Properties

Label 252.4.b.b
Level $252$
Weight $4$
Character orbit 252.b
Analytic conductor $14.868$
Analytic rank $0$
Dimension $4$
CM discriminant -84
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 \beta_{1} q^{2} -8 q^{4} + \beta_{2} q^{5} -7 \beta_{3} q^{7} + 16 \beta_{1} q^{8} +O(q^{10})\) \( q -2 \beta_{1} q^{2} -8 q^{4} + \beta_{2} q^{5} -7 \beta_{3} q^{7} + 16 \beta_{1} q^{8} + 4 \beta_{3} q^{10} -31 \beta_{1} q^{11} + 14 \beta_{2} q^{14} + 64 q^{16} + 37 \beta_{2} q^{17} + 58 \beta_{3} q^{19} -8 \beta_{2} q^{20} -124 q^{22} + 95 \beta_{1} q^{23} + 111 q^{25} + 56 \beta_{3} q^{28} + 76 \beta_{3} q^{31} -128 \beta_{1} q^{32} + 148 \beta_{3} q^{34} -49 \beta_{1} q^{35} -376 q^{37} -116 \beta_{2} q^{38} -32 \beta_{3} q^{40} + 109 \beta_{2} q^{41} + 248 \beta_{1} q^{44} + 380 q^{46} + 343 q^{49} -222 \beta_{1} q^{50} + 62 \beta_{3} q^{55} -112 \beta_{2} q^{56} -152 \beta_{2} q^{62} -512 q^{64} -296 \beta_{2} q^{68} -196 q^{70} -319 \beta_{1} q^{71} + 752 \beta_{1} q^{74} -464 \beta_{3} q^{76} + 217 \beta_{2} q^{77} + 64 \beta_{2} q^{80} + 436 \beta_{3} q^{82} -518 q^{85} + 992 q^{88} + 415 \beta_{2} q^{89} -760 \beta_{1} q^{92} + 406 \beta_{1} q^{95} -686 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 32q^{4} + O(q^{10}) \) \( 4q - 32q^{4} + 256q^{16} - 496q^{22} + 444q^{25} - 1504q^{37} + 1520q^{46} + 1372q^{49} - 2048q^{64} - 784q^{70} - 2072q^{85} + 3968q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 8 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 5 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 11 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 4\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{2} + 11 \beta_{1}\)\()/2\)

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
2.57794i
1.16372i
1.16372i
2.57794i
2.82843i 0 −8.00000 3.74166i 0 18.5203 22.6274i 0 −10.5830
55.2 2.82843i 0 −8.00000 3.74166i 0 −18.5203 22.6274i 0 10.5830
55.3 2.82843i 0 −8.00000 3.74166i 0 −18.5203 22.6274i 0 10.5830
55.4 2.82843i 0 −8.00000 3.74166i 0 18.5203 22.6274i 0 −10.5830
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.b.b 4
3.b odd 2 1 inner 252.4.b.b 4
4.b odd 2 1 inner 252.4.b.b 4
7.b odd 2 1 inner 252.4.b.b 4
12.b even 2 1 inner 252.4.b.b 4
21.c even 2 1 inner 252.4.b.b 4
28.d even 2 1 inner 252.4.b.b 4
84.h odd 2 1 CM 252.4.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.b.b 4 1.a even 1 1 trivial
252.4.b.b 4 3.b odd 2 1 inner
252.4.b.b 4 4.b odd 2 1 inner
252.4.b.b 4 7.b odd 2 1 inner
252.4.b.b 4 12.b even 2 1 inner
252.4.b.b 4 21.c even 2 1 inner
252.4.b.b 4 28.d even 2 1 inner
252.4.b.b 4 84.h odd 2 1 CM

Hecke kernels

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

\( T_{5}^{2} + 14 \)
\( T_{11}^{2} + 1922 \)
\( T_{19}^{2} - 23548 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 8 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( 14 + T^{2} )^{2} \)
$7$ \( ( -343 + T^{2} )^{2} \)
$11$ \( ( 1922 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( 19166 + T^{2} )^{2} \)
$19$ \( ( -23548 + T^{2} )^{2} \)
$23$ \( ( 18050 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( ( -40432 + T^{2} )^{2} \)
$37$ \( ( 376 + T )^{4} \)
$41$ \( ( 166334 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( ( 203522 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( 2411150 + T^{2} )^{2} \)
$97$ \( T^{4} \)
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