Properties

Label 4680.2.l.f
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: 8.0.1698758656.6
Defining polynomial: \(x^{8} + 18 x^{6} + 97 x^{4} + 176 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + ( \beta_{2} + \beta_{3} ) q^{5} -\beta_{4} q^{7} +O(q^{10})\) \( q + ( \beta_{2} + \beta_{3} ) q^{5} -\beta_{4} q^{7} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{11} + \beta_{2} q^{13} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{17} + ( 4 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{19} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{23} + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{25} + ( 2 - \beta_{6} ) q^{29} + ( -4 + 2 \beta_{6} ) q^{31} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} + \beta_{6} ) q^{35} + ( -2 \beta_{2} + 2 \beta_{5} + 2 \beta_{7} ) q^{37} + ( -\beta_{5} + \beta_{7} ) q^{41} + ( \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{43} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{47} - q^{49} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{53} + ( -5 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{55} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{59} + ( 4 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{61} -\beta_{7} q^{65} + ( 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{67} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{7} ) q^{71} + ( 6 \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{73} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{77} + ( 6 + \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{5} - 3 \beta_{7} ) q^{79} + ( 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{83} + ( -6 + 3 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{85} + ( -4 - 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{89} + \beta_{6} q^{91} + ( 4 - \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{95} + ( -2 \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 16q^{11} + 24q^{19} - 4q^{25} + 16q^{29} - 32q^{31} - 8q^{35} + 8q^{41} - 8q^{49} - 36q^{55} - 16q^{59} + 24q^{61} - 4q^{65} + 24q^{71} + 24q^{79} - 40q^{85} - 8q^{89} + 32q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 18 x^{6} + 97 x^{4} + 176 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} - 10 \nu^{5} + 20 \nu^{4} + 15 \nu^{3} + 2 \nu^{2} + 184 \nu - 80 \)\()/64\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{7} - 46 \nu^{5} - 179 \nu^{3} - 168 \nu \)\()/64\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - \nu^{6} + 18 \nu^{5} - 10 \nu^{4} + 97 \nu^{3} - \nu^{2} + 176 \nu + 40 \)\()/32\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 18 \nu^{5} - 89 \nu^{3} - 104 \nu \)\()/16\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + 6 \nu^{6} - 10 \nu^{5} + 92 \nu^{4} + 15 \nu^{3} + 358 \nu^{2} + 120 \nu + 272 \)\()/64\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} - 14 \nu^{4} - 37 \nu^{2} + 8 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} - 6 \nu^{6} - 10 \nu^{5} - 92 \nu^{4} + 15 \nu^{3} - 358 \nu^{2} + 120 \nu - 272 \)\()/64\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + \beta_{1} - 8\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(9 \beta_{7} + 9 \beta_{5} + 4 \beta_{4} - 5 \beta_{3} - 13 \beta_{2} - 5 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(9 \beta_{7} - 22 \beta_{6} - 9 \beta_{5} + 17 \beta_{3} + 17 \beta_{2} - 17 \beta_{1} + 56\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-81 \beta_{7} - 81 \beta_{5} - 56 \beta_{4} + 37 \beta_{3} + 141 \beta_{2} + 37 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-89 \beta_{7} + 218 \beta_{6} + 89 \beta_{5} - 201 \beta_{3} - 201 \beta_{2} + 201 \beta_{1} - 472\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(761 \beta_{7} + 761 \beta_{5} + 620 \beta_{4} - 325 \beta_{3} - 1485 \beta_{2} - 325 \beta_{1}\)\()/2\)

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
2.16053i
2.16053i
1.69230i
1.69230i
0.692297i
0.692297i
3.16053i
3.16053i
0 0 0 −2.23483 0.0743018i 0 2.82843i 0 0 0
2809.2 0 0 0 −2.23483 + 0.0743018i 0 2.82843i 0 0 0
2809.3 0 0 0 −0.489528 2.18183i 0 2.82843i 0 0 0
2809.4 0 0 0 −0.489528 + 2.18183i 0 2.82843i 0 0 0
2809.5 0 0 0 1.19663 1.88893i 0 2.82843i 0 0 0
2809.6 0 0 0 1.19663 + 1.88893i 0 2.82843i 0 0 0
2809.7 0 0 0 1.52773 1.63280i 0 2.82843i 0 0 0
2809.8 0 0 0 1.52773 + 1.63280i 0 2.82843i 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.f 8
3.b odd 2 1 1560.2.l.e 8
5.b even 2 1 inner 4680.2.l.f 8
12.b even 2 1 3120.2.l.o 8
15.d odd 2 1 1560.2.l.e 8
15.e even 4 1 7800.2.a.bv 4
15.e even 4 1 7800.2.a.bw 4
60.h even 2 1 3120.2.l.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.e 8 3.b odd 2 1
1560.2.l.e 8 15.d odd 2 1
3120.2.l.o 8 12.b even 2 1
3120.2.l.o 8 60.h even 2 1
4680.2.l.f 8 1.a even 1 1 trivial
4680.2.l.f 8 5.b even 2 1 inner
7800.2.a.bv 4 15.e even 4 1
7800.2.a.bw 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 \(S_{2}^{\mathrm{new}}(4680, [\chi])\):

\( T_{7}^{2} + 8 \)
\( T_{11}^{4} - 8 T_{11}^{3} + 6 T_{11}^{2} + 24 T_{11} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 625 + 50 T^{2} + 80 T^{3} + 2 T^{4} + 16 T^{5} + 2 T^{6} + T^{8} \)
$7$ \( ( 8 + T^{2} )^{4} \)
$11$ \( ( 8 + 24 T + 6 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$13$ \( ( 1 + T^{2} )^{4} \)
$17$ \( 50176 + 30464 T^{2} + 3280 T^{4} + 104 T^{6} + T^{8} \)
$19$ \( ( -448 + 224 T + 12 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$23$ \( 50176 + 30464 T^{2} + 3280 T^{4} + 104 T^{6} + T^{8} \)
$29$ \( ( -4 - 4 T + T^{2} )^{4} \)
$31$ \( ( -16 + 8 T + T^{2} )^{4} \)
$37$ \( 256 + 145920 T^{2} + 8416 T^{4} + 160 T^{6} + T^{8} \)
$41$ \( ( -8 + 56 T - 14 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$43$ \( 262144 + 77824 T^{2} + 6672 T^{4} + 152 T^{6} + T^{8} \)
$47$ \( 1048576 + 200704 T^{2} + 11140 T^{4} + 204 T^{6} + T^{8} \)
$53$ \( 246016 + 104960 T^{2} + 7392 T^{4} + 160 T^{6} + T^{8} \)
$59$ \( ( 8 - 24 T + 6 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$61$ \( ( -1472 + 736 T - 44 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$67$ \( 16384 + 38912 T^{2} + 15168 T^{4} + 304 T^{6} + T^{8} \)
$71$ \( ( 512 + 576 T - 46 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$73$ \( 25080064 + 1711616 T^{2} + 39392 T^{4} + 352 T^{6} + T^{8} \)
$79$ \( ( 6272 + 896 T - 132 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$83$ \( 19219456 + 4186496 T^{2} + 92420 T^{4} + 556 T^{6} + T^{8} \)
$89$ \( ( 184 - 1288 T - 190 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$97$ \( 256 + 145920 T^{2} + 8416 T^{4} + 160 T^{6} + T^{8} \)
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