Properties

Label 2001.2.a.n
Level $2001$
Weight $2$
Character orbit 2001.a
Self dual yes
Analytic conductor $15.978$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.9780654445\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + 5778 x^{8} - 5124 x^{7} - 9405 x^{6} + 8288 x^{5} + 6405 x^{4} - 6032 x^{3} - 400 x^{2} + 1088 x - 192\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} -\beta_{8} q^{5} -\beta_{1} q^{6} + ( 1 - \beta_{10} ) q^{7} + ( -\beta_{1} - \beta_{3} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} -\beta_{8} q^{5} -\beta_{1} q^{6} + ( 1 - \beta_{10} ) q^{7} + ( -\beta_{1} - \beta_{3} ) q^{8} + q^{9} + ( -\beta_{1} - \beta_{13} - \beta_{15} ) q^{10} + ( 1 + \beta_{13} ) q^{11} + ( 2 + \beta_{2} ) q^{12} + ( 1 - \beta_{2} - \beta_{7} ) q^{13} + ( 1 - \beta_{1} + \beta_{6} + \beta_{10} ) q^{14} -\beta_{8} q^{15} + ( 3 + \beta_{2} + \beta_{8} + \beta_{12} - \beta_{14} ) q^{16} + ( \beta_{1} + \beta_{5} - \beta_{8} - \beta_{12} ) q^{17} -\beta_{1} q^{18} + ( 1 - \beta_{5} + \beta_{15} ) q^{19} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{20} + ( 1 - \beta_{10} ) q^{21} + ( -\beta_{1} - \beta_{5} + \beta_{8} - \beta_{11} ) q^{22} - q^{23} + ( -\beta_{1} - \beta_{3} ) q^{24} + ( 1 - \beta_{4} + \beta_{14} ) q^{25} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{6} ) q^{26} + q^{27} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} ) q^{28} - q^{29} + ( -\beta_{1} - \beta_{13} - \beta_{15} ) q^{30} + ( 2 + \beta_{3} - \beta_{9} ) q^{31} + ( -2 - 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{9} - \beta_{11} + \beta_{14} ) q^{32} + ( 1 + \beta_{13} ) q^{33} + ( -1 + \beta_{3} + \beta_{4} + \beta_{9} ) q^{34} + ( -2 + \beta_{4} + \beta_{7} - 2 \beta_{8} + \beta_{14} ) q^{35} + ( 2 + \beta_{2} ) q^{36} + ( 2 + \beta_{8} + \beta_{11} ) q^{37} + ( 1 - 2 \beta_{1} - \beta_{4} + 2 \beta_{8} - \beta_{9} ) q^{38} + ( 1 - \beta_{2} - \beta_{7} ) q^{39} + ( -2 + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{40} + ( -1 + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{15} ) q^{41} + ( 1 - \beta_{1} + \beta_{6} + \beta_{10} ) q^{42} + ( 2 + \beta_{8} - \beta_{11} + \beta_{12} ) q^{43} + ( 1 - \beta_{5} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{44} -\beta_{8} q^{45} + \beta_{1} q^{46} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{9} + \beta_{11} ) q^{47} + ( 3 + \beta_{2} + \beta_{8} + \beta_{12} - \beta_{14} ) q^{48} + ( 3 + 3 \beta_{1} - \beta_{4} + \beta_{6} + \beta_{9} - 2 \beta_{10} - \beta_{14} ) q^{49} + ( 1 - 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} + \beta_{11} - \beta_{14} ) q^{50} + ( \beta_{1} + \beta_{5} - \beta_{8} - \beta_{12} ) q^{51} + ( -2 + 2 \beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{12} + \beta_{14} ) q^{52} + ( -2 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} ) q^{53} -\beta_{1} q^{54} + ( \beta_{1} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{11} - \beta_{12} ) q^{55} + ( 3 + \beta_{1} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} + \beta_{15} ) q^{56} + ( 1 - \beta_{5} + \beta_{15} ) q^{57} + \beta_{1} q^{58} + ( -\beta_{2} + \beta_{8} + \beta_{9} - \beta_{12} - \beta_{14} ) q^{59} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{60} + ( 2 + \beta_{4} + \beta_{8} + \beta_{9} - \beta_{14} ) q^{61} + ( -4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{14} + \beta_{15} ) q^{62} + ( 1 - \beta_{10} ) q^{63} + ( 4 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{64} + ( 1 - \beta_{1} + \beta_{4} - \beta_{5} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{65} + ( -\beta_{1} - \beta_{5} + \beta_{8} - \beta_{11} ) q^{66} + ( 2 + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{13} ) q^{67} + ( -2 - 2 \beta_{2} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{14} - \beta_{15} ) q^{68} - q^{69} + ( 2 + 2 \beta_{1} + \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{70} + ( 2 - \beta_{2} + \beta_{6} + \beta_{7} - 2 \beta_{14} - \beta_{15} ) q^{71} + ( -\beta_{1} - \beta_{3} ) q^{72} + ( 2 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{9} + \beta_{11} - \beta_{15} ) q^{73} + ( -1 - 2 \beta_{4} + \beta_{5} + \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{74} + ( 1 - \beta_{4} + \beta_{14} ) q^{75} + ( 5 + 2 \beta_{2} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{76} + ( -2 + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{11} + \beta_{15} ) q^{77} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{6} ) q^{78} + ( 3 + \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{10} + \beta_{14} ) q^{79} + ( \beta_{1} + 2 \beta_{3} + \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{10} + \beta_{11} - 2 \beta_{13} ) q^{80} + q^{81} + ( \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{10} - \beta_{12} + \beta_{14} ) q^{82} + ( 2 - \beta_{1} + \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{11} - \beta_{14} - \beta_{15} ) q^{83} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} ) q^{84} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} + \beta_{9} + 2 \beta_{13} ) q^{85} + ( -3 \beta_{1} - \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{9} - \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{86} - q^{87} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} - \beta_{11} + 2 \beta_{13} ) q^{88} + ( -2 \beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} - \beta_{15} ) q^{89} + ( -\beta_{1} - \beta_{13} - \beta_{15} ) q^{90} + ( -2 - 4 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{13} ) q^{91} + ( -2 - \beta_{2} ) q^{92} + ( 2 + \beta_{3} - \beta_{9} ) q^{93} + ( -2 - \beta_{1} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{94} + ( 2 + 4 \beta_{1} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{95} + ( -2 - 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{9} - \beta_{11} + \beta_{14} ) q^{96} + ( 2 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} - \beta_{10} ) q^{97} + ( -9 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{98} + ( 1 + \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - q^{2} + 16q^{3} + 25q^{4} + 3q^{5} - q^{6} + 13q^{7} + 16q^{9} + O(q^{10}) \) \( 16q - q^{2} + 16q^{3} + 25q^{4} + 3q^{5} - q^{6} + 13q^{7} + 16q^{9} + 11q^{10} + 8q^{11} + 25q^{12} + 19q^{13} + 16q^{14} + 3q^{15} + 31q^{16} - 4q^{17} - q^{18} + 19q^{19} + 16q^{20} + 13q^{21} + 6q^{22} - 16q^{23} + 23q^{25} - 15q^{26} + 16q^{27} + 18q^{28} - 16q^{29} + 11q^{30} + 24q^{31} - 21q^{32} + 8q^{33} - 9q^{34} - 13q^{35} + 25q^{36} + 26q^{37} + 19q^{39} - 22q^{40} - 15q^{41} + 16q^{42} + 33q^{43} + 6q^{44} + 3q^{45} + q^{46} + 13q^{47} + 31q^{48} + 41q^{49} + 13q^{50} - 4q^{51} - 26q^{52} + 5q^{53} - q^{54} + 9q^{55} + 40q^{56} + 19q^{57} + q^{58} + 2q^{59} + 16q^{60} + 29q^{61} - 32q^{62} + 13q^{63} + 28q^{64} + 18q^{65} + 6q^{66} + 32q^{67} - 26q^{68} - 16q^{69} + 18q^{70} + 29q^{71} + 19q^{73} - 16q^{74} + 23q^{75} + 64q^{76} - 21q^{77} - 15q^{78} + 56q^{79} + 16q^{81} + 14q^{82} + 5q^{83} + 18q^{84} + 16q^{85} - 20q^{86} - 16q^{87} + q^{88} + 7q^{89} + 11q^{90} - 6q^{91} - 25q^{92} + 24q^{93} - 11q^{94} + 39q^{95} - 21q^{96} + 35q^{97} - 109q^{98} + 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + 5778 x^{8} - 5124 x^{7} - 9405 x^{6} + 8288 x^{5} + 6405 x^{4} - 6032 x^{3} - 400 x^{2} + 1088 x - 192\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\((\)\( -35 \nu^{15} - 86 \nu^{14} + 850 \nu^{13} + 2133 \nu^{12} - 7813 \nu^{11} - 20506 \nu^{10} + 33795 \nu^{9} + 95926 \nu^{8} - 69732 \nu^{7} - 222160 \nu^{6} + 62103 \nu^{5} + 219845 \nu^{4} - 18368 \nu^{3} - 50704 \nu^{2} + 1952 \nu + 4864 \)\()/3904\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{15} + 40 \nu^{14} + 226 \nu^{13} - 985 \nu^{12} - 3713 \nu^{11} + 9288 \nu^{10} + 29681 \nu^{9} - 42120 \nu^{8} - 124816 \nu^{7} + 95352 \nu^{6} + 269185 \nu^{5} - 103543 \nu^{4} - 251870 \nu^{3} + 48528 \nu^{2} + 50752 \nu - 7392 \)\()/1952\)
\(\beta_{6}\)\(=\)\((\)\( 19 \nu^{15} - 30 \nu^{14} - 566 \nu^{13} + 815 \nu^{12} + 6765 \nu^{11} - 8674 \nu^{10} - 41247 \nu^{9} + 45498 \nu^{8} + 134604 \nu^{7} - 120680 \nu^{6} - 224535 \nu^{5} + 146511 \nu^{4} + 162520 \nu^{3} - 61040 \nu^{2} - 29768 \nu + 7008 \)\()/976\)
\(\beta_{7}\)\(=\)\((\)\(-141 \nu^{15} + 30 \nu^{14} + 3982 \nu^{13} - 693 \nu^{12} - 45683 \nu^{11} + 6722 \nu^{10} + 272925 \nu^{9} - 35982 \nu^{8} - 900764 \nu^{7} + 114336 \nu^{6} + 1586665 \nu^{5} - 208365 \nu^{4} - 1269304 \nu^{3} + 182064 \nu^{2} + 249856 \nu - 36288\)\()/3904\)
\(\beta_{8}\)\(=\)\((\)\(147 \nu^{15} - 78 \nu^{14} - 4302 \nu^{13} + 2119 \nu^{12} + 51261 \nu^{11} - 23626 \nu^{10} - 318351 \nu^{9} + 138498 \nu^{8} + 1089876 \nu^{7} - 451872 \nu^{6} - 1976055 \nu^{5} + 791727 \nu^{4} + 1595216 \nu^{3} - 632064 \nu^{2} - 290848 \nu + 113088\)\()/3904\)
\(\beta_{9}\)\(=\)\((\)\(207 \nu^{15} - 314 \nu^{14} - 5794 \nu^{13} + 7839 \nu^{12} + 65561 \nu^{11} - 77254 \nu^{10} - 382455 \nu^{9} + 381394 \nu^{8} + 1211508 \nu^{7} - 988048 \nu^{6} - 1996211 \nu^{5} + 1282479 \nu^{4} + 1439312 \nu^{3} - 715664 \nu^{2} - 236192 \nu + 108096\)\()/3904\)
\(\beta_{10}\)\(=\)\((\)\(-71 \nu^{15} + 19 \nu^{14} + 1977 \nu^{13} - 506 \nu^{12} - 22127 \nu^{11} + 5583 \nu^{10} + 127194 \nu^{9} - 32268 \nu^{8} - 397008 \nu^{7} + 101400 \nu^{6} + 648475 \nu^{5} - 164508 \nu^{4} - 472752 \nu^{3} + 118040 \nu^{2} + 85400 \nu - 21616\)\()/976\)
\(\beta_{11}\)\(=\)\((\)\(163 \nu^{15} - 84 \nu^{14} - 4830 \nu^{13} + 2343 \nu^{12} + 58043 \nu^{11} - 26532 \nu^{10} - 361895 \nu^{9} + 155796 \nu^{8} + 1238992 \nu^{7} - 501384 \nu^{6} - 2245015 \nu^{5} + 853469 \nu^{4} + 1823774 \nu^{3} - 654520 \nu^{2} - 346480 \nu + 116832\)\()/1952\)
\(\beta_{12}\)\(=\)\((\)\(411 \nu^{15} - 238 \nu^{14} - 11550 \nu^{13} + 6303 \nu^{12} + 130773 \nu^{11} - 67610 \nu^{10} - 762327 \nu^{9} + 374322 \nu^{8} + 2419716 \nu^{7} - 1126080 \nu^{6} - 4037823 \nu^{5} + 1774663 \nu^{4} + 3041408 \nu^{3} - 1272208 \nu^{2} - 597312 \nu + 236352\)\()/3904\)
\(\beta_{13}\)\(=\)\((\)\(551 \nu^{15} - 382 \nu^{14} - 15438 \nu^{13} + 9971 \nu^{12} + 174713 \nu^{11} - 105146 \nu^{10} - 1021947 \nu^{9} + 571338 \nu^{8} + 3272532 \nu^{7} - 1684848 \nu^{6} - 5542347 \nu^{5} + 2591083 \nu^{4} + 4237280 \nu^{3} - 1775040 \nu^{2} - 800320 \nu + 298880\)\()/3904\)
\(\beta_{14}\)\(=\)\((\)\(279 \nu^{15} - 158 \nu^{14} - 7926 \nu^{13} + 4211 \nu^{12} + 91017 \nu^{11} - 45618 \nu^{10} - 540339 \nu^{9} + 256410 \nu^{8} + 1754796 \nu^{7} - 788976 \nu^{6} - 3006939 \nu^{5} + 1281243 \nu^{4} + 2318312 \nu^{3} - 938472 \nu^{2} - 444080 \nu + 164960\)\()/1952\)
\(\beta_{15}\)\(=\)\((\)\(-155 \nu^{15} + 142 \nu^{14} + 4322 \nu^{13} - 3695 \nu^{12} - 48613 \nu^{11} + 38438 \nu^{10} + 282051 \nu^{9} - 202962 \nu^{8} - 893472 \nu^{7} + 569592 \nu^{6} + 1492239 \nu^{5} - 811191 \nu^{4} - 1122980 \nu^{3} + 501772 \nu^{2} + 210816 \nu - 81776\)\()/976\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{14} + \beta_{12} + \beta_{8} + 7 \beta_{2} + 23\)
\(\nu^{5}\)\(=\)\(-\beta_{14} + \beta_{11} + \beta_{9} + \beta_{5} - \beta_{4} + 9 \beta_{3} + 30 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(\beta_{15} - 11 \beta_{14} + 2 \beta_{13} + 10 \beta_{12} + 10 \beta_{8} + \beta_{6} + \beta_{3} + 47 \beta_{2} + 2 \beta_{1} + 146\)
\(\nu^{7}\)\(=\)\(-12 \beta_{14} + \beta_{12} + 13 \beta_{11} + 2 \beta_{10} + 12 \beta_{9} - 3 \beta_{8} - 2 \beta_{7} + 13 \beta_{5} - 12 \beta_{4} + 68 \beta_{3} + 2 \beta_{2} + 194 \beta_{1} + 28\)
\(\nu^{8}\)\(=\)\(16 \beta_{15} - 93 \beta_{14} + 30 \beta_{13} + 80 \beta_{12} - \beta_{11} + 2 \beta_{10} + \beta_{9} + 80 \beta_{8} + 16 \beta_{6} - \beta_{5} - 3 \beta_{4} + 15 \beta_{3} + 317 \beta_{2} + 35 \beta_{1} + 972\)
\(\nu^{9}\)\(=\)\(\beta_{15} - 107 \beta_{14} - 2 \beta_{13} + 16 \beta_{12} + 124 \beta_{11} + 34 \beta_{10} + 112 \beta_{9} - 46 \beta_{8} - 34 \beta_{7} + 3 \beta_{6} + 128 \beta_{5} - 112 \beta_{4} + 490 \beta_{3} + 34 \beta_{2} + 1309 \beta_{1} + 296\)
\(\nu^{10}\)\(=\)\(174 \beta_{15} - 721 \beta_{14} + 310 \beta_{13} + 602 \beta_{12} - 19 \beta_{11} + 40 \beta_{10} + 20 \beta_{9} + 602 \beta_{8} - 6 \beta_{7} + 180 \beta_{6} - 19 \beta_{5} - 62 \beta_{4} + 157 \beta_{3} + 2157 \beta_{2} + 415 \beta_{1} + 6647\)
\(\nu^{11}\)\(=\)\(20 \beta_{15} - 859 \beta_{14} - 38 \beta_{13} + 177 \beta_{12} + 1050 \beta_{11} + 400 \beta_{10} + 956 \beta_{9} - 481 \beta_{8} - 402 \beta_{7} + 66 \beta_{6} + 1134 \beta_{5} - 962 \beta_{4} + 3480 \beta_{3} + 394 \beta_{2} + 9065 \beta_{1} + 2797\)
\(\nu^{12}\)\(=\)\(1614 \beta_{15} - 5389 \beta_{14} + 2758 \beta_{13} + 4436 \beta_{12} - 229 \beta_{11} + 532 \beta_{10} + 263 \beta_{9} + 4434 \beta_{8} - 138 \beta_{7} + 1758 \beta_{6} - 217 \beta_{5} - 833 \beta_{4} + 1430 \beta_{3} + 14805 \beta_{2} + 4183 \beta_{1} + 46238\)
\(\nu^{13}\)\(=\)\(265 \beta_{15} - 6590 \beta_{14} - 456 \beta_{13} + 1693 \beta_{12} + 8376 \beta_{11} + 4048 \beta_{10} + 7808 \beta_{9} - 4293 \beta_{8} - 4086 \beta_{7} + 933 \beta_{6} + 9516 \beta_{5} - 7970 \beta_{4} + 24630 \beta_{3} + 3888 \beta_{2} + 63859 \beta_{1} + 24859\)
\(\nu^{14}\)\(=\)\(13794 \beta_{15} - 39596 \beta_{14} + 22738 \beta_{13} + 32438 \beta_{12} - 2242 \beta_{11} + 5914 \beta_{10} + 2891 \beta_{9} + 32364 \beta_{8} - 2028 \beta_{7} + 15942 \beta_{6} - 1918 \beta_{5} - 9257 \beta_{4} + 12171 \beta_{3} + 102405 \beta_{2} + 38678 \beta_{1} + 325551\)
\(\nu^{15}\)\(=\)\(2965 \beta_{15} - 49525 \beta_{14} - 4410 \beta_{13} + 15062 \beta_{12} + 64576 \beta_{11} + 37798 \beta_{10} + 62174 \beta_{9} - 35264 \beta_{8} - 38250 \beta_{7} + 10833 \beta_{6} + 77308 \beta_{5} - 64850 \beta_{4} + 174439 \beta_{3} + 35202 \beta_{2} + 455351 \beta_{1} + 212580\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.77164
2.63319
2.18082
2.03156
1.78406
1.40306
0.563182
0.386616
0.300211
−0.510814
−1.44866
−1.53756
−2.00482
−2.39038
−2.50224
−2.65986
−2.77164 1.00000 5.68196 2.29773 −2.77164 −5.27354 −10.2051 1.00000 −6.36848
1.2 −2.63319 1.00000 4.93371 0.107081 −2.63319 4.37350 −7.72501 1.00000 −0.281966
1.3 −2.18082 1.00000 2.75597 −3.64804 −2.18082 2.94825 −1.64864 1.00000 7.95571
1.4 −2.03156 1.00000 2.12722 −2.52723 −2.03156 −2.32870 −0.258453 1.00000 5.13421
1.5 −1.78406 1.00000 1.18288 3.42296 −1.78406 3.39829 1.45779 1.00000 −6.10677
1.6 −1.40306 1.00000 −0.0314276 1.13298 −1.40306 −1.90949 2.85021 1.00000 −1.58963
1.7 −0.563182 1.00000 −1.68283 −1.59433 −0.563182 4.95339 2.07410 1.00000 0.897897
1.8 −0.386616 1.00000 −1.85053 0.287925 −0.386616 −2.57107 1.48868 1.00000 −0.111317
1.9 −0.300211 1.00000 −1.90987 −3.45593 −0.300211 −1.15345 1.17379 1.00000 1.03751
1.10 0.510814 1.00000 −1.73907 2.52081 0.510814 1.21289 −1.90997 1.00000 1.28766
1.11 1.44866 1.00000 0.0986233 2.08765 1.44866 4.31885 −2.75445 1.00000 3.02430
1.12 1.53756 1.00000 0.364091 −2.97145 1.53756 1.52213 −2.51531 1.00000 −4.56878
1.13 2.00482 1.00000 2.01932 3.94433 2.00482 −2.43681 0.0387322 1.00000 7.90768
1.14 2.39038 1.00000 3.71391 3.15865 2.39038 2.42988 4.09689 1.00000 7.55038
1.15 2.50224 1.00000 4.26120 0.499377 2.50224 0.377265 5.65806 1.00000 1.24956
1.16 2.65986 1.00000 5.07484 −2.26251 2.65986 3.13860 8.17864 1.00000 −6.01796
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(1\)
\(29\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2001.2.a.n 16
3.b odd 2 1 6003.2.a.r 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.n 16 1.a even 1 1 trivial
6003.2.a.r 16 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2001))\):

\(T_{2}^{16} + \cdots\)
\(T_{5}^{16} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -192 - 1088 T - 400 T^{2} + 6032 T^{3} + 6405 T^{4} - 8288 T^{5} - 9405 T^{6} + 5124 T^{7} + 5778 T^{8} - 1665 T^{9} - 1835 T^{10} + 295 T^{11} + 316 T^{12} - 27 T^{13} - 28 T^{14} + T^{15} + T^{16} \)
$3$ \( ( -1 + T )^{16} \)
$5$ \( 3072 - 47632 T + 206600 T^{2} - 271940 T^{3} - 82804 T^{4} + 313392 T^{5} - 45374 T^{6} - 127665 T^{7} + 32879 T^{8} + 25446 T^{9} - 7691 T^{10} - 2674 T^{11} + 862 T^{12} + 142 T^{13} - 47 T^{14} - 3 T^{15} + T^{16} \)
$7$ \( 843776 - 2825600 T + 376192 T^{2} + 4054912 T^{3} - 1903280 T^{4} - 1863536 T^{5} + 1226632 T^{6} + 315492 T^{7} - 329424 T^{8} + 572 T^{9} + 41418 T^{10} - 5895 T^{11} - 2113 T^{12} + 561 T^{13} + 8 T^{14} - 13 T^{15} + T^{16} \)
$11$ \( 6912 - 37184 T + 6432 T^{2} + 224240 T^{3} - 286240 T^{4} - 246892 T^{5} + 630306 T^{6} - 259579 T^{7} - 126636 T^{8} + 94997 T^{9} + 3918 T^{10} - 10913 T^{11} + 685 T^{12} + 501 T^{13} - 52 T^{14} - 8 T^{15} + T^{16} \)
$13$ \( -79048 + 443732 T - 554162 T^{2} - 690925 T^{3} + 1698743 T^{4} - 339225 T^{5} - 1168330 T^{6} + 785635 T^{7} + 44723 T^{8} - 185290 T^{9} + 47010 T^{10} + 7077 T^{11} - 4515 T^{12} + 391 T^{13} + 90 T^{14} - 19 T^{15} + T^{16} \)
$17$ \( -1542733824 - 1176928256 T + 982781952 T^{2} + 929163008 T^{3} - 146704448 T^{4} - 254793856 T^{5} - 12417992 T^{6} + 30163813 T^{7} + 4012949 T^{8} - 1736451 T^{9} - 308523 T^{10} + 50659 T^{11} + 10524 T^{12} - 721 T^{13} - 167 T^{14} + 4 T^{15} + T^{16} \)
$19$ \( -474324992 + 773967872 T + 351935488 T^{2} - 1464427296 T^{3} + 1105468952 T^{4} - 248136236 T^{5} - 91170946 T^{6} + 62416401 T^{7} - 8957955 T^{8} - 2054468 T^{9} + 795824 T^{10} - 46894 T^{11} - 14278 T^{12} + 2240 T^{13} + 2 T^{14} - 19 T^{15} + T^{16} \)
$23$ \( ( 1 + T )^{16} \)
$29$ \( ( 1 + T )^{16} \)
$31$ \( -664365824 - 1752439040 T + 97970208 T^{2} + 1948179124 T^{3} - 405715976 T^{4} - 547946720 T^{5} + 176626616 T^{6} + 46889597 T^{7} - 22344412 T^{8} - 574735 T^{9} + 1115560 T^{10} - 81285 T^{11} - 20435 T^{12} + 2863 T^{13} + 50 T^{14} - 24 T^{15} + T^{16} \)
$37$ \( 692734264 + 778463232 T - 971322508 T^{2} - 1771887089 T^{3} - 608985598 T^{4} + 229226233 T^{5} + 142132304 T^{6} - 12181410 T^{7} - 13191163 T^{8} + 612038 T^{9} + 659618 T^{10} - 44247 T^{11} - 16780 T^{12} + 1869 T^{13} + 126 T^{14} - 26 T^{15} + T^{16} \)
$41$ \( 15815731488 + 20375331376 T - 19415440888 T^{2} - 23080929516 T^{3} - 533411588 T^{4} + 5171875808 T^{5} + 1346715450 T^{6} - 247932811 T^{7} - 146831847 T^{8} - 14036096 T^{9} + 2387025 T^{10} + 501964 T^{11} + 2628 T^{12} - 4892 T^{13} - 229 T^{14} + 15 T^{15} + T^{16} \)
$43$ \( 5860667264 - 38601853376 T + 93772357728 T^{2} - 100749522672 T^{3} + 42675839064 T^{4} - 153424804 T^{5} - 5381543966 T^{6} + 1502111039 T^{7} - 45638723 T^{8} - 43635324 T^{9} + 6701799 T^{10} + 61418 T^{11} - 91140 T^{12} + 6312 T^{13} + 161 T^{14} - 33 T^{15} + T^{16} \)
$47$ \( -27131904 + 68922368 T + 70816000 T^{2} - 190397888 T^{3} - 67771200 T^{4} + 142405056 T^{5} + 18104192 T^{6} - 39074196 T^{7} - 611360 T^{8} + 4057544 T^{9} - 178140 T^{10} - 157745 T^{11} + 9810 T^{12} + 2461 T^{13} - 175 T^{14} - 13 T^{15} + T^{16} \)
$53$ \( -537119225856 + 253534057984 T + 215130851328 T^{2} - 123556520192 T^{3} - 19323536448 T^{4} + 18505531552 T^{5} - 566397984 T^{6} - 1068277456 T^{7} + 107689460 T^{8} + 27206324 T^{9} - 3757804 T^{10} - 337441 T^{11} + 56317 T^{12} + 2036 T^{13} - 389 T^{14} - 5 T^{15} + T^{16} \)
$59$ \( 962614616064 + 1306867229696 T - 366986400896 T^{2} - 381132048224 T^{3} + 80876415504 T^{4} + 37670484048 T^{5} - 8076350292 T^{6} - 1601832542 T^{7} + 366243073 T^{8} + 32428904 T^{9} - 8161404 T^{10} - 313688 T^{11} + 91062 T^{12} + 1360 T^{13} - 488 T^{14} - 2 T^{15} + T^{16} \)
$61$ \( -262090689152 - 224939210624 T + 249216778752 T^{2} + 127482032048 T^{3} - 96033381848 T^{4} - 2752063968 T^{5} + 8375525988 T^{6} - 495272045 T^{7} - 309410835 T^{8} + 30578287 T^{9} + 5370058 T^{10} - 712632 T^{11} - 36982 T^{12} + 7477 T^{13} - 32 T^{14} - 29 T^{15} + T^{16} \)
$67$ \( 37036651159552 - 24480908677120 T - 1771799802880 T^{2} + 4358019097728 T^{3} - 605078723512 T^{4} - 229149747304 T^{5} + 59947967040 T^{6} + 2372328105 T^{7} - 1860208985 T^{8} + 89872173 T^{9} + 22135687 T^{10} - 2157067 T^{11} - 76756 T^{12} + 14779 T^{13} - 173 T^{14} - 32 T^{15} + T^{16} \)
$71$ \( 412266061824 - 1310724164864 T - 696405063288 T^{2} + 1268212316019 T^{3} - 291302321277 T^{4} - 85177308597 T^{5} + 33628781948 T^{6} + 423349429 T^{7} - 1179003325 T^{8} + 77114130 T^{9} + 15362584 T^{10} - 1750315 T^{11} - 56311 T^{12} + 13027 T^{13} - 204 T^{14} - 29 T^{15} + T^{16} \)
$73$ \( 3421018126336 + 4221327640064 T + 497195116544 T^{2} - 847237175936 T^{3} - 145212648160 T^{4} + 76663192592 T^{5} + 9517988856 T^{6} - 3821900132 T^{7} - 212384104 T^{8} + 100770924 T^{9} + 725114 T^{10} - 1368301 T^{11} + 31876 T^{12} + 8731 T^{13} - 369 T^{14} - 19 T^{15} + T^{16} \)
$79$ \( -180646877312 + 30152592064 T + 403617024800 T^{2} - 25833118288 T^{3} - 192235358296 T^{4} + 59937740580 T^{5} + 18622064006 T^{6} - 14244394771 T^{7} + 3319587728 T^{8} - 326657019 T^{9} - 2191964 T^{10} + 3644227 T^{11} - 309179 T^{12} + 3151 T^{13} + 956 T^{14} - 56 T^{15} + T^{16} \)
$83$ \( 9511719825408 + 11000142724096 T - 279061993216 T^{2} - 3095984342720 T^{3} - 23161041984 T^{4} + 348664984032 T^{5} - 26238150704 T^{6} - 12296938876 T^{7} + 1277507500 T^{8} + 188364308 T^{9} - 22739124 T^{10} - 1366563 T^{11} + 187053 T^{12} + 4436 T^{13} - 709 T^{14} - 5 T^{15} + T^{16} \)
$89$ \( 1186015739904 + 13469670272000 T + 5218648397952 T^{2} - 8943012172496 T^{3} + 1011328453296 T^{4} + 699183503400 T^{5} - 104863254596 T^{6} - 21164031171 T^{7} + 3186569707 T^{8} + 287810203 T^{9} - 42450014 T^{10} - 1884040 T^{11} + 274236 T^{12} + 5863 T^{13} - 846 T^{14} - 7 T^{15} + T^{16} \)
$97$ \( 1415545856 - 4708312576 T + 577031168 T^{2} + 5685060736 T^{3} - 319741440 T^{4} - 1552597984 T^{5} + 140413408 T^{6} + 167343272 T^{7} - 23933076 T^{8} - 7603404 T^{9} + 1609070 T^{10} + 96873 T^{11} - 42435 T^{12} + 1739 T^{13} + 326 T^{14} - 35 T^{15} + T^{16} \)
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