Properties

Label 3332.2.b.c
Level $3332$
Weight $2$
Character orbit 3332.b
Analytic conductor $26.606$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3332,2,Mod(2549,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, 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("3332.2549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3332.b (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: \(26.6061539535\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{7} q^{5} + ( - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_{7} q^{5} + ( - \beta_1 - 1) q^{9} + \beta_{5} q^{11} - \beta_{3} q^{13} + q^{15} + ( - \beta_{7} - \beta_{6} - \beta_{2}) q^{17} + ( - \beta_{6} - \beta_{3}) q^{19} + ( - \beta_{5} + \beta_{4}) q^{23} + (\beta_1 + 2) q^{25} + ( - \beta_{7} - \beta_{2}) q^{27} + ( - \beta_{5} + 2 \beta_{4}) q^{29} + (\beta_{7} + 2 \beta_{2}) q^{31} + ( - \beta_{6} + \beta_{3}) q^{33} + ( - \beta_{5} - \beta_{4}) q^{37} + (\beta_{5} + 2 \beta_{4}) q^{39} - \beta_{2} q^{41} + (\beta_1 + 3) q^{43} + (3 \beta_{7} + \beta_{2}) q^{45} - \beta_{6} q^{47} + ( - \beta_{5} - \beta_{4} + \beta_1 + 3) q^{51} + (\beta_1 + 2) q^{53} + (2 \beta_{6} + \beta_{3}) q^{55} + \beta_{4} q^{57} - 2 \beta_{6} q^{59} + (4 \beta_{7} + 5 \beta_{2}) q^{61} + (\beta_{5} - \beta_{4}) q^{65} + (3 \beta_1 + 4) q^{67} + \beta_{3} q^{69} + (\beta_{5} + \beta_{4}) q^{71} + (4 \beta_{7} - \beta_{2}) q^{73} + (\beta_{7} + 5 \beta_{2}) q^{75} + ( - \beta_{5} + 2 \beta_{4}) q^{79} - 2 \beta_1 q^{81} + (2 \beta_{6} + 3 \beta_{3}) q^{83} + (2 \beta_{5} - \beta_{4} - \beta_1 + 2) q^{85} + ( - \beta_{6} + 3 \beta_{3}) q^{87} - 3 \beta_{6} q^{89} + ( - 2 \beta_1 - 7) q^{93} + (3 \beta_{5} - 2 \beta_{4}) q^{95} - 3 \beta_{7} q^{97} + (\beta_{5} - 3 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{9} + 8 q^{15} + 12 q^{25} + 20 q^{43} + 20 q^{51} + 12 q^{53} + 20 q^{67} + 8 q^{81} + 20 q^{85} - 48 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{6} - 31 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{7} + 8\nu^{5} - 16\nu^{3} + \nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} + 8\nu^{5} - 24\nu^{3} + 17\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{6} - 16\nu^{4} + 32\nu^{2} - 7 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{7} + 4\nu^{5} - 10\nu^{3} + 7\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\nu^{7} - 40\nu^{5} + 104\nu^{3} - 5\nu ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{7} + 3\beta_{3} + 4\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{5} - 3\beta_{4} + 2\beta _1 + 10 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 5\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{5} - 3\beta_{4} - 2\beta _1 - 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{7} + 9\beta_{6} - 15\beta_{3} + 26\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -8\beta _1 - 31 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -10\beta_{7} + 24\beta_{6} - 39\beta_{3} - 68\beta_{2} ) / 12 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2549.1
−1.40126 0.809017i
1.40126 0.809017i
0.535233 + 0.309017i
−0.535233 + 0.309017i
−0.535233 0.309017i
0.535233 0.309017i
1.40126 + 0.809017i
−1.40126 + 0.809017i
0 2.61803i 0 0.381966i 0 0 0 −3.85410 0
2549.2 0 2.61803i 0 0.381966i 0 0 0 −3.85410 0
2549.3 0 0.381966i 0 2.61803i 0 0 0 2.85410 0
2549.4 0 0.381966i 0 2.61803i 0 0 0 2.85410 0
2549.5 0 0.381966i 0 2.61803i 0 0 0 2.85410 0
2549.6 0 0.381966i 0 2.61803i 0 0 0 2.85410 0
2549.7 0 2.61803i 0 0.381966i 0 0 0 −3.85410 0
2549.8 0 2.61803i 0 0.381966i 0 0 0 −3.85410 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2549.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
17.b even 2 1 inner
119.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.2.b.c 8
7.b odd 2 1 inner 3332.2.b.c 8
17.b even 2 1 inner 3332.2.b.c 8
119.d odd 2 1 inner 3332.2.b.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.2.b.c 8 1.a even 1 1 trivial
3332.2.b.c 8 7.b odd 2 1 inner
3332.2.b.c 8 17.b even 2 1 inner
3332.2.b.c 8 119.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 \(S_{2}^{\mathrm{new}}(3332, [\chi])\):

\( T_{3}^{4} + 7T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{13}^{4} - 36T_{13}^{2} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 7 T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 7 T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 36 T^{2} + 144)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 14 T^{2} + 289)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 36 T^{2} + 144)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 36 T^{2} + 144)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 27 T^{2} + 81)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 84 T^{2} + 144)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 7 T^{2} + 1)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 5 T - 5)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 3 T - 9)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 207 T^{2} + 9801)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 5 T - 95)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 84 T^{2} + 144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 135 T^{2} + 2025)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 276 T^{2} + 17424)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 63 T^{2} + 81)^{2} \) Copy content Toggle raw display
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