# Properties

 Label 575.2.a.a Level $575$ Weight $2$ Character orbit 575.a Self dual yes Analytic conductor $4.591$ 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] = [575,2,Mod(1,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("575.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$575 = 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 575.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: $$4.59139811622$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 115) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 4 q^{6} - q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 4 * q^6 - q^7 + q^9 $$q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 4 q^{6} - q^{7} + q^{9} - 4 q^{12} - 2 q^{13} + 2 q^{14} - 4 q^{16} + 5 q^{17} - 2 q^{18} + 8 q^{19} + 2 q^{21} + q^{23} + 4 q^{26} + 4 q^{27} - 2 q^{28} - 5 q^{29} - 5 q^{31} + 8 q^{32} - 10 q^{34} + 2 q^{36} - 7 q^{37} - 16 q^{38} + 4 q^{39} - 7 q^{41} - 4 q^{42} - 4 q^{43} - 2 q^{46} + 2 q^{47} + 8 q^{48} - 6 q^{49} - 10 q^{51} - 4 q^{52} + q^{53} - 8 q^{54} - 16 q^{57} + 10 q^{58} + 3 q^{59} - 6 q^{61} + 10 q^{62} - q^{63} - 8 q^{64} - 13 q^{67} + 10 q^{68} - 2 q^{69} + 13 q^{71} - 8 q^{73} + 14 q^{74} + 16 q^{76} - 8 q^{78} - 14 q^{79} - 11 q^{81} + 14 q^{82} + 3 q^{83} + 4 q^{84} + 8 q^{86} + 10 q^{87} - 14 q^{89} + 2 q^{91} + 2 q^{92} + 10 q^{93} - 4 q^{94} - 16 q^{96} - 14 q^{97} + 12 q^{98}+O(q^{100})$$ q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 4 * q^6 - q^7 + q^9 - 4 * q^12 - 2 * q^13 + 2 * q^14 - 4 * q^16 + 5 * q^17 - 2 * q^18 + 8 * q^19 + 2 * q^21 + q^23 + 4 * q^26 + 4 * q^27 - 2 * q^28 - 5 * q^29 - 5 * q^31 + 8 * q^32 - 10 * q^34 + 2 * q^36 - 7 * q^37 - 16 * q^38 + 4 * q^39 - 7 * q^41 - 4 * q^42 - 4 * q^43 - 2 * q^46 + 2 * q^47 + 8 * q^48 - 6 * q^49 - 10 * q^51 - 4 * q^52 + q^53 - 8 * q^54 - 16 * q^57 + 10 * q^58 + 3 * q^59 - 6 * q^61 + 10 * q^62 - q^63 - 8 * q^64 - 13 * q^67 + 10 * q^68 - 2 * q^69 + 13 * q^71 - 8 * q^73 + 14 * q^74 + 16 * q^76 - 8 * q^78 - 14 * q^79 - 11 * q^81 + 14 * q^82 + 3 * q^83 + 4 * q^84 + 8 * q^86 + 10 * q^87 - 14 * q^89 + 2 * q^91 + 2 * q^92 + 10 * q^93 - 4 * q^94 - 16 * q^96 - 14 * q^97 + 12 * 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
−2.00000 −2.00000 2.00000 0 4.00000 −1.00000 0 1.00000 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
$$5$$ $$-1$$
$$23$$ $$-1$$

## 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 575.2.a.a 1
3.b odd 2 1 5175.2.a.z 1
4.b odd 2 1 9200.2.a.bg 1
5.b even 2 1 575.2.a.e 1
5.c odd 4 2 115.2.b.a 2
15.d odd 2 1 5175.2.a.a 1
15.e even 4 2 1035.2.b.a 2
20.d odd 2 1 9200.2.a.g 1
20.e even 4 2 1840.2.e.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.b.a 2 5.c odd 4 2
575.2.a.a 1 1.a even 1 1 trivial
575.2.a.e 1 5.b even 2 1
1035.2.b.a 2 15.e even 4 2
1840.2.e.b 2 20.e even 4 2
5175.2.a.a 1 15.d odd 2 1
5175.2.a.z 1 3.b odd 2 1
9200.2.a.g 1 20.d odd 2 1
9200.2.a.bg 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(575))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{3} + 2$$ T3 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T + 2$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T$$
$13$ $$T + 2$$
$17$ $$T - 5$$
$19$ $$T - 8$$
$23$ $$T - 1$$
$29$ $$T + 5$$
$31$ $$T + 5$$
$37$ $$T + 7$$
$41$ $$T + 7$$
$43$ $$T + 4$$
$47$ $$T - 2$$
$53$ $$T - 1$$
$59$ $$T - 3$$
$61$ $$T + 6$$
$67$ $$T + 13$$
$71$ $$T - 13$$
$73$ $$T + 8$$
$79$ $$T + 14$$
$83$ $$T - 3$$
$89$ $$T + 14$$
$97$ $$T + 14$$