Properties

Label 4680.2.l.g
Level $4680$
Weight $2$
Character orbit 4680.l
Analytic conductor $37.370$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4680.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(37.3699881460\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.57815240704.2
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 89x^{4} - 170x^{3} + 162x^{2} - 72x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{5} + (2 \beta_{2} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{5} + (2 \beta_{2} - \beta_1) q^{7} + ( - \beta_{5} - \beta_{4} - \beta_{3}) q^{11} + \beta_{2} q^{13} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_1) q^{17} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1) q^{23} + ( - \beta_{6} - \beta_{5} - 2 \beta_{3} - \beta_{2}) q^{25} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 2) q^{29} + ( - 2 \beta_{5} - 2 \beta_{4} - \beta_1 + 2) q^{35} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_1) q^{37} + (\beta_{7} + \beta_{6} + \beta_{3} - 2) q^{41} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{43} + ( - \beta_{7} + \beta_{6} - 2 \beta_{2}) q^{47} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_{3} - 3) q^{49} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_1) q^{53} + ( - \beta_{7} + 2 \beta_{6} - \beta_{4} - \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 1) q^{55} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 2) q^{59} + (2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2) q^{61} - \beta_{5} q^{65} + ( - 2 \beta_{7} + 2 \beta_{6} + 4 \beta_{2}) q^{67} + ( - \beta_{7} - \beta_{6} - 3 \beta_{3} - 4) q^{71} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{73} + ( - 3 \beta_{7} + 3 \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{2} + \beta_1) q^{77} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{3}) q^{79} + (\beta_{7} - \beta_{6} + 6 \beta_{2} - 4 \beta_1) q^{83} + (\beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - \beta_1 + 2) q^{85} + (\beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 2) q^{89} + ( - \beta_{3} - 2) q^{91} + ( - 2 \beta_{7} + 2 \beta_{6} + 5 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{5} - 2 q^{11} + 16 q^{29} + 8 q^{35} - 14 q^{41} - 18 q^{49} + 10 q^{55} + 4 q^{59} + 22 q^{61} - 2 q^{65} - 30 q^{71} + 2 q^{79} + 24 q^{85} + 18 q^{89} - 14 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 89x^{4} - 170x^{3} + 162x^{2} - 72x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 72\nu^{7} + 747\nu^{6} + 76\nu^{5} + 32\nu^{4} + 6804\nu^{3} + 63783\nu^{2} + 9448\nu - 2736 ) / 18170 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -1881\nu^{7} + 2970\nu^{6} - 2894\nu^{5} - 836\nu^{4} - 167761\nu^{3} + 244926\nu^{2} - 234110\nu + 67844 ) / 36340 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -1611\nu^{7} + 1683\nu^{6} - 792\nu^{5} - 716\nu^{4} - 144063\nu^{3} + 136611\nu^{2} - 60588\nu + 28512 ) / 18170 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 358\nu^{7} - 374\nu^{6} + 176\nu^{5} + 361\nu^{4} + 32014\nu^{3} - 30358\nu^{2} + 11647\nu + 2749 ) / 1817 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1988\nu^{7} - 2087\nu^{6} + 1089\nu^{5} + 1893\nu^{4} + 178781\nu^{3} - 169443\nu^{2} + 84217\nu + 6221 ) / 9085 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1665\nu^{7} - 3394\nu^{6} + 2666\nu^{5} + 740\nu^{4} + 147349\nu^{3} - 289098\nu^{2} + 213034\nu - 59636 ) / 7268 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 9117 \nu^{7} + 17838 \nu^{6} - 14166 \nu^{5} - 4052 \nu^{4} - 811589 \nu^{3} + 1516102 \nu^{2} - 1096418 \nu + 328276 ) / 36340 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 10\beta_{2} - 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{7} - 9\beta_{6} + 9\beta_{5} - 9\beta_{4} + 2\beta_{3} + 4\beta_{2} - 2\beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{7} + 9\beta_{6} + 9\beta_{5} + 9\beta_{4} + 40\beta_{3} - 90 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -85\beta_{7} - 85\beta_{6} - 85\beta_{5} + 85\beta_{4} + 22\beta_{3} - 36\beta_{2} + 22\beta _1 + 36 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -85\beta_{7} + 85\beta_{6} + 77\beta_{5} - 77\beta_{4} + 850\beta_{2} + 384\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 801\beta_{7} + 809\beta_{6} - 809\beta_{5} + 801\beta_{4} - 230\beta_{3} - 300\beta_{2} + 230\beta _1 - 300 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(2081\) \(2341\) \(3511\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2809.1
−2.15569 2.15569i
−2.15569 + 2.15569i
0.353624 + 0.353624i
0.353624 0.353624i
0.594137 0.594137i
0.594137 + 0.594137i
2.20793 + 2.20793i
2.20793 2.20793i
0 0 0 −2.15569 0.594137i 0 0.633776i 0 0 0
2809.2 0 0 0 −2.15569 + 0.594137i 0 0.633776i 0 0 0
2809.3 0 0 0 0.353624 2.20793i 0 3.09417i 0 0 0
2809.4 0 0 0 0.353624 + 2.20793i 0 3.09417i 0 0 0
2809.5 0 0 0 0.594137 2.15569i 0 4.92778i 0 0 0
2809.6 0 0 0 0.594137 + 2.15569i 0 4.92778i 0 0 0
2809.7 0 0 0 2.20793 0.353624i 0 1.65573i 0 0 0
2809.8 0 0 0 2.20793 + 0.353624i 0 1.65573i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2809.8
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 4680.2.l.g 8
3.b odd 2 1 1560.2.l.d 8
5.b even 2 1 inner 4680.2.l.g 8
12.b even 2 1 3120.2.l.n 8
15.d odd 2 1 1560.2.l.d 8
15.e even 4 1 7800.2.a.bt 4
15.e even 4 1 7800.2.a.by 4
60.h even 2 1 3120.2.l.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.d 8 3.b odd 2 1
1560.2.l.d 8 15.d odd 2 1
3120.2.l.n 8 12.b even 2 1
3120.2.l.n 8 60.h even 2 1
4680.2.l.g 8 1.a even 1 1 trivial
4680.2.l.g 8 5.b even 2 1 inner
7800.2.a.bt 4 15.e even 4 1
7800.2.a.by 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4680, [\chi])\):

\( T_{7}^{8} + 37T_{7}^{6} + 340T_{7}^{4} + 768T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{4} + T_{11}^{3} - 20T_{11}^{2} + 26T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + 2 T^{6} + 6 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 37 T^{6} + 340 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{4} + T^{3} - 20 T^{2} + 26 T - 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + 61 T^{6} + 908 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} + 65 T^{6} + 1056 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} - 20 T^{2} + 144 T + 256)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 133 T^{6} + 5068 T^{4} + \cdots + 141376 \) Copy content Toggle raw display
$41$ \( (T^{4} + 7 T^{3} - 8 T^{2} - 22 T - 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 104 T^{6} + 3216 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$47$ \( T^{8} + 64 T^{6} + 1108 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$53$ \( T^{8} + 133 T^{6} + 5068 T^{4} + \cdots + 141376 \) Copy content Toggle raw display
$59$ \( (T^{4} - 2 T^{3} - 86 T^{2} + 36 T + 1072)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 11 T^{3} - 22 T^{2} + 128 T + 32)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 192 T^{6} + 5440 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$71$ \( (T^{4} + 15 T^{3} - 30 T^{2} - 530 T + 1256)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 124 T^{6} + 3568 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$79$ \( (T^{4} - T^{3} - 176 T^{2} + 52 T + 7328)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 464 T^{6} + 65620 T^{4} + \cdots + 7311616 \) Copy content Toggle raw display
$89$ \( (T^{4} - 9 T^{3} - 136 T^{2} + 378 T + 268)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 545 T^{6} + \cdots + 41783296 \) Copy content Toggle raw display
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