Properties

Label 320.6.f
Level 320
Weight 6
Character orbit f
Rep. character \(\chi_{320}(289,\cdot)\)
Character field \(\Q\)
Dimension 60
Newform subspaces 4
Sturm bound 288
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 40 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(288\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(320, [\chi])\).

Total New Old
Modular forms 252 60 192
Cusp forms 228 60 168
Eisenstein series 24 0 24

Trace form

\( 60q + 4860q^{9} + O(q^{10}) \) \( 60q + 4860q^{9} - 9348q^{25} + 14856q^{41} - 144060q^{49} - 94584q^{65} + 124620q^{81} + 229656q^{89} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(320, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
320.6.f.a \(4\) \(51.323\) \(\Q(i, \sqrt{5})\) \(\Q(\sqrt{-10}) \) \(0\) \(0\) \(0\) \(0\) \(q+5\beta _{3}q^{5}-11\beta _{2}q^{7}-3^{5}q^{9}+401\beta _{1}q^{11}+\cdots\)
320.6.f.b \(8\) \(51.323\) 8.0.\(\cdots\).45 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{4}q^{3}-5\beta _{5}q^{5}-11\beta _{2}q^{7}+123q^{9}+\cdots\)
320.6.f.c \(16\) \(51.323\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{7}q^{3}-\beta _{10}q^{5}+(-\beta _{8}+\beta _{12})q^{7}+\cdots\)
320.6.f.d \(32\) \(51.323\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{6}^{\mathrm{old}}(320, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(320, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 + 243 T^{2} )^{4} \))(\( ( 1 + 120 T^{2} + 59049 T^{4} )^{4} \))(\( ( 1 + 248 T^{2} + 69330 T^{4} + 14644152 T^{6} + 3486784401 T^{8} )^{4} \))
$5$ (\( ( 1 - 3125 T^{2} )^{2} \))(\( ( 1 - 5950 T^{2} + 9765625 T^{4} )^{2} \))(\( ( 1 - 220 T^{2} - 6114250 T^{4} - 2148437500 T^{6} + 95367431640625 T^{8} )^{2} \))
$7$ (\( ( 1 + 26886 T^{2} + 282475249 T^{4} )^{2} \))(\( ( 1 - 18852 T^{2} + 282475249 T^{4} )^{4} \))(\( ( 1 - 49648 T^{2} + 1179577874 T^{4} - 14024331162352 T^{6} + 79792266297612001 T^{8} )^{4} \))
$11$ (\( ( 1 + 321102 T^{2} + 25937424601 T^{4} )^{2} \))(\( ( 1 - 233298 T^{2} + 25937424601 T^{4} )^{4} \))(\( ( 1 - 577284 T^{2} + 134071408310 T^{4} - 14973260223363684 T^{6} + \)\(67\!\cdots\!01\)\( T^{8} )^{4} \))
$13$ (\( ( 1 + 682086 T^{2} + 137858491849 T^{4} )^{2} \))(\( ( 1 + 506394 T^{2} + 137858491849 T^{4} )^{4} \))(\( ( 1 + 1011892 T^{2} + 531287817014 T^{4} + 139497905034068308 T^{6} + \)\(19\!\cdots\!01\)\( T^{8} )^{4} \))
$17$ (\( ( 1 - 1419857 T^{2} )^{4} \))(\( ( 1 - 2662570 T^{2} + 2015993900449 T^{4} )^{4} \))(\( ( 1 + 554828 T^{2} + 3105096833510 T^{4} + 1118529863798317772 T^{6} + \)\(40\!\cdots\!01\)\( T^{8} )^{4} \))
$19$ (\( ( 1 - 1288802 T^{2} + 6131066257801 T^{4} )^{2} \))(\( ( 1 + 1673278 T^{2} + 6131066257801 T^{4} )^{4} \))(\( ( 1 - 8279076 T^{2} + 29313371734870 T^{4} - 50759563509370071876 T^{6} + \)\(37\!\cdots\!01\)\( T^{8} )^{4} \))
$23$ (\( ( 1 - 1772186 T^{2} + 41426511213649 T^{4} )^{2} \))(\( ( 1 - 12357236 T^{2} + 41426511213649 T^{4} )^{4} \))(\( ( 1 - 265392 T^{2} + 72532989580114 T^{4} - 10994264664012735408 T^{6} + \)\(17\!\cdots\!01\)\( T^{8} )^{4} \))
$29$ (\( ( 1 - 20511149 T^{2} )^{4} \))(\( ( 1 - 27077290 T^{2} + 420707233300201 T^{4} )^{4} \))(\( ( 1 - 18262996 T^{2} + 467877918484406 T^{4} - \)\(76\!\cdots\!96\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} )^{4} \))
$31$ (\( ( 1 + 28629151 T^{2} )^{4} \))(\( ( 1 + 18464654 T^{2} + 819628286980801 T^{4} )^{4} \))(\( ( 1 + 39407004 T^{2} + 689994929185606 T^{4} + \)\(32\!\cdots\!04\)\( T^{6} + \)\(67\!\cdots\!01\)\( T^{8} )^{4} \))
$37$ (\( ( 1 - 112652586 T^{2} + 4808584372417849 T^{4} )^{2} \))(\( ( 1 + 105952386 T^{2} + 4808584372417849 T^{4} )^{4} \))(\( ( 1 + 171738148 T^{2} + 16366808202619574 T^{4} + \)\(82\!\cdots\!52\)\( T^{6} + \)\(23\!\cdots\!01\)\( T^{8} )^{4} \))
$41$ (\( ( 1 - 15202 T + 115856201 T^{2} )^{4} \))(\( ( 1 - 11396 T + 115856201 T^{2} )^{8} \))(\( ( 1 + 12544 T + 129356290 T^{2} + 1453300185344 T^{3} + 13422659310152401 T^{4} )^{8} \))
$43$ (\( ( 1 + 147008443 T^{2} )^{4} \))(\( ( 1 - 72715480 T^{2} + 21611482313284249 T^{4} )^{4} \))(\( ( 1 + 515310072 T^{2} + 108506205300479794 T^{4} + \)\(11\!\cdots\!28\)\( T^{6} + \)\(46\!\cdots\!01\)\( T^{8} )^{4} \))
$47$ (\( ( 1 - 222705514 T^{2} + 52599132235830049 T^{4} )^{2} \))(\( ( 1 - 453462436 T^{2} + 52599132235830049 T^{4} )^{4} \))(\( ( 1 - 112031408 T^{2} + 100831510268064114 T^{4} - \)\(58\!\cdots\!92\)\( T^{6} + \)\(27\!\cdots\!01\)\( T^{8} )^{4} \))
$53$ (\( ( 1 + 552886486 T^{2} + 174887470365513049 T^{4} )^{2} \))(\( ( 1 + 539491786 T^{2} + 174887470365513049 T^{4} )^{4} \))(\( ( 1 + 823212052 T^{2} + 458582260389137174 T^{4} + \)\(14\!\cdots\!48\)\( T^{6} + \)\(30\!\cdots\!01\)\( T^{8} )^{4} \))
$59$ (\( ( 1 - 646632402 T^{2} + 511116753300641401 T^{4} )^{2} \))(\( ( 1 - 1161609714 T^{2} + 511116753300641401 T^{4} )^{4} \))(\( ( 1 - 2409567364 T^{2} + 2462718757251169526 T^{4} - \)\(12\!\cdots\!64\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} )^{4} \))
$61$ (\( ( 1 - 844596301 T^{2} )^{4} \))(\( ( 1 + 686048530 T^{2} + 713342911662882601 T^{4} )^{4} \))(\( ( 1 - 2000262204 T^{2} + 2103804668157499606 T^{4} - \)\(14\!\cdots\!04\)\( T^{6} + \)\(50\!\cdots\!01\)\( T^{8} )^{4} \))
$67$ (\( ( 1 + 1350125107 T^{2} )^{4} \))(\( ( 1 + 2629714328 T^{2} + 1822837804551761449 T^{4} )^{4} \))(\( ( 1 + 3461519512 T^{2} + 6000732802449327570 T^{4} + \)\(63\!\cdots\!88\)\( T^{6} + \)\(33\!\cdots\!01\)\( T^{8} )^{4} \))
$71$ (\( ( 1 + 1804229351 T^{2} )^{4} \))(\( ( 1 + 2762141854 T^{2} + 3255243551009881201 T^{4} )^{4} \))(\( ( 1 + 823411004 T^{2} + 3069578968927394406 T^{4} + \)\(26\!\cdots\!04\)\( T^{6} + \)\(10\!\cdots\!01\)\( T^{8} )^{4} \))
$73$ (\( ( 1 - 2073071593 T^{2} )^{4} \))(\( ( 1 - 4145966042 T^{2} + 4297625829703557649 T^{4} )^{4} \))(\( ( 1 - 4148991508 T^{2} + 8650887997979942790 T^{4} - \)\(17\!\cdots\!92\)\( T^{6} + \)\(18\!\cdots\!01\)\( T^{8} )^{4} \))
$79$ (\( ( 1 + 3077056399 T^{2} )^{4} \))(\( ( 1 + 5182544750 T^{2} + 9468276082626847201 T^{4} )^{4} \))(\( ( 1 + 2847963996 T^{2} + 15373907451301926406 T^{4} + \)\(26\!\cdots\!96\)\( T^{6} + \)\(89\!\cdots\!01\)\( T^{8} )^{4} \))
$83$ (\( ( 1 + 3939040643 T^{2} )^{4} \))(\( ( 1 - 3415955864 T^{2} + 15516041187205853449 T^{4} )^{4} \))(\( ( 1 + 7032005368 T^{2} + 39846447958779448850 T^{4} + \)\(10\!\cdots\!32\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} )^{4} \))
$89$ (\( ( 1 - 128786 T + 5584059449 T^{2} )^{4} \))(\( ( 1 + 99362 T + 5584059449 T^{2} )^{8} \))(\( ( 1 + 57412 T + 10662063350 T^{2} + 320592021085988 T^{3} + 31181719929966183601 T^{4} )^{8} \))
$97$ (\( ( 1 - 8587340257 T^{2} )^{4} \))(\( ( 1 + 15344227702 T^{2} + 73742412689492826049 T^{4} )^{4} \))(\( ( 1 - 31837059828 T^{2} + \)\(40\!\cdots\!94\)\( T^{4} - \)\(23\!\cdots\!72\)\( T^{6} + \)\(54\!\cdots\!01\)\( T^{8} )^{4} \))
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