Properties

Label 325.4.b
Level $325$
Weight $4$
Character orbit 325.b
Rep. character $\chi_{325}(274,\cdot)$
Character field $\Q$
Dimension $54$
Newform subspaces $9$
Sturm bound $140$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 325.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(140\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(325, [\chi])\).

Total New Old
Modular forms 110 54 56
Cusp forms 98 54 44
Eisenstein series 12 0 12

Trace form

\( 54 q - 240 q^{4} - 32 q^{6} - 494 q^{9} + O(q^{10}) \) \( 54 q - 240 q^{4} - 32 q^{6} - 494 q^{9} - 32 q^{11} + 44 q^{14} + 864 q^{16} + 104 q^{19} - 56 q^{21} + 716 q^{24} - 156 q^{26} - 40 q^{29} - 244 q^{31} + 200 q^{34} + 2720 q^{36} + 312 q^{39} - 1544 q^{41} + 1032 q^{44} - 20 q^{46} - 4310 q^{49} - 1128 q^{51} - 80 q^{54} + 1492 q^{56} + 868 q^{59} + 1928 q^{61} - 6792 q^{64} + 4556 q^{66} + 2856 q^{69} - 628 q^{71} - 3780 q^{74} - 7580 q^{76} + 2088 q^{79} + 7638 q^{81} - 1016 q^{84} + 2852 q^{86} - 1072 q^{89} - 1352 q^{91} + 6468 q^{94} - 18124 q^{96} - 2516 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(325, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
325.4.b.a 325.b 5.b $2$ $19.176$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+5iq^{2}-2iq^{3}-17q^{4}+10q^{6}+\cdots\)
325.4.b.b 325.b 5.b $2$ $19.176$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+5iq^{2}-7iq^{3}-17q^{4}+35q^{6}+\cdots\)
325.4.b.c 325.b 5.b $4$ $19.176$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(2\zeta_{12}+\zeta_{12}^{2})q^{2}+(5\zeta_{12}+3\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
325.4.b.d 325.b 5.b $4$ $19.176$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}+\beta _{2})q^{2}+(-2\beta _{1}+\beta _{2})q^{3}+(1+\cdots)q^{4}+\cdots\)
325.4.b.e 325.b 5.b $4$ $19.176$ \(\Q(i, \sqrt{17})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(3\beta _{1}+4\beta _{2})q^{3}+(3+\beta _{3})q^{4}+\cdots\)
325.4.b.f 325.b 5.b $4$ $19.176$ \(\Q(i, \sqrt{17})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-4\beta _{1}-2\beta _{2})q^{3}+(3+\beta _{3})q^{4}+\cdots\)
325.4.b.g 325.b 5.b $10$ $19.176$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(\beta _{1}+2\beta _{2}-\beta _{7})q^{3}+(-7+\cdots)q^{4}+\cdots\)
325.4.b.h 325.b 5.b $10$ $19.176$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(\beta _{1}-\beta _{8})q^{3}+(-5+\beta _{5}+\cdots)q^{4}+\cdots\)
325.4.b.i 325.b 5.b $14$ $19.176$ \(\mathbb{Q}[x]/(x^{14} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-\beta _{1}+\beta _{8})q^{3}+(-7+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(325, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(325, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 2}\)