Properties

Label 7.6.a
Level 7
Weight 6
Character orbit a
Rep. character \(\chi_{7}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 4
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 7.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(4\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(7))\).

Total New Old
Modular forms 5 3 2
Cusp forms 3 3 0
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(7\)Dim.
\(+\)\(1\)
\(-\)\(2\)

Trace form

\( 3q - q^{2} - 20q^{3} + 73q^{4} - 74q^{5} - 58q^{6} + 49q^{7} - 369q^{8} + 511q^{9} + O(q^{10}) \) \( 3q - q^{2} - 20q^{3} + 73q^{4} - 74q^{5} - 58q^{6} + 49q^{7} - 369q^{8} + 511q^{9} + 764q^{10} + 628q^{11} - 2506q^{12} - 490q^{13} + 931q^{14} - 872q^{15} + 1537q^{16} + 78q^{17} + 4007q^{18} - 3364q^{19} - 1288q^{20} + 392q^{21} - 4072q^{22} + 3912q^{23} + 3186q^{24} - 3227q^{25} + 3416q^{26} - 2312q^{27} - 3087q^{28} + 10114q^{29} - 14608q^{30} - 7664q^{31} - 8849q^{32} + 16768q^{33} + 23154q^{34} + 1862q^{35} + 7433q^{36} - 4166q^{37} - 14230q^{38} - 18536q^{39} + 23376q^{40} - 24010q^{41} - 16562q^{42} + 7860q^{43} - 15040q^{44} + 7870q^{45} + 7344q^{46} + 21024q^{47} + 21278q^{48} + 7203q^{49} - 19811q^{50} + 31704q^{51} + 21924q^{52} + 11730q^{53} - 78508q^{54} - 51896q^{55} + 17199q^{56} + 14248q^{57} + 3134q^{58} - 46668q^{59} + 55328q^{60} - 39106q^{61} + 91740q^{62} + 29645q^{63} - 89151q^{64} + 46900q^{65} + 99296q^{66} - 23620q^{67} - 132090q^{68} - 128928q^{69} - 17444q^{70} + 38856q^{71} + 25695q^{72} + 85534q^{73} + 147414q^{74} + 40340q^{75} - 19446q^{76} + 8036q^{77} - 92680q^{78} + 83040q^{79} - 150016q^{80} - 106493q^{81} + 121282q^{82} + 97020q^{83} - 29498q^{84} + 58572q^{85} - 258848q^{86} - 111032q^{87} - 124176q^{88} + 33694q^{89} + 67532q^{90} - 10290q^{91} + 274944q^{92} + 14736q^{93} + 66228q^{94} + 29752q^{95} + 293986q^{96} - 37730q^{97} - 2401q^{98} - 27644q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(7))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 7
7.6.a.a \(1\) \(1.123\) \(\Q\) None \(-10\) \(-14\) \(-56\) \(-49\) \(+\) \(q-10q^{2}-14q^{3}+68q^{4}-56q^{5}+\cdots\)
7.6.a.b \(2\) \(1.123\) \(\Q(\sqrt{57}) \) None \(9\) \(-6\) \(-18\) \(98\) \(-\) \(q+(5-\beta )q^{2}+(-6+6\beta )q^{3}+(7-9\beta )q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 10 T + 32 T^{2} \))(\( 1 - 9 T + 70 T^{2} - 288 T^{3} + 1024 T^{4} \))
$3$ (\( 1 + 14 T + 243 T^{2} \))(\( 1 + 6 T - 18 T^{2} + 1458 T^{3} + 59049 T^{4} \))
$5$ (\( 1 + 56 T + 3125 T^{2} \))(\( 1 + 18 T + 4906 T^{2} + 56250 T^{3} + 9765625 T^{4} \))
$7$ (\( 1 + 49 T \))(\( ( 1 - 49 T )^{2} \))
$11$ (\( 1 - 232 T + 161051 T^{2} \))(\( 1 - 396 T + 142198 T^{2} - 63776196 T^{3} + 25937424601 T^{4} \))
$13$ (\( 1 + 140 T + 371293 T^{2} \))(\( 1 + 350 T + 546978 T^{2} + 129952550 T^{3} + 137858491849 T^{4} \))
$17$ (\( 1 + 1722 T + 1419857 T^{2} \))(\( 1 - 1800 T + 3567406 T^{2} - 2555742600 T^{3} + 2015993900449 T^{4} \))
$19$ (\( 1 + 98 T + 2476099 T^{2} \))(\( 1 + 3266 T + 7614270 T^{2} + 8086939334 T^{3} + 6131066257801 T^{4} \))
$23$ (\( 1 - 1824 T + 6436343 T^{2} \))(\( 1 - 2088 T + 9365230 T^{2} - 13439084184 T^{3} + 41426511213649 T^{4} \))
$29$ (\( 1 - 3418 T + 20511149 T^{2} \))(\( 1 - 6696 T + 51326470 T^{2} - 137342653704 T^{3} + 420707233300201 T^{4} \))
$31$ (\( 1 + 7644 T + 28629151 T^{2} \))(\( 1 + 20 T + 53103102 T^{2} + 572583020 T^{3} + 819628286980801 T^{4} \))
$37$ (\( 1 + 10398 T + 69343957 T^{2} \))(\( 1 - 6232 T + 144242070 T^{2} - 432151540024 T^{3} + 4808584372417849 T^{4} \))
$41$ (\( 1 + 17962 T + 115856201 T^{2} \))(\( 1 + 6048 T + 223864366 T^{2} + 700698303648 T^{3} + 13422659310152401 T^{4} \))
$43$ (\( 1 - 10880 T + 147008443 T^{2} \))(\( 1 + 3020 T - 30383466 T^{2} + 443965497860 T^{3} + 21611482313284249 T^{4} \))
$47$ (\( 1 - 9324 T + 229345007 T^{2} \))(\( 1 - 11700 T + 292735582 T^{2} - 2683336581900 T^{3} + 52599132235830049 T^{4} \))
$53$ (\( 1 - 2262 T + 418195493 T^{2} \))(\( 1 - 9468 T + 858185230 T^{2} - 3959474927724 T^{3} + 174887470365513049 T^{4} \))
$59$ (\( 1 + 2730 T + 714924299 T^{2} \))(\( 1 + 43938 T + 1852599934 T^{2} + 31412343849462 T^{3} + 511116753300641401 T^{4} \))
$61$ (\( 1 - 25648 T + 844596301 T^{2} \))(\( 1 + 64754 T + 2408321418 T^{2} + 54690988874954 T^{3} + 713342911662882601 T^{4} \))
$67$ (\( 1 + 48404 T + 1350125107 T^{2} \))(\( 1 - 24784 T + 2799959190 T^{2} - 33461500651888 T^{3} + 1822837804551761449 T^{4} \))
$71$ (\( 1 + 58560 T + 1804229351 T^{2} \))(\( 1 - 97416 T + 5729557966 T^{2} - 175760806457016 T^{3} + 3255243551009881201 T^{4} \))
$73$ (\( 1 - 68082 T + 2073071593 T^{2} \))(\( 1 - 17452 T + 3828622374 T^{2} - 36179245441036 T^{3} + 4297625829703557649 T^{4} \))
$79$ (\( 1 - 31784 T + 3077056399 T^{2} \))(\( 1 - 51256 T + 3645565854 T^{2} - 157717602787144 T^{3} + 9468276082626847201 T^{4} \))
$83$ (\( 1 + 20538 T + 3939040643 T^{2} \))(\( 1 - 117558 T + 7798161502 T^{2} - 463065739909794 T^{3} + 15516041187205853449 T^{4} \))
$89$ (\( 1 + 50582 T + 5584059449 T^{2} \))(\( 1 - 84276 T + 5915697430 T^{2} - 470602194123924 T^{3} + 31181719929966183601 T^{4} \))
$97$ (\( 1 + 58506 T + 8587340257 T^{2} \))(\( 1 - 20776 T + 16174049358 T^{2} - 178410581179432 T^{3} + 73742412689492826049 T^{4} \))
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