Properties

Label 2646.2.h.s
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: 8.0.3317760000.3
Defining polynomial: \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 7\nu^{5} + 28\nu^{3} - 120\nu ) / 63 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{6} + 7\nu^{4} - 28\nu^{2} + 144 ) / 63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} - 7\nu^{5} - 35\nu^{3} + 180\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} + 2\nu^{2} + 18 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -10\nu^{6} - 14\nu^{4} - 7\nu^{2} + 234 ) / 63 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - \nu^{5} + 4\nu^{3} - 15\nu ) / 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{4} + 2\beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 2\beta_{5} - 6\beta_{3} + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + \beta_{4} + 10\beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{6} + 4\beta_{5} + 3\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{7} - 31\beta_{4} + 5\beta_{2} - 10\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -14\beta_{6} - 7\beta_{5} + 66 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 29\beta_{7} - 5\beta_{4} - 5\beta_{2} - 29\beta_1 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-\beta_{3}\) \(-1 + \beta_{3}\)

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
−1.01575 1.40294i
−1.72286 0.178197i
1.72286 + 0.178197i
1.01575 + 1.40294i
−1.01575 + 1.40294i
−1.72286 + 0.178197i
1.72286 0.178197i
1.01575 1.40294i
0.500000 0.866025i 0 −0.500000 0.866025i −3.44572 0 0 −1.00000 0 −1.72286 + 2.98408i
361.2 0.500000 0.866025i 0 −0.500000 0.866025i −2.03151 0 0 −1.00000 0 −1.01575 + 1.75934i
361.3 0.500000 0.866025i 0 −0.500000 0.866025i 2.03151 0 0 −1.00000 0 1.01575 1.75934i
361.4 0.500000 0.866025i 0 −0.500000 0.866025i 3.44572 0 0 −1.00000 0 1.72286 2.98408i
667.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −3.44572 0 0 −1.00000 0 −1.72286 2.98408i
667.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.03151 0 0 −1.00000 0 −1.01575 1.75934i
667.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.03151 0 0 −1.00000 0 1.01575 + 1.75934i
667.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 3.44572 0 0 −1.00000 0 1.72286 + 2.98408i
\(n\): 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.s 8
3.b odd 2 1 882.2.h.r 8
7.b odd 2 1 inner 2646.2.h.s 8
7.c even 3 1 2646.2.e.r 8
7.c even 3 1 2646.2.f.s 8
7.d odd 6 1 2646.2.e.r 8
7.d odd 6 1 2646.2.f.s 8
9.c even 3 1 2646.2.e.r 8
9.d odd 6 1 882.2.e.t 8
21.c even 2 1 882.2.h.r 8
21.g even 6 1 882.2.e.t 8
21.g even 6 1 882.2.f.p 8
21.h odd 6 1 882.2.e.t 8
21.h odd 6 1 882.2.f.p 8
63.g even 3 1 inner 2646.2.h.s 8
63.g even 3 1 7938.2.a.cd 4
63.h even 3 1 2646.2.f.s 8
63.i even 6 1 882.2.f.p 8
63.j odd 6 1 882.2.f.p 8
63.k odd 6 1 inner 2646.2.h.s 8
63.k odd 6 1 7938.2.a.cd 4
63.l odd 6 1 2646.2.e.r 8
63.n odd 6 1 882.2.h.r 8
63.n odd 6 1 7938.2.a.cu 4
63.o even 6 1 882.2.e.t 8
63.s even 6 1 882.2.h.r 8
63.s even 6 1 7938.2.a.cu 4
63.t odd 6 1 2646.2.f.s 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.e.t 8 9.d odd 6 1
882.2.e.t 8 21.g even 6 1
882.2.e.t 8 21.h odd 6 1
882.2.e.t 8 63.o even 6 1
882.2.f.p 8 21.g even 6 1
882.2.f.p 8 21.h odd 6 1
882.2.f.p 8 63.i even 6 1
882.2.f.p 8 63.j odd 6 1
882.2.h.r 8 3.b odd 2 1
882.2.h.r 8 21.c even 2 1
882.2.h.r 8 63.n odd 6 1
882.2.h.r 8 63.s even 6 1
2646.2.e.r 8 7.c even 3 1
2646.2.e.r 8 7.d odd 6 1
2646.2.e.r 8 9.c even 3 1
2646.2.e.r 8 63.l odd 6 1
2646.2.f.s 8 7.c even 3 1
2646.2.f.s 8 7.d odd 6 1
2646.2.f.s 8 63.h even 3 1
2646.2.f.s 8 63.t odd 6 1
2646.2.h.s 8 1.a even 1 1 trivial
2646.2.h.s 8 7.b odd 2 1 inner
2646.2.h.s 8 63.g even 3 1 inner
2646.2.h.s 8 63.k odd 6 1 inner
7938.2.a.cd 4 63.g even 3 1
7938.2.a.cd 4 63.k odd 6 1
7938.2.a.cu 4 63.n odd 6 1
7938.2.a.cu 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 \(S_{2}^{\mathrm{new}}(2646, [\chi])\):

\( T_{5}^{4} - 16T_{5}^{2} + 49 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{4} + 18T_{13}^{2} + 324 \) 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
$5$ \( (T^{4} - 16 T^{2} + 49)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T + 4)^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 40 T^{6} + 1575 T^{4} + \cdots + 625 \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T - 59)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 10 T^{3} + 90 T^{2} - 100 T + 100)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 30 T^{2} + 900)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{3} + 72 T^{2} + 224 T + 3136)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 10 T^{3} + 90 T^{2} + 100 T + 100)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 64 T^{6} + 3312 T^{4} + \cdots + 614656 \) Copy content Toggle raw display
$53$ \( (T^{4} - 2 T^{3} + 138 T^{2} + 268 T + 17956)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 76 T^{6} + 5292 T^{4} + \cdots + 234256 \) Copy content Toggle raw display
$61$ \( T^{8} + 40 T^{6} + 1575 T^{4} + \cdots + 625 \) Copy content Toggle raw display
$67$ \( (T^{4} - 6 T^{3} + 42 T^{2} + 36 T + 36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 12 T + 21)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + 256 T^{6} + \cdots + 21381376 \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + 27 T^{2} - 44 T + 121)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 76 T^{6} + 5292 T^{4} + \cdots + 234256 \) Copy content Toggle raw display
$89$ \( (T^{4} + 50 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 544 T^{6} + \cdots + 5158686976 \) Copy content Toggle raw display
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