# Properties

 Label 392.6.i.k Level $392$ Weight $6$ Character orbit 392.i Analytic conductor $62.870$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{193})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 49x^{2} + 48x + 2304$$ x^4 - x^3 + 49*x^2 + 48*x + 2304 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 7 \beta_1) q^{3} + ( - 5 \beta_{3} - 5 \beta_{2} + \cdots - 21) q^{5}+ \cdots + ( - 14 \beta_{3} - 14 \beta_{2} + \cdots + 1) q^{9}+O(q^{10})$$ q + (-b2 + 7*b1) * q^3 + (-5*b3 - 5*b2 + 21*b1 - 21) * q^5 + (-14*b3 - 14*b2 - b1 + 1) * q^9 $$q + ( - \beta_{2} + 7 \beta_1) q^{3} + ( - 5 \beta_{3} - 5 \beta_{2} + \cdots - 21) q^{5}+ \cdots + ( - 4998 \beta_{3} - 37470) q^{99}+O(q^{100})$$ q + (-b2 + 7*b1) * q^3 + (-5*b3 - 5*b2 + 21*b1 - 21) * q^5 + (-14*b3 - 14*b2 - b1 + 1) * q^9 + (-14*b2 + 358*b1) * q^11 + (-17*b3 - 357) * q^13 + (-56*b3 - 1112) * q^15 + (-54*b2 + 672*b1) * q^17 + (-93*b3 - 93*b2 - 973*b1 + 973) * q^19 + (-112*b3 - 112*b2 - 896*b1 + 896) * q^23 + (210*b2 - 2141*b1) * q^25 + (146*b3 - 994) * q^27 + (-546*b3 - 600) * q^29 + (-446*b2 + 3402*b1) * q^31 + (-456*b3 - 456*b2 + 5208*b1 - 5208) * q^33 + (154*b3 + 154*b2 + 7320*b1 - 7320) * q^37 + (476*b2 - 5780*b1) * q^39 + (-442*b3 + 3948) * q^41 + (518*b3 + 262) * q^43 + (289*b2 - 13489*b1) * q^45 + (746*b3 + 746*b2 - 9198*b1 + 9198) * q^47 + (-1050*b3 - 1050*b2 + 15126*b1 - 15126) * q^51 + (1008*b2 - 22566*b1) * q^53 + (-2084*b3 - 21028) * q^55 + (322*b3 - 11138) * q^57 + (365*b2 - 11291*b1) * q^59 + (-1117*b3 - 1117*b2 - 26411*b1 + 26411) * q^61 + (2142*b3 + 2142*b2 - 23902*b1 + 23902) * q^65 + (224*b2 - 4924*b1) * q^67 + (112*b3 - 15344) * q^69 + (4060*b3 - 420) * q^71 + (-1204*b2 + 61026*b1) * q^73 + (3611*b3 + 3611*b2 - 55517*b1 + 55517) * q^75 + (1372*b3 + 1372*b2 + 15852*b1 - 15852) * q^79 + (-3430*b2 + 20977*b1) * q^81 + (4185*b3 + 18487) * q^83 + (-4494*b3 - 66222) * q^85 + (4422*b2 - 109578*b1) * q^87 + (-1512*b3 - 1512*b2 - 105294*b1 + 105294) * q^89 + (-6524*b3 - 6524*b2 + 109892*b1 - 109892) * q^93 + (-2912*b2 - 69312*b1) * q^95 + (8882*b3 - 22120) * q^97 + (-4998*b3 - 37470) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 14 q^{3} - 42 q^{5} + 2 q^{9}+O(q^{10})$$ 4 * q + 14 * q^3 - 42 * q^5 + 2 * q^9 $$4 q + 14 q^{3} - 42 q^{5} + 2 q^{9} + 716 q^{11} - 1428 q^{13} - 4448 q^{15} + 1344 q^{17} + 1946 q^{19} + 1792 q^{23} - 4282 q^{25} - 3976 q^{27} - 2400 q^{29} + 6804 q^{31} - 10416 q^{33} - 14640 q^{37} - 11560 q^{39} + 15792 q^{41} + 1048 q^{43} - 26978 q^{45} + 18396 q^{47} - 30252 q^{51} - 45132 q^{53} - 84112 q^{55} - 44552 q^{57} - 22582 q^{59} + 52822 q^{61} + 47804 q^{65} - 9848 q^{67} - 61376 q^{69} - 1680 q^{71} + 122052 q^{73} + 111034 q^{75} - 31704 q^{79} + 41954 q^{81} + 73948 q^{83} - 264888 q^{85} - 219156 q^{87} + 210588 q^{89} - 219784 q^{93} - 138624 q^{95} - 88480 q^{97} - 149880 q^{99}+O(q^{100})$$ 4 * q + 14 * q^3 - 42 * q^5 + 2 * q^9 + 716 * q^11 - 1428 * q^13 - 4448 * q^15 + 1344 * q^17 + 1946 * q^19 + 1792 * q^23 - 4282 * q^25 - 3976 * q^27 - 2400 * q^29 + 6804 * q^31 - 10416 * q^33 - 14640 * q^37 - 11560 * q^39 + 15792 * q^41 + 1048 * q^43 - 26978 * q^45 + 18396 * q^47 - 30252 * q^51 - 45132 * q^53 - 84112 * q^55 - 44552 * q^57 - 22582 * q^59 + 52822 * q^61 + 47804 * q^65 - 9848 * q^67 - 61376 * q^69 - 1680 * q^71 + 122052 * q^73 + 111034 * q^75 - 31704 * q^79 + 41954 * q^81 + 73948 * q^83 - 264888 * q^85 - 219156 * q^87 + 210588 * q^89 - 219784 * q^93 - 138624 * q^95 - 88480 * q^97 - 149880 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 49x^{2} + 48x + 2304$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 49\nu^{2} - 49\nu + 2304 ) / 2352$$ (-v^3 + 49*v^2 - 49*v + 2304) / 2352 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 49\nu^{2} + 4753\nu - 2304 ) / 2352$$ (v^3 - 49*v^2 + 4753*v - 2304) / 2352 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} + 145 ) / 49$$ (2*v^3 + 145) / 49
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 97\beta _1 - 97 ) / 2$$ (b3 + b2 + 97*b1 - 97) / 2 $$\nu^{3}$$ $$=$$ $$( 49\beta_{3} - 145 ) / 2$$ (49*b3 - 145) / 2

## Character values

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

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

## 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.

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
 3.72311 + 6.44862i −3.22311 − 5.58259i 3.72311 − 6.44862i −3.22311 + 5.58259i
0 −3.44622 5.96903i 0 24.2311 41.9695i 0 0 0 97.7471 169.303i 0
177.2 0 10.4462 + 18.0934i 0 −45.2311 + 78.3426i 0 0 0 −96.7471 + 167.571i 0
361.1 0 −3.44622 + 5.96903i 0 24.2311 + 41.9695i 0 0 0 97.7471 + 169.303i 0
361.2 0 10.4462 18.0934i 0 −45.2311 78.3426i 0 0 0 −96.7471 167.571i 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.k 4
7.b odd 2 1 392.6.i.h 4
7.c even 3 1 56.6.a.d 2
7.c even 3 1 inner 392.6.i.k 4
7.d odd 6 1 392.6.a.e 2
7.d odd 6 1 392.6.i.h 4
21.h odd 6 1 504.6.a.m 2
28.f even 6 1 784.6.a.q 2
28.g odd 6 1 112.6.a.j 2
56.k odd 6 1 448.6.a.r 2
56.p even 6 1 448.6.a.x 2
84.n even 6 1 1008.6.a.bi 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.d 2 7.c even 3 1
112.6.a.j 2 28.g odd 6 1
392.6.a.e 2 7.d odd 6 1
392.6.i.h 4 7.b odd 2 1
392.6.i.h 4 7.d odd 6 1
392.6.i.k 4 1.a even 1 1 trivial
392.6.i.k 4 7.c even 3 1 inner
448.6.a.r 2 56.k odd 6 1
448.6.a.x 2 56.p even 6 1
504.6.a.m 2 21.h odd 6 1
784.6.a.q 2 28.f even 6 1
1008.6.a.bi 2 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 14T_{3}^{3} + 340T_{3}^{2} + 2016T_{3} + 20736$$ acting on $$S_{6}^{\mathrm{new}}(392, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 14 T^{3} + \cdots + 20736$$
$5$ $$T^{4} + 42 T^{3} + \cdots + 19219456$$
$7$ $$T^{4}$$
$11$ $$T^{4} + \cdots + 8160592896$$
$13$ $$(T^{2} + 714 T + 71672)^{2}$$
$17$ $$T^{4} + \cdots + 12366329616$$
$19$ $$T^{4} + \cdots + 522046710784$$
$23$ $$T^{4} + \cdots + 2618493566976$$
$29$ $$(T^{2} + 1200 T - 57176388)^{2}$$
$31$ $$T^{4} + \cdots + 719161357689856$$
$37$ $$T^{4} + \cdots + 24\!\cdots\!44$$
$41$ $$(T^{2} - 7896 T - 22118548)^{2}$$
$43$ $$(T^{2} - 524 T - 51717888)^{2}$$
$47$ $$T^{4} + \cdots + 520039929619456$$
$53$ $$T^{4} + \cdots + 98\!\cdots\!16$$
$59$ $$T^{4} + \cdots + 10\!\cdots\!36$$
$61$ $$T^{4} + \cdots + 20\!\cdots\!36$$
$67$ $$T^{4} + \cdots + 212046252228864$$
$71$ $$(T^{2} + 840 T - 3181158400)^{2}$$
$73$ $$T^{4} + \cdots + 11\!\cdots\!44$$
$79$ $$T^{4} + \cdots + 12\!\cdots\!64$$
$83$ $$(T^{2} - 36974 T - 3038476256)^{2}$$
$89$ $$T^{4} + \cdots + 11\!\cdots\!36$$
$97$ $$(T^{2} + 44240 T - 14736460932)^{2}$$