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Properties

Label 352.2.g.b
Level $352$
Weight $2$
Character orbit 352.g
Analytic conductor $2.811$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [352,2,Mod(175,352)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("352.175");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 352.g (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: \(8\)
Coefficient field: 8.0.10070523904.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 6x^{4} + 16 \) 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} + \beta_{6} q^{5} - \beta_{7} q^{7} + 2 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{3} + \beta_{6} q^{5} - \beta_{7} q^{7} + 2 \beta_1 q^{9} + ( - \beta_{5} + \beta_1 - 1) q^{11} + (\beta_{7} - \beta_{3}) q^{13} + (\beta_{6} - \beta_{4}) q^{15} + ( - \beta_{5} - \beta_{2}) q^{17} + \beta_{5} q^{19} + ( - \beta_{7} + \beta_{3}) q^{21} + (\beta_{6} + \beta_{4}) q^{23} - 2 q^{25} + ( - \beta_1 + 1) q^{27} - 2 \beta_{7} q^{29} + ( - \beta_{6} + \beta_{4}) q^{31} + ( - \beta_{5} + \beta_{2} + 1) q^{33} + ( - \beta_{5} - 2 \beta_{2}) q^{35} + ( - \beta_{6} + 2 \beta_{4}) q^{37} + (3 \beta_{7} - 2 \beta_{3}) q^{39} + (3 \beta_{5} + \beta_{2}) q^{41} + 2 \beta_{2} q^{43} - 2 \beta_{4} q^{45} + ( - 2 \beta_1 + 1) q^{49} + \beta_{5} q^{51} - 2 \beta_{4} q^{53} + ( - \beta_{7} - \beta_{6} + \cdots + 2 \beta_{3}) q^{55}+ \cdots + (2 \beta_{2} - 2 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} - 8 q^{11} - 16 q^{25} + 8 q^{27} + 8 q^{33} + 8 q^{49} + 8 q^{59} - 8 q^{67} - 16 q^{75} - 8 q^{81} - 40 q^{89} - 32 q^{91} - 8 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 6x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 2\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 6\nu^{3} + 8\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 6\nu^{3} + 8\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 10\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 2\nu^{5} - 2\nu^{3} + 4\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{4} + 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 2\nu^{5} - 2\nu^{3} - 4\nu ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + \beta_{5} - \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{7} + 2\beta_{5} - \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{4} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -3\beta_{7} - 3\beta_{5} + \beta_{3} - \beta_{2} \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
175.1
1.16342 − 0.804019i
−1.16342 + 0.804019i
1.16342 + 0.804019i
−1.16342 − 0.804019i
−0.804019 − 1.16342i
0.804019 + 1.16342i
−0.804019 + 1.16342i
0.804019 − 1.16342i
0 −0.414214 0 − 2.64575i 0 −3.29066 0 −2.82843 0
175.2 0 −0.414214 0 − 2.64575i 0 3.29066 0 −2.82843 0
175.3 0 −0.414214 0 2.64575i 0 −3.29066 0 −2.82843 0
175.4 0 −0.414214 0 2.64575i 0 3.29066 0 −2.82843 0
175.5 0 2.41421 0 − 2.64575i 0 −2.27411 0 2.82843 0
175.6 0 2.41421 0 − 2.64575i 0 2.27411 0 2.82843 0
175.7 0 2.41421 0 2.64575i 0 −2.27411 0 2.82843 0
175.8 0 2.41421 0 2.64575i 0 2.27411 0 2.82843 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 175.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
11.b odd 2 1 inner
88.g 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.g.b 8
3.b odd 2 1 3168.2.h.g 8
4.b odd 2 1 88.2.g.b ✓ 8
8.b even 2 1 88.2.g.b ✓ 8
8.d odd 2 1 inner 352.2.g.b 8
11.b odd 2 1 inner 352.2.g.b 8
12.b even 2 1 792.2.h.g 8
16.e even 4 2 2816.2.e.o 16
16.f odd 4 2 2816.2.e.o 16
24.f even 2 1 3168.2.h.g 8
24.h odd 2 1 792.2.h.g 8
33.d even 2 1 3168.2.h.g 8
44.c even 2 1 88.2.g.b ✓ 8
44.g even 10 4 968.2.k.g 32
44.h odd 10 4 968.2.k.g 32
88.b odd 2 1 88.2.g.b ✓ 8
88.g even 2 1 inner 352.2.g.b 8
88.o even 10 4 968.2.k.g 32
88.p odd 10 4 968.2.k.g 32
132.d odd 2 1 792.2.h.g 8
176.i even 4 2 2816.2.e.o 16
176.l odd 4 2 2816.2.e.o 16
264.m even 2 1 792.2.h.g 8
264.p odd 2 1 3168.2.h.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.g.b ✓ 8 4.b odd 2 1
88.2.g.b ✓ 8 8.b even 2 1
88.2.g.b ✓ 8 44.c even 2 1
88.2.g.b ✓ 8 88.b odd 2 1
352.2.g.b 8 1.a even 1 1 trivial
352.2.g.b 8 8.d odd 2 1 inner
352.2.g.b 8 11.b odd 2 1 inner
352.2.g.b 8 88.g even 2 1 inner
792.2.h.g 8 12.b even 2 1
792.2.h.g 8 24.h odd 2 1
792.2.h.g 8 132.d odd 2 1
792.2.h.g 8 264.m even 2 1
968.2.k.g 32 44.g even 10 4
968.2.k.g 32 44.h odd 10 4
968.2.k.g 32 88.o even 10 4
968.2.k.g 32 88.p odd 10 4
2816.2.e.o 16 16.e even 4 2
2816.2.e.o 16 16.f odd 4 2
2816.2.e.o 16 176.i even 4 2
2816.2.e.o 16 176.l odd 4 2
3168.2.h.g 8 3.b odd 2 1
3168.2.h.g 8 24.f even 2 1
3168.2.h.g 8 33.d even 2 1
3168.2.h.g 8 264.p odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(352, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2 T - 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 7)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 16 T^{2} + 56)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 32 T^{2} + 56)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 32 T^{2} + 56)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 16 T^{2} + 56)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 42 T^{2} + 49)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 64 T^{2} + 896)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 42 T^{2} + 49)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 126 T^{2} + 2401)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 128 T^{2} + 2744)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 128 T^{2} + 3584)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{2} + 56)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 2 T - 17)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} - 64 T^{2} + 896)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T - 49)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 42 T^{2} + 49)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 64 T^{2} + 56)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 128 T^{2} + 3584)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 64 T^{2} + 896)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 10 T - 7)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T - 71)^{4} \) Copy content Toggle raw display
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