# Properties

 Label 1323.2.s.c Level $1323$ Weight $2$ Character orbit 1323.s Analytic conductor $10.564$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.5642081874$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 7 x^{10} + 37 x^{8} - 78 x^{6} + 123 x^{4} - 36 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{2} + ( -\beta_{3} - \beta_{5} - \beta_{8} ) q^{4} -\beta_{7} q^{5} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{8} ) q^{8} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{2} + ( -\beta_{3} - \beta_{5} - \beta_{8} ) q^{4} -\beta_{7} q^{5} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{8} ) q^{8} + ( \beta_{9} + \beta_{11} ) q^{10} + ( \beta_{1} + \beta_{4} + 2 \beta_{8} ) q^{11} + ( \beta_{7} + \beta_{9} - \beta_{10} ) q^{13} + ( -\beta_{3} + \beta_{4} + \beta_{8} ) q^{16} + ( 2 \beta_{6} - \beta_{11} ) q^{17} + ( -\beta_{2} + \beta_{6} - 2 \beta_{7} - \beta_{10} ) q^{19} + ( \beta_{2} - 2 \beta_{9} ) q^{20} + ( -2 \beta_{1} - 2 \beta_{4} - \beta_{5} - 2 \beta_{8} ) q^{22} + ( 1 - \beta_{3} - 3 \beta_{5} ) q^{23} + ( 1 - 2 \beta_{1} + \beta_{3} + \beta_{5} ) q^{25} + ( 2 \beta_{2} + \beta_{6} - 4 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{26} + ( -2 + \beta_{1} - 2 \beta_{5} ) q^{29} + ( 2 \beta_{6} + 2 \beta_{7} + \beta_{10} ) q^{31} + ( -1 - \beta_{1} + 2 \beta_{3} - 4 \beta_{4} + \beta_{5} - 2 \beta_{8} ) q^{32} + ( -2 \beta_{7} - \beta_{10} ) q^{34} + ( 1 - \beta_{1} + \beta_{3} - \beta_{8} ) q^{37} + ( 2 \beta_{2} + \beta_{6} + 2 \beta_{9} + \beta_{11} ) q^{38} + ( -\beta_{2} + 2 \beta_{6} - \beta_{7} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{40} + ( -2 \beta_{2} - 2 \beta_{7} + \beta_{9} - 2 \beta_{10} ) q^{41} + ( 1 - 3 \beta_{1} + 5 \beta_{3} + 4 \beta_{5} - \beta_{8} ) q^{43} + ( 2 + 3 \beta_{1} + 2 \beta_{5} ) q^{44} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{8} ) q^{46} + ( 2 \beta_{2} + 2 \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{47} + ( 4 - \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{8} ) q^{50} + ( -5 \beta_{2} - \beta_{6} - \beta_{7} + 5 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{52} + ( -2 + \beta_{1} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{8} ) q^{53} + ( -2 \beta_{2} + \beta_{6} + 2 \beta_{9} - \beta_{11} ) q^{55} + ( -2 + 3 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{58} + ( -2 \beta_{2} + \beta_{6} + 4 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{59} + ( \beta_{9} + \beta_{11} ) q^{61} + ( -\beta_{2} - \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{11} ) q^{62} + ( -2 + 7 \beta_{1} - 4 \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{64} + ( -10 + 2 \beta_{1} - \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{8} ) q^{65} + ( 1 + 2 \beta_{1} - \beta_{3} - 2 \beta_{5} + 3 \beta_{8} ) q^{67} + ( \beta_{2} - \beta_{6} + \beta_{9} - \beta_{11} ) q^{68} + ( -2 - 2 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} + 7 \beta_{5} - 4 \beta_{8} ) q^{71} + ( -\beta_{7} + 3 \beta_{9} + \beta_{10} + 3 \beta_{11} ) q^{73} + ( 1 + \beta_{3} - \beta_{5} ) q^{74} + ( -\beta_{7} - 4 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{76} + ( -\beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{8} ) q^{79} + ( 2 \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} ) q^{80} + ( 5 \beta_{2} + 2 \beta_{6} + 2 \beta_{7} + \beta_{10} ) q^{82} + ( -2 \beta_{2} + \beta_{6} + 4 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{83} + ( -6 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} + 6 \beta_{8} ) q^{85} + ( 5 - 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - 11 \beta_{5} - 4 \beta_{8} ) q^{86} + ( -4 - 3 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{88} + ( -3 \beta_{2} + 6 \beta_{9} ) q^{89} + ( -1 - 4 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} + \beta_{5} - 2 \beta_{8} ) q^{92} + ( -2 \beta_{2} + \beta_{6} - 4 \beta_{7} - 2 \beta_{10} ) q^{94} + ( 6 - 3 \beta_{1} - 3 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} + 3 \beta_{8} ) q^{95} + ( \beta_{6} + 2 \beta_{7} + \beta_{10} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 6q^{2} + 2q^{4} + O(q^{10})$$ $$12q - 6q^{2} + 2q^{4} + 2q^{16} - 10q^{22} - 30q^{29} - 12q^{32} + 2q^{37} - 10q^{43} + 54q^{44} + 20q^{46} + 36q^{50} - 12q^{53} - 4q^{58} + 16q^{64} - 78q^{65} + 12q^{67} - 6q^{79} - 6q^{85} - 68q^{88} - 30q^{92} + 72q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 7 x^{10} + 37 x^{8} - 78 x^{6} + 123 x^{4} - 36 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-28 \nu^{10} + 148 \nu^{8} + 446 \nu^{6} - 3807 \nu^{4} + 17052 \nu^{2} + 4446$$$$)/12897$$ $$\beta_{2}$$ $$=$$ $$($$$$-49 \nu^{11} + 259 \nu^{9} - 1369 \nu^{7} + 861 \nu^{5} - 252 \nu^{3} - 15864 \nu$$$$)/4299$$ $$\beta_{3}$$ $$=$$ $$($$$$-164 \nu^{10} + 1481 \nu^{8} - 7214 \nu^{6} + 17007 \nu^{4} - 16197 \nu^{2} - 3438$$$$)/12897$$ $$\beta_{4}$$ $$=$$ $$($$$$-175 \nu^{10} + 925 \nu^{8} - 3661 \nu^{6} - 1224 \nu^{4} + 16296 \nu^{2} - 30249$$$$)/12897$$ $$\beta_{5}$$ $$=$$ $$($$$$148 \nu^{10} - 987 \nu^{8} + 5217 \nu^{6} - 10175 \nu^{4} + 17343 \nu^{2} - 777$$$$)/4299$$ $$\beta_{6}$$ $$=$$ $$($$$$70 \nu^{11} - 370 \nu^{9} + 1751 \nu^{7} - 1230 \nu^{5} + 360 \nu^{3} + 8947 \nu$$$$)/1433$$ $$\beta_{7}$$ $$=$$ $$($$$$-730 \nu^{11} + 5701 \nu^{9} - 30748 \nu^{7} + 76698 \nu^{5} - 122898 \nu^{3} + 66168 \nu$$$$)/12897$$ $$\beta_{8}$$ $$=$$ $$($$$$-877 \nu^{10} + 6478 \nu^{8} - 34855 \nu^{6} + 79281 \nu^{4} - 123654 \nu^{2} + 31473$$$$)/12897$$ $$\beta_{9}$$ $$=$$ $$($$$$543 \nu^{11} - 3689 \nu^{9} + 19499 \nu^{7} - 39839 \nu^{5} + 64821 \nu^{3} - 18972 \nu$$$$)/4299$$ $$\beta_{10}$$ $$=$$ $$($$$$2237 \nu^{11} - 15509 \nu^{9} + 81362 \nu^{7} - 167049 \nu^{5} + 249792 \nu^{3} - 42912 \nu$$$$)/12897$$ $$\beta_{11}$$ $$=$$ $$($$$$-890 \nu^{11} + 6342 \nu^{9} - 33522 \nu^{7} + 71935 \nu^{5} - 111438 \nu^{3} + 32616 \nu$$$$)/4299$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{10} - 2 \beta_{7} + \beta_{6} - \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} + 2 \beta_{5} + \beta_{4} - \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{11} - 8 \beta_{10} + \beta_{9} - 4 \beta_{7} + 4 \beta_{6} - \beta_{2}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{8} + 6 \beta_{5} - 6 \beta_{3} + 5 \beta_{1} - 12$$ $$\nu^{5}$$ $$=$$ $$($$$$-19 \beta_{11} - 16 \beta_{10} - 2 \beta_{9} + 16 \beta_{7}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$-7 \beta_{5} - 16 \beta_{4} - 7 \beta_{3} + 30 \beta_{1} - 51$$ $$\nu^{7}$$ $$=$$ $$($$$$67 \beta_{10} + 134 \beta_{7} - 88 \beta_{6} - 23 \beta_{2}$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$-67 \beta_{8} - 118 \beta_{5} - 67 \beta_{4} + 104 \beta_{3} + 37 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$($$$$400 \beta_{11} + 578 \beta_{10} + 134 \beta_{9} + 289 \beta_{7} - 400 \beta_{6} - 134 \beta_{2}$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$-289 \beta_{8} - 333 \beta_{5} + 645 \beta_{3} - 467 \beta_{1} + 978$$ $$\nu^{11}$$ $$=$$ $$($$$$1801 \beta_{11} + 1267 \beta_{10} + 668 \beta_{9} - 1267 \beta_{7}$$$$)/3$$

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$\beta_{5}$$ $$1 - \beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
656.1
 −1.29589 − 0.748185i 1.29589 + 0.748185i 0.474636 + 0.274031i −0.474636 − 0.274031i −1.82904 − 1.05600i 1.82904 + 1.05600i −1.29589 + 0.748185i 1.29589 − 0.748185i 0.474636 − 0.274031i −0.474636 + 0.274031i −1.82904 + 1.05600i 1.82904 − 1.05600i
−1.97141 + 1.13819i 0 1.59097 2.75564i −1.43429 0 0 2.69056i 0 2.82757 1.63250i
656.2 −1.97141 + 1.13819i 0 1.59097 2.75564i 1.43429 0 0 2.69056i 0 −2.82757 + 1.63250i
656.3 −0.555632 + 0.320794i 0 −0.794182 + 1.37556i −2.21105 0 0 2.30225i 0 1.22853 0.709292i
656.4 −0.555632 + 0.320794i 0 −0.794182 + 1.37556i 2.21105 0 0 2.30225i 0 −1.22853 + 0.709292i
656.5 1.02704 0.592963i 0 −0.296790 + 0.514055i −2.83797 0 0 3.07579i 0 −2.91472 + 1.68281i
656.6 1.02704 0.592963i 0 −0.296790 + 0.514055i 2.83797 0 0 3.07579i 0 2.91472 1.68281i
962.1 −1.97141 1.13819i 0 1.59097 + 2.75564i −1.43429 0 0 2.69056i 0 2.82757 + 1.63250i
962.2 −1.97141 1.13819i 0 1.59097 + 2.75564i 1.43429 0 0 2.69056i 0 −2.82757 1.63250i
962.3 −0.555632 0.320794i 0 −0.794182 1.37556i −2.21105 0 0 2.30225i 0 1.22853 + 0.709292i
962.4 −0.555632 0.320794i 0 −0.794182 1.37556i 2.21105 0 0 2.30225i 0 −1.22853 0.709292i
962.5 1.02704 + 0.592963i 0 −0.296790 0.514055i −2.83797 0 0 3.07579i 0 −2.91472 1.68281i
962.6 1.02704 + 0.592963i 0 −0.296790 0.514055i 2.83797 0 0 3.07579i 0 2.91472 + 1.68281i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 962.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.n odd 6 1 inner
63.s even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.s.c 12
3.b odd 2 1 441.2.s.c 12
7.b odd 2 1 inner 1323.2.s.c 12
7.c even 3 1 189.2.o.a 12
7.c even 3 1 1323.2.i.c 12
7.d odd 6 1 189.2.o.a 12
7.d odd 6 1 1323.2.i.c 12
9.c even 3 1 441.2.i.c 12
9.d odd 6 1 1323.2.i.c 12
21.c even 2 1 441.2.s.c 12
21.g even 6 1 63.2.o.a 12
21.g even 6 1 441.2.i.c 12
21.h odd 6 1 63.2.o.a 12
21.h odd 6 1 441.2.i.c 12
28.f even 6 1 3024.2.cc.a 12
28.g odd 6 1 3024.2.cc.a 12
63.g even 3 1 441.2.s.c 12
63.g even 3 1 567.2.c.c 12
63.h even 3 1 63.2.o.a 12
63.i even 6 1 189.2.o.a 12
63.j odd 6 1 189.2.o.a 12
63.k odd 6 1 441.2.s.c 12
63.k odd 6 1 567.2.c.c 12
63.l odd 6 1 441.2.i.c 12
63.n odd 6 1 567.2.c.c 12
63.n odd 6 1 inner 1323.2.s.c 12
63.o even 6 1 1323.2.i.c 12
63.s even 6 1 567.2.c.c 12
63.s even 6 1 inner 1323.2.s.c 12
63.t odd 6 1 63.2.o.a 12
84.j odd 6 1 1008.2.cc.a 12
84.n even 6 1 1008.2.cc.a 12
252.r odd 6 1 3024.2.cc.a 12
252.u odd 6 1 1008.2.cc.a 12
252.bb even 6 1 3024.2.cc.a 12
252.bj even 6 1 1008.2.cc.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.o.a 12 21.g even 6 1
63.2.o.a 12 21.h odd 6 1
63.2.o.a 12 63.h even 3 1
63.2.o.a 12 63.t odd 6 1
189.2.o.a 12 7.c even 3 1
189.2.o.a 12 7.d odd 6 1
189.2.o.a 12 63.i even 6 1
189.2.o.a 12 63.j odd 6 1
441.2.i.c 12 9.c even 3 1
441.2.i.c 12 21.g even 6 1
441.2.i.c 12 21.h odd 6 1
441.2.i.c 12 63.l odd 6 1
441.2.s.c 12 3.b odd 2 1
441.2.s.c 12 21.c even 2 1
441.2.s.c 12 63.g even 3 1
441.2.s.c 12 63.k odd 6 1
567.2.c.c 12 63.g even 3 1
567.2.c.c 12 63.k odd 6 1
567.2.c.c 12 63.n odd 6 1
567.2.c.c 12 63.s even 6 1
1008.2.cc.a 12 84.j odd 6 1
1008.2.cc.a 12 84.n even 6 1
1008.2.cc.a 12 252.u odd 6 1
1008.2.cc.a 12 252.bj even 6 1
1323.2.i.c 12 7.c even 3 1
1323.2.i.c 12 7.d odd 6 1
1323.2.i.c 12 9.d odd 6 1
1323.2.i.c 12 63.o even 6 1
1323.2.s.c 12 1.a even 1 1 trivial
1323.2.s.c 12 7.b odd 2 1 inner
1323.2.s.c 12 63.n odd 6 1 inner
1323.2.s.c 12 63.s even 6 1 inner
3024.2.cc.a 12 28.f even 6 1
3024.2.cc.a 12 28.g odd 6 1
3024.2.cc.a 12 252.r odd 6 1
3024.2.cc.a 12 252.bb even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 3 T_{2}^{5} + T_{2}^{4} - 6 T_{2}^{3} + T_{2}^{2} + 6 T_{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(1323, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 3 + 6 T + T^{2} - 6 T^{3} + T^{4} + 3 T^{5} + T^{6} )^{2}$$
$3$ $$T^{12}$$
$5$ $$( -81 + 66 T^{2} - 15 T^{4} + T^{6} )^{2}$$
$7$ $$T^{12}$$
$11$ $$( 3 + 121 T^{2} + 22 T^{4} + T^{6} )^{2}$$
$13$ $$1750329 - 718389 T^{2} + 235314 T^{4} - 21789 T^{6} + 1482 T^{8} - 45 T^{10} + T^{12}$$
$17$ $$531441 + 492075 T^{2} + 416259 T^{4} + 34992 T^{6} + 2241 T^{8} + 54 T^{10} + T^{12}$$
$19$ $$4782969 - 2125764 T^{2} + 807003 T^{4} - 56862 T^{6} + 2997 T^{8} - 63 T^{10} + T^{12}$$
$23$ $$( 27 + 97 T^{2} + 22 T^{4} + T^{6} )^{2}$$
$29$ $$( 243 + 594 T + 619 T^{2} + 330 T^{3} + 97 T^{4} + 15 T^{5} + T^{6} )^{2}$$
$31$ $$729 - 43578 T^{2} + 2601513 T^{4} - 208152 T^{6} + 15027 T^{8} - 129 T^{10} + T^{12}$$
$37$ $$( 1 - 4 T + 17 T^{2} + 2 T^{3} + 5 T^{4} - T^{5} + T^{6} )^{2}$$
$41$ $$531441 + 684531 T^{2} + 829233 T^{4} + 66150 T^{6} + 4245 T^{8} + 72 T^{10} + T^{12}$$
$43$ $$( 38809 + 9062 T + 3101 T^{2} + 164 T^{3} + 71 T^{4} + 5 T^{5} + T^{6} )^{2}$$
$47$ $$6561 + 108864 T^{2} + 1798803 T^{4} + 124830 T^{6} + 7305 T^{8} + 93 T^{10} + T^{12}$$
$53$ $$( 20667 + 13197 T + 2311 T^{2} - 318 T^{3} - 41 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$59$ $$18539817921 + 1785887676 T^{2} + 142210197 T^{4} + 2600082 T^{6} + 34845 T^{8} + 219 T^{10} + T^{12}$$
$61$ $$59049 - 40095 T^{2} + 21393 T^{4} - 3474 T^{6} + 411 T^{8} - 24 T^{10} + T^{12}$$
$67$ $$( 49 + 105 T + 183 T^{2} + 104 T^{3} + 51 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$71$ $$( 363 + 2194 T^{2} + 163 T^{4} + T^{6} )^{2}$$
$73$ $$4782969 - 26453952 T^{2} + 145703043 T^{4} - 3370410 T^{6} + 65745 T^{8} - 279 T^{10} + T^{12}$$
$79$ $$( 6241 + 1896 T + 813 T^{2} + 86 T^{3} + 33 T^{4} + 3 T^{5} + T^{6} )^{2}$$
$83$ $$132211504881 + 6087905487 T^{2} + 195243543 T^{4} + 3190644 T^{6} + 38013 T^{8} + 234 T^{10} + T^{12}$$
$89$ $$282429536481 + 15109399071 T^{2} + 636134877 T^{4} + 8148762 T^{6} + 76545 T^{8} + 324 T^{10} + T^{12}$$
$97$ $$10673289 - 2979504 T^{2} + 606321 T^{4} - 56394 T^{6} + 3849 T^{8} - 69 T^{10} + T^{12}$$