Properties

Label 300.2.a
Level $300$
Weight $2$
Character orbit 300.a
Rep. character $\chi_{300}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $4$
Sturm bound $120$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 300.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(120\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(300))\).

Total New Old
Modular forms 78 4 74
Cusp forms 43 4 39
Eisenstein series 35 0 35

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

\(2\)\(3\)\(5\)FrickeDim.
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(3\)

Trace form

\( 4q + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{9} + 4q^{11} + 10q^{19} + 6q^{21} - 24q^{29} + 6q^{31} + 10q^{39} - 20q^{41} + 6q^{49} - 4q^{51} - 4q^{59} - 22q^{61} - 4q^{69} - 8q^{79} + 4q^{81} - 20q^{89} - 10q^{91} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(300))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5
300.2.a.a \(1\) \(2.396\) \(\Q\) None \(0\) \(-1\) \(0\) \(-4\) \(-\) \(+\) \(-\) \(q-q^{3}-4q^{7}+q^{9}-4q^{11}-4q^{17}+\cdots\)
300.2.a.b \(1\) \(2.396\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(+\) \(q-q^{3}+q^{7}+q^{9}+6q^{11}-5q^{13}+\cdots\)
300.2.a.c \(1\) \(2.396\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(q+q^{3}-q^{7}+q^{9}+6q^{11}+5q^{13}+\cdots\)
300.2.a.d \(1\) \(2.396\) \(\Q\) None \(0\) \(1\) \(0\) \(4\) \(-\) \(-\) \(-\) \(q+q^{3}+4q^{7}+q^{9}-4q^{11}+4q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(300))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(300)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))
$5$ 1
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))
$13$ (\( 1 + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))
$17$ (\( 1 + 4 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))
$19$ (\( 1 + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))
$23$ (\( 1 + 4 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 + T + 31 T^{2} \))(\( 1 + T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))
$37$ (\( 1 - 8 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))
$41$ (\( 1 + 10 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))
$53$ (\( 1 - 12 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))
$59$ (\( 1 - 4 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( 1 + 13 T + 61 T^{2} \))(\( 1 + 13 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))
$67$ (\( 1 - 4 T + 67 T^{2} \))(\( 1 + 11 T + 67 T^{2} \))(\( 1 - 11 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))
$73$ (\( 1 - 8 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 8 T + 73 T^{2} \))
$79$ (\( 1 + 12 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 12 T + 79 T^{2} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))
$89$ (\( 1 + 10 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))
$97$ (\( 1 + 8 T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))
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