# Properties

 Label 252.2.n.a Level 252 Weight 2 Character orbit 252.n 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.n (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: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.366025 1.36603i 1.73205 −1.73205 + 1.00000i 3.46410i −0.633975 2.36603i −1.73205 + 2.00000i 2.00000 + 2.00000i 3.00000 4.73205 1.26795i
31.2 1.36603 0.366025i −1.73205 1.73205 1.00000i 3.46410i −2.36603 + 0.633975i 1.73205 2.00000i 2.00000 2.00000i 3.00000 1.26795 + 4.73205i
187.1 −0.366025 + 1.36603i 1.73205 −1.73205 1.00000i 3.46410i −0.633975 + 2.36603i −1.73205 2.00000i 2.00000 2.00000i 3.00000 4.73205 + 1.26795i
187.2 1.36603 + 0.366025i −1.73205 1.73205 + 1.00000i 3.46410i −2.36603 0.633975i 1.73205 + 2.00000i 2.00000 + 2.00000i 3.00000 1.26795 4.73205i
 $$n$$: 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.k odd 6 1 inner
252.n 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.n.a 4
3.b odd 2 1 756.2.n.a 4
4.b odd 2 1 inner 252.2.n.a 4
7.d odd 6 1 252.2.bj.a yes 4
9.c even 3 1 252.2.bj.a yes 4
9.d odd 6 1 756.2.bj.a 4
12.b even 2 1 756.2.n.a 4
21.g even 6 1 756.2.bj.a 4
28.f even 6 1 252.2.bj.a yes 4
36.f odd 6 1 252.2.bj.a yes 4
36.h even 6 1 756.2.bj.a 4
63.k odd 6 1 inner 252.2.n.a 4
63.s even 6 1 756.2.n.a 4
84.j odd 6 1 756.2.bj.a 4
252.n even 6 1 inner 252.2.n.a 4
252.bn odd 6 1 756.2.n.a 4

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

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4}$$
$3$ $$( 1 - 3 T^{2} )^{2}$$
$5$ $$( 1 + 2 T^{2} + 25 T^{4} )^{2}$$
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 18 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 7 T + 13 T^{2} )^{2}( 1 - 2 T + 13 T^{2} )^{2}$$
$17$ $$( 1 + 9 T + 44 T^{2} + 153 T^{3} + 289 T^{4} )^{2}$$
$19$ $$1 - 35 T^{2} + 864 T^{4} - 12635 T^{6} + 130321 T^{8}$$
$23$ $$( 1 - 30 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 5 T - 4 T^{2} + 145 T^{3} + 841 T^{4} )^{2}$$
$31$ $$1 - 35 T^{2} + 264 T^{4} - 33635 T^{6} + 923521 T^{8}$$
$37$ $$( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 3 T + 44 T^{2} - 123 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 - 35 T^{2} - 624 T^{4} - 64715 T^{6} + 3418801 T^{8}$$
$47$ $$1 - 91 T^{2} + 6072 T^{4} - 201019 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 + T - 52 T^{2} + 53 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 - 9 T + 88 T^{2} - 549 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 + 53 T^{2} - 1680 T^{4} + 237917 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 138 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$1 + 149 T^{2} + 15960 T^{4} + 929909 T^{6} + 38950081 T^{8}$$
$83$ $$1 - 139 T^{2} + 12432 T^{4} - 957571 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 - 15 T + 164 T^{2} - 1335 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 3 T + 100 T^{2} - 291 T^{3} + 9409 T^{4} )^{2}$$