Properties

Label 8016.2.a.bc
Level 8016
Weight 2
Character orbit 8016.a
Self dual Yes
Analytic conductor 64.008
Analytic rank 0
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8016.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{3}\) \( + \beta_{1} q^{5} \) \( + ( \beta_{3} + \beta_{6} ) q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{3}\) \( + \beta_{1} q^{5} \) \( + ( \beta_{3} + \beta_{6} ) q^{7} \) \(+ q^{9}\) \( + ( 1 + \beta_{7} ) q^{11} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{13} \) \( + \beta_{1} q^{15} \) \( + ( 1 + \beta_{1} - \beta_{8} ) q^{17} \) \( + ( 1 + \beta_{3} - \beta_{8} ) q^{19} \) \( + ( \beta_{3} + \beta_{6} ) q^{21} \) \( -\beta_{2} q^{23} \) \( + ( 2 + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{25} \) \(+ q^{27}\) \( + ( 1 + \beta_{2} + \beta_{3} + \beta_{8} ) q^{29} \) \( + ( -3 - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{31} \) \( + ( 1 + \beta_{7} ) q^{33} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{35} \) \( + ( 2 + \beta_{1} - \beta_{5} ) q^{37} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{39} \) \( + ( 2 + \beta_{3} + \beta_{4} + \beta_{6} ) q^{41} \) \( + ( \beta_{3} + \beta_{4} ) q^{43} \) \( + \beta_{1} q^{45} \) \( + ( 1 + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{47} \) \( + ( 3 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} ) q^{49} \) \( + ( 1 + \beta_{1} - \beta_{8} ) q^{51} \) \( + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} ) q^{53} \) \( + ( -1 + 4 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{55} \) \( + ( 1 + \beta_{3} - \beta_{8} ) q^{57} \) \( + ( 2 - \beta_{2} + \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{59} \) \( + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{7} + \beta_{8} ) q^{61} \) \( + ( \beta_{3} + \beta_{6} ) q^{63} \) \( + ( 3 - \beta_{6} + \beta_{8} ) q^{65} \) \( + ( -1 + 2 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{8} ) q^{67} \) \( -\beta_{2} q^{69} \) \( + ( -1 - \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{71} \) \( + ( 5 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} ) q^{73} \) \( + ( 2 + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{75} \) \( + ( 2 + \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{77} \) \( + ( -2 + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{79} \) \(+ q^{81}\) \( + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{83} \) \( + ( 5 + 2 \beta_{1} + 2 \beta_{4} + \beta_{7} - 2 \beta_{8} ) q^{85} \) \( + ( 1 + \beta_{2} + \beta_{3} + \beta_{8} ) q^{87} \) \( + ( 5 - 2 \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{8} ) q^{89} \) \( + ( -7 + \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{91} \) \( + ( -3 - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{93} \) \( + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} ) q^{95} \) \( + ( 6 - \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{97} \) \( + ( 1 + \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(9q \) \(\mathstrut +\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut +\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut 9q^{11} \) \(\mathstrut +\mathstrut 10q^{13} \) \(\mathstrut +\mathstrut q^{15} \) \(\mathstrut +\mathstrut 7q^{17} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut 2q^{21} \) \(\mathstrut +\mathstrut 3q^{23} \) \(\mathstrut +\mathstrut 18q^{25} \) \(\mathstrut +\mathstrut 9q^{27} \) \(\mathstrut +\mathstrut 5q^{29} \) \(\mathstrut -\mathstrut 12q^{31} \) \(\mathstrut +\mathstrut 9q^{33} \) \(\mathstrut +\mathstrut 6q^{35} \) \(\mathstrut +\mathstrut 15q^{37} \) \(\mathstrut +\mathstrut 10q^{39} \) \(\mathstrut +\mathstrut 14q^{41} \) \(\mathstrut -\mathstrut 6q^{43} \) \(\mathstrut +\mathstrut q^{45} \) \(\mathstrut +\mathstrut 3q^{47} \) \(\mathstrut +\mathstrut 27q^{49} \) \(\mathstrut +\mathstrut 7q^{51} \) \(\mathstrut +\mathstrut 9q^{53} \) \(\mathstrut -\mathstrut 19q^{55} \) \(\mathstrut +\mathstrut 2q^{57} \) \(\mathstrut +\mathstrut 9q^{59} \) \(\mathstrut +\mathstrut 30q^{61} \) \(\mathstrut -\mathstrut 2q^{63} \) \(\mathstrut +\mathstrut 28q^{65} \) \(\mathstrut -\mathstrut 16q^{67} \) \(\mathstrut +\mathstrut 3q^{69} \) \(\mathstrut +\mathstrut 3q^{71} \) \(\mathstrut +\mathstrut 32q^{73} \) \(\mathstrut +\mathstrut 18q^{75} \) \(\mathstrut +\mathstrut 18q^{77} \) \(\mathstrut -\mathstrut 24q^{79} \) \(\mathstrut +\mathstrut 9q^{81} \) \(\mathstrut +\mathstrut 3q^{83} \) \(\mathstrut +\mathstrut 37q^{85} \) \(\mathstrut +\mathstrut 5q^{87} \) \(\mathstrut +\mathstrut 46q^{89} \) \(\mathstrut -\mathstrut 33q^{91} \) \(\mathstrut -\mathstrut 12q^{93} \) \(\mathstrut -\mathstrut 11q^{95} \) \(\mathstrut +\mathstrut 43q^{97} \) \(\mathstrut +\mathstrut 9q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(x^{8}\mathstrut -\mathstrut \) \(31\) \(x^{7}\mathstrut +\mathstrut \) \(24\) \(x^{6}\mathstrut +\mathstrut \) \(293\) \(x^{5}\mathstrut -\mathstrut \) \(101\) \(x^{4}\mathstrut -\mathstrut \) \(864\) \(x^{3}\mathstrut -\mathstrut \) \(278\) \(x^{2}\mathstrut +\mathstrut \) \(24\) \(x\mathstrut +\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 454 \nu^{8} + 294 \nu^{7} - 14998 \nu^{6} - 10080 \nu^{5} + 154841 \nu^{4} + 125150 \nu^{3} - 499492 \nu^{2} - 541382 \nu - 14306 \)\()/27853\)
\(\beta_{3}\)\(=\)\((\)\( -575 \nu^{8} + 924 \nu^{7} + 16824 \nu^{6} - 25452 \nu^{5} - 142113 \nu^{4} + 175571 \nu^{3} + 337666 \nu^{2} - 190010 \nu - 38190 \)\()/27853\)
\(\beta_{4}\)\(=\)\((\)\( -2488 \nu^{8} + 2646 \nu^{7} + 78500 \nu^{6} - 64078 \nu^{5} - 762703 \nu^{4} + 284359 \nu^{3} + 2349144 \nu^{2} + 659583 \nu - 147092 \)\()/27853\)
\(\beta_{5}\)\(=\)\((\)\( 3597 \nu^{8} - 3731 \nu^{7} - 111199 \nu^{6} + 90552 \nu^{5} + 1044226 \nu^{4} - 410322 \nu^{3} - 3026506 \nu^{2} - 792549 \nu + 6960 \)\()/27853\)
\(\beta_{6}\)\(=\)\((\)\( -577 \nu^{8} + 611 \nu^{7} + 18018 \nu^{6} - 14597 \nu^{5} - 172292 \nu^{4} + 60797 \nu^{3} + 516238 \nu^{2} + 174672 \nu - 3088 \)\()/3979\)
\(\beta_{7}\)\(=\)\((\)\( -5148 \nu^{8} + 5362 \nu^{7} + 158825 \nu^{6} - 128653 \nu^{5} - 1487567 \nu^{4} + 551542 \nu^{3} + 4318881 \nu^{2} + 1355670 \nu - 76455 \)\()/27853\)
\(\beta_{8}\)\(=\)\((\)\( -5294 \nu^{8} + 4830 \nu^{7} + 163120 \nu^{6} - 116641 \nu^{5} - 1527615 \nu^{4} + 492437 \nu^{3} + 4438630 \nu^{2} + 1359676 \nu - 110393 \)\()/27853\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(-\)\(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(-\)\(4\) \(\beta_{8}\mathstrut +\mathstrut \) \(16\) \(\beta_{7}\mathstrut -\mathstrut \) \(16\) \(\beta_{6}\mathstrut +\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(14\) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut -\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(90\)
\(\nu^{5}\)\(=\)\(-\)\(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(6\) \(\beta_{7}\mathstrut -\mathstrut \) \(42\) \(\beta_{6}\mathstrut -\mathstrut \) \(42\) \(\beta_{5}\mathstrut +\mathstrut \) \(17\) \(\beta_{4}\mathstrut -\mathstrut \) \(50\) \(\beta_{3}\mathstrut +\mathstrut \) \(22\) \(\beta_{2}\mathstrut +\mathstrut \) \(187\) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)
\(\nu^{6}\)\(=\)\(-\)\(101\) \(\beta_{8}\mathstrut +\mathstrut \) \(246\) \(\beta_{7}\mathstrut -\mathstrut \) \(255\) \(\beta_{6}\mathstrut +\mathstrut \) \(99\) \(\beta_{5}\mathstrut +\mathstrut \) \(218\) \(\beta_{4}\mathstrut +\mathstrut \) \(96\) \(\beta_{3}\mathstrut -\mathstrut \) \(125\) \(\beta_{2}\mathstrut +\mathstrut \) \(31\) \(\beta_{1}\mathstrut +\mathstrut \) \(1271\)
\(\nu^{7}\)\(=\)\(-\)\(91\) \(\beta_{8}\mathstrut +\mathstrut \) \(150\) \(\beta_{7}\mathstrut -\mathstrut \) \(711\) \(\beta_{6}\mathstrut -\mathstrut \) \(746\) \(\beta_{5}\mathstrut +\mathstrut \) \(253\) \(\beta_{4}\mathstrut -\mathstrut \) \(943\) \(\beta_{3}\mathstrut +\mathstrut \) \(417\) \(\beta_{2}\mathstrut +\mathstrut \) \(2807\) \(\beta_{1}\mathstrut -\mathstrut \) \(57\)
\(\nu^{8}\)\(=\)\(-\)\(1980\) \(\beta_{8}\mathstrut +\mathstrut \) \(3806\) \(\beta_{7}\mathstrut -\mathstrut \) \(3988\) \(\beta_{6}\mathstrut +\mathstrut \) \(1062\) \(\beta_{5}\mathstrut +\mathstrut \) \(3465\) \(\beta_{4}\mathstrut +\mathstrut \) \(1859\) \(\beta_{3}\mathstrut -\mathstrut \) \(2420\) \(\beta_{2}\mathstrut +\mathstrut \) \(285\) \(\beta_{1}\mathstrut +\mathstrut \) \(18844\)

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
−4.06168
−2.94765
−1.74256
−0.419033
−0.0546093
0.111665
2.94172
3.25977
3.91239
0 1.00000 0 −4.06168 0 1.52198 0 1.00000 0
1.2 0 1.00000 0 −2.94765 0 −4.65088 0 1.00000 0
1.3 0 1.00000 0 −1.74256 0 −2.35372 0 1.00000 0
1.4 0 1.00000 0 −0.419033 0 4.23221 0 1.00000 0
1.5 0 1.00000 0 −0.0546093 0 −3.75281 0 1.00000 0
1.6 0 1.00000 0 0.111665 0 3.78430 0 1.00000 0
1.7 0 1.00000 0 2.94172 0 2.78729 0 1.00000 0
1.8 0 1.00000 0 3.25977 0 −1.95900 0 1.00000 0
1.9 0 1.00000 0 3.91239 0 −1.60938 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(167\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\(T_{5}^{9} - \cdots\)
\(T_{7}^{9} + \cdots\)
\(T_{11}^{9} - \cdots\)
\(T_{13}^{9} - \cdots\)