# Properties

 Label 1620.2.r.h Level $1620$ Weight $2$ Character orbit 1620.r Analytic conductor $12.936$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1620.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.9357651274$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 3 x^{14} - 11 x^{12} - 90 x^{10} - 450 x^{8} - 2250 x^{6} - 6875 x^{4} + 46875 x^{2} + 390625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{14} q^{5} + ( \beta_{2} - \beta_{12} ) q^{7} +O(q^{10})$$ $$q + \beta_{14} q^{5} + ( \beta_{2} - \beta_{12} ) q^{7} + ( -\beta_{1} - \beta_{4} - \beta_{8} - \beta_{14} ) q^{11} + ( -\beta_{3} + \beta_{9} + \beta_{13} ) q^{13} + ( -\beta_{4} + \beta_{7} + \beta_{10} + \beta_{15} ) q^{17} + ( -1 - \beta_{9} + \beta_{11} + \beta_{13} ) q^{19} + ( 2 \beta_{1} - 2 \beta_{10} - 2 \beta_{14} ) q^{23} + ( -\beta_{2} - \beta_{6} - \beta_{11} + \beta_{12} ) q^{25} + 3 \beta_{8} q^{29} + ( 1 + \beta_{3} - \beta_{6} - \beta_{9} + \beta_{13} ) q^{31} + ( -\beta_{7} + 2 \beta_{10} - \beta_{15} ) q^{35} + ( -\beta_{2} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{37} + ( \beta_{1} + \beta_{5} - \beta_{10} + \beta_{14} ) q^{41} + ( -\beta_{2} + \beta_{12} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{14} + \beta_{15} ) q^{47} + ( 7 + 2 \beta_{3} - 7 \beta_{6} - 2 \beta_{9} + 2 \beta_{13} ) q^{49} + ( -\beta_{7} - \beta_{15} ) q^{53} + ( 5 + \beta_{2} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{55} + ( \beta_{1} - 3 \beta_{5} - \beta_{10} + \beta_{14} ) q^{59} + ( \beta_{3} - 4 \beta_{6} - \beta_{11} ) q^{61} + ( -3 \beta_{1} - 5 \beta_{8} - \beta_{15} ) q^{65} + ( -2 \beta_{3} + 2 \beta_{9} + 2 \beta_{12} + 2 \beta_{13} ) q^{67} + ( 3 \beta_{4} - \beta_{5} - \beta_{8} + 3 \beta_{10} ) q^{71} + ( \beta_{2} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{73} + ( 2 \beta_{1} + \beta_{7} - 2 \beta_{10} - 2 \beta_{14} ) q^{77} + 6 \beta_{6} q^{79} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{14} - \beta_{15} ) q^{83} + ( -2 + 2 \beta_{6} - 2 \beta_{12} + 3 \beta_{13} ) q^{85} + ( 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{8} + 2 \beta_{10} ) q^{89} + ( 4 - 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{13} ) q^{91} + ( 3 \beta_{1} + 5 \beta_{5} - \beta_{7} - 3 \beta_{10} - 3 \beta_{14} ) q^{95} + ( -\beta_{2} - 2 \beta_{3} - 2 \beta_{11} + \beta_{12} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 24q^{19} - 6q^{25} + 4q^{31} + 48q^{49} + 80q^{55} - 28q^{61} + 48q^{79} - 22q^{85} + 48q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 3 x^{14} - 11 x^{12} - 90 x^{10} - 450 x^{8} - 2250 x^{6} - 6875 x^{4} + 46875 x^{2} + 390625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{12} - 72 \nu^{6} - 4625$$$$)/4500$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{14} + 178 \nu^{12} + 639 \nu^{10} + 1485 \nu^{8} + 7425 \nu^{6} - 64125 \nu^{4} - 597500 \nu^{2} - 2140625$$$$)/562500$$ $$\beta_{4}$$ $$=$$ $$($$$$-7 \nu^{15} + 204 \nu^{13} + 1377 \nu^{11} + 9405 \nu^{9} - 11475 \nu^{7} - 57375 \nu^{5} - 317500 \nu^{3} - 3984375 \nu$$$$)/5625000$$ $$\beta_{5}$$ $$=$$ $$($$$$7 \nu^{15} - 304 \nu^{13} - 927 \nu^{11} - 3555 \nu^{9} - 17775 \nu^{7} + 113625 \nu^{5} + 1048750 \nu^{3} + 4390625 \nu$$$$)/5625000$$ $$\beta_{6}$$ $$=$$ $$($$$$9 \nu^{14} + 52 \nu^{12} + 351 \nu^{10} - 585 \nu^{8} - 2925 \nu^{6} - 14625 \nu^{4} - 146250 \nu^{2} + 109375$$$$)/1125000$$ $$\beta_{7}$$ $$=$$ $$($$$$-8 \nu^{15} - 149 \nu^{13} + 963 \nu^{11} - 405 \nu^{9} - 2025 \nu^{7} - 235125 \nu^{5} + 476875 \nu^{3} + 1187500 \nu$$$$)/2812500$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{15} + 8 \nu^{13} + 9 \nu^{11} + 45 \nu^{9} + 945 \nu^{7} - 3375 \nu^{5} - 12500 \nu^{3} - 94375 \nu$$$$)/225000$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{14} + 8 \nu^{12} + 9 \nu^{10} + 45 \nu^{8} + 945 \nu^{6} - 3375 \nu^{4} - 12500 \nu^{2} - 49375$$$$)/45000$$ $$\beta_{10}$$ $$=$$ $$($$$$-9 \nu^{15} - 52 \nu^{13} - 351 \nu^{11} + 585 \nu^{9} + 2925 \nu^{7} + 14625 \nu^{5} + 146250 \nu^{3} + 1015625 \nu$$$$)/1125000$$ $$\beta_{11}$$ $$=$$ $$($$$$31 \nu^{14} + 18 \nu^{12} - 441 \nu^{10} - 3465 \nu^{8} - 17325 \nu^{6} - 176625 \nu^{4} + 40000 \nu^{2} + 2390625$$$$)/1125000$$ $$\beta_{12}$$ $$=$$ $$($$$$-16 \nu^{14} + 52 \nu^{12} + 351 \nu^{10} + 1215 \nu^{8} - 2925 \nu^{6} - 14625 \nu^{4} + 194375 \nu^{2} - 1015625$$$$)/562500$$ $$\beta_{13}$$ $$=$$ $$($$$$17 \nu^{14} + 76 \nu^{12} + 513 \nu^{10} + 3195 \nu^{8} - 4275 \nu^{6} - 21375 \nu^{4} - 426250 \nu^{2} - 1484375$$$$)/562500$$ $$\beta_{14}$$ $$=$$ $$($$$$\nu^{15} + 3 \nu^{13} - 11 \nu^{11} - 90 \nu^{9} - 450 \nu^{7} - 2250 \nu^{5} - 6875 \nu^{3} + 46875 \nu$$$$)/78125$$ $$\beta_{15}$$ $$=$$ $$($$$$-13 \nu^{15} + 41 \nu^{13} - 117 \nu^{11} - 585 \nu^{9} - 225 \nu^{7} + 43875 \nu^{5} + 162500 \nu^{3} - 456250 \nu$$$$)/562500$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{12} + \beta_{9} + \beta_{6} - \beta_{3} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} + 5 \beta_{8} + \beta_{7} + 5 \beta_{5} + 2 \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{12} - 5 \beta_{11} + 7 \beta_{6} - 2 \beta_{3} + 3 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$-9 \beta_{14} - 10 \beta_{10} - 8 \beta_{7} + 10 \beta_{5} + 10 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-10 \beta_{13} - 10 \beta_{11} + 26 \beta_{9} - 14 \beta_{2} + 9$$ $$\nu^{7}$$ $$=$$ $$-4 \beta_{15} - 58 \beta_{14} + 130 \beta_{8} - 58 \beta_{4} + 45 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$130 \beta_{13} + 57 \beta_{12} + 37 \beta_{9} - 383 \beta_{6} - 37 \beta_{3} + 383$$ $$\nu^{9}$$ $$=$$ $$-73 \beta_{15} + 290 \beta_{10} + 185 \beta_{8} - 73 \beta_{7} + 185 \beta_{5} + 504 \beta_{4}$$ $$\nu^{10}$$ $$=$$ $$509 \beta_{12} - 185 \beta_{11} + 2479 \beta_{6} - 144 \beta_{3} - 509 \beta_{2}$$ $$\nu^{11}$$ $$=$$ $$-1573 \beta_{14} - 2520 \beta_{10} + 324 \beta_{7} + 720 \beta_{5} + 2520 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-720 \beta_{13} - 720 \beta_{11} + 1872 \beta_{9} + 3492 \beta_{2} + 5273$$ $$\nu^{13}$$ $$=$$ $$4212 \beta_{15} + 4824 \beta_{14} + 9360 \beta_{8} + 4824 \beta_{4} + 7865 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$9360 \beta_{13} - 4771 \beta_{12} + 16289 \beta_{9} + 31049 \beta_{6} - 16289 \beta_{3} - 31049$$ $$\nu^{15}$$ $$=$$ $$-14131 \beta_{15} - 24120 \beta_{10} + 81445 \beta_{8} - 14131 \beta_{7} + 81445 \beta_{5} + 18538 \beta_{4}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$$.

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1 + \beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 2.23368 + 0.103355i 1.76348 + 1.37482i 1.02733 − 1.98610i 0.308893 + 2.21463i −0.308893 − 2.21463i −1.02733 + 1.98610i −1.76348 − 1.37482i −2.23368 − 0.103355i 2.23368 − 0.103355i 1.76348 − 1.37482i 1.02733 + 1.98610i 0.308893 − 2.21463i −0.308893 + 2.21463i −1.02733 − 1.98610i −1.76348 + 1.37482i −2.23368 + 0.103355i
0 0 0 −2.23368 + 0.103355i 0 −1.10979 + 0.640739i 0 0 0
109.2 0 0 0 −1.76348 + 1.37482i 0 −4.27415 + 2.46768i 0 0 0
109.3 0 0 0 −1.02733 1.98610i 0 1.10979 0.640739i 0 0 0
109.4 0 0 0 −0.308893 + 2.21463i 0 4.27415 2.46768i 0 0 0
109.5 0 0 0 0.308893 2.21463i 0 4.27415 2.46768i 0 0 0
109.6 0 0 0 1.02733 + 1.98610i 0 1.10979 0.640739i 0 0 0
109.7 0 0 0 1.76348 1.37482i 0 −4.27415 + 2.46768i 0 0 0
109.8 0 0 0 2.23368 0.103355i 0 −1.10979 + 0.640739i 0 0 0
1189.1 0 0 0 −2.23368 0.103355i 0 −1.10979 0.640739i 0 0 0
1189.2 0 0 0 −1.76348 1.37482i 0 −4.27415 2.46768i 0 0 0
1189.3 0 0 0 −1.02733 + 1.98610i 0 1.10979 + 0.640739i 0 0 0
1189.4 0 0 0 −0.308893 2.21463i 0 4.27415 + 2.46768i 0 0 0
1189.5 0 0 0 0.308893 + 2.21463i 0 4.27415 + 2.46768i 0 0 0
1189.6 0 0 0 1.02733 1.98610i 0 1.10979 + 0.640739i 0 0 0
1189.7 0 0 0 1.76348 + 1.37482i 0 −4.27415 2.46768i 0 0 0
1189.8 0 0 0 2.23368 + 0.103355i 0 −1.10979 0.640739i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1189.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.d odd 2 1 inner
45.h odd 6 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.r.h 16
3.b odd 2 1 inner 1620.2.r.h 16
5.b even 2 1 inner 1620.2.r.h 16
9.c even 3 1 1620.2.d.e 8
9.c even 3 1 inner 1620.2.r.h 16
9.d odd 6 1 1620.2.d.e 8
9.d odd 6 1 inner 1620.2.r.h 16
15.d odd 2 1 inner 1620.2.r.h 16
45.h odd 6 1 1620.2.d.e 8
45.h odd 6 1 inner 1620.2.r.h 16
45.j even 6 1 1620.2.d.e 8
45.j even 6 1 inner 1620.2.r.h 16
45.k odd 12 2 8100.2.a.be 8
45.l even 12 2 8100.2.a.be 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.d.e 8 9.c even 3 1
1620.2.d.e 8 9.d odd 6 1
1620.2.d.e 8 45.h odd 6 1
1620.2.d.e 8 45.j even 6 1
1620.2.r.h 16 1.a even 1 1 trivial
1620.2.r.h 16 3.b odd 2 1 inner
1620.2.r.h 16 5.b even 2 1 inner
1620.2.r.h 16 9.c even 3 1 inner
1620.2.r.h 16 9.d odd 6 1 inner
1620.2.r.h 16 15.d odd 2 1 inner
1620.2.r.h 16 45.h odd 6 1 inner
1620.2.r.h 16 45.j even 6 1 inner
8100.2.a.be 8 45.k odd 12 2
8100.2.a.be 8 45.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1620, [\chi])$$:

 $$T_{7}^{8} - 26 T_{7}^{6} + 636 T_{7}^{4} - 1040 T_{7}^{2} + 1600$$ $$T_{11}^{8} + 23 T_{11}^{6} + 429 T_{11}^{4} + 2300 T_{11}^{2} + 10000$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$390625 + 46875 T^{2} - 6875 T^{4} - 2250 T^{6} - 450 T^{8} - 90 T^{10} - 11 T^{12} + 3 T^{14} + T^{16}$$
$7$ $$( 1600 - 1040 T^{2} + 636 T^{4} - 26 T^{6} + T^{8} )^{2}$$
$11$ $$( 10000 + 2300 T^{2} + 429 T^{4} + 23 T^{6} + T^{8} )^{2}$$
$13$ $$( 129600 - 18360 T^{2} + 2241 T^{4} - 51 T^{6} + T^{8} )^{2}$$
$17$ $$( 1690 + 83 T^{2} + T^{4} )^{4}$$
$19$ $$( -30 + 3 T + T^{2} )^{8}$$
$23$ $$( 409600 - 43520 T^{2} + 3984 T^{4} - 68 T^{6} + T^{8} )^{2}$$
$29$ $$( 729 + 27 T^{2} + T^{4} )^{4}$$
$31$ $$( 1024 + 32 T + 33 T^{2} - T^{3} + T^{4} )^{4}$$
$37$ $$( 1690 + 89 T^{2} + T^{4} )^{4}$$
$41$ $$( 256 + 560 T^{2} + 1209 T^{4} + 35 T^{6} + T^{8} )^{2}$$
$43$ $$( 1600 - 1040 T^{2} + 636 T^{4} - 26 T^{6} + T^{8} )^{2}$$
$47$ $$( 16000000 - 680000 T^{2} + 24900 T^{4} - 170 T^{6} + T^{8} )^{2}$$
$53$ $$( 360 + 78 T^{2} + T^{4} )^{4}$$
$59$ $$( 4096 + 3776 T^{2} + 3417 T^{4} + 59 T^{6} + T^{8} )^{2}$$
$61$ $$( 400 - 140 T + 69 T^{2} + 7 T^{3} + T^{4} )^{4}$$
$67$ $$( 16000000 - 1040000 T^{2} + 63600 T^{4} - 260 T^{6} + T^{8} )^{2}$$
$71$ $$( 6084 - 231 T^{2} + T^{4} )^{4}$$
$73$ $$( 250 + 65 T^{2} + T^{4} )^{4}$$
$79$ $$( 36 - 6 T + T^{2} )^{8}$$
$83$ $$( 6553600 - 312320 T^{2} + 12324 T^{4} - 122 T^{6} + T^{8} )^{2}$$
$89$ $$( 961 - 110 T^{2} + T^{4} )^{4}$$
$97$ $$( 25600 - 32960 T^{2} + 42276 T^{4} - 206 T^{6} + T^{8} )^{2}$$