Properties

Label 252.2.l.a
Level 252
Weight 2
Character orbit 252.l
Analytic conductor 2.012
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 252.l (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.01223013094\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + 2 q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + 2 q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} -3 q^{9} + 4 q^{11} -3 \zeta_{6} q^{13} + ( 2 - 4 \zeta_{6} ) q^{15} -7 \zeta_{6} q^{17} + ( -5 + 5 \zeta_{6} ) q^{19} + ( 5 - 4 \zeta_{6} ) q^{21} + 4 q^{23} - q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 1 - \zeta_{6} ) q^{29} + ( 3 - 3 \zeta_{6} ) q^{31} + ( 4 - 8 \zeta_{6} ) q^{33} + ( 2 + 4 \zeta_{6} ) q^{35} + ( -11 + 11 \zeta_{6} ) q^{37} + ( -6 + 3 \zeta_{6} ) q^{39} + 9 \zeta_{6} q^{41} + ( -5 + 5 \zeta_{6} ) q^{43} -6 q^{45} -3 \zeta_{6} q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( -14 + 7 \zeta_{6} ) q^{51} -3 \zeta_{6} q^{53} + 8 q^{55} + ( 5 + 5 \zeta_{6} ) q^{57} + ( 7 - 7 \zeta_{6} ) q^{59} -3 \zeta_{6} q^{61} + ( -3 - 6 \zeta_{6} ) q^{63} -6 \zeta_{6} q^{65} + ( -13 + 13 \zeta_{6} ) q^{67} + ( 4 - 8 \zeta_{6} ) q^{69} -8 q^{71} -7 \zeta_{6} q^{73} + ( -1 + 2 \zeta_{6} ) q^{75} + ( 4 + 8 \zeta_{6} ) q^{77} + 9 \zeta_{6} q^{79} + 9 q^{81} + ( -1 + \zeta_{6} ) q^{83} -14 \zeta_{6} q^{85} + ( -1 - \zeta_{6} ) q^{87} + ( -15 + 15 \zeta_{6} ) q^{89} + ( 6 - 9 \zeta_{6} ) q^{91} + ( -3 - 3 \zeta_{6} ) q^{93} + ( -10 + 10 \zeta_{6} ) q^{95} + ( 17 - 17 \zeta_{6} ) q^{97} -12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} + 4q^{7} - 6q^{9} + O(q^{10}) \) \( 2q + 4q^{5} + 4q^{7} - 6q^{9} + 8q^{11} - 3q^{13} - 7q^{17} - 5q^{19} + 6q^{21} + 8q^{23} - 2q^{25} + q^{29} + 3q^{31} + 8q^{35} - 11q^{37} - 9q^{39} + 9q^{41} - 5q^{43} - 12q^{45} - 3q^{47} + 2q^{49} - 21q^{51} - 3q^{53} + 16q^{55} + 15q^{57} + 7q^{59} - 3q^{61} - 12q^{63} - 6q^{65} - 13q^{67} - 16q^{71} - 7q^{73} + 16q^{77} + 9q^{79} + 18q^{81} - q^{83} - 14q^{85} - 3q^{87} - 15q^{89} + 3q^{91} - 9q^{93} - 10q^{95} + 17q^{97} - 24q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-\zeta_{6}\) \(-1 + \zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.73205i 0 2.00000 0 2.00000 1.73205i 0 −3.00000 0
205.1 0 1.73205i 0 2.00000 0 2.00000 + 1.73205i 0 −3.00000 0
\(n\): 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 252.2.l.a yes 2
3.b odd 2 1 756.2.l.a 2
4.b odd 2 1 1008.2.t.b 2
7.b odd 2 1 1764.2.l.b 2
7.c even 3 1 252.2.i.a 2
7.c even 3 1 1764.2.j.c 2
7.d odd 6 1 1764.2.i.b 2
7.d odd 6 1 1764.2.j.a 2
9.c even 3 1 252.2.i.a 2
9.c even 3 1 2268.2.k.a 2
9.d odd 6 1 756.2.i.a 2
9.d odd 6 1 2268.2.k.b 2
12.b even 2 1 3024.2.t.b 2
21.c even 2 1 5292.2.l.b 2
21.g even 6 1 5292.2.i.b 2
21.g even 6 1 5292.2.j.b 2
21.h odd 6 1 756.2.i.a 2
21.h odd 6 1 5292.2.j.c 2
28.g odd 6 1 1008.2.q.f 2
36.f odd 6 1 1008.2.q.f 2
36.h even 6 1 3024.2.q.e 2
63.g even 3 1 inner 252.2.l.a yes 2
63.h even 3 1 1764.2.j.c 2
63.h even 3 1 2268.2.k.a 2
63.i even 6 1 5292.2.j.b 2
63.j odd 6 1 2268.2.k.b 2
63.j odd 6 1 5292.2.j.c 2
63.k odd 6 1 1764.2.l.b 2
63.l odd 6 1 1764.2.i.b 2
63.n odd 6 1 756.2.l.a 2
63.o even 6 1 5292.2.i.b 2
63.s even 6 1 5292.2.l.b 2
63.t odd 6 1 1764.2.j.a 2
84.n even 6 1 3024.2.q.e 2
252.o even 6 1 3024.2.t.b 2
252.bl odd 6 1 1008.2.t.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.a 2 7.c even 3 1
252.2.i.a 2 9.c even 3 1
252.2.l.a yes 2 1.a even 1 1 trivial
252.2.l.a yes 2 63.g even 3 1 inner
756.2.i.a 2 9.d odd 6 1
756.2.i.a 2 21.h odd 6 1
756.2.l.a 2 3.b odd 2 1
756.2.l.a 2 63.n odd 6 1
1008.2.q.f 2 28.g odd 6 1
1008.2.q.f 2 36.f odd 6 1
1008.2.t.b 2 4.b odd 2 1
1008.2.t.b 2 252.bl odd 6 1
1764.2.i.b 2 7.d odd 6 1
1764.2.i.b 2 63.l odd 6 1
1764.2.j.a 2 7.d odd 6 1
1764.2.j.a 2 63.t odd 6 1
1764.2.j.c 2 7.c even 3 1
1764.2.j.c 2 63.h even 3 1
1764.2.l.b 2 7.b odd 2 1
1764.2.l.b 2 63.k odd 6 1
2268.2.k.a 2 9.c even 3 1
2268.2.k.a 2 63.h even 3 1
2268.2.k.b 2 9.d odd 6 1
2268.2.k.b 2 63.j odd 6 1
3024.2.q.e 2 36.h even 6 1
3024.2.q.e 2 84.n even 6 1
3024.2.t.b 2 12.b even 2 1
3024.2.t.b 2 252.o even 6 1
5292.2.i.b 2 21.g even 6 1
5292.2.i.b 2 63.o even 6 1
5292.2.j.b 2 21.g even 6 1
5292.2.j.b 2 63.i even 6 1
5292.2.j.c 2 21.h odd 6 1
5292.2.j.c 2 63.j odd 6 1
5292.2.l.b 2 21.c even 2 1
5292.2.l.b 2 63.s even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 2 \) acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 3 T^{2} \)
$5$ \( ( 1 - 2 T + 5 T^{2} )^{2} \)
$7$ \( 1 - 4 T + 7 T^{2} \)
$11$ \( ( 1 - 4 T + 11 T^{2} )^{2} \)
$13$ \( 1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4} \)
$17$ \( 1 + 7 T + 32 T^{2} + 119 T^{3} + 289 T^{4} \)
$19$ \( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \)
$23$ \( ( 1 - 4 T + 23 T^{2} )^{2} \)
$29$ \( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \)
$31$ \( 1 - 3 T - 22 T^{2} - 93 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \)
$41$ \( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \)
$47$ \( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 7 T - 10 T^{2} - 413 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 3 T - 52 T^{2} + 183 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 8 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \)
$79$ \( 1 - 9 T + 2 T^{2} - 711 T^{3} + 6241 T^{4} \)
$83$ \( 1 + T - 82 T^{2} + 83 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4} \)
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