# Properties

 Label 180.3.f.h Level $180$ Weight $3$ Character orbit 180.f Analytic conductor $4.905$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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## 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: $$8$$ Coefficient field: 8.0.389136420864.4 Defining polynomial: $$x^{8} + 5 x^{6} + 24 x^{4} + 80 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} + \beta_{4} ) q^{5} + ( -\beta_{1} - \beta_{5} ) q^{7} + ( -2 \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} + \beta_{4} ) q^{5} + ( -\beta_{1} - \beta_{5} ) q^{7} + ( -2 \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{8} + ( -5 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{10} + ( 4 \beta_{2} + \beta_{4} + \beta_{7} ) q^{11} + ( 2 \beta_{1} + \beta_{4} - \beta_{7} ) q^{13} + ( -2 - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{14} + ( -7 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{16} + ( -\beta_{4} - 2 \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{17} + 2 \beta_{3} q^{19} + ( -6 - 6 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{6} - 2 \beta_{7} ) q^{20} + ( -4 \beta_{1} + 6 \beta_{4} + 2 \beta_{6} - 6 \beta_{7} ) q^{22} + ( 5 \beta_{1} - 3 \beta_{4} - \beta_{5} + 3 \beta_{7} ) q^{23} + ( 5 + 2 \beta_{1} - \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{25} + ( -10 + 2 \beta_{2} ) q^{26} + ( -2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{28} + ( 26 - 3 \beta_{4} - 3 \beta_{7} ) q^{29} + ( 8 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{31} + ( -6 \beta_{1} + 3 \beta_{4} + 4 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{32} + ( 6 - 6 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} - 6 \beta_{7} ) q^{34} + ( -5 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{7} ) q^{35} + ( -6 \beta_{1} - 3 \beta_{4} + 3 \beta_{7} ) q^{37} + ( 4 \beta_{4} + 8 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{38} + ( -3 - 4 \beta_{1} - 5 \beta_{2} + \beta_{3} - 5 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 7 \beta_{7} ) q^{40} + ( 26 + 6 \beta_{4} + 6 \beta_{7} ) q^{41} + ( 13 \beta_{1} - 5 \beta_{4} + 3 \beta_{5} + 5 \beta_{7} ) q^{43} + ( -46 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{44} + ( 16 + 4 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{46} + ( 17 \beta_{1} - 9 \beta_{4} - \beta_{5} + 9 \beta_{7} ) q^{47} + ( -9 + 6 \beta_{4} + 6 \beta_{7} ) q^{49} + ( -4 + 5 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} + 2 \beta_{6} - 4 \beta_{7} ) q^{50} + ( -12 \beta_{1} + 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{52} + ( 6 \beta_{1} + 3 \beta_{4} - 3 \beta_{7} ) q^{53} + ( -16 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} - 9 \beta_{7} ) q^{55} + ( 26 + 6 \beta_{2} - 2 \beta_{3} + 6 \beta_{4} + 6 \beta_{7} ) q^{56} + ( 26 \beta_{1} - 6 \beta_{4} + 6 \beta_{6} + 6 \beta_{7} ) q^{58} + ( -4 \beta_{2} - 8 \beta_{3} - \beta_{4} - \beta_{7} ) q^{59} + 38 q^{61} + ( -8 \beta_{1} + 8 \beta_{4} - 8 \beta_{5} - 8 \beta_{7} ) q^{62} + ( -5 - 3 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} + 3 \beta_{7} ) q^{64} + ( -20 + 2 \beta_{1} - \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{65} + ( 29 \beta_{1} - 15 \beta_{4} - \beta_{5} + 15 \beta_{7} ) q^{67} + ( 12 \beta_{1} - 22 \beta_{4} - 8 \beta_{5} + 2 \beta_{6} + 22 \beta_{7} ) q^{68} + ( -12 + 4 \beta_{1} - 8 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} + 8 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{70} + ( 24 \beta_{2} + 4 \beta_{3} + 6 \beta_{4} + 6 \beta_{7} ) q^{71} + ( 2 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} - 2 \beta_{7} ) q^{73} + ( 30 - 6 \beta_{2} ) q^{74} + ( -12 + 12 \beta_{2} - 4 \beta_{3} + 12 \beta_{4} + 12 \beta_{7} ) q^{76} + ( -16 \beta_{1} - 6 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} + 6 \beta_{7} ) q^{77} + ( -8 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{79} + ( 42 + 2 \beta_{1} - 10 \beta_{2} + 2 \beta_{3} - 7 \beta_{4} + 4 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} ) q^{80} + ( 26 \beta_{1} + 12 \beta_{4} - 12 \beta_{6} - 12 \beta_{7} ) q^{82} + ( -13 \beta_{1} + 9 \beta_{4} + 5 \beta_{5} - 9 \beta_{7} ) q^{83} + ( 12 + 24 \beta_{1} + 5 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} - 23 \beta_{7} ) q^{85} + ( 36 + 16 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{7} ) q^{86} + ( -44 \beta_{1} + 6 \beta_{4} + 8 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} ) q^{88} + ( -66 - 4 \beta_{4} - 4 \beta_{7} ) q^{89} + ( -8 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{91} + ( 12 \beta_{1} + 4 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} - 4 \beta_{7} ) q^{92} + ( 52 + 16 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{94} + ( -24 \beta_{2} + 2 \beta_{3} + 12 \beta_{5} - 12 \beta_{7} ) q^{95} + ( -44 \beta_{1} - 24 \beta_{4} - 4 \beta_{5} - 8 \beta_{6} + 24 \beta_{7} ) q^{97} + ( -9 \beta_{1} + 12 \beta_{4} - 12 \beta_{6} - 12 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 10q^{4} - 4q^{5} + O(q^{10})$$ $$8q - 10q^{4} - 4q^{5} - 42q^{10} - 20q^{14} - 46q^{16} - 52q^{20} + 32q^{25} - 84q^{26} + 184q^{29} + 12q^{34} - 6q^{40} + 256q^{41} - 348q^{44} + 112q^{46} - 24q^{49} - 72q^{50} + 244q^{56} + 304q^{61} - 10q^{64} - 168q^{65} - 104q^{70} + 252q^{74} - 24q^{76} + 308q^{80} + 24q^{85} + 280q^{86} - 560q^{89} + 376q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{6} + 24 x^{4} + 80 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} + 3 \nu^{4} + 16$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 4 \nu^{6} + 5 \nu^{5} + 20 \nu^{4} + 24 \nu^{3} + 32 \nu^{2} + 80 \nu + 192$$$$)/64$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 3 \nu^{5} - 16 \nu^{3} - 16 \nu$$$$)/32$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} - 5 \nu^{5} + 8 \nu^{3} - 16 \nu$$$$)/32$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{6} - 5 \nu^{5} + 20 \nu^{4} - 24 \nu^{3} + 32 \nu^{2} - 80 \nu + 192$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$-\beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + \beta_{4} + \beta_{3} - \beta_{2} - 7$$ $$\nu^{5}$$ $$=$$ $$-3 \beta_{7} - \beta_{6} + 4 \beta_{5} + 3 \beta_{4} - 6 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$3 \beta_{7} + 3 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 5$$ $$\nu^{7}$$ $$=$$ $$7 \beta_{7} - 19 \beta_{6} - 20 \beta_{5} - 7 \beta_{4} - 2 \beta_{1}$$

## 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
 −1.52274 − 1.29664i −1.52274 + 1.29664i −0.656712 − 1.88911i −0.656712 + 1.88911i 0.656712 − 1.88911i 0.656712 + 1.88911i 1.52274 − 1.29664i 1.52274 + 1.29664i
−1.52274 1.29664i 0 0.637459 + 3.94888i −4.27492 2.59328i 0 −0.837253 4.14959 6.83966i 0 3.14704 + 9.49190i
19.2 −1.52274 + 1.29664i 0 0.637459 3.94888i −4.27492 + 2.59328i 0 −0.837253 4.14959 + 6.83966i 0 3.14704 9.49190i
19.3 −0.656712 1.88911i 0 −3.13746 + 2.48120i 3.27492 3.77822i 0 9.55505 6.74766 + 4.29756i 0 −9.28814 3.70547i
19.4 −0.656712 + 1.88911i 0 −3.13746 2.48120i 3.27492 + 3.77822i 0 9.55505 6.74766 4.29756i 0 −9.28814 + 3.70547i
19.5 0.656712 1.88911i 0 −3.13746 2.48120i 3.27492 3.77822i 0 −9.55505 −6.74766 + 4.29756i 0 −4.98678 8.66787i
19.6 0.656712 + 1.88911i 0 −3.13746 + 2.48120i 3.27492 + 3.77822i 0 −9.55505 −6.74766 4.29756i 0 −4.98678 + 8.66787i
19.7 1.52274 1.29664i 0 0.637459 3.94888i −4.27492 2.59328i 0 0.837253 −4.14959 6.83966i 0 −9.87212 + 1.59414i
19.8 1.52274 + 1.29664i 0 0.637459 + 3.94888i −4.27492 + 2.59328i 0 0.837253 −4.14959 + 6.83966i 0 −9.87212 1.59414i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.8 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
5.b even 2 1 inner
20.d 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.f.h 8
3.b odd 2 1 60.3.f.b 8
4.b odd 2 1 inner 180.3.f.h 8
5.b even 2 1 inner 180.3.f.h 8
5.c odd 4 2 900.3.c.r 8
12.b even 2 1 60.3.f.b 8
15.d odd 2 1 60.3.f.b 8
15.e even 4 2 300.3.c.f 8
20.d odd 2 1 inner 180.3.f.h 8
20.e even 4 2 900.3.c.r 8
24.f even 2 1 960.3.j.e 8
24.h odd 2 1 960.3.j.e 8
60.h even 2 1 60.3.f.b 8
60.l odd 4 2 300.3.c.f 8
120.i odd 2 1 960.3.j.e 8
120.m even 2 1 960.3.j.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.b 8 3.b odd 2 1
60.3.f.b 8 12.b even 2 1
60.3.f.b 8 15.d odd 2 1
60.3.f.b 8 60.h even 2 1
180.3.f.h 8 1.a even 1 1 trivial
180.3.f.h 8 4.b odd 2 1 inner
180.3.f.h 8 5.b even 2 1 inner
180.3.f.h 8 20.d odd 2 1 inner
300.3.c.f 8 15.e even 4 2
300.3.c.f 8 60.l odd 4 2
900.3.c.r 8 5.c odd 4 2
900.3.c.r 8 20.e even 4 2
960.3.j.e 8 24.f even 2 1
960.3.j.e 8 24.h odd 2 1
960.3.j.e 8 120.i odd 2 1
960.3.j.e 8 120.m even 2 1

## Hecke kernels

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

 $$T_{7}^{4} - 92 T_{7}^{2} + 64$$ $$T_{13}^{4} + 84 T_{13}^{2} + 1536$$ $$T_{23}^{4} - 368 T_{23}^{2} + 1024$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 + 80 T^{2} + 24 T^{4} + 5 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 625 + 50 T - 6 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$7$ $$( 64 - 92 T^{2} + T^{4} )^{2}$$
$11$ $$( 24576 + 348 T^{2} + T^{4} )^{2}$$
$13$ $$( 1536 + 84 T^{2} + T^{4} )^{2}$$
$17$ $$( 221184 + 1044 T^{2} + T^{4} )^{2}$$
$19$ $$( 221184 + 1008 T^{2} + T^{4} )^{2}$$
$23$ $$( 1024 - 368 T^{2} + T^{4} )^{2}$$
$29$ $$( 16 - 46 T + T^{2} )^{4}$$
$31$ $$( 393216 + 2304 T^{2} + T^{4} )^{2}$$
$37$ $$( 124416 + 756 T^{2} + T^{4} )^{2}$$
$41$ $$( -1028 - 64 T + T^{2} )^{4}$$
$43$ $$( 1364224 - 2528 T^{2} + T^{4} )^{2}$$
$47$ $$( 614656 - 3296 T^{2} + T^{4} )^{2}$$
$53$ $$( 124416 + 756 T^{2} + T^{4} )^{2}$$
$59$ $$( 69033984 + 16668 T^{2} + T^{4} )^{2}$$
$61$ $$( -38 + T )^{8}$$
$67$ $$( 7573504 - 9392 T^{2} + T^{4} )^{2}$$
$71$ $$( 884736 + 17136 T^{2} + T^{4} )^{2}$$
$73$ $$( 3538944 + 4176 T^{2} + T^{4} )^{2}$$
$79$ $$( 393216 + 2304 T^{2} + T^{4} )^{2}$$
$83$ $$( 2027776 - 4064 T^{2} + T^{4} )^{2}$$
$89$ $$( 3988 + 140 T + T^{2} )^{4}$$
$97$ $$( 495550464 + 45888 T^{2} + T^{4} )^{2}$$
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