# 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: $$x^{4} - x^{2} + 1$$ 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.

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

## 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

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