# Properties

 Label 231.2.c.a Level 231 Weight 2 Character orbit 231.c Analytic conductor 1.845 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 231.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{13}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{2} -\beta_{8} q^{3} + ( -1 + \beta_{2} ) q^{4} -\beta_{4} q^{5} -\beta_{12} q^{6} + ( \beta_{3} + \beta_{13} ) q^{7} + \beta_{3} q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{6} q^{2} -\beta_{8} q^{3} + ( -1 + \beta_{2} ) q^{4} -\beta_{4} q^{5} -\beta_{12} q^{6} + ( \beta_{3} + \beta_{13} ) q^{7} + \beta_{3} q^{8} - q^{9} + ( 2 \beta_{10} + \beta_{15} ) q^{10} + ( 1 - \beta_{6} - \beta_{7} + \beta_{9} ) q^{11} + ( \beta_{5} + \beta_{8} ) q^{12} + ( -\beta_{10} - \beta_{12} - \beta_{15} ) q^{13} + ( 1 + \beta_{1} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{14} + ( -1 - \beta_{1} ) q^{15} + ( \beta_{1} + \beta_{2} ) q^{16} + 2 \beta_{15} q^{17} + \beta_{6} q^{18} + ( \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{19} + ( \beta_{4} + \beta_{5} + 2 \beta_{8} + 2 \beta_{11} - \beta_{13} + \beta_{14} ) q^{20} + ( -\beta_{10} + \beta_{14} ) q^{21} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{22} + ( 2 - 2 \beta_{2} + 2 \beta_{7} - \beta_{13} + \beta_{14} ) q^{23} -\beta_{10} q^{24} + ( -2 - \beta_{1} + 2 \beta_{7} - \beta_{13} + \beta_{14} ) q^{25} + ( -2 \beta_{4} + \beta_{5} + 2 \beta_{8} - 2 \beta_{11} + \beta_{13} - \beta_{14} ) q^{26} + \beta_{8} q^{27} + ( -\beta_{3} - 2 \beta_{9} + 2 \beta_{12} + \beta_{13} + \beta_{15} ) q^{28} + ( -\beta_{3} + \beta_{6} - \beta_{9} + \beta_{13} - \beta_{14} ) q^{29} + ( 2 \beta_{3} + \beta_{9} ) q^{30} + ( 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{8} ) q^{31} + ( \beta_{3} + 2 \beta_{6} - \beta_{9} ) q^{32} + ( -\beta_{8} - \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{33} + ( 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} ) q^{34} + ( -\beta_{3} + \beta_{6} + \beta_{9} - 2 \beta_{10} + 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{35} + ( 1 - \beta_{2} ) q^{36} + ( 1 + \beta_{1} - 2 \beta_{7} + \beta_{13} - \beta_{14} ) q^{37} + ( -\beta_{5} - 4 \beta_{8} ) q^{38} + ( -\beta_{3} + \beta_{6} - \beta_{9} ) q^{39} + ( -\beta_{10} + 3 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{40} + ( 2 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{41} + ( 1 - \beta_{2} - \beta_{4} + \beta_{7} - \beta_{11} ) q^{42} + ( -4 \beta_{3} + 2 \beta_{6} - \beta_{13} + \beta_{14} ) q^{43} + ( -3 - 2 \beta_{1} - 2 \beta_{3} + \beta_{6} + \beta_{7} - 2 \beta_{13} + 2 \beta_{14} ) q^{44} + \beta_{4} q^{45} + ( -4 \beta_{6} + 2 \beta_{9} + \beta_{13} - \beta_{14} ) q^{46} + ( \beta_{4} - 2 \beta_{5} + 2 \beta_{8} + 2 \beta_{11} - \beta_{13} + \beta_{14} ) q^{47} + ( -\beta_{4} + \beta_{5} + \beta_{8} ) q^{48} + ( -1 - 2 \beta_{1} + 2 \beta_{8} - 2 \beta_{11} + \beta_{13} - \beta_{14} ) q^{49} + ( 4 \beta_{3} + \beta_{6} + 3 \beta_{9} + \beta_{13} - \beta_{14} ) q^{50} + 2 \beta_{9} q^{51} + ( 3 \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{52} + ( 2 - 2 \beta_{2} ) q^{53} + \beta_{12} q^{54} + ( -2 \beta_{4} - 2 \beta_{5} + \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{55} + ( -3 - \beta_{1} + \beta_{4} + \beta_{7} - 3 \beta_{8} - \beta_{11} ) q^{56} + ( \beta_{3} - \beta_{6} - \beta_{9} + \beta_{13} - \beta_{14} ) q^{57} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{58} + ( \beta_{4} - 2 \beta_{5} - 2 \beta_{8} - 2 \beta_{11} + \beta_{13} - \beta_{14} ) q^{59} + ( 3 + \beta_{1} - \beta_{2} - 2 \beta_{7} + \beta_{13} - \beta_{14} ) q^{60} -4 \beta_{12} q^{61} + ( -2 \beta_{10} - 6 \beta_{12} - 2 \beta_{15} ) q^{62} + ( -\beta_{3} - \beta_{13} ) q^{63} + ( 7 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{7} - \beta_{13} + \beta_{14} ) q^{64} + ( -\beta_{3} - 3 \beta_{6} - \beta_{9} ) q^{65} + ( -\beta_{4} + \beta_{5} + 2 \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{66} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{7} - \beta_{13} + \beta_{14} ) q^{67} + ( -6 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{68} + ( -2 \beta_{5} - 2 \beta_{8} + 2 \beta_{11} - \beta_{13} + \beta_{14} ) q^{69} + ( 2 + \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - 6 \beta_{8} - 2 \beta_{11} + 2 \beta_{13} - 2 \beta_{14} ) q^{70} + ( -4 - 2 \beta_{1} - 2 \beta_{7} + \beta_{13} - \beta_{14} ) q^{71} -\beta_{3} q^{72} + ( -5 \beta_{10} + 3 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} ) q^{73} + ( -4 \beta_{3} - 3 \beta_{9} - \beta_{13} + \beta_{14} ) q^{74} + ( \beta_{4} + \beta_{8} + 2 \beta_{11} - \beta_{13} + \beta_{14} ) q^{75} + ( 3 \beta_{10} - 3 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{76} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} + 2 \beta_{12} ) q^{77} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{7} - \beta_{13} + \beta_{14} ) q^{78} + ( 6 \beta_{6} - 2 \beta_{9} + 3 \beta_{13} - 3 \beta_{14} ) q^{79} + ( \beta_{5} - 4 \beta_{8} ) q^{80} + q^{81} + ( 2 \beta_{5} + 4 \beta_{8} ) q^{82} + ( 4 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{83} + ( -2 \beta_{6} + \beta_{9} + \beta_{10} + \beta_{14} + 2 \beta_{15} ) q^{84} + ( 4 \beta_{3} + 2 \beta_{9} - 2 \beta_{13} + 2 \beta_{14} ) q^{85} + ( -4 \beta_{1} + 2 \beta_{7} - \beta_{13} + \beta_{14} ) q^{86} + ( \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{87} + ( 3 \beta_{3} - \beta_{6} + \beta_{7} + \beta_{9} - \beta_{13} + \beta_{14} ) q^{88} + ( 2 \beta_{4} - 2 \beta_{8} + 2 \beta_{11} - \beta_{13} + \beta_{14} ) q^{89} + ( -2 \beta_{10} - \beta_{15} ) q^{90} + ( 1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{7} + 4 \beta_{8} - \beta_{13} + \beta_{14} ) q^{91} + ( -8 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{7} + \beta_{13} - \beta_{14} ) q^{92} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{93} + ( -2 \beta_{10} + \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{94} + ( -\beta_{3} + \beta_{6} - 3 \beta_{9} + 2 \beta_{13} - 2 \beta_{14} ) q^{95} + ( -\beta_{10} + 2 \beta_{12} + \beta_{15} ) q^{96} + ( -2 \beta_{4} + 6 \beta_{5} + 8 \beta_{8} ) q^{97} + ( 4 \beta_{3} - \beta_{6} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{98} + ( -1 + \beta_{6} + \beta_{7} - \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 12q^{4} - 16q^{9} + O(q^{10})$$ $$16q - 12q^{4} - 16q^{9} + 16q^{11} + 8q^{14} - 8q^{15} - 4q^{16} - 20q^{22} + 24q^{23} - 24q^{25} + 12q^{36} + 8q^{37} + 12q^{42} - 32q^{44} + 24q^{53} - 40q^{56} - 12q^{58} + 36q^{60} + 88q^{64} - 32q^{67} + 36q^{70} - 48q^{71} + 12q^{78} + 16q^{81} + 32q^{86} + 16q^{91} - 128q^{92} - 40q^{93} - 16q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + x^{14} - 4 x^{12} - 49 x^{10} + 11 x^{8} + 395 x^{6} + 900 x^{4} + 1125 x^{2} + 625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-75631 \nu^{14} - 26336 \nu^{12} + 423219 \nu^{10} + 3371514 \nu^{8} - 2776621 \nu^{6} - 34025725 \nu^{4} - 47891850 \nu^{2} - 47672250$$$$)/15616375$$ $$\beta_{2}$$ $$=$$ $$($$$$-122987 \nu^{14} - 97172 \nu^{12} + 701513 \nu^{10} + 5799603 \nu^{8} - 3932417 \nu^{6} - 54634025 \nu^{4} - 77779875 \nu^{2} - 52412250$$$$)/15616375$$ $$\beta_{3}$$ $$=$$ $$($$$$-1660556 \nu^{14} + 460994 \nu^{12} + 5462774 \nu^{10} + 77606794 \nu^{8} - 106849066 \nu^{6} - 503635320 \nu^{4} - 927067650 \nu^{2} - 854571625$$$$)/ 171780125$$ $$\beta_{4}$$ $$=$$ $$($$$$-739864 \nu^{15} + 254886 \nu^{13} + 6858931 \nu^{11} + 26926811 \nu^{9} - 104639904 \nu^{7} - 341075305 \nu^{5} + 72330250 \nu^{3} + 1014860875 \nu$$$$)/ 858900625$$ $$\beta_{5}$$ $$=$$ $$($$$$1157122 \nu^{15} - 1428203 \nu^{13} - 8678488 \nu^{11} - 43919978 \nu^{9} + 137248467 \nu^{7} + 474206065 \nu^{5} - 222668250 \nu^{3} - 1705295750 \nu$$$$)/ 858900625$$ $$\beta_{6}$$ $$=$$ $$($$$$-4995212 \nu^{14} + 2030168 \nu^{12} + 16902678 \nu^{10} + 220892193 \nu^{8} - 365647252 \nu^{6} - 1473068360 \nu^{4} - 2412896875 \nu^{2} - 2203374500$$$$)/ 171780125$$ $$\beta_{7}$$ $$=$$ $$($$$$5423411 \nu^{14} - 2714684 \nu^{12} - 18323989 \nu^{10} - 234916384 \nu^{8} + 402906301 \nu^{6} + 1537657175 \nu^{4} + 2367403750 \nu^{2} + 2546363000$$$$)/ 171780125$$ $$\beta_{8}$$ $$=$$ $$($$$$-2293 \nu^{15} - 486 \nu^{13} + 12284 \nu^{11} + 101834 \nu^{9} - 117086 \nu^{7} - 906528 \nu^{5} - 1102430 \nu^{3} - 955700 \nu$$$$)/633875$$ $$\beta_{9}$$ $$=$$ $$($$$$691353 \nu^{14} - 217862 \nu^{12} - 2440727 \nu^{10} - 30839287 \nu^{8} + 47497193 \nu^{6} + 206180345 \nu^{4} + 360618175 \nu^{2} + 329532250$$$$)/15616375$$ $$\beta_{10}$$ $$=$$ $$($$$$631743 \nu^{15} - 346072 \nu^{13} - 1947962 \nu^{11} - 27635822 \nu^{9} + 50248808 \nu^{7} + 172281470 \nu^{5} + 262942100 \nu^{3} + 255027250 \nu$$$$)/78081875$$ $$\beta_{11}$$ $$=$$ $$($$$$276139 \nu^{15} + 1073735 \nu^{14} + 41490 \nu^{13} - 529596 \nu^{12} - 1383000 \nu^{11} - 3783171 \nu^{10} - 12036710 \nu^{9} - 46979616 \nu^{8} + 14341710 \nu^{7} + 81866729 \nu^{6} + 109481046 \nu^{5} + 310606859 \nu^{4} + 132560550 \nu^{3} + 477661630 \nu^{2} + 115019500 \nu + 434987125$$$$)/34356025$$ $$\beta_{12}$$ $$=$$ $$($$$$-1923341 \nu^{15} + 737049 \nu^{13} + 6650629 \nu^{11} + 85453649 \nu^{9} - 139204361 \nu^{7} - 570693280 \nu^{5} - 950462600 \nu^{3} - 861041125 \nu$$$$)/78081875$$ $$\beta_{13}$$ $$=$$ $$($$$$23106644 \nu^{15} + 26843375 \nu^{14} - 6020806 \nu^{13} - 13239900 \nu^{12} - 81123251 \nu^{11} - 94579275 \nu^{10} - 1034055231 \nu^{9} - 1174490400 \nu^{8} + 1552651034 \nu^{7} + 2046668225 \nu^{6} + 7031765455 \nu^{5} + 7765171475 \nu^{4} + 12171549500 \nu^{3} + 11941540750 \nu^{2} + 10753920375 \nu + 10874678125$$$$)/ 858900625$$ $$\beta_{14}$$ $$=$$ $$($$$$23106644 \nu^{15} - 26843375 \nu^{14} - 6020806 \nu^{13} + 13239900 \nu^{12} - 81123251 \nu^{11} + 94579275 \nu^{10} - 1034055231 \nu^{9} + 1174490400 \nu^{8} + 1552651034 \nu^{7} - 2046668225 \nu^{6} + 7031765455 \nu^{5} - 7765171475 \nu^{4} + 12171549500 \nu^{3} - 11941540750 \nu^{2} + 10753920375 \nu - 10874678125$$$$)/ 858900625$$ $$\beta_{15}$$ $$=$$ $$($$$$-2876824 \nu^{15} + 1459141 \nu^{13} + 9708086 \nu^{11} + 124820466 \nu^{9} - 221224649 \nu^{7} - 811428665 \nu^{5} - 1284311600 \nu^{3} - 1203761250 \nu$$$$)/78081875$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{14} - \beta_{13} - 2 \beta_{12} - 2 \beta_{8} - 2 \beta_{5}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{9} - \beta_{7} + \beta_{3} + \beta_{2} - \beta_{1} - 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} - 9 \beta_{10} + 7 \beta_{8} - 2 \beta_{4}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{14} - \beta_{13} - 6 \beta_{9} - 2 \beta_{7} - 12 \beta_{6} - 2 \beta_{3} + 2 \beta_{2} - 8 \beta_{1}$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$6 \beta_{15} + 5 \beta_{14} - 15 \beta_{13} - 18 \beta_{12} + 20 \beta_{11} + 2 \beta_{10} + 36 \beta_{8}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-6 \beta_{14} + 6 \beta_{13} - \beta_{9} - 29 \beta_{7} - 31 \beta_{6} + 29 \beta_{3} - 4 \beta_{2} + 14 \beta_{1} + 75$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$11 \beta_{15} - 4 \beta_{14} - 15 \beta_{13} - 35 \beta_{12} + 11 \beta_{11} + 11 \beta_{10} + 35 \beta_{8} - 46 \beta_{5} - 76 \beta_{4}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$25 \beta_{14} - 25 \beta_{13} + 19 \beta_{9} + 19 \beta_{7} - 34 \beta_{6} + 91 \beta_{3} + 15 \beta_{2} - 27 \beta_{1} - 65$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-113 \beta_{15} + 134 \beta_{14} - 57 \beta_{13} + 191 \beta_{12} + 191 \beta_{11} - 191 \beta_{10} + 419 \beta_{8} - 76 \beta_{5} - 114 \beta_{4}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$265 \beta_{14} - 265 \beta_{13} - 248 \beta_{9} + 170 \beta_{7} - 744 \beta_{6} - 56 \beta_{3} - 398$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$247 \beta_{15} + 494 \beta_{14} - 363 \beta_{13} + 53 \beta_{12} + 857 \beta_{11} + 857 \beta_{10} + 1923 \beta_{8} - 38 \beta_{5} - 78 \beta_{4}$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$305 \beta_{14} - 305 \beta_{13} + 77 \beta_{9} - 77 \beta_{7} - 1065 \beta_{6} + 1337 \beta_{3} - 988 \beta_{2} + 1598 \beta_{1} + 1793$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$169 \beta_{15} + 571 \beta_{14} + 740 \beta_{13} + 2039 \beta_{12} - 169 \beta_{11} + 2601 \beta_{10} - 393 \beta_{8} - 1870 \beta_{5} - 3012 \beta_{4}$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$3376 \beta_{14} - 3376 \beta_{13} + 1749 \beta_{9} + 5903 \beta_{7} - 287 \beta_{6} + 5903 \beta_{3} - 338 \beta_{2} + 562 \beta_{1} - 12655$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$-1546 \beta_{15} + 1660 \beta_{14} + 620 \beta_{13} + 4638 \beta_{12} + 1040 \beta_{11} - 367 \beta_{10} + 2334 \beta_{8}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/231\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 −0.917186 − 1.66637i 0.917186 + 1.66637i 0.0566033 − 1.17421i −0.0566033 + 1.17421i −1.86824 + 0.357358i 1.86824 − 0.357358i 0.644389 + 0.983224i −0.644389 − 0.983224i −0.644389 + 0.983224i 0.644389 − 0.983224i 1.86824 + 0.357358i −1.86824 − 0.357358i −0.0566033 − 1.17421i 0.0566033 + 1.17421i 0.917186 − 1.66637i −0.917186 + 1.66637i
2.30927i 1.00000i −3.33275 3.77447i −2.30927 −0.474903 + 2.60278i 3.07768i −1.00000 −8.71628
76.2 2.30927i 1.00000i −3.33275 3.77447i 2.30927 0.474903 + 2.60278i 3.07768i −1.00000 8.71628
76.3 2.08529i 1.00000i −2.34841 0.833366i −2.08529 −2.19849 1.47195i 0.726543i −1.00000 1.73781
76.4 2.08529i 1.00000i −2.34841 0.833366i 2.08529 2.19849 1.47195i 0.726543i −1.00000 −1.73781
76.5 1.13370i 1.00000i 0.714715 2.77447i −1.13370 2.60278 0.474903i 3.07768i −1.00000 3.14542
76.6 1.13370i 1.00000i 0.714715 2.77447i 1.13370 −2.60278 0.474903i 3.07768i −1.00000 −3.14542
76.7 0.183172i 1.00000i 1.96645 1.83337i −0.183172 −1.47195 2.19849i 0.726543i −1.00000 −0.335821
76.8 0.183172i 1.00000i 1.96645 1.83337i 0.183172 1.47195 2.19849i 0.726543i −1.00000 0.335821
76.9 0.183172i 1.00000i 1.96645 1.83337i 0.183172 1.47195 + 2.19849i 0.726543i −1.00000 0.335821
76.10 0.183172i 1.00000i 1.96645 1.83337i −0.183172 −1.47195 + 2.19849i 0.726543i −1.00000 −0.335821
76.11 1.13370i 1.00000i 0.714715 2.77447i 1.13370 −2.60278 + 0.474903i 3.07768i −1.00000 −3.14542
76.12 1.13370i 1.00000i 0.714715 2.77447i −1.13370 2.60278 + 0.474903i 3.07768i −1.00000 3.14542
76.13 2.08529i 1.00000i −2.34841 0.833366i 2.08529 2.19849 + 1.47195i 0.726543i −1.00000 −1.73781
76.14 2.08529i 1.00000i −2.34841 0.833366i −2.08529 −2.19849 + 1.47195i 0.726543i −1.00000 1.73781
76.15 2.30927i 1.00000i −3.33275 3.77447i 2.30927 0.474903 2.60278i 3.07768i −1.00000 8.71628
76.16 2.30927i 1.00000i −3.33275 3.77447i −2.30927 −0.474903 2.60278i 3.07768i −1.00000 −8.71628
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 76.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.c.a 16
3.b odd 2 1 693.2.c.e 16
4.b odd 2 1 3696.2.q.e 16
7.b odd 2 1 inner 231.2.c.a 16
11.b odd 2 1 inner 231.2.c.a 16
21.c even 2 1 693.2.c.e 16
28.d even 2 1 3696.2.q.e 16
33.d even 2 1 693.2.c.e 16
44.c even 2 1 3696.2.q.e 16
77.b even 2 1 inner 231.2.c.a 16
231.h odd 2 1 693.2.c.e 16
308.g odd 2 1 3696.2.q.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.c.a 16 1.a even 1 1 trivial
231.2.c.a 16 7.b odd 2 1 inner
231.2.c.a 16 11.b odd 2 1 inner
231.2.c.a 16 77.b even 2 1 inner
693.2.c.e 16 3.b odd 2 1
693.2.c.e 16 21.c even 2 1
693.2.c.e 16 33.d even 2 1
693.2.c.e 16 231.h odd 2 1
3696.2.q.e 16 4.b odd 2 1
3696.2.q.e 16 28.d even 2 1
3696.2.q.e 16 44.c even 2 1
3696.2.q.e 16 308.g odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(231, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 5 T^{2} + 16 T^{4} - 45 T^{6} + 101 T^{8} - 180 T^{10} + 256 T^{12} - 320 T^{14} + 256 T^{16} )^{2}$$
$3$ $$( 1 + T^{2} )^{8}$$
$5$ $$( 1 - 14 T^{2} + 121 T^{4} - 774 T^{6} + 4196 T^{8} - 19350 T^{10} + 75625 T^{12} - 218750 T^{14} + 390625 T^{16} )^{2}$$
$7$ $$1 - 4 T^{4} - 314 T^{8} - 9604 T^{12} + 5764801 T^{16}$$
$11$ $$( 1 - 4 T + 6 T^{2} - 44 T^{3} + 121 T^{4} )^{4}$$
$13$ $$( 1 + 70 T^{2} + 2381 T^{4} + 51950 T^{6} + 796636 T^{8} + 8779550 T^{10} + 68003741 T^{12} + 337876630 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 + 20 T^{2} + 1076 T^{4} + 16300 T^{6} + 457366 T^{8} + 4710700 T^{10} + 89868596 T^{12} + 482751380 T^{14} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 + 102 T^{2} + 5093 T^{4} + 161814 T^{6} + 3616780 T^{8} + 58414854 T^{10} + 663724853 T^{12} + 4798679862 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 6 T + 48 T^{2} - 270 T^{3} + 1086 T^{4} - 6210 T^{5} + 25392 T^{6} - 73002 T^{7} + 279841 T^{8} )^{4}$$
$29$ $$( 1 - 158 T^{2} + 12293 T^{4} - 609286 T^{6} + 20954380 T^{8} - 512409526 T^{10} + 8694605333 T^{12} - 93982084718 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 - 108 T^{2} + 7028 T^{4} - 324436 T^{6} + 11297430 T^{8} - 311782996 T^{10} + 6490505588 T^{12} - 95850397548 T^{14} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 - 2 T + 97 T^{2} - 250 T^{3} + 4636 T^{4} - 9250 T^{5} + 132793 T^{6} - 101306 T^{7} + 1874161 T^{8} )^{4}$$
$41$ $$( 1 + 192 T^{2} + 16508 T^{4} + 888384 T^{6} + 38165830 T^{8} + 1493373504 T^{10} + 46647662588 T^{12} + 912020014272 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 - 140 T^{2} + 13796 T^{4} - 867860 T^{6} + 43988886 T^{8} - 1604673140 T^{10} + 47165778596 T^{12} - 884990826860 T^{14} + 11688200277601 T^{16} )^{2}$$
$47$ $$( 1 - 230 T^{2} + 28521 T^{4} - 2266390 T^{6} + 126631796 T^{8} - 5006455510 T^{10} + 139173381801 T^{12} - 2479219525670 T^{14} + 23811286661761 T^{16} )^{2}$$
$53$ $$( 1 - 6 T + 188 T^{2} - 850 T^{3} + 14486 T^{4} - 45050 T^{5} + 528092 T^{6} - 893262 T^{7} + 7890481 T^{8} )^{4}$$
$59$ $$( 1 - 278 T^{2} + 40073 T^{4} - 3829126 T^{6} + 263019940 T^{8} - 13329187606 T^{10} + 485579007353 T^{12} - 11726188352198 T^{14} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 + 312 T^{2} + 48988 T^{4} + 5009224 T^{6} + 360566630 T^{8} + 18639322504 T^{10} + 678280058908 T^{12} + 16074356800632 T^{14} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 + 8 T + 207 T^{2} + 1600 T^{3} + 18936 T^{4} + 107200 T^{5} + 929223 T^{6} + 2406104 T^{7} + 20151121 T^{8} )^{4}$$
$71$ $$( 1 + 12 T + 248 T^{2} + 1804 T^{3} + 23150 T^{4} + 128084 T^{5} + 1250168 T^{6} + 4294932 T^{7} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 - 10 T^{2} + 18381 T^{4} - 178370 T^{6} + 140146796 T^{8} - 950533730 T^{10} + 521988067821 T^{12} - 1513342262890 T^{14} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 - 168 T^{2} + 16988 T^{4} - 1063576 T^{6} + 82499910 T^{8} - 6637777816 T^{10} + 661683976028 T^{12} - 40838692527528 T^{14} + 1517108809906561 T^{16} )^{2}$$
$83$ $$( 1 + 180 T^{2} + 21956 T^{4} + 2097260 T^{6} + 156627606 T^{8} + 14448024140 T^{10} + 1041994895876 T^{12} + 58849267206420 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 - 528 T^{2} + 133148 T^{4} - 20911856 T^{6} + 2232438150 T^{8} - 165642811376 T^{10} + 8354003904668 T^{12} - 262406121627408 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 - 60 T^{2} + 15076 T^{4} - 1727620 T^{6} + 110798006 T^{8} - 16255176580 T^{10} + 1334667440356 T^{12} - 49978320295740 T^{14} + 7837433594376961 T^{16} )^{2}$$