Properties

Label 2646.2.h.t
Level $2646$
Weight $2$
Character orbit 2646.h
Analytic conductor $21.128$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 882)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - \beta_1 q^{4} + ( - 2 \beta_{6} - 2 \beta_{5}) q^{5} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} - \beta_1 q^{4} + ( - 2 \beta_{6} - 2 \beta_{5}) q^{5} - q^{8} - 2 \beta_{7} q^{10} + (\beta_{4} + 2) q^{11} + ( - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3}) q^{13} + (\beta_1 - 1) q^{16} + (5 \beta_{7} - 3 \beta_{5} + 3 \beta_{3}) q^{17} + ( - 3 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} - 2 \beta_{3}) q^{19} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5}) q^{20} + (\beta_{4} - \beta_{2} - 2 \beta_1 + 2) q^{22} + (2 \beta_{4} - 2) q^{23} + (4 \beta_{4} + 3) q^{25} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3}) q^{26} - 4 \beta_1 q^{29} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{3}) q^{31} + \beta_1 q^{32} + (5 \beta_{7} - 5 \beta_{6} - 5 \beta_{5} + 3 \beta_{3}) q^{34} + (2 \beta_{2} - 4 \beta_1) q^{37} + (3 \beta_{6} + \beta_{5}) q^{38} + (2 \beta_{6} + 2 \beta_{5}) q^{40} + (3 \beta_{7} - 5 \beta_{5} + 5 \beta_{3}) q^{41} + ( - \beta_{2} - 2 \beta_1) q^{43} + ( - \beta_{2} - 2 \beta_1) q^{44} + (2 \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 2) q^{46} + (4 \beta_{7} + 2 \beta_{5} - 2 \beta_{3}) q^{47} + (4 \beta_{4} - 4 \beta_{2} - 3 \beta_1 + 3) q^{50} + (2 \beta_{6} + 4 \beta_{5}) q^{52} + ( - 4 \beta_{4} + 4 \beta_{2} - 4 \beta_1 + 4) q^{53} + ( - 6 \beta_{6} - 8 \beta_{5}) q^{55} - 4 q^{58} + (3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 2 \beta_{3}) q^{59} + ( - 8 \beta_{7} + 6 \beta_{5} - 6 \beta_{3}) q^{61} + (2 \beta_{6} - 2 \beta_{5}) q^{62} + q^{64} + (8 \beta_{4} - 8 \beta_{2} - 12 \beta_1 + 12) q^{65} + (2 \beta_{2} - 9 \beta_1) q^{67} + ( - 5 \beta_{6} - 2 \beta_{5}) q^{68} + (2 \beta_{4} - 6) q^{71} + (5 \beta_{7} - 2 \beta_{5} + 2 \beta_{3}) q^{73} + (2 \beta_{4} - 4) q^{74} + (3 \beta_{7} - 2 \beta_{5} + 2 \beta_{3}) q^{76} + (4 \beta_{4} - 4 \beta_{2} + 2 \beta_1 - 2) q^{79} + 2 \beta_{7} q^{80} + (3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + 5 \beta_{3}) q^{82} + (10 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} + \beta_{3}) q^{83} + ( - 4 \beta_{4} + 4 \beta_{2} + 14 \beta_1 - 14) q^{85} + ( - \beta_{4} - 2) q^{86} + ( - \beta_{4} - 2) q^{88} + 5 \beta_{3} q^{89} + ( - 2 \beta_{2} + 2 \beta_1) q^{92} + (4 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} - 2 \beta_{3}) q^{94} + ( - 2 \beta_{2} - 8 \beta_1) q^{95} + (5 \beta_{7} - 5 \beta_{6} - 5 \beta_{5} + 11 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} + 16 q^{11} - 4 q^{16} + 8 q^{22} - 16 q^{23} + 24 q^{25} - 16 q^{29} + 4 q^{32} - 16 q^{37} - 8 q^{43} - 8 q^{44} - 8 q^{46} + 12 q^{50} + 16 q^{53} - 32 q^{58} + 8 q^{64} + 48 q^{65} - 36 q^{67} - 48 q^{71} - 32 q^{74} - 8 q^{79} - 56 q^{85} - 16 q^{86} - 16 q^{88} + 8 q^{92} - 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{6} + \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -2\beta_{7} + \beta_{6} + 3\beta_{5} - \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{4} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{3} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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
0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
−0.965926 0.258819i
0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
−0.965926 + 0.258819i
0.500000 0.866025i 0 −0.500000 0.866025i −3.86370 0 0 −1.00000 0 −1.93185 + 3.34607i
361.2 0.500000 0.866025i 0 −0.500000 0.866025i −1.03528 0 0 −1.00000 0 −0.517638 + 0.896575i
361.3 0.500000 0.866025i 0 −0.500000 0.866025i 1.03528 0 0 −1.00000 0 0.517638 0.896575i
361.4 0.500000 0.866025i 0 −0.500000 0.866025i 3.86370 0 0 −1.00000 0 1.93185 3.34607i
667.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −3.86370 0 0 −1.00000 0 −1.93185 3.34607i
667.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.03528 0 0 −1.00000 0 −0.517638 0.896575i
667.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.03528 0 0 −1.00000 0 0.517638 + 0.896575i
667.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 3.86370 0 0 −1.00000 0 1.93185 + 3.34607i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 667.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.g even 3 1 inner
63.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2646.2.h.t 8
3.b odd 2 1 882.2.h.q 8
7.b odd 2 1 inner 2646.2.h.t 8
7.c even 3 1 2646.2.e.q 8
7.c even 3 1 2646.2.f.r 8
7.d odd 6 1 2646.2.e.q 8
7.d odd 6 1 2646.2.f.r 8
9.c even 3 1 2646.2.e.q 8
9.d odd 6 1 882.2.e.s 8
21.c even 2 1 882.2.h.q 8
21.g even 6 1 882.2.e.s 8
21.g even 6 1 882.2.f.q 8
21.h odd 6 1 882.2.e.s 8
21.h odd 6 1 882.2.f.q 8
63.g even 3 1 inner 2646.2.h.t 8
63.g even 3 1 7938.2.a.ci 4
63.h even 3 1 2646.2.f.r 8
63.i even 6 1 882.2.f.q 8
63.j odd 6 1 882.2.f.q 8
63.k odd 6 1 inner 2646.2.h.t 8
63.k odd 6 1 7938.2.a.ci 4
63.l odd 6 1 2646.2.e.q 8
63.n odd 6 1 882.2.h.q 8
63.n odd 6 1 7938.2.a.cp 4
63.o even 6 1 882.2.e.s 8
63.s even 6 1 882.2.h.q 8
63.s even 6 1 7938.2.a.cp 4
63.t odd 6 1 2646.2.f.r 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.e.s 8 9.d odd 6 1
882.2.e.s 8 21.g even 6 1
882.2.e.s 8 21.h odd 6 1
882.2.e.s 8 63.o even 6 1
882.2.f.q 8 21.g even 6 1
882.2.f.q 8 21.h odd 6 1
882.2.f.q 8 63.i even 6 1
882.2.f.q 8 63.j odd 6 1
882.2.h.q 8 3.b odd 2 1
882.2.h.q 8 21.c even 2 1
882.2.h.q 8 63.n odd 6 1
882.2.h.q 8 63.s even 6 1
2646.2.e.q 8 7.c even 3 1
2646.2.e.q 8 7.d odd 6 1
2646.2.e.q 8 9.c even 3 1
2646.2.e.q 8 63.l odd 6 1
2646.2.f.r 8 7.c even 3 1
2646.2.f.r 8 7.d odd 6 1
2646.2.f.r 8 63.h even 3 1
2646.2.f.r 8 63.t odd 6 1
2646.2.h.t 8 1.a even 1 1 trivial
2646.2.h.t 8 7.b odd 2 1 inner
2646.2.h.t 8 63.g even 3 1 inner
2646.2.h.t 8 63.k odd 6 1 inner
7938.2.a.ci 4 63.g even 3 1
7938.2.a.ci 4 63.k odd 6 1
7938.2.a.cp 4 63.n odd 6 1
7938.2.a.cp 4 63.s even 6 1

Hecke kernels

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

\( T_{5}^{4} - 16T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} + 1 \) Copy content Toggle raw display
\( T_{13}^{8} + 48T_{13}^{6} + 2160T_{13}^{4} + 6912T_{13}^{2} + 20736 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 16 T^{2} + 16)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 48 T^{6} + 2160 T^{4} + \cdots + 20736 \) Copy content Toggle raw display
$17$ \( T^{8} + 76 T^{6} + 4407 T^{4} + \cdots + 1874161 \) Copy content Toggle raw display
$19$ \( T^{8} + 28 T^{6} + 615 T^{4} + \cdots + 28561 \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T - 8)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + 48 T^{6} + 2160 T^{4} + \cdots + 20736 \) Copy content Toggle raw display
$37$ \( (T^{4} + 8 T^{3} + 60 T^{2} + 32 T + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 76 T^{6} + 5655 T^{4} + \cdots + 14641 \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{3} + 15 T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 112 T^{6} + 12480 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( (T^{4} - 8 T^{3} + 96 T^{2} + 256 T + 1024)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 76 T^{6} + 5655 T^{4} + \cdots + 14641 \) Copy content Toggle raw display
$61$ \( T^{8} + 208 T^{6} + \cdots + 59969536 \) Copy content Toggle raw display
$67$ \( (T^{4} + 18 T^{3} + 255 T^{2} + 1242 T + 4761)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 24)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + 76 T^{6} + 4407 T^{4} + \cdots + 1874161 \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + 60 T^{2} - 176 T + 1936)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 364 T^{6} + \cdots + 193877776 \) Copy content Toggle raw display
$89$ \( (T^{4} + 50 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 364 T^{6} + \cdots + 131079601 \) Copy content Toggle raw display
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