Properties

Label 8048.2.a.p
Level 8048
Weight 2
Character orbit 8048.a
Self dual Yes
Analytic conductor 64.264
Analytic rank 1
Dimension 10
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8048.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2636035467\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1 - \beta_{1} ) q^{3} \) \( -\beta_{2} q^{5} \) \( + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{9} ) q^{7} \) \( + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 - \beta_{1} ) q^{3} \) \( -\beta_{2} q^{5} \) \( + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{9} ) q^{7} \) \( + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{9} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{11} \) \( + ( -2 + \beta_{3} + \beta_{7} ) q^{13} \) \( -\beta_{3} q^{15} \) \( + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{17} \) \( + ( 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{19} \) \( + ( 1 + \beta_{2} + \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{9} ) q^{21} \) \( + ( 1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{23} \) \( + ( -2 + \beta_{2} + \beta_{6} - \beta_{7} ) q^{25} \) \( + ( -1 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{27} \) \( + ( -2 + \beta_{2} - \beta_{3} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{29} \) \( + ( 2 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{31} \) \( + ( \beta_{1} - \beta_{4} - 2 \beta_{9} ) q^{33} \) \( + ( 1 - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{35} \) \( + ( -6 + 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{37} \) \( + ( -2 + 3 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{39} \) \( + ( -2 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{8} - \beta_{9} ) q^{41} \) \( + ( 2 + \beta_{1} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{9} ) q^{43} \) \( + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{45} \) \( + ( -2 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{47} \) \( + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} ) q^{49} \) \( + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{51} \) \( + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{53} \) \( + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{9} ) q^{55} \) \( + ( -2 - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} + 2 \beta_{9} ) q^{57} \) \( + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} ) q^{59} \) \( + ( 3 - 3 \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 6 \beta_{7} + \beta_{8} + \beta_{9} ) q^{61} \) \( + ( -1 + 2 \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{63} \) \( + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{65} \) \( + ( -4 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + \beta_{5} - 5 \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{67} \) \( + ( 1 - 4 \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} ) q^{69} \) \( + ( 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - 6 \beta_{9} ) q^{71} \) \( + ( -4 + 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} - 4 \beta_{9} ) q^{73} \) \( + ( -4 + 4 \beta_{1} + \beta_{3} + \beta_{5} ) q^{75} \) \( + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{9} ) q^{77} \) \( + ( 4 - 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 6 \beta_{8} + 4 \beta_{9} ) q^{79} \) \( + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - \beta_{9} ) q^{81} \) \( + ( -5 - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} - 5 \beta_{6} + \beta_{7} + 4 \beta_{8} + 3 \beta_{9} ) q^{83} \) \( + ( 2 \beta_{2} - 2 \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{85} \) \( + ( 2 - 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} + \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} ) q^{87} \) \( + ( -4 + 4 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - 5 \beta_{6} + 3 \beta_{7} - 3 \beta_{9} ) q^{89} \) \( + ( 1 - 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{4} - 2 \beta_{5} - \beta_{8} + \beta_{9} ) q^{91} \) \( + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{8} ) q^{93} \) \( + ( -5 + \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{95} \) \( + ( -7 - 2 \beta_{1} + \beta_{2} - 3 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 5 \beta_{9} ) q^{97} \) \( + ( -4 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{6} - 4 \beta_{7} + 4 \beta_{8} + \beta_{9} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut +\mathstrut 5q^{7} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut +\mathstrut 5q^{7} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 3q^{11} \) \(\mathstrut -\mathstrut 18q^{13} \) \(\mathstrut +\mathstrut 2q^{15} \) \(\mathstrut -\mathstrut 11q^{17} \) \(\mathstrut +\mathstrut q^{21} \) \(\mathstrut +\mathstrut 2q^{23} \) \(\mathstrut -\mathstrut 27q^{25} \) \(\mathstrut +\mathstrut 2q^{27} \) \(\mathstrut -\mathstrut 9q^{29} \) \(\mathstrut +\mathstrut 22q^{31} \) \(\mathstrut -\mathstrut 10q^{33} \) \(\mathstrut +\mathstrut 6q^{35} \) \(\mathstrut -\mathstrut 35q^{37} \) \(\mathstrut -\mathstrut 8q^{39} \) \(\mathstrut -\mathstrut 4q^{41} \) \(\mathstrut +\mathstrut 20q^{43} \) \(\mathstrut +\mathstrut 2q^{45} \) \(\mathstrut -\mathstrut 7q^{47} \) \(\mathstrut -\mathstrut 27q^{49} \) \(\mathstrut -\mathstrut 9q^{51} \) \(\mathstrut -\mathstrut 24q^{53} \) \(\mathstrut +\mathstrut 11q^{55} \) \(\mathstrut -\mathstrut 23q^{57} \) \(\mathstrut -\mathstrut 17q^{59} \) \(\mathstrut -\mathstrut 4q^{61} \) \(\mathstrut -\mathstrut 10q^{63} \) \(\mathstrut -\mathstrut 16q^{65} \) \(\mathstrut +\mathstrut 6q^{67} \) \(\mathstrut -\mathstrut 2q^{69} \) \(\mathstrut +\mathstrut q^{71} \) \(\mathstrut -\mathstrut 31q^{73} \) \(\mathstrut -\mathstrut 30q^{75} \) \(\mathstrut +\mathstrut 3q^{77} \) \(\mathstrut +\mathstrut 10q^{79} \) \(\mathstrut -\mathstrut 6q^{81} \) \(\mathstrut -\mathstrut 22q^{83} \) \(\mathstrut -\mathstrut 6q^{85} \) \(\mathstrut -\mathstrut 25q^{87} \) \(\mathstrut +\mathstrut q^{89} \) \(\mathstrut -\mathstrut 10q^{91} \) \(\mathstrut -\mathstrut 6q^{93} \) \(\mathstrut -\mathstrut 39q^{95} \) \(\mathstrut -\mathstrut 57q^{97} \) \(\mathstrut -\mathstrut 35q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(2\) \(x^{9}\mathstrut -\mathstrut \) \(9\) \(x^{8}\mathstrut +\mathstrut \) \(14\) \(x^{7}\mathstrut +\mathstrut \) \(27\) \(x^{6}\mathstrut -\mathstrut \) \(27\) \(x^{5}\mathstrut -\mathstrut \) \(34\) \(x^{4}\mathstrut +\mathstrut \) \(14\) \(x^{3}\mathstrut +\mathstrut \) \(17\) \(x^{2}\mathstrut +\mathstrut \) \(x\mathstrut -\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( 3 \nu^{9} - 8 \nu^{8} - 22 \nu^{7} + 57 \nu^{6} + 46 \nu^{5} - 113 \nu^{4} - 34 \nu^{3} + 65 \nu^{2} + 12 \nu - 5 \)
\(\beta_{3}\)\(=\)\( 5 \nu^{9} - 13 \nu^{8} - 37 \nu^{7} + 92 \nu^{6} + 78 \nu^{5} - 181 \nu^{4} - 57 \nu^{3} + 104 \nu^{2} + 20 \nu - 8 \)
\(\beta_{4}\)\(=\)\( -6 \nu^{9} + 15 \nu^{8} + 47 \nu^{7} - 108 \nu^{6} - 113 \nu^{5} + 221 \nu^{4} + 108 \nu^{3} - 139 \nu^{2} - 44 \nu + 13 \)
\(\beta_{5}\)\(=\)\( 10 \nu^{9} - 25 \nu^{8} - 78 \nu^{7} + 180 \nu^{6} + 184 \nu^{5} - 368 \nu^{4} - 165 \nu^{3} + 230 \nu^{2} + 60 \nu - 21 \)
\(\beta_{6}\)\(=\)\( -11 \nu^{9} + 28 \nu^{8} + 84 \nu^{7} - 200 \nu^{6} - 191 \nu^{5} + 402 \nu^{4} + 165 \nu^{3} - 244 \nu^{2} - 63 \nu + 22 \)
\(\beta_{7}\)\(=\)\( -18 \nu^{9} + 45 \nu^{8} + 139 \nu^{7} - 321 \nu^{6} - 321 \nu^{5} + 645 \nu^{4} + 278 \nu^{3} - 394 \nu^{2} - 102 \nu + 38 \)
\(\beta_{8}\)\(=\)\( -21 \nu^{9} + 53 \nu^{8} + 162 \nu^{7} - 380 \nu^{6} - 375 \nu^{5} + 770 \nu^{4} + 331 \nu^{3} - 475 \nu^{2} - 127 \nu + 45 \)
\(\beta_{9}\)\(=\)\( -22 \nu^{9} + 55 \nu^{8} + 170 \nu^{7} - 392 \nu^{6} - 394 \nu^{5} + 785 \nu^{4} + 346 \nu^{3} - 473 \nu^{2} - 130 \nu + 42 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{3}\)\(=\)\(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(8\) \(\beta_{4}\mathstrut -\mathstrut \) \(6\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{9}\mathstrut +\mathstrut \) \(8\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(22\) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{4}\mathstrut -\mathstrut \) \(9\) \(\beta_{3}\mathstrut -\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(34\) \(\beta_{1}\mathstrut -\mathstrut \) \(12\)
\(\nu^{6}\)\(=\)\(-\)\(8\) \(\beta_{9}\mathstrut +\mathstrut \) \(12\) \(\beta_{8}\mathstrut +\mathstrut \) \(19\) \(\beta_{7}\mathstrut -\mathstrut \) \(73\) \(\beta_{6}\mathstrut +\mathstrut \) \(22\) \(\beta_{5}\mathstrut +\mathstrut \) \(58\) \(\beta_{4}\mathstrut -\mathstrut \) \(37\) \(\beta_{3}\mathstrut -\mathstrut \) \(24\) \(\beta_{2}\mathstrut +\mathstrut \) \(83\) \(\beta_{1}\mathstrut -\mathstrut \) \(28\)
\(\nu^{7}\)\(=\)\(-\)\(12\) \(\beta_{9}\mathstrut +\mathstrut \) \(58\) \(\beta_{8}\mathstrut +\mathstrut \) \(37\) \(\beta_{7}\mathstrut -\mathstrut \) \(192\) \(\beta_{6}\mathstrut +\mathstrut \) \(73\) \(\beta_{5}\mathstrut +\mathstrut \) \(113\) \(\beta_{4}\mathstrut -\mathstrut \) \(71\) \(\beta_{3}\mathstrut -\mathstrut \) \(63\) \(\beta_{2}\mathstrut +\mathstrut \) \(252\) \(\beta_{1}\mathstrut -\mathstrut \) \(107\)
\(\nu^{8}\)\(=\)\(-\)\(58\) \(\beta_{9}\mathstrut +\mathstrut \) \(113\) \(\beta_{8}\mathstrut +\mathstrut \) \(152\) \(\beta_{7}\mathstrut -\mathstrut \) \(578\) \(\beta_{6}\mathstrut +\mathstrut \) \(192\) \(\beta_{5}\mathstrut +\mathstrut \) \(425\) \(\beta_{4}\mathstrut -\mathstrut \) \(247\) \(\beta_{3}\mathstrut -\mathstrut \) \(220\) \(\beta_{2}\mathstrut +\mathstrut \) \(661\) \(\beta_{1}\mathstrut -\mathstrut \) \(278\)
\(\nu^{9}\)\(=\)\(-\)\(113\) \(\beta_{9}\mathstrut +\mathstrut \) \(425\) \(\beta_{8}\mathstrut +\mathstrut \) \(345\) \(\beta_{7}\mathstrut -\mathstrut \) \(1565\) \(\beta_{6}\mathstrut +\mathstrut \) \(578\) \(\beta_{5}\mathstrut +\mathstrut \) \(967\) \(\beta_{4}\mathstrut -\mathstrut \) \(554\) \(\beta_{3}\mathstrut -\mathstrut \) \(591\) \(\beta_{2}\mathstrut +\mathstrut \) \(1920\) \(\beta_{1}\mathstrut -\mathstrut \) \(879\)

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
2.78533
1.95007
1.31567
1.07636
0.208270
−0.489003
−0.510671
−0.858231
−1.40552
−2.07227
0 −1.78533 0 −0.701114 0 2.02991 0 0.187388 0
1.2 0 −0.950069 0 −2.28693 0 −2.71022 0 −2.09737 0
1.3 0 −0.315672 0 2.25024 0 3.20647 0 −2.90035 0
1.4 0 −0.0763625 0 1.17276 0 −0.469303 0 −2.99417 0
1.5 0 0.791730 0 0.178789 0 0.0809018 0 −2.37316 0
1.6 0 1.48900 0 −1.79865 0 −0.552233 0 −0.782869 0
1.7 0 1.51067 0 −2.23445 0 3.60329 0 −0.717874 0
1.8 0 1.85823 0 1.44291 0 1.96509 0 0.453023 0
1.9 0 2.40552 0 0.590303 0 −1.95900 0 2.78655 0
1.10 0 3.07227 0 0.386144 0 −0.194914 0 6.43884 0
\(n\): 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.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(503\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\(T_{3}^{10} - \cdots\)
\(T_{5}^{10} + \cdots\)
\(T_{7}^{10} - \cdots\)
\(T_{13}^{10} + \cdots\)