Properties

Label 100.6.c
Level 100
Weight 6
Character orbit c
Rep. character \(\chi_{100}(49,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 3
Sturm bound 90
Trace bound 9

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Defining parameters

Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 100.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(90\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(100, [\chi])\).

Total New Old
Modular forms 84 8 76
Cusp forms 66 8 58
Eisenstein series 18 0 18

Trace form

\( 8q - 1348q^{9} + O(q^{10}) \) \( 8q - 1348q^{9} - 3448q^{19} + 1088q^{21} - 16968q^{29} + 9448q^{31} - 40328q^{39} - 19212q^{41} - 36576q^{49} + 32496q^{51} + 112224q^{59} - 4664q^{61} + 265056q^{69} - 147336q^{71} + 123776q^{79} + 273232q^{81} - 203892q^{89} - 470008q^{91} - 120600q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(100, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
100.6.c.a \(2\) \(16.038\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+11iq^{3}-109iq^{7}-241q^{9}-480q^{11}+\cdots\)
100.6.c.b \(2\) \(16.038\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+6iq^{3}-44iq^{7}+99q^{9}+540q^{11}+\cdots\)
100.6.c.c \(4\) \(16.038\) \(\Q(i, \sqrt{409})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{1}+2\beta _{2})q^{3}+(-6\beta _{1}+4\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(100, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(100, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 - 2 T^{2} + 59049 T^{4} \))(\( 1 - 342 T^{2} + 59049 T^{4} \))(\( 1 + 46 T^{2} - 44973 T^{4} + 2716254 T^{6} + 3486784401 T^{8} \))
$5$ 1
$7$ (\( 1 + 13910 T^{2} + 282475249 T^{4} \))(\( 1 - 25870 T^{2} + 282475249 T^{4} \))(\( 1 - 36980 T^{2} + 883272198 T^{4} - 10445934708020 T^{6} + 79792266297612001 T^{8} \))
$11$ (\( ( 1 + 480 T + 161051 T^{2} )^{2} \))(\( ( 1 - 540 T + 161051 T^{2} )^{2} \))(\( ( 1 + 60 T + 230977 T^{2} + 9663060 T^{3} + 25937424601 T^{4} )^{2} \))
$13$ (\( 1 - 355702 T^{2} + 137858491849 T^{4} \))(\( 1 - 567862 T^{2} + 137858491849 T^{4} \))(\( 1 - 590804 T^{2} + 163581027702 T^{4} - 81447348418356596 T^{6} + \)\(19\!\cdots\!01\)\( T^{8} \))
$17$ (\( 1 - 2805118 T^{2} + 2015993900449 T^{4} \))(\( 1 - 2486878 T^{2} + 2015993900449 T^{4} \))(\( 1 - 1327586 T^{2} + 3973871730147 T^{4} - 2676405278321486114 T^{6} + \)\(40\!\cdots\!01\)\( T^{8} \))
$19$ (\( ( 1 - 1204 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 836 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 2092 T + 5218089 T^{2} + 5179999108 T^{3} + 6131066257801 T^{4} )^{2} \))
$23$ (\( 1 - 2722090 T^{2} + 41426511213649 T^{4} \))(\( 1 + 3970130 T^{2} + 41426511213649 T^{4} \))(\( 1 - 10160180 T^{2} + 108548175292998 T^{4} - \)\(42\!\cdots\!20\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} \))
$29$ (\( ( 1 + 5526 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 594 T + 20511149 T^{2} )^{2} \))(\( ( 1 + 3552 T + 20618074 T^{2} + 72855601248 T^{3} + 420707233300201 T^{4} )^{2} \))
$31$ (\( ( 1 - 9356 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 4256 T + 28629151 T^{2} )^{2} \))(\( ( 1 + 8888 T + 67804938 T^{2} + 254455894088 T^{3} + 819628286980801 T^{4} )^{2} \))
$37$ (\( 1 - 107125990 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 138599110 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 33594380 T^{2} - 2634705186058602 T^{4} - \)\(16\!\cdots\!20\)\( T^{6} + \)\(23\!\cdots\!01\)\( T^{8} \))
$41$ (\( ( 1 + 14394 T + 115856201 T^{2} )^{2} \))(\( ( 1 - 17226 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 12438 T + 217381963 T^{2} + 1441019428038 T^{3} + 13422659310152401 T^{4} )^{2} \))
$43$ (\( 1 - 293879986 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 147606886 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 540244172 T^{2} + 116157205788519894 T^{4} - \)\(11\!\cdots\!28\)\( T^{6} + \)\(46\!\cdots\!01\)\( T^{8} \))
$47$ (\( 1 - 197996698 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 457010398 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 902407196 T^{2} + 308772689280765702 T^{4} - \)\(47\!\cdots\!04\)\( T^{6} + \)\(27\!\cdots\!01\)\( T^{8} \))
$53$ (\( 1 - 817259110 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 456374950 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 1189716620 T^{2} + 656395125486896598 T^{4} - \)\(20\!\cdots\!80\)\( T^{6} + \)\(30\!\cdots\!01\)\( T^{8} \))
$59$ (\( ( 1 - 11748 T + 714924299 T^{2} )^{2} \))(\( ( 1 - 7668 T + 714924299 T^{2} )^{2} \))(\( ( 1 - 36696 T + 1234961302 T^{2} - 26234862076104 T^{3} + 511116753300641401 T^{4} )^{2} \))
$61$ (\( ( 1 - 13202 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 34738 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 19204 T + 627029406 T^{2} - 16219627364404 T^{3} + 713342911662882601 T^{4} )^{2} \))
$67$ (\( 1 - 2567032450 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 2224486870 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 974988050 T^{2} + 2516736182389071123 T^{4} - \)\(17\!\cdots\!50\)\( T^{6} + \)\(33\!\cdots\!01\)\( T^{8} \))
$71$ (\( ( 1 + 29532 T + 1804229351 T^{2} )^{2} \))(\( ( 1 + 46872 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 2736 T + 2006886526 T^{2} - 4936371504336 T^{3} + 3255243551009881201 T^{4} )^{2} \))
$73$ (\( 1 - 3010587982 T^{2} + 4297625829703557649 T^{4} \))(\( 1 + 418480658 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 8204978114 T^{2} + 25425197509477462947 T^{4} - \)\(35\!\cdots\!86\)\( T^{6} + \)\(18\!\cdots\!01\)\( T^{8} \))
$79$ (\( ( 1 + 31208 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 76912 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 16184 T + 6157384362 T^{2} - 49799080761416 T^{3} + 9468276082626847201 T^{4} )^{2} \))
$83$ (\( 1 - 6398448130 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 3292624630 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 7195965410 T^{2} + 40258768525685533923 T^{4} - \)\(11\!\cdots\!90\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} \))
$89$ (\( ( 1 + 119514 T + 5584059449 T^{2} )^{2} \))(\( ( 1 + 29754 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 47322 T - 1006825781 T^{2} - 264248861245578 T^{3} + 31181719929966183601 T^{4} )^{2} \))
$97$ (\( 1 - 8214543550 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 2193410110 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 28221621500 T^{2} + \)\(34\!\cdots\!98\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{6} + \)\(54\!\cdots\!01\)\( T^{8} \))
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