# Properties

 Label 392.3.g.h Level 392 Weight 3 Character orbit 392.g Analytic conductor 10.681 Analytic rank 0 Dimension 4 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: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-7})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -2 - \beta_{2} ) q^{3} + ( -3 + \beta_{3} ) q^{4} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{5} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{6} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{8} + ( -3 + 4 \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -2 - \beta_{2} ) q^{3} + ( -3 + \beta_{3} ) q^{4} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{5} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{6} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{8} + ( -3 + 4 \beta_{2} ) q^{9} + ( 7 - 2 \beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{10} + ( 4 - 6 \beta_{2} ) q^{11} + ( 6 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{12} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{13} -2 \beta_{3} q^{15} + ( 2 - 6 \beta_{3} ) q^{16} + ( -18 - 4 \beta_{2} ) q^{17} + ( 4 + 3 \beta_{1} + 4 \beta_{3} ) q^{18} + ( -2 + 19 \beta_{2} ) q^{19} + ( -14 - 6 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{20} + ( -6 - 4 \beta_{1} - 6 \beta_{3} ) q^{22} + ( 16 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{23} + ( 10 - 4 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} ) q^{24} + ( -17 + 28 \beta_{2} ) q^{25} + ( -7 + 2 \beta_{1} + 8 \beta_{2} + \beta_{3} ) q^{26} + ( 16 + 4 \beta_{2} ) q^{27} + ( -12 \beta_{1} - 6 \beta_{2} ) q^{29} + ( 2 \beta_{1} + 8 \beta_{2} ) q^{30} + ( 8 \beta_{1} + 4 \beta_{2} - 12 \beta_{3} ) q^{31} + ( 4 \beta_{1} + 24 \beta_{2} ) q^{32} + ( 4 + 8 \beta_{2} ) q^{33} + ( -4 + 18 \beta_{1} - 4 \beta_{3} ) q^{34} + ( 9 - 8 \beta_{1} - 16 \beta_{2} - 3 \beta_{3} ) q^{36} + ( -20 \beta_{1} - 10 \beta_{2} + 8 \beta_{3} ) q^{37} + ( 19 + 2 \beta_{1} + 19 \beta_{3} ) q^{38} + 2 \beta_{3} q^{39} + ( -14 + 20 \beta_{1} + 24 \beta_{2} + 10 \beta_{3} ) q^{40} + ( 10 + 12 \beta_{2} ) q^{41} + ( -20 + 2 \beta_{2} ) q^{43} + ( -12 + 12 \beta_{1} + 24 \beta_{2} + 4 \beta_{3} ) q^{44} + ( -22 \beta_{1} - 11 \beta_{2} - 14 \beta_{3} ) q^{45} + ( 56 + 2 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{46} + ( 8 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{47} + ( -4 - 12 \beta_{1} - 8 \beta_{2} + 12 \beta_{3} ) q^{48} + ( 28 + 17 \beta_{1} + 28 \beta_{3} ) q^{50} + ( 44 + 26 \beta_{2} ) q^{51} + ( 14 + 6 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} ) q^{52} + ( 24 \beta_{1} + 12 \beta_{2} - 20 \beta_{3} ) q^{53} + ( 4 - 16 \beta_{1} + 4 \beta_{3} ) q^{54} + ( 32 \beta_{1} + 16 \beta_{2} + 20 \beta_{3} ) q^{55} + ( -34 - 36 \beta_{2} ) q^{57} + ( -42 + 6 \beta_{3} ) q^{58} + ( -46 - 11 \beta_{2} ) q^{59} + ( 14 + 6 \beta_{3} ) q^{60} + ( 6 \beta_{1} + 3 \beta_{2} - 10 \beta_{3} ) q^{61} + ( 28 + 12 \beta_{1} + 48 \beta_{2} - 4 \beta_{3} ) q^{62} + ( 36 + 20 \beta_{3} ) q^{64} + ( 42 - 28 \beta_{2} ) q^{65} + ( 8 - 4 \beta_{1} + 8 \beta_{3} ) q^{66} + ( -56 + 16 \beta_{2} ) q^{67} + ( 54 + 8 \beta_{1} + 16 \beta_{2} - 18 \beta_{3} ) q^{68} + ( -36 \beta_{1} - 18 \beta_{2} + 20 \beta_{3} ) q^{69} + ( 32 \beta_{1} + 16 \beta_{2} - 16 \beta_{3} ) q^{71} + ( -40 - 6 \beta_{1} + 12 \beta_{2} - 8 \beta_{3} ) q^{72} + ( -58 - 8 \beta_{2} ) q^{73} + ( -70 - 8 \beta_{1} - 32 \beta_{2} + 10 \beta_{3} ) q^{74} + ( -22 - 39 \beta_{2} ) q^{75} + ( 6 - 38 \beta_{1} - 76 \beta_{2} - 2 \beta_{3} ) q^{76} + ( -2 \beta_{1} - 8 \beta_{2} ) q^{78} + ( 32 \beta_{1} + 16 \beta_{2} + 8 \beta_{3} ) q^{79} + ( 84 + 4 \beta_{1} - 40 \beta_{2} + 4 \beta_{3} ) q^{80} + ( -13 - 60 \beta_{2} ) q^{81} + ( 12 - 10 \beta_{1} + 12 \beta_{3} ) q^{82} + ( -22 + 13 \beta_{2} ) q^{83} + ( -20 \beta_{1} - 10 \beta_{2} - 28 \beta_{3} ) q^{85} + ( 2 + 20 \beta_{1} + 2 \beta_{3} ) q^{86} + ( 24 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{87} + ( 60 + 8 \beta_{1} - 16 \beta_{2} + 12 \beta_{3} ) q^{88} + ( -78 + 24 \beta_{2} ) q^{89} + ( -77 + 14 \beta_{1} + 56 \beta_{2} + 11 \beta_{3} ) q^{90} + ( 14 - 48 \beta_{1} + 32 \beta_{2} + 6 \beta_{3} ) q^{92} + ( -40 \beta_{1} - 20 \beta_{2} + 32 \beta_{3} ) q^{93} + ( 28 + 8 \beta_{1} + 32 \beta_{2} - 4 \beta_{3} ) q^{94} + ( -80 \beta_{1} - 40 \beta_{2} - 42 \beta_{3} ) q^{95} + ( -44 - 8 \beta_{1} - 48 \beta_{2} + 4 \beta_{3} ) q^{96} + ( 34 - 92 \beta_{2} ) q^{97} + ( -60 + 34 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{3} - 12q^{4} - 4q^{6} - 12q^{9} + O(q^{10})$$ $$4q - 8q^{3} - 12q^{4} - 4q^{6} - 12q^{9} + 28q^{10} + 16q^{11} + 24q^{12} + 8q^{16} - 72q^{17} + 16q^{18} - 8q^{19} - 56q^{20} - 24q^{22} + 40q^{24} - 68q^{25} - 28q^{26} + 64q^{27} + 16q^{33} - 16q^{34} + 36q^{36} + 76q^{38} - 56q^{40} + 40q^{41} - 80q^{43} - 48q^{44} + 224q^{46} - 16q^{48} + 112q^{50} + 176q^{51} + 56q^{52} + 16q^{54} - 136q^{57} - 168q^{58} - 184q^{59} + 56q^{60} + 112q^{62} + 144q^{64} + 168q^{65} + 32q^{66} - 224q^{67} + 216q^{68} - 160q^{72} - 232q^{73} - 280q^{74} - 88q^{75} + 24q^{76} + 336q^{80} - 52q^{81} + 48q^{82} - 88q^{83} + 8q^{86} + 240q^{88} - 312q^{89} - 308q^{90} + 56q^{92} + 112q^{94} - 176q^{96} + 136q^{97} - 240q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 6 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 3$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} - 2 \beta_{1}$$

## 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
 0.707107 + 1.87083i 0.707107 − 1.87083i −0.707107 + 1.87083i −0.707107 − 1.87083i
−0.707107 1.87083i −0.585786 −3.00000 + 2.64575i 9.03316i 0.414214 + 1.09591i 0 7.07107 + 3.74166i −8.65685 16.8995 6.38741i
99.2 −0.707107 + 1.87083i −0.585786 −3.00000 2.64575i 9.03316i 0.414214 1.09591i 0 7.07107 3.74166i −8.65685 16.8995 + 6.38741i
99.3 0.707107 1.87083i −3.41421 −3.00000 2.64575i 1.54985i −2.41421 + 6.38741i 0 −7.07107 + 3.74166i 2.65685 −2.89949 1.09591i
99.4 0.707107 + 1.87083i −3.41421 −3.00000 + 2.64575i 1.54985i −2.41421 6.38741i 0 −7.07107 3.74166i 2.65685 −2.89949 + 1.09591i
 $$n$$: 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.h 4
4.b odd 2 1 1568.3.g.h 4
7.b odd 2 1 56.3.g.a 4
7.c even 3 2 392.3.k.j 8
7.d odd 6 2 392.3.k.i 8
8.b even 2 1 1568.3.g.h 4
8.d odd 2 1 inner 392.3.g.h 4
21.c even 2 1 504.3.g.a 4
28.d even 2 1 224.3.g.a 4
56.e even 2 1 56.3.g.a 4
56.h odd 2 1 224.3.g.a 4
56.k odd 6 2 392.3.k.j 8
56.m even 6 2 392.3.k.i 8
84.h odd 2 1 2016.3.g.a 4
112.j even 4 2 1792.3.d.g 8
112.l odd 4 2 1792.3.d.g 8
168.e odd 2 1 504.3.g.a 4
168.i even 2 1 2016.3.g.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.a 4 7.b odd 2 1
56.3.g.a 4 56.e even 2 1
224.3.g.a 4 28.d even 2 1
224.3.g.a 4 56.h odd 2 1
392.3.g.h 4 1.a even 1 1 trivial
392.3.g.h 4 8.d odd 2 1 inner
392.3.k.i 8 7.d odd 6 2
392.3.k.i 8 56.m even 6 2
392.3.k.j 8 7.c even 3 2
392.3.k.j 8 56.k odd 6 2
504.3.g.a 4 21.c even 2 1
504.3.g.a 4 168.e odd 2 1
1568.3.g.h 4 4.b odd 2 1
1568.3.g.h 4 8.b even 2 1
1792.3.d.g 8 112.j even 4 2
1792.3.d.g 8 112.l odd 4 2
2016.3.g.a 4 84.h odd 2 1
2016.3.g.a 4 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 4 T_{3} + 2$$ acting on $$S_{3}^{\mathrm{new}}(392, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T^{2} + 16 T^{4}$$
$3$ $$( 1 + 4 T + 20 T^{2} + 36 T^{3} + 81 T^{4} )^{2}$$
$5$ $$1 - 16 T^{2} - 254 T^{4} - 10000 T^{6} + 390625 T^{8}$$
$7$ 1
$11$ $$( 1 - 8 T + 186 T^{2} - 968 T^{3} + 14641 T^{4} )^{2}$$
$13$ $$1 - 592 T^{2} + 143170 T^{4} - 16908112 T^{6} + 815730721 T^{8}$$
$17$ $$( 1 + 36 T + 870 T^{2} + 10404 T^{3} + 83521 T^{4} )^{2}$$
$19$ $$( 1 + 4 T + 4 T^{2} + 1444 T^{3} + 130321 T^{4} )^{2}$$
$23$ $$1 - 268 T^{2} + 477286 T^{4} - 74997388 T^{6} + 78310985281 T^{8}$$
$29$ $$( 1 - 1178 T^{2} + 707281 T^{4} )^{2}$$
$31$ $$1 - 1380 T^{2} + 1419974 T^{4} - 1274458980 T^{6} + 852891037441 T^{8}$$
$37$ $$1 - 1780 T^{2} + 2031622 T^{4} - 3336006580 T^{6} + 3512479453921 T^{8}$$
$41$ $$( 1 - 20 T + 3174 T^{2} - 33620 T^{3} + 2825761 T^{4} )^{2}$$
$43$ $$( 1 + 40 T + 4090 T^{2} + 73960 T^{3} + 3418801 T^{4} )^{2}$$
$47$ $$1 - 7492 T^{2} + 23390470 T^{4} - 36558570052 T^{6} + 23811286661761 T^{8}$$
$53$ $$1 - 1604 T^{2} - 6155034 T^{4} - 12656331524 T^{6} + 62259690411361 T^{8}$$
$59$ $$( 1 + 92 T + 8836 T^{2} + 320252 T^{3} + 12117361 T^{4} )^{2}$$
$61$ $$1 - 13232 T^{2} + 71110338 T^{4} - 183208168112 T^{6} + 191707312997281 T^{8}$$
$67$ $$( 1 + 112 T + 11602 T^{2} + 502768 T^{3} + 20151121 T^{4} )^{2}$$
$71$ $$1 - 9412 T^{2} + 47279686 T^{4} - 239174741572 T^{6} + 645753531245761 T^{8}$$
$73$ $$( 1 + 116 T + 13894 T^{2} + 618164 T^{3} + 28398241 T^{4} )^{2}$$
$79$ $$1 - 16900 T^{2} + 142880134 T^{4} - 658256368900 T^{6} + 1517108809906561 T^{8}$$
$83$ $$( 1 + 44 T + 13924 T^{2} + 303116 T^{3} + 47458321 T^{4} )^{2}$$
$89$ $$( 1 + 156 T + 20774 T^{2} + 1235676 T^{3} + 62742241 T^{4} )^{2}$$
$97$ $$( 1 - 68 T + 3046 T^{2} - 639812 T^{3} + 88529281 T^{4} )^{2}$$