Properties

Label 1155.2.i.c
Level 1155
Weight 2
Character orbit 1155.i
Analytic conductor 9.223
Analytic rank 0
Dimension 16
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1155.i (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16} \)
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{8} q^{2} \) \( + \beta_{9} q^{3} \) \( + ( -2 - \beta_{3} ) q^{4} \) \( -\beta_{9} q^{5} \) \( -\beta_{14} q^{6} \) \( + ( -\beta_{7} + \beta_{14} ) q^{7} \) \( + ( -\beta_{4} + \beta_{8} ) q^{8} \) \(- q^{9}\) \(+O(q^{10})\) \( q\) \( -\beta_{8} q^{2} \) \( + \beta_{9} q^{3} \) \( + ( -2 - \beta_{3} ) q^{4} \) \( -\beta_{9} q^{5} \) \( -\beta_{14} q^{6} \) \( + ( -\beta_{7} + \beta_{14} ) q^{7} \) \( + ( -\beta_{4} + \beta_{8} ) q^{8} \) \(- q^{9}\) \( + \beta_{14} q^{10} \) \( + ( 1 + \beta_{1} - \beta_{10} ) q^{11} \) \( + ( \beta_{5} - 2 \beta_{9} ) q^{12} \) \( + ( -\beta_{11} + 2 \beta_{15} ) q^{13} \) \( + ( 1 - \beta_{1} - \beta_{5} + 4 \beta_{9} ) q^{14} \) \(+ q^{15}\) \( + ( 2 + 2 \beta_{1} + \beta_{3} ) q^{16} \) \( + ( 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{17} \) \( + \beta_{8} q^{18} \) \( + ( -2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{19} \) \( + ( -\beta_{5} + 2 \beta_{9} ) q^{20} \) \( + ( -\beta_{8} - \beta_{13} ) q^{21} \) \( + ( -\beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} ) q^{22} \) \( + ( 3 + \beta_{3} ) q^{23} \) \( + ( -\beta_{11} + \beta_{14} ) q^{24} \) \(- q^{25}\) \( + ( -2 \beta_{2} - 2 \beta_{5} + 2 \beta_{9} - 4 \beta_{12} ) q^{26} \) \( -\beta_{9} q^{27} \) \( + ( -\beta_{4} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} - 3 \beta_{14} ) q^{28} \) \( + ( -\beta_{4} + \beta_{8} - \beta_{10} ) q^{29} \) \( -\beta_{8} q^{30} \) \( + ( -\beta_{2} - \beta_{5} - 2 \beta_{9} - \beta_{12} ) q^{31} \) \( + ( \beta_{4} - 2 \beta_{7} - \beta_{8} - 2 \beta_{10} ) q^{32} \) \( + ( \beta_{9} - \beta_{12} - \beta_{15} ) q^{33} \) \( + ( -\beta_{2} - \beta_{5} + 2 \beta_{9} - 3 \beta_{12} ) q^{34} \) \( + ( \beta_{8} + \beta_{13} ) q^{35} \) \( + ( 2 + \beta_{3} ) q^{36} \) \( + ( -2 + \beta_{1} + \beta_{3} - \beta_{6} ) q^{37} \) \( + ( -\beta_{2} + \beta_{5} - 2 \beta_{9} + \beta_{12} ) q^{38} \) \( + ( \beta_{4} - 2 \beta_{10} ) q^{39} \) \( + ( \beta_{11} - \beta_{14} ) q^{40} \) \( + ( \beta_{11} - 2 \beta_{14} + 2 \beta_{15} ) q^{41} \) \( + ( -4 - \beta_{3} + \beta_{9} + \beta_{12} ) q^{42} \) \( + ( -3 \beta_{4} - \beta_{8} + \beta_{10} ) q^{43} \) \( + ( -3 - 2 \beta_{1} - 3 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} + 2 \beta_{10} ) q^{44} \) \( + \beta_{9} q^{45} \) \( + ( \beta_{4} - 4 \beta_{8} ) q^{46} \) \( + ( -\beta_{2} - \beta_{5} + 8 \beta_{9} - \beta_{12} ) q^{47} \) \( + ( -\beta_{5} + 2 \beta_{9} - 2 \beta_{12} ) q^{48} \) \( + ( 1 + 2 \beta_{3} - 2 \beta_{9} - 2 \beta_{12} ) q^{49} \) \( + \beta_{8} q^{50} \) \( + ( -2 \beta_{7} - \beta_{8} - \beta_{10} ) q^{51} \) \( + ( 4 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} - 6 \beta_{15} ) q^{52} \) \( + ( -1 - 2 \beta_{1} - 3 \beta_{3} + 2 \beta_{6} ) q^{53} \) \( + \beta_{14} q^{54} \) \( + ( -\beta_{9} + \beta_{12} + \beta_{15} ) q^{55} \) \( + ( 1 + \beta_{3} + 3 \beta_{5} - \beta_{6} - 6 \beta_{9} + 2 \beta_{12} ) q^{56} \) \( + ( 2 \beta_{7} + \beta_{8} - \beta_{10} ) q^{57} \) \( + ( 6 + \beta_{1} + 3 \beta_{3} + \beta_{6} ) q^{58} \) \( + ( \beta_{2} - 3 \beta_{9} + \beta_{12} ) q^{59} \) \( + ( -2 - \beta_{3} ) q^{60} \) \( + ( -2 \beta_{11} - 2 \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{61} \) \( + ( 2 \beta_{11} + 2 \beta_{14} - 4 \beta_{15} ) q^{62} \) \( + ( \beta_{7} - \beta_{14} ) q^{63} \) \( + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{6} ) q^{64} \) \( + ( -\beta_{4} + 2 \beta_{10} ) q^{65} \) \( + ( \beta_{2} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{66} \) \( + ( 4 + 2 \beta_{1} - 2 \beta_{3} ) q^{67} \) \( + ( 4 \beta_{11} + 2 \beta_{13} - 4 \beta_{15} ) q^{68} \) \( + ( -\beta_{5} + 3 \beta_{9} ) q^{69} \) \( + ( 4 + \beta_{3} - \beta_{9} - \beta_{12} ) q^{70} \) \( + ( -2 - 3 \beta_{1} - \beta_{3} + \beta_{6} ) q^{71} \) \( + ( \beta_{4} - \beta_{8} ) q^{72} \) \( + ( -2 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{73} \) \( + ( 2 \beta_{4} + 2 \beta_{8} - 4 \beta_{10} ) q^{74} \) \( -\beta_{9} q^{75} \) \( + ( -2 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} ) q^{76} \) \( + ( -\beta_{2} - \beta_{4} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{77} \) \( + ( -2 - 4 \beta_{1} - 2 \beta_{3} + 2 \beta_{6} ) q^{78} \) \( + ( \beta_{4} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{10} ) q^{79} \) \( + ( \beta_{5} - 2 \beta_{9} + 2 \beta_{12} ) q^{80} \) \(+ q^{81}\) \( + ( -2 \beta_{2} + 4 \beta_{5} - 10 \beta_{9} ) q^{82} \) \( + ( 2 \beta_{11} + 2 \beta_{13} + \beta_{14} - 5 \beta_{15} ) q^{83} \) \( + ( -\beta_{4} + 3 \beta_{8} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{84} \) \( + ( 2 \beta_{7} + \beta_{8} + \beta_{10} ) q^{85} \) \( + ( 2 + 7 \beta_{1} + 5 \beta_{3} - \beta_{6} ) q^{86} \) \( + ( -\beta_{11} + \beta_{14} - \beta_{15} ) q^{87} \) \( + ( -2 + 2 \beta_{1} + \beta_{3} - 3 \beta_{4} - \beta_{7} + 3 \beta_{8} + 3 \beta_{10} ) q^{88} \) \( + ( \beta_{2} - 3 \beta_{9} - \beta_{12} ) q^{89} \) \( -\beta_{14} q^{90} \) \( + ( 2 + 4 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} - 2 \beta_{6} + 2 \beta_{9} + \beta_{12} ) q^{91} \) \( + ( -12 - 2 \beta_{1} - 4 \beta_{3} ) q^{92} \) \( + ( 2 - \beta_{1} - \beta_{3} + \beta_{6} ) q^{93} \) \( + ( 2 \beta_{11} - 8 \beta_{14} - 4 \beta_{15} ) q^{94} \) \( + ( -2 \beta_{7} - \beta_{8} + \beta_{10} ) q^{95} \) \( + ( \beta_{11} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{96} \) \( + ( \beta_{2} + 2 \beta_{5} + 5 \beta_{9} + \beta_{12} ) q^{97} \) \( + ( 2 \beta_{4} - 3 \beta_{8} + 2 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{98} \) \( + ( -1 - \beta_{1} + \beta_{10} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut -\mathstrut 24q^{4} \) \(\mathstrut -\mathstrut 16q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut -\mathstrut 24q^{4} \) \(\mathstrut -\mathstrut 16q^{9} \) \(\mathstrut +\mathstrut 16q^{11} \) \(\mathstrut +\mathstrut 16q^{14} \) \(\mathstrut +\mathstrut 16q^{15} \) \(\mathstrut +\mathstrut 24q^{16} \) \(\mathstrut +\mathstrut 40q^{23} \) \(\mathstrut -\mathstrut 16q^{25} \) \(\mathstrut +\mathstrut 24q^{36} \) \(\mathstrut -\mathstrut 40q^{37} \) \(\mathstrut -\mathstrut 56q^{42} \) \(\mathstrut -\mathstrut 24q^{44} \) \(\mathstrut +\mathstrut 8q^{53} \) \(\mathstrut +\mathstrut 8q^{56} \) \(\mathstrut +\mathstrut 72q^{58} \) \(\mathstrut -\mathstrut 24q^{60} \) \(\mathstrut +\mathstrut 8q^{64} \) \(\mathstrut +\mathstrut 80q^{67} \) \(\mathstrut +\mathstrut 56q^{70} \) \(\mathstrut -\mathstrut 24q^{71} \) \(\mathstrut -\mathstrut 16q^{78} \) \(\mathstrut +\mathstrut 16q^{81} \) \(\mathstrut -\mathstrut 8q^{86} \) \(\mathstrut -\mathstrut 40q^{88} \) \(\mathstrut +\mathstrut 16q^{91} \) \(\mathstrut -\mathstrut 160q^{92} \) \(\mathstrut +\mathstrut 40q^{93} \) \(\mathstrut -\mathstrut 16q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -4976 \nu^{14} + 6064 \nu^{12} - 53806 \nu^{10} + 1664 \nu^{8} + 584304 \nu^{6} + 1397920 \nu^{4} + 1900400 \nu^{2} - 33766125 \)\()/15616375\)
\(\beta_{2}\)\(=\)\((\)\( 52907 \nu^{15} + 3769822 \nu^{13} + 7571087 \nu^{11} - 23218303 \nu^{9} - 220485308 \nu^{7} - 61141995 \nu^{5} + 2011459400 \nu^{3} + 5720425875 \nu \)\()/\)\(858900625\)
\(\beta_{3}\)\(=\)\((\)\( 75631 \nu^{14} + 26336 \nu^{12} - 423219 \nu^{10} - 3371514 \nu^{8} + 2776621 \nu^{6} + 34025725 \nu^{4} + 47891850 \nu^{2} + 32055875 \)\()/15616375\)
\(\beta_{4}\)\(=\)\((\)\( 1098448 \nu^{14} - 1820672 \nu^{12} - 3507762 \nu^{10} - 38951322 \nu^{8} + 145662708 \nu^{6} + 254483540 \nu^{4} - 92662550 \nu^{2} - 101116250 \)\()/\)\(171780125\)
\(\beta_{5}\)\(=\)\((\)\( 739864 \nu^{15} - 254886 \nu^{13} - 6858931 \nu^{11} - 26926811 \nu^{9} + 104639904 \nu^{7} + 341075305 \nu^{5} - 72330250 \nu^{3} - 1014860875 \nu \)\()/\)\(858900625\)
\(\beta_{6}\)\(=\)\((\)\( 64321 \nu^{14} + 44136 \nu^{12} - 365249 \nu^{10} - 2994144 \nu^{8} + 2128291 \nu^{6} + 28658755 \nu^{4} + 40690320 \nu^{2} + 30499350 \)\()/3123275\)
\(\beta_{7}\)\(=\)\((\)\( -3708119 \nu^{14} + 2186986 \nu^{12} + 13453081 \nu^{10} + 157291286 \nu^{8} - 302484579 \nu^{6} - 1049398975 \nu^{4} - 1461240500 \nu^{2} - 1320364000 \)\()/\)\(171780125\)
\(\beta_{8}\)\(=\)\((\)\( 1073735 \nu^{14} - 529596 \nu^{12} - 3783171 \nu^{10} - 46979616 \nu^{8} + 81866729 \nu^{6} + 310606859 \nu^{4} + 477661630 \nu^{2} + 434987125 \)\()/34356025\)
\(\beta_{9}\)\(=\)\((\)\( 2293 \nu^{15} + 486 \nu^{13} - 12284 \nu^{11} - 101834 \nu^{9} + 117086 \nu^{7} + 906528 \nu^{5} + 1102430 \nu^{3} + 955700 \nu \)\()/633875\)
\(\beta_{10}\)\(=\)\((\)\( -1446284 \nu^{14} + 355734 \nu^{12} + 4966964 \nu^{10} + 65045254 \nu^{8} - 95756546 \nu^{6} - 437603572 \nu^{4} - 798277730 \nu^{2} - 730658725 \)\()/34356025\)
\(\beta_{11}\)\(=\)\((\)\( 515378 \nu^{15} + 520818 \nu^{13} - 2369422 \nu^{11} - 27002382 \nu^{9} + 8880998 \nu^{7} + 226234550 \nu^{5} + 509713400 \nu^{3} + 379881750 \nu \)\()/78081875\)
\(\beta_{12}\)\(=\)\((\)\( 599 \nu^{15} + 90 \nu^{13} - 3000 \nu^{11} - 26110 \nu^{9} + 31110 \nu^{7} + 237486 \nu^{5} + 287550 \nu^{3} + 249500 \nu \)\()/74525\)
\(\beta_{13}\)\(=\)\((\)\( -1468861 \nu^{15} + 201274 \nu^{13} + 5426879 \nu^{11} + 66369199 \nu^{9} - 90901286 \nu^{7} - 466969935 \nu^{5} - 843562400 \nu^{3} - 722601875 \nu \)\()/78081875\)
\(\beta_{14}\)\(=\)\((\)\( 2100604 \nu^{15} - 547346 \nu^{13} - 7374841 \nu^{11} - 94005021 \nu^{9} + 141150094 \nu^{7} + 639251405 \nu^{5} + 1106504500 \nu^{3} + 977629125 \nu \)\()/78081875\)
\(\beta_{15}\)\(=\)\((\)\( -2699561 \nu^{15} + 1648844 \nu^{13} + 8983874 \nu^{11} + 116269094 \nu^{9} - 219278916 \nu^{7} - 742870540 \nu^{5} - 1128269700 \nu^{3} - 1087173250 \nu \)\()/78081875\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(\beta_{15}\mathstrut -\mathstrut \) \(7\) \(\beta_{14}\mathstrut -\mathstrut \) \(9\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(7\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(3\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(11\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(9\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(3\) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(7\) \(\beta_{13}\mathstrut +\mathstrut \) \(10\) \(\beta_{12}\mathstrut +\mathstrut \) \(6\) \(\beta_{11}\mathstrut -\mathstrut \) \(18\) \(\beta_{9}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(15\) \(\beta_{10}\mathstrut -\mathstrut \) \(31\) \(\beta_{8}\mathstrut -\mathstrut \) \(13\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(16\) \(\beta_{4}\mathstrut -\mathstrut \) \(16\) \(\beta_{3}\mathstrut +\mathstrut \) \(29\) \(\beta_{1}\mathstrut +\mathstrut \) \(59\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(12\) \(\beta_{15}\mathstrut +\mathstrut \) \(4\) \(\beta_{14}\mathstrut +\mathstrut \) \(34\) \(\beta_{13}\mathstrut +\mathstrut \) \(11\) \(\beta_{12}\mathstrut +\mathstrut \) \(23\) \(\beta_{11}\mathstrut -\mathstrut \) \(58\) \(\beta_{9}\mathstrut +\mathstrut \) \(99\) \(\beta_{5}\mathstrut +\mathstrut \) \(23\) \(\beta_{2}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(53\) \(\beta_{10}\mathstrut -\mathstrut \) \(191\) \(\beta_{8}\mathstrut -\mathstrut \) \(167\) \(\beta_{7}\mathstrut -\mathstrut \) \(15\) \(\beta_{6}\mathstrut +\mathstrut \) \(15\) \(\beta_{4}\mathstrut +\mathstrut \) \(69\) \(\beta_{3}\mathstrut -\mathstrut \) \(38\) \(\beta_{1}\mathstrut -\mathstrut \) \(61\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(39\) \(\beta_{15}\mathstrut -\mathstrut \) \(153\) \(\beta_{14}\mathstrut -\mathstrut \) \(343\) \(\beta_{13}\mathstrut +\mathstrut \) \(191\) \(\beta_{12}\mathstrut -\mathstrut \) \(152\) \(\beta_{11}\mathstrut -\mathstrut \) \(457\) \(\beta_{9}\mathstrut +\mathstrut \) \(152\) \(\beta_{5}\mathstrut +\mathstrut \) \(38\) \(\beta_{2}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(124\) \(\beta_{10}\mathstrut -\mathstrut \) \(28\) \(\beta_{8}\mathstrut +\mathstrut \) \(276\) \(\beta_{7}\mathstrut +\mathstrut \) \(248\) \(\beta_{4}\mathstrut -\mathstrut \) \(85\) \(\beta_{1}\mathstrut -\mathstrut \) \(199\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(150\) \(\beta_{15}\mathstrut +\mathstrut \) \(838\) \(\beta_{14}\mathstrut +\mathstrut \) \(954\) \(\beta_{13}\mathstrut +\mathstrut \) \(857\) \(\beta_{12}\mathstrut +\mathstrut \) \(97\) \(\beta_{11}\mathstrut -\mathstrut \) \(1942\) \(\beta_{9}\mathstrut +\mathstrut \) \(97\) \(\beta_{5}\mathstrut +\mathstrut \) \(19\) \(\beta_{2}\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(-\)\(571\) \(\beta_{10}\mathstrut -\mathstrut \) \(1453\) \(\beta_{8}\mathstrut -\mathstrut \) \(843\) \(\beta_{7}\mathstrut +\mathstrut \) \(494\) \(\beta_{6}\mathstrut +\mathstrut \) \(494\) \(\beta_{4}\mathstrut -\mathstrut \) \(2092\) \(\beta_{3}\mathstrut +\mathstrut \) \(77\) \(\beta_{1}\mathstrut -\mathstrut \) \(299\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(1104\) \(\beta_{15}\mathstrut +\mathstrut \) \(2808\) \(\beta_{14}\mathstrut +\mathstrut \) \(1666\) \(\beta_{13}\mathstrut -\mathstrut \) \(169\) \(\beta_{12}\mathstrut -\mathstrut \) \(935\) \(\beta_{11}\mathstrut -\mathstrut \) \(542\) \(\beta_{9}\mathstrut +\mathstrut \) \(3947\) \(\beta_{5}\mathstrut +\mathstrut \) \(935\) \(\beta_{2}\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(-\)\(1018\) \(\beta_{10}\mathstrut -\mathstrut \) \(5734\) \(\beta_{8}\mathstrut -\mathstrut \) \(6634\) \(\beta_{7}\mathstrut +\mathstrut \) \(169\) \(\beta_{6}\mathstrut -\mathstrut \) \(731\) \(\beta_{4}\mathstrut -\mathstrut \) \(731\) \(\beta_{3}\mathstrut -\mathstrut \) \(5903\) \(\beta_{1}\mathstrut -\mathstrut \) \(13386\)\()/4\)
\(\nu^{15}\)\(=\)\(1546\) \(\beta_{15}\mathstrut +\mathstrut \) \(367\) \(\beta_{14}\mathstrut -\mathstrut \) \(3459\) \(\beta_{13}\mathstrut +\mathstrut \) \(1040\) \(\beta_{12}\mathstrut -\mathstrut \) \(3092\) \(\beta_{11}\mathstrut -\mathstrut \) \(2334\) \(\beta_{9}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
−1.86824 + 0.357358i
1.86824 0.357358i
−0.0566033 1.17421i
0.0566033 + 1.17421i
−0.644389 + 0.983224i
0.644389 0.983224i
0.917186 1.66637i
−0.917186 + 1.66637i
−0.917186 1.66637i
0.917186 + 1.66637i
0.644389 + 0.983224i
−0.644389 0.983224i
0.0566033 1.17421i
−0.0566033 + 1.17421i
1.86824 + 0.357358i
−1.86824 0.357358i
2.60278i 1.00000i −4.77447 1.00000i −2.60278 2.60278 0.474903i 7.22133i −1.00000 2.60278
76.2 2.60278i 1.00000i −4.77447 1.00000i 2.60278 −2.60278 0.474903i 7.22133i −1.00000 −2.60278
76.3 2.19849i 1.00000i −2.83337 1.00000i −2.19849 2.19849 + 1.47195i 1.83215i −1.00000 2.19849
76.4 2.19849i 1.00000i −2.83337 1.00000i 2.19849 −2.19849 + 1.47195i 1.83215i −1.00000 −2.19849
76.5 1.47195i 1.00000i −0.166634 1.00000i −1.47195 1.47195 + 2.19849i 2.69862i −1.00000 1.47195
76.6 1.47195i 1.00000i −0.166634 1.00000i 1.47195 −1.47195 + 2.19849i 2.69862i −1.00000 −1.47195
76.7 0.474903i 1.00000i 1.77447 1.00000i −0.474903 0.474903 2.60278i 1.79251i −1.00000 0.474903
76.8 0.474903i 1.00000i 1.77447 1.00000i 0.474903 −0.474903 2.60278i 1.79251i −1.00000 −0.474903
76.9 0.474903i 1.00000i 1.77447 1.00000i 0.474903 −0.474903 + 2.60278i 1.79251i −1.00000 −0.474903
76.10 0.474903i 1.00000i 1.77447 1.00000i −0.474903 0.474903 + 2.60278i 1.79251i −1.00000 0.474903
76.11 1.47195i 1.00000i −0.166634 1.00000i 1.47195 −1.47195 2.19849i 2.69862i −1.00000 −1.47195
76.12 1.47195i 1.00000i −0.166634 1.00000i −1.47195 1.47195 2.19849i 2.69862i −1.00000 1.47195
76.13 2.19849i 1.00000i −2.83337 1.00000i 2.19849 −2.19849 1.47195i 1.83215i −1.00000 −2.19849
76.14 2.19849i 1.00000i −2.83337 1.00000i −2.19849 2.19849 1.47195i 1.83215i −1.00000 2.19849
76.15 2.60278i 1.00000i −4.77447 1.00000i 2.60278 −2.60278 + 0.474903i 7.22133i −1.00000 −2.60278
76.16 2.60278i 1.00000i −4.77447 1.00000i −2.60278 2.60278 + 0.474903i 7.22133i −1.00000 2.60278
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.16
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.b Odd 1 yes
11.b Odd 1 yes
77.b Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{8} \) \(\mathstrut +\mathstrut 14 T_{2}^{6} \) \(\mathstrut +\mathstrut 61 T_{2}^{4} \) \(\mathstrut +\mathstrut 84 T_{2}^{2} \) \(\mathstrut +\mathstrut 16 \) acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\).