Properties

Label 1638.2.c.k
Level $1638$
Weight $2$
Character orbit 1638.c
Analytic conductor $13.079$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1638,2,Mod(883,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.883");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.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: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.3356224.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 8x^{4} + 16x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{4} + ( - \beta_{4} - \beta_1) q^{5} - \beta_1 q^{7} - \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{4} + ( - \beta_{4} - \beta_1) q^{5} - \beta_1 q^{7} - \beta_1 q^{8} + (\beta_{3} + 1) q^{10} + ( - \beta_{5} + \beta_1) q^{11} + ( - \beta_{5} + \beta_{3} - 1) q^{13} + q^{14} + q^{16} + (\beta_{3} - 1) q^{17} + (\beta_{4} - 3 \beta_1) q^{19} + (\beta_{4} + \beta_1) q^{20} + (\beta_{2} - 1) q^{22} + ( - 2 \beta_{3} - \beta_{2} - 1) q^{23} + ( - 2 \beta_{3} - \beta_{2}) q^{25} + (\beta_{4} + \beta_{2} - \beta_1) q^{26} + \beta_1 q^{28} + ( - 2 \beta_{3} + \beta_{2} + 3) q^{29} + (2 \beta_{5} - 2 \beta_1) q^{31} + \beta_1 q^{32} + (\beta_{4} - \beta_1) q^{34} + ( - \beta_{3} - 1) q^{35} + ( - \beta_{5} + 4 \beta_{4} + 3 \beta_1) q^{37} + ( - \beta_{3} + 3) q^{38} + ( - \beta_{3} - 1) q^{40} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{41} + (2 \beta_{3} - 3 \beta_{2} - 3) q^{43} + (\beta_{5} - \beta_1) q^{44} + ( - \beta_{5} - 2 \beta_{4} - \beta_1) q^{46} - 2 \beta_{5} q^{47} - q^{49} + ( - \beta_{5} - 2 \beta_{4}) q^{50} + (\beta_{5} - \beta_{3} + 1) q^{52} + 6 q^{53} + ( - \beta_{3} - 2 \beta_{2} + 1) q^{55} - q^{56} + (\beta_{5} - 2 \beta_{4} + 3 \beta_1) q^{58} + ( - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_1) q^{59} + (\beta_{3} - 2 \beta_{2} + 3) q^{61} + ( - 2 \beta_{2} + 2) q^{62} - q^{64} + ( - \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{65} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_1) q^{67} + ( - \beta_{3} + 1) q^{68} + ( - \beta_{4} - \beta_1) q^{70} + ( - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_1) q^{71} + (2 \beta_{5} + 3 \beta_{4} - 5 \beta_1) q^{73} + ( - 4 \beta_{3} + \beta_{2} - 3) q^{74} + ( - \beta_{4} + 3 \beta_1) q^{76} + ( - \beta_{2} + 1) q^{77} + ( - 2 \beta_{3} - 2 \beta_{2} + 4) q^{79} + ( - \beta_{4} - \beta_1) q^{80} + ( - 2 \beta_{3} + 2 \beta_{2} + 2) q^{82} + ( - 4 \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{83} + ( - \beta_{5} - 3 \beta_1) q^{85} + ( - 3 \beta_{5} + 2 \beta_{4} - 3 \beta_1) q^{86} + ( - \beta_{2} + 1) q^{88} + 2 \beta_{4} q^{89} + ( - \beta_{4} - \beta_{2} + \beta_1) q^{91} + (2 \beta_{3} + \beta_{2} + 1) q^{92} + 2 \beta_{2} q^{94} + ( - 2 \beta_{3} + \beta_{2} + 1) q^{95} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_1) q^{97} - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} + 8 q^{10} - 4 q^{13} + 6 q^{14} + 6 q^{16} - 4 q^{17} - 4 q^{22} - 12 q^{23} - 6 q^{25} + 2 q^{26} + 16 q^{29} - 8 q^{35} + 16 q^{38} - 8 q^{40} - 20 q^{43} - 6 q^{49} + 4 q^{52} + 36 q^{53} - 6 q^{56} + 16 q^{61} + 8 q^{62} - 6 q^{64} - 8 q^{65} + 4 q^{68} - 24 q^{74} + 4 q^{77} + 16 q^{79} + 12 q^{82} + 4 q^{88} - 2 q^{91} + 12 q^{92} + 4 q^{94} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 8x^{4} + 16x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{4} - 3\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} + 5\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} + 7\nu^{3} + 11\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 7\nu^{3} + 13\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{5} + 2\beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{3} - 5\beta_{2} + 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 17\beta_{5} - 15\beta_{4} - 14\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
1.86081i
2.11491i
0.254102i
0.254102i
2.11491i
1.86081i
1.00000i 0 −1.00000 1.32340i 0 1.00000i 1.00000i 0 −1.32340
883.2 1.00000i 0 −1.00000 1.64207i 0 1.00000i 1.00000i 0 1.64207
883.3 1.00000i 0 −1.00000 3.68133i 0 1.00000i 1.00000i 0 3.68133
883.4 1.00000i 0 −1.00000 3.68133i 0 1.00000i 1.00000i 0 3.68133
883.5 1.00000i 0 −1.00000 1.64207i 0 1.00000i 1.00000i 0 1.64207
883.6 1.00000i 0 −1.00000 1.32340i 0 1.00000i 1.00000i 0 −1.32340
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 883.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.c.k yes 6
3.b odd 2 1 1638.2.c.j 6
13.b even 2 1 inner 1638.2.c.k yes 6
39.d odd 2 1 1638.2.c.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1638.2.c.j 6 3.b odd 2 1
1638.2.c.j 6 39.d odd 2 1
1638.2.c.k yes 6 1.a even 1 1 trivial
1638.2.c.k yes 6 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\):

\( T_{5}^{6} + 18T_{5}^{4} + 65T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{6} + 26T_{11}^{4} + 105T_{11}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{3} + 2T_{17}^{2} - 5T_{17} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 18 T^{4} + 65 T^{2} + 64 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 26 T^{4} + 105 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{6} + 4 T^{5} + 19 T^{4} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( (T^{3} + 2 T^{2} - 5 T - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 34 T^{4} + 161 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( (T^{3} + 6 T^{2} - 31 T + 28)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 8 T^{2} - 11 T + 14)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 104 T^{4} + 1680 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$37$ \( T^{6} + 254 T^{4} + 20833 T^{2} + \cdots + 550564 \) Copy content Toggle raw display
$41$ \( T^{6} + 140 T^{4} + 4912 T^{2} + \cdots + 33856 \) Copy content Toggle raw display
$43$ \( (T^{3} + 10 T^{2} - 87 T - 772)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 100 T^{4} + 2816 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$53$ \( (T - 6)^{6} \) Copy content Toggle raw display
$59$ \( T^{6} + 192 T^{4} + 11264 T^{2} + \cdots + 200704 \) Copy content Toggle raw display
$61$ \( (T^{3} - 8 T^{2} - 29 T - 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 140 T^{4} + 3376 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$71$ \( T^{6} + 192 T^{4} + 11264 T^{2} + \cdots + 200704 \) Copy content Toggle raw display
$73$ \( T^{6} + 278 T^{4} + 25641 T^{2} + \cdots + 784996 \) Copy content Toggle raw display
$79$ \( (T^{3} - 8 T^{2} - 64 T + 448)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 424 T^{4} + 57488 T^{2} + \cdots + 2458624 \) Copy content Toggle raw display
$89$ \( T^{6} + 52 T^{4} + 704 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$97$ \( T^{6} + 140 T^{4} + 4912 T^{2} + \cdots + 33856 \) Copy content Toggle raw display
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