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Properties

Label 4.13.b
Level $4$
Weight $13$
Character orbit 4.b
Rep. character $\chi_{4}(3,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $2$
Sturm bound $6$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(6\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{13}(4, [\chi])\).

Total New Old
Modular forms 7 7 0
Cusp forms 5 5 0
Eisenstein series 2 2 0

Trace form

\( 5 q + 44 q^{2} + 2576 q^{4} + 5146 q^{5} + 15744 q^{6} + 188864 q^{8} - 834123 q^{9} + O(q^{10}) \) \( 5 q + 44 q^{2} + 2576 q^{4} + 5146 q^{5} + 15744 q^{6} + 188864 q^{8} - 834123 q^{9} - 554984 q^{10} + 6919680 q^{12} - 409862 q^{13} - 18365184 q^{14} + 56172800 q^{16} + 11951818 q^{17} - 176665236 q^{18} + 259330336 q^{20} + 39226368 q^{21} - 408017280 q^{22} + 708028416 q^{24} - 121460289 q^{25} - 491723624 q^{26} - 4992000 q^{28} - 614460998 q^{29} + 898855680 q^{30} - 2276402176 q^{32} + 1746286080 q^{33} + 4897652056 q^{34} - 8728844016 q^{36} + 148433818 q^{37} + 7026341760 q^{38} - 6821646464 q^{40} + 610959850 q^{41} + 7485849600 q^{42} - 1860503040 q^{44} - 15090499974 q^{45} - 7050967296 q^{46} + 37817180160 q^{48} + 31516086437 q^{49} - 44609247804 q^{50} + 47103731488 q^{52} - 58411039142 q^{53} - 68093883648 q^{54} + 45300363264 q^{56} + 111679050240 q^{57} - 72766023272 q^{58} + 32055168000 q^{60} - 131609948870 q^{61} + 52348584960 q^{62} - 95850655744 q^{64} + 243506825492 q^{65} + 215300997120 q^{66} - 186954993632 q^{68} - 561988451328 q^{69} + 242172664320 q^{70} - 396528138816 q^{72} + 684251630698 q^{73} + 424199567896 q^{74} - 251463467520 q^{76} - 839111961600 q^{77} + 70820954880 q^{78} - 23691390464 q^{80} + 1371041884773 q^{81} - 501400690472 q^{82} + 679181647872 q^{84} - 1296906788428 q^{85} - 464014452096 q^{86} + 817476802560 q^{88} + 1634577955882 q^{89} - 1552028227944 q^{90} + 1152180771840 q^{92} - 3122417602560 q^{93} - 1545503476224 q^{94} + 2213836652544 q^{96} + 2934122003338 q^{97} + 588933129644 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{13}^{\mathrm{new}}(4, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
4.13.b.a 4.b 4.b $1$ $3.656$ \(\Q\) \(\Q(\sqrt{-1}) \) \(-64\) \(0\) \(23506\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2^{6}q^{2}+2^{12}q^{4}+23506q^{5}-2^{18}q^{8}+\cdots\)
4.13.b.b 4.b 4.b $4$ $3.656$ 4.0.8546467905.1 None \(108\) \(0\) \(-18360\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(3^{3}+\beta _{1})q^{2}+(-\beta _{1}+\beta _{3})q^{3}+(-380+\cdots)q^{4}+\cdots\)