Properties

Label 2240.1.bt.b
Level $2240$
Weight $1$
Character orbit 2240.bt
Analytic conductor $1.118$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -20
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.11790562830\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.980.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} + \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{3} + \zeta_{6} q^{5} -\zeta_{6}^{2} q^{7} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{3} + \zeta_{6} q^{5} -\zeta_{6}^{2} q^{7} + q^{15} -\zeta_{6} q^{21} -\zeta_{6} q^{23} + \zeta_{6}^{2} q^{25} + q^{27} + q^{29} + q^{35} - q^{41} - q^{43} + 2 \zeta_{6} q^{47} -\zeta_{6} q^{49} -\zeta_{6} q^{61} -\zeta_{6}^{2} q^{67} - q^{69} + \zeta_{6} q^{75} -\zeta_{6}^{2} q^{81} - q^{83} -\zeta_{6}^{2} q^{87} + \zeta_{6} q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{5} + q^{7} + O(q^{10}) \) \( 2 q + q^{3} + q^{5} + q^{7} + 2 q^{15} - q^{21} - q^{23} - q^{25} + 2 q^{27} + 2 q^{29} + 2 q^{35} - 2 q^{41} - 2 q^{43} + 2 q^{47} - q^{49} - q^{61} + q^{67} - 2 q^{69} + q^{75} + q^{81} - 2 q^{83} + q^{87} + q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 0 0
639.1 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 0 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 \(\Q(\sqrt{-5}) \)
7.c even 3 1 inner
140.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.1.bt.b 2
4.b odd 2 1 2240.1.bt.a 2
5.b even 2 1 2240.1.bt.a 2
7.c even 3 1 inner 2240.1.bt.b 2
8.b even 2 1 140.1.p.b yes 2
8.d odd 2 1 140.1.p.a 2
20.d odd 2 1 CM 2240.1.bt.b 2
24.f even 2 1 1260.1.ci.b 2
24.h odd 2 1 1260.1.ci.a 2
28.g odd 6 1 2240.1.bt.a 2
35.j even 6 1 2240.1.bt.a 2
40.e odd 2 1 140.1.p.b yes 2
40.f even 2 1 140.1.p.a 2
40.i odd 4 2 700.1.u.a 4
40.k even 4 2 700.1.u.a 4
56.e even 2 1 980.1.p.a 2
56.h odd 2 1 980.1.p.b 2
56.j odd 6 1 980.1.f.a 1
56.j odd 6 1 980.1.p.b 2
56.k odd 6 1 140.1.p.a 2
56.k odd 6 1 980.1.f.c 1
56.m even 6 1 980.1.f.d 1
56.m even 6 1 980.1.p.a 2
56.p even 6 1 140.1.p.b yes 2
56.p even 6 1 980.1.f.b 1
120.i odd 2 1 1260.1.ci.b 2
120.m even 2 1 1260.1.ci.a 2
140.p odd 6 1 inner 2240.1.bt.b 2
168.s odd 6 1 1260.1.ci.a 2
168.v even 6 1 1260.1.ci.b 2
280.c odd 2 1 980.1.p.a 2
280.n even 2 1 980.1.p.b 2
280.ba even 6 1 980.1.f.a 1
280.ba even 6 1 980.1.p.b 2
280.bf even 6 1 140.1.p.a 2
280.bf even 6 1 980.1.f.c 1
280.bi odd 6 1 140.1.p.b yes 2
280.bi odd 6 1 980.1.f.b 1
280.bk odd 6 1 980.1.f.d 1
280.bk odd 6 1 980.1.p.a 2
280.br even 12 2 700.1.u.a 4
280.bt odd 12 2 700.1.u.a 4
840.cg odd 6 1 1260.1.ci.b 2
840.cv even 6 1 1260.1.ci.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.p.a 2 8.d odd 2 1
140.1.p.a 2 40.f even 2 1
140.1.p.a 2 56.k odd 6 1
140.1.p.a 2 280.bf even 6 1
140.1.p.b yes 2 8.b even 2 1
140.1.p.b yes 2 40.e odd 2 1
140.1.p.b yes 2 56.p even 6 1
140.1.p.b yes 2 280.bi odd 6 1
700.1.u.a 4 40.i odd 4 2
700.1.u.a 4 40.k even 4 2
700.1.u.a 4 280.br even 12 2
700.1.u.a 4 280.bt odd 12 2
980.1.f.a 1 56.j odd 6 1
980.1.f.a 1 280.ba even 6 1
980.1.f.b 1 56.p even 6 1
980.1.f.b 1 280.bi odd 6 1
980.1.f.c 1 56.k odd 6 1
980.1.f.c 1 280.bf even 6 1
980.1.f.d 1 56.m even 6 1
980.1.f.d 1 280.bk odd 6 1
980.1.p.a 2 56.e even 2 1
980.1.p.a 2 56.m even 6 1
980.1.p.a 2 280.c odd 2 1
980.1.p.a 2 280.bk odd 6 1
980.1.p.b 2 56.h odd 2 1
980.1.p.b 2 56.j odd 6 1
980.1.p.b 2 280.n even 2 1
980.1.p.b 2 280.ba even 6 1
1260.1.ci.a 2 24.h odd 2 1
1260.1.ci.a 2 120.m even 2 1
1260.1.ci.a 2 168.s odd 6 1
1260.1.ci.a 2 840.cv even 6 1
1260.1.ci.b 2 24.f even 2 1
1260.1.ci.b 2 120.i odd 2 1
1260.1.ci.b 2 168.v even 6 1
1260.1.ci.b 2 840.cg odd 6 1
2240.1.bt.a 2 4.b odd 2 1
2240.1.bt.a 2 5.b even 2 1
2240.1.bt.a 2 28.g odd 6 1
2240.1.bt.a 2 35.j even 6 1
2240.1.bt.b 2 1.a even 1 1 trivial
2240.1.bt.b 2 7.c even 3 1 inner
2240.1.bt.b 2 20.d odd 2 1 CM
2240.1.bt.b 2 140.p odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2240, [\chi])\).

Hecke characteristic polynomials

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