# Properties

 Label 950.2.l.g Level $950$ Weight $2$ Character orbit 950.l Analytic conductor $7.586$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.l (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 24 x^{10} + 264 x^{8} - 1511 x^{6} + 4812 x^{4} - 7788 x^{2} + 5329$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{8} q^{2} + \beta_{4} q^{3} + \beta_{10} q^{4} + ( \beta_{6} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{6} + ( -\beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{11} ) q^{7} + ( 1 - \beta_{11} ) q^{8} + ( -\beta_{1} + \beta_{2} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{11} ) q^{9} +O(q^{10})$$ $$q + \beta_{8} q^{2} + \beta_{4} q^{3} + \beta_{10} q^{4} + ( \beta_{6} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{6} + ( -\beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{11} ) q^{7} + ( 1 - \beta_{11} ) q^{8} + ( -\beta_{1} + \beta_{2} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{11} ) q^{9} + ( -\beta_{3} + \beta_{4} + \beta_{9} + \beta_{10} ) q^{11} + ( -\beta_{2} - \beta_{9} ) q^{12} + ( 2 - \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{10} ) q^{13} + ( -1 - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{14} + ( \beta_{8} + \beta_{9} ) q^{16} + ( -2 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{17} + ( -1 - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{18} + ( \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{10} ) q^{19} + ( -2 + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{21} + ( 1 - \beta_{1} + \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{22} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{8} - 2 \beta_{10} ) q^{23} + ( -\beta_{5} + \beta_{11} ) q^{24} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{26} + ( -2 + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{27} + ( \beta_{1} - \beta_{2} - \beta_{5} ) q^{28} + ( 4 - 2 \beta_{8} - 4 \beta_{9} - 2 \beta_{11} ) q^{29} + ( -2 \beta_{5} + 2 \beta_{6} - 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{31} + ( \beta_{7} + \beta_{10} ) q^{32} + ( -\beta_{1} + \beta_{3} + 2 \beta_{5} + 2 \beta_{7} - 3 \beta_{9} - 4 \beta_{11} ) q^{33} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{10} + 2 \beta_{11} ) q^{34} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{7} - 3 \beta_{8} - 2 \beta_{11} ) q^{36} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{10} ) q^{37} + ( -2 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{38} + ( 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} + 6 \beta_{9} - 4 \beta_{10} - 2 \beta_{11} ) q^{39} + ( 1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + 3 \beta_{10} - 2 \beta_{11} ) q^{41} + ( -2 - 2 \beta_{3} - 4 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{42} + ( \beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{9} + 2 \beta_{11} ) q^{43} + ( -\beta_{2} + \beta_{4} + \beta_{8} - \beta_{11} ) q^{44} + ( -2 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} ) q^{46} + ( 4 + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 8 \beta_{11} ) q^{47} + ( \beta_{3} + \beta_{6} - \beta_{9} + \beta_{11} ) q^{48} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{7} + 2 \beta_{9} - \beta_{11} ) q^{49} + ( -4 + 3 \beta_{6} - \beta_{8} - 5 \beta_{9} + 5 \beta_{11} ) q^{51} + ( -2 + \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{11} ) q^{52} + ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{4} - 3 \beta_{5} + 4 \beta_{8} + 4 \beta_{9} + \beta_{11} ) q^{53} + ( 2 - \beta_{2} - \beta_{3} + 2 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{54} + ( \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{10} + 2 \beta_{11} ) q^{56} + ( 4 + 2 \beta_{1} - \beta_{2} - 2 \beta_{5} - 6 \beta_{7} + 4 \beta_{8} + 5 \beta_{9} - 6 \beta_{10} ) q^{57} + ( -4 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{58} + ( -4 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{6} + 2 \beta_{7} - 2 \beta_{10} + 3 \beta_{11} ) q^{59} + ( 4 + 2 \beta_{3} + 6 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} ) q^{61} + ( 2 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} + 2 \beta_{9} - 2 \beta_{10} ) q^{62} + ( -1 + \beta_{2} - \beta_{4} + 2 \beta_{6} + 8 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 8 \beta_{10} + 3 \beta_{11} ) q^{63} -\beta_{11} q^{64} + ( -2 + \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 5 \beta_{9} - 2 \beta_{11} ) q^{66} + ( -\beta_{1} + \beta_{2} + \beta_{5} - 7 \beta_{8} - 7 \beta_{11} ) q^{67} + ( 2 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{10} - 2 \beta_{11} ) q^{68} + ( 4 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 6 \beta_{9} + 2 \beta_{10} - 6 \beta_{11} ) q^{69} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 6 \beta_{11} ) q^{71} + ( 2 - \beta_{1} + \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{72} + ( -2 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 3 \beta_{6} - 2 \beta_{7} - 6 \beta_{10} + 3 \beta_{11} ) q^{73} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 3 \beta_{10} - 2 \beta_{11} ) q^{74} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{76} + ( -5 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + 4 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{77} + ( -2 + 2 \beta_{2} + 2 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} + 2 \beta_{10} + 6 \beta_{11} ) q^{78} + ( -4 + 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{79} + ( 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 3 \beta_{9} - 3 \beta_{11} ) q^{81} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - \beta_{9} + 4 \beta_{10} ) q^{82} + ( -2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - 7 \beta_{9} + 2 \beta_{11} ) q^{83} + ( 2 - 2 \beta_{1} - 4 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{84} + ( 2 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{9} ) q^{86} + ( -4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{87} + ( -\beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{88} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - 6 \beta_{7} - 2 \beta_{8} - 6 \beta_{10} + \beta_{11} ) q^{89} + ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{4} - 3 \beta_{5} - 2 \beta_{8} + 6 \beta_{9} - 4 \beta_{10} + 9 \beta_{11} ) q^{91} + ( -1 - \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - 4 \beta_{8} - 4 \beta_{9} + \beta_{11} ) q^{92} + ( -4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + 4 \beta_{8} - 4 \beta_{10} - 2 \beta_{11} ) q^{93} + ( 3 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} - 4 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{94} + ( \beta_{1} - \beta_{2} + \beta_{8} ) q^{96} + ( \beta_{1} - \beta_{2} + \beta_{3} + 4 \beta_{7} + 4 \beta_{8} + 4 \beta_{10} + 4 \beta_{11} ) q^{97} + ( 1 - \beta_{1} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{98} + ( 1 + \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{8} + 4 \beta_{9} + 6 \beta_{10} + 3 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 3q^{3} + 3q^{6} + 6q^{7} + 6q^{8} + 9q^{9} + O(q^{10})$$ $$12q - 3q^{3} + 3q^{6} + 6q^{7} + 6q^{8} + 9q^{9} - 6q^{11} + 18q^{13} - 6q^{14} - 12q^{17} - 24q^{18} + 6q^{19} - 36q^{21} + 9q^{22} - 3q^{23} + 3q^{24} - 3q^{26} - 15q^{27} - 3q^{28} + 36q^{29} - 24q^{31} - 15q^{33} - 6q^{34} + 9q^{36} - 24q^{37} - 15q^{38} - 12q^{39} - 12q^{41} - 18q^{42} + 12q^{43} - 9q^{44} - 18q^{46} + 6q^{48} - 27q^{51} - 18q^{52} + 36q^{53} + 9q^{54} + 12q^{56} + 42q^{57} - 27q^{59} + 54q^{61} + 24q^{62} + 3q^{63} - 6q^{64} - 39q^{66} - 39q^{67} + 15q^{68} - 24q^{69} + 24q^{71} + 18q^{72} + 15q^{74} + 9q^{76} - 78q^{77} + 6q^{78} - 36q^{79} - 9q^{81} + 12q^{82} + 12q^{84} + 24q^{86} - 18q^{87} + 6q^{88} + 18q^{89} + 12q^{91} - 12q^{92} - 54q^{93} + 18q^{94} + 27q^{97} + 18q^{98} + 18q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 24 x^{10} + 264 x^{8} - 1511 x^{6} + 4812 x^{4} - 7788 x^{2} + 5329$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-25 \nu^{10} + 510 \nu^{8} - 4764 \nu^{6} + 21478 \nu^{4} - 53220 \nu^{2} - 4267 \nu + 54312$$$$)/8534$$ $$\beta_{2}$$ $$=$$ $$($$$$-25 \nu^{10} + 510 \nu^{8} - 4764 \nu^{6} + 21478 \nu^{4} - 53220 \nu^{2} + 4267 \nu + 54312$$$$)/8534$$ $$\beta_{3}$$ $$=$$ $$($$$$8448 \nu^{11} - 93513 \nu^{10} - 223995 \nu^{9} + 2303442 \nu^{8} + 2546362 \nu^{7} - 24775981 \nu^{6} - 16062484 \nu^{5} + 132143140 \nu^{4} + 55183959 \nu^{3} - 327536692 \nu^{2} - 93428707 \nu + 302962702$$$$)/44231722$$ $$\beta_{4}$$ $$=$$ $$($$$$-8990 \nu^{11} + 119136 \nu^{10} + 302119 \nu^{9} - 2818822 \nu^{8} - 4351295 \nu^{7} + 29611793 \nu^{6} + 31424214 \nu^{5} - 152541749 \nu^{4} - 109794343 \nu^{3} + 387894990 \nu^{2} + 126631241 \nu - 386103862$$$$)/44231722$$ $$\beta_{5}$$ $$=$$ $$($$$$-8448 \nu^{11} - 93513 \nu^{10} + 223995 \nu^{9} + 2303442 \nu^{8} - 2546362 \nu^{7} - 24775981 \nu^{6} + 16062484 \nu^{5} + 132143140 \nu^{4} - 55183959 \nu^{3} - 327536692 \nu^{2} + 93428707 \nu + 302962702$$$$)/44231722$$ $$\beta_{6}$$ $$=$$ $$($$$$8990 \nu^{11} + 119136 \nu^{10} - 302119 \nu^{9} - 2818822 \nu^{8} + 4351295 \nu^{7} + 29611793 \nu^{6} - 31424214 \nu^{5} - 152541749 \nu^{4} + 109794343 \nu^{3} + 387894990 \nu^{2} - 126631241 \nu - 386103862$$$$)/44231722$$ $$\beta_{7}$$ $$=$$ $$($$$$18753 \nu^{11} - 79059 \nu^{10} - 398826 \nu^{9} + 1645785 \nu^{8} + 3920032 \nu^{7} - 15276564 \nu^{6} - 18664159 \nu^{5} + 64129040 \nu^{4} + 49442218 \nu^{3} - 120030688 \nu^{2} - 47447629 \nu + 48113424$$$$)/44231722$$ $$\beta_{8}$$ $$=$$ $$($$$$18753 \nu^{11} + 79059 \nu^{10} - 398826 \nu^{9} - 1645785 \nu^{8} + 3920032 \nu^{7} + 15276564 \nu^{6} - 18664159 \nu^{5} - 64129040 \nu^{4} + 49442218 \nu^{3} + 120030688 \nu^{2} - 47447629 \nu - 48113424$$$$)/44231722$$ $$\beta_{9}$$ $$=$$ $$($$$$43673 \nu^{11} - 25623 \nu^{10} - 1033698 \nu^{9} + 515380 \nu^{8} + 10872015 \nu^{7} - 4835812 \nu^{6} - 56490486 \nu^{5} + 20398609 \nu^{4} + 142140376 \nu^{3} - 38242437 \nu^{2} - 132619320 \nu - 5322284$$$$)/44231722$$ $$\beta_{10}$$ $$=$$ $$($$$$43673 \nu^{11} + 25623 \nu^{10} - 1033698 \nu^{9} - 515380 \nu^{8} + 10872015 \nu^{7} + 4835812 \nu^{6} - 56490486 \nu^{5} - 20398609 \nu^{4} + 142140376 \nu^{3} + 38242437 \nu^{2} - 132619320 \nu + 5322284$$$$)/44231722$$ $$\beta_{11}$$ $$=$$ $$($$$$-744 \nu^{11} + 16031 \nu^{9} - 159186 \nu^{7} + 776412 \nu^{5} - 2012234 \nu^{3} + 1909212 \nu + 311491$$$$)/622982$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{2} - \beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{10} + \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 4$$ $$\nu^{3}$$ $$=$$ $$6 \beta_{11} + 8 \beta_{8} + 8 \beta_{7} - \beta_{5} + \beta_{3} + 3 \beta_{2} - 3 \beta_{1} - 3$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 11 \beta_{6} + 11 \beta_{5} + 11 \beta_{4} + 11 \beta_{3} + 3 \beta_{2} + 3 \beta_{1} + 8$$ $$\nu^{5}$$ $$=$$ $$68 \beta_{11} + 5 \beta_{10} + 5 \beta_{9} + 85 \beta_{8} + 85 \beta_{7} + \beta_{6} + 3 \beta_{5} - \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 3 \beta_{1} - 34$$ $$\nu^{6}$$ $$=$$ $$116 \beta_{10} - 116 \beta_{9} - 16 \beta_{8} + 16 \beta_{7} + 82 \beta_{6} + 77 \beta_{5} + 82 \beta_{4} + 77 \beta_{3} + 33 \beta_{2} + 33 \beta_{1} - 106$$ $$\nu^{7}$$ $$=$$ $$392 \beta_{11} + 32 \beta_{10} + 32 \beta_{9} + 514 \beta_{8} + 514 \beta_{7} + 49 \beta_{6} + 133 \beta_{5} - 49 \beta_{4} - 133 \beta_{3} - 188 \beta_{2} + 188 \beta_{1} - 196$$ $$\nu^{8}$$ $$=$$ $$1976 \beta_{10} - 1976 \beta_{9} - 543 \beta_{8} + 543 \beta_{7} + 326 \beta_{6} + 294 \beta_{5} + 326 \beta_{4} + 294 \beta_{3} + 147 \beta_{2} + 147 \beta_{1} - 1864$$ $$\nu^{9}$$ $$=$$ $$-386 \beta_{11} - 300 \beta_{10} - 300 \beta_{9} + 536 \beta_{8} + 536 \beta_{7} + 690 \beta_{6} + 1580 \beta_{5} - 690 \beta_{4} - 1580 \beta_{3} - 2190 \beta_{2} + 2190 \beta_{1} + 193$$ $$\nu^{10}$$ $$=$$ $$18616 \beta_{10} - 18616 \beta_{9} - 6310 \beta_{8} + 6310 \beta_{7} - 1654 \beta_{6} - 1354 \beta_{5} - 1654 \beta_{4} - 1354 \beta_{3} - 883 \beta_{2} - 883 \beta_{1} - 17296$$ $$\nu^{11}$$ $$=$$ $$-38292 \beta_{11} - 8093 \beta_{10} - 8093 \beta_{9} - 31360 \beta_{8} - 31360 \beta_{7} + 5427 \beta_{6} + 11423 \beta_{5} - 5427 \beta_{4} - 11423 \beta_{3} - 15642 \beta_{2} + 15642 \beta_{1} + 19146$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$1$$ $$\beta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 1.97287 + 0.642788i −1.97287 + 0.642788i −1.34865 + 0.342020i 1.34865 + 0.342020i 1.97287 − 0.642788i −1.97287 − 0.642788i −2.79086 − 0.984808i 2.79086 − 0.984808i −1.34865 − 0.342020i 1.34865 − 0.342020i −2.79086 + 0.984808i 2.79086 + 0.984808i
−0.766044 + 0.642788i −2.57374 0.936765i 0.173648 0.984808i 0 2.57374 0.936765i 1.92448 3.33331i 0.500000 + 0.866025i 3.44848 + 2.89362i 0
101.2 −0.766044 + 0.642788i 1.13405 + 0.412760i 0.173648 0.984808i 0 −1.13405 + 0.412760i −1.09813 + 1.90202i 0.500000 + 0.866025i −1.18244 0.992183i 0
251.1 0.939693 + 0.342020i −0.397366 2.25357i 0.766044 + 0.642788i 0 0.397366 2.25357i 1.38429 2.39766i 0.500000 + 0.866025i −2.10161 + 0.764925i 0
251.2 0.939693 + 0.342020i 0.0710139 + 0.402740i 0.766044 + 0.642788i 0 −0.0710139 + 0.402740i −1.15033 + 1.99244i 0.500000 + 0.866025i 2.66192 0.968860i 0
301.1 −0.766044 0.642788i −2.57374 + 0.936765i 0.173648 + 0.984808i 0 2.57374 + 0.936765i 1.92448 + 3.33331i 0.500000 0.866025i 3.44848 2.89362i 0
301.2 −0.766044 0.642788i 1.13405 0.412760i 0.173648 + 0.984808i 0 −1.13405 0.412760i −1.09813 1.90202i 0.500000 0.866025i −1.18244 + 0.992183i 0
351.1 −0.173648 0.984808i −2.00490 + 1.68231i −0.939693 + 0.342020i 0 2.00490 + 1.68231i 0.485218 0.840422i 0.500000 + 0.866025i 0.668514 3.79133i 0
351.2 −0.173648 0.984808i 2.27095 1.90555i −0.939693 + 0.342020i 0 −2.27095 1.90555i 1.45447 2.51922i 0.500000 + 0.866025i 1.00513 5.70040i 0
651.1 0.939693 0.342020i −0.397366 + 2.25357i 0.766044 0.642788i 0 0.397366 + 2.25357i 1.38429 + 2.39766i 0.500000 0.866025i −2.10161 0.764925i 0
651.2 0.939693 0.342020i 0.0710139 0.402740i 0.766044 0.642788i 0 −0.0710139 0.402740i −1.15033 1.99244i 0.500000 0.866025i 2.66192 + 0.968860i 0
701.1 −0.173648 + 0.984808i −2.00490 1.68231i −0.939693 0.342020i 0 2.00490 1.68231i 0.485218 + 0.840422i 0.500000 0.866025i 0.668514 + 3.79133i 0
701.2 −0.173648 + 0.984808i 2.27095 + 1.90555i −0.939693 0.342020i 0 −2.27095 + 1.90555i 1.45447 + 2.51922i 0.500000 0.866025i 1.00513 + 5.70040i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 701.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.l.g 12
5.b even 2 1 190.2.k.c 12
5.c odd 4 2 950.2.u.f 24
19.e even 9 1 inner 950.2.l.g 12
95.o odd 18 1 3610.2.a.bd 6
95.p even 18 1 190.2.k.c 12
95.p even 18 1 3610.2.a.bf 6
95.q odd 36 2 950.2.u.f 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.k.c 12 5.b even 2 1
190.2.k.c 12 95.p even 18 1
950.2.l.g 12 1.a even 1 1 trivial
950.2.l.g 12 19.e even 9 1 inner
950.2.u.f 24 5.c odd 4 2
950.2.u.f 24 95.q odd 36 2
3610.2.a.bd 6 95.o odd 18 1
3610.2.a.bf 6 95.p even 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{3} + T^{6} )^{2}$$
$3$ $$576 - 864 T + 3600 T^{2} - 2064 T^{3} - 936 T^{4} + 210 T^{5} + 391 T^{6} + 189 T^{7} + 51 T^{8} - 4 T^{9} + 3 T^{11} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$23104 - 25536 T + 34608 T^{2} - 12096 T^{3} + 12804 T^{4} - 4566 T^{5} + 3139 T^{6} - 861 T^{7} + 429 T^{8} - 108 T^{9} + 39 T^{10} - 6 T^{11} + T^{12}$$
$11$ $$81 - 324 T + 1215 T^{2} - 1152 T^{3} + 1683 T^{4} + 792 T^{5} + 1828 T^{6} + 420 T^{7} + 321 T^{8} + 56 T^{9} + 42 T^{10} + 6 T^{11} + T^{12}$$
$13$ $$64 - 2016 T + 21312 T^{2} - 70288 T^{3} + 118944 T^{4} - 88056 T^{5} + 54993 T^{6} - 20646 T^{7} + 5202 T^{8} - 1015 T^{9} + 162 T^{10} - 18 T^{11} + T^{12}$$
$17$ $$46656 + 116640 T + 419904 T^{2} + 544968 T^{3} + 340524 T^{4} + 72252 T^{5} - 10575 T^{6} - 6156 T^{7} - 504 T^{8} + 123 T^{9} + 72 T^{10} + 12 T^{11} + T^{12}$$
$19$ $$47045881 - 14856594 T + 10165038 T^{2} - 3402064 T^{3} + 1117656 T^{4} - 324900 T^{5} + 74997 T^{6} - 17100 T^{7} + 3096 T^{8} - 496 T^{9} + 78 T^{10} - 6 T^{11} + T^{12}$$
$23$ $$5184 - 2592 T + 18144 T^{2} + 1368 T^{3} + 684 T^{4} - 8244 T^{5} + 3133 T^{6} - 2055 T^{7} + 2505 T^{8} - 388 T^{9} - 30 T^{10} + 3 T^{11} + T^{12}$$
$29$ $$( 5184 - 2592 T + 1296 T^{2} - 576 T^{3} + 144 T^{4} - 18 T^{5} + T^{6} )^{2}$$
$31$ $$1910738944 - 310530048 T + 381978624 T^{2} + 116822016 T^{3} + 46632192 T^{4} + 8385024 T^{5} + 1777408 T^{6} + 261888 T^{7} + 42288 T^{8} + 4608 T^{9} + 444 T^{10} + 24 T^{11} + T^{12}$$
$37$ $$( -7624 + 3792 T + 1230 T^{2} - 549 T^{3} - 45 T^{4} + 12 T^{5} + T^{6} )^{2}$$
$41$ $$1794623769 + 2103450039 T + 1053195777 T^{2} + 301768578 T^{3} + 58495932 T^{4} + 8049294 T^{5} + 709083 T^{6} + 69930 T^{7} + 16371 T^{8} + 1860 T^{9} + 171 T^{10} + 12 T^{11} + T^{12}$$
$43$ $$27625536 - 33302016 T + 19865088 T^{2} - 5221008 T^{3} + 1516032 T^{4} - 247086 T^{5} + 19765 T^{6} + 16338 T^{7} + 1092 T^{8} + 145 T^{9} + 6 T^{10} - 12 T^{11} + T^{12}$$
$47$ $$39575532096 - 2642665824 T + 5723291520 T^{2} + 497912832 T^{3} - 55890972 T^{4} - 10338678 T^{5} + 1459701 T^{6} + 120042 T^{7} - 5994 T^{8} - 585 T^{9} - 36 T^{10} + T^{12}$$
$53$ $$59962296384 - 15136028064 T - 304479648 T^{2} + 858339432 T^{3} - 63749772 T^{4} - 10693332 T^{5} + 4098313 T^{6} - 792963 T^{7} + 107172 T^{8} - 10277 T^{9} + 729 T^{10} - 36 T^{11} + T^{12}$$
$59$ $$18429849 + 16691184 T + 3923478 T^{2} + 164025 T^{3} + 995814 T^{4} + 802629 T^{5} + 301158 T^{6} + 77274 T^{7} + 19845 T^{8} + 2997 T^{9} + 342 T^{10} + 27 T^{11} + T^{12}$$
$61$ $$437981184 - 584644608 T + 500299776 T^{2} - 293964288 T^{3} + 119964672 T^{4} - 34591872 T^{5} + 7402624 T^{6} - 1253568 T^{7} + 172176 T^{8} - 17944 T^{9} + 1284 T^{10} - 54 T^{11} + T^{12}$$
$67$ $$437981184 - 2778568704 T + 7432434432 T^{2} + 1402236480 T^{3} + 92195712 T^{4} + 12802020 T^{5} + 3653149 T^{6} + 752745 T^{7} + 115785 T^{8} + 11488 T^{9} + 798 T^{10} + 39 T^{11} + T^{12}$$
$71$ $$2985984 + 16422912 T + 40310784 T^{2} + 23307264 T^{3} + 11467008 T^{4} + 2571264 T^{5} - 521280 T^{6} - 165024 T^{7} + 42048 T^{8} - 4008 T^{9} + 360 T^{10} - 24 T^{11} + T^{12}$$
$73$ $$345847495744 + 51022514880 T + 5823556992 T^{2} - 620891440 T^{3} - 188745984 T^{4} - 15747750 T^{5} + 3404127 T^{6} + 418959 T^{7} + 1026 T^{8} - 331 T^{9} + 99 T^{10} + T^{12}$$
$79$ $$162205696 + 414480384 T + 491467776 T^{2} + 292608000 T^{3} + 104345088 T^{4} + 25758720 T^{5} + 4950784 T^{6} + 769248 T^{7} + 95808 T^{8} + 9360 T^{9} + 696 T^{10} + 36 T^{11} + T^{12}$$
$83$ $$5184 - 59616 T + 627264 T^{2} - 831240 T^{3} + 1563552 T^{4} + 1265814 T^{5} + 1065691 T^{6} + 245013 T^{7} + 48771 T^{8} + 2230 T^{9} + 219 T^{10} + T^{12}$$
$89$ $$101062809 - 223749621 T + 176170464 T^{2} - 54327582 T^{3} + 9690660 T^{4} - 400446 T^{5} - 77129 T^{6} + 48852 T^{7} - 306 T^{8} - 298 T^{9} + 171 T^{10} - 18 T^{11} + T^{12}$$
$97$ $$168169024 - 377213184 T + 389769552 T^{2} - 220667544 T^{3} + 75475512 T^{4} - 17358966 T^{5} + 3132235 T^{6} - 469953 T^{7} + 57303 T^{8} - 5508 T^{9} + 432 T^{10} - 27 T^{11} + T^{12}$$