# Properties

 Label 2475.4.a.e 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 33) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - 7 q^{4} + 26 q^{7} + 15 q^{8}+O(q^{10})$$ q - q^2 - 7 * q^4 + 26 * q^7 + 15 * q^8 $$q - q^{2} - 7 q^{4} + 26 q^{7} + 15 q^{8} - 11 q^{11} + 32 q^{13} - 26 q^{14} + 41 q^{16} + 74 q^{17} - 60 q^{19} + 11 q^{22} - 182 q^{23} - 32 q^{26} - 182 q^{28} + 90 q^{29} - 8 q^{31} - 161 q^{32} - 74 q^{34} + 66 q^{37} + 60 q^{38} - 422 q^{41} - 408 q^{43} + 77 q^{44} + 182 q^{46} - 506 q^{47} + 333 q^{49} - 224 q^{52} + 348 q^{53} + 390 q^{56} - 90 q^{58} + 200 q^{59} + 132 q^{61} + 8 q^{62} - 167 q^{64} + 1036 q^{67} - 518 q^{68} - 762 q^{71} + 542 q^{73} - 66 q^{74} + 420 q^{76} - 286 q^{77} - 550 q^{79} + 422 q^{82} - 132 q^{83} + 408 q^{86} - 165 q^{88} - 570 q^{89} + 832 q^{91} + 1274 q^{92} + 506 q^{94} - 14 q^{97} - 333 q^{98}+O(q^{100})$$ q - q^2 - 7 * q^4 + 26 * q^7 + 15 * q^8 - 11 * q^11 + 32 * q^13 - 26 * q^14 + 41 * q^16 + 74 * q^17 - 60 * q^19 + 11 * q^22 - 182 * q^23 - 32 * q^26 - 182 * q^28 + 90 * q^29 - 8 * q^31 - 161 * q^32 - 74 * q^34 + 66 * q^37 + 60 * q^38 - 422 * q^41 - 408 * q^43 + 77 * q^44 + 182 * q^46 - 506 * q^47 + 333 * q^49 - 224 * q^52 + 348 * q^53 + 390 * q^56 - 90 * q^58 + 200 * q^59 + 132 * q^61 + 8 * q^62 - 167 * q^64 + 1036 * q^67 - 518 * q^68 - 762 * q^71 + 542 * q^73 - 66 * q^74 + 420 * q^76 - 286 * q^77 - 550 * q^79 + 422 * q^82 - 132 * q^83 + 408 * q^86 - 165 * q^88 - 570 * q^89 + 832 * q^91 + 1274 * q^92 + 506 * q^94 - 14 * q^97 - 333 * 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
−1.00000 0 −7.00000 0 0 26.0000 15.0000 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.e 1
3.b odd 2 1 825.4.a.f 1
5.b even 2 1 99.4.a.a 1
15.d odd 2 1 33.4.a.b 1
15.e even 4 2 825.4.c.f 2
20.d odd 2 1 1584.4.a.l 1
55.d odd 2 1 1089.4.a.e 1
60.h even 2 1 528.4.a.h 1
105.g even 2 1 1617.4.a.d 1
120.i odd 2 1 2112.4.a.u 1
120.m even 2 1 2112.4.a.h 1
165.d even 2 1 363.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.b 1 15.d odd 2 1
99.4.a.a 1 5.b even 2 1
363.4.a.d 1 165.d even 2 1
528.4.a.h 1 60.h even 2 1
825.4.a.f 1 3.b odd 2 1
825.4.c.f 2 15.e even 4 2
1089.4.a.e 1 55.d odd 2 1
1584.4.a.l 1 20.d odd 2 1
1617.4.a.d 1 105.g even 2 1
2112.4.a.h 1 120.m even 2 1
2112.4.a.u 1 120.i odd 2 1
2475.4.a.e 1 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{2} + 1$$ T2 + 1 $$T_{7} - 26$$ T7 - 26 $$T_{29} - 90$$ T29 - 90

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 26$$
$11$ $$T + 11$$
$13$ $$T - 32$$
$17$ $$T - 74$$
$19$ $$T + 60$$
$23$ $$T + 182$$
$29$ $$T - 90$$
$31$ $$T + 8$$
$37$ $$T - 66$$
$41$ $$T + 422$$
$43$ $$T + 408$$
$47$ $$T + 506$$
$53$ $$T - 348$$
$59$ $$T - 200$$
$61$ $$T - 132$$
$67$ $$T - 1036$$
$71$ $$T + 762$$
$73$ $$T - 542$$
$79$ $$T + 550$$
$83$ $$T + 132$$
$89$ $$T + 570$$
$97$ $$T + 14$$