# Properties

 Label 931.2.a.p Level $931$ Weight $2$ Character orbit 931.a Self dual yes Analytic conductor $7.434$ Analytic rank $1$ Dimension $10$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$931 = 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 931.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.43407242818$$ Analytic rank: $$1$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 2 x^{9} - 13 x^{8} + 24 x^{7} + 57 x^{6} - 98 x^{5} - 93 x^{4} + 152 x^{3} + 39 x^{2} - 58 x - 7$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} -\beta_{4} q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -2 - \beta_{9} ) q^{5} + ( \beta_{5} + \beta_{7} + \beta_{9} ) q^{6} + ( -1 - \beta_{2} - \beta_{6} - \beta_{7} ) q^{8} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} -\beta_{4} q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -2 - \beta_{9} ) q^{5} + ( \beta_{5} + \beta_{7} + \beta_{9} ) q^{6} + ( -1 - \beta_{2} - \beta_{6} - \beta_{7} ) q^{8} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} ) q^{9} + ( 1 + 2 \beta_{1} + \beta_{3} - \beta_{8} ) q^{10} + ( -\beta_{2} + \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{11} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} ) q^{12} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} ) q^{13} + ( \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{15} + ( \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{16} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{17} + ( -1 + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{18} - q^{19} + ( -4 - 2 \beta_{2} - \beta_{7} - \beta_{9} ) q^{20} + ( \beta_{1} - \beta_{3} + 2 \beta_{5} - \beta_{8} ) q^{22} + ( 1 + \beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{23} + ( 2 \beta_{1} - \beta_{2} - \beta_{7} - \beta_{8} ) q^{24} + ( 3 - \beta_{4} + \beta_{5} - \beta_{8} + 3 \beta_{9} ) q^{25} + ( -2 - 2 \beta_{2} - \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{9} ) q^{26} + ( -1 + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{27} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{29} + ( -2 - \beta_{2} - \beta_{4} - 5 \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{30} + ( -\beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{31} + ( -3 - 2 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{8} ) q^{32} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{33} + ( -4 + 4 \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{34} + ( 1 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} ) q^{36} + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{37} + \beta_{1} q^{38} + ( -2 + \beta_{2} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{9} ) q^{39} + ( 2 + \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{40} + ( -5 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{41} + ( -1 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{9} ) q^{43} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{44} + ( -2 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{45} + ( -4 - 3 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{9} ) q^{46} + ( -2 - \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{47} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{48} + ( -3 - 3 \beta_{1} - 3 \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} + 4 \beta_{8} ) q^{50} + ( -3 - \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{8} ) q^{51} + ( 5 + \beta_{1} + 3 \beta_{2} - \beta_{4} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{52} + ( 2 \beta_{2} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - 4 \beta_{9} ) q^{53} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} - \beta_{9} ) q^{54} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} - 2 \beta_{8} ) q^{55} + \beta_{4} q^{57} + ( -1 + \beta_{1} - \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} ) q^{58} + ( -4 - \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{9} ) q^{59} + ( 4 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - 3 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{60} + ( 2 - 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} + 3 \beta_{7} - 2 \beta_{8} ) q^{61} + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{62} + ( 2 + 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{64} + ( 1 + \beta_{1} - 3 \beta_{2} - \beta_{4} - \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{65} + ( -2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - \beta_{7} + 3 \beta_{8} - 2 \beta_{9} ) q^{66} + ( -2 - \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{67} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 3 \beta_{8} - \beta_{9} ) q^{68} + ( -2 - 5 \beta_{1} + 3 \beta_{2} + \beta_{3} - 4 \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{69} + ( -4 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{9} ) q^{71} + ( 4 + \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} ) q^{72} + ( -2 + \beta_{1} - \beta_{3} + 3 \beta_{4} - 4 \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{73} + ( 4 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} ) q^{74} + ( 1 - 4 \beta_{2} - \beta_{3} + \beta_{4} - 6 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} + \beta_{8} ) q^{75} + ( -1 - \beta_{2} ) q^{76} + ( 1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} ) q^{78} + ( -1 + \beta_{3} - 3 \beta_{4} - \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{79} + ( 2 - 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + 4 \beta_{9} ) q^{80} + ( 1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{8} + \beta_{9} ) q^{81} + ( 2 + 4 \beta_{1} - \beta_{2} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{82} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{9} ) q^{83} + ( 6 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{8} + 3 \beta_{9} ) q^{85} + ( -7 + 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} - 2 \beta_{9} ) q^{86} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} + 5 \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{9} ) q^{87} + ( 2 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{88} + ( -5 - 2 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{8} ) q^{89} + ( 8 + \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 5 \beta_{8} + 4 \beta_{9} ) q^{90} + ( 4 + 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} + 5 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{92} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{93} + ( 2 + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{94} + ( 2 + \beta_{9} ) q^{95} + ( -3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} - 4 \beta_{9} ) q^{96} + ( 1 - 4 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{7} + 2 \beta_{8} + 3 \beta_{9} ) q^{97} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 6 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 2 q^{2} - 4 q^{3} + 10 q^{4} - 16 q^{5} - 8 q^{6} - 6 q^{8} + 10 q^{9} + O(q^{10})$$ $$10 q - 2 q^{2} - 4 q^{3} + 10 q^{4} - 16 q^{5} - 8 q^{6} - 6 q^{8} + 10 q^{9} + 12 q^{10} - 12 q^{12} - 12 q^{13} + 2 q^{16} - 16 q^{17} + 2 q^{18} - 10 q^{19} - 32 q^{20} + 4 q^{22} + 12 q^{23} + 8 q^{24} + 14 q^{25} - 24 q^{26} - 16 q^{27} - 12 q^{29} - 12 q^{30} - 8 q^{31} - 34 q^{32} + 4 q^{33} - 16 q^{34} - 6 q^{36} + 4 q^{37} + 2 q^{38} - 4 q^{39} + 20 q^{40} - 40 q^{41} + 4 q^{43} - 20 q^{44} - 24 q^{45} - 32 q^{46} - 16 q^{47} - 12 q^{48} - 34 q^{50} - 28 q^{51} + 40 q^{52} - 8 q^{54} - 16 q^{55} + 4 q^{57} - 8 q^{58} - 36 q^{59} + 32 q^{60} - 16 q^{61} + 16 q^{62} + 18 q^{64} + 8 q^{65} - 8 q^{66} - 28 q^{67} - 40 q^{68} - 48 q^{69} - 12 q^{71} + 34 q^{72} + 24 q^{74} + 32 q^{75} - 10 q^{76} + 28 q^{78} - 8 q^{79} - 8 q^{80} + 14 q^{81} + 8 q^{82} + 40 q^{85} - 52 q^{86} + 8 q^{87} - 4 q^{88} - 48 q^{89} + 64 q^{90} + 28 q^{92} + 40 q^{93} + 36 q^{94} + 16 q^{95} + 8 q^{96} + 16 q^{97} - 8 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} - 13 x^{8} + 24 x^{7} + 57 x^{6} - 98 x^{5} - 93 x^{4} + 152 x^{3} + 39 x^{2} - 58 x - 7$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{9} - 91 \nu^{8} + 169 \nu^{7} + 995 \nu^{6} - 1285 \nu^{5} - 3248 \nu^{4} + 2542 \nu^{3} + 2964 \nu^{2} - 581 \nu - 574$$$$)/229$$ $$\beta_{4}$$ $$=$$ $$($$$$27 \nu^{9} - 31 \nu^{8} - 335 \nu^{7} + 193 \nu^{6} + 1432 \nu^{5} + 109 \nu^{4} - 2486 \nu^{3} - 1542 \nu^{2} + 1546 \nu + 879$$$$)/229$$ $$\beta_{5}$$ $$=$$ $$($$$$47 \nu^{9} - 37 \nu^{8} - 651 \nu^{7} + 319 \nu^{6} + 3061 \nu^{5} - 845 \nu^{4} - 5464 \nu^{3} + 878 \nu^{2} + 2776 \nu - 22$$$$)/229$$ $$\beta_{6}$$ $$=$$ $$($$$$-57 \nu^{9} + 40 \nu^{8} + 809 \nu^{7} - 382 \nu^{6} - 3761 \nu^{5} + 1093 \nu^{4} + 6266 \nu^{3} - 943 \nu^{2} - 2704 \nu - 329$$$$)/229$$ $$\beta_{7}$$ $$=$$ $$($$$$57 \nu^{9} - 40 \nu^{8} - 809 \nu^{7} + 382 \nu^{6} + 3761 \nu^{5} - 1093 \nu^{4} - 6037 \nu^{3} + 714 \nu^{2} + 1788 \nu + 787$$$$)/229$$ $$\beta_{8}$$ $$=$$ $$($$$$67 \nu^{9} - 43 \nu^{8} - 967 \nu^{7} + 445 \nu^{6} + 4690 \nu^{5} - 1570 \nu^{4} - 8671 \nu^{3} + 1924 \nu^{2} + 4693 \nu + 222$$$$)/229$$ $$\beta_{9}$$ $$=$$ $$($$$$-81 \nu^{9} + 93 \nu^{8} + 1005 \nu^{7} - 808 \nu^{6} - 4067 \nu^{5} + 1963 \nu^{4} + 5855 \nu^{3} - 1099 \nu^{2} - 2119 \nu - 576$$$$)/229$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} + \beta_{2} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{4} + 7 \beta_{2} + \beta_{1} + 14$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{8} + 8 \beta_{7} + 10 \beta_{6} - \beta_{5} + \beta_{4} + 10 \beta_{2} + 20 \beta_{1} + 11$$ $$\nu^{6}$$ $$=$$ $$-\beta_{9} + 12 \beta_{8} + 11 \beta_{7} + 13 \beta_{6} - 21 \beta_{5} + 8 \beta_{4} + 48 \beta_{2} + 13 \beta_{1} + 78$$ $$\nu^{7}$$ $$=$$ $$-4 \beta_{9} + 26 \beta_{8} + 56 \beta_{7} + 81 \beta_{6} - 20 \beta_{5} + 11 \beta_{4} - \beta_{3} + 85 \beta_{2} + 115 \beta_{1} + 97$$ $$\nu^{8}$$ $$=$$ $$-17 \beta_{9} + 110 \beta_{8} + 95 \beta_{7} + 130 \beta_{6} - 176 \beta_{5} + 56 \beta_{4} - 4 \beta_{3} + 337 \beta_{2} + 127 \beta_{1} + 480$$ $$\nu^{9}$$ $$=$$ $$-62 \beta_{9} + 253 \beta_{8} + 389 \beta_{7} + 619 \beta_{6} - 239 \beta_{5} + 95 \beta_{4} - 17 \beta_{3} + 689 \beta_{2} + 726 \beta_{1} + 788$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.76052 2.20114 1.58984 1.31483 0.897685 −0.115985 −0.686476 −1.74841 −1.94026 −2.27289
−2.76052 0.148409 5.62048 −2.53457 −0.409686 0 −9.99440 −2.97797 6.99673
1.2 −2.20114 −2.62563 2.84501 −4.23731 5.77938 0 −1.86000 3.89394 9.32690
1.3 −1.58984 1.33784 0.527606 0.921074 −2.12696 0 2.34088 −1.21017 −1.46436
1.4 −1.31483 3.01444 −0.271216 −4.02523 −3.96349 0 2.98627 6.08688 5.29250
1.5 −0.897685 −0.315265 −1.19416 −1.27247 0.283008 0 2.86735 −2.90061 1.14227
1.6 0.115985 −2.98171 −1.98655 −0.455421 −0.345835 0 −0.462382 5.89060 −0.0528223
1.7 0.686476 1.12233 −1.52875 0.737311 0.770452 0 −2.42240 −1.74038 0.506146
1.8 1.74841 0.931300 1.05694 −4.09480 1.62830 0 −1.64885 −2.13268 −7.15939
1.9 1.94026 −2.74246 1.76460 0.683496 −5.32108 0 −0.456739 4.52109 1.32616
1.10 2.27289 −1.88926 3.16604 −1.72209 −4.29408 0 2.65028 0.569302 −3.91413
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$1$$
$$19$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.a.p 10
3.b odd 2 1 8379.2.a.ct 10
7.b odd 2 1 931.2.a.q yes 10
7.c even 3 2 931.2.f.r 20
7.d odd 6 2 931.2.f.q 20
21.c even 2 1 8379.2.a.cs 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
931.2.a.p 10 1.a even 1 1 trivial
931.2.a.q yes 10 7.b odd 2 1
931.2.f.q 20 7.d odd 6 2
931.2.f.r 20 7.c even 3 2
8379.2.a.cs 10 21.c even 2 1
8379.2.a.ct 10 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(931))$$:

 $$T_{2}^{10} + \cdots$$ $$T_{3}^{10} + \cdots$$ $$T_{5}^{10} + \cdots$$ $$T_{13}^{10} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-7 + 58 T + 39 T^{2} - 152 T^{3} - 93 T^{4} + 98 T^{5} + 57 T^{6} - 24 T^{7} - 13 T^{8} + 2 T^{9} + T^{10}$$
$3$ $$-8 + 40 T + 135 T^{2} - 272 T^{3} - 74 T^{4} + 208 T^{5} + 33 T^{6} - 56 T^{7} - 12 T^{8} + 4 T^{9} + T^{10}$$
$5$ $$-82 - 64 T + 437 T^{2} + 232 T^{3} - 578 T^{4} - 432 T^{5} + 154 T^{6} + 248 T^{7} + 96 T^{8} + 16 T^{9} + T^{10}$$
$7$ $$T^{10}$$
$11$ $$6368 - 39296 T + 31073 T^{2} + 8016 T^{3} - 10770 T^{4} - 400 T^{5} + 1279 T^{6} - 62 T^{8} + T^{10}$$
$13$ $$1022 - 7084 T - 16485 T^{2} + 408 T^{3} + 10570 T^{4} + 2088 T^{5} - 1564 T^{6} - 512 T^{7} - 4 T^{8} + 12 T^{9} + T^{10}$$
$17$ $$37646 - 86136 T - 52123 T^{2} + 38920 T^{3} + 25028 T^{4} - 2376 T^{5} - 3745 T^{6} - 632 T^{7} + 34 T^{8} + 16 T^{9} + T^{10}$$
$19$ $$( 1 + T )^{10}$$
$23$ $$-917392 - 976824 T + 271465 T^{2} + 300916 T^{3} - 24612 T^{4} - 28124 T^{5} + 1630 T^{6} + 1004 T^{7} - 68 T^{8} - 12 T^{9} + T^{10}$$
$29$ $$168952 + 147672 T - 134185 T^{2} - 102612 T^{3} + 30228 T^{4} + 16736 T^{5} - 1257 T^{6} - 980 T^{7} - 48 T^{8} + 12 T^{9} + T^{10}$$
$31$ $$-281848 - 149408 T + 948007 T^{2} - 438968 T^{3} - 129392 T^{4} + 41276 T^{5} + 7051 T^{6} - 1044 T^{7} - 146 T^{8} + 8 T^{9} + T^{10}$$
$37$ $$-792136 - 1528392 T + 497839 T^{2} + 629344 T^{3} - 160340 T^{4} - 44740 T^{5} + 9740 T^{6} + 880 T^{7} - 188 T^{8} - 4 T^{9} + T^{10}$$
$41$ $$-10740226 + 3091340 T + 6208111 T^{2} + 1045572 T^{3} - 502798 T^{4} - 188368 T^{5} - 14421 T^{6} + 2572 T^{7} + 568 T^{8} + 40 T^{9} + T^{10}$$
$43$ $$-1282048 + 1401344 T + 1114256 T^{2} - 142880 T^{3} - 157136 T^{4} - 336 T^{5} + 7800 T^{6} + 280 T^{7} - 148 T^{8} - 4 T^{9} + T^{10}$$
$47$ $$-4289848 - 9533544 T - 4652271 T^{2} + 180112 T^{3} + 536702 T^{4} + 80408 T^{5} - 10194 T^{6} - 2992 T^{7} - 106 T^{8} + 16 T^{9} + T^{10}$$
$53$ $$-288453476 + 439001964 T + 145733335 T^{2} - 11459484 T^{3} - 5189604 T^{4} + 94736 T^{5} + 71609 T^{6} - 240 T^{7} - 440 T^{8} + T^{10}$$
$59$ $$-1952528 - 14890728 T - 4183273 T^{2} + 4179764 T^{3} + 1074834 T^{4} - 85564 T^{5} - 39648 T^{6} - 1700 T^{7} + 338 T^{8} + 36 T^{9} + T^{10}$$
$61$ $$238217966 + 68954976 T - 47792443 T^{2} - 15635296 T^{3} + 1031354 T^{4} + 552160 T^{5} + 12898 T^{6} - 5600 T^{7} - 284 T^{8} + 16 T^{9} + T^{10}$$
$67$ $$1231484 + 4265436 T + 4683359 T^{2} + 2148020 T^{3} + 324904 T^{4} - 55880 T^{5} - 22957 T^{6} - 1664 T^{7} + 172 T^{8} + 28 T^{9} + T^{10}$$
$71$ $$7611772 + 7686108 T - 1603897 T^{2} - 1546548 T^{3} + 72308 T^{4} + 103128 T^{5} + 2316 T^{6} - 2476 T^{7} - 168 T^{8} + 12 T^{9} + T^{10}$$
$73$ $$30259838 - 19245752 T - 13825831 T^{2} + 9694512 T^{3} - 1150574 T^{4} - 203872 T^{5} + 38887 T^{6} + 1000 T^{7} - 352 T^{8} + T^{10}$$
$79$ $$-358185536 + 31936064 T + 86709040 T^{2} - 4965952 T^{3} - 5167968 T^{4} + 247328 T^{5} + 81268 T^{6} - 2680 T^{7} - 484 T^{8} + 8 T^{9} + T^{10}$$
$83$ $$328888 + 1624448 T + 1471729 T^{2} - 679032 T^{3} - 258800 T^{4} + 48784 T^{5} + 15955 T^{6} - 520 T^{7} - 254 T^{8} + T^{10}$$
$89$ $$457135072 + 420585344 T + 137098928 T^{2} + 12515552 T^{3} - 3129544 T^{4} - 907568 T^{5} - 71508 T^{6} + 2600 T^{7} + 786 T^{8} + 48 T^{9} + T^{10}$$
$97$ $$-223920578 - 36400420 T + 43591331 T^{2} + 9221908 T^{3} - 1694602 T^{4} - 363804 T^{5} + 30540 T^{6} + 4748 T^{7} - 312 T^{8} - 16 T^{9} + T^{10}$$