Properties

Label 1680.2.bg.a
Level $1680$
Weight $2$
Character orbit 1680.bg
Analytic conductor $13.415$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.bg (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( -2 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( -2 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -2 + 2 \zeta_{6} ) q^{11} + q^{13} + q^{15} + ( 4 - 4 \zeta_{6} ) q^{17} -\zeta_{6} q^{19} + ( 3 - 2 \zeta_{6} ) q^{21} + 4 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + q^{27} + ( -5 + 5 \zeta_{6} ) q^{31} -2 \zeta_{6} q^{33} + ( -1 + 3 \zeta_{6} ) q^{35} + 5 \zeta_{6} q^{37} + ( -1 + \zeta_{6} ) q^{39} + 2 q^{41} + 9 q^{43} + ( -1 + \zeta_{6} ) q^{45} -2 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + 4 \zeta_{6} q^{51} + ( -12 + 12 \zeta_{6} ) q^{53} + 2 q^{55} + q^{57} + ( -8 + 8 \zeta_{6} ) q^{59} + 14 \zeta_{6} q^{61} + ( -1 + 3 \zeta_{6} ) q^{63} -\zeta_{6} q^{65} + ( 9 - 9 \zeta_{6} ) q^{67} -4 q^{69} -2 q^{71} + ( -1 + \zeta_{6} ) q^{73} -\zeta_{6} q^{75} + ( 6 - 4 \zeta_{6} ) q^{77} -3 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 18 q^{83} -4 q^{85} + 4 \zeta_{6} q^{89} + ( -2 - \zeta_{6} ) q^{91} -5 \zeta_{6} q^{93} + ( -1 + \zeta_{6} ) q^{95} + 10 q^{97} + 2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - q^{5} - 5q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{3} - q^{5} - 5q^{7} - q^{9} - 2q^{11} + 2q^{13} + 2q^{15} + 4q^{17} - q^{19} + 4q^{21} + 4q^{23} - q^{25} + 2q^{27} - 5q^{31} - 2q^{33} + q^{35} + 5q^{37} - q^{39} + 4q^{41} + 18q^{43} - q^{45} - 2q^{47} + 11q^{49} + 4q^{51} - 12q^{53} + 4q^{55} + 2q^{57} - 8q^{59} + 14q^{61} + q^{63} - q^{65} + 9q^{67} - 8q^{69} - 4q^{71} - q^{73} - q^{75} + 8q^{77} - 3q^{79} - q^{81} + 36q^{83} - 8q^{85} + 4q^{89} - 5q^{91} - 5q^{93} - q^{95} + 20q^{97} + 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 −2.50000 + 0.866025i 0 −0.500000 + 0.866025i 0
1201.1 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 −2.50000 0.866025i 0 −0.500000 0.866025i 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 1680.2.bg.a 2
4.b odd 2 1 420.2.q.a 2
7.c even 3 1 inner 1680.2.bg.a 2
12.b even 2 1 1260.2.s.d 2
20.d odd 2 1 2100.2.q.a 2
20.e even 4 2 2100.2.bc.c 4
28.d even 2 1 2940.2.q.h 2
28.f even 6 1 2940.2.a.h 1
28.f even 6 1 2940.2.q.h 2
28.g odd 6 1 420.2.q.a 2
28.g odd 6 1 2940.2.a.d 1
84.j odd 6 1 8820.2.a.y 1
84.n even 6 1 1260.2.s.d 2
84.n even 6 1 8820.2.a.j 1
140.p odd 6 1 2100.2.q.a 2
140.w even 12 2 2100.2.bc.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.a 2 4.b odd 2 1
420.2.q.a 2 28.g odd 6 1
1260.2.s.d 2 12.b even 2 1
1260.2.s.d 2 84.n even 6 1
1680.2.bg.a 2 1.a even 1 1 trivial
1680.2.bg.a 2 7.c even 3 1 inner
2100.2.q.a 2 20.d odd 2 1
2100.2.q.a 2 140.p odd 6 1
2100.2.bc.c 4 20.e even 4 2
2100.2.bc.c 4 140.w even 12 2
2940.2.a.d 1 28.g odd 6 1
2940.2.a.h 1 28.f even 6 1
2940.2.q.h 2 28.d even 2 1
2940.2.q.h 2 28.f even 6 1
8820.2.a.j 1 84.n even 6 1
8820.2.a.y 1 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1680, [\chi])\):

\( T_{11}^{2} + 2 T_{11} + 4 \)
\( T_{13} - 1 \)
\( T_{17}^{2} - 4 T_{17} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 7 + 5 T + T^{2} \)
$11$ \( 4 + 2 T + T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( 16 - 4 T + T^{2} \)
$19$ \( 1 + T + T^{2} \)
$23$ \( 16 - 4 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( 25 + 5 T + T^{2} \)
$37$ \( 25 - 5 T + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( ( -9 + T )^{2} \)
$47$ \( 4 + 2 T + T^{2} \)
$53$ \( 144 + 12 T + T^{2} \)
$59$ \( 64 + 8 T + T^{2} \)
$61$ \( 196 - 14 T + T^{2} \)
$67$ \( 81 - 9 T + T^{2} \)
$71$ \( ( 2 + T )^{2} \)
$73$ \( 1 + T + T^{2} \)
$79$ \( 9 + 3 T + T^{2} \)
$83$ \( ( -18 + T )^{2} \)
$89$ \( 16 - 4 T + T^{2} \)
$97$ \( ( -10 + T )^{2} \)
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