Properties

Label 280.2.bv.a
Level $280$
Weight $2$
Character orbit 280.bv
Analytic conductor $2.236$
Analytic rank $1$
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.bv (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.23581125660\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

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

Character values

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

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(1\) \(-1\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
−1.00000 + 1.00000i −0.500000 0.133975i 2.00000i −0.133975 + 2.23205i 0.633975 0.366025i −2.50000 0.866025i 2.00000 + 2.00000i −2.36603 1.36603i −2.09808 2.36603i
157.1 −1.00000 + 1.00000i −0.500000 1.86603i 2.00000i −1.86603 1.23205i 2.36603 + 1.36603i −2.50000 + 0.866025i 2.00000 + 2.00000i −0.633975 + 0.366025i 3.09808 0.633975i
173.1 −1.00000 1.00000i −0.500000 + 1.86603i 2.00000i −1.86603 + 1.23205i 2.36603 1.36603i −2.50000 0.866025i 2.00000 2.00000i −0.633975 0.366025i 3.09808 + 0.633975i
213.1 −1.00000 1.00000i −0.500000 + 0.133975i 2.00000i −0.133975 2.23205i 0.633975 + 0.366025i −2.50000 + 0.866025i 2.00000 2.00000i −2.36603 + 1.36603i −2.09808 + 2.36603i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
280.bv even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.bv.a 4
5.c odd 4 1 280.2.bv.d yes 4
7.d odd 6 1 280.2.bv.c yes 4
8.b even 2 1 280.2.bv.b yes 4
35.k even 12 1 280.2.bv.b yes 4
40.i odd 4 1 280.2.bv.c yes 4
56.j odd 6 1 280.2.bv.d yes 4
280.bv even 12 1 inner 280.2.bv.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bv.a 4 1.a even 1 1 trivial
280.2.bv.a 4 280.bv even 12 1 inner
280.2.bv.b yes 4 8.b even 2 1
280.2.bv.b yes 4 35.k even 12 1
280.2.bv.c yes 4 7.d odd 6 1
280.2.bv.c yes 4 40.i odd 4 1
280.2.bv.d yes 4 5.c odd 4 1
280.2.bv.d yes 4 56.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 2 T_{3}^{3} + 5 T_{3}^{2} + 4 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(280, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + 2 T + T^{2} )^{2} \)
$3$ \( 1 + 4 T + 5 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( 25 + 20 T + 11 T^{2} + 4 T^{3} + T^{4} \)
$7$ \( ( 7 + 5 T + T^{2} )^{2} \)
$11$ \( 4 - 12 T + 14 T^{2} - 6 T^{3} + T^{4} \)
$13$ \( ( 18 + 6 T + T^{2} )^{2} \)
$17$ \( 324 + 108 T + 18 T^{2} + 6 T^{3} + T^{4} \)
$19$ \( ( 12 + 6 T + T^{2} )^{2} \)
$23$ \( 81 + 108 T + 45 T^{2} + 6 T^{3} + T^{4} \)
$29$ \( ( -11 - 2 T + T^{2} )^{2} \)
$31$ \( 4 + 36 T + 110 T^{2} + 18 T^{3} + T^{4} \)
$37$ \( 144 + 36 T^{2} + 12 T^{3} + T^{4} \)
$41$ \( 9 + 42 T^{2} + T^{4} \)
$43$ \( 9 + 18 T + 18 T^{2} + 6 T^{3} + T^{4} \)
$47$ \( 36 - 108 T + 90 T^{2} - 6 T^{3} + T^{4} \)
$53$ \( 1296 + 864 T + 180 T^{2} + 12 T^{3} + T^{4} \)
$59$ \( 1936 + 1056 T + 236 T^{2} + 24 T^{3} + T^{4} \)
$61$ \( 1521 + 234 T + 75 T^{2} - 6 T^{3} + T^{4} \)
$67$ \( 1089 + 396 T + 117 T^{2} + 18 T^{3} + T^{4} \)
$71$ \( ( 132 - 24 T + T^{2} )^{2} \)
$73$ \( 8836 + 2444 T + 290 T^{2} + 22 T^{3} + T^{4} \)
$79$ \( 144 + 288 T + 204 T^{2} + 24 T^{3} + T^{4} \)
$83$ \( 4761 + 414 T + 18 T^{2} - 6 T^{3} + T^{4} \)
$89$ \( 13689 + 2808 T + 459 T^{2} + 24 T^{3} + T^{4} \)
$97$ \( 14884 - 3904 T + 512 T^{2} - 32 T^{3} + T^{4} \)
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