# Properties

 Label 300.2.a.a Level $300$ Weight $2$ Character orbit 300.a Self dual yes Analytic conductor $2.396$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,2,Mod(1,300)]

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([0, 0, 0]))

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

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

chi := DirichletCharacter("300.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.a (trivial)

## 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: yes Analytic conductor: $$2.39551206064$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 60) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{3} - 4 q^{7} + q^{9}+O(q^{10})$$ q - q^3 - 4 * q^7 + q^9 $$q - q^{3} - 4 q^{7} + q^{9} - 4 q^{11} - 4 q^{17} + 4 q^{21} - 4 q^{23} - q^{27} - 6 q^{29} + 4 q^{31} + 4 q^{33} + 8 q^{37} - 10 q^{41} - 4 q^{43} + 4 q^{47} + 9 q^{49} + 4 q^{51} + 12 q^{53} + 4 q^{59} + 2 q^{61} - 4 q^{63} + 4 q^{67} + 4 q^{69} + 8 q^{73} + 16 q^{77} - 12 q^{79} + q^{81} - 4 q^{83} + 6 q^{87} - 10 q^{89} - 4 q^{93} - 8 q^{97} - 4 q^{99}+O(q^{100})$$ q - q^3 - 4 * q^7 + q^9 - 4 * q^11 - 4 * q^17 + 4 * q^21 - 4 * q^23 - q^27 - 6 * q^29 + 4 * q^31 + 4 * q^33 + 8 * q^37 - 10 * q^41 - 4 * q^43 + 4 * q^47 + 9 * q^49 + 4 * q^51 + 12 * q^53 + 4 * q^59 + 2 * q^61 - 4 * q^63 + 4 * q^67 + 4 * q^69 + 8 * q^73 + 16 * q^77 - 12 * q^79 + q^81 - 4 * q^83 + 6 * q^87 - 10 * q^89 - 4 * q^93 - 8 * q^97 - 4 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −1.00000 0 0 0 −4.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.a.a 1
3.b odd 2 1 900.2.a.a 1
4.b odd 2 1 1200.2.a.s 1
5.b even 2 1 300.2.a.d 1
5.c odd 4 2 60.2.d.a 2
8.b even 2 1 4800.2.a.bn 1
8.d odd 2 1 4800.2.a.bf 1
12.b even 2 1 3600.2.a.bm 1
15.d odd 2 1 900.2.a.h 1
15.e even 4 2 180.2.d.a 2
20.d odd 2 1 1200.2.a.a 1
20.e even 4 2 240.2.f.b 2
35.f even 4 2 2940.2.k.c 2
35.k even 12 4 2940.2.bb.e 4
35.l odd 12 4 2940.2.bb.d 4
40.e odd 2 1 4800.2.a.bk 1
40.f even 2 1 4800.2.a.bj 1
40.i odd 4 2 960.2.f.f 2
40.k even 4 2 960.2.f.c 2
45.k odd 12 4 1620.2.r.c 4
45.l even 12 4 1620.2.r.d 4
60.h even 2 1 3600.2.a.d 1
60.l odd 4 2 720.2.f.c 2
80.i odd 4 2 3840.2.d.r 2
80.j even 4 2 3840.2.d.be 2
80.s even 4 2 3840.2.d.b 2
80.t odd 4 2 3840.2.d.o 2
120.q odd 4 2 2880.2.f.p 2
120.w even 4 2 2880.2.f.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.d.a 2 5.c odd 4 2
180.2.d.a 2 15.e even 4 2
240.2.f.b 2 20.e even 4 2
300.2.a.a 1 1.a even 1 1 trivial
300.2.a.d 1 5.b even 2 1
720.2.f.c 2 60.l odd 4 2
900.2.a.a 1 3.b odd 2 1
900.2.a.h 1 15.d odd 2 1
960.2.f.c 2 40.k even 4 2
960.2.f.f 2 40.i odd 4 2
1200.2.a.a 1 20.d odd 2 1
1200.2.a.s 1 4.b odd 2 1
1620.2.r.c 4 45.k odd 12 4
1620.2.r.d 4 45.l even 12 4
2880.2.f.l 2 120.w even 4 2
2880.2.f.p 2 120.q odd 4 2
2940.2.k.c 2 35.f even 4 2
2940.2.bb.d 4 35.l odd 12 4
2940.2.bb.e 4 35.k even 12 4
3600.2.a.d 1 60.h even 2 1
3600.2.a.bm 1 12.b even 2 1
3840.2.d.b 2 80.s even 4 2
3840.2.d.o 2 80.t odd 4 2
3840.2.d.r 2 80.i odd 4 2
3840.2.d.be 2 80.j even 4 2
4800.2.a.bf 1 8.d odd 2 1
4800.2.a.bj 1 40.f even 2 1
4800.2.a.bk 1 40.e odd 2 1
4800.2.a.bn 1 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} + 4$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(300))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T + 4$$
$11$ $$T + 4$$
$13$ $$T$$
$17$ $$T + 4$$
$19$ $$T$$
$23$ $$T + 4$$
$29$ $$T + 6$$
$31$ $$T - 4$$
$37$ $$T - 8$$
$41$ $$T + 10$$
$43$ $$T + 4$$
$47$ $$T - 4$$
$53$ $$T - 12$$
$59$ $$T - 4$$
$61$ $$T - 2$$
$67$ $$T - 4$$
$71$ $$T$$
$73$ $$T - 8$$
$79$ $$T + 12$$
$83$ $$T + 4$$
$89$ $$T + 10$$
$97$ $$T + 8$$