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Properties

Label 8.13.d
Level $8$
Weight $13$
Character orbit 8.d
Rep. character $\chi_{8}(3,\cdot)$
Character field $\Q$
Dimension $11$
Newform subspaces $2$
Sturm bound $13$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 8.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(13\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{13}(8, [\chi])\).

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

Trace form

\( 11 q - 46 q^{2} - 2 q^{3} + 1652 q^{4} - 88676 q^{6} + 289304 q^{8} + 1594321 q^{9} + O(q^{10}) \) \( 11 q - 46 q^{2} - 2 q^{3} + 1652 q^{4} - 88676 q^{6} + 289304 q^{8} + 1594321 q^{9} + 1873200 q^{10} - 2668322 q^{11} - 2732552 q^{12} + 12728736 q^{14} - 21516784 q^{16} - 2419562 q^{17} + 1926742 q^{18} - 51868610 q^{19} + 17615520 q^{20} - 103924772 q^{22} + 341593744 q^{24} - 462805765 q^{25} + 151295184 q^{26} + 967927996 q^{27} - 132075840 q^{28} + 456064800 q^{30} - 1577462176 q^{32} + 787396076 q^{33} + 1555715044 q^{34} - 2838470400 q^{35} - 4478101220 q^{36} - 3983433572 q^{38} + 8180322240 q^{40} - 3710988362 q^{41} + 14692585920 q^{42} + 18050876638 q^{43} - 20073482120 q^{44} - 16813594656 q^{46} + 24349522528 q^{48} - 25691199637 q^{49} + 40513425650 q^{50} + 6952490492 q^{51} - 36172521120 q^{52} - 100535014088 q^{54} + 154450364544 q^{56} + 22549744556 q^{57} + 193270394640 q^{58} - 100809712226 q^{59} - 347360715840 q^{60} - 299237961600 q^{62} + 344932668992 q^{64} + 96485235840 q^{65} + 567267337352 q^{66} + 125184923518 q^{67} - 482345621912 q^{68} - 799057954560 q^{70} + 1051757829832 q^{72} - 9696103562 q^{73} + 742739480496 q^{74} - 91148212130 q^{75} - 1059904664072 q^{76} - 1795838526240 q^{78} + 1981932232320 q^{80} + 39118873879 q^{81} + 2206318374628 q^{82} - 339177211202 q^{83} - 3144240693120 q^{84} - 2711857667492 q^{86} + 2674492705168 q^{88} + 47015069110 q^{89} + 3983485096080 q^{90} + 361546645248 q^{91} - 2649411172800 q^{92} - 2517413216064 q^{94} + 3543110668864 q^{96} + 182261693398 q^{97} + 3263732873234 q^{98} - 906543916678 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{13}^{\mathrm{new}}(8, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
8.13.d.a 8.d 8.d $1$ $7.312$ \(\Q\) \(\Q(\sqrt{-2}) \) \(64\) \(658\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{6}q^{2}+658q^{3}+2^{12}q^{4}+42112q^{6}+\cdots\)
8.13.d.b 8.d 8.d $10$ $7.312$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-110\) \(-660\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-11-\beta _{1})q^{2}+(-66+3\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots\)