# Properties

 Label 392.6.i.e Level $392$ Weight $6$ Character orbit 392.i Analytic conductor $62.870$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$62.8704573667$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - 20 \zeta_{6} + 20) q^{3} - 74 \zeta_{6} q^{5} - 157 \zeta_{6} q^{9} +O(q^{10})$$ q + (-20*z + 20) * q^3 - 74*z * q^5 - 157*z * q^9 $$q + ( - 20 \zeta_{6} + 20) q^{3} - 74 \zeta_{6} q^{5} - 157 \zeta_{6} q^{9} + (124 \zeta_{6} - 124) q^{11} - 478 q^{13} - 1480 q^{15} + (1198 \zeta_{6} - 1198) q^{17} + 3044 \zeta_{6} q^{19} - 184 \zeta_{6} q^{23} + (2351 \zeta_{6} - 2351) q^{25} + 1720 q^{27} - 3282 q^{29} + (5728 \zeta_{6} - 5728) q^{31} + 2480 \zeta_{6} q^{33} - 10326 \zeta_{6} q^{37} + (9560 \zeta_{6} - 9560) q^{39} + 8886 q^{41} - 9188 q^{43} + (11618 \zeta_{6} - 11618) q^{45} + 23664 \zeta_{6} q^{47} + 23960 \zeta_{6} q^{51} + (11686 \zeta_{6} - 11686) q^{53} + 9176 q^{55} + 60880 q^{57} + ( - 16876 \zeta_{6} + 16876) q^{59} - 18482 \zeta_{6} q^{61} + 35372 \zeta_{6} q^{65} + ( - 15532 \zeta_{6} + 15532) q^{67} - 3680 q^{69} - 31960 q^{71} + (4886 \zeta_{6} - 4886) q^{73} + 47020 \zeta_{6} q^{75} - 44560 \zeta_{6} q^{79} + ( - 72551 \zeta_{6} + 72551) q^{81} - 67364 q^{83} + 88652 q^{85} + (65640 \zeta_{6} - 65640) q^{87} + 71994 \zeta_{6} q^{89} + 114560 \zeta_{6} q^{93} + ( - 225256 \zeta_{6} + 225256) q^{95} - 48866 q^{97} + 19468 q^{99} +O(q^{100})$$ q + (-20*z + 20) * q^3 - 74*z * q^5 - 157*z * q^9 + (124*z - 124) * q^11 - 478 * q^13 - 1480 * q^15 + (1198*z - 1198) * q^17 + 3044*z * q^19 - 184*z * q^23 + (2351*z - 2351) * q^25 + 1720 * q^27 - 3282 * q^29 + (5728*z - 5728) * q^31 + 2480*z * q^33 - 10326*z * q^37 + (9560*z - 9560) * q^39 + 8886 * q^41 - 9188 * q^43 + (11618*z - 11618) * q^45 + 23664*z * q^47 + 23960*z * q^51 + (11686*z - 11686) * q^53 + 9176 * q^55 + 60880 * q^57 + (-16876*z + 16876) * q^59 - 18482*z * q^61 + 35372*z * q^65 + (-15532*z + 15532) * q^67 - 3680 * q^69 - 31960 * q^71 + (4886*z - 4886) * q^73 + 47020*z * q^75 - 44560*z * q^79 + (-72551*z + 72551) * q^81 - 67364 * q^83 + 88652 * q^85 + (65640*z - 65640) * q^87 + 71994*z * q^89 + 114560*z * q^93 + (-225256*z + 225256) * q^95 - 48866 * q^97 + 19468 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 20 q^{3} - 74 q^{5} - 157 q^{9}+O(q^{10})$$ 2 * q + 20 * q^3 - 74 * q^5 - 157 * q^9 $$2 q + 20 q^{3} - 74 q^{5} - 157 q^{9} - 124 q^{11} - 956 q^{13} - 2960 q^{15} - 1198 q^{17} + 3044 q^{19} - 184 q^{23} - 2351 q^{25} + 3440 q^{27} - 6564 q^{29} - 5728 q^{31} + 2480 q^{33} - 10326 q^{37} - 9560 q^{39} + 17772 q^{41} - 18376 q^{43} - 11618 q^{45} + 23664 q^{47} + 23960 q^{51} - 11686 q^{53} + 18352 q^{55} + 121760 q^{57} + 16876 q^{59} - 18482 q^{61} + 35372 q^{65} + 15532 q^{67} - 7360 q^{69} - 63920 q^{71} - 4886 q^{73} + 47020 q^{75} - 44560 q^{79} + 72551 q^{81} - 134728 q^{83} + 177304 q^{85} - 65640 q^{87} + 71994 q^{89} + 114560 q^{93} + 225256 q^{95} - 97732 q^{97} + 38936 q^{99}+O(q^{100})$$ 2 * q + 20 * q^3 - 74 * q^5 - 157 * q^9 - 124 * q^11 - 956 * q^13 - 2960 * q^15 - 1198 * q^17 + 3044 * q^19 - 184 * q^23 - 2351 * q^25 + 3440 * q^27 - 6564 * q^29 - 5728 * q^31 + 2480 * q^33 - 10326 * q^37 - 9560 * q^39 + 17772 * q^41 - 18376 * q^43 - 11618 * q^45 + 23664 * q^47 + 23960 * q^51 - 11686 * q^53 + 18352 * q^55 + 121760 * q^57 + 16876 * q^59 - 18482 * q^61 + 35372 * q^65 + 15532 * q^67 - 7360 * q^69 - 63920 * q^71 - 4886 * q^73 + 47020 * q^75 - 44560 * q^79 + 72551 * q^81 - 134728 * q^83 + 177304 * q^85 - 65640 * q^87 + 71994 * q^89 + 114560 * q^93 + 225256 * q^95 - 97732 * q^97 + 38936 * q^99

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
177.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 10.0000 + 17.3205i 0 −37.0000 + 64.0859i 0 0 0 −78.5000 + 135.966i 0
361.1 0 10.0000 17.3205i 0 −37.0000 64.0859i 0 0 0 −78.5000 135.966i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.6.i.e 2
7.b odd 2 1 392.6.i.b 2
7.c even 3 1 392.6.a.b 1
7.c even 3 1 inner 392.6.i.e 2
7.d odd 6 1 8.6.a.a 1
7.d odd 6 1 392.6.i.b 2
21.g even 6 1 72.6.a.f 1
28.f even 6 1 16.6.a.a 1
28.g odd 6 1 784.6.a.l 1
35.i odd 6 1 200.6.a.a 1
35.k even 12 2 200.6.c.a 2
56.j odd 6 1 64.6.a.a 1
56.m even 6 1 64.6.a.g 1
77.i even 6 1 968.6.a.a 1
84.j odd 6 1 144.6.a.k 1
112.v even 12 2 256.6.b.d 2
112.x odd 12 2 256.6.b.f 2
140.s even 6 1 400.6.a.l 1
140.x odd 12 2 400.6.c.d 2
168.ba even 6 1 576.6.a.g 1
168.be odd 6 1 576.6.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.a.a 1 7.d odd 6 1
16.6.a.a 1 28.f even 6 1
64.6.a.a 1 56.j odd 6 1
64.6.a.g 1 56.m even 6 1
72.6.a.f 1 21.g even 6 1
144.6.a.k 1 84.j odd 6 1
200.6.a.a 1 35.i odd 6 1
200.6.c.a 2 35.k even 12 2
256.6.b.d 2 112.v even 12 2
256.6.b.f 2 112.x odd 12 2
392.6.a.b 1 7.c even 3 1
392.6.i.b 2 7.b odd 2 1
392.6.i.b 2 7.d odd 6 1
392.6.i.e 2 1.a even 1 1 trivial
392.6.i.e 2 7.c even 3 1 inner
400.6.a.l 1 140.s even 6 1
400.6.c.d 2 140.x odd 12 2
576.6.a.g 1 168.ba even 6 1
576.6.a.h 1 168.be odd 6 1
784.6.a.l 1 28.g odd 6 1
968.6.a.a 1 77.i even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 20T_{3} + 400$$ acting on $$S_{6}^{\mathrm{new}}(392, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 20T + 400$$
$5$ $$T^{2} + 74T + 5476$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 124T + 15376$$
$13$ $$(T + 478)^{2}$$
$17$ $$T^{2} + 1198 T + 1435204$$
$19$ $$T^{2} - 3044 T + 9265936$$
$23$ $$T^{2} + 184T + 33856$$
$29$ $$(T + 3282)^{2}$$
$31$ $$T^{2} + 5728 T + 32809984$$
$37$ $$T^{2} + 10326 T + 106626276$$
$41$ $$(T - 8886)^{2}$$
$43$ $$(T + 9188)^{2}$$
$47$ $$T^{2} - 23664 T + 559984896$$
$53$ $$T^{2} + 11686 T + 136562596$$
$59$ $$T^{2} - 16876 T + 284799376$$
$61$ $$T^{2} + 18482 T + 341584324$$
$67$ $$T^{2} - 15532 T + 241243024$$
$71$ $$(T + 31960)^{2}$$
$73$ $$T^{2} + 4886 T + 23872996$$
$79$ $$T^{2} + 44560 T + 1985593600$$
$83$ $$(T + 67364)^{2}$$
$89$ $$T^{2} - 71994 T + 5183136036$$
$97$ $$(T + 48866)^{2}$$