Properties

Label 2240.2.n.h
Level $2240$
Weight $2$
Character orbit 2240.n
Analytic conductor $17.886$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.n (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1731891456.1
Defining polynomial: \(x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 2 + \beta_{5} ) q^{5} + ( 2 + \beta_{2} ) q^{7} + ( 1 + \beta_{1} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 2 + \beta_{5} ) q^{5} + ( 2 + \beta_{2} ) q^{7} + ( 1 + \beta_{1} ) q^{9} + ( 2 \beta_{3} - \beta_{7} ) q^{11} + ( \beta_{4} + \beta_{5} ) q^{13} + ( 2 \beta_{1} - \beta_{4} + \beta_{5} ) q^{15} -\beta_{7} q^{17} + ( -2 \beta_{4} - 2 \beta_{5} ) q^{19} + ( 2 \beta_{1} - \beta_{6} ) q^{21} + ( 4 - 2 \beta_{1} ) q^{23} + ( 3 + 4 \beta_{5} ) q^{25} + ( 4 - \beta_{1} ) q^{27} -\beta_{6} q^{29} + 2 \beta_{7} q^{31} + ( -4 \beta_{3} + \beta_{7} ) q^{33} + ( 4 + 2 \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{35} + 2 \beta_{7} q^{37} + ( -\beta_{4} - 3 \beta_{5} ) q^{39} + 2 \beta_{6} q^{41} + ( -4 \beta_{2} - 2 \beta_{6} ) q^{43} + ( 2 + 2 \beta_{1} - \beta_{4} + 2 \beta_{5} ) q^{45} + ( 2 \beta_{2} + \beta_{6} ) q^{47} + ( 1 + 4 \beta_{2} ) q^{49} + ( -4 \beta_{3} - \beta_{7} ) q^{51} + ( 2 \beta_{2} + 4 \beta_{3} + \beta_{6} - 2 \beta_{7} ) q^{55} + ( 2 \beta_{4} + 6 \beta_{5} ) q^{57} -4 \beta_{5} q^{59} + ( 4 - 2 \beta_{1} ) q^{61} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{6} ) q^{63} + ( -2 + \beta_{1} + 2 \beta_{4} + 2 \beta_{5} ) q^{65} -4 \beta_{2} q^{67} + ( -8 + 2 \beta_{1} ) q^{69} + ( 4 \beta_{4} - 8 \beta_{5} ) q^{71} -4 \beta_{3} q^{73} + ( 3 \beta_{1} - 4 \beta_{4} + 4 \beta_{5} ) q^{75} + ( 4 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} ) q^{77} + ( -3 \beta_{4} + 3 \beta_{5} ) q^{79} -7 q^{81} -4 q^{83} + ( \beta_{6} - 2 \beta_{7} ) q^{85} + ( 4 \beta_{2} - \beta_{6} ) q^{87} -2 \beta_{6} q^{89} + ( -2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{91} + ( 8 \beta_{3} + 2 \beta_{7} ) q^{93} + ( 4 - 2 \beta_{1} - 4 \beta_{4} - 4 \beta_{5} ) q^{95} + ( -8 \beta_{3} + \beta_{7} ) q^{97} -2 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{3} + 16q^{5} + 16q^{7} + 12q^{9} + O(q^{10}) \) \( 8q + 4q^{3} + 16q^{5} + 16q^{7} + 12q^{9} + 8q^{15} + 8q^{21} + 24q^{23} + 24q^{25} + 28q^{27} + 32q^{35} + 24q^{45} + 8q^{49} + 24q^{61} + 24q^{63} - 12q^{65} - 56q^{69} + 12q^{75} - 56q^{81} - 32q^{83} + 24q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 181 \)\()/65\)
\(\beta_{2}\)\(=\)\((\)\( 9 \nu^{6} - 65 \nu^{4} + 585 \nu^{2} - 776 \)\()/520\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 25 \nu^{5} - 145 \nu^{3} + 544 \nu \)\()/320\)
\(\beta_{4}\)\(=\)\((\)\( -9 \nu^{7} + 65 \nu^{5} - 585 \nu^{3} + 256 \nu \)\()/1040\)
\(\beta_{5}\)\(=\)\((\)\( 9 \nu^{7} - 65 \nu^{5} + 377 \nu^{3} - 256 \nu \)\()/832\)
\(\beta_{6}\)\(=\)\((\)\( -37 \nu^{6} + 325 \nu^{4} - 1885 \nu^{2} + 2728 \)\()/520\)
\(\beta_{7}\)\(=\)\((\)\( -49 \nu^{7} + 585 \nu^{5} - 4225 \nu^{3} + 16416 \nu \)\()/4160\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{4} - \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + 5 \beta_{2} - \beta_{1} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\(-4 \beta_{5} - 5 \beta_{4}\)
\(\nu^{4}\)\(=\)\((\)\(9 \beta_{6} + 29 \beta_{2} + 9 \beta_{1} - 29\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-29 \beta_{7} - 36 \beta_{5} - 29 \beta_{4} + 65 \beta_{3}\)\()/2\)
\(\nu^{6}\)\(=\)\(65 \beta_{1} - 181\)
\(\nu^{7}\)\(=\)\((\)\(-181 \beta_{7} + 260 \beta_{5} + 181 \beta_{4} + 441 \beta_{3}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1119.1
−2.21837 + 1.28078i
2.21837 + 1.28078i
2.21837 1.28078i
−2.21837 1.28078i
1.35234 0.780776i
−1.35234 0.780776i
−1.35234 + 0.780776i
1.35234 + 0.780776i
0 −1.56155 0 2.00000 1.00000i 0 2.00000 1.73205i 0 −0.561553 0
1119.2 0 −1.56155 0 2.00000 1.00000i 0 2.00000 + 1.73205i 0 −0.561553 0
1119.3 0 −1.56155 0 2.00000 + 1.00000i 0 2.00000 1.73205i 0 −0.561553 0
1119.4 0 −1.56155 0 2.00000 + 1.00000i 0 2.00000 + 1.73205i 0 −0.561553 0
1119.5 0 2.56155 0 2.00000 1.00000i 0 2.00000 1.73205i 0 3.56155 0
1119.6 0 2.56155 0 2.00000 1.00000i 0 2.00000 + 1.73205i 0 3.56155 0
1119.7 0 2.56155 0 2.00000 + 1.00000i 0 2.00000 1.73205i 0 3.56155 0
1119.8 0 2.56155 0 2.00000 + 1.00000i 0 2.00000 + 1.73205i 0 3.56155 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1119.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 inner
56.h odd 2 1 inner
280.n 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.n.h yes 8
4.b odd 2 1 2240.2.n.b yes 8
5.b even 2 1 2240.2.n.b yes 8
7.b odd 2 1 2240.2.n.a 8
8.b even 2 1 2240.2.n.a 8
8.d odd 2 1 2240.2.n.g yes 8
20.d odd 2 1 inner 2240.2.n.h yes 8
28.d even 2 1 2240.2.n.g yes 8
35.c odd 2 1 2240.2.n.g yes 8
40.e odd 2 1 2240.2.n.a 8
40.f even 2 1 2240.2.n.g yes 8
56.e even 2 1 2240.2.n.b yes 8
56.h odd 2 1 inner 2240.2.n.h yes 8
140.c even 2 1 2240.2.n.a 8
280.c odd 2 1 2240.2.n.b yes 8
280.n even 2 1 inner 2240.2.n.h yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.n.a 8 7.b odd 2 1
2240.2.n.a 8 8.b even 2 1
2240.2.n.a 8 40.e odd 2 1
2240.2.n.a 8 140.c even 2 1
2240.2.n.b yes 8 4.b odd 2 1
2240.2.n.b yes 8 5.b even 2 1
2240.2.n.b yes 8 56.e even 2 1
2240.2.n.b yes 8 280.c odd 2 1
2240.2.n.g yes 8 8.d odd 2 1
2240.2.n.g yes 8 28.d even 2 1
2240.2.n.g yes 8 35.c odd 2 1
2240.2.n.g yes 8 40.f even 2 1
2240.2.n.h yes 8 1.a even 1 1 trivial
2240.2.n.h yes 8 20.d odd 2 1 inner
2240.2.n.h yes 8 56.h odd 2 1 inner
2240.2.n.h yes 8 280.n even 2 1 inner

Hecke kernels

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

\( T_{3}^{2} - T_{3} - 4 \)
\( T_{11}^{4} - 39 T_{11}^{2} + 36 \)
\( T_{17}^{4} - 27 T_{17}^{2} + 144 \)
\( T_{23}^{2} - 6 T_{23} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( -4 - T + T^{2} )^{4} \)
$5$ \( ( 5 - 4 T + T^{2} )^{4} \)
$7$ \( ( 7 - 4 T + T^{2} )^{4} \)
$11$ \( ( 36 - 39 T^{2} + T^{4} )^{2} \)
$13$ \( ( 4 + 13 T^{2} + T^{4} )^{2} \)
$17$ \( ( 144 - 27 T^{2} + T^{4} )^{2} \)
$19$ \( ( 64 + 52 T^{2} + T^{4} )^{2} \)
$23$ \( ( -8 - 6 T + T^{2} )^{4} \)
$29$ \( ( 144 + 27 T^{2} + T^{4} )^{2} \)
$31$ \( ( 2304 - 108 T^{2} + T^{4} )^{2} \)
$37$ \( ( 2304 - 108 T^{2} + T^{4} )^{2} \)
$41$ \( ( 2304 + 108 T^{2} + T^{4} )^{2} \)
$43$ \( ( 576 + 156 T^{2} + T^{4} )^{2} \)
$47$ \( ( 36 + 39 T^{2} + T^{4} )^{2} \)
$53$ \( T^{8} \)
$59$ \( ( 16 + T^{2} )^{4} \)
$61$ \( ( -8 - 6 T + T^{2} )^{4} \)
$67$ \( ( 48 + T^{2} )^{4} \)
$71$ \( ( 1024 + 208 T^{2} + T^{4} )^{2} \)
$73$ \( ( -48 + T^{2} )^{4} \)
$79$ \( ( 1296 + 81 T^{2} + T^{4} )^{2} \)
$83$ \( ( 4 + T )^{8} \)
$89$ \( ( 2304 + 108 T^{2} + T^{4} )^{2} \)
$97$ \( ( 24336 - 363 T^{2} + T^{4} )^{2} \)
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