Properties

Label 6003.2.a.q
Level 6003
Weight 2
Character orbit 6003.a
Self dual Yes
Analytic conductor 47.934
Analytic rank 1
Dimension 16
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6003.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 3 x^{15} - 22 x^{14} + 68 x^{13} + 187 x^{12} - 597 x^{11} - 795 x^{10} + 2592 x^{9} + 1860 x^{8} - 5877 x^{7} - 2496 x^{6} + 6612 x^{5} + 1842 x^{4} - 3011 x^{3} - 505 x^{2} + 336 x + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-124169 \nu^{15} + 194810 \nu^{14} + 2801696 \nu^{13} - 4124788 \nu^{12} - 24534615 \nu^{11} + 32910414 \nu^{10} + 106985521 \nu^{9} - 125237015 \nu^{8} - 247933907 \nu^{7} + 238939362 \nu^{6} + 295392450 \nu^{5} - 218624434 \nu^{4} - 150942644 \nu^{3} + 82287639 \nu^{2} + 14985488 \nu - 5285248\)\()/1247936\)
\(\beta_{4}\)\(=\)\((\)\(-320937 \nu^{15} + 55514 \nu^{14} + 7810176 \nu^{13} - 279828 \nu^{12} - 74718487 \nu^{11} - 9187314 \nu^{10} + 357397585 \nu^{9} + 105825449 \nu^{8} - 891751731 \nu^{7} - 417755646 \nu^{6} + 1076238434 \nu^{5} + 659418286 \nu^{4} - 486230356 \nu^{3} - 310735913 \nu^{2} + 70510224 \nu + 26486912\)\()/1247936\)
\(\beta_{5}\)\(=\)\((\)\(-399265 \nu^{15} + 390970 \nu^{14} + 9549392 \nu^{13} - 8005092 \nu^{12} - 89853839 \nu^{11} + 59863870 \nu^{10} + 424971913 \nu^{9} - 199268623 \nu^{8} - 1062555835 \nu^{7} + 281768162 \nu^{6} + 1319983010 \nu^{5} - 121294226 \nu^{4} - 647488676 \nu^{3} + 18055599 \nu^{2} + 82151376 \nu + 9101888\)\()/1247936\)
\(\beta_{6}\)\(=\)\((\)\(412871 \nu^{15} - 286326 \nu^{14} - 9954704 \nu^{13} + 5268764 \nu^{12} + 94577321 \nu^{11} - 31443954 \nu^{10} - 451780703 \nu^{9} + 50528745 \nu^{8} + 1135959741 \nu^{7} + 125345330 \nu^{6} - 1394129102 \nu^{5} - 412374178 \nu^{4} + 636921244 \nu^{3} + 221078775 \nu^{2} - 83614256 \nu - 22861184\)\()/1247936\)
\(\beta_{7}\)\(=\)\((\)\(524711 \nu^{15} - 423750 \nu^{14} - 12681760 \nu^{13} + 8084172 \nu^{12} + 121020601 \nu^{11} - 52379826 \nu^{10} - 582618367 \nu^{9} + 118277337 \nu^{8} + 1484699485 \nu^{7} + 41030562 \nu^{6} - 1866965854 \nu^{5} - 405092882 \nu^{4} + 902289932 \nu^{3} + 240369543 \nu^{2} - 128155856 \nu - 29967232\)\()/1247936\)
\(\beta_{8}\)\(=\)\((\)\(17613 \nu^{15} - 15970 \nu^{14} - 416256 \nu^{13} + 315428 \nu^{12} + 3856563 \nu^{11} - 2217254 \nu^{10} - 17880021 \nu^{9} + 6505683 \nu^{8} + 43543519 \nu^{7} - 6272874 \nu^{6} - 51974890 \nu^{5} - 2466278 \nu^{4} + 23503844 \nu^{3} + 2851149 \nu^{2} - 2582832 \nu - 482240\)\()/40256\)
\(\beta_{9}\)\(=\)\((\)\(767889 \nu^{15} - 741754 \nu^{14} - 18434160 \nu^{13} + 14936836 \nu^{12} + 174451711 \nu^{11} - 108143006 \nu^{10} - 832081593 \nu^{9} + 334114239 \nu^{8} + 2103849867 \nu^{7} - 371571234 \nu^{6} - 2644257954 \nu^{5} - 37531118 \nu^{4} + 1301242372 \nu^{3} + 105743457 \nu^{2} - 165493584 \nu - 25501376\)\()/1247936\)
\(\beta_{10}\)\(=\)\((\)\(781759 \nu^{15} - 787334 \nu^{14} - 18711408 \nu^{13} + 15967900 \nu^{12} + 176274769 \nu^{11} - 117193986 \nu^{10} - 834663895 \nu^{9} + 373891409 \nu^{8} + 2083463077 \nu^{7} - 467484126 \nu^{6} - 2550207070 \nu^{5} + 95845550 \nu^{4} + 1166520668 \nu^{3} + 4288591 \nu^{2} - 120432048 \nu - 9694400\)\()/1247936\)
\(\beta_{11}\)\(=\)\((\)\(-785211 \nu^{15} + 785182 \nu^{14} + 18568432 \nu^{13} - 15860300 \nu^{12} - 172213973 \nu^{11} + 116053706 \nu^{10} + 799917747 \nu^{9} - 371328085 \nu^{8} - 1954600569 \nu^{7} + 478390486 \nu^{6} + 2346845462 \nu^{5} - 130836902 \nu^{4} - 1069127244 \nu^{3} - 5825419 \nu^{2} + 112498544 \nu + 12619200\)\()/1247936\)
\(\beta_{12}\)\(=\)\((\)\(-25785 \nu^{15} + 22922 \nu^{14} + 619504 \nu^{13} - 451684 \nu^{12} - 5863511 \nu^{11} + 3136078 \nu^{10} + 27912865 \nu^{9} - 8752791 \nu^{8} - 70040835 \nu^{7} + 6098098 \nu^{6} + 86000690 \nu^{5} + 8993566 \nu^{4} - 39174116 \nu^{3} - 6494857 \nu^{2} + 4130640 \nu + 1090240\)\()/40256\)
\(\beta_{13}\)\(=\)\((\)\(-203897 \nu^{15} + 168786 \nu^{14} + 4840280 \nu^{13} - 3291564 \nu^{12} - 45092591 \nu^{11} + 22541886 \nu^{10} + 210501617 \nu^{9} - 61808511 \nu^{8} - 517413267 \nu^{7} + 41832298 \nu^{6} + 627687626 \nu^{5} + 64022150 \nu^{4} - 296442140 \nu^{3} - 51140001 \nu^{2} + 37586128 \nu + 8409984\)\()/311984\)
\(\beta_{14}\)\(=\)\((\)\(-268687 \nu^{15} + 248146 \nu^{14} + 6403548 \nu^{13} - 4997680 \nu^{12} - 59968933 \nu^{11} + 36315186 \nu^{10} + 281841111 \nu^{9} - 114014061 \nu^{8} - 698535541 \nu^{7} + 137042970 \nu^{6} + 855515802 \nu^{5} - 15085386 \nu^{4} - 406520288 \nu^{3} - 17897027 \nu^{2} + 48368920 \nu + 5982064\)\()/311984\)
\(\beta_{15}\)\(=\)\((\)\(743697 \nu^{15} - 561258 \nu^{14} - 17803568 \nu^{13} + 10903780 \nu^{12} + 167505279 \nu^{11} - 74079278 \nu^{10} - 790224857 \nu^{9} + 198902831 \nu^{8} + 1960071611 \nu^{7} - 119133362 \nu^{6} - 2384502434 \nu^{5} - 224585838 \nu^{4} + 1101186116 \nu^{3} + 139655233 \nu^{2} - 120861728 \nu - 23649344\)\()/623968\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{13} - \beta_{11} - \beta_{10} + \beta_{6} - \beta_{5} + 6 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 8 \beta_{2} - \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(\beta_{14} + 10 \beta_{13} - \beta_{12} - 10 \beta_{11} - 10 \beta_{10} + \beta_{9} - \beta_{7} + 9 \beta_{6} - 10 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 39 \beta_{1} + 7\)
\(\nu^{6}\)\(=\)\(\beta_{14} - \beta_{13} + 11 \beta_{12} + 12 \beta_{11} + 11 \beta_{10} + 12 \beta_{9} + 12 \beta_{8} - 13 \beta_{7} + 13 \beta_{6} + 10 \beta_{5} + \beta_{4} + \beta_{3} + 59 \beta_{2} - 13 \beta_{1} + 76\)
\(\nu^{7}\)\(=\)\(-\beta_{15} + 13 \beta_{14} + 83 \beta_{13} - 13 \beta_{12} - 80 \beta_{11} - 81 \beta_{10} + 11 \beta_{9} + 2 \beta_{8} - 12 \beta_{7} + 69 \beta_{6} - 87 \beta_{5} - 15 \beta_{4} - 16 \beta_{3} - 25 \beta_{2} + 265 \beta_{1} + 42\)
\(\nu^{8}\)\(=\)\(16 \beta_{14} - 19 \beta_{13} + 96 \beta_{12} + 112 \beta_{11} + 100 \beta_{10} + 108 \beta_{9} + 106 \beta_{8} - 124 \beta_{7} + 123 \beta_{6} + 81 \beta_{5} + 14 \beta_{4} + 12 \beta_{3} + 429 \beta_{2} - 125 \beta_{1} + 452\)
\(\nu^{9}\)\(=\)\(-17 \beta_{15} + 125 \beta_{14} + 651 \beta_{13} - 131 \beta_{12} - 597 \beta_{11} - 618 \beta_{10} + 88 \beta_{9} + 34 \beta_{8} - 110 \beta_{7} + 509 \beta_{6} - 722 \beta_{5} - 159 \beta_{4} - 180 \beta_{3} - 237 \beta_{2} + 1852 \beta_{1} + 250\)
\(\nu^{10}\)\(=\)\(-3 \beta_{15} + 171 \beta_{14} - 237 \beta_{13} + 773 \beta_{12} + 963 \beta_{11} + 854 \beta_{10} + 878 \beta_{9} + 843 \beta_{8} - 1061 \beta_{7} + 1039 \beta_{6} + 622 \beta_{5} + 139 \beta_{4} + 100 \beta_{3} + 3121 \beta_{2} - 1076 \beta_{1} + 2853\)
\(\nu^{11}\)\(=\)\(-196 \beta_{15} + 1071 \beta_{14} + 4994 \beta_{13} - 1199 \beta_{12} - 4347 \beta_{11} - 4624 \beta_{10} + 619 \beta_{9} + 379 \beta_{8} - 912 \beta_{7} + 3716 \beta_{6} - 5836 \beta_{5} - 1472 \beta_{4} - 1758 \beta_{3} - 2040 \beta_{2} + 13193 \beta_{1} + 1529\)
\(\nu^{12}\)\(=\)\(-63 \beta_{15} + 1545 \beta_{14} - 2457 \beta_{13} + 6004 \beta_{12} + 7979 \beta_{11} + 7061 \beta_{10} + 6817 \beta_{9} + 6418 \beta_{8} - 8638 \beta_{7} + 8344 \beta_{6} + 4706 \beta_{5} + 1215 \beta_{4} + 708 \beta_{3} + 22820 \beta_{2} - 8807 \beta_{1} + 18746\)
\(\nu^{13}\)\(=\)\(-1927 \beta_{15} + 8665 \beta_{14} + 37940 \beta_{13} - 10419 \beta_{12} - 31431 \beta_{11} - 34406 \beta_{10} + 4041 \beta_{9} + 3535 \beta_{8} - 7203 \beta_{7} + 27095 \beta_{6} - 46361 \beta_{5} - 12728 \beta_{4} - 15911 \beta_{3} - 16793 \beta_{2} + 95276 \beta_{1} + 9651\)
\(\nu^{14}\)\(=\)\(-838 \beta_{15} + 12813 \beta_{14} - 23034 \beta_{13} + 45821 \beta_{12} + 64739 \beta_{11} + 57186 \beta_{10} + 51735 \beta_{9} + 47943 \beta_{8} - 68494 \beta_{7} + 65322 \beta_{6} + 35510 \beta_{5} + 10007 \beta_{4} + 4501 \beta_{3} + 167840 \beta_{2} - 70290 \beta_{1} + 126701\)
\(\nu^{15}\)\(=\)\(-17420 \beta_{15} + 67905 \beta_{14} + 286953 \beta_{13} - 87569 \beta_{12} - 227366 \beta_{11} - 255923 \beta_{10} + 24932 \beta_{9} + 30047 \beta_{8} - 55434 \beta_{7} + 197977 \beta_{6} - 363846 \beta_{5} - 105776 \beta_{4} - 137308 \beta_{3} - 134974 \beta_{2} + 695043 \beta_{1} + 62508\)

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
2.73896
2.51658
2.14844
2.13862
1.88851
1.84222
0.745705
0.445942
−0.208883
−0.319955
−1.03000
−1.28888
−1.63671
−1.80038
−2.42429
−2.75586
−2.73896 0 5.50188 1.91750 0 −0.472052 −9.59151 0 −5.25195
1.2 −2.51658 0 4.33319 −2.87619 0 −3.18638 −5.87166 0 7.23817
1.3 −2.14844 0 2.61580 1.42439 0 3.61731 −1.32302 0 −3.06021
1.4 −2.13862 0 2.57368 0.343330 0 −2.49131 −1.22687 0 −0.734251
1.5 −1.88851 0 1.56646 −3.37829 0 1.01841 0.818750 0 6.37991
1.6 −1.84222 0 1.39376 −2.07719 0 4.15058 1.11683 0 3.82664
1.7 −0.745705 0 −1.44392 2.81935 0 3.27962 2.56815 0 −2.10240
1.8 −0.445942 0 −1.80114 −2.46971 0 −2.81741 1.69509 0 1.10135
1.9 0.208883 0 −1.95637 2.17278 0 −2.92851 −0.826420 0 0.453858
1.10 0.319955 0 −1.89763 −1.26502 0 1.17463 −1.24707 0 −0.404751
1.11 1.03000 0 −0.939097 −3.95819 0 4.48231 −3.02727 0 −4.07695
1.12 1.28888 0 −0.338793 −4.18846 0 −3.67888 −3.01442 0 −5.39841
1.13 1.63671 0 0.678809 1.90242 0 0.625725 −2.16240 0 3.11371
1.14 1.80038 0 1.24138 −2.67126 0 1.66993 −1.36580 0 −4.80929
1.15 2.42429 0 3.87720 0.275174 0 −5.16310 4.55089 0 0.667103
1.16 2.75586 0 5.59478 −3.97063 0 1.71912 9.90674 0 −10.9425
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

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