Properties

Label 2880.2.t.d
Level $2880$
Weight $2$
Character orbit 2880.t
Analytic conductor $22.997$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

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: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 2 x^{18} - 4 x^{17} + 7 x^{16} + 16 x^{15} + 6 x^{14} - 36 x^{13} - 42 x^{12} + 40 x^{11} + 136 x^{10} + 80 x^{9} - 168 x^{8} - 288 x^{7} + 96 x^{6} + 512 x^{5} + 448 x^{4} - 512 x^{3} - 512 x^{2} + 1024\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{5} -\beta_{13} q^{7} +O(q^{10})\) \( q + \beta_{5} q^{5} -\beta_{13} q^{7} + ( -\beta_{5} + \beta_{8} ) q^{11} + \beta_{14} q^{13} + ( 1 + \beta_{19} ) q^{17} + ( -\beta_{6} - \beta_{9} + \beta_{10} - \beta_{12} ) q^{19} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{16} - \beta_{17} ) q^{23} -\beta_{3} q^{25} + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{13} - \beta_{14} - \beta_{16} - \beta_{18} - \beta_{19} ) q^{29} + ( -1 - \beta_{1} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{14} + \beta_{15} + \beta_{18} ) q^{31} -\beta_{16} q^{35} + ( 1 + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{8} + \beta_{17} - \beta_{18} ) q^{37} + ( \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{13} ) q^{41} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{15} - \beta_{17} - \beta_{19} ) q^{43} + ( -\beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{47} + ( -2 - 2 \beta_{1} + 2 \beta_{5} + \beta_{7} + \beta_{10} + \beta_{14} + \beta_{15} + \beta_{18} ) q^{49} + ( 1 - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{13} + \beta_{15} - \beta_{17} + \beta_{18} ) q^{53} + ( \beta_{2} + \beta_{3} ) q^{55} + ( -1 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} - \beta_{18} ) q^{59} + ( -2 \beta_{1} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} - \beta_{16} ) q^{61} -\beta_{10} q^{65} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{13} + \beta_{14} + \beta_{16} + \beta_{19} ) q^{67} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{71} + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{16} - \beta_{17} ) q^{73} + ( 3 + \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} + 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{16} - \beta_{18} ) q^{77} + ( -3 + \beta_{1} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{10} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{79} + ( -1 - 4 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{7} - \beta_{9} + \beta_{10} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{16} + \beta_{18} ) q^{83} + ( \beta_{5} + \beta_{11} ) q^{85} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{89} + ( 2 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{13} - 2 \beta_{15} - 2 \beta_{18} - 2 \beta_{19} ) q^{91} + ( -\beta_{14} - \beta_{15} - \beta_{18} - \beta_{19} ) q^{95} + ( 4 + 2 \beta_{7} - \beta_{14} - \beta_{15} - 2 \beta_{16} - 2 \beta_{17} - 2 \beta_{19} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + O(q^{10}) \) \( 20q + 8q^{11} + 24q^{17} + 4q^{19} - 16q^{29} + 16q^{37} + 8q^{43} - 52q^{49} + 16q^{53} - 16q^{59} - 4q^{61} + 8q^{67} + 40q^{77} - 56q^{79} - 48q^{83} + 4q^{85} + 8q^{91} + 56q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 2 x^{18} - 4 x^{17} + 7 x^{16} + 16 x^{15} + 6 x^{14} - 36 x^{13} - 42 x^{12} + 40 x^{11} + 136 x^{10} + 80 x^{9} - 168 x^{8} - 288 x^{7} + 96 x^{6} + 512 x^{5} + 448 x^{4} - 512 x^{3} - 512 x^{2} + 1024\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 18 \nu^{19} + 33 \nu^{18} + 8 \nu^{17} - 104 \nu^{16} - 94 \nu^{15} + 199 \nu^{14} + 616 \nu^{13} + 364 \nu^{12} - 768 \nu^{11} - 1426 \nu^{10} + 152 \nu^{9} + 3332 \nu^{8} + 3808 \nu^{7} - 1024 \nu^{6} - 5536 \nu^{5} - 2976 \nu^{4} + 7040 \nu^{3} + 8448 \nu^{2} + 2816 \nu - 5632 \)\()/512\)
\(\beta_{2}\)\(=\)\((\)\(5 \nu^{19} - 22 \nu^{18} - 84 \nu^{17} - 112 \nu^{16} + 107 \nu^{15} + 334 \nu^{14} + 24 \nu^{13} - 1216 \nu^{12} - 1906 \nu^{11} - 100 \nu^{10} + 3180 \nu^{9} + 2992 \nu^{8} - 4208 \nu^{7} - 11728 \nu^{6} - 7520 \nu^{5} + 6912 \nu^{4} + 13952 \nu^{3} - 3456 \nu^{2} - 22016 \nu - 23552\)\()/512\)
\(\beta_{3}\)\(=\)\((\)\(-21 \nu^{19} + 10 \nu^{18} + 104 \nu^{17} + 192 \nu^{16} - 123 \nu^{15} - 594 \nu^{14} - 364 \nu^{13} + 1376 \nu^{12} + 2786 \nu^{11} + 652 \nu^{10} - 4436 \nu^{9} - 5696 \nu^{8} + 3232 \nu^{7} + 15248 \nu^{6} + 11424 \nu^{5} - 7936 \nu^{4} - 21760 \nu^{3} - 128 \nu^{2} + 24064 \nu + 28672\)\()/512\)
\(\beta_{4}\)\(=\)\((\)\( 16 \nu^{19} + 10 \nu^{18} - 47 \nu^{17} - 138 \nu^{16} + 20 \nu^{15} + 394 \nu^{14} + 479 \nu^{13} - 518 \nu^{12} - 1776 \nu^{11} - 1048 \nu^{10} + 2342 \nu^{9} + 4684 \nu^{8} + 492 \nu^{7} - 8104 \nu^{6} - 8944 \nu^{5} + 2336 \nu^{4} + 14816 \nu^{3} + 6080 \nu^{2} - 9856 \nu - 16896 \)\()/256\)
\(\beta_{5}\)\(=\)\((\)\( -31 \nu^{19} - 39 \nu^{18} + 30 \nu^{17} + 204 \nu^{16} + 67 \nu^{15} - 497 \nu^{14} - 946 \nu^{13} + 64 \nu^{12} + 2082 \nu^{11} + 2086 \nu^{10} - 2008 \nu^{9} - 6636 \nu^{8} - 3800 \nu^{7} + 7264 \nu^{6} + 11680 \nu^{5} + 736 \nu^{4} - 17728 \nu^{3} - 11776 \nu^{2} + 5120 \nu + 17920 \)\()/512\)
\(\beta_{6}\)\(=\)\((\)\(62 \nu^{19} + 3 \nu^{18} - 252 \nu^{17} - 544 \nu^{16} + 222 \nu^{15} + 1605 \nu^{14} + 1324 \nu^{13} - 3188 \nu^{12} - 7456 \nu^{11} - 2566 \nu^{10} + 11248 \nu^{9} + 16412 \nu^{8} - 5808 \nu^{7} - 39264 \nu^{6} - 32352 \nu^{5} + 19232 \nu^{4} + 59392 \nu^{3} + 5632 \nu^{2} - 61696 \nu - 76288\)\()/512\)
\(\beta_{7}\)\(=\)\((\)\( -23 \nu^{19} - 44 \nu^{18} - 6 \nu^{17} + 140 \nu^{16} + 135 \nu^{15} - 276 \nu^{14} - 854 \nu^{13} - 516 \nu^{12} + 1070 \nu^{11} + 2064 \nu^{10} - 112 \nu^{9} - 4608 \nu^{8} - 5544 \nu^{7} + 1120 \nu^{6} + 7872 \nu^{5} + 4672 \nu^{4} - 8768 \nu^{3} - 12416 \nu^{2} - 4608 \nu + 6144 \)\()/256\)
\(\beta_{8}\)\(=\)\((\)\(11 \nu^{19} - 65 \nu^{18} - 202 \nu^{17} - 220 \nu^{16} + 337 \nu^{15} + 777 \nu^{14} - 210 \nu^{13} - 3024 \nu^{12} - 4042 \nu^{11} + 698 \nu^{10} + 7648 \nu^{9} + 5452 \nu^{8} - 11480 \nu^{7} - 26464 \nu^{6} - 13216 \nu^{5} + 18336 \nu^{4} + 28864 \nu^{3} - 13312 \nu^{2} - 48640 \nu - 45568\)\()/512\)
\(\beta_{9}\)\(=\)\((\)\( -12 \nu^{19} + 9 \nu^{18} + 67 \nu^{17} + 116 \nu^{16} - 84 \nu^{15} - 365 \nu^{14} - 179 \nu^{13} + 920 \nu^{12} + 1724 \nu^{11} + 290 \nu^{10} - 2854 \nu^{9} - 3332 \nu^{8} + 2516 \nu^{7} + 9824 \nu^{6} + 6736 \nu^{5} - 5568 \nu^{4} - 13408 \nu^{3} + 1088 \nu^{2} + 16256 \nu + 17920 \)\()/128\)
\(\beta_{10}\)\(=\)\((\)\(-31 \nu^{19} - 50 \nu^{18} + 2 \nu^{17} + 184 \nu^{16} + 135 \nu^{15} - 382 \nu^{14} - 1022 \nu^{13} - 456 \nu^{12} + 1494 \nu^{11} + 2380 \nu^{10} - 560 \nu^{9} - 5808 \nu^{8} - 5992 \nu^{7} + 2704 \nu^{6} + 9664 \nu^{5} + 4736 \nu^{4} - 12096 \nu^{3} - 13824 \nu^{2} - 3840 \nu + 9728\)\()/256\)
\(\beta_{11}\)\(=\)\((\)\(-66 \nu^{19} + 37 \nu^{18} + 350 \nu^{17} + 636 \nu^{16} - 418 \nu^{15} - 1989 \nu^{14} - 1182 \nu^{13} + 4712 \nu^{12} + 9368 \nu^{11} + 2094 \nu^{10} - 15092 \nu^{9} - 19124 \nu^{8} + 11400 \nu^{7} + 51984 \nu^{6} + 38656 \nu^{5} - 27232 \nu^{4} - 73536 \nu^{3} - 128 \nu^{2} + 82944 \nu + 98304\)\()/512\)
\(\beta_{12}\)\(=\)\((\)\(79 \nu^{19} - 21 \nu^{18} - 364 \nu^{17} - 736 \nu^{16} + 389 \nu^{15} + 2197 \nu^{14} + 1568 \nu^{13} - 4812 \nu^{12} - 10386 \nu^{11} - 3062 \nu^{10} + 16004 \nu^{9} + 21892 \nu^{8} - 10352 \nu^{7} - 55984 \nu^{6} - 44224 \nu^{5} + 27488 \nu^{4} + 81920 \nu^{3} + 2944 \nu^{2} - 88320 \nu - 108544\)\()/512\)
\(\beta_{13}\)\(=\)\((\)\(-33 \nu^{19} + 22 \nu^{18} + 184 \nu^{17} + 328 \nu^{16} - 223 \nu^{15} - 1006 \nu^{14} - 556 \nu^{13} + 2504 \nu^{12} + 4842 \nu^{11} + 996 \nu^{10} - 7812 \nu^{9} - 9568 \nu^{8} + 6496 \nu^{7} + 27280 \nu^{6} + 19872 \nu^{5} - 14592 \nu^{4} - 38016 \nu^{3} + 1536 \nu^{2} + 44800 \nu + 52736\)\()/256\)
\(\beta_{14}\)\(=\)\((\)\(-41 \nu^{19} - 75 \nu^{18} - 24 \nu^{17} + 228 \nu^{16} + 221 \nu^{15} - 421 \nu^{14} - 1428 \nu^{13} - 968 \nu^{12} + 1598 \nu^{11} + 3366 \nu^{10} + 44 \nu^{9} - 7452 \nu^{8} - 9488 \nu^{7} + 960 \nu^{6} + 12128 \nu^{5} + 8352 \nu^{4} - 14208 \nu^{3} - 20352 \nu^{2} - 10496 \nu + 10240\)\()/256\)
\(\beta_{15}\)\(=\)\((\)\(-56 \nu^{19} - 73 \nu^{18} + 46 \nu^{17} + 368 \nu^{16} + 148 \nu^{15} - 855 \nu^{14} - 1734 \nu^{13} - 52 \nu^{12} + 3588 \nu^{11} + 3930 \nu^{10} - 3068 \nu^{9} - 11716 \nu^{8} - 7560 \nu^{7} + 11776 \nu^{6} + 20704 \nu^{5} + 2720 \nu^{4} - 30144 \nu^{3} - 21632 \nu^{2} + 7168 \nu + 30720\)\()/256\)
\(\beta_{16}\)\(=\)\((\)\(-57 \nu^{19} - 99 \nu^{18} - 6 \nu^{17} + 336 \nu^{16} + 269 \nu^{15} - 693 \nu^{14} - 1926 \nu^{13} - 932 \nu^{12} + 2694 \nu^{11} + 4478 \nu^{10} - 1000 \nu^{9} - 10836 \nu^{8} - 11400 \nu^{7} + 4816 \nu^{6} + 18272 \nu^{5} + 8352 \nu^{4} - 22976 \nu^{3} - 26368 \nu^{2} - 6144 \nu + 18432\)\()/256\)
\(\beta_{17}\)\(=\)\((\)\(-70 \nu^{19} - 99 \nu^{18} + 48 \nu^{17} + 456 \nu^{16} + 218 \nu^{15} - 1045 \nu^{14} - 2240 \nu^{13} - 244 \nu^{12} + 4368 \nu^{11} + 5110 \nu^{10} - 3480 \nu^{9} - 14668 \nu^{8} - 10336 \nu^{7} + 13440 \nu^{6} + 25888 \nu^{5} + 4512 \nu^{4} - 36736 \nu^{3} - 28800 \nu^{2} + 6400 \nu + 36352\)\()/256\)
\(\beta_{18}\)\(=\)\((\)\(38 \nu^{19} + 57 \nu^{18} - 16 \nu^{17} - 234 \nu^{16} - 130 \nu^{15} + 527 \nu^{14} + 1208 \nu^{13} + 286 \nu^{12} - 2120 \nu^{11} - 2730 \nu^{10} + 1504 \nu^{9} + 7560 \nu^{8} + 6080 \nu^{7} - 5832 \nu^{6} - 12864 \nu^{5} - 3168 \nu^{4} + 18112 \nu^{3} + 15680 \nu^{2} - 896 \nu - 16512\)\()/128\)
\(\beta_{19}\)\(=\)\((\)\(177 \nu^{19} + 256 \nu^{18} - 86 \nu^{17} - 1108 \nu^{16} - 577 \nu^{15} + 2496 \nu^{14} + 5594 \nu^{13} + 1004 \nu^{12} - 10306 \nu^{11} - 12712 \nu^{10} + 7728 \nu^{9} + 35792 \nu^{8} + 27064 \nu^{7} - 30432 \nu^{6} - 62400 \nu^{5} - 13312 \nu^{4} + 88512 \nu^{3} + 72704 \nu^{2} - 9472 \nu - 86528\)\()/512\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{19} + \beta_{16} + \beta_{15} + \beta_{8} - \beta_{7} + \beta_{3} - \beta_{2} + \beta_{1} - 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{17} - \beta_{15} + \beta_{14} - \beta_{12} - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{2} - 2 \beta_{1} + 1\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{16} + \beta_{15} - 2 \beta_{13} + 2 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{4} - \beta_{2} + 3 \beta_{1} + 2\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{18} - 2 \beta_{17} + \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} + \beta_{8} + 2 \beta_{6} + 4 \beta_{5} - \beta_{3} - 2 \beta_{2} - 2 \beta_{1} - 4\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{19} + \beta_{18} + \beta_{17} - \beta_{14} - \beta_{13} + 2 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 2 \beta_{2} - 6 \beta_{1} - 6\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-4 \beta_{19} - 5 \beta_{17} - 2 \beta_{16} + \beta_{15} - \beta_{14} + 2 \beta_{13} + \beta_{12} + 2 \beta_{10} + \beta_{9} - 2 \beta_{8} + 3 \beta_{7} - \beta_{6} + 6 \beta_{5} - 14 \beta_{3} + \beta_{2} + 8 \beta_{1} - 9\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-2 \beta_{19} - 4 \beta_{18} + 10 \beta_{17} - 9 \beta_{16} - \beta_{15} - 6 \beta_{14} - 4 \beta_{13} - 2 \beta_{12} + 4 \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + 8 \beta_{6} - 12 \beta_{5} + 5 \beta_{4} - \beta_{2} - 5 \beta_{1} - 8\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(7 \beta_{18} + 2 \beta_{17} + 3 \beta_{16} + \beta_{15} - 3 \beta_{14} + 9 \beta_{13} + 4 \beta_{12} - 9 \beta_{11} + \beta_{10} + 6 \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 28 \beta_{5} - 5 \beta_{3} + 4 \beta_{2} + 10 \beta_{1} + 2\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-3 \beta_{19} + 5 \beta_{18} + 5 \beta_{17} - 14 \beta_{16} + 6 \beta_{15} - \beta_{14} - \beta_{13} - 8 \beta_{12} - 6 \beta_{11} + 17 \beta_{10} - 7 \beta_{9} + 4 \beta_{8} - 15 \beta_{6} - 37 \beta_{5} - 2 \beta_{4} - 8 \beta_{3} + 12 \beta_{2} - 4 \beta_{1} - 8\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(-4 \beta_{19} - 12 \beta_{18} + 5 \beta_{17} + 2 \beta_{16} - 9 \beta_{15} + \beta_{14} + 6 \beta_{13} - 5 \beta_{12} - 12 \beta_{11} - 10 \beta_{10} + 19 \beta_{9} + 2 \beta_{8} + \beta_{7} + 17 \beta_{6} - 34 \beta_{5} - 8 \beta_{4} - 10 \beta_{3} - \beta_{2} + 44 \beta_{1} + 37\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(6 \beta_{19} + 16 \beta_{18} + 6 \beta_{17} - 11 \beta_{16} + 5 \beta_{15} - 2 \beta_{14} - 20 \beta_{13} - 18 \beta_{12} - 24 \beta_{11} + 25 \beta_{10} - 5 \beta_{9} + 9 \beta_{8} - 5 \beta_{7} - 20 \beta_{6} + 72 \beta_{5} + 7 \beta_{4} + 44 \beta_{3} - 15 \beta_{2} + 13 \beta_{1} + 32\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(16 \beta_{19} + 21 \beta_{18} - 2 \beta_{17} + 53 \beta_{16} - 25 \beta_{15} + 43 \beta_{14} + 19 \beta_{13} + 4 \beta_{12} + \beta_{11} + 19 \beta_{10} - 10 \beta_{9} + 17 \beta_{8} - 46 \beta_{7} - 42 \beta_{6} - 108 \beta_{5} - 40 \beta_{4} - 15 \beta_{3} + 24 \beta_{2} - 2 \beta_{1} + 2\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(-37 \beta_{19} - 41 \beta_{18} - 73 \beta_{17} + 10 \beta_{16} - 18 \beta_{15} - 27 \beta_{14} - 27 \beta_{13} - 16 \beta_{12} - 18 \beta_{11} - 29 \beta_{10} + 11 \beta_{9} - 16 \beta_{8} + 36 \beta_{7} + 43 \beta_{6} - 15 \beta_{5} - 22 \beta_{4} + 124 \beta_{3} - 40 \beta_{2} - 72 \beta_{1} + 148\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(28 \beta_{19} - 12 \beta_{18} - 21 \beta_{17} + 46 \beta_{16} - 15 \beta_{15} + 63 \beta_{14} + 10 \beta_{13} + 37 \beta_{12} + 4 \beta_{11} - 22 \beta_{10} - 19 \beta_{9} - 2 \beta_{8} - 57 \beta_{7} - 33 \beta_{6} + 82 \beta_{5} - 8 \beta_{4} + 34 \beta_{3} - 63 \beta_{2} + 92 \beta_{1} - 269\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(-46 \beta_{19} + 48 \beta_{18} - 38 \beta_{17} + 3 \beta_{16} - 85 \beta_{15} + 18 \beta_{14} - 36 \beta_{13} + 34 \beta_{12} + 64 \beta_{11} + 7 \beta_{10} + 21 \beta_{9} + 7 \beta_{8} + 69 \beta_{7} - 36 \beta_{6} - 24 \beta_{5} - 7 \beta_{4} + 52 \beta_{3} + 7 \beta_{2} - 317 \beta_{1} + 112\)\()/4\)
\(\nu^{16}\)\(=\)\((\)\(64 \beta_{19} - 69 \beta_{18} - 30 \beta_{17} + 123 \beta_{16} + 105 \beta_{15} - 35 \beta_{14} + 29 \beta_{13} + 156 \beta_{12} + 87 \beta_{11} - 155 \beta_{10} + 42 \beta_{9} - 145 \beta_{8} - 98 \beta_{7} + 138 \beta_{6} + 44 \beta_{5} - 32 \beta_{4} - 49 \beta_{3} - 72 \beta_{2} - 230 \beta_{1} - 218\)\()/4\)
\(\nu^{17}\)\(=\)\((\)\(-51 \beta_{19} - 175 \beta_{18} - 71 \beta_{17} - 170 \beta_{16} - 110 \beta_{15} - 29 \beta_{14} + 227 \beta_{13} + 144 \beta_{12} - 38 \beta_{11} - 27 \beta_{10} - 11 \beta_{9} - 32 \beta_{8} + 244 \beta_{7} + 53 \beta_{6} + 47 \beta_{5} + 22 \beta_{4} - 12 \beta_{3} + 32 \beta_{2} - 48 \beta_{1} - 268\)\()/4\)
\(\nu^{18}\)\(=\)\((\)\(124 \beta_{19} + 292 \beta_{18} + 357 \beta_{17} - 158 \beta_{16} + 375 \beta_{15} + 81 \beta_{14} - 194 \beta_{13} + 115 \beta_{12} + 172 \beta_{11} - 146 \beta_{10} + 195 \beta_{9} + 130 \beta_{8} + 65 \beta_{7} - 15 \beta_{6} - 74 \beta_{5} + 216 \beta_{4} + 126 \beta_{3} + 39 \beta_{2} - 12 \beta_{1} - 139\)\()/4\)
\(\nu^{19}\)\(=\)\((\)\(254 \beta_{19} - 112 \beta_{18} + 62 \beta_{17} - 171 \beta_{16} + 101 \beta_{15} - 58 \beta_{14} + 228 \beta_{13} - 26 \beta_{12} - 16 \beta_{11} + 177 \beta_{10} + 59 \beta_{9} - 223 \beta_{8} + 43 \beta_{7} - 36 \beta_{6} + 776 \beta_{5} - 129 \beta_{4} - 1748 \beta_{3} + 185 \beta_{2} - 171 \beta_{1} + 392\)\()/4\)

Character values

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

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{3}\) \(1\)

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
1.18701 0.768775i
−0.491956 + 1.32589i
−0.720859 1.21670i
1.15787 + 0.811989i
−1.13207 + 0.847599i
1.19834 + 0.750988i
−0.0861743 + 1.41159i
−1.04932 0.948122i
−1.38431 + 0.289262i
1.32147 0.503713i
−1.13207 0.847599i
1.15787 0.811989i
−0.720859 + 1.21670i
−0.491956 1.32589i
1.18701 + 0.768775i
1.32147 + 0.503713i
−1.38431 0.289262i
−1.04932 + 0.948122i
−0.0861743 1.41159i
1.19834 0.750988i
0 0 0 −0.707107 + 0.707107i 0 4.92824i 0 0 0
721.2 0 0 0 −0.707107 + 0.707107i 0 3.46600i 0 0 0
721.3 0 0 0 −0.707107 + 0.707107i 0 0.0588949i 0 0 0
721.4 0 0 0 −0.707107 + 0.707107i 0 2.18060i 0 0 0
721.5 0 0 0 −0.707107 + 0.707107i 0 4.27253i 0 0 0
721.6 0 0 0 0.707107 0.707107i 0 3.79862i 0 0 0
721.7 0 0 0 0.707107 0.707107i 0 2.76462i 0 0 0
721.8 0 0 0 0.707107 0.707107i 0 0.740019i 0 0 0
721.9 0 0 0 0.707107 0.707107i 0 2.60796i 0 0 0
721.10 0 0 0 0.707107 0.707107i 0 2.69529i 0 0 0
2161.1 0 0 0 −0.707107 0.707107i 0 4.27253i 0 0 0
2161.2 0 0 0 −0.707107 0.707107i 0 2.18060i 0 0 0
2161.3 0 0 0 −0.707107 0.707107i 0 0.0588949i 0 0 0
2161.4 0 0 0 −0.707107 0.707107i 0 3.46600i 0 0 0
2161.5 0 0 0 −0.707107 0.707107i 0 4.92824i 0 0 0
2161.6 0 0 0 0.707107 + 0.707107i 0 2.69529i 0 0 0
2161.7 0 0 0 0.707107 + 0.707107i 0 2.60796i 0 0 0
2161.8 0 0 0 0.707107 + 0.707107i 0 0.740019i 0 0 0
2161.9 0 0 0 0.707107 + 0.707107i 0 2.76462i 0 0 0
2161.10 0 0 0 0.707107 + 0.707107i 0 3.79862i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2161.10
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.d 20
3.b odd 2 1 960.2.s.c 20
4.b odd 2 1 720.2.t.d 20
12.b even 2 1 240.2.s.c 20
16.e even 4 1 inner 2880.2.t.d 20
16.f odd 4 1 720.2.t.d 20
24.f even 2 1 1920.2.s.e 20
24.h odd 2 1 1920.2.s.f 20
48.i odd 4 1 960.2.s.c 20
48.i odd 4 1 1920.2.s.f 20
48.k even 4 1 240.2.s.c 20
48.k even 4 1 1920.2.s.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.s.c 20 12.b even 2 1
240.2.s.c 20 48.k even 4 1
720.2.t.d 20 4.b odd 2 1
720.2.t.d 20 16.f odd 4 1
960.2.s.c 20 3.b odd 2 1
960.2.s.c 20 48.i odd 4 1
1920.2.s.e 20 24.f even 2 1
1920.2.s.e 20 48.k even 4 1
1920.2.s.f 20 24.h odd 2 1
1920.2.s.f 20 48.i odd 4 1
2880.2.t.d 20 1.a even 1 1 trivial
2880.2.t.d 20 16.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{20} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(2880, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( T^{20} \)
$5$ \( ( 1 + T^{4} )^{5} \)
$7$ \( 262144 + 76283904 T^{2} + 204587008 T^{4} + 145248256 T^{6} + 49590528 T^{8} + 9697664 T^{10} + 1161104 T^{12} + 86368 T^{14} + 3880 T^{16} + 96 T^{18} + T^{20} \)
$11$ \( 1048576 + 115343360 T + 6343884800 T^{2} - 15595208704 T^{3} + 18952028160 T^{4} - 11886264320 T^{5} + 4560683008 T^{6} - 1115578368 T^{7} + 251479040 T^{8} - 84426752 T^{9} + 30799872 T^{10} - 7170048 T^{11} + 1093696 T^{12} - 173952 T^{13} + 57984 T^{14} - 13696 T^{15} + 1808 T^{16} - 112 T^{17} + 32 T^{18} - 8 T^{19} + T^{20} \)
$13$ \( 167981940736 - 262517686272 T + 205127811072 T^{2} - 90035355648 T^{3} + 25928249344 T^{4} - 7034568704 T^{5} + 3460349952 T^{6} - 1511043072 T^{7} + 413145856 T^{8} - 60063232 T^{9} + 12435968 T^{10} - 5323392 T^{11} + 1636752 T^{12} - 175680 T^{13} + 10368 T^{14} - 4192 T^{15} + 2088 T^{16} - 144 T^{17} + T^{20} \)
$17$ \( ( -20032 - 151040 T - 145472 T^{2} + 106144 T^{3} + 25976 T^{4} - 15824 T^{5} - 676 T^{6} + 792 T^{7} - 34 T^{8} - 12 T^{9} + T^{10} )^{2} \)
$19$ \( 19716653056 - 59527397376 T + 89860866048 T^{2} - 68702330880 T^{3} + 29049219072 T^{4} - 7396435968 T^{5} + 9631844352 T^{6} - 7963704832 T^{7} + 3224447552 T^{8} + 73408640 T^{9} + 28386432 T^{10} - 20915520 T^{11} + 7892400 T^{12} + 209856 T^{13} + 29888 T^{14} - 17152 T^{15} + 5164 T^{16} + 72 T^{17} + 8 T^{18} - 4 T^{19} + T^{20} \)
$23$ \( 217558810624 + 1211058880512 T^{2} + 1734629007360 T^{4} + 505220718592 T^{6} + 64474080320 T^{8} + 4370425280 T^{10} + 169125872 T^{12} + 3828512 T^{14} + 49980 T^{16} + 348 T^{18} + T^{20} \)
$29$ \( 19723262623744 + 2510316109824 T + 159752650752 T^{2} + 1868828770304 T^{3} + 2276071636992 T^{4} + 779917459456 T^{5} + 169368092672 T^{6} + 67031334912 T^{7} + 46757634048 T^{8} + 18135728128 T^{9} + 4155932672 T^{10} + 541822976 T^{11} + 52284928 T^{12} + 7973888 T^{13} + 1863680 T^{14} + 231168 T^{15} + 15632 T^{16} + 576 T^{17} + 128 T^{18} + 16 T^{19} + T^{20} \)
$31$ \( ( -28698368 + 4962560 T + 6148800 T^{2} - 621056 T^{3} - 468112 T^{4} + 19536 T^{5} + 14932 T^{6} - 168 T^{7} - 204 T^{8} + T^{10} )^{2} \)
$37$ \( 18939904 + 171573248 T + 777125888 T^{2} + 1848475648 T^{3} + 2665852928 T^{4} + 2241855488 T^{5} + 1128316928 T^{6} + 422612992 T^{7} + 451001088 T^{8} + 394871296 T^{9} + 184033792 T^{10} + 12409216 T^{11} - 2713200 T^{12} - 581952 T^{13} + 678528 T^{14} - 81440 T^{15} + 4904 T^{16} - 80 T^{17} + 128 T^{18} - 16 T^{19} + T^{20} \)
$41$ \( 3311118843904 + 11279143534592 T^{2} + 4549133533184 T^{4} + 806185730048 T^{6} + 78819631104 T^{8} + 4616884224 T^{10} + 166691328 T^{12} + 3697152 T^{14} + 48592 T^{16} + 344 T^{18} + T^{20} \)
$43$ \( 3288334336 - 31474057216 T + 150625845248 T^{2} - 354645180416 T^{3} + 430934327296 T^{4} - 72320286720 T^{5} + 76928778240 T^{6} - 118577430528 T^{7} + 83002048512 T^{8} - 32131072000 T^{9} + 7850893312 T^{10} - 1219307520 T^{11} + 121272576 T^{12} - 8994816 T^{13} + 1246208 T^{14} - 208640 T^{15} + 20960 T^{16} - 704 T^{17} + 32 T^{18} - 8 T^{19} + T^{20} \)
$47$ \( ( -12544 + 145152 T + 283712 T^{2} + 35072 T^{3} - 97448 T^{4} - 11520 T^{5} + 7980 T^{6} + 160 T^{7} - 166 T^{8} + T^{10} )^{2} \)
$53$ \( 4398046511104 - 37383395344384 T + 158879430213632 T^{2} - 217797791580160 T^{3} + 164036679303168 T^{4} - 64499805585408 T^{5} + 15258273972224 T^{6} - 2110865276928 T^{7} + 243227688960 T^{8} - 46965587968 T^{9} + 11259117568 T^{10} - 1463160832 T^{11} + 113316096 T^{12} - 8978432 T^{13} + 2199552 T^{14} - 279040 T^{15} + 18720 T^{16} - 512 T^{17} + 128 T^{18} - 16 T^{19} + T^{20} \)
$59$ \( 16482977321058304 - 15913855322423296 T + 7682191945367552 T^{2} + 225773976551424 T^{3} - 19284448182272 T^{4} - 8752961814528 T^{5} + 18984867758080 T^{6} + 1690337058816 T^{7} + 60533785600 T^{8} + 4783368192 T^{9} + 14857752576 T^{10} + 1869032960 T^{11} + 117256256 T^{12} + 208256 T^{13} + 2543744 T^{14} + 332800 T^{15} + 21776 T^{16} - 112 T^{17} + 128 T^{18} + 16 T^{19} + T^{20} \)
$61$ \( 74350019584 - 90727790592 T + 55356622848 T^{2} + 405824488448 T^{3} + 635588871424 T^{4} + 400694880256 T^{5} + 145373775872 T^{6} + 32054490112 T^{7} + 8596122240 T^{8} + 3542995712 T^{9} + 1229804288 T^{10} + 246492544 T^{11} + 31363232 T^{12} + 3663488 T^{13} + 926848 T^{14} + 186688 T^{15} + 21588 T^{16} + 840 T^{17} + 8 T^{18} + 4 T^{19} + T^{20} \)
$67$ \( 210453397504 - 2284922601472 T + 12403865550848 T^{2} - 27655294418944 T^{3} + 28263770488832 T^{4} + 13407377948672 T^{5} + 5667880435712 T^{6} - 2867528204288 T^{7} + 724321959936 T^{8} + 392953856 T^{9} + 1266810880 T^{10} - 1069056000 T^{11} + 402550784 T^{12} + 10567680 T^{13} + 430080 T^{14} - 164864 T^{15} + 42176 T^{16} + 960 T^{17} + 32 T^{18} - 8 T^{19} + T^{20} \)
$71$ \( 84783728164864 + 286334329552896 T^{2} + 119444957298688 T^{4} + 19751533281280 T^{6} + 1633300643840 T^{8} + 72970174464 T^{10} + 1802900480 T^{12} + 24765952 T^{14} + 184320 T^{16} + 688 T^{18} + T^{20} \)
$73$ \( 1224592353918976 + 9210388927217664 T^{2} + 5934864196173824 T^{4} + 620748281806848 T^{6} + 26910405677056 T^{8} + 619035629568 T^{10} + 8344750848 T^{12} + 68179456 T^{14} + 332272 T^{16} + 888 T^{18} + T^{20} \)
$79$ \( ( 46268416 + 77168640 T - 50255872 T^{2} - 3687424 T^{3} + 3393408 T^{4} + 266432 T^{5} - 70364 T^{6} - 9352 T^{7} - 104 T^{8} + 28 T^{9} + T^{10} )^{2} \)
$83$ \( 761801736963751936 + 875555940458299392 T + 503148107752538112 T^{2} + 158089032079769600 T^{3} + 31122539714969600 T^{4} + 4222033294524416 T^{5} + 700248334794752 T^{6} + 152230226034688 T^{7} + 27414010302464 T^{8} + 3259538489344 T^{9} + 322663907328 T^{10} + 40555229184 T^{11} + 6124317440 T^{12} + 684328960 T^{13} + 51499008 T^{14} + 2781696 T^{15} + 181424 T^{16} + 15936 T^{17} + 1152 T^{18} + 48 T^{19} + T^{20} \)
$89$ \( 1518596177800462336 + 469438536991899648 T^{2} + 52248775078248448 T^{4} + 2876984841338880 T^{6} + 88667808735232 T^{8} + 1624662126592 T^{10} + 18159960576 T^{12} + 123870720 T^{14} + 500432 T^{16} + 1096 T^{18} + T^{20} \)
$97$ \( ( -37943296 + 87896064 T - 52876288 T^{2} + 5846528 T^{3} + 2172480 T^{4} - 392640 T^{5} - 22480 T^{6} + 6080 T^{7} - 44 T^{8} - 28 T^{9} + T^{10} )^{2} \)
show more
show less