# Properties

 Label 810.4.e.c Level $810$ Weight $4$ Character orbit 810.e Analytic conductor $47.792$ 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] = [810,4,Mod(271,810)]

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([2, 0]))

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

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

chi := DirichletCharacter("810.271");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$810 = 2 \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 810.e (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: $$47.7915471046$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + 8 q^{8} +O(q^{10})$$ q + (2*z - 2) * q^2 - 4*z * q^4 - 5*z * q^5 + (-4*z + 4) * q^7 + 8 * q^8 $$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + 8 q^{8} + 10 q^{10} + (12 \zeta_{6} - 12) q^{11} + 58 \zeta_{6} q^{13} + 8 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} + 66 q^{17} - 100 q^{19} + (20 \zeta_{6} - 20) q^{20} - 24 \zeta_{6} q^{22} - 132 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} - 116 q^{26} - 16 q^{28} + ( - 90 \zeta_{6} + 90) q^{29} - 152 \zeta_{6} q^{31} - 32 \zeta_{6} q^{32} + (132 \zeta_{6} - 132) q^{34} - 20 q^{35} - 34 q^{37} + ( - 200 \zeta_{6} + 200) q^{38} - 40 \zeta_{6} q^{40} + 438 \zeta_{6} q^{41} + (32 \zeta_{6} - 32) q^{43} + 48 q^{44} + 264 q^{46} + ( - 204 \zeta_{6} + 204) q^{47} + 327 \zeta_{6} q^{49} - 50 \zeta_{6} q^{50} + ( - 232 \zeta_{6} + 232) q^{52} + 222 q^{53} + 60 q^{55} + ( - 32 \zeta_{6} + 32) q^{56} + 180 \zeta_{6} q^{58} - 420 \zeta_{6} q^{59} + (902 \zeta_{6} - 902) q^{61} + 304 q^{62} + 64 q^{64} + ( - 290 \zeta_{6} + 290) q^{65} + 1024 \zeta_{6} q^{67} - 264 \zeta_{6} q^{68} + ( - 40 \zeta_{6} + 40) q^{70} + 432 q^{71} + 362 q^{73} + ( - 68 \zeta_{6} + 68) q^{74} + 400 \zeta_{6} q^{76} + 48 \zeta_{6} q^{77} + ( - 160 \zeta_{6} + 160) q^{79} + 80 q^{80} - 876 q^{82} + (72 \zeta_{6} - 72) q^{83} - 330 \zeta_{6} q^{85} - 64 \zeta_{6} q^{86} + (96 \zeta_{6} - 96) q^{88} + 810 q^{89} + 232 q^{91} + (528 \zeta_{6} - 528) q^{92} + 408 \zeta_{6} q^{94} + 500 \zeta_{6} q^{95} + (1106 \zeta_{6} - 1106) q^{97} - 654 q^{98} +O(q^{100})$$ q + (2*z - 2) * q^2 - 4*z * q^4 - 5*z * q^5 + (-4*z + 4) * q^7 + 8 * q^8 + 10 * q^10 + (12*z - 12) * q^11 + 58*z * q^13 + 8*z * q^14 + (16*z - 16) * q^16 + 66 * q^17 - 100 * q^19 + (20*z - 20) * q^20 - 24*z * q^22 - 132*z * q^23 + (25*z - 25) * q^25 - 116 * q^26 - 16 * q^28 + (-90*z + 90) * q^29 - 152*z * q^31 - 32*z * q^32 + (132*z - 132) * q^34 - 20 * q^35 - 34 * q^37 + (-200*z + 200) * q^38 - 40*z * q^40 + 438*z * q^41 + (32*z - 32) * q^43 + 48 * q^44 + 264 * q^46 + (-204*z + 204) * q^47 + 327*z * q^49 - 50*z * q^50 + (-232*z + 232) * q^52 + 222 * q^53 + 60 * q^55 + (-32*z + 32) * q^56 + 180*z * q^58 - 420*z * q^59 + (902*z - 902) * q^61 + 304 * q^62 + 64 * q^64 + (-290*z + 290) * q^65 + 1024*z * q^67 - 264*z * q^68 + (-40*z + 40) * q^70 + 432 * q^71 + 362 * q^73 + (-68*z + 68) * q^74 + 400*z * q^76 + 48*z * q^77 + (-160*z + 160) * q^79 + 80 * q^80 - 876 * q^82 + (72*z - 72) * q^83 - 330*z * q^85 - 64*z * q^86 + (96*z - 96) * q^88 + 810 * q^89 + 232 * q^91 + (528*z - 528) * q^92 + 408*z * q^94 + 500*z * q^95 + (1106*z - 1106) * q^97 - 654 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} - 5 q^{5} + 4 q^{7} + 16 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^4 - 5 * q^5 + 4 * q^7 + 16 * q^8 $$2 q - 2 q^{2} - 4 q^{4} - 5 q^{5} + 4 q^{7} + 16 q^{8} + 20 q^{10} - 12 q^{11} + 58 q^{13} + 8 q^{14} - 16 q^{16} + 132 q^{17} - 200 q^{19} - 20 q^{20} - 24 q^{22} - 132 q^{23} - 25 q^{25} - 232 q^{26} - 32 q^{28} + 90 q^{29} - 152 q^{31} - 32 q^{32} - 132 q^{34} - 40 q^{35} - 68 q^{37} + 200 q^{38} - 40 q^{40} + 438 q^{41} - 32 q^{43} + 96 q^{44} + 528 q^{46} + 204 q^{47} + 327 q^{49} - 50 q^{50} + 232 q^{52} + 444 q^{53} + 120 q^{55} + 32 q^{56} + 180 q^{58} - 420 q^{59} - 902 q^{61} + 608 q^{62} + 128 q^{64} + 290 q^{65} + 1024 q^{67} - 264 q^{68} + 40 q^{70} + 864 q^{71} + 724 q^{73} + 68 q^{74} + 400 q^{76} + 48 q^{77} + 160 q^{79} + 160 q^{80} - 1752 q^{82} - 72 q^{83} - 330 q^{85} - 64 q^{86} - 96 q^{88} + 1620 q^{89} + 464 q^{91} - 528 q^{92} + 408 q^{94} + 500 q^{95} - 1106 q^{97} - 1308 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^4 - 5 * q^5 + 4 * q^7 + 16 * q^8 + 20 * q^10 - 12 * q^11 + 58 * q^13 + 8 * q^14 - 16 * q^16 + 132 * q^17 - 200 * q^19 - 20 * q^20 - 24 * q^22 - 132 * q^23 - 25 * q^25 - 232 * q^26 - 32 * q^28 + 90 * q^29 - 152 * q^31 - 32 * q^32 - 132 * q^34 - 40 * q^35 - 68 * q^37 + 200 * q^38 - 40 * q^40 + 438 * q^41 - 32 * q^43 + 96 * q^44 + 528 * q^46 + 204 * q^47 + 327 * q^49 - 50 * q^50 + 232 * q^52 + 444 * q^53 + 120 * q^55 + 32 * q^56 + 180 * q^58 - 420 * q^59 - 902 * q^61 + 608 * q^62 + 128 * q^64 + 290 * q^65 + 1024 * q^67 - 264 * q^68 + 40 * q^70 + 864 * q^71 + 724 * q^73 + 68 * q^74 + 400 * q^76 + 48 * q^77 + 160 * q^79 + 160 * q^80 - 1752 * q^82 - 72 * q^83 - 330 * q^85 - 64 * q^86 - 96 * q^88 + 1620 * q^89 + 464 * q^91 - 528 * q^92 + 408 * q^94 + 500 * q^95 - 1106 * q^97 - 1308 * q^98

## Character values

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

 $$n$$ $$487$$ $$731$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 1.73205i 0 −2.00000 + 3.46410i −2.50000 + 4.33013i 0 2.00000 + 3.46410i 8.00000 0 10.0000
541.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −2.50000 4.33013i 0 2.00000 3.46410i 8.00000 0 10.0000
 $$n$$: 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.4.e.c 2
3.b odd 2 1 810.4.e.w 2
9.c even 3 1 10.4.a.a 1
9.c even 3 1 inner 810.4.e.c 2
9.d odd 6 1 90.4.a.a 1
9.d odd 6 1 810.4.e.w 2
36.f odd 6 1 80.4.a.f 1
36.h even 6 1 720.4.a.j 1
45.h odd 6 1 450.4.a.q 1
45.j even 6 1 50.4.a.c 1
45.k odd 12 2 50.4.b.a 2
45.l even 12 2 450.4.c.d 2
63.g even 3 1 490.4.e.i 2
63.h even 3 1 490.4.e.i 2
63.k odd 6 1 490.4.e.a 2
63.l odd 6 1 490.4.a.o 1
63.t odd 6 1 490.4.e.a 2
72.n even 6 1 320.4.a.m 1
72.p odd 6 1 320.4.a.b 1
99.h odd 6 1 1210.4.a.b 1
117.t even 6 1 1690.4.a.a 1
144.v odd 12 2 1280.4.d.g 2
144.x even 12 2 1280.4.d.j 2
180.p odd 6 1 400.4.a.b 1
180.x even 12 2 400.4.c.c 2
315.bg odd 6 1 2450.4.a.b 1
360.z odd 6 1 1600.4.a.bx 1
360.bk even 6 1 1600.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 9.c even 3 1
50.4.a.c 1 45.j even 6 1
50.4.b.a 2 45.k odd 12 2
80.4.a.f 1 36.f odd 6 1
90.4.a.a 1 9.d odd 6 1
320.4.a.b 1 72.p odd 6 1
320.4.a.m 1 72.n even 6 1
400.4.a.b 1 180.p odd 6 1
400.4.c.c 2 180.x even 12 2
450.4.a.q 1 45.h odd 6 1
450.4.c.d 2 45.l even 12 2
490.4.a.o 1 63.l odd 6 1
490.4.e.a 2 63.k odd 6 1
490.4.e.a 2 63.t odd 6 1
490.4.e.i 2 63.g even 3 1
490.4.e.i 2 63.h even 3 1
720.4.a.j 1 36.h even 6 1
810.4.e.c 2 1.a even 1 1 trivial
810.4.e.c 2 9.c even 3 1 inner
810.4.e.w 2 3.b odd 2 1
810.4.e.w 2 9.d odd 6 1
1210.4.a.b 1 99.h odd 6 1
1280.4.d.g 2 144.v odd 12 2
1280.4.d.j 2 144.x even 12 2
1600.4.a.d 1 360.bk even 6 1
1600.4.a.bx 1 360.z odd 6 1
1690.4.a.a 1 117.t even 6 1
2450.4.a.b 1 315.bg odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(810, [\chi])$$:

 $$T_{7}^{2} - 4T_{7} + 16$$ T7^2 - 4*T7 + 16 $$T_{11}^{2} + 12T_{11} + 144$$ T11^2 + 12*T11 + 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 5T + 25$$
$7$ $$T^{2} - 4T + 16$$
$11$ $$T^{2} + 12T + 144$$
$13$ $$T^{2} - 58T + 3364$$
$17$ $$(T - 66)^{2}$$
$19$ $$(T + 100)^{2}$$
$23$ $$T^{2} + 132T + 17424$$
$29$ $$T^{2} - 90T + 8100$$
$31$ $$T^{2} + 152T + 23104$$
$37$ $$(T + 34)^{2}$$
$41$ $$T^{2} - 438T + 191844$$
$43$ $$T^{2} + 32T + 1024$$
$47$ $$T^{2} - 204T + 41616$$
$53$ $$(T - 222)^{2}$$
$59$ $$T^{2} + 420T + 176400$$
$61$ $$T^{2} + 902T + 813604$$
$67$ $$T^{2} - 1024 T + 1048576$$
$71$ $$(T - 432)^{2}$$
$73$ $$(T - 362)^{2}$$
$79$ $$T^{2} - 160T + 25600$$
$83$ $$T^{2} + 72T + 5184$$
$89$ $$(T - 810)^{2}$$
$97$ $$T^{2} + 1106 T + 1223236$$