Properties

Label 1232.4.a
Level $1232$
Weight $4$
Character orbit 1232.a
Rep. character $\chi_{1232}(1,\cdot)$
Character field $\Q$
Dimension $90$
Newform subspaces $30$
Sturm bound $768$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1232.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 30 \)
Sturm bound: \(768\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1232))\).

Total New Old
Modular forms 588 90 498
Cusp forms 564 90 474
Eisenstein series 24 0 24

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

\(2\)\(7\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(11\)
\(+\)\(+\)\(-\)\(-\)\(12\)
\(+\)\(-\)\(+\)\(-\)\(9\)
\(+\)\(-\)\(-\)\(+\)\(14\)
\(-\)\(+\)\(+\)\(-\)\(10\)
\(-\)\(+\)\(-\)\(+\)\(12\)
\(-\)\(-\)\(+\)\(+\)\(12\)
\(-\)\(-\)\(-\)\(-\)\(10\)
Plus space\(+\)\(49\)
Minus space\(-\)\(41\)

Trace form

\( 90 q - 4 q^{5} + 810 q^{9} + O(q^{10}) \) \( 90 q - 4 q^{5} + 810 q^{9} + 66 q^{11} + 92 q^{13} - 132 q^{15} + 52 q^{17} - 48 q^{19} - 276 q^{23} + 2206 q^{25} - 708 q^{27} + 284 q^{29} + 528 q^{31} - 660 q^{37} + 600 q^{39} - 236 q^{41} - 180 q^{45} - 1164 q^{47} + 4410 q^{49} + 1488 q^{51} + 572 q^{53} - 848 q^{57} + 2040 q^{59} + 2604 q^{61} + 232 q^{65} + 804 q^{67} + 1056 q^{69} - 668 q^{71} + 740 q^{73} - 5476 q^{75} + 2832 q^{79} + 7594 q^{81} + 456 q^{83} + 2696 q^{85} + 2768 q^{87} - 900 q^{89} + 1092 q^{91} + 4272 q^{93} + 1896 q^{95} - 1396 q^{97} + 1782 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\) 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 7 11
1232.4.a.a 1232.a 1.a $1$ $72.690$ \(\Q\) None \(0\) \(-9\) \(15\) \(7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-9q^{3}+15q^{5}+7q^{7}+54q^{9}-11q^{11}+\cdots\)
1232.4.a.b 1232.a 1.a $1$ $72.690$ \(\Q\) None \(0\) \(-7\) \(3\) \(-7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-7q^{3}+3q^{5}-7q^{7}+22q^{9}+11q^{11}+\cdots\)
1232.4.a.c 1232.a 1.a $1$ $72.690$ \(\Q\) None \(0\) \(-4\) \(-12\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{3}-12q^{5}-7q^{7}-11q^{9}-11q^{11}+\cdots\)
1232.4.a.d 1232.a 1.a $1$ $72.690$ \(\Q\) None \(0\) \(-4\) \(12\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{3}+12q^{5}-7q^{7}-11q^{9}-11q^{11}+\cdots\)
1232.4.a.e 1232.a 1.a $1$ $72.690$ \(\Q\) None \(0\) \(0\) \(2\) \(7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{5}+7q^{7}-3^{3}q^{9}-11q^{11}+26q^{13}+\cdots\)
1232.4.a.f 1232.a 1.a $1$ $72.690$ \(\Q\) None \(0\) \(2\) \(18\) \(-7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+18q^{5}-7q^{7}-23q^{9}+11q^{11}+\cdots\)
1232.4.a.g 1232.a 1.a $1$ $72.690$ \(\Q\) None \(0\) \(5\) \(-1\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{3}-q^{5}+7q^{7}-2q^{9}+11q^{11}+\cdots\)
1232.4.a.h 1232.a 1.a $1$ $72.690$ \(\Q\) None \(0\) \(7\) \(-1\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+7q^{3}-q^{5}-7q^{7}+22q^{9}-11q^{11}+\cdots\)
1232.4.a.i 1232.a 1.a $1$ $72.690$ \(\Q\) None \(0\) \(10\) \(-14\) \(-7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+10q^{3}-14q^{5}-7q^{7}+73q^{9}+11q^{11}+\cdots\)
1232.4.a.j 1232.a 1.a $2$ $72.690$ \(\Q(\sqrt{37}) \) None \(0\) \(-6\) \(26\) \(-14\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-3-\beta )q^{3}+(13-\beta )q^{5}-7q^{7}+\cdots\)
1232.4.a.k 1232.a 1.a $2$ $72.690$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(-8\) \(14\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+3\beta )q^{3}+(-4+2\beta )q^{5}+7q^{7}+\cdots\)
1232.4.a.l 1232.a 1.a $2$ $72.690$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(6\) \(14\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+3\beta )q^{3}+(3+\beta )q^{5}+7q^{7}+(19+\cdots)q^{9}+\cdots\)
1232.4.a.m 1232.a 1.a $2$ $72.690$ \(\Q(\sqrt{2}) \) None \(0\) \(4\) \(-4\) \(-14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+(-2-3\beta )q^{5}-7q^{7}-23q^{9}+\cdots\)
1232.4.a.n 1232.a 1.a $2$ $72.690$ \(\Q(\sqrt{37}) \) None \(0\) \(4\) \(0\) \(14\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-\beta q^{5}+7q^{7}-23q^{9}-11q^{11}+\cdots\)
1232.4.a.o 1232.a 1.a $2$ $72.690$ \(\Q(\sqrt{57}) \) None \(0\) \(5\) \(-17\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(3-\beta )q^{3}+(-7-3\beta )q^{5}+7q^{7}+\cdots\)
1232.4.a.p 1232.a 1.a $2$ $72.690$ \(\Q(\sqrt{137}) \) None \(0\) \(5\) \(-7\) \(-14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(3-\beta )q^{3}+(-3-\beta )q^{5}-7q^{7}+(2^{4}+\cdots)q^{9}+\cdots\)
1232.4.a.q 1232.a 1.a $3$ $72.690$ 3.3.7636.1 None \(0\) \(-6\) \(26\) \(21\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{3}+(9-\beta _{2})q^{5}+7q^{7}+\cdots\)
1232.4.a.r 1232.a 1.a $3$ $72.690$ 3.3.5925.1 None \(0\) \(-5\) \(9\) \(21\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{3}+(2+3\beta _{1})q^{5}+7q^{7}+\cdots\)
1232.4.a.s 1232.a 1.a $4$ $72.690$ 4.4.522072.1 None \(0\) \(-14\) \(10\) \(28\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-4-\beta _{2})q^{3}+(4-\beta _{1}+2\beta _{2})q^{5}+\cdots\)
1232.4.a.t 1232.a 1.a $4$ $72.690$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(3\) \(-19\) \(28\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(-5-\beta _{2})q^{5}+7q^{7}+\cdots\)
1232.4.a.u 1232.a 1.a $4$ $72.690$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(3\) \(-13\) \(-28\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(-4+\beta _{1}-\beta _{2})q^{5}+\cdots\)
1232.4.a.v 1232.a 1.a $4$ $72.690$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(3\) \(1\) \(28\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+\beta _{2}q^{5}+7q^{7}+(-5+\cdots)q^{9}+\cdots\)
1232.4.a.w 1232.a 1.a $4$ $72.690$ 4.4.509800.1 None \(0\) \(12\) \(-18\) \(28\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(3-\beta _{2})q^{3}+(-5+\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots\)
1232.4.a.x 1232.a 1.a $5$ $72.690$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-5\) \(15\) \(-35\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(3+\beta _{2})q^{5}-7q^{7}+\cdots\)
1232.4.a.y 1232.a 1.a $5$ $72.690$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-2\) \(-24\) \(-35\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(-4+\beta _{2}+\beta _{3})q^{5}-7q^{7}+\cdots\)
1232.4.a.z 1232.a 1.a $5$ $72.690$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-2\) \(-14\) \(35\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-2-\beta _{1}-\beta _{2})q^{5}+7q^{7}+\cdots\)
1232.4.a.ba 1232.a 1.a $6$ $72.690$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(-14\) \(-42\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+(-2-\beta _{3})q^{5}-7q^{7}+(4+\cdots)q^{9}+\cdots\)
1232.4.a.bb 1232.a 1.a $6$ $72.690$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(2\) \(-42\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{4}q^{3}+(-\beta _{2}+\beta _{4})q^{5}-7q^{7}+(11+\cdots)q^{9}+\cdots\)
1232.4.a.bc 1232.a 1.a $7$ $72.690$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(-3\) \(11\) \(-49\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(2-\beta _{2})q^{5}-7q^{7}+(15+\beta _{3}+\cdots)q^{9}+\cdots\)
1232.4.a.bd 1232.a 1.a $7$ $72.690$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(0\) \(6\) \(49\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1-\beta _{2})q^{5}+7q^{7}+(15-\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1232))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1232)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(308))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(616))\)\(^{\oplus 2}\)