# Properties

 Label 1323.2.c.f Level $1323$ Weight $2$ Character orbit 1323.c Analytic conductor $10.564$ Analytic rank $0$ Dimension $16$ 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: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 24 x^{14} + 212 x^{12} + 872 x^{10} + 1815 x^{8} + 1928 x^{6} + 996 x^{4} + 200 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} + ( -1 + \beta_{13} ) q^{4} -\beta_{2} q^{5} + ( -\beta_{1} - \beta_{5} + \beta_{12} - \beta_{15} ) q^{8} +O(q^{10})$$ $$q + \beta_{5} q^{2} + ( -1 + \beta_{13} ) q^{4} -\beta_{2} q^{5} + ( -\beta_{1} - \beta_{5} + \beta_{12} - \beta_{15} ) q^{8} + ( -\beta_{4} - \beta_{6} - \beta_{10} - \beta_{14} ) q^{10} + ( -\beta_{5} - \beta_{12} - \beta_{15} ) q^{11} + ( \beta_{4} - \beta_{6} + \beta_{10} + \beta_{14} ) q^{13} + ( 3 + \beta_{7} + 3 \beta_{8} - \beta_{13} ) q^{16} + ( \beta_{9} + 2 \beta_{11} ) q^{17} + ( \beta_{4} - \beta_{6} + \beta_{10} ) q^{19} + ( 4 \beta_{2} + \beta_{3} - 3 \beta_{9} + 2 \beta_{11} ) q^{20} + ( 4 - \beta_{8} - 2 \beta_{13} ) q^{22} + ( -2 \beta_{1} - \beta_{12} + \beta_{15} ) q^{23} + ( -1 + \beta_{7} + \beta_{8} - \beta_{13} ) q^{25} + ( -4 \beta_{2} - 3 \beta_{3} - \beta_{9} - 2 \beta_{11} ) q^{26} + ( 2 \beta_{1} + \beta_{12} - 2 \beta_{15} ) q^{29} + ( 3 \beta_{4} + \beta_{6} - 2 \beta_{10} + 2 \beta_{14} ) q^{31} + ( 4 \beta_{1} + 2 \beta_{5} - 2 \beta_{12} - 2 \beta_{15} ) q^{32} + ( 2 \beta_{4} - \beta_{6} - 3 \beta_{10} ) q^{34} + ( 2 + 2 \beta_{7} + 3 \beta_{8} - \beta_{13} ) q^{37} + ( -\beta_{2} - 2 \beta_{3} - \beta_{9} - \beta_{11} ) q^{38} + ( 5 \beta_{4} + 4 \beta_{6} + 3 \beta_{10} + 2 \beta_{14} ) q^{40} + ( 2 \beta_{3} - \beta_{9} - \beta_{11} ) q^{41} + ( -1 - \beta_{7} + 4 \beta_{8} ) q^{43} + ( \beta_{1} + 6 \beta_{5} - 3 \beta_{12} ) q^{44} + ( 1 + 2 \beta_{7} + 3 \beta_{8} - \beta_{13} ) q^{46} + ( \beta_{2} - 2 \beta_{9} + 3 \beta_{11} ) q^{47} + ( 4 \beta_{1} - 2 \beta_{12} ) q^{50} + ( -7 \beta_{4} - 2 \beta_{6} + \beta_{10} - 2 \beta_{14} ) q^{52} + ( -2 \beta_{1} + 5 \beta_{5} + \beta_{12} - 2 \beta_{15} ) q^{53} + ( 2 \beta_{4} + \beta_{6} + 3 \beta_{10} + \beta_{14} ) q^{55} + ( -2 \beta_{7} - 3 \beta_{8} ) q^{58} + ( -3 \beta_{2} - 4 \beta_{3} + 2 \beta_{9} - 3 \beta_{11} ) q^{59} + ( -5 \beta_{4} - \beta_{6} - 4 \beta_{10} + \beta_{14} ) q^{61} + ( -8 \beta_{2} - 4 \beta_{3} - 5 \beta_{11} ) q^{62} + ( -2 - 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{13} ) q^{64} + ( -2 \beta_{1} - 3 \beta_{5} - 2 \beta_{12} + 2 \beta_{15} ) q^{65} + ( 3 - 2 \beta_{7} + \beta_{8} + \beta_{13} ) q^{67} + ( 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{9} + 2 \beta_{11} ) q^{68} + ( \beta_{1} - 3 \beta_{5} + 5 \beta_{12} - \beta_{15} ) q^{71} + ( \beta_{4} - 2 \beta_{6} + \beta_{10} - \beta_{14} ) q^{73} + ( 8 \beta_{1} + 2 \beta_{5} - 4 \beta_{12} - \beta_{15} ) q^{74} + ( -2 \beta_{4} + 3 \beta_{10} - \beta_{14} ) q^{76} + ( -4 - \beta_{7} + 3 \beta_{8} + 3 \beta_{13} ) q^{79} + ( -10 \beta_{2} - \beta_{3} + 5 \beta_{9} - 3 \beta_{11} ) q^{80} + ( \beta_{4} - \beta_{6} + 2 \beta_{10} ) q^{82} + 5 \beta_{3} q^{83} + ( 4 + 2 \beta_{7} - \beta_{8} - \beta_{13} ) q^{85} + ( 2 \beta_{1} - 4 \beta_{12} + \beta_{15} ) q^{86} + ( -11 - \beta_{7} - 7 \beta_{8} + 3 \beta_{13} ) q^{88} + ( 2 \beta_{2} - 3 \beta_{3} - 5 \beta_{9} - 2 \beta_{11} ) q^{89} + ( 4 \beta_{1} + \beta_{5} - 6 \beta_{12} + \beta_{15} ) q^{92} + ( 4 \beta_{4} + 3 \beta_{6} + \beta_{14} ) q^{94} -\beta_{12} q^{95} + ( 4 \beta_{4} + 5 \beta_{6} - 6 \beta_{10} + 2 \beta_{14} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 16q^{4} + O(q^{10})$$ $$16q - 16q^{4} + 48q^{16} + 64q^{22} - 16q^{25} + 32q^{37} - 16q^{43} + 16q^{46} - 32q^{64} + 48q^{67} - 64q^{79} + 64q^{85} - 176q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 24 x^{14} + 212 x^{12} + 872 x^{10} + 1815 x^{8} + 1928 x^{6} + 996 x^{4} + 200 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3566 \nu^{15} - 82183 \nu^{13} - 677054 \nu^{11} - 2451006 \nu^{9} - 4029525 \nu^{7} - 2653934 \nu^{5} - 400750 \nu^{3} + 102636 \nu$$$$)/7231$$ $$\beta_{2}$$ $$=$$ $$($$$$4109 \nu^{14} + 95961 \nu^{12} + 809596 \nu^{10} + 3069810 \nu^{8} + 5537952 \nu^{6} + 4485192 \nu^{4} + 1280801 \nu^{2} - 9938$$$$)/7231$$ $$\beta_{3}$$ $$=$$ $$($$$$-4213 \nu^{14} - 98695 \nu^{12} - 836669 \nu^{10} - 3196410 \nu^{8} - 5829935 \nu^{6} - 4816716 \nu^{4} - 1459060 \nu^{2} - 8680$$$$)/7231$$ $$\beta_{4}$$ $$=$$ $$($$$$11880 \nu^{15} + 277026 \nu^{13} + 2330454 \nu^{11} + 8786075 \nu^{9} + 15693672 \nu^{7} + 12631032 \nu^{5} + 3918733 \nu^{3} + 206322 \nu$$$$)/7231$$ $$\beta_{5}$$ $$=$$ $$($$$$12411 \nu^{15} + 289535 \nu^{13} + 2437027 \nu^{11} + 9191538 \nu^{9} + 16395526 \nu^{7} + 13061987 \nu^{5} + 3821459 \nu^{3} + 89795 \nu$$$$)/7231$$ $$\beta_{6}$$ $$=$$ $$($$$$-12661 \nu^{15} - 294796 \nu^{13} - 2472050 \nu^{11} - 9248819 \nu^{9} - 16190613 \nu^{7} - 12258167 \nu^{5} - 2969225 \nu^{3} + 184528 \nu$$$$)/7231$$ $$\beta_{7}$$ $$=$$ $$($$$$-15611 \nu^{14} - 362454 \nu^{12} - 3026016 \nu^{10} - 11245322 \nu^{8} - 19531408 \nu^{6} - 14787914 \nu^{4} - 3904397 \nu^{2} + 16978$$$$)/7231$$ $$\beta_{8}$$ $$=$$ $$($$$$-2432 \nu^{14} - 56782 \nu^{12} - 478539 \nu^{10} - 1808300 \nu^{8} - 3232355 \nu^{6} - 2573576 \nu^{4} - 736177 \nu^{2} - 5170$$$$)/1033$$ $$\beta_{9}$$ $$=$$ $$($$$$20 \nu^{14} + 467 \nu^{12} + 3936 \nu^{10} + 14872 \nu^{8} + 26564 \nu^{6} + 21096 \nu^{4} + 5980 \nu^{2} + 18$$$$)/7$$ $$\beta_{10}$$ $$=$$ $$($$$$-20975 \nu^{15} - 489639 \nu^{13} - 4125450 \nu^{11} - 15583888 \nu^{9} - 27854760 \nu^{7} - 22235265 \nu^{5} - 6487208 \nu^{3} - 109968 \nu$$$$)/7231$$ $$\beta_{11}$$ $$=$$ $$($$$$24141 \nu^{14} + 562578 \nu^{12} + 4726427 \nu^{10} + 17764742 \nu^{8} + 31483261 \nu^{6} + 24752870 \nu^{4} + 7015238 \nu^{2} + 62386$$$$)/7231$$ $$\beta_{12}$$ $$=$$ $$($$$$25023 \nu^{15} + 585089 \nu^{13} + 4943611 \nu^{11} + 18772057 \nu^{9} + 33876484 \nu^{7} + 27549033 \nu^{5} + 8361354 \nu^{3} + 246543 \nu$$$$)/7231$$ $$\beta_{13}$$ $$=$$ $$($$$$-25833 \nu^{14} - 602052 \nu^{12} - 5058697 \nu^{10} - 19017892 \nu^{8} - 33715265 \nu^{6} - 26500974 \nu^{4} - 7440426 \nu^{2} - 19311$$$$)/7231$$ $$\beta_{14}$$ $$=$$ $$($$$$25926 \nu^{15} + 603881 \nu^{13} + 5068077 \nu^{11} + 18998083 \nu^{9} + 33405881 \nu^{7} + 25519016 \nu^{5} + 6225156 \nu^{3} - 437333 \nu$$$$)/7231$$ $$\beta_{15}$$ $$=$$ $$($$$$33297 \nu^{15} + 775702 \nu^{13} + 6512858 \nu^{11} + 24444196 \nu^{9} + 43160577 \nu^{7} + 33528527 \nu^{5} + 8990368 \nu^{3} - 173250 \nu$$$$)/7231$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{10} - \beta_{6} + \beta_{4} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} + \beta_{7} + 3 \beta_{3} + \beta_{2} - 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} - 5 \beta_{10} + 6 \beta_{6} + 6 \beta_{5} - 8 \beta_{4} - 7 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{13} - 9 \beta_{11} - 3 \beta_{9} - 6 \beta_{8} - 10 \beta_{7} - 27 \beta_{3} - 10 \beta_{2} + 38$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{15} + 3 \beta_{14} - \beta_{12} + 33 \beta_{10} - 47 \beta_{6} - 62 \beta_{5} + 71 \beta_{4} + 55 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-18 \beta_{13} + 86 \beta_{11} + 42 \beta_{9} + 83 \beta_{8} + 90 \beta_{7} + 229 \beta_{3} + 91 \beta_{2} - 294$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-11 \beta_{15} - 51 \beta_{14} + 34 \beta_{12} - 219 \beta_{10} + 385 \beta_{6} + 559 \beta_{5} - 630 \beta_{4} - 467 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$113 \beta_{13} - 820 \beta_{11} - 476 \beta_{9} - 909 \beta_{8} - 787 \beta_{7} - 1948 \beta_{3} - 832 \beta_{2} + 2417$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-145 \beta_{15} + 630 \beta_{14} - 552 \beta_{12} + 1387 \beta_{10} - 3204 \beta_{6} - 4908 \beta_{5} + 5582 \beta_{4} + 4085 \beta_{1}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-514 \beta_{13} + 7743 \beta_{11} + 4970 \beta_{9} + 9227 \beta_{8} + 6847 \beta_{7} + 16682 \beta_{3} + 7634 \beta_{2} - 20336$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$3077 \beta_{15} - 6896 \beta_{14} + 6970 \beta_{12} - 8084 \beta_{10} + 26945 \beta_{6} + 42906 \beta_{5} - 49477 \beta_{4} - 36116 \beta_{1}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$252 \beta_{13} - 36212 \beta_{11} - 24831 \beta_{9} - 45170 \beta_{8} - 29832 \beta_{7} - 71870 \beta_{3} - 34986 \beta_{2} + 86566$$ $$\nu^{13}$$ $$=$$ $$($$$$-40723 \beta_{15} + 70975 \beta_{14} - 78066 \beta_{12} + 39791 \beta_{10} - 228507 \beta_{6} - 375611 \beta_{5} + 439060 \beta_{4} + 320677 \beta_{1}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$27158 \beta_{13} + 671999 \beta_{11} + 482846 \beta_{9} + 865586 \beta_{8} + 521589 \beta_{7} + 1245269 \beta_{3} + 639533 \beta_{2} - 1485588$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$462786 \beta_{15} - 704028 \beta_{14} + 817509 \beta_{12} - 112000 \beta_{10} + 1951782 \beta_{6} + 3297345 \beta_{5} - 3901487 \beta_{4} - 2853438 \beta_{1}$$$$)/2$$

## 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
 − 1.54124i − 1.70654i − 0.829521i 2.99366i − 0.964652i − 0.0716223i 2.73296i − 0.810833i − 2.73296i 0.810833i 0.964652i 0.0716223i 0.829521i − 2.99366i 1.54124i 1.70654i
2.67093i 0 −5.13386 −3.20318 0 0 8.37032i 0 8.55547i
1322.2 2.67093i 0 −5.13386 3.20318 0 0 8.37032i 0 8.55547i
1322.3 2.02428i 0 −2.09771 −0.182734 0 0 0.197790i 0 0.369905i
1322.4 2.02428i 0 −2.09771 0.182734 0 0 0.197790i 0 0.369905i
1322.5 0.698626i 0 1.51192 −2.26676 0 0 2.45352i 0 1.58361i
1322.6 0.698626i 0 1.51192 2.26676 0 0 2.45352i 0 1.58361i
1322.7 0.529484i 0 1.71965 −0.753692 0 0 1.96949i 0 0.399068i
1322.8 0.529484i 0 1.71965 0.753692 0 0 1.96949i 0 0.399068i
1322.9 0.529484i 0 1.71965 −0.753692 0 0 1.96949i 0 0.399068i
1322.10 0.529484i 0 1.71965 0.753692 0 0 1.96949i 0 0.399068i
1322.11 0.698626i 0 1.51192 −2.26676 0 0 2.45352i 0 1.58361i
1322.12 0.698626i 0 1.51192 2.26676 0 0 2.45352i 0 1.58361i
1322.13 2.02428i 0 −2.09771 −0.182734 0 0 0.197790i 0 0.369905i
1322.14 2.02428i 0 −2.09771 0.182734 0 0 0.197790i 0 0.369905i
1322.15 2.67093i 0 −5.13386 −3.20318 0 0 8.37032i 0 8.55547i
1322.16 2.67093i 0 −5.13386 3.20318 0 0 8.37032i 0 8.55547i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1322.16 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.f 16
3.b odd 2 1 inner 1323.2.c.f 16
7.b odd 2 1 inner 1323.2.c.f 16
21.c even 2 1 inner 1323.2.c.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.2.c.f 16 1.a even 1 1 trivial
1323.2.c.f 16 3.b odd 2 1 inner
1323.2.c.f 16 7.b odd 2 1 inner
1323.2.c.f 16 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 12 T_{2}^{6} + 38 T_{2}^{4} + 24 T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1323, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 24 T^{2} + 38 T^{4} + 12 T^{6} + T^{8} )^{2}$$
$3$ $$T^{16}$$
$5$ $$( 1 - 32 T^{2} + 62 T^{4} - 16 T^{6} + T^{8} )^{2}$$
$7$ $$T^{16}$$
$11$ $$( 2116 + 7128 T^{2} + 1142 T^{4} + 60 T^{6} + T^{8} )^{2}$$
$13$ $$( 20164 + 12920 T^{2} + 1790 T^{4} + 76 T^{6} + T^{8} )^{2}$$
$17$ $$( 134689 - 36124 T^{2} + 2918 T^{4} - 92 T^{6} + T^{8} )^{2}$$
$19$ $$( 4 + 152 T^{2} + 206 T^{4} + 28 T^{6} + T^{8} )^{2}$$
$23$ $$( 148996 + 33712 T^{2} + 2676 T^{4} + 88 T^{6} + T^{8} )^{2}$$
$29$ $$( 4 + 10392 T^{2} + 2774 T^{4} + 108 T^{6} + T^{8} )^{2}$$
$31$ $$( 9604 + 176792 T^{2} + 9662 T^{4} + 172 T^{6} + T^{8} )^{2}$$
$37$ $$( 391 + 476 T - 88 T^{2} - 8 T^{3} + T^{4} )^{4}$$
$41$ $$( 529 - 1664 T^{2} + 878 T^{4} - 112 T^{6} + T^{8} )^{2}$$
$43$ $$( 943 + 16 T - 96 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$47$ $$( 18769 - 56036 T^{2} + 6806 T^{4} - 196 T^{6} + T^{8} )^{2}$$
$53$ $$( 6791236 + 1315376 T^{2} + 37044 T^{4} + 344 T^{6} + T^{8} )^{2}$$
$59$ $$( 24990001 - 1458692 T^{2} + 31286 T^{4} - 292 T^{6} + T^{8} )^{2}$$
$61$ $$( 5827396 + 3061832 T^{2} + 63062 T^{4} + 436 T^{6} + T^{8} )^{2}$$
$67$ $$( 612 + 504 T - 42 T^{2} - 12 T^{3} + T^{4} )^{4}$$
$71$ $$( 68029504 + 3723648 T^{2} + 63824 T^{4} + 432 T^{6} + T^{8} )^{2}$$
$73$ $$( 103684 + 43216 T^{2} + 3572 T^{4} + 104 T^{6} + T^{8} )^{2}$$
$79$ $$( -6167 - 1744 T - 58 T^{2} + 16 T^{3} + T^{4} )^{4}$$
$83$ $$( 390625 - 500000 T^{2} + 38750 T^{4} - 400 T^{6} + T^{8} )^{2}$$
$89$ $$( 2155024 - 1096496 T^{2} + 109580 T^{4} - 652 T^{6} + T^{8} )^{2}$$
$97$ $$( 36409156 + 7276072 T^{2} + 113798 T^{4} + 596 T^{6} + T^{8} )^{2}$$