Properties

Label 2240.2.e.d
Level $2240$
Weight $2$
Character orbit 2240.e
Analytic conductor $17.886$
Analytic rank $0$
Dimension $8$
CM discriminant -35
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: 8.0.121550625.1
Defining polynomial: \(x^{8} - x^{7} - 4 x^{6} - 9 x^{5} + 23 x^{4} + 18 x^{3} - 16 x^{2} + 8 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 560)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + \beta_{2} q^{5} + ( \beta_{1} - \beta_{6} ) q^{7} + ( -3 + \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{6} q^{3} + \beta_{2} q^{5} + ( \beta_{1} - \beta_{6} ) q^{7} + ( -3 + \beta_{5} ) q^{9} + ( \beta_{4} + \beta_{7} ) q^{11} + ( -\beta_{2} + \beta_{3} ) q^{13} + ( 3 \beta_{4} - \beta_{7} ) q^{15} + ( \beta_{2} + 3 \beta_{3} ) q^{17} + ( 4 - \beta_{5} ) q^{21} + 5 q^{25} + ( 2 \beta_{1} - 5 \beta_{6} ) q^{27} + ( -4 - \beta_{5} ) q^{29} + ( -\beta_{2} - \beta_{3} ) q^{33} + ( -\beta_{4} + 2 \beta_{7} ) q^{35} + ( -7 \beta_{4} + 3 \beta_{7} ) q^{39} + 5 \beta_{3} q^{45} + ( -4 \beta_{1} - \beta_{6} ) q^{47} -7 q^{49} + ( -9 \beta_{4} + 5 \beta_{7} ) q^{51} + ( 4 \beta_{1} - \beta_{6} ) q^{55} + ( \beta_{1} + 6 \beta_{6} ) q^{63} + ( -8 + \beta_{5} ) q^{65} + ( -2 \beta_{4} + 4 \beta_{7} ) q^{71} -6 \beta_{2} q^{73} + 5 \beta_{6} q^{75} + ( -5 \beta_{2} - 3 \beta_{3} ) q^{77} + ( -\beta_{4} + 3 \beta_{7} ) q^{79} + ( 17 - 2 \beta_{5} ) q^{81} + ( -6 \beta_{1} + 6 \beta_{6} ) q^{83} + ( -4 + 3 \beta_{5} ) q^{85} + ( -2 \beta_{1} + \beta_{6} ) q^{87} + ( 5 \beta_{4} - 3 \beta_{7} ) q^{91} + ( 5 \beta_{2} + 7 \beta_{3} ) q^{97} + ( 4 \beta_{4} + 2 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 20q^{9} + O(q^{10}) \) \( 8q - 20q^{9} + 28q^{21} + 40q^{25} - 36q^{29} - 56q^{49} - 60q^{65} + 128q^{81} - 20q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 4 x^{6} - 9 x^{5} + 23 x^{4} + 18 x^{3} - 16 x^{2} + 8 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -7 \nu^{7} - 93 \nu^{6} + 244 \nu^{5} + 547 \nu^{4} + 659 \nu^{3} - 3622 \nu^{2} - 1884 \nu + 1240 \)\()/1224\)
\(\beta_{2}\)\(=\)\((\)\( 25 \nu^{7} - 3 \nu^{6} - 70 \nu^{5} - 205 \nu^{4} - 95 \nu^{3} + 40 \nu^{2} - 120 \nu + 2624 \)\()/1224\)
\(\beta_{3}\)\(=\)\((\)\( -89 \nu^{7} + 129 \nu^{6} + 392 \nu^{5} + 485 \nu^{4} - 2375 \nu^{3} - 1550 \nu^{2} + 3324 \nu - 1720 \)\()/1224\)
\(\beta_{4}\)\(=\)\((\)\( 41 \nu^{7} - 111 \nu^{6} - 74 \nu^{5} - 173 \nu^{4} + 1517 \nu^{3} - 1036 \nu^{2} - 768 \nu + 1072 \)\()/408\)
\(\beta_{5}\)\(=\)\((\)\( -65 \nu^{7} + 69 \nu^{6} + 284 \nu^{5} + 533 \nu^{4} - 1691 \nu^{3} - 1226 \nu^{2} + 2556 \nu + 32 \)\()/408\)
\(\beta_{6}\)\(=\)\((\)\( 313 \nu^{7} - 621 \nu^{6} - 652 \nu^{5} - 2077 \nu^{4} + 9235 \nu^{3} - 3110 \nu^{2} - 3420 \nu + 5696 \)\()/1224\)
\(\beta_{7}\)\(=\)\((\)\( -107 \nu^{7} + 123 \nu^{6} + 320 \nu^{5} + 959 \nu^{4} - 2429 \nu^{3} - 722 \nu^{2} + 228 \nu - 1096 \)\()/408\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + 3 \beta_{6} + \beta_{5} - 5 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 4\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{7} - 2 \beta_{6} + \beta_{4} - 5 \beta_{2} + 2 \beta_{1} + 10\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-9 \beta_{7} - 9 \beta_{6} + 9 \beta_{5} + \beta_{4} - 21 \beta_{3} - 9 \beta_{2} + 12 \beta_{1} - 8\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{7} + 33 \beta_{6} + 5 \beta_{5} - 63 \beta_{4} - 11 \beta_{3} - 33 \beta_{2} + 22 \beta_{1} + 58\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-40 \beta_{7} - 45 \beta_{6} + 20 \beta_{4} - 36 \beta_{2} + 45 \beta_{1} + 81\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-91 \beta_{7} - 17 \beta_{6} + 91 \beta_{5} - 143 \beta_{4} - 203 \beta_{3} - 17 \beta_{2} + 186 \beta_{1} - 234\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2239.1
−1.44918 + 1.77086i
−0.862555 0.141174i
2.25820 0.369600i
0.553538 + 0.676408i
2.25820 + 0.369600i
0.553538 0.676408i
−1.44918 1.77086i
−0.862555 + 0.141174i
0 3.25937i 0 −2.23607 0 2.64575i 0 −7.62348 0
2239.2 0 3.25937i 0 2.23607 0 2.64575i 0 −7.62348 0
2239.3 0 0.613616i 0 −2.23607 0 2.64575i 0 2.62348 0
2239.4 0 0.613616i 0 2.23607 0 2.64575i 0 2.62348 0
2239.5 0 0.613616i 0 −2.23607 0 2.64575i 0 2.62348 0
2239.6 0 0.613616i 0 2.23607 0 2.64575i 0 2.62348 0
2239.7 0 3.25937i 0 −2.23607 0 2.64575i 0 −7.62348 0
2239.8 0 3.25937i 0 2.23607 0 2.64575i 0 −7.62348 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
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
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
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.d 8
4.b odd 2 1 inner 2240.2.e.d 8
5.b even 2 1 inner 2240.2.e.d 8
7.b odd 2 1 inner 2240.2.e.d 8
8.b even 2 1 560.2.e.c 8
8.d odd 2 1 560.2.e.c 8
20.d odd 2 1 inner 2240.2.e.d 8
28.d even 2 1 inner 2240.2.e.d 8
35.c odd 2 1 CM 2240.2.e.d 8
40.e odd 2 1 560.2.e.c 8
40.f even 2 1 560.2.e.c 8
40.i odd 4 2 2800.2.k.p 8
40.k even 4 2 2800.2.k.p 8
56.e even 2 1 560.2.e.c 8
56.h odd 2 1 560.2.e.c 8
140.c even 2 1 inner 2240.2.e.d 8
280.c odd 2 1 560.2.e.c 8
280.n even 2 1 560.2.e.c 8
280.s even 4 2 2800.2.k.p 8
280.y odd 4 2 2800.2.k.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.e.c 8 8.b even 2 1
560.2.e.c 8 8.d odd 2 1
560.2.e.c 8 40.e odd 2 1
560.2.e.c 8 40.f even 2 1
560.2.e.c 8 56.e even 2 1
560.2.e.c 8 56.h odd 2 1
560.2.e.c 8 280.c odd 2 1
560.2.e.c 8 280.n even 2 1
2240.2.e.d 8 1.a even 1 1 trivial
2240.2.e.d 8 4.b odd 2 1 inner
2240.2.e.d 8 5.b even 2 1 inner
2240.2.e.d 8 7.b odd 2 1 inner
2240.2.e.d 8 20.d odd 2 1 inner
2240.2.e.d 8 28.d even 2 1 inner
2240.2.e.d 8 35.c odd 2 1 CM
2240.2.e.d 8 140.c even 2 1 inner
2800.2.k.p 8 40.i odd 4 2
2800.2.k.p 8 40.k even 4 2
2800.2.k.p 8 280.s even 4 2
2800.2.k.p 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 \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{4} + 11 T_{3}^{2} + 4 \)
\( T_{11}^{4} + 31 T_{11}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 4 + 11 T^{2} + T^{4} )^{2} \)
$5$ \( ( -5 + T^{2} )^{4} \)
$7$ \( ( 7 + T^{2} )^{4} \)
$11$ \( ( 4 + 31 T^{2} + T^{4} )^{2} \)
$13$ \( ( 36 - 33 T^{2} + T^{4} )^{2} \)
$17$ \( ( 2116 - 97 T^{2} + T^{4} )^{2} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( ( -6 + 9 T + T^{2} )^{4} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( T^{8} \)
$47$ \( ( 6084 + 219 T^{2} + T^{4} )^{2} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( T^{8} \)
$67$ \( T^{8} \)
$71$ \( ( 140 + T^{2} )^{4} \)
$73$ \( ( -180 + T^{2} )^{4} \)
$79$ \( ( 6084 + 159 T^{2} + T^{4} )^{2} \)
$83$ \( ( 252 + T^{2} )^{4} \)
$89$ \( T^{8} \)
$97$ \( ( 60516 - 537 T^{2} + T^{4} )^{2} \)
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