↑

Properties

Label 1472.2.j.a
Level $1472$
Weight $2$
Character orbit 1472.j
Analytic conductor $11.754$
Analytic rank $1$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1472,2,Mod(369,1472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1472.369"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1472 = 2^{6} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1472.j (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.7539791775\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 368)
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.

\(f(q)\) \(=\) \( q + (i - 1) q^{3} + 4 i q^{7} + i q^{9} + ( - 4 i - 4) q^{11} + (3 i - 3) q^{13} - 2 q^{17} + ( - 4 i - 4) q^{21} - i q^{23} - 5 i q^{25} + ( - 4 i - 4) q^{27} + ( - 7 i + 7) q^{29} - 2 q^{31} + 8 q^{33} + \cdots + ( - 4 i + 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 8 q^{11} - 6 q^{13} - 4 q^{17} - 8 q^{21} - 8 q^{27} + 14 q^{29} - 4 q^{31} + 16 q^{33} + 8 q^{37} + 8 q^{43} - 18 q^{49} + 4 q^{51} - 16 q^{53} - 14 q^{59} + 4 q^{61} - 8 q^{63} + 8 q^{67}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1472\mathbb{Z}\right)^\times\).

\(n\) \(645\) \(833\) \(1151\)
\(\chi(n)\) \(i\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
369.1
1.00000i
− 1.00000i
0 −1.00000 + 1.00000i 0 0 0 4.00000i 0 1.00000i 0
1105.1 0 −1.00000 − 1.00000i 0 0 0 − 4.00000i 0 − 1.00000i 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
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1472.2.j.a 2
4.b odd 2 1 368.2.j.a ✓ 2
16.e even 4 1 inner 1472.2.j.a 2
16.f odd 4 1 368.2.j.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
368.2.j.a ✓ 2 4.b odd 2 1
368.2.j.a ✓ 2 16.f odd 4 1
1472.2.j.a 2 1.a even 1 1 trivial
1472.2.j.a 2 16.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(1472, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 8T + 32 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1 \) Copy content Toggle raw display
$29$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 8T + 32 \) Copy content Toggle raw display
$41$ \( T^{2} + 100 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T + 32 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 16T + 128 \) Copy content Toggle raw display
$59$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T + 32 \) Copy content Toggle raw display
$71$ \( T^{2} + 196 \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 8T + 32 \) Copy content Toggle raw display
$89$ \( T^{2} + 36 \) Copy content Toggle raw display
$97$ \( (T + 14)^{2} \) Copy content Toggle raw display
show more
show less