# Properties

 Label 378.2.g.f 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: $$x^{2} - x + 1$$ 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.

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

## 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.f yes 2
3.b odd 2 1 378.2.g.a 2
7.c even 3 1 inner 378.2.g.f yes 2
7.c even 3 1 2646.2.a.b 1
7.d odd 6 1 2646.2.a.m 1
9.c even 3 1 1134.2.e.f 2
9.c even 3 1 1134.2.h.k 2
9.d odd 6 1 1134.2.e.j 2
9.d odd 6 1 1134.2.h.g 2
21.g even 6 1 2646.2.a.r 1
21.h odd 6 1 378.2.g.a 2
21.h odd 6 1 2646.2.a.bc 1
63.g even 3 1 1134.2.e.f 2
63.h even 3 1 1134.2.h.k 2
63.j odd 6 1 1134.2.h.g 2
63.n odd 6 1 1134.2.e.j 2

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

## Hecke kernels

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

 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9 $$T_{11}$$ T11

## Hecke characteristic polynomials

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