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Properties

Label 5194.2.a
Level $5194$
Weight $2$
Character orbit 5194.a
Rep. character $\chi_{5194}(1,\cdot)$
Character field $\Q$
Dimension $179$
Newform subspaces $52$
Sturm bound $1512$
Trace bound $15$

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Defining parameters

Level: \( N \) \(=\) \( 5194 = 2 \cdot 7^{2} \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5194.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 52 \)
Sturm bound: \(1512\)
Trace bound: \(15\)
Distinguishing \(T_p\): \(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5194))\).

Total New Old
Modular forms 772 179 593
Cusp forms 741 179 562
Eisenstein series 31 0 31

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

\(2\)\(7\)\(53\)FrickeDim
\(+\)\(+\)\(+\)$+$\(23\)
\(+\)\(+\)\(-\)$-$\(23\)
\(+\)\(-\)\(+\)$-$\(22\)
\(+\)\(-\)\(-\)$+$\(22\)
\(-\)\(+\)\(+\)$-$\(27\)
\(-\)\(+\)\(-\)$+$\(15\)
\(-\)\(-\)\(+\)$+$\(19\)
\(-\)\(-\)\(-\)$-$\(28\)
Plus space\(+\)\(79\)
Minus space\(-\)\(100\)

Trace form

\( 179 q - q^{2} + 179 q^{4} + 6 q^{5} + 2 q^{6} - q^{8} + 193 q^{9} + O(q^{10}) \) \( 179 q - q^{2} + 179 q^{4} + 6 q^{5} + 2 q^{6} - q^{8} + 193 q^{9} + 10 q^{11} + 8 q^{15} + 179 q^{16} - 2 q^{17} + 3 q^{18} + 6 q^{20} + 16 q^{23} + 2 q^{24} + 191 q^{25} + 2 q^{26} + 24 q^{27} + 8 q^{29} + 28 q^{30} + 4 q^{31} - q^{32} + 4 q^{33} - 2 q^{34} + 193 q^{36} - 16 q^{37} + 2 q^{38} - 20 q^{39} + 34 q^{41} - 26 q^{43} + 10 q^{44} + 26 q^{45} + 12 q^{46} - 12 q^{47} + 25 q^{50} - 3 q^{53} - 10 q^{54} + 12 q^{55} - 42 q^{57} + 2 q^{58} - 22 q^{59} + 8 q^{60} + 42 q^{61} + 14 q^{62} + 179 q^{64} + 32 q^{65} - 24 q^{66} - 56 q^{67} - 2 q^{68} + 6 q^{69} + 4 q^{71} + 3 q^{72} + 22 q^{73} - 22 q^{74} + 16 q^{75} - 18 q^{78} - 48 q^{79} + 6 q^{80} + 243 q^{81} + 10 q^{82} + 12 q^{83} + 60 q^{85} - 20 q^{86} - 4 q^{87} - 20 q^{89} - 28 q^{90} + 16 q^{92} + 4 q^{93} + 44 q^{95} + 2 q^{96} + 38 q^{97} - 18 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5194))\) 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 53
5194.2.a.a 5194.a 1.a $1$ $41.474$ \(\Q\) None \(-1\) \(-3\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}+q^{4}+3q^{6}-q^{8}+6q^{9}+\cdots\)
5194.2.a.b 5194.a 1.a $1$ $41.474$ \(\Q\) None \(-1\) \(-2\) \(-4\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}-4q^{5}+2q^{6}-q^{8}+\cdots\)
5194.2.a.c 5194.a 1.a $1$ $41.474$ \(\Q\) None \(-1\) \(-2\) \(-1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}-q^{8}+\cdots\)
5194.2.a.d 5194.a 1.a $1$ $41.474$ \(\Q\) None \(-1\) \(-2\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots\)
5194.2.a.e 5194.a 1.a $1$ $41.474$ \(\Q\) None \(-1\) \(-2\) \(4\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+4q^{5}+2q^{6}-q^{8}+\cdots\)
5194.2.a.f 5194.a 1.a $1$ $41.474$ \(\Q\) None \(-1\) \(0\) \(-4\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-4q^{5}-q^{8}-3q^{9}+4q^{10}+\cdots\)
5194.2.a.g 5194.a 1.a $1$ $41.474$ \(\Q\) None \(-1\) \(0\) \(-3\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-3q^{5}-q^{8}-3q^{9}+3q^{10}+\cdots\)
5194.2.a.h 5194.a 1.a $1$ $41.474$ \(\Q\) None \(-1\) \(0\) \(1\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-q^{8}-3q^{9}-q^{10}+\cdots\)
5194.2.a.i 5194.a 1.a $1$ $41.474$ \(\Q\) None \(-1\) \(0\) \(4\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+4q^{5}-q^{8}-3q^{9}-4q^{10}+\cdots\)
5194.2.a.j 5194.a 1.a $1$ $41.474$ \(\Q\) None \(-1\) \(1\) \(4\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+4q^{5}-q^{6}-q^{8}+\cdots\)
5194.2.a.k 5194.a 1.a $1$ $41.474$ \(\Q\) None \(-1\) \(2\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{8}+q^{9}+\cdots\)
5194.2.a.l 5194.a 1.a $1$ $41.474$ \(\Q\) None \(1\) \(-3\) \(2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}+q^{4}+2q^{5}-3q^{6}+q^{8}+\cdots\)
5194.2.a.m 5194.a 1.a $1$ $41.474$ \(\Q\) None \(1\) \(-2\) \(-4\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-4q^{5}-2q^{6}+q^{8}+\cdots\)
5194.2.a.n 5194.a 1.a $1$ $41.474$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}-2q^{9}+\cdots\)
5194.2.a.o 5194.a 1.a $1$ $41.474$ \(\Q\) None \(1\) \(1\) \(2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+2q^{5}+q^{6}+q^{8}+\cdots\)
5194.2.a.p 5194.a 1.a $1$ $41.474$ \(\Q\) None \(1\) \(2\) \(-3\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}-3q^{5}+2q^{6}+q^{8}+\cdots\)
5194.2.a.q 5194.a 1.a $1$ $41.474$ \(\Q\) None \(1\) \(2\) \(4\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+4q^{5}+2q^{6}+q^{8}+\cdots\)
5194.2.a.r 5194.a 1.a $2$ $41.474$ \(\Q(\sqrt{2}) \) None \(-2\) \(-4\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-2+\beta )q^{3}+q^{4}+2\beta q^{5}+\cdots\)
5194.2.a.s 5194.a 1.a $2$ $41.474$ \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(-4\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+(-2+\beta )q^{5}+q^{6}+\cdots\)
5194.2.a.t 5194.a 1.a $2$ $41.474$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-\beta q^{5}-q^{8}-3q^{9}+\beta q^{10}+\cdots\)
5194.2.a.u 5194.a 1.a $2$ $41.474$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}+2\beta q^{5}-\beta q^{6}+\cdots\)
5194.2.a.v 5194.a 1.a $2$ $41.474$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{5}-\beta q^{6}-q^{8}+\cdots\)
5194.2.a.w 5194.a 1.a $2$ $41.474$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+2\beta q^{3}+q^{4}+\beta q^{5}-2\beta q^{6}+\cdots\)
5194.2.a.x 5194.a 1.a $2$ $41.474$ \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta )q^{3}+q^{4}+\beta q^{5}+(-1+\cdots)q^{6}+\cdots\)
5194.2.a.y 5194.a 1.a $2$ $41.474$ \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta )q^{3}+q^{4}-\beta q^{5}+(-1+\cdots)q^{6}+\cdots\)
5194.2.a.z 5194.a 1.a $2$ $41.474$ \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(4\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+(2+\beta )q^{5}-q^{6}+\cdots\)
5194.2.a.ba 5194.a 1.a $2$ $41.474$ \(\Q(\sqrt{2}) \) None \(-2\) \(4\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(2+\beta )q^{3}+q^{4}+2\beta q^{5}+(-2+\cdots)q^{6}+\cdots\)
5194.2.a.bb 5194.a 1.a $2$ $41.474$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(-4\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+(-2+\beta )q^{5}+\beta q^{6}+\cdots\)
5194.2.a.bc 5194.a 1.a $2$ $41.474$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+q^{8}-q^{9}+\cdots\)
5194.2.a.bd 5194.a 1.a $3$ $41.474$ 3.3.257.1 None \(-3\) \(-2\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+q^{4}-\beta _{2}q^{5}+\cdots\)
5194.2.a.be 5194.a 1.a $3$ $41.474$ 3.3.257.1 None \(-3\) \(2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}+\beta _{2}q^{5}+\cdots\)
5194.2.a.bf 5194.a 1.a $3$ $41.474$ 3.3.940.1 None \(-3\) \(3\) \(2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1-\beta _{1})q^{3}+q^{4}+(1+\beta _{2})q^{5}+\cdots\)
5194.2.a.bg 5194.a 1.a $3$ $41.474$ 3.3.316.1 None \(3\) \(4\) \(5\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta _{1})q^{3}+q^{4}+(2-\beta _{1})q^{5}+\cdots\)
5194.2.a.bh 5194.a 1.a $4$ $41.474$ 4.4.147069.1 None \(-4\) \(-4\) \(5\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+(1+\beta _{1})q^{5}+q^{6}+\cdots\)
5194.2.a.bi 5194.a 1.a $4$ $41.474$ \(\Q(\zeta_{24})^+\) None \(-4\) \(0\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+2\beta _{1}q^{5}-\beta _{1}q^{6}+\cdots\)
5194.2.a.bj 5194.a 1.a $4$ $41.474$ \(\Q(\sqrt{2}, \sqrt{7})\) None \(-4\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+(-2\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
5194.2.a.bk 5194.a 1.a $4$ $41.474$ \(\Q(\sqrt{2}, \sqrt{7})\) None \(-4\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{2}q^{5}-\beta _{1}q^{6}+\cdots\)
5194.2.a.bl 5194.a 1.a $4$ $41.474$ \(\Q(\zeta_{24})^+\) None \(-4\) \(0\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(2\beta _{1}+2\beta _{3})q^{3}+q^{4}+(\beta _{1}+2\beta _{3})q^{5}+\cdots\)
5194.2.a.bm 5194.a 1.a $4$ $41.474$ 4.4.147069.1 None \(-4\) \(4\) \(-5\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+(-1-\beta _{1})q^{5}-q^{6}+\cdots\)
5194.2.a.bn 5194.a 1.a $4$ $41.474$ \(\Q(\zeta_{24})^+\) None \(4\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+q^{8}+\cdots\)
5194.2.a.bo 5194.a 1.a $4$ $41.474$ 4.4.9248.1 None \(4\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+(-\beta _{1}+\beta _{3})q^{5}+\cdots\)
5194.2.a.bp 5194.a 1.a $6$ $41.474$ 6.6.1229312.1 None \(-6\) \(0\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-\beta _{4}-\beta _{5})q^{3}+q^{4}+\beta _{5}q^{5}+\cdots\)
5194.2.a.bq 5194.a 1.a $6$ $41.474$ 6.6.118210688.1 None \(-6\) \(0\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{4}q^{3}+q^{4}+(-\beta _{1}+\beta _{5})q^{5}+\cdots\)
5194.2.a.br 5194.a 1.a $6$ $41.474$ 6.6.111663536.1 None \(6\) \(-2\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{2}q^{5}-\beta _{1}q^{6}+\cdots\)
5194.2.a.bs 5194.a 1.a $7$ $41.474$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(-2\) \(-1\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{2}q^{3}+q^{4}-\beta _{1}q^{5}-\beta _{2}q^{6}+\cdots\)
5194.2.a.bt 5194.a 1.a $7$ $41.474$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(2\) \(1\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{2}q^{3}+q^{4}+\beta _{1}q^{5}+\beta _{2}q^{6}+\cdots\)
5194.2.a.bu 5194.a 1.a $8$ $41.474$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(-2\) \(-5\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+(-1+\beta _{3})q^{5}+\cdots\)
5194.2.a.bv 5194.a 1.a $8$ $41.474$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(2\) \(5\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+(1-\beta _{3})q^{5}+\beta _{1}q^{6}+\cdots\)
5194.2.a.bw 5194.a 1.a $9$ $41.474$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(-2\) \(-9\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+(-1+\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
5194.2.a.bx 5194.a 1.a $9$ $41.474$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(2\) \(9\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+(1-\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
5194.2.a.by 5194.a 1.a $10$ $41.474$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(10\) \(0\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{4}q^{5}+\beta _{1}q^{6}+\cdots\)
5194.2.a.bz 5194.a 1.a $18$ $41.474$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(18\) \(0\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{14}q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5194))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5194)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(53))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(106))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(371))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(742))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2597))\)\(^{\oplus 2}\)