Properties

Label 2304.4.a
Level $2304$
Weight $4$
Character orbit 2304.a
Rep. character $\chi_{2304}(1,\cdot)$
Character field $\Q$
Dimension $118$
Newform subspaces $56$
Sturm bound $1536$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2304.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 56 \)
Sturm bound: \(1536\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2304))\).

Total New Old
Modular forms 1200 122 1078
Cusp forms 1104 118 986
Eisenstein series 96 4 92

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

\(2\)\(3\)FrickeDim.
\(+\)\(+\)\(+\)\(26\)
\(+\)\(-\)\(-\)\(34\)
\(-\)\(+\)\(-\)\(22\)
\(-\)\(-\)\(+\)\(36\)
Plus space\(+\)\(62\)
Minus space\(-\)\(56\)

Trace form

\( 118 q + O(q^{10}) \) \( 118 q + 4 q^{17} + 2754 q^{25} - 4 q^{41} + 3750 q^{49} - 1472 q^{65} + 276 q^{73} + 172 q^{89} + 3164 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2304))\) 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 3
2304.4.a.a 2304.a 1.a $1$ $135.940$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-22\) \(0\) $-$ $+$ $N(\mathrm{U}(1))$ \(q-22q^{5}-92q^{13}+104q^{17}+359q^{25}+\cdots\)
2304.4.a.b 2304.a 1.a $1$ $135.940$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-22\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-22q^{5}+92q^{13}-104q^{17}+359q^{25}+\cdots\)
2304.4.a.c 2304.a 1.a $1$ $135.940$ \(\Q\) None \(0\) \(0\) \(-12\) \(-32\) $+$ $-$ $\mathrm{SU}(2)$ \(q-12q^{5}-2^{5}q^{7}-8q^{11}-20q^{13}+\cdots\)
2304.4.a.d 2304.a 1.a $1$ $135.940$ \(\Q\) None \(0\) \(0\) \(-12\) \(32\) $+$ $-$ $\mathrm{SU}(2)$ \(q-12q^{5}+2^{5}q^{7}+8q^{11}-20q^{13}+\cdots\)
2304.4.a.e 2304.a 1.a $1$ $135.940$ \(\Q\) None \(0\) \(0\) \(-8\) \(-12\) $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{5}-12q^{7}-12q^{11}-20q^{13}+\cdots\)
2304.4.a.f 2304.a 1.a $1$ $135.940$ \(\Q\) None \(0\) \(0\) \(-8\) \(12\) $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{5}+12q^{7}+12q^{11}-20q^{13}+\cdots\)
2304.4.a.g 2304.a 1.a $1$ $135.940$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-4\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q-4q^{5}-92q^{13}-94q^{17}-109q^{25}+\cdots\)
2304.4.a.h 2304.a 1.a $1$ $135.940$ \(\Q\) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q-18q^{11}-90q^{17}-106q^{19}-5^{3}q^{25}+\cdots\)
2304.4.a.i 2304.a 1.a $1$ $135.940$ \(\Q\) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+18q^{11}-90q^{17}+106q^{19}-5^{3}q^{25}+\cdots\)
2304.4.a.j 2304.a 1.a $1$ $135.940$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(4\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+4q^{5}+92q^{13}-94q^{17}-109q^{25}+\cdots\)
2304.4.a.k 2304.a 1.a $1$ $135.940$ \(\Q\) None \(0\) \(0\) \(8\) \(-12\) $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{5}-12q^{7}+12q^{11}+20q^{13}+\cdots\)
2304.4.a.l 2304.a 1.a $1$ $135.940$ \(\Q\) None \(0\) \(0\) \(8\) \(12\) $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{5}+12q^{7}-12q^{11}+20q^{13}+\cdots\)
2304.4.a.m 2304.a 1.a $1$ $135.940$ \(\Q\) None \(0\) \(0\) \(12\) \(-32\) $-$ $-$ $\mathrm{SU}(2)$ \(q+12q^{5}-2^{5}q^{7}+8q^{11}+20q^{13}+\cdots\)
2304.4.a.n 2304.a 1.a $1$ $135.940$ \(\Q\) None \(0\) \(0\) \(12\) \(32\) $-$ $-$ $\mathrm{SU}(2)$ \(q+12q^{5}+2^{5}q^{7}-8q^{11}+20q^{13}+\cdots\)
2304.4.a.o 2304.a 1.a $1$ $135.940$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(22\) \(0\) $-$ $+$ $N(\mathrm{U}(1))$ \(q+22q^{5}-92q^{13}-104q^{17}+359q^{25}+\cdots\)
2304.4.a.p 2304.a 1.a $1$ $135.940$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(22\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q+22q^{5}+92q^{13}+104q^{17}+359q^{25}+\cdots\)
2304.4.a.q 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(-12\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-6q^{5}-\beta q^{7}-2\beta q^{11}-20q^{13}+\cdots\)
2304.4.a.r 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(-12\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-6q^{5}-\beta q^{7}-2\beta q^{11}+20q^{13}+\cdots\)
2304.4.a.s 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(-8\) \(-16\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-4-\beta )q^{5}+(-8-\beta )q^{7}+(-4+\cdots)q^{11}+\cdots\)
2304.4.a.t 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(-8\) \(16\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-4-\beta )q^{5}+(8+\beta )q^{7}+(4-4\beta )q^{11}+\cdots\)
2304.4.a.u 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-6}) \) \(0\) \(0\) \(0\) \(-68\) $-$ $+$ $N(\mathrm{U}(1))$ \(q+7\beta q^{5}-34q^{7}+2\beta q^{11}+267q^{25}+\cdots\)
2304.4.a.v 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(-16\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2\beta q^{5}-8q^{7}-3\beta q^{11}-10\beta q^{13}+\cdots\)
2304.4.a.w 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+4\beta q^{5}-\beta q^{7}-48q^{11}+6\beta q^{13}+\cdots\)
2304.4.a.x 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+7\beta q^{7}-48q^{11}-12\beta q^{13}+\cdots\)
2304.4.a.y 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{11}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+\beta q^{7}-48q^{11}-4\beta q^{13}+\cdots\)
2304.4.a.z 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4\beta q^{5}-\beta q^{7}-48q^{11}-6\beta q^{13}+\cdots\)
2304.4.a.ba 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}-2\beta q^{7}-42q^{11}+3\beta q^{13}+\cdots\)
2304.4.a.bb 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}-5\beta q^{7}-20q^{11}-14\beta q^{13}+\cdots\)
2304.4.a.bc 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q+\beta q^{7}+2\beta q^{13}-56q^{19}-5^{3}q^{25}+\cdots\)
2304.4.a.bd 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $-$ $+$ $N(\mathrm{U}(1))$ \(q-\beta q^{7}+2\beta q^{13}+56q^{19}-5^{3}q^{25}+\cdots\)
2304.4.a.be 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3\beta q^{5}+\beta q^{7}+2^{4}\beta q^{13}+90q^{17}+\cdots\)
2304.4.a.bf 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q+5^{2}\beta q^{11}+90q^{17}+45\beta q^{19}-5^{3}q^{25}+\cdots\)
2304.4.a.bg 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3\beta q^{5}-\beta q^{7}+2^{4}\beta q^{13}+90q^{17}+\cdots\)
2304.4.a.bh 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+5\beta q^{7}+20q^{11}-14\beta q^{13}+\cdots\)
2304.4.a.bi 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+2\beta q^{7}+42q^{11}+3\beta q^{13}+\cdots\)
2304.4.a.bj 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4\beta q^{5}+\beta q^{7}+48q^{11}+6\beta q^{13}+\cdots\)
2304.4.a.bk 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}-7\beta q^{7}+48q^{11}-12\beta q^{13}+\cdots\)
2304.4.a.bl 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{11}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}-\beta q^{7}+48q^{11}-4\beta q^{13}+\cdots\)
2304.4.a.bm 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+4\beta q^{5}+\beta q^{7}+48q^{11}-6\beta q^{13}+\cdots\)
2304.4.a.bn 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(16\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2\beta q^{5}+8q^{7}+3\beta q^{11}-10\beta q^{13}+\cdots\)
2304.4.a.bo 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-6}) \) \(0\) \(0\) \(0\) \(68\) $+$ $+$ $N(\mathrm{U}(1))$ \(q+7\beta q^{5}+34q^{7}-2\beta q^{11}+267q^{25}+\cdots\)
2304.4.a.bp 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(8\) \(-16\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(4+\beta )q^{5}+(-8-\beta )q^{7}+(4-4\beta )q^{11}+\cdots\)
2304.4.a.bq 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(8\) \(16\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(4+\beta )q^{5}+(8+\beta )q^{7}+(-4+4\beta )q^{11}+\cdots\)
2304.4.a.br 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(12\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+6q^{5}-\beta q^{7}+2\beta q^{11}-20q^{13}+\cdots\)
2304.4.a.bs 2304.a 1.a $2$ $135.940$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(12\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+6q^{5}-\beta q^{7}+2\beta q^{11}+20q^{13}+\cdots\)
2304.4.a.bt 2304.a 1.a $3$ $135.940$ 3.3.1436.1 None \(0\) \(0\) \(-10\) \(-14\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta _{2})q^{5}+(-5+\beta _{1})q^{7}+(-2\beta _{1}+\cdots)q^{11}+\cdots\)
2304.4.a.bu 2304.a 1.a $3$ $135.940$ 3.3.1436.1 None \(0\) \(0\) \(-10\) \(14\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta _{2})q^{5}+(5-\beta _{1})q^{7}+(2\beta _{1}+\cdots)q^{11}+\cdots\)
2304.4.a.bv 2304.a 1.a $3$ $135.940$ 3.3.1436.1 None \(0\) \(0\) \(10\) \(-14\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta _{2})q^{5}+(-5+\beta _{1})q^{7}+(2\beta _{1}+\cdots)q^{11}+\cdots\)
2304.4.a.bw 2304.a 1.a $3$ $135.940$ 3.3.1436.1 None \(0\) \(0\) \(10\) \(14\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta _{2})q^{5}+(5-\beta _{1})q^{7}+(-2\beta _{1}+\cdots)q^{11}+\cdots\)
2304.4.a.bx 2304.a 1.a $4$ $135.940$ \(\Q(\sqrt{10}, \sqrt{22})\) None \(0\) \(0\) \(0\) \(-40\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{5}-10q^{7}+6\beta _{1}q^{11}-\beta _{3}q^{13}+\cdots\)
2304.4.a.by 2304.a 1.a $4$ $135.940$ 4.4.9792.1 None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{5}+(-\beta _{1}+\beta _{3})q^{7}+(-12-\beta _{2}+\cdots)q^{11}+\cdots\)
2304.4.a.bz 2304.a 1.a $4$ $135.940$ \(\Q(\sqrt{2}, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{5}-\beta _{2}q^{7}+7\beta _{1}q^{11}-\beta _{3}q^{13}+\cdots\)
2304.4.a.ca 2304.a 1.a $4$ $135.940$ \(\Q(\zeta_{24})^+\) \(\Q(\sqrt{-6}) \) \(0\) \(0\) \(0\) \(0\) $-$ $+$ $N(\mathrm{U}(1))$ \(q+\beta _{3}q^{5}-\beta _{2}q^{7}-\beta _{1}q^{11}-17q^{25}+\cdots\)
2304.4.a.cb 2304.a 1.a $4$ $135.940$ 4.4.9792.1 None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{5}+(\beta _{1}-\beta _{3})q^{7}+(12+\beta _{2})q^{11}+\cdots\)
2304.4.a.cc 2304.a 1.a $4$ $135.940$ \(\Q(\sqrt{10}, \sqrt{22})\) None \(0\) \(0\) \(0\) \(40\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{5}+10q^{7}-6\beta _{1}q^{11}-\beta _{3}q^{13}+\cdots\)
2304.4.a.cd 2304.a 1.a $8$ $135.940$ 8.8.\(\cdots\).7 None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{5}-\beta _{3}q^{7}-\beta _{5}q^{11}-\beta _{6}q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2304))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2304)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 14}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(384))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(576))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(768))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1152))\)\(^{\oplus 2}\)