Properties

Label 49.26.a
Level $49$
Weight $26$
Character orbit 49.a
Rep. character $\chi_{49}(1,\cdot)$
Character field $\Q$
Dimension $83$
Newform subspaces $8$
Sturm bound $121$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(121\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{26}(\Gamma_0(49))\).

Total New Old
Modular forms 120 88 32
Cusp forms 112 83 29
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.
\(+\)\(40\)
\(-\)\(43\)

Trace form

\( 83 q - 8144 q^{2} + 867080 q^{3} + 1306459308 q^{4} + 146507724 q^{5} + 6801730 q^{6} - 494183007228 q^{8} + 22455444819655 q^{9} + O(q^{10}) \) \( 83 q - 8144 q^{2} + 867080 q^{3} + 1306459308 q^{4} + 146507724 q^{5} + 6801730 q^{6} - 494183007228 q^{8} + 22455444819655 q^{9} + 7873423401036 q^{10} - 1168566239300 q^{11} - 18182387354390 q^{12} + 152233820234300 q^{13} - 225697387314640 q^{15} + 17783592509970228 q^{16} + 890345466512100 q^{17} - 6661163139315388 q^{18} - 8222563048978552 q^{19} + 18331778107463688 q^{20} + 103267723632755044 q^{22} - 173235273361559272 q^{23} + 293908768579856238 q^{24} + 4589971616015626313 q^{25} - 584933009554077408 q^{26} + 3922769538393328880 q^{27} - 1653899647118544898 q^{29} - 12881757765054475572 q^{30} - 7191126577509927328 q^{31} - 3390236171695043860 q^{32} + 11797797235791095360 q^{33} - 57620208465763991490 q^{34} + 316571700449333046172 q^{36} + 162213600404568252614 q^{37} + 34281788220574494030 q^{38} - 319669285263961999856 q^{39} - 13774896472154977776 q^{40} - 74639141051069370636 q^{41} + 21880606701669855788 q^{43} + 342079202355870278264 q^{44} + 1849727082326150851580 q^{45} - 1211789332647294240268 q^{46} + 176687454969987802560 q^{47} - 292048067755920217310 q^{48} - 2531626331384740719236 q^{50} + 178178103353853436496 q^{51} + 19643028411331763906300 q^{52} - 7784079190046396805790 q^{53} + 14423658066652703591836 q^{54} - 5113512347035525777104 q^{55} - 26622472058336512716480 q^{57} + 70437929997675180614192 q^{58} + 68701556695461618893976 q^{59} - 61423674900813202042016 q^{60} + 27789702050561441964476 q^{61} - 141816375593651405908620 q^{62} + 122827899506880129646412 q^{64} - 90880241641221930706872 q^{65} + 347565087599545720188736 q^{66} - 267512909420332350739956 q^{67} - 111004747401091477031910 q^{68} + 211946616798164944594848 q^{69} - 63042190997582968619744 q^{71} - 500448882194398084005036 q^{72} + 288411834517664662148180 q^{73} + 953763078815455362349908 q^{74} + 95112763570807869127960 q^{75} + 269301610828345798698422 q^{76} + 1761234973935500883466080 q^{78} + 930453336305837253798088 q^{79} - 3006846818225797803319584 q^{80} + 5072447872357986759046315 q^{81} - 465914977222579382195250 q^{82} + 1386608526842439411380520 q^{83} + 2895045087076613380439352 q^{85} - 7893134303609154538620408 q^{86} - 5688079512723372766591120 q^{87} + 8654329549883550963546888 q^{88} - 4678759556391325813573068 q^{89} + 9107560490792403483133468 q^{90} - 4527747471884315744934008 q^{92} + 10815228144330691594561760 q^{93} - 15021706035709683891600708 q^{94} - 5398195779378148397526576 q^{95} - 8034509136321444224352898 q^{96} - 4604276024318696624956540 q^{97} + 10158316560830407922354092 q^{99} + O(q^{100}) \)

Decomposition of \(S_{26}^{\mathrm{new}}(\Gamma_0(49))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 7
49.26.a.a 49.a 1.a $1$ $194.038$ \(\Q\) None \(-48\) \(195804\) \(741989850\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-48q^{2}+195804q^{3}-33552128q^{4}+\cdots\)
49.26.a.b 49.a 1.a $1$ $194.038$ \(\Q\) \(\Q(\sqrt{-7}) \) \(4411\) \(0\) \(0\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q+4411q^{2}-14097511q^{4}-210192720573q^{8}+\cdots\)
49.26.a.c 49.a 1.a $6$ $194.038$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-4230\) \(72104\) \(-110161332\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+(-705-\beta _{1})q^{2}+(12017-2\beta _{1}+\cdots)q^{3}+\cdots\)
49.26.a.d 49.a 1.a $7$ $194.038$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(8373\) \(599172\) \(-485320794\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+(1196+\beta _{1})q^{2}+(85596-3\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
49.26.a.e 49.a 1.a $12$ $194.038$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-8460\) \(0\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+(-705+\beta _{2})q^{2}-\beta _{1}q^{3}+(13487358+\cdots)q^{4}+\cdots\)
49.26.a.f 49.a 1.a $16$ $194.038$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(4050\) \(-531440\) \(-288173088\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+(253-\beta _{1})q^{2}+(-33214+8\beta _{1}+\cdots)q^{3}+\cdots\)
49.26.a.g 49.a 1.a $16$ $194.038$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(4050\) \(531440\) \(288173088\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+(253-\beta _{1})q^{2}+(33214-8\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots\)
49.26.a.h 49.a 1.a $24$ $194.038$ None \(-16290\) \(0\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$

Decomposition of \(S_{26}^{\mathrm{old}}(\Gamma_0(49))\) into lower level spaces

\( S_{26}^{\mathrm{old}}(\Gamma_0(49)) \cong \) \(S_{26}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)