# Properties

 Label 392.3.g.e Level 392 Weight 3 Character orbit 392.g 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.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

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

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{2} + q^{3} -4 \zeta_{6} q^{4} + ( 3 - 6 \zeta_{6} ) q^{5} + ( 2 - 2 \zeta_{6} ) q^{6} -8 q^{8} -8 q^{9} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{2} + q^{3} -4 \zeta_{6} q^{4} + ( 3 - 6 \zeta_{6} ) q^{5} + ( 2 - 2 \zeta_{6} ) q^{6} -8 q^{8} -8 q^{9} + ( -6 - 6 \zeta_{6} ) q^{10} + 17 q^{11} -4 \zeta_{6} q^{12} + ( 8 - 16 \zeta_{6} ) q^{13} + ( 3 - 6 \zeta_{6} ) q^{15} + ( -16 + 16 \zeta_{6} ) q^{16} -25 q^{17} + ( -16 + 16 \zeta_{6} ) q^{18} -7 q^{19} + ( -24 + 12 \zeta_{6} ) q^{20} + ( 34 - 34 \zeta_{6} ) q^{22} + ( -3 + 6 \zeta_{6} ) q^{23} -8 q^{24} -2 q^{25} + ( -16 - 16 \zeta_{6} ) q^{26} -17 q^{27} + ( -8 + 16 \zeta_{6} ) q^{29} + ( -6 - 6 \zeta_{6} ) q^{30} + ( 19 - 38 \zeta_{6} ) q^{31} + 32 \zeta_{6} q^{32} + 17 q^{33} + ( -50 + 50 \zeta_{6} ) q^{34} + 32 \zeta_{6} q^{36} + ( -5 + 10 \zeta_{6} ) q^{37} + ( -14 + 14 \zeta_{6} ) q^{38} + ( 8 - 16 \zeta_{6} ) q^{39} + ( -24 + 48 \zeta_{6} ) q^{40} + 26 q^{41} + 14 q^{43} -68 \zeta_{6} q^{44} + ( -24 + 48 \zeta_{6} ) q^{45} + ( 6 + 6 \zeta_{6} ) q^{46} + ( 29 - 58 \zeta_{6} ) q^{47} + ( -16 + 16 \zeta_{6} ) q^{48} + ( -4 + 4 \zeta_{6} ) q^{50} -25 q^{51} + ( -64 + 32 \zeta_{6} ) q^{52} + ( 53 - 106 \zeta_{6} ) q^{53} + ( -34 + 34 \zeta_{6} ) q^{54} + ( 51 - 102 \zeta_{6} ) q^{55} -7 q^{57} + ( 16 + 16 \zeta_{6} ) q^{58} -55 q^{59} + ( -24 + 12 \zeta_{6} ) q^{60} + ( -13 + 26 \zeta_{6} ) q^{61} + ( -38 - 38 \zeta_{6} ) q^{62} + 64 q^{64} -72 q^{65} + ( 34 - 34 \zeta_{6} ) q^{66} + 17 q^{67} + 100 \zeta_{6} q^{68} + ( -3 + 6 \zeta_{6} ) q^{69} + 64 q^{72} + 119 q^{73} + ( 10 + 10 \zeta_{6} ) q^{74} -2 q^{75} + 28 \zeta_{6} q^{76} + ( -16 - 16 \zeta_{6} ) q^{78} + ( -43 + 86 \zeta_{6} ) q^{79} + ( 48 + 48 \zeta_{6} ) q^{80} + 55 q^{81} + ( 52 - 52 \zeta_{6} ) q^{82} + 110 q^{83} + ( -75 + 150 \zeta_{6} ) q^{85} + ( 28 - 28 \zeta_{6} ) q^{86} + ( -8 + 16 \zeta_{6} ) q^{87} -136 q^{88} + 71 q^{89} + ( 48 + 48 \zeta_{6} ) q^{90} + ( 24 - 12 \zeta_{6} ) q^{92} + ( 19 - 38 \zeta_{6} ) q^{93} + ( -58 - 58 \zeta_{6} ) q^{94} + ( -21 + 42 \zeta_{6} ) q^{95} + 32 \zeta_{6} q^{96} -22 q^{97} -136 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{3} - 4q^{4} + 2q^{6} - 16q^{8} - 16q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{3} - 4q^{4} + 2q^{6} - 16q^{8} - 16q^{9} - 18q^{10} + 34q^{11} - 4q^{12} - 16q^{16} - 50q^{17} - 16q^{18} - 14q^{19} - 36q^{20} + 34q^{22} - 16q^{24} - 4q^{25} - 48q^{26} - 34q^{27} - 18q^{30} + 32q^{32} + 34q^{33} - 50q^{34} + 32q^{36} - 14q^{38} + 52q^{41} + 28q^{43} - 68q^{44} + 18q^{46} - 16q^{48} - 4q^{50} - 50q^{51} - 96q^{52} - 34q^{54} - 14q^{57} + 48q^{58} - 110q^{59} - 36q^{60} - 114q^{62} + 128q^{64} - 144q^{65} + 34q^{66} + 34q^{67} + 100q^{68} + 128q^{72} + 238q^{73} + 30q^{74} - 4q^{75} + 28q^{76} - 48q^{78} + 144q^{80} + 110q^{81} + 52q^{82} + 220q^{83} + 28q^{86} - 272q^{88} + 142q^{89} + 144q^{90} + 36q^{92} - 174q^{94} + 32q^{96} - 44q^{97} - 272q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$197$$ $$295$$ $$297$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 1.73205i 1.00000 −2.00000 3.46410i 5.19615i 1.00000 1.73205i 0 −8.00000 −8.00000 −9.00000 5.19615i
99.2 1.00000 + 1.73205i 1.00000 −2.00000 + 3.46410i 5.19615i 1.00000 + 1.73205i 0 −8.00000 −8.00000 −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
8.d odd 2 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.k.a 2 7.c even 3 1
56.3.k.a 2 56.k odd 6 1
56.3.k.b yes 2 7.c even 3 1
56.3.k.b yes 2 56.k odd 6 1
224.3.o.a 2 28.g odd 6 1
224.3.o.a 2 56.p even 6 1
224.3.o.b 2 28.g odd 6 1
224.3.o.b 2 56.p even 6 1
392.3.g.d 2 7.b odd 2 1
392.3.g.d 2 56.e even 2 1
392.3.g.e 2 1.a even 1 1 trivial
392.3.g.e 2 8.d odd 2 1 inner
392.3.k.a 2 7.d odd 6 1
392.3.k.a 2 56.m even 6 1
392.3.k.c 2 7.d odd 6 1
392.3.k.c 2 56.m even 6 1
1568.3.g.c 2 4.b odd 2 1
1568.3.g.c 2 8.b even 2 1
1568.3.g.f 2 28.d even 2 1
1568.3.g.f 2 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 1$$ acting on $$S_{3}^{\mathrm{new}}(392, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 4 T^{2}$$
$3$ $$( 1 - T + 9 T^{2} )^{2}$$
$5$ $$1 - 23 T^{2} + 625 T^{4}$$
$7$ 1
$11$ $$( 1 - 17 T + 121 T^{2} )^{2}$$
$13$ $$( 1 - 22 T + 169 T^{2} )( 1 + 22 T + 169 T^{2} )$$
$17$ $$( 1 + 25 T + 289 T^{2} )^{2}$$
$19$ $$( 1 + 7 T + 361 T^{2} )^{2}$$
$23$ $$1 - 1031 T^{2} + 279841 T^{4}$$
$29$ $$1 - 1490 T^{2} + 707281 T^{4}$$
$31$ $$1 - 839 T^{2} + 923521 T^{4}$$
$37$ $$1 - 2663 T^{2} + 1874161 T^{4}$$
$41$ $$( 1 - 26 T + 1681 T^{2} )^{2}$$
$43$ $$( 1 - 14 T + 1849 T^{2} )^{2}$$
$47$ $$1 - 1895 T^{2} + 4879681 T^{4}$$
$53$ $$( 1 - 53 T + 2809 T^{2} )( 1 + 53 T + 2809 T^{2} )$$
$59$ $$( 1 + 55 T + 3481 T^{2} )^{2}$$
$61$ $$1 - 6935 T^{2} + 13845841 T^{4}$$
$67$ $$( 1 - 17 T + 4489 T^{2} )^{2}$$
$71$ $$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$
$73$ $$( 1 - 119 T + 5329 T^{2} )^{2}$$
$79$ $$1 - 6935 T^{2} + 38950081 T^{4}$$
$83$ $$( 1 - 110 T + 6889 T^{2} )^{2}$$
$89$ $$( 1 - 71 T + 7921 T^{2} )^{2}$$
$97$ $$( 1 + 22 T + 9409 T^{2} )^{2}$$