$251$	12.4115	0.783406	0.391703	+	0.920092i	$0.371886\pi$
$252$	0	0
$253$	−1.23808	−0.0778375
$254$	0	0
$255$	−1.13154	−0.0708599
$256$	0	0
$257$	−28.1032	−1.75303	−0.876514	−	0.481376i	$-0.840137\pi$
$257$	−28.1032	−1.75303	−0.876514	+	0.481376i	$0.840137\pi$
$258$	0	0
$259$	0.381783	0.0237228
$260$	0	0
$261$	26.6122	1.64726
$262$	0	0
$263$	26.8874	1.65795	0.828975	−	0.559286i	$-0.188925\pi$
$263$	26.8874	1.65795	0.828975	+	0.559286i	$0.188925\pi$
$264$	0	0
$265$	−7.69355	−0.472611
$266$	0	0
$267$	−9.43292	−0.577286
$268$	0	0
$269$	−6.73976	−0.410930	−0.205465	−	0.978664i	$-0.565871\pi$
$269$	−6.73976	−0.410930	−0.205465	+	0.978664i	$0.565871\pi$
$270$	0	0
$271$	22.3557	1.35801	0.679006	−	0.734133i	$-0.262411\pi$
$271$	22.3557	1.35801	0.679006	+	0.734133i	$0.262411\pi$
$272$	0	0
$273$	50.2278	3.03992
$274$	0	0
$275$	−0.955941	−0.0576454
$276$	0	0
$277$	−27.6721	−1.66266	−0.831328	−	0.555782i	$-0.812419\pi$
$277$	−27.6721	−1.66266	−0.831328	+	0.555782i	$0.812419\pi$
$278$	0	0
$279$	−15.1543	−0.907265
$280$	0	0
$281$	7.69541	0.459070	0.229535	−	0.973300i	$-0.426279\pi$
$281$	7.69541	0.459070	0.229535	+	0.973300i	$0.426279\pi$
$282$	0	0
$283$	6.35867	0.377984	0.188992	−	0.981979i	$-0.439478\pi$
$283$	6.35867	0.377984	0.188992	+	0.981979i	$0.439478\pi$
$284$	0	0
$285$	−11.2211	−0.664681
$286$	0	0
$287$	13.6148	0.803655
$288$	0	0
$289$	−16.7708	−0.986515
$290$	0	0
$291$	−47.5896	−2.78975
$292$	0	0
$293$	24.3290	1.42131	0.710657	−	0.703539i	$-0.248398\pi$
$293$	24.3290	1.42131	0.710657	+	0.703539i	$0.248398\pi$
$294$	0	0
$295$	1.09711	0.0638762
$296$	0	0
$297$	−1.75958	−0.102101
$298$	0	0
$299$	−35.7142	−2.06541
$300$	0	0
$301$	−13.9333	−0.803102
$302$	0	0
$303$	20.3846	1.17107
$304$	0	0
$305$	−9.38487	−0.537376
$306$	0	0
$307$	−15.6001	−0.890347	−0.445173	−	0.895444i	$-0.646858\pi$
$307$	−15.6001	−0.890347	−0.445173	+	0.895444i	$0.646858\pi$
$308$	0	0
$309$	−24.2984	−1.38229
$310$	0	0
$311$	0.144936	0.00821857	0.00410929	−	0.999992i	$-0.498692\pi$
$311$	0.144936	0.00821857	0.00410929	+	0.999992i	$0.498692\pi$
$312$	0	0
$313$	−4.15141	−0.234651	−0.117326	−	0.993093i	$-0.537432\pi$
$313$	−4.15141	−0.234651	−0.117326	+	0.993093i	$0.537432\pi$
$314$	0	0
$315$	12.3078	0.693467
$316$	0	0
$317$	2.78821	0.156601	0.0783006	−	0.996930i	$-0.475051\pi$
$317$	2.78821	0.156601	0.0783006	+	0.996930i	$0.475051\pi$
$318$	0	0
$319$	1.02065	0.0571454
$320$	0	0
$321$	−40.4145	−2.25571
$322$	0	0
$323$	2.27330	0.126490
$324$	0	0
$325$	−27.5755	−1.52961
$326$	0	0
$327$	−8.50499	−0.470327
$328$	0	0
$329$	−0.0628718	−0.00346624
$330$	0	0
$331$	−5.76568	−0.316911	−0.158455	−	0.987366i	$-0.550651\pi$
$331$	−5.76568	−0.316911	−0.158455	+	0.987366i	$0.550651\pi$
$332$	0	0
$333$	−0.811702	−0.0444810
$334$	0	0
$335$	9.03276	0.493512
$336$	0	0
$337$	27.0361	1.47275	0.736376	−	0.676573i	$-0.236536\pi$
$337$	27.0361	1.47275	0.736376	+	0.676573i	$0.236536\pi$
$338$	0	0
$339$	30.1134	1.63554
$340$	0	0
$341$	−0.581208	−0.0314742
$342$	0	0
$343$	18.2003	0.982726
$344$	0	0
$345$	−13.3432	−0.718374
$346$	0	0
$347$	−10.4704	−0.562079	−0.281040	−	0.959696i	$-0.590679\pi$
$347$	−10.4704	−0.562079	−0.281040	+	0.959696i	$0.590679\pi$
$348$	0	0
$349$	−3.86175	−0.206715	−0.103357	−	0.994644i	$-0.532959\pi$
$349$	−3.86175	−0.206715	−0.103357	+	0.994644i	$0.532959\pi$
$350$	0	0
$351$	−50.7576	−2.70924
$352$	0	0
$353$	5.54004	0.294867	0.147433	−	0.989072i	$-0.452899\pi$
$353$	5.54004	0.294867	0.147433	+	0.989072i	$0.452899\pi$
$354$	0	0
$355$	−7.08944	−0.376269
$356$	0	0
$357$	−3.80175	−0.201210
$358$	0	0
$359$	−23.3192	−1.23074	−0.615371	−	0.788238i	$-0.710994\pi$
$359$	−23.3192	−1.23074	−0.615371	+	0.788238i	$0.710994\pi$
$360$	0	0
$361$	3.54350	0.186500
$362$	0	0
$363$	32.3363	1.69721
$364$	0	0
$365$	−4.39949	−0.230280
$366$	0	0
$367$	5.82004	0.303804	0.151902	−	0.988396i	$-0.451460\pi$
$367$	5.82004	0.303804	0.151902	+	0.988396i	$0.451460\pi$
$368$	0	0
$369$	−28.9462	−1.50688
$370$	0	0
$371$	−25.8488	−1.34200
$372$	0	0
$373$	13.6005	0.704208	0.352104	−	0.935961i	$-0.385466\pi$
$373$	13.6005	0.704208	0.352104	+	0.935961i	$0.385466\pi$
$374$	0	0
$375$	−22.1192	−1.14223
$376$	0	0
$377$	29.4421	1.51635
$378$	0	0
$379$	−18.1652	−0.933082	−0.466541	−	0.884500i	$-0.654500\pi$
$379$	−18.1652	−0.933082	−0.466541	+	0.884500i	$0.654500\pi$
$380$	0	0
$381$	15.2992	0.783800
$382$	0	0
$383$	−24.5075	−1.25228	−0.626138	−	0.779712i	$-0.715365\pi$
$383$	−24.5075	−1.25228	−0.626138	+	0.779712i	$0.715365\pi$
$384$	0	0
$385$	0.472038	0.0240573
$386$	0	0
$387$	29.6234	1.50584
$388$	0	0
$389$	31.2291	1.58338	0.791689	−	0.610925i	$-0.209202\pi$
$389$	31.2291	1.58338	0.791689	+	0.610925i	$0.209202\pi$
$390$	0	0
$391$	2.70322	0.136708
$392$	0	0
$393$	−19.3365	−0.975396
$394$	0	0
$395$	−12.2930	−0.618527
$396$	0	0
$397$	−7.11024	−0.356853	−0.178426	−	0.983953i	$-0.557101\pi$
$397$	−7.11024	−0.356853	−0.178426	+	0.983953i	$0.557101\pi$
$398$	0	0
$399$	−37.7006	−1.88739
$400$	0	0
$401$	18.2628	0.912003	0.456002	−	0.889979i	$-0.349281\pi$
$401$	18.2628	0.912003	0.456002	+	0.889979i	$0.349281\pi$
$402$	0	0
$403$	−16.7658	−0.835163
$404$	0	0
$405$	−5.23376	−0.260067
$406$	0	0
$407$	−0.0311309	−0.00154310
$408$	0	0
$409$	−34.3419	−1.69810	−0.849048	−	0.528315i	$-0.822824\pi$
$409$	−34.3419	−1.69810	−0.849048	+	0.528315i	$0.822824\pi$
$410$	0	0
$411$	38.4403	1.89612
$412$	0	0
$413$	3.68606	0.181379
$414$	0	0
$415$	−7.12489	−0.349747
$416$	0	0
$417$	40.2354	1.97034
$418$	0	0
$419$	−8.49633	−0.415073	−0.207537	−	0.978227i	$-0.566545\pi$
$419$	−8.49633	−0.415073	−0.207537	+	0.978227i	$0.566545\pi$
$420$	0	0
$421$	−19.1431	−0.932976	−0.466488	−	0.884528i	$-0.654481\pi$
$421$	−19.1431	−0.932976	−0.466488	+	0.884528i	$0.654481\pi$
$422$	0	0
$423$	0.133671	0.00649929
$424$	0	0
$425$	2.08720	0.101244
$426$	0	0
$427$	−31.5313	−1.52590
$428$	0	0
$429$	−4.09562	−0.197738
$430$	0	0
$431$	1.82975	0.0881358	0.0440679	−	0.999029i	$-0.485968\pi$
$431$	1.82975	0.0881358	0.0440679	+	0.999029i	$0.485968\pi$
$432$	0	0
$433$	−37.1059	−1.78320	−0.891598	−	0.452828i	$-0.850415\pi$
$433$	−37.1059	−1.78320	−0.891598	+	0.452828i	$0.850415\pi$
$434$	0	0
$435$	10.9999	0.527404
$436$	0	0
$437$	26.8069	1.28235
$438$	0	0
$439$	26.5074	1.26513	0.632565	−	0.774507i	$-0.282002\pi$
$439$	26.5074	1.26513	0.632565	+	0.774507i	$0.282002\pi$
$440$	0	0
$441$	1.32816	0.0632455
$442$	0	0
$443$	2.85657	0.135720	0.0678600	−	0.997695i	$-0.478383\pi$
$443$	2.85657	0.135720	0.0678600	+	0.997695i	$0.478383\pi$
$444$	0	0
$445$	−2.55724	−0.121225
$446$	0	0
$447$	42.8288	2.02573
$448$	0	0
$449$	−21.8857	−1.03285	−0.516426	−	0.856332i	$-0.672738\pi$
$449$	−21.8857	−1.03285	−0.516426	+	0.856332i	$0.672738\pi$
$450$	0	0
$451$	−1.11016	−0.0522755
$452$	0	0
$453$	55.9845	2.63038
$454$	0	0
$455$	13.6166	0.638356
$456$	0	0
$457$	22.7408	1.06377	0.531885	−	0.846817i	$-0.321484\pi$
$457$	22.7408	1.06377	0.531885	+	0.846817i	$0.321484\pi$
$458$	0	0
$459$	3.84185	0.179322
$460$	0	0
$461$	−13.1815	−0.613922	−0.306961	−	0.951722i	$-0.599312\pi$
$461$	−13.1815	−0.613922	−0.306961	+	0.951722i	$0.599312\pi$
$462$	0	0
$463$	−22.7481	−1.05719	−0.528596	−	0.848873i	$-0.677281\pi$
$463$	−22.7481	−1.05719	−0.528596	+	0.848873i	$0.677281\pi$
$464$	0	0
$465$	−6.26387	−0.290480
$466$	0	0
$467$	22.7694	1.05364	0.526822	−	0.849976i	$-0.323384\pi$
$467$	22.7694	1.05364	0.526822	+	0.849976i	$0.323384\pi$
$468$	0	0
$469$	30.3482	1.40135
$470$	0	0
$471$	32.0133	1.47510
$472$	0	0
$473$	1.13613	0.0522395
$474$	0	0
$475$	20.6980	0.949689
$476$	0	0
$477$	54.9566	2.51629
$478$	0	0
$479$	14.6159	0.667819	0.333909	−	0.942605i	$-0.391632\pi$
$479$	14.6159	0.667819	0.333909	+	0.942605i	$0.391632\pi$
$480$	0	0
$481$	−0.898017	−0.0409460
$482$	0	0
$483$	−44.8304	−2.03986
$484$	0	0
$485$	−12.9014	−0.585823
$486$	0	0
$487$	−8.21352	−0.372190	−0.186095	−	0.982532i	$-0.559583\pi$
$487$	−8.21352	−0.372190	−0.186095	+	0.982532i	$0.559583\pi$
$488$	0	0
$489$	37.7512	1.70717
$490$	0	0
$491$	−25.7906	−1.16391	−0.581956	−	0.813220i	$-0.697712\pi$
$491$	−25.7906	−1.16391	−0.581956	+	0.813220i	$0.697712\pi$
$492$	0	0
$493$	−2.22848	−0.100366
$494$	0	0
$495$	−1.00359	−0.0451081
$496$	0	0
$497$	−23.8191	−1.06843
$498$	0	0
$499$	6.07420	0.271918	0.135959	−	0.990714i	$-0.456588\pi$
$499$	6.07420	0.271918	0.135959	+	0.990714i	$0.456588\pi$
$500$	0	0
$501$	−55.0592	−2.45986
$502$	0	0
$503$	−1.49689	−0.0667430	−0.0333715	−	0.999443i	$-0.510624\pi$
$503$	−1.49689	−0.0667430	−0.0333715	+	0.999443i	$0.510624\pi$
$504$	0	0
$505$	5.52622	0.245913
$506$	0	0
$507$	−79.7607	−3.54230
$508$	0	0
$509$	−21.0548	−0.933239	−0.466620	−	0.884458i	$-0.654528\pi$
$509$	−21.0548	−0.933239	−0.466620	+	0.884458i	$0.654528\pi$
$510$	0	0
$511$	−14.7814	−0.653890
$512$	0	0
$513$	38.0983	1.68208
$514$	0	0
$515$	−6.58722	−0.290268
$516$	0	0
$517$	0.00512663	0.000225469 0
$518$	0	0
$519$	52.7542	2.31565
$520$	0	0
$521$	28.0654	1.22957	0.614783	−	0.788696i	$-0.289243\pi$
$521$	28.0654	1.22957	0.614783	+	0.788696i	$0.289243\pi$
$522$	0	0
$523$	−30.2556	−1.32298	−0.661492	−	0.749952i	$-0.730077\pi$
$523$	−30.2556	−1.32298	−0.661492	+	0.749952i	$0.730077\pi$
$524$	0	0
$525$	−34.6143	−1.51069
$526$	0	0
$527$	1.26901	0.0552788
$528$	0	0
$529$	8.87648	0.385934
$530$	0	0
$531$	−7.83688	−0.340092
$532$	0	0
$533$	−32.0242	−1.38712
$534$	0	0
$535$	−10.9562	−0.473680
$536$	0	0
$537$	38.3044	1.65296
$538$	0	0
$539$	0.0509383	0.00219407
$540$	0	0
$541$	−27.1930	−1.16912	−0.584559	−	0.811351i	$-0.698732\pi$
$541$	−27.1930	−1.16912	−0.584559	+	0.811351i	$0.698732\pi$
$542$	0	0
$543$	42.3170	1.81599
$544$	0	0
$545$	−2.30568	−0.0987645
$546$	0	0
$547$	−24.7625	−1.05877	−0.529385	−	0.848382i	$-0.677577\pi$
$547$	−24.7625	−1.05877	−0.529385	+	0.848382i	$0.677577\pi$
$548$	0	0
$549$	67.0381	2.86112
$550$	0	0
$551$	−22.0990	−0.941451
$552$	0	0
$553$	−41.3019	−1.75634
$554$	0	0
$555$	−0.335508	−0.0142415
$556$	0	0
$557$	−32.9881	−1.39775	−0.698875	−	0.715243i	$-0.746316\pi$
$557$	−32.9881	−1.39775	−0.698875	+	0.715243i	$0.746316\pi$
$558$	0	0
$559$	32.7734	1.38617
$560$	0	0
$561$	0.309998	0.0130881
$562$	0	0
$563$	41.0159	1.72861	0.864307	−	0.502965i	$-0.167758\pi$
$563$	41.0159	1.72861	0.864307	+	0.502965i	$0.167758\pi$
$564$	0	0
$565$	8.16366	0.343448
$566$	0	0
$567$	−17.5844	−0.738473
$568$	0	0
$569$	−27.5664	−1.15564	−0.577822	−	0.816163i	$-0.696097\pi$
$569$	−27.5664	−1.15564	−0.577822	+	0.816163i	$0.696097\pi$
$570$	0	0
$571$	−14.7526	−0.617379	−0.308689	−	0.951163i	$-0.599890\pi$
$571$	−14.7526	−0.617379	−0.308689	+	0.951163i	$0.599890\pi$
$572$	0	0
$573$	−34.6790	−1.44873
$574$	0	0
$575$	24.6123	1.02640
$576$	0	0
$577$	14.1699	0.589902	0.294951	−	0.955512i	$-0.404697\pi$
$577$	14.1699	0.589902	0.294951	+	0.955512i	$0.404697\pi$
$578$	0	0
$579$	−48.0156	−1.99546
$580$	0	0
$581$	−23.9382	−0.993123
$582$	0	0
$583$	2.10773	0.0872934
$584$	0	0
$585$	−28.9500	−1.19694
$586$	0	0
$587$	27.4876	1.13453	0.567267	−	0.823534i	$-0.308001\pi$
$587$	27.4876	1.13453	0.567267	+	0.823534i	$0.308001\pi$
$588$	0	0
$589$	12.5843	0.518527
$590$	0	0
$591$	−56.4433	−2.32177
$592$	0	0
$593$	6.68082	0.274348	0.137174	−	0.990547i	$-0.456198\pi$
$593$	6.68082	0.274348	0.137174	+	0.990547i	$0.456198\pi$
$594$	0	0
$595$	−1.03064	−0.0422523
$596$	0	0
$597$	−23.0613	−0.943837
$598$	0	0
$599$	−36.3935	−1.48700	−0.743500	−	0.668736i	$-0.766836\pi$
$599$	−36.3935	−1.48700	−0.743500	+	0.668736i	$0.766836\pi$
$600$	0	0
$601$	0.225101	0.00918205	0.00459102	−	0.999989i	$-0.498539\pi$
$601$	0.225101	0.00918205	0.00459102	+	0.999989i	$0.498539\pi$
$602$	0	0
$603$	−64.5229	−2.62758
$604$	0	0
$605$	8.76627	0.356400
$606$	0	0
$607$	13.6722	0.554939	0.277469	−	0.960734i	$-0.410504\pi$
$607$	13.6722	0.554939	0.277469	+	0.960734i	$0.410504\pi$
$608$	0	0
$609$	36.9573	1.49759
$610$	0	0
$611$	0.147885	0.00598279
$612$	0	0
$613$	15.2182	0.614655	0.307328	−	0.951604i	$-0.400565\pi$
$613$	15.2182	0.614655	0.307328	+	0.951604i	$0.400565\pi$
$614$	0	0
$615$	−11.9646	−0.482458
$616$	0	0
$617$	−0.218166	−0.00878303	−0.00439151	−	0.999990i	$-0.501398\pi$
$617$	−0.218166	−0.00878303	−0.00439151	+	0.999990i	$0.501398\pi$
$618$	0	0
$619$	23.1715	0.931341	0.465671	−	0.884958i	$-0.345813\pi$
$619$	23.1715	0.931341	0.465671	+	0.884958i	$0.345813\pi$
$620$	0	0
$621$	45.3033	1.81796
$622$	0	0
$623$	−8.59180	−0.344223
$624$	0	0
$625$	15.8001	0.632004
$626$	0	0
$627$	3.07415	0.122770
$628$	0	0
$629$	0.0679711	0.00271018
$630$	0	0
$631$	29.1651	1.16104	0.580522	−	0.814244i	$-0.302848\pi$
$631$	29.1651	1.16104	0.580522	+	0.814244i	$0.302848\pi$
$632$	0	0
$633$	45.0274	1.78968
$634$	0	0
$635$	4.14756	0.164591
$636$	0	0
$637$	1.46939	0.0582193
$638$	0	0
$639$	50.6414	2.00334
$640$	0	0
$641$	−12.6725	−0.500533	−0.250266	−	0.968177i	$-0.580518\pi$
$641$	−12.6725	−0.500533	−0.250266	+	0.968177i	$0.580518\pi$
$642$	0	0
$643$	−38.9407	−1.53567	−0.767835	−	0.640647i	$-0.778666\pi$
$643$	−38.9407	−1.53567	−0.767835	+	0.640647i	$0.778666\pi$
$644$	0	0
$645$	12.2445	0.482127
$646$	0	0
$647$	−37.3946	−1.47013	−0.735066	−	0.677995i	$-0.762849\pi$
$647$	−37.3946	−1.47013	−0.735066	+	0.677995i	$0.762849\pi$
$648$	0	0
$649$	−0.300565	−0.0117982
$650$	0	0
$651$	−21.0453	−0.824832
$652$	0	0
$653$	−2.19751	−0.0859952	−0.0429976	−	0.999075i	$-0.513691\pi$
$653$	−2.19751	−0.0859952	−0.0429976	+	0.999075i	$0.513691\pi$
$654$	0	0
$655$	−5.24206	−0.204824
$656$	0	0
$657$	31.4265	1.22606
$658$	0	0
$659$	−14.4177	−0.561632	−0.280816	−	0.959762i	$-0.590605\pi$
$659$	−14.4177	−0.561632	−0.280816	+	0.959762i	$0.590605\pi$
$660$	0	0
$661$	17.7059	0.688680	0.344340	−	0.938845i	$-0.388103\pi$
$661$	17.7059	0.688680	0.344340	+	0.938845i	$0.388103\pi$
$662$	0	0
$663$	8.94235	0.347292
$664$	0	0
$665$	−10.2205	−0.396335
$666$	0	0
$667$	−26.2783	−1.01750
$668$	0	0
$669$	−50.7408	−1.96175
$670$	0	0
$671$	2.57109	0.0992558
$672$	0	0
$673$	5.87929	0.226630	0.113315	−	0.993559i	$-0.463853\pi$
$673$	5.87929	0.226630	0.113315	+	0.993559i	$0.463853\pi$
$674$	0	0
$675$	34.9794	1.34636
$676$	0	0
$677$	18.4600	0.709474	0.354737	−	0.934966i	$-0.384570\pi$
$677$	18.4600	0.709474	0.354737	+	0.934966i	$0.384570\pi$
$678$	0	0
$679$	−43.3461	−1.66347
$680$	0	0
$681$	72.7958	2.78954
$682$	0	0
$683$	32.0097	1.22482	0.612409	−	0.790541i	$-0.290201\pi$
$683$	32.0097	1.22482	0.612409	+	0.790541i	$0.290201\pi$
$684$	0	0
$685$	10.4211	0.398168
$686$	0	0
$687$	4.80119	0.183177
$688$	0	0
$689$	60.8006	2.31632
$690$	0	0
$691$	−25.0089	−0.951383	−0.475692	−	0.879612i	$-0.657802\pi$
$691$	−25.0089	−0.951383	−0.475692	+	0.879612i	$0.657802\pi$
$692$	0	0
$693$	−3.37186	−0.128087
$694$	0	0
$695$	10.9077	0.413753
$696$	0	0
$697$	2.42392	0.0918125
$698$	0	0
$699$	5.57542	0.210882
$700$	0	0
$701$	−33.5545	−1.26733	−0.633667	−	0.773606i	$-0.718451\pi$
$701$	−33.5545	−1.26733	−0.633667	+	0.773606i	$0.718451\pi$
$702$	0	0
$703$	0.674045	0.0254221
$704$	0	0
$705$	0.0552514	0.00208089
$706$	0	0
$707$	18.5670	0.698283
$708$	0	0
$709$	−3.58215	−0.134531	−0.0672653	−	0.997735i	$-0.521427\pi$
$709$	−3.58215	−0.134531	−0.0672653	+	0.997735i	$0.521427\pi$
$710$	0	0
$711$	87.8113	3.29318
$712$	0	0
$713$	14.9642	0.560413
$714$	0	0
$715$	−1.11031	−0.0415233
$716$	0	0
$717$	50.6047	1.88987
$718$	0	0
$719$	−28.9579	−1.07995	−0.539974	−	0.841681i	$-0.681566\pi$
$719$	−28.9579	−1.07995	−0.539974	+	0.841681i	$0.681566\pi$
$720$	0	0
$721$	−22.1317	−0.824228
$722$	0	0
$723$	50.4754	1.87720
$724$	0	0
$725$	−20.2899	−0.753549
$726$	0	0
$727$	−33.6996	−1.24985	−0.624925	−	0.780685i	$-0.714870\pi$
$727$	−33.6996	−1.24985	−0.624925	+	0.780685i	$0.714870\pi$
$728$	0	0
$729$	−33.6891	−1.24774
$730$	0	0
$731$	−2.48063	−0.0917494
$732$	0	0
$733$	−45.6957	−1.68781	−0.843904	−	0.536494i	$-0.819748\pi$
$733$	−45.6957	−1.68781	−0.843904	+	0.536494i	$0.819748\pi$
$734$	0	0
$735$	0.548979	0.0202494
$736$	0	0
$737$	−2.47462	−0.0911540
$738$	0	0
$739$	36.4631	1.34132	0.670658	−	0.741767i	$-0.266012\pi$
$739$	36.4631	1.34132	0.670658	+	0.741767i	$0.266012\pi$
$740$	0	0
$741$	88.6782	3.25767
$742$	0	0
$743$	−2.62869	−0.0964374	−0.0482187	−	0.998837i	$-0.515354\pi$
$743$	−2.62869	−0.0964374	−0.0482187	+	0.998837i	$0.515354\pi$
$744$	0	0
$745$	11.6108	0.425386
$746$	0	0
$747$	50.8946	1.86214
$748$	0	0
$749$	−36.8107	−1.34504
$750$	0	0
$751$	−0.0572874	−0.00209045	−0.00104522	−	0.999999i	$-0.500333\pi$
$751$	−0.0572874	−0.00209045	−0.00104522	+	0.999999i	$0.500333\pi$
$752$	0	0
$753$	−36.6457	−1.33545
$754$	0	0
$755$	15.1772	0.552356
$756$	0	0
$757$	−34.6865	−1.26070	−0.630351	−	0.776311i	$-0.717089\pi$
$757$	−34.6865	−1.26070	−0.630351	+	0.776311i	$0.717089\pi$
$758$	0	0
$759$	3.65552	0.132687
$760$	0	0
$761$	48.1173	1.74425	0.872125	−	0.489282i	$-0.162741\pi$
$761$	48.1173	1.74425	0.872125	+	0.489282i	$0.162741\pi$
$762$	0	0
$763$	−7.74661	−0.280446
$764$	0	0
$765$	2.19124	0.0792243
$766$	0	0
$767$	−8.67024	−0.313064
$768$	0	0
$769$	9.07473	0.327243	0.163622	−	0.986523i	$-0.447682\pi$
$769$	9.07473	0.327243	0.163622	+	0.986523i	$0.447682\pi$
$770$	0	0
$771$	82.9766	2.98833
$772$	0	0
$773$	−30.5902	−1.10025	−0.550126	−	0.835081i	$-0.685420\pi$
$773$	−30.5902	−1.10025	−0.550126	+	0.835081i	$0.685420\pi$
$774$	0	0
$775$	11.5541	0.415035
$776$	0	0
$777$	−1.12724	−0.0404395
$778$	0	0
$779$	24.0372	0.861221
$780$	0	0
$781$	1.94223	0.0694985
$782$	0	0
$783$	−37.3472	−1.33468
$784$	0	0
$785$	8.67872	0.309757
$786$	0	0
$787$	−37.8721	−1.35000	−0.674998	−	0.737820i	$-0.735856\pi$
$787$	−37.8721	−1.35000	−0.674998	+	0.737820i	$0.735856\pi$
$788$	0	0
$789$	−79.3870	−2.82625
$790$	0	0
$791$	27.4282	0.975236
$792$	0	0
$793$	74.1668	2.63374
$794$	0	0
$795$	22.7157	0.805644
$796$	0	0
$797$	16.6057	0.588203	0.294101	−	0.955774i	$-0.404980\pi$
$797$	16.6057	0.588203	0.294101	+	0.955774i	$0.404980\pi$
$798$	0	0
$799$	−0.0111934	−0.000395996 0
$800$	0	0
$801$	18.2669	0.645429
$802$	0	0
$803$	1.20529	0.0425337
$804$	0	0
$805$	−12.1534	−0.428351
$806$	0	0
$807$	19.8996	0.700499
$808$	0	0
$809$	−18.3667	−0.645740	−0.322870	−	0.946443i	$-0.604648\pi$
$809$	−18.3667	−0.645740	−0.322870	+	0.946443i	$0.604648\pi$
$810$	0	0
$811$	−27.3691	−0.961058	−0.480529	−	0.876979i	$-0.659555\pi$
$811$	−27.3691	−0.961058	−0.480529	+	0.876979i	$0.659555\pi$
$812$	0	0
$813$	−66.0067	−2.31496
$814$	0	0
$815$	10.2342	0.358490
$816$	0	0
$817$	−24.5995	−0.860629
$818$	0	0
$819$	−97.2663	−3.39876
$820$	0	0
$821$	−22.4592	−0.783833	−0.391917	−	0.920001i	$-0.628188\pi$
$821$	−22.4592	−0.783833	−0.391917	+	0.920001i	$0.628188\pi$
$822$	0	0
$823$	−36.2686	−1.26424	−0.632122	−	0.774869i	$-0.717816\pi$
$823$	−36.2686	−1.26424	−0.632122	+	0.774869i	$0.717816\pi$
$824$	0	0
$825$	2.82248	0.0982662
$826$	0	0
$827$	−39.3806	−1.36940	−0.684699	−	0.728826i	$-0.740067\pi$
$827$	−39.3806	−1.36940	−0.684699	+	0.728826i	$0.740067\pi$
$828$	0	0
$829$	−38.8659	−1.34987	−0.674933	−	0.737879i	$-0.735828\pi$
$829$	−38.8659	−1.34987	−0.674933	+	0.737879i	$0.735828\pi$
$830$	0	0
$831$	81.7038	2.83427
$832$	0	0
$833$	−0.111218	−0.00385349
$834$	0	0
$835$	−14.9264	−0.516549
$836$	0	0
$837$	21.2673	0.735107
$838$	0	0
$839$	4.79295	0.165471	0.0827356	−	0.996572i	$-0.473634\pi$
$839$	4.79295	0.165471	0.0827356	+	0.996572i	$0.473634\pi$
$840$	0	0
$841$	−7.33665	−0.252988
$842$	0	0
$843$	−22.7212	−0.782561
$844$	0	0
$845$	−21.6229	−0.743850
$846$	0	0
$847$	29.4529	1.01201
$848$	0	0
$849$	−18.7744	−0.644336
$850$	0	0
$851$	0.801518	0.0274757
$852$	0	0
$853$	21.3960	0.732586	0.366293	−	0.930499i	$-0.380627\pi$
$853$	21.3960	0.732586	0.366293	+	0.930499i	$0.380627\pi$
$854$	0	0
$855$	21.7297	0.743140
$856$	0	0
$857$	1.49451	0.0510516	0.0255258	−	0.999674i	$-0.491874\pi$
$857$	1.49451	0.0510516	0.0255258	+	0.999674i	$0.491874\pi$
$858$	0	0
$859$	−23.0394	−0.786094	−0.393047	−	0.919518i	$-0.628579\pi$
$859$	−23.0394	−0.786094	−0.393047	+	0.919518i	$0.628579\pi$
$860$	0	0
$861$	−40.1986	−1.36996
$862$	0	0
$863$	−57.8263	−1.96843	−0.984216	−	0.176972i	$-0.943370\pi$
$863$	−57.8263	−1.96843	−0.984216	+	0.176972i	$0.943370\pi$
$864$	0	0
$865$	14.3015	0.486266
$866$	0	0
$867$	49.5168	1.68168
$868$	0	0
$869$	3.36780	0.114245
$870$	0	0
$871$	−71.3841	−2.41876
$872$	0	0
$873$	92.1575	3.11906
$874$	0	0
$875$	−20.1468	−0.681087
$876$	0	0
$877$	−3.96930	−0.134034	−0.0670170	−	0.997752i	$-0.521348\pi$
$877$	−3.96930	−0.134034	−0.0670170	+	0.997752i	$0.521348\pi$
$878$	0	0
$879$	−71.8329	−2.42287
$880$	0	0
$881$	−46.4309	−1.56430	−0.782149	−	0.623092i	$-0.785876\pi$
$881$	−46.4309	−1.56430	−0.782149	+	0.623092i	$0.785876\pi$
$882$	0	0
$883$	10.5473	0.354946	0.177473	−	0.984126i	$-0.443208\pi$
$883$	10.5473	0.354946	0.177473	+	0.984126i	$0.443208\pi$
$884$	0	0
$885$	−3.23929	−0.108888
$886$	0	0
$887$	40.8051	1.37010	0.685051	−	0.728495i	$-0.259780\pi$
$887$	40.8051	1.37010	0.685051	+	0.728495i	$0.259780\pi$
$888$	0	0
$889$	13.9350	0.467363
$890$	0	0
$891$	1.43385	0.0480356
$892$	0	0
$893$	−0.111001	−0.00371452
$894$	0	0
$895$	10.3842	0.347106
$896$	0	0
$897$	105.449	3.52083
$898$	0	0
$899$	−12.3362	−0.411435
$900$	0	0
$901$	−4.60201	−0.153315
$902$	0	0
$903$	41.1390	1.36902
$904$	0	0
$905$	11.4720	0.381342
$906$	0	0
$907$	−27.8958	−0.926266	−0.463133	−	0.886289i	$-0.653275\pi$
$907$	−27.8958	−0.926266	−0.463133	+	0.886289i	$0.653275\pi$
$908$	0	0
$909$	−39.4749	−1.30930
$910$	0	0
$911$	−40.7620	−1.35050	−0.675252	−	0.737587i	$-0.735965\pi$
$911$	−40.7620	−1.35050	−0.675252	+	0.737587i	$0.735965\pi$
$912$	0	0
$913$	1.95194	0.0645999
$914$	0	0
$915$	27.7095	0.916047
$916$	0	0
$917$	−17.6123	−0.581608
$918$	0	0
$919$	−2.68347	−0.0885196	−0.0442598	−	0.999020i	$-0.514093\pi$
$919$	−2.68347	−0.0885196	−0.0442598	+	0.999020i	$0.514093\pi$
$920$	0	0
$921$	46.0605	1.51774
$922$	0	0
$923$	56.0265	1.84413
$924$	0	0
$925$	0.618865	0.0203481
$926$	0	0
$927$	47.0539	1.54545
$928$	0	0
$929$	28.2636	0.927297	0.463649	−	0.886019i	$-0.346540\pi$
$929$	28.2636	0.927297	0.463649	+	0.886019i	$0.346540\pi$
$930$	0	0
$931$	−1.10291	−0.0361465
$932$	0	0
$933$	−0.427934	−0.0140099
$934$	0	0
$935$	0.0840397	0.00274839
$936$	0	0
$937$	−3.17189	−0.103621	−0.0518106	−	0.998657i	$-0.516499\pi$
$937$	−3.17189	−0.103621	−0.0518106	+	0.998657i	$0.516499\pi$
$938$	0	0
$939$	12.2573	0.400002
$940$	0	0
$941$	−11.2160	−0.365631	−0.182816	−	0.983147i	$-0.558521\pi$
$941$	−11.2160	−0.365631	−0.182816	+	0.983147i	$0.558521\pi$
$942$	0	0
$943$	28.5830	0.930790
$944$	0	0
$945$	−17.2726	−0.561878
$946$	0	0
$947$	9.13417	0.296821	0.148410	−	0.988926i	$-0.452584\pi$
$947$	9.13417	0.296821	0.148410	+	0.988926i	$0.452584\pi$
$948$	0	0
$949$	34.7683	1.12863
$950$	0	0
$951$	−8.23237	−0.266953
$952$	0	0
$953$	−9.25112	−0.299673	−0.149837	−	0.988711i	$-0.547875\pi$
$953$	−9.25112	−0.299673	−0.149837	+	0.988711i	$0.547875\pi$
$954$	0	0
$955$	−9.40137	−0.304221
$956$	0	0
$957$	−3.01354	−0.0974138
$958$	0	0
$959$	35.0126	1.13062
$960$	0	0
$961$	−23.9752	−0.773392
$962$	0	0
$963$	78.2627	2.52198
$964$	0	0
$965$	−13.0169	−0.419029
$966$	0	0
$967$	58.3316	1.87582	0.937910	−	0.346880i	$-0.112759\pi$
$967$	58.3316	1.87582	0.937910	+	0.346880i	$0.112759\pi$
$968$	0	0
$969$	−6.71207	−0.215623
$970$	0	0
$971$	−37.8058	−1.21325	−0.606623	−	0.794990i	$-0.707476\pi$
$971$	−37.8058	−1.21325	−0.606623	+	0.794990i	$0.707476\pi$
$972$	0	0
$973$	36.6477	1.17487
$974$	0	0
$975$	81.4186	2.60748
$976$	0	0
$977$	−29.3901	−0.940273	−0.470137	−	0.882594i	$-0.655795\pi$
$977$	−29.3901	−0.940273	−0.470137	+	0.882594i	$0.655795\pi$
$978$	0	0
$979$	0.700584	0.0223908
$980$	0	0
$981$	16.4700	0.525845
$982$	0	0
$983$	15.5243	0.495149	0.247574	−	0.968869i	$-0.420367\pi$
$983$	15.5243	0.495149	0.247574	+	0.968869i	$0.420367\pi$
$984$	0	0
$985$	−15.3016	−0.487551
$986$	0	0
$987$	0.185633	0.00590877
$988$	0	0
$989$	−29.2517	−0.930150
$990$	0	0
$991$	−33.8641	−1.07573	−0.537864	−	0.843032i	$-0.680768\pi$
$991$	−33.8641	−1.07573	−0.537864	+	0.843032i	$0.680768\pi$
$992$	0	0
$993$	17.0236	0.540227
$994$	0	0
$995$	−6.25186	−0.198197
$996$	0	0
$997$	49.3357	1.56248	0.781239	−	0.624232i	$-0.214588\pi$
$997$	49.3357	1.56248	0.781239	+	0.624232i	$0.214588\pi$
$998$	0	0
$999$	1.13913	0.0360405

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.a.1.7	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.a.1.7	✓	63	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:	\( N \)	\(=\)	\( 6044 = 2^{2} \cdot 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6044.a (trivial)

\(f(q)\)	\(=\)	\(q-2.95257 q^{3} -0.800433 q^{5} -2.68929 q^{7} +5.71766 q^{9} +O(q^{10})\)	\(q-2.95257 q^{3} -0.800433 q^{5} -2.68929 q^{7} +5.71766 q^{9} +0.219287 q^{11} +6.32566 q^{13} +2.36333 q^{15} -0.478791 q^{17} -4.74800 q^{19} +7.94032 q^{21} -5.64593 q^{23} -4.35931 q^{25} -8.02408 q^{27} +4.65439 q^{29} -2.65044 q^{31} -0.647461 q^{33} +2.15260 q^{35} -0.141964 q^{37} -18.6770 q^{39} -5.06259 q^{41} +5.18103 q^{43} -4.57660 q^{45} +0.0233786 q^{47} +0.232290 q^{49} +1.41366 q^{51} +9.61174 q^{53} -0.175525 q^{55} +14.0188 q^{57} -1.37065 q^{59} +11.7247 q^{61} -15.3765 q^{63} -5.06327 q^{65} -11.2848 q^{67} +16.6700 q^{69} +8.85701 q^{71} +5.49638 q^{73} +12.8712 q^{75} -0.589728 q^{77} +15.3579 q^{79} +6.53866 q^{81} +8.90129 q^{83} +0.383240 q^{85} -13.7424 q^{87} +3.19482 q^{89} -17.0116 q^{91} +7.82561 q^{93} +3.80046 q^{95} +16.1180 q^{97} +1.25381 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * q^99

Introduction

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