Properties

Label 252.2.b.b
Level 252
Weight 2
Character orbit 252.b
Analytic conductor 2.012
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: \( k \) = \( 2 \)
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: \(\Q(\sqrt{-2}, \sqrt{7})\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -2 q^{4} -\beta_{2} q^{5} + \beta_{3} q^{7} -2 \beta_{1} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} -2 q^{4} -\beta_{2} q^{5} + \beta_{3} q^{7} -2 \beta_{1} q^{8} + 2 \beta_{3} q^{10} -\beta_{1} q^{11} + \beta_{2} q^{14} + 4 q^{16} -\beta_{2} q^{17} + 2 \beta_{3} q^{19} + 2 \beta_{2} q^{20} + 2 q^{22} + 5 \beta_{1} q^{23} -9 q^{25} -2 \beta_{3} q^{28} -4 \beta_{3} q^{31} + 4 \beta_{1} q^{32} + 2 \beta_{3} q^{34} -7 \beta_{1} q^{35} + 8 q^{37} + 2 \beta_{2} q^{38} -4 \beta_{3} q^{40} -\beta_{2} q^{41} + 2 \beta_{1} q^{44} -10 q^{46} + 7 q^{49} -9 \beta_{1} q^{50} -2 \beta_{3} q^{55} -2 \beta_{2} q^{56} -4 \beta_{2} q^{62} -8 q^{64} + 2 \beta_{2} q^{68} + 14 q^{70} + 11 \beta_{1} q^{71} + 8 \beta_{1} q^{74} -4 \beta_{3} q^{76} -\beta_{2} q^{77} -4 \beta_{2} q^{80} + 2 \beta_{3} q^{82} -14 q^{85} -4 q^{88} + 5 \beta_{2} q^{89} -10 \beta_{1} q^{92} -14 \beta_{1} q^{95} + 7 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + O(q^{10}) \) \( 4q - 8q^{4} + 16q^{16} + 8q^{22} - 36q^{25} + 32q^{37} - 40q^{46} + 28q^{49} - 32q^{64} + 56q^{70} - 56q^{85} - 16q^{88} + O(q^{100}) \)

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
1.41421i 0 −2.00000 3.74166i 0 −2.64575 2.82843i 0 −5.29150
55.2 1.41421i 0 −2.00000 3.74166i 0 2.64575 2.82843i 0 5.29150
55.3 1.41421i 0 −2.00000 3.74166i 0 2.64575 2.82843i 0 5.29150
55.4 1.41421i 0 −2.00000 3.74166i 0 −2.64575 2.82843i 0 −5.29150
\(n\): 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 252.2.b.b 4
3.b odd 2 1 inner 252.2.b.b 4
4.b odd 2 1 inner 252.2.b.b 4
7.b odd 2 1 inner 252.2.b.b 4
8.b even 2 1 4032.2.b.k 4
8.d odd 2 1 4032.2.b.k 4
12.b even 2 1 inner 252.2.b.b 4
21.c even 2 1 inner 252.2.b.b 4
24.f even 2 1 4032.2.b.k 4
24.h odd 2 1 4032.2.b.k 4
28.d even 2 1 inner 252.2.b.b 4
56.e even 2 1 4032.2.b.k 4
56.h odd 2 1 4032.2.b.k 4
84.h odd 2 1 CM 252.2.b.b 4
168.e odd 2 1 4032.2.b.k 4
168.i even 2 1 4032.2.b.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.b.b 4 1.a even 1 1 trivial
252.2.b.b 4 3.b odd 2 1 inner
252.2.b.b 4 4.b odd 2 1 inner
252.2.b.b 4 7.b odd 2 1 inner
252.2.b.b 4 12.b even 2 1 inner
252.2.b.b 4 21.c even 2 1 inner
252.2.b.b 4 28.d even 2 1 inner
252.2.b.b 4 84.h odd 2 1 CM
4032.2.b.k 4 8.b even 2 1
4032.2.b.k 4 8.d odd 2 1
4032.2.b.k 4 24.f even 2 1
4032.2.b.k 4 24.h odd 2 1
4032.2.b.k 4 56.e even 2 1
4032.2.b.k 4 56.h odd 2 1
4032.2.b.k 4 168.e odd 2 1
4032.2.b.k 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 \(S_{2}^{\mathrm{new}}(252, [\chi])\):

\( T_{5}^{2} + 14 \)
\( T_{11}^{2} + 2 \)
\( T_{19}^{2} - 28 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} )^{2} \)
$3$ \( \)
$5$ \( ( 1 + 4 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( ( 1 - 20 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 13 T^{2} )^{4} \)
$17$ \( ( 1 - 20 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 10 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 4 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 29 T^{2} )^{4} \)
$31$ \( ( 1 - 50 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 8 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 68 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 43 T^{2} )^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( ( 1 + 53 T^{2} )^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{4} \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 - 67 T^{2} )^{4} \)
$71$ \( ( 1 + 100 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{4} \)
$79$ \( ( 1 - 79 T^{2} )^{4} \)
$83$ \( ( 1 + 83 T^{2} )^{4} \)
$89$ \( ( 1 + 172 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 97 T^{2} )^{4} \)
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