# Properties

 Label 300.3.b.d Level $300$ Weight $3$ Character orbit 300.b Analytic conductor $8.174$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,3,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, 3, names="a")

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

chi := DirichletCharacter("300.149");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ 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: $$8.17440793081$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{35})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 17x^{2} + 81$$ x^4 - 17*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - 8 \beta_{2} q^{7} + (\beta_{3} + 9) q^{9}+O(q^{10})$$ q + b1 * q^3 - 8*b2 * q^7 + (b3 + 9) * q^9 $$q + \beta_1 q^{3} - 8 \beta_{2} q^{7} + (\beta_{3} + 9) q^{9} + ( - 6 \beta_{3} - 3) q^{11} + 2 \beta_{2} q^{13} + ( - 3 \beta_{2} + 6 \beta_1) q^{17} - 11 q^{19} - 8 \beta_{3} q^{21} + ( - 6 \beta_{2} + 12 \beta_1) q^{23} + (9 \beta_{2} + 8 \beta_1) q^{27} + (12 \beta_{3} + 6) q^{29} - 46 q^{31} + ( - 54 \beta_{2} + 3 \beta_1) q^{33} + 16 \beta_{2} q^{37} + 2 \beta_{3} q^{39} + ( - 18 \beta_{3} - 9) q^{41} + 62 \beta_{2} q^{43} + ( - 6 \beta_{2} + 12 \beta_1) q^{47} - 15 q^{49} + (3 \beta_{3} + 54) q^{51} + (6 \beta_{2} - 12 \beta_1) q^{53} - 11 \beta_1 q^{57} + ( - 24 \beta_{3} - 12) q^{59} - 16 q^{61} + ( - 72 \beta_{2} + 8 \beta_1) q^{63} - 113 \beta_{2} q^{67} + (6 \beta_{3} + 108) q^{69} + (36 \beta_{3} + 18) q^{71} + 101 \beta_{2} q^{73} + (24 \beta_{2} - 48 \beta_1) q^{77} - 68 q^{79} + (17 \beta_{3} + 72) q^{81} + (3 \beta_{2} - 6 \beta_1) q^{83} + (108 \beta_{2} - 6 \beta_1) q^{87} + (18 \beta_{3} + 9) q^{89} + 16 q^{91} - 46 \beta_1 q^{93} + 22 \beta_{2} q^{97} + ( - 51 \beta_{3} + 27) q^{99}+O(q^{100})$$ q + b1 * q^3 - 8*b2 * q^7 + (b3 + 9) * q^9 + (-6*b3 - 3) * q^11 + 2*b2 * q^13 + (-3*b2 + 6*b1) * q^17 - 11 * q^19 - 8*b3 * q^21 + (-6*b2 + 12*b1) * q^23 + (9*b2 + 8*b1) * q^27 + (12*b3 + 6) * q^29 - 46 * q^31 + (-54*b2 + 3*b1) * q^33 + 16*b2 * q^37 + 2*b3 * q^39 + (-18*b3 - 9) * q^41 + 62*b2 * q^43 + (-6*b2 + 12*b1) * q^47 - 15 * q^49 + (3*b3 + 54) * q^51 + (6*b2 - 12*b1) * q^53 - 11*b1 * q^57 + (-24*b3 - 12) * q^59 - 16 * q^61 + (-72*b2 + 8*b1) * q^63 - 113*b2 * q^67 + (6*b3 + 108) * q^69 + (36*b3 + 18) * q^71 + 101*b2 * q^73 + (24*b2 - 48*b1) * q^77 - 68 * q^79 + (17*b3 + 72) * q^81 + (3*b2 - 6*b1) * q^83 + (108*b2 - 6*b1) * q^87 + (18*b3 + 9) * q^89 + 16 * q^91 - 46*b1 * q^93 + 22*b2 * q^97 + (-51*b3 + 27) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 34 q^{9}+O(q^{10})$$ 4 * q + 34 * q^9 $$4 q + 34 q^{9} - 44 q^{19} + 16 q^{21} - 184 q^{31} - 4 q^{39} - 60 q^{49} + 210 q^{51} - 64 q^{61} + 420 q^{69} - 272 q^{79} + 254 q^{81} + 64 q^{91} + 210 q^{99}+O(q^{100})$$ 4 * q + 34 * q^9 - 44 * q^19 + 16 * q^21 - 184 * q^31 - 4 * q^39 - 60 * q^49 + 210 * q^51 - 64 * q^61 + 420 * q^69 - 272 * q^79 + 254 * q^81 + 64 * q^91 + 210 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 17x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 8\nu ) / 9$$ (v^3 - 8*v) / 9 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 9$$ v^2 - 9
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 9$$ b3 + 9 $$\nu^{3}$$ $$=$$ $$9\beta_{2} + 8\beta_1$$ 9*b2 + 8*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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 −2.95804 − 0.500000i −2.95804 + 0.500000i 2.95804 − 0.500000i 2.95804 + 0.500000i
0 −2.95804 0.500000i 0 0 0 8.00000i 0 8.50000 + 2.95804i 0
149.2 0 −2.95804 + 0.500000i 0 0 0 8.00000i 0 8.50000 2.95804i 0
149.3 0 2.95804 0.500000i 0 0 0 8.00000i 0 8.50000 2.95804i 0
149.4 0 2.95804 + 0.500000i 0 0 0 8.00000i 0 8.50000 + 2.95804i 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 inner
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 300.3.b.d 4
3.b odd 2 1 inner 300.3.b.d 4
4.b odd 2 1 1200.3.c.j 4
5.b even 2 1 inner 300.3.b.d 4
5.c odd 4 1 300.3.g.f 2
5.c odd 4 1 300.3.g.g yes 2
12.b even 2 1 1200.3.c.j 4
15.d odd 2 1 inner 300.3.b.d 4
15.e even 4 1 300.3.g.f 2
15.e even 4 1 300.3.g.g yes 2
20.d odd 2 1 1200.3.c.j 4
20.e even 4 1 1200.3.l.k 2
20.e even 4 1 1200.3.l.m 2
60.h even 2 1 1200.3.c.j 4
60.l odd 4 1 1200.3.l.k 2
60.l odd 4 1 1200.3.l.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.b.d 4 1.a even 1 1 trivial
300.3.b.d 4 3.b odd 2 1 inner
300.3.b.d 4 5.b even 2 1 inner
300.3.b.d 4 15.d odd 2 1 inner
300.3.g.f 2 5.c odd 4 1
300.3.g.f 2 15.e even 4 1
300.3.g.g yes 2 5.c odd 4 1
300.3.g.g yes 2 15.e even 4 1
1200.3.c.j 4 4.b odd 2 1
1200.3.c.j 4 12.b even 2 1
1200.3.c.j 4 20.d odd 2 1
1200.3.c.j 4 60.h even 2 1
1200.3.l.k 2 20.e even 4 1
1200.3.l.k 2 60.l odd 4 1
1200.3.l.m 2 20.e even 4 1
1200.3.l.m 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7}^{2} + 64$$ T7^2 + 64 $$T_{11}^{2} + 315$$ T11^2 + 315

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 17T^{2} + 81$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 64)^{2}$$
$11$ $$(T^{2} + 315)^{2}$$
$13$ $$(T^{2} + 4)^{2}$$
$17$ $$(T^{2} - 315)^{2}$$
$19$ $$(T + 11)^{4}$$
$23$ $$(T^{2} - 1260)^{2}$$
$29$ $$(T^{2} + 1260)^{2}$$
$31$ $$(T + 46)^{4}$$
$37$ $$(T^{2} + 256)^{2}$$
$41$ $$(T^{2} + 2835)^{2}$$
$43$ $$(T^{2} + 3844)^{2}$$
$47$ $$(T^{2} - 1260)^{2}$$
$53$ $$(T^{2} - 1260)^{2}$$
$59$ $$(T^{2} + 5040)^{2}$$
$61$ $$(T + 16)^{4}$$
$67$ $$(T^{2} + 12769)^{2}$$
$71$ $$(T^{2} + 11340)^{2}$$
$73$ $$(T^{2} + 10201)^{2}$$
$79$ $$(T + 68)^{4}$$
$83$ $$(T^{2} - 315)^{2}$$
$89$ $$(T^{2} + 2835)^{2}$$
$97$ $$(T^{2} + 484)^{2}$$