Properties

Label 350.2.h.b
Level 350
Weight 2
Character orbit 350.h
Analytic conductor 2.795
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 350.h (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{11} + 6 x^{10} + x^{9} - 14 x^{8} + 10 x^{7} + 35 x^{6} - 110 x^{5} + 230 x^{4} - 325 x^{3} + 300 x^{2} - 250 x + 125\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -8500 \nu^{11} + 27691 \nu^{10} - 20739 \nu^{9} - 55289 \nu^{8} + 111206 \nu^{7} + 70641 \nu^{6} - 396950 \nu^{5} + 602875 \nu^{4} - 982560 \nu^{3} + 1174480 \nu^{2} - 484350 \nu + 675625 \)\()/374525\)
\(\beta_{2}\)\(=\)\((\)\( 8797 \nu^{11} - 32897 \nu^{10} + 41413 \nu^{9} + 21743 \nu^{8} - 123267 \nu^{7} + 62971 \nu^{6} + 289305 \nu^{5} - 899230 \nu^{4} + 1845250 \nu^{3} - 2461995 \nu^{2} + 1672325 \nu - 1238725 \)\()/374525\)
\(\beta_{3}\)\(=\)\((\)\(40926 \nu^{11} - 112175 \nu^{10} + 101300 \nu^{9} + 175115 \nu^{8} - 345870 \nu^{7} - 63756 \nu^{6} + 1348465 \nu^{5} - 2678090 \nu^{4} + 5881015 \nu^{3} - 5965005 \nu^{2} + 4836950 \nu - 4105300\)\()/374525\)
\(\beta_{4}\)\(=\)\((\)\(-42604 \nu^{11} + 116457 \nu^{10} - 103013 \nu^{9} - 180213 \nu^{8} + 353992 \nu^{7} + 75931 \nu^{6} - 1393785 \nu^{5} + 2785270 \nu^{4} - 6054855 \nu^{3} + 6127255 \nu^{2} - 4983050 \nu + 4795375\)\()/374525\)
\(\beta_{5}\)\(=\)\((\)\(67540 \nu^{11} - 184923 \nu^{10} + 163897 \nu^{9} + 284782 \nu^{8} - 564428 \nu^{7} - 116598 \nu^{6} + 2231415 \nu^{5} - 4422365 \nu^{4} + 9666380 \nu^{3} - 9790840 \nu^{2} + 7942075 \nu - 7323600\)\()/374525\)
\(\beta_{6}\)\(=\)\((\)\(102504 \nu^{11} - 303172 \nu^{10} + 308798 \nu^{9} + 405938 \nu^{8} - 1025517 \nu^{7} + 57319 \nu^{6} + 3586600 \nu^{5} - 7705180 \nu^{4} + 15939430 \nu^{3} - 16813455 \nu^{2} + 13232475 \nu - 11045025\)\()/374525\)
\(\beta_{7}\)\(=\)\((\)\(132868 \nu^{11} - 375458 \nu^{10} + 364552 \nu^{9} + 534927 \nu^{8} - 1217313 \nu^{7} - 53541 \nu^{6} + 4501195 \nu^{5} - 9364890 \nu^{4} + 19877275 \nu^{3} - 20424280 \nu^{2} + 16911225 \nu - 14403300\)\()/374525\)
\(\beta_{8}\)\(=\)\((\)\(136311 \nu^{11} - 386732 \nu^{10} + 370398 \nu^{9} + 564408 \nu^{8} - 1256242 \nu^{7} - 81583 \nu^{6} + 4691080 \nu^{5} - 9591220 \nu^{4} + 20149035 \nu^{3} - 20853240 \nu^{2} + 16917725 \nu - 14601675\)\()/374525\)
\(\beta_{9}\)\(=\)\((\)\(-154033 \nu^{11} + 447428 \nu^{10} - 442987 \nu^{9} - 633032 \nu^{8} + 1498533 \nu^{7} + 26626 \nu^{6} - 5410370 \nu^{5} + 11237145 \nu^{4} - 23275375 \nu^{3} + 24254305 \nu^{2} - 19733200 \nu + 16535300\)\()/374525\)
\(\beta_{10}\)\(=\)\((\)\(155098 \nu^{11} - 444623 \nu^{10} + 433467 \nu^{9} + 643337 \nu^{8} - 1489778 \nu^{7} - 32386 \nu^{6} + 5432570 \nu^{5} - 11273445 \nu^{4} + 23226025 \nu^{3} - 23612030 \nu^{2} + 19064875 \nu - 16066775\)\()/374525\)
\(\beta_{11}\)\(=\)\((\)\(-158566 \nu^{11} + 460173 \nu^{10} - 458557 \nu^{9} - 645092 \nu^{8} + 1543428 \nu^{7} - 10416 \nu^{6} - 5566125 \nu^{5} + 11700245 \nu^{4} - 24130845 \nu^{3} + 24912970 \nu^{2} - 19893000 \nu + 16705350\)\()/374525\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{11} + \beta_{10} - 4 \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 4\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{11} + \beta_{10} - 4 \beta_{9} - 6 \beta_{8} - \beta_{7} + 3 \beta_{6} + \beta_{5} - 6 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 2 \beta_{1} + 4\)\()/5\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{11} + \beta_{10} - 4 \beta_{9} - 11 \beta_{8} - \beta_{7} - 2 \beta_{6} + 16 \beta_{5} + 19 \beta_{4} + 8 \beta_{3} + \beta_{2} + 2 \beta_{1} - 16\)\()/5\)
\(\nu^{4}\)\(=\)\((\)\(17 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + 4 \beta_{7} + 23 \beta_{6} + 11 \beta_{5} + 29 \beta_{4} + 13 \beta_{3} - 14 \beta_{2} - 3 \beta_{1} - 21\)\()/5\)
\(\nu^{5}\)\(=\)\((\)\(-8 \beta_{11} - 9 \beta_{10} + \beta_{9} + 14 \beta_{8} - 6 \beta_{7} - 2 \beta_{6} + 46 \beta_{5} + 69 \beta_{4} - 17 \beta_{3} - 9 \beta_{2} + 2 \beta_{1} - 21\)\()/5\)
\(\nu^{6}\)\(=\)\((\)\(57 \beta_{11} + 31 \beta_{10} + 6 \beta_{9} + 44 \beta_{8} - 36 \beta_{7} + 48 \beta_{6} + \beta_{5} + 24 \beta_{4} + 8 \beta_{3} - 14 \beta_{2} + 27 \beta_{1} - 26\)\()/5\)
\(\nu^{7}\)\(=\)\((\)\(-103 \beta_{11} - 69 \beta_{10} - 44 \beta_{9} + 19 \beta_{8} - 96 \beta_{7} - 52 \beta_{6} + 11 \beta_{5} - 101 \beta_{4} - 32 \beta_{3} - 9 \beta_{2} + 67 \beta_{1} + 99\)\()/5\)
\(\nu^{8}\)\(=\)\((\)\(-63 \beta_{11} + 21 \beta_{10} - 89 \beta_{9} - \beta_{8} - 141 \beta_{7} - 142 \beta_{6} - 119 \beta_{5} - 46 \beta_{4} + 303 \beta_{3} + 96 \beta_{2} + 82 \beta_{1} - 256\)\()/5\)
\(\nu^{9}\)\(=\)\((\)\(-333 \beta_{11} - 299 \beta_{10} + 76 \beta_{9} - 26 \beta_{8} + 154 \beta_{7} - 52 \beta_{6} - 539 \beta_{5} - 371 \beta_{4} + 363 \beta_{3} - 164 \beta_{2} - 103 \beta_{1} - 121\)\()/5\)
\(\nu^{10}\)\(=\)\((\)\(-508 \beta_{11} - 84 \beta_{10} + 576 \beta_{9} + 264 \beta_{8} + 669 \beta_{7} - 677 \beta_{6} - 454 \beta_{5} + 944 \beta_{4} + 733 \beta_{3} + 366 \beta_{2} - 398 \beta_{1} - 1321\)\()/5\)
\(\nu^{11}\)\(=\)\((\)\(782 \beta_{11} + 131 \beta_{10} + 1906 \beta_{9} + 1319 \beta_{8} + 1764 \beta_{7} + 1048 \beta_{6} - 2374 \beta_{5} - 976 \beta_{4} - 192 \beta_{3} - 564 \beta_{2} - 1073 \beta_{1} + 799\)\()/5\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
0.0830190 1.17264i
−0.258306 + 1.14684i
1.17529 + 0.0257946i
1.70682 + 0.839517i
−1.74662 + 0.753236i
1.03979 1.59275i
1.70682 0.839517i
−1.74662 0.753236i
1.03979 + 1.59275i
0.0830190 + 1.17264i
−0.258306 1.14684i
1.17529 0.0257946i
−0.309017 + 0.951057i −2.43768 1.77108i −0.809017 0.587785i 0.539075 + 2.17011i 2.43768 1.77108i −1.00000 0.809017 0.587785i 1.87852 + 5.78148i −2.23049 0.157911i
71.2 −0.309017 + 0.951057i 0.408688 + 0.296929i −0.809017 0.587785i −0.206815 2.22648i −0.408688 + 0.296929i −1.00000 0.809017 0.587785i −0.848192 2.61047i 2.18142 + 0.491328i
71.3 −0.309017 + 0.951057i 1.71998 + 1.24964i −0.809017 0.587785i −2.14128 + 0.644154i −1.71998 + 1.24964i −1.00000 0.809017 0.587785i 0.469677 + 1.44552i 0.0490643 2.23553i
141.1 0.809017 + 0.587785i −0.197419 0.607592i 0.309017 + 0.951057i −0.380957 2.20338i 0.197419 0.607592i −1.00000 −0.309017 + 0.951057i 2.09686 1.52346i 0.986912 2.00649i
141.2 0.809017 + 0.587785i 0.241548 + 0.743409i 0.309017 + 0.951057i 1.92325 + 1.14066i −0.241548 + 0.743409i −1.00000 −0.309017 + 0.951057i 1.93274 1.40422i 0.885482 + 2.05327i
141.3 0.809017 + 0.587785i 0.764888 + 2.35408i 0.309017 + 0.951057i −2.23328 + 0.111663i −0.764888 + 2.35408i −1.00000 −0.309017 + 0.951057i −2.52960 + 1.83786i −1.87239 1.22235i
211.1 0.809017 0.587785i −0.197419 + 0.607592i 0.309017 0.951057i −0.380957 + 2.20338i 0.197419 + 0.607592i −1.00000 −0.309017 0.951057i 2.09686 + 1.52346i 0.986912 + 2.00649i
211.2 0.809017 0.587785i 0.241548 0.743409i 0.309017 0.951057i 1.92325 1.14066i −0.241548 0.743409i −1.00000 −0.309017 0.951057i 1.93274 + 1.40422i 0.885482 2.05327i
211.3 0.809017 0.587785i 0.764888 2.35408i 0.309017 0.951057i −2.23328 0.111663i −0.764888 2.35408i −1.00000 −0.309017 0.951057i −2.52960 1.83786i −1.87239 + 1.22235i
281.1 −0.309017 0.951057i −2.43768 + 1.77108i −0.809017 + 0.587785i 0.539075 2.17011i 2.43768 + 1.77108i −1.00000 0.809017 + 0.587785i 1.87852 5.78148i −2.23049 + 0.157911i
281.2 −0.309017 0.951057i 0.408688 0.296929i −0.809017 + 0.587785i −0.206815 + 2.22648i −0.408688 0.296929i −1.00000 0.809017 + 0.587785i −0.848192 + 2.61047i 2.18142 0.491328i
281.3 −0.309017 0.951057i 1.71998 1.24964i −0.809017 + 0.587785i −2.14128 0.644154i −1.71998 1.24964i −1.00000 0.809017 + 0.587785i 0.469677 1.44552i 0.0490643 + 2.23553i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 281.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.h.b 12
25.d even 5 1 inner 350.2.h.b 12
25.d even 5 1 8750.2.a.p 6
25.e even 10 1 8750.2.a.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.h.b 12 1.a even 1 1 trivial
350.2.h.b 12 25.d even 5 1 inner
8750.2.a.p 6 25.d even 5 1
8750.2.a.q 6 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{12} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{3} \)
$3$ \( 1 - T - 7 T^{2} + 27 T^{4} + 54 T^{5} - 85 T^{6} - 282 T^{7} + 38 T^{8} + 744 T^{9} + 1035 T^{10} - 917 T^{11} - 4667 T^{12} - 2751 T^{13} + 9315 T^{14} + 20088 T^{15} + 3078 T^{16} - 68526 T^{17} - 61965 T^{18} + 118098 T^{19} + 177147 T^{20} - 413343 T^{22} - 177147 T^{23} + 531441 T^{24} \)
$5$ \( 1 + 5 T + 15 T^{2} + 45 T^{3} + 80 T^{4} + 125 T^{5} + 325 T^{6} + 625 T^{7} + 2000 T^{8} + 5625 T^{9} + 9375 T^{10} + 15625 T^{11} + 15625 T^{12} \)
$7$ \( ( 1 + T )^{12} \)
$11$ \( 1 - 7 T + 15 T^{2} - 48 T^{3} + 369 T^{4} - 1218 T^{5} + 3391 T^{6} - 18040 T^{7} + 68194 T^{8} - 176652 T^{9} + 566355 T^{10} - 2220705 T^{11} + 7824965 T^{12} - 24427755 T^{13} + 68528955 T^{14} - 235123812 T^{15} + 998428354 T^{16} - 2905360040 T^{17} + 6007363351 T^{18} - 23735374278 T^{19} + 79098427089 T^{20} - 113181489168 T^{21} + 389061369015 T^{22} - 1997181694277 T^{23} + 3138428376721 T^{24} \)
$13$ \( 1 - 3 T - 20 T^{2} + 118 T^{3} + 161 T^{4} - 2778 T^{5} + 507 T^{6} + 49992 T^{7} - 117950 T^{8} - 558088 T^{9} + 2845345 T^{10} + 2678925 T^{11} - 39869651 T^{12} + 34826025 T^{13} + 480863305 T^{14} - 1226119336 T^{15} - 3368769950 T^{16} + 18561679656 T^{17} + 2447192163 T^{18} - 174315380226 T^{19} + 131332646081 T^{20} + 1251330926014 T^{21} - 2757169836980 T^{22} - 5376481182111 T^{23} + 23298085122481 T^{24} \)
$17$ \( 1 - 4 T - 52 T^{2} + 212 T^{3} + 1204 T^{4} - 8304 T^{5} - 1499 T^{6} + 234288 T^{7} - 622968 T^{8} - 4319236 T^{9} + 20524788 T^{10} + 33120788 T^{11} - 413458971 T^{12} + 563053396 T^{13} + 5931663732 T^{14} - 21220406468 T^{15} - 52030910328 T^{16} + 332655456816 T^{17} - 36182215931 T^{18} - 3407452340592 T^{19} + 8398811958964 T^{20} + 25140629817364 T^{21} - 104831682823348 T^{22} - 137087585230532 T^{23} + 582622237229761 T^{24} \)
$19$ \( 1 - 4 T - 55 T^{2} + 322 T^{3} + 786 T^{4} - 10430 T^{5} + 19782 T^{6} + 121858 T^{7} - 830761 T^{8} + 935330 T^{9} + 9745914 T^{10} - 23118154 T^{11} - 60429309 T^{12} - 439244926 T^{13} + 3518274954 T^{14} + 6415428470 T^{15} - 108265604281 T^{16} + 301732471942 T^{17} + 930661617942 T^{18} - 9323082237770 T^{19} + 13349080550226 T^{20} + 103905438684838 T^{21} - 337208644179055 T^{22} - 465961035592876 T^{23} + 2213314919066161 T^{24} \)
$23$ \( 1 + T - 53 T^{2} - 306 T^{3} + 975 T^{4} + 11712 T^{5} + 26649 T^{6} - 153426 T^{7} - 961598 T^{8} - 2533552 T^{9} - 218007 T^{10} + 52959247 T^{11} + 441957817 T^{12} + 1218062681 T^{13} - 115325703 T^{14} - 30825727184 T^{15} - 269094545918 T^{16} - 987502361118 T^{17} + 3945008405961 T^{18} + 39877315635264 T^{19} + 76353210648975 T^{20} - 551152714407678 T^{21} - 2195605094323397 T^{22} + 952809757913927 T^{23} + 21914624432020321 T^{24} \)
$29$ \( 1 - 22 T + 156 T^{2} + 68 T^{3} - 6774 T^{4} + 30632 T^{5} + 46573 T^{6} - 1017678 T^{7} + 5929708 T^{8} - 26575028 T^{9} + 50923194 T^{10} + 552546070 T^{11} - 5342883971 T^{12} + 16023836030 T^{13} + 42826406154 T^{14} - 648138357892 T^{15} + 4193969803948 T^{16} - 20873745092022 T^{17} + 27702706528933 T^{18} + 528398211097288 T^{19} - 3388669201397814 T^{20} + 986485926359092 T^{21} + 65630328394831356 T^{22} - 268411214845528238 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 - 31 T + 453 T^{2} - 4225 T^{3} + 29693 T^{4} - 192850 T^{5} + 1377427 T^{6} - 10076394 T^{7} + 65664126 T^{8} - 379719432 T^{9} + 2153666263 T^{10} - 12728315052 T^{11} + 73593545547 T^{12} - 394577766612 T^{13} + 2069673278743 T^{14} - 11312221598712 T^{15} + 60642199307646 T^{16} - 288478605361494 T^{17} + 1222471532808787 T^{18} - 5305807631306350 T^{19} + 25324893574735613 T^{20} - 111707403628834975 T^{21} + 371291614002302853 T^{22} - 787662783788549761 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 - 9 T - 17 T^{2} + 47 T^{3} + 2009 T^{4} + 17896 T^{5} - 183604 T^{6} - 200352 T^{7} - 1312908 T^{8} + 29143544 T^{9} + 202793238 T^{10} - 1300954802 T^{11} - 1160573366 T^{12} - 48135327674 T^{13} + 277623942822 T^{14} + 1476207934232 T^{15} - 2460600970188 T^{16} - 13893200472864 T^{17} - 471077631598036 T^{18} + 1698900873172168 T^{19} + 7056571222927289 T^{20} + 6108201770368619 T^{21} - 81745934331103433 T^{22} - 1601258596015143717 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 + 19 T + 68 T^{2} - 655 T^{3} - 1387 T^{4} + 63285 T^{5} + 319397 T^{6} - 2378554 T^{7} - 20541144 T^{8} + 66494623 T^{9} + 976204363 T^{10} - 1644832272 T^{11} - 49442383823 T^{12} - 67438123152 T^{13} + 1640999534203 T^{14} + 4582875911783 T^{15} - 58044363610584 T^{16} - 275570230313354 T^{17} + 1517169044262677 T^{18} + 12325024222559085 T^{19} - 11075091292790827 T^{20} - 214435167028044455 T^{21} + 912740833090363268 T^{22} + 10456251602608720379 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( ( 1 - 25 T + 439 T^{2} - 5258 T^{3} + 52746 T^{4} - 426931 T^{5} + 3057464 T^{6} - 18358033 T^{7} + 97527354 T^{8} - 418047806 T^{9} + 1500853639 T^{10} - 3675211075 T^{11} + 6321363049 T^{12} )^{2} \)
$47$ \( 1 + 24 T + 115 T^{2} - 2376 T^{3} - 35394 T^{4} - 95696 T^{5} + 1682298 T^{6} + 17182896 T^{7} + 34195735 T^{8} - 445798096 T^{9} - 3285259630 T^{10} - 981842400 T^{11} + 79567059979 T^{12} - 46146592800 T^{13} - 7257138522670 T^{14} - 46284095721008 T^{15} + 166864278360535 T^{16} + 3940811403400272 T^{17} + 18133852389546042 T^{18} - 48481806135827248 T^{19} - 842776680106368834 T^{20} - 2659054004092174392 T^{21} + 6048900207120455635 T^{22} + 59331821162016295272 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 - 35 T + 588 T^{2} - 6446 T^{3} + 52523 T^{4} - 302216 T^{5} + 600290 T^{6} + 9127215 T^{7} - 111906848 T^{8} + 601751953 T^{9} - 6692433 T^{10} - 37037958071 T^{11} + 396587073686 T^{12} - 1963011777763 T^{13} - 18799044297 T^{14} + 89587025506781 T^{15} - 882998857913888 T^{16} + 3816960176641995 T^{17} + 13305044342127410 T^{18} - 355016501836978792 T^{19} + 3270065719475913803 T^{20} - 21270276112756549318 T^{21} + \)\(10\!\cdots\!12\)\( T^{22} - \)\(32\!\cdots\!95\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( 1 - T - 45 T^{2} - 377 T^{3} + 5021 T^{4} - 8190 T^{5} - 34303 T^{6} - 1922028 T^{7} + 23142924 T^{8} - 85445370 T^{9} + 25142239 T^{10} - 5172383726 T^{11} + 106812734191 T^{12} - 305170639834 T^{13} + 87520133959 T^{14} - 17548684645230 T^{15} + 280431164703564 T^{16} - 1374104520558372 T^{17} - 1446918845487223 T^{18} - 20382055660667610 T^{19} + 737235627211295741 T^{20} - 3265949423632912003 T^{21} - 23000253898528863045 T^{22} - 30155888444737842659 T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 - 8 T - 144 T^{2} + 1210 T^{3} + 15692 T^{4} - 91008 T^{5} - 1749157 T^{6} + 5810908 T^{7} + 166130048 T^{8} - 350338768 T^{9} - 12151497572 T^{10} + 9558414952 T^{11} + 772713424737 T^{12} + 583063312072 T^{13} - 45215722465412 T^{14} - 79520243899408 T^{15} + 2300210229930368 T^{16} + 4907871402251308 T^{17} - 90117223456163677 T^{18} - 286014740020599168 T^{19} + 3008271155553333452 T^{20} + 14149916772329310610 T^{21} - \)\(10\!\cdots\!44\)\( T^{22} - \)\(34\!\cdots\!88\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 + 36 T + 448 T^{2} + 882 T^{3} - 30116 T^{4} - 288924 T^{5} - 1370359 T^{6} - 12802922 T^{7} - 99303768 T^{8} + 312986874 T^{9} + 9836955068 T^{10} + 57716972648 T^{11} + 232196539349 T^{12} + 3867037167416 T^{13} + 44158091300252 T^{14} + 94134871184862 T^{15} - 2001082244723928 T^{16} - 17285546435162654 T^{17} - 123960458130728671 T^{18} - 1751085039856342452 T^{19} - 12229134177295800356 T^{20} + 23996163337532143254 T^{21} + \)\(81\!\cdots\!52\)\( T^{22} + \)\(43\!\cdots\!88\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 - T - 272 T^{2} + 1515 T^{3} + 31793 T^{4} - 390805 T^{5} - 1127933 T^{6} + 51435256 T^{7} - 189880844 T^{8} - 3895432897 T^{9} + 35948452553 T^{10} + 120778460288 T^{11} - 3267845694113 T^{12} + 8575270680448 T^{13} + 181216149319673 T^{14} - 1394218283598167 T^{15} - 4825191435738764 T^{16} + 92800998551398856 T^{17} - 144488537543865293 T^{18} - 3554418433499994755 T^{19} + 20530442018896479473 T^{20} + 69460478588450281965 T^{21} - \)\(88\!\cdots\!72\)\( T^{22} - \)\(23\!\cdots\!71\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 + 31 T + 157 T^{2} - 4351 T^{3} - 56165 T^{4} + 21012 T^{5} + 4960409 T^{6} + 40392564 T^{7} - 66672508 T^{8} - 3587879652 T^{9} - 16845025467 T^{10} + 109824364092 T^{11} + 1752697474667 T^{12} + 8017178578716 T^{13} - 89767140713643 T^{14} - 1395746178582084 T^{15} - 1893381950258428 T^{16} + 83736676996834452 T^{17} + 750679658091992201 T^{18} + 232127937683266164 T^{19} - 45294831061231059365 T^{20} - \)\(25\!\cdots\!63\)\( T^{21} + \)\(67\!\cdots\!93\)\( T^{22} + \)\(97\!\cdots\!87\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 - 2 T + 286 T^{2} - 1942 T^{3} + 45506 T^{4} - 458068 T^{5} + 6694253 T^{6} - 59915278 T^{7} + 860339148 T^{8} - 6564781658 T^{9} + 84094060634 T^{10} - 659242015430 T^{11} + 6879140123889 T^{12} - 52080119218970 T^{13} + 524831032416794 T^{14} - 3236693383878662 T^{15} + 33510279502070988 T^{16} - 184362689567763922 T^{17} + 1627288928383820813 T^{18} - 8796696181471880812 T^{19} + 69037553503607964866 T^{20} - \)\(23\!\cdots\!98\)\( T^{21} + \)\(27\!\cdots\!86\)\( T^{22} - \)\(14\!\cdots\!58\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 + 19 T + 3 T^{2} - 2230 T^{3} - 9853 T^{4} + 260324 T^{5} + 3150455 T^{6} - 5556832 T^{7} - 239317072 T^{8} + 68226904 T^{9} + 18813056155 T^{10} + 47128776483 T^{11} - 758831244287 T^{12} + 3911688448089 T^{13} + 129603143851795 T^{14} + 39011256757448 T^{15} - 11357586423756112 T^{16} - 21888587094322976 T^{17} + 1030010933982232895 T^{18} + 7064165337823659148 T^{19} - 22191835363265970973 T^{20} - \)\(41\!\cdots\!90\)\( T^{21} + 46548123561617560347 T^{22} + \)\(24\!\cdots\!73\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 - 40 T + 783 T^{2} - 11380 T^{3} + 136646 T^{4} - 1233320 T^{5} + 8628700 T^{6} - 61036520 T^{7} + 518583425 T^{8} - 6988908095 T^{9} + 118589802798 T^{10} - 1538822973165 T^{11} + 15615590135134 T^{12} - 136955244611685 T^{13} + 939349827962958 T^{14} - 4926963550824055 T^{15} + 32537086229955425 T^{16} - 340831556240077480 T^{17} + 4288302465315180700 T^{18} - 54551389953353826280 T^{19} + \)\(53\!\cdots\!26\)\( T^{20} - \)\(39\!\cdots\!20\)\( T^{21} + \)\(24\!\cdots\!83\)\( T^{22} - \)\(11\!\cdots\!60\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 - 28 T + 403 T^{2} - 4628 T^{3} + 55650 T^{4} - 570136 T^{5} + 5049586 T^{6} - 55109928 T^{7} + 751828627 T^{8} - 8579658424 T^{9} + 86105944162 T^{10} - 959504128864 T^{11} + 10485381483807 T^{12} - 93071900499808 T^{13} + 810170828620258 T^{14} - 7830422592807352 T^{15} + 66558847783527187 T^{16} - 473247703274771496 T^{17} + 4206163774481409394 T^{18} - 46066010719213433368 T^{19} + \)\(43\!\cdots\!50\)\( T^{20} - \)\(35\!\cdots\!76\)\( T^{21} + \)\(29\!\cdots\!47\)\( T^{22} - \)\(20\!\cdots\!84\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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