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Properties

Label 1716.2.a
Level $1716$
Weight $2$
Character orbit 1716.a
Rep. character $\chi_{1716}(1,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $9$
Sturm bound $672$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1716.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(672\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1716))\).

Total New Old
Modular forms 348 20 328
Cusp forms 325 20 305
Eisenstein series 23 0 23

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

\(2\)\(3\)\(11\)\(13\)FrickeDim.
\(-\)\(+\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(4\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(6\)
Minus space\(-\)\(14\)

Trace form

\( 20q + 20q^{9} + O(q^{10}) \) \( 20q + 20q^{9} + 8q^{17} + 8q^{19} + 16q^{23} + 28q^{25} + 32q^{29} + 16q^{31} - 4q^{33} - 8q^{35} - 8q^{37} + 8q^{41} - 16q^{43} + 24q^{47} + 36q^{49} + 8q^{53} - 16q^{59} + 8q^{61} + 8q^{65} - 8q^{67} - 24q^{71} + 16q^{73} + 16q^{75} + 8q^{77} - 8q^{79} + 20q^{81} - 32q^{83} + 56q^{85} + 40q^{89} - 8q^{91} + 8q^{93} + 8q^{95} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1716))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 11 13
1716.2.a.a \(1\) \(13.702\) \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}-2q^{7}+q^{9}-q^{11}+q^{13}-2q^{17}+\cdots\)
1716.2.a.b \(1\) \(13.702\) \(\Q\) None \(0\) \(1\) \(-2\) \(-2\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}-2q^{5}-2q^{7}+q^{9}+q^{11}+q^{13}+\cdots\)
1716.2.a.c \(1\) \(13.702\) \(\Q\) None \(0\) \(1\) \(0\) \(2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+2q^{7}+q^{9}-q^{11}+q^{13}+6q^{17}+\cdots\)
1716.2.a.d \(2\) \(13.702\) \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(-2\) \(0\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}+(-1+\beta )q^{5}+q^{9}+q^{11}-q^{13}+\cdots\)
1716.2.a.e \(2\) \(13.702\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(-4\) \(0\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}+(-2+\beta )q^{5}-2\beta q^{7}+q^{9}+\cdots\)
1716.2.a.f \(3\) \(13.702\) 3.3.1620.1 None \(0\) \(-3\) \(0\) \(0\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}+\beta _{1}q^{5}+\beta _{2}q^{7}+q^{9}-q^{11}+\cdots\)
1716.2.a.g \(3\) \(13.702\) 3.3.564.1 None \(0\) \(3\) \(2\) \(4\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}+(1-\beta _{1})q^{5}+(1-\beta _{2})q^{7}+q^{9}+\cdots\)
1716.2.a.h \(3\) \(13.702\) 3.3.404.1 None \(0\) \(3\) \(4\) \(-4\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+(1+2\beta _{1}-\beta _{2})q^{5}+(-2+\beta _{1}+\cdots)q^{7}+\cdots\)
1716.2.a.i \(4\) \(13.702\) 4.4.90996.1 None \(0\) \(-4\) \(2\) \(2\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}-\beta _{1}q^{5}+(1+\beta _{2})q^{7}+q^{9}+q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1716))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1716)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(143))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(286))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(429))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(572))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(858))\)\(^{\oplus 2}\)