# Properties

 Label 392.2.i.h Level 392 Weight 2 Character orbit 392.i Analytic conductor 3.130 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.13013575923$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{5} -\beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{5} -\beta_{2} q^{9} + ( -6 - 6 \beta_{2} ) q^{11} -4 \beta_{3} q^{13} + 4 q^{15} + \beta_{1} q^{17} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{19} + 4 \beta_{2} q^{23} + ( -3 - 3 \beta_{2} ) q^{25} -4 \beta_{3} q^{27} -6 q^{29} + 2 \beta_{1} q^{31} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{33} + 2 \beta_{2} q^{37} + ( 8 + 8 \beta_{2} ) q^{39} -\beta_{3} q^{41} + 10 q^{43} -2 \beta_{1} q^{45} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{47} + 2 \beta_{2} q^{51} + ( 2 + 2 \beta_{2} ) q^{53} + 12 \beta_{3} q^{55} -6 q^{57} + \beta_{1} q^{59} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{61} -16 \beta_{2} q^{65} + ( -4 - 4 \beta_{2} ) q^{67} + 4 \beta_{3} q^{69} -12 q^{71} -7 \beta_{1} q^{73} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{75} -4 \beta_{2} q^{79} + ( 5 + 5 \beta_{2} ) q^{81} + \beta_{3} q^{83} + 4 q^{85} -6 \beta_{1} q^{87} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{89} + 4 \beta_{2} q^{93} + ( 12 + 12 \beta_{2} ) q^{95} + 9 \beta_{3} q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{9} + O(q^{10})$$ $$4q + 2q^{9} - 12q^{11} + 16q^{15} - 8q^{23} - 6q^{25} - 24q^{29} - 4q^{37} + 16q^{39} + 40q^{43} - 4q^{51} + 4q^{53} - 24q^{57} + 32q^{65} - 8q^{67} - 48q^{71} + 8q^{79} + 10q^{81} + 16q^{85} - 8q^{93} + 24q^{95} - 24q^{99} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
177.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 −0.707107 1.22474i 0 −1.41421 + 2.44949i 0 0 0 0.500000 0.866025i 0
177.2 0 0.707107 + 1.22474i 0 1.41421 2.44949i 0 0 0 0.500000 0.866025i 0
361.1 0 −0.707107 + 1.22474i 0 −1.41421 2.44949i 0 0 0 0.500000 + 0.866025i 0
361.2 0 0.707107 1.22474i 0 1.41421 + 2.44949i 0 0 0 0.500000 + 0.866025i 0
 $$n$$: 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
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.2.i.h 4
3.b odd 2 1 3528.2.s.bj 4
4.b odd 2 1 784.2.i.n 4
7.b odd 2 1 inner 392.2.i.h 4
7.c even 3 1 392.2.a.g 2
7.c even 3 1 inner 392.2.i.h 4
7.d odd 6 1 392.2.a.g 2
7.d odd 6 1 inner 392.2.i.h 4
21.c even 2 1 3528.2.s.bj 4
21.g even 6 1 3528.2.a.be 2
21.g even 6 1 3528.2.s.bj 4
21.h odd 6 1 3528.2.a.be 2
21.h odd 6 1 3528.2.s.bj 4
28.d even 2 1 784.2.i.n 4
28.f even 6 1 784.2.a.k 2
28.f even 6 1 784.2.i.n 4
28.g odd 6 1 784.2.a.k 2
28.g odd 6 1 784.2.i.n 4
35.i odd 6 1 9800.2.a.bv 2
35.j even 6 1 9800.2.a.bv 2
56.j odd 6 1 3136.2.a.bk 2
56.k odd 6 1 3136.2.a.bp 2
56.m even 6 1 3136.2.a.bp 2
56.p even 6 1 3136.2.a.bk 2
84.j odd 6 1 7056.2.a.ct 2
84.n even 6 1 7056.2.a.ct 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.a.g 2 7.c even 3 1
392.2.a.g 2 7.d odd 6 1
392.2.i.h 4 1.a even 1 1 trivial
392.2.i.h 4 7.b odd 2 1 inner
392.2.i.h 4 7.c even 3 1 inner
392.2.i.h 4 7.d odd 6 1 inner
784.2.a.k 2 28.f even 6 1
784.2.a.k 2 28.g odd 6 1
784.2.i.n 4 4.b odd 2 1
784.2.i.n 4 28.d even 2 1
784.2.i.n 4 28.f even 6 1
784.2.i.n 4 28.g odd 6 1
3136.2.a.bk 2 56.j odd 6 1
3136.2.a.bk 2 56.p even 6 1
3136.2.a.bp 2 56.k odd 6 1
3136.2.a.bp 2 56.m even 6 1
3528.2.a.be 2 21.g even 6 1
3528.2.a.be 2 21.h odd 6 1
3528.2.s.bj 4 3.b odd 2 1
3528.2.s.bj 4 21.c even 2 1
3528.2.s.bj 4 21.g even 6 1
3528.2.s.bj 4 21.h odd 6 1
7056.2.a.ct 2 84.j odd 6 1
7056.2.a.ct 2 84.n even 6 1
9800.2.a.bv 2 35.i odd 6 1
9800.2.a.bv 2 35.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(392, [\chi])$$:

 $$T_{3}^{4} + 2 T_{3}^{2} + 4$$ $$T_{5}^{4} + 8 T_{5}^{2} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 4 T^{2} + 7 T^{4} - 36 T^{6} + 81 T^{8}$$
$5$ $$( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} )$$
$7$ 1
$11$ $$( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 6 T^{2} + 169 T^{4} )^{2}$$
$17$ $$1 - 32 T^{2} + 735 T^{4} - 9248 T^{6} + 83521 T^{8}$$
$19$ $$1 - 20 T^{2} + 39 T^{4} - 7220 T^{6} + 130321 T^{8}$$
$23$ $$( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{4}$$
$31$ $$1 - 54 T^{2} + 1955 T^{4} - 51894 T^{6} + 923521 T^{8}$$
$37$ $$( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 80 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 10 T + 43 T^{2} )^{4}$$
$47$ $$1 - 86 T^{2} + 5187 T^{4} - 189974 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8}$$
$61$ $$1 - 50 T^{2} - 1221 T^{4} - 186050 T^{6} + 13845841 T^{8}$$
$67$ $$( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 + 12 T + 71 T^{2} )^{4}$$
$73$ $$1 - 48 T^{2} - 3025 T^{4} - 255792 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 - 17 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2}$$
$83$ $$( 1 + 164 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$1 - 160 T^{2} + 17679 T^{4} - 1267360 T^{6} + 62742241 T^{8}$$
$97$ $$( 1 + 32 T^{2} + 9409 T^{4} )^{2}$$