Properties

Label 3800.2.a
Level $3800$
Weight $2$
Character orbit 3800.a
Rep. character $\chi_{3800}(1,\cdot)$
Character field $\Q$
Dimension $85$
Newform subspaces $31$
Sturm bound $1200$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 31 \)
Sturm bound: \(1200\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3800))\).

Total New Old
Modular forms 624 85 539
Cusp forms 577 85 492
Eisenstein series 47 0 47

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

\(2\)\(5\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(9\)
\(+\)\(+\)\(-\)\(-\)\(12\)
\(+\)\(-\)\(+\)\(-\)\(11\)
\(+\)\(-\)\(-\)\(+\)\(11\)
\(-\)\(+\)\(+\)\(-\)\(13\)
\(-\)\(+\)\(-\)\(+\)\(7\)
\(-\)\(-\)\(+\)\(+\)\(8\)
\(-\)\(-\)\(-\)\(-\)\(14\)
Plus space\(+\)\(35\)
Minus space\(-\)\(50\)

Trace form

\( 85 q - 4 q^{7} + 79 q^{9} + O(q^{10}) \) \( 85 q - 4 q^{7} + 79 q^{9} + 6 q^{11} - 2 q^{13} + 6 q^{17} + 3 q^{19} + 16 q^{21} + 6 q^{23} + 30 q^{29} - 12 q^{31} + 20 q^{33} + 14 q^{37} + 22 q^{39} + 14 q^{41} + 2 q^{43} + 6 q^{47} + 81 q^{49} + 20 q^{51} + 14 q^{53} + 2 q^{57} - 8 q^{59} + 16 q^{61} + 6 q^{63} - 4 q^{67} - 4 q^{69} + 40 q^{71} + 18 q^{73} - 14 q^{77} - 36 q^{79} + 101 q^{81} - 20 q^{83} - 18 q^{87} + 30 q^{89} + 12 q^{93} - 6 q^{97} - 14 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3800))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5 19
3800.2.a.a 3800.a 1.a $1$ $30.343$ \(\Q\) None \(0\) \(-3\) \(0\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+q^{7}+6q^{9}+4q^{11}-q^{13}+\cdots\)
3800.2.a.b 3800.a 1.a $1$ $30.343$ \(\Q\) None \(0\) \(-2\) \(0\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-4q^{7}+q^{9}-4q^{11}-6q^{17}+\cdots\)
3800.2.a.c 3800.a 1.a $1$ $30.343$ \(\Q\) None \(0\) \(-2\) \(0\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+2q^{7}+q^{9}+4q^{11}+8q^{17}+\cdots\)
3800.2.a.d 3800.a 1.a $1$ $30.343$ \(\Q\) None \(0\) \(-1\) \(0\) \(-3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{7}-2q^{9}+2q^{11}-q^{13}+\cdots\)
3800.2.a.e 3800.a 1.a $1$ $30.343$ \(\Q\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{9}-4q^{11}+6q^{13}+6q^{17}-q^{19}+\cdots\)
3800.2.a.f 3800.a 1.a $1$ $30.343$ \(\Q\) None \(0\) \(2\) \(0\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-4q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
3800.2.a.g 3800.a 1.a $1$ $30.343$ \(\Q\) None \(0\) \(2\) \(0\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-2q^{7}+q^{9}+4q^{11}-8q^{17}+\cdots\)
3800.2.a.h 3800.a 1.a $1$ $30.343$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}-4q^{11}-4q^{13}+2q^{17}+\cdots\)
3800.2.a.i 3800.a 1.a $1$ $30.343$ \(\Q\) None \(0\) \(2\) \(0\) \(3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+3q^{7}+q^{9}-3q^{11}+4q^{13}+\cdots\)
3800.2.a.j 3800.a 1.a $2$ $30.343$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+(1-2\beta )q^{9}+2q^{11}+\cdots\)
3800.2.a.k 3800.a 1.a $2$ $30.343$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+2\beta q^{7}+(3+2\beta )q^{9}+\cdots\)
3800.2.a.l 3800.a 1.a $2$ $30.343$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+(1-\beta )q^{7}-2\beta q^{9}+\cdots\)
3800.2.a.m 3800.a 1.a $2$ $30.343$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(-2+2\beta )q^{7}-q^{9}+(2-2\beta )q^{11}+\cdots\)
3800.2.a.n 3800.a 1.a $2$ $30.343$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+2\beta q^{7}-q^{9}+(2-2\beta )q^{11}+\cdots\)
3800.2.a.o 3800.a 1.a $2$ $30.343$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(2+2\beta )q^{7}-q^{9}+(2+2\beta )q^{11}+\cdots\)
3800.2.a.p 3800.a 1.a $2$ $30.343$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(-1-\beta )q^{7}+2\beta q^{9}+\cdots\)
3800.2.a.q 3800.a 1.a $2$ $30.343$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-2\beta q^{7}+(3+2\beta )q^{9}+(-1+\cdots)q^{13}+\cdots\)
3800.2.a.r 3800.a 1.a $3$ $30.343$ 3.3.961.1 None \(0\) \(-1\) \(0\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-2+\beta _{1}-\beta _{2})q^{7}+(5-\beta _{1}+\cdots)q^{9}+\cdots\)
3800.2.a.s 3800.a 1.a $3$ $30.343$ \(\Q(\zeta_{18})^+\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(\beta _{1}-2\beta _{2})q^{7}+(-1+\beta _{2})q^{9}+\cdots\)
3800.2.a.t 3800.a 1.a $3$ $30.343$ \(\Q(\zeta_{18})^+\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-\beta _{1}+2\beta _{2})q^{7}+(-1+\beta _{2})q^{9}+\cdots\)
3800.2.a.u 3800.a 1.a $3$ $30.343$ 3.3.257.1 None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}-\beta _{2}q^{7}+(\beta _{1}-\beta _{2})q^{9}+(-1+\cdots)q^{11}+\cdots\)
3800.2.a.v 3800.a 1.a $3$ $30.343$ 3.3.257.1 None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+\beta _{2}q^{7}+(\beta _{1}-\beta _{2})q^{9}+(-1+\cdots)q^{11}+\cdots\)
3800.2.a.w 3800.a 1.a $3$ $30.343$ 3.3.316.1 None \(0\) \(1\) \(0\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1-2\beta _{1}+\beta _{2})q^{7}+\beta _{2}q^{9}+\cdots\)
3800.2.a.x 3800.a 1.a $3$ $30.343$ 3.3.229.1 None \(0\) \(1\) \(0\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}-\beta _{1}q^{7}+(1-\beta _{1})q^{9}+(-2+\cdots)q^{13}+\cdots\)
3800.2.a.y 3800.a 1.a $3$ $30.343$ 3.3.568.1 None \(0\) \(1\) \(0\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(2+\beta _{2})q^{7}+(1+2\beta _{1}+\beta _{2})q^{9}+\cdots\)
3800.2.a.z 3800.a 1.a $6$ $30.343$ 6.6.253565184.1 None \(0\) \(-2\) \(0\) \(-6\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-1-\beta _{1}-\beta _{3})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
3800.2.a.ba 3800.a 1.a $6$ $30.343$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-2\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-\beta _{1}-\beta _{4})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
3800.2.a.bb 3800.a 1.a $6$ $30.343$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-2\) \(0\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+\beta _{3}q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
3800.2.a.bc 3800.a 1.a $6$ $30.343$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(2\) \(0\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{4})q^{7}+(1+\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
3800.2.a.bd 3800.a 1.a $6$ $30.343$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(2\) \(0\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-\beta _{3}q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
3800.2.a.be 3800.a 1.a $6$ $30.343$ 6.6.253565184.1 None \(0\) \(2\) \(0\) \(6\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1+\beta _{1}+\beta _{3})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3800))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3800)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(380))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(475))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(760))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(950))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1900))\)\(^{\oplus 2}\)