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Properties

Label 252.12.a
Level $252$
Weight $12$
Character orbit 252.a
Rep. character $\chi_{252}(1,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $9$
Sturm bound $576$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 252.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(576\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(252))\).

Total New Old
Modular forms 540 28 512
Cusp forms 516 28 488
Eisenstein series 24 0 24

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

\(2\)\(3\)\(7\)FrickeDim
\(-\)\(+\)\(+\)$-$\(6\)
\(-\)\(+\)\(-\)$+$\(6\)
\(-\)\(-\)\(+\)$+$\(8\)
\(-\)\(-\)\(-\)$-$\(8\)
Plus space\(+\)\(14\)
Minus space\(-\)\(14\)

Trace form

\( 28 q + 5832 q^{5} + O(q^{10}) \) \( 28 q + 5832 q^{5} - 883188 q^{11} + 370216 q^{13} - 3073176 q^{17} + 2278352 q^{19} - 8803548 q^{23} + 256972812 q^{25} - 297312840 q^{29} + 274778488 q^{31} - 206322732 q^{35} - 545782576 q^{37} - 1607493384 q^{41} + 74602592 q^{43} - 4513494888 q^{47} + 7909306972 q^{49} - 5050154376 q^{53} + 17234799496 q^{55} + 9673522848 q^{59} - 9208148888 q^{61} + 27766843488 q^{65} + 2478950040 q^{67} + 22845545388 q^{71} - 41300224968 q^{73} - 8488879560 q^{77} - 46976967080 q^{79} + 74973071304 q^{83} + 5559731560 q^{85} - 21534571224 q^{89} + 68899153624 q^{91} - 133767695232 q^{95} + 76939430184 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(252))\) 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 7
252.12.a.a 252.a 1.a $1$ $193.622$ \(\Q\) None \(0\) \(0\) \(2130\) \(16807\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2130q^{5}+7^{5}q^{7}+704196q^{11}+\cdots\)
252.12.a.b 252.a 1.a $2$ $193.622$ \(\Q(\sqrt{1435009}) \) None \(0\) \(0\) \(-5496\) \(33614\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2748-\beta )q^{5}+7^{5}q^{7}+(-562782+\cdots)q^{11}+\cdots\)
252.12.a.c 252.a 1.a $2$ $193.622$ \(\Q(\sqrt{37321}) \) None \(0\) \(0\) \(5130\) \(33614\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(2565-5\beta )q^{5}+7^{5}q^{7}+(295515+\cdots)q^{11}+\cdots\)
252.12.a.d 252.a 1.a $2$ $193.622$ \(\Q(\sqrt{1000465}) \) None \(0\) \(0\) \(8910\) \(-33614\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(4455-5\beta )q^{5}-7^{5}q^{7}+(-166617+\cdots)q^{11}+\cdots\)
252.12.a.e 252.a 1.a $3$ $193.622$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(0\) \(-4986\) \(50421\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1662-\beta _{2})q^{5}+7^{5}q^{7}+(-287932+\cdots)q^{11}+\cdots\)
252.12.a.f 252.a 1.a $3$ $193.622$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(0\) \(-4762\) \(-50421\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1585-5\beta _{1}-2\beta _{2})q^{5}-7^{5}q^{7}+\cdots\)
252.12.a.g 252.a 1.a $3$ $193.622$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(0\) \(4906\) \(-50421\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1635+\beta _{1})q^{5}-7^{5}q^{7}+(74880+\cdots)q^{11}+\cdots\)
252.12.a.h 252.a 1.a $6$ $193.622$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(-100842\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{5}-7^{5}q^{7}+(21\beta _{1}-\beta _{2})q^{11}+\cdots\)
252.12.a.i 252.a 1.a $6$ $193.622$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(100842\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{5}+7^{5}q^{7}+(15\beta _{1}+\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(252))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(252)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 18}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 9}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)