Properties

Label 1680.2.f.a
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: \(\Q(\sqrt{-3}) \)
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: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 + \zeta_{6} ) q^{3} + q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})\) \( q + ( -2 + \zeta_{6} ) q^{3} + q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} + ( 3 - 6 \zeta_{6} ) q^{11} + ( 1 - 2 \zeta_{6} ) q^{13} + ( -2 + \zeta_{6} ) q^{15} -3 q^{17} + ( -2 + 4 \zeta_{6} ) q^{19} + ( 4 + \zeta_{6} ) q^{21} + q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -3 + 6 \zeta_{6} ) q^{29} + ( -6 + 12 \zeta_{6} ) q^{31} + 9 \zeta_{6} q^{33} + ( -1 - 2 \zeta_{6} ) q^{35} -8 q^{37} + 3 \zeta_{6} q^{39} -6 q^{41} -10 q^{43} + ( 3 - 3 \zeta_{6} ) q^{45} + 3 q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( 6 - 3 \zeta_{6} ) q^{51} + ( 6 - 12 \zeta_{6} ) q^{53} + ( 3 - 6 \zeta_{6} ) q^{55} -6 \zeta_{6} q^{57} + 6 q^{59} + ( -4 + 8 \zeta_{6} ) q^{61} + ( -9 + 3 \zeta_{6} ) q^{63} + ( 1 - 2 \zeta_{6} ) q^{65} -2 q^{67} + ( 6 - 12 \zeta_{6} ) q^{71} + ( -4 + 8 \zeta_{6} ) q^{73} + ( -2 + \zeta_{6} ) q^{75} + ( -15 + 12 \zeta_{6} ) q^{77} + 13 q^{79} -9 \zeta_{6} q^{81} -12 q^{83} -3 q^{85} -9 \zeta_{6} q^{87} + ( -5 + 4 \zeta_{6} ) q^{91} -18 \zeta_{6} q^{93} + ( -2 + 4 \zeta_{6} ) q^{95} + ( 1 - 2 \zeta_{6} ) q^{97} + ( -9 - 9 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} + 2q^{5} - 4q^{7} + 3q^{9} + O(q^{10}) \) \( 2q - 3q^{3} + 2q^{5} - 4q^{7} + 3q^{9} - 3q^{15} - 6q^{17} + 9q^{21} + 2q^{25} + 9q^{33} - 4q^{35} - 16q^{37} + 3q^{39} - 12q^{41} - 20q^{43} + 3q^{45} + 6q^{47} + 2q^{49} + 9q^{51} - 6q^{57} + 12q^{59} - 15q^{63} - 4q^{67} - 3q^{75} - 18q^{77} + 26q^{79} - 9q^{81} - 24q^{83} - 6q^{85} - 9q^{87} - 6q^{91} - 18q^{93} - 27q^{99} + O(q^{100}) \)

Character values

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

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

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 −1.50000 0.866025i 0 1.00000 0 −2.00000 + 1.73205i 0 1.50000 + 2.59808i 0
881.2 0 −1.50000 + 0.866025i 0 1.00000 0 −2.00000 1.73205i 0 1.50000 2.59808i 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.a 2
3.b odd 2 1 1680.2.f.d 2
4.b odd 2 1 420.2.d.b yes 2
7.b odd 2 1 1680.2.f.d 2
12.b even 2 1 420.2.d.a 2
20.d odd 2 1 2100.2.d.a 2
20.e even 4 2 2100.2.f.d 4
21.c even 2 1 inner 1680.2.f.a 2
28.d even 2 1 420.2.d.a 2
60.h even 2 1 2100.2.d.e 2
60.l odd 4 2 2100.2.f.c 4
84.h odd 2 1 420.2.d.b yes 2
140.c even 2 1 2100.2.d.e 2
140.j odd 4 2 2100.2.f.c 4
420.o odd 2 1 2100.2.d.a 2
420.w even 4 2 2100.2.f.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.d.a 2 12.b even 2 1
420.2.d.a 2 28.d even 2 1
420.2.d.b yes 2 4.b odd 2 1
420.2.d.b yes 2 84.h odd 2 1
1680.2.f.a 2 1.a even 1 1 trivial
1680.2.f.a 2 21.c even 2 1 inner
1680.2.f.d 2 3.b odd 2 1
1680.2.f.d 2 7.b odd 2 1
2100.2.d.a 2 20.d odd 2 1
2100.2.d.a 2 420.o odd 2 1
2100.2.d.e 2 60.h even 2 1
2100.2.d.e 2 140.c even 2 1
2100.2.f.c 4 60.l odd 4 2
2100.2.f.c 4 140.j odd 4 2
2100.2.f.d 4 20.e even 4 2
2100.2.f.d 4 420.w even 4 2

Hecke kernels

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

\( T_{11}^{2} + 27 \)
\( T_{17} + 3 \)
\( T_{41} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 + 3 T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( 7 + 4 T + T^{2} \)
$11$ \( 27 + T^{2} \)
$13$ \( 3 + T^{2} \)
$17$ \( ( 3 + T )^{2} \)
$19$ \( 12 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( 27 + T^{2} \)
$31$ \( 108 + T^{2} \)
$37$ \( ( 8 + T )^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( ( 10 + T )^{2} \)
$47$ \( ( -3 + T )^{2} \)
$53$ \( 108 + T^{2} \)
$59$ \( ( -6 + T )^{2} \)
$61$ \( 48 + T^{2} \)
$67$ \( ( 2 + T )^{2} \)
$71$ \( 108 + T^{2} \)
$73$ \( 48 + T^{2} \)
$79$ \( ( -13 + T )^{2} \)
$83$ \( ( 12 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( 3 + T^{2} \)
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