# Properties

 Label 120.2.w.a Level $120$ Weight $2$ Character orbit 120.w Analytic conductor $0.958$ Analytic rank $0$ Dimension $4$ CM discriminant -24 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [120,2,Mod(53,120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(120, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 2, 3]))

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

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

chi := DirichletCharacter("120.53");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$120 = 2^{3} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 120.w (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: $$0.958204824255$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 1) q^{2} + \beta_1 q^{3} - 2 \beta_{2} q^{4} + ( - \beta_{3} + \beta_{2} + 1) q^{5} + (\beta_{3} - \beta_1) q^{6} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{7} + (2 \beta_{2} + 2) q^{8} + 3 \beta_{2} q^{9}+O(q^{10})$$ q + (b2 - 1) * q^2 + b1 * q^3 - 2*b2 * q^4 + (-b3 + b2 + 1) * q^5 + (b3 - b1) * q^6 + (-2*b3 + b2 - 1) * q^7 + (2*b2 + 2) * q^8 + 3*b2 * q^9 $$q + (\beta_{2} - 1) q^{2} + \beta_1 q^{3} - 2 \beta_{2} q^{4} + ( - \beta_{3} + \beta_{2} + 1) q^{5} + (\beta_{3} - \beta_1) q^{6} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{7} + (2 \beta_{2} + 2) q^{8} + 3 \beta_{2} q^{9} + (\beta_{3} + \beta_1 - 2) q^{10} + (\beta_{3} - \beta_1 - 4) q^{11} - 2 \beta_{3} q^{12} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{14} + (\beta_{3} + \beta_1 + 3) q^{15} - 4 q^{16} + ( - 3 \beta_{2} - 3) q^{18} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{20} + (\beta_{3} - \beta_1 + 6) q^{21} + ( - 2 \beta_{3} - 4 \beta_{2} + 4) q^{22} + (2 \beta_{3} + 2 \beta_1) q^{24} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{25} + 3 \beta_{3} q^{27} + (2 \beta_{2} - 4 \beta_1 + 2) q^{28} + ( - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{29} + (3 \beta_{2} - 2 \beta_1 - 3) q^{30} + (2 \beta_{3} - 2 \beta_1) q^{31} + ( - 4 \beta_{2} + 4) q^{32} + ( - 3 \beta_{2} - 4 \beta_1 - 3) q^{33} + ( - \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 2) q^{35} + 6 q^{36} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{40} + ( - 2 \beta_{3} + 6 \beta_{2} - 6) q^{42} + (2 \beta_{3} + 8 \beta_{2} + 2 \beta_1) q^{44} + (3 \beta_{2} + 3 \beta_1 - 3) q^{45} - 4 \beta_1 q^{48} + (4 \beta_{3} - 7 \beta_{2} + 4 \beta_1) q^{49} + (4 \beta_{3} + \beta_{2} + 1) q^{50} + 2 \beta_1 q^{53} + ( - 3 \beta_{3} - 3 \beta_1) q^{54} + (4 \beta_{3} - \beta_{2} - 2 \beta_1 - 7) q^{55} + ( - 4 \beta_{3} + 4 \beta_1 - 4) q^{56} + (2 \beta_{2} + 6 \beta_1 + 2) q^{58} + ( - 3 \beta_{3} + 8 \beta_{2} - 3 \beta_1) q^{59} + ( - 2 \beta_{3} - 6 \beta_{2} + 2 \beta_1) q^{60} - 4 \beta_{3} q^{62} + ( - 3 \beta_{2} + 6 \beta_1 - 3) q^{63} + 8 \beta_{2} q^{64} + ( - 4 \beta_{3} + 4 \beta_1 + 6) q^{66} + (4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 8) q^{70} + (6 \beta_{2} - 6) q^{72} + (7 \beta_{2} - 4 \beta_1 + 7) q^{73} + ( - \beta_{3} + 6 \beta_{2} + 6) q^{75} + (6 \beta_{3} + 2 \beta_{2} - 2) q^{77} + ( - 6 \beta_{3} - 6 \beta_1) q^{79} + (4 \beta_{3} - 4 \beta_{2} - 4) q^{80} - 9 q^{81} + (4 \beta_{2} + 4) q^{83} + (2 \beta_{3} - 12 \beta_{2} + 2 \beta_1) q^{84} + ( - 2 \beta_{3} - 9 \beta_{2} + 9) q^{87} + ( - 8 \beta_{2} - 4 \beta_1 - 8) q^{88} + (3 \beta_{3} - 6 \beta_{2} - 3 \beta_1) q^{90} + ( - 6 \beta_{2} - 6) q^{93} + ( - 4 \beta_{3} + 4 \beta_1) q^{96} + (8 \beta_{3} + \beta_{2} - 1) q^{97} + (7 \beta_{2} - 8 \beta_1 + 7) q^{98} + ( - 3 \beta_{3} - 12 \beta_{2} - 3 \beta_1) q^{99}+O(q^{100})$$ q + (b2 - 1) * q^2 + b1 * q^3 - 2*b2 * q^4 + (-b3 + b2 + 1) * q^5 + (b3 - b1) * q^6 + (-2*b3 + b2 - 1) * q^7 + (2*b2 + 2) * q^8 + 3*b2 * q^9 + (b3 + b1 - 2) * q^10 + (b3 - b1 - 4) * q^11 - 2*b3 * q^12 + (2*b3 - 2*b2 + 2*b1) * q^14 + (b3 + b1 + 3) * q^15 - 4 * q^16 + (-3*b2 - 3) * q^18 + (-2*b2 - 2*b1 + 2) * q^20 + (b3 - b1 + 6) * q^21 + (-2*b3 - 4*b2 + 4) * q^22 + (2*b3 + 2*b1) * q^24 + (-2*b3 - b2 + 2*b1) * q^25 + 3*b3 * q^27 + (2*b2 - 4*b1 + 2) * q^28 + (-3*b3 - 2*b2 - 3*b1) * q^29 + (3*b2 - 2*b1 - 3) * q^30 + (2*b3 - 2*b1) * q^31 + (-4*b2 + 4) * q^32 + (-3*b2 - 4*b1 - 3) * q^33 + (-b3 - 6*b2 + 3*b1 - 2) * q^35 + 6 * q^36 + (-2*b3 + 4*b2 + 2*b1) * q^40 + (-2*b3 + 6*b2 - 6) * q^42 + (2*b3 + 8*b2 + 2*b1) * q^44 + (3*b2 + 3*b1 - 3) * q^45 - 4*b1 * q^48 + (4*b3 - 7*b2 + 4*b1) * q^49 + (4*b3 + b2 + 1) * q^50 + 2*b1 * q^53 + (-3*b3 - 3*b1) * q^54 + (4*b3 - b2 - 2*b1 - 7) * q^55 + (-4*b3 + 4*b1 - 4) * q^56 + (2*b2 + 6*b1 + 2) * q^58 + (-3*b3 + 8*b2 - 3*b1) * q^59 + (-2*b3 - 6*b2 + 2*b1) * q^60 - 4*b3 * q^62 + (-3*b2 + 6*b1 - 3) * q^63 + 8*b2 * q^64 + (-4*b3 + 4*b1 + 6) * q^66 + (4*b3 + 4*b2 - 2*b1 + 8) * q^70 + (6*b2 - 6) * q^72 + (7*b2 - 4*b1 + 7) * q^73 + (-b3 + 6*b2 + 6) * q^75 + (6*b3 + 2*b2 - 2) * q^77 + (-6*b3 - 6*b1) * q^79 + (4*b3 - 4*b2 - 4) * q^80 - 9 * q^81 + (4*b2 + 4) * q^83 + (2*b3 - 12*b2 + 2*b1) * q^84 + (-2*b3 - 9*b2 + 9) * q^87 + (-8*b2 - 4*b1 - 8) * q^88 + (3*b3 - 6*b2 - 3*b1) * q^90 + (-6*b2 - 6) * q^93 + (-4*b3 + 4*b1) * q^96 + (8*b3 + b2 - 1) * q^97 + (7*b2 - 8*b1 + 7) * q^98 + (-3*b3 - 12*b2 - 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 4 q^{5} - 4 q^{7} + 8 q^{8}+O(q^{10})$$ 4 * q - 4 * q^2 + 4 * q^5 - 4 * q^7 + 8 * q^8 $$4 q - 4 q^{2} + 4 q^{5} - 4 q^{7} + 8 q^{8} - 8 q^{10} - 16 q^{11} + 12 q^{15} - 16 q^{16} - 12 q^{18} + 8 q^{20} + 24 q^{21} + 16 q^{22} + 8 q^{28} - 12 q^{30} + 16 q^{32} - 12 q^{33} - 8 q^{35} + 24 q^{36} - 24 q^{42} - 12 q^{45} + 4 q^{50} - 28 q^{55} - 16 q^{56} + 8 q^{58} - 12 q^{63} + 24 q^{66} + 32 q^{70} - 24 q^{72} + 28 q^{73} + 24 q^{75} - 8 q^{77} - 16 q^{80} - 36 q^{81} + 16 q^{83} + 36 q^{87} - 32 q^{88} - 24 q^{93} - 4 q^{97} + 28 q^{98}+O(q^{100})$$ 4 * q - 4 * q^2 + 4 * q^5 - 4 * q^7 + 8 * q^8 - 8 * q^10 - 16 * q^11 + 12 * q^15 - 16 * q^16 - 12 * q^18 + 8 * q^20 + 24 * q^21 + 16 * q^22 + 8 * q^28 - 12 * q^30 + 16 * q^32 - 12 * q^33 - 8 * q^35 + 24 * q^36 - 24 * q^42 - 12 * q^45 + 4 * q^50 - 28 * q^55 - 16 * q^56 + 8 * q^58 - 12 * q^63 + 24 * q^66 + 32 * q^70 - 24 * q^72 + 28 * q^73 + 24 * q^75 - 8 * q^77 - 16 * q^80 - 36 * q^81 + 16 * q^83 + 36 * q^87 - 32 * q^88 - 24 * q^93 - 4 * q^97 + 28 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/120\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$41$$ $$61$$ $$97$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
−1.00000 + 1.00000i −1.22474 1.22474i 2.00000i −0.224745 + 2.22474i 2.44949 −3.44949 + 3.44949i 2.00000 + 2.00000i 3.00000i −2.00000 2.44949i
53.2 −1.00000 + 1.00000i 1.22474 + 1.22474i 2.00000i 2.22474 0.224745i −2.44949 1.44949 1.44949i 2.00000 + 2.00000i 3.00000i −2.00000 + 2.44949i
77.1 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i −0.224745 2.22474i 2.44949 −3.44949 3.44949i 2.00000 2.00000i 3.00000i −2.00000 + 2.44949i
77.2 −1.00000 1.00000i 1.22474 1.22474i 2.00000i 2.22474 + 0.224745i −2.44949 1.44949 + 1.44949i 2.00000 2.00000i 3.00000i −2.00000 2.44949i
 $$n$$: 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
5.c odd 4 1 inner
120.w even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.2.w.a 4
3.b odd 2 1 120.2.w.b yes 4
4.b odd 2 1 480.2.bi.b 4
5.b even 2 1 600.2.w.h 4
5.c odd 4 1 inner 120.2.w.a 4
5.c odd 4 1 600.2.w.h 4
8.b even 2 1 120.2.w.b yes 4
8.d odd 2 1 480.2.bi.a 4
12.b even 2 1 480.2.bi.a 4
15.d odd 2 1 600.2.w.b 4
15.e even 4 1 120.2.w.b yes 4
15.e even 4 1 600.2.w.b 4
20.e even 4 1 480.2.bi.b 4
24.f even 2 1 480.2.bi.b 4
24.h odd 2 1 CM 120.2.w.a 4
40.f even 2 1 600.2.w.b 4
40.i odd 4 1 120.2.w.b yes 4
40.i odd 4 1 600.2.w.b 4
40.k even 4 1 480.2.bi.a 4
60.l odd 4 1 480.2.bi.a 4
120.i odd 2 1 600.2.w.h 4
120.q odd 4 1 480.2.bi.b 4
120.w even 4 1 inner 120.2.w.a 4
120.w even 4 1 600.2.w.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.w.a 4 1.a even 1 1 trivial
120.2.w.a 4 5.c odd 4 1 inner
120.2.w.a 4 24.h odd 2 1 CM
120.2.w.a 4 120.w even 4 1 inner
120.2.w.b yes 4 3.b odd 2 1
120.2.w.b yes 4 8.b even 2 1
120.2.w.b yes 4 15.e even 4 1
120.2.w.b yes 4 40.i odd 4 1
480.2.bi.a 4 8.d odd 2 1
480.2.bi.a 4 12.b even 2 1
480.2.bi.a 4 40.k even 4 1
480.2.bi.a 4 60.l odd 4 1
480.2.bi.b 4 4.b odd 2 1
480.2.bi.b 4 20.e even 4 1
480.2.bi.b 4 24.f even 2 1
480.2.bi.b 4 120.q odd 4 1
600.2.w.b 4 15.d odd 2 1
600.2.w.b 4 15.e even 4 1
600.2.w.b 4 40.f even 2 1
600.2.w.b 4 40.i odd 4 1
600.2.w.h 4 5.b even 2 1
600.2.w.h 4 5.c odd 4 1
600.2.w.h 4 120.i odd 2 1
600.2.w.h 4 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(120, [\chi])$$:

 $$T_{7}^{4} + 4T_{7}^{3} + 8T_{7}^{2} - 40T_{7} + 100$$ T7^4 + 4*T7^3 + 8*T7^2 - 40*T7 + 100 $$T_{11}^{2} + 8T_{11} + 10$$ T11^2 + 8*T11 + 10

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T + 2)^{2}$$
$3$ $$T^{4} + 9$$
$5$ $$T^{4} - 4 T^{3} + \cdots + 25$$
$7$ $$T^{4} + 4 T^{3} + \cdots + 100$$
$11$ $$(T^{2} + 8 T + 10)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4} + 116T^{2} + 2500$$
$31$ $$(T^{2} - 24)^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 144$$
$59$ $$T^{4} + 236T^{2} + 100$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} - 28 T^{3} + \cdots + 2500$$
$79$ $$(T^{2} + 216)^{2}$$
$83$ $$(T^{2} - 8 T + 32)^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4} + 4 T^{3} + \cdots + 36100$$