# Properties

 Label 363.2.e.d Level $363$ Weight $2$ Character orbit 363.e Analytic conductor $2.899$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,2,Mod(124,363)]

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([0, 8]))

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

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

chi := DirichletCharacter("363.124");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 363.e (of order $$5$$, 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: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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 $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{2} - \zeta_{10}^{2} q^{3} - 2 \zeta_{10}^{3} q^{4} - 4 \zeta_{10} q^{5} + 2 \zeta_{10} q^{6} + \zeta_{10}^{3} q^{7} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} +O(q^{10})$$ q + (2*z^3 - 2*z^2 + 2*z - 2) * q^2 - z^2 * q^3 - 2*z^3 * q^4 - 4*z * q^5 + 2*z * q^6 + z^3 * q^7 + (z^3 - z^2 + z - 1) * q^9 $$q + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{2} - \zeta_{10}^{2} q^{3} - 2 \zeta_{10}^{3} q^{4} - 4 \zeta_{10} q^{5} + 2 \zeta_{10} q^{6} + \zeta_{10}^{3} q^{7} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} + 8 q^{10} - 2 q^{12} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{13} - 2 \zeta_{10}^{2} q^{14} + 4 \zeta_{10}^{3} q^{15} + 4 \zeta_{10} q^{16} + 4 \zeta_{10} q^{17} - 2 \zeta_{10}^{3} q^{18} + 3 \zeta_{10}^{2} q^{19} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} - 8) q^{20} + q^{21} + 2 q^{23} + 11 \zeta_{10}^{2} q^{25} - 4 \zeta_{10}^{3} q^{26} + \zeta_{10} q^{27} + 2 \zeta_{10} q^{28} + 6 \zeta_{10}^{3} q^{29} - 8 \zeta_{10}^{2} q^{30} + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 5) q^{31} - 8 q^{32} - 8 q^{34} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{35} + 2 \zeta_{10}^{2} q^{36} - 3 \zeta_{10}^{3} q^{37} - 6 \zeta_{10} q^{38} + 2 \zeta_{10} q^{39} + 2 \zeta_{10}^{2} q^{41} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{42} - 12 q^{43} + 4 q^{45} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{46} + 2 \zeta_{10}^{2} q^{47} - 4 \zeta_{10}^{3} q^{48} + 6 \zeta_{10} q^{49} - 22 \zeta_{10} q^{50} - 4 \zeta_{10}^{3} q^{51} + 4 \zeta_{10}^{2} q^{52} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 6) q^{53} - 2 q^{54} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{57} - 12 \zeta_{10}^{2} q^{58} + 10 \zeta_{10}^{3} q^{59} + 8 \zeta_{10} q^{60} + 3 \zeta_{10} q^{61} + 10 \zeta_{10}^{3} q^{62} - \zeta_{10}^{2} q^{63} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{64} + 8 q^{65} - q^{67} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{68} - 2 \zeta_{10}^{2} q^{69} + 8 \zeta_{10}^{3} q^{70} - 11 \zeta_{10}^{3} q^{73} + 6 \zeta_{10}^{2} q^{74} + ( - 11 \zeta_{10}^{3} + 11 \zeta_{10}^{2} - 11 \zeta_{10} + 11) q^{75} + 6 q^{76} - 4 q^{78} + ( - 11 \zeta_{10}^{3} + 11 \zeta_{10}^{2} - 11 \zeta_{10} + 11) q^{79} - 16 \zeta_{10}^{2} q^{80} - \zeta_{10}^{3} q^{81} - 4 \zeta_{10} q^{82} + 6 \zeta_{10} q^{83} - 2 \zeta_{10}^{3} q^{84} - 16 \zeta_{10}^{2} q^{85} + ( - 24 \zeta_{10}^{3} + 24 \zeta_{10}^{2} - 24 \zeta_{10} + 24) q^{86} + 6 q^{87} + 12 q^{89} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} - 8) q^{90} - 2 \zeta_{10}^{2} q^{91} - 4 \zeta_{10}^{3} q^{92} - 5 \zeta_{10} q^{93} - 4 \zeta_{10} q^{94} - 12 \zeta_{10}^{3} q^{95} + 8 \zeta_{10}^{2} q^{96} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 5) q^{97} - 12 q^{98} +O(q^{100})$$ q + (2*z^3 - 2*z^2 + 2*z - 2) * q^2 - z^2 * q^3 - 2*z^3 * q^4 - 4*z * q^5 + 2*z * q^6 + z^3 * q^7 + (z^3 - z^2 + z - 1) * q^9 + 8 * q^10 - 2 * q^12 + (2*z^3 - 2*z^2 + 2*z - 2) * q^13 - 2*z^2 * q^14 + 4*z^3 * q^15 + 4*z * q^16 + 4*z * q^17 - 2*z^3 * q^18 + 3*z^2 * q^19 + (8*z^3 - 8*z^2 + 8*z - 8) * q^20 + q^21 + 2 * q^23 + 11*z^2 * q^25 - 4*z^3 * q^26 + z * q^27 + 2*z * q^28 + 6*z^3 * q^29 - 8*z^2 * q^30 + (-5*z^3 + 5*z^2 - 5*z + 5) * q^31 - 8 * q^32 - 8 * q^34 + (-4*z^3 + 4*z^2 - 4*z + 4) * q^35 + 2*z^2 * q^36 - 3*z^3 * q^37 - 6*z * q^38 + 2*z * q^39 + 2*z^2 * q^41 + (2*z^3 - 2*z^2 + 2*z - 2) * q^42 - 12 * q^43 + 4 * q^45 + (4*z^3 - 4*z^2 + 4*z - 4) * q^46 + 2*z^2 * q^47 - 4*z^3 * q^48 + 6*z * q^49 - 22*z * q^50 - 4*z^3 * q^51 + 4*z^2 * q^52 + (6*z^3 - 6*z^2 + 6*z - 6) * q^53 - 2 * q^54 + (-3*z^3 + 3*z^2 - 3*z + 3) * q^57 - 12*z^2 * q^58 + 10*z^3 * q^59 + 8*z * q^60 + 3*z * q^61 + 10*z^3 * q^62 - z^2 * q^63 + (-8*z^3 + 8*z^2 - 8*z + 8) * q^64 + 8 * q^65 - q^67 + (-8*z^3 + 8*z^2 - 8*z + 8) * q^68 - 2*z^2 * q^69 + 8*z^3 * q^70 - 11*z^3 * q^73 + 6*z^2 * q^74 + (-11*z^3 + 11*z^2 - 11*z + 11) * q^75 + 6 * q^76 - 4 * q^78 + (-11*z^3 + 11*z^2 - 11*z + 11) * q^79 - 16*z^2 * q^80 - z^3 * q^81 - 4*z * q^82 + 6*z * q^83 - 2*z^3 * q^84 - 16*z^2 * q^85 + (-24*z^3 + 24*z^2 - 24*z + 24) * q^86 + 6 * q^87 + 12 * q^89 + (8*z^3 - 8*z^2 + 8*z - 8) * q^90 - 2*z^2 * q^91 - 4*z^3 * q^92 - 5*z * q^93 - 4*z * q^94 - 12*z^3 * q^95 + 8*z^2 * q^96 + (5*z^3 - 5*z^2 + 5*z - 5) * q^97 - 12 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + q^{3} - 2 q^{4} - 4 q^{5} + 2 q^{6} + q^{7} - q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + q^3 - 2 * q^4 - 4 * q^5 + 2 * q^6 + q^7 - q^9 $$4 q - 2 q^{2} + q^{3} - 2 q^{4} - 4 q^{5} + 2 q^{6} + q^{7} - q^{9} + 32 q^{10} - 8 q^{12} - 2 q^{13} + 2 q^{14} + 4 q^{15} + 4 q^{16} + 4 q^{17} - 2 q^{18} - 3 q^{19} - 8 q^{20} + 4 q^{21} + 8 q^{23} - 11 q^{25} - 4 q^{26} + q^{27} + 2 q^{28} + 6 q^{29} + 8 q^{30} + 5 q^{31} - 32 q^{32} - 32 q^{34} + 4 q^{35} - 2 q^{36} - 3 q^{37} - 6 q^{38} + 2 q^{39} - 2 q^{41} - 2 q^{42} - 48 q^{43} + 16 q^{45} - 4 q^{46} - 2 q^{47} - 4 q^{48} + 6 q^{49} - 22 q^{50} - 4 q^{51} - 4 q^{52} - 6 q^{53} - 8 q^{54} + 3 q^{57} + 12 q^{58} + 10 q^{59} + 8 q^{60} + 3 q^{61} + 10 q^{62} + q^{63} + 8 q^{64} + 32 q^{65} - 4 q^{67} + 8 q^{68} + 2 q^{69} + 8 q^{70} - 11 q^{73} - 6 q^{74} + 11 q^{75} + 24 q^{76} - 16 q^{78} + 11 q^{79} + 16 q^{80} - q^{81} - 4 q^{82} + 6 q^{83} - 2 q^{84} + 16 q^{85} + 24 q^{86} + 24 q^{87} + 48 q^{89} - 8 q^{90} + 2 q^{91} - 4 q^{92} - 5 q^{93} - 4 q^{94} - 12 q^{95} - 8 q^{96} - 5 q^{97} - 48 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 + q^3 - 2 * q^4 - 4 * q^5 + 2 * q^6 + q^7 - q^9 + 32 * q^10 - 8 * q^12 - 2 * q^13 + 2 * q^14 + 4 * q^15 + 4 * q^16 + 4 * q^17 - 2 * q^18 - 3 * q^19 - 8 * q^20 + 4 * q^21 + 8 * q^23 - 11 * q^25 - 4 * q^26 + q^27 + 2 * q^28 + 6 * q^29 + 8 * q^30 + 5 * q^31 - 32 * q^32 - 32 * q^34 + 4 * q^35 - 2 * q^36 - 3 * q^37 - 6 * q^38 + 2 * q^39 - 2 * q^41 - 2 * q^42 - 48 * q^43 + 16 * q^45 - 4 * q^46 - 2 * q^47 - 4 * q^48 + 6 * q^49 - 22 * q^50 - 4 * q^51 - 4 * q^52 - 6 * q^53 - 8 * q^54 + 3 * q^57 + 12 * q^58 + 10 * q^59 + 8 * q^60 + 3 * q^61 + 10 * q^62 + q^63 + 8 * q^64 + 32 * q^65 - 4 * q^67 + 8 * q^68 + 2 * q^69 + 8 * q^70 - 11 * q^73 - 6 * q^74 + 11 * q^75 + 24 * q^76 - 16 * q^78 + 11 * q^79 + 16 * q^80 - q^81 - 4 * q^82 + 6 * q^83 - 2 * q^84 + 16 * q^85 + 24 * q^86 + 24 * q^87 + 48 * q^89 - 8 * q^90 + 2 * q^91 - 4 * q^92 - 5 * q^93 - 4 * q^94 - 12 * q^95 - 8 * q^96 - 5 * q^97 - 48 * q^98

## Character values

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

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
124.1
 0.809017 + 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 − 0.587785i
−1.61803 + 1.17557i −0.309017 0.951057i 0.618034 1.90211i −3.23607 2.35114i 1.61803 + 1.17557i −0.309017 + 0.951057i 0 −0.809017 + 0.587785i 8.00000
130.1 0.618034 1.90211i 0.809017 0.587785i −1.61803 1.17557i 1.23607 + 3.80423i −0.618034 1.90211i 0.809017 + 0.587785i 0 0.309017 0.951057i 8.00000
148.1 0.618034 + 1.90211i 0.809017 + 0.587785i −1.61803 + 1.17557i 1.23607 3.80423i −0.618034 + 1.90211i 0.809017 0.587785i 0 0.309017 + 0.951057i 8.00000
202.1 −1.61803 1.17557i −0.309017 + 0.951057i 0.618034 + 1.90211i −3.23607 + 2.35114i 1.61803 1.17557i −0.309017 0.951057i 0 −0.809017 0.587785i 8.00000
 $$n$$: 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
11.c even 5 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.e.d 4
11.b odd 2 1 363.2.e.i 4
11.c even 5 1 363.2.a.c yes 1
11.c even 5 3 inner 363.2.e.d 4
11.d odd 10 1 363.2.a.a 1
11.d odd 10 3 363.2.e.i 4
33.f even 10 1 1089.2.a.k 1
33.h odd 10 1 1089.2.a.a 1
44.g even 10 1 5808.2.a.bh 1
44.h odd 10 1 5808.2.a.bi 1
55.h odd 10 1 9075.2.a.t 1
55.j even 10 1 9075.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.a 1 11.d odd 10 1
363.2.a.c yes 1 11.c even 5 1
363.2.e.d 4 1.a even 1 1 trivial
363.2.e.d 4 11.c even 5 3 inner
363.2.e.i 4 11.b odd 2 1
363.2.e.i 4 11.d odd 10 3
1089.2.a.a 1 33.h odd 10 1
1089.2.a.k 1 33.f even 10 1
5808.2.a.bh 1 44.g even 10 1
5808.2.a.bi 1 44.h odd 10 1
9075.2.a.b 1 55.j even 10 1
9075.2.a.t 1 55.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 2T_{2}^{3} + 4T_{2}^{2} + 8T_{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(363, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$3$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$5$ $$T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256$$
$7$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$17$ $$T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256$$
$19$ $$T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81$$
$23$ $$(T - 2)^{4}$$
$29$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$31$ $$T^{4} - 5 T^{3} + 25 T^{2} - 125 T + 625$$
$37$ $$T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81$$
$41$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$43$ $$(T + 12)^{4}$$
$47$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$53$ $$T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$59$ $$T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 10000$$
$61$ $$T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81$$
$67$ $$(T + 1)^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 11 T^{3} + 121 T^{2} + \cdots + 14641$$
$79$ $$T^{4} - 11 T^{3} + 121 T^{2} + \cdots + 14641$$
$83$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$89$ $$(T - 12)^{4}$$
$97$ $$T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625$$