Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.bl (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: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + 2 \zeta_{12} q^{4} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{5} + ( -1 - 2 \zeta_{12}^{2} ) q^{7} + ( -2 - 2 \zeta_{12}^{3} ) q^{8} -3 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + 2 \zeta_{12} q^{4} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{5} + ( -1 - 2 \zeta_{12}^{2} ) q^{7} + ( -2 - 2 \zeta_{12}^{3} ) q^{8} -3 \zeta_{12}^{2} q^{9} + ( -1 - \zeta_{12} + \zeta_{12}^{2} ) q^{10} + 3 \zeta_{12} q^{11} -\zeta_{12}^{3} q^{13} + ( -2 + 3 \zeta_{12} + 3 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{14} + 4 \zeta_{12}^{2} q^{16} + ( 6 - 6 \zeta_{12}^{2} ) q^{17} + ( -3 + 3 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{18} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{19} + 2 q^{20} + ( -3 - 3 \zeta_{12}^{3} ) q^{22} -3 \zeta_{12}^{2} q^{23} + ( 1 - \zeta_{12}^{2} ) q^{25} + ( -\zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{26} + ( -2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{28} -10 \zeta_{12}^{3} q^{29} + ( 4 - 4 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{32} + ( -6 + 6 \zeta_{12}^{3} ) q^{34} + ( -3 \zeta_{12} + \zeta_{12}^{3} ) q^{35} -6 \zeta_{12}^{3} q^{36} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{37} + ( 1 + \zeta_{12} - \zeta_{12}^{2} ) q^{38} + ( -2 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{40} + 9 q^{41} + 6 \zeta_{12}^{3} q^{43} + 6 \zeta_{12}^{2} q^{44} -3 \zeta_{12} q^{45} + ( -3 + 3 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{46} -5 \zeta_{12}^{2} q^{47} + ( -3 + 8 \zeta_{12}^{2} ) q^{49} + ( -1 + \zeta_{12}^{3} ) q^{50} + ( 2 - 2 \zeta_{12}^{2} ) q^{52} + 9 \zeta_{12} q^{53} + 3 q^{55} + ( 2 - 4 \zeta_{12} + 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{56} + ( -10 \zeta_{12} + 10 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{58} + 8 \zeta_{12} q^{59} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{61} + ( -6 + 9 \zeta_{12}^{2} ) q^{63} + 8 \zeta_{12}^{3} q^{64} -\zeta_{12}^{2} q^{65} -2 \zeta_{12} q^{67} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{68} + ( 3 + \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{70} -10 q^{71} + ( -6 \zeta_{12} + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{72} + ( 8 - 8 \zeta_{12}^{2} ) q^{73} + ( 5 + 5 \zeta_{12} - 5 \zeta_{12}^{2} ) q^{74} -2 q^{76} + ( -3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{77} + 4 \zeta_{12}^{2} q^{79} + 4 \zeta_{12} q^{80} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + ( -9 \zeta_{12} - 9 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{82} + 16 \zeta_{12}^{3} q^{83} -6 \zeta_{12}^{3} q^{85} + ( 6 \zeta_{12} - 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{86} + ( 6 - 6 \zeta_{12} - 6 \zeta_{12}^{2} ) q^{88} + 2 \zeta_{12}^{2} q^{89} + ( 3 + 3 \zeta_{12}^{3} ) q^{90} + ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{91} -6 \zeta_{12}^{3} q^{92} + ( -5 + 5 \zeta_{12} + 5 \zeta_{12}^{2} ) q^{94} + ( -1 + \zeta_{12}^{2} ) q^{95} + 8 q^{97} + ( 8 - 5 \zeta_{12} - 5 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{98} -9 \zeta_{12}^{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 8q^{7} - 8q^{8} - 6q^{9} + O(q^{10}) \) \( 4q - 2q^{2} - 8q^{7} - 8q^{8} - 6q^{9} - 2q^{10} - 2q^{14} + 8q^{16} + 12q^{17} - 6q^{18} + 8q^{20} - 12q^{22} - 6q^{23} + 2q^{25} + 2q^{26} + 8q^{32} - 24q^{34} + 2q^{38} - 4q^{40} + 36q^{41} + 12q^{44} - 6q^{46} - 10q^{47} + 4q^{49} - 4q^{50} + 4q^{52} + 12q^{55} + 16q^{56} + 20q^{58} - 6q^{63} - 2q^{65} + 10q^{70} - 40q^{71} + 12q^{72} + 16q^{73} + 10q^{74} - 8q^{76} + 8q^{79} - 18q^{81} - 18q^{82} - 12q^{86} + 12q^{88} + 4q^{89} + 12q^{90} - 10q^{94} - 2q^{95} + 32q^{97} + 22q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−1.36603 0.366025i 0 1.73205 + 1.00000i 0.866025 0.500000i 0 −2.00000 1.73205i −2.00000 2.00000i −1.50000 2.59808i −1.36603 + 0.366025i
221.2 0.366025 1.36603i 0 −1.73205 1.00000i −0.866025 + 0.500000i 0 −2.00000 1.73205i −2.00000 + 2.00000i −1.50000 2.59808i 0.366025 + 1.36603i
261.1 −1.36603 + 0.366025i 0 1.73205 1.00000i 0.866025 + 0.500000i 0 −2.00000 + 1.73205i −2.00000 + 2.00000i −1.50000 + 2.59808i −1.36603 0.366025i
261.2 0.366025 + 1.36603i 0 −1.73205 + 1.00000i −0.866025 0.500000i 0 −2.00000 + 1.73205i −2.00000 2.00000i −1.50000 + 2.59808i 0.366025 1.36603i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.b even 2 1 inner
56.p 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.bl.a 4
4.b odd 2 1 1120.2.cb.a 4
7.c even 3 1 inner 280.2.bl.a 4
8.b even 2 1 inner 280.2.bl.a 4
8.d odd 2 1 1120.2.cb.a 4
28.g odd 6 1 1120.2.cb.a 4
56.k odd 6 1 1120.2.cb.a 4
56.p even 6 1 inner 280.2.bl.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bl.a 4 1.a even 1 1 trivial
280.2.bl.a 4 7.c even 3 1 inner
280.2.bl.a 4 8.b even 2 1 inner
280.2.bl.a 4 56.p even 6 1 inner
1120.2.cb.a 4 4.b odd 2 1
1120.2.cb.a 4 8.d odd 2 1
1120.2.cb.a 4 28.g odd 6 1
1120.2.cb.a 4 56.k odd 6 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

$p$ $F_p(T)$
$2$ \( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 1 - T^{2} + T^{4} \)
$7$ \( ( 7 + 4 T + T^{2} )^{2} \)
$11$ \( 81 - 9 T^{2} + T^{4} \)
$13$ \( ( 1 + T^{2} )^{2} \)
$17$ \( ( 36 - 6 T + T^{2} )^{2} \)
$19$ \( 1 - T^{2} + T^{4} \)
$23$ \( ( 9 + 3 T + T^{2} )^{2} \)
$29$ \( ( 100 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( 625 - 25 T^{2} + T^{4} \)
$41$ \( ( -9 + T )^{4} \)
$43$ \( ( 36 + T^{2} )^{2} \)
$47$ \( ( 25 + 5 T + T^{2} )^{2} \)
$53$ \( 6561 - 81 T^{2} + T^{4} \)
$59$ \( 4096 - 64 T^{2} + T^{4} \)
$61$ \( 4096 - 64 T^{2} + T^{4} \)
$67$ \( 16 - 4 T^{2} + T^{4} \)
$71$ \( ( 10 + T )^{4} \)
$73$ \( ( 64 - 8 T + T^{2} )^{2} \)
$79$ \( ( 16 - 4 T + T^{2} )^{2} \)
$83$ \( ( 256 + T^{2} )^{2} \)
$89$ \( ( 4 - 2 T + T^{2} )^{2} \)
$97$ \( ( -8 + T )^{4} \)
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