Properties

Label 6003.2.a.p
Level 6003
Weight 2
Character orbit 6003.a
Self dual Yes
Analytic conductor 47.934
Analytic rank 0
Dimension 14
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: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( -\beta_{11} q^{5} \) \( + ( \beta_{3} - \beta_{11} - \beta_{13} ) q^{7} \) \( + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( -\beta_{11} q^{5} \) \( + ( \beta_{3} - \beta_{11} - \beta_{13} ) q^{7} \) \( + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{8} \) \( + ( -1 + \beta_{10} - \beta_{11} ) q^{10} \) \( + ( 1 + \beta_{6} ) q^{11} \) \( + ( 1 - \beta_{8} ) q^{13} \) \( + ( -\beta_{1} + 2 \beta_{2} + \beta_{5} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{14} \) \( + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{16} \) \( + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{11} - \beta_{13} ) q^{17} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{19} \) \( + ( -2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{20} \) \( + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} + \beta_{13} ) q^{22} \) \(+ q^{23}\) \( + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{13} ) q^{25} \) \( + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} + \beta_{13} ) q^{26} \) \( + ( 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{10} - \beta_{11} ) q^{28} \) \(+ q^{29}\) \( + ( -3 - \beta_{3} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{31} \) \( + ( 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{13} ) q^{32} \) \( + ( \beta_{2} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{34} \) \( + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{13} ) q^{35} \) \( + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{37} \) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} ) q^{38} \) \( + ( -1 - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{40} \) \( + ( 3 + \beta_{2} + \beta_{4} - \beta_{10} + \beta_{11} + \beta_{12} ) q^{41} \) \( + ( 1 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{43} \) \( + ( 3 - \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{9} ) q^{44} \) \( + \beta_{1} q^{46} \) \( + ( 2 + \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{47} \) \( + ( 2 - \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{11} + \beta_{12} ) q^{49} \) \( + ( 3 - 2 \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{13} ) q^{50} \) \( + ( 2 + 3 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{8} - \beta_{13} ) q^{52} \) \( + ( 2 - 2 \beta_{1} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{53} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{55} \) \( + ( 5 + \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{56} \) \( + \beta_{1} q^{58} \) \( + ( 3 - \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{59} \) \( + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{61} \) \( + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{7} + \beta_{8} + 2 \beta_{12} + \beta_{13} ) q^{62} \) \( + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{64} \) \( + ( -\beta_{3} + \beta_{6} + \beta_{10} - 2 \beta_{11} + \beta_{13} ) q^{65} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{67} \) \( + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{68} \) \( + ( 1 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{11} - \beta_{12} - 3 \beta_{13} ) q^{70} \) \( + ( 4 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{10} + \beta_{13} ) q^{71} \) \( + ( 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{73} \) \( + ( -1 - 2 \beta_{2} - \beta_{3} + \beta_{6} + 2 \beta_{8} + \beta_{13} ) q^{74} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{12} + \beta_{13} ) q^{76} \) \( + ( 4 + \beta_{1} - 3 \beta_{2} + 4 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{10} - \beta_{13} ) q^{77} \) \( + ( -5 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{8} - \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{79} \) \( + ( -2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{10} - \beta_{12} - \beta_{13} ) q^{80} \) \( + ( -1 + 5 \beta_{1} + 3 \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{9} + 4 \beta_{11} + \beta_{13} ) q^{82} \) \( + ( -3 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{83} \) \( + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 3 \beta_{10} - \beta_{11} + \beta_{12} ) q^{85} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} - \beta_{12} - 2 \beta_{13} ) q^{86} \) \( + ( 7 \beta_{1} - \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{88} \) \( + ( -2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - 3 \beta_{7} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{89} \) \( + ( -1 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{7} + 3 \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{91} \) \( + ( 1 + \beta_{2} ) q^{92} \) \( + ( 2 + 2 \beta_{1} - \beta_{4} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{13} ) q^{94} \) \( + ( 5 + \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{95} \) \( + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{13} ) q^{97} \) \( + ( -1 + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(14q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut -\mathstrut 5q^{10} \) \(\mathstrut +\mathstrut 12q^{11} \) \(\mathstrut +\mathstrut 13q^{13} \) \(\mathstrut +\mathstrut 9q^{14} \) \(\mathstrut +\mathstrut 14q^{17} \) \(\mathstrut -\mathstrut 9q^{19} \) \(\mathstrut +\mathstrut 2q^{20} \) \(\mathstrut -\mathstrut 9q^{22} \) \(\mathstrut +\mathstrut 14q^{23} \) \(\mathstrut +\mathstrut 13q^{25} \) \(\mathstrut +\mathstrut 16q^{26} \) \(\mathstrut +\mathstrut 3q^{28} \) \(\mathstrut +\mathstrut 14q^{29} \) \(\mathstrut -\mathstrut 28q^{31} \) \(\mathstrut +\mathstrut 4q^{32} \) \(\mathstrut +\mathstrut 14q^{34} \) \(\mathstrut +\mathstrut 9q^{35} \) \(\mathstrut -\mathstrut 12q^{37} \) \(\mathstrut -\mathstrut 2q^{38} \) \(\mathstrut -\mathstrut 20q^{40} \) \(\mathstrut +\mathstrut 25q^{41} \) \(\mathstrut +\mathstrut 5q^{43} \) \(\mathstrut +\mathstrut 37q^{44} \) \(\mathstrut +\mathstrut 2q^{46} \) \(\mathstrut +\mathstrut 17q^{47} \) \(\mathstrut +\mathstrut 17q^{49} \) \(\mathstrut +\mathstrut 44q^{50} \) \(\mathstrut +\mathstrut 25q^{52} \) \(\mathstrut +\mathstrut 17q^{53} \) \(\mathstrut +\mathstrut q^{55} \) \(\mathstrut +\mathstrut 54q^{56} \) \(\mathstrut +\mathstrut 2q^{58} \) \(\mathstrut +\mathstrut 18q^{59} \) \(\mathstrut -\mathstrut 13q^{61} \) \(\mathstrut +\mathstrut 8q^{62} \) \(\mathstrut +\mathstrut 20q^{64} \) \(\mathstrut +\mathstrut 16q^{65} \) \(\mathstrut +\mathstrut 2q^{67} \) \(\mathstrut +\mathstrut 19q^{68} \) \(\mathstrut +\mathstrut 14q^{70} \) \(\mathstrut +\mathstrut 55q^{71} \) \(\mathstrut +\mathstrut 19q^{73} \) \(\mathstrut -\mathstrut 4q^{74} \) \(\mathstrut -\mathstrut 32q^{76} \) \(\mathstrut +\mathstrut 19q^{77} \) \(\mathstrut -\mathstrut 68q^{79} \) \(\mathstrut +\mathstrut 2q^{80} \) \(\mathstrut -\mathstrut 12q^{82} \) \(\mathstrut +\mathstrut 21q^{83} \) \(\mathstrut +\mathstrut 16q^{85} \) \(\mathstrut +\mathstrut 22q^{86} \) \(\mathstrut -\mathstrut 25q^{88} \) \(\mathstrut +\mathstrut 17q^{89} \) \(\mathstrut -\mathstrut 30q^{91} \) \(\mathstrut +\mathstrut 12q^{92} \) \(\mathstrut +\mathstrut 16q^{94} \) \(\mathstrut +\mathstrut 55q^{95} \) \(\mathstrut +\mathstrut 25q^{97} \) \(\mathstrut +\mathstrut 31q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(2\) \(x^{13}\mathstrut -\mathstrut \) \(18\) \(x^{12}\mathstrut +\mathstrut \) \(34\) \(x^{11}\mathstrut +\mathstrut \) \(124\) \(x^{10}\mathstrut -\mathstrut \) \(216\) \(x^{9}\mathstrut -\mathstrut \) \(420\) \(x^{8}\mathstrut +\mathstrut \) \(647\) \(x^{7}\mathstrut +\mathstrut \) \(750\) \(x^{6}\mathstrut -\mathstrut \) \(939\) \(x^{5}\mathstrut -\mathstrut \) \(717\) \(x^{4}\mathstrut +\mathstrut \) \(604\) \(x^{3}\mathstrut +\mathstrut \) \(352\) \(x^{2}\mathstrut -\mathstrut \) \(128\) \(x\mathstrut -\mathstrut \) \(64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{13} - 2 \nu^{12} - 18 \nu^{11} + 34 \nu^{10} + 124 \nu^{9} - 216 \nu^{8} - 420 \nu^{7} + 647 \nu^{6} + 750 \nu^{5} - 939 \nu^{4} - 701 \nu^{3} + 604 \nu^{2} + 256 \nu - 128 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{13} + 2 \nu^{12} + 18 \nu^{11} - 34 \nu^{10} - 124 \nu^{9} + 216 \nu^{8} + 420 \nu^{7} - 647 \nu^{6} - 750 \nu^{5} + 939 \nu^{4} + 717 \nu^{3} - 604 \nu^{2} - 336 \nu + 112 \)\()/16\)
\(\beta_{5}\)\(=\)\((\)\( -21 \nu^{13} + 142 \nu^{12} + 210 \nu^{11} - 2450 \nu^{10} + 108 \nu^{9} + 15640 \nu^{8} - 7004 \nu^{7} - 46019 \nu^{6} + 25206 \nu^{5} + 63135 \nu^{4} - 30667 \nu^{3} - 37648 \nu^{2} + 10560 \nu + 8496 \)\()/368\)
\(\beta_{6}\)\(=\)\((\)\( 11 \nu^{13} - 47 \nu^{12} - 156 \nu^{11} + 808 \nu^{10} + 686 \nu^{9} - 5152 \nu^{8} - 664 \nu^{7} + 15225 \nu^{6} - 1989 \nu^{5} - 21091 \nu^{4} + 3544 \nu^{3} + 12241 \nu^{2} - 800 \nu - 2124 \)\()/92\)
\(\beta_{7}\)\(=\)\((\)\( -14 \nu^{13} - 5 \nu^{12} + 278 \nu^{11} + 130 \nu^{10} - 2090 \nu^{9} - 1196 \nu^{8} + 7444 \nu^{7} + 4986 \nu^{6} - 12843 \nu^{5} - 9660 \nu^{4} + 9877 \nu^{3} + 7661 \nu^{2} - 2620 \nu - 1604 \)\()/92\)
\(\beta_{8}\)\(=\)\((\)\( 49 \nu^{13} - 86 \nu^{12} - 858 \nu^{11} + 1362 \nu^{10} + 5636 \nu^{9} - 7728 \nu^{8} - 17452 \nu^{7} + 19119 \nu^{6} + 25826 \nu^{5} - 19251 \nu^{4} - 16273 \nu^{3} + 5168 \nu^{2} + 3144 \nu + 48 \)\()/184\)
\(\beta_{9}\)\(=\)\((\)\( 97 \nu^{13} - 266 \nu^{12} - 1522 \nu^{11} + 4386 \nu^{10} + 8412 \nu^{9} - 26680 \nu^{8} - 19396 \nu^{7} + 74967 \nu^{6} + 15014 \nu^{5} - 97451 \nu^{4} + 2763 \nu^{3} + 49860 \nu^{2} - 3040 \nu - 6544 \)\()/368\)
\(\beta_{10}\)\(=\)\((\)\( 10 \nu^{13} - 26 \nu^{12} - 169 \nu^{11} + 446 \nu^{10} + 1046 \nu^{9} - 2852 \nu^{8} - 2912 \nu^{7} + 8530 \nu^{6} + 3486 \nu^{5} - 12121 \nu^{4} - 1282 \nu^{3} + 7283 \nu^{2} - 31 \nu - 1266 \)\()/46\)
\(\beta_{11}\)\(=\)\((\)\( 12 \nu^{13} - 22 \nu^{12} - 212 \nu^{11} + 365 \nu^{10} + 1407 \nu^{9} - 2231 \nu^{8} - 4419 \nu^{7} + 6257 \nu^{6} + 6727 \nu^{5} - 8027 \nu^{4} - 4510 \nu^{3} + 4020 \nu^{2} + 961 \nu - 544 \)\()/46\)
\(\beta_{12}\)\(=\)\((\)\( 157 \nu^{13} - 146 \nu^{12} - 2858 \nu^{11} + 2002 \nu^{10} + 19748 \nu^{9} - 8832 \nu^{8} - 65388 \nu^{7} + 12435 \nu^{6} + 106310 \nu^{5} + 4761 \nu^{4} - 77057 \nu^{3} - 17164 \nu^{2} + 18072 \nu + 5088 \)\()/368\)
\(\beta_{13}\)\(=\)\((\)\( -177 \nu^{13} + 198 \nu^{12} + 3242 \nu^{11} - 2986 \nu^{10} - 22484 \nu^{9} + 15640 \nu^{8} + 74340 \nu^{7} - 33175 \nu^{6} - 119906 \nu^{5} + 21091 \nu^{4} + 85233 \nu^{3} + 8992 \nu^{2} - 18608 \nu - 6144 \)\()/368\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(30\) \(\beta_{1}\mathstrut +\mathstrut \) \(8\)
\(\nu^{6}\)\(=\)\(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(11\) \(\beta_{6}\mathstrut +\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(45\) \(\beta_{2}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(78\)
\(\nu^{7}\)\(=\)\(14\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(3\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(13\) \(\beta_{8}\mathstrut -\mathstrut \) \(12\) \(\beta_{7}\mathstrut +\mathstrut \) \(13\) \(\beta_{6}\mathstrut +\mathstrut \) \(13\) \(\beta_{5}\mathstrut +\mathstrut \) \(70\) \(\beta_{4}\mathstrut +\mathstrut \) \(65\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(194\) \(\beta_{1}\mathstrut +\mathstrut \) \(58\)
\(\nu^{8}\)\(=\)\(16\) \(\beta_{13}\mathstrut +\mathstrut \) \(13\) \(\beta_{12}\mathstrut +\mathstrut \) \(17\) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(14\) \(\beta_{9}\mathstrut +\mathstrut \) \(3\) \(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(94\) \(\beta_{6}\mathstrut +\mathstrut \) \(80\) \(\beta_{5}\mathstrut +\mathstrut \) \(96\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(289\) \(\beta_{2}\mathstrut +\mathstrut \) \(112\) \(\beta_{1}\mathstrut +\mathstrut \) \(475\)
\(\nu^{9}\)\(=\)\(137\) \(\beta_{13}\mathstrut +\mathstrut \) \(16\) \(\beta_{12}\mathstrut +\mathstrut \) \(50\) \(\beta_{11}\mathstrut -\mathstrut \) \(4\) \(\beta_{10}\mathstrut -\mathstrut \) \(16\) \(\beta_{9}\mathstrut +\mathstrut \) \(120\) \(\beta_{8}\mathstrut -\mathstrut \) \(105\) \(\beta_{7}\mathstrut +\mathstrut \) \(124\) \(\beta_{6}\mathstrut +\mathstrut \) \(122\) \(\beta_{5}\mathstrut +\mathstrut \) \(520\) \(\beta_{4}\mathstrut +\mathstrut \) \(445\) \(\beta_{3}\mathstrut +\mathstrut \) \(17\) \(\beta_{2}\mathstrut +\mathstrut \) \(1299\) \(\beta_{1}\mathstrut +\mathstrut \) \(418\)
\(\nu^{10}\)\(=\)\(177\) \(\beta_{13}\mathstrut +\mathstrut \) \(121\) \(\beta_{12}\mathstrut +\mathstrut \) \(195\) \(\beta_{11}\mathstrut -\mathstrut \) \(34\) \(\beta_{10}\mathstrut -\mathstrut \) \(136\) \(\beta_{9}\mathstrut +\mathstrut \) \(51\) \(\beta_{8}\mathstrut -\mathstrut \) \(38\) \(\beta_{7}\mathstrut +\mathstrut \) \(736\) \(\beta_{6}\mathstrut +\mathstrut \) \(601\) \(\beta_{5}\mathstrut +\mathstrut \) \(777\) \(\beta_{4}\mathstrut +\mathstrut \) \(22\) \(\beta_{3}\mathstrut +\mathstrut \) \(1877\) \(\beta_{2}\mathstrut +\mathstrut \) \(953\) \(\beta_{1}\mathstrut +\mathstrut \) \(3043\)
\(\nu^{11}\)\(=\)\(1168\) \(\beta_{13}\mathstrut +\mathstrut \) \(174\) \(\beta_{12}\mathstrut +\mathstrut \) \(558\) \(\beta_{11}\mathstrut -\mathstrut \) \(74\) \(\beta_{10}\mathstrut -\mathstrut \) \(172\) \(\beta_{9}\mathstrut +\mathstrut \) \(970\) \(\beta_{8}\mathstrut -\mathstrut \) \(826\) \(\beta_{7}\mathstrut +\mathstrut \) \(1057\) \(\beta_{6}\mathstrut +\mathstrut \) \(1015\) \(\beta_{5}\mathstrut +\mathstrut \) \(3789\) \(\beta_{4}\mathstrut +\mathstrut \) \(3004\) \(\beta_{3}\mathstrut +\mathstrut \) \(204\) \(\beta_{2}\mathstrut +\mathstrut \) \(8860\) \(\beta_{1}\mathstrut +\mathstrut \) \(3035\)
\(\nu^{12}\)\(=\)\(1673\) \(\beta_{13}\mathstrut +\mathstrut \) \(998\) \(\beta_{12}\mathstrut +\mathstrut \) \(1878\) \(\beta_{11}\mathstrut -\mathstrut \) \(384\) \(\beta_{10}\mathstrut -\mathstrut \) \(1144\) \(\beta_{9}\mathstrut +\mathstrut \) \(575\) \(\beta_{8}\mathstrut -\mathstrut \) \(467\) \(\beta_{7}\mathstrut +\mathstrut \) \(5533\) \(\beta_{6}\mathstrut +\mathstrut \) \(4407\) \(\beta_{5}\mathstrut +\mathstrut \) \(6070\) \(\beta_{4}\mathstrut +\mathstrut \) \(306\) \(\beta_{3}\mathstrut +\mathstrut \) \(12339\) \(\beta_{2}\mathstrut +\mathstrut \) \(7743\) \(\beta_{1}\mathstrut +\mathstrut \) \(20110\)
\(\nu^{13}\)\(=\)\(9303\) \(\beta_{13}\mathstrut +\mathstrut \) \(1611\) \(\beta_{12}\mathstrut +\mathstrut \) \(5255\) \(\beta_{11}\mathstrut -\mathstrut \) \(880\) \(\beta_{10}\mathstrut -\mathstrut \) \(1573\) \(\beta_{9}\mathstrut +\mathstrut \) \(7354\) \(\beta_{8}\mathstrut -\mathstrut \) \(6212\) \(\beta_{7}\mathstrut +\mathstrut \) \(8528\) \(\beta_{6}\mathstrut +\mathstrut \) \(7981\) \(\beta_{5}\mathstrut +\mathstrut \) \(27353\) \(\beta_{4}\mathstrut +\mathstrut \) \(20239\) \(\beta_{3}\mathstrut +\mathstrut \) \(2122\) \(\beta_{2}\mathstrut +\mathstrut \) \(61084\) \(\beta_{1}\mathstrut +\mathstrut \) \(22213\)

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
−2.55124
−1.98240
−1.86019
−1.37642
−0.894954
−0.656871
−0.429717
0.630703
1.14192
1.35292
1.57408
2.05175
2.31598
2.68441
−2.55124 0 4.50880 1.64740 0 −2.91756 −6.40055 0 −4.20290
1.2 −1.98240 0 1.92989 0.252916 0 −1.21499 0.138982 0 −0.501379
1.3 −1.86019 0 1.46029 −0.608499 0 0.188796 1.00396 0 1.13192
1.4 −1.37642 0 −0.105481 1.48313 0 4.92788 2.89802 0 −2.04140
1.5 −0.894954 0 −1.19906 −0.474904 0 −1.99579 2.86301 0 0.425017
1.6 −0.656871 0 −1.56852 −3.35329 0 −2.47996 2.34406 0 2.20268
1.7 −0.429717 0 −1.81534 3.93237 0 0.264416 1.63952 0 −1.68981
1.8 0.630703 0 −1.60221 −2.56371 0 2.03354 −2.27193 0 −1.61694
1.9 1.14192 0 −0.696014 4.45694 0 −4.52974 −3.07864 0 5.08948
1.10 1.35292 0 −0.169600 0.738177 0 3.44152 −2.93530 0 0.998696
1.11 1.57408 0 0.477741 −0.564555 0 −3.58089 −2.39616 0 −0.888657
1.12 2.05175 0 2.20969 −3.75134 0 −2.35999 0.430233 0 −7.69683
1.13 2.31598 0 3.36376 2.86683 0 3.60131 3.15843 0 6.63952
1.14 2.68441 0 5.20606 −1.06146 0 1.62145 8.60637 0 −2.84940
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\(T_{2}^{14} - \cdots\)
\(T_{5}^{14} - \cdots\)