# Properties

 Label 392.2.a Level 392 Weight 2 Character orbit a Rep. character $$\chi_{392}(1,\cdot)$$ Character field $$\Q$$ Dimension 10 Newform subspaces 8 Sturm bound 112 Trace bound 9

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$392 = 2^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 392.a (trivial) Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$112$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(392))$$.

Total New Old
Modular forms 72 10 62
Cusp forms 41 10 31
Eisenstein series 31 0 31

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

$$2$$$$7$$FrickeDim.
$$+$$$$+$$$$+$$$$1$$
$$+$$$$-$$$$-$$$$4$$
$$-$$$$+$$$$-$$$$3$$
$$-$$$$-$$$$+$$$$2$$
Plus space$$+$$$$3$$
Minus space$$-$$$$7$$

## Trace form

 $$10q - 2q^{3} + 2q^{5} + 14q^{9} + O(q^{10})$$ $$10q - 2q^{3} + 2q^{5} + 14q^{9} + 4q^{11} - 2q^{13} + 12q^{15} + 8q^{17} - 6q^{19} - 4q^{23} + 6q^{25} + 4q^{27} - 8q^{29} - 12q^{31} + 12q^{37} - 8q^{39} + 10q^{45} + 12q^{47} - 4q^{51} - 16q^{53} + 8q^{55} - 4q^{57} - 6q^{59} + 2q^{61} - 36q^{65} + 20q^{67} - 16q^{69} + 4q^{73} - 22q^{75} - 4q^{79} + 10q^{81} - 14q^{83} - 24q^{85} - 4q^{87} - 4q^{89} - 12q^{93} - 28q^{95} + 8q^{97} - 64q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(392))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 7
392.2.a.a $$1$$ $$3.130$$ $$\Q$$ None $$0$$ $$-3$$ $$1$$ $$0$$ $$-$$ $$-$$ $$q-3q^{3}+q^{5}+6q^{9}-q^{11}-2q^{13}+\cdots$$
392.2.a.b $$1$$ $$3.130$$ $$\Q$$ None $$0$$ $$-2$$ $$4$$ $$0$$ $$+$$ $$-$$ $$q-2q^{3}+4q^{5}+q^{9}-8q^{15}+2q^{17}+\cdots$$
392.2.a.c $$1$$ $$3.130$$ $$\Q$$ None $$0$$ $$-1$$ $$-1$$ $$0$$ $$+$$ $$+$$ $$q-q^{3}-q^{5}-2q^{9}+3q^{11}-6q^{13}+\cdots$$
392.2.a.d $$1$$ $$3.130$$ $$\Q$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$-$$ $$-$$ $$q-2q^{5}-3q^{9}-4q^{11}-2q^{13}+6q^{17}+\cdots$$
392.2.a.e $$1$$ $$3.130$$ $$\Q$$ None $$0$$ $$1$$ $$1$$ $$0$$ $$+$$ $$-$$ $$q+q^{3}+q^{5}-2q^{9}+3q^{11}+6q^{13}+\cdots$$
392.2.a.f $$1$$ $$3.130$$ $$\Q$$ None $$0$$ $$3$$ $$-1$$ $$0$$ $$-$$ $$+$$ $$q+3q^{3}-q^{5}+6q^{9}-q^{11}+2q^{13}+\cdots$$
392.2.a.g $$2$$ $$3.130$$ $$\Q(\sqrt{2})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$-$$ $$+$$ $$q+\beta q^{3}+2\beta q^{5}-q^{9}+6q^{11}-4\beta q^{13}+\cdots$$
392.2.a.h $$2$$ $$3.130$$ $$\Q(\sqrt{2})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$+$$ $$-$$ $$q+\beta q^{3}+\beta q^{5}+5q^{9}-4q^{11}-\beta q^{13}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(392))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(392)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(14))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(49))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(56))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(98))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(196))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + 3 T + 3 T^{2}$$)($$1 + 2 T + 3 T^{2}$$)($$1 + T + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 - T + 3 T^{2}$$)($$1 - 3 T + 3 T^{2}$$)($$1 + 4 T^{2} + 9 T^{4}$$)($$1 - 2 T^{2} + 9 T^{4}$$)
$5$ ($$1 - T + 5 T^{2}$$)($$1 - 4 T + 5 T^{2}$$)($$1 + T + 5 T^{2}$$)($$1 + 2 T + 5 T^{2}$$)($$1 - T + 5 T^{2}$$)($$1 + T + 5 T^{2}$$)($$1 + 2 T^{2} + 25 T^{4}$$)($$1 + 2 T^{2} + 25 T^{4}$$)
$7$ 1
$11$ ($$1 + T + 11 T^{2}$$)($$1 + 11 T^{2}$$)($$1 - 3 T + 11 T^{2}$$)($$1 + 4 T + 11 T^{2}$$)($$1 - 3 T + 11 T^{2}$$)($$1 + T + 11 T^{2}$$)($$( 1 - 6 T + 11 T^{2} )^{2}$$)($$( 1 + 4 T + 11 T^{2} )^{2}$$)
$13$ ($$1 + 2 T + 13 T^{2}$$)($$1 + 13 T^{2}$$)($$1 + 6 T + 13 T^{2}$$)($$1 + 2 T + 13 T^{2}$$)($$1 - 6 T + 13 T^{2}$$)($$1 - 2 T + 13 T^{2}$$)($$1 - 6 T^{2} + 169 T^{4}$$)($$1 + 18 T^{2} + 169 T^{4}$$)
$17$ ($$1 + 3 T + 17 T^{2}$$)($$1 - 2 T + 17 T^{2}$$)($$1 + 5 T + 17 T^{2}$$)($$1 - 6 T + 17 T^{2}$$)($$1 - 5 T + 17 T^{2}$$)($$1 - 3 T + 17 T^{2}$$)($$1 + 32 T^{2} + 289 T^{4}$$)($$1 + 2 T^{2} + 289 T^{4}$$)
$19$ ($$1 + 5 T + 19 T^{2}$$)($$1 - 2 T + 19 T^{2}$$)($$1 - T + 19 T^{2}$$)($$1 + 8 T + 19 T^{2}$$)($$1 + T + 19 T^{2}$$)($$1 - 5 T + 19 T^{2}$$)($$1 + 20 T^{2} + 361 T^{4}$$)($$1 + 30 T^{2} + 361 T^{4}$$)
$23$ ($$1 + 3 T + 23 T^{2}$$)($$1 - 8 T + 23 T^{2}$$)($$1 + 7 T + 23 T^{2}$$)($$1 + 23 T^{2}$$)($$1 + 7 T + 23 T^{2}$$)($$1 + 3 T + 23 T^{2}$$)($$( 1 - 4 T + 23 T^{2} )^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)
$29$ ($$1 + 6 T + 29 T^{2}$$)($$1 - 2 T + 29 T^{2}$$)($$1 - 2 T + 29 T^{2}$$)($$1 - 6 T + 29 T^{2}$$)($$1 - 2 T + 29 T^{2}$$)($$1 + 6 T + 29 T^{2}$$)($$( 1 + 6 T + 29 T^{2} )^{2}$$)($$( 1 - 2 T + 29 T^{2} )^{2}$$)
$31$ ($$1 - T + 31 T^{2}$$)($$1 + 4 T + 31 T^{2}$$)($$1 + 5 T + 31 T^{2}$$)($$1 + 8 T + 31 T^{2}$$)($$1 - 5 T + 31 T^{2}$$)($$1 + T + 31 T^{2}$$)($$1 + 54 T^{2} + 961 T^{4}$$)($$1 + 30 T^{2} + 961 T^{4}$$)
$37$ ($$1 + 5 T + 37 T^{2}$$)($$1 + 6 T + 37 T^{2}$$)($$1 - 3 T + 37 T^{2}$$)($$1 + 2 T + 37 T^{2}$$)($$1 - 3 T + 37 T^{2}$$)($$1 + 5 T + 37 T^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 - 10 T + 37 T^{2} )^{2}$$)
$41$ ($$1 - 10 T + 41 T^{2}$$)($$1 - 2 T + 41 T^{2}$$)($$1 + 2 T + 41 T^{2}$$)($$1 + 2 T + 41 T^{2}$$)($$1 - 2 T + 41 T^{2}$$)($$1 + 10 T + 41 T^{2}$$)($$1 + 80 T^{2} + 1681 T^{4}$$)($$1 + 50 T^{2} + 1681 T^{4}$$)
$43$ ($$1 + 4 T + 43 T^{2}$$)($$1 - 8 T + 43 T^{2}$$)($$1 + 4 T + 43 T^{2}$$)($$1 + 4 T + 43 T^{2}$$)($$1 + 4 T + 43 T^{2}$$)($$1 + 4 T + 43 T^{2}$$)($$( 1 - 10 T + 43 T^{2} )^{2}$$)($$( 1 + 4 T + 43 T^{2} )^{2}$$)
$47$ ($$1 + T + 47 T^{2}$$)($$1 - 4 T + 47 T^{2}$$)($$1 - 5 T + 47 T^{2}$$)($$1 - 8 T + 47 T^{2}$$)($$1 + 5 T + 47 T^{2}$$)($$1 - T + 47 T^{2}$$)($$1 + 86 T^{2} + 2209 T^{4}$$)($$1 + 62 T^{2} + 2209 T^{4}$$)
$53$ ($$1 + 9 T + 53 T^{2}$$)($$1 + 10 T + 53 T^{2}$$)($$1 + T + 53 T^{2}$$)($$1 - 6 T + 53 T^{2}$$)($$1 + T + 53 T^{2}$$)($$1 + 9 T + 53 T^{2}$$)($$( 1 + 2 T + 53 T^{2} )^{2}$$)($$( 1 - 6 T + 53 T^{2} )^{2}$$)
$59$ ($$1 + 3 T + 59 T^{2}$$)($$1 + 6 T + 59 T^{2}$$)($$1 - 15 T + 59 T^{2}$$)($$1 + 59 T^{2}$$)($$1 + 15 T + 59 T^{2}$$)($$1 - 3 T + 59 T^{2}$$)($$1 + 116 T^{2} + 3481 T^{4}$$)($$1 + 110 T^{2} + 3481 T^{4}$$)
$61$ ($$1 + 3 T + 61 T^{2}$$)($$1 + 4 T + 61 T^{2}$$)($$1 + 5 T + 61 T^{2}$$)($$1 - 6 T + 61 T^{2}$$)($$1 - 5 T + 61 T^{2}$$)($$1 - 3 T + 61 T^{2}$$)($$1 + 50 T^{2} + 3721 T^{4}$$)($$1 - 78 T^{2} + 3721 T^{4}$$)
$67$ ($$1 - 11 T + 67 T^{2}$$)($$1 + 12 T + 67 T^{2}$$)($$1 + 9 T + 67 T^{2}$$)($$1 + 4 T + 67 T^{2}$$)($$1 + 9 T + 67 T^{2}$$)($$1 - 11 T + 67 T^{2}$$)($$( 1 - 4 T + 67 T^{2} )^{2}$$)($$( 1 - 12 T + 67 T^{2} )^{2}$$)
$71$ ($$1 - 16 T + 71 T^{2}$$)($$1 + 71 T^{2}$$)($$1 + 71 T^{2}$$)($$1 + 8 T + 71 T^{2}$$)($$1 + 71 T^{2}$$)($$1 - 16 T + 71 T^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)
$73$ ($$1 + 7 T + 73 T^{2}$$)($$1 - 14 T + 73 T^{2}$$)($$1 - 7 T + 73 T^{2}$$)($$1 + 10 T + 73 T^{2}$$)($$1 + 7 T + 73 T^{2}$$)($$1 - 7 T + 73 T^{2}$$)($$1 + 48 T^{2} + 5329 T^{4}$$)($$( 1 + 73 T^{2} )^{2}$$)
$79$ ($$1 + 11 T + 79 T^{2}$$)($$1 + 8 T + 79 T^{2}$$)($$1 - T + 79 T^{2}$$)($$1 - 16 T + 79 T^{2}$$)($$1 - T + 79 T^{2}$$)($$1 + 11 T + 79 T^{2}$$)($$( 1 + 4 T + 79 T^{2} )^{2}$$)($$( 1 - 8 T + 79 T^{2} )^{2}$$)
$83$ ($$1 - 4 T + 83 T^{2}$$)($$1 + 6 T + 83 T^{2}$$)($$1 - 12 T + 83 T^{2}$$)($$1 + 8 T + 83 T^{2}$$)($$1 + 12 T + 83 T^{2}$$)($$1 + 4 T + 83 T^{2}$$)($$1 + 164 T^{2} + 6889 T^{4}$$)($$1 - 34 T^{2} + 6889 T^{4}$$)
$89$ ($$1 - 9 T + 89 T^{2}$$)($$1 + 10 T + 89 T^{2}$$)($$1 - 7 T + 89 T^{2}$$)($$1 - 6 T + 89 T^{2}$$)($$1 + 7 T + 89 T^{2}$$)($$1 + 9 T + 89 T^{2}$$)($$1 + 160 T^{2} + 7921 T^{4}$$)($$( 1 + 89 T^{2} )^{2}$$)
$97$ ($$1 + 6 T + 97 T^{2}$$)($$1 - 2 T + 97 T^{2}$$)($$1 + 2 T + 97 T^{2}$$)($$1 - 6 T + 97 T^{2}$$)($$1 - 2 T + 97 T^{2}$$)($$1 - 6 T + 97 T^{2}$$)($$1 + 32 T^{2} + 9409 T^{4}$$)($$1 + 162 T^{2} + 9409 T^{4}$$)