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Properties

Label 13.12.a
Level $13$
Weight $12$
Character orbit 13.a
Rep. character $\chi_{13}(1,\cdot)$
Character field $\Q$
Dimension $11$
Newform subspaces $2$
Sturm bound $14$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 13.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(14\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(13))\).

Total New Old
Modular forms 15 11 4
Cusp forms 13 11 2
Eisenstein series 2 0 2

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

\(13\)Dim
\(+\)\(6\)
\(-\)\(5\)

Trace form

\( 11 q + 14 q^{2} - 20 q^{3} + 18302 q^{4} + 770 q^{5} - 22356 q^{6} - 40472 q^{7} - 41544 q^{8} + 1048863 q^{9} + O(q^{10}) \) \( 11 q + 14 q^{2} - 20 q^{3} + 18302 q^{4} + 770 q^{5} - 22356 q^{6} - 40472 q^{7} - 41544 q^{8} + 1048863 q^{9} - 6558 q^{10} - 1443852 q^{11} + 278982 q^{12} - 371293 q^{13} + 3197926 q^{14} - 9404984 q^{15} + 15961490 q^{16} + 12448086 q^{17} - 24241138 q^{18} - 4602356 q^{19} + 75833300 q^{20} - 37638800 q^{21} - 16924608 q^{22} - 1353800 q^{23} - 154650708 q^{24} + 144966677 q^{25} - 35644128 q^{26} + 189892144 q^{27} - 283857056 q^{28} - 224809782 q^{29} - 274325902 q^{30} - 172350848 q^{31} + 1139629712 q^{32} - 818480720 q^{33} - 755306916 q^{34} + 622506648 q^{35} + 3169855192 q^{36} - 369863390 q^{37} + 1769119492 q^{38} - 360896796 q^{39} - 873537294 q^{40} - 2578116626 q^{41} - 7013954238 q^{42} - 794814044 q^{43} + 830321236 q^{44} + 8689697546 q^{45} + 1817442660 q^{46} - 8015746656 q^{47} + 6283092766 q^{48} + 9258535155 q^{49} - 10817398534 q^{50} + 13339519280 q^{51} - 4572844588 q^{52} + 4511046890 q^{53} - 26339445228 q^{54} - 4629796776 q^{55} + 4056068978 q^{56} - 14931881088 q^{57} + 22177924776 q^{58} + 3853933924 q^{59} - 2270304432 q^{60} + 2078626882 q^{61} - 561722056 q^{62} + 361546328 q^{63} - 15176929966 q^{64} - 2173549222 q^{65} + 24816629664 q^{66} + 38915233708 q^{67} - 14369028870 q^{68} + 387900928 q^{69} + 25212520260 q^{70} - 58066820168 q^{71} - 16884990288 q^{72} - 6608698322 q^{73} - 19505266742 q^{74} - 14978188484 q^{75} + 146173699240 q^{76} - 20579410408 q^{77} + 34618616734 q^{78} - 97023255680 q^{79} + 69870018508 q^{80} + 134641759827 q^{81} - 48818256084 q^{82} + 2601089100 q^{83} - 303502689380 q^{84} - 49120712172 q^{85} + 109680301404 q^{86} + 5242109864 q^{87} - 163780604808 q^{88} + 86246478958 q^{89} - 476289754080 q^{90} - 11925931160 q^{91} + 355165655260 q^{92} + 332006816880 q^{93} + 245017459950 q^{94} + 133983808456 q^{95} - 184167544724 q^{96} - 270561627674 q^{97} + 317151283102 q^{98} - 271122388524 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(13))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 13
13.12.a.a 13.a 1.a $5$ $9.988$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 13.12.a.a \(-41\) \(-496\) \(-2542\) \(-36296\) $-$ $\mathrm{SU}(2)$ \(q+(-8-\beta _{1})q^{2}+(-99-2\beta _{1}-\beta _{4})q^{3}+\cdots\)
13.12.a.b 13.a 1.a $6$ $9.988$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 13.12.a.b \(55\) \(476\) \(3312\) \(-4176\) $+$ $\mathrm{SU}(2)$ \(q+(9+\beta _{1})q^{2}+(80-2\beta _{1}+\beta _{4})q^{3}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(13))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(13)) \simeq \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)