Properties

Label 672.2.i
Level 672
Weight 2
Character orbit i
Rep. character \(\chi_{672}(209,\cdot)\)
Character field \(\Q\)
Dimension 28
Newform subspaces 5
Sturm bound 256
Trace bound 7

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Defining parameters

Level: \( N \) = \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 672.i (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 168 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(256\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(672, [\chi])\).

Total New Old
Modular forms 144 36 108
Cusp forms 112 28 84
Eisenstein series 32 8 24

Trace form

\( 28q + 4q^{7} - 4q^{9} + O(q^{10}) \) \( 28q + 4q^{7} - 4q^{9} - 8q^{15} - 20q^{25} + 16q^{39} + 4q^{49} - 16q^{57} + 36q^{63} - 24q^{79} + 12q^{81} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(672, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
672.2.i.a \(4\) \(5.366\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(-4\) \(0\) \(8\) \(q+(-1-\beta _{1})q^{3}+\beta _{1}q^{5}+(2-\beta _{2})q^{7}+\cdots\)
672.2.i.b \(4\) \(5.366\) \(\Q(\sqrt{2}, \sqrt{-3})\) \(\Q(\sqrt{-6}) \) \(0\) \(0\) \(0\) \(-4\) \(q+\beta _{2}q^{3}+2\beta _{2}q^{5}+(-1+\beta _{3})q^{7}+\cdots\)
672.2.i.c \(4\) \(5.366\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(4\) \(0\) \(8\) \(q+(1-\beta _{1})q^{3}+\beta _{1}q^{5}+(2-\beta _{2})q^{7}+\cdots\)
672.2.i.d \(8\) \(5.366\) 8.0.3317760000.1 None \(0\) \(0\) \(0\) \(-8\) \(q+(-\beta _{2}-\beta _{3})q^{3}-\beta _{3}q^{5}+(-1-\beta _{6}+\cdots)q^{7}+\cdots\)
672.2.i.e \(8\) \(5.366\) 8.0.\(\cdots\).11 \(\Q(\sqrt{-14}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}+(\beta _{6}-\beta _{7})q^{5}+\beta _{3}q^{7}+(\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(672, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(672, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 3}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 + 2 T + 3 T^{2} )^{2} \))(\( ( 1 + 3 T^{2} )^{2} \))(\( ( 1 - 2 T + 3 T^{2} )^{2} \))(\( ( 1 - 4 T^{2} + 9 T^{4} )^{2} \))(\( 1 - 10 T^{4} + 81 T^{8} \))
$5$ (\( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 25 T^{4} )^{4} \))(\( ( 1 + 22 T^{4} + 625 T^{8} )^{2} \))
$7$ (\( ( 1 - 4 T + 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 4 T + 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{4} \))(\( ( 1 - 7 T^{2} )^{4} \))
$11$ (\( ( 1 + 16 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 16 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 121 T^{4} )^{4} \))(\( ( 1 + 11 T^{2} )^{8} \))
$13$ (\( ( 1 + 2 T + 13 T^{2} )^{4} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 - 2 T + 13 T^{2} )^{4} \))(\( ( 1 + 16 T^{2} + 169 T^{4} )^{4} \))(\( ( 1 + 310 T^{4} + 28561 T^{8} )^{2} \))
$17$ (\( ( 1 - 20 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 17 T^{2} )^{4} \))(\( ( 1 - 20 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 17 T^{2} )^{8} \))(\( ( 1 + 17 T^{2} )^{8} \))
$19$ (\( ( 1 + 4 T + 19 T^{2} )^{4} \))(\( ( 1 + 19 T^{2} )^{4} \))(\( ( 1 - 4 T + 19 T^{2} )^{4} \))(\( ( 1 + 28 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 - 650 T^{4} + 130321 T^{8} )^{2} \))
$23$ (\( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 23 T^{2} )^{4} \))(\( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 529 T^{4} )^{4} \))(\( ( 1 - 6 T + 23 T^{2} )^{4}( 1 + 6 T + 23 T^{2} )^{4} \))
$29$ (\( ( 1 + 34 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 50 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 34 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 + 29 T^{2} )^{8} \))
$31$ (\( ( 1 - 14 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 10 T + 31 T^{2} )^{2}( 1 + 10 T + 31 T^{2} )^{2} \))(\( ( 1 - 14 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 10 T + 31 T^{2} )^{4}( 1 + 10 T + 31 T^{2} )^{4} \))(\( ( 1 - 31 T^{2} )^{8} \))
$37$ (\( ( 1 + 34 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} )^{4} \))(\( ( 1 + 34 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} )^{8} \))(\( ( 1 - 37 T^{2} )^{8} \))
$41$ (\( ( 1 + 76 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 76 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 + 41 T^{2} )^{8} \))
$43$ (\( ( 1 - 74 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 43 T^{2} )^{4} \))(\( ( 1 - 74 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 1849 T^{4} )^{4} \))(\( ( 1 - 43 T^{2} )^{8} \))
$47$ (\( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 2209 T^{4} )^{4} \))(\( ( 1 + 47 T^{2} )^{8} \))
$53$ (\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 - 94 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 + 53 T^{2} )^{8} \))(\( ( 1 + 53 T^{2} )^{8} \))
$59$ (\( ( 1 - 86 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 86 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 3481 T^{4} )^{4} \))(\( ( 1 - 1130 T^{4} + 12117361 T^{8} )^{2} \))
$61$ (\( ( 1 - 10 T + 61 T^{2} )^{4} \))(\( ( 1 + 61 T^{2} )^{4} \))(\( ( 1 + 10 T + 61 T^{2} )^{4} \))(\( ( 1 + 112 T^{2} + 3721 T^{4} )^{4} \))(\( ( 1 - 7370 T^{4} + 13845841 T^{8} )^{2} \))
$67$ (\( ( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2} \))(\( ( 1 - 67 T^{2} )^{4} \))(\( ( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2} \))(\( ( 1 - 74 T^{2} + 4489 T^{4} )^{4} \))(\( ( 1 - 67 T^{2} )^{8} \))
$71$ (\( ( 1 - 140 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 140 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 62 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 - 110 T^{2} + 5041 T^{4} )^{4} \))
$73$ (\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 + 70 T^{2} + 5329 T^{4} )^{4} \))(\( ( 1 - 73 T^{2} )^{8} \))
$79$ (\( ( 1 + 8 T + 79 T^{2} )^{4} \))(\( ( 1 + 10 T + 79 T^{2} )^{4} \))(\( ( 1 + 8 T + 79 T^{2} )^{4} \))(\( ( 1 - 10 T + 79 T^{2} )^{8} \))(\( ( 1 + 130 T^{2} + 6241 T^{4} )^{4} \))
$83$ (\( ( 1 - 38 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 134 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 116 T^{2} + 6889 T^{4} )^{4} \))(\( ( 1 - 13130 T^{4} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 + 124 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 124 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 89 T^{2} )^{8} \))(\( ( 1 + 89 T^{2} )^{8} \))
$97$ (\( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2}( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 170 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 - 97 T^{2} )^{8} \))
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