# Properties

 Label 361.2.e.d Level $361$ Weight $2$ Character orbit 361.e Analytic conductor $2.883$ Analytic rank $0$ Dimension $6$ Inner twists $6$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [361,2,Mod(28,361)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(361, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([8]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("361.28");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$361 = 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 361.e (of order $$9$$, degree $$6$$, 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: $$2.88259951297$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-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 $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{18} q^{3} + 2 \zeta_{18}^{5} q^{4} + 3 \zeta_{18}^{4} q^{5} + ( - \zeta_{18}^{3} + 1) q^{7} + \zeta_{18}^{2} q^{9}+O(q^{10})$$ q + 2*z * q^3 + 2*z^5 * q^4 + 3*z^4 * q^5 + (-z^3 + 1) * q^7 + z^2 * q^9 $$q + 2 \zeta_{18} q^{3} + 2 \zeta_{18}^{5} q^{4} + 3 \zeta_{18}^{4} q^{5} + ( - \zeta_{18}^{3} + 1) q^{7} + \zeta_{18}^{2} q^{9} - 3 \zeta_{18}^{3} q^{11} + (4 \zeta_{18}^{3} - 4) q^{12} + ( - 4 \zeta_{18}^{5} + 4 \zeta_{18}^{2}) q^{13} + 6 \zeta_{18}^{5} q^{15} - 4 \zeta_{18} q^{16} + (3 \zeta_{18}^{4} - 3 \zeta_{18}) q^{17} - 6 q^{20} + ( - 2 \zeta_{18}^{4} + 2 \zeta_{18}) q^{21} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{2}) q^{25} - 4 \zeta_{18}^{3} q^{27} + 2 \zeta_{18}^{2} q^{28} + 6 \zeta_{18}^{2} q^{29} + ( - 4 \zeta_{18}^{3} + 4) q^{31} - 6 \zeta_{18}^{4} q^{33} + 3 \zeta_{18} q^{35} + (2 \zeta_{18}^{4} - 2 \zeta_{18}) q^{36} + 2 q^{37} + 8 q^{39} + 6 \zeta_{18} q^{41} - \zeta_{18}^{4} q^{43} + ( - 6 \zeta_{18}^{5} + 6 \zeta_{18}^{2}) q^{44} + (3 \zeta_{18}^{3} - 3) q^{45} - 3 \zeta_{18}^{2} q^{47} - 8 \zeta_{18}^{2} q^{48} + 6 \zeta_{18}^{3} q^{49} + (6 \zeta_{18}^{5} - 6 \zeta_{18}^{2}) q^{51} + 8 \zeta_{18}^{4} q^{52} - 12 \zeta_{18}^{5} q^{53} + ( - 9 \zeta_{18}^{4} + 9 \zeta_{18}) q^{55} + (6 \zeta_{18}^{4} - 6 \zeta_{18}) q^{59} - 12 \zeta_{18} q^{60} + \zeta_{18}^{5} q^{61} + ( - \zeta_{18}^{5} + \zeta_{18}^{2}) q^{63} + ( - 8 \zeta_{18}^{3} + 8) q^{64} + 12 \zeta_{18}^{3} q^{65} - 4 \zeta_{18}^{2} q^{67} - 6 \zeta_{18}^{3} q^{68} + 6 \zeta_{18}^{4} q^{71} + 7 \zeta_{18} q^{73} - 8 q^{75} - 3 q^{77} - 8 \zeta_{18} q^{79} - 12 \zeta_{18}^{5} q^{80} - 11 \zeta_{18}^{4} q^{81} + (12 \zeta_{18}^{3} - 12) q^{83} + 4 \zeta_{18}^{3} q^{84} - 9 \zeta_{18}^{2} q^{85} + 12 \zeta_{18}^{3} q^{87} + (12 \zeta_{18}^{5} - 12 \zeta_{18}^{2}) q^{89} - 4 \zeta_{18}^{5} q^{91} + ( - 8 \zeta_{18}^{4} + 8 \zeta_{18}) q^{93} + ( - 8 \zeta_{18}^{4} + 8 \zeta_{18}) q^{97} - 3 \zeta_{18}^{5} q^{99} +O(q^{100})$$ q + 2*z * q^3 + 2*z^5 * q^4 + 3*z^4 * q^5 + (-z^3 + 1) * q^7 + z^2 * q^9 - 3*z^3 * q^11 + (4*z^3 - 4) * q^12 + (-4*z^5 + 4*z^2) * q^13 + 6*z^5 * q^15 - 4*z * q^16 + (3*z^4 - 3*z) * q^17 - 6 * q^20 + (-2*z^4 + 2*z) * q^21 + (4*z^5 - 4*z^2) * q^25 - 4*z^3 * q^27 + 2*z^2 * q^28 + 6*z^2 * q^29 + (-4*z^3 + 4) * q^31 - 6*z^4 * q^33 + 3*z * q^35 + (2*z^4 - 2*z) * q^36 + 2 * q^37 + 8 * q^39 + 6*z * q^41 - z^4 * q^43 + (-6*z^5 + 6*z^2) * q^44 + (3*z^3 - 3) * q^45 - 3*z^2 * q^47 - 8*z^2 * q^48 + 6*z^3 * q^49 + (6*z^5 - 6*z^2) * q^51 + 8*z^4 * q^52 - 12*z^5 * q^53 + (-9*z^4 + 9*z) * q^55 + (6*z^4 - 6*z) * q^59 - 12*z * q^60 + z^5 * q^61 + (-z^5 + z^2) * q^63 + (-8*z^3 + 8) * q^64 + 12*z^3 * q^65 - 4*z^2 * q^67 - 6*z^3 * q^68 + 6*z^4 * q^71 + 7*z * q^73 - 8 * q^75 - 3 * q^77 - 8*z * q^79 - 12*z^5 * q^80 - 11*z^4 * q^81 + (12*z^3 - 12) * q^83 + 4*z^3 * q^84 - 9*z^2 * q^85 + 12*z^3 * q^87 + (12*z^5 - 12*z^2) * q^89 - 4*z^5 * q^91 + (-8*z^4 + 8*z) * q^93 + (-8*z^4 + 8*z) * q^97 - 3*z^5 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{7}+O(q^{10})$$ 6 * q + 3 * q^7 $$6 q + 3 q^{7} - 9 q^{11} - 12 q^{12} - 36 q^{20} - 12 q^{27} + 12 q^{31} + 12 q^{37} + 48 q^{39} - 9 q^{45} + 18 q^{49} + 24 q^{64} + 36 q^{65} - 18 q^{68} - 48 q^{75} - 18 q^{77} - 36 q^{83} + 12 q^{84} + 36 q^{87}+O(q^{100})$$ 6 * q + 3 * q^7 - 9 * q^11 - 12 * q^12 - 36 * q^20 - 12 * q^27 + 12 * q^31 + 12 * q^37 + 48 * q^39 - 9 * q^45 + 18 * q^49 + 24 * q^64 + 36 * q^65 - 18 * q^68 - 48 * q^75 - 18 * q^77 - 36 * q^83 + 12 * q^84 + 36 * q^87

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/361\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{18}^{5}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1
 −0.766044 + 0.642788i 0.939693 − 0.342020i −0.173648 + 0.984808i −0.173648 − 0.984808i 0.939693 + 0.342020i −0.766044 − 0.642788i
0 −1.53209 + 1.28558i 1.87939 0.684040i −2.81908 1.02606i 0 0.500000 0.866025i 0 0.173648 0.984808i 0
54.1 0 1.87939 0.684040i −0.347296 1.96962i 0.520945 2.95442i 0 0.500000 + 0.866025i 0 0.766044 0.642788i 0
62.1 0 −0.347296 + 1.96962i −1.53209 + 1.28558i 2.29813 + 1.92836i 0 0.500000 + 0.866025i 0 −0.939693 0.342020i 0
99.1 0 −0.347296 1.96962i −1.53209 1.28558i 2.29813 1.92836i 0 0.500000 0.866025i 0 −0.939693 + 0.342020i 0
234.1 0 1.87939 + 0.684040i −0.347296 + 1.96962i 0.520945 + 2.95442i 0 0.500000 0.866025i 0 0.766044 + 0.642788i 0
245.1 0 −1.53209 1.28558i 1.87939 + 0.684040i −2.81908 + 1.02606i 0 0.500000 + 0.866025i 0 0.173648 + 0.984808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 28.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 2 inner
19.e even 9 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.2.e.d 6
19.b odd 2 1 361.2.e.e 6
19.c even 3 2 inner 361.2.e.d 6
19.d odd 6 2 361.2.e.e 6
19.e even 9 1 19.2.a.a 1
19.e even 9 2 361.2.c.c 2
19.e even 9 3 inner 361.2.e.d 6
19.f odd 18 1 361.2.a.b 1
19.f odd 18 2 361.2.c.a 2
19.f odd 18 3 361.2.e.e 6
57.j even 18 1 3249.2.a.d 1
57.l odd 18 1 171.2.a.b 1
76.k even 18 1 5776.2.a.c 1
76.l odd 18 1 304.2.a.f 1
95.o odd 18 1 9025.2.a.d 1
95.p even 18 1 475.2.a.b 1
95.q odd 36 2 475.2.b.a 2
133.u even 9 1 931.2.f.c 2
133.w even 9 1 931.2.f.c 2
133.x odd 18 1 931.2.f.b 2
133.y odd 18 1 931.2.a.a 1
133.z odd 18 1 931.2.f.b 2
152.t even 18 1 1216.2.a.o 1
152.u odd 18 1 1216.2.a.b 1
209.q odd 18 1 2299.2.a.b 1
228.v even 18 1 2736.2.a.c 1
247.bn even 18 1 3211.2.a.a 1
285.bd odd 18 1 4275.2.a.i 1
323.s even 18 1 5491.2.a.b 1
380.ba odd 18 1 7600.2.a.c 1
399.cj even 18 1 8379.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.a.a 1 19.e even 9 1
171.2.a.b 1 57.l odd 18 1
304.2.a.f 1 76.l odd 18 1
361.2.a.b 1 19.f odd 18 1
361.2.c.a 2 19.f odd 18 2
361.2.c.c 2 19.e even 9 2
361.2.e.d 6 1.a even 1 1 trivial
361.2.e.d 6 19.c even 3 2 inner
361.2.e.d 6 19.e even 9 3 inner
361.2.e.e 6 19.b odd 2 1
361.2.e.e 6 19.d odd 6 2
361.2.e.e 6 19.f odd 18 3
475.2.a.b 1 95.p even 18 1
475.2.b.a 2 95.q odd 36 2
931.2.a.a 1 133.y odd 18 1
931.2.f.b 2 133.x odd 18 1
931.2.f.b 2 133.z odd 18 1
931.2.f.c 2 133.u even 9 1
931.2.f.c 2 133.w even 9 1
1216.2.a.b 1 152.u odd 18 1
1216.2.a.o 1 152.t even 18 1
2299.2.a.b 1 209.q odd 18 1
2736.2.a.c 1 228.v even 18 1
3211.2.a.a 1 247.bn even 18 1
3249.2.a.d 1 57.j even 18 1
4275.2.a.i 1 285.bd odd 18 1
5491.2.a.b 1 323.s even 18 1
5776.2.a.c 1 76.k even 18 1
7600.2.a.c 1 380.ba odd 18 1
8379.2.a.j 1 399.cj even 18 1
9025.2.a.d 1 95.o odd 18 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(361, [\chi])$$:

 $$T_{2}$$ T2 $$T_{3}^{6} - 8T_{3}^{3} + 64$$ T3^6 - 8*T3^3 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - 8T^{3} + 64$$
$5$ $$T^{6} + 27T^{3} + 729$$
$7$ $$(T^{2} - T + 1)^{3}$$
$11$ $$(T^{2} + 3 T + 9)^{3}$$
$13$ $$T^{6} - 64T^{3} + 4096$$
$17$ $$T^{6} - 27T^{3} + 729$$
$19$ $$T^{6}$$
$23$ $$T^{6}$$
$29$ $$T^{6} + 216 T^{3} + 46656$$
$31$ $$(T^{2} - 4 T + 16)^{3}$$
$37$ $$(T - 2)^{6}$$
$41$ $$T^{6} - 216 T^{3} + 46656$$
$43$ $$T^{6} - T^{3} + 1$$
$47$ $$T^{6} - 27T^{3} + 729$$
$53$ $$T^{6} + 1728 T^{3} + 2985984$$
$59$ $$T^{6} - 216 T^{3} + 46656$$
$61$ $$T^{6} - T^{3} + 1$$
$67$ $$T^{6} - 64T^{3} + 4096$$
$71$ $$T^{6} + 216 T^{3} + 46656$$
$73$ $$T^{6} - 343 T^{3} + 117649$$
$79$ $$T^{6} + 512 T^{3} + 262144$$
$83$ $$(T^{2} + 12 T + 144)^{3}$$
$89$ $$T^{6} + 1728 T^{3} + 2985984$$
$97$ $$T^{6} + 512 T^{3} + 262144$$