# Properties

 Label 378.2.k.a Level $378$ Weight $2$ Character orbit 378.k Analytic conductor $3.018$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.k (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.01834519640$$ 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -1 + 3 \zeta_{12}^{2} ) q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -1 + 3 \zeta_{12}^{2} ) q^{7} + \zeta_{12}^{3} q^{8} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{11} + ( -2 + 4 \zeta_{12}^{2} ) q^{13} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + 3 q^{22} + 5 \zeta_{12}^{2} q^{25} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{26} + ( -3 + 2 \zeta_{12}^{2} ) q^{28} -9 \zeta_{12}^{3} q^{29} + ( -2 + \zeta_{12}^{2} ) q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + ( 8 - 8 \zeta_{12}^{2} ) q^{37} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{41} + 4 q^{43} + 3 \zeta_{12} q^{44} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{47} + ( -8 + 3 \zeta_{12}^{2} ) q^{49} + 5 \zeta_{12}^{3} q^{50} + ( -4 + 2 \zeta_{12}^{2} ) q^{52} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{53} + ( -3 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{56} + ( 9 - 9 \zeta_{12}^{2} ) q^{58} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{59} + ( -8 - 8 \zeta_{12}^{2} ) q^{61} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{62} - q^{64} -2 \zeta_{12}^{2} q^{67} -12 \zeta_{12}^{3} q^{71} + ( 6 - 3 \zeta_{12}^{2} ) q^{73} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{74} + ( 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{77} + ( 13 - 13 \zeta_{12}^{2} ) q^{79} + ( -6 - 6 \zeta_{12}^{2} ) q^{82} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{83} + 4 \zeta_{12} q^{86} + 3 \zeta_{12}^{2} q^{88} + ( -6 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{89} + ( -10 + 2 \zeta_{12}^{2} ) q^{91} + ( 12 - 6 \zeta_{12}^{2} ) q^{94} + ( 5 - 10 \zeta_{12}^{2} ) q^{97} + ( -8 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + 2q^{7} + O(q^{10})$$ $$4q + 2q^{4} + 2q^{7} - 2q^{16} + 12q^{22} + 10q^{25} - 8q^{28} - 6q^{31} + 16q^{37} + 16q^{43} - 26q^{49} - 12q^{52} + 18q^{58} - 48q^{61} - 4q^{64} - 4q^{67} + 18q^{73} + 26q^{79} - 36q^{82} + 6q^{88} - 36q^{91} + 36q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
215.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 0.500000 + 2.59808i 1.00000i 0 0
215.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 0.500000 + 2.59808i 1.00000i 0 0
269.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 0.500000 2.59808i 1.00000i 0 0
269.2 0.866025 0.500000i 0 0.500000 0.866025i 0 0 0.500000 2.59808i 1.00000i 0 0
 $$n$$: 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
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.k.a 4
3.b odd 2 1 inner 378.2.k.a 4
7.c even 3 1 2646.2.d.c 4
7.d odd 6 1 inner 378.2.k.a 4
7.d odd 6 1 2646.2.d.c 4
9.c even 3 1 1134.2.l.d 4
9.c even 3 1 1134.2.t.a 4
9.d odd 6 1 1134.2.l.d 4
9.d odd 6 1 1134.2.t.a 4
21.g even 6 1 inner 378.2.k.a 4
21.g even 6 1 2646.2.d.c 4
21.h odd 6 1 2646.2.d.c 4
63.i even 6 1 1134.2.t.a 4
63.k odd 6 1 1134.2.l.d 4
63.s even 6 1 1134.2.l.d 4
63.t odd 6 1 1134.2.t.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.k.a 4 1.a even 1 1 trivial
378.2.k.a 4 3.b odd 2 1 inner
378.2.k.a 4 7.d odd 6 1 inner
378.2.k.a 4 21.g even 6 1 inner
1134.2.l.d 4 9.c even 3 1
1134.2.l.d 4 9.d odd 6 1
1134.2.l.d 4 63.k odd 6 1
1134.2.l.d 4 63.s even 6 1
1134.2.t.a 4 9.c even 3 1
1134.2.t.a 4 9.d odd 6 1
1134.2.t.a 4 63.i even 6 1
1134.2.t.a 4 63.t odd 6 1
2646.2.d.c 4 7.c even 3 1
2646.2.d.c 4 7.d odd 6 1
2646.2.d.c 4 21.g even 6 1
2646.2.d.c 4 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{2}^{\mathrm{new}}(378, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 7 - T + T^{2} )^{2}$$
$11$ $$81 - 9 T^{2} + T^{4}$$
$13$ $$( 12 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( 81 + T^{2} )^{2}$$
$31$ $$( 3 + 3 T + T^{2} )^{2}$$
$37$ $$( 64 - 8 T + T^{2} )^{2}$$
$41$ $$( -108 + T^{2} )^{2}$$
$43$ $$( -4 + T )^{4}$$
$47$ $$11664 + 108 T^{2} + T^{4}$$
$53$ $$1296 - 36 T^{2} + T^{4}$$
$59$ $$729 + 27 T^{2} + T^{4}$$
$61$ $$( 192 + 24 T + T^{2} )^{2}$$
$67$ $$( 4 + 2 T + T^{2} )^{2}$$
$71$ $$( 144 + T^{2} )^{2}$$
$73$ $$( 27 - 9 T + T^{2} )^{2}$$
$79$ $$( 169 - 13 T + T^{2} )^{2}$$
$83$ $$( -27 + T^{2} )^{2}$$
$89$ $$11664 + 108 T^{2} + T^{4}$$
$97$ $$( 75 + T^{2} )^{2}$$