Properties

Label 8040.2.a.bc
Level 8040
Weight 2
Character orbit 8040.a
Self dual Yes
Analytic conductor 64.200
Analytic rank 0
Dimension 10
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8040.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1997232251\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(3\) \(x^{9}\mathstrut -\mathstrut \) \(37\) \(x^{8}\mathstrut +\mathstrut \) \(132\) \(x^{7}\mathstrut +\mathstrut \) \(358\) \(x^{6}\mathstrut -\mathstrut \) \(1708\) \(x^{5}\mathstrut -\mathstrut \) \(92\) \(x^{4}\mathstrut +\mathstrut \) \(5969\) \(x^{3}\mathstrut -\mathstrut \) \(3864\) \(x^{2}\mathstrut -\mathstrut \) \(4752\) \(x\mathstrut +\mathstrut \) \(3524\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(1820505\) \(\nu^{9}\mathstrut +\mathstrut \) \(648324\) \(\nu^{8}\mathstrut -\mathstrut \) \(58839604\) \(\nu^{7}\mathstrut +\mathstrut \) \(40103260\) \(\nu^{6}\mathstrut +\mathstrut \) \(547588947\) \(\nu^{5}\mathstrut -\mathstrut \) \(1158625792\) \(\nu^{4}\mathstrut -\mathstrut \) \(1218757803\) \(\nu^{3}\mathstrut +\mathstrut \) \(5781898709\) \(\nu^{2}\mathstrut +\mathstrut \) \(470820744\) \(\nu\mathstrut -\mathstrut \) \(6029040447\)\()/\)\(712963777\)
\(\beta_{2}\)\(=\)\((\)\(-\)\(9643245\) \(\nu^{9}\mathstrut +\mathstrut \) \(13049510\) \(\nu^{8}\mathstrut +\mathstrut \) \(405600855\) \(\nu^{7}\mathstrut -\mathstrut \) \(622469771\) \(\nu^{6}\mathstrut -\mathstrut \) \(5438935403\) \(\nu^{5}\mathstrut +\mathstrut \) \(8604438743\) \(\nu^{4}\mathstrut +\mathstrut \) \(24641558311\) \(\nu^{3}\mathstrut -\mathstrut \) \(32331453390\) \(\nu^{2}\mathstrut -\mathstrut \) \(34441127948\) \(\nu\mathstrut +\mathstrut \) \(24517354298\)\()/\)\(1425927554\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(26086723\) \(\nu^{9}\mathstrut -\mathstrut \) \(252840\) \(\nu^{8}\mathstrut +\mathstrut \) \(996434659\) \(\nu^{7}\mathstrut -\mathstrut \) \(442082665\) \(\nu^{6}\mathstrut -\mathstrut \) \(11857995913\) \(\nu^{5}\mathstrut +\mathstrut \) \(9509001303\) \(\nu^{4}\mathstrut +\mathstrut \) \(44010196469\) \(\nu^{3}\mathstrut -\mathstrut \) \(36860976572\) \(\nu^{2}\mathstrut -\mathstrut \) \(42982345244\) \(\nu\mathstrut +\mathstrut \) \(28946015068\)\()/\)\(1425927554\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(18725497\) \(\nu^{9}\mathstrut +\mathstrut \) \(12691238\) \(\nu^{8}\mathstrut +\mathstrut \) \(723832368\) \(\nu^{7}\mathstrut -\mathstrut \) \(798514962\) \(\nu^{6}\mathstrut -\mathstrut \) \(8641020986\) \(\nu^{5}\mathstrut +\mathstrut \) \(12215412133\) \(\nu^{4}\mathstrut +\mathstrut \) \(31178512266\) \(\nu^{3}\mathstrut -\mathstrut \) \(42939304662\) \(\nu^{2}\mathstrut -\mathstrut \) \(29564834748\) \(\nu\mathstrut +\mathstrut \) \(28921908461\)\()/\)\(712963777\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(39985935\) \(\nu^{9}\mathstrut +\mathstrut \) \(21312224\) \(\nu^{8}\mathstrut +\mathstrut \) \(1524928207\) \(\nu^{7}\mathstrut -\mathstrut \) \(1501477001\) \(\nu^{6}\mathstrut -\mathstrut \) \(17859830925\) \(\nu^{5}\mathstrut +\mathstrut \) \(23732873701\) \(\nu^{4}\mathstrut +\mathstrut \) \(62299880455\) \(\nu^{3}\mathstrut -\mathstrut \) \(81281718972\) \(\nu^{2}\mathstrut -\mathstrut \) \(56651824652\) \(\nu\mathstrut +\mathstrut \) \(50050926048\)\()/\)\(1425927554\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(53420509\) \(\nu^{9}\mathstrut +\mathstrut \) \(49948754\) \(\nu^{8}\mathstrut +\mathstrut \) \(2092831555\) \(\nu^{7}\mathstrut -\mathstrut \) \(2752544585\) \(\nu^{6}\mathstrut -\mathstrut \) \(25227887457\) \(\nu^{5}\mathstrut +\mathstrut \) \(40011083555\) \(\nu^{4}\mathstrut +\mathstrut \) \(90980987213\) \(\nu^{3}\mathstrut -\mathstrut \) \(140121203272\) \(\nu^{2}\mathstrut -\mathstrut \) \(85015093680\) \(\nu\mathstrut +\mathstrut \) \(97540858780\)\()/\)\(1425927554\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(27977725\) \(\nu^{9}\mathstrut +\mathstrut \) \(22939251\) \(\nu^{8}\mathstrut +\mathstrut \) \(1085047513\) \(\nu^{7}\mathstrut -\mathstrut \) \(1328495831\) \(\nu^{6}\mathstrut -\mathstrut \) \(12902838205\) \(\nu^{5}\mathstrut +\mathstrut \) \(19656566495\) \(\nu^{4}\mathstrut +\mathstrut \) \(45461921568\) \(\nu^{3}\mathstrut -\mathstrut \) \(67762156460\) \(\nu^{2}\mathstrut -\mathstrut \) \(42694551972\) \(\nu\mathstrut +\mathstrut \) \(44873983953\)\()/\)\(712963777\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(29521266\) \(\nu^{9}\mathstrut +\mathstrut \) \(37943916\) \(\nu^{8}\mathstrut +\mathstrut \) \(1166227892\) \(\nu^{7}\mathstrut -\mathstrut \) \(1880005231\) \(\nu^{6}\mathstrut -\mathstrut \) \(14093727028\) \(\nu^{5}\mathstrut +\mathstrut \) \(25929421256\) \(\nu^{4}\mathstrut +\mathstrut \) \(50178947424\) \(\nu^{3}\mathstrut -\mathstrut \) \(89483442459\) \(\nu^{2}\mathstrut -\mathstrut \) \(46084337074\) \(\nu\mathstrut +\mathstrut \) \(60525448260\)\()/\)\(712963777\)
\(\beta_{9}\)\(=\)\((\)\(45139551\) \(\nu^{9}\mathstrut -\mathstrut \) \(37028569\) \(\nu^{8}\mathstrut -\mathstrut \) \(1746534425\) \(\nu^{7}\mathstrut +\mathstrut \) \(2162038384\) \(\nu^{6}\mathstrut +\mathstrut \) \(20720606549\) \(\nu^{5}\mathstrut -\mathstrut \) \(32209590738\) \(\nu^{4}\mathstrut -\mathstrut \) \(72543332013\) \(\nu^{3}\mathstrut +\mathstrut \) \(112947152263\) \(\nu^{2}\mathstrut +\mathstrut \) \(63721166168\) \(\nu\mathstrut -\mathstrut \) \(75217331604\)\()/\)\(712963777\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(17\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(5\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(8\) \(\beta_{7}\mathstrut +\mathstrut \) \(15\) \(\beta_{6}\mathstrut +\mathstrut \) \(14\) \(\beta_{5}\mathstrut -\mathstrut \) \(15\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\mathstrut -\mathstrut \) \(17\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(57\) \(\beta_{9}\mathstrut -\mathstrut \) \(13\) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{7}\mathstrut -\mathstrut \) \(67\) \(\beta_{6}\mathstrut -\mathstrut \) \(12\) \(\beta_{5}\mathstrut -\mathstrut \) \(13\) \(\beta_{4}\mathstrut -\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(36\) \(\beta_{2}\mathstrut +\mathstrut \) \(32\) \(\beta_{1}\mathstrut +\mathstrut \) \(245\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(119\) \(\beta_{9}\mathstrut +\mathstrut \) \(17\) \(\beta_{8}\mathstrut -\mathstrut \) \(96\) \(\beta_{7}\mathstrut +\mathstrut \) \(287\) \(\beta_{6}\mathstrut +\mathstrut \) \(222\) \(\beta_{5}\mathstrut -\mathstrut \) \(265\) \(\beta_{4}\mathstrut +\mathstrut \) \(27\) \(\beta_{3}\mathstrut -\mathstrut \) \(22\) \(\beta_{2}\mathstrut -\mathstrut \) \(92\) \(\beta_{1}\mathstrut -\mathstrut \) \(469\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(1003\) \(\beta_{9}\mathstrut -\mathstrut \) \(139\) \(\beta_{8}\mathstrut +\mathstrut \) \(110\) \(\beta_{7}\mathstrut -\mathstrut \) \(1365\) \(\beta_{6}\mathstrut -\mathstrut \) \(326\) \(\beta_{5}\mathstrut -\mathstrut \) \(91\) \(\beta_{4}\mathstrut -\mathstrut \) \(113\) \(\beta_{3}\mathstrut +\mathstrut \) \(606\) \(\beta_{2}\mathstrut +\mathstrut \) \(558\) \(\beta_{1}\mathstrut +\mathstrut \) \(4161\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(2423\) \(\beta_{9}\mathstrut +\mathstrut \) \(145\) \(\beta_{8}\mathstrut -\mathstrut \) \(1406\) \(\beta_{7}\mathstrut +\mathstrut \) \(5743\) \(\beta_{6}\mathstrut +\mathstrut \) \(3766\) \(\beta_{5}\mathstrut -\mathstrut \) \(4859\) \(\beta_{4}\mathstrut +\mathstrut \) \(505\) \(\beta_{3}\mathstrut -\mathstrut \) \(644\) \(\beta_{2}\mathstrut -\mathstrut \) \(1782\) \(\beta_{1}\mathstrut -\mathstrut \) \(10369\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(17727\) \(\beta_{9}\mathstrut -\mathstrut \) \(927\) \(\beta_{8}\mathstrut +\mathstrut \) \(2254\) \(\beta_{7}\mathstrut -\mathstrut \) \(27377\) \(\beta_{6}\mathstrut -\mathstrut \) \(7094\) \(\beta_{5}\mathstrut +\mathstrut \) \(1069\) \(\beta_{4}\mathstrut -\mathstrut \) \(1091\) \(\beta_{3}\mathstrut +\mathstrut \) \(10348\) \(\beta_{2}\mathstrut +\mathstrut \) \(10156\) \(\beta_{1}\mathstrut +\mathstrut \) \(75113\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(47525\) \(\beta_{9}\mathstrut -\mathstrut \) \(1463\) \(\beta_{8}\mathstrut -\mathstrut \) \(22344\) \(\beta_{7}\mathstrut +\mathstrut \) \(115779\) \(\beta_{6}\mathstrut +\mathstrut \) \(66128\) \(\beta_{5}\mathstrut -\mathstrut \) \(90593\) \(\beta_{4}\mathstrut +\mathstrut \) \(7923\) \(\beta_{3}\mathstrut -\mathstrut \) \(14672\) \(\beta_{2}\mathstrut -\mathstrut \) \(33712\) \(\beta_{1}\mathstrut -\mathstrut \) \(215295\)\()/2\)

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
−1.93365
1.90235
2.61555
4.04435
−1.02133
−4.09581
0.731185
3.07629
−4.47145
2.15252
0 −1.00000 0 1.00000 0 −4.46032 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −4.35860 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 −3.08369 0 1.00000 0
1.4 0 −1.00000 0 1.00000 0 −1.71989 0 1.00000 0
1.5 0 −1.00000 0 1.00000 0 −1.55723 0 1.00000 0
1.6 0 −1.00000 0 1.00000 0 1.83837 0 1.00000 0
1.7 0 −1.00000 0 1.00000 0 1.87892 0 1.00000 0
1.8 0 −1.00000 0 1.00000 0 3.78235 0 1.00000 0
1.9 0 −1.00000 0 1.00000 0 3.93512 0 1.00000 0
1.10 0 −1.00000 0 1.00000 0 4.74497 0 1.00000 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
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(67\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8040))\):

\(T_{7}^{10} - \cdots\)
\(T_{11}^{10} - \cdots\)
\(T_{13}^{10} + \cdots\)
\(T_{17}^{10} - \cdots\)