Properties

Label 14.8.a
Level 14
Weight 8
Character orbit a
Rep. character \(\chi_{14}(1,\cdot)\)
Character field \(\Q\)
Dimension 4
Newform subspaces 3
Sturm bound 16
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 14.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(16\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(14))\).

Total New Old
Modular forms 16 4 12
Cusp forms 12 4 8
Eisenstein series 4 0 4

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

\(2\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(3\)
Minus space\(-\)\(1\)

Trace form

\( 4q + 16q^{2} - 78q^{3} + 256q^{4} + 174q^{5} + 688q^{6} + 1024q^{8} + 8720q^{9} + O(q^{10}) \) \( 4q + 16q^{2} - 78q^{3} + 256q^{4} + 174q^{5} + 688q^{6} + 1024q^{8} + 8720q^{9} - 5776q^{10} - 972q^{11} - 4992q^{12} - 3734q^{13} + 5488q^{14} - 41368q^{15} + 16384q^{16} + 516q^{17} - 2832q^{18} - 53098q^{19} + 11136q^{20} + 74774q^{21} - 46304q^{22} - 28152q^{23} + 44032q^{24} + 375120q^{25} - 143728q^{26} + 1548q^{27} - 54852q^{29} + 256832q^{30} - 31940q^{31} + 65536q^{32} - 815984q^{33} - 35648q^{34} + 26754q^{35} + 558080q^{36} - 465412q^{37} - 1008496q^{38} + 230672q^{39} - 369664q^{40} + 242460q^{41} + 148176q^{42} + 1642172q^{43} - 62208q^{44} - 1189082q^{45} - 19136q^{46} + 1906020q^{47} - 319488q^{48} + 470596q^{49} + 1039696q^{50} - 4313924q^{51} - 238976q^{52} + 1400616q^{53} + 3095584q^{54} + 5313016q^{55} + 351232q^{56} - 501468q^{57} + 970688q^{58} - 7505418q^{59} - 2647552q^{60} + 734830q^{61} - 4777696q^{62} - 1609356q^{63} + 1048576q^{64} - 2132844q^{65} - 3368576q^{66} + 1643016q^{67} + 33024q^{68} + 10154464q^{69} + 2672656q^{70} - 692616q^{71} - 181248q^{72} + 1957104q^{73} - 290752q^{74} - 13947610q^{75} - 3398272q^{76} - 2012724q^{77} + 11181568q^{78} + 5696344q^{79} + 712704q^{80} + 8410712q^{81} + 4193664q^{82} - 901650q^{83} + 4785536q^{84} - 12721028q^{85} + 12553952q^{86} + 30574484q^{87} - 2963456q^{88} - 10146264q^{89} - 42033872q^{90} - 3108266q^{91} - 1801728q^{92} - 19258120q^{93} + 3778272q^{94} + 23488776q^{95} + 2818048q^{96} + 17718564q^{97} + 1882384q^{98} - 27165244q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(14))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7
14.8.a.a \(1\) \(4.373\) \(\Q\) None \(-8\) \(-82\) \(448\) \(-343\) \(+\) \(+\) \(q-8q^{2}-82q^{3}+2^{6}q^{4}+448q^{5}+\cdots\)
14.8.a.b \(1\) \(4.373\) \(\Q\) None \(8\) \(-66\) \(-400\) \(-343\) \(-\) \(+\) \(q+8q^{2}-66q^{3}+2^{6}q^{4}-20^{2}q^{5}+\cdots\)
14.8.a.c \(2\) \(4.373\) \(\Q(\sqrt{1969}) \) None \(16\) \(70\) \(126\) \(686\) \(-\) \(-\) \(q+8q^{2}+(35-\beta )q^{3}+2^{6}q^{4}+(63+9\beta )q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(14))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(14)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 8 T \))(\( 1 - 8 T \))(\( ( 1 - 8 T )^{2} \))
$3$ (\( 1 + 82 T + 2187 T^{2} \))(\( 1 + 66 T + 2187 T^{2} \))(\( 1 - 70 T + 3630 T^{2} - 153090 T^{3} + 4782969 T^{4} \))
$5$ (\( 1 - 448 T + 78125 T^{2} \))(\( 1 + 400 T + 78125 T^{2} \))(\( 1 - 126 T + 730 T^{2} - 9843750 T^{3} + 6103515625 T^{4} \))
$7$ (\( 1 + 343 T \))(\( 1 + 343 T \))(\( ( 1 - 343 T )^{2} \))
$11$ (\( 1 - 2408 T + 19487171 T^{2} \))(\( 1 - 40 T + 19487171 T^{2} \))(\( 1 + 3420 T + 10638598 T^{2} + 66646124820 T^{3} + 379749833583241 T^{4} \))
$13$ (\( 1 - 7116 T + 62748517 T^{2} \))(\( 1 + 4452 T + 62748517 T^{2} \))(\( 1 + 6398 T + 65395986 T^{2} + 401465011766 T^{3} + 3937376385699289 T^{4} \))
$17$ (\( 1 - 2486 T + 410338673 T^{2} \))(\( 1 - 36502 T + 410338673 T^{2} \))(\( 1 + 38472 T + 1174752142 T^{2} + 15786549427656 T^{3} + 168377826559400929 T^{4} \))
$19$ (\( 1 - 36482 T + 893871739 T^{2} \))(\( 1 + 46222 T + 893871739 T^{2} \))(\( 1 + 43358 T + 2141455038 T^{2} + 38756490859562 T^{3} + 799006685782884121 T^{4} \))
$23$ (\( 1 + 12880 T + 3404825447 T^{2} \))(\( 1 + 105200 T + 3404825447 T^{2} \))(\( 1 - 89928 T + 8706372814 T^{2} - 306189142797816 T^{3} + 11592836324538749809 T^{4} \))
$29$ (\( 1 + 88094 T + 17249876309 T^{2} \))(\( 1 + 126334 T + 17249876309 T^{2} \))(\( 1 - 159576 T + 29581073878 T^{2} - 2752666261884984 T^{3} + \)\(29\!\cdots\!81\)\( T^{4} \))
$31$ (\( 1 - 282636 T + 27512614111 T^{2} \))(\( 1 + 170964 T + 27512614111 T^{2} \))(\( 1 + 143612 T + 59486595774 T^{2} + 3951141537708932 T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))
$37$ (\( 1 + 214534 T + 94931877133 T^{2} \))(\( 1 - 20954 T + 94931877133 T^{2} \))(\( 1 + 271832 T + 32500290822 T^{2} + 25805522024817656 T^{3} + \)\(90\!\cdots\!89\)\( T^{4} \))
$41$ (\( 1 + 140874 T + 194754273881 T^{2} \))(\( 1 - 318486 T + 194754273881 T^{2} \))(\( 1 - 64848 T + 315504978094 T^{2} - 12629425152635088 T^{3} + \)\(37\!\cdots\!61\)\( T^{4} \))
$43$ (\( 1 - 36464 T + 271818611107 T^{2} \))(\( 1 - 77744 T + 271818611107 T^{2} \))(\( 1 - 1527964 T + 1127024379942 T^{2} - 415329052301496148 T^{3} + \)\(73\!\cdots\!49\)\( T^{4} \))
$47$ (\( 1 - 716868 T + 506623120463 T^{2} \))(\( 1 - 703716 T + 506623120463 T^{2} \))(\( 1 - 485436 T + 473142464734 T^{2} - 245933101105076868 T^{3} + \)\(25\!\cdots\!69\)\( T^{4} \))
$53$ (\( 1 + 56946 T + 1174711139837 T^{2} \))(\( 1 - 1603278 T + 1174711139837 T^{2} \))(\( 1 + 145716 T + 2270203949662 T^{2} + 171174208452488292 T^{3} + \)\(13\!\cdots\!69\)\( T^{4} \))
$59$ (\( 1 + 2149862 T + 2488651484819 T^{2} \))(\( 1 + 1171894 T + 2488651484819 T^{2} \))(\( 1 + 4183662 T + 9350646993118 T^{2} + 10411676648280827178 T^{3} + \)\(61\!\cdots\!61\)\( T^{4} \))
$61$ (\( 1 - 3084360 T + 3142742836021 T^{2} \))(\( 1 + 2068872 T + 3142742836021 T^{2} \))(\( 1 + 280658 T + 5246022775002 T^{2} + 882035918871981818 T^{3} + \)\(98\!\cdots\!41\)\( T^{4} \))
$67$ (\( 1 + 3034364 T + 6060711605323 T^{2} \))(\( 1 + 994268 T + 6060711605323 T^{2} \))(\( 1 - 5671648 T + 17884478342022 T^{2} - 34374222854906982304 T^{3} + \)\(36\!\cdots\!29\)\( T^{4} \))
$71$ (\( 1 + 106624 T + 9095120158391 T^{2} \))(\( 1 - 33280 T + 9095120158391 T^{2} \))(\( 1 + 619272 T + 17339691732334 T^{2} + 5632353250727111352 T^{3} + \)\(82\!\cdots\!81\)\( T^{4} \))
$73$ (\( 1 - 988930 T + 11047398519097 T^{2} \))(\( 1 + 2971454 T + 11047398519097 T^{2} \))(\( 1 - 3939628 T + 25580837567814 T^{2} - 43522640532993075916 T^{3} + \)\(12\!\cdots\!09\)\( T^{4} \))
$79$ (\( 1 - 3415896 T + 19203908986159 T^{2} \))(\( 1 + 2376168 T + 19203908986159 T^{2} \))(\( 1 - 4656616 T + 21455665606878 T^{2} - 89425229847491777944 T^{3} + \)\(36\!\cdots\!81\)\( T^{4} \))
$83$ (\( 1 + 15142 T + 27136050989627 T^{2} \))(\( 1 + 2122358 T + 27136050989627 T^{2} \))(\( 1 - 1235850 T + 18764123455390 T^{2} - 33536088615530527950 T^{3} + \)\(73\!\cdots\!29\)\( T^{4} \))
$89$ (\( 1 - 174810 T + 44231334895529 T^{2} \))(\( 1 - 6920346 T + 44231334895529 T^{2} \))(\( 1 + 17241420 T + 151950390368758 T^{2} + \)\(76\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \))
$97$ (\( 1 - 13506790 T + 80798284478113 T^{2} \))(\( 1 - 4952710 T + 80798284478113 T^{2} \))(\( 1 + 740936 T + 158749094330286 T^{2} + 59866357708075133768 T^{3} + \)\(65\!\cdots\!69\)\( T^{4} \))
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