Properties

Label 2240.1.p.e
Level $2240$
Weight $1$
Character orbit 2240.p
Analytic conductor $1.118$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -20
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.11790562830\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1120)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.19600.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{2} q^{5} -\zeta_{8} q^{7} + q^{9} +O(q^{10})\) \( q + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{2} q^{5} -\zeta_{8} q^{7} + q^{9} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{15} + ( 1 + \zeta_{8}^{2} ) q^{21} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{23} - q^{25} -2 q^{29} -\zeta_{8}^{3} q^{35} -2 \zeta_{8}^{2} q^{41} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{43} + \zeta_{8}^{2} q^{45} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{47} + \zeta_{8}^{2} q^{49} -2 \zeta_{8}^{2} q^{61} -\zeta_{8} q^{63} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{67} -2 \zeta_{8}^{2} q^{69} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{75} - q^{81} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{83} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{87} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{9} + 4 q^{21} - 4 q^{25} - 8 q^{29} - 4 q^{81} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0 −1.41421 0 1.00000i 0 −0.707107 + 0.707107i 0 1.00000 0
769.2 0 −1.41421 0 1.00000i 0 −0.707107 0.707107i 0 1.00000 0
769.3 0 1.41421 0 1.00000i 0 0.707107 0.707107i 0 1.00000 0
769.4 0 1.41421 0 1.00000i 0 0.707107 + 0.707107i 0 1.00000 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 2240.1.p.e 4
4.b odd 2 1 inner 2240.1.p.e 4
5.b even 2 1 inner 2240.1.p.e 4
7.b odd 2 1 inner 2240.1.p.e 4
8.b even 2 1 1120.1.p.a 4
8.d odd 2 1 1120.1.p.a 4
20.d odd 2 1 CM 2240.1.p.e 4
28.d even 2 1 inner 2240.1.p.e 4
35.c odd 2 1 inner 2240.1.p.e 4
40.e odd 2 1 1120.1.p.a 4
40.f even 2 1 1120.1.p.a 4
56.e even 2 1 1120.1.p.a 4
56.h odd 2 1 1120.1.p.a 4
140.c even 2 1 inner 2240.1.p.e 4
280.c odd 2 1 1120.1.p.a 4
280.n even 2 1 1120.1.p.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.1.p.a 4 8.b even 2 1
1120.1.p.a 4 8.d odd 2 1
1120.1.p.a 4 40.e odd 2 1
1120.1.p.a 4 40.f even 2 1
1120.1.p.a 4 56.e even 2 1
1120.1.p.a 4 56.h odd 2 1
1120.1.p.a 4 280.c odd 2 1
1120.1.p.a 4 280.n even 2 1
2240.1.p.e 4 1.a even 1 1 trivial
2240.1.p.e 4 4.b odd 2 1 inner
2240.1.p.e 4 5.b even 2 1 inner
2240.1.p.e 4 7.b odd 2 1 inner
2240.1.p.e 4 20.d odd 2 1 CM
2240.1.p.e 4 28.d even 2 1 inner
2240.1.p.e 4 35.c odd 2 1 inner
2240.1.p.e 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 \(S_{1}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{2} - 2 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -2 + T^{2} )^{2} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( 1 + T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( 2 + T^{2} )^{2} \)
$29$ \( ( 2 + T )^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 4 + T^{2} )^{2} \)
$43$ \( ( 2 + T^{2} )^{2} \)
$47$ \( ( -2 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 4 + T^{2} )^{2} \)
$67$ \( ( 2 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( -2 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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