Properties

Label 3231.1.d.a
Level $3231$
Weight $1$
Character orbit 3231.d
Self dual yes
Analytic conductor $1.612$
Analytic rank $0$
Dimension $9$
Projective image $D_{19}$
CM discriminant -359
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3231,1,Mod(2512,3231)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3231, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3231.2512");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3231 = 3^{2} \cdot 359 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3231.d (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: yes
Analytic conductor: \(1.61247905582\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\Q(\zeta_{38})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 8x^{7} + 7x^{6} + 21x^{5} - 15x^{4} - 20x^{3} + 10x^{2} + 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 359)
Projective image: \(D_{19}\)
Projective field: Galois closure of 19.1.99048986760825351881639.1

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{2} + ( - \beta_1 + 1) q^{4} - \beta_{4} q^{5} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{2} + ( - \beta_1 + 1) q^{4} - \beta_{4} q^{5} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{8} + (\beta_{6} - \beta_{5}) q^{10} - \beta_{6} q^{11} + (\beta_{2} - \beta_1 + 1) q^{16} - \beta_{2} q^{17} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{20} + (\beta_{4} - \beta_{3}) q^{22} + \beta_1 q^{23} + (\beta_{8} + 1) q^{25} + (\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{32} + (\beta_{8} - \beta_{7}) q^{34} - \beta_{3} q^{37} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4}) q^{40} + \beta_{3} q^{41} + (\beta_{7} - \beta_{6} + \beta_{5}) q^{44} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{46} + \beta_{5} q^{47} + q^{49} + (\beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 1) q^{50} + ( - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{55} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{64} + (\beta_{3} - \beta_{2} + \beta_1) q^{68} + ( - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{73} + (\beta_{7} - \beta_{6}) q^{74} + \beta_{4} q^{79} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}) q^{80} + ( - \beta_{7} + \beta_{6}) q^{82} + (\beta_{6} + \beta_{2}) q^{85} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{88} + ( - \beta_{2} + \beta_1 - 2) q^{92} + ( - \beta_{5} + \beta_{4}) q^{94} + (\beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{2} + 8 q^{4} + q^{5} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + q^{2} + 8 q^{4} + q^{5} + 2 q^{8} - 2 q^{10} + q^{11} + 7 q^{16} + q^{17} + 3 q^{20} - 2 q^{22} + q^{23} + 8 q^{25} + 3 q^{32} - 2 q^{34} - q^{37} - 4 q^{40} + q^{41} + 3 q^{44} - 2 q^{46} + q^{47} + 9 q^{49} + 3 q^{50} - 2 q^{55} + 6 q^{64} + 3 q^{68} - q^{73} + 2 q^{74} - q^{79} + 5 q^{80} - 2 q^{82} - 2 q^{85} - 4 q^{88} - 16 q^{92} - 2 q^{94} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{38} + \zeta_{38}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) 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
\(\beta_{8}\)\(=\) \( \nu^{8} - 8\nu^{6} + 20\nu^{4} - 16\nu^{2} + 2 \) 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
\(\nu^{8}\)\(=\) \( \beta_{8} + 8\beta_{6} + 28\beta_{4} + 56\beta_{2} + 70 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3231\mathbb{Z}\right)^\times\).

\(n\) \(2161\) \(2873\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2512.1
−1.57828
−0.490971
0.803391
1.75895
1.97272
1.35456
0.165159
−1.09390
−1.89163
−1.89163 0 2.57828 1.75895 0 0 −2.98553 0 −3.32729
2512.2 −1.57828 0 1.49097 −1.09390 0 0 −0.774890 0 1.72648
2512.3 −1.09390 0 0.196609 0.165159 0 0 0.878826 0 −0.180666
2512.4 −0.490971 0 −0.758948 0.803391 0 0 0.863592 0 −0.394442
2512.5 0.165159 0 −0.972723 −1.57828 0 0 −0.325812 0 −0.260667
2512.6 0.803391 0 −0.354563 1.97272 0 0 −1.08824 0 1.58487
2512.7 1.35456 0 0.834841 −1.89163 0 0 −0.223718 0 −2.56234
2512.8 1.75895 0 2.09390 1.35456 0 0 1.92411 0 2.38261
2512.9 1.97272 0 2.89163 −0.490971 0 0 3.73167 0 −0.968550
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2512.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
359.b odd 2 1 CM by \(\Q(\sqrt{-359}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3231.1.d.a 9
3.b odd 2 1 359.1.b.a 9
359.b odd 2 1 CM 3231.1.d.a 9
1077.b even 2 1 359.1.b.a 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
359.1.b.a 9 3.b odd 2 1
359.1.b.a 9 1077.b even 2 1
3231.1.d.a 9 1.a even 1 1 trivial
3231.1.d.a 9 359.b odd 2 1 CM

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3231, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - T^{8} - 8 T^{7} + 7 T^{6} + 21 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{9} \) Copy content Toggle raw display
$5$ \( T^{9} - T^{8} - 8 T^{7} + 7 T^{6} + 21 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$7$ \( T^{9} \) Copy content Toggle raw display
$11$ \( T^{9} - T^{8} - 8 T^{7} + 7 T^{6} + 21 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} - T^{8} - 8 T^{7} + 7 T^{6} + 21 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$19$ \( T^{9} \) Copy content Toggle raw display
$23$ \( T^{9} - T^{8} - 8 T^{7} + 7 T^{6} + 21 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$29$ \( T^{9} \) Copy content Toggle raw display
$31$ \( T^{9} \) Copy content Toggle raw display
$37$ \( T^{9} + T^{8} - 8 T^{7} - 7 T^{6} + 21 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{9} - T^{8} - 8 T^{7} + 7 T^{6} + 21 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$43$ \( T^{9} \) Copy content Toggle raw display
$47$ \( T^{9} - T^{8} - 8 T^{7} + 7 T^{6} + 21 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$53$ \( T^{9} \) Copy content Toggle raw display
$59$ \( T^{9} \) Copy content Toggle raw display
$61$ \( T^{9} \) Copy content Toggle raw display
$67$ \( T^{9} \) Copy content Toggle raw display
$71$ \( T^{9} \) Copy content Toggle raw display
$73$ \( T^{9} + T^{8} - 8 T^{7} - 7 T^{6} + 21 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{9} + T^{8} - 8 T^{7} - 7 T^{6} + 21 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{9} \) Copy content Toggle raw display
$89$ \( T^{9} \) Copy content Toggle raw display
$97$ \( T^{9} \) Copy content Toggle raw display
show more
show less