# Properties

 Label 240.2.h.b Level $240$ Weight $2$ Character orbit 240.h Analytic conductor $1.916$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 240.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.91640964851$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} - \beta_1 q^{5} + 2 \beta_{2} q^{7} + 3 q^{9}+O(q^{10})$$ q - b3 * q^3 - b1 * q^5 + 2*b2 * q^7 + 3 * q^9 $$q - \beta_{3} q^{3} - \beta_1 q^{5} + 2 \beta_{2} q^{7} + 3 q^{9} + 2 \beta_{3} q^{11} + 4 q^{13} + \beta_{2} q^{15} + 6 \beta_1 q^{17} - 2 \beta_{2} q^{19} - 6 \beta_1 q^{21} + 2 \beta_{3} q^{23} - q^{25} - 3 \beta_{3} q^{27} + 6 \beta_1 q^{29} + 2 \beta_{2} q^{31} - 6 q^{33} + 2 \beta_{3} q^{35} - 4 q^{37} - 4 \beta_{3} q^{39} - 12 \beta_1 q^{41} - 4 \beta_{2} q^{43} - 3 \beta_1 q^{45} - 2 \beta_{3} q^{47} - 5 q^{49} - 6 \beta_{2} q^{51} + 6 \beta_1 q^{53} - 2 \beta_{2} q^{55} + 6 \beta_1 q^{57} + 2 \beta_{3} q^{59} - 10 q^{61} + 6 \beta_{2} q^{63} - 4 \beta_1 q^{65} - 4 \beta_{2} q^{67} - 6 q^{69} + 8 \beta_{3} q^{71} - 2 q^{73} + \beta_{3} q^{75} + 12 \beta_1 q^{77} + 6 \beta_{2} q^{79} + 9 q^{81} - 6 \beta_{3} q^{83} + 6 q^{85} - 6 \beta_{2} q^{87} + 8 \beta_{2} q^{91} - 6 \beta_1 q^{93} - 2 \beta_{3} q^{95} + 10 q^{97} + 6 \beta_{3} q^{99}+O(q^{100})$$ q - b3 * q^3 - b1 * q^5 + 2*b2 * q^7 + 3 * q^9 + 2*b3 * q^11 + 4 * q^13 + b2 * q^15 + 6*b1 * q^17 - 2*b2 * q^19 - 6*b1 * q^21 + 2*b3 * q^23 - q^25 - 3*b3 * q^27 + 6*b1 * q^29 + 2*b2 * q^31 - 6 * q^33 + 2*b3 * q^35 - 4 * q^37 - 4*b3 * q^39 - 12*b1 * q^41 - 4*b2 * q^43 - 3*b1 * q^45 - 2*b3 * q^47 - 5 * q^49 - 6*b2 * q^51 + 6*b1 * q^53 - 2*b2 * q^55 + 6*b1 * q^57 + 2*b3 * q^59 - 10 * q^61 + 6*b2 * q^63 - 4*b1 * q^65 - 4*b2 * q^67 - 6 * q^69 + 8*b3 * q^71 - 2 * q^73 + b3 * q^75 + 12*b1 * q^77 + 6*b2 * q^79 + 9 * q^81 - 6*b3 * q^83 + 6 * q^85 - 6*b2 * q^87 + 8*b2 * q^91 - 6*b1 * q^93 - 2*b3 * q^95 + 10 * q^97 + 6*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{9}+O(q^{10})$$ 4 * q + 12 * q^9 $$4 q + 12 q^{9} + 16 q^{13} - 4 q^{25} - 24 q^{33} - 16 q^{37} - 20 q^{49} - 40 q^{61} - 24 q^{69} - 8 q^{73} + 36 q^{81} + 24 q^{85} + 40 q^{97}+O(q^{100})$$ 4 * q + 12 * q^9 + 16 * q^13 - 4 * q^25 - 24 * q^33 - 16 * q^37 - 20 * q^49 - 40 * q^61 - 24 * q^69 - 8 * q^73 + 36 * q^81 + 24 * q^85 + 40 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$161$$ $$181$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i
0 −1.73205 0 1.00000i 0 3.46410i 0 3.00000 0
191.2 0 −1.73205 0 1.00000i 0 3.46410i 0 3.00000 0
191.3 0 1.73205 0 1.00000i 0 3.46410i 0 3.00000 0
191.4 0 1.73205 0 1.00000i 0 3.46410i 0 3.00000 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
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.2.h.b 4
3.b odd 2 1 inner 240.2.h.b 4
4.b odd 2 1 inner 240.2.h.b 4
5.b even 2 1 1200.2.h.m 4
5.c odd 4 1 1200.2.o.a 4
5.c odd 4 1 1200.2.o.b 4
8.b even 2 1 960.2.h.d 4
8.d odd 2 1 960.2.h.d 4
12.b even 2 1 inner 240.2.h.b 4
15.d odd 2 1 1200.2.h.m 4
15.e even 4 1 1200.2.o.a 4
15.e even 4 1 1200.2.o.b 4
20.d odd 2 1 1200.2.h.m 4
20.e even 4 1 1200.2.o.a 4
20.e even 4 1 1200.2.o.b 4
24.f even 2 1 960.2.h.d 4
24.h odd 2 1 960.2.h.d 4
60.h even 2 1 1200.2.h.m 4
60.l odd 4 1 1200.2.o.a 4
60.l odd 4 1 1200.2.o.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.h.b 4 1.a even 1 1 trivial
240.2.h.b 4 3.b odd 2 1 inner
240.2.h.b 4 4.b odd 2 1 inner
240.2.h.b 4 12.b even 2 1 inner
960.2.h.d 4 8.b even 2 1
960.2.h.d 4 8.d odd 2 1
960.2.h.d 4 24.f even 2 1
960.2.h.d 4 24.h odd 2 1
1200.2.h.m 4 5.b even 2 1
1200.2.h.m 4 15.d odd 2 1
1200.2.h.m 4 20.d odd 2 1
1200.2.h.m 4 60.h even 2 1
1200.2.o.a 4 5.c odd 4 1
1200.2.o.a 4 15.e even 4 1
1200.2.o.a 4 20.e even 4 1
1200.2.o.a 4 60.l odd 4 1
1200.2.o.b 4 5.c odd 4 1
1200.2.o.b 4 15.e even 4 1
1200.2.o.b 4 20.e even 4 1
1200.2.o.b 4 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 12$$ acting on $$S_{2}^{\mathrm{new}}(240, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 3)^{2}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$(T^{2} + 12)^{2}$$
$11$ $$(T^{2} - 12)^{2}$$
$13$ $$(T - 4)^{4}$$
$17$ $$(T^{2} + 36)^{2}$$
$19$ $$(T^{2} + 12)^{2}$$
$23$ $$(T^{2} - 12)^{2}$$
$29$ $$(T^{2} + 36)^{2}$$
$31$ $$(T^{2} + 12)^{2}$$
$37$ $$(T + 4)^{4}$$
$41$ $$(T^{2} + 144)^{2}$$
$43$ $$(T^{2} + 48)^{2}$$
$47$ $$(T^{2} - 12)^{2}$$
$53$ $$(T^{2} + 36)^{2}$$
$59$ $$(T^{2} - 12)^{2}$$
$61$ $$(T + 10)^{4}$$
$67$ $$(T^{2} + 48)^{2}$$
$71$ $$(T^{2} - 192)^{2}$$
$73$ $$(T + 2)^{4}$$
$79$ $$(T^{2} + 108)^{2}$$
$83$ $$(T^{2} - 108)^{2}$$
$89$ $$T^{4}$$
$97$ $$(T - 10)^{4}$$