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.

\(5\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(54\)
\(+\)\(-\)\(-\)\(57\)
\(-\)\(+\)\(-\)\(62\)
\(-\)\(-\)\(+\)\(56\)
Plus space\(+\)\(110\)
Minus space\(-\)\(119\)

Trace form

\( 229 q - 3 q^{2} + 213 q^{4} + 8 q^{6} + 4 q^{7} - 3 q^{8} + 193 q^{9} + O(q^{10}) \) \( 229 q - 3 q^{2} + 213 q^{4} + 8 q^{6} + 4 q^{7} - 3 q^{8} + 193 q^{9} - 8 q^{12} + 10 q^{14} + 173 q^{16} - 11 q^{18} + 4 q^{19} + 20 q^{21} - 6 q^{22} - 2 q^{23} + 28 q^{24} + 18 q^{27} + 28 q^{28} + 6 q^{29} + 24 q^{31} + q^{32} - 20 q^{33} + 26 q^{34} + 171 q^{36} - 10 q^{37} + 36 q^{38} - 10 q^{41} + 2 q^{43} - 32 q^{44} + 4 q^{46} + 12 q^{47} + 30 q^{48} + 147 q^{49} - 58 q^{51} + 6 q^{53} + 4 q^{54} + 12 q^{56} - 4 q^{57} + 2 q^{58} - 20 q^{59} + 24 q^{61} - 6 q^{62} - 4 q^{63} + 119 q^{64} + 26 q^{66} - 16 q^{67} + 70 q^{68} - 12 q^{71} + 21 q^{72} - 14 q^{73} + 32 q^{74} + 60 q^{76} - 4 q^{77} + 22 q^{79} + 53 q^{81} + 28 q^{82} - 40 q^{83} + 64 q^{84} - 16 q^{86} + 44 q^{87} - 34 q^{88} - 2 q^{89} + 40 q^{92} + 60 q^{93} - 30 q^{94} + 36 q^{96} - 2 q^{97} - 51 q^{98} - 80 q^{99} + O(q^{100}) \)

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

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