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Properties

Label 350.10.a
Level $350$
Weight $10$
Character orbit 350.a
Rep. character $\chi_{350}(1,\cdot)$
Character field $\Q$
Dimension $86$
Newform subspaces $24$
Sturm bound $600$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 350.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(600\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(350))\).

Total New Old
Modular forms 552 86 466
Cusp forms 528 86 442
Eisenstein series 24 0 24

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

\(2\)\(5\)\(7\)FrickeDim
\(+\)\(+\)\(+\)$+$\(10\)
\(+\)\(+\)\(-\)$-$\(11\)
\(+\)\(-\)\(+\)$-$\(12\)
\(+\)\(-\)\(-\)$+$\(10\)
\(-\)\(+\)\(+\)$-$\(10\)
\(-\)\(+\)\(-\)$+$\(9\)
\(-\)\(-\)\(+\)$+$\(11\)
\(-\)\(-\)\(-\)$-$\(13\)
Plus space\(+\)\(40\)
Minus space\(-\)\(46\)

Trace form

\( 86 q - 442 q^{3} + 22016 q^{4} + 3040 q^{6} + 552678 q^{9} + O(q^{10}) \) \( 86 q - 442 q^{3} + 22016 q^{4} + 3040 q^{6} + 552678 q^{9} + 36528 q^{11} - 113152 q^{12} + 217662 q^{13} + 76832 q^{14} + 5636096 q^{16} - 432680 q^{17} + 19520 q^{18} + 1848578 q^{19} + 427378 q^{21} + 247744 q^{22} + 1249824 q^{23} + 778240 q^{24} + 5466336 q^{26} - 15416740 q^{27} + 12648992 q^{29} - 9823676 q^{31} + 10689000 q^{33} - 37304448 q^{34} + 141485568 q^{36} + 47012080 q^{37} - 31595488 q^{38} - 32233568 q^{39} - 51910140 q^{41} - 6223392 q^{42} - 43983644 q^{43} + 9351168 q^{44} + 92511360 q^{46} - 105407996 q^{47} - 28966912 q^{48} + 495772886 q^{49} - 363754504 q^{51} + 55721472 q^{52} + 88933796 q^{53} + 331929088 q^{54} + 19668992 q^{56} - 93824748 q^{57} - 128310464 q^{58} + 687082598 q^{59} - 232329722 q^{61} + 154825536 q^{62} - 48356140 q^{63} + 1442840576 q^{64} + 203034432 q^{66} - 283265000 q^{67} - 110766080 q^{68} - 958235160 q^{69} + 48422040 q^{71} + 4997120 q^{72} + 735090764 q^{73} + 300447936 q^{74} + 473235968 q^{76} - 401917796 q^{77} - 1710429056 q^{78} - 763111392 q^{79} + 5567528802 q^{81} + 1710835776 q^{82} + 160701298 q^{83} + 109408768 q^{84} - 388510016 q^{86} + 3205091476 q^{87} + 63422464 q^{88} + 1438826432 q^{89} - 893772250 q^{91} + 319954944 q^{92} + 2031921632 q^{93} - 1704789056 q^{94} + 199229440 q^{96} - 3140619112 q^{97} - 2399054932 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(350))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5 7
350.10.a.a 350.a 1.a $1$ $180.263$ \(\Q\) None \(-16\) \(-170\) \(0\) \(2401\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}-170q^{3}+2^{8}q^{4}+2720q^{6}+\cdots\)
350.10.a.b 350.a 1.a $1$ $180.263$ \(\Q\) None \(16\) \(-120\) \(0\) \(-2401\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}-120q^{3}+2^{8}q^{4}-1920q^{6}+\cdots\)
350.10.a.c 350.a 1.a $1$ $180.263$ \(\Q\) None \(16\) \(6\) \(0\) \(2401\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+6q^{3}+2^{8}q^{4}+96q^{6}+7^{4}q^{7}+\cdots\)
350.10.a.d 350.a 1.a $1$ $180.263$ \(\Q\) None \(16\) \(87\) \(0\) \(-2401\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+87q^{3}+2^{8}q^{4}+1392q^{6}+\cdots\)
350.10.a.e 350.a 1.a $2$ $180.263$ \(\Q(\sqrt{457}) \) None \(-32\) \(41\) \(0\) \(4802\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(20-\beta )q^{3}+2^{8}q^{4}+(-320+\cdots)q^{6}+\cdots\)
350.10.a.f 350.a 1.a $2$ $180.263$ \(\Q(\sqrt{2473}) \) None \(-32\) \(143\) \(0\) \(-4802\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(71-\beta )q^{3}+2^{8}q^{4}+(-1136+\cdots)q^{6}+\cdots\)
350.10.a.g 350.a 1.a $2$ $180.263$ \(\Q(\sqrt{5881}) \) None \(32\) \(-135\) \(0\) \(4802\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(-68-\beta )q^{3}+2^{8}q^{4}+(-1088+\cdots)q^{6}+\cdots\)
350.10.a.h 350.a 1.a $2$ $180.263$ \(\Q(\sqrt{3061}) \) None \(32\) \(-110\) \(0\) \(4802\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(-55-\beta )q^{3}+2^{8}q^{4}+(-880+\cdots)q^{6}+\cdots\)
350.10.a.i 350.a 1.a $2$ $180.263$ \(\Q(\sqrt{541}) \) None \(32\) \(-58\) \(0\) \(-4802\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(-29-\beta )q^{3}+2^{8}q^{4}+(-464+\cdots)q^{6}+\cdots\)
350.10.a.j 350.a 1.a $2$ $180.263$ \(\Q(\sqrt{2305}) \) None \(32\) \(14\) \(0\) \(-4802\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(7+\beta )q^{3}+2^{8}q^{4}+(112+\cdots)q^{6}+\cdots\)
350.10.a.k 350.a 1.a $3$ $180.263$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-48\) \(-206\) \(0\) \(-7203\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(-69+\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
350.10.a.l 350.a 1.a $3$ $180.263$ 3.3.2997373.1 None \(-48\) \(66\) \(0\) \(7203\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(22+\beta _{2})q^{3}+2^{8}q^{4}+(-352+\cdots)q^{6}+\cdots\)
350.10.a.m 350.a 1.a $4$ $180.263$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-64\) \(-161\) \(0\) \(9604\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(-40-\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
350.10.a.n 350.a 1.a $4$ $180.263$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-64\) \(-7\) \(0\) \(-9604\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(-2+\beta _{1})q^{3}+2^{8}q^{4}+(2^{5}+\cdots)q^{6}+\cdots\)
350.10.a.o 350.a 1.a $4$ $180.263$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(64\) \(7\) \(0\) \(9604\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(2-\beta _{1})q^{3}+2^{8}q^{4}+(2^{5}+\cdots)q^{6}+\cdots\)
350.10.a.p 350.a 1.a $4$ $180.263$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(64\) \(161\) \(0\) \(-9604\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(40+\beta _{1})q^{3}+2^{8}q^{4}+(640+\cdots)q^{6}+\cdots\)
350.10.a.q 350.a 1.a $5$ $180.263$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-80\) \(-96\) \(0\) \(-12005\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(-19-\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
350.10.a.r 350.a 1.a $5$ $180.263$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-80\) \(74\) \(0\) \(12005\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(15-\beta _{1})q^{3}+2^{8}q^{4}+(-240+\cdots)q^{6}+\cdots\)
350.10.a.s 350.a 1.a $5$ $180.263$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(80\) \(-74\) \(0\) \(-12005\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(-15+\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
350.10.a.t 350.a 1.a $5$ $180.263$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(80\) \(96\) \(0\) \(12005\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(19+\beta _{1})q^{3}+2^{8}q^{4}+(304+\cdots)q^{6}+\cdots\)
350.10.a.u 350.a 1.a $6$ $180.263$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-96\) \(77\) \(0\) \(14406\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(13+\beta _{1})q^{3}+2^{8}q^{4}+(-208+\cdots)q^{6}+\cdots\)
350.10.a.v 350.a 1.a $6$ $180.263$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(96\) \(-77\) \(0\) \(-14406\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(-13-\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
350.10.a.w 350.a 1.a $8$ $180.263$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-128\) \(-77\) \(0\) \(-19208\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(-10+\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
350.10.a.x 350.a 1.a $8$ $180.263$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(128\) \(77\) \(0\) \(19208\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(10-\beta _{1})q^{3}+2^{8}q^{4}+(160+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(350))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(350)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)