# Properties

 Label 2205.2.a.e Level $2205$ Weight $2$ Character orbit 2205.a Self dual yes Analytic conductor $17.607$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$17.6070136457$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{4} - q^{5}+O(q^{10})$$ q - 2 * q^4 - q^5 $$q - 2 q^{4} - q^{5} + 3 q^{11} - 5 q^{13} + 4 q^{16} + 3 q^{17} - 2 q^{19} + 2 q^{20} + 6 q^{23} + q^{25} - 3 q^{29} + 4 q^{31} + 2 q^{37} - 12 q^{41} - 10 q^{43} - 6 q^{44} + 9 q^{47} + 10 q^{52} - 12 q^{53} - 3 q^{55} - 8 q^{61} - 8 q^{64} + 5 q^{65} - 4 q^{67} - 6 q^{68} - 2 q^{73} + 4 q^{76} - q^{79} - 4 q^{80} + 12 q^{83} - 3 q^{85} - 12 q^{89} - 12 q^{92} + 2 q^{95} + q^{97}+O(q^{100})$$ q - 2 * q^4 - q^5 + 3 * q^11 - 5 * q^13 + 4 * q^16 + 3 * q^17 - 2 * q^19 + 2 * q^20 + 6 * q^23 + q^25 - 3 * q^29 + 4 * q^31 + 2 * q^37 - 12 * q^41 - 10 * q^43 - 6 * q^44 + 9 * q^47 + 10 * q^52 - 12 * q^53 - 3 * q^55 - 8 * q^61 - 8 * q^64 + 5 * q^65 - 4 * q^67 - 6 * q^68 - 2 * q^73 + 4 * q^76 - q^79 - 4 * q^80 + 12 * q^83 - 3 * q^85 - 12 * q^89 - 12 * q^92 + 2 * q^95 + 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.

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
0 0 −2.00000 −1.00000 0 0 0 0 0
 $$n$$: 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 2205.2.a.e 1
3.b odd 2 1 245.2.a.c 1
7.b odd 2 1 315.2.a.b 1
12.b even 2 1 3920.2.a.ba 1
15.d odd 2 1 1225.2.a.e 1
15.e even 4 2 1225.2.b.d 2
21.c even 2 1 35.2.a.a 1
21.g even 6 2 245.2.e.a 2
21.h odd 6 2 245.2.e.b 2
28.d even 2 1 5040.2.a.v 1
35.c odd 2 1 1575.2.a.f 1
35.f even 4 2 1575.2.d.c 2
84.h odd 2 1 560.2.a.b 1
105.g even 2 1 175.2.a.b 1
105.k odd 4 2 175.2.b.a 2
168.e odd 2 1 2240.2.a.u 1
168.i even 2 1 2240.2.a.k 1
231.h odd 2 1 4235.2.a.c 1
273.g even 2 1 5915.2.a.f 1
420.o odd 2 1 2800.2.a.z 1
420.w even 4 2 2800.2.g.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.a 1 21.c even 2 1
175.2.a.b 1 105.g even 2 1
175.2.b.a 2 105.k odd 4 2
245.2.a.c 1 3.b odd 2 1
245.2.e.a 2 21.g even 6 2
245.2.e.b 2 21.h odd 6 2
315.2.a.b 1 7.b odd 2 1
560.2.a.b 1 84.h odd 2 1
1225.2.a.e 1 15.d odd 2 1
1225.2.b.d 2 15.e even 4 2
1575.2.a.f 1 35.c odd 2 1
1575.2.d.c 2 35.f even 4 2
2205.2.a.e 1 1.a even 1 1 trivial
2240.2.a.k 1 168.i even 2 1
2240.2.a.u 1 168.e odd 2 1
2800.2.a.z 1 420.o odd 2 1
2800.2.g.l 2 420.w even 4 2
3920.2.a.ba 1 12.b even 2 1
4235.2.a.c 1 231.h odd 2 1
5040.2.a.v 1 28.d even 2 1
5915.2.a.f 1 273.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2205))$$:

 $$T_{2}$$ T2 $$T_{11} - 3$$ T11 - 3 $$T_{13} + 5$$ T13 + 5 $$T_{17} - 3$$ T17 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T - 3$$
$13$ $$T + 5$$
$17$ $$T - 3$$
$19$ $$T + 2$$
$23$ $$T - 6$$
$29$ $$T + 3$$
$31$ $$T - 4$$
$37$ $$T - 2$$
$41$ $$T + 12$$
$43$ $$T + 10$$
$47$ $$T - 9$$
$53$ $$T + 12$$
$59$ $$T$$
$61$ $$T + 8$$
$67$ $$T + 4$$
$71$ $$T$$
$73$ $$T + 2$$
$79$ $$T + 1$$
$83$ $$T - 12$$
$89$ $$T + 12$$
$97$ $$T - 1$$
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