Properties

Label 3332.2.b.a
Level $3332$
Weight $2$
Character orbit 3332.b
Analytic conductor $26.606$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) 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)
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 2 \beta q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + 2 \beta q^{5} + q^{9} + \beta q^{11} + 4 q^{13} - 4 q^{15} + ( - 2 \beta - 3) q^{17} + 4 q^{19} + \beta q^{23} - 3 q^{25} + 4 \beta q^{27} - 2 \beta q^{29} + 3 \beta q^{31} - 2 q^{33} + 6 \beta q^{37} + 4 \beta q^{39} + 8 \beta q^{41} + 8 q^{43} + 2 \beta q^{45} + 12 q^{47} + ( - 3 \beta + 4) q^{51} - 6 q^{53} - 4 q^{55} + 4 \beta q^{57} - 6 \beta q^{61} + 8 \beta q^{65} - 4 q^{67} - 2 q^{69} - 5 \beta q^{71} - 3 \beta q^{75} - 3 \beta q^{79} - 5 q^{81} + ( - 6 \beta + 8) q^{85} + 4 q^{87} - 12 q^{89} - 6 q^{93} + 8 \beta q^{95} + \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{9} + 8 q^{13} - 8 q^{15} - 6 q^{17} + 8 q^{19} - 6 q^{25} - 4 q^{33} + 16 q^{43} + 24 q^{47} + 8 q^{51} - 12 q^{53} - 8 q^{55} - 8 q^{67} - 4 q^{69} - 10 q^{81} + 16 q^{85} + 8 q^{87} - 24 q^{89} - 12 q^{93}+O(q^{100}) \) 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
1.41421i
1.41421i
0 1.41421i 0 2.82843i 0 0 0 1.00000 0
2549.2 0 1.41421i 0 2.82843i 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 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.a 2
7.b odd 2 1 68.2.b.a 2
17.b even 2 1 inner 3332.2.b.a 2
21.c even 2 1 612.2.b.a 2
28.d even 2 1 272.2.b.c 2
35.c odd 2 1 1700.2.c.a 2
35.f even 4 2 1700.2.g.a 4
56.e even 2 1 1088.2.b.f 2
56.h odd 2 1 1088.2.b.e 2
84.h odd 2 1 2448.2.c.d 2
119.d odd 2 1 68.2.b.a 2
119.f odd 4 2 1156.2.a.c 2
119.l odd 8 2 1156.2.e.a 2
119.l odd 8 2 1156.2.e.b 2
119.p even 16 8 1156.2.h.d 8
357.c even 2 1 612.2.b.a 2
476.e even 2 1 272.2.b.c 2
476.k even 4 2 4624.2.a.n 2
595.b odd 2 1 1700.2.c.a 2
595.p even 4 2 1700.2.g.a 4
952.e odd 2 1 1088.2.b.e 2
952.k even 2 1 1088.2.b.f 2
1428.b odd 2 1 2448.2.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.b.a 2 7.b odd 2 1
68.2.b.a 2 119.d odd 2 1
272.2.b.c 2 28.d even 2 1
272.2.b.c 2 476.e even 2 1
612.2.b.a 2 21.c even 2 1
612.2.b.a 2 357.c even 2 1
1088.2.b.e 2 56.h odd 2 1
1088.2.b.e 2 952.e odd 2 1
1088.2.b.f 2 56.e even 2 1
1088.2.b.f 2 952.k even 2 1
1156.2.a.c 2 119.f odd 4 2
1156.2.e.a 2 119.l odd 8 2
1156.2.e.b 2 119.l odd 8 2
1156.2.h.d 8 119.p even 16 8
1700.2.c.a 2 35.c odd 2 1
1700.2.c.a 2 595.b odd 2 1
1700.2.g.a 4 35.f even 4 2
1700.2.g.a 4 595.p even 4 2
2448.2.c.d 2 84.h odd 2 1
2448.2.c.d 2 1428.b odd 2 1
3332.2.b.a 2 1.a even 1 1 trivial
3332.2.b.a 2 17.b even 2 1 inner
4624.2.a.n 2 476.k even 4 2

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2 \) Copy content Toggle raw display
$5$ \( T^{2} + 8 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 2 \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 17 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2 \) Copy content Toggle raw display
$29$ \( T^{2} + 8 \) Copy content Toggle raw display
$31$ \( T^{2} + 18 \) Copy content Toggle raw display
$37$ \( T^{2} + 72 \) Copy content Toggle raw display
$41$ \( T^{2} + 128 \) Copy content Toggle raw display
$43$ \( (T - 8)^{2} \) Copy content Toggle raw display
$47$ \( (T - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 72 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 50 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 18 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T + 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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