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

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

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

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 7
1225.4.a.a \(1\) \(72.277\) \(\Q\) None \(-3\) \(-2\) \(0\) \(0\) \(+\) \(-\) \(q-3q^{2}-2q^{3}+q^{4}+6q^{6}+21q^{8}+\cdots\)
1225.4.a.b \(1\) \(72.277\) \(\Q\) None \(-3\) \(2\) \(0\) \(0\) \(+\) \(+\) \(q-3q^{2}+2q^{3}+q^{4}-6q^{6}+21q^{8}+\cdots\)
1225.4.a.c \(1\) \(72.277\) \(\Q\) None \(-2\) \(-7\) \(0\) \(0\) \(+\) \(+\) \(q-2q^{2}-7q^{3}-4q^{4}+14q^{6}+24q^{8}+\cdots\)
1225.4.a.d \(1\) \(72.277\) \(\Q\) None \(-2\) \(7\) \(0\) \(0\) \(+\) \(-\) \(q-2q^{2}+7q^{3}-4q^{4}-14q^{6}+24q^{8}+\cdots\)
1225.4.a.e \(1\) \(72.277\) \(\Q\) None \(-1\) \(-8\) \(0\) \(0\) \(+\) \(-\) \(q-q^{2}-8q^{3}-7q^{4}+8q^{6}+15q^{8}+\cdots\)
1225.4.a.f \(1\) \(72.277\) \(\Q\) None \(-1\) \(-6\) \(0\) \(0\) \(+\) \(-\) \(q-q^{2}-6q^{3}-7q^{4}+6q^{6}+15q^{8}+\cdots\)
1225.4.a.g \(1\) \(72.277\) \(\Q\) None \(-1\) \(6\) \(0\) \(0\) \(+\) \(-\) \(q-q^{2}+6q^{3}-7q^{4}-6q^{6}+15q^{8}+\cdots\)
1225.4.a.h \(1\) \(72.277\) \(\Q\) None \(-1\) \(7\) \(0\) \(0\) \(-\) \(-\) \(q-q^{2}+7q^{3}-7q^{4}-7q^{6}+15q^{8}+\cdots\)
1225.4.a.i \(1\) \(72.277\) \(\Q\) None \(1\) \(-7\) \(0\) \(0\) \(+\) \(-\) \(q+q^{2}-7q^{3}-7q^{4}-7q^{6}-15q^{8}+\cdots\)
1225.4.a.j \(1\) \(72.277\) \(\Q\) None \(1\) \(-2\) \(0\) \(0\) \(+\) \(-\) \(q+q^{2}-2q^{3}-7q^{4}-2q^{6}-15q^{8}+\cdots\)
1225.4.a.k \(1\) \(72.277\) \(\Q\) None \(4\) \(2\) \(0\) \(0\) \(+\) \(-\) \(q+4q^{2}+2q^{3}+8q^{4}+8q^{6}-23q^{9}+\cdots\)
1225.4.a.l \(1\) \(72.277\) \(\Q\) \(\Q(\sqrt{-7}) \) \(5\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+5q^{2}+17q^{4}+45q^{8}-3^{3}q^{9}-68q^{11}+\cdots\)
1225.4.a.m \(2\) \(72.277\) \(\Q(\sqrt{2}) \) None \(-8\) \(2\) \(0\) \(0\) \(+\) \(-\) \(q+(-4+\beta )q^{2}+(1+4\beta )q^{3}+(10-8\beta )q^{4}+\cdots\)
1225.4.a.n \(2\) \(72.277\) \(\Q(\sqrt{2}) \) None \(-6\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-3q^{2}-6\beta q^{3}+q^{4}+18\beta q^{6}+21q^{8}+\cdots\)
1225.4.a.o \(2\) \(72.277\) \(\Q(\sqrt{21}) \) \(\Q(\sqrt{-7}) \) \(-5\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+(-2-\beta )q^{2}+(1+5\beta )q^{4}+(-11+\cdots)q^{8}+\cdots\)
1225.4.a.p \(2\) \(72.277\) \(\Q(\sqrt{11}) \) None \(-2\) \(-10\) \(0\) \(0\) \(+\) \(-\) \(q+(-1+\beta )q^{2}-5q^{3}+(4-2\beta )q^{4}+\cdots\)
1225.4.a.q \(2\) \(72.277\) \(\Q(\sqrt{11}) \) None \(-2\) \(10\) \(0\) \(0\) \(+\) \(-\) \(q+(-1+\beta )q^{2}+5q^{3}+(4-2\beta )q^{4}+\cdots\)
1225.4.a.r \(2\) \(72.277\) \(\Q(\sqrt{41}) \) None \(-1\) \(-5\) \(0\) \(0\) \(-\) \(-\) \(q-\beta q^{2}+(-2-\beta )q^{3}+(2+\beta )q^{4}+(10+\cdots)q^{6}+\cdots\)
1225.4.a.s \(2\) \(72.277\) \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-35}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+2\beta q^{3}-8q^{4}+53q^{9}+72q^{11}+\cdots\)
1225.4.a.t \(2\) \(72.277\) \(\Q(\sqrt{41}) \) None \(1\) \(5\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{2}+(2+\beta )q^{3}+(2+\beta )q^{4}+(10+\cdots)q^{6}+\cdots\)
1225.4.a.u \(2\) \(72.277\) \(\Q(\sqrt{21}) \) \(\Q(\sqrt{-7}) \) \(5\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+(3-\beta )q^{2}+(6-5\beta )q^{4}+(19-8\beta )q^{8}+\cdots\)
1225.4.a.v \(2\) \(72.277\) \(\Q(\sqrt{2}) \) None \(6\) \(-2\) \(0\) \(0\) \(+\) \(-\) \(q+(3+\beta )q^{2}+(-1+3\beta )q^{3}+(3+6\beta )q^{4}+\cdots\)
1225.4.a.w \(2\) \(72.277\) \(\Q(\sqrt{2}) \) None \(6\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+3q^{2}+6\beta q^{3}+q^{4}+18\beta q^{6}-21q^{8}+\cdots\)
1225.4.a.x \(2\) \(72.277\) \(\Q(\sqrt{2}) \) None \(6\) \(2\) \(0\) \(0\) \(+\) \(+\) \(q+(3+\beta )q^{2}+(1-3\beta )q^{3}+(3+6\beta )q^{4}+\cdots\)
1225.4.a.y \(3\) \(72.277\) 3.3.14360.1 None \(3\) \(2\) \(0\) \(0\) \(+\) \(-\) \(q+(1-\beta _{1})q^{2}+(1+\beta _{1}-\beta _{2})q^{3}+(4+\cdots)q^{4}+\cdots\)
1225.4.a.z \(4\) \(72.277\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-4\) \(-3\) \(0\) \(0\) \(+\) \(-\) \(q+(-1+\beta _{1})q^{2}+(-1+\beta _{3})q^{3}+(9+\cdots)q^{4}+\cdots\)
1225.4.a.ba \(4\) \(72.277\) 4.4.23265040.2 None \(-2\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta _{2}q^{2}+\beta _{1}q^{3}+(2-\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{6}+\cdots\)
1225.4.a.bb \(4\) \(72.277\) \(\Q(\sqrt{2}, \sqrt{65})\) None \(-2\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+(-1-\beta _{1})q^{2}+\beta _{2}q^{3}+(9+\beta _{1})q^{4}+\cdots\)
1225.4.a.bc \(4\) \(72.277\) 4.4.23265040.2 None \(2\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q-\beta _{2}q^{2}+\beta _{1}q^{3}+(2-\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
1225.4.a.bd \(4\) \(72.277\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(4\) \(3\) \(0\) \(0\) \(-\) \(-\) \(q+(1-\beta _{1})q^{2}+(1-\beta _{3})q^{3}+(9+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.be \(5\) \(72.277\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-4\) \(10\) \(0\) \(0\) \(-\) \(-\) \(q+(-1+\beta _{1})q^{2}+(2-\beta _{3})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)
1225.4.a.bf \(5\) \(72.277\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-1\) \(-8\) \(0\) \(0\) \(+\) \(-\) \(q-\beta _{1}q^{2}+(-2+\beta _{2})q^{3}+(7+\beta _{3})q^{4}+\cdots\)
1225.4.a.bg \(5\) \(72.277\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-1\) \(8\) \(0\) \(0\) \(+\) \(+\) \(q-\beta _{1}q^{2}+(2-\beta _{2})q^{3}+(7+\beta _{3})q^{4}+\cdots\)
1225.4.a.bh \(5\) \(72.277\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(4\) \(-10\) \(0\) \(0\) \(-\) \(-\) \(q+(1-\beta _{1})q^{2}+(-2+\beta _{3})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)
1225.4.a.bi \(6\) \(72.277\) 6.6.1163891200.1 None \(2\) \(-16\) \(0\) \(0\) \(+\) \(+\) \(q+\beta _{1}q^{2}+(-3+\beta _{5})q^{3}+(2+3\beta _{1}+\cdots)q^{4}+\cdots\)
1225.4.a.bj \(6\) \(72.277\) 6.6.1163891200.1 None \(2\) \(16\) \(0\) \(0\) \(+\) \(+\) \(q+\beta _{1}q^{2}+(3-\beta _{5})q^{3}+(2+3\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bk \(8\) \(72.277\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-1\) \(-6\) \(0\) \(0\) \(+\) \(-\) \(q-\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(3+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bl \(8\) \(72.277\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-1\) \(6\) \(0\) \(0\) \(+\) \(+\) \(q-\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(3+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bm \(8\) \(72.277\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-\beta _{6}q^{2}-\beta _{4}q^{3}+(9-\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
1225.4.a.bn \(8\) \(72.277\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(1\) \(-6\) \(0\) \(0\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(3+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bo \(8\) \(72.277\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(1\) \(6\) \(0\) \(0\) \(-\) \(-\) \(q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(3+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bp \(10\) \(72.277\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+\beta _{1}q^{2}+\beta _{4}q^{3}+(3+\beta _{2})q^{4}+(-5+\cdots)q^{6}+\cdots\)
1225.4.a.bq \(10\) \(72.277\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta _{1}q^{2}-\beta _{4}q^{3}+(3+\beta _{2})q^{4}+(5+\beta _{2}+\cdots)q^{6}+\cdots\)
1225.4.a.br \(12\) \(72.277\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-10\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+(-1+\beta _{2})q^{2}+\beta _{3}q^{3}+(5-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
1225.4.a.bs \(12\) \(72.277\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-\beta _{5}q^{2}+\beta _{2}q^{3}+(4+\beta _{3})q^{4}+(\beta _{8}+\cdots)q^{6}+\cdots\)
1225.4.a.bt \(12\) \(72.277\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(10\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+(1-\beta _{2})q^{2}+\beta _{3}q^{3}+(5-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)

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

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