Properties

Label 2352.4.a.ba
Level $2352$
Weight $4$
Character orbit 2352.a
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} - 2q^{5} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} - 2q^{5} + 9q^{9} + 8q^{11} + 42q^{13} - 6q^{15} + 2q^{17} - 124q^{19} - 76q^{23} - 121q^{25} + 27q^{27} + 254q^{29} - 72q^{31} + 24q^{33} + 398q^{37} + 126q^{39} - 462q^{41} - 212q^{43} - 18q^{45} - 264q^{47} + 6q^{51} - 162q^{53} - 16q^{55} - 372q^{57} - 772q^{59} - 30q^{61} - 84q^{65} + 764q^{67} - 228q^{69} + 236q^{71} - 418q^{73} - 363q^{75} - 552q^{79} + 81q^{81} + 1036q^{83} - 4q^{85} + 762q^{87} - 30q^{89} - 216q^{93} + 248q^{95} + 1190q^{97} + 72q^{99} + O(q^{100}) \)

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
0
0 3.00000 0 −2.00000 0 0 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.4.a.ba 1
4.b odd 2 1 294.4.a.h 1
7.b odd 2 1 336.4.a.d 1
12.b even 2 1 882.4.a.d 1
21.c even 2 1 1008.4.a.j 1
28.d even 2 1 42.4.a.b 1
28.f even 6 2 294.4.e.a 2
28.g odd 6 2 294.4.e.d 2
56.e even 2 1 1344.4.a.f 1
56.h odd 2 1 1344.4.a.t 1
84.h odd 2 1 126.4.a.c 1
84.j odd 6 2 882.4.g.s 2
84.n even 6 2 882.4.g.r 2
140.c even 2 1 1050.4.a.d 1
140.j odd 4 2 1050.4.g.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.b 1 28.d even 2 1
126.4.a.c 1 84.h odd 2 1
294.4.a.h 1 4.b odd 2 1
294.4.e.a 2 28.f even 6 2
294.4.e.d 2 28.g odd 6 2
336.4.a.d 1 7.b odd 2 1
882.4.a.d 1 12.b even 2 1
882.4.g.r 2 84.n even 6 2
882.4.g.s 2 84.j odd 6 2
1008.4.a.j 1 21.c even 2 1
1050.4.a.d 1 140.c even 2 1
1050.4.g.n 2 140.j odd 4 2
1344.4.a.f 1 56.e even 2 1
1344.4.a.t 1 56.h odd 2 1
2352.4.a.ba 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} + 2 \)
\( T_{11} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( 2 + T \)
$7$ \( T \)
$11$ \( -8 + T \)
$13$ \( -42 + T \)
$17$ \( -2 + T \)
$19$ \( 124 + T \)
$23$ \( 76 + T \)
$29$ \( -254 + T \)
$31$ \( 72 + T \)
$37$ \( -398 + T \)
$41$ \( 462 + T \)
$43$ \( 212 + T \)
$47$ \( 264 + T \)
$53$ \( 162 + T \)
$59$ \( 772 + T \)
$61$ \( 30 + T \)
$67$ \( -764 + T \)
$71$ \( -236 + T \)
$73$ \( 418 + T \)
$79$ \( 552 + T \)
$83$ \( -1036 + T \)
$89$ \( 30 + T \)
$97$ \( -1190 + T \)
show more
show less