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Properties

Label 2340.2.bp.c
Level $2340$
Weight $2$
Character orbit 2340.bp
Analytic conductor $18.685$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2340,2,Mod(1477,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.1477"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2340.bp (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,4,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.6849940730\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content 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: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (i + 2) q^{5} - 4 q^{7} + (2 i - 2) q^{11} + (2 i - 3) q^{13} + ( - 3 i + 3) q^{17} + ( - 2 i + 2) q^{19} + (2 i + 2) q^{23} + (4 i + 3) q^{25} + ( - 4 i - 8) q^{35} - 4 q^{37} + ( - 7 i - 7) q^{41}+ \cdots + 12 i q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 8 q^{7} - 4 q^{11} - 6 q^{13} + 6 q^{17} + 4 q^{19} + 4 q^{23} + 6 q^{25} - 16 q^{35} - 8 q^{37} - 14 q^{41} - 16 q^{43} - 8 q^{47} + 18 q^{49} - 14 q^{53} - 12 q^{55} - 20 q^{59} - 16 q^{65}+ \cdots + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(1171\) \(2081\)
\(\chi(n)\) \(i\) \(i\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1477.1
1.00000i
− 1.00000i
0 0 0 2.00000 + 1.00000i 0 −4.00000 0 0 0
1513.1 0 0 0 2.00000 − 1.00000i 0 −4.00000 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
65.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2340.2.bp.c yes 2
3.b odd 2 1 2340.2.bp.a yes 2
5.c odd 4 1 2340.2.u.b yes 2
13.d odd 4 1 2340.2.u.b yes 2
15.e even 4 1 2340.2.u.a ✓ 2
39.f even 4 1 2340.2.u.a ✓ 2
65.f even 4 1 inner 2340.2.bp.c yes 2
195.u odd 4 1 2340.2.bp.a yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2340.2.u.a ✓ 2 15.e even 4 1
2340.2.u.a ✓ 2 39.f even 4 1
2340.2.u.b yes 2 5.c odd 4 1
2340.2.u.b yes 2 13.d odd 4 1
2340.2.bp.a yes 2 3.b odd 2 1
2340.2.bp.a yes 2 195.u odd 4 1
2340.2.bp.c yes 2 1.a even 1 1 trivial
2340.2.bp.c yes 2 65.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2340, [\chi])\):

\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} + 8 \) 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
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$43$ \( T^{2} + 16T + 128 \) Copy content Toggle raw display
$47$ \( (T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$59$ \( T^{2} + 20T + 200 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 16T + 128 \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( T^{2} + 16 \) Copy content Toggle raw display
$83$ \( (T + 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$97$ \( T^{2} + 144 \) Copy content Toggle raw display
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