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Properties

Label 45.22.a
Level $45$
Weight $22$
Character orbit 45.a
Rep. character $\chi_{45}(1,\cdot)$
Character field $\Q$
Dimension $35$
Newform subspaces $9$
Sturm bound $132$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 45.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(132\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(45))\).

Total New Old
Modular forms 130 35 95
Cusp forms 122 35 87
Eisenstein series 8 0 8

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
\(+\)\(+\)$+$\(7\)
\(+\)\(-\)$-$\(7\)
\(-\)\(+\)$-$\(11\)
\(-\)\(-\)$+$\(10\)
Plus space\(+\)\(17\)
Minus space\(-\)\(18\)

Trace form

\( 35 q - 450 q^{2} + 37092448 q^{4} - 9765625 q^{5} - 825625194 q^{7} - 10439782308 q^{8} + O(q^{10}) \) \( 35 q - 450 q^{2} + 37092448 q^{4} - 9765625 q^{5} - 825625194 q^{7} - 10439782308 q^{8} - 21230468750 q^{10} + 100485153300 q^{11} - 771284128714 q^{13} + 3510480095124 q^{14} + 40556399561668 q^{16} - 9518707770798 q^{17} + 730015670924 q^{19} + 18385507812500 q^{20} + 14352924285416 q^{22} - 591220071153858 q^{23} + 3337860107421875 q^{25} - 2270292757080192 q^{26} + 1337553253512672 q^{28} - 7662999717941826 q^{29} + 13261742321338260 q^{31} - 64493257842014196 q^{32} - 8262558565939588 q^{34} - 23746140136718750 q^{35} - 9133121046033914 q^{37} - 41383039467522816 q^{38} - 62747867460937500 q^{40} + 98648163093462762 q^{41} + 542216520751755842 q^{43} + 432620205660044196 q^{44} - 1396186420442520192 q^{46} + 1048798256818902906 q^{47} + 2603943793271938591 q^{49} - 42915344238281250 q^{50} - 4747693247402316032 q^{52} - 4019753680926973638 q^{53} + 1728133135703125000 q^{55} + 3974171369020241820 q^{56} + 13397477002725251212 q^{58} - 7566133238356424592 q^{59} + 9523645469526409142 q^{61} - 28425687883203539112 q^{62} - 6310329210315245620 q^{64} - 9132682625996093750 q^{65} + 45026308453023095958 q^{67} + 16538501771571473280 q^{68} - 21843674255859375000 q^{70} - 15004564733533811412 q^{71} - 230993478586578680842 q^{73} + 116156289541873487304 q^{74} + 123387326479548794440 q^{76} + 186178714042885457568 q^{77} - 128709406486079080296 q^{79} + 183119713208750000000 q^{80} - 578127545066324355068 q^{82} + 233092163491091175750 q^{83} + 14778862034160156250 q^{85} + 732519126065257635840 q^{86} - 66725519083197387552 q^{88} - 521251596848590130538 q^{89} + 363611801309982930228 q^{91} - 537521639101351507296 q^{92} - 610487157069372734680 q^{94} - 569833023472695312500 q^{95} + 1023495057611653535398 q^{97} - 3435459248900910336834 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(45))\) 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
45.22.a.a 45.a 1.a $1$ $125.765$ \(\Q\) None \(-544\) \(0\) \(9765625\) \(1277698380\) $-$ $-$ $\mathrm{SU}(2)$ \(q-544q^{2}-1801216q^{4}+5^{10}q^{5}+\cdots\)
45.22.a.b 45.a 1.a $3$ $125.765$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-803\) \(0\) \(29296875\) \(-1577598316\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-268+\beta _{1})q^{2}+(2238318+120\beta _{1}+\cdots)q^{4}+\cdots\)
45.22.a.c 45.a 1.a $3$ $125.765$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-702\) \(0\) \(29296875\) \(-2072418204\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-234-\beta _{1})q^{2}+(1748076+1030\beta _{1}+\cdots)q^{4}+\cdots\)
45.22.a.d 45.a 1.a $3$ $125.765$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(1312\) \(0\) \(29296875\) \(684416558\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(437-\beta _{1})q^{2}+(330323-190\beta _{1}+\cdots)q^{4}+\cdots\)
45.22.a.e 45.a 1.a $3$ $125.765$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(2300\) \(0\) \(-29296875\) \(465666872\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(767+\beta _{1})q^{2}+(421908+1324\beta _{1}+\cdots)q^{4}+\cdots\)
45.22.a.f 45.a 1.a $4$ $125.765$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-2910\) \(0\) \(-39062500\) \(512613800\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-728+\beta _{1})q^{2}+(2291466-298\beta _{1}+\cdots)q^{4}+\cdots\)
45.22.a.g 45.a 1.a $4$ $125.765$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(897\) \(0\) \(-39062500\) \(-234577504\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(224+\beta _{1})q^{2}+(-290854+294\beta _{1}+\cdots)q^{4}+\cdots\)
45.22.a.h 45.a 1.a $7$ $125.765$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-575\) \(0\) \(68359375\) \(59286610\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-82-\beta _{1})q^{2}+(1191192-4\beta _{1}+\cdots)q^{4}+\cdots\)
45.22.a.i 45.a 1.a $7$ $125.765$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(575\) \(0\) \(-68359375\) \(59286610\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(82+\beta _{1})q^{2}+(1191192-4\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(45))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(45)) \cong \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)