Properties

Label 98.10.a
Level $98$
Weight $10$
Character orbit 98.a
Rep. character $\chi_{98}(1,\cdot)$
Character field $\Q$
Dimension $31$
Newform subspaces $12$
Sturm bound $140$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(140\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(98))\).

Total New Old
Modular forms 134 31 103
Cusp forms 118 31 87
Eisenstein series 16 0 16

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

\(2\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(9\)
\(-\)\(-\)\(+\)\(7\)
Plus space\(+\)\(14\)
Minus space\(-\)\(17\)

Trace form

\( 31 q + 16 q^{2} + 6 q^{3} + 7936 q^{4} + 756 q^{5} - 544 q^{6} + 4096 q^{8} + 164295 q^{9} + O(q^{10}) \) \( 31 q + 16 q^{2} + 6 q^{3} + 7936 q^{4} + 756 q^{5} - 544 q^{6} + 4096 q^{8} + 164295 q^{9} - 57344 q^{10} + 47360 q^{11} + 1536 q^{12} - 149328 q^{13} - 574204 q^{15} + 2031616 q^{16} + 1133226 q^{17} + 817616 q^{18} - 364822 q^{19} + 193536 q^{20} + 310208 q^{22} + 381988 q^{23} - 139264 q^{24} + 19653841 q^{25} - 2801728 q^{26} + 1068372 q^{27} + 316342 q^{29} + 4605312 q^{30} - 2041188 q^{31} + 1048576 q^{32} - 41272000 q^{33} - 9521056 q^{34} + 42059520 q^{36} + 61320506 q^{37} + 18670112 q^{38} + 33141280 q^{39} - 14680064 q^{40} - 36076902 q^{41} - 161828564 q^{43} + 12124160 q^{44} + 112432292 q^{45} + 39999872 q^{46} - 24343308 q^{47} + 393216 q^{48} + 130681520 q^{50} - 3882188 q^{51} - 38227968 q^{52} - 89805274 q^{53} - 35188672 q^{54} + 12306224 q^{55} + 283722084 q^{57} - 199875936 q^{58} - 3712782 q^{59} - 146996224 q^{60} - 43059924 q^{61} - 111019072 q^{62} + 520093696 q^{64} + 124521768 q^{65} + 114458624 q^{66} + 176737592 q^{67} + 290105856 q^{68} + 286188224 q^{69} + 335540328 q^{71} + 209309696 q^{72} - 113029370 q^{73} + 97442592 q^{74} + 1419702850 q^{75} - 93394432 q^{76} + 315704192 q^{78} - 375697700 q^{79} + 49545216 q^{80} + 997890439 q^{81} + 732392544 q^{82} + 591878946 q^{83} + 1429760996 q^{85} + 879180224 q^{86} + 1233966460 q^{87} + 79413248 q^{88} - 124823298 q^{89} - 2088905728 q^{90} + 97788928 q^{92} - 1867544788 q^{93} - 1081162944 q^{94} - 332382196 q^{95} - 35651584 q^{96} + 792186490 q^{97} + 1215212156 q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(98))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7
98.10.a.a \(1\) \(50.474\) \(\Q\) None \(-16\) \(6\) \(-560\) \(0\) \(+\) \(-\) \(q-2^{4}q^{2}+6q^{3}+2^{8}q^{4}-560q^{5}+\cdots\)
98.10.a.b \(1\) \(50.474\) \(\Q\) None \(16\) \(-170\) \(-544\) \(0\) \(-\) \(-\) \(q+2^{4}q^{2}-170q^{3}+2^{8}q^{4}-544q^{5}+\cdots\)
98.10.a.c \(1\) \(50.474\) \(\Q\) None \(16\) \(156\) \(-870\) \(0\) \(-\) \(-\) \(q+2^{4}q^{2}+156q^{3}+2^{8}q^{4}-870q^{5}+\cdots\)
98.10.a.d \(2\) \(50.474\) \(\Q(\sqrt{43}) \) None \(-32\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q-2^{4}q^{2}+\beta q^{3}+2^{8}q^{4}+35\beta q^{5}+\cdots\)
98.10.a.e \(2\) \(50.474\) \(\Q(\sqrt{2305}) \) None \(-32\) \(14\) \(2730\) \(0\) \(+\) \(-\) \(q-2^{4}q^{2}+(7+5\beta )q^{3}+2^{8}q^{4}+(1365+\cdots)q^{5}+\cdots\)
98.10.a.f \(2\) \(50.474\) \(\Q(\sqrt{106}) \) None \(32\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+2^{4}q^{2}+3\beta q^{3}+2^{8}q^{4}-31\beta q^{5}+\cdots\)
98.10.a.g \(3\) \(50.474\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-48\) \(-71\) \(1085\) \(0\) \(+\) \(+\) \(q-2^{4}q^{2}+(-24-\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
98.10.a.h \(3\) \(50.474\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-48\) \(71\) \(-1085\) \(0\) \(+\) \(-\) \(q-2^{4}q^{2}+(24+\beta _{1})q^{3}+2^{8}q^{4}+(-363+\cdots)q^{5}+\cdots\)
98.10.a.i \(3\) \(50.474\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(48\) \(-233\) \(-733\) \(0\) \(-\) \(-\) \(q+2^{4}q^{2}+(-78-\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
98.10.a.j \(3\) \(50.474\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(48\) \(233\) \(733\) \(0\) \(-\) \(+\) \(q+2^{4}q^{2}+(78+\beta _{1})q^{3}+2^{8}q^{4}+(246+\cdots)q^{5}+\cdots\)
98.10.a.k \(4\) \(50.474\) \(\Q(\sqrt{2}, \sqrt{4817})\) None \(-64\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q-2^{4}q^{2}+(5\beta _{1}+\beta _{3})q^{3}+2^{8}q^{4}+(104\beta _{1}+\cdots)q^{5}+\cdots\)
98.10.a.l \(6\) \(50.474\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(96\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+2^{4}q^{2}+(\beta _{2}+\beta _{3})q^{3}+2^{8}q^{4}+(-2\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(98))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(98)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)