# Properties

 Label 1764.4.t.a Level $1764$ Weight $4$ Character orbit 1764.t Analytic conductor $104.079$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1764.t (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$104.079369250$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 42 x^{12} + 1139 x^{8} + 26250 x^{4} + 390625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{24}\cdot 3^{16}$$ Twist minimal: no (minimal twist has level 252) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{11} q^{5} +O(q^{10})$$ $$q + \beta_{11} q^{5} + ( -\beta_{2} - \beta_{4} ) q^{11} + ( \beta_{12} - \beta_{14} ) q^{13} + ( \beta_{7} + \beta_{13} - \beta_{15} ) q^{17} + ( -3 \beta_{9} + 2 \beta_{10} - 2 \beta_{14} ) q^{19} + ( 5 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} ) q^{23} + ( -5 \beta_{1} - 4 \beta_{8} ) q^{25} + ( 10 \beta_{4} + \beta_{6} ) q^{29} + ( 6 \beta_{9} - 6 \beta_{12} ) q^{31} + ( -64 - 64 \beta_{1} + 6 \beta_{5} ) q^{37} + ( 29 \beta_{7} - 29 \beta_{11} - 3 \beta_{13} ) q^{41} + ( 4 - 5 \beta_{5} + 5 \beta_{8} ) q^{43} + ( -32 \beta_{11} + 3 \beta_{15} ) q^{47} + ( -18 \beta_{2} - 13 \beta_{3} - 18 \beta_{4} ) q^{53} + ( -7 \beta_{12} - 4 \beta_{14} ) q^{55} + ( -12 \beta_{7} - 2 \beta_{13} + 2 \beta_{15} ) q^{59} + ( -20 \beta_{9} + 10 \beta_{10} - 10 \beta_{14} ) q^{61} + ( 8 \beta_{2} - 24 \beta_{3} - 24 \beta_{6} ) q^{65} + ( -292 \beta_{1} + 4 \beta_{8} ) q^{67} + ( -19 \beta_{4} - 10 \beta_{6} ) q^{71} + ( -3 \beta_{9} - 11 \beta_{10} + 3 \beta_{12} ) q^{73} + ( -224 - 224 \beta_{1} - 12 \beta_{5} ) q^{79} + ( -28 \beta_{7} + 28 \beta_{11} + 9 \beta_{13} ) q^{83} + ( -168 - 4 \beta_{5} + 4 \beta_{8} ) q^{85} + ( 19 \beta_{11} - 5 \beta_{15} ) q^{89} + ( 56 \beta_{2} - 88 \beta_{3} + 56 \beta_{4} ) q^{95} + ( -\beta_{12} - 15 \beta_{14} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q + 40q^{25} - 512q^{37} + 64q^{43} + 2336q^{67} - 1792q^{79} - 2688q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 42 x^{12} + 1139 x^{8} + 26250 x^{4} + 390625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$42 \nu^{12} + 1139 \nu^{8} + 47838 \nu^{4} + 390625$$$$)/711875$$ $$\beta_{2}$$ $$=$$ $$($$$$-771 \nu^{12} - 71757 \nu^{8} - 878169 \nu^{4} - 20238750$$$$)/1423750$$ $$\beta_{3}$$ $$=$$ $$($$$$801 \nu^{14} + 174267 \nu^{10} + 912339 \nu^{6} + 21026250 \nu^{2}$$$$)/35593750$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{12} - 6993$$$$)/2278$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{14} - 150189 \nu^{2}$$$$)/28475$$ $$\beta_{6}$$ $$=$$ $$($$$$-378 \nu^{14} - 10251 \nu^{10} - 53667 \nu^{6} - 3515625 \nu^{2}$$$$)/2093750$$ $$\beta_{7}$$ $$=$$ $$($$$$-1028 \nu^{15} - 21000 \nu^{13} - 95676 \nu^{11} - 569500 \nu^{9} - 2594642 \nu^{7} - 23919000 \nu^{5} - 86782500 \nu^{3} - 373281250 \nu$$$$)/88984375$$ $$\beta_{8}$$ $$=$$ $$($$$$-6567 \nu^{14} - 228939 \nu^{10} - 7479813 \nu^{6} - 172383750 \nu^{2}$$$$)/17796875$$ $$\beta_{9}$$ $$=$$ $$($$$$-2861 \nu^{15} - 7600 \nu^{13} - 94537 \nu^{11} - 1053575 \nu^{9} - 4682429 \nu^{7} - 26453275 \nu^{5} + 43070000 \nu^{3} - 857984375 \nu$$$$)/88984375$$ $$\beta_{10}$$ $$=$$ $$($$$$-3304 \nu^{15} - 1850 \nu^{13} + 384982 \nu^{11} + 797300 \nu^{9} + 6202994 \nu^{7} - 37700900 \nu^{5} + 153883750 \nu^{3} - 297718750 \nu$$$$)/88984375$$ $$\beta_{11}$$ $$=$$ $$($$$$4778 \nu^{15} - 10500 \nu^{13} + 95676 \nu^{11} - 284750 \nu^{9} + 2594642 \nu^{7} - 11959500 \nu^{5} + 5827500 \nu^{3} - 453593750 \nu$$$$)/88984375$$ $$\beta_{12}$$ $$=$$ $$($$$$-7597 \nu^{15} + 54475 \nu^{13} - 189074 \nu^{11} + 1053575 \nu^{9} - 9364858 \nu^{7} + 26453275 \nu^{5} - 140335625 \nu^{3} + 113000000 \nu$$$$)/88984375$$ $$\beta_{13}$$ $$=$$ $$($$$$-734 \nu^{15} + 13050 \nu^{13} - 45828 \nu^{11} + 204350 \nu^{9} + 252724 \nu^{7} + 2301450 \nu^{5} - 4946250 \nu^{3} + 87125000 \nu$$$$)/5234375$$ $$\beta_{14}$$ $$=$$ $$($$$$19642 \nu^{15} + 45950 \nu^{13} + 769964 \nu^{11} + 398650 \nu^{9} + 12405988 \nu^{7} - 18850450 \nu^{5} + 274988750 \nu^{3} + 707875000 \nu$$$$)/88984375$$ $$\beta_{15}$$ $$=$$ $$($$$$-30614 \nu^{15} + 59400 \nu^{13} - 389538 \nu^{11} - 3473950 \nu^{9} + 2148154 \nu^{7} - 39124650 \nu^{5} - 316695000 \nu^{3} - 611968750 \nu$$$$)/88984375$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{14} - 2 \beta_{12} - 12 \beta_{11} - \beta_{10} - 2 \beta_{9} + 6 \beta_{7}$$$$)/72$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{6} - 3 \beta_{5} - 2 \beta_{3}$$$$)/18$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{15} - \beta_{14} + 2 \beta_{13} - 14 \beta_{12} - 26 \beta_{11} + 2 \beta_{10} + 28 \beta_{9} - 26 \beta_{7}$$$$)/72$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{4} + 2 \beta_{2} + 63 \beta_{1}$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$10 \beta_{15} - 37 \beta_{14} - 20 \beta_{13} + 28 \beta_{12} + 122 \beta_{11} - 37 \beta_{10} - 14 \beta_{9} - 244 \beta_{7}$$$$)/72$$ $$\nu^{6}$$ $$=$$ $$($$$$-51 \beta_{8} + 134 \beta_{6} + 51 \beta_{5}$$$$)/18$$ $$\nu^{7}$$ $$=$$ $$($$$$84 \beta_{15} - 166 \beta_{14} + 84 \beta_{13} - 338 \beta_{12} + 684 \beta_{11} + 83 \beta_{10} - 338 \beta_{9} - 342 \beta_{7}$$$$)/72$$ $$\nu^{8}$$ $$=$$ $$-28 \beta_{2} - 257 \beta_{1} - 257$$ $$\nu^{9}$$ $$=$$ $$($$$$-840 \beta_{15} - 929 \beta_{14} + 420 \beta_{13} + 662 \beta_{12} + 1374 \beta_{11} + 1858 \beta_{10} - 1324 \beta_{9} + 1374 \beta_{7}$$$$)/72$$ $$\nu^{10}$$ $$=$$ $$($$$$267 \beta_{8} + 4378 \beta_{3}$$$$)/18$$ $$\nu^{11}$$ $$=$$ $$($$$$2278 \beta_{15} + 4111 \beta_{14} - 4556 \beta_{13} + 10892 \beta_{12} + 1886 \beta_{11} + 4111 \beta_{10} - 5446 \beta_{9} - 3772 \beta_{7}$$$$)/72$$ $$\nu^{12}$$ $$=$$ $$($$$$-2278 \beta_{4} - 6993$$$$)/3$$ $$\nu^{13}$$ $$=$$ $$($$$$11390 \beta_{15} + 31786 \beta_{14} + 11390 \beta_{13} + 36554 \beta_{12} + 37084 \beta_{11} - 15893 \beta_{10} + 36554 \beta_{9} - 18542 \beta_{7}$$$$)/72$$ $$\nu^{14}$$ $$=$$ $$($$$$-100126 \beta_{6} + 20661 \beta_{5} - 100126 \beta_{3}$$$$)/18$$ $$\nu^{15}$$ $$=$$ $$($$$$-86352 \beta_{15} + 120787 \beta_{14} + 43176 \beta_{13} - 17482 \beta_{12} + 292962 \beta_{11} - 241574 \beta_{10} + 34964 \beta_{9} + 292962 \beta_{7}$$$$)/72$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1 + \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
521.1
 2.07637 + 0.829861i 1.75687 + 1.38326i −0.829861 + 2.07637i 1.38326 − 1.75687i 0.829861 − 2.07637i −1.38326 + 1.75687i −2.07637 − 0.829861i −1.75687 − 1.38326i 2.07637 − 0.829861i 1.75687 − 1.38326i −0.829861 − 2.07637i 1.38326 + 1.75687i 0.829861 + 2.07637i −1.38326 − 1.75687i −2.07637 + 0.829861i −1.75687 + 1.38326i
0 0 0 −7.66648 13.2787i 0 0 0 0 0
521.2 0 0 0 −7.66648 13.2787i 0 0 0 0 0
521.3 0 0 0 −1.10680 1.91704i 0 0 0 0 0
521.4 0 0 0 −1.10680 1.91704i 0 0 0 0 0
521.5 0 0 0 1.10680 + 1.91704i 0 0 0 0 0
521.6 0 0 0 1.10680 + 1.91704i 0 0 0 0 0
521.7 0 0 0 7.66648 + 13.2787i 0 0 0 0 0
521.8 0 0 0 7.66648 + 13.2787i 0 0 0 0 0
1097.1 0 0 0 −7.66648 + 13.2787i 0 0 0 0 0
1097.2 0 0 0 −7.66648 + 13.2787i 0 0 0 0 0
1097.3 0 0 0 −1.10680 + 1.91704i 0 0 0 0 0
1097.4 0 0 0 −1.10680 + 1.91704i 0 0 0 0 0
1097.5 0 0 0 1.10680 1.91704i 0 0 0 0 0
1097.6 0 0 0 1.10680 1.91704i 0 0 0 0 0
1097.7 0 0 0 7.66648 13.2787i 0 0 0 0 0
1097.8 0 0 0 7.66648 13.2787i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1097.8 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
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.4.t.a 16
3.b odd 2 1 inner 1764.4.t.a 16
7.b odd 2 1 inner 1764.4.t.a 16
7.c even 3 1 252.4.f.a 8
7.c even 3 1 inner 1764.4.t.a 16
7.d odd 6 1 252.4.f.a 8
7.d odd 6 1 inner 1764.4.t.a 16
21.c even 2 1 inner 1764.4.t.a 16
21.g even 6 1 252.4.f.a 8
21.g even 6 1 inner 1764.4.t.a 16
21.h odd 6 1 252.4.f.a 8
21.h odd 6 1 inner 1764.4.t.a 16
28.f even 6 1 1008.4.k.d 8
28.g odd 6 1 1008.4.k.d 8
84.j odd 6 1 1008.4.k.d 8
84.n even 6 1 1008.4.k.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.f.a 8 7.c even 3 1
252.4.f.a 8 7.d odd 6 1
252.4.f.a 8 21.g even 6 1
252.4.f.a 8 21.h odd 6 1
1008.4.k.d 8 28.f even 6 1
1008.4.k.d 8 28.g odd 6 1
1008.4.k.d 8 84.j odd 6 1
1008.4.k.d 8 84.n even 6 1
1764.4.t.a 16 1.a even 1 1 trivial
1764.4.t.a 16 3.b odd 2 1 inner
1764.4.t.a 16 7.b odd 2 1 inner
1764.4.t.a 16 7.c even 3 1 inner
1764.4.t.a 16 7.d odd 6 1 inner
1764.4.t.a 16 21.c even 2 1 inner
1764.4.t.a 16 21.g even 6 1 inner
1764.4.t.a 16 21.h odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 240 T_{5}^{6} + 56448 T_{5}^{4} + 276480 T_{5}^{2} + 1327104$$ acting on $$S_{4}^{\mathrm{new}}(1764, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$( 1327104 + 276480 T^{2} + 56448 T^{4} + 240 T^{6} + T^{8} )^{2}$$
$7$ $$T^{16}$$
$11$ $$( 171396 - 414 T^{2} + T^{4} )^{4}$$
$13$ $$( 3875328 + 5088 T^{2} + T^{4} )^{4}$$
$17$ $$( 37903417344 + 3186653184 T^{2} + 267716736 T^{4} + 16368 T^{6} + T^{8} )^{2}$$
$19$ $$( 2767192326144 - 36250730496 T^{2} + 473227776 T^{4} - 21792 T^{6} + T^{8} )^{2}$$
$23$ $$( 8860231742470416 - 2070457172784 T^{2} + 389695212 T^{4} - 21996 T^{6} + T^{8} )^{2}$$
$29$ $$( 1700572644 + 83124 T^{2} + T^{4} )^{4}$$
$31$ $$( 85630228705050624 - 11124486438912 T^{2} + 1152589824 T^{4} - 38016 T^{6} + T^{8} )^{2}$$
$37$ $$( 661106944 - 3291136 T + 42096 T^{2} + 128 T^{3} + T^{4} )^{4}$$
$41$ $$( 22577275008 - 328560 T^{2} + T^{4} )^{4}$$
$43$ $$( -20684 - 8 T + T^{2} )^{8}$$
$47$ $$($$$$17\!\cdots\!24$$$$+ 16927197799514112 T^{2} + 124623489024 T^{4} + 407616 T^{6} + T^{8} )^{2}$$
$53$ $$($$$$12\!\cdots\!96$$$$- 3681637515747792 T^{2} + 92949818220 T^{4} - 323028 T^{6} + T^{8} )^{2}$$
$59$ $$( 3932242550345170944 + 204073290694656 T^{2} + 8607891456 T^{4} + 102912 T^{6} + T^{8} )^{2}$$
$61$ $$($$$$49\!\cdots\!00$$$$- 15843852288000000 T^{2} + 482365440000 T^{4} - 710400 T^{6} + T^{8} )^{2}$$
$67$ $$( 5186304256 - 42057344 T + 269040 T^{2} - 584 T^{3} + T^{4} )^{4}$$
$71$ $$( 17756628516 + 331308 T^{2} + T^{4} )^{4}$$
$73$ $$($$$$28\!\cdots\!04$$$$- 24883615693651968 T^{2} + 161233519104 T^{4} - 463584 T^{6} + T^{8} )^{2}$$
$79$ $$( 4768731136 - 30937088 T + 269760 T^{2} + 448 T^{3} + T^{4} )^{4}$$
$83$ $$( 328711292928 - 1430592 T^{2} + T^{4} )^{4}$$
$89$ $$($$$$58\!\cdots\!84$$$$+ 12166978337372160 T^{2} + 229115392128 T^{4} + 503280 T^{6} + T^{8} )^{2}$$
$97$ $$( 149346427392 + 816096 T^{2} + T^{4} )^{4}$$