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Properties

Label 675.3.d.k
Level $675$
Weight $3$
Character orbit 675.d
Analytic conductor $18.392$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [675,3,Mod(674,675)] 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("675.674"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 675.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,6,0,14,0,0,0,42,0,0,0,0,0,0,0,46,84,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.3924178443\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.60217600.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 16x^{4} + 64x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{2} + ( - \beta_{4} - 2 \beta_{2} + 2) q^{4} + (\beta_{5} - \beta_{3}) q^{7} + ( - 3 \beta_{4} - 3 \beta_{2} + 6) q^{8} + ( - \beta_{5} + 2 \beta_{3} - \beta_1) q^{11} + ( - 2 \beta_{5} - \beta_{3} - \beta_1) q^{13}+ \cdots + (9 \beta_{4} + 23 \beta_{2} - 50) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 14 q^{4} + 42 q^{8} + 46 q^{16} + 84 q^{17} + 16 q^{19} + 102 q^{23} - 56 q^{31} + 174 q^{32} + 80 q^{34} + 96 q^{38} + 234 q^{46} + 138 q^{47} - 74 q^{49} + 120 q^{53} - 46 q^{61} - 36 q^{62}+ \cdots - 318 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 16x^{4} + 64x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5\nu^{3} + 40\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 8\nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} - 12\nu^{3} - 26\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 12\nu^{3} + 36\nu ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{3} ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{5} - 8\beta_{3} + 2\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{4} + 2\beta_{2} + 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 70\beta_{5} + 60\beta_{3} - 24\beta_1 ) / 5 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
674.1
− 2.94600i
2.94600i
− 0.252000i
0.252000i
2.69399i
− 2.69399i
−1.94600 0 −0.213103 0 0 − 6.41178i 8.19868 0 0
674.2 −1.94600 0 −0.213103 0 0 6.41178i 8.19868 0 0
674.3 1.25200 0 −2.43250 0 0 − 7.62099i −8.05349 0 0
674.4 1.25200 0 −2.43250 0 0 7.62099i −8.05349 0 0
674.5 3.69399 0 9.64560 0 0 − 9.20921i 20.8548 0 0
674.6 3.69399 0 9.64560 0 0 9.20921i 20.8548 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 674.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.3.d.k 6
3.b odd 2 1 675.3.d.j 6
5.b even 2 1 675.3.d.j 6
5.c odd 4 1 675.3.c.r ✓ 6
5.c odd 4 1 675.3.c.s yes 6
15.d odd 2 1 inner 675.3.d.k 6
15.e even 4 1 675.3.c.r ✓ 6
15.e even 4 1 675.3.c.s yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
675.3.c.r ✓ 6 5.c odd 4 1
675.3.c.r ✓ 6 15.e even 4 1
675.3.c.s yes 6 5.c odd 4 1
675.3.c.s yes 6 15.e even 4 1
675.3.d.j 6 3.b odd 2 1
675.3.d.j 6 5.b even 2 1
675.3.d.k 6 1.a even 1 1 trivial
675.3.d.k 6 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(675, [\chi])\):

\( T_{2}^{3} - 3T_{2}^{2} - 5T_{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{6} + 184T_{7}^{4} + 10800T_{7}^{2} + 202500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{3} - 3 T^{2} - 5 T + 9)^{2} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 184 T^{4} + \cdots + 202500 \) Copy content Toggle raw display
$11$ \( T^{6} + 499 T^{4} + \cdots + 455625 \) Copy content Toggle raw display
$13$ \( T^{6} + 951 T^{4} + \cdots + 20025625 \) Copy content Toggle raw display
$17$ \( (T^{3} - 42 T^{2} + \cdots + 738)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 8 T^{2} + \cdots - 766)^{2} \) Copy content Toggle raw display
$23$ \( (T^{3} - 51 T^{2} + \cdots + 1215)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 544 T^{4} + \cdots + 202500 \) Copy content Toggle raw display
$31$ \( (T^{3} + 28 T^{2} + \cdots - 20484)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 1305015625 \) Copy content Toggle raw display
$41$ \( T^{6} + 6904 T^{4} + \cdots + 590490000 \) Copy content Toggle raw display
$43$ \( T^{6} + 4716 T^{4} + \cdots + 52562500 \) Copy content Toggle raw display
$47$ \( (T^{3} - 69 T^{2} + \cdots + 174303)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 60 T^{2} + \cdots + 391374)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 10115330625 \) Copy content Toggle raw display
$61$ \( (T^{3} + 23 T^{2} + \cdots + 63331)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 8236 T^{4} + \cdots + 58522500 \) Copy content Toggle raw display
$71$ \( T^{6} + 16939 T^{4} + \cdots + 14630625 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 25824490000 \) Copy content Toggle raw display
$79$ \( (T^{3} + 158 T^{2} + \cdots - 141930)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 162 T^{2} + \cdots - 16524)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 1117354702500 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 74133675625 \) Copy content Toggle raw display
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