# Properties

 Label 378.2.g.c 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: $$\Q(\sqrt{-3})$$ 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.

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

## 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.c 2
3.b odd 2 1 378.2.g.d yes 2
7.c even 3 1 inner 378.2.g.c 2
7.c even 3 1 2646.2.a.t 1
7.d odd 6 1 2646.2.a.bb 1
9.c even 3 1 1134.2.e.o 2
9.c even 3 1 1134.2.h.b 2
9.d odd 6 1 1134.2.e.b 2
9.d odd 6 1 1134.2.h.o 2
21.g even 6 1 2646.2.a.c 1
21.h odd 6 1 378.2.g.d yes 2
21.h odd 6 1 2646.2.a.k 1
63.g even 3 1 1134.2.e.o 2
63.h even 3 1 1134.2.h.b 2
63.j odd 6 1 1134.2.h.o 2
63.n odd 6 1 1134.2.e.b 2

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

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ 
$5$ $$1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4}$$
$7$ $$1 - T + 7 T^{2}$$
$11$ $$1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 6 T + 13 T^{2} )^{2}$$
$17$ $$1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4}$$
$19$ $$1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 7 T + 29 T^{2} )^{2}$$
$31$ $$1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4}$$
$37$ $$1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{2}$$
$47$ $$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$
$53$ $$1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4}$$
$59$ $$1 + 7 T - 10 T^{2} + 413 T^{3} + 3481 T^{4}$$
$61$ $$1 - 61 T^{2} + 3721 T^{4}$$
$67$ $$1 + 10 T + 33 T^{2} + 670 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 4 T + 71 T^{2} )^{2}$$
$73$ $$1 + 13 T + 96 T^{2} + 949 T^{3} + 5329 T^{4}$$
$79$ $$1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 7 T + 83 T^{2} )^{2}$$
$89$ $$1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 5 T + 97 T^{2} )^{2}$$