Properties

Label 3332.2.a.h
Level $3332$
Weight $2$
Character orbit 3332.a
Self dual yes
Analytic conductor $26.606$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3332,2,Mod(1,3332)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3332.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3332.a (trivial)

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: \(26.6061539535\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 68)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{3} - 2 \beta q^{5} + ( - 2 \beta + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{3} - 2 \beta q^{5} + ( - 2 \beta + 1) q^{9} + ( - \beta - 3) q^{11} + (2 \beta - 2) q^{13} + (2 \beta - 6) q^{15} + q^{17} + ( - 2 \beta - 2) q^{19} + ( - \beta - 3) q^{23} + 7 q^{25} - 4 q^{27} - 2 \beta q^{29} + (3 \beta + 1) q^{31} - 2 \beta q^{33} + (2 \beta + 8) q^{37} + ( - 4 \beta + 8) q^{39} + 6 q^{41} + (6 \beta + 2) q^{43} + ( - 2 \beta + 12) q^{45} - 4 \beta q^{47} + (\beta - 1) q^{51} + ( - 4 \beta + 6) q^{53} + (6 \beta + 6) q^{55} - 4 q^{57} + ( - 2 \beta - 6) q^{59} + (2 \beta + 4) q^{61} + (4 \beta - 12) q^{65} + ( - 4 \beta + 8) q^{67} - 2 \beta q^{69} + (3 \beta - 3) q^{71} - 2 q^{73} + (7 \beta - 7) q^{75} + (3 \beta - 7) q^{79} + (2 \beta + 1) q^{81} + (2 \beta + 6) q^{83} - 2 \beta q^{85} + (2 \beta - 6) q^{87} + ( - 2 \beta - 6) q^{89} + ( - 2 \beta + 8) q^{93} + (4 \beta + 12) q^{95} + ( - 4 \beta - 2) q^{97} + (5 \beta + 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{9} - 6 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 4 q^{19} - 6 q^{23} + 14 q^{25} - 8 q^{27} + 2 q^{31} + 16 q^{37} + 16 q^{39} + 12 q^{41} + 4 q^{43} + 24 q^{45} - 2 q^{51} + 12 q^{53} + 12 q^{55} - 8 q^{57} - 12 q^{59} + 8 q^{61} - 24 q^{65} + 16 q^{67} - 6 q^{71} - 4 q^{73} - 14 q^{75} - 14 q^{79} + 2 q^{81} + 12 q^{83} - 12 q^{87} - 12 q^{89} + 16 q^{93} + 24 q^{95} - 4 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
0 −2.73205 0 3.46410 0 0 0 4.46410 0
1.2 0 0.732051 0 −3.46410 0 0 0 −2.46410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(17\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.2.a.h 2
7.b odd 2 1 68.2.a.a 2
21.c even 2 1 612.2.a.e 2
28.d even 2 1 272.2.a.e 2
35.c odd 2 1 1700.2.a.d 2
35.f even 4 2 1700.2.e.c 4
56.e even 2 1 1088.2.a.t 2
56.h odd 2 1 1088.2.a.p 2
77.b even 2 1 8228.2.a.k 2
84.h odd 2 1 2448.2.a.y 2
119.d odd 2 1 1156.2.a.a 2
119.f odd 4 2 1156.2.b.c 4
119.l odd 8 4 1156.2.e.d 8
119.p even 16 8 1156.2.h.f 16
140.c even 2 1 6800.2.a.bh 2
168.e odd 2 1 9792.2.a.cs 2
168.i even 2 1 9792.2.a.cr 2
476.e even 2 1 4624.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.a.a 2 7.b odd 2 1
272.2.a.e 2 28.d even 2 1
612.2.a.e 2 21.c even 2 1
1088.2.a.p 2 56.h odd 2 1
1088.2.a.t 2 56.e even 2 1
1156.2.a.a 2 119.d odd 2 1
1156.2.b.c 4 119.f odd 4 2
1156.2.e.d 8 119.l odd 8 4
1156.2.h.f 16 119.p even 16 8
1700.2.a.d 2 35.c odd 2 1
1700.2.e.c 4 35.f even 4 2
2448.2.a.y 2 84.h odd 2 1
3332.2.a.h 2 1.a even 1 1 trivial
4624.2.a.x 2 476.e even 2 1
6800.2.a.bh 2 140.c even 2 1
8228.2.a.k 2 77.b even 2 1
9792.2.a.cr 2 168.i even 2 1
9792.2.a.cs 2 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3332))\):

\( T_{3}^{2} + 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$5$ \( T^{2} - 12 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$17$ \( (T - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$29$ \( T^{2} - 12 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T - 26 \) Copy content Toggle raw display
$37$ \( T^{2} - 16T + 52 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 104 \) Copy content Toggle raw display
$47$ \( T^{2} - 48 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T - 12 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 16T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T - 18 \) Copy content Toggle raw display
$73$ \( (T + 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 22 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T + 24 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
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