LMFDB - Embedded newform 5733.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5733,2,Mod(1,5733)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5733, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5733.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5733 = 3^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5733.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:	$45.7782354788$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.7168.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 7 $ x^4 - 6*x^2 + 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.25928$ of defining polynomial
Character	$\chi$	$=$	5733.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.25928 q^{2} -0.414214 q^{4} -1.78089 q^{5} +3.04017 q^{8} +O(q^{10})$	$q-1.25928 q^{2} -0.414214 q^{4} -1.78089 q^{5} +3.04017 q^{8} +2.24264 q^{10} +1.78089 q^{11} +1.00000 q^{13} -3.00000 q^{16} -6.08034 q^{17} +0.737669 q^{20} -2.24264 q^{22} +6.08034 q^{23} -1.82843 q^{25} -1.25928 q^{26} +3.56178 q^{29} -10.8284 q^{31} -2.30250 q^{32} +7.65685 q^{34} +10.4853 q^{37} -5.41421 q^{40} +6.81801 q^{41} -9.65685 q^{43} -0.737669 q^{44} -7.65685 q^{46} +9.33657 q^{47} +2.30250 q^{50} -0.414214 q^{52} -3.17157 q^{55} -4.48528 q^{58} -12.8984 q^{59} -7.65685 q^{61} +13.6360 q^{62} +8.89949 q^{64} -1.78089 q^{65} +8.48528 q^{67} +2.51856 q^{68} -6.81801 q^{71} +12.1421 q^{73} -13.2039 q^{74} -6.00000 q^{79} +5.34267 q^{80} -8.58579 q^{82} +0.737669 q^{83} +10.8284 q^{85} +12.1607 q^{86} +5.41421 q^{88} -0.305553 q^{89} -2.51856 q^{92} -11.7574 q^{94} -7.65685 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4}+O(q^{10}) $ 4 * q + 4 * q^4	$ 4 q + 4 q^{4} - 8 q^{10} + 4 q^{13} - 12 q^{16} + 8 q^{22} + 4 q^{25} - 32 q^{31} + 8 q^{34} + 8 q^{37} - 16 q^{40} - 16 q^{43} - 8 q^{46} + 4 q^{52} - 24 q^{55} + 16 q^{58} - 8 q^{61} - 4 q^{64} - 8 q^{73} - 24 q^{79} - 40 q^{82} + 32 q^{85} + 16 q^{88} - 64 q^{94} - 8 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 - 8 * q^10 + 4 * q^13 - 12 * q^16 + 8 * q^22 + 4 * q^25 - 32 * q^31 + 8 * q^34 + 8 * q^37 - 16 * q^40 - 16 * q^43 - 8 * q^46 + 4 * q^52 - 24 * q^55 + 16 * q^58 - 8 * q^61 - 4 * q^64 - 8 * q^73 - 24 * q^79 - 40 * q^82 + 32 * q^85 + 16 * q^88 - 64 * q^94 - 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.25928	−0.890446	−0.445223	−	0.895420i	$-0.646876\pi$
$2$	−1.25928	−0.890446	−0.445223	+	0.895420i	$0.646876\pi$
$3$	0	0
$3$	0	0
$4$	−0.414214	−0.207107
$5$	−1.78089	−0.796439	−0.398219	−	0.917290i	$-0.630372\pi$
$5$	−1.78089	−0.796439	−0.398219	+	0.917290i	$0.630372\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	3.04017	1.07486
$9$	0	0
$10$	2.24264	0.709185
$11$	1.78089	0.536959	0.268479	−	0.963285i	$-0.413479\pi$
$11$	1.78089	0.536959	0.268479	+	0.963285i	$0.413479\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	−3.00000	−0.750000
$17$	−6.08034	−1.47470	−0.737350	−	0.675511i	$-0.763923\pi$
$17$	−6.08034	−1.47470	−0.737350	+	0.675511i	$0.763923\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0.737669	0.164948
$21$	0	0
$22$	−2.24264	−0.478133
$23$	6.08034	1.26784	0.633920	−	0.773399i	$-0.281445\pi$
$23$	6.08034	1.26784	0.633920	+	0.773399i	$0.281445\pi$
$24$	0	0
$25$	−1.82843	−0.365685
$26$	−1.25928	−0.246965
$27$	0	0
$28$	0	0
$29$	3.56178	0.661406	0.330703	−	0.943735i	$-0.392714\pi$
$29$	3.56178	0.661406	0.330703	+	0.943735i	$0.392714\pi$
$30$	0	0
$31$	−10.8284	−1.94484	−0.972421	−	0.233231i	$-0.925070\pi$
$31$	−10.8284	−1.94484	−0.972421	+	0.233231i	$0.925070\pi$
$32$	−2.30250	−0.407029
$33$	0	0
$34$	7.65685	1.31314
$35$	0	0
$36$	0	0
$37$	10.4853	1.72377	0.861885	−	0.507104i	$-0.169284\pi$
$37$	10.4853	1.72377	0.861885	+	0.507104i	$0.169284\pi$
$38$	0	0
$39$	0	0
$40$	−5.41421	−0.856062
$41$	6.81801	1.06479	0.532397	−	0.846495i	$-0.321291\pi$
$41$	6.81801	1.06479	0.532397	+	0.846495i	$0.321291\pi$
$42$	0	0
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	−0.737669	−0.111208
$45$	0	0
$46$	−7.65685	−1.12894
$47$	9.33657	1.36188	0.680939	−	0.732340i	$-0.261572\pi$
$47$	9.33657	1.36188	0.680939	+	0.732340i	$0.261572\pi$
$48$	0	0
$49$	0	0
$50$	2.30250	0.325623
$51$	0	0
$52$	−0.414214	−0.0574411
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−3.17157	−0.427655
$56$	0	0
$57$	0	0
$58$	−4.48528	−0.588946
$59$	−12.8984	−1.67922	−0.839611	−	0.543188i	$-0.817217\pi$
$59$	−12.8984	−1.67922	−0.839611	+	0.543188i	$0.817217\pi$
$60$	0	0
$61$	−7.65685	−0.980360	−0.490180	−	0.871621i	$-0.663069\pi$
$61$	−7.65685	−0.980360	−0.490180	+	0.871621i	$0.663069\pi$
$62$	13.6360	1.73178
$63$	0	0
$64$	8.89949	1.11244
$65$	−1.78089	−0.220892
$66$	0	0
$67$	8.48528	1.03664	0.518321	−	0.855186i	$-0.326557\pi$
$67$	8.48528	1.03664	0.518321	+	0.855186i	$0.326557\pi$
$68$	2.51856	0.305420
$69$	0	0
$70$	0	0
$71$	−6.81801	−0.809149	−0.404575	−	0.914505i	$-0.632580\pi$
$71$	−6.81801	−0.809149	−0.404575	+	0.914505i	$0.632580\pi$
$72$	0	0
$73$	12.1421	1.42113	0.710565	−	0.703632i	$-0.248440\pi$
$73$	12.1421	1.42113	0.710565	+	0.703632i	$0.248440\pi$
$74$	−13.2039	−1.53492
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	5.34267	0.597329
$81$	0	0
$82$	−8.58579	−0.948141
$83$	0.737669	0.0809697	0.0404849	−	0.999180i	$-0.487110\pi$
$83$	0.737669	0.0809697	0.0404849	+	0.999180i	$0.487110\pi$
$84$	0	0
$85$	10.8284	1.17451
$86$	12.1607	1.31132
$87$	0	0
$88$	5.41421	0.577157
$89$	−0.305553	−0.0323885	−0.0161943	−	0.999869i	$-0.505155\pi$
$89$	−0.305553	−0.0323885	−0.0161943	+	0.999869i	$0.505155\pi$
$90$	0	0
$91$	0	0
$92$	−2.51856	−0.262578
$93$	0	0
$94$	−11.7574	−1.21268
$95$	0	0
$96$	0	0
$97$	−7.65685	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$97$	−7.65685	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$98$	0	0
$99$	0	0
$100$	0.757359	0.0757359
$101$	4.60500	0.458215	0.229108	−	0.973401i	$-0.426419\pi$
$101$	4.60500	0.458215	0.229108	+	0.973401i	$0.426419\pi$
$102$	0	0
$103$	−18.4853	−1.82141	−0.910704	−	0.413059i	$-0.864460\pi$
$103$	−18.4853	−1.82141	−0.910704	+	0.413059i	$0.864460\pi$
$104$	3.04017	0.298113
$105$	0	0
$106$	0	0
$107$	6.08034	0.587809	0.293904	−	0.955835i	$-0.405045\pi$
$107$	6.08034	0.587809	0.293904	+	0.955835i	$0.405045\pi$
$108$	0	0
$109$	3.65685	0.350263	0.175132	−	0.984545i	$-0.443965\pi$
$109$	3.65685	0.350263	0.175132	+	0.984545i	$0.443965\pi$
$110$	3.99390	0.380803
$111$	0	0
$112$	0	0
$113$	13.6360	1.28277	0.641385	−	0.767220i	$-0.278360\pi$
$113$	13.6360	1.28277	0.641385	+	0.767220i	$0.278360\pi$
$114$	0	0
$115$	−10.8284	−1.00976
$116$	−1.47534	−0.136982
$117$	0	0
$118$	16.2426	1.49526
$119$	0	0
$120$	0	0
$121$	−7.82843	−0.711675
$122$	9.64212	0.872957
$123$	0	0
$124$	4.48528	0.402790
$125$	12.1607	1.08768
$126$	0	0
$127$	−4.34315	−0.385392	−0.192696	−	0.981259i	$-0.561723\pi$
$127$	−4.34315	−0.385392	−0.192696	+	0.981259i	$0.561723\pi$
$128$	−6.60195	−0.583536
$129$	0	0
$130$	2.24264	0.196693
$131$	15.7225	1.37368	0.686839	−	0.726809i	$-0.258998\pi$
$131$	15.7225	1.37368	0.686839	+	0.726809i	$0.258998\pi$
$132$	0	0
$133$	0	0
$134$	−10.6853	−0.923073
$135$	0	0
$136$	−18.4853	−1.58510
$137$	11.4230	0.975934	0.487967	−	0.872862i	$-0.337739\pi$
$137$	11.4230	0.975934	0.487967	+	0.872862i	$0.337739\pi$
$138$	0	0
$139$	12.1421	1.02988	0.514941	−	0.857225i	$-0.327814\pi$
$139$	12.1421	1.02988	0.514941	+	0.857225i	$0.327814\pi$
$140$	0	0
$141$	0	0
$142$	8.58579	0.720503
$143$	1.78089	0.148926
$144$	0	0
$145$	−6.34315	−0.526770
$146$	−15.2904	−1.26544
$147$	0	0
$148$	−4.34315	−0.357004
$149$	−12.8984	−1.05667	−0.528337	−	0.849035i	$-0.677184\pi$
$149$	−12.8984	−1.05667	−0.528337	+	0.849035i	$0.677184\pi$
$150$	0	0
$151$	0.485281	0.0394916	0.0197458	−	0.999805i	$-0.493714\pi$
$151$	0.485281	0.0394916	0.0197458	+	0.999805i	$0.493714\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	19.2842	1.54895
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	7.55568	0.601098
$159$	0	0
$160$	4.10051	0.324173
$161$	0	0
$162$	0	0
$163$	1.65685	0.129775	0.0648874	−	0.997893i	$-0.479331\pi$
$163$	1.65685	0.129775	0.0648874	+	0.997893i	$0.479331\pi$
$164$	−2.82411	−0.220526
$165$	0	0
$166$	−0.928932	−0.0720991
$167$	12.8984	0.998105	0.499052	−	0.866572i	$-0.333682\pi$
$167$	12.8984	0.998105	0.499052	+	0.866572i	$0.333682\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−13.6360	−1.04584
$171$	0	0
$172$	4.00000	0.304997
$173$	12.5928	0.957413	0.478706	−	0.877975i	$-0.341106\pi$
$173$	12.5928	0.957413	0.478706	+	0.877975i	$0.341106\pi$
$174$	0	0
$175$	0	0
$176$	−5.34267	−0.402719
$177$	0	0
$178$	0.384776	0.0288402
$179$	16.7657	1.25313	0.626563	−	0.779371i	$-0.284461\pi$
$179$	16.7657	1.25313	0.626563	+	0.779371i	$0.284461\pi$
$180$	0	0
$181$	15.3137	1.13826	0.569129	−	0.822248i	$-0.307280\pi$
$181$	15.3137	1.13826	0.569129	+	0.822248i	$0.307280\pi$
$182$	0	0
$183$	0	0
$184$	18.4853	1.36275
$185$	−18.6731	−1.37288
$186$	0	0
$187$	−10.8284	−0.791853
$188$	−3.86733	−0.282054
$189$	0	0
$190$	0	0
$191$	7.55568	0.546710	0.273355	−	0.961913i	$-0.411867\pi$
$191$	7.55568	0.546710	0.273355	+	0.961913i	$0.411867\pi$
$192$	0	0
$193$	−7.17157	−0.516221	−0.258111	−	0.966115i	$-0.583100\pi$
$193$	−7.17157	−0.516221	−0.258111	+	0.966115i	$0.583100\pi$
$194$	9.64212	0.692264
$195$	0	0
$196$	0	0
$197$	−25.0590	−1.78538	−0.892691	−	0.450669i	$-0.851186\pi$
$197$	−25.0590	−1.78538	−0.892691	+	0.450669i	$0.851186\pi$
$198$	0	0
$199$	−8.97056	−0.635906	−0.317953	−	0.948106i	$-0.602995\pi$
$199$	−8.97056	−0.635906	−0.317953	+	0.948106i	$0.602995\pi$
$200$	−5.55873	−0.393062
$201$	0	0
$202$	−5.79899	−0.408016
$203$	0	0
$204$	0	0
$205$	−12.1421	−0.848044
$206$	23.2781	1.62187
$207$	0	0
$208$	−3.00000	−0.208013
$209$	0	0
$210$	0	0
$211$	−10.9706	−0.755245	−0.377622	−	0.925960i	$-0.623258\pi$
$211$	−10.9706	−0.755245	−0.377622	+	0.925960i	$0.623258\pi$
$212$	0	0
$213$	0	0
$214$	−7.65685	−0.523412
$215$	17.1978	1.17288
$216$	0	0
$217$	0	0
$218$	−4.60500	−0.311890
$219$	0	0
$220$	1.31371	0.0885702
$221$	−6.08034	−0.409008
$222$	0	0
$223$	−8.97056	−0.600713	−0.300357	−	0.953827i	$-0.597106\pi$
$223$	−8.97056	−0.600713	−0.300357	+	0.953827i	$0.597106\pi$
$224$	0	0
$225$	0	0
$226$	−17.1716	−1.14224
$227$	7.86123	0.521768	0.260884	−	0.965370i	$-0.415986\pi$
$227$	7.86123	0.521768	0.260884	+	0.965370i	$0.415986\pi$
$228$	0	0
$229$	−18.4853	−1.22154	−0.610771	−	0.791807i	$-0.709140\pi$
$229$	−18.4853	−1.22154	−0.610771	+	0.791807i	$0.709140\pi$
$230$	13.6360	0.899133
$231$	0	0
$232$	10.8284	0.710921
$233$	−22.8460	−1.49669	−0.748347	−	0.663308i	$-0.769152\pi$
$233$	−22.8460	−1.49669	−0.748347	+	0.663308i	$0.769152\pi$
$234$	0	0
$235$	−16.6274	−1.08465
$236$	5.34267	0.347778
$237$	0	0
$238$	0	0
$239$	6.81801	0.441021	0.220510	−	0.975385i	$-0.429228\pi$
$239$	6.81801	0.441021	0.220510	+	0.975385i	$0.429228\pi$
$240$	0	0
$241$	−27.4558	−1.76859	−0.884293	−	0.466932i	$-0.845359\pi$
$241$	−27.4558	−1.76859	−0.884293	+	0.466932i	$0.845359\pi$
$242$	9.85818	0.633708
$243$	0	0
$244$	3.17157	0.203039
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−32.9203	−2.09044
$249$	0	0
$250$	−15.3137	−0.968524
$251$	10.6853	0.674453	0.337227	−	0.941424i	$-0.390511\pi$
$251$	10.6853	0.674453	0.337227	+	0.941424i	$0.390511\pi$
$252$	0	0
$253$	10.8284	0.680777
$254$	5.46924	0.343170
$255$	0	0
$256$	−9.48528	−0.592830
$257$	−7.55568	−0.471310	−0.235655	−	0.971837i	$-0.575724\pi$
$257$	−7.55568	−0.471310	−0.235655	+	0.971837i	$0.575724\pi$
$258$	0	0
$259$	0	0
$260$	0.737669	0.0457483
$261$	0	0
$262$	−19.7990	−1.22319
$263$	−18.2410	−1.12479	−0.562395	−	0.826869i	$-0.690120\pi$
$263$	−18.2410	−1.12479	−0.562395	+	0.826869i	$0.690120\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−3.51472	−0.214696
$269$	−7.55568	−0.460678	−0.230339	−	0.973110i	$-0.573984\pi$
$269$	−7.55568	−0.460678	−0.230339	+	0.973110i	$0.573984\pi$
$270$	0	0
$271$	−17.1716	−1.04310	−0.521549	−	0.853221i	$-0.674646\pi$
$271$	−17.1716	−1.04310	−0.521549	+	0.853221i	$0.674646\pi$
$272$	18.2410	1.10602
$273$	0	0
$274$	−14.3848	−0.869016
$275$	−3.25623	−0.196358
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	−15.2904	−0.917054
$279$	0	0
$280$	0	0
$281$	−12.8984	−0.769451	−0.384726	−	0.923031i	$-0.625704\pi$
$281$	−12.8984	−0.769451	−0.384726	+	0.923031i	$0.625704\pi$
$282$	0	0
$283$	−15.3137	−0.910305	−0.455153	−	0.890413i	$-0.650415\pi$
$283$	−15.3137	−0.910305	−0.455153	+	0.890413i	$0.650415\pi$
$284$	2.82411	0.167580
$285$	0	0
$286$	−2.24264	−0.132610
$287$	0	0
$288$	0	0
$289$	19.9706	1.17474
$290$	7.98780	0.469060
$291$	0	0
$292$	−5.02944	−0.294326
$293$	−31.1394	−1.81918	−0.909591	−	0.415505i	$-0.863605\pi$
$293$	−31.1394	−1.81918	−0.909591	+	0.415505i	$0.863605\pi$
$294$	0	0
$295$	22.9706	1.33740
$296$	31.8771	1.85282
$297$	0	0
$298$	16.2426	0.940911
$299$	6.08034	0.351635
$300$	0	0
$301$	0	0
$302$	−0.611105	−0.0351652
$303$	0	0
$304$	0	0
$305$	13.6360	0.780796
$306$	0	0
$307$	8.97056	0.511977	0.255989	−	0.966680i	$-0.417599\pi$
$307$	8.97056	0.511977	0.255989	+	0.966680i	$0.417599\pi$
$308$	0	0
$309$	0	0
$310$	−24.2843	−1.37925
$311$	−20.7596	−1.17717	−0.588584	−	0.808436i	$-0.700314\pi$
$311$	−20.7596	−1.17717	−0.588584	+	0.808436i	$0.700314\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	0	0
$315$	0	0
$316$	2.48528	0.139808
$317$	−22.1084	−1.24173	−0.620865	−	0.783918i	$-0.713218\pi$
$317$	−22.1084	−1.24173	−0.620865	+	0.783918i	$0.713218\pi$
$318$	0	0
$319$	6.34315	0.355148
$320$	−15.8490	−0.885988
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1.82843	−0.101423
$326$	−2.08644	−0.115557
$327$	0	0
$328$	20.7279	1.14451
$329$	0	0
$330$	0	0
$331$	13.6569	0.750649	0.375324	−	0.926894i	$-0.377531\pi$
$331$	13.6569	0.750649	0.375324	+	0.926894i	$0.377531\pi$
$332$	−0.305553	−0.0167694
$333$	0	0
$334$	−16.2426	−0.888758
$335$	−15.1114	−0.825622
$336$	0	0
$337$	−29.4558	−1.60456	−0.802281	−	0.596947i	$-0.796380\pi$
$337$	−29.4558	−1.60456	−0.802281	+	0.596947i	$0.796380\pi$
$338$	−1.25928	−0.0684958
$339$	0	0
$340$	−4.48528	−0.243249
$341$	−19.2842	−1.04430
$342$	0	0
$343$	0	0
$344$	−29.3585	−1.58290
$345$	0	0
$346$	−15.8579	−0.852524
$347$	−12.5928	−0.676017	−0.338008	−	0.941143i	$-0.609753\pi$
$347$	−12.5928	−0.676017	−0.338008	+	0.941143i	$0.609753\pi$
$348$	0	0
$349$	−1.31371	−0.0703212	−0.0351606	−	0.999382i	$-0.511194\pi$
$349$	−1.31371	−0.0703212	−0.0351606	+	0.999382i	$0.511194\pi$
$350$	0	0
$351$	0	0
$352$	−4.10051	−0.218558
$353$	−6.81801	−0.362886	−0.181443	−	0.983401i	$-0.558077\pi$
$353$	−6.81801	−0.362886	−0.181443	+	0.983401i	$0.558077\pi$
$354$	0	0
$355$	12.1421	0.644438
$356$	0.126564	0.00670788
$357$	0	0
$358$	−21.1127	−1.11584
$359$	16.0280	0.845927	0.422963	−	0.906147i	$-0.360990\pi$
$359$	16.0280	0.845927	0.422963	+	0.906147i	$0.360990\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−19.2842	−1.01356
$363$	0	0
$364$	0	0
$365$	−21.6238	−1.13184
$366$	0	0
$367$	27.4558	1.43318	0.716592	−	0.697493i	$-0.245701\pi$
$367$	27.4558	1.43318	0.716592	+	0.697493i	$0.245701\pi$
$368$	−18.2410	−0.950879
$369$	0	0
$370$	23.5147	1.22247
$371$	0	0
$372$	0	0
$373$	12.4853	0.646463	0.323232	−	0.946320i	$-0.395231\pi$
$373$	12.4853	0.646463	0.323232	+	0.946320i	$0.395231\pi$
$374$	13.6360	0.705102
$375$	0	0
$376$	28.3848	1.46383
$377$	3.56178	0.183441
$378$	0	0
$379$	−11.5147	−0.591471	−0.295736	−	0.955270i	$-0.595565\pi$
$379$	−11.5147	−0.591471	−0.295736	+	0.955270i	$0.595565\pi$
$380$	0	0
$381$	0	0
$382$	−9.51472	−0.486815
$383$	−28.6208	−1.46246	−0.731228	−	0.682133i	$-0.761052\pi$
$383$	−28.6208	−1.46246	−0.731228	+	0.682133i	$0.761052\pi$
$384$	0	0
$385$	0	0
$386$	9.03102	0.459667
$387$	0	0
$388$	3.17157	0.161012
$389$	−7.12356	−0.361179	−0.180590	−	0.983559i	$-0.557801\pi$
$389$	−7.12356	−0.361179	−0.180590	+	0.983559i	$0.557801\pi$
$390$	0	0
$391$	−36.9706	−1.86968
$392$	0	0
$393$	0	0
$394$	31.5563	1.58979
$395$	10.6853	0.537638
$396$	0	0
$397$	−31.1716	−1.56446	−0.782228	−	0.622992i	$-0.785917\pi$
$397$	−31.1716	−1.56446	−0.782228	+	0.622992i	$0.785917\pi$
$398$	11.2965	0.566240
$399$	0	0
$400$	5.48528	0.274264
$401$	−7.86123	−0.392571	−0.196286	−	0.980547i	$-0.562888\pi$
$401$	−7.86123	−0.392571	−0.196286	+	0.980547i	$0.562888\pi$
$402$	0	0
$403$	−10.8284	−0.539402
$404$	−1.90746	−0.0948994
$405$	0	0
$406$	0	0
$407$	18.6731	0.925593
$408$	0	0
$409$	5.02944	0.248690	0.124345	−	0.992239i	$-0.460317\pi$
$409$	5.02944	0.248690	0.124345	+	0.992239i	$0.460317\pi$
$410$	15.2904	0.755137
$411$	0	0
$412$	7.65685	0.377226
$413$	0	0
$414$	0	0
$415$	−1.31371	−0.0644874
$416$	−2.30250	−0.112889
$417$	0	0
$418$	0	0
$419$	−32.9203	−1.60826	−0.804130	−	0.594453i	$-0.797369\pi$
$419$	−32.9203	−1.60826	−0.804130	+	0.594453i	$0.797369\pi$
$420$	0	0
$421$	6.48528	0.316073	0.158037	−	0.987433i	$-0.449484\pi$
$421$	6.48528	0.316073	0.158037	+	0.987433i	$0.449484\pi$
$422$	13.8150	0.672504
$423$	0	0
$424$	0	0
$425$	11.1175	0.539276
$426$	0	0
$427$	0	0
$428$	−2.51856	−0.121739
$429$	0	0
$430$	−21.6569	−1.04439
$431$	16.0280	0.772043	0.386021	−	0.922490i	$-0.373849\pi$
$431$	16.0280	0.772043	0.386021	+	0.922490i	$0.373849\pi$
$432$	0	0
$433$	−22.9706	−1.10389	−0.551947	−	0.833879i	$-0.686115\pi$
$433$	−22.9706	−1.10389	−0.551947	+	0.833879i	$0.686115\pi$
$434$	0	0
$435$	0	0
$436$	−1.51472	−0.0725419
$437$	0	0
$438$	0	0
$439$	−27.4558	−1.31040	−0.655198	−	0.755457i	$-0.727415\pi$
$439$	−27.4558	−1.31040	−0.655198	+	0.755457i	$0.727415\pi$
$440$	−9.64212	−0.459670
$441$	0	0
$442$	7.65685	0.364199
$443$	−38.3895	−1.82394	−0.911970	−	0.410256i	$-0.865439\pi$
$443$	−38.3895	−1.82394	−0.911970	+	0.410256i	$0.865439\pi$
$444$	0	0
$445$	0.544156	0.0257955
$446$	11.2965	0.534902
$447$	0	0
$448$	0	0
$449$	38.6951	1.82613	0.913066	−	0.407811i	$-0.133708\pi$
$449$	38.6951	1.82613	0.913066	+	0.407811i	$0.133708\pi$
$450$	0	0
$451$	12.1421	0.571751
$452$	−5.64823	−0.265670
$453$	0	0
$454$	−9.89949	−0.464606
$455$	0	0
$456$	0	0
$457$	−2.97056	−0.138957	−0.0694785	−	0.997583i	$-0.522134\pi$
$457$	−2.97056	−0.138957	−0.0694785	+	0.997583i	$0.522134\pi$
$458$	23.2781	1.08772
$459$	0	0
$460$	4.48528	0.209127
$461$	−15.4169	−0.718037	−0.359019	−	0.933330i	$-0.616889\pi$
$461$	−15.4169	−0.718037	−0.359019	+	0.933330i	$0.616889\pi$
$462$	0	0
$463$	20.4853	0.952032	0.476016	−	0.879437i	$-0.342080\pi$
$463$	20.4853	0.952032	0.476016	+	0.879437i	$0.342080\pi$
$464$	−10.6853	−0.496055
$465$	0	0
$466$	28.7696	1.33272
$467$	3.56178	0.164820	0.0824098	−	0.996599i	$-0.473738\pi$
$467$	3.56178	0.164820	0.0824098	+	0.996599i	$0.473738\pi$
$468$	0	0
$469$	0	0
$470$	20.9386	0.965824
$471$	0	0
$472$	−39.2132	−1.80493
$473$	−17.1978	−0.790756
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−8.58579	−0.392705
$479$	−9.33657	−0.426599	−0.213299	−	0.976987i	$-0.568421\pi$
$479$	−9.33657	−0.426599	−0.213299	+	0.976987i	$0.568421\pi$
$480$	0	0
$481$	10.4853	0.478088
$482$	34.5746	1.57483
$483$	0	0
$484$	3.24264	0.147393
$485$	13.6360	0.619180
$486$	0	0
$487$	0.485281	0.0219902	0.0109951	−	0.999940i	$-0.496500\pi$
$487$	0.485281	0.0219902	0.0109951	+	0.999940i	$0.496500\pi$
$488$	−23.2781	−1.05375
$489$	0	0
$490$	0	0
$491$	9.03102	0.407564	0.203782	−	0.979016i	$-0.434677\pi$
$491$	9.03102	0.407564	0.203782	+	0.979016i	$0.434677\pi$
$492$	0	0
$493$	−21.6569	−0.975376
$494$	0	0
$495$	0	0
$496$	32.4853	1.45863
$497$	0	0
$498$	0	0
$499$	−32.2843	−1.44524	−0.722621	−	0.691244i	$-0.757063\pi$
$499$	−32.2843	−1.44524	−0.722621	+	0.691244i	$0.757063\pi$
$500$	−5.03712	−0.225267
$501$	0	0
$502$	−13.4558	−0.600564
$503$	−16.5867	−0.739564	−0.369782	−	0.929118i	$-0.620568\pi$
$503$	−16.5867	−0.739564	−0.369782	+	0.929118i	$0.620568\pi$
$504$	0	0
$505$	−8.20101	−0.364940
$506$	−13.6360	−0.606195
$507$	0	0
$508$	1.79899	0.0798173
$509$	−36.1765	−1.60350	−0.801748	−	0.597663i	$-0.796096\pi$
$509$	−36.1765	−1.60350	−0.801748	+	0.597663i	$0.796096\pi$
$510$	0	0
$511$	0	0
$512$	25.1485	1.11142
$513$	0	0
$514$	9.51472	0.419676
$515$	32.9203	1.45064
$516$	0	0
$517$	16.6274	0.731273
$518$	0	0
$519$	0	0
$520$	−5.41421	−0.237429
$521$	14.6792	0.643109	0.321555	−	0.946891i	$-0.395795\pi$
$521$	14.6792	0.643109	0.321555	+	0.946891i	$0.395795\pi$
$522$	0	0
$523$	−9.51472	−0.416050	−0.208025	−	0.978124i	$-0.566703\pi$
$523$	−9.51472	−0.416050	−0.208025	+	0.978124i	$0.566703\pi$
$524$	−6.51246	−0.284498
$525$	0	0
$526$	22.9706	1.00156
$527$	65.8405	2.86806
$528$	0	0
$529$	13.9706	0.607416
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.81801	0.295321
$534$	0	0
$535$	−10.8284	−0.468154
$536$	25.7967	1.11425
$537$	0	0
$538$	9.51472	0.410209
$539$	0	0
$540$	0	0
$541$	−19.4558	−0.836472	−0.418236	−	0.908338i	$-0.637352\pi$
$541$	−19.4558	−0.836472	−0.418236	+	0.908338i	$0.637352\pi$
$542$	21.6238	0.928823
$543$	0	0
$544$	14.0000	0.600245
$545$	−6.51246	−0.278963
$546$	0	0
$547$	31.3137	1.33888	0.669439	−	0.742867i	$-0.266535\pi$
$547$	31.3137	1.33888	0.669439	+	0.742867i	$0.266535\pi$
$548$	−4.73157	−0.202123
$549$	0	0
$550$	4.10051	0.174846
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−22.6670	−0.963030
$555$	0	0
$556$	−5.02944	−0.213296
$557$	7.25013	0.307198	0.153599	−	0.988133i	$-0.450914\pi$
$557$	7.25013	0.307198	0.153599	+	0.988133i	$0.450914\pi$
$558$	0	0
$559$	−9.65685	−0.408441
$560$	0	0
$561$	0	0
$562$	16.2426	0.685154
$563$	−25.1856	−1.06145	−0.530723	−	0.847545i	$-0.678080\pi$
$563$	−25.1856	−1.06145	−0.530723	+	0.847545i	$0.678080\pi$
$564$	0	0
$565$	−24.2843	−1.02165
$566$	19.2842	0.810577
$567$	0	0
$568$	−20.7279	−0.869724
$569$	30.8338	1.29262	0.646311	−	0.763074i	$-0.276311\pi$
$569$	30.8338	1.29262	0.646311	+	0.763074i	$0.276311\pi$
$570$	0	0
$571$	18.0000	0.753277	0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000	0.753277	0.376638	+	0.926360i	$0.377080\pi$
$572$	−0.737669	−0.0308435
$573$	0	0
$574$	0	0
$575$	−11.1175	−0.463630
$576$	0	0
$577$	40.1421	1.67114	0.835569	−	0.549385i	$-0.185138\pi$
$577$	40.1421	1.67114	0.835569	+	0.549385i	$0.185138\pi$
$578$	−25.1485	−1.04604
$579$	0	0
$580$	2.62742	0.109098
$581$	0	0
$582$	0	0
$583$	0	0
$584$	36.9142	1.52752
$585$	0	0
$586$	39.2132	1.61988
$587$	−3.68835	−0.152234	−0.0761172	−	0.997099i	$-0.524252\pi$
$587$	−3.68835	−0.152234	−0.0761172	+	0.997099i	$0.524252\pi$
$588$	0	0
$589$	0	0
$590$	−28.9264	−1.19088
$591$	0	0
$592$	−31.4558	−1.29283
$593$	40.3494	1.65695	0.828475	−	0.560025i	$-0.189209\pi$
$593$	40.3494	1.65695	0.828475	+	0.560025i	$0.189209\pi$
$594$	0	0
$595$	0	0
$596$	5.34267	0.218844
$597$	0	0
$598$	−7.65685	−0.313112
$599$	1.90746	0.0779365	0.0389683	−	0.999240i	$-0.487593\pi$
$599$	1.90746	0.0779365	0.0389683	+	0.999240i	$0.487593\pi$
$600$	0	0
$601$	22.9706	0.936989	0.468494	−	0.883466i	$-0.344797\pi$
$601$	22.9706	0.936989	0.468494	+	0.883466i	$0.344797\pi$
$602$	0	0
$603$	0	0
$604$	−0.201010	−0.00817899
$605$	13.9416	0.566806
$606$	0	0
$607$	24.8284	1.00775	0.503877	−	0.863775i	$-0.331906\pi$
$607$	24.8284	1.00775	0.503877	+	0.863775i	$0.331906\pi$
$608$	0	0
$609$	0	0
$610$	−17.1716	−0.695257
$611$	9.33657	0.377717
$612$	0	0
$613$	42.2843	1.70785	0.853923	−	0.520400i	$-0.174217\pi$
$613$	42.2843	1.70785	0.853923	+	0.520400i	$0.174217\pi$
$614$	−11.2965	−0.455888
$615$	0	0
$616$	0	0
$617$	−12.2872	−0.494666	−0.247333	−	0.968931i	$-0.579554\pi$
$617$	−12.2872	−0.494666	−0.247333	+	0.968931i	$0.579554\pi$
$618$	0	0
$619$	−1.85786	−0.0746739	−0.0373369	−	0.999303i	$-0.511887\pi$
$619$	−1.85786	−0.0746739	−0.0373369	+	0.999303i	$0.511887\pi$
$620$	−7.98780	−0.320798
$621$	0	0
$622$	26.1421	1.04820
$623$	0	0
$624$	0	0
$625$	−12.5147	−0.500589
$626$	17.6299	0.704633
$627$	0	0
$628$	0	0
$629$	−63.7541	−2.54204
$630$	0	0
$631$	−8.28427	−0.329792	−0.164896	−	0.986311i	$-0.552729\pi$
$631$	−8.28427	−0.329792	−0.164896	+	0.986311i	$0.552729\pi$
$632$	−18.2410	−0.725589
$633$	0	0
$634$	27.8406	1.10569
$635$	7.73467	0.306941
$636$	0	0
$637$	0	0
$638$	−7.98780	−0.316240
$639$	0	0
$640$	11.7574	0.464750
$641$	−26.4078	−1.04305	−0.521523	−	0.853237i	$-0.674636\pi$
$641$	−26.4078	−1.04305	−0.521523	+	0.853237i	$0.674636\pi$
$642$	0	0
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.21001	−0.362083	−0.181041	−	0.983475i	$-0.557947\pi$
$647$	−9.21001	−0.362083	−0.181041	+	0.983475i	$0.557947\pi$
$648$	0	0
$649$	−22.9706	−0.901673
$650$	2.30250	0.0903116
$651$	0	0
$652$	−0.686292	−0.0268772
$653$	1.47534	0.0577345	0.0288672	−	0.999583i	$-0.490810\pi$
$653$	1.47534	0.0577345	0.0288672	+	0.999583i	$0.490810\pi$
$654$	0	0
$655$	−28.0000	−1.09405
$656$	−20.4540	−0.798596
$657$	0	0
$658$	0	0
$659$	17.6299	0.686764	0.343382	−	0.939196i	$-0.388427\pi$
$659$	17.6299	0.686764	0.343382	+	0.939196i	$0.388427\pi$
$660$	0	0
$661$	7.65685	0.297817	0.148909	−	0.988851i	$-0.452424\pi$
$661$	7.65685	0.297817	0.148909	+	0.988851i	$0.452424\pi$
$662$	−17.1978	−0.668412
$663$	0	0
$664$	2.24264	0.0870313
$665$	0	0
$666$	0	0
$667$	21.6569	0.838557
$668$	−5.34267	−0.206714
$669$	0	0
$670$	19.0294	0.735171
$671$	−13.6360	−0.526413
$672$	0	0
$673$	6.97056	0.268695	0.134348	−	0.990934i	$-0.457106\pi$
$673$	6.97056	0.268695	0.134348	+	0.990934i	$0.457106\pi$
$674$	37.0932	1.42878
$675$	0	0
$676$	−0.414214	−0.0159313
$677$	6.08034	0.233687	0.116843	−	0.993150i	$-0.462722\pi$
$677$	6.08034	0.233687	0.116843	+	0.993150i	$0.462722\pi$
$678$	0	0
$679$	0	0
$680$	32.9203	1.26243
$681$	0	0
$682$	24.2843	0.929893
$683$	−21.9294	−0.839104	−0.419552	−	0.907731i	$-0.637813\pi$
$683$	−21.9294	−0.839104	−0.419552	+	0.907731i	$0.637813\pi$
$684$	0	0
$685$	−20.3431	−0.777272
$686$	0	0
$687$	0	0
$688$	28.9706	1.10449
$689$	0	0
$690$	0	0
$691$	4.48528	0.170628	0.0853141	−	0.996354i	$-0.472811\pi$
$691$	4.48528	0.170628	0.0853141	+	0.996354i	$0.472811\pi$
$692$	−5.21611	−0.198287
$693$	0	0
$694$	15.8579	0.601956
$695$	−21.6238	−0.820238
$696$	0	0
$697$	−41.4558	−1.57025
$698$	1.65433	0.0626172
$699$	0	0
$700$	0	0
$701$	5.03712	0.190249	0.0951247	−	0.995465i	$-0.469675\pi$
$701$	5.03712	0.190249	0.0951247	+	0.995465i	$0.469675\pi$
$702$	0	0
$703$	0	0
$704$	15.8490	0.597333
$705$	0	0
$706$	8.58579	0.323130
$707$	0	0
$708$	0	0
$709$	−43.6569	−1.63957	−0.819784	−	0.572673i	$-0.805906\pi$
$709$	−43.6569	−1.63957	−0.819784	+	0.572673i	$0.805906\pi$
$710$	−15.2904	−0.573837
$711$	0	0
$712$	−0.928932	−0.0348132
$713$	−65.8405	−2.46575
$714$	0	0
$715$	−3.17157	−0.118610
$716$	−6.94458	−0.259531
$717$	0	0
$718$	−20.1838	−0.753251
$719$	−28.7474	−1.07210	−0.536048	−	0.844187i	$-0.680084\pi$
$719$	−28.7474	−1.07210	−0.536048	+	0.844187i	$0.680084\pi$
$720$	0	0
$721$	0	0
$722$	23.9263	0.890446
$723$	0	0
$724$	−6.34315	−0.235741
$725$	−6.51246	−0.241867
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	0	0
$729$	0	0
$730$	27.2304	1.00784
$731$	58.7170	2.17173
$732$	0	0
$733$	−9.51472	−0.351434	−0.175717	−	0.984441i	$-0.556224\pi$
$733$	−9.51472	−0.351434	−0.175717	+	0.984441i	$0.556224\pi$
$734$	−34.5746	−1.27617
$735$	0	0
$736$	−14.0000	−0.516047
$737$	15.1114	0.556634
$738$	0	0
$739$	−13.1716	−0.484524	−0.242262	−	0.970211i	$-0.577889\pi$
$739$	−13.1716	−0.484524	−0.242262	+	0.970211i	$0.577889\pi$
$740$	7.73467	0.284332
$741$	0	0
$742$	0	0
$743$	−42.6890	−1.56611	−0.783053	−	0.621955i	$-0.786339\pi$
$743$	−42.6890	−1.56611	−0.783053	+	0.621955i	$0.786339\pi$
$744$	0	0
$745$	22.9706	0.841576
$746$	−15.7225	−0.575640
$747$	0	0
$748$	4.48528	0.163998
$749$	0	0
$750$	0	0
$751$	44.2843	1.61596	0.807978	−	0.589213i	$-0.200562\pi$
$751$	44.2843	1.61596	0.807978	+	0.589213i	$0.200562\pi$
$752$	−28.0097	−1.02141
$753$	0	0
$754$	−4.48528	−0.163344
$755$	−0.864233	−0.0314527
$756$	0	0
$757$	7.51472	0.273127	0.136564	−	0.990631i	$-0.456394\pi$
$757$	7.51472	0.273127	0.136564	+	0.990631i	$0.456394\pi$
$758$	14.5003	0.526673
$759$	0	0
$760$	0	0
$761$	−13.3305	−0.483229	−0.241615	−	0.970372i	$-0.577677\pi$
$761$	−13.3305	−0.483229	−0.241615	+	0.970372i	$0.577677\pi$
$762$	0	0
$763$	0	0
$764$	−3.12967	−0.113227
$765$	0	0
$766$	36.0416	1.30224
$767$	−12.8984	−0.465733
$768$	0	0
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	0	0
$772$	2.97056	0.106913
$773$	14.5527	0.523424	0.261712	−	0.965146i	$-0.415713\pi$
$773$	14.5527	0.523424	0.261712	+	0.965146i	$0.415713\pi$
$774$	0	0
$775$	19.7990	0.711201
$776$	−23.2781	−0.835637
$777$	0	0
$778$	8.97056	0.321610
$779$	0	0
$780$	0	0
$781$	−12.1421	−0.434480
$782$	46.5563	1.66485
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.85786	−0.0662257	−0.0331129	−	0.999452i	$-0.510542\pi$
$787$	−1.85786	−0.0662257	−0.0331129	+	0.999452i	$0.510542\pi$
$788$	10.3798	0.369765
$789$	0	0
$790$	−13.4558	−0.478737
$791$	0	0
$792$	0	0
$793$	−7.65685	−0.271903
$794$	39.2537	1.39306
$795$	0	0
$796$	3.71573	0.131701
$797$	25.3646	0.898460	0.449230	−	0.893416i	$-0.351698\pi$
$797$	25.3646	0.898460	0.449230	+	0.893416i	$0.351698\pi$
$798$	0	0
$799$	−56.7696	−2.00836
$800$	4.20996	0.148844
$801$	0	0
$802$	9.89949	0.349563
$803$	21.6238	0.763088
$804$	0	0
$805$	0	0
$806$	13.6360	0.480308
$807$	0	0
$808$	14.0000	0.492518
$809$	−35.0067	−1.23077	−0.615385	−	0.788226i	$-0.711000\pi$
$809$	−35.0067	−1.23077	−0.615385	+	0.788226i	$0.711000\pi$
$810$	0	0
$811$	4.48528	0.157500	0.0787498	−	0.996894i	$-0.474907\pi$
$811$	4.48528	0.157500	0.0787498	+	0.996894i	$0.474907\pi$
$812$	0	0
$813$	0	0
$814$	−23.5147	−0.824190
$815$	−2.95068	−0.103358
$816$	0	0
$817$	0	0
$818$	−6.33347	−0.221445
$819$	0	0
$820$	5.02944	0.175636
$821$	8.47234	0.295687	0.147843	−	0.989011i	$-0.452767\pi$
$821$	8.47234	0.295687	0.147843	+	0.989011i	$0.452767\pi$
$822$	0	0
$823$	−20.2843	−0.707065	−0.353533	−	0.935422i	$-0.615020\pi$
$823$	−20.2843	−0.707065	−0.353533	+	0.935422i	$0.615020\pi$
$824$	−56.1984	−1.95776
$825$	0	0
$826$	0	0
$827$	−52.5101	−1.82595	−0.912977	−	0.408011i	$-0.866223\pi$
$827$	−52.5101	−1.82595	−0.912977	+	0.408011i	$0.866223\pi$
$828$	0	0
$829$	26.6863	0.926853	0.463427	−	0.886135i	$-0.346620\pi$
$829$	26.6863	0.926853	0.463427	+	0.886135i	$0.346620\pi$
$830$	1.65433	0.0574225
$831$	0	0
$832$	8.89949	0.308534
$833$	0	0
$834$	0	0
$835$	−22.9706	−0.794929
$836$	0	0
$837$	0	0
$838$	41.4558	1.43207
$839$	28.6208	0.988100	0.494050	−	0.869433i	$-0.335516\pi$
$839$	28.6208	0.988100	0.494050	+	0.869433i	$0.335516\pi$
$840$	0	0
$841$	−16.3137	−0.562542
$842$	−8.16679	−0.281446
$843$	0	0
$844$	4.54416	0.156416
$845$	−1.78089	−0.0612645
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	−14.0000	−0.480196
$851$	63.7541	2.18546
$852$	0	0
$853$	−27.4558	−0.940070	−0.470035	−	0.882648i	$-0.655759\pi$
$853$	−27.4558	−0.940070	−0.470035	+	0.882648i	$0.655759\pi$
$854$	0	0
$855$	0	0
$856$	18.4853	0.631814
$857$	−38.1364	−1.30271	−0.651357	−	0.758771i	$-0.725800\pi$
$857$	−38.1364	−1.30271	−0.651357	+	0.758771i	$0.725800\pi$
$858$	0	0
$859$	−24.2843	−0.828569	−0.414284	−	0.910148i	$-0.635968\pi$
$859$	−24.2843	−0.828569	−0.414284	+	0.910148i	$0.635968\pi$
$860$	−7.12356	−0.242912
$861$	0	0
$862$	−20.1838	−0.687462
$863$	26.7134	0.909334	0.454667	−	0.890662i	$-0.349758\pi$
$863$	26.7134	0.909334	0.454667	+	0.890662i	$0.349758\pi$
$864$	0	0
$865$	−22.4264	−0.762521
$866$	28.9264	0.982958
$867$	0	0
$868$	0	0
$869$	−10.6853	−0.362476
$870$	0	0
$871$	8.48528	0.287513
$872$	11.1175	0.376485
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	34.5746	1.16684
$879$	0	0
$880$	9.51472	0.320741
$881$	7.55568	0.254557	0.127279	−	0.991867i	$-0.459376\pi$
$881$	7.55568	0.254557	0.127279	+	0.991867i	$0.459376\pi$
$882$	0	0
$883$	19.6569	0.661506	0.330753	−	0.943717i	$-0.392697\pi$
$883$	19.6569	0.661506	0.330753	+	0.943717i	$0.392697\pi$
$884$	2.51856	0.0847083
$885$	0	0
$886$	48.3431	1.62412
$887$	−48.6427	−1.63326	−0.816632	−	0.577159i	$-0.804161\pi$
$887$	−48.6427	−1.63326	−0.816632	+	0.577159i	$0.804161\pi$
$888$	0	0
$889$	0	0
$890$	−0.685245	−0.0229695
$891$	0	0
$892$	3.71573	0.124412
$893$	0	0
$894$	0	0
$895$	−29.8579	−0.998038
$896$	0	0
$897$	0	0
$898$	−48.7279	−1.62607
$899$	−38.5685	−1.28633
$900$	0	0
$901$	0	0
$902$	−15.2904	−0.509113
$903$	0	0
$904$	41.4558	1.37880
$905$	−27.2720	−0.906553
$906$	0	0
$907$	27.9411	0.927770	0.463885	−	0.885895i	$-0.346455\pi$
$907$	27.9411	0.927770	0.463885	+	0.885895i	$0.346455\pi$
$908$	−3.25623	−0.108062
$909$	0	0
$910$	0	0
$911$	−49.9391	−1.65456	−0.827278	−	0.561793i	$-0.810112\pi$
$911$	−49.9391	−1.65456	−0.827278	+	0.561793i	$0.810112\pi$
$912$	0	0
$913$	1.31371	0.0434774
$914$	3.74077	0.123734
$915$	0	0
$916$	7.65685	0.252990
$917$	0	0
$918$	0	0
$919$	−15.6569	−0.516472	−0.258236	−	0.966082i	$-0.583141\pi$
$919$	−15.6569	−0.516472	−0.258236	+	0.966082i	$0.583141\pi$
$920$	−32.9203	−1.08535
$921$	0	0
$922$	19.4142	0.639373
$923$	−6.81801	−0.224418
$924$	0	0
$925$	−19.1716	−0.630357
$926$	−25.7967	−0.847732
$927$	0	0
$928$	−8.20101	−0.269211
$929$	35.5654	1.16686	0.583431	−	0.812162i	$-0.301710\pi$
$929$	35.5654	1.16686	0.583431	+	0.812162i	$0.301710\pi$
$930$	0	0
$931$	0	0
$932$	9.46314	0.309975
$933$	0	0
$934$	−4.48528	−0.146763
$935$	19.2842	0.630662
$936$	0	0
$937$	44.6274	1.45791	0.728957	−	0.684559i	$-0.240005\pi$
$937$	44.6274	1.45791	0.728957	+	0.684559i	$0.240005\pi$
$938$	0	0
$939$	0	0
$940$	6.88730	0.224639
$941$	5.34267	0.174166	0.0870831	−	0.996201i	$-0.472245\pi$
$941$	5.34267	0.174166	0.0870831	+	0.996201i	$0.472245\pi$
$942$	0	0
$943$	41.4558	1.34999
$944$	38.6951	1.25942
$945$	0	0
$946$	21.6569	0.704125
$947$	−46.8618	−1.52281	−0.761403	−	0.648279i	$-0.775489\pi$
$947$	−46.8618	−1.52281	−0.761403	+	0.648279i	$0.775489\pi$
$948$	0	0
$949$	12.1421	0.394150
$950$	0	0
$951$	0	0
$952$	0	0
$953$	50.9823	1.65148	0.825740	−	0.564052i	$-0.190758\pi$
$953$	50.9823	1.65148	0.825740	+	0.564052i	$0.190758\pi$
$954$	0	0
$955$	−13.4558	−0.435421
$956$	−2.82411	−0.0913383
$957$	0	0
$958$	11.7574	0.379863
$959$	0	0
$960$	0	0
$961$	86.2548	2.78241
$962$	−13.2039	−0.425711
$963$	0	0
$964$	11.3726	0.366286
$965$	12.7718	0.411138
$966$	0	0
$967$	33.9411	1.09147	0.545737	−	0.837957i	$-0.316250\pi$
$967$	33.9411	1.09147	0.545737	+	0.837957i	$0.316250\pi$
$968$	−23.7998	−0.764953
$969$	0	0
$970$	−17.1716	−0.551346
$971$	−43.6056	−1.39937	−0.699685	−	0.714451i	$-0.746677\pi$
$971$	−43.6056	−1.39937	−0.699685	+	0.714451i	$0.746677\pi$
$972$	0	0
$973$	0	0
$974$	−0.611105	−0.0195811
$975$	0	0
$976$	22.9706	0.735270
$977$	−22.1084	−0.707309	−0.353655	−	0.935376i	$-0.615061\pi$
$977$	−22.1084	−0.707309	−0.353655	+	0.935376i	$0.615061\pi$
$978$	0	0
$979$	−0.544156	−0.0173913
$980$	0	0
$981$	0	0
$982$	−11.3726	−0.362914
$983$	−17.3244	−0.552562	−0.276281	−	0.961077i	$-0.589102\pi$
$983$	−17.3244	−0.552562	−0.276281	+	0.961077i	$0.589102\pi$
$984$	0	0
$985$	44.6274	1.42195
$986$	27.2720	0.868519
$987$	0	0
$988$	0	0
$989$	−58.7170	−1.86709
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	24.9325	0.791607
$993$	0	0
$994$	0	0
$995$	15.9756	0.506460
$996$	0	0
$997$	19.0294	0.602668	0.301334	−	0.953519i	$-0.402568\pi$
$997$	19.0294	0.602668	0.301334	+	0.953519i	$0.402568\pi$
$998$	40.6549	1.28691
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.bg.1.2	✓	4
3.2	odd	2	inner	5733.2.a.bg.1.3	yes	4
7.6	odd	2		5733.2.a.bh.1.2	yes	4
21.20	even	2		5733.2.a.bh.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5733.2.a.bg.1.2	✓	4	1.1	even	1	trivial
5733.2.a.bg.1.3	yes	4	3.2	odd	2	inner
5733.2.a.bh.1.2	yes	4	7.6	odd	2
5733.2.a.bh.1.3	yes	4	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5733.a (trivial)

\(f(q)\)	\(=\)	\(q-1.25928 q^{2} -0.414214 q^{4} -1.78089 q^{5} +3.04017 q^{8} +O(q^{10})\)	\(q-1.25928 q^{2} -0.414214 q^{4} -1.78089 q^{5} +3.04017 q^{8} +2.24264 q^{10} +1.78089 q^{11} +1.00000 q^{13} -3.00000 q^{16} -6.08034 q^{17} +0.737669 q^{20} -2.24264 q^{22} +6.08034 q^{23} -1.82843 q^{25} -1.25928 q^{26} +3.56178 q^{29} -10.8284 q^{31} -2.30250 q^{32} +7.65685 q^{34} +10.4853 q^{37} -5.41421 q^{40} +6.81801 q^{41} -9.65685 q^{43} -0.737669 q^{44} -7.65685 q^{46} +9.33657 q^{47} +2.30250 q^{50} -0.414214 q^{52} -3.17157 q^{55} -4.48528 q^{58} -12.8984 q^{59} -7.65685 q^{61} +13.6360 q^{62} +8.89949 q^{64} -1.78089 q^{65} +8.48528 q^{67} +2.51856 q^{68} -6.81801 q^{71} +12.1421 q^{73} -13.2039 q^{74} -6.00000 q^{79} +5.34267 q^{80} -8.58579 q^{82} +0.737669 q^{83} +10.8284 q^{85} +12.1607 q^{86} +5.41421 q^{88} -0.305553 q^{89} -2.51856 q^{92} -11.7574 q^{94} -7.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4}+O(q^{10}) \) 4 * q + 4 * q^4	\( 4 q + 4 q^{4} - 8 q^{10} + 4 q^{13} - 12 q^{16} + 8 q^{22} + 4 q^{25} - 32 q^{31} + 8 q^{34} + 8 q^{37} - 16 q^{40} - 16 q^{43} - 8 q^{46} + 4 q^{52} - 24 q^{55} + 16 q^{58} - 8 q^{61} - 4 q^{64} - 8 q^{73} - 24 q^{79} - 40 q^{82} + 32 q^{85} + 16 q^{88} - 64 q^{94} - 8 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 - 8 * q^10 + 4 * q^13 - 12 * q^16 + 8 * q^22 + 4 * q^25 - 32 * q^31 + 8 * q^34 + 8 * q^37 - 16 * q^40 - 16 * q^43 - 8 * q^46 + 4 * q^52 - 24 * q^55 + 16 * q^58 - 8 * q^61 - 4 * q^64 - 8 * q^73 - 24 * q^79 - 40 * q^82 + 32 * q^85 + 16 * q^88 - 64 * q^94 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more