# Properties

 Label 378.2.g.d Level $378$ Weight $2$ Character orbit 378.g Analytic conductor $3.018$ Analytic rank $0$ 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: $$0$$ 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.

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

## 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 −1.00000 1.73205i 0 0.500000 2.59808i −1.00000 0 1.00000 1.73205i
163.1 0.500000 0.866025i 0 −0.500000 0.866025i −1.00000 + 1.73205i 0 0.500000 + 2.59808i −1.00000 0 1.00000 + 1.73205i
 $$n$$: 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
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.d yes 2
3.b odd 2 1 378.2.g.c 2
7.c even 3 1 inner 378.2.g.d yes 2
7.c even 3 1 2646.2.a.k 1
7.d odd 6 1 2646.2.a.c 1
9.c even 3 1 1134.2.e.b 2
9.c even 3 1 1134.2.h.o 2
9.d odd 6 1 1134.2.e.o 2
9.d odd 6 1 1134.2.h.b 2
21.g even 6 1 2646.2.a.bb 1
21.h odd 6 1 378.2.g.c 2
21.h odd 6 1 2646.2.a.t 1
63.g even 3 1 1134.2.e.b 2
63.h even 3 1 1134.2.h.o 2
63.j odd 6 1 1134.2.h.b 2
63.n odd 6 1 1134.2.e.o 2

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

## Hecke kernels

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

 $$T_{5}^{2} + 2 T_{5} + 4$$ $$T_{11}^{2} - 5 T_{11} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 + 2 T + T^{2}$$
$7$ $$7 - T + T^{2}$$
$11$ $$25 - 5 T + T^{2}$$
$13$ $$( -6 + T )^{2}$$
$17$ $$16 + 4 T + T^{2}$$
$19$ $$16 - 4 T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$( 7 + T )^{2}$$
$31$ $$9 + 3 T + T^{2}$$
$37$ $$64 + 8 T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$36 + 6 T + T^{2}$$
$59$ $$49 - 7 T + T^{2}$$
$61$ $$T^{2}$$
$67$ $$100 + 10 T + T^{2}$$
$71$ $$( -4 + T )^{2}$$
$73$ $$169 + 13 T + T^{2}$$
$79$ $$9 - 3 T + T^{2}$$
$83$ $$( -7 + T )^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$( 5 + T )^{2}$$