Properties

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

Related objects

Downloads

Learn more about

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.b 2
3.b odd 2 1 882.2.h.h 2
7.b odd 2 1 2646.2.h.c 2
7.c even 3 1 378.2.f.b 2
7.c even 3 1 2646.2.e.i 2
7.d odd 6 1 2646.2.e.h 2
7.d odd 6 1 2646.2.f.b 2
9.c even 3 1 2646.2.e.i 2
9.d odd 6 1 882.2.e.a 2
21.c even 2 1 882.2.h.g 2
21.g even 6 1 882.2.e.e 2
21.g even 6 1 882.2.f.f 2
21.h odd 6 1 126.2.f.b 2
21.h odd 6 1 882.2.e.a 2
28.g odd 6 1 3024.2.r.c 2
63.g even 3 1 1134.2.a.f 1
63.g even 3 1 inner 2646.2.h.b 2
63.h even 3 1 378.2.f.b 2
63.i even 6 1 882.2.f.f 2
63.j odd 6 1 126.2.f.b 2
63.k odd 6 1 2646.2.h.c 2
63.k odd 6 1 7938.2.a.bb 1
63.l odd 6 1 2646.2.e.h 2
63.n odd 6 1 882.2.h.h 2
63.n odd 6 1 1134.2.a.c 1
63.o even 6 1 882.2.e.e 2
63.s even 6 1 882.2.h.g 2
63.s even 6 1 7938.2.a.e 1
63.t odd 6 1 2646.2.f.b 2
84.n even 6 1 1008.2.r.a 2
252.o even 6 1 9072.2.a.t 1
252.u odd 6 1 3024.2.r.c 2
252.bb even 6 1 1008.2.r.a 2
252.bl odd 6 1 9072.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.b 2 21.h odd 6 1
126.2.f.b 2 63.j odd 6 1
378.2.f.b 2 7.c even 3 1
378.2.f.b 2 63.h even 3 1
882.2.e.a 2 9.d odd 6 1
882.2.e.a 2 21.h odd 6 1
882.2.e.e 2 21.g even 6 1
882.2.e.e 2 63.o even 6 1
882.2.f.f 2 21.g even 6 1
882.2.f.f 2 63.i even 6 1
882.2.h.g 2 21.c even 2 1
882.2.h.g 2 63.s even 6 1
882.2.h.h 2 3.b odd 2 1
882.2.h.h 2 63.n odd 6 1
1008.2.r.a 2 84.n even 6 1
1008.2.r.a 2 252.bb even 6 1
1134.2.a.c 1 63.n odd 6 1
1134.2.a.f 1 63.g even 3 1
2646.2.e.h 2 7.d odd 6 1
2646.2.e.h 2 63.l odd 6 1
2646.2.e.i 2 7.c even 3 1
2646.2.e.i 2 9.c even 3 1
2646.2.f.b 2 7.d odd 6 1
2646.2.f.b 2 63.t odd 6 1
2646.2.h.b 2 1.a even 1 1 trivial
2646.2.h.b 2 63.g even 3 1 inner
2646.2.h.c 2 7.b odd 2 1
2646.2.h.c 2 63.k odd 6 1
3024.2.r.c 2 28.g odd 6 1
3024.2.r.c 2 252.u odd 6 1
7938.2.a.e 1 63.s even 6 1
7938.2.a.bb 1 63.k odd 6 1
9072.2.a.f 1 252.bl odd 6 1
9072.2.a.t 1 252.o even 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ 1
$5$ \( ( 1 + 2 T + 5 T^{2} )^{2} \)
$7$ 1
$11$ \( ( 1 + T + 11 T^{2} )^{2} \)
$13$ \( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} \)
$17$ \( 1 + 5 T + 8 T^{2} + 85 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \)
$23$ \( ( 1 + 4 T + 23 T^{2} )^{2} \)
$29$ \( 1 + 4 T - 13 T^{2} + 116 T^{3} + 841 T^{4} \)
$31$ \( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} \)
$37$ \( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \)
$43$ \( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 7 T - 10 T^{2} + 413 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 12 T + 83 T^{2} - 732 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 8 T + 71 T^{2} )^{2} \)
$73$ \( 1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 6 T - 43 T^{2} - 474 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 16 T + 173 T^{2} - 1328 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 19 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \)
show more
show less