Properties

Label 252.2.b.c
Level $252$
Weight $2$
Character orbit 252.b
Analytic conductor $2.012$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.01223013094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
Defining polynomial: \(x^{4} - 3 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + 2 \beta_{2} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + 2 \beta_{2} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} -2 \beta_{3} q^{11} + ( -\beta_{1} + 2 \beta_{3} ) q^{14} + ( -1 + 3 \beta_{2} ) q^{16} + ( 4 - 4 \beta_{2} ) q^{22} -4 \beta_{3} q^{23} + 5 q^{25} + ( -5 + 3 \beta_{2} ) q^{28} + ( -8 \beta_{1} + 2 \beta_{3} ) q^{29} + ( -\beta_{1} + 3 \beta_{3} ) q^{32} -6 q^{37} + ( 2 - 4 \beta_{2} ) q^{43} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{44} + ( 8 - 8 \beta_{2} ) q^{46} -7 q^{49} + 5 \beta_{1} q^{50} + ( 8 \beta_{1} - 2 \beta_{3} ) q^{53} + ( -5 \beta_{1} + 3 \beta_{3} ) q^{56} + ( -12 - 4 \beta_{2} ) q^{58} + ( -7 + 5 \beta_{2} ) q^{64} + ( 6 - 12 \beta_{2} ) q^{67} + 8 \beta_{3} q^{71} -6 \beta_{1} q^{74} + ( 8 \beta_{1} - 2 \beta_{3} ) q^{77} + ( -6 + 12 \beta_{2} ) q^{79} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{86} + ( 12 - 4 \beta_{2} ) q^{88} + ( 8 \beta_{1} - 8 \beta_{3} ) q^{92} -7 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{4} + O(q^{10}) \) \( 4 q + 6 q^{4} + 2 q^{16} + 8 q^{22} + 20 q^{25} - 14 q^{28} - 24 q^{37} + 16 q^{46} - 28 q^{49} - 56 q^{58} - 18 q^{64} + 40 q^{88} + O(q^{100}) \)

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

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

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 0.500000i 0 1.50000 + 1.32288i 0 0 2.64575i −1.32288 2.50000i 0 0
55.2 −1.32288 + 0.500000i 0 1.50000 1.32288i 0 0 2.64575i −1.32288 + 2.50000i 0 0
55.3 1.32288 0.500000i 0 1.50000 1.32288i 0 0 2.64575i 1.32288 2.50000i 0 0
55.4 1.32288 + 0.500000i 0 1.50000 + 1.32288i 0 0 2.64575i 1.32288 + 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 252.2.b.c 4
3.b odd 2 1 inner 252.2.b.c 4
4.b odd 2 1 inner 252.2.b.c 4
7.b odd 2 1 CM 252.2.b.c 4
8.b even 2 1 4032.2.b.m 4
8.d odd 2 1 4032.2.b.m 4
12.b even 2 1 inner 252.2.b.c 4
21.c even 2 1 inner 252.2.b.c 4
24.f even 2 1 4032.2.b.m 4
24.h odd 2 1 4032.2.b.m 4
28.d even 2 1 inner 252.2.b.c 4
56.e even 2 1 4032.2.b.m 4
56.h odd 2 1 4032.2.b.m 4
84.h odd 2 1 inner 252.2.b.c 4
168.e odd 2 1 4032.2.b.m 4
168.i even 2 1 4032.2.b.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.b.c 4 1.a even 1 1 trivial
252.2.b.c 4 3.b odd 2 1 inner
252.2.b.c 4 4.b odd 2 1 inner
252.2.b.c 4 7.b odd 2 1 CM
252.2.b.c 4 12.b even 2 1 inner
252.2.b.c 4 21.c even 2 1 inner
252.2.b.c 4 28.d even 2 1 inner
252.2.b.c 4 84.h odd 2 1 inner
4032.2.b.m 4 8.b even 2 1
4032.2.b.m 4 8.d odd 2 1
4032.2.b.m 4 24.f even 2 1
4032.2.b.m 4 24.h odd 2 1
4032.2.b.m 4 56.e even 2 1
4032.2.b.m 4 56.h odd 2 1
4032.2.b.m 4 168.e odd 2 1
4032.2.b.m 4 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\):

\( T_{5} \)
\( T_{11}^{2} + 16 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 3 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 7 + T^{2} )^{2} \)
$11$ \( ( 16 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( 64 + T^{2} )^{2} \)
$29$ \( ( -112 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( 6 + T )^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 28 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( ( -112 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( 252 + T^{2} )^{2} \)
$71$ \( ( 256 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( ( 252 + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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