Properties

Label 2240.2.e.a
Level $2240$
Weight $2$
Character orbit 2240.e
Analytic conductor $17.886$
Analytic rank $0$
Dimension $4$
CM discriminant -20
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: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \beta_{1} + \beta_{2} ) q^{3} -\beta_{3} q^{5} + ( \beta_{1} + 2 \beta_{2} ) q^{7} -7 q^{9} +O(q^{10})\) \( q + ( 2 \beta_{1} + \beta_{2} ) q^{3} -\beta_{3} q^{5} + ( \beta_{1} + 2 \beta_{2} ) q^{7} -7 q^{9} -5 \beta_{2} q^{15} + ( -5 - 3 \beta_{3} ) q^{21} + \beta_{2} q^{23} -5 q^{25} + ( -8 \beta_{1} - 4 \beta_{2} ) q^{27} -6 q^{29} + ( 3 \beta_{1} - \beta_{2} ) q^{35} -2 \beta_{3} q^{41} -9 \beta_{2} q^{43} + 7 \beta_{3} q^{45} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{47} + ( 2 - 3 \beta_{3} ) q^{49} -6 \beta_{3} q^{61} + ( -7 \beta_{1} - 14 \beta_{2} ) q^{63} + 3 \beta_{2} q^{67} -2 \beta_{3} q^{69} + ( -10 \beta_{1} - 5 \beta_{2} ) q^{75} + 19 q^{81} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{83} + ( -12 \beta_{1} - 6 \beta_{2} ) q^{87} -8 \beta_{3} q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 28q^{9} + O(q^{10}) \) \( 4q - 28q^{9} - 20q^{21} - 20q^{25} - 24q^{29} + 8q^{49} + 76q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 9\):

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

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
−0.707107 1.58114i
0.707107 1.58114i
0.707107 + 1.58114i
−0.707107 + 1.58114i
0 3.16228i 0 2.23607i 0 2.12132 1.58114i 0 −7.00000 0
2239.2 0 3.16228i 0 2.23607i 0 −2.12132 1.58114i 0 −7.00000 0
2239.3 0 3.16228i 0 2.23607i 0 −2.12132 + 1.58114i 0 −7.00000 0
2239.4 0 3.16228i 0 2.23607i 0 2.12132 + 1.58114i 0 −7.00000 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
7.b 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.a 4
4.b odd 2 1 inner 2240.2.e.a 4
5.b even 2 1 inner 2240.2.e.a 4
7.b odd 2 1 inner 2240.2.e.a 4
8.b even 2 1 140.2.c.a 4
8.d odd 2 1 140.2.c.a 4
20.d odd 2 1 CM 2240.2.e.a 4
28.d even 2 1 inner 2240.2.e.a 4
35.c odd 2 1 inner 2240.2.e.a 4
40.e odd 2 1 140.2.c.a 4
40.f even 2 1 140.2.c.a 4
40.i odd 4 2 700.2.g.d 4
40.k even 4 2 700.2.g.d 4
56.e even 2 1 140.2.c.a 4
56.h odd 2 1 140.2.c.a 4
56.j odd 6 2 980.2.s.b 8
56.k odd 6 2 980.2.s.b 8
56.m even 6 2 980.2.s.b 8
56.p even 6 2 980.2.s.b 8
140.c even 2 1 inner 2240.2.e.a 4
280.c odd 2 1 140.2.c.a 4
280.n even 2 1 140.2.c.a 4
280.s even 4 2 700.2.g.d 4
280.y odd 4 2 700.2.g.d 4
280.ba even 6 2 980.2.s.b 8
280.bf even 6 2 980.2.s.b 8
280.bi odd 6 2 980.2.s.b 8
280.bk odd 6 2 980.2.s.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.c.a 4 8.b even 2 1
140.2.c.a 4 8.d odd 2 1
140.2.c.a 4 40.e odd 2 1
140.2.c.a 4 40.f even 2 1
140.2.c.a 4 56.e even 2 1
140.2.c.a 4 56.h odd 2 1
140.2.c.a 4 280.c odd 2 1
140.2.c.a 4 280.n even 2 1
700.2.g.d 4 40.i odd 4 2
700.2.g.d 4 40.k even 4 2
700.2.g.d 4 280.s even 4 2
700.2.g.d 4 280.y odd 4 2
980.2.s.b 8 56.j odd 6 2
980.2.s.b 8 56.k odd 6 2
980.2.s.b 8 56.m even 6 2
980.2.s.b 8 56.p even 6 2
980.2.s.b 8 280.ba even 6 2
980.2.s.b 8 280.bf even 6 2
980.2.s.b 8 280.bi odd 6 2
980.2.s.b 8 280.bk odd 6 2
2240.2.e.a 4 1.a even 1 1 trivial
2240.2.e.a 4 4.b odd 2 1 inner
2240.2.e.a 4 5.b even 2 1 inner
2240.2.e.a 4 7.b odd 2 1 inner
2240.2.e.a 4 20.d odd 2 1 CM
2240.2.e.a 4 28.d even 2 1 inner
2240.2.e.a 4 35.c odd 2 1 inner
2240.2.e.a 4 140.c 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 \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{2} + 10 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 10 + T^{2} )^{2} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( 49 - 4 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( -2 + T^{2} )^{2} \)
$29$ \( ( 6 + T )^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 20 + T^{2} )^{2} \)
$43$ \( ( -162 + T^{2} )^{2} \)
$47$ \( ( 90 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 180 + T^{2} )^{2} \)
$67$ \( ( -18 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 90 + T^{2} )^{2} \)
$89$ \( ( 320 + T^{2} )^{2} \)
$97$ \( T^{4} \)
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