# Properties

 Label 30.3.b.a Level $30$ Weight $3$ Character orbit 30.b Analytic conductor $0.817$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [30,3,Mod(29,30)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(30, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("30.29");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$30 = 2 \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 30.b (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: $$0.817440793081$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 16x^{2} + 81$$ x^4 + 16*x^2 + 81 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( - \beta_{2} - \beta_1) q^{3} + 2 q^{4} + ( - \beta_{3} - 2 \beta_{2}) q^{5} + (\beta_{3} - 1) q^{6} + (\beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} + ( - \beta_{3} - 8) q^{9}+O(q^{10})$$ q + b2 * q^2 + (-b2 - b1) * q^3 + 2 * q^4 + (-b3 - 2*b2) * q^5 + (b3 - 1) * q^6 + (b2 + 2*b1) * q^7 + 2*b2 * q^8 + (-b3 - 8) * q^9 $$q + \beta_{2} q^{2} + ( - \beta_{2} - \beta_1) q^{3} + 2 q^{4} + ( - \beta_{3} - 2 \beta_{2}) q^{5} + (\beta_{3} - 1) q^{6} + (\beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} + ( - \beta_{3} - 8) q^{9} + (\beta_{2} + 2 \beta_1 - 4) q^{10} + 4 \beta_{3} q^{11} + ( - 2 \beta_{2} - 2 \beta_1) q^{12} - 2 \beta_{3} q^{14} + ( - 2 \beta_{3} + 8 \beta_{2} + \cdots + 2) q^{15}+ \cdots + ( - 32 \beta_{3} + 68) q^{99}+O(q^{100})$$ q + b2 * q^2 + (-b2 - b1) * q^3 + 2 * q^4 + (-b3 - 2*b2) * q^5 + (b3 - 1) * q^6 + (b2 + 2*b1) * q^7 + 2*b2 * q^8 + (-b3 - 8) * q^9 + (b2 + 2*b1 - 4) * q^10 + 4*b3 * q^11 + (-2*b2 - 2*b1) * q^12 - 2*b3 * q^14 + (-2*b3 + 8*b2 - b1 + 2) * q^15 + 4 * q^16 - 8*b2 * q^17 + (-7*b2 + 2*b1) * q^18 + 12 * q^19 + (-2*b3 - 4*b2) * q^20 + (b3 + 17) * q^21 + (-4*b2 - 8*b1) * q^22 + 17*b2 * q^23 + (2*b3 - 2) * q^24 + (-4*b2 - 8*b1 - 9) * q^25 + (16*b2 + 7*b1) * q^27 + (2*b2 + 4*b1) * q^28 + (b3 + 4*b2 + 4*b1 + 17) * q^30 - 32 * q^31 + 4*b2 * q^32 + (-32*b2 + 4*b1) * q^33 - 16 * q^34 + (4*b3 - 17*b2) * q^35 + (-2*b3 - 16) * q^36 + (4*b2 + 8*b1) * q^37 + 12*b2 * q^38 + (2*b2 + 4*b1 - 8) * q^40 - 14*b3 * q^41 + (16*b2 - 2*b1) * q^42 + (-7*b2 - 14*b1) * q^43 + 8*b3 * q^44 + (8*b3 + 14*b2 - 4*b1 - 17) * q^45 + 34 * q^46 - 25*b2 * q^47 + (-4*b2 - 4*b1) * q^48 + 15 * q^49 + (8*b3 - 9*b2) * q^50 + (-8*b3 + 8) * q^51 + 48*b2 * q^53 + (-7*b3 + 25) * q^54 + (8*b2 + 16*b1 + 68) * q^55 - 4*b3 * q^56 + (-12*b2 - 12*b1) * q^57 + 4*b3 * q^59 + (-4*b3 + 16*b2 - 2*b1 + 4) * q^60 - 16 * q^61 - 32*b2 * q^62 + (-25*b2 - 16*b1) * q^63 + 8 * q^64 + (-4*b3 - 68) * q^66 + (-b2 - 2*b1) * q^67 - 16*b2 * q^68 + (17*b3 - 17) * q^69 + (-4*b2 - 8*b1 - 34) * q^70 + (-14*b2 + 4*b1) * q^72 + (20*b2 + 40*b1) * q^73 - 8*b3 * q^74 + (-4*b3 + 9*b2 + 9*b1 - 68) * q^75 + 24 * q^76 + 68*b2 * q^77 - 72 * q^79 + (-4*b3 - 8*b2) * q^80 + (16*b3 + 47) * q^81 + (14*b2 + 28*b1) * q^82 - 31*b2 * q^83 + (2*b3 + 34) * q^84 + (-8*b2 - 16*b1 + 32) * q^85 + 14*b3 * q^86 + (-8*b2 - 16*b1) * q^88 - 16*b3 * q^89 + (4*b3 - 25*b2 - 16*b1 + 32) * q^90 + 34*b2 * q^92 + (32*b2 + 32*b1) * q^93 - 50 * q^94 + (-12*b3 - 24*b2) * q^95 + (4*b3 - 4) * q^96 + (-28*b2 - 56*b1) * q^97 + 15*b2 * q^98 + (-32*b3 + 68) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4} - 4 q^{6} - 32 q^{9}+O(q^{10})$$ 4 * q + 8 * q^4 - 4 * q^6 - 32 * q^9 $$4 q + 8 q^{4} - 4 q^{6} - 32 q^{9} - 16 q^{10} + 8 q^{15} + 16 q^{16} + 48 q^{19} + 68 q^{21} - 8 q^{24} - 36 q^{25} + 68 q^{30} - 128 q^{31} - 64 q^{34} - 64 q^{36} - 32 q^{40} - 68 q^{45} + 136 q^{46} + 60 q^{49} + 32 q^{51} + 100 q^{54} + 272 q^{55} + 16 q^{60} - 64 q^{61} + 32 q^{64} - 272 q^{66} - 68 q^{69} - 136 q^{70} - 272 q^{75} + 96 q^{76} - 288 q^{79} + 188 q^{81} + 136 q^{84} + 128 q^{85} + 128 q^{90} - 200 q^{94} - 16 q^{96} + 272 q^{99}+O(q^{100})$$ 4 * q + 8 * q^4 - 4 * q^6 - 32 * q^9 - 16 * q^10 + 8 * q^15 + 16 * q^16 + 48 * q^19 + 68 * q^21 - 8 * q^24 - 36 * q^25 + 68 * q^30 - 128 * q^31 - 64 * q^34 - 64 * q^36 - 32 * q^40 - 68 * q^45 + 136 * q^46 + 60 * q^49 + 32 * q^51 + 100 * q^54 + 272 * q^55 + 16 * q^60 - 64 * q^61 + 32 * q^64 - 272 * q^66 - 68 * q^69 - 136 * q^70 - 272 * q^75 + 96 * q^76 - 288 * q^79 + 188 * q^81 + 136 * q^84 + 128 * q^85 + 128 * q^90 - 200 * q^94 - 16 * q^96 + 272 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 16x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 7\nu ) / 9$$ (v^3 + 7*v) / 9 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 8$$ v^2 + 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 8$$ b3 - 8 $$\nu^{3}$$ $$=$$ $$9\beta_{2} - 7\beta_1$$ 9*b2 - 7*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/30\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 0.707107 + 2.91548i 0.707107 − 2.91548i −0.707107 + 2.91548i −0.707107 − 2.91548i
−1.41421 0.707107 2.91548i 2.00000 2.82843 4.12311i −1.00000 + 4.12311i 5.83095i −2.82843 −8.00000 4.12311i −4.00000 + 5.83095i
29.2 −1.41421 0.707107 + 2.91548i 2.00000 2.82843 + 4.12311i −1.00000 4.12311i 5.83095i −2.82843 −8.00000 + 4.12311i −4.00000 5.83095i
29.3 1.41421 −0.707107 2.91548i 2.00000 −2.82843 + 4.12311i −1.00000 4.12311i 5.83095i 2.82843 −8.00000 + 4.12311i −4.00000 + 5.83095i
29.4 1.41421 −0.707107 + 2.91548i 2.00000 −2.82843 4.12311i −1.00000 + 4.12311i 5.83095i 2.82843 −8.00000 4.12311i −4.00000 5.83095i
 $$n$$: 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 30.3.b.a 4
3.b odd 2 1 inner 30.3.b.a 4
4.b odd 2 1 240.3.c.c 4
5.b even 2 1 inner 30.3.b.a 4
5.c odd 4 2 150.3.d.d 4
8.b even 2 1 960.3.c.f 4
8.d odd 2 1 960.3.c.e 4
9.c even 3 2 810.3.j.c 8
9.d odd 6 2 810.3.j.c 8
12.b even 2 1 240.3.c.c 4
15.d odd 2 1 inner 30.3.b.a 4
15.e even 4 2 150.3.d.d 4
20.d odd 2 1 240.3.c.c 4
20.e even 4 2 1200.3.l.t 4
24.f even 2 1 960.3.c.e 4
24.h odd 2 1 960.3.c.f 4
40.e odd 2 1 960.3.c.e 4
40.f even 2 1 960.3.c.f 4
45.h odd 6 2 810.3.j.c 8
45.j even 6 2 810.3.j.c 8
60.h even 2 1 240.3.c.c 4
60.l odd 4 2 1200.3.l.t 4
120.i odd 2 1 960.3.c.f 4
120.m even 2 1 960.3.c.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.b.a 4 1.a even 1 1 trivial
30.3.b.a 4 3.b odd 2 1 inner
30.3.b.a 4 5.b even 2 1 inner
30.3.b.a 4 15.d odd 2 1 inner
150.3.d.d 4 5.c odd 4 2
150.3.d.d 4 15.e even 4 2
240.3.c.c 4 4.b odd 2 1
240.3.c.c 4 12.b even 2 1
240.3.c.c 4 20.d odd 2 1
240.3.c.c 4 60.h even 2 1
810.3.j.c 8 9.c even 3 2
810.3.j.c 8 9.d odd 6 2
810.3.j.c 8 45.h odd 6 2
810.3.j.c 8 45.j even 6 2
960.3.c.e 4 8.d odd 2 1
960.3.c.e 4 24.f even 2 1
960.3.c.e 4 40.e odd 2 1
960.3.c.e 4 120.m even 2 1
960.3.c.f 4 8.b even 2 1
960.3.c.f 4 24.h odd 2 1
960.3.c.f 4 40.f even 2 1
960.3.c.f 4 120.i odd 2 1
1200.3.l.t 4 20.e even 4 2
1200.3.l.t 4 60.l odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(30, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{2}$$
$3$ $$T^{4} + 16T^{2} + 81$$
$5$ $$T^{4} + 18T^{2} + 625$$
$7$ $$(T^{2} + 34)^{2}$$
$11$ $$(T^{2} + 272)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 128)^{2}$$
$19$ $$(T - 12)^{4}$$
$23$ $$(T^{2} - 578)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T + 32)^{4}$$
$37$ $$(T^{2} + 544)^{2}$$
$41$ $$(T^{2} + 3332)^{2}$$
$43$ $$(T^{2} + 1666)^{2}$$
$47$ $$(T^{2} - 1250)^{2}$$
$53$ $$(T^{2} - 4608)^{2}$$
$59$ $$(T^{2} + 272)^{2}$$
$61$ $$(T + 16)^{4}$$
$67$ $$(T^{2} + 34)^{2}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 13600)^{2}$$
$79$ $$(T + 72)^{4}$$
$83$ $$(T^{2} - 1922)^{2}$$
$89$ $$(T^{2} + 4352)^{2}$$
$97$ $$(T^{2} + 26656)^{2}$$