# Properties

 Label 378.2.g.h Level $378$ Weight $2$ Character orbit 378.g Analytic conductor $3.018$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{2} + \beta_{2} q^{4} + ( - \beta_{2} + \beta_1 - 1) q^{5} - \beta_{3} q^{7} - q^{8}+O(q^{10})$$ q + (b2 + 1) * q^2 + b2 * q^4 + (-b2 + b1 - 1) * q^5 - b3 * q^7 - q^8 $$q + (\beta_{2} + 1) q^{2} + \beta_{2} q^{4} + ( - \beta_{2} + \beta_1 - 1) q^{5} - \beta_{3} q^{7} - q^{8} + (\beta_{3} - \beta_{2} + \beta_1) q^{10} + (\beta_{3} - \beta_{2} + \beta_1) q^{11} + ( - \beta_{3} - 2) q^{13} + \beta_1 q^{14} + ( - \beta_{2} - 1) q^{16} + (\beta_{3} - \beta_{2} + \beta_1) q^{17} + ( - 2 \beta_{2} - 2) q^{19} + (\beta_{3} + 1) q^{20} + (\beta_{3} + 1) q^{22} + (4 \beta_{2} + 2 \beta_1 + 4) q^{23} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{25} + ( - 2 \beta_{2} + \beta_1 - 2) q^{26} + (\beta_{3} + \beta_1) q^{28} + (\beta_{3} - 5) q^{29} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{31} - \beta_{2} q^{32} + (\beta_{3} + 1) q^{34} + (7 \beta_{2} - \beta_1 + 7) q^{35} + (4 \beta_{2} - 3 \beta_1 + 4) q^{37} - 2 \beta_{2} q^{38} + (\beta_{2} - \beta_1 + 1) q^{40} + ( - 3 \beta_{3} - 3) q^{41} + 5 q^{43} + (\beta_{2} - \beta_1 + 1) q^{44} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{46} + ( - 3 \beta_{2} - 3 \beta_1 - 3) q^{47} + 7 q^{49} + ( - 2 \beta_{3} - 3) q^{50} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{52} - 6 \beta_{2} q^{53} + ( - 2 \beta_{3} - 8) q^{55} + \beta_{3} q^{56} + ( - 5 \beta_{2} - \beta_1 - 5) q^{58} + ( - \beta_{3} - 11 \beta_{2} - \beta_1) q^{59} + ( - 10 \beta_{2} - \beta_1 - 10) q^{61} + (\beta_{3} + 2) q^{62} + q^{64} + (9 \beta_{2} - 3 \beta_1 + 9) q^{65} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{67} + (\beta_{2} - \beta_1 + 1) q^{68} + ( - \beta_{3} + 7 \beta_{2} - \beta_1) q^{70} + ( - \beta_{3} - 13) q^{71} + ( - 4 \beta_{3} - 4 \beta_1) q^{73} + ( - 3 \beta_{3} + 4 \beta_{2} - 3 \beta_1) q^{74} + 2 q^{76} + ( - \beta_{3} + 7 \beta_{2} - \beta_1) q^{77} + (2 \beta_{2} + 5 \beta_1 + 2) q^{79} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{80} + ( - 3 \beta_{2} + 3 \beta_1 - 3) q^{82} + (2 \beta_{3} + 8) q^{83} + ( - 2 \beta_{3} - 8) q^{85} + (5 \beta_{2} + 5) q^{86} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{88} + ( - 3 \beta_{2} - 3 \beta_1 - 3) q^{89} + (2 \beta_{3} + 7) q^{91} + (2 \beta_{3} - 4) q^{92} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{94} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{95} + (4 \beta_{3} + 3) q^{97} + (7 \beta_{2} + 7) q^{98}+O(q^{100})$$ q + (b2 + 1) * q^2 + b2 * q^4 + (-b2 + b1 - 1) * q^5 - b3 * q^7 - q^8 + (b3 - b2 + b1) * q^10 + (b3 - b2 + b1) * q^11 + (-b3 - 2) * q^13 + b1 * q^14 + (-b2 - 1) * q^16 + (b3 - b2 + b1) * q^17 + (-2*b2 - 2) * q^19 + (b3 + 1) * q^20 + (b3 + 1) * q^22 + (4*b2 + 2*b1 + 4) * q^23 + (-2*b3 + 3*b2 - 2*b1) * q^25 + (-2*b2 + b1 - 2) * q^26 + (b3 + b1) * q^28 + (b3 - 5) * q^29 + (b3 - 2*b2 + b1) * q^31 - b2 * q^32 + (b3 + 1) * q^34 + (7*b2 - b1 + 7) * q^35 + (4*b2 - 3*b1 + 4) * q^37 - 2*b2 * q^38 + (b2 - b1 + 1) * q^40 + (-3*b3 - 3) * q^41 + 5 * q^43 + (b2 - b1 + 1) * q^44 + (2*b3 + 4*b2 + 2*b1) * q^46 + (-3*b2 - 3*b1 - 3) * q^47 + 7 * q^49 + (-2*b3 - 3) * q^50 + (b3 - 2*b2 + b1) * q^52 - 6*b2 * q^53 + (-2*b3 - 8) * q^55 + b3 * q^56 + (-5*b2 - b1 - 5) * q^58 + (-b3 - 11*b2 - b1) * q^59 + (-10*b2 - b1 - 10) * q^61 + (b3 + 2) * q^62 + q^64 + (9*b2 - 3*b1 + 9) * q^65 + (2*b3 + 3*b2 + 2*b1) * q^67 + (b2 - b1 + 1) * q^68 + (-b3 + 7*b2 - b1) * q^70 + (-b3 - 13) * q^71 + (-4*b3 - 4*b1) * q^73 + (-3*b3 + 4*b2 - 3*b1) * q^74 + 2 * q^76 + (-b3 + 7*b2 - b1) * q^77 + (2*b2 + 5*b1 + 2) * q^79 + (-b3 + b2 - b1) * q^80 + (-3*b2 + 3*b1 - 3) * q^82 + (2*b3 + 8) * q^83 + (-2*b3 - 8) * q^85 + (5*b2 + 5) * q^86 + (-b3 + b2 - b1) * q^88 + (-3*b2 - 3*b1 - 3) * q^89 + (2*b3 + 7) * q^91 + (2*b3 - 4) * q^92 + (-3*b3 - 3*b2 - 3*b1) * q^94 + (-2*b3 + 2*b2 - 2*b1) * q^95 + (4*b3 + 3) * q^97 + (7*b2 + 7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{4} - 2 q^{5} - 4 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^4 - 2 * q^5 - 4 * q^8 $$4 q + 2 q^{2} - 2 q^{4} - 2 q^{5} - 4 q^{8} + 2 q^{10} + 2 q^{11} - 8 q^{13} - 2 q^{16} + 2 q^{17} - 4 q^{19} + 4 q^{20} + 4 q^{22} + 8 q^{23} - 6 q^{25} - 4 q^{26} - 20 q^{29} + 4 q^{31} + 2 q^{32} + 4 q^{34} + 14 q^{35} + 8 q^{37} + 4 q^{38} + 2 q^{40} - 12 q^{41} + 20 q^{43} + 2 q^{44} - 8 q^{46} - 6 q^{47} + 28 q^{49} - 12 q^{50} + 4 q^{52} + 12 q^{53} - 32 q^{55} - 10 q^{58} + 22 q^{59} - 20 q^{61} + 8 q^{62} + 4 q^{64} + 18 q^{65} - 6 q^{67} + 2 q^{68} - 14 q^{70} - 52 q^{71} - 8 q^{74} + 8 q^{76} - 14 q^{77} + 4 q^{79} - 2 q^{80} - 6 q^{82} + 32 q^{83} - 32 q^{85} + 10 q^{86} - 2 q^{88} - 6 q^{89} + 28 q^{91} - 16 q^{92} + 6 q^{94} - 4 q^{95} + 12 q^{97} + 14 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^4 - 2 * q^5 - 4 * q^8 + 2 * q^10 + 2 * q^11 - 8 * q^13 - 2 * q^16 + 2 * q^17 - 4 * q^19 + 4 * q^20 + 4 * q^22 + 8 * q^23 - 6 * q^25 - 4 * q^26 - 20 * q^29 + 4 * q^31 + 2 * q^32 + 4 * q^34 + 14 * q^35 + 8 * q^37 + 4 * q^38 + 2 * q^40 - 12 * q^41 + 20 * q^43 + 2 * q^44 - 8 * q^46 - 6 * q^47 + 28 * q^49 - 12 * q^50 + 4 * q^52 + 12 * q^53 - 32 * q^55 - 10 * q^58 + 22 * q^59 - 20 * q^61 + 8 * q^62 + 4 * q^64 + 18 * q^65 - 6 * q^67 + 2 * q^68 - 14 * q^70 - 52 * q^71 - 8 * q^74 + 8 * q^76 - 14 * q^77 + 4 * q^79 - 2 * q^80 - 6 * q^82 + 32 * q^83 - 32 * q^85 + 10 * q^86 - 2 * q^88 - 6 * q^89 + 28 * q^91 - 16 * q^92 + 6 * q^94 - 4 * q^95 + 12 * q^97 + 14 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## Character values

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

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 −1.32288 − 2.29129i 1.32288 + 2.29129i −1.32288 + 2.29129i 1.32288 − 2.29129i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.82288 3.15731i 0 −2.64575 −1.00000 0 1.82288 3.15731i
109.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.822876 + 1.42526i 0 2.64575 −1.00000 0 −0.822876 + 1.42526i
163.1 0.500000 0.866025i 0 −0.500000 0.866025i −1.82288 + 3.15731i 0 −2.64575 −1.00000 0 1.82288 + 3.15731i
163.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.822876 1.42526i 0 2.64575 −1.00000 0 −0.822876 1.42526i
 $$n$$: 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.h yes 4
3.b odd 2 1 378.2.g.g 4
7.c even 3 1 inner 378.2.g.h yes 4
7.c even 3 1 2646.2.a.bi 2
7.d odd 6 1 2646.2.a.bf 2
9.c even 3 1 1134.2.e.q 4
9.c even 3 1 1134.2.h.t 4
9.d odd 6 1 1134.2.e.t 4
9.d odd 6 1 1134.2.h.q 4
21.g even 6 1 2646.2.a.bo 2
21.h odd 6 1 378.2.g.g 4
21.h odd 6 1 2646.2.a.bl 2
63.g even 3 1 1134.2.e.q 4
63.h even 3 1 1134.2.h.t 4
63.j odd 6 1 1134.2.h.q 4
63.n odd 6 1 1134.2.e.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.g.g 4 3.b odd 2 1
378.2.g.g 4 21.h odd 6 1
378.2.g.h yes 4 1.a even 1 1 trivial
378.2.g.h yes 4 7.c even 3 1 inner
1134.2.e.q 4 9.c even 3 1
1134.2.e.q 4 63.g even 3 1
1134.2.e.t 4 9.d odd 6 1
1134.2.e.t 4 63.n odd 6 1
1134.2.h.q 4 9.d odd 6 1
1134.2.h.q 4 63.j odd 6 1
1134.2.h.t 4 9.c even 3 1
1134.2.h.t 4 63.h even 3 1
2646.2.a.bf 2 7.d odd 6 1
2646.2.a.bi 2 7.c even 3 1
2646.2.a.bl 2 21.h odd 6 1
2646.2.a.bo 2 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 $$S_{2}^{\mathrm{new}}(378, [\chi])$$:

 $$T_{5}^{4} + 2T_{5}^{3} + 10T_{5}^{2} - 12T_{5} + 36$$ T5^4 + 2*T5^3 + 10*T5^2 - 12*T5 + 36 $$T_{11}^{4} - 2T_{11}^{3} + 10T_{11}^{2} + 12T_{11} + 36$$ T11^4 - 2*T11^3 + 10*T11^2 + 12*T11 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36$$
$7$ $$(T^{2} - 7)^{2}$$
$11$ $$T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36$$
$13$ $$(T^{2} + 4 T - 3)^{2}$$
$17$ $$T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36$$
$19$ $$(T^{2} + 2 T + 4)^{2}$$
$23$ $$T^{4} - 8 T^{3} + 76 T^{2} + 96 T + 144$$
$29$ $$(T^{2} + 10 T + 18)^{2}$$
$31$ $$T^{4} - 4 T^{3} + 19 T^{2} + 12 T + 9$$
$37$ $$T^{4} - 8 T^{3} + 111 T^{2} + \cdots + 2209$$
$41$ $$(T^{2} + 6 T - 54)^{2}$$
$43$ $$(T - 5)^{4}$$
$47$ $$T^{4} + 6 T^{3} + 90 T^{2} + \cdots + 2916$$
$53$ $$(T^{2} - 6 T + 36)^{2}$$
$59$ $$T^{4} - 22 T^{3} + 370 T^{2} + \cdots + 12996$$
$61$ $$T^{4} + 20 T^{3} + 307 T^{2} + \cdots + 8649$$
$67$ $$T^{4} + 6 T^{3} + 55 T^{2} - 114 T + 361$$
$71$ $$(T^{2} + 26 T + 162)^{2}$$
$73$ $$T^{4} + 112 T^{2} + 12544$$
$79$ $$T^{4} - 4 T^{3} + 187 T^{2} + \cdots + 29241$$
$83$ $$(T^{2} - 16 T + 36)^{2}$$
$89$ $$T^{4} + 6 T^{3} + 90 T^{2} + \cdots + 2916$$
$97$ $$(T^{2} - 6 T - 103)^{2}$$