# Properties

 Label 300.2.i.a Level $300$ Weight $2$ Character orbit 300.i Analytic conductor $2.396$ 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] = [300,2,Mod(257,300)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1]))

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

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

chi := DirichletCharacter("300.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.i (of order $$4$$, degree $$2$$, 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: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ 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: $$2$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - 1) q^{3} + ( - \beta_{2} + 1) q^{7} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10})$$ q + (b3 - 1) * q^3 + (-b2 + 1) * q^7 + (-b3 - 2*b2 + b1 + 1) * q^9 $$q + (\beta_{3} - 1) q^{3} + ( - \beta_{2} + 1) q^{7} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{9} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{11} + (3 \beta_{2} + 3) q^{13} + (\beta_{2} + 2 \beta_1 + 1) q^{17} + 2 \beta_{2} q^{19} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{21} + (2 \beta_{3} + \beta_{2} - 1) q^{23} + (4 \beta_{2} + \beta_1 - 3) q^{27} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{29} + 4 q^{31} + ( - 2 \beta_{3} - 6 \beta_{2} - 4) q^{33} + ( - 3 \beta_{2} + 3) q^{37} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 3) q^{39} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{41} + (3 \beta_{2} + 3) q^{43} + ( - 3 \beta_{2} - 6 \beta_1 - 3) q^{47} + 5 \beta_{2} q^{49} + ( - \beta_{3} - \beta_{2} - \beta_1 - 5) q^{51} + ( - 2 \beta_{3} - \beta_{2} + 1) q^{53} + ( - 2 \beta_{2} - 2 \beta_1) q^{57} + (4 \beta_{3} - 4 \beta_1 - 4) q^{59} - 6 q^{61} + ( - 2 \beta_{3} - 3 \beta_{2} - 1) q^{63} + ( - \beta_{2} + 1) q^{67} + ( - \beta_{3} - 5 \beta_{2} + \beta_1 + 1) q^{69} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{71} + ( - \beta_{2} - 1) q^{73} + (2 \beta_{2} + 4 \beta_1 + 2) q^{77} + 6 \beta_{2} q^{79} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 1) q^{81} + ( - 6 \beta_{3} - 3 \beta_{2} + 3) q^{83} + (4 \beta_{2} - 2 \beta_1 - 6) q^{87} + (2 \beta_{3} - 2 \beta_1 - 2) q^{89} + 6 q^{91} + (4 \beta_{3} - 4) q^{93} + (9 \beta_{2} - 9) q^{97} + ( - 4 \beta_{3} + 10 \beta_{2} + 4 \beta_1 + 4) q^{99}+O(q^{100})$$ q + (b3 - 1) * q^3 + (-b2 + 1) * q^7 + (-b3 - 2*b2 + b1 + 1) * q^9 + (2*b3 + 2*b2 + 2*b1) * q^11 + (3*b2 + 3) * q^13 + (b2 + 2*b1 + 1) * q^17 + 2*b2 * q^19 + (b3 + b2 + b1 - 1) * q^21 + (2*b3 + b2 - 1) * q^23 + (4*b2 + b1 - 3) * q^27 + (-2*b3 + 2*b1 + 2) * q^29 + 4 * q^31 + (-2*b3 - 6*b2 - 4) * q^33 + (-3*b2 + 3) * q^37 + (3*b3 - 3*b2 - 3*b1 - 3) * q^39 + (-4*b3 - 4*b2 - 4*b1) * q^41 + (3*b2 + 3) * q^43 + (-3*b2 - 6*b1 - 3) * q^47 + 5*b2 * q^49 + (-b3 - b2 - b1 - 5) * q^51 + (-2*b3 - b2 + 1) * q^53 + (-2*b2 - 2*b1) * q^57 + (4*b3 - 4*b1 - 4) * q^59 - 6 * q^61 + (-2*b3 - 3*b2 - 1) * q^63 + (-b2 + 1) * q^67 + (-b3 - 5*b2 + b1 + 1) * q^69 + (-2*b3 - 2*b2 - 2*b1) * q^71 + (-b2 - 1) * q^73 + (2*b2 + 4*b1 + 2) * q^77 + 6*b2 * q^79 + (-4*b3 - 4*b2 - 4*b1 + 1) * q^81 + (-6*b3 - 3*b2 + 3) * q^83 + (4*b2 - 2*b1 - 6) * q^87 + (2*b3 - 2*b1 - 2) * q^89 + 6 * q^91 + (4*b3 - 4) * q^93 + (9*b2 - 9) * q^97 + (-4*b3 + 10*b2 + 4*b1 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 4 q^{7}+O(q^{10})$$ 4 * q - 2 * q^3 + 4 * q^7 $$4 q - 2 q^{3} + 4 q^{7} + 12 q^{13} - 4 q^{21} - 14 q^{27} + 16 q^{31} - 20 q^{33} + 12 q^{37} + 12 q^{43} - 20 q^{51} + 4 q^{57} - 24 q^{61} - 8 q^{63} + 4 q^{67} - 4 q^{73} + 4 q^{81} - 20 q^{87} + 24 q^{91} - 8 q^{93} - 36 q^{97}+O(q^{100})$$ 4 * q - 2 * q^3 + 4 * q^7 + 12 * q^13 - 4 * q^21 - 14 * q^27 + 16 * q^31 - 20 * q^33 + 12 * q^37 + 12 * q^43 - 20 * q^51 + 4 * q^57 - 24 * q^61 - 8 * q^63 + 4 * q^67 - 4 * q^73 + 4 * q^81 - 20 * q^87 + 24 * q^91 - 8 * q^93 - 36 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu + 1$$ v^2 + v + 1 $$\beta_{2}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v $$\beta_{3}$$ $$=$$ $$-\nu^{2} + \nu - 1$$ -v^2 + v - 1
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + \beta _1 - 2 ) / 2$$ (-b3 + b1 - 2) / 2 $$\nu^{3}$$ $$=$$ $$-\beta_{3} + \beta_{2} - \beta_1$$ -b3 + b2 - b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 − 0.618034i 1.61803i 0.618034i − 1.61803i
0 −1.61803 0.618034i 0 0 0 1.00000 + 1.00000i 0 2.23607 + 2.00000i 0
257.2 0 0.618034 + 1.61803i 0 0 0 1.00000 + 1.00000i 0 −2.23607 + 2.00000i 0
293.1 0 −1.61803 + 0.618034i 0 0 0 1.00000 1.00000i 0 2.23607 2.00000i 0
293.2 0 0.618034 1.61803i 0 0 0 1.00000 1.00000i 0 −2.23607 2.00000i 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.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.i.a 4
3.b odd 2 1 inner 300.2.i.a 4
4.b odd 2 1 1200.2.v.i 4
5.b even 2 1 60.2.i.a 4
5.c odd 4 1 60.2.i.a 4
5.c odd 4 1 inner 300.2.i.a 4
12.b even 2 1 1200.2.v.i 4
15.d odd 2 1 60.2.i.a 4
15.e even 4 1 60.2.i.a 4
15.e even 4 1 inner 300.2.i.a 4
20.d odd 2 1 240.2.v.b 4
20.e even 4 1 240.2.v.b 4
20.e even 4 1 1200.2.v.i 4
40.e odd 2 1 960.2.v.h 4
40.f even 2 1 960.2.v.e 4
40.i odd 4 1 960.2.v.e 4
40.k even 4 1 960.2.v.h 4
45.h odd 6 2 1620.2.x.b 8
45.j even 6 2 1620.2.x.b 8
45.k odd 12 2 1620.2.x.b 8
45.l even 12 2 1620.2.x.b 8
60.h even 2 1 240.2.v.b 4
60.l odd 4 1 240.2.v.b 4
60.l odd 4 1 1200.2.v.i 4
120.i odd 2 1 960.2.v.e 4
120.m even 2 1 960.2.v.h 4
120.q odd 4 1 960.2.v.h 4
120.w even 4 1 960.2.v.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.i.a 4 5.b even 2 1
60.2.i.a 4 5.c odd 4 1
60.2.i.a 4 15.d odd 2 1
60.2.i.a 4 15.e even 4 1
240.2.v.b 4 20.d odd 2 1
240.2.v.b 4 20.e even 4 1
240.2.v.b 4 60.h even 2 1
240.2.v.b 4 60.l odd 4 1
300.2.i.a 4 1.a even 1 1 trivial
300.2.i.a 4 3.b odd 2 1 inner
300.2.i.a 4 5.c odd 4 1 inner
300.2.i.a 4 15.e even 4 1 inner
960.2.v.e 4 40.f even 2 1
960.2.v.e 4 40.i odd 4 1
960.2.v.e 4 120.i odd 2 1
960.2.v.e 4 120.w even 4 1
960.2.v.h 4 40.e odd 2 1
960.2.v.h 4 40.k even 4 1
960.2.v.h 4 120.m even 2 1
960.2.v.h 4 120.q odd 4 1
1200.2.v.i 4 4.b odd 2 1
1200.2.v.i 4 12.b even 2 1
1200.2.v.i 4 20.e even 4 1
1200.2.v.i 4 60.l odd 4 1
1620.2.x.b 8 45.h odd 6 2
1620.2.x.b 8 45.j even 6 2
1620.2.x.b 8 45.k odd 12 2
1620.2.x.b 8 45.l even 12 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 2 T^{3} + 2 T^{2} + 6 T + 9$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 2 T + 2)^{2}$$
$11$ $$(T^{2} + 20)^{2}$$
$13$ $$(T^{2} - 6 T + 18)^{2}$$
$17$ $$T^{4} + 100$$
$19$ $$(T^{2} + 4)^{2}$$
$23$ $$T^{4} + 100$$
$29$ $$(T^{2} - 20)^{2}$$
$31$ $$(T - 4)^{4}$$
$37$ $$(T^{2} - 6 T + 18)^{2}$$
$41$ $$(T^{2} + 80)^{2}$$
$43$ $$(T^{2} - 6 T + 18)^{2}$$
$47$ $$T^{4} + 8100$$
$53$ $$T^{4} + 100$$
$59$ $$(T^{2} - 80)^{2}$$
$61$ $$(T + 6)^{4}$$
$67$ $$(T^{2} - 2 T + 2)^{2}$$
$71$ $$(T^{2} + 20)^{2}$$
$73$ $$(T^{2} + 2 T + 2)^{2}$$
$79$ $$(T^{2} + 36)^{2}$$
$83$ $$T^{4} + 8100$$
$89$ $$(T^{2} - 20)^{2}$$
$97$ $$(T^{2} + 18 T + 162)^{2}$$