# Properties

 Label 378.2.g.a Level $378$ Weight $2$ Character orbit 378.g Analytic conductor $3.018$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.01834519640$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} -3 \zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + q^{8} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} -3 \zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + q^{8} + ( -3 + 3 \zeta_{6} ) q^{10} -4 q^{13} + ( 2 + \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} + 3 q^{20} -6 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + 4 \zeta_{6} q^{26} + ( 1 - 3 \zeta_{6} ) q^{28} -3 q^{29} + ( -8 + 8 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} + 6 q^{34} + ( 6 + 3 \zeta_{6} ) q^{35} -8 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} -3 \zeta_{6} q^{40} -6 q^{41} + 8 q^{43} + ( -6 + 6 \zeta_{6} ) q^{46} -6 \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + 4 q^{50} + ( 4 - 4 \zeta_{6} ) q^{52} + ( -9 + 9 \zeta_{6} ) q^{53} + ( -3 + 2 \zeta_{6} ) q^{56} + 3 \zeta_{6} q^{58} + ( 3 - 3 \zeta_{6} ) q^{59} + 10 \zeta_{6} q^{61} + 8 q^{62} + q^{64} + 12 \zeta_{6} q^{65} + ( 10 - 10 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{68} + ( 3 - 9 \zeta_{6} ) q^{70} + 6 q^{71} + ( 7 - 7 \zeta_{6} ) q^{73} + ( -8 + 8 \zeta_{6} ) q^{74} -4 q^{76} -17 \zeta_{6} q^{79} + ( -3 + 3 \zeta_{6} ) q^{80} + 6 \zeta_{6} q^{82} -12 q^{83} + 18 q^{85} -8 \zeta_{6} q^{86} -6 \zeta_{6} q^{89} + ( 12 - 8 \zeta_{6} ) q^{91} + 6 q^{92} + ( -6 + 6 \zeta_{6} ) q^{94} + ( 12 - 12 \zeta_{6} ) q^{95} -10 q^{97} + ( -8 + 3 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} - 3q^{5} - 4q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} - 3q^{5} - 4q^{7} + 2q^{8} - 3q^{10} - 8q^{13} + 5q^{14} - q^{16} - 6q^{17} + 4q^{19} + 6q^{20} - 6q^{23} - 4q^{25} + 4q^{26} - q^{28} - 6q^{29} - 8q^{31} - q^{32} + 12q^{34} + 15q^{35} - 8q^{37} + 4q^{38} - 3q^{40} - 12q^{41} + 16q^{43} - 6q^{46} - 6q^{47} + 2q^{49} + 8q^{50} + 4q^{52} - 9q^{53} - 4q^{56} + 3q^{58} + 3q^{59} + 10q^{61} + 16q^{62} + 2q^{64} + 12q^{65} + 10q^{67} - 6q^{68} - 3q^{70} + 12q^{71} + 7q^{73} - 8q^{74} - 8q^{76} - 17q^{79} - 3q^{80} + 6q^{82} - 24q^{83} + 36q^{85} - 8q^{86} - 6q^{89} + 16q^{91} + 12q^{92} - 6q^{94} + 12q^{95} - 20q^{97} - 13q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −1.50000 2.59808i 0 −2.00000 + 1.73205i 1.00000 0 −1.50000 + 2.59808i
163.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.50000 + 2.59808i 0 −2.00000 1.73205i 1.00000 0 −1.50000 2.59808i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.g.a 2
3.b odd 2 1 378.2.g.f yes 2
7.c even 3 1 inner 378.2.g.a 2
7.c even 3 1 2646.2.a.bc 1
7.d odd 6 1 2646.2.a.r 1
9.c even 3 1 1134.2.e.j 2
9.c even 3 1 1134.2.h.g 2
9.d odd 6 1 1134.2.e.f 2
9.d odd 6 1 1134.2.h.k 2
21.g even 6 1 2646.2.a.m 1
21.h odd 6 1 378.2.g.f yes 2
21.h odd 6 1 2646.2.a.b 1
63.g even 3 1 1134.2.e.j 2
63.h even 3 1 1134.2.h.g 2
63.j odd 6 1 1134.2.h.k 2
63.n odd 6 1 1134.2.e.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.g.a 2 1.a even 1 1 trivial
378.2.g.a 2 7.c even 3 1 inner
378.2.g.f yes 2 3.b odd 2 1
378.2.g.f yes 2 21.h odd 6 1
1134.2.e.f 2 9.d odd 6 1
1134.2.e.f 2 63.n odd 6 1
1134.2.e.j 2 9.c even 3 1
1134.2.e.j 2 63.g even 3 1
1134.2.h.g 2 9.c even 3 1
1134.2.h.g 2 63.h even 3 1
1134.2.h.k 2 9.d odd 6 1
1134.2.h.k 2 63.j odd 6 1
2646.2.a.b 1 21.h odd 6 1
2646.2.a.m 1 21.g even 6 1
2646.2.a.r 1 7.d odd 6 1
2646.2.a.bc 1 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(378, [\chi])$$:

 $$T_{5}^{2} + 3 T_{5} + 9$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 + 3 T + T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$16 - 4 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$( 3 + T )^{2}$$
$31$ $$64 + 8 T + T^{2}$$
$37$ $$64 + 8 T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$81 + 9 T + T^{2}$$
$59$ $$9 - 3 T + T^{2}$$
$61$ $$100 - 10 T + T^{2}$$
$67$ $$100 - 10 T + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$49 - 7 T + T^{2}$$
$79$ $$289 + 17 T + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$( 10 + T )^{2}$$