# Properties

 Label 2205.4.a.c 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 15) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{2} + q^{4} - 5 q^{5} + 21 q^{8}+O(q^{10})$$ q - 3 * q^2 + q^4 - 5 * q^5 + 21 * q^8 $$q - 3 q^{2} + q^{4} - 5 q^{5} + 21 q^{8} + 15 q^{10} + 24 q^{11} - 74 q^{13} - 71 q^{16} + 54 q^{17} + 124 q^{19} - 5 q^{20} - 72 q^{22} + 120 q^{23} + 25 q^{25} + 222 q^{26} + 78 q^{29} - 200 q^{31} + 45 q^{32} - 162 q^{34} - 70 q^{37} - 372 q^{38} - 105 q^{40} + 330 q^{41} + 92 q^{43} + 24 q^{44} - 360 q^{46} - 24 q^{47} - 75 q^{50} - 74 q^{52} - 450 q^{53} - 120 q^{55} - 234 q^{58} + 24 q^{59} + 322 q^{61} + 600 q^{62} + 433 q^{64} + 370 q^{65} - 196 q^{67} + 54 q^{68} + 288 q^{71} + 430 q^{73} + 210 q^{74} + 124 q^{76} - 520 q^{79} + 355 q^{80} - 990 q^{82} + 156 q^{83} - 270 q^{85} - 276 q^{86} + 504 q^{88} + 1026 q^{89} + 120 q^{92} + 72 q^{94} - 620 q^{95} + 286 q^{97}+O(q^{100})$$ q - 3 * q^2 + q^4 - 5 * q^5 + 21 * q^8 + 15 * q^10 + 24 * q^11 - 74 * q^13 - 71 * q^16 + 54 * q^17 + 124 * q^19 - 5 * q^20 - 72 * q^22 + 120 * q^23 + 25 * q^25 + 222 * q^26 + 78 * q^29 - 200 * q^31 + 45 * q^32 - 162 * q^34 - 70 * q^37 - 372 * q^38 - 105 * q^40 + 330 * q^41 + 92 * q^43 + 24 * q^44 - 360 * q^46 - 24 * q^47 - 75 * q^50 - 74 * q^52 - 450 * q^53 - 120 * q^55 - 234 * q^58 + 24 * q^59 + 322 * q^61 + 600 * q^62 + 433 * q^64 + 370 * q^65 - 196 * q^67 + 54 * q^68 + 288 * q^71 + 430 * q^73 + 210 * q^74 + 124 * q^76 - 520 * q^79 + 355 * q^80 - 990 * q^82 + 156 * q^83 - 270 * q^85 - 276 * q^86 + 504 * q^88 + 1026 * q^89 + 120 * q^92 + 72 * q^94 - 620 * q^95 + 286 * 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
−3.00000 0 1.00000 −5.00000 0 0 21.0000 0 15.0000
 $$n$$: 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.c 1
3.b odd 2 1 735.4.a.i 1
7.b odd 2 1 45.4.a.b 1
21.c even 2 1 15.4.a.b 1
28.d even 2 1 720.4.a.r 1
35.c odd 2 1 225.4.a.g 1
35.f even 4 2 225.4.b.d 2
63.l odd 6 2 405.4.e.k 2
63.o even 6 2 405.4.e.d 2
84.h odd 2 1 240.4.a.f 1
105.g even 2 1 75.4.a.a 1
105.k odd 4 2 75.4.b.a 2
168.e odd 2 1 960.4.a.l 1
168.i even 2 1 960.4.a.bi 1
231.h odd 2 1 1815.4.a.a 1
420.o odd 2 1 1200.4.a.o 1
420.w even 4 2 1200.4.f.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 21.c even 2 1
45.4.a.b 1 7.b odd 2 1
75.4.a.a 1 105.g even 2 1
75.4.b.a 2 105.k odd 4 2
225.4.a.g 1 35.c odd 2 1
225.4.b.d 2 35.f even 4 2
240.4.a.f 1 84.h odd 2 1
405.4.e.d 2 63.o even 6 2
405.4.e.k 2 63.l odd 6 2
720.4.a.r 1 28.d even 2 1
735.4.a.i 1 3.b odd 2 1
960.4.a.l 1 168.e odd 2 1
960.4.a.bi 1 168.i even 2 1
1200.4.a.o 1 420.o odd 2 1
1200.4.f.m 2 420.w even 4 2
1815.4.a.a 1 231.h odd 2 1
2205.4.a.c 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2205))$$:

 $$T_{2} + 3$$ T2 + 3 $$T_{11} - 24$$ T11 - 24 $$T_{13} + 74$$ T13 + 74 $$T_{17} - 54$$ T17 - 54

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 3$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T$$
$11$ $$T - 24$$
$13$ $$T + 74$$
$17$ $$T - 54$$
$19$ $$T - 124$$
$23$ $$T - 120$$
$29$ $$T - 78$$
$31$ $$T + 200$$
$37$ $$T + 70$$
$41$ $$T - 330$$
$43$ $$T - 92$$
$47$ $$T + 24$$
$53$ $$T + 450$$
$59$ $$T - 24$$
$61$ $$T - 322$$
$67$ $$T + 196$$
$71$ $$T - 288$$
$73$ $$T - 430$$
$79$ $$T + 520$$
$83$ $$T - 156$$
$89$ $$T - 1026$$
$97$ $$T - 286$$