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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Newspace parameters

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

Self dual: no
Analytic conductor: \(2.23581125660\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 9\)
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

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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + 2 q^{4} + \beta_{3} q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} -2 \beta_{1} q^{8} -3 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + 2 q^{4} + \beta_{3} q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} -2 \beta_{1} q^{8} -3 q^{9} + \beta_{2} q^{10} -2 q^{11} -2 \beta_{3} q^{13} + ( 2 + \beta_{2} ) q^{14} + 4 q^{16} + 3 \beta_{1} q^{18} -2 \beta_{2} q^{19} + 2 \beta_{3} q^{20} + 2 \beta_{1} q^{22} -6 \beta_{1} q^{23} -5 q^{25} -2 \beta_{2} q^{26} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{28} -4 \beta_{1} q^{32} + ( -5 + \beta_{2} ) q^{35} -6 q^{36} + 8 \beta_{1} q^{37} -4 \beta_{3} q^{38} + 2 \beta_{2} q^{40} -4 \beta_{2} q^{41} -4 q^{44} -3 \beta_{3} q^{45} + 12 q^{46} + 6 \beta_{3} q^{47} + ( -3 + 2 \beta_{2} ) q^{49} + 5 \beta_{1} q^{50} -4 \beta_{3} q^{52} + 4 \beta_{1} q^{53} -2 \beta_{3} q^{55} + ( 4 + 2 \beta_{2} ) q^{56} + 2 \beta_{2} q^{59} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{63} + 8 q^{64} + 10 q^{65} + ( 5 \beta_{1} + 2 \beta_{3} ) q^{70} + 6 \beta_{1} q^{72} -16 q^{74} -4 \beta_{2} q^{76} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{77} + 4 \beta_{3} q^{80} + 9 q^{81} -8 \beta_{3} q^{82} + 4 \beta_{1} q^{88} + 4 \beta_{2} q^{89} -3 \beta_{2} q^{90} + ( 10 - 2 \beta_{2} ) q^{91} -12 \beta_{1} q^{92} + 6 \beta_{2} q^{94} -10 \beta_{1} q^{95} + ( 3 \beta_{1} + 4 \beta_{3} ) q^{98} + 6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} - 12q^{9} + O(q^{10}) \) \( 4q + 8q^{4} - 12q^{9} - 8q^{11} + 8q^{14} + 16q^{16} - 20q^{25} - 20q^{35} - 24q^{36} - 16q^{44} + 48q^{46} - 12q^{49} + 16q^{56} + 32q^{64} + 40q^{65} - 64q^{74} + 36q^{81} + 40q^{91} + 24q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 7 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{2} + 7 \beta_{1}\)\()/2\)

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.

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 \(S_{2}^{\mathrm{new}}(280, [\chi])\).

Hecke characteristic polynomials

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