# Properties

 Label 2300.2.c.b Level $2300$ Weight $2$ Character orbit 2300.c Analytic conductor $18.366$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1749,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2300.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2300 = 2^{2} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2300.c (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: $$18.3655924649$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 92) 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} - 4 i q^{7} - 6 q^{9} +O(q^{10})$$ q + 3*i * q^3 - 4*i * q^7 - 6 * q^9 $$q + 3 i q^{3} - 4 i q^{7} - 6 q^{9} + 2 q^{11} + 5 i q^{13} + 4 i q^{17} + 2 q^{19} + 12 q^{21} - i q^{23} - 9 i q^{27} + 7 q^{29} - 3 q^{31} + 6 i q^{33} + 2 i q^{37} - 15 q^{39} - 9 q^{41} + 8 i q^{43} + 9 i q^{47} - 9 q^{49} - 12 q^{51} - 2 i q^{53} + 6 i q^{57} - 2 q^{61} + 24 i q^{63} + 14 i q^{67} + 3 q^{69} - 3 q^{71} + 3 i q^{73} - 8 i q^{77} + 6 q^{79} + 9 q^{81} - 8 i q^{83} + 21 i q^{87} - 12 q^{89} + 20 q^{91} - 9 i q^{93} - 12 q^{99} +O(q^{100})$$ q + 3*i * q^3 - 4*i * q^7 - 6 * q^9 + 2 * q^11 + 5*i * q^13 + 4*i * q^17 + 2 * q^19 + 12 * q^21 - i * q^23 - 9*i * q^27 + 7 * q^29 - 3 * q^31 + 6*i * q^33 + 2*i * q^37 - 15 * q^39 - 9 * q^41 + 8*i * q^43 + 9*i * q^47 - 9 * q^49 - 12 * q^51 - 2*i * q^53 + 6*i * q^57 - 2 * q^61 + 24*i * q^63 + 14*i * q^67 + 3 * q^69 - 3 * q^71 + 3*i * q^73 - 8*i * q^77 + 6 * q^79 + 9 * q^81 - 8*i * q^83 + 21*i * q^87 - 12 * q^89 + 20 * q^91 - 9*i * q^93 - 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 12 q^{9}+O(q^{10})$$ 2 * q - 12 * q^9 $$2 q - 12 q^{9} + 4 q^{11} + 4 q^{19} + 24 q^{21} + 14 q^{29} - 6 q^{31} - 30 q^{39} - 18 q^{41} - 18 q^{49} - 24 q^{51} - 4 q^{61} + 6 q^{69} - 6 q^{71} + 12 q^{79} + 18 q^{81} - 24 q^{89} + 40 q^{91} - 24 q^{99}+O(q^{100})$$ 2 * q - 12 * q^9 + 4 * q^11 + 4 * q^19 + 24 * q^21 + 14 * q^29 - 6 * q^31 - 30 * q^39 - 18 * q^41 - 18 * q^49 - 24 * q^51 - 4 * q^61 + 6 * q^69 - 6 * q^71 + 12 * q^79 + 18 * q^81 - 24 * q^89 + 40 * q^91 - 24 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1749.1
 − 1.00000i 1.00000i
0 3.00000i 0 0 0 4.00000i 0 −6.00000 0
1749.2 0 3.00000i 0 0 0 4.00000i 0 −6.00000 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 2300.2.c.b 2
5.b even 2 1 inner 2300.2.c.b 2
5.c odd 4 1 92.2.a.a 1
5.c odd 4 1 2300.2.a.h 1
15.e even 4 1 828.2.a.c 1
20.e even 4 1 368.2.a.g 1
20.e even 4 1 9200.2.a.b 1
35.f even 4 1 4508.2.a.d 1
40.i odd 4 1 1472.2.a.n 1
40.k even 4 1 1472.2.a.b 1
60.l odd 4 1 3312.2.a.q 1
115.e even 4 1 2116.2.a.a 1
460.k odd 4 1 8464.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.2.a.a 1 5.c odd 4 1
368.2.a.g 1 20.e even 4 1
828.2.a.c 1 15.e even 4 1
1472.2.a.b 1 40.k even 4 1
1472.2.a.n 1 40.i odd 4 1
2116.2.a.a 1 115.e even 4 1
2300.2.a.h 1 5.c odd 4 1
2300.2.c.b 2 1.a even 1 1 trivial
2300.2.c.b 2 5.b even 2 1 inner
3312.2.a.q 1 60.l odd 4 1
4508.2.a.d 1 35.f even 4 1
8464.2.a.s 1 460.k odd 4 1
9200.2.a.b 1 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2300, [\chi])$$:

 $$T_{3}^{2} + 9$$ T3^2 + 9 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11} - 2$$ T11 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 25$$
$17$ $$T^{2} + 16$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$(T - 7)^{2}$$
$31$ $$(T + 3)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 9)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 81$$
$53$ $$T^{2} + 4$$
$59$ $$T^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 196$$
$71$ $$(T + 3)^{2}$$
$73$ $$T^{2} + 9$$
$79$ $$(T - 6)^{2}$$
$83$ $$T^{2} + 64$$
$89$ $$(T + 12)^{2}$$
$97$ $$T^{2}$$