# Properties

 Label 300.9.b.b Level $300$ Weight $9$ Character orbit 300.b Analytic conductor $122.214$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,9,Mod(149,300)]

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, 1, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("300.149");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 300.b (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: $$122.213583018$$ 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 12) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 81 i q^{3} + 4034 i q^{7} - 6561 q^{9} +O(q^{10})$$ q - 81*i * q^3 + 4034*i * q^7 - 6561 * q^9 $$q - 81 i q^{3} + 4034 i q^{7} - 6561 q^{9} + 35806 i q^{13} + 258526 q^{19} + 326754 q^{21} + 531441 i q^{27} - 1809406 q^{31} + 503522 i q^{37} + 2900286 q^{39} - 3492194 i q^{43} - 10508355 q^{49} - 20940606 i q^{57} - 23826526 q^{61} - 26467074 i q^{63} - 5421406 i q^{67} - 16169282 i q^{73} + 18887038 q^{79} + 43046721 q^{81} - 144441404 q^{91} + 146561886 i q^{93} + 176908034 i q^{97} +O(q^{100})$$ q - 81*i * q^3 + 4034*i * q^7 - 6561 * q^9 + 35806*i * q^13 + 258526 * q^19 + 326754 * q^21 + 531441*i * q^27 - 1809406 * q^31 + 503522*i * q^37 + 2900286 * q^39 - 3492194*i * q^43 - 10508355 * q^49 - 20940606*i * q^57 - 23826526 * q^61 - 26467074*i * q^63 - 5421406*i * q^67 - 16169282*i * q^73 + 18887038 * q^79 + 43046721 * q^81 - 144441404 * q^91 + 146561886*i * q^93 + 176908034*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 13122 q^{9}+O(q^{10})$$ 2 * q - 13122 * q^9 $$2 q - 13122 q^{9} + 517052 q^{19} + 653508 q^{21} - 3618812 q^{31} + 5800572 q^{39} - 21016710 q^{49} - 47653052 q^{61} + 37774076 q^{79} + 86093442 q^{81} - 288882808 q^{91}+O(q^{100})$$ 2 * q - 13122 * q^9 + 517052 * q^19 + 653508 * q^21 - 3618812 * q^31 + 5800572 * q^39 - 21016710 * q^49 - 47653052 * q^61 + 37774076 * q^79 + 86093442 * q^81 - 288882808 * q^91

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 1.00000i − 1.00000i
0 81.0000i 0 0 0 4034.00i 0 −6561.00 0
149.2 0 81.0000i 0 0 0 4034.00i 0 −6561.00 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
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.9.b.b 2
3.b odd 2 1 CM 300.9.b.b 2
5.b even 2 1 inner 300.9.b.b 2
5.c odd 4 1 12.9.c.a 1
5.c odd 4 1 300.9.g.a 1
15.d odd 2 1 inner 300.9.b.b 2
15.e even 4 1 12.9.c.a 1
15.e even 4 1 300.9.g.a 1
20.e even 4 1 48.9.e.a 1
40.i odd 4 1 192.9.e.a 1
40.k even 4 1 192.9.e.b 1
45.k odd 12 2 324.9.g.a 2
45.l even 12 2 324.9.g.a 2
60.l odd 4 1 48.9.e.a 1
120.q odd 4 1 192.9.e.b 1
120.w even 4 1 192.9.e.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.9.c.a 1 5.c odd 4 1
12.9.c.a 1 15.e even 4 1
48.9.e.a 1 20.e even 4 1
48.9.e.a 1 60.l odd 4 1
192.9.e.a 1 40.i odd 4 1
192.9.e.a 1 120.w even 4 1
192.9.e.b 1 40.k even 4 1
192.9.e.b 1 120.q odd 4 1
300.9.b.b 2 1.a even 1 1 trivial
300.9.b.b 2 3.b odd 2 1 CM
300.9.b.b 2 5.b even 2 1 inner
300.9.b.b 2 15.d odd 2 1 inner
300.9.g.a 1 5.c odd 4 1
300.9.g.a 1 15.e even 4 1
324.9.g.a 2 45.k odd 12 2
324.9.g.a 2 45.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 16273156$$ acting on $$S_{9}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 6561$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16273156$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 1282069636$$
$17$ $$T^{2}$$
$19$ $$(T - 258526)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T + 1809406)^{2}$$
$37$ $$T^{2} + 253534404484$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 12195418933636$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 23826526)^{2}$$
$67$ $$T^{2} + 29391643016836$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 261445680395524$$
$79$ $$(T - 18887038)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 31\!\cdots\!56$$