Properties

Label 1694.4.a
Level $1694$
Weight $4$
Character orbit 1694.a
Rep. character $\chi_{1694}(1,\cdot)$
Character field $\Q$
Dimension $164$
Newform subspaces $39$
Sturm bound $1056$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1694 = 2 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1694.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 39 \)
Sturm bound: \(1056\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1694))\).

Total New Old
Modular forms 816 164 652
Cusp forms 768 164 604
Eisenstein series 48 0 48

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

\(2\)\(7\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(23\)
\(+\)\(+\)\(-\)\(-\)\(19\)
\(+\)\(-\)\(+\)\(-\)\(17\)
\(+\)\(-\)\(-\)\(+\)\(24\)
\(-\)\(+\)\(+\)\(-\)\(19\)
\(-\)\(+\)\(-\)\(+\)\(21\)
\(-\)\(-\)\(+\)\(+\)\(25\)
\(-\)\(-\)\(-\)\(-\)\(16\)
Plus space\(+\)\(93\)
Minus space\(-\)\(71\)

Trace form

\( 164 q - 4 q^{2} - 6 q^{3} + 656 q^{4} + 6 q^{5} + 20 q^{6} - 16 q^{8} + 1516 q^{9} + O(q^{10}) \) \( 164 q - 4 q^{2} - 6 q^{3} + 656 q^{4} + 6 q^{5} + 20 q^{6} - 16 q^{8} + 1516 q^{9} + 4 q^{10} - 24 q^{12} - 150 q^{13} + 28 q^{14} + 176 q^{15} + 2624 q^{16} + 100 q^{17} - 44 q^{18} - 2 q^{19} + 24 q^{20} + 70 q^{21} + 288 q^{23} + 80 q^{24} + 3876 q^{25} - 4 q^{26} - 180 q^{27} - 276 q^{29} - 544 q^{30} - 308 q^{31} - 64 q^{32} - 304 q^{34} - 126 q^{35} + 6064 q^{36} + 112 q^{37} + 428 q^{38} + 696 q^{39} + 16 q^{40} + 908 q^{41} + 84 q^{42} + 1068 q^{43} + 1742 q^{45} + 336 q^{46} + 1092 q^{47} - 96 q^{48} + 8036 q^{49} - 1348 q^{50} + 1204 q^{51} - 600 q^{52} + 36 q^{53} + 1160 q^{54} + 112 q^{56} + 1604 q^{57} - 664 q^{58} - 154 q^{59} + 704 q^{60} - 730 q^{61} + 360 q^{62} - 420 q^{63} + 10496 q^{64} - 196 q^{65} + 2508 q^{67} + 400 q^{68} + 72 q^{69} + 644 q^{70} - 200 q^{71} - 176 q^{72} + 144 q^{73} - 1712 q^{74} - 258 q^{75} - 8 q^{76} - 1344 q^{78} + 1536 q^{79} + 96 q^{80} + 13632 q^{81} + 3296 q^{82} + 4358 q^{83} + 280 q^{84} + 3284 q^{85} + 944 q^{86} + 1308 q^{87} - 4008 q^{89} + 868 q^{90} - 742 q^{91} + 1152 q^{92} + 1216 q^{93} + 1592 q^{94} - 88 q^{95} + 320 q^{96} + 2644 q^{97} - 196 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1694))\) 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 7 11
1694.4.a.a 1694.a 1.a $1$ $99.949$ \(\Q\) None \(-2\) \(-10\) \(-14\) \(-7\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-10q^{3}+4q^{4}-14q^{5}+20q^{6}+\cdots\)
1694.4.a.b 1694.a 1.a $1$ $99.949$ \(\Q\) None \(-2\) \(-2\) \(-12\) \(-7\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-2q^{3}+4q^{4}-12q^{5}+4q^{6}+\cdots\)
1694.4.a.c 1694.a 1.a $1$ $99.949$ \(\Q\) None \(-2\) \(-2\) \(18\) \(-7\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-2q^{3}+4q^{4}+18q^{5}+4q^{6}+\cdots\)
1694.4.a.d 1694.a 1.a $1$ $99.949$ \(\Q\) None \(-2\) \(7\) \(3\) \(-7\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+7q^{3}+4q^{4}+3q^{5}-14q^{6}+\cdots\)
1694.4.a.e 1694.a 1.a $1$ $99.949$ \(\Q\) None \(2\) \(-5\) \(-1\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-5q^{3}+4q^{4}-q^{5}-10q^{6}+\cdots\)
1694.4.a.f 1694.a 1.a $1$ $99.949$ \(\Q\) None \(2\) \(0\) \(2\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+2q^{5}+7q^{7}+8q^{8}+\cdots\)
1694.4.a.g 1694.a 1.a $1$ $99.949$ \(\Q\) None \(2\) \(8\) \(-14\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+8q^{3}+4q^{4}-14q^{5}+2^{4}q^{6}+\cdots\)
1694.4.a.h 1694.a 1.a $2$ $99.949$ \(\Q(\sqrt{57}) \) None \(-4\) \(-5\) \(-17\) \(14\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-2-\beta )q^{3}+4q^{4}+(-10+\cdots)q^{5}+\cdots\)
1694.4.a.i 1694.a 1.a $2$ $99.949$ \(\Q(\sqrt{133}) \) None \(-4\) \(-5\) \(-2\) \(-14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-2-\beta )q^{3}+4q^{4}-q^{5}+\cdots\)
1694.4.a.j 1694.a 1.a $2$ $99.949$ \(\Q(\sqrt{5}) \) None \(-4\) \(-4\) \(-13\) \(-14\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1-6\beta )q^{3}+4q^{4}+(-12+\cdots)q^{5}+\cdots\)
1694.4.a.k 1694.a 1.a $2$ $99.949$ \(\Q(\sqrt{5}) \) None \(-4\) \(6\) \(2\) \(-14\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(2+2\beta )q^{3}+4q^{4}+(6-10\beta )q^{5}+\cdots\)
1694.4.a.l 1694.a 1.a $2$ $99.949$ \(\Q(\sqrt{137}) \) None \(4\) \(-5\) \(-7\) \(-14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-2-\beta )q^{3}+4q^{4}+(-4+\cdots)q^{5}+\cdots\)
1694.4.a.m 1694.a 1.a $2$ $99.949$ \(\Q(\sqrt{133}) \) None \(4\) \(-5\) \(-2\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-2-\beta )q^{3}+4q^{4}-q^{5}+\cdots\)
1694.4.a.n 1694.a 1.a $2$ $99.949$ \(\Q(\sqrt{5}) \) None \(4\) \(-4\) \(-13\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1-6\beta )q^{3}+4q^{4}+(-12+\cdots)q^{5}+\cdots\)
1694.4.a.o 1694.a 1.a $2$ $99.949$ \(\Q(\sqrt{5}) \) None \(4\) \(6\) \(2\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(2+2\beta )q^{3}+4q^{4}+(6-10\beta )q^{5}+\cdots\)
1694.4.a.p 1694.a 1.a $2$ $99.949$ \(\Q(\sqrt{37}) \) None \(4\) \(6\) \(26\) \(-14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(3+\beta )q^{3}+4q^{4}+(13-\beta )q^{5}+\cdots\)
1694.4.a.q 1694.a 1.a $3$ $99.949$ 3.3.27093.1 None \(-6\) \(-5\) \(-5\) \(-21\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-2+\beta _{1})q^{3}+4q^{4}+(-1+\cdots)q^{5}+\cdots\)
1694.4.a.r 1694.a 1.a $3$ $99.949$ 3.3.46700.1 None \(-6\) \(-5\) \(5\) \(21\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-2+\beta _{1})q^{3}+4q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
1694.4.a.s 1694.a 1.a $3$ $99.949$ 3.3.7636.1 None \(-6\) \(6\) \(26\) \(21\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(2-\beta _{1})q^{3}+4q^{4}+(9-\beta _{2})q^{5}+\cdots\)
1694.4.a.t 1694.a 1.a $3$ $99.949$ 3.3.27093.1 None \(6\) \(-5\) \(-5\) \(21\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-2+\beta _{1})q^{3}+4q^{4}+(-1+\cdots)q^{5}+\cdots\)
1694.4.a.u 1694.a 1.a $3$ $99.949$ 3.3.46700.1 None \(6\) \(-5\) \(5\) \(-21\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-2+\beta _{1})q^{3}+4q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
1694.4.a.v 1694.a 1.a $4$ $99.949$ 4.4.4388525.1 None \(-8\) \(-10\) \(2\) \(-28\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1+3\beta _{2})q^{3}+4q^{4}+(3+\cdots)q^{5}+\cdots\)
1694.4.a.w 1694.a 1.a $4$ $99.949$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(1\) \(4\) \(28\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta _{1}q^{3}+4q^{4}+(1+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
1694.4.a.x 1694.a 1.a $4$ $99.949$ 4.4.4388525.1 None \(8\) \(-10\) \(2\) \(28\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1+3\beta _{2})q^{3}+4q^{4}+(2+\cdots)q^{5}+\cdots\)
1694.4.a.y 1694.a 1.a $4$ $99.949$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(1\) \(4\) \(-28\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+\beta _{1}q^{3}+4q^{4}+(1+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
1694.4.a.z 1694.a 1.a $5$ $99.949$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-10\) \(1\) \(9\) \(35\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta _{1}q^{3}+4q^{4}+(2-\beta _{2})q^{5}+\cdots\)
1694.4.a.ba 1694.a 1.a $5$ $99.949$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-10\) \(1\) \(15\) \(-35\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta _{1}q^{3}+4q^{4}+(3-\beta _{2})q^{5}+\cdots\)
1694.4.a.bb 1694.a 1.a $5$ $99.949$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(10\) \(1\) \(9\) \(-35\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+\beta _{1}q^{3}+4q^{4}+(2-\beta _{2})q^{5}+\cdots\)
1694.4.a.bc 1694.a 1.a $5$ $99.949$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(10\) \(1\) \(15\) \(35\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+\beta _{1}q^{3}+4q^{4}+(3-\beta _{2})q^{5}+\cdots\)
1694.4.a.bd 1694.a 1.a $6$ $99.949$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-12\) \(-10\) \(-8\) \(42\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-2+\beta _{1})q^{3}+4q^{4}+(-1+\cdots)q^{5}+\cdots\)
1694.4.a.be 1694.a 1.a $6$ $99.949$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(12\) \(-10\) \(-8\) \(-42\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-2+\beta _{1})q^{3}+4q^{4}+(-1+\cdots)q^{5}+\cdots\)
1694.4.a.bf 1694.a 1.a $8$ $99.949$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-16\) \(12\) \(-7\) \(56\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1-\beta _{2})q^{3}+4q^{4}+(-1+\beta _{2}+\cdots)q^{5}+\cdots\)
1694.4.a.bg 1694.a 1.a $8$ $99.949$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(16\) \(12\) \(-7\) \(-56\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1-\beta _{2})q^{3}+4q^{4}+(-1+\beta _{2}+\cdots)q^{5}+\cdots\)
1694.4.a.bh 1694.a 1.a $10$ $99.949$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-20\) \(-3\) \(-27\) \(70\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta _{1}q^{3}+4q^{4}+(-3-\beta _{4}+\cdots)q^{5}+\cdots\)
1694.4.a.bi 1694.a 1.a $10$ $99.949$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-20\) \(2\) \(12\) \(-70\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta _{1}q^{3}+4q^{4}+(1+2\beta _{2}-\beta _{7}+\cdots)q^{5}+\cdots\)
1694.4.a.bj 1694.a 1.a $10$ $99.949$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-20\) \(17\) \(11\) \(-70\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(2-\beta _{1})q^{3}+4q^{4}+(1-\beta _{5}+\cdots)q^{5}+\cdots\)
1694.4.a.bk 1694.a 1.a $10$ $99.949$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(20\) \(-3\) \(-27\) \(-70\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+\beta _{1}q^{3}+4q^{4}+(-3-\beta _{4}+\cdots)q^{5}+\cdots\)
1694.4.a.bl 1694.a 1.a $10$ $99.949$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(20\) \(2\) \(12\) \(70\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+\beta _{1}q^{3}+4q^{4}+(1+2\beta _{2}-\beta _{7}+\cdots)q^{5}+\cdots\)
1694.4.a.bm 1694.a 1.a $10$ $99.949$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(20\) \(17\) \(11\) \(70\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(2-\beta _{1})q^{3}+4q^{4}+(1-\beta _{5}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1694))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1694)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(847))\)\(^{\oplus 2}\)