Properties

Label 35.8.a
Level $35$
Weight $8$
Character orbit 35.a
Rep. character $\chi_{35}(1,\cdot)$
Character field $\Q$
Dimension $14$
Newform subspaces $4$
Sturm bound $32$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 35.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(32\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(35))\).

Total New Old
Modular forms 30 14 16
Cusp forms 26 14 12
Eisenstein series 4 0 4

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

\(5\)\(7\)FrickeDim
\(+\)\(+\)$+$\(4\)
\(+\)\(-\)$-$\(3\)
\(-\)\(+\)$-$\(2\)
\(-\)\(-\)$+$\(5\)
Plus space\(+\)\(9\)
Minus space\(-\)\(5\)

Trace form

\( 14 q + 2 q^{2} - 52 q^{3} + 1026 q^{4} + 1616 q^{6} + 686 q^{7} - 3342 q^{8} + 4072 q^{9} + O(q^{10}) \) \( 14 q + 2 q^{2} - 52 q^{3} + 1026 q^{4} + 1616 q^{6} + 686 q^{7} - 3342 q^{8} + 4072 q^{9} + 6500 q^{10} - 2246 q^{11} + 14052 q^{12} - 9572 q^{13} - 8918 q^{14} + 15250 q^{15} + 113426 q^{16} + 81220 q^{17} + 43814 q^{18} - 87448 q^{19} + 28126 q^{21} - 255316 q^{22} + 244384 q^{23} + 421772 q^{24} + 218750 q^{25} - 164772 q^{26} - 675364 q^{27} + 187278 q^{28} + 33238 q^{29} - 370000 q^{30} + 539148 q^{31} - 105894 q^{32} + 176180 q^{33} + 85768 q^{34} + 171500 q^{35} - 372286 q^{36} - 379888 q^{37} - 1810300 q^{38} - 515862 q^{39} - 234000 q^{40} - 1955020 q^{41} + 296352 q^{42} + 1663428 q^{43} - 1637880 q^{44} - 701000 q^{45} + 63712 q^{46} + 971812 q^{47} - 631948 q^{48} + 1647086 q^{49} + 31250 q^{50} - 2854498 q^{51} - 1725368 q^{52} + 1522108 q^{53} + 8978644 q^{54} + 811000 q^{55} - 1524978 q^{56} + 1509396 q^{57} - 7935328 q^{58} + 6278284 q^{59} - 1212500 q^{60} + 2888752 q^{61} - 4179912 q^{62} + 275086 q^{63} + 4856658 q^{64} + 777250 q^{65} + 9144388 q^{66} + 4612844 q^{67} + 13506692 q^{68} - 5536356 q^{69} + 686000 q^{70} + 663688 q^{71} - 5066990 q^{72} + 9970600 q^{73} - 2388372 q^{74} - 812500 q^{75} + 7372772 q^{76} + 9609488 q^{77} - 39153804 q^{78} + 1213230 q^{79} + 8284000 q^{80} + 18161366 q^{81} - 27849468 q^{82} - 13081312 q^{83} - 9347436 q^{84} - 6395250 q^{85} - 35129544 q^{86} + 22708596 q^{87} - 25552408 q^{88} - 3428952 q^{89} + 8858000 q^{90} + 3481450 q^{91} - 9105048 q^{92} + 8963244 q^{93} - 9630484 q^{94} + 21126500 q^{95} - 35450012 q^{96} - 24698356 q^{97} + 235298 q^{98} + 13137812 q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(35))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 7
35.8.a.a 35.a 1.a $2$ $10.933$ \(\Q(\sqrt{11}) \) None 35.8.a.a \(16\) \(-30\) \(250\) \(-686\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(8+\beta )q^{2}+(-15-6\beta )q^{3}+(-20+\cdots)q^{4}+\cdots\)
35.8.a.b 35.a 1.a $3$ $10.933$ 3.3.2268428.1 None 35.8.a.b \(-23\) \(-50\) \(-375\) \(1029\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-8+\beta _{1})q^{2}+(-17+2\beta _{1}-\beta _{2})q^{3}+\cdots\)
35.8.a.c 35.a 1.a $4$ $10.933$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 35.8.a.c \(-2\) \(-37\) \(-500\) \(-1372\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+(-11-\beta _{1}+2\beta _{2}+\cdots)q^{3}+\cdots\)
35.8.a.d 35.a 1.a $5$ $10.933$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 35.8.a.d \(11\) \(65\) \(625\) \(1715\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(2+\beta _{1})q^{2}+(13-\beta _{2})q^{3}+(111-\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(35))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(35)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)