# Properties

 Label 5225.2.a.bc Level $5225$ Weight $2$ Character orbit 5225.a Self dual yes Analytic conductor $41.722$ Analytic rank $0$ Dimension $30$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5225 = 5^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5225.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: $$41.7218350561$$ Analytic rank: $$0$$ Dimension: $$30$$ Twist minimal: no (minimal twist has level 1045) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30 q + 42 q^{4} + 12 q^{6} + 40 q^{9}+O(q^{10})$$ 30 * q + 42 * q^4 + 12 * q^6 + 40 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$30 q + 42 q^{4} + 12 q^{6} + 40 q^{9} + 30 q^{11} - 4 q^{14} + 66 q^{16} + 30 q^{19} + 14 q^{21} + 22 q^{24} + 30 q^{29} + 26 q^{31} + 12 q^{34} + 78 q^{36} + 64 q^{39} + 22 q^{41} + 42 q^{44} + 28 q^{46} + 60 q^{49} + 64 q^{51} + 62 q^{54} - 32 q^{56} - 14 q^{59} + 78 q^{61} + 90 q^{64} + 12 q^{66} - 28 q^{69} + 20 q^{71} + 42 q^{74} + 42 q^{76} + 102 q^{79} + 42 q^{81} + 98 q^{84} - 52 q^{86} - 8 q^{89} + 56 q^{91} + 40 q^{94} - 74 q^{96} + 40 q^{99}+O(q^{100})$$ 30 * q + 42 * q^4 + 12 * q^6 + 40 * q^9 + 30 * q^11 - 4 * q^14 + 66 * q^16 + 30 * q^19 + 14 * q^21 + 22 * q^24 + 30 * q^29 + 26 * q^31 + 12 * q^34 + 78 * q^36 + 64 * q^39 + 22 * q^41 + 42 * q^44 + 28 * q^46 + 60 * q^49 + 64 * q^51 + 62 * q^54 - 32 * q^56 - 14 * q^59 + 78 * q^61 + 90 * q^64 + 12 * q^66 - 28 * q^69 + 20 * q^71 + 42 * q^74 + 42 * q^76 + 102 * q^79 + 42 * q^81 + 98 * q^84 - 52 * q^86 - 8 * q^89 + 56 * q^91 + 40 * q^94 - 74 * q^96 + 40 * q^99

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −2.79219 −1.83665 5.79631 0 5.12828 0.971452 −10.6000 0.373298 0
1.2 −2.77451 2.74433 5.69792 0 −7.61418 3.20844 −10.2599 4.53135 0
1.3 −2.44393 −0.661177 3.97280 0 1.61587 −2.05193 −4.82139 −2.56284 0
1.4 −2.39831 0.881728 3.75187 0 −2.11465 −2.42713 −4.20151 −2.22256 0
1.5 −2.39110 −3.31399 3.71736 0 7.92409 0.907652 −4.10637 7.98255 0
1.6 −2.14697 −2.93729 2.60947 0 6.30628 −3.64055 −1.30851 5.62770 0
1.7 −1.98136 2.35379 1.92578 0 −4.66371 −1.46477 0.147060 2.54034 0
1.8 −1.66433 1.25863 0.769991 0 −2.09478 4.13429 2.04714 −1.41584 0
1.9 −1.65738 −1.46266 0.746921 0 2.42418 5.18504 2.07683 −0.860633 0
1.10 −1.45151 0.0791235 0.106883 0 −0.114849 −2.96150 2.74788 −2.99374 0
1.11 −0.961626 −0.510054 −1.07528 0 0.490481 −0.688271 2.95726 −2.73984 0
1.12 −0.661223 3.01131 −1.56278 0 −1.99115 0.592449 2.35579 6.06799 0
1.13 −0.598278 1.31473 −1.64206 0 −0.786572 −0.976473 2.17897 −1.27149 0
1.14 −0.379302 −2.60759 −1.85613 0 0.989065 −4.15654 1.46264 3.79954 0
1.15 −0.202376 −2.47875 −1.95904 0 0.501638 5.06314 0.801215 3.14418 0
1.16 0.202376 2.47875 −1.95904 0 0.501638 −5.06314 −0.801215 3.14418 0
1.17 0.379302 2.60759 −1.85613 0 0.989065 4.15654 −1.46264 3.79954 0
1.18 0.598278 −1.31473 −1.64206 0 −0.786572 0.976473 −2.17897 −1.27149 0
1.19 0.661223 −3.01131 −1.56278 0 −1.99115 −0.592449 −2.35579 6.06799 0
1.20 0.961626 0.510054 −1.07528 0 0.490481 0.688271 −2.95726 −2.73984 0
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.30 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$11$$ $$-1$$
$$19$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5225.2.a.bc 30
5.b even 2 1 inner 5225.2.a.bc 30
5.c odd 4 2 1045.2.b.e 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.e 30 5.c odd 4 2
5225.2.a.bc 30 1.a even 1 1 trivial
5225.2.a.bc 30 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5225))$$:

 $$T_{2}^{30} - 51 T_{2}^{28} + 1161 T_{2}^{26} - 15571 T_{2}^{24} + 136754 T_{2}^{22} - 826847 T_{2}^{20} + \cdots - 2916$$ T2^30 - 51*T2^28 + 1161*T2^26 - 15571*T2^24 + 136754*T2^22 - 826847*T2^20 + 3521942*T2^18 - 10632298*T2^16 + 22578273*T2^14 - 33035145*T2^12 + 32143513*T2^10 - 19735815*T2^8 + 7123036*T2^6 - 1368509*T2^4 + 116568*T2^2 - 2916 $$T_{7}^{30} - 135 T_{7}^{28} + 7918 T_{7}^{26} - 265743 T_{7}^{24} + 5660203 T_{7}^{22} + \cdots - 1597441024$$ T7^30 - 135*T7^28 + 7918*T7^26 - 265743*T7^24 + 5660203*T7^22 - 80248794*T7^20 + 773258086*T7^18 - 5086884825*T7^16 + 22690237315*T7^14 - 67611340281*T7^12 + 132013622516*T7^10 - 165510791412*T7^8 + 129891767104*T7^6 - 60878175104*T7^4 + 15391012864*T7^2 - 1597441024