# Properties

 Label 300.2.e.a Level $300$ Weight $2$ Character orbit 300.e Analytic conductor $2.396$ Analytic rank $0$ Dimension $4$ CM discriminant -15 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,2,Mod(251,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("300.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.e (of order $$2$$, degree $$1$$, 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: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + x^{2} + 4$$ x^4 + x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{3} + \beta_1) q^{3} + \beta_{2} q^{4} + (\beta_{2} + 2) q^{6} + (2 \beta_{3} - \beta_1) q^{8} + 3 q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b3 + b1) * q^3 + b2 * q^4 + (b2 + 2) * q^6 + (2*b3 - b1) * q^8 + 3 * q^9 $$q + \beta_1 q^{2} + ( - \beta_{3} + \beta_1) q^{3} + \beta_{2} q^{4} + (\beta_{2} + 2) q^{6} + (2 \beta_{3} - \beta_1) q^{8} + 3 q^{9} + (2 \beta_{3} + \beta_1) q^{12} + ( - \beta_{2} - 4) q^{16} + (2 \beta_{3} + 2 \beta_1) q^{17} + 3 \beta_1 q^{18} + ( - 4 \beta_{2} - 2) q^{19} + (2 \beta_{3} - 2 \beta_1) q^{23} + (\beta_{2} - 4) q^{24} + ( - 3 \beta_{3} + 3 \beta_1) q^{27} + ( - 4 \beta_{2} - 2) q^{31} + ( - 2 \beta_{3} - 3 \beta_1) q^{32} + (2 \beta_{2} - 4) q^{34} + 3 \beta_{2} q^{36} + ( - 8 \beta_{3} + 2 \beta_1) q^{38} + ( - 2 \beta_{2} - 4) q^{46} + (6 \beta_{3} - 6 \beta_1) q^{47} + (2 \beta_{3} - 5 \beta_1) q^{48} + 7 q^{49} + (4 \beta_{2} + 2) q^{51} + (2 \beta_{3} + 2 \beta_1) q^{53} + (3 \beta_{2} + 6) q^{54} + ( - 6 \beta_{3} - 6 \beta_1) q^{57} - 2 q^{61} + ( - 8 \beta_{3} + 2 \beta_1) q^{62} + ( - 3 \beta_{2} + 4) q^{64} + (4 \beta_{3} - 6 \beta_1) q^{68} - 6 q^{69} + (6 \beta_{3} - 3 \beta_1) q^{72} + (2 \beta_{2} + 16) q^{76} + (4 \beta_{2} + 2) q^{79} + 9 q^{81} + ( - 2 \beta_{3} + 2 \beta_1) q^{83} + ( - 4 \beta_{3} - 2 \beta_1) q^{92} + ( - 6 \beta_{3} - 6 \beta_1) q^{93} + ( - 6 \beta_{2} - 12) q^{94} + ( - 5 \beta_{2} - 4) q^{96} + 7 \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + (-b3 + b1) * q^3 + b2 * q^4 + (b2 + 2) * q^6 + (2*b3 - b1) * q^8 + 3 * q^9 + (2*b3 + b1) * q^12 + (-b2 - 4) * q^16 + (2*b3 + 2*b1) * q^17 + 3*b1 * q^18 + (-4*b2 - 2) * q^19 + (2*b3 - 2*b1) * q^23 + (b2 - 4) * q^24 + (-3*b3 + 3*b1) * q^27 + (-4*b2 - 2) * q^31 + (-2*b3 - 3*b1) * q^32 + (2*b2 - 4) * q^34 + 3*b2 * q^36 + (-8*b3 + 2*b1) * q^38 + (-2*b2 - 4) * q^46 + (6*b3 - 6*b1) * q^47 + (2*b3 - 5*b1) * q^48 + 7 * q^49 + (4*b2 + 2) * q^51 + (2*b3 + 2*b1) * q^53 + (3*b2 + 6) * q^54 + (-6*b3 - 6*b1) * q^57 - 2 * q^61 + (-8*b3 + 2*b1) * q^62 + (-3*b2 + 4) * q^64 + (4*b3 - 6*b1) * q^68 - 6 * q^69 + (6*b3 - 3*b1) * q^72 + (2*b2 + 16) * q^76 + (4*b2 + 2) * q^79 + 9 * q^81 + (-2*b3 + 2*b1) * q^83 + (-4*b3 - 2*b1) * q^92 + (-6*b3 - 6*b1) * q^93 + (-6*b2 - 12) * q^94 + (-5*b2 - 4) * q^96 + 7*b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{4} + 6 q^{6} + 12 q^{9}+O(q^{10})$$ 4 * q - 2 * q^4 + 6 * q^6 + 12 * q^9 $$4 q - 2 q^{4} + 6 q^{6} + 12 q^{9} - 14 q^{16} - 18 q^{24} - 20 q^{34} - 6 q^{36} - 12 q^{46} + 28 q^{49} + 18 q^{54} - 8 q^{61} + 22 q^{64} - 24 q^{69} + 60 q^{76} + 36 q^{81} - 36 q^{94} - 6 q^{96}+O(q^{100})$$ 4 * q - 2 * q^4 + 6 * q^6 + 12 * q^9 - 14 * q^16 - 18 * q^24 - 20 * q^34 - 6 * q^36 - 12 * q^46 + 28 * q^49 + 18 * q^54 - 8 * q^61 + 22 * q^64 - 24 * q^69 + 60 * q^76 + 36 * q^81 - 36 * q^94 - 6 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + x^{2} + 4$$ :

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

## Character values

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

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 −0.866025 − 1.11803i −0.866025 + 1.11803i 0.866025 − 1.11803i 0.866025 + 1.11803i
−0.866025 1.11803i −1.73205 −0.500000 + 1.93649i 0 1.50000 + 1.93649i 0 2.59808 1.11803i 3.00000 0
251.2 −0.866025 + 1.11803i −1.73205 −0.500000 1.93649i 0 1.50000 1.93649i 0 2.59808 + 1.11803i 3.00000 0
251.3 0.866025 1.11803i 1.73205 −0.500000 1.93649i 0 1.50000 1.93649i 0 −2.59808 1.11803i 3.00000 0
251.4 0.866025 + 1.11803i 1.73205 −0.500000 + 1.93649i 0 1.50000 + 1.93649i 0 −2.59808 + 1.11803i 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.e.a 4
3.b odd 2 1 inner 300.2.e.a 4
4.b odd 2 1 inner 300.2.e.a 4
5.b even 2 1 inner 300.2.e.a 4
5.c odd 4 2 60.2.h.b 4
12.b even 2 1 inner 300.2.e.a 4
15.d odd 2 1 CM 300.2.e.a 4
15.e even 4 2 60.2.h.b 4
20.d odd 2 1 inner 300.2.e.a 4
20.e even 4 2 60.2.h.b 4
40.i odd 4 2 960.2.o.a 4
40.k even 4 2 960.2.o.a 4
60.h even 2 1 inner 300.2.e.a 4
60.l odd 4 2 60.2.h.b 4
120.q odd 4 2 960.2.o.a 4
120.w even 4 2 960.2.o.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.h.b 4 5.c odd 4 2
60.2.h.b 4 15.e even 4 2
60.2.h.b 4 20.e even 4 2
60.2.h.b 4 60.l odd 4 2
300.2.e.a 4 1.a even 1 1 trivial
300.2.e.a 4 3.b odd 2 1 inner
300.2.e.a 4 4.b odd 2 1 inner
300.2.e.a 4 5.b even 2 1 inner
300.2.e.a 4 12.b even 2 1 inner
300.2.e.a 4 15.d odd 2 1 CM
300.2.e.a 4 20.d odd 2 1 inner
300.2.e.a 4 60.h even 2 1 inner
960.2.o.a 4 40.i odd 4 2
960.2.o.a 4 40.k even 4 2
960.2.o.a 4 120.q odd 4 2
960.2.o.a 4 120.w even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7}$$ T7 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{2} + 4$$
$3$ $$(T^{2} - 3)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 20)^{2}$$
$19$ $$(T^{2} + 60)^{2}$$
$23$ $$(T^{2} - 12)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T^{2} + 60)^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} - 108)^{2}$$
$53$ $$(T^{2} + 20)^{2}$$
$59$ $$T^{4}$$
$61$ $$(T + 2)^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} + 60)^{2}$$
$83$ $$(T^{2} - 12)^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$