# Properties

 Label 392.6.i.i 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: $$\Q(\sqrt{-3}, \sqrt{-115})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 28x^{2} - 29x + 841$$ x^4 - x^3 - 28*x^2 - 29*x + 841 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_{2} + 3 \beta_1) q^{3} + (3 \beta_{3} + 41 \beta_1 + 41) q^{5} + (6 \beta_{3} - 111 \beta_1 - 111) q^{9}+O(q^{10})$$ q + (-b3 + b2 + 3*b1) * q^3 + (3*b3 + 41*b1 + 41) * q^5 + (6*b3 - 111*b1 - 111) * q^9 $$q + ( - \beta_{3} + \beta_{2} + 3 \beta_1) q^{3} + (3 \beta_{3} + 41 \beta_1 + 41) q^{5} + (6 \beta_{3} - 111 \beta_1 - 111) q^{9} + (6 \beta_{3} - 6 \beta_{2} + 170 \beta_1) q^{11} + (39 \beta_{2} - 455) q^{13} + (32 \beta_{2} + 912) q^{15} + ( - 6 \beta_{3} + 6 \beta_{2} - 1608 \beta_1) q^{17} + ( - 45 \beta_{3} - 337 \beta_1 - 337) q^{19} + ( - 72 \beta_{3} + 552 \beta_1 + 552) q^{23} + (246 \beta_{3} - 246 \beta_{2} + 1661 \beta_1) q^{25} + (114 \beta_{2} + 1674) q^{27} + (114 \beta_{2} + 4032) q^{29} + (210 \beta_{3} - 210 \beta_{2} + 3106 \beta_1) q^{31} + (152 \beta_{3} + 1560 \beta_1 + 1560) q^{33} + (390 \beta_{3} + 4256 \beta_1 + 4256) q^{37} + (572 \beta_{3} - 572 \beta_{2} - 14820 \beta_1) q^{39} + (678 \beta_{2} + 652) q^{41} + ( - 798 \beta_{2} - 5002) q^{43} + ( - 87 \beta_{3} + 87 \beta_{2} + 1659 \beta_1) q^{45} + (714 \beta_{3} - 6374 \beta_1 - 6374) q^{47} + ( - 1590 \beta_{3} + 2754 \beta_1 + 2754) q^{51} + (768 \beta_{3} - 768 \beta_{2} - 5610 \beta_1) q^{53} + ( - 756 \beta_{2} - 13180) q^{55} + ( - 202 \beta_{2} - 14514) q^{57} + (2253 \beta_{3} - 2253 \beta_{2} + 6009 \beta_1) q^{59} + (75 \beta_{3} + 51369 \beta_1 + 51369) q^{61} + (234 \beta_{3} + 21710 \beta_1 + 21710) q^{65} + ( - 1944 \beta_{3} + \cdots + 12068 \beta_1) q^{67}+ \cdots + ( - 354 \beta_{2} + 6450) q^{99}+O(q^{100})$$ q + (-b3 + b2 + 3*b1) * q^3 + (3*b3 + 41*b1 + 41) * q^5 + (6*b3 - 111*b1 - 111) * q^9 + (6*b3 - 6*b2 + 170*b1) * q^11 + (39*b2 - 455) * q^13 + (32*b2 + 912) * q^15 + (-6*b3 + 6*b2 - 1608*b1) * q^17 + (-45*b3 - 337*b1 - 337) * q^19 + (-72*b3 + 552*b1 + 552) * q^23 + (246*b3 - 246*b2 + 1661*b1) * q^25 + (114*b2 + 1674) * q^27 + (114*b2 + 4032) * q^29 + (210*b3 - 210*b2 + 3106*b1) * q^31 + (152*b3 + 1560*b1 + 1560) * q^33 + (390*b3 + 4256*b1 + 4256) * q^37 + (572*b3 - 572*b2 - 14820*b1) * q^39 + (678*b2 + 652) * q^41 + (-798*b2 - 5002) * q^43 + (-87*b3 + 87*b2 + 1659*b1) * q^45 + (714*b3 - 6374*b1 - 6374) * q^47 + (-1590*b3 + 2754*b1 + 2754) * q^51 + (768*b3 - 768*b2 - 5610*b1) * q^53 + (-756*b2 - 13180) * q^55 + (-202*b2 - 14514) * q^57 + (2253*b3 - 2253*b2 + 6009*b1) * q^59 + (75*b3 + 51369*b1 + 51369) * q^61 + (234*b3 + 21710*b1 + 21710) * q^65 + (-1944*b3 + 1944*b2 + 12068*b1) * q^67 + (768*b2 - 26496) * q^69 + (-1740*b2 + 44860) * q^71 + (-1716*b3 + 1716*b2 + 27794*b1) * q^73 + (923*b3 + 79887*b1 + 79887) * q^75 + (1956*b3 - 24412*b1 - 24412) * q^79 + (126*b3 - 126*b2 - 61281*b1) * q^81 + (-3447*b2 - 17891) * q^83 + (5070*b2 + 72138) * q^85 + (-3690*b3 + 3690*b2 - 27234*b1) * q^87 + (-4728*b3 - 9150*b1 - 9150) * q^89 + (2476*b3 + 63132*b1 + 63132) * q^93 + (-2856*b3 + 2856*b2 - 60392*b1) * q^95 + (-462*b2 + 34992) * q^97 + (-354*b2 + 6450) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{3} + 82 q^{5} - 222 q^{9}+O(q^{10})$$ 4 * q - 6 * q^3 + 82 * q^5 - 222 * q^9 $$4 q - 6 q^{3} + 82 q^{5} - 222 q^{9} - 340 q^{11} - 1820 q^{13} + 3648 q^{15} + 3216 q^{17} - 674 q^{19} + 1104 q^{23} - 3322 q^{25} + 6696 q^{27} + 16128 q^{29} - 6212 q^{31} + 3120 q^{33} + 8512 q^{37} + 29640 q^{39} + 2608 q^{41} - 20008 q^{43} - 3318 q^{45} - 12748 q^{47} + 5508 q^{51} + 11220 q^{53} - 52720 q^{55} - 58056 q^{57} - 12018 q^{59} + 102738 q^{61} + 43420 q^{65} - 24136 q^{67} - 105984 q^{69} + 179440 q^{71} - 55588 q^{73} + 159774 q^{75} - 48824 q^{79} + 122562 q^{81} - 71564 q^{83} + 288552 q^{85} + 54468 q^{87} - 18300 q^{89} + 126264 q^{93} + 120784 q^{95} + 139968 q^{97} + 25800 q^{99}+O(q^{100})$$ 4 * q - 6 * q^3 + 82 * q^5 - 222 * q^9 - 340 * q^11 - 1820 * q^13 + 3648 * q^15 + 3216 * q^17 - 674 * q^19 + 1104 * q^23 - 3322 * q^25 + 6696 * q^27 + 16128 * q^29 - 6212 * q^31 + 3120 * q^33 + 8512 * q^37 + 29640 * q^39 + 2608 * q^41 - 20008 * q^43 - 3318 * q^45 - 12748 * q^47 + 5508 * q^51 + 11220 * q^53 - 52720 * q^55 - 58056 * q^57 - 12018 * q^59 + 102738 * q^61 + 43420 * q^65 - 24136 * q^67 - 105984 * q^69 + 179440 * q^71 - 55588 * q^73 + 159774 * q^75 - 48824 * q^79 + 122562 * q^81 - 71564 * q^83 + 288552 * q^85 + 54468 * q^87 - 18300 * q^89 + 126264 * q^93 + 120784 * q^95 + 139968 * q^97 + 25800 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 28x^{2} - 29x + 841$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 28\nu^{2} - 28\nu - 841 ) / 812$$ (v^3 + 28*v^2 - 28*v - 841) / 812 $$\beta_{2}$$ $$=$$ $$( -2\nu^{3} + 2\nu^{2} + 114\nu + 29 ) / 29$$ (-2*v^3 + 2*v^2 + 114*v + 29) / 29 $$\beta_{3}$$ $$=$$ $$( 59\nu^{3} + 28\nu^{2} + 1596\nu - 3335 ) / 812$$ (59*v^3 + 28*v^2 + 1596*v - 3335) / 812
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 3\beta_1 ) / 6$$ (b3 + b2 - 3*b1) / 6 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 171\beta _1 + 171 ) / 6$$ (-b3 + 2*b2 + 171*b1 + 171) / 6 $$\nu^{3}$$ $$=$$ $$( 28\beta_{3} - 14\beta_{2} + 129 ) / 3$$ (28*b3 - 14*b2 + 129) / 3

## Character values

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

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

## 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
 −4.39354 + 3.11396i 4.89354 − 2.24794i −4.39354 − 3.11396i 4.89354 + 2.24794i
0 −10.7871 18.6838i 0 −7.36126 + 12.7501i 0 0 0 −111.223 + 192.643i 0
177.2 0 7.78709 + 13.4876i 0 48.3613 83.7642i 0 0 0 0.222527 0.385428i 0
361.1 0 −10.7871 + 18.6838i 0 −7.36126 12.7501i 0 0 0 −111.223 192.643i 0
361.2 0 7.78709 13.4876i 0 48.3613 + 83.7642i 0 0 0 0.222527 + 0.385428i 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.i 4
7.b odd 2 1 392.6.i.j 4
7.c even 3 1 392.6.a.d 2
7.c even 3 1 inner 392.6.i.i 4
7.d odd 6 1 56.6.a.e 2
7.d odd 6 1 392.6.i.j 4
21.g even 6 1 504.6.a.i 2
28.f even 6 1 112.6.a.i 2
28.g odd 6 1 784.6.a.u 2
56.j odd 6 1 448.6.a.v 2
56.m even 6 1 448.6.a.t 2
84.j odd 6 1 1008.6.a.bd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.e 2 7.d odd 6 1
112.6.a.i 2 28.f even 6 1
392.6.a.d 2 7.c even 3 1
392.6.i.i 4 1.a even 1 1 trivial
392.6.i.i 4 7.c even 3 1 inner
392.6.i.j 4 7.b odd 2 1
392.6.i.j 4 7.d odd 6 1
448.6.a.t 2 56.m even 6 1
448.6.a.v 2 56.j odd 6 1
504.6.a.i 2 21.g even 6 1
784.6.a.u 2 28.g odd 6 1
1008.6.a.bd 2 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 6T_{3}^{3} + 372T_{3}^{2} - 2016T_{3} + 112896$$ acting on $$S_{6}^{\mathrm{new}}(392, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 6 T^{3} + \cdots + 112896$$
$5$ $$T^{4} - 82 T^{3} + \cdots + 2027776$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 340 T^{3} + \cdots + 271590400$$
$13$ $$(T^{2} + 910 T - 317720)^{2}$$
$17$ $$T^{4} + \cdots + 6621584683536$$
$19$ $$T^{4} + \cdots + 342290523136$$
$23$ $$T^{4} + \cdots + 2201591218176$$
$29$ $$(T^{2} - 8064 T + 11773404)^{2}$$
$31$ $$T^{4} + \cdots + 30994428445696$$
$37$ $$T^{4} + \cdots + 11\!\cdots\!96$$
$41$ $$(T^{2} - 1304 T - 158165876)^{2}$$
$43$ $$(T^{2} + 10004 T - 194677376)^{2}$$
$47$ $$T^{4} + \cdots + 18\!\cdots\!36$$
$53$ $$T^{4} + \cdots + 29\!\cdots\!00$$
$59$ $$T^{4} + \cdots + 29\!\cdots\!76$$
$61$ $$T^{4} + \cdots + 69\!\cdots\!96$$
$67$ $$T^{4} + \cdots + 13\!\cdots\!16$$
$71$ $$(T^{2} - 89720 T + 967897600)^{2}$$
$73$ $$T^{4} + \cdots + 59\!\cdots\!56$$
$79$ $$T^{4} + \cdots + 52\!\cdots\!76$$
$83$ $$(T^{2} + 35782 T - 3779136224)^{2}$$
$89$ $$T^{4} + \cdots + 58\!\cdots\!00$$
$97$ $$(T^{2} - 69984 T + 1150801884)^{2}$$