Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [352,2,Mod(97,352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(352, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("352.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 352.m (of order \(5\), degree \(4\), 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: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} + \zeta_{10}) q^{3} + ( - 2 \zeta_{10}^{2} - 2) q^{5} + ( - 2 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{7} + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} + \zeta_{10}) q^{3} + ( - 2 \zeta_{10}^{2} - 2) q^{5} + ( - 2 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{7} + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} + \cdots + 1) q^{9}+ \cdots + (3 \zeta_{10}^{2} - 6 \zeta_{10} + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 6 q^{5} + 10 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 6 q^{5} + 10 q^{7} - q^{9} + 11 q^{11} - 4 q^{13} + 2 q^{15} - 6 q^{17} + 9 q^{19} + 20 q^{21} + 12 q^{23} + 9 q^{25} + 11 q^{27} - 6 q^{29} - 6 q^{31} + 3 q^{33} - 20 q^{35} - 2 q^{37} - 2 q^{39} - 18 q^{41} - 6 q^{43} + 4 q^{45} + 8 q^{47} - 13 q^{49} - 3 q^{51} - 12 q^{53} - 24 q^{55} + 17 q^{57} - 9 q^{59} - 8 q^{61} - 20 q^{63} + 16 q^{65} - 10 q^{67} + 6 q^{69} - 12 q^{71} - 14 q^{73} - 13 q^{75} + 40 q^{77} + 6 q^{79} - 14 q^{81} + q^{83} - 6 q^{85} - 28 q^{87} + 30 q^{89} - 20 q^{91} + 2 q^{93} - 26 q^{95} - 7 q^{97} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(133\) \(287\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
0 0.500000 0.363271i 0 −0.381966 1.17557i 0 3.61803 + 2.62866i 0 −0.809017 + 2.48990i 0
225.1 0 0.500000 + 0.363271i 0 −0.381966 + 1.17557i 0 3.61803 2.62866i 0 −0.809017 2.48990i 0
257.1 0 0.500000 1.53884i 0 −2.61803 + 1.90211i 0 1.38197 + 4.25325i 0 0.309017 + 0.224514i 0
289.1 0 0.500000 + 1.53884i 0 −2.61803 1.90211i 0 1.38197 4.25325i 0 0.309017 0.224514i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 352.2.m.b yes 4
4.b odd 2 1 352.2.m.a 4
8.b even 2 1 704.2.m.c 4
8.d odd 2 1 704.2.m.f 4
11.c even 5 1 inner 352.2.m.b yes 4
11.c even 5 1 3872.2.a.y 2
11.d odd 10 1 3872.2.a.z 2
44.g even 10 1 3872.2.a.o 2
44.h odd 10 1 352.2.m.a 4
44.h odd 10 1 3872.2.a.n 2
88.k even 10 1 7744.2.a.co 2
88.l odd 10 1 704.2.m.f 4
88.l odd 10 1 7744.2.a.cp 2
88.o even 10 1 704.2.m.c 4
88.o even 10 1 7744.2.a.ca 2
88.p odd 10 1 7744.2.a.bz 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
352.2.m.a 4 4.b odd 2 1
352.2.m.a 4 44.h odd 10 1
352.2.m.b yes 4 1.a even 1 1 trivial
352.2.m.b yes 4 11.c even 5 1 inner
704.2.m.c 4 8.b even 2 1
704.2.m.c 4 88.o even 10 1
704.2.m.f 4 8.d odd 2 1
704.2.m.f 4 88.l odd 10 1
3872.2.a.n 2 44.h odd 10 1
3872.2.a.o 2 44.g even 10 1
3872.2.a.y 2 11.c even 5 1
3872.2.a.z 2 11.d odd 10 1
7744.2.a.bz 2 88.p odd 10 1
7744.2.a.ca 2 88.o even 10 1
7744.2.a.co 2 88.k even 10 1
7744.2.a.cp 2 88.l odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2T_{3}^{3} + 4T_{3}^{2} - 3T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(352, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{4} - 10 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$11$ \( T^{4} - 11 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{4} - 9 T^{3} + \cdots + 1681 \) Copy content Toggle raw display
$23$ \( (T^{2} - 6 T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$31$ \( T^{4} + 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$41$ \( T^{4} + 18 T^{3} + \cdots + 3721 \) Copy content Toggle raw display
$43$ \( (T^{2} + 3 T - 29)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{4} + 12 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( T^{4} + 9 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$61$ \( T^{4} + 8 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$67$ \( (T^{2} + 5 T + 5)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 12 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$73$ \( T^{4} + 14 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$79$ \( T^{4} - 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$83$ \( T^{4} - T^{3} + \cdots + 121 \) Copy content Toggle raw display
$89$ \( (T^{2} - 15 T + 55)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 7 T^{3} + \cdots + 361 \) Copy content Toggle raw display
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