# Properties

 Label 363.4.a.d 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: no (minimal twist has level 33) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - 3 q^{3} - 7 q^{4} - 4 q^{5} - 3 q^{6} + 26 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10})$$ q + q^2 - 3 * q^3 - 7 * q^4 - 4 * q^5 - 3 * q^6 + 26 * q^7 - 15 * q^8 + 9 * q^9 $$q + q^{2} - 3 q^{3} - 7 q^{4} - 4 q^{5} - 3 q^{6} + 26 q^{7} - 15 q^{8} + 9 q^{9} - 4 q^{10} + 21 q^{12} + 32 q^{13} + 26 q^{14} + 12 q^{15} + 41 q^{16} - 74 q^{17} + 9 q^{18} + 60 q^{19} + 28 q^{20} - 78 q^{21} - 182 q^{23} + 45 q^{24} - 109 q^{25} + 32 q^{26} - 27 q^{27} - 182 q^{28} + 90 q^{29} + 12 q^{30} - 8 q^{31} + 161 q^{32} - 74 q^{34} - 104 q^{35} - 63 q^{36} - 66 q^{37} + 60 q^{38} - 96 q^{39} + 60 q^{40} - 422 q^{41} - 78 q^{42} - 408 q^{43} - 36 q^{45} - 182 q^{46} - 506 q^{47} - 123 q^{48} + 333 q^{49} - 109 q^{50} + 222 q^{51} - 224 q^{52} + 348 q^{53} - 27 q^{54} - 390 q^{56} - 180 q^{57} + 90 q^{58} - 200 q^{59} - 84 q^{60} - 132 q^{61} - 8 q^{62} + 234 q^{63} - 167 q^{64} - 128 q^{65} - 1036 q^{67} + 518 q^{68} + 546 q^{69} - 104 q^{70} + 762 q^{71} - 135 q^{72} + 542 q^{73} - 66 q^{74} + 327 q^{75} - 420 q^{76} - 96 q^{78} + 550 q^{79} - 164 q^{80} + 81 q^{81} - 422 q^{82} + 132 q^{83} + 546 q^{84} + 296 q^{85} - 408 q^{86} - 270 q^{87} + 570 q^{89} - 36 q^{90} + 832 q^{91} + 1274 q^{92} + 24 q^{93} - 506 q^{94} - 240 q^{95} - 483 q^{96} + 14 q^{97} + 333 q^{98}+O(q^{100})$$ q + q^2 - 3 * q^3 - 7 * q^4 - 4 * q^5 - 3 * q^6 + 26 * q^7 - 15 * q^8 + 9 * q^9 - 4 * q^10 + 21 * q^12 + 32 * q^13 + 26 * q^14 + 12 * q^15 + 41 * q^16 - 74 * q^17 + 9 * q^18 + 60 * q^19 + 28 * q^20 - 78 * q^21 - 182 * q^23 + 45 * q^24 - 109 * q^25 + 32 * q^26 - 27 * q^27 - 182 * q^28 + 90 * q^29 + 12 * q^30 - 8 * q^31 + 161 * q^32 - 74 * q^34 - 104 * q^35 - 63 * q^36 - 66 * q^37 + 60 * q^38 - 96 * q^39 + 60 * q^40 - 422 * q^41 - 78 * q^42 - 408 * q^43 - 36 * q^45 - 182 * q^46 - 506 * q^47 - 123 * q^48 + 333 * q^49 - 109 * q^50 + 222 * q^51 - 224 * q^52 + 348 * q^53 - 27 * q^54 - 390 * q^56 - 180 * q^57 + 90 * q^58 - 200 * q^59 - 84 * q^60 - 132 * q^61 - 8 * q^62 + 234 * q^63 - 167 * q^64 - 128 * q^65 - 1036 * q^67 + 518 * q^68 + 546 * q^69 - 104 * q^70 + 762 * q^71 - 135 * q^72 + 542 * q^73 - 66 * q^74 + 327 * q^75 - 420 * q^76 - 96 * q^78 + 550 * q^79 - 164 * q^80 + 81 * q^81 - 422 * q^82 + 132 * q^83 + 546 * q^84 + 296 * q^85 - 408 * q^86 - 270 * q^87 + 570 * q^89 - 36 * q^90 + 832 * q^91 + 1274 * q^92 + 24 * q^93 - 506 * q^94 - 240 * q^95 - 483 * q^96 + 14 * q^97 + 333 * 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 −4.00000 −3.00000 26.0000 −15.0000 9.00000 −4.00000
 $$n$$: 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.d 1
3.b odd 2 1 1089.4.a.e 1
11.b odd 2 1 33.4.a.b 1
33.d even 2 1 99.4.a.a 1
44.c even 2 1 528.4.a.h 1
55.d odd 2 1 825.4.a.f 1
55.e even 4 2 825.4.c.f 2
77.b even 2 1 1617.4.a.d 1
88.b odd 2 1 2112.4.a.u 1
88.g even 2 1 2112.4.a.h 1
132.d odd 2 1 1584.4.a.l 1
165.d even 2 1 2475.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.b 1 11.b odd 2 1
99.4.a.a 1 33.d even 2 1
363.4.a.d 1 1.a even 1 1 trivial
528.4.a.h 1 44.c even 2 1
825.4.a.f 1 55.d odd 2 1
825.4.c.f 2 55.e even 4 2
1089.4.a.e 1 3.b odd 2 1
1584.4.a.l 1 132.d odd 2 1
1617.4.a.d 1 77.b even 2 1
2112.4.a.h 1 88.g even 2 1
2112.4.a.u 1 88.b odd 2 1
2475.4.a.e 1 165.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 $$T_{5} + 4$$ T5 + 4 $$T_{7} - 26$$ T7 - 26

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 3$$
$5$ $$T + 4$$
$7$ $$T - 26$$
$11$ $$T$$
$13$ $$T - 32$$
$17$ $$T + 74$$
$19$ $$T - 60$$
$23$ $$T + 182$$
$29$ $$T - 90$$
$31$ $$T + 8$$
$37$ $$T + 66$$
$41$ $$T + 422$$
$43$ $$T + 408$$
$47$ $$T + 506$$
$53$ $$T - 348$$
$59$ $$T + 200$$
$61$ $$T + 132$$
$67$ $$T + 1036$$
$71$ $$T - 762$$
$73$ $$T - 542$$
$79$ $$T - 550$$
$83$ $$T - 132$$
$89$ $$T - 570$$
$97$ $$T - 14$$