# Properties

 Label 80.9.p.c Level $80$ Weight $9$ Character orbit 80.p Analytic conductor $32.590$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 80.p (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.5902888049$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} - 30 x^{3} + 1089 x^{2} - 3168 x + 4608$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{7}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 5) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 12 - 12 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{3} + ( 31 - 33 \beta_{1} + 18 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{5} + ( 371 + 371 \beta_{1} + 56 \beta_{2} + 7 \beta_{3} - 56 \beta_{5} ) q^{7} + ( -3105 \beta_{1} - 3 \beta_{3} + 3 \beta_{4} - 288 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( 12 - 12 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{3} + ( 31 - 33 \beta_{1} + 18 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{5} + ( 371 + 371 \beta_{1} + 56 \beta_{2} + 7 \beta_{3} - 56 \beta_{5} ) q^{7} + ( -3105 \beta_{1} - 3 \beta_{3} + 3 \beta_{4} - 288 \beta_{5} ) q^{9} + ( -3902 - 60 \beta_{2} + 85 \beta_{3} + 85 \beta_{4} ) q^{11} + ( -19778 + 19778 \beta_{1} - 317 \beta_{2} + 80 \beta_{4} - 317 \beta_{5} ) q^{13} + ( -40737 - 22209 \beta_{1} + 1564 \beta_{2} + 108 \beta_{3} - 181 \beta_{4} + 202 \beta_{5} ) q^{15} + ( -44567 - 44567 \beta_{1} + 1416 \beta_{2} - 466 \beta_{3} - 1416 \beta_{5} ) q^{17} + ( 54018 \beta_{1} - 684 \beta_{3} + 684 \beta_{4} - 1614 \beta_{5} ) q^{19} + ( 37884 + 3206 \beta_{2} - 511 \beta_{3} - 511 \beta_{4} ) q^{21} + ( -4861 + 4861 \beta_{1} + 1720 \beta_{2} - 1581 \beta_{4} + 1720 \beta_{5} ) q^{23} + ( -59555 + 23490 \beta_{1} + 5460 \beta_{2} + 2245 \beta_{3} + 785 \beta_{4} + 12530 \beta_{5} ) q^{25} + ( -68724 - 68724 \beta_{1} + 10308 \beta_{2} - 156 \beta_{3} - 10308 \beta_{5} ) q^{27} + ( 432678 \beta_{1} + 2386 \beta_{3} - 2386 \beta_{4} - 13594 \beta_{5} ) q^{29} + ( 137098 - 31380 \beta_{2} - 3045 \beta_{3} - 3045 \beta_{4} ) q^{31} + ( 718296 - 718296 \beta_{1} - 13478 \beta_{2} - 6862 \beta_{4} - 13478 \beta_{5} ) q^{33} + ( 815864 + 220598 \beta_{1} + 37667 \beta_{2} + 3549 \beta_{3} - 2968 \beta_{4} - 6069 \beta_{5} ) q^{35} + ( -77174 - 77174 \beta_{1} + 3015 \beta_{2} + 1506 \beta_{3} - 3015 \beta_{5} ) q^{37} + ( -413502 \beta_{1} + 17417 \beta_{3} - 17417 \beta_{4} - 3368 \beta_{5} ) q^{39} + ( 411182 + 36180 \beta_{2} - 12505 \beta_{3} - 12505 \beta_{4} ) q^{41} + ( -145474 + 145474 \beta_{1} + 45421 \beta_{2} - 5323 \beta_{4} + 45421 \beta_{5} ) q^{43} + ( 14787 + 2531709 \beta_{1} + 100461 \beta_{2} - 6258 \beta_{3} - 33219 \beta_{4} + 12798 \beta_{5} ) q^{45} + ( 2536385 + 2536385 \beta_{1} + 43258 \beta_{2} + 4263 \beta_{3} - 43258 \beta_{5} ) q^{47} + ( -1423499 \beta_{1} + 8575 \beta_{3} - 8575 \beta_{4} - 210700 \beta_{5} ) q^{49} + ( -5992656 + 299558 \beta_{2} - 52723 \beta_{3} - 52723 \beta_{4} ) q^{51} + ( -2296052 + 2296052 \beta_{1} + 137781 \beta_{2} - 4186 \beta_{4} + 137781 \beta_{5} ) q^{53} + ( -756542 - 5272794 \beta_{1} + 61524 \beta_{2} + 25703 \beta_{3} - 41621 \beta_{4} - 73418 \beta_{5} ) q^{55} + ( -5837922 - 5837922 \beta_{1} + 137874 \beta_{2} + 63132 \beta_{3} - 137874 \beta_{5} ) q^{57} + ( -1850946 \beta_{1} - 30152 \beta_{3} + 30152 \beta_{4} - 538642 \beta_{5} ) q^{59} + ( 3894782 + 562500 \beta_{2} - 95625 \beta_{3} - 95625 \beta_{4} ) q^{61} + ( -7407225 + 7407225 \beta_{1} - 169932 \beta_{2} - 27237 \beta_{4} - 169932 \beta_{5} ) q^{63} + ( -5122253 + 1965779 \beta_{1} - 230034 \beta_{2} + 137927 \beta_{3} - 12939 \beta_{4} - 454412 \beta_{5} ) q^{65} + ( 5437022 + 5437022 \beta_{1} - 851 \beta_{2} + 103661 \beta_{3} + 851 \beta_{5} ) q^{67} + ( 15024180 \beta_{1} - 7561 \beta_{3} + 7561 \beta_{4} + 439894 \beta_{5} ) q^{69} + ( 2041738 + 1027500 \beta_{2} - 78125 \beta_{3} - 78125 \beta_{4} ) q^{71} + ( 18724411 - 18724411 \beta_{1} - 256170 \beta_{2} + 12756 \beta_{4} - 256170 \beta_{5} ) q^{73} + ( 21371610 - 673230 \beta_{1} - 675295 \beta_{2} + 154260 \beta_{3} - 197195 \beta_{4} + 775565 \beta_{5} ) q^{75} + ( 4388258 + 4388258 \beta_{1} + 76048 \beta_{2} - 110754 \beta_{3} - 76048 \beta_{5} ) q^{77} + ( -21443688 \beta_{1} - 122456 \beta_{3} + 122456 \beta_{4} + 1191224 \beta_{5} ) q^{79} + ( 11351097 - 776952 \beta_{2} - 179613 \beta_{3} - 179613 \beta_{4} ) q^{81} + ( 2471480 - 2471480 \beta_{1} - 243037 \beta_{2} + 212813 \beta_{4} - 243037 \beta_{5} ) q^{83} + ( -3162344 + 26696542 \beta_{1} - 283607 \beta_{2} + 200271 \beta_{3} - 286147 \beta_{4} + 1625149 \beta_{5} ) q^{85} + ( 22223538 + 22223538 \beta_{1} - 217746 \beta_{2} + 150972 \beta_{3} + 217746 \beta_{5} ) q^{87} + ( -37413756 \beta_{1} - 442572 \beta_{3} + 442572 \beta_{4} + 224388 \beta_{5} ) q^{89} + ( -26889716 - 2653462 \beta_{2} - 224203 \beta_{3} - 224203 \beta_{4} ) q^{91} + ( -16007064 + 16007064 \beta_{1} - 1056838 \beta_{2} + 679018 \beta_{4} - 1056838 \beta_{5} ) q^{93} + ( -40465800 + 17098650 \beta_{1} + 942600 \beta_{2} + 410700 \beta_{3} + 213600 \beta_{4} - 100950 \beta_{5} ) q^{95} + ( -31155095 - 31155095 \beta_{1} - 724088 \beta_{2} + 861272 \beta_{3} + 724088 \beta_{5} ) q^{97} + ( 9673830 \beta_{1} - 486789 \beta_{3} + 486789 \beta_{4} - 1102644 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 72q^{3} + 220q^{5} + 2352q^{7} + O(q^{10})$$ $$6q + 72q^{3} + 220q^{5} + 2352q^{7} - 23192q^{11} - 119142q^{13} - 241440q^{15} - 265502q^{17} + 231672q^{21} - 28888q^{23} - 340350q^{25} - 392040q^{27} + 747648q^{31} + 4269096q^{33} + 4971680q^{35} - 454002q^{37} + 2489432q^{41} - 792648q^{43} + 210690q^{45} + 15313352q^{47} - 35567712q^{51} - 13509122q^{53} - 4448040q^{55} - 34625520q^{57} + 24111192q^{61} - 44837688q^{63} - 30943610q^{65} + 32827752q^{67} + 13992928q^{71} + 111859638q^{73} + 126793200q^{75} + 26260136q^{77} + 65834226q^{81} + 14768432q^{83} - 19713030q^{85} + 133207680q^{87} - 167542032q^{91} - 96798024q^{93} - 239661000q^{95} - 186656202q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} - 30 x^{3} + 1089 x^{2} - 3168 x + 4608$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$39 \nu^{5} - 22 \nu^{4} - 10 \nu^{3} + 790 \nu^{2} + 41631 \nu - 62928$$$$)/66000$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{5} + 44 \nu^{4} + 70 \nu^{3} - 30 \nu^{2} - 192 \nu + 27831$$$$)/825$$ $$\beta_{3}$$ $$=$$ $$($$$$687 \nu^{5} + 1474 \nu^{4} + 670 \nu^{3} - 52930 \nu^{2} + 1893423 \nu - 205824$$$$)/66000$$ $$\beta_{4}$$ $$=$$ $$($$$$-1647 \nu^{5} + 3806 \nu^{4} - 34270 \nu^{3} + 67330 \nu^{2} - 1801263 \nu + 5168544$$$$)/66000$$ $$\beta_{5}$$ $$=$$ $$($$$$-1721 \nu^{5} + 858 \nu^{4} + 4390 \nu^{3} + 93190 \nu^{2} - 1703409 \nu + 2598192$$$$)/66000$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + 2 \beta_{3} - \beta_{2} + 13 \beta_{1} + 13$$$$)/40$$ $$\nu^{2}$$ $$=$$ $$($$$$9 \beta_{5} + \beta_{4} - \beta_{3} + 457 \beta_{1}$$$$)/20$$ $$\nu^{3}$$ $$=$$ $$($$$$51 \beta_{5} - 62 \beta_{4} + 51 \beta_{2} - 577 \beta_{1} + 577$$$$)/40$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{4} + 5 \beta_{3} + 30 \beta_{2} - 1388$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-1419 \beta_{5} - 2038 \beta_{3} + 1419 \beta_{2} + 35153 \beta_{1} + 35153$$$$)/40$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 −4.23471 − 4.23471i 1.52966 + 1.52966i 3.70505 + 3.70505i −4.23471 + 4.23471i 1.52966 − 1.52966i 3.70505 − 3.70505i
0 −75.2981 + 75.2981i 0 −14.1685 + 624.839i 0 −730.992 730.992i 0 4778.60i 0
17.2 0 20.3321 20.3321i 0 558.542 280.457i 0 2415.21 + 2415.21i 0 5734.21i 0
17.3 0 90.9660 90.9660i 0 −434.373 449.383i 0 −508.219 508.219i 0 9988.61i 0
33.1 0 −75.2981 75.2981i 0 −14.1685 624.839i 0 −730.992 + 730.992i 0 4778.60i 0
33.2 0 20.3321 + 20.3321i 0 558.542 + 280.457i 0 2415.21 2415.21i 0 5734.21i 0
33.3 0 90.9660 + 90.9660i 0 −434.373 + 449.383i 0 −508.219 + 508.219i 0 9988.61i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 33.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.9.p.c 6
4.b odd 2 1 5.9.c.a 6
5.c odd 4 1 inner 80.9.p.c 6
12.b even 2 1 45.9.g.a 6
20.d odd 2 1 25.9.c.b 6
20.e even 4 1 5.9.c.a 6
20.e even 4 1 25.9.c.b 6
60.l odd 4 1 45.9.g.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.9.c.a 6 4.b odd 2 1
5.9.c.a 6 20.e even 4 1
25.9.c.b 6 20.d odd 2 1
25.9.c.b 6 20.e even 4 1
45.9.g.a 6 12.b even 2 1
45.9.g.a 6 60.l odd 4 1
80.9.p.c 6 1.a even 1 1 trivial
80.9.p.c 6 5.c odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 72 T_{3}^{5} + 2592 T_{3}^{4} + 383400 T_{3}^{3} + 170615844 T_{3}^{2} - 7276369968 T_{3} + 155160150048$$ acting on $$S_{9}^{\mathrm{new}}(80, [\chi])$$.