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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 $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(1791,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.1791");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.116319195136.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 77x^{4} + 4 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} - \beta_{3} q^{5} + (\beta_{6} - \beta_{4}) q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} - \beta_{3} q^{5} + (\beta_{6} - \beta_{4}) q^{7} + (\beta_{2} + 1) q^{9} + (\beta_{6} + \beta_{4} + \beta_1) q^{11} + \beta_{5} q^{13} - \beta_{6} q^{15} + \beta_{5} q^{17} + (2 \beta_{7} + 2 \beta_{4} - 2 \beta_1) q^{19} + ( - \beta_{5} + 3 \beta_{3} + 1) q^{21} + (2 \beta_{6} - \beta_{4} - \beta_1) q^{23} - q^{25} + (3 \beta_{7} + \beta_{4} - \beta_1) q^{27} + \beta_{2} q^{29} - 2 \beta_{7} q^{31} + ( - \beta_{5} + 6 \beta_{3}) q^{33} + (\beta_{7} - \beta_1) q^{35} + (2 \beta_{2} + 2) q^{37} + ( - 5 \beta_{6} + \beta_{4} + \beta_1) q^{39} - 6 \beta_{3} q^{41} + (3 \beta_{4} + 3 \beta_1) q^{43} + (\beta_{5} - \beta_{3}) q^{45} + (\beta_{7} + 2 \beta_{4} - 2 \beta_1) q^{47} + ( - \beta_{5} - 3 \beta_{3} - \beta_{2} - 2) q^{49} + ( - 5 \beta_{6} + \beta_{4} + \beta_1) q^{51} + 6 q^{53} + (\beta_{7} - \beta_{4} + \beta_1) q^{55} + (2 \beta_{2} + 4) q^{57} + ( - 2 \beta_{7} + 2 \beta_{4} - 2 \beta_1) q^{59} + ( - 2 \beta_{5} - 6 \beta_{3}) q^{61} + (\beta_{7} + 5 \beta_{6} + \cdots - \beta_1) q^{63}+ \cdots + (8 \beta_{6} - 4 \beta_{4} - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{9} + 8 q^{21} - 8 q^{25} + 4 q^{29} + 24 q^{37} - 20 q^{49} + 48 q^{53} + 40 q^{57} - 4 q^{65} - 8 q^{77} + 56 q^{81} - 4 q^{85} - 72 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 43 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 79\nu^{2} ) / 18 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 79\nu^{3} ) / 18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{6} - 377\nu^{2} ) / 18 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{7} + 2\nu^{5} + 693\nu^{3} + 158\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -9\nu^{7} + 2\nu^{5} - 693\nu^{3} + 158\nu ) / 36 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + 9\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{2} - 43 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{7} + 9\beta_{6} - 79\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -79\beta_{5} - 377\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -79\beta_{7} + 79\beta_{6} - 693\beta_{4} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{4} - 9T_{3}^{2} + 2 \) Copy content Toggle raw display
\( T_{19}^{4} - 76T_{19}^{2} + 1152 \) Copy content Toggle raw display

Hecke characteristic polynomials

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