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Properties

Label 56.3.c.a
Level $56$
Weight $3$
Character orbit 56.c
Analytic conductor $1.526$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [56,3,Mod(41,56)]
 
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, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("56.41");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 56.c (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: no
Analytic conductor: \(1.52588948042\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + (2 \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + (2 \beta_{2} + 1) q^{9} + ( - 4 \beta_{2} + 2) q^{11} + (\beta_{3} - 5 \beta_1) q^{13} + ( - 2 \beta_{2} - 8) q^{15} - 2 \beta_{3} q^{17} + ( - 2 \beta_{3} - 7 \beta_1) q^{19} + (\beta_{3} + 6 \beta_{2} - \beta_1 - 8) q^{21} + (8 \beta_{2} - 6) q^{23} + ( - 14 \beta_{2} - 15) q^{25} + (2 \beta_{3} + 6 \beta_1) q^{27} + ( - 4 \beta_{2} + 26) q^{29} + (2 \beta_{3} - 4 \beta_1) q^{31} + ( - 4 \beta_{3} + 10 \beta_1) q^{33} + (4 \beta_{3} + 10 \beta_{2} + \cdots + 24) q^{35}+ \cdots - 62 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{7} + 4 q^{9} + 8 q^{11} - 32 q^{15} - 32 q^{21} - 24 q^{23} - 60 q^{25} + 104 q^{29} + 96 q^{35} + 40 q^{37} + 160 q^{39} - 184 q^{43} - 124 q^{49} - 152 q^{53} + 224 q^{57} + 68 q^{63} + 32 q^{65} + 8 q^{67} - 248 q^{71} - 120 q^{77} - 248 q^{79} - 156 q^{81} + 256 q^{85} + 288 q^{91} + 128 q^{93} + 480 q^{95} - 248 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{3} + 12\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} - 6\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
− 1.84776i
− 0.765367i
0.765367i
1.84776i
0 − 3.69552i 0 − 0.634051i 0 −1.82843 − 6.75699i 0 −4.65685 0
41.2 0 − 1.53073i 0 − 8.92177i 0 3.82843 + 5.86030i 0 6.65685 0
41.3 0 1.53073i 0 8.92177i 0 3.82843 − 5.86030i 0 6.65685 0
41.4 0 3.69552i 0 0.634051i 0 −1.82843 + 6.75699i 0 −4.65685 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
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 56.3.c.a ✓ 4
3.b odd 2 1 504.3.f.a 4
4.b odd 2 1 112.3.c.c 4
5.b even 2 1 1400.3.f.a 4
5.c odd 4 2 1400.3.p.a 8
7.b odd 2 1 inner 56.3.c.a ✓ 4
7.c even 3 2 392.3.o.b 8
7.d odd 6 2 392.3.o.b 8
8.b even 2 1 448.3.c.f 4
8.d odd 2 1 448.3.c.e 4
12.b even 2 1 1008.3.f.h 4
21.c even 2 1 504.3.f.a 4
28.d even 2 1 112.3.c.c 4
28.f even 6 2 784.3.s.f 8
28.g odd 6 2 784.3.s.f 8
35.c odd 2 1 1400.3.f.a 4
35.f even 4 2 1400.3.p.a 8
56.e even 2 1 448.3.c.e 4
56.h odd 2 1 448.3.c.f 4
84.h odd 2 1 1008.3.f.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.c.a ✓ 4 1.a even 1 1 trivial
56.3.c.a ✓ 4 7.b odd 2 1 inner
112.3.c.c 4 4.b odd 2 1
112.3.c.c 4 28.d even 2 1
392.3.o.b 8 7.c even 3 2
392.3.o.b 8 7.d odd 6 2
448.3.c.e 4 8.d odd 2 1
448.3.c.e 4 56.e even 2 1
448.3.c.f 4 8.b even 2 1
448.3.c.f 4 56.h odd 2 1
504.3.f.a 4 3.b odd 2 1
504.3.f.a 4 21.c even 2 1
784.3.s.f 8 28.f even 6 2
784.3.s.f 8 28.g odd 6 2
1008.3.f.h 4 12.b even 2 1
1008.3.f.h 4 84.h odd 2 1
1400.3.f.a 4 5.b even 2 1
1400.3.f.a 4 35.c odd 2 1
1400.3.p.a 8 5.c odd 4 2
1400.3.p.a 8 35.f even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(56, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 16T^{2} + 32 \) Copy content Toggle raw display
$5$ \( T^{4} + 80T^{2} + 32 \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T - 124)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 464T^{2} + 32 \) Copy content Toggle raw display
$17$ \( T^{4} + 256T^{2} + 8192 \) Copy content Toggle raw display
$19$ \( T^{4} + 1040 T^{2} + 253472 \) Copy content Toggle raw display
$23$ \( (T^{2} + 12 T - 476)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 52 T + 548)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 512 T^{2} + 32768 \) Copy content Toggle raw display
$37$ \( (T^{2} - 20 T - 1052)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 5440 T^{2} + 2580992 \) Copy content Toggle raw display
$43$ \( (T + 46)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 7424 T^{2} + 8192 \) Copy content Toggle raw display
$53$ \( (T^{2} + 76 T + 932)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 3600 T^{2} + 749088 \) Copy content Toggle raw display
$61$ \( T^{4} + 7632 T^{2} + 749088 \) Copy content Toggle raw display
$67$ \( (T^{2} - 4 T - 508)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 124 T + 3556)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 3136 T^{2} + 1229312 \) Copy content Toggle raw display
$79$ \( (T^{2} + 124 T + 3044)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 11152 T^{2} + 30921248 \) Copy content Toggle raw display
$89$ \( T^{4} + 6208 T^{2} + 25088 \) Copy content Toggle raw display
$97$ \( T^{4} + 33280 T^{2} + 204505088 \) Copy content Toggle raw display
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