Properties

Label 360.2.m.c
Level $360$
Weight $2$
Character orbit 360.m
Analytic conductor $2.875$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 17 x^{12} - 104 x^{10} + 713 x^{8} + 238 x^{6} + 1004 x^{4} - 152 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 - \beta_{2} ) q^{4} -\beta_{12} q^{5} + ( -\beta_{6} - \beta_{8} ) q^{7} + ( -\beta_{3} + \beta_{13} + \beta_{15} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 - \beta_{2} ) q^{4} -\beta_{12} q^{5} + ( -\beta_{6} - \beta_{8} ) q^{7} + ( -\beta_{3} + \beta_{13} + \beta_{15} ) q^{8} + ( -\beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{14} ) q^{10} + ( -\beta_{5} - \beta_{13} + \beta_{15} ) q^{11} + ( \beta_{6} + \beta_{8} + \beta_{10} + \beta_{14} ) q^{13} + ( \beta_{5} + \beta_{7} - \beta_{12} - \beta_{15} ) q^{14} + ( -\beta_{2} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{14} ) q^{16} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{13} + \beta_{15} ) q^{17} + ( 2 \beta_{2} + \beta_{10} - \beta_{14} ) q^{19} + ( \beta_{1} + 2 \beta_{5} - \beta_{7} + 2 \beta_{13} ) q^{20} + ( -1 - 2 \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{22} + ( -2 \beta_{1} + 2 \beta_{4} ) q^{23} + ( 3 + \beta_{2} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{14} ) q^{25} + ( -\beta_{5} + \beta_{7} + \beta_{12} + \beta_{15} ) q^{26} + ( 1 + \beta_{8} + \beta_{9} + \beta_{11} ) q^{28} + ( 2 \beta_{5} - 4 \beta_{7} + 2 \beta_{12} + \beta_{13} + \beta_{15} ) q^{29} + ( -2 \beta_{9} + 2 \beta_{11} ) q^{31} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{13} + \beta_{15} ) q^{32} + ( -1 + \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{14} ) q^{34} + ( -2 \beta_{1} + 2 \beta_{3} + \beta_{5} - \beta_{13} - \beta_{15} ) q^{35} + ( \beta_{6} + \beta_{8} + 2 \beta_{10} + 2 \beta_{14} ) q^{37} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{38} + ( -1 + \beta_{2} + 2 \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{14} ) q^{40} + ( -\beta_{5} - 2 \beta_{13} + 2 \beta_{15} ) q^{41} + ( 2 \beta_{6} - 2 \beta_{8} ) q^{43} + ( \beta_{5} + \beta_{7} - 3 \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{44} + ( -2 - 4 \beta_{2} - 2 \beta_{9} + 2 \beta_{11} ) q^{46} + ( -\beta_{1} + \beta_{4} - 3 \beta_{13} - 3 \beta_{15} ) q^{47} + ( -3 + 2 \beta_{2} + \beta_{10} - \beta_{14} ) q^{49} + ( -3 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{12} + 2 \beta_{13} - \beta_{15} ) q^{50} + ( -1 + \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} + 2 \beta_{14} ) q^{52} + ( 3 \beta_{1} - 3 \beta_{4} ) q^{53} + ( 2 \beta_{2} - \beta_{6} - \beta_{8} - 2 \beta_{10} ) q^{55} + ( 3 \beta_{5} - \beta_{7} + 3 \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{56} + ( -1 + 2 \beta_{6} - \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{14} ) q^{58} + ( -5 \beta_{5} + \beta_{13} - \beta_{15} ) q^{59} + ( -2 \beta_{1} - 4 \beta_{13} - 4 \beta_{15} ) q^{62} + ( 2 + 3 \beta_{2} + 3 \beta_{9} - \beta_{10} - 3 \beta_{11} + \beta_{14} ) q^{64} + ( 3 \beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{15} ) q^{65} + ( -4 - 2 \beta_{6} + 2 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - 4 \beta_{11} + 2 \beta_{14} ) q^{67} + ( 2 \beta_{1} - 4 \beta_{4} - 2 \beta_{13} - 2 \beta_{15} ) q^{68} + ( 1 + \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{14} ) q^{70} + ( -2 \beta_{5} + 4 \beta_{7} ) q^{71} + ( 2 - 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{14} ) q^{73} + ( -\beta_{5} + 3 \beta_{7} + \beta_{12} + \beta_{15} ) q^{74} + ( -4 - 2 \beta_{2} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{14} ) q^{76} + ( 3 \beta_{1} - 3 \beta_{4} - \beta_{13} - \beta_{15} ) q^{77} + ( -4 \beta_{2} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{14} ) q^{79} + ( 2 \beta_{1} + \beta_{3} - \beta_{5} - 3 \beta_{7} + 3 \beta_{12} + 3 \beta_{13} + 2 \beta_{15} ) q^{80} + ( -2 - 4 \beta_{6} - \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{82} + ( 2 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} ) q^{83} + ( \beta_{2} + 5 \beta_{6} + 5 \beta_{8} + \beta_{9} - \beta_{11} + \beta_{14} ) q^{85} + ( -6 \beta_{5} + 2 \beta_{7} - 2 \beta_{12} - 2 \beta_{15} ) q^{86} + ( 1 - 4 \beta_{6} - \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} - 2 \beta_{14} ) q^{88} + ( \beta_{5} + 4 \beta_{13} - 4 \beta_{15} ) q^{89} + ( -2 \beta_{2} - \beta_{10} + \beta_{14} ) q^{91} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{92} + ( 2 - 2 \beta_{2} - 4 \beta_{9} + 4 \beta_{11} ) q^{94} + ( -\beta_{1} + \beta_{4} - 2 \beta_{5} + 4 \beta_{7} - 4 \beta_{12} - 3 \beta_{13} - 3 \beta_{15} ) q^{95} + ( -2 - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{14} ) q^{97} + ( \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 12q^{4} + O(q^{10}) \) \( 16q + 12q^{4} - 8q^{10} - 12q^{16} + 16q^{19} + 40q^{25} - 32q^{34} - 28q^{40} - 48q^{46} - 32q^{49} + 36q^{64} + 32q^{70} - 56q^{76} - 16q^{91} + 24q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 17 x^{12} - 104 x^{10} + 713 x^{8} + 238 x^{6} + 1004 x^{4} - 152 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-49720141 \nu^{14} - 25545398 \nu^{12} + 871193385 \nu^{10} + 5556489734 \nu^{8} - 33063326753 \nu^{6} - 31333405668 \nu^{4} - 32657858676 \nu^{2} - 35294863008\)\()/ 26277512192 \)
\(\beta_{2}\)\(=\)\((\)\(-107882115 \nu^{14} - 77610858 \nu^{12} + 1877392743 \nu^{10} + 12516347930 \nu^{8} - 70055295983 \nu^{6} - 85825593884 \nu^{4} - 89210141260 \nu^{2} - 17971784544\)\()/ 26277512192 \)
\(\beta_{3}\)\(=\)\((\)\(172551585 \nu^{14} - 95959474 \nu^{12} - 2917951085 \nu^{10} - 16219192030 \nu^{8} + 132913218629 \nu^{6} - 29977594636 \nu^{4} + 141071725764 \nu^{2} - 68095836896\)\()/ 26277512192 \)
\(\beta_{4}\)\(=\)\((\)\(-174161853 \nu^{14} - 18352086 \nu^{12} + 2890619609 \nu^{10} + 18336673830 \nu^{8} - 121116108369 \nu^{6} - 48061754788 \nu^{4} - 222085788724 \nu^{2} - 21115026592\)\()/ 26277512192 \)
\(\beta_{5}\)\(=\)\((\)\( 74983 \nu^{15} + 31874 \nu^{13} - 1281915 \nu^{11} - 8349682 \nu^{9} + 50321507 \nu^{7} + 41578380 \nu^{5} + 78299292 \nu^{3} + 9807584 \nu \)\()/8515072\)
\(\beta_{6}\)\(=\)\((\)\(-281117679 \nu^{15} - 123908562 \nu^{13} + 4827183075 \nu^{11} + 31323608386 \nu^{9} - 188355497355 \nu^{7} - 159644286348 \nu^{5} - 274289473788 \nu^{3} - 65561067232 \nu\)\()/ 26277512192 \)
\(\beta_{7}\)\(=\)\((\)\(-294063601 \nu^{15} + 120583890 \nu^{13} + 5015106685 \nu^{11} + 28521843902 \nu^{9} - 222445799381 \nu^{7} + 15093694540 \nu^{5} - 256609656068 \nu^{3} + 167281353952 \nu\)\()/ 26277512192 \)
\(\beta_{8}\)\(=\)\((\)\(354912961 \nu^{15} + 119115470 \nu^{13} - 6034586637 \nu^{11} - 39017889374 \nu^{9} + 240529045221 \nu^{7} + 170796167476 \nu^{5} + 394053088964 \nu^{3} + 16283111712 \nu\)\()/ 26277512192 \)
\(\beta_{9}\)\(=\)\((\)\(-234112727 \nu^{15} + 231790677 \nu^{14} + 51115902 \nu^{13} + 29428326 \nu^{12} + 3952863147 \nu^{11} - 3964762513 \nu^{10} + 23429258866 \nu^{9} - 24597755702 \nu^{8} - 171780953875 \nu^{7} + 162793574729 \nu^{6} - 16484030124 \nu^{5} + 77872917956 \nu^{4} - 239347722396 \nu^{3} + 223778607892 \nu^{2} + 46805651488 \nu - 26297259104\)\()/ 26277512192 \)
\(\beta_{10}\)\(=\)\((\)\(-293670462 \nu^{15} + 157920899 \nu^{14} + 51649052 \nu^{13} - 22764694 \nu^{12} + 5006333862 \nu^{11} - 2755245415 \nu^{10} + 29691206852 \nu^{9} - 15956221978 \nu^{8} - 214944395254 \nu^{7} + 116401708015 \nu^{6} - 35344268184 \nu^{5} + 28168725020 \nu^{4} - 274462663288 \nu^{3} + 100553570892 \nu^{2} + 110717890624 \nu - 26166657184\)\()/ 26277512192 \)
\(\beta_{11}\)\(=\)\((\)\(-234112727 \nu^{15} - 231790677 \nu^{14} + 51115902 \nu^{13} - 29428326 \nu^{12} + 3952863147 \nu^{11} + 3964762513 \nu^{10} + 23429258866 \nu^{9} + 24597755702 \nu^{8} - 171780953875 \nu^{7} - 162793574729 \nu^{6} - 16484030124 \nu^{5} - 77872917956 \nu^{4} - 239347722396 \nu^{3} - 223778607892 \nu^{2} + 46805651488 \nu + 19746912\)\()/ 26277512192 \)
\(\beta_{12}\)\(=\)\((\)\(-70157797 \nu^{15} - 42874683 \nu^{14} + 14792042 \nu^{13} - 5353898 \nu^{12} + 1211877025 \nu^{11} + 733594303 \nu^{10} + 7037062950 \nu^{9} + 4552132442 \nu^{8} - 51884284441 \nu^{7} - 30063289735 \nu^{6} - 7908584276 \nu^{5} - 14639454572 \nu^{4} - 53117003316 \nu^{3} - 40621691148 \nu^{2} + 21188795360 \nu + 2357445536\)\()/ 6569378048 \)
\(\beta_{13}\)\(=\)\((\)\(-73637523 \nu^{15} + 42874683 \nu^{14} - 15623210 \nu^{13} + 5353898 \nu^{12} + 1248695415 \nu^{11} - 733594303 \nu^{10} + 7924300122 \nu^{9} - 4552132442 \nu^{8} - 50816039743 \nu^{7} + 30063289735 \nu^{6} - 28229063676 \nu^{5} + 14639454572 \nu^{4} - 79607779148 \nu^{3} + 40621691148 \nu^{2} - 8897479776 \nu - 2357445536\)\()/ 6569378048 \)
\(\beta_{14}\)\(=\)\((\)\(-293670462 \nu^{15} - 157920899 \nu^{14} + 51649052 \nu^{13} + 22764694 \nu^{12} + 5006333862 \nu^{11} + 2755245415 \nu^{10} + 29691206852 \nu^{9} + 15956221978 \nu^{8} - 214944395254 \nu^{7} - 116401708015 \nu^{6} - 35344268184 \nu^{5} - 28168725020 \nu^{4} - 274462663288 \nu^{3} - 100553570892 \nu^{2} + 110717890624 \nu + 26166657184\)\()/ 26277512192 \)
\(\beta_{15}\)\(=\)\((\)\(73637523 \nu^{15} + 42874683 \nu^{14} + 15623210 \nu^{13} + 5353898 \nu^{12} - 1248695415 \nu^{11} - 733594303 \nu^{10} - 7924300122 \nu^{9} - 4552132442 \nu^{8} + 50816039743 \nu^{7} + 30063289735 \nu^{6} + 28229063676 \nu^{5} + 14639454572 \nu^{4} + 79607779148 \nu^{3} + 40621691148 \nu^{2} + 8897479776 \nu - 2357445536\)\()/ 6569378048 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{13} - \beta_{12} + \beta_{7} + \beta_{6}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{15} - \beta_{14} - \beta_{13} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_{1} - 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{15} + 5 \beta_{14} - 4 \beta_{12} - 2 \beta_{11} + 5 \beta_{10} - 2 \beta_{9} + 6 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} - 2\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(10 \beta_{15} - 5 \beta_{14} + 10 \beta_{13} + 11 \beta_{11} + 5 \beta_{10} - 11 \beta_{9} - 6 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 14 \beta_{1} + 7\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-14 \beta_{15} - 12 \beta_{14} + 10 \beta_{13} - 4 \beta_{12} - 11 \beta_{11} - 12 \beta_{10} - 11 \beta_{9} + 4 \beta_{8} + 28 \beta_{7} - 2 \beta_{6} - 11\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-44 \beta_{15} - \beta_{14} - 44 \beta_{13} - 26 \beta_{11} + \beta_{10} + 26 \beta_{9} + 4 \beta_{4} + 12 \beta_{3} - 42 \beta_{2} + 64 \beta_{1} + 88\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(10 \beta_{15} + 59 \beta_{14} - 40 \beta_{13} - 30 \beta_{12} + 3 \beta_{11} + 59 \beta_{10} + 3 \beta_{9} - 66 \beta_{8} - 60 \beta_{7} + 136 \beta_{6} + 246 \beta_{5} + 3\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-120 \beta_{15} - 206 \beta_{14} - 120 \beta_{13} + 109 \beta_{11} + 206 \beta_{10} - 109 \beta_{9} - 328 \beta_{4} - 176 \beta_{3} + 86 \beta_{2} - 184 \beta_{1} - 693\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-588 \beta_{15} + 393 \beta_{14} + 686 \beta_{13} + 98 \beta_{12} - 534 \beta_{11} + 393 \beta_{10} - 534 \beta_{9} + 438 \beta_{8} - 260 \beta_{7} - 574 \beta_{6} - 42 \beta_{5} - 534\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(816 \beta_{15} + 357 \beta_{14} + 816 \beta_{13} + 1241 \beta_{11} - 357 \beta_{10} - 1241 \beta_{9} - 132 \beta_{4} + 1260 \beta_{3} - 2440 \beta_{2} + 1920 \beta_{1} + 2701\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(1974 \beta_{15} - 3674 \beta_{14} + 622 \beta_{13} + 2596 \beta_{12} + 225 \beta_{11} - 3674 \beta_{10} + 225 \beta_{9} - 4960 \beta_{8} + 2808 \beta_{7} - 778 \beta_{6} + 2788 \beta_{5} + 225\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(-13520 \beta_{15} + 293 \beta_{14} - 13520 \beta_{13} - 8456 \beta_{11} - 293 \beta_{10} + 8456 \beta_{9} - 768 \beta_{4} + 1400 \beta_{3} - 794 \beta_{2} - 7016 \beta_{1} - 8774\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(1986 \beta_{15} + 20283 \beta_{14} + 1324 \beta_{13} + 3310 \beta_{12} - 475 \beta_{11} + 20283 \beta_{10} - 475 \beta_{9} - 3594 \beta_{8} - 30988 \beta_{7} + 3780 \beta_{6} + 24418 \beta_{5} - 475\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(31448 \beta_{15} - 12444 \beta_{14} + 31448 \beta_{13} + 49285 \beta_{11} + 12444 \beta_{10} - 49285 \beta_{9} - 36796 \beta_{4} - 2456 \beta_{3} - 3598 \beta_{2} - 36412 \beta_{1} - 89305\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(-27344 \beta_{15} - 65833 \beta_{14} + 107890 \beta_{13} + 80546 \beta_{12} - 49240 \beta_{11} - 65833 \beta_{10} - 49240 \beta_{9} + 1314 \beta_{8} + 58924 \beta_{7} - 173294 \beta_{6} - 134346 \beta_{5} - 49240\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1
0.877859 + 2.23141i
−0.877859 2.23141i
0.877859 2.23141i
−0.877859 + 2.23141i
−2.15532 0.457057i
2.15532 + 0.457057i
−2.15532 + 0.457057i
2.15532 0.457057i
−0.645096 0.854135i
0.645096 + 0.854135i
−0.645096 + 0.854135i
0.645096 0.854135i
0.409646 0.286988i
−0.409646 + 0.286988i
0.409646 + 0.286988i
−0.409646 0.286988i
−1.37491 0.331077i 0 1.78078 + 0.910404i −1.64901 1.51022i 0 −0.936426 −2.14700 1.84130i 0 1.76724 + 2.62238i
179.2 −1.37491 0.331077i 0 1.78078 + 0.910404i 1.64901 1.51022i 0 0.936426 −2.14700 1.84130i 0 −2.76724 + 1.53048i
179.3 −1.37491 + 0.331077i 0 1.78078 0.910404i −1.64901 + 1.51022i 0 −0.936426 −2.14700 + 1.84130i 0 1.76724 2.62238i
179.4 −1.37491 + 0.331077i 0 1.78078 0.910404i 1.64901 + 1.51022i 0 0.936426 −2.14700 + 1.84130i 0 −2.76724 1.53048i
179.5 −0.927153 1.06789i 0 −0.280776 + 1.98019i −2.18650 0.468213i 0 3.02045 2.37495 1.53610i 0 1.52722 + 2.76904i
179.6 −0.927153 1.06789i 0 −0.280776 + 1.98019i 2.18650 0.468213i 0 −3.02045 2.37495 1.53610i 0 −2.52722 1.90083i
179.7 −0.927153 + 1.06789i 0 −0.280776 1.98019i −2.18650 + 0.468213i 0 3.02045 2.37495 + 1.53610i 0 1.52722 2.76904i
179.8 −0.927153 + 1.06789i 0 −0.280776 1.98019i 2.18650 + 0.468213i 0 −3.02045 2.37495 + 1.53610i 0 −2.52722 + 1.90083i
179.9 0.927153 1.06789i 0 −0.280776 1.98019i −2.18650 0.468213i 0 −3.02045 −2.37495 1.53610i 0 −2.52722 + 1.90083i
179.10 0.927153 1.06789i 0 −0.280776 1.98019i 2.18650 0.468213i 0 3.02045 −2.37495 1.53610i 0 1.52722 2.76904i
179.11 0.927153 + 1.06789i 0 −0.280776 + 1.98019i −2.18650 + 0.468213i 0 −3.02045 −2.37495 + 1.53610i 0 −2.52722 1.90083i
179.12 0.927153 + 1.06789i 0 −0.280776 + 1.98019i 2.18650 + 0.468213i 0 3.02045 −2.37495 + 1.53610i 0 1.52722 + 2.76904i
179.13 1.37491 0.331077i 0 1.78078 0.910404i −1.64901 1.51022i 0 0.936426 2.14700 1.84130i 0 −2.76724 1.53048i
179.14 1.37491 0.331077i 0 1.78078 0.910404i 1.64901 1.51022i 0 −0.936426 2.14700 1.84130i 0 1.76724 2.62238i
179.15 1.37491 + 0.331077i 0 1.78078 + 0.910404i −1.64901 + 1.51022i 0 0.936426 2.14700 + 1.84130i 0 −2.76724 + 1.53048i
179.16 1.37491 + 0.331077i 0 1.78078 + 0.910404i 1.64901 + 1.51022i 0 −0.936426 2.14700 + 1.84130i 0 1.76724 + 2.62238i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
8.d odd 2 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
40.e odd 2 1 inner
120.m even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.m.c 16
3.b odd 2 1 inner 360.2.m.c 16
4.b odd 2 1 1440.2.m.c 16
5.b even 2 1 inner 360.2.m.c 16
5.c odd 4 2 1800.2.b.g 16
8.b even 2 1 1440.2.m.c 16
8.d odd 2 1 inner 360.2.m.c 16
12.b even 2 1 1440.2.m.c 16
15.d odd 2 1 inner 360.2.m.c 16
15.e even 4 2 1800.2.b.g 16
20.d odd 2 1 1440.2.m.c 16
20.e even 4 2 7200.2.b.i 16
24.f even 2 1 inner 360.2.m.c 16
24.h odd 2 1 1440.2.m.c 16
40.e odd 2 1 inner 360.2.m.c 16
40.f even 2 1 1440.2.m.c 16
40.i odd 4 2 7200.2.b.i 16
40.k even 4 2 1800.2.b.g 16
60.h even 2 1 1440.2.m.c 16
60.l odd 4 2 7200.2.b.i 16
120.i odd 2 1 1440.2.m.c 16
120.m even 2 1 inner 360.2.m.c 16
120.q odd 4 2 1800.2.b.g 16
120.w even 4 2 7200.2.b.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.m.c 16 1.a even 1 1 trivial
360.2.m.c 16 3.b odd 2 1 inner
360.2.m.c 16 5.b even 2 1 inner
360.2.m.c 16 8.d odd 2 1 inner
360.2.m.c 16 15.d odd 2 1 inner
360.2.m.c 16 24.f even 2 1 inner
360.2.m.c 16 40.e odd 2 1 inner
360.2.m.c 16 120.m even 2 1 inner
1440.2.m.c 16 4.b odd 2 1
1440.2.m.c 16 8.b even 2 1
1440.2.m.c 16 12.b even 2 1
1440.2.m.c 16 20.d odd 2 1
1440.2.m.c 16 24.h odd 2 1
1440.2.m.c 16 40.f even 2 1
1440.2.m.c 16 60.h even 2 1
1440.2.m.c 16 120.i odd 2 1
1800.2.b.g 16 5.c odd 4 2
1800.2.b.g 16 15.e even 4 2
1800.2.b.g 16 40.k even 4 2
1800.2.b.g 16 120.q odd 4 2
7200.2.b.i 16 20.e even 4 2
7200.2.b.i 16 40.i odd 4 2
7200.2.b.i 16 60.l odd 4 2
7200.2.b.i 16 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 10 T_{7}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 - 12 T^{2} + 6 T^{4} - 3 T^{6} + T^{8} )^{2} \)
$3$ \( T^{16} \)
$5$ \( ( 625 - 250 T^{2} + 58 T^{4} - 10 T^{6} + T^{8} )^{2} \)
$7$ \( ( 8 - 10 T^{2} + T^{4} )^{4} \)
$11$ \( ( 64 + 18 T^{2} + T^{4} )^{4} \)
$13$ \( ( 32 - 14 T^{2} + T^{4} )^{4} \)
$17$ \( ( 104 - 46 T^{2} + T^{4} )^{4} \)
$19$ \( ( -16 - 2 T + T^{2} )^{8} \)
$23$ \( ( 512 + 56 T^{2} + T^{4} )^{4} \)
$29$ \( ( 3328 - 118 T^{2} + T^{4} )^{4} \)
$31$ \( ( 208 + 72 T^{2} + T^{4} )^{4} \)
$37$ \( ( 8 - 58 T^{2} + T^{4} )^{4} \)
$41$ \( ( 34 + T^{2} )^{8} \)
$43$ \( ( 1664 + 88 T^{2} + T^{4} )^{4} \)
$47$ \( ( 32 + 116 T^{2} + T^{4} )^{4} \)
$53$ \( ( 2592 + 126 T^{2} + T^{4} )^{4} \)
$59$ \( ( 2704 + 138 T^{2} + T^{4} )^{4} \)
$61$ \( T^{16} \)
$67$ \( ( 26624 + 328 T^{2} + T^{4} )^{4} \)
$71$ \( ( 832 - 144 T^{2} + T^{4} )^{4} \)
$73$ \( ( 6656 + 244 T^{2} + T^{4} )^{4} \)
$79$ \( ( 208 + 200 T^{2} + T^{4} )^{4} \)
$83$ \( ( 6656 - 232 T^{2} + T^{4} )^{4} \)
$89$ \( ( 17956 + 276 T^{2} + T^{4} )^{4} \)
$97$ \( ( 416 + 92 T^{2} + T^{4} )^{4} \)
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