Properties

Label 280.2.s.a
Level $280$
Weight $2$
Character orbit 280.s
Analytic conductor $2.236$
Analytic rank $0$
Dimension $8$
CM discriminant -56
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.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.23581125660\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.40282095616.8
Defining polynomial: \(x^{8} - 4 x^{6} + 8 x^{4} - 36 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{3} ) q^{2} + ( -\beta_{1} - \beta_{4} ) q^{3} + 2 \beta_{3} q^{4} + ( \beta_{1} + \beta_{4} - \beta_{5} ) q^{5} + ( \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{6} + \beta_{7} q^{7} + ( 2 - 2 \beta_{3} ) q^{8} + ( \beta_{2} - \beta_{3} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{3} ) q^{2} + ( -\beta_{1} - \beta_{4} ) q^{3} + 2 \beta_{3} q^{4} + ( \beta_{1} + \beta_{4} - \beta_{5} ) q^{5} + ( \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{6} + \beta_{7} q^{7} + ( 2 - 2 \beta_{3} ) q^{8} + ( \beta_{2} - \beta_{3} - \beta_{7} ) q^{9} + ( -\beta_{1} - 2 \beta_{4} - \beta_{6} ) q^{10} + ( -2 \beta_{5} - 2 \beta_{6} ) q^{12} + ( -\beta_{1} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{13} + ( \beta_{2} - \beta_{7} ) q^{14} + ( 2 + 3 \beta_{3} + \beta_{7} ) q^{15} -4 q^{16} + ( -1 - 2 \beta_{2} + \beta_{3} ) q^{18} + ( 2 \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{19} + ( 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{20} + ( -2 \beta_{1} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{21} + ( -3 - 2 \beta_{2} + 3 \beta_{3} ) q^{23} + ( -2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{24} + ( -3 - \beta_{2} - 3 \beta_{3} ) q^{25} + ( 3 \beta_{1} - \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{26} + ( \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{27} -2 \beta_{2} q^{28} + ( 1 + \beta_{2} - 5 \beta_{3} - \beta_{7} ) q^{30} + ( 4 + 4 \beta_{3} ) q^{32} + ( 2 \beta_{1} + \beta_{5} - 2 \beta_{6} ) q^{35} + ( 2 + 2 \beta_{2} + 2 \beta_{7} ) q^{36} + ( -\beta_{1} - \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{38} + ( 3 \beta_{2} - 8 \beta_{3} - 3 \beta_{7} ) q^{39} + ( 2 \beta_{1} - 4 \beta_{5} - 2 \beta_{6} ) q^{40} + ( \beta_{1} - \beta_{4} + 3 \beta_{5} + 3 \beta_{6} ) q^{42} + ( -4 \beta_{1} - 2 \beta_{5} - \beta_{6} ) q^{45} + ( 6 + 2 \beta_{2} + 2 \beta_{7} ) q^{46} + ( 4 \beta_{1} + 4 \beta_{4} ) q^{48} + 7 \beta_{3} q^{49} + ( \beta_{2} + 6 \beta_{3} + \beta_{7} ) q^{50} + ( -4 \beta_{1} + 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{52} + ( 2 \beta_{4} - 2 \beta_{6} ) q^{54} + ( 2 \beta_{2} + 2 \beta_{7} ) q^{56} + ( -1 - \beta_{3} - 2 \beta_{7} ) q^{57} + ( -7 \beta_{1} - 2 \beta_{4} + 7 \beta_{5} + 2 \beta_{6} ) q^{59} + ( -6 - 2 \beta_{2} + 4 \beta_{3} ) q^{60} + ( \beta_{1} + 6 \beta_{4} + \beta_{5} + 6 \beta_{6} ) q^{61} + ( 7 + \beta_{2} - 7 \beta_{3} ) q^{63} -8 \beta_{3} q^{64} + ( 4 - 2 \beta_{2} - \beta_{3} + 5 \beta_{7} ) q^{65} + ( 5 \beta_{1} + 7 \beta_{4} - 5 \beta_{5} - 7 \beta_{6} ) q^{69} + ( -4 \beta_{1} + \beta_{4} - \beta_{5} ) q^{70} + ( -4 - 3 \beta_{2} - 3 \beta_{7} ) q^{71} + ( -2 - 2 \beta_{3} - 4 \beta_{7} ) q^{72} + ( 4 \beta_{1} + 5 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{75} + ( -2 \beta_{1} + 4 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} ) q^{76} + ( -8 - 6 \beta_{2} + 8 \beta_{3} ) q^{78} + ( \beta_{2} + 12 \beta_{3} - \beta_{7} ) q^{79} + ( -4 \beta_{1} - 4 \beta_{4} + 4 \beta_{5} ) q^{80} + ( -3 + \beta_{2} + \beta_{7} ) q^{81} + ( 4 \beta_{1} + 4 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{83} + ( 2 \beta_{1} + 4 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} ) q^{84} + ( 3 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 5 \beta_{6} ) q^{90} + ( -4 \beta_{1} - 9 \beta_{4} - 4 \beta_{5} - 9 \beta_{6} ) q^{91} + ( -6 - 6 \beta_{3} - 4 \beta_{7} ) q^{92} + ( -4 + \beta_{2} - 5 \beta_{3} + 4 \beta_{7} ) q^{95} + ( -4 \beta_{1} - 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} ) q^{96} + ( 7 - 7 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{2} + 16q^{8} + O(q^{10}) \) \( 8q - 8q^{2} + 16q^{8} + 16q^{15} - 32q^{16} - 8q^{18} - 24q^{23} - 24q^{25} + 8q^{30} + 32q^{32} + 16q^{36} + 48q^{46} - 8q^{57} - 48q^{60} + 56q^{63} + 32q^{65} - 32q^{71} - 16q^{72} - 64q^{78} - 24q^{81} - 48q^{92} - 32q^{95} + 56q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{6} + 8 x^{4} - 36 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{6} + \nu^{4} + 43 \nu^{2} - 81 \)\()/45\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{6} - \nu^{4} + 2 \nu^{2} + 36 \)\()/45\)
\(\beta_{4}\)\(=\)\((\)\( 4 \nu^{7} + 2 \nu^{5} + 41 \nu^{3} - 162 \nu \)\()/135\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} - 8 \nu^{3} + 36 \nu \)\()/27\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{7} - \nu^{5} + 2 \nu^{3} + 36 \nu \)\()/45\)
\(\beta_{7}\)\(=\)\((\)\( -7 \nu^{6} + 19 \nu^{4} - 38 \nu^{2} + 171 \)\()/45\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 1\)
\(\nu^{3}\)\(=\)\(2 \beta_{6} + 3 \beta_{4} + 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{7} - 5 \beta_{3} + 2 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-\beta_{6} + 6 \beta_{5} + 6 \beta_{4}\)
\(\nu^{6}\)\(=\)\(-\beta_{7} - 19 \beta_{3} + 19\)
\(\nu^{7}\)\(=\)\(-20 \beta_{6} - 3 \beta_{5} + 20 \beta_{1}\)

Character values

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

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
1.71331 + 0.254137i
−1.03179 + 1.39119i
1.03179 1.39119i
−1.71331 0.254137i
1.71331 0.254137i
−1.03179 1.39119i
1.03179 + 1.39119i
−1.71331 + 0.254137i
−1.00000 + 1.00000i −1.96744 1.96744i 2.00000i 0.254137 + 2.22158i 3.93488 1.87083 1.87083i 2.00000 + 2.00000i 4.74166i −2.47572 1.96744i
13.2 −1.00000 + 1.00000i −0.359404 0.359404i 2.00000i 1.39119 + 1.75060i 0.718808 −1.87083 + 1.87083i 2.00000 + 2.00000i 2.74166i −3.14179 0.359404i
13.3 −1.00000 + 1.00000i 0.359404 + 0.359404i 2.00000i −1.39119 1.75060i −0.718808 −1.87083 + 1.87083i 2.00000 + 2.00000i 2.74166i 3.14179 + 0.359404i
13.4 −1.00000 + 1.00000i 1.96744 + 1.96744i 2.00000i −0.254137 2.22158i −3.93488 1.87083 1.87083i 2.00000 + 2.00000i 4.74166i 2.47572 + 1.96744i
237.1 −1.00000 1.00000i −1.96744 + 1.96744i 2.00000i 0.254137 2.22158i 3.93488 1.87083 + 1.87083i 2.00000 2.00000i 4.74166i −2.47572 + 1.96744i
237.2 −1.00000 1.00000i −0.359404 + 0.359404i 2.00000i 1.39119 1.75060i 0.718808 −1.87083 1.87083i 2.00000 2.00000i 2.74166i −3.14179 + 0.359404i
237.3 −1.00000 1.00000i 0.359404 0.359404i 2.00000i −1.39119 + 1.75060i −0.718808 −1.87083 1.87083i 2.00000 2.00000i 2.74166i 3.14179 0.359404i
237.4 −1.00000 1.00000i 1.96744 1.96744i 2.00000i −0.254137 + 2.22158i −3.93488 1.87083 + 1.87083i 2.00000 2.00000i 4.74166i 2.47572 1.96744i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 237.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
5.c odd 4 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
35.f even 4 1 inner
40.i odd 4 1 inner
280.s even 4 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 60 T_{3}^{4} + 4 \) acting on \(S_{2}^{\mathrm{new}}(280, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + 2 T + T^{2} )^{4} \)
$3$ \( 4 + 60 T^{4} + T^{8} \)
$5$ \( 625 + 300 T^{2} + 72 T^{4} + 12 T^{6} + T^{8} \)
$7$ \( ( 49 + T^{4} )^{2} \)
$11$ \( T^{8} \)
$13$ \( 3694084 + 3900 T^{4} + T^{8} \)
$17$ \( T^{8} \)
$19$ \( ( 338 + 64 T^{2} + T^{4} )^{2} \)
$23$ \( ( 100 - 120 T + 72 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( ( 25538 + 344 T^{2} + T^{4} )^{2} \)
$61$ \( ( 8978 - 256 T^{2} + T^{4} )^{2} \)
$67$ \( T^{8} \)
$71$ \( ( -110 + 8 T + T^{2} )^{4} \)
$73$ \( T^{8} \)
$79$ \( ( 16900 + 316 T^{2} + T^{4} )^{2} \)
$83$ \( 416241604 + 94620 T^{4} + T^{8} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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