Properties

Label 4400.2.a
Level $4400$
Weight $2$
Character orbit 4400.a
Rep. character $\chi_{4400}(1,\cdot)$
Character field $\Q$
Dimension $95$
Newform subspaces $57$
Sturm bound $1440$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 57 \)
Sturm bound: \(1440\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4400))\).

Total New Old
Modular forms 756 95 661
Cusp forms 685 95 590
Eisenstein series 71 0 71

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

\(2\)\(5\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(10\)
\(+\)\(+\)\(-\)\(-\)\(13\)
\(+\)\(-\)\(+\)\(-\)\(15\)
\(+\)\(-\)\(-\)\(+\)\(9\)
\(-\)\(+\)\(+\)\(-\)\(11\)
\(-\)\(+\)\(-\)\(+\)\(11\)
\(-\)\(-\)\(+\)\(+\)\(13\)
\(-\)\(-\)\(-\)\(-\)\(13\)
Plus space\(+\)\(43\)
Minus space\(-\)\(52\)

Trace form

\( 95 q + 2 q^{3} + 4 q^{7} + 95 q^{9} + O(q^{10}) \) \( 95 q + 2 q^{3} + 4 q^{7} + 95 q^{9} - 3 q^{11} + 2 q^{13} - 2 q^{17} + 12 q^{19} + 8 q^{21} - 6 q^{23} - 10 q^{27} + 10 q^{29} + 30 q^{31} + 10 q^{37} - 8 q^{39} - 2 q^{41} + 16 q^{43} - 4 q^{47} + 103 q^{49} - 12 q^{51} + 2 q^{53} + 24 q^{57} - 22 q^{59} - 6 q^{61} - 40 q^{63} + 22 q^{67} - 8 q^{69} + 34 q^{71} - 2 q^{73} + 60 q^{79} + 103 q^{81} - 8 q^{83} + 16 q^{87} - 22 q^{89} + 64 q^{91} + 24 q^{93} + 2 q^{97} - 9 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4400))\) 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 11
4400.2.a.a 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(-3\) \(0\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-2q^{7}+6q^{9}+q^{11}+6q^{17}+\cdots\)
4400.2.a.b 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(-3\) \(0\) \(-1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-q^{7}+6q^{9}+q^{11}+5q^{17}+\cdots\)
4400.2.a.c 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(-2\) \(0\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-4q^{7}+q^{9}-q^{11}-6q^{13}+\cdots\)
4400.2.a.d 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(-2\) \(0\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-4q^{7}+q^{9}+q^{11}-5q^{13}+\cdots\)
4400.2.a.e 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(-2\) \(0\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-4q^{7}+q^{9}+q^{11}+4q^{13}+\cdots\)
4400.2.a.f 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(-2\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}-q^{11}-2q^{13}+6q^{17}+\cdots\)
4400.2.a.g 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(-2\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}-q^{11}+3q^{13}-4q^{17}+\cdots\)
4400.2.a.h 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(-2\) \(0\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+2q^{7}+q^{9}+q^{11}+4q^{17}+\cdots\)
4400.2.a.i 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{7}-2q^{9}-q^{11}-4q^{13}+\cdots\)
4400.2.a.j 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}-2q^{9}+q^{11}-q^{17}-q^{19}+\cdots\)
4400.2.a.k 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(-1\) \(0\) \(3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{7}-2q^{9}-q^{11}-4q^{13}+\cdots\)
4400.2.a.l 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(-1\) \(0\) \(3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{7}-2q^{9}-q^{11}+6q^{13}+\cdots\)
4400.2.a.m 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(0\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{7}-3q^{9}-q^{11}+4q^{13}+4q^{17}+\cdots\)
4400.2.a.n 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(0\) \(0\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{7}-3q^{9}+q^{11}-6q^{13}-2q^{17}+\cdots\)
4400.2.a.o 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(0\) \(0\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{7}-3q^{9}+q^{11}+8q^{19}-8q^{23}+\cdots\)
4400.2.a.p 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{9}+q^{11}-2q^{13}-6q^{17}+4q^{19}+\cdots\)
4400.2.a.q 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(0\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{7}-3q^{9}+q^{11}+6q^{13}+2q^{17}+\cdots\)
4400.2.a.r 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(0\) \(0\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{7}-3q^{9}+q^{11}-6q^{13}+6q^{17}+\cdots\)
4400.2.a.s 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(1\) \(0\) \(-3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{7}-2q^{9}-q^{11}+4q^{13}+\cdots\)
4400.2.a.t 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}-2q^{9}+q^{11}-2q^{13}+\cdots\)
4400.2.a.u 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}-2q^{9}+q^{11}+q^{17}-q^{19}+\cdots\)
4400.2.a.v 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(1\) \(0\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{7}-2q^{9}+q^{11}+4q^{13}+\cdots\)
4400.2.a.w 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(1\) \(0\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+5q^{7}-2q^{9}-q^{11}-2q^{13}+\cdots\)
4400.2.a.x 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(2\) \(0\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}-2q^{7}+q^{9}+q^{11}-4q^{17}+\cdots\)
4400.2.a.y 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}-q^{11}-3q^{13}+4q^{17}+\cdots\)
4400.2.a.z 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}-q^{11}+4q^{17}+4q^{19}+\cdots\)
4400.2.a.ba 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}-q^{11}+2q^{13}-6q^{17}+\cdots\)
4400.2.a.bb 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(2\) \(0\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+4q^{7}+q^{9}-q^{11}+6q^{13}+\cdots\)
4400.2.a.bc 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(2\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+4q^{7}+q^{9}+q^{11}+5q^{13}+\cdots\)
4400.2.a.bd 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(3\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{7}+6q^{9}+q^{11}-5q^{17}+\cdots\)
4400.2.a.be 4400.a 1.a $1$ $35.134$ \(\Q\) None \(0\) \(3\) \(0\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{7}+6q^{9}+q^{11}+6q^{13}+\cdots\)
4400.2.a.bf 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{13}) \) None \(0\) \(-3\) \(0\) \(-5\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+(-2-\beta )q^{7}+(1+3\beta )q^{9}+\cdots\)
4400.2.a.bg 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{5}) \) None \(0\) \(-3\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+(-2+3\beta )q^{7}+(-1+\cdots)q^{9}+\cdots\)
4400.2.a.bh 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{13}) \) None \(0\) \(-1\) \(0\) \(-5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(-3+\beta )q^{7}+\beta q^{9}+q^{11}+\cdots\)
4400.2.a.bi 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{21}) \) None \(0\) \(-1\) \(0\) \(-5\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(-3+\beta )q^{7}+(2+\beta )q^{9}+q^{11}+\cdots\)
4400.2.a.bj 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(0\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}-\beta q^{7}+(1+\beta )q^{9}-q^{11}-2q^{13}+\cdots\)
4400.2.a.bk 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{5}) \) None \(0\) \(-1\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(1-\beta )q^{7}+(-2+\beta )q^{9}+q^{11}+\cdots\)
4400.2.a.bl 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{33}) \) None \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+\beta q^{7}+(5+\beta )q^{9}+q^{11}-2q^{13}+\cdots\)
4400.2.a.bm 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{5}) \) None \(0\) \(-1\) \(0\) \(3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(1+\beta )q^{7}+(-2+\beta )q^{9}-q^{11}+\cdots\)
4400.2.a.bn 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-2q^{7}+5q^{9}-q^{11}+(4+\beta )q^{13}+\cdots\)
4400.2.a.bo 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{5}) \) None \(0\) \(1\) \(0\) \(-3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(-1-\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
4400.2.a.bp 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(0\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-2\beta q^{7}+(1+\beta )q^{9}+q^{11}+\cdots\)
4400.2.a.bq 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{5}) \) None \(0\) \(1\) \(0\) \(-1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(-1+\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
4400.2.a.br 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(0\) \(3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(2-\beta )q^{7}+(1+\beta )q^{9}-q^{11}+\cdots\)
4400.2.a.bs 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{13}) \) None \(0\) \(1\) \(0\) \(5\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(3-\beta )q^{7}+\beta q^{9}+q^{11}-5q^{13}+\cdots\)
4400.2.a.bt 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(0\) \(5\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(2+\beta )q^{7}+(1+\beta )q^{9}+q^{11}+\cdots\)
4400.2.a.bu 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{21}) \) None \(0\) \(1\) \(0\) \(5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(3-\beta )q^{7}+(2+\beta )q^{9}+q^{11}+\cdots\)
4400.2.a.bv 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{5}) \) None \(0\) \(3\) \(0\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(2-3\beta )q^{7}+(-1+3\beta )q^{9}+\cdots\)
4400.2.a.bw 4400.a 1.a $2$ $35.134$ \(\Q(\sqrt{13}) \) None \(0\) \(3\) \(0\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(2+\beta )q^{7}+(1+3\beta )q^{9}+\cdots\)
4400.2.a.bx 4400.a 1.a $3$ $35.134$ 3.3.837.1 None \(0\) \(-3\) \(0\) \(-3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(-1+\beta _{2})q^{7}+(2+\cdots)q^{9}+\cdots\)
4400.2.a.by 4400.a 1.a $3$ $35.134$ 3.3.1229.1 None \(0\) \(-1\) \(0\) \(-3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-1-\beta _{2})q^{7}+(2+\beta _{2})q^{9}+\cdots\)
4400.2.a.bz 4400.a 1.a $3$ $35.134$ 3.3.1229.1 None \(0\) \(1\) \(0\) \(3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1+\beta _{2})q^{7}+(2+\beta _{2})q^{9}+\cdots\)
4400.2.a.ca 4400.a 1.a $3$ $35.134$ 3.3.837.1 None \(0\) \(3\) \(0\) \(3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(1-\beta _{2})q^{7}+(2-2\beta _{1}+\cdots)q^{9}+\cdots\)
4400.2.a.cb 4400.a 1.a $4$ $35.134$ 4.4.54764.1 None \(0\) \(-1\) \(0\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-\beta _{3}q^{7}+(1+\beta _{1}+\beta _{2}+\beta _{3})q^{9}+\cdots\)
4400.2.a.cc 4400.a 1.a $4$ $35.134$ \(\Q(\sqrt{3}, \sqrt{11})\) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
4400.2.a.cd 4400.a 1.a $4$ $35.134$ \(\Q(\sqrt{3}, \sqrt{19})\) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{7}+(3+\beta _{3})q^{9}+\cdots\)
4400.2.a.ce 4400.a 1.a $4$ $35.134$ 4.4.54764.1 None \(0\) \(1\) \(0\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+\beta _{3}q^{7}+(1+\beta _{1}+\beta _{2}+\beta _{3})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4400))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4400)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(440))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(550))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(880))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2200))\)\(^{\oplus 2}\)