Properties

Label 6021.2.a.l
Level 6021
Weight 2
Character orbit 6021.a
Self dual Yes
Analytic conductor 48.078
Analytic rank 0
Dimension 10
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 6021 = 3^{3} \cdot 223 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6021.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.077927057\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 2 + \beta_{3} + \beta_{6} + \beta_{8} ) q^{4} \) \( + ( \beta_{4} - \beta_{5} ) q^{5} \) \( + \beta_{6} q^{7} \) \( + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 2 + \beta_{3} + \beta_{6} + \beta_{8} ) q^{4} \) \( + ( \beta_{4} - \beta_{5} ) q^{5} \) \( + \beta_{6} q^{7} \) \( + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{8} \) \( + ( -2 - \beta_{3} - \beta_{8} + 2 \beta_{9} ) q^{10} \) \( + ( -\beta_{1} + \beta_{2} - 2 \beta_{7} ) q^{11} \) \( + ( -\beta_{8} + \beta_{9} ) q^{13} \) \( + ( 2 \beta_{2} + \beta_{4} ) q^{14} \) \( + ( 3 - \beta_{3} + 3 \beta_{6} + 2 \beta_{8} ) q^{16} \) \( + ( \beta_{2} + \beta_{5} ) q^{17} \) \( + ( 3 + \beta_{3} + \beta_{8} ) q^{19} \) \( + ( -\beta_{1} + 3 \beta_{2} - \beta_{5} - 2 \beta_{7} ) q^{20} \) \( + ( -5 - \beta_{6} - 3 \beta_{8} + 4 \beta_{9} ) q^{22} \) \( + ( -\beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{23} \) \( + ( 3 - 2 \beta_{6} - \beta_{9} ) q^{25} \) \( + ( \beta_{1} + \beta_{2} - 2 \beta_{5} - 2 \beta_{7} ) q^{26} \) \( + ( 2 - 2 \beta_{3} + 3 \beta_{6} + \beta_{8} + \beta_{9} ) q^{28} \) \( + ( 3 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{7} ) q^{29} \) \( + ( -1 + 2 \beta_{3} - \beta_{6} + \beta_{8} ) q^{31} \) \( + ( 5 \beta_{2} + 4 \beta_{4} + 3 \beta_{7} ) q^{32} \) \( + ( 3 + 3 \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{34} \) \( + ( -\beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{35} \) \( + ( 3 + 2 \beta_{3} + \beta_{6} - \beta_{9} ) q^{37} \) \( + ( 4 \beta_{1} - \beta_{2} + \beta_{5} ) q^{38} \) \( + ( -1 - \beta_{3} + 2 \beta_{6} - 3 \beta_{8} + \beta_{9} ) q^{40} \) \( + ( -3 \beta_{1} + \beta_{2} + \beta_{4} ) q^{41} \) \( + ( -\beta_{3} - 2 \beta_{6} - \beta_{8} + \beta_{9} ) q^{43} \) \( + ( \beta_{1} - 7 \beta_{5} - 3 \beta_{7} ) q^{44} \) \( + ( -2 - \beta_{6} + 4 \beta_{8} - 4 \beta_{9} ) q^{46} \) \( + ( -\beta_{1} - 4 \beta_{2} + 2 \beta_{5} + 2 \beta_{7} ) q^{47} \) \( + ( -4 - \beta_{3} + \beta_{6} + \beta_{8} ) q^{49} \) \( + ( 2 \beta_{1} - 5 \beta_{2} - 3 \beta_{4} + \beta_{5} + \beta_{7} ) q^{50} \) \( + ( -1 - \beta_{6} - 3 \beta_{8} + 4 \beta_{9} ) q^{52} \) \( + ( -\beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} ) q^{53} \) \( + ( 9 + 7 \beta_{3} + 2 \beta_{6} + \beta_{8} - 4 \beta_{9} ) q^{55} \) \( + ( \beta_{1} + 5 \beta_{2} + 5 \beta_{4} + 2 \beta_{7} ) q^{56} \) \( + ( 10 + 5 \beta_{3} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{58} \) \( + ( 3 \beta_{1} - 3 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{59} \) \( + ( 4 + \beta_{3} + 2 \beta_{8} ) q^{61} \) \( + ( \beta_{1} - 4 \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{62} \) \( + ( 2 - 6 \beta_{3} + 11 \beta_{6} + 3 \beta_{8} - 2 \beta_{9} ) q^{64} \) \( + ( 3 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{65} \) \( + ( 3 - 3 \beta_{6} - \beta_{8} - \beta_{9} ) q^{67} \) \( + ( 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{4} + \beta_{5} + 3 \beta_{7} ) q^{68} \) \( + ( 3 + 3 \beta_{3} + 2 \beta_{8} - \beta_{9} ) q^{70} \) \( + ( 4 \beta_{1} + \beta_{2} + \beta_{4} - 3 \beta_{5} ) q^{71} \) \( + ( 5 - 2 \beta_{6} - \beta_{8} - 3 \beta_{9} ) q^{73} \) \( + ( 4 \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{74} \) \( + ( 11 + 4 \beta_{3} + 3 \beta_{6} + 4 \beta_{8} - \beta_{9} ) q^{76} \) \( + ( 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{5} + \beta_{7} ) q^{77} \) \( + ( -3 - \beta_{3} + \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{79} \) \( + ( \beta_{1} + \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{80} \) \( + ( -11 - 4 \beta_{3} - 2 \beta_{8} + \beta_{9} ) q^{82} \) \( + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{7} ) q^{83} \) \( + ( -4 + \beta_{3} + 2 \beta_{6} ) q^{85} \) \( + ( -2 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{86} \) \( + ( -3 - 3 \beta_{3} - 7 \beta_{6} - 10 \beta_{8} + 5 \beta_{9} ) q^{88} \) \( + ( -\beta_{1} + \beta_{2} - 4 \beta_{4} - \beta_{7} ) q^{89} \) \( + ( 2 + \beta_{3} - \beta_{6} ) q^{91} \) \( + ( -4 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} + 6 \beta_{5} + 4 \beta_{7} ) q^{92} \) \( + ( -2 + 3 \beta_{3} - 5 \beta_{6} + 5 \beta_{8} - 6 \beta_{9} ) q^{94} \) \( + ( -\beta_{1} + 3 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} - \beta_{7} ) q^{95} \) \( + ( -1 + 2 \beta_{6} - 3 \beta_{9} ) q^{97} \) \( + ( -5 \beta_{1} + 3 \beta_{2} + 3 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut +\mathstrut 20q^{4} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut +\mathstrut 20q^{4} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 10q^{10} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 44q^{16} \) \(\mathstrut +\mathstrut 28q^{19} \) \(\mathstrut -\mathstrut 42q^{22} \) \(\mathstrut +\mathstrut 22q^{25} \) \(\mathstrut +\mathstrut 40q^{28} \) \(\mathstrut -\mathstrut 18q^{31} \) \(\mathstrut +\mathstrut 36q^{34} \) \(\mathstrut +\mathstrut 20q^{37} \) \(\mathstrut -\mathstrut 4q^{40} \) \(\mathstrut +\mathstrut 2q^{43} \) \(\mathstrut -\mathstrut 30q^{46} \) \(\mathstrut -\mathstrut 32q^{49} \) \(\mathstrut -\mathstrut 2q^{52} \) \(\mathstrut +\mathstrut 52q^{55} \) \(\mathstrut +\mathstrut 84q^{58} \) \(\mathstrut +\mathstrut 40q^{61} \) \(\mathstrut +\mathstrut 64q^{64} \) \(\mathstrut +\mathstrut 18q^{67} \) \(\mathstrut +\mathstrut 18q^{70} \) \(\mathstrut +\mathstrut 32q^{73} \) \(\mathstrut +\mathstrut 104q^{76} \) \(\mathstrut -\mathstrut 16q^{79} \) \(\mathstrut -\mathstrut 94q^{82} \) \(\mathstrut -\mathstrut 40q^{85} \) \(\mathstrut -\mathstrut 32q^{88} \) \(\mathstrut +\mathstrut 14q^{91} \) \(\mathstrut -\mathstrut 56q^{94} \) \(\mathstrut -\mathstrut 18q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(20\) \(x^{8}\mathstrut +\mathstrut \) \(139\) \(x^{6}\mathstrut -\mathstrut \) \(384\) \(x^{4}\mathstrut +\mathstrut \) \(331\) \(x^{2}\mathstrut -\mathstrut \) \(63\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{9} - 14 \nu^{7} + 52 \nu^{5} - 30 \nu^{3} - 2 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{8} - 29 \nu^{6} + 116 \nu^{4} - 95 \nu^{2} + 2 \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( 8 \nu^{9} - 121 \nu^{7} + 539 \nu^{5} - 675 \nu^{3} + 173 \nu \)\()/15\)
\(\beta_{5}\)\(=\)\((\)\( -13 \nu^{9} + 191 \nu^{7} - 799 \nu^{5} + 840 \nu^{3} - 238 \nu \)\()/15\)
\(\beta_{6}\)\(=\)\((\)\( 6 \nu^{8} - 87 \nu^{6} + 353 \nu^{4} - 325 \nu^{2} + 51 \)\()/5\)
\(\beta_{7}\)\(=\)\((\)\( -19 \nu^{9} + 278 \nu^{7} - 1147 \nu^{5} + 1110 \nu^{3} - 154 \nu \)\()/15\)
\(\beta_{8}\)\(=\)\((\)\( -8 \nu^{8} + 116 \nu^{6} - 469 \nu^{4} + 425 \nu^{2} - 73 \)\()/5\)
\(\beta_{9}\)\(=\)\( 3 \nu^{8} - 44 \nu^{6} + 183 \nu^{4} - 185 \nu^{2} + 38 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(8\) \(\beta_{8}\mathstrut +\mathstrut \) \(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(23\)
\(\nu^{5}\)\(=\)\(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(8\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(13\) \(\beta_{2}\mathstrut +\mathstrut \) \(28\) \(\beta_{1}\)
\(\nu^{6}\)\(=\)\(-\)\(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(59\) \(\beta_{8}\mathstrut +\mathstrut \) \(77\) \(\beta_{6}\mathstrut +\mathstrut \) \(20\) \(\beta_{3}\mathstrut +\mathstrut \) \(144\)
\(\nu^{7}\)\(=\)\(41\) \(\beta_{7}\mathstrut +\mathstrut \) \(61\) \(\beta_{5}\mathstrut +\mathstrut \) \(114\) \(\beta_{4}\mathstrut +\mathstrut \) \(132\) \(\beta_{2}\mathstrut +\mathstrut \) \(162\) \(\beta_{1}\)
\(\nu^{8}\)\(=\)\(-\)\(29\) \(\beta_{9}\mathstrut +\mathstrut \) \(439\) \(\beta_{8}\mathstrut +\mathstrut \) \(642\) \(\beta_{6}\mathstrut +\mathstrut \) \(50\) \(\beta_{3}\mathstrut +\mathstrut \) \(943\)
\(\nu^{9}\)\(=\)\(418\) \(\beta_{7}\mathstrut +\mathstrut \) \(468\) \(\beta_{5}\mathstrut +\mathstrut \) \(1002\) \(\beta_{4}\mathstrut +\mathstrut \) \(1205\) \(\beta_{2}\mathstrut +\mathstrut \) \(964\) \(\beta_{1}\)

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
−2.81643
−2.32314
−2.31657
−1.02020
−0.513295
0.513295
1.02020
2.31657
2.32314
2.81643
−2.81643 0 5.93226 −0.124866 0 3.22829 −11.0749 0 0.351677
1.2 −2.32314 0 3.39697 3.53749 0 −1.40218 −3.24535 0 −8.21807
1.3 −2.31657 0 3.36649 −1.78959 0 0.694189 −3.16557 0 4.14570
1.4 −1.02020 0 −0.959195 2.96046 0 0.816945 3.01897 0 −3.02025
1.5 −0.513295 0 −1.73653 −3.39171 0 −2.33725 1.91794 0 1.74095
1.6 0.513295 0 −1.73653 3.39171 0 −2.33725 −1.91794 0 1.74095
1.7 1.02020 0 −0.959195 −2.96046 0 0.816945 −3.01897 0 −3.02025
1.8 2.31657 0 3.36649 1.78959 0 0.694189 3.16557 0 4.14570
1.9 2.32314 0 3.39697 −3.53749 0 −1.40218 3.24535 0 −8.21807
1.10 2.81643 0 5.93226 0.124866 0 3.22829 11.0749 0 0.351677
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(223\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6021))\):

\(T_{2}^{10} \) \(\mathstrut -\mathstrut 20 T_{2}^{8} \) \(\mathstrut +\mathstrut 139 T_{2}^{6} \) \(\mathstrut -\mathstrut 384 T_{2}^{4} \) \(\mathstrut +\mathstrut 331 T_{2}^{2} \) \(\mathstrut -\mathstrut 63 \)
\(T_{5}^{10} \) \(\mathstrut -\mathstrut 36 T_{5}^{8} \) \(\mathstrut +\mathstrut 460 T_{5}^{6} \) \(\mathstrut -\mathstrut 2404 T_{5}^{4} \) \(\mathstrut +\mathstrut 4078 T_{5}^{2} \) \(\mathstrut -\mathstrut 63 \)