Properties

Label 325.6.b
Level $325$
Weight $6$
Character orbit 325.b
Rep. character $\chi_{325}(274,\cdot)$
Character field $\Q$
Dimension $90$
Newform subspaces $9$
Sturm bound $210$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(210\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(325, [\chi])\).

Total New Old
Modular forms 182 90 92
Cusp forms 170 90 80
Eisenstein series 12 0 12

Trace form

\( 90 q - 1400 q^{4} + 664 q^{6} - 8174 q^{9} + O(q^{10}) \) \( 90 q - 1400 q^{4} + 664 q^{6} - 8174 q^{9} + 652 q^{11} + 2588 q^{14} + 16768 q^{16} - 7608 q^{19} - 11540 q^{21} + 1100 q^{24} + 4056 q^{26} - 15004 q^{29} + 28632 q^{31} + 76760 q^{34} + 116936 q^{36} - 12168 q^{39} + 4816 q^{41} - 19464 q^{44} - 23476 q^{46} - 146134 q^{49} - 39648 q^{51} + 21784 q^{54} - 208076 q^{56} - 80384 q^{59} + 69100 q^{61} - 203848 q^{64} - 159340 q^{66} + 286500 q^{69} + 149876 q^{71} - 433092 q^{74} + 399380 q^{76} + 308868 q^{79} + 528618 q^{81} - 250904 q^{84} - 242380 q^{86} - 29296 q^{89} + 140608 q^{91} - 644140 q^{94} + 1068020 q^{96} + 450724 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(325, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
325.6.b.a 325.b 5.b $2$ $52.125$ \(\Q(\sqrt{-1}) \) None 65.6.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+5iq^{2}-6iq^{3}+7q^{4}+30q^{6}-244iq^{7}+\cdots\)
325.6.b.b 325.b 5.b $4$ $52.125$ \(\Q(i, \sqrt{17})\) None 13.6.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}+3\beta _{2})q^{2}+(6\beta _{1}-11\beta _{2})q^{3}+\cdots\)
325.6.b.c 325.b 5.b $6$ $52.125$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None 13.6.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}+2\beta _{2})q^{2}+(\beta _{1}-3\beta _{2}-\beta _{5})q^{3}+\cdots\)
325.6.b.d 325.b 5.b $6$ $52.125$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None 65.6.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}+\beta _{2})q^{2}+(-5\beta _{2}-\beta _{5})q^{3}+(4+\cdots)q^{4}+\cdots\)
325.6.b.e 325.b 5.b $8$ $52.125$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 65.6.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(2\beta _{1}+\beta _{2}-\beta _{3})q^{2}+(-\beta _{1}-\beta _{3}+\cdots)q^{3}+\cdots\)
325.6.b.f 325.b 5.b $12$ $52.125$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 65.6.a.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-\beta _{7}-3\beta _{8})q^{3}+(-23+\cdots)q^{4}+\cdots\)
325.6.b.g 325.b 5.b $12$ $52.125$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 65.6.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-\beta _{1}+6\beta _{3}-\beta _{7})q^{3}+(-22+\cdots)q^{4}+\cdots\)
325.6.b.h 325.b 5.b $18$ $52.125$ \(\mathbb{Q}[x]/(x^{18} + \cdots)\) None 325.6.a.h \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}-\beta _{10})q^{2}+(\beta _{10}-\beta _{12})q^{3}+(-11+\cdots)q^{4}+\cdots\)
325.6.b.i 325.b 5.b $22$ $52.125$ None 325.6.a.j \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{6}^{\mathrm{old}}(325, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(325, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 2}\)