# Properties

 Label 392.6.i.a Level $392$ Weight $6$ Character orbit 392.i Analytic conductor $62.870$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,6,Mod(177,392)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("392.177");

S:= CuspForms(chi, 6);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$62.8704573667$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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 56) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + (30 \zeta_{6} - 30) q^{3} - 32 \zeta_{6} q^{5} - 657 \zeta_{6} q^{9} +O(q^{10})$$ q + (30*z - 30) * q^3 - 32*z * q^5 - 657*z * q^9 $$q + (30 \zeta_{6} - 30) q^{3} - 32 \zeta_{6} q^{5} - 657 \zeta_{6} q^{9} + ( - 624 \zeta_{6} + 624) q^{11} - 708 q^{13} + 960 q^{15} + (934 \zeta_{6} - 934) q^{17} - 1858 \zeta_{6} q^{19} + 1120 \zeta_{6} q^{23} + ( - 2101 \zeta_{6} + 2101) q^{25} + 12420 q^{27} - 1174 q^{29} + (2908 \zeta_{6} - 2908) q^{31} + 18720 \zeta_{6} q^{33} + 12462 \zeta_{6} q^{37} + ( - 21240 \zeta_{6} + 21240) q^{39} + 2662 q^{41} - 7144 q^{43} + (21024 \zeta_{6} - 21024) q^{45} + 7468 \zeta_{6} q^{47} - 28020 \zeta_{6} q^{51} + ( - 27274 \zeta_{6} + 27274) q^{53} - 19968 q^{55} + 55740 q^{57} + (2490 \zeta_{6} - 2490) q^{59} + 11096 \zeta_{6} q^{61} + 22656 \zeta_{6} q^{65} + (39756 \zeta_{6} - 39756) q^{67} - 33600 q^{69} - 69888 q^{71} + (16450 \zeta_{6} - 16450) q^{73} + 63030 \zeta_{6} q^{75} - 78376 \zeta_{6} q^{79} + (212949 \zeta_{6} - 212949) q^{81} + 109818 q^{83} + 29888 q^{85} + ( - 35220 \zeta_{6} + 35220) q^{87} + 56966 \zeta_{6} q^{89} - 87240 \zeta_{6} q^{93} + (59456 \zeta_{6} - 59456) q^{95} - 115946 q^{97} - 409968 q^{99} +O(q^{100})$$ q + (30*z - 30) * q^3 - 32*z * q^5 - 657*z * q^9 + (-624*z + 624) * q^11 - 708 * q^13 + 960 * q^15 + (934*z - 934) * q^17 - 1858*z * q^19 + 1120*z * q^23 + (-2101*z + 2101) * q^25 + 12420 * q^27 - 1174 * q^29 + (2908*z - 2908) * q^31 + 18720*z * q^33 + 12462*z * q^37 + (-21240*z + 21240) * q^39 + 2662 * q^41 - 7144 * q^43 + (21024*z - 21024) * q^45 + 7468*z * q^47 - 28020*z * q^51 + (-27274*z + 27274) * q^53 - 19968 * q^55 + 55740 * q^57 + (2490*z - 2490) * q^59 + 11096*z * q^61 + 22656*z * q^65 + (39756*z - 39756) * q^67 - 33600 * q^69 - 69888 * q^71 + (16450*z - 16450) * q^73 + 63030*z * q^75 - 78376*z * q^79 + (212949*z - 212949) * q^81 + 109818 * q^83 + 29888 * q^85 + (-35220*z + 35220) * q^87 + 56966*z * q^89 - 87240*z * q^93 + (59456*z - 59456) * q^95 - 115946 * q^97 - 409968 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 30 q^{3} - 32 q^{5} - 657 q^{9}+O(q^{10})$$ 2 * q - 30 * q^3 - 32 * q^5 - 657 * q^9 $$2 q - 30 q^{3} - 32 q^{5} - 657 q^{9} + 624 q^{11} - 1416 q^{13} + 1920 q^{15} - 934 q^{17} - 1858 q^{19} + 1120 q^{23} + 2101 q^{25} + 24840 q^{27} - 2348 q^{29} - 2908 q^{31} + 18720 q^{33} + 12462 q^{37} + 21240 q^{39} + 5324 q^{41} - 14288 q^{43} - 21024 q^{45} + 7468 q^{47} - 28020 q^{51} + 27274 q^{53} - 39936 q^{55} + 111480 q^{57} - 2490 q^{59} + 11096 q^{61} + 22656 q^{65} - 39756 q^{67} - 67200 q^{69} - 139776 q^{71} - 16450 q^{73} + 63030 q^{75} - 78376 q^{79} - 212949 q^{81} + 219636 q^{83} + 59776 q^{85} + 35220 q^{87} + 56966 q^{89} - 87240 q^{93} - 59456 q^{95} - 231892 q^{97} - 819936 q^{99}+O(q^{100})$$ 2 * q - 30 * q^3 - 32 * q^5 - 657 * q^9 + 624 * q^11 - 1416 * q^13 + 1920 * q^15 - 934 * q^17 - 1858 * q^19 + 1120 * q^23 + 2101 * q^25 + 24840 * q^27 - 2348 * q^29 - 2908 * q^31 + 18720 * q^33 + 12462 * q^37 + 21240 * q^39 + 5324 * q^41 - 14288 * q^43 - 21024 * q^45 + 7468 * q^47 - 28020 * q^51 + 27274 * q^53 - 39936 * q^55 + 111480 * q^57 - 2490 * q^59 + 11096 * q^61 + 22656 * q^65 - 39756 * q^67 - 67200 * q^69 - 139776 * q^71 - 16450 * q^73 + 63030 * q^75 - 78376 * q^79 - 212949 * q^81 + 219636 * q^83 + 59776 * q^85 + 35220 * q^87 + 56966 * q^89 - 87240 * q^93 - 59456 * q^95 - 231892 * q^97 - 819936 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 −15.0000 25.9808i 0 −16.0000 + 27.7128i 0 0 0 −328.500 + 568.979i 0
361.1 0 −15.0000 + 25.9808i 0 −16.0000 27.7128i 0 0 0 −328.500 568.979i 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.a 2
7.b odd 2 1 392.6.i.f 2
7.c even 3 1 56.6.a.b 1
7.c even 3 1 inner 392.6.i.a 2
7.d odd 6 1 392.6.a.a 1
7.d odd 6 1 392.6.i.f 2
21.h odd 6 1 504.6.a.b 1
28.f even 6 1 784.6.a.n 1
28.g odd 6 1 112.6.a.a 1
56.k odd 6 1 448.6.a.p 1
56.p even 6 1 448.6.a.a 1
84.n even 6 1 1008.6.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.b 1 7.c even 3 1
112.6.a.a 1 28.g odd 6 1
392.6.a.a 1 7.d odd 6 1
392.6.i.a 2 1.a even 1 1 trivial
392.6.i.a 2 7.c even 3 1 inner
392.6.i.f 2 7.b odd 2 1
392.6.i.f 2 7.d odd 6 1
448.6.a.a 1 56.p even 6 1
448.6.a.p 1 56.k odd 6 1
504.6.a.b 1 21.h odd 6 1
784.6.a.n 1 28.f even 6 1
1008.6.a.h 1 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 30T_{3} + 900$$ acting on $$S_{6}^{\mathrm{new}}(392, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 30T + 900$$
$5$ $$T^{2} + 32T + 1024$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 624T + 389376$$
$13$ $$(T + 708)^{2}$$
$17$ $$T^{2} + 934T + 872356$$
$19$ $$T^{2} + 1858 T + 3452164$$
$23$ $$T^{2} - 1120 T + 1254400$$
$29$ $$(T + 1174)^{2}$$
$31$ $$T^{2} + 2908 T + 8456464$$
$37$ $$T^{2} - 12462 T + 155301444$$
$41$ $$(T - 2662)^{2}$$
$43$ $$(T + 7144)^{2}$$
$47$ $$T^{2} - 7468 T + 55771024$$
$53$ $$T^{2} - 27274 T + 743871076$$
$59$ $$T^{2} + 2490 T + 6200100$$
$61$ $$T^{2} - 11096 T + 123121216$$
$67$ $$T^{2} + \cdots + 1580539536$$
$71$ $$(T + 69888)^{2}$$
$73$ $$T^{2} + 16450 T + 270602500$$
$79$ $$T^{2} + \cdots + 6142797376$$
$83$ $$(T - 109818)^{2}$$
$89$ $$T^{2} + \cdots + 3245125156$$
$97$ $$(T + 115946)^{2}$$