Properties

Label 6018.2.a.y
Level 6018
Weight 2
Character orbit 6018.a
Self dual yes
Analytic conductor 48.054
Analytic rank 1
Dimension 10
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6018.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0539719364\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} - 33 x^{8} + 53 x^{7} + 356 x^{6} - 433 x^{5} - 1296 x^{4} + 1135 x^{3} + 930 x^{2} - 186 x - 104\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-43951 \nu^{9} - 76913 \nu^{8} + 988626 \nu^{7} + 8141512 \nu^{6} - 20812677 \nu^{5} - 127966190 \nu^{4} + 261039766 \nu^{3} + 516802760 \nu^{2} - 917083841 \nu - 232805875\)\()/51474769\)
\(\beta_{3}\)\(=\)\((\)\(214570 \nu^{9} - 1275880 \nu^{8} - 6472015 \nu^{7} + 39620633 \nu^{6} + 61025321 \nu^{5} - 392668405 \nu^{4} - 154873115 \nu^{3} + 1263523452 \nu^{2} - 135541142 \nu - 459680139\)\()/51474769\)
\(\beta_{4}\)\(=\)\((\)\(-249465 \nu^{9} + 1418601 \nu^{8} + 4174377 \nu^{7} - 28056148 \nu^{6} - 13181246 \nu^{5} + 145009646 \nu^{4} - 29021559 \nu^{3} - 194173638 \nu^{2} + 117499426 \nu - 47082704\)\()/51474769\)
\(\beta_{5}\)\(=\)\((\)\(414280 \nu^{9} - 956844 \nu^{8} - 14645292 \nu^{7} + 33222269 \nu^{6} + 160178219 \nu^{5} - 349088916 \nu^{4} - 537665586 \nu^{3} + 1070383049 \nu^{2} + 174956104 \nu - 154245468\)\()/51474769\)
\(\beta_{6}\)\(=\)\((\)\(417990 \nu^{9} - 662240 \nu^{8} - 15536862 \nu^{7} + 20221183 \nu^{6} + 181938909 \nu^{5} - 182686833 \nu^{4} - 684929521 \nu^{3} + 504776496 \nu^{2} + 448652800 \nu - 87407927\)\()/51474769\)
\(\beta_{7}\)\(=\)\((\)\(743180 \nu^{9} - 1756250 \nu^{8} - 21953329 \nu^{7} + 43199010 \nu^{6} + 209998564 \nu^{5} - 329491644 \nu^{4} - 657244249 \nu^{3} + 833201672 \nu^{2} + 274790504 \nu - 191137786\)\()/51474769\)
\(\beta_{8}\)\(=\)\((\)\(-1112920 \nu^{9} + 3475010 \nu^{8} + 31167323 \nu^{7} - 82936201 \nu^{6} - 285693609 \nu^{5} + 595528719 \nu^{4} + 862414939 \nu^{3} - 1317064691 \nu^{2} - 250454498 \nu + 114968772\)\()/51474769\)
\(\beta_{9}\)\(=\)\((\)\(1849040 \nu^{9} - 4265675 \nu^{8} - 59471294 \nu^{7} + 114940606 \nu^{6} + 618418870 \nu^{5} - 964601918 \nu^{4} - 2064959828 \nu^{3} + 2587254209 \nu^{2} + 760020540 \nu - 338810025\)\()/51474769\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{8} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 7\)
\(\nu^{3}\)\(=\)\(2 \beta_{9} - 2 \beta_{8} - 3 \beta_{7} - 3 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + \beta_{3} + 10 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-21 \beta_{8} - 5 \beta_{7} - 2 \beta_{6} - 31 \beta_{5} + 36 \beta_{4} + 14 \beta_{3} + 2 \beta_{1} + 90\)
\(\nu^{5}\)\(=\)\(38 \beta_{9} - 46 \beta_{8} - 59 \beta_{7} - 60 \beta_{6} - 86 \beta_{5} + 91 \beta_{4} + 26 \beta_{3} - 5 \beta_{2} + 117 \beta_{1} + 67\)
\(\nu^{6}\)\(=\)\(14 \beta_{9} - 349 \beta_{8} - 117 \beta_{7} - 70 \beta_{6} - 475 \beta_{5} + 585 \beta_{4} + 207 \beta_{3} - 5 \beta_{2} + 65 \beta_{1} + 1256\)
\(\nu^{7}\)\(=\)\(596 \beta_{9} - 866 \beta_{8} - 952 \beta_{7} - 1005 \beta_{6} - 1522 \beta_{5} + 1682 \beta_{4} + 499 \beta_{3} - 110 \beta_{2} + 1516 \beta_{1} + 1552\)
\(\nu^{8}\)\(=\)\(501 \beta_{9} - 5491 \beta_{8} - 2178 \beta_{7} - 1690 \beta_{6} - 7398 \beta_{5} + 9322 \beta_{4} + 3127 \beta_{3} - 160 \beta_{2} + 1492 \beta_{1} + 18164\)
\(\nu^{9}\)\(=\)\(9007 \beta_{9} - 15231 \beta_{8} - 14597 \beta_{7} - 16198 \beta_{6} - 25570 \beta_{5} + 29253 \beta_{4} + 8721 \beta_{3} - 1924 \beta_{2} + 20830 \beta_{1} + 30910\)

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
4.05993
3.94189
2.38622
1.40255
0.390474
−0.313984
−0.556815
−2.34533
−3.46791
−3.49703
−1.00000 1.00000 1.00000 −4.05993 −1.00000 4.32251 −1.00000 1.00000 4.05993
1.2 −1.00000 1.00000 1.00000 −3.94189 −1.00000 −4.15692 −1.00000 1.00000 3.94189
1.3 −1.00000 1.00000 1.00000 −2.38622 −1.00000 0.0691424 −1.00000 1.00000 2.38622
1.4 −1.00000 1.00000 1.00000 −1.40255 −1.00000 0.699671 −1.00000 1.00000 1.40255
1.5 −1.00000 1.00000 1.00000 −0.390474 −1.00000 −3.35881 −1.00000 1.00000 0.390474
1.6 −1.00000 1.00000 1.00000 0.313984 −1.00000 2.10092 −1.00000 1.00000 −0.313984
1.7 −1.00000 1.00000 1.00000 0.556815 −1.00000 0.727556 −1.00000 1.00000 −0.556815
1.8 −1.00000 1.00000 1.00000 2.34533 −1.00000 −0.244087 −1.00000 1.00000 −2.34533
1.9 −1.00000 1.00000 1.00000 3.46791 −1.00000 −1.61634 −1.00000 1.00000 −3.46791
1.10 −1.00000 1.00000 1.00000 3.49703 −1.00000 −4.54363 −1.00000 1.00000 −3.49703
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

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 6018.2.a.y 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.y 10 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(17\) \(1\)
\(59\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\(T_{5}^{10} + \cdots\)
\(T_{7}^{10} + \cdots\)