Properties

Label 252.4.b.a
Level $252$
Weight $4$
Character orbit 252.b
Analytic conductor $14.868$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Defining polynomial: \(x^{2} - x + 2\)
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.

\(f(q)\) \(=\) \( q + ( -3 + \beta ) q^{2} + ( 7 - 5 \beta ) q^{4} + ( -7 + 14 \beta ) q^{7} + ( -11 + 17 \beta ) q^{8} +O(q^{10})\) \( q + ( -3 + \beta ) q^{2} + ( 7 - 5 \beta ) q^{4} + ( -7 + 14 \beta ) q^{7} + ( -11 + 17 \beta ) q^{8} + ( 10 - 20 \beta ) q^{11} + ( -7 - 35 \beta ) q^{14} + ( -1 - 45 \beta ) q^{16} + ( 10 + 50 \beta ) q^{22} + ( -82 + 164 \beta ) q^{23} + 125 q^{25} + ( 91 + 63 \beta ) q^{28} -166 q^{29} + ( 93 + 89 \beta ) q^{32} -450 q^{37} + ( -202 + 404 \beta ) q^{43} + ( -130 - 90 \beta ) q^{44} + ( -82 - 410 \beta ) q^{46} -343 q^{49} + ( -375 + 125 \beta ) q^{50} -590 q^{53} + ( -399 - 35 \beta ) q^{56} + ( 498 - 166 \beta ) q^{58} + ( -457 - 85 \beta ) q^{64} + ( -306 + 612 \beta ) q^{67} + ( -370 + 740 \beta ) q^{71} + ( 1350 - 450 \beta ) q^{74} + 490 q^{77} + ( 90 - 180 \beta ) q^{79} + ( -202 - 1010 \beta ) q^{86} + ( 570 + 50 \beta ) q^{88} + ( 1066 + 738 \beta ) q^{92} + ( 1029 - 343 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 5q^{2} + 9q^{4} - 5q^{8} + O(q^{10}) \) \( 2q - 5q^{2} + 9q^{4} - 5q^{8} - 49q^{14} - 47q^{16} + 70q^{22} + 250q^{25} + 245q^{28} - 332q^{29} + 275q^{32} - 900q^{37} - 350q^{44} - 574q^{46} - 686q^{49} - 625q^{50} - 1180q^{53} - 833q^{56} + 830q^{58} - 999q^{64} + 2250q^{74} + 980q^{77} - 1414q^{86} + 1190q^{88} + 2870q^{92} + 1715q^{98} + O(q^{100}) \)

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
−2.50000 1.32288i 0 4.50000 + 6.61438i 0 0 18.5203i −2.50000 22.4889i 0 0
55.2 −2.50000 + 1.32288i 0 4.50000 6.61438i 0 0 18.5203i −2.50000 + 22.4889i 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 252.4.b.a 2
3.b odd 2 1 28.4.d.a 2
4.b odd 2 1 inner 252.4.b.a 2
7.b odd 2 1 CM 252.4.b.a 2
12.b even 2 1 28.4.d.a 2
21.c even 2 1 28.4.d.a 2
21.g even 6 2 196.4.f.a 4
21.h odd 6 2 196.4.f.a 4
24.f even 2 1 448.4.f.a 2
24.h odd 2 1 448.4.f.a 2
28.d even 2 1 inner 252.4.b.a 2
84.h odd 2 1 28.4.d.a 2
84.j odd 6 2 196.4.f.a 4
84.n even 6 2 196.4.f.a 4
168.e odd 2 1 448.4.f.a 2
168.i even 2 1 448.4.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.d.a 2 3.b odd 2 1
28.4.d.a 2 12.b even 2 1
28.4.d.a 2 21.c even 2 1
28.4.d.a 2 84.h odd 2 1
196.4.f.a 4 21.g even 6 2
196.4.f.a 4 21.h odd 6 2
196.4.f.a 4 84.j odd 6 2
196.4.f.a 4 84.n even 6 2
252.4.b.a 2 1.a even 1 1 trivial
252.4.b.a 2 4.b odd 2 1 inner
252.4.b.a 2 7.b odd 2 1 CM
252.4.b.a 2 28.d even 2 1 inner
448.4.f.a 2 24.f even 2 1
448.4.f.a 2 24.h odd 2 1
448.4.f.a 2 168.e odd 2 1
448.4.f.a 2 168.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 \(S_{4}^{\mathrm{new}}(252, [\chi])\):

\( T_{5} \)
\( T_{11}^{2} + 700 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8 + 5 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 343 + T^{2} \)
$11$ \( 700 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( 47068 + T^{2} \)
$29$ \( ( 166 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 450 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 285628 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( 590 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( T^{2} \)
$67$ \( 655452 + T^{2} \)
$71$ \( 958300 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( 56700 + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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