# Properties

 Label 273.2.bj.c Level $273$ Weight $2$ Character orbit 273.bj Analytic conductor $2.180$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.bj (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 11 x^{14} + 88 x^{12} - 303 x^{10} + 758 x^{8} - 968 x^{6} + 867 x^{4} - 30 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{9} ) q^{2} -\beta_{10} q^{3} + ( \beta_{10} - \beta_{11} ) q^{4} + ( -\beta_{8} + \beta_{15} ) q^{5} -\beta_{9} q^{6} + ( \beta_{1} - \beta_{7} + \beta_{14} ) q^{7} + ( \beta_{7} + \beta_{9} - \beta_{13} - \beta_{14} ) q^{8} + ( -1 + \beta_{10} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{9} ) q^{2} -\beta_{10} q^{3} + ( \beta_{10} - \beta_{11} ) q^{4} + ( -\beta_{8} + \beta_{15} ) q^{5} -\beta_{9} q^{6} + ( \beta_{1} - \beta_{7} + \beta_{14} ) q^{7} + ( \beta_{7} + \beta_{9} - \beta_{13} - \beta_{14} ) q^{8} + ( -1 + \beta_{10} ) q^{9} -\beta_{5} q^{10} -\beta_{6} q^{11} + ( 1 - \beta_{2} - \beta_{10} + \beta_{11} ) q^{12} + ( -\beta_{2} - \beta_{3} - \beta_{15} ) q^{13} + ( 3 - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{10} ) q^{14} -\beta_{15} q^{15} + ( -2 + \beta_{2} - \beta_{4} + 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{16} + ( \beta_{5} - 2 \beta_{10} - \beta_{12} ) q^{17} -\beta_{1} q^{18} + ( -\beta_{1} - \beta_{8} - \beta_{9} + \beta_{15} ) q^{19} + ( -\beta_{7} - \beta_{9} + \beta_{14} - \beta_{15} ) q^{20} + ( -\beta_{1} + \beta_{7} - \beta_{9} ) q^{21} + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{22} + ( \beta_{3} + \beta_{4} - \beta_{5} + \beta_{10} - \beta_{12} ) q^{23} + ( \beta_{1} + \beta_{6} - \beta_{7} ) q^{24} + ( -\beta_{10} + \beta_{11} ) q^{25} + ( -1 + 2 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} + 2 \beta_{9} - \beta_{13} + \beta_{15} ) q^{26} + q^{27} + ( 3 \beta_{1} + \beta_{6} + \beta_{7} + \beta_{8} + 3 \beta_{9} - \beta_{13} - \beta_{14} ) q^{28} + ( -2 + \beta_{2} + \beta_{3} ) q^{29} + ( -1 - \beta_{3} + \beta_{5} ) q^{30} + ( -2 \beta_{1} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{31} + ( -2 \beta_{1} - \beta_{6} ) q^{32} + ( \beta_{6} - \beta_{13} ) q^{33} + ( -\beta_{9} + 2 \beta_{13} - \beta_{15} ) q^{34} + ( -1 + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{35} + ( -1 + \beta_{2} ) q^{36} + ( \beta_{1} + \beta_{6} + \beta_{9} - \beta_{13} - 2 \beta_{14} ) q^{37} + ( -\beta_{5} - 3 \beta_{10} + \beta_{11} ) q^{38} + ( \beta_{5} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{39} + ( -\beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{10} + \beta_{11} ) q^{40} + ( \beta_{7} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{41} + ( -\beta_{3} - 3 \beta_{10} + \beta_{11} ) q^{42} + ( 4 + 3 \beta_{3} + \beta_{4} ) q^{43} + ( -5 \beta_{1} - 2 \beta_{6} + \beta_{8} - 5 \beta_{9} + 2 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{44} + \beta_{8} q^{45} + ( 2 \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{46} + ( 3 \beta_{1} - \beta_{6} - 3 \beta_{8} + 3 \beta_{9} + \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{47} + ( 2 - \beta_{2} + \beta_{4} ) q^{48} + ( -1 - 2 \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{10} + \beta_{11} ) q^{49} + ( -\beta_{7} - 3 \beta_{9} + \beta_{13} + \beta_{14} ) q^{50} + ( -1 + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{10} + \beta_{12} ) q^{51} + ( -\beta_{1} + \beta_{7} + \beta_{8} + 6 \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{52} + ( 3 \beta_{5} + 3 \beta_{10} ) q^{53} + ( \beta_{1} + \beta_{9} ) q^{54} + ( -1 - 2 \beta_{3} + \beta_{4} ) q^{55} + ( 1 - 2 \beta_{3} + 4 \beta_{10} - 3 \beta_{11} + \beta_{12} ) q^{56} + ( \beta_{9} - \beta_{15} ) q^{57} + ( -4 \beta_{1} - \beta_{6} + \beta_{8} - 4 \beta_{9} + \beta_{13} - \beta_{15} ) q^{58} + ( -\beta_{1} - \beta_{6} - 3 \beta_{7} ) q^{59} + ( -\beta_{1} + \beta_{7} + \beta_{8} ) q^{60} + ( 1 - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{61} + ( -6 + 4 \beta_{2} + \beta_{3} - \beta_{4} ) q^{62} + ( \beta_{9} - \beta_{14} ) q^{63} + ( -3 + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{64} + ( 4 + \beta_{1} + \beta_{2} - \beta_{6} + \beta_{8} + \beta_{9} - 4 \beta_{10} - \beta_{11} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{65} + ( -\beta_{5} + 2 \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{66} + ( -\beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{67} + ( 2 - 5 \beta_{2} - \beta_{3} + \beta_{5} - 3 \beta_{10} + 5 \beta_{11} ) q^{68} + ( -\beta_{3} - \beta_{4} ) q^{69} + ( -3 \beta_{1} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{14} + \beta_{15} ) q^{70} + ( -\beta_{7} + 5 \beta_{9} + \beta_{13} + \beta_{14} ) q^{71} + ( -\beta_{1} - \beta_{6} - \beta_{9} + \beta_{13} + \beta_{14} ) q^{72} + ( -\beta_{6} + 3 \beta_{7} + 3 \beta_{8} ) q^{73} + ( \beta_{5} + 3 \beta_{10} - 3 \beta_{11} + \beta_{12} ) q^{74} + ( -1 + \beta_{2} + \beta_{10} - \beta_{11} ) q^{75} + ( -2 \beta_{7} - 4 \beta_{9} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{76} + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{10} - 3 \beta_{11} + \beta_{12} ) q^{77} + ( 1 + \beta_{3} - 2 \beta_{9} + \beta_{13} - \beta_{15} ) q^{78} + ( -1 + 2 \beta_{2} - 4 \beta_{3} + \beta_{4} + 4 \beta_{5} - 3 \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{79} + ( \beta_{6} + 3 \beta_{7} ) q^{80} -\beta_{10} q^{81} + ( 2 - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{82} + ( 2 \beta_{7} - \beta_{9} - \beta_{13} - 2 \beta_{14} + 3 \beta_{15} ) q^{83} + ( -\beta_{7} - \beta_{8} - 3 \beta_{9} + \beta_{13} + \beta_{15} ) q^{84} + ( -3 \beta_{7} - 3 \beta_{9} - 2 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{85} + ( 4 \beta_{1} + 2 \beta_{6} + 3 \beta_{8} + 4 \beta_{9} - 2 \beta_{13} - 4 \beta_{14} - 3 \beta_{15} ) q^{86} + ( -\beta_{5} + 3 \beta_{10} - \beta_{11} ) q^{87} + ( -\beta_{5} - 13 \beta_{10} + 5 \beta_{11} ) q^{88} + ( -\beta_{1} - \beta_{8} - \beta_{9} + \beta_{14} + \beta_{15} ) q^{89} + ( 1 + \beta_{3} ) q^{90} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{13} + 2 \beta_{15} ) q^{91} + ( 3 - 4 \beta_{2} - \beta_{3} ) q^{92} + ( 2 \beta_{1} + \beta_{6} - \beta_{8} + 2 \beta_{9} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{93} + ( -\beta_{5} + 6 \beta_{10} - \beta_{11} - \beta_{12} ) q^{94} + ( \beta_{5} + 4 \beta_{10} + \beta_{11} ) q^{95} + ( 2 \beta_{1} + \beta_{6} + 2 \beta_{9} - \beta_{13} ) q^{96} + ( -\beta_{9} - \beta_{13} ) q^{97} + ( 3 \beta_{1} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{13} + \beta_{15} ) q^{98} + \beta_{13} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 8q^{3} + 6q^{4} - 8q^{9} + O(q^{10})$$ $$16q - 8q^{3} + 6q^{4} - 8q^{9} - 4q^{10} + 6q^{12} + 4q^{13} + 40q^{14} - 10q^{16} - 8q^{17} - 8q^{22} - 8q^{23} - 6q^{25} - 4q^{26} + 16q^{27} - 36q^{29} - 4q^{30} - 14q^{35} - 12q^{36} - 26q^{38} - 2q^{39} + 6q^{40} - 14q^{42} + 32q^{43} + 20q^{48} - 46q^{49} - 8q^{51} + 40q^{52} + 36q^{53} - 8q^{55} + 54q^{56} + 12q^{61} - 80q^{62} - 56q^{64} + 34q^{65} + 4q^{66} + 10q^{68} + 16q^{69} + 18q^{74} - 6q^{75} - 22q^{77} + 8q^{78} + 8q^{79} - 8q^{81} + 12q^{82} + 18q^{87} - 98q^{88} + 8q^{90} + 16q^{91} + 40q^{92} + 46q^{94} + 38q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 11 x^{14} + 88 x^{12} - 303 x^{10} + 758 x^{8} - 968 x^{6} + 867 x^{4} - 30 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$84975 \nu^{14} - 847968 \nu^{12} + 6630625 \nu^{10} - 18977750 \nu^{8} + 45035320 \nu^{6} - 24480525 \nu^{4} + 847175 \nu^{2} + 141812432$$$$)/47832247$$ $$\beta_{3}$$ $$=$$ $$($$$$493812 \nu^{14} - 5094945 \nu^{12} + 38532300 \nu^{10} - 110284680 \nu^{8} + 203384572 \nu^{6} - 142262748 \nu^{4} + 4923156 \nu^{2} - 24499860$$$$)/47832247$$ $$\beta_{4}$$ $$=$$ $$($$$$508068 \nu^{14} - 5088601 \nu^{12} + 39644700 \nu^{10} - 113468520 \nu^{8} + 257471965 \nu^{6} - 146369772 \nu^{4} + 5065284 \nu^{2} + 227456047$$$$)/47832247$$ $$\beta_{5}$$ $$=$$ $$($$$$-516836 \nu^{14} + 6587845 \nu^{12} - 54823490 \nu^{10} + 227035283 \nu^{8} - 593353298 \nu^{6} + 862031072 \nu^{4} - 708141783 \nu^{2} + 24504217$$$$)/47832247$$ $$\beta_{6}$$ $$=$$ $$($$$$593043 \nu^{15} - 5936569 \nu^{13} + 46275325 \nu^{11} - 132446270 \nu^{9} + 302507285 \nu^{7} - 170850297 \nu^{5} + 5912459 \nu^{3} + 464932973 \nu$$$$)/47832247$$ $$\beta_{7}$$ $$=$$ $$($$$$678018 \nu^{15} - 6784537 \nu^{13} + 52905950 \nu^{11} - 151424020 \nu^{9} + 347542605 \nu^{7} - 195330822 \nu^{5} + 6759634 \nu^{3} + 702409899 \nu$$$$)/47832247$$ $$\beta_{8}$$ $$=$$ $$($$$$1171830 \nu^{15} - 11879482 \nu^{13} + 91438250 \nu^{11} - 261708700 \nu^{9} + 550927177 \nu^{7} - 337593570 \nu^{5} + 11682790 \nu^{3} + 677910039 \nu$$$$)/47832247$$ $$\beta_{9}$$ $$=$$ $$($$$$-1684309 \nu^{15} + 18442424 \nu^{13} - 147371224 \nu^{11} + 503715002 \nu^{9} - 1257728472 \nu^{7} + 1585375792 \nu^{5} - 1435815378 \nu^{3} + 1849848 \nu$$$$)/47832247$$ $$\beta_{10}$$ $$=$$ $$($$$$-1684309 \nu^{14} + 18442424 \nu^{12} - 147371224 \nu^{10} + 503715002 \nu^{8} - 1257728472 \nu^{6} + 1585375792 \nu^{4} - 1435815378 \nu^{2} + 49682095$$$$)/47832247$$ $$\beta_{11}$$ $$=$$ $$($$$$-4967952 \nu^{14} + 54479304 \nu^{12} - 435483047 \nu^{10} + 1492167256 \nu^{8} - 3728150096 \nu^{6} + 4731646851 \nu^{4} - 4258766712 \nu^{2} + 147361976$$$$)/47832247$$ $$\beta_{12}$$ $$=$$ $$($$$$-7826720 \nu^{14} + 86276344 \nu^{12} - 690441745 \nu^{10} + 2385730760 \nu^{8} - 5973395120 \nu^{6} + 7707683038 \nu^{4} - 6838320936 \nu^{2} + 236620312$$$$)/47832247$$ $$\beta_{13}$$ $$=$$ $$($$$$16756333 \nu^{15} - 183577065 \nu^{13} + 1466942565 \nu^{11} - 5017774290 \nu^{9} + 12519509445 \nu^{7} - 15780931770 \nu^{5} + 13974630863 \nu^{3} - 18413505 \nu$$$$)/47832247$$ $$\beta_{14}$$ $$=$$ $$($$$$-24499860 \nu^{15} + 269004648 \nu^{13} - 2150892735 \nu^{11} + 7384925280 \nu^{9} - 18460609200 \nu^{7} + 23512479908 \nu^{5} - 21099115872 \nu^{3} + 730072644 \nu$$$$)/47832247$$ $$\beta_{15}$$ $$=$$ $$($$$$24504217 \nu^{15} - 269029551 \nu^{13} + 2149783251 \nu^{11} - 7369954261 \nu^{9} + 18347161203 \nu^{7} - 23126728758 \nu^{5} + 20383125067 \nu^{3} - 26984727 \nu$$$$)/47832247$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{11} - 3 \beta_{10} - \beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{14} + \beta_{13} - 5 \beta_{9} - \beta_{7}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{12} + 7 \beta_{11} - 16 \beta_{10}$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{14} + 9 \beta_{13} - 30 \beta_{9} - 9 \beta_{6} - 30 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-9 \beta_{4} + \beta_{3} + 48 \beta_{2} - 99$$ $$\nu^{7}$$ $$=$$ $$\beta_{8} + 56 \beta_{7} - 66 \beta_{6} - 195 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$66 \beta_{12} - 327 \beta_{11} + 661 \beta_{10} - 11 \beta_{5} - 66 \beta_{4} + 11 \beta_{3} + 327 \beta_{2} - 650$$ $$\nu^{9}$$ $$=$$ $$11 \beta_{15} - 382 \beta_{14} - 459 \beta_{13} + 1304 \beta_{9} + 382 \beta_{7}$$ $$\nu^{10}$$ $$=$$ $$459 \beta_{12} - 2222 \beta_{11} + 4448 \beta_{10} - 88 \beta_{5}$$ $$\nu^{11}$$ $$=$$ $$88 \beta_{15} - 2593 \beta_{14} - 3140 \beta_{13} + 8804 \beta_{9} - 88 \beta_{8} + 3140 \beta_{6} + 8804 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$3140 \beta_{4} - 635 \beta_{3} - 15084 \beta_{2} + 29464$$ $$\nu^{13}$$ $$=$$ $$-635 \beta_{8} - 17589 \beta_{7} + 21364 \beta_{6} + 59632 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-21364 \beta_{12} + 102360 \beta_{11} - 204035 \beta_{10} + 4410 \beta_{5} + 21364 \beta_{4} - 4410 \beta_{3} - 102360 \beta_{2} + 199625$$ $$\nu^{15}$$ $$=$$ $$-4410 \beta_{15} + 119314 \beta_{14} + 145088 \beta_{13} - 404345 \beta_{9} - 119314 \beta_{7}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1 + \beta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 −2.25575 − 1.30236i −1.34967 − 0.779232i −1.14630 − 0.661815i −0.161178 − 0.0930563i 0.161178 + 0.0930563i 1.14630 + 0.661815i 1.34967 + 0.779232i 2.25575 + 1.30236i −2.25575 + 1.30236i −1.34967 + 0.779232i −1.14630 + 0.661815i −0.161178 + 0.0930563i 0.161178 − 0.0930563i 1.14630 − 0.661815i 1.34967 − 0.779232i 2.25575 − 1.30236i
−2.25575 + 1.30236i −0.500000 + 0.866025i 2.39228 4.14355i 0.401974 0.232080i 2.60472i −2.25575 1.38260i 7.25298i −0.500000 0.866025i −0.604503 + 1.04703i
25.2 −1.34967 + 0.779232i −0.500000 + 0.866025i 0.214404 0.371358i 1.85159 1.06902i 1.55846i −1.34967 2.27561i 2.44865i −0.500000 0.866025i −1.66602 + 2.88564i
25.3 −1.14630 + 0.661815i −0.500000 + 0.866025i −0.124000 + 0.214775i −1.98394 + 1.14543i 1.32363i −1.14630 + 2.38453i 2.97552i −0.500000 0.866025i 1.51612 2.62600i
25.4 −0.161178 + 0.0930563i −0.500000 + 0.866025i −0.982681 + 1.70205i 2.28561 1.31960i 0.186113i −0.161178 + 2.64084i 0.738004i −0.500000 0.866025i −0.245594 + 0.425381i
25.5 0.161178 0.0930563i −0.500000 + 0.866025i −0.982681 + 1.70205i −2.28561 + 1.31960i 0.186113i 0.161178 2.64084i 0.738004i −0.500000 0.866025i −0.245594 + 0.425381i
25.6 1.14630 0.661815i −0.500000 + 0.866025i −0.124000 + 0.214775i 1.98394 1.14543i 1.32363i 1.14630 2.38453i 2.97552i −0.500000 0.866025i 1.51612 2.62600i
25.7 1.34967 0.779232i −0.500000 + 0.866025i 0.214404 0.371358i −1.85159 + 1.06902i 1.55846i 1.34967 + 2.27561i 2.44865i −0.500000 0.866025i −1.66602 + 2.88564i
25.8 2.25575 1.30236i −0.500000 + 0.866025i 2.39228 4.14355i −0.401974 + 0.232080i 2.60472i 2.25575 + 1.38260i 7.25298i −0.500000 0.866025i −0.604503 + 1.04703i
142.1 −2.25575 1.30236i −0.500000 0.866025i 2.39228 + 4.14355i 0.401974 + 0.232080i 2.60472i −2.25575 + 1.38260i 7.25298i −0.500000 + 0.866025i −0.604503 1.04703i
142.2 −1.34967 0.779232i −0.500000 0.866025i 0.214404 + 0.371358i 1.85159 + 1.06902i 1.55846i −1.34967 + 2.27561i 2.44865i −0.500000 + 0.866025i −1.66602 2.88564i
142.3 −1.14630 0.661815i −0.500000 0.866025i −0.124000 0.214775i −1.98394 1.14543i 1.32363i −1.14630 2.38453i 2.97552i −0.500000 + 0.866025i 1.51612 + 2.62600i
142.4 −0.161178 0.0930563i −0.500000 0.866025i −0.982681 1.70205i 2.28561 + 1.31960i 0.186113i −0.161178 2.64084i 0.738004i −0.500000 + 0.866025i −0.245594 0.425381i
142.5 0.161178 + 0.0930563i −0.500000 0.866025i −0.982681 1.70205i −2.28561 1.31960i 0.186113i 0.161178 + 2.64084i 0.738004i −0.500000 + 0.866025i −0.245594 0.425381i
142.6 1.14630 + 0.661815i −0.500000 0.866025i −0.124000 0.214775i 1.98394 + 1.14543i 1.32363i 1.14630 + 2.38453i 2.97552i −0.500000 + 0.866025i 1.51612 + 2.62600i
142.7 1.34967 + 0.779232i −0.500000 0.866025i 0.214404 + 0.371358i −1.85159 1.06902i 1.55846i 1.34967 2.27561i 2.44865i −0.500000 + 0.866025i −1.66602 2.88564i
142.8 2.25575 + 1.30236i −0.500000 0.866025i 2.39228 + 4.14355i −0.401974 0.232080i 2.60472i 2.25575 1.38260i 7.25298i −0.500000 + 0.866025i −0.604503 1.04703i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 142.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.bj.c 16
3.b odd 2 1 819.2.dl.f 16
7.c even 3 1 inner 273.2.bj.c 16
7.c even 3 1 1911.2.c.n 8
7.d odd 6 1 1911.2.c.k 8
13.b even 2 1 inner 273.2.bj.c 16
21.h odd 6 1 819.2.dl.f 16
39.d odd 2 1 819.2.dl.f 16
91.r even 6 1 inner 273.2.bj.c 16
91.r even 6 1 1911.2.c.n 8
91.s odd 6 1 1911.2.c.k 8
273.w odd 6 1 819.2.dl.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bj.c 16 1.a even 1 1 trivial
273.2.bj.c 16 7.c even 3 1 inner
273.2.bj.c 16 13.b even 2 1 inner
273.2.bj.c 16 91.r even 6 1 inner
819.2.dl.f 16 3.b odd 2 1
819.2.dl.f 16 21.h odd 6 1
819.2.dl.f 16 39.d odd 2 1
819.2.dl.f 16 273.w odd 6 1
1911.2.c.k 8 7.d odd 6 1
1911.2.c.k 8 91.s odd 6 1
1911.2.c.n 8 7.c even 3 1
1911.2.c.n 8 91.r even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 30 T^{2} + 867 T^{4} - 968 T^{6} + 758 T^{8} - 303 T^{10} + 88 T^{12} - 11 T^{14} + T^{16}$$
$3$ $$( 1 + T + T^{2} )^{8}$$
$5$ $$1296 - 6732 T^{2} + 31513 T^{4} - 16728 T^{6} + 6001 T^{8} - 1258 T^{10} + 193 T^{12} - 17 T^{14} + T^{16}$$
$7$ $$5764801 + 2705927 T^{2} + 679483 T^{4} + 120491 T^{6} + 17749 T^{8} + 2459 T^{10} + 283 T^{12} + 23 T^{14} + T^{16}$$
$11$ $$10556001 - 10802925 T^{2} + 8180260 T^{4} - 2578737 T^{6} + 593776 T^{8} - 42910 T^{10} + 2251 T^{12} - 56 T^{14} + T^{16}$$
$13$ $$( 28561 - 4394 T - 2704 T^{2} - 182 T^{3} + 382 T^{4} - 14 T^{5} - 16 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$17$ $$( 28561 + 41405 T + 51744 T^{2} + 13357 T^{3} + 3550 T^{4} + 294 T^{5} + 65 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$19$ $$923521 - 1268520 T^{2} + 1413738 T^{4} - 389936 T^{6} + 73763 T^{8} - 8304 T^{10} + 682 T^{12} - 32 T^{14} + T^{16}$$
$23$ $$( 244036 - 28158 T + 27455 T^{2} - 1159 T^{3} + 2135 T^{4} - 82 T^{5} + 65 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$29$ $$( 11 - 32 T + 13 T^{2} + 9 T^{3} + T^{4} )^{4}$$
$31$ $$11316496 - 30191900 T^{2} + 74344045 T^{4} - 15832251 T^{6} + 2431361 T^{8} - 181310 T^{10} + 9819 T^{12} - 108 T^{14} + T^{16}$$
$37$ $$454371856 - 559736844 T^{2} + 622858633 T^{4} - 77747056 T^{6} + 7058391 T^{8} - 269666 T^{10} + 7481 T^{12} - 103 T^{14} + T^{16}$$
$41$ $$( 7396 + 30663 T^{2} + 3089 T^{4} + 100 T^{6} + T^{8} )^{2}$$
$43$ $$( -3552 + 1363 T - 115 T^{2} - 8 T^{3} + T^{4} )^{4}$$
$47$ $$1442919878656 - 465745077248 T^{2} + 127831823872 T^{4} - 6650300736 T^{6} + 250815968 T^{8} - 4001204 T^{10} + 46293 T^{12} - 255 T^{14} + T^{16}$$
$53$ $$( 5517801 - 1585575 T + 519048 T^{2} - 66339 T^{3} + 15228 T^{4} - 1836 T^{5} + 297 T^{6} - 18 T^{7} + T^{8} )^{2}$$
$59$ $$909087685468561 - 51157912003239 T^{2} + 1831316357778 T^{4} - 40376042321 T^{6} + 654335516 T^{8} - 7307406 T^{10} + 60121 T^{12} - 308 T^{14} + T^{16}$$
$61$ $$( 1168561 - 610765 T + 248960 T^{2} - 49697 T^{3} + 8696 T^{4} - 740 T^{5} + 101 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$67$ $$1387488001 - 1026768685 T^{2} + 552687536 T^{4} - 139879325 T^{6} + 25925772 T^{8} - 945850 T^{10} + 26839 T^{12} - 180 T^{14} + T^{16}$$
$71$ $$( 4004001 + 1283443 T^{2} + 35851 T^{4} + 332 T^{6} + T^{8} )^{2}$$
$73$ $$2667616624656 - 505323370044 T^{2} + 66818563933 T^{4} - 4547587215 T^{6} + 223683481 T^{8} - 4407166 T^{10} + 62959 T^{12} - 284 T^{14} + T^{16}$$
$79$ $$( 199487376 + 15183300 T + 5180965 T^{2} - 193383 T^{3} + 71401 T^{4} - 1010 T^{5} + 301 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$83$ $$( 26244 + 57631 T^{2} + 16139 T^{4} + 252 T^{6} + T^{8} )^{2}$$
$89$ $$1296 - 155340 T^{2} + 18588193 T^{4} - 3715714 T^{6} + 514313 T^{8} - 37056 T^{10} + 1947 T^{12} - 53 T^{14} + T^{16}$$
$97$ $$( 3364 + 2723 T^{2} + 708 T^{4} + 63 T^{6} + T^{8} )^{2}$$
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