# Properties

 Label 1089.3.b.i Level $1089$ Weight $3$ Character orbit 1089.b Analytic conductor $29.673$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1089.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.6731007888$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 48 x^{14} + 921 x^{12} + 8986 x^{10} + 46812 x^{8} + 125072 x^{6} + 152129 x^{4} + 65614 x^{2} + 5041$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 99) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -2 + \beta_{2} ) q^{4} + \beta_{13} q^{5} + ( -1 + \beta_{3} - \beta_{5} + \beta_{9} ) q^{7} + ( -2 \beta_{1} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -2 + \beta_{2} ) q^{4} + \beta_{13} q^{5} + ( -1 + \beta_{3} - \beta_{5} + \beta_{9} ) q^{7} + ( -2 \beta_{1} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{8} + ( 1 + \beta_{3} + \beta_{6} - \beta_{9} ) q^{10} + ( 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{8} ) q^{13} + ( 3 \beta_{1} + \beta_{4} - 2 \beta_{10} - \beta_{11} + 5 \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{14} + ( -1 - \beta_{2} + 5 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{16} + ( 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{17} + ( -3 + 2 \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{7} - \beta_{8} ) q^{19} + ( 2 \beta_{4} + 2 \beta_{10} + 2 \beta_{12} - \beta_{14} - \beta_{15} ) q^{20} + ( -5 \beta_{1} - \beta_{4} + 2 \beta_{10} - 2 \beta_{11} - 4 \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{23} + ( -7 - 2 \beta_{2} + 10 \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{25} + ( -5 \beta_{1} + \beta_{4} - 3 \beta_{10} + 3 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{26} + ( 11 + 5 \beta_{2} - 16 \beta_{3} + 6 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{28} + ( 3 \beta_{1} - \beta_{4} - \beta_{10} + \beta_{11} + \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{29} + ( 6 - 8 \beta_{3} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{31} + ( -2 \beta_{1} + \beta_{4} + 3 \beta_{10} + 4 \beta_{12} - 3 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{32} + ( 15 - 6 \beta_{2} - 16 \beta_{3} - 2 \beta_{5} + 3 \beta_{7} - 3 \beta_{8} + 5 \beta_{9} ) q^{34} + ( 3 \beta_{1} + \beta_{10} + 3 \beta_{11} - 4 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 4 \beta_{15} ) q^{35} + ( -13 - 4 \beta_{2} + 4 \beta_{3} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{37} + ( -13 \beta_{1} - \beta_{10} + \beta_{11} + \beta_{12} - 5 \beta_{13} + 2 \beta_{14} + 5 \beta_{15} ) q^{38} + ( 17 + \beta_{2} - 8 \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{40} + ( -6 \beta_{1} + \beta_{4} + 2 \beta_{10} + 2 \beta_{11} - 7 \beta_{12} + 3 \beta_{13} - \beta_{14} ) q^{41} + ( 12 - \beta_{2} + 10 \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{43} + ( 2 + \beta_{2} + 21 \beta_{3} - 6 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} ) q^{46} + ( -16 \beta_{1} - 6 \beta_{4} + \beta_{11} + 2 \beta_{12} - \beta_{13} - 3 \beta_{14} + 5 \beta_{15} ) q^{47} + ( 12 - 2 \beta_{2} + 16 \beta_{3} + 3 \beta_{5} + 3 \beta_{6} + \beta_{7} - 5 \beta_{8} - 2 \beta_{9} ) q^{49} + ( 11 \beta_{1} - \beta_{4} + 13 \beta_{10} - 3 \beta_{11} + 14 \beta_{12} + 4 \beta_{13} - 2 \beta_{14} - 8 \beta_{15} ) q^{50} + ( 22 - 10 \beta_{2} + 17 \beta_{3} - \beta_{5} + 5 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} + 5 \beta_{9} ) q^{52} + ( -9 \beta_{1} - 6 \beta_{4} + 5 \beta_{10} + 2 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{53} + ( -15 \beta_{1} + 3 \beta_{4} + 6 \beta_{10} + 3 \beta_{11} - 17 \beta_{12} + 4 \beta_{13} - 3 \beta_{14} - 12 \beta_{15} ) q^{56} + ( -8 - 3 \beta_{2} + 11 \beta_{3} + 4 \beta_{5} + 8 \beta_{6} - \beta_{7} - \beta_{8} - 5 \beta_{9} ) q^{58} + ( -13 \beta_{1} - 3 \beta_{4} - 9 \beta_{10} - 6 \beta_{11} - 13 \beta_{12} + 7 \beta_{13} - 2 \beta_{14} + 7 \beta_{15} ) q^{59} + ( 23 - 3 \beta_{2} - 16 \beta_{3} - 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 6 \beta_{9} ) q^{61} + ( -14 \beta_{1} + \beta_{4} + 16 \beta_{10} - \beta_{11} - 16 \beta_{12} - 6 \beta_{13} - 6 \beta_{14} - 6 \beta_{15} ) q^{62} + ( 32 - \beta_{2} - 4 \beta_{3} + 6 \beta_{5} - \beta_{6} - 4 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{64} + ( -3 \beta_{1} - \beta_{4} + 12 \beta_{10} + 2 \beta_{11} + 4 \beta_{12} + 10 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{65} + ( 14 - \beta_{2} - 29 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - 5 \beta_{9} ) q^{67} + ( 14 \beta_{1} - \beta_{4} + \beta_{10} - 6 \beta_{11} - 3 \beta_{12} + 4 \beta_{13} + 6 \beta_{14} + 12 \beta_{15} ) q^{68} + ( -46 - \beta_{2} + 11 \beta_{3} - 4 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{9} ) q^{70} + ( -20 \beta_{1} - 3 \beta_{4} + 12 \beta_{10} + \beta_{11} - 15 \beta_{12} + 8 \beta_{13} - 6 \beta_{14} + 5 \beta_{15} ) q^{71} + ( 17 - \beta_{2} - 37 \beta_{3} - 3 \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{9} ) q^{73} + ( -12 \beta_{1} - \beta_{4} + 6 \beta_{10} - 10 \beta_{11} - 3 \beta_{12} + 2 \beta_{14} + 12 \beta_{15} ) q^{74} + ( 68 - 4 \beta_{2} - \beta_{3} + 6 \beta_{5} - 3 \beta_{7} + \beta_{8} - 4 \beta_{9} ) q^{76} + ( 1 + \beta_{2} + 28 \beta_{3} + 10 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 6 \beta_{9} ) q^{79} + ( 3 \beta_{1} + 9 \beta_{4} - \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - 8 \beta_{13} - 2 \beta_{14} + 3 \beta_{15} ) q^{80} + ( -10 \beta_{2} + 42 \beta_{3} - 9 \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{82} + ( -3 \beta_{1} + 2 \beta_{4} + 7 \beta_{10} - 2 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + 10 \beta_{14} - 4 \beta_{15} ) q^{83} + ( 29 - \beta_{2} - 7 \beta_{3} - 11 \beta_{6} + \beta_{7} + \beta_{9} ) q^{85} + ( 25 \beta_{1} - \beta_{4} + 9 \beta_{10} - 6 \beta_{11} + 14 \beta_{12} + 6 \beta_{13} + 3 \beta_{14} ) q^{86} + ( 8 \beta_{1} - \beta_{4} - \beta_{10} + 2 \beta_{11} + 17 \beta_{12} - 3 \beta_{13} + 9 \beta_{14} - 16 \beta_{15} ) q^{89} + ( -9 + 15 \beta_{2} + 2 \beta_{3} + 19 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{91} + ( 24 \beta_{1} - 4 \beta_{4} - 23 \beta_{10} - 2 \beta_{11} + 28 \beta_{12} - \beta_{13} + 11 \beta_{14} + \beta_{15} ) q^{92} + ( 92 - 14 \beta_{2} - 39 \beta_{3} + \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{94} + ( -\beta_{1} + 6 \beta_{4} + 7 \beta_{10} - 4 \beta_{11} + 11 \beta_{12} + 9 \beta_{13} - 2 \beta_{14} - 11 \beta_{15} ) q^{95} + ( -6 + 2 \beta_{2} + 3 \beta_{3} - 14 \beta_{5} - \beta_{6} + \beta_{7} - 5 \beta_{8} + 4 \beta_{9} ) q^{97} + ( 3 \beta_{1} - 6 \beta_{4} + 24 \beta_{10} - 8 \beta_{11} - 6 \beta_{12} - 3 \beta_{13} - 4 \beta_{14} + 7 \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 32 q^{4} - 8 q^{7} + O(q^{10})$$ $$16 q - 32 q^{4} - 8 q^{7} + 24 q^{10} + 4 q^{13} + 28 q^{16} - 20 q^{19} - 44 q^{25} + 16 q^{28} + 28 q^{31} + 148 q^{34} - 148 q^{37} + 224 q^{40} + 272 q^{43} + 208 q^{46} + 348 q^{49} + 520 q^{52} - 44 q^{58} + 224 q^{61} + 436 q^{64} + 24 q^{67} - 664 q^{70} - 4 q^{73} + 1052 q^{76} + 216 q^{79} + 348 q^{82} + 416 q^{85} - 168 q^{91} + 1140 q^{94} - 44 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 48 x^{14} + 921 x^{12} + 8986 x^{10} + 46812 x^{8} + 125072 x^{6} + 152129 x^{4} + 65614 x^{2} + 5041$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 6$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{14} - 38 \nu^{12} - 532 \nu^{10} - 3324 \nu^{8} - 8784 \nu^{6} - 7244 \nu^{4} + 1095 \nu^{2} + 1448$$$$)/792$$ $$\beta_{4}$$ $$=$$ $$($$$$-115 \nu^{15} + 373 \nu^{13} + 131438 \nu^{11} + 2591160 \nu^{9} + 20689524 \nu^{7} + 74808340 \nu^{5} + 112624593 \nu^{3} + 48462527 \nu$$$$)/1237104$$ $$\beta_{5}$$ $$=$$ $$($$$$4 \nu^{14} + 161 \nu^{12} + 2470 \nu^{10} + 18084 \nu^{8} + 65124 \nu^{6} + 109760 \nu^{4} + 72840 \nu^{2} + 8977$$$$)/792$$ $$\beta_{6}$$ $$=$$ $$($$$$-127 \nu^{14} - 5015 \nu^{12} - 74458 \nu^{10} - 513480 \nu^{8} - 1642716 \nu^{6} - 2151404 \nu^{4} - 740667 \nu^{2} + 72971$$$$)/17424$$ $$\beta_{7}$$ $$=$$ $$($$$$103 \nu^{14} + 4139 \nu^{12} + 63418 \nu^{10} + 465168 \nu^{8} + 1696932 \nu^{6} + 2964164 \nu^{4} + 1951707 \nu^{2} + 145345$$$$)/17424$$ $$\beta_{8}$$ $$=$$ $$($$$$-41 \nu^{14} - 1612 \nu^{12} - 23810 \nu^{10} - 163086 \nu^{8} - 515538 \nu^{6} - 653278 \nu^{4} - 200715 \nu^{2} - 6692$$$$)/4356$$ $$\beta_{9}$$ $$=$$ $$($$$$161 \nu^{14} + 6289 \nu^{12} + 92078 \nu^{10} + 624624 \nu^{8} + 1979532 \nu^{6} + 2767276 \nu^{4} + 1646349 \nu^{2} + 339227$$$$)/17424$$ $$\beta_{10}$$ $$=$$ $$($$$$127 \nu^{15} + 5741 \nu^{13} + 102838 \nu^{11} + 926376 \nu^{9} + 4389372 \nu^{7} + 10531028 \nu^{5} + 12165003 \nu^{3} + 6703315 \nu$$$$)/412368$$ $$\beta_{11}$$ $$=$$ $$($$$$83 \nu^{15} + 3629 \nu^{13} + 62314 \nu^{11} + 534116 \nu^{9} + 2407744 \nu^{7} + 5555816 \nu^{5} + 5961795 \nu^{3} + 2563575 \nu$$$$)/68728$$ $$\beta_{12}$$ $$=$$ $$($$$$-\nu^{15} - 38 \nu^{13} - 532 \nu^{11} - 3324 \nu^{9} - 8784 \nu^{7} - 7244 \nu^{5} + 1095 \nu^{3} + 656 \nu$$$$)/792$$ $$\beta_{13}$$ $$=$$ $$($$$$1528 \nu^{15} + 62339 \nu^{13} + 976318 \nu^{11} + 7403088 \nu^{9} + 28530000 \nu^{7} + 55659908 \nu^{5} + 52182408 \nu^{3} + 16374319 \nu$$$$)/618552$$ $$\beta_{14}$$ $$=$$ $$($$$$592 \nu^{15} + 21245 \nu^{13} + 265918 \nu^{11} + 1252548 \nu^{9} + 449556 \nu^{7} - 9680008 \nu^{5} - 15397068 \nu^{3} - 2699819 \nu$$$$)/206184$$ $$\beta_{15}$$ $$=$$ $$($$$$607 \nu^{15} + 25373 \nu^{13} + 411154 \nu^{11} + 3268980 \nu^{9} + 13372824 \nu^{7} + 27252464 \nu^{5} + 24058479 \nu^{3} + 6633991 \nu$$$$)/206184$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 6$$ $$\nu^{3}$$ $$=$$ $$-\beta_{13} - \beta_{12} + \beta_{11} - 10 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + 5 \beta_{3} - 13 \beta_{2} + 55$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{15} + 2 \beta_{14} + 13 \beta_{13} + 20 \beta_{12} - 16 \beta_{11} + 3 \beta_{10} + \beta_{4} + 110 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-19 \beta_{9} + 23 \beta_{8} - 4 \beta_{7} - 21 \beta_{6} + 26 \beta_{5} - 104 \beta_{3} + 163 \beta_{2} - 556$$ $$\nu^{7}$$ $$=$$ $$-58 \beta_{15} - 49 \beta_{14} - 144 \beta_{13} - 323 \beta_{12} + 226 \beta_{11} - 52 \beta_{10} - 26 \beta_{4} - 1262 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$300 \beta_{9} - 385 \beta_{8} + 107 \beta_{7} + 337 \beta_{6} - 479 \beta_{5} + 1705 \beta_{3} - 2050 \beta_{2} + 5876$$ $$\nu^{9}$$ $$=$$ $$1113 \beta_{15} + 886 \beta_{14} + 1561 \beta_{13} + 4853 \beta_{12} - 3072 \beta_{11} + 670 \beta_{10} + 490 \beta_{4} + 14948 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-4446 \beta_{9} + 5741 \beta_{8} - 1999 \beta_{7} - 4857 \beta_{6} + 7738 \beta_{5} - 25702 \beta_{3} + 25947 \beta_{2} - 64004$$ $$\nu^{11}$$ $$=$$ $$-18183 \beta_{15} - 14183 \beta_{14} - 17131 \beta_{13} - 70235 \beta_{12} + 40991 \beta_{11} - 7641 \beta_{10} - 7969 \beta_{4} - 181597 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$63680 \beta_{9} - 80998 \beta_{8} + 32366 \beta_{7} + 66278 \beta_{6} - 116784 \beta_{5} + 371616 \beta_{3} - 330394 \beta_{2} + 714871$$ $$\nu^{13}$$ $$=$$ $$274142 \beta_{15} + 212830 \beta_{14} + 192314 \beta_{13} + 992430 \beta_{12} - 541350 \beta_{11} + 80342 \beta_{10} + 119710 \beta_{4} + 2250309 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-892116 \beta_{9} + 1108664 \beta_{8} - 486972 \beta_{7} - 877608 \beta_{6} + 1692232 \beta_{5} - 5238840 \beta_{3} + 4228843 \beta_{2} - 8166432$$ $$\nu^{15}$$ $$=$$ $$-3948668 \beta_{15} - 3071320 \beta_{14} - 2213375 \beta_{13} - 13788227 \beta_{12} + 7107231 \beta_{11} - 780028 \beta_{10} - 1717092 \beta_{4} - 28311016 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$$.

 $$n$$ $$244$$ $$848$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
485.1
 − 3.62039i − 3.25431i − 2.99491i − 2.98036i − 1.96467i − 1.35141i − 0.816689i − 0.311356i 0.311356i 0.816689i 1.35141i 1.96467i 2.98036i 2.99491i 3.25431i 3.62039i
3.62039i 0 −9.10725 0.165198i 0 10.2989 18.4903i 0 0.598083
485.2 3.25431i 0 −6.59053 1.59288i 0 −10.8798 8.43039i 0 5.18373
485.3 2.99491i 0 −4.96950 1.61173i 0 −3.32981 2.90358i 0 4.82698
485.4 2.98036i 0 −4.88257 7.85680i 0 2.80793 2.63038i 0 −23.4161
485.5 1.96467i 0 0.140085 9.62261i 0 −5.23056 8.13389i 0 18.9052
485.6 1.35141i 0 2.17369 5.10469i 0 −13.1679 8.34318i 0 6.89852
485.7 0.816689i 0 3.33302 1.04696i 0 6.83027 5.98880i 0 0.855043
485.8 0.311356i 0 3.90306 5.94641i 0 8.67101 2.46067i 0 −1.85145
485.9 0.311356i 0 3.90306 5.94641i 0 8.67101 2.46067i 0 −1.85145
485.10 0.816689i 0 3.33302 1.04696i 0 6.83027 5.98880i 0 0.855043
485.11 1.35141i 0 2.17369 5.10469i 0 −13.1679 8.34318i 0 6.89852
485.12 1.96467i 0 0.140085 9.62261i 0 −5.23056 8.13389i 0 18.9052
485.13 2.98036i 0 −4.88257 7.85680i 0 2.80793 2.63038i 0 −23.4161
485.14 2.99491i 0 −4.96950 1.61173i 0 −3.32981 2.90358i 0 4.82698
485.15 3.25431i 0 −6.59053 1.59288i 0 −10.8798 8.43039i 0 5.18373
485.16 3.62039i 0 −9.10725 0.165198i 0 10.2989 18.4903i 0 0.598083
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 485.16 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.3.b.i 16
3.b odd 2 1 inner 1089.3.b.i 16
11.b odd 2 1 1089.3.b.j 16
11.c even 5 2 99.3.l.a 32
33.d even 2 1 1089.3.b.j 16
33.h odd 10 2 99.3.l.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.3.l.a 32 11.c even 5 2
99.3.l.a 32 33.h odd 10 2
1089.3.b.i 16 1.a even 1 1 trivial
1089.3.b.i 16 3.b odd 2 1 inner
1089.3.b.j 16 11.b odd 2 1
1089.3.b.j 16 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1089, [\chi])$$:

 $$T_{2}^{16} + \cdots$$ $$T_{7}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$5041 + 65614 T^{2} + 152129 T^{4} + 125072 T^{6} + 46812 T^{8} + 8986 T^{10} + 921 T^{12} + 48 T^{14} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$1038361 + 39902020 T^{2} + 68981402 T^{4} + 39192842 T^{6} + 8554743 T^{8} + 596782 T^{10} + 17478 T^{12} + 222 T^{14} + T^{16}$$
$7$ $$( 4273551 - 237648 T - 659963 T^{2} + 14910 T^{3} + 23990 T^{4} - 566 T^{5} - 275 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$11$ $$T^{16}$$
$13$ $$( -14305556 - 15402616 T - 3793354 T^{2} + 249464 T^{3} + 103543 T^{4} + 478 T^{5} - 634 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$17$ $$988191873299187216 + 88956782736863184 T^{2} + 2815770055538188 T^{4} + 42045393510188 T^{6} + 321722976021 T^{8} + 1271166384 T^{10} + 2593719 T^{12} + 2590 T^{14} + T^{16}$$
$19$ $$( 918027484 + 4588056 T - 52171778 T^{2} + 2105132 T^{3} + 462695 T^{4} - 12422 T^{5} - 1347 T^{6} + 10 T^{7} + T^{8} )^{2}$$
$23$ $$18547148744209285776 + 1651801037608554912 T^{2} + 47020940271422136 T^{4} + 495753991863264 T^{6} + 2442623588257 T^{8} + 6098160558 T^{10} + 7705814 T^{12} + 4566 T^{14} + T^{16}$$
$29$ $$4743235673649349776 + 1397243462894994384 T^{2} + 86039870134538860 T^{4} + 1187458426780036 T^{6} + 5989981087765 T^{8} + 13497697678 T^{10} + 14168543 T^{12} + 6366 T^{14} + T^{16}$$
$31$ $$( 37407528581 + 2015766806 T - 522872230 T^{2} - 16015196 T^{3} + 2405359 T^{4} + 32104 T^{5} - 3710 T^{6} - 14 T^{7} + T^{8} )^{2}$$
$37$ $$( -36000957284 - 7271365184 T + 148394438 T^{2} + 76068428 T^{3} + 444755 T^{4} - 181274 T^{5} - 2110 T^{6} + 74 T^{7} + T^{8} )^{2}$$
$41$ $$32\!\cdots\!96$$$$+$$$$11\!\cdots\!88$$$$T^{2} + 1410079256549870072 T^{4} + 7996418083525856 T^{6} + 22734834371505 T^{8} + 32873957338 T^{10} + 23904510 T^{12} + 8058 T^{14} + T^{16}$$
$43$ $$( -2337129036 + 4784421048 T - 2366066722 T^{2} + 331841496 T^{3} - 17221231 T^{4} + 276312 T^{5} + 3372 T^{6} - 136 T^{7} + T^{8} )^{2}$$
$47$ $$97\!\cdots\!16$$$$+$$$$68\!\cdots\!60$$$$T^{2} +$$$$18\!\cdots\!64$$$$T^{4} + 2423116878813450044 T^{6} + 1777899931712661 T^{8} + 744074072308 T^{10} + 173998203 T^{12} + 20934 T^{14} + T^{16}$$
$53$ $$20\!\cdots\!61$$$$+$$$$66\!\cdots\!94$$$$T^{2} +$$$$20\!\cdots\!94$$$$T^{4} + 2625908418278667542 T^{6} + 1834489911568167 T^{8} + 737394203506 T^{10} + 169500426 T^{12} + 20508 T^{14} + T^{16}$$
$59$ $$159750408613276321 +$$$$84\!\cdots\!20$$$$T^{2} +$$$$48\!\cdots\!34$$$$T^{4} + 9362698814372259374 T^{6} + 7538589501913491 T^{8} + 2673639622498 T^{10} + 450419218 T^{12} + 35014 T^{14} + T^{16}$$
$61$ $$( -99578003184 - 22327751952 T + 6905272212 T^{2} + 734842476 T^{3} - 68150765 T^{4} + 1590388 T^{5} - 8275 T^{6} - 112 T^{7} + T^{8} )^{2}$$
$67$ $$( 40427575967196 + 8718737372664 T - 369279935096 T^{2} - 2615104428 T^{3} + 157123947 T^{4} + 255596 T^{5} - 22299 T^{6} - 12 T^{7} + T^{8} )^{2}$$
$71$ $$15\!\cdots\!76$$$$+$$$$14\!\cdots\!92$$$$T^{2} +$$$$22\!\cdots\!64$$$$T^{4} +$$$$14\!\cdots\!20$$$$T^{6} + 48664634585595645 T^{8} + 9033072975030 T^{10} + 919959159 T^{12} + 47992 T^{14} + T^{16}$$
$73$ $$( 4747380152636 - 519621322572 T - 40452065486 T^{2} + 534604604 T^{3} + 43260471 T^{4} - 153894 T^{5} - 13751 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$79$ $$( -8796984423319 + 4708662432224 T - 192198437005 T^{2} - 5395477866 T^{3} + 99208530 T^{4} + 1482650 T^{5} - 16783 T^{6} - 108 T^{7} + T^{8} )^{2}$$
$83$ $$69\!\cdots\!01$$$$+$$$$34\!\cdots\!96$$$$T^{2} +$$$$63\!\cdots\!33$$$$T^{4} + 58539845456899006570 T^{6} + 28492321794308980 T^{8} + 7396389987862 T^{10} + 961787465 T^{12} + 53388 T^{14} + T^{16}$$
$89$ $$12\!\cdots\!76$$$$+$$$$35\!\cdots\!12$$$$T^{2} +$$$$35\!\cdots\!68$$$$T^{4} +$$$$14\!\cdots\!24$$$$T^{6} + 315663585882179761 T^{8} + 36561369884750 T^{10} + 2335709534 T^{12} + 76598 T^{14} + T^{16}$$
$97$ $$( -29900401075089 - 3118339354944 T + 60857936534 T^{2} + 8639347618 T^{3} + 121736947 T^{4} - 1333926 T^{5} - 25864 T^{6} + 22 T^{7} + T^{8} )^{2}$$