# Properties

 Label 49.22.a Level $49$ Weight $22$ Character orbit 49.a Rep. character $\chi_{49}(1,\cdot)$ Character field $\Q$ Dimension $69$ Newform subspaces $8$ Sturm bound $102$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 49.a (trivial) Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$102$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$, $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{22}(\Gamma_0(49))$$.

Total New Old
Modular forms 102 74 28
Cusp forms 94 69 25
Eisenstein series 8 5 3

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

$$7$$Dim
$$+$$$$33$$
$$-$$$$36$$

## Trace form

 $$69 q + 288 q^{2} - 128842 q^{3} + 68796140 q^{4} + 41640914 q^{5} - 57397886 q^{6} - 6326056668 q^{8} + 198780124867 q^{9} + O(q^{10})$$ $$69 q + 288 q^{2} - 128842 q^{3} + 68796140 q^{4} + 41640914 q^{5} - 57397886 q^{6} - 6326056668 q^{8} + 198780124867 q^{9} - 38036595604 q^{10} + 22346871674 q^{11} - 465998656214 q^{12} + 254283852998 q^{13} - 777783445750 q^{15} + 63688572427572 q^{16} + 2483071056396 q^{17} + 30697464728180 q^{18} + 13415723563298 q^{19} + 225488051615848 q^{20} - 272796343028028 q^{22} - 377409266701662 q^{23} - 166593058901682 q^{24} + 4956861206855553 q^{25} + 697088818177696 q^{26} - 3219173916471100 q^{27} - 3063043625462270 q^{29} + 14411245755170988 q^{30} + 8790568781116276 q^{31} - 5801120849031508 q^{32} + 14576486478194816 q^{33} - 1367743002110562 q^{34} + 30544510070393116 q^{36} + 39037880926076180 q^{37} + 202986820597254190 q^{38} - 80288059479880700 q^{39} + 143841217819931184 q^{40} - 8963531927668892 q^{41} - 301040838544717700 q^{43} + 188043750573718872 q^{44} + 1430155808841222890 q^{45} - 529241139654628556 q^{46} + 912252609604559916 q^{47} - 2697167689781413982 q^{48} + 175812809150646764 q^{50} + 2959642068676244402 q^{51} - 194286838693817604 q^{52} - 2823697892252236656 q^{53} - 2626336074996853604 q^{54} + 568804826451770776 q^{55} - 4981039943015243082 q^{57} - 5345937811930847696 q^{58} - 1208095872867738414 q^{59} + 15116231207662864384 q^{60} - 560942239522196350 q^{61} + 22046242285381503252 q^{62} + 120949572424887993548 q^{64} + 4293532657346501268 q^{65} - 114570303320241509216 q^{66} + 28634340347289436698 q^{67} + 22520606424157123002 q^{68} - 50260567954512185760 q^{69} - 21422328180050613664 q^{71} - 3261564781434553452 q^{72} + 17919746668505420368 q^{73} - 76711991875949438828 q^{74} - 602719016400413150 q^{75} + 168296942370576272022 q^{76} + 70623843862870583040 q^{78} - 47346203742046316358 q^{79} + 275757509990677004416 q^{80} + 497103600861438627517 q^{81} - 203591541535398480338 q^{82} - 228323000393364620982 q^{83} - 260468633940798287518 q^{85} - 286122576596686611640 q^{86} + 1123854421404845115020 q^{87} + 788496585315109090344 q^{88} + 938419369137921644792 q^{89} - 292380858026228411972 q^{90} - 1336502469896420871576 q^{92} + 270710738783316581186 q^{93} + 168155190652975397340 q^{94} - 1032114481900198545706 q^{95} + 1432591235788044923486 q^{96} - 330756189337511158004 q^{97} + 4326731834103672995440 q^{99} + O(q^{100})$$

## Decomposition of $$S_{22}^{\mathrm{new}}(\Gamma_0(49))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 7
49.22.a.a $1$ $136.944$ $$\Q$$ None $$-288$$ $$128844$$ $$-21640950$$ $$0$$ $-$ $$q-288q^{2}+128844q^{3}-2014208q^{4}+\cdots$$
49.22.a.b $1$ $136.944$ $$\Q$$ $$\Q(\sqrt{-7})$$ $$2795$$ $$0$$ $$0$$ $$0$$ $-$ $$q+2795q^{2}+5714873q^{4}+10111530195q^{8}+\cdots$$
49.22.a.c $5$ $136.944$ $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ None $$-2278$$ $$5810$$ $$60216716$$ $$0$$ $-$ $$q+(-456+\beta _{1})q^{2}+(1155+18\beta _{1}+\cdots)q^{3}+\cdots$$
49.22.a.d $6$ $136.944$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$2565$$ $$-263496$$ $$3065148$$ $$0$$ $-$ $$q+(428-\beta _{1})q^{2}+(-43924+15\beta _{1}+\cdots)q^{3}+\cdots$$
49.22.a.e $10$ $136.944$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$-460$$ $$0$$ $$0$$ $$0$$ $-$ $$q+(-46+\beta _{2})q^{2}-\beta _{1}q^{3}+(524612+\cdots)q^{4}+\cdots$$
49.22.a.f $13$ $136.944$ $$\mathbb{Q}[x]/(x^{13} - \cdots)$$ None $$-286$$ $$-118097$$ $$-19296893$$ $$0$$ $+$ $$q+(-22-\beta _{1})q^{2}+(-9084+6\beta _{1}+\cdots)q^{3}+\cdots$$
49.22.a.g $13$ $136.944$ $$\mathbb{Q}[x]/(x^{13} - \cdots)$$ None $$-286$$ $$118097$$ $$19296893$$ $$0$$ $-$ $$q+(-22-\beta _{1})q^{2}+(9084-6\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots$$
49.22.a.h $20$ $136.944$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$-1474$$ $$0$$ $$0$$ $$0$$ $+$ $$q+(-74+\beta _{1})q^{2}-\beta _{3}q^{3}+(1023984+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{22}^{\mathrm{old}}(\Gamma_0(49))$$ into lower level spaces

$$S_{22}^{\mathrm{old}}(\Gamma_0(49)) \cong$$ $$S_{22}^{\mathrm{new}}(\Gamma_0(1))$$$$^{\oplus 3}$$$$\oplus$$$$S_{22}^{\mathrm{new}}(\Gamma_0(7))$$$$^{\oplus 2}$$