# Properties

 Label 1183.2.e.f Level $1183$ Weight $2$ Character orbit 1183.e Analytic conductor $9.446$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# 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: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - x^{9} + 8 x^{8} + 7 x^{7} + 41 x^{6} + 18 x^{5} + 58 x^{4} + 28 x^{3} + 64 x^{2} + 16 x + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{7} ) q^{2} + ( -\beta_{4} + \beta_{9} ) q^{3} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{9} ) q^{4} + \beta_{8} q^{5} + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} ) q^{6} + ( -\beta_{4} - \beta_{6} + \beta_{9} ) q^{7} + ( -2 - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{8} + ( -\beta_{7} + \beta_{8} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{7} ) q^{2} + ( -\beta_{4} + \beta_{9} ) q^{3} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{9} ) q^{4} + \beta_{8} q^{5} + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} ) q^{6} + ( -\beta_{4} - \beta_{6} + \beta_{9} ) q^{7} + ( -2 - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{8} + ( -\beta_{7} + \beta_{8} ) q^{9} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{10} + ( 2 + \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{11} + ( -\beta_{1} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{12} + ( 1 - \beta_{1} - 2 \beta_{2} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} ) q^{14} + ( 2 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{15} + ( 3 \beta_{1} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{16} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{17} + ( 2 + \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{18} + ( 2 \beta_{1} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{19} + ( -1 - \beta_{2} + 3 \beta_{3} ) q^{20} + ( 2 + 2 \beta_{2} + \beta_{4} - 4 \beta_{7} + \beta_{8} ) q^{21} + ( 2 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{22} + ( -2 \beta_{7} - \beta_{9} ) q^{23} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{24} + ( -1 - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{25} + ( 2 \beta_{2} + \beta_{5} ) q^{27} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{28} + ( -1 - 2 \beta_{2} - \beta_{4} ) q^{29} + ( -3 \beta_{1} - \beta_{6} + 3 \beta_{7} + \beta_{8} + 4 \beta_{9} ) q^{30} + ( -2 + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{31} + ( 5 + \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{5} + 4 \beta_{6} - 5 \beta_{7} - \beta_{8} ) q^{32} + ( -2 \beta_{1} + 2 \beta_{7} + 3 \beta_{9} ) q^{33} + ( 5 + 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{34} + ( -2 - 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{35} + ( 1 + 2 \beta_{3} + \beta_{4} ) q^{36} + ( 2 \beta_{1} + 2 \beta_{6} + \beta_{9} ) q^{37} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{4} + \beta_{6} - 2 \beta_{7} - 4 \beta_{9} ) q^{38} + ( 2 \beta_{1} + 2 \beta_{6} - 6 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{40} + ( -4 - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{41} + ( 5 + 3 \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{42} + ( 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{43} + ( -\beta_{1} + \beta_{7} - 2 \beta_{8} + 3 \beta_{9} ) q^{44} + ( -6 - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 6 \beta_{7} - 2 \beta_{9} ) q^{45} + ( 1 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{46} + ( \beta_{6} - 4 \beta_{9} ) q^{47} + ( -5 - 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{48} + ( -1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{49} + ( -2 + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{50} + ( 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{51} + ( -3 + 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{9} ) q^{53} + ( -\beta_{1} - 3 \beta_{6} + 5 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{54} + ( -2 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} ) q^{55} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 5 \beta_{7} + \beta_{9} ) q^{56} + ( 4 + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{57} + ( 2 \beta_{1} + \beta_{6} - 6 \beta_{7} - 3 \beta_{9} ) q^{58} + ( 2 + \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{59} + ( -7 - 5 \beta_{1} - 5 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{6} + 7 \beta_{7} + 2 \beta_{9} ) q^{60} + ( -4 \beta_{6} + 3 \beta_{7} + 2 \beta_{9} ) q^{61} + ( -5 + \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{62} + ( -2 - 2 \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{63} + ( 3 + 4 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{64} + ( -5 - 5 \beta_{1} - 5 \beta_{2} - \beta_{3} - 5 \beta_{4} + \beta_{6} + 5 \beta_{7} + 5 \beta_{9} ) q^{66} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{67} + ( -7 \beta_{1} - 3 \beta_{6} + 8 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{68} + ( 4 + 2 \beta_{4} - \beta_{5} ) q^{69} + ( -7 - 3 \beta_{1} - 5 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} + 4 \beta_{9} ) q^{70} + ( -2 + 4 \beta_{2} - \beta_{3} + \beta_{4} ) q^{71} + ( \beta_{6} - 4 \beta_{7} + 3 \beta_{9} ) q^{72} + ( 2 \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{8} ) q^{73} + ( 7 + \beta_{1} + \beta_{2} - 5 \beta_{3} - \beta_{4} + 2 \beta_{5} + 5 \beta_{6} - 7 \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{74} + ( 2 \beta_{1} + 4 \beta_{7} - \beta_{8} - \beta_{9} ) q^{75} + ( -1 + 3 \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{76} + ( -5 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} + 3 \beta_{9} ) q^{77} + ( -2 \beta_{1} + 2 \beta_{6} - \beta_{8} - 2 \beta_{9} ) q^{79} + ( 11 + 5 \beta_{1} + 5 \beta_{2} - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - 11 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} ) q^{80} + ( 5 - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 5 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{81} + ( 8 \beta_{1} - 8 \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{82} + ( -2 + 2 \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{83} + ( -2 - 7 \beta_{1} - 4 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + 9 \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{84} + ( 4 - 4 \beta_{4} + \beta_{5} ) q^{85} + ( 2 \beta_{1} - 2 \beta_{6} + 6 \beta_{7} - \beta_{8} + 3 \beta_{9} ) q^{86} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{87} + ( 1 - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + 2 \beta_{9} ) q^{88} + ( 2 \beta_{1} - 2 \beta_{6} - 2 \beta_{8} + 3 \beta_{9} ) q^{89} + ( -8 - 2 \beta_{2} + 2 \beta_{5} ) q^{90} + ( 5 + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{92} + ( 4 \beta_{1} + 4 \beta_{7} - 3 \beta_{8} - 6 \beta_{9} ) q^{93} + ( -4 + 5 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{5} - 3 \beta_{6} + 4 \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{94} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 6 \beta_{9} ) q^{95} + ( 9 \beta_{1} + \beta_{6} - 9 \beta_{7} - \beta_{8} - \beta_{9} ) q^{96} + ( 2 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{97} + ( -4 - 3 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 9 \beta_{7} + \beta_{8} + 6 \beta_{9} ) q^{98} + ( -4 - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 4 q^{2} - 8 q^{4} + 2 q^{5} + 10 q^{6} - q^{7} - 18 q^{8} - 3 q^{9} + O(q^{10})$$ $$10 q + 4 q^{2} - 8 q^{4} + 2 q^{5} + 10 q^{6} - q^{7} - 18 q^{8} - 3 q^{9} + 5 q^{10} + 11 q^{11} - 5 q^{12} + 10 q^{14} - 10 q^{16} + 5 q^{17} + 9 q^{18} + 9 q^{19} - 2 q^{20} - 2 q^{21} + 16 q^{22} - 10 q^{23} - 9 q^{25} - 37 q^{28} - 6 q^{29} + 13 q^{30} - 6 q^{31} + 22 q^{32} + 8 q^{33} + 44 q^{34} - 4 q^{35} + 14 q^{36} + 4 q^{37} + 10 q^{38} - 28 q^{40} - 28 q^{41} + 52 q^{42} + 4 q^{43} - 32 q^{45} + 3 q^{46} + q^{47} - 46 q^{48} - 11 q^{49} - 18 q^{50} + 8 q^{51} - 17 q^{53} + 23 q^{54} - 21 q^{56} + 32 q^{57} - 27 q^{58} + 11 q^{59} - 29 q^{60} + 11 q^{61} - 46 q^{62} - 5 q^{63} + 18 q^{64} - 21 q^{66} + 13 q^{67} + 32 q^{68} + 36 q^{69} - 49 q^{70} - 30 q^{71} - 19 q^{72} + 33 q^{74} + 20 q^{75} - 16 q^{76} - 46 q^{77} - 2 q^{79} + 55 q^{80} + 19 q^{81} - 34 q^{82} - 12 q^{83} + 23 q^{84} + 44 q^{85} + 28 q^{86} + 8 q^{87} + 3 q^{88} - 4 q^{89} - 68 q^{90} + 42 q^{92} + 18 q^{93} - 20 q^{94} + 12 q^{95} - 37 q^{96} + 24 q^{97} + 7 q^{98} - 22 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} + 8 x^{8} + 7 x^{7} + 41 x^{6} + 18 x^{5} + 58 x^{4} + 28 x^{3} + 64 x^{2} + 16 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-364 \nu^{9} + 176 \nu^{8} - 220 \nu^{7} - 5913 \nu^{6} + 880 \nu^{5} + 6908 \nu^{4} + 84549 \nu^{3} + 9416 \nu^{2} + 2376 \nu + 30518$$$$)/118350$$ $$\beta_{3}$$ $$=$$ $$($$$$-983 \nu^{9} - 7328 \nu^{8} + 9160 \nu^{7} - 87336 \nu^{6} - 36640 \nu^{5} - 287624 \nu^{4} - 39747 \nu^{3} - 392048 \nu^{2} - 98928 \nu - 22604$$$$)/118350$$ $$\beta_{4}$$ $$=$$ $$($$$$-1159 \nu^{9} - 9844 \nu^{8} + 12305 \nu^{7} - 109053 \nu^{6} - 49220 \nu^{5} - 386377 \nu^{4} + 25194 \nu^{3} - 526654 \nu^{2} - 132894 \nu - 348592$$$$)/118350$$ $$\beta_{5}$$ $$=$$ $$($$$$916 \nu^{9} - 3044 \nu^{8} + 3805 \nu^{7} - 3978 \nu^{6} - 15220 \nu^{5} - 119477 \nu^{4} - 129531 \nu^{3} - 162854 \nu^{2} - 41094 \nu - 180842$$$$)/59175$$ $$\beta_{6}$$ $$=$$ $$($$$$2249 \nu^{9} - 9541 \nu^{8} + 26720 \nu^{7} - 34617 \nu^{6} + 31195 \nu^{5} - 152578 \nu^{4} + 39066 \nu^{3} - 46906 \nu^{2} - 20316 \nu - 3988$$$$)/78900$$ $$\beta_{7}$$ $$=$$ $$($$$$-15259 \nu^{9} + 14531 \nu^{8} - 121720 \nu^{7} - 107253 \nu^{6} - 637445 \nu^{5} - 272902 \nu^{4} - 871206 \nu^{3} - 258154 \nu^{2} - 957744 \nu - 2692$$$$)/236700$$ $$\beta_{8}$$ $$=$$ $$($$$$18139 \nu^{9} - 21776 \nu^{8} + 145570 \nu^{7} + 96813 \nu^{6} + 719570 \nu^{5} + 92092 \nu^{4} + 981276 \nu^{3} + 255184 \nu^{2} + 1362924 \nu + 532$$$$)/118350$$ $$\beta_{9}$$ $$=$$ $$($$$$1058 \nu^{9} - 1552 \nu^{8} + 9041 \nu^{7} + 3726 \nu^{6} + 39580 \nu^{5} + 2993 \nu^{4} + 53832 \nu^{3} + 11648 \nu^{2} + 50058 \nu - 136$$$$)/4734$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} + 3 \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} - 4$$ $$\nu^{4}$$ $$=$$ $$-8 \beta_{9} + 2 \beta_{8} - 19 \beta_{7} + 9 \beta_{6} - 13 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-22 \beta_{9} + 9 \beta_{8} - 45 \beta_{7} + 23 \beta_{6} - 9 \beta_{5} + 22 \beta_{4} - 23 \beta_{3} - 47 \beta_{2} - 47 \beta_{1} + 45$$ $$\nu^{6}$$ $$=$$ $$-23 \beta_{5} + 70 \beta_{4} - 78 \beta_{3} - 128 \beta_{2} + 154$$ $$\nu^{7}$$ $$=$$ $$206 \beta_{9} - 78 \beta_{8} + 431 \beta_{7} - 221 \beta_{6} + 407 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$628 \beta_{9} - 221 \beta_{8} + 1349 \beta_{7} - 691 \beta_{6} + 221 \beta_{5} - 628 \beta_{4} + 691 \beta_{3} + 1187 \beta_{2} + 1187 \beta_{1} - 1349$$ $$\nu^{9}$$ $$=$$ $$691 \beta_{5} - 1878 \beta_{4} + 2036 \beta_{3} + 3634 \beta_{2} - 3968$$

## Character values

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

 $$n$$ $$339$$ $$1016$$ $$\chi(n)$$ $$-\beta_{7}$$ $$1$$

## 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
 1.50426 − 2.60546i 0.597828 − 1.03547i −0.132804 + 0.230024i −0.606661 + 1.05077i −0.862625 + 1.49411i 1.50426 + 2.60546i 0.597828 + 1.03547i −0.132804 − 0.230024i −0.606661 − 1.05077i −0.862625 − 1.49411i
−1.00426 + 1.73943i −0.879528 1.52339i −1.01709 1.76164i 0.452861 0.784378i 3.53311 −0.237709 2.63505i 0.0686323 −0.0471392 + 0.0816475i 0.909582 + 1.57544i
170.2 −0.0978281 + 0.169443i 0.129894 + 0.224983i 0.980859 + 1.69890i 1.96625 3.40565i −0.0508292 −1.12324 + 2.39548i −0.775135 1.46625 2.53963i 0.384710 + 0.666337i
170.3 0.632804 1.09605i 1.31364 + 2.27529i 0.199118 + 0.344882i −1.45130 + 2.51373i 3.32511 1.29536 + 2.30696i 3.03523 −1.95130 + 3.37975i 1.83678 + 3.18139i
170.4 1.10666 1.91679i −1.23721 2.14292i −1.44940 2.51043i −1.06140 + 1.83839i −5.47671 −2.63169 + 0.272389i −1.98932 −1.56140 + 2.70442i 2.34921 + 4.06896i
170.5 1.36263 2.36014i 0.673208 + 1.16603i −2.71349 4.69991i 1.09358 1.89414i 3.66932 2.19729 1.47375i −9.33940 0.593582 1.02811i −2.98028 5.16200i
508.1 −1.00426 1.73943i −0.879528 + 1.52339i −1.01709 + 1.76164i 0.452861 + 0.784378i 3.53311 −0.237709 + 2.63505i 0.0686323 −0.0471392 0.0816475i 0.909582 1.57544i
508.2 −0.0978281 0.169443i 0.129894 0.224983i 0.980859 1.69890i 1.96625 + 3.40565i −0.0508292 −1.12324 2.39548i −0.775135 1.46625 + 2.53963i 0.384710 0.666337i
508.3 0.632804 + 1.09605i 1.31364 2.27529i 0.199118 0.344882i −1.45130 2.51373i 3.32511 1.29536 2.30696i 3.03523 −1.95130 3.37975i 1.83678 3.18139i
508.4 1.10666 + 1.91679i −1.23721 + 2.14292i −1.44940 + 2.51043i −1.06140 1.83839i −5.47671 −2.63169 0.272389i −1.98932 −1.56140 2.70442i 2.34921 4.06896i
508.5 1.36263 + 2.36014i 0.673208 1.16603i −2.71349 + 4.69991i 1.09358 + 1.89414i 3.66932 2.19729 + 1.47375i −9.33940 0.593582 + 1.02811i −2.98028 + 5.16200i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 508.5 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.e.f 10
7.c even 3 1 inner 1183.2.e.f 10
7.c even 3 1 8281.2.a.bw 5
7.d odd 6 1 8281.2.a.bx 5
13.b even 2 1 91.2.e.c 10
39.d odd 2 1 819.2.j.h 10
52.b odd 2 1 1456.2.r.p 10
91.b odd 2 1 637.2.e.m 10
91.r even 6 1 91.2.e.c 10
91.r even 6 1 637.2.a.l 5
91.s odd 6 1 637.2.a.k 5
91.s odd 6 1 637.2.e.m 10
273.w odd 6 1 819.2.j.h 10
273.w odd 6 1 5733.2.a.bl 5
273.ba even 6 1 5733.2.a.bm 5
364.bl odd 6 1 1456.2.r.p 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.c 10 13.b even 2 1
91.2.e.c 10 91.r even 6 1
637.2.a.k 5 91.s odd 6 1
637.2.a.l 5 91.r even 6 1
637.2.e.m 10 91.b odd 2 1
637.2.e.m 10 91.s odd 6 1
819.2.j.h 10 39.d odd 2 1
819.2.j.h 10 273.w odd 6 1
1183.2.e.f 10 1.a even 1 1 trivial
1183.2.e.f 10 7.c even 3 1 inner
1456.2.r.p 10 52.b odd 2 1
1456.2.r.p 10 364.bl odd 6 1
5733.2.a.bl 5 273.w odd 6 1
5733.2.a.bm 5 273.ba even 6 1
8281.2.a.bw 5 7.c even 3 1
8281.2.a.bx 5 7.d odd 6 1

## Hecke kernels

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

 $$T_{2}^{10} - \cdots$$ $$T_{3}^{10} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 36 T + 195 T^{2} - 210 T^{3} + 265 T^{4} - 116 T^{5} + 81 T^{6} - 30 T^{7} + 17 T^{8} - 4 T^{9} + T^{10}$$
$3$ $$16 - 64 T + 256 T^{2} - 72 T^{3} + 144 T^{4} - 4 T^{5} + 65 T^{6} + 9 T^{8} + T^{10}$$
$5$ $$2304 - 2304 T + 3264 T^{2} - 480 T^{3} + 1024 T^{4} - 156 T^{5} + 217 T^{6} - 10 T^{7} + 19 T^{8} - 2 T^{9} + T^{10}$$
$7$ $$16807 + 2401 T + 2058 T^{2} + 833 T^{3} + 119 T^{4} + 204 T^{5} + 17 T^{6} + 17 T^{7} + 6 T^{8} + T^{9} + T^{10}$$
$11$ $$1089 - 1485 T + 2751 T^{2} - 1386 T^{3} + 2467 T^{4} - 1749 T^{5} + 1099 T^{6} - 352 T^{7} + 85 T^{8} - 11 T^{9} + T^{10}$$
$13$ $$T^{10}$$
$17$ $$184041 - 39897 T + 54123 T^{2} - 9018 T^{3} + 11137 T^{4} - 1831 T^{5} + 921 T^{6} - 102 T^{7} + 47 T^{8} - 5 T^{9} + T^{10}$$
$19$ $$49729 - 38579 T + 69177 T^{2} + 24204 T^{3} + 31391 T^{4} + 427 T^{5} + 1607 T^{6} - 226 T^{7} + 95 T^{8} - 9 T^{9} + T^{10}$$
$23$ $$144 + 144 T + 456 T^{2} + 432 T^{3} + 1168 T^{4} + 1034 T^{5} + 713 T^{6} + 258 T^{7} + 69 T^{8} + 10 T^{9} + T^{10}$$
$29$ $$( -108 + 144 T - 19 T^{2} - 25 T^{3} + 3 T^{4} + T^{5} )^{2}$$
$31$ $$126736 - 180848 T + 221752 T^{2} - 95248 T^{3} + 43528 T^{4} - 230 T^{5} + 3825 T^{6} - 162 T^{7} + 97 T^{8} + 6 T^{9} + T^{10}$$
$37$ $$49505296 - 4643760 T + 5206008 T^{2} - 1114512 T^{3} + 504800 T^{4} - 77014 T^{5} + 14373 T^{6} - 912 T^{7} + 127 T^{8} - 4 T^{9} + T^{10}$$
$41$ $$( 1584 - 2544 T - 940 T^{2} - 28 T^{3} + 14 T^{4} + T^{5} )^{2}$$
$43$ $$( 64 - 288 T + 308 T^{2} - 72 T^{3} - 2 T^{4} + T^{5} )^{2}$$
$47$ $$26718561 - 14530059 T + 8036115 T^{2} - 1208826 T^{3} + 344071 T^{4} - 2771 T^{5} + 12591 T^{6} + 72 T^{7} + 125 T^{8} - T^{9} + T^{10}$$
$53$ $$398361681 + 254656881 T + 114371547 T^{2} + 27999402 T^{3} + 5280613 T^{4} + 593371 T^{5} + 59477 T^{6} + 3594 T^{7} + 363 T^{8} + 17 T^{9} + T^{10}$$
$59$ $$1089 - 1485 T + 2751 T^{2} - 1386 T^{3} + 2467 T^{4} - 1749 T^{5} + 1099 T^{6} - 352 T^{7} + 85 T^{8} - 11 T^{9} + T^{10}$$
$61$ $$71588521 + 49759141 T + 28105035 T^{2} + 6569330 T^{3} + 1397309 T^{4} + 44391 T^{5} + 17429 T^{6} - 190 T^{7} + 243 T^{8} - 11 T^{9} + T^{10}$$
$67$ $$515244601 + 13415109 T + 49379121 T^{2} - 8631036 T^{3} + 4274771 T^{4} - 387985 T^{5} + 54915 T^{6} - 2214 T^{7} + 331 T^{8} - 13 T^{9} + T^{10}$$
$71$ $$( 6336 - 456 T - 853 T^{2} - 25 T^{3} + 15 T^{4} + T^{5} )^{2}$$
$73$ $$506944 - 498400 T + 519904 T^{2} - 77400 T^{3} + 54264 T^{4} - 3862 T^{5} + 4925 T^{6} - 84 T^{7} + 75 T^{8} + T^{10}$$
$79$ $$1000000 + 1500000 T + 2060000 T^{2} + 559000 T^{3} + 239600 T^{4} - 31030 T^{5} + 16889 T^{6} - 654 T^{7} + 141 T^{8} + 2 T^{9} + T^{10}$$
$83$ $$( 7488 + 2688 T - 308 T^{2} - 124 T^{3} + 6 T^{4} + T^{5} )^{2}$$
$89$ $$59166864 + 16522416 T + 9952152 T^{2} + 893808 T^{3} + 783808 T^{4} + 98078 T^{5} + 24653 T^{6} + 768 T^{7} + 171 T^{8} + 4 T^{9} + T^{10}$$
$97$ $$( 2384 - 2240 T + 612 T^{2} - 16 T^{3} - 12 T^{4} + T^{5} )^{2}$$