Properties

Label 8048.2.a.r
Level 8048
Weight 2
Character orbit 8048.a
Self dual Yes
Analytic conductor 64.264
Analytic rank 0
Dimension 12
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8048.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2636035467\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{4} q^{3} \) \( + ( \beta_{7} - \beta_{10} ) q^{5} \) \( + ( -\beta_{2} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{7} \) \( + ( 2 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{4} q^{3} \) \( + ( \beta_{7} - \beta_{10} ) q^{5} \) \( + ( -\beta_{2} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{7} \) \( + ( 2 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} ) q^{9} \) \( + ( -2 + \beta_{6} - \beta_{9} - \beta_{11} ) q^{11} \) \( + ( -1 + \beta_{1} - \beta_{3} + \beta_{7} - \beta_{10} ) q^{13} \) \( + ( \beta_{1} - \beta_{3} - \beta_{10} + \beta_{11} ) q^{15} \) \( + ( 1 + \beta_{2} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{17} \) \( + ( -\beta_{3} + \beta_{4} - \beta_{6} - \beta_{11} ) q^{19} \) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{21} \) \( + ( 1 - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{23} \) \( + ( 1 - \beta_{1} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{25} \) \( + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{27} \) \( + ( 2 - \beta_{5} + \beta_{6} + \beta_{9} ) q^{29} \) \( + ( 1 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{31} \) \( + ( 2 + \beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{6} - 2 \beta_{7} - 3 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{33} \) \( + ( -3 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{35} \) \( + ( -3 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{37} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{39} \) \( + ( 3 + \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{41} \) \( + ( 3 \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} ) q^{43} \) \( + ( 1 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - 3 \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{45} \) \( + ( -2 + \beta_{1} - \beta_{3} - 2 \beta_{5} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{47} \) \( + ( 4 + \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{49} \) \( + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{51} \) \( + ( \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{53} \) \( + ( 1 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} ) q^{55} \) \( + ( 2 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + 3 \beta_{10} + 2 \beta_{11} ) q^{57} \) \( + ( -5 - 2 \beta_{2} + 2 \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{9} - 2 \beta_{11} ) q^{59} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{61} \) \( + ( -1 + 6 \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} - 3 \beta_{8} - 4 \beta_{9} + \beta_{10} + \beta_{11} ) q^{63} \) \( + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{8} - 2 \beta_{10} ) q^{65} \) \( + ( 3 - 3 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{67} \) \( + ( 2 - 5 \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{69} \) \( + ( -3 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{71} \) \( + ( 1 - 3 \beta_{1} + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{73} \) \( + ( 1 + 2 \beta_{1} - 3 \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{75} \) \( + ( 2 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{77} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{79} \) \( + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{81} \) \( + ( -3 - 2 \beta_{1} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{9} - 4 \beta_{11} ) q^{83} \) \( + ( 2 - 3 \beta_{1} + \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{85} \) \( + ( 2 - 5 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{87} \) \( + ( 2 - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{89} \) \( + ( -\beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{91} \) \( + ( 1 + 7 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{93} \) \( + ( -2 + 7 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - 5 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{95} \) \( + ( 3 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{97} \) \( + ( -7 + \beta_{1} - \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - 6 \beta_{9} + 5 \beta_{10} - 3 \beta_{11} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 27q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 27q^{9} \) \(\mathstrut -\mathstrut 18q^{11} \) \(\mathstrut -\mathstrut 4q^{13} \) \(\mathstrut +\mathstrut 2q^{15} \) \(\mathstrut +\mathstrut 12q^{17} \) \(\mathstrut +\mathstrut 7q^{21} \) \(\mathstrut +\mathstrut 9q^{23} \) \(\mathstrut +\mathstrut 25q^{25} \) \(\mathstrut +\mathstrut 18q^{27} \) \(\mathstrut +\mathstrut 34q^{29} \) \(\mathstrut +\mathstrut 11q^{31} \) \(\mathstrut +\mathstrut 4q^{33} \) \(\mathstrut -\mathstrut 21q^{35} \) \(\mathstrut -\mathstrut 22q^{37} \) \(\mathstrut -\mathstrut 13q^{39} \) \(\mathstrut +\mathstrut 32q^{41} \) \(\mathstrut +\mathstrut 8q^{43} \) \(\mathstrut +\mathstrut 13q^{45} \) \(\mathstrut -\mathstrut 24q^{47} \) \(\mathstrut +\mathstrut 36q^{49} \) \(\mathstrut -\mathstrut 16q^{51} \) \(\mathstrut -\mathstrut 2q^{53} \) \(\mathstrut +\mathstrut 12q^{55} \) \(\mathstrut +\mathstrut 26q^{57} \) \(\mathstrut -\mathstrut 26q^{59} \) \(\mathstrut +\mathstrut 12q^{61} \) \(\mathstrut -\mathstrut 5q^{63} \) \(\mathstrut +\mathstrut 66q^{65} \) \(\mathstrut +\mathstrut 21q^{67} \) \(\mathstrut +\mathstrut 20q^{69} \) \(\mathstrut -\mathstrut 50q^{71} \) \(\mathstrut +\mathstrut 17q^{73} \) \(\mathstrut +\mathstrut 14q^{75} \) \(\mathstrut +\mathstrut 25q^{77} \) \(\mathstrut +\mathstrut 9q^{79} \) \(\mathstrut +\mathstrut 48q^{81} \) \(\mathstrut -\mathstrut 25q^{83} \) \(\mathstrut +\mathstrut 24q^{85} \) \(\mathstrut +\mathstrut 10q^{87} \) \(\mathstrut +\mathstrut 21q^{89} \) \(\mathstrut +\mathstrut 9q^{91} \) \(\mathstrut +\mathstrut 31q^{93} \) \(\mathstrut -\mathstrut 22q^{95} \) \(\mathstrut +\mathstrut 18q^{97} \) \(\mathstrut -\mathstrut 102q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(4\) \(x^{11}\mathstrut -\mathstrut \) \(22\) \(x^{10}\mathstrut +\mathstrut \) \(67\) \(x^{9}\mathstrut +\mathstrut \) \(180\) \(x^{8}\mathstrut -\mathstrut \) \(383\) \(x^{7}\mathstrut -\mathstrut \) \(674\) \(x^{6}\mathstrut +\mathstrut \) \(875\) \(x^{5}\mathstrut +\mathstrut \) \(1077\) \(x^{4}\mathstrut -\mathstrut \) \(704\) \(x^{3}\mathstrut -\mathstrut \) \(467\) \(x^{2}\mathstrut +\mathstrut \) \(210\) \(x\mathstrut +\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(75810347\) \(\nu^{11}\mathstrut -\mathstrut \) \(750391815\) \(\nu^{10}\mathstrut +\mathstrut \) \(1111655415\) \(\nu^{9}\mathstrut +\mathstrut \) \(8946386707\) \(\nu^{8}\mathstrut -\mathstrut \) \(26000581657\) \(\nu^{7}\mathstrut -\mathstrut \) \(23597601417\) \(\nu^{6}\mathstrut +\mathstrut \) \(123118084521\) \(\nu^{5}\mathstrut -\mathstrut \) \(14637536823\) \(\nu^{4}\mathstrut -\mathstrut \) \(181271492783\) \(\nu^{3}\mathstrut +\mathstrut \) \(60217693337\) \(\nu^{2}\mathstrut +\mathstrut \) \(55678620842\) \(\nu\mathstrut -\mathstrut \) \(21112206658\)\()/\)\(7685328593\)
\(\beta_{2}\)\(=\)\((\)\(85937175\) \(\nu^{11}\mathstrut -\mathstrut \) \(183905264\) \(\nu^{10}\mathstrut -\mathstrut \) \(2549240479\) \(\nu^{9}\mathstrut +\mathstrut \) \(2711003021\) \(\nu^{8}\mathstrut +\mathstrut \) \(25020049519\) \(\nu^{7}\mathstrut -\mathstrut \) \(12327447567\) \(\nu^{6}\mathstrut -\mathstrut \) \(102725962309\) \(\nu^{5}\mathstrut +\mathstrut \) \(14587858375\) \(\nu^{4}\mathstrut +\mathstrut \) \(174655180495\) \(\nu^{3}\mathstrut +\mathstrut \) \(9429164566\) \(\nu^{2}\mathstrut -\mathstrut \) \(94016465624\) \(\nu\mathstrut -\mathstrut \) \(7917809651\)\()/\)\(7685328593\)
\(\beta_{3}\)\(=\)\((\)\(106132083\) \(\nu^{11}\mathstrut -\mathstrut \) \(842730380\) \(\nu^{10}\mathstrut +\mathstrut \) \(89329233\) \(\nu^{9}\mathstrut +\mathstrut \) \(11763798682\) \(\nu^{8}\mathstrut -\mathstrut \) \(16228654654\) \(\nu^{7}\mathstrut -\mathstrut \) \(49547560230\) \(\nu^{6}\mathstrut +\mathstrut \) \(88737164877\) \(\nu^{5}\mathstrut +\mathstrut \) \(67227676657\) \(\nu^{4}\mathstrut -\mathstrut \) \(136892879041\) \(\nu^{3}\mathstrut +\mathstrut \) \(1500080205\) \(\nu^{2}\mathstrut +\mathstrut \) \(44971372794\) \(\nu\mathstrut -\mathstrut \) \(34213508272\)\()/\)\(7685328593\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(146823889\) \(\nu^{11}\mathstrut +\mathstrut \) \(471833772\) \(\nu^{10}\mathstrut +\mathstrut \) \(3883703902\) \(\nu^{9}\mathstrut -\mathstrut \) \(8276421068\) \(\nu^{8}\mathstrut -\mathstrut \) \(36270871889\) \(\nu^{7}\mathstrut +\mathstrut \) \(47108213047\) \(\nu^{6}\mathstrut +\mathstrut \) \(145148416284\) \(\nu^{5}\mathstrut -\mathstrut \) \(87396024562\) \(\nu^{4}\mathstrut -\mathstrut \) \(229152895577\) \(\nu^{3}\mathstrut +\mathstrut \) \(8295739074\) \(\nu^{2}\mathstrut +\mathstrut \) \(73229034680\) \(\nu\mathstrut +\mathstrut \) \(8817559953\)\()/\)\(7685328593\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(159236033\) \(\nu^{11}\mathstrut +\mathstrut \) \(320476659\) \(\nu^{10}\mathstrut +\mathstrut \) \(4957239696\) \(\nu^{9}\mathstrut -\mathstrut \) \(5371489680\) \(\nu^{8}\mathstrut -\mathstrut \) \(49463115679\) \(\nu^{7}\mathstrut +\mathstrut \) \(29554239942\) \(\nu^{6}\mathstrut +\mathstrut \) \(198500196358\) \(\nu^{5}\mathstrut -\mathstrut \) \(49365470596\) \(\nu^{4}\mathstrut -\mathstrut \) \(296222896280\) \(\nu^{3}\mathstrut -\mathstrut \) \(12686086641\) \(\nu^{2}\mathstrut +\mathstrut \) \(72207935258\) \(\nu\mathstrut +\mathstrut \) \(3243266922\)\()/\)\(7685328593\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(194027177\) \(\nu^{11}\mathstrut +\mathstrut \) \(1001308266\) \(\nu^{10}\mathstrut +\mathstrut \) \(2733553715\) \(\nu^{9}\mathstrut -\mathstrut \) \(14135059823\) \(\nu^{8}\mathstrut -\mathstrut \) \(13861957626\) \(\nu^{7}\mathstrut +\mathstrut \) \(60776636563\) \(\nu^{6}\mathstrut +\mathstrut \) \(45916807651\) \(\nu^{5}\mathstrut -\mathstrut \) \(85839095249\) \(\nu^{4}\mathstrut -\mathstrut \) \(99449849249\) \(\nu^{3}\mathstrut +\mathstrut \) \(19870122683\) \(\nu^{2}\mathstrut +\mathstrut \) \(45163386928\) \(\nu\mathstrut -\mathstrut \) \(485767656\)\()/\)\(7685328593\)
\(\beta_{7}\)\(=\)\((\)\(206991323\) \(\nu^{11}\mathstrut -\mathstrut \) \(573051401\) \(\nu^{10}\mathstrut -\mathstrut \) \(5843045880\) \(\nu^{9}\mathstrut +\mathstrut \) \(9800401503\) \(\nu^{8}\mathstrut +\mathstrut \) \(56958916010\) \(\nu^{7}\mathstrut -\mathstrut \) \(54164896713\) \(\nu^{6}\mathstrut -\mathstrut \) \(235069553745\) \(\nu^{5}\mathstrut +\mathstrut \) \(92778145999\) \(\nu^{4}\mathstrut +\mathstrut \) \(381576916307\) \(\nu^{3}\mathstrut +\mathstrut \) \(8225672939\) \(\nu^{2}\mathstrut -\mathstrut \) \(134259974164\) \(\nu\mathstrut +\mathstrut \) \(11406893253\)\()/\)\(7685328593\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(210621704\) \(\nu^{11}\mathstrut +\mathstrut \) \(1410256576\) \(\nu^{10}\mathstrut +\mathstrut \) \(1550724992\) \(\nu^{9}\mathstrut -\mathstrut \) \(21641699784\) \(\nu^{8}\mathstrut +\mathstrut \) \(8025640067\) \(\nu^{7}\mathstrut +\mathstrut \) \(109632558981\) \(\nu^{6}\mathstrut -\mathstrut \) \(76381105265\) \(\nu^{5}\mathstrut -\mathstrut \) \(223152185298\) \(\nu^{4}\mathstrut +\mathstrut \) \(152245136301\) \(\nu^{3}\mathstrut +\mathstrut \) \(170615805006\) \(\nu^{2}\mathstrut -\mathstrut \) \(84649349814\) \(\nu\mathstrut -\mathstrut \) \(34320821610\)\()/\)\(7685328593\)
\(\beta_{9}\)\(=\)\((\)\(246499891\) \(\nu^{11}\mathstrut -\mathstrut \) \(1029238772\) \(\nu^{10}\mathstrut -\mathstrut \) \(4971149628\) \(\nu^{9}\mathstrut +\mathstrut \) \(16303798652\) \(\nu^{8}\mathstrut +\mathstrut \) \(36169318226\) \(\nu^{7}\mathstrut -\mathstrut \) \(84855835740\) \(\nu^{6}\mathstrut -\mathstrut \) \(115834448573\) \(\nu^{5}\mathstrut +\mathstrut \) \(164114746145\) \(\nu^{4}\mathstrut +\mathstrut \) \(146657810613\) \(\nu^{3}\mathstrut -\mathstrut \) \(94071613172\) \(\nu^{2}\mathstrut -\mathstrut \) \(30477383004\) \(\nu\mathstrut +\mathstrut \) \(24583412642\)\()/\)\(7685328593\)
\(\beta_{10}\)\(=\)\((\)\(566115680\) \(\nu^{11}\mathstrut -\mathstrut \) \(2117638831\) \(\nu^{10}\mathstrut -\mathstrut \) \(12926378732\) \(\nu^{9}\mathstrut +\mathstrut \) \(34046046658\) \(\nu^{8}\mathstrut +\mathstrut \) \(110177243468\) \(\nu^{7}\mathstrut -\mathstrut \) \(180551433551\) \(\nu^{6}\mathstrut -\mathstrut \) \(428670181367\) \(\nu^{5}\mathstrut +\mathstrut \) \(350202803716\) \(\nu^{4}\mathstrut +\mathstrut \) \(697102611922\) \(\nu^{3}\mathstrut -\mathstrut \) \(169392543143\) \(\nu^{2}\mathstrut -\mathstrut \) \(272671761634\) \(\nu\mathstrut +\mathstrut \) \(37969929527\)\()/\)\(7685328593\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(678429148\) \(\nu^{11}\mathstrut +\mathstrut \) \(3136864086\) \(\nu^{10}\mathstrut +\mathstrut \) \(11829253654\) \(\nu^{9}\mathstrut -\mathstrut \) \(46947000260\) \(\nu^{8}\mathstrut -\mathstrut \) \(76727815363\) \(\nu^{7}\mathstrut +\mathstrut \) \(222220060954\) \(\nu^{6}\mathstrut +\mathstrut \) \(246875385165\) \(\nu^{5}\mathstrut -\mathstrut \) \(359864002008\) \(\nu^{4}\mathstrut -\mathstrut \) \(366514343268\) \(\nu^{3}\mathstrut +\mathstrut \) \(96578599536\) \(\nu^{2}\mathstrut +\mathstrut \) \(114496122092\) \(\nu\mathstrut -\mathstrut \) \(915307019\)\()/\)\(7685328593\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(12\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(6\) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(13\) \(\beta_{9}\mathstrut -\mathstrut \) \(18\) \(\beta_{8}\mathstrut -\mathstrut \) \(5\) \(\beta_{7}\mathstrut +\mathstrut \) \(5\) \(\beta_{6}\mathstrut -\mathstrut \) \(12\) \(\beta_{5}\mathstrut -\mathstrut \) \(18\) \(\beta_{4}\mathstrut -\mathstrut \) \(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(23\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(28\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(4\) \(\beta_{11}\mathstrut -\mathstrut \) \(14\) \(\beta_{10}\mathstrut -\mathstrut \) \(83\) \(\beta_{9}\mathstrut -\mathstrut \) \(78\) \(\beta_{8}\mathstrut -\mathstrut \) \(49\) \(\beta_{7}\mathstrut +\mathstrut \) \(21\) \(\beta_{6}\mathstrut -\mathstrut \) \(50\) \(\beta_{5}\mathstrut -\mathstrut \) \(98\) \(\beta_{4}\mathstrut -\mathstrut \) \(47\) \(\beta_{3}\mathstrut +\mathstrut \) \(105\) \(\beta_{2}\mathstrut +\mathstrut \) \(33\) \(\beta_{1}\mathstrut +\mathstrut \) \(184\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(54\) \(\beta_{11}\mathstrut -\mathstrut \) \(80\) \(\beta_{10}\mathstrut -\mathstrut \) \(413\) \(\beta_{9}\mathstrut -\mathstrut \) \(442\) \(\beta_{8}\mathstrut -\mathstrut \) \(173\) \(\beta_{7}\mathstrut +\mathstrut \) \(121\) \(\beta_{6}\mathstrut -\mathstrut \) \(266\) \(\beta_{5}\mathstrut -\mathstrut \) \(464\) \(\beta_{4}\mathstrut -\mathstrut \) \(239\) \(\beta_{3}\mathstrut +\mathstrut \) \(575\) \(\beta_{2}\mathstrut +\mathstrut \) \(203\) \(\beta_{1}\mathstrut +\mathstrut \) \(744\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(114\) \(\beta_{11}\mathstrut -\mathstrut \) \(454\) \(\beta_{10}\mathstrut -\mathstrut \) \(2185\) \(\beta_{9}\mathstrut -\mathstrut \) \(2162\) \(\beta_{8}\mathstrut -\mathstrut \) \(1033\) \(\beta_{7}\mathstrut +\mathstrut \) \(533\) \(\beta_{6}\mathstrut -\mathstrut \) \(1274\) \(\beta_{5}\mathstrut -\mathstrut \) \(2436\) \(\beta_{4}\mathstrut -\mathstrut \) \(1295\) \(\beta_{3}\mathstrut +\mathstrut \) \(2861\) \(\beta_{2}\mathstrut +\mathstrut \) \(869\) \(\beta_{1}\mathstrut +\mathstrut \) \(4060\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(804\) \(\beta_{11}\mathstrut -\mathstrut \) \(2316\) \(\beta_{10}\mathstrut -\mathstrut \) \(11057\) \(\beta_{9}\mathstrut -\mathstrut \) \(11262\) \(\beta_{8}\mathstrut -\mathstrut \) \(4717\) \(\beta_{7}\mathstrut +\mathstrut \) \(2873\) \(\beta_{6}\mathstrut -\mathstrut \) \(6574\) \(\beta_{5}\mathstrut -\mathstrut \) \(12044\) \(\beta_{4}\mathstrut -\mathstrut \) \(6515\) \(\beta_{3}\mathstrut +\mathstrut \) \(14747\) \(\beta_{2}\mathstrut +\mathstrut \) \(4655\) \(\beta_{1}\mathstrut +\mathstrut \) \(19366\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(3012\) \(\beta_{11}\mathstrut -\mathstrut \) \(12194\) \(\beta_{10}\mathstrut -\mathstrut \) \(56555\) \(\beta_{9}\mathstrut -\mathstrut \) \(56556\) \(\beta_{8}\mathstrut -\mathstrut \) \(24987\) \(\beta_{7}\mathstrut +\mathstrut \) \(13785\) \(\beta_{6}\mathstrut -\mathstrut \) \(32756\) \(\beta_{5}\mathstrut -\mathstrut \) \(61798\) \(\beta_{4}\mathstrut -\mathstrut \) \(33615\) \(\beta_{3}\mathstrut +\mathstrut \) \(74493\) \(\beta_{2}\mathstrut +\mathstrut \) \(22409\) \(\beta_{1}\mathstrut +\mathstrut \) \(100286\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(16918\) \(\beta_{11}\mathstrut -\mathstrut \) \(61660\) \(\beta_{10}\mathstrut -\mathstrut \) \(286719\) \(\beta_{9}\mathstrut -\mathstrut \) \(288812\) \(\beta_{8}\mathstrut -\mathstrut \) \(123013\) \(\beta_{7}\mathstrut +\mathstrut \) \(71435\) \(\beta_{6}\mathstrut -\mathstrut \) \(167096\) \(\beta_{5}\mathstrut -\mathstrut \) \(310766\) \(\beta_{4}\mathstrut -\mathstrut \) \(169825\) \(\beta_{3}\mathstrut +\mathstrut \) \(379179\) \(\beta_{2}\mathstrut +\mathstrut \) \(115501\) \(\beta_{1}\mathstrut +\mathstrut \) \(500102\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(78302\) \(\beta_{11}\mathstrut -\mathstrut \) \(316360\) \(\beta_{10}\mathstrut -\mathstrut \) \(1456547\) \(\beta_{9}\mathstrut -\mathstrut \) \(1460328\) \(\beta_{8}\mathstrut -\mathstrut \) \(631427\) \(\beta_{7}\mathstrut +\mathstrut \) \(355657\) \(\beta_{6}\mathstrut -\mathstrut \) \(842796\) \(\beta_{5}\mathstrut -\mathstrut \) \(1581578\) \(\beta_{4}\mathstrut -\mathstrut \) \(865185\) \(\beta_{3}\mathstrut +\mathstrut \) \(1920919\) \(\beta_{2}\mathstrut +\mathstrut \) \(576653\) \(\beta_{1}\mathstrut +\mathstrut \) \(2551408\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(409448\) \(\beta_{11}\mathstrut -\mathstrut \) \(1601440\) \(\beta_{10}\mathstrut -\mathstrut \) \(7385239\) \(\beta_{9}\mathstrut -\mathstrut \) \(7418958\) \(\beta_{8}\mathstrut -\mathstrut \) \(3174907\) \(\beta_{7}\mathstrut +\mathstrut \) \(1816091\) \(\beta_{6}\mathstrut -\mathstrut \) \(4281724\) \(\beta_{5}\mathstrut -\mathstrut \) \(7998624\) \(\beta_{4}\mathstrut -\mathstrut \) \(4381093\) \(\beta_{3}\mathstrut +\mathstrut \) \(9749363\) \(\beta_{2}\mathstrut +\mathstrut \) \(2938403\) \(\beta_{1}\mathstrut +\mathstrut \) \(12876656\)\()/2\)

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
2.60861
−2.75875
0.753640
1.84513
−1.33502
5.07033
−0.00933287
−0.875563
−1.97018
−2.01674
0.396940
2.29094
0 −2.92999 0 −1.53308 0 4.18455 0 5.58485 0
1.2 0 −2.73829 0 −1.48375 0 −2.26909 0 4.49825 0
1.3 0 −2.49671 0 2.79920 0 −1.31469 0 3.23355 0
1.4 0 −1.99694 0 2.93695 0 −3.10313 0 0.987768 0
1.5 0 −0.656123 0 4.07967 0 0.827612 0 −2.56950 0
1.6 0 0.163781 0 −2.50185 0 −0.0804128 0 −2.97318 0
1.7 0 1.05851 0 −3.62237 0 4.42054 0 −1.87955 0
1.8 0 1.36355 0 2.23842 0 −4.41277 0 −1.14074 0
1.9 0 1.67724 0 1.77374 0 3.87316 0 −0.186857 0
1.10 0 3.12415 0 −0.847598 0 2.65658 0 6.76030 0
1.11 0 3.15470 0 4.23481 0 1.79853 0 6.95214 0
1.12 0 3.27612 0 −1.07414 0 −4.58087 0 7.73297 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(503\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\(T_{3}^{12} - \cdots\)
\(T_{5}^{12} - \cdots\)
\(T_{7}^{12} - \cdots\)
\(T_{13}^{12} + \cdots\)