Properties

Label 252.4.a.f
Level $252$
Weight $4$
Character orbit 252.a
Self dual yes
Analytic conductor $14.868$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} + 7 q^{7} +O(q^{10})\) \( q + \beta q^{5} + 7 q^{7} + \beta q^{11} + 26 q^{13} -5 \beta q^{17} + 68 q^{19} + 3 \beta q^{23} + 127 q^{25} -16 \beta q^{29} + 212 q^{31} + 7 \beta q^{35} + 218 q^{37} + 25 \beta q^{41} + 260 q^{43} + 26 \beta q^{47} + 49 q^{49} -30 \beta q^{53} + 252 q^{55} -18 \beta q^{59} -322 q^{61} + 26 \beta q^{65} + 356 q^{67} -71 \beta q^{71} -226 q^{73} + 7 \beta q^{77} + 440 q^{79} -16 \beta q^{83} -1260 q^{85} + 13 \beta q^{89} + 182 q^{91} + 68 \beta q^{95} -1330 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 14q^{7} + O(q^{10}) \) \( 2q + 14q^{7} + 52q^{13} + 136q^{19} + 254q^{25} + 424q^{31} + 436q^{37} + 520q^{43} + 98q^{49} + 504q^{55} - 644q^{61} + 712q^{67} - 452q^{73} + 880q^{79} - 2520q^{85} + 364q^{91} - 2660q^{97} + O(q^{100}) \)

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
−2.64575
2.64575
0 0 0 −15.8745 0 7.00000 0 0 0
1.2 0 0 0 15.8745 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.a.f 2
3.b odd 2 1 inner 252.4.a.f 2
4.b odd 2 1 1008.4.a.bc 2
7.b odd 2 1 1764.4.a.s 2
7.c even 3 2 1764.4.k.v 4
7.d odd 6 2 1764.4.k.u 4
12.b even 2 1 1008.4.a.bc 2
21.c even 2 1 1764.4.a.s 2
21.g even 6 2 1764.4.k.u 4
21.h odd 6 2 1764.4.k.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.a.f 2 1.a even 1 1 trivial
252.4.a.f 2 3.b odd 2 1 inner
1008.4.a.bc 2 4.b odd 2 1
1008.4.a.bc 2 12.b even 2 1
1764.4.a.s 2 7.b odd 2 1
1764.4.a.s 2 21.c even 2 1
1764.4.k.u 4 7.d odd 6 2
1764.4.k.u 4 21.g even 6 2
1764.4.k.v 4 7.c even 3 2
1764.4.k.v 4 21.h odd 6 2

Hecke kernels

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

\( T_{5}^{2} - 252 \)
\( T_{11}^{2} - 252 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -252 + T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( -252 + T^{2} \)
$13$ \( ( -26 + T )^{2} \)
$17$ \( -6300 + T^{2} \)
$19$ \( ( -68 + T )^{2} \)
$23$ \( -2268 + T^{2} \)
$29$ \( -64512 + T^{2} \)
$31$ \( ( -212 + T )^{2} \)
$37$ \( ( -218 + T )^{2} \)
$41$ \( -157500 + T^{2} \)
$43$ \( ( -260 + T )^{2} \)
$47$ \( -170352 + T^{2} \)
$53$ \( -226800 + T^{2} \)
$59$ \( -81648 + T^{2} \)
$61$ \( ( 322 + T )^{2} \)
$67$ \( ( -356 + T )^{2} \)
$71$ \( -1270332 + T^{2} \)
$73$ \( ( 226 + T )^{2} \)
$79$ \( ( -440 + T )^{2} \)
$83$ \( -64512 + T^{2} \)
$89$ \( -42588 + T^{2} \)
$97$ \( ( 1330 + T )^{2} \)
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