Properties

Label 2240.2.a.bd
Level $2240$
Weight $2$
Character orbit 2240.a
Self dual yes
Analytic conductor $17.886$
Analytic rank $1$
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: \(1\)
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 35)
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.

\(f(q)\) \(=\) \( q -\beta q^{3} - q^{5} + q^{7} + ( 1 + \beta ) q^{9} +O(q^{10})\) \( q -\beta q^{3} - q^{5} + q^{7} + ( 1 + \beta ) q^{9} + \beta q^{11} + ( -2 - \beta ) q^{13} + \beta q^{15} + ( -2 - \beta ) q^{17} + ( -4 + 2 \beta ) q^{19} -\beta q^{21} + 2 \beta q^{23} + q^{25} + ( -4 + \beta ) q^{27} + ( -2 + 3 \beta ) q^{29} + ( -4 - \beta ) q^{33} - q^{35} -6 q^{37} + ( 4 + 3 \beta ) q^{39} + ( 2 - 2 \beta ) q^{41} + ( 4 + 2 \beta ) q^{43} + ( -1 - \beta ) q^{45} + ( 4 - 3 \beta ) q^{47} + q^{49} + ( 4 + 3 \beta ) q^{51} + ( 2 - 2 \beta ) q^{53} -\beta q^{55} + ( -8 + 2 \beta ) q^{57} -4 q^{59} + ( -6 + 6 \beta ) q^{61} + ( 1 + \beta ) q^{63} + ( 2 + \beta ) q^{65} + ( 4 - 4 \beta ) q^{67} + ( -8 - 2 \beta ) q^{69} -8 q^{71} + ( -6 + 4 \beta ) q^{73} -\beta q^{75} + \beta q^{77} + ( 4 + \beta ) q^{79} -7 q^{81} + 4 q^{83} + ( 2 + \beta ) q^{85} + ( -12 - \beta ) q^{87} + ( 2 + 2 \beta ) q^{89} + ( -2 - \beta ) q^{91} + ( 4 - 2 \beta ) q^{95} + ( -2 - 5 \beta ) q^{97} + ( 4 + 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - 2q^{5} + 2q^{7} + 3q^{9} + O(q^{10}) \) \( 2q - q^{3} - 2q^{5} + 2q^{7} + 3q^{9} + q^{11} - 5q^{13} + q^{15} - 5q^{17} - 6q^{19} - q^{21} + 2q^{23} + 2q^{25} - 7q^{27} - q^{29} - 9q^{33} - 2q^{35} - 12q^{37} + 11q^{39} + 2q^{41} + 10q^{43} - 3q^{45} + 5q^{47} + 2q^{49} + 11q^{51} + 2q^{53} - q^{55} - 14q^{57} - 8q^{59} - 6q^{61} + 3q^{63} + 5q^{65} + 4q^{67} - 18q^{69} - 16q^{71} - 8q^{73} - q^{75} + q^{77} + 9q^{79} - 14q^{81} + 8q^{83} + 5q^{85} - 25q^{87} + 6q^{89} - 5q^{91} + 6q^{95} - 9q^{97} + 10q^{99} + O(q^{100}) \)

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
2.56155
−1.56155
0 −2.56155 0 −1.00000 0 1.00000 0 3.56155 0
1.2 0 1.56155 0 −1.00000 0 1.00000 0 −0.561553 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.bd 2
4.b odd 2 1 2240.2.a.bh 2
8.b even 2 1 560.2.a.i 2
8.d odd 2 1 35.2.a.b 2
24.f even 2 1 315.2.a.e 2
24.h odd 2 1 5040.2.a.bt 2
40.e odd 2 1 175.2.a.f 2
40.f even 2 1 2800.2.a.bi 2
40.i odd 4 2 2800.2.g.t 4
40.k even 4 2 175.2.b.b 4
56.e even 2 1 245.2.a.d 2
56.h odd 2 1 3920.2.a.bs 2
56.k odd 6 2 245.2.e.i 4
56.m even 6 2 245.2.e.h 4
88.g even 2 1 4235.2.a.m 2
104.h odd 2 1 5915.2.a.l 2
120.m even 2 1 1575.2.a.p 2
120.q odd 4 2 1575.2.d.e 4
168.e odd 2 1 2205.2.a.x 2
280.n even 2 1 1225.2.a.s 2
280.y odd 4 2 1225.2.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 8.d odd 2 1
175.2.a.f 2 40.e odd 2 1
175.2.b.b 4 40.k even 4 2
245.2.a.d 2 56.e even 2 1
245.2.e.h 4 56.m even 6 2
245.2.e.i 4 56.k odd 6 2
315.2.a.e 2 24.f even 2 1
560.2.a.i 2 8.b even 2 1
1225.2.a.s 2 280.n even 2 1
1225.2.b.f 4 280.y odd 4 2
1575.2.a.p 2 120.m even 2 1
1575.2.d.e 4 120.q odd 4 2
2205.2.a.x 2 168.e odd 2 1
2240.2.a.bd 2 1.a even 1 1 trivial
2240.2.a.bh 2 4.b odd 2 1
2800.2.a.bi 2 40.f even 2 1
2800.2.g.t 4 40.i odd 4 2
3920.2.a.bs 2 56.h odd 2 1
4235.2.a.m 2 88.g even 2 1
5040.2.a.bt 2 24.h odd 2 1
5915.2.a.l 2 104.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2240))\):

\( T_{3}^{2} + T_{3} - 4 \)
\( T_{11}^{2} - T_{11} - 4 \)
\( T_{13}^{2} + 5 T_{13} + 2 \)
\( T_{19}^{2} + 6 T_{19} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 + T + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -4 - T + T^{2} \)
$13$ \( 2 + 5 T + T^{2} \)
$17$ \( 2 + 5 T + T^{2} \)
$19$ \( -8 + 6 T + T^{2} \)
$23$ \( -16 - 2 T + T^{2} \)
$29$ \( -38 + T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 6 + T )^{2} \)
$41$ \( -16 - 2 T + T^{2} \)
$43$ \( 8 - 10 T + T^{2} \)
$47$ \( -32 - 5 T + T^{2} \)
$53$ \( -16 - 2 T + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( -144 + 6 T + T^{2} \)
$67$ \( -64 - 4 T + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( -52 + 8 T + T^{2} \)
$79$ \( 16 - 9 T + T^{2} \)
$83$ \( ( -4 + T )^{2} \)
$89$ \( -8 - 6 T + T^{2} \)
$97$ \( -86 + 9 T + T^{2} \)
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