LMFDB - Embedded newform 819.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,2,Mod(1,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, 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("819.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 819 = 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	819.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:	$6.53974792554$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 4 $ x^4 - 7*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.792287$ of defining polynomial
Character	$\chi$	$=$	819.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.792287 q^{2} -1.37228 q^{4} -2.52434 q^{5} +1.00000 q^{7} +2.67181 q^{8} +O(q^{10})$	\(q-0.792287 q^{2} -1.37228 q^{4} -2.52434 q^{5} +1.00000 q^{7} +2.67181 q^{8} +2.00000 q^{10} -3.46410 q^{11} -1.00000 q^{13} -0.792287 q^{14} +0.627719 q^{16} -1.58457 q^{17} +1.62772 q^{19} +3.46410 q^{20} +2.74456 q^{22} -4.40387 q^{23} +1.37228 q^{25} +0.792287 q^{26} -1.37228 q^{28} +5.98844 q^{29} +6.37228 q^{31} -5.84096 q^{32} +1.25544 q^{34} -2.52434 q^{35} +6.74456 q^{37} -1.28962 q^{38} -6.74456 q^{40} +1.58457 q^{41} +11.1168 q^{43} +4.75372 q^{44} +3.48913 q^{46} +9.15759 q^{47} +1.00000 q^{49} -1.08724 q^{50} +1.37228 q^{52} +0.939764 q^{53} +8.74456 q^{55} +2.67181 q^{56} -4.74456 q^{58} +5.04868 q^{59} +6.00000 q^{61} -5.04868 q^{62} +3.37228 q^{64} +2.52434 q^{65} -8.74456 q^{67} +2.17448 q^{68} +2.00000 q^{70} -1.58457 q^{71} +8.37228 q^{73} -5.34363 q^{74} -2.23369 q^{76} -3.46410 q^{77} -11.8614 q^{79} -1.58457 q^{80} -1.25544 q^{82} +12.3267 q^{83} +4.00000 q^{85} -8.80773 q^{86} -9.25544 q^{88} -2.52434 q^{89} -1.00000 q^{91} +6.04334 q^{92} -7.25544 q^{94} -4.10891 q^{95} +3.62772 q^{97} -0.792287 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4} + 4 q^{7}+O(q^{10}) $ 4 * q + 6 * q^4 + 4 * q^7	$ 4 q + 6 q^{4} + 4 q^{7} + 8 q^{10} - 4 q^{13} + 14 q^{16} + 18 q^{19} - 12 q^{22} - 6 q^{25} + 6 q^{28} + 14 q^{31} + 28 q^{34} + 4 q^{37} - 4 q^{40} + 10 q^{43} - 32 q^{46} + 4 q^{49} - 6 q^{52} + 12 q^{55} + 4 q^{58} + 24 q^{61} + 2 q^{64} - 12 q^{67} + 8 q^{70} + 22 q^{73} + 60 q^{76} + 10 q^{79} - 28 q^{82} + 16 q^{85} - 60 q^{88} - 4 q^{91} - 52 q^{94} + 26 q^{97}+O(q^{100}) $ 4 * q + 6 * q^4 + 4 * q^7 + 8 * q^10 - 4 * q^13 + 14 * q^16 + 18 * q^19 - 12 * q^22 - 6 * q^25 + 6 * q^28 + 14 * q^31 + 28 * q^34 + 4 * q^37 - 4 * q^40 + 10 * q^43 - 32 * q^46 + 4 * q^49 - 6 * q^52 + 12 * q^55 + 4 * q^58 + 24 * q^61 + 2 * q^64 - 12 * q^67 + 8 * q^70 + 22 * q^73 + 60 * q^76 + 10 * q^79 - 28 * q^82 + 16 * q^85 - 60 * q^88 - 4 * q^91 - 52 * q^94 + 26 * 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 $

$2$	−0.792287	−0.560232	−0.280116	−	0.959966i	$-0.590373\pi$
$2$	−0.792287	−0.560232	−0.280116	+	0.959966i	$0.590373\pi$
$3$	0	0
$3$	0	0
$4$	−1.37228	−0.686141
$5$	−2.52434	−1.12892	−0.564459	−	0.825461i	$-0.690915\pi$
$5$	−2.52434	−1.12892	−0.564459	+	0.825461i	$0.690915\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.67181	0.944629
$9$	0	0
$10$	2.00000	0.632456
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−0.792287	−0.211748
$15$	0	0
$16$	0.627719	0.156930
$17$	−1.58457	−0.384316	−0.192158	−	0.981364i	$-0.561549\pi$
$17$	−1.58457	−0.384316	−0.192158	+	0.981364i	$0.561549\pi$
$18$	0	0
$19$	1.62772	0.373424	0.186712	−	0.982415i	$-0.440217\pi$
$19$	1.62772	0.373424	0.186712	+	0.982415i	$0.440217\pi$
$20$	3.46410	0.774597
$21$	0	0
$22$	2.74456	0.585143
$23$	−4.40387	−0.918269	−0.459135	−	0.888367i	$-0.651840\pi$
$23$	−4.40387	−0.918269	−0.459135	+	0.888367i	$0.651840\pi$
$24$	0	0
$25$	1.37228	0.274456
$26$	0.792287	0.155380
$27$	0	0
$28$	−1.37228	−0.259337
$29$	5.98844	1.11203	0.556013	−	0.831174i	$-0.312331\pi$
$29$	5.98844	1.11203	0.556013	+	0.831174i	$0.312331\pi$
$30$	0	0
$31$	6.37228	1.14450	0.572248	−	0.820081i	$-0.306072\pi$
$31$	6.37228	1.14450	0.572248	+	0.820081i	$0.306072\pi$
$32$	−5.84096	−1.03255
$33$	0	0
$34$	1.25544	0.215306
$35$	−2.52434	−0.426691
$36$	0	0
$37$	6.74456	1.10880	0.554400	−	0.832251i	$-0.312948\pi$
$37$	6.74456	1.10880	0.554400	+	0.832251i	$0.312948\pi$
$38$	−1.28962	−0.209204
$39$	0	0
$40$	−6.74456	−1.06641
$41$	1.58457	0.247469	0.123734	−	0.992315i	$-0.460513\pi$
$41$	1.58457	0.247469	0.123734	+	0.992315i	$0.460513\pi$
$42$	0	0
$43$	11.1168	1.69530	0.847651	−	0.530554i	$-0.178016\pi$
$43$	11.1168	1.69530	0.847651	+	0.530554i	$0.178016\pi$
$44$	4.75372	0.716651
$45$	0	0
$46$	3.48913	0.514443
$47$	9.15759	1.33577	0.667886	−	0.744264i	$-0.267199\pi$
$47$	9.15759	1.33577	0.667886	+	0.744264i	$0.267199\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.08724	−0.153759
$51$	0	0
$52$	1.37228	0.190301
$53$	0.939764	0.129086	0.0645432	−	0.997915i	$-0.479441\pi$
$53$	0.939764	0.129086	0.0645432	+	0.997915i	$0.479441\pi$
$54$	0	0
$55$	8.74456	1.17912
$56$	2.67181	0.357036
$57$	0	0
$58$	−4.74456	−0.622992
$59$	5.04868	0.657282	0.328641	−	0.944455i	$-0.393409\pi$
$59$	5.04868	0.657282	0.328641	+	0.944455i	$0.393409\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−5.04868	−0.641182
$63$	0	0
$64$	3.37228	0.421535
$65$	2.52434	0.313106
$66$	0	0
$67$	−8.74456	−1.06832	−0.534159	−	0.845384i	$-0.679372\pi$
$67$	−8.74456	−1.06832	−0.534159	+	0.845384i	$0.679372\pi$
$68$	2.17448	0.263695
$69$	0	0
$70$	2.00000	0.239046
$71$	−1.58457	−0.188054	−0.0940272	−	0.995570i	$-0.529974\pi$
$71$	−1.58457	−0.188054	−0.0940272	+	0.995570i	$0.529974\pi$
$72$	0	0
$73$	8.37228	0.979901	0.489951	−	0.871750i	$-0.337015\pi$
$73$	8.37228	0.979901	0.489951	+	0.871750i	$0.337015\pi$
$74$	−5.34363	−0.621184
$75$	0	0
$76$	−2.23369	−0.256222
$77$	−3.46410	−0.394771
$78$	0	0
$79$	−11.8614	−1.33451	−0.667256	−	0.744828i	$-0.732531\pi$
$79$	−11.8614	−1.33451	−0.667256	+	0.744828i	$0.732531\pi$
$80$	−1.58457	−0.177161
$81$	0	0
$82$	−1.25544	−0.138640
$83$	12.3267	1.35303	0.676517	−	0.736427i	$-0.263488\pi$
$83$	12.3267	1.35303	0.676517	+	0.736427i	$0.263488\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	−8.80773	−0.949762
$87$	0	0
$88$	−9.25544	−0.986633
$89$	−2.52434	−0.267579	−0.133790	−	0.991010i	$-0.542715\pi$
$89$	−2.52434	−0.267579	−0.133790	+	0.991010i	$0.542715\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	6.04334	0.630062
$93$	0	0
$94$	−7.25544	−0.748341
$95$	−4.10891	−0.421565
$96$	0	0
$97$	3.62772	0.368339	0.184170	−	0.982894i	$-0.441040\pi$
$97$	3.62772	0.368339	0.184170	+	0.982894i	$0.441040\pi$
$98$	−0.792287	−0.0800331
$99$	0	0
$100$	−1.88316	−0.188316
$101$	−17.3205	−1.72345	−0.861727	−	0.507371i	$-0.830617\pi$
$101$	−17.3205	−1.72345	−0.861727	+	0.507371i	$0.830617\pi$
$102$	0	0
$103$	−13.4891	−1.32912	−0.664562	−	0.747234i	$-0.731382\pi$
$103$	−13.4891	−1.32912	−0.664562	+	0.747234i	$0.731382\pi$
$104$	−2.67181	−0.261993
$105$	0	0
$106$	−0.744563	−0.0723183
$107$	16.7306	1.61741	0.808704	−	0.588216i	$-0.200169\pi$
$107$	16.7306	1.61741	0.808704	+	0.588216i	$0.200169\pi$
$108$	0	0
$109$	−7.48913	−0.717328	−0.358664	−	0.933467i	$-0.616768\pi$
$109$	−7.48913	−0.717328	−0.358664	+	0.933467i	$0.616768\pi$
$110$	−6.92820	−0.660578
$111$	0	0
$112$	0.627719	0.0593138
$113$	−5.98844	−0.563345	−0.281672	−	0.959511i	$-0.590889\pi$
$113$	−5.98844	−0.563345	−0.281672	+	0.959511i	$0.590889\pi$
$114$	0	0
$115$	11.1168	1.03665
$116$	−8.21782	−0.763006
$117$	0	0
$118$	−4.00000	−0.368230
$119$	−1.58457	−0.145258
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.75372	−0.430382
$123$	0	0
$124$	−8.74456	−0.785285
$125$	9.15759	0.819080
$126$	0	0
$127$	1.48913	0.132139	0.0660693	−	0.997815i	$-0.478954\pi$
$127$	1.48913	0.132139	0.0660693	+	0.997815i	$0.478954\pi$
$128$	9.01011	0.796389
$129$	0	0
$130$	−2.00000	−0.175412
$131$	10.0974	0.882210	0.441105	−	0.897456i	$-0.354587\pi$
$131$	10.0974	0.882210	0.441105	+	0.897456i	$0.354587\pi$
$132$	0	0
$133$	1.62772	0.141141
$134$	6.92820	0.598506
$135$	0	0
$136$	−4.23369	−0.363036
$137$	17.0256	1.45459	0.727296	−	0.686324i	$-0.240777\pi$
$137$	17.0256	1.45459	0.727296	+	0.686324i	$0.240777\pi$
$138$	0	0
$139$	4.74456	0.402429	0.201214	−	0.979547i	$-0.435511\pi$
$139$	4.74456	0.402429	0.201214	+	0.979547i	$0.435511\pi$
$140$	3.46410	0.292770
$141$	0	0
$142$	1.25544	0.105354
$143$	3.46410	0.289683
$144$	0	0
$145$	−15.1168	−1.25539
$146$	−6.63325	−0.548972
$147$	0	0
$148$	−9.25544	−0.760792
$149$	15.1460	1.24081	0.620405	−	0.784281i	$-0.286968\pi$
$149$	15.1460	1.24081	0.620405	+	0.784281i	$0.286968\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	4.34896	0.352747
$153$	0	0
$154$	2.74456	0.221163
$155$	−16.0858	−1.29204
$156$	0	0
$157$	1.25544	0.100195	0.0500974	−	0.998744i	$-0.484047\pi$
$157$	1.25544	0.100195	0.0500974	+	0.998744i	$0.484047\pi$
$158$	9.39764	0.747636
$159$	0	0
$160$	14.7446	1.16566
$161$	−4.40387	−0.347073
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−2.17448	−0.169798
$165$	0	0
$166$	−9.76631	−0.758013
$167$	12.9166	0.999520	0.499760	−	0.866164i	$-0.333422\pi$
$167$	12.9166	0.999520	0.499760	+	0.866164i	$0.333422\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−3.16915	−0.243063
$171$	0	0
$172$	−15.2554	−1.16322
$173$	−11.6819	−0.888160	−0.444080	−	0.895987i	$-0.646469\pi$
$173$	−11.6819	−0.888160	−0.444080	+	0.895987i	$0.646469\pi$
$174$	0	0
$175$	1.37228	0.103735
$176$	−2.17448	−0.163908
$177$	0	0
$178$	2.00000	0.149906
$179$	13.2116	0.987481	0.493741	−	0.869609i	$-0.335629\pi$
$179$	13.2116	0.987481	0.493741	+	0.869609i	$0.335629\pi$
$180$	0	0
$181$	11.4891	0.853980	0.426990	−	0.904256i	$-0.359574\pi$
$181$	11.4891	0.853980	0.426990	+	0.904256i	$0.359574\pi$
$182$	0.792287	0.0587282
$183$	0	0
$184$	−11.7663	−0.867424
$185$	−17.0256	−1.25174
$186$	0	0
$187$	5.48913	0.401405
$188$	−12.5668	−0.916527
$189$	0	0
$190$	3.25544	0.236174
$191$	6.63325	0.479965	0.239983	−	0.970777i	$-0.422858\pi$
$191$	6.63325	0.479965	0.239983	+	0.970777i	$0.422858\pi$
$192$	0	0
$193$	−11.4891	−0.827005	−0.413503	−	0.910503i	$-0.635695\pi$
$193$	−11.4891	−0.827005	−0.413503	+	0.910503i	$0.635695\pi$
$194$	−2.87419	−0.206355
$195$	0	0
$196$	−1.37228	−0.0980201
$197$	−11.3870	−0.811288	−0.405644	−	0.914031i	$-0.632953\pi$
$197$	−11.3870	−0.811288	−0.405644	+	0.914031i	$0.632953\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	3.66648	0.259259
$201$	0	0
$202$	13.7228	0.965534
$203$	5.98844	0.420306
$204$	0	0
$205$	−4.00000	−0.279372
$206$	10.6873	0.744617
$207$	0	0
$208$	−0.627719	−0.0435245
$209$	−5.63858	−0.390029
$210$	0	0
$211$	−20.6060	−1.41857	−0.709287	−	0.704920i	$-0.750983\pi$
$211$	−20.6060	−1.41857	−0.709287	+	0.704920i	$0.750983\pi$
$212$	−1.28962	−0.0885715
$213$	0	0
$214$	−13.2554	−0.906123
$215$	−28.0627	−1.91386
$216$	0	0
$217$	6.37228	0.432579
$218$	5.93354	0.401870
$219$	0	0
$220$	−12.0000	−0.809040
$221$	1.58457	0.106590
$222$	0	0
$223$	−16.6060	−1.11202	−0.556009	−	0.831176i	$-0.687668\pi$
$223$	−16.6060	−1.11202	−0.556009	+	0.831176i	$0.687668\pi$
$224$	−5.84096	−0.390266
$225$	0	0
$226$	4.74456	0.315604
$227$	−11.3870	−0.755780	−0.377890	−	0.925851i	$-0.623350\pi$
$227$	−11.3870	−0.755780	−0.377890	+	0.925851i	$0.623350\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	−8.80773	−0.580765
$231$	0	0
$232$	16.0000	1.05045
$233$	−21.1345	−1.38456	−0.692282	−	0.721627i	$-0.743395\pi$
$233$	−21.1345	−1.38456	−0.692282	+	0.721627i	$0.743395\pi$
$234$	0	0
$235$	−23.1168	−1.50798
$236$	−6.92820	−0.450988
$237$	0	0
$238$	1.25544	0.0813779
$239$	−2.17448	−0.140656	−0.0703278	−	0.997524i	$-0.522405\pi$
$239$	−2.17448	−0.140656	−0.0703278	+	0.997524i	$0.522405\pi$
$240$	0	0
$241$	25.8614	1.66588	0.832940	−	0.553364i	$-0.186656\pi$
$241$	25.8614	1.66588	0.832940	+	0.553364i	$0.186656\pi$
$242$	−0.792287	−0.0509301
$243$	0	0
$244$	−8.23369	−0.527108
$245$	−2.52434	−0.161274
$246$	0	0
$247$	−1.62772	−0.103569
$248$	17.0256	1.08112
$249$	0	0
$250$	−7.25544	−0.458874
$251$	−25.2434	−1.59335	−0.796674	−	0.604409i	$-0.793409\pi$
$251$	−25.2434	−1.59335	−0.796674	+	0.604409i	$0.793409\pi$
$252$	0	0
$253$	15.2554	0.959101
$254$	−1.17981	−0.0740282
$255$	0	0
$256$	−13.8832	−0.867697
$257$	−6.63325	−0.413771	−0.206885	−	0.978365i	$-0.566333\pi$
$257$	−6.63325	−0.413771	−0.206885	+	0.978365i	$0.566333\pi$
$258$	0	0
$259$	6.74456	0.419087
$260$	−3.46410	−0.214834
$261$	0	0
$262$	−8.00000	−0.494242
$263$	−14.5012	−0.894183	−0.447092	−	0.894488i	$-0.647540\pi$
$263$	−14.5012	−0.894183	−0.447092	+	0.894488i	$0.647540\pi$
$264$	0	0
$265$	−2.37228	−0.145728
$266$	−1.28962	−0.0790717
$267$	0	0
$268$	12.0000	0.733017
$269$	19.2000	1.17065	0.585323	−	0.810800i	$-0.300968\pi$
$269$	19.2000	1.17065	0.585323	+	0.810800i	$0.300968\pi$
$270$	0	0
$271$	13.4891	0.819406	0.409703	−	0.912219i	$-0.365632\pi$
$271$	13.4891	0.819406	0.409703	+	0.912219i	$0.365632\pi$
$272$	−0.994667	−0.0603105
$273$	0	0
$274$	−13.4891	−0.814908
$275$	−4.75372	−0.286660
$276$	0	0
$277$	25.8614	1.55386	0.776931	−	0.629586i	$-0.216776\pi$
$277$	25.8614	1.55386	0.776931	+	0.629586i	$0.216776\pi$
$278$	−3.75906	−0.225453
$279$	0	0
$280$	−6.74456	−0.403065
$281$	18.9051	1.12778	0.563891	−	0.825849i	$-0.309304\pi$
$281$	18.9051	1.12778	0.563891	+	0.825849i	$0.309304\pi$
$282$	0	0
$283$	−22.9783	−1.36592	−0.682958	−	0.730458i	$-0.739307\pi$
$283$	−22.9783	−1.36592	−0.682958	+	0.730458i	$0.739307\pi$
$284$	2.17448	0.129032
$285$	0	0
$286$	−2.74456	−0.162289
$287$	1.58457	0.0935344
$288$	0	0
$289$	−14.4891	−0.852301
$290$	11.9769	0.703307
$291$	0	0
$292$	−11.4891	−0.672350
$293$	16.3807	0.956973	0.478487	−	0.878095i	$-0.341186\pi$
$293$	16.3807	0.956973	0.478487	+	0.878095i	$0.341186\pi$
$294$	0	0
$295$	−12.7446	−0.742017
$296$	18.0202	1.04740
$297$	0	0
$298$	−12.0000	−0.695141
$299$	4.40387	0.254682
$300$	0	0
$301$	11.1168	0.640764
$302$	6.33830	0.364728
$303$	0	0
$304$	1.02175	0.0586013
$305$	−15.1460	−0.867259
$306$	0	0
$307$	19.1168	1.09106	0.545528	−	0.838093i	$-0.316329\pi$
$307$	19.1168	1.09106	0.545528	+	0.838093i	$0.316329\pi$
$308$	4.75372	0.270868
$309$	0	0
$310$	12.7446	0.723843
$311$	32.7615	1.85773	0.928867	−	0.370414i	$-0.120784\pi$
$311$	32.7615	1.85773	0.928867	+	0.370414i	$0.120784\pi$
$312$	0	0
$313$	−4.51087	−0.254970	−0.127485	−	0.991841i	$-0.540690\pi$
$313$	−4.51087	−0.254970	−0.127485	+	0.991841i	$0.540690\pi$
$314$	−0.994667	−0.0561323
$315$	0	0
$316$	16.2772	0.915663
$317$	30.8820	1.73450	0.867252	−	0.497870i	$-0.165884\pi$
$317$	30.8820	1.73450	0.867252	+	0.497870i	$0.165884\pi$
$318$	0	0
$319$	−20.7446	−1.16147
$320$	−8.51278	−0.475879
$321$	0	0
$322$	3.48913	0.194441
$323$	−2.57924	−0.143513
$324$	0	0
$325$	−1.37228	−0.0761205
$326$	9.50744	0.526569
$327$	0	0
$328$	4.23369	0.233766
$329$	9.15759	0.504874
$330$	0	0
$331$	8.74456	0.480645	0.240322	−	0.970693i	$-0.422747\pi$
$331$	8.74456	0.480645	0.240322	+	0.970693i	$0.422747\pi$
$332$	−16.9157	−0.928372
$333$	0	0
$334$	−10.2337	−0.559962
$335$	22.0742	1.20604
$336$	0	0
$337$	−1.11684	−0.0608384	−0.0304192	−	0.999537i	$-0.509684\pi$
$337$	−1.11684	−0.0608384	−0.0304192	+	0.999537i	$0.509684\pi$
$338$	−0.792287	−0.0430947
$339$	0	0
$340$	−5.48913	−0.297690
$341$	−22.0742	−1.19539
$342$	0	0
$343$	1.00000	0.0539949
$344$	29.7021	1.60143
$345$	0	0
$346$	9.25544	0.497575
$347$	−4.05401	−0.217631	−0.108815	−	0.994062i	$-0.534706\pi$
$347$	−4.05401	−0.217631	−0.108815	+	0.994062i	$0.534706\pi$
$348$	0	0
$349$	24.3723	1.30462	0.652309	−	0.757953i	$-0.273800\pi$
$349$	24.3723	1.30462	0.652309	+	0.757953i	$0.273800\pi$
$350$	−1.08724	−0.0581155
$351$	0	0
$352$	20.2337	1.07846
$353$	−28.1176	−1.49655	−0.748274	−	0.663390i	$-0.769117\pi$
$353$	−28.1176	−1.49655	−0.748274	+	0.663390i	$0.769117\pi$
$354$	0	0
$355$	4.00000	0.212298
$356$	3.46410	0.183597
$357$	0	0
$358$	−10.4674	−0.553218
$359$	−19.2000	−1.01334	−0.506670	−	0.862140i	$-0.669124\pi$
$359$	−19.2000	−1.01334	−0.506670	+	0.862140i	$0.669124\pi$
$360$	0	0
$361$	−16.3505	−0.860554
$362$	−9.10268	−0.478426
$363$	0	0
$364$	1.37228	0.0719271
$365$	−21.1345	−1.10623
$366$	0	0
$367$	14.2337	0.742992	0.371496	−	0.928434i	$-0.378845\pi$
$367$	14.2337	0.742992	0.371496	+	0.928434i	$0.378845\pi$
$368$	−2.76439	−0.144104
$369$	0	0
$370$	13.4891	0.701266
$371$	0.939764	0.0487901
$372$	0	0
$373$	−0.510875	−0.0264521	−0.0132260	−	0.999913i	$-0.504210\pi$
$373$	−0.510875	−0.0264521	−0.0132260	+	0.999913i	$0.504210\pi$
$374$	−4.34896	−0.224880
$375$	0	0
$376$	24.4674	1.26181
$377$	−5.98844	−0.308420
$378$	0	0
$379$	18.2337	0.936602	0.468301	−	0.883569i	$-0.344866\pi$
$379$	18.2337	0.936602	0.468301	+	0.883569i	$0.344866\pi$
$380$	5.63858	0.289253
$381$	0	0
$382$	−5.25544	−0.268892
$383$	24.6535	1.25973	0.629867	−	0.776703i	$-0.283109\pi$
$383$	24.6535	1.25973	0.629867	+	0.776703i	$0.283109\pi$
$384$	0	0
$385$	8.74456	0.445664
$386$	9.10268	0.463314
$387$	0	0
$388$	−4.97825	−0.252732
$389$	26.5330	1.34528	0.672638	−	0.739972i	$-0.265161\pi$
$389$	26.5330	1.34528	0.672638	+	0.739972i	$0.265161\pi$
$390$	0	0
$391$	6.97825	0.352905
$392$	2.67181	0.134947
$393$	0	0
$394$	9.02175	0.454509
$395$	29.9422	1.50656
$396$	0	0
$397$	2.13859	0.107333	0.0536665	−	0.998559i	$-0.482909\pi$
$397$	2.13859	0.107333	0.0536665	+	0.998559i	$0.482909\pi$
$398$	−12.6766	−0.635420
$399$	0	0
$400$	0.861407	0.0430703
$401$	−17.0256	−0.850216	−0.425108	−	0.905143i	$-0.639764\pi$
$401$	−17.0256	−0.850216	−0.425108	+	0.905143i	$0.639764\pi$
$402$	0	0
$403$	−6.37228	−0.317426
$404$	23.7686	1.18253
$405$	0	0
$406$	−4.74456	−0.235469
$407$	−23.3639	−1.15810
$408$	0	0
$409$	32.0951	1.58700	0.793500	−	0.608570i	$-0.208257\pi$
$409$	32.0951	1.58700	0.793500	+	0.608570i	$0.208257\pi$
$410$	3.16915	0.156513
$411$	0	0
$412$	18.5109	0.911965
$413$	5.04868	0.248429
$414$	0	0
$415$	−31.1168	−1.52747
$416$	5.84096	0.286377
$417$	0	0
$418$	4.46738	0.218506
$419$	7.62792	0.372648	0.186324	−	0.982488i	$-0.440343\pi$
$419$	7.62792	0.372648	0.186324	+	0.982488i	$0.440343\pi$
$420$	0	0
$421$	26.7446	1.30345	0.651725	−	0.758455i	$-0.274046\pi$
$421$	26.7446	1.30345	0.651725	+	0.758455i	$0.274046\pi$
$422$	16.3258	0.794730
$423$	0	0
$424$	2.51087	0.121939
$425$	−2.17448	−0.105478
$426$	0	0
$427$	6.00000	0.290360
$428$	−22.9591	−1.10977
$429$	0	0
$430$	22.2337	1.07220
$431$	−23.6588	−1.13960	−0.569802	−	0.821782i	$-0.692980\pi$
$431$	−23.6588	−1.13960	−0.569802	+	0.821782i	$0.692980\pi$
$432$	0	0
$433$	17.7228	0.851704	0.425852	−	0.904793i	$-0.359974\pi$
$433$	17.7228	0.851704	0.425852	+	0.904793i	$0.359974\pi$
$434$	−5.04868	−0.242344
$435$	0	0
$436$	10.2772	0.492188
$437$	−7.16825	−0.342904
$438$	0	0
$439$	−4.74456	−0.226446	−0.113223	−	0.993570i	$-0.536117\pi$
$439$	−4.74456	−0.226446	−0.113223	+	0.993570i	$0.536117\pi$
$440$	23.3639	1.11383
$441$	0	0
$442$	−1.25544	−0.0597151
$443$	−32.8164	−1.55915	−0.779577	−	0.626307i	$-0.784566\pi$
$443$	−32.8164	−1.55915	−0.779577	+	0.626307i	$0.784566\pi$
$444$	0	0
$445$	6.37228	0.302075
$446$	13.1567	0.622987
$447$	0	0
$448$	3.37228	0.159325
$449$	7.51811	0.354802	0.177401	−	0.984139i	$-0.443231\pi$
$449$	7.51811	0.354802	0.177401	+	0.984139i	$0.443231\pi$
$450$	0	0
$451$	−5.48913	−0.258473
$452$	8.21782	0.386534
$453$	0	0
$454$	9.02175	0.423412
$455$	2.52434	0.118343
$456$	0	0
$457$	−30.7446	−1.43817	−0.719085	−	0.694922i	$-0.755439\pi$
$457$	−30.7446	−1.43817	−0.719085	+	0.694922i	$0.755439\pi$
$458$	−7.92287	−0.370211
$459$	0	0
$460$	−15.2554	−0.711288
$461$	36.2256	1.68719	0.843597	−	0.536977i	$-0.180434\pi$
$461$	36.2256	1.68719	0.843597	+	0.536977i	$0.180434\pi$
$462$	0	0
$463$	−27.2554	−1.26667	−0.633334	−	0.773879i	$-0.718314\pi$
$463$	−27.2554	−1.26667	−0.633334	+	0.773879i	$0.718314\pi$
$464$	3.75906	0.174510
$465$	0	0
$466$	16.7446	0.775677
$467$	5.04868	0.233625	0.116812	−	0.993154i	$-0.462732\pi$
$467$	5.04868	0.233625	0.116812	+	0.993154i	$0.462732\pi$
$468$	0	0
$469$	−8.74456	−0.403786
$470$	18.3152	0.844816
$471$	0	0
$472$	13.4891	0.620887
$473$	−38.5099	−1.77069
$474$	0	0
$475$	2.23369	0.102489
$476$	2.17448	0.0996672
$477$	0	0
$478$	1.72281	0.0787996
$479$	−33.1113	−1.51290	−0.756448	−	0.654054i	$-0.773067\pi$
$479$	−33.1113	−1.51290	−0.756448	+	0.654054i	$0.773067\pi$
$480$	0	0
$481$	−6.74456	−0.307526
$482$	−20.4897	−0.933278
$483$	0	0
$484$	−1.37228	−0.0623764
$485$	−9.15759	−0.415825
$486$	0	0
$487$	28.7446	1.30254	0.651270	−	0.758846i	$-0.274236\pi$
$487$	28.7446	1.30254	0.651270	+	0.758846i	$0.274236\pi$
$488$	16.0309	0.725684
$489$	0	0
$490$	2.00000	0.0903508
$491$	10.9822	0.495620	0.247810	−	0.968809i	$-0.420289\pi$
$491$	10.9822	0.495620	0.247810	+	0.968809i	$0.420289\pi$
$492$	0	0
$493$	−9.48913	−0.427369
$494$	1.28962	0.0580228
$495$	0	0
$496$	4.00000	0.179605
$497$	−1.58457	−0.0710779
$498$	0	0
$499$	14.9783	0.670519	0.335259	−	0.942126i	$-0.391176\pi$
$499$	14.9783	0.670519	0.335259	+	0.942126i	$0.391176\pi$
$500$	−12.5668	−0.562004
$501$	0	0
$502$	20.0000	0.892644
$503$	−32.1716	−1.43446	−0.717230	−	0.696837i	$-0.754590\pi$
$503$	−32.1716	−1.43446	−0.717230	+	0.696837i	$0.754590\pi$
$504$	0	0
$505$	43.7228	1.94564
$506$	−12.0867	−0.537319
$507$	0	0
$508$	−2.04350	−0.0906656
$509$	25.8882	1.14747	0.573737	−	0.819040i	$-0.305493\pi$
$509$	25.8882	1.14747	0.573737	+	0.819040i	$0.305493\pi$
$510$	0	0
$511$	8.37228	0.370368
$512$	−7.02078	−0.310277
$513$	0	0
$514$	5.25544	0.231807
$515$	34.0511	1.50047
$516$	0	0
$517$	−31.7228	−1.39517
$518$	−5.34363	−0.234786
$519$	0	0
$520$	6.74456	0.295769
$521$	26.1282	1.14470	0.572349	−	0.820010i	$-0.306032\pi$
$521$	26.1282	1.14470	0.572349	+	0.820010i	$0.306032\pi$
$522$	0	0
$523$	−23.7228	−1.03733	−0.518663	−	0.854979i	$-0.673570\pi$
$523$	−23.7228	−1.03733	−0.518663	+	0.854979i	$0.673570\pi$
$524$	−13.8564	−0.605320
$525$	0	0
$526$	11.4891	0.500950
$527$	−10.0974	−0.439848
$528$	0	0
$529$	−3.60597	−0.156781
$530$	1.87953	0.0816415
$531$	0	0
$532$	−2.23369	−0.0968427
$533$	−1.58457	−0.0686355
$534$	0	0
$535$	−42.2337	−1.82592
$536$	−23.3639	−1.00916
$537$	0	0
$538$	−15.2119	−0.655833
$539$	−3.46410	−0.149209
$540$	0	0
$541$	28.2337	1.21386	0.606931	−	0.794755i	$-0.292401\pi$
$541$	28.2337	1.21386	0.606931	+	0.794755i	$0.292401\pi$
$542$	−10.6873	−0.459057
$543$	0	0
$544$	9.25544	0.396824
$545$	18.9051	0.809805
$546$	0	0
$547$	−39.8614	−1.70435	−0.852175	−	0.523256i	$-0.824717\pi$
$547$	−39.8614	−1.70435	−0.852175	+	0.523256i	$0.824717\pi$
$548$	−23.3639	−0.998054
$549$	0	0
$550$	3.76631	0.160596
$551$	9.74749	0.415257
$552$	0	0
$553$	−11.8614	−0.504398
$554$	−20.4897	−0.870522
$555$	0	0
$556$	−6.51087	−0.276123
$557$	−6.92820	−0.293557	−0.146779	−	0.989169i	$-0.546891\pi$
$557$	−6.92820	−0.293557	−0.146779	+	0.989169i	$0.546891\pi$
$558$	0	0
$559$	−11.1168	−0.470192
$560$	−1.58457	−0.0669605
$561$	0	0
$562$	−14.9783	−0.631819
$563$	17.0256	0.717542	0.358771	−	0.933426i	$-0.383196\pi$
$563$	17.0256	0.717542	0.358771	+	0.933426i	$0.383196\pi$
$564$	0	0
$565$	15.1168	0.635970
$566$	18.2054	0.765229
$567$	0	0
$568$	−4.23369	−0.177642
$569$	−28.6526	−1.20118	−0.600589	−	0.799558i	$-0.705067\pi$
$569$	−28.6526	−1.20118	−0.600589	+	0.799558i	$0.705067\pi$
$570$	0	0
$571$	3.11684	0.130436	0.0652179	−	0.997871i	$-0.479226\pi$
$571$	3.11684	0.130436	0.0652179	+	0.997871i	$0.479226\pi$
$572$	−4.75372	−0.198763
$573$	0	0
$574$	−1.25544	−0.0524009
$575$	−6.04334	−0.252025
$576$	0	0
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	11.4795	0.477486
$579$	0	0
$580$	20.7446	0.861371
$581$	12.3267	0.511399
$582$	0	0
$583$	−3.25544	−0.134826
$584$	22.3692	0.925643
$585$	0	0
$586$	−12.9783	−0.536127
$587$	−23.0140	−0.949889	−0.474945	−	0.880016i	$-0.657532\pi$
$587$	−23.0140	−0.949889	−0.474945	+	0.880016i	$0.657532\pi$
$588$	0	0
$589$	10.3723	0.427382
$590$	10.0974	0.415701
$591$	0	0
$592$	4.23369	0.174004
$593$	35.3956	1.45352	0.726762	−	0.686889i	$-0.241024\pi$
$593$	35.3956	1.45352	0.726762	+	0.686889i	$0.241024\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	−20.7846	−0.851371
$597$	0	0
$598$	−3.48913	−0.142681
$599$	−22.1291	−0.904172	−0.452086	−	0.891974i	$-0.649320\pi$
$599$	−22.1291	−0.904172	−0.452086	+	0.891974i	$0.649320\pi$
$600$	0	0
$601$	12.9783	0.529394	0.264697	−	0.964332i	$-0.414728\pi$
$601$	12.9783	0.529394	0.264697	+	0.964332i	$0.414728\pi$
$602$	−8.80773	−0.358976
$603$	0	0
$604$	10.9783	0.446699
$605$	−2.52434	−0.102629
$606$	0	0
$607$	6.23369	0.253018	0.126509	−	0.991965i	$-0.459623\pi$
$607$	6.23369	0.253018	0.126509	+	0.991965i	$0.459623\pi$
$608$	−9.50744	−0.385578
$609$	0	0
$610$	12.0000	0.485866
$611$	−9.15759	−0.370476
$612$	0	0
$613$	4.23369	0.170997	0.0854985	−	0.996338i	$-0.472752\pi$
$613$	4.23369	0.170997	0.0854985	+	0.996338i	$0.472752\pi$
$614$	−15.1460	−0.611244
$615$	0	0
$616$	−9.25544	−0.372912
$617$	18.9051	0.761090	0.380545	−	0.924762i	$-0.375736\pi$
$617$	18.9051	0.761090	0.380545	+	0.924762i	$0.375736\pi$
$618$	0	0
$619$	1.48913	0.0598530	0.0299265	−	0.999552i	$-0.490473\pi$
$619$	1.48913	0.0598530	0.0299265	+	0.999552i	$0.490473\pi$
$620$	22.0742	0.886522
$621$	0	0
$622$	−25.9565	−1.04076
$623$	−2.52434	−0.101135
$624$	0	0
$625$	−29.9783	−1.19913
$626$	3.57391	0.142842
$627$	0	0
$628$	−1.72281	−0.0687477
$629$	−10.6873	−0.426129
$630$	0	0
$631$	−12.7446	−0.507353	−0.253677	−	0.967289i	$-0.581640\pi$
$631$	−12.7446	−0.507353	−0.253677	+	0.967289i	$0.581640\pi$
$632$	−31.6915	−1.26062
$633$	0	0
$634$	−24.4674	−0.971724
$635$	−3.75906	−0.149174
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	16.4356	0.650694
$639$	0	0
$640$	−22.7446	−0.899058
$641$	6.57835	0.259829	0.129915	−	0.991525i	$-0.458530\pi$
$641$	6.57835	0.259829	0.129915	+	0.991525i	$0.458530\pi$
$642$	0	0
$643$	−21.4891	−0.847448	−0.423724	−	0.905791i	$-0.639277\pi$
$643$	−21.4891	−0.847448	−0.423724	+	0.905791i	$0.639277\pi$
$644$	6.04334	0.238141
$645$	0	0
$646$	2.04350	0.0804004
$647$	7.62792	0.299884	0.149942	−	0.988695i	$-0.452091\pi$
$647$	7.62792	0.299884	0.149942	+	0.988695i	$0.452091\pi$
$648$	0	0
$649$	−17.4891	−0.686508
$650$	1.08724	0.0426451
$651$	0	0
$652$	16.4674	0.644912
$653$	47.9075	1.87477	0.937383	−	0.348300i	$-0.113241\pi$
$653$	47.9075	1.87477	0.937383	+	0.348300i	$0.113241\pi$
$654$	0	0
$655$	−25.4891	−0.995943
$656$	0.994667	0.0388352
$657$	0	0
$658$	−7.25544	−0.282846
$659$	−33.4063	−1.30132	−0.650662	−	0.759367i	$-0.725509\pi$
$659$	−33.4063	−1.30132	−0.650662	+	0.759367i	$0.725509\pi$
$660$	0	0
$661$	22.8832	0.890052	0.445026	−	0.895518i	$-0.353195\pi$
$661$	22.8832	0.890052	0.445026	+	0.895518i	$0.353195\pi$
$662$	−6.92820	−0.269272
$663$	0	0
$664$	32.9348	1.27812
$665$	−4.10891	−0.159337
$666$	0	0
$667$	−26.3723	−1.02114
$668$	−17.7253	−0.685811
$669$	0	0
$670$	−17.4891	−0.675664
$671$	−20.7846	−0.802381
$672$	0	0
$673$	−36.3723	−1.40205	−0.701024	−	0.713137i	$-0.747274\pi$
$673$	−36.3723	−1.40205	−0.701024	+	0.713137i	$0.747274\pi$
$674$	0.884861	0.0340836
$675$	0	0
$676$	−1.37228	−0.0527801
$677$	47.0227	1.80723	0.903614	−	0.428348i	$-0.140904\pi$
$677$	47.0227	1.80723	0.903614	+	0.428348i	$0.140904\pi$
$678$	0	0
$679$	3.62772	0.139219
$680$	10.6873	0.409838
$681$	0	0
$682$	17.4891	0.669693
$683$	15.4410	0.590833	0.295416	−	0.955369i	$-0.404542\pi$
$683$	15.4410	0.590833	0.295416	+	0.955369i	$0.404542\pi$
$684$	0	0
$685$	−42.9783	−1.64211
$686$	−0.792287	−0.0302497
$687$	0	0
$688$	6.97825	0.266043
$689$	−0.939764	−0.0358022
$690$	0	0
$691$	−23.1168	−0.879406	−0.439703	−	0.898143i	$-0.644916\pi$
$691$	−23.1168	−0.879406	−0.439703	+	0.898143i	$0.644916\pi$
$692$	16.0309	0.609403
$693$	0	0
$694$	3.21194	0.121924
$695$	−11.9769	−0.454309
$696$	0	0
$697$	−2.51087	−0.0951062
$698$	−19.3098	−0.730888
$699$	0	0
$700$	−1.88316	−0.0711766
$701$	14.2063	0.536563	0.268282	−	0.963341i	$-0.413544\pi$
$701$	14.2063	0.536563	0.268282	+	0.963341i	$0.413544\pi$
$702$	0	0
$703$	10.9783	0.414053
$704$	−11.6819	−0.440279
$705$	0	0
$706$	22.2772	0.838413
$707$	−17.3205	−0.651405
$708$	0	0
$709$	−34.4674	−1.29445	−0.647225	−	0.762299i	$-0.724070\pi$
$709$	−34.4674	−1.29445	−0.647225	+	0.762299i	$0.724070\pi$
$710$	−3.16915	−0.118936
$711$	0	0
$712$	−6.74456	−0.252763
$713$	−28.0627	−1.05096
$714$	0	0
$715$	−8.74456	−0.327028
$716$	−18.1300	−0.677551
$717$	0	0
$718$	15.2119	0.567705
$719$	52.3663	1.95293	0.976466	−	0.215669i	$-0.0691934\pi$
$719$	52.3663	1.95293	0.976466	+	0.215669i	$0.0691934\pi$
$720$	0	0
$721$	−13.4891	−0.502361
$722$	12.9543	0.482110
$723$	0	0
$724$	−15.7663	−0.585950
$725$	8.21782	0.305202
$726$	0	0
$727$	−15.2554	−0.565793	−0.282896	−	0.959150i	$-0.591295\pi$
$727$	−15.2554	−0.565793	−0.282896	+	0.959150i	$0.591295\pi$
$728$	−2.67181	−0.0990240
$729$	0	0
$730$	16.7446	0.619744
$731$	−17.6155	−0.651531
$732$	0	0
$733$	26.8832	0.992952	0.496476	−	0.868050i	$-0.334627\pi$
$733$	26.8832	0.992952	0.496476	+	0.868050i	$0.334627\pi$
$734$	−11.2772	−0.416248
$735$	0	0
$736$	25.7228	0.948155
$737$	30.2921	1.11582
$738$	0	0
$739$	15.2554	0.561180	0.280590	−	0.959828i	$-0.409470\pi$
$739$	15.2554	0.561180	0.280590	+	0.959828i	$0.409470\pi$
$740$	23.3639	0.858872
$741$	0	0
$742$	−0.744563	−0.0273338
$743$	22.3692	0.820646	0.410323	−	0.911940i	$-0.365416\pi$
$743$	22.3692	0.820646	0.410323	+	0.911940i	$0.365416\pi$
$744$	0	0
$745$	−38.2337	−1.40077
$746$	0.404759	0.0148193
$747$	0	0
$748$	−7.53262	−0.275420
$749$	16.7306	0.611323
$750$	0	0
$751$	−18.3723	−0.670414	−0.335207	−	0.942144i	$-0.608806\pi$
$751$	−18.3723	−0.670414	−0.335207	+	0.942144i	$0.608806\pi$
$752$	5.74839	0.209622
$753$	0	0
$754$	4.74456	0.172787
$755$	20.1947	0.734960
$756$	0	0
$757$	20.3723	0.740443	0.370222	−	0.928943i	$-0.379282\pi$
$757$	20.3723	0.740443	0.370222	+	0.928943i	$0.379282\pi$
$758$	−14.4463	−0.524714
$759$	0	0
$760$	−10.9783	−0.398223
$761$	−25.2983	−0.917062	−0.458531	−	0.888678i	$-0.651624\pi$
$761$	−25.2983	−0.917062	−0.458531	+	0.888678i	$0.651624\pi$
$762$	0	0
$763$	−7.48913	−0.271125
$764$	−9.10268	−0.329324
$765$	0	0
$766$	−19.5326	−0.705742
$767$	−5.04868	−0.182297
$768$	0	0
$769$	−0.372281	−0.0134248	−0.00671240	−	0.999977i	$-0.502137\pi$
$769$	−0.372281	−0.0134248	−0.00671240	+	0.999977i	$0.502137\pi$
$770$	−6.92820	−0.249675
$771$	0	0
$772$	15.7663	0.567442
$773$	−45.7330	−1.64490	−0.822451	−	0.568835i	$-0.807394\pi$
$773$	−45.7330	−1.64490	−0.822451	+	0.568835i	$0.807394\pi$
$774$	0	0
$775$	8.74456	0.314114
$776$	9.69259	0.347944
$777$	0	0
$778$	−21.0217	−0.753666
$779$	2.57924	0.0924109
$780$	0	0
$781$	5.48913	0.196416
$782$	−5.52878	−0.197709
$783$	0	0
$784$	0.627719	0.0224185
$785$	−3.16915	−0.113112
$786$	0	0
$787$	−0.883156	−0.0314811	−0.0157406	−	0.999876i	$-0.505011\pi$
$787$	−0.883156	−0.0314811	−0.0157406	+	0.999876i	$0.505011\pi$
$788$	15.6261	0.556658
$789$	0	0
$790$	−23.7228	−0.844020
$791$	−5.98844	−0.212924
$792$	0	0
$793$	−6.00000	−0.213066
$794$	−1.69438	−0.0601313
$795$	0	0
$796$	−21.9565	−0.778228
$797$	−9.80240	−0.347219	−0.173609	−	0.984815i	$-0.555543\pi$
$797$	−9.80240	−0.347219	−0.173609	+	0.984815i	$0.555543\pi$
$798$	0	0
$799$	−14.5109	−0.513358
$800$	−8.01544	−0.283389
$801$	0	0
$802$	13.4891	0.476318
$803$	−29.0024	−1.02347
$804$	0	0
$805$	11.1168	0.391817
$806$	5.04868	0.177832
$807$	0	0
$808$	−46.2772	−1.62803
$809$	42.6188	1.49840	0.749198	−	0.662346i	$-0.230439\pi$
$809$	42.6188	1.49840	0.749198	+	0.662346i	$0.230439\pi$
$810$	0	0
$811$	−28.4674	−0.999625	−0.499812	−	0.866134i	$-0.666598\pi$
$811$	−28.4674	−0.999625	−0.499812	+	0.866134i	$0.666598\pi$
$812$	−8.21782	−0.288389
$813$	0	0
$814$	18.5109	0.648806
$815$	30.2921	1.06108
$816$	0	0
$817$	18.0951	0.633067
$818$	−25.4285	−0.889088
$819$	0	0
$820$	5.48913	0.191689
$821$	−25.2434	−0.881000	−0.440500	−	0.897753i	$-0.645199\pi$
$821$	−25.2434	−0.881000	−0.440500	+	0.897753i	$0.645199\pi$
$822$	0	0
$823$	18.9783	0.661540	0.330770	−	0.943711i	$-0.392692\pi$
$823$	18.9783	0.661540	0.330770	+	0.943711i	$0.392692\pi$
$824$	−36.0404	−1.25553
$825$	0	0
$826$	−4.00000	−0.139178
$827$	−9.10268	−0.316531	−0.158266	−	0.987397i	$-0.550590\pi$
$827$	−9.10268	−0.316531	−0.158266	+	0.987397i	$0.550590\pi$
$828$	0	0
$829$	30.4674	1.05818	0.529088	−	0.848567i	$-0.322534\pi$
$829$	30.4674	1.05818	0.529088	+	0.848567i	$0.322534\pi$
$830$	24.6535	0.855734
$831$	0	0
$832$	−3.37228	−0.116913
$833$	−1.58457	−0.0549022
$834$	0	0
$835$	−32.6060	−1.12838
$836$	7.73772	0.267615
$837$	0	0
$838$	−6.04350	−0.208769
$839$	−39.0998	−1.34987	−0.674937	−	0.737875i	$-0.735829\pi$
$839$	−39.0998	−1.34987	−0.674937	+	0.737875i	$0.735829\pi$
$840$	0	0
$841$	6.86141	0.236600
$842$	−21.1894	−0.730234
$843$	0	0
$844$	28.2772	0.973341
$845$	−2.52434	−0.0868399
$846$	0	0
$847$	1.00000	0.0343604
$848$	0.589907	0.0202575
$849$	0	0
$850$	1.72281	0.0590920
$851$	−29.7021	−1.01818
$852$	0	0
$853$	−53.5842	−1.83469	−0.917344	−	0.398095i	$-0.869672\pi$
$853$	−53.5842	−1.83469	−0.917344	+	0.398095i	$0.869672\pi$
$854$	−4.75372	−0.162669
$855$	0	0
$856$	44.7011	1.52785
$857$	−0.294954	−0.0100754	−0.00503771	−	0.999987i	$-0.501604\pi$
$857$	−0.294954	−0.0100754	−0.00503771	+	0.999987i	$0.501604\pi$
$858$	0	0
$859$	16.7446	0.571317	0.285659	−	0.958331i	$-0.407788\pi$
$859$	16.7446	0.571317	0.285659	+	0.958331i	$0.407788\pi$
$860$	38.5099	1.31318
$861$	0	0
$862$	18.7446	0.638442
$863$	45.7330	1.55677	0.778385	−	0.627787i	$-0.216039\pi$
$863$	45.7330	1.55677	0.778385	+	0.627787i	$0.216039\pi$
$864$	0	0
$865$	29.4891	1.00266
$866$	−14.0416	−0.477151
$867$	0	0
$868$	−8.74456	−0.296810
$869$	41.0891	1.39385
$870$	0	0
$871$	8.74456	0.296298
$872$	−20.0096	−0.677609
$873$	0	0
$874$	5.67931	0.192106
$875$	9.15759	0.309583
$876$	0	0
$877$	2.74456	0.0926773	0.0463386	−	0.998926i	$-0.485245\pi$
$877$	2.74456	0.0926773	0.0463386	+	0.998926i	$0.485245\pi$
$878$	3.75906	0.126862
$879$	0	0
$880$	5.48913	0.185038
$881$	−9.80240	−0.330251	−0.165126	−	0.986273i	$-0.552803\pi$
$881$	−9.80240	−0.330251	−0.165126	+	0.986273i	$0.552803\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	−2.17448	−0.0731357
$885$	0	0
$886$	26.0000	0.873487
$887$	−10.6873	−0.358843	−0.179422	−	0.983772i	$-0.557423\pi$
$887$	−10.6873	−0.358843	−0.179422	+	0.983772i	$0.557423\pi$
$888$	0	0
$889$	1.48913	0.0499437
$890$	−5.04868	−0.169232
$891$	0	0
$892$	22.7881	0.763001
$893$	14.9060	0.498809
$894$	0	0
$895$	−33.3505	−1.11479
$896$	9.01011	0.301007
$897$	0	0
$898$	−5.95650	−0.198771
$899$	38.1600	1.27271
$900$	0	0
$901$	−1.48913	−0.0496100
$902$	4.34896	0.144805
$903$	0	0
$904$	−16.0000	−0.532152
$905$	−29.0024	−0.964074
$906$	0	0
$907$	−8.13859	−0.270238	−0.135119	−	0.990829i	$-0.543142\pi$
$907$	−8.13859	−0.270238	−0.135119	+	0.990829i	$0.543142\pi$
$908$	15.6261	0.518571
$909$	0	0
$910$	−2.00000	−0.0662994
$911$	8.86263	0.293632	0.146816	−	0.989164i	$-0.453097\pi$
$911$	8.86263	0.293632	0.146816	+	0.989164i	$0.453097\pi$
$912$	0	0
$913$	−42.7011	−1.41320
$914$	24.3585	0.805708
$915$	0	0
$916$	−13.7228	−0.453415
$917$	10.0974	0.333444
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	29.7021	0.979251
$921$	0	0
$922$	−28.7011	−0.945219
$923$	1.58457	0.0521569
$924$	0	0
$925$	9.25544	0.304317
$926$	21.5941	0.709627
$927$	0	0
$928$	−34.9783	−1.14822
$929$	3.81396	0.125132	0.0625660	−	0.998041i	$-0.480072\pi$
$929$	3.81396	0.125132	0.0625660	+	0.998041i	$0.480072\pi$
$930$	0	0
$931$	1.62772	0.0533463
$932$	29.0024	0.950006
$933$	0	0
$934$	−4.00000	−0.130884
$935$	−13.8564	−0.453153
$936$	0	0
$937$	−50.7446	−1.65775	−0.828876	−	0.559432i	$-0.811019\pi$
$937$	−50.7446	−1.65775	−0.828876	+	0.559432i	$0.811019\pi$
$938$	6.92820	0.226214
$939$	0	0
$940$	31.7228	1.03468
$941$	−32.8164	−1.06978	−0.534892	−	0.844921i	$-0.679648\pi$
$941$	−32.8164	−1.06978	−0.534892	+	0.844921i	$0.679648\pi$
$942$	0	0
$943$	−6.97825	−0.227243
$944$	3.16915	0.103147
$945$	0	0
$946$	30.5109	0.991994
$947$	36.9253	1.19991	0.599956	−	0.800033i	$-0.295185\pi$
$947$	36.9253	1.19991	0.599956	+	0.800033i	$0.295185\pi$
$948$	0	0
$949$	−8.37228	−0.271776
$950$	−1.76972	−0.0574174
$951$	0	0
$952$	−4.23369	−0.137215
$953$	−24.3036	−0.787271	−0.393636	−	0.919267i	$-0.628783\pi$
$953$	−24.3036	−0.787271	−0.393636	+	0.919267i	$0.628783\pi$
$954$	0	0
$955$	−16.7446	−0.541841
$956$	2.98400	0.0965095
$957$	0	0
$958$	26.2337	0.847572
$959$	17.0256	0.549784
$960$	0	0
$961$	9.60597	0.309870
$962$	5.34363	0.172286
$963$	0	0
$964$	−35.4891	−1.14303
$965$	29.0024	0.933621
$966$	0	0
$967$	−55.7228	−1.79192	−0.895962	−	0.444130i	$-0.853513\pi$
$967$	−55.7228	−1.79192	−0.895962	+	0.444130i	$0.853513\pi$
$968$	2.67181	0.0858754
$969$	0	0
$970$	7.25544	0.232958
$971$	−24.6535	−0.791168	−0.395584	−	0.918430i	$-0.629458\pi$
$971$	−24.6535	−0.791168	−0.395584	+	0.918430i	$0.629458\pi$
$972$	0	0
$973$	4.74456	0.152104
$974$	−22.7739	−0.729724
$975$	0	0
$976$	3.76631	0.120557
$977$	7.62792	0.244039	0.122019	−	0.992528i	$-0.461063\pi$
$977$	7.62792	0.244039	0.122019	+	0.992528i	$0.461063\pi$
$978$	0	0
$979$	8.74456	0.279477
$980$	3.46410	0.110657
$981$	0	0
$982$	−8.70106	−0.277662
$983$	33.7013	1.07490	0.537452	−	0.843295i	$-0.319387\pi$
$983$	33.7013	1.07490	0.537452	+	0.843295i	$0.319387\pi$
$984$	0	0
$985$	28.7446	0.915878
$986$	7.51811	0.239425
$987$	0	0
$988$	2.23369	0.0710631
$989$	−48.9571	−1.55674
$990$	0	0
$991$	52.4674	1.66668	0.833341	−	0.552760i	$-0.186425\pi$
$991$	52.4674	1.66668	0.833341	+	0.552760i	$0.186425\pi$
$992$	−37.2203	−1.18174
$993$	0	0
$994$	1.25544	0.0398201
$995$	−40.3894	−1.28043
$996$	0	0
$997$	1.25544	0.0397601	0.0198800	−	0.999802i	$-0.493672\pi$
$997$	1.25544	0.0397601	0.0198800	+	0.999802i	$0.493672\pi$
$998$	−11.8671	−0.375646
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.a.l.1.2	✓	4
3.2	odd	2	inner	819.2.a.l.1.3	yes	4
7.6	odd	2		5733.2.a.bi.1.2		4
21.20	even	2		5733.2.a.bi.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.a.l.1.2	✓	4	1.1	even	1	trivial
819.2.a.l.1.3	yes	4	3.2	odd	2	inner
5733.2.a.bi.1.2		4	7.6	odd	2
5733.2.a.bi.1.3		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:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.a (trivial)

\(f(q)\)	\(=\)	\(q-0.792287 q^{2} -1.37228 q^{4} -2.52434 q^{5} +1.00000 q^{7} +2.67181 q^{8} +O(q^{10})\)	\(q-0.792287 q^{2} -1.37228 q^{4} -2.52434 q^{5} +1.00000 q^{7} +2.67181 q^{8} +2.00000 q^{10} -3.46410 q^{11} -1.00000 q^{13} -0.792287 q^{14} +0.627719 q^{16} -1.58457 q^{17} +1.62772 q^{19} +3.46410 q^{20} +2.74456 q^{22} -4.40387 q^{23} +1.37228 q^{25} +0.792287 q^{26} -1.37228 q^{28} +5.98844 q^{29} +6.37228 q^{31} -5.84096 q^{32} +1.25544 q^{34} -2.52434 q^{35} +6.74456 q^{37} -1.28962 q^{38} -6.74456 q^{40} +1.58457 q^{41} +11.1168 q^{43} +4.75372 q^{44} +3.48913 q^{46} +9.15759 q^{47} +1.00000 q^{49} -1.08724 q^{50} +1.37228 q^{52} +0.939764 q^{53} +8.74456 q^{55} +2.67181 q^{56} -4.74456 q^{58} +5.04868 q^{59} +6.00000 q^{61} -5.04868 q^{62} +3.37228 q^{64} +2.52434 q^{65} -8.74456 q^{67} +2.17448 q^{68} +2.00000 q^{70} -1.58457 q^{71} +8.37228 q^{73} -5.34363 q^{74} -2.23369 q^{76} -3.46410 q^{77} -11.8614 q^{79} -1.58457 q^{80} -1.25544 q^{82} +12.3267 q^{83} +4.00000 q^{85} -8.80773 q^{86} -9.25544 q^{88} -2.52434 q^{89} -1.00000 q^{91} +6.04334 q^{92} -7.25544 q^{94} -4.10891 q^{95} +3.62772 q^{97} -0.792287 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4} + 4 q^{7}+O(q^{10}) \) 4 * q + 6 * q^4 + 4 * q^7	\( 4 q + 6 q^{4} + 4 q^{7} + 8 q^{10} - 4 q^{13} + 14 q^{16} + 18 q^{19} - 12 q^{22} - 6 q^{25} + 6 q^{28} + 14 q^{31} + 28 q^{34} + 4 q^{37} - 4 q^{40} + 10 q^{43} - 32 q^{46} + 4 q^{49} - 6 q^{52} + 12 q^{55} + 4 q^{58} + 24 q^{61} + 2 q^{64} - 12 q^{67} + 8 q^{70} + 22 q^{73} + 60 q^{76} + 10 q^{79} - 28 q^{82} + 16 q^{85} - 60 q^{88} - 4 q^{91} - 52 q^{94} + 26 q^{97}+O(q^{100}) \) 4 * q + 6 * q^4 + 4 * q^7 + 8 * q^10 - 4 * q^13 + 14 * q^16 + 18 * q^19 - 12 * q^22 - 6 * q^25 + 6 * q^28 + 14 * q^31 + 28 * q^34 + 4 * q^37 - 4 * q^40 + 10 * q^43 - 32 * q^46 + 4 * q^49 - 6 * q^52 + 12 * q^55 + 4 * q^58 + 24 * q^61 + 2 * q^64 - 12 * q^67 + 8 * q^70 + 22 * q^73 + 60 * q^76 + 10 * q^79 - 28 * q^82 + 16 * q^85 - 60 * q^88 - 4 * q^91 - 52 * q^94 + 26 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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