# Properties

 Label 378.2.d.c 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: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - q^{4} - 2 \beta_{3} q^{5} + ( - \beta_{2} + 2) q^{7} - \beta_1 q^{8}+O(q^{10})$$ q + b1 * q^2 - q^4 - 2*b3 * q^5 + (-b2 + 2) * q^7 - b1 * q^8 $$q + \beta_1 q^{2} - q^{4} - 2 \beta_{3} q^{5} + ( - \beta_{2} + 2) q^{7} - \beta_1 q^{8} - 2 \beta_{2} q^{10} - 6 \beta_1 q^{11} + \beta_{2} q^{13} + (\beta_{3} + 2 \beta_1) q^{14} + q^{16} + \beta_{3} q^{17} - 4 \beta_{2} q^{19} + 2 \beta_{3} q^{20} + 6 q^{22} + 3 \beta_1 q^{23} + 7 q^{25} - \beta_{3} q^{26} + (\beta_{2} - 2) q^{28} - 3 \beta_1 q^{29} - 3 \beta_{2} q^{31} + \beta_1 q^{32} + \beta_{2} q^{34} + ( - 4 \beta_{3} + 6 \beta_1) q^{35} - 2 q^{37} + 4 \beta_{3} q^{38} + 2 \beta_{2} q^{40} - 4 \beta_{3} q^{41} - 11 q^{43} + 6 \beta_1 q^{44} - 3 q^{46} + 4 \beta_{3} q^{47} + ( - 4 \beta_{2} + 1) q^{49} + 7 \beta_1 q^{50} - \beta_{2} q^{52} - 3 \beta_1 q^{53} + 12 \beta_{2} q^{55} + ( - \beta_{3} - 2 \beta_1) q^{56} + 3 q^{58} + 5 \beta_{3} q^{59} + 8 \beta_{2} q^{61} + 3 \beta_{3} q^{62} - q^{64} - 6 \beta_1 q^{65} - 7 q^{67} - \beta_{3} q^{68} + ( - 4 \beta_{2} - 6) q^{70} - 3 \beta_1 q^{71} - 4 \beta_{2} q^{73} - 2 \beta_1 q^{74} + 4 \beta_{2} q^{76} + ( - 6 \beta_{3} - 12 \beta_1) q^{77} + 8 q^{79} - 2 \beta_{3} q^{80} - 4 \beta_{2} q^{82} + 2 \beta_{3} q^{83} - 6 q^{85} - 11 \beta_1 q^{86} - 6 q^{88} - 3 \beta_{3} q^{89} + (2 \beta_{2} + 3) q^{91} - 3 \beta_1 q^{92} + 4 \beta_{2} q^{94} + 24 \beta_1 q^{95} + 4 \beta_{2} q^{97} + (4 \beta_{3} + \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 - q^4 - 2*b3 * q^5 + (-b2 + 2) * q^7 - b1 * q^8 - 2*b2 * q^10 - 6*b1 * q^11 + b2 * q^13 + (b3 + 2*b1) * q^14 + q^16 + b3 * q^17 - 4*b2 * q^19 + 2*b3 * q^20 + 6 * q^22 + 3*b1 * q^23 + 7 * q^25 - b3 * q^26 + (b2 - 2) * q^28 - 3*b1 * q^29 - 3*b2 * q^31 + b1 * q^32 + b2 * q^34 + (-4*b3 + 6*b1) * q^35 - 2 * q^37 + 4*b3 * q^38 + 2*b2 * q^40 - 4*b3 * q^41 - 11 * q^43 + 6*b1 * q^44 - 3 * q^46 + 4*b3 * q^47 + (-4*b2 + 1) * q^49 + 7*b1 * q^50 - b2 * q^52 - 3*b1 * q^53 + 12*b2 * q^55 + (-b3 - 2*b1) * q^56 + 3 * q^58 + 5*b3 * q^59 + 8*b2 * q^61 + 3*b3 * q^62 - q^64 - 6*b1 * q^65 - 7 * q^67 - b3 * q^68 + (-4*b2 - 6) * q^70 - 3*b1 * q^71 - 4*b2 * q^73 - 2*b1 * q^74 + 4*b2 * q^76 + (-6*b3 - 12*b1) * q^77 + 8 * q^79 - 2*b3 * q^80 - 4*b2 * q^82 + 2*b3 * q^83 - 6 * q^85 - 11*b1 * q^86 - 6 * q^88 - 3*b3 * q^89 + (2*b2 + 3) * q^91 - 3*b1 * q^92 + 4*b2 * q^94 + 24*b1 * q^95 + 4*b2 * q^97 + (4*b3 + b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 8 q^{7}+O(q^{10})$$ 4 * q - 4 * q^4 + 8 * q^7 $$4 q - 4 q^{4} + 8 q^{7} + 4 q^{16} + 24 q^{22} + 28 q^{25} - 8 q^{28} - 8 q^{37} - 44 q^{43} - 12 q^{46} + 4 q^{49} + 12 q^{58} - 4 q^{64} - 28 q^{67} - 24 q^{70} + 32 q^{79} - 24 q^{85} - 24 q^{88} + 12 q^{91}+O(q^{100})$$ 4 * q - 4 * q^4 + 8 * q^7 + 4 * q^16 + 24 * q^22 + 28 * q^25 - 8 * q^28 - 8 * q^37 - 44 * q^43 - 12 * q^46 + 4 * q^49 + 12 * q^58 - 4 * q^64 - 28 * q^67 - 24 * q^70 + 32 * q^79 - 24 * q^85 - 24 * q^88 + 12 * q^91

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## 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 −3.46410 0 2.00000 + 1.73205i 1.00000i 0 3.46410i
377.2 1.00000i 0 −1.00000 3.46410 0 2.00000 1.73205i 1.00000i 0 3.46410i
377.3 1.00000i 0 −1.00000 −3.46410 0 2.00000 1.73205i 1.00000i 0 3.46410i
377.4 1.00000i 0 −1.00000 3.46410 0 2.00000 + 1.73205i 1.00000i 0 3.46410i
 $$n$$: 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.c 4
3.b odd 2 1 inner 378.2.d.c 4
4.b odd 2 1 3024.2.k.f 4
7.b odd 2 1 inner 378.2.d.c 4
9.c even 3 1 1134.2.m.a 4
9.c even 3 1 1134.2.m.d 4
9.d odd 6 1 1134.2.m.a 4
9.d odd 6 1 1134.2.m.d 4
12.b even 2 1 3024.2.k.f 4
21.c even 2 1 inner 378.2.d.c 4
28.d even 2 1 3024.2.k.f 4
63.l odd 6 1 1134.2.m.a 4
63.l odd 6 1 1134.2.m.d 4
63.o even 6 1 1134.2.m.a 4
63.o even 6 1 1134.2.m.d 4
84.h odd 2 1 3024.2.k.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.d.c 4 1.a even 1 1 trivial
378.2.d.c 4 3.b odd 2 1 inner
378.2.d.c 4 7.b odd 2 1 inner
378.2.d.c 4 21.c even 2 1 inner
1134.2.m.a 4 9.c even 3 1
1134.2.m.a 4 9.d odd 6 1
1134.2.m.a 4 63.l odd 6 1
1134.2.m.a 4 63.o even 6 1
1134.2.m.d 4 9.c even 3 1
1134.2.m.d 4 9.d odd 6 1
1134.2.m.d 4 63.l odd 6 1
1134.2.m.d 4 63.o even 6 1
3024.2.k.f 4 4.b odd 2 1
3024.2.k.f 4 12.b even 2 1
3024.2.k.f 4 28.d even 2 1
3024.2.k.f 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 $$S_{2}^{\mathrm{new}}(378, [\chi])$$:

 $$T_{5}^{2} - 12$$ T5^2 - 12 $$T_{13}^{2} + 3$$ T13^2 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 12)^{2}$$
$7$ $$(T^{2} - 4 T + 7)^{2}$$
$11$ $$(T^{2} + 36)^{2}$$
$13$ $$(T^{2} + 3)^{2}$$
$17$ $$(T^{2} - 3)^{2}$$
$19$ $$(T^{2} + 48)^{2}$$
$23$ $$(T^{2} + 9)^{2}$$
$29$ $$(T^{2} + 9)^{2}$$
$31$ $$(T^{2} + 27)^{2}$$
$37$ $$(T + 2)^{4}$$
$41$ $$(T^{2} - 48)^{2}$$
$43$ $$(T + 11)^{4}$$
$47$ $$(T^{2} - 48)^{2}$$
$53$ $$(T^{2} + 9)^{2}$$
$59$ $$(T^{2} - 75)^{2}$$
$61$ $$(T^{2} + 192)^{2}$$
$67$ $$(T + 7)^{4}$$
$71$ $$(T^{2} + 9)^{2}$$
$73$ $$(T^{2} + 48)^{2}$$
$79$ $$(T - 8)^{4}$$
$83$ $$(T^{2} - 12)^{2}$$
$89$ $$(T^{2} - 27)^{2}$$
$97$ $$(T^{2} + 48)^{2}$$