Properties

Label 1680.2.bg.q
Level $1680$
Weight $2$
Character orbit 1680.bg
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.bg (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{3} + \beta_{2} q^{5} + (\beta_{3} + \beta_1) q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{3} + \beta_{2} q^{5} + (\beta_{3} + \beta_1) q^{7} + \beta_{2} q^{9} + (\beta_{2} + \beta_1 + 1) q^{11} - \beta_{3} q^{13} - q^{15} + ( - \beta_{2} - \beta_1 - 1) q^{17} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{19} + \beta_{3} q^{21} + (\beta_{3} + \beta_{2} + \beta_1) q^{23} + ( - \beta_{2} - 1) q^{25} - q^{27} + (\beta_{3} + 5) q^{29} + (\beta_{2} + 2 \beta_1 + 1) q^{31} + (\beta_{3} + \beta_{2} + \beta_1) q^{33} - \beta_1 q^{35} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{37} + \beta_1 q^{39} + (3 \beta_{3} - 3) q^{41} + (3 \beta_{3} - 2) q^{43} + ( - \beta_{2} - 1) q^{45} - 6 \beta_{2} q^{47} + ( - 7 \beta_{2} - 7) q^{49} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{51} + (2 \beta_{2} + 2 \beta_1 + 2) q^{53} + (\beta_{3} - 1) q^{55} + (2 \beta_{3} + 3) q^{57} + (3 \beta_{2} - 3 \beta_1 + 3) q^{59} + 8 \beta_{2} q^{61} - \beta_1 q^{63} + (\beta_{3} + \beta_1) q^{65} + ( - 2 \beta_{2} - \beta_1 - 2) q^{67} + (\beta_{3} - 1) q^{69} + ( - \beta_{3} - 11) q^{71} + 5 \beta_1 q^{73} - \beta_{2} q^{75} + (\beta_{3} - 7) q^{77} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{79} + ( - \beta_{2} - 1) q^{81} + (3 \beta_{3} + 3) q^{83} + ( - \beta_{3} + 1) q^{85} + (5 \beta_{2} - \beta_1 + 5) q^{87} + (5 \beta_{3} - \beta_{2} + 5 \beta_1) q^{89} + 7 \beta_{2} q^{91} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{93} + (3 \beta_{2} - 2 \beta_1 + 3) q^{95} + 8 q^{97} + (\beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 2 q^{5} - 2 q^{9} + 2 q^{11} - 4 q^{15} - 2 q^{17} + 6 q^{19} - 2 q^{23} - 2 q^{25} - 4 q^{27} + 20 q^{29} + 2 q^{31} - 2 q^{33} + 4 q^{37} - 12 q^{41} - 8 q^{43} - 2 q^{45} + 12 q^{47} - 14 q^{49} + 2 q^{51} + 4 q^{53} - 4 q^{55} + 12 q^{57} + 6 q^{59} - 16 q^{61} - 4 q^{67} - 4 q^{69} - 44 q^{71} + 2 q^{75} - 28 q^{77} - 6 q^{79} - 2 q^{81} + 12 q^{83} + 4 q^{85} + 10 q^{87} + 2 q^{89} - 14 q^{91} - 2 q^{93} + 6 q^{95} + 32 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
1.32288 + 2.29129i
−1.32288 2.29129i
1.32288 2.29129i
−1.32288 + 2.29129i
0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 −1.32288 + 2.29129i 0 −0.500000 + 0.866025i 0
961.2 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 1.32288 2.29129i 0 −0.500000 + 0.866025i 0
1201.1 0 0.500000 0.866025i 0 −0.500000 0.866025i 0 −1.32288 2.29129i 0 −0.500000 0.866025i 0
1201.2 0 0.500000 0.866025i 0 −0.500000 0.866025i 0 1.32288 + 2.29129i 0 −0.500000 0.866025i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.bg.q 4
4.b odd 2 1 420.2.q.c 4
7.c even 3 1 inner 1680.2.bg.q 4
12.b even 2 1 1260.2.s.f 4
20.d odd 2 1 2100.2.q.h 4
20.e even 4 2 2100.2.bc.e 8
28.d even 2 1 2940.2.q.t 4
28.f even 6 1 2940.2.a.m 2
28.f even 6 1 2940.2.q.t 4
28.g odd 6 1 420.2.q.c 4
28.g odd 6 1 2940.2.a.s 2
84.j odd 6 1 8820.2.a.bj 2
84.n even 6 1 1260.2.s.f 4
84.n even 6 1 8820.2.a.be 2
140.p odd 6 1 2100.2.q.h 4
140.w even 12 2 2100.2.bc.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.c 4 4.b odd 2 1
420.2.q.c 4 28.g odd 6 1
1260.2.s.f 4 12.b even 2 1
1260.2.s.f 4 84.n even 6 1
1680.2.bg.q 4 1.a even 1 1 trivial
1680.2.bg.q 4 7.c even 3 1 inner
2100.2.q.h 4 20.d odd 2 1
2100.2.q.h 4 140.p odd 6 1
2100.2.bc.e 8 20.e even 4 2
2100.2.bc.e 8 140.w even 12 2
2940.2.a.m 2 28.f even 6 1
2940.2.a.s 2 28.g odd 6 1
2940.2.q.t 4 28.d even 2 1
2940.2.q.t 4 28.f even 6 1
8820.2.a.be 2 84.n even 6 1
8820.2.a.bj 2 84.j odd 6 1

Hecke kernels

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

\( T_{11}^{4} - 2T_{11}^{3} + 10T_{11}^{2} + 12T_{11} + 36 \) Copy content Toggle raw display
\( T_{13}^{2} - 7 \) Copy content Toggle raw display
\( T_{17}^{4} + 2T_{17}^{3} + 10T_{17}^{2} - 12T_{17} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36 \) Copy content Toggle raw display
$13$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36 \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + 55 T^{2} + 114 T + 361 \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} - 10 T + 18)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 2 T^{3} + 31 T^{2} + 54 T + 729 \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} + 19 T^{2} + 12 T + 9 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 54)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 59)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 4 T^{3} + 40 T^{2} + 96 T + 576 \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} + 90 T^{2} + \cdots + 2916 \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 4 T^{3} + 19 T^{2} - 12 T + 9 \) Copy content Toggle raw display
$71$ \( (T^{2} + 22 T + 114)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 175 T^{2} + 30625 \) Copy content Toggle raw display
$79$ \( T^{4} + 6 T^{3} + 55 T^{2} - 114 T + 361 \) Copy content Toggle raw display
$83$ \( (T^{2} - 6 T - 54)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 2 T^{3} + 178 T^{2} + \cdots + 30276 \) Copy content Toggle raw display
$97$ \( (T - 8)^{4} \) Copy content Toggle raw display
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