Properties

Label 3024.2.r.f
Level 3024
Weight 2
Character orbit 3024.r
Analytic conductor 24.147
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3024.r (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{3} ) q^{5} + ( 1 - \beta_{1} ) q^{7} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{3} ) q^{5} + ( 1 - \beta_{1} ) q^{7} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{11} -2 \beta_{1} q^{13} + ( 1 - \beta_{2} ) q^{17} -5 q^{19} + ( 5 \beta_{1} - \beta_{3} ) q^{23} + ( -7 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{25} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{29} + 2 \beta_{1} q^{31} + ( 2 + \beta_{2} ) q^{35} + 2 q^{37} + ( 8 \beta_{1} - \beta_{3} ) q^{41} + ( -1 - 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{43} -\beta_{1} q^{49} + ( 4 + 2 \beta_{2} ) q^{53} + 6 q^{55} + 3 \beta_{3} q^{59} + ( 4 - 7 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{61} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{65} + ( 8 \beta_{1} - 3 \beta_{3} ) q^{67} -3 \beta_{2} q^{71} + ( 5 + 3 \beta_{2} ) q^{73} + ( 2 \beta_{1} - \beta_{3} ) q^{77} + ( 2 - 5 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{79} + ( -8 + 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{83} -6 \beta_{1} q^{85} + ( -10 - 2 \beta_{2} ) q^{89} -2 q^{91} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{95} + ( 1 + 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 3q^{5} + 2q^{7} + O(q^{10}) \) \( 4q + 3q^{5} + 2q^{7} - 3q^{11} - 4q^{13} + 6q^{17} - 20q^{19} + 9q^{23} - 11q^{25} - 6q^{29} + 4q^{31} + 6q^{35} + 8q^{37} + 15q^{41} + q^{43} - 2q^{49} + 12q^{53} + 24q^{55} + 3q^{59} + 11q^{61} + 6q^{65} + 13q^{67} + 6q^{71} + 14q^{73} + 3q^{77} + 7q^{79} - 12q^{83} - 12q^{85} - 36q^{89} - 8q^{91} - 15q^{95} - q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} + \nu^{2} + 2 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(1\) \(1\)

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
−1.18614 + 1.26217i
1.68614 0.396143i
−1.18614 1.26217i
1.68614 + 0.396143i
0 0 0 −0.686141 + 1.18843i 0 0.500000 + 0.866025i 0 0 0
1009.2 0 0 0 2.18614 3.78651i 0 0.500000 + 0.866025i 0 0 0
2017.1 0 0 0 −0.686141 1.18843i 0 0.500000 0.866025i 0 0 0
2017.2 0 0 0 2.18614 + 3.78651i 0 0.500000 0.866025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.r.f 4
3.b odd 2 1 1008.2.r.f 4
4.b odd 2 1 378.2.f.c 4
9.c even 3 1 inner 3024.2.r.f 4
9.c even 3 1 9072.2.a.bb 2
9.d odd 6 1 1008.2.r.f 4
9.d odd 6 1 9072.2.a.bm 2
12.b even 2 1 126.2.f.d 4
28.d even 2 1 2646.2.f.j 4
28.f even 6 1 2646.2.e.m 4
28.f even 6 1 2646.2.h.l 4
28.g odd 6 1 2646.2.e.n 4
28.g odd 6 1 2646.2.h.k 4
36.f odd 6 1 378.2.f.c 4
36.f odd 6 1 1134.2.a.n 2
36.h even 6 1 126.2.f.d 4
36.h even 6 1 1134.2.a.k 2
84.h odd 2 1 882.2.f.k 4
84.j odd 6 1 882.2.e.k 4
84.j odd 6 1 882.2.h.n 4
84.n even 6 1 882.2.e.l 4
84.n even 6 1 882.2.h.m 4
252.n even 6 1 2646.2.e.m 4
252.o even 6 1 882.2.e.l 4
252.r odd 6 1 882.2.h.n 4
252.s odd 6 1 882.2.f.k 4
252.s odd 6 1 7938.2.a.bh 2
252.u odd 6 1 2646.2.h.k 4
252.bb even 6 1 882.2.h.m 4
252.bi even 6 1 2646.2.f.j 4
252.bi even 6 1 7938.2.a.bs 2
252.bj even 6 1 2646.2.h.l 4
252.bl odd 6 1 2646.2.e.n 4
252.bn odd 6 1 882.2.e.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.d 4 12.b even 2 1
126.2.f.d 4 36.h even 6 1
378.2.f.c 4 4.b odd 2 1
378.2.f.c 4 36.f odd 6 1
882.2.e.k 4 84.j odd 6 1
882.2.e.k 4 252.bn odd 6 1
882.2.e.l 4 84.n even 6 1
882.2.e.l 4 252.o even 6 1
882.2.f.k 4 84.h odd 2 1
882.2.f.k 4 252.s odd 6 1
882.2.h.m 4 84.n even 6 1
882.2.h.m 4 252.bb even 6 1
882.2.h.n 4 84.j odd 6 1
882.2.h.n 4 252.r odd 6 1
1008.2.r.f 4 3.b odd 2 1
1008.2.r.f 4 9.d odd 6 1
1134.2.a.k 2 36.h even 6 1
1134.2.a.n 2 36.f odd 6 1
2646.2.e.m 4 28.f even 6 1
2646.2.e.m 4 252.n even 6 1
2646.2.e.n 4 28.g odd 6 1
2646.2.e.n 4 252.bl odd 6 1
2646.2.f.j 4 28.d even 2 1
2646.2.f.j 4 252.bi even 6 1
2646.2.h.k 4 28.g odd 6 1
2646.2.h.k 4 252.u odd 6 1
2646.2.h.l 4 28.f even 6 1
2646.2.h.l 4 252.bj even 6 1
3024.2.r.f 4 1.a even 1 1 trivial
3024.2.r.f 4 9.c even 3 1 inner
7938.2.a.bh 2 252.s odd 6 1
7938.2.a.bs 2 252.bi even 6 1
9072.2.a.bb 2 9.c even 3 1
9072.2.a.bm 2 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3024, [\chi])\):

\( T_{5}^{4} - 3 T_{5}^{3} + 15 T_{5}^{2} + 18 T_{5} + 36 \)
\( T_{11}^{4} + 3 T_{11}^{3} + 15 T_{11}^{2} - 18 T_{11} + 36 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 3 T + 5 T^{2} )^{2}( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} ) \)
$7$ \( ( 1 - T + T^{2} )^{2} \)
$11$ \( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 198 T^{5} - 847 T^{6} + 3993 T^{7} + 14641 T^{8} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 - 3 T + 28 T^{2} - 51 T^{3} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 5 T + 19 T^{2} )^{4} \)
$23$ \( ( 1 - 9 T + 23 T^{2} )^{2}( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} ) \)
$29$ \( 1 + 6 T + 2 T^{2} - 144 T^{3} - 729 T^{4} - 4176 T^{5} + 1682 T^{6} + 146334 T^{7} + 707281 T^{8} \)
$31$ \( ( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{4} \)
$41$ \( 1 - 15 T + 95 T^{2} - 720 T^{3} + 5994 T^{4} - 29520 T^{5} + 159695 T^{6} - 1033815 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - T - 11 T^{2} + 74 T^{3} - 1748 T^{4} + 3182 T^{5} - 20339 T^{6} - 79507 T^{7} + 3418801 T^{8} \)
$47$ \( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 6 T + 82 T^{2} - 318 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( 1 - 3 T - 37 T^{2} + 216 T^{3} - 1896 T^{4} + 12744 T^{5} - 128797 T^{6} - 616137 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 11 T + 43 T^{2} + 484 T^{3} - 5018 T^{4} + 29524 T^{5} + 160003 T^{6} - 2496791 T^{7} + 13845841 T^{8} \)
$67$ \( ( 1 - 13 T + 67 T^{2} )^{2}( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} ) \)
$71$ \( ( 1 - 3 T + 70 T^{2} - 213 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 7 T + 84 T^{2} - 511 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( 1 - 7 T - 47 T^{2} + 434 T^{3} - 896 T^{4} + 34286 T^{5} - 293327 T^{6} - 3451273 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 12 T + 74 T^{2} - 1152 T^{3} - 13941 T^{4} - 95616 T^{5} + 509786 T^{6} + 6861444 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 + 18 T + 226 T^{2} + 1602 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 + T - 119 T^{2} - 74 T^{3} + 4894 T^{4} - 7178 T^{5} - 1119671 T^{6} + 912673 T^{7} + 88529281 T^{8} \)
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