# Properties

 Label 300.3.b.c 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: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 60) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_1) q^{3} + \beta_1 q^{7} + ( - 2 \beta_{2} + 1) q^{9}+O(q^{10})$$ q + (b3 - b1) * q^3 + b1 * q^7 + (-2*b2 + 1) * q^9 $$q + (\beta_{3} - \beta_1) q^{3} + \beta_1 q^{7} + ( - 2 \beta_{2} + 1) q^{9} - 3 \beta_{2} q^{11} - 4 \beta_1 q^{13} + 6 \beta_{3} q^{17} + 34 q^{19} + (\beta_{2} + 4) q^{21} - 18 \beta_{3} q^{23} + ( - 7 \beta_{3} - 11 \beta_1) q^{27} - 9 \beta_{2} q^{29} + 14 q^{31} + ( - 12 \beta_{3} - 15 \beta_1) q^{33} + 28 \beta_1 q^{37} + ( - 4 \beta_{2} - 16) q^{39} + 6 \beta_{2} q^{41} - 4 \beta_1 q^{43} - 18 \beta_{3} q^{47} + 45 q^{49} + ( - 6 \beta_{2} + 30) q^{51} + 18 \beta_{3} q^{53} + (34 \beta_{3} - 34 \beta_1) q^{57} + 3 \beta_{2} q^{59} - 46 q^{61} + (8 \beta_{3} + \beta_1) q^{63} + 16 \beta_1 q^{67} + (18 \beta_{2} - 90) q^{69} - 12 \beta_{2} q^{71} + 53 \beta_1 q^{73} + 12 \beta_{3} q^{77} + 22 q^{79} + ( - 4 \beta_{2} - 79) q^{81} + 54 \beta_{3} q^{83} + ( - 36 \beta_{3} - 45 \beta_1) q^{87} + 24 \beta_{2} q^{89} + 16 q^{91} + (14 \beta_{3} - 14 \beta_1) q^{93} + 61 \beta_1 q^{97} + ( - 3 \beta_{2} - 120) q^{99}+O(q^{100})$$ q + (b3 - b1) * q^3 + b1 * q^7 + (-2*b2 + 1) * q^9 - 3*b2 * q^11 - 4*b1 * q^13 + 6*b3 * q^17 + 34 * q^19 + (b2 + 4) * q^21 - 18*b3 * q^23 + (-7*b3 - 11*b1) * q^27 - 9*b2 * q^29 + 14 * q^31 + (-12*b3 - 15*b1) * q^33 + 28*b1 * q^37 + (-4*b2 - 16) * q^39 + 6*b2 * q^41 - 4*b1 * q^43 - 18*b3 * q^47 + 45 * q^49 + (-6*b2 + 30) * q^51 + 18*b3 * q^53 + (34*b3 - 34*b1) * q^57 + 3*b2 * q^59 - 46 * q^61 + (8*b3 + b1) * q^63 + 16*b1 * q^67 + (18*b2 - 90) * q^69 - 12*b2 * q^71 + 53*b1 * q^73 + 12*b3 * q^77 + 22 * q^79 + (-4*b2 - 79) * q^81 + 54*b3 * q^83 + (-36*b3 - 45*b1) * q^87 + 24*b2 * q^89 + 16 * q^91 + (14*b3 - 14*b1) * q^93 + 61*b1 * q^97 + (-3*b2 - 120) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^9 $$4 q + 4 q^{9} + 136 q^{19} + 16 q^{21} + 56 q^{31} - 64 q^{39} + 180 q^{49} + 120 q^{51} - 184 q^{61} - 360 q^{69} + 88 q^{79} - 316 q^{81} + 64 q^{91} - 480 q^{99}+O(q^{100})$$ 4 * q + 4 * q^9 + 136 * q^19 + 16 * q^21 + 56 * q^31 - 64 * q^39 + 180 * q^49 + 120 * q^51 - 184 * q^61 - 360 * q^69 + 88 * q^79 - 316 * q^81 + 64 * q^91 - 480 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu^{3} + 4\nu$$ 2*v^3 + 4*v $$\beta_{2}$$ $$=$$ $$2\nu^{3} + 8\nu$$ 2*v^3 + 8*v $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 3$$ 2*v^2 + 3
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 4$$ (b2 - b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 3 ) / 2$$ (b3 - 3) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{2} + 2\beta_1 ) / 2$$ (-b2 + 2*b1) / 2

## 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
 − 1.61803i 1.61803i 0.618034i − 0.618034i
0 −2.23607 2.00000i 0 0 0 2.00000i 0 1.00000 + 8.94427i 0
149.2 0 −2.23607 + 2.00000i 0 0 0 2.00000i 0 1.00000 8.94427i 0
149.3 0 2.23607 2.00000i 0 0 0 2.00000i 0 1.00000 8.94427i 0
149.4 0 2.23607 + 2.00000i 0 0 0 2.00000i 0 1.00000 + 8.94427i 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.c 4
3.b odd 2 1 inner 300.3.b.c 4
4.b odd 2 1 1200.3.c.e 4
5.b even 2 1 inner 300.3.b.c 4
5.c odd 4 1 60.3.g.a 2
5.c odd 4 1 300.3.g.d 2
12.b even 2 1 1200.3.c.e 4
15.d odd 2 1 inner 300.3.b.c 4
15.e even 4 1 60.3.g.a 2
15.e even 4 1 300.3.g.d 2
20.d odd 2 1 1200.3.c.e 4
20.e even 4 1 240.3.l.a 2
20.e even 4 1 1200.3.l.r 2
40.i odd 4 1 960.3.l.a 2
40.k even 4 1 960.3.l.d 2
45.k odd 12 2 1620.3.o.b 4
45.l even 12 2 1620.3.o.b 4
60.h even 2 1 1200.3.c.e 4
60.l odd 4 1 240.3.l.a 2
60.l odd 4 1 1200.3.l.r 2
120.q odd 4 1 960.3.l.d 2
120.w even 4 1 960.3.l.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.g.a 2 5.c odd 4 1
60.3.g.a 2 15.e even 4 1
240.3.l.a 2 20.e even 4 1
240.3.l.a 2 60.l odd 4 1
300.3.b.c 4 1.a even 1 1 trivial
300.3.b.c 4 3.b odd 2 1 inner
300.3.b.c 4 5.b even 2 1 inner
300.3.b.c 4 15.d odd 2 1 inner
300.3.g.d 2 5.c odd 4 1
300.3.g.d 2 15.e even 4 1
960.3.l.a 2 40.i odd 4 1
960.3.l.a 2 120.w even 4 1
960.3.l.d 2 40.k even 4 1
960.3.l.d 2 120.q odd 4 1
1200.3.c.e 4 4.b odd 2 1
1200.3.c.e 4 12.b even 2 1
1200.3.c.e 4 20.d odd 2 1
1200.3.c.e 4 60.h even 2 1
1200.3.l.r 2 20.e even 4 1
1200.3.l.r 2 60.l odd 4 1
1620.3.o.b 4 45.k odd 12 2
1620.3.o.b 4 45.l even 12 2

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}^{2} + 180$$ T11^2 + 180

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 2T^{2} + 81$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T^{2} + 180)^{2}$$
$13$ $$(T^{2} + 64)^{2}$$
$17$ $$(T^{2} - 180)^{2}$$
$19$ $$(T - 34)^{4}$$
$23$ $$(T^{2} - 1620)^{2}$$
$29$ $$(T^{2} + 1620)^{2}$$
$31$ $$(T - 14)^{4}$$
$37$ $$(T^{2} + 3136)^{2}$$
$41$ $$(T^{2} + 720)^{2}$$
$43$ $$(T^{2} + 64)^{2}$$
$47$ $$(T^{2} - 1620)^{2}$$
$53$ $$(T^{2} - 1620)^{2}$$
$59$ $$(T^{2} + 180)^{2}$$
$61$ $$(T + 46)^{4}$$
$67$ $$(T^{2} + 1024)^{2}$$
$71$ $$(T^{2} + 2880)^{2}$$
$73$ $$(T^{2} + 11236)^{2}$$
$79$ $$(T - 22)^{4}$$
$83$ $$(T^{2} - 14580)^{2}$$
$89$ $$(T^{2} + 11520)^{2}$$
$97$ $$(T^{2} + 14884)^{2}$$