Properties

Label 2240.2.h.c
Level $2240$
Weight $2$
Character orbit 2240.h
Analytic conductor $17.886$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + q^{5} + ( \beta_{1} + \beta_{2} ) q^{7} - q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + q^{5} + ( \beta_{1} + \beta_{2} ) q^{7} - q^{9} -2 q^{13} -\beta_{1} q^{15} + \beta_{3} q^{17} + \beta_{1} q^{19} + ( 2 - \beta_{3} ) q^{21} -3 \beta_{1} q^{23} + q^{25} -2 \beta_{1} q^{27} -2 \beta_{3} q^{29} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{31} + ( \beta_{1} + \beta_{2} ) q^{35} + \beta_{3} q^{37} + 2 \beta_{1} q^{39} + ( \beta_{1} + 2 \beta_{2} ) q^{43} - q^{45} + ( \beta_{1} + 2 \beta_{2} ) q^{47} + ( 5 + \beta_{3} ) q^{49} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{51} -\beta_{3} q^{53} + 4 q^{57} -3 \beta_{1} q^{59} -10 q^{61} + ( -\beta_{1} - \beta_{2} ) q^{63} -2 q^{65} + ( 3 \beta_{1} + 6 \beta_{2} ) q^{67} -12 q^{69} + 3 \beta_{1} q^{71} -\beta_{3} q^{73} -\beta_{1} q^{75} -5 \beta_{1} q^{79} -11 q^{81} + 3 \beta_{1} q^{83} + \beta_{3} q^{85} + ( -4 \beta_{1} - 8 \beta_{2} ) q^{87} + 2 \beta_{3} q^{89} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{91} -4 \beta_{3} q^{93} + \beta_{1} q^{95} + \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{5} - 4q^{9} + O(q^{10}) \) \( 4q + 4q^{5} - 4q^{9} - 8q^{13} + 8q^{21} + 4q^{25} - 4q^{45} + 20q^{49} + 16q^{57} - 40q^{61} - 8q^{65} - 48q^{69} - 44q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
671.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
0 2.00000i 0 1.00000 0 −2.44949 + 1.00000i 0 −1.00000 0
671.2 0 2.00000i 0 1.00000 0 2.44949 + 1.00000i 0 −1.00000 0
671.3 0 2.00000i 0 1.00000 0 −2.44949 1.00000i 0 −1.00000 0
671.4 0 2.00000i 0 1.00000 0 2.44949 1.00000i 0 −1.00000 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
56.e even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.h.c yes 4
4.b odd 2 1 inner 2240.2.h.c yes 4
7.b odd 2 1 2240.2.h.a 4
8.b even 2 1 2240.2.h.a 4
8.d odd 2 1 2240.2.h.a 4
28.d even 2 1 2240.2.h.a 4
56.e even 2 1 inner 2240.2.h.c yes 4
56.h odd 2 1 inner 2240.2.h.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.h.a 4 7.b odd 2 1
2240.2.h.a 4 8.b even 2 1
2240.2.h.a 4 8.d odd 2 1
2240.2.h.a 4 28.d even 2 1
2240.2.h.c yes 4 1.a even 1 1 trivial
2240.2.h.c yes 4 4.b odd 2 1 inner
2240.2.h.c yes 4 56.e even 2 1 inner
2240.2.h.c yes 4 56.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 4 + T^{2} )^{2} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( 49 - 10 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( 2 + T )^{4} \)
$17$ \( ( 24 + T^{2} )^{2} \)
$19$ \( ( 4 + T^{2} )^{2} \)
$23$ \( ( 36 + T^{2} )^{2} \)
$29$ \( ( 96 + T^{2} )^{2} \)
$31$ \( ( -96 + T^{2} )^{2} \)
$37$ \( ( 24 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( -24 + T^{2} )^{2} \)
$47$ \( ( -24 + T^{2} )^{2} \)
$53$ \( ( 24 + T^{2} )^{2} \)
$59$ \( ( 36 + T^{2} )^{2} \)
$61$ \( ( 10 + T )^{4} \)
$67$ \( ( -216 + T^{2} )^{2} \)
$71$ \( ( 36 + T^{2} )^{2} \)
$73$ \( ( 24 + T^{2} )^{2} \)
$79$ \( ( 100 + T^{2} )^{2} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( ( 96 + T^{2} )^{2} \)
$97$ \( ( 24 + T^{2} )^{2} \)
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