# Properties

 Label 252.2.bj.a Level $252$ Weight $2$ Character orbit 252.bj Analytic conductor $2.012$ Analytic rank $1$ 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.bj (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$1$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{12}^{3} ) q^{2} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} -2 \zeta_{12}^{3} q^{4} + ( -2 - 2 \zeta_{12}^{2} ) q^{5} + ( 2 + \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{6} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{12}^{3} ) q^{2} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} -2 \zeta_{12}^{3} q^{4} + ( -2 - 2 \zeta_{12}^{2} ) q^{5} + ( 2 + \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{6} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( 2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{10} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{11} + ( -4 + 2 \zeta_{12}^{2} ) q^{12} + ( -6 + 3 \zeta_{12}^{2} ) q^{13} + ( -2 - 3 \zeta_{12} + 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{14} + 6 \zeta_{12}^{3} q^{15} -4 q^{16} + ( -3 - 3 \zeta_{12}^{2} ) q^{17} + ( 3 - 3 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{18} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{19} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{20} + ( 1 - 5 \zeta_{12}^{2} ) q^{21} + ( 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{22} + 4 \zeta_{12} q^{23} + ( 4 - 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{24} + 7 \zeta_{12}^{2} q^{25} + ( 6 - 3 \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{26} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( 4 - 6 \zeta_{12}^{2} ) q^{28} + ( -5 + 5 \zeta_{12}^{2} ) q^{29} + ( -6 - 6 \zeta_{12}^{3} ) q^{30} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{31} + ( 4 - 4 \zeta_{12}^{3} ) q^{32} + ( 2 + 2 \zeta_{12}^{2} ) q^{33} + ( 3 + 3 \zeta_{12} + 3 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{34} + ( -8 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{35} + 6 \zeta_{12} q^{36} -3 \zeta_{12}^{2} q^{37} + ( -2 - \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{38} + 9 \zeta_{12} q^{39} + ( -4 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{40} + ( -2 + \zeta_{12}^{2} ) q^{41} + ( -1 + 5 \zeta_{12} + 5 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{42} -11 \zeta_{12} q^{43} + 4 \zeta_{12}^{2} q^{44} + ( 12 - 6 \zeta_{12}^{2} ) q^{45} + ( -4 - 4 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{46} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{47} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{48} + ( 5 + 3 \zeta_{12}^{2} ) q^{49} + ( -7 \zeta_{12} - 7 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{50} + 9 \zeta_{12}^{3} q^{51} + ( 6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{52} + ( -1 + \zeta_{12}^{2} ) q^{53} + ( -3 - 6 \zeta_{12} + 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{54} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{55} + ( -4 + 6 \zeta_{12} + 6 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{56} + ( 3 - 3 \zeta_{12}^{2} ) q^{57} + ( 5 - 5 \zeta_{12} - 5 \zeta_{12}^{2} ) q^{58} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{59} + 12 q^{60} + ( 3 - 6 \zeta_{12}^{2} ) q^{61} + ( 3 + 6 \zeta_{12} - 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{62} + ( -6 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{63} + 8 \zeta_{12}^{3} q^{64} + 18 q^{65} + ( -2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{66} -9 \zeta_{12}^{3} q^{67} + ( -6 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{68} + ( 4 - 8 \zeta_{12}^{2} ) q^{69} + ( 10 + 8 \zeta_{12} - 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{70} + 2 \zeta_{12}^{3} q^{71} + ( -6 - 6 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{72} + ( -7 - 7 \zeta_{12}^{2} ) q^{73} + ( 3 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{74} + ( 7 \zeta_{12} - 14 \zeta_{12}^{3} ) q^{75} + ( 4 - 2 \zeta_{12}^{2} ) q^{76} + ( -6 + 2 \zeta_{12}^{2} ) q^{77} + ( -9 - 9 \zeta_{12} + 9 \zeta_{12}^{2} ) q^{78} + 3 \zeta_{12}^{3} q^{79} + ( 8 + 8 \zeta_{12}^{2} ) q^{80} -9 \zeta_{12}^{2} q^{81} + ( 2 - \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{82} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{83} + ( -10 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{84} + 18 \zeta_{12}^{2} q^{85} + ( 11 + 11 \zeta_{12} - 11 \zeta_{12}^{2} ) q^{86} + ( 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{87} + ( -4 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{88} + ( 10 - 5 \zeta_{12}^{2} ) q^{89} + ( -12 + 6 \zeta_{12} + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{90} + ( -15 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{91} + ( 8 - 8 \zeta_{12}^{2} ) q^{92} + 9 \zeta_{12}^{2} q^{93} + ( 1 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{94} -6 \zeta_{12}^{3} q^{95} + ( -8 - 4 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{96} + ( -1 - \zeta_{12}^{2} ) q^{97} + ( -5 - 3 \zeta_{12} - 3 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{98} -6 \zeta_{12}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 12q^{5} + 6q^{6} + 8q^{8} - 6q^{9} + O(q^{10})$$ $$4q - 4q^{2} - 12q^{5} + 6q^{6} + 8q^{8} - 6q^{9} + 12q^{10} - 12q^{12} - 18q^{13} - 2q^{14} - 16q^{16} - 18q^{17} + 6q^{18} - 6q^{21} - 4q^{22} + 12q^{24} + 14q^{25} + 18q^{26} + 4q^{28} - 10q^{29} - 24q^{30} + 16q^{32} + 12q^{33} + 18q^{34} - 6q^{37} - 6q^{38} - 24q^{40} - 6q^{41} + 6q^{42} + 8q^{44} + 36q^{45} - 8q^{46} + 26q^{49} - 14q^{50} - 2q^{53} - 4q^{56} + 6q^{57} + 10q^{58} + 48q^{60} + 72q^{65} - 12q^{66} + 24q^{70} - 12q^{72} - 42q^{73} + 6q^{74} + 12q^{76} - 20q^{77} - 18q^{78} + 48q^{80} - 18q^{81} + 6q^{82} + 36q^{85} + 22q^{86} - 8q^{88} + 30q^{89} - 36q^{90} + 16q^{92} + 18q^{93} - 24q^{96} - 6q^{97} - 26q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$-\zeta_{12}^{2}$$ $$\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
103.1
 0.866025 − 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−1.00000 1.00000i −0.866025 + 1.50000i 2.00000i −3.00000 + 1.73205i 2.36603 0.633975i 2.59808 0.500000i 2.00000 2.00000i −1.50000 2.59808i 4.73205 + 1.26795i
103.2 −1.00000 + 1.00000i 0.866025 1.50000i 2.00000i −3.00000 + 1.73205i 0.633975 + 2.36603i −2.59808 + 0.500000i 2.00000 + 2.00000i −1.50000 2.59808i 1.26795 4.73205i
115.1 −1.00000 1.00000i 0.866025 + 1.50000i 2.00000i −3.00000 1.73205i 0.633975 2.36603i −2.59808 0.500000i 2.00000 2.00000i −1.50000 + 2.59808i 1.26795 + 4.73205i
115.2 −1.00000 + 1.00000i −0.866025 1.50000i 2.00000i −3.00000 1.73205i 2.36603 + 0.633975i 2.59808 + 0.500000i 2.00000 + 2.00000i −1.50000 + 2.59808i 4.73205 1.26795i
 $$n$$: 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
63.t odd 6 1 inner
252.bj 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.bj.a yes 4
3.b odd 2 1 756.2.bj.a 4
4.b odd 2 1 inner 252.2.bj.a yes 4
7.d odd 6 1 252.2.n.a 4
9.c even 3 1 252.2.n.a 4
9.d odd 6 1 756.2.n.a 4
12.b even 2 1 756.2.bj.a 4
21.g even 6 1 756.2.n.a 4
28.f even 6 1 252.2.n.a 4
36.f odd 6 1 252.2.n.a 4
36.h even 6 1 756.2.n.a 4
63.i even 6 1 756.2.bj.a 4
63.t odd 6 1 inner 252.2.bj.a yes 4
84.j odd 6 1 756.2.n.a 4
252.r odd 6 1 756.2.bj.a 4
252.bj even 6 1 inner 252.2.bj.a yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.n.a 4 7.d odd 6 1
252.2.n.a 4 9.c even 3 1
252.2.n.a 4 28.f even 6 1
252.2.n.a 4 36.f odd 6 1
252.2.bj.a yes 4 1.a even 1 1 trivial
252.2.bj.a yes 4 4.b odd 2 1 inner
252.2.bj.a yes 4 63.t odd 6 1 inner
252.2.bj.a yes 4 252.bj even 6 1 inner
756.2.n.a 4 9.d odd 6 1
756.2.n.a 4 21.g even 6 1
756.2.n.a 4 36.h even 6 1
756.2.n.a 4 84.j odd 6 1
756.2.bj.a 4 3.b odd 2 1
756.2.bj.a 4 12.b even 2 1
756.2.bj.a 4 63.i even 6 1
756.2.bj.a 4 252.r odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 6 T_{5} + 12$$ acting on $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T + 2 T^{2} )^{2}$$
$3$ $$1 + 3 T^{2} + 9 T^{4}$$
$5$ $$( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} )^{2}$$
$7$ $$1 - 13 T^{2} + 49 T^{4}$$
$11$ $$1 + 18 T^{2} + 203 T^{4} + 2178 T^{6} + 14641 T^{8}$$
$13$ $$( 1 + 2 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2}$$
$17$ $$( 1 + 9 T + 44 T^{2} + 153 T^{3} + 289 T^{4} )^{2}$$
$19$ $$1 - 35 T^{2} + 864 T^{4} - 12635 T^{6} + 130321 T^{8}$$
$23$ $$1 + 30 T^{2} + 371 T^{4} + 15870 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 5 T - 4 T^{2} + 145 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 35 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 3 T + 44 T^{2} + 123 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 - 35 T^{2} - 624 T^{4} - 64715 T^{6} + 3418801 T^{8}$$
$47$ $$( 1 + 91 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 + T - 52 T^{2} + 53 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 115 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 95 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 53 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 138 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 149 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$1 - 139 T^{2} + 12432 T^{4} - 957571 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 - 15 T + 164 T^{2} - 1335 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 3 T + 100 T^{2} + 291 T^{3} + 9409 T^{4} )^{2}$$