# Properties

 Label 2205.4.a.t 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

## Newspace parameters

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2205.a (trivial)

## Newform invariants

 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 45) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 5 q^{2} + 17 q^{4} + 5 q^{5} + 45 q^{8} + O(q^{10})$$ $$q + 5 q^{2} + 17 q^{4} + 5 q^{5} + 45 q^{8} + 25 q^{10} + 50 q^{11} + 20 q^{13} + 89 q^{16} + 10 q^{17} + 44 q^{19} + 85 q^{20} + 250 q^{22} + 120 q^{23} + 25 q^{25} + 100 q^{26} - 50 q^{29} - 108 q^{31} + 85 q^{32} + 50 q^{34} - 40 q^{37} + 220 q^{38} + 225 q^{40} - 400 q^{41} + 280 q^{43} + 850 q^{44} + 600 q^{46} + 280 q^{47} + 125 q^{50} + 340 q^{52} - 610 q^{53} + 250 q^{55} - 250 q^{58} - 50 q^{59} + 518 q^{61} - 540 q^{62} - 287 q^{64} + 100 q^{65} - 180 q^{67} + 170 q^{68} + 700 q^{71} + 410 q^{73} - 200 q^{74} + 748 q^{76} - 516 q^{79} + 445 q^{80} - 2000 q^{82} - 660 q^{83} + 50 q^{85} + 1400 q^{86} + 2250 q^{88} + 1500 q^{89} + 2040 q^{92} + 1400 q^{94} + 220 q^{95} + 1630 q^{97} + O(q^{100})$$

## 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.

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
5.00000 0 17.0000 5.00000 0 0 45.0000 0 25.0000
 $$n$$: 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.t 1
3.b odd 2 1 2205.4.a.a 1
7.b odd 2 1 45.4.a.e yes 1
21.c even 2 1 45.4.a.a 1
28.d even 2 1 720.4.a.o 1
35.c odd 2 1 225.4.a.a 1
35.f even 4 2 225.4.b.b 2
63.l odd 6 2 405.4.e.b 2
63.o even 6 2 405.4.e.n 2
84.h odd 2 1 720.4.a.bc 1
105.g even 2 1 225.4.a.h 1
105.k odd 4 2 225.4.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.a.a 1 21.c even 2 1
45.4.a.e yes 1 7.b odd 2 1
225.4.a.a 1 35.c odd 2 1
225.4.a.h 1 105.g even 2 1
225.4.b.a 2 105.k odd 4 2
225.4.b.b 2 35.f even 4 2
405.4.e.b 2 63.l odd 6 2
405.4.e.n 2 63.o even 6 2
720.4.a.o 1 28.d even 2 1
720.4.a.bc 1 84.h odd 2 1
2205.4.a.a 1 3.b odd 2 1
2205.4.a.t 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2205))$$:

 $$T_{2} - 5$$ $$T_{11} - 50$$ $$T_{13} - 20$$ $$T_{17} - 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-5 + T$$
$3$ $$T$$
$5$ $$-5 + T$$
$7$ $$T$$
$11$ $$-50 + T$$
$13$ $$-20 + T$$
$17$ $$-10 + T$$
$19$ $$-44 + T$$
$23$ $$-120 + T$$
$29$ $$50 + T$$
$31$ $$108 + T$$
$37$ $$40 + T$$
$41$ $$400 + T$$
$43$ $$-280 + T$$
$47$ $$-280 + T$$
$53$ $$610 + T$$
$59$ $$50 + T$$
$61$ $$-518 + T$$
$67$ $$180 + T$$
$71$ $$-700 + T$$
$73$ $$-410 + T$$
$79$ $$516 + T$$
$83$ $$660 + T$$
$89$ $$-1500 + T$$
$97$ $$-1630 + T$$