Properties

Label 1617.4.a
Level $1617$
Weight $4$
Character orbit 1617.a
Rep. character $\chi_{1617}(1,\cdot)$
Character field $\Q$
Dimension $206$
Newform subspaces $32$
Sturm bound $896$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1617.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 32 \)
Sturm bound: \(896\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1617))\).

Total New Old
Modular forms 688 206 482
Cusp forms 656 206 450
Eisenstein series 32 0 32

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

\(3\)\(7\)\(11\)FrickeDim
\(+\)\(+\)\(+\)$+$\(26\)
\(+\)\(+\)\(-\)$-$\(24\)
\(+\)\(-\)\(+\)$-$\(25\)
\(+\)\(-\)\(-\)$+$\(28\)
\(-\)\(+\)\(+\)$-$\(26\)
\(-\)\(+\)\(-\)$+$\(28\)
\(-\)\(-\)\(+\)$+$\(26\)
\(-\)\(-\)\(-\)$-$\(23\)
Plus space\(+\)\(108\)
Minus space\(-\)\(98\)

Trace form

\( 206 q - 4 q^{2} + 844 q^{4} - 16 q^{5} - 12 q^{6} - 132 q^{8} + 1854 q^{9} + O(q^{10}) \) \( 206 q - 4 q^{2} + 844 q^{4} - 16 q^{5} - 12 q^{6} - 132 q^{8} + 1854 q^{9} - 112 q^{10} + 24 q^{12} - 172 q^{13} + 60 q^{15} + 3300 q^{16} + 152 q^{17} - 36 q^{18} + 120 q^{19} + 180 q^{20} + 44 q^{22} - 108 q^{23} - 324 q^{24} + 4890 q^{25} - 204 q^{26} - 632 q^{29} - 336 q^{30} - 728 q^{31} - 1260 q^{32} - 66 q^{33} - 528 q^{34} + 7596 q^{36} + 1188 q^{37} + 1032 q^{38} + 1176 q^{39} - 1632 q^{40} - 632 q^{41} + 576 q^{43} + 616 q^{44} - 144 q^{45} - 384 q^{46} - 196 q^{47} + 816 q^{48} + 2132 q^{50} - 864 q^{51} - 2984 q^{52} + 1944 q^{53} - 108 q^{54} - 440 q^{55} - 756 q^{57} + 232 q^{58} + 888 q^{59} - 372 q^{60} + 1548 q^{61} + 24 q^{62} + 15900 q^{64} + 832 q^{65} - 264 q^{66} + 1928 q^{67} + 3976 q^{68} + 516 q^{69} - 44 q^{71} - 1188 q^{72} - 52 q^{73} - 848 q^{74} + 216 q^{75} + 1456 q^{76} + 204 q^{78} - 2936 q^{79} + 8444 q^{80} + 16686 q^{81} + 5544 q^{82} + 1352 q^{83} - 5120 q^{85} + 1976 q^{86} - 504 q^{87} + 924 q^{88} - 2356 q^{89} - 1008 q^{90} - 2372 q^{92} - 1128 q^{93} - 824 q^{94} + 3528 q^{95} + 2076 q^{96} - 420 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 7 11
1617.4.a.a 1617.a 1.a $1$ $95.406$ \(\Q\) None 33.4.a.a \(-5\) \(-3\) \(14\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{2}-3q^{3}+17q^{4}+14q^{5}+15q^{6}+\cdots\)
1617.4.a.b 1617.a 1.a $1$ $95.406$ \(\Q\) None 231.4.a.a \(-3\) \(3\) \(4\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}+3q^{3}+q^{4}+4q^{5}-9q^{6}+\cdots\)
1617.4.a.c 1617.a 1.a $1$ $95.406$ \(\Q\) None 231.4.a.b \(-2\) \(-3\) \(-1\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}-4q^{4}-q^{5}+6q^{6}+\cdots\)
1617.4.a.d 1617.a 1.a $1$ $95.406$ \(\Q\) None 33.4.a.b \(-1\) \(3\) \(4\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}-7q^{4}+4q^{5}-3q^{6}+\cdots\)
1617.4.a.e 1617.a 1.a $1$ $95.406$ \(\Q\) None 231.4.a.c \(2\) \(3\) \(-11\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}-4q^{4}-11q^{5}+6q^{6}+\cdots\)
1617.4.a.f 1617.a 1.a $1$ $95.406$ \(\Q\) None 231.4.a.d \(3\) \(-3\) \(14\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}-3q^{3}+q^{4}+14q^{5}-9q^{6}+\cdots\)
1617.4.a.g 1617.a 1.a $1$ $95.406$ \(\Q\) None 231.4.a.e \(5\) \(-3\) \(6\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+5q^{2}-3q^{3}+17q^{4}+6q^{5}-15q^{6}+\cdots\)
1617.4.a.h 1617.a 1.a $2$ $95.406$ \(\Q(\sqrt{17}) \) None 231.4.a.g \(-3\) \(-6\) \(-25\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}-3q^{3}+(-3+3\beta )q^{4}+\cdots\)
1617.4.a.i 1617.a 1.a $2$ $95.406$ \(\Q(\sqrt{17}) \) None 231.4.a.f \(-3\) \(-6\) \(19\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}-3q^{3}+(-3+3\beta )q^{4}+\cdots\)
1617.4.a.j 1617.a 1.a $2$ $95.406$ \(\Q(\sqrt{33}) \) None 33.4.a.d \(1\) \(-6\) \(-16\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-3q^{3}+\beta q^{4}+(-10+4\beta )q^{5}+\cdots\)
1617.4.a.k 1617.a 1.a $2$ $95.406$ \(\Q(\sqrt{97}) \) None 33.4.a.c \(1\) \(6\) \(14\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+3q^{3}+(2^{4}+\beta )q^{4}+(6+2\beta )q^{5}+\cdots\)
1617.4.a.l 1617.a 1.a $2$ $95.406$ \(\Q(\sqrt{37}) \) None 231.4.a.i \(3\) \(-6\) \(-2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-3q^{3}+(2+3\beta )q^{4}+(1+\cdots)q^{5}+\cdots\)
1617.4.a.m 1617.a 1.a $2$ $95.406$ \(\Q(\sqrt{17}) \) None 231.4.a.h \(3\) \(6\) \(-1\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+3q^{3}+(-3+3\beta )q^{4}+\cdots\)
1617.4.a.n 1617.a 1.a $5$ $95.406$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 231.4.a.k \(-1\) \(15\) \(-21\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(4+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1617.4.a.o 1617.a 1.a $5$ $95.406$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 231.4.a.j \(-1\) \(15\) \(-7\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(4+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
1617.4.a.p 1617.a 1.a $5$ $95.406$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 231.4.a.l \(5\) \(-15\) \(-7\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-3q^{3}+(4-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1617.4.a.q 1617.a 1.a $7$ $95.406$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 1617.4.a.q \(-4\) \(-21\) \(20\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-3q^{3}+(5-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1617.4.a.r 1617.a 1.a $7$ $95.406$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 1617.4.a.q \(-4\) \(21\) \(-20\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+3q^{3}+(5-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1617.4.a.s 1617.a 1.a $7$ $95.406$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 1617.4.a.s \(0\) \(-21\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1617.4.a.t 1617.a 1.a $7$ $95.406$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 1617.4.a.s \(0\) \(21\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(4+\beta _{2})q^{4}+(\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots\)
1617.4.a.u 1617.a 1.a $8$ $95.406$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 231.4.i.a \(-4\) \(-24\) \(20\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(3+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1617.4.a.v 1617.a 1.a $8$ $95.406$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 231.4.i.a \(-4\) \(24\) \(-20\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(3+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1617.4.a.w 1617.a 1.a $10$ $95.406$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 231.4.i.c \(-4\) \(-30\) \(20\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1617.4.a.x 1617.a 1.a $10$ $95.406$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 231.4.i.c \(-4\) \(30\) \(-20\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1617.4.a.y 1617.a 1.a $10$ $95.406$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 231.4.i.b \(4\) \(-30\) \(-20\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1617.4.a.z 1617.a 1.a $10$ $95.406$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 231.4.i.b \(4\) \(30\) \(20\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1617.4.a.ba 1617.a 1.a $12$ $95.406$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 231.4.i.d \(4\) \(-36\) \(-20\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(7+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
1617.4.a.bb 1617.a 1.a $12$ $95.406$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 231.4.i.d \(4\) \(36\) \(20\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(7+\beta _{2})q^{4}+(2-\beta _{1}+\cdots)q^{5}+\cdots\)
1617.4.a.bc 1617.a 1.a $16$ $95.406$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 1617.4.a.bc \(-4\) \(-48\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(5+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1617.4.a.bd 1617.a 1.a $16$ $95.406$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 1617.4.a.bc \(-4\) \(48\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(5+\beta _{2})q^{4}+(\beta _{1}-\beta _{5}+\cdots)q^{5}+\cdots\)
1617.4.a.be 1617.a 1.a $16$ $95.406$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 1617.4.a.be \(4\) \(-48\) \(-40\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(5+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
1617.4.a.bf 1617.a 1.a $16$ $95.406$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 1617.4.a.be \(4\) \(48\) \(40\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(5+\beta _{2})q^{4}+(2+\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1617))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1617)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(231))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(539))\)\(^{\oplus 2}\)