Properties

Label 525.3.s.h
Level 525
Weight 3
Character orbit 525.s
Analytic conductor 14.305
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3052138789\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 105)
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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} + ( -\beta_{10} + 2 \beta_{13} ) q^{3} + ( 1 - \beta_{2} + \beta_{3} - \beta_{6} - \beta_{8} ) q^{4} + ( -1 + 2 \beta_{3} + \beta_{5} ) q^{6} + ( -\beta_{1} + \beta_{9} - 3 \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{7} + ( -\beta_{1} - \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} + 4 \beta_{13} + \beta_{14} ) q^{8} -3 \beta_{6} q^{9} +O(q^{10})\) \( q + \beta_{11} q^{2} + ( -\beta_{10} + 2 \beta_{13} ) q^{3} + ( 1 - \beta_{2} + \beta_{3} - \beta_{6} - \beta_{8} ) q^{4} + ( -1 + 2 \beta_{3} + \beta_{5} ) q^{6} + ( -\beta_{1} + \beta_{9} - 3 \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{7} + ( -\beta_{1} - \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} + 4 \beta_{13} + \beta_{14} ) q^{8} -3 \beta_{6} q^{9} + ( 6 - \beta_{3} + 2 \beta_{4} - 7 \beta_{6} + \beta_{7} - \beta_{8} ) q^{11} + ( -2 \beta_{1} + \beta_{10} + \beta_{11} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{12} + ( 2 \beta_{1} - 4 \beta_{9} + 2 \beta_{11} - 2 \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{13} + ( 10 - \beta_{2} + \beta_{3} - 5 \beta_{5} - 11 \beta_{6} + \beta_{7} - \beta_{8} ) q^{14} + ( -7 + 2 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} - 4 \beta_{7} + 3 \beta_{8} ) q^{16} + ( -4 \beta_{1} - \beta_{9} + 3 \beta_{10} + 8 \beta_{11} - \beta_{12} - 7 \beta_{13} ) q^{17} -3 \beta_{1} q^{18} + ( 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} ) q^{19} + ( 6 - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + 3 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{21} + ( -3 \beta_{1} + \beta_{9} + \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - 4 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} ) q^{22} + ( 8 \beta_{9} + 16 \beta_{10} - 3 \beta_{11} + 4 \beta_{12} - 12 \beta_{13} + 4 \beta_{14} - 6 \beta_{15} ) q^{23} + ( -7 + 3 \beta_{2} + \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{24} + ( 20 - 9 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 11 \beta_{6} - 5 \beta_{8} ) q^{26} + ( 6 \beta_{10} - 3 \beta_{13} ) q^{27} + ( -5 \beta_{1} - \beta_{9} - 9 \beta_{10} + 8 \beta_{11} - 5 \beta_{12} + 17 \beta_{13} - 2 \beta_{15} ) q^{28} + ( 10 - 4 \beta_{2} + 5 \beta_{4} - 5 \beta_{7} ) q^{29} + ( -10 + \beta_{2} + \beta_{3} + 2 \beta_{5} - 12 \beta_{6} + 4 \beta_{7} - 5 \beta_{8} ) q^{31} + ( \beta_{1} - 3 \beta_{9} + 11 \beta_{10} + 3 \beta_{12} - 3 \beta_{13} + 2 \beta_{14} ) q^{32} + ( 2 \beta_{1} + 7 \beta_{10} - \beta_{11} + 3 \beta_{12} + 4 \beta_{13} - \beta_{14} - \beta_{15} ) q^{33} + ( 16 - 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 36 \beta_{6} - 2 \beta_{7} - 8 \beta_{8} ) q^{34} + ( -6 + 3 \beta_{2} + 3 \beta_{5} ) q^{36} + ( 8 \beta_{9} + 26 \beta_{10} + 8 \beta_{11} + 4 \beta_{12} - 22 \beta_{13} + 4 \beta_{14} - \beta_{15} ) q^{37} + ( 7 \beta_{1} + 8 \beta_{9} + 10 \beta_{10} - 14 \beta_{11} + 8 \beta_{12} - 12 \beta_{13} - 2 \beta_{14} + 4 \beta_{15} ) q^{38} + ( -2 + 5 \beta_{2} + 6 \beta_{3} - 8 \beta_{4} + 6 \beta_{6} - 4 \beta_{7} + 9 \beta_{8} ) q^{39} + ( -2 + 5 \beta_{2} + 6 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + 10 \beta_{8} ) q^{41} + ( -7 \beta_{1} - 2 \beta_{9} + 16 \beta_{10} + 11 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{15} ) q^{42} + ( 5 \beta_{1} + 5 \beta_{9} + 5 \beta_{10} - 5 \beta_{11} + 10 \beta_{12} - 44 \beta_{13} + 2 \beta_{14} - 7 \beta_{15} ) q^{43} + ( 7 - 7 \beta_{3} - 7 \beta_{5} + \beta_{6} - \beta_{8} ) q^{44} + ( -11 + 11 \beta_{2} - 17 \beta_{3} + 4 \beta_{4} + 9 \beta_{6} + 2 \beta_{7} + 9 \beta_{8} ) q^{46} + ( -2 \beta_{1} - 6 \beta_{10} + \beta_{11} + \beta_{12} - 7 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{47} + ( -5 \beta_{1} + 6 \beta_{9} - 4 \beta_{10} - 5 \beta_{11} + 5 \beta_{13} + 3 \beta_{14} - 3 \beta_{15} ) q^{48} + ( 9 \beta_{3} + 2 \beta_{4} - 9 \beta_{5} - 6 \beta_{6} + 7 \beta_{8} ) q^{49} + ( -13 + \beta_{2} + 12 \beta_{3} - \beta_{4} + 12 \beta_{5} + 12 \beta_{6} - 2 \beta_{7} ) q^{51} + ( -15 \beta_{1} + 3 \beta_{9} + 13 \beta_{10} + 30 \beta_{11} + 3 \beta_{12} - 23 \beta_{13} + 6 \beta_{14} - 12 \beta_{15} ) q^{52} + ( 3 \beta_{1} - 5 \beta_{9} + 23 \beta_{10} + 5 \beta_{12} - 5 \beta_{13} - 12 \beta_{14} ) q^{53} + ( -3 - 3 \beta_{3} + 3 \beta_{5} ) q^{54} + ( 18 + \beta_{2} + 14 \beta_{3} - 6 \beta_{4} + 11 \beta_{5} - 6 \beta_{6} - 8 \beta_{7} - 6 \beta_{8} ) q^{56} + ( 6 \beta_{1} + 4 \beta_{9} + 4 \beta_{10} - 6 \beta_{11} + 8 \beta_{12} + \beta_{13} - 4 \beta_{14} ) q^{57} + ( -8 \beta_{9} + 4 \beta_{10} + 32 \beta_{11} - 4 \beta_{12} - 8 \beta_{13} - 4 \beta_{14} + 14 \beta_{15} ) q^{58} + ( 25 + 5 \beta_{2} - 14 \beta_{3} - 28 \beta_{5} - 2 \beta_{6} - \beta_{7} - 4 \beta_{8} ) q^{59} + ( -66 - 6 \beta_{2} + 4 \beta_{3} - 10 \beta_{4} - 4 \beta_{5} + 40 \beta_{6} + 2 \beta_{8} ) q^{61} + ( -20 \beta_{1} + 7 \beta_{9} + 3 \beta_{10} - 20 \beta_{11} + 2 \beta_{13} + 3 \beta_{14} - 4 \beta_{15} ) q^{62} + ( -3 \beta_{1} - 3 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} + 12 \beta_{13} ) q^{63} + ( 4 - 9 \beta_{2} - 6 \beta_{4} - \beta_{5} + 6 \beta_{7} ) q^{64} + ( -1 - 5 \beta_{2} + 3 \beta_{3} + 6 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} ) q^{66} + ( 5 \beta_{1} - \beta_{9} + 49 \beta_{10} + \beta_{12} - \beta_{13} + 9 \beta_{14} ) q^{67} + ( -40 \beta_{1} + 10 \beta_{10} + 20 \beta_{11} - 12 \beta_{12} + 22 \beta_{13} + 8 \beta_{14} + 2 \beta_{15} ) q^{68} + ( -5 - 6 \beta_{2} - 6 \beta_{3} + 12 \beta_{4} - 3 \beta_{5} + 16 \beta_{6} + 12 \beta_{7} - 12 \beta_{8} ) q^{69} + ( 32 + 4 \beta_{2} + 5 \beta_{4} - 28 \beta_{5} - 5 \beta_{7} ) q^{71} + ( 6 \beta_{9} + 12 \beta_{10} - 3 \beta_{11} + 3 \beta_{12} - 9 \beta_{13} + 3 \beta_{14} ) q^{72} + ( -11 \beta_{1} - 7 \beta_{9} + 31 \beta_{10} + 22 \beta_{11} - 7 \beta_{12} - 69 \beta_{13} + 3 \beta_{14} - 6 \beta_{15} ) q^{73} + ( 54 - 31 \beta_{3} + 14 \beta_{4} - 61 \beta_{6} + 7 \beta_{7} - 7 \beta_{8} ) q^{74} + ( -2 + 5 \beta_{2} - 46 \beta_{3} + 6 \beta_{4} - 23 \beta_{5} + 50 \beta_{6} + 6 \beta_{7} + 10 \beta_{8} ) q^{76} + ( -9 \beta_{1} + 6 \beta_{9} - 10 \beta_{10} - 4 \beta_{11} + 11 \beta_{12} + 23 \beta_{13} - 7 \beta_{14} - 4 \beta_{15} ) q^{77} + ( 3 \beta_{1} + \beta_{9} + \beta_{10} - 3 \beta_{11} + 2 \beta_{12} + 31 \beta_{13} + 14 \beta_{14} - 15 \beta_{15} ) q^{78} + ( -1 + 6 \beta_{2} - 5 \beta_{3} - 6 \beta_{4} - 5 \beta_{5} + 58 \beta_{6} - 12 \beta_{7} + 3 \beta_{8} ) q^{79} + ( -9 + 9 \beta_{6} ) q^{81} + ( 10 \beta_{1} + 24 \beta_{10} - 5 \beta_{11} + 9 \beta_{12} + 15 \beta_{13} + 7 \beta_{14} - 8 \beta_{15} ) q^{82} + ( -12 \beta_{1} - 5 \beta_{9} - 13 \beta_{10} - 12 \beta_{11} + 4 \beta_{13} + 9 \beta_{14} + 14 \beta_{15} ) q^{83} + ( -18 - \beta_{2} + 11 \beta_{3} + 3 \beta_{4} + 13 \beta_{5} - 20 \beta_{6} - 6 \beta_{7} - 8 \beta_{8} ) q^{84} + ( 51 - 3 \beta_{2} - 48 \beta_{3} + 3 \beta_{4} - 48 \beta_{5} + 16 \beta_{6} + 6 \beta_{7} + 12 \beta_{8} ) q^{86} + ( 15 \beta_{9} - 5 \beta_{10} + 15 \beta_{12} + 25 \beta_{13} + 14 \beta_{14} - 28 \beta_{15} ) q^{87} + ( 17 \beta_{1} + 5 \beta_{9} - 25 \beta_{10} - 5 \beta_{12} + 5 \beta_{13} + 10 \beta_{14} ) q^{88} + ( -47 - 11 \beta_{2} + 10 \beta_{3} - 11 \beta_{4} - 10 \beta_{5} + 34 \beta_{6} ) q^{89} + ( 11 - 27 \beta_{2} - 15 \beta_{3} + 14 \beta_{4} - 30 \beta_{5} - 10 \beta_{6} + 20 \beta_{7} - 41 \beta_{8} ) q^{91} + ( 53 \beta_{1} - 3 \beta_{9} - 3 \beta_{10} - 53 \beta_{11} - 6 \beta_{12} - 8 \beta_{13} - 29 \beta_{14} + 32 \beta_{15} ) q^{92} + ( -8 \beta_{9} + 28 \beta_{10} - 3 \beta_{11} - 4 \beta_{12} - 32 \beta_{13} - 4 \beta_{14} + 15 \beta_{15} ) q^{93} + ( -3 + 3 \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} - 4 \beta_{7} + \beta_{8} ) q^{94} + ( -18 + 7 \beta_{2} + \beta_{3} - 9 \beta_{4} - \beta_{5} + 14 \beta_{6} + 8 \beta_{8} ) q^{96} + ( -14 \beta_{1} - 10 \beta_{9} + 22 \beta_{10} - 14 \beta_{11} - 16 \beta_{13} - 12 \beta_{14} - 2 \beta_{15} ) q^{97} + ( 10 \beta_{1} - 7 \beta_{9} - 29 \beta_{10} + 9 \beta_{11} + 9 \beta_{12} + 83 \beta_{13} - 16 \beta_{15} ) q^{98} + ( -12 - 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 12q^{4} - 24q^{9} + O(q^{10}) \) \( 16q + 12q^{4} - 24q^{9} + 40q^{11} + 32q^{14} - 4q^{16} + 96q^{21} - 96q^{24} + 240q^{26} + 200q^{29} - 252q^{31} - 72q^{36} + 24q^{39} + 36q^{44} - 164q^{46} - 76q^{49} + 36q^{51} - 36q^{54} + 392q^{56} + 108q^{59} - 792q^{61} + 8q^{64} + 48q^{66} + 328q^{71} + 280q^{74} + 412q^{79} - 72q^{81} - 264q^{84} + 356q^{86} - 564q^{89} - 228q^{91} - 60q^{94} - 216q^{96} - 240q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 22 x^{14} + 343 x^{12} - 2542 x^{10} + 13621 x^{8} - 35080 x^{6} + 64300 x^{4} - 28000 x^{2} + 10000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -3243 \nu^{14} + 70156 \nu^{12} - 972969 \nu^{10} + 6085386 \nu^{8} - 17251573 \nu^{6} + 31891800 \nu^{4} - 13938000 \nu^{2} + 1162228000 \)\()/ 298803000 \)
\(\beta_{3}\)\(=\)\((\)\( 2097 \nu^{14} - 57414 \nu^{12} + 952781 \nu^{10} - 8714814 \nu^{8} + 49729797 \nu^{6} - 160507590 \nu^{4} + 245765100 \nu^{2} - 107196000 \)\()/ 117007000 \)
\(\beta_{4}\)\(=\)\((\)\( -86391 \nu^{14} + 1089782 \nu^{12} - 13023473 \nu^{10} - 31499288 \nu^{8} + 476316679 \nu^{6} - 5048799890 \nu^{4} + 9068677900 \nu^{2} - 14205490000 \)\()/ 2457147000 \)
\(\beta_{5}\)\(=\)\((\)\(9374291 \nu^{14} - 190524872 \nu^{12} + 2812486753 \nu^{10} - 17590557882 \nu^{8} + 77297768001 \nu^{6} - 92187176600 \nu^{4} + 40289506000 \nu^{2} + 411726170000\)\()/ 253086141000 \)
\(\beta_{6}\)\(=\)\((\)\(6119178 \nu^{14} - 131308181 \nu^{12} + 2033326684 \nu^{10} - 14560759471 \nu^{8} + 77131205568 \nu^{6} - 183317921405 \nu^{4} + 360875734400 \nu^{2} - 30551903500\)\()/ 126543070500 \)
\(\beta_{7}\)\(=\)\((\)\(3255576 \nu^{14} - 69896422 \nu^{12} + 1071617318 \nu^{10} - 7533395317 \nu^{8} + 37675335571 \nu^{6} - 75410643205 \nu^{4} + 83443297550 \nu^{2} + 62550453500\)\()/ 18077581500 \)
\(\beta_{8}\)\(=\)\((\)\(-23640839 \nu^{14} + 499185228 \nu^{12} - 7729957392 \nu^{10} + 54583447073 \nu^{8} - 293224368384 \nu^{6} + 696907060140 \nu^{4} - 1391895892850 \nu^{2} + 116147058000\)\()/ 126543070500 \)
\(\beta_{9}\)\(=\)\((\)\(-4776241 \nu^{15} + 115310592 \nu^{13} - 1863758843 \nu^{11} + 15788353442 \nu^{9} - 93939460491 \nu^{7} + 347196076720 \nu^{5} - 896995494300 \nu^{3} + 1505969628000 \nu\)\()/ 180775815000 \)
\(\beta_{10}\)\(=\)\((\)\(73224919 \nu^{15} - 1471847198 \nu^{13} + 21969033677 \nu^{11} - 137404223538 \nu^{9} + 632989616059 \nu^{7} - 720096969400 \nu^{5} + 314711354000 \nu^{3} + 3668496441000 \nu\)\()/ 2530861410000 \)
\(\beta_{11}\)\(=\)\((\)\(-6119178 \nu^{15} + 131308181 \nu^{13} - 2033326684 \nu^{11} + 14560759471 \nu^{9} - 77131205568 \nu^{7} + 183317921405 \nu^{5} - 360875734400 \nu^{3} + 157094974000 \nu\)\()/ 126543070500 \)
\(\beta_{12}\)\(=\)\((\)\(-226365739 \nu^{15} + 4362137888 \nu^{13} - 65222132237 \nu^{11} + 386625669928 \nu^{9} - 1857250226079 \nu^{7} + 1651261681150 \nu^{5} - 2802829493500 \nu^{3} - 19091613416000 \nu\)\()/ 2530861410000 \)
\(\beta_{13}\)\(=\)\((\)\( 53598 \nu^{15} - 1168671 \nu^{13} + 18097044 \nu^{11} - 131482211 \nu^{9} + 686484288 \nu^{7} - 1631568855 \nu^{5} + 2643813450 \nu^{3} - 271918500 \nu \)\()/ 585035000 \)
\(\beta_{14}\)\(=\)\((\)\(-54476337 \nu^{15} + 1090797454 \nu^{13} - 16344060171 \nu^{11} + 102223107774 \nu^{9} - 478394080057 \nu^{7} + 535722616200 \nu^{5} - 234132342000 \nu^{3} - 2845044101000 \nu\)\()/ 506172282000 \)
\(\beta_{15}\)\(=\)\((\)\(25018339 \nu^{15} - 562043572 \nu^{13} + 8812748265 \nu^{11} - 67088899600 \nu^{9} + 362617097947 \nu^{7} - 982088475474 \nu^{5} + 1723148574100 \nu^{3} - 750541216000 \nu\)\()/ 101234456400 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + 5 \beta_{6} - \beta_{5} - \beta_{3} + 1\)
\(\nu^{3}\)\(=\)\(-\beta_{14} - 4 \beta_{13} + 2 \beta_{12} - 9 \beta_{11} + \beta_{10} + \beta_{9} + 9 \beta_{1}\)
\(\nu^{4}\)\(=\)\(9 \beta_{8} + 2 \beta_{7} + 39 \beta_{6} + 4 \beta_{4} - 17 \beta_{3} + 11 \beta_{2} - 41\)
\(\nu^{5}\)\(=\)\(8 \beta_{15} + 13 \beta_{14} - 59 \beta_{13} + 13 \beta_{12} - 95 \beta_{11} + 72 \beta_{10} + 26 \beta_{9}\)
\(\nu^{6}\)\(=\)\(-34 \beta_{7} + 243 \beta_{5} + 34 \beta_{4} + 155 \beta_{2} - 684\)
\(\nu^{7}\)\(=\)\(466 \beta_{14} + 155 \beta_{13} - 155 \beta_{12} + 905 \beta_{10} + 155 \beta_{9} - 1081 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-925 \beta_{8} - 932 \beta_{7} - 4007 \beta_{6} + 3229 \beta_{5} - 466 \beta_{4} + 3229 \beta_{3} + 466 \beta_{2} - 3695\)
\(\nu^{9}\)\(=\)\(-2304 \beta_{15} + 4161 \beta_{14} + 14288 \beta_{13} - 3714 \beta_{12} + 12807 \beta_{11} - 1857 \beta_{10} - 1857 \beta_{9} - 12807 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-10503 \beta_{8} - 6018 \beta_{7} - 45981 \beta_{6} - 12036 \beta_{4} + 41435 \beta_{3} - 16521 \beta_{2} + 51999\)
\(\nu^{11}\)\(=\)\(-30932 \beta_{15} - 22539 \beta_{14} + 162097 \beta_{13} - 22539 \beta_{12} + 155033 \beta_{11} - 184636 \beta_{10} - 45078 \beta_{9}\)
\(\nu^{12}\)\(=\)\(76010 \beta_{7} - 522621 \beta_{5} - 76010 \beta_{4} - 276121 \beta_{2} + 1221776\)
\(\nu^{13}\)\(=\)\(-950762 \beta_{14} - 276121 \beta_{13} + 276121 \beta_{12} - 2060863 \beta_{10} - 276121 \beta_{9} + 1898119 \beta_{1}\)
\(\nu^{14}\)\(=\)\(1499599 \beta_{8} + 1901524 \beta_{7} + 6638309 \beta_{6} - 6535147 \beta_{5} + 950762 \beta_{4} - 6535147 \beta_{3} - 950762 \beta_{2} + 7485909\)
\(\nu^{15}\)\(=\)\(5035548 \beta_{15} - 8436671 \beta_{14} - 29274612 \beta_{13} + 6802246 \beta_{12} - 23376825 \beta_{11} + 3401123 \beta_{10} + 3401123 \beta_{9} + 23376825 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1
−3.04878 + 1.76021i
−2.18275 + 1.26021i
−1.44926 + 0.836732i
−0.583237 + 0.336732i
0.583237 0.336732i
1.44926 0.836732i
2.18275 1.26021i
3.04878 1.76021i
−3.04878 1.76021i
−2.18275 1.26021i
−1.44926 0.836732i
−0.583237 0.336732i
0.583237 + 0.336732i
1.44926 + 0.836732i
2.18275 + 1.26021i
3.04878 + 1.76021i
−3.04878 1.76021i −0.866025 1.50000i 4.19671 + 7.26891i 0 6.09756i −6.99575 0.244004i 15.4667i −1.50000 + 2.59808i 0
124.2 −2.18275 1.26021i 0.866025 + 1.50000i 1.17628 + 2.03737i 0 4.36551i 3.28656 6.18050i 4.15226i −1.50000 + 2.59808i 0
124.3 −1.44926 0.836732i −0.866025 1.50000i −0.599760 1.03881i 0 2.89852i −5.13152 4.76104i 8.70121i −1.50000 + 2.59808i 0
124.4 −0.583237 0.336732i 0.866025 + 1.50000i −1.77322 3.07131i 0 1.16647i −1.55742 6.82455i 5.08226i −1.50000 + 2.59808i 0
124.5 0.583237 + 0.336732i −0.866025 1.50000i −1.77322 3.07131i 0 1.16647i 1.55742 + 6.82455i 5.08226i −1.50000 + 2.59808i 0
124.6 1.44926 + 0.836732i 0.866025 + 1.50000i −0.599760 1.03881i 0 2.89852i 5.13152 + 4.76104i 8.70121i −1.50000 + 2.59808i 0
124.7 2.18275 + 1.26021i −0.866025 1.50000i 1.17628 + 2.03737i 0 4.36551i −3.28656 + 6.18050i 4.15226i −1.50000 + 2.59808i 0
124.8 3.04878 + 1.76021i 0.866025 + 1.50000i 4.19671 + 7.26891i 0 6.09756i 6.99575 + 0.244004i 15.4667i −1.50000 + 2.59808i 0
199.1 −3.04878 + 1.76021i −0.866025 + 1.50000i 4.19671 7.26891i 0 6.09756i −6.99575 + 0.244004i 15.4667i −1.50000 2.59808i 0
199.2 −2.18275 + 1.26021i 0.866025 1.50000i 1.17628 2.03737i 0 4.36551i 3.28656 + 6.18050i 4.15226i −1.50000 2.59808i 0
199.3 −1.44926 + 0.836732i −0.866025 + 1.50000i −0.599760 + 1.03881i 0 2.89852i −5.13152 + 4.76104i 8.70121i −1.50000 2.59808i 0
199.4 −0.583237 + 0.336732i 0.866025 1.50000i −1.77322 + 3.07131i 0 1.16647i −1.55742 + 6.82455i 5.08226i −1.50000 2.59808i 0
199.5 0.583237 0.336732i −0.866025 + 1.50000i −1.77322 + 3.07131i 0 1.16647i 1.55742 6.82455i 5.08226i −1.50000 2.59808i 0
199.6 1.44926 0.836732i 0.866025 1.50000i −0.599760 + 1.03881i 0 2.89852i 5.13152 4.76104i 8.70121i −1.50000 2.59808i 0
199.7 2.18275 1.26021i −0.866025 + 1.50000i 1.17628 2.03737i 0 4.36551i −3.28656 6.18050i 4.15226i −1.50000 2.59808i 0
199.8 3.04878 1.76021i 0.866025 1.50000i 4.19671 7.26891i 0 6.09756i 6.99575 0.244004i 15.4667i −1.50000 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.s.h 16
5.b even 2 1 inner 525.3.s.h 16
5.c odd 4 1 105.3.n.a 8
5.c odd 4 1 525.3.o.l 8
7.d odd 6 1 inner 525.3.s.h 16
15.e even 4 1 315.3.w.a 8
35.i odd 6 1 inner 525.3.s.h 16
35.k even 12 1 105.3.n.a 8
35.k even 12 1 525.3.o.l 8
35.k even 12 1 735.3.h.a 8
35.l odd 12 1 735.3.h.a 8
105.w odd 12 1 315.3.w.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.n.a 8 5.c odd 4 1
105.3.n.a 8 35.k even 12 1
315.3.w.a 8 15.e even 4 1
315.3.w.a 8 105.w odd 12 1
525.3.o.l 8 5.c odd 4 1
525.3.o.l 8 35.k even 12 1
525.3.s.h 16 1.a even 1 1 trivial
525.3.s.h 16 5.b even 2 1 inner
525.3.s.h 16 7.d odd 6 1 inner
525.3.s.h 16 35.i odd 6 1 inner
735.3.h.a 8 35.k even 12 1
735.3.h.a 8 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\):

\(T_{2}^{16} - \cdots\)
\(T_{11}^{8} - \cdots\)
\( T_{13}^{8} - 1164 T_{13}^{6} + 420414 T_{13}^{4} - 47028060 T_{13}^{2} + 1230957225 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 10 T^{2} + 39 T^{4} + 50 T^{6} - 235 T^{8} - 1560 T^{10} - 5684 T^{12} - 25760 T^{14} - 116976 T^{16} - 412160 T^{18} - 1455104 T^{20} - 6389760 T^{22} - 15400960 T^{24} + 52428800 T^{26} + 654311424 T^{28} + 2684354560 T^{30} + 4294967296 T^{32} \)
$3$ \( ( 1 + 3 T^{2} + 9 T^{4} )^{4} \)
$5$ 1
$7$ \( 1 + 38 T^{2} + 121 T^{4} - 132202 T^{6} - 7481516 T^{8} - 317417002 T^{10} + 697540921 T^{12} + 525968913638 T^{14} + 33232930569601 T^{16} \)
$11$ \( ( 1 - 20 T - 147 T^{2} + 2960 T^{3} + 57131 T^{4} - 519480 T^{5} - 8882912 T^{6} + 8003440 T^{7} + 1602642534 T^{8} + 968416240 T^{9} - 130054714592 T^{10} - 920290508280 T^{11} + 12246537230411 T^{12} + 76774776818960 T^{13} - 461348971377987 T^{14} - 7594996671664820 T^{15} + 45949729863572161 T^{16} )^{2} \)
$13$ \( ( 1 + 188 T^{2} + 39826 T^{4} + 8798048 T^{6} + 2113175419 T^{8} + 251281048928 T^{10} + 32487291694546 T^{12} + 4380040003026428 T^{14} + 665416609183179841 T^{16} )^{2} \)
$17$ \( 1 - 1208 T^{2} + 685248 T^{4} - 238401232 T^{6} + 61817983778 T^{8} - 17147293777272 T^{10} + 6777126590126848 T^{12} - 2780839422001632392 T^{14} + \)\(91\!\cdots\!83\)\( T^{16} - \)\(23\!\cdots\!32\)\( T^{18} + \)\(47\!\cdots\!68\)\( T^{20} - \)\(99\!\cdots\!92\)\( T^{22} + \)\(30\!\cdots\!18\)\( T^{24} - \)\(96\!\cdots\!32\)\( T^{26} + \)\(23\!\cdots\!08\)\( T^{28} - \)\(34\!\cdots\!28\)\( T^{30} + \)\(23\!\cdots\!61\)\( T^{32} \)
$19$ \( ( 1 + 598 T^{2} + 183481 T^{4} + 682560 T^{5} - 48973562 T^{6} + 501474240 T^{7} - 32269961996 T^{8} + 181032200640 T^{9} - 6382283573402 T^{10} + 32111636535360 T^{11} + 3116161130325721 T^{12} + 1323562321601564278 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} )^{2} \)
$23$ \( 1 + 850 T^{2} + 592959 T^{4} + 52476230 T^{6} - 67877437675 T^{8} - 90481750443780 T^{10} + 6471161742853546 T^{12} + 26894707124495548840 T^{14} + \)\(27\!\cdots\!14\)\( T^{16} + \)\(75\!\cdots\!40\)\( T^{18} + \)\(50\!\cdots\!26\)\( T^{20} - \)\(19\!\cdots\!80\)\( T^{22} - \)\(41\!\cdots\!75\)\( T^{24} + \)\(90\!\cdots\!30\)\( T^{26} + \)\(28\!\cdots\!19\)\( T^{28} + \)\(11\!\cdots\!50\)\( T^{30} + \)\(37\!\cdots\!21\)\( T^{32} \)
$29$ \( ( 1 - 50 T + 1234 T^{2} + 15850 T^{3} - 1164374 T^{4} + 13329850 T^{5} + 872784754 T^{6} - 29741166050 T^{7} + 500246412961 T^{8} )^{4} \)
$31$ \( ( 1 + 126 T + 9883 T^{2} + 578466 T^{3} + 27206317 T^{4} + 1079090100 T^{5} + 38160094402 T^{6} + 1243487527488 T^{7} + 38998740329170 T^{8} + 1194991513915968 T^{9} + 35241648542229442 T^{10} + 957696435880658100 T^{11} + 23204023931078714797 T^{12} + \)\(47\!\cdots\!66\)\( T^{13} + \)\(77\!\cdots\!63\)\( T^{14} + \)\(95\!\cdots\!46\)\( T^{15} + \)\(72\!\cdots\!81\)\( T^{16} )^{2} \)
$37$ \( 1 + 4012 T^{2} + 3715446 T^{4} + 1106164568 T^{6} + 16621668655361 T^{8} + 31297439550751656 T^{10} + 8237655295148593894 T^{12} + \)\(26\!\cdots\!24\)\( T^{14} + \)\(95\!\cdots\!52\)\( T^{16} + \)\(50\!\cdots\!64\)\( T^{18} + \)\(28\!\cdots\!74\)\( T^{20} + \)\(20\!\cdots\!36\)\( T^{22} + \)\(20\!\cdots\!01\)\( T^{24} + \)\(25\!\cdots\!68\)\( T^{26} + \)\(16\!\cdots\!06\)\( T^{28} + \)\(32\!\cdots\!52\)\( T^{30} + \)\(15\!\cdots\!81\)\( T^{32} \)
$41$ \( ( 1 - 10106 T^{2} + 48877645 T^{4} - 146585251874 T^{6} + 296639674915264 T^{8} - 414214887920726114 T^{10} + \)\(39\!\cdots\!45\)\( T^{12} - \)\(22\!\cdots\!86\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} )^{2} \)
$43$ \( ( 1 - 3058 T^{2} + 2023873 T^{4} - 9897938386 T^{6} + 38630411096740 T^{8} - 33839081651995186 T^{10} + 23655432960429168673 T^{12} - \)\(12\!\cdots\!58\)\( T^{14} + \)\(13\!\cdots\!01\)\( T^{16} )^{2} \)
$47$ \( 1 - 16718 T^{2} + 155339283 T^{4} - 1005045086002 T^{6} + 4999251300672953 T^{8} - 20050901794507340652 T^{10} + \)\(66\!\cdots\!18\)\( T^{12} - \)\(18\!\cdots\!72\)\( T^{14} + \)\(44\!\cdots\!78\)\( T^{16} - \)\(91\!\cdots\!32\)\( T^{18} + \)\(15\!\cdots\!98\)\( T^{20} - \)\(23\!\cdots\!32\)\( T^{22} + \)\(28\!\cdots\!13\)\( T^{24} - \)\(27\!\cdots\!02\)\( T^{26} + \)\(20\!\cdots\!23\)\( T^{28} - \)\(11\!\cdots\!98\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \)
$53$ \( 1 + 11914 T^{2} + 81330459 T^{4} + 353520432038 T^{6} + 995406028387193 T^{8} + 1125584826270060372 T^{10} - \)\(50\!\cdots\!38\)\( T^{12} - \)\(36\!\cdots\!48\)\( T^{14} - \)\(12\!\cdots\!02\)\( T^{16} - \)\(28\!\cdots\!88\)\( T^{18} - \)\(31\!\cdots\!18\)\( T^{20} + \)\(55\!\cdots\!52\)\( T^{22} + \)\(38\!\cdots\!53\)\( T^{24} + \)\(10\!\cdots\!38\)\( T^{26} + \)\(19\!\cdots\!79\)\( T^{28} + \)\(22\!\cdots\!54\)\( T^{30} + \)\(15\!\cdots\!41\)\( T^{32} \)
$59$ \( ( 1 - 54 T + 7198 T^{2} - 336204 T^{3} + 19932742 T^{4} + 202333950 T^{5} - 35478676088 T^{6} + 7641841019598 T^{7} - 385856896323245 T^{8} + 26601248589220638 T^{9} - 429907925960363768 T^{10} + 8534553984691411950 T^{11} + \)\(29\!\cdots\!82\)\( T^{12} - \)\(17\!\cdots\!04\)\( T^{13} + \)\(12\!\cdots\!38\)\( T^{14} - \)\(33\!\cdots\!94\)\( T^{15} + \)\(21\!\cdots\!41\)\( T^{16} )^{2} \)
$61$ \( ( 1 + 396 T + 83164 T^{2} + 12233232 T^{3} + 1413738778 T^{4} + 136250283708 T^{5} + 11318984386192 T^{6} + 825650586150588 T^{7} + 53403008176121923 T^{8} + 3072245831066337948 T^{9} + \)\(15\!\cdots\!72\)\( T^{10} + \)\(70\!\cdots\!88\)\( T^{11} + \)\(27\!\cdots\!18\)\( T^{12} + \)\(87\!\cdots\!32\)\( T^{13} + \)\(22\!\cdots\!44\)\( T^{14} + \)\(39\!\cdots\!36\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} )^{2} \)
$67$ \( 1 + 21394 T^{2} + 239151459 T^{4} + 1707944285678 T^{6} + 8446046232738713 T^{8} + 30181218213405933492 T^{10} + \)\(84\!\cdots\!02\)\( T^{12} + \)\(24\!\cdots\!52\)\( T^{14} + \)\(90\!\cdots\!38\)\( T^{16} + \)\(48\!\cdots\!92\)\( T^{18} + \)\(34\!\cdots\!82\)\( T^{20} + \)\(24\!\cdots\!12\)\( T^{22} + \)\(13\!\cdots\!53\)\( T^{24} + \)\(56\!\cdots\!78\)\( T^{26} + \)\(16\!\cdots\!39\)\( T^{28} + \)\(28\!\cdots\!54\)\( T^{30} + \)\(27\!\cdots\!61\)\( T^{32} \)
$71$ \( ( 1 - 82 T + 12166 T^{2} - 846262 T^{3} + 94594474 T^{4} - 4266006742 T^{5} + 309158511046 T^{6} - 10504223281522 T^{7} + 645753531245761 T^{8} )^{4} \)
$73$ \( 1 - 15422 T^{2} + 48442059 T^{4} - 65292445378 T^{6} + 4729429873870505 T^{8} - 29601647342638120044 T^{10} - \)\(89\!\cdots\!26\)\( T^{12} - \)\(37\!\cdots\!20\)\( T^{14} + \)\(66\!\cdots\!90\)\( T^{16} - \)\(10\!\cdots\!20\)\( T^{18} - \)\(72\!\cdots\!06\)\( T^{20} - \)\(67\!\cdots\!24\)\( T^{22} + \)\(30\!\cdots\!05\)\( T^{24} - \)\(12\!\cdots\!78\)\( T^{26} + \)\(25\!\cdots\!19\)\( T^{28} - \)\(22\!\cdots\!82\)\( T^{30} + \)\(42\!\cdots\!21\)\( T^{32} \)
$79$ \( ( 1 - 206 T + 5583 T^{2} + 659438 T^{3} + 124066817 T^{4} - 19494076044 T^{5} + 428008398310 T^{6} - 33090623674568 T^{7} + 7605703397631354 T^{8} - 206518582352978888 T^{9} + 16670961782854763110 T^{10} - \)\(47\!\cdots\!24\)\( T^{11} + \)\(18\!\cdots\!37\)\( T^{12} + \)\(62\!\cdots\!38\)\( T^{13} + \)\(32\!\cdots\!03\)\( T^{14} - \)\(75\!\cdots\!86\)\( T^{15} + \)\(23\!\cdots\!21\)\( T^{16} )^{2} \)
$83$ \( ( 1 + 20672 T^{2} + 223804480 T^{4} + 2182268545136 T^{6} + 17948924233578718 T^{8} + \)\(10\!\cdots\!56\)\( T^{10} + \)\(50\!\cdots\!80\)\( T^{12} + \)\(22\!\cdots\!92\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} )^{2} \)
$89$ \( ( 1 + 282 T + 59686 T^{2} + 9356196 T^{3} + 1240796086 T^{4} + 138656838366 T^{5} + 14271061565800 T^{6} + 1337157406377822 T^{7} + 121622616146107507 T^{8} + 10591623815918728062 T^{9} + \)\(89\!\cdots\!00\)\( T^{10} + \)\(68\!\cdots\!26\)\( T^{11} + \)\(48\!\cdots\!66\)\( T^{12} + \)\(29\!\cdots\!96\)\( T^{13} + \)\(14\!\cdots\!06\)\( T^{14} + \)\(55\!\cdots\!62\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} )^{2} \)
$97$ \( ( 1 + 44576 T^{2} + 925514428 T^{4} + 12414040936928 T^{6} + 128325632901816454 T^{8} + \)\(10\!\cdots\!68\)\( T^{10} + \)\(72\!\cdots\!08\)\( T^{12} + \)\(30\!\cdots\!16\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} )^{2} \)
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