# Properties

 Label 225.2.k Level 225 Weight 2 Character orbit k Rep. character $$\chi_{225}(49,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 32 Newform subspaces 3 Sturm bound 60 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 225.k (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$45$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$3$$ Sturm bound: $$60$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(225, [\chi])$$.

Total New Old
Modular forms 72 40 32
Cusp forms 48 32 16
Eisenstein series 24 8 16

## Trace form

 $$32q + 16q^{4} + 8q^{6} - 2q^{9} + O(q^{10})$$ $$32q + 16q^{4} + 8q^{6} - 2q^{9} + 10q^{11} - 18q^{14} - 16q^{16} + 8q^{19} - 30q^{21} + 18q^{24} - 56q^{26} - 14q^{29} - 8q^{31} + 18q^{34} + 4q^{36} - 10q^{39} + 26q^{41} - 8q^{44} - 6q^{49} - 10q^{51} - 44q^{54} + 60q^{56} + 2q^{59} + 10q^{61} - 44q^{64} - 32q^{66} - 42q^{69} - 64q^{71} + 40q^{74} - 14q^{76} - 22q^{79} - 22q^{81} + 18q^{84} + 28q^{86} + 132q^{89} - 4q^{91} - 42q^{94} + 124q^{96} + 2q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(225, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
225.2.k.a $$4$$ $$1.797$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}+(-\zeta_{12}-\zeta_{12}^{3})q^{3}-\zeta_{12}^{2}q^{4}+\cdots$$
225.2.k.b $$12$$ $$1.797$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{6}-\beta _{7})q^{2}+\beta _{6}q^{3}+(2+2\beta _{8}+\cdots)q^{4}+\cdots$$
225.2.k.c $$16$$ $$1.797$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-\beta _{10}+\beta _{14})q^{3}+(1-\beta _{3}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(225, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(225, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8}$$)($$1 + T^{2} - 2 T^{4} - 19 T^{6} - 14 T^{8} + 25 T^{10} + 197 T^{12} + 100 T^{14} - 224 T^{16} - 1216 T^{18} - 512 T^{20} + 1024 T^{22} + 4096 T^{24}$$)($$1 + 4 T^{2} + 6 T^{4} + 10 T^{6} + 19 T^{8} + 42 T^{10} + 95 T^{12} + 11 T^{14} - 267 T^{16} + 44 T^{18} + 1520 T^{20} + 2688 T^{22} + 4864 T^{24} + 10240 T^{26} + 24576 T^{28} + 65536 T^{30} + 65536 T^{32}$$)
$3$ ($$1 + 3 T^{2} + 9 T^{4}$$)($$1 - 7 T^{2} + 34 T^{4} - 123 T^{6} + 306 T^{8} - 567 T^{10} + 729 T^{12}$$)($$1 + 5 T^{2} + 4 T^{4} + 15 T^{6} + 135 T^{8} + 135 T^{10} + 324 T^{12} + 3645 T^{14} + 6561 T^{16}$$)
$5$ ()()()
$7$ ($$1 + 5 T^{2} - 24 T^{4} + 245 T^{6} + 2401 T^{8}$$)($$1 + 23 T^{2} + 287 T^{4} + 1944 T^{6} + 5053 T^{8} - 51983 T^{10} - 624746 T^{12} - 2547167 T^{14} + 12132253 T^{16} + 228709656 T^{18} + 1654497887 T^{20} + 6496930727 T^{22} + 13841287201 T^{24}$$)($$1 + 31 T^{2} + 465 T^{4} + 4686 T^{6} + 37554 T^{8} + 261471 T^{10} + 1619330 T^{12} + 9344750 T^{14} + 58806225 T^{16} + 457892750 T^{18} + 3888011330 T^{20} + 30761801679 T^{22} + 216491336754 T^{24} + 1323679016814 T^{26} + 6436198548465 T^{28} + 21024915258319 T^{30} + 33232930569601 T^{32}$$)
$11$ ($$( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2}$$)($$( 1 - 2 T - 21 T^{2} + 14 T^{3} + 286 T^{4} + 58 T^{5} - 3673 T^{6} + 638 T^{7} + 34606 T^{8} + 18634 T^{9} - 307461 T^{10} - 322102 T^{11} + 1771561 T^{12} )^{2}$$)($$( 1 - T - 18 T^{2} + 129 T^{3} + 120 T^{4} - 1886 T^{5} + 6087 T^{6} + 14944 T^{7} - 88007 T^{8} + 164384 T^{9} + 736527 T^{10} - 2510266 T^{11} + 1756920 T^{12} + 20775579 T^{13} - 31888098 T^{14} - 19487171 T^{15} + 214358881 T^{16} )^{2}$$)
$13$ ($$( 1 - T^{2} + 169 T^{4} )( 1 + 23 T^{2} + 169 T^{4} )$$)($$1 + 54 T^{2} + 1581 T^{4} + 30418 T^{6} + 431466 T^{8} + 4991934 T^{10} + 59312613 T^{12} + 843636846 T^{14} + 12323100426 T^{16} + 146821876162 T^{18} + 1289670269901 T^{20} + 7444358559846 T^{22} + 23298085122481 T^{24}$$)($$1 + 40 T^{2} + 726 T^{4} + 8330 T^{6} + 50411 T^{8} - 581655 T^{10} - 20871557 T^{12} - 333628385 T^{14} - 4436735157 T^{16} - 56383197065 T^{18} - 596112539477 T^{20} - 2807537588895 T^{22} + 41121801376331 T^{24} + 1148361237102170 T^{26} + 16914409798921206 T^{28} + 157495055427971560 T^{30} + 665416609183179841 T^{32}$$)
$17$ ($$( 1 - 18 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 82 T^{2} + 3087 T^{4} - 67244 T^{6} + 892143 T^{8} - 6848722 T^{10} + 24137569 T^{12} )^{2}$$)($$( 1 - 55 T^{2} + 2044 T^{4} - 50325 T^{6} + 994815 T^{8} - 14543925 T^{10} + 170716924 T^{12} - 1327566295 T^{14} + 6975757441 T^{16} )^{2}$$)
$19$ ($$( 1 - 8 T + 19 T^{2} )^{4}$$)($$( 1 + 4 T + 53 T^{2} + 148 T^{3} + 1007 T^{4} + 1444 T^{5} + 6859 T^{6} )^{4}$$)($$( 1 + 2 T + 49 T^{2} + 34 T^{3} + 1115 T^{4} + 646 T^{5} + 17689 T^{6} + 13718 T^{7} + 130321 T^{8} )^{4}$$)
$23$ ($$1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8}$$)($$1 + 63 T^{2} + 1143 T^{4} + 30064 T^{6} + 1830141 T^{8} + 38353041 T^{10} + 434719014 T^{12} + 20288758689 T^{14} + 512148487581 T^{16} + 4450550966896 T^{18} + 89509456176183 T^{20} + 2609870206459887 T^{22} + 21914624432020321 T^{24}$$)($$1 + 73 T^{2} + 2802 T^{4} + 80831 T^{6} + 1616966 T^{8} + 14420082 T^{10} - 392881007 T^{12} - 25025527652 T^{14} - 736058803599 T^{16} - 13238504127908 T^{18} - 109944213879887 T^{20} + 2134689658322898 T^{22} + 126626200625877446 T^{24} + 3348546327910462319 T^{26} + 61404777658520939442 T^{28} +$$$$84\!\cdots\!57$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$)
$29$ ($$( 1 + T - 28 T^{2} + 29 T^{3} + 841 T^{4} )^{2}$$)($$( 1 + 7 T - 9 T^{2} - 304 T^{3} - 803 T^{4} + 2929 T^{5} + 36038 T^{6} + 84941 T^{7} - 675323 T^{8} - 7414256 T^{9} - 6365529 T^{10} + 143578043 T^{11} + 594823321 T^{12} )^{2}$$)($$( 1 - T - 75 T^{2} - 186 T^{3} + 3234 T^{4} + 10969 T^{5} - 64920 T^{6} - 187910 T^{7} + 1140565 T^{8} - 5449390 T^{9} - 54597720 T^{10} + 267522941 T^{11} + 2287346754 T^{12} - 3815073714 T^{13} - 44611749075 T^{14} - 17249876309 T^{15} + 500246412961 T^{16} )^{2}$$)
$31$ ($$( 1 - 31 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 8 T + 31 T^{2} + 208 T^{3} - 158 T^{4} - 7756 T^{5} - 34897 T^{6} - 240436 T^{7} - 151838 T^{8} + 6196528 T^{9} + 28629151 T^{10} + 229033208 T^{11} + 887503681 T^{12} )^{2}$$)($$( 1 - 4 T - 66 T^{2} + 362 T^{3} + 1939 T^{4} - 11745 T^{5} - 45361 T^{6} + 144883 T^{7} + 1472277 T^{8} + 4491373 T^{9} - 43591921 T^{10} - 349895295 T^{11} + 1790707219 T^{12} + 10363752662 T^{13} - 58575242946 T^{14} - 110050456444 T^{15} + 852891037441 T^{16} )^{2}$$)
$37$ ($$( 1 - 58 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 162 T^{2} + 11847 T^{4} - 534412 T^{6} + 16218543 T^{8} - 303614082 T^{10} + 2565726409 T^{12} )^{2}$$)($$( 1 - 97 T^{2} + 3667 T^{4} - 33124 T^{6} - 1092101 T^{8} - 45346756 T^{10} + 6872548387 T^{12} - 248875461673 T^{14} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$( 1 + 5 T - 16 T^{2} + 205 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - 13 T + 27 T^{2} + 292 T^{3} + 445 T^{4} - 22279 T^{5} + 169790 T^{6} - 913439 T^{7} + 748045 T^{8} + 20124932 T^{9} + 76295547 T^{10} - 1506130613 T^{11} + 4750104241 T^{12} )^{2}$$)($$( 1 - 5 T - 114 T^{2} + 213 T^{3} + 9222 T^{4} - 3430 T^{5} - 507867 T^{6} + 107426 T^{7} + 20984173 T^{8} + 4404466 T^{9} - 853724427 T^{10} - 236399030 T^{11} + 26059167942 T^{12} + 24677370813 T^{13} - 541511883474 T^{14} - 973771369405 T^{15} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$( 1 - 61 T^{2} + 1849 T^{4} )( 1 + 83 T^{2} + 1849 T^{4} )$$)($$1 + 150 T^{2} + 13245 T^{4} + 587794 T^{6} + 8508330 T^{8} - 916818210 T^{10} - 62473985643 T^{12} - 1695196870290 T^{14} + 29088287112330 T^{16} + 3715659272023906 T^{18} + 154810212676825245 T^{20} + 3241722346992637350 T^{22} + 39959630797262576401 T^{24}$$)($$1 + 148 T^{2} + 8022 T^{4} + 218216 T^{6} + 12046721 T^{8} + 1190020152 T^{10} + 65107245958 T^{12} + 1936378110508 T^{14} + 52508765640996 T^{16} + 3580363126329292 T^{18} + 222588717588456358 T^{20} + 7522549416418163448 T^{22} +$$$$14\!\cdots\!21$$$$T^{24} +$$$$47\!\cdots\!84$$$$T^{26} +$$$$32\!\cdots\!22$$$$T^{28} +$$$$10\!\cdots\!52$$$$T^{30} +$$$$13\!\cdots\!01$$$$T^{32}$$)
$47$ ($$1 + 45 T^{2} - 184 T^{4} + 99405 T^{6} + 4879681 T^{8}$$)($$1 + 91 T^{2} + 1339 T^{4} - 12268 T^{6} + 3554941 T^{8} - 111607535 T^{10} - 21624150754 T^{12} - 246541044815 T^{14} + 17346978053821 T^{16} - 132239413656172 T^{18} + 31883312840097979 T^{20} + 4786521033460534459 T^{22} +$$$$11\!\cdots\!41$$$$T^{24}$$)($$1 + 190 T^{2} + 18813 T^{4} + 1189930 T^{6} + 47942596 T^{8} + 692067060 T^{10} - 71396434957 T^{12} - 7544792481460 T^{14} - 431982830702721 T^{16} - 16666446591545140 T^{18} - 348391827127408717 T^{20} + 7459939861847962740 T^{22} +$$$$11\!\cdots\!56$$$$T^{24} +$$$$62\!\cdots\!70$$$$T^{26} +$$$$21\!\cdots\!33$$$$T^{28} +$$$$48\!\cdots\!10$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$)
$53$ ($$( 1 - 102 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 274 T^{2} + 33303 T^{4} - 2287964 T^{6} + 93548127 T^{8} - 2161991794 T^{10} + 22164361129 T^{12} )^{2}$$)($$( 1 - 196 T^{2} + 19762 T^{4} - 1503447 T^{6} + 91143129 T^{8} - 4223182623 T^{10} + 155931685522 T^{12} - 4344214781284 T^{14} + 62259690411361 T^{16} )^{2}$$)
$59$ ($$( 1 + 14 T + 137 T^{2} + 826 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 2 T - 153 T^{2} - 110 T^{3} + 14962 T^{4} + 3194 T^{5} - 1012513 T^{6} + 188446 T^{7} + 52082722 T^{8} - 22591690 T^{9} - 1853956233 T^{10} + 1429848598 T^{11} + 42180533641 T^{12} )^{2}$$)($$( 1 - 17 T + 51 T^{2} + 144 T^{3} + 2220 T^{4} + 17291 T^{5} - 531846 T^{6} + 1855556 T^{7} + 4239943 T^{8} + 109477804 T^{9} - 1851355926 T^{10} + 3551208289 T^{11} + 26900541420 T^{12} + 102949099056 T^{13} + 2151207215691 T^{14} - 42307075241923 T^{15} + 146830437604321 T^{16} )^{2}$$)
$61$ ($$( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + T - 145 T^{2} - 240 T^{3} + 12217 T^{4} + 14087 T^{5} - 812786 T^{6} + 859307 T^{7} + 45459457 T^{8} - 54475440 T^{9} - 2007646945 T^{10} + 844596301 T^{11} + 51520374361 T^{12} )^{2}$$)($$( 1 - 13 T - 72 T^{2} + 1443 T^{3} + 6378 T^{4} - 120042 T^{5} - 78037 T^{6} + 2420938 T^{7} + 9620433 T^{8} + 147677218 T^{9} - 290375677 T^{10} - 27247253202 T^{11} + 88308773898 T^{12} + 1218752462343 T^{13} - 3709466953992 T^{14} - 40855656868273 T^{15} + 191707312997281 T^{16} )^{2}$$)
$67$ ($$1 + 125 T^{2} + 11136 T^{4} + 561125 T^{6} + 20151121 T^{8}$$)($$1 + 203 T^{2} + 14531 T^{4} + 1222044 T^{6} + 166308757 T^{8} + 11407412521 T^{10} + 552166785598 T^{12} + 51207874806769 T^{14} + 3351307885666597 T^{16} + 110544123179333436 T^{18} + 5900569422575550371 T^{20} +$$$$37\!\cdots\!47$$$$T^{22} +$$$$81\!\cdots\!61$$$$T^{24}$$)($$1 + 319 T^{2} + 51321 T^{4} + 5350950 T^{6} + 412923954 T^{8} + 28055592807 T^{10} + 2076453240410 T^{12} + 167751222063326 T^{14} + 12304355330991753 T^{16} + 753035235842270414 T^{18} + 41842860498343999610 T^{20} +$$$$25\!\cdots\!83$$$$T^{22} +$$$$16\!\cdots\!14$$$$T^{24} +$$$$97\!\cdots\!50$$$$T^{26} +$$$$41\!\cdots\!81$$$$T^{28} +$$$$11\!\cdots\!51$$$$T^{30} +$$$$16\!\cdots\!81$$$$T^{32}$$)
$71$ ($$( 1 - 2 T + 71 T^{2} )^{4}$$)($$( 1 + 10 T + 121 T^{2} + 712 T^{3} + 8591 T^{4} + 50410 T^{5} + 357911 T^{6} )^{4}$$)($$( 1 + 8 T + 244 T^{2} + 1441 T^{3} + 24947 T^{4} + 102311 T^{5} + 1230004 T^{6} + 2863288 T^{7} + 25411681 T^{8} )^{4}$$)
$73$ ($$( 1 - 130 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 246 T^{2} + 30015 T^{4} - 2521972 T^{6} + 159949935 T^{8} - 6985967286 T^{10} + 151334226289 T^{12} )^{2}$$)($$( 1 - 388 T^{2} + 71842 T^{4} - 8513491 T^{6} + 722371129 T^{8} - 45368393539 T^{10} + 2040186429922 T^{12} - 58717679800132 T^{14} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$( 1 + 6 T - 43 T^{2} + 474 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 2 T - 149 T^{2} + 374 T^{3} + 10642 T^{4} - 16154 T^{5} - 772597 T^{6} - 1276166 T^{7} + 66416722 T^{8} + 184396586 T^{9} - 5803562069 T^{10} - 6154112798 T^{11} + 243087455521 T^{12} )^{2}$$)($$( 1 + 7 T - 234 T^{2} - 1199 T^{3} + 36520 T^{4} + 121434 T^{5} - 3999001 T^{6} - 3702556 T^{7} + 362101113 T^{8} - 292501924 T^{9} - 24957765241 T^{10} + 59871697926 T^{11} + 1422456958120 T^{12} - 3689390622401 T^{13} - 56882464591914 T^{14} + 134427362903113 T^{15} + 1517108809906561 T^{16} )^{2}$$)
$83$ ($$1 + 85 T^{2} + 336 T^{4} + 585565 T^{6} + 47458321 T^{8}$$)($$1 + 327 T^{2} + 57207 T^{4} + 6488176 T^{6} + 532197333 T^{8} + 34415866233 T^{10} + 2415401371734 T^{12} + 237090902479137 T^{14} + 25257191864857893 T^{16} + 2121246683923784944 T^{18} +$$$$12\!\cdots\!87$$$$T^{20} +$$$$50\!\cdots\!23$$$$T^{22} +$$$$10\!\cdots\!61$$$$T^{24}$$)($$1 + 340 T^{2} + 56358 T^{4} + 5615330 T^{6} + 356102531 T^{8} + 15268102785 T^{10} + 850761191503 T^{12} + 106079060220415 T^{14} + 11086543633870179 T^{16} + 730778645858438935 T^{18} + 40375697720691846463 T^{20} +$$$$49\!\cdots\!65$$$$T^{22} +$$$$80\!\cdots\!71$$$$T^{24} +$$$$87\!\cdots\!70$$$$T^{26} +$$$$60\!\cdots\!38$$$$T^{28} +$$$$25\!\cdots\!60$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$)
$89$ ($$( 1 - 15 T + 89 T^{2} )^{4}$$)($$( 1 - 3 T + 89 T^{2} )^{12}$$)($$( 1 - 9 T + 257 T^{2} - 1998 T^{3} + 31929 T^{4} - 177822 T^{5} + 2035697 T^{6} - 6344721 T^{7} + 62742241 T^{8} )^{4}$$)
$97$ ($$1 + 190 T^{2} + 26691 T^{4} + 1787710 T^{6} + 88529281 T^{8}$$)($$1 + 186 T^{2} - 1131 T^{4} - 466418 T^{6} + 278880666 T^{8} + 18455439810 T^{10} - 762781062579 T^{12} + 173647233172290 T^{14} + 24689104845781146 T^{16} - 388513136594974322 T^{18} - 8864137395240342891 T^{20} +$$$$13\!\cdots\!14$$$$T^{22} +$$$$69\!\cdots\!41$$$$T^{24}$$)($$1 + 577 T^{2} + 174522 T^{4} + 37353959 T^{6} + 6361534046 T^{8} + 912884200698 T^{10} + 114408541714153 T^{12} + 12837001127592472 T^{14} + 1305837592754805081 T^{16} +$$$$12\!\cdots\!48$$$$T^{18} +$$$$10\!\cdots\!93$$$$T^{20} +$$$$76\!\cdots\!42$$$$T^{22} +$$$$49\!\cdots\!06$$$$T^{24} +$$$$27\!\cdots\!91$$$$T^{26} +$$$$12\!\cdots\!02$$$$T^{28} +$$$$37\!\cdots\!13$$$$T^{30} +$$$$61\!\cdots\!21$$$$T^{32}$$)