Properties

Label 2240.2.n.c
Level $2240$
Weight $2$
Character orbit 2240.n
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.n (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( 1 + \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{7} -3 q^{9} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{11} + ( -2 + 4 \zeta_{24}^{4} ) q^{13} -4 q^{17} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{19} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{23} + ( -1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{25} + ( 4 - 8 \zeta_{24}^{4} ) q^{29} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{31} + ( -4 \zeta_{24} - 2 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{35} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{37} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{41} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{43} + ( -3 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 6 \zeta_{24}^{4} + 3 \zeta_{24}^{5} ) q^{45} -10 \zeta_{24}^{6} q^{47} + ( 5 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{49} + ( 8 \zeta_{24} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} ) q^{53} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 6 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{55} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{59} + ( -10 \zeta_{24} - 10 \zeta_{24}^{3} + 10 \zeta_{24}^{5} ) q^{61} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{63} + ( 6 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{65} + ( 10 \zeta_{24} - 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} ) q^{67} + 4 \zeta_{24}^{6} q^{71} -12 q^{73} + ( -2 - 6 \zeta_{24} - 6 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 6 \zeta_{24}^{5} ) q^{77} + 12 \zeta_{24}^{6} q^{79} + 9 q^{81} + ( -16 \zeta_{24}^{2} + 8 \zeta_{24}^{6} ) q^{83} + ( -4 - 4 \zeta_{24} - 4 \zeta_{24}^{3} + 8 \zeta_{24}^{4} + 4 \zeta_{24}^{5} ) q^{85} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{89} + ( 6 \zeta_{24} - 4 \zeta_{24}^{2} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{91} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{95} -4 q^{97} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 24q^{9} + O(q^{10}) \) \( 8q - 24q^{9} - 32q^{17} - 8q^{25} + 40q^{49} + 48q^{65} - 96q^{73} + 72q^{81} - 32q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1119.1
0.258819 0.965926i
−0.965926 0.258819i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
0.965926 0.258819i
−0.258819 0.965926i
0 0 0 −1.41421 1.73205i 0 −2.44949 1.00000i 0 −3.00000 0
1119.2 0 0 0 −1.41421 1.73205i 0 2.44949 + 1.00000i 0 −3.00000 0
1119.3 0 0 0 −1.41421 + 1.73205i 0 −2.44949 + 1.00000i 0 −3.00000 0
1119.4 0 0 0 −1.41421 + 1.73205i 0 2.44949 1.00000i 0 −3.00000 0
1119.5 0 0 0 1.41421 1.73205i 0 −2.44949 + 1.00000i 0 −3.00000 0
1119.6 0 0 0 1.41421 1.73205i 0 2.44949 1.00000i 0 −3.00000 0
1119.7 0 0 0 1.41421 + 1.73205i 0 −2.44949 1.00000i 0 −3.00000 0
1119.8 0 0 0 1.41421 + 1.73205i 0 2.44949 + 1.00000i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1119.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner
280.c odd 2 1 inner
280.n 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.n.c 8
4.b odd 2 1 inner 2240.2.n.c 8
5.b even 2 1 2240.2.n.e yes 8
7.b odd 2 1 2240.2.n.e yes 8
8.b even 2 1 inner 2240.2.n.c 8
8.d odd 2 1 inner 2240.2.n.c 8
20.d odd 2 1 2240.2.n.e yes 8
28.d even 2 1 2240.2.n.e yes 8
35.c odd 2 1 inner 2240.2.n.c 8
40.e odd 2 1 2240.2.n.e yes 8
40.f even 2 1 2240.2.n.e yes 8
56.e even 2 1 2240.2.n.e yes 8
56.h odd 2 1 2240.2.n.e yes 8
140.c even 2 1 inner 2240.2.n.c 8
280.c odd 2 1 inner 2240.2.n.c 8
280.n even 2 1 inner 2240.2.n.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.n.c 8 1.a even 1 1 trivial
2240.2.n.c 8 4.b odd 2 1 inner
2240.2.n.c 8 8.b even 2 1 inner
2240.2.n.c 8 8.d odd 2 1 inner
2240.2.n.c 8 35.c odd 2 1 inner
2240.2.n.c 8 140.c even 2 1 inner
2240.2.n.c 8 280.c odd 2 1 inner
2240.2.n.c 8 280.n even 2 1 inner
2240.2.n.e yes 8 5.b even 2 1
2240.2.n.e yes 8 7.b odd 2 1
2240.2.n.e yes 8 20.d odd 2 1
2240.2.n.e yes 8 28.d even 2 1
2240.2.n.e yes 8 40.e odd 2 1
2240.2.n.e yes 8 40.f even 2 1
2240.2.n.e yes 8 56.e even 2 1
2240.2.n.e yes 8 56.h odd 2 1

Hecke kernels

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

\( T_{3} \)
\( T_{11}^{2} - 12 \)
\( T_{17} + 4 \)
\( T_{23}^{2} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 25 + 2 T^{2} + T^{4} )^{2} \)
$7$ \( ( 49 - 10 T^{2} + T^{4} )^{2} \)
$11$ \( ( -12 + T^{2} )^{4} \)
$13$ \( ( 12 + T^{2} )^{4} \)
$17$ \( ( 4 + T )^{8} \)
$19$ \( ( 8 + T^{2} )^{4} \)
$23$ \( ( -24 + T^{2} )^{4} \)
$29$ \( ( 48 + T^{2} )^{4} \)
$31$ \( ( -96 + T^{2} )^{4} \)
$37$ \( ( -32 + T^{2} )^{4} \)
$41$ \( ( 96 + T^{2} )^{4} \)
$43$ \( ( 8 + T^{2} )^{4} \)
$47$ \( ( 100 + T^{2} )^{4} \)
$53$ \( ( -128 + T^{2} )^{4} \)
$59$ \( ( 72 + T^{2} )^{4} \)
$61$ \( ( -200 + T^{2} )^{4} \)
$67$ \( ( 200 + T^{2} )^{4} \)
$71$ \( ( 16 + T^{2} )^{4} \)
$73$ \( ( 12 + T )^{8} \)
$79$ \( ( 144 + T^{2} )^{4} \)
$83$ \( ( -192 + T^{2} )^{4} \)
$89$ \( ( 96 + T^{2} )^{4} \)
$97$ \( ( 4 + T )^{8} \)
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