Properties

Label 48.2.k
Level 48
Weight 2
Character orbit k
Rep. character \(\chi_{48}(11,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 12
Newform subspaces 1
Sturm bound 16
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 48.k (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 48 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(16\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(48, [\chi])\).

Total New Old
Modular forms 20 20 0
Cusp forms 12 12 0
Eisenstein series 8 8 0

Trace form

\( 12q - 2q^{3} - 4q^{4} - 8q^{6} - 8q^{7} + O(q^{10}) \) \( 12q - 2q^{3} - 4q^{4} - 8q^{6} - 8q^{7} - 8q^{12} - 4q^{13} + 16q^{16} + 4q^{18} - 12q^{19} - 8q^{21} + 16q^{22} + 24q^{24} + 10q^{27} - 8q^{28} + 28q^{30} - 4q^{33} - 8q^{34} + 20q^{36} - 4q^{37} + 20q^{39} - 40q^{40} - 24q^{42} + 12q^{43} - 12q^{45} - 40q^{46} - 48q^{48} - 20q^{49} + 24q^{51} - 16q^{52} - 52q^{54} + 24q^{55} + 32q^{58} - 16q^{60} + 12q^{61} + 56q^{64} + 28q^{66} + 28q^{67} + 4q^{69} + 40q^{70} + 40q^{72} - 34q^{75} + 56q^{76} + 60q^{78} - 4q^{81} - 16q^{82} + 16q^{84} + 32q^{85} - 60q^{87} - 64q^{88} - 16q^{90} - 56q^{91} + 28q^{93} - 48q^{94} - 56q^{96} - 8q^{97} - 52q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(48, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
48.2.k.a \(12\) \(0.383\) 12.0.\(\cdots\).2 None \(0\) \(-2\) \(0\) \(-8\) \(q+\beta _{6}q^{2}-\beta _{10}q^{3}+(-\beta _{1}+\beta _{10}+\beta _{11})q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} - 2 T^{4} - 16 T^{6} - 8 T^{8} + 32 T^{10} + 64 T^{12} \)
$3$ \( 1 + 2 T + 2 T^{2} - 2 T^{3} - 5 T^{4} - 20 T^{5} - 28 T^{6} - 60 T^{7} - 45 T^{8} - 54 T^{9} + 162 T^{10} + 486 T^{11} + 729 T^{12} \)
$5$ \( 1 - 30 T^{4} - 49 T^{8} + 12796 T^{12} - 30625 T^{16} - 11718750 T^{20} + 244140625 T^{24} \)
$7$ \( ( 1 + 2 T + 15 T^{2} + 20 T^{3} + 105 T^{4} + 98 T^{5} + 343 T^{6} )^{4} \)
$11$ \( 1 - 62 T^{4} + 11023 T^{8} - 2631620 T^{12} + 161387743 T^{16} - 13290250622 T^{20} + 3138428376721 T^{24} \)
$13$ \( ( 1 + 2 T + 2 T^{2} - 6 T^{3} - 25 T^{4} + 412 T^{5} + 892 T^{6} + 5356 T^{7} - 4225 T^{8} - 13182 T^{9} + 57122 T^{10} + 742586 T^{11} + 4826809 T^{12} )^{2} \)
$17$ \( ( 1 - 62 T^{2} + 1903 T^{4} - 38180 T^{6} + 549967 T^{8} - 5178302 T^{10} + 24137569 T^{12} )^{2} \)
$19$ \( ( 1 + 6 T + 18 T^{2} + 82 T^{3} + 539 T^{4} + 2636 T^{5} + 9476 T^{6} + 50084 T^{7} + 194579 T^{8} + 562438 T^{9} + 2345778 T^{10} + 14856594 T^{11} + 47045881 T^{12} )^{2} \)
$23$ \( ( 1 - 86 T^{2} + 3791 T^{4} - 105684 T^{6} + 2005439 T^{8} - 24066326 T^{10} + 148035889 T^{12} )^{2} \)
$29$ \( 1 - 830 T^{4} + 2253679 T^{8} - 1165110596 T^{12} + 1593984336799 T^{16} - 415204522757630 T^{20} + 353814783205469041 T^{24} \)
$31$ \( ( 1 - 150 T^{2} + 10019 T^{4} - 392444 T^{6} + 9628259 T^{8} - 138528150 T^{10} + 887503681 T^{12} )^{2} \)
$37$ \( ( 1 + 2 T + 2 T^{2} - 54 T^{3} + 567 T^{4} + 8764 T^{5} + 17852 T^{6} + 324268 T^{7} + 776223 T^{8} - 2735262 T^{9} + 3748322 T^{10} + 138687914 T^{11} + 2565726409 T^{12} )^{2} \)
$41$ \( ( 1 + 138 T^{2} + 7887 T^{4} + 320492 T^{6} + 13258047 T^{8} + 389955018 T^{10} + 4750104241 T^{12} )^{2} \)
$43$ \( ( 1 - 6 T + 18 T^{2} - 226 T^{3} + 4235 T^{4} - 18188 T^{5} + 58436 T^{6} - 782084 T^{7} + 7830515 T^{8} - 17968582 T^{9} + 61538418 T^{10} - 882050658 T^{11} + 6321363049 T^{12} )^{2} \)
$47$ \( ( 1 + 170 T^{2} + 15791 T^{4} + 908172 T^{6} + 34882319 T^{8} + 829545770 T^{10} + 10779215329 T^{12} )^{2} \)
$53$ \( 1 + 7714 T^{4} + 19237903 T^{8} + 30633057916 T^{12} + 151796308101343 T^{16} + 480271251833238754 T^{20} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( ( 1 - 30 T + 450 T^{2} - 3458 T^{3} + 3915 T^{4} + 231740 T^{5} - 2735068 T^{6} + 13672660 T^{7} + 13628115 T^{8} - 710200582 T^{9} + 5452812450 T^{10} - 21447728970 T^{11} + 42180533641 T^{12} )( 1 + 30 T + 450 T^{2} + 3458 T^{3} + 3915 T^{4} - 231740 T^{5} - 2735068 T^{6} - 13672660 T^{7} + 13628115 T^{8} + 710200582 T^{9} + 5452812450 T^{10} + 21447728970 T^{11} + 42180533641 T^{12} ) \)
$61$ \( ( 1 - 6 T + 18 T^{2} - 430 T^{3} - 121 T^{4} + 30796 T^{5} - 90148 T^{6} + 1878556 T^{7} - 450241 T^{8} - 97601830 T^{9} + 249225138 T^{10} - 5067577806 T^{11} + 51520374361 T^{12} )^{2} \)
$67$ \( ( 1 - 14 T + 98 T^{2} - 706 T^{3} + 9435 T^{4} - 112164 T^{5} + 894884 T^{6} - 7514988 T^{7} + 42353715 T^{8} - 212338678 T^{9} + 1974809858 T^{10} - 18901751498 T^{11} + 90458382169 T^{12} )^{2} \)
$71$ \( ( 1 - 230 T^{2} + 30127 T^{4} - 2527028 T^{6} + 151870207 T^{8} - 5844686630 T^{10} + 128100283921 T^{12} )^{2} \)
$73$ \( ( 1 - 166 T^{2} + 19007 T^{4} - 1414164 T^{6} + 101288303 T^{8} - 4714108006 T^{10} + 151334226289 T^{12} )^{2} \)
$79$ \( ( 1 - 358 T^{2} + 58915 T^{4} - 5817628 T^{6} + 367688515 T^{8} - 13944128998 T^{10} + 243087455521 T^{12} )^{2} \)
$83$ \( 1 - 1374 T^{4} + 18563631 T^{8} - 336062521604 T^{12} + 880998758923551 T^{16} - 3094649526959042334 T^{20} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( ( 1 + 322 T^{2} + 51919 T^{4} + 5548348 T^{6} + 411250399 T^{8} + 20203001602 T^{10} + 496981290961 T^{12} )^{2} \)
$97$ \( ( 1 + 2 T + 163 T^{2} - 220 T^{3} + 15811 T^{4} + 18818 T^{5} + 912673 T^{6} )^{4} \)
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