Properties

Label 2240.2.g.h
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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.

\(f(q)\) \(=\) \( q + i q^{3} + ( 2 + i ) q^{5} -i q^{7} + 2 q^{9} +O(q^{10})\) \( q + i q^{3} + ( 2 + i ) q^{5} -i q^{7} + 2 q^{9} + 3 q^{11} + i q^{13} + ( -1 + 2 i ) q^{15} + 7 i q^{17} + q^{21} -6 i q^{23} + ( 3 + 4 i ) q^{25} + 5 i q^{27} -5 q^{29} + 2 q^{31} + 3 i q^{33} + ( 1 - 2 i ) q^{35} -2 i q^{37} - q^{39} + 2 q^{41} -4 i q^{43} + ( 4 + 2 i ) q^{45} -3 i q^{47} - q^{49} -7 q^{51} + 6 i q^{53} + ( 6 + 3 i ) q^{55} + 10 q^{59} + 8 q^{61} -2 i q^{63} + ( -1 + 2 i ) q^{65} -2 i q^{67} + 6 q^{69} -8 q^{71} -6 i q^{73} + ( -4 + 3 i ) q^{75} -3 i q^{77} + 5 q^{79} + q^{81} -4 i q^{83} + ( -7 + 14 i ) q^{85} -5 i q^{87} + q^{91} + 2 i q^{93} + 7 i q^{97} + 6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} + 4q^{9} + O(q^{10}) \) \( 2q + 4q^{5} + 4q^{9} + 6q^{11} - 2q^{15} + 2q^{21} + 6q^{25} - 10q^{29} + 4q^{31} + 2q^{35} - 2q^{39} + 4q^{41} + 8q^{45} - 2q^{49} - 14q^{51} + 12q^{55} + 20q^{59} + 16q^{61} - 2q^{65} + 12q^{69} - 16q^{71} - 8q^{75} + 10q^{79} + 2q^{81} - 14q^{85} + 2q^{91} + 12q^{99} + O(q^{100}) \)

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 1.00000i 0 2.00000 1.00000i 0 1.00000i 0 2.00000 0
449.2 0 1.00000i 0 2.00000 + 1.00000i 0 1.00000i 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
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.h 2
4.b odd 2 1 2240.2.g.g 2
5.b even 2 1 inner 2240.2.g.h 2
8.b even 2 1 35.2.b.a 2
8.d odd 2 1 560.2.g.b 2
20.d odd 2 1 2240.2.g.g 2
24.f even 2 1 5040.2.t.p 2
24.h odd 2 1 315.2.d.a 2
40.e odd 2 1 560.2.g.b 2
40.f even 2 1 35.2.b.a 2
40.i odd 4 1 175.2.a.a 1
40.i odd 4 1 175.2.a.c 1
40.k even 4 1 2800.2.a.l 1
40.k even 4 1 2800.2.a.w 1
56.h odd 2 1 245.2.b.a 2
56.j odd 6 2 245.2.j.d 4
56.p even 6 2 245.2.j.e 4
120.i odd 2 1 315.2.d.a 2
120.m even 2 1 5040.2.t.p 2
120.w even 4 1 1575.2.a.a 1
120.w even 4 1 1575.2.a.k 1
168.i even 2 1 2205.2.d.b 2
280.c odd 2 1 245.2.b.a 2
280.s even 4 1 1225.2.a.a 1
280.s even 4 1 1225.2.a.i 1
280.bf even 6 2 245.2.j.e 4
280.bk odd 6 2 245.2.j.d 4
840.u even 2 1 2205.2.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 8.b even 2 1
35.2.b.a 2 40.f even 2 1
175.2.a.a 1 40.i odd 4 1
175.2.a.c 1 40.i odd 4 1
245.2.b.a 2 56.h odd 2 1
245.2.b.a 2 280.c odd 2 1
245.2.j.d 4 56.j odd 6 2
245.2.j.d 4 280.bk odd 6 2
245.2.j.e 4 56.p even 6 2
245.2.j.e 4 280.bf even 6 2
315.2.d.a 2 24.h odd 2 1
315.2.d.a 2 120.i odd 2 1
560.2.g.b 2 8.d odd 2 1
560.2.g.b 2 40.e odd 2 1
1225.2.a.a 1 280.s even 4 1
1225.2.a.i 1 280.s even 4 1
1575.2.a.a 1 120.w even 4 1
1575.2.a.k 1 120.w even 4 1
2205.2.d.b 2 168.i even 2 1
2205.2.d.b 2 840.u even 2 1
2240.2.g.g 2 4.b odd 2 1
2240.2.g.g 2 20.d odd 2 1
2240.2.g.h 2 1.a even 1 1 trivial
2240.2.g.h 2 5.b even 2 1 inner
2800.2.a.l 1 40.k even 4 1
2800.2.a.w 1 40.k even 4 1
5040.2.t.p 2 24.f even 2 1
5040.2.t.p 2 120.m even 2 1

Hecke kernels

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

\( T_{3}^{2} + 1 \)
\( T_{11} - 3 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 5 - 4 T + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 49 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( ( -2 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 9 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( -10 + T )^{2} \)
$61$ \( ( -8 + T )^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( -5 + T )^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 49 + T^{2} \)
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