Properties

Label 63.2.h.a
Level $63$
Weight $2$
Character orbit 63.h
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.h (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: \(\Z[a_1, \ldots, a_{5}]\)
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.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 1.73205i −1.00000 0.500000 0.866025i 1.73205i 2.00000 1.73205i −3.00000 −3.00000 0.500000 0.866025i
58.1 1.00000 1.73205i −1.00000 0.500000 + 0.866025i 1.73205i 2.00000 + 1.73205i −3.00000 −3.00000 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.h 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.h.a yes 2
3.b odd 2 1 189.2.h.a 2
4.b odd 2 1 1008.2.q.c 2
7.b odd 2 1 441.2.h.a 2
7.c even 3 1 63.2.g.a 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.g.a 2
9.c even 3 1 63.2.g.a 2
9.c even 3 1 567.2.e.a 2
9.d odd 6 1 189.2.g.a 2
9.d odd 6 1 567.2.e.b 2
12.b even 2 1 3024.2.q.b 2
21.c even 2 1 1323.2.h.a 2
21.g even 6 1 1323.2.f.b 2
21.g even 6 1 1323.2.g.a 2
21.h odd 6 1 189.2.g.a 2
21.h odd 6 1 1323.2.f.a 2
28.g odd 6 1 1008.2.t.d 2
36.f odd 6 1 1008.2.t.d 2
36.h even 6 1 3024.2.t.d 2
63.g even 3 1 441.2.f.b 2
63.g even 3 1 567.2.e.a 2
63.h even 3 1 inner 63.2.h.a yes 2
63.h even 3 1 3969.2.a.d 1
63.i even 6 1 1323.2.h.a 2
63.i even 6 1 3969.2.a.a 1
63.j odd 6 1 189.2.h.a 2
63.j odd 6 1 3969.2.a.c 1
63.k odd 6 1 441.2.f.a 2
63.l odd 6 1 441.2.g.a 2
63.n odd 6 1 567.2.e.b 2
63.n odd 6 1 1323.2.f.a 2
63.o even 6 1 1323.2.g.a 2
63.s even 6 1 1323.2.f.b 2
63.t odd 6 1 441.2.h.a 2
63.t odd 6 1 3969.2.a.f 1
84.n even 6 1 3024.2.t.d 2
252.u odd 6 1 1008.2.q.c 2
252.bb even 6 1 3024.2.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a 2 7.c even 3 1
63.2.g.a 2 9.c even 3 1
63.2.h.a yes 2 1.a even 1 1 trivial
63.2.h.a yes 2 63.h even 3 1 inner
189.2.g.a 2 9.d odd 6 1
189.2.g.a 2 21.h odd 6 1
189.2.h.a 2 3.b odd 2 1
189.2.h.a 2 63.j odd 6 1
441.2.f.a 2 7.d odd 6 1
441.2.f.a 2 63.k odd 6 1
441.2.f.b 2 7.c even 3 1
441.2.f.b 2 63.g even 3 1
441.2.g.a 2 7.d odd 6 1
441.2.g.a 2 63.l odd 6 1
441.2.h.a 2 7.b odd 2 1
441.2.h.a 2 63.t odd 6 1
567.2.e.a 2 9.c even 3 1
567.2.e.a 2 63.g even 3 1
567.2.e.b 2 9.d odd 6 1
567.2.e.b 2 63.n odd 6 1
1008.2.q.c 2 4.b odd 2 1
1008.2.q.c 2 252.u odd 6 1
1008.2.t.d 2 28.g odd 6 1
1008.2.t.d 2 36.f odd 6 1
1323.2.f.a 2 21.h odd 6 1
1323.2.f.a 2 63.n odd 6 1
1323.2.f.b 2 21.g even 6 1
1323.2.f.b 2 63.s even 6 1
1323.2.g.a 2 21.g even 6 1
1323.2.g.a 2 63.o even 6 1
1323.2.h.a 2 21.c even 2 1
1323.2.h.a 2 63.i even 6 1
3024.2.q.b 2 12.b even 2 1
3024.2.q.b 2 252.bb even 6 1
3024.2.t.d 2 36.h even 6 1
3024.2.t.d 2 84.n even 6 1
3969.2.a.a 1 63.i even 6 1
3969.2.a.c 1 63.j odd 6 1
3969.2.a.d 1 63.h even 3 1
3969.2.a.f 1 63.t odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + 2 T^{2} )^{2} \)
$3$ \( 1 + 3 T^{2} \)
$5$ \( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \)
$7$ \( 1 - 4 T + 7 T^{2} \)
$11$ \( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \)
$29$ \( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4} \)
$43$ \( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 14 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 4 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 3 T - 64 T^{2} + 219 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 13 T + 80 T^{2} - 1157 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 9 T - 16 T^{2} - 873 T^{3} + 9409 T^{4} \)
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