Properties

Label 63.2.g.a
Level 63
Weight 2
Character orbit 63.g
Analytic conductor 0.503
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 63.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.503057532734\)
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: yes
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( -1 - \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} - q^{5} + ( -1 + 2 \zeta_{6} ) q^{6} + ( 2 - 3 \zeta_{6} ) q^{7} -3 q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( -1 - \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} - q^{5} + ( -1 + 2 \zeta_{6} ) q^{6} + ( 2 - 3 \zeta_{6} ) q^{7} -3 q^{8} + 3 \zeta_{6} q^{9} + \zeta_{6} q^{10} + 5 q^{11} + ( -2 + \zeta_{6} ) q^{12} + 5 \zeta_{6} q^{13} + ( -3 + \zeta_{6} ) q^{14} + ( 1 + \zeta_{6} ) q^{15} + \zeta_{6} q^{16} -3 \zeta_{6} q^{17} + ( 3 - 3 \zeta_{6} ) q^{18} + ( -1 + \zeta_{6} ) q^{19} + ( -1 + \zeta_{6} ) q^{20} + ( -5 + 4 \zeta_{6} ) q^{21} -5 \zeta_{6} q^{22} + 3 q^{23} + ( 3 + 3 \zeta_{6} ) q^{24} -4 q^{25} + ( 5 - 5 \zeta_{6} ) q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -1 - 2 \zeta_{6} ) q^{28} + ( 1 - \zeta_{6} ) q^{29} + ( 1 - 2 \zeta_{6} ) q^{30} + ( -5 + 5 \zeta_{6} ) q^{32} + ( -5 - 5 \zeta_{6} ) q^{33} + ( -3 + 3 \zeta_{6} ) q^{34} + ( -2 + 3 \zeta_{6} ) q^{35} + 3 q^{36} + ( -3 + 3 \zeta_{6} ) q^{37} + q^{38} + ( 5 - 10 \zeta_{6} ) q^{39} + 3 q^{40} + 5 \zeta_{6} q^{41} + ( 4 + \zeta_{6} ) q^{42} + ( 1 - \zeta_{6} ) q^{43} + ( 5 - 5 \zeta_{6} ) q^{44} -3 \zeta_{6} q^{45} -3 \zeta_{6} q^{46} + ( 1 - 2 \zeta_{6} ) q^{48} + ( -5 - 3 \zeta_{6} ) q^{49} + 4 \zeta_{6} q^{50} + ( -3 + 6 \zeta_{6} ) q^{51} + 5 q^{52} + 9 \zeta_{6} q^{53} + ( -6 + 3 \zeta_{6} ) q^{54} -5 q^{55} + ( -6 + 9 \zeta_{6} ) q^{56} + ( 2 - \zeta_{6} ) q^{57} - q^{58} + ( 2 - \zeta_{6} ) q^{60} + 14 \zeta_{6} q^{61} + ( 9 - 3 \zeta_{6} ) q^{63} + 7 q^{64} -5 \zeta_{6} q^{65} + ( -5 + 10 \zeta_{6} ) q^{66} + ( -4 + 4 \zeta_{6} ) q^{67} -3 q^{68} + ( -3 - 3 \zeta_{6} ) q^{69} + ( 3 - \zeta_{6} ) q^{70} -12 q^{71} -9 \zeta_{6} q^{72} -3 \zeta_{6} q^{73} + 3 q^{74} + ( 4 + 4 \zeta_{6} ) q^{75} + \zeta_{6} q^{76} + ( 10 - 15 \zeta_{6} ) q^{77} + ( -10 + 5 \zeta_{6} ) q^{78} -8 \zeta_{6} q^{79} -\zeta_{6} q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 5 - 5 \zeta_{6} ) q^{82} + ( 9 - 9 \zeta_{6} ) q^{83} + ( -1 + 5 \zeta_{6} ) q^{84} + 3 \zeta_{6} q^{85} - q^{86} + ( -2 + \zeta_{6} ) q^{87} -15 q^{88} + ( 13 - 13 \zeta_{6} ) q^{89} + ( -3 + 3 \zeta_{6} ) q^{90} + ( 15 - 5 \zeta_{6} ) q^{91} + ( 3 - 3 \zeta_{6} ) q^{92} + ( 1 - \zeta_{6} ) q^{95} + ( 10 - 5 \zeta_{6} ) q^{96} + ( 9 - 9 \zeta_{6} ) q^{97} + ( -3 + 8 \zeta_{6} ) q^{98} + 15 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 3q^{3} + q^{4} - 2q^{5} + q^{7} - 6q^{8} + 3q^{9} + O(q^{10}) \) \( 2q - q^{2} - 3q^{3} + q^{4} - 2q^{5} + q^{7} - 6q^{8} + 3q^{9} + q^{10} + 10q^{11} - 3q^{12} + 5q^{13} - 5q^{14} + 3q^{15} + q^{16} - 3q^{17} + 3q^{18} - q^{19} - q^{20} - 6q^{21} - 5q^{22} + 6q^{23} + 9q^{24} - 8q^{25} + 5q^{26} - 4q^{28} + q^{29} - 5q^{32} - 15q^{33} - 3q^{34} - q^{35} + 6q^{36} - 3q^{37} + 2q^{38} + 6q^{40} + 5q^{41} + 9q^{42} + q^{43} + 5q^{44} - 3q^{45} - 3q^{46} - 13q^{49} + 4q^{50} + 10q^{52} + 9q^{53} - 9q^{54} - 10q^{55} - 3q^{56} + 3q^{57} - 2q^{58} + 3q^{60} + 14q^{61} + 15q^{63} + 14q^{64} - 5q^{65} - 4q^{67} - 6q^{68} - 9q^{69} + 5q^{70} - 24q^{71} - 9q^{72} - 3q^{73} + 6q^{74} + 12q^{75} + q^{76} + 5q^{77} - 15q^{78} - 8q^{79} - q^{80} - 9q^{81} + 5q^{82} + 9q^{83} + 3q^{84} + 3q^{85} - 2q^{86} - 3q^{87} - 30q^{88} + 13q^{89} - 3q^{90} + 25q^{91} + 3q^{92} + q^{95} + 15q^{96} + 9q^{97} + 2q^{98} + 15q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i −1.50000 + 0.866025i 0.500000 + 0.866025i −1.00000 1.73205i 0.500000 + 2.59808i −3.00000 1.50000 2.59808i 0.500000 0.866025i
16.1 −0.500000 0.866025i −1.50000 0.866025i 0.500000 0.866025i −1.00000 1.73205i 0.500000 2.59808i −3.00000 1.50000 + 2.59808i 0.500000 + 0.866025i
\(n\): 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 63.2.g.a 2
3.b odd 2 1 189.2.g.a 2
4.b odd 2 1 1008.2.t.d 2
7.b odd 2 1 441.2.g.a 2
7.c even 3 1 63.2.h.a yes 2
7.c even 3 1 441.2.f.b 2
7.d odd 6 1 441.2.f.a 2
7.d odd 6 1 441.2.h.a 2
9.c even 3 1 63.2.h.a yes 2
9.c even 3 1 567.2.e.a 2
9.d odd 6 1 189.2.h.a 2
9.d odd 6 1 567.2.e.b 2
12.b even 2 1 3024.2.t.d 2
21.c even 2 1 1323.2.g.a 2
21.g even 6 1 1323.2.f.b 2
21.g even 6 1 1323.2.h.a 2
21.h odd 6 1 189.2.h.a 2
21.h odd 6 1 1323.2.f.a 2
28.g odd 6 1 1008.2.q.c 2
36.f odd 6 1 1008.2.q.c 2
36.h even 6 1 3024.2.q.b 2
63.g even 3 1 inner 63.2.g.a 2
63.g even 3 1 3969.2.a.d 1
63.h even 3 1 441.2.f.b 2
63.h even 3 1 567.2.e.a 2
63.i even 6 1 1323.2.f.b 2
63.j odd 6 1 567.2.e.b 2
63.j odd 6 1 1323.2.f.a 2
63.k odd 6 1 441.2.g.a 2
63.k odd 6 1 3969.2.a.f 1
63.l odd 6 1 441.2.h.a 2
63.n odd 6 1 189.2.g.a 2
63.n odd 6 1 3969.2.a.c 1
63.o even 6 1 1323.2.h.a 2
63.s even 6 1 1323.2.g.a 2
63.s even 6 1 3969.2.a.a 1
63.t odd 6 1 441.2.f.a 2
84.n even 6 1 3024.2.q.b 2
252.o even 6 1 3024.2.t.d 2
252.bl odd 6 1 1008.2.t.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a 2 1.a even 1 1 trivial
63.2.g.a 2 63.g even 3 1 inner
63.2.h.a yes 2 7.c even 3 1
63.2.h.a yes 2 9.c even 3 1
189.2.g.a 2 3.b odd 2 1
189.2.g.a 2 63.n odd 6 1
189.2.h.a 2 9.d odd 6 1
189.2.h.a 2 21.h odd 6 1
441.2.f.a 2 7.d odd 6 1
441.2.f.a 2 63.t odd 6 1
441.2.f.b 2 7.c even 3 1
441.2.f.b 2 63.h even 3 1
441.2.g.a 2 7.b odd 2 1
441.2.g.a 2 63.k odd 6 1
441.2.h.a 2 7.d odd 6 1
441.2.h.a 2 63.l odd 6 1
567.2.e.a 2 9.c even 3 1
567.2.e.a 2 63.h even 3 1
567.2.e.b 2 9.d odd 6 1
567.2.e.b 2 63.j odd 6 1
1008.2.q.c 2 28.g odd 6 1
1008.2.q.c 2 36.f odd 6 1
1008.2.t.d 2 4.b odd 2 1
1008.2.t.d 2 252.bl odd 6 1
1323.2.f.a 2 21.h odd 6 1
1323.2.f.a 2 63.j odd 6 1
1323.2.f.b 2 21.g even 6 1
1323.2.f.b 2 63.i even 6 1
1323.2.g.a 2 21.c even 2 1
1323.2.g.a 2 63.s even 6 1
1323.2.h.a 2 21.g even 6 1
1323.2.h.a 2 63.o even 6 1
3024.2.q.b 2 36.h even 6 1
3024.2.q.b 2 84.n even 6 1
3024.2.t.d 2 12.b even 2 1
3024.2.t.d 2 252.o even 6 1
3969.2.a.a 1 63.s even 6 1
3969.2.a.c 1 63.n odd 6 1
3969.2.a.d 1 63.g even 3 1
3969.2.a.f 1 63.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(63, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \)
$3$ \( 1 + 3 T + 3 T^{2} \)
$5$ \( ( 1 + T + 5 T^{2} )^{2} \)
$7$ \( 1 - T + 7 T^{2} \)
$11$ \( ( 1 - 5 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( ( 1 - 3 T + 23 T^{2} )^{2} \)
$29$ \( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \)
$31$ \( 1 - 31 T^{2} + 961 T^{4} \)
$37$ \( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4} \)
$43$ \( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \)
$67$ \( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 3 T - 64 T^{2} + 219 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 13 T + 80 T^{2} - 1157 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 9 T - 16 T^{2} - 873 T^{3} + 9409 T^{4} \)
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