# Properties

 Label 2475.4.a.a Level $2475$ Weight $4$ Character orbit 2475.a Self dual yes Analytic conductor $146.030$ Analytic rank $1$ 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] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2475.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: $$146.029727264$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 825) 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)

 $$f(q)$$ $$=$$ $$q - 5 q^{2} + 17 q^{4} + 3 q^{7} - 45 q^{8}+O(q^{10})$$ q - 5 * q^2 + 17 * q^4 + 3 * q^7 - 45 * q^8 $$q - 5 q^{2} + 17 q^{4} + 3 q^{7} - 45 q^{8} + 11 q^{11} + 32 q^{13} - 15 q^{14} + 89 q^{16} - 33 q^{17} + 47 q^{19} - 55 q^{22} - 113 q^{23} - 160 q^{26} + 51 q^{28} + 54 q^{29} + 178 q^{31} - 85 q^{32} + 165 q^{34} + 19 q^{37} - 235 q^{38} - 139 q^{41} - 308 q^{43} + 187 q^{44} + 565 q^{46} - 195 q^{47} - 334 q^{49} + 544 q^{52} - 152 q^{53} - 135 q^{56} - 270 q^{58} + 625 q^{59} + 320 q^{61} - 890 q^{62} - 287 q^{64} + 200 q^{67} - 561 q^{68} + 947 q^{71} - 448 q^{73} - 95 q^{74} + 799 q^{76} + 33 q^{77} - 721 q^{79} + 695 q^{82} - 142 q^{83} + 1540 q^{86} - 495 q^{88} - 404 q^{89} + 96 q^{91} - 1921 q^{92} + 975 q^{94} + 79 q^{97} + 1670 q^{98}+O(q^{100})$$ q - 5 * q^2 + 17 * q^4 + 3 * q^7 - 45 * q^8 + 11 * q^11 + 32 * q^13 - 15 * q^14 + 89 * q^16 - 33 * q^17 + 47 * q^19 - 55 * q^22 - 113 * q^23 - 160 * q^26 + 51 * q^28 + 54 * q^29 + 178 * q^31 - 85 * q^32 + 165 * q^34 + 19 * q^37 - 235 * q^38 - 139 * q^41 - 308 * q^43 + 187 * q^44 + 565 * q^46 - 195 * q^47 - 334 * q^49 + 544 * q^52 - 152 * q^53 - 135 * q^56 - 270 * q^58 + 625 * q^59 + 320 * q^61 - 890 * q^62 - 287 * q^64 + 200 * q^67 - 561 * q^68 + 947 * q^71 - 448 * q^73 - 95 * q^74 + 799 * q^76 + 33 * q^77 - 721 * q^79 + 695 * q^82 - 142 * q^83 + 1540 * q^86 - 495 * q^88 - 404 * q^89 + 96 * q^91 - 1921 * q^92 + 975 * q^94 + 79 * q^97 + 1670 * q^98

## 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 0 17.0000 0 0 3.00000 −45.0000 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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$-1$$

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2475))$$:

 $$T_{2} + 5$$ T2 + 5 $$T_{7} - 3$$ T7 - 3 $$T_{29} - 54$$ T29 - 54

## Hecke characteristic polynomials

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