↑

Properties

Label 352.2.c.a
Level $352$
Weight $2$
Character orbit 352.c
Analytic conductor $2.811$
Analytic rank $0$
Dimension $10$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [352,2,Mod(177,352)]
 
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([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("352.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 352.c (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: \(2.81073415115\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.578281160704.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 2x^{8} - 2x^{7} - 3x^{6} - 6x^{5} - 6x^{4} - 8x^{3} + 16x^{2} + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 88)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{6} - \beta_{5}) q^{5} + ( - \beta_{4} - \beta_{2}) q^{7} + (\beta_{9} - \beta_{4} - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{6} - \beta_{5}) q^{5} + ( - \beta_{4} - \beta_{2}) q^{7} + (\beta_{9} - \beta_{4} - \beta_{3} - 1) q^{9} + \beta_{6} q^{11} + ( - \beta_{8} + 2 \beta_{6} + \beta_1) q^{13} + (\beta_{4} - \beta_{3} - \beta_{2} - 1) q^{15} + (\beta_{9} + \beta_{3}) q^{17} + ( - 2 \beta_{8} - \beta_{7} - \beta_{5}) q^{19} + ( - \beta_{8} - 2 \beta_{7} + \cdots - \beta_1) q^{21}+ \cdots + ( - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{9} - 8 q^{15} - 4 q^{17} + 12 q^{23} - 6 q^{25} + 4 q^{31} - 24 q^{39} + 4 q^{41} + 4 q^{47} - 6 q^{49} + 8 q^{55} + 16 q^{57} + 40 q^{63} + 16 q^{65} + 12 q^{71} - 4 q^{73} - 16 q^{79} - 6 q^{81} - 32 q^{87} - 4 q^{89} - 24 q^{95} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 2x^{8} - 2x^{7} - 3x^{6} - 6x^{5} - 6x^{4} - 8x^{3} + 16x^{2} + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{8} + 2\nu^{6} - 2\nu^{5} - 3\nu^{4} - 6\nu^{3} + 2\nu^{2} - 8\nu + 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{8} + 2\nu^{5} + 3\nu^{4} + 2\nu^{3} + 4\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{9} + 4\nu^{8} - 2\nu^{7} + 6\nu^{6} + 9\nu^{5} - 2\nu^{4} - 18\nu^{3} - 72\nu + 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{9} + 6\nu^{8} - 2\nu^{7} + 2\nu^{6} + 5\nu^{5} - 8\nu^{4} - 14\nu^{3} + 28\nu^{2} - 72\nu + 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{9} + 4\nu^{8} + 10\nu^{7} + 10\nu^{6} + 7\nu^{5} - 6\nu^{4} - 22\nu^{3} - 64\nu^{2} - 56\nu - 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{9} - 4\nu^{8} - 2\nu^{7} - 2\nu^{6} + 9\nu^{5} + 22\nu^{4} + 14\nu^{3} + 24\nu^{2} - 8\nu - 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{9} + \nu^{8} - 2\nu^{6} - 8\nu^{5} - 7\nu^{4} - 6\nu^{3} - 6\nu^{2} + 24\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{9} - 4\nu^{8} + 10\nu^{7} + 10\nu^{6} + 31\nu^{5} + 34\nu^{4} - 6\nu^{3} - 16\nu^{2} - 72\nu - 176 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -4\nu^{9} + 7\nu^{8} - 4\nu^{7} + 6\nu^{6} + 6\nu^{5} - 5\nu^{4} - 14\nu^{3} + 22\nu^{2} - 80\nu + 72 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} + \beta_{7} - 2\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} - \beta_{8} - 3\beta_{7} + \beta_{5} - \beta_{4} - 3\beta_{3} - \beta_{2} - \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{9} + 2\beta_{7} + 3\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{9} + \beta_{8} + \beta_{7} - 6\beta_{6} - \beta_{5} + 3\beta_{4} - \beta_{3} + 9\beta_{2} - 5\beta _1 + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{9} - 2\beta_{7} - 2\beta_{6} - 3\beta_{4} - 3\beta_{3} + 3\beta_{2} + 5\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -\beta_{9} + 5\beta_{8} - \beta_{7} - 4\beta_{6} - \beta_{5} + 5\beta_{4} - 9\beta_{3} - 7\beta_{2} + \beta _1 + 26 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 7\beta_{9} + 2\beta_{8} + 6\beta_{7} - \beta_{6} + \beta_{5} - 3\beta_{4} - 9\beta_{3} + 2\beta_{2} - 8\beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - \beta_{9} - 13 \beta_{8} + 3 \beta_{7} + 18 \beta_{6} + 17 \beta_{5} + 17 \beta_{4} - 11 \beta_{3} + \cdots + 13 \beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
−1.28245 + 0.596081i
0.437403 − 1.34487i
−0.329042 + 1.37540i
−0.239536 + 1.39378i
1.41363 − 0.0406696i
1.41363 + 0.0406696i
−0.239536 − 1.39378i
−0.329042 − 1.37540i
0.437403 + 1.34487i
−1.28245 − 0.596081i
0 − 3.05779i 0 0.699283i 0 −3.27803 0 −6.35006 0
177.2 0 − 2.35300i 0 − 4.16794i 0 −0.933222 0 −2.53661 0
177.3 0 − 1.81026i 0 0.282461i 0 3.84939 0 −0.277041 0
177.4 0 − 1.33544i 0 1.93119i 0 1.83930 0 1.21660 0
177.5 0 − 0.229967i 0 2.51595i 0 −1.47743 0 2.94712 0
177.6 0 0.229967i 0 − 2.51595i 0 −1.47743 0 2.94712 0
177.7 0 1.33544i 0 − 1.93119i 0 1.83930 0 1.21660 0
177.8 0 1.81026i 0 − 0.282461i 0 3.84939 0 −0.277041 0
177.9 0 2.35300i 0 4.16794i 0 −0.933222 0 −2.53661 0
177.10 0 3.05779i 0 − 0.699283i 0 −3.27803 0 −6.35006 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.10
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 352.2.c.a 10
3.b odd 2 1 3168.2.f.g 10
4.b odd 2 1 88.2.c.a ✓ 10
8.b even 2 1 inner 352.2.c.a 10
8.d odd 2 1 88.2.c.a ✓ 10
11.b odd 2 1 3872.2.c.f 10
12.b even 2 1 792.2.f.g 10
16.e even 4 1 2816.2.a.p 5
16.e even 4 1 2816.2.a.q 5
16.f odd 4 1 2816.2.a.o 5
16.f odd 4 1 2816.2.a.r 5
24.f even 2 1 792.2.f.g 10
24.h odd 2 1 3168.2.f.g 10
44.c even 2 1 968.2.c.d 10
44.g even 10 4 968.2.o.h 40
44.h odd 10 4 968.2.o.g 40
88.b odd 2 1 3872.2.c.f 10
88.g even 2 1 968.2.c.d 10
88.k even 10 4 968.2.o.h 40
88.l odd 10 4 968.2.o.g 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.c.a ✓ 10 4.b odd 2 1
88.2.c.a ✓ 10 8.d odd 2 1
352.2.c.a 10 1.a even 1 1 trivial
352.2.c.a 10 8.b even 2 1 inner
792.2.f.g 10 12.b even 2 1
792.2.f.g 10 24.f even 2 1
968.2.c.d 10 44.c even 2 1
968.2.c.d 10 88.g even 2 1
968.2.o.g 40 44.h odd 10 4
968.2.o.g 40 88.l odd 10 4
968.2.o.h 40 44.g even 10 4
968.2.o.h 40 88.k even 10 4
2816.2.a.o 5 16.f odd 4 1
2816.2.a.p 5 16.e even 4 1
2816.2.a.q 5 16.e even 4 1
2816.2.a.r 5 16.f odd 4 1
3168.2.f.g 10 3.b odd 2 1
3168.2.f.g 10 24.h odd 2 1
3872.2.c.f 10 11.b odd 2 1
3872.2.c.f 10 88.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 20 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{10} + 28 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{5} - 16 T^{3} + \cdots + 32)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{10} + 52 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( (T^{5} + 2 T^{4} + \cdots + 464)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 96 T^{8} + \cdots + 262144 \) Copy content Toggle raw display
$23$ \( (T^{5} - 6 T^{4} + \cdots - 314)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + 132 T^{8} + \cdots + 262144 \) Copy content Toggle raw display
$31$ \( (T^{5} - 2 T^{4} + \cdots + 226)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 140 T^{8} + \cdots + 179776 \) Copy content Toggle raw display
$41$ \( (T^{5} - 2 T^{4} + \cdots - 464)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 100962304 \) Copy content Toggle raw display
$47$ \( (T^{5} - 2 T^{4} + \cdots + 224)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 448 T^{8} + \cdots + 44302336 \) Copy content Toggle raw display
$59$ \( T^{10} + 148 T^{8} + \cdots + 183184 \) Copy content Toggle raw display
$61$ \( T^{10} + 260 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$67$ \( T^{10} + 180 T^{8} + \cdots + 446224 \) Copy content Toggle raw display
$71$ \( (T^{5} - 6 T^{4} + \cdots - 83746)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 2 T^{4} + \cdots + 12752)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + 8 T^{4} + \cdots - 11008)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 400 T^{8} + \cdots + 802816 \) Copy content Toggle raw display
$89$ \( (T^{5} + 2 T^{4} + \cdots + 3566)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + 10 T^{4} + \cdots + 20462)^{2} \) Copy content Toggle raw display
show more
show less