Properties

Label 1680.2.f.h
Level $1680$
Weight $2$
Character orbit 1680.f
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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} - q^{5} + ( 2 - \beta_{2} ) q^{7} + ( 1 + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} - q^{5} + ( 2 - \beta_{2} ) q^{7} + ( 1 + \beta_{2} - \beta_{3} ) q^{9} + ( -\beta_{1} - \beta_{3} ) q^{11} + ( 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{13} -\beta_{1} q^{15} + ( -2 + \beta_{1} - \beta_{3} ) q^{17} + 2 \beta_{2} q^{19} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{21} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{23} + q^{25} + ( 4 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{27} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{29} -2 \beta_{2} q^{31} + ( 2 - \beta_{2} + \beta_{3} ) q^{33} + ( -2 + \beta_{2} ) q^{35} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{37} + ( -4 + 2 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{39} + 6 q^{41} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{43} + ( -1 - \beta_{2} + \beta_{3} ) q^{45} + ( -4 - \beta_{1} + \beta_{3} ) q^{47} + ( 1 - 4 \beta_{2} ) q^{49} + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{51} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{53} + ( \beta_{1} + \beta_{3} ) q^{55} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{57} + ( -4 + 2 \beta_{1} - 2 \beta_{3} ) q^{59} + 4 \beta_{2} q^{61} + ( 6 - 2 \beta_{1} - 3 \beta_{3} ) q^{63} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{65} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{67} + ( -6 - 2 \beta_{1} - 6 \beta_{3} ) q^{69} + ( -4 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{71} + 4 \beta_{2} q^{73} + \beta_{1} q^{75} + ( 2 - 3 \beta_{1} - \beta_{3} ) q^{77} + ( -\beta_{1} + \beta_{3} ) q^{79} + ( -3 + 5 \beta_{1} + 3 \beta_{2} ) q^{81} + ( -8 + 4 \beta_{1} - 4 \beta_{3} ) q^{83} + ( 2 - \beta_{1} + \beta_{3} ) q^{85} + ( 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{87} + ( -10 + 2 \beta_{1} - 2 \beta_{3} ) q^{89} + ( 9 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{91} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{93} -2 \beta_{2} q^{95} + ( 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{97} + ( -4 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{3} - 4q^{5} + 8q^{7} + 5q^{9} + O(q^{10}) \) \( 4q + q^{3} - 4q^{5} + 8q^{7} + 5q^{9} - q^{15} - 6q^{17} - q^{21} + 4q^{25} + 16q^{27} + 7q^{33} - 8q^{35} - 4q^{37} - 15q^{39} + 24q^{41} + 4q^{43} - 5q^{45} - 18q^{47} + 4q^{49} + 15q^{51} + 6q^{57} - 12q^{59} + 25q^{63} + 4q^{67} - 20q^{69} + q^{75} + 6q^{77} - 2q^{79} - 7q^{81} - 24q^{83} + 6q^{85} - q^{87} - 36q^{89} + 6q^{91} - 6q^{93} - 13q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 6 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 2 \nu - 3 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + 1\)
\(\nu^{3}\)\(=\)\(2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 4\)

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
−1.18614 1.26217i
−1.18614 + 1.26217i
1.68614 0.396143i
1.68614 + 0.396143i
0 −1.18614 1.26217i 0 −1.00000 0 2.00000 1.73205i 0 −0.186141 + 2.99422i 0
881.2 0 −1.18614 + 1.26217i 0 −1.00000 0 2.00000 + 1.73205i 0 −0.186141 2.99422i 0
881.3 0 1.68614 0.396143i 0 −1.00000 0 2.00000 + 1.73205i 0 2.68614 1.33591i 0
881.4 0 1.68614 + 0.396143i 0 −1.00000 0 2.00000 1.73205i 0 2.68614 + 1.33591i 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.h 4
3.b odd 2 1 1680.2.f.g 4
4.b odd 2 1 105.2.b.c 4
7.b odd 2 1 1680.2.f.g 4
12.b even 2 1 105.2.b.d yes 4
20.d odd 2 1 525.2.b.g 4
20.e even 4 2 525.2.g.e 8
21.c even 2 1 inner 1680.2.f.h 4
28.d even 2 1 105.2.b.d yes 4
28.f even 6 1 735.2.s.g 4
28.f even 6 1 735.2.s.j 4
28.g odd 6 1 735.2.s.h 4
28.g odd 6 1 735.2.s.i 4
60.h even 2 1 525.2.b.e 4
60.l odd 4 2 525.2.g.d 8
84.h odd 2 1 105.2.b.c 4
84.j odd 6 1 735.2.s.h 4
84.j odd 6 1 735.2.s.i 4
84.n even 6 1 735.2.s.g 4
84.n even 6 1 735.2.s.j 4
140.c even 2 1 525.2.b.e 4
140.j odd 4 2 525.2.g.d 8
420.o odd 2 1 525.2.b.g 4
420.w even 4 2 525.2.g.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.c 4 4.b odd 2 1
105.2.b.c 4 84.h odd 2 1
105.2.b.d yes 4 12.b even 2 1
105.2.b.d yes 4 28.d even 2 1
525.2.b.e 4 60.h even 2 1
525.2.b.e 4 140.c even 2 1
525.2.b.g 4 20.d odd 2 1
525.2.b.g 4 420.o odd 2 1
525.2.g.d 8 60.l odd 4 2
525.2.g.d 8 140.j odd 4 2
525.2.g.e 8 20.e even 4 2
525.2.g.e 8 420.w even 4 2
735.2.s.g 4 28.f even 6 1
735.2.s.g 4 84.n even 6 1
735.2.s.h 4 28.g odd 6 1
735.2.s.h 4 84.j odd 6 1
735.2.s.i 4 28.g odd 6 1
735.2.s.i 4 84.j odd 6 1
735.2.s.j 4 28.f even 6 1
735.2.s.j 4 84.n even 6 1
1680.2.f.g 4 3.b odd 2 1
1680.2.f.g 4 7.b odd 2 1
1680.2.f.h 4 1.a even 1 1 trivial
1680.2.f.h 4 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 \(S_{2}^{\mathrm{new}}(1680, [\chi])\):

\( T_{11}^{4} + 7 T_{11}^{2} + 4 \)
\( T_{17}^{2} + 3 T_{17} - 6 \)
\( T_{41} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 - 3 T - 2 T^{2} - T^{3} + T^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( ( 7 - 4 T + T^{2} )^{2} \)
$11$ \( 4 + 7 T^{2} + T^{4} \)
$13$ \( 576 + 51 T^{2} + T^{4} \)
$17$ \( ( -6 + 3 T + T^{2} )^{2} \)
$19$ \( ( 12 + T^{2} )^{2} \)
$23$ \( 256 + 76 T^{2} + T^{4} \)
$29$ \( 16 + 19 T^{2} + T^{4} \)
$31$ \( ( 12 + T^{2} )^{2} \)
$37$ \( ( -32 + 2 T + T^{2} )^{2} \)
$41$ \( ( -6 + T )^{4} \)
$43$ \( ( -32 - 2 T + T^{2} )^{2} \)
$47$ \( ( 12 + 9 T + T^{2} )^{2} \)
$53$ \( 256 + 76 T^{2} + T^{4} \)
$59$ \( ( -24 + 6 T + T^{2} )^{2} \)
$61$ \( ( 48 + T^{2} )^{2} \)
$67$ \( ( -32 - 2 T + T^{2} )^{2} \)
$71$ \( 16 + 184 T^{2} + T^{4} \)
$73$ \( ( 48 + T^{2} )^{2} \)
$79$ \( ( -8 + T + T^{2} )^{2} \)
$83$ \( ( -96 + 12 T + T^{2} )^{2} \)
$89$ \( ( 48 + 18 T + T^{2} )^{2} \)
$97$ \( 144 + 123 T^{2} + T^{4} \)
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