# Properties

 Label 363.4.a.k Level $363$ Weight $4$ Character orbit 363.a Self dual yes Analytic conductor $21.418$ 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] = [363,4,Mod(1,363)]

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([0, 0]))

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

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

chi := DirichletCharacter("363.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 363.a (trivial)

## 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: yes Analytic conductor: $$21.4176933321$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} - 3 q^{3} - 3 q^{4} - 2 q^{5} + 3 \beta q^{6} + 10 \beta q^{7} + 11 \beta q^{8} + 9 q^{9} +O(q^{10})$$ q - b * q^2 - 3 * q^3 - 3 * q^4 - 2 * q^5 + 3*b * q^6 + 10*b * q^7 + 11*b * q^8 + 9 * q^9 $$q - \beta q^{2} - 3 q^{3} - 3 q^{4} - 2 q^{5} + 3 \beta q^{6} + 10 \beta q^{7} + 11 \beta q^{8} + 9 q^{9} + 2 \beta q^{10} + 9 q^{12} - 12 \beta q^{13} - 50 q^{14} + 6 q^{15} - 31 q^{16} - 38 \beta q^{17} - 9 \beta q^{18} - 50 \beta q^{19} + 6 q^{20} - 30 \beta q^{21} + 112 q^{23} - 33 \beta q^{24} - 121 q^{25} + 60 q^{26} - 27 q^{27} - 30 \beta q^{28} + 82 \beta q^{29} - 6 \beta q^{30} + 120 q^{31} - 57 \beta q^{32} + 190 q^{34} - 20 \beta q^{35} - 27 q^{36} + 386 q^{37} + 250 q^{38} + 36 \beta q^{39} - 22 \beta q^{40} + 62 \beta q^{41} + 150 q^{42} + 74 \beta q^{43} - 18 q^{45} - 112 \beta q^{46} - 236 q^{47} + 93 q^{48} + 157 q^{49} + 121 \beta q^{50} + 114 \beta q^{51} + 36 \beta q^{52} - 78 q^{53} + 27 \beta q^{54} + 550 q^{56} + 150 \beta q^{57} - 410 q^{58} + 840 q^{59} - 18 q^{60} - 320 \beta q^{61} - 120 \beta q^{62} + 90 \beta q^{63} + 533 q^{64} + 24 \beta q^{65} - 276 q^{67} + 114 \beta q^{68} - 336 q^{69} + 100 q^{70} + 572 q^{71} + 99 \beta q^{72} - 484 \beta q^{73} - 386 \beta q^{74} + 363 q^{75} + 150 \beta q^{76} - 180 q^{78} + 330 \beta q^{79} + 62 q^{80} + 81 q^{81} - 310 q^{82} + 460 \beta q^{83} + 90 \beta q^{84} + 76 \beta q^{85} - 370 q^{86} - 246 \beta q^{87} + 914 q^{89} + 18 \beta q^{90} - 600 q^{91} - 336 q^{92} - 360 q^{93} + 236 \beta q^{94} + 100 \beta q^{95} + 171 \beta q^{96} + 386 q^{97} - 157 \beta q^{98} +O(q^{100})$$ q - b * q^2 - 3 * q^3 - 3 * q^4 - 2 * q^5 + 3*b * q^6 + 10*b * q^7 + 11*b * q^8 + 9 * q^9 + 2*b * q^10 + 9 * q^12 - 12*b * q^13 - 50 * q^14 + 6 * q^15 - 31 * q^16 - 38*b * q^17 - 9*b * q^18 - 50*b * q^19 + 6 * q^20 - 30*b * q^21 + 112 * q^23 - 33*b * q^24 - 121 * q^25 + 60 * q^26 - 27 * q^27 - 30*b * q^28 + 82*b * q^29 - 6*b * q^30 + 120 * q^31 - 57*b * q^32 + 190 * q^34 - 20*b * q^35 - 27 * q^36 + 386 * q^37 + 250 * q^38 + 36*b * q^39 - 22*b * q^40 + 62*b * q^41 + 150 * q^42 + 74*b * q^43 - 18 * q^45 - 112*b * q^46 - 236 * q^47 + 93 * q^48 + 157 * q^49 + 121*b * q^50 + 114*b * q^51 + 36*b * q^52 - 78 * q^53 + 27*b * q^54 + 550 * q^56 + 150*b * q^57 - 410 * q^58 + 840 * q^59 - 18 * q^60 - 320*b * q^61 - 120*b * q^62 + 90*b * q^63 + 533 * q^64 + 24*b * q^65 - 276 * q^67 + 114*b * q^68 - 336 * q^69 + 100 * q^70 + 572 * q^71 + 99*b * q^72 - 484*b * q^73 - 386*b * q^74 + 363 * q^75 + 150*b * q^76 - 180 * q^78 + 330*b * q^79 + 62 * q^80 + 81 * q^81 - 310 * q^82 + 460*b * q^83 + 90*b * q^84 + 76*b * q^85 - 370 * q^86 - 246*b * q^87 + 914 * q^89 + 18*b * q^90 - 600 * q^91 - 336 * q^92 - 360 * q^93 + 236*b * q^94 + 100*b * q^95 + 171*b * q^96 + 386 * q^97 - 157*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 6 q^{4} - 4 q^{5} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 6 * q^4 - 4 * q^5 + 18 * q^9 $$2 q - 6 q^{3} - 6 q^{4} - 4 q^{5} + 18 q^{9} + 18 q^{12} - 100 q^{14} + 12 q^{15} - 62 q^{16} + 12 q^{20} + 224 q^{23} - 242 q^{25} + 120 q^{26} - 54 q^{27} + 240 q^{31} + 380 q^{34} - 54 q^{36} + 772 q^{37} + 500 q^{38} + 300 q^{42} - 36 q^{45} - 472 q^{47} + 186 q^{48} + 314 q^{49} - 156 q^{53} + 1100 q^{56} - 820 q^{58} + 1680 q^{59} - 36 q^{60} + 1066 q^{64} - 552 q^{67} - 672 q^{69} + 200 q^{70} + 1144 q^{71} + 726 q^{75} - 360 q^{78} + 124 q^{80} + 162 q^{81} - 620 q^{82} - 740 q^{86} + 1828 q^{89} - 1200 q^{91} - 672 q^{92} - 720 q^{93} + 772 q^{97}+O(q^{100})$$ 2 * q - 6 * q^3 - 6 * q^4 - 4 * q^5 + 18 * q^9 + 18 * q^12 - 100 * q^14 + 12 * q^15 - 62 * q^16 + 12 * q^20 + 224 * q^23 - 242 * q^25 + 120 * q^26 - 54 * q^27 + 240 * q^31 + 380 * q^34 - 54 * q^36 + 772 * q^37 + 500 * q^38 + 300 * q^42 - 36 * q^45 - 472 * q^47 + 186 * q^48 + 314 * q^49 - 156 * q^53 + 1100 * q^56 - 820 * q^58 + 1680 * q^59 - 36 * q^60 + 1066 * q^64 - 552 * q^67 - 672 * q^69 + 200 * q^70 + 1144 * q^71 + 726 * q^75 - 360 * q^78 + 124 * q^80 + 162 * q^81 - 620 * q^82 - 740 * q^86 + 1828 * q^89 - 1200 * q^91 - 672 * q^92 - 720 * q^93 + 772 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.61803 −0.618034
−2.23607 −3.00000 −3.00000 −2.00000 6.70820 22.3607 24.5967 9.00000 4.47214
1.2 2.23607 −3.00000 −3.00000 −2.00000 −6.70820 −22.3607 −24.5967 9.00000 −4.47214
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$11$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.4.a.k 2
3.b odd 2 1 1089.4.a.o 2
11.b odd 2 1 inner 363.4.a.k 2
33.d even 2 1 1089.4.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.4.a.k 2 1.a even 1 1 trivial
363.4.a.k 2 11.b odd 2 1 inner
1089.4.a.o 2 3.b odd 2 1
1089.4.a.o 2 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(363))$$:

 $$T_{2}^{2} - 5$$ T2^2 - 5 $$T_{5} + 2$$ T5 + 2 $$T_{7}^{2} - 500$$ T7^2 - 500

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 5$$
$3$ $$(T + 3)^{2}$$
$5$ $$(T + 2)^{2}$$
$7$ $$T^{2} - 500$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 720$$
$17$ $$T^{2} - 7220$$
$19$ $$T^{2} - 12500$$
$23$ $$(T - 112)^{2}$$
$29$ $$T^{2} - 33620$$
$31$ $$(T - 120)^{2}$$
$37$ $$(T - 386)^{2}$$
$41$ $$T^{2} - 19220$$
$43$ $$T^{2} - 27380$$
$47$ $$(T + 236)^{2}$$
$53$ $$(T + 78)^{2}$$
$59$ $$(T - 840)^{2}$$
$61$ $$T^{2} - 512000$$
$67$ $$(T + 276)^{2}$$
$71$ $$(T - 572)^{2}$$
$73$ $$T^{2} - 1171280$$
$79$ $$T^{2} - 544500$$
$83$ $$T^{2} - 1058000$$
$89$ $$(T - 914)^{2}$$
$97$ $$(T - 386)^{2}$$