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Properties

Label 3822.2.c.j
Level $3822$
Weight $2$
Character orbit 3822.c
Analytic conductor $30.519$
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] = [3822,2,Mod(883,3822)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3822, 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("3822.883");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.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: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.9144576.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 546)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - q^{3} - q^{4} + \beta_1 q^{5} - \beta_{2} q^{6} - \beta_{2} q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - q^{3} - q^{4} + \beta_1 q^{5} - \beta_{2} q^{6} - \beta_{2} q^{8} + q^{9} + \beta_{3} q^{10} + ( - \beta_{2} + \beta_1) q^{11} + q^{12} + (\beta_{5} - \beta_{3} - 1) q^{13} - \beta_1 q^{15} + q^{16} + (\beta_{4} + \beta_{3} + 3) q^{17} + \beta_{2} q^{18} + ( - 2 \beta_{5} + \beta_{2}) q^{19} - \beta_1 q^{20} + (\beta_{3} + 1) q^{22} + ( - 2 \beta_{4} - \beta_{3}) q^{23} + \beta_{2} q^{24} + (\beta_{4} + 1) q^{25} + ( - \beta_{4} - \beta_{3} + \cdots + \beta_1) q^{26}+ \cdots + ( - \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{12} - 6 q^{13} + 6 q^{16} + 18 q^{17} + 6 q^{22} + 6 q^{25} - 6 q^{27} + 6 q^{29} - 6 q^{36} - 6 q^{38} + 6 q^{39} + 12 q^{43} - 6 q^{48} - 18 q^{51} + 6 q^{52} + 18 q^{53} - 24 q^{55} + 6 q^{61} + 24 q^{62} - 6 q^{64} + 12 q^{65} - 6 q^{66} - 18 q^{68} - 6 q^{75} - 24 q^{79} + 6 q^{81} - 12 q^{82} - 6 q^{87} - 6 q^{88} + 6 q^{94} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 12x^{4} + 36x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 6\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 6\nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 10\nu^{3} + 22\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{4} + 2\beta_{3} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} - 20\beta_{2} + 38\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3822\mathbb{Z}\right)^\times\).

\(n\) \(1471\) \(2549\) \(3433\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
− 2.26180i
− 0.339877i
2.60168i
− 2.60168i
0.339877i
2.26180i
− 1.00000i −1.00000 −1.00000 − 2.26180i 1.00000i 0 1.00000i 1.00000 −2.26180
883.2 − 1.00000i −1.00000 −1.00000 − 0.339877i 1.00000i 0 1.00000i 1.00000 −0.339877
883.3 − 1.00000i −1.00000 −1.00000 2.60168i 1.00000i 0 1.00000i 1.00000 2.60168
883.4 1.00000i −1.00000 −1.00000 − 2.60168i − 1.00000i 0 − 1.00000i 1.00000 2.60168
883.5 1.00000i −1.00000 −1.00000 0.339877i − 1.00000i 0 − 1.00000i 1.00000 −0.339877
883.6 1.00000i −1.00000 −1.00000 2.26180i − 1.00000i 0 − 1.00000i 1.00000 −2.26180
\(n\): 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 3822.2.c.j 6
7.b odd 2 1 3822.2.c.k 6
7.d odd 6 2 546.2.bk.b ✓ 12
13.b even 2 1 inner 3822.2.c.j 6
21.g even 6 2 1638.2.dm.c 12
91.b odd 2 1 3822.2.c.k 6
91.s odd 6 2 546.2.bk.b ✓ 12
273.ba even 6 2 1638.2.dm.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bk.b ✓ 12 7.d odd 6 2
546.2.bk.b ✓ 12 91.s odd 6 2
1638.2.dm.c 12 21.g even 6 2
1638.2.dm.c 12 273.ba even 6 2
3822.2.c.j 6 1.a even 1 1 trivial
3822.2.c.j 6 13.b even 2 1 inner
3822.2.c.k 6 7.b odd 2 1
3822.2.c.k 6 91.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3822, [\chi])\):

\( T_{5}^{6} + 12T_{5}^{4} + 36T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{6} + 15T_{11}^{4} + 27T_{11}^{2} + 9 \) Copy content Toggle raw display
\( T_{17}^{3} - 9T_{17}^{2} + 15T_{17} + 7 \) Copy content Toggle raw display
\( T_{19}^{6} + 99T_{19}^{4} + 2403T_{19}^{2} + 3969 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 12 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 15 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{6} + 6 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( (T^{3} - 9 T^{2} + 15 T + 7)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 99 T^{4} + \cdots + 3969 \) Copy content Toggle raw display
$23$ \( (T^{3} - 42 T + 98)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 3 T^{2} - 45 T - 77)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 16)^{3} \) Copy content Toggle raw display
$37$ \( T^{6} + 24 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$41$ \( T^{6} + 108 T^{4} + \cdots + 676 \) Copy content Toggle raw display
$43$ \( (T^{3} - 6 T^{2} + \cdots + 104)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 195 T^{4} + \cdots + 91809 \) Copy content Toggle raw display
$53$ \( (T - 3)^{6} \) Copy content Toggle raw display
$59$ \( T^{6} + 171 T^{4} + \cdots + 44521 \) Copy content Toggle raw display
$61$ \( (T^{3} - 3 T^{2} - 45 T - 67)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 195 T^{4} + \cdots + 12769 \) Copy content Toggle raw display
$71$ \( T^{6} + 51 T^{4} + \cdots + 441 \) Copy content Toggle raw display
$73$ \( T^{6} + 324 T^{4} + \cdots + 1214404 \) Copy content Toggle raw display
$79$ \( (T^{3} + 12 T^{2} + \cdots - 528)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 372 T^{4} + \cdots + 712336 \) Copy content Toggle raw display
$89$ \( T^{6} + 216 T^{4} + \cdots + 171396 \) Copy content Toggle raw display
$97$ \( T^{6} + 324 T^{4} + \cdots + 324 \) Copy content Toggle raw display
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