Properties

Label 8034.2.a.bb
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 6 x^{13} - 29 x^{12} + 207 x^{11} + 269 x^{10} - 2601 x^{9} - 847 x^{8} + 14851 x^{7} + 678 x^{6} - 39390 x^{5} - 3280 x^{4} + 42456 x^{3} + 10816 x^{2} - 7296 x - 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} -\beta_{1} q^{5} - q^{6} + \beta_{6} q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} + q^{3} + q^{4} -\beta_{1} q^{5} - q^{6} + \beta_{6} q^{7} - q^{8} + q^{9} + \beta_{1} q^{10} + ( -1 + \beta_{9} ) q^{11} + q^{12} - q^{13} -\beta_{6} q^{14} -\beta_{1} q^{15} + q^{16} + ( -1 - \beta_{6} - \beta_{10} ) q^{17} - q^{18} + ( \beta_{1} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{19} -\beta_{1} q^{20} + \beta_{6} q^{21} + ( 1 - \beta_{9} ) q^{22} + ( -1 - \beta_{5} - \beta_{6} ) q^{23} - q^{24} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{11} + \beta_{13} ) q^{25} + q^{26} + q^{27} + \beta_{6} q^{28} + ( -\beta_{7} + \beta_{11} - \beta_{12} ) q^{29} + \beta_{1} q^{30} + ( \beta_{2} + \beta_{5} + \beta_{6} + \beta_{11} ) q^{31} - q^{32} + ( -1 + \beta_{9} ) q^{33} + ( 1 + \beta_{6} + \beta_{10} ) q^{34} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{35} + q^{36} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} - \beta_{11} + \beta_{12} ) q^{37} + ( -\beta_{1} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{38} - q^{39} + \beta_{1} q^{40} + ( -3 + \beta_{1} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{41} -\beta_{6} q^{42} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{13} ) q^{43} + ( -1 + \beta_{9} ) q^{44} -\beta_{1} q^{45} + ( 1 + \beta_{5} + \beta_{6} ) q^{46} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{47} + q^{48} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{49} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{11} - \beta_{13} ) q^{50} + ( -1 - \beta_{6} - \beta_{10} ) q^{51} - q^{52} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{13} ) q^{53} - q^{54} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{12} + \beta_{13} ) q^{55} -\beta_{6} q^{56} + ( \beta_{1} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{57} + ( \beta_{7} - \beta_{11} + \beta_{12} ) q^{58} + ( 1 - \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{59} -\beta_{1} q^{60} + ( -1 + \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{61} + ( -\beta_{2} - \beta_{5} - \beta_{6} - \beta_{11} ) q^{62} + \beta_{6} q^{63} + q^{64} + \beta_{1} q^{65} + ( 1 - \beta_{9} ) q^{66} + ( \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{67} + ( -1 - \beta_{6} - \beta_{10} ) q^{68} + ( -1 - \beta_{5} - \beta_{6} ) q^{69} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{70} + ( -2 + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{8} - 3 \beta_{9} - \beta_{11} + 3 \beta_{12} + \beta_{13} ) q^{71} - q^{72} + ( -2 + 2 \beta_{1} - \beta_{4} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{73} + ( 1 - \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{74} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{11} + \beta_{13} ) q^{75} + ( \beta_{1} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{76} + ( -\beta_{7} - \beta_{8} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{77} + q^{78} + ( -1 + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{7} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{79} -\beta_{1} q^{80} + q^{81} + ( 3 - \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{82} + ( -3 + \beta_{1} + \beta_{3} + \beta_{8} - \beta_{9} - \beta_{11} + \beta_{13} ) q^{83} + \beta_{6} q^{84} + ( -1 - \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{5} + 2 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{13} ) q^{85} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{13} ) q^{86} + ( -\beta_{7} + \beta_{11} - \beta_{12} ) q^{87} + ( 1 - \beta_{9} ) q^{88} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{11} + \beta_{12} ) q^{89} + \beta_{1} q^{90} -\beta_{6} q^{91} + ( -1 - \beta_{5} - \beta_{6} ) q^{92} + ( \beta_{2} + \beta_{5} + \beta_{6} + \beta_{11} ) q^{93} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{94} + ( -2 + \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{95} - q^{96} + ( 3 - \beta_{1} + \beta_{4} - \beta_{7} + \beta_{9} + 2 \beta_{10} - 2 \beta_{12} ) q^{97} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{98} + ( -1 + \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + 6q^{10} - 8q^{11} + 14q^{12} - 14q^{13} + 4q^{14} - 6q^{15} + 14q^{16} - 4q^{17} - 14q^{18} - q^{19} - 6q^{20} - 4q^{21} + 8q^{22} - 9q^{23} - 14q^{24} + 24q^{25} + 14q^{26} + 14q^{27} - 4q^{28} - 10q^{29} + 6q^{30} - 5q^{31} - 14q^{32} - 8q^{33} + 4q^{34} - 16q^{35} + 14q^{36} - 4q^{37} + q^{38} - 14q^{39} + 6q^{40} - 24q^{41} + 4q^{42} - 8q^{44} - 6q^{45} + 9q^{46} - 32q^{47} + 14q^{48} + 24q^{49} - 24q^{50} - 4q^{51} - 14q^{52} - 5q^{53} - 14q^{54} - 8q^{55} + 4q^{56} - q^{57} + 10q^{58} - 13q^{59} - 6q^{60} + 2q^{61} + 5q^{62} - 4q^{63} + 14q^{64} + 6q^{65} + 8q^{66} - 16q^{67} - 4q^{68} - 9q^{69} + 16q^{70} - 29q^{71} - 14q^{72} + 4q^{74} + 24q^{75} - q^{76} - 9q^{77} + 14q^{78} - 21q^{79} - 6q^{80} + 14q^{81} + 24q^{82} - 40q^{83} - 4q^{84} - 7q^{85} - 10q^{87} + 8q^{88} - 48q^{89} + 6q^{90} + 4q^{91} - 9q^{92} - 5q^{93} + 32q^{94} - 26q^{95} - 14q^{96} + 18q^{97} - 24q^{98} - 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 6 x^{13} - 29 x^{12} + 207 x^{11} + 269 x^{10} - 2601 x^{9} - 847 x^{8} + 14851 x^{7} + 678 x^{6} - 39390 x^{5} - 3280 x^{4} + 42456 x^{3} + 10816 x^{2} - 7296 x - 2048\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(17211124339 \nu^{13} + 48838930211 \nu^{12} - 1083388884247 \nu^{11} - 1278700159484 \nu^{10} + 21896289725388 \nu^{9} + 8353374678764 \nu^{8} - 175798632679800 \nu^{7} + 21533806125636 \nu^{6} + 467406199485803 \nu^{5} - 351416224121502 \nu^{4} + 243900222298282 \nu^{3} + 769318123450260 \nu^{2} - 1166460932605168 \nu + 118321024199296\)\()/ 165587393151104 \)
\(\beta_{3}\)\(=\)\((\)\(-88374380939 \nu^{13} + 341475795883 \nu^{12} + 3738713297463 \nu^{11} - 13680291423778 \nu^{10} - 60950374089966 \nu^{9} + 203902612897310 \nu^{8} + 480284475172790 \nu^{7} - 1384162777559138 \nu^{6} - 1866871026854413 \nu^{5} + 4166130863706398 \nu^{4} + 3191603596475186 \nu^{3} - 4474034344329852 \nu^{2} - 1493240738430736 \nu + 823088103896320\)\()/ 165587393151104 \)
\(\beta_{4}\)\(=\)\((\)\(-406907271083 \nu^{13} + 2304692170158 \nu^{12} + 11004973389543 \nu^{11} - 75016193619169 \nu^{10} - 82526311620203 \nu^{9} + 861495599563087 \nu^{8} + 21324319463753 \nu^{7} - 4294335680262885 \nu^{6} + 1434806545518450 \nu^{5} + 9639491681032898 \nu^{4} - 3273111940991512 \nu^{3} - 10523459905050216 \nu^{2} + 2930798928626144 \nu + 3910063812049920\)\()/ 662349572604416 \)
\(\beta_{5}\)\(=\)\((\)\(113401884803 \nu^{13} - 883549115785 \nu^{12} - 2323893522087 \nu^{11} + 29836393456848 \nu^{10} - 852019922224 \nu^{9} - 364296872907848 \nu^{8} + 263847579785132 \nu^{7} + 2001792019065248 \nu^{6} - 1727833814125777 \nu^{5} - 5028642484757934 \nu^{4} + 3738614769502962 \nu^{3} + 5009595167008196 \nu^{2} - 2681914257270768 \nu - 1049016746216704\)\()/ 165587393151104 \)
\(\beta_{6}\)\(=\)\((\)\(736531555801 \nu^{13} - 3946026327910 \nu^{12} - 23999796454509 \nu^{11} + 138185242548231 \nu^{10} + 283131019369069 \nu^{9} - 1765360355157129 \nu^{8} - 1550184318259343 \nu^{7} + 10222081364164867 \nu^{6} + 4328705466068094 \nu^{5} - 27089707111959846 \nu^{4} - 6353469675911824 \nu^{3} + 27368166174349864 \nu^{2} + 3859293469715616 \nu - 3240111054661120\)\()/ 662349572604416 \)
\(\beta_{7}\)\(=\)\((\)\(1012895827421 \nu^{13} - 5401357076630 \nu^{12} - 30269645318433 \nu^{11} + 182412406997691 \nu^{10} + 294059953736009 \nu^{9} - 2216963035494389 \nu^{8} - 971532240127523 \nu^{7} + 11978958293225447 \nu^{6} + 149793013139294 \nu^{5} - 28653365696975870 \nu^{4} + 2303549139588352 \nu^{3} + 24319897537607016 \nu^{2} - 796238968943584 \nu - 804113198053376\)\()/ 662349572604416 \)
\(\beta_{8}\)\(=\)\((\)\(-161946406606 \nu^{13} + 993100465781 \nu^{12} + 4568699998010 \nu^{11} - 33836085473159 \nu^{10} - 39248816007673 \nu^{9} + 415409487608205 \nu^{8} + 80934994695967 \nu^{7} - 2274108863937751 \nu^{6} + 260702734356941 \nu^{5} + 5633078350696404 \nu^{4} - 672606967463486 \nu^{3} - 5548224775659980 \nu^{2} - 339551833683136 \nu + 821129125068864\)\()/ 82793696575552 \)
\(\beta_{9}\)\(=\)\((\)\(-339042867519 \nu^{13} + 2193728871065 \nu^{12} + 8513285647275 \nu^{11} - 73385354065404 \nu^{10} - 47754006020300 \nu^{9} + 882287181466036 \nu^{8} - 215439634992760 \nu^{7} - 4744161357889636 \nu^{6} + 2237814174755737 \nu^{5} + 11755366222327694 \nu^{4} - 3969078366908066 \nu^{3} - 12224135566363796 \nu^{2} + 525183514160720 \nu + 2100534074540032\)\()/ 165587393151104 \)
\(\beta_{10}\)\(=\)\((\)\(-2206595272859 \nu^{13} + 14412109193718 \nu^{12} + 54849889967511 \nu^{11} - 474790047743417 \nu^{10} - 307896247879923 \nu^{9} + 5544743002953591 \nu^{8} - 1212936140282751 \nu^{7} - 28137851717749085 \nu^{6} + 12427081549409178 \nu^{5} + 62275142320830146 \nu^{4} - 19724650983886152 \nu^{3} - 53935123111369352 \nu^{2} + 943866195323488 \nu + 6787275004993536\)\()/ 662349572604416 \)
\(\beta_{11}\)\(=\)\((\)\(-2312493819663 \nu^{13} + 15946662074178 \nu^{12} + 56053557441147 \nu^{11} - 536833570831273 \nu^{10} - 265156769412035 \nu^{9} + 6499156388583719 \nu^{8} - 2119780149955167 \nu^{7} - 35077355093355085 \nu^{6} + 18558566709942790 \nu^{5} + 85869291262886602 \nu^{4} - 35303846107753504 \nu^{3} - 83761116115370808 \nu^{2} + 9764388104189088 \nu + 10136073926334976\)\()/ 662349572604416 \)
\(\beta_{12}\)\(=\)\((\)\(-155979666211 \nu^{13} + 1092105641867 \nu^{12} + 3542117432817 \nu^{11} - 36139437565114 \nu^{10} - 10207803685344 \nu^{9} + 425935692445960 \nu^{8} - 227141704110620 \nu^{7} - 2204586185386584 \nu^{6} + 1611592294003569 \nu^{5} + 5085371693596568 \nu^{4} - 2867839136841206 \nu^{3} - 4698153606819416 \nu^{2} + 774956085111656 \nu + 564375014073344\)\()/ 41396848287776 \)
\(\beta_{13}\)\(=\)\((\)\(2734835840775 \nu^{13} - 17117209536866 \nu^{12} - 75341966167987 \nu^{11} + 586439935888449 \nu^{10} + 596543424673451 \nu^{9} - 7281353341457903 \nu^{8} - 504552281455193 \nu^{7} + 40700141428094181 \nu^{6} - 9221457804581926 \nu^{5} - 103939479614198202 \nu^{4} + 23513032611045888 \nu^{3} + 105396875559095672 \nu^{2} - 9119618453491232 \nu - 17591589253353984\)\()/ 662349572604416 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{13} + \beta_{11} + \beta_{3} - \beta_{2} + \beta_{1} + 7\)
\(\nu^{3}\)\(=\)\(\beta_{13} - 2 \beta_{12} + 3 \beta_{11} + \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + 10 \beta_{1} + 7\)
\(\nu^{4}\)\(=\)\(13 \beta_{13} - 4 \beta_{12} + 19 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + 5 \beta_{6} + 4 \beta_{5} + 4 \beta_{4} + 14 \beta_{3} - 14 \beta_{2} + 17 \beta_{1} + 88\)
\(\nu^{5}\)\(=\)\(26 \beta_{13} - 37 \beta_{12} + 65 \beta_{11} - 3 \beta_{10} - 5 \beta_{9} + 18 \beta_{8} - 20 \beta_{7} + 33 \beta_{6} + 16 \beta_{5} + 31 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + 123 \beta_{1} + 148\)
\(\nu^{6}\)\(=\)\(167 \beta_{13} - 103 \beta_{12} + 335 \beta_{11} + 18 \beta_{10} - 41 \beta_{9} - 37 \beta_{8} - 17 \beta_{7} + 111 \beta_{6} + 113 \beta_{5} + 108 \beta_{4} + 177 \beta_{3} - 177 \beta_{2} + 276 \beta_{1} + 1263\)
\(\nu^{7}\)\(=\)\(479 \beta_{13} - 611 \beta_{12} + 1200 \beta_{11} - 93 \beta_{10} - 162 \beta_{9} + 242 \beta_{8} - 350 \beta_{7} + 483 \beta_{6} + 315 \beta_{5} + 679 \beta_{4} + 107 \beta_{3} - 18 \beta_{2} + 1682 \beta_{1} + 2702\)
\(\nu^{8}\)\(=\)\(2222 \beta_{13} - 2060 \beta_{12} + 5775 \beta_{11} + 203 \beta_{10} - 1006 \beta_{9} - 885 \beta_{8} - 542 \beta_{7} + 1884 \beta_{6} + 2415 \beta_{5} + 2350 \beta_{4} + 2137 \beta_{3} - 2159 \beta_{2} + 4443 \beta_{1} + 19087\)
\(\nu^{9}\)\(=\)\(7842 \beta_{13} - 10023 \beta_{12} + 21287 \beta_{11} - 2124 \beta_{10} - 3977 \beta_{9} + 2676 \beta_{8} - 5991 \beta_{7} + 6985 \beta_{6} + 6668 \beta_{5} + 13401 \beta_{4} + 1459 \beta_{3} + 285 \beta_{2} + 24370 \beta_{1} + 47413\)
\(\nu^{10}\)\(=\)\(30616 \beta_{13} - 37773 \beta_{12} + 98893 \beta_{11} + 986 \beta_{10} - 21437 \beta_{9} - 18052 \beta_{8} - 12588 \beta_{7} + 29338 \beta_{6} + 46972 \beta_{5} + 47547 \beta_{4} + 24544 \beta_{3} - 25202 \beta_{2} + 71435 \beta_{1} + 296729\)
\(\nu^{11}\)\(=\)\(122139 \beta_{13} - 166504 \beta_{12} + 373041 \beta_{11} - 43026 \beta_{10} - 86704 \beta_{9} + 20835 \beta_{8} - 102258 \beta_{7} + 101217 \beta_{6} + 138595 \beta_{5} + 252965 \beta_{4} + 12729 \beta_{3} + 16621 \beta_{2} + 365621 \beta_{1} + 819867\)
\(\nu^{12}\)\(=\)\(434104 \beta_{13} - 667453 \beta_{12} + 1693145 \beta_{11} - 27689 \beta_{10} - 430192 \beta_{9} - 343448 \beta_{8} - 259336 \beta_{7} + 440956 \beta_{6} + 876463 \beta_{5} + 925564 \beta_{4} + 258691 \beta_{3} - 270723 \beta_{2} + 1151467 \beta_{1} + 4704078\)
\(\nu^{13}\)\(=\)\(1860057 \beta_{13} - 2802582 \beta_{12} + 6513435 \beta_{11} - 823424 \beta_{10} - 1769865 \beta_{9} - 35291 \beta_{8} - 1747648 \beta_{7} + 1471030 \beta_{6} + 2780674 \beta_{5} + 4665904 \beta_{4} - 25839 \beta_{3} + 487738 \beta_{2} + 5617405 \beta_{1} + 14098951\)

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
4.19123
3.81560
3.75223
2.57230
2.22430
1.73527
0.453790
−0.290663
−0.480706
−1.34813
−1.50270
−2.36221
−3.07028
−3.69005
−1.00000 1.00000 1.00000 −4.19123 −1.00000 −2.48485 −1.00000 1.00000 4.19123
1.2 −1.00000 1.00000 1.00000 −3.81560 −1.00000 4.94203 −1.00000 1.00000 3.81560
1.3 −1.00000 1.00000 1.00000 −3.75223 −1.00000 1.91373 −1.00000 1.00000 3.75223
1.4 −1.00000 1.00000 1.00000 −2.57230 −1.00000 −4.81484 −1.00000 1.00000 2.57230
1.5 −1.00000 1.00000 1.00000 −2.22430 −1.00000 1.37131 −1.00000 1.00000 2.22430
1.6 −1.00000 1.00000 1.00000 −1.73527 −1.00000 −3.07502 −1.00000 1.00000 1.73527
1.7 −1.00000 1.00000 1.00000 −0.453790 −1.00000 3.87720 −1.00000 1.00000 0.453790
1.8 −1.00000 1.00000 1.00000 0.290663 −1.00000 −3.15490 −1.00000 1.00000 −0.290663
1.9 −1.00000 1.00000 1.00000 0.480706 −1.00000 0.765483 −1.00000 1.00000 −0.480706
1.10 −1.00000 1.00000 1.00000 1.34813 −1.00000 2.71387 −1.00000 1.00000 −1.34813
1.11 −1.00000 1.00000 1.00000 1.50270 −1.00000 0.210711 −1.00000 1.00000 −1.50270
1.12 −1.00000 1.00000 1.00000 2.36221 −1.00000 −4.26027 −1.00000 1.00000 −2.36221
1.13 −1.00000 1.00000 1.00000 3.07028 −1.00000 −1.20257 −1.00000 1.00000 −3.07028
1.14 −1.00000 1.00000 1.00000 3.69005 −1.00000 −0.801879 −1.00000 1.00000 −3.69005
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(13\) \(1\)
\(103\) \(-1\)

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 8034.2.a.bb 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.bb 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{14} + \cdots\)
\(T_{7}^{14} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{14} \)
$3$ \( ( 1 - T )^{14} \)
$5$ \( 1 + 6 T + 41 T^{2} + 183 T^{3} + 804 T^{4} + 2916 T^{5} + 10253 T^{6} + 32069 T^{7} + 97548 T^{8} + 272380 T^{9} + 743535 T^{10} + 1893519 T^{11} + 4723591 T^{12} + 11092331 T^{13} + 25527862 T^{14} + 55461655 T^{15} + 118089775 T^{16} + 236689875 T^{17} + 464709375 T^{18} + 851187500 T^{19} + 1524187500 T^{20} + 2505390625 T^{21} + 4005078125 T^{22} + 5695312500 T^{23} + 7851562500 T^{24} + 8935546875 T^{25} + 10009765625 T^{26} + 7324218750 T^{27} + 6103515625 T^{28} \)
$7$ \( 1 + 4 T + 45 T^{2} + 150 T^{3} + 970 T^{4} + 2730 T^{5} + 13363 T^{6} + 32257 T^{7} + 134875 T^{8} + 283916 T^{9} + 1104598 T^{10} + 2080278 T^{11} + 8060734 T^{12} + 14244643 T^{13} + 56608556 T^{14} + 99712501 T^{15} + 394975966 T^{16} + 713535354 T^{17} + 2652139798 T^{18} + 4771776212 T^{19} + 15867908875 T^{20} + 26565026551 T^{21} + 77035035763 T^{22} + 110165347110 T^{23} + 274000991530 T^{24} + 296599011450 T^{25} + 622857924045 T^{26} + 387556041628 T^{27} + 678223072849 T^{28} \)
$11$ \( 1 + 8 T + 115 T^{2} + 693 T^{3} + 5771 T^{4} + 28541 T^{5} + 177373 T^{6} + 755945 T^{7} + 3867618 T^{8} + 14680639 T^{9} + 65285057 T^{10} + 226078711 T^{11} + 905329918 T^{12} + 2903978747 T^{13} + 10710168638 T^{14} + 31943766217 T^{15} + 109544920078 T^{16} + 300910764341 T^{17} + 955838519537 T^{18} + 2364331591589 T^{19} + 6851721211698 T^{20} + 14731229481595 T^{21} + 38021477799613 T^{22} + 67298185048831 T^{23} + 149684877372371 T^{24} + 197720987733423 T^{25} + 360919263322915 T^{26} + 276181697151448 T^{27} + 379749833583241 T^{28} \)
$13$ \( ( 1 + T )^{14} \)
$17$ \( 1 + 4 T + 112 T^{2} + 455 T^{3} + 6661 T^{4} + 27062 T^{5} + 277881 T^{6} + 1107307 T^{7} + 8993960 T^{8} + 34454734 T^{9} + 236670370 T^{10} + 857297371 T^{11} + 5189048050 T^{12} + 17526492347 T^{13} + 95917900138 T^{14} + 297950369899 T^{15} + 1499634886450 T^{16} + 4211901983723 T^{17} + 19766945972770 T^{18} + 48920795253038 T^{19} + 217092330083240 T^{20} + 454370884983611 T^{21} + 1938430453462521 T^{22} + 3209225113761814 T^{23} + 13428535370890789 T^{24} + 15593712819973015 T^{25} + 65253690569733232 T^{26} + 39618312131623748 T^{27} + 168377826559400929 T^{28} \)
$19$ \( 1 + T + 106 T^{2} - 17 T^{3} + 5586 T^{4} - 6827 T^{5} + 211256 T^{6} - 402664 T^{7} + 6693371 T^{8} - 14345340 T^{9} + 182347476 T^{10} - 400413994 T^{11} + 4240436866 T^{12} - 9324043833 T^{13} + 85718030132 T^{14} - 177156832827 T^{15} + 1530797708626 T^{16} - 2746439584846 T^{17} + 23763705419796 T^{18} - 35520482028660 T^{19} + 314895535554851 T^{20} - 359929969912696 T^{21} + 3587879593789496 T^{22} - 2202988912737233 T^{23} + 34248136116076386 T^{24} - 1980334401269723 T^{25} + 234611381421013066 T^{26} + 42052983462257059 T^{27} + 799006685782884121 T^{28} \)
$23$ \( 1 + 9 T + 218 T^{2} + 1394 T^{3} + 19904 T^{4} + 92422 T^{5} + 1060630 T^{6} + 3513688 T^{7} + 39696593 T^{8} + 92499915 T^{9} + 1227382805 T^{10} + 2173065758 T^{11} + 34402667030 T^{12} + 53128614126 T^{13} + 854793008294 T^{14} + 1221958124898 T^{15} + 18199010858870 T^{16} + 26439691077586 T^{17} + 343472031534005 T^{18} + 595361180410845 T^{19} + 5876520435026177 T^{20} + 11963494315218536 T^{21} + 83058980318587030 T^{22} + 166466131277733386 T^{23} + 824553279196469696 T^{24} + 1328216802532014238 T^{25} + 4777388126180429978 T^{26} + 4536327257428206447 T^{27} + 11592836324538749809 T^{28} \)
$29$ \( 1 + 10 T + 276 T^{2} + 2229 T^{3} + 36268 T^{4} + 252689 T^{5} + 3101986 T^{6} + 19180659 T^{7} + 194296543 T^{8} + 1081823807 T^{9} + 9461951626 T^{10} + 47796897838 T^{11} + 370298779492 T^{12} + 1700390057896 T^{13} + 11845573980640 T^{14} + 49311311678984 T^{15} + 311421273552772 T^{16} + 1165718541370982 T^{17} + 6692258607988906 T^{18} + 22189449297124243 T^{19} + 115572114966079303 T^{20} + 330863995275107631 T^{21} + 1551757369555240546 T^{22} + 3665796209496361741 T^{23} + 15258209937331689868 T^{24} + 27194936267758292841 T^{25} + 97652880164709455316 T^{26} + \)\(10\!\cdots\!90\)\( T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \)
$31$ \( 1 + 5 T + 246 T^{2} + 1015 T^{3} + 29204 T^{4} + 93459 T^{5} + 2215336 T^{6} + 4982146 T^{7} + 121083335 T^{8} + 156826740 T^{9} + 5165689838 T^{10} + 2248011146 T^{11} + 185126820172 T^{12} - 20406816505 T^{13} + 5962515707672 T^{14} - 632611311655 T^{15} + 177906874185292 T^{16} + 66970500050486 T^{17} + 4770623044879598 T^{18} + 4489816420297740 T^{19} + 107461905520256135 T^{20} + 137071860342662206 T^{21} + 1889440219320395176 T^{22} + 2471020647514150989 T^{23} + 23936424492987312404 T^{24} + 25789604049850903465 T^{25} + \)\(19\!\cdots\!06\)\( T^{26} + \)\(12\!\cdots\!55\)\( T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \)
$37$ \( 1 + 4 T + 265 T^{2} + 469 T^{3} + 34658 T^{4} + 11966 T^{5} + 3187605 T^{6} - 1583648 T^{7} + 229272679 T^{8} - 225871475 T^{9} + 13348935918 T^{10} - 16395454920 T^{11} + 644409391740 T^{12} - 813518143628 T^{13} + 26053793698036 T^{14} - 30100171314236 T^{15} + 882196457292060 T^{16} - 830478978062760 T^{17} + 25018055089014798 T^{18} - 15662821849926575 T^{19} + 588250967372479711 T^{20} - 150338677357921184 T^{21} + 11196397069715849205 T^{22} + 1555122178387891382 T^{23} + \)\(16\!\cdots\!42\)\( T^{24} + 83443364614566933697 T^{25} + \)\(17\!\cdots\!65\)\( T^{26} + \)\(97\!\cdots\!88\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \)
$41$ \( 1 + 24 T + 430 T^{2} + 4986 T^{3} + 50235 T^{4} + 414972 T^{5} + 3441888 T^{6} + 26808105 T^{7} + 209107874 T^{8} + 1471196358 T^{9} + 9760386156 T^{10} + 62059530618 T^{11} + 405109179442 T^{12} + 2730537791825 T^{13} + 17825882963660 T^{14} + 111952049464825 T^{15} + 680988530642002 T^{16} + 4277204909723178 T^{17} + 27580518544564716 T^{18} + 170447220962915958 T^{19} + 993284199113893634 T^{20} + 5220993023400605505 T^{21} + 27483218327008820448 T^{22} + \)\(13\!\cdots\!92\)\( T^{23} + \)\(67\!\cdots\!35\)\( T^{24} + \)\(27\!\cdots\!26\)\( T^{25} + \)\(97\!\cdots\!30\)\( T^{26} + \)\(22\!\cdots\!04\)\( T^{27} + \)\(37\!\cdots\!61\)\( T^{28} \)
$43$ \( 1 + 371 T^{2} + 72 T^{3} + 63145 T^{4} + 33658 T^{5} + 6443937 T^{6} + 6995121 T^{7} + 433134162 T^{8} + 870799568 T^{9} + 20088493843 T^{10} + 72850576834 T^{11} + 701555046604 T^{12} + 4321181093061 T^{13} + 25278722069618 T^{14} + 185810787001623 T^{15} + 1297175281170796 T^{16} + 5792130812340838 T^{17} + 68678562838942243 T^{18} + 128014888656752624 T^{19} + 2737998286926379938 T^{20} + 1901404074745408947 T^{21} + 75318026232243355137 T^{22} + 16916262132570261694 T^{23} + \)\(13\!\cdots\!05\)\( T^{24} + 66909149241928034904 T^{25} + \)\(14\!\cdots\!71\)\( T^{26} + \)\(73\!\cdots\!49\)\( T^{28} \)
$47$ \( 1 + 32 T + 937 T^{2} + 18332 T^{3} + 325258 T^{4} + 4713494 T^{5} + 62869397 T^{6} + 732375533 T^{7} + 7957211532 T^{8} + 77953219786 T^{9} + 719664123905 T^{10} + 6098481944010 T^{11} + 49065402157493 T^{12} + 365915276110075 T^{13} + 2601395262021930 T^{14} + 17198017977173525 T^{15} + 108385473365902037 T^{16} + 633162690872950230 T^{17} + 3511731351800874305 T^{18} + 17878181737492708502 T^{19} + 85772496521829974028 T^{20} + \)\(37\!\cdots\!79\)\( T^{21} + \)\(14\!\cdots\!17\)\( T^{22} + \)\(52\!\cdots\!98\)\( T^{23} + \)\(17\!\cdots\!42\)\( T^{24} + \)\(45\!\cdots\!96\)\( T^{25} + \)\(10\!\cdots\!17\)\( T^{26} + \)\(17\!\cdots\!64\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \)
$53$ \( 1 + 5 T + 310 T^{2} + 514 T^{3} + 45257 T^{4} - 53879 T^{5} + 4716903 T^{6} - 15081766 T^{7} + 425253352 T^{8} - 1670614529 T^{9} + 33658762096 T^{10} - 134303865386 T^{11} + 2251304821622 T^{12} - 8826593113975 T^{13} + 128261303595494 T^{14} - 467809435040675 T^{15} + 6323915243936198 T^{16} - 19994756567071522 T^{17} + 265583822802008176 T^{18} - 698643466568117797 T^{19} + 9425468865045754408 T^{20} - 17716718528614912142 T^{21} + \)\(29\!\cdots\!83\)\( T^{22} - \)\(17\!\cdots\!07\)\( T^{23} + \)\(79\!\cdots\!93\)\( T^{24} + \)\(47\!\cdots\!58\)\( T^{25} + \)\(15\!\cdots\!10\)\( T^{26} + \)\(13\!\cdots\!65\)\( T^{27} + \)\(13\!\cdots\!69\)\( T^{28} \)
$59$ \( 1 + 13 T + 295 T^{2} + 2438 T^{3} + 43865 T^{4} + 332795 T^{5} + 5032470 T^{6} + 32139538 T^{7} + 443875251 T^{8} + 2666335917 T^{9} + 34772734577 T^{10} + 188346686888 T^{11} + 2323971058803 T^{12} + 12131936012011 T^{13} + 145893820387140 T^{14} + 715784224708649 T^{15} + 8089743255693243 T^{16} + 38682454206370552 T^{17} + 421353777826691297 T^{18} + 1906228336359747183 T^{19} + 18722894957212818891 T^{20} + 79984108965096673622 T^{21} + \)\(73\!\cdots\!70\)\( T^{22} + \)\(28\!\cdots\!05\)\( T^{23} + \)\(22\!\cdots\!65\)\( T^{24} + \)\(73\!\cdots\!42\)\( T^{25} + \)\(52\!\cdots\!95\)\( T^{26} + \)\(13\!\cdots\!27\)\( T^{27} + \)\(61\!\cdots\!61\)\( T^{28} \)
$61$ \( 1 - 2 T + 531 T^{2} - 666 T^{3} + 138337 T^{4} - 79100 T^{5} + 23661761 T^{6} - 525585 T^{7} + 2997903312 T^{8} + 1049709712 T^{9} + 299443738513 T^{10} + 162692252758 T^{11} + 24337760215294 T^{12} + 14417559647767 T^{13} + 1631695419459942 T^{14} + 879471138513787 T^{15} + 90560805761108974 T^{16} + 36928050223263598 T^{17} + 4146050391896574433 T^{18} + 886580939878975312 T^{19} + \)\(15\!\cdots\!32\)\( T^{20} - 1651778493470097285 T^{21} + \)\(45\!\cdots\!41\)\( T^{22} - \)\(92\!\cdots\!00\)\( T^{23} + \)\(98\!\cdots\!37\)\( T^{24} - \)\(28\!\cdots\!26\)\( T^{25} + \)\(14\!\cdots\!51\)\( T^{26} - \)\(32\!\cdots\!62\)\( T^{27} + \)\(98\!\cdots\!41\)\( T^{28} \)
$67$ \( 1 + 16 T + 519 T^{2} + 6672 T^{3} + 124668 T^{4} + 1350481 T^{5} + 19154411 T^{6} + 184611695 T^{7} + 2223612699 T^{8} + 19913080426 T^{9} + 215047481933 T^{10} + 1818762266026 T^{11} + 17963409855248 T^{12} + 142652630232610 T^{13} + 1296849511237522 T^{14} + 9557726225584870 T^{15} + 80637746840208272 T^{16} + 547016395416777838 T^{17} + 4333447829177196893 T^{18} + 26885149840852855582 T^{19} + \)\(20\!\cdots\!31\)\( T^{20} + \)\(11\!\cdots\!85\)\( T^{21} + \)\(77\!\cdots\!51\)\( T^{22} + \)\(36\!\cdots\!07\)\( T^{23} + \)\(22\!\cdots\!32\)\( T^{24} + \)\(81\!\cdots\!76\)\( T^{25} + \)\(42\!\cdots\!59\)\( T^{26} + \)\(87\!\cdots\!92\)\( T^{27} + \)\(36\!\cdots\!29\)\( T^{28} \)
$71$ \( 1 + 29 T + 705 T^{2} + 12086 T^{3} + 191896 T^{4} + 2638398 T^{5} + 34300098 T^{6} + 408688847 T^{7} + 4632044190 T^{8} + 49512503143 T^{9} + 506040747423 T^{10} + 4940191614446 T^{11} + 46145162701177 T^{12} + 414037169596841 T^{13} + 3556392873560764 T^{14} + 29396639041375711 T^{15} + 232617765176633257 T^{16} + 1768148920917982306 T^{17} + 12859346046514848063 T^{18} + 89331911412080350193 T^{19} + \)\(59\!\cdots\!90\)\( T^{20} + \)\(37\!\cdots\!77\)\( T^{21} + \)\(22\!\cdots\!78\)\( T^{22} + \)\(12\!\cdots\!38\)\( T^{23} + \)\(62\!\cdots\!96\)\( T^{24} + \)\(27\!\cdots\!06\)\( T^{25} + \)\(11\!\cdots\!05\)\( T^{26} + \)\(33\!\cdots\!19\)\( T^{27} + \)\(82\!\cdots\!81\)\( T^{28} \)
$73$ \( 1 + 594 T^{2} - 544 T^{3} + 163050 T^{4} - 301055 T^{5} + 27556752 T^{6} - 79492103 T^{7} + 3229566043 T^{8} - 13470215088 T^{9} + 284293736642 T^{10} - 1649952824040 T^{11} + 20797805686594 T^{12} - 154352743930394 T^{13} + 1472400999295624 T^{14} - 11267750306918762 T^{15} + 110831506503859426 T^{16} - 641859697749568680 T^{17} + 8073442047950046722 T^{18} - 27924720250532795184 T^{19} + \)\(48\!\cdots\!27\)\( T^{20} - \)\(87\!\cdots\!91\)\( T^{21} + \)\(22\!\cdots\!12\)\( T^{22} - \)\(17\!\cdots\!15\)\( T^{23} + \)\(70\!\cdots\!50\)\( T^{24} - \)\(17\!\cdots\!88\)\( T^{25} + \)\(13\!\cdots\!74\)\( T^{26} + \)\(12\!\cdots\!09\)\( T^{28} \)
$79$ \( 1 + 21 T + 775 T^{2} + 13429 T^{3} + 287452 T^{4} + 4258277 T^{5} + 68796789 T^{6} + 891779172 T^{7} + 11978875046 T^{8} + 138054011422 T^{9} + 1613060027341 T^{10} + 16698475768546 T^{11} + 173602203340189 T^{12} + 1622757633749715 T^{13} + 15178391862412142 T^{14} + 128197853066227485 T^{15} + 1083451351046119549 T^{16} + 8232999794448151294 T^{17} + 62828818722794164621 T^{18} + \)\(42\!\cdots\!78\)\( T^{19} + \)\(29\!\cdots\!66\)\( T^{20} + \)\(17\!\cdots\!48\)\( T^{21} + \)\(10\!\cdots\!29\)\( T^{22} + \)\(51\!\cdots\!63\)\( T^{23} + \)\(27\!\cdots\!52\)\( T^{24} + \)\(10\!\cdots\!91\)\( T^{25} + \)\(45\!\cdots\!75\)\( T^{26} + \)\(98\!\cdots\!19\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \)
$83$ \( 1 + 40 T + 1489 T^{2} + 36375 T^{3} + 815996 T^{4} + 14737335 T^{5} + 247530180 T^{6} + 3587969428 T^{7} + 49040298776 T^{8} + 598999223560 T^{9} + 6997481642155 T^{10} + 74730470575341 T^{11} + 774154667115923 T^{12} + 7447656544893681 T^{13} + 70193975652690912 T^{14} + 618155493226175523 T^{15} + 5333151501761593547 T^{16} + 42729911578862504367 T^{17} + \)\(33\!\cdots\!55\)\( T^{18} + \)\(23\!\cdots\!80\)\( T^{19} + \)\(16\!\cdots\!44\)\( T^{20} + \)\(97\!\cdots\!56\)\( T^{21} + \)\(55\!\cdots\!80\)\( T^{22} + \)\(27\!\cdots\!05\)\( T^{23} + \)\(12\!\cdots\!04\)\( T^{24} + \)\(46\!\cdots\!25\)\( T^{25} + \)\(15\!\cdots\!29\)\( T^{26} + \)\(35\!\cdots\!20\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \)
$89$ \( 1 + 48 T + 1746 T^{2} + 47342 T^{3} + 1100728 T^{4} + 22059597 T^{5} + 396813094 T^{6} + 6440871499 T^{7} + 95971599569 T^{8} + 1317084657840 T^{9} + 16797056961134 T^{10} + 199376003767338 T^{11} + 2213791916248950 T^{12} + 22996361109544056 T^{13} + 224121753113991348 T^{14} + 2046676138749420984 T^{15} + 17535445768607932950 T^{16} + \)\(14\!\cdots\!22\)\( T^{17} + \)\(10\!\cdots\!94\)\( T^{18} + \)\(73\!\cdots\!60\)\( T^{19} + \)\(47\!\cdots\!09\)\( T^{20} + \)\(28\!\cdots\!71\)\( T^{21} + \)\(15\!\cdots\!14\)\( T^{22} + \)\(77\!\cdots\!73\)\( T^{23} + \)\(34\!\cdots\!28\)\( T^{24} + \)\(13\!\cdots\!38\)\( T^{25} + \)\(43\!\cdots\!66\)\( T^{26} + \)\(10\!\cdots\!12\)\( T^{27} + \)\(19\!\cdots\!41\)\( T^{28} \)
$97$ \( 1 - 18 T + 601 T^{2} - 9834 T^{3} + 207402 T^{4} - 2942663 T^{5} + 49182557 T^{6} - 631164431 T^{7} + 9013761651 T^{8} - 105528788834 T^{9} + 1343957536229 T^{10} - 14495773419210 T^{11} + 167194987366978 T^{12} - 1664560628547234 T^{13} + 17619370405805322 T^{14} - 161462380969081698 T^{15} + 1573137636135896002 T^{16} - 13229901013830648330 T^{17} + \)\(11\!\cdots\!49\)\( T^{18} - \)\(90\!\cdots\!38\)\( T^{19} + \)\(75\!\cdots\!79\)\( T^{20} - \)\(50\!\cdots\!03\)\( T^{21} + \)\(38\!\cdots\!77\)\( T^{22} - \)\(22\!\cdots\!71\)\( T^{23} + \)\(15\!\cdots\!98\)\( T^{24} - \)\(70\!\cdots\!02\)\( T^{25} + \)\(41\!\cdots\!41\)\( T^{26} - \)\(12\!\cdots\!86\)\( T^{27} + \)\(65\!\cdots\!69\)\( T^{28} \)
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