Properties

Label 170.2.h
Level $170$
Weight $2$
Character orbit 170.h
Rep. character $\chi_{170}(21,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $12$
Newform subspaces $2$
Sturm bound $54$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.h (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 17 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(54\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(170, [\chi])\).

Total New Old
Modular forms 60 12 48
Cusp forms 44 12 32
Eisenstein series 16 0 16

Trace form

\( 12q + 4q^{3} - 12q^{4} + 4q^{6} + O(q^{10}) \) \( 12q + 4q^{3} - 12q^{4} + 4q^{6} + 12q^{11} - 4q^{12} - 8q^{13} - 8q^{14} + 12q^{16} - 16q^{17} - 28q^{18} + 16q^{21} + 4q^{22} - 24q^{23} - 4q^{24} + 16q^{27} + 16q^{29} + 8q^{30} - 8q^{31} + 24q^{33} + 20q^{34} - 8q^{35} - 8q^{37} + 16q^{38} - 4q^{41} - 12q^{44} - 16q^{45} - 8q^{46} + 16q^{47} + 4q^{48} - 4q^{50} - 12q^{51} + 8q^{52} + 8q^{54} - 32q^{55} + 8q^{56} + 48q^{57} - 32q^{58} - 8q^{61} - 16q^{62} - 80q^{63} - 12q^{64} - 8q^{65} + 8q^{67} + 16q^{68} + 16q^{69} - 32q^{71} + 28q^{72} + 36q^{73} + 32q^{74} - 4q^{75} + 8q^{78} + 24q^{79} - 4q^{81} + 4q^{82} - 16q^{84} + 8q^{85} + 56q^{86} - 4q^{88} - 8q^{89} - 40q^{91} + 24q^{92} + 8q^{95} + 4q^{96} + 52q^{97} + 52q^{98} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(170, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
170.2.h.a \(4\) \(1.357\) \(\Q(\zeta_{8})\) None \(0\) \(4\) \(0\) \(4\) \(q-\zeta_{8}^{2}q^{2}+(1+\zeta_{8}+\zeta_{8}^{2})q^{3}-q^{4}+\cdots\)
170.2.h.b \(8\) \(1.357\) 8.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(-4\) \(q-\beta _{2}q^{2}-\beta _{7}q^{3}-q^{4}+\beta _{4}q^{5}-\beta _{6}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(170, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(170, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(34, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{4} \))
$3$ (\( 1 - 4 T + 8 T^{2} - 16 T^{3} + 31 T^{4} - 48 T^{5} + 72 T^{6} - 108 T^{7} + 81 T^{8} \))(\( 1 - 4 T^{3} - 19 T^{4} - 16 T^{5} + 8 T^{6} + 52 T^{7} + 244 T^{8} + 156 T^{9} + 72 T^{10} - 432 T^{11} - 1539 T^{12} - 972 T^{13} + 6561 T^{16} \))
$5$ (\( 1 + T^{4} \))(\( ( 1 + T^{4} )^{2} \))
$7$ (\( 1 - 4 T + 8 T^{2} - 20 T^{3} + 46 T^{4} - 140 T^{5} + 392 T^{6} - 1372 T^{7} + 2401 T^{8} \))(\( 1 + 4 T + 8 T^{2} - 12 T^{3} - 208 T^{4} - 524 T^{5} - 360 T^{6} + 3076 T^{7} + 16158 T^{8} + 21532 T^{9} - 17640 T^{10} - 179732 T^{11} - 499408 T^{12} - 201684 T^{13} + 941192 T^{14} + 3294172 T^{15} + 5764801 T^{16} \))
$11$ (\( ( 1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 4 T + 8 T^{2} + 12 T^{3} - 178 T^{4} + 132 T^{5} + 968 T^{6} - 5324 T^{7} + 14641 T^{8} )^{2} \))
$13$ (\( ( 1 - T + 13 T^{2} )^{4} \))(\( ( 1 + 6 T + 49 T^{2} + 206 T^{3} + 932 T^{4} + 2678 T^{5} + 8281 T^{6} + 13182 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4} \))(\( 1 + 8 T + 66 T^{2} + 344 T^{3} + 1730 T^{4} + 5848 T^{5} + 19074 T^{6} + 39304 T^{7} + 83521 T^{8} \))
$19$ (\( 1 - 70 T^{2} + 1939 T^{4} - 25270 T^{6} + 130321 T^{8} \))(\( 1 - 66 T^{2} + 2465 T^{4} - 63130 T^{6} + 1314132 T^{8} - 22789930 T^{10} + 321241265 T^{12} - 3105028146 T^{14} + 16983563041 T^{16} \))
$23$ (\( 1 + 8 T + 32 T^{2} - 40 T^{3} - 734 T^{4} - 920 T^{5} + 16928 T^{6} + 97336 T^{7} + 279841 T^{8} \))(\( ( 1 + 8 T + 32 T^{2} + 216 T^{3} + 1442 T^{4} + 4968 T^{5} + 16928 T^{6} + 97336 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( 1 + 8 T + 32 T^{2} - 96 T^{3} - 1393 T^{4} - 2784 T^{5} + 26912 T^{6} + 195112 T^{7} + 707281 T^{8} \))(\( 1 - 24 T + 288 T^{2} - 2640 T^{3} + 22065 T^{4} - 164976 T^{5} + 1089504 T^{6} - 6624840 T^{7} + 37336208 T^{8} - 192120360 T^{9} + 916272864 T^{10} - 4023599664 T^{11} + 15606155265 T^{12} - 54149433360 T^{13} + 171309116448 T^{14} - 413997031416 T^{15} + 500246412961 T^{16} \))
$31$ (\( 1 + 12 T + 72 T^{2} + 576 T^{3} + 4319 T^{4} + 17856 T^{5} + 69192 T^{6} + 357492 T^{7} + 923521 T^{8} \))(\( 1 - 4 T + 8 T^{2} + 240 T^{3} + 257 T^{4} - 12200 T^{5} + 75544 T^{6} + 16212 T^{7} - 1964000 T^{8} + 502572 T^{9} + 72597784 T^{10} - 363450200 T^{11} + 237344897 T^{12} + 6870996240 T^{13} + 7100029448 T^{14} - 110050456444 T^{15} + 852891037441 T^{16} \))
$37$ (\( 1 - 12 T + 72 T^{2} - 612 T^{3} + 5006 T^{4} - 22644 T^{5} + 98568 T^{6} - 607836 T^{7} + 1874161 T^{8} \))(\( 1 + 20 T + 200 T^{2} + 1484 T^{3} + 9888 T^{4} + 60164 T^{5} + 326808 T^{6} + 1410492 T^{7} + 6339550 T^{8} + 52188204 T^{9} + 447400152 T^{10} + 3047487092 T^{11} + 18531703968 T^{12} + 102906432188 T^{13} + 513145281800 T^{14} + 1898637542660 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 + 4 T + 8 T^{2} - 84 T^{3} - 2962 T^{4} - 3444 T^{5} + 13448 T^{6} + 275684 T^{7} + 2825761 T^{8} \))(\( 1 - 4420 T^{4} + 10011334 T^{8} - 12489863620 T^{12} + 7984925229121 T^{16} \))
$43$ (\( 1 - 64 T^{2} + 2130 T^{4} - 118336 T^{6} + 3418801 T^{8} \))(\( 1 - 228 T^{2} + 25352 T^{4} - 1829068 T^{6} + 92968878 T^{8} - 3381946732 T^{10} + 86673442952 T^{12} - 1441270775172 T^{14} + 11688200277601 T^{16} \))
$47$ (\( ( 1 + 2 T + 77 T^{2} + 94 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 - 10 T + 179 T^{2} - 1178 T^{3} + 12016 T^{4} - 55366 T^{5} + 395411 T^{6} - 1038230 T^{7} + 4879681 T^{8} )^{2} \))
$53$ (\( 1 - 18 T^{2} - 1501 T^{4} - 50562 T^{6} + 7890481 T^{8} \))(\( 1 - 366 T^{2} + 60833 T^{4} - 6024462 T^{6} + 390030964 T^{8} - 16922713758 T^{10} + 480001630673 T^{12} - 8112156173214 T^{14} + 62259690411361 T^{16} \))
$59$ (\( 1 - 38 T^{2} + 6931 T^{4} - 132278 T^{6} + 12117361 T^{8} \))(\( 1 - 250 T^{2} + 33137 T^{4} - 3007930 T^{6} + 202771380 T^{8} - 10470604330 T^{10} + 401532991457 T^{12} - 10545133410250 T^{14} + 146830437604321 T^{16} \))
$61$ (\( 1 - 8 T + 32 T^{2} + 416 T^{3} - 7361 T^{4} + 25376 T^{5} + 119072 T^{6} - 1815848 T^{7} + 13845841 T^{8} \))(\( 1 + 16 T + 128 T^{2} + 616 T^{3} + 2273 T^{4} + 43536 T^{5} + 595360 T^{6} + 5565640 T^{7} + 48746896 T^{8} + 339504040 T^{9} + 2215334560 T^{10} + 9881844816 T^{11} + 31471596593 T^{12} + 520271321416 T^{13} + 6594607918208 T^{14} + 50283885376336 T^{15} + 191707312997281 T^{16} \))
$67$ (\( ( 1 - 12 T + 162 T^{2} - 804 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 8 T + 156 T^{2} + 1288 T^{3} + 12678 T^{4} + 86296 T^{5} + 700284 T^{6} + 2406104 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( 1 + 36 T + 648 T^{2} + 8352 T^{3} + 81727 T^{4} + 592992 T^{5} + 3266568 T^{6} + 12884796 T^{7} + 25411681 T^{8} \))(\( 1 - 4 T + 8 T^{2} + 488 T^{3} - 2335 T^{4} - 45368 T^{5} + 319224 T^{6} - 1238244 T^{7} - 10911776 T^{8} - 87915324 T^{9} + 1609208184 T^{10} - 16237706248 T^{11} - 59336275135 T^{12} + 880463923288 T^{13} + 1024802271368 T^{14} - 36380480633564 T^{15} + 645753531245761 T^{16} \))
$73$ (\( 1 - 8 T + 32 T^{2} - 448 T^{3} + 5983 T^{4} - 32704 T^{5} + 170528 T^{6} - 3112136 T^{7} + 28398241 T^{8} \))(\( 1 - 28 T + 392 T^{2} - 4460 T^{3} + 37581 T^{4} - 178712 T^{5} + 217984 T^{6} + 8414576 T^{7} - 122737516 T^{8} + 614264048 T^{9} + 1161636736 T^{10} - 69522006104 T^{11} + 1067234295021 T^{12} - 9245899304780 T^{13} + 59323016705288 T^{14} - 309327158534716 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 - 16 T + 128 T^{2} - 1200 T^{3} + 11234 T^{4} - 94800 T^{5} + 798848 T^{6} - 7888624 T^{7} + 38950081 T^{8} \))(\( 1 - 8 T + 32 T^{2} + 232 T^{3} + 548 T^{4} + 19080 T^{5} - 143264 T^{6} + 3853144 T^{7} - 12081722 T^{8} + 304398376 T^{9} - 894110624 T^{10} + 9407184120 T^{11} + 21344644388 T^{12} + 713877084568 T^{13} + 7778798576672 T^{14} - 153631271889272 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( ( 1 - 68 T^{2} + 6889 T^{4} )^{2} \))(\( 1 - 300 T^{2} + 56408 T^{4} - 7222532 T^{6} + 693023310 T^{8} - 49756022948 T^{10} + 2677028970968 T^{12} - 98082112010700 T^{14} + 2252292232139041 T^{16} \))
$89$ (\( ( 1 - 18 T + 227 T^{2} - 1602 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 22 T + 213 T^{2} + 26 T^{3} - 10296 T^{4} + 2314 T^{5} + 1687173 T^{6} + 15509318 T^{7} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 - 32 T + 512 T^{2} - 6912 T^{3} + 79151 T^{4} - 670464 T^{5} + 4817408 T^{6} - 29205536 T^{7} + 88529281 T^{8} \))(\( 1 - 20 T + 200 T^{2} - 1676 T^{3} + 9693 T^{4} - 76136 T^{5} + 988608 T^{6} - 10834248 T^{7} + 115952980 T^{8} - 1050922056 T^{9} + 9301812672 T^{10} - 69487271528 T^{11} + 858114320733 T^{12} - 14392382270732 T^{13} + 166594400985800 T^{14} - 1615965689562260 T^{15} + 7837433594376961 T^{16} \))
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