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Properties

Label 15.22.a
Level $15$
Weight $22$
Character orbit 15.a
Rep. character $\chi_{15}(1,\cdot)$
Character field $\Q$
Dimension $14$
Newform subspaces $5$
Sturm bound $44$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 15.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(44\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(15))\).

Total New Old
Modular forms 44 14 30
Cusp forms 40 14 26
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.

\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(6\)
Minus space\(-\)\(8\)

Trace form

\( 14 q - 1148 q^{2} + 10259362 q^{4} - 44759142 q^{6} - 2141228772 q^{7} + 4153143708 q^{8} + 48814981614 q^{9} + O(q^{10}) \) \( 14 q - 1148 q^{2} + 10259362 q^{4} - 44759142 q^{6} - 2141228772 q^{7} + 4153143708 q^{8} + 48814981614 q^{9} - 51230468750 q^{10} - 137400742484 q^{11} + 123834728448 q^{12} - 410657492608 q^{13} - 3952487784132 q^{14} - 1153300781250 q^{15} + 16260202649170 q^{16} + 8785177804120 q^{17} - 4002828492348 q^{18} + 11136533133968 q^{19} - 98211015625000 q^{20} + 146014161365556 q^{21} + 222984794950340 q^{22} + 202674565846944 q^{23} - 1117366702083486 q^{24} + 1335144042968750 q^{25} + 2664215836742908 q^{26} - 7565957152173204 q^{28} + 5457848739985736 q^{29} - 1180980000000000 q^{30} - 6668517790368696 q^{31} + 58470949366219988 q^{32} - 10832491935747684 q^{33} - 69131404824926356 q^{34} + 25423901445312500 q^{35} + 35772183385812162 q^{36} - 28686580722503192 q^{37} + 3206258211838216 q^{38} + 34382671448455332 q^{39} - 114054546269531250 q^{40} + 102242145449701244 q^{41} + 312526808176412844 q^{42} - 388347618030448672 q^{43} - 58429634412420244 q^{44} + 971666306863384944 q^{46} - 1095842467005437144 q^{47} + 526920774318389232 q^{48} + 3810944887065963790 q^{49} - 109481811523437500 q^{50} + 1677438006475922532 q^{51} - 2581591185787477040 q^{52} + 6552126185902649104 q^{53} - 156065478127743942 q^{54} - 1311408809726562500 q^{55} - 7553861529875629980 q^{56} - 398813838523700184 q^{57} + 17934889139663005012 q^{58} - 1777264658856319988 q^{59} + 1590340652402343750 q^{60} + 15960969972623920460 q^{61} - 107399874882393624 q^{62} - 7466003081181985572 q^{63} + 19741548635175671978 q^{64} - 7173052057460937500 q^{65} - 15765178480672478436 q^{66} - 26009068613169366840 q^{67} - 25587960871526474536 q^{68} + 4986833646861373968 q^{69} - 35662561274648437500 q^{70} + 5284854230456043608 q^{71} + 14481116696165698908 q^{72} - 18476109254494484716 q^{73} + 35360481945402749348 q^{74} + 116089182200247345472 q^{76} - 578876957845515800064 q^{77} + 356850064032219050400 q^{78} + 353538794575561173480 q^{79} - 374990521121406250000 q^{80} + 170207316426797003214 q^{81} - 889089034247003234984 q^{82} + 91154100691595731896 q^{83} + 860815464698526501132 q^{84} - 301623211291835937500 q^{85} - 423760136111166072584 q^{86} - 545854478001321153348 q^{87} + 646315436617343605548 q^{88} + 1040836518703304009268 q^{89} - 178629599293417968750 q^{90} + 148622454897966057624 q^{91} - 2920822411631352313632 q^{92} - 1036071074698188207048 q^{93} + 2476941558932770009352 q^{94} - 505877326697421875000 q^{95} - 970502262100848020214 q^{96} - 1987505821845263522660 q^{97} + 4462427630084347544948 q^{98} - 479086765579029192084 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(15))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5
15.22.a.a 15.a 1.a $1$ $41.922$ \(\Q\) None \(544\) \(59049\) \(-9765625\) \(1277698380\) $-$ $+$ $\mathrm{SU}(2)$ \(q+544q^{2}+3^{10}q^{3}-1801216q^{4}+\cdots\)
15.22.a.b 15.a 1.a $3$ $41.922$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-2300\) \(177147\) \(29296875\) \(465666872\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-767-\beta _{1})q^{2}+3^{10}q^{3}+(421908+\cdots)q^{4}+\cdots\)
15.22.a.c 15.a 1.a $3$ $41.922$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(702\) \(-177147\) \(-29296875\) \(-2072418204\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(234+\beta _{1})q^{2}-3^{10}q^{3}+(1748076+\cdots)q^{4}+\cdots\)
15.22.a.d 15.a 1.a $3$ $41.922$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(803\) \(177147\) \(-29296875\) \(-1577598316\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(268-\beta _{1})q^{2}+3^{10}q^{3}+(2238318+\cdots)q^{4}+\cdots\)
15.22.a.e 15.a 1.a $4$ $41.922$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-897\) \(-236196\) \(39062500\) \(-234577504\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-224-\beta _{1})q^{2}-3^{10}q^{3}+(-290854+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(15))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(15)) \cong \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)