Properties

Label 2240.2.k.c
Level $2240$
Weight $2$
Character orbit 2240.k
Analytic conductor $17.886$
Analytic rank $0$
Dimension $8$
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.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.303595776.1
Defining polynomial: \(x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 560)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{6} q^{3} + \beta_{4} q^{5} + ( -\beta_{1} + \beta_{5} + \beta_{6} ) q^{7} -\beta_{7} q^{9} +O(q^{10})\) \( q -\beta_{6} q^{3} + \beta_{4} q^{5} + ( -\beta_{1} + \beta_{5} + \beta_{6} ) q^{7} -\beta_{7} q^{9} -\beta_{1} q^{11} + ( \beta_{2} + 3 \beta_{4} ) q^{13} -\beta_{1} q^{15} + ( -\beta_{2} - \beta_{4} ) q^{17} + 2 \beta_{3} q^{19} + ( -3 + 2 \beta_{4} + \beta_{7} ) q^{21} + ( -2 \beta_{1} - 2 \beta_{5} ) q^{23} - q^{25} + ( -2 \beta_{3} + \beta_{6} ) q^{27} + ( 3 + \beta_{7} ) q^{29} + ( -4 \beta_{3} + 2 \beta_{6} ) q^{31} + ( -\beta_{2} + 3 \beta_{4} ) q^{33} + ( \beta_{1} - \beta_{3} + \beta_{6} ) q^{35} + ( -4 + 2 \beta_{7} ) q^{37} + ( -\beta_{1} + 2 \beta_{5} ) q^{39} + ( -2 \beta_{2} + 2 \beta_{4} ) q^{41} + 2 \beta_{5} q^{43} -\beta_{2} q^{45} + ( -6 \beta_{3} + \beta_{6} ) q^{47} + ( -1 - 4 \beta_{4} - 2 \beta_{7} ) q^{49} + ( -\beta_{1} - 2 \beta_{5} ) q^{51} + ( -2 - 4 \beta_{7} ) q^{53} + \beta_{6} q^{55} + ( -2 + 2 \beta_{7} ) q^{57} + ( -2 \beta_{3} + 4 \beta_{6} ) q^{59} + ( 2 \beta_{2} - 4 \beta_{4} ) q^{61} + ( \beta_{1} + 2 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} ) q^{63} + ( -3 - \beta_{7} ) q^{65} + ( -4 \beta_{1} + 2 \beta_{5} ) q^{67} + ( -4 \beta_{2} + 8 \beta_{4} ) q^{69} + ( 4 \beta_{1} - 2 \beta_{5} ) q^{71} + ( -2 \beta_{2} + 2 \beta_{4} ) q^{73} + \beta_{6} q^{75} + ( -2 + \beta_{2} - 3 \beta_{4} ) q^{77} + ( -9 \beta_{1} + 4 \beta_{5} ) q^{79} + ( -1 + 2 \beta_{7} ) q^{81} -6 \beta_{3} q^{83} + ( 1 + \beta_{7} ) q^{85} + ( 2 \beta_{3} - \beta_{6} ) q^{87} + ( 4 \beta_{2} + 8 \beta_{4} ) q^{89} + ( \beta_{1} - 6 \beta_{3} - 2 \beta_{5} + 4 \beta_{6} ) q^{91} + ( -2 - 2 \beta_{7} ) q^{93} + 2 \beta_{5} q^{95} + ( 3 \beta_{2} - 5 \beta_{4} ) q^{97} + ( -2 \beta_{1} - 2 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{9} + O(q^{10}) \) \( 8q + 4q^{9} - 28q^{21} - 8q^{25} + 20q^{29} - 40q^{37} - 24q^{57} - 20q^{65} - 16q^{77} - 16q^{81} + 4q^{85} - 8q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{6} + 4 \nu^{4} + 20 \nu^{2} + 27 \)\()/36\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 8 \nu^{5} + 40 \nu^{3} + 165 \nu \)\()/72\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} + 2 \nu^{3} - 9 \nu \)\()/54\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{7} + 16 \nu^{5} + 8 \nu^{3} + 81 \nu \)\()/216\)
\(\beta_{5}\)\(=\)\((\)\( 5 \nu^{6} + 16 \nu^{4} + 80 \nu^{2} + 153 \)\()/72\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} + 5 \nu^{5} + 16 \nu^{3} + 18 \nu \)\()/27\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{6} + 5 \nu^{4} + 7 \nu^{2} + 18 \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{6} + \beta_{4} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} + 2 \beta_{5} + \beta_{1} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{6} - 4 \beta_{4} - \beta_{3}\)
\(\nu^{4}\)\(=\)\((\)\(5 \beta_{7} - 6 \beta_{5} + 5 \beta_{1} - 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{6} + 15 \beta_{4} + 16 \beta_{3} - \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(8 \beta_{5} - 16 \beta_{1} - 5\)
\(\nu^{7}\)\(=\)\((\)\(13 \beta_{6} + 35 \beta_{4} - 48 \beta_{3} - 13 \beta_{2}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1791.1
−1.26217 + 1.18614i
−1.26217 1.18614i
−0.396143 1.68614i
−0.396143 + 1.68614i
0.396143 1.68614i
0.396143 + 1.68614i
1.26217 + 1.18614i
1.26217 1.18614i
0 −2.52434 0 1.00000i 0 2.52434 + 0.792287i 0 3.37228 0
1791.2 0 −2.52434 0 1.00000i 0 2.52434 0.792287i 0 3.37228 0
1791.3 0 −0.792287 0 1.00000i 0 0.792287 + 2.52434i 0 −2.37228 0
1791.4 0 −0.792287 0 1.00000i 0 0.792287 2.52434i 0 −2.37228 0
1791.5 0 0.792287 0 1.00000i 0 −0.792287 2.52434i 0 −2.37228 0
1791.6 0 0.792287 0 1.00000i 0 −0.792287 + 2.52434i 0 −2.37228 0
1791.7 0 2.52434 0 1.00000i 0 −2.52434 0.792287i 0 3.37228 0
1791.8 0 2.52434 0 1.00000i 0 −2.52434 + 0.792287i 0 3.37228 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1791.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d 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.k.c 8
4.b odd 2 1 inner 2240.2.k.c 8
7.b odd 2 1 inner 2240.2.k.c 8
8.b even 2 1 560.2.k.a 8
8.d odd 2 1 560.2.k.a 8
24.f even 2 1 5040.2.d.e 8
24.h odd 2 1 5040.2.d.e 8
28.d even 2 1 inner 2240.2.k.c 8
40.e odd 2 1 2800.2.k.l 8
40.f even 2 1 2800.2.k.l 8
40.i odd 4 1 2800.2.e.i 8
40.i odd 4 1 2800.2.e.j 8
40.k even 4 1 2800.2.e.i 8
40.k even 4 1 2800.2.e.j 8
56.e even 2 1 560.2.k.a 8
56.h odd 2 1 560.2.k.a 8
168.e odd 2 1 5040.2.d.e 8
168.i even 2 1 5040.2.d.e 8
280.c odd 2 1 2800.2.k.l 8
280.n even 2 1 2800.2.k.l 8
280.s even 4 1 2800.2.e.i 8
280.s even 4 1 2800.2.e.j 8
280.y odd 4 1 2800.2.e.i 8
280.y odd 4 1 2800.2.e.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.k.a 8 8.b even 2 1
560.2.k.a 8 8.d odd 2 1
560.2.k.a 8 56.e even 2 1
560.2.k.a 8 56.h odd 2 1
2240.2.k.c 8 1.a even 1 1 trivial
2240.2.k.c 8 4.b odd 2 1 inner
2240.2.k.c 8 7.b odd 2 1 inner
2240.2.k.c 8 28.d even 2 1 inner
2800.2.e.i 8 40.i odd 4 1
2800.2.e.i 8 40.k even 4 1
2800.2.e.i 8 280.s even 4 1
2800.2.e.i 8 280.y odd 4 1
2800.2.e.j 8 40.i odd 4 1
2800.2.e.j 8 40.k even 4 1
2800.2.e.j 8 280.s even 4 1
2800.2.e.j 8 280.y odd 4 1
2800.2.k.l 8 40.e odd 2 1
2800.2.k.l 8 40.f even 2 1
2800.2.k.l 8 280.c odd 2 1
2800.2.k.l 8 280.n even 2 1
5040.2.d.e 8 24.f even 2 1
5040.2.d.e 8 24.h odd 2 1
5040.2.d.e 8 168.e odd 2 1
5040.2.d.e 8 168.i even 2 1

Hecke kernels

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

\( T_{3}^{4} - 7 T_{3}^{2} + 4 \)
\( T_{19}^{2} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 4 - 7 T^{2} + T^{4} )^{2} \)
$5$ \( ( 1 + T^{2} )^{4} \)
$7$ \( 2401 - 34 T^{4} + T^{8} \)
$11$ \( ( 4 + 7 T^{2} + T^{4} )^{2} \)
$13$ \( ( 4 + 29 T^{2} + T^{4} )^{2} \)
$17$ \( ( 64 + 17 T^{2} + T^{4} )^{2} \)
$19$ \( ( -12 + T^{2} )^{4} \)
$23$ \( ( 256 + 76 T^{2} + T^{4} )^{2} \)
$29$ \( ( -2 - 5 T + T^{2} )^{4} \)
$31$ \( ( 256 - 76 T^{2} + T^{4} )^{2} \)
$37$ \( ( -8 + 10 T + T^{2} )^{4} \)
$41$ \( ( 576 + 84 T^{2} + T^{4} )^{2} \)
$43$ \( ( 12 + T^{2} )^{4} \)
$47$ \( ( 7744 - 187 T^{2} + T^{4} )^{2} \)
$53$ \( ( -132 + T^{2} )^{4} \)
$59$ \( ( -44 + T^{2} )^{4} \)
$61$ \( ( 64 + 116 T^{2} + T^{4} )^{2} \)
$67$ \( ( 44 + T^{2} )^{4} \)
$71$ \( ( 44 + T^{2} )^{4} \)
$73$ \( ( 576 + 84 T^{2} + T^{4} )^{2} \)
$79$ \( ( 49284 + 447 T^{2} + T^{4} )^{2} \)
$83$ \( ( -108 + T^{2} )^{4} \)
$89$ \( ( 9216 + 336 T^{2} + T^{4} )^{2} \)
$97$ \( ( 1024 + 233 T^{2} + T^{4} )^{2} \)
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