Properties

Label 2240.2.k.e
Level $2240$
Weight $2$
Character orbit 2240.k
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.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.342102016.5
Defining polynomial: \(x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 140)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{2} - \beta_{5} ) q^{3} + \beta_{1} q^{5} + ( -\beta_{2} - \beta_{3} - \beta_{5} ) q^{7} + ( 3 - \beta_{1} + \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{2} - \beta_{5} ) q^{3} + \beta_{1} q^{5} + ( -\beta_{2} - \beta_{3} - \beta_{5} ) q^{7} + ( 3 - \beta_{1} + \beta_{6} - \beta_{7} ) q^{9} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{11} + 2 \beta_{1} q^{13} + ( -\beta_{3} - \beta_{4} ) q^{15} + ( 3 \beta_{1} - \beta_{6} - \beta_{7} ) q^{17} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{19} + ( 5 + 2 \beta_{1} + \beta_{6} ) q^{21} + ( -2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{23} - q^{25} + ( -2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{27} + 2 q^{29} + ( 2 \beta_{3} - 2 \beta_{4} ) q^{31} + ( 6 \beta_{1} + 2 \beta_{6} + 2 \beta_{7} ) q^{33} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{35} -2 q^{37} + ( -2 \beta_{3} - 2 \beta_{4} ) q^{39} + ( -3 \beta_{1} + \beta_{6} + \beta_{7} ) q^{41} + ( -2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{43} + ( 2 \beta_{1} + \beta_{6} + \beta_{7} ) q^{45} + ( -3 \beta_{3} + 3 \beta_{4} ) q^{47} + ( 4 + 5 \beta_{1} + \beta_{7} ) q^{49} + ( -2 \beta_{2} + 2 \beta_{5} ) q^{51} -2 q^{53} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{55} + ( -8 + 2 \beta_{1} - 2 \beta_{6} + 2 \beta_{7} ) q^{57} + ( -\beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{59} + 2 \beta_{1} q^{61} + ( -2 \beta_{2} - 5 \beta_{4} - 4 \beta_{5} ) q^{63} -2 q^{65} + ( -4 \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{5} ) q^{67} + ( -7 \beta_{1} - 3 \beta_{6} - 3 \beta_{7} ) q^{69} + ( -\beta_{2} - 3 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{71} + ( -3 \beta_{1} - 3 \beta_{6} - 3 \beta_{7} ) q^{73} + ( \beta_{2} + \beta_{5} ) q^{75} + ( -4 + 3 \beta_{1} + \beta_{6} + 3 \beta_{7} ) q^{77} + ( \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{79} + ( 7 - \beta_{1} + \beta_{6} - \beta_{7} ) q^{81} + ( -\beta_{2} - \beta_{5} ) q^{83} + ( -2 - \beta_{1} + \beta_{6} - \beta_{7} ) q^{85} + ( -2 \beta_{2} - 2 \beta_{5} ) q^{87} -12 \beta_{1} q^{89} + ( -2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{91} + ( 4 - 2 \beta_{1} + 2 \beta_{6} - 2 \beta_{7} ) q^{93} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{95} + ( -3 \beta_{1} + \beta_{6} + \beta_{7} ) q^{97} + ( \beta_{2} - 9 \beta_{3} - 9 \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 16q^{9} + O(q^{10}) \) \( 8q + 16q^{9} + 36q^{21} - 8q^{25} + 16q^{29} - 16q^{37} + 36q^{49} - 16q^{53} - 48q^{57} - 16q^{65} - 24q^{77} + 48q^{81} - 24q^{85} + 16q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} + \nu^{5} + 2 \nu^{3} \)\()/16\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} - 3 \nu^{5} - 5 \nu^{4} - 2 \nu^{3} + 2 \nu^{2} - 8 \)\()/16\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + \nu^{6} - \nu^{5} + 3 \nu^{4} + 6 \nu^{3} + 10 \nu^{2} - 8 \nu + 8 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} + \nu^{5} + 3 \nu^{4} - 6 \nu^{3} + 10 \nu^{2} + 8 \nu + 8 \)\()/16\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} - \nu^{6} - 3 \nu^{5} + 5 \nu^{4} - 2 \nu^{3} - 2 \nu^{2} + 8 \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{6} + 3 \nu^{5} - 4 \nu^{4} + 10 \nu^{3} + 24 \nu - 16 \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{6} + \nu^{5} + 2 \nu^{4} + 4 \nu^{3} + 12 \nu + 8 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} + \beta_{6} - \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} + 7 \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{7} - \beta_{6} + 5 \beta_{5} + \beta_{4} + \beta_{3} - 5 \beta_{2} + \beta_{1} - 8\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{7} - \beta_{6} - 9 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} - 9 \beta_{2} + 9 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(7 \beta_{7} - 7 \beta_{6} - 5 \beta_{5} - \beta_{4} - \beta_{3} + 5 \beta_{2} + 7 \beta_{1} - 8\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(\beta_{7} + \beta_{6} - 7 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 7 \beta_{2} - 41 \beta_{1}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1791.1
1.17915 0.780776i
1.17915 + 0.780776i
−0.599676 + 1.28078i
−0.599676 1.28078i
0.599676 + 1.28078i
0.599676 1.28078i
−1.17915 0.780776i
−1.17915 + 0.780776i
0 −3.02045 0 1.00000i 0 −2.17238 + 1.51022i 0 6.12311 0
1791.2 0 −3.02045 0 1.00000i 0 −2.17238 1.51022i 0 6.12311 0
1791.3 0 −0.936426 0 1.00000i 0 −2.60399 + 0.468213i 0 −2.12311 0
1791.4 0 −0.936426 0 1.00000i 0 −2.60399 0.468213i 0 −2.12311 0
1791.5 0 0.936426 0 1.00000i 0 2.60399 0.468213i 0 −2.12311 0
1791.6 0 0.936426 0 1.00000i 0 2.60399 + 0.468213i 0 −2.12311 0
1791.7 0 3.02045 0 1.00000i 0 2.17238 1.51022i 0 6.12311 0
1791.8 0 3.02045 0 1.00000i 0 2.17238 + 1.51022i 0 6.12311 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1791.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d 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.k.e 8
4.b odd 2 1 inner 2240.2.k.e 8
7.b odd 2 1 inner 2240.2.k.e 8
8.b even 2 1 140.2.g.c 8
8.d odd 2 1 140.2.g.c 8
24.f even 2 1 1260.2.c.c 8
24.h odd 2 1 1260.2.c.c 8
28.d even 2 1 inner 2240.2.k.e 8
40.e odd 2 1 700.2.g.j 8
40.f even 2 1 700.2.g.j 8
40.i odd 4 1 700.2.c.i 8
40.i odd 4 1 700.2.c.j 8
40.k even 4 1 700.2.c.i 8
40.k even 4 1 700.2.c.j 8
56.e even 2 1 140.2.g.c 8
56.h odd 2 1 140.2.g.c 8
56.j odd 6 2 980.2.o.e 16
56.k odd 6 2 980.2.o.e 16
56.m even 6 2 980.2.o.e 16
56.p even 6 2 980.2.o.e 16
168.e odd 2 1 1260.2.c.c 8
168.i even 2 1 1260.2.c.c 8
280.c odd 2 1 700.2.g.j 8
280.n even 2 1 700.2.g.j 8
280.s even 4 1 700.2.c.i 8
280.s even 4 1 700.2.c.j 8
280.y odd 4 1 700.2.c.i 8
280.y odd 4 1 700.2.c.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.c 8 8.b even 2 1
140.2.g.c 8 8.d odd 2 1
140.2.g.c 8 56.e even 2 1
140.2.g.c 8 56.h odd 2 1
700.2.c.i 8 40.i odd 4 1
700.2.c.i 8 40.k even 4 1
700.2.c.i 8 280.s even 4 1
700.2.c.i 8 280.y odd 4 1
700.2.c.j 8 40.i odd 4 1
700.2.c.j 8 40.k even 4 1
700.2.c.j 8 280.s even 4 1
700.2.c.j 8 280.y odd 4 1
700.2.g.j 8 40.e odd 2 1
700.2.g.j 8 40.f even 2 1
700.2.g.j 8 280.c odd 2 1
700.2.g.j 8 280.n even 2 1
980.2.o.e 16 56.j odd 6 2
980.2.o.e 16 56.k odd 6 2
980.2.o.e 16 56.m even 6 2
980.2.o.e 16 56.p even 6 2
1260.2.c.c 8 24.f even 2 1
1260.2.c.c 8 24.h odd 2 1
1260.2.c.c 8 168.e odd 2 1
1260.2.c.c 8 168.i even 2 1
2240.2.k.e 8 1.a even 1 1 trivial
2240.2.k.e 8 4.b odd 2 1 inner
2240.2.k.e 8 7.b odd 2 1 inner
2240.2.k.e 8 28.d 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 \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{4} - 10 T_{3}^{2} + 8 \)
\( T_{19}^{4} - 28 T_{19}^{2} + 128 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 8 - 10 T^{2} + T^{4} )^{2} \)
$5$ \( ( 1 + T^{2} )^{4} \)
$7$ \( 2401 - 882 T^{2} + 162 T^{4} - 18 T^{6} + T^{8} \)
$11$ \( ( 128 + 28 T^{2} + T^{4} )^{2} \)
$13$ \( ( 4 + T^{2} )^{4} \)
$17$ \( ( 64 + 52 T^{2} + T^{4} )^{2} \)
$19$ \( ( 128 - 28 T^{2} + T^{4} )^{2} \)
$23$ \( ( 1352 + 74 T^{2} + T^{4} )^{2} \)
$29$ \( ( -2 + T )^{8} \)
$31$ \( ( 512 - 56 T^{2} + T^{4} )^{2} \)
$37$ \( ( 2 + T )^{8} \)
$41$ \( ( 64 + 52 T^{2} + T^{4} )^{2} \)
$43$ \( ( 8 + 58 T^{2} + T^{4} )^{2} \)
$47$ \( ( 2592 - 126 T^{2} + T^{4} )^{2} \)
$53$ \( ( 2 + T )^{8} \)
$59$ \( ( 512 - 124 T^{2} + T^{4} )^{2} \)
$61$ \( ( 4 + T^{2} )^{4} \)
$67$ \( ( 2888 + 218 T^{2} + T^{4} )^{2} \)
$71$ \( ( 2048 + 92 T^{2} + T^{4} )^{2} \)
$73$ \( ( 20736 + 324 T^{2} + T^{4} )^{2} \)
$79$ \( ( 32 + 20 T^{2} + T^{4} )^{2} \)
$83$ \( ( 8 - 10 T^{2} + T^{4} )^{2} \)
$89$ \( ( 144 + T^{2} )^{4} \)
$97$ \( ( 64 + 52 T^{2} + T^{4} )^{2} \)
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