Properties

Label 252.2.k.a
Level 252
Weight 2
Character orbit 252.k
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.k (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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
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.

\(f(q)\) \(=\) \( q + ( -2 + 2 \zeta_{6} ) q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -2 + 2 \zeta_{6} ) q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{11} -3 q^{13} + 8 \zeta_{6} q^{17} + ( 1 - \zeta_{6} ) q^{19} + ( 8 - 8 \zeta_{6} ) q^{23} + \zeta_{6} q^{25} -4 q^{29} -3 \zeta_{6} q^{31} + ( -4 - 2 \zeta_{6} ) q^{35} + ( 1 - \zeta_{6} ) q^{37} -6 q^{41} + 11 q^{43} + ( 6 - 6 \zeta_{6} ) q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} -12 \zeta_{6} q^{53} -4 q^{55} + 4 \zeta_{6} q^{59} + ( 6 - 6 \zeta_{6} ) q^{61} + ( 6 - 6 \zeta_{6} ) q^{65} -13 \zeta_{6} q^{67} + 10 q^{71} + 11 \zeta_{6} q^{73} + ( -6 + 4 \zeta_{6} ) q^{77} + ( 3 - 3 \zeta_{6} ) q^{79} -2 q^{83} -16 q^{85} + ( 3 - 9 \zeta_{6} ) q^{91} + 2 \zeta_{6} q^{95} + 10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + q^{7} + O(q^{10}) \) \( 2q - 2q^{5} + q^{7} + 2q^{11} - 6q^{13} + 8q^{17} + q^{19} + 8q^{23} + q^{25} - 8q^{29} - 3q^{31} - 10q^{35} + q^{37} - 12q^{41} + 22q^{43} + 6q^{47} - 13q^{49} - 12q^{53} - 8q^{55} + 4q^{59} + 6q^{61} + 6q^{65} - 13q^{67} + 20q^{71} + 11q^{73} - 8q^{77} + 3q^{79} - 4q^{83} - 32q^{85} - 3q^{91} + 2q^{95} + 20q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −1.00000 + 1.73205i 0 0.500000 + 2.59808i 0 0 0
109.1 0 0 0 −1.00000 1.73205i 0 0.500000 2.59808i 0 0 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
7.c 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.k.a 2
3.b odd 2 1 84.2.i.a 2
4.b odd 2 1 1008.2.s.c 2
7.b odd 2 1 1764.2.k.j 2
7.c even 3 1 inner 252.2.k.a 2
7.c even 3 1 1764.2.a.h 1
7.d odd 6 1 1764.2.a.c 1
7.d odd 6 1 1764.2.k.j 2
9.c even 3 1 2268.2.i.b 2
9.c even 3 1 2268.2.l.g 2
9.d odd 6 1 2268.2.i.g 2
9.d odd 6 1 2268.2.l.b 2
12.b even 2 1 336.2.q.c 2
15.d odd 2 1 2100.2.q.b 2
15.e even 4 2 2100.2.bc.a 4
21.c even 2 1 588.2.i.b 2
21.g even 6 1 588.2.a.f 1
21.g even 6 1 588.2.i.b 2
21.h odd 6 1 84.2.i.a 2
21.h odd 6 1 588.2.a.a 1
24.f even 2 1 1344.2.q.n 2
24.h odd 2 1 1344.2.q.b 2
28.f even 6 1 7056.2.a.o 1
28.g odd 6 1 1008.2.s.c 2
28.g odd 6 1 7056.2.a.bs 1
63.g even 3 1 2268.2.i.b 2
63.h even 3 1 2268.2.l.g 2
63.j odd 6 1 2268.2.l.b 2
63.n odd 6 1 2268.2.i.g 2
84.h odd 2 1 2352.2.q.q 2
84.j odd 6 1 2352.2.a.k 1
84.j odd 6 1 2352.2.q.q 2
84.n even 6 1 336.2.q.c 2
84.n even 6 1 2352.2.a.o 1
105.o odd 6 1 2100.2.q.b 2
105.x even 12 2 2100.2.bc.a 4
168.s odd 6 1 1344.2.q.b 2
168.s odd 6 1 9408.2.a.cx 1
168.v even 6 1 1344.2.q.n 2
168.v even 6 1 9408.2.a.bi 1
168.ba even 6 1 9408.2.a.i 1
168.be odd 6 1 9408.2.a.bx 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.i.a 2 3.b odd 2 1
84.2.i.a 2 21.h odd 6 1
252.2.k.a 2 1.a even 1 1 trivial
252.2.k.a 2 7.c even 3 1 inner
336.2.q.c 2 12.b even 2 1
336.2.q.c 2 84.n even 6 1
588.2.a.a 1 21.h odd 6 1
588.2.a.f 1 21.g even 6 1
588.2.i.b 2 21.c even 2 1
588.2.i.b 2 21.g even 6 1
1008.2.s.c 2 4.b odd 2 1
1008.2.s.c 2 28.g odd 6 1
1344.2.q.b 2 24.h odd 2 1
1344.2.q.b 2 168.s odd 6 1
1344.2.q.n 2 24.f even 2 1
1344.2.q.n 2 168.v even 6 1
1764.2.a.c 1 7.d odd 6 1
1764.2.a.h 1 7.c even 3 1
1764.2.k.j 2 7.b odd 2 1
1764.2.k.j 2 7.d odd 6 1
2100.2.q.b 2 15.d odd 2 1
2100.2.q.b 2 105.o odd 6 1
2100.2.bc.a 4 15.e even 4 2
2100.2.bc.a 4 105.x even 12 2
2268.2.i.b 2 9.c even 3 1
2268.2.i.b 2 63.g even 3 1
2268.2.i.g 2 9.d odd 6 1
2268.2.i.g 2 63.n odd 6 1
2268.2.l.b 2 9.d odd 6 1
2268.2.l.b 2 63.j odd 6 1
2268.2.l.g 2 9.c even 3 1
2268.2.l.g 2 63.h even 3 1
2352.2.a.k 1 84.j odd 6 1
2352.2.a.o 1 84.n even 6 1
2352.2.q.q 2 84.h odd 2 1
2352.2.q.q 2 84.j odd 6 1
7056.2.a.o 1 28.f even 6 1
7056.2.a.bs 1 28.g odd 6 1
9408.2.a.i 1 168.ba even 6 1
9408.2.a.bi 1 168.v even 6 1
9408.2.a.bx 1 168.be odd 6 1
9408.2.a.cx 1 168.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 2 T_{5} + 4 \) acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ \( 1 - T + 7 T^{2} \)
$11$ \( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 3 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 4 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 11 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 10 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 2 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 89 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 - 10 T + 97 T^{2} )^{2} \)
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