Properties

Label 280.2.ba.a
Level $280$
Weight $2$
Character orbit 280.ba
Analytic conductor $2.236$
Analytic rank $0$
Dimension $8$
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.ba (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.23581125660\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.3
Defining polynomial: \(x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 14 \nu^{4} + 7 \nu^{2} - 36 \)\()/63\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{7} + 7 \nu^{5} + 35 \nu^{3} + 81 \nu \)\()/189\)
\(\beta_{3}\)\(=\)\((\)\( -4 \nu^{6} + 7 \nu^{4} - 28 \nu^{2} + 144 \)\()/63\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{7} - 7 \nu^{5} - 35 \nu^{3} + 180 \nu \)\()/189\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 13 \nu \)\()/21\)
\(\beta_{6}\)\(=\)\((\)\( -8 \nu^{6} + 14 \nu^{4} + 7 \nu^{2} + 162 \)\()/63\)
\(\beta_{7}\)\(=\)\((\)\( 19 \nu^{7} - 49 \nu^{5} + 133 \nu^{3} - 684 \nu \)\()/189\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{6} - 2 \beta_{3} + 2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + \beta_{5} + 7 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\(\beta_{3} + 4 \beta_{1}\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{7} - 19 \beta_{4}\)\()/2\)
\(\nu^{6}\)\(=\)\(-7 \beta_{6} + 7 \beta_{1} + 22\)
\(\nu^{7}\)\(=\)\((\)\(29 \beta_{5} - 13 \beta_{4} - 13 \beta_{2}\)\()/2\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.01575 + 1.40294i
1.72286 0.178197i
1.01575 1.40294i
−1.72286 + 0.178197i
−1.01575 1.40294i
1.72286 + 0.178197i
1.01575 + 1.40294i
−1.72286 0.178197i
−0.707107 + 1.22474i 0 −1.00000 1.73205i −1.93649 1.11803i 0 1.41421 + 2.23607i 2.82843 1.50000 2.59808i 2.73861 1.58114i
19.2 −0.707107 + 1.22474i 0 −1.00000 1.73205i 1.93649 + 1.11803i 0 1.41421 2.23607i 2.82843 1.50000 2.59808i −2.73861 + 1.58114i
19.3 0.707107 1.22474i 0 −1.00000 1.73205i −1.93649 1.11803i 0 −1.41421 + 2.23607i −2.82843 1.50000 2.59808i −2.73861 + 1.58114i
19.4 0.707107 1.22474i 0 −1.00000 1.73205i 1.93649 + 1.11803i 0 −1.41421 2.23607i −2.82843 1.50000 2.59808i 2.73861 1.58114i
59.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i −1.93649 + 1.11803i 0 1.41421 2.23607i 2.82843 1.50000 + 2.59808i 2.73861 + 1.58114i
59.2 −0.707107 1.22474i 0 −1.00000 + 1.73205i 1.93649 1.11803i 0 1.41421 + 2.23607i 2.82843 1.50000 + 2.59808i −2.73861 1.58114i
59.3 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −1.93649 + 1.11803i 0 −1.41421 2.23607i −2.82843 1.50000 + 2.59808i −2.73861 1.58114i
59.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 1.93649 1.11803i 0 −1.41421 + 2.23607i −2.82843 1.50000 + 2.59808i 2.73861 + 1.58114i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.4
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.d odd 6 1 inner
8.d odd 2 1 inner
35.i odd 6 1 inner
56.m even 6 1 inner
280.ba 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.ba.a 8
4.b odd 2 1 1120.2.bq.a 8
5.b even 2 1 inner 280.2.ba.a 8
7.d odd 6 1 inner 280.2.ba.a 8
8.b even 2 1 1120.2.bq.a 8
8.d odd 2 1 inner 280.2.ba.a 8
20.d odd 2 1 1120.2.bq.a 8
28.f even 6 1 1120.2.bq.a 8
35.i odd 6 1 inner 280.2.ba.a 8
40.e odd 2 1 CM 280.2.ba.a 8
40.f even 2 1 1120.2.bq.a 8
56.j odd 6 1 1120.2.bq.a 8
56.m even 6 1 inner 280.2.ba.a 8
140.s even 6 1 1120.2.bq.a 8
280.ba even 6 1 inner 280.2.ba.a 8
280.bk odd 6 1 1120.2.bq.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.ba.a 8 1.a even 1 1 trivial
280.2.ba.a 8 5.b even 2 1 inner
280.2.ba.a 8 7.d odd 6 1 inner
280.2.ba.a 8 8.d odd 2 1 inner
280.2.ba.a 8 35.i odd 6 1 inner
280.2.ba.a 8 40.e odd 2 1 CM
280.2.ba.a 8 56.m even 6 1 inner
280.2.ba.a 8 280.ba even 6 1 inner
1120.2.bq.a 8 4.b odd 2 1
1120.2.bq.a 8 8.b even 2 1
1120.2.bq.a 8 20.d odd 2 1
1120.2.bq.a 8 28.f even 6 1
1120.2.bq.a 8 40.f even 2 1
1120.2.bq.a 8 56.j odd 6 1
1120.2.bq.a 8 140.s even 6 1
1120.2.bq.a 8 280.bk 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

$p$ $F_p(T)$
$2$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( ( 25 - 5 T^{2} + T^{4} )^{2} \)
$7$ \( ( 49 + 6 T^{2} + T^{4} )^{2} \)
$11$ \( ( 841 - 58 T + 33 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$13$ \( ( 361 + 58 T^{2} + T^{4} )^{2} \)
$17$ \( T^{8} \)
$19$ \( ( 289 - 306 T + 125 T^{2} - 18 T^{3} + T^{4} )^{2} \)
$23$ \( 81 + 594 T^{2} + 4347 T^{4} + 66 T^{6} + T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( 83521 + 27166 T^{2} + 8547 T^{4} + 94 T^{6} + T^{8} \)
$41$ \( ( 1369 + 86 T^{2} + T^{4} )^{2} \)
$43$ \( T^{8} \)
$47$ \( 2313441 - 155142 T^{2} + 8883 T^{4} - 102 T^{6} + T^{8} \)
$53$ \( 260144641 + 4612894 T^{2} + 65667 T^{4} + 286 T^{6} + T^{8} \)
$59$ \( ( 1600 - 40 T^{2} + T^{4} )^{2} \)
$61$ \( T^{8} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( ( 25600 - 160 T^{2} + T^{4} )^{2} \)
$97$ \( T^{8} \)
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