# Properties

 Label 300.2.e.b Level $300$ Weight $2$ Character orbit 300.e Analytic conductor $2.396$ Analytic rank $0$ Dimension $4$ CM discriminant -20 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: $$\Q(\sqrt{2}, \sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 4x^{2} + 9$$ x^4 + 4*x^2 + 9 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( - \beta_{2} - \beta_1) q^{3} + 2 q^{4} + (\beta_{3} - 1) q^{6} + ( - \beta_{2} - 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} + ( - \beta_{3} - 2) q^{9}+O(q^{10})$$ q + b2 * q^2 + (-b2 - b1) * q^3 + 2 * q^4 + (b3 - 1) * q^6 + (-b2 - 2*b1) * q^7 + 2*b2 * q^8 + (-b3 - 2) * q^9 $$q + \beta_{2} q^{2} + ( - \beta_{2} - \beta_1) q^{3} + 2 q^{4} + (\beta_{3} - 1) q^{6} + ( - \beta_{2} - 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} + ( - \beta_{3} - 2) q^{9} + ( - 2 \beta_{2} - 2 \beta_1) q^{12} + 2 \beta_{3} q^{14} + 4 q^{16} + ( - \beta_{2} + 2 \beta_1) q^{18} + ( - \beta_{3} - 5) q^{21} + \beta_{2} q^{23} + (2 \beta_{3} - 2) q^{24} + (4 \beta_{2} + \beta_1) q^{27} + ( - 2 \beta_{2} - 4 \beta_1) q^{28} - 4 \beta_{3} q^{29} + 4 \beta_{2} q^{32} + ( - 2 \beta_{3} - 4) q^{36} - 2 \beta_{3} q^{41} + ( - 4 \beta_{2} + 2 \beta_1) q^{42} + (\beta_{2} + 2 \beta_1) q^{43} + 2 q^{46} - 7 \beta_{2} q^{47} + ( - 4 \beta_{2} - 4 \beta_1) q^{48} - 3 q^{49} + ( - \beta_{3} + 7) q^{54} + 4 \beta_{3} q^{56} + (4 \beta_{2} + 8 \beta_1) q^{58} + 8 q^{61} + (7 \beta_{2} + 4 \beta_1) q^{63} + 8 q^{64} + (5 \beta_{2} + 10 \beta_1) q^{67} + (\beta_{3} - 1) q^{69} + ( - 2 \beta_{2} + 4 \beta_1) q^{72} + (4 \beta_{3} - 1) q^{81} + (2 \beta_{2} + 4 \beta_1) q^{82} - 11 \beta_{2} q^{83} + ( - 2 \beta_{3} - 10) q^{84} - 2 \beta_{3} q^{86} + (8 \beta_{2} - 4 \beta_1) q^{87} + 8 \beta_{3} q^{89} + 2 \beta_{2} q^{92} - 14 q^{94} + (4 \beta_{3} - 4) q^{96} - 3 \beta_{2} q^{98}+O(q^{100})$$ q + b2 * q^2 + (-b2 - b1) * q^3 + 2 * q^4 + (b3 - 1) * q^6 + (-b2 - 2*b1) * q^7 + 2*b2 * q^8 + (-b3 - 2) * q^9 + (-2*b2 - 2*b1) * q^12 + 2*b3 * q^14 + 4 * q^16 + (-b2 + 2*b1) * q^18 + (-b3 - 5) * q^21 + b2 * q^23 + (2*b3 - 2) * q^24 + (4*b2 + b1) * q^27 + (-2*b2 - 4*b1) * q^28 - 4*b3 * q^29 + 4*b2 * q^32 + (-2*b3 - 4) * q^36 - 2*b3 * q^41 + (-4*b2 + 2*b1) * q^42 + (b2 + 2*b1) * q^43 + 2 * q^46 - 7*b2 * q^47 + (-4*b2 - 4*b1) * q^48 - 3 * q^49 + (-b3 + 7) * q^54 + 4*b3 * q^56 + (4*b2 + 8*b1) * q^58 + 8 * q^61 + (7*b2 + 4*b1) * q^63 + 8 * q^64 + (5*b2 + 10*b1) * q^67 + (b3 - 1) * q^69 + (-2*b2 + 4*b1) * q^72 + (4*b3 - 1) * q^81 + (2*b2 + 4*b1) * q^82 - 11*b2 * q^83 + (-2*b3 - 10) * q^84 - 2*b3 * q^86 + (8*b2 - 4*b1) * q^87 + 8*b3 * q^89 + 2*b2 * q^92 - 14 * q^94 + (4*b3 - 4) * q^96 - 3*b2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4} - 4 q^{6} - 8 q^{9}+O(q^{10})$$ 4 * q + 8 * q^4 - 4 * q^6 - 8 * q^9 $$4 q + 8 q^{4} - 4 q^{6} - 8 q^{9} + 16 q^{16} - 20 q^{21} - 8 q^{24} - 16 q^{36} + 8 q^{46} - 12 q^{49} + 28 q^{54} + 32 q^{61} + 32 q^{64} - 4 q^{69} - 4 q^{81} - 40 q^{84} - 56 q^{94} - 16 q^{96}+O(q^{100})$$ 4 * q + 8 * q^4 - 4 * q^6 - 8 * q^9 + 16 * q^16 - 20 * q^21 - 8 * q^24 - 16 * q^36 + 8 * q^46 - 12 * q^49 + 28 * q^54 + 32 * q^61 + 32 * q^64 - 4 * q^69 - 4 * q^81 - 40 * q^84 - 56 * q^94 - 16 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + \nu ) / 3$$ (v^3 + v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 2$$ b3 - 2 $$\nu^{3}$$ $$=$$ $$3\beta_{2} - \beta_1$$ 3*b2 - 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
 0.707107 + 1.58114i 0.707107 − 1.58114i −0.707107 + 1.58114i −0.707107 − 1.58114i
−1.41421 0.707107 1.58114i 2.00000 0 −1.00000 + 2.23607i 3.16228i −2.82843 −2.00000 2.23607i 0
251.2 −1.41421 0.707107 + 1.58114i 2.00000 0 −1.00000 2.23607i 3.16228i −2.82843 −2.00000 + 2.23607i 0
251.3 1.41421 −0.707107 1.58114i 2.00000 0 −1.00000 2.23607i 3.16228i 2.82843 −2.00000 + 2.23607i 0
251.4 1.41421 −0.707107 + 1.58114i 2.00000 0 −1.00000 + 2.23607i 3.16228i 2.82843 −2.00000 2.23607i 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.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.b 4
3.b odd 2 1 inner 300.2.e.b 4
4.b odd 2 1 inner 300.2.e.b 4
5.b even 2 1 inner 300.2.e.b 4
5.c odd 4 2 60.2.h.a 4
12.b even 2 1 inner 300.2.e.b 4
15.d odd 2 1 inner 300.2.e.b 4
15.e even 4 2 60.2.h.a 4
20.d odd 2 1 CM 300.2.e.b 4
20.e even 4 2 60.2.h.a 4
40.i odd 4 2 960.2.o.c 4
40.k even 4 2 960.2.o.c 4
60.h even 2 1 inner 300.2.e.b 4
60.l odd 4 2 60.2.h.a 4
120.q odd 4 2 960.2.o.c 4
120.w even 4 2 960.2.o.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.h.a 4 5.c odd 4 2
60.2.h.a 4 15.e even 4 2
60.2.h.a 4 20.e even 4 2
60.2.h.a 4 60.l odd 4 2
300.2.e.b 4 1.a even 1 1 trivial
300.2.e.b 4 3.b odd 2 1 inner
300.2.e.b 4 4.b odd 2 1 inner
300.2.e.b 4 5.b even 2 1 inner
300.2.e.b 4 12.b even 2 1 inner
300.2.e.b 4 15.d odd 2 1 inner
300.2.e.b 4 20.d odd 2 1 CM
300.2.e.b 4 60.h even 2 1 inner
960.2.o.c 4 40.i odd 4 2
960.2.o.c 4 40.k even 4 2
960.2.o.c 4 120.q odd 4 2
960.2.o.c 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 $$S_{2}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7}^{2} + 10$$ T7^2 + 10 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{2}$$
$3$ $$T^{4} + 4T^{2} + 9$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 10)^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} - 2)^{2}$$
$29$ $$(T^{2} + 80)^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 20)^{2}$$
$43$ $$(T^{2} + 10)^{2}$$
$47$ $$(T^{2} - 98)^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T - 8)^{4}$$
$67$ $$(T^{2} + 250)^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} - 242)^{2}$$
$89$ $$(T^{2} + 320)^{2}$$
$97$ $$T^{4}$$