Properties

Label 3375.1.c.a
Level $3375$
Weight $1$
Character orbit 3375.c
Analytic conductor $1.684$
Analytic rank $0$
Dimension $8$
Projective image $D_{15}$
CM discriminant -15
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3375,1,Mod(1376,3375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3375, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3375.1376");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3375 = 3^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3375.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.68434441764\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.324000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{15}\)
Projective field: Galois closure of 15.1.2463153133392333984375.1

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{3} + \beta_1) q^{2} + (\beta_{6} - \beta_{2} - 2) q^{4} + ( - \beta_{5} - 2 \beta_{3} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{3} + \beta_1) q^{2} + (\beta_{6} - \beta_{2} - 2) q^{4} + ( - \beta_{5} - 2 \beta_{3} - \beta_1) q^{8} + ( - \beta_{4} + \beta_{2} + 2) q^{16} + ( - \beta_{7} - \beta_{3}) q^{17} - \beta_{4} q^{19} - \beta_{7} q^{23} - \beta_{2} q^{31} + (2 \beta_{3} + \beta_1) q^{32} + (\beta_{6} + \beta_{2}) q^{34} + ( - \beta_{7} - 2 \beta_{5}) q^{38} + (\beta_{6} + \beta_{4} - 1) q^{46} + ( - \beta_{5} - \beta_{3}) q^{47} - q^{49} + \beta_1 q^{53} + (\beta_{6} - \beta_{4}) q^{61} + ( - \beta_{7} - \beta_{5} - 2 \beta_{3}) q^{62} + (\beta_{6} + \beta_{4} - \beta_{2} - 2) q^{64} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{3}) q^{68} + (\beta_{6} + 2 \beta_{4} - 1) q^{76} + ( - \beta_{6} + \beta_{4}) q^{79} + \beta_1 q^{83} + (\beta_{7} + 2 \beta_{5} - \beta_1) q^{92} + (\beta_{2} + 1) q^{94} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{4} + 12 q^{16} - 2 q^{19} + 2 q^{31} + 2 q^{34} - 2 q^{46} - 8 q^{49} + 2 q^{61} - 8 q^{64} - 2 q^{79} + 6 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 4\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 5\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} + 6\nu^{4} + 9\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} + 7\nu^{5} + 14\nu^{3} + 7\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 4\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 5\beta_{3} + 10\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} - 6\beta_{4} + 15\beta_{2} - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} - 7\beta_{5} + 21\beta_{3} - 35\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3375\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(2377\)
\(\chi(n)\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1376.1
1.33826i
0.209057i
1.95630i
1.82709i
1.82709i
1.95630i
0.209057i
1.33826i
1.95630i 0 −2.82709 0 0 0 3.57433i 0 0
1376.2 1.82709i 0 −2.33826 0 0 0 2.44512i 0 0
1376.3 1.33826i 0 −0.790943 0 0 0 0.279773i 0 0
1376.4 0.209057i 0 0.956295 0 0 0 0.408977i 0 0
1376.5 0.209057i 0 0.956295 0 0 0 0.408977i 0 0
1376.6 1.33826i 0 −0.790943 0 0 0 0.279773i 0 0
1376.7 1.82709i 0 −2.33826 0 0 0 2.44512i 0 0
1376.8 1.95630i 0 −2.82709 0 0 0 3.57433i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1376.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3375.1.c.a 8
3.b odd 2 1 inner 3375.1.c.a 8
5.b even 2 1 inner 3375.1.c.a 8
5.c odd 4 1 3375.1.d.a 4
5.c odd 4 1 3375.1.d.b 4
15.d odd 2 1 CM 3375.1.c.a 8
15.e even 4 1 3375.1.d.a 4
15.e even 4 1 3375.1.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3375.1.c.a 8 1.a even 1 1 trivial
3375.1.c.a 8 3.b odd 2 1 inner
3375.1.c.a 8 5.b even 2 1 inner
3375.1.c.a 8 15.d odd 2 1 CM
3375.1.d.a 4 5.c odd 4 1
3375.1.d.a 4 15.e even 4 1
3375.1.d.b 4 5.c odd 4 1
3375.1.d.b 4 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3375, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 9 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 9 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( (T^{4} + T^{3} - 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 9 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - T^{3} - 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 9 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} - T^{3} - 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + T^{3} - 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 9 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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