Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{3} + ( 2 - i ) q^{5} + i q^{7} -6 q^{9} +O(q^{10})\) \( q + 3 i q^{3} + ( 2 - i ) q^{5} + i q^{7} -6 q^{9} + 3 q^{11} -i q^{13} + ( 3 + 6 i ) q^{15} + 5 i q^{17} + 8 q^{19} -3 q^{21} -2 i q^{23} + ( 3 - 4 i ) q^{25} -9 i q^{27} - q^{29} + 2 q^{31} + 9 i q^{33} + ( 1 + 2 i ) q^{35} + 10 i q^{37} + 3 q^{39} -6 q^{41} -4 i q^{43} + ( -12 + 6 i ) q^{45} + 11 i q^{47} - q^{49} -15 q^{51} -6 i q^{53} + ( 6 - 3 i ) q^{55} + 24 i q^{57} + 10 q^{59} -6 i q^{63} + ( -1 - 2 i ) q^{65} + 10 i q^{67} + 6 q^{69} -10 i q^{73} + ( 12 + 9 i ) q^{75} + 3 i q^{77} -7 q^{79} + 9 q^{81} + 12 i q^{83} + ( 5 + 10 i ) q^{85} -3 i q^{87} -8 q^{89} + q^{91} + 6 i q^{93} + ( 16 - 8 i ) q^{95} -3 i q^{97} -18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} - 12q^{9} + O(q^{10}) \) \( 2q + 4q^{5} - 12q^{9} + 6q^{11} + 6q^{15} + 16q^{19} - 6q^{21} + 6q^{25} - 2q^{29} + 4q^{31} + 2q^{35} + 6q^{39} - 12q^{41} - 24q^{45} - 2q^{49} - 30q^{51} + 12q^{55} + 20q^{59} - 2q^{65} + 12q^{69} + 24q^{75} - 14q^{79} + 18q^{81} + 10q^{85} - 16q^{89} + 2q^{91} + 32q^{95} - 36q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.00000i
1.00000i
0 3.00000i 0 2.00000 + 1.00000i 0 1.00000i 0 −6.00000 0
449.2 0 3.00000i 0 2.00000 1.00000i 0 1.00000i 0 −6.00000 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
5.b 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.g.f 2
4.b odd 2 1 2240.2.g.e 2
5.b even 2 1 inner 2240.2.g.f 2
8.b even 2 1 560.2.g.a 2
8.d odd 2 1 140.2.e.a 2
20.d odd 2 1 2240.2.g.e 2
24.f even 2 1 1260.2.k.c 2
24.h odd 2 1 5040.2.t.s 2
40.e odd 2 1 140.2.e.a 2
40.f even 2 1 560.2.g.a 2
40.i odd 4 1 2800.2.a.a 1
40.i odd 4 1 2800.2.a.bf 1
40.k even 4 1 700.2.a.a 1
40.k even 4 1 700.2.a.j 1
56.e even 2 1 980.2.e.b 2
56.k odd 6 2 980.2.q.f 4
56.m even 6 2 980.2.q.c 4
120.i odd 2 1 5040.2.t.s 2
120.m even 2 1 1260.2.k.c 2
120.q odd 4 1 6300.2.a.c 1
120.q odd 4 1 6300.2.a.t 1
280.n even 2 1 980.2.e.b 2
280.y odd 4 1 4900.2.a.b 1
280.y odd 4 1 4900.2.a.w 1
280.ba even 6 2 980.2.q.c 4
280.bi odd 6 2 980.2.q.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.a 2 8.d odd 2 1
140.2.e.a 2 40.e odd 2 1
560.2.g.a 2 8.b even 2 1
560.2.g.a 2 40.f even 2 1
700.2.a.a 1 40.k even 4 1
700.2.a.j 1 40.k even 4 1
980.2.e.b 2 56.e even 2 1
980.2.e.b 2 280.n even 2 1
980.2.q.c 4 56.m even 6 2
980.2.q.c 4 280.ba even 6 2
980.2.q.f 4 56.k odd 6 2
980.2.q.f 4 280.bi odd 6 2
1260.2.k.c 2 24.f even 2 1
1260.2.k.c 2 120.m even 2 1
2240.2.g.e 2 4.b odd 2 1
2240.2.g.e 2 20.d odd 2 1
2240.2.g.f 2 1.a even 1 1 trivial
2240.2.g.f 2 5.b even 2 1 inner
2800.2.a.a 1 40.i odd 4 1
2800.2.a.bf 1 40.i odd 4 1
4900.2.a.b 1 280.y odd 4 1
4900.2.a.w 1 280.y odd 4 1
5040.2.t.s 2 24.h odd 2 1
5040.2.t.s 2 120.i odd 2 1
6300.2.a.c 1 120.q odd 4 1
6300.2.a.t 1 120.q odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 9 \)
\( T_{11} - 3 \)
\( T_{19} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( 5 - 4 T + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 25 + T^{2} \)
$19$ \( ( -8 + T )^{2} \)
$23$ \( 4 + T^{2} \)
$29$ \( ( 1 + T )^{2} \)
$31$ \( ( -2 + T )^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 121 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( -10 + T )^{2} \)
$61$ \( T^{2} \)
$67$ \( 100 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( ( 7 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( 8 + T )^{2} \)
$97$ \( 9 + T^{2} \)
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