# Properties

 Label 363.2.f.c Level $363$ Weight $2$ Character orbit 363.f Analytic conductor $2.899$ Analytic rank $0$ Dimension $8$ CM discriminant -11 Inner twists $16$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,2,Mod(161,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 7]))

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

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

chi := DirichletCharacter("363.161");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 363.f (of order $$10$$, degree $$4$$, 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: $$2.89856959337$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: 8.0.228765625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{7} - 2x^{6} + 5x^{5} + x^{4} + 15x^{3} - 18x^{2} - 27x + 81$$ x^8 - x^7 - 2*x^6 + 5*x^5 + x^4 + 15*x^3 - 18*x^2 - 27*x + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} - 2 \beta_{7} q^{4} + ( - 2 \beta_{5} - \beta_{4}) q^{5} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 2) q^{9}+O(q^{10})$$ q - b1 * q^3 - 2*b7 * q^4 + (-2*b5 - b4) * q^5 + (2*b7 + b6 - b5 - 3*b4 - 3*b3 - b2 + b1 + 2) * q^9 $$q - \beta_1 q^{3} - 2 \beta_{7} q^{4} + ( - 2 \beta_{5} - \beta_{4}) q^{5} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 2) q^{9}+ \cdots + ( - 17 \beta_{7} + 17 \beta_{4} + \cdots - 17) q^{97}+O(q^{100})$$ q - b1 * q^3 - 2*b7 * q^4 + (-2*b5 - b4) * q^5 + (2*b7 + b6 - b5 - 3*b4 - 3*b3 - b2 + b1 + 2) * q^9 + (2*b6 - 2) * q^12 + (5*b7 - b2) * q^15 - 4*b4 * q^16 + (-2*b7 + 4*b6 - 4*b5 - 2*b4 - 2*b3 - 4*b2 + 4*b1 - 2) * q^20 + (2*b6 - 1) * q^23 + 6*b3 * q^25 + (-2*b5 + 3*b4) * q^27 + (-5*b7 + 5*b4 + 5*b3 - 5) * q^31 + (-6*b3 + 2*b1) * q^36 - 7*b7 * q^37 + (-5*b6 + 8) * q^45 + (2*b3 - 4*b1) * q^47 + (4*b7 + 4*b2) * q^48 - 7*b4 * q^49 + (4*b7 - 8*b6 + 8*b5 + 4*b4 + 4*b3 + 8*b2 - 8*b1 + 4) * q^53 + (-b7 - 2*b2) * q^59 + (2*b5 + 12*b4) * q^60 + (8*b7 - 8*b4 - 8*b3 + 8) * q^64 - 13 * q^67 + (-6*b3 + b1) * q^69 + (10*b5 + 5*b4) * q^71 + (6*b7 - 6*b6 + 6*b5 + 6*b2 - 6*b1 + 6) * q^75 + (-4*b3 + 8*b1) * q^80 + (b7 - 5*b2) * q^81 + (-10*b6 + 5) * q^89 + (2*b7 + 4*b2) * q^92 + 5*b5 * q^93 + (-17*b7 + 17*b4 + 17*b3 - 17) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - q^{3} + 4 q^{4} + 5 q^{9}+O(q^{10})$$ 8 * q - q^3 + 4 * q^4 + 5 * q^9 $$8 q - q^{3} + 4 q^{4} + 5 q^{9} - 8 q^{12} - 11 q^{15} - 8 q^{16} + 12 q^{25} + 8 q^{27} - 10 q^{31} - 10 q^{36} + 14 q^{37} + 44 q^{45} - 4 q^{48} - 14 q^{49} + 22 q^{60} + 16 q^{64} - 104 q^{67} - 11 q^{69} + 6 q^{75} - 7 q^{81} - 5 q^{93} - 34 q^{97}+O(q^{100})$$ 8 * q - q^3 + 4 * q^4 + 5 * q^9 - 8 * q^12 - 11 * q^15 - 8 * q^16 + 12 * q^25 + 8 * q^27 - 10 * q^31 - 10 * q^36 + 14 * q^37 + 44 * q^45 - 4 * q^48 - 14 * q^49 + 22 * q^60 + 16 * q^64 - 104 * q^67 - 11 * q^69 + 6 * q^75 - 7 * q^81 - 5 * q^93 - 34 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 2x^{6} + 5x^{5} + x^{4} + 15x^{3} - 18x^{2} - 27x + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{7} + \nu^{6} + 2\nu^{5} - 5\nu^{4} - \nu^{3} - 15\nu^{2} + 18\nu + 27 ) / 27$$ (-v^7 + v^6 + 2*v^5 - 5*v^4 - v^3 - 15*v^2 + 18*v + 27) / 27 $$\beta_{3}$$ $$=$$ $$( \nu^{6} + 16\nu ) / 3$$ (v^6 + 16*v) / 3 $$\beta_{4}$$ $$=$$ $$( -2\nu^{7} - 4\nu^{6} + 10\nu^{5} + 2\nu^{4} - 5\nu^{3} - 36\nu^{2} - 54\nu + 162 ) / 27$$ (-2*v^7 - 4*v^6 + 10*v^5 + 2*v^4 - 5*v^3 - 36*v^2 - 54*v + 162) / 27 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} - 2\nu^{6} + 5\nu^{5} + \nu^{4} + 2\nu^{3} - 18\nu^{2} - 27\nu + 81 ) / 9$$ (-v^7 - 2*v^6 + 5*v^5 + v^4 + 2*v^3 - 18*v^2 - 27*v + 81) / 9 $$\beta_{6}$$ $$=$$ $$\nu^{5} + 16$$ v^5 + 16 $$\beta_{7}$$ $$=$$ $$( -5\nu^{7} + 5\nu^{6} + 10\nu^{5} + 2\nu^{4} - 5\nu^{3} - 75\nu^{2} + 90\nu + 135 ) / 27$$ (-5*v^7 + 5*v^6 + 10*v^5 + 2*v^4 - 5*v^3 - 75*v^2 + 90*v + 135) / 27
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{7} + \beta_{6} - \beta_{5} - 3\beta_{4} - 3\beta_{3} - \beta_{2} + \beta _1 + 2$$ 2*b7 + b6 - b5 - 3*b4 - 3*b3 - b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$2\beta_{5} - 3\beta_{4}$$ 2*b5 - 3*b4 $$\nu^{4}$$ $$=$$ $$\beta_{7} - 5\beta_{2}$$ b7 - 5*b2 $$\nu^{5}$$ $$=$$ $$\beta_{6} - 16$$ b6 - 16 $$\nu^{6}$$ $$=$$ $$3\beta_{3} - 16\beta_1$$ 3*b3 - 16*b1 $$\nu^{7}$$ $$=$$ $$-35\beta_{7} - 13\beta_{6} + 13\beta_{5} + 48\beta_{4} + 48\beta_{3} + 13\beta_{2} - 13\beta _1 - 35$$ -35*b7 - 13*b6 + 13*b5 + 48*b4 + 48*b3 + 13*b2 - 13*b1 - 35

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$-1$$ $$-\beta_{7}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 1.42264 + 0.987975i −1.73166 − 0.0369185i 1.37924 − 1.04771i −0.570223 + 1.63550i 1.37924 + 1.04771i −0.570223 − 1.63550i 1.42264 − 0.987975i −1.73166 + 0.0369185i
0 −1.42264 0.987975i −0.618034 1.90211i −1.94946 2.68321i 0 0 0 1.04781 + 2.81107i 0
161.2 0 1.73166 + 0.0369185i −0.618034 1.90211i 1.94946 + 2.68321i 0 0 0 2.99727 + 0.127860i 0
215.1 0 −1.37924 + 1.04771i 1.61803 1.17557i 3.15430 + 1.02489i 0 0 0 0.804606 2.89009i 0
215.2 0 0.570223 1.63550i 1.61803 1.17557i −3.15430 1.02489i 0 0 0 −2.34969 1.86519i 0
233.1 0 −1.37924 1.04771i 1.61803 + 1.17557i 3.15430 1.02489i 0 0 0 0.804606 + 2.89009i 0
233.2 0 0.570223 + 1.63550i 1.61803 + 1.17557i −3.15430 + 1.02489i 0 0 0 −2.34969 + 1.86519i 0
239.1 0 −1.42264 + 0.987975i −0.618034 + 1.90211i −1.94946 + 2.68321i 0 0 0 1.04781 2.81107i 0
239.2 0 1.73166 0.0369185i −0.618034 + 1.90211i 1.94946 2.68321i 0 0 0 2.99727 0.127860i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 161.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
3.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
33.d even 2 1 inner
33.f even 10 3 inner
33.h odd 10 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.f.c 8
3.b odd 2 1 inner 363.2.f.c 8
11.b odd 2 1 CM 363.2.f.c 8
11.c even 5 1 33.2.d.a 2
11.c even 5 3 inner 363.2.f.c 8
11.d odd 10 1 33.2.d.a 2
11.d odd 10 3 inner 363.2.f.c 8
33.d even 2 1 inner 363.2.f.c 8
33.f even 10 1 33.2.d.a 2
33.f even 10 3 inner 363.2.f.c 8
33.h odd 10 1 33.2.d.a 2
33.h odd 10 3 inner 363.2.f.c 8
44.g even 10 1 528.2.b.a 2
44.h odd 10 1 528.2.b.a 2
55.h odd 10 1 825.2.f.a 2
55.j even 10 1 825.2.f.a 2
55.k odd 20 2 825.2.d.a 4
55.l even 20 2 825.2.d.a 4
88.k even 10 1 2112.2.b.f 2
88.l odd 10 1 2112.2.b.f 2
88.o even 10 1 2112.2.b.e 2
88.p odd 10 1 2112.2.b.e 2
99.m even 15 2 891.2.g.a 4
99.n odd 30 2 891.2.g.a 4
99.o odd 30 2 891.2.g.a 4
99.p even 30 2 891.2.g.a 4
132.n odd 10 1 528.2.b.a 2
132.o even 10 1 528.2.b.a 2
165.o odd 10 1 825.2.f.a 2
165.r even 10 1 825.2.f.a 2
165.u odd 20 2 825.2.d.a 4
165.v even 20 2 825.2.d.a 4
264.r odd 10 1 2112.2.b.f 2
264.t odd 10 1 2112.2.b.e 2
264.u even 10 1 2112.2.b.e 2
264.w even 10 1 2112.2.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.d.a 2 11.c even 5 1
33.2.d.a 2 11.d odd 10 1
33.2.d.a 2 33.f even 10 1
33.2.d.a 2 33.h odd 10 1
363.2.f.c 8 1.a even 1 1 trivial
363.2.f.c 8 3.b odd 2 1 inner
363.2.f.c 8 11.b odd 2 1 CM
363.2.f.c 8 11.c even 5 3 inner
363.2.f.c 8 11.d odd 10 3 inner
363.2.f.c 8 33.d even 2 1 inner
363.2.f.c 8 33.f even 10 3 inner
363.2.f.c 8 33.h odd 10 3 inner
528.2.b.a 2 44.g even 10 1
528.2.b.a 2 44.h odd 10 1
528.2.b.a 2 132.n odd 10 1
528.2.b.a 2 132.o even 10 1
825.2.d.a 4 55.k odd 20 2
825.2.d.a 4 55.l even 20 2
825.2.d.a 4 165.u odd 20 2
825.2.d.a 4 165.v even 20 2
825.2.f.a 2 55.h odd 10 1
825.2.f.a 2 55.j even 10 1
825.2.f.a 2 165.o odd 10 1
825.2.f.a 2 165.r even 10 1
891.2.g.a 4 99.m even 15 2
891.2.g.a 4 99.n odd 30 2
891.2.g.a 4 99.o odd 30 2
891.2.g.a 4 99.p even 30 2
2112.2.b.e 2 88.o even 10 1
2112.2.b.e 2 88.p odd 10 1
2112.2.b.e 2 264.t odd 10 1
2112.2.b.e 2 264.u even 10 1
2112.2.b.f 2 88.k even 10 1
2112.2.b.f 2 88.l odd 10 1
2112.2.b.f 2 264.r odd 10 1
2112.2.b.f 2 264.w even 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(363, [\chi])$$:

 $$T_{2}$$ T2 $$T_{5}^{8} - 11T_{5}^{6} + 121T_{5}^{4} - 1331T_{5}^{2} + 14641$$ T5^8 - 11*T5^6 + 121*T5^4 - 1331*T5^2 + 14641 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + T^{7} + \cdots + 81$$
$5$ $$T^{8} - 11 T^{6} + \cdots + 14641$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$(T^{2} + 11)^{4}$$
$29$ $$T^{8}$$
$31$ $$(T^{4} + 5 T^{3} + \cdots + 625)^{2}$$
$37$ $$(T^{4} - 7 T^{3} + \cdots + 2401)^{2}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8} - 44 T^{6} + \cdots + 3748096$$
$53$ $$T^{8} - 176 T^{6} + \cdots + 959512576$$
$59$ $$T^{8} - 11 T^{6} + \cdots + 14641$$
$61$ $$T^{8}$$
$67$ $$(T + 13)^{8}$$
$71$ $$T^{8} + \cdots + 5719140625$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$(T^{2} + 275)^{4}$$
$97$ $$(T^{4} + 17 T^{3} + \cdots + 83521)^{2}$$