Properties

 Label 1323.2.h.g Level $1323$ Weight $2$ Character orbit 1323.h 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.h (of order $$3$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$10.5642081874$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{5}$$ Twist minimal: no (minimal twist has level 441) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{1} + \beta_{5} ) q^{4} + \beta_{2} q^{5} + ( -2 - \beta_{1} - \beta_{5} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{1} + \beta_{5} ) q^{4} + \beta_{2} q^{5} + ( -2 - \beta_{1} - \beta_{5} ) q^{8} + ( -2 \beta_{2} - \beta_{10} ) q^{10} + ( -\beta_{4} - \beta_{5} - \beta_{7} ) q^{11} + ( -\beta_{9} + \beta_{11} ) q^{13} + ( 2 \beta_{1} - \beta_{5} ) q^{16} + ( -\beta_{2} - \beta_{10} + \beta_{11} ) q^{17} + ( \beta_{2} + \beta_{3} + \beta_{8} - \beta_{10} ) q^{19} + ( 3 \beta_{2} + 2 \beta_{10} + \beta_{11} ) q^{20} + \beta_{4} q^{22} + ( 1 + \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{23} + ( -3 \beta_{1} + \beta_{4} + 3 \beta_{6} ) q^{25} + ( \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{26} + ( 4 + 4 \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{29} + ( \beta_{3} - 2 \beta_{8} ) q^{31} + ( -3 + \beta_{1} ) q^{32} + ( 3 \beta_{2} + 2 \beta_{11} ) q^{34} + ( -2 \beta_{1} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{37} + ( -\beta_{2} - \beta_{3} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{38} + ( -4 \beta_{2} - 2 \beta_{10} - \beta_{11} ) q^{40} + ( \beta_{2} + \beta_{3} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{41} + ( -2 - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{43} + ( \beta_{1} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{44} + ( -1 - \beta_{4} - 4 \beta_{6} + \beta_{7} ) q^{46} + ( \beta_{3} - \beta_{8} - 2 \beta_{9} ) q^{47} + ( 4 \beta_{1} - 9 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{50} + ( \beta_{2} + \beta_{3} + \beta_{8} - \beta_{10} ) q^{52} + ( 5 + 5 \beta_{4} - 3 \beta_{6} + 2 \beta_{7} ) q^{53} + ( -\beta_{3} - \beta_{8} - \beta_{9} ) q^{55} + ( -1 - \beta_{4} - 7 \beta_{6} + \beta_{7} ) q^{58} + ( -2 \beta_{3} - \beta_{8} + \beta_{9} ) q^{59} + ( -2 \beta_{3} + \beta_{8} - 2 \beta_{9} ) q^{61} + ( -4 \beta_{3} + \beta_{8} + 2 \beta_{9} ) q^{62} + ( -3 - 2 \beta_{1} + \beta_{5} ) q^{64} -3 \beta_{5} q^{65} + ( -2 + 3 \beta_{1} - 3 \beta_{5} ) q^{67} + ( -4 \beta_{2} - 3 \beta_{10} ) q^{68} + ( -6 - \beta_{1} ) q^{71} + ( \beta_{2} - \beta_{10} ) q^{73} + ( -\beta_{1} - 7 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{74} + ( \beta_{2} + \beta_{3} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{76} + ( -1 + 3 \beta_{1} + 3 \beta_{5} ) q^{79} + ( 4 \beta_{2} + \beta_{10} - \beta_{11} ) q^{80} + ( -4 \beta_{2} - 4 \beta_{3} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{82} + ( -3 \beta_{2} - \beta_{10} + \beta_{11} ) q^{83} + ( 6 \beta_{1} - 3 \beta_{4} - 6 \beta_{6} ) q^{85} + ( -7 - 7 \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{86} + ( -\beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{88} + ( 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{89} + ( 9 + 9 \beta_{4} + 4 \beta_{6} ) q^{92} + ( -3 \beta_{3} + 3 \beta_{8} - \beta_{9} ) q^{94} + ( -9 + 3 \beta_{5} ) q^{95} + ( -2 \beta_{2} - \beta_{10} - \beta_{11} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 4q^{2} + 12q^{4} - 24q^{8} + O(q^{10})$$ $$12q - 4q^{2} + 12q^{4} - 24q^{8} + 8q^{11} + 12q^{16} - 6q^{22} + 4q^{23} - 12q^{25} + 22q^{29} - 32q^{32} + 6q^{37} - 6q^{43} - 14q^{44} - 12q^{46} + 56q^{50} + 28q^{53} - 18q^{58} - 48q^{64} + 12q^{65} - 76q^{71} + 36q^{74} - 12q^{79} + 30q^{85} - 36q^{86} + 6q^{88} + 62q^{92} - 120q^{95} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-49 \nu^{10} + 259 \nu^{8} - 1369 \nu^{6} + 861 \nu^{4} - 252 \nu^{2} - 7266$$$$)/4299$$ $$\beta_{2}$$ $$=$$ $$($$$$-148 \nu^{11} + 987 \nu^{9} - 5217 \nu^{7} + 10175 \nu^{5} - 17343 \nu^{3} - 3522 \nu$$$$)/4299$$ $$\beta_{3}$$ $$=$$ $$($$$$-148 \nu^{11} + 987 \nu^{9} - 5217 \nu^{7} + 10175 \nu^{5} - 17343 \nu^{3} + 9375 \nu$$$$)/4299$$ $$\beta_{4}$$ $$=$$ $$($$$$148 \nu^{10} - 987 \nu^{8} + 5217 \nu^{6} - 10175 \nu^{4} + 17343 \nu^{2} - 5076$$$$)/4299$$ $$\beta_{5}$$ $$=$$ $$($$$$161 \nu^{10} - 851 \nu^{8} + 3884 \nu^{6} - 2829 \nu^{4} + 828 \nu^{2} + 6678$$$$)/4299$$ $$\beta_{6}$$ $$=$$ $$($$$$-296 \nu^{10} + 1974 \nu^{8} - 10434 \nu^{6} + 20350 \nu^{4} - 30387 \nu^{2} + 1554$$$$)/4299$$ $$\beta_{7}$$ $$=$$ $$($$$$-120 \nu^{10} + 839 \nu^{8} - 4230 \nu^{6} + 8250 \nu^{4} - 10034 \nu^{2} + 630$$$$)/1433$$ $$\beta_{8}$$ $$=$$ $$($$$$-494 \nu^{11} + 3430 \nu^{9} - 18130 \nu^{7} + 38978 \nu^{5} - 64569 \nu^{3} + 34836 \nu$$$$)/4299$$ $$\beta_{9}$$ $$=$$ $$($$$$532 \nu^{11} - 4245 \nu^{9} + 23052 \nu^{7} - 58070 \nu^{5} + 93015 \nu^{3} - 50082 \nu$$$$)/4299$$ $$\beta_{10}$$ $$=$$ $$($$$$641 \nu^{11} - 4207 \nu^{9} + 22237 \nu^{7} - 41561 \nu^{5} + 65325 \nu^{3} + 12756 \nu$$$$)/4299$$ $$\beta_{11}$$ $$=$$ $$($$$$-1162 \nu^{11} + 7575 \nu^{9} - 38811 \nu^{7} + 69140 \nu^{5} - 96255 \nu^{3} - 17544 \nu$$$$)/4299$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + 2 \beta_{4} + 2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{10} + \beta_{8} - 4 \beta_{3} - 8 \beta_{2}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$-\beta_{7} + 5 \beta_{6} - \beta_{5} + 7 \beta_{4} - 5 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{11} - 6 \beta_{10} + 2 \beta_{9} + 12 \beta_{8} - 34 \beta_{3} - 17 \beta_{2}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$-7 \beta_{5} - 23 \beta_{1} - 28$$ $$\nu^{7}$$ $$=$$ $$($$$$7 \beta_{11} + 30 \beta_{10} + 7 \beta_{9} + 30 \beta_{8} - 74 \beta_{3} + 74 \beta_{2}$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$37 \beta_{7} - 104 \beta_{6} - 118 \beta_{4} - 118$$ $$\nu^{9}$$ $$=$$ $$($$$$74 \beta_{11} + 282 \beta_{10} - 37 \beta_{9} - 141 \beta_{8} + 326 \beta_{3} + 652 \beta_{2}$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$178 \beta_{7} - 467 \beta_{6} + 178 \beta_{5} - 511 \beta_{4} + 467 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$($$$$178 \beta_{11} + 645 \beta_{10} - 356 \beta_{9} - 1290 \beta_{8} + 2890 \beta_{3} + 1445 \beta_{2}$$$$)/3$$

Character values

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

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

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
 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 1.29589 − 0.748185i −1.29589 + 0.748185i 0.474636 − 0.274031i −0.474636 + 0.274031i
−2.46050 0 4.05408 −1.82904 3.16799i 0 0 −5.05408 0 4.50036 + 7.79485i
226.2 −2.46050 0 4.05408 1.82904 + 3.16799i 0 0 −5.05408 0 −4.50036 7.79485i
226.3 −0.239123 0 −1.94282 −1.29589 2.24456i 0 0 0.942820 0 0.309879 + 0.536725i
226.4 −0.239123 0 −1.94282 1.29589 + 2.24456i 0 0 0.942820 0 −0.309879 0.536725i
226.5 1.69963 0 0.888736 −0.474636 0.822093i 0 0 −1.88874 0 −0.806704 1.39725i
226.6 1.69963 0 0.888736 0.474636 + 0.822093i 0 0 −1.88874 0 0.806704 + 1.39725i
802.1 −2.46050 0 4.05408 −1.82904 + 3.16799i 0 0 −5.05408 0 4.50036 7.79485i
802.2 −2.46050 0 4.05408 1.82904 3.16799i 0 0 −5.05408 0 −4.50036 + 7.79485i
802.3 −0.239123 0 −1.94282 −1.29589 + 2.24456i 0 0 0.942820 0 0.309879 0.536725i
802.4 −0.239123 0 −1.94282 1.29589 2.24456i 0 0 0.942820 0 −0.309879 + 0.536725i
802.5 1.69963 0 0.888736 −0.474636 + 0.822093i 0 0 −1.88874 0 −0.806704 + 1.39725i
802.6 1.69963 0 0.888736 0.474636 0.822093i 0 0 −1.88874 0 0.806704 1.39725i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 802.6 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.h even 3 1 inner
63.t odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.h.g 12
3.b odd 2 1 441.2.h.g 12
7.b odd 2 1 inner 1323.2.h.g 12
7.c even 3 1 1323.2.f.g 12
7.c even 3 1 1323.2.g.g 12
7.d odd 6 1 1323.2.f.g 12
7.d odd 6 1 1323.2.g.g 12
9.c even 3 1 1323.2.g.g 12
9.d odd 6 1 441.2.g.g 12
21.c even 2 1 441.2.h.g 12
21.g even 6 1 441.2.f.g 12
21.g even 6 1 441.2.g.g 12
21.h odd 6 1 441.2.f.g 12
21.h odd 6 1 441.2.g.g 12
63.g even 3 1 1323.2.f.g 12
63.h even 3 1 inner 1323.2.h.g 12
63.h even 3 1 3969.2.a.bd 6
63.i even 6 1 441.2.h.g 12
63.i even 6 1 3969.2.a.be 6
63.j odd 6 1 441.2.h.g 12
63.j odd 6 1 3969.2.a.be 6
63.k odd 6 1 1323.2.f.g 12
63.l odd 6 1 1323.2.g.g 12
63.n odd 6 1 441.2.f.g 12
63.o even 6 1 441.2.g.g 12
63.s even 6 1 441.2.f.g 12
63.t odd 6 1 inner 1323.2.h.g 12
63.t odd 6 1 3969.2.a.bd 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.f.g 12 21.g even 6 1
441.2.f.g 12 21.h odd 6 1
441.2.f.g 12 63.n odd 6 1
441.2.f.g 12 63.s even 6 1
441.2.g.g 12 9.d odd 6 1
441.2.g.g 12 21.g even 6 1
441.2.g.g 12 21.h odd 6 1
441.2.g.g 12 63.o even 6 1
441.2.h.g 12 3.b odd 2 1
441.2.h.g 12 21.c even 2 1
441.2.h.g 12 63.i even 6 1
441.2.h.g 12 63.j odd 6 1
1323.2.f.g 12 7.c even 3 1
1323.2.f.g 12 7.d odd 6 1
1323.2.f.g 12 63.g even 3 1
1323.2.f.g 12 63.k odd 6 1
1323.2.g.g 12 7.c even 3 1
1323.2.g.g 12 7.d odd 6 1
1323.2.g.g 12 9.c even 3 1
1323.2.g.g 12 63.l odd 6 1
1323.2.h.g 12 1.a even 1 1 trivial
1323.2.h.g 12 7.b odd 2 1 inner
1323.2.h.g 12 63.h even 3 1 inner
1323.2.h.g 12 63.t odd 6 1 inner
3969.2.a.bd 6 63.h even 3 1
3969.2.a.bd 6 63.t odd 6 1
3969.2.a.be 6 63.i even 6 1
3969.2.a.be 6 63.j odd 6 1

Hecke kernels

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

 $$T_{2}^{3} + T_{2}^{2} - 4 T_{2} - 1$$ $$T_{5}^{12} + 21 T_{5}^{10} + 333 T_{5}^{8} + 2106 T_{5}^{6} + 9963 T_{5}^{4} + 8748 T_{5}^{2} + 6561$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 - 4 T + T^{2} + T^{3} )^{4}$$
$3$ $$T^{12}$$
$5$ $$6561 + 8748 T^{2} + 9963 T^{4} + 2106 T^{6} + 333 T^{8} + 21 T^{10} + T^{12}$$
$7$ $$T^{12}$$
$11$ $$( 1 - T + 5 T^{2} + 2 T^{3} + 17 T^{4} - 4 T^{5} + T^{6} )^{2}$$
$13$ $$6561 + 28431 T^{2} + 120042 T^{4} + 13527 T^{6} + 1170 T^{8} + 39 T^{10} + T^{12}$$
$17$ $$15752961 + 6322617 T^{2} + 2204253 T^{4} + 125874 T^{6} + 5463 T^{8} + 84 T^{10} + T^{12}$$
$19$ $$15752961 + 5143824 T^{2} + 1381941 T^{4} + 89262 T^{6} + 4329 T^{8} + 75 T^{10} + T^{12}$$
$23$ $$( 3481 - 1475 T + 743 T^{2} - 68 T^{3} + 29 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$29$ $$( 7921 + 1246 T + 1175 T^{2} - 332 T^{3} + 107 T^{4} - 11 T^{5} + T^{6} )^{2}$$
$31$ $$( -77841 + 5508 T^{2} - 129 T^{4} + T^{6} )^{2}$$
$37$ $$( 729 + 648 T + 495 T^{2} + 126 T^{3} + 33 T^{4} - 3 T^{5} + T^{6} )^{2}$$
$41$ $$43046721 + 39858075 T^{2} + 35842743 T^{4} + 971028 T^{6} + 20169 T^{8} + 162 T^{10} + T^{12}$$
$43$ $$( 729 - 648 T + 495 T^{2} - 126 T^{3} + 33 T^{4} + 3 T^{5} + T^{6} )^{2}$$
$47$ $$( -194481 + 10584 T^{2} - 183 T^{4} + T^{6} )^{2}$$
$53$ $$( 69169 + 2893 T + 3803 T^{2} - 680 T^{3} + 185 T^{4} - 14 T^{5} + T^{6} )^{2}$$
$59$ $$( -149769 + 9450 T^{2} - 183 T^{4} + T^{6} )^{2}$$
$61$ $$( -558009 + 21519 T^{2} - 264 T^{4} + T^{6} )^{2}$$
$67$ $$( 353 - 111 T + T^{3} )^{4}$$
$71$ $$( 227 + 116 T + 19 T^{2} + T^{3} )^{4}$$
$73$ $$15752961 + 5143824 T^{2} + 1381941 T^{4} + 89262 T^{6} + 4329 T^{8} + 75 T^{10} + T^{12}$$
$79$ $$( -107 - 78 T + 3 T^{2} + T^{3} )^{4}$$
$83$ $$51769445841 + 3040925085 T^{2} + 126746613 T^{4} + 2592162 T^{6} + 38619 T^{8} + 228 T^{10} + T^{12}$$
$89$ $$15752961 + 22825719 T^{2} + 32097627 T^{4} + 1406808 T^{6} + 54765 T^{8} + 246 T^{10} + T^{12}$$
$97$ $$96059601 + 19582398 T^{2} + 2904093 T^{4} + 202176 T^{6} + 10323 T^{8} + 111 T^{10} + T^{12}$$