Properties

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

Newform invariants

Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{5} q^{5} - \beta_1 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{5} q^{5} - \beta_1 q^{7} - q^{9} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2}) q^{11} + ( - \beta_{4} + \beta_{3}) q^{13} + \beta_{3} q^{15} + (\beta_{5} + \beta_{2}) q^{17} + ( - \beta_{5} + \beta_{2}) q^{19} + q^{21} + (\beta_{4} - \beta_{3} + 2 \beta_1) q^{23} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2} + 2 \beta_1 - 1) q^{25} - \beta_1 q^{27} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2) q^{29} + ( - \beta_{5} + \beta_{2}) q^{31} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}) q^{33} - \beta_{3} q^{35} + ( - 2 \beta_{4} + 2 \beta_{3}) q^{37} + ( - \beta_{5} + \beta_{2}) q^{39} + ( - \beta_{5} + \beta_{2} + 4) q^{41} + (2 \beta_{5} + 2 \beta_{2} + 4 \beta_1) q^{43} - \beta_{5} q^{45} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{47} - q^{49} + (\beta_{4} + \beta_{3}) q^{51} + (2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{53} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 2) q^{55} + (\beta_{4} - \beta_{3}) q^{57} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 4) q^{59} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2) q^{61} + \beta_1 q^{63} + (\beta_{4} - \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{65} + (\beta_{5} - \beta_{2} - 2) q^{69} + ( - 3 \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2}) q^{71} + (2 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{73} + (\beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{75} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{77} + ( - 2 \beta_{5} + 2 \beta_{2} - 8) q^{79} + q^{81} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{83} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2} + 2 \beta_1 - 6) q^{85} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{87} + (\beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{89} + (\beta_{5} - \beta_{2}) q^{91} + (\beta_{4} - \beta_{3}) q^{93} + (\beta_{5} + 2 \beta_{4} - \beta_{2} - 2 \beta_1 - 4) q^{95} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} - 4 \beta_1) q^{97} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{5} - 6 q^{9} - 4 q^{11} + 4 q^{19} + 6 q^{21} - 2 q^{25} - 4 q^{29} + 4 q^{31} + 4 q^{39} + 28 q^{41} + 2 q^{45} - 6 q^{49} + 20 q^{55} + 16 q^{59} + 4 q^{61} - 8 q^{65} - 16 q^{69} + 12 q^{71} - 8 q^{75} - 40 q^{79} + 6 q^{81} - 32 q^{85} - 4 q^{89} - 4 q^{91} - 28 q^{95} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -7\nu^{5} + 10\nu^{4} - 5\nu^{3} - 30\nu^{2} - 32\nu + 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{5} + 3\nu^{4} + 10\nu^{3} - 32\nu^{2} - 74\nu - 3 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -10\nu^{5} + 11\nu^{4} - 17\nu^{3} - 10\nu^{2} - 72\nu - 11 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\nu^{5} - 27\nu^{4} + 25\nu^{3} + 12\nu^{2} + 68\nu - 65 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -19\nu^{5} + 37\nu^{4} - 30\nu^{3} - 42\nu^{2} - 54\nu + 55 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{2} - 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - \beta_{4} - 3\beta_{3} + 3\beta_{2} - 4\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} - 5\beta_{4} - 5\beta_{3} + \beta_{2} - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -11\beta_{5} - 11\beta_{4} - 5\beta_{3} - 5\beta_{2} + 18\beta _1 - 18 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1
−0.854638 0.854638i
1.45161 + 1.45161i
0.403032 + 0.403032i
−0.854638 + 0.854638i
1.45161 1.45161i
0.403032 0.403032i
0 1.00000i 0 −2.17009 0.539189i 0 1.00000i 0 −1.00000 0
1009.2 0 1.00000i 0 −0.311108 + 2.21432i 0 1.00000i 0 −1.00000 0
1009.3 0 1.00000i 0 1.48119 1.67513i 0 1.00000i 0 −1.00000 0
1009.4 0 1.00000i 0 −2.17009 + 0.539189i 0 1.00000i 0 −1.00000 0
1009.5 0 1.00000i 0 −0.311108 2.21432i 0 1.00000i 0 −1.00000 0
1009.6 0 1.00000i 0 1.48119 + 1.67513i 0 1.00000i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1009.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b 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.t.i 6
3.b odd 2 1 5040.2.t.ba 6
4.b odd 2 1 840.2.t.e 6
5.b even 2 1 inner 1680.2.t.i 6
5.c odd 4 1 8400.2.a.dh 3
5.c odd 4 1 8400.2.a.dk 3
12.b even 2 1 2520.2.t.j 6
15.d odd 2 1 5040.2.t.ba 6
20.d odd 2 1 840.2.t.e 6
20.e even 4 1 4200.2.a.bo 3
20.e even 4 1 4200.2.a.bq 3
60.h even 2 1 2520.2.t.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.e 6 4.b odd 2 1
840.2.t.e 6 20.d odd 2 1
1680.2.t.i 6 1.a even 1 1 trivial
1680.2.t.i 6 5.b even 2 1 inner
2520.2.t.j 6 12.b even 2 1
2520.2.t.j 6 60.h even 2 1
4200.2.a.bo 3 20.e even 4 1
4200.2.a.bq 3 20.e even 4 1
5040.2.t.ba 6 3.b odd 2 1
5040.2.t.ba 6 15.d odd 2 1
8400.2.a.dh 3 5.c odd 4 1
8400.2.a.dk 3 5.c odd 4 1

Hecke kernels

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

\( T_{11}^{3} + 2T_{11}^{2} - 20T_{11} - 8 \) Copy content Toggle raw display
\( T_{13}^{6} + 28T_{13}^{4} + 176T_{13}^{2} + 64 \) Copy content Toggle raw display
\( T_{19}^{3} - 2T_{19}^{2} - 12T_{19} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} + 3 T^{4} + 12 T^{3} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( (T^{3} + 2 T^{2} - 20 T - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 28 T^{4} + 176 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{6} + 32 T^{4} + 256 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{3} - 2 T^{2} - 12 T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 48 T^{4} + 320 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{3} + 2 T^{2} - 84 T - 104)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 2 T^{2} - 12 T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 112 T^{4} + 2816 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( (T^{3} - 14 T^{2} + 52 T - 40)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 176 T^{4} + 7936 T^{2} + \cdots + 102400 \) Copy content Toggle raw display
$47$ \( T^{6} + 192 T^{4} + 8192 T^{2} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( T^{6} + 252 T^{4} + 14000 T^{2} + \cdots + 222784 \) Copy content Toggle raw display
$59$ \( (T^{3} - 8 T^{2} - 64 T + 256)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 2 T^{2} - 148 T - 536)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} \) Copy content Toggle raw display
$71$ \( (T^{3} - 6 T^{2} - 100 T - 200)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 284 T^{4} + 24496 T^{2} + \cdots + 577600 \) Copy content Toggle raw display
$79$ \( (T^{3} + 20 T^{2} + 80 T - 64)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 192 T^{4} + 8192 T^{2} + \cdots + 65536 \) Copy content Toggle raw display
$89$ \( (T^{3} + 2 T^{2} - 60 T - 200)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 188 T^{4} + 9648 T^{2} + \cdots + 118336 \) Copy content Toggle raw display
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