Properties

Label 2240.2.h.d
Level $2240$
Weight $2$
Character orbit 2240.h
Analytic conductor $17.886$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12}^{3} q^{3} + q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 2 q^{9} +O(q^{10})\) \( q + \zeta_{12}^{3} q^{3} + q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 2 q^{9} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{11} + q^{13} + \zeta_{12}^{3} q^{15} + ( 1 - 2 \zeta_{12}^{2} ) q^{17} + 2 \zeta_{12}^{3} q^{19} + ( 1 + 2 \zeta_{12}^{2} ) q^{21} + q^{25} + 5 \zeta_{12}^{3} q^{27} + ( 1 - 2 \zeta_{12}^{2} ) q^{29} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{31} + ( -3 + 6 \zeta_{12}^{2} ) q^{33} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{35} + ( 4 - 8 \zeta_{12}^{2} ) q^{37} + \zeta_{12}^{3} q^{39} + ( 6 - 12 \zeta_{12}^{2} ) q^{41} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{43} + 2 q^{45} + ( -14 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{47} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{51} + ( 2 - 4 \zeta_{12}^{2} ) q^{53} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{55} -2 q^{57} + 6 \zeta_{12}^{3} q^{59} + 8 q^{61} + ( 4 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{63} + q^{65} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{67} + ( -4 + 8 \zeta_{12}^{2} ) q^{73} + \zeta_{12}^{3} q^{75} + ( 15 - 12 \zeta_{12}^{2} ) q^{77} -\zeta_{12}^{3} q^{79} + q^{81} + 12 \zeta_{12}^{3} q^{83} + ( 1 - 2 \zeta_{12}^{2} ) q^{85} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{87} + ( -4 + 8 \zeta_{12}^{2} ) q^{89} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{91} + ( 2 - 4 \zeta_{12}^{2} ) q^{93} + 2 \zeta_{12}^{3} q^{95} + ( -5 + 10 \zeta_{12}^{2} ) q^{97} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{5} + 8q^{9} + O(q^{10}) \) \( 4q + 4q^{5} + 8q^{9} + 4q^{13} + 8q^{21} + 4q^{25} + 8q^{45} - 4q^{49} - 8q^{57} + 32q^{61} + 4q^{65} + 36q^{77} + 4q^{81} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
671.1
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0 1.00000i 0 1.00000 0 −1.73205 + 2.00000i 0 2.00000 0
671.2 0 1.00000i 0 1.00000 0 1.73205 + 2.00000i 0 2.00000 0
671.3 0 1.00000i 0 1.00000 0 −1.73205 2.00000i 0 2.00000 0
671.4 0 1.00000i 0 1.00000 0 1.73205 2.00000i 0 2.00000 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
4.b odd 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.h.d yes 4
4.b odd 2 1 inner 2240.2.h.d yes 4
7.b odd 2 1 2240.2.h.b 4
8.b even 2 1 2240.2.h.b 4
8.d odd 2 1 2240.2.h.b 4
28.d even 2 1 2240.2.h.b 4
56.e even 2 1 inner 2240.2.h.d yes 4
56.h odd 2 1 inner 2240.2.h.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.h.b 4 7.b odd 2 1
2240.2.h.b 4 8.b even 2 1
2240.2.h.b 4 8.d odd 2 1
2240.2.h.b 4 28.d even 2 1
2240.2.h.d yes 4 1.a even 1 1 trivial
2240.2.h.d yes 4 4.b odd 2 1 inner
2240.2.h.d yes 4 56.e even 2 1 inner
2240.2.h.d yes 4 56.h odd 2 1 inner

Hecke kernels

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

\( T_{3}^{2} + 1 \)
\( T_{13} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( 49 + 2 T^{2} + T^{4} \)
$11$ \( ( -27 + T^{2} )^{2} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( ( 3 + T^{2} )^{2} \)
$19$ \( ( 4 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( ( 3 + T^{2} )^{2} \)
$31$ \( ( -12 + T^{2} )^{2} \)
$37$ \( ( 48 + T^{2} )^{2} \)
$41$ \( ( 108 + T^{2} )^{2} \)
$43$ \( ( -12 + T^{2} )^{2} \)
$47$ \( ( -147 + T^{2} )^{2} \)
$53$ \( ( 12 + T^{2} )^{2} \)
$59$ \( ( 36 + T^{2} )^{2} \)
$61$ \( ( -8 + T )^{4} \)
$67$ \( ( -108 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( ( 48 + T^{2} )^{2} \)
$79$ \( ( 1 + T^{2} )^{2} \)
$83$ \( ( 144 + T^{2} )^{2} \)
$89$ \( ( 48 + T^{2} )^{2} \)
$97$ \( ( 75 + T^{2} )^{2} \)
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