# Properties

 Label 1350.2.q.h Level 1350 Weight 2 Character orbit 1350.q Analytic conductor 10.780 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1350.q (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.7798042729$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: 16.0.9349208943630483456.9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 90) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + 9297 x^{8} - 11276 x^{7} + 11224 x^{6} - 9024 x^{5} + 5736 x^{4} - 2780 x^{3} + 972 x^{2} - 220 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$1548 \nu^{15} - 13188 \nu^{14} + 76652 \nu^{13} - 318997 \nu^{12} + 1019601 \nu^{11} - 2665197 \nu^{10} + 5595909 \nu^{9} - 9912750 \nu^{8} + 14273792 \nu^{7} - 17359125 \nu^{6} + 16940634 \nu^{5} - 13645786 \nu^{4} + 8382900 \nu^{3} - 3952842 \nu^{2} + 1197759 \nu - 204955$$$$)/17095$$ $$\beta_{2}$$ $$=$$ $$($$$$-3456 \nu^{15} + 25920 \nu^{14} - 151876 \nu^{13} + 594074 \nu^{12} - 1879372 \nu^{11} + 4666596 \nu^{10} - 9554736 \nu^{9} + 15945783 \nu^{8} - 21928484 \nu^{7} + 24527176 \nu^{6} - 22138664 \nu^{5} + 15739188 \nu^{4} - 8598668 \nu^{3} + 3418428 \nu^{2} - 929924 \nu + 142555$$$$)/17095$$ $$\beta_{3}$$ $$=$$ $$($$$$62 \nu^{14} - 434 \nu^{13} + 2541 \nu^{12} - 9604 \nu^{11} + 30137 \nu^{10} - 72992 \nu^{9} + 147747 \nu^{8} - 240948 \nu^{7} + 326020 \nu^{6} - 354283 \nu^{5} + 310030 \nu^{4} - 208129 \nu^{3} + 103708 \nu^{2} - 33855 \nu + 5945$$$$)/65$$ $$\beta_{4}$$ $$=$$ $$($$$$-1548 \nu^{15} + 26338 \nu^{14} - 168702 \nu^{13} + 849994 \nu^{12} - 3008933 \nu^{11} + 8792571 \nu^{10} - 20191094 \nu^{9} + 38713617 \nu^{8} - 60027902 \nu^{7} + 77090107 \nu^{6} - 79360265 \nu^{5} + 65574610 \nu^{4} - 41344164 \nu^{3} + 19292843 \nu^{2} - 5858382 \nu + 979490$$$$)/17095$$ $$\beta_{5}$$ $$=$$ $$($$$$15207 \nu^{15} - 93144 \nu^{14} + 525827 \nu^{13} - 1795712 \nu^{12} + 5254311 \nu^{11} - 11330101 \nu^{10} + 20458668 \nu^{9} - 28148715 \nu^{8} + 30952836 \nu^{7} - 24081648 \nu^{6} + 11893278 \nu^{5} - 253940 \nu^{4} - 4590360 \nu^{3} + 3846345 \nu^{2} - 1513502 \nu + 281130$$$$)/17095$$ $$\beta_{6}$$ $$=$$ $$($$$$15977 \nu^{15} - 98393 \nu^{14} + 556086 \nu^{13} - 1902441 \nu^{12} + 5567337 \nu^{11} - 11970872 \nu^{10} + 21486123 \nu^{9} - 29032660 \nu^{8} + 30783293 \nu^{7} - 21374907 \nu^{6} + 6423019 \nu^{5} + 7076841 \nu^{4} - 11188252 \nu^{3} + 8068367 \nu^{2} - 3136623 \nu + 610830$$$$)/17095$$ $$\beta_{7}$$ $$=$$ $$($$$$15977 \nu^{15} - 141262 \nu^{14} + 856169 \nu^{13} - 3642449 \nu^{12} + 12106306 \nu^{11} - 32236863 \nu^{10} + 70027507 \nu^{9} - 125576015 \nu^{8} + 185376534 \nu^{7} - 225133368 \nu^{6} + 221440773 \nu^{5} - 173497381 \nu^{4} + 104335813 \nu^{3} - 45658062 \nu^{2} + 12999216 \nu - 1883725$$$$)/17095$$ $$\beta_{8}$$ $$=$$ $$($$$$15207 \nu^{15} - 151267 \nu^{14} + 932688 \nu^{13} - 4151403 \nu^{12} + 14099264 \nu^{11} - 38692358 \nu^{10} + 85888071 \nu^{9} - 157951050 \nu^{8} + 238257063 \nu^{7} - 296346875 \nu^{6} + 298087511 \nu^{5} - 239467694 \nu^{4} + 147674805 \nu^{3} - 66643178 \nu^{2} + 19588566 \nu - 2984015$$$$)/17095$$ $$\beta_{9}$$ $$=$$ $$($$$$26630 \nu^{15} - 182630 \nu^{14} + 1056740 \nu^{13} - 3923413 \nu^{12} + 12097498 \nu^{11} - 28666706 \nu^{10} + 56590650 \nu^{9} - 89528958 \nu^{8} + 116711850 \nu^{7} - 121073088 \nu^{6} + 99711814 \nu^{5} - 61611091 \nu^{4} + 27082796 \nu^{3} - 6949514 \nu^{2} + 510382 \nu + 218885$$$$)/17095$$ $$\beta_{10}$$ $$=$$ $$($$$$49663 \nu^{15} - 358139 \nu^{14} + 2089429 \nu^{13} - 8006091 \nu^{12} + 25109764 \nu^{11} - 61357045 \nu^{10} + 124293850 \nu^{9} - 204357372 \nu^{8} + 277503791 \nu^{7} - 305524016 \nu^{6} + 271172221 \nu^{5} - 188678893 \nu^{4} + 99647214 \nu^{3} - 37587574 \nu^{2} + 9111403 \nu - 1090565$$$$)/17095$$ $$\beta_{11}$$ $$=$$ $$($$$$46358 \nu^{15} - 350841 \nu^{14} + 2068305 \nu^{13} - 8157610 \nu^{12} + 26035188 \nu^{11} - 65341929 \nu^{10} + 135478704 \nu^{9} - 229565910 \nu^{8} + 321502876 \nu^{7} - 367712403 \nu^{6} + 340522371 \nu^{5} - 249596834 \nu^{4} + 140137971 \nu^{3} - 57050598 \nu^{2} + 15149982 \nu - 2064105$$$$)/17095$$ $$\beta_{12}$$ $$=$$ $$($$$$-46358 \nu^{15} + 373985 \nu^{14} - 2230313 \nu^{13} + 9095994 \nu^{12} - 29559388 \nu^{11} + 76252747 \nu^{10} - 161588818 \nu^{9} + 281449761 \nu^{8} - 404524612 \nu^{7} + 477125400 \nu^{6} - 456031708 \nu^{5} + 346814258 \nu^{4} - 202568648 \nu^{3} + 86298565 \nu^{2} - 24026495 \nu + 3439595$$$$)/17095$$ $$\beta_{13}$$ $$=$$ $$($$$$-54184 \nu^{15} + 399016 \nu^{14} - 2335837 \nu^{13} + 9056462 \nu^{12} - 28575749 \nu^{11} + 70523459 \nu^{10} - 143951776 \nu^{9} + 239041170 \nu^{8} - 327539991 \nu^{7} + 364427829 \nu^{6} - 326718848 \nu^{5} + 229677473 \nu^{4} - 122374471 \nu^{3} + 46393946 \nu^{2} - 11232929 \nu + 1312670$$$$)/17095$$ $$\beta_{14}$$ $$=$$ $$($$$$10652 \nu^{15} - 79890 \nu^{14} + 470562 \nu^{13} - 1846988 \nu^{12} + 5882478 \nu^{11} - 14702424 \nu^{10} + 30390552 \nu^{9} - 51252471 \nu^{8} + 71445138 \nu^{7} - 81160006 \nu^{6} + 74558856 \nu^{5} - 53991081 \nu^{4} + 29816774 \nu^{3} - 11804598 \nu^{2} + 3003630 \nu - 370592$$$$)/3419$$ $$\beta_{15}$$ $$=$$ $$($$$$-54184 \nu^{15} + 413744 \nu^{14} - 2438933 \nu^{13} + 9652683 \nu^{12} - 30812827 \nu^{11} + 77432206 \nu^{10} - 160446084 \nu^{9} + 271670265 \nu^{8} - 379486173 \nu^{7} + 432289456 \nu^{6} - 397583146 \nu^{5} + 288232582 \nu^{4} - 159062971 \nu^{3} + 62844859 \nu^{2} - 15915907 \nu + 1951760$$$$)/17095$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + 2 \beta_{10} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{15} - \beta_{12} + \beta_{10} - \beta_{9} - \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{1} - 2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{15} + 7 \beta_{14} - 5 \beta_{13} + 4 \beta_{12} - 7 \beta_{11} - 7 \beta_{10} - 3 \beta_{9} - \beta_{8} + 3 \beta_{7} + 3 \beta_{6} + 6 \beta_{5} - 7 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} - 5 \beta_{1} - 5$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-10 \beta_{15} + 3 \beta_{14} + 2 \beta_{13} + 8 \beta_{12} - 4 \beta_{11} - 8 \beta_{10} + 6 \beta_{9} + 6 \beta_{6} + 6 \beta_{5} - 16 \beta_{4} - 10 \beta_{3} - 3 \beta_{2} - 14 \beta_{1} + 9$$ $$\nu^{5}$$ $$=$$ $$($$$$-21 \beta_{15} - 46 \beta_{14} + 44 \beta_{13} - 19 \beta_{12} + 44 \beta_{11} + 19 \beta_{10} + 35 \beta_{9} + 3 \beta_{8} - 35 \beta_{7} - 5 \beta_{6} - 26 \beta_{5} + 19 \beta_{4} - \beta_{3} + 25 \beta_{2} + 3 \beta_{1} + 36$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$60 \beta_{15} - 43 \beta_{14} - 5 \beta_{13} - 70 \beta_{12} + 55 \beta_{11} + 49 \beta_{10} - 30 \beta_{9} - 10 \beta_{8} - 15 \beta_{7} - 60 \beta_{6} - 40 \beta_{5} + 122 \beta_{4} + 69 \beta_{3} + 45 \beta_{2} + 93 \beta_{1} - 50$$ $$\nu^{7}$$ $$=$$ $$($$$$321 \beta_{15} + 267 \beta_{14} - 365 \beta_{13} + 17 \beta_{12} - 213 \beta_{11} - 28 \beta_{10} - 336 \beta_{9} - 60 \beta_{8} + 294 \beta_{7} - 126 \beta_{6} + 136 \beta_{5} + 84 \beta_{4} + 179 \beta_{3} - 126 \beta_{2} + 162 \beta_{1} - 307$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$-256 \beta_{15} + 446 \beta_{14} - 108 \beta_{13} + 544 \beta_{12} - 528 \beta_{11} - 304 \beta_{10} + 74 \beta_{9} + 84 \beta_{8} + 280 \beta_{7} + 420 \beta_{6} + 316 \beta_{5} - 868 \beta_{4} - 392 \beta_{3} - 469 \beta_{2} - 572 \beta_{1} + 267$$ $$\nu^{9}$$ $$=$$ $$($$$$-3026 \beta_{15} - 1225 \beta_{14} + 2701 \beta_{13} + 906 \beta_{12} + 525 \beta_{11} - 297 \beta_{10} + 2835 \beta_{9} + 795 \beta_{8} - 1899 \beta_{7} + 2007 \beta_{6} - 606 \beta_{5} - 2265 \beta_{4} - 2202 \beta_{3} + 9 \beta_{2} - 2277 \beta_{1} + 2725$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$274 \beta_{15} - 4001 \beta_{14} + 2070 \beta_{13} - 3737 \beta_{12} + 4287 \beta_{11} + 1922 \beta_{10} + 846 \beta_{9} - 300 \beta_{8} - 3285 \beta_{7} - 2250 \beta_{6} - 2609 \beta_{5} + 5530 \beta_{4} + 1694 \beta_{3} + 3870 \beta_{2} + 3067 \beta_{1} - 932$$ $$\nu^{11}$$ $$=$$ $$($$$$23673 \beta_{15} + 1926 \beta_{14} - 17346 \beta_{13} - 14155 \beta_{12} + 4640 \beta_{11} + 5251 \beta_{10} - 20779 \beta_{9} - 7509 \beta_{8} + 8569 \beta_{7} - 20801 \beta_{6} + 506 \beta_{5} + 28021 \beta_{4} + 20095 \beta_{3} + 8261 \beta_{2} + 22843 \beta_{1} - 22758$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$10280 \beta_{15} + 31947 \beta_{14} - 24436 \beta_{13} + 21968 \beta_{12} - 30552 \beta_{11} - 11620 \beta_{10} - 17265 \beta_{9} - 1650 \beta_{8} + 30690 \beta_{7} + 6798 \beta_{6} + 20444 \beta_{5} - 28982 \beta_{4} - 2254 \beta_{3} - 26565 \beta_{2} - 11868 \beta_{1} - 3138$$ $$\nu^{13}$$ $$=$$ $$($$$$-160655 \beta_{15} + 46695 \beta_{14} + 88659 \beta_{13} + 152123 \beta_{12} - 96769 \beta_{11} - 59234 \beta_{10} + 128544 \beta_{9} + 57480 \beta_{8} - 4992 \beta_{7} + 177762 \beta_{6} + 32656 \beta_{5} - 273318 \beta_{4} - 157219 \beta_{3} - 120978 \beta_{2} - 196060 \beta_{1} + 171575$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$-160979 \beta_{15} - 225841 \beta_{14} + 234065 \beta_{13} - 95851 \beta_{12} + 187408 \beta_{11} + 60319 \beta_{10} + 200859 \beta_{9} + 43732 \beta_{8} - 244699 \beta_{7} + 38129 \beta_{6} - 144445 \beta_{5} + 90048 \beta_{4} - 62947 \beta_{3} + 146874 \beta_{2} - 5512 \beta_{1} + 106657$$ $$\nu^{15}$$ $$=$$ $$($$$$914760 \beta_{15} - 806121 \beta_{14} - 233687 \beta_{13} - 1368388 \beta_{12} + 1121513 \beta_{11} + 561279 \beta_{10} - 600297 \beta_{9} - 364797 \beta_{8} - 458997 \beta_{7} - 1313487 \beta_{6} - 528672 \beta_{5} + 2298683 \beta_{4} + 1078190 \beta_{3} + 1251033 \beta_{2} + 1491905 \beta_{1} - 1132093$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1027$$ $$\chi(n)$$ $$\beta_{2}$$ $$-\beta_{14}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 0.5 − 0.410882i 0.5 − 2.00333i 0.5 + 1.00333i 0.5 − 0.589118i 0.5 + 0.410882i 0.5 + 2.00333i 0.5 − 1.00333i 0.5 + 0.589118i 0.5 + 1.33108i 0.5 − 1.74530i 0.5 − 0.331082i 0.5 + 2.74530i 0.5 − 1.33108i 0.5 + 1.74530i 0.5 + 0.331082i 0.5 − 2.74530i
−0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 0.622279 + 2.32238i −0.707107 + 0.707107i 0 0
143.2 −0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 1.00635 + 3.75574i −0.707107 + 0.707107i 0 0
143.3 0.965926 0.258819i 0 0.866025 0.500000i 0 0 −0.686453 2.56188i 0.707107 0.707107i 0 0
143.4 0.965926 0.258819i 0 0.866025 0.500000i 0 0 0.521929 + 1.94786i 0.707107 0.707107i 0 0
557.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 0.622279 2.32238i −0.707107 0.707107i 0 0
557.2 −0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 1.00635 3.75574i −0.707107 0.707107i 0 0
557.3 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 −0.686453 + 2.56188i 0.707107 + 0.707107i 0 0
557.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 0.521929 1.94786i 0.707107 + 0.707107i 0 0
1007.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 −3.75574 + 1.00635i 0.707107 + 0.707107i 0 0
1007.2 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 −2.32238 + 0.622279i 0.707107 + 0.707107i 0 0
1007.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 −1.94786 + 0.521929i −0.707107 0.707107i 0 0
1007.4 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 2.56188 0.686453i −0.707107 0.707107i 0 0
1043.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 −3.75574 1.00635i 0.707107 0.707107i 0 0
1043.2 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 −2.32238 0.622279i 0.707107 0.707107i 0 0
1043.3 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 −1.94786 0.521929i −0.707107 + 0.707107i 0 0
1043.4 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 2.56188 + 0.686453i −0.707107 + 0.707107i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1043.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.d odd 6 1 inner
45.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.q.h 16
3.b odd 2 1 450.2.p.h 16
5.b even 2 1 270.2.m.b 16
5.c odd 4 1 270.2.m.b 16
5.c odd 4 1 inner 1350.2.q.h 16
9.c even 3 1 450.2.p.h 16
9.d odd 6 1 inner 1350.2.q.h 16
15.d odd 2 1 90.2.l.b 16
15.e even 4 1 90.2.l.b 16
15.e even 4 1 450.2.p.h 16
45.h odd 6 1 270.2.m.b 16
45.h odd 6 1 810.2.f.c 16
45.j even 6 1 90.2.l.b 16
45.j even 6 1 810.2.f.c 16
45.k odd 12 1 90.2.l.b 16
45.k odd 12 1 450.2.p.h 16
45.k odd 12 1 810.2.f.c 16
45.l even 12 1 270.2.m.b 16
45.l even 12 1 810.2.f.c 16
45.l even 12 1 inner 1350.2.q.h 16
60.h even 2 1 720.2.cu.b 16
60.l odd 4 1 720.2.cu.b 16
180.p odd 6 1 720.2.cu.b 16
180.x even 12 1 720.2.cu.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.l.b 16 15.d odd 2 1
90.2.l.b 16 15.e even 4 1
90.2.l.b 16 45.j even 6 1
90.2.l.b 16 45.k odd 12 1
270.2.m.b 16 5.b even 2 1
270.2.m.b 16 5.c odd 4 1
270.2.m.b 16 45.h odd 6 1
270.2.m.b 16 45.l even 12 1
450.2.p.h 16 3.b odd 2 1
450.2.p.h 16 9.c even 3 1
450.2.p.h 16 15.e even 4 1
450.2.p.h 16 45.k odd 12 1
720.2.cu.b 16 60.h even 2 1
720.2.cu.b 16 60.l odd 4 1
720.2.cu.b 16 180.p odd 6 1
720.2.cu.b 16 180.x even 12 1
810.2.f.c 16 45.h odd 6 1
810.2.f.c 16 45.j even 6 1
810.2.f.c 16 45.k odd 12 1
810.2.f.c 16 45.l even 12 1
1350.2.q.h 16 1.a even 1 1 trivial
1350.2.q.h 16 5.c odd 4 1 inner
1350.2.q.h 16 9.d odd 6 1 inner
1350.2.q.h 16 45.l even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1350, [\chi])$$:

 $$T_{7}^{16} + \cdots$$ $$T_{11}^{8} - 22 T_{11}^{6} + 441 T_{11}^{4} - 528 T_{11}^{3} - 754 T_{11}^{2} + 1032 T_{11} + 1849$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{4} + T^{8} )^{2}$$
$3$ 
$5$ 
$7$ $$1 + 8 T + 32 T^{2} + 32 T^{3} - 168 T^{4} - 752 T^{5} - 128 T^{6} + 4696 T^{7} + 9614 T^{8} - 32960 T^{9} - 159808 T^{10} - 211120 T^{11} + 686208 T^{12} + 2864720 T^{13} + 4398240 T^{14} - 5724096 T^{15} - 29956541 T^{16} - 40068672 T^{17} + 215513760 T^{18} + 982598960 T^{19} + 1647585408 T^{20} - 3548293840 T^{21} - 18801251392 T^{22} - 27143977280 T^{23} + 55422796814 T^{24} + 189500538472 T^{25} - 36156831872 T^{26} - 1486949710736 T^{27} - 2325336249768 T^{28} + 3100448333024 T^{29} + 21703138331168 T^{30} + 37980492079544 T^{31} + 33232930569601 T^{32}$$
$11$ $$( 1 + 22 T^{2} + 199 T^{4} + 264 T^{5} + 1138 T^{6} + 6840 T^{7} + 10132 T^{8} + 75240 T^{9} + 137698 T^{10} + 351384 T^{11} + 2913559 T^{12} + 38974342 T^{14} + 214358881 T^{16} )^{2}$$
$13$ $$1 - 48 T^{3} - 208 T^{4} - 648 T^{5} + 1152 T^{6} + 768 T^{7} + 58078 T^{8} - 38784 T^{9} + 412704 T^{10} + 164616 T^{11} + 6437888 T^{12} - 1602600 T^{13} + 36461952 T^{14} - 584300928 T^{15} - 665621597 T^{16} - 7595912064 T^{17} + 6162069888 T^{18} - 3520912200 T^{19} + 183872519168 T^{20} + 61120768488 T^{21} + 1992043381536 T^{22} - 2433638483328 T^{23} + 47376008814238 T^{24} + 8144255518464 T^{25} + 158812982610048 T^{26} - 1161319935335976 T^{27} - 4846001705476048 T^{28} - 14538005116428144 T^{29} + 665416609183179841 T^{32}$$
$17$ $$1 - 356 T^{4} + 24298 T^{8} + 31798576 T^{12} - 15547912973 T^{16} + 2655848866096 T^{20} + 169496954301418 T^{24} - 207413516453794916 T^{28} + 48661191875666868481 T^{32}$$
$19$ $$( 1 - 100 T^{2} + 5038 T^{4} - 162424 T^{6} + 3656035 T^{8} - 58635064 T^{10} + 656557198 T^{12} - 4704588100 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$1 + 24 T + 288 T^{2} + 2304 T^{3} + 14956 T^{4} + 93528 T^{5} + 591552 T^{6} + 3510696 T^{7} + 18949162 T^{8} + 98431632 T^{9} + 520135776 T^{10} + 2742734952 T^{11} + 13919721136 T^{12} + 68549757432 T^{13} + 335617355232 T^{14} + 1641179221488 T^{15} + 7932015899923 T^{16} + 37747122094224 T^{17} + 177541580917728 T^{18} + 834044898675144 T^{19} + 3895308682419376 T^{20} + 17653182909160536 T^{21} + 76998762000864864 T^{22} + 335142525423339504 T^{23} + 1483927546469284522 T^{24} + 6323299443987508248 T^{25} + 24505935561456493248 T^{26} + 89114391038173764456 T^{27} +$$$$32\!\cdots\!76$$$$T^{28} +$$$$11\!\cdots\!32$$$$T^{29} +$$$$33\!\cdots\!92$$$$T^{30} +$$$$63\!\cdots\!68$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$1 - 152 T^{2} + 12036 T^{4} - 646864 T^{6} + 26386346 T^{8} - 870522600 T^{10} + 24736791952 T^{12} - 660065237288 T^{14} + 18313327590963 T^{16} - 555114864559208 T^{18} + 17495862948602512 T^{20} - 517807143937554600 T^{22} + 13199674937647830506 T^{24} -$$$$27\!\cdots\!64$$$$T^{26} +$$$$42\!\cdots\!76$$$$T^{28} -$$$$45\!\cdots\!12$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$( 1 + 4 T - 36 T^{2} - 280 T^{3} - 244 T^{4} + 4476 T^{5} + 26680 T^{6} + 13564 T^{7} - 525585 T^{8} + 420484 T^{9} + 25639480 T^{10} + 133344516 T^{11} - 225339124 T^{12} - 8016162280 T^{13} - 31950132516 T^{14} + 110050456444 T^{15} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 + 576 T^{3} + 1060 T^{4} - 8640 T^{5} + 165888 T^{6} + 634752 T^{7} - 2979546 T^{8} + 23485824 T^{9} + 227100672 T^{10} - 437641920 T^{11} + 1986610660 T^{12} + 39942119232 T^{13} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 + 12 T + 154 T^{2} + 1272 T^{3} + 9001 T^{4} + 62400 T^{5} + 437338 T^{6} + 2866716 T^{7} + 21170932 T^{8} + 117535356 T^{9} + 735165178 T^{10} + 4300670400 T^{11} + 25434674761 T^{12} + 147369087672 T^{13} + 731516053114 T^{14} + 2337051286572 T^{15} + 7984925229121 T^{16} )^{2}$$
$43$ $$1 - 96 T^{3} + 854 T^{4} - 1872 T^{5} + 4608 T^{6} - 53616 T^{7} + 4490401 T^{8} + 2573760 T^{9} + 2964096 T^{10} - 348774432 T^{11} - 9048958570 T^{12} - 6137398128 T^{13} + 6878498688 T^{14} + 451002697968 T^{15} - 494432039036 T^{16} + 19393116012624 T^{17} + 12718344074112 T^{18} - 487966112962896 T^{19} - 30936588608074570 T^{20} - 51272786206529376 T^{21} + 18737126928088704 T^{22} + 699595868522752320 T^{23} + 52484706214739808001 T^{24} - 26947005481605774288 T^{25} + 99585710499613819392 T^{26} -$$$$17\!\cdots\!04$$$$T^{27} +$$$$34\!\cdots\!54$$$$T^{28} -$$$$16\!\cdots\!28$$$$T^{29} +$$$$13\!\cdots\!01$$$$T^{32}$$
$47$ $$1 - 48 T + 1152 T^{2} - 18432 T^{3} + 221336 T^{4} - 2122632 T^{5} + 16776576 T^{6} - 110138352 T^{7} + 584847502 T^{8} - 2167705440 T^{9} - 101278944 T^{10} + 103905054792 T^{11} - 1313297330560 T^{12} + 11748806482152 T^{13} - 89054821472256 T^{14} + 622050377830560 T^{15} - 4255385068735805 T^{16} + 29236367758036320 T^{17} - 196722100632213504 T^{18} + 1219796335396467096 T^{19} - 6408472031284351360 T^{20} + 23830105518606623544 T^{21} - 1091707545669732576 T^{22} -$$$$10\!\cdots\!20$$$$T^{23} +$$$$13\!\cdots\!22$$$$T^{24} -$$$$12\!\cdots\!84$$$$T^{25} +$$$$88\!\cdots\!24$$$$T^{26} -$$$$52\!\cdots\!96$$$$T^{27} +$$$$25\!\cdots\!76$$$$T^{28} -$$$$10\!\cdots\!64$$$$T^{29} +$$$$29\!\cdots\!88$$$$T^{30} -$$$$57\!\cdots\!64$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 4192 T^{4} + 3754180 T^{8} - 35115635936 T^{12} - 139198621476794 T^{16} - 277079258155925216 T^{20} +$$$$23\!\cdots\!80$$$$T^{24} +$$$$20\!\cdots\!72$$$$T^{28} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 - 428 T^{2} + 100818 T^{4} - 16624408 T^{6} + 2120510801 T^{8} - 219902811120 T^{10} + 19093571371522 T^{12} - 1411764424766636 T^{14} + 89719050225623076 T^{16} - 4914351962612659916 T^{18} +$$$$23\!\cdots\!42$$$$T^{20} -$$$$92\!\cdots\!20$$$$T^{22} +$$$$31\!\cdots\!21$$$$T^{24} -$$$$84\!\cdots\!08$$$$T^{26} +$$$$17\!\cdots\!58$$$$T^{28} -$$$$26\!\cdots\!08$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$( 1 + 12 T + 8 T^{2} + 1176 T^{3} + 13384 T^{4} - 12108 T^{5} + 574064 T^{6} + 5555196 T^{7} - 14824577 T^{8} + 338866956 T^{9} + 2136092144 T^{10} - 2748285948 T^{11} + 185312735944 T^{12} + 993245249976 T^{13} + 412162994888 T^{14} + 37712914032252 T^{15} + 191707312997281 T^{16} )^{2}$$
$67$ $$1 - 16 T + 128 T^{2} + 128 T^{3} - 5562 T^{4} + 6640 T^{5} + 613888 T^{6} - 5277488 T^{7} + 23820881 T^{8} - 40207664 T^{9} + 355221248 T^{10} - 2724412528 T^{11} + 197658905958 T^{12} - 3038720900128 T^{13} + 23981818128000 T^{14} + 20130978078192 T^{15} - 1000499923621916 T^{16} + 1348775531238864 T^{17} + 107654381576592000 T^{18} - 913934814085197664 T^{19} + 3983048530687278918 T^{20} - 3678297755878140496 T^{21} + 32132739406133126912 T^{22} -$$$$24\!\cdots\!72$$$$T^{23} +$$$$96\!\cdots\!21$$$$T^{24} -$$$$14\!\cdots\!36$$$$T^{25} +$$$$11\!\cdots\!12$$$$T^{26} +$$$$81\!\cdots\!20$$$$T^{27} -$$$$45\!\cdots\!82$$$$T^{28} +$$$$70\!\cdots\!36$$$$T^{29} +$$$$47\!\cdots\!12$$$$T^{30} -$$$$39\!\cdots\!88$$$$T^{31} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$( 1 - 296 T^{2} + 51256 T^{4} - 5830568 T^{6} + 486270130 T^{8} - 29391893288 T^{10} + 1302501121336 T^{12} - 37917684040616 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 + 8 T + 32 T^{2} + 736 T^{3} + 2366 T^{4} - 45704 T^{5} - 170496 T^{6} - 3682392 T^{7} - 78261341 T^{8} - 268814616 T^{9} - 908573184 T^{10} - 17779632968 T^{11} + 67190238206 T^{12} + 1525780692448 T^{13} + 4842695241248 T^{14} + 88379188152776 T^{15} + 806460091894081 T^{16} )^{2}$$
$79$ $$1 + 200 T^{2} + 7464 T^{4} - 683600 T^{6} + 16634510 T^{8} + 10268111400 T^{10} + 468563441536 T^{12} + 2801113391000 T^{14} + 270299704808259 T^{16} + 17481748673231000 T^{18} + 18250584001465964416 T^{20} +$$$$24\!\cdots\!00$$$$T^{22} +$$$$25\!\cdots\!10$$$$T^{24} -$$$$64\!\cdots\!00$$$$T^{26} +$$$$44\!\cdots\!24$$$$T^{28} +$$$$73\!\cdots\!00$$$$T^{30} +$$$$23\!\cdots\!21$$$$T^{32}$$
$83$ $$1 - 48 T + 1152 T^{2} - 18432 T^{3} + 215516 T^{4} - 1906416 T^{5} + 13102848 T^{6} - 74216976 T^{7} + 433250986 T^{8} - 3712707360 T^{9} + 40409712000 T^{10} - 415323735120 T^{11} + 3731885962352 T^{12} - 29403431286576 T^{13} + 213762329771904 T^{14} - 1567513821393504 T^{15} + 12987012407281267 T^{16} - 130103647175660832 T^{17} + 1472608689798646656 T^{18} - 16812499765057431312 T^{19} +$$$$17\!\cdots\!92$$$$T^{20} -$$$$16\!\cdots\!60$$$$T^{21} +$$$$13\!\cdots\!00$$$$T^{22} -$$$$10\!\cdots\!20$$$$T^{23} +$$$$97\!\cdots\!26$$$$T^{24} -$$$$13\!\cdots\!28$$$$T^{25} +$$$$20\!\cdots\!52$$$$T^{26} -$$$$24\!\cdots\!72$$$$T^{27} +$$$$23\!\cdots\!76$$$$T^{28} -$$$$16\!\cdots\!16$$$$T^{29} +$$$$84\!\cdots\!08$$$$T^{30} -$$$$29\!\cdots\!36$$$$T^{31} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$( 1 + 328 T^{2} + 42642 T^{4} + 2598088 T^{6} + 62742241 T^{8} )^{4}$$
$97$ $$1 - 48 T + 1152 T^{2} - 15456 T^{3} + 79442 T^{4} + 1258944 T^{5} - 32502528 T^{6} + 320382528 T^{7} - 602207375 T^{8} - 26997703632 T^{9} + 412336144512 T^{10} - 2856970349424 T^{11} + 3870433274354 T^{12} + 127396070447952 T^{13} - 1412601730375680 T^{14} + 7780535897090304 T^{15} - 41112146089827164 T^{16} + 754711982017759488 T^{17} - 13291169681104773120 T^{18} +$$$$11\!\cdots\!96$$$$T^{19} +$$$$34\!\cdots\!74$$$$T^{20} -$$$$24\!\cdots\!68$$$$T^{21} +$$$$34\!\cdots\!48$$$$T^{22} -$$$$21\!\cdots\!16$$$$T^{23} -$$$$47\!\cdots\!75$$$$T^{24} +$$$$24\!\cdots\!76$$$$T^{25} -$$$$23\!\cdots\!72$$$$T^{26} +$$$$90\!\cdots\!32$$$$T^{27} +$$$$55\!\cdots\!22$$$$T^{28} -$$$$10\!\cdots\!12$$$$T^{29} +$$$$75\!\cdots\!88$$$$T^{30} -$$$$30\!\cdots\!64$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$