# Properties

 Label 5625.2.a.bf Level $5625$ Weight $2$ Character orbit 5625.a Self dual yes Analytic conductor $44.916$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5625 = 3^{2} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5625.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.9158511370$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: no (minimal twist has level 225) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 32 q^{4}+O(q^{10})$$ 24 * q + 32 * q^4 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 32 q^{4} + 56 q^{16} + 36 q^{19} + 52 q^{31} + 60 q^{34} + 60 q^{46} + 72 q^{49} + 68 q^{61} + 108 q^{64} + 88 q^{76} + 84 q^{79} + 80 q^{91} + 100 q^{94}+O(q^{100})$$ 24 * q + 32 * q^4 + 56 * q^16 + 36 * q^19 + 52 * q^31 + 60 * q^34 + 60 * q^46 + 72 * q^49 + 68 * q^61 + 108 * q^64 + 88 * q^76 + 84 * q^79 + 80 * q^91 + 100 * q^94

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −2.64061 0 4.97280 0 0 −4.34551 −7.84999 0 0
1.2 −2.64061 0 4.97280 0 0 4.34551 −7.84999 0 0
1.3 −2.61097 0 4.81718 0 0 −1.21018 −7.35558 0 0
1.4 −2.61097 0 4.81718 0 0 1.21018 −7.35558 0 0
1.5 −1.71285 0 0.933844 0 0 −3.13880 1.82616 0 0
1.6 −1.71285 0 0.933844 0 0 3.13880 1.82616 0 0
1.7 −1.54077 0 0.373979 0 0 −3.37636 2.50533 0 0
1.8 −1.54077 0 0.373979 0 0 3.37636 2.50533 0 0
1.9 −0.899356 0 −1.19116 0 0 −3.86000 2.86999 0 0
1.10 −0.899356 0 −1.19116 0 0 3.86000 2.86999 0 0
1.11 −0.305545 0 −1.90664 0 0 −1.87098 1.19366 0 0
1.12 −0.305545 0 −1.90664 0 0 1.87098 1.19366 0 0
1.13 0.305545 0 −1.90664 0 0 −1.87098 −1.19366 0 0
1.14 0.305545 0 −1.90664 0 0 1.87098 −1.19366 0 0
1.15 0.899356 0 −1.19116 0 0 −3.86000 −2.86999 0 0
1.16 0.899356 0 −1.19116 0 0 3.86000 −2.86999 0 0
1.17 1.54077 0 0.373979 0 0 −3.37636 −2.50533 0 0
1.18 1.54077 0 0.373979 0 0 3.37636 −2.50533 0 0
1.19 1.71285 0 0.933844 0 0 −3.13880 −1.82616 0 0
1.20 1.71285 0 0.933844 0 0 3.13880 −1.82616 0 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5625.2.a.bf 24
3.b odd 2 1 inner 5625.2.a.bf 24
5.b even 2 1 inner 5625.2.a.bf 24
15.d odd 2 1 inner 5625.2.a.bf 24
25.f odd 20 2 225.2.m.c 24
75.l even 20 2 225.2.m.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.m.c 24 25.f odd 20 2
225.2.m.c 24 75.l even 20 2
5625.2.a.bf 24 1.a even 1 1 trivial
5625.2.a.bf 24 3.b odd 2 1 inner
5625.2.a.bf 24 5.b even 2 1 inner
5625.2.a.bf 24 15.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5625))$$:

 $$T_{2}^{12} - 20T_{2}^{10} + 145T_{2}^{8} - 465T_{2}^{6} + 655T_{2}^{4} - 325T_{2}^{2} + 25$$ T2^12 - 20*T2^10 + 145*T2^8 - 465*T2^6 + 655*T2^4 - 325*T2^2 + 25 $$T_{7}^{12} - 60T_{7}^{10} + 1390T_{7}^{8} - 15575T_{7}^{6} + 85825T_{7}^{4} - 207000T_{7}^{2} + 162000$$ T7^12 - 60*T7^10 + 1390*T7^8 - 15575*T7^6 + 85825*T7^4 - 207000*T7^2 + 162000