Properties

Label 6003.2.a.m
Level 6003
Weight 2
Character orbit 6003.a
Self dual Yes
Analytic conductor 47.934
Analytic rank 1
Dimension 11
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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 2 + \beta_{2} ) q^{4} \) \( + \beta_{4} q^{5} \) \( -\beta_{6} q^{7} \) \( + ( -1 - \beta_{1} - \beta_{3} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 2 + \beta_{2} ) q^{4} \) \( + \beta_{4} q^{5} \) \( -\beta_{6} q^{7} \) \( + ( -1 - \beta_{1} - \beta_{3} ) q^{8} \) \( + ( 1 - \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{10} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} ) q^{11} \) \( + \beta_{7} q^{13} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{10} ) q^{14} \) \( + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} ) q^{16} \) \( + ( -2 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{10} ) q^{17} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{9} - \beta_{10} ) q^{19} \) \( + ( -2 + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{10} ) q^{20} \) \( + ( -3 - \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} ) q^{22} \) \(- q^{23}\) \( + ( 1 + \beta_{2} - \beta_{3} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{25} \) \( + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} ) q^{26} \) \( + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{10} ) q^{28} \) \(+ q^{29}\) \( + ( 4 - \beta_{3} + \beta_{5} + \beta_{6} ) q^{31} \) \( + ( -2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{32} \) \( + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{34} \) \( + ( -2 - 2 \beta_{1} + \beta_{3} - \beta_{6} - \beta_{8} ) q^{35} \) \( + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{37} \) \( + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 3 \beta_{8} - \beta_{9} + \beta_{10} ) q^{38} \) \( + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{40} \) \( + ( \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{7} - 2 \beta_{9} ) q^{41} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{43} \) \( + ( -4 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + 4 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{44} \) \( + \beta_{1} q^{46} \) \( + ( -1 + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{47} \) \( + ( 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{49} \) \( + ( -\beta_{1} + 2 \beta_{2} + 3 \beta_{4} - \beta_{6} + \beta_{8} + \beta_{10} ) q^{50} \) \( + ( -3 - \beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{52} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{6} + \beta_{9} + \beta_{10} ) q^{53} \) \( + ( -1 - \beta_{1} - 2 \beta_{3} - 2 \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{55} \) \( + ( -5 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{56} \) \( -\beta_{1} q^{58} \) \( + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{59} \) \( + ( -2 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{61} \) \( + ( 3 - 4 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + \beta_{10} ) q^{62} \) \( + ( 3 + 4 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{64} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} ) q^{65} \) \( + ( -4 - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{9} ) q^{67} \) \( + ( -6 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{10} ) q^{68} \) \( + ( 3 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{9} - \beta_{10} ) q^{70} \) \( + ( -5 - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{71} \) \( + ( \beta_{1} - \beta_{2} + 4 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{73} \) \( + ( -3 + 4 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - \beta_{10} ) q^{74} \) \( + ( -2 + 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} + 8 \beta_{6} + \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} ) q^{76} \) \( + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} ) q^{77} \) \( + ( 4 + 5 \beta_{1} - \beta_{2} + \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{79} \) \( + ( -7 - 2 \beta_{1} - \beta_{2} + \beta_{4} - 3 \beta_{5} - 5 \beta_{6} - 3 \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{80} \) \( + ( -2 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{6} + 4 \beta_{8} + \beta_{9} + \beta_{10} ) q^{82} \) \( + ( 2 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{83} \) \( + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} ) q^{85} \) \( + ( -5 - 2 \beta_{2} - 2 \beta_{3} + \beta_{7} ) q^{86} \) \( + ( -5 - 2 \beta_{1} - \beta_{3} + 5 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} - \beta_{7} - 3 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} ) q^{88} \) \( + ( -4 + 3 \beta_{1} + \beta_{3} + \beta_{6} + 2 \beta_{9} ) q^{89} \) \( + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{91} \) \( + ( -2 - \beta_{2} ) q^{92} \) \( + ( 3 \beta_{1} + \beta_{2} - \beta_{3} + 6 \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{94} \) \( + ( -3 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} - 3 \beta_{8} - \beta_{9} - \beta_{10} ) q^{95} \) \( + ( -4 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} ) q^{97} \) \( + ( -9 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 7 \beta_{4} + \beta_{6} - 3 \beta_{7} - \beta_{9} - \beta_{10} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(11q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 18q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut 18q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(11q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 18q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut 18q^{8} \) \(\mathstrut +\mathstrut 14q^{10} \) \(\mathstrut -\mathstrut 11q^{11} \) \(\mathstrut -\mathstrut 5q^{13} \) \(\mathstrut -\mathstrut 17q^{14} \) \(\mathstrut +\mathstrut 20q^{16} \) \(\mathstrut -\mathstrut 15q^{17} \) \(\mathstrut -\mathstrut 6q^{19} \) \(\mathstrut -\mathstrut 21q^{20} \) \(\mathstrut -\mathstrut 10q^{22} \) \(\mathstrut -\mathstrut 11q^{23} \) \(\mathstrut +\mathstrut 3q^{25} \) \(\mathstrut +\mathstrut 5q^{26} \) \(\mathstrut +\mathstrut 7q^{28} \) \(\mathstrut +\mathstrut 11q^{29} \) \(\mathstrut +\mathstrut 35q^{31} \) \(\mathstrut -\mathstrut 28q^{32} \) \(\mathstrut +\mathstrut 28q^{34} \) \(\mathstrut -\mathstrut 15q^{35} \) \(\mathstrut -\mathstrut 28q^{37} \) \(\mathstrut +\mathstrut 2q^{38} \) \(\mathstrut -\mathstrut q^{40} \) \(\mathstrut -\mathstrut 10q^{41} \) \(\mathstrut -\mathstrut 6q^{43} \) \(\mathstrut -\mathstrut 18q^{44} \) \(\mathstrut +\mathstrut 2q^{46} \) \(\mathstrut -\mathstrut 15q^{47} \) \(\mathstrut +\mathstrut 22q^{49} \) \(\mathstrut -\mathstrut 15q^{50} \) \(\mathstrut -\mathstrut 36q^{52} \) \(\mathstrut +\mathstrut 7q^{53} \) \(\mathstrut -\mathstrut 12q^{55} \) \(\mathstrut -\mathstrut 56q^{56} \) \(\mathstrut -\mathstrut 2q^{58} \) \(\mathstrut +\mathstrut 20q^{59} \) \(\mathstrut -\mathstrut 20q^{61} \) \(\mathstrut +\mathstrut 11q^{62} \) \(\mathstrut +\mathstrut 36q^{64} \) \(\mathstrut -\mathstrut 11q^{65} \) \(\mathstrut -\mathstrut 39q^{67} \) \(\mathstrut -\mathstrut 35q^{68} \) \(\mathstrut +\mathstrut 38q^{70} \) \(\mathstrut -\mathstrut 49q^{71} \) \(\mathstrut -\mathstrut 3q^{73} \) \(\mathstrut -\mathstrut 37q^{74} \) \(\mathstrut -\mathstrut 18q^{76} \) \(\mathstrut -\mathstrut 25q^{77} \) \(\mathstrut +\mathstrut 41q^{79} \) \(\mathstrut -\mathstrut 51q^{80} \) \(\mathstrut -\mathstrut 19q^{82} \) \(\mathstrut -\mathstrut 13q^{83} \) \(\mathstrut -\mathstrut 62q^{86} \) \(\mathstrut -\mathstrut 40q^{88} \) \(\mathstrut -\mathstrut 34q^{89} \) \(\mathstrut +\mathstrut 2q^{91} \) \(\mathstrut -\mathstrut 18q^{92} \) \(\mathstrut -\mathstrut 14q^{94} \) \(\mathstrut -\mathstrut 25q^{95} \) \(\mathstrut -\mathstrut 11q^{97} \) \(\mathstrut -\mathstrut 53q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11}\mathstrut -\mathstrut \) \(2\) \(x^{10}\mathstrut -\mathstrut \) \(18\) \(x^{9}\mathstrut +\mathstrut \) \(30\) \(x^{8}\mathstrut +\mathstrut \) \(124\) \(x^{7}\mathstrut -\mathstrut \) \(152\) \(x^{6}\mathstrut -\mathstrut \) \(408\) \(x^{5}\mathstrut +\mathstrut \) \(285\) \(x^{4}\mathstrut +\mathstrut \) \(634\) \(x^{3}\mathstrut -\mathstrut \) \(93\) \(x^{2}\mathstrut -\mathstrut \) \(369\) \(x\mathstrut -\mathstrut \) \(108\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu - 1 \)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{10} + 5 \nu^{9} + 9 \nu^{8} - 63 \nu^{7} - 13 \nu^{6} + 251 \nu^{5} - 27 \nu^{4} - 360 \nu^{3} - 4 \nu^{2} + 165 \nu + 48 \)\()/6\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{10} + 7 \nu^{9} + 21 \nu^{8} - 87 \nu^{7} - 59 \nu^{6} + 343 \nu^{5} + 63 \nu^{4} - 498 \nu^{3} - 170 \nu^{2} + 261 \nu + 153 \)\()/9\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{10} - \nu^{9} + 24 \nu^{8} + 6 \nu^{7} - 187 \nu^{6} + 23 \nu^{5} + 567 \nu^{4} - 150 \nu^{3} - 598 \nu^{2} + 117 \nu + 162 \)\()/9\)
\(\beta_{7}\)\(=\)\( \nu^{8} - 2 \nu^{7} - 12 \nu^{6} + 22 \nu^{5} + 44 \nu^{4} - 68 \nu^{3} - 57 \nu^{2} + 52 \nu + 31 \)
\(\beta_{8}\)\(=\)\((\)\( -4 \nu^{10} + 14 \nu^{9} + 42 \nu^{8} - 174 \nu^{7} - 109 \nu^{6} + 686 \nu^{5} + 18 \nu^{4} - 996 \nu^{3} - 7 \nu^{2} + 504 \nu + 135 \)\()/9\)
\(\beta_{9}\)\(=\)\((\)\( 2 \nu^{10} - 7 \nu^{9} - 30 \nu^{8} + 105 \nu^{7} + 167 \nu^{6} - 541 \nu^{5} - 450 \nu^{4} + 1101 \nu^{3} + 620 \nu^{2} - 693 \nu - 369 \)\()/9\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{10} + 5 \nu^{9} + 15 \nu^{8} - 75 \nu^{7} - 91 \nu^{6} + 389 \nu^{5} + 303 \nu^{4} - 822 \nu^{3} - 538 \nu^{2} + 591 \nu + 354 \)\()/6\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(22\)
\(\nu^{5}\)\(=\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(29\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)
\(\nu^{6}\)\(=\)\(12\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(12\) \(\beta_{7}\mathstrut +\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(47\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\) \(\beta_{1}\mathstrut +\mathstrut \) \(135\)
\(\nu^{7}\)\(=\)\(12\) \(\beta_{10}\mathstrut +\mathstrut \) \(16\) \(\beta_{9}\mathstrut +\mathstrut \) \(12\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(10\) \(\beta_{5}\mathstrut -\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(82\) \(\beta_{3}\mathstrut +\mathstrut \) \(25\) \(\beta_{2}\mathstrut +\mathstrut \) \(184\) \(\beta_{1}\mathstrut +\mathstrut \) \(89\)
\(\nu^{8}\)\(=\)\(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(110\) \(\beta_{9}\mathstrut +\mathstrut \) \(14\) \(\beta_{8}\mathstrut +\mathstrut \) \(107\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(78\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(112\) \(\beta_{3}\mathstrut +\mathstrut \) \(319\) \(\beta_{2}\mathstrut +\mathstrut \) \(142\) \(\beta_{1}\mathstrut +\mathstrut \) \(875\)
\(\nu^{9}\)\(=\)\(105\) \(\beta_{10}\mathstrut +\mathstrut \) \(172\) \(\beta_{9}\mathstrut +\mathstrut \) \(106\) \(\beta_{8}\mathstrut +\mathstrut \) \(53\) \(\beta_{7}\mathstrut +\mathstrut \) \(15\) \(\beta_{6}\mathstrut -\mathstrut \) \(73\) \(\beta_{5}\mathstrut -\mathstrut \) \(71\) \(\beta_{4}\mathstrut +\mathstrut \) \(629\) \(\beta_{3}\mathstrut +\mathstrut \) \(240\) \(\beta_{2}\mathstrut +\mathstrut \) \(1229\) \(\beta_{1}\mathstrut +\mathstrut \) \(752\)
\(\nu^{10}\)\(=\)\(38\) \(\beta_{10}\mathstrut +\mathstrut \) \(910\) \(\beta_{9}\mathstrut +\mathstrut \) \(138\) \(\beta_{8}\mathstrut +\mathstrut \) \(856\) \(\beta_{7}\mathstrut +\mathstrut \) \(30\) \(\beta_{6}\mathstrut +\mathstrut \) \(559\) \(\beta_{5}\mathstrut +\mathstrut \) \(36\) \(\beta_{4}\mathstrut +\mathstrut \) \(954\) \(\beta_{3}\mathstrut +\mathstrut \) \(2194\) \(\beta_{2}\mathstrut +\mathstrut \) \(1266\) \(\beta_{1}\mathstrut +\mathstrut \) \(5861\)

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
2.72479
2.70316
2.24285
1.76484
1.17662
−0.467085
−0.661934
−1.05971
−1.94502
−1.96728
−2.51124
−2.72479 0 5.42450 1.02492 0 4.32726 −9.33107 0 −2.79269
1.2 −2.70316 0 5.30708 −2.79988 0 0.880738 −8.93955 0 7.56852
1.3 −2.24285 0 3.03039 −3.33222 0 0.336371 −2.31101 0 7.47368
1.4 −1.76484 0 1.11465 2.54815 0 −4.97592 1.56250 0 −4.49707
1.5 −1.17662 0 −0.615555 −2.85352 0 3.62966 3.07753 0 3.35752
1.6 0.467085 0 −1.78183 0.0105419 0 −1.85912 −1.76644 0 0.00492395
1.7 0.661934 0 −1.56184 1.16115 0 4.80000 −2.35770 0 0.768607
1.8 1.05971 0 −0.877023 1.30384 0 0.720797 −3.04880 0 1.38169
1.9 1.94502 0 1.78310 −0.890641 0 −3.69089 −0.421884 0 −1.73231
1.10 1.96728 0 1.87020 3.90206 0 −0.839519 −0.255353 0 7.67646
1.11 2.51124 0 4.30634 −2.07440 0 −0.329384 5.79178 0 −5.20933
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\(T_{2}^{11} + \cdots\)
\(T_{5}^{11} + \cdots\)