# 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: $$\Q(\sqrt{-3}, \sqrt{7})$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$

## 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 $$S_{2}^{\mathrm{new}}(378, [\chi])$$:

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ 
$5$ $$1 + 2 T - 12 T^{3} - 29 T^{4} - 60 T^{5} + 250 T^{7} + 625 T^{8}$$
$7$ $$( 1 - 7 T^{2} )^{2}$$
$11$ $$1 - 2 T - 12 T^{2} + 12 T^{3} + 91 T^{4} + 132 T^{5} - 1452 T^{6} - 2662 T^{7} + 14641 T^{8}$$
$13$ $$( 1 + 4 T + 23 T^{2} + 52 T^{3} + 169 T^{4} )^{2}$$
$17$ $$1 - 2 T - 24 T^{2} + 12 T^{3} + 427 T^{4} + 204 T^{5} - 6936 T^{6} - 9826 T^{7} + 83521 T^{8}$$
$19$ $$( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 - 8 T + 30 T^{2} + 96 T^{3} - 845 T^{4} + 2208 T^{5} + 15870 T^{6} - 97336 T^{7} + 279841 T^{8}$$
$29$ $$( 1 + 10 T + 76 T^{2} + 290 T^{3} + 841 T^{4} )^{2}$$
$31$ $$1 - 4 T - 43 T^{2} + 12 T^{3} + 2024 T^{4} + 372 T^{5} - 41323 T^{6} - 119164 T^{7} + 923521 T^{8}$$
$37$ $$( 1 - 8 T + 37 T^{2} )^{2}( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} )$$
$41$ $$( 1 + 6 T + 28 T^{2} + 246 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 5 T + 43 T^{2} )^{4}$$
$47$ $$1 + 6 T - 4 T^{2} - 324 T^{3} - 2301 T^{4} - 15228 T^{5} - 8836 T^{6} + 622938 T^{7} + 4879681 T^{8}$$
$53$ $$( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 - 22 T + 252 T^{2} - 2508 T^{3} + 21787 T^{4} - 147972 T^{5} + 877212 T^{6} - 4518338 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 20 T + 185 T^{2} + 1860 T^{3} + 18104 T^{4} + 113460 T^{5} + 688385 T^{6} + 4539620 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 6 T - 79 T^{2} - 114 T^{3} + 6324 T^{4} - 7638 T^{5} - 354631 T^{6} + 1804578 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 + 26 T + 304 T^{2} + 1846 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$1 - 34 T^{2} - 4173 T^{4} - 181186 T^{6} + 28398241 T^{8}$$
$79$ $$1 - 4 T + 29 T^{2} + 684 T^{3} - 7336 T^{4} + 54036 T^{5} + 180989 T^{6} - 1972156 T^{7} + 38950081 T^{8}$$
$83$ $$( 1 - 16 T + 202 T^{2} - 1328 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$1 + 6 T - 88 T^{2} - 324 T^{3} + 4251 T^{4} - 28836 T^{5} - 697048 T^{6} + 4229814 T^{7} + 62742241 T^{8}$$
$97$ $$( 1 - 6 T + 91 T^{2} - 582 T^{3} + 9409 T^{4} )^{2}$$