Properties

Label 2240.2.a.bj
Level $2240$
Weight $2$
Character orbit 2240.a
Self dual yes
Analytic conductor $17.886$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - q^{5} + q^{7} + ( 1 + \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{3} - q^{5} + q^{7} + ( 1 + \beta ) q^{9} + \beta q^{11} + ( -2 + \beta ) q^{13} -\beta q^{15} + ( 2 + \beta ) q^{17} + \beta q^{21} + q^{25} + ( 4 - \beta ) q^{27} + ( -2 + 3 \beta ) q^{29} -2 \beta q^{31} + ( 4 + \beta ) q^{33} - q^{35} + ( -2 - 2 \beta ) q^{37} + ( 4 - \beta ) q^{39} + 2 q^{41} + 4 q^{43} + ( -1 - \beta ) q^{45} + ( 4 + 3 \beta ) q^{47} + q^{49} + ( 4 + 3 \beta ) q^{51} + ( -10 + 2 \beta ) q^{53} -\beta q^{55} -4 \beta q^{59} + 2 q^{61} + ( 1 + \beta ) q^{63} + ( 2 - \beta ) q^{65} + ( 4 - 4 \beta ) q^{67} + 4 q^{71} + ( 6 + 4 \beta ) q^{73} + \beta q^{75} + \beta q^{77} + 5 \beta q^{79} -7 q^{81} + ( 4 - 4 \beta ) q^{83} + ( -2 - \beta ) q^{85} + ( 12 + \beta ) q^{87} + ( 2 + 4 \beta ) q^{89} + ( -2 + \beta ) q^{91} + ( -8 - 2 \beta ) q^{93} + ( 2 - 7 \beta ) q^{97} + ( 4 + 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - 2q^{5} + 2q^{7} + 3q^{9} + O(q^{10}) \) \( 2q + q^{3} - 2q^{5} + 2q^{7} + 3q^{9} + q^{11} - 3q^{13} - q^{15} + 5q^{17} + q^{21} + 2q^{25} + 7q^{27} - q^{29} - 2q^{31} + 9q^{33} - 2q^{35} - 6q^{37} + 7q^{39} + 4q^{41} + 8q^{43} - 3q^{45} + 11q^{47} + 2q^{49} + 11q^{51} - 18q^{53} - q^{55} - 4q^{59} + 4q^{61} + 3q^{63} + 3q^{65} + 4q^{67} + 8q^{71} + 16q^{73} + q^{75} + q^{77} + 5q^{79} - 14q^{81} + 4q^{83} - 5q^{85} + 25q^{87} + 8q^{89} - 3q^{91} - 18q^{93} - 3q^{97} + 10q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.56155
2.56155
0 −1.56155 0 −1.00000 0 1.00000 0 −0.561553 0
1.2 0 2.56155 0 −1.00000 0 1.00000 0 3.56155 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.a.bj 2
4.b odd 2 1 2240.2.a.bc 2
8.b even 2 1 1120.2.a.r 2
8.d odd 2 1 1120.2.a.t yes 2
40.e odd 2 1 5600.2.a.ba 2
40.f even 2 1 5600.2.a.bh 2
56.e even 2 1 7840.2.a.bc 2
56.h odd 2 1 7840.2.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.a.r 2 8.b even 2 1
1120.2.a.t yes 2 8.d odd 2 1
2240.2.a.bc 2 4.b odd 2 1
2240.2.a.bj 2 1.a even 1 1 trivial
5600.2.a.ba 2 40.e odd 2 1
5600.2.a.bh 2 40.f even 2 1
7840.2.a.bc 2 56.e even 2 1
7840.2.a.bh 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2240))\):

\( T_{3}^{2} - T_{3} - 4 \)
\( T_{11}^{2} - T_{11} - 4 \)
\( T_{13}^{2} + 3 T_{13} - 2 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 - T + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -4 - T + T^{2} \)
$13$ \( -2 + 3 T + T^{2} \)
$17$ \( 2 - 5 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( -38 + T + T^{2} \)
$31$ \( -16 + 2 T + T^{2} \)
$37$ \( -8 + 6 T + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( -8 - 11 T + T^{2} \)
$53$ \( 64 + 18 T + T^{2} \)
$59$ \( -64 + 4 T + T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( -64 - 4 T + T^{2} \)
$71$ \( ( -4 + T )^{2} \)
$73$ \( -4 - 16 T + T^{2} \)
$79$ \( -100 - 5 T + T^{2} \)
$83$ \( -64 - 4 T + T^{2} \)
$89$ \( -52 - 8 T + T^{2} \)
$97$ \( -206 + 3 T + T^{2} \)
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