# Properties

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

# Related objects

## 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: $$\Q(\sqrt{-3}, \sqrt{-7})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( 1 + \beta_{1} - \beta_{2} ) q^{4} + ( -2 + \beta_{2} ) q^{5} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{7} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( 1 + \beta_{1} - \beta_{2} ) q^{4} + ( -2 + \beta_{2} ) q^{5} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{7} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{8} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{10} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{11} + ( 2 - 4 \beta_{2} ) q^{13} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{14} + ( -2 \beta_{2} + 3 \beta_{3} ) q^{16} + ( -4 - 4 \beta_{2} ) q^{17} + ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{20} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{22} + ( 2 \beta_{2} - 4 \beta_{3} ) q^{23} + ( -2 + 2 \beta_{2} ) q^{25} + ( 4 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{26} + ( 2 \beta_{2} + 3 \beta_{3} ) q^{28} -5 q^{29} + ( -1 + 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{31} + ( 5 + \beta_{1} - 5 \beta_{2} ) q^{32} + ( 4 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{34} + ( 1 - 2 \beta_{1} - 2 \beta_{3} ) q^{35} + ( -6 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{40} + ( -2 + 4 \beta_{2} ) q^{41} + ( 4 - 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{43} + ( 2 \beta_{2} + 3 \beta_{3} ) q^{44} + ( -6 - 2 \beta_{1} + 6 \beta_{2} ) q^{46} + ( -4 + 8 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{47} + 7 \beta_{2} q^{49} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{50} + ( -2 - 2 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{52} + ( 7 - 7 \beta_{2} ) q^{53} + ( 1 - 2 \beta_{1} - 2 \beta_{3} ) q^{55} + ( 1 + 5 \beta_{1} - \beta_{2} ) q^{56} -5 \beta_{3} q^{58} + ( 3 - 6 \beta_{1} - 9 \beta_{2} + 12 \beta_{3} ) q^{59} + ( -12 + 6 \beta_{2} ) q^{61} + ( -3 - \beta_{1} + 7 \beta_{2} - \beta_{3} ) q^{62} + ( 7 - 5 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{64} + 6 \beta_{2} q^{65} + ( -8 - 8 \beta_{1} + 4 \beta_{3} ) q^{68} + ( -6 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{70} + ( 2 - 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{71} + ( -4 - 4 \beta_{2} ) q^{73} + 7 \beta_{2} q^{77} + ( 3 \beta_{2} - 6 \beta_{3} ) q^{79} + ( -1 + 3 \beta_{1} + 5 \beta_{2} - 6 \beta_{3} ) q^{80} + ( -4 + 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{82} + ( -1 + 2 \beta_{1} + 2 \beta_{3} ) q^{83} + 12 q^{85} + ( -16 \beta_{2} + 4 \beta_{3} ) q^{86} + ( 1 + 5 \beta_{1} - \beta_{2} ) q^{88} + ( 12 - 6 \beta_{2} ) q^{89} + ( 2 - 4 \beta_{1} - 6 \beta_{2} + 8 \beta_{3} ) q^{91} + ( -10 + 6 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{92} + ( 6 + 2 \beta_{1} + 10 \beta_{2} - 4 \beta_{3} ) q^{94} + ( 5 - 10 \beta_{2} ) q^{97} + ( -7 + 7 \beta_{1} + 7 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{2} + 3q^{4} - 6q^{5} + 10q^{8} + O(q^{10})$$ $$4q + q^{2} + 3q^{4} - 6q^{5} + 10q^{8} - 3q^{10} + 14q^{14} - q^{16} - 24q^{17} + 14q^{22} - 4q^{25} + 6q^{26} + 7q^{28} - 20q^{29} + 11q^{32} - 15q^{40} + 7q^{44} - 14q^{46} + 14q^{49} - 4q^{50} - 18q^{52} + 14q^{53} + 7q^{56} - 5q^{58} - 36q^{61} + 18q^{64} + 12q^{65} - 36q^{68} - 21q^{70} - 24q^{73} + 14q^{77} + 3q^{80} - 6q^{82} + 48q^{85} - 28q^{86} + 7q^{88} + 36q^{89} - 28q^{92} + 42q^{94} - 7q^{98} + O(q^{100})$$

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

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

## Character values

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

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

## 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
 −0.895644 + 1.09445i 1.39564 − 0.228425i −0.895644 − 1.09445i 1.39564 + 0.228425i
−0.895644 1.09445i 0 −0.395644 + 1.96048i −1.50000 0.866025i 0 −2.29129 + 1.32288i 2.50000 1.32288i 0 0.395644 + 2.41733i
19.2 1.39564 + 0.228425i 0 1.89564 + 0.637600i −1.50000 0.866025i 0 2.29129 1.32288i 2.50000 + 1.32288i 0 −1.89564 1.55130i
199.1 −0.895644 + 1.09445i 0 −0.395644 1.96048i −1.50000 + 0.866025i 0 −2.29129 1.32288i 2.50000 + 1.32288i 0 0.395644 2.41733i
199.2 1.39564 0.228425i 0 1.89564 0.637600i −1.50000 + 0.866025i 0 2.29129 + 1.32288i 2.50000 1.32288i 0 −1.89564 + 1.55130i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 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.d yes 4
3.b odd 2 1 252.2.bf.a 4
4.b odd 2 1 inner 252.2.bf.d yes 4
7.c even 3 1 1764.2.b.b 4
7.d odd 6 1 inner 252.2.bf.d yes 4
7.d odd 6 1 1764.2.b.b 4
12.b even 2 1 252.2.bf.a 4
21.g even 6 1 252.2.bf.a 4
21.g even 6 1 1764.2.b.h 4
21.h odd 6 1 1764.2.b.h 4
28.f even 6 1 inner 252.2.bf.d yes 4
28.f even 6 1 1764.2.b.b 4
28.g odd 6 1 1764.2.b.b 4
84.j odd 6 1 252.2.bf.a 4
84.j odd 6 1 1764.2.b.h 4
84.n even 6 1 1764.2.b.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.bf.a 4 3.b odd 2 1
252.2.bf.a 4 12.b even 2 1
252.2.bf.a 4 21.g even 6 1
252.2.bf.a 4 84.j odd 6 1
252.2.bf.d yes 4 1.a even 1 1 trivial
252.2.bf.d yes 4 4.b odd 2 1 inner
252.2.bf.d yes 4 7.d odd 6 1 inner
252.2.bf.d yes 4 28.f even 6 1 inner
1764.2.b.b 4 7.c even 3 1
1764.2.b.b 4 7.d odd 6 1
1764.2.b.b 4 28.f even 6 1
1764.2.b.b 4 28.g odd 6 1
1764.2.b.h 4 21.g even 6 1
1764.2.b.h 4 21.h odd 6 1
1764.2.b.h 4 84.j odd 6 1
1764.2.b.h 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 $$S_{2}^{\mathrm{new}}(252, [\chi])$$:

 $$T_{5}^{2} + 3 T_{5} + 3$$ $$T_{11}^{4} - 7 T_{11}^{2} + 49$$ $$T_{19}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T - T^{2} - 2 T^{3} + 4 T^{4}$$
$3$ 
$5$ $$( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} )^{2}$$
$7$ $$1 - 7 T^{2} + 49 T^{4}$$
$11$ $$1 + 15 T^{2} + 104 T^{4} + 1815 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - 14 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 12 T + 65 T^{2} + 204 T^{3} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 19 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} )( 1 + 8 T + 41 T^{2} + 184 T^{3} + 529 T^{4} )$$
$29$ $$( 1 + 5 T + 29 T^{2} )^{4}$$
$31$ $$1 - 41 T^{2} + 720 T^{4} - 39401 T^{6} + 923521 T^{8}$$
$37$ $$( 1 - 37 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 70 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 26 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 - 10 T^{2} - 2109 T^{4} - 22090 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 7 T - 4 T^{2} - 371 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 + 71 T^{2} + 1560 T^{4} + 247151 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 + 18 T + 169 T^{2} + 1098 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 67 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 16 T + 71 T^{2} )^{2}( 1 + 16 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$1 + 95 T^{2} + 2784 T^{4} + 592895 T^{6} + 38950081 T^{8}$$
$83$ $$( 1 + 145 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 18 T + 197 T^{2} - 1602 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 119 T^{2} + 9409 T^{4} )^{2}$$