# Properties

 Label 363.3.b.d Level $363$ Weight $3$ Character orbit 363.b Analytic conductor $9.891$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,3,Mod(122,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("363.122");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 363.b (of order $$2$$, degree $$1$$, 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: $$9.89103359628$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 3$$ x^2 - x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = \sqrt{-11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + 3 q^{3} - 7 q^{4} - 2 \beta q^{5} - 3 \beta q^{6} + 8 q^{7} + 3 \beta q^{8} + 9 q^{9} +O(q^{10})$$ q - b * q^2 + 3 * q^3 - 7 * q^4 - 2*b * q^5 - 3*b * q^6 + 8 * q^7 + 3*b * q^8 + 9 * q^9 $$q - \beta q^{2} + 3 q^{3} - 7 q^{4} - 2 \beta q^{5} - 3 \beta q^{6} + 8 q^{7} + 3 \beta q^{8} + 9 q^{9} - 22 q^{10} - 21 q^{12} - 4 q^{13} - 8 \beta q^{14} - 6 \beta q^{15} + 5 q^{16} - 4 \beta q^{17} - 9 \beta q^{18} + 6 q^{19} + 14 \beta q^{20} + 24 q^{21} + 2 \beta q^{23} + 9 \beta q^{24} - 19 q^{25} + 4 \beta q^{26} + 27 q^{27} - 56 q^{28} + 12 \beta q^{29} - 66 q^{30} - 26 q^{31} + 7 \beta q^{32} - 44 q^{34} - 16 \beta q^{35} - 63 q^{36} + 30 q^{37} - 6 \beta q^{38} - 12 q^{39} + 66 q^{40} - 4 \beta q^{41} - 24 \beta q^{42} - 42 q^{43} - 18 \beta q^{45} + 22 q^{46} + 26 \beta q^{47} + 15 q^{48} + 15 q^{49} + 19 \beta q^{50} - 12 \beta q^{51} + 28 q^{52} - 18 \beta q^{53} - 27 \beta q^{54} + 24 \beta q^{56} + 18 q^{57} + 132 q^{58} + 20 \beta q^{59} + 42 \beta q^{60} - 12 q^{61} + 26 \beta q^{62} + 72 q^{63} + 97 q^{64} + 8 \beta q^{65} + 2 q^{67} + 28 \beta q^{68} + 6 \beta q^{69} - 176 q^{70} - 18 \beta q^{71} + 27 \beta q^{72} + 74 q^{73} - 30 \beta q^{74} - 57 q^{75} - 42 q^{76} + 12 \beta q^{78} + 40 q^{79} - 10 \beta q^{80} + 81 q^{81} - 44 q^{82} + 12 \beta q^{83} - 168 q^{84} - 88 q^{85} + 42 \beta q^{86} + 36 \beta q^{87} - 36 \beta q^{89} - 198 q^{90} - 32 q^{91} - 14 \beta q^{92} - 78 q^{93} + 286 q^{94} - 12 \beta q^{95} + 21 \beta q^{96} + 62 q^{97} - 15 \beta q^{98} +O(q^{100})$$ q - b * q^2 + 3 * q^3 - 7 * q^4 - 2*b * q^5 - 3*b * q^6 + 8 * q^7 + 3*b * q^8 + 9 * q^9 - 22 * q^10 - 21 * q^12 - 4 * q^13 - 8*b * q^14 - 6*b * q^15 + 5 * q^16 - 4*b * q^17 - 9*b * q^18 + 6 * q^19 + 14*b * q^20 + 24 * q^21 + 2*b * q^23 + 9*b * q^24 - 19 * q^25 + 4*b * q^26 + 27 * q^27 - 56 * q^28 + 12*b * q^29 - 66 * q^30 - 26 * q^31 + 7*b * q^32 - 44 * q^34 - 16*b * q^35 - 63 * q^36 + 30 * q^37 - 6*b * q^38 - 12 * q^39 + 66 * q^40 - 4*b * q^41 - 24*b * q^42 - 42 * q^43 - 18*b * q^45 + 22 * q^46 + 26*b * q^47 + 15 * q^48 + 15 * q^49 + 19*b * q^50 - 12*b * q^51 + 28 * q^52 - 18*b * q^53 - 27*b * q^54 + 24*b * q^56 + 18 * q^57 + 132 * q^58 + 20*b * q^59 + 42*b * q^60 - 12 * q^61 + 26*b * q^62 + 72 * q^63 + 97 * q^64 + 8*b * q^65 + 2 * q^67 + 28*b * q^68 + 6*b * q^69 - 176 * q^70 - 18*b * q^71 + 27*b * q^72 + 74 * q^73 - 30*b * q^74 - 57 * q^75 - 42 * q^76 + 12*b * q^78 + 40 * q^79 - 10*b * q^80 + 81 * q^81 - 44 * q^82 + 12*b * q^83 - 168 * q^84 - 88 * q^85 + 42*b * q^86 + 36*b * q^87 - 36*b * q^89 - 198 * q^90 - 32 * q^91 - 14*b * q^92 - 78 * q^93 + 286 * q^94 - 12*b * q^95 + 21*b * q^96 + 62 * q^97 - 15*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 14 q^{4} + 16 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 14 * q^4 + 16 * q^7 + 18 * q^9 $$2 q + 6 q^{3} - 14 q^{4} + 16 q^{7} + 18 q^{9} - 44 q^{10} - 42 q^{12} - 8 q^{13} + 10 q^{16} + 12 q^{19} + 48 q^{21} - 38 q^{25} + 54 q^{27} - 112 q^{28} - 132 q^{30} - 52 q^{31} - 88 q^{34} - 126 q^{36} + 60 q^{37} - 24 q^{39} + 132 q^{40} - 84 q^{43} + 44 q^{46} + 30 q^{48} + 30 q^{49} + 56 q^{52} + 36 q^{57} + 264 q^{58} - 24 q^{61} + 144 q^{63} + 194 q^{64} + 4 q^{67} - 352 q^{70} + 148 q^{73} - 114 q^{75} - 84 q^{76} + 80 q^{79} + 162 q^{81} - 88 q^{82} - 336 q^{84} - 176 q^{85} - 396 q^{90} - 64 q^{91} - 156 q^{93} + 572 q^{94} + 124 q^{97}+O(q^{100})$$ 2 * q + 6 * q^3 - 14 * q^4 + 16 * q^7 + 18 * q^9 - 44 * q^10 - 42 * q^12 - 8 * q^13 + 10 * q^16 + 12 * q^19 + 48 * q^21 - 38 * q^25 + 54 * q^27 - 112 * q^28 - 132 * q^30 - 52 * q^31 - 88 * q^34 - 126 * q^36 + 60 * q^37 - 24 * q^39 + 132 * q^40 - 84 * q^43 + 44 * q^46 + 30 * q^48 + 30 * q^49 + 56 * q^52 + 36 * q^57 + 264 * q^58 - 24 * q^61 + 144 * q^63 + 194 * q^64 + 4 * q^67 - 352 * q^70 + 148 * q^73 - 114 * q^75 - 84 * q^76 + 80 * q^79 + 162 * q^81 - 88 * q^82 - 336 * q^84 - 176 * q^85 - 396 * q^90 - 64 * q^91 - 156 * q^93 + 572 * q^94 + 124 * q^97

## Character values

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

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
122.1
 0.5 + 1.65831i 0.5 − 1.65831i
3.31662i 3.00000 −7.00000 6.63325i 9.94987i 8.00000 9.94987i 9.00000 −22.0000
122.2 3.31662i 3.00000 −7.00000 6.63325i 9.94987i 8.00000 9.94987i 9.00000 −22.0000
 $$n$$: 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.b.d 2
3.b odd 2 1 inner 363.3.b.d 2
11.b odd 2 1 33.3.b.a 2
11.c even 5 4 363.3.h.d 8
11.d odd 10 4 363.3.h.e 8
33.d even 2 1 33.3.b.a 2
33.f even 10 4 363.3.h.e 8
33.h odd 10 4 363.3.h.d 8
44.c even 2 1 528.3.i.a 2
132.d odd 2 1 528.3.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.b.a 2 11.b odd 2 1
33.3.b.a 2 33.d even 2 1
363.3.b.d 2 1.a even 1 1 trivial
363.3.b.d 2 3.b odd 2 1 inner
363.3.h.d 8 11.c even 5 4
363.3.h.d 8 33.h odd 10 4
363.3.h.e 8 11.d odd 10 4
363.3.h.e 8 33.f even 10 4
528.3.i.a 2 44.c even 2 1
528.3.i.a 2 132.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(363, [\chi])$$:

 $$T_{2}^{2} + 11$$ T2^2 + 11 $$T_{5}^{2} + 44$$ T5^2 + 44 $$T_{7} - 8$$ T7 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 11$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} + 44$$
$7$ $$(T - 8)^{2}$$
$11$ $$T^{2}$$
$13$ $$(T + 4)^{2}$$
$17$ $$T^{2} + 176$$
$19$ $$(T - 6)^{2}$$
$23$ $$T^{2} + 44$$
$29$ $$T^{2} + 1584$$
$31$ $$(T + 26)^{2}$$
$37$ $$(T - 30)^{2}$$
$41$ $$T^{2} + 176$$
$43$ $$(T + 42)^{2}$$
$47$ $$T^{2} + 7436$$
$53$ $$T^{2} + 3564$$
$59$ $$T^{2} + 4400$$
$61$ $$(T + 12)^{2}$$
$67$ $$(T - 2)^{2}$$
$71$ $$T^{2} + 3564$$
$73$ $$(T - 74)^{2}$$
$79$ $$(T - 40)^{2}$$
$83$ $$T^{2} + 1584$$
$89$ $$T^{2} + 14256$$
$97$ $$(T - 62)^{2}$$