# Properties

 Label 300.2.i.b Level $300$ Weight $2$ Character orbit 300.i Analytic conductor $2.396$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,2,Mod(257,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1]))

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

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

chi := DirichletCharacter("300.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.i (of order $$4$$, degree $$2$$, 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: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + 2 \beta_{3} q^{7} + 3 \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^3 + 2*b3 * q^7 + 3*b2 * q^9 $$q + \beta_1 q^{3} + 2 \beta_{3} q^{7} + 3 \beta_{2} q^{9} + 4 \beta_1 q^{13} - 8 \beta_{2} q^{19} - 6 q^{21} + 3 \beta_{3} q^{27} + 4 q^{31} - 4 \beta_{3} q^{37} + 12 \beta_{2} q^{39} - 6 \beta_1 q^{43} - 5 \beta_{2} q^{49} - 8 \beta_{3} q^{57} + 14 q^{61} - 6 \beta_1 q^{63} + 2 \beta_{3} q^{67} - 8 \beta_1 q^{73} - 4 \beta_{2} q^{79} - 9 q^{81} - 24 q^{91} + 4 \beta_1 q^{93} - 8 \beta_{3} q^{97}+O(q^{100})$$ q + b1 * q^3 + 2*b3 * q^7 + 3*b2 * q^9 + 4*b1 * q^13 - 8*b2 * q^19 - 6 * q^21 + 3*b3 * q^27 + 4 * q^31 - 4*b3 * q^37 + 12*b2 * q^39 - 6*b1 * q^43 - 5*b2 * q^49 - 8*b3 * q^57 + 14 * q^61 - 6*b1 * q^63 + 2*b3 * q^67 - 8*b1 * q^73 - 4*b2 * q^79 - 9 * q^81 - 24 * q^91 + 4*b1 * q^93 - 8*b3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 24 q^{21} + 16 q^{31} + 56 q^{61} - 36 q^{81} - 96 q^{91}+O(q^{100})$$ 4 * q - 24 * q^21 + 16 * q^31 + 56 * q^61 - 36 * q^81 - 96 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 3$$ (v^3) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$3\beta_{3}$$ 3*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 −1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i
0 −1.22474 + 1.22474i 0 0 0 2.44949 + 2.44949i 0 3.00000i 0
257.2 0 1.22474 1.22474i 0 0 0 −2.44949 2.44949i 0 3.00000i 0
293.1 0 −1.22474 1.22474i 0 0 0 2.44949 2.44949i 0 3.00000i 0
293.2 0 1.22474 + 1.22474i 0 0 0 −2.44949 + 2.44949i 0 3.00000i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.i.b 4
3.b odd 2 1 CM 300.2.i.b 4
4.b odd 2 1 1200.2.v.e 4
5.b even 2 1 inner 300.2.i.b 4
5.c odd 4 2 inner 300.2.i.b 4
12.b even 2 1 1200.2.v.e 4
15.d odd 2 1 inner 300.2.i.b 4
15.e even 4 2 inner 300.2.i.b 4
20.d odd 2 1 1200.2.v.e 4
20.e even 4 2 1200.2.v.e 4
60.h even 2 1 1200.2.v.e 4
60.l odd 4 2 1200.2.v.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.i.b 4 1.a even 1 1 trivial
300.2.i.b 4 3.b odd 2 1 CM
300.2.i.b 4 5.b even 2 1 inner
300.2.i.b 4 5.c odd 4 2 inner
300.2.i.b 4 15.d odd 2 1 inner
300.2.i.b 4 15.e even 4 2 inner
1200.2.v.e 4 4.b odd 2 1
1200.2.v.e 4 12.b even 2 1
1200.2.v.e 4 20.d odd 2 1
1200.2.v.e 4 20.e even 4 2
1200.2.v.e 4 60.h even 2 1
1200.2.v.e 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 144$$ acting on $$S_{2}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 144$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 2304$$
$17$ $$T^{4}$$
$19$ $$(T^{2} + 64)^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$(T - 4)^{4}$$
$37$ $$T^{4} + 2304$$
$41$ $$T^{4}$$
$43$ $$T^{4} + 11664$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T - 14)^{4}$$
$67$ $$T^{4} + 144$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 36864$$
$79$ $$(T^{2} + 16)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} + 36864$$