LMFDB - Embedded newform 6040.2.a.n.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 14 x^{11} + 70 x^{10} + 41 x^{9} - 403 x^{8} + 109 x^{7} + 870 x^{6} - 444 x^{5} + \cdots + 11 $ x^13 - 4x^12 - 14x^11 + 70x^10 + 41x^9 - 403x^8 + 109x^7 + 870x^6 - 444x^5 - 708x^4 + 373x^3 + 108x^2 - 82x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-2.51561$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.51561 q^{3} -1.00000 q^{5} +0.120976 q^{7} +3.32830 q^{9} +O(q^{10})$	$q+2.51561 q^{3} -1.00000 q^{5} +0.120976 q^{7} +3.32830 q^{9} -4.09936 q^{11} +3.60234 q^{13} -2.51561 q^{15} -4.30373 q^{17} +0.636881 q^{19} +0.304329 q^{21} -4.42838 q^{23} +1.00000 q^{25} +0.825887 q^{27} -9.18966 q^{29} -1.42586 q^{31} -10.3124 q^{33} -0.120976 q^{35} +7.49323 q^{37} +9.06209 q^{39} +1.44319 q^{41} +5.74287 q^{43} -3.32830 q^{45} -0.925327 q^{47} -6.98536 q^{49} -10.8265 q^{51} -9.31866 q^{53} +4.09936 q^{55} +1.60215 q^{57} +0.847329 q^{59} -8.83873 q^{61} +0.402645 q^{63} -3.60234 q^{65} +0.304633 q^{67} -11.1401 q^{69} -7.18871 q^{71} -0.396378 q^{73} +2.51561 q^{75} -0.495924 q^{77} +5.51447 q^{79} -7.90730 q^{81} -9.53699 q^{83} +4.30373 q^{85} -23.1176 q^{87} +0.440584 q^{89} +0.435797 q^{91} -3.58691 q^{93} -0.636881 q^{95} +0.658888 q^{97} -13.6439 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9}+O(q^{10}) $ 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9	$ 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9} - 14 q^{11} + 5 q^{13} + 4 q^{15} - 8 q^{17} + 16 q^{19} - 5 q^{21} - 4 q^{23} + 13 q^{25} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 19 q^{33} + 6 q^{37} + 7 q^{39} - 18 q^{41} + 7 q^{43} - 5 q^{45} - 22 q^{47} - q^{49} + 12 q^{51} - 17 q^{53} + 14 q^{55} - 16 q^{57} - 6 q^{59} + 10 q^{61} - 5 q^{65} + 12 q^{67} + 13 q^{69} - 16 q^{71} - 24 q^{73} - 4 q^{75} - 11 q^{77} + 36 q^{79} - 19 q^{81} + q^{83} + 8 q^{85} - 8 q^{87} - 53 q^{89} + 23 q^{91} - 9 q^{93} - 16 q^{95} - 21 q^{97} - 9 q^{99}+O(q^{100}) $ 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9 - 14 * q^11 + 5 * q^13 + 4 * q^15 - 8 * q^17 + 16 * q^19 - 5 * q^21 - 4 * q^23 + 13 * q^25 + 2 * q^27 - 6 * q^29 + 11 * q^31 - 19 * q^33 + 6 * q^37 + 7 * q^39 - 18 * q^41 + 7 * q^43 - 5 * q^45 - 22 * q^47 - q^49 + 12 * q^51 - 17 * q^53 + 14 * q^55 - 16 * q^57 - 6 * q^59 + 10 * q^61 - 5 * q^65 + 12 * q^67 + 13 * q^69 - 16 * q^71 - 24 * q^73 - 4 * q^75 - 11 * q^77 + 36 * q^79 - 19 * q^81 + q^83 + 8 * q^85 - 8 * q^87 - 53 * q^89 + 23 * q^91 - 9 * q^93 - 16 * q^95 - 21 * q^97 - 9 * 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
$3$	2.51561	1.45239	0.726195	−	0.687489i	$-0.241287\pi$
$3$	2.51561	1.45239	0.726195	+	0.687489i	$0.241287\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.120976	0.0457246	0.0228623	−	0.999739i	$-0.492722\pi$
$7$	0.120976	0.0457246	0.0228623	+	0.999739i	$0.492722\pi$
$8$	0	0
$9$	3.32830	1.10943
$10$	0	0
$11$	−4.09936	−1.23600	−0.618001	−	0.786177i	$-0.712057\pi$
$11$	−4.09936	−1.23600	−0.618001	+	0.786177i	$0.712057\pi$
$12$	0	0
$13$	3.60234	0.999110	0.499555	−	0.866282i	$-0.333497\pi$
$13$	3.60234	0.999110	0.499555	+	0.866282i	$0.333497\pi$
$14$	0	0
$15$	−2.51561	−0.649528
$16$	0	0
$17$	−4.30373	−1.04381	−0.521904	−	0.853004i	$-0.674778\pi$
$17$	−4.30373	−1.04381	−0.521904	+	0.853004i	$0.674778\pi$
$18$	0	0
$19$	0.636881	0.146111	0.0730553	−	0.997328i	$-0.476725\pi$
$19$	0.636881	0.146111	0.0730553	+	0.997328i	$0.476725\pi$
$20$	0	0
$21$	0.304329	0.0664100
$22$	0	0
$23$	−4.42838	−0.923380	−0.461690	−	0.887041i	$-0.652757\pi$
$23$	−4.42838	−0.923380	−0.461690	+	0.887041i	$0.652757\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.825887	0.158942
$28$	0	0
$29$	−9.18966	−1.70648	−0.853239	−	0.521521i	$-0.825365\pi$
$29$	−9.18966	−1.70648	−0.853239	+	0.521521i	$0.825365\pi$
$30$	0	0
$31$	−1.42586	−0.256092	−0.128046	−	0.991768i	$-0.540870\pi$
$31$	−1.42586	−0.256092	−0.128046	+	0.991768i	$0.540870\pi$
$32$	0	0
$33$	−10.3124	−1.79516
$34$	0	0
$35$	−0.120976	−0.0204487
$36$	0	0
$37$	7.49323	1.23188	0.615940	−	0.787793i	$-0.288776\pi$
$37$	7.49323	1.23188	0.615940	+	0.787793i	$0.288776\pi$
$38$	0	0
$39$	9.06209	1.45110
$40$	0	0
$41$	1.44319	0.225389	0.112694	−	0.993630i	$-0.464052\pi$
$41$	1.44319	0.225389	0.112694	+	0.993630i	$0.464052\pi$
$42$	0	0
$43$	5.74287	0.875780	0.437890	−	0.899029i	$-0.355726\pi$
$43$	5.74287	0.875780	0.437890	+	0.899029i	$0.355726\pi$
$44$	0	0
$45$	−3.32830	−0.496154
$46$	0	0
$47$	−0.925327	−0.134973	−0.0674864	−	0.997720i	$-0.521498\pi$
$47$	−0.925327	−0.134973	−0.0674864	+	0.997720i	$0.521498\pi$
$48$	0	0
$49$	−6.98536	−0.997909
$50$	0	0
$51$	−10.8265	−1.51602
$52$	0	0
$53$	−9.31866	−1.28002	−0.640009	−	0.768368i	$-0.721069\pi$
$53$	−9.31866	−1.28002	−0.640009	+	0.768368i	$0.721069\pi$
$54$	0	0
$55$	4.09936	0.552757
$56$	0	0
$57$	1.60215	0.212209
$58$	0	0
$59$	0.847329	0.110313	0.0551564	−	0.998478i	$-0.482434\pi$
$59$	0.847329	0.110313	0.0551564	+	0.998478i	$0.482434\pi$
$60$	0	0
$61$	−8.83873	−1.13168	−0.565842	−	0.824514i	$-0.691449\pi$
$61$	−8.83873	−1.13168	−0.565842	+	0.824514i	$0.691449\pi$
$62$	0	0
$63$	0.402645	0.0507285
$64$	0	0
$65$	−3.60234	−0.446815
$66$	0	0
$67$	0.304633	0.0372168	0.0186084	−	0.999827i	$-0.494076\pi$
$67$	0.304633	0.0372168	0.0186084	+	0.999827i	$0.494076\pi$
$68$	0	0
$69$	−11.1401	−1.34111
$70$	0	0
$71$	−7.18871	−0.853143	−0.426571	−	0.904454i	$-0.640279\pi$
$71$	−7.18871	−0.853143	−0.426571	+	0.904454i	$0.640279\pi$
$72$	0	0
$73$	−0.396378	−0.0463925	−0.0231963	−	0.999731i	$-0.507384\pi$
$73$	−0.396378	−0.0463925	−0.0231963	+	0.999731i	$0.507384\pi$
$74$	0	0
$75$	2.51561	0.290478
$76$	0	0
$77$	−0.495924	−0.0565157
$78$	0	0
$79$	5.51447	0.620427	0.310213	−	0.950667i	$-0.399599\pi$
$79$	5.51447	0.620427	0.310213	+	0.950667i	$0.399599\pi$
$80$	0	0
$81$	−7.90730	−0.878589
$82$	0	0
$83$	−9.53699	−1.04682	−0.523410	−	0.852081i	$-0.675340\pi$
$83$	−9.53699	−1.04682	−0.523410	+	0.852081i	$0.675340\pi$
$84$	0	0
$85$	4.30373	0.466805
$86$	0	0
$87$	−23.1176	−2.47847
$88$	0	0
$89$	0.440584	0.0467018	0.0233509	−	0.999727i	$-0.492567\pi$
$89$	0.440584	0.0467018	0.0233509	+	0.999727i	$0.492567\pi$
$90$	0	0
$91$	0.435797	0.0456839
$92$	0	0
$93$	−3.58691	−0.371945
$94$	0	0
$95$	−0.636881	−0.0653426
$96$	0	0
$97$	0.658888	0.0668999	0.0334499	−	0.999440i	$-0.489351\pi$
$97$	0.658888	0.0668999	0.0334499	+	0.999440i	$0.489351\pi$
$98$	0	0
$99$	−13.6439	−1.37126
$100$	0	0
$101$	8.46980	0.842776	0.421388	−	0.906880i	$-0.361543\pi$
$101$	8.46980	0.842776	0.421388	+	0.906880i	$0.361543\pi$
$102$	0	0
$103$	1.40490	0.138429	0.0692147	−	0.997602i	$-0.477951\pi$
$103$	1.40490	0.138429	0.0692147	+	0.997602i	$0.477951\pi$
$104$	0	0
$105$	−0.304329	−0.0296994
$106$	0	0
$107$	−0.00336729	−0.000325529 0	−0.000162764	−	1.00000i	$-0.500052\pi$
$107$	−0.00336729	−0.000325529 0	−0.000162764		1.00000i	$0.500052\pi$
$108$	0	0
$109$	−5.27063	−0.504835	−0.252417	−	0.967618i	$-0.581226\pi$
$109$	−5.27063	−0.504835	−0.252417	+	0.967618i	$0.581226\pi$
$110$	0	0
$111$	18.8501	1.78917
$112$	0	0
$113$	−13.8059	−1.29875	−0.649375	−	0.760468i	$-0.724969\pi$
$113$	−13.8059	−1.29875	−0.649375	+	0.760468i	$0.724969\pi$
$114$	0	0
$115$	4.42838	0.412948
$116$	0	0
$117$	11.9897	1.10845
$118$	0	0
$119$	−0.520648	−0.0477278
$120$	0	0
$121$	5.80472	0.527702
$122$	0	0
$123$	3.63051	0.327352
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.84458	0.784830	0.392415	−	0.919788i	$-0.371640\pi$
$127$	8.84458	0.784830	0.392415	+	0.919788i	$0.371640\pi$
$128$	0	0
$129$	14.4468	1.27197
$130$	0	0
$131$	−14.8722	−1.29939	−0.649695	−	0.760195i	$-0.725103\pi$
$131$	−14.8722	−1.29939	−0.649695	+	0.760195i	$0.725103\pi$
$132$	0	0
$133$	0.0770473	0.00668085
$134$	0	0
$135$	−0.825887	−0.0710810
$136$	0	0
$137$	2.03127	0.173543	0.0867714	−	0.996228i	$-0.472345\pi$
$137$	2.03127	0.173543	0.0867714	+	0.996228i	$0.472345\pi$
$138$	0	0
$139$	−3.58846	−0.304369	−0.152185	−	0.988352i	$-0.548631\pi$
$139$	−3.58846	−0.304369	−0.152185	+	0.988352i	$0.548631\pi$
$140$	0	0
$141$	−2.32776	−0.196033
$142$	0	0
$143$	−14.7673	−1.23490
$144$	0	0
$145$	9.18966	0.763160
$146$	0	0
$147$	−17.5725	−1.44935
$148$	0	0
$149$	−4.74259	−0.388528	−0.194264	−	0.980949i	$-0.562232\pi$
$149$	−4.74259	−0.388528	−0.194264	+	0.980949i	$0.562232\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−14.3241	−1.15804
$154$	0	0
$155$	1.42586	0.114528
$156$	0	0
$157$	17.4246	1.39063	0.695316	−	0.718704i	$-0.255264\pi$
$157$	17.4246	1.39063	0.695316	+	0.718704i	$0.255264\pi$
$158$	0	0
$159$	−23.4421	−1.85908
$160$	0	0
$161$	−0.535727	−0.0422212
$162$	0	0
$163$	−14.5656	−1.14087	−0.570434	−	0.821344i	$-0.693225\pi$
$163$	−14.5656	−1.14087	−0.570434	+	0.821344i	$0.693225\pi$
$164$	0	0
$165$	10.3124	0.802818
$166$	0	0
$167$	12.4847	0.966095	0.483047	−	0.875594i	$-0.339530\pi$
$167$	12.4847	0.966095	0.483047	+	0.875594i	$0.339530\pi$
$168$	0	0
$169$	−0.0231425	−0.00178019
$170$	0	0
$171$	2.11973	0.162100
$172$	0	0
$173$	−3.15825	−0.240118	−0.120059	−	0.992767i	$-0.538308\pi$
$173$	−3.15825	−0.240118	−0.120059	+	0.992767i	$0.538308\pi$
$174$	0	0
$175$	0.120976	0.00914493
$176$	0	0
$177$	2.13155	0.160217
$178$	0	0
$179$	−4.51177	−0.337225	−0.168613	−	0.985682i	$-0.553929\pi$
$179$	−4.51177	−0.337225	−0.168613	+	0.985682i	$0.553929\pi$
$180$	0	0
$181$	−9.12447	−0.678217	−0.339108	−	0.940747i	$-0.610125\pi$
$181$	−9.12447	−0.678217	−0.339108	+	0.940747i	$0.610125\pi$
$182$	0	0
$183$	−22.2348	−1.64364
$184$	0	0
$185$	−7.49323	−0.550913
$186$	0	0
$187$	17.6425	1.29015
$188$	0	0
$189$	0.0999125	0.00726756
$190$	0	0
$191$	14.5069	1.04968	0.524842	−	0.851200i	$-0.324124\pi$
$191$	14.5069	1.04968	0.524842	+	0.851200i	$0.324124\pi$
$192$	0	0
$193$	22.0257	1.58544	0.792722	−	0.609584i	$-0.208663\pi$
$193$	22.0257	1.58544	0.792722	+	0.609584i	$0.208663\pi$
$194$	0	0
$195$	−9.06209	−0.648950
$196$	0	0
$197$	10.4788	0.746582	0.373291	−	0.927714i	$-0.378229\pi$
$197$	10.4788	0.746582	0.373291	+	0.927714i	$0.378229\pi$
$198$	0	0
$199$	8.18979	0.580559	0.290279	−	0.956942i	$-0.406252\pi$
$199$	8.18979	0.580559	0.290279	+	0.956942i	$0.406252\pi$
$200$	0	0
$201$	0.766338	0.0540533
$202$	0	0
$203$	−1.11173	−0.0780280
$204$	0	0
$205$	−1.44319	−0.100797
$206$	0	0
$207$	−14.7390	−1.02443
$208$	0	0
$209$	−2.61080	−0.180593
$210$	0	0
$211$	−2.20845	−0.152036	−0.0760179	−	0.997106i	$-0.524221\pi$
$211$	−2.20845	−0.152036	−0.0760179	+	0.997106i	$0.524221\pi$
$212$	0	0
$213$	−18.0840	−1.23910
$214$	0	0
$215$	−5.74287	−0.391661
$216$	0	0
$217$	−0.172495	−0.0117097
$218$	0	0
$219$	−0.997133	−0.0673800
$220$	0	0
$221$	−15.5035	−1.04288
$222$	0	0
$223$	−6.51747	−0.436442	−0.218221	−	0.975899i	$-0.570025\pi$
$223$	−6.51747	−0.436442	−0.218221	+	0.975899i	$0.570025\pi$
$224$	0	0
$225$	3.32830	0.221887
$226$	0	0
$227$	−6.61265	−0.438897	−0.219449	−	0.975624i	$-0.570426\pi$
$227$	−6.61265	−0.438897	−0.219449	+	0.975624i	$0.570426\pi$
$228$	0	0
$229$	−12.7722	−0.844010	−0.422005	−	0.906593i	$-0.638674\pi$
$229$	−12.7722	−0.844010	−0.422005	+	0.906593i	$0.638674\pi$
$230$	0	0
$231$	−1.24755	−0.0820829
$232$	0	0
$233$	−4.04152	−0.264769	−0.132384	−	0.991198i	$-0.542263\pi$
$233$	−4.04152	−0.264769	−0.132384	+	0.991198i	$0.542263\pi$
$234$	0	0
$235$	0.925327	0.0603617
$236$	0	0
$237$	13.8723	0.901101
$238$	0	0
$239$	−7.41028	−0.479331	−0.239666	−	0.970855i	$-0.577038\pi$
$239$	−7.41028	−0.479331	−0.239666	+	0.970855i	$0.577038\pi$
$240$	0	0
$241$	−15.8819	−1.02304	−0.511521	−	0.859271i	$-0.670918\pi$
$241$	−15.8819	−1.02304	−0.511521	+	0.859271i	$0.670918\pi$
$242$	0	0
$243$	−22.3694	−1.43500
$244$	0	0
$245$	6.98536	0.446279
$246$	0	0
$247$	2.29426	0.145980
$248$	0	0
$249$	−23.9914	−1.52039
$250$	0	0
$251$	−3.79442	−0.239502	−0.119751	−	0.992804i	$-0.538210\pi$
$251$	−3.79442	−0.239502	−0.119751	+	0.992804i	$0.538210\pi$
$252$	0	0
$253$	18.1535	1.14130
$254$	0	0
$255$	10.8265	0.677983
$256$	0	0
$257$	0.670538	0.0418270	0.0209135	−	0.999781i	$-0.493343\pi$
$257$	0.670538	0.0418270	0.0209135	+	0.999781i	$0.493343\pi$
$258$	0	0
$259$	0.906501	0.0563272
$260$	0	0
$261$	−30.5860	−1.89323
$262$	0	0
$263$	−23.8994	−1.47370	−0.736851	−	0.676055i	$-0.763688\pi$
$263$	−23.8994	−1.47370	−0.736851	+	0.676055i	$0.763688\pi$
$264$	0	0
$265$	9.31866	0.572441
$266$	0	0
$267$	1.10834	0.0678292
$268$	0	0
$269$	13.3917	0.816507	0.408254	−	0.912869i	$-0.366138\pi$
$269$	13.3917	0.816507	0.408254	+	0.912869i	$0.366138\pi$
$270$	0	0
$271$	2.30006	0.139719	0.0698595	−	0.997557i	$-0.477745\pi$
$271$	2.30006	0.139719	0.0698595	+	0.997557i	$0.477745\pi$
$272$	0	0
$273$	1.09630	0.0663508
$274$	0	0
$275$	−4.09936	−0.247200
$276$	0	0
$277$	9.68844	0.582122	0.291061	−	0.956705i	$-0.405992\pi$
$277$	9.68844	0.582122	0.291061	+	0.956705i	$0.405992\pi$
$278$	0	0
$279$	−4.74569	−0.284117
$280$	0	0
$281$	3.95303	0.235818	0.117909	−	0.993024i	$-0.462381\pi$
$281$	3.95303	0.235818	0.117909	+	0.993024i	$0.462381\pi$
$282$	0	0
$283$	1.28379	0.0763134	0.0381567	−	0.999272i	$-0.487851\pi$
$283$	1.28379	0.0763134	0.0381567	+	0.999272i	$0.487851\pi$
$284$	0	0
$285$	−1.60215	−0.0949029
$286$	0	0
$287$	0.174592	0.0103058
$288$	0	0
$289$	1.52212	0.0895362
$290$	0	0
$291$	1.65751	0.0971647
$292$	0	0
$293$	15.2621	0.891619	0.445809	−	0.895128i	$-0.352916\pi$
$293$	15.2621	0.891619	0.445809	+	0.895128i	$0.352916\pi$
$294$	0	0
$295$	−0.847329	−0.0493334
$296$	0	0
$297$	−3.38560	−0.196453
$298$	0	0
$299$	−15.9525	−0.922558
$300$	0	0
$301$	0.694750	0.0400447
$302$	0	0
$303$	21.3067	1.22404
$304$	0	0
$305$	8.83873	0.506104
$306$	0	0
$307$	13.2319	0.755183	0.377592	−	0.925972i	$-0.376752\pi$
$307$	13.2319	0.755183	0.377592	+	0.925972i	$0.376752\pi$
$308$	0	0
$309$	3.53419	0.201053
$310$	0	0
$311$	−5.81027	−0.329470	−0.164735	−	0.986338i	$-0.552677\pi$
$311$	−5.81027	−0.329470	−0.164735	+	0.986338i	$0.552677\pi$
$312$	0	0
$313$	−2.80645	−0.158630	−0.0793149	−	0.996850i	$-0.525273\pi$
$313$	−2.80645	−0.158630	−0.0793149	+	0.996850i	$0.525273\pi$
$314$	0	0
$315$	−0.402645	−0.0226865
$316$	0	0
$317$	29.7962	1.67352	0.836760	−	0.547570i	$-0.184447\pi$
$317$	29.7962	1.67352	0.836760	+	0.547570i	$0.184447\pi$
$318$	0	0
$319$	37.6717	2.10921
$320$	0	0
$321$	−0.00847081	−0.000472794 0
$322$	0	0
$323$	−2.74097	−0.152511
$324$	0	0
$325$	3.60234	0.199822
$326$	0	0
$327$	−13.2589	−0.733216
$328$	0	0
$329$	−0.111942	−0.00617158
$330$	0	0
$331$	2.64819	0.145558	0.0727788	−	0.997348i	$-0.476813\pi$
$331$	2.64819	0.145558	0.0727788	+	0.997348i	$0.476813\pi$
$332$	0	0
$333$	24.9398	1.36669
$334$	0	0
$335$	−0.304633	−0.0166439
$336$	0	0
$337$	−32.4167	−1.76585	−0.882925	−	0.469513i	$-0.844429\pi$
$337$	−32.4167	−1.76585	−0.882925	+	0.469513i	$0.844429\pi$
$338$	0	0
$339$	−34.7303	−1.88629
$340$	0	0
$341$	5.84510	0.316530
$342$	0	0
$343$	−1.69189	−0.0913537
$344$	0	0
$345$	11.1401	0.599762
$346$	0	0
$347$	−0.270962	−0.0145460	−0.00727300	−	0.999974i	$-0.502315\pi$
$347$	−0.270962	−0.0145460	−0.00727300	+	0.999974i	$0.502315\pi$
$348$	0	0
$349$	−23.6114	−1.26389	−0.631946	−	0.775013i	$-0.717743\pi$
$349$	−23.6114	−1.26389	−0.631946	+	0.775013i	$0.717743\pi$
$350$	0	0
$351$	2.97512	0.158800
$352$	0	0
$353$	8.96218	0.477009	0.238504	−	0.971141i	$-0.423343\pi$
$353$	8.96218	0.477009	0.238504	+	0.971141i	$0.423343\pi$
$354$	0	0
$355$	7.18871	0.381537
$356$	0	0
$357$	−1.30975	−0.0693193
$358$	0	0
$359$	16.4328	0.867289	0.433644	−	0.901084i	$-0.357227\pi$
$359$	16.4328	0.867289	0.433644	+	0.901084i	$0.357227\pi$
$360$	0	0
$361$	−18.5944	−0.978652
$362$	0	0
$363$	14.6024	0.766428
$364$	0	0
$365$	0.396378	0.0207474
$366$	0	0
$367$	19.0244	0.993063	0.496532	−	0.868019i	$-0.334607\pi$
$367$	19.0244	0.993063	0.496532	+	0.868019i	$0.334607\pi$
$368$	0	0
$369$	4.80339	0.250054
$370$	0	0
$371$	−1.12733	−0.0585283
$372$	0	0
$373$	−14.5752	−0.754674	−0.377337	−	0.926076i	$-0.623160\pi$
$373$	−14.5752	−0.754674	−0.377337	+	0.926076i	$0.623160\pi$
$374$	0	0
$375$	−2.51561	−0.129906
$376$	0	0
$377$	−33.1043	−1.70496
$378$	0	0
$379$	0.991197	0.0509144	0.0254572	−	0.999676i	$-0.491896\pi$
$379$	0.991197	0.0509144	0.0254572	+	0.999676i	$0.491896\pi$
$380$	0	0
$381$	22.2495	1.13988
$382$	0	0
$383$	14.2715	0.729240	0.364620	−	0.931156i	$-0.381199\pi$
$383$	14.2715	0.729240	0.364620	+	0.931156i	$0.381199\pi$
$384$	0	0
$385$	0.495924	0.0252746
$386$	0	0
$387$	19.1140	0.971621
$388$	0	0
$389$	−2.88429	−0.146239	−0.0731196	−	0.997323i	$-0.523295\pi$
$389$	−2.88429	−0.146239	−0.0731196	+	0.997323i	$0.523295\pi$
$390$	0	0
$391$	19.0585	0.963832
$392$	0	0
$393$	−37.4127	−1.88722
$394$	0	0
$395$	−5.51447	−0.277463
$396$	0	0
$397$	−16.6276	−0.834515	−0.417258	−	0.908788i	$-0.637009\pi$
$397$	−16.6276	−0.834515	−0.417258	+	0.908788i	$0.637009\pi$
$398$	0	0
$399$	0.193821	0.00970320
$400$	0	0
$401$	−14.8206	−0.740104	−0.370052	−	0.929011i	$-0.620660\pi$
$401$	−14.8206	−0.740104	−0.370052	+	0.929011i	$0.620660\pi$
$402$	0	0
$403$	−5.13643	−0.255864
$404$	0	0
$405$	7.90730	0.392917
$406$	0	0
$407$	−30.7174	−1.52261
$408$	0	0
$409$	−17.4854	−0.864598	−0.432299	−	0.901730i	$-0.642297\pi$
$409$	−17.4854	−0.864598	−0.432299	+	0.901730i	$0.642297\pi$
$410$	0	0
$411$	5.10988	0.252052
$412$	0	0
$413$	0.102506	0.00504401
$414$	0	0
$415$	9.53699	0.468152
$416$	0	0
$417$	−9.02717	−0.442063
$418$	0	0
$419$	11.3225	0.553143	0.276571	−	0.960993i	$-0.410802\pi$
$419$	11.3225	0.553143	0.276571	+	0.960993i	$0.410802\pi$
$420$	0	0
$421$	8.94405	0.435906	0.217953	−	0.975959i	$-0.430062\pi$
$421$	8.94405	0.435906	0.217953	+	0.975959i	$0.430062\pi$
$422$	0	0
$423$	−3.07977	−0.149744
$424$	0	0
$425$	−4.30373	−0.208762
$426$	0	0
$427$	−1.06927	−0.0517458
$428$	0	0
$429$	−37.1487	−1.79356
$430$	0	0
$431$	27.1240	1.30652	0.653259	−	0.757135i	$-0.273401\pi$
$431$	27.1240	1.30652	0.653259	+	0.757135i	$0.273401\pi$
$432$	0	0
$433$	7.89203	0.379267	0.189633	−	0.981855i	$-0.439270\pi$
$433$	7.89203	0.379267	0.189633	+	0.981855i	$0.439270\pi$
$434$	0	0
$435$	23.1176	1.10841
$436$	0	0
$437$	−2.82035	−0.134916
$438$	0	0
$439$	−24.5124	−1.16991	−0.584956	−	0.811065i	$-0.698888\pi$
$439$	−24.5124	−1.16991	−0.584956	+	0.811065i	$0.698888\pi$
$440$	0	0
$441$	−23.2494	−1.10712
$442$	0	0
$443$	5.91264	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$443$	5.91264	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$444$	0	0
$445$	−0.440584	−0.0208857
$446$	0	0
$447$	−11.9305	−0.564294
$448$	0	0
$449$	−3.69181	−0.174227	−0.0871136	−	0.996198i	$-0.527764\pi$
$449$	−3.69181	−0.174227	−0.0871136	+	0.996198i	$0.527764\pi$
$450$	0	0
$451$	−5.91616	−0.278581
$452$	0	0
$453$	2.51561	0.118194
$454$	0	0
$455$	−0.435797	−0.0204305
$456$	0	0
$457$	5.67253	0.265350	0.132675	−	0.991160i	$-0.457643\pi$
$457$	5.67253	0.265350	0.132675	+	0.991160i	$0.457643\pi$
$458$	0	0
$459$	−3.55440	−0.165905
$460$	0	0
$461$	−0.711224	−0.0331250	−0.0165625	−	0.999863i	$-0.505272\pi$
$461$	−0.711224	−0.0331250	−0.0165625	+	0.999863i	$0.505272\pi$
$462$	0	0
$463$	−18.1087	−0.841583	−0.420791	−	0.907157i	$-0.638248\pi$
$463$	−18.1087	−0.841583	−0.420791	+	0.907157i	$0.638248\pi$
$464$	0	0
$465$	3.58691	0.166339
$466$	0	0
$467$	3.19225	0.147720	0.0738599	−	0.997269i	$-0.476468\pi$
$467$	3.19225	0.147720	0.0738599	+	0.997269i	$0.476468\pi$
$468$	0	0
$469$	0.0368533	0.00170173
$470$	0	0
$471$	43.8335	2.01974
$472$	0	0
$473$	−23.5421	−1.08247
$474$	0	0
$475$	0.636881	0.0292221
$476$	0	0
$477$	−31.0154	−1.42010
$478$	0	0
$479$	−10.6250	−0.485467	−0.242733	−	0.970093i	$-0.578044\pi$
$479$	−10.6250	−0.485467	−0.242733	+	0.970093i	$0.578044\pi$
$480$	0	0
$481$	26.9932	1.23078
$482$	0	0
$483$	−1.34768	−0.0613216
$484$	0	0
$485$	−0.658888	−0.0299185
$486$	0	0
$487$	−23.9411	−1.08487	−0.542436	−	0.840097i	$-0.682498\pi$
$487$	−23.9411	−1.08487	−0.542436	+	0.840097i	$0.682498\pi$
$488$	0	0
$489$	−36.6414	−1.65698
$490$	0	0
$491$	9.03622	0.407799	0.203899	−	0.978992i	$-0.434638\pi$
$491$	9.03622	0.407799	0.203899	+	0.978992i	$0.434638\pi$
$492$	0	0
$493$	39.5498	1.78124
$494$	0	0
$495$	13.6439	0.613248
$496$	0	0
$497$	−0.869661	−0.0390096
$498$	0	0
$499$	16.1135	0.721340	0.360670	−	0.932694i	$-0.382548\pi$
$499$	16.1135	0.721340	0.360670	+	0.932694i	$0.382548\pi$
$500$	0	0
$501$	31.4067	1.40315
$502$	0	0
$503$	42.6168	1.90019	0.950094	−	0.311965i	$-0.100987\pi$
$503$	42.6168	1.90019	0.950094	+	0.311965i	$0.100987\pi$
$504$	0	0
$505$	−8.46980	−0.376901
$506$	0	0
$507$	−0.0582175	−0.00258553
$508$	0	0
$509$	7.46980	0.331093	0.165547	−	0.986202i	$-0.447061\pi$
$509$	7.46980	0.331093	0.165547	+	0.986202i	$0.447061\pi$
$510$	0	0
$511$	−0.0479522	−0.00212128
$512$	0	0
$513$	0.525992	0.0232231
$514$	0	0
$515$	−1.40490	−0.0619075
$516$	0	0
$517$	3.79325	0.166827
$518$	0	0
$519$	−7.94494	−0.348744
$520$	0	0
$521$	16.1225	0.706339	0.353170	−	0.935559i	$-0.385104\pi$
$521$	16.1225	0.706339	0.353170	+	0.935559i	$0.385104\pi$
$522$	0	0
$523$	−21.4075	−0.936084	−0.468042	−	0.883706i	$-0.655040\pi$
$523$	−21.4075	−0.936084	−0.468042	+	0.883706i	$0.655040\pi$
$524$	0	0
$525$	0.304329	0.0132820
$526$	0	0
$527$	6.13651	0.267311
$528$	0	0
$529$	−3.38949	−0.147369
$530$	0	0
$531$	2.82017	0.122385
$532$	0	0
$533$	5.19887	0.225188
$534$	0	0
$535$	0.00336729	0.000145581 0
$536$	0	0
$537$	−11.3499	−0.489782
$538$	0	0
$539$	28.6355	1.23342
$540$	0	0
$541$	−36.3312	−1.56200	−0.781000	−	0.624531i	$-0.785290\pi$
$541$	−36.3312	−1.56200	−0.781000	+	0.624531i	$0.785290\pi$
$542$	0	0
$543$	−22.9536	−0.985035
$544$	0	0
$545$	5.27063	0.225769
$546$	0	0
$547$	−18.2833	−0.781738	−0.390869	−	0.920446i	$-0.627826\pi$
$547$	−18.2833	−0.781738	−0.390869	+	0.920446i	$0.627826\pi$
$548$	0	0
$549$	−29.4180	−1.25553
$550$	0	0
$551$	−5.85272	−0.249334
$552$	0	0
$553$	0.667119	0.0283688
$554$	0	0
$555$	−18.8501	−0.800141
$556$	0	0
$557$	41.2598	1.74824	0.874118	−	0.485714i	$-0.161440\pi$
$557$	41.2598	1.74824	0.874118	+	0.485714i	$0.161440\pi$
$558$	0	0
$559$	20.6878	0.875000
$560$	0	0
$561$	44.3818	1.87380
$562$	0	0
$563$	34.7003	1.46244	0.731221	−	0.682140i	$-0.238951\pi$
$563$	34.7003	1.46244	0.731221	+	0.682140i	$0.238951\pi$
$564$	0	0
$565$	13.8059	0.580819
$566$	0	0
$567$	−0.956594	−0.0401732
$568$	0	0
$569$	−26.1792	−1.09749	−0.548745	−	0.835990i	$-0.684894\pi$
$569$	−26.1792	−1.09749	−0.548745	+	0.835990i	$0.684894\pi$
$570$	0	0
$571$	35.5647	1.48834	0.744169	−	0.667991i	$-0.232846\pi$
$571$	35.5647	1.48834	0.744169	+	0.667991i	$0.232846\pi$
$572$	0	0
$573$	36.4938	1.52455
$574$	0	0
$575$	−4.42838	−0.184676
$576$	0	0
$577$	5.65334	0.235352	0.117676	−	0.993052i	$-0.462456\pi$
$577$	5.65334	0.235352	0.117676	+	0.993052i	$0.462456\pi$
$578$	0	0
$579$	55.4081	2.30268
$580$	0	0
$581$	−1.15375	−0.0478655
$582$	0	0
$583$	38.2005	1.58210
$584$	0	0
$585$	−11.9897	−0.495713
$586$	0	0
$587$	−5.85673	−0.241733	−0.120867	−	0.992669i	$-0.538567\pi$
$587$	−5.85673	−0.241733	−0.120867	+	0.992669i	$0.538567\pi$
$588$	0	0
$589$	−0.908103	−0.0374177
$590$	0	0
$591$	26.3605	1.08433
$592$	0	0
$593$	−18.8028	−0.772139	−0.386070	−	0.922470i	$-0.626168\pi$
$593$	−18.8028	−0.772139	−0.386070	+	0.922470i	$0.626168\pi$
$594$	0	0
$595$	0.520648	0.0213445
$596$	0	0
$597$	20.6023	0.843198
$598$	0	0
$599$	23.7978	0.972350	0.486175	−	0.873862i	$-0.338392\pi$
$599$	23.7978	0.972350	0.486175	+	0.873862i	$0.338392\pi$
$600$	0	0
$601$	20.0169	0.816506	0.408253	−	0.912869i	$-0.366138\pi$
$601$	20.0169	0.816506	0.408253	+	0.912869i	$0.366138\pi$
$602$	0	0
$603$	1.01391	0.0412896
$604$	0	0
$605$	−5.80472	−0.235995
$606$	0	0
$607$	24.9059	1.01090	0.505449	−	0.862856i	$-0.331327\pi$
$607$	24.9059	1.01090	0.505449	+	0.862856i	$0.331327\pi$
$608$	0	0
$609$	−2.79668	−0.113327
$610$	0	0
$611$	−3.33334	−0.134853
$612$	0	0
$613$	21.3670	0.863006	0.431503	−	0.902112i	$-0.357983\pi$
$613$	21.3670	0.863006	0.431503	+	0.902112i	$0.357983\pi$
$614$	0	0
$615$	−3.63051	−0.146396
$616$	0	0
$617$	−35.1179	−1.41379	−0.706897	−	0.707316i	$-0.749906\pi$
$617$	−35.1179	−1.41379	−0.706897	+	0.707316i	$0.749906\pi$
$618$	0	0
$619$	10.3946	0.417794	0.208897	−	0.977938i	$-0.433013\pi$
$619$	10.3946	0.417794	0.208897	+	0.977938i	$0.433013\pi$
$620$	0	0
$621$	−3.65734	−0.146764
$622$	0	0
$623$	0.0533001	0.00213542
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−6.56777	−0.262291
$628$	0	0
$629$	−32.2489	−1.28585
$630$	0	0
$631$	25.9852	1.03445	0.517226	−	0.855849i	$-0.326965\pi$
$631$	25.9852	1.03445	0.517226	+	0.855849i	$0.326965\pi$
$632$	0	0
$633$	−5.55560	−0.220815
$634$	0	0
$635$	−8.84458	−0.350987
$636$	0	0
$637$	−25.1637	−0.997021
$638$	0	0
$639$	−23.9262	−0.946506
$640$	0	0
$641$	8.17697	0.322971	0.161485	−	0.986875i	$-0.448372\pi$
$641$	8.17697	0.322971	0.161485	+	0.986875i	$0.448372\pi$
$642$	0	0
$643$	34.6237	1.36543	0.682713	−	0.730687i	$-0.260800\pi$
$643$	34.6237	1.36543	0.682713	+	0.730687i	$0.260800\pi$
$644$	0	0
$645$	−14.4468	−0.568844
$646$	0	0
$647$	47.9076	1.88344	0.941722	−	0.336393i	$-0.109207\pi$
$647$	47.9076	1.88344	0.941722	+	0.336393i	$0.109207\pi$
$648$	0	0
$649$	−3.47350	−0.136347
$650$	0	0
$651$	−0.433930	−0.0170070
$652$	0	0
$653$	−13.4293	−0.525529	−0.262765	−	0.964860i	$-0.584634\pi$
$653$	−13.4293	−0.525529	−0.262765	+	0.964860i	$0.584634\pi$
$654$	0	0
$655$	14.8722	0.581105
$656$	0	0
$657$	−1.31927	−0.0514695
$658$	0	0
$659$	45.3073	1.76492	0.882462	−	0.470384i	$-0.155885\pi$
$659$	45.3073	1.76492	0.882462	+	0.470384i	$0.155885\pi$
$660$	0	0
$661$	−8.57109	−0.333377	−0.166688	−	0.986010i	$-0.553307\pi$
$661$	−8.57109	−0.333377	−0.166688	+	0.986010i	$0.553307\pi$
$662$	0	0
$663$	−39.0008	−1.51467
$664$	0	0
$665$	−0.0770473	−0.00298777
$666$	0	0
$667$	40.6953	1.57573
$668$	0	0
$669$	−16.3954	−0.633884
$670$	0	0
$671$	36.2331	1.39876
$672$	0	0
$673$	3.70972	0.142999	0.0714996	−	0.997441i	$-0.477222\pi$
$673$	3.70972	0.142999	0.0714996	+	0.997441i	$0.477222\pi$
$674$	0	0
$675$	0.825887	0.0317884
$676$	0	0
$677$	18.8045	0.722715	0.361357	−	0.932427i	$-0.382313\pi$
$677$	18.8045	0.722715	0.361357	+	0.932427i	$0.382313\pi$
$678$	0	0
$679$	0.0797096	0.00305897
$680$	0	0
$681$	−16.6349	−0.637449
$682$	0	0
$683$	3.38397	0.129484	0.0647420	−	0.997902i	$-0.479378\pi$
$683$	3.38397	0.129484	0.0647420	+	0.997902i	$0.479378\pi$
$684$	0	0
$685$	−2.03127	−0.0776107
$686$	0	0
$687$	−32.1299	−1.22583
$688$	0	0
$689$	−33.5690	−1.27888
$690$	0	0
$691$	−39.1019	−1.48751	−0.743753	−	0.668455i	$-0.766956\pi$
$691$	−39.1019	−1.48751	−0.743753	+	0.668455i	$0.766956\pi$
$692$	0	0
$693$	−1.65058	−0.0627005
$694$	0	0
$695$	3.58846	0.136118
$696$	0	0
$697$	−6.21112	−0.235263
$698$	0	0
$699$	−10.1669	−0.384547
$700$	0	0
$701$	28.1289	1.06241	0.531206	−	0.847243i	$-0.321739\pi$
$701$	28.1289	1.06241	0.531206	+	0.847243i	$0.321739\pi$
$702$	0	0
$703$	4.77230	0.179991
$704$	0	0
$705$	2.32776	0.0876687
$706$	0	0
$707$	1.02464	0.0385356
$708$	0	0
$709$	−50.3732	−1.89181	−0.945903	−	0.324451i	$-0.894821\pi$
$709$	−50.3732	−1.89181	−0.945903	+	0.324451i	$0.894821\pi$
$710$	0	0
$711$	18.3538	0.688323
$712$	0	0
$713$	6.31424	0.236470
$714$	0	0
$715$	14.7673	0.552265
$716$	0	0
$717$	−18.6414	−0.696176
$718$	0	0
$719$	−15.1127	−0.563610	−0.281805	−	0.959472i	$-0.590933\pi$
$719$	−15.1127	−0.563610	−0.281805	+	0.959472i	$0.590933\pi$
$720$	0	0
$721$	0.169960	0.00632963
$722$	0	0
$723$	−39.9526	−1.48585
$724$	0	0
$725$	−9.18966	−0.341295
$726$	0	0
$727$	5.30945	0.196917	0.0984583	−	0.995141i	$-0.468609\pi$
$727$	5.30945	0.196917	0.0984583	+	0.995141i	$0.468609\pi$
$728$	0	0
$729$	−32.5507	−1.20558
$730$	0	0
$731$	−24.7158	−0.914147
$732$	0	0
$733$	1.62995	0.0602037	0.0301018	−	0.999547i	$-0.490417\pi$
$733$	1.62995	0.0602037	0.0301018	+	0.999547i	$0.490417\pi$
$734$	0	0
$735$	17.5725	0.648170
$736$	0	0
$737$	−1.24880	−0.0460001
$738$	0	0
$739$	38.2300	1.40631	0.703157	−	0.711034i	$-0.251773\pi$
$739$	38.2300	1.40631	0.703157	+	0.711034i	$0.251773\pi$
$740$	0	0
$741$	5.77148	0.212020
$742$	0	0
$743$	−10.6144	−0.389405	−0.194702	−	0.980862i	$-0.562374\pi$
$743$	−10.6144	−0.389405	−0.194702	+	0.980862i	$0.562374\pi$
$744$	0	0
$745$	4.74259	0.173755
$746$	0	0
$747$	−31.7420	−1.16138
$748$	0	0
$749$	−0.000407362 0	−1.48847e−5 0
$750$	0	0
$751$	−2.66539	−0.0972616	−0.0486308	−	0.998817i	$-0.515486\pi$
$751$	−2.66539	−0.0972616	−0.0486308	+	0.998817i	$0.515486\pi$
$752$	0	0
$753$	−9.54529	−0.347850
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	−14.9741	−0.544243	−0.272121	−	0.962263i	$-0.587725\pi$
$757$	−14.9741	−0.544243	−0.272121	+	0.962263i	$0.587725\pi$
$758$	0	0
$759$	45.6671	1.65761
$760$	0	0
$761$	4.66445	0.169086	0.0845431	−	0.996420i	$-0.473057\pi$
$761$	4.66445	0.169086	0.0845431	+	0.996420i	$0.473057\pi$
$762$	0	0
$763$	−0.637619	−0.0230834
$764$	0	0
$765$	14.3241	0.517890
$766$	0	0
$767$	3.05237	0.110215
$768$	0	0
$769$	−34.2946	−1.23670	−0.618348	−	0.785905i	$-0.712198\pi$
$769$	−34.2946	−1.23670	−0.618348	+	0.785905i	$0.712198\pi$
$770$	0	0
$771$	1.68681	0.0607491
$772$	0	0
$773$	16.4163	0.590452	0.295226	−	0.955428i	$-0.404605\pi$
$773$	16.4163	0.590452	0.295226	+	0.955428i	$0.404605\pi$
$774$	0	0
$775$	−1.42586	−0.0512184
$776$	0	0
$777$	2.28041	0.0818091
$778$	0	0
$779$	0.919142	0.0329317
$780$	0	0
$781$	29.4691	1.05449
$782$	0	0
$783$	−7.58962	−0.271231
$784$	0	0
$785$	−17.4246	−0.621910
$786$	0	0
$787$	−48.6856	−1.73545	−0.867726	−	0.497042i	$-0.834419\pi$
$787$	−48.6856	−1.73545	−0.867726	+	0.497042i	$0.834419\pi$
$788$	0	0
$789$	−60.1217	−2.14039
$790$	0	0
$791$	−1.67018	−0.0593849
$792$	0	0
$793$	−31.8401	−1.13068
$794$	0	0
$795$	23.4421	0.831407
$796$	0	0
$797$	10.3942	0.368183	0.184092	−	0.982909i	$-0.441066\pi$
$797$	10.3942	0.368183	0.184092	+	0.982909i	$0.441066\pi$
$798$	0	0
$799$	3.98236	0.140886
$800$	0	0
$801$	1.46640	0.0518126
$802$	0	0
$803$	1.62489	0.0573412
$804$	0	0
$805$	0.535727	0.0188819
$806$	0	0
$807$	33.6884	1.18589
$808$	0	0
$809$	−15.2145	−0.534915	−0.267457	−	0.963570i	$-0.586183\pi$
$809$	−15.2145	−0.534915	−0.267457	+	0.963570i	$0.586183\pi$
$810$	0	0
$811$	−42.6762	−1.49856	−0.749281	−	0.662252i	$-0.769601\pi$
$811$	−42.6762	−1.49856	−0.749281	+	0.662252i	$0.769601\pi$
$812$	0	0
$813$	5.78607	0.202926
$814$	0	0
$815$	14.5656	0.510211
$816$	0	0
$817$	3.65753	0.127961
$818$	0	0
$819$	1.45046	0.0506833
$820$	0	0
$821$	−0.279852	−0.00976690	−0.00488345	−	0.999988i	$-0.501554\pi$
$821$	−0.279852	−0.00976690	−0.00488345	+	0.999988i	$0.501554\pi$
$822$	0	0
$823$	−9.55514	−0.333071	−0.166536	−	0.986035i	$-0.553258\pi$
$823$	−9.55514	−0.333071	−0.166536	+	0.986035i	$0.553258\pi$
$824$	0	0
$825$	−10.3124	−0.359031
$826$	0	0
$827$	−24.1438	−0.839561	−0.419780	−	0.907626i	$-0.637893\pi$
$827$	−24.1438	−0.839561	−0.419780	+	0.907626i	$0.637893\pi$
$828$	0	0
$829$	−29.5515	−1.02637	−0.513183	−	0.858279i	$-0.671534\pi$
$829$	−29.5515	−1.02637	−0.513183	+	0.858279i	$0.671534\pi$
$830$	0	0
$831$	24.3724	0.845468
$832$	0	0
$833$	30.0631	1.04163
$834$	0	0
$835$	−12.4847	−0.432051
$836$	0	0
$837$	−1.17760	−0.0407037
$838$	0	0
$839$	−38.4070	−1.32596	−0.662978	−	0.748639i	$-0.730708\pi$
$839$	−38.4070	−1.32596	−0.662978	+	0.748639i	$0.730708\pi$
$840$	0	0
$841$	55.4499	1.91206
$842$	0	0
$843$	9.94430	0.342500
$844$	0	0
$845$	0.0231425	0.000796126 0
$846$	0	0
$847$	0.702232	0.0241290
$848$	0	0
$849$	3.22952	0.110837
$850$	0	0
$851$	−33.1828	−1.13749
$852$	0	0
$853$	−55.1455	−1.88815	−0.944073	−	0.329736i	$-0.893040\pi$
$853$	−55.1455	−1.88815	−0.944073	+	0.329736i	$0.893040\pi$
$854$	0	0
$855$	−2.11973	−0.0724934
$856$	0	0
$857$	−47.2442	−1.61383	−0.806916	−	0.590666i	$-0.798865\pi$
$857$	−47.2442	−1.61383	−0.806916	+	0.590666i	$0.798865\pi$
$858$	0	0
$859$	−34.0848	−1.16296	−0.581480	−	0.813561i	$-0.697526\pi$
$859$	−34.0848	−1.16296	−0.581480	+	0.813561i	$0.697526\pi$
$860$	0	0
$861$	0.439205	0.0149681
$862$	0	0
$863$	54.5112	1.85558	0.927792	−	0.373099i	$-0.121705\pi$
$863$	54.5112	1.85558	0.927792	+	0.373099i	$0.121705\pi$
$864$	0	0
$865$	3.15825	0.107384
$866$	0	0
$867$	3.82905	0.130041
$868$	0	0
$869$	−22.6058	−0.766849
$870$	0	0
$871$	1.09739	0.0371837
$872$	0	0
$873$	2.19298	0.0742211
$874$	0	0
$875$	−0.120976	−0.00408973
$876$	0	0
$877$	48.0161	1.62139	0.810695	−	0.585469i	$-0.199090\pi$
$877$	48.0161	1.62139	0.810695	+	0.585469i	$0.199090\pi$
$878$	0	0
$879$	38.3934	1.29498
$880$	0	0
$881$	27.0196	0.910312	0.455156	−	0.890412i	$-0.349583\pi$
$881$	27.0196	0.910312	0.455156	+	0.890412i	$0.349583\pi$
$882$	0	0
$883$	−13.0017	−0.437541	−0.218770	−	0.975776i	$-0.570205\pi$
$883$	−13.0017	−0.437541	−0.218770	+	0.975776i	$0.570205\pi$
$884$	0	0
$885$	−2.13155	−0.0716513
$886$	0	0
$887$	2.60376	0.0874258	0.0437129	−	0.999044i	$-0.486081\pi$
$887$	2.60376	0.0874258	0.0437129	+	0.999044i	$0.486081\pi$
$888$	0	0
$889$	1.06998	0.0358861
$890$	0	0
$891$	32.4148	1.08594
$892$	0	0
$893$	−0.589323	−0.0197210
$894$	0	0
$895$	4.51177	0.150812
$896$	0	0
$897$	−40.1303	−1.33991
$898$	0	0
$899$	13.1032	0.437015
$900$	0	0
$901$	40.1050	1.33609
$902$	0	0
$903$	1.74772	0.0581605
$904$	0	0
$905$	9.12447	0.303308
$906$	0	0
$907$	13.8124	0.458634	0.229317	−	0.973352i	$-0.426351\pi$
$907$	13.8124	0.458634	0.229317	+	0.973352i	$0.426351\pi$
$908$	0	0
$909$	28.1901	0.935005
$910$	0	0
$911$	18.8681	0.625129	0.312564	−	0.949897i	$-0.398812\pi$
$911$	18.8681	0.625129	0.312564	+	0.949897i	$0.398812\pi$
$912$	0	0
$913$	39.0955	1.29387
$914$	0	0
$915$	22.2348	0.735060
$916$	0	0
$917$	−1.79918	−0.0594141
$918$	0	0
$919$	33.9687	1.12052	0.560261	−	0.828316i	$-0.310701\pi$
$919$	33.9687	1.12052	0.560261	+	0.828316i	$0.310701\pi$
$920$	0	0
$921$	33.2863	1.09682
$922$	0	0
$923$	−25.8962	−0.852383
$924$	0	0
$925$	7.49323	0.246376
$926$	0	0
$927$	4.67595	0.153578
$928$	0	0
$929$	−16.3605	−0.536772	−0.268386	−	0.963311i	$-0.586490\pi$
$929$	−16.3605	−0.536772	−0.268386	+	0.963311i	$0.586490\pi$
$930$	0	0
$931$	−4.44885	−0.145805
$932$	0	0
$933$	−14.6164	−0.478519
$934$	0	0
$935$	−17.6425	−0.576973
$936$	0	0
$937$	10.9262	0.356943	0.178471	−	0.983945i	$-0.442885\pi$
$937$	10.9262	0.356943	0.178471	+	0.983945i	$0.442885\pi$
$938$	0	0
$939$	−7.05993	−0.230392
$940$	0	0
$941$	10.2262	0.333365	0.166683	−	0.986011i	$-0.446694\pi$
$941$	10.2262	0.333365	0.166683	+	0.986011i	$0.446694\pi$
$942$	0	0
$943$	−6.39100	−0.208120
$944$	0	0
$945$	−0.0999125	−0.00325015
$946$	0	0
$947$	12.5167	0.406739	0.203370	−	0.979102i	$-0.434811\pi$
$947$	12.5167	0.406739	0.203370	+	0.979102i	$0.434811\pi$
$948$	0	0
$949$	−1.42789	−0.0463512
$950$	0	0
$951$	74.9556	2.43060
$952$	0	0
$953$	17.9462	0.581335	0.290667	−	0.956824i	$-0.406123\pi$
$953$	17.9462	0.581335	0.290667	+	0.956824i	$0.406123\pi$
$954$	0	0
$955$	−14.5069	−0.469433
$956$	0	0
$957$	94.7674	3.06339
$958$	0	0
$959$	0.245734	0.00793518
$960$	0	0
$961$	−28.9669	−0.934417
$962$	0	0
$963$	−0.0112074	−0.000361153 0
$964$	0	0
$965$	−22.0257	−0.709032
$966$	0	0
$967$	38.4828	1.23752	0.618762	−	0.785578i	$-0.287634\pi$
$967$	38.4828	1.23752	0.618762	+	0.785578i	$0.287634\pi$
$968$	0	0
$969$	−6.89521	−0.221506
$970$	0	0
$971$	9.40194	0.301723	0.150861	−	0.988555i	$-0.451795\pi$
$971$	9.40194	0.301723	0.150861	+	0.988555i	$0.451795\pi$
$972$	0	0
$973$	−0.434118	−0.0139172
$974$	0	0
$975$	9.06209	0.290219
$976$	0	0
$977$	−4.12695	−0.132033	−0.0660165	−	0.997819i	$-0.521029\pi$
$977$	−4.12695	−0.132033	−0.0660165	+	0.997819i	$0.521029\pi$
$978$	0	0
$979$	−1.80611	−0.0577235
$980$	0	0
$981$	−17.5423	−0.560081
$982$	0	0
$983$	−40.6878	−1.29774	−0.648870	−	0.760899i	$-0.724758\pi$
$983$	−40.6878	−1.29774	−0.648870	+	0.760899i	$0.724758\pi$
$984$	0	0
$985$	−10.4788	−0.333882
$986$	0	0
$987$	−0.281604	−0.00896354
$988$	0	0
$989$	−25.4316	−0.808678
$990$	0	0
$991$	11.7512	0.373290	0.186645	−	0.982427i	$-0.440239\pi$
$991$	11.7512	0.373290	0.186645	+	0.982427i	$0.440239\pi$
$992$	0	0
$993$	6.66181	0.211406
$994$	0	0
$995$	−8.18979	−0.259634
$996$	0	0
$997$	−45.1247	−1.42911	−0.714557	−	0.699577i	$-0.753372\pi$
$997$	−45.1247	−1.42911	−0.714557	+	0.699577i	$0.753372\pi$
$998$	0	0
$999$	6.18856	0.195797

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.n.1.12	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.n.1.12	✓	13	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q+2.51561 q^{3} -1.00000 q^{5} +0.120976 q^{7} +3.32830 q^{9} +O(q^{10})\)	\(q+2.51561 q^{3} -1.00000 q^{5} +0.120976 q^{7} +3.32830 q^{9} -4.09936 q^{11} +3.60234 q^{13} -2.51561 q^{15} -4.30373 q^{17} +0.636881 q^{19} +0.304329 q^{21} -4.42838 q^{23} +1.00000 q^{25} +0.825887 q^{27} -9.18966 q^{29} -1.42586 q^{31} -10.3124 q^{33} -0.120976 q^{35} +7.49323 q^{37} +9.06209 q^{39} +1.44319 q^{41} +5.74287 q^{43} -3.32830 q^{45} -0.925327 q^{47} -6.98536 q^{49} -10.8265 q^{51} -9.31866 q^{53} +4.09936 q^{55} +1.60215 q^{57} +0.847329 q^{59} -8.83873 q^{61} +0.402645 q^{63} -3.60234 q^{65} +0.304633 q^{67} -11.1401 q^{69} -7.18871 q^{71} -0.396378 q^{73} +2.51561 q^{75} -0.495924 q^{77} +5.51447 q^{79} -7.90730 q^{81} -9.53699 q^{83} +4.30373 q^{85} -23.1176 q^{87} +0.440584 q^{89} +0.435797 q^{91} -3.58691 q^{93} -0.636881 q^{95} +0.658888 q^{97} -13.6439 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9}+O(q^{10}) \) 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9	\( 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9} - 14 q^{11} + 5 q^{13} + 4 q^{15} - 8 q^{17} + 16 q^{19} - 5 q^{21} - 4 q^{23} + 13 q^{25} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 19 q^{33} + 6 q^{37} + 7 q^{39} - 18 q^{41} + 7 q^{43} - 5 q^{45} - 22 q^{47} - q^{49} + 12 q^{51} - 17 q^{53} + 14 q^{55} - 16 q^{57} - 6 q^{59} + 10 q^{61} - 5 q^{65} + 12 q^{67} + 13 q^{69} - 16 q^{71} - 24 q^{73} - 4 q^{75} - 11 q^{77} + 36 q^{79} - 19 q^{81} + q^{83} + 8 q^{85} - 8 q^{87} - 53 q^{89} + 23 q^{91} - 9 q^{93} - 16 q^{95} - 21 q^{97} - 9 q^{99}+O(q^{100}) \) 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9 - 14 * q^11 + 5 * q^13 + 4 * q^15 - 8 * q^17 + 16 * q^19 - 5 * q^21 - 4 * q^23 + 13 * q^25 + 2 * q^27 - 6 * q^29 + 11 * q^31 - 19 * q^33 + 6 * q^37 + 7 * q^39 - 18 * q^41 + 7 * q^43 - 5 * q^45 - 22 * q^47 - q^49 + 12 * q^51 - 17 * q^53 + 14 * q^55 - 16 * q^57 - 6 * q^59 + 10 * q^61 - 5 * q^65 + 12 * q^67 + 13 * q^69 - 16 * q^71 - 24 * q^73 - 4 * q^75 - 11 * q^77 + 36 * q^79 - 19 * q^81 + q^83 + 8 * q^85 - 8 * q^87 - 53 * q^89 + 23 * q^91 - 9 * q^93 - 16 * q^95 - 21 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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