Properties

Label 240.2.a.d
Level $240$
Weight $2$
Character orbit 240.a
Self dual yes
Analytic conductor $1.916$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + q^{9} + O(q^{10}) \) \( q + q^{3} + q^{5} + q^{9} + 4q^{11} - 2q^{13} + q^{15} + 2q^{17} - 4q^{19} + q^{25} + q^{27} - 2q^{29} + 4q^{33} - 10q^{37} - 2q^{39} + 10q^{41} - 4q^{43} + q^{45} - 8q^{47} - 7q^{49} + 2q^{51} - 10q^{53} + 4q^{55} - 4q^{57} + 4q^{59} - 2q^{61} - 2q^{65} - 12q^{67} + 8q^{71} + 10q^{73} + q^{75} + q^{81} - 12q^{83} + 2q^{85} - 2q^{87} - 6q^{89} - 4q^{95} + 2q^{97} + 4q^{99} + O(q^{100}) \)

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 1.00000 0 1.00000 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)

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 240.2.a.d 1
3.b odd 2 1 720.2.a.c 1
4.b odd 2 1 15.2.a.a 1
5.b even 2 1 1200.2.a.e 1
5.c odd 4 2 1200.2.f.h 2
8.b even 2 1 960.2.a.a 1
8.d odd 2 1 960.2.a.l 1
12.b even 2 1 45.2.a.a 1
15.d odd 2 1 3600.2.a.u 1
15.e even 4 2 3600.2.f.e 2
16.e even 4 2 3840.2.k.r 2
16.f odd 4 2 3840.2.k.m 2
20.d odd 2 1 75.2.a.b 1
20.e even 4 2 75.2.b.b 2
24.f even 2 1 2880.2.a.y 1
24.h odd 2 1 2880.2.a.bc 1
28.d even 2 1 735.2.a.c 1
28.f even 6 2 735.2.i.d 2
28.g odd 6 2 735.2.i.e 2
36.f odd 6 2 405.2.e.f 2
36.h even 6 2 405.2.e.c 2
40.e odd 2 1 4800.2.a.t 1
40.f even 2 1 4800.2.a.bz 1
40.i odd 4 2 4800.2.f.c 2
40.k even 4 2 4800.2.f.bf 2
44.c even 2 1 1815.2.a.d 1
52.b odd 2 1 2535.2.a.j 1
60.h even 2 1 225.2.a.b 1
60.l odd 4 2 225.2.b.b 2
68.d odd 2 1 4335.2.a.c 1
76.d even 2 1 5415.2.a.j 1
84.h odd 2 1 2205.2.a.i 1
92.b even 2 1 7935.2.a.d 1
132.d odd 2 1 5445.2.a.c 1
140.c even 2 1 3675.2.a.j 1
156.h even 2 1 7605.2.a.g 1
220.g even 2 1 9075.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 4.b odd 2 1
45.2.a.a 1 12.b even 2 1
75.2.a.b 1 20.d odd 2 1
75.2.b.b 2 20.e even 4 2
225.2.a.b 1 60.h even 2 1
225.2.b.b 2 60.l odd 4 2
240.2.a.d 1 1.a even 1 1 trivial
405.2.e.c 2 36.h even 6 2
405.2.e.f 2 36.f odd 6 2
720.2.a.c 1 3.b odd 2 1
735.2.a.c 1 28.d even 2 1
735.2.i.d 2 28.f even 6 2
735.2.i.e 2 28.g odd 6 2
960.2.a.a 1 8.b even 2 1
960.2.a.l 1 8.d odd 2 1
1200.2.a.e 1 5.b even 2 1
1200.2.f.h 2 5.c odd 4 2
1815.2.a.d 1 44.c even 2 1
2205.2.a.i 1 84.h odd 2 1
2535.2.a.j 1 52.b odd 2 1
2880.2.a.y 1 24.f even 2 1
2880.2.a.bc 1 24.h odd 2 1
3600.2.a.u 1 15.d odd 2 1
3600.2.f.e 2 15.e even 4 2
3675.2.a.j 1 140.c even 2 1
3840.2.k.m 2 16.f odd 4 2
3840.2.k.r 2 16.e even 4 2
4335.2.a.c 1 68.d odd 2 1
4800.2.a.t 1 40.e odd 2 1
4800.2.a.bz 1 40.f even 2 1
4800.2.f.c 2 40.i odd 4 2
4800.2.f.bf 2 40.k even 4 2
5415.2.a.j 1 76.d even 2 1
5445.2.a.c 1 132.d odd 2 1
7605.2.a.g 1 156.h even 2 1
7935.2.a.d 1 92.b even 2 1
9075.2.a.g 1 220.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(240))\):

\( T_{7} \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( -1 + T \)
$7$ \( T \)
$11$ \( -4 + T \)
$13$ \( 2 + T \)
$17$ \( -2 + T \)
$19$ \( 4 + T \)
$23$ \( T \)
$29$ \( 2 + T \)
$31$ \( T \)
$37$ \( 10 + T \)
$41$ \( -10 + T \)
$43$ \( 4 + T \)
$47$ \( 8 + T \)
$53$ \( 10 + T \)
$59$ \( -4 + T \)
$61$ \( 2 + T \)
$67$ \( 12 + T \)
$71$ \( -8 + T \)
$73$ \( -10 + T \)
$79$ \( T \)
$83$ \( 12 + T \)
$89$ \( 6 + T \)
$97$ \( -2 + T \)
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