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Properties

Label 1225.4.a
Level $1225$
Weight $4$
Character orbit 1225.a
Rep. character $\chi_{1225}(1,\cdot)$
Character field $\Q$
Dimension $187$
Newform subspaces $46$
Sturm bound $560$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 46 \)
Sturm bound: \(560\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\), \(3\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1225))\).

Total New Old
Modular forms 444 202 242
Cusp forms 396 187 209
Eisenstein series 48 15 33

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

\(5\)\(7\)FrickeDim
\(+\)\(+\)$+$\(45\)
\(+\)\(-\)$-$\(45\)
\(-\)\(+\)$-$\(46\)
\(-\)\(-\)$+$\(51\)
Plus space\(+\)\(96\)
Minus space\(-\)\(91\)

Trace form

\( 187 q - 4 q^{3} + 714 q^{4} + 16 q^{6} + 48 q^{8} + 1571 q^{9} + O(q^{10}) \) \( 187 q - 4 q^{3} + 714 q^{4} + 16 q^{6} + 48 q^{8} + 1571 q^{9} - 20 q^{11} - 174 q^{12} + 6 q^{13} + 2650 q^{16} - 116 q^{17} - 232 q^{18} + 80 q^{19} + 44 q^{22} + 196 q^{23} + 180 q^{24} - 264 q^{26} + 176 q^{27} + 50 q^{29} + 36 q^{31} + 1184 q^{32} + 376 q^{33} - 32 q^{34} + 5500 q^{36} + 488 q^{37} - 390 q^{38} - 112 q^{39} + 286 q^{41} + 454 q^{43} - 918 q^{44} + 672 q^{46} - 734 q^{47} - 2110 q^{48} - 292 q^{51} + 1876 q^{52} - 948 q^{53} + 2830 q^{54} - 1694 q^{57} + 1692 q^{58} - 330 q^{59} - 434 q^{61} - 564 q^{62} + 11238 q^{64} + 3542 q^{66} - 644 q^{67} + 2314 q^{68} - 88 q^{69} - 404 q^{71} - 1852 q^{72} - 512 q^{73} - 748 q^{74} + 3840 q^{76} + 3816 q^{78} - 1542 q^{79} + 12263 q^{81} - 690 q^{82} - 3504 q^{83} - 672 q^{86} - 1464 q^{87} + 344 q^{88} - 1550 q^{89} + 2780 q^{92} + 5686 q^{93} + 3488 q^{94} - 4 q^{96} + 4548 q^{97} - 1472 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1225))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 7
1225.4.a.a 1225.a 1.a $1$ $72.277$ \(\Q\) None \(-3\) \(-2\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}-2q^{3}+q^{4}+6q^{6}+21q^{8}+\cdots\)
1225.4.a.b 1225.a 1.a $1$ $72.277$ \(\Q\) None \(-3\) \(2\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}+2q^{3}+q^{4}-6q^{6}+21q^{8}+\cdots\)
1225.4.a.c 1225.a 1.a $1$ $72.277$ \(\Q\) None \(-2\) \(-7\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-7q^{3}-4q^{4}+14q^{6}+24q^{8}+\cdots\)
1225.4.a.d 1225.a 1.a $1$ $72.277$ \(\Q\) None \(-2\) \(7\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+7q^{3}-4q^{4}-14q^{6}+24q^{8}+\cdots\)
1225.4.a.e 1225.a 1.a $1$ $72.277$ \(\Q\) None \(-1\) \(-8\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-8q^{3}-7q^{4}+8q^{6}+15q^{8}+\cdots\)
1225.4.a.f 1225.a 1.a $1$ $72.277$ \(\Q\) None \(-1\) \(-6\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-6q^{3}-7q^{4}+6q^{6}+15q^{8}+\cdots\)
1225.4.a.g 1225.a 1.a $1$ $72.277$ \(\Q\) None \(-1\) \(6\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+6q^{3}-7q^{4}-6q^{6}+15q^{8}+\cdots\)
1225.4.a.h 1225.a 1.a $1$ $72.277$ \(\Q\) None \(-1\) \(7\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+7q^{3}-7q^{4}-7q^{6}+15q^{8}+\cdots\)
1225.4.a.i 1225.a 1.a $1$ $72.277$ \(\Q\) None \(1\) \(-7\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-7q^{3}-7q^{4}-7q^{6}-15q^{8}+\cdots\)
1225.4.a.j 1225.a 1.a $1$ $72.277$ \(\Q\) None \(1\) \(-2\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}-7q^{4}-2q^{6}-15q^{8}+\cdots\)
1225.4.a.k 1225.a 1.a $1$ $72.277$ \(\Q\) None \(4\) \(2\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2q^{3}+8q^{4}+8q^{6}-23q^{9}+\cdots\)
1225.4.a.l 1225.a 1.a $1$ $72.277$ \(\Q\) \(\Q(\sqrt{-7}) \) \(5\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+5q^{2}+17q^{4}+45q^{8}-3^{3}q^{9}-68q^{11}+\cdots\)
1225.4.a.m 1225.a 1.a $2$ $72.277$ \(\Q(\sqrt{2}) \) None \(-8\) \(2\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-4+\beta )q^{2}+(1+4\beta )q^{3}+(10-8\beta )q^{4}+\cdots\)
1225.4.a.n 1225.a 1.a $2$ $72.277$ \(\Q(\sqrt{2}) \) None \(-6\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}-6\beta q^{3}+q^{4}+18\beta q^{6}+21q^{8}+\cdots\)
1225.4.a.o 1225.a 1.a $2$ $72.277$ \(\Q(\sqrt{21}) \) \(\Q(\sqrt{-7}) \) \(-5\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+(-2-\beta )q^{2}+(1+5\beta )q^{4}+(-11+\cdots)q^{8}+\cdots\)
1225.4.a.p 1225.a 1.a $2$ $72.277$ \(\Q(\sqrt{11}) \) None \(-2\) \(-10\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}-5q^{3}+(4-2\beta )q^{4}+\cdots\)
1225.4.a.q 1225.a 1.a $2$ $72.277$ \(\Q(\sqrt{11}) \) None \(-2\) \(10\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+5q^{3}+(4-2\beta )q^{4}+\cdots\)
1225.4.a.r 1225.a 1.a $2$ $72.277$ \(\Q(\sqrt{41}) \) None \(-1\) \(-5\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-2-\beta )q^{3}+(2+\beta )q^{4}+(10+\cdots)q^{6}+\cdots\)
1225.4.a.s 1225.a 1.a $2$ $72.277$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-35}) \) \(0\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q+2\beta q^{3}-8q^{4}+53q^{9}+72q^{11}+\cdots\)
1225.4.a.t 1225.a 1.a $2$ $72.277$ \(\Q(\sqrt{41}) \) None \(1\) \(5\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(2+\beta )q^{3}+(2+\beta )q^{4}+(10+\cdots)q^{6}+\cdots\)
1225.4.a.u 1225.a 1.a $2$ $72.277$ \(\Q(\sqrt{21}) \) \(\Q(\sqrt{-7}) \) \(5\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q+(3-\beta )q^{2}+(6-5\beta )q^{4}+(19-8\beta )q^{8}+\cdots\)
1225.4.a.v 1225.a 1.a $2$ $72.277$ \(\Q(\sqrt{2}) \) None \(6\) \(-2\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta )q^{2}+(-1+3\beta )q^{3}+(3+6\beta )q^{4}+\cdots\)
1225.4.a.w 1225.a 1.a $2$ $72.277$ \(\Q(\sqrt{2}) \) None \(6\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}+6\beta q^{3}+q^{4}+18\beta q^{6}-21q^{8}+\cdots\)
1225.4.a.x 1225.a 1.a $2$ $72.277$ \(\Q(\sqrt{2}) \) None \(6\) \(2\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(3+\beta )q^{2}+(1-3\beta )q^{3}+(3+6\beta )q^{4}+\cdots\)
1225.4.a.y 1225.a 1.a $3$ $72.277$ 3.3.14360.1 None \(3\) \(2\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(1+\beta _{1}-\beta _{2})q^{3}+(4+\cdots)q^{4}+\cdots\)
1225.4.a.z 1225.a 1.a $4$ $72.277$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-4\) \(-3\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(-1+\beta _{3})q^{3}+(9+\cdots)q^{4}+\cdots\)
1225.4.a.ba 1225.a 1.a $4$ $72.277$ 4.4.23265040.2 None \(-2\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+\beta _{1}q^{3}+(2-\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{6}+\cdots\)
1225.4.a.bb 1225.a 1.a $4$ $72.277$ \(\Q(\sqrt{2}, \sqrt{65})\) None \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+\beta _{2}q^{3}+(9+\beta _{1})q^{4}+\cdots\)
1225.4.a.bc 1225.a 1.a $4$ $72.277$ 4.4.23265040.2 None \(2\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+\beta _{1}q^{3}+(2-\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
1225.4.a.bd 1225.a 1.a $4$ $72.277$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(4\) \(3\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(1-\beta _{3})q^{3}+(9+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.be 1225.a 1.a $5$ $72.277$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-4\) \(10\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(2-\beta _{3})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)
1225.4.a.bf 1225.a 1.a $5$ $72.277$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-1\) \(-8\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-2+\beta _{2})q^{3}+(7+\beta _{3})q^{4}+\cdots\)
1225.4.a.bg 1225.a 1.a $5$ $72.277$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-1\) \(8\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2-\beta _{2})q^{3}+(7+\beta _{3})q^{4}+\cdots\)
1225.4.a.bh 1225.a 1.a $5$ $72.277$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(4\) \(-10\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(-2+\beta _{3})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)
1225.4.a.bi 1225.a 1.a $6$ $72.277$ 6.6.1163891200.1 None \(2\) \(-16\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-3+\beta _{5})q^{3}+(2+3\beta _{1}+\cdots)q^{4}+\cdots\)
1225.4.a.bj 1225.a 1.a $6$ $72.277$ 6.6.1163891200.1 None \(2\) \(16\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3-\beta _{5})q^{3}+(2+3\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bk 1225.a 1.a $8$ $72.277$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-1\) \(-6\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(3+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bl 1225.a 1.a $8$ $72.277$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-1\) \(6\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(3+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bm 1225.a 1.a $8$ $72.277$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{6}q^{2}-\beta _{4}q^{3}+(9-\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
1225.4.a.bn 1225.a 1.a $8$ $72.277$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(1\) \(-6\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(3+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bo 1225.a 1.a $8$ $72.277$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(1\) \(6\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(3+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bp 1225.a 1.a $10$ $72.277$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{4}q^{3}+(3+\beta _{2})q^{4}+(-5+\cdots)q^{6}+\cdots\)
1225.4.a.bq 1225.a 1.a $10$ $72.277$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-\beta _{4}q^{3}+(3+\beta _{2})q^{4}+(5+\beta _{2}+\cdots)q^{6}+\cdots\)
1225.4.a.br 1225.a 1.a $12$ $72.277$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-10\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{2})q^{2}+\beta _{3}q^{3}+(5-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bs 1225.a 1.a $12$ $72.277$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{5}q^{2}+\beta _{2}q^{3}+(4+\beta _{3})q^{4}+(\beta _{8}+\cdots)q^{6}+\cdots\)
1225.4.a.bt 1225.a 1.a $12$ $72.277$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(10\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{2})q^{2}+\beta _{3}q^{3}+(5-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1225))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1225)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 2}\)