Properties

Label 280.2.bj.c
Level $280$
Weight $2$
Character orbit 280.bj
Analytic conductor $2.236$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.bj (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.23581125660\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + \beta_{2} q^{5} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{6} + ( 3 - \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + \beta_{2} q^{5} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{6} + ( 3 - \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{9} + \beta_{3} q^{10} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{11} + ( 2 - 4 \beta_{2} + 2 \beta_{3} ) q^{12} + ( 2 + 2 \beta_{1} - \beta_{3} ) q^{13} + ( 3 \beta_{1} - \beta_{3} ) q^{14} + ( 1 - 2 \beta_{2} + \beta_{3} ) q^{15} + ( -4 + 4 \beta_{2} ) q^{16} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{17} + ( 4 - 8 \beta_{2} + 2 \beta_{3} ) q^{18} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{19} + ( -2 + 2 \beta_{2} ) q^{20} + ( -4 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{21} + ( -8 + 4 \beta_{2} ) q^{22} + ( -2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{23} + ( -4 + 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{24} + ( -1 + \beta_{2} ) q^{25} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{26} + ( 3 - 6 \beta_{2} + 5 \beta_{3} ) q^{27} + ( 2 + 4 \beta_{2} ) q^{28} + ( 5 - 10 \beta_{2} + \beta_{3} ) q^{29} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{30} + ( 4 - \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{31} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} + ( -8 + 6 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{33} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{34} + ( 1 + 2 \beta_{2} ) q^{35} + ( -4 + 4 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{36} + ( 4 - 2 \beta_{2} ) q^{37} + 6 q^{38} -\beta_{1} q^{39} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{40} + ( 3 - 6 \beta_{2} - 2 \beta_{3} ) q^{41} + ( 2 - 4 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{42} + ( -7 - 2 \beta_{1} + \beta_{3} ) q^{43} + ( -8 \beta_{1} + 4 \beta_{3} ) q^{44} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{45} + ( -4 - 2 \beta_{1} + \beta_{3} ) q^{46} -6 \beta_{2} q^{47} + ( 8 - 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{48} + ( 8 - 5 \beta_{2} ) q^{49} + ( -\beta_{1} + \beta_{3} ) q^{50} + ( -\beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{51} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{52} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{53} + ( -10 + 3 \beta_{1} + 10 \beta_{2} - 6 \beta_{3} ) q^{54} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{55} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{56} + ( 6 - 6 \beta_{1} + 3 \beta_{3} ) q^{57} + ( -2 + 5 \beta_{1} + 2 \beta_{2} - 10 \beta_{3} ) q^{58} + ( -2 + 5 \beta_{1} - 2 \beta_{2} ) q^{59} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{60} + ( -3 \beta_{1} + 7 \beta_{2} - 3 \beta_{3} ) q^{61} + ( -4 + 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{62} + ( 2 - 8 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{63} -8 q^{64} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{65} + ( 12 - 8 \beta_{1} + 4 \beta_{3} ) q^{66} + ( -5 - 3 \beta_{1} + 5 \beta_{2} + 6 \beta_{3} ) q^{67} + ( 4 - 8 \beta_{2} - 2 \beta_{3} ) q^{68} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{69} + ( \beta_{1} + 2 \beta_{3} ) q^{70} + \beta_{3} q^{71} + ( 16 - 4 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{72} + ( -4 - 3 \beta_{1} - 4 \beta_{2} ) q^{73} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{74} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{75} + 6 \beta_{1} q^{76} + ( -2 \beta_{1} + 10 \beta_{3} ) q^{77} -2 \beta_{2} q^{78} + ( 4 + 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{79} -4 q^{80} + ( -13 + 2 \beta_{1} + 13 \beta_{2} - 4 \beta_{3} ) q^{81} + ( 4 + 3 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} ) q^{82} + ( -3 + 6 \beta_{2} - 5 \beta_{3} ) q^{83} + ( 2 + 2 \beta_{1} - 10 \beta_{2} + 4 \beta_{3} ) q^{84} + ( 2 - 4 \beta_{2} - \beta_{3} ) q^{85} + ( -2 - 7 \beta_{1} - 2 \beta_{2} ) q^{86} + ( -17 + 6 \beta_{1} + 17 \beta_{2} - 12 \beta_{3} ) q^{87} + ( -8 - 8 \beta_{2} ) q^{88} + ( 6 + 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{89} + ( 8 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{90} + ( 6 + 5 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{91} + ( -2 - 4 \beta_{1} - 2 \beta_{2} ) q^{92} + ( -12 + 7 \beta_{1} + 6 \beta_{2} - 7 \beta_{3} ) q^{93} -6 \beta_{3} q^{94} + 3 \beta_{1} q^{95} + ( -8 + 8 \beta_{1} - 4 \beta_{3} ) q^{96} + ( -2 + 4 \beta_{2} + 6 \beta_{3} ) q^{97} + ( 8 \beta_{1} - 5 \beta_{3} ) q^{98} + ( 24 - 8 \beta_{1} + 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{3} + 4q^{4} + 2q^{5} + 4q^{6} + 10q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 6q^{3} + 4q^{4} + 2q^{5} + 4q^{6} + 10q^{7} + 4q^{9} + 8q^{13} - 8q^{16} - 12q^{17} - 4q^{20} - 18q^{21} - 24q^{22} - 6q^{23} - 8q^{24} - 2q^{25} + 12q^{26} + 16q^{28} - 4q^{30} + 8q^{31} - 24q^{33} - 4q^{34} + 8q^{35} - 8q^{36} + 12q^{37} + 24q^{38} + 16q^{42} - 28q^{43} - 4q^{45} - 16q^{46} - 12q^{47} + 24q^{48} + 22q^{49} + 8q^{51} + 8q^{52} + 12q^{53} - 20q^{54} + 24q^{57} - 4q^{58} - 12q^{59} + 12q^{60} + 14q^{61} - 12q^{62} + 16q^{63} - 32q^{64} + 4q^{65} + 48q^{66} - 10q^{67} - 4q^{69} + 48q^{72} - 24q^{73} + 6q^{75} - 4q^{78} + 12q^{79} - 16q^{80} - 26q^{81} + 8q^{82} - 12q^{84} - 12q^{86} - 34q^{87} - 48q^{88} + 18q^{89} + 24q^{90} + 20q^{91} - 12q^{92} - 36q^{93} - 32q^{96} + 96q^{99} + O(q^{100}) \)

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

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

Character values

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

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
131.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i −2.72474 + 1.57313i 1.00000 1.73205i 0.500000 0.866025i 2.22474 3.85337i 2.50000 + 0.866025i 2.82843i 3.44949 5.97469i 1.41421i
131.2 1.22474 0.707107i −0.275255 + 0.158919i 1.00000 1.73205i 0.500000 0.866025i −0.224745 + 0.389270i 2.50000 + 0.866025i 2.82843i −1.44949 + 2.51059i 1.41421i
171.1 −1.22474 0.707107i −2.72474 1.57313i 1.00000 + 1.73205i 0.500000 + 0.866025i 2.22474 + 3.85337i 2.50000 0.866025i 2.82843i 3.44949 + 5.97469i 1.41421i
171.2 1.22474 + 0.707107i −0.275255 0.158919i 1.00000 + 1.73205i 0.500000 + 0.866025i −0.224745 0.389270i 2.50000 0.866025i 2.82843i −1.44949 2.51059i 1.41421i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.bj.c yes 4
4.b odd 2 1 1120.2.bz.c 4
7.d odd 6 1 280.2.bj.b 4
8.b even 2 1 1120.2.bz.b 4
8.d odd 2 1 280.2.bj.b 4
28.f even 6 1 1120.2.bz.b 4
56.j odd 6 1 1120.2.bz.c 4
56.m even 6 1 inner 280.2.bj.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bj.b 4 7.d odd 6 1
280.2.bj.b 4 8.d odd 2 1
280.2.bj.c yes 4 1.a even 1 1 trivial
280.2.bj.c yes 4 56.m even 6 1 inner
1120.2.bz.b 4 8.b even 2 1
1120.2.bz.b 4 28.f even 6 1
1120.2.bz.c 4 4.b odd 2 1
1120.2.bz.c 4 56.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(280, [\chi])\):

\( T_{3}^{4} + 6 T_{3}^{3} + 13 T_{3}^{2} + 6 T_{3} + 1 \)
\( T_{13}^{2} - 4 T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T^{2} + T^{4} \)
$3$ \( 1 + 6 T + 13 T^{2} + 6 T^{3} + T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( ( 7 - 5 T + T^{2} )^{2} \)
$11$ \( 576 + 24 T^{2} + T^{4} \)
$13$ \( ( -2 - 4 T + T^{2} )^{2} \)
$17$ \( 100 + 120 T + 58 T^{2} + 12 T^{3} + T^{4} \)
$19$ \( 324 - 18 T^{2} + T^{4} \)
$23$ \( 25 - 30 T + 7 T^{2} + 6 T^{3} + T^{4} \)
$29$ \( 5329 + 154 T^{2} + T^{4} \)
$31$ \( 100 - 80 T + 54 T^{2} - 8 T^{3} + T^{4} \)
$37$ \( ( 12 - 6 T + T^{2} )^{2} \)
$41$ \( 361 + 70 T^{2} + T^{4} \)
$43$ \( ( 43 + 14 T + T^{2} )^{2} \)
$47$ \( ( 36 + 6 T + T^{2} )^{2} \)
$53$ \( 100 - 120 T + 58 T^{2} - 12 T^{3} + T^{4} \)
$59$ \( 1444 - 456 T + 10 T^{2} + 12 T^{3} + T^{4} \)
$61$ \( 25 + 70 T + 201 T^{2} - 14 T^{3} + T^{4} \)
$67$ \( 841 - 290 T + 129 T^{2} + 10 T^{3} + T^{4} \)
$71$ \( ( 2 + T^{2} )^{2} \)
$73$ \( 900 + 720 T + 222 T^{2} + 24 T^{3} + T^{4} \)
$79$ \( 36 + 72 T + 42 T^{2} - 12 T^{3} + T^{4} \)
$83$ \( 529 + 154 T^{2} + T^{4} \)
$89$ \( 25 + 90 T + 103 T^{2} - 18 T^{3} + T^{4} \)
$97$ \( 3600 + 168 T^{2} + T^{4} \)
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