# Properties

 Label 378.2.k.d Level $378$ Weight $2$ Character orbit 378.k Analytic conductor $3.018$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{6} + \beta_{5}) q^{5} + (\beta_{7} - 1) q^{7} + \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + b2 * q^4 + (-b6 + b5) * q^5 + (b7 - 1) * q^7 + b3 * q^8 $$q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{6} + \beta_{5}) q^{5} + (\beta_{7} - 1) q^{7} + \beta_{3} q^{8} + (\beta_{7} - \beta_{4}) q^{10} + ( - 2 \beta_{6} + \beta_{5}) q^{11} + ( - \beta_{7} - 2 \beta_{2} + 1) q^{13} + ( - \beta_{6} + \beta_{5} - \beta_1) q^{14} + (\beta_{2} - 1) q^{16} - \beta_{5} q^{17} + 2 \beta_{4} q^{19} - \beta_{6} q^{20} + (\beta_{7} - 2 \beta_{4}) q^{22} + 6 \beta_1 q^{23} - \beta_{2} q^{25} + (\beta_{6} - \beta_{5} - 2 \beta_{3} + \beta_1) q^{26} + (\beta_{7} - \beta_{4} - \beta_{2}) q^{28} + ( - \beta_{6} + 2 \beta_{5} + 6 \beta_{3}) q^{29} + ( - 3 \beta_{7} + 3 \beta_{4} + \beta_{2} - 2) q^{31} + (\beta_{3} - \beta_1) q^{32} - \beta_{7} q^{34} + (\beta_{6} - \beta_{5} - 6 \beta_1) q^{35} + ( - 2 \beta_{7} + \beta_{4} + \beta_{2} - 1) q^{37} + 2 \beta_{5} q^{38} - \beta_{4} q^{40} - \beta_{6} q^{41} + 7 q^{43} + ( - \beta_{6} - \beta_{5}) q^{44} + 6 \beta_{2} q^{46} + ( - \beta_{6} + \beta_{5} - 12 \beta_{3} + 6 \beta_1) q^{47} + ( - 2 \beta_{7} - 5) q^{49} - \beta_{3} q^{50} + ( - \beta_{7} + \beta_{4} - \beta_{2} + 2) q^{52} + (4 \beta_{6} - 2 \beta_{5} + 6 \beta_{3} - 6 \beta_1) q^{53} + ( - 12 \beta_{2} + 6) q^{55} + ( - \beta_{6} - \beta_{3}) q^{56} + (2 \beta_{7} - \beta_{4} + 6 \beta_{2} - 6) q^{58} + \beta_{5} q^{59} + ( - \beta_{4} + \beta_{2} + 1) q^{61} + (3 \beta_{6} + \beta_{3} - 2 \beta_1) q^{62} - q^{64} + (\beta_{6} + \beta_{5} + 6 \beta_1) q^{65} + (2 \beta_{7} + 2 \beta_{4} - 5 \beta_{2}) q^{67} + (\beta_{6} - \beta_{5}) q^{68} + ( - \beta_{7} + \beta_{4} - 6 \beta_{2}) q^{70} + (3 \beta_{6} - 6 \beta_{5}) q^{71} + (2 \beta_{7} - 2 \beta_{4} + 6 \beta_{2} - 12) q^{73} + (2 \beta_{6} - \beta_{5} + \beta_{3} - \beta_1) q^{74} + 2 \beta_{7} q^{76} + (2 \beta_{6} - \beta_{5} - 6 \beta_{3} - 6 \beta_1) q^{77} + (2 \beta_{7} - \beta_{4} + 5 \beta_{2} - 5) q^{79} - \beta_{5} q^{80} - \beta_{4} q^{82} + (2 \beta_{6} + 6 \beta_{3} - 12 \beta_1) q^{83} + 6 q^{85} + 7 \beta_1 q^{86} + ( - \beta_{7} - \beta_{4}) q^{88} + (3 \beta_{6} - 3 \beta_{5} - 12 \beta_{3} + 6 \beta_1) q^{89} + (2 \beta_{4} + 2 \beta_{2} + 5) q^{91} + 6 \beta_{3} q^{92} + (\beta_{7} - \beta_{4} - 6 \beta_{2} + 12) q^{94} + (12 \beta_{3} - 12 \beta_1) q^{95} + (2 \beta_{7} + 2 \beta_{2} - 1) q^{97} + (2 \beta_{6} - 2 \beta_{5} - 5 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + b2 * q^4 + (-b6 + b5) * q^5 + (b7 - 1) * q^7 + b3 * q^8 + (b7 - b4) * q^10 + (-2*b6 + b5) * q^11 + (-b7 - 2*b2 + 1) * q^13 + (-b6 + b5 - b1) * q^14 + (b2 - 1) * q^16 - b5 * q^17 + 2*b4 * q^19 - b6 * q^20 + (b7 - 2*b4) * q^22 + 6*b1 * q^23 - b2 * q^25 + (b6 - b5 - 2*b3 + b1) * q^26 + (b7 - b4 - b2) * q^28 + (-b6 + 2*b5 + 6*b3) * q^29 + (-3*b7 + 3*b4 + b2 - 2) * q^31 + (b3 - b1) * q^32 - b7 * q^34 + (b6 - b5 - 6*b1) * q^35 + (-2*b7 + b4 + b2 - 1) * q^37 + 2*b5 * q^38 - b4 * q^40 - b6 * q^41 + 7 * q^43 + (-b6 - b5) * q^44 + 6*b2 * q^46 + (-b6 + b5 - 12*b3 + 6*b1) * q^47 + (-2*b7 - 5) * q^49 - b3 * q^50 + (-b7 + b4 - b2 + 2) * q^52 + (4*b6 - 2*b5 + 6*b3 - 6*b1) * q^53 + (-12*b2 + 6) * q^55 + (-b6 - b3) * q^56 + (2*b7 - b4 + 6*b2 - 6) * q^58 + b5 * q^59 + (-b4 + b2 + 1) * q^61 + (3*b6 + b3 - 2*b1) * q^62 - q^64 + (b6 + b5 + 6*b1) * q^65 + (2*b7 + 2*b4 - 5*b2) * q^67 + (b6 - b5) * q^68 + (-b7 + b4 - 6*b2) * q^70 + (3*b6 - 6*b5) * q^71 + (2*b7 - 2*b4 + 6*b2 - 12) * q^73 + (2*b6 - b5 + b3 - b1) * q^74 + 2*b7 * q^76 + (2*b6 - b5 - 6*b3 - 6*b1) * q^77 + (2*b7 - b4 + 5*b2 - 5) * q^79 - b5 * q^80 - b4 * q^82 + (2*b6 + 6*b3 - 12*b1) * q^83 + 6 * q^85 + 7*b1 * q^86 + (-b7 - b4) * q^88 + (3*b6 - 3*b5 - 12*b3 + 6*b1) * q^89 + (2*b4 + 2*b2 + 5) * q^91 + 6*b3 * q^92 + (b7 - b4 - 6*b2 + 12) * q^94 + (12*b3 - 12*b1) * q^95 + (2*b7 + 2*b2 - 1) * q^97 + (2*b6 - 2*b5 - 5*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{4} - 8 q^{7}+O(q^{10})$$ 8 * q + 4 * q^4 - 8 * q^7 $$8 q + 4 q^{4} - 8 q^{7} - 4 q^{16} - 4 q^{25} - 4 q^{28} - 12 q^{31} - 4 q^{37} + 56 q^{43} + 24 q^{46} - 40 q^{49} + 12 q^{52} - 24 q^{58} + 12 q^{61} - 8 q^{64} - 20 q^{67} - 24 q^{70} - 72 q^{73} - 20 q^{79} + 48 q^{85} + 48 q^{91} + 72 q^{94}+O(q^{100})$$ 8 * q + 4 * q^4 - 8 * q^7 - 4 * q^16 - 4 * q^25 - 4 * q^28 - 12 * q^31 - 4 * q^37 + 56 * q^43 + 24 * q^46 - 40 * q^49 + 12 * q^52 - 24 * q^58 + 12 * q^61 - 8 * q^64 - 20 * q^67 - 24 * q^70 - 72 * q^73 - 20 * q^79 + 48 * q^85 + 48 * q^91 + 72 * q^94

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{4}$$ $$=$$ $$\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24}$$ v^7 - v^5 + v^3 + 2*v $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - \zeta_{24}$$ -v^7 + 2*v^5 + 2*v^3 - v $$\beta_{6}$$ $$=$$ $$-2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -2*v^7 + v^5 + v^3 + v $$\beta_{7}$$ $$=$$ $$2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}$$ 2*v^7 + v^5 - v^3 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} ) / 6$$ (b7 + 2*b6 - b5 + b4) / 6 $$\zeta_{24}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{3}$$ $$=$$ $$( -\beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{4} ) / 6$$ (-b7 - b6 + 2*b5 + 2*b4) / 6 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{24}^{5}$$ $$=$$ $$( 2\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} ) / 6$$ (2*b7 + b6 + b5 - b4) / 6 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_{3}$$ b3 $$\zeta_{24}^{7}$$ $$=$$ $$( \beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} ) / 6$$ (b7 - 2*b6 + b5 + b4) / 6

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
215.1
 0.258819 − 0.965926i −0.258819 + 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i
−0.866025 0.500000i 0 0.500000 + 0.866025i −1.22474 + 2.12132i 0 −1.00000 2.44949i 1.00000i 0 2.12132 1.22474i
215.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.22474 2.12132i 0 −1.00000 + 2.44949i 1.00000i 0 −2.12132 + 1.22474i
215.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.22474 + 2.12132i 0 −1.00000 + 2.44949i 1.00000i 0 −2.12132 + 1.22474i
215.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.22474 2.12132i 0 −1.00000 2.44949i 1.00000i 0 2.12132 1.22474i
269.1 −0.866025 + 0.500000i 0 0.500000 0.866025i −1.22474 2.12132i 0 −1.00000 + 2.44949i 1.00000i 0 2.12132 + 1.22474i
269.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.22474 + 2.12132i 0 −1.00000 2.44949i 1.00000i 0 −2.12132 1.22474i
269.3 0.866025 0.500000i 0 0.500000 0.866025i −1.22474 2.12132i 0 −1.00000 2.44949i 1.00000i 0 −2.12132 1.22474i
269.4 0.866025 0.500000i 0 0.500000 0.866025i 1.22474 + 2.12132i 0 −1.00000 + 2.44949i 1.00000i 0 2.12132 + 1.22474i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 269.4 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.d 8
3.b odd 2 1 inner 378.2.k.d 8
7.c even 3 1 2646.2.d.d 8
7.d odd 6 1 inner 378.2.k.d 8
7.d odd 6 1 2646.2.d.d 8
9.c even 3 1 1134.2.l.e 8
9.c even 3 1 1134.2.t.f 8
9.d odd 6 1 1134.2.l.e 8
9.d odd 6 1 1134.2.t.f 8
21.g even 6 1 inner 378.2.k.d 8
21.g even 6 1 2646.2.d.d 8
21.h odd 6 1 2646.2.d.d 8
63.i even 6 1 1134.2.t.f 8
63.k odd 6 1 1134.2.l.e 8
63.s even 6 1 1134.2.l.e 8
63.t odd 6 1 1134.2.t.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.k.d 8 1.a even 1 1 trivial
378.2.k.d 8 3.b odd 2 1 inner
378.2.k.d 8 7.d odd 6 1 inner
378.2.k.d 8 21.g even 6 1 inner
1134.2.l.e 8 9.c even 3 1
1134.2.l.e 8 9.d odd 6 1
1134.2.l.e 8 63.k odd 6 1
1134.2.l.e 8 63.s even 6 1
1134.2.t.f 8 9.c even 3 1
1134.2.t.f 8 9.d odd 6 1
1134.2.t.f 8 63.i even 6 1
1134.2.t.f 8 63.t odd 6 1
2646.2.d.d 8 7.c even 3 1
2646.2.d.d 8 7.d odd 6 1
2646.2.d.d 8 21.g even 6 1
2646.2.d.d 8 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 6T_{5}^{2} + 36$$ acting on $$S_{2}^{\mathrm{new}}(378, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{2}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 6 T^{2} + 36)^{2}$$
$7$ $$(T^{2} + 2 T + 7)^{4}$$
$11$ $$(T^{4} - 18 T^{2} + 324)^{2}$$
$13$ $$(T^{4} + 18 T^{2} + 9)^{2}$$
$17$ $$(T^{4} + 6 T^{2} + 36)^{2}$$
$19$ $$(T^{4} - 24 T^{2} + 576)^{2}$$
$23$ $$(T^{4} - 36 T^{2} + 1296)^{2}$$
$29$ $$(T^{4} + 108 T^{2} + 324)^{2}$$
$31$ $$(T^{4} + 6 T^{3} - 39 T^{2} - 306 T + 2601)^{2}$$
$37$ $$(T^{4} + 2 T^{3} + 21 T^{2} - 34 T + 289)^{2}$$
$41$ $$(T^{2} - 6)^{4}$$
$43$ $$(T - 7)^{8}$$
$47$ $$T^{8} + 228 T^{6} + \cdots + 108243216$$
$53$ $$T^{8} - 216 T^{6} + 45360 T^{4} + \cdots + 1679616$$
$59$ $$(T^{4} + 6 T^{2} + 36)^{2}$$
$61$ $$(T^{4} - 6 T^{3} + 9 T^{2} + 18 T + 9)^{2}$$
$67$ $$(T^{4} + 10 T^{3} + 147 T^{2} - 470 T + 2209)^{2}$$
$71$ $$(T^{2} + 162)^{4}$$
$73$ $$(T^{4} + 36 T^{3} + 516 T^{2} + 3024 T + 7056)^{2}$$
$79$ $$(T^{4} + 10 T^{3} + 93 T^{2} + 70 T + 49)^{2}$$
$83$ $$(T^{4} - 264 T^{2} + 7056)^{2}$$
$89$ $$T^{8} + 324 T^{6} + 102060 T^{4} + \cdots + 8503056$$
$97$ $$(T^{4} + 54 T^{2} + 441)^{2}$$