Properties

Label 350.2.j
Level 350
Weight 2
Character orbit j
Rep. character \(\chi_{350}(149,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 24
Newform subspaces 6
Sturm bound 120
Trace bound 11

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Defining parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 350.j (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 35 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 6 \)
Sturm bound: \(120\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(350, [\chi])\).

Total New Old
Modular forms 144 24 120
Cusp forms 96 24 72
Eisenstein series 48 0 48

Trace form

\( 24q + 12q^{4} + 16q^{6} + 8q^{9} + O(q^{10}) \) \( 24q + 12q^{4} + 16q^{6} + 8q^{9} + 8q^{11} + 4q^{14} - 12q^{16} + 4q^{19} - 20q^{21} + 8q^{24} - 16q^{26} - 24q^{29} - 12q^{31} - 16q^{34} + 16q^{36} - 32q^{39} - 8q^{41} - 8q^{44} - 24q^{46} + 48q^{49} + 12q^{51} + 20q^{54} - 4q^{56} + 4q^{59} + 16q^{61} - 24q^{64} - 8q^{71} + 12q^{74} + 8q^{76} - 28q^{79} + 28q^{81} - 4q^{84} + 16q^{86} + 16q^{89} - 4q^{91} - 4q^{94} - 8q^{96} - 48q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(350, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
350.2.j.a \(4\) \(2.795\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{2}+(-2\zeta_{12}+2\zeta_{12}^{3})q^{3}+\cdots\)
350.2.j.b \(4\) \(2.795\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{2}+(-\zeta_{12}+\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
350.2.j.c \(4\) \(2.795\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(3\zeta_{12}-2\zeta_{12}^{3})q^{7}+\cdots\)
350.2.j.d \(4\) \(2.795\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{2}+(2\zeta_{12}-2\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
350.2.j.e \(4\) \(2.795\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{2}+(2\zeta_{12}-2\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
350.2.j.f \(4\) \(2.795\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{2}+(3\zeta_{12}-3\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(350, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(350, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))
$3$ (\( 1 + 2 T^{2} - 5 T^{4} + 18 T^{6} + 81 T^{8} \))(\( 1 + 5 T^{2} + 16 T^{4} + 45 T^{6} + 81 T^{8} \))(\( ( 1 + 3 T^{2} + 9 T^{4} )^{2} \))(\( 1 + 2 T^{2} - 5 T^{4} + 18 T^{6} + 81 T^{8} \))(\( 1 + 2 T^{2} - 5 T^{4} + 18 T^{6} + 81 T^{8} \))(\( ( 1 - 3 T^{2} )^{2}( 1 + 3 T^{2} + 9 T^{4} ) \))
$5$ 1
$7$ (\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))
$11$ (\( ( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \))
$13$ (\( ( 1 - T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{4} \))(\( ( 1 - 25 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 13 T^{2} )^{4} \))
$17$ (\( 1 - 2 T^{2} - 285 T^{4} - 578 T^{6} + 83521 T^{8} \))(\( ( 1 + 17 T^{2} + 289 T^{4} )^{2} \))(\( 1 - 15 T^{2} - 64 T^{4} - 4335 T^{6} + 83521 T^{8} \))(\( 1 - 2 T^{2} - 285 T^{4} - 578 T^{6} + 83521 T^{8} \))(\( 1 + 25 T^{2} + 336 T^{4} + 7225 T^{6} + 83521 T^{8} \))(\( 1 + 18 T^{2} + 35 T^{4} + 5202 T^{6} + 83521 T^{8} \))
$19$ (\( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 19 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 7 T + 19 T^{2} )^{2}( 1 - T + 19 T^{2} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8} \))(\( 1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8} \))(\( 1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8} \))(\( 1 - 35 T^{2} + 696 T^{4} - 18515 T^{6} + 279841 T^{8} \))(\( 1 - 35 T^{2} + 696 T^{4} - 18515 T^{6} + 279841 T^{8} \))(\( 1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 - 6 T + 29 T^{2} )^{4} \))(\( ( 1 - 3 T + 29 T^{2} )^{4} \))(\( ( 1 + 6 T + 29 T^{2} )^{4} \))(\( ( 1 + 6 T + 29 T^{2} )^{4} \))(\( ( 1 - 6 T + 29 T^{2} )^{4} \))(\( ( 1 + 9 T + 29 T^{2} )^{4} \))
$31$ (\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))
$37$ (\( ( 1 - 73 T^{2} + 1369 T^{4} )( 1 + 26 T^{2} + 1369 T^{4} ) \))(\( 1 + 58 T^{2} + 1995 T^{4} + 79402 T^{6} + 1874161 T^{8} \))(\( 1 + 58 T^{2} + 1995 T^{4} + 79402 T^{6} + 1874161 T^{8} \))(\( 1 + 25 T^{2} - 744 T^{4} + 34225 T^{6} + 1874161 T^{8} \))(\( 1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8} \))(\( 1 + 58 T^{2} + 1995 T^{4} + 79402 T^{6} + 1874161 T^{8} \))
$41$ (\( ( 1 - 3 T + 41 T^{2} )^{4} \))(\( ( 1 - 9 T + 41 T^{2} )^{4} \))(\( ( 1 + 7 T + 41 T^{2} )^{4} \))(\( ( 1 - 3 T + 41 T^{2} )^{4} \))(\( ( 1 + 3 T + 41 T^{2} )^{4} \))(\( ( 1 + 7 T + 41 T^{2} )^{4} \))
$43$ (\( ( 1 + 14 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 82 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 61 T^{2} + 1849 T^{4} )^{2} \))
$47$ (\( 1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8} \))(\( ( 1 + 47 T^{2} + 2209 T^{4} )^{2} \))(\( 1 + 45 T^{2} - 184 T^{4} + 99405 T^{6} + 4879681 T^{8} \))(\( 1 + 13 T^{2} - 2040 T^{4} + 28717 T^{6} + 4879681 T^{8} \))(\( 1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8} \))(\( 1 + 30 T^{2} - 1309 T^{4} + 66270 T^{6} + 4879681 T^{8} \))
$53$ (\( 1 + 97 T^{2} + 6600 T^{4} + 272473 T^{6} + 7890481 T^{8} \))(\( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} \))(\( ( 1 - 14 T + 143 T^{2} - 742 T^{3} + 2809 T^{4} )( 1 + 14 T + 143 T^{2} + 742 T^{3} + 2809 T^{4} ) \))(\( 1 + 25 T^{2} - 2184 T^{4} + 70225 T^{6} + 7890481 T^{8} \))(\( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} \))(\( 1 + 102 T^{2} + 7595 T^{4} + 286518 T^{6} + 7890481 T^{8} \))
$59$ (\( ( 1 - 59 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 14 T + 137 T^{2} + 826 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 59 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} - 590 T^{3} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 13 T + 61 T^{2} )^{2}( 1 - T + 61 T^{2} )^{2} \))(\( ( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 13 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \))
$67$ (\( 1 + 118 T^{2} + 9435 T^{4} + 529702 T^{6} + 20151121 T^{8} \))(\( ( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \))(\( 1 - 10 T^{2} - 4389 T^{4} - 44890 T^{6} + 20151121 T^{8} \))(\( 1 + 70 T^{2} + 411 T^{4} + 314230 T^{6} + 20151121 T^{8} \))(\( 1 + 130 T^{2} + 12411 T^{4} + 583570 T^{6} + 20151121 T^{8} \))(\( 1 + 53 T^{2} - 1680 T^{4} + 237917 T^{6} + 20151121 T^{8} \))
$71$ (\( ( 1 - 12 T + 71 T^{2} )^{4} \))(\( ( 1 + 6 T + 71 T^{2} )^{4} \))(\( ( 1 + T + 71 T^{2} )^{4} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 + 9 T + 71 T^{2} )^{4} \))(\( ( 1 - 2 T + 71 T^{2} )^{4} \))
$73$ (\( 1 + 130 T^{2} + 11571 T^{4} + 692770 T^{6} + 28398241 T^{8} \))(\( ( 1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4} )( 1 + 6 T - 37 T^{2} + 438 T^{3} + 5329 T^{4} ) \))(\( 1 - 50 T^{2} - 2829 T^{4} - 266450 T^{6} + 28398241 T^{8} \))(\( 1 + 130 T^{2} + 11571 T^{4} + 692770 T^{6} + 28398241 T^{8} \))(\( ( 1 - 97 T^{2} + 5329 T^{4} )( 1 + 143 T^{2} + 5329 T^{4} ) \))(\( 1 + 130 T^{2} + 11571 T^{4} + 692770 T^{6} + 28398241 T^{8} \))
$79$ (\( ( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 2 T - 75 T^{2} - 158 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 5 T - 54 T^{2} - 395 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( ( 1 - 22 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 157 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 30 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( ( 1 - 130 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 117 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 7 T - 40 T^{2} - 623 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - T - 88 T^{2} - 89 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 + 2 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 145 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 94 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 169 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 9409 T^{4} )^{2} \))
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