Properties

Label 252.2.bf.c
Level 252
Weight 2
Character orbit 252.bf
Analytic conductor 2.012
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 252.bf (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.01223013094\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{5} + ( 2 - 3 \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{5} + ( 2 - 3 \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} + 4 q^{10} -4 \beta_{1} q^{11} + ( -3 + 6 \beta_{2} ) q^{13} + ( 2 \beta_{1} - 3 \beta_{3} ) q^{14} + ( -4 + 4 \beta_{2} ) q^{16} -2 \beta_{1} q^{17} + 5 \beta_{2} q^{19} + 4 \beta_{1} q^{20} -8 \beta_{2} q^{22} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{23} + ( 3 - 3 \beta_{2} ) q^{25} + ( -3 \beta_{1} + 6 \beta_{3} ) q^{26} + ( 6 - 2 \beta_{2} ) q^{28} + ( -1 + \beta_{2} ) q^{31} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} -4 \beta_{2} q^{34} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{35} -5 \beta_{2} q^{37} + 5 \beta_{3} q^{38} + 8 \beta_{2} q^{40} -4 \beta_{3} q^{41} + ( 3 - 6 \beta_{2} ) q^{43} -8 \beta_{3} q^{44} -4 q^{46} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{47} + ( -5 - 3 \beta_{2} ) q^{49} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{50} + ( -12 + 6 \beta_{2} ) q^{52} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{53} -16 q^{55} + ( 6 \beta_{1} - 2 \beta_{3} ) q^{56} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{59} + ( 8 - 4 \beta_{2} ) q^{61} + ( -\beta_{1} + \beta_{3} ) q^{62} -8 q^{64} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{65} + ( 5 + 5 \beta_{2} ) q^{67} -4 \beta_{3} q^{68} + ( 8 - 12 \beta_{2} ) q^{70} -2 \beta_{3} q^{71} + ( 1 + \beta_{2} ) q^{73} -5 \beta_{3} q^{74} + ( -10 + 10 \beta_{2} ) q^{76} + ( -8 \beta_{1} + 12 \beta_{3} ) q^{77} + ( -2 + \beta_{2} ) q^{79} + 8 \beta_{3} q^{80} + ( 8 - 8 \beta_{2} ) q^{82} + ( 12 \beta_{1} - 6 \beta_{3} ) q^{83} -8 q^{85} + ( 3 \beta_{1} - 6 \beta_{3} ) q^{86} + ( 16 - 16 \beta_{2} ) q^{88} + ( 8 \beta_{1} - 8 \beta_{3} ) q^{89} + ( 12 + 3 \beta_{2} ) q^{91} -4 \beta_{1} q^{92} + ( -4 + 8 \beta_{2} ) q^{94} + 10 \beta_{1} q^{95} + ( -5 \beta_{1} - 3 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} + 2q^{7} + O(q^{10}) \) \( 4q + 4q^{4} + 2q^{7} + 16q^{10} - 8q^{16} + 10q^{19} - 16q^{22} + 6q^{25} + 20q^{28} - 2q^{31} - 8q^{34} - 10q^{37} + 16q^{40} - 16q^{46} - 26q^{49} - 36q^{52} - 64q^{55} + 24q^{61} - 32q^{64} + 30q^{67} + 8q^{70} + 6q^{73} - 20q^{76} - 6q^{79} + 16q^{82} - 32q^{85} + 32q^{88} + 54q^{91} + O(q^{100}) \)

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

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

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i −2.44949 1.41421i 0 0.500000 + 2.59808i 2.82843i 0 4.00000
19.2 1.22474 0.707107i 0 1.00000 1.73205i 2.44949 + 1.41421i 0 0.500000 + 2.59808i 2.82843i 0 4.00000
199.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −2.44949 + 1.41421i 0 0.500000 2.59808i 2.82843i 0 4.00000
199.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 2.44949 1.41421i 0 0.500000 2.59808i 2.82843i 0 4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
28.f even 6 1 inner
84.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.bf.c yes 4
3.b odd 2 1 inner 252.2.bf.c yes 4
4.b odd 2 1 252.2.bf.b 4
7.c even 3 1 1764.2.b.c 4
7.d odd 6 1 252.2.bf.b 4
7.d odd 6 1 1764.2.b.d 4
12.b even 2 1 252.2.bf.b 4
21.g even 6 1 252.2.bf.b 4
21.g even 6 1 1764.2.b.d 4
21.h odd 6 1 1764.2.b.c 4
28.f even 6 1 inner 252.2.bf.c yes 4
28.f even 6 1 1764.2.b.c 4
28.g odd 6 1 1764.2.b.d 4
84.j odd 6 1 inner 252.2.bf.c yes 4
84.j odd 6 1 1764.2.b.c 4
84.n even 6 1 1764.2.b.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.bf.b 4 4.b odd 2 1
252.2.bf.b 4 7.d odd 6 1
252.2.bf.b 4 12.b even 2 1
252.2.bf.b 4 21.g even 6 1
252.2.bf.c yes 4 1.a even 1 1 trivial
252.2.bf.c yes 4 3.b odd 2 1 inner
252.2.bf.c yes 4 28.f even 6 1 inner
252.2.bf.c yes 4 84.j odd 6 1 inner
1764.2.b.c 4 7.c even 3 1
1764.2.b.c 4 21.h odd 6 1
1764.2.b.c 4 28.f even 6 1
1764.2.b.c 4 84.j odd 6 1
1764.2.b.d 4 7.d odd 6 1
1764.2.b.d 4 21.g even 6 1
1764.2.b.d 4 28.g odd 6 1
1764.2.b.d 4 84.n even 6 1

Hecke kernels

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

\( T_{5}^{4} - 8 T_{5}^{2} + 64 \)
\( T_{11}^{4} - 32 T_{11}^{2} + 1024 \)
\( T_{19}^{2} - 5 T_{19} + 25 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 4 T^{4} \)
$3$ \( \)
$5$ \( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \)
$7$ \( ( 1 - T + 7 T^{2} )^{2} \)
$11$ \( 1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 26 T^{2} + 387 T^{4} + 7514 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 + 38 T^{2} + 915 T^{4} + 20102 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 29 T^{2} )^{4} \)
$31$ \( ( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 50 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 59 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 70 T^{2} + 2691 T^{4} - 154630 T^{6} + 4879681 T^{8} \)
$53$ \( 1 - 82 T^{2} + 3915 T^{4} - 230338 T^{6} + 7890481 T^{8} \)
$59$ \( 1 - 94 T^{2} + 5355 T^{4} - 327214 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )^{2}( 1 + T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 134 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 10 T + 73 T^{2} )^{2}( 1 + 7 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 3 T + 82 T^{2} + 237 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 50 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( 1 + 50 T^{2} - 5421 T^{4} + 396050 T^{6} + 62742241 T^{8} \)
$97$ \( ( 1 - 97 T^{2} )^{4} \)
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