# Properties

 Label 120.2.m.a Level $120$ Weight $2$ Character orbit 120.m Analytic conductor $0.958$ Analytic rank $0$ Dimension $4$ CM discriminant -15 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$120 = 2^{3} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 120.m (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.958204824255$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{-5})$$ Defining polynomial: $$x^{4} + x^{2} + 4$$ x^4 + x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} - \beta_1) q^{3} + \beta_{2} q^{4} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{2} - 2) q^{6} + (2 \beta_{3} - \beta_1) q^{8} + 3 q^{9}+O(q^{10})$$ q + b1 * q^2 + (b3 - b1) * q^3 + b2 * q^4 + (b3 + b1) * q^5 + (-b2 - 2) * q^6 + (2*b3 - b1) * q^8 + 3 * q^9 $$q + \beta_1 q^{2} + (\beta_{3} - \beta_1) q^{3} + \beta_{2} q^{4} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{2} - 2) q^{6} + (2 \beta_{3} - \beta_1) q^{8} + 3 q^{9} + (\beta_{2} - 2) q^{10} + ( - 2 \beta_{3} - \beta_1) q^{12} + ( - 2 \beta_{2} - 1) q^{15} + ( - \beta_{2} - 4) q^{16} + ( - 4 \beta_{3} + 4 \beta_1) q^{17} + 3 \beta_1 q^{18} + 4 q^{19} + (2 \beta_{3} - 3 \beta_1) q^{20} + ( - 4 \beta_{3} - 4 \beta_1) q^{23} + ( - \beta_{2} + 4) q^{24} - 5 q^{25} + (3 \beta_{3} - 3 \beta_1) q^{27} + ( - 4 \beta_{3} + \beta_1) q^{30} + (4 \beta_{2} + 2) q^{31} + ( - 2 \beta_{3} - 3 \beta_1) q^{32} + (4 \beta_{2} + 8) q^{34} + 3 \beta_{2} q^{36} + 4 \beta_1 q^{38} + ( - 3 \beta_{2} - 4) q^{40} + (3 \beta_{3} + 3 \beta_1) q^{45} + ( - 4 \beta_{2} + 8) q^{46} + (4 \beta_{3} + 4 \beta_1) q^{47} + ( - 2 \beta_{3} + 5 \beta_1) q^{48} - 7 q^{49} - 5 \beta_1 q^{50} - 12 q^{51} + ( - 2 \beta_{3} - 2 \beta_1) q^{53} + ( - 3 \beta_{2} - 6) q^{54} + (4 \beta_{3} - 4 \beta_1) q^{57} + (\beta_{2} + 8) q^{60} + ( - 8 \beta_{2} - 4) q^{61} + (8 \beta_{3} - 2 \beta_1) q^{62} + ( - 3 \beta_{2} + 4) q^{64} + (8 \beta_{3} + 4 \beta_1) q^{68} + (8 \beta_{2} + 4) q^{69} + (6 \beta_{3} - 3 \beta_1) q^{72} + ( - 5 \beta_{3} + 5 \beta_1) q^{75} + 4 \beta_{2} q^{76} + ( - 4 \beta_{2} - 2) q^{79} + ( - 6 \beta_{3} - \beta_1) q^{80} + 9 q^{81} + ( - 2 \beta_{3} + 2 \beta_1) q^{83} + (8 \beta_{2} + 4) q^{85} + (3 \beta_{2} - 6) q^{90} + ( - 8 \beta_{3} + 12 \beta_1) q^{92} + ( - 6 \beta_{3} - 6 \beta_1) q^{93} + (4 \beta_{2} - 8) q^{94} + (4 \beta_{3} + 4 \beta_1) q^{95} + (5 \beta_{2} + 4) q^{96} - 7 \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 - b1) * q^3 + b2 * q^4 + (b3 + b1) * q^5 + (-b2 - 2) * q^6 + (2*b3 - b1) * q^8 + 3 * q^9 + (b2 - 2) * q^10 + (-2*b3 - b1) * q^12 + (-2*b2 - 1) * q^15 + (-b2 - 4) * q^16 + (-4*b3 + 4*b1) * q^17 + 3*b1 * q^18 + 4 * q^19 + (2*b3 - 3*b1) * q^20 + (-4*b3 - 4*b1) * q^23 + (-b2 + 4) * q^24 - 5 * q^25 + (3*b3 - 3*b1) * q^27 + (-4*b3 + b1) * q^30 + (4*b2 + 2) * q^31 + (-2*b3 - 3*b1) * q^32 + (4*b2 + 8) * q^34 + 3*b2 * q^36 + 4*b1 * q^38 + (-3*b2 - 4) * q^40 + (3*b3 + 3*b1) * q^45 + (-4*b2 + 8) * q^46 + (4*b3 + 4*b1) * q^47 + (-2*b3 + 5*b1) * q^48 - 7 * q^49 - 5*b1 * q^50 - 12 * q^51 + (-2*b3 - 2*b1) * q^53 + (-3*b2 - 6) * q^54 + (4*b3 - 4*b1) * q^57 + (b2 + 8) * q^60 + (-8*b2 - 4) * q^61 + (8*b3 - 2*b1) * q^62 + (-3*b2 + 4) * q^64 + (8*b3 + 4*b1) * q^68 + (8*b2 + 4) * q^69 + (6*b3 - 3*b1) * q^72 + (-5*b3 + 5*b1) * q^75 + 4*b2 * q^76 + (-4*b2 - 2) * q^79 + (-6*b3 - b1) * q^80 + 9 * q^81 + (-2*b3 + 2*b1) * q^83 + (8*b2 + 4) * q^85 + (3*b2 - 6) * q^90 + (-8*b3 + 12*b1) * q^92 + (-6*b3 - 6*b1) * q^93 + (4*b2 - 8) * q^94 + (4*b3 + 4*b1) * q^95 + (5*b2 + 4) * q^96 - 7*b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{4} - 6 q^{6} + 12 q^{9}+O(q^{10})$$ 4 * q - 2 * q^4 - 6 * q^6 + 12 * q^9 $$4 q - 2 q^{4} - 6 q^{6} + 12 q^{9} - 10 q^{10} - 14 q^{16} + 16 q^{19} + 18 q^{24} - 20 q^{25} + 24 q^{34} - 6 q^{36} - 10 q^{40} + 40 q^{46} - 28 q^{49} - 48 q^{51} - 18 q^{54} + 30 q^{60} + 22 q^{64} - 8 q^{76} + 36 q^{81} - 30 q^{90} - 40 q^{94} + 6 q^{96}+O(q^{100})$$ 4 * q - 2 * q^4 - 6 * q^6 + 12 * q^9 - 10 * q^10 - 14 * q^16 + 16 * q^19 + 18 * q^24 - 20 * q^25 + 24 * q^34 - 6 * q^36 - 10 * q^40 + 40 * q^46 - 28 * q^49 - 48 * q^51 - 18 * q^54 + 30 * q^60 + 22 * q^64 - 8 * q^76 + 36 * q^81 - 30 * q^90 - 40 * q^94 + 6 * q^96

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

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

## Character values

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

 $$n$$ $$31$$ $$41$$ $$61$$ $$97$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1
 −0.866025 − 1.11803i −0.866025 + 1.11803i 0.866025 − 1.11803i 0.866025 + 1.11803i
−0.866025 1.11803i 1.73205 −0.500000 + 1.93649i 2.23607i −1.50000 1.93649i 0 2.59808 1.11803i 3.00000 −2.50000 + 1.93649i
59.2 −0.866025 + 1.11803i 1.73205 −0.500000 1.93649i 2.23607i −1.50000 + 1.93649i 0 2.59808 + 1.11803i 3.00000 −2.50000 1.93649i
59.3 0.866025 1.11803i −1.73205 −0.500000 1.93649i 2.23607i −1.50000 + 1.93649i 0 −2.59808 1.11803i 3.00000 −2.50000 1.93649i
59.4 0.866025 + 1.11803i −1.73205 −0.500000 + 1.93649i 2.23607i −1.50000 1.93649i 0 −2.59808 + 1.11803i 3.00000 −2.50000 + 1.93649i
 $$n$$: 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner
40.e odd 2 1 inner
120.m even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.2.m.a 4
3.b odd 2 1 inner 120.2.m.a 4
4.b odd 2 1 480.2.m.a 4
5.b even 2 1 inner 120.2.m.a 4
5.c odd 4 2 600.2.b.c 4
8.b even 2 1 480.2.m.a 4
8.d odd 2 1 inner 120.2.m.a 4
12.b even 2 1 480.2.m.a 4
15.d odd 2 1 CM 120.2.m.a 4
15.e even 4 2 600.2.b.c 4
20.d odd 2 1 480.2.m.a 4
20.e even 4 2 2400.2.b.c 4
24.f even 2 1 inner 120.2.m.a 4
24.h odd 2 1 480.2.m.a 4
40.e odd 2 1 inner 120.2.m.a 4
40.f even 2 1 480.2.m.a 4
40.i odd 4 2 2400.2.b.c 4
40.k even 4 2 600.2.b.c 4
60.h even 2 1 480.2.m.a 4
60.l odd 4 2 2400.2.b.c 4
120.i odd 2 1 480.2.m.a 4
120.m even 2 1 inner 120.2.m.a 4
120.q odd 4 2 600.2.b.c 4
120.w even 4 2 2400.2.b.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.m.a 4 1.a even 1 1 trivial
120.2.m.a 4 3.b odd 2 1 inner
120.2.m.a 4 5.b even 2 1 inner
120.2.m.a 4 8.d odd 2 1 inner
120.2.m.a 4 15.d odd 2 1 CM
120.2.m.a 4 24.f even 2 1 inner
120.2.m.a 4 40.e odd 2 1 inner
120.2.m.a 4 120.m even 2 1 inner
480.2.m.a 4 4.b odd 2 1
480.2.m.a 4 8.b even 2 1
480.2.m.a 4 12.b even 2 1
480.2.m.a 4 20.d odd 2 1
480.2.m.a 4 24.h odd 2 1
480.2.m.a 4 40.f even 2 1
480.2.m.a 4 60.h even 2 1
480.2.m.a 4 120.i odd 2 1
600.2.b.c 4 5.c odd 4 2
600.2.b.c 4 15.e even 4 2
600.2.b.c 4 40.k even 4 2
600.2.b.c 4 120.q odd 4 2
2400.2.b.c 4 20.e even 4 2
2400.2.b.c 4 40.i odd 4 2
2400.2.b.c 4 60.l odd 4 2
2400.2.b.c 4 120.w even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}$$ acting on $$S_{2}^{\mathrm{new}}(120, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{2} + 4$$
$3$ $$(T^{2} - 3)^{2}$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 48)^{2}$$
$19$ $$(T - 4)^{4}$$
$23$ $$(T^{2} + 80)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T^{2} + 60)^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} + 80)^{2}$$
$53$ $$(T^{2} + 20)^{2}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + 240)^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} + 60)^{2}$$
$83$ $$(T^{2} - 12)^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$