Properties

Label 2704.2.f.o
Level $2704$
Weight $2$
Character orbit 2704.f
Analytic conductor $21.592$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2704,2,Mod(337,2704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2704, 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("2704.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2704 = 2^{4} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2704.f (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: \(21.5915487066\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 169)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{2}) q^{3} + (\beta_{5} - \beta_{3}) q^{5} + (2 \beta_{5} + \beta_{3} - 2 \beta_1) q^{7} + (\beta_{4} + 2 \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_{2}) q^{3} + (\beta_{5} - \beta_{3}) q^{5} + (2 \beta_{5} + \beta_{3} - 2 \beta_1) q^{7} + (\beta_{4} + 2 \beta_{2} - 2) q^{9} + (3 \beta_{5} + \beta_{3}) q^{11} + ( - \beta_{5} + \beta_1) q^{15} + (\beta_{4} - 2 \beta_{2} + 1) q^{17} + (\beta_{3} - 3 \beta_1) q^{19} + (2 \beta_{5} + 3 \beta_{3} - \beta_1) q^{21} + (2 \beta_{4} + 2 \beta_{2} - 3) q^{23} + (3 \beta_{4} + 2 \beta_{2}) q^{25} + ( - 3 \beta_{4} - \beta_{2} + 1) q^{27} + (5 \beta_{4} + 3 \beta_{2} - 3) q^{29} + ( - 2 \beta_{5} - 3 \beta_{3} - 2 \beta_1) q^{31} + (5 \beta_{5} + 4 \beta_{3} - \beta_1) q^{33} + (4 \beta_{4} + \beta_{2} - 3) q^{35} + (4 \beta_{5} + \beta_{3} + \beta_1) q^{37} + (5 \beta_{5} + 6 \beta_{3} - 2 \beta_1) q^{41} + ( - 2 \beta_{4} - 3 \beta_{2} + 6) q^{43} + ( - 3 \beta_{5} + 2 \beta_{3}) q^{45} + (7 \beta_{5} + \beta_{3} - 2 \beta_1) q^{47} + (\beta_{4} + 1) q^{49} + ( - \beta_{4} - 3 \beta_{2} + 1) q^{51} + ( - 4 \beta_{4} - 7 \beta_{2} + 4) q^{53} + (\beta_{4} + 2 \beta_{2} - 3) q^{55} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{57} + (5 \beta_{5} + 4 \beta_1) q^{59} + (6 \beta_{4} + \beta_{2} - 1) q^{61} + (\beta_{5} + 2 \beta_{3} + 3 \beta_1) q^{63} + (\beta_{5} + 5 \beta_{3} + \beta_1) q^{67} + ( - \beta_{4} + \beta_{2} + 2) q^{69} + ( - 10 \beta_{5} + 3 \beta_1) q^{71} + (7 \beta_{5} + 3 \beta_{3} - 9 \beta_1) q^{73} + (2 \beta_{4} + 4 \beta_{2} + 3) q^{75} + ( - 5 \beta_{2} - 1) q^{77} + (\beta_{4} - 8 \beta_{2} + 4) q^{79} + ( - 3 \beta_{4} - 7 \beta_{2} + 3) q^{81} + ( - 4 \beta_{5} + 2 \beta_{3} + 7 \beta_1) q^{83} + ( - \beta_{3} + 4 \beta_1) q^{85} + (3 \beta_{2} + 5) q^{87} + ( - 6 \beta_{5} + 7 \beta_1) q^{89} + ( - 10 \beta_{5} - 5 \beta_{3} + 3 \beta_1) q^{93} + (4 \beta_{4} - \beta_{2}) q^{95} + (2 \beta_{5} + 7 \beta_{3} - 6 \beta_1) q^{97} + (3 \beta_{5} + 6 \beta_{3} - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} - 6 q^{9} + 4 q^{17} - 10 q^{23} + 10 q^{25} - 2 q^{27} - 2 q^{29} - 8 q^{35} + 26 q^{43} + 8 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} + 8 q^{61} + 12 q^{69} + 30 q^{75} - 16 q^{77} + 10 q^{79} - 2 q^{81} + 36 q^{87} + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 5x^{4} + 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 3\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 4\nu^{3} + 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 3\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{3} + 9\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1185\) \(2367\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.24698i
1.24698i
1.80194i
1.80194i
0.445042i
0.445042i
0 −0.801938 0 2.80194i 0 2.69202i 0 −2.35690 0
337.2 0 −0.801938 0 2.80194i 0 2.69202i 0 −2.35690 0
337.3 0 0.554958 0 1.44504i 0 2.04892i 0 −2.69202 0
337.4 0 0.554958 0 1.44504i 0 2.04892i 0 −2.69202 0
337.5 0 2.24698 0 0.246980i 0 2.35690i 0 2.04892 0
337.6 0 2.24698 0 0.246980i 0 2.35690i 0 2.04892 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.6
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 2704.2.f.o 6
4.b odd 2 1 169.2.b.b 6
12.b even 2 1 1521.2.b.l 6
13.b even 2 1 inner 2704.2.f.o 6
13.d odd 4 1 2704.2.a.z 3
13.d odd 4 1 2704.2.a.ba 3
52.b odd 2 1 169.2.b.b 6
52.f even 4 1 169.2.a.b 3
52.f even 4 1 169.2.a.c yes 3
52.i odd 6 2 169.2.e.b 12
52.j odd 6 2 169.2.e.b 12
52.l even 12 2 169.2.c.b 6
52.l even 12 2 169.2.c.c 6
156.h even 2 1 1521.2.b.l 6
156.l odd 4 1 1521.2.a.o 3
156.l odd 4 1 1521.2.a.r 3
260.u even 4 1 4225.2.a.bb 3
260.u even 4 1 4225.2.a.bg 3
364.p odd 4 1 8281.2.a.bf 3
364.p odd 4 1 8281.2.a.bj 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.2.a.b 3 52.f even 4 1
169.2.a.c yes 3 52.f even 4 1
169.2.b.b 6 4.b odd 2 1
169.2.b.b 6 52.b odd 2 1
169.2.c.b 6 52.l even 12 2
169.2.c.c 6 52.l even 12 2
169.2.e.b 12 52.i odd 6 2
169.2.e.b 12 52.j odd 6 2
1521.2.a.o 3 156.l odd 4 1
1521.2.a.r 3 156.l odd 4 1
1521.2.b.l 6 12.b even 2 1
1521.2.b.l 6 156.h even 2 1
2704.2.a.z 3 13.d odd 4 1
2704.2.a.ba 3 13.d odd 4 1
2704.2.f.o 6 1.a even 1 1 trivial
2704.2.f.o 6 13.b even 2 1 inner
4225.2.a.bb 3 260.u even 4 1
4225.2.a.bg 3 260.u even 4 1
8281.2.a.bf 3 364.p odd 4 1
8281.2.a.bj 3 364.p odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2704, [\chi])\):

\( T_{3}^{3} - 2T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{6} + 10T_{5}^{4} + 17T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{6} + 26T_{11}^{4} + 153T_{11}^{2} + 169 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} - 2 T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} + 10 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} + 17 T^{4} + \cdots + 169 \) Copy content Toggle raw display
$11$ \( T^{6} + 26 T^{4} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} - 2 T^{2} - 15 T - 13)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 38 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{3} + 5 T^{2} - T - 13)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + T^{2} - 44 T + 83)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 97 T^{4} + \cdots + 27889 \) Copy content Toggle raw display
$37$ \( T^{6} + 62 T^{4} + \cdots + 841 \) Copy content Toggle raw display
$41$ \( T^{6} + 147 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$43$ \( (T^{3} - 13 T^{2} + \cdots + 13)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 122 T^{4} + \cdots + 27889 \) Copy content Toggle raw display
$53$ \( (T^{3} - T^{2} - 86 T + 337)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 195 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( (T^{3} - 4 T^{2} + \cdots + 239)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 145 T^{4} + \cdots + 1681 \) Copy content Toggle raw display
$71$ \( T^{6} + 285 T^{4} + \cdots + 299209 \) Copy content Toggle raw display
$73$ \( T^{6} + 321 T^{4} + \cdots + 829921 \) Copy content Toggle raw display
$79$ \( (T^{3} - 5 T^{2} + \cdots - 127)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 329 T^{4} + \cdots + 41209 \) Copy content Toggle raw display
$89$ \( T^{6} + 269 T^{4} + \cdots + 78961 \) Copy content Toggle raw display
$97$ \( T^{6} + 217 T^{4} + \cdots + 90601 \) Copy content Toggle raw display
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