↑

Properties

Label 42.12.e
Level $42$
Weight $12$
Character orbit 42.e
Rep. character $\chi_{42}(25,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $28$
Newform subspaces $4$
Sturm bound $96$
Trace bound $2$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 42.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(96\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(42, [\chi])\).

Total New Old
Modular forms 184 28 156
Cusp forms 168 28 140
Eisenstein series 16 0 16

Trace form

\( 28 q - 14336 q^{4} - 10124 q^{5} + 31104 q^{6} - 72418 q^{7} - 826686 q^{9} + O(q^{10}) \) \( 28 q - 14336 q^{4} - 10124 q^{5} + 31104 q^{6} - 72418 q^{7} - 826686 q^{9} - 318272 q^{10} - 1638040 q^{11} + 1978584 q^{13} - 335360 q^{14} + 4555764 q^{15} - 14680064 q^{16} + 1559804 q^{17} - 5097220 q^{19} + 20733952 q^{20} + 29958012 q^{21} - 51113344 q^{22} - 20646748 q^{23} - 15925248 q^{24} + 65193944 q^{25} - 67420544 q^{26} + 126420992 q^{28} + 276341656 q^{29} - 115675776 q^{30} - 371468682 q^{31} - 67370778 q^{33} + 844796672 q^{34} + 150672680 q^{35} + 1693052928 q^{36} - 1351393444 q^{37} - 891388032 q^{38} + 251218260 q^{39} - 325910528 q^{40} + 5451177312 q^{41} + 24960960 q^{42} - 3530671024 q^{43} - 1677352960 q^{44} - 597812076 q^{45} + 3449698688 q^{46} - 3475255764 q^{47} + 10706104834 q^{49} - 5634512384 q^{50} - 2024725572 q^{51} - 1013035008 q^{52} + 9444322068 q^{53} - 918330048 q^{54} + 13463802380 q^{55} + 7451312128 q^{56} + 11007159336 q^{57} - 1514752448 q^{58} - 26063526872 q^{59} - 2332551168 q^{60} - 20964535920 q^{61} + 24035449600 q^{62} - 3013860960 q^{63} + 30064771072 q^{64} - 40538303292 q^{65} - 10018660608 q^{66} + 21217199788 q^{67} + 1597239296 q^{68} + 10716278616 q^{69} + 1230037184 q^{70} - 42719023432 q^{71} - 43156310732 q^{73} + 3430345344 q^{74} - 14813488008 q^{75} + 10439106560 q^{76} + 9063313036 q^{77} + 27329156352 q^{78} + 57995267574 q^{79} - 10615783424 q^{80} - 48814981614 q^{81} - 927854592 q^{82} - 238907176288 q^{83} - 22924394496 q^{84} + 213787952168 q^{85} + 30817020288 q^{86} - 20620471158 q^{87} + 26170032128 q^{88} + 187906684896 q^{89} + 37587286656 q^{90} - 358553662500 q^{91} + 42284539904 q^{92} + 124846567812 q^{93} - 3636794112 q^{94} - 401515267132 q^{95} - 16307453952 q^{96} - 242632624124 q^{97} + 36348195072 q^{98} + 193449247920 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{12}^{\mathrm{new}}(42, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
42.12.e.a 42.e 7.c $6$ $32.270$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-96\) \(729\) \(1045\) \(45731\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2^{5}-2^{5}\beta _{2})q^{2}-3^{5}\beta _{2}q^{3}+2^{10}\beta _{2}q^{4}+\cdots\)
42.12.e.b 42.e 7.c $6$ $32.270$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(96\) \(-729\) \(1331\) \(-83545\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2^{5}+2^{5}\beta _{1})q^{2}+3^{5}\beta _{1}q^{3}+2^{10}\beta _{1}q^{4}+\cdots\)
42.12.e.c 42.e 7.c $8$ $32.270$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-128\) \(-972\) \(-11080\) \(31758\) $\mathrm{SU}(2)[C_{3}]$ \(q-2^{5}\beta _{1}q^{2}+(-3^{5}+3^{5}\beta _{1})q^{3}+(-2^{10}+\cdots)q^{4}+\cdots\)
42.12.e.d 42.e 7.c $8$ $32.270$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(128\) \(972\) \(-1420\) \(-66362\) $\mathrm{SU}(2)[C_{3}]$ \(q+2^{5}\beta _{1}q^{2}+(3^{5}-3^{5}\beta _{1})q^{3}+(-2^{10}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(42, [\chi])\) into lower level spaces

\( S_{12}^{\mathrm{old}}(42, [\chi]) \cong \) \(S_{12}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)