Properties

Label 352.2.e.a
Level $352$
Weight $2$
Character orbit 352.e
Analytic conductor $2.811$
Analytic rank $0$
Dimension $12$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [352,2,Mod(351,352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("352.351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 352.e (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: \(2.81073415115\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.653473922154496.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{15} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - \beta_{5} q^{5} + \beta_{8} q^{7} + \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - \beta_{5} q^{5} + \beta_{8} q^{7} + \beta_{3} q^{9} + \beta_{7} q^{11} - \beta_{10} q^{13} + (\beta_{4} - \beta_{2}) q^{15} - \beta_{11} q^{17} + ( - \beta_{9} - \beta_{7}) q^{19} + (\beta_{11} - \beta_{10} - \beta_1) q^{21} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{23}+ \cdots + ( - \beta_{9} - 2 \beta_{8} + \cdots + 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{9} - 4 q^{25} - 16 q^{33} - 24 q^{45} + 44 q^{49} + 8 q^{53} + 8 q^{69} + 32 q^{77} + 12 q^{81} - 88 q^{93} - 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 2\nu^{5} + 5\nu^{3} - 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} - 6\nu^{8} + 9\nu^{6} - 14\nu^{4} ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} - 4\nu^{6} + 9\nu^{4} - 12\nu^{2} + 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{10} - 10\nu^{8} + 27\nu^{6} - 34\nu^{4} + 64\nu^{2} - 32 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} - 2\nu^{8} + 5\nu^{6} - 10\nu^{4} + 12\nu^{2} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3 \nu^{11} + 3 \nu^{10} + 2 \nu^{9} - 18 \nu^{8} - 11 \nu^{7} + 43 \nu^{6} + 26 \nu^{5} - 106 \nu^{4} + \cdots - 224 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3 \nu^{11} + 3 \nu^{10} - 2 \nu^{9} - 18 \nu^{8} + 11 \nu^{7} + 43 \nu^{6} - 26 \nu^{5} - 106 \nu^{4} + \cdots - 224 ) / 64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{11} + 6\nu^{9} - 17\nu^{7} + 30\nu^{5} - 40\nu^{3} + 48\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5 \nu^{11} - 3 \nu^{10} - 2 \nu^{9} + 18 \nu^{8} + 3 \nu^{7} - 43 \nu^{6} - 26 \nu^{5} + 106 \nu^{4} + \cdots + 224 ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{11} - 4\nu^{9} + 9\nu^{7} - 20\nu^{5} + 32\nu^{3} - 24\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} - 2\nu^{9} + 5\nu^{7} - 2\nu^{5} + 12\nu^{3} + 8\nu ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{10} - \beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{6} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} + \beta_{10} + \beta_{9} + 2\beta_{8} + \beta_{6} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{5} + 3\beta_{4} + \beta_{3} - \beta_{2} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8\beta_{11} - 5\beta_{10} + 5\beta_{9} - \beta_{8} + 2\beta_{7} + 3\beta_{6} - \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{7} - 5\beta_{6} - 3\beta_{5} + 11\beta_{4} - 5\beta_{3} - 6\beta_{2} - 7 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3\beta_{11} - 9\beta_{10} - \beta_{9} - 10\beta_{8} - \beta_{6} + 3\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -4\beta_{7} - 4\beta_{6} + 6\beta_{5} + \beta_{4} - 5\beta_{3} - 15\beta_{2} - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -8\beta_{11} - 19\beta_{10} - 13\beta_{9} - 7\beta_{8} + 14\beta_{7} - 27\beta_{6} + 25\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -3\beta_{7} - 3\beta_{6} + 43\beta_{5} - 3\beta_{4} + 13\beta_{3} - 26\beta_{2} - 33 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 5\beta_{11} + 5\beta_{10} - 11\beta_{9} + 14\beta_{8} + 24\beta_{7} - 35\beta_{6} - 7\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/352\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(287\) \(321\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1
0.892524 1.09700i
−0.892524 + 1.09700i
−1.35489 + 0.405301i
1.35489 0.405301i
−1.16947 0.795191i
1.16947 + 0.795191i
−1.16947 + 0.795191i
1.16947 0.795191i
−1.35489 0.405301i
1.35489 + 0.405301i
0.892524 + 1.09700i
−0.892524 1.09700i
0 2.81361i 0 0.289169 0 −4.38799 0 −4.91638 0
351.2 0 2.81361i 0 0.289169 0 4.38799 0 −4.91638 0
351.3 0 1.34292i 0 2.48929 0 −1.62121 0 1.19656 0
351.4 0 1.34292i 0 2.48929 0 1.62121 0 1.19656 0
351.5 0 0.529317i 0 −2.77846 0 −3.18077 0 2.71982 0
351.6 0 0.529317i 0 −2.77846 0 3.18077 0 2.71982 0
351.7 0 0.529317i 0 −2.77846 0 −3.18077 0 2.71982 0
351.8 0 0.529317i 0 −2.77846 0 3.18077 0 2.71982 0
351.9 0 1.34292i 0 2.48929 0 −1.62121 0 1.19656 0
351.10 0 1.34292i 0 2.48929 0 1.62121 0 1.19656 0
351.11 0 2.81361i 0 0.289169 0 −4.38799 0 −4.91638 0
351.12 0 2.81361i 0 0.289169 0 4.38799 0 −4.91638 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 351.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.b odd 2 1 inner
44.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 352.2.e.a 12
3.b odd 2 1 3168.2.o.e 12
4.b odd 2 1 inner 352.2.e.a 12
8.b even 2 1 704.2.e.d 12
8.d odd 2 1 704.2.e.d 12
11.b odd 2 1 inner 352.2.e.a 12
12.b even 2 1 3168.2.o.e 12
16.e even 4 1 2816.2.g.d 12
16.e even 4 1 2816.2.g.i 12
16.f odd 4 1 2816.2.g.d 12
16.f odd 4 1 2816.2.g.i 12
33.d even 2 1 3168.2.o.e 12
44.c even 2 1 inner 352.2.e.a 12
88.b odd 2 1 704.2.e.d 12
88.g even 2 1 704.2.e.d 12
132.d odd 2 1 3168.2.o.e 12
176.i even 4 1 2816.2.g.d 12
176.i even 4 1 2816.2.g.i 12
176.l odd 4 1 2816.2.g.d 12
176.l odd 4 1 2816.2.g.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
352.2.e.a 12 1.a even 1 1 trivial
352.2.e.a 12 4.b odd 2 1 inner
352.2.e.a 12 11.b odd 2 1 inner
352.2.e.a 12 44.c even 2 1 inner
704.2.e.d 12 8.b even 2 1
704.2.e.d 12 8.d odd 2 1
704.2.e.d 12 88.b odd 2 1
704.2.e.d 12 88.g even 2 1
2816.2.g.d 12 16.e even 4 1
2816.2.g.d 12 16.f odd 4 1
2816.2.g.d 12 176.i even 4 1
2816.2.g.d 12 176.l odd 4 1
2816.2.g.i 12 16.e even 4 1
2816.2.g.i 12 16.f odd 4 1
2816.2.g.i 12 176.i even 4 1
2816.2.g.i 12 176.l odd 4 1
3168.2.o.e 12 3.b odd 2 1
3168.2.o.e 12 12.b even 2 1
3168.2.o.e 12 33.d even 2 1
3168.2.o.e 12 132.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(352, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 10 T^{4} + 17 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{3} - 7 T + 2)^{4} \) Copy content Toggle raw display
$7$ \( (T^{6} - 32 T^{4} + \cdots - 512)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{10} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( (T^{6} + 40 T^{4} + \cdots + 128)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 64 T^{4} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 104 T^{4} + \cdots - 15488)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 50 T^{4} + \cdots + 3844)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 136 T^{4} + \cdots + 32768)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 98 T^{4} + \cdots + 1156)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 7 T + 2)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + 160 T^{4} + \cdots + 147968)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 72 T^{4} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 156 T^{4} + \cdots + 118336)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 2 T^{2} - 68 T + 8)^{4} \) Copy content Toggle raw display
$59$ \( (T^{6} + 186 T^{4} + \cdots + 1156)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 136 T^{4} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 202 T^{4} + \cdots + 131044)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 242 T^{4} + \cdots + 24964)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 160 T^{4} + \cdots + 147968)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 128)^{6} \) Copy content Toggle raw display
$83$ \( (T^{6} - 264 T^{4} + \cdots - 131072)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 187 T + 358)^{4} \) Copy content Toggle raw display
$97$ \( (T^{3} + 16 T^{2} + \cdots - 722)^{4} \) Copy content Toggle raw display
show more
show less