Properties

Label 7.7.d
Level 7
Weight 7
Character orbit d
Rep. character \(\chi_{7}(3,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 6
Newform subspaces 2
Sturm bound 4
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 7.d (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(4\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(7, [\chi])\).

Total New Old
Modular forms 10 10 0
Cusp forms 6 6 0
Eisenstein series 4 4 0

Trace form

\( 6q + 4q^{2} - 3q^{3} - 20q^{4} + 165q^{5} - 406q^{7} - 1312q^{8} + 42q^{9} + O(q^{10}) \) \( 6q + 4q^{2} - 3q^{3} - 20q^{4} + 165q^{5} - 406q^{7} - 1312q^{8} + 42q^{9} + 2580q^{10} + 403q^{11} + 5964q^{12} - 5936q^{14} - 20910q^{15} + 3064q^{16} + 8229q^{17} + 6672q^{18} + 29985q^{19} - 8967q^{21} - 63584q^{22} + 8383q^{23} + 28872q^{24} + 19950q^{25} + 33768q^{26} - 18004q^{28} - 26588q^{29} - 10560q^{30} - 39399q^{31} - 39072q^{32} - 36711q^{33} + 46305q^{35} + 175488q^{36} + 17707q^{37} + 60960q^{38} + 16548q^{39} - 137280q^{40} - 22344q^{42} + 285388q^{43} - 122652q^{44} - 370530q^{45} - 184592q^{46} - 407823q^{47} + 375438q^{49} + 578800q^{50} + 64809q^{51} + 128856q^{52} + 165043q^{53} - 44496q^{54} - 214480q^{56} - 141534q^{57} - 11696q^{58} - 436299q^{59} - 207900q^{60} + 141201q^{61} + 380898q^{63} - 778720q^{64} + 10500q^{65} + 1025532q^{66} + 701271q^{67} + 799596q^{68} - 1634640q^{70} - 1277684q^{71} + 141072q^{72} - 421599q^{73} - 688900q^{74} + 826050q^{75} + 1149967q^{77} - 1268736q^{78} + 573759q^{79} + 2654040q^{80} + 872505q^{81} + 2585016q^{82} - 1982148q^{84} - 1677390q^{85} - 947680q^{86} - 1909746q^{87} - 556648q^{88} - 2990823q^{89} + 711480q^{91} + 1178376q^{92} + 551337q^{93} + 223584q^{94} + 958485q^{95} + 972720q^{96} + 1050364q^{98} + 1879428q^{99} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(7, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
7.7.d.a \(2\) \(1.610\) \(\Q(\sqrt{-3}) \) None \(12\) \(-21\) \(315\) \(-686\) \(q+12\zeta_{6}q^{2}+(-7-7\zeta_{6})q^{3}+(-80+\cdots)q^{4}+\cdots\)
7.7.d.b \(4\) \(1.610\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(-8\) \(18\) \(-150\) \(280\) \(q+(-4-4\beta _{1}+\beta _{2}+\beta _{3})q^{2}+(6+3\beta _{1}+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 12 T + 80 T^{2} - 768 T^{3} + 4096 T^{4} \))(\( 1 + 8 T - 62 T^{2} - 16 T^{3} + 7684 T^{4} - 1024 T^{5} - 253952 T^{6} + 2097152 T^{7} + 16777216 T^{8} \))
$3$ (\( 1 + 21 T + 876 T^{2} + 15309 T^{3} + 531441 T^{4} \))(\( 1 - 18 T + 579 T^{2} - 8478 T^{3} - 230868 T^{4} - 6180462 T^{5} + 307704339 T^{6} - 6973568802 T^{7} + 282429536481 T^{8} \))
$5$ (\( 1 - 315 T + 48700 T^{2} - 4921875 T^{3} + 244140625 T^{4} \))(\( ( 1 + 50 T + 15625 T^{2} )^{2}( 1 + 50 T - 13125 T^{2} + 781250 T^{3} + 244140625 T^{4} ) \))
$7$ (\( ( 1 + 343 T )^{2} \))(\( 1 - 280 T - 30870 T^{2} - 32941720 T^{3} + 13841287201 T^{4} \))
$11$ (\( 1 + 1479 T + 415880 T^{2} + 2620138719 T^{3} + 3138428376721 T^{4} \))(\( 1 - 1882 T - 296981 T^{2} - 556663606 T^{3} + 5324028798940 T^{4} - 986163534508966 T^{5} - 932053597746979301 T^{6} - \)\(10\!\cdots\!42\)\( T^{7} + \)\(98\!\cdots\!41\)\( T^{8} \))
$13$ (\( 1 - 9418418 T^{2} + 23298085122481 T^{4} \))(\( 1 - 5662108 T^{2} + 45798091584678 T^{4} - \)\(13\!\cdots\!48\)\( T^{6} + \)\(54\!\cdots\!61\)\( T^{8} \))
$17$ (\( 1 + 5229 T + 33251716 T^{2} + 126215348301 T^{3} + 582622237229761 T^{4} \))(\( 1 - 13458 T + 123704369 T^{2} - 852319108698 T^{3} + 4885539755960772 T^{4} - 20572911296216475162 T^{5} + \)\(72\!\cdots\!09\)\( T^{6} - \)\(18\!\cdots\!22\)\( T^{7} + \)\(33\!\cdots\!21\)\( T^{8} \))
$19$ (\( 1 - 11907 T + 94304764 T^{2} - 560175305067 T^{3} + 2213314919066161 T^{4} \))(\( 1 - 18078 T + 229836563 T^{2} - 2185603715730 T^{3} + 17528226347742732 T^{4} - \)\(10\!\cdots\!30\)\( T^{5} + \)\(50\!\cdots\!43\)\( T^{6} - \)\(18\!\cdots\!98\)\( T^{7} + \)\(48\!\cdots\!21\)\( T^{8} \))
$23$ (\( 1 - 5913 T - 113072320 T^{2} - 875336211657 T^{3} + 21914624432020321 T^{4} \))(\( 1 - 2470 T - 79604405 T^{2} + 519605188310 T^{3} - 15472377282077396 T^{4} + 76920215980483257590 T^{5} - \)\(17\!\cdots\!05\)\( T^{6} - \)\(80\!\cdots\!30\)\( T^{7} + \)\(48\!\cdots\!41\)\( T^{8} \))
$29$ (\( ( 1 - 3978 T + 594823321 T^{2} )^{2} \))(\( ( 1 + 17272 T + 1034818938 T^{2} + 10273788400312 T^{3} + 353814783205469041 T^{4} )^{2} \))
$31$ (\( 1 + 22197 T + 1051739284 T^{2} + 19699919207157 T^{3} + 787662783788549761 T^{4} \))(\( 1 + 17202 T + 1848915731 T^{2} + 30108307322526 T^{3} + 2363355465741121116 T^{4} + \)\(26\!\cdots\!06\)\( T^{5} + \)\(14\!\cdots\!91\)\( T^{6} + \)\(12\!\cdots\!82\)\( T^{7} + \)\(62\!\cdots\!21\)\( T^{8} \))
$37$ (\( 1 - 61577 T + 1226000520 T^{2} - 157989735086993 T^{3} + 6582952005840035281 T^{4} \))(\( 1 + 43870 T - 2981741615 T^{2} - 9876641872610 T^{3} + 12551071586310417844 T^{4} - \)\(25\!\cdots\!90\)\( T^{5} - \)\(19\!\cdots\!15\)\( T^{6} + \)\(74\!\cdots\!30\)\( T^{7} + \)\(43\!\cdots\!61\)\( T^{8} \))
$41$ (\( 1 + 2726428318 T^{2} + 22563490300366186081 T^{4} \))(\( 1 - 17928625852 T^{2} + \)\(12\!\cdots\!86\)\( T^{4} - \)\(40\!\cdots\!12\)\( T^{6} + \)\(50\!\cdots\!61\)\( T^{8} \))
$43$ (\( ( 1 + 17414 T + 6321363049 T^{2} )^{2} \))(\( ( 1 - 160108 T + 18917216814 T^{2} - 1012100795049292 T^{3} + 39959630797262576401 T^{4} )^{2} \))
$47$ (\( 1 + 53109 T + 11719403956 T^{2} + 572473346907861 T^{3} + \)\(11\!\cdots\!41\)\( T^{4} \))(\( 1 + 354714 T + 72542959067 T^{2} + 10855058969376390 T^{3} + \)\(12\!\cdots\!12\)\( T^{4} + \)\(11\!\cdots\!10\)\( T^{5} + \)\(84\!\cdots\!47\)\( T^{6} + \)\(44\!\cdots\!46\)\( T^{7} + \)\(13\!\cdots\!81\)\( T^{8} \))
$53$ (\( 1 - 60513 T - 18502537960 T^{2} - 1341231984999177 T^{3} + \)\(49\!\cdots\!41\)\( T^{4} \))(\( 1 - 104530 T + 4615583617 T^{2} + 3973999063436750 T^{3} - \)\(69\!\cdots\!52\)\( T^{4} + \)\(88\!\cdots\!50\)\( T^{5} + \)\(22\!\cdots\!97\)\( T^{6} - \)\(11\!\cdots\!70\)\( T^{7} + \)\(24\!\cdots\!81\)\( T^{8} \))
$59$ (\( 1 + 373653 T + 88719388444 T^{2} + 15760882936560573 T^{3} + \)\(17\!\cdots\!81\)\( T^{4} \))(\( 1 + 62646 T + 79736932091 T^{2} + 4913247993652074 T^{3} + \)\(44\!\cdots\!32\)\( T^{4} + \)\(20\!\cdots\!34\)\( T^{5} + \)\(14\!\cdots\!71\)\( T^{6} + \)\(47\!\cdots\!66\)\( T^{7} + \)\(31\!\cdots\!61\)\( T^{8} \))
$61$ (\( 1 - 281883 T + 78006382924 T^{2} - 14522717686001763 T^{3} + \)\(26\!\cdots\!21\)\( T^{4} \))(\( 1 + 140682 T + 107922721457 T^{2} + 14254685210248818 T^{3} + \)\(79\!\cdots\!68\)\( T^{4} + \)\(73\!\cdots\!98\)\( T^{5} + \)\(28\!\cdots\!97\)\( T^{6} + \)\(19\!\cdots\!42\)\( T^{7} + \)\(70\!\cdots\!41\)\( T^{8} \))
$67$ (\( 1 - 268777 T - 18217306440 T^{2} - 24313132584237313 T^{3} + \)\(81\!\cdots\!61\)\( T^{4} \))(\( 1 - 432494 T - 31495708061 T^{2} - 16274750845744946 T^{3} + \)\(22\!\cdots\!64\)\( T^{4} - \)\(14\!\cdots\!74\)\( T^{5} - \)\(25\!\cdots\!21\)\( T^{6} - \)\(32\!\cdots\!46\)\( T^{7} + \)\(66\!\cdots\!21\)\( T^{8} \))
$71$ (\( ( 1 - 101922 T + 128100283921 T^{2} )^{2} \))(\( ( 1 + 740764 T + 392433088158 T^{2} + 94892078718455644 T^{3} + \)\(16\!\cdots\!41\)\( T^{4} )^{2} \))
$73$ (\( 1 - 550179 T + 252233203636 T^{2} - 83260913285455731 T^{3} + \)\(22\!\cdots\!21\)\( T^{4} \))(\( 1 + 971778 T + 542579371577 T^{2} + 221366374699952922 T^{3} + \)\(76\!\cdots\!72\)\( T^{4} + \)\(33\!\cdots\!58\)\( T^{5} + \)\(12\!\cdots\!17\)\( T^{6} + \)\(33\!\cdots\!82\)\( T^{7} + \)\(52\!\cdots\!41\)\( T^{8} \))
$79$ (\( 1 + 362231 T - 111876158160 T^{2} + 88053812100827351 T^{3} + \)\(59\!\cdots\!41\)\( T^{4} \))(\( 1 - 935990 T + 192740690395 T^{2} - 184541359611781370 T^{3} + \)\(19\!\cdots\!84\)\( T^{4} - \)\(44\!\cdots\!70\)\( T^{5} + \)\(11\!\cdots\!95\)\( T^{6} - \)\(13\!\cdots\!90\)\( T^{7} + \)\(34\!\cdots\!81\)\( T^{8} \))
$83$ (\( 1 - 606885669938 T^{2} + \)\(10\!\cdots\!61\)\( T^{4} \))(\( 1 - 467743512772 T^{2} + \)\(19\!\cdots\!90\)\( T^{4} - \)\(49\!\cdots\!92\)\( T^{6} + \)\(11\!\cdots\!21\)\( T^{8} \))
$89$ (\( 1 + 2311533 T + 2278042894324 T^{2} + 1148788654438953213 T^{3} + \)\(24\!\cdots\!21\)\( T^{4} \))(\( 1 + 679290 T + 854922243761 T^{2} + 476257425629046690 T^{3} + \)\(32\!\cdots\!00\)\( T^{4} + \)\(23\!\cdots\!90\)\( T^{5} + \)\(21\!\cdots\!81\)\( T^{6} + \)\(83\!\cdots\!90\)\( T^{7} + \)\(61\!\cdots\!41\)\( T^{8} \))
$97$ (\( 1 + 626523106942 T^{2} + \)\(69\!\cdots\!41\)\( T^{4} \))(\( 1 - 2133147299644 T^{2} + \)\(22\!\cdots\!54\)\( T^{4} - \)\(14\!\cdots\!04\)\( T^{6} + \)\(48\!\cdots\!81\)\( T^{8} \))
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