# Properties

 Label 252.2.bf.b 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} -2 q^{4} -2 \beta_{1} q^{5} + ( 1 - 3 \beta_{2} ) q^{7} -2 \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} -2 q^{4} -2 \beta_{1} q^{5} + ( 1 - 3 \beta_{2} ) q^{7} -2 \beta_{3} q^{8} + ( 4 - 4 \beta_{2} ) q^{10} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{11} + ( 3 - 6 \beta_{2} ) q^{13} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{14} + 4 q^{16} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{17} + ( -5 + 5 \beta_{2} ) q^{19} + 4 \beta_{1} q^{20} -8 \beta_{2} q^{22} -2 \beta_{1} q^{23} + 3 \beta_{2} q^{25} + ( 6 \beta_{1} - 3 \beta_{3} ) q^{26} + ( -2 + 6 \beta_{2} ) q^{28} + \beta_{2} q^{31} + 4 \beta_{3} q^{32} + 4 \beta_{2} q^{34} + ( -2 \beta_{1} + 6 \beta_{3} ) q^{35} + ( -5 + 5 \beta_{2} ) q^{37} -5 \beta_{1} q^{38} + ( -8 + 8 \beta_{2} ) q^{40} -4 \beta_{3} q^{41} + ( 3 - 6 \beta_{2} ) q^{43} + ( 8 \beta_{1} - 8 \beta_{3} ) q^{44} + ( 4 - 4 \beta_{2} ) q^{46} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{47} + ( -8 + 3 \beta_{2} ) q^{49} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{50} + ( -6 + 12 \beta_{2} ) q^{52} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{53} + 16 q^{55} + ( -6 \beta_{1} + 4 \beta_{3} ) q^{56} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{59} + ( 4 + 4 \beta_{2} ) q^{61} + ( -\beta_{1} + \beta_{3} ) q^{62} -8 q^{64} + ( -6 \beta_{1} + 12 \beta_{3} ) q^{65} + ( -10 + 5 \beta_{2} ) q^{67} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{68} + ( -8 - 4 \beta_{2} ) q^{70} + 2 \beta_{3} q^{71} + ( 2 - \beta_{2} ) q^{73} -5 \beta_{1} q^{74} + ( 10 - 10 \beta_{2} ) q^{76} + ( 8 \beta_{1} + 4 \beta_{3} ) q^{77} + ( 1 + \beta_{2} ) q^{79} -8 \beta_{1} q^{80} + 8 q^{82} + ( 12 \beta_{1} - 6 \beta_{3} ) q^{83} -8 q^{85} + ( 6 \beta_{1} - 3 \beta_{3} ) q^{86} + 16 \beta_{2} q^{88} -8 \beta_{1} q^{89} + ( -15 + 3 \beta_{2} ) q^{91} + 4 \beta_{1} q^{92} + ( 4 + 4 \beta_{2} ) q^{94} + ( 10 \beta_{1} - 10 \beta_{3} ) q^{95} + ( -3 \beta_{1} - 5 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} - 2q^{7} + O(q^{10})$$ $$4q - 8q^{4} - 2q^{7} + 8q^{10} + 16q^{16} - 10q^{19} - 16q^{22} + 6q^{25} + 4q^{28} + 2q^{31} + 8q^{34} - 10q^{37} - 16q^{40} + 8q^{46} - 26q^{49} + 64q^{55} + 24q^{61} - 32q^{64} - 30q^{67} - 40q^{70} + 6q^{73} + 20q^{76} + 6q^{79} + 32q^{82} - 32q^{85} + 32q^{88} - 54q^{91} + 24q^{94} + O(q^{100})$$

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 $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

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

## 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
1.41421i 0 −2.00000 2.44949 + 1.41421i 0 −0.500000 2.59808i 2.82843i 0 2.00000 3.46410i
19.2 1.41421i 0 −2.00000 −2.44949 1.41421i 0 −0.500000 2.59808i 2.82843i 0 2.00000 3.46410i
199.1 1.41421i 0 −2.00000 −2.44949 + 1.41421i 0 −0.500000 + 2.59808i 2.82843i 0 2.00000 + 3.46410i
199.2 1.41421i 0 −2.00000 2.44949 1.41421i 0 −0.500000 + 2.59808i 2.82843i 0 2.00000 + 3.46410i
 $$n$$: 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
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.b 4
3.b odd 2 1 inner 252.2.bf.b 4
4.b odd 2 1 252.2.bf.c yes 4
7.c even 3 1 1764.2.b.d 4
7.d odd 6 1 252.2.bf.c yes 4
7.d odd 6 1 1764.2.b.c 4
12.b even 2 1 252.2.bf.c yes 4
21.g even 6 1 252.2.bf.c yes 4
21.g even 6 1 1764.2.b.c 4
21.h odd 6 1 1764.2.b.d 4
28.f even 6 1 inner 252.2.bf.b 4
28.f even 6 1 1764.2.b.d 4
28.g odd 6 1 1764.2.b.c 4
84.j odd 6 1 inner 252.2.bf.b 4
84.j odd 6 1 1764.2.b.d 4
84.n even 6 1 1764.2.b.c 4

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

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