LMFDB - Embedded newform 5733.2.a.v.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5733,2,Mod(1,5733)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,3,0,3,0,0,0,6,0,-5,6,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$45.7782354788$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+0.381966 q^{2} -1.85410 q^{4} +2.23607 q^{5} -1.47214 q^{8} +0.854102 q^{10} +3.00000 q^{11} -1.00000 q^{13} +3.14590 q^{16} +7.47214 q^{17} +3.00000 q^{19} -4.14590 q^{20} +1.14590 q^{22} +3.76393 q^{23} -0.381966 q^{26} +4.47214 q^{29} +5.00000 q^{31} +4.14590 q^{32} +2.85410 q^{34} -8.70820 q^{37} +1.14590 q^{38} -3.29180 q^{40} -4.47214 q^{41} -8.00000 q^{43} -5.56231 q^{44} +1.43769 q^{46} -1.47214 q^{47} +1.85410 q^{52} -1.47214 q^{53} +6.70820 q^{55} +1.70820 q^{58} -7.47214 q^{59} +3.00000 q^{61} +1.90983 q^{62} -4.70820 q^{64} -2.23607 q^{65} -3.00000 q^{67} -13.8541 q^{68} -8.94427 q^{71} +10.7082 q^{73} -3.32624 q^{74} -5.56231 q^{76} +10.7082 q^{79} +7.03444 q^{80} -1.70820 q^{82} +16.7082 q^{85} -3.05573 q^{86} -4.41641 q^{88} +2.23607 q^{89} -6.97871 q^{92} -0.562306 q^{94} +6.70820 q^{95} -17.4164 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 3 q^{4} + 6 q^{8} - 5 q^{10} + 6 q^{11} - 2 q^{13} + 13 q^{16} + 6 q^{17} + 6 q^{19} - 15 q^{20} + 9 q^{22} + 12 q^{23} - 3 q^{26} + 10 q^{31} + 15 q^{32} - q^{34} - 4 q^{37} + 9 q^{38}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^8 - 5 * q^10 + 6 * q^11 - 2 * q^13 + 13 * q^16 + 6 * q^17 + 6 * q^19 - 15 * q^20 + 9 * q^22 + 12 * q^23 - 3 * q^26 + 10 * q^31 + 15 * q^32 - q^34 - 4 * q^37 + 9 * q^38 - 20 * q^40 - 16 * q^43 + 9 * q^44 + 23 * q^46 + 6 * q^47 - 3 * q^52 + 6 * q^53 - 10 * q^58 - 6 * q^59 + 6 * q^61 + 15 * q^62 + 4 * q^64 - 6 * q^67 - 21 * q^68 + 8 * q^73 + 9 * q^74 + 9 * q^76 + 8 * q^79 - 15 * q^80 + 10 * q^82 + 20 * q^85 - 24 * q^86 + 18 * q^88 + 33 * q^92 + 19 * q^94 - 8 * q^97

Coefficient data

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

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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 $

$2$	0.381966	0.270091	0.135045	−	0.990839i	$-0.456882\pi$
$2$	0.381966	0.270091	0.135045	+	0.990839i	$0.456882\pi$
$3$	0	0
$3$	0	0
$4$	−1.85410	−0.927051
$5$	2.23607	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$5$	2.23607	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.47214	−0.520479
$9$	0	0
$10$	0.854102	0.270091
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	3.14590	0.786475
$17$	7.47214	1.81226	0.906130	−	0.423000i	$-0.139023\pi$
$17$	7.47214	1.81226	0.906130	+	0.423000i	$0.139023\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	−4.14590	−0.927051
$21$	0	0
$22$	1.14590	0.244306
$23$	3.76393	0.784834	0.392417	−	0.919787i	$-0.371639\pi$
$23$	3.76393	0.784834	0.392417	+	0.919787i	$0.371639\pi$
$24$	0	0
$25$	0	0
$26$	−0.381966	−0.0749097
$27$	0	0
$28$	0	0
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	4.14590	0.732898
$33$	0	0
$34$	2.85410	0.489474
$35$	0	0
$36$	0	0
$37$	−8.70820	−1.43162	−0.715810	−	0.698295i	$-0.753942\pi$
$37$	−8.70820	−1.43162	−0.715810	+	0.698295i	$0.753942\pi$
$38$	1.14590	0.185889
$39$	0	0
$40$	−3.29180	−0.520479
$41$	−4.47214	−0.698430	−0.349215	−	0.937043i	$-0.613552\pi$
$41$	−4.47214	−0.698430	−0.349215	+	0.937043i	$0.613552\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−5.56231	−0.838549
$45$	0	0
$46$	1.43769	0.211976
$47$	−1.47214	−0.214733	−0.107367	−	0.994220i	$-0.534242\pi$
$47$	−1.47214	−0.214733	−0.107367	+	0.994220i	$0.534242\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	1.85410	0.257118
$53$	−1.47214	−0.202213	−0.101107	−	0.994876i	$-0.532238\pi$
$53$	−1.47214	−0.202213	−0.101107	+	0.994876i	$0.532238\pi$
$54$	0	0
$55$	6.70820	0.904534
$56$	0	0
$57$	0	0
$58$	1.70820	0.224298
$59$	−7.47214	−0.972789	−0.486395	−	0.873739i	$-0.661688\pi$
$59$	−7.47214	−0.972789	−0.486395	+	0.873739i	$0.661688\pi$
$60$	0	0
$61$	3.00000	0.384111	0.192055	−	0.981384i	$-0.438485\pi$
$61$	3.00000	0.384111	0.192055	+	0.981384i	$0.438485\pi$
$62$	1.90983	0.242549
$63$	0	0
$64$	−4.70820	−0.588525
$65$	−2.23607	−0.277350
$66$	0	0
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	−13.8541	−1.68006
$69$	0	0
$70$	0	0
$71$	−8.94427	−1.06149	−0.530745	−	0.847532i	$-0.678088\pi$
$71$	−8.94427	−1.06149	−0.530745	+	0.847532i	$0.678088\pi$
$72$	0	0
$73$	10.7082	1.25330	0.626650	−	0.779301i	$-0.284425\pi$
$73$	10.7082	1.25330	0.626650	+	0.779301i	$0.284425\pi$
$74$	−3.32624	−0.386667
$75$	0	0
$76$	−5.56231	−0.638040
$77$	0	0
$78$	0	0
$79$	10.7082	1.20477	0.602384	−	0.798207i	$-0.294218\pi$
$79$	10.7082	1.20477	0.602384	+	0.798207i	$0.294218\pi$
$80$	7.03444	0.786475
$81$	0	0
$82$	−1.70820	−0.188640
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	16.7082	1.81226
$86$	−3.05573	−0.329508
$87$	0	0
$88$	−4.41641	−0.470791
$89$	2.23607	0.237023	0.118511	−	0.992953i	$-0.462188\pi$
$89$	2.23607	0.237023	0.118511	+	0.992953i	$0.462188\pi$
$90$	0	0
$91$	0	0
$92$	−6.97871	−0.727581
$93$	0	0
$94$	−0.562306	−0.0579974
$95$	6.70820	0.688247
$96$	0	0
$97$	−17.4164	−1.76837	−0.884184	−	0.467139i	$-0.845285\pi$
$97$	−17.4164	−1.76837	−0.884184	+	0.467139i	$0.845285\pi$
$98$	0	0
$99$	0	0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.v.1.1		2
3.2	odd	2		637.2.a.f.1.2		2
7.2	even	3		819.2.j.c.235.2		4
7.4	even	3		819.2.j.c.352.2		4
7.6	odd	2		5733.2.a.w.1.1		2
21.2	odd	6		91.2.e.b.53.1	✓	4
21.5	even	6		637.2.e.h.508.1		4
21.11	odd	6		91.2.e.b.79.1	yes	4
21.17	even	6		637.2.e.h.79.1		4
21.20	even	2		637.2.a.e.1.2		2
39.38	odd	2		8281.2.a.z.1.1		2
84.11	even	6		1456.2.r.j.625.2		4
84.23	even	6		1456.2.r.j.417.2		4
273.116	odd	6		1183.2.e.d.170.2		4
273.233	odd	6		1183.2.e.d.508.2		4
273.272	even	2		8281.2.a.ba.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.e.b.53.1	✓	4	21.2	odd	6
91.2.e.b.79.1	yes	4	21.11	odd	6
637.2.a.e.1.2		2	21.20	even	2
637.2.a.f.1.2		2	3.2	odd	2
637.2.e.h.79.1		4	21.17	even	6
637.2.e.h.508.1		4	21.5	even	6
819.2.j.c.235.2		4	7.2	even	3
819.2.j.c.352.2		4	7.4	even	3
1183.2.e.d.170.2		4	273.116	odd	6
1183.2.e.d.508.2		4	273.233	odd	6
1456.2.r.j.417.2		4	84.23	even	6
1456.2.r.j.625.2		4	84.11	even	6
5733.2.a.v.1.1		2	1.1	even	1	trivial
5733.2.a.w.1.1		2	7.6	odd	2
8281.2.a.z.1.1		2	39.38	odd	2
8281.2.a.ba.1.1		2	273.272	even	2

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Belyi maps

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

\(f(q)\)	\(=\)	\(q+0.381966 q^{2} -1.85410 q^{4} +2.23607 q^{5} -1.47214 q^{8} +0.854102 q^{10} +3.00000 q^{11} -1.00000 q^{13} +3.14590 q^{16} +7.47214 q^{17} +3.00000 q^{19} -4.14590 q^{20} +1.14590 q^{22} +3.76393 q^{23} -0.381966 q^{26} +4.47214 q^{29} +5.00000 q^{31} +4.14590 q^{32} +2.85410 q^{34} -8.70820 q^{37} +1.14590 q^{38} -3.29180 q^{40} -4.47214 q^{41} -8.00000 q^{43} -5.56231 q^{44} +1.43769 q^{46} -1.47214 q^{47} +1.85410 q^{52} -1.47214 q^{53} +6.70820 q^{55} +1.70820 q^{58} -7.47214 q^{59} +3.00000 q^{61} +1.90983 q^{62} -4.70820 q^{64} -2.23607 q^{65} -3.00000 q^{67} -13.8541 q^{68} -8.94427 q^{71} +10.7082 q^{73} -3.32624 q^{74} -5.56231 q^{76} +10.7082 q^{79} +7.03444 q^{80} -1.70820 q^{82} +16.7082 q^{85} -3.05573 q^{86} -4.41641 q^{88} +2.23607 q^{89} -6.97871 q^{92} -0.562306 q^{94} +6.70820 q^{95} -17.4164 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 3 q^{4} + 6 q^{8} - 5 q^{10} + 6 q^{11} - 2 q^{13} + 13 q^{16} + 6 q^{17} + 6 q^{19} - 15 q^{20} + 9 q^{22} + 12 q^{23} - 3 q^{26} + 10 q^{31} + 15 q^{32} - q^{34} - 4 q^{37} + 9 q^{38}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^8 - 5 * q^10 + 6 * q^11 - 2 * q^13 + 13 * q^16 + 6 * q^17 + 6 * q^19 - 15 * q^20 + 9 * q^22 + 12 * q^23 - 3 * q^26 + 10 * q^31 + 15 * q^32 - q^34 - 4 * q^37 + 9 * q^38 - 20 * q^40 - 16 * q^43 + 9 * q^44 + 23 * q^46 + 6 * q^47 - 3 * q^52 + 6 * q^53 - 10 * q^58 - 6 * q^59 + 6 * q^61 + 15 * q^62 + 4 * q^64 - 6 * q^67 - 21 * q^68 + 8 * q^73 + 9 * q^74 + 9 * q^76 + 8 * q^79 - 15 * q^80 + 10 * q^82 + 20 * q^85 - 24 * q^86 + 18 * q^88 + 33 * q^92 + 19 * q^94 - 8 * q^97

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