Properties

Label 2240.2.b.f
Level $2240$
Weight $2$
Character orbit 2240.b
Analytic conductor $17.886$
Analytic rank $0$
Dimension $6$
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.3534400.1
Defining polynomial: \(x^{6} - 2 x^{5} - 3 x^{4} + 16 x^{3} + x^{2} - 12 x + 40\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{5} q^{3} -\beta_{4} q^{5} + q^{7} + ( -2 - \beta_{1} - \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{5} q^{3} -\beta_{4} q^{5} + q^{7} + ( -2 - \beta_{1} - \beta_{3} ) q^{9} + ( -\beta_{2} - \beta_{4} - \beta_{5} ) q^{11} + ( -\beta_{2} - \beta_{4} + \beta_{5} ) q^{13} + \beta_{1} q^{15} + ( -\beta_{1} + 2 \beta_{3} ) q^{17} -2 \beta_{5} q^{19} -\beta_{5} q^{21} + ( -2 - 2 \beta_{1} - 2 \beta_{3} ) q^{23} - q^{25} + ( -4 \beta_{4} + \beta_{5} ) q^{27} + ( 2 \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{29} + ( -2 + 2 \beta_{3} ) q^{31} + ( -4 - \beta_{1} ) q^{33} -\beta_{4} q^{35} + ( 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{37} + ( 6 + \beta_{1} + 2 \beta_{3} ) q^{39} + ( -6 - 2 \beta_{1} ) q^{41} -2 \beta_{4} q^{43} + ( -\beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{45} + ( 3 + \beta_{1} - \beta_{3} ) q^{47} + q^{49} + ( 3 \beta_{2} - 7 \beta_{4} - \beta_{5} ) q^{51} + ( -2 \beta_{2} + 6 \beta_{4} ) q^{53} + ( -1 + \beta_{1} + \beta_{3} ) q^{55} + ( -10 - 2 \beta_{1} - 2 \beta_{3} ) q^{57} + ( 2 \beta_{2} - 6 \beta_{4} ) q^{59} + ( -2 \beta_{2} + 8 \beta_{4} - 2 \beta_{5} ) q^{61} + ( -2 - \beta_{1} - \beta_{3} ) q^{63} + ( -1 - \beta_{1} + \beta_{3} ) q^{65} + ( -6 \beta_{4} - 2 \beta_{5} ) q^{67} + ( -8 \beta_{4} + 6 \beta_{5} ) q^{69} + ( -8 - 2 \beta_{1} ) q^{71} + ( -8 - 2 \beta_{1} - 2 \beta_{3} ) q^{73} + \beta_{5} q^{75} + ( -\beta_{2} - \beta_{4} - \beta_{5} ) q^{77} + ( 2 - 5 \beta_{1} - 2 \beta_{3} ) q^{79} + ( -1 + 2 \beta_{1} - 2 \beta_{3} ) q^{81} -2 \beta_{5} q^{83} + ( 2 \beta_{2} - \beta_{5} ) q^{85} + ( 3 + 5 \beta_{1} - \beta_{3} ) q^{87} + ( -8 + 2 \beta_{1} + 2 \beta_{3} ) q^{89} + ( -\beta_{2} - \beta_{4} + \beta_{5} ) q^{91} + ( 2 \beta_{2} - 2 \beta_{4} ) q^{93} + 2 \beta_{1} q^{95} + ( 4 + \beta_{1} - 2 \beta_{3} ) q^{97} + ( -2 \beta_{2} - 8 \beta_{4} + 2 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{7} - 10q^{9} + O(q^{10}) \) \( 6q + 6q^{7} - 10q^{9} - 4q^{17} - 8q^{23} - 6q^{25} - 16q^{31} - 24q^{33} + 32q^{39} - 36q^{41} + 20q^{47} + 6q^{49} - 8q^{55} - 56q^{57} - 10q^{63} - 8q^{65} - 48q^{71} - 44q^{73} + 16q^{79} - 2q^{81} + 20q^{87} - 52q^{89} + 28q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 3 x^{4} + 16 x^{3} + x^{2} - 12 x + 40\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -11 \nu^{5} - 30 \nu^{4} + 53 \nu^{3} + 115 \nu^{2} - 236 \nu - 660 \)\()/445\)
\(\beta_{2}\)\(=\)\((\)\( -27 \nu^{5} + 250 \nu^{4} - 679 \nu^{3} + 80 \nu^{2} + 2293 \nu - 2510 \)\()/890\)
\(\beta_{3}\)\(=\)\((\)\( -17 \nu^{5} + 75 \nu^{4} + \nu^{3} - 510 \nu^{2} + 768 \nu + 315 \)\()/445\)
\(\beta_{4}\)\(=\)\((\)\( -9 \nu^{5} + 24 \nu^{4} + 11 \nu^{3} - 92 \nu^{2} - 7 \nu - 6 \)\()/178\)
\(\beta_{5}\)\(=\)\((\)\( -28 \nu^{5} + 45 \nu^{4} + 54 \nu^{3} - 395 \nu^{2} - 358 \nu + 100 \)\()/445\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{5} + \beta_{3} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{5} + 4 \beta_{4} - \beta_{3} + \beta_{1} + 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-8 \beta_{5} + 11 \beta_{4} + \beta_{3} - 3 \beta_{2} - 7\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-13 \beta_{5} + 30 \beta_{4} - 7 \beta_{3} - 2 \beta_{2} - 15 \beta_{1} - 19\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-13 \beta_{5} + 13 \beta_{4} - 8 \beta_{3} - 9 \beta_{2} - 51 \beta_{1} - 92\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1121.1
2.19082 1.44755i
0.627553 1.14620i
−1.81837 0.301352i
−1.81837 + 0.301352i
0.627553 + 1.14620i
2.19082 + 1.44755i
0 2.89511i 0 1.00000i 0 1.00000 0 −5.38164 0
1121.2 0 2.29240i 0 1.00000i 0 1.00000 0 −2.25511 0
1121.3 0 0.602705i 0 1.00000i 0 1.00000 0 2.63675 0
1121.4 0 0.602705i 0 1.00000i 0 1.00000 0 2.63675 0
1121.5 0 2.29240i 0 1.00000i 0 1.00000 0 −2.25511 0
1121.6 0 2.89511i 0 1.00000i 0 1.00000 0 −5.38164 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1121.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.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.b.f yes 6
4.b odd 2 1 2240.2.b.e 6
8.b even 2 1 inner 2240.2.b.f yes 6
8.d odd 2 1 2240.2.b.e 6
16.e even 4 1 8960.2.a.bh 3
16.e even 4 1 8960.2.a.bn 3
16.f odd 4 1 8960.2.a.bk 3
16.f odd 4 1 8960.2.a.bq 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.b.e 6 4.b odd 2 1
2240.2.b.e 6 8.d odd 2 1
2240.2.b.f yes 6 1.a even 1 1 trivial
2240.2.b.f yes 6 8.b even 2 1 inner
8960.2.a.bh 3 16.e even 4 1
8960.2.a.bk 3 16.f odd 4 1
8960.2.a.bn 3 16.e even 4 1
8960.2.a.bq 3 16.f odd 4 1

Hecke kernels

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

\( T_{3}^{6} + 14 T_{3}^{4} + 49 T_{3}^{2} + 16 \)
\( T_{23}^{3} + 4 T_{23}^{2} - 60 T_{23} - 160 \)
\( T_{31}^{3} + 8 T_{31}^{2} - 24 T_{31} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 16 + 49 T^{2} + 14 T^{4} + T^{6} \)
$5$ \( ( 1 + T^{2} )^{3} \)
$7$ \( ( -1 + T )^{6} \)
$11$ \( 100 + 201 T^{2} + 38 T^{4} + T^{6} \)
$13$ \( 3364 + 689 T^{2} + 46 T^{4} + T^{6} \)
$17$ \( ( -106 - 55 T + 2 T^{2} + T^{3} )^{2} \)
$19$ \( 1024 + 784 T^{2} + 56 T^{4} + T^{6} \)
$23$ \( ( -160 - 60 T + 4 T^{2} + T^{3} )^{2} \)
$29$ \( 35344 + 3353 T^{2} + 102 T^{4} + T^{6} \)
$31$ \( ( -32 - 24 T + 8 T^{2} + T^{3} )^{2} \)
$37$ \( 25600 + 4496 T^{2} + 168 T^{4} + T^{6} \)
$41$ \( ( 80 + 80 T + 18 T^{2} + T^{3} )^{2} \)
$43$ \( ( 4 + T^{2} )^{3} \)
$47$ \( ( 64 + 13 T - 10 T^{2} + T^{3} )^{2} \)
$53$ \( 25600 + 6720 T^{2} + 176 T^{4} + T^{6} \)
$59$ \( 25600 + 6720 T^{2} + 176 T^{4} + T^{6} \)
$61$ \( 25600 + 16256 T^{2} + 292 T^{4} + T^{6} \)
$67$ \( 256 + 5824 T^{2} + 164 T^{4} + T^{6} \)
$71$ \( ( 320 + 164 T + 24 T^{2} + T^{3} )^{2} \)
$73$ \( ( -160 + 96 T + 22 T^{2} + T^{3} )^{2} \)
$79$ \( ( 1472 - 179 T - 8 T^{2} + T^{3} )^{2} \)
$83$ \( 1024 + 784 T^{2} + 56 T^{4} + T^{6} \)
$89$ \( ( 160 + 160 T + 26 T^{2} + T^{3} )^{2} \)
$97$ \( ( 230 + 9 T - 14 T^{2} + T^{3} )^{2} \)
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