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Properties

Label 280.2.n.a
Level $280$
Weight $2$
Character orbit 280.n
Analytic conductor $2.236$
Analytic rank $0$
Dimension $4$
CM discriminant -40
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [280,2,Mod(139,280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("280.139");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.n (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: \(2.23581125660\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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)
 

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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 2 q^{4} + \beta_{3} q^{5} + (\beta_{3} - \beta_1) q^{7} - 2 \beta_1 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 2 q^{4} + \beta_{3} q^{5} + (\beta_{3} - \beta_1) q^{7} - 2 \beta_1 q^{8} - 3 q^{9} + \beta_{2} q^{10} - 2 q^{11} - 2 \beta_{3} q^{13} + (\beta_{2} + 2) q^{14} + 4 q^{16} + 3 \beta_1 q^{18} - 2 \beta_{2} q^{19} + 2 \beta_{3} q^{20} + 2 \beta_1 q^{22} - 6 \beta_1 q^{23} - 5 q^{25} - 2 \beta_{2} q^{26} + (2 \beta_{3} - 2 \beta_1) q^{28} - 4 \beta_1 q^{32} + (\beta_{2} - 5) q^{35} - 6 q^{36} + 8 \beta_1 q^{37} - 4 \beta_{3} q^{38} + 2 \beta_{2} q^{40} - 4 \beta_{2} q^{41} - 4 q^{44} - 3 \beta_{3} q^{45} + 12 q^{46} + 6 \beta_{3} q^{47} + (2 \beta_{2} - 3) q^{49} + 5 \beta_1 q^{50} - 4 \beta_{3} q^{52} + 4 \beta_1 q^{53} - 2 \beta_{3} q^{55} + (2 \beta_{2} + 4) q^{56} + 2 \beta_{2} q^{59} + ( - 3 \beta_{3} + 3 \beta_1) q^{63} + 8 q^{64} + 10 q^{65} + (2 \beta_{3} + 5 \beta_1) q^{70} + 6 \beta_1 q^{72} - 16 q^{74} - 4 \beta_{2} q^{76} + ( - 2 \beta_{3} + 2 \beta_1) q^{77} + 4 \beta_{3} q^{80} + 9 q^{81} - 8 \beta_{3} q^{82} + 4 \beta_1 q^{88} + 4 \beta_{2} q^{89} - 3 \beta_{2} q^{90} + ( - 2 \beta_{2} + 10) q^{91} - 12 \beta_1 q^{92} + 6 \beta_{2} q^{94} - 10 \beta_1 q^{95} + (4 \beta_{3} + 3 \beta_1) q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} - 12 q^{9} - 8 q^{11} + 8 q^{14} + 16 q^{16} - 20 q^{25} - 20 q^{35} - 24 q^{36} - 16 q^{44} + 48 q^{46} - 12 q^{49} + 16 q^{56} + 32 q^{64} + 40 q^{65} - 64 q^{74} + 36 q^{81} + 40 q^{91} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 7\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 7\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1
−0.707107 + 1.58114i
−0.707107 − 1.58114i
0.707107 − 1.58114i
0.707107 + 1.58114i
−1.41421 0 2.00000 − 2.23607i 0 −1.41421 − 2.23607i −2.82843 −3.00000 3.16228i
139.2 −1.41421 0 2.00000 2.23607i 0 −1.41421 + 2.23607i −2.82843 −3.00000 − 3.16228i
139.3 1.41421 0 2.00000 − 2.23607i 0 1.41421 − 2.23607i 2.82843 −3.00000 − 3.16228i
139.4 1.41421 0 2.00000 2.23607i 0 1.41421 + 2.23607i 2.82843 −3.00000 3.16228i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
5.b even 2 1 inner
7.b odd 2 1 inner
8.d odd 2 1 inner
35.c odd 2 1 inner
56.e even 2 1 inner
280.n even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.n.a ✓ 4
4.b odd 2 1 1120.2.n.a 4
5.b even 2 1 inner 280.2.n.a ✓ 4
7.b odd 2 1 inner 280.2.n.a ✓ 4
8.b even 2 1 1120.2.n.a 4
8.d odd 2 1 inner 280.2.n.a ✓ 4
20.d odd 2 1 1120.2.n.a 4
28.d even 2 1 1120.2.n.a 4
35.c odd 2 1 inner 280.2.n.a ✓ 4
40.e odd 2 1 CM 280.2.n.a ✓ 4
40.f even 2 1 1120.2.n.a 4
56.e even 2 1 inner 280.2.n.a ✓ 4
56.h odd 2 1 1120.2.n.a 4
140.c even 2 1 1120.2.n.a 4
280.c odd 2 1 1120.2.n.a 4
280.n even 2 1 inner 280.2.n.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.n.a ✓ 4 1.a even 1 1 trivial
280.2.n.a ✓ 4 5.b even 2 1 inner
280.2.n.a ✓ 4 7.b odd 2 1 inner
280.2.n.a ✓ 4 8.d odd 2 1 inner
280.2.n.a ✓ 4 35.c odd 2 1 inner
280.2.n.a ✓ 4 40.e odd 2 1 CM
280.2.n.a ✓ 4 56.e even 2 1 inner
280.2.n.a ✓ 4 280.n even 2 1 inner
1120.2.n.a 4 4.b odd 2 1
1120.2.n.a 4 8.b even 2 1
1120.2.n.a 4 20.d odd 2 1
1120.2.n.a 4 28.d even 2 1
1120.2.n.a 4 40.f even 2 1
1120.2.n.a 4 56.h odd 2 1
1120.2.n.a 4 140.c even 2 1
1120.2.n.a 4 280.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(280, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 6T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T + 2)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 40)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 160)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 180)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 40)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 160)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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