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 $8$

Related objects

Downloads

Learn more

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: \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - 2 \beta_{3} q^{4} + \beta_{6} q^{5} + ( - \beta_{6} + \beta_{2} + \beta_1) q^{7} + 2 \beta_{2} q^{8} + ( - 3 \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - 2 \beta_{3} q^{4} + \beta_{6} q^{5} + ( - \beta_{6} + \beta_{2} + \beta_1) q^{7} + 2 \beta_{2} q^{8} + ( - 3 \beta_{3} + 3) q^{9} - \beta_{5} q^{10} + (2 \beta_{7} + \beta_{5} - \beta_{3}) q^{11} + ( - \beta_{6} + 4 \beta_{4} + 2 \beta_{2} + \beta_1) q^{13} + ( - \beta_{7} + 2 \beta_{3} - 2) q^{14} + (4 \beta_{3} - 4) q^{16} + (3 \beta_{4} + 3 \beta_{2}) q^{18} + ( - \beta_{7} + 3 \beta_{3} + 3) q^{19} + ( - 2 \beta_{6} + 2 \beta_1) q^{20} + (2 \beta_{6} + \beta_{2} + 2 \beta_1) q^{22} + (\beta_{6} - 3 \beta_{4} - 2 \beta_1) q^{23} + 5 \beta_{3} q^{25} + ( - \beta_{7} - 4 \beta_{3} - 4) q^{26} + ( - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{28} + ( - 4 \beta_{4} - 4 \beta_{2}) q^{32} + ( - \beta_{7} - 5 \beta_{3} + 5) q^{35} - 6 q^{36} + (\beta_{6} + 4 \beta_{4} - 2 \beta_1) q^{37} + (3 \beta_{4} - 3 \beta_{2} - 2 \beta_1) q^{38} - 2 \beta_{7} q^{40} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} - 1) q^{41} + ( - 2 \beta_{7} - 4 \beta_{5} + 2 \beta_{3} - 2) q^{44} + 3 \beta_1 q^{45} + (2 \beta_{7} + \beta_{5} + 6 \beta_{3}) q^{46} + ( - 3 \beta_{6} - \beta_{4} - 2 \beta_{2}) q^{47} + (2 \beta_{7} + 2 \beta_{5} - 3) q^{49} - 5 \beta_{2} q^{50} + ( - 4 \beta_{4} + 4 \beta_{2} - 2 \beta_1) q^{52} + (6 \beta_{6} - 2 \beta_{4} - 2 \beta_{2} - 3 \beta_1) q^{53} + ( - \beta_{6} - 10 \beta_{4} - 5 \beta_{2} + \beta_1) q^{55} + (2 \beta_{7} + 2 \beta_{5} + 4) q^{56} - 2 \beta_{5} q^{59} + ( - 3 \beta_{6} - 3 \beta_{4}) q^{63} + 8 q^{64} + ( - 2 \beta_{7} - 4 \beta_{5} - 5 \beta_{3} + 5) q^{65} + (5 \beta_{4} + 5 \beta_{2} - 2 \beta_1) q^{70} - 6 \beta_{4} q^{72} + (2 \beta_{7} + \beta_{5} - 8 \beta_{3}) q^{74} + (2 \beta_{7} + 2 \beta_{5} - 12 \beta_{3} + 6) q^{76} + ( - 4 \beta_{6} + 4 \beta_{4} - 6 \beta_{2} + \beta_1) q^{77} - 4 \beta_1 q^{80} - 9 \beta_{3} q^{81} + (4 \beta_{6} - \beta_{4} - 2 \beta_{2}) q^{82} + ( - 8 \beta_{6} - 2 \beta_{4} - 2 \beta_{2} + 4 \beta_1) q^{88} - 4 \beta_{7} q^{89} + ( - 3 \beta_{7} - 3 \beta_{5}) q^{90} + ( - \beta_{7} + 3 \beta_{5} + 8 \beta_{3} - 9) q^{91} + (2 \beta_{6} - 6 \beta_{2} + 2 \beta_1) q^{92} + (3 \beta_{5} - 2 \beta_{3} + 4) q^{94} + (6 \beta_{6} + 5 \beta_{4} + 5 \beta_{2} - 3 \beta_1) q^{95} + (4 \beta_{6} - 3 \beta_{4}) q^{98} + (3 \beta_{7} - 3 \beta_{5} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 12 q^{9} - 4 q^{11} - 8 q^{14} - 16 q^{16} + 36 q^{19} + 20 q^{25} - 48 q^{26} + 20 q^{35} - 48 q^{36} - 8 q^{44} + 24 q^{46} - 24 q^{49} + 32 q^{56} + 64 q^{64} + 20 q^{65} - 32 q^{74} - 36 q^{81} - 40 q^{91} + 24 q^{94} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 14\nu^{4} + 7\nu^{2} - 36 ) / 63 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{6} + 7\nu^{4} - 28\nu^{2} + 144 ) / 63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} - 7\nu^{5} - 35\nu^{3} + 180\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 13\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{6} + 14\nu^{4} + 7\nu^{2} + 162 ) / 63 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\nu^{7} - 49\nu^{5} + 133\nu^{3} - 684\nu ) / 189 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + \beta_{5} + 7\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{7} - 19\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{6} + 7\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 29\beta_{5} - 13\beta_{4} - 13\beta_{2} ) / 2 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(280, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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