Properties

Label 1680.2.bg.u
Level $1680$
Weight $2$
Character orbit 1680.bg
Analytic conductor $13.415$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.38363328.2
Defining polynomial: \( x^{6} - x^{5} - 3x^{4} - 2x^{3} - 21x^{2} - 49x + 343 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{3} + \beta_{3} q^{5} + ( - \beta_{4} + \beta_1) q^{7} + \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{3} + \beta_{3} q^{5} + ( - \beta_{4} + \beta_1) q^{7} + \beta_{3} q^{9} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{4} + \beta_{2}) q^{13} - q^{15} + ( - \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{17} + (2 \beta_{5} + \beta_{3} - 2 \beta_1) q^{19} + \beta_1 q^{21} + (\beta_{5} + \beta_{4} - 3 \beta_{3} + \beta_{2} - 2 \beta_1) q^{23} + ( - \beta_{3} - 1) q^{25} - q^{27} + ( - 2 \beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{29} + (\beta_{5} + \beta_{4} + \beta_{3} + 1) q^{31} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{33} + \beta_{4} q^{35} + ( - \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - \beta_1) q^{37} + ( - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1) q^{39} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_{2} - \beta_1 + 3) q^{41} + (\beta_{5} - \beta_{4} + \beta_1) q^{43} + ( - \beta_{3} - 1) q^{45} + ( - \beta_{4} + 4 \beta_{3} - \beta_{2} + \beta_1) q^{47} + ( - 2 \beta_{5} - \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{49} + ( - \beta_{5} - 3 \beta_{3} + \beta_1) q^{51} + ( - 3 \beta_{5} - 3 \beta_{4} + 2 \beta_{2} + 2 \beta_1) q^{53} + ( - \beta_{5} + \beta_{4} - \beta_1 + 1) q^{55} + ( - 2 \beta_{4} + 2 \beta_{2} - 1) q^{57} + (\beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{59} + (2 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{61} + \beta_{4} q^{63} + ( - \beta_{5} + \beta_1) q^{65} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 2) q^{67} + ( - \beta_{5} - \beta_{4} + 2 \beta_{2} - \beta_1 + 3) q^{69} + ( - 2 \beta_{5} - \beta_{4} + 3 \beta_{2} - 2 \beta_1 + 1) q^{71} + (2 \beta_{5} + 2 \beta_{4} - \beta_{2} - \beta_1) q^{73} - \beta_{3} q^{75} + ( - 3 \beta_{5} + 6 \beta_{3} + \beta_{2} + 9) q^{77} + ( - 2 \beta_{5} + \beta_{4} - 7 \beta_{3} + \beta_{2} + \beta_1) q^{79} + ( - \beta_{3} - 1) q^{81} + (2 \beta_{5} - 3 \beta_{4} + \beta_{2} + 2 \beta_1 - 3) q^{83} + (\beta_{4} - \beta_{2} + 3) q^{85} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{87} + ( - \beta_{5} + 7 \beta_{3} + \beta_1) q^{89} + (\beta_{5} - \beta_{4} + 5 \beta_{3} + 2 \beta_{2} + 4) q^{91} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{93} + ( - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{95} + ( - 2 \beta_{4} + 2 \beta_{2} + 2) q^{97} + ( - \beta_{5} + \beta_{4} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 3 q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} - 3 q^{5} + 2 q^{7} - 3 q^{9} - q^{11} + 2 q^{13} - 6 q^{15} - 8 q^{17} - q^{19} + q^{21} + 9 q^{23} - 3 q^{25} - 6 q^{27} + 4 q^{31} + q^{33} - q^{35} - 15 q^{37} + q^{39} + 10 q^{41} + 4 q^{43} - 3 q^{45} - 11 q^{47} + 4 q^{49} + 8 q^{51} + q^{53} + 2 q^{55} - 2 q^{57} + 12 q^{59} - 4 q^{61} - q^{63} - q^{65} - 10 q^{67} + 18 q^{69} + 4 q^{71} + 3 q^{75} + 31 q^{77} + 18 q^{79} - 3 q^{81} - 8 q^{83} + 16 q^{85} - 22 q^{89} + 14 q^{91} - 4 q^{93} - q^{95} + 16 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 3x^{4} - 2x^{3} - 21x^{2} - 49x + 343 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 22\nu^{4} - 31\nu^{3} - 233\nu^{2} + 168\nu + 833 ) / 784 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} - 4\nu^{4} - 33\nu^{3} + 27\nu^{2} + 126\nu - 49 ) / 784 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 6\nu^{4} + 3\nu^{3} + 9\nu^{2} + 84\nu + 147 ) / 112 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -15\nu^{5} - 6\nu^{4} + 17\nu^{3} - 201\nu^{2} + 504\nu + 1617 ) / 784 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + 2\beta_{3} - 2\beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} + 3\beta_{4} - 18\beta_{3} - 3\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{5} - 15\beta_{4} + 2\beta_{3} - 9\beta_{2} + 9\beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -39\beta_{5} - 4\beta_{4} - 48\beta_{3} + 27\beta_{2} + 30\beta _1 + 54 \) Copy content Toggle raw display

Character values

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

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\) \(1\) \(1\)

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
−2.33916 1.23625i
2.59174 0.531877i
0.247423 + 2.63416i
−2.33916 + 1.23625i
2.59174 + 0.531877i
0.247423 2.63416i
0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 −2.24021 + 1.40765i 0 −0.500000 + 0.866025i 0
961.2 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0.835250 2.51045i 0 −0.500000 + 0.866025i 0
961.3 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 2.40496 + 1.10280i 0 −0.500000 + 0.866025i 0
1201.1 0 0.500000 0.866025i 0 −0.500000 0.866025i 0 −2.24021 1.40765i 0 −0.500000 0.866025i 0
1201.2 0 0.500000 0.866025i 0 −0.500000 0.866025i 0 0.835250 + 2.51045i 0 −0.500000 0.866025i 0
1201.3 0 0.500000 0.866025i 0 −0.500000 0.866025i 0 2.40496 1.10280i 0 −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1201.3
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.u 6
4.b odd 2 1 840.2.bg.i 6
7.c even 3 1 inner 1680.2.bg.u 6
12.b even 2 1 2520.2.bi.o 6
28.f even 6 1 5880.2.a.bt 3
28.g odd 6 1 840.2.bg.i 6
28.g odd 6 1 5880.2.a.bw 3
84.n even 6 1 2520.2.bi.o 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.bg.i 6 4.b odd 2 1
840.2.bg.i 6 28.g odd 6 1
1680.2.bg.u 6 1.a even 1 1 trivial
1680.2.bg.u 6 7.c even 3 1 inner
2520.2.bi.o 6 12.b even 2 1
2520.2.bi.o 6 84.n even 6 1
5880.2.a.bt 3 28.f even 6 1
5880.2.a.bw 3 28.g odd 6 1

Hecke kernels

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

\( T_{11}^{6} + T_{11}^{5} + 23T_{11}^{4} - 50T_{11}^{3} + 470T_{11}^{2} - 308T_{11} + 196 \) Copy content Toggle raw display
\( T_{13}^{3} - T_{13}^{2} - 15T_{13} + 3 \) Copy content Toggle raw display
\( T_{17}^{6} + 8T_{17}^{5} + 58T_{17}^{4} + 96T_{17}^{3} + 228T_{17}^{2} - 144T_{17} + 576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} - 2 T^{5} + 8 T^{3} - 98 T + 343 \) Copy content Toggle raw display
$11$ \( T^{6} + T^{5} + 23 T^{4} - 50 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$13$ \( (T^{3} - T^{2} - 15 T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 8 T^{5} + 58 T^{4} + 96 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$19$ \( T^{6} + T^{5} + 62 T^{4} + 13 T^{3} + \cdots + 1369 \) Copy content Toggle raw display
$23$ \( T^{6} - 9 T^{5} + 111 T^{4} + \cdots + 93636 \) Copy content Toggle raw display
$29$ \( (T^{3} - 78 T - 256)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 4 T^{5} + 35 T^{4} + \cdots + 1156 \) Copy content Toggle raw display
$37$ \( T^{6} + 15 T^{5} + 228 T^{4} + \cdots + 271441 \) Copy content Toggle raw display
$41$ \( (T^{3} - 5 T^{2} - 102 T + 618)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 2 T^{2} - 21 T + 36)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 11 T^{5} + 105 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$53$ \( T^{6} - T^{5} + 121 T^{4} + \cdots + 254016 \) Copy content Toggle raw display
$59$ \( T^{6} - 12 T^{5} + 174 T^{4} + \cdots + 254016 \) Copy content Toggle raw display
$61$ \( T^{6} + 4 T^{5} + 72 T^{4} + \cdots + 9216 \) Copy content Toggle raw display
$67$ \( T^{6} + 10 T^{5} + 177 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$71$ \( (T^{3} - 2 T^{2} - 146 T + 588)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 57 T^{4} + 324 T^{3} + \cdots + 26244 \) Copy content Toggle raw display
$79$ \( T^{6} - 18 T^{5} + 267 T^{4} + \cdots + 44944 \) Copy content Toggle raw display
$83$ \( (T^{3} + 4 T^{2} - 126 T - 408)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 22 T^{5} + 338 T^{4} + \cdots + 80656 \) Copy content Toggle raw display
$97$ \( (T^{3} - 8 T^{2} - 40 T + 128)^{2} \) Copy content Toggle raw display
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