Properties

Label 4020.2.q.k
Level 4020
Weight 2
Character orbit 4020.q
Analytic conductor 32.100
Analytic rank 0
Dimension 14
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4020.q (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} -\beta_{6} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{5} -\beta_{6} q^{7} + q^{9} + ( 1 - \beta_{5} - \beta_{10} - \beta_{12} ) q^{11} + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{13} - q^{15} + ( -\beta_{2} - 2 \beta_{5} - \beta_{11} + \beta_{13} ) q^{17} + ( -\beta_{2} + 2 \beta_{5} - \beta_{11} ) q^{19} + \beta_{6} q^{21} + ( -\beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{23} + q^{25} - q^{27} + ( -1 - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{29} + ( 1 + \beta_{1} - \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{31} + ( -1 + \beta_{5} + \beta_{10} + \beta_{12} ) q^{33} -\beta_{6} q^{35} + ( -2 \beta_{2} + \beta_{9} - 2 \beta_{11} - \beta_{12} ) q^{37} + ( \beta_{1} + \beta_{3} - \beta_{5} ) q^{39} + ( -2 - \beta_{4} + 2 \beta_{5} + \beta_{12} + \beta_{13} ) q^{41} + ( 1 - \beta_{2} - 2 \beta_{8} + 2 \beta_{9} ) q^{43} + q^{45} + ( 2 - 2 \beta_{1} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{47} + ( -\beta_{2} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{49} + ( \beta_{2} + 2 \beta_{5} + \beta_{11} - \beta_{13} ) q^{51} + ( \beta_{2} + \beta_{3} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{53} + ( 1 - \beta_{5} - \beta_{10} - \beta_{12} ) q^{55} + ( \beta_{2} - 2 \beta_{5} + \beta_{11} ) q^{57} + ( -1 - \beta_{3} + 2 \beta_{8} + \beta_{9} ) q^{59} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{12} + \beta_{13} ) q^{61} -\beta_{6} q^{63} + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{65} + ( 3 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{67} + ( \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{69} + ( 4 - \beta_{1} + 2 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} + \beta_{10} + 2 \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{71} + ( -\beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{73} - q^{75} + ( \beta_{1} + \beta_{3} + 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} + 2 \beta_{9} - 2 \beta_{12} + \beta_{13} ) q^{77} + ( 1 + \beta_{1} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{79} + q^{81} + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} - \beta_{13} ) q^{83} + ( -\beta_{2} - 2 \beta_{5} - \beta_{11} + \beta_{13} ) q^{85} + ( 1 + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{87} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} ) q^{89} + ( -4 - \beta_{3} + 2 \beta_{4} + \beta_{8} ) q^{91} + ( -1 - \beta_{1} + \beta_{5} + \beta_{6} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{93} + ( -\beta_{2} + 2 \beta_{5} - \beta_{11} ) q^{95} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} + \beta_{11} - \beta_{13} ) q^{97} + ( 1 - \beta_{5} - \beta_{10} - \beta_{12} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{3} + 14q^{5} + 3q^{7} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{3} + 14q^{5} + 3q^{7} + 14q^{9} + 6q^{11} + 9q^{13} - 14q^{15} - 12q^{17} + 14q^{19} - 3q^{21} + 6q^{23} + 14q^{25} - 14q^{27} - q^{29} + 7q^{31} - 6q^{33} + 3q^{35} - 2q^{37} - 9q^{39} - 18q^{41} - 6q^{43} + 14q^{45} + 7q^{47} + 12q^{51} - 12q^{53} + 6q^{55} - 14q^{57} - 2q^{59} + 3q^{63} + 9q^{65} + 25q^{67} - 6q^{69} + 22q^{71} - 15q^{73} - 14q^{75} + q^{77} + 9q^{79} + 14q^{81} - q^{83} - 12q^{85} + q^{87} - 12q^{89} - 38q^{91} - 7q^{93} + 14q^{95} + 16q^{97} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - x^{13} + 11 x^{12} - 8 x^{11} + 88 x^{10} - 57 x^{9} + 270 x^{8} + 17 x^{7} + 458 x^{6} - 101 x^{5} + 189 x^{4} - 30 x^{3} + 54 x^{2} - 12 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\(-2680061185 \nu^{13} + 4118856992 \nu^{12} - 29747186705 \nu^{11} + 35610122429 \nu^{10} - 234940121607 \nu^{9} + 266952713308 \nu^{8} - 708369775075 \nu^{7} + 250820920885 \nu^{6} - 934856019808 \nu^{5} + 984764699106 \nu^{4} - 308324347672 \nu^{3} + 320958040428 \nu^{2} - 76957145272 \nu + 970249939441\)\()/ 325648644013 \)
\(\beta_{3}\)\(=\)\((\)\(3347996299 \nu^{13} - 8708118669 \nu^{12} + 45065673273 \nu^{11} - 86278343802 \nu^{10} + 365843919170 \nu^{9} - 660716032257 \nu^{8} + 1437864427346 \nu^{7} - 1359823613067 \nu^{6} + 2035024146712 \nu^{5} - 2207859665815 \nu^{4} + 2602300698723 \nu^{3} - 717088584314 \nu^{2} + 171410592976 \nu - 194090246132\)\()/ 325648644013 \)
\(\beta_{4}\)\(=\)\((\)\(-10892153560 \nu^{13} + 9758640678 \nu^{12} - 105683331260 \nu^{11} + 64289364146 \nu^{10} - 815634907035 \nu^{9} + 455144021782 \nu^{8} - 1814707096630 \nu^{7} - 908790740124 \nu^{6} - 2130418252432 \nu^{5} + 2120608715454 \nu^{4} + 2831612352657 \nu^{3} + 695977413972 \nu^{2} - 167893036528 \nu + 861786952374\)\()/ 325648644013 \)
\(\beta_{5}\)\(=\)\((\)\(-48522561533 \nu^{13} + 45174565234 \nu^{12} - 525040058194 \nu^{11} + 343114818991 \nu^{10} - 4183707071102 \nu^{9} + 2399942088211 \nu^{8} - 12440375581653 \nu^{7} - 2262747973407 \nu^{6} - 20863509569047 \nu^{5} + 2865754568121 \nu^{4} - 6962904463922 \nu^{3} - 1146623852733 \nu^{2} - 1903129738468 \nu + 410860145420\)\()/ 651297288026 \)
\(\beta_{6}\)\(=\)\((\)\(-809674271 \nu^{13} - 252390382 \nu^{12} - 8122293536 \nu^{11} - 4713678201 \nu^{10} - 65999965440 \nu^{9} - 42770485013 \nu^{8} - 184014082317 \nu^{7} - 266207177033 \nu^{6} - 473551890151 \nu^{5} - 353375346337 \nu^{4} - 160280487010 \nu^{3} - 54504988027 \nu^{2} - 43501677200 \nu - 7944658082\)\()/ 8458406338 \)
\(\beta_{7}\)\(=\)\((\)\(37194535803 \nu^{13} - 47919288537 \nu^{12} + 395084796401 \nu^{11} - 400339609502 \nu^{10} + 3098569859104 \nu^{9} - 2969882527405 \nu^{8} + 8589017027582 \nu^{7} - 1664041174406 \nu^{6} + 11026504067640 \nu^{5} - 11470995916307 \nu^{4} - 2133066776665 \nu^{3} - 3744294132514 \nu^{2} + 898967472816 \nu - 841108812233\)\()/ 325648644013 \)
\(\beta_{8}\)\(=\)\((\)\(-45711237420 \nu^{13} + 74034574851 \nu^{12} - 518899664240 \nu^{11} + 668330017670 \nu^{10} - 4117338117621 \nu^{9} + 5030477527030 \nu^{8} - 12888553943330 \nu^{7} + 6408314158958 \nu^{6} - 17209864321680 \nu^{5} + 18222273890390 \nu^{4} - 5135681407065 \nu^{3} + 5935414947940 \nu^{2} - 1422383351760 \nu + 974875365383\)\()/ 325648644013 \)
\(\beta_{9}\)\(=\)\((\)\(-58435287502 \nu^{13} + 104277787850 \nu^{12} - 683843870764 \nu^{11} + 962775171505 \nu^{10} - 5449809996738 \nu^{9} + 7279578337109 \nu^{8} - 17910517000443 \nu^{7} + 10738605919044 \nu^{6} - 24249792781624 \nu^{5} + 25830617586941 \nu^{4} - 12925259206893 \nu^{3} + 8407638741310 \nu^{2} - 2013572275960 \nu + 2201747399207\)\()/ 325648644013 \)
\(\beta_{10}\)\(=\)\((\)\(126618741863 \nu^{13} - 49216475610 \nu^{12} + 1335886940970 \nu^{11} - 183514685193 \nu^{10} + 10733169032550 \nu^{9} - 585966489615 \nu^{8} + 31403882537963 \nu^{7} + 21731699467035 \nu^{6} + 63503082991395 \nu^{5} + 22467398449373 \nu^{4} + 21347976231450 \nu^{3} + 5457745711665 \nu^{2} + 2242764960296 \nu + 903683328390\)\()/ 651297288026 \)
\(\beta_{11}\)\(=\)\((\)\(-20029651747 \nu^{13} + 18183711674 \nu^{12} - 216517971596 \nu^{11} + 136874887445 \nu^{10} - 1725891567156 \nu^{9} + 952274405431 \nu^{8} - 5129198170687 \nu^{7} - 1041412251713 \nu^{6} - 8674402381075 \nu^{5} + 946819186593 \nu^{4} - 2896009242346 \nu^{3} - 490070050147 \nu^{2} - 793639274980 \nu - 101131348946\)\()/ 93042469718 \)
\(\beta_{12}\)\(=\)\((\)\(156907511177 \nu^{13} - 63210384410 \nu^{12} + 1644341611148 \nu^{11} - 250245646301 \nu^{10} + 13207909689396 \nu^{9} - 942157914055 \nu^{8} + 38225859873625 \nu^{7} + 26110314425969 \nu^{6} + 77748959678947 \nu^{5} + 23844336362505 \nu^{4} + 26131870142026 \nu^{3} + 6616282893811 \nu^{2} + 10832044605312 \nu + 1100661461810\)\()/ 651297288026 \)
\(\beta_{13}\)\(=\)\((\)\(-87738052524 \nu^{13} + 71009258532 \nu^{12} - 933363507764 \nu^{11} + 502153407798 \nu^{10} - 7424278972227 \nu^{9} + 3401605949923 \nu^{8} - 21440487873381 \nu^{7} - 6897421130211 \nu^{6} - 36599366732449 \nu^{5} + 1482620818260 \nu^{4} - 8695369936605 \nu^{3} - 1496669907681 \nu^{2} - 2171714647326 \nu + 438504860088\)\()/ 325648644013 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(\beta_{11} - 3 \beta_{5} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{9} - \beta_{8} + 3 \beta_{3} - \beta_{2} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{13} - \beta_{12} - 8 \beta_{11} - \beta_{6} + 18 \beta_{5} - \beta_{4} + \beta_{1} - 18\)
\(\nu^{5}\)\(=\)\(8 \beta_{12} + 10 \beta_{11} + \beta_{10} - 8 \beta_{9} + 9 \beta_{8} + \beta_{7} + 9 \beta_{6} - 12 \beta_{5} - 20 \beta_{3} + 10 \beta_{2} - 20 \beta_{1}\)
\(\nu^{6}\)\(=\)\(10 \beta_{9} - 10 \beta_{8} + \beta_{7} + 10 \beta_{4} + 13 \beta_{3} - 58 \beta_{2} + 119\)
\(\nu^{7}\)\(=\)\(-\beta_{13} - 58 \beta_{12} - 84 \beta_{11} - 11 \beta_{10} - 68 \beta_{6} + 117 \beta_{5} + \beta_{4} + 136 \beta_{1} - 117\)
\(\nu^{8}\)\(=\)\(-80 \beta_{13} + 84 \beta_{12} + 415 \beta_{11} - 12 \beta_{10} - 84 \beta_{9} + 83 \beta_{8} - 12 \beta_{7} + 83 \beta_{6} - 812 \beta_{5} - 127 \beta_{3} + 415 \beta_{2} - 127 \beta_{1}\)
\(\nu^{9}\)\(=\)\(415 \beta_{9} - 495 \beta_{8} - 92 \beta_{7} - 9 \beta_{4} + 939 \beta_{3} - 673 \beta_{2} + 1027\)
\(\nu^{10}\)\(=\)\(596 \beta_{13} - 673 \beta_{12} - 2975 \beta_{11} + 101 \beta_{10} - 664 \beta_{6} + 5649 \beta_{5} - 596 \beta_{4} + 1114 \beta_{1} - 5649\)
\(\nu^{11}\)\(=\)\(33 \beta_{13} + 2975 \beta_{12} + 5280 \beta_{11} + 697 \beta_{10} - 2975 \beta_{9} + 3571 \beta_{8} + 697 \beta_{7} + 3571 \beta_{6} - 8509 \beta_{5} - 6568 \beta_{3} + 5280 \beta_{2} - 6568 \beta_{1}\)
\(\nu^{12}\)\(=\)\(5280 \beta_{9} - 5247 \beta_{8} + 730 \beta_{7} + 4301 \beta_{4} + 9250 \beta_{3} - 21424 \beta_{2} + 39857\)
\(\nu^{13}\)\(=\)\(216 \beta_{13} - 21424 \beta_{12} - 40903 \beta_{11} - 5031 \beta_{10} - 25725 \beta_{6} + 68139 \beta_{5} - 216 \beta_{4} + 46436 \beta_{1} - 68139\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(-1 + \beta_{5}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
841.1
0.295122 0.511167i
−0.288886 + 0.500366i
1.10425 1.91261i
0.150493 0.260662i
1.24942 2.16407i
−1.36264 + 2.36015i
−0.647766 + 1.12196i
0.295122 + 0.511167i
−0.288886 0.500366i
1.10425 + 1.91261i
0.150493 + 0.260662i
1.24942 + 2.16407i
−1.36264 2.36015i
−0.647766 1.12196i
0 −1.00000 0 1.00000 0 −1.59463 + 2.76199i 0 1.00000 0
841.2 0 −1.00000 0 1.00000 0 −1.13810 + 1.97125i 0 1.00000 0
841.3 0 −1.00000 0 1.00000 0 −0.952306 + 1.64944i 0 1.00000 0
841.4 0 −1.00000 0 1.00000 0 0.827889 1.43395i 0 1.00000 0
841.5 0 −1.00000 0 1.00000 0 1.07806 1.86725i 0 1.00000 0
841.6 0 −1.00000 0 1.00000 0 1.26463 2.19041i 0 1.00000 0
841.7 0 −1.00000 0 1.00000 0 2.01447 3.48916i 0 1.00000 0
3781.1 0 −1.00000 0 1.00000 0 −1.59463 2.76199i 0 1.00000 0
3781.2 0 −1.00000 0 1.00000 0 −1.13810 1.97125i 0 1.00000 0
3781.3 0 −1.00000 0 1.00000 0 −0.952306 1.64944i 0 1.00000 0
3781.4 0 −1.00000 0 1.00000 0 0.827889 + 1.43395i 0 1.00000 0
3781.5 0 −1.00000 0 1.00000 0 1.07806 + 1.86725i 0 1.00000 0
3781.6 0 −1.00000 0 1.00000 0 1.26463 + 2.19041i 0 1.00000 0
3781.7 0 −1.00000 0 1.00000 0 2.01447 + 3.48916i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3781.7
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
67.c Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4020, [\chi])\):

\(T_{7}^{14} - \cdots\)
\(T_{11}^{14} - \cdots\)
\(T_{17}^{14} + \cdots\)