# Properties

 Label 1183.2.e.i Level $1183$ Weight $2$ Character orbit 1183.e Analytic conductor $9.446$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1183.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.44630255912$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 11 x^{14} + 85 x^{12} + 334 x^{10} + 952 x^{8} + 1050 x^{6} + 853 x^{4} + 93 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{9} q^{2} + ( -1 - \beta_{4} - \beta_{6} ) q^{3} + ( -1 - \beta_{6} - \beta_{10} ) q^{4} + \beta_{13} q^{5} + ( -\beta_{1} + \beta_{8} - \beta_{9} - \beta_{12} - \beta_{13} ) q^{6} + ( \beta_{9} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{7} -\beta_{12} q^{8} + ( 1 + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{11} ) q^{9} +O(q^{10})$$ $$q + \beta_{9} q^{2} + ( -1 - \beta_{4} - \beta_{6} ) q^{3} + ( -1 - \beta_{6} - \beta_{10} ) q^{4} + \beta_{13} q^{5} + ( -\beta_{1} + \beta_{8} - \beta_{9} - \beta_{12} - \beta_{13} ) q^{6} + ( \beta_{9} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{7} -\beta_{12} q^{8} + ( 1 + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{11} ) q^{9} + ( -2 \beta_{4} + \beta_{10} + \beta_{11} ) q^{10} + ( -\beta_{1} - \beta_{3} - \beta_{8} - \beta_{14} + \beta_{15} ) q^{11} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{12} + ( -2 - \beta_{2} - \beta_{4} + \beta_{10} + \beta_{11} ) q^{14} + ( -2 \beta_{1} + \beta_{8} - 2 \beta_{9} - \beta_{13} ) q^{15} + ( 1 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{10} ) q^{16} + ( -1 - \beta_{6} - \beta_{11} ) q^{17} + ( 3 \beta_{1} - \beta_{3} - 4 \beta_{8} - \beta_{14} + \beta_{15} ) q^{18} + ( -\beta_{3} + 2 \beta_{9} - \beta_{12} ) q^{19} + ( -\beta_{1} + \beta_{3} + 3 \beta_{8} - 2 \beta_{9} + \beta_{12} - 3 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{20} + ( -\beta_{3} - \beta_{9} - 2 \beta_{12} - \beta_{13} - \beta_{14} ) q^{21} + ( -3 - \beta_{2} - \beta_{5} - \beta_{7} ) q^{22} + ( 1 + \beta_{4} + \beta_{5} - \beta_{6} ) q^{23} + ( 3 \beta_{1} - 2 \beta_{8} - \beta_{14} + \beta_{15} ) q^{24} + ( \beta_{4} - 2 \beta_{10} - \beta_{11} ) q^{25} + ( 2 + 2 \beta_{2} - \beta_{5} - \beta_{7} ) q^{27} + ( \beta_{1} + 2 \beta_{8} - \beta_{9} - \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{28} + ( -3 - 2 \beta_{2} - 3 \beta_{5} ) q^{29} + ( 2 - \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{30} + ( -\beta_{1} - \beta_{14} + \beta_{15} ) q^{31} + ( -\beta_{1} + 2 \beta_{3} - \beta_{8} ) q^{32} + ( \beta_{1} + \beta_{3} + \beta_{9} + 2 \beta_{14} + \beta_{15} ) q^{33} + ( \beta_{1} - \beta_{3} - 3 \beta_{8} + 2 \beta_{9} - 2 \beta_{12} + 3 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{34} + ( -1 + \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} - 2 \beta_{10} - \beta_{11} ) q^{35} + ( 1 - 5 \beta_{5} - 2 \beta_{7} ) q^{36} + ( -\beta_{1} - \beta_{3} - 3 \beta_{9} - 2 \beta_{14} - \beta_{15} ) q^{37} + ( -4 + \beta_{4} - 4 \beta_{6} - \beta_{10} ) q^{38} + ( 1 + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{11} ) q^{40} + ( \beta_{3} - \beta_{8} - \beta_{9} + 2 \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{41} + ( 6 - \beta_{2} + 4 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} + 2 \beta_{10} - \beta_{11} ) q^{42} + ( -2 - 2 \beta_{5} + \beta_{7} ) q^{43} + ( -\beta_{1} - \beta_{3} - 2 \beta_{9} + 2 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{44} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{45} + ( -\beta_{1} - \beta_{3} - \beta_{8} ) q^{46} + ( -\beta_{1} + 2 \beta_{3} - \beta_{9} + 3 \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{47} + ( 2 - \beta_{2} ) q^{48} + ( 2 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} + \beta_{7} - 2 \beta_{10} ) q^{49} + ( -\beta_{3} - 4 \beta_{8} + \beta_{9} - 3 \beta_{12} + 4 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{50} + ( -1 + 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{10} ) q^{51} + ( -3 - 3 \beta_{6} - 2 \beta_{10} ) q^{53} + ( \beta_{1} + 2 \beta_{3} + 8 \beta_{9} + \beta_{12} + 4 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{54} + ( 4 + 4 \beta_{2} + 3 \beta_{5} + \beta_{7} ) q^{55} + ( 5 + 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{56} + ( 2 \beta_{1} - \beta_{3} + 3 \beta_{9} - 4 \beta_{12} + \beta_{14} + 2 \beta_{15} ) q^{57} + ( \beta_{3} - 2 \beta_{9} + \beta_{12} + 3 \beta_{13} ) q^{58} + ( -3 \beta_{1} - \beta_{3} + 3 \beta_{8} + \beta_{14} - \beta_{15} ) q^{59} + ( 4 \beta_{1} - 3 \beta_{3} - 3 \beta_{8} - \beta_{14} + \beta_{15} ) q^{60} + ( 2 \beta_{2} + \beta_{6} - 2 \beta_{10} ) q^{61} + ( -2 - \beta_{2} + 2 \beta_{5} ) q^{62} + ( 7 \beta_{1} - 3 \beta_{8} + 3 \beta_{9} + 3 \beta_{13} + \beta_{15} ) q^{63} + ( -5 - 2 \beta_{2} + 2 \beta_{5} - \beta_{7} ) q^{64} -\beta_{10} q^{66} + ( -3 \beta_{3} - 2 \beta_{8} ) q^{67} + ( -4 - 3 \beta_{2} - 4 \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{10} + \beta_{11} ) q^{68} + ( 4 + \beta_{5} - \beta_{7} ) q^{69} + ( -4 \beta_{1} - \beta_{3} - 4 \beta_{8} - 4 \beta_{9} - 2 \beta_{12} - \beta_{14} + \beta_{15} ) q^{70} + ( -4 \beta_{1} - \beta_{8} - 4 \beta_{9} + 3 \beta_{12} + \beta_{13} ) q^{71} + ( 3 \beta_{3} + 2 \beta_{9} + 3 \beta_{12} + 3 \beta_{13} ) q^{72} + ( 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{8} - 2 \beta_{14} + 2 \beta_{15} ) q^{73} + ( 6 + 6 \beta_{6} + 3 \beta_{10} ) q^{74} + ( -2 - 2 \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{11} ) q^{75} + ( -\beta_{1} - \beta_{8} - \beta_{9} - 2 \beta_{12} + \beta_{13} ) q^{76} + ( 2 - 2 \beta_{2} - 3 \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} + 4 \beta_{10} ) q^{77} + ( -1 - \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} + \beta_{11} ) q^{79} + ( 2 \beta_{1} + \beta_{3} + 2 \beta_{8} + \beta_{14} - \beta_{15} ) q^{80} + ( -5 - 2 \beta_{4} - 5 \beta_{6} - 4 \beta_{10} ) q^{81} + ( -4 - 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - \beta_{7} + \beta_{11} ) q^{82} + ( 6 \beta_{1} - \beta_{3} - \beta_{8} + 7 \beta_{9} - 2 \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{83} + ( 8 \beta_{1} - 5 \beta_{8} + 4 \beta_{9} + \beta_{12} + 3 \beta_{13} - \beta_{14} ) q^{84} + ( 2 \beta_{1} + 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{12} - 3 \beta_{13} ) q^{85} + ( -\beta_{1} + 3 \beta_{3} - 3 \beta_{9} + 4 \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{86} + ( -8 - 8 \beta_{6} + 2 \beta_{10} + \beta_{11} ) q^{87} + ( -1 - 2 \beta_{4} - \beta_{6} + 2 \beta_{10} ) q^{88} + ( -\beta_{1} - 3 \beta_{3} + 3 \beta_{9} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{89} + ( 14 + 6 \beta_{2} - 2 \beta_{5} ) q^{90} + ( -\beta_{2} - \beta_{5} - \beta_{7} ) q^{92} + ( 2 \beta_{3} - 6 \beta_{9} + 2 \beta_{12} - 3 \beta_{13} ) q^{93} + ( -6 - 5 \beta_{4} - 6 \beta_{6} - \beta_{10} + \beta_{11} ) q^{94} + ( 1 - 3 \beta_{4} + \beta_{6} + 2 \beta_{10} + \beta_{11} ) q^{95} + ( -2 \beta_{1} + \beta_{3} - 9 \beta_{9} + 3 \beta_{12} - 4 \beta_{13} - 4 \beta_{14} - 2 \beta_{15} ) q^{96} + ( 2 \beta_{1} - \beta_{3} - \beta_{8} + 3 \beta_{9} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{97} + ( -3 \beta_{1} - 3 \beta_{8} - 6 \beta_{9} + 2 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{98} + ( -2 \beta_{1} - \beta_{3} - 2 \beta_{8} - \beta_{9} - \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 4q^{3} - 6q^{4} - 12q^{9} + O(q^{10})$$ $$16q - 4q^{3} - 6q^{4} - 12q^{9} + 6q^{10} - 18q^{12} - 26q^{14} + 2q^{16} - 8q^{17} - 36q^{22} + 12q^{23} + 32q^{27} - 16q^{29} - 38q^{30} + 56q^{36} - 34q^{38} - 4q^{40} + 16q^{42} - 16q^{43} + 36q^{48} - 40q^{49} - 16q^{51} - 20q^{53} + 24q^{55} + 36q^{56} - 12q^{61} - 44q^{62} - 88q^{64} + 2q^{66} - 2q^{68} + 56q^{69} + 42q^{74} - 8q^{75} + 76q^{77} + 20q^{79} - 24q^{81} + 16q^{82} - 68q^{87} - 4q^{88} + 216q^{90} + 12q^{92} - 26q^{94} + 16q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 11 x^{14} + 85 x^{12} + 334 x^{10} + 952 x^{8} + 1050 x^{6} + 853 x^{4} + 93 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$24498 \nu^{14} + 246060 \nu^{12} + 1852321 \nu^{10} + 6411671 \nu^{8} + 17193085 \nu^{6} + 6321845 \nu^{4} + 690027 \nu^{2} - 40872159$$$$)/14163622$$ $$\beta_{3}$$ $$=$$ $$($$$$-24498 \nu^{15} - 246060 \nu^{13} - 1852321 \nu^{11} - 6411671 \nu^{9} - 17193085 \nu^{7} - 6321845 \nu^{5} - 690027 \nu^{3} + 55035781 \nu$$$$)/14163622$$ $$\beta_{4}$$ $$=$$ $$($$$$172099 \nu^{14} + 2170865 \nu^{12} + 17340370 \nu^{10} + 78484018 \nu^{8} + 236538400 \nu^{6} + 377649654 \nu^{4} + 218482087 \nu^{2} + 23829231$$$$)/42490866$$ $$\beta_{5}$$ $$=$$ $$($$$$99072 \nu^{14} + 1000291 \nu^{12} + 7490944 \nu^{10} + 25929344 \nu^{8} + 66564370 \nu^{6} + 25566080 \nu^{4} + 2790528 \nu^{2} - 20454191$$$$)/14163622$$ $$\beta_{6}$$ $$=$$ $$($$$$539569 \nu^{14} + 5861765 \nu^{12} + 45125185 \nu^{10} + 174659083 \nu^{8} + 494434675 \nu^{6} + 514968195 \nu^{4} + 441286822 \nu^{2} + 5618970$$$$)/42490866$$ $$\beta_{7}$$ $$=$$ $$($$$$-102312 \nu^{14} - 1048444 \nu^{12} - 7735924 \nu^{10} - 26777324 \nu^{8} - 67022271 \nu^{6} - 26402180 \nu^{4} - 2881788 \nu^{2} + 23341327$$$$)/7081811$$ $$\beta_{8}$$ $$=$$ $$($$$$123570 \nu^{15} + 1246351 \nu^{13} + 9343265 \nu^{11} + 32341015 \nu^{9} + 83757455 \nu^{7} + 31887925 \nu^{5} + 3480555 \nu^{3} - 61326350 \nu$$$$)/14163622$$ $$\beta_{9}$$ $$=$$ $$($$$$-539569 \nu^{15} - 5861765 \nu^{13} - 45125185 \nu^{11} - 174659083 \nu^{9} - 494434675 \nu^{7} - 514968195 \nu^{5} - 441286822 \nu^{3} - 48109836 \nu$$$$)/42490866$$ $$\beta_{10}$$ $$=$$ $$($$$$-515071 \nu^{14} - 5615705 \nu^{12} - 43272864 \nu^{10} - 168247412 \nu^{8} - 477241590 \nu^{6} - 508646350 \nu^{4} - 426433173 \nu^{2} - 46491129$$$$)/14163622$$ $$\beta_{11}$$ $$=$$ $$($$$$3598 \nu^{14} + 40712 \nu^{12} + 317920 \nu^{10} + 1287475 \nu^{8} + 3722089 \nu^{6} + 4460568 \nu^{4} + 3361729 \nu^{2} + 366555$$$$)/95271$$ $$\beta_{12}$$ $$=$$ $$($$$$1079138 \nu^{15} + 11723530 \nu^{13} + 90250370 \nu^{11} + 349318166 \nu^{9} + 988869350 \nu^{7} + 1029936390 \nu^{5} + 861328211 \nu^{3} + 11237940 \nu$$$$)/21245433$$ $$\beta_{13}$$ $$=$$ $$($$$$-205171 \nu^{15} - 2261795 \nu^{13} - 17480377 \nu^{11} - 68898667 \nu^{9} - 196608895 \nu^{7} - 219868809 \nu^{5} - 176278948 \nu^{3} - 19219314 \nu$$$$)/3862806$$ $$\beta_{14}$$ $$=$$ $$($$$$1137374 \nu^{15} + 12639109 \nu^{13} + 98087264 \nu^{11} + 390741062 \nu^{9} + 1125399698 \nu^{7} + 1313214930 \nu^{5} + 1094093084 \nu^{3} + 167644965 \nu$$$$)/11588418$$ $$\beta_{15}$$ $$=$$ $$($$$$17689120 \nu^{15} + 191553668 \nu^{13} + 1470474691 \nu^{11} + 5653350919 \nu^{9} + 15847248841 \nu^{7} + 15781577445 \nu^{5} + 12180871093 \nu^{3} - 359333319 \nu$$$$)/ 127472598$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{10} + 3 \beta_{6} - \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{12} - 4 \beta_{9} - 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{10} - 14 \beta_{6} - \beta_{4} - 14$$ $$\nu^{5}$$ $$=$$ $$\beta_{13} + 6 \beta_{12} + 19 \beta_{9} + 6 \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$-\beta_{7} - 8 \beta_{5} + 24 \beta_{2} + 61$$ $$\nu^{7}$$ $$=$$ $$\beta_{15} - \beta_{14} - 11 \beta_{8} - 32 \beta_{3} + 94 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-11 \beta_{11} + 115 \beta_{10} + 11 \beta_{7} + 345 \beta_{6} + 52 \beta_{5} + 52 \beta_{4} - 115 \beta_{2} + 52$$ $$\nu^{9}$$ $$=$$ $$-22 \beta_{15} - 11 \beta_{14} - 85 \beta_{13} - 145 \beta_{12} - 493 \beta_{9} + 85 \beta_{8} + 11 \beta_{3} - 482 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$85 \beta_{11} - 553 \beta_{10} - 1736 \beta_{6} - 315 \beta_{4} - 1736$$ $$\nu^{11}$$ $$=$$ $$85 \beta_{15} + 170 \beta_{14} + 570 \beta_{13} + 698 \beta_{12} + 2544 \beta_{9} + 783 \beta_{3} + 85 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-570 \beta_{7} - 1838 \beta_{5} + 2672 \beta_{2} + 6935$$ $$\nu^{13}$$ $$=$$ $$570 \beta_{15} - 570 \beta_{14} - 3548 \beta_{8} - 4510 \beta_{3} + 12015 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-3548 \beta_{11} + 12977 \beta_{10} + 3548 \beta_{7} + 44495 \beta_{6} + 10466 \beta_{5} + 10466 \beta_{4} - 12977 \beta_{2} + 10466$$ $$\nu^{15}$$ $$=$$ $$-7096 \beta_{15} - 3548 \beta_{14} - 21110 \beta_{13} - 16347 \beta_{12} - 68116 \beta_{9} + 21110 \beta_{8} + 3548 \beta_{3} - 64568 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times$$.

 $$n$$ $$339$$ $$1016$$ $$\chi(n)$$ $$\beta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
170.1
 −1.14241 − 1.97871i −1.06275 − 1.84073i −0.536527 − 0.929293i −0.166188 − 0.287846i 0.166188 + 0.287846i 0.536527 + 0.929293i 1.06275 + 1.84073i 1.14241 + 1.97871i −1.14241 + 1.97871i −1.06275 + 1.84073i −0.536527 + 0.929293i −0.166188 + 0.287846i 0.166188 − 0.287846i 0.536527 − 0.929293i 1.06275 − 1.84073i 1.14241 − 1.97871i
−1.14241 + 1.97871i −1.57521 2.72835i −1.61019 2.78892i −1.06250 + 1.84030i 7.19813 0.331665 + 2.62488i 2.78832 −3.46258 + 5.99736i −2.42760 4.20473i
170.2 −1.06275 + 1.84073i 0.0894272 + 0.154892i −1.25885 2.18040i 1.80301 3.12291i −0.380153 2.35320 1.20931i 1.10038 1.48401 2.57037i 3.83229 + 6.63772i
170.3 −0.536527 + 0.929293i 1.21570 + 2.10566i 0.424277 + 0.734868i 0.312716 0.541640i −2.60903 −1.21561 + 2.34996i −3.05665 −1.45586 + 2.52163i 0.335561 + 0.581209i
170.4 −0.166188 + 0.287846i −0.729919 1.26426i 0.944763 + 1.63638i −0.722811 + 1.25195i 0.485214 −1.36920 2.26391i −1.29278 0.434437 0.752468i −0.240245 0.416116i
170.5 0.166188 0.287846i −0.729919 1.26426i 0.944763 + 1.63638i 0.722811 1.25195i −0.485214 1.36920 + 2.26391i 1.29278 0.434437 0.752468i −0.240245 0.416116i
170.6 0.536527 0.929293i 1.21570 + 2.10566i 0.424277 + 0.734868i −0.312716 + 0.541640i 2.60903 1.21561 2.34996i 3.05665 −1.45586 + 2.52163i 0.335561 + 0.581209i
170.7 1.06275 1.84073i 0.0894272 + 0.154892i −1.25885 2.18040i −1.80301 + 3.12291i 0.380153 −2.35320 + 1.20931i −1.10038 1.48401 2.57037i 3.83229 + 6.63772i
170.8 1.14241 1.97871i −1.57521 2.72835i −1.61019 2.78892i 1.06250 1.84030i −7.19813 −0.331665 2.62488i −2.78832 −3.46258 + 5.99736i −2.42760 4.20473i
508.1 −1.14241 1.97871i −1.57521 + 2.72835i −1.61019 + 2.78892i −1.06250 1.84030i 7.19813 0.331665 2.62488i 2.78832 −3.46258 5.99736i −2.42760 + 4.20473i
508.2 −1.06275 1.84073i 0.0894272 0.154892i −1.25885 + 2.18040i 1.80301 + 3.12291i −0.380153 2.35320 + 1.20931i 1.10038 1.48401 + 2.57037i 3.83229 6.63772i
508.3 −0.536527 0.929293i 1.21570 2.10566i 0.424277 0.734868i 0.312716 + 0.541640i −2.60903 −1.21561 2.34996i −3.05665 −1.45586 2.52163i 0.335561 0.581209i
508.4 −0.166188 0.287846i −0.729919 + 1.26426i 0.944763 1.63638i −0.722811 1.25195i 0.485214 −1.36920 + 2.26391i −1.29278 0.434437 + 0.752468i −0.240245 + 0.416116i
508.5 0.166188 + 0.287846i −0.729919 + 1.26426i 0.944763 1.63638i 0.722811 + 1.25195i −0.485214 1.36920 2.26391i 1.29278 0.434437 + 0.752468i −0.240245 + 0.416116i
508.6 0.536527 + 0.929293i 1.21570 2.10566i 0.424277 0.734868i −0.312716 0.541640i 2.60903 1.21561 + 2.34996i 3.05665 −1.45586 2.52163i 0.335561 0.581209i
508.7 1.06275 + 1.84073i 0.0894272 0.154892i −1.25885 + 2.18040i −1.80301 3.12291i 0.380153 −2.35320 1.20931i −1.10038 1.48401 + 2.57037i 3.83229 6.63772i
508.8 1.14241 + 1.97871i −1.57521 + 2.72835i −1.61019 + 2.78892i 1.06250 + 1.84030i −7.19813 −0.331665 + 2.62488i −2.78832 −3.46258 5.99736i −2.42760 + 4.20473i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 508.8 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.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.e.i 16
7.c even 3 1 inner 1183.2.e.i 16
7.c even 3 1 8281.2.a.ck 8
7.d odd 6 1 8281.2.a.cj 8
13.b even 2 1 inner 1183.2.e.i 16
13.d odd 4 2 91.2.r.a 16
39.f even 4 2 819.2.dl.e 16
91.i even 4 2 637.2.r.f 16
91.r even 6 1 inner 1183.2.e.i 16
91.r even 6 1 8281.2.a.ck 8
91.s odd 6 1 8281.2.a.cj 8
91.z odd 12 2 91.2.r.a 16
91.z odd 12 2 637.2.c.f 8
91.bb even 12 2 637.2.c.e 8
91.bb even 12 2 637.2.r.f 16
273.cd even 12 2 819.2.dl.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.r.a 16 13.d odd 4 2
91.2.r.a 16 91.z odd 12 2
637.2.c.e 8 91.bb even 12 2
637.2.c.f 8 91.z odd 12 2
637.2.r.f 16 91.i even 4 2
637.2.r.f 16 91.bb even 12 2
819.2.dl.e 16 39.f even 4 2
819.2.dl.e 16 273.cd even 12 2
1183.2.e.i 16 1.a even 1 1 trivial
1183.2.e.i 16 7.c even 3 1 inner
1183.2.e.i 16 13.b even 2 1 inner
1183.2.e.i 16 91.r even 6 1 inner
8281.2.a.cj 8 7.d odd 6 1
8281.2.a.cj 8 91.s odd 6 1
8281.2.a.ck 8 7.c even 3 1
8281.2.a.ck 8 91.r even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1183, [\chi])$$:

 $$T_{2}^{16} + \cdots$$ $$T_{3}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 93 T^{2} + 853 T^{4} + 1050 T^{6} + 952 T^{8} + 334 T^{10} + 85 T^{12} + 11 T^{14} + T^{16}$$
$3$ $$( 4 - 20 T + 114 T^{2} + 62 T^{3} + 67 T^{4} + 6 T^{5} + 11 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$5$ $$2304 + 7680 T^{2} + 20656 T^{4} + 14560 T^{6} + 7361 T^{8} + 1740 T^{10} + 297 T^{12} + 20 T^{14} + T^{16}$$
$7$ $$5764801 + 2352980 T^{2} + 521017 T^{4} + 78988 T^{6} + 10652 T^{8} + 1612 T^{10} + 217 T^{12} + 20 T^{14} + T^{16}$$
$11$ $$729 + 9180 T^{2} + 99508 T^{4} + 199832 T^{6} + 337509 T^{8} + 30312 T^{10} + 2108 T^{12} + 52 T^{14} + T^{16}$$
$13$ $$T^{16}$$
$17$ $$( 15129 - 6396 T + 5164 T^{2} + 56 T^{3} + 485 T^{4} + 24 T^{5} + 36 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$19$ $$10673289 + 7618644 T^{2} + 3674044 T^{4} + 971784 T^{6} + 185725 T^{8} + 19096 T^{10} + 1396 T^{12} + 44 T^{14} + T^{16}$$
$23$ $$( 36 - 60 T + 130 T^{2} - 22 T^{3} + 91 T^{4} - 50 T^{5} + 31 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$29$ $$( 624 - 208 T - 63 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$31$ $$1136229264 + 657036336 T^{2} + 309454636 T^{4} + 35364492 T^{6} + 2779213 T^{8} + 128296 T^{10} + 4309 T^{12} + 80 T^{14} + T^{16}$$
$37$ $$76527504 + 433655856 T^{2} + 2419355628 T^{4} + 213389964 T^{6} + 12939021 T^{8} + 422496 T^{10} + 10053 T^{12} + 120 T^{14} + T^{16}$$
$41$ $$( 292032 - 88192 T^{2} + 5732 T^{4} - 132 T^{6} + T^{8} )^{2}$$
$43$ $$( -104 + 156 T - 66 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$47$ $$57728231289 + 58599199164 T^{2} + 56549167060 T^{4} + 2884224440 T^{6} + 101089845 T^{8} + 1905768 T^{10} + 26204 T^{12} + 196 T^{14} + T^{16}$$
$53$ $$( 7569 - 11310 T + 16900 T^{2} - 1740 T^{3} + 1213 T^{4} + 260 T^{5} + 100 T^{6} + 10 T^{7} + T^{8} )^{2}$$
$59$ $$12487392009 + 18073959780 T^{2} + 24989166028 T^{4} + 1652371368 T^{6} + 79227709 T^{8} + 1646008 T^{10} + 24868 T^{12} + 188 T^{14} + T^{16}$$
$61$ $$( 49729 - 20962 T + 14188 T^{2} - 420 T^{3} + 917 T^{4} + 44 T^{5} + 60 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$67$ $$66330457209 + 73439011956 T^{2} + 75732974260 T^{4} + 6027737800 T^{6} + 387569525 T^{8} + 5578872 T^{10} + 59004 T^{12} + 284 T^{14} + T^{16}$$
$71$ $$( 397488 - 253084 T^{2} + 19829 T^{4} - 292 T^{6} + T^{8} )^{2}$$
$73$ $$8437677133824 + 1371747640320 T^{2} + 164987876800 T^{4} + 7922514640 T^{6} + 273313457 T^{8} + 4249020 T^{10} + 47625 T^{12} + 260 T^{14} + T^{16}$$
$79$ $$( 64 - 480 T + 3672 T^{2} + 380 T^{3} + 689 T^{4} - 210 T^{5} + 91 T^{6} - 10 T^{7} + T^{8} )^{2}$$
$83$ $$( 5483712 - 959920 T^{2} + 28692 T^{4} - 296 T^{6} + T^{8} )^{2}$$
$89$ $$58102628210064 + 22401057000432 T^{2} + 8157120819724 T^{4} + 178140025756 T^{6} + 2655587933 T^{8} + 21797952 T^{10} + 130701 T^{12} + 440 T^{14} + T^{16}$$
$97$ $$( 192 - 4816 T^{2} + 2740 T^{4} - 104 T^{6} + T^{8} )^{2}$$