Properties

Label 252.4.k.b
Level $252$
Weight $4$
Character orbit 252.k
Analytic conductor $14.868$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + 19 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -1 + 19 \zeta_{6} ) q^{7} -19 q^{13} + ( -107 + 107 \zeta_{6} ) q^{19} + 125 \zeta_{6} q^{25} + 289 \zeta_{6} q^{31} + ( -323 + 323 \zeta_{6} ) q^{37} + 71 q^{43} + ( -360 + 323 \zeta_{6} ) q^{49} + ( -182 + 182 \zeta_{6} ) q^{61} + 127 \zeta_{6} q^{67} + 271 \zeta_{6} q^{73} + ( 1387 - 1387 \zeta_{6} ) q^{79} + ( 19 - 361 \zeta_{6} ) q^{91} -1330 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 17q^{7} + O(q^{10}) \) \( 2q + 17q^{7} - 38q^{13} - 107q^{19} + 125q^{25} + 289q^{31} - 323q^{37} + 142q^{43} - 397q^{49} - 182q^{61} + 127q^{67} + 271q^{73} + 1387q^{79} - 323q^{91} - 2660q^{97} + O(q^{100}) \)

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 0 0 8.50000 + 16.4545i 0 0 0
109.1 0 0 0 0 0 8.50000 16.4545i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.k.b 2
3.b odd 2 1 CM 252.4.k.b 2
7.b odd 2 1 1764.4.k.i 2
7.c even 3 1 inner 252.4.k.b 2
7.c even 3 1 1764.4.a.f 1
7.d odd 6 1 1764.4.a.g 1
7.d odd 6 1 1764.4.k.i 2
21.c even 2 1 1764.4.k.i 2
21.g even 6 1 1764.4.a.g 1
21.g even 6 1 1764.4.k.i 2
21.h odd 6 1 inner 252.4.k.b 2
21.h odd 6 1 1764.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.k.b 2 1.a even 1 1 trivial
252.4.k.b 2 3.b odd 2 1 CM
252.4.k.b 2 7.c even 3 1 inner
252.4.k.b 2 21.h odd 6 1 inner
1764.4.a.f 1 7.c even 3 1
1764.4.a.f 1 21.h odd 6 1
1764.4.a.g 1 7.d odd 6 1
1764.4.a.g 1 21.g even 6 1
1764.4.k.i 2 7.b odd 2 1
1764.4.k.i 2 7.d odd 6 1
1764.4.k.i 2 21.c even 2 1
1764.4.k.i 2 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(252, [\chi])\):

\( T_{5} \)
\( T_{13} + 19 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 343 - 17 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 19 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 11449 + 107 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 83521 - 289 T + T^{2} \)
$37$ \( 104329 + 323 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -71 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 33124 + 182 T + T^{2} \)
$67$ \( 16129 - 127 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 73441 - 271 T + T^{2} \)
$79$ \( 1923769 - 1387 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 1330 + T )^{2} \)
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