Properties

Label 336.2.bc.f
Level 336
Weight 2
Character orbit 336.bc
Analytic conductor 2.683
Analytic rank 0
Dimension 16
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 336.bc (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{5} + \beta_{8} ) q^{3} -\beta_{14} q^{5} + \beta_{10} q^{7} -\beta_{1} q^{9} +O(q^{10})\) \( q + ( -\beta_{5} + \beta_{8} ) q^{3} -\beta_{14} q^{5} + \beta_{10} q^{7} -\beta_{1} q^{9} + ( \beta_{9} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{11} + ( \beta_{3} + \beta_{6} - \beta_{8} + \beta_{13} ) q^{13} + ( \beta_{1} - \beta_{2} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{15} + ( 2 \beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{11} - \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{17} + ( \beta_{2} + \beta_{6} - \beta_{10} ) q^{19} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{21} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} - \beta_{13} ) q^{23} + ( \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{13} ) q^{25} + ( 1 - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{27} + ( -\beta_{1} + 2 \beta_{3} - 2 \beta_{5} - \beta_{7} + \beta_{8} + 4 \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{14} - \beta_{15} ) q^{29} + ( 4 - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{13} ) q^{31} + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{9} - \beta_{10} + \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{33} + ( -\beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{13} + 2 \beta_{15} ) q^{35} + ( -1 + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{13} ) q^{37} + ( -\beta_{2} + 3 \beta_{4} - \beta_{6} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{15} ) q^{39} + ( 2 \beta_{3} + 2 \beta_{8} ) q^{41} + ( -2 + 2 \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{13} ) q^{43} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - \beta_{11} - \beta_{14} + \beta_{15} ) q^{45} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{47} + ( -3 + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{49} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} - 3 \beta_{11} - \beta_{14} ) q^{51} + ( -2 \beta_{9} - \beta_{11} - 2 \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{53} + ( -2 + \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - \beta_{6} - 2 \beta_{8} + \beta_{10} - \beta_{13} ) q^{55} + ( -\beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} ) q^{57} + ( 2 - 2 \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} - 4 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} + 3 \beta_{14} ) q^{59} + ( 1 + \beta_{2} + \beta_{4} + \beta_{6} - 2 \beta_{10} + \beta_{13} ) q^{61} + ( 3 - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - 3 \beta_{9} - \beta_{10} - 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{63} + ( 1 - \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} - 3 \beta_{7} - 6 \beta_{8} + \beta_{11} + \beta_{13} + 2 \beta_{14} ) q^{65} + ( -\beta_{2} - 2 \beta_{4} - \beta_{10} + \beta_{13} ) q^{67} + ( 2 - \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{12} ) q^{69} + ( 1 + 3 \beta_{1} - \beta_{2} - 2 \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} - \beta_{12} + 3 \beta_{15} ) q^{71} + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{13} ) q^{73} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} + 3 \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{75} + ( \beta_{1} - 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - 3 \beta_{11} + \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{77} + ( -4 + \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{9} - \beta_{13} ) q^{79} + ( -2 \beta_{3} + 3 \beta_{4} + \beta_{6} - \beta_{10} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{81} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{8} - \beta_{10} - 3 \beta_{11} + 2 \beta_{13} - 2 \beta_{15} ) q^{83} + ( 1 - \beta_{2} + 2 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 4 \beta_{9} + \beta_{10} + 2 \beta_{13} ) q^{85} + ( -8 - 4 \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} + 2 \beta_{11} + 2 \beta_{14} - 2 \beta_{15} ) q^{87} + ( 1 - \beta_{2} + 2 \beta_{3} + \beta_{4} + 4 \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 4 \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{89} + ( -4 + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{8} - 3 \beta_{9} - \beta_{10} - 2 \beta_{13} ) q^{91} + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 6 \beta_{8} - \beta_{9} - \beta_{10} - 3 \beta_{13} - \beta_{14} ) q^{93} + ( -2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{95} + ( -2 + \beta_{2} + \beta_{3} + 4 \beta_{4} - 3 \beta_{6} - \beta_{8} + \beta_{10} - 3 \beta_{13} ) q^{97} + ( -3 - 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{6} - \beta_{7} + 4 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} + 4 \beta_{14} - 3 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 4q^{7} + 2q^{9} + O(q^{10}) \) \( 16q - 4q^{7} + 2q^{9} - 8q^{15} + 6q^{19} + 14q^{21} - 18q^{25} + 48q^{31} - 12q^{33} - 2q^{37} + 22q^{39} - 20q^{43} - 42q^{45} - 28q^{49} - 6q^{51} - 8q^{57} + 36q^{61} + 32q^{63} - 14q^{67} + 30q^{73} - 54q^{75} - 28q^{79} + 30q^{81} + 16q^{85} - 78q^{87} - 66q^{91} + 16q^{93} - 20q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 65 x^{12} - 48 x^{11} - 94 x^{10} + 444 x^{9} - 962 x^{8} + 1332 x^{7} - 846 x^{6} - 1296 x^{5} + 5265 x^{4} - 10206 x^{3} + 13851 x^{2} - 13122 x + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{15} - 15 \nu^{14} + 71 \nu^{13} + 48 \nu^{12} - 110 \nu^{11} + 384 \nu^{10} - 266 \nu^{9} - 252 \nu^{8} + 854 \nu^{7} - 1272 \nu^{6} + 378 \nu^{5} + 2160 \nu^{4} - 14985 \nu^{3} + 21627 \nu^{2} + 28431 \nu - 52488 \)\()/34992\)
\(\beta_{3}\)\(=\)\((\)\( -11 \nu^{15} - 30 \nu^{14} + 142 \nu^{13} - 363 \nu^{12} + 662 \nu^{11} - 258 \nu^{10} - 1288 \nu^{9} + 3546 \nu^{8} - 8138 \nu^{7} + 7392 \nu^{6} + 846 \nu^{5} - 28890 \nu^{4} + 41067 \nu^{3} - 38880 \nu^{2} + 34992 \nu + 19683 \)\()/69984\)
\(\beta_{4}\)\(=\)\((\)\( -22 \nu^{15} + 75 \nu^{14} - 238 \nu^{13} + 489 \nu^{12} - 737 \nu^{11} + 429 \nu^{10} + 1483 \nu^{9} - 5787 \nu^{8} + 11651 \nu^{7} - 14727 \nu^{6} + 5787 \nu^{5} + 23193 \nu^{4} - 70227 \nu^{3} + 126846 \nu^{2} - 147987 \nu + 131220 \)\()/34992\)
\(\beta_{5}\)\(=\)\((\)\( -25 \nu^{15} + 111 \nu^{14} - 295 \nu^{13} + 606 \nu^{12} - 770 \nu^{11} - 66 \nu^{10} + 2656 \nu^{9} - 7812 \nu^{8} + 13268 \nu^{7} - 13818 \nu^{6} - 2826 \nu^{5} + 39366 \nu^{4} - 88047 \nu^{3} + 133893 \nu^{2} - 131949 \nu + 69984 \)\()/23328\)
\(\beta_{6}\)\(=\)\((\)\( -19 \nu^{15} + 95 \nu^{14} - 274 \nu^{13} + 500 \nu^{12} - 653 \nu^{11} + 145 \nu^{10} + 2131 \nu^{9} - 6443 \nu^{8} + 11435 \nu^{7} - 11899 \nu^{6} - 513 \nu^{5} + 29421 \nu^{4} - 73926 \nu^{3} + 110970 \nu^{2} - 131463 \nu + 85293 \)\()/11664\)
\(\beta_{7}\)\(=\)\((\)\(-125 \nu^{15} + 669 \nu^{14} - 2069 \nu^{13} + 4116 \nu^{12} - 5794 \nu^{11} + 1302 \nu^{10} + 16736 \nu^{9} - 53232 \nu^{8} + 92296 \nu^{7} - 97758 \nu^{6} + 3510 \nu^{5} + 239166 \nu^{4} - 587979 \nu^{3} + 857547 \nu^{2} - 888651 \nu + 599238\)\()/69984\)
\(\beta_{8}\)\(=\)\((\)\( 19 \nu^{15} - 82 \nu^{14} + 226 \nu^{13} - 433 \nu^{12} + 542 \nu^{11} - 74 \nu^{10} - 1912 \nu^{9} + 5482 \nu^{8} - 9482 \nu^{7} + 10040 \nu^{6} - 42 \nu^{5} - 24354 \nu^{4} + 60237 \nu^{3} - 95580 \nu^{2} + 105948 \nu - 75087 \)\()/7776\)
\(\beta_{9}\)\(=\)\((\)\(193 \nu^{15} - 753 \nu^{14} + 1975 \nu^{13} - 3624 \nu^{12} + 4292 \nu^{11} - 84 \nu^{10} - 16018 \nu^{9} + 46218 \nu^{8} - 81374 \nu^{7} + 85788 \nu^{6} + 756 \nu^{5} - 197316 \nu^{4} + 515565 \nu^{3} - 849285 \nu^{2} + 958635 \nu - 708588\)\()/69984\)
\(\beta_{10}\)\(=\)\((\)\(100 \nu^{15} - 429 \nu^{14} + 1207 \nu^{13} - 2301 \nu^{12} + 2648 \nu^{11} + 510 \nu^{10} - 10624 \nu^{9} + 29190 \nu^{8} - 48824 \nu^{7} + 45918 \nu^{6} + 9288 \nu^{5} - 142074 \nu^{4} + 312012 \nu^{3} - 474093 \nu^{2} + 502281 \nu - 347733\)\()/34992\)
\(\beta_{11}\)\(=\)\((\)\( -61 \nu^{15} + 302 \nu^{14} - 784 \nu^{13} + 1463 \nu^{12} - 1772 \nu^{11} + 100 \nu^{10} + 6358 \nu^{9} - 18728 \nu^{8} + 32084 \nu^{7} - 33490 \nu^{6} - 1116 \nu^{5} + 85716 \nu^{4} - 197181 \nu^{3} + 331290 \nu^{2} - 374220 \nu + 255879 \)\()/23328\)
\(\beta_{12}\)\(=\)\((\)\(-74 \nu^{15} + 351 \nu^{14} - 1037 \nu^{13} + 2043 \nu^{12} - 2470 \nu^{11} + 234 \nu^{10} + 8450 \nu^{9} - 24816 \nu^{8} + 42964 \nu^{7} - 43500 \nu^{6} - 3870 \nu^{5} + 111618 \nu^{4} - 271836 \nu^{3} + 402489 \nu^{2} - 438615 \nu + 321489\)\()/23328\)
\(\beta_{13}\)\(=\)\((\)\( -41 \nu^{15} + 179 \nu^{14} - 482 \nu^{13} + 962 \nu^{12} - 1225 \nu^{11} + 205 \nu^{10} + 3863 \nu^{9} - 11879 \nu^{8} + 20887 \nu^{7} - 22783 \nu^{6} + 1875 \nu^{5} + 53433 \nu^{4} - 134892 \nu^{3} + 214650 \nu^{2} - 231579 \nu + 165483 \)\()/11664\)
\(\beta_{14}\)\(=\)\((\)\(-275 \nu^{15} + 1101 \nu^{14} - 2921 \nu^{13} + 5466 \nu^{12} - 7066 \nu^{11} + 6 \nu^{10} + 24788 \nu^{9} - 70872 \nu^{8} + 123088 \nu^{7} - 124578 \nu^{6} - 5346 \nu^{5} + 321678 \nu^{4} - 799713 \nu^{3} + 1259955 \nu^{2} - 1358127 \nu + 883548\)\()/69984\)
\(\beta_{15}\)\(=\)\((\)\( -2 \nu^{15} + 8 \nu^{14} - 21 \nu^{13} + 38 \nu^{12} - 41 \nu^{11} - 17 \nu^{10} + 195 \nu^{9} - 497 \nu^{8} + 779 \nu^{7} - 661 \nu^{6} - 349 \nu^{5} + 2499 \nu^{4} - 5247 \nu^{3} + 7371 \nu^{2} - 7452 \nu + 4617 \)\()/432\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} - 2 \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - 4 \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - 3 \beta_{3} + 2 \beta_{1} + 2\)\()/4\)
\(\nu^{2}\)\(=\)\(\beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{15} - 6 \beta_{13} + \beta_{12} + 6 \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - 6 \beta_{6} + 10 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_{1} - 5\)\()/4\)
\(\nu^{4}\)\(=\)\(\beta_{15} - 2 \beta_{14} + \beta_{13} - 2 \beta_{12} - \beta_{10} + \beta_{6} + 3 \beta_{4} - 2 \beta_{3}\)
\(\nu^{5}\)\(=\)\((\)\(-2 \beta_{15} - 6 \beta_{14} + 17 \beta_{13} - 14 \beta_{12} - 7 \beta_{11} - 16 \beta_{10} - 6 \beta_{9} + 7 \beta_{7} - \beta_{6} - 25 \beta_{5} - 15 \beta_{4} - 7 \beta_{3} - \beta_{2} - \beta_{1} - 15\)\()/4\)
\(\nu^{6}\)\(=\)\(-2 \beta_{15} + 8 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} - 6 \beta_{10} + 8 \beta_{9} + 6 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 12 \beta_{5} + 4 \beta_{3} + 6 \beta_{2} - 2 \beta_{1} + 3\)
\(\nu^{7}\)\(=\)\((\)\(-13 \beta_{15} + 2 \beta_{14} - 37 \beta_{13} - 11 \beta_{12} + 13 \beta_{11} - 37 \beta_{10} - 92 \beta_{9} + 51 \beta_{8} + 22 \beta_{7} - 11 \beta_{6} - 51 \beta_{5} - 11 \beta_{4} - 81 \beta_{3} + 48 \beta_{2} - 26 \beta_{1} + 22\)\()/4\)
\(\nu^{8}\)\(=\)\(-2 \beta_{14} + 12 \beta_{13} - 18 \beta_{11} + 4 \beta_{10} - 14 \beta_{9} - 4 \beta_{8} - 14 \beta_{7} + 8 \beta_{6} + 2 \beta_{5} - 12 \beta_{4} - 42 \beta_{3} - 8 \beta_{2} + 15 \beta_{1} + 12\)
\(\nu^{9}\)\(=\)\((\)\(-37 \beta_{15} - 10 \beta_{13} - 37 \beta_{12} - 94 \beta_{11} - 21 \beta_{10} - 299 \beta_{8} - 37 \beta_{7} - 10 \beta_{6} - 194 \beta_{4} - 102 \beta_{3} - 21 \beta_{2} + 37 \beta_{1} + 97\)\()/4\)
\(\nu^{10}\)\(=\)\(23 \beta_{15} - 28 \beta_{14} - 19 \beta_{13} - 12 \beta_{12} + 16 \beta_{11} + 19 \beta_{10} + 4 \beta_{9} - 44 \beta_{8} + 25 \beta_{6} - 44 \beta_{5} + 51 \beta_{4} - 16 \beta_{3} + 44 \beta_{2}\)
\(\nu^{11}\)\(=\)\((\)\(-6 \beta_{15} - 682 \beta_{14} + 179 \beta_{13} - 346 \beta_{12} - 173 \beta_{11} - 432 \beta_{10} - 794 \beta_{9} + 173 \beta_{7} + 253 \beta_{6} - 259 \beta_{5} - 349 \beta_{4} - 173 \beta_{3} + 253 \beta_{2} - 3 \beta_{1} - 349\)\()/4\)
\(\nu^{12}\)\(=\)\(-44 \beta_{15} + 48 \beta_{14} + 64 \beta_{13} + 172 \beta_{12} + 24 \beta_{11} - 32 \beta_{10} + 200 \beta_{9} - 8 \beta_{8} - 172 \beta_{7} - 64 \beta_{6} + 16 \beta_{5} + 100 \beta_{3} + 32 \beta_{2} - 44 \beta_{1} - 135\)
\(\nu^{13}\)\(=\)\((\)\(-855 \beta_{15} + 1182 \beta_{14} - 615 \beta_{13} + 327 \beta_{12} + 855 \beta_{11} - 615 \beta_{10} + 1068 \beta_{9} - 647 \beta_{8} - 654 \beta_{7} - 681 \beta_{6} + 647 \beta_{5} - 1209 \beta_{4} + 741 \beta_{3} + 1296 \beta_{2} - 1710 \beta_{1} + 2418\)\()/4\)
\(\nu^{14}\)\(=\)\(196 \beta_{14} + 324 \beta_{13} + 372 \beta_{11} + 460 \beta_{10} - 224 \beta_{9} + 416 \beta_{8} - 20 \beta_{7} - 136 \beta_{6} - 208 \beta_{5} - 984 \beta_{4} - 468 \beta_{3} + 136 \beta_{2} - 191 \beta_{1} + 984\)
\(\nu^{15}\)\(=\)\((\)\(-2919 \beta_{15} + 2858 \beta_{13} + 153 \beta_{12} - 490 \beta_{11} - 1463 \beta_{10} - 9 \beta_{8} + 153 \beta_{7} + 2858 \beta_{6} - 13798 \beta_{4} - 962 \beta_{3} - 1463 \beta_{2} + 2919 \beta_{1} + 6899\)\()/4\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1 - \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.247636 + 1.71426i
−0.601642 1.62420i
1.60841 0.642670i
−1.70742 + 0.291063i
1.73018 + 0.0805675i
1.22961 1.21986i
0.934861 + 1.45809i
−0.441628 + 1.67480i
0.247636 1.71426i
−0.601642 + 1.62420i
1.60841 + 0.642670i
−1.70742 0.291063i
1.73018 0.0805675i
1.22961 + 1.21986i
0.934861 1.45809i
−0.441628 1.67480i
0 −1.71426 + 0.247636i 0 −1.28955 2.23357i 0 0.203402 + 2.63792i 0 2.87735 0.849022i 0
17.2 0 −1.62420 + 0.601642i 0 0.0726693 + 0.125867i 0 −1.05451 2.42652i 0 2.27605 1.95437i 0
17.3 0 −0.642670 1.60841i 0 1.28955 + 2.23357i 0 0.203402 + 2.63792i 0 −2.17395 + 2.06735i 0
17.4 0 −0.291063 1.70742i 0 −0.0726693 0.125867i 0 −1.05451 2.42652i 0 −2.83056 + 0.993934i 0
17.5 0 −0.0805675 + 1.73018i 0 1.90017 + 3.29119i 0 −2.23495 + 1.41598i 0 −2.98702 0.278792i 0
17.6 0 1.21986 + 1.22961i 0 −1.40397 2.43175i 0 2.08606 1.62738i 0 −0.0238727 + 2.99991i 0
17.7 0 1.45809 0.934861i 0 −1.90017 3.29119i 0 −2.23495 + 1.41598i 0 1.25207 2.72623i 0
17.8 0 1.67480 + 0.441628i 0 1.40397 + 2.43175i 0 2.08606 1.62738i 0 2.60993 + 1.47928i 0
257.1 0 −1.71426 0.247636i 0 −1.28955 + 2.23357i 0 0.203402 2.63792i 0 2.87735 + 0.849022i 0
257.2 0 −1.62420 0.601642i 0 0.0726693 0.125867i 0 −1.05451 + 2.42652i 0 2.27605 + 1.95437i 0
257.3 0 −0.642670 + 1.60841i 0 1.28955 2.23357i 0 0.203402 2.63792i 0 −2.17395 2.06735i 0
257.4 0 −0.291063 + 1.70742i 0 −0.0726693 + 0.125867i 0 −1.05451 + 2.42652i 0 −2.83056 0.993934i 0
257.5 0 −0.0805675 1.73018i 0 1.90017 3.29119i 0 −2.23495 1.41598i 0 −2.98702 + 0.278792i 0
257.6 0 1.21986 1.22961i 0 −1.40397 + 2.43175i 0 2.08606 + 1.62738i 0 −0.0238727 2.99991i 0
257.7 0 1.45809 + 0.934861i 0 −1.90017 + 3.29119i 0 −2.23495 1.41598i 0 1.25207 + 2.72623i 0
257.8 0 1.67480 0.441628i 0 1.40397 2.43175i 0 2.08606 + 1.62738i 0 2.60993 1.47928i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
7.d Odd 1 yes
21.g Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\):

\(T_{5}^{16} + \cdots\)
\( T_{13}^{8} + 55 T_{13}^{6} + 836 T_{13}^{4} + 3584 T_{13}^{2} + 4096 \)