Properties

Label 6003.2.a.n
Level 6003
Weight 2
Character orbit 6003.a
Self dual Yes
Analytic conductor 47.934
Analytic rank 0
Dimension 12
CM No
Inner twists 1

Related objects

Downloads

Learn more about

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: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(3\) \(x^{11}\mathstrut -\mathstrut \) \(13\) \(x^{10}\mathstrut +\mathstrut \) \(41\) \(x^{9}\mathstrut +\mathstrut \) \(54\) \(x^{8}\mathstrut -\mathstrut \) \(188\) \(x^{7}\mathstrut -\mathstrut \) \(77\) \(x^{6}\mathstrut +\mathstrut \) \(342\) \(x^{5}\mathstrut +\mathstrut \) \(13\) \(x^{4}\mathstrut -\mathstrut \) \(215\) \(x^{3}\mathstrut +\mathstrut \) \(9\) \(x^{2}\mathstrut +\mathstrut \) \(37\) \(x\mathstrut -\mathstrut \) \(5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 69 \nu^{11} - 2268 \nu^{10} + 4126 \nu^{9} + 30343 \nu^{8} - 62480 \nu^{7} - 131017 \nu^{6} + 277030 \nu^{5} + 194815 \nu^{4} - 433403 \nu^{3} - 31183 \nu^{2} + 158295 \nu - 20718 \)\()/6469\)
\(\beta_{3}\)\(=\)\((\)\( -127 \nu^{11} - 607 \nu^{10} + 4875 \nu^{9} + 7810 \nu^{8} - 51132 \nu^{7} - 30082 \nu^{6} + 209382 \nu^{5} + 24036 \nu^{4} - 328362 \nu^{3} + 49707 \nu^{2} + 117880 \nu - 29932 \)\()/6469\)
\(\beta_{4}\)\(=\)\((\)\( -353 \nu^{11} - 210 \nu^{10} + 8049 \nu^{9} + 773 \nu^{8} - 63527 \nu^{7} + 13625 \nu^{6} + 210856 \nu^{5} - 84627 \nu^{4} - 276488 \nu^{3} + 135717 \nu^{2} + 84019 \nu - 25638 \)\()/6469\)
\(\beta_{5}\)\(=\)\((\)\( 549 \nu^{11} - 1451 \nu^{10} - 8798 \nu^{9} + 21760 \nu^{8} + 52179 \nu^{7} - 114560 \nu^{6} - 143208 \nu^{5} + 255406 \nu^{4} + 177916 \nu^{3} - 216607 \nu^{2} - 75949 \nu + 34852 \)\()/6469\)
\(\beta_{6}\)\(=\)\((\)\( -874 \nu^{11} + 2852 \nu^{10} + 10271 \nu^{9} - 37175 \nu^{8} - 32306 \nu^{7} + 154428 \nu^{6} + 1464 \nu^{5} - 218601 \nu^{4} + 86000 \nu^{3} + 34877 \nu^{2} - 19087 \nu + 23075 \)\()/6469\)
\(\beta_{7}\)\(=\)\((\)\( 1153 \nu^{11} - 2741 \nu^{10} - 16651 \nu^{9} + 36674 \nu^{8} + 84275 \nu^{7} - 163016 \nu^{6} - 182412 \nu^{5} + 282647 \nu^{4} + 156809 \nu^{3} - 153652 \nu^{2} - 43159 \nu + 10438 \)\()/6469\)
\(\beta_{8}\)\(=\)\((\)\( -1227 \nu^{11} + 2642 \nu^{10} + 18320 \nu^{9} - 36402 \nu^{8} - 95833 \nu^{7} + 168053 \nu^{6} + 212320 \nu^{5} - 303228 \nu^{4} - 190488 \nu^{3} + 177063 \nu^{2} + 64932 \nu - 21970 \)\()/6469\)
\(\beta_{9}\)\(=\)\((\)\( 1231 \nu^{11} - 3336 \nu^{10} - 16487 \nu^{9} + 44255 \nu^{8} + 73554 \nu^{7} - 190180 \nu^{6} - 130539 \nu^{5} + 291927 \nu^{4} + 89329 \nu^{3} - 89336 \nu^{2} - 43380 \nu - 10451 \)\()/6469\)
\(\beta_{10}\)\(=\)\((\)\( 2257 \nu^{11} - 6684 \nu^{10} - 28263 \nu^{9} + 88739 \nu^{8} + 106697 \nu^{7} - 383278 \nu^{6} - 103569 \nu^{5} + 605079 \nu^{4} - 88693 \nu^{3} - 238564 \nu^{2} + 63686 \nu + 8869 \)\()/6469\)
\(\beta_{11}\)\(=\)\((\)\( -2687 \nu^{11} + 10130 \nu^{10} + 28520 \nu^{9} - 134015 \nu^{8} - 63186 \nu^{7} + 578643 \nu^{6} - 126167 \nu^{5} - 926609 \nu^{4} + 441457 \nu^{3} + 391735 \nu^{2} - 161659 \nu - 20106 \)\()/6469\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(7\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(7\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(6\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut -\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(30\) \(\beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{11}\mathstrut +\mathstrut \) \(11\) \(\beta_{10}\mathstrut -\mathstrut \) \(8\) \(\beta_{9}\mathstrut +\mathstrut \) \(47\) \(\beta_{8}\mathstrut -\mathstrut \) \(10\) \(\beta_{7}\mathstrut -\mathstrut \) \(45\) \(\beta_{6}\mathstrut +\mathstrut \) \(11\) \(\beta_{5}\mathstrut -\mathstrut \) \(35\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(86\)
\(\nu^{7}\)\(=\)\(10\) \(\beta_{11}\mathstrut +\mathstrut \) \(21\) \(\beta_{10}\mathstrut +\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(24\) \(\beta_{8}\mathstrut +\mathstrut \) \(20\) \(\beta_{7}\mathstrut +\mathstrut \) \(22\) \(\beta_{6}\mathstrut +\mathstrut \) \(78\) \(\beta_{5}\mathstrut +\mathstrut \) \(76\) \(\beta_{4}\mathstrut +\mathstrut \) \(78\) \(\beta_{3}\mathstrut -\mathstrut \) \(68\) \(\beta_{2}\mathstrut +\mathstrut \) \(192\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{8}\)\(=\)\(12\) \(\beta_{11}\mathstrut +\mathstrut \) \(90\) \(\beta_{10}\mathstrut -\mathstrut \) \(50\) \(\beta_{9}\mathstrut +\mathstrut \) \(314\) \(\beta_{8}\mathstrut -\mathstrut \) \(77\) \(\beta_{7}\mathstrut -\mathstrut \) \(283\) \(\beta_{6}\mathstrut +\mathstrut \) \(94\) \(\beta_{5}\mathstrut -\mathstrut \) \(209\) \(\beta_{4}\mathstrut +\mathstrut \) \(19\) \(\beta_{3}\mathstrut -\mathstrut \) \(81\) \(\beta_{2}\mathstrut +\mathstrut \) \(32\) \(\beta_{1}\mathstrut +\mathstrut \) \(527\)
\(\nu^{9}\)\(=\)\(74\) \(\beta_{11}\mathstrut +\mathstrut \) \(167\) \(\beta_{10}\mathstrut +\mathstrut \) \(43\) \(\beta_{9}\mathstrut +\mathstrut \) \(214\) \(\beta_{8}\mathstrut +\mathstrut \) \(148\) \(\beta_{7}\mathstrut +\mathstrut \) \(178\) \(\beta_{6}\mathstrut +\mathstrut \) \(559\) \(\beta_{5}\mathstrut +\mathstrut \) \(524\) \(\beta_{4}\mathstrut +\mathstrut \) \(565\) \(\beta_{3}\mathstrut -\mathstrut \) \(488\) \(\beta_{2}\mathstrut +\mathstrut \) \(1261\) \(\beta_{1}\mathstrut +\mathstrut \) \(40\)
\(\nu^{10}\)\(=\)\(99\) \(\beta_{11}\mathstrut +\mathstrut \) \(661\) \(\beta_{10}\mathstrut -\mathstrut \) \(284\) \(\beta_{9}\mathstrut +\mathstrut \) \(2098\) \(\beta_{8}\mathstrut -\mathstrut \) \(546\) \(\beta_{7}\mathstrut -\mathstrut \) \(1773\) \(\beta_{6}\mathstrut +\mathstrut \) \(724\) \(\beta_{5}\mathstrut -\mathstrut \) \(1281\) \(\beta_{4}\mathstrut +\mathstrut \) \(227\) \(\beta_{3}\mathstrut -\mathstrut \) \(622\) \(\beta_{2}\mathstrut +\mathstrut \) \(351\) \(\beta_{1}\mathstrut +\mathstrut \) \(3354\)
\(\nu^{11}\)\(=\)\(491\) \(\beta_{11}\mathstrut +\mathstrut \) \(1212\) \(\beta_{10}\mathstrut +\mathstrut \) \(431\) \(\beta_{9}\mathstrut +\mathstrut \) \(1715\) \(\beta_{8}\mathstrut +\mathstrut \) \(980\) \(\beta_{7}\mathstrut +\mathstrut \) \(1286\) \(\beta_{6}\mathstrut +\mathstrut \) \(3859\) \(\beta_{5}\mathstrut +\mathstrut \) \(3450\) \(\beta_{4}\mathstrut +\mathstrut \) \(3981\) \(\beta_{3}\mathstrut -\mathstrut \) \(3436\) \(\beta_{2}\mathstrut +\mathstrut \) \(8380\) \(\beta_{1}\mathstrut +\mathstrut \) \(515\)

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
−2.52122
−1.89304
−1.65670
−0.724122
−0.613795
0.147789
0.431373
1.08419
1.49364
1.99588
2.58646
2.66955
−2.52122 0 4.35654 3.14716 0 −3.79117 −5.94135 0 −7.93467
1.2 −1.89304 0 1.58359 4.13664 0 2.17357 0.788273 0 −7.83081
1.3 −1.65670 0 0.744648 −1.18716 0 −3.31827 2.07974 0 1.96676
1.4 −0.724122 0 −1.47565 1.17886 0 −4.09140 2.51679 0 −0.853636
1.5 −0.613795 0 −1.62326 0.782514 0 3.32687 2.22394 0 −0.480303
1.6 0.147789 0 −1.97816 −0.429801 0 0.404424 −0.587929 0 −0.0635200
1.7 0.431373 0 −1.81392 3.65141 0 2.43060 −1.64522 0 1.57512
1.8 1.08419 0 −0.824542 −0.786014 0 −4.63453 −3.06233 0 −0.852185
1.9 1.49364 0 0.230961 1.29771 0 1.16738 −2.64231 0 1.93831
1.10 1.99588 0 1.98352 −2.38535 0 −0.101071 −0.0328940 0 −4.76086
1.11 2.58646 0 4.68978 3.69848 0 −0.298143 6.95702 0 9.56597
1.12 2.66955 0 5.12647 2.89555 0 −0.268248 8.34627 0 7.72982
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\(T_{2}^{12} - \cdots\)
\(T_{5}^{12} - \cdots\)