Properties

Label 12.12.a
Level 12
Weight 12
Character orbit a
Rep. character \(\chi_{12}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newform subspaces 2
Sturm bound 24
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 12.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(24\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(12))\).

Total New Old
Modular forms 25 2 23
Cusp forms 19 2 17
Eisenstein series 6 0 6

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

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

Trace form

\( 2q + 12852q^{5} - 77000q^{7} + 118098q^{9} + O(q^{10}) \) \( 2q + 12852q^{5} - 77000q^{7} + 118098q^{9} - 137808q^{11} + 1092700q^{13} - 1732104q^{15} - 2702700q^{17} + 170800q^{19} + 23147208q^{21} - 25261200q^{23} + 10334894q^{25} - 226450188q^{29} + 374152408q^{31} + 358230600q^{33} - 834294384q^{35} - 135236900q^{37} + 732207600q^{39} - 1031482620q^{41} + 1563677200q^{43} + 758897748q^{45} - 4139780400q^{47} + 3546699282q^{49} + 1436386608q^{51} - 2251057500q^{53} - 6139603008q^{55} - 153527400q^{57} + 14284492992q^{59} - 700025396q^{61} - 4546773000q^{63} - 3717354600q^{65} - 1966587200q^{67} - 14765714208q^{69} + 25293783600q^{71} - 13777935500q^{73} - 22261000608q^{75} + 75518805600q^{77} - 35976619976q^{79} + 6973568802q^{81} - 3501565200q^{83} - 38434553784q^{85} - 13860768600q^{87} - 55164907980q^{89} + 101443739600q^{91} + 22426810200q^{93} + 3349296000q^{95} + 93399168100q^{97} - 8137424592q^{99} + O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(12))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3
12.12.a.a \(1\) \(9.220\) \(\Q\) None \(0\) \(-243\) \(9990\) \(-86128\) \(-\) \(+\) \(q-3^{5}q^{3}+9990q^{5}-86128q^{7}+3^{10}q^{9}+\cdots\)
12.12.a.b \(1\) \(9.220\) \(\Q\) None \(0\) \(243\) \(2862\) \(9128\) \(-\) \(-\) \(q+3^{5}q^{3}+2862q^{5}+9128q^{7}+3^{10}q^{9}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(12))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(12)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 243 T \))(\( 1 - 243 T \))
$5$ (\( 1 - 9990 T + 48828125 T^{2} \))(\( 1 - 2862 T + 48828125 T^{2} \))
$7$ (\( 1 + 86128 T + 1977326743 T^{2} \))(\( 1 - 9128 T + 1977326743 T^{2} \))
$11$ (\( 1 + 806004 T + 285311670611 T^{2} \))(\( 1 - 668196 T + 285311670611 T^{2} \))
$13$ (\( 1 + 960250 T + 1792160394037 T^{2} \))(\( 1 - 2052950 T + 1792160394037 T^{2} \))
$17$ (\( 1 + 4306878 T + 34271896307633 T^{2} \))(\( 1 - 1604178 T + 34271896307633 T^{2} \))
$19$ (\( 1 - 401300 T + 116490258898219 T^{2} \))(\( 1 + 230500 T + 116490258898219 T^{2} \))
$23$ (\( 1 - 17751528 T + 952809757913927 T^{2} \))(\( 1 + 43012728 T + 952809757913927 T^{2} \))
$29$ (\( 1 + 84704994 T + 12200509765705829 T^{2} \))(\( 1 + 141745194 T + 12200509765705829 T^{2} \))
$31$ (\( 1 - 140930504 T + 25408476896404831 T^{2} \))(\( 1 - 233221904 T + 25408476896404831 T^{2} \))
$37$ (\( 1 + 413506594 T + 177917621779460413 T^{2} \))(\( 1 - 278269694 T + 177917621779460413 T^{2} \))
$41$ (\( 1 - 150094890 T + 550329031716248441 T^{2} \))(\( 1 + 1181577510 T + 550329031716248441 T^{2} \))
$43$ (\( 1 - 706702028 T + 929293739471222707 T^{2} \))(\( 1 - 856975172 T + 929293739471222707 T^{2} \))
$47$ (\( 1 + 2475725472 T + 2472159215084012303 T^{2} \))(\( 1 + 1664054928 T + 2472159215084012303 T^{2} \))
$53$ (\( 1 - 1600124166 T + 9269035929372191597 T^{2} \))(\( 1 + 3851181666 T + 9269035929372191597 T^{2} \))
$59$ (\( 1 - 3945492396 T + 30155888444737842659 T^{2} \))(\( 1 - 10339000596 T + 30155888444737842659 T^{2} \))
$61$ (\( 1 + 885973498 T + 43513917611435838661 T^{2} \))(\( 1 - 185948102 T + 43513917611435838661 T^{2} \))
$67$ (\( 1 + 4881597772 T + \)\(12\!\cdots\!83\)\( T^{2} \))(\( 1 - 2915010572 T + \)\(12\!\cdots\!83\)\( T^{2} \))
$71$ (\( 1 - 12631469400 T + \)\(23\!\cdots\!71\)\( T^{2} \))(\( 1 - 12662314200 T + \)\(23\!\cdots\!71\)\( T^{2} \))
$73$ (\( 1 - 1423335194 T + \)\(31\!\cdots\!77\)\( T^{2} \))(\( 1 + 15201270694 T + \)\(31\!\cdots\!77\)\( T^{2} \))
$79$ (\( 1 - 667407512 T + \)\(74\!\cdots\!79\)\( T^{2} \))(\( 1 + 36644027488 T + \)\(74\!\cdots\!79\)\( T^{2} \))
$83$ (\( 1 - 5716071828 T + \)\(12\!\cdots\!67\)\( T^{2} \))(\( 1 + 9217637028 T + \)\(12\!\cdots\!67\)\( T^{2} \))
$89$ (\( 1 + 85738736790 T + \)\(27\!\cdots\!89\)\( T^{2} \))(\( 1 - 30573828810 T + \)\(27\!\cdots\!89\)\( T^{2} \))
$97$ (\( 1 + 52302647806 T + \)\(71\!\cdots\!53\)\( T^{2} \))(\( 1 - 145701815906 T + \)\(71\!\cdots\!53\)\( T^{2} \))
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