↑

Properties

Label 675.4.a.l
Level $675$
Weight $4$
Character orbit 675.a
Self dual yes
Analytic conductor $39.826$
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] = [675,4,Mod(1,675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,10,0,0,-18] 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: yes
Analytic conductor: \(39.8262892539\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + 5 q^{4} - 9 q^{7} + 3 \beta q^{8} - 4 \beta q^{11} + 34 q^{13} + 9 \beta q^{14} - 79 q^{16} + 2 \beta q^{17} - 101 q^{19} + 52 q^{22} + 30 \beta q^{23} - 34 \beta q^{26} - 45 q^{28} + 46 \beta q^{29} + \cdots + 262 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{4} - 18 q^{7} + 68 q^{13} - 158 q^{16} - 202 q^{19} + 104 q^{22} - 90 q^{28} - 6 q^{31} - 52 q^{34} - 134 q^{37} - 274 q^{43} - 780 q^{46} - 524 q^{49} + 340 q^{52} - 1196 q^{58} - 1126 q^{61}+ \cdots - 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.30278
−1.30278
−3.60555 0 5.00000 0 0 −9.00000 10.8167 0 0
1.2 3.60555 0 5.00000 0 0 −9.00000 −10.8167 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.4.a.l ✓ 2
3.b odd 2 1 inner 675.4.a.l ✓ 2
5.b even 2 1 675.4.a.m yes 2
5.c odd 4 2 675.4.b.j 4
15.d odd 2 1 675.4.a.m yes 2
15.e even 4 2 675.4.b.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
675.4.a.l ✓ 2 1.a even 1 1 trivial
675.4.a.l ✓ 2 3.b odd 2 1 inner
675.4.a.m yes 2 5.b even 2 1
675.4.a.m yes 2 15.d odd 2 1
675.4.b.j 4 5.c odd 4 2
675.4.b.j 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(675))\):

\( T_{2}^{2} - 13 \) Copy content Toggle raw display
\( T_{7} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 13 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 9)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 208 \) Copy content Toggle raw display
$13$ \( (T - 34)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 52 \) Copy content Toggle raw display
$19$ \( (T + 101)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 11700 \) Copy content Toggle raw display
$29$ \( T^{2} - 27508 \) Copy content Toggle raw display
$31$ \( (T + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T + 67)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 40768 \) Copy content Toggle raw display
$43$ \( (T + 137)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 13312 \) Copy content Toggle raw display
$53$ \( T^{2} - 459472 \) Copy content Toggle raw display
$59$ \( T^{2} - 761332 \) Copy content Toggle raw display
$61$ \( (T + 563)^{2} \) Copy content Toggle raw display
$67$ \( (T - 1044)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 617812 \) Copy content Toggle raw display
$73$ \( (T + 503)^{2} \) Copy content Toggle raw display
$79$ \( (T - 615)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 56628 \) Copy content Toggle raw display
$89$ \( T^{2} - 947700 \) Copy content Toggle raw display
$97$ \( (T + 19)^{2} \) Copy content Toggle raw display
show more
show less