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: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} - 22 x^{10} + 67 x^{9} + 180 x^{8} - 383 x^{7} - 674 x^{6} + 875 x^{5} + 1077 x^{4} - 704 x^{3} - 467 x^{2} + 210 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1006)
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 3q^{3} + 7q^{5} + 2q^{7} + 27q^{9} + O(q^{10}) \) \( 12q + 3q^{3} + 7q^{5} + 2q^{7} + 27q^{9} - 18q^{11} - 4q^{13} + 2q^{15} + 12q^{17} + 7q^{21} + 9q^{23} + 25q^{25} + 18q^{27} + 34q^{29} + 11q^{31} + 4q^{33} - 21q^{35} - 22q^{37} - 13q^{39} + 32q^{41} + 8q^{43} + 13q^{45} - 24q^{47} + 36q^{49} - 16q^{51} - 2q^{53} + 12q^{55} + 26q^{57} - 26q^{59} + 12q^{61} - 5q^{63} + 66q^{65} + 21q^{67} + 20q^{69} - 50q^{71} + 17q^{73} + 14q^{75} + 25q^{77} + 9q^{79} + 48q^{81} - 25q^{83} + 24q^{85} + 10q^{87} + 21q^{89} + 9q^{91} + 31q^{93} - 22q^{95} + 18q^{97} - 102q^{99} + O(q^{100}) \)

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

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

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8048.2.a.r 12
4.b odd 2 1 1006.2.a.i 12
12.b even 2 1 9054.2.a.bj 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1006.2.a.i 12 4.b odd 2 1
8048.2.a.r 12 1.a even 1 1 trivial
9054.2.a.bj 12 12.b even 2 1

Atkin-Lehner signs

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

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 3 T + 9 T^{2} - 21 T^{3} + 51 T^{4} - 125 T^{5} + 293 T^{6} - 592 T^{7} + 1250 T^{8} - 2291 T^{9} + 4498 T^{10} - 8234 T^{11} + 14628 T^{12} - 24702 T^{13} + 40482 T^{14} - 61857 T^{15} + 101250 T^{16} - 143856 T^{17} + 213597 T^{18} - 273375 T^{19} + 334611 T^{20} - 413343 T^{21} + 531441 T^{22} - 531441 T^{23} + 531441 T^{24} \)
$5$ \( 1 - 7 T + 42 T^{2} - 190 T^{3} + 781 T^{4} - 2772 T^{5} + 9074 T^{6} - 27190 T^{7} + 76806 T^{8} - 202490 T^{9} + 509798 T^{10} - 1215619 T^{11} + 2791764 T^{12} - 6078095 T^{13} + 12744950 T^{14} - 25311250 T^{15} + 48003750 T^{16} - 84968750 T^{17} + 141781250 T^{18} - 216562500 T^{19} + 305078125 T^{20} - 371093750 T^{21} + 410156250 T^{22} - 341796875 T^{23} + 244140625 T^{24} \)
$7$ \( 1 - 2 T + 26 T^{2} - 41 T^{3} + 387 T^{4} - 495 T^{5} + 4456 T^{6} - 4830 T^{7} + 43901 T^{8} - 43875 T^{9} + 380095 T^{10} - 361109 T^{11} + 2860403 T^{12} - 2527763 T^{13} + 18624655 T^{14} - 15049125 T^{15} + 105406301 T^{16} - 81177810 T^{17} + 524243944 T^{18} - 407653785 T^{19} + 2230977987 T^{20} - 1654497887 T^{21} + 7344356474 T^{22} - 3954653486 T^{23} + 13841287201 T^{24} \)
$11$ \( 1 + 18 T + 206 T^{2} + 1751 T^{3} + 12433 T^{4} + 76100 T^{5} + 416305 T^{6} + 2060807 T^{7} + 9385744 T^{8} + 39508541 T^{9} + 155009377 T^{10} + 567389449 T^{11} + 1945839084 T^{12} + 6241283939 T^{13} + 18756134617 T^{14} + 52585868071 T^{15} + 137416677904 T^{16} + 331895028157 T^{17} + 737509702105 T^{18} + 1482973713100 T^{19} + 2665123967473 T^{20} + 4128766406941 T^{21} + 5343109467806 T^{22} + 5135610070998 T^{23} + 3138428376721 T^{24} \)
$13$ \( 1 + 4 T + 65 T^{2} + 248 T^{3} + 2225 T^{4} + 7917 T^{5} + 54689 T^{6} + 179986 T^{7} + 1073280 T^{8} + 3265684 T^{9} + 17414630 T^{10} + 49451901 T^{11} + 242072156 T^{12} + 642874713 T^{13} + 2943072470 T^{14} + 7174707748 T^{15} + 30653950080 T^{16} + 66827541898 T^{17} + 263973357401 T^{18} + 496780009089 T^{19} + 1815000854225 T^{20} + 2629915844504 T^{21} + 8960801970185 T^{22} + 7168641576148 T^{23} + 23298085122481 T^{24} \)
$17$ \( 1 - 12 T + 166 T^{2} - 1221 T^{3} + 9798 T^{4} - 51318 T^{5} + 298418 T^{6} - 1123183 T^{7} + 5047911 T^{8} - 11448777 T^{9} + 42672488 T^{10} + 1695375 T^{11} + 253046308 T^{12} + 28821375 T^{13} + 12332349032 T^{14} - 56247841401 T^{15} + 421606574631 T^{16} - 1594759244831 T^{17} + 7203085065842 T^{18} - 21057760021014 T^{19} + 68348471406918 T^{20} - 144795797202837 T^{21} + 334654987474534 T^{22} - 411262755691596 T^{23} + 582622237229761 T^{24} \)
$19$ \( 1 + 59 T^{2} + 67 T^{3} + 2447 T^{4} + 3079 T^{5} + 74847 T^{6} + 136062 T^{7} + 1860811 T^{8} + 3815153 T^{9} + 41316958 T^{10} + 91606659 T^{11} + 798996362 T^{12} + 1740526521 T^{13} + 14915421838 T^{14} + 26168134427 T^{15} + 242502750331 T^{16} + 336902982138 T^{17} + 3521243055207 T^{18} + 2752231084381 T^{19} + 41558778761327 T^{20} + 21620075751193 T^{21} + 361732909210259 T^{22} + 2213314919066161 T^{24} \)
$23$ \( 1 - 9 T + 157 T^{2} - 1198 T^{3} + 12814 T^{4} - 84379 T^{5} + 700261 T^{6} - 4074957 T^{7} + 28422478 T^{8} - 148320623 T^{9} + 904244314 T^{10} - 4251116175 T^{11} + 23114338603 T^{12} - 97775672025 T^{13} + 478345242106 T^{14} - 1804617020041 T^{15} + 7953774665998 T^{16} - 26227820962251 T^{17} + 103663759667029 T^{18} - 287295766392413 T^{19} + 1003476965390734 T^{20} - 2157780888432674 T^{21} + 6503962260542893 T^{22} - 8575287821225343 T^{23} + 21914624432020321 T^{24} \)
$29$ \( 1 - 34 T + 717 T^{2} - 11125 T^{3} + 141257 T^{4} - 1531538 T^{5} + 14628117 T^{6} - 125171225 T^{7} + 971974667 T^{8} - 6905339085 T^{9} + 45156961894 T^{10} - 272841031757 T^{11} + 1526411184342 T^{12} - 7912389920953 T^{13} + 37977004952854 T^{14} - 168414314944065 T^{15} + 687459214450427 T^{16} - 2567405646487525 T^{17} + 8701145133916557 T^{18} - 26418841062533242 T^{19} + 70663307555631977 T^{20} - 161391998981542625 T^{21} + 301647086276244117 T^{22} - 414817332033998186 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 - 11 T + 246 T^{2} - 2242 T^{3} + 28987 T^{4} - 229197 T^{5} + 2208928 T^{6} - 15528844 T^{7} + 122865191 T^{8} - 777995717 T^{9} + 5304569738 T^{10} - 30338429999 T^{11} + 183095285754 T^{12} - 940491329969 T^{13} + 5097691518218 T^{14} - 23177270405147 T^{15} + 113468584057511 T^{16} - 444577619731444 T^{17} + 1960431731063968 T^{18} - 6305808616398867 T^{19} + 24722752502302267 T^{20} - 59277632884224382 T^{21} + 201628558597277046 T^{22} - 279493245860453141 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 + 22 T + 455 T^{2} + 6559 T^{3} + 85475 T^{4} + 942209 T^{5} + 9511859 T^{6} + 86045010 T^{7} + 722793987 T^{8} + 5570654269 T^{9} + 40242008154 T^{10} + 269557457559 T^{11} + 1700095666794 T^{12} + 9973625929683 T^{13} + 55091309162826 T^{14} + 282170350687657 T^{15} + 1354632301469907 T^{16} + 5966701473504570 T^{17} + 24404827834984331 T^{18} + 89445669021606797 T^{19} + 300229181323897475 T^{20} + 852419051315910043 T^{21} + 2187905889450121295 T^{22} + 3914187679148129086 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 - 32 T + 770 T^{2} - 13419 T^{3} + 199542 T^{4} - 2512554 T^{5} + 28288052 T^{6} - 283801421 T^{7} + 2605368715 T^{8} - 21811184839 T^{9} + 169130978058 T^{10} - 1209112436159 T^{11} + 8049974165820 T^{12} - 49573609882519 T^{13} + 284309174115498 T^{14} - 1503248670288719 T^{15} + 7362149305467115 T^{16} - 32880154475461621 T^{17} + 134371195774828532 T^{18} - 489330629856802074 T^{19} + 1593327950069262582 T^{20} - 4393138177632562659 T^{21} + 10335447668817348770 T^{22} - 17610529014919950112 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 - 8 T + 213 T^{2} - 1770 T^{3} + 23231 T^{4} - 166041 T^{5} + 1560689 T^{6} - 8765026 T^{7} + 63390606 T^{8} - 259839552 T^{9} + 1507551598 T^{10} - 3483878321 T^{11} + 34544523436 T^{12} - 149806767803 T^{13} + 2787462904702 T^{14} - 20659063260864 T^{15} + 216719867183406 T^{16} - 1288532825114518 T^{17} + 9865681775580761 T^{18} - 45133034006817387 T^{19} + 271528580648948831 T^{20} - 889588923128212110 T^{21} + 4603245732729545037 T^{22} - 7434349915769781656 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 + 24 T + 624 T^{2} + 9813 T^{3} + 154353 T^{4} + 1867001 T^{5} + 22286272 T^{6} + 222278140 T^{7} + 2182329723 T^{8} + 18581477729 T^{9} + 155728331067 T^{10} + 1151472566375 T^{11} + 8382970589145 T^{12} + 54119210619625 T^{13} + 344003883327003 T^{14} + 1929184762257967 T^{15} + 10649072885058363 T^{16} + 50978381574246980 T^{17} + 240228524768663488 T^{18} + 945865872527541463 T^{19} + 3675343530102795633 T^{20} + 10982027332557452571 T^{21} + 32821858515157950576 T^{22} + 59331821162016295272 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 + 2 T + 395 T^{2} + 573 T^{3} + 79159 T^{4} + 85111 T^{5} + 10523539 T^{6} + 8593086 T^{7} + 1028184511 T^{8} + 664497415 T^{9} + 77483123342 T^{10} + 41921174393 T^{11} + 4607531907962 T^{12} + 2221822242829 T^{13} + 217650093467678 T^{14} + 98928381652955 T^{15} + 8112870348539791 T^{16} + 3593589836161398 T^{17} + 233247518751115531 T^{18} + 99980839822666907 T^{19} + 4928414833272925399 T^{20} + 1890764538102622209 T^{21} + 69080550794377654355 T^{22} + 18538071858744383194 T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( 1 + 26 T + 437 T^{2} + 5246 T^{3} + 56014 T^{4} + 561929 T^{5} + 5627753 T^{6} + 53428741 T^{7} + 487722895 T^{8} + 4192620027 T^{9} + 35209729306 T^{10} + 283290463699 T^{11} + 2227511236260 T^{12} + 16714137358241 T^{13} + 122565067714186 T^{14} + 861076108525233 T^{15} + 5909914386680095 T^{16} + 38197505205877559 T^{17} + 237381624739738673 T^{18} + 1398445440212855851 T^{19} + 8224560131968436494 T^{20} + 45446076064663809994 T^{21} + \)\(22\!\cdots\!37\)\( T^{22} + \)\(78\!\cdots\!34\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 - 12 T + 461 T^{2} - 4767 T^{3} + 106261 T^{4} - 975590 T^{5} + 16121299 T^{6} - 133141815 T^{7} + 1794097610 T^{8} - 13394248577 T^{9} + 154375149996 T^{10} - 1039460652715 T^{11} + 10534585951880 T^{12} - 63407099815615 T^{13} + 574429933135116 T^{14} - 3040239936256037 T^{15} + 24840790246540010 T^{16} - 112451084457426315 T^{17} + 830575359665614939 T^{18} - 3066028483393727390 T^{19} + 20371010786404076341 T^{20} - 55745994424540350147 T^{21} + \)\(32\!\cdots\!61\)\( T^{22} - \)\(52\!\cdots\!32\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 - 21 T + 458 T^{2} - 6321 T^{3} + 87583 T^{4} - 965771 T^{5} + 10922727 T^{6} - 107004290 T^{7} + 1094708152 T^{8} - 9997299639 T^{9} + 94636033095 T^{10} - 802402093488 T^{11} + 6948793830000 T^{12} - 53760940263696 T^{13} + 424821152563455 T^{14} - 3006817831324557 T^{15} + 22059596430638392 T^{16} - 144469178485709030 T^{17} + 988052213293654863 T^{18} - 5853259507784399033 T^{19} + 35564625403443288703 T^{20} - \)\(17\!\cdots\!87\)\( T^{21} + \)\(83\!\cdots\!42\)\( T^{22} - \)\(25\!\cdots\!43\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 + 50 T + 1711 T^{2} + 43305 T^{3} + 908171 T^{4} + 16189750 T^{5} + 253701007 T^{6} + 3535881037 T^{7} + 44474886435 T^{8} + 507739217863 T^{9} + 5299361943762 T^{10} + 50676244010449 T^{11} + 445479921732642 T^{12} + 3598013324741879 T^{13} + 26714083558504242 T^{14} + 181725451204564193 T^{15} + 1130181626597447235 T^{16} + 6379540348599716987 T^{17} + 32499171027743608447 T^{18} + \)\(14\!\cdots\!50\)\( T^{19} + \)\(58\!\cdots\!31\)\( T^{20} + \)\(19\!\cdots\!55\)\( T^{21} + \)\(55\!\cdots\!11\)\( T^{22} + \)\(11\!\cdots\!50\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 - 17 T + 610 T^{2} - 8290 T^{3} + 174647 T^{4} - 2005546 T^{5} + 31885540 T^{6} - 318801846 T^{7} + 4202591870 T^{8} - 37297650102 T^{9} + 427206769000 T^{10} - 3402453895951 T^{11} + 34693832766584 T^{12} - 248379134404423 T^{13} + 2276584872001000 T^{14} - 14509419949729734 T^{15} + 119346216748900670 T^{16} - 660899050738560678 T^{17} + 4825373525706961060 T^{18} - 22156065910380911962 T^{19} + \)\(14\!\cdots\!07\)\( T^{20} - \)\(48\!\cdots\!70\)\( T^{21} + \)\(26\!\cdots\!90\)\( T^{22} - \)\(53\!\cdots\!09\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 - 9 T + 648 T^{2} - 6279 T^{3} + 204926 T^{4} - 2056730 T^{5} + 42044625 T^{6} - 420858912 T^{7} + 6250252261 T^{8} - 60111649704 T^{9} + 709782021291 T^{10} - 6315539448032 T^{11} + 63127054087280 T^{12} - 498927616394528 T^{13} + 4429749594877131 T^{14} - 29637387658410456 T^{15} + 243447831836383141 T^{16} - 1295006608245777888 T^{17} + 10220520909584624625 T^{18} - 39497255729102800070 T^{19} + \)\(31\!\cdots\!86\)\( T^{20} - \)\(75\!\cdots\!01\)\( T^{21} + \)\(61\!\cdots\!48\)\( T^{22} - \)\(67\!\cdots\!11\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 + 25 T + 697 T^{2} + 12233 T^{3} + 208949 T^{4} + 2854575 T^{5} + 37114827 T^{6} + 419442950 T^{7} + 4525988754 T^{8} + 44654078439 T^{9} + 428913306760 T^{10} + 3945221485244 T^{11} + 36292751560520 T^{12} + 327453383275252 T^{13} + 2954783770269640 T^{14} + 25532621548400493 T^{15} + 214795827129722034 T^{16} + 1652202827469816850 T^{17} + 12134335396905842163 T^{18} + 77461892753714493525 T^{19} + \)\(47\!\cdots\!09\)\( T^{20} + \)\(22\!\cdots\!99\)\( T^{21} + \)\(10\!\cdots\!53\)\( T^{22} + \)\(32\!\cdots\!75\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 - 21 T + 781 T^{2} - 13992 T^{3} + 297410 T^{4} - 4532053 T^{5} + 72309901 T^{6} - 948480268 T^{7} + 12500242567 T^{8} - 143122686371 T^{9} + 1625460925398 T^{10} - 16402737491831 T^{11} + 163663794940716 T^{12} - 1459843636772959 T^{13} + 12875275990077558 T^{14} - 100897057088277499 T^{15} + 784293231697172647 T^{16} - 5296370202715452332 T^{17} + 35936667948242104861 T^{18} - \)\(20\!\cdots\!37\)\( T^{19} + \)\(11\!\cdots\!10\)\( T^{20} - \)\(49\!\cdots\!28\)\( T^{21} + \)\(24\!\cdots\!81\)\( T^{22} - \)\(58\!\cdots\!69\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 - 18 T + 591 T^{2} - 8155 T^{3} + 166085 T^{4} - 1930089 T^{5} + 31869251 T^{6} - 331651870 T^{7} + 4795815957 T^{8} - 45773961233 T^{9} + 596253361156 T^{10} - 5240536538331 T^{11} + 62648823242718 T^{12} - 508332044218107 T^{13} + 5610147875116804 T^{14} - 41776658520405809 T^{15} + 424570138481536917 T^{16} - 2848007454560330590 T^{17} + 26546193901055538179 T^{18} - \)\(15\!\cdots\!57\)\( T^{19} + \)\(13\!\cdots\!85\)\( T^{20} - \)\(61\!\cdots\!35\)\( T^{21} + \)\(43\!\cdots\!59\)\( T^{22} - \)\(12\!\cdots\!54\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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