Properties

Label 1680.2.k.a
Level 1680
Weight 2
Character orbit 1680.k
Analytic conductor 13.415
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1680.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} ) q^{3} + ( -\beta_{1} + \beta_{2} ) q^{5} + ( 1 - \beta_{3} ) q^{7} + ( -1 + 2 \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{1} ) q^{3} + ( -\beta_{1} + \beta_{2} ) q^{5} + ( 1 - \beta_{3} ) q^{7} + ( -1 + 2 \beta_{1} ) q^{9} -2 \beta_{1} q^{11} + 4 q^{13} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{15} -2 \beta_{1} q^{17} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{21} + 2 \beta_{2} q^{23} + ( 1 - 2 \beta_{3} ) q^{25} + ( 5 - \beta_{1} ) q^{27} -4 \beta_{1} q^{29} + 4 \beta_{3} q^{31} + ( -4 + 2 \beta_{1} ) q^{33} + ( -4 \beta_{1} - \beta_{2} ) q^{35} + ( -4 - 4 \beta_{1} ) q^{39} + 2 \beta_{2} q^{41} -2 \beta_{3} q^{43} + ( 4 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{45} + 2 \beta_{1} q^{47} + ( -5 - 2 \beta_{3} ) q^{49} + ( -4 + 2 \beta_{1} ) q^{51} + ( -4 - 2 \beta_{3} ) q^{55} + 4 \beta_{2} q^{59} + 4 \beta_{3} q^{61} + ( -1 + 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{63} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{65} + 2 \beta_{3} q^{67} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{69} -2 \beta_{1} q^{71} -8 q^{73} + ( -1 - \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{75} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{77} -8 q^{79} + ( -7 - 4 \beta_{1} ) q^{81} + 2 \beta_{1} q^{83} + ( -4 - 2 \beta_{3} ) q^{85} + ( -8 + 4 \beta_{1} ) q^{87} -6 \beta_{2} q^{89} + ( 4 - 4 \beta_{3} ) q^{91} + ( 8 \beta_{2} - 4 \beta_{3} ) q^{93} -8 q^{97} + ( 8 + 2 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 4q^{7} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 4q^{7} - 4q^{9} + 16q^{13} - 8q^{15} - 4q^{21} + 4q^{25} + 20q^{27} - 16q^{33} - 16q^{39} + 16q^{45} - 20q^{49} - 16q^{51} - 16q^{55} - 4q^{63} - 32q^{73} - 4q^{75} - 32q^{79} - 28q^{81} - 16q^{85} - 32q^{87} + 16q^{91} - 32q^{97} + 32q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 3 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{3} + 5 \beta_{1}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
1.93185i
0.517638i
1.93185i
0.517638i
0 −1.00000 1.41421i 0 −1.73205 1.41421i 0 1.00000 + 2.44949i 0 −1.00000 + 2.82843i 0
209.2 0 −1.00000 1.41421i 0 1.73205 1.41421i 0 1.00000 2.44949i 0 −1.00000 + 2.82843i 0
209.3 0 −1.00000 + 1.41421i 0 −1.73205 + 1.41421i 0 1.00000 2.44949i 0 −1.00000 2.82843i 0
209.4 0 −1.00000 + 1.41421i 0 1.73205 + 1.41421i 0 1.00000 + 2.44949i 0 −1.00000 2.82843i 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
3.b odd 2 1 inner
35.c odd 2 1 inner
105.g 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.k.a 4
3.b odd 2 1 inner 1680.2.k.a 4
4.b odd 2 1 105.2.g.c yes 4
5.b even 2 1 1680.2.k.c 4
7.b odd 2 1 1680.2.k.c 4
12.b even 2 1 105.2.g.c yes 4
15.d odd 2 1 1680.2.k.c 4
20.d odd 2 1 105.2.g.a 4
20.e even 4 2 525.2.b.j 8
21.c even 2 1 1680.2.k.c 4
28.d even 2 1 105.2.g.a 4
28.f even 6 2 735.2.p.c 8
28.g odd 6 2 735.2.p.a 8
35.c odd 2 1 inner 1680.2.k.a 4
60.h even 2 1 105.2.g.a 4
60.l odd 4 2 525.2.b.j 8
84.h odd 2 1 105.2.g.a 4
84.j odd 6 2 735.2.p.c 8
84.n even 6 2 735.2.p.a 8
105.g even 2 1 inner 1680.2.k.a 4
140.c even 2 1 105.2.g.c yes 4
140.j odd 4 2 525.2.b.j 8
140.p odd 6 2 735.2.p.c 8
140.s even 6 2 735.2.p.a 8
420.o odd 2 1 105.2.g.c yes 4
420.w even 4 2 525.2.b.j 8
420.ba even 6 2 735.2.p.c 8
420.be odd 6 2 735.2.p.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.g.a 4 20.d odd 2 1
105.2.g.a 4 28.d even 2 1
105.2.g.a 4 60.h even 2 1
105.2.g.a 4 84.h odd 2 1
105.2.g.c yes 4 4.b odd 2 1
105.2.g.c yes 4 12.b even 2 1
105.2.g.c yes 4 140.c even 2 1
105.2.g.c yes 4 420.o odd 2 1
525.2.b.j 8 20.e even 4 2
525.2.b.j 8 60.l odd 4 2
525.2.b.j 8 140.j odd 4 2
525.2.b.j 8 420.w even 4 2
735.2.p.a 8 28.g odd 6 2
735.2.p.a 8 84.n even 6 2
735.2.p.a 8 140.s even 6 2
735.2.p.a 8 420.be odd 6 2
735.2.p.c 8 28.f even 6 2
735.2.p.c 8 84.j odd 6 2
735.2.p.c 8 140.p odd 6 2
735.2.p.c 8 420.ba even 6 2
1680.2.k.a 4 1.a even 1 1 trivial
1680.2.k.a 4 3.b odd 2 1 inner
1680.2.k.a 4 35.c odd 2 1 inner
1680.2.k.a 4 105.g even 2 1 inner
1680.2.k.c 4 5.b even 2 1
1680.2.k.c 4 7.b odd 2 1
1680.2.k.c 4 15.d odd 2 1
1680.2.k.c 4 21.c even 2 1

Hecke kernels

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

\( T_{11}^{2} + 8 \)
\( T_{13} - 4 \)
\( T_{23}^{2} - 12 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + 2 T + 3 T^{2} )^{2} \)
$5$ \( 1 - 2 T^{2} + 25 T^{4} \)
$7$ \( ( 1 - 2 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )^{4} \)
$17$ \( ( 1 - 26 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 19 T^{2} )^{4} \)
$23$ \( ( 1 + 34 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 26 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 34 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 37 T^{2} )^{4} \)
$41$ \( ( 1 + 70 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 62 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 86 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 53 T^{2} )^{4} \)
$59$ \( ( 1 + 70 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 26 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 110 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 134 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 8 T + 73 T^{2} )^{4} \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 - 18 T + 83 T^{2} )^{2}( 1 + 18 T + 83 T^{2} )^{2} \)
$89$ \( ( 1 + 70 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 8 T + 97 T^{2} )^{4} \)
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