Properties

Label 350.2.g
Level 350
Weight 2
Character orbit g
Rep. character \(\chi_{350}(293,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 24
Newform subspaces 2
Sturm bound 120
Trace bound 1

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Defining parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 350.g (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 35 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(120\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

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

\( 24q + 8q^{7} + O(q^{10}) \) \( 24q + 8q^{7} - 24q^{16} - 8q^{18} + 8q^{21} + 16q^{22} + 8q^{23} - 8q^{28} - 56q^{36} - 32q^{37} - 32q^{43} + 32q^{46} + 64q^{51} + 32q^{53} + 16q^{56} + 8q^{57} + 16q^{58} - 16q^{67} + 32q^{71} - 8q^{72} + 16q^{77} + 40q^{78} - 88q^{81} - 16q^{88} - 112q^{91} + 8q^{92} + 16q^{93} - 32q^{98} + O(q^{100}) \)

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.g.a \(8\) \(2.795\) \(\Q(\zeta_{16})\) None \(0\) \(0\) \(0\) \(8\) \(q+\zeta_{16}q^{2}+\zeta_{16}^{2}q^{3}+\zeta_{16}^{3}q^{4}+(\zeta_{16}^{4}+\cdots)q^{6}+\cdots\)
350.2.g.b \(16\) \(2.795\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{2}+\beta _{3}q^{3}+\beta _{10}q^{4}+\beta _{15}q^{6}+\cdots\)

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

$p$ $F_p(T)$
$2$ (\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{4} \))
$3$ (\( ( 1 - 8 T^{2} + 32 T^{4} - 72 T^{6} + 81 T^{8} )( 1 + 8 T^{2} + 32 T^{4} + 72 T^{6} + 81 T^{8} ) \))(\( ( 1 - 10 T^{4} + 139 T^{8} - 810 T^{12} + 6561 T^{16} )^{2} \))
$5$ 1
$7$ (\( 1 - 8 T + 32 T^{2} - 88 T^{3} + 226 T^{4} - 616 T^{5} + 1568 T^{6} - 2744 T^{7} + 2401 T^{8} \))(\( 1 + 28 T^{4} + 3270 T^{8} + 67228 T^{12} + 5764801 T^{16} \))
$11$ (\( ( 1 + 14 T^{2} + 121 T^{4} )^{4} \))(\( ( 1 + 19 T^{2} + 121 T^{4} )^{8} \))
$13$ (\( ( 1 - 240 T^{4} + 28561 T^{8} )^{2} \))(\( ( 1 - 260 T^{4} + 30822 T^{8} - 7425860 T^{12} + 815730721 T^{16} )^{2} \))
$17$ (\( 1 - 252 T^{4} + 17030 T^{8} - 21047292 T^{12} + 6975757441 T^{16} \))(\( ( 1 - 602 T^{4} + 184635 T^{8} - 50279642 T^{12} + 6975757441 T^{16} )^{2} \))
$19$ (\( ( 1 + 24 T^{2} + 288 T^{4} + 8664 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 + 34 T^{2} + 903 T^{4} + 12274 T^{6} + 130321 T^{8} )^{4} \))
$23$ (\( ( 1 - 4 T + 8 T^{2} - 84 T^{3} + 878 T^{4} - 1932 T^{5} + 4232 T^{6} - 48668 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 + 28 T^{4} + 171078 T^{8} + 7835548 T^{12} + 78310985281 T^{16} )^{2} \))
$29$ (\( ( 1 - 92 T^{2} + 3670 T^{4} - 77372 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 92 T^{2} + 3690 T^{4} - 77372 T^{6} + 707281 T^{8} )^{4} \))
$31$ (\( ( 1 - 108 T^{2} + 4806 T^{4} - 103788 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 28 T^{2} + 1146 T^{4} - 26908 T^{6} + 923521 T^{8} )^{4} \))
$37$ (\( ( 1 + 16 T + 128 T^{2} + 1040 T^{3} + 7666 T^{4} + 38480 T^{5} + 175232 T^{6} + 810448 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 - 644 T^{4} - 1762266 T^{8} - 1206959684 T^{12} + 3512479453921 T^{16} )^{2} \))
$41$ (\( ( 1 - 148 T^{2} + 8806 T^{4} - 248788 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 38 T^{2} + 2751 T^{4} - 63878 T^{6} + 2825761 T^{8} )^{4} \))
$43$ (\( ( 1 + 8 T + 32 T^{2} + 344 T^{3} + 1849 T^{4} )^{4} \))(\( ( 1 + 1778 T^{4} + 3418801 T^{8} )^{4} \))
$47$ (\( 1 + 6020 T^{4} + 17872774 T^{8} + 29375679620 T^{12} + 23811286661761 T^{16} \))(\( ( 1 - 2660 T^{4} + 3301254 T^{8} - 12979951460 T^{12} + 23811286661761 T^{16} )^{2} \))
$53$ (\( ( 1 - 16 T + 128 T^{2} - 784 T^{3} + 4786 T^{4} - 41552 T^{5} + 359552 T^{6} - 2382032 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 + 6268 T^{4} + 25462950 T^{8} + 49457534908 T^{12} + 62259690411361 T^{16} )^{2} \))
$59$ (\( ( 1 + 136 T^{2} + 10336 T^{4} + 473416 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 59 T^{2} )^{16} \))
$61$ (\( ( 1 - 96 T^{2} + 8688 T^{4} - 357216 T^{6} + 13845841 T^{8} )^{2} \))(\( ( 1 - 76 T^{2} + 7158 T^{4} - 282796 T^{6} + 13845841 T^{8} )^{4} \))
$67$ (\( ( 1 + 8 T + 32 T^{2} - 552 T^{3} - 8974 T^{4} - 36984 T^{5} + 143648 T^{6} + 2406104 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 - 2642 T^{4} + 1066035 T^{8} - 53239261682 T^{12} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 + 4 T + 144 T^{2} + 284 T^{3} + 5041 T^{4} )^{4} \))(\( ( 1 - 6 T + 124 T^{2} - 426 T^{3} + 5041 T^{4} )^{8} \))
$73$ (\( 1 + 1220 T^{4} + 55730374 T^{8} + 34645854020 T^{12} + 806460091894081 T^{16} \))(\( ( 1 - 11930 T^{4} + 77910459 T^{8} - 338791015130 T^{12} + 806460091894081 T^{16} )^{2} \))
$79$ (\( ( 1 - 208 T^{2} + 22498 T^{4} - 1298128 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 154 T^{2} + 6241 T^{4} )^{8} \))
$83$ (\( 1 - 4224 T^{4} + 99283874 T^{8} - 200463947904 T^{12} + 2252292232139041 T^{16} \))(\( ( 1 - 4394 T^{4} + 58495659 T^{8} - 208531862474 T^{12} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( ( 1 - 60 T^{2} + 10470 T^{4} - 475260 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 + 230 T^{2} + 28095 T^{4} + 1821830 T^{6} + 62742241 T^{8} )^{4} \))
$97$ (\( 1 - 11900 T^{4} + 108781062 T^{8} - 1053498443900 T^{12} + 7837433594376961 T^{16} \))(\( ( 1 - 380 T^{4} + 60281862 T^{8} - 33641126780 T^{12} + 7837433594376961 T^{16} )^{2} \))
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