Properties

Label 1680.2.bg.o
Level $1680$
Weight $2$
Character orbit 1680.bg
Analytic conductor $13.415$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{12}^{2} q^{3} + ( -1 + \zeta_{12}^{2} ) q^{5} + ( -\zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + ( -1 + \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q -\zeta_{12}^{2} q^{3} + ( -1 + \zeta_{12}^{2} ) q^{5} + ( -\zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + ( -1 + \zeta_{12}^{2} ) q^{9} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{11} + ( 4 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{13} + q^{15} + ( \zeta_{12} - 5 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{17} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{19} + ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{21} + ( -3 - \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{23} -\zeta_{12}^{2} q^{25} + q^{27} + ( 1 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{29} + ( -2 \zeta_{12} + 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{31} + ( -1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{33} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{35} + ( -2 + 3 \zeta_{12} + 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{37} + ( \zeta_{12} - 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{39} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{41} + ( 2 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{43} -\zeta_{12}^{2} q^{45} + ( 2 - 2 \zeta_{12}^{2} ) q^{47} + ( -8 + 5 \zeta_{12}^{2} ) q^{49} + ( -5 + \zeta_{12} + 5 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{51} + ( -6 \zeta_{12} - 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{53} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{55} + ( -1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{57} + ( -3 \zeta_{12} + 5 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{59} + ( -4 + 4 \zeta_{12}^{2} ) q^{61} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{63} + ( -4 + \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{65} + ( -5 \zeta_{12} - 6 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{67} + ( 3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{69} + ( -1 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{71} + ( -5 \zeta_{12} - 4 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{73} + ( -1 + \zeta_{12}^{2} ) q^{75} + ( 5 - 2 \zeta_{12} - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{77} + ( 3 - 6 \zeta_{12} - 3 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{79} -\zeta_{12}^{2} q^{81} + ( -3 - 14 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{83} + ( 5 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{85} + ( 3 \zeta_{12} - \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{87} + ( -3 - 7 \zeta_{12} + 3 \zeta_{12}^{2} + 14 \zeta_{12}^{3} ) q^{89} + ( -1 - 4 \zeta_{12} + 5 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{91} + ( 3 - 2 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{93} + ( 2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{95} + ( 8 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{97} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} - 2q^{5} - 2q^{9} + O(q^{10}) \) \( 4q - 2q^{3} - 2q^{5} - 2q^{9} - 2q^{11} + 16q^{13} + 4q^{15} - 10q^{17} + 2q^{19} - 6q^{23} - 2q^{25} + 4q^{27} + 4q^{29} + 6q^{31} - 2q^{33} - 4q^{37} - 8q^{39} + 4q^{41} + 8q^{43} - 2q^{45} + 4q^{47} - 22q^{49} - 10q^{51} - 4q^{53} + 4q^{55} - 4q^{57} + 10q^{59} - 8q^{61} - 8q^{65} - 12q^{67} + 12q^{69} - 4q^{71} - 8q^{73} - 2q^{75} + 12q^{77} + 6q^{79} - 2q^{81} - 12q^{83} + 20q^{85} - 2q^{87} - 6q^{89} + 6q^{91} + 6q^{93} + 2q^{95} + 32q^{97} + 4q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(1\) \(1\) \(1\) \(1\)

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
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 −0.866025 2.50000i 0 −0.500000 + 0.866025i 0
961.2 0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0.866025 + 2.50000i 0 −0.500000 + 0.866025i 0
1201.1 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 −0.866025 + 2.50000i 0 −0.500000 0.866025i 0
1201.2 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0.866025 2.50000i 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.o 4
4.b odd 2 1 105.2.i.d 4
7.c even 3 1 inner 1680.2.bg.o 4
12.b even 2 1 315.2.j.c 4
20.d odd 2 1 525.2.i.f 4
20.e even 4 1 525.2.r.a 4
20.e even 4 1 525.2.r.f 4
28.d even 2 1 735.2.i.l 4
28.f even 6 1 735.2.a.h 2
28.f even 6 1 735.2.i.l 4
28.g odd 6 1 105.2.i.d 4
28.g odd 6 1 735.2.a.g 2
84.j odd 6 1 2205.2.a.ba 2
84.n even 6 1 315.2.j.c 4
84.n even 6 1 2205.2.a.z 2
140.p odd 6 1 525.2.i.f 4
140.p odd 6 1 3675.2.a.bg 2
140.s even 6 1 3675.2.a.be 2
140.w even 12 1 525.2.r.a 4
140.w even 12 1 525.2.r.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.d 4 4.b odd 2 1
105.2.i.d 4 28.g odd 6 1
315.2.j.c 4 12.b even 2 1
315.2.j.c 4 84.n even 6 1
525.2.i.f 4 20.d odd 2 1
525.2.i.f 4 140.p odd 6 1
525.2.r.a 4 20.e even 4 1
525.2.r.a 4 140.w even 12 1
525.2.r.f 4 20.e even 4 1
525.2.r.f 4 140.w even 12 1
735.2.a.g 2 28.g odd 6 1
735.2.a.h 2 28.f even 6 1
735.2.i.l 4 28.d even 2 1
735.2.i.l 4 28.f even 6 1
1680.2.bg.o 4 1.a even 1 1 trivial
1680.2.bg.o 4 7.c even 3 1 inner
2205.2.a.z 2 84.n even 6 1
2205.2.a.ba 2 84.j odd 6 1
3675.2.a.be 2 140.s even 6 1
3675.2.a.bg 2 140.p odd 6 1

Hecke kernels

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

\( T_{11}^{4} + 2 T_{11}^{3} + 6 T_{11}^{2} - 4 T_{11} + 4 \)
\( T_{13}^{2} - 8 T_{13} + 13 \)
\( T_{17}^{4} + 10 T_{17}^{3} + 78 T_{17}^{2} + 220 T_{17} + 484 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( 49 + 11 T^{2} + T^{4} \)
$11$ \( 4 - 4 T + 6 T^{2} + 2 T^{3} + T^{4} \)
$13$ \( ( 13 - 8 T + T^{2} )^{2} \)
$17$ \( 484 + 220 T + 78 T^{2} + 10 T^{3} + T^{4} \)
$19$ \( 121 + 22 T + 15 T^{2} - 2 T^{3} + T^{4} \)
$23$ \( 36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4} \)
$29$ \( ( -26 - 2 T + T^{2} )^{2} \)
$31$ \( 9 + 18 T + 39 T^{2} - 6 T^{3} + T^{4} \)
$37$ \( 529 - 92 T + 39 T^{2} + 4 T^{3} + T^{4} \)
$41$ \( ( -2 - 2 T + T^{2} )^{2} \)
$43$ \( ( -23 - 4 T + T^{2} )^{2} \)
$47$ \( ( 4 - 2 T + T^{2} )^{2} \)
$53$ \( 10816 - 416 T + 120 T^{2} + 4 T^{3} + T^{4} \)
$59$ \( 4 + 20 T + 102 T^{2} - 10 T^{3} + T^{4} \)
$61$ \( ( 16 + 4 T + T^{2} )^{2} \)
$67$ \( 1521 - 468 T + 183 T^{2} + 12 T^{3} + T^{4} \)
$71$ \( ( -26 + 2 T + T^{2} )^{2} \)
$73$ \( 3481 - 472 T + 123 T^{2} + 8 T^{3} + T^{4} \)
$79$ \( 9801 + 594 T + 135 T^{2} - 6 T^{3} + T^{4} \)
$83$ \( ( -138 + 6 T + T^{2} )^{2} \)
$89$ \( 19044 - 828 T + 174 T^{2} + 6 T^{3} + T^{4} \)
$97$ \( ( 16 - 16 T + T^{2} )^{2} \)
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