Properties

Label 252.4.b.c
Level $252$
Weight $4$
Character orbit 252.b
Analytic conductor $14.868$
Analytic rank $0$
Dimension $4$
CM discriminant -7
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: \(\Q(i, \sqrt{7})\)
Defining polynomial: \(x^{4} - 3 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} + ( -2 - 5 \beta_{2} ) q^{4} + ( -7 + 14 \beta_{2} ) q^{7} + ( 10 \beta_{1} + 7 \beta_{3} ) q^{8} +O(q^{10})\) \( q -\beta_{3} q^{2} + ( -2 - 5 \beta_{2} ) q^{4} + ( -7 + 14 \beta_{2} ) q^{7} + ( 10 \beta_{1} + 7 \beta_{3} ) q^{8} + ( -17 \beta_{1} + 17 \beta_{3} ) q^{11} + ( -28 \beta_{1} - 7 \beta_{3} ) q^{14} + ( -46 + 45 \beta_{2} ) q^{16} + ( 136 + 68 \beta_{2} ) q^{22} + ( -10 \beta_{1} + 10 \beta_{3} ) q^{23} + 125 q^{25} + ( 154 - 63 \beta_{2} ) q^{28} + ( 125 \beta_{1} + 75 \beta_{3} ) q^{29} + ( -90 \beta_{1} + \beta_{3} ) q^{32} + 450 q^{37} + ( -202 + 404 \beta_{2} ) q^{43} + ( -136 \beta_{1} - 204 \beta_{3} ) q^{44} + ( 80 + 40 \beta_{2} ) q^{46} -343 q^{49} -125 \beta_{3} q^{50} + ( 235 \beta_{1} + 141 \beta_{3} ) q^{53} + ( 126 \beta_{1} - 91 \beta_{3} ) q^{56} + ( -600 + 500 \beta_{2} ) q^{58} + ( 542 - 85 \beta_{2} ) q^{64} + ( 306 - 612 \beta_{2} ) q^{67} + ( -172 \beta_{1} + 172 \beta_{3} ) q^{71} -450 \beta_{3} q^{74} + ( 595 \beta_{1} + 357 \beta_{3} ) q^{77} + ( -90 + 180 \beta_{2} ) q^{79} + ( -808 \beta_{1} - 202 \beta_{3} ) q^{86} + ( 408 - 1156 \beta_{2} ) q^{88} + ( -80 \beta_{1} - 120 \beta_{3} ) q^{92} + 343 \beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 18q^{4} + O(q^{10}) \) \( 4q - 18q^{4} - 94q^{16} + 680q^{22} + 500q^{25} + 490q^{28} + 1800q^{37} + 400q^{46} - 1372q^{49} - 1400q^{58} + 1998q^{64} - 680q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 3 x^{2} + 4\):

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

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
1.32288 0.500000i
1.32288 + 0.500000i
−1.32288 0.500000i
−1.32288 + 0.500000i
−1.32288 2.50000i 0 −4.50000 + 6.61438i 0 0 18.5203i 22.4889 + 2.50000i 0 0
55.2 −1.32288 + 2.50000i 0 −4.50000 6.61438i 0 0 18.5203i 22.4889 2.50000i 0 0
55.3 1.32288 2.50000i 0 −4.50000 6.61438i 0 0 18.5203i −22.4889 + 2.50000i 0 0
55.4 1.32288 + 2.50000i 0 −4.50000 + 6.61438i 0 0 18.5203i −22.4889 2.50000i 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.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h 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.b.c 4
3.b odd 2 1 inner 252.4.b.c 4
4.b odd 2 1 inner 252.4.b.c 4
7.b odd 2 1 CM 252.4.b.c 4
12.b even 2 1 inner 252.4.b.c 4
21.c even 2 1 inner 252.4.b.c 4
28.d even 2 1 inner 252.4.b.c 4
84.h odd 2 1 inner 252.4.b.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.b.c 4 1.a even 1 1 trivial
252.4.b.c 4 3.b odd 2 1 inner
252.4.b.c 4 4.b odd 2 1 inner
252.4.b.c 4 7.b odd 2 1 CM
252.4.b.c 4 12.b even 2 1 inner
252.4.b.c 4 21.c even 2 1 inner
252.4.b.c 4 28.d even 2 1 inner
252.4.b.c 4 84.h odd 2 1 inner

Hecke kernels

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

\( T_{5} \)
\( T_{11}^{2} + 4624 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 64 + 9 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 343 + T^{2} )^{2} \)
$11$ \( ( 4624 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( 1600 + T^{2} )^{2} \)
$29$ \( ( -70000 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( -450 + T )^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 285628 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( ( -247408 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( 655452 + T^{2} )^{2} \)
$71$ \( ( 473344 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( ( 56700 + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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