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Properties

Label 441.6.a
Level $441$
Weight $6$
Character orbit 441.a
Rep. character $\chi_{441}(1,\cdot)$
Character field $\Q$
Dimension $83$
Newform subspaces $31$
Sturm bound $336$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 31 \)
Sturm bound: \(336\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(441))\).

Total New Old
Modular forms 296 88 208
Cusp forms 264 83 181
Eisenstein series 32 5 27

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

\(3\)\(7\)FrickeDim
\(+\)\(+\)$+$\(15\)
\(+\)\(-\)$-$\(19\)
\(-\)\(+\)$-$\(25\)
\(-\)\(-\)$+$\(24\)
Plus space\(+\)\(39\)
Minus space\(-\)\(44\)

Trace form

\( 83 q + 2 q^{2} + 1296 q^{4} - 36 q^{5} + 132 q^{8} + O(q^{10}) \) \( 83 q + 2 q^{2} + 1296 q^{4} - 36 q^{5} + 132 q^{8} + 192 q^{10} + 200 q^{11} + 104 q^{13} + 18876 q^{16} - 2208 q^{17} - 4312 q^{19} - 7764 q^{20} - 4964 q^{22} + 6304 q^{23} + 46469 q^{25} + 5544 q^{26} + 7618 q^{29} + 5576 q^{31} + 5836 q^{32} + 18522 q^{34} + 17366 q^{37} - 26526 q^{38} + 26136 q^{40} - 13008 q^{41} - 26896 q^{43} + 30376 q^{44} + 50612 q^{46} - 8832 q^{47} + 70322 q^{50} + 64532 q^{52} - 29558 q^{53} - 14472 q^{55} + 93092 q^{58} - 58176 q^{59} + 23432 q^{61} + 131004 q^{62} + 238484 q^{64} - 47580 q^{65} - 172056 q^{67} - 105270 q^{68} + 150668 q^{71} + 1964 q^{73} + 280596 q^{74} - 207970 q^{76} + 37540 q^{79} - 498564 q^{80} + 137586 q^{82} + 172560 q^{83} - 159324 q^{85} - 635940 q^{86} - 468000 q^{88} - 77016 q^{89} + 562352 q^{92} + 646404 q^{94} + 321228 q^{95} - 147724 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\) 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 7
441.6.a.a 441.a 1.a $1$ $70.729$ \(\Q\) \(\Q(\sqrt{-7}) \) \(-11\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q-11q^{2}+89q^{4}-627q^{8}+76q^{11}+\cdots\)
441.6.a.b 441.a 1.a $1$ $70.729$ \(\Q\) None \(-10\) \(0\) \(-106\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-10q^{2}+68q^{4}-106q^{5}-360q^{8}+\cdots\)
441.6.a.c 441.a 1.a $1$ $70.729$ \(\Q\) None \(-5\) \(0\) \(94\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-5q^{2}-7q^{4}+94q^{5}+195q^{8}-470q^{10}+\cdots\)
441.6.a.d 441.a 1.a $1$ $70.729$ \(\Q\) None \(-1\) \(0\) \(-34\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-31q^{4}-34q^{5}+63q^{8}+34q^{10}+\cdots\)
441.6.a.e 441.a 1.a $1$ $70.729$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-2^{5}q^{4}-427q^{13}+2^{10}q^{16}+3143q^{19}+\cdots\)
441.6.a.f 441.a 1.a $1$ $70.729$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-2^{5}q^{4}+427q^{13}+2^{10}q^{16}-3143q^{19}+\cdots\)
441.6.a.g 441.a 1.a $1$ $70.729$ \(\Q\) None \(2\) \(0\) \(-11\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-28q^{4}-11q^{5}-120q^{8}+\cdots\)
441.6.a.h 441.a 1.a $1$ $70.729$ \(\Q\) None \(2\) \(0\) \(11\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-28q^{4}+11q^{5}-120q^{8}+\cdots\)
441.6.a.i 441.a 1.a $1$ $70.729$ \(\Q\) None \(6\) \(0\) \(6\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+6q^{2}+4q^{4}+6q^{5}-168q^{8}+6^{2}q^{10}+\cdots\)
441.6.a.j 441.a 1.a $1$ $70.729$ \(\Q\) None \(6\) \(0\) \(78\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+6q^{2}+4q^{4}+78q^{5}-168q^{8}+468q^{10}+\cdots\)
441.6.a.k 441.a 1.a $1$ $70.729$ \(\Q\) None \(10\) \(0\) \(-56\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+10q^{2}+68q^{4}-56q^{5}+360q^{8}+\cdots\)
441.6.a.l 441.a 1.a $2$ $70.729$ \(\Q(\sqrt{57}) \) None \(-9\) \(0\) \(-18\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-4-\beta )q^{2}+(-2+9\beta )q^{4}+(-14+\cdots)q^{5}+\cdots\)
441.6.a.m 441.a 1.a $2$ $70.729$ \(\Q(\sqrt{37}) \) None \(-2\) \(0\) \(-38\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(6+2\beta )q^{4}+(-19+\cdots)q^{5}+\cdots\)
441.6.a.n 441.a 1.a $2$ $70.729$ \(\Q(\sqrt{37}) \) None \(-2\) \(0\) \(38\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(6+2\beta )q^{4}+(19+10\beta )q^{5}+\cdots\)
441.6.a.o 441.a 1.a $2$ $70.729$ \(\Q(\sqrt{7}) \) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+\beta q^{2}-5^{2}q^{4}-57\beta q^{8}-302\beta q^{11}+\cdots\)
441.6.a.p 441.a 1.a $2$ $70.729$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-4q^{4}+7\beta q^{5}-6^{2}\beta q^{8}+14^{2}q^{10}+\cdots\)
441.6.a.q 441.a 1.a $2$ $70.729$ \(\Q(\sqrt{193}) \) None \(3\) \(0\) \(-72\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(17+3\beta )q^{4}-6^{2}q^{5}+\cdots\)
441.6.a.r 441.a 1.a $2$ $70.729$ \(\Q(\sqrt{193}) \) None \(3\) \(0\) \(72\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(17+3\beta )q^{4}+6^{2}q^{5}+\cdots\)
441.6.a.s 441.a 1.a $2$ $70.729$ \(\Q(\sqrt{249}) \) None \(3\) \(0\) \(-33\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(31+3\beta )q^{4}+(-13-7\beta )q^{5}+\cdots\)
441.6.a.t 441.a 1.a $2$ $70.729$ \(\Q(\sqrt{249}) \) None \(3\) \(0\) \(33\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(31+3\beta )q^{4}+(13+7\beta )q^{5}+\cdots\)
441.6.a.u 441.a 1.a $2$ $70.729$ \(\Q(\sqrt{39}) \) None \(4\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-28q^{4}-\beta q^{5}-120q^{8}-2\beta q^{10}+\cdots\)
441.6.a.v 441.a 1.a $4$ $70.729$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-3\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(18-\beta _{1}+\beta _{2})q^{4}+\cdots\)
441.6.a.w 441.a 1.a $4$ $70.729$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-3\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(18-\beta _{1}+\beta _{2})q^{4}+\cdots\)
441.6.a.x 441.a 1.a $4$ $70.729$ 4.4.358541904.1 None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(24+\beta _{3})q^{4}+(-3\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
441.6.a.y 441.a 1.a $4$ $70.729$ \(\Q(\sqrt{19}, \sqrt{69})\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+44q^{4}+\beta _{2}q^{5}+12\beta _{1}q^{8}+\cdots\)
441.6.a.z 441.a 1.a $4$ $70.729$ \(\Q(\sqrt{2}, \sqrt{113})\) None \(10\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{2}-5\beta _{1}q^{4}+(8\beta _{2}-2\beta _{3})q^{5}+\cdots\)
441.6.a.ba 441.a 1.a $6$ $70.729$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-2\) \(0\) \(-100\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+(5^{2}-\beta _{1}+\beta _{4})q^{4}+(-2^{4}+\cdots)q^{5}+\cdots\)
441.6.a.bb 441.a 1.a $6$ $70.729$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-2\) \(0\) \(100\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+(5^{2}-\beta _{1}+\beta _{4})q^{4}+(2^{4}+2\beta _{1}+\cdots)q^{5}+\cdots\)
441.6.a.bc 441.a 1.a $6$ $70.729$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(30+\beta _{3})q^{4}+(-2\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
441.6.a.bd 441.a 1.a $6$ $70.729$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(30+\beta _{3})q^{4}+(2\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
441.6.a.be 441.a 1.a $8$ $70.729$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{4}q^{2}+(3-\beta _{2})q^{4}+\beta _{1}q^{5}+(-\beta _{4}+\cdots)q^{8}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(441))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(441)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)