Properties

Label 3240.2.f.d
Level $3240$
Weight $2$
Character orbit 3240.f
Analytic conductor $25.872$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3240,2,Mod(649,3240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3240.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3240.f (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: \(25.8715302549\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 1) q^{5} + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 1) q^{5} + \beta q^{7} - 5 q^{11} - 2 \beta q^{13} + 3 \beta q^{17} + 7 q^{19} + 2 \beta q^{23} + ( - 2 \beta - 3) q^{25} + 5 q^{29} - 3 q^{31} + (\beta + 4) q^{35} - \beta q^{37} + 7 q^{41} - 3 \beta q^{43} + 3 \beta q^{47} + 3 q^{49} - 5 \beta q^{53} + (5 \beta - 5) q^{55} + 15 q^{59} + 14 q^{61} + ( - 2 \beta - 8) q^{65} - 2 \beta q^{67} - 5 q^{71} - 7 \beta q^{73} - 5 \beta q^{77} + 8 q^{79} + 7 \beta q^{83} + (3 \beta + 12) q^{85} - 3 q^{89} + 8 q^{91} + ( - 7 \beta + 7) q^{95} - \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 10 q^{11} + 14 q^{19} - 6 q^{25} + 10 q^{29} - 6 q^{31} + 8 q^{35} + 14 q^{41} + 6 q^{49} - 10 q^{55} + 30 q^{59} + 28 q^{61} - 16 q^{65} - 10 q^{71} + 16 q^{79} + 24 q^{85} - 6 q^{89} + 16 q^{91} + 14 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(3161\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.00000i
1.00000i
0 0 0 1.00000 2.00000i 0 2.00000i 0 0 0
649.2 0 0 0 1.00000 + 2.00000i 0 2.00000i 0 0 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3240.2.f.d yes 2
3.b odd 2 1 3240.2.f.c 2
5.b even 2 1 inner 3240.2.f.d yes 2
15.d odd 2 1 3240.2.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3240.2.f.c 2 3.b odd 2 1
3240.2.f.c 2 15.d odd 2 1
3240.2.f.d yes 2 1.a even 1 1 trivial
3240.2.f.d yes 2 5.b 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 \(S_{2}^{\mathrm{new}}(3240, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T + 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T - 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T - 5)^{2} \) Copy content Toggle raw display
$31$ \( (T + 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 7)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 100 \) Copy content Toggle raw display
$59$ \( (T - 15)^{2} \) Copy content Toggle raw display
$61$ \( (T - 14)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T + 5)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 196 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 196 \) Copy content Toggle raw display
$89$ \( (T + 3)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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