LMFDB - Embedded newform 5733.2.a.br.1.1

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:	$6$
Coefficient field:	6.6.4507648.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.758419$ 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)

$f(q)$	$=$	$q-2.18322 q^{2} +2.76645 q^{4} -2.11065 q^{5} -1.67333 q^{8} +O(q^{10})$	$q-2.18322 q^{2} +2.76645 q^{4} -2.11065 q^{5} -1.67333 q^{8} +4.60802 q^{10} +5.76889 q^{11} +1.00000 q^{13} -1.87966 q^{16} -1.64082 q^{17} -2.67077 q^{19} -5.83901 q^{20} -12.5948 q^{22} -6.42469 q^{23} -0.545141 q^{25} -2.18322 q^{26} +6.04973 q^{29} +5.12202 q^{31} +7.45036 q^{32} +3.58227 q^{34} +5.74772 q^{37} +5.83087 q^{38} +3.53181 q^{40} +7.14100 q^{41} -4.47061 q^{43} +15.9593 q^{44} +14.0265 q^{46} -11.7910 q^{47} +1.19016 q^{50} +2.76645 q^{52} -3.44959 q^{53} -12.1761 q^{55} -13.2079 q^{58} -13.1805 q^{59} +6.24666 q^{61} -11.1825 q^{62} -12.5065 q^{64} -2.11065 q^{65} +7.74216 q^{67} -4.53925 q^{68} -13.6372 q^{71} -15.5041 q^{73} -12.5485 q^{74} -7.38854 q^{76} +1.12214 q^{79} +3.96731 q^{80} -15.5904 q^{82} -4.96925 q^{83} +3.46321 q^{85} +9.76032 q^{86} -9.65324 q^{88} -1.14630 q^{89} -17.7736 q^{92} +25.7423 q^{94} +5.63707 q^{95} -6.97223 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{4} - 6 q^{5}+O(q^{10}) $ 6 * q + 4 * q^4 - 6 * q^5	$ 6 q + 4 q^{4} - 6 q^{5} + 4 q^{10} - 4 q^{11} + 6 q^{13} - 16 q^{17} + 2 q^{19} - 16 q^{20} - 12 q^{22} + 6 q^{23} - 4 q^{25} + 6 q^{29} + 6 q^{31} + 20 q^{32} - 8 q^{38} + 4 q^{40} + 8 q^{41} + 2 q^{43} + 4 q^{44} + 8 q^{46} - 30 q^{47} - 8 q^{50} + 4 q^{52} + 14 q^{53} - 8 q^{55} - 8 q^{58} - 24 q^{59} - 28 q^{62} - 20 q^{64} - 6 q^{65} + 16 q^{67} - 28 q^{68} - 8 q^{71} - 6 q^{73} + 12 q^{74} - 16 q^{76} - 22 q^{79} + 28 q^{80} - 40 q^{82} - 50 q^{83} - 8 q^{85} + 16 q^{86} - 44 q^{88} - 26 q^{89} - 20 q^{92} - 32 q^{94} + 6 q^{95} - 14 q^{97}+O(q^{100}) $ 6 * q + 4 * q^4 - 6 * q^5 + 4 * q^10 - 4 * q^11 + 6 * q^13 - 16 * q^17 + 2 * q^19 - 16 * q^20 - 12 * q^22 + 6 * q^23 - 4 * q^25 + 6 * q^29 + 6 * q^31 + 20 * q^32 - 8 * q^38 + 4 * q^40 + 8 * q^41 + 2 * q^43 + 4 * q^44 + 8 * q^46 - 30 * q^47 - 8 * q^50 + 4 * q^52 + 14 * q^53 - 8 * q^55 - 8 * q^58 - 24 * q^59 - 28 * q^62 - 20 * q^64 - 6 * q^65 + 16 * q^67 - 28 * q^68 - 8 * q^71 - 6 * q^73 + 12 * q^74 - 16 * q^76 - 22 * q^79 + 28 * q^80 - 40 * q^82 - 50 * q^83 - 8 * q^85 + 16 * q^86 - 44 * q^88 - 26 * q^89 - 20 * q^92 - 32 * q^94 + 6 * q^95 - 14 * 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 $

$2$	−2.18322	−1.54377	−0.771885	−	0.635762i	$-0.780686\pi$
$2$	−2.18322	−1.54377	−0.771885	+	0.635762i	$0.780686\pi$
$3$	0	0
$3$	0	0
$4$	2.76645	1.38322
$5$	−2.11065	−0.943913	−0.471957	−	0.881622i	$-0.656452\pi$
$5$	−2.11065	−0.943913	−0.471957	+	0.881622i	$0.656452\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.67333	−0.591610
$9$	0	0
$10$	4.60802	1.45718
$11$	5.76889	1.73939	0.869694	−	0.493592i	$-0.164316\pi$
$11$	5.76889	1.73939	0.869694	+	0.493592i	$0.164316\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	−1.87966	−0.469915
$17$	−1.64082	−0.397958	−0.198979	−	0.980004i	$-0.563763\pi$
$17$	−1.64082	−0.397958	−0.198979	+	0.980004i	$0.563763\pi$
$18$	0	0
$19$	−2.67077	−0.612716	−0.306358	−	0.951916i	$-0.599110\pi$
$19$	−2.67077	−0.612716	−0.306358	+	0.951916i	$0.599110\pi$
$20$	−5.83901	−1.30564
$21$	0	0
$22$	−12.5948	−2.68521
$23$	−6.42469	−1.33964	−0.669820	−	0.742523i	$-0.733629\pi$
$23$	−6.42469	−1.33964	−0.669820	+	0.742523i	$0.733629\pi$
$24$	0	0
$25$	−0.545141	−0.109028
$26$	−2.18322	−0.428165
$27$	0	0
$28$	0	0
$29$	6.04973	1.12341	0.561704	−	0.827338i	$-0.310146\pi$
$29$	6.04973	1.12341	0.561704	+	0.827338i	$0.310146\pi$
$30$	0	0
$31$	5.12202	0.919942	0.459971	−	0.887934i	$-0.347860\pi$
$31$	5.12202	0.919942	0.459971	+	0.887934i	$0.347860\pi$
$32$	7.45036	1.31705
$33$	0	0
$34$	3.58227	0.614355
$35$	0	0
$36$	0	0
$37$	5.74772	0.944919	0.472460	−	0.881352i	$-0.343366\pi$
$37$	5.74772	0.944919	0.472460	+	0.881352i	$0.343366\pi$
$38$	5.83087	0.945892
$39$	0	0
$40$	3.53181	0.558428
$41$	7.14100	1.11524	0.557619	−	0.830097i	$-0.311715\pi$
$41$	7.14100	1.11524	0.557619	+	0.830097i	$0.311715\pi$
$42$	0	0
$43$	−4.47061	−0.681761	−0.340881	−	0.940107i	$-0.610725\pi$
$43$	−4.47061	−0.681761	−0.340881	+	0.940107i	$0.610725\pi$
$44$	15.9593	2.40596
$45$	0	0
$46$	14.0265	2.06810
$47$	−11.7910	−1.71989	−0.859947	−	0.510384i	$-0.829503\pi$
$47$	−11.7910	−1.71989	−0.859947	+	0.510384i	$0.829503\pi$
$48$	0	0
$49$	0	0
$50$	1.19016	0.168314
$51$	0	0
$52$	2.76645	0.383637
$53$	−3.44959	−0.473838	−0.236919	−	0.971529i	$-0.576138\pi$
$53$	−3.44959	−0.473838	−0.236919	+	0.971529i	$0.576138\pi$
$54$	0	0
$55$	−12.1761	−1.64183
$56$	0	0
$57$	0	0
$58$	−13.2079	−1.73428
$59$	−13.1805	−1.71596	−0.857981	−	0.513682i	$-0.828281\pi$
$59$	−13.1805	−1.71596	−0.857981	+	0.513682i	$0.828281\pi$
$60$	0	0
$61$	6.24666	0.799803	0.399901	−	0.916558i	$-0.369044\pi$
$61$	6.24666	0.799803	0.399901	+	0.916558i	$0.369044\pi$
$62$	−11.1825	−1.42018
$63$	0	0
$64$	−12.5065	−1.56331
$65$	−2.11065	−0.261794
$66$	0	0
$67$	7.74216	0.945856	0.472928	−	0.881101i	$-0.343197\pi$
$67$	7.74216	0.945856	0.472928	+	0.881101i	$0.343197\pi$
$68$	−4.53925	−0.550465
$69$	0	0
$70$	0	0
$71$	−13.6372	−1.61844	−0.809221	−	0.587504i	$-0.800111\pi$
$71$	−13.6372	−1.61844	−0.809221	+	0.587504i	$0.800111\pi$
$72$	0	0
$73$	−15.5041	−1.81462	−0.907308	−	0.420467i	$-0.861866\pi$
$73$	−15.5041	−1.81462	−0.907308	+	0.420467i	$0.861866\pi$
$74$	−12.5485	−1.45874
$75$	0	0
$76$	−7.38854	−0.847524
$77$	0	0
$78$	0	0
$79$	1.12214	0.126251	0.0631254	−	0.998006i	$-0.479893\pi$
$79$	1.12214	0.126251	0.0631254	+	0.998006i	$0.479893\pi$
$80$	3.96731	0.443559
$81$	0	0
$82$	−15.5904	−1.72167
$83$	−4.96925	−0.545446	−0.272723	−	0.962093i	$-0.587924\pi$
$83$	−4.96925	−0.545446	−0.272723	+	0.962093i	$0.587924\pi$
$84$	0	0
$85$	3.46321	0.375638
$86$	9.76032	1.05248
$87$	0	0
$88$	−9.65324	−1.02904
$89$	−1.14630	−0.121507	−0.0607535	−	0.998153i	$-0.519350\pi$
$89$	−1.14630	−0.121507	−0.0607535	+	0.998153i	$0.519350\pi$
$90$	0	0
$91$	0	0
$92$	−17.7736	−1.85302
$93$	0	0
$94$	25.7423	2.65512
$95$	5.63707	0.578351
$96$	0	0
$97$	−6.97223	−0.707923	−0.353961	−	0.935260i	$-0.615166\pi$
$97$	−6.97223	−0.707923	−0.353961	+	0.935260i	$0.615166\pi$
$98$	0	0
$99$	0	0
$100$	−1.50810	−0.150810
$101$	6.49919	0.646694	0.323347	−	0.946280i	$-0.395192\pi$
$101$	6.49919	0.646694	0.323347	+	0.946280i	$0.395192\pi$
$102$	0	0
$103$	−0.578048	−0.0569568	−0.0284784	−	0.999594i	$-0.509066\pi$
$103$	−0.578048	−0.0569568	−0.0284784	+	0.999594i	$0.509066\pi$
$104$	−1.67333	−0.164083
$105$	0	0
$106$	7.53122	0.731497
$107$	16.2573	1.57165	0.785824	−	0.618450i	$-0.212239\pi$
$107$	16.2573	1.57165	0.785824	+	0.618450i	$0.212239\pi$
$108$	0	0
$109$	1.78191	0.170676	0.0853378	−	0.996352i	$-0.472803\pi$
$109$	1.78191	0.170676	0.0853378	+	0.996352i	$0.472803\pi$
$110$	26.5832	2.53461
$111$	0	0
$112$	0	0
$113$	7.52215	0.707624	0.353812	−	0.935316i	$-0.384885\pi$
$113$	7.52215	0.707624	0.353812	+	0.935316i	$0.384885\pi$
$114$	0	0
$115$	13.5603	1.26450
$116$	16.7363	1.55392
$117$	0	0
$118$	28.7760	2.64905
$119$	0	0
$120$	0	0
$121$	22.2801	2.02547
$122$	−13.6378	−1.23471
$123$	0	0
$124$	14.1698	1.27249
$125$	11.7039	1.04683
$126$	0	0
$127$	−19.3056	−1.71310	−0.856549	−	0.516067i	$-0.827396\pi$
$127$	−19.3056	−1.71310	−0.856549	+	0.516067i	$0.827396\pi$
$128$	12.4036	1.09634
$129$	0	0
$130$	4.60802	0.404150
$131$	−9.69703	−0.847234	−0.423617	−	0.905842i	$-0.639240\pi$
$131$	−9.69703	−0.847234	−0.423617	+	0.905842i	$0.639240\pi$
$132$	0	0
$133$	0	0
$134$	−16.9028	−1.46018
$135$	0	0
$136$	2.74563	0.235436
$137$	15.4493	1.31993	0.659963	−	0.751298i	$-0.270572\pi$
$137$	15.4493	1.31993	0.659963	+	0.751298i	$0.270572\pi$
$138$	0	0
$139$	3.84912	0.326478	0.163239	−	0.986587i	$-0.447806\pi$
$139$	3.84912	0.326478	0.163239	+	0.986587i	$0.447806\pi$
$140$	0	0
$141$	0	0
$142$	29.7731	2.49850
$143$	5.76889	0.482419
$144$	0	0
$145$	−12.7689	−1.06040
$146$	33.8488	2.80135
$147$	0	0
$148$	15.9008	1.30704
$149$	1.04028	0.0852231	0.0426115	−	0.999092i	$-0.486432\pi$
$149$	1.04028	0.0852231	0.0426115	+	0.999092i	$0.486432\pi$
$150$	0	0
$151$	−4.34336	−0.353458	−0.176729	−	0.984260i	$-0.556552\pi$
$151$	−4.34336	−0.353458	−0.176729	+	0.984260i	$0.556552\pi$
$152$	4.46906	0.362489
$153$	0	0
$154$	0	0
$155$	−10.8108	−0.868345
$156$	0	0
$157$	0.336000	0.0268157	0.0134079	−	0.999910i	$-0.495732\pi$
$157$	0.336000	0.0268157	0.0134079	+	0.999910i	$0.495732\pi$
$158$	−2.44988	−0.194902
$159$	0	0
$160$	−15.7251	−1.24318
$161$	0	0
$162$	0	0
$163$	6.79919	0.532553	0.266277	−	0.963897i	$-0.414207\pi$
$163$	6.79919	0.532553	0.266277	+	0.963897i	$0.414207\pi$
$164$	19.7552	1.54262
$165$	0	0
$166$	10.8490	0.842043
$167$	−0.668649	−0.0517416	−0.0258708	−	0.999665i	$-0.508236\pi$
$167$	−0.668649	−0.0517416	−0.0258708	+	0.999665i	$0.508236\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−7.56094	−0.579898
$171$	0	0
$172$	−12.3677	−0.943029
$173$	−25.1178	−1.90967	−0.954837	−	0.297131i	$-0.903970\pi$
$173$	−25.1178	−1.90967	−0.954837	+	0.297131i	$0.903970\pi$
$174$	0	0
$175$	0	0
$176$	−10.8436	−0.817364
$177$	0	0
$178$	2.50262	0.187579
$179$	1.89527	0.141659	0.0708294	−	0.997488i	$-0.477435\pi$
$179$	1.89527	0.141659	0.0708294	+	0.997488i	$0.477435\pi$
$180$	0	0
$181$	11.3595	0.844347	0.422174	−	0.906515i	$-0.361267\pi$
$181$	11.3595	0.844347	0.422174	+	0.906515i	$0.361267\pi$
$182$	0	0
$183$	0	0
$184$	10.7506	0.792544
$185$	−12.1314	−0.891922
$186$	0	0
$187$	−9.46573	−0.692203
$188$	−32.6192	−2.37900
$189$	0	0
$190$	−12.3070	−0.892840
$191$	23.4407	1.69611	0.848055	−	0.529909i	$-0.177774\pi$
$191$	23.4407	1.69611	0.848055	+	0.529909i	$0.177774\pi$
$192$	0	0
$193$	−1.85210	−0.133317	−0.0666584	−	0.997776i	$-0.521234\pi$
$193$	−1.85210	−0.133317	−0.0666584	+	0.997776i	$0.521234\pi$
$194$	15.2219	1.09287
$195$	0	0
$196$	0	0
$197$	9.87082	0.703267	0.351634	−	0.936138i	$-0.385626\pi$
$197$	9.87082	0.703267	0.351634	+	0.936138i	$0.385626\pi$
$198$	0	0
$199$	12.2492	0.868324	0.434162	−	0.900835i	$-0.357044\pi$
$199$	12.2492	0.868324	0.434162	+	0.900835i	$0.357044\pi$
$200$	0.912199	0.0645022
$201$	0	0
$202$	−14.1892	−0.998347
$203$	0	0
$204$	0	0
$205$	−15.0722	−1.05269
$206$	1.26201	0.0879281
$207$	0	0
$208$	−1.87966	−0.130331
$209$	−15.4074	−1.06575
$210$	0	0
$211$	−0.739899	−0.0509368	−0.0254684	−	0.999676i	$-0.508108\pi$
$211$	−0.739899	−0.0509368	−0.0254684	+	0.999676i	$0.508108\pi$
$212$	−9.54312	−0.655424
$213$	0	0
$214$	−35.4932	−2.42626
$215$	9.43591	0.643523
$216$	0	0
$217$	0	0
$218$	−3.89029	−0.263484
$219$	0	0
$220$	−33.6847	−2.27102
$221$	−1.64082	−0.110374
$222$	0	0
$223$	8.30577	0.556196	0.278098	−	0.960553i	$-0.410296\pi$
$223$	8.30577	0.556196	0.278098	+	0.960553i	$0.410296\pi$
$224$	0	0
$225$	0	0
$226$	−16.4225	−1.09241
$227$	−13.2917	−0.882198	−0.441099	−	0.897458i	$-0.645411\pi$
$227$	−13.2917	−0.882198	−0.441099	+	0.897458i	$0.645411\pi$
$228$	0	0
$229$	17.9207	1.18423	0.592115	−	0.805853i	$-0.298293\pi$
$229$	17.9207	1.18423	0.592115	+	0.805853i	$0.298293\pi$
$230$	−29.6051	−1.95210
$231$	0	0
$232$	−10.1232	−0.664619
$233$	−6.29968	−0.412705	−0.206353	−	0.978478i	$-0.566159\pi$
$233$	−6.29968	−0.412705	−0.206353	+	0.978478i	$0.566159\pi$
$234$	0	0
$235$	24.8867	1.62343
$236$	−36.4633	−2.37356
$237$	0	0
$238$	0	0
$239$	−9.41783	−0.609189	−0.304594	−	0.952482i	$-0.598521\pi$
$239$	−9.41783	−0.609189	−0.304594	+	0.952482i	$0.598521\pi$
$240$	0	0
$241$	−19.9493	−1.28505	−0.642524	−	0.766266i	$-0.722113\pi$
$241$	−19.9493	−1.28505	−0.642524	+	0.766266i	$0.722113\pi$
$242$	−48.6424	−3.12685
$243$	0	0
$244$	17.2811	1.10631
$245$	0	0
$246$	0	0
$247$	−2.67077	−0.169937
$248$	−8.57080	−0.544246
$249$	0	0
$250$	−25.5521	−1.61606
$251$	−1.22202	−0.0771330	−0.0385665	−	0.999256i	$-0.512279\pi$
$251$	−1.22202	−0.0771330	−0.0385665	+	0.999256i	$0.512279\pi$
$252$	0	0
$253$	−37.0633	−2.33015
$254$	42.1484	2.64463
$255$	0	0
$256$	−2.06691	−0.129182
$257$	22.4240	1.39877	0.699386	−	0.714745i	$-0.253457\pi$
$257$	22.4240	1.39877	0.699386	+	0.714745i	$0.253457\pi$
$258$	0	0
$259$	0	0
$260$	−5.83901	−0.362120
$261$	0	0
$262$	21.1707	1.30793
$263$	−8.99048	−0.554377	−0.277188	−	0.960816i	$-0.589403\pi$
$263$	−8.99048	−0.554377	−0.277188	+	0.960816i	$0.589403\pi$
$264$	0	0
$265$	7.28090	0.447262
$266$	0	0
$267$	0	0
$268$	21.4183	1.30833
$269$	22.2054	1.35389	0.676945	−	0.736034i	$-0.263303\pi$
$269$	22.2054	1.35389	0.676945	+	0.736034i	$0.263303\pi$
$270$	0	0
$271$	−10.8423	−0.658624	−0.329312	−	0.944221i	$-0.606817\pi$
$271$	−10.8423	−0.658624	−0.329312	+	0.944221i	$0.606817\pi$
$272$	3.08419	0.187006
$273$	0	0
$274$	−33.7293	−2.03766
$275$	−3.14486	−0.189642
$276$	0	0
$277$	5.88828	0.353792	0.176896	−	0.984230i	$-0.443394\pi$
$277$	5.88828	0.353792	0.176896	+	0.984230i	$0.443394\pi$
$278$	−8.40347	−0.504006
$279$	0	0
$280$	0	0
$281$	−26.9071	−1.60514	−0.802572	−	0.596556i	$-0.796535\pi$
$281$	−26.9071	−1.60514	−0.802572	+	0.596556i	$0.796535\pi$
$282$	0	0
$283$	−3.49717	−0.207885	−0.103943	−	0.994583i	$-0.533146\pi$
$283$	−3.49717	−0.207885	−0.103943	+	0.994583i	$0.533146\pi$
$284$	−37.7267	−2.23867
$285$	0	0
$286$	−12.5948	−0.744744
$287$	0	0
$288$	0	0
$289$	−14.3077	−0.841630
$290$	27.8773	1.63701
$291$	0	0
$292$	−42.8913	−2.51002
$293$	−1.00509	−0.0587179	−0.0293589	−	0.999569i	$-0.509347\pi$
$293$	−1.00509	−0.0587179	−0.0293589	+	0.999569i	$0.509347\pi$
$294$	0	0
$295$	27.8196	1.61972
$296$	−9.61780	−0.559024
$297$	0	0
$298$	−2.27116	−0.131565
$299$	−6.42469	−0.371549
$300$	0	0
$301$	0	0
$302$	9.48252	0.545657
$303$	0	0
$304$	5.02013	0.287924
$305$	−13.1845	−0.754944
$306$	0	0
$307$	4.25772	0.243001	0.121501	−	0.992591i	$-0.461229\pi$
$307$	4.25772	0.243001	0.121501	+	0.992591i	$0.461229\pi$
$308$	0	0
$309$	0	0
$310$	23.6024	1.34052
$311$	6.94043	0.393556	0.196778	−	0.980448i	$-0.436952\pi$
$311$	6.94043	0.393556	0.196778	+	0.980448i	$0.436952\pi$
$312$	0	0
$313$	−9.96299	−0.563142	−0.281571	−	0.959540i	$-0.590855\pi$
$313$	−9.96299	−0.563142	−0.281571	+	0.959540i	$0.590855\pi$
$314$	−0.733563	−0.0413973
$315$	0	0
$316$	3.10435	0.174633
$317$	−3.75345	−0.210815	−0.105407	−	0.994429i	$-0.533615\pi$
$317$	−3.75345	−0.210815	−0.105407	+	0.994429i	$0.533615\pi$
$318$	0	0
$319$	34.9003	1.95404
$320$	26.3968	1.47563
$321$	0	0
$322$	0	0
$323$	4.38225	0.243835
$324$	0	0
$325$	−0.545141	−0.0302390
$326$	−14.8441	−0.822140
$327$	0	0
$328$	−11.9492	−0.659785
$329$	0	0
$330$	0	0
$331$	−19.9812	−1.09827	−0.549134	−	0.835734i	$-0.685042\pi$
$331$	−19.9812	−1.09827	−0.549134	+	0.835734i	$0.685042\pi$
$332$	−13.7472	−0.754474
$333$	0	0
$334$	1.45981	0.0798771
$335$	−16.3410	−0.892805
$336$	0	0
$337$	18.4887	1.00714	0.503571	−	0.863954i	$-0.332019\pi$
$337$	18.4887	1.00714	0.503571	+	0.863954i	$0.332019\pi$
$338$	−2.18322	−0.118751
$339$	0	0
$340$	9.58078	0.519591
$341$	29.5484	1.60013
$342$	0	0
$343$	0	0
$344$	7.48078	0.403337
$345$	0	0
$346$	54.8377	2.94810
$347$	0.796100	0.0427369	0.0213684	−	0.999772i	$-0.493198\pi$
$347$	0.796100	0.0427369	0.0213684	+	0.999772i	$0.493198\pi$
$348$	0	0
$349$	−11.3725	−0.608754	−0.304377	−	0.952552i	$-0.598448\pi$
$349$	−11.3725	−0.608754	−0.304377	+	0.952552i	$0.598448\pi$
$350$	0	0
$351$	0	0
$352$	42.9803	2.29086
$353$	−24.1068	−1.28307	−0.641537	−	0.767092i	$-0.721703\pi$
$353$	−24.1068	−1.28307	−0.641537	+	0.767092i	$0.721703\pi$
$354$	0	0
$355$	28.7835	1.52767
$356$	−3.17117	−0.168072
$357$	0	0
$358$	−4.13778	−0.218689
$359$	−9.07182	−0.478792	−0.239396	−	0.970922i	$-0.576949\pi$
$359$	−9.07182	−0.478792	−0.239396	+	0.970922i	$0.576949\pi$
$360$	0	0
$361$	−11.8670	−0.624579
$362$	−24.8003	−1.30348
$363$	0	0
$364$	0	0
$365$	32.7238	1.71284
$366$	0	0
$367$	−31.5507	−1.64693	−0.823467	−	0.567364i	$-0.807963\pi$
$367$	−31.5507	−1.64693	−0.823467	+	0.567364i	$0.807963\pi$
$368$	12.0762	0.629517
$369$	0	0
$370$	26.4856	1.37692
$371$	0	0
$372$	0	0
$373$	−16.0581	−0.831457	−0.415728	−	0.909489i	$-0.636473\pi$
$373$	−16.0581	−0.831457	−0.415728	+	0.909489i	$0.636473\pi$
$374$	20.6658	1.06860
$375$	0	0
$376$	19.7302	1.01751
$377$	6.04973	0.311577
$378$	0	0
$379$	−5.36895	−0.275785	−0.137892	−	0.990447i	$-0.544033\pi$
$379$	−5.36895	−0.275785	−0.137892	+	0.990447i	$0.544033\pi$
$380$	15.5947	0.799989
$381$	0	0
$382$	−51.1762	−2.61840
$383$	−34.5075	−1.76325	−0.881625	−	0.471951i	$-0.843550\pi$
$383$	−34.5075	−1.76325	−0.881625	+	0.471951i	$0.843550\pi$
$384$	0	0
$385$	0	0
$386$	4.04353	0.205810
$387$	0	0
$388$	−19.2883	−0.979216
$389$	−22.9210	−1.16214	−0.581070	−	0.813854i	$-0.697366\pi$
$389$	−22.9210	−1.16214	−0.581070	+	0.813854i	$0.697366\pi$
$390$	0	0
$391$	10.5418	0.533120
$392$	0	0
$393$	0	0
$394$	−21.5502	−1.08568
$395$	−2.36845	−0.119170
$396$	0	0
$397$	10.8294	0.543510	0.271755	−	0.962366i	$-0.412396\pi$
$397$	10.8294	0.543510	0.271755	+	0.962366i	$0.412396\pi$
$398$	−26.7427	−1.34049
$399$	0	0
$400$	1.02468	0.0512340
$401$	−37.5415	−1.87473	−0.937367	−	0.348342i	$-0.886745\pi$
$401$	−37.5415	−1.87473	−0.937367	+	0.348342i	$0.886745\pi$
$402$	0	0
$403$	5.12202	0.255146
$404$	17.9797	0.894523
$405$	0	0
$406$	0	0
$407$	33.1580	1.64358
$408$	0	0
$409$	−20.4727	−1.01231	−0.506155	−	0.862443i	$-0.668934\pi$
$409$	−20.4727	−1.01231	−0.506155	+	0.862443i	$0.668934\pi$
$410$	32.9059	1.62511
$411$	0	0
$412$	−1.59914	−0.0787840
$413$	0	0
$414$	0	0
$415$	10.4884	0.514854
$416$	7.45036	0.365284
$417$	0	0
$418$	33.6377	1.64527
$419$	15.4980	0.757127	0.378564	−	0.925575i	$-0.376418\pi$
$419$	15.4980	0.757127	0.378564	+	0.925575i	$0.376418\pi$
$420$	0	0
$421$	17.9390	0.874293	0.437147	−	0.899390i	$-0.355989\pi$
$421$	17.9390	0.874293	0.437147	+	0.899390i	$0.355989\pi$
$422$	1.61536	0.0786346
$423$	0	0
$424$	5.77229	0.280327
$425$	0.894480	0.0433886
$426$	0	0
$427$	0	0
$428$	44.9749	2.17394
$429$	0	0
$430$	−20.6007	−0.993452
$431$	−21.4438	−1.03291	−0.516456	−	0.856313i	$-0.672749\pi$
$431$	−21.4438	−1.03291	−0.516456	+	0.856313i	$0.672749\pi$
$432$	0	0
$433$	−14.9365	−0.717800	−0.358900	−	0.933376i	$-0.616848\pi$
$433$	−14.9365	−0.717800	−0.358900	+	0.933376i	$0.616848\pi$
$434$	0	0
$435$	0	0
$436$	4.92955	0.236083
$437$	17.1589	0.820819
$438$	0	0
$439$	−3.59323	−0.171495	−0.0857476	−	0.996317i	$-0.527328\pi$
$439$	−3.59323	−0.171495	−0.0857476	+	0.996317i	$0.527328\pi$
$440$	20.3746	0.971323
$441$	0	0
$442$	3.58227	0.170391
$443$	13.6534	0.648694	0.324347	−	0.945938i	$-0.394855\pi$
$443$	13.6534	0.648694	0.324347	+	0.945938i	$0.394855\pi$
$444$	0	0
$445$	2.41943	0.114692
$446$	−18.1333	−0.858638
$447$	0	0
$448$	0	0
$449$	8.72412	0.411717	0.205858	−	0.978582i	$-0.434001\pi$
$449$	8.72412	0.411717	0.205858	+	0.978582i	$0.434001\pi$
$450$	0	0
$451$	41.1957	1.93983
$452$	20.8096	0.978803
$453$	0	0
$454$	29.0186	1.36191
$455$	0	0
$456$	0	0
$457$	−30.4659	−1.42514	−0.712568	−	0.701603i	$-0.752468\pi$
$457$	−30.4659	−1.42514	−0.712568	+	0.701603i	$0.752468\pi$
$458$	−39.1247	−1.82818
$459$	0	0
$460$	37.5139	1.74909
$461$	−5.22253	−0.243237	−0.121619	−	0.992577i	$-0.538809\pi$
$461$	−5.22253	−0.243237	−0.121619	+	0.992577i	$0.538809\pi$
$462$	0	0
$463$	20.6220	0.958386	0.479193	−	0.877709i	$-0.340929\pi$
$463$	20.6220	0.958386	0.479193	+	0.877709i	$0.340929\pi$
$464$	−11.3714	−0.527906
$465$	0	0
$466$	13.7536	0.637122
$467$	−1.67149	−0.0773474	−0.0386737	−	0.999252i	$-0.512313\pi$
$467$	−1.67149	−0.0773474	−0.0386737	+	0.999252i	$0.512313\pi$
$468$	0	0
$469$	0	0
$470$	−54.3332	−2.50620
$471$	0	0
$472$	22.0553	1.01518
$473$	−25.7905	−1.18585
$474$	0	0
$475$	1.45595	0.0668034
$476$	0	0
$477$	0	0
$478$	20.5612	0.940447
$479$	−10.6439	−0.486333	−0.243166	−	0.969985i	$-0.578186\pi$
$479$	−10.6439	−0.486333	−0.243166	+	0.969985i	$0.578186\pi$
$480$	0	0
$481$	5.74772	0.262073
$482$	43.5537	1.98382
$483$	0	0
$484$	61.6369	2.80168
$485$	14.7160	0.668217
$486$	0	0
$487$	−31.8312	−1.44241	−0.721206	−	0.692721i	$-0.756412\pi$
$487$	−31.8312	−1.44241	−0.721206	+	0.692721i	$0.756412\pi$
$488$	−10.4527	−0.473171
$489$	0	0
$490$	0	0
$491$	−6.87077	−0.310074	−0.155037	−	0.987909i	$-0.549550\pi$
$491$	−6.87077	−0.310074	−0.155037	+	0.987909i	$0.549550\pi$
$492$	0	0
$493$	−9.92654	−0.447069
$494$	5.83087	0.262343
$495$	0	0
$496$	−9.62765	−0.432294
$497$	0	0
$498$	0	0
$499$	0.344335	0.0154145	0.00770727	−	0.999970i	$-0.497547\pi$
$499$	0.344335	0.0154145	0.00770727	+	0.999970i	$0.497547\pi$
$500$	32.3782	1.44800
$501$	0	0
$502$	2.66793	0.119076
$503$	−30.0808	−1.34123	−0.670617	−	0.741803i	$-0.733971\pi$
$503$	−30.0808	−1.34123	−0.670617	+	0.741803i	$0.733971\pi$
$504$	0	0
$505$	−13.7175	−0.610423
$506$	80.9174	3.59722
$507$	0	0
$508$	−53.4080	−2.36960
$509$	−33.3105	−1.47646	−0.738232	−	0.674547i	$-0.764339\pi$
$509$	−33.3105	−1.47646	−0.738232	+	0.674547i	$0.764339\pi$
$510$	0	0
$511$	0	0
$512$	−20.2947	−0.896908
$513$	0	0
$514$	−48.9565	−2.15938
$515$	1.22006	0.0537622
$516$	0	0
$517$	−68.0210	−2.99156
$518$	0	0
$519$	0	0
$520$	3.53181	0.154880
$521$	−32.7726	−1.43579	−0.717897	−	0.696149i	$-0.754895\pi$
$521$	−32.7726	−1.43579	−0.717897	+	0.696149i	$0.754895\pi$
$522$	0	0
$523$	25.2616	1.10461	0.552307	−	0.833641i	$-0.313748\pi$
$523$	25.2616	1.10461	0.552307	+	0.833641i	$0.313748\pi$
$524$	−26.8263	−1.17191
$525$	0	0
$526$	19.6282	0.855830
$527$	−8.40432	−0.366098
$528$	0	0
$529$	18.2766	0.794636
$530$	−15.8958	−0.690469
$531$	0	0
$532$	0	0
$533$	7.14100	0.309311
$534$	0	0
$535$	−34.3134	−1.48350
$536$	−12.9552	−0.559578
$537$	0	0
$538$	−48.4794	−2.09009
$539$	0	0
$540$	0	0
$541$	13.1630	0.565920	0.282960	−	0.959132i	$-0.408684\pi$
$541$	13.1630	0.565920	0.282960	+	0.959132i	$0.408684\pi$
$542$	23.6712	1.01676
$543$	0	0
$544$	−12.2247	−0.524130
$545$	−3.76099	−0.161103
$546$	0	0
$547$	−2.79349	−0.119441	−0.0597204	−	0.998215i	$-0.519021\pi$
$547$	−2.79349	−0.119441	−0.0597204	+	0.998215i	$0.519021\pi$
$548$	42.7398	1.82575
$549$	0	0
$550$	6.86592	0.292764
$551$	−16.1574	−0.688330
$552$	0	0
$553$	0	0
$554$	−12.8554	−0.546174
$555$	0	0
$556$	10.6484	0.451592
$557$	4.61284	0.195452	0.0977262	−	0.995213i	$-0.468843\pi$
$557$	4.61284	0.195452	0.0977262	+	0.995213i	$0.468843\pi$
$558$	0	0
$559$	−4.47061	−0.189087
$560$	0	0
$561$	0	0
$562$	58.7441	2.47797
$563$	−40.2805	−1.69762	−0.848811	−	0.528697i	$-0.822681\pi$
$563$	−40.2805	−1.69762	−0.848811	+	0.528697i	$0.822681\pi$
$564$	0	0
$565$	−15.8767	−0.667936
$566$	7.63510	0.320927
$567$	0	0
$568$	22.8195	0.957486
$569$	−7.19869	−0.301785	−0.150892	−	0.988550i	$-0.548215\pi$
$569$	−7.19869	−0.301785	−0.150892	+	0.988550i	$0.548215\pi$
$570$	0	0
$571$	11.3820	0.476323	0.238161	−	0.971226i	$-0.423455\pi$
$571$	11.3820	0.476323	0.238161	+	0.971226i	$0.423455\pi$
$572$	15.9593	0.667294
$573$	0	0
$574$	0	0
$575$	3.50236	0.146059
$576$	0	0
$577$	−35.6257	−1.48312	−0.741560	−	0.670887i	$-0.765914\pi$
$577$	−35.6257	−1.48312	−0.741560	+	0.670887i	$0.765914\pi$
$578$	31.2369	1.29928
$579$	0	0
$580$	−35.3245	−1.46677
$581$	0	0
$582$	0	0
$583$	−19.9003	−0.824188
$584$	25.9434	1.07354
$585$	0	0
$586$	2.19433	0.0906469
$587$	−4.14755	−0.171188	−0.0855938	−	0.996330i	$-0.527279\pi$
$587$	−4.14755	−0.171188	−0.0855938	+	0.996330i	$0.527279\pi$
$588$	0	0
$589$	−13.6797	−0.563663
$590$	−60.7362	−2.50047
$591$	0	0
$592$	−10.8038	−0.444032
$593$	24.1313	0.990953	0.495476	−	0.868621i	$-0.334994\pi$
$593$	24.1313	0.990953	0.495476	+	0.868621i	$0.334994\pi$
$594$	0	0
$595$	0	0
$596$	2.87788	0.117883
$597$	0	0
$598$	14.0265	0.573587
$599$	48.5607	1.98414	0.992068	−	0.125703i	$-0.0401185\pi$
$599$	48.5607	1.98414	0.992068	+	0.125703i	$0.0401185\pi$
$600$	0	0
$601$	−33.2069	−1.35454	−0.677270	−	0.735735i	$-0.736837\pi$
$601$	−33.2069	−1.35454	−0.677270	+	0.735735i	$0.736837\pi$
$602$	0	0
$603$	0	0
$604$	−12.0157	−0.488911
$605$	−47.0257	−1.91186
$606$	0	0
$607$	−21.7946	−0.884616	−0.442308	−	0.896863i	$-0.645840\pi$
$607$	−21.7946	−0.884616	−0.442308	+	0.896863i	$0.645840\pi$
$608$	−19.8982	−0.806978
$609$	0	0
$610$	28.7847	1.16546
$611$	−11.7910	−0.477013
$612$	0	0
$613$	−22.7991	−0.920847	−0.460423	−	0.887700i	$-0.652302\pi$
$613$	−22.7991	−0.920847	−0.460423	+	0.887700i	$0.652302\pi$
$614$	−9.29554	−0.375138
$615$	0	0
$616$	0	0
$617$	42.5433	1.71273	0.856364	−	0.516373i	$-0.172718\pi$
$617$	42.5433	1.71273	0.856364	+	0.516373i	$0.172718\pi$
$618$	0	0
$619$	−44.1285	−1.77367	−0.886836	−	0.462084i	$-0.847102\pi$
$619$	−44.1285	−1.77367	−0.886836	+	0.462084i	$0.847102\pi$
$620$	−29.9075	−1.20112
$621$	0	0
$622$	−15.1525	−0.607559
$623$	0	0
$624$	0	0
$625$	−21.9771	−0.879085
$626$	21.7514	0.869361
$627$	0	0
$628$	0.929528	0.0370922
$629$	−9.43098	−0.376038
$630$	0	0
$631$	9.09226	0.361957	0.180979	−	0.983487i	$-0.442074\pi$
$631$	9.09226	0.361957	0.180979	+	0.983487i	$0.442074\pi$
$632$	−1.87771	−0.0746912
$633$	0	0
$634$	8.19461	0.325449
$635$	40.7475	1.61701
$636$	0	0
$637$	0	0
$638$	−76.1950	−3.01659
$639$	0	0
$640$	−26.1797	−1.03485
$641$	22.6642	0.895181	0.447591	−	0.894239i	$-0.352282\pi$
$641$	22.6642	0.895181	0.447591	+	0.894239i	$0.352282\pi$
$642$	0	0
$643$	24.3792	0.961422	0.480711	−	0.876879i	$-0.340379\pi$
$643$	24.3792	0.961422	0.480711	+	0.876879i	$0.340379\pi$
$644$	0	0
$645$	0	0
$646$	−9.56742	−0.376425
$647$	37.0364	1.45605	0.728026	−	0.685549i	$-0.240438\pi$
$647$	37.0364	1.45605	0.728026	+	0.685549i	$0.240438\pi$
$648$	0	0
$649$	−76.0372	−2.98472
$650$	1.19016	0.0466820
$651$	0	0
$652$	18.8096	0.736641
$653$	−18.7341	−0.733122	−0.366561	−	0.930394i	$-0.619465\pi$
$653$	−18.7341	−0.733122	−0.366561	+	0.930394i	$0.619465\pi$
$654$	0	0
$655$	20.4671	0.799715
$656$	−13.4227	−0.524067
$657$	0	0
$658$	0	0
$659$	28.5206	1.11100	0.555502	−	0.831515i	$-0.312526\pi$
$659$	28.5206	1.11100	0.555502	+	0.831515i	$0.312526\pi$
$660$	0	0
$661$	−19.9758	−0.776968	−0.388484	−	0.921455i	$-0.627001\pi$
$661$	−19.9758	−0.776968	−0.388484	+	0.921455i	$0.627001\pi$
$662$	43.6234	1.69547
$663$	0	0
$664$	8.31517	0.322691
$665$	0	0
$666$	0	0
$667$	−38.8677	−1.50496
$668$	−1.84978	−0.0715702
$669$	0	0
$670$	35.6760	1.37829
$671$	36.0363	1.39117
$672$	0	0
$673$	−4.18849	−0.161454	−0.0807272	−	0.996736i	$-0.525724\pi$
$673$	−4.18849	−0.161454	−0.0807272	+	0.996736i	$0.525724\pi$
$674$	−40.3649	−1.55480
$675$	0	0
$676$	2.76645	0.106402
$677$	38.3396	1.47351	0.736755	−	0.676159i	$-0.236357\pi$
$677$	38.3396	1.47351	0.736755	+	0.676159i	$0.236357\pi$
$678$	0	0
$679$	0	0
$680$	−5.79507	−0.222231
$681$	0	0
$682$	−64.5106	−2.47024
$683$	2.61207	0.0999482	0.0499741	−	0.998751i	$-0.484086\pi$
$683$	2.61207	0.0999482	0.0499741	+	0.998751i	$0.484086\pi$
$684$	0	0
$685$	−32.6082	−1.24589
$686$	0	0
$687$	0	0
$688$	8.40322	0.320370
$689$	−3.44959	−0.131419
$690$	0	0
$691$	−40.9185	−1.55661	−0.778307	−	0.627884i	$-0.783921\pi$
$691$	−40.9185	−1.55661	−0.778307	+	0.627884i	$0.783921\pi$
$692$	−69.4872	−2.64151
$693$	0	0
$694$	−1.73806	−0.0659759
$695$	−8.12415	−0.308167
$696$	0	0
$697$	−11.7171	−0.443817
$698$	24.8286	0.939776
$699$	0	0
$700$	0	0
$701$	0.762896	0.0288142	0.0144071	−	0.999896i	$-0.495414\pi$
$701$	0.762896	0.0288142	0.0144071	+	0.999896i	$0.495414\pi$
$702$	0	0
$703$	−15.3508	−0.578967
$704$	−72.1484	−2.71920
$705$	0	0
$706$	52.6304	1.98077
$707$	0	0
$708$	0	0
$709$	2.65281	0.0996284	0.0498142	−	0.998759i	$-0.484137\pi$
$709$	2.65281	0.0996284	0.0498142	+	0.998759i	$0.484137\pi$
$710$	−62.8407	−2.35837
$711$	0	0
$712$	1.91813	0.0718848
$713$	−32.9074	−1.23239
$714$	0	0
$715$	−12.1761	−0.455362
$716$	5.24315	0.195946
$717$	0	0
$718$	19.8058	0.739145
$719$	−11.0443	−0.411883	−0.205942	−	0.978564i	$-0.566026\pi$
$719$	−11.0443	−0.411883	−0.205942	+	0.978564i	$0.566026\pi$
$720$	0	0
$721$	0	0
$722$	25.9083	0.964206
$723$	0	0
$724$	31.4255	1.16792
$725$	−3.29796	−0.122483
$726$	0	0
$727$	3.46566	0.128534	0.0642672	−	0.997933i	$-0.479529\pi$
$727$	3.46566	0.128534	0.0642672	+	0.997933i	$0.479529\pi$
$728$	0	0
$729$	0	0
$730$	−71.4432	−2.64423
$731$	7.33547	0.271312
$732$	0	0
$733$	−29.0332	−1.07236	−0.536182	−	0.844102i	$-0.680134\pi$
$733$	−29.0332	−1.07236	−0.536182	+	0.844102i	$0.680134\pi$
$734$	68.8822	2.54249
$735$	0	0
$736$	−47.8663	−1.76437
$737$	44.6637	1.64521
$738$	0	0
$739$	47.5522	1.74924	0.874618	−	0.484813i	$-0.161112\pi$
$739$	47.5522	1.74924	0.874618	+	0.484813i	$0.161112\pi$
$740$	−33.5610	−1.23373
$741$	0	0
$742$	0	0
$743$	−12.6122	−0.462697	−0.231349	−	0.972871i	$-0.574314\pi$
$743$	−12.6122	−0.462697	−0.231349	+	0.972871i	$0.574314\pi$
$744$	0	0
$745$	−2.19567	−0.0804432
$746$	35.0584	1.28358
$747$	0	0
$748$	−26.1864	−0.957471
$749$	0	0
$750$	0	0
$751$	−7.70781	−0.281262	−0.140631	−	0.990062i	$-0.544913\pi$
$751$	−7.70781	−0.281262	−0.140631	+	0.990062i	$0.544913\pi$
$752$	22.1631	0.808204
$753$	0	0
$754$	−13.2079	−0.481003
$755$	9.16734	0.333633
$756$	0	0
$757$	2.39454	0.0870311	0.0435156	−	0.999053i	$-0.486144\pi$
$757$	2.39454	0.0870311	0.0435156	+	0.999053i	$0.486144\pi$
$758$	11.7216	0.425748
$759$	0	0
$760$	−9.43265	−0.342158
$761$	41.3296	1.49820	0.749098	−	0.662459i	$-0.230487\pi$
$761$	41.3296	1.49820	0.749098	+	0.662459i	$0.230487\pi$
$762$	0	0
$763$	0	0
$764$	64.8475	2.34610
$765$	0	0
$766$	75.3374	2.72205
$767$	−13.1805	−0.475922
$768$	0	0
$769$	36.7414	1.32493	0.662464	−	0.749094i	$-0.269511\pi$
$769$	36.7414	1.32493	0.662464	+	0.749094i	$0.269511\pi$
$770$	0	0
$771$	0	0
$772$	−5.12373	−0.184407
$773$	23.9672	0.862040	0.431020	−	0.902342i	$-0.358154\pi$
$773$	23.9672	0.862040	0.431020	+	0.902342i	$0.358154\pi$
$774$	0	0
$775$	−2.79222	−0.100300
$776$	11.6668	0.418814
$777$	0	0
$778$	50.0415	1.79408
$779$	−19.0720	−0.683324
$780$	0	0
$781$	−78.6718	−2.81510
$782$	−23.0150	−0.823015
$783$	0	0
$784$	0	0
$785$	−0.709180	−0.0253117
$786$	0	0
$787$	36.0134	1.28374	0.641870	−	0.766814i	$-0.278159\pi$
$787$	36.0134	1.28374	0.641870	+	0.766814i	$0.278159\pi$
$788$	27.3071	0.972776
$789$	0	0
$790$	5.17085	0.183971
$791$	0	0
$792$	0	0
$793$	6.24666	0.221825
$794$	−23.6429	−0.839054
$795$	0	0
$796$	33.8868	1.20109
$797$	18.8155	0.666478	0.333239	−	0.942842i	$-0.391858\pi$
$797$	18.8155	0.666478	0.333239	+	0.942842i	$0.391858\pi$
$798$	0	0
$799$	19.3469	0.684445
$800$	−4.06150	−0.143596
$801$	0	0
$802$	81.9614	2.89416
$803$	−89.4414	−3.15632
$804$	0	0
$805$	0	0
$806$	−11.1825	−0.393886
$807$	0	0
$808$	−10.8753	−0.382591
$809$	47.3322	1.66411	0.832056	−	0.554692i	$-0.187164\pi$
$809$	47.3322	1.66411	0.832056	+	0.554692i	$0.187164\pi$
$810$	0	0
$811$	19.2555	0.676153	0.338077	−	0.941119i	$-0.390224\pi$
$811$	19.2555	0.676153	0.338077	+	0.941119i	$0.390224\pi$
$812$	0	0
$813$	0	0
$814$	−72.3912	−2.53731
$815$	−14.3507	−0.502684
$816$	0	0
$817$	11.9400	0.417726
$818$	44.6964	1.56277
$819$	0	0
$820$	−41.6964	−1.45610
$821$	−41.5165	−1.44894	−0.724468	−	0.689308i	$-0.757915\pi$
$821$	−41.5165	−1.44894	−0.724468	+	0.689308i	$0.757915\pi$
$822$	0	0
$823$	24.5627	0.856201	0.428100	−	0.903731i	$-0.359183\pi$
$823$	24.5627	0.856201	0.428100	+	0.903731i	$0.359183\pi$
$824$	0.967262	0.0336962
$825$	0	0
$826$	0	0
$827$	0.580957	0.0202019	0.0101009	−	0.999949i	$-0.496785\pi$
$827$	0.580957	0.0202019	0.0101009	+	0.999949i	$0.496785\pi$
$828$	0	0
$829$	14.6630	0.509268	0.254634	−	0.967038i	$-0.418045\pi$
$829$	14.6630	0.509268	0.254634	+	0.967038i	$0.418045\pi$
$830$	−22.8984	−0.794815
$831$	0	0
$832$	−12.5065	−0.433583
$833$	0	0
$834$	0	0
$835$	1.41129	0.0488396
$836$	−42.6237	−1.47417
$837$	0	0
$838$	−33.8356	−1.16883
$839$	−25.6151	−0.884332	−0.442166	−	0.896933i	$-0.645790\pi$
$839$	−25.6151	−0.884332	−0.442166	+	0.896933i	$0.645790\pi$
$840$	0	0
$841$	7.59929	0.262044
$842$	−39.1648	−1.34971
$843$	0	0
$844$	−2.04689	−0.0704569
$845$	−2.11065	−0.0726087
$846$	0	0
$847$	0	0
$848$	6.48406	0.222664
$849$	0	0
$850$	−1.95285	−0.0669820
$851$	−36.9273	−1.26585
$852$	0	0
$853$	−0.587340	−0.0201101	−0.0100551	−	0.999949i	$-0.503201\pi$
$853$	−0.587340	−0.0201101	−0.0100551	+	0.999949i	$0.503201\pi$
$854$	0	0
$855$	0	0
$856$	−27.2037	−0.929803
$857$	−31.1680	−1.06468	−0.532340	−	0.846531i	$-0.678687\pi$
$857$	−31.1680	−1.06468	−0.532340	+	0.846531i	$0.678687\pi$
$858$	0	0
$859$	−22.7944	−0.777735	−0.388867	−	0.921294i	$-0.627134\pi$
$859$	−22.7944	−0.777735	−0.388867	+	0.921294i	$0.627134\pi$
$860$	26.1039	0.890137
$861$	0	0
$862$	46.8166	1.59458
$863$	−50.6249	−1.72329	−0.861646	−	0.507510i	$-0.830566\pi$
$863$	−50.6249	−1.72329	−0.861646	+	0.507510i	$0.830566\pi$
$864$	0	0
$865$	53.0150	1.80257
$866$	32.6096	1.10812
$867$	0	0
$868$	0	0
$869$	6.47352	0.219599
$870$	0	0
$871$	7.74216	0.262333
$872$	−2.98171	−0.100973
$873$	0	0
$874$	−37.4615	−1.26716
$875$	0	0
$876$	0	0
$877$	11.9495	0.403505	0.201753	−	0.979436i	$-0.435336\pi$
$877$	11.9495	0.403505	0.201753	+	0.979436i	$0.435336\pi$
$878$	7.84480	0.264749
$879$	0	0
$880$	22.8870	0.771520
$881$	−54.6400	−1.84087	−0.920434	−	0.390898i	$-0.872165\pi$
$881$	−54.6400	−1.84087	−0.920434	+	0.390898i	$0.872165\pi$
$882$	0	0
$883$	−26.7256	−0.899388	−0.449694	−	0.893183i	$-0.648467\pi$
$883$	−26.7256	−0.899388	−0.449694	+	0.893183i	$0.648467\pi$
$884$	−4.53925	−0.152671
$885$	0	0
$886$	−29.8084	−1.00143
$887$	5.82278	0.195510	0.0977549	−	0.995211i	$-0.468834\pi$
$887$	5.82278	0.195510	0.0977549	+	0.995211i	$0.468834\pi$
$888$	0	0
$889$	0	0
$890$	−5.28215	−0.177058
$891$	0	0
$892$	22.9775	0.769343
$893$	31.4910	1.05381
$894$	0	0
$895$	−4.00025	−0.133714
$896$	0	0
$897$	0	0
$898$	−19.0467	−0.635595
$899$	30.9868	1.03347
$900$	0	0
$901$	5.66017	0.188568
$902$	−89.9392	−2.99465
$903$	0	0
$904$	−12.5870	−0.418638
$905$	−23.9760	−0.796990
$906$	0	0
$907$	−54.4672	−1.80855	−0.904276	−	0.426948i	$-0.859589\pi$
$907$	−54.4672	−1.80855	−0.904276	+	0.426948i	$0.859589\pi$
$908$	−36.7707	−1.22028
$909$	0	0
$910$	0	0
$911$	−5.94815	−0.197071	−0.0985355	−	0.995134i	$-0.531416\pi$
$911$	−5.94815	−0.197071	−0.0985355	+	0.995134i	$0.531416\pi$
$912$	0	0
$913$	−28.6671	−0.948742
$914$	66.5138	2.20008
$915$	0	0
$916$	49.5766	1.63806
$917$	0	0
$918$	0	0
$919$	13.6480	0.450207	0.225103	−	0.974335i	$-0.427728\pi$
$919$	13.6480	0.450207	0.225103	+	0.974335i	$0.427728\pi$
$920$	−22.6908	−0.748093
$921$	0	0
$922$	11.4019	0.375503
$923$	−13.6372	−0.448875
$924$	0	0
$925$	−3.13332	−0.103023
$926$	−45.0224	−1.47953
$927$	0	0
$928$	45.0727	1.47958
$929$	−36.3062	−1.19117	−0.595585	−	0.803293i	$-0.703080\pi$
$929$	−36.3062	−1.19117	−0.595585	+	0.803293i	$0.703080\pi$
$930$	0	0
$931$	0	0
$932$	−17.4277	−0.570864
$933$	0	0
$934$	3.64923	0.119406
$935$	19.9789	0.653379
$936$	0	0
$937$	27.7384	0.906175	0.453087	−	0.891466i	$-0.350323\pi$
$937$	27.7384	0.906175	0.453087	+	0.891466i	$0.350323\pi$
$938$	0	0
$939$	0	0
$940$	68.8478	2.24557
$941$	31.9163	1.04044	0.520222	−	0.854031i	$-0.325849\pi$
$941$	31.9163	1.04044	0.520222	+	0.854031i	$0.325849\pi$
$942$	0	0
$943$	−45.8787	−1.49402
$944$	24.7750	0.806356
$945$	0	0
$946$	56.3063	1.83067
$947$	−23.1657	−0.752785	−0.376392	−	0.926460i	$-0.622836\pi$
$947$	−23.1657	−0.752785	−0.376392	+	0.926460i	$0.622836\pi$
$948$	0	0
$949$	−15.5041	−0.503284
$950$	−3.17865	−0.103129
$951$	0	0
$952$	0	0
$953$	−25.7340	−0.833607	−0.416803	−	0.908997i	$-0.636850\pi$
$953$	−25.7340	−0.833607	−0.416803	+	0.908997i	$0.636850\pi$
$954$	0	0
$955$	−49.4752	−1.60098
$956$	−26.0539	−0.842645
$957$	0	0
$958$	23.2380	0.750786
$959$	0	0
$960$	0	0
$961$	−4.76494	−0.153708
$962$	−12.5485	−0.404581
$963$	0	0
$964$	−55.1887	−1.77751
$965$	3.90913	0.125839
$966$	0	0
$967$	16.2544	0.522706	0.261353	−	0.965243i	$-0.415831\pi$
$967$	16.2544	0.522706	0.261353	+	0.965243i	$0.415831\pi$
$968$	−37.2819	−1.19829
$969$	0	0
$970$	−32.1282	−1.03157
$971$	−39.7465	−1.27553	−0.637763	−	0.770232i	$-0.720140\pi$
$971$	−39.7465	−1.27553	−0.637763	+	0.770232i	$0.720140\pi$
$972$	0	0
$973$	0	0
$974$	69.4946	2.22675
$975$	0	0
$976$	−11.7416	−0.375839
$977$	26.1884	0.837842	0.418921	−	0.908023i	$-0.362408\pi$
$977$	26.1884	0.837842	0.418921	+	0.908023i	$0.362408\pi$
$978$	0	0
$979$	−6.61286	−0.211348
$980$	0	0
$981$	0	0
$982$	15.0004	0.478682
$983$	−12.2915	−0.392039	−0.196020	−	0.980600i	$-0.562802\pi$
$983$	−12.2915	−0.392039	−0.196020	+	0.980600i	$0.562802\pi$
$984$	0	0
$985$	−20.8339	−0.663823
$986$	21.6718	0.690171
$987$	0	0
$988$	−7.38854	−0.235061
$989$	28.7223	0.913315
$990$	0	0
$991$	11.4681	0.364298	0.182149	−	0.983271i	$-0.441695\pi$
$991$	11.4681	0.364298	0.182149	+	0.983271i	$0.441695\pi$
$992$	38.1609	1.21161
$993$	0	0
$994$	0	0
$995$	−25.8539	−0.819623
$996$	0	0
$997$	−5.30214	−0.167920	−0.0839602	−	0.996469i	$-0.526757\pi$
$997$	−5.30214	−0.167920	−0.0839602	+	0.996469i	$0.526757\pi$
$998$	−0.751758	−0.0237965
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.br.1.1		6
3.2	odd	2		637.2.a.n.1.6	yes	6
7.6	odd	2		5733.2.a.bu.1.1		6
21.2	odd	6		637.2.e.n.508.1		12
21.5	even	6		637.2.e.o.508.1		12
21.11	odd	6		637.2.e.n.79.1		12
21.17	even	6		637.2.e.o.79.1		12
21.20	even	2		637.2.a.m.1.6	✓	6
39.38	odd	2		8281.2.a.cd.1.1		6
273.272	even	2		8281.2.a.cc.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.a.m.1.6	✓	6	21.20	even	2
637.2.a.n.1.6	yes	6	3.2	odd	2
637.2.e.n.79.1		12	21.11	odd	6
637.2.e.n.508.1		12	21.2	odd	6
637.2.e.o.79.1		12	21.17	even	6
637.2.e.o.508.1		12	21.5	even	6
5733.2.a.br.1.1		6	1.1	even	1	trivial
5733.2.a.bu.1.1		6	7.6	odd	2
8281.2.a.cc.1.1		6	273.272	even	2
8281.2.a.cd.1.1		6	39.38	odd	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)

\(f(q)\)	\(=\)	\(q-2.18322 q^{2} +2.76645 q^{4} -2.11065 q^{5} -1.67333 q^{8} +O(q^{10})\)	\(q-2.18322 q^{2} +2.76645 q^{4} -2.11065 q^{5} -1.67333 q^{8} +4.60802 q^{10} +5.76889 q^{11} +1.00000 q^{13} -1.87966 q^{16} -1.64082 q^{17} -2.67077 q^{19} -5.83901 q^{20} -12.5948 q^{22} -6.42469 q^{23} -0.545141 q^{25} -2.18322 q^{26} +6.04973 q^{29} +5.12202 q^{31} +7.45036 q^{32} +3.58227 q^{34} +5.74772 q^{37} +5.83087 q^{38} +3.53181 q^{40} +7.14100 q^{41} -4.47061 q^{43} +15.9593 q^{44} +14.0265 q^{46} -11.7910 q^{47} +1.19016 q^{50} +2.76645 q^{52} -3.44959 q^{53} -12.1761 q^{55} -13.2079 q^{58} -13.1805 q^{59} +6.24666 q^{61} -11.1825 q^{62} -12.5065 q^{64} -2.11065 q^{65} +7.74216 q^{67} -4.53925 q^{68} -13.6372 q^{71} -15.5041 q^{73} -12.5485 q^{74} -7.38854 q^{76} +1.12214 q^{79} +3.96731 q^{80} -15.5904 q^{82} -4.96925 q^{83} +3.46321 q^{85} +9.76032 q^{86} -9.65324 q^{88} -1.14630 q^{89} -17.7736 q^{92} +25.7423 q^{94} +5.63707 q^{95} -6.97223 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{4} - 6 q^{5}+O(q^{10}) \) 6 * q + 4 * q^4 - 6 * q^5	\( 6 q + 4 q^{4} - 6 q^{5} + 4 q^{10} - 4 q^{11} + 6 q^{13} - 16 q^{17} + 2 q^{19} - 16 q^{20} - 12 q^{22} + 6 q^{23} - 4 q^{25} + 6 q^{29} + 6 q^{31} + 20 q^{32} - 8 q^{38} + 4 q^{40} + 8 q^{41} + 2 q^{43} + 4 q^{44} + 8 q^{46} - 30 q^{47} - 8 q^{50} + 4 q^{52} + 14 q^{53} - 8 q^{55} - 8 q^{58} - 24 q^{59} - 28 q^{62} - 20 q^{64} - 6 q^{65} + 16 q^{67} - 28 q^{68} - 8 q^{71} - 6 q^{73} + 12 q^{74} - 16 q^{76} - 22 q^{79} + 28 q^{80} - 40 q^{82} - 50 q^{83} - 8 q^{85} + 16 q^{86} - 44 q^{88} - 26 q^{89} - 20 q^{92} - 32 q^{94} + 6 q^{95} - 14 q^{97}+O(q^{100}) \) 6 * q + 4 * q^4 - 6 * q^5 + 4 * q^10 - 4 * q^11 + 6 * q^13 - 16 * q^17 + 2 * q^19 - 16 * q^20 - 12 * q^22 + 6 * q^23 - 4 * q^25 + 6 * q^29 + 6 * q^31 + 20 * q^32 - 8 * q^38 + 4 * q^40 + 8 * q^41 + 2 * q^43 + 4 * q^44 + 8 * q^46 - 30 * q^47 - 8 * q^50 + 4 * q^52 + 14 * q^53 - 8 * q^55 - 8 * q^58 - 24 * q^59 - 28 * q^62 - 20 * q^64 - 6 * q^65 + 16 * q^67 - 28 * q^68 - 8 * q^71 - 6 * q^73 + 12 * q^74 - 16 * q^76 - 22 * q^79 + 28 * q^80 - 40 * q^82 - 50 * q^83 - 8 * q^85 + 16 * q^86 - 44 * q^88 - 26 * q^89 - 20 * q^92 - 32 * q^94 + 6 * q^95 - 14 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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