Properties

Label 240.4.a.f
Level $240$
Weight $4$
Character orbit 240.a
Self dual yes
Analytic conductor $14.160$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.1604584014\)
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

\(f(q)\) \(=\) \( q + 3 q^{3} - 5 q^{5} - 20 q^{7} + 9 q^{9} + O(q^{10}) \) \( q + 3 q^{3} - 5 q^{5} - 20 q^{7} + 9 q^{9} + 24 q^{11} + 74 q^{13} - 15 q^{15} + 54 q^{17} + 124 q^{19} - 60 q^{21} + 120 q^{23} + 25 q^{25} + 27 q^{27} - 78 q^{29} - 200 q^{31} + 72 q^{33} + 100 q^{35} - 70 q^{37} + 222 q^{39} + 330 q^{41} - 92 q^{43} - 45 q^{45} + 24 q^{47} + 57 q^{49} + 162 q^{51} + 450 q^{53} - 120 q^{55} + 372 q^{57} - 24 q^{59} - 322 q^{61} - 180 q^{63} - 370 q^{65} + 196 q^{67} + 360 q^{69} + 288 q^{71} - 430 q^{73} + 75 q^{75} - 480 q^{77} + 520 q^{79} + 81 q^{81} - 156 q^{83} - 270 q^{85} - 234 q^{87} + 1026 q^{89} - 1480 q^{91} - 600 q^{93} - 620 q^{95} - 286 q^{97} + 216 q^{99} + O(q^{100}) \)

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 −5.00000 0 −20.0000 0 9.00000 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 240.4.a.f 1
3.b odd 2 1 720.4.a.r 1
4.b odd 2 1 15.4.a.b 1
5.b even 2 1 1200.4.a.o 1
5.c odd 4 2 1200.4.f.m 2
8.b even 2 1 960.4.a.l 1
8.d odd 2 1 960.4.a.bi 1
12.b even 2 1 45.4.a.b 1
20.d odd 2 1 75.4.a.a 1
20.e even 4 2 75.4.b.a 2
28.d even 2 1 735.4.a.i 1
36.f odd 6 2 405.4.e.d 2
36.h even 6 2 405.4.e.k 2
44.c even 2 1 1815.4.a.a 1
60.h even 2 1 225.4.a.g 1
60.l odd 4 2 225.4.b.d 2
84.h odd 2 1 2205.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 4.b odd 2 1
45.4.a.b 1 12.b even 2 1
75.4.a.a 1 20.d odd 2 1
75.4.b.a 2 20.e even 4 2
225.4.a.g 1 60.h even 2 1
225.4.b.d 2 60.l odd 4 2
240.4.a.f 1 1.a even 1 1 trivial
405.4.e.d 2 36.f odd 6 2
405.4.e.k 2 36.h even 6 2
720.4.a.r 1 3.b odd 2 1
735.4.a.i 1 28.d even 2 1
960.4.a.l 1 8.b even 2 1
960.4.a.bi 1 8.d odd 2 1
1200.4.a.o 1 5.b even 2 1
1200.4.f.m 2 5.c odd 4 2
1815.4.a.a 1 44.c even 2 1
2205.4.a.c 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(240))\):

\( T_{7} + 20 \)
\( T_{11} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( 5 + T \)
$7$ \( 20 + T \)
$11$ \( -24 + T \)
$13$ \( -74 + T \)
$17$ \( -54 + T \)
$19$ \( -124 + T \)
$23$ \( -120 + T \)
$29$ \( 78 + T \)
$31$ \( 200 + T \)
$37$ \( 70 + T \)
$41$ \( -330 + T \)
$43$ \( 92 + T \)
$47$ \( -24 + T \)
$53$ \( -450 + T \)
$59$ \( 24 + T \)
$61$ \( 322 + T \)
$67$ \( -196 + T \)
$71$ \( -288 + T \)
$73$ \( 430 + T \)
$79$ \( -520 + T \)
$83$ \( 156 + T \)
$89$ \( -1026 + T \)
$97$ \( 286 + T \)
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