# Properties

 Label 847.2.f.u Level 847 Weight 2 Character orbit 847.f Analytic conductor 6.763 Analytic rank 0 Dimension 12 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 847.f (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.76332905120$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} ) q^{2} -\beta_{8} q^{3} + ( 3 \beta_{3} + \beta_{5} ) q^{4} + \beta_{9} q^{5} + ( -\beta_{6} + 4 \beta_{7} + \beta_{9} ) q^{6} -\beta_{3} q^{7} + ( \beta_{8} + 3 \beta_{10} + 2 \beta_{11} ) q^{8} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} ) q^{2} -\beta_{8} q^{3} + ( 3 \beta_{3} + \beta_{5} ) q^{4} + \beta_{9} q^{5} + ( -\beta_{6} + 4 \beta_{7} + \beta_{9} ) q^{6} -\beta_{3} q^{7} + ( \beta_{8} + 3 \beta_{10} + 2 \beta_{11} ) q^{8} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{9} + ( 2 + 2 \beta_{4} ) q^{10} + ( 2 + 3 \beta_{2} + \beta_{4} ) q^{12} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{13} + ( -\beta_{8} - \beta_{10} ) q^{14} + ( -2 \beta_{3} - \beta_{5} ) q^{15} + ( -2 \beta_{6} - 5 \beta_{7} - 3 \beta_{9} ) q^{16} + ( -\beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{17} + ( -3 \beta_{1} + 5 \beta_{3} ) q^{18} + ( 2 \beta_{8} - 2 \beta_{11} ) q^{19} + ( 6 + 2 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 6 \beta_{10} - 2 \beta_{11} ) q^{20} -\beta_{2} q^{21} + ( 2 - \beta_{4} ) q^{23} + ( 8 + \beta_{1} + \beta_{2} + 8 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - \beta_{6} - 8 \beta_{7} - \beta_{8} - 3 \beta_{9} - 8 \beta_{10} - 3 \beta_{11} ) q^{24} + ( -2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{25} + ( 3 \beta_{1} - 4 \beta_{3} - 3 \beta_{5} ) q^{26} + ( -2 \beta_{6} + 6 \beta_{7} + \beta_{9} ) q^{27} + ( 3 \beta_{7} + \beta_{9} ) q^{28} + ( 2 \beta_{1} - 2 \beta_{5} ) q^{29} + ( -2 \beta_{8} - 4 \beta_{10} - 2 \beta_{11} ) q^{30} + ( 4 - \beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{6} - 4 \beta_{7} + \beta_{8} - 4 \beta_{10} ) q^{31} + ( -13 - \beta_{2} - 4 \beta_{4} ) q^{32} + ( -4 + 3 \beta_{2} - 3 \beta_{4} ) q^{34} + ( -\beta_{4} - \beta_{5} + \beta_{9} + \beta_{11} ) q^{35} + ( 4 \beta_{8} - 5 \beta_{10} - \beta_{11} ) q^{36} + ( -2 \beta_{1} - 6 \beta_{3} + \beta_{5} ) q^{37} + ( 2 \beta_{6} - 4 \beta_{7} + 2 \beta_{9} ) q^{38} + ( 4 \beta_{6} - 6 \beta_{7} - 2 \beta_{9} ) q^{39} + ( 4 \beta_{1} + 14 \beta_{3} + 2 \beta_{5} ) q^{40} + ( \beta_{8} + 6 \beta_{10} + \beta_{11} ) q^{41} + ( -4 + \beta_{1} + \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + 4 \beta_{7} - \beta_{8} + \beta_{9} + 4 \beta_{10} + \beta_{11} ) q^{42} + ( -2 - 2 \beta_{4} ) q^{43} + ( -2 - 2 \beta_{2} + 2 \beta_{4} ) q^{45} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{46} + ( -\beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{47} + ( \beta_{1} + 14 \beta_{3} + 5 \beta_{5} ) q^{48} -\beta_{7} q^{49} + ( -\beta_{6} + 7 \beta_{7} ) q^{50} + ( -4 \beta_{1} + 6 \beta_{3} + 2 \beta_{5} ) q^{51} + ( -5 \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{52} + ( 4 + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{7} - 2 \beta_{9} - 4 \beta_{10} - 2 \beta_{11} ) q^{53} + 8 \beta_{2} q^{54} + ( 3 + \beta_{2} + 2 \beta_{4} ) q^{56} + ( 4 - 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{6} - 4 \beta_{7} + 4 \beta_{8} - 4 \beta_{10} ) q^{57} + ( -2 \beta_{8} + 4 \beta_{10} - 2 \beta_{11} ) q^{58} -\beta_{1} q^{59} + ( 2 \beta_{6} + 12 \beta_{7} + 4 \beta_{9} ) q^{60} + ( -\beta_{6} - 6 \beta_{7} - \beta_{9} ) q^{61} + ( 5 \beta_{1} - \beta_{5} ) q^{62} + ( -2 \beta_{8} + \beta_{10} + \beta_{11} ) q^{63} + ( -15 - 8 \beta_{1} - 8 \beta_{2} - 15 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 8 \beta_{6} + 15 \beta_{7} + 8 \beta_{8} + 3 \beta_{9} + 15 \beta_{10} + 3 \beta_{11} ) q^{64} + ( -8 - 2 \beta_{2} + 2 \beta_{4} ) q^{65} + ( -2 - 4 \beta_{2} + \beta_{4} ) q^{67} + ( -2 - 5 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 5 \beta_{6} + 2 \beta_{7} + 5 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{68} + ( -2 \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{69} + ( -2 \beta_{3} - 2 \beta_{5} ) q^{70} + ( 4 \beta_{6} - 2 \beta_{7} - 5 \beta_{9} ) q^{71} + ( 3 \beta_{6} + \beta_{7} - 2 \beta_{9} ) q^{72} + ( -\beta_{1} - 6 \beta_{3} - \beta_{5} ) q^{73} + ( -4 \beta_{8} - 12 \beta_{10} ) q^{74} + ( -6 + 3 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} + 6 \beta_{7} - 3 \beta_{8} + \beta_{9} + 6 \beta_{10} + \beta_{11} ) q^{75} + ( 8 - 2 \beta_{2} + 2 \beta_{4} ) q^{76} + ( 6 - 10 \beta_{2} ) q^{78} + ( 10 + 10 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 10 \beta_{7} - 2 \beta_{9} - 10 \beta_{10} - 2 \beta_{11} ) q^{79} + ( 6 \beta_{8} + 22 \beta_{10} + 4 \beta_{11} ) q^{80} + ( -4 \beta_{1} + 3 \beta_{3} - 2 \beta_{5} ) q^{81} + ( -5 \beta_{6} - 12 \beta_{7} - 3 \beta_{9} ) q^{82} + ( 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{9} ) q^{83} + ( -3 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{84} + ( 2 \beta_{8} + 8 \beta_{10} - 2 \beta_{11} ) q^{85} + ( -6 - 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + 6 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} + 6 \beta_{10} + 4 \beta_{11} ) q^{86} + ( 4 - 4 \beta_{2} ) q^{87} + ( -8 - 2 \beta_{2} - \beta_{4} ) q^{89} + ( -6 - 6 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 6 \beta_{7} - 2 \beta_{9} + 6 \beta_{10} - 2 \beta_{11} ) q^{90} + ( \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{91} -2 \beta_{1} q^{92} + ( 6 \beta_{6} - 4 \beta_{7} - \beta_{9} ) q^{93} + ( \beta_{6} + 8 \beta_{7} + 3 \beta_{9} ) q^{94} + ( -4 \beta_{1} - 8 \beta_{3} + 4 \beta_{5} ) q^{95} + ( 11 \beta_{8} + 12 \beta_{10} + 5 \beta_{11} ) q^{96} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - 4 \beta_{10} - 3 \beta_{11} ) q^{97} + ( -1 - \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 2q^{2} + q^{3} - 8q^{4} - q^{5} + 12q^{6} + 3q^{7} + 6q^{8} - 4q^{9} + O(q^{10})$$ $$12q + 2q^{2} + q^{3} - 8q^{4} - q^{5} + 12q^{6} + 3q^{7} + 6q^{8} - 4q^{9} + 16q^{10} + 8q^{12} + 8q^{13} - 2q^{14} + 5q^{15} - 10q^{16} + 8q^{17} - 18q^{18} + 14q^{20} + 4q^{21} + 28q^{23} + 20q^{24} - 2q^{25} + 12q^{26} + 19q^{27} + 8q^{28} - 8q^{30} + 13q^{31} - 136q^{32} - 48q^{34} + q^{35} - 18q^{36} + 17q^{37} - 16q^{38} - 20q^{39} - 36q^{40} + 16q^{41} - 12q^{42} - 16q^{43} - 24q^{45} - 4q^{47} - 36q^{48} - 3q^{49} + 22q^{50} - 20q^{51} + 10q^{53} - 32q^{54} + 24q^{56} + 16q^{57} + 16q^{58} - q^{59} + 30q^{60} - 16q^{61} + 4q^{62} + 4q^{63} - 34q^{64} - 96q^{65} - 12q^{67} + 7q^{69} + 4q^{70} - 5q^{71} + 2q^{72} + 16q^{73} - 32q^{74} - 20q^{75} + 96q^{76} + 112q^{78} + 28q^{79} + 56q^{80} - 15q^{81} - 28q^{82} + 8q^{83} + 2q^{84} + 24q^{85} - 12q^{86} + 64q^{87} - 84q^{89} - 20q^{90} - 8q^{91} - 2q^{92} - 17q^{93} + 20q^{94} + 24q^{95} + 20q^{96} + 11q^{97} - 8q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} + 7 x^{10} - 15 x^{9} + 59 x^{8} + 118 x^{7} + 266 x^{6} + 324 x^{5} + 1036 x^{4} + 376 x^{3} + 136 x^{2} + 48 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$15 \nu^{10} + 5251 \nu^{5} + 24224$$$$)/66764$$ $$\beta_{3}$$ $$=$$ $$($$$$-59 \nu^{11} - 16203 \nu^{6} + 367616 \nu$$$$)/133528$$ $$\beta_{4}$$ $$=$$ $$($$$$37 \nu^{10} + 10727 \nu^{5} - 104932$$$$)/33382$$ $$\beta_{5}$$ $$=$$ $$($$$$133 \nu^{11} + 37657 \nu^{6} - 577480 \nu$$$$)/66764$$ $$\beta_{6}$$ $$=$$ $$($$$$59 \nu^{11} - 413 \nu^{10} + 885 \nu^{9} - 3481 \nu^{8} + 9241 \nu^{7} - 15694 \nu^{6} - 19116 \nu^{5} - 61124 \nu^{4} - 22184 \nu^{3} - 375640 \nu^{2} - 2832 \nu - 944$$$$)/133528$$ $$\beta_{7}$$ $$=$$ $$($$$$48 \nu^{11} - 336 \nu^{10} + 720 \nu^{9} - 2832 \nu^{8} + 7801 \nu^{7} - 12768 \nu^{6} - 15552 \nu^{5} - 49728 \nu^{4} - 18048 \nu^{3} - 259493 \nu^{2} - 2304 \nu - 768$$$$)/33382$$ $$\beta_{8}$$ $$=$$ $$($$$$1093 \nu^{11} - 1093 \nu^{10} + 7563 \nu^{9} - 16395 \nu^{8} + 64487 \nu^{7} + 128974 \nu^{6} + 290738 \nu^{5} + 332228 \nu^{4} + 1132348 \nu^{3} + 410968 \nu^{2} + 148648 \nu + 52464$$$$)/133528$$ $$\beta_{9}$$ $$=$$ $$($$$$-325 \nu^{11} + 2275 \nu^{10} - 4875 \nu^{9} + 19175 \nu^{8} - 53167 \nu^{7} + 86450 \nu^{6} + 105300 \nu^{5} + 336700 \nu^{4} + 122200 \nu^{3} + 1633540 \nu^{2} + 15600 \nu + 5200$$$$)/66764$$ $$\beta_{10}$$ $$=$$ $$($$$$1514 \nu^{11} - 1514 \nu^{10} + 10583 \nu^{9} - 22710 \nu^{8} + 89326 \nu^{7} + 178652 \nu^{6} + 402724 \nu^{5} + 485285 \nu^{4} + 1568504 \nu^{3} + 569264 \nu^{2} + 205904 \nu + 72672$$$$)/66764$$ $$\beta_{11}$$ $$=$$ $$($$$$-4771 \nu^{11} + 4771 \nu^{10} - 33471 \nu^{9} + 71565 \nu^{8} - 281489 \nu^{7} - 562978 \nu^{6} - 1269086 \nu^{5} - 1567258 \nu^{4} - 4942756 \nu^{3} - 1793896 \nu^{2} - 648856 \nu - 229008$$$$)/66764$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{9} - 4 \beta_{7} + 2 \beta_{6}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{11} - 6 \beta_{10} - \beta_{9} + 8 \beta_{8} - 6 \beta_{7} + 8 \beta_{6} + \beta_{5} + \beta_{4} + 6 \beta_{3} - 8 \beta_{2} - 8 \beta_{1} + 6$$ $$\nu^{4}$$ $$=$$ $$-7 \beta_{11} - 30 \beta_{10} + 22 \beta_{8}$$ $$\nu^{5}$$ $$=$$ $$-15 \beta_{4} + 74 \beta_{2} - 74$$ $$\nu^{6}$$ $$=$$ $$59 \beta_{5} + 266 \beta_{3} - 222 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$163 \beta_{9} + 770 \beta_{7} - 710 \beta_{6}$$ $$\nu^{8}$$ $$=$$ $$547 \beta_{11} + 2514 \beta_{10} + 547 \beta_{9} - 2190 \beta_{8} + 2514 \beta_{7} - 2190 \beta_{6} - 547 \beta_{5} - 547 \beta_{4} - 2514 \beta_{3} + 2190 \beta_{2} + 2190 \beta_{1} - 2514$$ $$\nu^{9}$$ $$=$$ $$1643 \beta_{11} + 7666 \beta_{10} - 6894 \beta_{8}$$ $$\nu^{10}$$ $$=$$ $$5251 \beta_{4} - 21454 \beta_{2} + 24290$$ $$\nu^{11}$$ $$=$$ $$-16203 \beta_{5} - 75314 \beta_{3} + 67198 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$365$$ $$\chi(n)$$ $$1$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
148.1
 −1.42513 + 1.03542i −0.293939 + 0.213559i 2.52809 − 1.83676i −0.965643 − 2.97194i 0.112275 + 0.345546i 0.544351 + 1.67534i −1.42513 − 1.03542i −0.293939 − 0.213559i 2.52809 + 1.83676i −0.965643 + 2.97194i 0.112275 − 0.345546i 0.544351 − 1.67534i
−0.853368 2.62640i −1.42513 1.03542i −4.55169 + 3.30700i −0.811540 + 2.49766i −1.50326 + 4.62655i 0.809017 0.587785i 8.10146 + 5.88605i 0.0318546 + 0.0980384i 7.25240
148.2 −0.421292 1.29660i −0.293939 0.213559i 0.114343 0.0830753i 0.970726 2.98759i −0.153067 + 0.471092i 0.809017 0.587785i −2.36180 1.71595i −0.886258 2.72762i −4.28267
148.3 0.656626 + 2.02089i 2.52809 + 1.83676i −2.03479 + 1.47836i 0.149831 0.461131i −2.05188 + 6.31504i 0.809017 0.587785i −0.885558 0.643395i 2.09047 + 6.43381i 1.03028
323.1 −1.71907 1.24898i −0.965643 + 2.97194i 0.777220 + 2.39204i −0.392262 + 0.284995i 5.37189 3.90291i −0.309017 0.951057i 0.338253 1.04104i −5.47293 3.97631i 1.03028
323.2 1.10296 + 0.801344i 0.112275 0.345546i −0.0436753 0.134419i −2.54139 + 1.84643i 0.400735 0.291151i −0.309017 0.951057i 0.902127 2.77646i 2.32025 + 1.68576i −4.28267
323.3 2.23415 + 1.62320i 0.544351 1.67534i 1.73859 + 5.35083i 2.12464 1.54364i 3.93558 2.85936i −0.309017 0.951057i −3.09448 + 9.52384i −0.0833965 0.0605911i 7.25240
372.1 −0.853368 + 2.62640i −1.42513 + 1.03542i −4.55169 3.30700i −0.811540 2.49766i −1.50326 4.62655i 0.809017 + 0.587785i 8.10146 5.88605i 0.0318546 0.0980384i 7.25240
372.2 −0.421292 + 1.29660i −0.293939 + 0.213559i 0.114343 + 0.0830753i 0.970726 + 2.98759i −0.153067 0.471092i 0.809017 + 0.587785i −2.36180 + 1.71595i −0.886258 + 2.72762i −4.28267
372.3 0.656626 2.02089i 2.52809 1.83676i −2.03479 1.47836i 0.149831 + 0.461131i −2.05188 6.31504i 0.809017 + 0.587785i −0.885558 + 0.643395i 2.09047 6.43381i 1.03028
729.1 −1.71907 + 1.24898i −0.965643 2.97194i 0.777220 2.39204i −0.392262 0.284995i 5.37189 + 3.90291i −0.309017 + 0.951057i 0.338253 + 1.04104i −5.47293 + 3.97631i 1.03028
729.2 1.10296 0.801344i 0.112275 + 0.345546i −0.0436753 + 0.134419i −2.54139 1.84643i 0.400735 + 0.291151i −0.309017 + 0.951057i 0.902127 + 2.77646i 2.32025 1.68576i −4.28267
729.3 2.23415 1.62320i 0.544351 + 1.67534i 1.73859 5.35083i 2.12464 + 1.54364i 3.93558 + 2.85936i −0.309017 + 0.951057i −3.09448 9.52384i −0.0833965 + 0.0605911i 7.25240
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 729.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.f.u 12
11.b odd 2 1 847.2.f.t 12
11.c even 5 1 847.2.a.i 3
11.c even 5 3 inner 847.2.f.u 12
11.d odd 10 1 847.2.a.j yes 3
11.d odd 10 3 847.2.f.t 12
33.f even 10 1 7623.2.a.bz 3
33.h odd 10 1 7623.2.a.ce 3
77.j odd 10 1 5929.2.a.t 3
77.l even 10 1 5929.2.a.y 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.i 3 11.c even 5 1
847.2.a.j yes 3 11.d odd 10 1
847.2.f.t 12 11.b odd 2 1
847.2.f.t 12 11.d odd 10 3
847.2.f.u 12 1.a even 1 1 trivial
847.2.f.u 12 11.c even 5 3 inner
5929.2.a.t 3 77.j odd 10 1
5929.2.a.y 3 77.l even 10 1
7623.2.a.bz 3 33.f even 10 1
7623.2.a.ce 3 33.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(847, [\chi])$$:

 $$T_{2}^{12} - \cdots$$ $$T_{3}^{12} - \cdots$$ $$T_{13}^{12} - \cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 3 T^{2} - 4 T^{3} + 3 T^{4} + 16 T^{5} - 33 T^{6} + 50 T^{7} - 65 T^{8} + 56 T^{9} + 118 T^{10} - 256 T^{11} + 388 T^{12} - 512 T^{13} + 472 T^{14} + 448 T^{15} - 1040 T^{16} + 1600 T^{17} - 2112 T^{18} + 2048 T^{19} + 768 T^{20} - 2048 T^{21} + 3072 T^{22} - 4096 T^{23} + 4096 T^{24}$$
$3$ $$1 - T - 2 T^{2} - 3 T^{3} + 8 T^{4} - 32 T^{5} + 32 T^{6} + 72 T^{7} + 97 T^{8} - 272 T^{9} + 712 T^{10} - 744 T^{11} - 1601 T^{12} - 2232 T^{13} + 6408 T^{14} - 7344 T^{15} + 7857 T^{16} + 17496 T^{17} + 23328 T^{18} - 69984 T^{19} + 52488 T^{20} - 59049 T^{21} - 118098 T^{22} - 177147 T^{23} + 531441 T^{24}$$
$5$ $$1 + T - 6 T^{2} - 7 T^{3} + 6 T^{4} + 28 T^{5} + 54 T^{6} - 136 T^{7} + 19 T^{8} + 1340 T^{9} + 3418 T^{10} - 3238 T^{11} - 29939 T^{12} - 16190 T^{13} + 85450 T^{14} + 167500 T^{15} + 11875 T^{16} - 425000 T^{17} + 843750 T^{18} + 2187500 T^{19} + 2343750 T^{20} - 13671875 T^{21} - 58593750 T^{22} + 48828125 T^{23} + 244140625 T^{24}$$
$7$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{3}$$
$11$ 
$13$ $$1 - 8 T + 23 T^{2} - 324 T^{4} - 760 T^{5} + 15724 T^{6} - 61496 T^{7} + 78885 T^{8} + 469128 T^{9} - 888268 T^{10} - 10299304 T^{11} + 57810247 T^{12} - 133890952 T^{13} - 150117292 T^{14} + 1030674216 T^{15} + 2253034485 T^{16} - 22833034328 T^{17} + 75896744716 T^{18} - 47688872920 T^{19} - 264296753604 T^{20} + 3170745312527 T^{22} - 14337283152296 T^{23} + 23298085122481 T^{24}$$
$17$ $$1 - 8 T + 11 T^{2} + 128 T^{3} - 844 T^{4} - 1016 T^{5} + 29908 T^{6} - 111320 T^{7} - 72771 T^{8} + 2499144 T^{9} - 6443380 T^{10} - 29627592 T^{11} + 234698431 T^{12} - 503669064 T^{13} - 1862136820 T^{14} + 12278294472 T^{15} - 6077906691 T^{16} - 158058481240 T^{17} + 721906413652 T^{18} - 416904091768 T^{19} - 5887539280204 T^{20} + 15179248191616 T^{21} + 22175932904939 T^{22} - 274175170461064 T^{23} + 582622237229761 T^{24}$$
$19$ $$1 - 17 T^{2} - 64 T^{3} - 34 T^{4} - 3264 T^{5} + 3306 T^{6} + 78336 T^{7} + 280279 T^{8} - 50048 T^{9} + 3108178 T^{10} - 20001792 T^{11} - 162841925 T^{12} - 380034048 T^{13} + 1122052258 T^{14} - 343279232 T^{15} + 36526239559 T^{16} + 193967691264 T^{17} + 155533682586 T^{18} - 2917597356096 T^{19} - 577441143394 T^{20} - 20652012657856 T^{21} - 104228126382617 T^{22} + 2213314919066161 T^{24}$$
$23$ $$( 1 - 7 T + 77 T^{2} - 314 T^{3} + 1771 T^{4} - 3703 T^{5} + 12167 T^{6} )^{4}$$
$29$ $$1 - 47 T^{2} + 64 T^{3} + 846 T^{4} + 9024 T^{5} + 4006 T^{6} - 457216 T^{7} + 382439 T^{8} + 7884928 T^{9} + 61357278 T^{10} + 56983552 T^{11} - 2998123365 T^{12} + 1652523008 T^{13} + 51601470798 T^{14} + 192305508992 T^{15} + 270491838359 T^{16} - 9378025501184 T^{17} + 2382862223926 T^{18} + 155662883812416 T^{19} + 423208465365006 T^{20} + 928457342455616 T^{21} - 19773239965109447 T^{22} + 353814783205469041 T^{24}$$
$31$ $$1 - 13 T + 26 T^{2} + 657 T^{3} - 5460 T^{4} + 15964 T^{5} + 22960 T^{6} - 713908 T^{7} + 7426341 T^{8} - 36420348 T^{9} + 3945344 T^{10} + 1142793028 T^{11} - 8639431253 T^{12} + 35426583868 T^{13} + 3791475584 T^{14} - 1084998587268 T^{15} + 6858381866661 T^{16} - 20438579932108 T^{17} + 20377084515760 T^{18} + 439211371668004 T^{19} - 4656785064427860 T^{20} + 17370831759560847 T^{21} + 21310335461500826 T^{22} - 330310199653262803 T^{23} + 787662783788549761 T^{24}$$
$37$ $$1 - 17 T + 106 T^{2} + 35 T^{3} - 5106 T^{4} + 24264 T^{5} + 104582 T^{6} - 1555984 T^{7} + 4789839 T^{8} + 22847880 T^{9} - 147468394 T^{10} - 903867842 T^{11} + 11727294013 T^{12} - 33443110154 T^{13} - 201884231386 T^{14} + 1157313665640 T^{15} + 8976929450079 T^{16} - 107898087588688 T^{17} + 268328799306038 T^{18} + 2303427066755112 T^{19} - 17934720091720626 T^{20} + 4548660892827695 T^{21} + 509709943476291994 T^{22} - 3024599570250827021 T^{23} + 6582952005840035281 T^{24}$$
$41$ $$1 - 16 T + 67 T^{2} + 560 T^{3} - 7100 T^{4} + 28000 T^{5} - 43388 T^{6} + 246672 T^{7} + 616013 T^{8} - 60687280 T^{9} + 471922140 T^{10} + 387365904 T^{11} - 19332075665 T^{12} + 15882002064 T^{13} + 793301117340 T^{14} - 4182628024880 T^{15} + 1740705510893 T^{16} + 28578480813072 T^{17} - 206097522808508 T^{18} + 5453119668668000 T^{19} - 56692969126759100 T^{20} + 183333883260618160 T^{21} + 899318173780210867 T^{22} - 8805264507459975056 T^{23} + 22563490300366186081 T^{24}$$
$43$ $$( 1 + 4 T + 101 T^{2} + 312 T^{3} + 4343 T^{4} + 7396 T^{5} + 79507 T^{6} )^{4}$$
$47$ $$1 + 4 T - 111 T^{2} - 568 T^{3} + 7392 T^{4} + 49396 T^{5} - 365232 T^{6} - 2901388 T^{7} + 16129633 T^{8} + 122188172 T^{9} - 498402320 T^{10} - 2354973940 T^{11} + 16802495119 T^{12} - 110683775180 T^{13} - 1100970724880 T^{14} + 12685942581556 T^{15} + 78707463687073 T^{16} - 665418851169716 T^{17} - 3936914373041328 T^{18} + 25025155658390348 T^{19} + 176013031003737312 T^{20} - 635666108722371656 T^{21} - 5838503678177135439 T^{22} + 9888636860336049212 T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$
$53$ $$1 - 10 T - 59 T^{2} + 1184 T^{3} - 926 T^{4} - 69722 T^{5} + 294406 T^{6} + 2232546 T^{7} - 16364317 T^{8} - 47271754 T^{9} + 615669366 T^{10} + 954847470 T^{11} - 28997636753 T^{12} + 50606915910 T^{13} + 1729415249094 T^{14} - 7037676920258 T^{15} - 129122332366477 T^{16} + 933640675115178 T^{17} + 6525320902544374 T^{18} - 81903210091715314 T^{19} - 57652473320920286 T^{20} + 3906920092693725472 T^{21} - 10318360751565269891 T^{22} - 92690359293721915970 T^{23} +$$$$49\!\cdots\!41$$$$T^{24}$$
$59$ $$1 + T - 170 T^{2} - 221 T^{3} + 18880 T^{4} + 19352 T^{5} - 1722416 T^{6} - 1333488 T^{7} + 139186697 T^{8} + 76390936 T^{9} - 9605185704 T^{10} - 2482296192 T^{11} + 594022989415 T^{12} - 146455475328 T^{13} - 33435651435624 T^{14} + 15689094044744 T^{15} + 1686575453946617 T^{16} - 953342973624912 T^{17} - 72652426031796656 T^{18} + 48160383534217288 T^{19} + 2772158661969580480 T^{20} - 1914522075922741519 T^{21} - 86889848061109038170 T^{22} + 30155888444737842659 T^{23} +$$$$17\!\cdots\!81$$$$T^{24}$$
$61$ $$1 + 16 T + 7 T^{2} - 1840 T^{3} - 13860 T^{4} + 4320 T^{5} + 400652 T^{6} + 1904048 T^{7} + 38865093 T^{8} + 488718000 T^{9} + 1383027860 T^{10} - 26636392784 T^{11} - 345384133145 T^{12} - 1624819959824 T^{13} + 5146246667060 T^{14} + 110929700358000 T^{15} + 538119898128213 T^{16} + 1608151897726448 T^{17} + 20641741028483372 T^{18} + 13576649051610720 T^{19} - 2657063358142314660 T^{20} - 21517228810814819440 T^{21} + 4993400381640178207 T^{22} +$$$$69\!\cdots\!76$$$$T^{23} +$$$$26\!\cdots\!21$$$$T^{24}$$
$67$ $$( 1 + 3 T + 113 T^{2} - 22 T^{3} + 7571 T^{4} + 13467 T^{5} + 300763 T^{6} )^{4}$$
$71$ $$1 + 5 T + 20 T^{2} - 695 T^{3} - 7780 T^{4} - 68170 T^{5} - 5786 T^{6} + 5263790 T^{7} + 56924985 T^{8} + 339614790 T^{9} - 118215790 T^{10} - 23737080060 T^{11} - 265061379329 T^{12} - 1685332684260 T^{13} - 595925797390 T^{14} + 121551869103690 T^{15} + 1446559559749785 T^{16} + 9497084415500290 T^{17} - 741188242766906 T^{18} - 620014341197514470 T^{19} - 5023962473092020580 T^{20} - 31864707999322076545 T^{21} + 65104871020197624020 T^{22} +$$$$11\!\cdots\!55$$$$T^{23} +$$$$16\!\cdots\!41$$$$T^{24}$$
$73$ $$1 - 16 T - 29 T^{2} + 2608 T^{3} - 15612 T^{4} - 54432 T^{5} + 597956 T^{6} - 1898096 T^{7} + 105180429 T^{8} - 1137548208 T^{9} + 139658396 T^{10} + 77531528720 T^{11} - 791045249105 T^{12} + 5659801596560 T^{13} + 744239592284 T^{14} - 442525591231536 T^{15} + 2986939171225389 T^{16} - 3934888898386928 T^{17} + 90491208614865284 T^{18} - 601331996191487904 T^{19} - 12590454954650392572 T^{20} +$$$$15\!\cdots\!04$$$$T^{21} -$$$$12\!\cdots\!21$$$$T^{22} -$$$$50\!\cdots\!32$$$$T^{23} +$$$$22\!\cdots\!21$$$$T^{24}$$
$79$ $$1 - 28 T + 319 T^{2} - 848 T^{3} - 23118 T^{4} + 368100 T^{5} - 2689762 T^{6} + 7469044 T^{7} + 64811091 T^{8} - 1075382940 T^{9} + 8371511678 T^{10} - 40089262324 T^{11} + 177958244563 T^{12} - 3167051723596 T^{13} + 52246604382398 T^{14} - 530205729354660 T^{15} + 2524397244148371 T^{16} + 22982669634612556 T^{17} - 653847400537076002 T^{18} + 7068958897805127900 T^{19} - 35072521467419877198 T^{20} -$$$$10\!\cdots\!12$$$$T^{21} +$$$$30\!\cdots\!19$$$$T^{22} -$$$$20\!\cdots\!12$$$$T^{23} +$$$$59\!\cdots\!41$$$$T^{24}$$
$83$ $$1 - 8 T - 129 T^{2} + 1184 T^{3} + 10542 T^{4} - 294632 T^{5} + 609882 T^{6} + 30027704 T^{7} - 108165497 T^{8} - 2317255144 T^{9} + 27636921250 T^{10} + 30542384840 T^{11} - 2623552045061 T^{12} + 2535017941720 T^{13} + 190390750491250 T^{14} - 1324976367022328 T^{15} - 5133352877750537 T^{16} + 118280346471973672 T^{17} + 199395048791032458 T^{18} - 7995148975175782264 T^{19} + 23743664711209770222 T^{20} +$$$$22\!\cdots\!52$$$$T^{21} -$$$$20\!\cdots\!21$$$$T^{22} -$$$$10\!\cdots\!36$$$$T^{23} +$$$$10\!\cdots\!61$$$$T^{24}$$
$89$ $$( 1 + 21 T + 371 T^{2} + 3838 T^{3} + 33019 T^{4} + 166341 T^{5} + 704969 T^{6} )^{4}$$
$97$ $$1 - 11 T - 138 T^{2} + 2685 T^{3} - 414 T^{4} - 183880 T^{5} + 1306526 T^{6} - 9183508 T^{7} - 69630505 T^{8} + 3158826216 T^{9} - 21575809142 T^{10} - 200368131722 T^{11} + 3986418389017 T^{12} - 19435708777034 T^{13} - 203006788217078 T^{14} + 2882975399035368 T^{15} - 6164338543316905 T^{16} - 78861907948881556 T^{17} + 1088299581711866654 T^{18} - 14857188549835418440 T^{19} - 3244697508072061854 T^{20} +$$$$20\!\cdots\!45$$$$T^{21} -$$$$10\!\cdots\!62$$$$T^{22} -$$$$78\!\cdots\!83$$$$T^{23} +$$$$69\!\cdots\!41$$$$T^{24}$$