# Properties

 Label 2475.4.a.j Level $2475$ Weight $4$ Character orbit 2475.a Self dual yes Analytic conductor $146.030$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 8 q^{4} + 21 q^{7}+O(q^{10})$$ q + 4 * q^2 + 8 * q^4 + 21 * q^7 $$q + 4 q^{2} + 8 q^{4} + 21 q^{7} - 11 q^{11} - 68 q^{13} + 84 q^{14} - 64 q^{16} - 21 q^{17} + 125 q^{19} - 44 q^{22} - 137 q^{23} - 272 q^{26} + 168 q^{28} + 150 q^{29} + 292 q^{31} - 256 q^{32} - 84 q^{34} - 349 q^{37} + 500 q^{38} - 497 q^{41} - 208 q^{43} - 88 q^{44} - 548 q^{46} + 369 q^{47} + 98 q^{49} - 544 q^{52} - 542 q^{53} + 600 q^{58} - 235 q^{59} + 482 q^{61} + 1168 q^{62} - 512 q^{64} - 734 q^{67} - 168 q^{68} - 587 q^{71} - 518 q^{73} - 1396 q^{74} + 1000 q^{76} - 231 q^{77} - 1045 q^{79} - 1988 q^{82} + 608 q^{83} - 832 q^{86} + 770 q^{89} - 1428 q^{91} - 1096 q^{92} + 1476 q^{94} + 1541 q^{97} + 392 q^{98}+O(q^{100})$$ q + 4 * q^2 + 8 * q^4 + 21 * q^7 - 11 * q^11 - 68 * q^13 + 84 * q^14 - 64 * q^16 - 21 * q^17 + 125 * q^19 - 44 * q^22 - 137 * q^23 - 272 * q^26 + 168 * q^28 + 150 * q^29 + 292 * q^31 - 256 * q^32 - 84 * q^34 - 349 * q^37 + 500 * q^38 - 497 * q^41 - 208 * q^43 - 88 * q^44 - 548 * q^46 + 369 * q^47 + 98 * q^49 - 544 * q^52 - 542 * q^53 + 600 * q^58 - 235 * q^59 + 482 * q^61 + 1168 * q^62 - 512 * q^64 - 734 * q^67 - 168 * q^68 - 587 * q^71 - 518 * q^73 - 1396 * q^74 + 1000 * q^76 - 231 * q^77 - 1045 * q^79 - 1988 * q^82 + 608 * q^83 - 832 * q^86 + 770 * q^89 - 1428 * q^91 - 1096 * q^92 + 1476 * q^94 + 1541 * q^97 + 392 * 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
4.00000 0 8.00000 0 0 21.0000 0 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.j 1
3.b odd 2 1 825.4.a.b 1
5.b even 2 1 2475.4.a.c 1
15.d odd 2 1 825.4.a.h yes 1
15.e even 4 2 825.4.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.b 1 3.b odd 2 1
825.4.a.h yes 1 15.d odd 2 1
825.4.c.c 2 15.e even 4 2
2475.4.a.c 1 5.b even 2 1
2475.4.a.j 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2475))$$:

 $$T_{2} - 4$$ T2 - 4 $$T_{7} - 21$$ T7 - 21 $$T_{29} - 150$$ T29 - 150

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 21$$
$11$ $$T + 11$$
$13$ $$T + 68$$
$17$ $$T + 21$$
$19$ $$T - 125$$
$23$ $$T + 137$$
$29$ $$T - 150$$
$31$ $$T - 292$$
$37$ $$T + 349$$
$41$ $$T + 497$$
$43$ $$T + 208$$
$47$ $$T - 369$$
$53$ $$T + 542$$
$59$ $$T + 235$$
$61$ $$T - 482$$
$67$ $$T + 734$$
$71$ $$T + 587$$
$73$ $$T + 518$$
$79$ $$T + 1045$$
$83$ $$T - 608$$
$89$ $$T - 770$$
$97$ $$T - 1541$$
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