# Properties

 Label 392.3.j.a Level 392 Weight 3 Character orbit 392.j Analytic conductor 10.681 Analytic rank 0 Dimension 4 CM discriminant -56 Inner twists 8

# Learn more about

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$10.6812263629$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{2} q^{2} + \beta_{1} q^{3} + ( -4 - 4 \beta_{2} ) q^{4} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{5} + 2 \beta_{3} q^{6} + 8 q^{8} -\beta_{2} q^{9} +O(q^{10})$$ $$q + 2 \beta_{2} q^{2} + \beta_{1} q^{3} + ( -4 - 4 \beta_{2} ) q^{4} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{5} + 2 \beta_{3} q^{6} + 8 q^{8} -\beta_{2} q^{9} -6 \beta_{1} q^{10} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{12} + 9 \beta_{3} q^{13} -24 q^{15} + 16 \beta_{2} q^{16} + ( 2 + 2 \beta_{2} ) q^{18} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{19} -12 \beta_{3} q^{20} -10 \beta_{2} q^{23} + 8 \beta_{1} q^{24} + ( -47 - 47 \beta_{2} ) q^{25} + ( -18 \beta_{1} - 18 \beta_{3} ) q^{26} -10 \beta_{3} q^{27} -48 \beta_{2} q^{30} + ( -32 - 32 \beta_{2} ) q^{32} -4 q^{36} -6 \beta_{1} q^{38} + ( -72 - 72 \beta_{2} ) q^{39} + ( 24 \beta_{1} + 24 \beta_{3} ) q^{40} + 3 \beta_{1} q^{45} + ( 20 + 20 \beta_{2} ) q^{46} + 16 \beta_{3} q^{48} + 94 q^{50} + 36 \beta_{1} q^{52} + ( 20 \beta_{1} + 20 \beta_{3} ) q^{54} -24 q^{57} -27 \beta_{1} q^{59} + ( 96 + 96 \beta_{2} ) q^{60} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{61} + 64 q^{64} -216 \beta_{2} q^{65} -10 \beta_{3} q^{69} -110 q^{71} -8 \beta_{2} q^{72} + ( -47 \beta_{1} - 47 \beta_{3} ) q^{75} -12 \beta_{3} q^{76} + 144 q^{78} + 130 \beta_{2} q^{79} -48 \beta_{1} q^{80} + ( 71 + 71 \beta_{2} ) q^{81} + 9 \beta_{3} q^{83} + 6 \beta_{3} q^{90} -40 q^{92} + ( -72 - 72 \beta_{2} ) q^{95} + ( -32 \beta_{1} - 32 \beta_{3} ) q^{96} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 8q^{4} + 32q^{8} + 2q^{9} + O(q^{10})$$ $$4q - 4q^{2} - 8q^{4} + 32q^{8} + 2q^{9} - 96q^{15} - 32q^{16} + 4q^{18} + 20q^{23} - 94q^{25} + 96q^{30} - 64q^{32} - 16q^{36} - 144q^{39} + 40q^{46} + 376q^{50} - 96q^{57} + 192q^{60} + 256q^{64} + 432q^{65} - 440q^{71} + 16q^{72} + 576q^{78} - 260q^{79} + 142q^{81} - 160q^{92} - 144q^{95} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3}$$

## Character values

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

 $$n$$ $$197$$ $$295$$ $$297$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
117.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−1.00000 + 1.73205i −1.41421 2.44949i −2.00000 3.46410i 4.24264 7.34847i 5.65685 0 8.00000 0.500000 0.866025i 8.48528 + 14.6969i
117.2 −1.00000 + 1.73205i 1.41421 + 2.44949i −2.00000 3.46410i −4.24264 + 7.34847i −5.65685 0 8.00000 0.500000 0.866025i −8.48528 14.6969i
325.1 −1.00000 1.73205i −1.41421 + 2.44949i −2.00000 + 3.46410i 4.24264 + 7.34847i 5.65685 0 8.00000 0.500000 + 0.866025i 8.48528 14.6969i
325.2 −1.00000 1.73205i 1.41421 2.44949i −2.00000 + 3.46410i −4.24264 7.34847i −5.65685 0 8.00000 0.500000 + 0.866025i −8.48528 + 14.6969i
 $$n$$: 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.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
56.j odd 6 1 inner
56.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.3.j.a 4
7.b odd 2 1 inner 392.3.j.a 4
7.c even 3 1 56.3.h.c 2
7.c even 3 1 inner 392.3.j.a 4
7.d odd 6 1 56.3.h.c 2
7.d odd 6 1 inner 392.3.j.a 4
8.b even 2 1 inner 392.3.j.a 4
21.g even 6 1 504.3.l.a 2
21.h odd 6 1 504.3.l.a 2
28.f even 6 1 224.3.h.c 2
28.g odd 6 1 224.3.h.c 2
56.h odd 2 1 CM 392.3.j.a 4
56.j odd 6 1 56.3.h.c 2
56.j odd 6 1 inner 392.3.j.a 4
56.k odd 6 1 224.3.h.c 2
56.m even 6 1 224.3.h.c 2
56.p even 6 1 56.3.h.c 2
56.p even 6 1 inner 392.3.j.a 4
84.j odd 6 1 2016.3.l.c 2
84.n even 6 1 2016.3.l.c 2
168.s odd 6 1 504.3.l.a 2
168.v even 6 1 2016.3.l.c 2
168.ba even 6 1 504.3.l.a 2
168.be odd 6 1 2016.3.l.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.h.c 2 7.c even 3 1
56.3.h.c 2 7.d odd 6 1
56.3.h.c 2 56.j odd 6 1
56.3.h.c 2 56.p even 6 1
224.3.h.c 2 28.f even 6 1
224.3.h.c 2 28.g odd 6 1
224.3.h.c 2 56.k odd 6 1
224.3.h.c 2 56.m even 6 1
392.3.j.a 4 1.a even 1 1 trivial
392.3.j.a 4 7.b odd 2 1 inner
392.3.j.a 4 7.c even 3 1 inner
392.3.j.a 4 7.d odd 6 1 inner
392.3.j.a 4 8.b even 2 1 inner
392.3.j.a 4 56.h odd 2 1 CM
392.3.j.a 4 56.j odd 6 1 inner
392.3.j.a 4 56.p even 6 1 inner
504.3.l.a 2 21.g even 6 1
504.3.l.a 2 21.h odd 6 1
504.3.l.a 2 168.s odd 6 1
504.3.l.a 2 168.ba even 6 1
2016.3.l.c 2 84.j odd 6 1
2016.3.l.c 2 84.n even 6 1
2016.3.l.c 2 168.v even 6 1
2016.3.l.c 2 168.be odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 8 T_{3}^{2} + 64$$ acting on $$S_{3}^{\mathrm{new}}(392, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T + 4 T^{2} )^{2}$$
$3$ $$1 - 10 T^{2} + 19 T^{4} - 810 T^{6} + 6561 T^{8}$$
$5$ $$1 + 22 T^{2} - 141 T^{4} + 13750 T^{6} + 390625 T^{8}$$
$7$ 1
$11$ $$( 1 - 11 T + 121 T^{2} )^{2}( 1 + 11 T + 121 T^{2} )^{2}$$
$13$ $$( 1 - 310 T^{2} + 28561 T^{4} )^{2}$$
$17$ $$( 1 - 17 T + 289 T^{2} )^{2}( 1 + 17 T + 289 T^{2} )^{2}$$
$19$ $$1 - 650 T^{2} + 292179 T^{4} - 84708650 T^{6} + 16983563041 T^{8}$$
$23$ $$( 1 - 10 T - 429 T^{2} - 5290 T^{3} + 279841 T^{4} )^{2}$$
$29$ $$( 1 - 29 T )^{4}( 1 + 29 T )^{4}$$
$31$ $$( 1 - 31 T + 961 T^{2} )^{2}( 1 + 31 T + 961 T^{2} )^{2}$$
$37$ $$( 1 - 37 T + 1369 T^{2} )^{2}( 1 + 37 T + 1369 T^{2} )^{2}$$
$41$ $$( 1 - 41 T )^{4}( 1 + 41 T )^{4}$$
$43$ $$( 1 - 43 T )^{4}( 1 + 43 T )^{4}$$
$47$ $$( 1 - 47 T + 2209 T^{2} )^{2}( 1 + 47 T + 2209 T^{2} )^{2}$$
$53$ $$( 1 - 53 T + 2809 T^{2} )^{2}( 1 + 53 T + 2809 T^{2} )^{2}$$
$59$ $$1 - 1130 T^{2} - 10840461 T^{4} - 13692617930 T^{6} + 146830437604321 T^{8}$$
$61$ $$1 - 7370 T^{2} + 40471059 T^{4} - 102043848170 T^{6} + 191707312997281 T^{8}$$
$67$ $$( 1 - 67 T + 4489 T^{2} )^{2}( 1 + 67 T + 4489 T^{2} )^{2}$$
$71$ $$( 1 + 110 T + 5041 T^{2} )^{4}$$
$73$ $$( 1 - 73 T + 5329 T^{2} )^{2}( 1 + 73 T + 5329 T^{2} )^{2}$$
$79$ $$( 1 + 130 T + 10659 T^{2} + 811330 T^{3} + 38950081 T^{4} )^{2}$$
$83$ $$( 1 + 13130 T^{2} + 47458321 T^{4} )^{2}$$
$89$ $$( 1 - 89 T + 7921 T^{2} )^{2}( 1 + 89 T + 7921 T^{2} )^{2}$$
$97$ $$( 1 - 97 T )^{4}( 1 + 97 T )^{4}$$
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