# Properties

 Label 363.4.a.f 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{2} + 3 q^{3} + q^{4} - 12 q^{5} + 9 q^{6} + 12 q^{7} - 21 q^{8} + 9 q^{9}+O(q^{10})$$ q + 3 * q^2 + 3 * q^3 + q^4 - 12 * q^5 + 9 * q^6 + 12 * q^7 - 21 * q^8 + 9 * q^9 $$q + 3 q^{2} + 3 q^{3} + q^{4} - 12 q^{5} + 9 q^{6} + 12 q^{7} - 21 q^{8} + 9 q^{9} - 36 q^{10} + 3 q^{12} - 66 q^{13} + 36 q^{14} - 36 q^{15} - 71 q^{16} - 114 q^{17} + 27 q^{18} + 42 q^{19} - 12 q^{20} + 36 q^{21} + 18 q^{23} - 63 q^{24} + 19 q^{25} - 198 q^{26} + 27 q^{27} + 12 q^{28} + 186 q^{29} - 108 q^{30} - 308 q^{31} - 45 q^{32} - 342 q^{34} - 144 q^{35} + 9 q^{36} - 146 q^{37} + 126 q^{38} - 198 q^{39} + 252 q^{40} + 42 q^{41} + 108 q^{42} - 366 q^{43} - 108 q^{45} + 54 q^{46} + 618 q^{47} - 213 q^{48} - 199 q^{49} + 57 q^{50} - 342 q^{51} - 66 q^{52} - 408 q^{53} + 81 q^{54} - 252 q^{56} + 126 q^{57} + 558 q^{58} - 132 q^{59} - 36 q^{60} + 630 q^{61} - 924 q^{62} + 108 q^{63} + 433 q^{64} + 792 q^{65} - 452 q^{67} - 114 q^{68} + 54 q^{69} - 432 q^{70} - 282 q^{71} - 189 q^{72} + 684 q^{73} - 438 q^{74} + 57 q^{75} + 42 q^{76} - 594 q^{78} + 1272 q^{79} + 852 q^{80} + 81 q^{81} + 126 q^{82} - 432 q^{83} + 36 q^{84} + 1368 q^{85} - 1098 q^{86} + 558 q^{87} + 954 q^{89} - 324 q^{90} - 792 q^{91} + 18 q^{92} - 924 q^{93} + 1854 q^{94} - 504 q^{95} - 135 q^{96} + 326 q^{97} - 597 q^{98}+O(q^{100})$$ q + 3 * q^2 + 3 * q^3 + q^4 - 12 * q^5 + 9 * q^6 + 12 * q^7 - 21 * q^8 + 9 * q^9 - 36 * q^10 + 3 * q^12 - 66 * q^13 + 36 * q^14 - 36 * q^15 - 71 * q^16 - 114 * q^17 + 27 * q^18 + 42 * q^19 - 12 * q^20 + 36 * q^21 + 18 * q^23 - 63 * q^24 + 19 * q^25 - 198 * q^26 + 27 * q^27 + 12 * q^28 + 186 * q^29 - 108 * q^30 - 308 * q^31 - 45 * q^32 - 342 * q^34 - 144 * q^35 + 9 * q^36 - 146 * q^37 + 126 * q^38 - 198 * q^39 + 252 * q^40 + 42 * q^41 + 108 * q^42 - 366 * q^43 - 108 * q^45 + 54 * q^46 + 618 * q^47 - 213 * q^48 - 199 * q^49 + 57 * q^50 - 342 * q^51 - 66 * q^52 - 408 * q^53 + 81 * q^54 - 252 * q^56 + 126 * q^57 + 558 * q^58 - 132 * q^59 - 36 * q^60 + 630 * q^61 - 924 * q^62 + 108 * q^63 + 433 * q^64 + 792 * q^65 - 452 * q^67 - 114 * q^68 + 54 * q^69 - 432 * q^70 - 282 * q^71 - 189 * q^72 + 684 * q^73 - 438 * q^74 + 57 * q^75 + 42 * q^76 - 594 * q^78 + 1272 * q^79 + 852 * q^80 + 81 * q^81 + 126 * q^82 - 432 * q^83 + 36 * q^84 + 1368 * q^85 - 1098 * q^86 + 558 * q^87 + 954 * q^89 - 324 * q^90 - 792 * q^91 + 18 * q^92 - 924 * q^93 + 1854 * q^94 - 504 * q^95 - 135 * q^96 + 326 * q^97 - 597 * 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
3.00000 3.00000 1.00000 −12.0000 9.00000 12.0000 −21.0000 9.00000 −36.0000
 $$n$$: 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.f yes 1
3.b odd 2 1 1089.4.a.c 1
11.b odd 2 1 363.4.a.b 1
33.d even 2 1 1089.4.a.i 1

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 3$$
$3$ $$T - 3$$
$5$ $$T + 12$$
$7$ $$T - 12$$
$11$ $$T$$
$13$ $$T + 66$$
$17$ $$T + 114$$
$19$ $$T - 42$$
$23$ $$T - 18$$
$29$ $$T - 186$$
$31$ $$T + 308$$
$37$ $$T + 146$$
$41$ $$T - 42$$
$43$ $$T + 366$$
$47$ $$T - 618$$
$53$ $$T + 408$$
$59$ $$T + 132$$
$61$ $$T - 630$$
$67$ $$T + 452$$
$71$ $$T + 282$$
$73$ $$T - 684$$
$79$ $$T - 1272$$
$83$ $$T + 432$$
$89$ $$T - 954$$
$97$ $$T - 326$$