Properties

Label 2240.2.n.d
Level $2240$
Weight $2$
Character orbit 2240.n
Analytic conductor $17.886$
Analytic rank $0$
Dimension $8$
CM discriminant -280
Inner twists $16$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.384160000.1
Defining polynomial: \(x^{8} - 9 x^{6} + 37 x^{4} - 36 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} -\beta_{3} q^{7} -3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{5} -\beta_{3} q^{7} -3 q^{9} -\beta_{2} q^{17} + \beta_{6} q^{19} + 5 q^{25} -\beta_{7} q^{35} + \beta_{5} q^{37} -2 \beta_{7} q^{43} -3 \beta_{1} q^{45} -2 \beta_{3} q^{47} -7 q^{49} + \beta_{5} q^{53} -3 \beta_{6} q^{59} + 2 \beta_{1} q^{61} + 3 \beta_{3} q^{63} -2 \beta_{7} q^{67} -\beta_{4} q^{71} + 3 \beta_{2} q^{73} + 3 \beta_{4} q^{79} + 9 q^{81} + \beta_{5} q^{85} -5 \beta_{4} q^{95} -\beta_{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 24q^{9} + O(q^{10}) \) \( 8q - 24q^{9} + 40q^{25} - 56q^{49} + 72q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 9 x^{6} + 37 x^{4} - 36 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{6} + 5 \nu^{4} - 5 \nu^{2} - 38 \)\()/18\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 13 \nu^{5} - 65 \nu^{3} + 104 \nu \)\()/12\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{6} - 16 \nu^{4} + 64 \nu^{2} - 35 \)\()/9\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{7} - 59 \nu^{5} + 215 \nu^{3} - 88 \nu \)\()/36\)
\(\beta_{5}\)\(=\)\((\)\( -11 \nu^{7} + 103 \nu^{5} - 451 \nu^{3} + 704 \nu \)\()/36\)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{7} - 41 \nu^{5} + 157 \nu^{3} - 64 \nu \)\()/12\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{6} + 9 \nu^{4} - 33 \nu^{2} + 18 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} + \beta_{5} - \beta_{4} - \beta_{2}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + 3 \beta_{3} + 3 \beta_{1} + 9\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{6} + \beta_{5} - 10 \beta_{4} - 2 \beta_{2}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(9 \beta_{7} + 21 \beta_{3} + 3 \beta_{1} + 7\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(55 \beta_{6} - 5 \beta_{5} - 121 \beta_{4} + 11 \beta_{2}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(20 \beta_{7} + 45 \beta_{3} - 36 \beta_{1} - 81\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(169 \beta_{6} - 91 \beta_{5} - 377 \beta_{4} + 203 \beta_{2}\)\()/8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1119.1
0.817582 + 0.309017i
−0.817582 0.309017i
0.817582 0.309017i
−0.817582 + 0.309017i
−2.14046 0.809017i
2.14046 + 0.809017i
−2.14046 + 0.809017i
2.14046 0.809017i
0 0 0 −2.23607 0 2.64575i 0 −3.00000 0
1119.2 0 0 0 −2.23607 0 2.64575i 0 −3.00000 0
1119.3 0 0 0 −2.23607 0 2.64575i 0 −3.00000 0
1119.4 0 0 0 −2.23607 0 2.64575i 0 −3.00000 0
1119.5 0 0 0 2.23607 0 2.64575i 0 −3.00000 0
1119.6 0 0 0 2.23607 0 2.64575i 0 −3.00000 0
1119.7 0 0 0 2.23607 0 2.64575i 0 −3.00000 0
1119.8 0 0 0 2.23607 0 2.64575i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1119.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
280.c odd 2 1 CM by \(\Q(\sqrt{-70}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner
140.c even 2 1 inner
280.n even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.n.d 8
4.b odd 2 1 inner 2240.2.n.d 8
5.b even 2 1 inner 2240.2.n.d 8
7.b odd 2 1 inner 2240.2.n.d 8
8.b even 2 1 inner 2240.2.n.d 8
8.d odd 2 1 inner 2240.2.n.d 8
20.d odd 2 1 inner 2240.2.n.d 8
28.d even 2 1 inner 2240.2.n.d 8
35.c odd 2 1 inner 2240.2.n.d 8
40.e odd 2 1 inner 2240.2.n.d 8
40.f even 2 1 inner 2240.2.n.d 8
56.e even 2 1 inner 2240.2.n.d 8
56.h odd 2 1 inner 2240.2.n.d 8
140.c even 2 1 inner 2240.2.n.d 8
280.c odd 2 1 CM 2240.2.n.d 8
280.n even 2 1 inner 2240.2.n.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.n.d 8 1.a even 1 1 trivial
2240.2.n.d 8 4.b odd 2 1 inner
2240.2.n.d 8 5.b even 2 1 inner
2240.2.n.d 8 7.b odd 2 1 inner
2240.2.n.d 8 8.b even 2 1 inner
2240.2.n.d 8 8.d odd 2 1 inner
2240.2.n.d 8 20.d odd 2 1 inner
2240.2.n.d 8 28.d even 2 1 inner
2240.2.n.d 8 35.c odd 2 1 inner
2240.2.n.d 8 40.e odd 2 1 inner
2240.2.n.d 8 40.f even 2 1 inner
2240.2.n.d 8 56.e even 2 1 inner
2240.2.n.d 8 56.h odd 2 1 inner
2240.2.n.d 8 140.c even 2 1 inner
2240.2.n.d 8 280.c odd 2 1 CM
2240.2.n.d 8 280.n even 2 1 inner

Hecke kernels

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

\( T_{3} \)
\( T_{11} \)
\( T_{17}^{2} - 28 \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( -5 + T^{2} )^{4} \)
$7$ \( ( 7 + T^{2} )^{4} \)
$11$ \( T^{8} \)
$13$ \( T^{8} \)
$17$ \( ( -28 + T^{2} )^{4} \)
$19$ \( ( 20 + T^{2} )^{4} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( ( -140 + T^{2} )^{4} \)
$41$ \( T^{8} \)
$43$ \( ( 140 + T^{2} )^{4} \)
$47$ \( ( 28 + T^{2} )^{4} \)
$53$ \( ( -140 + T^{2} )^{4} \)
$59$ \( ( 180 + T^{2} )^{4} \)
$61$ \( ( -20 + T^{2} )^{4} \)
$67$ \( ( 140 + T^{2} )^{4} \)
$71$ \( ( 4 + T^{2} )^{4} \)
$73$ \( ( -252 + T^{2} )^{4} \)
$79$ \( ( 36 + T^{2} )^{4} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( ( -28 + T^{2} )^{4} \)
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