# Properties

 Label 392.3.k.a Level 392 Weight 3 Character orbit 392.k Analytic conductor 10.681 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$392 = 2^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 392.k (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.6812263629$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + \zeta_{6} q^{3} + 4 q^{4} + ( 3 + 3 \zeta_{6} ) q^{5} -2 \zeta_{6} q^{6} -8 q^{8} + ( 8 - 8 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q -2 q^{2} + \zeta_{6} q^{3} + 4 q^{4} + ( 3 + 3 \zeta_{6} ) q^{5} -2 \zeta_{6} q^{6} -8 q^{8} + ( 8 - 8 \zeta_{6} ) q^{9} + ( -6 - 6 \zeta_{6} ) q^{10} -17 \zeta_{6} q^{11} + 4 \zeta_{6} q^{12} + ( 8 - 16 \zeta_{6} ) q^{13} + ( -3 + 6 \zeta_{6} ) q^{15} + 16 q^{16} -25 \zeta_{6} q^{17} + ( -16 + 16 \zeta_{6} ) q^{18} + ( -7 + 7 \zeta_{6} ) q^{19} + ( 12 + 12 \zeta_{6} ) q^{20} + 34 \zeta_{6} q^{22} + ( 3 + 3 \zeta_{6} ) q^{23} -8 \zeta_{6} q^{24} + 2 \zeta_{6} q^{25} + ( -16 + 32 \zeta_{6} ) q^{26} + 17 q^{27} + ( 8 - 16 \zeta_{6} ) q^{29} + ( 6 - 12 \zeta_{6} ) q^{30} + ( -38 + 19 \zeta_{6} ) q^{31} -32 q^{32} + ( 17 - 17 \zeta_{6} ) q^{33} + 50 \zeta_{6} q^{34} + ( 32 - 32 \zeta_{6} ) q^{36} + ( 5 + 5 \zeta_{6} ) q^{37} + ( 14 - 14 \zeta_{6} ) q^{38} + ( 16 - 8 \zeta_{6} ) q^{39} + ( -24 - 24 \zeta_{6} ) q^{40} -26 q^{41} + 14 q^{43} -68 \zeta_{6} q^{44} + ( 48 - 24 \zeta_{6} ) q^{45} + ( -6 - 6 \zeta_{6} ) q^{46} + ( 29 + 29 \zeta_{6} ) q^{47} + 16 \zeta_{6} q^{48} -4 \zeta_{6} q^{50} + ( 25 - 25 \zeta_{6} ) q^{51} + ( 32 - 64 \zeta_{6} ) q^{52} + ( 106 - 53 \zeta_{6} ) q^{53} -34 q^{54} + ( 51 - 102 \zeta_{6} ) q^{55} -7 q^{57} + ( -16 + 32 \zeta_{6} ) q^{58} -55 \zeta_{6} q^{59} + ( -12 + 24 \zeta_{6} ) q^{60} + ( -13 - 13 \zeta_{6} ) q^{61} + ( 76 - 38 \zeta_{6} ) q^{62} + 64 q^{64} + ( 72 - 72 \zeta_{6} ) q^{65} + ( -34 + 34 \zeta_{6} ) q^{66} -17 \zeta_{6} q^{67} -100 \zeta_{6} q^{68} + ( -3 + 6 \zeta_{6} ) q^{69} + ( -64 + 64 \zeta_{6} ) q^{72} + 119 \zeta_{6} q^{73} + ( -10 - 10 \zeta_{6} ) q^{74} + ( -2 + 2 \zeta_{6} ) q^{75} + ( -28 + 28 \zeta_{6} ) q^{76} + ( -32 + 16 \zeta_{6} ) q^{78} + ( 43 + 43 \zeta_{6} ) q^{79} + ( 48 + 48 \zeta_{6} ) q^{80} -55 \zeta_{6} q^{81} + 52 q^{82} -110 q^{83} + ( 75 - 150 \zeta_{6} ) q^{85} -28 q^{86} + ( 16 - 8 \zeta_{6} ) q^{87} + 136 \zeta_{6} q^{88} + ( 71 - 71 \zeta_{6} ) q^{89} + ( -96 + 48 \zeta_{6} ) q^{90} + ( 12 + 12 \zeta_{6} ) q^{92} + ( -19 - 19 \zeta_{6} ) q^{93} + ( -58 - 58 \zeta_{6} ) q^{94} + ( -42 + 21 \zeta_{6} ) q^{95} -32 \zeta_{6} q^{96} + 22 q^{97} -136 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + q^{3} + 8q^{4} + 9q^{5} - 2q^{6} - 16q^{8} + 8q^{9} + O(q^{10})$$ $$2q - 4q^{2} + q^{3} + 8q^{4} + 9q^{5} - 2q^{6} - 16q^{8} + 8q^{9} - 18q^{10} - 17q^{11} + 4q^{12} + 32q^{16} - 25q^{17} - 16q^{18} - 7q^{19} + 36q^{20} + 34q^{22} + 9q^{23} - 8q^{24} + 2q^{25} + 34q^{27} - 57q^{31} - 64q^{32} + 17q^{33} + 50q^{34} + 32q^{36} + 15q^{37} + 14q^{38} + 24q^{39} - 72q^{40} - 52q^{41} + 28q^{43} - 68q^{44} + 72q^{45} - 18q^{46} + 87q^{47} + 16q^{48} - 4q^{50} + 25q^{51} + 159q^{53} - 68q^{54} - 14q^{57} - 55q^{59} - 39q^{61} + 114q^{62} + 128q^{64} + 72q^{65} - 34q^{66} - 17q^{67} - 100q^{68} - 64q^{72} + 119q^{73} - 30q^{74} - 2q^{75} - 28q^{76} - 48q^{78} + 129q^{79} + 144q^{80} - 55q^{81} + 104q^{82} - 220q^{83} - 56q^{86} + 24q^{87} + 136q^{88} + 71q^{89} - 144q^{90} + 36q^{92} - 57q^{93} - 174q^{94} - 63q^{95} - 32q^{96} + 44q^{97} - 272q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$295$$ $$297$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1 + \zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 − 0.866025i 0.5 + 0.866025i
−2.00000 0.500000 0.866025i 4.00000 4.50000 2.59808i −1.00000 + 1.73205i 0 −8.00000 4.00000 + 6.92820i −9.00000 + 5.19615i
275.1 −2.00000 0.500000 + 0.866025i 4.00000 4.50000 + 2.59808i −1.00000 1.73205i 0 −8.00000 4.00000 6.92820i −9.00000 5.19615i
 $$n$$: 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
56.k odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.3.k.a 2
7.b odd 2 1 56.3.k.a 2
7.c even 3 1 392.3.g.d 2
7.c even 3 1 392.3.k.c 2
7.d odd 6 1 56.3.k.b yes 2
7.d odd 6 1 392.3.g.e 2
8.d odd 2 1 392.3.k.c 2
28.d even 2 1 224.3.o.a 2
28.f even 6 1 224.3.o.b 2
28.f even 6 1 1568.3.g.c 2
28.g odd 6 1 1568.3.g.f 2
56.e even 2 1 56.3.k.b yes 2
56.h odd 2 1 224.3.o.b 2
56.j odd 6 1 224.3.o.a 2
56.j odd 6 1 1568.3.g.c 2
56.k odd 6 1 392.3.g.d 2
56.k odd 6 1 inner 392.3.k.a 2
56.m even 6 1 56.3.k.a 2
56.m even 6 1 392.3.g.e 2
56.p even 6 1 1568.3.g.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.k.a 2 7.b odd 2 1
56.3.k.a 2 56.m even 6 1
56.3.k.b yes 2 7.d odd 6 1
56.3.k.b yes 2 56.e even 2 1
224.3.o.a 2 28.d even 2 1
224.3.o.a 2 56.j odd 6 1
224.3.o.b 2 28.f even 6 1
224.3.o.b 2 56.h odd 2 1
392.3.g.d 2 7.c even 3 1
392.3.g.d 2 56.k odd 6 1
392.3.g.e 2 7.d odd 6 1
392.3.g.e 2 56.m even 6 1
392.3.k.a 2 1.a even 1 1 trivial
392.3.k.a 2 56.k odd 6 1 inner
392.3.k.c 2 7.c even 3 1
392.3.k.c 2 8.d odd 2 1
1568.3.g.c 2 28.f even 6 1
1568.3.g.c 2 56.j odd 6 1
1568.3.g.f 2 28.g odd 6 1
1568.3.g.f 2 56.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(392, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{5}^{2} - 9 T_{5} + 27$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T )^{2}$$
$3$ $$1 - T - 8 T^{2} - 9 T^{3} + 81 T^{4}$$
$5$ $$1 - 9 T + 52 T^{2} - 225 T^{3} + 625 T^{4}$$
$7$ 1
$11$ $$1 + 17 T + 168 T^{2} + 2057 T^{3} + 14641 T^{4}$$
$13$ $$( 1 - 22 T + 169 T^{2} )( 1 + 22 T + 169 T^{2} )$$
$17$ $$1 + 25 T + 336 T^{2} + 7225 T^{3} + 83521 T^{4}$$
$19$ $$1 + 7 T - 312 T^{2} + 2527 T^{3} + 130321 T^{4}$$
$23$ $$1 - 9 T + 556 T^{2} - 4761 T^{3} + 279841 T^{4}$$
$29$ $$1 - 1490 T^{2} + 707281 T^{4}$$
$31$ $$1 + 57 T + 2044 T^{2} + 54777 T^{3} + 923521 T^{4}$$
$37$ $$1 - 15 T + 1444 T^{2} - 20535 T^{3} + 1874161 T^{4}$$
$41$ $$( 1 + 26 T + 1681 T^{2} )^{2}$$
$43$ $$( 1 - 14 T + 1849 T^{2} )^{2}$$
$47$ $$1 - 87 T + 4732 T^{2} - 192183 T^{3} + 4879681 T^{4}$$
$53$ $$( 1 - 53 T )^{2}( 1 - 53 T + 2809 T^{2} )$$
$59$ $$1 + 55 T - 456 T^{2} + 191455 T^{3} + 12117361 T^{4}$$
$61$ $$1 + 39 T + 4228 T^{2} + 145119 T^{3} + 13845841 T^{4}$$
$67$ $$1 + 17 T - 4200 T^{2} + 76313 T^{3} + 20151121 T^{4}$$
$71$ $$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$
$73$ $$1 - 119 T + 8832 T^{2} - 634151 T^{3} + 28398241 T^{4}$$
$79$ $$1 - 129 T + 11788 T^{2} - 805089 T^{3} + 38950081 T^{4}$$
$83$ $$( 1 + 110 T + 6889 T^{2} )^{2}$$
$89$ $$1 - 71 T - 2880 T^{2} - 562391 T^{3} + 62742241 T^{4}$$
$97$ $$( 1 - 22 T + 9409 T^{2} )^{2}$$