# Properties

 Label 378.4.g.b Level $378$ Weight $4$ Character orbit 378.g Analytic conductor $22.303$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(109,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("378.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 378.g (of order $$3$$, degree $$2$$, 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: $$22.3027219822$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.11184604443.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 43x^{4} - 210x^{3} + 1849x^{2} - 4515x + 11025$$ x^6 + 43*x^4 - 210*x^3 + 1849*x^2 - 4515*x + 11025 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta_1 q^{2} + (4 \beta_1 - 4) q^{4} + (\beta_{5} - \beta_{3} + 2 \beta_1) q^{5} + ( - 2 \beta_{5} - \beta_{2} - 2 \beta_1 + 3) q^{7} - 8 q^{8}+O(q^{10})$$ q + 2*b1 * q^2 + (4*b1 - 4) * q^4 + (b5 - b3 + 2*b1) * q^5 + (-2*b5 - b2 - 2*b1 + 3) * q^7 - 8 * q^8 $$q + 2 \beta_1 q^{2} + (4 \beta_1 - 4) q^{4} + (\beta_{5} - \beta_{3} + 2 \beta_1) q^{5} + ( - 2 \beta_{5} - \beta_{2} - 2 \beta_1 + 3) q^{7} - 8 q^{8} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots - 4) q^{10}+ \cdots + ( - 12 \beta_{5} - 70 \beta_{4} + \cdots + 74) q^{98}+O(q^{100})$$ q + 2*b1 * q^2 + (4*b1 - 4) * q^4 + (b5 - b3 + 2*b1) * q^5 + (-2*b5 - b2 - 2*b1 + 3) * q^7 - 8 * q^8 + (2*b4 - 2*b3 + 2*b2 + 4*b1 - 4) * q^10 + (b5 + 3*b4 - 2*b3 + 2*b2 + 11*b1 - 11) * q^11 + (-b5 + b4 + b2 + 3) * q^13 + (2*b5 - 6*b2 + 2*b1 + 4) * q^14 - 16*b1 * q^16 + (-b5 + 5*b4 - 6*b3 + 6*b2 + 14*b1 - 14) * q^17 + (4*b5 + 5*b4 - 4*b3 - 5*b2 + 22*b1) * q^19 + (-4*b5 + 4*b4 + 4*b2 - 8) * q^20 + (-4*b5 + 4*b4 + 2*b3 + 6*b2 - 22) * q^22 + (8*b5 + 9*b4 - 8*b3 - 9*b2 + b1) * q^23 + (5*b5 + 9*b4 - 4*b3 + 4*b2 - 40*b1 + 40) * q^25 + (-2*b5 + 2*b3 + 6*b1) * q^26 + (12*b5 - 8*b2 + 12*b1 - 4) * q^28 + (-20*b5 + 20*b4 - 11*b3 + 9*b2 + 61) * q^29 + (22*b5 + 7*b4 + 15*b3 - 15*b2 + 39*b1 - 39) * q^31 + (-32*b1 + 32) * q^32 + (-12*b5 + 12*b4 - 2*b3 + 10*b2 - 28) * q^34 + (7*b5 - 14*b4 - 7*b3 - 7*b2 - 77*b1 + 126) * q^35 + (-9*b5 + 15*b4 + 9*b3 - 15*b2 + 130*b1) * q^37 + (10*b5 + 8*b4 + 2*b3 - 2*b2 + 44*b1 - 44) * q^38 + (-8*b5 + 8*b3 - 16*b1) * q^40 + (-3*b5 + 3*b4 - 15*b3 - 12*b2 - 63) * q^41 + (-38*b5 + 38*b4 - b3 + 37*b2 - 133) * q^43 + (-12*b5 - 4*b4 + 12*b3 + 4*b2 - 44*b1) * q^44 + (18*b5 + 16*b4 + 2*b3 - 2*b2 + 2*b1 - 2) * q^46 + (7*b5 + 14*b4 - 7*b3 - 14*b2 - 54*b1) * q^47 + (-16*b5 + 21*b4 + 35*b3 + 6*b2 - 37*b1 + 66) * q^49 + (-8*b5 + 8*b4 + 10*b3 + 18*b2 + 80) * q^50 + (-4*b4 + 4*b3 - 4*b2 + 12*b1 - 12) * q^52 + (-22*b5 - 41*b4 + 19*b3 - 19*b2 + 111*b1 - 111) * q^53 + (-22*b5 + 22*b4 + 11*b3 + 33*b2 - 226) * q^55 + (16*b5 + 8*b2 + 16*b1 - 24) * q^56 + (-18*b5 + 22*b4 + 18*b3 - 22*b2 + 122*b1) * q^58 + (-7*b5 + 60*b4 - 67*b3 + 67*b2 + 160*b1 - 160) * q^59 + (16*b5 + 42*b4 - 16*b3 - 42*b2 - 123*b1) * q^61 + (30*b5 - 30*b4 + 44*b3 + 14*b2 - 78) * q^62 + 64 * q^64 + (-4*b5 - 5*b4 + 4*b3 + 5*b2 - 75*b1) * q^65 + (51*b5 + 46*b4 + 5*b3 - 5*b2 + 43*b1 - 43) * q^67 + (-20*b5 + 4*b4 + 20*b3 - 4*b2 - 56*b1) * q^68 + (14*b5 + 14*b4 - 42*b3 + 98*b1 + 154) * q^70 + (61*b5 - 61*b4 + 27*b3 - 34*b2 + 134) * q^71 + (7*b5 + 27*b4 - 20*b3 + 20*b2 - 113*b1 + 113) * q^73 + (30*b5 - 18*b4 + 48*b3 - 48*b2 + 260*b1 - 260) * q^74 + (4*b5 - 4*b4 + 20*b3 + 16*b2 - 88) * q^76 + (35*b5 - 49*b3 - 14*b2 + 280*b1 + 77) * q^77 + (76*b5 + 27*b4 - 76*b3 - 27*b2 + 489*b1) * q^79 + (-16*b4 + 16*b3 - 16*b2 - 32*b1 + 32) * q^80 + (24*b5 + 30*b4 - 24*b3 - 30*b2 - 126*b1) * q^82 + (53*b5 - 53*b4 - 28*b3 - 81*b2 + 62) * q^83 + (-19*b5 + 19*b4 + 29*b3 + 48*b2 - 472) * q^85 + (-74*b5 + 2*b4 + 74*b3 - 2*b2 - 266*b1) * q^86 + (-8*b5 - 24*b4 + 16*b3 - 16*b2 - 88*b1 + 88) * q^88 + (-42*b5 - 84*b4 + 42*b3 + 84*b2 - 453*b1) * q^89 + (-10*b5 + 21*b4 - 14*b3 + 2*b2 + 116*b1 - 34) * q^91 + (4*b5 - 4*b4 + 36*b3 + 32*b2 - 4) * q^92 + (28*b5 + 14*b4 + 14*b3 - 14*b2 - 108*b1 + 108) * q^94 + (55*b4 - 55*b3 + 55*b2 + 173*b1 - 173) * q^95 + (-6*b5 + 6*b4 - 38*b3 - 32*b2 + 220) * q^97 + (-12*b5 - 70*b4 + 112*b3 - 20*b2 + 58*b1 + 74) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{2} - 12 q^{4} + 5 q^{5} + 8 q^{7} - 48 q^{8}+O(q^{10})$$ 6 * q + 6 * q^2 - 12 * q^4 + 5 * q^5 + 8 * q^7 - 48 * q^8 $$6 q + 6 q^{2} - 12 q^{4} + 5 q^{5} + 8 q^{7} - 48 q^{8} - 10 q^{10} - 29 q^{11} + 20 q^{13} + 20 q^{14} - 48 q^{16} - 38 q^{17} + 57 q^{19} - 40 q^{20} - 116 q^{22} - 14 q^{23} + 134 q^{25} + 20 q^{26} + 8 q^{28} + 362 q^{29} - 88 q^{31} + 96 q^{32} - 152 q^{34} + 490 q^{35} + 384 q^{37} - 114 q^{38} - 40 q^{40} - 432 q^{41} - 726 q^{43} - 116 q^{44} + 28 q^{46} - 183 q^{47} + 372 q^{49} + 536 q^{50} - 40 q^{52} - 396 q^{53} - 1268 q^{55} - 64 q^{56} + 362 q^{58} - 427 q^{59} - 427 q^{61} - 352 q^{62} + 384 q^{64} - 216 q^{65} - 32 q^{67} - 152 q^{68} + 1162 q^{70} + 790 q^{71} + 373 q^{73} - 768 q^{74} - 456 q^{76} + 1211 q^{77} + 1364 q^{79} + 80 q^{80} - 432 q^{82} + 154 q^{83} - 2678 q^{85} - 726 q^{86} + 232 q^{88} - 1233 q^{89} + 131 q^{91} + 112 q^{92} + 366 q^{94} - 464 q^{95} + 1180 q^{97} + 720 q^{98}+O(q^{100})$$ 6 * q + 6 * q^2 - 12 * q^4 + 5 * q^5 + 8 * q^7 - 48 * q^8 - 10 * q^10 - 29 * q^11 + 20 * q^13 + 20 * q^14 - 48 * q^16 - 38 * q^17 + 57 * q^19 - 40 * q^20 - 116 * q^22 - 14 * q^23 + 134 * q^25 + 20 * q^26 + 8 * q^28 + 362 * q^29 - 88 * q^31 + 96 * q^32 - 152 * q^34 + 490 * q^35 + 384 * q^37 - 114 * q^38 - 40 * q^40 - 432 * q^41 - 726 * q^43 - 116 * q^44 + 28 * q^46 - 183 * q^47 + 372 * q^49 + 536 * q^50 - 40 * q^52 - 396 * q^53 - 1268 * q^55 - 64 * q^56 + 362 * q^58 - 427 * q^59 - 427 * q^61 - 352 * q^62 + 384 * q^64 - 216 * q^65 - 32 * q^67 - 152 * q^68 + 1162 * q^70 + 790 * q^71 + 373 * q^73 - 768 * q^74 - 456 * q^76 + 1211 * q^77 + 1364 * q^79 + 80 * q^80 - 432 * q^82 + 154 * q^83 - 2678 * q^85 - 726 * q^86 + 232 * q^88 - 1233 * q^89 + 131 * q^91 + 112 * q^92 + 366 * q^94 - 464 * q^95 + 1180 * q^97 + 720 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 43x^{4} - 210x^{3} + 1849x^{2} - 4515x + 11025$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} + 43\nu^{3} - 105\nu^{2} + 1849\nu ) / 4515$$ (v^5 + 43*v^3 - 105*v^2 + 1849*v) / 4515 $$\beta_{2}$$ $$=$$ $$( \nu^{4} + 4\nu^{3} + 43\nu^{2} - 62\nu + 827 ) / 43$$ (v^4 + 4*v^3 + 43*v^2 - 62*v + 827) / 43 $$\beta_{3}$$ $$=$$ $$( \nu^{4} + 3\nu^{3} + 43\nu^{2} - 148\nu + 932 ) / 43$$ (v^4 + 3*v^3 + 43*v^2 - 148*v + 932) / 43 $$\beta_{4}$$ $$=$$ $$( -29\nu^{5} - 827\nu^{3} + 7560\nu^{2} - 40076\nu + 86835 ) / 4515$$ (-29*v^5 - 827*v^3 + 7560*v^2 - 40076*v + 86835) / 4515 $$\beta_{5}$$ $$=$$ $$( -29\nu^{5} - 932\nu^{3} + 7560\nu^{2} - 35561\nu + 97860 ) / 4515$$ (-29*v^5 - 932*v^3 + 7560*v^2 - 35561*v + 97860) / 4515
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} ) / 3$$ (b5 - b4 - b3 + b2) / 3 $$\nu^{2}$$ $$=$$ $$( 5\beta_{5} - 2\beta_{4} + 7\beta_{3} - 7\beta_{2} + 87\beta _1 - 87 ) / 3$$ (5*b5 - 2*b4 + 7*b3 - 7*b2 + 87*b1 - 87) / 3 $$\nu^{3}$$ $$=$$ $$( -86\beta_{5} + 86\beta_{4} - 43\beta_{3} + 43\beta_{2} + 315 ) / 3$$ (-86*b5 + 86*b4 - 43*b3 + 43*b2 + 315) / 3 $$\nu^{4}$$ $$=$$ $$( 191\beta_{5} - 320\beta_{4} - 191\beta_{3} + 320\beta_{2} - 3741\beta_1 ) / 3$$ (191*b5 - 320*b4 - 191*b3 + 320*b2 - 3741*b1) / 3 $$\nu^{5}$$ $$=$$ $$( 2374\beta_{5} - 2059\beta_{4} + 4433\beta_{3} - 4433\beta_{2} + 22680\beta _1 - 22680 ) / 3$$ (2374*b5 - 2059*b4 + 4433*b3 - 4433*b2 + 22680*b1 - 22680) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$1$$ $$-\beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 −3.77219 − 6.53362i 1.60692 + 2.78326i 2.16527 + 3.75036i −3.77219 + 6.53362i 1.60692 − 2.78326i 2.16527 − 3.75036i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i −3.90212 6.75867i 0 16.3412 + 8.71578i −8.00000 0 7.80425 13.5173i
109.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −0.301070 0.521469i 0 −17.6665 + 5.55825i −8.00000 0 0.602141 1.04294i
109.3 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 6.70319 + 11.6103i 0 5.32531 17.7381i −8.00000 0 −13.4064 + 23.2205i
163.1 1.00000 1.73205i 0 −2.00000 3.46410i −3.90212 + 6.75867i 0 16.3412 8.71578i −8.00000 0 7.80425 + 13.5173i
163.2 1.00000 1.73205i 0 −2.00000 3.46410i −0.301070 + 0.521469i 0 −17.6665 5.55825i −8.00000 0 0.602141 + 1.04294i
163.3 1.00000 1.73205i 0 −2.00000 3.46410i 6.70319 11.6103i 0 5.32531 + 17.7381i −8.00000 0 −13.4064 23.2205i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.3 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 378.4.g.b yes 6
3.b odd 2 1 378.4.g.a 6
7.c even 3 1 inner 378.4.g.b yes 6
21.h odd 6 1 378.4.g.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.4.g.a 6 3.b odd 2 1
378.4.g.a 6 21.h odd 6 1
378.4.g.b yes 6 1.a even 1 1 trivial
378.4.g.b yes 6 7.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} - 5T_{5}^{5} + 133T_{5}^{4} + 666T_{5}^{3} + 11349T_{5}^{2} + 6804T_{5} + 3969$$ acting on $$S_{4}^{\mathrm{new}}(378, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 4)^{3}$$
$3$ $$T^{6}$$
$5$ $$T^{6} - 5 T^{5} + \cdots + 3969$$
$7$ $$T^{6} - 8 T^{5} + \cdots + 40353607$$
$11$ $$T^{6} + 29 T^{5} + \cdots + 4862025$$
$13$ $$(T^{3} - 10 T^{2} + \cdots + 603)^{2}$$
$17$ $$T^{6} + \cdots + 2114252361$$
$19$ $$T^{6} + \cdots + 19329618961$$
$23$ $$T^{6} + \cdots + 148635694089$$
$29$ $$(T^{3} - 181 T^{2} + \cdots - 862407)^{2}$$
$31$ $$T^{6} + \cdots + 27799667506521$$
$37$ $$T^{6} + \cdots + 1957974001$$
$41$ $$(T^{3} + 216 T^{2} + \cdots + 164025)^{2}$$
$43$ $$(T^{3} + 363 T^{2} + \cdots - 6611597)^{2}$$
$47$ $$T^{6} + \cdots + 2568361041$$
$53$ $$T^{6} + \cdots + 13\!\cdots\!61$$
$59$ $$T^{6} + \cdots + 15\!\cdots\!69$$
$61$ $$T^{6} + \cdots + 1548419366025$$
$67$ $$T^{6} + \cdots + 70\!\cdots\!25$$
$71$ $$(T^{3} - 395 T^{2} + \cdots + 115887969)^{2}$$
$73$ $$T^{6} + \cdots + 202747156022761$$
$79$ $$T^{6} + \cdots + 21738803675121$$
$83$ $$(T^{3} - 77 T^{2} + \cdots - 172613511)^{2}$$
$89$ $$T^{6} + \cdots + 37\!\cdots\!69$$
$97$ $$(T^{3} - 590 T^{2} + \cdots + 67884200)^{2}$$