# Properties

 Label 2169.2.a.h Level $2169$ Weight $2$ Character orbit 2169.a Self dual yes Analytic conductor $17.320$ Analytic rank $1$ Dimension $12$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2169 = 3^{2} \cdot 241$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2169.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.3195521984$$ Analytic rank: $$1$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} - 328 x^{3} - 45 x^{2} + 18 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 241) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{9} ) q^{5} + ( \beta_{4} - \beta_{5} ) q^{7} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{9} ) q^{5} + ( \beta_{4} - \beta_{5} ) q^{7} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{8} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{10} + ( -2 + \beta_{1} - \beta_{3} + \beta_{8} ) q^{11} + ( -\beta_{2} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{13} + ( -1 + \beta_{1} - \beta_{4} - \beta_{7} + \beta_{8} ) q^{14} + ( 1 + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{16} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{11} ) q^{17} + ( -1 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} ) q^{19} + ( -2 + 2 \beta_{1} - \beta_{3} - \beta_{6} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{20} + ( -3 + 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{10} + \beta_{11} ) q^{22} + ( -5 + \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{23} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{11} ) q^{25} + ( -3 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{26} + ( -1 + \beta_{7} ) q^{28} + ( 1 + \beta_{1} - \beta_{3} + \beta_{6} - \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{29} + ( \beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{6} - \beta_{9} - \beta_{10} ) q^{31} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{7} - 2 \beta_{9} + \beta_{10} - 3 \beta_{11} ) q^{32} + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{7} - \beta_{11} ) q^{34} + ( -2 + \beta_{1} + \beta_{2} - \beta_{5} - 2 \beta_{7} + \beta_{8} ) q^{35} + ( -1 + 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{37} + ( 3 + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} + 4 \beta_{10} - \beta_{11} ) q^{38} + ( -7 + 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{40} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{11} ) q^{41} + ( \beta_{1} + \beta_{3} - \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{43} + ( -6 + \beta_{1} - 3 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{44} + ( -2 + 2 \beta_{1} - 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{6} + 3 \beta_{8} + 4 \beta_{9} + 3 \beta_{10} ) q^{46} + ( -2 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{47} + ( 2 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} - 2 \beta_{11} ) q^{49} + ( 5 - \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - 4 \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{50} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{52} + ( -1 + \beta_{1} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} ) q^{53} + ( 1 - \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{55} + ( -\beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{7} - \beta_{8} - \beta_{10} ) q^{56} + ( -5 + 2 \beta_{1} - \beta_{3} + \beta_{5} - 2 \beta_{6} + 4 \beta_{8} - \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{58} + ( -4 - 2 \beta_{1} + 3 \beta_{3} - 3 \beta_{5} + \beta_{8} - 3 \beta_{10} + 2 \beta_{11} ) q^{59} + ( -1 + \beta_{3} + \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{61} + ( -3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} ) q^{62} + ( 2 + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} - 4 \beta_{8} + 2 \beta_{10} ) q^{64} + ( 3 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{65} + ( 3 - \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{67} + ( 2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{11} ) q^{68} + ( 1 + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{70} + ( -9 + 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{71} + ( -1 + 2 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{9} + 3 \beta_{11} ) q^{73} + ( -1 - 2 \beta_{1} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{74} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{76} + ( 2 - \beta_{1} - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + \beta_{7} - \beta_{8} ) q^{77} + ( -2 - \beta_{2} - 2 \beta_{5} + 2 \beta_{7} - 4 \beta_{10} ) q^{79} + ( -6 + 4 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} + 4 \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{80} + ( 5 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 4 \beta_{7} - \beta_{8} + 4 \beta_{9} + 5 \beta_{11} ) q^{82} + ( 1 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - \beta_{10} + 3 \beta_{11} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{85} + ( 3 + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - \beta_{7} - 2 \beta_{8} + 5 \beta_{10} - 2 \beta_{11} ) q^{86} + ( -6 + 5 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} - 5 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 5 \beta_{10} + 5 \beta_{11} ) q^{88} + ( 1 + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} + 3 \beta_{9} + 3 \beta_{10} ) q^{89} + ( -2 - 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{91} + ( -8 + 7 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} - 6 \beta_{4} - 4 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - 6 \beta_{10} + 3 \beta_{11} ) q^{92} + ( 4 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{94} + ( -3 - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{95} + ( -4 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{97} + ( -8 + 4 \beta_{1} + \beta_{2} + 4 \beta_{3} - 5 \beta_{4} - \beta_{5} - 5 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - 6 \beta_{10} + 4 \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 3q^{2} + 13q^{4} - 6q^{5} + 3q^{7} - 9q^{8} + O(q^{10})$$ $$12q - 3q^{2} + 13q^{4} - 6q^{5} + 3q^{7} - 9q^{8} - 7q^{10} - 22q^{11} - 5q^{13} - 6q^{14} + 15q^{16} + 4q^{17} - 6q^{19} - 10q^{20} - 12q^{22} - 32q^{23} + 4q^{25} - 8q^{26} - 11q^{28} - 6q^{29} + 8q^{31} - q^{32} - 19q^{34} - 15q^{35} - 8q^{37} + 10q^{38} - 52q^{40} + q^{41} - 2q^{43} - 42q^{44} - 25q^{46} - 34q^{47} - 9q^{49} + 27q^{50} - 41q^{52} - 5q^{53} - 3q^{55} - q^{56} - 33q^{58} - 26q^{59} - 26q^{61} + 17q^{62} + 13q^{64} + 25q^{65} + 6q^{67} + 35q^{68} - 4q^{70} - 94q^{71} - 22q^{73} - 26q^{74} - 20q^{76} + 7q^{77} + 9q^{79} - 19q^{80} + 15q^{82} + 8q^{83} + 4q^{85} - 9q^{86} + 6q^{88} + 3q^{89} - 20q^{91} - 36q^{92} + 48q^{94} - 33q^{95} - 29q^{97} - 28q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} - 328 x^{3} - 45 x^{2} + 18 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{11} + \nu^{10} - 23 \nu^{9} - 13 \nu^{8} + 184 \nu^{7} + 54 \nu^{6} - 611 \nu^{5} - 94 \nu^{4} + 768 \nu^{3} + 94 \nu^{2} - 213 \nu - 8$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{11} + 9 \nu^{10} + 57 \nu^{9} - 119 \nu^{8} - 273 \nu^{7} + 488 \nu^{6} + 538 \nu^{5} - 663 \nu^{4} - 422 \nu^{3} + 168 \nu^{2} + 34 \nu + 7$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$7 \nu^{11} - 10 \nu^{10} - 130 \nu^{9} + 172 \nu^{8} + 869 \nu^{7} - 1046 \nu^{6} - 2491 \nu^{5} + 2625 \nu^{4} + 2774 \nu^{3} - 2230 \nu^{2} - 745 \nu + 85$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$-13 \nu^{11} + 48 \nu^{10} + 160 \nu^{9} - 690 \nu^{8} - 569 \nu^{7} + 3330 \nu^{6} + 597 \nu^{5} - 6481 \nu^{4} - 470 \nu^{3} + 4634 \nu^{2} + 951 \nu - 169$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$13 \nu^{11} - 44 \nu^{10} - 180 \nu^{9} + 662 \nu^{8} + 829 \nu^{7} - 3426 \nu^{6} - 1629 \nu^{5} + 7333 \nu^{4} + 1798 \nu^{3} - 5698 \nu^{2} - 1335 \nu + 141$$$$)/16$$ $$\beta_{8}$$ $$=$$ $$($$$$11 \nu^{11} - 30 \nu^{10} - 158 \nu^{9} + 432 \nu^{8} + 773 \nu^{7} - 2086 \nu^{6} - 1631 \nu^{5} + 4025 \nu^{4} + 1654 \nu^{3} - 2750 \nu^{2} - 741 \nu + 93$$$$)/8$$ $$\beta_{9}$$ $$=$$ $$($$$$-11 \nu^{11} + 34 \nu^{10} + 150 \nu^{9} - 492 \nu^{8} - 669 \nu^{7} + 2398 \nu^{6} + 1223 \nu^{5} - 4717 \nu^{4} - 1178 \nu^{3} + 3354 \nu^{2} + 737 \nu - 109$$$$)/8$$ $$\beta_{10}$$ $$=$$ $$($$$$25 \nu^{11} - 76 \nu^{10} - 340 \nu^{9} + 1086 \nu^{8} + 1513 \nu^{7} - 5178 \nu^{6} - 2777 \nu^{5} + 9809 \nu^{4} + 2718 \nu^{3} - 6602 \nu^{2} - 1643 \nu + 201$$$$)/16$$ $$\beta_{11}$$ $$=$$ $$($$$$31 \nu^{11} - 90 \nu^{10} - 450 \nu^{9} + 1340 \nu^{8} + 2237 \nu^{7} - 6822 \nu^{6} - 4819 \nu^{5} + 14249 \nu^{4} + 5014 \nu^{3} - 10758 \nu^{2} - 2497 \nu + 381$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} - \beta_{10} + \beta_{9} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 5 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{11} + \beta_{9} - \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 8 \beta_{2} + 15$$ $$\nu^{5}$$ $$=$$ $$11 \beta_{11} - 9 \beta_{10} + 10 \beta_{9} - \beta_{7} - 8 \beta_{6} - 11 \beta_{5} - 9 \beta_{4} + 9 \beta_{3} + \beta_{2} + 29 \beta_{1} - 10$$ $$\nu^{6}$$ $$=$$ $$20 \beta_{11} + 2 \beta_{10} + 10 \beta_{9} - 14 \beta_{8} - 19 \beta_{7} - 10 \beta_{6} - 9 \beta_{5} - 2 \beta_{4} + 11 \beta_{3} + 56 \beta_{2} + 88$$ $$\nu^{7}$$ $$=$$ $$93 \beta_{11} - 67 \beta_{10} + 78 \beta_{9} - 3 \beta_{8} - 12 \beta_{7} - 54 \beta_{6} - 94 \beta_{5} - 68 \beta_{4} + 71 \beta_{3} + 11 \beta_{2} + 181 \beta_{1} - 77$$ $$\nu^{8}$$ $$=$$ $$163 \beta_{11} + 24 \beta_{10} + 76 \beta_{9} - 138 \beta_{8} - 146 \beta_{7} - 78 \beta_{6} - 71 \beta_{5} - 30 \beta_{4} + 98 \beta_{3} + 379 \beta_{2} + 4 \beta_{1} + 558$$ $$\nu^{9}$$ $$=$$ $$720 \beta_{11} - 478 \beta_{10} + 557 \beta_{9} - 59 \beta_{8} - 106 \beta_{7} - 355 \beta_{6} - 733 \beta_{5} - 499 \beta_{4} + 542 \beta_{3} + 88 \beta_{2} + 1179 \beta_{1} - 544$$ $$\nu^{10}$$ $$=$$ $$1256 \beta_{11} + 191 \beta_{10} + 522 \beta_{9} - 1189 \beta_{8} - 1056 \beta_{7} - 566 \beta_{6} - 561 \beta_{5} - 323 \beta_{4} + 822 \beta_{3} + 2542 \beta_{2} + 76 \beta_{1} + 3665$$ $$\nu^{11}$$ $$=$$ $$5372 \beta_{11} - 3384 \beta_{10} + 3821 \beta_{9} - 748 \beta_{8} - 845 \beta_{7} - 2351 \beta_{6} - 5486 \beta_{5} - 3655 \beta_{4} + 4093 \beta_{3} + 630 \beta_{2} + 7881 \beta_{1} - 3713$$

## 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
 2.70063 2.49073 2.01020 1.63125 1.54879 0.115670 0.0822506 −0.342147 −1.28632 −1.32986 −2.02418 −2.59703
−2.70063 0 5.29342 −0.533570 0 0.354992 −8.89432 0 1.44098
1.2 −2.49073 0 4.20371 3.14843 0 0.136122 −5.48885 0 −7.84189
1.3 −2.01020 0 2.04092 −1.92585 0 −0.852319 −0.0822476 0 3.87135
1.4 −1.63125 0 0.660992 −1.75438 0 5.06139 2.18426 0 2.86183
1.5 −1.54879 0 0.398765 −0.334961 0 −4.24623 2.47998 0 0.518786
1.6 −0.115670 0 −1.98662 1.31091 0 3.19647 0.461133 0 −0.151633
1.7 −0.0822506 0 −1.99323 −4.31963 0 0.690569 0.328446 0 0.355292
1.8 0.342147 0 −1.88294 0.548903 0 1.82459 −1.32853 0 0.187805
1.9 1.28632 0 −0.345373 −0.612768 0 1.03110 −3.01691 0 −0.788217
1.10 1.32986 0 −0.231473 3.40432 0 −3.83334 −2.96755 0 4.52727
1.11 2.02418 0 2.09729 −1.44091 0 0.381245 0.196936 0 −2.91665
1.12 2.59703 0 4.74454 −3.49051 0 −0.744578 7.12764 0 −9.06493
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$241$$ $$-1$$

## 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 2169.2.a.h 12
3.b odd 2 1 241.2.a.b 12
12.b even 2 1 3856.2.a.n 12
15.d odd 2 1 6025.2.a.h 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.2.a.b 12 3.b odd 2 1
2169.2.a.h 12 1.a even 1 1 trivial
3856.2.a.n 12 12.b even 2 1
6025.2.a.h 12 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2169))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 18 T - 45 T^{2} + 328 T^{3} + 105 T^{4} - 444 T^{5} - 123 T^{6} + 219 T^{7} + 65 T^{8} - 44 T^{9} - 14 T^{10} + 3 T^{11} + T^{12}$$
$3$ $$T^{12}$$
$5$ $$62 + 347 T + 339 T^{2} - 1071 T^{3} - 2193 T^{4} - 497 T^{5} + 1301 T^{6} + 797 T^{7} - 68 T^{8} - 134 T^{9} - 14 T^{10} + 6 T^{11} + T^{12}$$
$7$ $$4 - 53 T + 200 T^{2} - 131 T^{3} - 588 T^{4} + 855 T^{5} + 263 T^{6} - 854 T^{7} + 245 T^{8} + 96 T^{9} - 33 T^{10} - 3 T^{11} + T^{12}$$
$11$ $$128 + 1460 T + 5900 T^{2} + 9672 T^{3} + 3100 T^{4} - 9811 T^{5} - 12739 T^{6} - 5545 T^{7} - 215 T^{8} + 553 T^{9} + 177 T^{10} + 22 T^{11} + T^{12}$$
$13$ $$-52672 - 441248 T - 90512 T^{2} + 288472 T^{3} + 69802 T^{4} - 64645 T^{5} - 15049 T^{6} + 6470 T^{7} + 1425 T^{8} - 296 T^{9} - 62 T^{10} + 5 T^{11} + T^{12}$$
$17$ $$154144 + 302576 T - 261264 T^{2} - 295792 T^{3} + 159474 T^{4} + 87027 T^{5} - 35221 T^{6} - 9972 T^{7} + 2997 T^{8} + 370 T^{9} - 97 T^{10} - 4 T^{11} + T^{12}$$
$19$ $$-3556280 - 3617301 T + 903711 T^{2} + 1614089 T^{3} + 58969 T^{4} - 247081 T^{5} - 28947 T^{6} + 16891 T^{7} + 2538 T^{8} - 524 T^{9} - 86 T^{10} + 6 T^{11} + T^{12}$$
$23$ $$-116949436 - 276824423 T - 182523158 T^{2} - 31479187 T^{3} + 11455889 T^{4} + 5191210 T^{5} + 372702 T^{6} - 138011 T^{7} - 28306 T^{8} - 627 T^{9} + 304 T^{10} + 32 T^{11} + T^{12}$$
$29$ $$58109390 + 179077009 T + 164527164 T^{2} + 50865568 T^{3} - 3169769 T^{4} - 4236578 T^{5} - 355685 T^{6} + 116722 T^{7} + 15216 T^{8} - 1375 T^{9} - 213 T^{10} + 6 T^{11} + T^{12}$$
$31$ $$-318193616 + 468437780 T + 77270368 T^{2} - 108211396 T^{3} + 841016 T^{4} + 8192365 T^{5} - 688338 T^{6} - 208450 T^{7} + 22930 T^{8} + 2167 T^{9} - 262 T^{10} - 8 T^{11} + T^{12}$$
$37$ $$50796928 - 18033984 T - 29177888 T^{2} + 5067040 T^{3} + 4698614 T^{4} - 619307 T^{5} - 323567 T^{6} + 36249 T^{7} + 10466 T^{8} - 928 T^{9} - 159 T^{10} + 8 T^{11} + T^{12}$$
$41$ $$-63338 + 4034251 T - 930334 T^{2} - 9341574 T^{3} - 976432 T^{4} + 2920817 T^{5} - 471938 T^{6} - 92737 T^{7} + 21111 T^{8} + 708 T^{9} - 262 T^{10} - T^{11} + T^{12}$$
$43$ $$12503272 - 1488169 T - 25360675 T^{2} - 4519982 T^{3} + 6500883 T^{4} + 657869 T^{5} - 569920 T^{6} - 18272 T^{7} + 18808 T^{8} - 26 T^{9} - 237 T^{10} + 2 T^{11} + T^{12}$$
$47$ $$53297792 - 37689968 T - 69239488 T^{2} - 11362328 T^{3} + 11628792 T^{4} + 4772881 T^{5} + 233524 T^{6} - 179952 T^{7} - 34508 T^{8} - 851 T^{9} + 332 T^{10} + 34 T^{11} + T^{12}$$
$53$ $$-3014 - 298853 T + 6874728 T^{2} + 11787807 T^{3} + 1527793 T^{4} - 1538756 T^{5} - 270515 T^{6} + 65448 T^{7} + 12170 T^{8} - 1019 T^{9} - 195 T^{10} + 5 T^{11} + T^{12}$$
$59$ $$-25476160 - 96501648 T - 59605584 T^{2} + 36571744 T^{3} + 47289076 T^{4} + 17302787 T^{5} + 2412075 T^{6} - 67258 T^{7} - 57444 T^{8} - 5338 T^{9} + 22 T^{10} + 26 T^{11} + T^{12}$$
$61$ $$10893274 - 29191311 T + 16419914 T^{2} + 10617016 T^{3} - 8091392 T^{4} - 1560634 T^{5} + 801476 T^{6} + 117122 T^{7} - 22505 T^{8} - 3955 T^{9} + 20 T^{10} + 26 T^{11} + T^{12}$$
$67$ $$4538509504 + 4714883120 T - 678403520 T^{2} - 669419464 T^{3} + 75258180 T^{4} + 29459511 T^{5} - 3555172 T^{6} - 474596 T^{7} + 62307 T^{8} + 2947 T^{9} - 429 T^{10} - 6 T^{11} + T^{12}$$
$71$ $$-12017198348 - 39886445545 T - 42976926619 T^{2} - 21246049133 T^{3} - 5606812300 T^{4} - 801669175 T^{5} - 46063490 T^{6} + 3647013 T^{7} + 918997 T^{8} + 79770 T^{9} + 3737 T^{10} + 94 T^{11} + T^{12}$$
$73$ $$2219968 + 25741920 T + 64034288 T^{2} - 76860088 T^{3} - 83908922 T^{4} - 7168229 T^{5} + 3607447 T^{6} + 533877 T^{7} - 28097 T^{8} - 7860 T^{9} - 208 T^{10} + 22 T^{11} + T^{12}$$
$79$ $$-1277319040 + 3418562576 T + 1017318496 T^{2} - 2331826216 T^{3} + 163831840 T^{4} + 90986869 T^{5} - 7904508 T^{6} - 1163461 T^{7} + 109307 T^{8} + 5783 T^{9} - 581 T^{10} - 9 T^{11} + T^{12}$$
$83$ $$98860915136 + 10813065520 T - 10895951696 T^{2} - 1151271176 T^{3} + 452061900 T^{4} + 42544271 T^{5} - 9197429 T^{6} - 669374 T^{7} + 100342 T^{8} + 4386 T^{9} - 548 T^{10} - 8 T^{11} + T^{12}$$
$89$ $$-1500609440 + 3427797584 T - 829552176 T^{2} - 490376448 T^{3} + 124536598 T^{4} + 21479633 T^{5} - 5233031 T^{6} - 305681 T^{7} + 79002 T^{8} + 1663 T^{9} - 479 T^{10} - 3 T^{11} + T^{12}$$
$97$ $$107861318 - 194920563 T - 162786822 T^{2} + 64432913 T^{3} + 85302025 T^{4} + 27963948 T^{5} + 3390903 T^{6} - 118070 T^{7} - 70766 T^{8} - 5577 T^{9} + 85 T^{10} + 29 T^{11} + T^{12}$$