# Properties

 Label 363.4.a.m Level $363$ Weight $4$ Character orbit 363.a Self dual yes Analytic conductor $21.418$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 3 q^{3} - 5 q^{4} - 3 q^{5} + 3 \beta q^{6} + 2 \beta q^{7} - 13 \beta q^{8} + 9 q^{9} +O(q^{10})$$ q + b * q^2 + 3 * q^3 - 5 * q^4 - 3 * q^5 + 3*b * q^6 + 2*b * q^7 - 13*b * q^8 + 9 * q^9 $$q + \beta q^{2} + 3 q^{3} - 5 q^{4} - 3 q^{5} + 3 \beta q^{6} + 2 \beta q^{7} - 13 \beta q^{8} + 9 q^{9} - 3 \beta q^{10} - 15 q^{12} + \beta q^{13} + 6 q^{14} - 9 q^{15} + q^{16} + 7 \beta q^{17} + 9 \beta q^{18} - 76 \beta q^{19} + 15 q^{20} + 6 \beta q^{21} - 174 q^{23} - 39 \beta q^{24} - 116 q^{25} + 3 q^{26} + 27 q^{27} - 10 \beta q^{28} - 31 \beta q^{29} - 9 \beta q^{30} - 20 q^{31} + 105 \beta q^{32} + 21 q^{34} - 6 \beta q^{35} - 45 q^{36} - 299 q^{37} - 228 q^{38} + 3 \beta q^{39} + 39 \beta q^{40} - 175 \beta q^{41} + 18 q^{42} + 286 \beta q^{43} - 27 q^{45} - 174 \beta q^{46} - 384 q^{47} + 3 q^{48} - 331 q^{49} - 116 \beta q^{50} + 21 \beta q^{51} - 5 \beta q^{52} + 255 q^{53} + 27 \beta q^{54} - 78 q^{56} - 228 \beta q^{57} - 93 q^{58} + 570 q^{59} + 45 q^{60} + 216 \beta q^{61} - 20 \beta q^{62} + 18 \beta q^{63} + 307 q^{64} - 3 \beta q^{65} + 46 q^{67} - 35 \beta q^{68} - 522 q^{69} - 18 q^{70} - 630 q^{71} - 117 \beta q^{72} + 332 \beta q^{73} - 299 \beta q^{74} - 348 q^{75} + 380 \beta q^{76} + 9 q^{78} + 144 \beta q^{79} - 3 q^{80} + 81 q^{81} - 525 q^{82} + 744 \beta q^{83} - 30 \beta q^{84} - 21 \beta q^{85} + 858 q^{86} - 93 \beta q^{87} - 207 q^{89} - 27 \beta q^{90} + 6 q^{91} + 870 q^{92} - 60 q^{93} - 384 \beta q^{94} + 228 \beta q^{95} + 315 \beta q^{96} - 1615 q^{97} - 331 \beta q^{98} +O(q^{100})$$ q + b * q^2 + 3 * q^3 - 5 * q^4 - 3 * q^5 + 3*b * q^6 + 2*b * q^7 - 13*b * q^8 + 9 * q^9 - 3*b * q^10 - 15 * q^12 + b * q^13 + 6 * q^14 - 9 * q^15 + q^16 + 7*b * q^17 + 9*b * q^18 - 76*b * q^19 + 15 * q^20 + 6*b * q^21 - 174 * q^23 - 39*b * q^24 - 116 * q^25 + 3 * q^26 + 27 * q^27 - 10*b * q^28 - 31*b * q^29 - 9*b * q^30 - 20 * q^31 + 105*b * q^32 + 21 * q^34 - 6*b * q^35 - 45 * q^36 - 299 * q^37 - 228 * q^38 + 3*b * q^39 + 39*b * q^40 - 175*b * q^41 + 18 * q^42 + 286*b * q^43 - 27 * q^45 - 174*b * q^46 - 384 * q^47 + 3 * q^48 - 331 * q^49 - 116*b * q^50 + 21*b * q^51 - 5*b * q^52 + 255 * q^53 + 27*b * q^54 - 78 * q^56 - 228*b * q^57 - 93 * q^58 + 570 * q^59 + 45 * q^60 + 216*b * q^61 - 20*b * q^62 + 18*b * q^63 + 307 * q^64 - 3*b * q^65 + 46 * q^67 - 35*b * q^68 - 522 * q^69 - 18 * q^70 - 630 * q^71 - 117*b * q^72 + 332*b * q^73 - 299*b * q^74 - 348 * q^75 + 380*b * q^76 + 9 * q^78 + 144*b * q^79 - 3 * q^80 + 81 * q^81 - 525 * q^82 + 744*b * q^83 - 30*b * q^84 - 21*b * q^85 + 858 * q^86 - 93*b * q^87 - 207 * q^89 - 27*b * q^90 + 6 * q^91 + 870 * q^92 - 60 * q^93 - 384*b * q^94 + 228*b * q^95 + 315*b * q^96 - 1615 * q^97 - 331*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 10 q^{4} - 6 q^{5} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 10 * q^4 - 6 * q^5 + 18 * q^9 $$2 q + 6 q^{3} - 10 q^{4} - 6 q^{5} + 18 q^{9} - 30 q^{12} + 12 q^{14} - 18 q^{15} + 2 q^{16} + 30 q^{20} - 348 q^{23} - 232 q^{25} + 6 q^{26} + 54 q^{27} - 40 q^{31} + 42 q^{34} - 90 q^{36} - 598 q^{37} - 456 q^{38} + 36 q^{42} - 54 q^{45} - 768 q^{47} + 6 q^{48} - 662 q^{49} + 510 q^{53} - 156 q^{56} - 186 q^{58} + 1140 q^{59} + 90 q^{60} + 614 q^{64} + 92 q^{67} - 1044 q^{69} - 36 q^{70} - 1260 q^{71} - 696 q^{75} + 18 q^{78} - 6 q^{80} + 162 q^{81} - 1050 q^{82} + 1716 q^{86} - 414 q^{89} + 12 q^{91} + 1740 q^{92} - 120 q^{93} - 3230 q^{97}+O(q^{100})$$ 2 * q + 6 * q^3 - 10 * q^4 - 6 * q^5 + 18 * q^9 - 30 * q^12 + 12 * q^14 - 18 * q^15 + 2 * q^16 + 30 * q^20 - 348 * q^23 - 232 * q^25 + 6 * q^26 + 54 * q^27 - 40 * q^31 + 42 * q^34 - 90 * q^36 - 598 * q^37 - 456 * q^38 + 36 * q^42 - 54 * q^45 - 768 * q^47 + 6 * q^48 - 662 * q^49 + 510 * q^53 - 156 * q^56 - 186 * q^58 + 1140 * q^59 + 90 * q^60 + 614 * q^64 + 92 * q^67 - 1044 * q^69 - 36 * q^70 - 1260 * q^71 - 696 * q^75 + 18 * q^78 - 6 * q^80 + 162 * q^81 - 1050 * q^82 + 1716 * q^86 - 414 * q^89 + 12 * q^91 + 1740 * q^92 - 120 * q^93 - 3230 * 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
 −1.73205 1.73205
−1.73205 3.00000 −5.00000 −3.00000 −5.19615 −3.46410 22.5167 9.00000 5.19615
1.2 1.73205 3.00000 −5.00000 −3.00000 5.19615 3.46410 −22.5167 9.00000 −5.19615
 $$n$$: 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.m 2
3.b odd 2 1 1089.4.a.n 2
11.b odd 2 1 inner 363.4.a.m 2
33.d even 2 1 1089.4.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.4.a.m 2 1.a even 1 1 trivial
363.4.a.m 2 11.b odd 2 1 inner
1089.4.a.n 2 3.b odd 2 1
1089.4.a.n 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(363))$$:

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{5} + 3$$ T5 + 3 $$T_{7}^{2} - 12$$ T7^2 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$(T - 3)^{2}$$
$5$ $$(T + 3)^{2}$$
$7$ $$T^{2} - 12$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 3$$
$17$ $$T^{2} - 147$$
$19$ $$T^{2} - 17328$$
$23$ $$(T + 174)^{2}$$
$29$ $$T^{2} - 2883$$
$31$ $$(T + 20)^{2}$$
$37$ $$(T + 299)^{2}$$
$41$ $$T^{2} - 91875$$
$43$ $$T^{2} - 245388$$
$47$ $$(T + 384)^{2}$$
$53$ $$(T - 255)^{2}$$
$59$ $$(T - 570)^{2}$$
$61$ $$T^{2} - 139968$$
$67$ $$(T - 46)^{2}$$
$71$ $$(T + 630)^{2}$$
$73$ $$T^{2} - 330672$$
$79$ $$T^{2} - 62208$$
$83$ $$T^{2} - 1660608$$
$89$ $$(T + 207)^{2}$$
$97$ $$(T + 1615)^{2}$$