Properties

Label 70.2.e
Level 70
Weight 2
Character orbit e
Rep. character \(\chi_{70}(11,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 8
Newform subspaces 4
Sturm bound 24
Trace bound 3

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 70.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(24\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(70, [\chi])\).

Total New Old
Modular forms 32 8 24
Cusp forms 16 8 8
Eisenstein series 16 0 16

Trace form

\( 8q - 4q^{4} + 2q^{5} + 4q^{6} - 8q^{7} - 6q^{9} + O(q^{10}) \) \( 8q - 4q^{4} + 2q^{5} + 4q^{6} - 8q^{7} - 6q^{9} + 2q^{10} + 2q^{11} + 2q^{14} - 8q^{15} - 4q^{16} + 4q^{17} - 8q^{18} + 6q^{19} - 4q^{20} - 18q^{21} + 8q^{22} - 12q^{23} - 2q^{24} - 4q^{25} - 10q^{26} + 24q^{27} + 4q^{28} + 12q^{29} - 2q^{30} - 8q^{31} + 24q^{33} + 16q^{34} + 6q^{35} + 12q^{36} + 4q^{37} + 8q^{38} + 12q^{39} + 2q^{40} + 16q^{41} + 4q^{42} - 40q^{43} + 2q^{44} + 4q^{45} - 20q^{47} - 22q^{49} + 12q^{51} - 4q^{53} - 14q^{54} + 16q^{55} + 2q^{56} - 24q^{57} - 4q^{59} + 4q^{60} - 10q^{61} - 48q^{62} + 36q^{63} + 8q^{64} + 2q^{65} + 4q^{68} - 36q^{69} + 6q^{70} + 16q^{71} - 8q^{72} + 28q^{73} - 18q^{74} - 12q^{76} - 8q^{77} - 16q^{78} + 8q^{79} + 2q^{80} + 12q^{81} + 16q^{82} - 32q^{83} - 16q^{85} + 10q^{86} - 24q^{87} - 4q^{88} + 2q^{89} - 20q^{90} - 12q^{91} + 24q^{92} + 12q^{93} - 2q^{94} - 4q^{95} - 2q^{96} + 64q^{97} + 24q^{98} + 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(70, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
70.2.e.a \(2\) \(0.559\) \(\Q(\sqrt{-3}) \) None \(-1\) \(-3\) \(1\) \(1\) \(q-\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
70.2.e.b \(2\) \(0.559\) \(\Q(\sqrt{-3}) \) None \(-1\) \(2\) \(1\) \(-4\) \(q-\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
70.2.e.c \(2\) \(0.559\) \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(1\) \(-1\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
70.2.e.d \(2\) \(0.559\) \(\Q(\sqrt{-3}) \) None \(1\) \(2\) \(-1\) \(-4\) \(q+\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(70, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(70, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))
$3$ (\( ( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} \))
$5$ (\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))
$7$ (\( 1 - T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))
$11$ (\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))
$13$ (\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))
$17$ (\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))
$19$ (\( 1 - 6 T + 17 T^{2} - 114 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))
$23$ (\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \))
$29$ (\( ( 1 - 9 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 3 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))
$31$ (\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))
$37$ (\( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))
$41$ (\( ( 1 + 7 T + 41 T^{2} )^{2} \))(\( ( 1 - 3 T + 41 T^{2} )^{2} \))(\( ( 1 - 9 T + 41 T^{2} )^{2} \))(\( ( 1 - 3 T + 41 T^{2} )^{2} \))
$43$ (\( ( 1 + 5 T + 43 T^{2} )^{2} \))(\( ( 1 + 10 T + 43 T^{2} )^{2} \))(\( ( 1 + 7 T + 43 T^{2} )^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{2} \))
$47$ (\( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4} \))
$53$ (\( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} \))(\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))
$59$ (\( 1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))
$61$ (\( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))
$67$ (\( 1 - 9 T + 14 T^{2} - 603 T^{3} + 4489 T^{4} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))
$71$ (\( ( 1 - 2 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))
$73$ (\( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4} \))(\( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} \))(\( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} \))(\( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \))
$83$ (\( ( 1 + 7 T + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 3 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))
$89$ (\( 1 + T - 88 T^{2} + 89 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))
$97$ (\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))
show more
show less