Properties

Label 2240.2.e.e
Level $2240$
Weight $2$
Character orbit 2240.e
Analytic conductor $17.886$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12745506816.5
Defining polynomial: \(x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81\)
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.

\(f(q)\) \(=\) \( q -\beta_{5} q^{3} + \beta_{4} q^{5} -\beta_{1} q^{7} + q^{9} +O(q^{10})\) \( q -\beta_{5} q^{3} + \beta_{4} q^{5} -\beta_{1} q^{7} + q^{9} + 2 \beta_{3} q^{11} + ( \beta_{4} - \beta_{6} ) q^{13} + ( \beta_{1} + \beta_{3} ) q^{15} + ( -2 \beta_{4} + 2 \beta_{6} ) q^{17} + ( -2 \beta_{2} - \beta_{5} ) q^{19} + ( \beta_{4} + \beta_{6} ) q^{21} + ( -2 - \beta_{7} ) q^{25} -4 \beta_{5} q^{27} + 6 q^{29} + ( -2 \beta_{4} + 2 \beta_{6} ) q^{33} + ( \beta_{2} - 3 \beta_{5} ) q^{35} -2 \beta_{7} q^{37} + 2 \beta_{3} q^{39} + ( -2 \beta_{4} - 2 \beta_{6} ) q^{41} -2 \beta_{1} q^{43} + \beta_{4} q^{45} -2 \beta_{5} q^{47} + 7 q^{49} -4 \beta_{3} q^{51} + 2 \beta_{7} q^{53} + ( -2 \beta_{2} - 4 \beta_{5} ) q^{55} + 2 \beta_{7} q^{57} + ( -2 \beta_{2} - \beta_{5} ) q^{59} + ( -3 \beta_{4} - 3 \beta_{6} ) q^{61} -\beta_{1} q^{63} + ( 3 - \beta_{7} ) q^{65} + 2 \beta_{1} q^{67} + ( 4 \beta_{4} - 4 \beta_{6} ) q^{73} + ( -2 \beta_{2} + \beta_{5} ) q^{75} -2 \beta_{7} q^{77} -4 \beta_{3} q^{79} -5 q^{81} -7 \beta_{5} q^{83} + ( -6 + 2 \beta_{7} ) q^{85} -6 \beta_{5} q^{87} + ( 2 \beta_{4} + 2 \beta_{6} ) q^{89} + ( 2 \beta_{2} + \beta_{5} ) q^{91} + ( 3 \beta_{1} - 7 \beta_{3} ) q^{95} + ( 6 \beta_{4} - 6 \beta_{6} ) q^{97} + 2 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{9} - 16q^{25} + 48q^{29} + 56q^{49} + 24q^{65} - 40q^{81} - 48q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{6} - 148 \)\()/55\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 11 \nu^{5} - 88 \nu^{3} + 336 \nu \)\()/99\)
\(\beta_{3}\)\(=\)\((\)\( 16 \nu^{6} - 110 \nu^{4} + 880 \nu^{2} - 657 \)\()/495\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} - 5 \nu^{5} + 40 \nu^{3} + 48 \nu \)\()/45\)
\(\beta_{5}\)\(=\)\((\)\( -8 \nu^{7} + 55 \nu^{5} - 341 \nu^{3} + 81 \nu \)\()/297\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 8 \nu^{5} + 55 \nu^{3} - 45 \nu \)\()/27\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{6} + 8 \nu^{4} - 46 \nu^{2} + 36 \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} + \beta_{5} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + 4 \beta_{3} + \beta_{1} + 4\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{6} + 11 \beta_{5} + 5 \beta_{4}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(8 \beta_{7} + 23 \beta_{3} - 8 \beta_{1} - 23\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-24 \beta_{6} + 24 \beta_{5} + 55 \beta_{4} - 31 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(-55 \beta_{1} - 148\)
\(\nu^{7}\)\(=\)\((\)\(-368 \beta_{6} - 368 \beta_{5} + 165 \beta_{4} - 203 \beta_{2}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2239.1
−1.00781 0.581861i
2.23256 + 1.28897i
1.00781 0.581861i
−2.23256 + 1.28897i
2.23256 1.28897i
−1.00781 + 0.581861i
−2.23256 1.28897i
1.00781 + 0.581861i
0 1.41421i 0 −1.22474 1.87083i 0 2.64575 0 1.00000 0
2239.2 0 1.41421i 0 −1.22474 + 1.87083i 0 −2.64575 0 1.00000 0
2239.3 0 1.41421i 0 1.22474 1.87083i 0 2.64575 0 1.00000 0
2239.4 0 1.41421i 0 1.22474 + 1.87083i 0 −2.64575 0 1.00000 0
2239.5 0 1.41421i 0 −1.22474 1.87083i 0 −2.64575 0 1.00000 0
2239.6 0 1.41421i 0 −1.22474 + 1.87083i 0 2.64575 0 1.00000 0
2239.7 0 1.41421i 0 1.22474 1.87083i 0 −2.64575 0 1.00000 0
2239.8 0 1.41421i 0 1.22474 + 1.87083i 0 2.64575 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2239.8
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
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c 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.e.e 8
4.b odd 2 1 inner 2240.2.e.e 8
5.b even 2 1 inner 2240.2.e.e 8
7.b odd 2 1 inner 2240.2.e.e 8
8.b even 2 1 560.2.e.d 8
8.d odd 2 1 560.2.e.d 8
20.d odd 2 1 inner 2240.2.e.e 8
28.d even 2 1 inner 2240.2.e.e 8
35.c odd 2 1 inner 2240.2.e.e 8
40.e odd 2 1 560.2.e.d 8
40.f even 2 1 560.2.e.d 8
40.i odd 4 2 2800.2.k.k 8
40.k even 4 2 2800.2.k.k 8
56.e even 2 1 560.2.e.d 8
56.h odd 2 1 560.2.e.d 8
140.c even 2 1 inner 2240.2.e.e 8
280.c odd 2 1 560.2.e.d 8
280.n even 2 1 560.2.e.d 8
280.s even 4 2 2800.2.k.k 8
280.y odd 4 2 2800.2.k.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.e.d 8 8.b even 2 1
560.2.e.d 8 8.d odd 2 1
560.2.e.d 8 40.e odd 2 1
560.2.e.d 8 40.f even 2 1
560.2.e.d 8 56.e even 2 1
560.2.e.d 8 56.h odd 2 1
560.2.e.d 8 280.c odd 2 1
560.2.e.d 8 280.n even 2 1
2240.2.e.e 8 1.a even 1 1 trivial
2240.2.e.e 8 4.b odd 2 1 inner
2240.2.e.e 8 5.b even 2 1 inner
2240.2.e.e 8 7.b odd 2 1 inner
2240.2.e.e 8 20.d odd 2 1 inner
2240.2.e.e 8 28.d even 2 1 inner
2240.2.e.e 8 35.c odd 2 1 inner
2240.2.e.e 8 140.c even 2 1 inner
2800.2.k.k 8 40.i odd 4 2
2800.2.k.k 8 40.k even 4 2
2800.2.k.k 8 280.s even 4 2
2800.2.k.k 8 280.y odd 4 2

Hecke kernels

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

\( T_{3}^{2} + 2 \)
\( T_{11}^{2} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 2 + T^{2} )^{4} \)
$5$ \( ( 25 + 4 T^{2} + T^{4} )^{2} \)
$7$ \( ( -7 + T^{2} )^{4} \)
$11$ \( ( 12 + T^{2} )^{4} \)
$13$ \( ( -6 + T^{2} )^{4} \)
$17$ \( ( -24 + T^{2} )^{4} \)
$19$ \( ( -42 + T^{2} )^{4} \)
$23$ \( T^{8} \)
$29$ \( ( -6 + T )^{8} \)
$31$ \( T^{8} \)
$37$ \( ( 84 + T^{2} )^{4} \)
$41$ \( ( 56 + T^{2} )^{4} \)
$43$ \( ( -28 + T^{2} )^{4} \)
$47$ \( ( 8 + T^{2} )^{4} \)
$53$ \( ( 84 + T^{2} )^{4} \)
$59$ \( ( -42 + T^{2} )^{4} \)
$61$ \( ( 126 + T^{2} )^{4} \)
$67$ \( ( -28 + T^{2} )^{4} \)
$71$ \( T^{8} \)
$73$ \( ( -96 + T^{2} )^{4} \)
$79$ \( ( 48 + T^{2} )^{4} \)
$83$ \( ( 98 + T^{2} )^{4} \)
$89$ \( ( 56 + T^{2} )^{4} \)
$97$ \( ( -216 + T^{2} )^{4} \)
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