# Properties

 Label 180.3.f.f Level $180$ Weight $3$ Character orbit 180.f Analytic conductor $4.905$ Analytic rank $0$ Dimension $4$ CM discriminant -15 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 180.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.90464475849$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{15})$$ Defining polynomial: $$x^{4} - 7 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 4 + \beta_{3} ) q^{4} + 5 \beta_{2} q^{5} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 4 + \beta_{3} ) q^{4} + 5 \beta_{2} q^{5} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{8} + 5 \beta_{3} q^{10} + ( 12 + 7 \beta_{3} ) q^{16} + 14 \beta_{2} q^{17} + ( -8 - 16 \beta_{3} ) q^{19} + ( -5 \beta_{1} + 20 \beta_{2} ) q^{20} + ( 16 \beta_{1} - 8 \beta_{2} ) q^{23} -25 q^{25} + ( -16 - 32 \beta_{3} ) q^{31} + ( 5 \beta_{1} + 28 \beta_{2} ) q^{32} + 14 \beta_{3} q^{34} + ( 8 \beta_{1} - 64 \beta_{2} ) q^{38} + ( -20 + 15 \beta_{3} ) q^{40} + ( 64 + 8 \beta_{3} ) q^{46} + ( -48 \beta_{1} + 24 \beta_{2} ) q^{47} -49 q^{49} -25 \beta_{1} q^{50} -86 \beta_{2} q^{53} + 118 q^{61} + ( 16 \beta_{1} - 128 \beta_{2} ) q^{62} + ( 20 + 33 \beta_{3} ) q^{64} + ( -14 \beta_{1} + 56 \beta_{2} ) q^{68} + ( 32 - 56 \beta_{3} ) q^{76} + ( 32 + 64 \beta_{3} ) q^{79} + ( -35 \beta_{1} + 60 \beta_{2} ) q^{80} + ( 32 \beta_{1} - 16 \beta_{2} ) q^{83} -70 q^{85} + ( 56 \beta_{1} + 32 \beta_{2} ) q^{92} + ( -192 - 24 \beta_{3} ) q^{94} + ( 80 \beta_{1} - 40 \beta_{2} ) q^{95} -49 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 14q^{4} + O(q^{10})$$ $$4q + 14q^{4} - 10q^{10} + 34q^{16} - 100q^{25} - 28q^{34} - 110q^{40} + 240q^{46} - 196q^{49} + 472q^{61} + 14q^{64} + 240q^{76} - 280q^{85} - 720q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 7 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 3 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} + 3 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/180\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$91$$ $$101$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −1.93649 − 0.500000i −1.93649 + 0.500000i 1.93649 − 0.500000i 1.93649 + 0.500000i
−1.93649 0.500000i 0 3.50000 + 1.93649i 5.00000i 0 0 −5.80948 5.50000i 0 −2.50000 + 9.68246i
19.2 −1.93649 + 0.500000i 0 3.50000 1.93649i 5.00000i 0 0 −5.80948 + 5.50000i 0 −2.50000 9.68246i
19.3 1.93649 0.500000i 0 3.50000 1.93649i 5.00000i 0 0 5.80948 5.50000i 0 −2.50000 9.68246i
19.4 1.93649 + 0.500000i 0 3.50000 + 1.93649i 5.00000i 0 0 5.80948 + 5.50000i 0 −2.50000 + 9.68246i
 $$n$$: 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
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.f.f 4
3.b odd 2 1 inner 180.3.f.f 4
4.b odd 2 1 inner 180.3.f.f 4
5.b even 2 1 inner 180.3.f.f 4
5.c odd 4 1 900.3.c.g 2
5.c odd 4 1 900.3.c.i 2
12.b even 2 1 inner 180.3.f.f 4
15.d odd 2 1 CM 180.3.f.f 4
15.e even 4 1 900.3.c.g 2
15.e even 4 1 900.3.c.i 2
20.d odd 2 1 inner 180.3.f.f 4
20.e even 4 1 900.3.c.g 2
20.e even 4 1 900.3.c.i 2
60.h even 2 1 inner 180.3.f.f 4
60.l odd 4 1 900.3.c.g 2
60.l odd 4 1 900.3.c.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.f.f 4 1.a even 1 1 trivial
180.3.f.f 4 3.b odd 2 1 inner
180.3.f.f 4 4.b odd 2 1 inner
180.3.f.f 4 5.b even 2 1 inner
180.3.f.f 4 12.b even 2 1 inner
180.3.f.f 4 15.d odd 2 1 CM
180.3.f.f 4 20.d odd 2 1 inner
180.3.f.f 4 60.h even 2 1 inner
900.3.c.g 2 5.c odd 4 1
900.3.c.g 2 15.e even 4 1
900.3.c.g 2 20.e even 4 1
900.3.c.g 2 60.l odd 4 1
900.3.c.i 2 5.c odd 4 1
900.3.c.i 2 15.e even 4 1
900.3.c.i 2 20.e even 4 1
900.3.c.i 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(180, [\chi])$$:

 $$T_{7}$$ $$T_{13}$$ $$T_{23}^{2} - 960$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 7 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 25 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$( 196 + T^{2} )^{2}$$
$19$ $$( 960 + T^{2} )^{2}$$
$23$ $$( -960 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 3840 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$( -8640 + T^{2} )^{2}$$
$53$ $$( 7396 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( -118 + T )^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$( 15360 + T^{2} )^{2}$$
$83$ $$( -3840 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$