Properties

Label 56.5.h.b
Level $56$
Weight $5$
Character orbit 56.h
Self dual yes
Analytic conductor $5.789$
Analytic rank $0$
Dimension $1$
CM discriminant -56
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [56,5,Mod(13,56)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(56, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("56.13");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 56.h (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: \(5.78871793270\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 10 q^{3} + 16 q^{4} - 22 q^{5} + 40 q^{6} + 49 q^{7} + 64 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 10 q^{3} + 16 q^{4} - 22 q^{5} + 40 q^{6} + 49 q^{7} + 64 q^{8} + 19 q^{9} - 88 q^{10} + 160 q^{12} - 310 q^{13} + 196 q^{14} - 220 q^{15} + 256 q^{16} + 76 q^{18} + 650 q^{19} - 352 q^{20} + 490 q^{21} - 958 q^{23} + 640 q^{24} - 141 q^{25} - 1240 q^{26} - 620 q^{27} + 784 q^{28} - 880 q^{30} + 1024 q^{32} - 1078 q^{35} + 304 q^{36} + 2600 q^{38} - 3100 q^{39} - 1408 q^{40} + 1960 q^{42} - 418 q^{45} - 3832 q^{46} + 2560 q^{48} + 2401 q^{49} - 564 q^{50} - 4960 q^{52} - 2480 q^{54} + 3136 q^{56} + 6500 q^{57} + 1130 q^{59} - 3520 q^{60} + 7370 q^{61} + 931 q^{63} + 4096 q^{64} + 6820 q^{65} - 9580 q^{69} - 4312 q^{70} + 2018 q^{71} + 1216 q^{72} - 1410 q^{75} + 10400 q^{76} - 12400 q^{78} + 4418 q^{79} - 5632 q^{80} - 7739 q^{81} + 13130 q^{83} + 7840 q^{84} - 1672 q^{90} - 15190 q^{91} - 15328 q^{92} - 14300 q^{95} + 10240 q^{96} + 9604 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0
4.00000 10.0000 16.0000 −22.0000 40.0000 49.0000 64.0000 19.0000 −88.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 56.5.h.b yes 1
4.b odd 2 1 224.5.h.a 1
7.b odd 2 1 56.5.h.a 1
8.b even 2 1 56.5.h.a 1
8.d odd 2 1 224.5.h.b 1
28.d even 2 1 224.5.h.b 1
56.e even 2 1 224.5.h.a 1
56.h odd 2 1 CM 56.5.h.b yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.5.h.a 1 7.b odd 2 1
56.5.h.a 1 8.b even 2 1
56.5.h.b yes 1 1.a even 1 1 trivial
56.5.h.b yes 1 56.h odd 2 1 CM
224.5.h.a 1 4.b odd 2 1
224.5.h.a 1 56.e even 2 1
224.5.h.b 1 8.d odd 2 1
224.5.h.b 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 10 \) acting on \(S_{5}^{\mathrm{new}}(56, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T - 10 \) Copy content Toggle raw display
$5$ \( T + 22 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 310 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 650 \) Copy content Toggle raw display
$23$ \( T + 958 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T - 1130 \) Copy content Toggle raw display
$61$ \( T - 7370 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T - 2018 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T - 4418 \) Copy content Toggle raw display
$83$ \( T - 13130 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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