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Properties

Label 675.4.a.k
Level $675$
Weight $4$
Character orbit 675.a
Self dual yes
Analytic conductor $39.826$
Analytic rank $1$
Dimension $2$
CM discriminant -15
Inner twists $2$

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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 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);
 
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")
 
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,-9,0,27,0,0,0] 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: \(\Q(\sqrt{5}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 135)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 4) q^{2} + (9 \beta + 9) q^{4} + ( - 46 \beta - 13) q^{8} + (171 \beta + 26) q^{16} + (62 \beta - 49) q^{17} + ( - 18 \beta + 91) q^{19} + ( - 44 \beta - 149) q^{23} + ( - 198 \beta + 215) q^{31}+ \cdots + (343 \beta + 1372) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{2} + 27 q^{4} - 72 q^{8} + 223 q^{16} - 36 q^{17} + 164 q^{19} - 342 q^{23} + 232 q^{31} - 855 q^{32} + 7 q^{34} - 693 q^{38} + 1649 q^{46} - 686 q^{49} + 792 q^{53} + 358 q^{61} - 549 q^{62}+ \cdots + 3087 q^{98}+O(q^{100}) \) 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
1.61803
−0.618034
−5.61803 0 23.5623 0 0 0 −87.4296 0 0
1.2 −3.38197 0 3.43769 0 0 0 15.4296 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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.4.a.k 2
3.b odd 2 1 675.4.a.o 2
5.b even 2 1 675.4.a.o 2
5.c odd 4 2 135.4.b.a ✓ 4
15.d odd 2 1 CM 675.4.a.k 2
15.e even 4 2 135.4.b.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.b.a ✓ 4 5.c odd 4 2
135.4.b.a ✓ 4 15.e even 4 2
675.4.a.k 2 1.a even 1 1 trivial
675.4.a.k 2 15.d odd 2 1 CM
675.4.a.o 2 3.b odd 2 1
675.4.a.o 2 5.b even 2 1

Hecke kernels

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

\( T_{2}^{2} + 9T_{2} + 19 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 9T + 19 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 36T - 4481 \) Copy content Toggle raw display
$19$ \( T^{2} - 164T + 6319 \) Copy content Toggle raw display
$23$ \( T^{2} + 342T + 26821 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 232T - 35549 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 297680 \) Copy content Toggle raw display
$53$ \( T^{2} - 792T + 60211 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 358T - 552779 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 304 T - 1386701 \) Copy content Toggle raw display
$83$ \( T^{2} + 1422 T + 102241 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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