Properties

Label 3120.2.l.m
Level $3120$
Weight $2$
Character orbit 3120.l
Analytic conductor $24.913$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.9133254306\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{3} -\beta_{5} q^{5} - q^{9} +O(q^{10})\) \( q -\beta_{3} q^{3} -\beta_{5} q^{5} - q^{9} + ( 2 - \beta_{4} - \beta_{5} ) q^{11} + \beta_{3} q^{13} -\beta_{2} q^{15} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{17} + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{19} + ( -\beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{4} - \beta_{5} ) q^{23} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{25} + \beta_{3} q^{27} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -4 + 2 \beta_{4} + 2 \beta_{5} ) q^{31} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{33} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{37} + q^{39} + ( -4 - \beta_{1} + \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{41} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{43} + \beta_{5} q^{45} + ( 2 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - \beta_{4} + \beta_{5} ) q^{47} + 7 q^{49} + ( 2 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{51} + ( -2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{53} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{55} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{57} + ( -2 - 3 \beta_{4} - 3 \beta_{5} ) q^{59} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{61} + \beta_{2} q^{65} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{67} + ( 4 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{69} + ( -2 + \beta_{1} - \beta_{2} ) q^{71} + ( -4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{73} + ( 1 + \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{75} + ( 6 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{79} + q^{81} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{83} + ( -6 - \beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{85} + ( -2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{87} + ( 8 + \beta_{1} - \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{89} + ( 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{93} + ( -2 + 3 \beta_{1} + \beta_{2} + 6 \beta_{3} + \beta_{4} + \beta_{5} ) q^{95} + 6 \beta_{3} q^{97} + ( -2 + \beta_{4} + \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{9} + O(q^{10}) \) \( 6q - 6q^{9} + 12q^{11} - 2q^{15} - 4q^{19} - 10q^{25} + 20q^{29} - 24q^{31} + 6q^{39} - 20q^{41} + 42q^{49} + 8q^{51} + 20q^{55} - 12q^{59} + 4q^{61} + 2q^{65} + 20q^{69} - 16q^{71} + 4q^{75} + 40q^{79} + 6q^{81} - 32q^{85} + 44q^{89} - 16q^{95} - 12q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{5} + 25 \nu^{4} + 10 \nu^{3} - 4 \nu^{2} - 121 \nu + 323 \)\()/121\)
\(\beta_{2}\)\(=\)\((\)\( -7 \nu^{5} - 27 \nu^{4} - 35 \nu^{3} + 14 \nu^{2} - 121 \nu - 223 \)\()/121\)
\(\beta_{3}\)\(=\)\((\)\( -25 \nu^{5} - 10 \nu^{4} - 4 \nu^{3} + 50 \nu^{2} - 605 \nu + 258 \)\()/242\)
\(\beta_{4}\)\(=\)\((\)\( -65 \nu^{5} - 26 \nu^{4} + 38 \nu^{3} + 372 \nu^{2} - 1573 \nu + 574 \)\()/242\)
\(\beta_{5}\)\(=\)\((\)\( 75 \nu^{5} + 30 \nu^{4} + 12 \nu^{3} - 392 \nu^{2} + 1573 \nu - 774 \)\()/242\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{5} - \beta_{4} - \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{5} + \beta_{4} - 6 \beta_{3} + \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{5} + 5 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} - 5 \beta_{1} + 4\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-9 \beta_{5} - 9 \beta_{4} - 5 \beta_{2} + 5 \beta_{1} - 30\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(25 \beta_{5} + 29 \beta_{4} - 32 \beta_{3} + 29 \beta_{2} + 25 \beta_{1} + 32\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times\).

\(n\) \(1951\) \(2081\) \(2341\) \(2497\) \(2641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
−1.75233 + 1.75233i
0.432320 0.432320i
1.32001 1.32001i
−1.75233 1.75233i
0.432320 + 0.432320i
1.32001 + 1.32001i
0 1.00000i 0 −1.75233 1.38900i 0 0 0 −1.00000 0
1249.2 0 1.00000i 0 0.432320 + 2.19388i 0 0 0 −1.00000 0
1249.3 0 1.00000i 0 1.32001 1.80487i 0 0 0 −1.00000 0
1249.4 0 1.00000i 0 −1.75233 + 1.38900i 0 0 0 −1.00000 0
1249.5 0 1.00000i 0 0.432320 2.19388i 0 0 0 −1.00000 0
1249.6 0 1.00000i 0 1.32001 + 1.80487i 0 0 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.6
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 3120.2.l.m 6
4.b odd 2 1 1560.2.l.c 6
5.b even 2 1 inner 3120.2.l.m 6
12.b even 2 1 4680.2.l.e 6
20.d odd 2 1 1560.2.l.c 6
20.e even 4 1 7800.2.a.bj 3
20.e even 4 1 7800.2.a.bp 3
60.h even 2 1 4680.2.l.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.c 6 4.b odd 2 1
1560.2.l.c 6 20.d odd 2 1
3120.2.l.m 6 1.a even 1 1 trivial
3120.2.l.m 6 5.b even 2 1 inner
4680.2.l.e 6 12.b even 2 1
4680.2.l.e 6 60.h even 2 1
7800.2.a.bj 3 20.e even 4 1
7800.2.a.bp 3 20.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 \(S_{2}^{\mathrm{new}}(3120, [\chi])\):

\( T_{7} \)
\( T_{11}^{3} - 6 T_{11}^{2} + 2 T_{11} + 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( ( 1 + T^{2} )^{3} \)
$5$ \( 125 + 25 T^{2} + 8 T^{3} + 5 T^{4} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( ( 20 + 2 T - 6 T^{2} + T^{3} )^{2} \)
$13$ \( ( 1 + T^{2} )^{3} \)
$17$ \( 4096 + 912 T^{2} + 56 T^{4} + T^{6} \)
$19$ \( ( 16 - 24 T + 2 T^{2} + T^{3} )^{2} \)
$23$ \( 6400 + 1664 T^{2} + 84 T^{4} + T^{6} \)
$29$ \( ( 472 - 44 T - 10 T^{2} + T^{3} )^{2} \)
$31$ \( ( -160 + 8 T + 12 T^{2} + T^{3} )^{2} \)
$37$ \( 18496 + 2416 T^{2} + 92 T^{4} + T^{6} \)
$41$ \( ( -332 - 34 T + 10 T^{2} + T^{3} )^{2} \)
$43$ \( 4096 + 912 T^{2} + 56 T^{4} + T^{6} \)
$47$ \( 65536 + 9348 T^{2} + 188 T^{4} + T^{6} \)
$53$ \( 222784 + 11376 T^{2} + 188 T^{4} + T^{6} \)
$59$ \( ( 44 - 78 T + 6 T^{2} + T^{3} )^{2} \)
$61$ \( ( 32 - 32 T - 2 T^{2} + T^{3} )^{2} \)
$67$ \( 65536 + 18432 T^{2} + 272 T^{4} + T^{6} \)
$71$ \( ( -64 + 2 T + 8 T^{2} + T^{3} )^{2} \)
$73$ \( 678976 + 31792 T^{2} + 332 T^{4} + T^{6} \)
$79$ \( ( -160 + 108 T - 20 T^{2} + T^{3} )^{2} \)
$83$ \( 53824 + 5700 T^{2} + 140 T^{4} + T^{6} \)
$89$ \( ( 244 + 94 T - 22 T^{2} + T^{3} )^{2} \)
$97$ \( ( 36 + T^{2} )^{3} \)
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