Properties

Label 240.4.a.e
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

\(f(q)\) \(=\) \( q - 3q^{3} + 5q^{5} + 24q^{7} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} + 5q^{5} + 24q^{7} + 9q^{9} - 52q^{11} + 22q^{13} - 15q^{15} - 14q^{17} + 20q^{19} - 72q^{21} + 168q^{23} + 25q^{25} - 27q^{27} + 230q^{29} + 288q^{31} + 156q^{33} + 120q^{35} - 34q^{37} - 66q^{39} + 122q^{41} + 188q^{43} + 45q^{45} - 256q^{47} + 233q^{49} + 42q^{51} - 338q^{53} - 260q^{55} - 60q^{57} - 100q^{59} + 742q^{61} + 216q^{63} + 110q^{65} + 84q^{67} - 504q^{69} + 328q^{71} - 38q^{73} - 75q^{75} - 1248q^{77} + 240q^{79} + 81q^{81} - 1212q^{83} - 70q^{85} - 690q^{87} + 330q^{89} + 528q^{91} - 864q^{93} + 100q^{95} + 866q^{97} - 468q^{99} + O(q^{100}) \)

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 24.0000 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.e 1
3.b odd 2 1 720.4.a.n 1
4.b odd 2 1 15.4.a.a 1
5.b even 2 1 1200.4.a.t 1
5.c odd 4 2 1200.4.f.b 2
8.b even 2 1 960.4.a.ba 1
8.d odd 2 1 960.4.a.b 1
12.b even 2 1 45.4.a.c 1
20.d odd 2 1 75.4.a.b 1
20.e even 4 2 75.4.b.b 2
28.d even 2 1 735.4.a.e 1
36.f odd 6 2 405.4.e.g 2
36.h even 6 2 405.4.e.i 2
44.c even 2 1 1815.4.a.e 1
60.h even 2 1 225.4.a.f 1
60.l odd 4 2 225.4.b.e 2
84.h odd 2 1 2205.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 4.b odd 2 1
45.4.a.c 1 12.b even 2 1
75.4.a.b 1 20.d odd 2 1
75.4.b.b 2 20.e even 4 2
225.4.a.f 1 60.h even 2 1
225.4.b.e 2 60.l odd 4 2
240.4.a.e 1 1.a even 1 1 trivial
405.4.e.g 2 36.f odd 6 2
405.4.e.i 2 36.h even 6 2
720.4.a.n 1 3.b odd 2 1
735.4.a.e 1 28.d even 2 1
960.4.a.b 1 8.d odd 2 1
960.4.a.ba 1 8.b even 2 1
1200.4.a.t 1 5.b even 2 1
1200.4.f.b 2 5.c odd 4 2
1815.4.a.e 1 44.c even 2 1
2205.4.a.l 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(240))\):

\( T_{7} - 24 \)
\( T_{11} + 52 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( -5 + T \)
$7$ \( -24 + T \)
$11$ \( 52 + T \)
$13$ \( -22 + T \)
$17$ \( 14 + T \)
$19$ \( -20 + T \)
$23$ \( -168 + T \)
$29$ \( -230 + T \)
$31$ \( -288 + T \)
$37$ \( 34 + T \)
$41$ \( -122 + T \)
$43$ \( -188 + T \)
$47$ \( 256 + T \)
$53$ \( 338 + T \)
$59$ \( 100 + T \)
$61$ \( -742 + T \)
$67$ \( -84 + T \)
$71$ \( -328 + T \)
$73$ \( 38 + T \)
$79$ \( -240 + T \)
$83$ \( 1212 + T \)
$89$ \( -330 + T \)
$97$ \( -866 + T \)
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