Properties

Label 356.1.d.b
Level $356$
Weight $1$
Character orbit 356.d
Self dual yes
Analytic conductor $0.178$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -356
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [356,1,Mod(355,356)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(356, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("356.355");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 356 = 2^{2} \cdot 89 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 356.d (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: yes
Analytic conductor: \(0.177667144497\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.356.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.356.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 2 q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 2 q^{7} + q^{8} - q^{10} - q^{12} + 2 q^{14} + q^{15} + q^{16} - q^{17} - q^{19} - q^{20} - 2 q^{21} - q^{23} - q^{24} + q^{27} + 2 q^{28} + q^{30} - q^{31} + q^{32} - q^{34} - 2 q^{35} - q^{38} - q^{40} - 2 q^{42} - q^{43} - q^{46} - q^{48} + 3 q^{49} + q^{51} - q^{53} + q^{54} + 2 q^{56} + q^{57} + 2 q^{59} + q^{60} - q^{62} + q^{64} - q^{68} + q^{69} - 2 q^{70} - q^{73} - q^{76} - q^{80} - q^{81} + 2 q^{83} - 2 q^{84} + q^{85} - q^{86} + q^{89} - q^{92} + q^{93} + q^{95} - q^{96} - q^{97} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/356\mathbb{Z}\right)^\times\).

\(n\) \(179\) \(181\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
355.1
0
1.00000 −1.00000 1.00000 −1.00000 −1.00000 2.00000 1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
356.d odd 2 1 CM by \(\Q(\sqrt{-89}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 356.1.d.b 1
3.b odd 2 1 3204.1.h.b 1
4.b odd 2 1 356.1.d.c yes 1
12.b even 2 1 3204.1.h.a 1
89.b even 2 1 356.1.d.c yes 1
267.b odd 2 1 3204.1.h.a 1
356.d odd 2 1 CM 356.1.d.b 1
1068.e even 2 1 3204.1.h.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
356.1.d.b 1 1.a even 1 1 trivial
356.1.d.b 1 356.d odd 2 1 CM
356.1.d.c yes 1 4.b odd 2 1
356.1.d.c yes 1 89.b even 2 1
3204.1.h.a 1 12.b even 2 1
3204.1.h.a 1 267.b odd 2 1
3204.1.h.b 1 3.b odd 2 1
3204.1.h.b 1 1068.e even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(356, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 1 \) Copy content Toggle raw display
$19$ \( T + 1 \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 1 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 1 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 1 \) Copy content Toggle raw display
$59$ \( T - 2 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 1 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 2 \) Copy content Toggle raw display
$89$ \( T - 1 \) Copy content Toggle raw display
$97$ \( T + 1 \) Copy content Toggle raw display
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