Properties

Label 1680.2.f.c
Level 1680
Weight 2
Character orbit 1680.f
Analytic conductor 13.415
Analytic rank 1
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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} -3 q^{9} + ( 2 - 4 \zeta_{6} ) q^{11} + ( 1 - 2 \zeta_{6} ) q^{15} -6 q^{17} + ( 2 - 4 \zeta_{6} ) q^{19} + ( 1 + 4 \zeta_{6} ) q^{21} + ( -2 + 4 \zeta_{6} ) q^{23} + q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -4 + 8 \zeta_{6} ) q^{29} + ( -2 + 4 \zeta_{6} ) q^{31} -6 q^{33} + ( -3 + 2 \zeta_{6} ) q^{35} -2 q^{37} -6 q^{41} + 8 q^{43} -3 q^{45} -12 q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -6 + 12 \zeta_{6} ) q^{51} + ( 2 - 4 \zeta_{6} ) q^{55} -6 q^{57} -12 q^{59} + ( 4 - 8 \zeta_{6} ) q^{61} + ( 9 - 6 \zeta_{6} ) q^{63} -8 q^{67} + 6 q^{69} + ( -2 + 4 \zeta_{6} ) q^{71} + ( -4 + 8 \zeta_{6} ) q^{73} + ( 1 - 2 \zeta_{6} ) q^{75} + ( 2 + 8 \zeta_{6} ) q^{77} -8 q^{79} + 9 q^{81} -6 q^{85} + 12 q^{87} -6 q^{89} + 6 q^{93} + ( 2 - 4 \zeta_{6} ) q^{95} + ( -4 + 8 \zeta_{6} ) q^{97} + ( -6 + 12 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - 4q^{7} - 6q^{9} + O(q^{10}) \) \( 2q + 2q^{5} - 4q^{7} - 6q^{9} - 12q^{17} + 6q^{21} + 2q^{25} - 12q^{33} - 4q^{35} - 4q^{37} - 12q^{41} + 16q^{43} - 6q^{45} - 24q^{47} + 2q^{49} - 12q^{57} - 24q^{59} + 12q^{63} - 16q^{67} + 12q^{69} + 12q^{77} - 16q^{79} + 18q^{81} - 12q^{85} + 24q^{87} - 12q^{89} + 12q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 0 1.00000 0 −2.00000 + 1.73205i 0 −3.00000 0
881.2 0 1.73205i 0 1.00000 0 −2.00000 1.73205i 0 −3.00000 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
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.f.c 2
3.b odd 2 1 1680.2.f.b 2
4.b odd 2 1 105.2.b.b yes 2
7.b odd 2 1 1680.2.f.b 2
12.b even 2 1 105.2.b.a 2
20.d odd 2 1 525.2.b.b 2
20.e even 4 2 525.2.g.c 4
21.c even 2 1 inner 1680.2.f.c 2
28.d even 2 1 105.2.b.a 2
28.f even 6 1 735.2.s.b 2
28.f even 6 1 735.2.s.d 2
28.g odd 6 1 735.2.s.a 2
28.g odd 6 1 735.2.s.f 2
60.h even 2 1 525.2.b.a 2
60.l odd 4 2 525.2.g.b 4
84.h odd 2 1 105.2.b.b yes 2
84.j odd 6 1 735.2.s.a 2
84.j odd 6 1 735.2.s.f 2
84.n even 6 1 735.2.s.b 2
84.n even 6 1 735.2.s.d 2
140.c even 2 1 525.2.b.a 2
140.j odd 4 2 525.2.g.b 4
420.o odd 2 1 525.2.b.b 2
420.w even 4 2 525.2.g.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.a 2 12.b even 2 1
105.2.b.a 2 28.d even 2 1
105.2.b.b yes 2 4.b odd 2 1
105.2.b.b yes 2 84.h odd 2 1
525.2.b.a 2 60.h even 2 1
525.2.b.a 2 140.c even 2 1
525.2.b.b 2 20.d odd 2 1
525.2.b.b 2 420.o odd 2 1
525.2.g.b 4 60.l odd 4 2
525.2.g.b 4 140.j odd 4 2
525.2.g.c 4 20.e even 4 2
525.2.g.c 4 420.w even 4 2
735.2.s.a 2 28.g odd 6 1
735.2.s.a 2 84.j odd 6 1
735.2.s.b 2 28.f even 6 1
735.2.s.b 2 84.n even 6 1
735.2.s.d 2 28.f even 6 1
735.2.s.d 2 84.n even 6 1
735.2.s.f 2 28.g odd 6 1
735.2.s.f 2 84.j odd 6 1
1680.2.f.b 2 3.b odd 2 1
1680.2.f.b 2 7.b odd 2 1
1680.2.f.c 2 1.a even 1 1 trivial
1680.2.f.c 2 21.c even 2 1 inner

Hecke kernels

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

\( T_{11}^{2} + 12 \)
\( T_{17} + 6 \)
\( T_{41} + 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T^{2} \)
$5$ \( ( 1 - T )^{2} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 - 10 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 13 T^{2} )^{2} \)
$17$ \( ( 1 + 6 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 - 34 T^{2} + 529 T^{4} \)
$29$ \( 1 - 10 T^{2} + 841 T^{4} \)
$31$ \( 1 - 50 T^{2} + 961 T^{4} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 12 T + 47 T^{2} )^{2} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( ( 1 + 12 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \)
$67$ \( ( 1 + 8 T + 67 T^{2} )^{2} \)
$71$ \( 1 - 130 T^{2} + 5041 T^{4} \)
$73$ \( 1 - 98 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( ( 1 + 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 146 T^{2} + 9409 T^{4} \)
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