Properties

Label 2646.2.h.o
Level $2646$
Weight $2$
Character orbit 2646.h
Analytic conductor $21.128$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + (\beta_{4} - 1) q^{4} + \beta_{3} q^{5} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + (\beta_{4} - 1) q^{4} + \beta_{3} q^{5} + q^{8} + ( - \beta_{3} - \beta_{2}) q^{10} + \beta_{3} q^{11} + (2 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{13} - \beta_{4} q^{16} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{17} + (3 \beta_{5} - 2 \beta_{4} + 2) q^{19} + \beta_{2} q^{20} + ( - \beta_{3} - \beta_{2}) q^{22} + (\beta_1 - 2) q^{23} + ( - 2 \beta_{3} + \beta_1 - 1) q^{25} + ( - 2 \beta_{5} + 3 \beta_{4} - \beta_{2} - 3) q^{26} + ( - 3 \beta_{5} + 2 \beta_{2}) q^{29} + (\beta_{5} + 6 \beta_{4} - \beta_{2} - 6) q^{31} + (\beta_{4} - 1) q^{32} + ( - 2 \beta_{5} + 2 \beta_{4} - 2) q^{34} + ( - \beta_{4} + 1) q^{37} + ( - 3 \beta_1 - 2) q^{38} + \beta_{3} q^{40} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{41} + (3 \beta_{5} + 2 \beta_{4} + 3 \beta_{2} - 2) q^{43} + \beta_{2} q^{44} + (\beta_{5} + 2 \beta_{4} - \beta_1) q^{46} + ( - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{47} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{50} + ( - \beta_{3} + 2 \beta_1 + 3) q^{52} + ( - \beta_{5} - 5 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{53} + ( - 2 \beta_{3} + \beta_1 + 4) q^{55} + (2 \beta_{3} + 3 \beta_1) q^{58} + (\beta_{5} + 5 \beta_{4} + 2 \beta_{2} - 5) q^{59} + ( - \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{61} + ( - \beta_{3} - \beta_1 + 6) q^{62} + q^{64} + (\beta_{5} + 2 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} - \beta_1) q^{65} + (\beta_{5} - 2 \beta_{4} - 4 \beta_{2} + 2) q^{67} + (2 \beta_1 + 2) q^{68} + (2 \beta_{3} - 7 \beta_1 - 4) q^{71} + (\beta_{5} - 8 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - \beta_1) q^{73} - q^{74} + ( - 3 \beta_{5} + 2 \beta_{4} + 3 \beta_1) q^{76} + (4 \beta_{5} - \beta_{3} - \beta_{2} - 4 \beta_1) q^{79} + ( - \beta_{3} - \beta_{2}) q^{80} + (\beta_{5} - \beta_{4} - 2 \beta_{2} + 1) q^{82} + ( - 2 \beta_{5} - \beta_{4} - 3 \beta_{2} + 1) q^{83} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{85} + (3 \beta_{3} - 3 \beta_1 + 2) q^{86} + \beta_{3} q^{88} + (4 \beta_{5} + 2 \beta_{4} + \beta_{2} - 2) q^{89} + ( - \beta_{5} - 2 \beta_{4} + 2) q^{92} + (3 \beta_{5} + 3 \beta_{4} + 3 \beta_{2} - 3) q^{94} + (3 \beta_{5} - 3 \beta_{4} - 2 \beta_{2} + 3) q^{95} + (2 \beta_{5} + 8 \beta_{4} - 2 \beta_{2} - 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{4} - 2 q^{5} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} - 3 q^{4} - 2 q^{5} + 6 q^{8} + q^{10} - 2 q^{11} - 8 q^{13} - 3 q^{16} - 4 q^{17} + 3 q^{19} + q^{20} + q^{22} - 14 q^{23} - 4 q^{25} - 8 q^{26} + 5 q^{29} - 20 q^{31} - 3 q^{32} - 4 q^{34} + 3 q^{37} - 6 q^{38} - 2 q^{40} - 6 q^{43} + q^{44} + 7 q^{46} - 9 q^{47} + 2 q^{50} + 16 q^{52} - 15 q^{53} + 26 q^{55} - 10 q^{58} - 14 q^{59} - 8 q^{61} + 40 q^{62} + 6 q^{64} + 12 q^{65} + q^{67} + 8 q^{68} - 14 q^{71} - 19 q^{73} - 6 q^{74} + 3 q^{76} + 5 q^{79} + q^{80} + 2 q^{83} - 2 q^{85} + 12 q^{86} - 2 q^{88} - 9 q^{89} + 7 q^{92} - 9 q^{94} + 4 q^{95} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-\beta_{4}\) \(-1 + \beta_{4}\)

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
0.500000 1.41036i
0.500000 + 2.05195i
0.500000 + 0.224437i
0.500000 + 1.41036i
0.500000 2.05195i
0.500000 0.224437i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −3.18194 0 0 1.00000 0 1.59097 2.75564i
361.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.593579 0 0 1.00000 0 −0.296790 + 0.514055i
361.3 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.58836 0 0 1.00000 0 −0.794182 + 1.37556i
667.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −3.18194 0 0 1.00000 0 1.59097 + 2.75564i
667.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0.593579 0 0 1.00000 0 −0.296790 0.514055i
667.3 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.58836 0 0 1.00000 0 −0.794182 1.37556i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 667.3
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.o 6
3.b odd 2 1 882.2.h.p 6
7.b odd 2 1 378.2.h.c 6
7.c even 3 1 2646.2.e.p 6
7.c even 3 1 2646.2.f.m 6
7.d odd 6 1 378.2.e.d 6
7.d odd 6 1 2646.2.f.l 6
9.c even 3 1 2646.2.e.p 6
9.d odd 6 1 882.2.e.o 6
21.c even 2 1 126.2.h.d yes 6
21.g even 6 1 126.2.e.c 6
21.g even 6 1 882.2.f.n 6
21.h odd 6 1 882.2.e.o 6
21.h odd 6 1 882.2.f.o 6
28.d even 2 1 3024.2.t.h 6
28.f even 6 1 3024.2.q.g 6
63.g even 3 1 inner 2646.2.h.o 6
63.g even 3 1 7938.2.a.bz 3
63.h even 3 1 2646.2.f.m 6
63.i even 6 1 882.2.f.n 6
63.i even 6 1 1134.2.g.m 6
63.j odd 6 1 882.2.f.o 6
63.k odd 6 1 378.2.h.c 6
63.k odd 6 1 7938.2.a.ca 3
63.l odd 6 1 378.2.e.d 6
63.l odd 6 1 1134.2.g.l 6
63.n odd 6 1 882.2.h.p 6
63.n odd 6 1 7938.2.a.bw 3
63.o even 6 1 126.2.e.c 6
63.o even 6 1 1134.2.g.m 6
63.s even 6 1 126.2.h.d yes 6
63.s even 6 1 7938.2.a.bv 3
63.t odd 6 1 1134.2.g.l 6
63.t odd 6 1 2646.2.f.l 6
84.h odd 2 1 1008.2.t.h 6
84.j odd 6 1 1008.2.q.g 6
252.n even 6 1 3024.2.t.h 6
252.s odd 6 1 1008.2.q.g 6
252.bi even 6 1 3024.2.q.g 6
252.bn odd 6 1 1008.2.t.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.c 6 21.g even 6 1
126.2.e.c 6 63.o even 6 1
126.2.h.d yes 6 21.c even 2 1
126.2.h.d yes 6 63.s even 6 1
378.2.e.d 6 7.d odd 6 1
378.2.e.d 6 63.l odd 6 1
378.2.h.c 6 7.b odd 2 1
378.2.h.c 6 63.k odd 6 1
882.2.e.o 6 9.d odd 6 1
882.2.e.o 6 21.h odd 6 1
882.2.f.n 6 21.g even 6 1
882.2.f.n 6 63.i even 6 1
882.2.f.o 6 21.h odd 6 1
882.2.f.o 6 63.j odd 6 1
882.2.h.p 6 3.b odd 2 1
882.2.h.p 6 63.n odd 6 1
1008.2.q.g 6 84.j odd 6 1
1008.2.q.g 6 252.s odd 6 1
1008.2.t.h 6 84.h odd 2 1
1008.2.t.h 6 252.bn odd 6 1
1134.2.g.l 6 63.l odd 6 1
1134.2.g.l 6 63.t odd 6 1
1134.2.g.m 6 63.i even 6 1
1134.2.g.m 6 63.o even 6 1
2646.2.e.p 6 7.c even 3 1
2646.2.e.p 6 9.c even 3 1
2646.2.f.l 6 7.d odd 6 1
2646.2.f.l 6 63.t odd 6 1
2646.2.f.m 6 7.c even 3 1
2646.2.f.m 6 63.h even 3 1
2646.2.h.o 6 1.a even 1 1 trivial
2646.2.h.o 6 63.g even 3 1 inner
3024.2.q.g 6 28.f even 6 1
3024.2.q.g 6 252.bi even 6 1
3024.2.t.h 6 28.d even 2 1
3024.2.t.h 6 252.n even 6 1
7938.2.a.bv 3 63.s even 6 1
7938.2.a.bw 3 63.n odd 6 1
7938.2.a.bz 3 63.g even 3 1
7938.2.a.ca 3 63.k odd 6 1

Hecke kernels

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

\( T_{5}^{3} + T_{5}^{2} - 6T_{5} + 3 \) Copy content Toggle raw display
\( T_{11}^{3} + T_{11}^{2} - 6T_{11} + 3 \) Copy content Toggle raw display
\( T_{13}^{6} + 8T_{13}^{5} + 63T_{13}^{4} + 146T_{13}^{3} + 553T_{13}^{2} - 69T_{13} + 4761 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{3} + T^{2} - 6 T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} + T^{2} - 6 T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 8 T^{5} + 63 T^{4} + \cdots + 4761 \) Copy content Toggle raw display
$17$ \( T^{6} + 4 T^{5} + 28 T^{4} + 240 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$19$ \( T^{6} - 3 T^{5} + 45 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$23$ \( (T^{3} + 7 T^{2} + 12 T + 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} - 5 T^{5} + 91 T^{4} + \cdots + 131769 \) Copy content Toggle raw display
$31$ \( T^{6} + 20 T^{5} + 279 T^{4} + \cdots + 40401 \) Copy content Toggle raw display
$37$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$41$ \( T^{6} + 33 T^{4} + 18 T^{3} + 1089 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{6} + 6 T^{5} + 105 T^{4} + \cdots + 16129 \) Copy content Toggle raw display
$47$ \( T^{6} + 9 T^{5} + 135 T^{4} + \cdots + 35721 \) Copy content Toggle raw display
$53$ \( T^{6} + 15 T^{5} + 159 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$59$ \( T^{6} + 14 T^{5} + 157 T^{4} + \cdots + 3969 \) Copy content Toggle raw display
$61$ \( T^{6} + 8 T^{5} + 69 T^{4} + \cdots + 8649 \) Copy content Toggle raw display
$67$ \( T^{6} - T^{5} + 113 T^{4} + \cdots + 44521 \) Copy content Toggle raw display
$71$ \( (T^{3} + 7 T^{2} - 198 T - 1593)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 19 T^{5} + 353 T^{4} + \cdots + 398161 \) Copy content Toggle raw display
$79$ \( T^{6} - 5 T^{5} + 99 T^{4} + \cdots + 103041 \) Copy content Toggle raw display
$83$ \( T^{6} - 2 T^{5} + 67 T^{4} + \cdots + 21609 \) Copy content Toggle raw display
$89$ \( T^{6} + 9 T^{5} + 123 T^{4} - 396 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$97$ \( T^{6} + 28 T^{5} + 572 T^{4} + \cdots + 61504 \) Copy content Toggle raw display
show more
show less