Properties

Label 2646.2.h.c
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: \(x^{2} - x + 1\)
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.

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

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 2.00000 0 0 1.00000 0 −1.00000 + 1.73205i
667.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 2.00000 0 0 1.00000 0 −1.00000 1.73205i
\(n\): 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.c 2
3.b odd 2 1 882.2.h.g 2
7.b odd 2 1 2646.2.h.b 2
7.c even 3 1 2646.2.e.h 2
7.c even 3 1 2646.2.f.b 2
7.d odd 6 1 378.2.f.b 2
7.d odd 6 1 2646.2.e.i 2
9.c even 3 1 2646.2.e.h 2
9.d odd 6 1 882.2.e.e 2
21.c even 2 1 882.2.h.h 2
21.g even 6 1 126.2.f.b 2
21.g even 6 1 882.2.e.a 2
21.h odd 6 1 882.2.e.e 2
21.h odd 6 1 882.2.f.f 2
28.f even 6 1 3024.2.r.c 2
63.g even 3 1 inner 2646.2.h.c 2
63.g even 3 1 7938.2.a.bb 1
63.h even 3 1 2646.2.f.b 2
63.i even 6 1 126.2.f.b 2
63.j odd 6 1 882.2.f.f 2
63.k odd 6 1 1134.2.a.f 1
63.k odd 6 1 2646.2.h.b 2
63.l odd 6 1 2646.2.e.i 2
63.n odd 6 1 882.2.h.g 2
63.n odd 6 1 7938.2.a.e 1
63.o even 6 1 882.2.e.a 2
63.s even 6 1 882.2.h.h 2
63.s even 6 1 1134.2.a.c 1
63.t odd 6 1 378.2.f.b 2
84.j odd 6 1 1008.2.r.a 2
252.n even 6 1 9072.2.a.f 1
252.r odd 6 1 1008.2.r.a 2
252.bj even 6 1 3024.2.r.c 2
252.bn odd 6 1 9072.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.b 2 21.g even 6 1
126.2.f.b 2 63.i even 6 1
378.2.f.b 2 7.d odd 6 1
378.2.f.b 2 63.t odd 6 1
882.2.e.a 2 21.g even 6 1
882.2.e.a 2 63.o even 6 1
882.2.e.e 2 9.d odd 6 1
882.2.e.e 2 21.h odd 6 1
882.2.f.f 2 21.h odd 6 1
882.2.f.f 2 63.j odd 6 1
882.2.h.g 2 3.b odd 2 1
882.2.h.g 2 63.n odd 6 1
882.2.h.h 2 21.c even 2 1
882.2.h.h 2 63.s even 6 1
1008.2.r.a 2 84.j odd 6 1
1008.2.r.a 2 252.r odd 6 1
1134.2.a.c 1 63.s even 6 1
1134.2.a.f 1 63.k odd 6 1
2646.2.e.h 2 7.c even 3 1
2646.2.e.h 2 9.c even 3 1
2646.2.e.i 2 7.d odd 6 1
2646.2.e.i 2 63.l odd 6 1
2646.2.f.b 2 7.c even 3 1
2646.2.f.b 2 63.h even 3 1
2646.2.h.b 2 7.b odd 2 1
2646.2.h.b 2 63.k odd 6 1
2646.2.h.c 2 1.a even 1 1 trivial
2646.2.h.c 2 63.g even 3 1 inner
3024.2.r.c 2 28.f even 6 1
3024.2.r.c 2 252.bj even 6 1
7938.2.a.e 1 63.n odd 6 1
7938.2.a.bb 1 63.g even 3 1
9072.2.a.f 1 252.n even 6 1
9072.2.a.t 1 252.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 \(S_{2}^{\mathrm{new}}(2646, [\chi])\):

\( T_{5} - 2 \)
\( T_{11} + 1 \)
\( T_{13}^{2} + 6 T_{13} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -2 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( 36 + 6 T + T^{2} \)
$17$ \( 25 - 5 T + T^{2} \)
$19$ \( 49 + 7 T + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( 16 + 4 T + T^{2} \)
$31$ \( 36 + 6 T + T^{2} \)
$37$ \( 4 + 2 T + T^{2} \)
$41$ \( 9 + 3 T + T^{2} \)
$43$ \( 1 - T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 144 - 12 T + T^{2} \)
$59$ \( 49 - 7 T + T^{2} \)
$61$ \( 144 + 12 T + T^{2} \)
$67$ \( 169 + 13 T + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( 1 - T + T^{2} \)
$79$ \( 36 - 6 T + T^{2} \)
$83$ \( 256 + 16 T + T^{2} \)
$89$ \( 36 - 6 T + T^{2} \)
$97$ \( 25 + 5 T + T^{2} \)
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