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Properties

Label 56.2.e.b
Level $56$
Weight $2$
Character orbit 56.e
Analytic conductor $0.447$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [56,2,Mod(27,56)]
 
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([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("56.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 56.e (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: \(0.447162251319\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + \beta_{2} q^{3} - 2 \beta_1 q^{4} - \beta_{3} q^{5} + ( - \beta_{3} - \beta_{2}) q^{6} + (\beta_{3} + \beta_1) q^{7} + (2 \beta_1 + 2) q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + \beta_{2} q^{3} - 2 \beta_1 q^{4} - \beta_{3} q^{5} + ( - \beta_{3} - \beta_{2}) q^{6} + (\beta_{3} + \beta_1) q^{7} + (2 \beta_1 + 2) q^{8} - 3 q^{9} + (\beta_{3} - \beta_{2}) q^{10} + 2 q^{11} + 2 \beta_{3} q^{12} + \beta_{3} q^{13} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{14} - 6 \beta_1 q^{15} - 4 q^{16} - 2 \beta_{2} q^{17} + ( - 3 \beta_1 + 3) q^{18} - \beta_{2} q^{19} + 2 \beta_{2} q^{20} + ( - \beta_{3} + 6 \beta_1) q^{21} + (2 \beta_1 - 2) q^{22} + 4 \beta_1 q^{23} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{24} + q^{25} + ( - \beta_{3} + \beta_{2}) q^{26} + ( - 2 \beta_{2} + 2) q^{28} - 4 \beta_1 q^{29} + (6 \beta_1 + 6) q^{30} - 2 \beta_{3} q^{31} + ( - 4 \beta_1 + 4) q^{32} + 2 \beta_{2} q^{33} + (2 \beta_{3} + 2 \beta_{2}) q^{34} + ( - \beta_{2} - 6) q^{35} + 6 \beta_1 q^{36} - 8 \beta_1 q^{37} + (\beta_{3} + \beta_{2}) q^{38} + 6 \beta_1 q^{39} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{40} + (\beta_{3} - \beta_{2} - 6 \beta_1 - 6) q^{42} - 6 q^{43} - 4 \beta_1 q^{44} + 3 \beta_{3} q^{45} + ( - 4 \beta_1 - 4) q^{46} + 2 \beta_{3} q^{47} - 4 \beta_{2} q^{48} + (2 \beta_{2} + 5) q^{49} + (\beta_1 - 1) q^{50} + 12 q^{51} - 2 \beta_{2} q^{52} + 4 \beta_1 q^{53} - 2 \beta_{3} q^{55} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{56} + 6 q^{57} + (4 \beta_1 + 4) q^{58} - \beta_{2} q^{59} - 12 q^{60} + 3 \beta_{3} q^{61} + (2 \beta_{3} - 2 \beta_{2}) q^{62} + ( - 3 \beta_{3} - 3 \beta_1) q^{63} + 8 \beta_1 q^{64} - 6 q^{65} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{66} - 2 q^{67} - 4 \beta_{3} q^{68} - 4 \beta_{3} q^{69} + (\beta_{3} + \beta_{2} - 6 \beta_1 + 6) q^{70} - 10 \beta_1 q^{71} + ( - 6 \beta_1 - 6) q^{72} + 6 \beta_{2} q^{73} + (8 \beta_1 + 8) q^{74} + \beta_{2} q^{75} - 2 \beta_{3} q^{76} + (2 \beta_{3} + 2 \beta_1) q^{77} + ( - 6 \beta_1 - 6) q^{78} + 6 \beta_1 q^{79} + 4 \beta_{3} q^{80} - 9 q^{81} + \beta_{2} q^{83} + (2 \beta_{2} + 12) q^{84} + 12 \beta_1 q^{85} + ( - 6 \beta_1 + 6) q^{86} + 4 \beta_{3} q^{87} + (4 \beta_1 + 4) q^{88} - 6 \beta_{2} q^{89} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{90} + (\beta_{2} + 6) q^{91} + 8 q^{92} - 12 \beta_1 q^{93} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{94} + 6 \beta_1 q^{95} + (4 \beta_{3} + 4 \beta_{2}) q^{96} - 2 \beta_{2} q^{97} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots - 5) q^{98}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 8 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 8 q^{8} - 12 q^{9} + 8 q^{11} - 4 q^{14} - 16 q^{16} + 12 q^{18} - 8 q^{22} + 4 q^{25} + 8 q^{28} + 24 q^{30} + 16 q^{32} - 24 q^{35} - 24 q^{42} - 24 q^{43} - 16 q^{46} + 20 q^{49} - 4 q^{50} + 48 q^{51} - 8 q^{56} + 24 q^{57} + 16 q^{58} - 48 q^{60} - 24 q^{65} - 8 q^{67} + 24 q^{70} - 24 q^{72} + 32 q^{74} - 24 q^{78} - 36 q^{81} + 48 q^{84} + 24 q^{86} + 16 q^{88} + 24 q^{91} + 32 q^{92} - 20 q^{98} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

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

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

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

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
1.22474 − 1.22474i
−1.22474 + 1.22474i
−1.22474 − 1.22474i
1.22474 + 1.22474i
−1.00000 − 1.00000i − 2.44949i 2.00000i −2.44949 −2.44949 + 2.44949i 2.44949 − 1.00000i 2.00000 − 2.00000i −3.00000 2.44949 + 2.44949i
27.2 −1.00000 − 1.00000i 2.44949i 2.00000i 2.44949 2.44949 − 2.44949i −2.44949 − 1.00000i 2.00000 − 2.00000i −3.00000 −2.44949 − 2.44949i
27.3 −1.00000 + 1.00000i − 2.44949i − 2.00000i 2.44949 2.44949 + 2.44949i −2.44949 + 1.00000i 2.00000 + 2.00000i −3.00000 −2.44949 + 2.44949i
27.4 −1.00000 + 1.00000i 2.44949i − 2.00000i −2.44949 −2.44949 − 2.44949i 2.44949 + 1.00000i 2.00000 + 2.00000i −3.00000 2.44949 − 2.44949i
\(n\): 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
8.d odd 2 1 inner
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 56.2.e.b ✓ 4
3.b odd 2 1 504.2.p.f 4
4.b odd 2 1 224.2.e.b 4
7.b odd 2 1 inner 56.2.e.b ✓ 4
7.c even 3 2 392.2.m.f 8
7.d odd 6 2 392.2.m.f 8
8.b even 2 1 224.2.e.b 4
8.d odd 2 1 inner 56.2.e.b ✓ 4
12.b even 2 1 2016.2.p.e 4
16.e even 4 1 1792.2.f.e 4
16.e even 4 1 1792.2.f.f 4
16.f odd 4 1 1792.2.f.e 4
16.f odd 4 1 1792.2.f.f 4
21.c even 2 1 504.2.p.f 4
24.f even 2 1 504.2.p.f 4
24.h odd 2 1 2016.2.p.e 4
28.d even 2 1 224.2.e.b 4
28.f even 6 2 1568.2.q.e 8
28.g odd 6 2 1568.2.q.e 8
56.e even 2 1 inner 56.2.e.b ✓ 4
56.h odd 2 1 224.2.e.b 4
56.j odd 6 2 1568.2.q.e 8
56.k odd 6 2 392.2.m.f 8
56.m even 6 2 392.2.m.f 8
56.p even 6 2 1568.2.q.e 8
84.h odd 2 1 2016.2.p.e 4
112.j even 4 1 1792.2.f.e 4
112.j even 4 1 1792.2.f.f 4
112.l odd 4 1 1792.2.f.e 4
112.l odd 4 1 1792.2.f.f 4
168.e odd 2 1 504.2.p.f 4
168.i even 2 1 2016.2.p.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.e.b ✓ 4 1.a even 1 1 trivial
56.2.e.b ✓ 4 7.b odd 2 1 inner
56.2.e.b ✓ 4 8.d odd 2 1 inner
56.2.e.b ✓ 4 56.e even 2 1 inner
224.2.e.b 4 4.b odd 2 1
224.2.e.b 4 8.b even 2 1
224.2.e.b 4 28.d even 2 1
224.2.e.b 4 56.h odd 2 1
392.2.m.f 8 7.c even 3 2
392.2.m.f 8 7.d odd 6 2
392.2.m.f 8 56.k odd 6 2
392.2.m.f 8 56.m even 6 2
504.2.p.f 4 3.b odd 2 1
504.2.p.f 4 21.c even 2 1
504.2.p.f 4 24.f even 2 1
504.2.p.f 4 168.e odd 2 1
1568.2.q.e 8 28.f even 6 2
1568.2.q.e 8 28.g odd 6 2
1568.2.q.e 8 56.j odd 6 2
1568.2.q.e 8 56.p even 6 2
1792.2.f.e 4 16.e even 4 1
1792.2.f.e 4 16.f odd 4 1
1792.2.f.e 4 112.j even 4 1
1792.2.f.e 4 112.l odd 4 1
1792.2.f.f 4 16.e even 4 1
1792.2.f.f 4 16.f odd 4 1
1792.2.f.f 4 112.j even 4 1
1792.2.f.f 4 112.l odd 4 1
2016.2.p.e 4 12.b even 2 1
2016.2.p.e 4 24.h odd 2 1
2016.2.p.e 4 84.h odd 2 1
2016.2.p.e 4 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 6 \) acting on \(S_{2}^{\mathrm{new}}(56, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 10T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T - 2)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T + 6)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
$67$ \( (T + 2)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 216)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 216)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
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