Properties

Label 5408.2.a
Level $5408$
Weight $2$
Character orbit 5408.a
Rep. character $\chi_{5408}(1,\cdot)$
Character field $\Q$
Dimension $155$
Newform subspaces $47$
Sturm bound $1456$
Trace bound $21$

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Defining parameters

Level: \( N \) \(=\) \( 5408 = 2^{5} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5408.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 47 \)
Sturm bound: \(1456\)
Trace bound: \(21\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5408))\).

Total New Old
Modular forms 784 155 629
Cusp forms 673 155 518
Eisenstein series 111 0 111

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

\(2\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(35\)
\(+\)\(-\)\(-\)\(42\)
\(-\)\(+\)\(-\)\(42\)
\(-\)\(-\)\(+\)\(36\)
Plus space\(+\)\(71\)
Minus space\(-\)\(84\)

Trace form

\( 155 q + 2 q^{5} + 151 q^{9} - 10 q^{17} - 16 q^{21} + 141 q^{25} - 6 q^{29} + 2 q^{37} + 14 q^{41} + 26 q^{45} + 147 q^{49} + 34 q^{53} + 16 q^{57} + 10 q^{61} + 32 q^{69} - 34 q^{73} + 48 q^{77} + 195 q^{81}+ \cdots + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5408))\) 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 13
5408.2.a.a 5408.a 1.a $1$ $43.183$ \(\Q\) None 416.2.f.a \(0\) \(-3\) \(-3\) \(-1\) $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-3q^{5}-q^{7}+6q^{9}+4q^{11}+\cdots\)
5408.2.a.b 5408.a 1.a $1$ $43.183$ \(\Q\) None 416.2.f.a \(0\) \(-3\) \(3\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+3q^{5}+q^{7}+6q^{9}-4q^{11}+\cdots\)
5408.2.a.c 5408.a 1.a $1$ $43.183$ \(\Q\) None 416.2.i.a \(0\) \(-2\) \(-1\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-q^{5}+q^{9}+4q^{11}+2q^{15}+\cdots\)
5408.2.a.d 5408.a 1.a $1$ $43.183$ \(\Q\) None 416.2.i.a \(0\) \(-2\) \(1\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}+q^{9}-4q^{11}-2q^{15}+\cdots\)
5408.2.a.e 5408.a 1.a $1$ $43.183$ \(\Q\) None 416.2.a.a \(0\) \(-1\) \(-1\) \(3\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+3q^{7}-2q^{9}+2q^{11}+\cdots\)
5408.2.a.f 5408.a 1.a $1$ $43.183$ \(\Q\) \(\Q(\sqrt{-1}) \) 416.2.f.b \(0\) \(0\) \(-4\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q-4q^{5}-3q^{9}+2q^{17}+11q^{25}+10q^{29}+\cdots\)
5408.2.a.g 5408.a 1.a $1$ $43.183$ \(\Q\) \(\Q(\sqrt{-1}) \) 32.2.a.a \(0\) \(0\) \(2\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q+2q^{5}-3q^{9}+2q^{17}-q^{25}-10q^{29}+\cdots\)
5408.2.a.h 5408.a 1.a $1$ $43.183$ \(\Q\) \(\Q(\sqrt{-1}) \) 416.2.f.b \(0\) \(0\) \(4\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+4q^{5}-3q^{9}+2q^{17}+11q^{25}+10q^{29}+\cdots\)
5408.2.a.i 5408.a 1.a $1$ $43.183$ \(\Q\) None 416.2.a.a \(0\) \(1\) \(-1\) \(-3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-3q^{7}-2q^{9}-2q^{11}+\cdots\)
5408.2.a.j 5408.a 1.a $1$ $43.183$ \(\Q\) None 416.2.i.a \(0\) \(2\) \(-1\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{5}+q^{9}-4q^{11}-2q^{15}+\cdots\)
5408.2.a.k 5408.a 1.a $1$ $43.183$ \(\Q\) None 416.2.i.a \(0\) \(2\) \(1\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}+q^{9}+4q^{11}+2q^{15}+\cdots\)
5408.2.a.l 5408.a 1.a $1$ $43.183$ \(\Q\) None 416.2.f.a \(0\) \(3\) \(-3\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-3q^{5}+q^{7}+6q^{9}-4q^{11}+\cdots\)
5408.2.a.m 5408.a 1.a $1$ $43.183$ \(\Q\) None 416.2.f.a \(0\) \(3\) \(3\) \(-1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+3q^{5}-q^{7}+6q^{9}+4q^{11}+\cdots\)
5408.2.a.n 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{2}) \) None 416.2.i.c \(0\) \(-2\) \(0\) \(-6\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+2\beta q^{5}+(-3+\beta )q^{7}+\cdots\)
5408.2.a.o 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{2}) \) None 416.2.i.c \(0\) \(-2\) \(0\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}-2\beta q^{5}+(3-\beta )q^{7}+\cdots\)
5408.2.a.p 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{17}) \) None 416.2.a.c \(0\) \(-1\) \(3\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(2-\beta )q^{5}+(2-\beta )q^{7}+(1+\beta )q^{9}+\cdots\)
5408.2.a.q 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{5}) \) None 416.2.a.d \(0\) \(0\) \(-6\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}-3q^{5}+\beta q^{7}+2q^{9}-2\beta q^{11}+\cdots\)
5408.2.a.r 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-1}) \) 416.2.w.b \(0\) \(0\) \(-4\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+(-2-\beta )q^{5}-3q^{9}+(-1-4\beta )q^{17}+\cdots\)
5408.2.a.s 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-1}) \) 416.2.i.e \(0\) \(0\) \(-2\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q+(-1-\beta )q^{5}-3q^{9}+(-1+2\beta )q^{17}+\cdots\)
5408.2.a.t 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{5}) \) None 416.2.f.e \(0\) \(0\) \(-2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}-q^{5}-\beta q^{7}+2q^{9}+\beta q^{15}+\cdots\)
5408.2.a.u 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{3}) \) None 416.2.w.a \(0\) \(0\) \(0\) \(-6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+2\beta q^{5}-3q^{7}-5q^{11}+6q^{15}+\cdots\)
5408.2.a.v 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{13}) \) \(\Q(\sqrt{-13}) \) 416.2.f.d \(0\) \(0\) \(0\) \(-2\) $-$ $-$ $N(\mathrm{U}(1))$ \(q+(-1-\beta )q^{7}-3q^{9}+(3-\beta )q^{11}+\cdots\)
5408.2.a.w 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{11}) \) None 416.2.i.d \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-\beta q^{7}+8q^{9}-\beta q^{11}-3q^{17}+\cdots\)
5408.2.a.x 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{11}) \) None 416.2.i.d \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+\beta q^{7}+8q^{9}+\beta q^{11}-3q^{17}+\cdots\)
5408.2.a.y 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{13}) \) \(\Q(\sqrt{-13}) \) 416.2.f.d \(0\) \(0\) \(0\) \(2\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+(1+\beta )q^{7}-3q^{9}+(-3+\beta )q^{11}+\cdots\)
5408.2.a.z 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{3}) \) None 416.2.w.a \(0\) \(0\) \(0\) \(6\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-2\beta q^{5}+3q^{7}+5q^{11}-6q^{15}+\cdots\)
5408.2.a.ba 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-1}) \) 416.2.i.e \(0\) \(0\) \(2\) \(0\) $-$ $+$ $N(\mathrm{U}(1))$ \(q+(1-\beta )q^{5}-3q^{9}+(-1-2\beta )q^{17}+\cdots\)
5408.2.a.bb 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{5}) \) None 416.2.f.e \(0\) \(0\) \(2\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+q^{5}+\beta q^{7}+2q^{9}-\beta q^{15}+\cdots\)
5408.2.a.bc 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-1}) \) 416.2.w.b \(0\) \(0\) \(4\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q+(2-\beta )q^{5}-3q^{9}+(-1+4\beta )q^{17}+\cdots\)
5408.2.a.bd 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{17}) \) None 416.2.a.c \(0\) \(1\) \(3\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(2-\beta )q^{5}+(-2+\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
5408.2.a.be 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{2}) \) None 416.2.i.c \(0\) \(2\) \(0\) \(-6\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+2\beta q^{5}+(-3-\beta )q^{7}+\cdots\)
5408.2.a.bf 5408.a 1.a $2$ $43.183$ \(\Q(\sqrt{2}) \) None 416.2.i.c \(0\) \(2\) \(0\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-2\beta q^{5}+(3+\beta )q^{7}+2\beta q^{9}+\cdots\)
5408.2.a.bg 5408.a 1.a $4$ $43.183$ 4.4.7488.1 None 416.2.w.c \(0\) \(0\) \(-8\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-2q^{5}+(\beta _{1}+\beta _{3})q^{7}+(2-2\beta _{2}+\cdots)q^{9}+\cdots\)
5408.2.a.bh 5408.a 1.a $4$ $43.183$ 4.4.9248.1 None 416.2.i.g \(0\) \(0\) \(-2\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+\beta _{3}q^{5}+(-\beta _{1}+\beta _{2})q^{7}+\cdots\)
5408.2.a.bi 5408.a 1.a $4$ $43.183$ 4.4.9248.1 None 416.2.i.g \(0\) \(0\) \(2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}-\beta _{3}q^{5}+(\beta _{1}-\beta _{2})q^{7}-\beta _{3}q^{9}+\cdots\)
5408.2.a.bj 5408.a 1.a $4$ $43.183$ 4.4.13448.1 None 416.2.a.f \(0\) \(0\) \(2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}-\beta _{3}q^{5}+(-\beta _{1}+\beta _{2})q^{7}+\cdots\)
5408.2.a.bk 5408.a 1.a $4$ $43.183$ 4.4.7488.1 None 416.2.w.c \(0\) \(0\) \(8\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+2q^{5}+(-\beta _{1}-\beta _{3})q^{7}+(2+\cdots)q^{9}+\cdots\)
5408.2.a.bl 5408.a 1.a $6$ $43.183$ 6.6.1997632.1 None 5408.2.a.bl \(0\) \(0\) \(-4\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-1+\beta _{4})q^{5}+(\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
5408.2.a.bm 5408.a 1.a $6$ $43.183$ 6.6.134509248.1 None 416.2.w.d \(0\) \(0\) \(0\) \(-6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-\beta _{1}+\beta _{3})q^{5}+(-1-\beta _{2}+\cdots)q^{7}+\cdots\)
5408.2.a.bn 5408.a 1.a $6$ $43.183$ 6.6.134509248.1 None 416.2.w.d \(0\) \(0\) \(0\) \(6\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(\beta _{1}-\beta _{3})q^{5}+(1+\beta _{2})q^{7}+\cdots\)
5408.2.a.bo 5408.a 1.a $6$ $43.183$ 6.6.1997632.1 None 5408.2.a.bl \(0\) \(0\) \(4\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1-\beta _{4})q^{5}+(-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
5408.2.a.bp 5408.a 1.a $9$ $43.183$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 5408.2.a.bp \(0\) \(-7\) \(-6\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(-1-\beta _{3})q^{5}+(-\beta _{4}+\cdots)q^{7}+\cdots\)
5408.2.a.bq 5408.a 1.a $9$ $43.183$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 5408.2.a.bp \(0\) \(-7\) \(6\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(1+\beta _{3})q^{5}+(\beta _{4}-\beta _{6}+\cdots)q^{7}+\cdots\)
5408.2.a.br 5408.a 1.a $9$ $43.183$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 5408.2.a.bp \(0\) \(7\) \(-6\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(-1-\beta _{3})q^{5}+(\beta _{4}-\beta _{6}+\cdots)q^{7}+\cdots\)
5408.2.a.bs 5408.a 1.a $9$ $43.183$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 5408.2.a.bp \(0\) \(7\) \(6\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(1+\beta _{3})q^{5}+(-\beta _{4}+\beta _{6}+\cdots)q^{7}+\cdots\)
5408.2.a.bt 5408.a 1.a $12$ $43.183$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 5408.2.a.bt \(0\) \(0\) \(-8\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-1+\beta _{4}+\beta _{5}+\beta _{7})q^{5}+\cdots\)
5408.2.a.bu 5408.a 1.a $12$ $43.183$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 5408.2.a.bt \(0\) \(0\) \(8\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1-\beta _{4}-\beta _{5}-\beta _{7})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5408))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5408)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(416))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(676))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1352))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2704))\)\(^{\oplus 2}\)