Properties

Label 2880.2.t.c
Level $2880$
Weight $2$
Character orbit 2880.t
Analytic conductor $22.997$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.t (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} - 112 x^{6} - 64 x^{5} + 112 x^{4} + 256 x^{2} - 512 x + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{5} + ( -\beta_{1} + \beta_{4} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{7} +O(q^{10})\) \( q + \beta_{5} q^{5} + ( -\beta_{1} + \beta_{4} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{7} + ( -1 + \beta_{5} - \beta_{9} + \beta_{10} ) q^{11} + ( 1 - \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{13} + ( 1 - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{17} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{8} + \beta_{12} - \beta_{13} ) q^{19} + ( \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} + 3 \beta_{8} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{23} -\beta_{8} q^{25} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{29} + ( \beta_{1} + \beta_{3} - \beta_{6} - \beta_{7} + \beta_{12} + \beta_{13} ) q^{31} + ( \beta_{3} + \beta_{12} ) q^{35} + ( -1 - \beta_{2} - \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{11} + \beta_{14} ) q^{37} + ( \beta_{4} - \beta_{8} - \beta_{11} - 2 \beta_{12} ) q^{41} + ( -1 - 4 \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} ) q^{43} + ( -2 + \beta_{1} + \beta_{2} - 3 \beta_{5} + \beta_{6} + \beta_{9} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{47} + ( -4 + 4 \beta_{1} + \beta_{2} - 2 \beta_{6} - 2 \beta_{9} + 2 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} + 2 \beta_{15} ) q^{49} + ( -2 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{14} ) q^{53} + ( \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{55} + ( -2 + \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{11} + 2 \beta_{14} ) q^{59} + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{8} - 3 \beta_{11} + 2 \beta_{12} + 3 \beta_{14} + 2 \beta_{15} ) q^{61} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{10} + \beta_{12} + \beta_{14} + \beta_{15} ) q^{65} + ( -5 + 7 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - 2 \beta_{11} + 4 \beta_{12} - \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{67} + ( -6 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} - 4 \beta_{12} + 2 \beta_{13} - 2 \beta_{15} ) q^{71} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{15} ) q^{73} + ( -2 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - 2 \beta_{11} - 3 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{77} + ( -2 - 3 \beta_{1} - \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{79} + ( 1 + 3 \beta_{1} + \beta_{5} - \beta_{6} - \beta_{7} + 4 \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{83} + ( -2 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{85} + ( 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{10} + 4 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{15} ) q^{89} + ( -5 + \beta_{2} + \beta_{4} + 7 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{9} - 3 \beta_{10} + 4 \beta_{11} + 4 \beta_{14} ) q^{91} + ( 3 - \beta_{1} - \beta_{2} + \beta_{6} + \beta_{9} - \beta_{10} - \beta_{12} + \beta_{13} ) q^{95} + ( 1 + \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{9} - 3 \beta_{10} + \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 8q^{11} + 8q^{19} + 16q^{29} - 16q^{37} - 8q^{43} - 40q^{47} - 16q^{49} - 16q^{53} - 8q^{59} + 16q^{61} - 40q^{67} - 16q^{77} - 16q^{79} + 40q^{83} - 16q^{85} - 32q^{91} + 32q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} - 112 x^{6} - 64 x^{5} + 112 x^{4} + 256 x^{2} - 512 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -163 \nu^{15} + 58 \nu^{14} + 376 \nu^{13} + 568 \nu^{12} - 501 \nu^{11} - 2502 \nu^{10} - 632 \nu^{9} + 3284 \nu^{8} + 6101 \nu^{7} - 1962 \nu^{6} - 10212 \nu^{5} - 4496 \nu^{4} + 4544 \nu^{3} + 16768 \nu^{2} - 6656 \nu - 7296 \)\()/2688\)
\(\beta_{2}\)\(=\)\((\)\( -9 \nu^{15} - 80 \nu^{14} + 152 \nu^{13} + 168 \nu^{12} + 65 \nu^{11} - 740 \nu^{10} - 1008 \nu^{9} + 1260 \nu^{8} + 2359 \nu^{7} + 1444 \nu^{6} - 5092 \nu^{5} - 4728 \nu^{4} + 2944 \nu^{3} + 5856 \nu^{2} + 7168 \nu - 12288 \)\()/128\)
\(\beta_{3}\)\(=\)\((\)\( 41 \nu^{15} + 80 \nu^{14} - 264 \nu^{13} - 312 \nu^{12} + 47 \nu^{11} + 1380 \nu^{10} + 1312 \nu^{9} - 2204 \nu^{8} - 4071 \nu^{7} - 1316 \nu^{6} + 8052 \nu^{5} + 6376 \nu^{4} - 4208 \nu^{3} - 9952 \nu^{2} - 7680 \nu + 15488 \)\()/384\)
\(\beta_{4}\)\(=\)\((\)\(303 \nu^{15} - 1892 \nu^{14} + 1724 \nu^{13} + 1784 \nu^{12} + 3497 \nu^{11} - 6864 \nu^{10} - 19332 \nu^{9} + 11948 \nu^{8} + 27583 \nu^{7} + 39328 \nu^{6} - 63120 \nu^{5} - 81576 \nu^{4} + 32528 \nu^{3} + 58720 \nu^{2} + 166592 \nu - 209536\)\()/2688\)
\(\beta_{5}\)\(=\)\((\)\( 397 \nu^{15} - 502 \nu^{14} - 772 \nu^{13} - 664 \nu^{12} + 2523 \nu^{11} + 5226 \nu^{10} - 3292 \nu^{9} - 9932 \nu^{8} - 10139 \nu^{7} + 16854 \nu^{6} + 24480 \nu^{5} - 4720 \nu^{4} - 21776 \nu^{3} - 40576 \nu^{2} + 42176 \nu + 8832 \)\()/2688\)
\(\beta_{6}\)\(=\)\((\)\(-655 \nu^{15} + 60 \nu^{14} + 2516 \nu^{13} + 2480 \nu^{12} - 3641 \nu^{11} - 14664 \nu^{10} - 3308 \nu^{9} + 27372 \nu^{8} + 36625 \nu^{7} - 15880 \nu^{6} - 84768 \nu^{5} - 33296 \nu^{4} + 64416 \nu^{3} + 114624 \nu^{2} - 6592 \nu - 123008\)\()/2688\)
\(\beta_{7}\)\(=\)\((\)\(-715 \nu^{15} + 2724 \nu^{14} - 1240 \nu^{13} - 1648 \nu^{12} - 7037 \nu^{11} + 2592 \nu^{10} + 27040 \nu^{9} - 2292 \nu^{8} - 22475 \nu^{7} - 67264 \nu^{6} + 45252 \nu^{5} + 102304 \nu^{4} - 7248 \nu^{3} - 26112 \nu^{2} - 239872 \nu + 227968\)\()/2688\)
\(\beta_{8}\)\(=\)\((\)\( 396 \nu^{15} - 1201 \nu^{14} + 256 \nu^{13} + 460 \nu^{12} + 3508 \nu^{11} + 489 \nu^{10} - 11700 \nu^{9} - 2228 \nu^{8} + 6320 \nu^{7} + 32015 \nu^{6} - 9900 \nu^{5} - 42864 \nu^{4} - 5840 \nu^{3} - 1504 \nu^{2} + 108736 \nu - 83072 \)\()/1344\)
\(\beta_{9}\)\(=\)\((\)\( -137 \nu^{15} + 466 \nu^{14} - 144 \nu^{13} - 264 \nu^{12} - 1303 \nu^{11} + 106 \nu^{10} + 4672 \nu^{9} + 492 \nu^{8} - 3289 \nu^{7} - 12506 \nu^{6} + 4972 \nu^{5} + 17552 \nu^{4} + 1864 \nu^{3} - 1184 \nu^{2} - 41984 \nu + 32000 \)\()/448\)
\(\beta_{10}\)\(=\)\((\)\( -935 \nu^{15} + 3056 \nu^{14} - 676 \nu^{13} - 1552 \nu^{12} - 8961 \nu^{11} - 636 \nu^{10} + 30236 \nu^{9} + 5644 \nu^{8} - 17975 \nu^{7} - 81876 \nu^{6} + 25272 \nu^{5} + 108608 \nu^{4} + 16480 \nu^{3} + 1280 \nu^{2} - 270016 \nu + 203136 \)\()/2688\)
\(\beta_{11}\)\(=\)\((\)\( -1011 \nu^{15} + 2684 \nu^{14} - 164 \nu^{13} - 632 \nu^{12} - 8357 \nu^{11} - 3480 \nu^{10} + 25260 \nu^{9} + 9028 \nu^{8} - 7939 \nu^{7} - 73048 \nu^{6} + 8664 \nu^{5} + 88680 \nu^{4} + 19696 \nu^{3} + 21344 \nu^{2} - 240704 \nu + 166912 \)\()/2688\)
\(\beta_{12}\)\(=\)\((\)\(-1269 \nu^{15} + 3292 \nu^{14} + 236 \nu^{13} - 664 \nu^{12} - 11155 \nu^{11} - 6240 \nu^{10} + 31596 \nu^{9} + 17228 \nu^{8} - 5477 \nu^{7} - 95888 \nu^{6} - 4752 \nu^{5} + 107784 \nu^{4} + 43856 \nu^{3} + 44320 \nu^{2} - 296512 \nu + 176384\)\()/2688\)
\(\beta_{13}\)\(=\)\((\)\(-1583 \nu^{15} + 4222 \nu^{14} - 120 \nu^{13} - 1032 \nu^{12} - 13337 \nu^{11} - 5850 \nu^{10} + 40424 \nu^{9} + 15908 \nu^{8} - 12087 \nu^{7} - 117622 \nu^{6} + 9804 \nu^{5} + 141728 \nu^{4} + 39344 \nu^{3} + 38848 \nu^{2} - 383232 \nu + 252160\)\()/2688\)
\(\beta_{14}\)\(=\)\((\)\( 239 \nu^{15} - 554 \nu^{14} - 56 \nu^{13} + 16 \nu^{12} + 1833 \nu^{11} + 1254 \nu^{10} - 5096 \nu^{9} - 2764 \nu^{8} + 311 \nu^{7} + 15690 \nu^{6} + 1044 \nu^{5} - 17384 \nu^{4} - 6592 \nu^{3} - 8672 \nu^{2} + 50560 \nu - 29952 \)\()/384\)
\(\beta_{15}\)\(=\)\((\)\( -356 \nu^{15} + 1000 \nu^{14} - 139 \nu^{13} - 312 \nu^{12} - 3008 \nu^{11} - 856 \nu^{10} + 9619 \nu^{9} + 2572 \nu^{8} - 4104 \nu^{7} - 27068 \nu^{6} + 5701 \nu^{5} + 34644 \nu^{4} + 6408 \nu^{3} + 4944 \nu^{2} - 90864 \nu + 65248 \)\()/224\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} - 2 \beta_{14} - \beta_{12} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_{1} + 3\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{14} - \beta_{13} + \beta_{12} - 3 \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 3 \beta_{1} + 4\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{15} - 2 \beta_{13} + \beta_{12} - 2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{6} + \beta_{2} + 1\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{15} + \beta_{14} - 3 \beta_{13} + \beta_{12} - 3 \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_{1} + 2\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(3 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} + 9 \beta_{12} - 8 \beta_{11} + 2 \beta_{10} - 3 \beta_{9} + 5 \beta_{8} + \beta_{7} - 4 \beta_{6} - 3 \beta_{5} - \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 12 \beta_{1} - 13\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(2 \beta_{15} + 3 \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} - 4 \beta_{9} - 5 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + \beta_{2} - 2 \beta_{1} - 6\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(7 \beta_{15} + 4 \beta_{14} + 3 \beta_{12} - 6 \beta_{11} - 6 \beta_{10} - 3 \beta_{9} + 19 \beta_{8} + 9 \beta_{7} - 4 \beta_{6} + 3 \beta_{5} + \beta_{4} - 3 \beta_{3} + 5 \beta_{2} - 12 \beta_{1} - 19\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-2 \beta_{15} + 13 \beta_{14} + 3 \beta_{13} + 13 \beta_{12} + 13 \beta_{11} - 17 \beta_{10} - 3 \beta_{9} - 16 \beta_{8} + 8 \beta_{7} - 14 \beta_{6} - 11 \beta_{5} + 2 \beta_{4} + 11 \beta_{3} + 14 \beta_{2} + 11 \beta_{1} - 32\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-9 \beta_{15} - 6 \beta_{14} + 23 \beta_{13} - 16 \beta_{12} + 6 \beta_{11} - 7 \beta_{10} - 3 \beta_{9} - 16 \beta_{8} + 11 \beta_{7} - 2 \beta_{5} + 10 \beta_{4} - \beta_{3} - 3 \beta_{2} - 17 \beta_{1} - 13\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-4 \beta_{15} - 13 \beta_{14} + 17 \beta_{13} - 3 \beta_{12} + 13 \beta_{11} - 13 \beta_{10} + 3 \beta_{9} + 36 \beta_{8} - 10 \beta_{7} + 10 \beta_{6} + 7 \beta_{5} - 30 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - 51 \beta_{1} + 8\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(-31 \beta_{15} + 4 \beta_{14} + 84 \beta_{13} - 43 \beta_{12} + 30 \beta_{11} - 56 \beta_{10} + 35 \beta_{9} - 41 \beta_{8} + 69 \beta_{7} - 26 \beta_{6} - \beta_{5} + 37 \beta_{4} + 9 \beta_{3} + 31 \beta_{2} - 20 \beta_{1} + 23\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(-46 \beta_{15} - 44 \beta_{14} + 49 \beta_{13} - 52 \beta_{12} + 26 \beta_{11} - 14 \beta_{10} + 24 \beta_{9} - 88 \beta_{8} + \beta_{7} + 33 \beta_{6} + 6 \beta_{5} + \beta_{4} - 7 \beta_{3} - 28 \beta_{2} - 47 \beta_{1} + 23\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(-5 \beta_{15} - 38 \beta_{14} + 76 \beta_{13} - 17 \beta_{12} - 84 \beta_{11} + 88 \beta_{10} - 3 \beta_{9} + 139 \beta_{8} - 75 \beta_{7} + 36 \beta_{6} + 55 \beta_{5} - 75 \beta_{4} - 37 \beta_{3} - 33 \beta_{2} - 6 \beta_{1} + 139\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(-32 \beta_{15} + 9 \beta_{14} - 23 \beta_{13} - 81 \beta_{12} + 27 \beta_{11} - 9 \beta_{10} + 95 \beta_{9} - 178 \beta_{8} + 50 \beta_{7} - 18 \beta_{6} - 61 \beta_{5} - 14 \beta_{4} - 33 \beta_{3} + 66 \beta_{2} - 131 \beta_{1} + 68\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(-45 \beta_{15} + 4 \beta_{14} + 82 \beta_{13} - 85 \beta_{12} - 58 \beta_{11} + 11 \beta_{10} + 77 \beta_{9} - 108 \beta_{8} + 125 \beta_{6} + 24 \beta_{5} - 4 \beta_{4} - 32 \beta_{3} - 89 \beta_{2} + 32 \beta_{1} + 63\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{8}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
−0.966675 1.03225i
1.38652 0.278517i
−1.39563 + 0.228522i
1.26868 0.624862i
−0.530822 + 1.31081i
−0.296075 1.38287i
1.32070 + 0.505727i
1.21331 0.726558i
1.26868 + 0.624862i
−1.39563 0.228522i
1.38652 + 0.278517i
−0.966675 + 1.03225i
1.21331 + 0.726558i
1.32070 0.505727i
−0.296075 + 1.38287i
−0.530822 1.31081i
0 0 0 −0.707107 + 0.707107i 0 1.73696i 0 0 0
721.2 0 0 0 −0.707107 + 0.707107i 0 0.982011i 0 0 0
721.3 0 0 0 −0.707107 + 0.707107i 0 0.690576i 0 0 0
721.4 0 0 0 −0.707107 + 0.707107i 0 4.02840i 0 0 0
721.5 0 0 0 0.707107 0.707107i 0 2.73482i 0 0 0
721.6 0 0 0 0.707107 0.707107i 0 2.66881i 0 0 0
721.7 0 0 0 0.707107 0.707107i 0 2.89402i 0 0 0
721.8 0 0 0 0.707107 0.707107i 0 4.50961i 0 0 0
2161.1 0 0 0 −0.707107 0.707107i 0 4.02840i 0 0 0
2161.2 0 0 0 −0.707107 0.707107i 0 0.690576i 0 0 0
2161.3 0 0 0 −0.707107 0.707107i 0 0.982011i 0 0 0
2161.4 0 0 0 −0.707107 0.707107i 0 1.73696i 0 0 0
2161.5 0 0 0 0.707107 + 0.707107i 0 4.50961i 0 0 0
2161.6 0 0 0 0.707107 + 0.707107i 0 2.89402i 0 0 0
2161.7 0 0 0 0.707107 + 0.707107i 0 2.66881i 0 0 0
2161.8 0 0 0 0.707107 + 0.707107i 0 2.73482i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2161.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.t.c 16
3.b odd 2 1 320.2.l.a 16
4.b odd 2 1 720.2.t.c 16
12.b even 2 1 80.2.l.a 16
15.d odd 2 1 1600.2.l.i 16
15.e even 4 1 1600.2.q.g 16
15.e even 4 1 1600.2.q.h 16
16.e even 4 1 inner 2880.2.t.c 16
16.f odd 4 1 720.2.t.c 16
24.f even 2 1 640.2.l.b 16
24.h odd 2 1 640.2.l.a 16
48.i odd 4 1 320.2.l.a 16
48.i odd 4 1 640.2.l.a 16
48.k even 4 1 80.2.l.a 16
48.k even 4 1 640.2.l.b 16
60.h even 2 1 400.2.l.h 16
60.l odd 4 1 400.2.q.g 16
60.l odd 4 1 400.2.q.h 16
96.o even 8 1 5120.2.a.s 8
96.o even 8 1 5120.2.a.v 8
96.p odd 8 1 5120.2.a.t 8
96.p odd 8 1 5120.2.a.u 8
240.t even 4 1 400.2.l.h 16
240.z odd 4 1 400.2.q.g 16
240.bb even 4 1 1600.2.q.h 16
240.bd odd 4 1 400.2.q.h 16
240.bf even 4 1 1600.2.q.g 16
240.bm odd 4 1 1600.2.l.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.l.a 16 12.b even 2 1
80.2.l.a 16 48.k even 4 1
320.2.l.a 16 3.b odd 2 1
320.2.l.a 16 48.i odd 4 1
400.2.l.h 16 60.h even 2 1
400.2.l.h 16 240.t even 4 1
400.2.q.g 16 60.l odd 4 1
400.2.q.g 16 240.z odd 4 1
400.2.q.h 16 60.l odd 4 1
400.2.q.h 16 240.bd odd 4 1
640.2.l.a 16 24.h odd 2 1
640.2.l.a 16 48.i odd 4 1
640.2.l.b 16 24.f even 2 1
640.2.l.b 16 48.k even 4 1
720.2.t.c 16 4.b odd 2 1
720.2.t.c 16 16.f odd 4 1
1600.2.l.i 16 15.d odd 2 1
1600.2.l.i 16 240.bm odd 4 1
1600.2.q.g 16 15.e even 4 1
1600.2.q.g 16 240.bf even 4 1
1600.2.q.h 16 15.e even 4 1
1600.2.q.h 16 240.bb even 4 1
2880.2.t.c 16 1.a even 1 1 trivial
2880.2.t.c 16 16.e even 4 1 inner
5120.2.a.s 8 96.o even 8 1
5120.2.a.t 8 96.p odd 8 1
5120.2.a.u 8 96.p odd 8 1
5120.2.a.v 8 96.o even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(2880, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( ( 1 + T^{4} )^{4} \)
$7$ \( 204304 + 811008 T^{2} + 1033536 T^{4} + 549632 T^{6} + 145224 T^{8} + 20736 T^{10} + 1616 T^{12} + 64 T^{14} + T^{16} \)
$11$ \( 1290496 + 799744 T + 247808 T^{2} + 848384 T^{3} + 3958016 T^{4} + 3673856 T^{5} + 1795584 T^{6} + 446848 T^{7} + 139616 T^{8} + 82368 T^{9} + 40320 T^{10} + 9568 T^{11} + 1232 T^{12} + 80 T^{13} + 32 T^{14} + 8 T^{15} + T^{16} \)
$13$ \( 20647936 + 46530560 T + 52428800 T^{2} + 27869184 T^{3} + 9146368 T^{4} + 3137536 T^{5} + 2654208 T^{6} + 1400832 T^{7} + 415872 T^{8} + 49152 T^{9} + 8192 T^{10} + 4352 T^{11} + 1600 T^{12} + 128 T^{13} + T^{16} \)
$17$ \( ( 13888 + 5120 T - 7744 T^{2} - 1536 T^{3} + 1408 T^{4} + 64 T^{5} - 72 T^{6} + T^{8} )^{2} \)
$19$ \( 614656 - 4164608 T + 14108672 T^{2} - 26513920 T^{3} + 30308608 T^{4} - 19398912 T^{5} + 7595520 T^{6} - 1921408 T^{7} + 731744 T^{8} - 363072 T^{9} + 132480 T^{10} - 27488 T^{11} + 3216 T^{12} - 176 T^{13} + 32 T^{14} - 8 T^{15} + T^{16} \)
$23$ \( 1731856 + 5740288 T^{2} + 5719232 T^{4} + 2620928 T^{6} + 622088 T^{8} + 77504 T^{10} + 4784 T^{12} + 128 T^{14} + T^{16} \)
$29$ \( 3017085184 + 4042700800 T + 2708480000 T^{2} + 456489984 T^{3} - 5714688 T^{4} - 1816064 T^{5} + 37230592 T^{6} + 2768128 T^{7} - 199840 T^{8} - 351616 T^{9} + 198144 T^{10} - 18624 T^{11} + 1104 T^{12} - 288 T^{13} + 128 T^{14} - 16 T^{15} + T^{16} \)
$31$ \( ( -20224 - 58368 T - 26112 T^{2} + 4096 T^{3} + 2848 T^{4} - 64 T^{5} - 96 T^{6} + T^{8} )^{2} \)
$37$ \( 18939904 + 236191744 T + 1472724992 T^{2} + 2707357696 T^{3} + 2705047552 T^{4} + 1380728832 T^{5} + 381124608 T^{6} + 40185856 T^{7} + 22554112 T^{8} + 9370624 T^{9} + 2144256 T^{10} + 249088 T^{11} + 16320 T^{12} + 704 T^{13} + 128 T^{14} + 16 T^{15} + T^{16} \)
$41$ \( 110660014336 + 325786435584 T^{2} + 63304144128 T^{4} + 5024061440 T^{6} + 206041952 T^{8} + 4686848 T^{10} + 59088 T^{12} + 384 T^{14} + T^{16} \)
$43$ \( 53640976 - 331044800 T + 1021520000 T^{2} - 1500385792 T^{3} + 1191507520 T^{4} - 280880992 T^{5} + 26364032 T^{6} + 1854880 T^{7} + 5444360 T^{8} - 950192 T^{9} + 76832 T^{10} + 18816 T^{11} + 5328 T^{12} - 440 T^{13} + 32 T^{14} + 8 T^{15} + T^{16} \)
$47$ \( ( 575044 + 693376 T + 244768 T^{2} + 584 T^{3} - 14936 T^{4} - 2392 T^{5} - 8 T^{6} + 20 T^{7} + T^{8} )^{2} \)
$53$ \( 383725735936 + 196640112640 T + 50384076800 T^{2} + 10552958976 T^{3} + 26109784064 T^{4} + 13942091776 T^{5} + 3861454848 T^{6} + 498471936 T^{7} + 43316352 T^{8} + 6504448 T^{9} + 1892352 T^{10} + 236416 T^{11} + 15552 T^{12} + 448 T^{13} + 128 T^{14} + 16 T^{15} + T^{16} \)
$59$ \( 12227051776 - 30451745792 T + 37920376832 T^{2} - 27384356352 T^{3} + 12226302208 T^{4} - 3038184192 T^{5} + 314344960 T^{6} + 20658560 T^{7} + 7403616 T^{8} - 2562240 T^{9} + 291200 T^{10} + 65632 T^{11} + 8464 T^{12} - 272 T^{13} + 32 T^{14} + 8 T^{15} + T^{16} \)
$61$ \( 1393986371584 + 1092943347712 T + 428456542208 T^{2} - 95024578560 T^{3} + 27549499392 T^{4} + 9643622400 T^{5} + 2332164096 T^{6} - 240730112 T^{7} + 17876992 T^{8} + 4136960 T^{9} + 1425408 T^{10} - 176640 T^{11} + 11520 T^{12} + 384 T^{13} + 128 T^{14} - 16 T^{15} + T^{16} \)
$67$ \( 46120451769616 + 26534918246592 T + 7633293466752 T^{2} + 2087109786752 T^{3} + 1043974444608 T^{4} + 475604544352 T^{5} + 148072661120 T^{6} + 31795927072 T^{7} + 4867387016 T^{8} + 537576688 T^{9} + 43098400 T^{10} + 2604960 T^{11} + 147664 T^{12} + 10680 T^{13} + 800 T^{14} + 40 T^{15} + T^{16} \)
$71$ \( 3333516427264 + 2007385505792 T^{2} + 408856297472 T^{4} + 33695596544 T^{6} + 1144522752 T^{8} + 18817024 T^{10} + 157440 T^{12} + 640 T^{14} + T^{16} \)
$73$ \( 15847788544 + 28362989568 T^{2} + 17757564928 T^{4} + 4377603072 T^{6} + 321195136 T^{8} + 9035648 T^{10} + 108992 T^{12} + 560 T^{14} + T^{16} \)
$79$ \( ( 4352 - 31232 T - 61952 T^{2} + 4992 T^{3} + 5856 T^{4} - 352 T^{5} - 160 T^{6} + 8 T^{7} + T^{8} )^{2} \)
$83$ \( 2050640656 + 6041972416 T + 8900981888 T^{2} + 5315507456 T^{3} + 1230069056 T^{4} - 140621856 T^{5} + 1135671424 T^{6} + 379938272 T^{7} + 64272392 T^{8} - 9891472 T^{9} + 315168 T^{10} + 36928 T^{11} + 37520 T^{12} - 7976 T^{13} + 800 T^{14} - 40 T^{15} + T^{16} \)
$89$ \( 684153962496 + 380947267584 T^{2} + 69045698560 T^{4} + 5734359040 T^{6} + 244188672 T^{8} + 5576192 T^{10} + 68032 T^{12} + 416 T^{14} + T^{16} \)
$97$ \( ( -8549312 + 7621376 T - 1675968 T^{2} - 78720 T^{3} + 47936 T^{4} - 416 T^{5} - 440 T^{6} + T^{8} )^{2} \)
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