Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(6.53952634465\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-7})\)
Defining polynomial: \(x^{4} - 2 x^{3} - x^{2} + 2 x + 22\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -2 + \beta_{2} ) q^{5} + 2 \beta_{1} q^{7} + 3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -2 + \beta_{2} ) q^{5} + 2 \beta_{1} q^{7} + 3 q^{9} -2 \beta_{3} q^{11} -2 \beta_{2} q^{13} + ( -2 \beta_{1} - \beta_{3} ) q^{15} + 2 \beta_{2} q^{17} -4 \beta_{3} q^{19} + 6 q^{21} + 16 \beta_{1} q^{23} + ( -17 - 4 \beta_{2} ) q^{25} + 3 \beta_{1} q^{27} -8 q^{29} + 6 \beta_{2} q^{33} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{35} -10 \beta_{2} q^{37} + 2 \beta_{3} q^{39} + 50 q^{41} + 36 \beta_{1} q^{43} + ( -6 + 3 \beta_{2} ) q^{45} -28 \beta_{1} q^{47} -37 q^{49} -2 \beta_{3} q^{51} + 6 \beta_{2} q^{53} + ( -42 \beta_{1} + 4 \beta_{3} ) q^{55} + 12 \beta_{2} q^{57} + 2 \beta_{3} q^{59} -26 q^{61} + 6 \beta_{1} q^{63} + ( 42 + 4 \beta_{2} ) q^{65} -32 \beta_{1} q^{67} + 48 q^{69} + 12 \beta_{3} q^{71} -28 \beta_{2} q^{73} + ( -17 \beta_{1} + 4 \beta_{3} ) q^{75} + 12 \beta_{2} q^{77} + 16 \beta_{3} q^{79} + 9 q^{81} + 76 \beta_{1} q^{83} + ( -42 - 4 \beta_{2} ) q^{85} -8 \beta_{1} q^{87} -86 q^{89} + 4 \beta_{3} q^{91} + ( -84 \beta_{1} + 8 \beta_{3} ) q^{95} -24 \beta_{2} q^{97} -6 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{5} + 12 q^{9} + O(q^{10}) \) \( 4 q - 8 q^{5} + 12 q^{9} + 24 q^{21} - 68 q^{25} - 32 q^{29} + 200 q^{41} - 24 q^{45} - 148 q^{49} - 104 q^{61} + 168 q^{65} + 192 q^{69} + 36 q^{81} - 168 q^{85} - 344 q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - x^{2} + 2 x + 22\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{3} + 3 \nu^{2} + 13 \nu - 7 \)\()/19\)
\(\beta_{2}\)\(=\)\( -\nu^{2} + \nu + 1 \)
\(\beta_{3}\)\(=\)\((\)\( 12 \nu^{3} - 18 \nu^{2} + 36 \nu - 15 \)\()/19\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 6 \beta_{1} + 3\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 6 \beta_{2} + 6 \beta_{1} + 9\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(8 \beta_{3} - 9 \beta_{2} - 9 \beta_{1} + 12\)\()/6\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−1.23205 1.32288i
−1.23205 + 1.32288i
2.23205 + 1.32288i
2.23205 1.32288i
0 −1.73205 0 −2.00000 4.58258i 0 −3.46410 0 3.00000 0
79.2 0 −1.73205 0 −2.00000 + 4.58258i 0 −3.46410 0 3.00000 0
79.3 0 1.73205 0 −2.00000 4.58258i 0 3.46410 0 3.00000 0
79.4 0 1.73205 0 −2.00000 + 4.58258i 0 3.46410 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
5.b even 2 1 inner
20.d 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.j.b 4
3.b odd 2 1 720.3.j.f 4
4.b odd 2 1 inner 240.3.j.b 4
5.b even 2 1 inner 240.3.j.b 4
5.c odd 4 2 1200.3.e.l 4
8.b even 2 1 960.3.j.c 4
8.d odd 2 1 960.3.j.c 4
12.b even 2 1 720.3.j.f 4
15.d odd 2 1 720.3.j.f 4
15.e even 4 2 3600.3.e.be 4
20.d odd 2 1 inner 240.3.j.b 4
20.e even 4 2 1200.3.e.l 4
40.e odd 2 1 960.3.j.c 4
40.f even 2 1 960.3.j.c 4
60.h even 2 1 720.3.j.f 4
60.l odd 4 2 3600.3.e.be 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.3.j.b 4 1.a even 1 1 trivial
240.3.j.b 4 4.b odd 2 1 inner
240.3.j.b 4 5.b even 2 1 inner
240.3.j.b 4 20.d odd 2 1 inner
720.3.j.f 4 3.b odd 2 1
720.3.j.f 4 12.b even 2 1
720.3.j.f 4 15.d odd 2 1
720.3.j.f 4 60.h even 2 1
960.3.j.c 4 8.b even 2 1
960.3.j.c 4 8.d odd 2 1
960.3.j.c 4 40.e odd 2 1
960.3.j.c 4 40.f even 2 1
1200.3.e.l 4 5.c odd 4 2
1200.3.e.l 4 20.e even 4 2
3600.3.e.be 4 15.e even 4 2
3600.3.e.be 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{2} - 12 \)
\( T_{11}^{2} + 252 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( ( 25 + 4 T + T^{2} )^{2} \)
$7$ \( ( -12 + T^{2} )^{2} \)
$11$ \( ( 252 + T^{2} )^{2} \)
$13$ \( ( 84 + T^{2} )^{2} \)
$17$ \( ( 84 + T^{2} )^{2} \)
$19$ \( ( 1008 + T^{2} )^{2} \)
$23$ \( ( -768 + T^{2} )^{2} \)
$29$ \( ( 8 + T )^{4} \)
$31$ \( T^{4} \)
$37$ \( ( 2100 + T^{2} )^{2} \)
$41$ \( ( -50 + T )^{4} \)
$43$ \( ( -3888 + T^{2} )^{2} \)
$47$ \( ( -2352 + T^{2} )^{2} \)
$53$ \( ( 756 + T^{2} )^{2} \)
$59$ \( ( 252 + T^{2} )^{2} \)
$61$ \( ( 26 + T )^{4} \)
$67$ \( ( -3072 + T^{2} )^{2} \)
$71$ \( ( 9072 + T^{2} )^{2} \)
$73$ \( ( 16464 + T^{2} )^{2} \)
$79$ \( ( 16128 + T^{2} )^{2} \)
$83$ \( ( -17328 + T^{2} )^{2} \)
$89$ \( ( 86 + T )^{4} \)
$97$ \( ( 12096 + T^{2} )^{2} \)
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