Properties

Label 128.5.f.a
Level 128
Weight 5
Character orbit 128.f
Analytic conductor 13.231
Analytic rank 0
Dimension 14
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 128 = 2^{7} \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 128.f (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(13.2313552747\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{42} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{5} q^{3} \) \( + \beta_{8} q^{5} \) \( + \beta_{9} q^{7} \) \( + ( -19 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{12} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{5} q^{3} \) \( + \beta_{8} q^{5} \) \( + \beta_{9} q^{7} \) \( + ( -19 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{12} ) q^{9} \) \( + ( 7 + 7 \beta_{1} - \beta_{3} - \beta_{11} ) q^{11} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{13} \) \( + ( -24 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{8} + \beta_{10} - 2 \beta_{12} + \beta_{13} ) q^{15} \) \( + ( -2 + 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{17} \) \( + ( -49 + 49 \beta_{1} + 2 \beta_{4} - 2 \beta_{6} + \beta_{7} + 5 \beta_{8} - 2 \beta_{9} + 2 \beta_{12} + 2 \beta_{13} ) q^{19} \) \( + ( 13 - 13 \beta_{1} + 3 \beta_{4} + 15 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} - 3 \beta_{9} - 2 \beta_{12} + \beta_{13} ) q^{21} \) \( + ( -76 + 4 \beta_{2} + 8 \beta_{3} - 4 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} - 8 \beta_{8} - 3 \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{13} ) q^{23} \) \( + ( -47 \beta_{1} - 13 \beta_{2} + \beta_{3} + 6 \beta_{4} - 13 \beta_{5} - 3 \beta_{7} + \beta_{8} - \beta_{10} - 3 \beta_{11} - 4 \beta_{12} - \beta_{13} ) q^{25} \) \( + ( -115 - 115 \beta_{1} + 11 \beta_{2} + 17 \beta_{3} - 2 \beta_{4} - 6 \beta_{6} - 2 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} - 6 \beta_{12} ) q^{27} \) \( + ( -56 - 56 \beta_{1} - 36 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} - 2 \beta_{6} + 6 \beta_{9} - 4 \beta_{10} - 2 \beta_{12} ) q^{29} \) \( + ( 20 \beta_{1} + 4 \beta_{2} + 16 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} + 16 \beta_{8} + 3 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} ) q^{31} \) \( + ( 16 - 43 \beta_{2} - 4 \beta_{3} + 43 \beta_{5} + 3 \beta_{6} + 4 \beta_{8} - 13 \beta_{9} - 4 \beta_{10} + 4 \beta_{13} ) q^{33} \) \( + ( 102 - 102 \beta_{1} - 6 \beta_{4} + 8 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 34 \beta_{8} + 6 \beta_{9} - 2 \beta_{12} + 6 \beta_{13} ) q^{35} \) \( + ( 115 - 115 \beta_{1} - 13 \beta_{4} - 59 \beta_{5} + 8 \beta_{6} - 5 \beta_{7} - 2 \beta_{8} + 13 \beta_{9} - 8 \beta_{12} + 3 \beta_{13} ) q^{37} \) \( + ( -188 - 10 \beta_{2} - 26 \beta_{3} + 10 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} + 26 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - 6 \beta_{11} + 2 \beta_{13} ) q^{39} \) \( + ( -6 \beta_{1} + 67 \beta_{2} + 11 \beta_{3} - 12 \beta_{4} + 67 \beta_{5} - 5 \beta_{7} + 11 \beta_{8} - 3 \beta_{10} - 5 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} ) q^{41} \) \( + ( 112 + 112 \beta_{1} + 7 \beta_{2} - 40 \beta_{3} + 8 \beta_{4} - 4 \beta_{6} + 8 \beta_{9} + 4 \beta_{10} - 4 \beta_{12} ) q^{43} \) \( + ( -115 - 115 \beta_{1} + 129 \beta_{2} + 2 \beta_{3} - 11 \beta_{4} - 6 \beta_{6} - 11 \beta_{9} - \beta_{10} + 3 \beta_{11} - 6 \beta_{12} ) q^{45} \) \( + ( 400 \beta_{1} + 18 \beta_{2} - 34 \beta_{3} + 18 \beta_{5} - 4 \beta_{7} - 34 \beta_{8} - \beta_{10} - 4 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{47} \) \( + ( 15 + 110 \beta_{2} - 16 \beta_{3} - 110 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} + 16 \beta_{8} + 6 \beta_{9} - 4 \beta_{11} ) q^{49} \) \( + ( -229 + 229 \beta_{1} - 12 \beta_{4} + 5 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} + 53 \beta_{8} + 12 \beta_{9} - 4 \beta_{12} - 4 \beta_{13} ) q^{51} \) \( + ( 65 - 65 \beta_{1} + 3 \beta_{4} + 187 \beta_{5} - 2 \beta_{6} + 9 \beta_{7} + 4 \beta_{8} - 3 \beta_{9} + 2 \beta_{12} - 3 \beta_{13} ) q^{53} \) \( + ( 844 - 26 \beta_{2} + 38 \beta_{3} + 26 \beta_{5} + 16 \beta_{6} - 6 \beta_{7} - 38 \beta_{8} + 13 \beta_{9} + 4 \beta_{10} + 6 \beta_{11} - 4 \beta_{13} ) q^{55} \) \( + ( 56 \beta_{1} - 165 \beta_{2} + 4 \beta_{3} + \beta_{4} - 165 \beta_{5} + 12 \beta_{7} + 4 \beta_{8} + 4 \beta_{10} + 12 \beta_{11} + 9 \beta_{12} + 4 \beta_{13} ) q^{57} \) \( + ( -194 - 194 \beta_{1} - 39 \beta_{2} + 50 \beta_{3} + 6 \beta_{4} + 22 \beta_{6} + 6 \beta_{9} - 10 \beta_{10} + 6 \beta_{11} + 22 \beta_{12} ) q^{59} \) \( + ( 303 + 303 \beta_{1} - 261 \beta_{2} - 14 \beta_{3} - 5 \beta_{4} + 10 \beta_{6} - 5 \beta_{9} + 13 \beta_{10} - 7 \beta_{11} + 10 \beta_{12} ) q^{61} \) \( + ( -1424 \beta_{1} - 14 \beta_{2} + 30 \beta_{3} - 17 \beta_{4} - 14 \beta_{5} + 12 \beta_{7} + 30 \beta_{8} - 9 \beta_{10} + 12 \beta_{11} - 14 \beta_{12} - 9 \beta_{13} ) q^{63} \) \( + ( -108 - 177 \beta_{2} + 31 \beta_{3} + 177 \beta_{5} - 10 \beta_{6} - 7 \beta_{7} - 31 \beta_{8} + 36 \beta_{9} + 17 \beta_{10} + 7 \beta_{11} - 17 \beta_{13} ) q^{65} \) \( + ( 573 - 573 \beta_{1} + 44 \beta_{4} - 50 \beta_{5} - 16 \beta_{6} - 5 \beta_{7} - 5 \beta_{8} - 44 \beta_{9} + 16 \beta_{12} - 24 \beta_{13} ) q^{67} \) \( + ( -713 + 713 \beta_{1} + 49 \beta_{4} - 379 \beta_{5} - 42 \beta_{6} + 15 \beta_{7} - \beta_{8} - 49 \beta_{9} + 42 \beta_{12} - 13 \beta_{13} ) q^{69} \) \( + ( -1484 + 84 \beta_{2} - 8 \beta_{3} - 84 \beta_{5} + 10 \beta_{6} - 30 \beta_{7} + 8 \beta_{8} - 3 \beta_{9} + 7 \beta_{10} + 30 \beta_{11} - 7 \beta_{13} ) q^{71} \) \( + ( 242 \beta_{1} + 243 \beta_{2} - 80 \beta_{3} + 41 \beta_{4} + 243 \beta_{5} + 20 \beta_{7} - 80 \beta_{8} + 16 \beta_{10} + 20 \beta_{11} + 17 \beta_{12} + 16 \beta_{13} ) q^{73} \) \( + ( 1268 + 1268 \beta_{1} - 27 \beta_{2} - 24 \beta_{3} - 58 \beta_{4} + 14 \beta_{6} - 58 \beta_{9} - 18 \beta_{10} + 12 \beta_{11} + 14 \beta_{12} ) q^{75} \) \( + ( 595 + 595 \beta_{1} + 359 \beta_{2} - 3 \beta_{3} + 51 \beta_{4} + 38 \beta_{6} + 51 \beta_{9} + 17 \beta_{10} + 5 \beta_{11} + 38 \beta_{12} ) q^{77} \) \( + ( 2172 \beta_{1} - 174 \beta_{2} + 2 \beta_{3} + 20 \beta_{4} - 174 \beta_{5} + 22 \beta_{7} + 2 \beta_{8} - 10 \beta_{10} + 22 \beta_{11} + 28 \beta_{12} - 10 \beta_{13} ) q^{79} \) \( + ( 39 + 325 \beta_{2} + 88 \beta_{3} - 325 \beta_{5} - 5 \beta_{6} - 88 \beta_{8} - 37 \beta_{9} + 16 \beta_{10} - 16 \beta_{13} ) q^{81} \) \( + ( -1236 + 1236 \beta_{1} + 30 \beta_{4} - 3 \beta_{5} + 22 \beta_{6} - 4 \beta_{7} - 16 \beta_{8} - 30 \beta_{9} - 22 \beta_{12} - 14 \beta_{13} ) q^{83} \) \( + ( -649 + 649 \beta_{1} - 37 \beta_{4} + 377 \beta_{5} - 28 \beta_{6} + 15 \beta_{7} - 29 \beta_{8} + 37 \beta_{9} + 28 \beta_{12} - \beta_{13} ) q^{85} \) \( + ( 3460 + 84 \beta_{2} - 64 \beta_{3} - 84 \beta_{5} + 42 \beta_{6} - 6 \beta_{7} + 64 \beta_{8} + 5 \beta_{9} - \beta_{10} + 6 \beta_{11} + \beta_{13} ) q^{87} \) \( + ( 534 \beta_{1} - 295 \beta_{2} - 44 \beta_{3} - 81 \beta_{4} - 295 \beta_{5} - 44 \beta_{8} - 4 \beta_{10} + 31 \beta_{12} - 4 \beta_{13} ) q^{89} \) \( + ( -2044 - 2044 \beta_{1} - 30 \beta_{2} - 136 \beta_{3} + 22 \beta_{4} + 14 \beta_{6} + 22 \beta_{9} + 14 \beta_{10} + 28 \beta_{11} + 14 \beta_{12} ) q^{91} \) \( + ( -582 - 582 \beta_{1} - 310 \beta_{2} + 2 \beta_{3} - 70 \beta_{4} - 20 \beta_{6} - 70 \beta_{9} + 14 \beta_{10} + 30 \beta_{11} - 20 \beta_{12} ) q^{93} \) \( + ( -3064 \beta_{1} + 20 \beta_{2} - 108 \beta_{3} - 9 \beta_{4} + 20 \beta_{5} - 28 \beta_{7} - 108 \beta_{8} - 16 \beta_{10} - 28 \beta_{11} + 16 \beta_{12} - 16 \beta_{13} ) q^{95} \) \( + ( 74 - 248 \beta_{2} - 105 \beta_{3} + 248 \beta_{5} - 15 \beta_{6} + 37 \beta_{7} + 105 \beta_{8} + 23 \beta_{9} - 7 \beta_{10} - 37 \beta_{11} + 7 \beta_{13} ) q^{97} \) \( + ( 3450 - 3450 \beta_{1} - 122 \beta_{4} + 19 \beta_{5} + 34 \beta_{6} - 18 \beta_{7} + 106 \beta_{8} + 122 \beta_{9} - 34 \beta_{12} - 2 \beta_{13} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(14q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 94q^{11} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut -\mathstrut 706q^{19} \) \(\mathstrut +\mathstrut 164q^{21} \) \(\mathstrut -\mathstrut 1148q^{23} \) \(\mathstrut -\mathstrut 1664q^{27} \) \(\mathstrut -\mathstrut 862q^{29} \) \(\mathstrut -\mathstrut 4q^{33} \) \(\mathstrut +\mathstrut 1340q^{35} \) \(\mathstrut +\mathstrut 1826q^{37} \) \(\mathstrut -\mathstrut 2684q^{39} \) \(\mathstrut +\mathstrut 1694q^{43} \) \(\mathstrut -\mathstrut 1410q^{45} \) \(\mathstrut +\mathstrut 682q^{49} \) \(\mathstrut -\mathstrut 3012q^{51} \) \(\mathstrut +\mathstrut 482q^{53} \) \(\mathstrut +\mathstrut 11780q^{55} \) \(\mathstrut -\mathstrut 2786q^{59} \) \(\mathstrut +\mathstrut 3778q^{61} \) \(\mathstrut -\mathstrut 2020q^{65} \) \(\mathstrut +\mathstrut 7998q^{67} \) \(\mathstrut -\mathstrut 9628q^{69} \) \(\mathstrut -\mathstrut 19964q^{71} \) \(\mathstrut +\mathstrut 17570q^{75} \) \(\mathstrut +\mathstrut 9508q^{77} \) \(\mathstrut +\mathstrut 1454q^{81} \) \(\mathstrut -\mathstrut 17282q^{83} \) \(\mathstrut -\mathstrut 9948q^{85} \) \(\mathstrut +\mathstrut 49284q^{87} \) \(\mathstrut -\mathstrut 28036q^{91} \) \(\mathstrut -\mathstrut 8896q^{93} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut +\mathstrut 49214q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(4\) \(x^{13}\mathstrut +\mathstrut \) \(15\) \(x^{12}\mathstrut -\mathstrut \) \(34\) \(x^{11}\mathstrut +\mathstrut \) \(62\) \(x^{10}\mathstrut -\mathstrut \) \(312\) \(x^{9}\mathstrut +\mathstrut \) \(1432\) \(x^{8}\mathstrut -\mathstrut \) \(4960\) \(x^{7}\mathstrut +\mathstrut \) \(11456\) \(x^{6}\mathstrut -\mathstrut \) \(19968\) \(x^{5}\mathstrut +\mathstrut \) \(31744\) \(x^{4}\mathstrut -\mathstrut \) \(139264\) \(x^{3}\mathstrut +\mathstrut \) \(491520\) \(x^{2}\mathstrut -\mathstrut \) \(1048576\) \(x\mathstrut +\mathstrut \) \(2097152\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(2875\) \(\nu^{13}\mathstrut +\mathstrut \) \(13444\) \(\nu^{12}\mathstrut -\mathstrut \) \(26581\) \(\nu^{11}\mathstrut +\mathstrut \) \(16062\) \(\nu^{10}\mathstrut -\mathstrut \) \(24954\) \(\nu^{9}\mathstrut +\mathstrut \) \(748984\) \(\nu^{8}\mathstrut -\mathstrut \) \(4619080\) \(\nu^{7}\mathstrut +\mathstrut \) \(10623840\) \(\nu^{6}\mathstrut -\mathstrut \) \(11499840\) \(\nu^{5}\mathstrut +\mathstrut \) \(5960704\) \(\nu^{4}\mathstrut -\mathstrut \) \(51930112\) \(\nu^{3}\mathstrut +\mathstrut \) \(429211648\) \(\nu^{2}\mathstrut -\mathstrut \) \(1316257792\) \(\nu\mathstrut +\mathstrut \) \(1279524864\)\()/\)\(678952960\)
\(\beta_{2}\)\(=\)\((\)\(-\)\(11609\) \(\nu^{13}\mathstrut +\mathstrut \) \(81068\) \(\nu^{12}\mathstrut -\mathstrut \) \(216535\) \(\nu^{11}\mathstrut +\mathstrut \) \(92170\) \(\nu^{10}\mathstrut -\mathstrut \) \(445086\) \(\nu^{9}\mathstrut +\mathstrut \) \(4629992\) \(\nu^{8}\mathstrut -\mathstrut \) \(22287000\) \(\nu^{7}\mathstrut +\mathstrut \) \(63665440\) \(\nu^{6}\mathstrut -\mathstrut \) \(81223104\) \(\nu^{5}\mathstrut +\mathstrut \) \(58993664\) \(\nu^{4}\mathstrut -\mathstrut \) \(414198784\) \(\nu^{3}\mathstrut +\mathstrut \) \(2235252736\) \(\nu^{2}\mathstrut -\mathstrut \) \(5795840000\) \(\nu\mathstrut +\mathstrut \) \(7904428032\)\()/\)\(678952960\)
\(\beta_{3}\)\(=\)\((\)\(23279\) \(\nu^{13}\mathstrut +\mathstrut \) \(167756\) \(\nu^{12}\mathstrut -\mathstrut \) \(482591\) \(\nu^{11}\mathstrut +\mathstrut \) \(959162\) \(\nu^{10}\mathstrut -\mathstrut \) \(197678\) \(\nu^{9}\mathstrut +\mathstrut \) \(877992\) \(\nu^{8}\mathstrut -\mathstrut \) \(35554520\) \(\nu^{7}\mathstrut +\mathstrut \) \(170800160\) \(\nu^{6}\mathstrut -\mathstrut \) \(398883776\) \(\nu^{5}\mathstrut +\mathstrut \) \(133770240\) \(\nu^{4}\mathstrut -\mathstrut \) \(180005888\) \(\nu^{3}\mathstrut +\mathstrut \) \(2196488192\) \(\nu^{2}\mathstrut -\mathstrut \) \(17134354432\) \(\nu\mathstrut +\mathstrut \) \(45595492352\)\()/\)\(678952960\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(3499\) \(\nu^{13}\mathstrut +\mathstrut \) \(6652\) \(\nu^{12}\mathstrut +\mathstrut \) \(13019\) \(\nu^{11}\mathstrut -\mathstrut \) \(10778\) \(\nu^{10}\mathstrut +\mathstrut \) \(122070\) \(\nu^{9}\mathstrut +\mathstrut \) \(1232776\) \(\nu^{8}\mathstrut -\mathstrut \) \(3340360\) \(\nu^{7}\mathstrut -\mathstrut \) \(34400\) \(\nu^{6}\mathstrut +\mathstrut \) \(24398016\) \(\nu^{5}\mathstrut +\mathstrut \) \(5933568\) \(\nu^{4}\mathstrut -\mathstrut \) \(170023936\) \(\nu^{3}\mathstrut +\mathstrut \) \(637231104\) \(\nu^{2}\mathstrut -\mathstrut \) \(494927872\) \(\nu\mathstrut -\mathstrut \) \(3419275264\)\()/84869120\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(30299\) \(\nu^{13}\mathstrut +\mathstrut \) \(175428\) \(\nu^{12}\mathstrut -\mathstrut \) \(396085\) \(\nu^{11}\mathstrut +\mathstrut \) \(537790\) \(\nu^{10}\mathstrut -\mathstrut \) \(1159226\) \(\nu^{9}\mathstrut +\mathstrut \) \(9651512\) \(\nu^{8}\mathstrut -\mathstrut \) \(53625160\) \(\nu^{7}\mathstrut +\mathstrut \) \(150852960\) \(\nu^{6}\mathstrut -\mathstrut \) \(203316544\) \(\nu^{5}\mathstrut +\mathstrut \) \(134759424\) \(\nu^{4}\mathstrut -\mathstrut \) \(982549504\) \(\nu^{3}\mathstrut +\mathstrut \) \(4972855296\) \(\nu^{2}\mathstrut -\mathstrut \) \(16046653440\) \(\nu\mathstrut +\mathstrut \) \(20694433792\)\()/\)\(678952960\)
\(\beta_{6}\)\(=\)\((\)\(187\) \(\nu^{13}\mathstrut +\mathstrut \) \(588\) \(\nu^{12}\mathstrut -\mathstrut \) \(3179\) \(\nu^{11}\mathstrut +\mathstrut \) \(8306\) \(\nu^{10}\mathstrut -\mathstrut \) \(18854\) \(\nu^{9}\mathstrut +\mathstrut \) \(936\) \(\nu^{8}\mathstrut -\mathstrut \) \(206648\) \(\nu^{7}\mathstrut +\mathstrut \) \(1252384\) \(\nu^{6}\mathstrut -\mathstrut \) \(3000512\) \(\nu^{5}\mathstrut +\mathstrut \) \(5232640\) \(\nu^{4}\mathstrut -\mathstrut \) \(7527424\) \(\nu^{3}\mathstrut +\mathstrut \) \(7045120\) \(\nu^{2}\mathstrut -\mathstrut \) \(93421568\) \(\nu\mathstrut +\mathstrut \) \(359268352\)\()/2424832\)
\(\beta_{7}\)\(=\)\((\)\(54457\) \(\nu^{13}\mathstrut -\mathstrut \) \(666028\) \(\nu^{12}\mathstrut +\mathstrut \) \(2372471\) \(\nu^{11}\mathstrut -\mathstrut \) \(2827082\) \(\nu^{10}\mathstrut -\mathstrut \) \(5318178\) \(\nu^{9}\mathstrut -\mathstrut \) \(5702120\) \(\nu^{8}\mathstrut +\mathstrut \) \(151698840\) \(\nu^{7}\mathstrut -\mathstrut \) \(675003680\) \(\nu^{6}\mathstrut +\mathstrut \) \(1063292352\) \(\nu^{5}\mathstrut +\mathstrut \) \(531149824\) \(\nu^{4}\mathstrut -\mathstrut \) \(2507641856\) \(\nu^{3}\mathstrut -\mathstrut \) \(7324450816\) \(\nu^{2}\mathstrut +\mathstrut \) \(52640448512\) \(\nu\mathstrut -\mathstrut \) \(123104133120\)\()/\)\(678952960\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(63941\) \(\nu^{13}\mathstrut +\mathstrut \) \(248956\) \(\nu^{12}\mathstrut -\mathstrut \) \(325931\) \(\nu^{11}\mathstrut +\mathstrut \) \(303682\) \(\nu^{10}\mathstrut -\mathstrut \) \(1485958\) \(\nu^{9}\mathstrut +\mathstrut \) \(16189512\) \(\nu^{8}\mathstrut -\mathstrut \) \(70629560\) \(\nu^{7}\mathstrut +\mathstrut \) \(157086880\) \(\nu^{6}\mathstrut -\mathstrut \) \(67802816\) \(\nu^{5}\mathstrut +\mathstrut \) \(199142400\) \(\nu^{4}\mathstrut -\mathstrut \) \(922754048\) \(\nu^{3}\mathstrut +\mathstrut \) \(7022559232\) \(\nu^{2}\mathstrut -\mathstrut \) \(16313188352\) \(\nu\mathstrut +\mathstrut \) \(4344512512\)\()/\)\(678952960\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(1373\) \(\nu^{13}\mathstrut -\mathstrut \) \(1028\) \(\nu^{12}\mathstrut +\mathstrut \) \(16237\) \(\nu^{11}\mathstrut -\mathstrut \) \(30894\) \(\nu^{10}\mathstrut +\mathstrut \) \(21354\) \(\nu^{9}\mathstrut +\mathstrut \) \(156488\) \(\nu^{8}\mathstrut +\mathstrut \) \(410632\) \(\nu^{7}\mathstrut -\mathstrut \) \(4973408\) \(\nu^{6}\mathstrut +\mathstrut \) \(13415232\) \(\nu^{5}\mathstrut -\mathstrut \) \(9518080\) \(\nu^{4}\mathstrut -\mathstrut \) \(7715840\) \(\nu^{3}\mathstrut -\mathstrut \) \(46219264\) \(\nu^{2}\mathstrut +\mathstrut \) \(423264256\) \(\nu\mathstrut -\mathstrut \) \(1804861440\)\()/9699328\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(60737\) \(\nu^{13}\mathstrut -\mathstrut \) \(163572\) \(\nu^{12}\mathstrut +\mathstrut \) \(819089\) \(\nu^{11}\mathstrut -\mathstrut \) \(871398\) \(\nu^{10}\mathstrut -\mathstrut \) \(2291022\) \(\nu^{9}\mathstrut +\mathstrut \) \(15043880\) \(\nu^{8}\mathstrut +\mathstrut \) \(15561640\) \(\nu^{7}\mathstrut -\mathstrut \) \(288799200\) \(\nu^{6}\mathstrut +\mathstrut \) \(569406528\) \(\nu^{5}\mathstrut +\mathstrut \) \(16526336\) \(\nu^{4}\mathstrut -\mathstrut \) \(1198050304\) \(\nu^{3}\mathstrut +\mathstrut \) \(1135230976\) \(\nu^{2}\mathstrut +\mathstrut \) \(35844947968\) \(\nu\mathstrut -\mathstrut \) \(98412789760\)\()/\)\(339476480\)
\(\beta_{11}\)\(=\)\((\)\(191167\) \(\nu^{13}\mathstrut -\mathstrut \) \(198708\) \(\nu^{12}\mathstrut -\mathstrut \) \(780399\) \(\nu^{11}\mathstrut +\mathstrut \) \(1949018\) \(\nu^{10}\mathstrut +\mathstrut \) \(2939442\) \(\nu^{9}\mathstrut -\mathstrut \) \(59111000\) \(\nu^{8}\mathstrut +\mathstrut \) \(130720680\) \(\nu^{7}\mathstrut +\mathstrut \) \(55001120\) \(\nu^{6}\mathstrut -\mathstrut \) \(991562688\) \(\nu^{5}\mathstrut +\mathstrut \) \(198482944\) \(\nu^{4}\mathstrut +\mathstrut \) \(4742575104\) \(\nu^{3}\mathstrut -\mathstrut \) \(12649414656\) \(\nu^{2}\mathstrut +\mathstrut \) \(6744604672\) \(\nu\mathstrut +\mathstrut \) \(139540561920\)\()/\)\(678952960\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(107577\) \(\nu^{13}\mathstrut +\mathstrut \) \(484524\) \(\nu^{12}\mathstrut -\mathstrut \) \(1270135\) \(\nu^{11}\mathstrut +\mathstrut \) \(2145610\) \(\nu^{10}\mathstrut -\mathstrut \) \(3926878\) \(\nu^{9}\mathstrut +\mathstrut \) \(32206056\) \(\nu^{8}\mathstrut -\mathstrut \) \(152847000\) \(\nu^{7}\mathstrut +\mathstrut \) \(393864480\) \(\nu^{6}\mathstrut -\mathstrut \) \(547344832\) \(\nu^{5}\mathstrut +\mathstrut \) \(682642432\) \(\nu^{4}\mathstrut -\mathstrut \) \(2131233792\) \(\nu^{3}\mathstrut +\mathstrut \) \(12085280768\) \(\nu^{2}\mathstrut -\mathstrut \) \(41333391360\) \(\nu\mathstrut +\mathstrut \) \(33873985536\)\()/\)\(339476480\)
\(\beta_{13}\)\(=\)\((\)\(108593\) \(\nu^{13}\mathstrut -\mathstrut \) \(103372\) \(\nu^{12}\mathstrut -\mathstrut \) \(236161\) \(\nu^{11}\mathstrut +\mathstrut \) \(473862\) \(\nu^{10}\mathstrut +\mathstrut \) \(1410158\) \(\nu^{9}\mathstrut -\mathstrut \) \(32112040\) \(\nu^{8}\mathstrut +\mathstrut \) \(64026840\) \(\nu^{7}\mathstrut -\mathstrut \) \(34664480\) \(\nu^{6}\mathstrut -\mathstrut \) \(122103872\) \(\nu^{5}\mathstrut -\mathstrut \) \(344812544\) \(\nu^{4}\mathstrut +\mathstrut \) \(2863123456\) \(\nu^{3}\mathstrut -\mathstrut \) \(10052583424\) \(\nu^{2}\mathstrut +\mathstrut \) \(11285659648\) \(\nu\mathstrut +\mathstrut \) \(31120424960\)\()/\)\(339476480\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(9\) \(\beta_{1}\mathstrut +\mathstrut \) \(17\)\()/64\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(4\) \(\beta_{9}\mathstrut +\mathstrut \) \(7\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(3\) \(\beta_{6}\mathstrut +\mathstrut \) \(7\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(21\) \(\beta_{1}\mathstrut -\mathstrut \) \(63\)\()/64\)
\(\nu^{3}\)\(=\)\((\)\(3\) \(\beta_{12}\mathstrut -\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(12\) \(\beta_{9}\mathstrut +\mathstrut \) \(31\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(7\) \(\beta_{6}\mathstrut -\mathstrut \) \(65\) \(\beta_{5}\mathstrut -\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut -\mathstrut \) \(20\) \(\beta_{2}\mathstrut +\mathstrut \) \(167\) \(\beta_{1}\mathstrut -\mathstrut \) \(67\)\()/64\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(6\) \(\beta_{13}\mathstrut +\mathstrut \) \(11\) \(\beta_{12}\mathstrut +\mathstrut \) \(12\) \(\beta_{11}\mathstrut -\mathstrut \) \(5\) \(\beta_{10}\mathstrut +\mathstrut \) \(12\) \(\beta_{9}\mathstrut +\mathstrut \) \(45\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(29\) \(\beta_{6}\mathstrut -\mathstrut \) \(255\) \(\beta_{5}\mathstrut +\mathstrut \) \(28\) \(\beta_{4}\mathstrut -\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(82\) \(\beta_{2}\mathstrut +\mathstrut \) \(967\) \(\beta_{1}\mathstrut +\mathstrut \) \(121\)\()/64\)
\(\nu^{5}\)\(=\)\((\)\(44\) \(\beta_{13}\mathstrut +\mathstrut \) \(15\) \(\beta_{12}\mathstrut -\mathstrut \) \(34\) \(\beta_{11}\mathstrut -\mathstrut \) \(23\) \(\beta_{10}\mathstrut -\mathstrut \) \(36\) \(\beta_{9}\mathstrut +\mathstrut \) \(107\) \(\beta_{8}\mathstrut +\mathstrut \) \(5\) \(\beta_{7}\mathstrut -\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(29\) \(\beta_{5}\mathstrut +\mathstrut \) \(108\) \(\beta_{4}\mathstrut -\mathstrut \) \(118\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut -\mathstrut \) \(1909\) \(\beta_{1}\mathstrut +\mathstrut \) \(5561\)\()/64\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(38\) \(\beta_{13}\mathstrut -\mathstrut \) \(189\) \(\beta_{12}\mathstrut -\mathstrut \) \(108\) \(\beta_{11}\mathstrut -\mathstrut \) \(5\) \(\beta_{10}\mathstrut -\mathstrut \) \(132\) \(\beta_{9}\mathstrut +\mathstrut \) \(445\) \(\beta_{8}\mathstrut -\mathstrut \) \(157\) \(\beta_{7}\mathstrut +\mathstrut \) \(133\) \(\beta_{6}\mathstrut +\mathstrut \) \(657\) \(\beta_{5}\mathstrut -\mathstrut \) \(68\) \(\beta_{4}\mathstrut -\mathstrut \) \(316\) \(\beta_{3}\mathstrut -\mathstrut \) \(422\) \(\beta_{2}\mathstrut -\mathstrut \) \(8849\) \(\beta_{1}\mathstrut -\mathstrut \) \(12447\)\()/64\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(280\) \(\beta_{13}\mathstrut +\mathstrut \) \(7\) \(\beta_{12}\mathstrut +\mathstrut \) \(242\) \(\beta_{11}\mathstrut -\mathstrut \) \(291\) \(\beta_{10}\mathstrut -\mathstrut \) \(84\) \(\beta_{9}\mathstrut +\mathstrut \) \(1143\) \(\beta_{8}\mathstrut +\mathstrut \) \(89\) \(\beta_{7}\mathstrut -\mathstrut \) \(371\) \(\beta_{6}\mathstrut +\mathstrut \) \(1255\) \(\beta_{5}\mathstrut +\mathstrut \) \(156\) \(\beta_{4}\mathstrut -\mathstrut \) \(266\) \(\beta_{3}\mathstrut +\mathstrut \) \(1616\) \(\beta_{2}\mathstrut -\mathstrut \) \(44397\) \(\beta_{1}\mathstrut -\mathstrut \) \(10215\)\()/64\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(858\) \(\beta_{13}\mathstrut +\mathstrut \) \(611\) \(\beta_{12}\mathstrut -\mathstrut \) \(1136\) \(\beta_{11}\mathstrut -\mathstrut \) \(153\) \(\beta_{10}\mathstrut -\mathstrut \) \(1156\) \(\beta_{9}\mathstrut +\mathstrut \) \(409\) \(\beta_{8}\mathstrut -\mathstrut \) \(185\) \(\beta_{7}\mathstrut -\mathstrut \) \(867\) \(\beta_{6}\mathstrut -\mathstrut \) \(8315\) \(\beta_{5}\mathstrut -\mathstrut \) \(1284\) \(\beta_{4}\mathstrut +\mathstrut \) \(3976\) \(\beta_{3}\mathstrut +\mathstrut \) \(5494\) \(\beta_{2}\mathstrut -\mathstrut \) \(59217\) \(\beta_{1}\mathstrut +\mathstrut \) \(66329\)\()/64\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(2564\) \(\beta_{13}\mathstrut -\mathstrut \) \(153\) \(\beta_{12}\mathstrut +\mathstrut \) \(1158\) \(\beta_{11}\mathstrut -\mathstrut \) \(127\) \(\beta_{10}\mathstrut +\mathstrut \) \(3660\) \(\beta_{9}\mathstrut -\mathstrut \) \(6677\) \(\beta_{8}\mathstrut -\mathstrut \) \(3323\) \(\beta_{7}\mathstrut +\mathstrut \) \(3381\) \(\beta_{6}\mathstrut -\mathstrut \) \(23677\) \(\beta_{5}\mathstrut +\mathstrut \) \(8796\) \(\beta_{4}\mathstrut +\mathstrut \) \(4802\) \(\beta_{3}\mathstrut +\mathstrut \) \(4708\) \(\beta_{2}\mathstrut +\mathstrut \) \(108659\) \(\beta_{1}\mathstrut +\mathstrut \) \(89825\)\()/64\)
\(\nu^{10}\)\(=\)\((\)\(3434\) \(\beta_{13}\mathstrut +\mathstrut \) \(6691\) \(\beta_{12}\mathstrut +\mathstrut \) \(548\) \(\beta_{11}\mathstrut +\mathstrut \) \(2587\) \(\beta_{10}\mathstrut -\mathstrut \) \(4804\) \(\beta_{9}\mathstrut +\mathstrut \) \(3021\) \(\beta_{8}\mathstrut -\mathstrut \) \(1357\) \(\beta_{7}\mathstrut -\mathstrut \) \(5915\) \(\beta_{6}\mathstrut +\mathstrut \) \(53569\) \(\beta_{5}\mathstrut +\mathstrut \) \(21852\) \(\beta_{4}\mathstrut +\mathstrut \) \(7444\) \(\beta_{3}\mathstrut -\mathstrut \) \(121494\) \(\beta_{2}\mathstrut -\mathstrut \) \(580657\) \(\beta_{1}\mathstrut +\mathstrut \) \(628385\)\()/64\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(15088\) \(\beta_{13}\mathstrut -\mathstrut \) \(31849\) \(\beta_{12}\mathstrut -\mathstrut \) \(5174\) \(\beta_{11}\mathstrut +\mathstrut \) \(8437\) \(\beta_{10}\mathstrut +\mathstrut \) \(2892\) \(\beta_{9}\mathstrut +\mathstrut \) \(90719\) \(\beta_{8}\mathstrut -\mathstrut \) \(9359\) \(\beta_{7}\mathstrut +\mathstrut \) \(15405\) \(\beta_{6}\mathstrut +\mathstrut \) \(84255\) \(\beta_{5}\mathstrut -\mathstrut \) \(32196\) \(\beta_{4}\mathstrut +\mathstrut \) \(79198\) \(\beta_{3}\mathstrut -\mathstrut \) \(343064\) \(\beta_{2}\mathstrut -\mathstrut \) \(15805\) \(\beta_{1}\mathstrut -\mathstrut \) \(1111399\)\()/64\)
\(\nu^{12}\)\(=\)\((\)\(9886\) \(\beta_{13}\mathstrut +\mathstrut \) \(1059\) \(\beta_{12}\mathstrut +\mathstrut \) \(65352\) \(\beta_{11}\mathstrut -\mathstrut \) \(12833\) \(\beta_{10}\mathstrut +\mathstrut \) \(144956\) \(\beta_{9}\mathstrut +\mathstrut \) \(270289\) \(\beta_{8}\mathstrut +\mathstrut \) \(38927\) \(\beta_{7}\mathstrut -\mathstrut \) \(10163\) \(\beta_{6}\mathstrut -\mathstrut \) \(170835\) \(\beta_{5}\mathstrut +\mathstrut \) \(47548\) \(\beta_{4}\mathstrut +\mathstrut \) \(313424\) \(\beta_{3}\mathstrut +\mathstrut \) \(396334\) \(\beta_{2}\mathstrut -\mathstrut \) \(2474769\) \(\beta_{1}\mathstrut +\mathstrut \) \(1768521\)\()/64\)
\(\nu^{13}\)\(=\)\((\)\(124644\) \(\beta_{13}\mathstrut +\mathstrut \) \(4807\) \(\beta_{12}\mathstrut -\mathstrut \) \(241810\) \(\beta_{11}\mathstrut +\mathstrut \) \(41769\) \(\beta_{10}\mathstrut -\mathstrut \) \(273012\) \(\beta_{9}\mathstrut +\mathstrut \) \(107123\) \(\beta_{8}\mathstrut +\mathstrut \) \(93757\) \(\beta_{7}\mathstrut -\mathstrut \) \(97819\) \(\beta_{6}\mathstrut -\mathstrut \) \(1479205\) \(\beta_{5}\mathstrut +\mathstrut \) \(169308\) \(\beta_{4}\mathstrut +\mathstrut \) \(1013274\) \(\beta_{3}\mathstrut +\mathstrut \) \(1212684\) \(\beta_{2}\mathstrut +\mathstrut \) \(85715\) \(\beta_{1}\mathstrut -\mathstrut \) \(514479\)\()/64\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-\beta_{1}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.153862 2.82424i
0.336831 + 2.80830i
2.79265 + 0.448449i
−2.40693 + 1.48549i
−2.15805 1.82834i
2.24452 1.72109i
1.03712 + 2.63142i
0.153862 + 2.82424i
0.336831 2.80830i
2.79265 0.448449i
−2.40693 1.48549i
−2.15805 + 1.82834i
2.24452 + 1.72109i
1.03712 2.63142i
0 −9.42589 9.42589i 0 2.84710 + 2.84710i 0 76.7794 0 96.6949i 0
31.2 0 −7.86839 7.86839i 0 −27.2309 27.2309i 0 −50.3097 0 42.8233i 0
31.3 0 −4.63552 4.63552i 0 29.2002 + 29.2002i 0 −59.6196 0 38.0239i 0
31.4 0 −0.0461995 0.0461995i 0 8.04297 + 8.04297i 0 49.8797 0 80.9957i 0
31.5 0 3.91498 + 3.91498i 0 −4.72348 4.72348i 0 −45.3712 0 50.3458i 0
31.6 0 5.54016 + 5.54016i 0 −21.7374 21.7374i 0 6.62054 0 19.6133i 0
31.7 0 11.5209 + 11.5209i 0 14.6016 + 14.6016i 0 24.0210 0 184.461i 0
95.1 0 −9.42589 + 9.42589i 0 2.84710 2.84710i 0 76.7794 0 96.6949i 0
95.2 0 −7.86839 + 7.86839i 0 −27.2309 + 27.2309i 0 −50.3097 0 42.8233i 0
95.3 0 −4.63552 + 4.63552i 0 29.2002 29.2002i 0 −59.6196 0 38.0239i 0
95.4 0 −0.0461995 + 0.0461995i 0 8.04297 8.04297i 0 49.8797 0 80.9957i 0
95.5 0 3.91498 3.91498i 0 −4.72348 + 4.72348i 0 −45.3712 0 50.3458i 0
95.6 0 5.54016 5.54016i 0 −21.7374 + 21.7374i 0 6.62054 0 19.6133i 0
95.7 0 11.5209 11.5209i 0 14.6016 14.6016i 0 24.0210 0 184.461i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.7
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
16.f Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{14} + \cdots\) acting on \(S_{5}^{\mathrm{new}}(128, [\chi])\).