# Properties

 Label 2205.4.a.l Level $2205$ Weight $4$ Character orbit 2205.a Self dual yes Analytic conductor $130.099$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - q^{2} - 7q^{4} + 5q^{5} + 15q^{8} + O(q^{10})$$ $$q - q^{2} - 7q^{4} + 5q^{5} + 15q^{8} - 5q^{10} - 52q^{11} - 22q^{13} + 41q^{16} - 14q^{17} + 20q^{19} - 35q^{20} + 52q^{22} + 168q^{23} + 25q^{25} + 22q^{26} - 230q^{29} + 288q^{31} - 161q^{32} + 14q^{34} - 34q^{37} - 20q^{38} + 75q^{40} + 122q^{41} - 188q^{43} + 364q^{44} - 168q^{46} + 256q^{47} - 25q^{50} + 154q^{52} + 338q^{53} - 260q^{55} + 230q^{58} + 100q^{59} - 742q^{61} - 288q^{62} - 167q^{64} - 110q^{65} - 84q^{67} + 98q^{68} + 328q^{71} + 38q^{73} + 34q^{74} - 140q^{76} - 240q^{79} + 205q^{80} - 122q^{82} + 1212q^{83} - 70q^{85} + 188q^{86} - 780q^{88} + 330q^{89} - 1176q^{92} - 256q^{94} + 100q^{95} - 866q^{97} + O(q^{100})$$

## 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
−1.00000 0 −7.00000 5.00000 0 0 15.0000 0 −5.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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.l 1
3.b odd 2 1 735.4.a.e 1
7.b odd 2 1 45.4.a.c 1
21.c even 2 1 15.4.a.a 1
28.d even 2 1 720.4.a.n 1
35.c odd 2 1 225.4.a.f 1
35.f even 4 2 225.4.b.e 2
63.l odd 6 2 405.4.e.i 2
63.o even 6 2 405.4.e.g 2
84.h odd 2 1 240.4.a.e 1
105.g even 2 1 75.4.a.b 1
105.k odd 4 2 75.4.b.b 2
168.e odd 2 1 960.4.a.ba 1
168.i even 2 1 960.4.a.b 1
231.h odd 2 1 1815.4.a.e 1
420.o odd 2 1 1200.4.a.t 1
420.w even 4 2 1200.4.f.b 2

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

 $$T_{2} + 1$$ $$T_{11} + 52$$ $$T_{13} + 22$$ $$T_{17} + 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$-5 + T$$
$7$ $$T$$
$11$ $$52 + T$$
$13$ $$22 + T$$
$17$ $$14 + T$$
$19$ $$-20 + T$$
$23$ $$-168 + T$$
$29$ $$230 + T$$
$31$ $$-288 + T$$
$37$ $$34 + T$$
$41$ $$-122 + T$$
$43$ $$188 + T$$
$47$ $$-256 + T$$
$53$ $$-338 + T$$
$59$ $$-100 + T$$
$61$ $$742 + T$$
$67$ $$84 + T$$
$71$ $$-328 + T$$
$73$ $$-38 + T$$
$79$ $$240 + T$$
$83$ $$-1212 + T$$
$89$ $$-330 + T$$
$97$ $$866 + T$$