# Properties

 Label 1323.2.h.e Level 1323 Weight 2 Character orbit 1323.h Analytic conductor 10.564 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.h (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.5642081874$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 Defining polynomial: $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 - \beta_{1} + \beta_{3} ) q^{4} + ( \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{5} + ( 2 - \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 - \beta_{1} + \beta_{3} ) q^{4} + ( \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{5} + ( 2 - \beta_{1} + \beta_{3} ) q^{8} + ( -3 \beta_{1} - \beta_{4} + 3 \beta_{5} ) q^{10} + ( -1 - \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{11} + ( 1 - \beta_{4} ) q^{13} + ( -2 \beta_{1} - \beta_{3} ) q^{16} + ( \beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{17} + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} ) q^{19} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + 5 \beta_{4} + 2 \beta_{5} ) q^{20} + ( 5 - 2 \beta_{2} - 5 \beta_{4} - 2 \beta_{5} ) q^{22} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{23} + ( -3 + 2 \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{25} -\beta_{5} q^{26} + ( -\beta_{1} + \beta_{5} ) q^{29} + ( 2 + 2 \beta_{1} + \beta_{3} ) q^{31} + ( 3 + \beta_{1} ) q^{32} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{34} + 3 \beta_{2} q^{37} + ( -2 + \beta_{2} + 2 \beta_{4} + 3 \beta_{5} ) q^{38} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} ) q^{40} + ( 7 + \beta_{2} - 7 \beta_{4} ) q^{41} + ( -4 \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{5} ) q^{43} + ( 6 - 6 \beta_{4} - 5 \beta_{5} ) q^{44} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{46} + ( -3 + 3 \beta_{1} - 3 \beta_{3} ) q^{47} + ( 5 - \beta_{2} - 5 \beta_{4} + 4 \beta_{5} ) q^{50} + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} ) q^{52} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{53} + ( -5 \beta_{1} - \beta_{3} ) q^{55} + ( -\beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{58} + ( -4 - \beta_{1} - 2 \beta_{3} ) q^{59} + ( 1 - \beta_{1} - 2 \beta_{3} ) q^{61} + ( -7 - \beta_{1} - 2 \beta_{3} ) q^{62} + ( -3 + 2 \beta_{1} + \beta_{3} ) q^{64} + ( 2 + \beta_{3} ) q^{65} + ( 2 + 7 \beta_{1} - \beta_{3} ) q^{67} + ( -\beta_{1} + 4 \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{5} ) q^{68} + ( -2 + \beta_{1} + 2 \beta_{3} ) q^{71} + ( -\beta_{1} - 5 \beta_{2} - 5 \beta_{3} - \beta_{4} + \beta_{5} ) q^{73} + ( 3 - 3 \beta_{4} + 3 \beta_{5} ) q^{74} + ( -6 + \beta_{2} + 6 \beta_{4} + 4 \beta_{5} ) q^{76} + ( 3 - 5 \beta_{1} - \beta_{3} ) q^{79} + ( -7 \beta_{1} - 6 \beta_{4} + 7 \beta_{5} ) q^{80} + ( 1 - \beta_{4} - 6 \beta_{5} ) q^{82} + ( \beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{83} + ( -5 + 4 \beta_{2} + 5 \beta_{4} - 2 \beta_{5} ) q^{85} + ( -5 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 11 \beta_{4} + 5 \beta_{5} ) q^{86} + ( 5 - \beta_{2} - 5 \beta_{4} - 7 \beta_{5} ) q^{88} + ( -1 - \beta_{2} + \beta_{4} - 6 \beta_{5} ) q^{89} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{92} + ( -6 + 9 \beta_{1} - 3 \beta_{3} ) q^{94} + ( -5 + 2 \beta_{1} - \beta_{3} ) q^{95} + ( 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 2q^{2} + 6q^{4} + 5q^{5} + 12q^{8} + O(q^{10})$$ $$6q + 2q^{2} + 6q^{4} + 5q^{5} + 12q^{8} - 2q^{11} + 3q^{13} + 6q^{16} + 12q^{17} - 3q^{19} + 16q^{20} + 15q^{22} - 6q^{25} + q^{26} + q^{29} + 6q^{31} + 16q^{32} - 3q^{34} + 3q^{37} - 8q^{38} + 21q^{40} + 22q^{41} + 3q^{43} + 23q^{44} - 12q^{46} - 18q^{47} + 10q^{50} + 3q^{52} - 18q^{53} + 12q^{55} + 9q^{58} - 18q^{59} + 12q^{61} - 36q^{62} - 24q^{64} + 10q^{65} - 6q^{68} - 18q^{71} + 3q^{73} + 6q^{74} - 21q^{76} + 30q^{79} - 11q^{80} + 9q^{82} + 12q^{83} - 9q^{85} + 34q^{86} + 21q^{88} + 2q^{89} + 15q^{92} - 48q^{94} - 32q^{95} + 3q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 8 \nu^{3} + 5 \nu^{2} - 18 \nu + 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} + 6 \nu^{2} - 5 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 9$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 30 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_{1} - 4$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$7 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_{1} - 10$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$16 \beta_{5} + 11 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} - 17 \beta_{1} + 5$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-14 \beta_{5} - 16 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 23 \beta_{1} + 47$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-1 + \beta_{4}$$ $$-1 + \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 0.5 − 0.224437i 0.5 + 1.41036i 0.5 − 2.05195i 0.5 + 0.224437i 0.5 − 1.41036i 0.5 + 2.05195i
−1.69963 0 0.888736 1.79418 + 3.10761i 0 0 1.88874 0 −3.04944 5.28179i
226.2 0.239123 0 −1.94282 −0.590972 1.02359i 0 0 −0.942820 0 −0.141315 0.244765i
226.3 2.46050 0 4.05408 1.29679 + 2.24611i 0 0 5.05408 0 3.19076 + 5.52655i
802.1 −1.69963 0 0.888736 1.79418 3.10761i 0 0 1.88874 0 −3.04944 + 5.28179i
802.2 0.239123 0 −1.94282 −0.590972 + 1.02359i 0 0 −0.942820 0 −0.141315 + 0.244765i
802.3 2.46050 0 4.05408 1.29679 2.24611i 0 0 5.05408 0 3.19076 5.52655i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 802.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.h.e 6
3.b odd 2 1 441.2.h.b 6
7.b odd 2 1 1323.2.h.d 6
7.c even 3 1 1323.2.f.c 6
7.c even 3 1 1323.2.g.b 6
7.d odd 6 1 189.2.f.a 6
7.d odd 6 1 1323.2.g.c 6
9.c even 3 1 1323.2.g.b 6
9.d odd 6 1 441.2.g.d 6
21.c even 2 1 441.2.h.c 6
21.g even 6 1 63.2.f.b 6
21.g even 6 1 441.2.g.e 6
21.h odd 6 1 441.2.f.d 6
21.h odd 6 1 441.2.g.d 6
28.f even 6 1 3024.2.r.g 6
63.g even 3 1 1323.2.f.c 6
63.h even 3 1 inner 1323.2.h.e 6
63.h even 3 1 3969.2.a.p 3
63.i even 6 1 441.2.h.c 6
63.i even 6 1 567.2.a.d 3
63.j odd 6 1 441.2.h.b 6
63.j odd 6 1 3969.2.a.m 3
63.k odd 6 1 189.2.f.a 6
63.l odd 6 1 1323.2.g.c 6
63.n odd 6 1 441.2.f.d 6
63.o even 6 1 441.2.g.e 6
63.s even 6 1 63.2.f.b 6
63.t odd 6 1 567.2.a.g 3
63.t odd 6 1 1323.2.h.d 6
84.j odd 6 1 1008.2.r.k 6
252.n even 6 1 3024.2.r.g 6
252.r odd 6 1 9072.2.a.bq 3
252.bj even 6 1 9072.2.a.cd 3
252.bn odd 6 1 1008.2.r.k 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b 6 21.g even 6 1
63.2.f.b 6 63.s even 6 1
189.2.f.a 6 7.d odd 6 1
189.2.f.a 6 63.k odd 6 1
441.2.f.d 6 21.h odd 6 1
441.2.f.d 6 63.n odd 6 1
441.2.g.d 6 9.d odd 6 1
441.2.g.d 6 21.h odd 6 1
441.2.g.e 6 21.g even 6 1
441.2.g.e 6 63.o even 6 1
441.2.h.b 6 3.b odd 2 1
441.2.h.b 6 63.j odd 6 1
441.2.h.c 6 21.c even 2 1
441.2.h.c 6 63.i even 6 1
567.2.a.d 3 63.i even 6 1
567.2.a.g 3 63.t odd 6 1
1008.2.r.k 6 84.j odd 6 1
1008.2.r.k 6 252.bn odd 6 1
1323.2.f.c 6 7.c even 3 1
1323.2.f.c 6 63.g even 3 1
1323.2.g.b 6 7.c even 3 1
1323.2.g.b 6 9.c even 3 1
1323.2.g.c 6 7.d odd 6 1
1323.2.g.c 6 63.l odd 6 1
1323.2.h.d 6 7.b odd 2 1
1323.2.h.d 6 63.t odd 6 1
1323.2.h.e 6 1.a even 1 1 trivial
1323.2.h.e 6 63.h even 3 1 inner
3024.2.r.g 6 28.f even 6 1
3024.2.r.g 6 252.n even 6 1
3969.2.a.m 3 63.j odd 6 1
3969.2.a.p 3 63.h even 3 1
9072.2.a.bq 3 252.r odd 6 1
9072.2.a.cd 3 252.bj even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1323, [\chi])$$:

 $$T_{2}^{3} - T_{2}^{2} - 4 T_{2} + 1$$ $$T_{5}^{6} - 5 T_{5}^{5} + 23 T_{5}^{4} - 32 T_{5}^{3} + 59 T_{5}^{2} + 22 T_{5} + 121$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + 2 T^{2} - 3 T^{3} + 4 T^{4} - 4 T^{5} + 8 T^{6} )^{2}$$
$3$ 1
$5$ $$1 - 5 T + 8 T^{2} - 7 T^{3} + 9 T^{4} + 62 T^{5} - 299 T^{6} + 310 T^{7} + 225 T^{8} - 875 T^{9} + 5000 T^{10} - 15625 T^{11} + 15625 T^{12}$$
$7$ 1
$11$ $$1 + 2 T - 10 T^{2} + 34 T^{3} + 48 T^{4} - 416 T^{5} + 31 T^{6} - 4576 T^{7} + 5808 T^{8} + 45254 T^{9} - 146410 T^{10} + 322102 T^{11} + 1771561 T^{12}$$
$13$ $$( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} )^{3}$$
$17$ $$1 - 12 T + 54 T^{2} - 210 T^{3} + 1350 T^{4} - 5898 T^{5} + 19735 T^{6} - 100266 T^{7} + 390150 T^{8} - 1031730 T^{9} + 4510134 T^{10} - 17038284 T^{11} + 24137569 T^{12}$$
$19$ $$1 + 3 T - 42 T^{2} - 61 T^{3} + 1311 T^{4} + 726 T^{5} - 27501 T^{6} + 13794 T^{7} + 473271 T^{8} - 418399 T^{9} - 5473482 T^{10} + 7428297 T^{11} + 47045881 T^{12}$$
$23$ $$1 - 36 T^{2} + 18 T^{3} + 468 T^{4} - 324 T^{5} - 5393 T^{6} - 7452 T^{7} + 247572 T^{8} + 219006 T^{9} - 10074276 T^{10} + 148035889 T^{12}$$
$29$ $$1 - T - 82 T^{2} + 31 T^{3} + 4425 T^{4} - 758 T^{5} - 148595 T^{6} - 21982 T^{7} + 3721425 T^{8} + 756059 T^{9} - 57997042 T^{10} - 20511149 T^{11} + 594823321 T^{12}$$
$31$ $$( 1 - 3 T + 69 T^{2} - 213 T^{3} + 2139 T^{4} - 2883 T^{5} + 29791 T^{6} )^{2}$$
$37$ $$1 - 3 T - 48 T^{2} + 435 T^{3} + 231 T^{4} - 8724 T^{5} + 60581 T^{6} - 322788 T^{7} + 316239 T^{8} + 22034055 T^{9} - 89959728 T^{10} - 208031871 T^{11} + 2565726409 T^{12}$$
$41$ $$1 - 22 T + 206 T^{2} - 1802 T^{3} + 18432 T^{4} - 135116 T^{5} + 808243 T^{6} - 5539756 T^{7} + 30984192 T^{8} - 124195642 T^{9} + 582106766 T^{10} - 2548836422 T^{11} + 4750104241 T^{12}$$
$43$ $$1 - 3 T - 54 T^{2} + 569 T^{3} + 123 T^{4} - 13170 T^{5} + 115347 T^{6} - 566310 T^{7} + 227427 T^{8} + 45239483 T^{9} - 184615254 T^{10} - 441025329 T^{11} + 6321363049 T^{12}$$
$47$ $$( 1 + 9 T + 87 T^{2} + 657 T^{3} + 4089 T^{4} + 19881 T^{5} + 103823 T^{6} )^{2}$$
$53$ $$1 + 18 T + 90 T^{2} + 378 T^{3} + 7848 T^{4} + 52668 T^{5} + 160459 T^{6} + 2791404 T^{7} + 22045032 T^{8} + 56275506 T^{9} + 710143290 T^{10} + 7527518874 T^{11} + 22164361129 T^{12}$$
$59$ $$( 1 + 9 T + 171 T^{2} + 999 T^{3} + 10089 T^{4} + 31329 T^{5} + 205379 T^{6} )^{2}$$
$61$ $$( 1 - 6 T + 162 T^{2} - 665 T^{3} + 9882 T^{4} - 22326 T^{5} + 226981 T^{6} )^{2}$$
$67$ $$( 1 - 6 T^{2} + 683 T^{3} - 402 T^{4} + 300763 T^{6} )^{2}$$
$71$ $$( 1 + 9 T + 207 T^{2} + 1197 T^{3} + 14697 T^{4} + 45369 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$1 - 3 T - 42 T^{2} + 1209 T^{3} - 3165 T^{4} - 28380 T^{5} + 1003961 T^{6} - 2071740 T^{7} - 16866285 T^{8} + 470321553 T^{9} - 1192726122 T^{10} - 6219214779 T^{11} + 151334226289 T^{12}$$
$79$ $$( 1 - 15 T + 189 T^{2} - 1601 T^{3} + 14931 T^{4} - 93615 T^{5} + 493039 T^{6} )^{2}$$
$83$ $$1 - 12 T - 144 T^{2} + 582 T^{3} + 34812 T^{4} - 90444 T^{5} - 2656433 T^{6} - 7506852 T^{7} + 239819868 T^{8} + 332780034 T^{9} - 6833998224 T^{10} - 47268487716 T^{11} + 326940373369 T^{12}$$
$89$ $$1 - 2 T - 112 T^{2} + 1238 T^{3} + 1662 T^{4} - 59806 T^{5} + 720895 T^{6} - 5322734 T^{7} + 13164702 T^{8} + 872751622 T^{9} - 7027130992 T^{10} - 11168118898 T^{11} + 496981290961 T^{12}$$
$97$ $$1 - 3 T - 168 T^{2} - 573 T^{3} + 14223 T^{4} + 78504 T^{5} - 1297807 T^{6} + 7614888 T^{7} + 133824207 T^{8} - 522961629 T^{9} - 14872919208 T^{10} - 25762020771 T^{11} + 832972004929 T^{12}$$