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Properties

Label 1352.2.o.c
Level $1352$
Weight $2$
Character orbit 1352.o
Analytic conductor $10.796$
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] = [1352,2,Mod(361,1352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1352, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1352.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1352 = 2^{3} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1352.o (of order \(6\), degree \(2\), 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: \(10.7957743533\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.1731891456.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4} - \beta_{2}) q^{3} + ( - \beta_{7} + \beta_{6}) q^{5} + \beta_1 q^{7} + (\beta_{5} + 2 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{4} - \beta_{2}) q^{3} + ( - \beta_{7} + \beta_{6}) q^{5} + \beta_1 q^{7} + (\beta_{5} + 2 \beta_{2}) q^{9} + ( - \beta_{6} + \beta_1) q^{11} + (4 \beta_{7} - 2 \beta_{6} + \cdots + 2 \beta_1) q^{15}+ \cdots + ( - 4 \beta_{7} + 2 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} - 6 q^{9} - 20 q^{17} + 10 q^{23} - 12 q^{25} + 28 q^{27} + 20 q^{29} - 20 q^{35} - 18 q^{43} - 10 q^{49} + 20 q^{51} + 52 q^{53} + 20 q^{55} - 4 q^{61} - 46 q^{69} + 54 q^{75} + 36 q^{77} - 32 q^{79} + 28 q^{81} + 10 q^{87} + 20 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{6} - 65\nu^{4} + 585\nu^{2} - 1296 ) / 1040 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 181\nu ) / 260 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 116 ) / 65 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -29\nu^{6} + 325\nu^{4} - 1885\nu^{2} + 4176 ) / 1040 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{7} - 65\nu^{5} + 585\nu^{3} - 256\nu ) / 1040 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{7} - 65\nu^{5} + 377\nu^{3} - 256\nu ) / 832 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + 5\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{7} + 5\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{5} + 29\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -36\beta_{7} + 29\beta_{6} - 36\beta_{3} - 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 65\beta_{4} - 116 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -260\beta_{3} - 181\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1185\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−2.21837 − 1.28078i
2.21837 + 1.28078i
−1.35234 − 0.780776i
1.35234 + 0.780776i
2.21837 − 1.28078i
−2.21837 + 1.28078i
1.35234 − 0.780776i
−1.35234 + 0.780776i
0 −1.28078 − 2.21837i 0 − 3.56155i 0 −2.21837 − 1.28078i 0 −1.78078 + 3.08440i 0
361.2 0 −1.28078 − 2.21837i 0 3.56155i 0 2.21837 + 1.28078i 0 −1.78078 + 3.08440i 0
361.3 0 0.780776 + 1.35234i 0 − 0.561553i 0 −1.35234 − 0.780776i 0 0.280776 − 0.486319i 0
361.4 0 0.780776 + 1.35234i 0 0.561553i 0 1.35234 + 0.780776i 0 0.280776 − 0.486319i 0
1161.1 0 −1.28078 + 2.21837i 0 − 3.56155i 0 2.21837 − 1.28078i 0 −1.78078 − 3.08440i 0
1161.2 0 −1.28078 + 2.21837i 0 3.56155i 0 −2.21837 + 1.28078i 0 −1.78078 − 3.08440i 0
1161.3 0 0.780776 − 1.35234i 0 − 0.561553i 0 1.35234 − 0.780776i 0 0.280776 + 0.486319i 0
1161.4 0 0.780776 − 1.35234i 0 0.561553i 0 −1.35234 + 0.780776i 0 0.280776 + 0.486319i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1352.2.o.c 8
13.b even 2 1 inner 1352.2.o.c 8
13.c even 3 1 1352.2.f.d 4
13.c even 3 1 inner 1352.2.o.c 8
13.d odd 4 1 104.2.i.b ✓ 4
13.d odd 4 1 1352.2.i.e 4
13.e even 6 1 1352.2.f.d 4
13.e even 6 1 inner 1352.2.o.c 8
13.f odd 12 1 104.2.i.b ✓ 4
13.f odd 12 1 1352.2.a.f 2
13.f odd 12 1 1352.2.a.h 2
13.f odd 12 1 1352.2.i.e 4
39.f even 4 1 936.2.t.f 4
39.k even 12 1 936.2.t.f 4
52.f even 4 1 208.2.i.e 4
52.i odd 6 1 2704.2.f.l 4
52.j odd 6 1 2704.2.f.l 4
52.l even 12 1 208.2.i.e 4
52.l even 12 1 2704.2.a.q 2
52.l even 12 1 2704.2.a.r 2
104.j odd 4 1 832.2.i.o 4
104.m even 4 1 832.2.i.l 4
104.u even 12 1 832.2.i.l 4
104.x odd 12 1 832.2.i.o 4
156.l odd 4 1 1872.2.t.s 4
156.v odd 12 1 1872.2.t.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.i.b ✓ 4 13.d odd 4 1
104.2.i.b ✓ 4 13.f odd 12 1
208.2.i.e 4 52.f even 4 1
208.2.i.e 4 52.l even 12 1
832.2.i.l 4 104.m even 4 1
832.2.i.l 4 104.u even 12 1
832.2.i.o 4 104.j odd 4 1
832.2.i.o 4 104.x odd 12 1
936.2.t.f 4 39.f even 4 1
936.2.t.f 4 39.k even 12 1
1352.2.a.f 2 13.f odd 12 1
1352.2.a.h 2 13.f odd 12 1
1352.2.f.d 4 13.c even 3 1
1352.2.f.d 4 13.e even 6 1
1352.2.i.e 4 13.d odd 4 1
1352.2.i.e 4 13.f odd 12 1
1352.2.o.c 8 1.a even 1 1 trivial
1352.2.o.c 8 13.b even 2 1 inner
1352.2.o.c 8 13.c even 3 1 inner
1352.2.o.c 8 13.e even 6 1 inner
1872.2.t.s 4 156.l odd 4 1
1872.2.t.s 4 156.v odd 12 1
2704.2.a.q 2 52.l even 12 1
2704.2.a.r 2 52.l even 12 1
2704.2.f.l 4 52.i odd 6 1
2704.2.f.l 4 52.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1352, [\chi])\):

\( T_{3}^{4} + T_{3}^{3} + 5T_{3}^{2} - 4T_{3} + 16 \) Copy content Toggle raw display
\( T_{5}^{4} + 13T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{8} - 9T_{11}^{6} + 65T_{11}^{4} - 144T_{11}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + 5 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 13 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 9 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{8} - 9 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{2} + 5 T + 25)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} - 9 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T^{4} - 5 T^{3} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 5 T + 25)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 138 T^{6} + \cdots + 20151121 \) Copy content Toggle raw display
$43$ \( (T^{4} + 9 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 13 T + 38)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} - 9 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( (T^{4} + 2 T^{3} + \cdots + 4489)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 273 T^{6} + \cdots + 268435456 \) Copy content Toggle raw display
$71$ \( T^{8} - 81 T^{6} + \cdots + 1679616 \) Copy content Toggle raw display
$73$ \( (T^{4} + 189 T^{2} + 324)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 208 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 101 T^{6} + \cdots + 456976 \) Copy content Toggle raw display
$97$ \( T^{8} - 325 T^{6} + \cdots + 6250000 \) Copy content Toggle raw display
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