# Properties

 Label 943.1.n.b Level $943$ Weight $1$ Character orbit 943.n Analytic conductor $0.471$ Analytic rank $0$ Dimension $16$ Projective image $D_{60}$ CM discriminant -23 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$943 = 23 \cdot 41$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 943.n (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.470618306913$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{20})$$ Coefficient field: $$\Q(\zeta_{60})$$ Defining polynomial: $$x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{60}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{60} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{60}^{5} + \zeta_{60}^{19} ) q^{2} + ( \zeta_{60} + \zeta_{60}^{14} ) q^{3} + ( -\zeta_{60}^{8} + \zeta_{60}^{10} - \zeta_{60}^{24} ) q^{4} + ( -\zeta_{60}^{3} - \zeta_{60}^{6} - \zeta_{60}^{19} + \zeta_{60}^{20} ) q^{6} + ( \zeta_{60}^{13} - \zeta_{60}^{15} - \zeta_{60}^{27} + \zeta_{60}^{29} ) q^{8} + ( \zeta_{60}^{2} + \zeta_{60}^{15} + \zeta_{60}^{28} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{60}^{5} + \zeta_{60}^{19} ) q^{2} + ( \zeta_{60} + \zeta_{60}^{14} ) q^{3} + ( -\zeta_{60}^{8} + \zeta_{60}^{10} - \zeta_{60}^{24} ) q^{4} + ( -\zeta_{60}^{3} - \zeta_{60}^{6} - \zeta_{60}^{19} + \zeta_{60}^{20} ) q^{6} + ( \zeta_{60}^{13} - \zeta_{60}^{15} - \zeta_{60}^{27} + \zeta_{60}^{29} ) q^{8} + ( \zeta_{60}^{2} + \zeta_{60}^{15} + \zeta_{60}^{28} ) q^{9} + ( \zeta_{60}^{8} - \zeta_{60}^{9} + \zeta_{60}^{11} - \zeta_{60}^{22} + \zeta_{60}^{24} - \zeta_{60}^{25} ) q^{12} + ( -\zeta_{60} + \zeta_{60}^{8} ) q^{13} + ( -\zeta_{60}^{2} + \zeta_{60}^{4} + \zeta_{60}^{16} - \zeta_{60}^{18} + \zeta_{60}^{20} ) q^{16} + ( \zeta_{60}^{3} - \zeta_{60}^{4} - \zeta_{60}^{7} - \zeta_{60}^{17} - \zeta_{60}^{20} + \zeta_{60}^{21} ) q^{18} + \zeta_{60}^{12} q^{23} + ( -1 + \zeta_{60}^{11} - \zeta_{60}^{13} + \zeta_{60}^{14} - \zeta_{60}^{16} + \zeta_{60}^{27} - \zeta_{60}^{28} - \zeta_{60}^{29} ) q^{24} + \zeta_{60}^{18} q^{25} + ( \zeta_{60}^{6} - \zeta_{60}^{13} - \zeta_{60}^{20} + \zeta_{60}^{27} ) q^{26} + ( \zeta_{60}^{3} - \zeta_{60}^{12} + \zeta_{60}^{16} + \zeta_{60}^{29} ) q^{27} + ( -\zeta_{60}^{10} + \zeta_{60}^{23} ) q^{29} + ( \zeta_{60}^{5} + \zeta_{60}^{7} ) q^{31} + ( -\zeta_{60}^{5} + \zeta_{60}^{7} - \zeta_{60}^{9} - \zeta_{60}^{21} + \zeta_{60}^{23} - \zeta_{60}^{25} ) q^{32} + ( \zeta_{60}^{6} - \zeta_{60}^{8} + \zeta_{60}^{9} - \zeta_{60}^{10} + \zeta_{60}^{12} + \zeta_{60}^{22} - \zeta_{60}^{23} + \zeta_{60}^{25} - \zeta_{60}^{26} ) q^{36} + ( -\zeta_{60}^{2} + \zeta_{60}^{9} - \zeta_{60}^{15} + \zeta_{60}^{22} ) q^{39} -\zeta_{60}^{22} q^{41} + ( -\zeta_{60} - \zeta_{60}^{17} ) q^{46} + ( -\zeta_{60}^{14} + \zeta_{60}^{25} ) q^{47} + ( -1 + \zeta_{60}^{2} - \zeta_{60}^{3} - \zeta_{60}^{4} + \zeta_{60}^{5} - \zeta_{60}^{16} + \zeta_{60}^{17} + \zeta_{60}^{18} - \zeta_{60}^{19} + \zeta_{60}^{21} ) q^{48} -\zeta_{60}^{21} q^{49} + ( -\zeta_{60}^{7} - \zeta_{60}^{23} ) q^{50} + ( \zeta_{60}^{2} + \zeta_{60}^{9} - \zeta_{60}^{11} - \zeta_{60}^{16} + \zeta_{60}^{18} + \zeta_{60}^{25} ) q^{52} + ( \zeta_{60} + \zeta_{60}^{4} - \zeta_{60}^{5} - \zeta_{60}^{8} + \zeta_{60}^{17} - \zeta_{60}^{18} - \zeta_{60}^{21} + \zeta_{60}^{22} ) q^{54} + ( -\zeta_{60}^{12} + \zeta_{60}^{15} - \zeta_{60}^{28} - \zeta_{60}^{29} ) q^{58} + ( -\zeta_{60}^{6} - \zeta_{60}^{18} ) q^{59} + ( -\zeta_{60}^{10} - \zeta_{60}^{12} + \zeta_{60}^{24} + \zeta_{60}^{26} ) q^{62} + ( -1 + \zeta_{60}^{10} - \zeta_{60}^{12} + \zeta_{60}^{14} - \zeta_{60}^{24} + \zeta_{60}^{26} - \zeta_{60}^{28} ) q^{64} + ( \zeta_{60}^{13} + \zeta_{60}^{26} ) q^{69} + ( -\zeta_{60}^{28} + \zeta_{60}^{29} ) q^{71} + ( 1 - \zeta_{60} - \zeta_{60}^{11} + \zeta_{60}^{12} + \zeta_{60}^{13} - \zeta_{60}^{14} + \zeta_{60}^{15} - \zeta_{60}^{17} + \zeta_{60}^{25} - \zeta_{60}^{27} + \zeta_{60}^{28} - \zeta_{60}^{29} ) q^{72} + ( -\zeta_{60}^{4} - \zeta_{60}^{26} ) q^{73} + ( -\zeta_{60}^{2} + \zeta_{60}^{19} ) q^{75} + ( \zeta_{60}^{4} + \zeta_{60}^{7} - \zeta_{60}^{11} - \zeta_{60}^{14} + \zeta_{60}^{20} - \zeta_{60}^{21} - \zeta_{60}^{27} + \zeta_{60}^{28} ) q^{78} + ( -1 + \zeta_{60}^{4} - \zeta_{60}^{13} + \zeta_{60}^{17} - \zeta_{60}^{26} ) q^{81} + ( \zeta_{60}^{11} + \zeta_{60}^{27} ) q^{82} + ( -\zeta_{60}^{7} - \zeta_{60}^{11} ) q^{87} + ( \zeta_{60}^{6} - \zeta_{60}^{20} + \zeta_{60}^{22} ) q^{92} + ( \zeta_{60}^{6} + \zeta_{60}^{8} + \zeta_{60}^{19} + \zeta_{60}^{21} ) q^{93} + ( 1 + \zeta_{60}^{3} - \zeta_{60}^{14} + \zeta_{60}^{19} ) q^{94} + ( \zeta_{60}^{5} - \zeta_{60}^{6} - \zeta_{60}^{7} + \zeta_{60}^{8} + \zeta_{60}^{9} - \zeta_{60}^{10} - \zeta_{60}^{19} + \zeta_{60}^{21} - \zeta_{60}^{22} - \zeta_{60}^{23} + \zeta_{60}^{24} - \zeta_{60}^{26} ) q^{96} + ( \zeta_{60}^{10} + \zeta_{60}^{26} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 2q^{3} + 10q^{4} - 12q^{6} + O(q^{10})$$ $$16q - 2q^{3} + 10q^{4} - 12q^{6} + 2q^{13} - 6q^{16} + 6q^{18} - 4q^{23} - 22q^{24} + 4q^{25} + 12q^{26} + 6q^{27} - 8q^{29} - 10q^{36} + 2q^{41} + 2q^{47} - 18q^{48} - 6q^{54} + 2q^{58} - 8q^{59} - 10q^{62} - 6q^{64} - 2q^{69} - 2q^{71} + 16q^{72} + 2q^{75} - 2q^{78} - 12q^{81} + 10q^{92} + 6q^{93} + 18q^{94} - 10q^{96} + 6q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/943\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$534$$ $$\chi(n)$$ $$\zeta_{60}^{9}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
367.1
 0.994522 + 0.104528i −0.406737 − 0.913545i 0.406737 − 0.913545i −0.994522 + 0.104528i 0.207912 + 0.978148i 0.743145 − 0.669131i 0.207912 − 0.978148i 0.743145 + 0.669131i −0.743145 − 0.669131i −0.207912 + 0.978148i −0.743145 + 0.669131i −0.207912 − 0.978148i 0.994522 − 0.104528i −0.406737 + 0.913545i 0.406737 + 0.913545i −0.994522 − 0.104528i
−1.27276 + 0.413545i 1.09905 + 1.09905i 0.639886 0.464905i 0 −1.85334 0.944322i 0 0.164446 0.226341i 1.41582i 0
367.2 1.86055 0.604528i −1.32028 1.32028i 2.28716 1.66172i 0 −3.25460 1.65830i 0 2.10094 2.89169i 2.48629i 0
390.1 −1.86055 0.604528i −0.506809 0.506809i 2.28716 + 1.66172i 0 0.636561 + 1.24932i 0 −2.10094 2.89169i 0.486290i 0
390.2 1.27276 + 0.413545i −0.889993 0.889993i 0.639886 + 0.464905i 0 −0.764697 1.50080i 0 −0.164446 0.226341i 0.584177i 0
459.1 −0.122881 + 0.169131i 1.18606 + 1.18606i 0.295511 + 0.909491i 0 −0.346343 + 0.0548553i 0 −0.388960 0.126381i 1.81347i 0
459.2 1.07394 1.47815i 0.0740142 + 0.0740142i −0.722562 2.22382i 0 0.188891 0.0299173i 0 −2.32545 0.755585i 0.989044i 0
528.1 −0.122881 0.169131i 1.18606 1.18606i 0.295511 0.909491i 0 −0.346343 0.0548553i 0 −0.388960 + 0.126381i 1.81347i 0
528.2 1.07394 + 1.47815i 0.0740142 0.0740142i −0.722562 + 2.22382i 0 0.188891 + 0.0299173i 0 −2.32545 + 0.755585i 0.989044i 0
620.1 −1.07394 1.47815i −1.41228 1.41228i −0.722562 + 2.22382i 0 −0.570857 + 3.60425i 0 2.32545 0.755585i 2.98904i 0
620.2 0.122881 + 0.169131i 0.770236 + 0.770236i 0.295511 0.909491i 0 −0.0356234 + 0.224918i 0 0.388960 0.126381i 0.186527i 0
689.1 −1.07394 + 1.47815i −1.41228 + 1.41228i −0.722562 2.22382i 0 −0.570857 3.60425i 0 2.32545 + 0.755585i 2.98904i 0
689.2 0.122881 0.169131i 0.770236 0.770236i 0.295511 + 0.909491i 0 −0.0356234 0.224918i 0 0.388960 + 0.126381i 0.186527i 0
758.1 −1.27276 0.413545i 1.09905 1.09905i 0.639886 + 0.464905i 0 −1.85334 + 0.944322i 0 0.164446 + 0.226341i 1.41582i 0
758.2 1.86055 + 0.604528i −1.32028 + 1.32028i 2.28716 + 1.66172i 0 −3.25460 + 1.65830i 0 2.10094 + 2.89169i 2.48629i 0
781.1 −1.86055 + 0.604528i −0.506809 + 0.506809i 2.28716 1.66172i 0 0.636561 1.24932i 0 −2.10094 + 2.89169i 0.486290i 0
781.2 1.27276 0.413545i −0.889993 + 0.889993i 0.639886 0.464905i 0 −0.764697 + 1.50080i 0 −0.164446 + 0.226341i 0.584177i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 781.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by $$\Q(\sqrt{-23})$$
41.g even 20 1 inner
943.n odd 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 943.1.n.b 16
23.b odd 2 1 CM 943.1.n.b 16
41.g even 20 1 inner 943.1.n.b 16
943.n odd 20 1 inner 943.1.n.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
943.1.n.b 16 1.a even 1 1 trivial
943.1.n.b 16 23.b odd 2 1 CM
943.1.n.b 16 41.g even 20 1 inner
943.1.n.b 16 943.n odd 20 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - \cdots$$ acting on $$S_{1}^{\mathrm{new}}(943, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 13 T^{2} + 508 T^{4} - 589 T^{6} + 315 T^{8} - 89 T^{10} + 28 T^{12} - 7 T^{14} + T^{16}$$
$3$ $$1 - 12 T + 72 T^{2} + 122 T^{3} + 112 T^{4} - 60 T^{5} + 98 T^{6} + 126 T^{7} + 90 T^{8} - 26 T^{9} + 28 T^{10} + 30 T^{11} + 17 T^{12} - 2 T^{13} + 2 T^{14} + 2 T^{15} + T^{16}$$
$5$ $$T^{16}$$
$7$ $$T^{16}$$
$11$ $$T^{16}$$
$13$ $$1 - 18 T + 77 T^{2} + 38 T^{3} + 207 T^{4} - 50 T^{5} + 103 T^{6} - 106 T^{7} - 44 T^{9} + 23 T^{10} + 10 T^{11} + 7 T^{12} - 8 T^{13} + 2 T^{14} - 2 T^{15} + T^{16}$$
$17$ $$T^{16}$$
$19$ $$T^{16}$$
$23$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
$29$ $$1 + 22 T + 187 T^{2} + 658 T^{3} + 1302 T^{4} + 1840 T^{5} + 2213 T^{6} + 2344 T^{7} + 2135 T^{8} + 1676 T^{9} + 1133 T^{10} + 650 T^{11} + 312 T^{12} + 122 T^{13} + 37 T^{14} + 8 T^{15} + T^{16}$$
$31$ $$1 + 9 T^{2} + 32 T^{4} + 7 T^{6} + 75 T^{8} + 43 T^{10} + 12 T^{12} + T^{14} + T^{16}$$
$37$ $$T^{16}$$
$41$ $$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$
$43$ $$T^{16}$$
$47$ $$1 - 8 T + 87 T^{2} + 48 T^{3} + 142 T^{4} + 20 T^{5} - 87 T^{6} - 36 T^{7} + 15 T^{8} - 24 T^{9} + 33 T^{10} - 10 T^{11} - 8 T^{12} + 12 T^{13} - 3 T^{14} - 2 T^{15} + T^{16}$$
$53$ $$T^{16}$$
$59$ $$( 1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4} )^{4}$$
$61$ $$T^{16}$$
$67$ $$T^{16}$$
$71$ $$1 + 8 T + 117 T^{2} + 502 T^{3} + 1027 T^{4} + 1170 T^{5} + 783 T^{6} + 286 T^{7} - 76 T^{9} - 57 T^{10} - 30 T^{11} - 13 T^{12} - 2 T^{13} + 2 T^{14} + 2 T^{15} + T^{16}$$
$73$ $$( 1 + 8 T^{2} + 14 T^{4} + 7 T^{6} + T^{8} )^{2}$$
$79$ $$T^{16}$$
$83$ $$T^{16}$$
$89$ $$T^{16}$$
$97$ $$T^{16}$$