Properties

Label 2646.2.h.l
Level $2646$
Weight $2$
Character orbit 2646.h
Analytic conductor $21.128$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -1 + \beta_{1} ) q^{4} + ( 1 - \beta_{2} ) q^{5} + q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -1 + \beta_{1} ) q^{4} + ( 1 - \beta_{2} ) q^{5} + q^{8} + ( -2 \beta_{1} + \beta_{3} ) q^{10} + ( -2 - \beta_{2} ) q^{11} + 2 \beta_{1} q^{13} -\beta_{1} q^{16} + ( \beta_{1} + \beta_{3} ) q^{17} + ( 5 - 5 \beta_{1} ) q^{19} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{20} + ( \beta_{1} + \beta_{3} ) q^{22} + ( 5 + \beta_{2} ) q^{23} + ( 4 - 3 \beta_{2} ) q^{25} + ( 2 - 2 \beta_{1} ) q^{26} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{29} + ( 2 - 2 \beta_{1} ) q^{31} + ( -1 + \beta_{1} ) q^{32} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{34} + ( -2 + 2 \beta_{1} ) q^{37} -5 q^{38} + ( 1 - \beta_{2} ) q^{40} + ( -7 \beta_{1} - \beta_{3} ) q^{41} + ( -2 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{43} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{44} + ( -4 \beta_{1} - \beta_{3} ) q^{46} + ( -7 \beta_{1} + 3 \beta_{3} ) q^{50} -2 q^{52} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{53} + 6 q^{55} + ( 2 - 2 \beta_{2} ) q^{58} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{59} + ( -4 \beta_{1} - 3 \beta_{3} ) q^{61} -2 q^{62} + q^{64} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{65} + ( -8 + 5 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{67} + ( -2 - \beta_{2} ) q^{68} + ( -3 - 3 \beta_{2} ) q^{71} + ( 5 \beta_{1} - 3 \beta_{3} ) q^{73} + 2 q^{74} + 5 \beta_{1} q^{76} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{79} + ( -2 \beta_{1} + \beta_{3} ) q^{80} + ( -8 + 7 \beta_{1} - \beta_{2} + \beta_{3} ) q^{82} + ( -4 + 8 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{83} -6 \beta_{1} q^{85} + ( 2 + 3 \beta_{2} ) q^{86} + ( -2 - \beta_{2} ) q^{88} + ( -8 + 10 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{89} + ( -5 + 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{92} + ( 5 - 10 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{95} + ( 2 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 2q^{4} + 6q^{5} + 4q^{8} + O(q^{10}) \) \( 4q - 2q^{2} - 2q^{4} + 6q^{5} + 4q^{8} - 3q^{10} - 6q^{11} + 4q^{13} - 2q^{16} + 3q^{17} + 10q^{19} - 3q^{20} + 3q^{22} + 18q^{23} + 22q^{25} + 4q^{26} - 6q^{29} + 4q^{31} - 2q^{32} + 3q^{34} - 4q^{37} - 20q^{38} + 6q^{40} - 15q^{41} - q^{43} + 3q^{44} - 9q^{46} - 11q^{50} - 8q^{52} - 6q^{53} + 24q^{55} + 12q^{58} + 3q^{59} - 11q^{61} - 8q^{62} + 4q^{64} + 6q^{65} - 13q^{67} - 6q^{68} - 6q^{71} + 7q^{73} + 8q^{74} + 10q^{76} - 7q^{79} - 3q^{80} - 15q^{82} - 12q^{83} - 12q^{85} + 2q^{86} - 6q^{88} - 18q^{89} - 9q^{92} + 15q^{95} + q^{97} + O(q^{100}) \)

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 \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-\beta_{1}\) \(-1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
1.68614 0.396143i
−1.18614 + 1.26217i
1.68614 + 0.396143i
−1.18614 1.26217i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −1.37228 0 0 1.00000 0 0.686141 1.18843i
361.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 4.37228 0 0 1.00000 0 −2.18614 + 3.78651i
667.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.37228 0 0 1.00000 0 0.686141 + 1.18843i
667.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 4.37228 0 0 1.00000 0 −2.18614 3.78651i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2646.2.h.l 4
3.b odd 2 1 882.2.h.n 4
7.b odd 2 1 2646.2.h.k 4
7.c even 3 1 2646.2.e.m 4
7.c even 3 1 2646.2.f.j 4
7.d odd 6 1 378.2.f.c 4
7.d odd 6 1 2646.2.e.n 4
9.c even 3 1 2646.2.e.m 4
9.d odd 6 1 882.2.e.k 4
21.c even 2 1 882.2.h.m 4
21.g even 6 1 126.2.f.d 4
21.g even 6 1 882.2.e.l 4
21.h odd 6 1 882.2.e.k 4
21.h odd 6 1 882.2.f.k 4
28.f even 6 1 3024.2.r.f 4
63.g even 3 1 inner 2646.2.h.l 4
63.g even 3 1 7938.2.a.bs 2
63.h even 3 1 2646.2.f.j 4
63.i even 6 1 126.2.f.d 4
63.j odd 6 1 882.2.f.k 4
63.k odd 6 1 1134.2.a.n 2
63.k odd 6 1 2646.2.h.k 4
63.l odd 6 1 2646.2.e.n 4
63.n odd 6 1 882.2.h.n 4
63.n odd 6 1 7938.2.a.bh 2
63.o even 6 1 882.2.e.l 4
63.s even 6 1 882.2.h.m 4
63.s even 6 1 1134.2.a.k 2
63.t odd 6 1 378.2.f.c 4
84.j odd 6 1 1008.2.r.f 4
252.n even 6 1 9072.2.a.bb 2
252.r odd 6 1 1008.2.r.f 4
252.bj even 6 1 3024.2.r.f 4
252.bn odd 6 1 9072.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.d 4 21.g even 6 1
126.2.f.d 4 63.i even 6 1
378.2.f.c 4 7.d odd 6 1
378.2.f.c 4 63.t odd 6 1
882.2.e.k 4 9.d odd 6 1
882.2.e.k 4 21.h odd 6 1
882.2.e.l 4 21.g even 6 1
882.2.e.l 4 63.o even 6 1
882.2.f.k 4 21.h odd 6 1
882.2.f.k 4 63.j odd 6 1
882.2.h.m 4 21.c even 2 1
882.2.h.m 4 63.s even 6 1
882.2.h.n 4 3.b odd 2 1
882.2.h.n 4 63.n odd 6 1
1008.2.r.f 4 84.j odd 6 1
1008.2.r.f 4 252.r odd 6 1
1134.2.a.k 2 63.s even 6 1
1134.2.a.n 2 63.k odd 6 1
2646.2.e.m 4 7.c even 3 1
2646.2.e.m 4 9.c even 3 1
2646.2.e.n 4 7.d odd 6 1
2646.2.e.n 4 63.l odd 6 1
2646.2.f.j 4 7.c even 3 1
2646.2.f.j 4 63.h even 3 1
2646.2.h.k 4 7.b odd 2 1
2646.2.h.k 4 63.k odd 6 1
2646.2.h.l 4 1.a even 1 1 trivial
2646.2.h.l 4 63.g even 3 1 inner
3024.2.r.f 4 28.f even 6 1
3024.2.r.f 4 252.bj even 6 1
7938.2.a.bh 2 63.n odd 6 1
7938.2.a.bs 2 63.g even 3 1
9072.2.a.bb 2 252.n even 6 1
9072.2.a.bm 2 252.bn odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2646, [\chi])\):

\( T_{5}^{2} - 3 T_{5} - 6 \)
\( T_{11}^{2} + 3 T_{11} - 6 \)
\( T_{13}^{2} - 2 T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( -6 - 3 T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( -6 + 3 T + T^{2} )^{2} \)
$13$ \( ( 4 - 2 T + T^{2} )^{2} \)
$17$ \( 36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4} \)
$19$ \( ( 25 - 5 T + T^{2} )^{2} \)
$23$ \( ( 12 - 9 T + T^{2} )^{2} \)
$29$ \( 576 - 144 T + 60 T^{2} + 6 T^{3} + T^{4} \)
$31$ \( ( 4 - 2 T + T^{2} )^{2} \)
$37$ \( ( 4 + 2 T + T^{2} )^{2} \)
$41$ \( 2304 + 720 T + 177 T^{2} + 15 T^{3} + T^{4} \)
$43$ \( 5476 - 74 T + 75 T^{2} + T^{3} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( 576 - 144 T + 60 T^{2} + 6 T^{3} + T^{4} \)
$59$ \( 5184 + 216 T + 81 T^{2} - 3 T^{3} + T^{4} \)
$61$ \( 1936 - 484 T + 165 T^{2} + 11 T^{3} + T^{4} \)
$67$ \( 1024 - 416 T + 201 T^{2} + 13 T^{3} + T^{4} \)
$71$ \( ( -72 + 3 T + T^{2} )^{2} \)
$73$ \( 3844 + 434 T + 111 T^{2} - 7 T^{3} + T^{4} \)
$79$ \( 3844 - 434 T + 111 T^{2} + 7 T^{3} + T^{4} \)
$83$ \( 9216 - 1152 T + 240 T^{2} + 12 T^{3} + T^{4} \)
$89$ \( 2304 + 864 T + 276 T^{2} + 18 T^{3} + T^{4} \)
$97$ \( 5476 + 74 T + 75 T^{2} - T^{3} + T^{4} \)
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