Properties

Label 3328.2.b.g
Level $3328$
Weight $2$
Character orbit 3328.b
Analytic conductor $26.574$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3328,2,Mod(1665,3328)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3328, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3328.1665");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3328.b (of order \(2\), degree \(1\), not 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: \(26.5742137927\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{3} - i q^{5} - q^{7} - 6 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{3} - i q^{5} - q^{7} - 6 q^{9} - 2 i q^{11} + i q^{13} + 3 q^{15} - 3 q^{17} - 6 i q^{19} - 3 i q^{21} + 4 q^{23} + 4 q^{25} - 9 i q^{27} - 2 i q^{29} + 4 q^{31} + 6 q^{33} + i q^{35} + 3 i q^{37} - 3 q^{39} - 5 i q^{43} + 6 i q^{45} + 13 q^{47} - 6 q^{49} - 9 i q^{51} + 12 i q^{53} - 2 q^{55} + 18 q^{57} - 10 i q^{59} + 8 i q^{61} + 6 q^{63} + q^{65} + 2 i q^{67} + 12 i q^{69} + 5 q^{71} + 10 q^{73} + 12 i q^{75} + 2 i q^{77} - 4 q^{79} + 9 q^{81} + 3 i q^{85} + 6 q^{87} - 6 q^{89} - i q^{91} + 12 i q^{93} - 6 q^{95} + 14 q^{97} + 12 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{7} - 12 q^{9} + 6 q^{15} - 6 q^{17} + 8 q^{23} + 8 q^{25} + 8 q^{31} + 12 q^{33} - 6 q^{39} + 26 q^{47} - 12 q^{49} - 4 q^{55} + 36 q^{57} + 12 q^{63} + 2 q^{65} + 10 q^{71} + 20 q^{73} - 8 q^{79} + 18 q^{81} + 12 q^{87} - 12 q^{89} - 12 q^{95} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(769\) \(1535\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1665.1
1.00000i
1.00000i
0 3.00000i 0 1.00000i 0 −1.00000 0 −6.00000 0
1665.2 0 3.00000i 0 1.00000i 0 −1.00000 0 −6.00000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3328.2.b.g 2
4.b odd 2 1 3328.2.b.k 2
8.b even 2 1 inner 3328.2.b.g 2
8.d odd 2 1 3328.2.b.k 2
16.e even 4 1 26.2.a.b 1
16.e even 4 1 832.2.a.j 1
16.f odd 4 1 208.2.a.d 1
16.f odd 4 1 832.2.a.a 1
48.i odd 4 1 234.2.a.b 1
48.i odd 4 1 7488.2.a.w 1
48.k even 4 1 1872.2.a.m 1
48.k even 4 1 7488.2.a.v 1
80.i odd 4 1 650.2.b.a 2
80.k odd 4 1 5200.2.a.c 1
80.q even 4 1 650.2.a.g 1
80.t odd 4 1 650.2.b.a 2
112.l odd 4 1 1274.2.a.o 1
112.w even 12 2 1274.2.f.l 2
112.x odd 12 2 1274.2.f.a 2
144.w odd 12 2 2106.2.e.t 2
144.x even 12 2 2106.2.e.h 2
176.l odd 4 1 3146.2.a.a 1
208.l even 4 1 2704.2.f.j 2
208.m odd 4 1 338.2.b.a 2
208.o odd 4 1 2704.2.a.n 1
208.p even 4 1 338.2.a.a 1
208.r odd 4 1 338.2.b.a 2
208.s even 4 1 2704.2.f.j 2
208.be odd 12 2 338.2.e.d 4
208.bh even 12 2 338.2.c.g 2
208.bj even 12 2 338.2.c.c 2
208.bl odd 12 2 338.2.e.d 4
240.bb even 4 1 5850.2.e.v 2
240.bf even 4 1 5850.2.e.v 2
240.bm odd 4 1 5850.2.a.bn 1
272.r even 4 1 7514.2.a.i 1
304.j odd 4 1 9386.2.a.f 1
624.u even 4 1 3042.2.b.f 2
624.bi odd 4 1 3042.2.a.l 1
624.bm even 4 1 3042.2.b.f 2
1040.be even 4 1 8450.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 16.e even 4 1
208.2.a.d 1 16.f odd 4 1
234.2.a.b 1 48.i odd 4 1
338.2.a.a 1 208.p even 4 1
338.2.b.a 2 208.m odd 4 1
338.2.b.a 2 208.r odd 4 1
338.2.c.c 2 208.bj even 12 2
338.2.c.g 2 208.bh even 12 2
338.2.e.d 4 208.be odd 12 2
338.2.e.d 4 208.bl odd 12 2
650.2.a.g 1 80.q even 4 1
650.2.b.a 2 80.i odd 4 1
650.2.b.a 2 80.t odd 4 1
832.2.a.a 1 16.f odd 4 1
832.2.a.j 1 16.e even 4 1
1274.2.a.o 1 112.l odd 4 1
1274.2.f.a 2 112.x odd 12 2
1274.2.f.l 2 112.w even 12 2
1872.2.a.m 1 48.k even 4 1
2106.2.e.h 2 144.x even 12 2
2106.2.e.t 2 144.w odd 12 2
2704.2.a.n 1 208.o odd 4 1
2704.2.f.j 2 208.l even 4 1
2704.2.f.j 2 208.s even 4 1
3042.2.a.l 1 624.bi odd 4 1
3042.2.b.f 2 624.u even 4 1
3042.2.b.f 2 624.bm even 4 1
3146.2.a.a 1 176.l odd 4 1
3328.2.b.g 2 1.a even 1 1 trivial
3328.2.b.g 2 8.b even 2 1 inner
3328.2.b.k 2 4.b odd 2 1
3328.2.b.k 2 8.d odd 2 1
5200.2.a.c 1 80.k odd 4 1
5850.2.a.bn 1 240.bm odd 4 1
5850.2.e.v 2 240.bb even 4 1
5850.2.e.v 2 240.bf even 4 1
7488.2.a.v 1 48.k even 4 1
7488.2.a.w 1 48.i odd 4 1
7514.2.a.i 1 272.r even 4 1
8450.2.a.y 1 1040.be even 4 1
9386.2.a.f 1 304.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3328, [\chi])\):

\( T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 4 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 9 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 25 \) Copy content Toggle raw display
$47$ \( (T - 13)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 144 \) Copy content Toggle raw display
$59$ \( T^{2} + 100 \) Copy content Toggle raw display
$61$ \( T^{2} + 64 \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( (T - 5)^{2} \) Copy content Toggle raw display
$73$ \( (T - 10)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( (T - 14)^{2} \) Copy content Toggle raw display
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