# Properties

 Label 825.4.a.j Level $825$ Weight $4$ Character orbit 825.a Self dual yes Analytic conductor $48.677$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.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: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 5 q^{2} + 3 q^{3} + 17 q^{4} + 15 q^{6} + 3 q^{7} + 45 q^{8} + 9 q^{9}+O(q^{10})$$ q + 5 * q^2 + 3 * q^3 + 17 * q^4 + 15 * q^6 + 3 * q^7 + 45 * q^8 + 9 * q^9 $$q + 5 q^{2} + 3 q^{3} + 17 q^{4} + 15 q^{6} + 3 q^{7} + 45 q^{8} + 9 q^{9} - 11 q^{11} + 51 q^{12} + 32 q^{13} + 15 q^{14} + 89 q^{16} + 33 q^{17} + 45 q^{18} + 47 q^{19} + 9 q^{21} - 55 q^{22} + 113 q^{23} + 135 q^{24} + 160 q^{26} + 27 q^{27} + 51 q^{28} - 54 q^{29} + 178 q^{31} + 85 q^{32} - 33 q^{33} + 165 q^{34} + 153 q^{36} + 19 q^{37} + 235 q^{38} + 96 q^{39} + 139 q^{41} + 45 q^{42} - 308 q^{43} - 187 q^{44} + 565 q^{46} + 195 q^{47} + 267 q^{48} - 334 q^{49} + 99 q^{51} + 544 q^{52} + 152 q^{53} + 135 q^{54} + 135 q^{56} + 141 q^{57} - 270 q^{58} - 625 q^{59} + 320 q^{61} + 890 q^{62} + 27 q^{63} - 287 q^{64} - 165 q^{66} + 200 q^{67} + 561 q^{68} + 339 q^{69} - 947 q^{71} + 405 q^{72} - 448 q^{73} + 95 q^{74} + 799 q^{76} - 33 q^{77} + 480 q^{78} - 721 q^{79} + 81 q^{81} + 695 q^{82} + 142 q^{83} + 153 q^{84} - 1540 q^{86} - 162 q^{87} - 495 q^{88} + 404 q^{89} + 96 q^{91} + 1921 q^{92} + 534 q^{93} + 975 q^{94} + 255 q^{96} + 79 q^{97} - 1670 q^{98} - 99 q^{99}+O(q^{100})$$ q + 5 * q^2 + 3 * q^3 + 17 * q^4 + 15 * q^6 + 3 * q^7 + 45 * q^8 + 9 * q^9 - 11 * q^11 + 51 * q^12 + 32 * q^13 + 15 * q^14 + 89 * q^16 + 33 * q^17 + 45 * q^18 + 47 * q^19 + 9 * q^21 - 55 * q^22 + 113 * q^23 + 135 * q^24 + 160 * q^26 + 27 * q^27 + 51 * q^28 - 54 * q^29 + 178 * q^31 + 85 * q^32 - 33 * q^33 + 165 * q^34 + 153 * q^36 + 19 * q^37 + 235 * q^38 + 96 * q^39 + 139 * q^41 + 45 * q^42 - 308 * q^43 - 187 * q^44 + 565 * q^46 + 195 * q^47 + 267 * q^48 - 334 * q^49 + 99 * q^51 + 544 * q^52 + 152 * q^53 + 135 * q^54 + 135 * q^56 + 141 * q^57 - 270 * q^58 - 625 * q^59 + 320 * q^61 + 890 * q^62 + 27 * q^63 - 287 * q^64 - 165 * q^66 + 200 * q^67 + 561 * q^68 + 339 * q^69 - 947 * q^71 + 405 * q^72 - 448 * q^73 + 95 * q^74 + 799 * q^76 - 33 * q^77 + 480 * q^78 - 721 * q^79 + 81 * q^81 + 695 * q^82 + 142 * q^83 + 153 * q^84 - 1540 * q^86 - 162 * q^87 - 495 * q^88 + 404 * q^89 + 96 * q^91 + 1921 * q^92 + 534 * q^93 + 975 * q^94 + 255 * q^96 + 79 * q^97 - 1670 * q^98 - 99 * 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
5.00000 3.00000 17.0000 0 15.0000 3.00000 45.0000 9.00000 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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$1$$

## 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 825.4.a.j yes 1
3.b odd 2 1 2475.4.a.a 1
5.b even 2 1 825.4.a.a 1
5.c odd 4 2 825.4.c.b 2
15.d odd 2 1 2475.4.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.a 1 5.b even 2 1
825.4.a.j yes 1 1.a even 1 1 trivial
825.4.c.b 2 5.c odd 4 2
2475.4.a.a 1 3.b odd 2 1
2475.4.a.k 1 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(825))$$:

 $$T_{2} - 5$$ T2 - 5 $$T_{7} - 3$$ T7 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 5$$
$3$ $$T - 3$$
$5$ $$T$$
$7$ $$T - 3$$
$11$ $$T + 11$$
$13$ $$T - 32$$
$17$ $$T - 33$$
$19$ $$T - 47$$
$23$ $$T - 113$$
$29$ $$T + 54$$
$31$ $$T - 178$$
$37$ $$T - 19$$
$41$ $$T - 139$$
$43$ $$T + 308$$
$47$ $$T - 195$$
$53$ $$T - 152$$
$59$ $$T + 625$$
$61$ $$T - 320$$
$67$ $$T - 200$$
$71$ $$T + 947$$
$73$ $$T + 448$$
$79$ $$T + 721$$
$83$ $$T - 142$$
$89$ $$T - 404$$
$97$ $$T - 79$$