Properties

Label 1680.2.f.e
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: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 210)
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.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{2} ) q^{3} - q^{5} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{7} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{2} ) q^{3} - q^{5} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{7} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{9} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{11} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{13} + ( -\beta_{1} - \beta_{2} ) q^{15} + ( -2 + \beta_{1} - \beta_{3} ) q^{17} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{19} + ( -2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{21} + 4 \beta_{2} q^{23} + q^{25} + ( -4 - 3 \beta_{2} + \beta_{3} ) q^{27} + ( 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{29} + ( -\beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{31} + ( -6 - 2 \beta_{1} + 4 \beta_{2} ) q^{33} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{35} + ( 4 + \beta_{1} - \beta_{3} ) q^{37} + ( 4 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{39} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{41} + ( -8 - 4 \beta_{1} + 4 \beta_{3} ) q^{43} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{45} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{47} + ( -3 - 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{49} + ( 2 - 3 \beta_{1} + \beta_{3} ) q^{51} + ( -2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{53} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{55} + ( 2 + 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{57} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{59} + ( \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{61} + ( -5 - 6 \beta_{2} - \beta_{3} ) q^{63} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{65} + 12 q^{67} + ( -4 + 4 \beta_{3} ) q^{69} + ( -\beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{71} + ( -\beta_{1} + 8 \beta_{2} - \beta_{3} ) q^{73} + ( \beta_{1} + \beta_{2} ) q^{75} + ( -8 - 6 \beta_{1} + 2 \beta_{2} ) q^{77} + ( -4 - 4 \beta_{1} + 4 \beta_{3} ) q^{79} + ( 1 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{81} + ( -8 - 7 \beta_{1} + 7 \beta_{3} ) q^{83} + ( 2 - \beta_{1} + \beta_{3} ) q^{85} + ( -8 - 3 \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{87} + ( 12 + 2 \beta_{1} - 2 \beta_{3} ) q^{89} + ( 6 + 3 \beta_{1} - \beta_{3} ) q^{91} + ( 8 + \beta_{1} - 2 \beta_{2} - 5 \beta_{3} ) q^{93} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{95} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{97} + ( -4 - 4 \beta_{1} - 10 \beta_{2} + 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} - 4q^{5} - 6q^{7} + O(q^{10}) \) \( 4q - 2q^{3} - 4q^{5} - 6q^{7} + 2q^{15} - 12q^{17} + 4q^{25} - 14q^{27} - 20q^{33} + 6q^{35} + 12q^{37} + 12q^{39} - 8q^{41} - 16q^{43} - 8q^{47} + 16q^{51} + 12q^{57} - 22q^{63} + 48q^{67} - 8q^{69} - 2q^{75} - 20q^{77} + 4q^{81} - 4q^{83} + 12q^{85} - 28q^{87} + 40q^{89} + 16q^{91} + 20q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

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

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
1.61803i
1.61803i
0.618034i
0.618034i
0 −1.61803 0.618034i 0 −1.00000 0 −2.61803 + 0.381966i 0 2.23607 + 2.00000i 0
881.2 0 −1.61803 + 0.618034i 0 −1.00000 0 −2.61803 0.381966i 0 2.23607 2.00000i 0
881.3 0 0.618034 1.61803i 0 −1.00000 0 −0.381966 2.61803i 0 −2.23607 2.00000i 0
881.4 0 0.618034 + 1.61803i 0 −1.00000 0 −0.381966 + 2.61803i 0 −2.23607 + 2.00000i 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.e 4
3.b odd 2 1 1680.2.f.i 4
4.b odd 2 1 210.2.b.b yes 4
7.b odd 2 1 1680.2.f.i 4
12.b even 2 1 210.2.b.a 4
20.d odd 2 1 1050.2.b.a 4
20.e even 4 1 1050.2.d.a 4
20.e even 4 1 1050.2.d.f 4
21.c even 2 1 inner 1680.2.f.e 4
28.d even 2 1 210.2.b.a 4
60.h even 2 1 1050.2.b.c 4
60.l odd 4 1 1050.2.d.c 4
60.l odd 4 1 1050.2.d.d 4
84.h odd 2 1 210.2.b.b yes 4
140.c even 2 1 1050.2.b.c 4
140.j odd 4 1 1050.2.d.c 4
140.j odd 4 1 1050.2.d.d 4
420.o odd 2 1 1050.2.b.a 4
420.w even 4 1 1050.2.d.a 4
420.w even 4 1 1050.2.d.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.b.a 4 12.b even 2 1
210.2.b.a 4 28.d even 2 1
210.2.b.b yes 4 4.b odd 2 1
210.2.b.b yes 4 84.h odd 2 1
1050.2.b.a 4 20.d odd 2 1
1050.2.b.a 4 420.o odd 2 1
1050.2.b.c 4 60.h even 2 1
1050.2.b.c 4 140.c even 2 1
1050.2.d.a 4 20.e even 4 1
1050.2.d.a 4 420.w even 4 1
1050.2.d.c 4 60.l odd 4 1
1050.2.d.c 4 140.j odd 4 1
1050.2.d.d 4 60.l odd 4 1
1050.2.d.d 4 140.j odd 4 1
1050.2.d.f 4 20.e even 4 1
1050.2.d.f 4 420.w even 4 1
1680.2.f.e 4 1.a even 1 1 trivial
1680.2.f.e 4 21.c even 2 1 inner
1680.2.f.i 4 3.b odd 2 1
1680.2.f.i 4 7.b odd 2 1

Hecke kernels

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

\( T_{11}^{2} + 20 \)
\( T_{17}^{2} + 6 T_{17} + 4 \)
\( T_{41}^{2} + 4 T_{41} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + 6 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( 49 + 42 T + 18 T^{2} + 6 T^{3} + T^{4} \)
$11$ \( ( 20 + T^{2} )^{2} \)
$13$ \( 16 + 12 T^{2} + T^{4} \)
$17$ \( ( 4 + 6 T + T^{2} )^{2} \)
$19$ \( 16 + 72 T^{2} + T^{4} \)
$23$ \( ( 16 + T^{2} )^{2} \)
$29$ \( 1936 + 92 T^{2} + T^{4} \)
$31$ \( 400 + 60 T^{2} + T^{4} \)
$37$ \( ( 4 - 6 T + T^{2} )^{2} \)
$41$ \( ( -16 + 4 T + T^{2} )^{2} \)
$43$ \( ( -64 + 8 T + T^{2} )^{2} \)
$47$ \( ( -16 + 4 T + T^{2} )^{2} \)
$53$ \( 16 + 72 T^{2} + T^{4} \)
$59$ \( ( -20 + T^{2} )^{2} \)
$61$ \( 400 + 60 T^{2} + T^{4} \)
$67$ \( ( -12 + T )^{4} \)
$71$ \( 400 + 60 T^{2} + T^{4} \)
$73$ \( 5776 + 172 T^{2} + T^{4} \)
$79$ \( ( -80 + T^{2} )^{2} \)
$83$ \( ( -244 + 2 T + T^{2} )^{2} \)
$89$ \( ( 80 - 20 T + T^{2} )^{2} \)
$97$ \( 16 + 28 T^{2} + T^{4} \)
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