Properties

Label 4225.2.a
Level $4225$
Weight $2$
Character orbit 4225.a
Rep. character $\chi_{4225}(1,\cdot)$
Character field $\Q$
Dimension $229$
Newform subspaces $54$
Sturm bound $910$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 4225 = 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4225.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 54 \)
Sturm bound: \(910\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4225))\).

Total New Old
Modular forms 496 262 234
Cusp forms 413 229 184
Eisenstein series 83 33 50

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

\(5\)\(13\)FrickeTotalCuspEisenstein
AllNewOldAllNewOldAllNewOld
\(+\)\(+\)\(+\)\(119\)\(59\)\(60\)\(99\)\(54\)\(45\)\(20\)\(5\)\(15\)
\(+\)\(-\)\(-\)\(128\)\(63\)\(65\)\(107\)\(57\)\(50\)\(21\)\(6\)\(15\)
\(-\)\(+\)\(-\)\(126\)\(72\)\(54\)\(105\)\(62\)\(43\)\(21\)\(10\)\(11\)
\(-\)\(-\)\(+\)\(123\)\(68\)\(55\)\(102\)\(56\)\(46\)\(21\)\(12\)\(9\)
Plus space\(+\)\(242\)\(127\)\(115\)\(201\)\(110\)\(91\)\(41\)\(17\)\(24\)
Minus space\(-\)\(254\)\(135\)\(119\)\(212\)\(119\)\(93\)\(42\)\(16\)\(26\)

Trace form

\( 229 q - 3 q^{2} + 213 q^{4} + 8 q^{6} + 4 q^{7} - 3 q^{8} + 193 q^{9} - 8 q^{12} + 10 q^{14} + 173 q^{16} - 11 q^{18} + 4 q^{19} + 20 q^{21} - 6 q^{22} - 2 q^{23} + 28 q^{24} + 18 q^{27} + 28 q^{28} + 6 q^{29}+ \cdots - 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4225))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 13
4225.2.a.a 4225.a 1.a $1$ $33.737$ \(\Q\) None 325.2.c.c \(-2\) \(-1\) \(0\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+2q^{7}+\cdots\)
4225.2.a.b 4225.a 1.a $1$ $33.737$ \(\Q\) None 325.2.a.a \(-2\) \(1\) \(0\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+2q^{4}-2q^{6}-2q^{7}+\cdots\)
4225.2.a.c 4225.a 1.a $1$ $33.737$ \(\Q\) None 325.2.c.c \(-2\) \(1\) \(0\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+2q^{4}-2q^{6}+2q^{7}+\cdots\)
4225.2.a.d 4225.a 1.a $1$ $33.737$ \(\Q\) None 325.2.c.a \(-1\) \(-2\) \(0\) \(-5\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}-q^{4}+2q^{6}-5q^{7}+3q^{8}+\cdots\)
4225.2.a.e 4225.a 1.a $1$ $33.737$ \(\Q\) None 65.2.d.a \(-1\) \(-2\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}-q^{4}+2q^{6}+3q^{8}+q^{9}+\cdots\)
4225.2.a.f 4225.a 1.a $1$ $33.737$ \(\Q\) None 325.2.c.a \(-1\) \(2\) \(0\) \(-5\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}-q^{4}-2q^{6}-5q^{7}+3q^{8}+\cdots\)
4225.2.a.g 4225.a 1.a $1$ $33.737$ \(\Q\) None 65.2.a.a \(-1\) \(2\) \(0\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}-q^{4}-2q^{6}-4q^{7}+3q^{8}+\cdots\)
4225.2.a.h 4225.a 1.a $1$ $33.737$ \(\Q\) None 65.2.d.a \(-1\) \(2\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}-q^{4}-2q^{6}+3q^{8}+q^{9}+\cdots\)
4225.2.a.i 4225.a 1.a $1$ $33.737$ \(\Q\) None 325.2.a.b \(0\) \(-1\) \(0\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{4}-4q^{7}-2q^{9}+6q^{11}+\cdots\)
4225.2.a.j 4225.a 1.a $1$ $33.737$ \(\Q\) None 325.2.a.b \(0\) \(1\) \(0\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}+4q^{7}-2q^{9}+6q^{11}+\cdots\)
4225.2.a.k 4225.a 1.a $1$ $33.737$ \(\Q\) None 65.2.d.a \(1\) \(-2\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}-q^{4}-2q^{6}-3q^{8}+q^{9}+\cdots\)
4225.2.a.l 4225.a 1.a $1$ $33.737$ \(\Q\) None 325.2.c.a \(1\) \(-2\) \(0\) \(5\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}-q^{4}-2q^{6}+5q^{7}-3q^{8}+\cdots\)
4225.2.a.m 4225.a 1.a $1$ $33.737$ \(\Q\) None 65.2.d.a \(1\) \(2\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}-q^{4}+2q^{6}-3q^{8}+q^{9}+\cdots\)
4225.2.a.n 4225.a 1.a $1$ $33.737$ \(\Q\) None 325.2.c.a \(1\) \(2\) \(0\) \(5\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}-q^{4}+2q^{6}+5q^{7}-3q^{8}+\cdots\)
4225.2.a.o 4225.a 1.a $1$ $33.737$ \(\Q\) None 325.2.c.c \(2\) \(-1\) \(0\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+2q^{4}-2q^{6}-2q^{7}+\cdots\)
4225.2.a.p 4225.a 1.a $1$ $33.737$ \(\Q\) None 325.2.a.a \(2\) \(-1\) \(0\) \(2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+2q^{4}-2q^{6}+2q^{7}+\cdots\)
4225.2.a.q 4225.a 1.a $1$ $33.737$ \(\Q\) None 325.2.c.c \(2\) \(1\) \(0\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+q^{3}+2q^{4}+2q^{6}-2q^{7}+\cdots\)
4225.2.a.r 4225.a 1.a $2$ $33.737$ \(\Q(\sqrt{2}) \) None 65.2.a.b \(-2\) \(0\) \(0\) \(4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}-\beta q^{3}+(1-2\beta )q^{4}+\cdots\)
4225.2.a.s 4225.a 1.a $2$ $33.737$ \(\Q(\sqrt{2}) \) None 325.2.a.f \(-2\) \(0\) \(0\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+2\beta q^{3}+(1-2\beta )q^{4}+\cdots\)
4225.2.a.t 4225.a 1.a $2$ $33.737$ \(\Q(\sqrt{13}) \) None 65.2.e.b \(-1\) \(-2\) \(0\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+(1+\beta )q^{4}+\beta q^{6}-q^{7}+\cdots\)
4225.2.a.u 4225.a 1.a $2$ $33.737$ \(\Q(\sqrt{5}) \) None 65.2.e.a \(-1\) \(0\) \(0\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
4225.2.a.v 4225.a 1.a $2$ $33.737$ \(\Q(\sqrt{3}) \) None 13.2.e.a \(0\) \(-4\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-2q^{3}+q^{4}-2\beta q^{6}-\beta q^{8}+\cdots\)
4225.2.a.w 4225.a 1.a $2$ $33.737$ \(\Q(\sqrt{3}) \) None 65.2.a.c \(0\) \(-2\) \(0\) \(4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{3}+q^{4}+(3-\beta )q^{6}+\cdots\)
4225.2.a.x 4225.a 1.a $2$ $33.737$ \(\Q(\sqrt{13}) \) None 65.2.e.b \(1\) \(-2\) \(0\) \(2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{3}+(1+\beta )q^{4}-\beta q^{6}+q^{7}+\cdots\)
4225.2.a.y 4225.a 1.a $2$ $33.737$ \(\Q(\sqrt{5}) \) None 65.2.e.a \(1\) \(0\) \(0\) \(4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
4225.2.a.z 4225.a 1.a $2$ $33.737$ \(\Q(\sqrt{2}) \) None 325.2.a.f \(2\) \(0\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+2\beta q^{3}+(1+2\beta )q^{4}+(4+\cdots)q^{6}+\cdots\)
4225.2.a.ba 4225.a 1.a $3$ $33.737$ 3.3.148.1 None 65.2.b.a \(-3\) \(4\) \(0\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}+(1+\beta _{1})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)
4225.2.a.bb 4225.a 1.a $3$ $33.737$ \(\Q(\zeta_{14})^+\) None 169.2.a.b \(-2\) \(2\) \(0\) \(-3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}+(\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
4225.2.a.bc 4225.a 1.a $3$ $33.737$ 3.3.564.1 None 65.2.c.a \(-1\) \(2\) \(0\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1-\beta _{1})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
4225.2.a.bd 4225.a 1.a $3$ $33.737$ \(\Q(\zeta_{14})^+\) None 845.2.a.h \(-1\) \(5\) \(0\) \(-5\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{1}-\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
4225.2.a.be 4225.a 1.a $3$ $33.737$ 3.3.564.1 None 65.2.c.a \(1\) \(2\) \(0\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1-\beta _{1})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
4225.2.a.bf 4225.a 1.a $3$ $33.737$ \(\Q(\zeta_{14})^+\) None 845.2.a.h \(1\) \(5\) \(0\) \(5\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{1}-\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
4225.2.a.bg 4225.a 1.a $3$ $33.737$ \(\Q(\zeta_{14})^+\) None 169.2.a.b \(2\) \(2\) \(0\) \(3\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(1+\beta _{2})q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots\)
4225.2.a.bh 4225.a 1.a $3$ $33.737$ 3.3.148.1 None 65.2.b.a \(3\) \(-4\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{2}+(-1-\beta _{1})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)
4225.2.a.bi 4225.a 1.a $4$ $33.737$ 4.4.4752.1 None 65.2.m.a \(-2\) \(2\) \(0\) \(-10\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+\beta _{3}q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
4225.2.a.bj 4225.a 1.a $4$ $33.737$ \(\Q(\sqrt{3}, \sqrt{7})\) None 65.2.l.a \(0\) \(-4\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}-q^{3}-\beta _{2}q^{4}-\beta _{3}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
4225.2.a.bk 4225.a 1.a $4$ $33.737$ \(\Q(\sqrt{3}, \sqrt{7})\) None 65.2.l.a \(0\) \(4\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}+q^{3}-\beta _{2}q^{4}+\beta _{3}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
4225.2.a.bl 4225.a 1.a $4$ $33.737$ 4.4.4752.1 None 65.2.m.a \(2\) \(2\) \(0\) \(10\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(\beta _{1}+\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{6}+\cdots\)
4225.2.a.bm 4225.a 1.a $5$ $33.737$ 5.5.1068321.1 None 325.2.e.c \(0\) \(-3\) \(0\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bn 4225.a 1.a $5$ $33.737$ 5.5.1068321.1 None 325.2.e.c \(0\) \(-3\) \(0\) \(2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bo 4225.a 1.a $5$ $33.737$ 5.5.1068321.1 None 325.2.e.c \(0\) \(3\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bp 4225.a 1.a $5$ $33.737$ 5.5.1068321.1 None 325.2.e.c \(0\) \(3\) \(0\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bq 4225.a 1.a $6$ $33.737$ 6.6.199374400.1 None 65.2.n.a \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-\beta _{1}+\beta _{5})q^{3}+(1+\beta _{2}+\cdots)q^{4}+\cdots\)
4225.2.a.br 4225.a 1.a $6$ $33.737$ 6.6.199374400.1 None 65.2.n.a \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(\beta _{1}-\beta _{5})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bs 4225.a 1.a $9$ $33.737$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 845.2.a.n \(-3\) \(-7\) \(0\) \(-7\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}-\beta _{6})q^{2}+(-1+\beta _{4})q^{3}+(2+\cdots)q^{4}+\cdots\)
4225.2.a.bt 4225.a 1.a $9$ $33.737$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 845.2.a.n \(3\) \(-7\) \(0\) \(7\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(\beta _{1}+\beta _{6})q^{2}+(-1+\beta _{4})q^{3}+(2-\beta _{3}+\cdots)q^{4}+\cdots\)
4225.2.a.bu 4225.a 1.a $10$ $33.737$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 325.2.n.e \(0\) \(-6\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-1-\beta _{7})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bv 4225.a 1.a $10$ $33.737$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 325.2.n.e \(0\) \(6\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{7})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4225.2.a.bw 4225.a 1.a $12$ $33.737$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 4225.2.a.bw \(-7\) \(-1\) \(0\) \(-12\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+\beta _{7}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
4225.2.a.bx 4225.a 1.a $12$ $33.737$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 4225.2.a.bw \(-7\) \(1\) \(0\) \(-12\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-\beta _{7}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
4225.2.a.by 4225.a 1.a $12$ $33.737$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 4225.2.a.bw \(7\) \(-1\) \(0\) \(12\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+\beta _{7}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
4225.2.a.bz 4225.a 1.a $12$ $33.737$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 4225.2.a.bw \(7\) \(1\) \(0\) \(12\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-\beta _{7}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
4225.2.a.ca 4225.a 1.a $18$ $33.737$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None 845.2.b.g \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-\beta _{14}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{6}+\cdots\)
4225.2.a.cb 4225.a 1.a $18$ $33.737$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None 845.2.b.g \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{14}q^{3}+(1+\beta _{2})q^{4}+(1+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4225))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4225)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(325))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(845))\)\(^{\oplus 2}\)