# Properties

 Label 575.2.b.b Level $575$ Weight $2$ Character orbit 575.b Analytic conductor $4.591$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,2,Mod(24,575)]

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([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("575.24");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$575 = 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 575.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$4.59139811622$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{4} - i q^{7} + 3 i q^{8} + 3 q^{9} +O(q^{10})$$ q + i * q^2 + q^4 - i * q^7 + 3*i * q^8 + 3 * q^9 $$q + i q^{2} + q^{4} - i q^{7} + 3 i q^{8} + 3 q^{9} - q^{11} + i q^{13} + q^{14} - q^{16} + 3 i q^{18} + 5 q^{19} - i q^{22} + i q^{23} - q^{26} - i q^{28} + 5 q^{29} - 2 q^{31} + 5 i q^{32} + 3 q^{36} + 4 i q^{37} + 5 i q^{38} - 5 q^{41} - 9 i q^{43} - q^{44} - q^{46} + 6 i q^{47} + 6 q^{49} + i q^{52} + 2 i q^{53} + 3 q^{56} + 5 i q^{58} - 8 q^{59} - 8 q^{61} - 2 i q^{62} - 3 i q^{63} - 7 q^{64} - 8 i q^{67} - 10 q^{71} + 9 i q^{72} - 3 i q^{73} - 4 q^{74} + 5 q^{76} + i q^{77} + 3 q^{79} + 9 q^{81} - 5 i q^{82} + 3 i q^{83} + 9 q^{86} - 3 i q^{88} - 10 q^{89} + q^{91} + i q^{92} - 6 q^{94} + 2 i q^{97} + 6 i q^{98} - 3 q^{99} +O(q^{100})$$ q + i * q^2 + q^4 - i * q^7 + 3*i * q^8 + 3 * q^9 - q^11 + i * q^13 + q^14 - q^16 + 3*i * q^18 + 5 * q^19 - i * q^22 + i * q^23 - q^26 - i * q^28 + 5 * q^29 - 2 * q^31 + 5*i * q^32 + 3 * q^36 + 4*i * q^37 + 5*i * q^38 - 5 * q^41 - 9*i * q^43 - q^44 - q^46 + 6*i * q^47 + 6 * q^49 + i * q^52 + 2*i * q^53 + 3 * q^56 + 5*i * q^58 - 8 * q^59 - 8 * q^61 - 2*i * q^62 - 3*i * q^63 - 7 * q^64 - 8*i * q^67 - 10 * q^71 + 9*i * q^72 - 3*i * q^73 - 4 * q^74 + 5 * q^76 + i * q^77 + 3 * q^79 + 9 * q^81 - 5*i * q^82 + 3*i * q^83 + 9 * q^86 - 3*i * q^88 - 10 * q^89 + q^91 + i * q^92 - 6 * q^94 + 2*i * q^97 + 6*i * q^98 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 + 6 * q^9 $$2 q + 2 q^{4} + 6 q^{9} - 2 q^{11} + 2 q^{14} - 2 q^{16} + 10 q^{19} - 2 q^{26} + 10 q^{29} - 4 q^{31} + 6 q^{36} - 10 q^{41} - 2 q^{44} - 2 q^{46} + 12 q^{49} + 6 q^{56} - 16 q^{59} - 16 q^{61} - 14 q^{64} - 20 q^{71} - 8 q^{74} + 10 q^{76} + 6 q^{79} + 18 q^{81} + 18 q^{86} - 20 q^{89} + 2 q^{91} - 12 q^{94} - 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 + 6 * q^9 - 2 * q^11 + 2 * q^14 - 2 * q^16 + 10 * q^19 - 2 * q^26 + 10 * q^29 - 4 * q^31 + 6 * q^36 - 10 * q^41 - 2 * q^44 - 2 * q^46 + 12 * q^49 + 6 * q^56 - 16 * q^59 - 16 * q^61 - 14 * q^64 - 20 * q^71 - 8 * q^74 + 10 * q^76 + 6 * q^79 + 18 * q^81 + 18 * q^86 - 20 * q^89 + 2 * q^91 - 12 * q^94 - 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/575\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$277$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
24.1
 − 1.00000i 1.00000i
1.00000i 0 1.00000 0 0 1.00000i 3.00000i 3.00000 0
24.2 1.00000i 0 1.00000 0 0 1.00000i 3.00000i 3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.2.b.b 2
5.b even 2 1 inner 575.2.b.b 2
5.c odd 4 1 575.2.a.c 1
5.c odd 4 1 575.2.a.d yes 1
15.e even 4 1 5175.2.a.e 1
15.e even 4 1 5175.2.a.u 1
20.e even 4 1 9200.2.a.r 1
20.e even 4 1 9200.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
575.2.a.c 1 5.c odd 4 1
575.2.a.d yes 1 5.c odd 4 1
575.2.b.b 2 1.a even 1 1 trivial
575.2.b.b 2 5.b even 2 1 inner
5175.2.a.e 1 15.e even 4 1
5175.2.a.u 1 15.e even 4 1
9200.2.a.r 1 20.e even 4 1
9200.2.a.u 1 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(575, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2}$$
$19$ $$(T - 5)^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$(T - 5)^{2}$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T + 5)^{2}$$
$43$ $$T^{2} + 81$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 4$$
$59$ $$(T + 8)^{2}$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T + 10)^{2}$$
$73$ $$T^{2} + 9$$
$79$ $$(T - 3)^{2}$$
$83$ $$T^{2} + 9$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} + 4$$