LMFDB - Embedded newform 5445.2.a.bw.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, 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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.74043072.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 21x^{2} - 12 $ x^6 - 9x^4 + 21x^2 - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.924607$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.924607 q^{2} -1.14510 q^{4} -1.00000 q^{5} +4.64003 q^{7} -2.90798 q^{8} +O(q^{10})$	$q+0.924607 q^{2} -1.14510 q^{4} -1.00000 q^{5} +4.64003 q^{7} -2.90798 q^{8} -0.924607 q^{10} +4.29021 q^{14} -0.398534 q^{16} -6.37208 q^{17} +4.64003 q^{19} +1.14510 q^{20} -6.74657 q^{23} +1.00000 q^{25} -5.31331 q^{28} -3.46410 q^{29} -2.20293 q^{31} +5.44748 q^{32} -5.89167 q^{34} -4.64003 q^{35} +3.45636 q^{37} +4.29021 q^{38} +2.90798 q^{40} -5.81597 q^{41} -0.234325 q^{43} -6.23792 q^{46} +9.32698 q^{47} +14.5299 q^{49} +0.924607 q^{50} -3.54364 q^{53} -13.4931 q^{56} -3.20293 q^{58} -14.5804 q^{59} -1.96638 q^{61} -2.03684 q^{62} +5.83384 q^{64} -1.45636 q^{67} +7.29669 q^{68} -4.29021 q^{70} +2.58041 q^{71} -13.6858 q^{73} +3.19578 q^{74} -5.31331 q^{76} -3.58126 q^{79} +0.398534 q^{80} -5.37748 q^{82} -7.16253 q^{83} +6.37208 q^{85} -0.216658 q^{86} +8.98627 q^{89} +7.72551 q^{92} +8.62379 q^{94} -4.64003 q^{95} -12.4426 q^{97} +13.4345 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{4} - 6 q^{5}+O(q^{10}) $ 6 * q + 6 * q^4 - 6 * q^5	$ 6 q + 6 q^{4} - 6 q^{5} - 6 q^{16} - 6 q^{20} - 24 q^{23} + 6 q^{25} - 6 q^{31} - 6 q^{34} + 30 q^{37} - 12 q^{47} + 12 q^{49} - 12 q^{53} - 48 q^{56} - 12 q^{58} - 36 q^{59} - 18 q^{67} - 36 q^{71} + 6 q^{80} + 12 q^{82} - 60 q^{86} - 12 q^{89} - 18 q^{92} - 18 q^{97}+O(q^{100}) $ 6 * q + 6 * q^4 - 6 * q^5 - 6 * q^16 - 6 * q^20 - 24 * q^23 + 6 * q^25 - 6 * q^31 - 6 * q^34 + 30 * q^37 - 12 * q^47 + 12 * q^49 - 12 * q^53 - 48 * q^56 - 12 * q^58 - 36 * q^59 - 18 * q^67 - 36 * q^71 + 6 * q^80 + 12 * q^82 - 60 * q^86 - 12 * q^89 - 18 * q^92 - 18 * 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 $

$2$	0.924607	0.653796	0.326898	−	0.945060i	$-0.393997\pi$
$2$	0.924607	0.653796	0.326898	+	0.945060i	$0.393997\pi$
$3$	0	0
$3$	0	0
$4$	−1.14510	−0.572551
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.64003	1.75377	0.876884	−	0.480702i	$-0.159618\pi$
$7$	4.64003	1.75377	0.876884	+	0.480702i	$0.159618\pi$
$8$	−2.90798	−1.02813
$9$	0	0
$10$	−0.924607	−0.292386
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	4.29021	1.14661
$15$	0	0
$16$	−0.398534	−0.0996336
$17$	−6.37208	−1.54546	−0.772729	−	0.634736i	$-0.781109\pi$
$17$	−6.37208	−1.54546	−0.772729	+	0.634736i	$0.781109\pi$
$18$	0	0
$19$	4.64003	1.06450	0.532248	−	0.846588i	$-0.321347\pi$
$19$	4.64003	1.06450	0.532248	+	0.846588i	$0.321347\pi$
$20$	1.14510	0.256053
$21$	0	0
$22$	0	0
$23$	−6.74657	−1.40676	−0.703378	−	0.710816i	$-0.748326\pi$
$23$	−6.74657	−1.40676	−0.703378	+	0.710816i	$0.748326\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−5.31331	−1.00412
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	−2.20293	−0.395658	−0.197829	−	0.980237i	$-0.563389\pi$
$31$	−2.20293	−0.395658	−0.197829	+	0.980237i	$0.563389\pi$
$32$	5.44748	0.962987
$33$	0	0
$34$	−5.89167	−1.01041
$35$	−4.64003	−0.784309
$36$	0	0
$37$	3.45636	0.568223	0.284111	−	0.958791i	$-0.408301\pi$
$37$	3.45636	0.568223	0.284111	+	0.958791i	$0.408301\pi$
$38$	4.29021	0.695963
$39$	0	0
$40$	2.90798	0.459792
$41$	−5.81597	−0.908301	−0.454151	−	0.890925i	$-0.650057\pi$
$41$	−5.81597	−0.908301	−0.454151	+	0.890925i	$0.650057\pi$
$42$	0	0
$43$	−0.234325	−0.0357342	−0.0178671	−	0.999840i	$-0.505688\pi$
$43$	−0.234325	−0.0357342	−0.0178671	+	0.999840i	$0.505688\pi$
$44$	0	0
$45$	0	0
$46$	−6.23792	−0.919731
$47$	9.32698	1.36048	0.680240	−	0.732990i	$-0.261876\pi$
$47$	9.32698	1.36048	0.680240	+	0.732990i	$0.261876\pi$
$48$	0	0
$49$	14.5299	2.07570
$50$	0.924607	0.130759
$51$	0	0
$52$	0	0
$53$	−3.54364	−0.486756	−0.243378	−	0.969932i	$-0.578256\pi$
$53$	−3.54364	−0.486756	−0.243378	+	0.969932i	$0.578256\pi$
$54$	0	0
$55$	0	0
$56$	−13.4931	−1.80310
$57$	0	0
$58$	−3.20293	−0.420565
$59$	−14.5804	−1.89821	−0.949104	−	0.314963i	$-0.898008\pi$
$59$	−14.5804	−1.89821	−0.949104	+	0.314963i	$0.898008\pi$
$60$	0	0
$61$	−1.96638	−0.251769	−0.125884	−	0.992045i	$-0.540177\pi$
$61$	−1.96638	−0.251769	−0.125884	+	0.992045i	$0.540177\pi$
$62$	−2.03684	−0.258680
$63$	0	0
$64$	5.83384	0.729230
$65$	0	0
$66$	0	0
$67$	−1.45636	−0.177923	−0.0889615	−	0.996035i	$-0.528355\pi$
$67$	−1.45636	−0.177923	−0.0889615	+	0.996035i	$0.528355\pi$
$68$	7.29669	0.884854
$69$	0	0
$70$	−4.29021	−0.512778
$71$	2.58041	0.306238	0.153119	−	0.988208i	$-0.451068\pi$
$71$	2.58041	0.306238	0.153119	+	0.988208i	$0.451068\pi$
$72$	0	0
$73$	−13.6858	−1.60180	−0.800899	−	0.598799i	$-0.795645\pi$
$73$	−13.6858	−1.60180	−0.800899	+	0.598799i	$0.795645\pi$
$74$	3.19578	0.371501
$75$	0	0
$76$	−5.31331	−0.609479
$77$	0	0
$78$	0	0
$79$	−3.58126	−0.402924	−0.201462	−	0.979496i	$-0.564569\pi$
$79$	−3.58126	−0.402924	−0.201462	+	0.979496i	$0.564569\pi$
$80$	0.398534	0.0445575
$81$	0	0
$82$	−5.37748	−0.593843
$83$	−7.16253	−0.786190	−0.393095	−	0.919498i	$-0.628596\pi$
$83$	−7.16253	−0.786190	−0.393095	+	0.919498i	$0.628596\pi$
$84$	0	0
$85$	6.37208	0.691150
$86$	−0.216658	−0.0233628
$87$	0	0
$88$	0	0
$89$	8.98627	0.952543	0.476272	−	0.879298i	$-0.341988\pi$
$89$	8.98627	0.952543	0.476272	+	0.879298i	$0.341988\pi$
$90$	0	0
$91$	0	0
$92$	7.72551	0.805440
$93$	0	0
$94$	8.62379	0.889476
$95$	−4.64003	−0.476057
$96$	0	0
$97$	−12.4426	−1.26336	−0.631679	−	0.775230i	$-0.717634\pi$
$97$	−12.4426	−1.26336	−0.631679	+	0.775230i	$0.717634\pi$
$98$	13.4345	1.35708
$99$	0	0
$100$	−1.14510	−0.114510
$101$	−15.0960	−1.50211	−0.751056	−	0.660239i	$-0.770455\pi$
$101$	−15.0960	−1.50211	−0.751056	+	0.660239i	$0.770455\pi$
$102$	0	0
$103$	19.0230	1.87440	0.937198	−	0.348797i	$-0.113410\pi$
$103$	19.0230	1.87440	0.937198	+	0.348797i	$0.113410\pi$
$104$	0	0
$105$	0	0
$106$	−3.27647	−0.318239
$107$	17.2670	1.66927	0.834634	−	0.550805i	$-0.185679\pi$
$107$	17.2670	1.66927	0.834634	+	0.550805i	$0.185679\pi$
$108$	0	0
$109$	−14.1544	−1.35575	−0.677874	−	0.735178i	$-0.737099\pi$
$109$	−14.1544	−1.35575	−0.677874	+	0.735178i	$0.737099\pi$
$110$	0	0
$111$	0	0
$112$	−1.84921	−0.174734
$113$	3.94950	0.371538	0.185769	−	0.982593i	$-0.440522\pi$
$113$	3.94950	0.371538	0.185769	+	0.982593i	$0.440522\pi$
$114$	0	0
$115$	6.74657	0.629121
$116$	3.96675	0.368304
$117$	0	0
$118$	−13.4811	−1.24104
$119$	−29.5667	−2.71037
$120$	0	0
$121$	0	0
$122$	−1.81812	−0.164605
$123$	0	0
$124$	2.52258	0.226535
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.5063	−1.46470	−0.732348	−	0.680930i	$-0.761576\pi$
$127$	−16.5063	−1.46470	−0.732348	+	0.680930i	$0.761576\pi$
$128$	−5.50095	−0.486220
$129$	0	0
$130$	0	0
$131$	15.9739	1.39565	0.697825	−	0.716268i	$-0.254151\pi$
$131$	15.9739	1.39565	0.697825	+	0.716268i	$0.254151\pi$
$132$	0	0
$133$	21.5299	1.86688
$134$	−1.34656	−0.116325
$135$	0	0
$136$	18.5299	1.58893
$137$	−5.03677	−0.430321	−0.215160	−	0.976579i	$-0.569027\pi$
$137$	−5.03677	−0.430321	−0.215160	+	0.976579i	$0.569027\pi$
$138$	0	0
$139$	18.8479	1.59866	0.799330	−	0.600892i	$-0.205188\pi$
$139$	18.8479	1.59866	0.799330	+	0.600892i	$0.205188\pi$
$140$	5.31331	0.449057
$141$	0	0
$142$	2.38586	0.200217
$143$	0	0
$144$	0	0
$145$	3.46410	0.287678
$146$	−12.6540	−1.04725
$147$	0	0
$148$	−3.95789	−0.325337
$149$	9.74872	0.798646	0.399323	−	0.916810i	$-0.369245\pi$
$149$	9.74872	0.798646	0.399323	+	0.916810i	$0.369245\pi$
$150$	0	0
$151$	−17.2670	−1.40517	−0.702586	−	0.711599i	$-0.747971\pi$
$151$	−17.2670	−1.40517	−0.702586	+	0.711599i	$0.747971\pi$
$152$	−13.4931	−1.09444
$153$	0	0
$154$	0	0
$155$	2.20293	0.176944
$156$	0	0
$157$	−6.61718	−0.528109	−0.264054	−	0.964508i	$-0.585060\pi$
$157$	−6.61718	−0.528109	−0.264054	+	0.964508i	$0.585060\pi$
$158$	−3.31126	−0.263430
$159$	0	0
$160$	−5.44748	−0.430661
$161$	−31.3043	−2.46712
$162$	0	0
$163$	−12.4426	−0.974582	−0.487291	−	0.873240i	$-0.662015\pi$
$163$	−12.4426	−0.974582	−0.487291	+	0.873240i	$0.662015\pi$
$164$	6.65988	0.520049
$165$	0	0
$166$	−6.62252	−0.514007
$167$	−2.17101	−0.167998	−0.0839988	−	0.996466i	$-0.526769\pi$
$167$	−2.17101	−0.167998	−0.0839988	+	0.996466i	$0.526769\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	5.89167	0.451871
$171$	0	0
$172$	0.268326	0.0204597
$173$	8.81142	0.669920	0.334960	−	0.942232i	$-0.391277\pi$
$173$	8.81142	0.669920	0.334960	+	0.942232i	$0.391277\pi$
$174$	0	0
$175$	4.64003	0.350754
$176$	0	0
$177$	0	0
$178$	8.30877	0.622768
$179$	−13.8990	−1.03886	−0.519430	−	0.854513i	$-0.673856\pi$
$179$	−13.8990	−1.03886	−0.519430	+	0.854513i	$0.673856\pi$
$180$	0	0
$181$	−20.6172	−1.53246	−0.766232	−	0.642564i	$-0.777870\pi$
$181$	−20.6172	−1.53246	−0.766232	+	0.642564i	$0.777870\pi$
$182$	0	0
$183$	0	0
$184$	19.6189	1.44632
$185$	−3.45636	−0.254117
$186$	0	0
$187$	0	0
$188$	−10.6803	−0.778944
$189$	0	0
$190$	−4.29021	−0.311244
$191$	−13.4931	−0.976329	−0.488165	−	0.872752i	$-0.662333\pi$
$191$	−13.4931	−0.976329	−0.488165	+	0.872752i	$0.662333\pi$
$192$	0	0
$193$	−18.7308	−1.34827	−0.674135	−	0.738608i	$-0.735483\pi$
$193$	−18.7308	−1.34827	−0.674135	+	0.738608i	$0.735483\pi$
$194$	−11.5045	−0.825978
$195$	0	0
$196$	−16.6382	−1.18845
$197$	13.8564	0.987228	0.493614	−	0.869681i	$-0.335676\pi$
$197$	13.8564	0.987228	0.493614	+	0.869681i	$0.335676\pi$
$198$	0	0
$199$	−19.3638	−1.37266	−0.686330	−	0.727290i	$-0.740779\pi$
$199$	−19.3638	−1.37266	−0.686330	+	0.727290i	$0.740779\pi$
$200$	−2.90798	−0.205625
$201$	0	0
$202$	−13.9579	−0.982074
$203$	−16.0735	−1.12814
$204$	0	0
$205$	5.81597	0.406205
$206$	17.5888	1.22547
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0.351487	0.0241974	0.0120987	−	0.999927i	$-0.496149\pi$
$211$	0.351487	0.0241974	0.0120987	+	0.999927i	$0.496149\pi$
$212$	4.05783	0.278693
$213$	0	0
$214$	15.9652	1.09136
$215$	0.234325	0.0159808
$216$	0	0
$217$	−10.2217	−0.693892
$218$	−13.0873	−0.886382
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	2.94950	0.197513	0.0987565	−	0.995112i	$-0.468513\pi$
$223$	2.94950	0.197513	0.0987565	+	0.995112i	$0.468513\pi$
$224$	25.2765	1.68886
$225$	0	0
$226$	3.65173	0.242910
$227$	10.5732	0.701765	0.350883	−	0.936419i	$-0.385882\pi$
$227$	10.5732	0.701765	0.350883	+	0.936419i	$0.385882\pi$
$228$	0	0
$229$	−11.9495	−0.789645	−0.394823	−	0.918757i	$-0.629194\pi$
$229$	−11.9495	−0.789645	−0.394823	+	0.918757i	$0.629194\pi$
$230$	6.23792	0.411316
$231$	0	0
$232$	10.0735	0.661361
$233$	−7.48432	−0.490314	−0.245157	−	0.969483i	$-0.578840\pi$
$233$	−7.48432	−0.490314	−0.245157	+	0.969483i	$0.578840\pi$
$234$	0	0
$235$	−9.32698	−0.608425
$236$	16.6961	1.08682
$237$	0	0
$238$	−27.3376	−1.77203
$239$	28.3088	1.83115	0.915574	−	0.402150i	$-0.131737\pi$
$239$	28.3088	1.83115	0.915574	+	0.402150i	$0.131737\pi$
$240$	0	0
$241$	6.13776	0.395368	0.197684	−	0.980266i	$-0.436658\pi$
$241$	6.13776	0.395368	0.197684	+	0.980266i	$0.436658\pi$
$242$	0	0
$243$	0	0
$244$	2.25170	0.144150
$245$	−14.5299	−0.928282
$246$	0	0
$247$	0	0
$248$	6.40609	0.406787
$249$	0	0
$250$	−0.924607	−0.0584773
$251$	2.98627	0.188492	0.0942459	−	0.995549i	$-0.469956\pi$
$251$	2.98627	0.188492	0.0942459	+	0.995549i	$0.469956\pi$
$252$	0	0
$253$	0	0
$254$	−15.2618	−0.957612
$255$	0	0
$256$	−16.7539	−1.04712
$257$	−24.1240	−1.50482	−0.752408	−	0.658697i	$-0.771108\pi$
$257$	−24.1240	−1.50482	−0.752408	+	0.658697i	$0.771108\pi$
$258$	0	0
$259$	16.0376	0.996530
$260$	0	0
$261$	0	0
$262$	14.7696	0.912470
$263$	−9.46092	−0.583386	−0.291693	−	0.956512i	$-0.594218\pi$
$263$	−9.46092	−0.583386	−0.291693	+	0.956512i	$0.594218\pi$
$264$	0	0
$265$	3.54364	0.217684
$266$	19.9067	1.22056
$267$	0	0
$268$	1.66769	0.101870
$269$	−1.08727	−0.0662923	−0.0331461	−	0.999451i	$-0.510553\pi$
$269$	−1.08727	−0.0662923	−0.0331461	+	0.999451i	$0.510553\pi$
$270$	0	0
$271$	6.10376	0.370777	0.185388	−	0.982665i	$-0.440646\pi$
$271$	6.10376	0.370777	0.185388	+	0.982665i	$0.440646\pi$
$272$	2.53950	0.153980
$273$	0	0
$274$	−4.65703	−0.281342
$275$	0	0
$276$	0	0
$277$	−26.4299	−1.58802	−0.794011	−	0.607904i	$-0.792011\pi$
$277$	−26.4299	−1.58802	−0.794011	+	0.607904i	$0.792011\pi$
$278$	17.4269	1.04520
$279$	0	0
$280$	13.4931	0.806369
$281$	−30.5333	−1.82147	−0.910733	−	0.412996i	$-0.864482\pi$
$281$	−30.5333	−1.82147	−0.910733	+	0.412996i	$0.864482\pi$
$282$	0	0
$283$	−2.28817	−0.136018	−0.0680088	−	0.997685i	$-0.521665\pi$
$283$	−2.28817	−0.136018	−0.0680088	+	0.997685i	$0.521665\pi$
$284$	−2.95484	−0.175337
$285$	0	0
$286$	0	0
$287$	−26.9863	−1.59295
$288$	0	0
$289$	23.6035	1.38844
$290$	3.20293	0.188083
$291$	0	0
$292$	15.6716	0.917112
$293$	−21.2338	−1.24049	−0.620246	−	0.784408i	$-0.712967\pi$
$293$	−21.2338	−1.24049	−0.620246	+	0.784408i	$0.712967\pi$
$294$	0	0
$295$	14.5804	0.848904
$296$	−10.0510	−0.584205
$297$	0	0
$298$	9.01373	0.522151
$299$	0	0
$300$	0	0
$301$	−1.08727	−0.0626694
$302$	−15.9652	−0.918695
$303$	0	0
$304$	−1.84921	−0.106060
$305$	1.96638	0.112594
$306$	0	0
$307$	−11.5682	−0.660234	−0.330117	−	0.943940i	$-0.607088\pi$
$307$	−11.5682	−0.660234	−0.330117	+	0.943940i	$0.607088\pi$
$308$	0	0
$309$	0	0
$310$	2.03684	0.115685
$311$	13.8990	0.788140	0.394070	−	0.919080i	$-0.371067\pi$
$311$	13.8990	0.788140	0.394070	+	0.919080i	$0.371067\pi$
$312$	0	0
$313$	12.1745	0.688146	0.344073	−	0.938943i	$-0.388193\pi$
$313$	12.1745	0.688146	0.344073	+	0.938943i	$0.388193\pi$
$314$	−6.11829	−0.345275
$315$	0	0
$316$	4.10091	0.230694
$317$	−26.7045	−1.49987	−0.749936	−	0.661510i	$-0.769916\pi$
$317$	−26.7045	−1.49987	−0.749936	+	0.661510i	$0.769916\pi$
$318$	0	0
$319$	0	0
$320$	−5.83384	−0.326122
$321$	0	0
$322$	−28.9442	−1.61300
$323$	−29.5667	−1.64513
$324$	0	0
$325$	0	0
$326$	−11.5045	−0.637178
$327$	0	0
$328$	16.9127	0.933849
$329$	43.2775	2.38597
$330$	0	0
$331$	−19.4648	−1.06988	−0.534940	−	0.844890i	$-0.679666\pi$
$331$	−19.4648	−1.06988	−0.534940	+	0.844890i	$0.679666\pi$
$332$	8.20183	0.450134
$333$	0	0
$334$	−2.00733	−0.109836
$335$	1.45636	0.0795696
$336$	0	0
$337$	16.5063	0.899155	0.449577	−	0.893241i	$-0.351575\pi$
$337$	16.5063	0.899155	0.449577	+	0.893241i	$0.351575\pi$
$338$	−12.0199	−0.653796
$339$	0	0
$340$	−7.29669	−0.395719
$341$	0	0
$342$	0	0
$343$	34.9390	1.88653
$344$	0.681412	0.0367393
$345$	0	0
$346$	8.14709	0.437991
$347$	1.05877	0.0568377	0.0284189	−	0.999596i	$-0.490953\pi$
$347$	1.05877	0.0568377	0.0284189	+	0.999596i	$0.490953\pi$
$348$	0	0
$349$	13.5983	0.727901	0.363950	−	0.931418i	$-0.381428\pi$
$349$	13.5983	0.727901	0.363950	+	0.931418i	$0.381428\pi$
$350$	4.29021	0.229321
$351$	0	0
$352$	0	0
$353$	−29.0093	−1.54401	−0.772005	−	0.635616i	$-0.780746\pi$
$353$	−29.0093	−1.54401	−0.772005	+	0.635616i	$0.780746\pi$
$354$	0	0
$355$	−2.58041	−0.136954
$356$	−10.2902	−0.545380
$357$	0	0
$358$	−12.8511	−0.679202
$359$	14.0907	0.743680	0.371840	−	0.928297i	$-0.378727\pi$
$359$	14.0907	0.743680	0.371840	+	0.928297i	$0.378727\pi$
$360$	0	0
$361$	2.52991	0.133153
$362$	−19.0628	−1.00192
$363$	0	0
$364$	0	0
$365$	13.6858	0.716346
$366$	0	0
$367$	5.01373	0.261714	0.130857	−	0.991401i	$-0.458227\pi$
$367$	5.01373	0.261714	0.130857	+	0.991401i	$0.458227\pi$
$368$	2.68874	0.140160
$369$	0	0
$370$	−3.19578	−0.166140
$371$	−16.4426	−0.853657
$372$	0	0
$373$	9.91935	0.513604	0.256802	−	0.966464i	$-0.417331\pi$
$373$	9.91935	0.513604	0.256802	+	0.966464i	$0.417331\pi$
$374$	0	0
$375$	0	0
$376$	−27.1227	−1.39875
$377$	0	0
$378$	0	0
$379$	4.92112	0.252781	0.126390	−	0.991981i	$-0.459661\pi$
$379$	4.92112	0.252781	0.126390	+	0.991981i	$0.459661\pi$
$380$	5.31331	0.272567
$381$	0	0
$382$	−12.4758	−0.638320
$383$	17.9725	0.918354	0.459177	−	0.888345i	$-0.348144\pi$
$383$	17.9725	0.918354	0.459177	+	0.888345i	$0.348144\pi$
$384$	0	0
$385$	0	0
$386$	−17.3186	−0.881493
$387$	0	0
$388$	14.2481	0.723337
$389$	13.0598	0.662159	0.331080	−	0.943603i	$-0.392587\pi$
$389$	13.0598	0.662159	0.331080	+	0.943603i	$0.392587\pi$
$390$	0	0
$391$	42.9897	2.17408
$392$	−42.2527	−2.13408
$393$	0	0
$394$	12.8117	0.645445
$395$	3.58126	0.180193
$396$	0	0
$397$	−27.7045	−1.39045	−0.695223	−	0.718794i	$-0.744695\pi$
$397$	−27.7045	−1.39045	−0.695223	+	0.718794i	$0.744695\pi$
$398$	−17.9039	−0.897439
$399$	0	0
$400$	−0.398534	−0.0199267
$401$	−31.0598	−1.55105	−0.775527	−	0.631315i	$-0.782516\pi$
$401$	−31.0598	−1.55105	−0.775527	+	0.631315i	$0.782516\pi$
$402$	0	0
$403$	0	0
$404$	17.2865	0.860036
$405$	0	0
$406$	−14.8617	−0.737574
$407$	0	0
$408$	0	0
$409$	25.8058	1.27602	0.638008	−	0.770030i	$-0.279759\pi$
$409$	25.8058	1.27602	0.638008	+	0.770030i	$0.279759\pi$
$410$	5.37748	0.265575
$411$	0	0
$412$	−21.7833	−1.07319
$413$	−67.6536	−3.32902
$414$	0	0
$415$	7.16253	0.351595
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−15.2618	−0.745589	−0.372794	−	0.927914i	$-0.621600\pi$
$419$	−15.2618	−0.745589	−0.372794	+	0.927914i	$0.621600\pi$
$420$	0	0
$421$	37.9220	1.84821	0.924104	−	0.382142i	$-0.124813\pi$
$421$	37.9220	1.84821	0.924104	+	0.382142i	$0.124813\pi$
$422$	0.324987	0.0158201
$423$	0	0
$424$	10.3048	0.500447
$425$	−6.37208	−0.309091
$426$	0	0
$427$	−9.12405	−0.441544
$428$	−19.7725	−0.955741
$429$	0	0
$430$	0.216658	0.0104482
$431$	−18.6671	−0.899161	−0.449581	−	0.893240i	$-0.648427\pi$
$431$	−18.6671	−0.899161	−0.449581	+	0.893240i	$0.648427\pi$
$432$	0	0
$433$	−18.4426	−0.886297	−0.443148	−	0.896448i	$-0.646138\pi$
$433$	−18.4426	−0.886297	−0.443148	+	0.896448i	$0.646138\pi$
$434$	−9.45103	−0.453664
$435$	0	0
$436$	16.2083	0.776235
$437$	−31.3043	−1.49749
$438$	0	0
$439$	1.46373	0.0698598	0.0349299	−	0.999390i	$-0.488879\pi$
$439$	1.46373	0.0698598	0.0349299	+	0.999390i	$0.488879\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.8853	0.802243	0.401122	−	0.916025i	$-0.368620\pi$
$443$	16.8853	0.802243	0.401122	+	0.916025i	$0.368620\pi$
$444$	0	0
$445$	−8.98627	−0.425990
$446$	2.72713	0.129133
$447$	0	0
$448$	27.0692	1.27890
$449$	15.2618	0.720250	0.360125	−	0.932904i	$-0.382734\pi$
$449$	15.2618	0.720250	0.360125	+	0.932904i	$0.382734\pi$
$450$	0	0
$451$	0	0
$452$	−4.52258	−0.212724
$453$	0	0
$454$	9.77601	0.458811
$455$	0	0
$456$	0	0
$457$	1.58089	0.0739508	0.0369754	−	0.999316i	$-0.488228\pi$
$457$	1.58089	0.0739508	0.0369754	+	0.999316i	$0.488228\pi$
$458$	−11.0486	−0.516267
$459$	0	0
$460$	−7.72551	−0.360204
$461$	−7.86550	−0.366333	−0.183166	−	0.983082i	$-0.558635\pi$
$461$	−7.86550	−0.366333	−0.183166	+	0.983082i	$0.558635\pi$
$462$	0	0
$463$	35.6677	1.65762	0.828809	−	0.559532i	$-0.189019\pi$
$463$	35.6677	1.65762	0.828809	+	0.559532i	$0.189019\pi$
$464$	1.38056	0.0640911
$465$	0	0
$466$	−6.92005	−0.320565
$467$	20.8937	0.966843	0.483422	−	0.875388i	$-0.339394\pi$
$467$	20.8937	0.966843	0.483422	+	0.875388i	$0.339394\pi$
$468$	0	0
$469$	−6.75757	−0.312036
$470$	−8.62379	−0.397786
$471$	0	0
$472$	42.3996	1.95160
$473$	0	0
$474$	0	0
$475$	4.64003	0.212899
$476$	33.8569	1.55183
$477$	0	0
$478$	26.1745	1.19720
$479$	10.2174	0.466843	0.233422	−	0.972376i	$-0.425008\pi$
$479$	10.2174	0.466843	0.233422	+	0.972376i	$0.425008\pi$
$480$	0	0
$481$	0	0
$482$	5.67501	0.258490
$483$	0	0
$484$	0	0
$485$	12.4426	0.564991
$486$	0	0
$487$	15.3186	0.694151	0.347076	−	0.937837i	$-0.387175\pi$
$487$	15.3186	0.694151	0.347076	+	0.937837i	$0.387175\pi$
$488$	5.71819	0.258850
$489$	0	0
$490$	−13.4345	−0.606907
$491$	34.5341	1.55850	0.779251	−	0.626712i	$-0.215600\pi$
$491$	34.5341	1.55850	0.779251	+	0.626712i	$0.215600\pi$
$492$	0	0
$493$	22.0735	0.994143
$494$	0	0
$495$	0	0
$496$	0.877944	0.0394208
$497$	11.9732	0.537071
$498$	0	0
$499$	29.6035	1.32523	0.662616	−	0.748959i	$-0.269446\pi$
$499$	29.6035	1.32523	0.662616	+	0.748959i	$0.269446\pi$
$500$	1.14510	0.0512105
$501$	0	0
$502$	2.76113	0.123235
$503$	3.81990	0.170321	0.0851604	−	0.996367i	$-0.472860\pi$
$503$	3.81990	0.170321	0.0851604	+	0.996367i	$0.472860\pi$
$504$	0	0
$505$	15.0960	0.671765
$506$	0	0
$507$	0	0
$508$	18.9014	0.838614
$509$	−33.7980	−1.49807	−0.749035	−	0.662530i	$-0.769483\pi$
$509$	−33.7980	−1.49807	−0.749035	+	0.662530i	$0.769483\pi$
$510$	0	0
$511$	−63.5025	−2.80918
$512$	−4.48887	−0.198382
$513$	0	0
$514$	−22.3053	−0.983843
$515$	−19.0230	−0.838256
$516$	0	0
$517$	0	0
$518$	14.8285	0.651527
$519$	0	0
$520$	0	0
$521$	3.26182	0.142903	0.0714515	−	0.997444i	$-0.477237\pi$
$521$	3.26182	0.142903	0.0714515	+	0.997444i	$0.477237\pi$
$522$	0	0
$523$	−10.8653	−0.475105	−0.237552	−	0.971375i	$-0.576345\pi$
$523$	−10.8653	−0.475105	−0.237552	+	0.971375i	$0.576345\pi$
$524$	−18.2918	−0.799081
$525$	0	0
$526$	−8.74763	−0.381415
$527$	14.0373	0.611473
$528$	0	0
$529$	22.5162	0.978964
$530$	3.27647	0.142321
$531$	0	0
$532$	−24.6540	−1.06888
$533$	0	0
$534$	0	0
$535$	−17.2670	−0.746519
$536$	4.23508	0.182928
$537$	0	0
$538$	−1.00530	−0.0433416
$539$	0	0
$540$	0	0
$541$	−20.3160	−0.873451	−0.436726	−	0.899595i	$-0.643862\pi$
$541$	−20.3160	−0.873451	−0.436726	+	0.899595i	$0.643862\pi$
$542$	5.64358	0.242412
$543$	0	0
$544$	−34.7118	−1.48826
$545$	14.1544	0.606309
$546$	0	0
$547$	23.1365	0.989244	0.494622	−	0.869108i	$-0.335306\pi$
$547$	23.1365	0.989244	0.494622	+	0.869108i	$0.335306\pi$
$548$	5.76762	0.246381
$549$	0	0
$550$	0	0
$551$	−16.0735	−0.684756
$552$	0	0
$553$	−16.6172	−0.706635
$554$	−24.4373	−1.03824
$555$	0	0
$556$	−21.5828	−0.915315
$557$	−0.449182	−0.0190324	−0.00951622	−	0.999955i	$-0.503029\pi$
$557$	−0.449182	−0.0190324	−0.00951622	+	0.999955i	$0.503029\pi$
$558$	0	0
$559$	0	0
$560$	1.84921	0.0781435
$561$	0	0
$562$	−28.2313	−1.19087
$563$	16.4426	0.692973	0.346486	−	0.938055i	$-0.387375\pi$
$563$	16.4426	0.692973	0.346486	+	0.938055i	$0.387375\pi$
$564$	0	0
$565$	−3.94950	−0.166157
$566$	−2.11566	−0.0889277
$567$	0	0
$568$	−7.50379	−0.314852
$569$	8.97774	0.376366	0.188183	−	0.982134i	$-0.439740\pi$
$569$	8.97774	0.376366	0.188183	+	0.982134i	$0.439740\pi$
$570$	0	0
$571$	21.7321	0.909461	0.454731	−	0.890629i	$-0.349735\pi$
$571$	21.7321	0.909461	0.454731	+	0.890629i	$0.349735\pi$
$572$	0	0
$573$	0	0
$574$	−24.9517	−1.04146
$575$	−6.74657	−0.281351
$576$	0	0
$577$	−16.4426	−0.684516	−0.342258	−	0.939606i	$-0.611192\pi$
$577$	−16.4426	−0.684516	−0.342258	+	0.939606i	$0.611192\pi$
$578$	21.8239	0.907755
$579$	0	0
$580$	−3.96675	−0.164710
$581$	−33.2344	−1.37879
$582$	0	0
$583$	0	0
$584$	39.7980	1.64685
$585$	0	0
$586$	−19.6329	−0.811028
$587$	−13.8338	−0.570984	−0.285492	−	0.958381i	$-0.592157\pi$
$587$	−13.8338	−0.570984	−0.285492	+	0.958381i	$0.592157\pi$
$588$	0	0
$589$	−10.2217	−0.421177
$590$	13.4811	0.555010
$591$	0	0
$592$	−1.37748	−0.0566141
$593$	18.0915	0.742928	0.371464	−	0.928447i	$-0.378856\pi$
$593$	18.0915	0.742928	0.371464	+	0.928447i	$0.378856\pi$
$594$	0	0
$595$	29.5667	1.21212
$596$	−11.1633	−0.457266
$597$	0	0
$598$	0	0
$599$	37.7138	1.54094	0.770472	−	0.637474i	$-0.220021\pi$
$599$	37.7138	1.54094	0.770472	+	0.637474i	$0.220021\pi$
$600$	0	0
$601$	8.98205	0.366385	0.183193	−	0.983077i	$-0.441357\pi$
$601$	8.98205	0.366385	0.183193	+	0.983077i	$0.441357\pi$
$602$	−1.00530	−0.0409730
$603$	0	0
$604$	19.7725	0.804533
$605$	0	0
$606$	0	0
$607$	−8.34277	−0.338623	−0.169311	−	0.985563i	$-0.554154\pi$
$607$	−8.34277	−0.338623	−0.169311	+	0.985563i	$0.554154\pi$
$608$	25.2765	1.02510
$609$	0	0
$610$	1.81812	0.0736137
$611$	0	0
$612$	0	0
$613$	32.5872	1.31618	0.658092	−	0.752938i	$-0.271364\pi$
$613$	32.5872	1.31618	0.658092	+	0.752938i	$0.271364\pi$
$614$	−10.6961	−0.431658
$615$	0	0
$616$	0	0
$617$	20.1745	0.812197	0.406098	−	0.913829i	$-0.366889\pi$
$617$	20.1745	0.812197	0.406098	+	0.913829i	$0.366889\pi$
$618$	0	0
$619$	−35.3354	−1.42025	−0.710124	−	0.704076i	$-0.751361\pi$
$619$	−35.3354	−1.42025	−0.710124	+	0.704076i	$0.751361\pi$
$620$	−2.52258	−0.101309
$621$	0	0
$622$	12.8511	0.515282
$623$	41.6966	1.67054
$624$	0	0
$625$	1.00000	0.0400000
$626$	11.2567	0.449907
$627$	0	0
$628$	7.57736	0.302369
$629$	−22.0242	−0.878164
$630$	0	0
$631$	−10.6456	−0.423793	−0.211897	−	0.977292i	$-0.567964\pi$
$631$	−10.6456	−0.423793	−0.211897	+	0.977292i	$0.567964\pi$
$632$	10.4143	0.414257
$633$	0	0
$634$	−24.6911	−0.980610
$635$	16.5063	0.655032
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	5.50095	0.217444
$641$	−4.94018	−0.195125	−0.0975627	−	0.995229i	$-0.531105\pi$
$641$	−4.94018	−0.195125	−0.0975627	+	0.995229i	$0.531105\pi$
$642$	0	0
$643$	41.1240	1.62177	0.810887	−	0.585203i	$-0.198985\pi$
$643$	41.1240	1.62177	0.810887	+	0.585203i	$0.198985\pi$
$644$	35.8466	1.41256
$645$	0	0
$646$	−27.3376	−1.07558
$647$	37.4005	1.47037	0.735183	−	0.677868i	$-0.237096\pi$
$647$	37.4005	1.47037	0.735183	+	0.677868i	$0.237096\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	14.2481	0.557998
$653$	15.2344	0.596167	0.298083	−	0.954540i	$-0.403653\pi$
$653$	15.2344	0.596167	0.298083	+	0.954540i	$0.403653\pi$
$654$	0	0
$655$	−15.9739	−0.624154
$656$	2.31786	0.0904973
$657$	0	0
$658$	40.0147	1.55993
$659$	3.05484	0.119000	0.0594998	−	0.998228i	$-0.481049\pi$
$659$	3.05484	0.119000	0.0594998	+	0.998228i	$0.481049\pi$
$660$	0	0
$661$	−34.4426	−1.33966	−0.669832	−	0.742513i	$-0.733634\pi$
$661$	−34.4426	−1.33966	−0.669832	+	0.742513i	$0.733634\pi$
$662$	−17.9972	−0.699483
$663$	0	0
$664$	20.8285	0.808303
$665$	−21.5299	−0.834894
$666$	0	0
$667$	23.3708	0.904921
$668$	2.48603	0.0961873
$669$	0	0
$670$	1.34656	0.0520223
$671$	0	0
$672$	0	0
$673$	37.9345	1.46227	0.731134	−	0.682234i	$-0.238992\pi$
$673$	37.9345	1.46227	0.731134	+	0.682234i	$0.238992\pi$
$674$	15.2618	0.587863
$675$	0	0
$676$	14.8863	0.572551
$677$	−9.15268	−0.351766	−0.175883	−	0.984411i	$-0.556278\pi$
$677$	−9.15268	−0.351766	−0.175883	+	0.984411i	$0.556278\pi$
$678$	0	0
$679$	−57.7342	−2.21564
$680$	−18.5299	−0.710590
$681$	0	0
$682$	0	0
$683$	−27.7980	−1.06366	−0.531830	−	0.846851i	$-0.678496\pi$
$683$	−27.7980	−1.06366	−0.531830	+	0.846851i	$0.678496\pi$
$684$	0	0
$685$	5.03677	0.192445
$686$	32.3049	1.23341
$687$	0	0
$688$	0.0933864	0.00356033
$689$	0	0
$690$	0	0
$691$	29.4373	1.11985	0.559924	−	0.828544i	$-0.310830\pi$
$691$	29.4373	1.11985	0.559924	+	0.828544i	$0.310830\pi$
$692$	−10.0900	−0.383563
$693$	0	0
$694$	0.978945	0.0371603
$695$	−18.8479	−0.714943
$696$	0	0
$697$	37.0598	1.40374
$698$	12.5731	0.475898
$699$	0	0
$700$	−5.31331	−0.200824
$701$	6.92820	0.261675	0.130837	−	0.991404i	$-0.458233\pi$
$701$	6.92820	0.261675	0.130837	+	0.991404i	$0.458233\pi$
$702$	0	0
$703$	16.0376	0.604871
$704$	0	0
$705$	0	0
$706$	−26.8222	−1.00947
$707$	−70.0461	−2.63435
$708$	0	0
$709$	−51.9956	−1.95274	−0.976368	−	0.216116i	$-0.930661\pi$
$709$	−51.9956	−1.95274	−0.976368	+	0.216116i	$0.930661\pi$
$710$	−2.38586	−0.0895399
$711$	0	0
$712$	−26.1319	−0.979335
$713$	14.8622	0.556595
$714$	0	0
$715$	0	0
$716$	15.9158	0.594801
$717$	0	0
$718$	13.0284	0.486215
$719$	27.3921	1.02155	0.510777	−	0.859713i	$-0.329358\pi$
$719$	27.3921	1.02155	0.510777	+	0.859713i	$0.329358\pi$
$720$	0	0
$721$	88.2676	3.28726
$722$	2.33917	0.0870549
$723$	0	0
$724$	23.6088	0.877414
$725$	−3.46410	−0.128654
$726$	0	0
$727$	48.9588	1.81578	0.907891	−	0.419206i	$-0.137692\pi$
$727$	48.9588	1.81578	0.907891	+	0.419206i	$0.137692\pi$
$728$	0	0
$729$	0	0
$730$	12.6540	0.468344
$731$	1.49314	0.0552256
$732$	0	0
$733$	−24.5510	−0.906813	−0.453407	−	0.891304i	$-0.649791\pi$
$733$	−24.5510	−0.906813	−0.453407	+	0.891304i	$0.649791\pi$
$734$	4.63572	0.171108
$735$	0	0
$736$	−36.7518	−1.35469
$737$	0	0
$738$	0	0
$739$	−20.5605	−0.756331	−0.378165	−	0.925738i	$-0.623445\pi$
$739$	−20.5605	−0.756331	−0.378165	+	0.925738i	$0.623445\pi$
$740$	3.95789	0.145495
$741$	0	0
$742$	−15.2029	−0.558117
$743$	29.3082	1.07521	0.537607	−	0.843195i	$-0.319328\pi$
$743$	29.3082	1.07521	0.537607	+	0.843195i	$0.319328\pi$
$744$	0	0
$745$	−9.74872	−0.357165
$746$	9.17149	0.335792
$747$	0	0
$748$	0	0
$749$	80.1196	2.92751
$750$	0	0
$751$	2.95789	0.107935	0.0539675	−	0.998543i	$-0.482813\pi$
$751$	2.95789	0.107935	0.0539675	+	0.998543i	$0.482813\pi$
$752$	−3.71712	−0.135550
$753$	0	0
$754$	0	0
$755$	17.2670	0.628412
$756$	0	0
$757$	2.36909	0.0861060	0.0430530	−	0.999073i	$-0.486292\pi$
$757$	2.36909	0.0861060	0.0430530	+	0.999073i	$0.486292\pi$
$758$	4.55010	0.165267
$759$	0	0
$760$	13.4931	0.489448
$761$	52.8599	1.91617	0.958085	−	0.286485i	$-0.0924869\pi$
$761$	52.8599	1.91617	0.958085	+	0.286485i	$0.0924869\pi$
$762$	0	0
$763$	−65.6770	−2.37767
$764$	15.4510	0.558999
$765$	0	0
$766$	16.6175	0.600416
$767$	0	0
$768$	0	0
$769$	8.48531	0.305988	0.152994	−	0.988227i	$-0.451108\pi$
$769$	8.48531	0.305988	0.152994	+	0.988227i	$0.451108\pi$
$770$	0	0
$771$	0	0
$772$	21.4486	0.771954
$773$	−39.7917	−1.43121	−0.715605	−	0.698506i	$-0.753849\pi$
$773$	−39.7917	−1.43121	−0.715605	+	0.698506i	$0.753849\pi$
$774$	0	0
$775$	−2.20293	−0.0791316
$776$	36.1830	1.29889
$777$	0	0
$778$	12.0752	0.432917
$779$	−26.9863	−0.966884
$780$	0	0
$781$	0	0
$782$	39.7486	1.42141
$783$	0	0
$784$	−5.79067	−0.206810
$785$	6.61718	0.236177
$786$	0	0
$787$	−14.0907	−0.502280	−0.251140	−	0.967951i	$-0.580806\pi$
$787$	−14.0907	−0.502280	−0.251140	+	0.967951i	$0.580806\pi$
$788$	−15.8670	−0.565239
$789$	0	0
$790$	3.31126	0.117809
$791$	18.3258	0.651591
$792$	0	0
$793$	0	0
$794$	−25.6157	−0.909068
$795$	0	0
$796$	22.1735	0.785918
$797$	37.0598	1.31273	0.656363	−	0.754445i	$-0.272094\pi$
$797$	37.0598	1.31273	0.656363	+	0.754445i	$0.272094\pi$
$798$	0	0
$799$	−59.4323	−2.10256
$800$	5.44748	0.192597
$801$	0	0
$802$	−28.7181	−1.01407
$803$	0	0
$804$	0	0
$805$	31.3043	1.10333
$806$	0	0
$807$	0	0
$808$	43.8990	1.54436
$809$	7.52424	0.264538	0.132269	−	0.991214i	$-0.457774\pi$
$809$	7.52424	0.264538	0.132269	+	0.991214i	$0.457774\pi$
$810$	0	0
$811$	40.0579	1.40662	0.703312	−	0.710881i	$-0.251704\pi$
$811$	40.0579	1.40662	0.703312	+	0.710881i	$0.251704\pi$
$812$	18.4059	0.645919
$813$	0	0
$814$	0	0
$815$	12.4426	0.435847
$816$	0	0
$817$	−1.08727	−0.0380389
$818$	23.8602	0.834253
$819$	0	0
$820$	−6.65988	−0.232573
$821$	42.4676	1.48213	0.741064	−	0.671434i	$-0.234321\pi$
$821$	42.4676	1.48213	0.741064	+	0.671434i	$0.234321\pi$
$822$	0	0
$823$	17.5299	0.611054	0.305527	−	0.952183i	$-0.401167\pi$
$823$	17.5299	0.611054	0.305527	+	0.952183i	$0.401167\pi$
$824$	−55.3187	−1.92712
$825$	0	0
$826$	−62.5530	−2.17650
$827$	−24.7854	−0.861872	−0.430936	−	0.902383i	$-0.641816\pi$
$827$	−24.7854	−0.861872	−0.430936	+	0.902383i	$0.641816\pi$
$828$	0	0
$829$	8.66769	0.301041	0.150521	−	0.988607i	$-0.451905\pi$
$829$	8.66769	0.301041	0.150521	+	0.988607i	$0.451905\pi$
$830$	6.62252	0.229871
$831$	0	0
$832$	0	0
$833$	−92.5858	−3.20791
$834$	0	0
$835$	2.17101	0.0751308
$836$	0	0
$837$	0	0
$838$	−14.1112	−0.487463
$839$	−43.4657	−1.50060	−0.750301	−	0.661096i	$-0.770091\pi$
$839$	−43.4657	−1.50060	−0.750301	+	0.661096i	$0.770091\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	35.0630	1.20835
$843$	0	0
$844$	−0.402489	−0.0138542
$845$	13.0000	0.447214
$846$	0	0
$847$	0	0
$848$	1.41226	0.0484973
$849$	0	0
$850$	−5.89167	−0.202083
$851$	−23.3186	−0.799351
$852$	0	0
$853$	−21.7262	−0.743891	−0.371946	−	0.928254i	$-0.621309\pi$
$853$	−21.7262	−0.743891	−0.371946	+	0.928254i	$0.621309\pi$
$854$	−8.43615	−0.288679
$855$	0	0
$856$	−50.2123	−1.71622
$857$	32.7977	1.12035	0.560174	−	0.828375i	$-0.310734\pi$
$857$	32.7977	1.12035	0.560174	+	0.828375i	$0.310734\pi$
$858$	0	0
$859$	−0.442636	−0.0151025	−0.00755127	−	0.999971i	$-0.502404\pi$
$859$	−0.442636	−0.0151025	−0.00755127	+	0.999971i	$0.502404\pi$
$860$	−0.268326	−0.00914983
$861$	0	0
$862$	−17.2597	−0.587868
$863$	−19.9265	−0.678304	−0.339152	−	0.940732i	$-0.610140\pi$
$863$	−19.9265	−0.678304	−0.339152	+	0.940732i	$0.610140\pi$
$864$	0	0
$865$	−8.81142	−0.299597
$866$	−17.0522	−0.579457
$867$	0	0
$868$	11.7049	0.397289
$869$	0	0
$870$	0	0
$871$	0	0
$872$	41.1608	1.39388
$873$	0	0
$874$	−28.9442	−0.979051
$875$	−4.64003	−0.156862
$876$	0	0
$877$	25.6590	0.866442	0.433221	−	0.901288i	$-0.357377\pi$
$877$	25.6590	0.866442	0.433221	+	0.901288i	$0.357377\pi$
$878$	1.35337	0.0456740
$879$	0	0
$880$	0	0
$881$	32.9863	1.11134	0.555668	−	0.831404i	$-0.312463\pi$
$881$	32.9863	1.11134	0.555668	+	0.831404i	$0.312463\pi$
$882$	0	0
$883$	44.5897	1.50056	0.750282	−	0.661118i	$-0.229918\pi$
$883$	44.5897	1.50056	0.750282	+	0.661118i	$0.229918\pi$
$884$	0	0
$885$	0	0
$886$	15.6122	0.524503
$887$	25.5953	0.859405	0.429703	−	0.902970i	$-0.358618\pi$
$887$	25.5953	0.859405	0.429703	+	0.902970i	$0.358618\pi$
$888$	0	0
$889$	−76.5897	−2.56874
$890$	−8.30877	−0.278511
$891$	0	0
$892$	−3.37748	−0.113086
$893$	43.2775	1.44823
$894$	0	0
$895$	13.8990	0.464592
$896$	−25.5246	−0.852716
$897$	0	0
$898$	14.1112	0.470896
$899$	7.63118	0.254514
$900$	0	0
$901$	22.5804	0.752261
$902$	0	0
$903$	0	0
$904$	−11.4851	−0.381988
$905$	20.6172	0.685338
$906$	0	0
$907$	−4.29554	−0.142631	−0.0713156	−	0.997454i	$-0.522720\pi$
$907$	−4.29554	−0.142631	−0.0713156	+	0.997454i	$0.522720\pi$
$908$	−12.1074	−0.401797
$909$	0	0
$910$	0	0
$911$	12.6814	0.420154	0.210077	−	0.977685i	$-0.432629\pi$
$911$	12.6814	0.420154	0.210077	+	0.977685i	$0.432629\pi$
$912$	0	0
$913$	0	0
$914$	1.46170	0.0483487
$915$	0	0
$916$	13.6834	0.452112
$917$	74.1196	2.44765
$918$	0	0
$919$	−18.6238	−0.614343	−0.307172	−	0.951654i	$-0.599383\pi$
$919$	−18.6238	−0.614343	−0.307172	+	0.951654i	$0.599383\pi$
$920$	−19.6189	−0.646816
$921$	0	0
$922$	−7.27249	−0.239507
$923$	0	0
$924$	0	0
$925$	3.45636	0.113645
$926$	32.9786	1.08374
$927$	0	0
$928$	−18.8706	−0.619458
$929$	−42.2206	−1.38521	−0.692607	−	0.721315i	$-0.743538\pi$
$929$	−42.2206	−1.38521	−0.692607	+	0.721315i	$0.743538\pi$
$930$	0	0
$931$	67.4193	2.20958
$932$	8.57032	0.280730
$933$	0	0
$934$	19.3184	0.632118
$935$	0	0
$936$	0	0
$937$	−14.0270	−0.458243	−0.229122	−	0.973398i	$-0.573585\pi$
$937$	−14.0270	−0.458243	−0.229122	+	0.973398i	$0.573585\pi$
$938$	−6.24810	−0.204008
$939$	0	0
$940$	10.6803	0.348355
$941$	−44.0485	−1.43594	−0.717970	−	0.696075i	$-0.754928\pi$
$941$	−44.0485	−1.43594	−0.717970	+	0.696075i	$0.754928\pi$
$942$	0	0
$943$	39.2378	1.27776
$944$	5.81080	0.189125
$945$	0	0
$946$	0	0
$947$	−30.7191	−0.998237	−0.499119	−	0.866534i	$-0.666343\pi$
$947$	−30.7191	−0.998237	−0.499119	+	0.866534i	$0.666343\pi$
$948$	0	0
$949$	0	0
$950$	4.29021	0.139193
$951$	0	0
$952$	85.9794	2.78661
$953$	31.9479	1.03489	0.517447	−	0.855715i	$-0.326883\pi$
$953$	31.9479	1.03489	0.517447	+	0.855715i	$0.326883\pi$
$954$	0	0
$955$	13.4931	0.436628
$956$	−32.4165	−1.04843
$957$	0	0
$958$	9.44704	0.305220
$959$	−23.3708	−0.754682
$960$	0	0
$961$	−26.1471	−0.843455
$962$	0	0
$963$	0	0
$964$	−7.02836	−0.226368
$965$	18.7308	0.602965
$966$	0	0
$967$	−34.5978	−1.11259	−0.556295	−	0.830985i	$-0.687777\pi$
$967$	−34.5978	−1.11259	−0.556295	+	0.830985i	$0.687777\pi$
$968$	0	0
$969$	0	0
$970$	11.5045	0.369389
$971$	32.9588	1.05770	0.528849	−	0.848716i	$-0.322624\pi$
$971$	32.9588	1.05770	0.528849	+	0.848716i	$0.322624\pi$
$972$	0	0
$973$	87.4550	2.80368
$974$	14.1637	0.453833
$975$	0	0
$976$	0.783668	0.0250846
$977$	39.3584	1.25919	0.629594	−	0.776925i	$-0.283221\pi$
$977$	39.3584	1.25919	0.629594	+	0.776925i	$0.283221\pi$
$978$	0	0
$979$	0	0
$980$	16.6382	0.531489
$981$	0	0
$982$	31.9304	1.01894
$983$	−40.3868	−1.28814	−0.644069	−	0.764967i	$-0.722755\pi$
$983$	−40.3868	−1.28814	−0.644069	+	0.764967i	$0.722755\pi$
$984$	0	0
$985$	−13.8564	−0.441502
$986$	20.4093	0.649966
$987$	0	0
$988$	0	0
$989$	1.58089	0.0502693
$990$	0	0
$991$	−4.67302	−0.148443	−0.0742217	−	0.997242i	$-0.523647\pi$
$991$	−4.67302	−0.148443	−0.0742217	+	0.997242i	$0.523647\pi$
$992$	−12.0004	−0.381014
$993$	0	0
$994$	11.0705	0.351135
$995$	19.3638	0.613872
$996$	0	0
$997$	28.6544	0.907495	0.453747	−	0.891130i	$-0.350087\pi$
$997$	28.6544	0.907495	0.453747	+	0.891130i	$0.350087\pi$
$998$	27.3716	0.866431
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bw.1.4	yes	6
3.2	odd	2		5445.2.a.by.1.3	yes	6
11.10	odd	2	inner	5445.2.a.bw.1.3	✓	6
33.32	even	2		5445.2.a.by.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5445.2.a.bw.1.3	✓	6	11.10	odd	2	inner
5445.2.a.bw.1.4	yes	6	1.1	even	1	trivial
5445.2.a.by.1.3	yes	6	3.2	odd	2
5445.2.a.by.1.4	yes	6	33.32	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q+0.924607 q^{2} -1.14510 q^{4} -1.00000 q^{5} +4.64003 q^{7} -2.90798 q^{8} +O(q^{10})\)	\(q+0.924607 q^{2} -1.14510 q^{4} -1.00000 q^{5} +4.64003 q^{7} -2.90798 q^{8} -0.924607 q^{10} +4.29021 q^{14} -0.398534 q^{16} -6.37208 q^{17} +4.64003 q^{19} +1.14510 q^{20} -6.74657 q^{23} +1.00000 q^{25} -5.31331 q^{28} -3.46410 q^{29} -2.20293 q^{31} +5.44748 q^{32} -5.89167 q^{34} -4.64003 q^{35} +3.45636 q^{37} +4.29021 q^{38} +2.90798 q^{40} -5.81597 q^{41} -0.234325 q^{43} -6.23792 q^{46} +9.32698 q^{47} +14.5299 q^{49} +0.924607 q^{50} -3.54364 q^{53} -13.4931 q^{56} -3.20293 q^{58} -14.5804 q^{59} -1.96638 q^{61} -2.03684 q^{62} +5.83384 q^{64} -1.45636 q^{67} +7.29669 q^{68} -4.29021 q^{70} +2.58041 q^{71} -13.6858 q^{73} +3.19578 q^{74} -5.31331 q^{76} -3.58126 q^{79} +0.398534 q^{80} -5.37748 q^{82} -7.16253 q^{83} +6.37208 q^{85} -0.216658 q^{86} +8.98627 q^{89} +7.72551 q^{92} +8.62379 q^{94} -4.64003 q^{95} -12.4426 q^{97} +13.4345 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{4} - 6 q^{5}+O(q^{10}) \) 6 * q + 6 * q^4 - 6 * q^5	\( 6 q + 6 q^{4} - 6 q^{5} - 6 q^{16} - 6 q^{20} - 24 q^{23} + 6 q^{25} - 6 q^{31} - 6 q^{34} + 30 q^{37} - 12 q^{47} + 12 q^{49} - 12 q^{53} - 48 q^{56} - 12 q^{58} - 36 q^{59} - 18 q^{67} - 36 q^{71} + 6 q^{80} + 12 q^{82} - 60 q^{86} - 12 q^{89} - 18 q^{92} - 18 q^{97}+O(q^{100}) \) 6 * q + 6 * q^4 - 6 * q^5 - 6 * q^16 - 6 * q^20 - 24 * q^23 + 6 * q^25 - 6 * q^31 - 6 * q^34 + 30 * q^37 - 12 * q^47 + 12 * q^49 - 12 * q^53 - 48 * q^56 - 12 * q^58 - 36 * q^59 - 18 * q^67 - 36 * q^71 + 6 * q^80 + 12 * q^82 - 60 * q^86 - 12 * q^89 - 18 * q^92 - 18 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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