# Properties

 Label 99.2.j.a Level 99 Weight 2 Character orbit 99.j Analytic conductor 0.791 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 99.j (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.790518980011$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{2} + ( -\beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} ) q^{4} + ( \beta_{1} - \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{5} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{9} ) q^{7} + ( 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{8} +O(q^{10})$$ $$q + ( \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{2} + ( -\beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} ) q^{4} + ( \beta_{1} - \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{5} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{9} ) q^{7} + ( 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{8} + ( -2 + 4 \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{9} ) q^{10} + ( \beta_{1} - \beta_{8} + 2 \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{11} + ( -\beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{9} ) q^{13} + ( -2 \beta_{1} + \beta_{8} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{14} + ( -1 - 2 \beta_{3} + \beta_{5} ) q^{16} + ( -2 \beta_{1} - \beta_{10} - \beta_{11} - 2 \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{17} + ( -2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{19} + ( \beta_{1} - 4 \beta_{8} - \beta_{11} - \beta_{12} + \beta_{13} + 5 \beta_{14} - \beta_{15} ) q^{20} + ( -2 + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{22} + ( -\beta_{1} + \beta_{11} + \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{23} + ( -4 \beta_{3} - 4 \beta_{4} + \beta_{6} + \beta_{7} ) q^{25} + ( \beta_{1} + \beta_{8} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 5 \beta_{14} + 4 \beta_{15} ) q^{26} + ( -1 - 2 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} + 5 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} ) q^{28} + ( -2 \beta_{8} - \beta_{10} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{29} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{31} + ( -\beta_{1} - \beta_{8} + 2 \beta_{10} - \beta_{11} + \beta_{14} ) q^{32} + ( 3 - 2 \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{34} + ( \beta_{8} + \beta_{10} + 3 \beta_{11} - 3 \beta_{12} + 3 \beta_{14} - 2 \beta_{15} ) q^{35} + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{37} + ( -2 \beta_{1} + 4 \beta_{8} - 5 \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} + 5 \beta_{15} ) q^{38} + ( 7 - 3 \beta_{2} - \beta_{3} - 4 \beta_{4} - 8 \beta_{5} - 3 \beta_{6} ) q^{40} + ( 2 \beta_{8} + 4 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{41} + ( -1 + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{9} ) q^{43} + ( \beta_{1} + \beta_{8} - \beta_{10} - 4 \beta_{12} - 5 \beta_{13} - 3 \beta_{14} - 4 \beta_{15} ) q^{44} + ( -2 + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 3 \beta_{9} ) q^{46} + ( 6 \beta_{1} + 2 \beta_{8} + \beta_{10} + 3 \beta_{12} + 3 \beta_{14} ) q^{47} + ( -2 + 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{7} ) q^{49} + ( 2 \beta_{1} - \beta_{8} + 7 \beta_{10} + 5 \beta_{11} + 2 \beta_{12} - 4 \beta_{15} ) q^{50} + ( -4 + 3 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{52} + ( -\beta_{1} - 3 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{53} + ( 3 - 2 \beta_{2} - 3 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 3 \beta_{9} ) q^{55} + ( 3 \beta_{1} - \beta_{8} - 3 \beta_{11} + 4 \beta_{12} - \beta_{13} - 2 \beta_{14} + 4 \beta_{15} ) q^{56} + ( -\beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{9} ) q^{58} + ( -\beta_{1} + \beta_{8} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{15} ) q^{59} + ( -2 + 3 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{9} ) q^{61} + ( 3 \beta_{1} - 2 \beta_{8} + 2 \beta_{10} - \beta_{12} - \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{62} + ( 7 - 6 \beta_{3} - 7 \beta_{4} - 6 \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{9} ) q^{64} + ( -4 \beta_{1} + 3 \beta_{8} - 6 \beta_{10} - 4 \beta_{11} - \beta_{13} - 7 \beta_{14} + 4 \beta_{15} ) q^{65} + ( 4 - 2 \beta_{2} + 4 \beta_{4} + 4 \beta_{5} + \beta_{7} - \beta_{9} ) q^{67} + ( -2 \beta_{1} + 4 \beta_{8} - 4 \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 6 \beta_{15} ) q^{68} + ( 6 - 2 \beta_{2} - 6 \beta_{3} + \beta_{4} + 3 \beta_{6} - \beta_{7} - \beta_{9} ) q^{70} + ( -\beta_{1} + 4 \beta_{8} - 3 \beta_{10} + \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 4 \beta_{15} ) q^{71} + ( -2 - \beta_{2} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 4 \beta_{7} - 4 \beta_{9} ) q^{73} + ( -5 \beta_{8} + 4 \beta_{10} + \beta_{11} - 6 \beta_{12} + \beta_{13} + 9 \beta_{14} - 8 \beta_{15} ) q^{74} + ( -1 + 2 \beta_{3} - 5 \beta_{4} + 7 \beta_{5} + \beta_{7} + 2 \beta_{9} ) q^{76} + ( -\beta_{1} + \beta_{8} - 4 \beta_{10} - 5 \beta_{11} + 3 \beta_{12} - \beta_{13} - 7 \beta_{14} + 5 \beta_{15} ) q^{77} + ( -3 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - 3 \beta_{9} ) q^{79} + ( -6 \beta_{1} + 3 \beta_{8} - 6 \beta_{10} - 4 \beta_{11} + \beta_{12} + 4 \beta_{13} - 3 \beta_{14} + 4 \beta_{15} ) q^{80} + ( 1 - 6 \beta_{3} - \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + 2 \beta_{9} ) q^{82} + ( \beta_{1} - 3 \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{14} + 4 \beta_{15} ) q^{83} + ( -10 + 2 \beta_{3} + 8 \beta_{4} + 5 \beta_{5} ) q^{85} + ( -2 \beta_{1} + \beta_{8} - 5 \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} ) q^{86} + ( -4 - 2 \beta_{2} + 5 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} - 6 \beta_{9} ) q^{88} + ( -3 \beta_{1} - 3 \beta_{8} + 3 \beta_{11} - 4 \beta_{12} + \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{89} + ( \beta_{2} - 6 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{9} ) q^{91} + ( 2 \beta_{1} - 5 \beta_{8} + 9 \beta_{10} + 10 \beta_{11} - 3 \beta_{12} + 3 \beta_{13} + 6 \beta_{14} - 12 \beta_{15} ) q^{92} + ( -1 + \beta_{2} - 3 \beta_{3} - 6 \beta_{4} - 7 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{94} + ( -5 \beta_{8} - 3 \beta_{12} - 3 \beta_{13} + \beta_{14} - 5 \beta_{15} ) q^{95} + ( 3 - 4 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - \beta_{6} + 6 \beta_{7} - 2 \beta_{9} ) q^{97} + ( 6 \beta_{1} - 8 \beta_{8} + 16 \beta_{10} + 6 \beta_{11} + 2 \beta_{13} + 12 \beta_{14} - 12 \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 4q^{4} + O(q^{10})$$ $$16q - 4q^{4} - 20q^{16} - 48q^{22} - 32q^{25} + 40q^{28} + 16q^{31} + 40q^{34} - 12q^{37} + 60q^{40} - 40q^{46} - 24q^{49} - 40q^{52} + 16q^{55} + 12q^{58} + 36q^{64} + 96q^{67} + 76q^{70} - 20q^{73} - 12q^{82} - 100q^{85} - 12q^{88} - 72q^{91} - 80q^{94} + 60q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$155 \nu^{14} - 1408 \nu^{12} - 4611 \nu^{10} + 9188 \nu^{8} + 65146 \nu^{6} - 26953 \nu^{4} + 145025 \nu^{2} + 108500$$$$)/176275$$ $$\beta_{3}$$ $$=$$ $$($$$$5996 \nu^{14} + 4807 \nu^{12} - 95306 \nu^{10} - 277252 \nu^{8} + 500966 \nu^{6} + 1108500 \nu^{4} + 1748000 \nu^{2} + 2787000$$$$)/881375$$ $$\beta_{4}$$ $$=$$ $$($$$$8474 \nu^{14} + 2743 \nu^{12} - 149594 \nu^{10} - 389798 \nu^{8} + 955934 \nu^{6} + 2009635 \nu^{4} + 3037000 \nu^{2} + 3175500$$$$)/881375$$ $$\beta_{5}$$ $$=$$ $$($$$$-8672 \nu^{14} - 3649 \nu^{12} + 153342 \nu^{10} + 402539 \nu^{8} - 938462 \nu^{6} - 1990750 \nu^{4} - 3034300 \nu^{2} - 3730750$$$$)/881375$$ $$\beta_{6}$$ $$=$$ $$($$$$-12916 \nu^{14} - 10637 \nu^{12} + 225196 \nu^{10} + 690882 \nu^{8} - 1135131 \nu^{6} - 3696115 \nu^{4} - 6351875 \nu^{2} - 6749625$$$$)/881375$$ $$\beta_{7}$$ $$=$$ $$($$$$-14403 \nu^{14} - 14916 \nu^{12} + 224078 \nu^{10} + 748351 \nu^{8} - 1023533 \nu^{6} - 3087765 \nu^{4} - 6177175 \nu^{2} - 6732875$$$$)/881375$$ $$\beta_{8}$$ $$=$$ $$($$$$-1496 \nu^{15} - 2572 \nu^{13} + 24401 \nu^{11} + 91992 \nu^{9} - 100886 \nu^{7} - 478015 \nu^{5} - 412275 \nu^{3} - 598500 \nu$$$$)/400625$$ $$\beta_{9}$$ $$=$$ $$($$$$-22098 \nu^{14} - 13706 \nu^{12} + 377023 \nu^{10} + 1116041 \nu^{8} - 1983428 \nu^{6} - 5540615 \nu^{4} - 10624800 \nu^{2} - 11238000$$$$)/881375$$ $$\beta_{10}$$ $$=$$ $$($$$$22194 \nu^{15} + 11523 \nu^{13} - 339409 \nu^{11} - 1024903 \nu^{9} + 1669899 \nu^{7} + 3769000 \nu^{5} + 11127550 \nu^{3} + 9814875 \nu$$$$)/4406875$$ $$\beta_{11}$$ $$=$$ $$($$$$25903 \nu^{15} - 3944 \nu^{13} - 422373 \nu^{11} - 826866 \nu^{9} + 3248903 \nu^{7} + 2357930 \nu^{5} + 4320250 \nu^{3} + 4286500 \nu$$$$)/4406875$$ $$\beta_{12}$$ $$=$$ $$($$$$8474 \nu^{15} + 2743 \nu^{13} - 149594 \nu^{11} - 389798 \nu^{9} + 955934 \nu^{7} + 2009635 \nu^{5} + 3037000 \nu^{3} + 2294125 \nu$$$$)/881375$$ $$\beta_{13}$$ $$=$$ $$($$$$-8672 \nu^{15} - 3649 \nu^{13} + 153342 \nu^{11} + 402539 \nu^{9} - 938462 \nu^{7} - 1990750 \nu^{5} - 3034300 \nu^{3} - 3730750 \nu$$$$)/881375$$ $$\beta_{14}$$ $$=$$ $$($$$$-50508 \nu^{15} - 41726 \nu^{13} + 817683 \nu^{11} + 2587261 \nu^{9} - 4065438 \nu^{7} - 11753915 \nu^{5} - 22984550 \nu^{3} - 22711875 \nu$$$$)/4406875$$ $$\beta_{15}$$ $$=$$ $$($$$$-11661 \nu^{15} - 8721 \nu^{13} + 202398 \nu^{11} + 611766 \nu^{9} - 1064278 \nu^{7} - 3191899 \nu^{5} - 5385345 \nu^{3} - 5486525 \nu$$$$)/881375$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} - \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{15} - 2 \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - 3 \beta_{10} + 2 \beta_{8} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{7} - 2 \beta_{6} + 8 \beta_{5} + 5 \beta_{4} + 5 \beta_{3}$$ $$\nu^{5}$$ $$=$$ $$-13 \beta_{15} + 13 \beta_{14} + 12 \beta_{13} + 7 \beta_{11} + 7 \beta_{10} - 5 \beta_{8} + 7 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$11 \beta_{9} - 11 \beta_{7} - 6 \beta_{6} + 11 \beta_{4} - 15 \beta_{3} + 5 \beta_{2} + 15$$ $$\nu^{7}$$ $$=$$ $$20 \beta_{15} - 43 \beta_{14} + 9 \beta_{12} - 4 \beta_{11} - 43 \beta_{10} + 20 \beta_{8} + 9 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$32 \beta_{9} - 52 \beta_{6} + 53 \beta_{5} + 18 \beta_{4} + 53 \beta_{3} + 32 \beta_{2} - 18$$ $$\nu^{9}$$ $$=$$ $$-209 \beta_{15} + 137 \beta_{14} + 137 \beta_{13} - 87 \beta_{12} + 137 \beta_{11} + 21 \beta_{10} - 21 \beta_{8}$$ $$\nu^{10}$$ $$=$$ $$21 \beta_{9} - 36 \beta_{7} + 275 \beta_{5} + 275 \beta_{4} + 166$$ $$\nu^{11}$$ $$=$$ $$78 \beta_{15} - 129 \beta_{14} + 239 \beta_{13} + 239 \beta_{12} - 78 \beta_{10} + 384 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$752 \beta_{9} - 462 \beta_{7} - 752 \beta_{6} - 110 \beta_{5} - 177 \beta_{3} + 462 \beta_{2} + 110$$ $$\nu^{13}$$ $$=$$ $$-1037 \beta_{15} - 639 \beta_{14} + 642 \beta_{13} - 1037 \beta_{12} + 1037 \beta_{11} - 1681 \beta_{10} + 639 \beta_{8} - 642 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$639 \beta_{7} - 639 \beta_{6} + 5431 \beta_{5} + 3360 \beta_{4} + 3360 \beta_{3} + 400 \beta_{2}$$ $$\nu^{15}$$ $$=$$ $$-6716 \beta_{15} + 6716 \beta_{14} + 7109 \beta_{13} + 4399 \beta_{11} + 4399 \beta_{10} - 2960 \beta_{8} + 4399 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$56$$ $$\chi(n)$$ $$\beta_{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1
 −0.0783900 − 1.17295i −0.752864 + 0.902863i 0.752864 − 0.902863i 0.0783900 + 1.17295i 0.556839 − 1.81878i 1.90184 + 0.0324487i −1.90184 − 0.0324487i −0.556839 + 1.81878i 0.556839 + 1.81878i 1.90184 − 0.0324487i −1.90184 + 0.0324487i −0.556839 − 1.81878i −0.0783900 + 1.17295i −0.752864 − 0.902863i 0.752864 + 0.902863i 0.0783900 − 1.17295i
−1.97102 + 1.43203i 0 1.21618 3.74302i 2.23109 3.07083i 0 −0.349790 0.113654i 1.45728 + 4.48505i 0 9.24768i
8.2 −0.205228 + 0.149107i 0 −0.598148 + 1.84091i −1.71735 + 2.36373i 0 2.58586 + 0.840196i −0.308515 0.949513i 0 0.741170i
8.3 0.205228 0.149107i 0 −0.598148 + 1.84091i 1.71735 2.36373i 0 2.58586 + 0.840196i 0.308515 + 0.949513i 0 0.741170i
8.4 1.97102 1.43203i 0 1.21618 3.74302i −2.23109 + 3.07083i 0 −0.349790 0.113654i −1.45728 4.48505i 0 9.24768i
17.1 −0.726437 2.23574i 0 −2.85280 + 2.07268i −2.13811 0.694712i 0 −2.38116 3.27739i 2.90269 + 2.10893i 0 5.28492i
17.2 −0.212694 0.654604i 0 1.23477 0.897110i 0.0381457 + 0.0123943i 0 0.145094 + 0.199704i −1.96356 1.42661i 0 0.0276065i
17.3 0.212694 + 0.654604i 0 1.23477 0.897110i −0.0381457 0.0123943i 0 0.145094 + 0.199704i 1.96356 + 1.42661i 0 0.0276065i
17.4 0.726437 + 2.23574i 0 −2.85280 + 2.07268i 2.13811 + 0.694712i 0 −2.38116 3.27739i −2.90269 2.10893i 0 5.28492i
35.1 −0.726437 + 2.23574i 0 −2.85280 2.07268i −2.13811 + 0.694712i 0 −2.38116 + 3.27739i 2.90269 2.10893i 0 5.28492i
35.2 −0.212694 + 0.654604i 0 1.23477 + 0.897110i 0.0381457 0.0123943i 0 0.145094 0.199704i −1.96356 + 1.42661i 0 0.0276065i
35.3 0.212694 0.654604i 0 1.23477 + 0.897110i −0.0381457 + 0.0123943i 0 0.145094 0.199704i 1.96356 1.42661i 0 0.0276065i
35.4 0.726437 2.23574i 0 −2.85280 2.07268i 2.13811 0.694712i 0 −2.38116 + 3.27739i −2.90269 + 2.10893i 0 5.28492i
62.1 −1.97102 1.43203i 0 1.21618 + 3.74302i 2.23109 + 3.07083i 0 −0.349790 + 0.113654i 1.45728 4.48505i 0 9.24768i
62.2 −0.205228 0.149107i 0 −0.598148 1.84091i −1.71735 2.36373i 0 2.58586 0.840196i −0.308515 + 0.949513i 0 0.741170i
62.3 0.205228 + 0.149107i 0 −0.598148 1.84091i 1.71735 + 2.36373i 0 2.58586 0.840196i 0.308515 0.949513i 0 0.741170i
62.4 1.97102 + 1.43203i 0 1.21618 + 3.74302i −2.23109 3.07083i 0 −0.349790 + 0.113654i −1.45728 + 4.48505i 0 9.24768i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 62.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.j.a 16
3.b odd 2 1 inner 99.2.j.a 16
4.b odd 2 1 1584.2.cd.c 16
9.c even 3 2 891.2.u.c 32
9.d odd 6 2 891.2.u.c 32
11.c even 5 1 1089.2.d.g 16
11.d odd 10 1 inner 99.2.j.a 16
11.d odd 10 1 1089.2.d.g 16
12.b even 2 1 1584.2.cd.c 16
33.f even 10 1 inner 99.2.j.a 16
33.f even 10 1 1089.2.d.g 16
33.h odd 10 1 1089.2.d.g 16
44.g even 10 1 1584.2.cd.c 16
99.o odd 30 2 891.2.u.c 32
99.p even 30 2 891.2.u.c 32
132.n odd 10 1 1584.2.cd.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.j.a 16 1.a even 1 1 trivial
99.2.j.a 16 3.b odd 2 1 inner
99.2.j.a 16 11.d odd 10 1 inner
99.2.j.a 16 33.f even 10 1 inner
891.2.u.c 32 9.c even 3 2
891.2.u.c 32 9.d odd 6 2
891.2.u.c 32 99.o odd 30 2
891.2.u.c 32 99.p even 30 2
1089.2.d.g 16 11.c even 5 1
1089.2.d.g 16 11.d odd 10 1
1089.2.d.g 16 33.f even 10 1
1089.2.d.g 16 33.h odd 10 1
1584.2.cd.c 16 4.b odd 2 1
1584.2.cd.c 16 12.b even 2 1
1584.2.cd.c 16 44.g even 10 1
1584.2.cd.c 16 132.n odd 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(99, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 5 T^{4} - 20 T^{6} + 30 T^{8} - 106 T^{10} + 212 T^{12} - 350 T^{14} + 1105 T^{16} - 1400 T^{18} + 3392 T^{20} - 6784 T^{22} + 7680 T^{24} - 20480 T^{26} + 20480 T^{28} - 32768 T^{30} + 65536 T^{32}$$
$3$ 1
$5$ $$1 + 26 T^{2} + 267 T^{4} + 1158 T^{6} - 345 T^{8} - 23178 T^{10} - 84123 T^{12} - 137046 T^{14} - 351264 T^{16} - 3426150 T^{18} - 52576875 T^{20} - 362156250 T^{22} - 134765625 T^{24} + 11308593750 T^{26} + 65185546875 T^{28} + 158691406250 T^{30} + 152587890625 T^{32}$$
$7$ $$( 1 + 13 T^{2} + 20 T^{3} + 80 T^{4} + 260 T^{5} + 783 T^{6} + 2170 T^{7} + 6819 T^{8} + 15190 T^{9} + 38367 T^{10} + 89180 T^{11} + 192080 T^{12} + 336140 T^{13} + 1529437 T^{14} + 5764801 T^{16} )^{2}$$
$11$ $$1 - 60 T^{2} + 1779 T^{4} - 33610 T^{6} + 439341 T^{8} - 4066810 T^{10} + 26046339 T^{12} - 106293660 T^{14} + 214358881 T^{16}$$
$13$ $$( 1 + 7 T^{2} + 80 T^{3} + 170 T^{4} + 560 T^{5} + 4587 T^{6} + 13570 T^{7} + 40179 T^{8} + 176410 T^{9} + 775203 T^{10} + 1230320 T^{11} + 4855370 T^{12} + 29703440 T^{13} + 33787663 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$1 - 58 T^{2} + 1230 T^{4} + 3270 T^{6} - 669405 T^{8} + 13275246 T^{10} - 49036968 T^{12} - 2836388280 T^{14} + 74928517905 T^{16} - 819716212920 T^{18} - 4095616604328 T^{20} + 320432166316974 T^{22} - 4669606909792605 T^{24} + 6592300054468230 T^{26} + 716625351792606030 T^{28} - 9765913940445253882 T^{30} + 48661191875666868481 T^{32}$$
$19$ $$( 1 + 34 T^{2} + 20 T^{3} + 395 T^{4} + 680 T^{5} + 1356 T^{6} + 48280 T^{7} - 47451 T^{8} + 917320 T^{9} + 489516 T^{10} + 4664120 T^{11} + 51476795 T^{12} + 49521980 T^{13} + 1599559954 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 102 T^{2} + 5430 T^{4} - 195472 T^{6} + 5177079 T^{8} - 103404688 T^{10} + 1519536630 T^{12} - 15099660678 T^{14} + 78310985281 T^{16} )^{2}$$
$29$ $$1 + 20 T^{2} + 338 T^{4} + 28080 T^{6} - 44437 T^{8} - 6939980 T^{10} + 33962216 T^{12} - 13624904800 T^{14} - 492015847995 T^{16} - 11458544936800 T^{18} + 24020830094696 T^{20} - 4128061951273580 T^{22} - 22229449852747957 T^{24} + 11813459111069644080 T^{26} +$$$$11\!\cdots\!58$$$$T^{28} +$$$$59\!\cdots\!20$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$( 1 - 8 T - 24 T^{2} + 88 T^{3} + 1681 T^{4} - 7064 T^{5} + 54924 T^{6} - 168376 T^{7} - 1019363 T^{8} - 5219656 T^{9} + 52781964 T^{10} - 210443624 T^{11} + 1552438801 T^{12} + 2519365288 T^{13} - 21300088344 T^{14} - 220100912888 T^{15} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 + 6 T - 7 T^{2} - 118 T^{3} - 254 T^{4} + 3396 T^{5} + 36665 T^{6} - 93294 T^{7} - 1522501 T^{8} - 3451878 T^{9} + 50194385 T^{10} + 172017588 T^{11} - 476036894 T^{12} - 8182586926 T^{13} - 17960084863 T^{14} + 569591262798 T^{15} + 3512479453921 T^{16} )^{2}$$
$41$ $$1 - 138 T^{2} + 4581 T^{4} + 242158 T^{6} - 19370244 T^{8} + 47020276 T^{10} + 27362707279 T^{12} - 302286948126 T^{14} - 26002484153453 T^{16} - 508144359799806 T^{18} + 77320471083414319 T^{20} + 223351212440590516 T^{22} -$$$$15\!\cdots\!24$$$$T^{24} +$$$$32\!\cdots\!58$$$$T^{26} +$$$$10\!\cdots\!61$$$$T^{28} -$$$$52\!\cdots\!18$$$$T^{30} +$$$$63\!\cdots\!41$$$$T^{32}$$
$43$ $$( 1 - 180 T^{2} + 12786 T^{4} - 463360 T^{6} + 14361051 T^{8} - 856752640 T^{10} + 43712789586 T^{12} - 1137845348820 T^{14} + 11688200277601 T^{16} )^{2}$$
$47$ $$1 + 192 T^{2} + 14115 T^{4} + 332470 T^{6} - 12112005 T^{8} - 594955574 T^{10} + 21788151037 T^{12} + 2123156175840 T^{14} + 93108257741080 T^{16} + 4690051992430560 T^{18} + 106319226640379197 T^{20} - 6413154243334793846 T^{22} -$$$$28\!\cdots\!05$$$$T^{24} +$$$$17\!\cdots\!30$$$$T^{26} +$$$$16\!\cdots\!15$$$$T^{28} +$$$$49\!\cdots\!48$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 166 T^{2} + 9174 T^{4} - 111674 T^{6} - 30841109 T^{8} - 525703922 T^{10} + 67446933816 T^{12} + 3591171944168 T^{14} + 110175435537457 T^{16} + 10087601991167912 T^{18} + 532188749783405496 T^{20} - 11651891574139647938 T^{22} -$$$$19\!\cdots\!49$$$$T^{24} -$$$$19\!\cdots\!26$$$$T^{26} +$$$$45\!\cdots\!34$$$$T^{28} +$$$$22\!\cdots\!54$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 + 170 T^{2} + 22518 T^{4} + 2498730 T^{6} + 233218683 T^{8} + 18919071570 T^{10} + 1419551700816 T^{12} + 95367637994400 T^{14} + 5833449282172425 T^{16} + 331974747858506400 T^{18} + 17201220416951466576 T^{20} +$$$$79\!\cdots\!70$$$$T^{22} +$$$$34\!\cdots\!43$$$$T^{24} +$$$$12\!\cdots\!30$$$$T^{26} +$$$$40\!\cdots\!58$$$$T^{28} +$$$$10\!\cdots\!70$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$( 1 + 83 T^{2} + 420 T^{3} + 3413 T^{4} + 34860 T^{5} + 227941 T^{6} + 4235840 T^{7} + 8009300 T^{8} + 258386240 T^{9} + 848168461 T^{10} + 7912557660 T^{11} + 47255855333 T^{12} + 354730446420 T^{13} + 4276191071963 T^{14} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 - 24 T + 429 T^{2} - 5008 T^{3} + 47859 T^{4} - 335536 T^{5} + 1925781 T^{6} - 7218312 T^{7} + 20151121 T^{8} )^{4}$$
$71$ $$1 + 338 T^{2} + 50091 T^{4} + 3709542 T^{6} + 63034791 T^{8} - 13931687706 T^{10} - 1397230758891 T^{12} - 60140798780574 T^{14} - 2187888491323968 T^{16} - 303169766652873534 T^{18} - 35505982328326005771 T^{20} -$$$$17\!\cdots\!26$$$$T^{22} +$$$$40\!\cdots\!51$$$$T^{24} +$$$$12\!\cdots\!42$$$$T^{26} +$$$$82\!\cdots\!31$$$$T^{28} +$$$$27\!\cdots\!78$$$$T^{30} +$$$$41\!\cdots\!21$$$$T^{32}$$
$73$ $$( 1 + 10 T + 276 T^{2} + 2170 T^{3} + 24127 T^{4} + 117430 T^{5} - 172242 T^{6} - 4912540 T^{7} - 124201175 T^{8} - 358615420 T^{9} - 917877618 T^{10} + 45682266310 T^{11} + 685164360607 T^{12} + 4498565356810 T^{13} + 41768246455764 T^{14} + 110473985190970 T^{15} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 + 203 T^{2} - 180 T^{3} + 12338 T^{4} - 36540 T^{5} - 807689 T^{6} - 4199410 T^{7} - 150806725 T^{8} - 331753390 T^{9} - 5040787049 T^{10} - 18015645060 T^{11} + 480566099378 T^{12} - 553870151820 T^{13} + 49346753470763 T^{14} + 1517108809906561 T^{16} )^{2}$$
$83$ $$1 - 208 T^{2} + 15045 T^{4} + 774660 T^{6} - 236047260 T^{8} + 17546406456 T^{10} - 74692835973 T^{12} - 112052982977490 T^{14} + 12843285776364315 T^{16} - 771932999731928610 T^{18} - 3544796586006981333 T^{20} +$$$$57\!\cdots\!64$$$$T^{22} -$$$$53\!\cdots\!60$$$$T^{24} +$$$$12\!\cdots\!40$$$$T^{26} +$$$$16\!\cdots\!45$$$$T^{28} -$$$$15\!\cdots\!32$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$( 1 - 582 T^{2} + 157158 T^{4} - 25689904 T^{6} + 2778331335 T^{8} - 203489729584 T^{10} + 9860445111078 T^{12} - 289243111339302 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 - 30 T + 411 T^{2} - 3150 T^{3} + 11217 T^{4} + 58230 T^{5} - 1794727 T^{6} + 27808950 T^{7} - 324085500 T^{8} + 2697468150 T^{9} - 16886586343 T^{10} + 53144948790 T^{11} + 993032944977 T^{12} - 27050121809550 T^{13} + 342351494025819 T^{14} - 2423948534343390 T^{15} + 7837433594376961 T^{16} )^{2}$$