Properties

Label 4016.2.a.k
Level 4016
Weight 2
Character orbit 4016.a
Self dual yes
Analytic conductor 32.068
Analytic rank 0
Dimension 17
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 251)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + \beta_{6} q^{5} + \beta_{13} q^{7} + ( 1 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{4} q^{3} + \beta_{6} q^{5} + \beta_{13} q^{7} + ( 1 + \beta_{3} ) q^{9} + ( 1 + \beta_{1} + \beta_{6} - \beta_{11} - \beta_{16} ) q^{11} + ( 1 - \beta_{1} + \beta_{2} - \beta_{6} - \beta_{10} ) q^{13} + ( \beta_{3} - \beta_{4} - \beta_{10} + \beta_{11} - \beta_{15} ) q^{15} + ( -1 - \beta_{1} + \beta_{5} + \beta_{6} - \beta_{15} ) q^{17} + ( -2 - \beta_{1} + \beta_{2} - \beta_{6} + \beta_{11} + \beta_{12} + \beta_{16} ) q^{19} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{16} ) q^{21} + ( -1 + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{14} ) q^{23} + ( 2 - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{15} ) q^{25} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{9} + \beta_{14} ) q^{27} + ( 2 + \beta_{7} + \beta_{8} - \beta_{14} ) q^{29} + ( \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{31} + ( \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} + 2 \beta_{15} ) q^{33} + ( 2 - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{11} + \beta_{13} - 2 \beta_{14} ) q^{35} + ( -\beta_{1} + 2 \beta_{4} + 3 \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{14} + \beta_{16} ) q^{37} + ( -\beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + \beta_{10} - \beta_{11} - \beta_{16} ) q^{39} + ( 1 + \beta_{1} - \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{41} + ( \beta_{1} + \beta_{3} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{12} + \beta_{14} ) q^{43} + ( -2 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - 3 \beta_{14} + \beta_{16} ) q^{45} + ( 2 - \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{13} - \beta_{14} ) q^{47} + ( 5 + 2 \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{8} + \beta_{9} - \beta_{11} + \beta_{15} - \beta_{16} ) q^{49} + ( -1 - \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{14} ) q^{51} + ( -\beta_{2} + \beta_{6} - \beta_{8} - \beta_{14} - \beta_{15} - \beta_{16} ) q^{53} + ( -1 + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{11} + \beta_{15} + \beta_{16} ) q^{55} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{14} - \beta_{16} ) q^{57} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{9} - \beta_{10} + \beta_{14} - 2 \beta_{15} + \beta_{16} ) q^{59} + ( 7 + 2 \beta_{1} - \beta_{3} - 3 \beta_{5} - 2 \beta_{6} - \beta_{8} + 3 \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{14} + \beta_{15} ) q^{61} + ( 5 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{13} + 3 \beta_{15} ) q^{63} + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} + \beta_{13} + \beta_{16} ) q^{65} + ( -\beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{67} + ( -1 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{8} - 2 \beta_{9} + \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} + \beta_{16} ) q^{69} + ( 4 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{9} - 3 \beta_{11} + \beta_{15} - \beta_{16} ) q^{71} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + \beta_{7} - 2 \beta_{11} - \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{73} + ( 2 + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{75} + ( -3 - \beta_{1} + 2 \beta_{4} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} + 2 \beta_{16} ) q^{77} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{13} - \beta_{15} ) q^{79} + ( \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 3 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{11} - 2 \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} ) q^{81} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{15} + \beta_{16} ) q^{83} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{85} + ( -2 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{9} - \beta_{12} + 2 \beta_{14} - 2 \beta_{15} ) q^{87} + ( 4 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{11} - \beta_{12} + 2 \beta_{14} - 2 \beta_{16} ) q^{89} + ( -1 - \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} + 3 \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{15} + \beta_{16} ) q^{91} + ( 4 + 2 \beta_{1} + \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} ) q^{93} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - 3 \beta_{8} + 2 \beta_{11} + \beta_{12} - 3 \beta_{14} ) q^{95} + ( -3 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - 3 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{14} + 2 \beta_{16} ) q^{97} + ( 6 + 4 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{16} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 3q^{5} - 3q^{7} + 25q^{9} + O(q^{10}) \) \( 17q + 3q^{5} - 3q^{7} + 25q^{9} + q^{11} + 22q^{13} + 8q^{15} - q^{17} - 13q^{19} + 25q^{21} + 2q^{23} + 32q^{25} + 15q^{27} + 28q^{29} - 12q^{31} - 16q^{33} + 15q^{35} + 27q^{37} - 13q^{39} - q^{41} - 9q^{43} - 7q^{45} + 20q^{47} + 32q^{49} + 2q^{51} + q^{53} + 11q^{55} - 24q^{57} + 20q^{59} + 59q^{61} + 41q^{63} - 14q^{65} - 15q^{67} + 38q^{69} + 26q^{71} + 8q^{73} + 20q^{75} - 33q^{79} + 29q^{81} + 67q^{85} + 11q^{87} + 11q^{89} + 2q^{91} + 28q^{93} + 8q^{95} - 10q^{97} + 36q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + 6186 x^{9} - 9216 x^{8} - 11921 x^{7} + 13680 x^{6} + 13752 x^{5} - 9400 x^{4} - 8800 x^{3} + 1920 x^{2} + 2240 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{2}\)\(=\)\((\)\(-22 \nu^{16} + 15 \nu^{15} + 647 \nu^{14} - 374 \nu^{13} - 7676 \nu^{12} + 3513 \nu^{11} + 46913 \nu^{10} - 14575 \nu^{9} - 156321 \nu^{8} + 19540 \nu^{7} + 276564 \nu^{6} + 28861 \nu^{5} - 231693 \nu^{4} - 78844 \nu^{3} + 59404 \nu^{2} + 34224 \nu + 4304\)\()/304\)
\(\beta_{3}\)\(=\)\((\)\(-85 \nu^{16} + 178 \nu^{15} + 2264 \nu^{14} - 4694 \nu^{13} - 23617 \nu^{12} + 48558 \nu^{11} + 120979 \nu^{10} - 246626 \nu^{9} - 312422 \nu^{8} + 626264 \nu^{7} + 384541 \nu^{6} - 723064 \nu^{5} - 234500 \nu^{4} + 335368 \nu^{3} + 66416 \nu^{2} - 42336 \nu - 3200\)\()/1216\)
\(\beta_{4}\)\(=\)\((\)\(-69 \nu^{16} + 212 \nu^{15} + 1752 \nu^{14} - 5638 \nu^{13} - 17157 \nu^{12} + 59424 \nu^{11} + 79979 \nu^{10} - 313888 \nu^{9} - 174414 \nu^{8} + 866052 \nu^{7} + 146661 \nu^{6} - 1198330 \nu^{5} - 54216 \nu^{4} + 779784 \nu^{3} + 55216 \nu^{2} - 183520 \nu - 27776\)\()/1216\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{16} - \nu^{15} + 31 \nu^{14} + 30 \nu^{13} - 391 \nu^{12} - 363 \nu^{11} + 2570 \nu^{10} + 2275 \nu^{9} - 9331 \nu^{8} - 7882 \nu^{7} + 18119 \nu^{6} + 14855 \nu^{5} - 16325 \nu^{4} - 13706 \nu^{3} + 4320 \nu^{2} + 4336 \nu + 448 \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\(-42 \nu^{16} - 37 \nu^{15} + 1306 \nu^{14} + 1148 \nu^{13} - 16568 \nu^{12} - 14297 \nu^{11} + 109772 \nu^{10} + 91723 \nu^{9} - 401886 \nu^{8} - 323534 \nu^{7} + 783066 \nu^{6} + 619761 \nu^{5} - 694624 \nu^{4} - 587700 \nu^{3} + 170912 \nu^{2} + 190336 \nu + 23776\)\()/608\)
\(\beta_{7}\)\(=\)\((\)\(-235 \nu^{16} + 244 \nu^{15} + 6608 \nu^{14} - 6066 \nu^{13} - 74651 \nu^{12} + 58072 \nu^{11} + 432413 \nu^{10} - 261584 \nu^{9} - 1360330 \nu^{8} + 517908 \nu^{7} + 2278651 \nu^{6} - 214814 \nu^{5} - 1860528 \nu^{4} - 368928 \nu^{3} + 508704 \nu^{2} + 220000 \nu + 19264\)\()/1216\)
\(\beta_{8}\)\(=\)\((\)\(-55 \nu^{16} + 66 \nu^{15} + 1551 \nu^{14} - 1676 \nu^{13} - 17575 \nu^{12} + 16544 \nu^{11} + 102092 \nu^{10} - 78608 \nu^{9} - 321823 \nu^{8} + 176682 \nu^{7} + 539429 \nu^{6} - 143716 \nu^{5} - 441435 \nu^{4} - 12012 \nu^{3} + 125672 \nu^{2} + 30688 \nu + 1184\)\()/304\)
\(\beta_{9}\)\(=\)\((\)\(217 \nu^{16} - 420 \nu^{15} - 5804 \nu^{14} + 10814 \nu^{13} + 61465 \nu^{12} - 109080 \nu^{11} - 326267 \nu^{10} + 539504 \nu^{9} + 908994 \nu^{8} - 1330756 \nu^{7} - 1298289 \nu^{6} + 1475730 \nu^{5} + 933932 \nu^{4} - 616832 \nu^{3} - 264000 \nu^{2} + 60736 \nu + 8384\)\()/1216\)
\(\beta_{10}\)\(=\)\((\)\(273 \nu^{16} - 54 \nu^{15} - 8052 \nu^{14} + 670 \nu^{13} + 96501 \nu^{12} + 3222 \nu^{11} - 601779 \nu^{10} - 92994 \nu^{9} + 2071842 \nu^{8} + 586448 \nu^{7} - 3830305 \nu^{6} - 1609452 \nu^{5} + 3345112 \nu^{4} + 1879504 \nu^{3} - 907248 \nu^{2} - 668096 \nu - 63040\)\()/1216\)
\(\beta_{11}\)\(=\)\((\)\(81 \nu^{16} - 120 \nu^{15} - 2231 \nu^{14} + 3068 \nu^{13} + 24529 \nu^{12} - 30574 \nu^{11} - 136930 \nu^{10} + 147646 \nu^{9} + 409343 \nu^{8} - 344458 \nu^{7} - 642219 \nu^{6} + 322886 \nu^{5} + 496127 \nu^{4} - 58112 \nu^{3} - 132396 \nu^{2} - 24512 \nu - 3424\)\()/304\)
\(\beta_{12}\)\(=\)\((\)\(-81 \nu^{16} + 120 \nu^{15} + 2231 \nu^{14} - 3068 \nu^{13} - 24529 \nu^{12} + 30574 \nu^{11} + 136930 \nu^{10} - 147646 \nu^{9} - 409343 \nu^{8} + 344458 \nu^{7} + 642219 \nu^{6} - 322886 \nu^{5} - 496127 \nu^{4} + 58112 \nu^{3} + 132396 \nu^{2} + 25120 \nu + 3424\)\()/304\)
\(\beta_{13}\)\(=\)\((\)\(-112 \nu^{16} + 85 \nu^{15} + 3204 \nu^{14} - 2018 \nu^{13} - 36974 \nu^{12} + 17893 \nu^{11} + 219968 \nu^{10} - 68557 \nu^{9} - 715256 \nu^{8} + 72676 \nu^{7} + 1242448 \nu^{6} + 185079 \nu^{5} - 1036154 \nu^{4} - 411582 \nu^{3} + 273948 \nu^{2} + 173264 \nu + 17904\)\()/304\)
\(\beta_{14}\)\(=\)\((\)\(455 \nu^{16} - 508 \nu^{15} - 12812 \nu^{14} + 12770 \nu^{13} + 144951 \nu^{12} - 124248 \nu^{11} - 840781 \nu^{10} + 576016 \nu^{9} + 2647622 \nu^{8} - 1224636 \nu^{7} - 4436367 \nu^{6} + 789678 \nu^{5} + 3625052 \nu^{4} + 418192 \nu^{3} - 1002880 \nu^{2} - 348832 \nu - 31296\)\()/1216\)
\(\beta_{15}\)\(=\)\((\)\(-483 \nu^{16} + 268 \nu^{15} + 13936 \nu^{14} - 5874 \nu^{13} - 162659 \nu^{12} + 44784 \nu^{11} + 982565 \nu^{10} - 107976 \nu^{9} - 3259674 \nu^{8} - 264940 \nu^{7} + 5799123 \nu^{6} + 1718170 \nu^{5} - 4930848 \nu^{4} - 2477584 \nu^{3} + 1307936 \nu^{2} + 940640 \nu + 98624\)\()/1216\)
\(\beta_{16}\)\(=\)\((\)\(-332 \nu^{16} + 159 \nu^{15} + 9636 \nu^{14} - 3364 \nu^{13} - 113202 \nu^{12} + 23535 \nu^{11} + 688604 \nu^{10} - 33845 \nu^{9} - 2300544 \nu^{8} - 307434 \nu^{7} + 4115096 \nu^{6} + 1400433 \nu^{5} - 3494786 \nu^{4} - 1908228 \nu^{3} + 903176 \nu^{2} + 714624 \nu + 80704\)\()/608\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{12} + \beta_{11}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1} + 4\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{12} + 3 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{6} - 2 \beta_{5} + 2\)\()/2\)
\(\nu^{4}\)\(=\)\(-\beta_{14} - \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + 7 \beta_{1} + 23\)
\(\nu^{5}\)\(=\)\((\)\(-2 \beta_{14} - 4 \beta_{13} + 29 \beta_{12} + 11 \beta_{11} - 22 \beta_{10} + 18 \beta_{9} - 20 \beta_{8} + 2 \beta_{7} - 18 \beta_{6} - 22 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + 4 \beta_{1} + 18\)\()/2\)
\(\nu^{6}\)\(=\)\(-\beta_{16} + \beta_{15} - 12 \beta_{14} + 2 \beta_{13} - 10 \beta_{11} - 12 \beta_{10} + 12 \beta_{9} - 24 \beta_{8} - 12 \beta_{7} - 12 \beta_{6} - 14 \beta_{5} + \beta_{3} + 48 \beta_{1} + 147\)
\(\nu^{7}\)\(=\)\((\)\(-2 \beta_{16} + 6 \beta_{15} - 30 \beta_{14} - 52 \beta_{13} + 177 \beta_{12} + 37 \beta_{11} - 192 \beta_{10} + 144 \beta_{9} - 174 \beta_{8} + 26 \beta_{7} - 144 \beta_{6} - 200 \beta_{5} - 52 \beta_{4} + 50 \beta_{3} + 4 \beta_{2} + 58 \beta_{1} + 162\)\()/2\)
\(\nu^{8}\)\(=\)\(-16 \beta_{16} + 18 \beta_{15} - 112 \beta_{14} + 28 \beta_{13} + 2 \beta_{12} - 82 \beta_{11} - 110 \beta_{10} + 114 \beta_{9} - 226 \beta_{8} - 110 \beta_{7} - 108 \beta_{6} - 144 \beta_{5} + 2 \beta_{4} + 14 \beta_{3} + 2 \beta_{2} + 339 \beta_{1} + 986\)
\(\nu^{9}\)\(=\)\((\)\(-30 \beta_{16} + 102 \beta_{15} - 322 \beta_{14} - 496 \beta_{13} + 1113 \beta_{12} + 67 \beta_{11} - 1546 \beta_{10} + 1126 \beta_{9} - 1448 \beta_{8} + 250 \beta_{7} - 1118 \beta_{6} - 1694 \beta_{5} - 484 \beta_{4} + 466 \beta_{3} + 68 \beta_{2} + 614 \beta_{1} + 1436\)\()/2\)
\(\nu^{10}\)\(=\)\(-176 \beta_{16} + 222 \beta_{15} - 953 \beta_{14} + 272 \beta_{13} + 34 \beta_{12} - 647 \beta_{11} - 923 \beta_{10} + 1007 \beta_{9} - 1956 \beta_{8} - 911 \beta_{7} - 885 \beta_{6} - 1321 \beta_{5} + 50 \beta_{4} + 146 \beta_{3} + 38 \beta_{2} + 2461 \beta_{1} + 6825\)
\(\nu^{11}\)\(=\)\((\)\(-314 \beta_{16} + 1214 \beta_{15} - 3060 \beta_{14} - 4220 \beta_{13} + 7161 \beta_{12} - 557 \beta_{11} - 11994 \beta_{10} + 8786 \beta_{9} - 11814 \beta_{8} + 2120 \beta_{7} - 8586 \beta_{6} - 13870 \beta_{5} - 3944 \beta_{4} + 3898 \beta_{3} + 784 \beta_{2} + 5778 \beta_{1} + 12424\)\()/2\)
\(\nu^{12}\)\(=\)\(-1665 \beta_{16} + 2343 \beta_{15} - 7772 \beta_{14} + 2270 \beta_{13} + 390 \beta_{12} - 5088 \beta_{11} - 7458 \beta_{10} + 8622 \beta_{9} - 16278 \beta_{8} - 7178 \beta_{7} - 6984 \beta_{6} - 11498 \beta_{5} + 734 \beta_{4} + 1375 \beta_{3} + 474 \beta_{2} + 18270 \beta_{1} + 48397\)
\(\nu^{13}\)\(=\)\((\)\(-2872 \beta_{16} + 12548 \beta_{15} - 27424 \beta_{14} - 33972 \beta_{13} + 46965 \beta_{12} - 9867 \beta_{11} - 91272 \beta_{10} + 68804 \beta_{9} - 95412 \beta_{8} + 16732 \beta_{7} - 65684 \beta_{6} - 111572 \beta_{5} - 29944 \beta_{4} + 30964 \beta_{3} + 7696 \beta_{2} + 51344 \beta_{1} + 105308\)\()/2\)
\(\nu^{14}\)\(=\)\(-14572 \beta_{16} + 22760 \beta_{15} - 62088 \beta_{14} + 17404 \beta_{13} + 3808 \beta_{12} - 40160 \beta_{11} - 59188 \beta_{10} + 72596 \beta_{9} - 132704 \beta_{8} - 55032 \beta_{7} - 54300 \beta_{6} - 97396 \beta_{5} + 8556 \beta_{4} + 12332 \beta_{3} + 4968 \beta_{2} + 138009 \beta_{1} + 349916\)
\(\nu^{15}\)\(=\)\((\)\(-24712 \beta_{16} + 120508 \beta_{15} - 237700 \beta_{14} - 265304 \beta_{13} + 313117 \beta_{12} - 106437 \beta_{11} - 687462 \beta_{10} + 541310 \beta_{9} - 766130 \beta_{8} + 125856 \beta_{7} - 502070 \beta_{6} - 888926 \beta_{5} - 217320 \beta_{4} + 239312 \beta_{3} + 69500 \beta_{2} + 441796 \beta_{1} + 879042\)\()/2\)
\(\nu^{16}\)\(=\)\(-121836 \beta_{16} + 210152 \beta_{15} - 491137 \beta_{14} + 125988 \beta_{13} + 34252 \beta_{12} - 318049 \beta_{11} - 465593 \beta_{10} + 604689 \beta_{9} - 1069530 \beta_{8} - 415321 \beta_{7} - 420437 \beta_{6} - 812357 \beta_{5} + 88088 \beta_{4} + 107424 \beta_{3} + 47652 \beta_{2} + 1056139 \beta_{1} + 2569939\)

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
−2.32547
−0.139956
1.37907
1.37191
−0.787554
2.64128
−2.27410
2.18124
−2.51582
1.84638
2.33247
−1.09599
−0.622810
−2.65791
2.82015
−0.932399
0.779516
0 −3.27059 0 −2.03057 0 −1.64874 0 7.69673 0
1.2 0 −2.55238 0 2.96478 0 −0.820250 0 3.51462 0
1.3 0 −2.46273 0 1.31548 0 3.48956 0 3.06502 0
1.4 0 −2.36731 0 −1.32016 0 −4.19189 0 2.60414 0
1.5 0 −1.89108 0 −0.652808 0 −1.12991 0 0.576182 0
1.6 0 −1.66988 0 −3.99830 0 2.26241 0 −0.211508 0
1.7 0 −0.935470 0 3.41593 0 −3.69332 0 −2.12490 0
1.8 0 −0.923498 0 1.20531 0 1.96022 0 −2.14715 0
1.9 0 0.505139 0 −3.97764 0 −1.36760 0 −2.74483 0
1.10 0 0.508487 0 3.51373 0 −0.924114 0 −2.74144 0
1.11 0 0.826533 0 −1.37815 0 −4.67534 0 −2.31684 0
1.12 0 1.16074 0 4.05107 0 4.08218 0 −1.65268 0
1.13 0 1.30185 0 −1.69081 0 −3.93205 0 −1.30520 0
1.14 0 2.62368 0 1.15029 0 2.51426 0 3.88372 0
1.15 0 2.95607 0 2.29008 0 2.82675 0 5.73835 0
1.16 0 3.04458 0 −3.52221 0 4.32039 0 6.26949 0
1.17 0 3.14584 0 1.66398 0 −2.07256 0 6.89631 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4016.2.a.k 17
4.b odd 2 1 251.2.a.b 17
12.b even 2 1 2259.2.a.k 17
20.d odd 2 1 6275.2.a.e 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
251.2.a.b 17 4.b odd 2 1
2259.2.a.k 17 12.b even 2 1
4016.2.a.k 17 1.a even 1 1 trivial
6275.2.a.e 17 20.d odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(251\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{17} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).