Properties

Label 252.2.b.a
Level 252
Weight 2
Character orbit 252.b
Analytic conductor 2.012
Analytic rank 0
Dimension 2
CM discriminant -7
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.01223013094\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-7})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( -2 + \beta ) q^{4} + ( -1 + 2 \beta ) q^{7} + ( -2 - \beta ) q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( -2 + \beta ) q^{4} + ( -1 + 2 \beta ) q^{7} + ( -2 - \beta ) q^{8} + ( -2 + 4 \beta ) q^{11} + ( -4 + \beta ) q^{14} + ( 2 - 3 \beta ) q^{16} + ( -8 + 2 \beta ) q^{22} + ( 2 - 4 \beta ) q^{23} + 5 q^{25} + ( -2 - 3 \beta ) q^{28} + 2 q^{29} + ( 6 - \beta ) q^{32} + 6 q^{37} + ( 2 - 4 \beta ) q^{43} + ( -4 - 6 \beta ) q^{44} + ( 8 - 2 \beta ) q^{46} -7 q^{49} + 5 \beta q^{50} + 10 q^{53} + ( 6 - 5 \beta ) q^{56} + 2 \beta q^{58} + ( 2 + 5 \beta ) q^{64} + ( -6 + 12 \beta ) q^{67} + ( 2 - 4 \beta ) q^{71} + 6 \beta q^{74} -14 q^{77} + ( 6 - 12 \beta ) q^{79} + ( 8 - 2 \beta ) q^{86} + ( 12 - 10 \beta ) q^{88} + ( 4 + 6 \beta ) q^{92} -7 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - 3q^{4} - 5q^{8} + O(q^{10}) \) \( 2q + q^{2} - 3q^{4} - 5q^{8} - 7q^{14} + q^{16} - 14q^{22} + 10q^{25} - 7q^{28} + 4q^{29} + 11q^{32} + 12q^{37} - 14q^{44} + 14q^{46} - 14q^{49} + 5q^{50} + 20q^{53} + 7q^{56} + 2q^{58} + 9q^{64} + 6q^{74} - 28q^{77} + 14q^{86} + 14q^{88} + 14q^{92} - 7q^{98} + O(q^{100}) \)

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
0.500000 1.32288i
0.500000 + 1.32288i
0.500000 1.32288i 0 −1.50000 1.32288i 0 0 2.64575i −2.50000 + 1.32288i 0 0
55.2 0.500000 + 1.32288i 0 −1.50000 + 1.32288i 0 0 2.64575i −2.50000 1.32288i 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 \(\Q(\sqrt{-7}) \)
4.b odd 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.a 2
3.b odd 2 1 28.2.d.a 2
4.b odd 2 1 inner 252.2.b.a 2
7.b odd 2 1 CM 252.2.b.a 2
8.b even 2 1 4032.2.b.e 2
8.d odd 2 1 4032.2.b.e 2
12.b even 2 1 28.2.d.a 2
15.d odd 2 1 700.2.g.a 2
15.e even 4 2 700.2.c.d 4
21.c even 2 1 28.2.d.a 2
21.g even 6 2 196.2.f.b 4
21.h odd 6 2 196.2.f.b 4
24.f even 2 1 448.2.f.b 2
24.h odd 2 1 448.2.f.b 2
28.d even 2 1 inner 252.2.b.a 2
48.i odd 4 2 1792.2.e.b 4
48.k even 4 2 1792.2.e.b 4
56.e even 2 1 4032.2.b.e 2
56.h odd 2 1 4032.2.b.e 2
60.h even 2 1 700.2.g.a 2
60.l odd 4 2 700.2.c.d 4
84.h odd 2 1 28.2.d.a 2
84.j odd 6 2 196.2.f.b 4
84.n even 6 2 196.2.f.b 4
105.g even 2 1 700.2.g.a 2
105.k odd 4 2 700.2.c.d 4
168.e odd 2 1 448.2.f.b 2
168.i even 2 1 448.2.f.b 2
336.v odd 4 2 1792.2.e.b 4
336.y even 4 2 1792.2.e.b 4
420.o odd 2 1 700.2.g.a 2
420.w even 4 2 700.2.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.d.a 2 3.b odd 2 1
28.2.d.a 2 12.b even 2 1
28.2.d.a 2 21.c even 2 1
28.2.d.a 2 84.h odd 2 1
196.2.f.b 4 21.g even 6 2
196.2.f.b 4 21.h odd 6 2
196.2.f.b 4 84.j odd 6 2
196.2.f.b 4 84.n even 6 2
252.2.b.a 2 1.a even 1 1 trivial
252.2.b.a 2 4.b odd 2 1 inner
252.2.b.a 2 7.b odd 2 1 CM
252.2.b.a 2 28.d even 2 1 inner
448.2.f.b 2 24.f even 2 1
448.2.f.b 2 24.h odd 2 1
448.2.f.b 2 168.e odd 2 1
448.2.f.b 2 168.i even 2 1
700.2.c.d 4 15.e even 4 2
700.2.c.d 4 60.l odd 4 2
700.2.c.d 4 105.k odd 4 2
700.2.c.d 4 420.w even 4 2
700.2.g.a 2 15.d odd 2 1
700.2.g.a 2 60.h even 2 1
700.2.g.a 2 105.g even 2 1
700.2.g.a 2 420.o odd 2 1
1792.2.e.b 4 48.i odd 4 2
1792.2.e.b 4 48.k even 4 2
1792.2.e.b 4 336.v odd 4 2
1792.2.e.b 4 336.y even 4 2
4032.2.b.e 2 8.b even 2 1
4032.2.b.e 2 8.d odd 2 1
4032.2.b.e 2 56.e even 2 1
4032.2.b.e 2 56.h odd 2 1

Hecke kernels

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 2 T^{2} \)
$3$ \( \)
$5$ \( ( 1 - 5 T^{2} )^{2} \)
$7$ \( 1 + 7 T^{2} \)
$11$ \( ( 1 - 4 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} ) \)
$13$ \( ( 1 - 13 T^{2} )^{2} \)
$17$ \( ( 1 - 17 T^{2} )^{2} \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} ) \)
$29$ \( ( 1 - 2 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 6 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 - 41 T^{2} )^{2} \)
$43$ \( ( 1 - 12 T + 43 T^{2} )( 1 + 12 T + 43 T^{2} ) \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 - 10 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 61 T^{2} )^{2} \)
$67$ \( ( 1 - 4 T + 67 T^{2} )( 1 + 4 T + 67 T^{2} ) \)
$71$ \( ( 1 - 16 T + 71 T^{2} )( 1 + 16 T + 71 T^{2} ) \)
$73$ \( ( 1 - 73 T^{2} )^{2} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} ) \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( ( 1 - 89 T^{2} )^{2} \)
$97$ \( ( 1 - 97 T^{2} )^{2} \)
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