# 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: $$\Q(\zeta_{12})$$ 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.

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

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

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ 
$5$ $$( 1 + 7 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 + 5 T + 7 T^{2} )^{2}$$
$11$ $$( 1 - 13 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 2 T + 13 T^{2} )^{2}( 1 + 2 T + 13 T^{2} )^{2}$$
$17$ $$( 1 - 14 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2}$$
$23$ $$( 1 - 10 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 22 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 35 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{4}$$
$41$ $$( 1 + 70 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 2 T + 43 T^{2} )^{4}$$
$47$ $$( 1 + 82 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 97 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 106 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2}$$
$67$ $$( 1 - 2 T + 67 T^{2} )^{4}$$
$71$ $$( 1 + 2 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{4}$$
$83$ $$( 1 + 163 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 70 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 47 T^{2} + 9409 T^{4} )^{2}$$