Properties

Label 7400.2.a
Level $7400$
Weight $2$
Character orbit 7400.a
Rep. character $\chi_{7400}(1,\cdot)$
Character field $\Q$
Dimension $171$
Newform subspaces $31$
Sturm bound $2280$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 7400 = 2^{3} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7400.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 31 \)
Sturm bound: \(2280\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(7400))\).

Total New Old
Modular forms 1164 171 993
Cusp forms 1117 171 946
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.

\(2\)\(5\)\(37\)FrickeDim
\(+\)\(+\)\(+\)$+$\(19\)
\(+\)\(+\)\(-\)$-$\(23\)
\(+\)\(-\)\(+\)$-$\(25\)
\(+\)\(-\)\(-\)$+$\(19\)
\(-\)\(+\)\(+\)$-$\(20\)
\(-\)\(+\)\(-\)$+$\(19\)
\(-\)\(-\)\(+\)$+$\(21\)
\(-\)\(-\)\(-\)$-$\(25\)
Plus space\(+\)\(78\)
Minus space\(-\)\(93\)

Trace form

\( 171 q + 2 q^{3} - 4 q^{7} + 161 q^{9} + O(q^{10}) \) \( 171 q + 2 q^{3} - 4 q^{7} + 161 q^{9} + 2 q^{11} + 2 q^{13} + 2 q^{17} + 12 q^{19} + 16 q^{21} + 8 q^{23} + 8 q^{27} + 10 q^{29} - 4 q^{33} + q^{37} + 16 q^{39} - 8 q^{41} + 8 q^{43} - 8 q^{47} + 175 q^{49} + 72 q^{51} - 2 q^{53} + 16 q^{57} + 32 q^{59} - 22 q^{61} + 12 q^{63} + 22 q^{67} - 24 q^{69} + 28 q^{71} - 12 q^{73} - 32 q^{77} - 28 q^{79} + 179 q^{81} + 12 q^{83} + 20 q^{87} - 18 q^{89} + 12 q^{91} - 48 q^{93} - 18 q^{97} + 64 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(7400))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5 37
7400.2.a.a 7400.a 1.a $1$ $59.089$ \(\Q\) None 1480.2.a.d \(0\) \(-2\) \(0\) \(-3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-3q^{7}+q^{9}+3q^{11}-5q^{17}+\cdots\)
7400.2.a.b 7400.a 1.a $1$ $59.089$ \(\Q\) None 1480.2.d.a \(0\) \(-2\) \(0\) \(4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+4q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
7400.2.a.c 7400.a 1.a $1$ $59.089$ \(\Q\) None 1480.2.a.c \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}-2q^{9}-3q^{11}+8q^{17}+\cdots\)
7400.2.a.d 7400.a 1.a $1$ $59.089$ \(\Q\) None 7400.2.a.d \(0\) \(-1\) \(0\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{7}-2q^{9}-3q^{11}+7q^{17}+\cdots\)
7400.2.a.e 7400.a 1.a $1$ $59.089$ \(\Q\) None 7400.2.a.d \(0\) \(1\) \(0\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{7}-2q^{9}-3q^{11}-7q^{17}+\cdots\)
7400.2.a.f 7400.a 1.a $1$ $59.089$ \(\Q\) None 296.2.a.a \(0\) \(1\) \(0\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}-2q^{9}+q^{11}+6q^{13}+\cdots\)
7400.2.a.g 7400.a 1.a $1$ $59.089$ \(\Q\) None 296.2.a.b \(0\) \(1\) \(0\) \(3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+3q^{7}-2q^{9}-3q^{11}-2q^{17}+\cdots\)
7400.2.a.h 7400.a 1.a $1$ $59.089$ \(\Q\) None 1480.2.d.a \(0\) \(2\) \(0\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}-4q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
7400.2.a.i 7400.a 1.a $1$ $59.089$ \(\Q\) None 1480.2.a.b \(0\) \(2\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-2q^{7}+q^{9}+6q^{13}+2q^{17}+\cdots\)
7400.2.a.j 7400.a 1.a $1$ $59.089$ \(\Q\) None 1480.2.a.a \(0\) \(3\) \(0\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+5q^{7}+6q^{9}-3q^{11}+4q^{13}+\cdots\)
7400.2.a.k 7400.a 1.a $3$ $59.089$ 3.3.229.1 None 296.2.a.c \(0\) \(-2\) \(0\) \(-7\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{3}+(-2-\beta _{1}+\beta _{2})q^{7}+\cdots\)
7400.2.a.l 7400.a 1.a $3$ $59.089$ 3.3.316.1 None 1480.2.a.e \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+\beta _{1}q^{7}+\beta _{2}q^{9}+(-1-\beta _{2})q^{11}+\cdots\)
7400.2.a.m 7400.a 1.a $3$ $59.089$ 3.3.568.1 None 1480.2.a.f \(0\) \(-1\) \(0\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+\beta _{1}q^{7}+(1+2\beta _{1}+\beta _{2})q^{9}+\cdots\)
7400.2.a.n 7400.a 1.a $4$ $59.089$ 4.4.48389.1 None 296.2.a.d \(0\) \(-2\) \(0\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(\beta _{2}-\beta _{3})q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)
7400.2.a.o 7400.a 1.a $5$ $59.089$ 5.5.998068.1 None 1480.2.a.j \(0\) \(-1\) \(0\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{4}q^{3}+\beta _{2}q^{7}+(2-\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{9}+\cdots\)
7400.2.a.p 7400.a 1.a $5$ $59.089$ 5.5.6397264.1 None 1480.2.a.i \(0\) \(0\) \(0\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-1-\beta _{1}+\beta _{3})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
7400.2.a.q 7400.a 1.a $5$ $59.089$ 5.5.935504.1 None 1480.2.a.h \(0\) \(1\) \(0\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-\beta _{3}q^{7}+(\beta _{1}+\beta _{2}-\beta _{3})q^{9}+\cdots\)
7400.2.a.r 7400.a 1.a $5$ $59.089$ 5.5.583504.1 None 1480.2.a.g \(0\) \(5\) \(0\) \(5\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(1+\beta _{2}-\beta _{4})q^{7}+(1+\cdots)q^{9}+\cdots\)
7400.2.a.s 7400.a 1.a $6$ $59.089$ 6.6.693982032.1 None 1480.2.a.k \(0\) \(-1\) \(0\) \(-8\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-1-\beta _{2})q^{7}+(2+\beta _{3}-\beta _{4}+\cdots)q^{9}+\cdots\)
7400.2.a.t 7400.a 1.a $8$ $59.089$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 7400.2.a.t \(0\) \(-4\) \(0\) \(-6\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-1+\beta _{7})q^{7}+(1+\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
7400.2.a.u 7400.a 1.a $8$ $59.089$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 7400.2.a.u \(0\) \(-1\) \(0\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-\beta _{7}q^{7}+(-1+\beta _{3}-\beta _{4}+\cdots)q^{9}+\cdots\)
7400.2.a.v 7400.a 1.a $8$ $59.089$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 7400.2.a.u \(0\) \(1\) \(0\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+\beta _{7}q^{7}+(-1+\beta _{3}-\beta _{4}+\cdots)q^{9}+\cdots\)
7400.2.a.w 7400.a 1.a $8$ $59.089$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 7400.2.a.t \(0\) \(4\) \(0\) \(6\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1-\beta _{7})q^{7}+(1+\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
7400.2.a.x 7400.a 1.a $9$ $59.089$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 7400.2.a.x \(0\) \(-2\) \(0\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-\beta _{7}q^{7}+(1+\beta _{1}+\beta _{3}+\beta _{4}+\cdots)q^{9}+\cdots\)
7400.2.a.y 7400.a 1.a $9$ $59.089$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 7400.2.a.x \(0\) \(2\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+\beta _{7}q^{7}+(1+\beta _{1}+\beta _{3}+\beta _{4}+\cdots)q^{9}+\cdots\)
7400.2.a.z 7400.a 1.a $10$ $59.089$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 7400.2.a.z \(0\) \(-4\) \(0\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-\beta _{5}q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
7400.2.a.ba 7400.a 1.a $10$ $59.089$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 1480.2.d.b \(0\) \(-4\) \(0\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(1-\beta _{4})q^{7}+(\beta _{2}+\beta _{3})q^{9}+\cdots\)
7400.2.a.bb 7400.a 1.a $10$ $59.089$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 1480.2.d.b \(0\) \(4\) \(0\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-1+\beta _{4})q^{7}+(\beta _{2}+\beta _{3}+\cdots)q^{9}+\cdots\)
7400.2.a.bc 7400.a 1.a $10$ $59.089$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 7400.2.a.z \(0\) \(4\) \(0\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+\beta _{5}q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
7400.2.a.bd 7400.a 1.a $16$ $59.089$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 1480.2.d.c \(0\) \(-2\) \(0\) \(-9\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-1-\beta _{8})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
7400.2.a.be 7400.a 1.a $16$ $59.089$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 1480.2.d.c \(0\) \(2\) \(0\) \(9\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1+\beta _{8})q^{7}+(1+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(7400))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(7400)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(148))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(296))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(370))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(740))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(925))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1480))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1850))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3700))\)\(^{\oplus 2}\)