Properties

Label 2646.2.h.d
Level $2646$
Weight $2$
Character orbit 2646.h
Analytic conductor $21.128$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + 3 q^{5} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + 3 q^{5} + q^{8} + (3 \zeta_{6} - 3) q^{10} + 3 q^{11} + (\zeta_{6} - 1) q^{13} + (\zeta_{6} - 1) q^{16} + (3 \zeta_{6} - 3) q^{17} - 7 \zeta_{6} q^{19} - 3 \zeta_{6} q^{20} + (3 \zeta_{6} - 3) q^{22} + 9 q^{23} + 4 q^{25} - \zeta_{6} q^{26} + 3 \zeta_{6} q^{29} + 8 \zeta_{6} q^{31} - \zeta_{6} q^{32} - 3 \zeta_{6} q^{34} + \zeta_{6} q^{37} + 7 q^{38} + 3 q^{40} + (3 \zeta_{6} - 3) q^{41} + \zeta_{6} q^{43} - 3 \zeta_{6} q^{44} + (9 \zeta_{6} - 9) q^{46} + (4 \zeta_{6} - 4) q^{50} + q^{52} + ( - 3 \zeta_{6} + 3) q^{53} + 9 q^{55} - 3 q^{58} + ( - 2 \zeta_{6} + 2) q^{61} - 8 q^{62} + q^{64} + (3 \zeta_{6} - 3) q^{65} + 4 \zeta_{6} q^{67} + 3 q^{68} - 12 q^{71} + ( - 11 \zeta_{6} + 11) q^{73} - q^{74} + (7 \zeta_{6} - 7) q^{76} + ( - 16 \zeta_{6} + 16) q^{79} + (3 \zeta_{6} - 3) q^{80} - 3 \zeta_{6} q^{82} + 9 \zeta_{6} q^{83} + (9 \zeta_{6} - 9) q^{85} - q^{86} + 3 q^{88} - 3 \zeta_{6} q^{89} - 9 \zeta_{6} q^{92} - 21 \zeta_{6} q^{95} - \zeta_{6} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 6 q^{5} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} + 6 q^{5} + 2 q^{8} - 3 q^{10} + 6 q^{11} - q^{13} - q^{16} - 3 q^{17} - 7 q^{19} - 3 q^{20} - 3 q^{22} + 18 q^{23} + 8 q^{25} - q^{26} + 3 q^{29} + 8 q^{31} - q^{32} - 3 q^{34} + q^{37} + 14 q^{38} + 6 q^{40} - 3 q^{41} + q^{43} - 3 q^{44} - 9 q^{46} - 4 q^{50} + 2 q^{52} + 3 q^{53} + 18 q^{55} - 6 q^{58} + 2 q^{61} - 16 q^{62} + 2 q^{64} - 3 q^{65} + 4 q^{67} + 6 q^{68} - 24 q^{71} + 11 q^{73} - 2 q^{74} - 7 q^{76} + 16 q^{79} - 3 q^{80} - 3 q^{82} + 9 q^{83} - 9 q^{85} - 2 q^{86} + 6 q^{88} - 3 q^{89} - 9 q^{92} - 21 q^{95} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(-\zeta_{6}\)

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 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 3.00000 0 0 1.00000 0 −1.50000 + 2.59808i
667.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 3.00000 0 0 1.00000 0 −1.50000 2.59808i
\(n\): 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.d 2
3.b odd 2 1 882.2.h.i 2
7.b odd 2 1 378.2.h.a 2
7.c even 3 1 2646.2.e.g 2
7.c even 3 1 2646.2.f.a 2
7.d odd 6 1 378.2.e.b 2
7.d odd 6 1 2646.2.f.d 2
9.c even 3 1 2646.2.e.g 2
9.d odd 6 1 882.2.e.c 2
21.c even 2 1 126.2.h.b yes 2
21.g even 6 1 126.2.e.a 2
21.g even 6 1 882.2.f.i 2
21.h odd 6 1 882.2.e.c 2
21.h odd 6 1 882.2.f.g 2
28.d even 2 1 3024.2.t.a 2
28.f even 6 1 3024.2.q.f 2
63.g even 3 1 inner 2646.2.h.d 2
63.g even 3 1 7938.2.a.be 1
63.h even 3 1 2646.2.f.a 2
63.i even 6 1 882.2.f.i 2
63.i even 6 1 1134.2.g.e 2
63.j odd 6 1 882.2.f.g 2
63.k odd 6 1 378.2.h.a 2
63.k odd 6 1 7938.2.a.t 1
63.l odd 6 1 378.2.e.b 2
63.l odd 6 1 1134.2.g.c 2
63.n odd 6 1 882.2.h.i 2
63.n odd 6 1 7938.2.a.b 1
63.o even 6 1 126.2.e.a 2
63.o even 6 1 1134.2.g.e 2
63.s even 6 1 126.2.h.b yes 2
63.s even 6 1 7938.2.a.m 1
63.t odd 6 1 1134.2.g.c 2
63.t odd 6 1 2646.2.f.d 2
84.h odd 2 1 1008.2.t.f 2
84.j odd 6 1 1008.2.q.a 2
252.n even 6 1 3024.2.t.a 2
252.s odd 6 1 1008.2.q.a 2
252.bi even 6 1 3024.2.q.f 2
252.bn odd 6 1 1008.2.t.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.a 2 21.g even 6 1
126.2.e.a 2 63.o even 6 1
126.2.h.b yes 2 21.c even 2 1
126.2.h.b yes 2 63.s even 6 1
378.2.e.b 2 7.d odd 6 1
378.2.e.b 2 63.l odd 6 1
378.2.h.a 2 7.b odd 2 1
378.2.h.a 2 63.k odd 6 1
882.2.e.c 2 9.d odd 6 1
882.2.e.c 2 21.h odd 6 1
882.2.f.g 2 21.h odd 6 1
882.2.f.g 2 63.j odd 6 1
882.2.f.i 2 21.g even 6 1
882.2.f.i 2 63.i even 6 1
882.2.h.i 2 3.b odd 2 1
882.2.h.i 2 63.n odd 6 1
1008.2.q.a 2 84.j odd 6 1
1008.2.q.a 2 252.s odd 6 1
1008.2.t.f 2 84.h odd 2 1
1008.2.t.f 2 252.bn odd 6 1
1134.2.g.c 2 63.l odd 6 1
1134.2.g.c 2 63.t odd 6 1
1134.2.g.e 2 63.i even 6 1
1134.2.g.e 2 63.o even 6 1
2646.2.e.g 2 7.c even 3 1
2646.2.e.g 2 9.c even 3 1
2646.2.f.a 2 7.c even 3 1
2646.2.f.a 2 63.h even 3 1
2646.2.f.d 2 7.d odd 6 1
2646.2.f.d 2 63.t odd 6 1
2646.2.h.d 2 1.a even 1 1 trivial
2646.2.h.d 2 63.g even 3 1 inner
3024.2.q.f 2 28.f even 6 1
3024.2.q.f 2 252.bi even 6 1
3024.2.t.a 2 28.d even 2 1
3024.2.t.a 2 252.n even 6 1
7938.2.a.b 1 63.n odd 6 1
7938.2.a.m 1 63.s even 6 1
7938.2.a.t 1 63.k odd 6 1
7938.2.a.be 1 63.g even 3 1

Hecke kernels

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

\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display
\( T_{13}^{2} + T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$23$ \( (T - 9)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$37$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$83$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$89$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$97$ \( T^{2} + T + 1 \) Copy content Toggle raw display
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