# Properties

 Label 175.2.k Level 175 Weight 2 Character orbit k Rep. character $$\chi_{175}(74,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 20 Newform subspaces 2 Sturm bound 40 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 175.k (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$35$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$2$$ Sturm bound: $$40$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(175, [\chi])$$.

Total New Old
Modular forms 52 28 24
Cusp forms 28 20 8
Eisenstein series 24 8 16

## Trace form

 $$20q + 10q^{4} - 16q^{6} + 8q^{9} + O(q^{10})$$ $$20q + 10q^{4} - 16q^{6} + 8q^{9} - 4q^{11} - 26q^{14} - 2q^{16} - 8q^{19} + 6q^{21} - 16q^{24} - 6q^{26} + 40q^{29} + 14q^{31} - 4q^{36} + 32q^{39} - 44q^{41} - 12q^{44} + 16q^{46} - 44q^{49} - 14q^{51} - 42q^{54} - 6q^{56} - 26q^{59} + 2q^{61} + 120q^{64} - 44q^{66} - 24q^{69} - 4q^{71} - 30q^{74} + 124q^{76} + 34q^{79} + 30q^{81} + 2q^{84} + 48q^{86} - 24q^{89} - 62q^{91} - 26q^{94} - 14q^{96} - 20q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(175, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
175.2.k.a $$8$$ $$1.397$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{24}^{3}q^{2}-\zeta_{24}^{4}q^{3}+(1-\zeta_{24}-\zeta_{24}^{6}+\cdots)q^{4}+\cdots$$
175.2.k.b $$12$$ $$1.397$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(\beta _{10}-\beta _{11})q^{3}+(1+\beta _{3}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(175, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(175, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 2 T^{2} + 3 T^{4} - 14 T^{6} - 31 T^{8} - 56 T^{10} + 48 T^{12} + 128 T^{14} + 256 T^{16}$$)($$1 + 3 T^{2} - T^{4} - 16 T^{6} - 9 T^{8} + 38 T^{10} + 113 T^{12} + 152 T^{14} - 144 T^{16} - 1024 T^{18} - 256 T^{20} + 3072 T^{22} + 4096 T^{24}$$)
$3$ ($$1 + 6 T^{2} + 17 T^{4} + 6 T^{6} - 44 T^{8} + 54 T^{10} + 1377 T^{12} + 4374 T^{14} + 6561 T^{16}$$)($$1 + 5 T^{2} - 5 T^{6} + 75 T^{8} - 220 T^{10} - 1967 T^{12} - 1980 T^{14} + 6075 T^{16} - 3645 T^{18} + 295245 T^{22} + 531441 T^{24}$$)
$5$ 1
$7$ ($$1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8}$$)($$1 + 12 T^{2} + 12 T^{4} - 349 T^{6} + 588 T^{8} + 28812 T^{10} + 117649 T^{12}$$)
$11$ ($$( 1 + 4 T - 2 T^{2} - 16 T^{3} + 27 T^{4} - 176 T^{5} - 242 T^{6} + 5324 T^{7} + 14641 T^{8} )^{2}$$)($$( 1 - 2 T - 10 T^{2} - 30 T^{3} + 44 T^{4} + 388 T^{5} + 23 T^{6} + 4268 T^{7} + 5324 T^{8} - 39930 T^{9} - 146410 T^{10} - 322102 T^{11} + 1771561 T^{12} )^{2}$$)
$13$ ($$( 1 - 28 T^{2} + 406 T^{4} - 4732 T^{6} + 28561 T^{8} )^{2}$$)($$( 1 - 32 T^{2} + 312 T^{4} - 1641 T^{6} + 52728 T^{8} - 913952 T^{10} + 4826809 T^{12} )^{2}$$)
$17$ ($$1 + 44 T^{2} + 1002 T^{4} + 15664 T^{6} + 229331 T^{8} + 4526896 T^{10} + 83688042 T^{12} + 1062053036 T^{14} + 6975757441 T^{16}$$)($$1 + 60 T^{2} + 1700 T^{4} + 35390 T^{6} + 702600 T^{8} + 13798760 T^{10} + 249192563 T^{12} + 3987841640 T^{14} + 58681854600 T^{16} + 854228566910 T^{18} + 11858787649700 T^{20} + 120959634026940 T^{22} + 582622237229761 T^{24}$$)
$19$ ($$( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 + 4 T - 24 T^{2} - 158 T^{3} + 204 T^{4} + 1724 T^{5} + 2823 T^{6} + 32756 T^{7} + 73644 T^{8} - 1083722 T^{9} - 3127704 T^{10} + 9904396 T^{11} + 47045881 T^{12} )^{2}$$)
$23$ ($$1 + 86 T^{2} + 4497 T^{4} + 158326 T^{6} + 4211876 T^{8} + 83754454 T^{10} + 1258444977 T^{12} + 12731086454 T^{14} + 78310985281 T^{16}$$)($$1 + 117 T^{2} + 7604 T^{4} + 351191 T^{6} + 12704151 T^{8} + 377097632 T^{10} + 9413788433 T^{12} + 199484647328 T^{14} + 3555142319991 T^{16} + 51988871893799 T^{18} + 595476732076724 T^{20} + 4846901811996933 T^{22} + 21914624432020321 T^{24}$$)
$29$ ($$( 1 - T + 29 T^{2} )^{8}$$)($$( 1 - 8 T + 82 T^{2} - 389 T^{3} + 2378 T^{4} - 6728 T^{5} + 24389 T^{6} )^{4}$$)
$31$ ($$( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} )^{4}$$)($$( 1 + 5 T - 50 T^{2} - 119 T^{3} + 2065 T^{4} + 80 T^{5} - 77809 T^{6} + 2480 T^{7} + 1984465 T^{8} - 3545129 T^{9} - 46176050 T^{10} + 143145755 T^{11} + 887503681 T^{12} )^{2}$$)
$37$ ($$( 1 + 37 T^{2} + 1369 T^{4} )^{4}$$)($$1 + 140 T^{2} + 9500 T^{4} + 496350 T^{6} + 23947600 T^{8} + 1024983040 T^{10} + 39266587843 T^{12} + 1403201781760 T^{14} + 44881657963600 T^{16} + 1273498303107150 T^{18} + 33368554812249500 T^{20} + 673201812138498860 T^{22} + 6582952005840035281 T^{24}$$)
$41$ ($$( 1 + 10 T + 99 T^{2} + 410 T^{3} + 1681 T^{4} )^{4}$$)($$( 1 + T + 77 T^{2} + 187 T^{3} + 3157 T^{4} + 1681 T^{5} + 68921 T^{6} )^{4}$$)
$43$ ($$( 1 - 118 T^{2} + 6979 T^{4} - 218182 T^{6} + 3418801 T^{8} )^{2}$$)($$( 1 - 137 T^{2} + 8277 T^{4} - 361401 T^{6} + 15304173 T^{8} - 468375737 T^{10} + 6321363049 T^{12} )^{2}$$)
$47$ ($$( 1 + 90 T^{2} + 5891 T^{4} + 198810 T^{6} + 4879681 T^{8} )^{2}$$)($$1 + 4 T^{2} - 3704 T^{4} + 16022 T^{6} + 5623204 T^{8} - 48345236 T^{10} - 9088343797 T^{12} - 106794626324 T^{14} + 27439441717924 T^{16} + 172704588001238 T^{18} - 88197005795162744 T^{20} + 210396528943320196 T^{22} +$$$$11\!\cdots\!41$$$$T^{24}$$)
$53$ ($$1 + 164 T^{2} + 15066 T^{4} + 1018768 T^{6} + 56507555 T^{8} + 2861719312 T^{10} + 118877986746 T^{12} + 3634955225156 T^{14} + 62259690411361 T^{16}$$)($$1 + 180 T^{2} + 16700 T^{4} + 954830 T^{6} + 33983400 T^{8} + 232503080 T^{10} - 28393979917 T^{12} + 653101151720 T^{14} + 268145372015400 T^{16} + 21163196936803070 T^{18} + 1039736829869728700 T^{20} + 31479744665792348820 T^{22} +$$$$49\!\cdots\!41$$$$T^{24}$$)
$59$ ($$( 1 + 8 T + 2 T^{2} - 448 T^{3} - 3413 T^{4} - 26432 T^{5} + 6962 T^{6} + 1643032 T^{7} + 12117361 T^{8} )^{2}$$)($$( 1 + 5 T - 90 T^{2} - 815 T^{3} + 2965 T^{4} + 29480 T^{5} + 26483 T^{6} + 1739320 T^{7} + 10321165 T^{8} - 167383885 T^{9} - 1090562490 T^{10} + 3574621495 T^{11} + 42180533641 T^{12} )^{2}$$)
$61$ ($$( 1 - 6 T - 23 T^{2} + 378 T^{3} - 2436 T^{4} + 23058 T^{5} - 85583 T^{6} - 1361886 T^{7} + 13845841 T^{8} )^{2}$$)($$( 1 + 5 T - 140 T^{2} - 269 T^{3} + 14425 T^{4} + 8210 T^{5} - 1000579 T^{6} + 500810 T^{7} + 53675425 T^{8} - 61057889 T^{9} - 1938417740 T^{10} + 4222981505 T^{11} + 51520374361 T^{12} )^{2}$$)
$67$ ($$1 + 22 T^{2} - 7647 T^{4} - 18634 T^{6} + 43789364 T^{8} - 83648026 T^{10} - 154095622287 T^{12} + 1990084407718 T^{14} + 406067677556641 T^{16}$$)($$1 + 314 T^{2} + 53101 T^{4} + 6531102 T^{6} + 648138474 T^{8} + 54464437634 T^{10} + 3935187653893 T^{12} + 244490860539026 T^{14} + 13060716814329354 T^{16} + 590792920700720238 T^{18} + 21562599745935193741 T^{20} +$$$$57\!\cdots\!86$$$$T^{22} +$$$$81\!\cdots\!61$$$$T^{24}$$)
$71$ ($$( 1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4} )^{4}$$)($$( 1 - 7 T + 124 T^{2} - 571 T^{3} + 8804 T^{4} - 35287 T^{5} + 357911 T^{6} )^{4}$$)
$73$ ($$1 + 268 T^{2} + 43338 T^{4} + 4777904 T^{6} + 400303859 T^{8} + 25461450416 T^{10} + 1230722968458 T^{12} + 40557572645452 T^{14} + 806460091894081 T^{16}$$)($$1 + 230 T^{2} + 21925 T^{4} + 1827570 T^{6} + 185266650 T^{8} + 12633670430 T^{10} + 701193878653 T^{12} + 67324829721470 T^{14} + 5261246975962650 T^{16} + 276573891938987730 T^{18} + 17681637514777725925 T^{20} +$$$$98\!\cdots\!70$$$$T^{22} +$$$$22\!\cdots\!21$$$$T^{24}$$)
$79$ ($$( 1 - 24 T + 282 T^{2} - 3264 T^{3} + 34691 T^{4} - 257856 T^{5} + 1759962 T^{6} - 11832936 T^{7} + 38950081 T^{8} )^{2}$$)($$( 1 + 7 T - 96 T^{2} - 1499 T^{3} + 771 T^{4} + 65582 T^{5} + 599343 T^{6} + 5180978 T^{7} + 4811811 T^{8} - 739065461 T^{9} - 3739207776 T^{10} + 21539394793 T^{11} + 243087455521 T^{12} )^{2}$$)
$83$ ($$( 1 - 6 T^{2} + 13139 T^{4} - 41334 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 - 220 T^{2} + 29892 T^{4} - 2915285 T^{6} + 205925988 T^{8} - 10440830620 T^{10} + 326940373369 T^{12} )^{2}$$)
$89$ ($$( 1 + 6 T - 119 T^{2} - 138 T^{3} + 12900 T^{4} - 12282 T^{5} - 942599 T^{6} + 4229814 T^{7} + 62742241 T^{8} )^{2}$$)($$( 1 + 6 T - 94 T^{2} + 198 T^{3} + 3584 T^{4} - 53484 T^{5} - 294637 T^{6} - 4760076 T^{7} + 28388864 T^{8} + 139583862 T^{9} - 5897770654 T^{10} + 33504356694 T^{11} + 496981290961 T^{12} )^{2}$$)
$97$ ($$( 1 - 252 T^{2} + 30086 T^{4} - 2371068 T^{6} + 88529281 T^{8} )^{2}$$)($$( 1 - 313 T^{2} + 45237 T^{4} - 4644209 T^{6} + 425634933 T^{8} - 27709664953 T^{10} + 832972004929 T^{12} )^{2}$$)