# Properties

 Label 80.6.n.b Level 80 Weight 6 Character orbit 80.n Analytic conductor 12.831 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 80.n (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.8307055850$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{195})$$ Defining polynomial: $$x^{4} - 97 x^{2} + 2401$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( -25 - 50 \beta_{1} ) q^{5} + 9 \beta_{3} q^{7} + 147 \beta_{1} q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( -25 - 50 \beta_{1} ) q^{5} + 9 \beta_{3} q^{7} + 147 \beta_{1} q^{9} + ( -15 \beta_{2} + 15 \beta_{3} ) q^{11} + ( -695 + 695 \beta_{1} ) q^{13} + ( 25 \beta_{2} - 50 \beta_{3} ) q^{15} + ( -95 - 95 \beta_{1} ) q^{17} + ( -30 \beta_{2} - 30 \beta_{3} ) q^{19} -3510 q^{21} + 183 \beta_{2} q^{23} + ( -1875 + 2500 \beta_{1} ) q^{25} -96 \beta_{3} q^{27} + 2876 \beta_{1} q^{29} + ( -345 \beta_{2} + 345 \beta_{3} ) q^{31} + ( -5850 + 5850 \beta_{1} ) q^{33} + ( -450 \beta_{2} - 225 \beta_{3} ) q^{35} + ( -5155 - 5155 \beta_{1} ) q^{37} + ( 695 \beta_{2} + 695 \beta_{3} ) q^{39} + 9218 q^{41} + 111 \beta_{2} q^{43} + ( 7350 - 3675 \beta_{1} ) q^{45} -903 \beta_{3} q^{47} -14783 \beta_{1} q^{49} + ( 95 \beta_{2} - 95 \beta_{3} ) q^{51} + ( 11345 - 11345 \beta_{1} ) q^{53} + ( -375 \beta_{2} - 1125 \beta_{3} ) q^{55} + ( 11700 + 11700 \beta_{1} ) q^{57} + ( 30 \beta_{2} + 30 \beta_{3} ) q^{59} + 18678 q^{61} + 1323 \beta_{2} q^{63} + ( 52125 + 17375 \beta_{1} ) q^{65} + 1245 \beta_{3} q^{67} -71370 \beta_{1} q^{69} + ( -2085 \beta_{2} + 2085 \beta_{3} ) q^{71} + ( -24055 + 24055 \beta_{1} ) q^{73} + ( 1875 \beta_{2} + 2500 \beta_{3} ) q^{75} + ( -52650 - 52650 \beta_{1} ) q^{77} + ( -3360 \beta_{2} - 3360 \beta_{3} ) q^{79} + 73161 q^{81} -2553 \beta_{2} q^{83} + ( -2375 + 7125 \beta_{1} ) q^{85} + 2876 \beta_{3} q^{87} + 74296 \beta_{1} q^{89} + ( 6255 \beta_{2} - 6255 \beta_{3} ) q^{91} + ( -134550 + 134550 \beta_{1} ) q^{93} + ( 2250 \beta_{2} - 750 \beta_{3} ) q^{95} + ( -49255 - 49255 \beta_{1} ) q^{97} + ( 2205 \beta_{2} + 2205 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 100q^{5} + O(q^{10})$$ $$4q - 100q^{5} - 2780q^{13} - 380q^{17} - 14040q^{21} - 7500q^{25} - 23400q^{33} - 20620q^{37} + 36872q^{41} + 29400q^{45} + 45380q^{53} + 46800q^{57} + 74712q^{61} + 208500q^{65} - 96220q^{73} - 210600q^{77} + 292644q^{81} - 9500q^{85} - 538200q^{93} - 197020q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 97 x^{2} + 2401$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 48 \nu$$$$)/49$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 98 \nu^{2} + 146 \nu - 4753$$$$)/49$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} - 98 \nu^{2} + 146 \nu + 4753$$$$)/49$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 2 \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} + 194$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$12 \beta_{3} + 12 \beta_{2} + 73 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$21$$ $$31$$ $$\chi(n)$$ $$\beta_{1}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 6.98212 + 0.500000i −6.98212 + 0.500000i 6.98212 − 0.500000i −6.98212 − 0.500000i
0 −13.9642 13.9642i 0 −25.0000 50.0000i 0 125.678 125.678i 0 147.000i 0
47.2 0 13.9642 + 13.9642i 0 −25.0000 50.0000i 0 −125.678 + 125.678i 0 147.000i 0
63.1 0 −13.9642 + 13.9642i 0 −25.0000 + 50.0000i 0 125.678 + 125.678i 0 147.000i 0
63.2 0 13.9642 13.9642i 0 −25.0000 + 50.0000i 0 −125.678 125.678i 0 147.000i 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
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.6.n.b 4
4.b odd 2 1 inner 80.6.n.b 4
5.b even 2 1 400.6.n.d 4
5.c odd 4 1 inner 80.6.n.b 4
5.c odd 4 1 400.6.n.d 4
20.d odd 2 1 400.6.n.d 4
20.e even 4 1 inner 80.6.n.b 4
20.e even 4 1 400.6.n.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.6.n.b 4 1.a even 1 1 trivial
80.6.n.b 4 4.b odd 2 1 inner
80.6.n.b 4 5.c odd 4 1 inner
80.6.n.b 4 20.e even 4 1 inner
400.6.n.d 4 5.b even 2 1
400.6.n.d 4 5.c odd 4 1
400.6.n.d 4 20.d odd 2 1
400.6.n.d 4 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 152100$$ acting on $$S_{6}^{\mathrm{new}}(80, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 108882 T^{4} + 3486784401 T^{8}$$
$5$ $$( 1 + 50 T + 3125 T^{2} )^{2}$$
$7$ $$1 - 560853922 T^{4} + 79792266297612001 T^{8}$$
$11$ $$( 1 - 146602 T^{2} + 25937424601 T^{4} )^{2}$$
$13$ $$( 1 + 1390 T + 966050 T^{2} + 516097270 T^{3} + 137858491849 T^{4} )^{2}$$
$17$ $$( 1 + 190 T + 18050 T^{2} + 269772830 T^{3} + 2015993900449 T^{4} )^{2}$$
$19$ $$( 1 + 4250198 T^{2} + 6131066257801 T^{4} )^{2}$$
$23$ $$1 - 82817669402722 T^{4} +$$$$17\!\cdots\!01$$$$T^{8}$$
$29$ $$( 1 - 32750922 T^{2} + 420707233300201 T^{4} )^{2}$$
$31$ $$( 1 + 35581198 T^{2} + 819628286980801 T^{4} )^{2}$$
$37$ $$( 1 + 10310 T + 53148050 T^{2} + 714936196670 T^{3} + 4808584372417849 T^{4} )^{2}$$
$41$ $$( 1 - 9218 T + 115856201 T^{2} )^{4}$$
$43$ $$1 + 40420440476627918 T^{4} +$$$$46\!\cdots\!01$$$$T^{8}$$
$47$ $$1 - 85407260265966082 T^{4} +$$$$27\!\cdots\!01$$$$T^{8}$$
$53$ $$( 1 - 22690 T + 257418050 T^{2} - 9488855736170 T^{3} + 174887470365513049 T^{4} )^{2}$$
$59$ $$( 1 + 1429146598 T^{2} + 511116753300641401 T^{4} )^{2}$$
$61$ $$( 1 - 18678 T + 844596301 T^{2} )^{4}$$
$67$ $$1 + 746452483343412398 T^{4} +$$$$33\!\cdots\!01$$$$T^{8}$$
$71$ $$( 1 - 217623202 T^{2} + 3255243551009881201 T^{4} )^{2}$$
$73$ $$( 1 + 48110 T + 1157286050 T^{2} + 99735474339230 T^{3} + 4297625829703557649 T^{4} )^{2}$$
$79$ $$( 1 - 2651775202 T^{2} + 9468276082626847201 T^{4} )^{2}$$
$83$ $$1 - 2557737354504584722 T^{4} +$$$$24\!\cdots\!01$$$$T^{8}$$
$89$ $$( 1 - 5648223282 T^{2} + 31181719929966183601 T^{4} )^{2}$$
$97$ $$( 1 + 98510 T + 4852110050 T^{2} + 845938888717070 T^{3} + 73742412689492826049 T^{4} )^{2}$$