Properties

Label 35.10.a
Level $35$
Weight $10$
Character orbit 35.a
Rep. character $\chi_{35}(1,\cdot)$
Character field $\Q$
Dimension $18$
Newform subspaces $5$
Sturm bound $40$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 35.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(40\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(35))\).

Total New Old
Modular forms 38 18 20
Cusp forms 34 18 16
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
\(+\)\(+\)$+$\(4\)
\(+\)\(-\)$-$\(5\)
\(-\)\(+\)$-$\(6\)
\(-\)\(-\)$+$\(3\)
Plus space\(+\)\(7\)
Minus space\(-\)\(11\)

Trace form

\( 18 q + 2 q^{2} - 292 q^{3} + 5122 q^{4} + 4304 q^{6} - 4802 q^{7} - 8766 q^{8} + 200470 q^{9} + O(q^{10}) \) \( 18 q + 2 q^{2} - 292 q^{3} + 5122 q^{4} + 4304 q^{6} - 4802 q^{7} - 8766 q^{8} + 200470 q^{9} + 22500 q^{10} + 37972 q^{11} - 426588 q^{12} + 94448 q^{13} + 24010 q^{14} - 335000 q^{15} + 1693970 q^{16} - 1264652 q^{17} + 2094806 q^{18} + 1823828 q^{19} - 19208 q^{21} - 659076 q^{22} - 1968584 q^{23} + 2183948 q^{24} + 7031250 q^{25} + 21507516 q^{26} - 2035408 q^{27} - 10453954 q^{28} + 6445312 q^{29} + 3050000 q^{30} - 10915824 q^{31} - 28934550 q^{32} + 45734648 q^{33} - 5360792 q^{34} - 6002500 q^{35} + 92406098 q^{36} + 2690372 q^{37} - 18091900 q^{38} - 73514484 q^{39} + 50310000 q^{40} + 76904348 q^{41} - 12446784 q^{42} - 21190792 q^{43} - 95101704 q^{44} + 33820000 q^{45} - 88935040 q^{46} + 5509312 q^{47} - 163653100 q^{48} + 103766418 q^{49} + 781250 q^{50} - 132500620 q^{51} - 50849208 q^{52} - 120104252 q^{53} - 46141964 q^{54} - 92205000 q^{55} + 19548942 q^{56} - 168899208 q^{57} + 576773296 q^{58} + 9290476 q^{59} - 397512500 q^{60} - 86369600 q^{61} - 537252648 q^{62} - 77970074 q^{63} + 674436530 q^{64} - 48065000 q^{65} - 1327317596 q^{66} + 497623456 q^{67} - 397539388 q^{68} + 593805240 q^{69} - 48020000 q^{70} + 73818616 q^{71} + 1737012370 q^{72} - 162943356 q^{73} + 302347548 q^{74} - 114062500 q^{75} - 1877635388 q^{76} - 75180112 q^{77} - 4816044 q^{78} + 279703380 q^{79} + 1636780000 q^{80} + 2109004130 q^{81} + 640444260 q^{82} - 332763892 q^{83} + 1378337268 q^{84} - 382997500 q^{85} + 2463754824 q^{86} + 3805847952 q^{87} + 2168058136 q^{88} - 3579693828 q^{89} + 244730000 q^{90} + 86522436 q^{91} - 3018039768 q^{92} - 1624114440 q^{93} - 3851452820 q^{94} - 1305242500 q^{95} - 805565564 q^{96} + 59279636 q^{97} + 11529602 q^{98} + 4488271832 q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(35))\) 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
35.10.a.a 35.a 1.a $1$ $18.026$ \(\Q\) None 35.10.a.a \(28\) \(-116\) \(625\) \(2401\) $-$ $-$ $\mathrm{SU}(2)$ \(q+28q^{2}-116q^{3}+272q^{4}+5^{4}q^{5}+\cdots\)
35.10.a.b 35.a 1.a $2$ $18.026$ \(\Q(\sqrt{2}) \) None 35.10.a.b \(-24\) \(-174\) \(1250\) \(4802\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-12+\beta )q^{2}+(-87+54\beta )q^{3}+\cdots\)
35.10.a.c 35.a 1.a $4$ $18.026$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 35.10.a.c \(-19\) \(-18\) \(-2500\) \(-9604\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-5-\beta _{1})q^{2}+(-4+\beta _{2})q^{3}+(435+\cdots)q^{4}+\cdots\)
35.10.a.d 35.a 1.a $5$ $18.026$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 35.10.a.d \(2\) \(140\) \(-3125\) \(12005\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(28+\beta _{2})q^{3}+(168-4\beta _{1}+\cdots)q^{4}+\cdots\)
35.10.a.e 35.a 1.a $6$ $18.026$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 35.10.a.e \(15\) \(-124\) \(3750\) \(-14406\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(3-\beta _{1})q^{2}+(-20-\beta _{1}+\beta _{2})q^{3}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(35))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(35)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)