# Properties

 Label 756.2.n.a Level $756$ Weight $2$ Character orbit 756.n Analytic conductor $6.037$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 756.n (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 252) 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$379$$ $$\chi(n)$$ $$-1 + \zeta_{12}^{2}$$ $$\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−1.36603 0.366025i 0 1.73205 + 1.00000i 3.46410i 0 1.73205 + 2.00000i −2.00000 2.00000i 0 1.26795 4.73205i
19.2 0.366025 1.36603i 0 −1.73205 1.00000i 3.46410i 0 −1.73205 2.00000i −2.00000 + 2.00000i 0 4.73205 + 1.26795i
199.1 −1.36603 + 0.366025i 0 1.73205 1.00000i 3.46410i 0 1.73205 2.00000i −2.00000 + 2.00000i 0 1.26795 + 4.73205i
199.2 0.366025 + 1.36603i 0 −1.73205 + 1.00000i 3.46410i 0 −1.73205 + 2.00000i −2.00000 2.00000i 0 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.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 756.2.n.a 4
3.b odd 2 1 252.2.n.a 4
4.b odd 2 1 inner 756.2.n.a 4
7.d odd 6 1 756.2.bj.a 4
9.c even 3 1 756.2.bj.a 4
9.d odd 6 1 252.2.bj.a yes 4
12.b even 2 1 252.2.n.a 4
21.g even 6 1 252.2.bj.a yes 4
28.f even 6 1 756.2.bj.a 4
36.f odd 6 1 756.2.bj.a 4
36.h even 6 1 252.2.bj.a yes 4
63.k odd 6 1 inner 756.2.n.a 4
63.s even 6 1 252.2.n.a 4
84.j odd 6 1 252.2.bj.a yes 4
252.n even 6 1 inner 756.2.n.a 4
252.bn odd 6 1 252.2.n.a 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 12$$ acting on $$S_{2}^{\mathrm{new}}(756, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 12 + T^{2} )^{2}$$
$7$ $$49 + 2 T^{2} + T^{4}$$
$11$ $$( 4 + T^{2} )^{2}$$
$13$ $$( 27 - 9 T + T^{2} )^{2}$$
$17$ $$( 27 - 9 T + T^{2} )^{2}$$
$19$ $$9 + 3 T^{2} + T^{4}$$
$23$ $$( 16 + T^{2} )^{2}$$
$29$ $$( 25 - 5 T + T^{2} )^{2}$$
$31$ $$729 + 27 T^{2} + T^{4}$$
$37$ $$( 9 + 3 T + T^{2} )^{2}$$
$41$ $$( 3 + 3 T + T^{2} )^{2}$$
$43$ $$14641 - 121 T^{2} + T^{4}$$
$47$ $$9 + 3 T^{2} + T^{4}$$
$53$ $$( 1 - T + T^{2} )^{2}$$
$59$ $$9 + 3 T^{2} + T^{4}$$
$61$ $$( 27 - 9 T + T^{2} )^{2}$$
$67$ $$6561 - 81 T^{2} + T^{4}$$
$71$ $$( 4 + T^{2} )^{2}$$
$73$ $$( 147 + 21 T + T^{2} )^{2}$$
$79$ $$81 - 9 T^{2} + T^{4}$$
$83$ $$729 + 27 T^{2} + T^{4}$$
$89$ $$( 75 + 15 T + T^{2} )^{2}$$
$97$ $$( 3 - 3 T + T^{2} )^{2}$$