Properties

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

Related objects

Downloads

Learn more

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: 8.0.116319195136.7
Defining polynomial: \(x^{8} + 77 x^{4} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 560)
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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} -\beta_{3} q^{5} + ( -\beta_{4} + \beta_{6} ) q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{7} q^{3} -\beta_{3} q^{5} + ( -\beta_{4} + \beta_{6} ) q^{7} + ( 1 + \beta_{2} ) q^{9} + ( \beta_{1} + \beta_{4} + \beta_{6} ) q^{11} + \beta_{5} q^{13} -\beta_{6} q^{15} + \beta_{5} q^{17} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} ) q^{19} + ( 1 + 3 \beta_{3} - \beta_{5} ) q^{21} + ( -\beta_{1} - \beta_{4} + 2 \beta_{6} ) q^{23} - q^{25} + ( -\beta_{1} + \beta_{4} + 3 \beta_{7} ) q^{27} + \beta_{2} q^{29} -2 \beta_{7} q^{31} + ( 6 \beta_{3} - \beta_{5} ) q^{33} + ( -\beta_{1} + \beta_{7} ) q^{35} + ( 2 + 2 \beta_{2} ) q^{37} + ( \beta_{1} + \beta_{4} - 5 \beta_{6} ) q^{39} -6 \beta_{3} q^{41} + ( 3 \beta_{1} + 3 \beta_{4} ) q^{43} + ( -\beta_{3} + \beta_{5} ) q^{45} + ( -2 \beta_{1} + 2 \beta_{4} + \beta_{7} ) q^{47} + ( -2 - \beta_{2} - 3 \beta_{3} - \beta_{5} ) q^{49} + ( \beta_{1} + \beta_{4} - 5 \beta_{6} ) q^{51} + 6 q^{53} + ( \beta_{1} - \beta_{4} + \beta_{7} ) q^{55} + ( 4 + 2 \beta_{2} ) q^{57} + ( -2 \beta_{1} + 2 \beta_{4} - 2 \beta_{7} ) q^{59} + ( -6 \beta_{3} - 2 \beta_{5} ) q^{61} + ( -\beta_{1} + 2 \beta_{4} + 5 \beta_{6} + \beta_{7} ) q^{63} -\beta_{2} q^{65} + ( \beta_{1} + \beta_{4} + 4 \beta_{6} ) q^{67} + ( 6 \beta_{3} - 2 \beta_{5} ) q^{69} + ( 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{6} ) q^{71} + 6 \beta_{3} q^{73} -\beta_{7} q^{75} + ( -2 \beta_{2} + 6 \beta_{3} + \beta_{5} ) q^{77} + ( 3 \beta_{1} + 3 \beta_{4} - 3 \beta_{6} ) q^{79} + 7 q^{81} + ( 3 \beta_{1} - 3 \beta_{4} ) q^{83} -\beta_{2} q^{85} + ( -\beta_{1} + \beta_{4} + 5 \beta_{7} ) q^{87} + 12 \beta_{3} q^{89} + ( 3 \beta_{1} + \beta_{4} - \beta_{6} + 4 \beta_{7} ) q^{91} + ( -8 - 2 \beta_{2} ) q^{93} + ( 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{6} ) q^{95} + ( -12 \beta_{3} + \beta_{5} ) q^{97} + ( -4 \beta_{1} - 4 \beta_{4} + 8 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 12q^{9} + O(q^{10}) \) \( 8q + 12q^{9} + 8q^{21} - 8q^{25} + 4q^{29} + 24q^{37} - 20q^{49} + 48q^{53} + 40q^{57} - 4q^{65} - 8q^{77} + 56q^{81} - 4q^{85} - 72q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 77 x^{4} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} + 43 \)\()/9\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 79 \nu^{2} \)\()/18\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 79 \nu^{3} \)\()/18\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{6} - 377 \nu^{2} \)\()/18\)
\(\beta_{6}\)\(=\)\((\)\( 9 \nu^{7} + 2 \nu^{5} + 693 \nu^{3} + 158 \nu \)\()/36\)
\(\beta_{7}\)\(=\)\((\)\( -9 \nu^{7} + 2 \nu^{5} - 693 \nu^{3} + 158 \nu \)\()/36\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + 5 \beta_{3}\)
\(\nu^{3}\)\(=\)\(\beta_{7} - \beta_{6} + 9 \beta_{4}\)
\(\nu^{4}\)\(=\)\(9 \beta_{2} - 43\)
\(\nu^{5}\)\(=\)\(9 \beta_{7} + 9 \beta_{6} - 79 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-79 \beta_{5} - 377 \beta_{3}\)
\(\nu^{7}\)\(=\)\(-79 \beta_{7} + 79 \beta_{6} - 693 \beta_{4}\)

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
−0.337637 0.337637i
−0.337637 + 0.337637i
−2.09428 2.09428i
−2.09428 + 2.09428i
2.09428 + 2.09428i
2.09428 2.09428i
0.337637 + 0.337637i
0.337637 0.337637i
0 −2.96176 0 1.00000i 0 −0.337637 2.62412i 0 5.77200 0
1791.2 0 −2.96176 0 1.00000i 0 −0.337637 + 2.62412i 0 5.77200 0
1791.3 0 −0.477491 0 1.00000i 0 −2.09428 + 1.61679i 0 −2.77200 0
1791.4 0 −0.477491 0 1.00000i 0 −2.09428 1.61679i 0 −2.77200 0
1791.5 0 0.477491 0 1.00000i 0 2.09428 1.61679i 0 −2.77200 0
1791.6 0 0.477491 0 1.00000i 0 2.09428 + 1.61679i 0 −2.77200 0
1791.7 0 2.96176 0 1.00000i 0 0.337637 + 2.62412i 0 5.77200 0
1791.8 0 2.96176 0 1.00000i 0 0.337637 2.62412i 0 5.77200 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.d 8
4.b odd 2 1 inner 2240.2.k.d 8
7.b odd 2 1 inner 2240.2.k.d 8
8.b even 2 1 560.2.k.b 8
8.d odd 2 1 560.2.k.b 8
24.f even 2 1 5040.2.d.d 8
24.h odd 2 1 5040.2.d.d 8
28.d even 2 1 inner 2240.2.k.d 8
40.e odd 2 1 2800.2.k.m 8
40.f even 2 1 2800.2.k.m 8
40.i odd 4 1 2800.2.e.g 8
40.i odd 4 1 2800.2.e.h 8
40.k even 4 1 2800.2.e.g 8
40.k even 4 1 2800.2.e.h 8
56.e even 2 1 560.2.k.b 8
56.h odd 2 1 560.2.k.b 8
168.e odd 2 1 5040.2.d.d 8
168.i even 2 1 5040.2.d.d 8
280.c odd 2 1 2800.2.k.m 8
280.n even 2 1 2800.2.k.m 8
280.s even 4 1 2800.2.e.g 8
280.s even 4 1 2800.2.e.h 8
280.y odd 4 1 2800.2.e.g 8
280.y odd 4 1 2800.2.e.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.k.b 8 8.b even 2 1
560.2.k.b 8 8.d odd 2 1
560.2.k.b 8 56.e even 2 1
560.2.k.b 8 56.h odd 2 1
2240.2.k.d 8 1.a even 1 1 trivial
2240.2.k.d 8 4.b odd 2 1 inner
2240.2.k.d 8 7.b odd 2 1 inner
2240.2.k.d 8 28.d even 2 1 inner
2800.2.e.g 8 40.i odd 4 1
2800.2.e.g 8 40.k even 4 1
2800.2.e.g 8 280.s even 4 1
2800.2.e.g 8 280.y odd 4 1
2800.2.e.h 8 40.i odd 4 1
2800.2.e.h 8 40.k even 4 1
2800.2.e.h 8 280.s even 4 1
2800.2.e.h 8 280.y odd 4 1
2800.2.k.m 8 40.e odd 2 1
2800.2.k.m 8 40.f even 2 1
2800.2.k.m 8 280.c odd 2 1
2800.2.k.m 8 280.n even 2 1
5040.2.d.d 8 24.f even 2 1
5040.2.d.d 8 24.h odd 2 1
5040.2.d.d 8 168.e odd 2 1
5040.2.d.d 8 168.i even 2 1

Hecke kernels

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

\( T_{3}^{4} - 9 T_{3}^{2} + 2 \)
\( T_{19}^{4} - 76 T_{19}^{2} + 1152 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 2 - 9 T^{2} + T^{4} )^{2} \)
$5$ \( ( 1 + T^{2} )^{4} \)
$7$ \( 2401 + 490 T^{2} + 50 T^{4} + 10 T^{6} + T^{8} \)
$11$ \( ( 288 + 35 T^{2} + T^{4} )^{2} \)
$13$ \( ( 324 + 37 T^{2} + T^{4} )^{2} \)
$17$ \( ( 324 + 37 T^{2} + T^{4} )^{2} \)
$19$ \( ( 1152 - 76 T^{2} + T^{4} )^{2} \)
$23$ \( ( 288 + 38 T^{2} + T^{4} )^{2} \)
$29$ \( ( -18 - T + T^{2} )^{4} \)
$31$ \( ( 32 - 36 T^{2} + T^{4} )^{2} \)
$37$ \( ( -64 - 6 T + T^{2} )^{4} \)
$41$ \( ( 36 + T^{2} )^{4} \)
$43$ \( ( 648 + 162 T^{2} + T^{4} )^{2} \)
$47$ \( ( 162 - 65 T^{2} + T^{4} )^{2} \)
$53$ \( ( -6 + T )^{8} \)
$59$ \( ( 4608 - 140 T^{2} + T^{4} )^{2} \)
$61$ \( ( 2304 + 196 T^{2} + T^{4} )^{2} \)
$67$ \( ( 5832 + 194 T^{2} + T^{4} )^{2} \)
$71$ \( ( 4608 + 140 T^{2} + T^{4} )^{2} \)
$73$ \( ( 36 + T^{2} )^{4} \)
$79$ \( ( 5832 + 171 T^{2} + T^{4} )^{2} \)
$83$ \( ( 648 - 162 T^{2} + T^{4} )^{2} \)
$89$ \( ( 144 + T^{2} )^{4} \)
$97$ \( ( 19044 + 349 T^{2} + T^{4} )^{2} \)
show more
show less