# Properties

 Label 810.3.j.c Level $810$ Weight $3$ Character orbit 810.j Analytic conductor $22.071$ Analytic rank $0$ Dimension $8$ Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,3,Mod(269,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([1, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("810.269");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$810 = 2 \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 810.j (of order $$6$$, 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: $$22.0709014132$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.443364212736.6 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 16x^{6} + 175x^{4} - 1296x^{2} + 6561$$ x^8 - 16*x^6 + 175*x^4 - 1296*x^2 + 6561 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 30) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{6} - \beta_{2}) q^{2} + (2 \beta_1 - 2) q^{4} + (\beta_{5} + \beta_{3} - 2 \beta_{2}) q^{5} - \beta_{4} q^{7} + 2 \beta_{6} q^{8}+O(q^{10})$$ q + (-b6 - b2) * q^2 + (2*b1 - 2) * q^4 + (b5 + b3 - 2*b2) * q^5 - b4 * q^7 + 2*b6 * q^8 $$q + ( - \beta_{6} - \beta_{2}) q^{2} + (2 \beta_1 - 2) q^{4} + (\beta_{5} + \beta_{3} - 2 \beta_{2}) q^{5} - \beta_{4} q^{7} + 2 \beta_{6} q^{8} + ( - \beta_{7} + \beta_{4} - 4) q^{10} + 4 \beta_{3} q^{11} + (2 \beta_{5} + 2 \beta_{3}) q^{14} - 4 \beta_1 q^{16} - 8 \beta_{6} q^{17} + 12 q^{19} + (4 \beta_{6} - 2 \beta_{3} + 4 \beta_{2}) q^{20} - 4 \beta_{7} q^{22} + 17 \beta_{2} q^{23} + (4 \beta_{4} + 9 \beta_1) q^{25} + ( - 2 \beta_{7} + 2 \beta_{4}) q^{28} + ( - 32 \beta_1 + 32) q^{31} + 4 \beta_{2} q^{32} + 16 \beta_1 q^{34} + ( - 17 \beta_{6} + 4 \beta_{5}) q^{35} + ( - 4 \beta_{7} + 4 \beta_{4}) q^{37} + ( - 12 \beta_{6} - 12 \beta_{2}) q^{38} + (2 \beta_{7} - 8 \beta_1 + 8) q^{40} + (14 \beta_{5} + 14 \beta_{3}) q^{41} + 7 \beta_{4} q^{43} + 8 \beta_{5} q^{44} + 34 q^{46} + (25 \beta_{6} + 25 \beta_{2}) q^{47} + (15 \beta_1 - 15) q^{49} + ( - 8 \beta_{5} - 8 \beta_{3} - 9 \beta_{2}) q^{50} + 48 \beta_{6} q^{53} + ( - 8 \beta_{7} + 8 \beta_{4} + 68) q^{55} - 4 \beta_{3} q^{56} + ( - 4 \beta_{5} - 4 \beta_{3}) q^{59} + 16 \beta_1 q^{61} - 32 \beta_{6} q^{62} + 8 q^{64} - \beta_{7} q^{67} - 16 \beta_{2} q^{68} + (4 \beta_{4} + 34 \beta_1) q^{70} + ( - 20 \beta_{7} + 20 \beta_{4}) q^{73} - 8 \beta_{3} q^{74} + (24 \beta_1 - 24) q^{76} + 68 \beta_{2} q^{77} + 72 \beta_1 q^{79} + ( - 8 \beta_{6} - 4 \beta_{5}) q^{80} + ( - 14 \beta_{7} + 14 \beta_{4}) q^{82} + (31 \beta_{6} + 31 \beta_{2}) q^{83} + ( - 8 \beta_{7} + 32 \beta_1 - 32) q^{85} + ( - 14 \beta_{5} - 14 \beta_{3}) q^{86} + 8 \beta_{4} q^{88} - 16 \beta_{5} q^{89} + ( - 34 \beta_{6} - 34 \beta_{2}) q^{92} + ( - 50 \beta_1 + 50) q^{94} + (12 \beta_{5} + 12 \beta_{3} - 24 \beta_{2}) q^{95} + 28 \beta_{4} q^{97} + 15 \beta_{6} q^{98}+O(q^{100})$$ q + (-b6 - b2) * q^2 + (2*b1 - 2) * q^4 + (b5 + b3 - 2*b2) * q^5 - b4 * q^7 + 2*b6 * q^8 + (-b7 + b4 - 4) * q^10 + 4*b3 * q^11 + (2*b5 + 2*b3) * q^14 - 4*b1 * q^16 - 8*b6 * q^17 + 12 * q^19 + (4*b6 - 2*b3 + 4*b2) * q^20 - 4*b7 * q^22 + 17*b2 * q^23 + (4*b4 + 9*b1) * q^25 + (-2*b7 + 2*b4) * q^28 + (-32*b1 + 32) * q^31 + 4*b2 * q^32 + 16*b1 * q^34 + (-17*b6 + 4*b5) * q^35 + (-4*b7 + 4*b4) * q^37 + (-12*b6 - 12*b2) * q^38 + (2*b7 - 8*b1 + 8) * q^40 + (14*b5 + 14*b3) * q^41 + 7*b4 * q^43 + 8*b5 * q^44 + 34 * q^46 + (25*b6 + 25*b2) * q^47 + (15*b1 - 15) * q^49 + (-8*b5 - 8*b3 - 9*b2) * q^50 + 48*b6 * q^53 + (-8*b7 + 8*b4 + 68) * q^55 - 4*b3 * q^56 + (-4*b5 - 4*b3) * q^59 + 16*b1 * q^61 - 32*b6 * q^62 + 8 * q^64 - b7 * q^67 - 16*b2 * q^68 + (4*b4 + 34*b1) * q^70 + (-20*b7 + 20*b4) * q^73 - 8*b3 * q^74 + (24*b1 - 24) * q^76 + 68*b2 * q^77 + 72*b1 * q^79 + (-8*b6 - 4*b5) * q^80 + (-14*b7 + 14*b4) * q^82 + (31*b6 + 31*b2) * q^83 + (-8*b7 + 32*b1 - 32) * q^85 + (-14*b5 - 14*b3) * q^86 + 8*b4 * q^88 - 16*b5 * q^89 + (-34*b6 - 34*b2) * q^92 + (-50*b1 + 50) * q^94 + (12*b5 + 12*b3 - 24*b2) * q^95 + 28*b4 * q^97 + 15*b6 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{4}+O(q^{10})$$ 8 * q - 8 * q^4 $$8 q - 8 q^{4} - 32 q^{10} - 16 q^{16} + 96 q^{19} + 36 q^{25} + 128 q^{31} + 64 q^{34} + 32 q^{40} + 272 q^{46} - 60 q^{49} + 544 q^{55} + 64 q^{61} + 64 q^{64} + 136 q^{70} - 96 q^{76} + 288 q^{79} - 128 q^{85} + 200 q^{94}+O(q^{100})$$ 8 * q - 8 * q^4 - 32 * q^10 - 16 * q^16 + 96 * q^19 + 36 * q^25 + 128 * q^31 + 64 * q^34 + 32 * q^40 + 272 * q^46 - 60 * q^49 + 544 * q^55 + 64 * q^61 + 64 * q^64 + 136 * q^70 - 96 * q^76 + 288 * q^79 - 128 * q^85 + 200 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 16x^{6} + 175x^{4} - 1296x^{2} + 6561$$ :

 $$\beta_{1}$$ $$=$$ $$( -16\nu^{6} + 175\nu^{4} - 2800\nu^{2} + 20736 ) / 14175$$ (-16*v^6 + 175*v^4 - 2800*v^2 + 20736) / 14175 $$\beta_{2}$$ $$=$$ $$( 31\nu^{7} - 1225\nu^{5} + 5425\nu^{3} - 40176\nu ) / 127575$$ (31*v^7 - 1225*v^5 + 5425*v^3 - 40176*v) / 127575 $$\beta_{3}$$ $$=$$ $$( -47\nu^{6} + 1400\nu^{4} - 8225\nu^{2} + 60912 ) / 14175$$ (-47*v^6 + 1400*v^4 - 8225*v^2 + 60912) / 14175 $$\beta_{4}$$ $$=$$ $$( \nu^{7} + 3079\nu ) / 1575$$ (v^7 + 3079*v) / 1575 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} - 104 ) / 175$$ (-v^6 - 104) / 175 $$\beta_{6}$$ $$=$$ $$( -16\nu^{7} + 175\nu^{5} - 775\nu^{3} + 6561\nu ) / 18225$$ (-16*v^7 + 175*v^5 - 775*v^3 + 6561*v) / 18225 $$\beta_{7}$$ $$=$$ $$( -319\nu^{7} + 4375\nu^{5} - 55825\nu^{3} + 413424\nu ) / 127575$$ (-319*v^7 + 4375*v^5 - 55825*v^3 + 413424*v) / 127575
 $$\nu$$ $$=$$ $$( \beta_{6} + \beta_{4} + \beta_{2} ) / 2$$ (b6 + b4 + b2) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{3} - 8\beta _1 + 8$$ b5 + b3 - 8*b1 + 8 $$\nu^{3}$$ $$=$$ $$( -7\beta_{7} + 25\beta_{6} + 7\beta_{4} ) / 2$$ (-7*b7 + 25*b6 + 7*b4) / 2 $$\nu^{4}$$ $$=$$ $$16\beta_{3} - 47\beta_1$$ 16*b3 - 47*b1 $$\nu^{5}$$ $$=$$ $$( -31\beta_{7} - 319\beta_{2} ) / 2$$ (-31*b7 - 319*b2) / 2 $$\nu^{6}$$ $$=$$ $$-175\beta_{5} - 104$$ -175*b5 - 104 $$\nu^{7}$$ $$=$$ $$( -3079\beta_{6} + 71\beta_{4} - 3079\beta_{2} ) / 2$$ (-3079*b6 + 71*b4 - 3079*b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$$.

 $$n$$ $$487$$ $$731$$ $$\chi(n)$$ $$-1$$ $$1 - \beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1
 −2.17132 − 2.07011i 2.87843 + 0.845366i 2.17132 + 2.07011i −2.87843 − 0.845366i −2.17132 + 2.07011i 2.87843 − 0.845366i 2.17132 − 2.07011i −2.87843 + 0.845366i
−0.707107 + 1.22474i 0 −1.00000 1.73205i −2.15650 + 4.51104i 0 5.04975 + 2.91548i 2.82843 0 −4.00000 5.83095i
269.2 −0.707107 + 1.22474i 0 −1.00000 1.73205i 4.98493 + 0.387937i 0 −5.04975 2.91548i 2.82843 0 −4.00000 + 5.83095i
269.3 0.707107 1.22474i 0 −1.00000 1.73205i −4.98493 0.387937i 0 −5.04975 2.91548i −2.82843 0 −4.00000 + 5.83095i
269.4 0.707107 1.22474i 0 −1.00000 1.73205i 2.15650 4.51104i 0 5.04975 + 2.91548i −2.82843 0 −4.00000 5.83095i
539.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i −2.15650 4.51104i 0 5.04975 2.91548i 2.82843 0 −4.00000 + 5.83095i
539.2 −0.707107 1.22474i 0 −1.00000 + 1.73205i 4.98493 0.387937i 0 −5.04975 + 2.91548i 2.82843 0 −4.00000 5.83095i
539.3 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −4.98493 + 0.387937i 0 −5.04975 + 2.91548i −2.82843 0 −4.00000 5.83095i
539.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 2.15650 + 4.51104i 0 5.04975 2.91548i −2.82843 0 −4.00000 + 5.83095i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 269.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.d odd 2 1 inner
45.h odd 6 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.3.j.c 8
3.b odd 2 1 inner 810.3.j.c 8
5.b even 2 1 inner 810.3.j.c 8
9.c even 3 1 30.3.b.a 4
9.c even 3 1 inner 810.3.j.c 8
9.d odd 6 1 30.3.b.a 4
9.d odd 6 1 inner 810.3.j.c 8
15.d odd 2 1 inner 810.3.j.c 8
36.f odd 6 1 240.3.c.c 4
36.h even 6 1 240.3.c.c 4
45.h odd 6 1 30.3.b.a 4
45.h odd 6 1 inner 810.3.j.c 8
45.j even 6 1 30.3.b.a 4
45.j even 6 1 inner 810.3.j.c 8
45.k odd 12 2 150.3.d.d 4
45.l even 12 2 150.3.d.d 4
72.j odd 6 1 960.3.c.f 4
72.l even 6 1 960.3.c.e 4
72.n even 6 1 960.3.c.f 4
72.p odd 6 1 960.3.c.e 4
180.n even 6 1 240.3.c.c 4
180.p odd 6 1 240.3.c.c 4
180.v odd 12 2 1200.3.l.t 4
180.x even 12 2 1200.3.l.t 4
360.z odd 6 1 960.3.c.e 4
360.bd even 6 1 960.3.c.e 4
360.bh odd 6 1 960.3.c.f 4
360.bk even 6 1 960.3.c.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.b.a 4 9.c even 3 1
30.3.b.a 4 9.d odd 6 1
30.3.b.a 4 45.h odd 6 1
30.3.b.a 4 45.j even 6 1
150.3.d.d 4 45.k odd 12 2
150.3.d.d 4 45.l even 12 2
240.3.c.c 4 36.f odd 6 1
240.3.c.c 4 36.h even 6 1
240.3.c.c 4 180.n even 6 1
240.3.c.c 4 180.p odd 6 1
810.3.j.c 8 1.a even 1 1 trivial
810.3.j.c 8 3.b odd 2 1 inner
810.3.j.c 8 5.b even 2 1 inner
810.3.j.c 8 9.c even 3 1 inner
810.3.j.c 8 9.d odd 6 1 inner
810.3.j.c 8 15.d odd 2 1 inner
810.3.j.c 8 45.h odd 6 1 inner
810.3.j.c 8 45.j even 6 1 inner
960.3.c.e 4 72.l even 6 1
960.3.c.e 4 72.p odd 6 1
960.3.c.e 4 360.z odd 6 1
960.3.c.e 4 360.bd even 6 1
960.3.c.f 4 72.j odd 6 1
960.3.c.f 4 72.n even 6 1
960.3.c.f 4 360.bh odd 6 1
960.3.c.f 4 360.bk even 6 1
1200.3.l.t 4 180.v odd 12 2
1200.3.l.t 4 180.x even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(810, [\chi])$$:

 $$T_{7}^{4} - 34T_{7}^{2} + 1156$$ T7^4 - 34*T7^2 + 1156 $$T_{17}^{2} - 128$$ T17^2 - 128

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 2 T^{2} + 4)^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 18 T^{6} + \cdots + 390625$$
$7$ $$(T^{4} - 34 T^{2} + 1156)^{2}$$
$11$ $$(T^{4} - 272 T^{2} + 73984)^{2}$$
$13$ $$T^{8}$$
$17$ $$(T^{2} - 128)^{4}$$
$19$ $$(T - 12)^{8}$$
$23$ $$(T^{4} + 578 T^{2} + 334084)^{2}$$
$29$ $$T^{8}$$
$31$ $$(T^{2} - 32 T + 1024)^{4}$$
$37$ $$(T^{2} + 544)^{4}$$
$41$ $$(T^{4} - 3332 T^{2} + 11102224)^{2}$$
$43$ $$(T^{4} - 1666 T^{2} + 2775556)^{2}$$
$47$ $$(T^{4} + 1250 T^{2} + 1562500)^{2}$$
$53$ $$(T^{2} - 4608)^{4}$$
$59$ $$(T^{4} - 272 T^{2} + 73984)^{2}$$
$61$ $$(T^{2} - 16 T + 256)^{4}$$
$67$ $$(T^{4} - 34 T^{2} + 1156)^{2}$$
$71$ $$T^{8}$$
$73$ $$(T^{2} + 13600)^{4}$$
$79$ $$(T^{2} - 72 T + 5184)^{4}$$
$83$ $$(T^{4} + 1922 T^{2} + 3694084)^{2}$$
$89$ $$(T^{2} + 4352)^{4}$$
$97$ $$(T^{4} - 26656 T^{2} + 710542336)^{2}$$