# Properties

 Label 2205.4.a.q Level $2205$ Weight $4$ Character orbit 2205.a Self dual yes Analytic conductor $130.099$ Analytic rank $0$ 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] = [2205,4,Mod(1,2205)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2205.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: $$130.099211563$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 5) 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 8 q^{4} - 5 q^{5}+O(q^{10})$$ q + 4 * q^2 + 8 * q^4 - 5 * q^5 $$q + 4 q^{2} + 8 q^{4} - 5 q^{5} - 20 q^{10} - 32 q^{11} + 38 q^{13} - 64 q^{16} + 26 q^{17} - 100 q^{19} - 40 q^{20} - 128 q^{22} + 78 q^{23} + 25 q^{25} + 152 q^{26} + 50 q^{29} + 108 q^{31} - 256 q^{32} + 104 q^{34} + 266 q^{37} - 400 q^{38} + 22 q^{41} + 442 q^{43} - 256 q^{44} + 312 q^{46} - 514 q^{47} + 100 q^{50} + 304 q^{52} - 2 q^{53} + 160 q^{55} + 200 q^{58} + 500 q^{59} + 518 q^{61} + 432 q^{62} - 512 q^{64} - 190 q^{65} + 126 q^{67} + 208 q^{68} - 412 q^{71} + 878 q^{73} + 1064 q^{74} - 800 q^{76} + 600 q^{79} + 320 q^{80} + 88 q^{82} + 282 q^{83} - 130 q^{85} + 1768 q^{86} - 150 q^{89} + 624 q^{92} - 2056 q^{94} + 500 q^{95} - 386 q^{97}+O(q^{100})$$ q + 4 * q^2 + 8 * q^4 - 5 * q^5 - 20 * q^10 - 32 * q^11 + 38 * q^13 - 64 * q^16 + 26 * q^17 - 100 * q^19 - 40 * q^20 - 128 * q^22 + 78 * q^23 + 25 * q^25 + 152 * q^26 + 50 * q^29 + 108 * q^31 - 256 * q^32 + 104 * q^34 + 266 * q^37 - 400 * q^38 + 22 * q^41 + 442 * q^43 - 256 * q^44 + 312 * q^46 - 514 * q^47 + 100 * q^50 + 304 * q^52 - 2 * q^53 + 160 * q^55 + 200 * q^58 + 500 * q^59 + 518 * q^61 + 432 * q^62 - 512 * q^64 - 190 * q^65 + 126 * q^67 + 208 * q^68 - 412 * q^71 + 878 * q^73 + 1064 * q^74 - 800 * q^76 + 600 * q^79 + 320 * q^80 + 88 * q^82 + 282 * q^83 - 130 * q^85 + 1768 * q^86 - 150 * q^89 + 624 * q^92 - 2056 * q^94 + 500 * q^95 - 386 * 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
 0
4.00000 0 8.00000 −5.00000 0 0 0 0 −20.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 2205.4.a.q 1
3.b odd 2 1 245.4.a.a 1
7.b odd 2 1 45.4.a.d 1
15.d odd 2 1 1225.4.a.k 1
21.c even 2 1 5.4.a.a 1
21.g even 6 2 245.4.e.f 2
21.h odd 6 2 245.4.e.g 2
28.d even 2 1 720.4.a.u 1
35.c odd 2 1 225.4.a.b 1
35.f even 4 2 225.4.b.c 2
63.l odd 6 2 405.4.e.c 2
63.o even 6 2 405.4.e.l 2
84.h odd 2 1 80.4.a.d 1
105.g even 2 1 25.4.a.c 1
105.k odd 4 2 25.4.b.a 2
168.e odd 2 1 320.4.a.h 1
168.i even 2 1 320.4.a.g 1
231.h odd 2 1 605.4.a.d 1
273.g even 2 1 845.4.a.b 1
336.v odd 4 2 1280.4.d.l 2
336.y even 4 2 1280.4.d.e 2
357.c even 2 1 1445.4.a.a 1
399.h odd 2 1 1805.4.a.h 1
420.o odd 2 1 400.4.a.m 1
420.w even 4 2 400.4.c.k 2
840.b odd 2 1 1600.4.a.s 1
840.u even 2 1 1600.4.a.bi 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 21.c even 2 1
25.4.a.c 1 105.g even 2 1
25.4.b.a 2 105.k odd 4 2
45.4.a.d 1 7.b odd 2 1
80.4.a.d 1 84.h odd 2 1
225.4.a.b 1 35.c odd 2 1
225.4.b.c 2 35.f even 4 2
245.4.a.a 1 3.b odd 2 1
245.4.e.f 2 21.g even 6 2
245.4.e.g 2 21.h odd 6 2
320.4.a.g 1 168.i even 2 1
320.4.a.h 1 168.e odd 2 1
400.4.a.m 1 420.o odd 2 1
400.4.c.k 2 420.w even 4 2
405.4.e.c 2 63.l odd 6 2
405.4.e.l 2 63.o even 6 2
605.4.a.d 1 231.h odd 2 1
720.4.a.u 1 28.d even 2 1
845.4.a.b 1 273.g even 2 1
1225.4.a.k 1 15.d odd 2 1
1280.4.d.e 2 336.y even 4 2
1280.4.d.l 2 336.v odd 4 2
1445.4.a.a 1 357.c even 2 1
1600.4.a.s 1 840.b odd 2 1
1600.4.a.bi 1 840.u even 2 1
1805.4.a.h 1 399.h odd 2 1
2205.4.a.q 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2205))$$:

 $$T_{2} - 4$$ T2 - 4 $$T_{11} + 32$$ T11 + 32 $$T_{13} - 38$$ T13 - 38 $$T_{17} - 26$$ T17 - 26

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T$$
$11$ $$T + 32$$
$13$ $$T - 38$$
$17$ $$T - 26$$
$19$ $$T + 100$$
$23$ $$T - 78$$
$29$ $$T - 50$$
$31$ $$T - 108$$
$37$ $$T - 266$$
$41$ $$T - 22$$
$43$ $$T - 442$$
$47$ $$T + 514$$
$53$ $$T + 2$$
$59$ $$T - 500$$
$61$ $$T - 518$$
$67$ $$T - 126$$
$71$ $$T + 412$$
$73$ $$T - 878$$
$79$ $$T - 600$$
$83$ $$T - 282$$
$89$ $$T + 150$$
$97$ $$T + 386$$