# Properties

 Label 363.4.a.e Level $363$ Weight $4$ Character orbit 363.a Self dual yes Analytic conductor $21.418$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - 3 q^{3} - 7 q^{4} + 7 q^{5} - 3 q^{6} + 4 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10})$$ q + q^2 - 3 * q^3 - 7 * q^4 + 7 * q^5 - 3 * q^6 + 4 * q^7 - 15 * q^8 + 9 * q^9 $$q + q^{2} - 3 q^{3} - 7 q^{4} + 7 q^{5} - 3 q^{6} + 4 q^{7} - 15 q^{8} + 9 q^{9} + 7 q^{10} + 21 q^{12} + 43 q^{13} + 4 q^{14} - 21 q^{15} + 41 q^{16} - 41 q^{17} + 9 q^{18} - 72 q^{19} - 49 q^{20} - 12 q^{21} + 104 q^{23} + 45 q^{24} - 76 q^{25} + 43 q^{26} - 27 q^{27} - 28 q^{28} - 273 q^{29} - 21 q^{30} - 272 q^{31} + 161 q^{32} - 41 q^{34} + 28 q^{35} - 63 q^{36} - 165 q^{37} - 72 q^{38} - 129 q^{39} - 105 q^{40} + 403 q^{41} - 12 q^{42} + 120 q^{43} + 63 q^{45} + 104 q^{46} - 220 q^{47} - 123 q^{48} - 327 q^{49} - 76 q^{50} + 123 q^{51} - 301 q^{52} - 741 q^{53} - 27 q^{54} - 60 q^{56} + 216 q^{57} - 273 q^{58} - 112 q^{59} + 147 q^{60} - 858 q^{61} - 272 q^{62} + 36 q^{63} - 167 q^{64} + 301 q^{65} + 284 q^{67} + 287 q^{68} - 312 q^{69} + 28 q^{70} - 624 q^{71} - 135 q^{72} + 586 q^{73} - 165 q^{74} + 228 q^{75} + 504 q^{76} - 129 q^{78} + 308 q^{79} + 287 q^{80} + 81 q^{81} + 403 q^{82} + 84 q^{84} - 287 q^{85} + 120 q^{86} + 819 q^{87} - 321 q^{89} + 63 q^{90} + 172 q^{91} - 728 q^{92} + 816 q^{93} - 220 q^{94} - 504 q^{95} - 483 q^{96} + 179 q^{97} - 327 q^{98}+O(q^{100})$$ q + q^2 - 3 * q^3 - 7 * q^4 + 7 * q^5 - 3 * q^6 + 4 * q^7 - 15 * q^8 + 9 * q^9 + 7 * q^10 + 21 * q^12 + 43 * q^13 + 4 * q^14 - 21 * q^15 + 41 * q^16 - 41 * q^17 + 9 * q^18 - 72 * q^19 - 49 * q^20 - 12 * q^21 + 104 * q^23 + 45 * q^24 - 76 * q^25 + 43 * q^26 - 27 * q^27 - 28 * q^28 - 273 * q^29 - 21 * q^30 - 272 * q^31 + 161 * q^32 - 41 * q^34 + 28 * q^35 - 63 * q^36 - 165 * q^37 - 72 * q^38 - 129 * q^39 - 105 * q^40 + 403 * q^41 - 12 * q^42 + 120 * q^43 + 63 * q^45 + 104 * q^46 - 220 * q^47 - 123 * q^48 - 327 * q^49 - 76 * q^50 + 123 * q^51 - 301 * q^52 - 741 * q^53 - 27 * q^54 - 60 * q^56 + 216 * q^57 - 273 * q^58 - 112 * q^59 + 147 * q^60 - 858 * q^61 - 272 * q^62 + 36 * q^63 - 167 * q^64 + 301 * q^65 + 284 * q^67 + 287 * q^68 - 312 * q^69 + 28 * q^70 - 624 * q^71 - 135 * q^72 + 586 * q^73 - 165 * q^74 + 228 * q^75 + 504 * q^76 - 129 * q^78 + 308 * q^79 + 287 * q^80 + 81 * q^81 + 403 * q^82 + 84 * q^84 - 287 * q^85 + 120 * q^86 + 819 * q^87 - 321 * q^89 + 63 * q^90 + 172 * q^91 - 728 * q^92 + 816 * q^93 - 220 * q^94 - 504 * q^95 - 483 * q^96 + 179 * q^97 - 327 * q^98

## 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
 0
1.00000 −3.00000 −7.00000 7.00000 −3.00000 4.00000 −15.0000 9.00000 7.00000
 $$n$$: 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.4.a.e yes 1
3.b odd 2 1 1089.4.a.d 1
11.b odd 2 1 363.4.a.c 1
33.d even 2 1 1089.4.a.h 1

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

 $$T_{2} - 1$$ T2 - 1 $$T_{5} - 7$$ T5 - 7 $$T_{7} - 4$$ T7 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 3$$
$5$ $$T - 7$$
$7$ $$T - 4$$
$11$ $$T$$
$13$ $$T - 43$$
$17$ $$T + 41$$
$19$ $$T + 72$$
$23$ $$T - 104$$
$29$ $$T + 273$$
$31$ $$T + 272$$
$37$ $$T + 165$$
$41$ $$T - 403$$
$43$ $$T - 120$$
$47$ $$T + 220$$
$53$ $$T + 741$$
$59$ $$T + 112$$
$61$ $$T + 858$$
$67$ $$T - 284$$
$71$ $$T + 624$$
$73$ $$T - 586$$
$79$ $$T - 308$$
$83$ $$T$$
$89$ $$T + 321$$
$97$ $$T - 179$$