# Properties

 Label 5175.2.a.u Level $5175$ Weight $2$ Character orbit 5175.a Self dual yes Analytic conductor $41.323$ 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] = [5175,2,Mod(1,5175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5175, 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("5175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5175 = 3^{2} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5175.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: $$41.3225830460$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 575) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} + q^{7} - 3 q^{8}+O(q^{10})$$ q + q^2 - q^4 + q^7 - 3 * q^8 $$q + q^{2} - q^{4} + q^{7} - 3 q^{8} + q^{11} + q^{13} + q^{14} - q^{16} - 5 q^{19} + q^{22} - q^{23} + q^{26} - q^{28} + 5 q^{29} - 2 q^{31} + 5 q^{32} - 4 q^{37} - 5 q^{38} + 5 q^{41} - 9 q^{43} - q^{44} - q^{46} + 6 q^{47} - 6 q^{49} - q^{52} - 2 q^{53} - 3 q^{56} + 5 q^{58} - 8 q^{59} - 8 q^{61} - 2 q^{62} + 7 q^{64} + 8 q^{67} + 10 q^{71} - 3 q^{73} - 4 q^{74} + 5 q^{76} + q^{77} - 3 q^{79} + 5 q^{82} - 3 q^{83} - 9 q^{86} - 3 q^{88} - 10 q^{89} + q^{91} + q^{92} + 6 q^{94} - 2 q^{97} - 6 q^{98}+O(q^{100})$$ q + q^2 - q^4 + q^7 - 3 * q^8 + q^11 + q^13 + q^14 - q^16 - 5 * q^19 + q^22 - q^23 + q^26 - q^28 + 5 * q^29 - 2 * q^31 + 5 * q^32 - 4 * q^37 - 5 * q^38 + 5 * q^41 - 9 * q^43 - q^44 - q^46 + 6 * q^47 - 6 * q^49 - q^52 - 2 * q^53 - 3 * q^56 + 5 * q^58 - 8 * q^59 - 8 * q^61 - 2 * q^62 + 7 * q^64 + 8 * q^67 + 10 * q^71 - 3 * q^73 - 4 * q^74 + 5 * q^76 + q^77 - 3 * q^79 + 5 * q^82 - 3 * q^83 - 9 * q^86 - 3 * q^88 - 10 * q^89 + q^91 + q^92 + 6 * q^94 - 2 * q^97 - 6 * 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 1.00000 −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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$1$$

## 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 5175.2.a.u 1
3.b odd 2 1 575.2.a.c 1
5.b even 2 1 5175.2.a.e 1
12.b even 2 1 9200.2.a.r 1
15.d odd 2 1 575.2.a.d yes 1
15.e even 4 2 575.2.b.b 2
60.h even 2 1 9200.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
575.2.a.c 1 3.b odd 2 1
575.2.a.d yes 1 15.d odd 2 1
575.2.b.b 2 15.e even 4 2
5175.2.a.e 1 5.b even 2 1
5175.2.a.u 1 1.a even 1 1 trivial
9200.2.a.r 1 12.b even 2 1
9200.2.a.u 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5175))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T - 1$$
$13$ $$T - 1$$
$17$ $$T$$
$19$ $$T + 5$$
$23$ $$T + 1$$
$29$ $$T - 5$$
$31$ $$T + 2$$
$37$ $$T + 4$$
$41$ $$T - 5$$
$43$ $$T + 9$$
$47$ $$T - 6$$
$53$ $$T + 2$$
$59$ $$T + 8$$
$61$ $$T + 8$$
$67$ $$T - 8$$
$71$ $$T - 10$$
$73$ $$T + 3$$
$79$ $$T + 3$$
$83$ $$T + 3$$
$89$ $$T + 10$$
$97$ $$T + 2$$