# Properties

 Label 378.2.k.b Level 378 Weight 2 Character orbit 378.k 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.k (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{5} + ( 3 - 2 \zeta_{12}^{2} ) q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{5} + ( 3 - 2 \zeta_{12}^{2} ) q^{7} + \zeta_{12}^{3} q^{8} + ( -4 + 2 \zeta_{12}^{2} ) q^{10} + ( -3 + 6 \zeta_{12}^{2} ) q^{13} + ( 3 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{17} + ( -2 - 2 \zeta_{12}^{2} ) q^{19} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{20} -6 \zeta_{12} q^{23} -7 \zeta_{12}^{2} q^{25} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{26} + ( 2 + \zeta_{12}^{2} ) q^{28} + ( 10 - 5 \zeta_{12}^{2} ) q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + ( -4 + 8 \zeta_{12}^{2} ) q^{34} + ( 2 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{35} + ( 5 - 5 \zeta_{12}^{2} ) q^{37} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{38} + ( -2 - 2 \zeta_{12}^{2} ) q^{40} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{41} + q^{43} -6 \zeta_{12}^{2} q^{46} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{47} + ( 5 - 8 \zeta_{12}^{2} ) q^{49} -7 \zeta_{12}^{3} q^{50} + ( -6 + 3 \zeta_{12}^{2} ) q^{52} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{53} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{56} + ( -4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{59} + ( -1 - \zeta_{12}^{2} ) q^{61} + ( 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{62} - q^{64} -18 \zeta_{12} q^{65} + 13 \zeta_{12}^{2} q^{67} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{68} + ( -8 + 10 \zeta_{12}^{2} ) q^{70} -6 \zeta_{12}^{3} q^{71} + ( -8 + 4 \zeta_{12}^{2} ) q^{73} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{74} + ( 2 - 4 \zeta_{12}^{2} ) q^{76} + ( 7 - 7 \zeta_{12}^{2} ) q^{79} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{80} + ( 4 + 4 \zeta_{12}^{2} ) q^{82} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{83} -24 q^{85} + \zeta_{12} q^{86} + ( -6 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{89} + ( 3 + 12 \zeta_{12}^{2} ) q^{91} -6 \zeta_{12}^{3} q^{92} + ( 8 - 4 \zeta_{12}^{2} ) q^{94} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{95} + ( 1 - 2 \zeta_{12}^{2} ) q^{97} + ( 5 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + 8q^{7} + O(q^{10})$$ $$4q + 2q^{4} + 8q^{7} - 12q^{10} - 2q^{16} - 12q^{19} - 14q^{25} + 10q^{28} + 30q^{31} + 10q^{37} - 12q^{40} + 4q^{43} - 12q^{46} + 4q^{49} - 18q^{52} - 6q^{61} - 4q^{64} + 26q^{67} - 12q^{70} - 24q^{73} + 14q^{79} + 24q^{82} - 96q^{85} + 36q^{91} + 24q^{94} + O(q^{100})$$

## Character values

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

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$-1$$ $$1 - \zeta_{12}^{2}$$

## 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
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i 1.73205 3.00000i 0 2.00000 1.73205i 1.00000i 0 −3.00000 + 1.73205i
215.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.73205 + 3.00000i 0 2.00000 1.73205i 1.00000i 0 −3.00000 + 1.73205i
269.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.73205 + 3.00000i 0 2.00000 + 1.73205i 1.00000i 0 −3.00000 1.73205i
269.2 0.866025 0.500000i 0 0.500000 0.866025i −1.73205 3.00000i 0 2.00000 + 1.73205i 1.00000i 0 −3.00000 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.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.b 4
3.b odd 2 1 inner 378.2.k.b 4
7.c even 3 1 2646.2.d.b 4
7.d odd 6 1 inner 378.2.k.b 4
7.d odd 6 1 2646.2.d.b 4
9.c even 3 1 1134.2.l.a 4
9.c even 3 1 1134.2.t.b 4
9.d odd 6 1 1134.2.l.a 4
9.d odd 6 1 1134.2.t.b 4
21.g even 6 1 inner 378.2.k.b 4
21.g even 6 1 2646.2.d.b 4
21.h odd 6 1 2646.2.d.b 4
63.i even 6 1 1134.2.t.b 4
63.k odd 6 1 1134.2.l.a 4
63.s even 6 1 1134.2.l.a 4
63.t odd 6 1 1134.2.t.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.k.b 4 1.a even 1 1 trivial
378.2.k.b 4 3.b odd 2 1 inner
378.2.k.b 4 7.d odd 6 1 inner
378.2.k.b 4 21.g even 6 1 inner
1134.2.l.a 4 9.c even 3 1
1134.2.l.a 4 9.d odd 6 1
1134.2.l.a 4 63.k odd 6 1
1134.2.l.a 4 63.s even 6 1
1134.2.t.b 4 9.c even 3 1
1134.2.t.b 4 9.d odd 6 1
1134.2.t.b 4 63.i even 6 1
1134.2.t.b 4 63.t odd 6 1
2646.2.d.b 4 7.c even 3 1
2646.2.d.b 4 7.d odd 6 1
2646.2.d.b 4 21.g even 6 1
2646.2.d.b 4 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 12 T_{5}^{2} + 144$$ acting on $$S_{2}^{\mathrm{new}}(378, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ 
$5$ $$1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8}$$
$7$ $$( 1 - 4 T + 7 T^{2} )^{2}$$
$11$ $$( 1 + 11 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 5 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2}$$
$17$ $$1 + 14 T^{2} - 93 T^{4} + 4046 T^{6} + 83521 T^{8}$$
$19$ $$( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2}$$
$23$ $$1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 29 T^{2} )^{4}$$
$31$ $$( 1 - 11 T + 31 T^{2} )^{2}( 1 - 4 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 34 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - T + 43 T^{2} )^{4}$$
$47$ $$1 - 46 T^{2} - 93 T^{4} - 101614 T^{6} + 4879681 T^{8}$$
$53$ $$1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8}$$
$59$ $$1 - 70 T^{2} + 1419 T^{4} - 243670 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 + 3 T + 64 T^{2} + 183 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 106 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 154 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8}$$
$97$ $$( 1 - 191 T^{2} + 9409 T^{4} )^{2}$$