LMFDB - Embedded newform 8044.2.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8044,2,Mod(1,8044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8044, 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("8044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8044 = 2^{2} \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8044.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:	$64.2316633859$
Analytic rank:	$1$
Dimension:	$80$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.15920 q^{3} -3.92220 q^{5} +2.93790 q^{7} +6.98053 q^{9} +O(q^{10})$	$q-3.15920 q^{3} -3.92220 q^{5} +2.93790 q^{7} +6.98053 q^{9} +4.02102 q^{11} +4.44954 q^{13} +12.3910 q^{15} -3.19063 q^{17} -0.637295 q^{19} -9.28139 q^{21} -8.89082 q^{23} +10.3836 q^{25} -12.5753 q^{27} +1.09511 q^{29} -0.879031 q^{31} -12.7032 q^{33} -11.5230 q^{35} +1.36117 q^{37} -14.0570 q^{39} -3.25173 q^{41} +0.622184 q^{43} -27.3790 q^{45} -0.945800 q^{47} +1.63123 q^{49} +10.0798 q^{51} -2.12301 q^{53} -15.7712 q^{55} +2.01334 q^{57} +1.80760 q^{59} +0.658742 q^{61} +20.5081 q^{63} -17.4520 q^{65} +14.2504 q^{67} +28.0878 q^{69} -7.23050 q^{71} -6.55655 q^{73} -32.8039 q^{75} +11.8133 q^{77} +5.06465 q^{79} +18.7862 q^{81} +0.308846 q^{83} +12.5143 q^{85} -3.45966 q^{87} -13.7678 q^{89} +13.0723 q^{91} +2.77703 q^{93} +2.49960 q^{95} -4.99052 q^{97} +28.0688 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * 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
$3$	−3.15920	−1.82396	−0.911982	−	0.410231i	$-0.865448\pi$
$3$	−3.15920	−1.82396	−0.911982	+	0.410231i	$0.865448\pi$
$4$	0	0
$5$	−3.92220	−1.75406	−0.877030	−	0.480436i	$-0.840479\pi$
$5$	−3.92220	−1.75406	−0.877030	+	0.480436i	$0.840479\pi$
$6$	0	0
$7$	2.93790	1.11042	0.555210	−	0.831710i	$-0.312638\pi$
$7$	2.93790	1.11042	0.555210	+	0.831710i	$0.312638\pi$
$8$	0	0
$9$	6.98053	2.32684
$10$	0	0
$11$	4.02102	1.21238	0.606191	−	0.795319i	$-0.292697\pi$
$11$	4.02102	1.21238	0.606191	+	0.795319i	$0.292697\pi$
$12$	0	0
$13$	4.44954	1.23408	0.617040	−	0.786932i	$-0.288332\pi$
$13$	4.44954	1.23408	0.617040	+	0.786932i	$0.288332\pi$
$14$	0	0
$15$	12.3910	3.19934
$16$	0	0
$17$	−3.19063	−0.773842	−0.386921	−	0.922113i	$-0.626461\pi$
$17$	−3.19063	−0.773842	−0.386921	+	0.922113i	$0.626461\pi$
$18$	0	0
$19$	−0.637295	−0.146206	−0.0731028	−	0.997324i	$-0.523290\pi$
$19$	−0.637295	−0.146206	−0.0731028	+	0.997324i	$0.523290\pi$
$20$	0	0
$21$	−9.28139	−2.02537
$22$	0	0
$23$	−8.89082	−1.85386	−0.926932	−	0.375230i	$-0.877564\pi$
$23$	−8.89082	−1.85386	−0.926932	+	0.375230i	$0.877564\pi$
$24$	0	0
$25$	10.3836	2.07673
$26$	0	0
$27$	−12.5753	−2.42011
$28$	0	0
$29$	1.09511	0.203356	0.101678	−	0.994817i	$-0.467579\pi$
$29$	1.09511	0.203356	0.101678	+	0.994817i	$0.467579\pi$
$30$	0	0
$31$	−0.879031	−0.157879	−0.0789393	−	0.996879i	$-0.525153\pi$
$31$	−0.879031	−0.157879	−0.0789393	+	0.996879i	$0.525153\pi$
$32$	0	0
$33$	−12.7032	−2.21134
$34$	0	0
$35$	−11.5230	−1.94774
$36$	0	0
$37$	1.36117	0.223775	0.111887	−	0.993721i	$-0.464310\pi$
$37$	1.36117	0.223775	0.111887	+	0.993721i	$0.464310\pi$
$38$	0	0
$39$	−14.0570	−2.25092
$40$	0	0
$41$	−3.25173	−0.507835	−0.253917	−	0.967226i	$-0.581719\pi$
$41$	−3.25173	−0.507835	−0.253917	+	0.967226i	$0.581719\pi$
$42$	0	0
$43$	0.622184	0.0948822	0.0474411	−	0.998874i	$-0.484893\pi$
$43$	0.622184	0.0948822	0.0474411	+	0.998874i	$0.484893\pi$
$44$	0	0
$45$	−27.3790	−4.08142
$46$	0	0
$47$	−0.945800	−0.137959	−0.0689796	−	0.997618i	$-0.521974\pi$
$47$	−0.945800	−0.137959	−0.0689796	+	0.997618i	$0.521974\pi$
$48$	0	0
$49$	1.63123	0.233033
$50$	0	0
$51$	10.0798	1.41146
$52$	0	0
$53$	−2.12301	−0.291618	−0.145809	−	0.989313i	$-0.546579\pi$
$53$	−2.12301	−0.291618	−0.145809	+	0.989313i	$0.546579\pi$
$54$	0	0
$55$	−15.7712	−2.12659
$56$	0	0
$57$	2.01334	0.266674
$58$	0	0
$59$	1.80760	0.235329	0.117665	−	0.993053i	$-0.462459\pi$
$59$	1.80760	0.235329	0.117665	+	0.993053i	$0.462459\pi$
$60$	0	0
$61$	0.658742	0.0843432	0.0421716	−	0.999110i	$-0.486572\pi$
$61$	0.658742	0.0843432	0.0421716	+	0.999110i	$0.486572\pi$
$62$	0	0
$63$	20.5081	2.58377
$64$	0	0
$65$	−17.4520	−2.16465
$66$	0	0
$67$	14.2504	1.74096	0.870480	−	0.492203i	$-0.163808\pi$
$67$	14.2504	1.74096	0.870480	+	0.492203i	$0.163808\pi$
$68$	0	0
$69$	28.0878	3.38138
$70$	0	0
$71$	−7.23050	−0.858103	−0.429052	−	0.903280i	$-0.641152\pi$
$71$	−7.23050	−0.858103	−0.429052	+	0.903280i	$0.641152\pi$
$72$	0	0
$73$	−6.55655	−0.767387	−0.383693	−	0.923461i	$-0.625348\pi$
$73$	−6.55655	−0.767387	−0.383693	+	0.923461i	$0.625348\pi$
$74$	0	0
$75$	−32.8039	−3.78787
$76$	0	0
$77$	11.8133	1.34625
$78$	0	0
$79$	5.06465	0.569817	0.284909	−	0.958555i	$-0.408037\pi$
$79$	5.06465	0.569817	0.284909	+	0.958555i	$0.408037\pi$
$80$	0	0
$81$	18.7862	2.08735
$82$	0	0
$83$	0.308846	0.0339003	0.0169501	−	0.999856i	$-0.494604\pi$
$83$	0.308846	0.0339003	0.0169501	+	0.999856i	$0.494604\pi$
$84$	0	0
$85$	12.5143	1.35737
$86$	0	0
$87$	−3.45966	−0.370914
$88$	0	0
$89$	−13.7678	−1.45938	−0.729692	−	0.683776i	$-0.760336\pi$
$89$	−13.7678	−1.45938	−0.729692	+	0.683776i	$0.760336\pi$
$90$	0	0
$91$	13.0723	1.37035
$92$	0	0
$93$	2.77703	0.287965
$94$	0	0
$95$	2.49960	0.256453
$96$	0	0
$97$	−4.99052	−0.506711	−0.253355	−	0.967373i	$-0.581534\pi$
$97$	−4.99052	−0.506711	−0.253355	+	0.967373i	$0.581534\pi$
$98$	0	0
$99$	28.0688	2.82102
$100$	0	0
$101$	1.41664	0.140961	0.0704807	−	0.997513i	$-0.477547\pi$
$101$	1.41664	0.140961	0.0704807	+	0.997513i	$0.477547\pi$
$102$	0	0
$103$	−6.26653	−0.617460	−0.308730	−	0.951150i	$-0.599904\pi$
$103$	−6.26653	−0.617460	−0.308730	+	0.951150i	$0.599904\pi$
$104$	0	0
$105$	36.4034	3.55261
$106$	0	0
$107$	11.6843	1.12956	0.564781	−	0.825241i	$-0.308961\pi$
$107$	11.6843	1.12956	0.564781	+	0.825241i	$0.308961\pi$
$108$	0	0
$109$	−13.3010	−1.27401	−0.637004	−	0.770861i	$-0.719826\pi$
$109$	−13.3010	−1.27401	−0.637004	+	0.770861i	$0.719826\pi$
$110$	0	0
$111$	−4.30020	−0.408157
$112$	0	0
$113$	0.0733398	0.00689923	0.00344961	−	0.999994i	$-0.498902\pi$
$113$	0.0733398	0.00689923	0.00344961	+	0.999994i	$0.498902\pi$
$114$	0	0
$115$	34.8715	3.25179
$116$	0	0
$117$	31.0601	2.87151
$118$	0	0
$119$	−9.37375	−0.859290
$120$	0	0
$121$	5.16858	0.469870
$122$	0	0
$123$	10.2729	0.926272
$124$	0	0
$125$	−21.1157	−1.88864
$126$	0	0
$127$	17.3247	1.53732	0.768661	−	0.639656i	$-0.220923\pi$
$127$	17.3247	1.53732	0.768661	+	0.639656i	$0.220923\pi$
$128$	0	0
$129$	−1.96560	−0.173062
$130$	0	0
$131$	−2.66068	−0.232464	−0.116232	−	0.993222i	$-0.537082\pi$
$131$	−2.66068	−0.232464	−0.116232	+	0.993222i	$0.537082\pi$
$132$	0	0
$133$	−1.87231	−0.162350
$134$	0	0
$135$	49.3227	4.24502
$136$	0	0
$137$	17.0212	1.45422	0.727109	−	0.686522i	$-0.240863\pi$
$137$	17.0212	1.45422	0.727109	+	0.686522i	$0.240863\pi$
$138$	0	0
$139$	−19.5149	−1.65523	−0.827616	−	0.561295i	$-0.810303\pi$
$139$	−19.5149	−1.65523	−0.827616	+	0.561295i	$0.810303\pi$
$140$	0	0
$141$	2.98797	0.251632
$142$	0	0
$143$	17.8917	1.49618
$144$	0	0
$145$	−4.29523	−0.356699
$146$	0	0
$147$	−5.15338	−0.425043
$148$	0	0
$149$	20.7777	1.70217	0.851087	−	0.525025i	$-0.175944\pi$
$149$	20.7777	1.70217	0.851087	+	0.525025i	$0.175944\pi$
$150$	0	0
$151$	−1.82064	−0.148161	−0.0740807	−	0.997252i	$-0.523602\pi$
$151$	−1.82064	−0.148161	−0.0740807	+	0.997252i	$0.523602\pi$
$152$	0	0
$153$	−22.2723	−1.80061
$154$	0	0
$155$	3.44773	0.276929
$156$	0	0
$157$	9.92303	0.791944	0.395972	−	0.918263i	$-0.370408\pi$
$157$	9.92303	0.791944	0.395972	+	0.918263i	$0.370408\pi$
$158$	0	0
$159$	6.70702	0.531901
$160$	0	0
$161$	−26.1203	−2.05857
$162$	0	0
$163$	7.82390	0.612815	0.306407	−	0.951900i	$-0.400873\pi$
$163$	7.82390	0.612815	0.306407	+	0.951900i	$0.400873\pi$
$164$	0	0
$165$	49.8244	3.87882
$166$	0	0
$167$	−19.8426	−1.53546	−0.767732	−	0.640771i	$-0.778615\pi$
$167$	−19.8426	−1.53546	−0.767732	+	0.640771i	$0.778615\pi$
$168$	0	0
$169$	6.79841	0.522954
$170$	0	0
$171$	−4.44866	−0.340197
$172$	0	0
$173$	22.6733	1.72382	0.861910	−	0.507061i	$-0.169268\pi$
$173$	22.6733	1.72382	0.861910	+	0.507061i	$0.169268\pi$
$174$	0	0
$175$	30.5060	2.30604
$176$	0	0
$177$	−5.71056	−0.429232
$178$	0	0
$179$	−22.9355	−1.71428	−0.857141	−	0.515082i	$-0.827761\pi$
$179$	−22.9355	−1.71428	−0.857141	+	0.515082i	$0.827761\pi$
$180$	0	0
$181$	1.22810	0.0912836	0.0456418	−	0.998958i	$-0.485467\pi$
$181$	1.22810	0.0912836	0.0456418	+	0.998958i	$0.485467\pi$
$182$	0	0
$183$	−2.08109	−0.153839
$184$	0	0
$185$	−5.33878	−0.392515
$186$	0	0
$187$	−12.8296	−0.938192
$188$	0	0
$189$	−36.9448	−2.68734
$190$	0	0
$191$	−10.3825	−0.751254	−0.375627	−	0.926771i	$-0.622573\pi$
$191$	−10.3825	−0.751254	−0.375627	+	0.926771i	$0.622573\pi$
$192$	0	0
$193$	26.3848	1.89922	0.949610	−	0.313433i	$-0.101479\pi$
$193$	26.3848	1.89922	0.949610	+	0.313433i	$0.101479\pi$
$194$	0	0
$195$	55.1342	3.94824
$196$	0	0
$197$	−13.0167	−0.927404	−0.463702	−	0.885991i	$-0.653479\pi$
$197$	−13.0167	−0.927404	−0.463702	+	0.885991i	$0.653479\pi$
$198$	0	0
$199$	7.72762	0.547796	0.273898	−	0.961759i	$-0.411687\pi$
$199$	7.72762	0.547796	0.273898	+	0.961759i	$0.411687\pi$
$200$	0	0
$201$	−45.0197	−3.17545
$202$	0	0
$203$	3.21731	0.225811
$204$	0	0
$205$	12.7539	0.890773
$206$	0	0
$207$	−62.0626	−4.31365
$208$	0	0
$209$	−2.56257	−0.177257
$210$	0	0
$211$	−22.5565	−1.55285	−0.776426	−	0.630209i	$-0.782969\pi$
$211$	−22.5565	−1.55285	−0.776426	+	0.630209i	$0.782969\pi$
$212$	0	0
$213$	22.8426	1.56515
$214$	0	0
$215$	−2.44033	−0.166429
$216$	0	0
$217$	−2.58250	−0.175312
$218$	0	0
$219$	20.7134	1.39968
$220$	0	0
$221$	−14.1969	−0.954983
$222$	0	0
$223$	8.79676	0.589075	0.294537	−	0.955640i	$-0.404834\pi$
$223$	8.79676	0.589075	0.294537	+	0.955640i	$0.404834\pi$
$224$	0	0
$225$	72.4832	4.83221
$226$	0	0
$227$	13.6460	0.905715	0.452857	−	0.891583i	$-0.350405\pi$
$227$	13.6460	0.905715	0.452857	+	0.891583i	$0.350405\pi$
$228$	0	0
$229$	−3.79782	−0.250967	−0.125483	−	0.992096i	$-0.540048\pi$
$229$	−3.79782	−0.250967	−0.125483	+	0.992096i	$0.540048\pi$
$230$	0	0
$231$	−37.3206	−2.45552
$232$	0	0
$233$	−18.8567	−1.23534	−0.617672	−	0.786436i	$-0.711924\pi$
$233$	−18.8567	−1.23534	−0.617672	+	0.786436i	$0.711924\pi$
$234$	0	0
$235$	3.70961	0.241989
$236$	0	0
$237$	−16.0002	−1.03933
$238$	0	0
$239$	−24.9836	−1.61605	−0.808026	−	0.589147i	$-0.799464\pi$
$239$	−24.9836	−1.61605	−0.808026	+	0.589147i	$0.799464\pi$
$240$	0	0
$241$	3.31242	0.213371	0.106686	−	0.994293i	$-0.465976\pi$
$241$	3.31242	0.213371	0.106686	+	0.994293i	$0.465976\pi$
$242$	0	0
$243$	−21.6234	−1.38714
$244$	0	0
$245$	−6.39800	−0.408753
$246$	0	0
$247$	−2.83567	−0.180429
$248$	0	0
$249$	−0.975706	−0.0618329
$250$	0	0
$251$	−4.90143	−0.309376	−0.154688	−	0.987963i	$-0.549437\pi$
$251$	−4.90143	−0.309376	−0.154688	+	0.987963i	$0.549437\pi$
$252$	0	0
$253$	−35.7501	−2.24759
$254$	0	0
$255$	−39.5351	−2.47578
$256$	0	0
$257$	31.7291	1.97921	0.989604	−	0.143822i	$-0.0459392\pi$
$257$	31.7291	1.97921	0.989604	+	0.143822i	$0.0459392\pi$
$258$	0	0
$259$	3.99897	0.248484
$260$	0	0
$261$	7.64443	0.473178
$262$	0	0
$263$	9.32170	0.574801	0.287400	−	0.957811i	$-0.407209\pi$
$263$	9.32170	0.574801	0.287400	+	0.957811i	$0.407209\pi$
$264$	0	0
$265$	8.32688	0.511516
$266$	0	0
$267$	43.4952	2.66186
$268$	0	0
$269$	−5.18881	−0.316368	−0.158184	−	0.987410i	$-0.550564\pi$
$269$	−5.18881	−0.316368	−0.158184	+	0.987410i	$0.550564\pi$
$270$	0	0
$271$	23.6581	1.43713	0.718563	−	0.695461i	$-0.244800\pi$
$271$	23.6581	1.43713	0.718563	+	0.695461i	$0.244800\pi$
$272$	0	0
$273$	−41.2979	−2.49946
$274$	0	0
$275$	41.7527	2.51779
$276$	0	0
$277$	1.48359	0.0891403	0.0445702	−	0.999006i	$-0.485808\pi$
$277$	1.48359	0.0891403	0.0445702	+	0.999006i	$0.485808\pi$
$278$	0	0
$279$	−6.13610	−0.367359
$280$	0	0
$281$	−18.6221	−1.11090	−0.555451	−	0.831550i	$-0.687454\pi$
$281$	−18.6221	−1.11090	−0.555451	+	0.831550i	$0.687454\pi$
$282$	0	0
$283$	−7.07013	−0.420276	−0.210138	−	0.977672i	$-0.567391\pi$
$283$	−7.07013	−0.420276	−0.210138	+	0.977672i	$0.567391\pi$
$284$	0	0
$285$	−7.89672	−0.467761
$286$	0	0
$287$	−9.55324	−0.563910
$288$	0	0
$289$	−6.81986	−0.401168
$290$	0	0
$291$	15.7660	0.924221
$292$	0	0
$293$	17.9304	1.04750	0.523752	−	0.851870i	$-0.324532\pi$
$293$	17.9304	1.04750	0.523752	+	0.851870i	$0.324532\pi$
$294$	0	0
$295$	−7.08976	−0.412782
$296$	0	0
$297$	−50.5654	−2.93410
$298$	0	0
$299$	−39.5601	−2.28782
$300$	0	0
$301$	1.82791	0.105359
$302$	0	0
$303$	−4.47546	−0.257108
$304$	0	0
$305$	−2.58371	−0.147943
$306$	0	0
$307$	−4.96624	−0.283438	−0.141719	−	0.989907i	$-0.545263\pi$
$307$	−4.96624	−0.283438	−0.141719	+	0.989907i	$0.545263\pi$
$308$	0	0
$309$	19.7972	1.12622
$310$	0	0
$311$	11.4073	0.646849	0.323425	−	0.946254i	$-0.395166\pi$
$311$	11.4073	0.646849	0.323425	+	0.946254i	$0.395166\pi$
$312$	0	0
$313$	−25.3879	−1.43501	−0.717506	−	0.696552i	$-0.754716\pi$
$313$	−25.3879	−1.43501	−0.717506	+	0.696552i	$0.754716\pi$
$314$	0	0
$315$	−80.4366	−4.53209
$316$	0	0
$317$	24.4921	1.37561	0.687806	−	0.725894i	$-0.258574\pi$
$317$	24.4921	1.37561	0.687806	+	0.725894i	$0.258574\pi$
$318$	0	0
$319$	4.40344	0.246546
$320$	0	0
$321$	−36.9129	−2.06028
$322$	0	0
$323$	2.03337	0.113140
$324$	0	0
$325$	46.2024	2.56285
$326$	0	0
$327$	42.0206	2.32374
$328$	0	0
$329$	−2.77866	−0.153193
$330$	0	0
$331$	3.22793	0.177423	0.0887115	−	0.996057i	$-0.471725\pi$
$331$	3.22793	0.177423	0.0887115	+	0.996057i	$0.471725\pi$
$332$	0	0
$333$	9.50168	0.520689
$334$	0	0
$335$	−55.8928	−3.05375
$336$	0	0
$337$	26.7263	1.45588	0.727938	−	0.685643i	$-0.240479\pi$
$337$	26.7263	1.45588	0.727938	+	0.685643i	$0.240479\pi$
$338$	0	0
$339$	−0.231695	−0.0125839
$340$	0	0
$341$	−3.53460	−0.191409
$342$	0	0
$343$	−15.7729	−0.851656
$344$	0	0
$345$	−110.166	−5.93114
$346$	0	0
$347$	−3.36910	−0.180863	−0.0904314	−	0.995903i	$-0.528825\pi$
$347$	−3.36910	−0.180863	−0.0904314	+	0.995903i	$0.528825\pi$
$348$	0	0
$349$	−2.76275	−0.147887	−0.0739433	−	0.997262i	$-0.523558\pi$
$349$	−2.76275	−0.147887	−0.0739433	+	0.997262i	$0.523558\pi$
$350$	0	0
$351$	−55.9542	−2.98661
$352$	0	0
$353$	−33.9813	−1.80864	−0.904321	−	0.426854i	$-0.859622\pi$
$353$	−33.9813	−1.80864	−0.904321	+	0.426854i	$0.859622\pi$
$354$	0	0
$355$	28.3595	1.50516
$356$	0	0
$357$	29.6135	1.56731
$358$	0	0
$359$	−22.2066	−1.17202	−0.586009	−	0.810304i	$-0.699302\pi$
$359$	−22.2066	−1.17202	−0.586009	+	0.810304i	$0.699302\pi$
$360$	0	0
$361$	−18.5939	−0.978624
$362$	0	0
$363$	−16.3285	−0.857026
$364$	0	0
$365$	25.7161	1.34604
$366$	0	0
$367$	−14.9070	−0.778137	−0.389069	−	0.921209i	$-0.627203\pi$
$367$	−14.9070	−0.778137	−0.389069	+	0.921209i	$0.627203\pi$
$368$	0	0
$369$	−22.6988	−1.18165
$370$	0	0
$371$	−6.23719	−0.323819
$372$	0	0
$373$	−4.90561	−0.254003	−0.127001	−	0.991903i	$-0.540535\pi$
$373$	−4.90561	−0.254003	−0.127001	+	0.991903i	$0.540535\pi$
$374$	0	0
$375$	66.7085	3.44481
$376$	0	0
$377$	4.87272	0.250958
$378$	0	0
$379$	−31.9485	−1.64108	−0.820542	−	0.571586i	$-0.806328\pi$
$379$	−31.9485	−1.64108	−0.820542	+	0.571586i	$0.806328\pi$
$380$	0	0
$381$	−54.7323	−2.80402
$382$	0	0
$383$	−23.7232	−1.21220	−0.606099	−	0.795389i	$-0.707267\pi$
$383$	−23.7232	−1.21220	−0.606099	+	0.795389i	$0.707267\pi$
$384$	0	0
$385$	−46.3342	−2.36141
$386$	0	0
$387$	4.34317	0.220776
$388$	0	0
$389$	−4.51455	−0.228897	−0.114448	−	0.993429i	$-0.536510\pi$
$389$	−4.51455	−0.228897	−0.114448	+	0.993429i	$0.536510\pi$
$390$	0	0
$391$	28.3673	1.43460
$392$	0	0
$393$	8.40560	0.424006
$394$	0	0
$395$	−19.8645	−0.999494
$396$	0	0
$397$	−38.4941	−1.93197	−0.965983	−	0.258606i	$-0.916737\pi$
$397$	−38.4941	−1.93197	−0.965983	+	0.258606i	$0.916737\pi$
$398$	0	0
$399$	5.91499	0.296120
$400$	0	0
$401$	−3.03666	−0.151643	−0.0758217	−	0.997121i	$-0.524158\pi$
$401$	−3.03666	−0.151643	−0.0758217	+	0.997121i	$0.524158\pi$
$402$	0	0
$403$	−3.91128	−0.194835
$404$	0	0
$405$	−73.6831	−3.66134
$406$	0	0
$407$	5.47329	0.271301
$408$	0	0
$409$	−8.66162	−0.428289	−0.214145	−	0.976802i	$-0.568696\pi$
$409$	−8.66162	−0.428289	−0.214145	+	0.976802i	$0.568696\pi$
$410$	0	0
$411$	−53.7733	−2.65244
$412$	0	0
$413$	5.31054	0.261315
$414$	0	0
$415$	−1.21136	−0.0594631
$416$	0	0
$417$	61.6514	3.01908
$418$	0	0
$419$	−14.6799	−0.717162	−0.358581	−	0.933499i	$-0.616739\pi$
$419$	−14.6799	−0.717162	−0.358581	+	0.933499i	$0.616739\pi$
$420$	0	0
$421$	13.9659	0.680658	0.340329	−	0.940306i	$-0.389462\pi$
$421$	13.9659	0.680658	0.340329	+	0.940306i	$0.389462\pi$
$422$	0	0
$423$	−6.60218	−0.321009
$424$	0	0
$425$	−33.1304	−1.60706
$426$	0	0
$427$	1.93531	0.0936564
$428$	0	0
$429$	−56.5233	−2.72897
$430$	0	0
$431$	32.3801	1.55969	0.779846	−	0.625971i	$-0.215297\pi$
$431$	32.3801	1.55969	0.779846	+	0.625971i	$0.215297\pi$
$432$	0	0
$433$	−1.96349	−0.0943593	−0.0471796	−	0.998886i	$-0.515023\pi$
$433$	−1.96349	−0.0943593	−0.0471796	+	0.998886i	$0.515023\pi$
$434$	0	0
$435$	13.5695	0.650606
$436$	0	0
$437$	5.66608	0.271045
$438$	0	0
$439$	−38.5425	−1.83953	−0.919767	−	0.392466i	$-0.871622\pi$
$439$	−38.5425	−1.83953	−0.919767	+	0.392466i	$0.871622\pi$
$440$	0	0
$441$	11.3868	0.542230
$442$	0	0
$443$	−1.30904	−0.0621943	−0.0310972	−	0.999516i	$-0.509900\pi$
$443$	−1.30904	−0.0621943	−0.0310972	+	0.999516i	$0.509900\pi$
$444$	0	0
$445$	54.0000	2.55985
$446$	0	0
$447$	−65.6408	−3.10470
$448$	0	0
$449$	20.1060	0.948860	0.474430	−	0.880293i	$-0.342654\pi$
$449$	20.1060	0.948860	0.474430	+	0.880293i	$0.342654\pi$
$450$	0	0
$451$	−13.0753	−0.615690
$452$	0	0
$453$	5.75176	0.270241
$454$	0	0
$455$	−51.2721	−2.40367
$456$	0	0
$457$	−13.0034	−0.608275	−0.304137	−	0.952628i	$-0.598368\pi$
$457$	−13.0034	−0.608275	−0.304137	+	0.952628i	$0.598368\pi$
$458$	0	0
$459$	40.1231	1.87278
$460$	0	0
$461$	38.4863	1.79249	0.896244	−	0.443562i	$-0.146285\pi$
$461$	38.4863	1.79249	0.896244	+	0.443562i	$0.146285\pi$
$462$	0	0
$463$	−6.05925	−0.281597	−0.140799	−	0.990038i	$-0.544967\pi$
$463$	−6.05925	−0.281597	−0.140799	+	0.990038i	$0.544967\pi$
$464$	0	0
$465$	−10.8921	−0.505108
$466$	0	0
$467$	−8.77533	−0.406074	−0.203037	−	0.979171i	$-0.565081\pi$
$467$	−8.77533	−0.406074	−0.203037	+	0.979171i	$0.565081\pi$
$468$	0	0
$469$	41.8661	1.93320
$470$	0	0
$471$	−31.3488	−1.44448
$472$	0	0
$473$	2.50181	0.115033
$474$	0	0
$475$	−6.61744	−0.303629
$476$	0	0
$477$	−14.8198	−0.678550
$478$	0	0
$479$	−26.2420	−1.19903	−0.599514	−	0.800364i	$-0.704639\pi$
$479$	−26.2420	−1.19903	−0.599514	+	0.800364i	$0.704639\pi$
$480$	0	0
$481$	6.05658	0.276156
$482$	0	0
$483$	82.5192	3.75475
$484$	0	0
$485$	19.5738	0.888801
$486$	0	0
$487$	−27.6326	−1.25215	−0.626075	−	0.779763i	$-0.715340\pi$
$487$	−27.6326	−1.25215	−0.626075	+	0.779763i	$0.715340\pi$
$488$	0	0
$489$	−24.7172	−1.11775
$490$	0	0
$491$	−15.0202	−0.677854	−0.338927	−	0.940813i	$-0.610064\pi$
$491$	−15.0202	−0.677854	−0.338927	+	0.940813i	$0.610064\pi$
$492$	0	0
$493$	−3.49409	−0.157366
$494$	0	0
$495$	−110.091	−4.94824
$496$	0	0
$497$	−21.2425	−0.952855
$498$	0	0
$499$	33.9050	1.51779	0.758897	−	0.651210i	$-0.225738\pi$
$499$	33.9050	1.51779	0.758897	+	0.651210i	$0.225738\pi$
$500$	0	0
$501$	62.6866	2.80063
$502$	0	0
$503$	6.24897	0.278628	0.139314	−	0.990248i	$-0.455510\pi$
$503$	6.24897	0.278628	0.139314	+	0.990248i	$0.455510\pi$
$504$	0	0
$505$	−5.55636	−0.247255
$506$	0	0
$507$	−21.4775	−0.953850
$508$	0	0
$509$	25.2412	1.11880	0.559398	−	0.828899i	$-0.311032\pi$
$509$	25.2412	1.11880	0.559398	+	0.828899i	$0.311032\pi$
$510$	0	0
$511$	−19.2625	−0.852121
$512$	0	0
$513$	8.01416	0.353834
$514$	0	0
$515$	24.5786	1.08306
$516$	0	0
$517$	−3.80308	−0.167259
$518$	0	0
$519$	−71.6295	−3.14418
$520$	0	0
$521$	20.6285	0.903750	0.451875	−	0.892081i	$-0.350755\pi$
$521$	20.6285	0.903750	0.451875	+	0.892081i	$0.350755\pi$
$522$	0	0
$523$	4.73418	0.207011	0.103506	−	0.994629i	$-0.466994\pi$
$523$	4.73418	0.207011	0.103506	+	0.994629i	$0.466994\pi$
$524$	0	0
$525$	−96.3745	−4.20613
$526$	0	0
$527$	2.80467	0.122173
$528$	0	0
$529$	56.0466	2.43681
$530$	0	0
$531$	12.6180	0.547574
$532$	0	0
$533$	−14.4687	−0.626709
$534$	0	0
$535$	−45.8281	−1.98132
$536$	0	0
$537$	72.4578	3.12679
$538$	0	0
$539$	6.55920	0.282525
$540$	0	0
$541$	−20.3288	−0.874005	−0.437003	−	0.899460i	$-0.643960\pi$
$541$	−20.3288	−0.874005	−0.437003	+	0.899460i	$0.643960\pi$
$542$	0	0
$543$	−3.87980	−0.166498
$544$	0	0
$545$	52.1693	2.23469
$546$	0	0
$547$	0.823463	0.0352088	0.0176044	−	0.999845i	$-0.494396\pi$
$547$	0.823463	0.0352088	0.0176044	+	0.999845i	$0.494396\pi$
$548$	0	0
$549$	4.59836	0.196253
$550$	0	0
$551$	−0.697907	−0.0297318
$552$	0	0
$553$	14.8794	0.632737
$554$	0	0
$555$	16.8662	0.715932
$556$	0	0
$557$	22.7450	0.963737	0.481869	−	0.876243i	$-0.339958\pi$
$557$	22.7450	0.963737	0.481869	+	0.876243i	$0.339958\pi$
$558$	0	0
$559$	2.76843	0.117092
$560$	0	0
$561$	40.5312	1.71123
$562$	0	0
$563$	−20.7525	−0.874612	−0.437306	−	0.899313i	$-0.644067\pi$
$563$	−20.7525	−0.874612	−0.437306	+	0.899313i	$0.644067\pi$
$564$	0	0
$565$	−0.287653	−0.0121017
$566$	0	0
$567$	55.1918	2.31784
$568$	0	0
$569$	−13.3805	−0.560939	−0.280469	−	0.959863i	$-0.590490\pi$
$569$	−13.3805	−0.560939	−0.280469	+	0.959863i	$0.590490\pi$
$570$	0	0
$571$	40.6282	1.70024	0.850118	−	0.526593i	$-0.176531\pi$
$571$	40.6282	1.70024	0.850118	+	0.526593i	$0.176531\pi$
$572$	0	0
$573$	32.8005	1.37026
$574$	0	0
$575$	−92.3190	−3.84997
$576$	0	0
$577$	12.5573	0.522767	0.261384	−	0.965235i	$-0.415821\pi$
$577$	12.5573	0.522767	0.261384	+	0.965235i	$0.415821\pi$
$578$	0	0
$579$	−83.3548	−3.46411
$580$	0	0
$581$	0.907358	0.0376435
$582$	0	0
$583$	−8.53667	−0.353553
$584$	0	0
$585$	−121.824	−5.03680
$586$	0	0
$587$	9.29059	0.383464	0.191732	−	0.981447i	$-0.438590\pi$
$587$	9.29059	0.383464	0.191732	+	0.981447i	$0.438590\pi$
$588$	0	0
$589$	0.560202	0.0230827
$590$	0	0
$591$	41.1224	1.69155
$592$	0	0
$593$	−21.2265	−0.871667	−0.435833	−	0.900027i	$-0.643546\pi$
$593$	−21.2265	−0.871667	−0.435833	+	0.900027i	$0.643546\pi$
$594$	0	0
$595$	36.7657	1.50725
$596$	0	0
$597$	−24.4131	−0.999160
$598$	0	0
$599$	33.5713	1.37169	0.685843	−	0.727749i	$-0.259434\pi$
$599$	33.5713	1.37169	0.685843	+	0.727749i	$0.259434\pi$
$600$	0	0
$601$	39.1106	1.59535	0.797677	−	0.603085i	$-0.206062\pi$
$601$	39.1106	1.59535	0.797677	+	0.603085i	$0.206062\pi$
$602$	0	0
$603$	99.4751	4.05094
$604$	0	0
$605$	−20.2722	−0.824181
$606$	0	0
$607$	−9.86751	−0.400510	−0.200255	−	0.979744i	$-0.564177\pi$
$607$	−9.86751	−0.400510	−0.200255	+	0.979744i	$0.564177\pi$
$608$	0	0
$609$	−10.1641	−0.411871
$610$	0	0
$611$	−4.20838	−0.170253
$612$	0	0
$613$	21.2223	0.857162	0.428581	−	0.903503i	$-0.359014\pi$
$613$	21.2223	0.857162	0.428581	+	0.903503i	$0.359014\pi$
$614$	0	0
$615$	−40.2922	−1.62474
$616$	0	0
$617$	33.6663	1.35535	0.677676	−	0.735360i	$-0.262987\pi$
$617$	33.6663	1.35535	0.677676	+	0.735360i	$0.262987\pi$
$618$	0	0
$619$	8.94664	0.359596	0.179798	−	0.983704i	$-0.442456\pi$
$619$	8.94664	0.359596	0.179798	+	0.983704i	$0.442456\pi$
$620$	0	0
$621$	111.804	4.48656
$622$	0	0
$623$	−40.4483	−1.62053
$624$	0	0
$625$	30.9016	1.23606
$626$	0	0
$627$	8.09568	0.323310
$628$	0	0
$629$	−4.34299	−0.173167
$630$	0	0
$631$	−6.46958	−0.257550	−0.128775	−	0.991674i	$-0.541104\pi$
$631$	−6.46958	−0.257550	−0.128775	+	0.991674i	$0.541104\pi$
$632$	0	0
$633$	71.2603	2.83234
$634$	0	0
$635$	−67.9511	−2.69656
$636$	0	0
$637$	7.25822	0.287581
$638$	0	0
$639$	−50.4727	−1.99667
$640$	0	0
$641$	−13.5311	−0.534448	−0.267224	−	0.963635i	$-0.586106\pi$
$641$	−13.5311	−0.534448	−0.267224	+	0.963635i	$0.586106\pi$
$642$	0	0
$643$	−43.9051	−1.73145	−0.865723	−	0.500523i	$-0.833141\pi$
$643$	−43.9051	−1.73145	−0.865723	+	0.500523i	$0.833141\pi$
$644$	0	0
$645$	7.70948	0.303560
$646$	0	0
$647$	−22.2530	−0.874854	−0.437427	−	0.899254i	$-0.644110\pi$
$647$	−22.2530	−0.874854	−0.437427	+	0.899254i	$0.644110\pi$
$648$	0	0
$649$	7.26839	0.285309
$650$	0	0
$651$	8.15863	0.319762
$652$	0	0
$653$	−31.1530	−1.21911	−0.609555	−	0.792744i	$-0.708652\pi$
$653$	−31.1530	−1.21911	−0.609555	+	0.792744i	$0.708652\pi$
$654$	0	0
$655$	10.4357	0.407756
$656$	0	0
$657$	−45.7682	−1.78559
$658$	0	0
$659$	−29.2869	−1.14085	−0.570427	−	0.821348i	$-0.693222\pi$
$659$	−29.2869	−1.14085	−0.570427	+	0.821348i	$0.693222\pi$
$660$	0	0
$661$	12.9540	0.503852	0.251926	−	0.967747i	$-0.418936\pi$
$661$	12.9540	0.503852	0.251926	+	0.967747i	$0.418936\pi$
$662$	0	0
$663$	44.8506	1.74185
$664$	0	0
$665$	7.34355	0.284771
$666$	0	0
$667$	−9.73640	−0.376995
$668$	0	0
$669$	−27.7907	−1.07445
$670$	0	0
$671$	2.64881	0.102256
$672$	0	0
$673$	19.0312	0.733598	0.366799	−	0.930300i	$-0.380454\pi$
$673$	19.0312	0.733598	0.366799	+	0.930300i	$0.380454\pi$
$674$	0	0
$675$	−130.577	−5.02591
$676$	0	0
$677$	42.8515	1.64692	0.823459	−	0.567375i	$-0.192041\pi$
$677$	42.8515	1.64692	0.823459	+	0.567375i	$0.192041\pi$
$678$	0	0
$679$	−14.6616	−0.562662
$680$	0	0
$681$	−43.1103	−1.65199
$682$	0	0
$683$	−46.7203	−1.78770	−0.893852	−	0.448363i	$-0.852007\pi$
$683$	−46.7203	−1.78770	−0.893852	+	0.448363i	$0.852007\pi$
$684$	0	0
$685$	−66.7604	−2.55079
$686$	0	0
$687$	11.9981	0.457754
$688$	0	0
$689$	−9.44643	−0.359881
$690$	0	0
$691$	1.83864	0.0699453	0.0349727	−	0.999388i	$-0.488866\pi$
$691$	1.83864	0.0699453	0.0349727	+	0.999388i	$0.488866\pi$
$692$	0	0
$693$	82.4632	3.13252
$694$	0	0
$695$	76.5412	2.90337
$696$	0	0
$697$	10.3751	0.392984
$698$	0	0
$699$	59.5720	2.25322
$700$	0	0
$701$	44.2176	1.67008	0.835038	−	0.550192i	$-0.185446\pi$
$701$	44.2176	1.67008	0.835038	+	0.550192i	$0.185446\pi$
$702$	0	0
$703$	−0.867467	−0.0327171
$704$	0	0
$705$	−11.7194	−0.441378
$706$	0	0
$707$	4.16195	0.156526
$708$	0	0
$709$	16.9725	0.637417	0.318709	−	0.947853i	$-0.396751\pi$
$709$	16.9725	0.637417	0.318709	+	0.947853i	$0.396751\pi$
$710$	0	0
$711$	35.3539	1.32588
$712$	0	0
$713$	7.81531	0.292686
$714$	0	0
$715$	−70.1747	−2.62438
$716$	0	0
$717$	78.9280	2.94762
$718$	0	0
$719$	4.28941	0.159968	0.0799840	−	0.996796i	$-0.474513\pi$
$719$	4.28941	0.159968	0.0799840	+	0.996796i	$0.474513\pi$
$720$	0	0
$721$	−18.4104	−0.685640
$722$	0	0
$723$	−10.4646	−0.389182
$724$	0	0
$725$	11.3712	0.422315
$726$	0	0
$727$	30.7623	1.14091	0.570455	−	0.821329i	$-0.306767\pi$
$727$	30.7623	1.14091	0.570455	+	0.821329i	$0.306767\pi$
$728$	0	0
$729$	11.9541	0.442745
$730$	0	0
$731$	−1.98516	−0.0734238
$732$	0	0
$733$	−35.9494	−1.32782	−0.663910	−	0.747812i	$-0.731104\pi$
$733$	−35.9494	−1.32782	−0.663910	+	0.747812i	$0.731104\pi$
$734$	0	0
$735$	20.2126	0.745551
$736$	0	0
$737$	57.3010	2.11071
$738$	0	0
$739$	32.9117	1.21068	0.605338	−	0.795968i	$-0.293038\pi$
$739$	32.9117	1.21068	0.605338	+	0.795968i	$0.293038\pi$
$740$	0	0
$741$	8.95844	0.329097
$742$	0	0
$743$	−37.3338	−1.36964	−0.684821	−	0.728711i	$-0.740120\pi$
$743$	−37.3338	−1.36964	−0.684821	+	0.728711i	$0.740120\pi$
$744$	0	0
$745$	−81.4941	−2.98571
$746$	0	0
$747$	2.15591	0.0788806
$748$	0	0
$749$	34.3272	1.25429
$750$	0	0
$751$	−45.3612	−1.65525	−0.827627	−	0.561278i	$-0.810310\pi$
$751$	−45.3612	−1.65525	−0.827627	+	0.561278i	$0.810310\pi$
$752$	0	0
$753$	15.4846	0.564290
$754$	0	0
$755$	7.14090	0.259884
$756$	0	0
$757$	−19.4501	−0.706925	−0.353462	−	0.935449i	$-0.614996\pi$
$757$	−19.4501	−0.706925	−0.353462	+	0.935449i	$0.614996\pi$
$758$	0	0
$759$	112.942	4.09952
$760$	0	0
$761$	54.0510	1.95935	0.979674	−	0.200596i	$-0.0642879\pi$
$761$	54.0510	1.95935	0.979674	+	0.200596i	$0.0642879\pi$
$762$	0	0
$763$	−39.0770	−1.41468
$764$	0	0
$765$	87.3563	3.15838
$766$	0	0
$767$	8.04299	0.290415
$768$	0	0
$769$	−16.3135	−0.588282	−0.294141	−	0.955762i	$-0.595033\pi$
$769$	−16.3135	−0.588282	−0.294141	+	0.955762i	$0.595033\pi$
$770$	0	0
$771$	−100.238	−3.61000
$772$	0	0
$773$	35.8153	1.28819	0.644094	−	0.764946i	$-0.277234\pi$
$773$	35.8153	1.28819	0.644094	+	0.764946i	$0.277234\pi$
$774$	0	0
$775$	−9.12753	−0.327871
$776$	0	0
$777$	−12.6335	−0.453226
$778$	0	0
$779$	2.07231	0.0742483
$780$	0	0
$781$	−29.0740	−1.04035
$782$	0	0
$783$	−13.7713	−0.492145
$784$	0	0
$785$	−38.9201	−1.38912
$786$	0	0
$787$	2.64531	0.0942950	0.0471475	−	0.998888i	$-0.484987\pi$
$787$	2.64531	0.0942950	0.0471475	+	0.998888i	$0.484987\pi$
$788$	0	0
$789$	−29.4491	−1.04842
$790$	0	0
$791$	0.215465	0.00766104
$792$	0	0
$793$	2.93110	0.104086
$794$	0	0
$795$	−26.3062	−0.932986
$796$	0	0
$797$	−50.3734	−1.78432	−0.892159	−	0.451722i	$-0.850810\pi$
$797$	−50.3734	−1.78432	−0.892159	+	0.451722i	$0.850810\pi$
$798$	0	0
$799$	3.01770	0.106759
$800$	0	0
$801$	−96.1065	−3.39576
$802$	0	0
$803$	−26.3640	−0.930366
$804$	0	0
$805$	102.449	3.61085
$806$	0	0
$807$	16.3925	0.577043
$808$	0	0
$809$	6.19301	0.217735	0.108867	−	0.994056i	$-0.465278\pi$
$809$	6.19301	0.217735	0.108867	+	0.994056i	$0.465278\pi$
$810$	0	0
$811$	−24.3958	−0.856652	−0.428326	−	0.903624i	$-0.640897\pi$
$811$	−24.3958	−0.856652	−0.428326	+	0.903624i	$0.640897\pi$
$812$	0	0
$813$	−74.7406	−2.62127
$814$	0	0
$815$	−30.6869	−1.07491
$816$	0	0
$817$	−0.396515	−0.0138723
$818$	0	0
$819$	91.2514	3.18858
$820$	0	0
$821$	−3.81348	−0.133091	−0.0665457	−	0.997783i	$-0.521198\pi$
$821$	−3.81348	−0.133091	−0.0665457	+	0.997783i	$0.521198\pi$
$822$	0	0
$823$	11.6474	0.406004	0.203002	−	0.979178i	$-0.434930\pi$
$823$	11.6474	0.406004	0.203002	+	0.979178i	$0.434930\pi$
$824$	0	0
$825$	−131.905	−4.59235
$826$	0	0
$827$	18.6366	0.648057	0.324029	−	0.946047i	$-0.394963\pi$
$827$	18.6366	0.648057	0.324029	+	0.946047i	$0.394963\pi$
$828$	0	0
$829$	2.06975	0.0718853	0.0359427	−	0.999354i	$-0.488557\pi$
$829$	2.06975	0.0718853	0.0359427	+	0.999354i	$0.488557\pi$
$830$	0	0
$831$	−4.68696	−0.162589
$832$	0	0
$833$	−5.20465	−0.180331
$834$	0	0
$835$	77.8265	2.69330
$836$	0	0
$837$	11.0541	0.382084
$838$	0	0
$839$	4.17282	0.144062	0.0720309	−	0.997402i	$-0.477052\pi$
$839$	4.17282	0.144062	0.0720309	+	0.997402i	$0.477052\pi$
$840$	0	0
$841$	−27.8007	−0.958646
$842$	0	0
$843$	58.8309	2.02624
$844$	0	0
$845$	−26.6647	−0.917293
$846$	0	0
$847$	15.1847	0.521754
$848$	0	0
$849$	22.3359	0.766567
$850$	0	0
$851$	−12.1019	−0.414848
$852$	0	0
$853$	−53.1216	−1.81885	−0.909424	−	0.415869i	$-0.863477\pi$
$853$	−53.1216	−1.81885	−0.909424	+	0.415869i	$0.863477\pi$
$854$	0	0
$855$	17.4485	0.596726
$856$	0	0
$857$	−17.7859	−0.607556	−0.303778	−	0.952743i	$-0.598248\pi$
$857$	−17.7859	−0.607556	−0.303778	+	0.952743i	$0.598248\pi$
$858$	0	0
$859$	11.3471	0.387158	0.193579	−	0.981085i	$-0.437990\pi$
$859$	11.3471	0.387158	0.193579	+	0.981085i	$0.437990\pi$
$860$	0	0
$861$	30.1806	1.02855
$862$	0	0
$863$	37.5610	1.27859	0.639296	−	0.768961i	$-0.279226\pi$
$863$	37.5610	1.27859	0.639296	+	0.768961i	$0.279226\pi$
$864$	0	0
$865$	−88.9292	−3.02368
$866$	0	0
$867$	21.5453	0.731716
$868$	0	0
$869$	20.3650	0.690836
$870$	0	0
$871$	63.4076	2.14849
$872$	0	0
$873$	−34.8365	−1.17904
$874$	0	0
$875$	−62.0356	−2.09719
$876$	0	0
$877$	−24.9962	−0.844060	−0.422030	−	0.906582i	$-0.638682\pi$
$877$	−24.9962	−0.844060	−0.422030	+	0.906582i	$0.638682\pi$
$878$	0	0
$879$	−56.6457	−1.91061
$880$	0	0
$881$	−46.4282	−1.56421	−0.782103	−	0.623150i	$-0.785853\pi$
$881$	−46.4282	−1.56421	−0.782103	+	0.623150i	$0.785853\pi$
$882$	0	0
$883$	−8.29498	−0.279148	−0.139574	−	0.990212i	$-0.544573\pi$
$883$	−8.29498	−0.279148	−0.139574	+	0.990212i	$0.544573\pi$
$884$	0	0
$885$	22.3980	0.752899
$886$	0	0
$887$	−0.734992	−0.0246786	−0.0123393	−	0.999924i	$-0.503928\pi$
$887$	−0.734992	−0.0246786	−0.0123393	+	0.999924i	$0.503928\pi$
$888$	0	0
$889$	50.8983	1.70707
$890$	0	0
$891$	75.5395	2.53067
$892$	0	0
$893$	0.602754	0.0201704
$894$	0	0
$895$	89.9576	3.00695
$896$	0	0
$897$	124.978	4.17289
$898$	0	0
$899$	−0.962633	−0.0321056
$900$	0	0
$901$	6.77376	0.225667
$902$	0	0
$903$	−5.77473	−0.192171
$904$	0	0
$905$	−4.81683	−0.160117
$906$	0	0
$907$	−20.5448	−0.682179	−0.341090	−	0.940031i	$-0.610796\pi$
$907$	−20.5448	−0.682179	−0.341090	+	0.940031i	$0.610796\pi$
$908$	0	0
$909$	9.88893	0.327995
$910$	0	0
$911$	13.1498	0.435671	0.217836	−	0.975985i	$-0.430100\pi$
$911$	13.1498	0.435671	0.217836	+	0.975985i	$0.430100\pi$
$912$	0	0
$913$	1.24188	0.0411001
$914$	0	0
$915$	8.16246	0.269843
$916$	0	0
$917$	−7.81679	−0.258133
$918$	0	0
$919$	−47.0204	−1.55106	−0.775529	−	0.631312i	$-0.782517\pi$
$919$	−47.0204	−1.55106	−0.775529	+	0.631312i	$0.782517\pi$
$920$	0	0
$921$	15.6893	0.516981
$922$	0	0
$923$	−32.1724	−1.05897
$924$	0	0
$925$	14.1339	0.464719
$926$	0	0
$927$	−43.7437	−1.43673
$928$	0	0
$929$	30.4260	0.998246	0.499123	−	0.866531i	$-0.333656\pi$
$929$	30.4260	0.998246	0.499123	+	0.866531i	$0.333656\pi$
$930$	0	0
$931$	−1.03957	−0.0340707
$932$	0	0
$933$	−36.0380	−1.17983
$934$	0	0
$935$	50.3202	1.64565
$936$	0	0
$937$	16.5997	0.542290	0.271145	−	0.962539i	$-0.412598\pi$
$937$	16.5997	0.542290	0.271145	+	0.962539i	$0.412598\pi$
$938$	0	0
$939$	80.2055	2.61741
$940$	0	0
$941$	41.8550	1.36444	0.682218	−	0.731149i	$-0.261016\pi$
$941$	41.8550	1.36444	0.682218	+	0.731149i	$0.261016\pi$
$942$	0	0
$943$	28.9105	0.941457
$944$	0	0
$945$	144.905	4.71376
$946$	0	0
$947$	−44.4937	−1.44585	−0.722926	−	0.690926i	$-0.757203\pi$
$947$	−44.4937	−1.44585	−0.722926	+	0.690926i	$0.757203\pi$
$948$	0	0
$949$	−29.1736	−0.947017
$950$	0	0
$951$	−77.3753	−2.50907
$952$	0	0
$953$	−34.5667	−1.11972	−0.559862	−	0.828586i	$-0.689146\pi$
$953$	−34.5667	−1.11972	−0.559862	+	0.828586i	$0.689146\pi$
$954$	0	0
$955$	40.7223	1.31774
$956$	0	0
$957$	−13.9113	−0.449690
$958$	0	0
$959$	50.0065	1.61479
$960$	0	0
$961$	−30.2273	−0.975074
$962$	0	0
$963$	81.5624	2.62831
$964$	0	0
$965$	−103.486	−3.33135
$966$	0	0
$967$	3.85664	0.124021	0.0620106	−	0.998075i	$-0.480249\pi$
$967$	3.85664	0.124021	0.0620106	+	0.998075i	$0.480249\pi$
$968$	0	0
$969$	−6.42383	−0.206363
$970$	0	0
$971$	−4.23621	−0.135946	−0.0679732	−	0.997687i	$-0.521653\pi$
$971$	−4.23621	−0.135946	−0.0679732	+	0.997687i	$0.521653\pi$
$972$	0	0
$973$	−57.3327	−1.83800
$974$	0	0
$975$	−145.962	−4.67454
$976$	0	0
$977$	−46.3018	−1.48133	−0.740663	−	0.671877i	$-0.765488\pi$
$977$	−46.3018	−1.48133	−0.740663	+	0.671877i	$0.765488\pi$
$978$	0	0
$979$	−55.3605	−1.76933
$980$	0	0
$981$	−92.8482	−2.96441
$982$	0	0
$983$	42.8675	1.36726	0.683630	−	0.729829i	$-0.260400\pi$
$983$	42.8675	1.36726	0.683630	+	0.729829i	$0.260400\pi$
$984$	0	0
$985$	51.0542	1.62672
$986$	0	0
$987$	8.77834	0.279418
$988$	0	0
$989$	−5.53173	−0.175899
$990$	0	0
$991$	−31.7560	−1.00876	−0.504381	−	0.863481i	$-0.668279\pi$
$991$	−31.7560	−1.00876	−0.504381	+	0.863481i	$0.668279\pi$
$992$	0	0
$993$	−10.1977	−0.323613
$994$	0	0
$995$	−30.3092	−0.960868
$996$	0	0
$997$	44.9129	1.42241	0.711203	−	0.702987i	$-0.248151\pi$
$997$	44.9129	1.42241	0.711203	+	0.702987i	$0.248151\pi$
$998$	0	0
$999$	−17.1171	−0.541560

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.a.1.3	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.a.1.3	✓	80	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:	\( N \)	\(=\)	\( 8044 = 2^{2} \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8044.a (trivial)

\(f(q)\)	\(=\)	\(q-3.15920 q^{3} -3.92220 q^{5} +2.93790 q^{7} +6.98053 q^{9} +O(q^{10})\)	\(q-3.15920 q^{3} -3.92220 q^{5} +2.93790 q^{7} +6.98053 q^{9} +4.02102 q^{11} +4.44954 q^{13} +12.3910 q^{15} -3.19063 q^{17} -0.637295 q^{19} -9.28139 q^{21} -8.89082 q^{23} +10.3836 q^{25} -12.5753 q^{27} +1.09511 q^{29} -0.879031 q^{31} -12.7032 q^{33} -11.5230 q^{35} +1.36117 q^{37} -14.0570 q^{39} -3.25173 q^{41} +0.622184 q^{43} -27.3790 q^{45} -0.945800 q^{47} +1.63123 q^{49} +10.0798 q^{51} -2.12301 q^{53} -15.7712 q^{55} +2.01334 q^{57} +1.80760 q^{59} +0.658742 q^{61} +20.5081 q^{63} -17.4520 q^{65} +14.2504 q^{67} +28.0878 q^{69} -7.23050 q^{71} -6.55655 q^{73} -32.8039 q^{75} +11.8133 q^{77} +5.06465 q^{79} +18.7862 q^{81} +0.308846 q^{83} +12.5143 q^{85} -3.45966 q^{87} -13.7678 q^{89} +13.0723 q^{91} +2.77703 q^{93} +2.49960 q^{95} -4.99052 q^{97} +28.0688 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more