Properties

Label 240.3.e.b
Level $240$
Weight $3$
Character orbit 240.e
Analytic conductor $6.540$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 240.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.53952634465\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} -\beta_{1} q^{5} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} -\beta_{1} q^{5} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{7} -3 q^{9} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{11} + ( 4 - 6 \beta_{1} ) q^{13} + \beta_{3} q^{15} + ( -12 - 6 \beta_{1} ) q^{17} + ( 8 \beta_{2} - 4 \beta_{3} ) q^{19} + ( 6 - 6 \beta_{1} ) q^{21} + ( 4 \beta_{2} + 4 \beta_{3} ) q^{23} + 5 q^{25} -3 \beta_{2} q^{27} + ( 12 - 18 \beta_{1} ) q^{29} + ( -8 \beta_{2} - 4 \beta_{3} ) q^{31} + ( 6 - 6 \beta_{1} ) q^{33} + ( -10 \beta_{2} - 2 \beta_{3} ) q^{35} + ( -40 + 6 \beta_{1} ) q^{37} + ( 4 \beta_{2} + 6 \beta_{3} ) q^{39} + 42 q^{41} + ( 4 \beta_{2} - 16 \beta_{3} ) q^{43} + 3 \beta_{1} q^{45} + ( 8 \beta_{2} + 8 \beta_{3} ) q^{47} + ( -23 + 24 \beta_{1} ) q^{49} + ( -12 \beta_{2} + 6 \beta_{3} ) q^{51} + ( -48 + 18 \beta_{1} ) q^{53} + ( -10 \beta_{2} - 2 \beta_{3} ) q^{55} + ( -24 - 12 \beta_{1} ) q^{57} + ( 22 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 86 + 12 \beta_{1} ) q^{61} + ( 6 \beta_{2} + 6 \beta_{3} ) q^{63} + ( 30 - 4 \beta_{1} ) q^{65} + ( 16 \beta_{2} - 12 \beta_{3} ) q^{67} + ( -12 + 12 \beta_{1} ) q^{69} + ( 4 \beta_{2} + 28 \beta_{3} ) q^{71} + ( 22 + 36 \beta_{1} ) q^{73} + 5 \beta_{2} q^{75} + ( -72 + 24 \beta_{1} ) q^{77} + 4 \beta_{3} q^{79} + 9 q^{81} + ( -48 \beta_{2} + 12 \beta_{3} ) q^{83} + ( 30 + 12 \beta_{1} ) q^{85} + ( 12 \beta_{2} + 18 \beta_{3} ) q^{87} + ( -66 + 12 \beta_{1} ) q^{89} + ( -68 \beta_{2} - 20 \beta_{3} ) q^{91} + ( 24 - 12 \beta_{1} ) q^{93} + ( -20 \beta_{2} + 8 \beta_{3} ) q^{95} + ( -86 + 24 \beta_{1} ) q^{97} + ( 6 \beta_{2} + 6 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + O(q^{10}) \) \( 4 q - 12 q^{9} + 16 q^{13} - 48 q^{17} + 24 q^{21} + 20 q^{25} + 48 q^{29} + 24 q^{33} - 160 q^{37} + 168 q^{41} - 92 q^{49} - 192 q^{53} - 96 q^{57} + 344 q^{61} + 120 q^{65} - 48 q^{69} + 88 q^{73} - 288 q^{77} + 36 q^{81} + 120 q^{85} - 264 q^{89} + 96 q^{93} - 344 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 2 \)
\(\beta_{2}\)\(=\)\( -\nu^{3} + 2 \nu^{2} - 2 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{3} - 2 \nu^{2} + 6 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 3 \beta_{2} + \beta_{1} - 3\)\()/4\)
\(\nu^{3}\)\(=\)\(\beta_{1} - 2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.309017 + 0.535233i
0.809017 1.40126i
−0.309017 0.535233i
0.809017 + 1.40126i
0 1.73205i 0 −2.23607 0 4.28187i 0 −3.00000 0
31.2 0 1.73205i 0 2.23607 0 11.2101i 0 −3.00000 0
31.3 0 1.73205i 0 −2.23607 0 4.28187i 0 −3.00000 0
31.4 0 1.73205i 0 2.23607 0 11.2101i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.e.b 4
3.b odd 2 1 720.3.e.e 4
4.b odd 2 1 inner 240.3.e.b 4
5.b even 2 1 1200.3.e.j 4
5.c odd 4 2 1200.3.j.e 8
8.b even 2 1 960.3.e.a 4
8.d odd 2 1 960.3.e.a 4
12.b even 2 1 720.3.e.e 4
15.d odd 2 1 3600.3.e.z 4
15.e even 4 2 3600.3.j.p 8
20.d odd 2 1 1200.3.e.j 4
20.e even 4 2 1200.3.j.e 8
24.f even 2 1 2880.3.e.d 4
24.h odd 2 1 2880.3.e.d 4
60.h even 2 1 3600.3.e.z 4
60.l odd 4 2 3600.3.j.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.3.e.b 4 1.a even 1 1 trivial
240.3.e.b 4 4.b odd 2 1 inner
720.3.e.e 4 3.b odd 2 1
720.3.e.e 4 12.b even 2 1
960.3.e.a 4 8.b even 2 1
960.3.e.a 4 8.d odd 2 1
1200.3.e.j 4 5.b even 2 1
1200.3.e.j 4 20.d odd 2 1
1200.3.j.e 8 5.c odd 4 2
1200.3.j.e 8 20.e even 4 2
2880.3.e.d 4 24.f even 2 1
2880.3.e.d 4 24.h odd 2 1
3600.3.e.z 4 15.d odd 2 1
3600.3.e.z 4 60.h even 2 1
3600.3.j.p 8 15.e even 4 2
3600.3.j.p 8 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{2} - 8 T_{13} - 164 \) acting on \(S_{3}^{\mathrm{new}}(240, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( ( -5 + T^{2} )^{2} \)
$7$ \( 2304 + 144 T^{2} + T^{4} \)
$11$ \( 2304 + 144 T^{2} + T^{4} \)
$13$ \( ( -164 - 8 T + T^{2} )^{2} \)
$17$ \( ( -36 + 24 T + T^{2} )^{2} \)
$19$ \( 2304 + 864 T^{2} + T^{4} \)
$23$ \( 36864 + 576 T^{2} + T^{4} \)
$29$ \( ( -1476 - 24 T + T^{2} )^{2} \)
$31$ \( 2304 + 864 T^{2} + T^{4} \)
$37$ \( ( 1420 + 80 T + T^{2} )^{2} \)
$41$ \( ( -42 + T )^{4} \)
$43$ \( 14379264 + 7776 T^{2} + T^{4} \)
$47$ \( 589824 + 2304 T^{2} + T^{4} \)
$53$ \( ( 684 + 96 T + T^{2} )^{2} \)
$59$ \( 1937664 + 3024 T^{2} + T^{4} \)
$61$ \( ( 6676 - 172 T + T^{2} )^{2} \)
$67$ \( 1937664 + 5856 T^{2} + T^{4} \)
$71$ \( 137170944 + 23616 T^{2} + T^{4} \)
$73$ \( ( -5996 - 44 T + T^{2} )^{2} \)
$79$ \( ( 240 + T^{2} )^{2} \)
$83$ \( 22581504 + 18144 T^{2} + T^{4} \)
$89$ \( ( 3636 + 132 T + T^{2} )^{2} \)
$97$ \( ( 4516 + 172 T + T^{2} )^{2} \)
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