Newform orbit 1521.4.a.o

• Label: 1521.4.a.o
• Level: $1521$
• Weight: $4$
• Character orbit: 1521.a
• Self dual: yes
• Analytic conductor: $89.742$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(89.7419051187\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + b * q^2 - 5 * q^4 + b * q^5 + 8*b * q^7 - 13*b * q^8 + 3 * q^10 - 8*b * q^11 + 24 * q^14 + q^16 + 117 * q^17 - 66*b * q^19 - 5*b * q^20 - 24 * q^22 - 78 * q^23 - 122 * q^25 - 40*b * q^28 + 141 * q^29 + 90*b * q^31 + 105*b * q^32 + 117*b * q^34 + 24 * q^35 - 83*b * q^37 - 198 * q^38 - 39 * q^40 + 157*b * q^41 - 104 * q^43 + 40*b * q^44 - 78*b * q^46 + 174*b * q^47 - 151 * q^49 - 122*b * q^50 - 93 * q^53 - 24 * q^55 - 312 * q^56 + 141*b * q^58 - 164*b * q^59 + 145 * q^61 + 270 * q^62 + 307 * q^64 + 454*b * q^67 - 585 * q^68 + 24*b * q^70 - 610*b * q^71 - 265*b * q^73 - 249 * q^74 + 330*b * q^76 - 192 * q^77 + 1276 * q^79 + b * q^80 + 471 * q^82 + 456*b * q^83 + 117*b * q^85 - 104*b * q^86 + 312 * q^88 + 564*b * q^89 + 390 * q^92 + 522 * q^94 - 198 * q^95 + 116*b * q^97 - 151*b * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 10 * q^4 + 6 * q^10 + 48 * q^14 + 2 * q^16 + 234 * q^17 - 48 * q^22 - 156 * q^23 - 244 * q^25 + 282 * q^29 + 48 * q^35 - 396 * q^38 - 78 * q^40 - 208 * q^43 - 302 * q^49 - 186 * q^53 - 48 * q^55 - 624 * q^56 + 290 * q^61 + 540 * q^62 + 614 * q^64 - 1170 * q^68 - 498 * q^74 - 384 * q^77 + 2552 * q^79 + 942 * q^82 + 624 * q^88 + 780 * q^92 + 1044 * q^94 - 396 * q^95

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−1.73205
1.73205

−1.73205

0

−5.00000

−1.73205

0

−13.8564

22.5167

0

3.00000

1.2

1.73205

0

−5.00000

1.73205

0

13.8564

−22.5167

0

3.00000

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\( -1 \)
\(13\)	\( -1 \)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1521.4.a.o		2
3.b	odd	2	1		169.4.a.i		2
13.b	even	2	1	inner	1521.4.a.o		2
13.f	odd	12	2		117.4.q.a		2
39.d	odd	2	1		169.4.a.i		2
39.f	even	4	2		169.4.b.d		2
39.h	odd	6	2		169.4.c.h		4
39.i	odd	6	2		169.4.c.h		4
39.k	even	12	2		13.4.e.b	✓	2
39.k	even	12	2		169.4.e.a		2
156.v	odd	12	2		208.4.w.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.4.e.b	✓	2	39.k	even	12	2
117.4.q.a		2	13.f	odd	12	2
169.4.a.i		2	3.b	odd	2	1
169.4.a.i		2	39.d	odd	2	1
169.4.b.d		2	39.f	even	4	2
169.4.c.h		4	39.h	odd	6	2
169.4.c.h		4	39.i	odd	6	2
169.4.e.a		2	39.k	even	12	2
208.4.w.b		2	156.v	odd	12	2
1521.4.a.o		2	1.a	even	1	1	trivial
1521.4.a.o		2	13.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2}^{2} - 3 \)

\( T_{5}^{2} - 3 \)

\( T_{7}^{2} - 192 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 3 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 3 \)
$7$	\( T^{2} - 192 \)
$11$	\( T^{2} - 192 \)