Properties

Label 280.2.q.d
Level $280$
Weight $2$
Character orbit 280.q
Analytic conductor $2.236$
Analytic rank $0$
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.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.23581125660\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1 + 1) q^{3} + \beta_{2} q^{5} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + (2 \beta_{3} + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1 + 1) q^{3} + \beta_{2} q^{5} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + (2 \beta_{3} + 2 \beta_1) q^{9} + (2 \beta_{2} + 2 \beta_1 + 2) q^{11} + 2 q^{13} + (\beta_{3} - 1) q^{15} + (2 \beta_{2} - 4 \beta_1 + 2) q^{17} + (4 \beta_{3} + 4 \beta_1) q^{19} + ( - 3 \beta_{2} - 2 \beta_1 - 4) q^{21} + ( - \beta_{3} - 7 \beta_{2} - \beta_1) q^{23} + ( - \beta_{2} - 1) q^{25} + ( - \beta_{3} - 1) q^{27} + ( - 2 \beta_{3} - 5) q^{29} + (2 \beta_{2} + 2 \beta_1 + 2) q^{31} + (4 \beta_{3} + 6 \beta_{2} + 4 \beta_1) q^{33} + ( - \beta_{3} - 2 \beta_1 + 1) q^{35} + ( - 4 \beta_{3} - 4 \beta_1) q^{37} + (2 \beta_{2} + 2 \beta_1 + 2) q^{39} + (2 \beta_{3} + 3) q^{41} + ( - 7 \beta_{3} + 3) q^{43} - 2 \beta_1 q^{45} + ( - 4 \beta_{3} + 6 \beta_{2} - 4 \beta_1) q^{47} + ( - 2 \beta_{3} - 5 \beta_{2} + 2 \beta_1) q^{49} + ( - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1) q^{51} - 4 \beta_1 q^{53} + (2 \beta_{3} - 2) q^{55} + (4 \beta_{3} - 8) q^{57} + (4 \beta_{2} + 4) q^{59} + ( - 4 \beta_{3} + \beta_{2} - 4 \beta_1) q^{61} + ( - 2 \beta_{3} - 8 \beta_{2} - 4) q^{63} + 2 \beta_{2} q^{65} + (3 \beta_{2} - 7 \beta_1 + 3) q^{67} + ( - 8 \beta_{3} + 9) q^{69} - 12 q^{71} + (2 \beta_{2} + 4 \beta_1 + 2) q^{73} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{75} + ( - 6 \beta_{2} - 4 \beta_1 - 8) q^{77} - 4 \beta_{2} q^{79} + (\beta_{2} + 6 \beta_1 + 1) q^{81} + (3 \beta_{3} - 9) q^{83} + ( - 4 \beta_{3} - 2) q^{85} + ( - \beta_{2} - 3 \beta_1 - 1) q^{87} + ( - 4 \beta_{3} - 11 \beta_{2} - 4 \beta_1) q^{89} + (4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{91} + (4 \beta_{3} + 6 \beta_{2} + 4 \beta_1) q^{93} - 4 \beta_1 q^{95} + 6 q^{97} + (4 \beta_{3} - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 4 q^{11} + 8 q^{13} - 4 q^{15} + 4 q^{17} - 10 q^{21} + 14 q^{23} - 2 q^{25} - 4 q^{27} - 20 q^{29} + 4 q^{31} - 12 q^{33} + 4 q^{35} + 4 q^{39} + 12 q^{41} + 12 q^{43} - 12 q^{47} + 10 q^{49} + 12 q^{51} - 8 q^{55} - 32 q^{57} + 8 q^{59} - 2 q^{61} - 4 q^{65} + 6 q^{67} + 36 q^{69} - 48 q^{71} + 4 q^{73} + 2 q^{75} - 20 q^{77} + 8 q^{79} + 2 q^{81} - 36 q^{83} - 8 q^{85} - 2 q^{87} + 22 q^{89} - 4 q^{91} - 12 q^{93} + 24 q^{97} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
0 −0.207107 + 0.358719i 0 −0.500000 0.866025i 0 1.62132 + 2.09077i 0 1.41421 + 2.44949i 0
81.2 0 1.20711 2.09077i 0 −0.500000 0.866025i 0 −2.62132 0.358719i 0 −1.41421 2.44949i 0
121.1 0 −0.207107 0.358719i 0 −0.500000 + 0.866025i 0 1.62132 2.09077i 0 1.41421 2.44949i 0
121.2 0 1.20711 + 2.09077i 0 −0.500000 + 0.866025i 0 −2.62132 + 0.358719i 0 −1.41421 + 2.44949i 0
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.q.d 4
3.b odd 2 1 2520.2.bi.k 4
4.b odd 2 1 560.2.q.j 4
5.b even 2 1 1400.2.q.h 4
5.c odd 4 2 1400.2.bh.g 8
7.b odd 2 1 1960.2.q.q 4
7.c even 3 1 inner 280.2.q.d 4
7.c even 3 1 1960.2.a.p 2
7.d odd 6 1 1960.2.a.t 2
7.d odd 6 1 1960.2.q.q 4
21.h odd 6 1 2520.2.bi.k 4
28.f even 6 1 3920.2.a.bp 2
28.g odd 6 1 560.2.q.j 4
28.g odd 6 1 3920.2.a.bz 2
35.i odd 6 1 9800.2.a.br 2
35.j even 6 1 1400.2.q.h 4
35.j even 6 1 9800.2.a.bz 2
35.l odd 12 2 1400.2.bh.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.d 4 1.a even 1 1 trivial
280.2.q.d 4 7.c even 3 1 inner
560.2.q.j 4 4.b odd 2 1
560.2.q.j 4 28.g odd 6 1
1400.2.q.h 4 5.b even 2 1
1400.2.q.h 4 35.j even 6 1
1400.2.bh.g 8 5.c odd 4 2
1400.2.bh.g 8 35.l odd 12 2
1960.2.a.p 2 7.c even 3 1
1960.2.a.t 2 7.d odd 6 1
1960.2.q.q 4 7.b odd 2 1
1960.2.q.q 4 7.d odd 6 1
2520.2.bi.k 4 3.b odd 2 1
2520.2.bi.k 4 21.h odd 6 1
3920.2.a.bp 2 28.f even 6 1
3920.2.a.bz 2 28.g odd 6 1
9800.2.a.br 2 35.i odd 6 1
9800.2.a.bz 2 35.j even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} - 3 T^{2} + 14 T + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + 20 T^{2} + 16 T + 16 \) Copy content Toggle raw display
$13$ \( (T - 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + 44 T^{2} + 112 T + 784 \) Copy content Toggle raw display
$19$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$23$ \( T^{4} - 14 T^{3} + 149 T^{2} + \cdots + 2209 \) Copy content Toggle raw display
$29$ \( (T^{2} + 10 T + 17)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + 20 T^{2} + 16 T + 16 \) Copy content Toggle raw display
$37$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$41$ \( (T^{2} - 6 T + 1)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 6 T - 89)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 12 T^{3} + 140 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$53$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$59$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + 35 T^{2} - 62 T + 961 \) Copy content Toggle raw display
$67$ \( T^{4} - 6 T^{3} + 125 T^{2} + \cdots + 7921 \) Copy content Toggle raw display
$71$ \( (T + 12)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} + 44 T^{2} + 112 T + 784 \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 18 T + 63)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 22 T^{3} + 395 T^{2} + \cdots + 7921 \) Copy content Toggle raw display
$97$ \( (T - 6)^{4} \) Copy content Toggle raw display
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