# Properties

 Label 4275.2.a.o Level $4275$ Weight $2$ Character orbit 4275.a Self dual yes Analytic conductor $34.136$ 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] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, 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("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4275 = 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4275.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: $$34.1360468641$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 285) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} + 2 q^{7} - 3 q^{8}+O(q^{10})$$ q + q^2 - q^4 + 2 * q^7 - 3 * q^8 $$q + q^{2} - q^{4} + 2 q^{7} - 3 q^{8} + 2 q^{11} + 4 q^{13} + 2 q^{14} - q^{16} + 2 q^{17} - q^{19} + 2 q^{22} - 4 q^{23} + 4 q^{26} - 2 q^{28} - 4 q^{29} + 5 q^{32} + 2 q^{34} - q^{38} + 10 q^{43} - 2 q^{44} - 4 q^{46} + 12 q^{47} - 3 q^{49} - 4 q^{52} - 2 q^{53} - 6 q^{56} - 4 q^{58} - 4 q^{59} + 2 q^{61} + 7 q^{64} + 16 q^{67} - 2 q^{68} + 2 q^{73} + q^{76} + 4 q^{77} - 8 q^{79} - 12 q^{83} + 10 q^{86} - 6 q^{88} + 8 q^{91} + 4 q^{92} + 12 q^{94} + 16 q^{97} - 3 q^{98}+O(q^{100})$$ q + q^2 - q^4 + 2 * q^7 - 3 * q^8 + 2 * q^11 + 4 * q^13 + 2 * q^14 - q^16 + 2 * q^17 - q^19 + 2 * q^22 - 4 * q^23 + 4 * q^26 - 2 * q^28 - 4 * q^29 + 5 * q^32 + 2 * q^34 - q^38 + 10 * q^43 - 2 * q^44 - 4 * q^46 + 12 * q^47 - 3 * q^49 - 4 * q^52 - 2 * q^53 - 6 * q^56 - 4 * q^58 - 4 * q^59 + 2 * q^61 + 7 * q^64 + 16 * q^67 - 2 * q^68 + 2 * q^73 + q^76 + 4 * q^77 - 8 * q^79 - 12 * q^83 + 10 * q^86 - 6 * q^88 + 8 * q^91 + 4 * q^92 + 12 * q^94 + 16 * q^97 - 3 * 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 −1.00000 0 0 2.00000 −3.00000 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
$$3$$ $$-1$$
$$5$$ $$+1$$
$$19$$ $$+1$$

## 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 4275.2.a.o 1
3.b odd 2 1 1425.2.a.d 1
5.b even 2 1 855.2.a.b 1
15.d odd 2 1 285.2.a.b 1
15.e even 4 2 1425.2.c.d 2
60.h even 2 1 4560.2.a.v 1
285.b even 2 1 5415.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.b 1 15.d odd 2 1
855.2.a.b 1 5.b even 2 1
1425.2.a.d 1 3.b odd 2 1
1425.2.c.d 2 15.e even 4 2
4275.2.a.o 1 1.a even 1 1 trivial
4560.2.a.v 1 60.h even 2 1
5415.2.a.c 1 285.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4275))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{7} - 2$$ T7 - 2 $$T_{11} - 2$$ T11 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 2$$
$11$ $$T - 2$$
$13$ $$T - 4$$
$17$ $$T - 2$$
$19$ $$T + 1$$
$23$ $$T + 4$$
$29$ $$T + 4$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T - 10$$
$47$ $$T - 12$$
$53$ $$T + 2$$
$59$ $$T + 4$$
$61$ $$T - 2$$
$67$ $$T - 16$$
$71$ $$T$$
$73$ $$T - 2$$
$79$ $$T + 8$$
$83$ $$T + 12$$
$89$ $$T$$
$97$ $$T - 16$$