LMFDB - Embedded newform 6044.2.a.a.1.11

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			1.11
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)

$f(q)$	$=$	$q-2.46300 q^{3} -1.99757 q^{5} +3.35645 q^{7} +3.06637 q^{9} +O(q^{10})$	$q-2.46300 q^{3} -1.99757 q^{5} +3.35645 q^{7} +3.06637 q^{9} +2.14622 q^{11} -3.15119 q^{13} +4.92002 q^{15} -6.40828 q^{17} +5.57506 q^{19} -8.26693 q^{21} +3.66075 q^{23} -1.00970 q^{25} -0.163464 q^{27} +1.20670 q^{29} -0.823109 q^{31} -5.28615 q^{33} -6.70475 q^{35} -3.98234 q^{37} +7.76138 q^{39} -4.51047 q^{41} -8.66655 q^{43} -6.12529 q^{45} +8.64365 q^{47} +4.26575 q^{49} +15.7836 q^{51} -4.95635 q^{53} -4.28724 q^{55} -13.7314 q^{57} +0.934526 q^{59} +1.10706 q^{61} +10.2921 q^{63} +6.29473 q^{65} -2.52141 q^{67} -9.01643 q^{69} +10.1170 q^{71} +7.58146 q^{73} +2.48690 q^{75} +7.20369 q^{77} -15.4356 q^{79} -8.79649 q^{81} +14.9854 q^{83} +12.8010 q^{85} -2.97210 q^{87} -2.21431 q^{89} -10.5768 q^{91} +2.02732 q^{93} -11.1366 q^{95} +13.1362 q^{97} +6.58111 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.46300	−1.42201	−0.711007	−	0.703185i	$-0.751760\pi$
$3$	−2.46300	−1.42201	−0.711007	+	0.703185i	$0.751760\pi$
$4$	0	0
$5$	−1.99757	−0.893342	−0.446671	−	0.894698i	$-0.647390\pi$
$5$	−1.99757	−0.893342	−0.446671	+	0.894698i	$0.647390\pi$
$6$	0	0
$7$	3.35645	1.26862	0.634309	−	0.773079i	$-0.281285\pi$
$7$	3.35645	1.26862	0.634309	+	0.773079i	$0.281285\pi$
$8$	0	0
$9$	3.06637	1.02212
$10$	0	0
$11$	2.14622	0.647111	0.323555	−	0.946209i	$-0.395122\pi$
$11$	2.14622	0.647111	0.323555	+	0.946209i	$0.395122\pi$
$12$	0	0
$13$	−3.15119	−0.873983	−0.436992	−	0.899466i	$-0.643956\pi$
$13$	−3.15119	−0.873983	−0.436992	+	0.899466i	$0.643956\pi$
$14$	0	0
$15$	4.92002	1.27034
$16$	0	0
$17$	−6.40828	−1.55424	−0.777119	−	0.629354i	$-0.783319\pi$
$17$	−6.40828	−1.55424	−0.777119	+	0.629354i	$0.783319\pi$
$18$	0	0
$19$	5.57506	1.27901	0.639503	−	0.768789i	$-0.279140\pi$
$19$	5.57506	1.27901	0.639503	+	0.768789i	$0.279140\pi$
$20$	0	0
$21$	−8.26693	−1.80399
$22$	0	0
$23$	3.66075	0.763319	0.381660	−	0.924303i	$-0.375353\pi$
$23$	3.66075	0.763319	0.381660	+	0.924303i	$0.375353\pi$
$24$	0	0
$25$	−1.00970	−0.201941
$26$	0	0
$27$	−0.163464	−0.0314586
$28$	0	0
$29$	1.20670	0.224078	0.112039	−	0.993704i	$-0.464262\pi$
$29$	1.20670	0.224078	0.112039	+	0.993704i	$0.464262\pi$
$30$	0	0
$31$	−0.823109	−0.147835	−0.0739174	−	0.997264i	$-0.523550\pi$
$31$	−0.823109	−0.147835	−0.0739174	+	0.997264i	$0.523550\pi$
$32$	0	0
$33$	−5.28615	−0.920201
$34$	0	0
$35$	−6.70475	−1.13331
$36$	0	0
$37$	−3.98234	−0.654693	−0.327346	−	0.944904i	$-0.606154\pi$
$37$	−3.98234	−0.654693	−0.327346	+	0.944904i	$0.606154\pi$
$38$	0	0
$39$	7.76138	1.24282
$40$	0	0
$41$	−4.51047	−0.704417	−0.352208	−	0.935922i	$-0.614569\pi$
$41$	−4.51047	−0.704417	−0.352208	+	0.935922i	$0.614569\pi$
$42$	0	0
$43$	−8.66655	−1.32164	−0.660818	−	0.750546i	$-0.729791\pi$
$43$	−8.66655	−1.32164	−0.660818	+	0.750546i	$0.729791\pi$
$44$	0	0
$45$	−6.12529	−0.913105
$46$	0	0
$47$	8.64365	1.26081	0.630403	−	0.776268i	$-0.282890\pi$
$47$	8.64365	1.26081	0.630403	+	0.776268i	$0.282890\pi$
$48$	0	0
$49$	4.26575	0.609393
$50$	0	0
$51$	15.7836	2.21015
$52$	0	0
$53$	−4.95635	−0.680807	−0.340404	−	0.940279i	$-0.610564\pi$
$53$	−4.95635	−0.680807	−0.340404	+	0.940279i	$0.610564\pi$
$54$	0	0
$55$	−4.28724	−0.578091
$56$	0	0
$57$	−13.7314	−1.81876
$58$	0	0
$59$	0.934526	0.121665	0.0608324	−	0.998148i	$-0.480624\pi$
$59$	0.934526	0.121665	0.0608324	+	0.998148i	$0.480624\pi$
$60$	0	0
$61$	1.10706	0.141745	0.0708724	−	0.997485i	$-0.477422\pi$
$61$	1.10706	0.141745	0.0708724	+	0.997485i	$0.477422\pi$
$62$	0	0
$63$	10.2921	1.29668
$64$	0	0
$65$	6.29473	0.780766
$66$	0	0
$67$	−2.52141	−0.308039	−0.154020	−	0.988068i	$-0.549222\pi$
$67$	−2.52141	−0.308039	−0.154020	+	0.988068i	$0.549222\pi$
$68$	0	0
$69$	−9.01643	−1.08545
$70$	0	0
$71$	10.1170	1.20067	0.600333	−	0.799750i	$-0.295035\pi$
$71$	10.1170	1.20067	0.600333	+	0.799750i	$0.295035\pi$
$72$	0	0
$73$	7.58146	0.887343	0.443671	−	0.896190i	$-0.353676\pi$
$73$	7.58146	0.887343	0.443671	+	0.896190i	$0.353676\pi$
$74$	0	0
$75$	2.48690	0.287163
$76$	0	0
$77$	7.20369	0.820937
$78$	0	0
$79$	−15.4356	−1.73664	−0.868322	−	0.496001i	$-0.834801\pi$
$79$	−15.4356	−1.73664	−0.868322	+	0.496001i	$0.834801\pi$
$80$	0	0
$81$	−8.79649	−0.977388
$82$	0	0
$83$	14.9854	1.64487	0.822433	−	0.568862i	$-0.192616\pi$
$83$	14.9854	1.64487	0.822433	+	0.568862i	$0.192616\pi$
$84$	0	0
$85$	12.8010	1.38846
$86$	0	0
$87$	−2.97210	−0.318642
$88$	0	0
$89$	−2.21431	−0.234717	−0.117358	−	0.993090i	$-0.537443\pi$
$89$	−2.21431	−0.234717	−0.117358	+	0.993090i	$0.537443\pi$
$90$	0	0
$91$	−10.5768	−1.10875
$92$	0	0
$93$	2.02732	0.210223
$94$	0	0
$95$	−11.1366	−1.14259
$96$	0	0
$97$	13.1362	1.33377	0.666887	−	0.745159i	$-0.267626\pi$
$97$	13.1362	1.33377	0.666887	+	0.745159i	$0.267626\pi$
$98$	0	0
$99$	6.58111	0.661427
$100$	0	0
$101$	0.768429	0.0764615	0.0382308	−	0.999269i	$-0.487828\pi$
$101$	0.768429	0.0764615	0.0382308	+	0.999269i	$0.487828\pi$
$102$	0	0
$103$	11.0465	1.08844	0.544220	−	0.838942i	$-0.316826\pi$
$103$	11.0465	1.08844	0.544220	+	0.838942i	$0.316826\pi$
$104$	0	0
$105$	16.5138	1.61158
$106$	0	0
$107$	5.75974	0.556815	0.278408	−	0.960463i	$-0.410193\pi$
$107$	5.75974	0.556815	0.278408	+	0.960463i	$0.410193\pi$
$108$	0	0
$109$	−8.94618	−0.856888	−0.428444	−	0.903568i	$-0.640938\pi$
$109$	−8.94618	−0.856888	−0.428444	+	0.903568i	$0.640938\pi$
$110$	0	0
$111$	9.80851	0.930982
$112$	0	0
$113$	0.407981	0.0383796	0.0191898	−	0.999816i	$-0.493891\pi$
$113$	0.407981	0.0383796	0.0191898	+	0.999816i	$0.493891\pi$
$114$	0	0
$115$	−7.31262	−0.681905
$116$	0	0
$117$	−9.66271	−0.893318
$118$	0	0
$119$	−21.5091	−1.97173
$120$	0	0
$121$	−6.39372	−0.581247
$122$	0	0
$123$	11.1093	1.00169
$124$	0	0
$125$	12.0048	1.07374
$126$	0	0
$127$	9.69173	0.860002	0.430001	−	0.902828i	$-0.358513\pi$
$127$	9.69173	0.860002	0.430001	+	0.902828i	$0.358513\pi$
$128$	0	0
$129$	21.3457	1.87939
$130$	0	0
$131$	−5.43655	−0.474994	−0.237497	−	0.971388i	$-0.576327\pi$
$131$	−5.43655	−0.474994	−0.237497	+	0.971388i	$0.576327\pi$
$132$	0	0
$133$	18.7124	1.62257
$134$	0	0
$135$	0.326530	0.0281033
$136$	0	0
$137$	20.8954	1.78522	0.892609	−	0.450832i	$-0.148873\pi$
$137$	20.8954	1.78522	0.892609	+	0.450832i	$0.148873\pi$
$138$	0	0
$139$	−20.6759	−1.75370	−0.876852	−	0.480760i	$-0.840361\pi$
$139$	−20.6759	−1.75370	−0.876852	+	0.480760i	$0.840361\pi$
$140$	0	0
$141$	−21.2893	−1.79288
$142$	0	0
$143$	−6.76316	−0.565564
$144$	0	0
$145$	−2.41047	−0.200178
$146$	0	0
$147$	−10.5065	−0.866565
$148$	0	0
$149$	−15.5865	−1.27689	−0.638446	−	0.769666i	$-0.720423\pi$
$149$	−15.5865	−1.27689	−0.638446	+	0.769666i	$0.720423\pi$
$150$	0	0
$151$	9.89166	0.804972	0.402486	−	0.915426i	$-0.368146\pi$
$151$	9.89166	0.804972	0.402486	+	0.915426i	$0.368146\pi$
$152$	0	0
$153$	−19.6502	−1.58862
$154$	0	0
$155$	1.64422	0.132067
$156$	0	0
$157$	6.51786	0.520182	0.260091	−	0.965584i	$-0.416247\pi$
$157$	6.51786	0.520182	0.260091	+	0.965584i	$0.416247\pi$
$158$	0	0
$159$	12.2075	0.968117
$160$	0	0
$161$	12.2871	0.968361
$162$	0	0
$163$	−21.4391	−1.67924	−0.839620	−	0.543175i	$-0.817222\pi$
$163$	−21.4391	−1.67924	−0.839620	+	0.543175i	$0.817222\pi$
$164$	0	0
$165$	10.5595	0.822053
$166$	0	0
$167$	16.5780	1.28284	0.641421	−	0.767189i	$-0.278345\pi$
$167$	16.5780	1.28284	0.641421	+	0.767189i	$0.278345\pi$
$168$	0	0
$169$	−3.06999	−0.236153
$170$	0	0
$171$	17.0952	1.30730
$172$	0	0
$173$	−17.2291	−1.30990	−0.654951	−	0.755671i	$-0.727311\pi$
$173$	−17.2291	−1.30990	−0.654951	+	0.755671i	$0.727311\pi$
$174$	0	0
$175$	−3.38902	−0.256186
$176$	0	0
$177$	−2.30174	−0.173009
$178$	0	0
$179$	−1.76579	−0.131981	−0.0659906	−	0.997820i	$-0.521021\pi$
$179$	−1.76579	−0.131981	−0.0659906	+	0.997820i	$0.521021\pi$
$180$	0	0
$181$	10.8636	0.807485	0.403743	−	0.914873i	$-0.367709\pi$
$181$	10.8636	0.807485	0.403743	+	0.914873i	$0.367709\pi$
$182$	0	0
$183$	−2.72670	−0.201563
$184$	0	0
$185$	7.95501	0.584864
$186$	0	0
$187$	−13.7536	−1.00576
$188$	0	0
$189$	−0.548657	−0.0399090
$190$	0	0
$191$	−17.4356	−1.26160	−0.630799	−	0.775946i	$-0.717273\pi$
$191$	−17.4356	−1.26160	−0.630799	+	0.775946i	$0.717273\pi$
$192$	0	0
$193$	−24.9168	−1.79355	−0.896774	−	0.442489i	$-0.854095\pi$
$193$	−24.9168	−1.79355	−0.896774	+	0.442489i	$0.854095\pi$
$194$	0	0
$195$	−15.5039	−1.11026
$196$	0	0
$197$	13.4074	0.955241	0.477620	−	0.878566i	$-0.341499\pi$
$197$	13.4074	0.955241	0.477620	+	0.878566i	$0.341499\pi$
$198$	0	0
$199$	19.1112	1.35476	0.677379	−	0.735634i	$-0.263116\pi$
$199$	19.1112	1.35476	0.677379	+	0.735634i	$0.263116\pi$
$200$	0	0
$201$	6.21023	0.438036
$202$	0	0
$203$	4.05022	0.284270
$204$	0	0
$205$	9.00999	0.629285
$206$	0	0
$207$	11.2252	0.780206
$208$	0	0
$209$	11.9653	0.827659
$210$	0	0
$211$	−26.5848	−1.83017	−0.915087	−	0.403256i	$-0.867879\pi$
$211$	−26.5848	−1.83017	−0.915087	+	0.403256i	$0.867879\pi$
$212$	0	0
$213$	−24.9182	−1.70736
$214$	0	0
$215$	17.3121	1.18067
$216$	0	0
$217$	−2.76273	−0.187546
$218$	0	0
$219$	−18.6731	−1.26181
$220$	0	0
$221$	20.1937	1.35838
$222$	0	0
$223$	−3.33473	−0.223310	−0.111655	−	0.993747i	$-0.535615\pi$
$223$	−3.33473	−0.223310	−0.111655	+	0.993747i	$0.535615\pi$
$224$	0	0
$225$	−3.09612	−0.206408
$226$	0	0
$227$	19.5367	1.29670	0.648348	−	0.761344i	$-0.275460\pi$
$227$	19.5367	1.29670	0.648348	+	0.761344i	$0.275460\pi$
$228$	0	0
$229$	−23.3686	−1.54424	−0.772122	−	0.635475i	$-0.780804\pi$
$229$	−23.3686	−1.54424	−0.772122	+	0.635475i	$0.780804\pi$
$230$	0	0
$231$	−17.7427	−1.16738
$232$	0	0
$233$	17.5320	1.14856	0.574280	−	0.818659i	$-0.305282\pi$
$233$	17.5320	1.14856	0.574280	+	0.818659i	$0.305282\pi$
$234$	0	0
$235$	−17.2663	−1.12633
$236$	0	0
$237$	38.0179	2.46953
$238$	0	0
$239$	9.99738	0.646676	0.323338	−	0.946283i	$-0.395195\pi$
$239$	9.99738	0.646676	0.323338	+	0.946283i	$0.395195\pi$
$240$	0	0
$241$	−12.8994	−0.830927	−0.415463	−	0.909610i	$-0.636381\pi$
$241$	−12.8994	−0.830927	−0.415463	+	0.909610i	$0.636381\pi$
$242$	0	0
$243$	22.1561	1.42132
$244$	0	0
$245$	−8.52115	−0.544396
$246$	0	0
$247$	−17.5681	−1.11783
$248$	0	0
$249$	−36.9091	−2.33902
$250$	0	0
$251$	−0.110009	−0.00694368	−0.00347184	−	0.999994i	$-0.501105\pi$
$251$	−0.110009	−0.00694368	−0.00347184	+	0.999994i	$0.501105\pi$
$252$	0	0
$253$	7.85679	0.493952
$254$	0	0
$255$	−31.5289	−1.97442
$256$	0	0
$257$	−26.5672	−1.65722	−0.828609	−	0.559828i	$-0.810867\pi$
$257$	−26.5672	−1.65722	−0.828609	+	0.559828i	$0.810867\pi$
$258$	0	0
$259$	−13.3665	−0.830556
$260$	0	0
$261$	3.70018	0.229035
$262$	0	0
$263$	5.00606	0.308687	0.154343	−	0.988017i	$-0.450674\pi$
$263$	5.00606	0.308687	0.154343	+	0.988017i	$0.450674\pi$
$264$	0	0
$265$	9.90067	0.608193
$266$	0	0
$267$	5.45385	0.333770
$268$	0	0
$269$	−20.5520	−1.25308	−0.626538	−	0.779391i	$-0.715529\pi$
$269$	−20.5520	−1.25308	−0.626538	+	0.779391i	$0.715529\pi$
$270$	0	0
$271$	2.08166	0.126452	0.0632260	−	0.997999i	$-0.479861\pi$
$271$	2.08166	0.126452	0.0632260	+	0.997999i	$0.479861\pi$
$272$	0	0
$273$	26.0507	1.57666
$274$	0	0
$275$	−2.16705	−0.130678
$276$	0	0
$277$	−15.9622	−0.959077	−0.479538	−	0.877521i	$-0.659196\pi$
$277$	−15.9622	−0.959077	−0.479538	+	0.877521i	$0.659196\pi$
$278$	0	0
$279$	−2.52396	−0.151105
$280$	0	0
$281$	14.6106	0.871595	0.435797	−	0.900045i	$-0.356466\pi$
$281$	14.6106	0.871595	0.435797	+	0.900045i	$0.356466\pi$
$282$	0	0
$283$	−21.1678	−1.25829	−0.629147	−	0.777286i	$-0.716596\pi$
$283$	−21.1678	−1.25829	−0.629147	+	0.777286i	$0.716596\pi$
$284$	0	0
$285$	27.4294	1.62478
$286$	0	0
$287$	−15.1392	−0.893636
$288$	0	0
$289$	24.0661	1.41565
$290$	0	0
$291$	−32.3543	−1.89664
$292$	0	0
$293$	−6.11336	−0.357147	−0.178573	−	0.983927i	$-0.557148\pi$
$293$	−6.11336	−0.357147	−0.178573	+	0.983927i	$0.557148\pi$
$294$	0	0
$295$	−1.86678	−0.108688
$296$	0	0
$297$	−0.350830	−0.0203572
$298$	0	0
$299$	−11.5357	−0.667128
$300$	0	0
$301$	−29.0888	−1.67665
$302$	0	0
$303$	−1.89264	−0.108729
$304$	0	0
$305$	−2.21144	−0.126627
$306$	0	0
$307$	2.76265	0.157673	0.0788364	−	0.996888i	$-0.474880\pi$
$307$	2.76265	0.157673	0.0788364	+	0.996888i	$0.474880\pi$
$308$	0	0
$309$	−27.2074	−1.54778
$310$	0	0
$311$	−9.22152	−0.522904	−0.261452	−	0.965216i	$-0.584201\pi$
$311$	−9.22152	−0.522904	−0.261452	+	0.965216i	$0.584201\pi$
$312$	0	0
$313$	7.20115	0.407033	0.203517	−	0.979072i	$-0.434763\pi$
$313$	7.20115	0.407033	0.203517	+	0.979072i	$0.434763\pi$
$314$	0	0
$315$	−20.5592	−1.15838
$316$	0	0
$317$	5.95692	0.334574	0.167287	−	0.985908i	$-0.446499\pi$
$317$	5.95692	0.334574	0.167287	+	0.985908i	$0.446499\pi$
$318$	0	0
$319$	2.58984	0.145003
$320$	0	0
$321$	−14.1862	−0.791799
$322$	0	0
$323$	−35.7265	−1.98788
$324$	0	0
$325$	3.18177	0.176493
$326$	0	0
$327$	22.0344	1.21851
$328$	0	0
$329$	29.0120	1.59948
$330$	0	0
$331$	−34.0972	−1.87415	−0.937077	−	0.349123i	$-0.886479\pi$
$331$	−34.0972	−1.87415	−0.937077	+	0.349123i	$0.886479\pi$
$332$	0	0
$333$	−12.2113	−0.669176
$334$	0	0
$335$	5.03670	0.275184
$336$	0	0
$337$	22.6488	1.23376	0.616879	−	0.787058i	$-0.288397\pi$
$337$	22.6488	1.23376	0.616879	+	0.787058i	$0.288397\pi$
$338$	0	0
$339$	−1.00486	−0.0545763
$340$	0	0
$341$	−1.76658	−0.0956655
$342$	0	0
$343$	−9.17737	−0.495531
$344$	0	0
$345$	18.0110	0.969678
$346$	0	0
$347$	−29.0528	−1.55964	−0.779818	−	0.626006i	$-0.784688\pi$
$347$	−29.0528	−1.55964	−0.779818	+	0.626006i	$0.784688\pi$
$348$	0	0
$349$	−27.0657	−1.44880	−0.724398	−	0.689382i	$-0.757882\pi$
$349$	−27.0657	−1.44880	−0.724398	+	0.689382i	$0.757882\pi$
$350$	0	0
$351$	0.515105	0.0274943
$352$	0	0
$353$	−36.7375	−1.95534	−0.977669	−	0.210149i	$-0.932605\pi$
$353$	−36.7375	−1.95534	−0.977669	+	0.210149i	$0.932605\pi$
$354$	0	0
$355$	−20.2094	−1.07261
$356$	0	0
$357$	52.9769	2.80383
$358$	0	0
$359$	−20.2626	−1.06942	−0.534709	−	0.845036i	$-0.679579\pi$
$359$	−20.2626	−1.06942	−0.534709	+	0.845036i	$0.679579\pi$
$360$	0	0
$361$	12.0812	0.635855
$362$	0	0
$363$	15.7477	0.826542
$364$	0	0
$365$	−15.1445	−0.792700
$366$	0	0
$367$	−2.34257	−0.122281	−0.0611407	−	0.998129i	$-0.519474\pi$
$367$	−2.34257	−0.122281	−0.0611407	+	0.998129i	$0.519474\pi$
$368$	0	0
$369$	−13.8308	−0.720000
$370$	0	0
$371$	−16.6357	−0.863685
$372$	0	0
$373$	−24.1913	−1.25258	−0.626290	−	0.779590i	$-0.715427\pi$
$373$	−24.1913	−1.25258	−0.626290	+	0.779590i	$0.715427\pi$
$374$	0	0
$375$	−29.5679	−1.52688
$376$	0	0
$377$	−3.80254	−0.195841
$378$	0	0
$379$	15.6946	0.806175	0.403088	−	0.915161i	$-0.367937\pi$
$379$	15.6946	0.806175	0.403088	+	0.915161i	$0.367937\pi$
$380$	0	0
$381$	−23.8707	−1.22293
$382$	0	0
$383$	2.39851	0.122558	0.0612790	−	0.998121i	$-0.480482\pi$
$383$	2.39851	0.122558	0.0612790	+	0.998121i	$0.480482\pi$
$384$	0	0
$385$	−14.3899	−0.733377
$386$	0	0
$387$	−26.5748	−1.35087
$388$	0	0
$389$	−35.6090	−1.80545	−0.902724	−	0.430220i	$-0.858436\pi$
$389$	−35.6090	−1.80545	−0.902724	+	0.430220i	$0.858436\pi$
$390$	0	0
$391$	−23.4591	−1.18638
$392$	0	0
$393$	13.3902	0.675448
$394$	0	0
$395$	30.8338	1.55142
$396$	0	0
$397$	20.3950	1.02360	0.511799	−	0.859105i	$-0.328979\pi$
$397$	20.3950	1.02360	0.511799	+	0.859105i	$0.328979\pi$
$398$	0	0
$399$	−46.0886	−2.30732
$400$	0	0
$401$	4.09397	0.204443	0.102222	−	0.994762i	$-0.467405\pi$
$401$	4.09397	0.204443	0.102222	+	0.994762i	$0.467405\pi$
$402$	0	0
$403$	2.59378	0.129205
$404$	0	0
$405$	17.5716	0.873141
$406$	0	0
$407$	−8.54700	−0.423659
$408$	0	0
$409$	−0.660050	−0.0326374	−0.0163187	−	0.999867i	$-0.505195\pi$
$409$	−0.660050	−0.0326374	−0.0163187	+	0.999867i	$0.505195\pi$
$410$	0	0
$411$	−51.4655	−2.53860
$412$	0	0
$413$	3.13669	0.154346
$414$	0	0
$415$	−29.9345	−1.46943
$416$	0	0
$417$	50.9247	2.49379
$418$	0	0
$419$	9.85468	0.481433	0.240716	−	0.970595i	$-0.422618\pi$
$419$	9.85468	0.481433	0.240716	+	0.970595i	$0.422618\pi$
$420$	0	0
$421$	−3.05114	−0.148703	−0.0743516	−	0.997232i	$-0.523689\pi$
$421$	−3.05114	−0.148703	−0.0743516	+	0.997232i	$0.523689\pi$
$422$	0	0
$423$	26.5046	1.28870
$424$	0	0
$425$	6.47047	0.313864
$426$	0	0
$427$	3.71580	0.179820
$428$	0	0
$429$	16.6577	0.804240
$430$	0	0
$431$	−19.0015	−0.915270	−0.457635	−	0.889140i	$-0.651303\pi$
$431$	−19.0015	−0.915270	−0.457635	+	0.889140i	$0.651303\pi$
$432$	0	0
$433$	11.5592	0.555500	0.277750	−	0.960653i	$-0.410411\pi$
$433$	11.5592	0.555500	0.277750	+	0.960653i	$0.410411\pi$
$434$	0	0
$435$	5.93698	0.284656
$436$	0	0
$437$	20.4089	0.976290
$438$	0	0
$439$	−24.0081	−1.14584	−0.572921	−	0.819610i	$-0.694190\pi$
$439$	−24.0081	−1.14584	−0.572921	+	0.819610i	$0.694190\pi$
$440$	0	0
$441$	13.0804	0.622874
$442$	0	0
$443$	−34.5714	−1.64254	−0.821268	−	0.570543i	$-0.806733\pi$
$443$	−34.5714	−1.64254	−0.821268	+	0.570543i	$0.806733\pi$
$444$	0	0
$445$	4.42325	0.209682
$446$	0	0
$447$	38.3894	1.81576
$448$	0	0
$449$	−16.2414	−0.766480	−0.383240	−	0.923649i	$-0.625192\pi$
$449$	−16.2414	−0.766480	−0.383240	+	0.923649i	$0.625192\pi$
$450$	0	0
$451$	−9.68048	−0.455836
$452$	0	0
$453$	−24.3632	−1.14468
$454$	0	0
$455$	21.1280	0.990494
$456$	0	0
$457$	−7.89934	−0.369516	−0.184758	−	0.982784i	$-0.559150\pi$
$457$	−7.89934	−0.369516	−0.184758	+	0.982784i	$0.559150\pi$
$458$	0	0
$459$	1.04752	0.0488941
$460$	0	0
$461$	−25.3883	−1.18245	−0.591225	−	0.806507i	$-0.701355\pi$
$461$	−25.3883	−1.18245	−0.591225	+	0.806507i	$0.701355\pi$
$462$	0	0
$463$	9.78116	0.454569	0.227285	−	0.973828i	$-0.427015\pi$
$463$	9.78116	0.454569	0.227285	+	0.973828i	$0.427015\pi$
$464$	0	0
$465$	−4.04972	−0.187801
$466$	0	0
$467$	31.1047	1.43935	0.719676	−	0.694310i	$-0.244290\pi$
$467$	31.1047	1.43935	0.719676	+	0.694310i	$0.244290\pi$
$468$	0	0
$469$	−8.46299	−0.390784
$470$	0	0
$471$	−16.0535	−0.739706
$472$	0	0
$473$	−18.6004	−0.855246
$474$	0	0
$475$	−5.62916	−0.258283
$476$	0	0
$477$	−15.1980	−0.695868
$478$	0	0
$479$	−34.3717	−1.57048	−0.785241	−	0.619191i	$-0.787461\pi$
$479$	−34.3717	−1.57048	−0.785241	+	0.619191i	$0.787461\pi$
$480$	0	0
$481$	12.5491	0.572191
$482$	0	0
$483$	−30.2632	−1.37702
$484$	0	0
$485$	−26.2404	−1.19152
$486$	0	0
$487$	−17.6861	−0.801436	−0.400718	−	0.916202i	$-0.631239\pi$
$487$	−17.6861	−0.801436	−0.400718	+	0.916202i	$0.631239\pi$
$488$	0	0
$489$	52.8045	2.38790
$490$	0	0
$491$	−4.42783	−0.199825	−0.0999126	−	0.994996i	$-0.531856\pi$
$491$	−4.42783	−0.199825	−0.0999126	+	0.994996i	$0.531856\pi$
$492$	0	0
$493$	−7.73286	−0.348271
$494$	0	0
$495$	−13.1462	−0.590880
$496$	0	0
$497$	33.9572	1.52319
$498$	0	0
$499$	26.9776	1.20768	0.603841	−	0.797105i	$-0.293636\pi$
$499$	26.9776	1.20768	0.603841	+	0.797105i	$0.293636\pi$
$500$	0	0
$501$	−40.8316	−1.82422
$502$	0	0
$503$	7.25828	0.323631	0.161815	−	0.986821i	$-0.448265\pi$
$503$	7.25828	0.323631	0.161815	+	0.986821i	$0.448265\pi$
$504$	0	0
$505$	−1.53499	−0.0683062
$506$	0	0
$507$	7.56139	0.335813
$508$	0	0
$509$	14.9618	0.663170	0.331585	−	0.943425i	$-0.392417\pi$
$509$	14.9618	0.663170	0.331585	+	0.943425i	$0.392417\pi$
$510$	0	0
$511$	25.4468	1.12570
$512$	0	0
$513$	−0.911319	−0.0402357
$514$	0	0
$515$	−22.0661	−0.972349
$516$	0	0
$517$	18.5512	0.815881
$518$	0	0
$519$	42.4352	1.86270
$520$	0	0
$521$	28.1997	1.23545	0.617725	−	0.786394i	$-0.288054\pi$
$521$	28.1997	1.23545	0.617725	+	0.786394i	$0.288054\pi$
$522$	0	0
$523$	23.8334	1.04216	0.521081	−	0.853507i	$-0.325529\pi$
$523$	23.8334	1.04216	0.521081	+	0.853507i	$0.325529\pi$
$524$	0	0
$525$	8.34716	0.364300
$526$	0	0
$527$	5.27472	0.229770
$528$	0	0
$529$	−9.59890	−0.417343
$530$	0	0
$531$	2.86560	0.124356
$532$	0	0
$533$	14.2133	0.615649
$534$	0	0
$535$	−11.5055	−0.497426
$536$	0	0
$537$	4.34914	0.187679
$538$	0	0
$539$	9.15526	0.394345
$540$	0	0
$541$	−17.4096	−0.748497	−0.374249	−	0.927328i	$-0.622099\pi$
$541$	−17.4096	−0.748497	−0.374249	+	0.927328i	$0.622099\pi$
$542$	0	0
$543$	−26.7571	−1.14826
$544$	0	0
$545$	17.8706	0.765494
$546$	0	0
$547$	−8.39816	−0.359079	−0.179540	−	0.983751i	$-0.557461\pi$
$547$	−8.39816	−0.359079	−0.179540	+	0.983751i	$0.557461\pi$
$548$	0	0
$549$	3.39466	0.144881
$550$	0	0
$551$	6.72741	0.286597
$552$	0	0
$553$	−51.8089	−2.20314
$554$	0	0
$555$	−19.5932	−0.831685
$556$	0	0
$557$	23.8977	1.01258	0.506290	−	0.862364i	$-0.331017\pi$
$557$	23.8977	1.01258	0.506290	+	0.862364i	$0.331017\pi$
$558$	0	0
$559$	27.3100	1.15509
$560$	0	0
$561$	33.8751	1.43021
$562$	0	0
$563$	29.7119	1.25221	0.626103	−	0.779740i	$-0.284649\pi$
$563$	29.7119	1.25221	0.626103	+	0.779740i	$0.284649\pi$
$564$	0	0
$565$	−0.814971	−0.0342861
$566$	0	0
$567$	−29.5250	−1.23993
$568$	0	0
$569$	5.44760	0.228375	0.114188	−	0.993459i	$-0.463573\pi$
$569$	5.44760	0.228375	0.114188	+	0.993459i	$0.463573\pi$
$570$	0	0
$571$	−23.9219	−1.00110	−0.500549	−	0.865708i	$-0.666869\pi$
$571$	−23.9219	−1.00110	−0.500549	+	0.865708i	$0.666869\pi$
$572$	0	0
$573$	42.9439	1.79401
$574$	0	0
$575$	−3.69628	−0.154145
$576$	0	0
$577$	−25.0471	−1.04273	−0.521363	−	0.853335i	$-0.674576\pi$
$577$	−25.0471	−1.04273	−0.521363	+	0.853335i	$0.674576\pi$
$578$	0	0
$579$	61.3700	2.55045
$580$	0	0
$581$	50.2979	2.08671
$582$	0	0
$583$	−10.6374	−0.440558
$584$	0	0
$585$	19.3020	0.798038
$586$	0	0
$587$	10.8473	0.447716	0.223858	−	0.974622i	$-0.428135\pi$
$587$	10.8473	0.447716	0.223858	+	0.974622i	$0.428135\pi$
$588$	0	0
$589$	−4.58888	−0.189082
$590$	0	0
$591$	−33.0225	−1.35837
$592$	0	0
$593$	−26.9787	−1.10788	−0.553941	−	0.832556i	$-0.686877\pi$
$593$	−26.9787	−1.10788	−0.553941	+	0.832556i	$0.686877\pi$
$594$	0	0
$595$	42.9659	1.76143
$596$	0	0
$597$	−47.0709	−1.92648
$598$	0	0
$599$	14.8391	0.606308	0.303154	−	0.952942i	$-0.401960\pi$
$599$	14.8391	0.606308	0.303154	+	0.952942i	$0.401960\pi$
$600$	0	0
$601$	−28.1141	−1.14680	−0.573398	−	0.819277i	$-0.694375\pi$
$601$	−28.1141	−1.14680	−0.573398	+	0.819277i	$0.694375\pi$
$602$	0	0
$603$	−7.73157	−0.314854
$604$	0	0
$605$	12.7719	0.519252
$606$	0	0
$607$	−12.2820	−0.498511	−0.249256	−	0.968438i	$-0.580186\pi$
$607$	−12.2820	−0.498511	−0.249256	+	0.968438i	$0.580186\pi$
$608$	0	0
$609$	−9.97569	−0.404235
$610$	0	0
$611$	−27.2378	−1.10192
$612$	0	0
$613$	−32.3243	−1.30557	−0.652784	−	0.757544i	$-0.726399\pi$
$613$	−32.3243	−1.30557	−0.652784	+	0.757544i	$0.726399\pi$
$614$	0	0
$615$	−22.1916	−0.894852
$616$	0	0
$617$	−32.3486	−1.30231	−0.651154	−	0.758946i	$-0.725715\pi$
$617$	−32.3486	−1.30231	−0.651154	+	0.758946i	$0.725715\pi$
$618$	0	0
$619$	−37.2827	−1.49852	−0.749260	−	0.662276i	$-0.769591\pi$
$619$	−37.2827	−1.49852	−0.749260	+	0.662276i	$0.769591\pi$
$620$	0	0
$621$	−0.598400	−0.0240130
$622$	0	0
$623$	−7.43223	−0.297766
$624$	0	0
$625$	−18.9320	−0.757279
$626$	0	0
$627$	−29.4706	−1.17694
$628$	0	0
$629$	25.5200	1.01755
$630$	0	0
$631$	35.0378	1.39483	0.697416	−	0.716666i	$-0.254333\pi$
$631$	35.0378	1.39483	0.697416	+	0.716666i	$0.254333\pi$
$632$	0	0
$633$	65.4784	2.60253
$634$	0	0
$635$	−19.3599	−0.768276
$636$	0	0
$637$	−13.4422	−0.532599
$638$	0	0
$639$	31.0224	1.22723
$640$	0	0
$641$	1.33053	0.0525526	0.0262763	−	0.999655i	$-0.491635\pi$
$641$	1.33053	0.0525526	0.0262763	+	0.999655i	$0.491635\pi$
$642$	0	0
$643$	27.9897	1.10380	0.551902	−	0.833909i	$-0.313902\pi$
$643$	27.9897	1.10380	0.551902	+	0.833909i	$0.313902\pi$
$644$	0	0
$645$	−42.6396	−1.67893
$646$	0	0
$647$	−32.5508	−1.27971	−0.639853	−	0.768498i	$-0.721005\pi$
$647$	−32.5508	−1.27971	−0.639853	+	0.768498i	$0.721005\pi$
$648$	0	0
$649$	2.00570	0.0787307
$650$	0	0
$651$	6.80459	0.266693
$652$	0	0
$653$	−22.5680	−0.883153	−0.441576	−	0.897224i	$-0.645581\pi$
$653$	−22.5680	−0.883153	−0.441576	+	0.897224i	$0.645581\pi$
$654$	0	0
$655$	10.8599	0.424332
$656$	0	0
$657$	23.2475	0.906973
$658$	0	0
$659$	27.6815	1.07832	0.539159	−	0.842204i	$-0.318742\pi$
$659$	27.6815	1.07832	0.539159	+	0.842204i	$0.318742\pi$
$660$	0	0
$661$	20.8762	0.811990	0.405995	−	0.913875i	$-0.366925\pi$
$661$	20.8762	0.811990	0.405995	+	0.913875i	$0.366925\pi$
$662$	0	0
$663$	−49.7371	−1.93163
$664$	0	0
$665$	−37.3794	−1.44951
$666$	0	0
$667$	4.41742	0.171043
$668$	0	0
$669$	8.21344	0.317550
$670$	0	0
$671$	2.37601	0.0917247
$672$	0	0
$673$	−10.0767	−0.388429	−0.194215	−	0.980959i	$-0.562216\pi$
$673$	−10.0767	−0.388429	−0.194215	+	0.980959i	$0.562216\pi$
$674$	0	0
$675$	0.165050	0.00635277
$676$	0	0
$677$	30.7816	1.18303	0.591516	−	0.806293i	$-0.298530\pi$
$677$	30.7816	1.18303	0.591516	+	0.806293i	$0.298530\pi$
$678$	0	0
$679$	44.0908	1.69205
$680$	0	0
$681$	−48.1189	−1.84392
$682$	0	0
$683$	10.0607	0.384962	0.192481	−	0.981301i	$-0.438347\pi$
$683$	10.0607	0.384962	0.192481	+	0.981301i	$0.438347\pi$
$684$	0	0
$685$	−41.7402	−1.59481
$686$	0	0
$687$	57.5570	2.19593
$688$	0	0
$689$	15.6184	0.595014
$690$	0	0
$691$	47.4509	1.80512	0.902559	−	0.430565i	$-0.141686\pi$
$691$	47.4509	1.80512	0.902559	+	0.430565i	$0.141686\pi$
$692$	0	0
$693$	22.0892	0.839098
$694$	0	0
$695$	41.3015	1.56666
$696$	0	0
$697$	28.9044	1.09483
$698$	0	0
$699$	−43.1813	−1.63327
$700$	0	0
$701$	−35.0618	−1.32427	−0.662134	−	0.749386i	$-0.730349\pi$
$701$	−35.0618	−1.32427	−0.662134	+	0.749386i	$0.730349\pi$
$702$	0	0
$703$	−22.2018	−0.837356
$704$	0	0
$705$	42.5269	1.60166
$706$	0	0
$707$	2.57919	0.0970005
$708$	0	0
$709$	−46.7249	−1.75479	−0.877395	−	0.479769i	$-0.840721\pi$
$709$	−46.7249	−1.75479	−0.877395	+	0.479769i	$0.840721\pi$
$710$	0	0
$711$	−47.3313	−1.77506
$712$	0	0
$713$	−3.01320	−0.112845
$714$	0	0
$715$	13.5099	0.505242
$716$	0	0
$717$	−24.6235	−0.919583
$718$	0	0
$719$	18.6771	0.696537	0.348268	−	0.937395i	$-0.386770\pi$
$719$	18.6771	0.696537	0.348268	+	0.937395i	$0.386770\pi$
$720$	0	0
$721$	37.0769	1.38082
$722$	0	0
$723$	31.7713	1.18159
$724$	0	0
$725$	−1.21841	−0.0452505
$726$	0	0
$727$	41.7205	1.54733	0.773664	−	0.633596i	$-0.218422\pi$
$727$	41.7205	1.54733	0.773664	+	0.633596i	$0.218422\pi$
$728$	0	0
$729$	−28.1811	−1.04374
$730$	0	0
$731$	55.5377	2.05414
$732$	0	0
$733$	26.2719	0.970373	0.485187	−	0.874411i	$-0.338752\pi$
$733$	26.2719	0.970373	0.485187	+	0.874411i	$0.338752\pi$
$734$	0	0
$735$	20.9876	0.774139
$736$	0	0
$737$	−5.41151	−0.199336
$738$	0	0
$739$	33.2962	1.22482	0.612410	−	0.790540i	$-0.290200\pi$
$739$	33.2962	1.22482	0.612410	+	0.790540i	$0.290200\pi$
$740$	0	0
$741$	43.2701	1.58957
$742$	0	0
$743$	25.7351	0.944129	0.472064	−	0.881564i	$-0.343509\pi$
$743$	25.7351	0.944129	0.472064	+	0.881564i	$0.343509\pi$
$744$	0	0
$745$	31.1351	1.14070
$746$	0	0
$747$	45.9509	1.68125
$748$	0	0
$749$	19.3323	0.706386
$750$	0	0
$751$	16.0652	0.586226	0.293113	−	0.956078i	$-0.405309\pi$
$751$	16.0652	0.586226	0.293113	+	0.956078i	$0.405309\pi$
$752$	0	0
$753$	0.270951	0.00987401
$754$	0	0
$755$	−19.7593	−0.719115
$756$	0	0
$757$	29.0009	1.05406	0.527029	−	0.849847i	$-0.323306\pi$
$757$	29.0009	1.05406	0.527029	+	0.849847i	$0.323306\pi$
$758$	0	0
$759$	−19.3513	−0.702407
$760$	0	0
$761$	9.07559	0.328990	0.164495	−	0.986378i	$-0.447401\pi$
$761$	9.07559	0.328990	0.164495	+	0.986378i	$0.447401\pi$
$762$	0	0
$763$	−30.0274	−1.08706
$764$	0	0
$765$	39.2526	1.41918
$766$	0	0
$767$	−2.94487	−0.106333
$768$	0	0
$769$	−19.9910	−0.720894	−0.360447	−	0.932780i	$-0.617376\pi$
$769$	−19.9910	−0.720894	−0.360447	+	0.932780i	$0.617376\pi$
$770$	0	0
$771$	65.4350	2.35659
$772$	0	0
$773$	11.0217	0.396424	0.198212	−	0.980159i	$-0.436486\pi$
$773$	11.0217	0.396424	0.198212	+	0.980159i	$0.436486\pi$
$774$	0	0
$775$	0.831097	0.0298539
$776$	0	0
$777$	32.9217	1.18106
$778$	0	0
$779$	−25.1461	−0.900953
$780$	0	0
$781$	21.7133	0.776964
$782$	0	0
$783$	−0.197251	−0.00704918
$784$	0	0
$785$	−13.0199	−0.464700
$786$	0	0
$787$	−54.2136	−1.93251	−0.966253	−	0.257594i	$-0.917070\pi$
$787$	−54.2136	−1.93251	−0.966253	+	0.257594i	$0.917070\pi$
$788$	0	0
$789$	−12.3299	−0.438957
$790$	0	0
$791$	1.36937	0.0486891
$792$	0	0
$793$	−3.48857	−0.123883
$794$	0	0
$795$	−24.3853	−0.864859
$796$	0	0
$797$	−19.6385	−0.695632	−0.347816	−	0.937563i	$-0.613077\pi$
$797$	−19.6385	−0.695632	−0.347816	+	0.937563i	$0.613077\pi$
$798$	0	0
$799$	−55.3910	−1.95959
$800$	0	0
$801$	−6.78990	−0.239909
$802$	0	0
$803$	16.2715	0.574209
$804$	0	0
$805$	−24.5444	−0.865077
$806$	0	0
$807$	50.6195	1.78189
$808$	0	0
$809$	26.6810	0.938054	0.469027	−	0.883184i	$-0.344605\pi$
$809$	26.6810	0.938054	0.469027	+	0.883184i	$0.344605\pi$
$810$	0	0
$811$	−22.4512	−0.788367	−0.394183	−	0.919032i	$-0.628973\pi$
$811$	−22.4512	−0.788367	−0.394183	+	0.919032i	$0.628973\pi$
$812$	0	0
$813$	−5.12713	−0.179816
$814$	0	0
$815$	42.8261	1.50013
$816$	0	0
$817$	−48.3165	−1.69038
$818$	0	0
$819$	−32.4324	−1.13328
$820$	0	0
$821$	43.8751	1.53125	0.765627	−	0.643285i	$-0.222429\pi$
$821$	43.8751	1.53125	0.765627	+	0.643285i	$0.222429\pi$
$822$	0	0
$823$	19.5229	0.680526	0.340263	−	0.940330i	$-0.389484\pi$
$823$	19.5229	0.680526	0.340263	+	0.940330i	$0.389484\pi$
$824$	0	0
$825$	5.33745	0.185826
$826$	0	0
$827$	−38.0189	−1.32205	−0.661023	−	0.750366i	$-0.729877\pi$
$827$	−38.0189	−1.32205	−0.661023	+	0.750366i	$0.729877\pi$
$828$	0	0
$829$	17.0599	0.592515	0.296257	−	0.955108i	$-0.404261\pi$
$829$	17.0599	0.592515	0.296257	+	0.955108i	$0.404261\pi$
$830$	0	0
$831$	39.3149	1.36382
$832$	0	0
$833$	−27.3361	−0.947141
$834$	0	0
$835$	−33.1157	−1.14602
$836$	0	0
$837$	0.134548	0.00465067
$838$	0	0
$839$	16.8520	0.581796	0.290898	−	0.956754i	$-0.406046\pi$
$839$	16.8520	0.581796	0.290898	+	0.956754i	$0.406046\pi$
$840$	0	0
$841$	−27.5439	−0.949789
$842$	0	0
$843$	−35.9859	−1.23942
$844$	0	0
$845$	6.13253	0.210966
$846$	0	0
$847$	−21.4602	−0.737381
$848$	0	0
$849$	52.1362	1.78931
$850$	0	0
$851$	−14.5784	−0.499740
$852$	0	0
$853$	−8.28695	−0.283740	−0.141870	−	0.989885i	$-0.545311\pi$
$853$	−8.28695	−0.283740	−0.141870	+	0.989885i	$0.545311\pi$
$854$	0	0
$855$	−34.1488	−1.16787
$856$	0	0
$857$	8.73482	0.298376	0.149188	−	0.988809i	$-0.452334\pi$
$857$	8.73482	0.298376	0.149188	+	0.988809i	$0.452334\pi$
$858$	0	0
$859$	−32.3158	−1.10260	−0.551301	−	0.834307i	$-0.685868\pi$
$859$	−32.3158	−1.10260	−0.551301	+	0.834307i	$0.685868\pi$
$860$	0	0
$861$	37.2877	1.27076
$862$	0	0
$863$	14.0683	0.478890	0.239445	−	0.970910i	$-0.423035\pi$
$863$	14.0683	0.478890	0.239445	+	0.970910i	$0.423035\pi$
$864$	0	0
$865$	34.4163	1.17019
$866$	0	0
$867$	−59.2748	−2.01308
$868$	0	0
$869$	−33.1283	−1.12380
$870$	0	0
$871$	7.94545	0.269221
$872$	0	0
$873$	40.2803	1.36328
$874$	0	0
$875$	40.2936	1.36217
$876$	0	0
$877$	−39.9315	−1.34839	−0.674195	−	0.738554i	$-0.735509\pi$
$877$	−39.9315	−1.34839	−0.674195	+	0.738554i	$0.735509\pi$
$878$	0	0
$879$	15.0572	0.507867
$880$	0	0
$881$	31.8686	1.07368	0.536840	−	0.843684i	$-0.319618\pi$
$881$	31.8686	1.07368	0.536840	+	0.843684i	$0.319618\pi$
$882$	0	0
$883$	36.8173	1.23900	0.619501	−	0.784996i	$-0.287335\pi$
$883$	36.8173	1.23900	0.619501	+	0.784996i	$0.287335\pi$
$884$	0	0
$885$	4.59789	0.154556
$886$	0	0
$887$	1.76947	0.0594130	0.0297065	−	0.999559i	$-0.490543\pi$
$887$	1.76947	0.0594130	0.0297065	+	0.999559i	$0.490543\pi$
$888$	0	0
$889$	32.5298	1.09101
$890$	0	0
$891$	−18.8792	−0.632479
$892$	0	0
$893$	48.1888	1.61258
$894$	0	0
$895$	3.52729	0.117904
$896$	0	0
$897$	28.4125	0.948666
$898$	0	0
$899$	−0.993244	−0.0331266
$900$	0	0
$901$	31.7617	1.05814
$902$	0	0
$903$	71.6458	2.38422
$904$	0	0
$905$	−21.7008	−0.721360
$906$	0	0
$907$	−1.71595	−0.0569770	−0.0284885	−	0.999594i	$-0.509069\pi$
$907$	−1.71595	−0.0569770	−0.0284885	+	0.999594i	$0.509069\pi$
$908$	0	0
$909$	2.35628	0.0781530
$910$	0	0
$911$	−34.3836	−1.13918	−0.569589	−	0.821929i	$-0.692898\pi$
$911$	−34.3836	−1.13918	−0.569589	+	0.821929i	$0.692898\pi$
$912$	0	0
$913$	32.1621	1.06441
$914$	0	0
$915$	5.44677	0.180065
$916$	0	0
$917$	−18.2475	−0.602586
$918$	0	0
$919$	37.0870	1.22339	0.611694	−	0.791095i	$-0.290488\pi$
$919$	37.0870	1.22339	0.611694	+	0.791095i	$0.290488\pi$
$920$	0	0
$921$	−6.80441	−0.224213
$922$	0	0
$923$	−31.8806	−1.04936
$924$	0	0
$925$	4.02099	0.132209
$926$	0	0
$927$	33.8725	1.11252
$928$	0	0
$929$	29.9504	0.982641	0.491320	−	0.870979i	$-0.336514\pi$
$929$	29.9504	0.982641	0.491320	+	0.870979i	$0.336514\pi$
$930$	0	0
$931$	23.7818	0.779417
$932$	0	0
$933$	22.7126	0.743577
$934$	0	0
$935$	27.4738	0.898491
$936$	0	0
$937$	−4.95660	−0.161925	−0.0809625	−	0.996717i	$-0.525799\pi$
$937$	−4.95660	−0.161925	−0.0809625	+	0.996717i	$0.525799\pi$
$938$	0	0
$939$	−17.7364	−0.578806
$940$	0	0
$941$	32.1974	1.04961	0.524803	−	0.851224i	$-0.324139\pi$
$941$	32.1974	1.04961	0.524803	+	0.851224i	$0.324139\pi$
$942$	0	0
$943$	−16.5117	−0.537695
$944$	0	0
$945$	1.09598	0.0356523
$946$	0	0
$947$	−45.2048	−1.46896	−0.734479	−	0.678631i	$-0.762574\pi$
$947$	−45.2048	−1.46896	−0.734479	+	0.678631i	$0.762574\pi$
$948$	0	0
$949$	−23.8906	−0.775523
$950$	0	0
$951$	−14.6719	−0.475768
$952$	0	0
$953$	21.4068	0.693435	0.346718	−	0.937970i	$-0.387296\pi$
$953$	21.4068	0.693435	0.346718	+	0.937970i	$0.387296\pi$
$954$	0	0
$955$	34.8289	1.12704
$956$	0	0
$957$	−6.37879	−0.206197
$958$	0	0
$959$	70.1345	2.26476
$960$	0	0
$961$	−30.3225	−0.978145
$962$	0	0
$963$	17.6615	0.569133
$964$	0	0
$965$	49.7730	1.60225
$966$	0	0
$967$	−24.2331	−0.779284	−0.389642	−	0.920966i	$-0.627401\pi$
$967$	−24.2331	−0.779284	−0.389642	+	0.920966i	$0.627401\pi$
$968$	0	0
$969$	87.9945	2.82679
$970$	0	0
$971$	5.36526	0.172180	0.0860898	−	0.996287i	$-0.472563\pi$
$971$	5.36526	0.172180	0.0860898	+	0.996287i	$0.472563\pi$
$972$	0	0
$973$	−69.3975	−2.22478
$974$	0	0
$975$	−7.83670	−0.250975
$976$	0	0
$977$	−27.8206	−0.890060	−0.445030	−	0.895516i	$-0.646807\pi$
$977$	−27.8206	−0.890060	−0.445030	+	0.895516i	$0.646807\pi$
$978$	0	0
$979$	−4.75241	−0.151888
$980$	0	0
$981$	−27.4323	−0.875845
$982$	0	0
$983$	−27.4426	−0.875284	−0.437642	−	0.899149i	$-0.644186\pi$
$983$	−27.4426	−0.875284	−0.437642	+	0.899149i	$0.644186\pi$
$984$	0	0
$985$	−26.7823	−0.853356
$986$	0	0
$987$	−71.4565	−2.27448
$988$	0	0
$989$	−31.7261	−1.00883
$990$	0	0
$991$	−35.6943	−1.13387	−0.566933	−	0.823764i	$-0.691870\pi$
$991$	−35.6943	−1.13387	−0.566933	+	0.823764i	$0.691870\pi$
$992$	0	0
$993$	83.9815	2.66507
$994$	0	0
$995$	−38.1760	−1.21026
$996$	0	0
$997$	−33.4741	−1.06014	−0.530068	−	0.847955i	$-0.677834\pi$
$997$	−33.4741	−1.06014	−0.530068	+	0.847955i	$0.677834\pi$
$998$	0	0
$999$	0.650968	0.0205957

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.a.1.11	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.a.1.11	✓	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)

\(f(q)\)	\(=\)	\(q-2.46300 q^{3} -1.99757 q^{5} +3.35645 q^{7} +3.06637 q^{9} +O(q^{10})\)	\(q-2.46300 q^{3} -1.99757 q^{5} +3.35645 q^{7} +3.06637 q^{9} +2.14622 q^{11} -3.15119 q^{13} +4.92002 q^{15} -6.40828 q^{17} +5.57506 q^{19} -8.26693 q^{21} +3.66075 q^{23} -1.00970 q^{25} -0.163464 q^{27} +1.20670 q^{29} -0.823109 q^{31} -5.28615 q^{33} -6.70475 q^{35} -3.98234 q^{37} +7.76138 q^{39} -4.51047 q^{41} -8.66655 q^{43} -6.12529 q^{45} +8.64365 q^{47} +4.26575 q^{49} +15.7836 q^{51} -4.95635 q^{53} -4.28724 q^{55} -13.7314 q^{57} +0.934526 q^{59} +1.10706 q^{61} +10.2921 q^{63} +6.29473 q^{65} -2.52141 q^{67} -9.01643 q^{69} +10.1170 q^{71} +7.58146 q^{73} +2.48690 q^{75} +7.20369 q^{77} -15.4356 q^{79} -8.79649 q^{81} +14.9854 q^{83} +12.8010 q^{85} -2.97210 q^{87} -2.21431 q^{89} -10.5768 q^{91} +2.02732 q^{93} -11.1366 q^{95} +13.1362 q^{97} +6.58111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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