# Properties

 Label 378.2.d.a Level $378$ Weight $2$ Character orbit 378.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{12}^{3} q^{2} - q^{4} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{5} + (\zeta_{12}^{2} - 3) q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ q - z^3 * q^2 - q^4 + (z^3 - 2*z) * q^5 + (z^2 - 3) * q^7 + z^3 * q^8 $$q - \zeta_{12}^{3} q^{2} - q^{4} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{5} + (\zeta_{12}^{2} - 3) q^{7} + \zeta_{12}^{3} q^{8} + (2 \zeta_{12}^{2} - 1) q^{10} - 3 \zeta_{12}^{3} q^{11} + (8 \zeta_{12}^{2} - 4) q^{13} + (2 \zeta_{12}^{3} + \zeta_{12}) q^{14} + q^{16} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{17} + (4 \zeta_{12}^{2} - 2) q^{19} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{20} - 3 q^{22} + 6 \zeta_{12}^{3} q^{23} - 2 q^{25} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{26} + ( - \zeta_{12}^{2} + 3) q^{28} - 6 \zeta_{12}^{3} q^{29} + ( - 6 \zeta_{12}^{2} + 3) q^{31} - \zeta_{12}^{3} q^{32} + (8 \zeta_{12}^{2} - 4) q^{34} + ( - 4 \zeta_{12}^{3} + 5 \zeta_{12}) q^{35} - 2 q^{37} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{38} + ( - 2 \zeta_{12}^{2} + 1) q^{40} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{41} - 2 q^{43} + 3 \zeta_{12}^{3} q^{44} + 6 q^{46} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{47} + ( - 5 \zeta_{12}^{2} + 8) q^{49} + 2 \zeta_{12}^{3} q^{50} + ( - 8 \zeta_{12}^{2} + 4) q^{52} + 3 \zeta_{12}^{3} q^{53} + (6 \zeta_{12}^{2} - 3) q^{55} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{56} - 6 q^{58} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{59} + ( - 8 \zeta_{12}^{2} + 4) q^{61} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{62} - q^{64} - 12 \zeta_{12}^{3} q^{65} + 2 q^{67} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{68} + ( - 5 \zeta_{12}^{2} + 1) q^{70} + 12 \zeta_{12}^{3} q^{71} + ( - 14 \zeta_{12}^{2} + 7) q^{73} + 2 \zeta_{12}^{3} q^{74} + ( - 4 \zeta_{12}^{2} + 2) q^{76} + (6 \zeta_{12}^{3} + 3 \zeta_{12}) q^{77} + 8 q^{79} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{80} + (4 \zeta_{12}^{2} - 2) q^{82} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{83} + 12 q^{85} + 2 \zeta_{12}^{3} q^{86} + 3 q^{88} + (6 \zeta_{12}^{3} - 12 \zeta_{12}) q^{89} + ( - 20 \zeta_{12}^{2} + 4) q^{91} - 6 \zeta_{12}^{3} q^{92} + ( - 4 \zeta_{12}^{2} + 2) q^{94} - 6 \zeta_{12}^{3} q^{95} + (14 \zeta_{12}^{2} - 7) q^{97} + ( - 3 \zeta_{12}^{3} - 5 \zeta_{12}) q^{98} +O(q^{100})$$ q - z^3 * q^2 - q^4 + (z^3 - 2*z) * q^5 + (z^2 - 3) * q^7 + z^3 * q^8 + (2*z^2 - 1) * q^10 - 3*z^3 * q^11 + (8*z^2 - 4) * q^13 + (2*z^3 + z) * q^14 + q^16 + (4*z^3 - 8*z) * q^17 + (4*z^2 - 2) * q^19 + (-z^3 + 2*z) * q^20 - 3 * q^22 + 6*z^3 * q^23 - 2 * q^25 + (-4*z^3 + 8*z) * q^26 + (-z^2 + 3) * q^28 - 6*z^3 * q^29 + (-6*z^2 + 3) * q^31 - z^3 * q^32 + (8*z^2 - 4) * q^34 + (-4*z^3 + 5*z) * q^35 - 2 * q^37 + (-2*z^3 + 4*z) * q^38 + (-2*z^2 + 1) * q^40 + (2*z^3 - 4*z) * q^41 - 2 * q^43 + 3*z^3 * q^44 + 6 * q^46 + (-2*z^3 + 4*z) * q^47 + (-5*z^2 + 8) * q^49 + 2*z^3 * q^50 + (-8*z^2 + 4) * q^52 + 3*z^3 * q^53 + (6*z^2 - 3) * q^55 + (-2*z^3 - z) * q^56 - 6 * q^58 + (2*z^3 - 4*z) * q^59 + (-8*z^2 + 4) * q^61 + (3*z^3 - 6*z) * q^62 - q^64 - 12*z^3 * q^65 + 2 * q^67 + (-4*z^3 + 8*z) * q^68 + (-5*z^2 + 1) * q^70 + 12*z^3 * q^71 + (-14*z^2 + 7) * q^73 + 2*z^3 * q^74 + (-4*z^2 + 2) * q^76 + (6*z^3 + 3*z) * q^77 + 8 * q^79 + (z^3 - 2*z) * q^80 + (4*z^2 - 2) * q^82 + (-z^3 + 2*z) * q^83 + 12 * q^85 + 2*z^3 * q^86 + 3 * q^88 + (6*z^3 - 12*z) * q^89 + (-20*z^2 + 4) * q^91 - 6*z^3 * q^92 + (-4*z^2 + 2) * q^94 - 6*z^3 * q^95 + (14*z^2 - 7) * q^97 + (-3*z^3 - 5*z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 10 q^{7}+O(q^{10})$$ 4 * q - 4 * q^4 - 10 * q^7 $$4 q - 4 q^{4} - 10 q^{7} + 4 q^{16} - 12 q^{22} - 8 q^{25} + 10 q^{28} - 8 q^{37} - 8 q^{43} + 24 q^{46} + 22 q^{49} - 24 q^{58} - 4 q^{64} + 8 q^{67} - 6 q^{70} + 32 q^{79} + 48 q^{85} + 12 q^{88} - 24 q^{91}+O(q^{100})$$ 4 * q - 4 * q^4 - 10 * q^7 + 4 * q^16 - 12 * q^22 - 8 * q^25 + 10 * q^28 - 8 * q^37 - 8 * q^43 + 24 * q^46 + 22 * q^49 - 24 * q^58 - 4 * q^64 + 8 * q^67 - 6 * q^70 + 32 * q^79 + 48 * q^85 + 12 * q^88 - 24 * q^91

## Character values

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

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
377.1
 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i
1.00000i 0 −1.00000 −1.73205 0 −2.50000 + 0.866025i 1.00000i 0 1.73205i
377.2 1.00000i 0 −1.00000 1.73205 0 −2.50000 0.866025i 1.00000i 0 1.73205i
377.3 1.00000i 0 −1.00000 −1.73205 0 −2.50000 0.866025i 1.00000i 0 1.73205i
377.4 1.00000i 0 −1.00000 1.73205 0 −2.50000 + 0.866025i 1.00000i 0 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
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.d.a 4
3.b odd 2 1 inner 378.2.d.a 4
4.b odd 2 1 3024.2.k.j 4
7.b odd 2 1 inner 378.2.d.a 4
9.c even 3 1 1134.2.m.e 4
9.c even 3 1 1134.2.m.f 4
9.d odd 6 1 1134.2.m.e 4
9.d odd 6 1 1134.2.m.f 4
12.b even 2 1 3024.2.k.j 4
21.c even 2 1 inner 378.2.d.a 4
28.d even 2 1 3024.2.k.j 4
63.l odd 6 1 1134.2.m.e 4
63.l odd 6 1 1134.2.m.f 4
63.o even 6 1 1134.2.m.e 4
63.o even 6 1 1134.2.m.f 4
84.h odd 2 1 3024.2.k.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.d.a 4 1.a even 1 1 trivial
378.2.d.a 4 3.b odd 2 1 inner
378.2.d.a 4 7.b odd 2 1 inner
378.2.d.a 4 21.c even 2 1 inner
1134.2.m.e 4 9.c even 3 1
1134.2.m.e 4 9.d odd 6 1
1134.2.m.e 4 63.l odd 6 1
1134.2.m.e 4 63.o even 6 1
1134.2.m.f 4 9.c even 3 1
1134.2.m.f 4 9.d odd 6 1
1134.2.m.f 4 63.l odd 6 1
1134.2.m.f 4 63.o even 6 1
3024.2.k.j 4 4.b odd 2 1
3024.2.k.j 4 12.b even 2 1
3024.2.k.j 4 28.d even 2 1
3024.2.k.j 4 84.h odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 3$$ T5^2 - 3 $$T_{13}^{2} + 48$$ T13^2 + 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 3)^{2}$$
$7$ $$(T^{2} + 5 T + 7)^{2}$$
$11$ $$(T^{2} + 9)^{2}$$
$13$ $$(T^{2} + 48)^{2}$$
$17$ $$(T^{2} - 48)^{2}$$
$19$ $$(T^{2} + 12)^{2}$$
$23$ $$(T^{2} + 36)^{2}$$
$29$ $$(T^{2} + 36)^{2}$$
$31$ $$(T^{2} + 27)^{2}$$
$37$ $$(T + 2)^{4}$$
$41$ $$(T^{2} - 12)^{2}$$
$43$ $$(T + 2)^{4}$$
$47$ $$(T^{2} - 12)^{2}$$
$53$ $$(T^{2} + 9)^{2}$$
$59$ $$(T^{2} - 12)^{2}$$
$61$ $$(T^{2} + 48)^{2}$$
$67$ $$(T - 2)^{4}$$
$71$ $$(T^{2} + 144)^{2}$$
$73$ $$(T^{2} + 147)^{2}$$
$79$ $$(T - 8)^{4}$$
$83$ $$(T^{2} - 3)^{2}$$
$89$ $$(T^{2} - 108)^{2}$$
$97$ $$(T^{2} + 147)^{2}$$