# Properties

 Label 2016.2.cr Level 2016 Weight 2 Character orbit cr Rep. character $$\chi_{2016}(1297,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 76 Newform subspaces 5 Sturm bound 768 Trace bound 25

# Related objects

## Defining parameters

 Level: $$N$$ = $$2016 = 2^{5} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 2016.cr (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$56$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$5$$ Sturm bound: $$768$$ Trace bound: $$25$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(2016, [\chi])$$.

Total New Old
Modular forms 832 84 748
Cusp forms 704 76 628
Eisenstein series 128 8 120

## Trace form

 $$76q + 4q^{7} + O(q^{10})$$ $$76q + 4q^{7} + 2q^{17} - 6q^{23} + 28q^{25} - 10q^{31} + 8q^{41} + 6q^{47} + 4q^{49} - 4q^{55} - 8q^{65} - 48q^{71} + 6q^{73} + 6q^{79} + 10q^{89} - 10q^{95} - 56q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(2016, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
2016.2.cr.a $$8$$ $$16.098$$ 8.0.12960000.1 None $$0$$ $$0$$ $$0$$ $$4$$ $$q+\beta _{4}q^{5}+(-1-3\beta _{2})q^{7}+\beta _{1}q^{11}+\cdots$$
2016.2.cr.b $$8$$ $$16.098$$ $$\Q(\zeta_{24})$$ $$\Q(\sqrt{-6})$$ $$0$$ $$0$$ $$0$$ $$4$$ $$q+(-\zeta_{24}^{2}+\zeta_{24}^{3}+2\zeta_{24}^{4})q^{5}+(1+\cdots)q^{7}+\cdots$$
2016.2.cr.c $$12$$ $$16.098$$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$4$$ $$q+(-\beta _{4}+\beta _{7})q^{5}+(\beta _{8}-\beta _{9})q^{7}+(\beta _{10}+\cdots)q^{11}+\cdots$$
2016.2.cr.d $$16$$ $$16.098$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+\beta _{1}q^{5}+(-1+\beta _{4}+2\beta _{6}-\beta _{12}+\cdots)q^{7}+\cdots$$
2016.2.cr.e $$32$$ $$16.098$$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(2016, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(2016, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(224, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(504, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(672, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ ($$( 1 + 5 T^{2} )^{4}( 1 - 5 T^{2} + 25 T^{4} )^{2}$$)($$( 1 - 2 T^{2} + 25 T^{4} )^{2}( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} )$$)($$1 + 17 T^{2} + 147 T^{4} + 788 T^{6} + 2793 T^{8} + 5371 T^{10} + 7494 T^{12} + 134275 T^{14} + 1745625 T^{16} + 12312500 T^{18} + 57421875 T^{20} + 166015625 T^{22} + 244140625 T^{24}$$)($$( 1 + 12 T^{2} + 66 T^{4} + 336 T^{6} + 1859 T^{8} + 8400 T^{10} + 41250 T^{12} + 187500 T^{14} + 390625 T^{16} )^{2}$$)
$7$ ($$( 1 - T + 7 T^{2} )^{4}$$)($$( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} )^{2}$$)($$( 1 - 2 T + 5 T^{2} - 32 T^{3} + 35 T^{4} - 98 T^{5} + 343 T^{6} )^{2}$$)($$( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} )^{4}$$)
$11$ ($$( 1 + 17 T^{2} + 168 T^{4} + 2057 T^{6} + 14641 T^{8} )^{2}$$)($$( 1 + 10 T^{2} + 121 T^{4} )^{2}( 1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8} )$$)($$1 + 29 T^{2} + 327 T^{4} + 2696 T^{6} + 24957 T^{8} - 44693 T^{10} - 3466650 T^{12} - 5407853 T^{14} + 365395437 T^{16} + 4776128456 T^{18} + 70095354087 T^{20} + 752185313429 T^{22} + 3138428376721 T^{24}$$)($$( 1 + 12 T^{2} - 6 T^{4} - 1104 T^{6} - 9565 T^{8} - 133584 T^{10} - 87846 T^{12} + 21258732 T^{14} + 214358881 T^{16} )^{2}$$)
$13$ ($$( 1 - 13 T^{2} )^{8}$$)($$( 1 - 13 T^{2} )^{8}$$)($$( 1 - 46 T^{2} + 1191 T^{4} - 18804 T^{6} + 201279 T^{8} - 1313806 T^{10} + 4826809 T^{12} )^{2}$$)($$( 1 + 10 T^{2} + 235 T^{4} + 1690 T^{6} + 28561 T^{8} )^{4}$$)
$17$ ($$( 1 - 22 T^{2} + 195 T^{4} - 6358 T^{6} + 83521 T^{8} )^{2}$$)($$( 1 - 17 T^{2} + 289 T^{4} )^{4}$$)($$( 1 - T - 33 T^{2} - 8 T^{3} + 565 T^{4} + 425 T^{5} - 9946 T^{6} + 7225 T^{7} + 163285 T^{8} - 39304 T^{9} - 2756193 T^{10} - 1419857 T^{11} + 24137569 T^{12} )^{2}$$)($$( 1 - 12 T^{2} + 178 T^{4} + 7344 T^{6} - 123981 T^{8} + 2122416 T^{10} + 14866738 T^{12} - 289650828 T^{14} + 6975757441 T^{16} )^{2}$$)
$19$ ($$( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} )^{2}( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} )^{2}$$)($$( 1 + 19 T^{2} + 361 T^{4} )^{4}$$)($$1 + 53 T^{2} + 1215 T^{4} + 18608 T^{6} + 219741 T^{8} - 1587413 T^{10} - 104375754 T^{12} - 573056093 T^{14} + 28636866861 T^{16} + 875429753648 T^{18} + 20635029094815 T^{20} + 324946511663453 T^{22} + 2213314919066161 T^{24}$$)($$( 1 + 66 T^{2} + 2553 T^{4} + 71346 T^{6} + 1547972 T^{8} + 25755906 T^{10} + 332709513 T^{12} + 3105028146 T^{14} + 16983563041 T^{16} )^{2}$$)
$23$ ($$( 1 + 2 T^{2} - 525 T^{4} + 1058 T^{6} + 279841 T^{8} )^{2}$$)($$( 1 - 23 T^{2} + 529 T^{4} )^{4}$$)($$( 1 - T - 61 T^{2} + 12 T^{3} + 2381 T^{4} + 29 T^{5} - 63238 T^{6} + 667 T^{7} + 1259549 T^{8} + 146004 T^{9} - 17070301 T^{10} - 6436343 T^{11} + 148035889 T^{12} )^{2}$$)($$( 1 - 68 T^{2} + 2418 T^{4} - 78064 T^{6} + 2140499 T^{8} - 41295856 T^{10} + 676655538 T^{12} - 10066440452 T^{14} + 78310985281 T^{16} )^{2}$$)
$29$ ($$( 1 - 53 T^{2} + 841 T^{4} )^{4}$$)($$( 1 + 50 T^{2} + 1659 T^{4} + 42050 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 102 T^{2} + 4567 T^{4} - 141988 T^{6} + 3840847 T^{8} - 72142662 T^{10} + 594823321 T^{12} )^{2}$$)($$( 1 - 52 T^{2} + 1846 T^{4} - 43732 T^{6} + 707281 T^{8} )^{4}$$)
$31$ ($$( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} )^{4}$$)($$( 1 - 10 T + 31 T^{2} )^{4}( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} )^{2}$$)($$( 1 + 5 T - 71 T^{2} - 126 T^{3} + 4725 T^{4} + 4133 T^{5} - 158330 T^{6} + 128123 T^{7} + 4540725 T^{8} - 3753666 T^{9} - 65569991 T^{10} + 143145755 T^{11} + 887503681 T^{12} )^{2}$$)($$( 1 + 10 T + 15 T^{2} + 230 T^{3} + 3164 T^{4} + 7130 T^{5} + 14415 T^{6} + 297910 T^{7} + 923521 T^{8} )^{4}$$)
$37$ ($$( 1 + 14 T^{2} - 1173 T^{4} + 19166 T^{6} + 1874161 T^{8} )^{2}$$)($$( 1 + 37 T^{2} + 1369 T^{4} )^{4}$$)($$1 + 161 T^{2} + 13515 T^{4} + 840428 T^{6} + 43791513 T^{8} + 2009438227 T^{10} + 80476513014 T^{12} + 2750920932763 T^{14} + 82072345795593 T^{16} + 2156308314463052 T^{18} + 47471159819742315 T^{20} + 774182083959273689 T^{22} + 6582952005840035281 T^{24}$$)($$( 1 + 6 T^{2} - 2583 T^{4} - 714 T^{6} + 4935716 T^{8} - 977466 T^{10} - 4840957863 T^{12} + 15394358454 T^{14} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$( 1 - 26 T^{2} + 1681 T^{4} )^{4}$$)($$( 1 + 41 T^{2} )^{8}$$)($$( 1 - 2 T + 83 T^{2} - 80 T^{3} + 3403 T^{4} - 3362 T^{5} + 68921 T^{6} )^{4}$$)($$( 1 + 68 T^{2} + 4390 T^{4} + 114308 T^{6} + 2825761 T^{8} )^{4}$$)
$43$ ($$( 1 - 43 T^{2} )^{8}$$)($$( 1 - 43 T^{2} )^{8}$$)($$( 1 - 94 T^{2} + 3543 T^{4} - 112068 T^{6} + 6551007 T^{8} - 321367294 T^{10} + 6321363049 T^{12} )^{2}$$)($$( 1 - 50 T^{2} + 1435 T^{4} - 92450 T^{6} + 3418801 T^{8} )^{4}$$)
$47$ ($$( 1 - 82 T^{2} + 4515 T^{4} - 181138 T^{6} + 4879681 T^{8} )^{2}$$)($$( 1 - 47 T^{2} + 2209 T^{4} )^{4}$$)($$( 1 - 15 T + 57 T^{2} - 78 T^{3} + 2013 T^{4} - 1383 T^{5} - 125066 T^{6} - 65001 T^{7} + 4446717 T^{8} - 8098194 T^{9} + 278141817 T^{10} - 3440175105 T^{11} + 10779215329 T^{12} )^{2}$$)($$( 1 + 28 T^{2} - 3182 T^{4} - 12656 T^{6} + 9117619 T^{8} - 27957104 T^{10} - 15527144942 T^{12} + 301818029212 T^{14} + 23811286661761 T^{16} )^{2}$$)
$53$ ($$( 1 - 19 T^{2} - 2448 T^{4} - 53371 T^{6} + 7890481 T^{8} )^{2}$$)($$( 1 + 94 T^{2} + 2809 T^{4} )^{2}( 1 - 94 T^{2} + 6027 T^{4} - 264046 T^{6} + 7890481 T^{8} )$$)($$1 + 161 T^{2} + 12171 T^{4} + 564332 T^{6} + 17660601 T^{8} + 173268691 T^{10} - 11132063562 T^{12} + 486711753019 T^{14} + 139350636639081 T^{16} + 12508058244650828 T^{18} + 757762691996674731 T^{20} + 28156882728847600889 T^{22} +$$$$49\!\cdots\!41$$$$T^{24}$$)($$( 1 + 108 T^{2} + 5442 T^{4} + 65232 T^{6} - 1941373 T^{8} + 183236688 T^{10} + 42939997602 T^{12} + 2393751001932 T^{14} + 62259690411361 T^{16} )^{2}$$)
$59$ ($$( 1 + 113 T^{2} + 9288 T^{4} + 393353 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 + 10 T^{2} + 3481 T^{4} )^{2}( 1 - 10 T^{2} - 3381 T^{4} - 34810 T^{6} + 12117361 T^{8} )$$)($$1 + 177 T^{2} + 16739 T^{4} + 834316 T^{6} + 7372221 T^{8} - 3081613133 T^{10} - 272281952338 T^{12} - 10727095315973 T^{14} + 89331863228781 T^{16} + 35191894105224556 T^{18} + 2457794695058729219 T^{20} + 90467665334213527977 T^{22} +$$$$17\!\cdots\!81$$$$T^{24}$$)($$( 1 + 100 T^{2} + 930 T^{4} + 210800 T^{6} + 35337539 T^{8} + 733794800 T^{10} + 11269145730 T^{12} + 4218053364100 T^{14} + 146830437604321 T^{16} )^{2}$$)
$61$ ($$( 1 + 62 T^{2} + 123 T^{4} + 230702 T^{6} + 13845841 T^{8} )^{2}$$)($$( 1 + 61 T^{2} + 3721 T^{4} )^{4}$$)($$1 + 73 T^{2} - 3941 T^{4} - 121236 T^{6} + 22314201 T^{8} - 326849717 T^{10} - 131320604138 T^{12} - 1216207796957 T^{14} + 308958879088041 T^{16} - 6246124106030196 T^{18} - 755518520522284421 T^{20} + 52074032551390429873 T^{22} +$$$$26\!\cdots\!21$$$$T^{24}$$)($$( 1 + 164 T^{2} + 13242 T^{4} + 1018768 T^{6} + 72505859 T^{8} + 3790835728 T^{10} + 183346626522 T^{12} + 8449341395204 T^{14} + 191707312997281 T^{16} )^{2}$$)
$67$ ($$( 1 + 74 T^{2} + 987 T^{4} + 332186 T^{6} + 20151121 T^{8} )^{2}$$)($$( 1 + 67 T^{2} + 4489 T^{4} )^{4}$$)($$1 + 361 T^{2} + 73723 T^{4} + 10542804 T^{6} + 1164287325 T^{8} + 103688059123 T^{10} + 7623484120510 T^{12} + 465455697403147 T^{14} + 23461694764841325 T^{16} + 953684993364861876 T^{18} + 29936527392508244443 T^{20} +$$$$65\!\cdots\!89$$$$T^{22} +$$$$81\!\cdots\!61$$$$T^{24}$$)($$( 1 + 98 T^{2} + 537 T^{4} + 8722 T^{6} + 18947012 T^{8} + 39153058 T^{10} + 10821151977 T^{12} + 8864921452562 T^{14} + 406067677556641 T^{16} )^{2}$$)
$71$ ($$( 1 + 34 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 + 71 T^{2} )^{8}$$)($$( 1 - 8 T + 157 T^{2} - 704 T^{3} + 11147 T^{4} - 40328 T^{5} + 357911 T^{6} )^{4}$$)($$( 1 + 36 T^{2} + 1694 T^{4} + 181476 T^{6} + 25411681 T^{8} )^{4}$$)
$73$ ($$( 1 - 17 T + 73 T^{2} )^{4}( 1 + 7 T + 73 T^{2} )^{4}$$)($$( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} )^{4}$$)($$( 1 + 5 T - 101 T^{2} + 52 T^{3} + 5233 T^{4} - 28921 T^{5} - 430618 T^{6} - 2111233 T^{7} + 27886657 T^{8} + 20228884 T^{9} - 2868222341 T^{10} + 10365357965 T^{11} + 151334226289 T^{12} )^{2}$$)($$( 1 - 3 T - 64 T^{2} - 219 T^{3} + 5329 T^{4} )^{8}$$)
$79$ ($$( 1 - 4 T + 79 T^{2} )^{4}( 1 + 17 T + 79 T^{2} )^{4}$$)($$( 1 - 10 T + 79 T^{2} )^{4}( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 11 T - 137 T^{2} + 656 T^{3} + 26965 T^{4} - 70973 T^{5} - 1975630 T^{6} - 5606867 T^{7} + 168288565 T^{8} + 323433584 T^{9} - 5336161097 T^{10} - 33847620389 T^{11} + 243087455521 T^{12} )^{2}$$)($$( 1 - 6 T - 113 T^{2} + 54 T^{3} + 13116 T^{4} + 4266 T^{5} - 705233 T^{6} - 2958234 T^{7} + 38950081 T^{8} )^{4}$$)
$83$ ($$( 1 - 41 T^{2} + 6889 T^{4} )^{4}$$)($$( 1 + 134 T^{2} + 11067 T^{4} + 923126 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 - 446 T^{2} + 86503 T^{4} - 9357636 T^{6} + 595919167 T^{8} - 21166411166 T^{10} + 326940373369 T^{12} )^{2}$$)($$( 1 - 132 T^{2} + 15822 T^{4} - 909348 T^{6} + 47458321 T^{8} )^{4}$$)
$89$ ($$( 1 + 14 T^{2} - 7725 T^{4} + 110894 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 - 89 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 - 5 T - 133 T^{2} + 1452 T^{3} + 5297 T^{4} - 76103 T^{5} + 246758 T^{6} - 6773167 T^{7} + 41957537 T^{8} + 1023614988 T^{9} - 8344718053 T^{10} - 27920297245 T^{11} + 496981290961 T^{12} )^{2}$$)($$( 1 - 260 T^{2} + 34986 T^{4} - 4360720 T^{6} + 465471155 T^{8} - 34541263120 T^{10} + 2195100043626 T^{12} - 129215135649860 T^{14} + 3936588805702081 T^{16} )^{2}$$)
$97$ ($$( 1 + T + 97 T^{2} )^{8}$$)($$( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} )^{4}$$)($$( 1 - 10 T + 255 T^{2} - 1968 T^{3} + 24735 T^{4} - 94090 T^{5} + 912673 T^{6} )^{4}$$)($$( 1 + 4 T + 190 T^{2} + 388 T^{3} + 9409 T^{4} )^{8}$$)