# Properties

 Label 180.3.c.a Level $180$ Weight $3$ Character orbit 180.c Analytic conductor $4.905$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.90464475849$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{10} q^{2} + 4 \zeta_{10}^{2} q^{4} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{5} + ( -4 + 8 \zeta_{10} - 2 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{7} + 8 \zeta_{10}^{3} q^{8} +O(q^{10})$$ $$q + 2 \zeta_{10} q^{2} + 4 \zeta_{10}^{2} q^{4} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{5} + ( -4 + 8 \zeta_{10} - 2 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{7} + 8 \zeta_{10}^{3} q^{8} + ( -4 + 2 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{10} + ( -8 + 16 \zeta_{10} - 12 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{11} + ( -2 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{13} + ( -12 + 4 \zeta_{10} + 4 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{14} + ( -16 + 16 \zeta_{10} - 16 \zeta_{10}^{2} + 16 \zeta_{10}^{3} ) q^{16} + ( 14 + 16 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{17} + ( -8 + 16 \zeta_{10} - 8 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{19} + ( -8 \zeta_{10} + 4 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{20} + ( -8 - 8 \zeta_{10} + 24 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{22} + ( 4 - 8 \zeta_{10} - 2 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{23} + 5 q^{25} + ( 8 - 12 \zeta_{10} + 8 \zeta_{10}^{2} ) q^{26} + ( -16 - 8 \zeta_{10} - 8 \zeta_{10}^{2} + 24 \zeta_{10}^{3} ) q^{28} + ( -2 - 8 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{29} + ( 8 - 16 \zeta_{10} - 12 \zeta_{10}^{2} - 28 \zeta_{10}^{3} ) q^{31} -32 q^{32} + ( 32 - 4 \zeta_{10} + 32 \zeta_{10}^{2} ) q^{34} + ( -10 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{35} + ( 14 + 20 \zeta_{10}^{2} - 20 \zeta_{10}^{3} ) q^{37} + ( -16 + 16 \zeta_{10}^{2} ) q^{38} + ( 16 - 16 \zeta_{10} - 8 \zeta_{10}^{3} ) q^{40} + ( 34 + 12 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{41} + ( 8 - 16 \zeta_{10} + 14 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{43} + ( 32 - 48 \zeta_{10} + 16 \zeta_{10}^{2} + 16 \zeta_{10}^{3} ) q^{44} + ( 20 - 12 \zeta_{10} + 4 \zeta_{10}^{2} - 24 \zeta_{10}^{3} ) q^{46} + ( -12 + 24 \zeta_{10} - 38 \zeta_{10}^{2} - 14 \zeta_{10}^{3} ) q^{47} + ( -11 - 20 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{49} + 10 \zeta_{10} q^{50} + ( 16 \zeta_{10} - 24 \zeta_{10}^{2} + 16 \zeta_{10}^{3} ) q^{52} + ( 34 - 20 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{53} + ( -16 + 32 \zeta_{10} - 28 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{55} + ( -48 + 16 \zeta_{10} - 64 \zeta_{10}^{2} + 32 \zeta_{10}^{3} ) q^{56} + ( -16 + 12 \zeta_{10} - 16 \zeta_{10}^{2} ) q^{58} + ( -24 + 48 \zeta_{10} + 48 \zeta_{10}^{3} ) q^{59} + ( 6 - 52 \zeta_{10}^{2} + 52 \zeta_{10}^{3} ) q^{61} + ( 56 - 40 \zeta_{10} + 24 \zeta_{10}^{2} - 80 \zeta_{10}^{3} ) q^{62} -64 \zeta_{10} q^{64} + ( -6 + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{65} + ( 40 - 80 \zeta_{10} + 18 \zeta_{10}^{2} - 62 \zeta_{10}^{3} ) q^{67} + ( 64 \zeta_{10} - 8 \zeta_{10}^{2} + 64 \zeta_{10}^{3} ) q^{68} + ( 20 - 20 \zeta_{10} + 20 \zeta_{10}^{2} - 40 \zeta_{10}^{3} ) q^{70} + ( 40 - 80 \zeta_{10} + 44 \zeta_{10}^{2} - 36 \zeta_{10}^{3} ) q^{71} + ( 98 + 64 \zeta_{10}^{2} - 64 \zeta_{10}^{3} ) q^{73} + ( 40 - 12 \zeta_{10} + 40 \zeta_{10}^{2} ) q^{74} + ( -32 \zeta_{10} + 32 \zeta_{10}^{3} ) q^{76} + ( -40 + 40 \zeta_{10}^{2} - 40 \zeta_{10}^{3} ) q^{77} + ( -32 + 64 \zeta_{10} + 8 \zeta_{10}^{2} + 72 \zeta_{10}^{3} ) q^{79} + ( 16 + 16 \zeta_{10} - 16 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{80} + ( 24 + 44 \zeta_{10} + 24 \zeta_{10}^{2} ) q^{82} + ( 24 - 48 \zeta_{10} + 50 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{83} + ( -46 - 12 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{85} + ( 4 + 12 \zeta_{10} - 28 \zeta_{10}^{2} + 24 \zeta_{10}^{3} ) q^{86} + ( -32 + 96 \zeta_{10} - 128 \zeta_{10}^{2} + 64 \zeta_{10}^{3} ) q^{88} + ( -18 - 80 \zeta_{10}^{2} + 80 \zeta_{10}^{3} ) q^{89} + ( 16 - 32 \zeta_{10} + 28 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{91} + ( 48 - 8 \zeta_{10} + 24 \zeta_{10}^{2} - 40 \zeta_{10}^{3} ) q^{92} + ( 28 - 52 \zeta_{10} + 76 \zeta_{10}^{2} - 104 \zeta_{10}^{3} ) q^{94} + ( -8 + 16 \zeta_{10} - 24 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{95} + ( -54 + 24 \zeta_{10}^{2} - 24 \zeta_{10}^{3} ) q^{97} + ( -40 + 18 \zeta_{10} - 40 \zeta_{10}^{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 4q^{4} + 8q^{8} + O(q^{10})$$ $$4q + 2q^{2} - 4q^{4} + 8q^{8} - 10q^{10} - 16q^{13} - 40q^{14} - 16q^{16} + 24q^{17} - 20q^{20} - 80q^{22} + 20q^{25} + 12q^{26} - 40q^{28} + 8q^{29} - 128q^{32} + 92q^{34} + 16q^{37} - 80q^{38} + 40q^{40} + 112q^{41} + 80q^{44} + 40q^{46} - 4q^{49} + 10q^{50} + 56q^{52} + 176q^{53} - 80q^{56} - 36q^{58} + 128q^{61} + 80q^{62} - 64q^{64} - 40q^{65} + 136q^{68} + 264q^{73} + 108q^{74} - 240q^{77} + 80q^{80} + 116q^{82} - 160q^{85} + 80q^{86} + 160q^{88} + 88q^{89} + 120q^{92} - 120q^{94} - 264q^{97} - 102q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1
 −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 − 0.587785i 0.809017 + 0.587785i
−0.618034 1.90211i 0 −3.23607 + 2.35114i 2.23607 0 5.25731i 6.47214 + 4.70228i 0 −1.38197 4.25325i
91.2 −0.618034 + 1.90211i 0 −3.23607 2.35114i 2.23607 0 5.25731i 6.47214 4.70228i 0 −1.38197 + 4.25325i
91.3 1.61803 1.17557i 0 1.23607 3.80423i −2.23607 0 8.50651i −2.47214 7.60845i 0 −3.61803 + 2.62866i
91.4 1.61803 + 1.17557i 0 1.23607 + 3.80423i −2.23607 0 8.50651i −2.47214 + 7.60845i 0 −3.61803 2.62866i
 $$n$$: 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.c.a 4
3.b odd 2 1 20.3.b.a 4
4.b odd 2 1 inner 180.3.c.a 4
5.b even 2 1 900.3.c.k 4
5.c odd 4 2 900.3.f.e 8
8.b even 2 1 2880.3.e.e 4
8.d odd 2 1 2880.3.e.e 4
12.b even 2 1 20.3.b.a 4
15.d odd 2 1 100.3.b.f 4
15.e even 4 2 100.3.d.b 8
20.d odd 2 1 900.3.c.k 4
20.e even 4 2 900.3.f.e 8
24.f even 2 1 320.3.b.c 4
24.h odd 2 1 320.3.b.c 4
48.i odd 4 2 1280.3.g.e 8
48.k even 4 2 1280.3.g.e 8
60.h even 2 1 100.3.b.f 4
60.l odd 4 2 100.3.d.b 8
120.i odd 2 1 1600.3.b.s 4
120.m even 2 1 1600.3.b.s 4
120.q odd 4 2 1600.3.h.n 8
120.w even 4 2 1600.3.h.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.b.a 4 3.b odd 2 1
20.3.b.a 4 12.b even 2 1
100.3.b.f 4 15.d odd 2 1
100.3.b.f 4 60.h even 2 1
100.3.d.b 8 15.e even 4 2
100.3.d.b 8 60.l odd 4 2
180.3.c.a 4 1.a even 1 1 trivial
180.3.c.a 4 4.b odd 2 1 inner
320.3.b.c 4 24.f even 2 1
320.3.b.c 4 24.h odd 2 1
900.3.c.k 4 5.b even 2 1
900.3.c.k 4 20.d odd 2 1
900.3.f.e 8 5.c odd 4 2
900.3.f.e 8 20.e even 4 2
1280.3.g.e 8 48.i odd 4 2
1280.3.g.e 8 48.k even 4 2
1600.3.b.s 4 120.i odd 2 1
1600.3.b.s 4 120.m even 2 1
1600.3.h.n 8 120.q odd 4 2
1600.3.h.n 8 120.w even 4 2
2880.3.e.e 4 8.b even 2 1
2880.3.e.e 4 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 100 T_{7}^{2} + 2000$$ acting on $$S_{3}^{\mathrm{new}}(180, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -5 + T^{2} )^{2}$$
$7$ $$2000 + 100 T^{2} + T^{4}$$
$11$ $$1280 + 400 T^{2} + T^{4}$$
$13$ $$( -4 + 8 T + T^{2} )^{2}$$
$17$ $$( -284 - 12 T + T^{2} )^{2}$$
$19$ $$20480 + 320 T^{2} + T^{4}$$
$23$ $$80 + 260 T^{2} + T^{4}$$
$29$ $$( -76 - 4 T + T^{2} )^{2}$$
$31$ $$154880 + 2320 T^{2} + T^{4}$$
$37$ $$( -484 - 8 T + T^{2} )^{2}$$
$41$ $$( 604 - 56 T + T^{2} )^{2}$$
$43$ $$2000 + 500 T^{2} + T^{4}$$
$47$ $$3561680 + 4100 T^{2} + T^{4}$$
$53$ $$( 1436 - 88 T + T^{2} )^{2}$$
$59$ $$1658880 + 5760 T^{2} + T^{4}$$
$61$ $$( -2356 - 64 T + T^{2} )^{2}$$
$67$ $$19920080 + 10420 T^{2} + T^{4}$$
$71$ $$10138880 + 8080 T^{2} + T^{4}$$
$73$ $$( -764 - 132 T + T^{2} )^{2}$$
$79$ $$2478080 + 13120 T^{2} + T^{4}$$
$83$ $$2620880 + 6260 T^{2} + T^{4}$$
$89$ $$( -7516 - 44 T + T^{2} )^{2}$$
$97$ $$( 3636 + 132 T + T^{2} )^{2}$$