Properties

Label 252.4.a.a
Level $252$
Weight $4$
Character orbit 252.a
Self dual yes
Analytic conductor $14.868$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 14q^{5} - 7q^{7} + O(q^{10}) \) \( q - 14q^{5} - 7q^{7} - 4q^{11} + 54q^{13} + 14q^{17} + 92q^{19} + 152q^{23} + 71q^{25} + 106q^{29} - 144q^{31} + 98q^{35} + 158q^{37} + 390q^{41} - 508q^{43} + 528q^{47} + 49q^{49} - 606q^{53} + 56q^{55} + 364q^{59} + 678q^{61} - 756q^{65} + 844q^{67} + 8q^{71} - 422q^{73} + 28q^{77} + 384q^{79} + 548q^{83} - 196q^{85} - 1194q^{89} - 378q^{91} - 1288q^{95} - 1502q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 −14.0000 0 −7.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.a.a 1
3.b odd 2 1 84.4.a.b 1
4.b odd 2 1 1008.4.a.d 1
7.b odd 2 1 1764.4.a.l 1
7.c even 3 2 1764.4.k.n 2
7.d odd 6 2 1764.4.k.c 2
12.b even 2 1 336.4.a.e 1
15.d odd 2 1 2100.4.a.g 1
15.e even 4 2 2100.4.k.g 2
21.c even 2 1 588.4.a.a 1
21.g even 6 2 588.4.i.h 2
21.h odd 6 2 588.4.i.a 2
24.f even 2 1 1344.4.a.p 1
24.h odd 2 1 1344.4.a.b 1
84.h odd 2 1 2352.4.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.b 1 3.b odd 2 1
252.4.a.a 1 1.a even 1 1 trivial
336.4.a.e 1 12.b even 2 1
588.4.a.a 1 21.c even 2 1
588.4.i.a 2 21.h odd 6 2
588.4.i.h 2 21.g even 6 2
1008.4.a.d 1 4.b odd 2 1
1344.4.a.b 1 24.h odd 2 1
1344.4.a.p 1 24.f even 2 1
1764.4.a.l 1 7.b odd 2 1
1764.4.k.c 2 7.d odd 6 2
1764.4.k.n 2 7.c even 3 2
2100.4.a.g 1 15.d odd 2 1
2100.4.k.g 2 15.e even 4 2
2352.4.a.v 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(252))\):

\( T_{5} + 14 \)
\( T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 14 + T \)
$7$ \( 7 + T \)
$11$ \( 4 + T \)
$13$ \( -54 + T \)
$17$ \( -14 + T \)
$19$ \( -92 + T \)
$23$ \( -152 + T \)
$29$ \( -106 + T \)
$31$ \( 144 + T \)
$37$ \( -158 + T \)
$41$ \( -390 + T \)
$43$ \( 508 + T \)
$47$ \( -528 + T \)
$53$ \( 606 + T \)
$59$ \( -364 + T \)
$61$ \( -678 + T \)
$67$ \( -844 + T \)
$71$ \( -8 + T \)
$73$ \( 422 + T \)
$79$ \( -384 + T \)
$83$ \( -548 + T \)
$89$ \( 1194 + T \)
$97$ \( 1502 + T \)
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