# Properties

 Label 300.3.g.e Level $300$ Weight $3$ Character orbit 300.g Analytic conductor $8.174$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,3,Mod(101,300)]

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, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("300.101");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 300.g (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: $$8.17440793081$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 5$$ x^2 + 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = \sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 2) q^{3} + 8 q^{7} + ( - 4 \beta - 1) q^{9}+O(q^{10})$$ q + (b - 2) * q^3 + 8 * q^7 + (-4*b - 1) * q^9 $$q + (\beta - 2) q^{3} + 8 q^{7} + ( - 4 \beta - 1) q^{9} - 4 \beta q^{11} + 12 q^{13} + 14 \beta q^{17} + 6 q^{19} + (8 \beta - 16) q^{21} + 2 \beta q^{23} + (7 \beta + 22) q^{27} + 12 \beta q^{29} + 34 q^{31} + (8 \beta + 20) q^{33} + 44 q^{37} + (12 \beta - 24) q^{39} + 8 \beta q^{41} - 28 q^{43} - 2 \beta q^{47} + 15 q^{49} + ( - 28 \beta - 70) q^{51} + 18 \beta q^{53} + (6 \beta - 12) q^{57} - 44 \beta q^{59} + 74 q^{61} + ( - 32 \beta - 8) q^{63} - 92 q^{67} + ( - 4 \beta - 10) q^{69} + 24 \beta q^{71} + 56 q^{73} - 32 \beta q^{77} + 78 q^{79} + (8 \beta - 79) q^{81} - 46 \beta q^{83} + ( - 24 \beta - 60) q^{87} + 8 \beta q^{89} + 96 q^{91} + (34 \beta - 68) q^{93} - 32 q^{97} + (4 \beta - 80) q^{99} +O(q^{100})$$ q + (b - 2) * q^3 + 8 * q^7 + (-4*b - 1) * q^9 - 4*b * q^11 + 12 * q^13 + 14*b * q^17 + 6 * q^19 + (8*b - 16) * q^21 + 2*b * q^23 + (7*b + 22) * q^27 + 12*b * q^29 + 34 * q^31 + (8*b + 20) * q^33 + 44 * q^37 + (12*b - 24) * q^39 + 8*b * q^41 - 28 * q^43 - 2*b * q^47 + 15 * q^49 + (-28*b - 70) * q^51 + 18*b * q^53 + (6*b - 12) * q^57 - 44*b * q^59 + 74 * q^61 + (-32*b - 8) * q^63 - 92 * q^67 + (-4*b - 10) * q^69 + 24*b * q^71 + 56 * q^73 - 32*b * q^77 + 78 * q^79 + (8*b - 79) * q^81 - 46*b * q^83 + (-24*b - 60) * q^87 + 8*b * q^89 + 96 * q^91 + (34*b - 68) * q^93 - 32 * q^97 + (4*b - 80) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} + 16 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^3 + 16 * q^7 - 2 * q^9 $$2 q - 4 q^{3} + 16 q^{7} - 2 q^{9} + 24 q^{13} + 12 q^{19} - 32 q^{21} + 44 q^{27} + 68 q^{31} + 40 q^{33} + 88 q^{37} - 48 q^{39} - 56 q^{43} + 30 q^{49} - 140 q^{51} - 24 q^{57} + 148 q^{61} - 16 q^{63} - 184 q^{67} - 20 q^{69} + 112 q^{73} + 156 q^{79} - 158 q^{81} - 120 q^{87} + 192 q^{91} - 136 q^{93} - 64 q^{97} - 160 q^{99}+O(q^{100})$$ 2 * q - 4 * q^3 + 16 * q^7 - 2 * q^9 + 24 * q^13 + 12 * q^19 - 32 * q^21 + 44 * q^27 + 68 * q^31 + 40 * q^33 + 88 * q^37 - 48 * q^39 - 56 * q^43 + 30 * q^49 - 140 * q^51 - 24 * q^57 + 148 * q^61 - 16 * q^63 - 184 * q^67 - 20 * q^69 + 112 * q^73 + 156 * q^79 - 158 * q^81 - 120 * q^87 + 192 * q^91 - 136 * q^93 - 64 * q^97 - 160 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 − 2.23607i 2.23607i
0 −2.00000 2.23607i 0 0 0 8.00000 0 −1.00000 + 8.94427i 0
101.2 0 −2.00000 + 2.23607i 0 0 0 8.00000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.g.e 2
3.b odd 2 1 inner 300.3.g.e 2
4.b odd 2 1 1200.3.l.q 2
5.b even 2 1 300.3.g.h 2
5.c odd 4 2 60.3.b.a 4
12.b even 2 1 1200.3.l.q 2
15.d odd 2 1 300.3.g.h 2
15.e even 4 2 60.3.b.a 4
20.d odd 2 1 1200.3.l.h 2
20.e even 4 2 240.3.c.d 4
40.i odd 4 2 960.3.c.h 4
40.k even 4 2 960.3.c.g 4
45.k odd 12 4 1620.3.t.b 8
45.l even 12 4 1620.3.t.b 8
60.h even 2 1 1200.3.l.h 2
60.l odd 4 2 240.3.c.d 4
120.q odd 4 2 960.3.c.g 4
120.w even 4 2 960.3.c.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.b.a 4 5.c odd 4 2
60.3.b.a 4 15.e even 4 2
240.3.c.d 4 20.e even 4 2
240.3.c.d 4 60.l odd 4 2
300.3.g.e 2 1.a even 1 1 trivial
300.3.g.e 2 3.b odd 2 1 inner
300.3.g.h 2 5.b even 2 1
300.3.g.h 2 15.d odd 2 1
960.3.c.g 4 40.k even 4 2
960.3.c.g 4 120.q odd 4 2
960.3.c.h 4 40.i odd 4 2
960.3.c.h 4 120.w even 4 2
1200.3.l.h 2 20.d odd 2 1
1200.3.l.h 2 60.h even 2 1
1200.3.l.q 2 4.b odd 2 1
1200.3.l.q 2 12.b even 2 1
1620.3.t.b 8 45.k odd 12 4
1620.3.t.b 8 45.l even 12 4

## Hecke kernels

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

 $$T_{7} - 8$$ T7 - 8 $$T_{11}^{2} + 80$$ T11^2 + 80

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4T + 9$$
$5$ $$T^{2}$$
$7$ $$(T - 8)^{2}$$
$11$ $$T^{2} + 80$$
$13$ $$(T - 12)^{2}$$
$17$ $$T^{2} + 980$$
$19$ $$(T - 6)^{2}$$
$23$ $$T^{2} + 20$$
$29$ $$T^{2} + 720$$
$31$ $$(T - 34)^{2}$$
$37$ $$(T - 44)^{2}$$
$41$ $$T^{2} + 320$$
$43$ $$(T + 28)^{2}$$
$47$ $$T^{2} + 20$$
$53$ $$T^{2} + 1620$$
$59$ $$T^{2} + 9680$$
$61$ $$(T - 74)^{2}$$
$67$ $$(T + 92)^{2}$$
$71$ $$T^{2} + 2880$$
$73$ $$(T - 56)^{2}$$
$79$ $$(T - 78)^{2}$$
$83$ $$T^{2} + 10580$$
$89$ $$T^{2} + 320$$
$97$ $$(T + 32)^{2}$$