Properties

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

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(671,2240)]
 
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, 1, 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("2240.671");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.h (of order \(2\), degree \(1\), 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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + q^{5} + (\beta_{3} - 2 \beta_1) q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + q^{5} + (\beta_{3} - 2 \beta_1) q^{7} + 2 q^{9} + 3 \beta_{3} q^{11} + q^{13} + \beta_1 q^{15} - \beta_{2} q^{17} + 2 \beta_1 q^{19} + (\beta_{2} + 2) q^{21} + q^{25} + 5 \beta_1 q^{27} - \beta_{2} q^{29} - 2 \beta_{3} q^{31} + 3 \beta_{2} q^{33} + (\beta_{3} - 2 \beta_1) q^{35} - 4 \beta_{2} q^{37} + \beta_1 q^{39} - 6 \beta_{2} q^{41} + 2 \beta_{3} q^{43} + 2 q^{45} - 7 \beta_{3} q^{47} + ( - 4 \beta_{2} - 1) q^{49} + \beta_{3} q^{51} - 2 \beta_{2} q^{53} + 3 \beta_{3} q^{55} - 2 q^{57} + 6 \beta_1 q^{59} + 8 q^{61} + (2 \beta_{3} - 4 \beta_1) q^{63} + q^{65} - 6 \beta_{3} q^{67} + 4 \beta_{2} q^{73} + \beta_1 q^{75} + ( - 6 \beta_{2} + 9) q^{77} - \beta_1 q^{79} + q^{81} + 12 \beta_1 q^{83} - \beta_{2} q^{85} + \beta_{3} q^{87} + 4 \beta_{2} q^{89} + (\beta_{3} - 2 \beta_1) q^{91} - 2 \beta_{2} q^{93} + 2 \beta_1 q^{95} + 5 \beta_{2} q^{97} + 6 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} + 8 q^{9} + 4 q^{13} + 8 q^{21} + 4 q^{25} + 8 q^{45} - 4 q^{49} - 8 q^{57} + 32 q^{61} + 4 q^{65} + 36 q^{77} + 4 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
671.1
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0 1.00000i 0 1.00000 0 −1.73205 + 2.00000i 0 2.00000 0
671.2 0 1.00000i 0 1.00000 0 1.73205 + 2.00000i 0 2.00000 0
671.3 0 1.00000i 0 1.00000 0 −1.73205 2.00000i 0 2.00000 0
671.4 0 1.00000i 0 1.00000 0 1.73205 2.00000i 0 2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.h.d yes 4
4.b odd 2 1 inner 2240.2.h.d yes 4
7.b odd 2 1 2240.2.h.b 4
8.b even 2 1 2240.2.h.b 4
8.d odd 2 1 2240.2.h.b 4
28.d even 2 1 2240.2.h.b 4
56.e even 2 1 inner 2240.2.h.d yes 4
56.h odd 2 1 inner 2240.2.h.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.h.b 4 7.b odd 2 1
2240.2.h.b 4 8.b even 2 1
2240.2.h.b 4 8.d odd 2 1
2240.2.h.b 4 28.d even 2 1
2240.2.h.d yes 4 1.a even 1 1 trivial
2240.2.h.d yes 4 4.b odd 2 1 inner
2240.2.h.d yes 4 56.e even 2 1 inner
2240.2.h.d yes 4 56.h odd 2 1 inner

Hecke kernels

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

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$13$ \( (T - 1)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 147)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$61$ \( (T - 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
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