# Properties

 Label 1323.2.c.d Level $1323$ Weight $2$ Character orbit 1323.c 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.5642081874$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 9 x^{10} + 59 x^{8} - 180 x^{6} + 403 x^{4} - 198 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 189) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{8} q^{2} + ( -1 - \beta_{6} ) q^{4} + \beta_{1} q^{5} + ( -\beta_{8} + \beta_{10} ) q^{8} +O(q^{10})$$ $$q + \beta_{8} q^{2} + ( -1 - \beta_{6} ) q^{4} + \beta_{1} q^{5} + ( -\beta_{8} + \beta_{10} ) q^{8} + ( \beta_{5} + \beta_{11} ) q^{10} + ( \beta_{8} - \beta_{10} ) q^{11} + ( \beta_{5} + \beta_{7} ) q^{13} + ( \beta_{6} + \beta_{9} ) q^{16} -\beta_{3} q^{17} + ( -\beta_{5} + \beta_{11} ) q^{19} -\beta_{2} q^{20} + ( -2 - 3 \beta_{6} - \beta_{9} ) q^{22} + ( \beta_{4} + \beta_{8} + \beta_{10} ) q^{23} + ( 5 - 2 \beta_{6} - \beta_{9} ) q^{25} + ( -\beta_{1} - 2 \beta_{3} ) q^{26} + ( \beta_{8} + \beta_{10} ) q^{29} + ( -\beta_{5} + \beta_{7} + 2 \beta_{11} ) q^{31} + \beta_{4} q^{32} + ( -\beta_{5} - 3 \beta_{7} + \beta_{11} ) q^{34} + ( -1 - \beta_{9} ) q^{37} + ( -\beta_{2} + 2 \beta_{3} ) q^{38} + ( 2 \beta_{5} - \beta_{11} ) q^{40} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{41} + ( 2 - \beta_{9} ) q^{43} + ( -\beta_{4} - 6 \beta_{8} + 2 \beta_{10} ) q^{44} + ( -2 - \beta_{9} ) q^{46} + ( -\beta_{1} + 2 \beta_{3} ) q^{47} + ( -\beta_{4} + \beta_{8} + 3 \beta_{10} ) q^{50} + ( -\beta_{5} - 4 \beta_{7} + \beta_{11} ) q^{52} + ( -\beta_{4} - 2 \beta_{8} - \beta_{10} ) q^{53} + ( -2 \beta_{5} + \beta_{11} ) q^{55} + ( -4 + \beta_{6} + \beta_{9} ) q^{58} + ( \beta_{1} + \beta_{3} ) q^{59} + ( \beta_{7} + 2 \beta_{11} ) q^{61} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{62} + ( 2 + \beta_{6} ) q^{64} + ( -\beta_{4} + 3 \beta_{8} + 3 \beta_{10} ) q^{65} + ( -5 - \beta_{6} - 2 \beta_{9} ) q^{67} + ( -\beta_{2} + 3 \beta_{3} ) q^{68} + ( \beta_{4} + \beta_{8} ) q^{71} + ( -2 \beta_{5} + 3 \beta_{7} - \beta_{11} ) q^{73} + ( -\beta_{4} - \beta_{8} + \beta_{10} ) q^{74} + ( 6 \beta_{7} - 3 \beta_{11} ) q^{76} + ( 7 - \beta_{6} - 2 \beta_{9} ) q^{79} + ( -\beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{80} + ( -3 \beta_{7} - \beta_{11} ) q^{82} + ( -\beta_{1} - 2 \beta_{3} ) q^{83} + ( -2 + \beta_{6} + 2 \beta_{9} ) q^{85} + ( -\beta_{4} + 2 \beta_{8} + \beta_{10} ) q^{86} + ( 10 + 5 \beta_{6} + 2 \beta_{9} ) q^{88} + ( -\beta_{2} - 2 \beta_{3} ) q^{89} + ( \beta_{4} + 3 \beta_{10} ) q^{92} + ( \beta_{5} + 6 \beta_{7} - 3 \beta_{11} ) q^{94} + ( -\beta_{4} - 3 \beta_{10} ) q^{95} + ( 2 \beta_{5} - 4 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 16q^{4} + O(q^{10})$$ $$12q - 16q^{4} + 8q^{16} - 40q^{22} + 48q^{25} - 16q^{37} + 20q^{43} - 28q^{46} - 40q^{58} + 28q^{64} - 72q^{67} + 72q^{79} - 12q^{85} + 148q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 9 x^{10} + 59 x^{8} - 180 x^{6} + 403 x^{4} - 198 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$1298 \nu^{11} - 10953 \nu^{9} + 71803 \nu^{7} - 202311 \nu^{5} + 490451 \nu^{3} - 438921 \nu$$$$)/197955$$ $$\beta_{2}$$ $$=$$ $$($$$$8 \nu^{11} - 117 \nu^{9} + 850 \nu^{7} - 3420 \nu^{5} + 7895 \nu^{3} - 7056 \nu$$$$)/747$$ $$\beta_{3}$$ $$=$$ $$($$$$-3734 \nu^{11} + 34254 \nu^{9} - 224554 \nu^{7} + 743958 \nu^{5} - 1731773 \nu^{3} + 1545408 \nu$$$$)/197955$$ $$\beta_{4}$$ $$=$$ $$($$$$1298 \nu^{11} - 10953 \nu^{9} + 71803 \nu^{7} - 202311 \nu^{5} + 490451 \nu^{3} - 43011 \nu$$$$)/65985$$ $$\beta_{5}$$ $$=$$ $$($$$$787 \nu^{10} - 10047 \nu^{8} + 58532 \nu^{6} - 191649 \nu^{4} + 281254 \nu^{2} - 89064$$$$)/65985$$ $$\beta_{6}$$ $$=$$ $$($$$$288 \nu^{10} - 1888 \nu^{8} + 9933 \nu^{6} - 12896 \nu^{4} + 6336 \nu^{2} + 68439$$$$)/21995$$ $$\beta_{7}$$ $$=$$ $$($$$$2596 \nu^{10} - 21906 \nu^{8} + 143606 \nu^{6} - 404622 \nu^{4} + 980902 \nu^{2} - 283977$$$$)/197955$$ $$\beta_{8}$$ $$=$$ $$($$$$5192 \nu^{11} - 43812 \nu^{9} + 287212 \nu^{7} - 809244 \nu^{5} + 1763849 \nu^{3} - 172044 \nu$$$$)/197955$$ $$\beta_{9}$$ $$=$$ $$($$$$-333 \nu^{10} + 2183 \nu^{8} - 9423 \nu^{6} + 14911 \nu^{4} - 7326 \nu^{2} + 48026$$$$)/21995$$ $$\beta_{10}$$ $$=$$ $$($$$$-88 \nu^{11} + 783 \nu^{9} - 4868 \nu^{7} + 13716 \nu^{5} - 26581 \nu^{3} + 2916 \nu$$$$)/2385$$ $$\beta_{11}$$ $$=$$ $$($$$$-2353 \nu^{10} + 20313 \nu^{8} - 133163 \nu^{6} + 393741 \nu^{4} - 843586 \nu^{2} + 248931$$$$)/65985$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} - 3 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{11} - \beta_{9} + 9 \beta_{7} - 2 \beta_{6} + 10$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{8} + 4 \beta_{4}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{11} + 2 \beta_{9} + 12 \beta_{7} + 3 \beta_{6} + \beta_{5} - 14$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-3 \beta_{10} - 18 \beta_{8} + 17 \beta_{4} + 18 \beta_{3} + 3 \beta_{2} + 51 \beta_{1}$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$($$$$32 \beta_{9} + 37 \beta_{6} - 185$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$27 \beta_{10} + 96 \beta_{8} - 74 \beta_{4} + 96 \beta_{3} + 27 \beta_{2} + 222 \beta_{1}$$$$)/6$$ $$\nu^{8}$$ $$=$$ $$($$$$-106 \beta_{11} + 55 \beta_{9} - 222 \beta_{7} + 51 \beta_{6} - 59 \beta_{5} - 277$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$177 \beta_{10} + 495 \beta_{8} - 328 \beta_{4}$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$-1479 \beta_{11} - 835 \beta_{9} - 2952 \beta_{7} - 644 \beta_{6} - 1026 \beta_{5} + 3787$$$$)/6$$ $$\nu^{11}$$ $$=$$ $$($$$$1026 \beta_{10} + 2505 \beta_{8} - 1477 \beta_{4} - 2505 \beta_{3} - 1026 \beta_{2} - 4431 \beta_{1}$$$$)/6$$

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1322.1
 0.617942 − 0.356769i −0.617942 − 0.356769i 1.90412 + 1.09935i −1.90412 + 1.09935i 1.65604 − 0.956115i −1.65604 − 0.956115i 1.65604 + 0.956115i −1.65604 + 0.956115i 1.90412 − 1.09935i −1.90412 − 1.09935i 0.617942 + 0.356769i −0.617942 + 0.356769i
2.49086i 0 −4.20440 −1.23588 0 0 5.49086i 0 3.07842i
1322.2 2.49086i 0 −4.20440 1.23588 0 0 5.49086i 0 3.07842i
1322.3 1.83424i 0 −1.36445 −3.80824 0 0 1.16576i 0 6.98525i
1322.4 1.83424i 0 −1.36445 3.80824 0 0 1.16576i 0 6.98525i
1322.5 0.656620i 0 1.56885 −3.31208 0 0 2.34338i 0 2.17478i
1322.6 0.656620i 0 1.56885 3.31208 0 0 2.34338i 0 2.17478i
1322.7 0.656620i 0 1.56885 −3.31208 0 0 2.34338i 0 2.17478i
1322.8 0.656620i 0 1.56885 3.31208 0 0 2.34338i 0 2.17478i
1322.9 1.83424i 0 −1.36445 −3.80824 0 0 1.16576i 0 6.98525i
1322.10 1.83424i 0 −1.36445 3.80824 0 0 1.16576i 0 6.98525i
1322.11 2.49086i 0 −4.20440 −1.23588 0 0 5.49086i 0 3.07842i
1322.12 2.49086i 0 −4.20440 1.23588 0 0 5.49086i 0 3.07842i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1322.12 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
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.c.d 12
3.b odd 2 1 inner 1323.2.c.d 12
7.b odd 2 1 inner 1323.2.c.d 12
7.c even 3 1 189.2.p.d 12
7.d odd 6 1 189.2.p.d 12
21.c even 2 1 inner 1323.2.c.d 12
21.g even 6 1 189.2.p.d 12
21.h odd 6 1 189.2.p.d 12
63.g even 3 1 567.2.s.f 12
63.h even 3 1 567.2.i.f 12
63.i even 6 1 567.2.i.f 12
63.j odd 6 1 567.2.i.f 12
63.k odd 6 1 567.2.s.f 12
63.n odd 6 1 567.2.s.f 12
63.s even 6 1 567.2.s.f 12
63.t odd 6 1 567.2.i.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.d 12 7.c even 3 1
189.2.p.d 12 7.d odd 6 1
189.2.p.d 12 21.g even 6 1
189.2.p.d 12 21.h odd 6 1
567.2.i.f 12 63.h even 3 1
567.2.i.f 12 63.i even 6 1
567.2.i.f 12 63.j odd 6 1
567.2.i.f 12 63.t odd 6 1
567.2.s.f 12 63.g even 3 1
567.2.s.f 12 63.k odd 6 1
567.2.s.f 12 63.n odd 6 1
567.2.s.f 12 63.s even 6 1
1323.2.c.d 12 1.a even 1 1 trivial
1323.2.c.d 12 3.b odd 2 1 inner
1323.2.c.d 12 7.b odd 2 1 inner
1323.2.c.d 12 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 10 T_{2}^{4} + 25 T_{2}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(1323, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 9 + 25 T^{2} + 10 T^{4} + T^{6} )^{2}$$
$3$ $$T^{12}$$
$5$ $$( -243 + 198 T^{2} - 27 T^{4} + T^{6} )^{2}$$
$7$ $$T^{12}$$
$11$ $$( 225 + 214 T^{2} + 37 T^{4} + T^{6} )^{2}$$
$13$ $$( 675 + 369 T^{2} + 42 T^{4} + T^{6} )^{2}$$
$17$ $$( -243 + 225 T^{2} - 30 T^{4} + T^{6} )^{2}$$
$19$ $$( 243 + 630 T^{2} + 51 T^{4} + T^{6} )^{2}$$
$23$ $$( 729 + 517 T^{2} + 94 T^{4} + T^{6} )^{2}$$
$29$ $$( 81 + 322 T^{2} + 37 T^{4} + T^{6} )^{2}$$
$31$ $$( 64827 + 5733 T^{2} + 138 T^{4} + T^{6} )^{2}$$
$37$ $$( -67 - 19 T + 4 T^{2} + T^{3} )^{4}$$
$41$ $$( -243 + 1953 T^{2} - 114 T^{4} + T^{6} )^{2}$$
$43$ $$( -1 - 16 T - 5 T^{2} + T^{3} )^{4}$$
$47$ $$( -19683 + 3870 T^{2} - 135 T^{4} + T^{6} )^{2}$$
$53$ $$( 9 + 1717 T^{2} + 118 T^{4} + T^{6} )^{2}$$
$59$ $$( -2187 + 738 T^{2} - 63 T^{4} + T^{6} )^{2}$$
$61$ $$( 49923 + 4923 T^{2} + 129 T^{4} + T^{6} )^{2}$$
$67$ $$( -677 + 15 T + 18 T^{2} + T^{3} )^{4}$$
$71$ $$( 19881 + 2290 T^{2} + 85 T^{4} + T^{6} )^{2}$$
$73$ $$( 177147 + 10782 T^{2} + 195 T^{4} + T^{6} )^{2}$$
$79$ $$( 7 + 15 T - 18 T^{2} + T^{3} )^{4}$$
$83$ $$( -6075 + 5814 T^{2} - 159 T^{4} + T^{6} )^{2}$$
$89$ $$( -87723 + 9675 T^{2} - 201 T^{4} + T^{6} )^{2}$$
$97$ $$( 1728 + 3744 T^{2} + 204 T^{4} + T^{6} )^{2}$$