# Properties

 Label 3150.2.bf Level 3150 Weight 2 Character orbit bf Rep. character $$\chi_{3150}(1151,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 104 Newform subspaces 6 Sturm bound 1440 Trace bound 11

# Related objects

## Defining parameters

 Level: $$N$$ = $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3150.bf (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$21$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$6$$ Sturm bound: $$1440$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$11$$, $$37$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3150, [\chi])$$.

Total New Old
Modular forms 1536 104 1432
Cusp forms 1344 104 1240
Eisenstein series 192 0 192

## Trace form

 $$104q + 52q^{4} - 4q^{7} + O(q^{10})$$ $$104q + 52q^{4} - 4q^{7} - 52q^{16} - 48q^{19} + 24q^{22} + 4q^{28} + 12q^{31} - 32q^{37} - 16q^{46} - 28q^{49} + 20q^{58} + 48q^{61} - 104q^{64} + 8q^{67} - 24q^{73} - 20q^{79} - 48q^{82} + 12q^{88} + 16q^{91} + 24q^{94} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3150, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
3150.2.bf.a $$8$$ $$25.153$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\zeta_{24}q^{2}+\zeta_{24}^{2}q^{4}+(-\zeta_{24}^{2}+\zeta_{24}^{4}+\cdots)q^{7}+\cdots$$
3150.2.bf.b $$8$$ $$25.153$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{24}^{2}q^{2}+\zeta_{24}^{4}q^{4}+(\zeta_{24}-3\zeta_{24}^{5}+\cdots)q^{7}+\cdots$$
3150.2.bf.c $$8$$ $$25.153$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}^{2}q^{2}+\zeta_{24}^{4}q^{4}+(-\zeta_{24}+3\zeta_{24}^{5}+\cdots)q^{7}+\cdots$$
3150.2.bf.d $$24$$ $$25.153$$ None $$0$$ $$0$$ $$0$$ $$-4$$
3150.2.bf.e $$24$$ $$25.153$$ None $$0$$ $$0$$ $$0$$ $$4$$
3150.2.bf.f $$32$$ $$25.153$$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3150, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3150, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(315, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(525, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(630, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1050, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1575, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)
$3$ 1
$5$ 1
$7$ ($$( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} )^{2}$$)($$1 - 94 T^{4} + 2401 T^{8}$$)($$1 - 94 T^{4} + 2401 T^{8}$$)
$11$ ($$( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} )^{2}$$)($$1 + 24 T + 304 T^{2} + 2688 T^{3} + 18481 T^{4} + 104232 T^{5} + 496816 T^{6} + 2036016 T^{7} + 7239280 T^{8} + 22396176 T^{9} + 60114736 T^{10} + 138732792 T^{11} + 270580321 T^{12} + 432905088 T^{13} + 538554544 T^{14} + 467692104 T^{15} + 214358881 T^{16}$$)($$1 - 24 T + 304 T^{2} - 2688 T^{3} + 18481 T^{4} - 104232 T^{5} + 496816 T^{6} - 2036016 T^{7} + 7239280 T^{8} - 22396176 T^{9} + 60114736 T^{10} - 138732792 T^{11} + 270580321 T^{12} - 432905088 T^{13} + 538554544 T^{14} - 467692104 T^{15} + 214358881 T^{16}$$)
$13$ ($$( 1 - 20 T^{2} + 169 T^{4} )^{4}$$)($$1 - 80 T^{2} + 3006 T^{4} - 69568 T^{6} + 1087139 T^{8} - 11756992 T^{10} + 85854366 T^{12} - 386144720 T^{14} + 815730721 T^{16}$$)($$1 - 80 T^{2} + 3006 T^{4} - 69568 T^{6} + 1087139 T^{8} - 11756992 T^{10} + 85854366 T^{12} - 386144720 T^{14} + 815730721 T^{16}$$)
$17$ ($$1 - 32 T^{2} + 478 T^{4} + 1024 T^{6} - 81341 T^{8} + 295936 T^{10} + 39923038 T^{12} - 772402208 T^{14} + 6975757441 T^{16}$$)($$1 - 28 T^{2} + 192 T^{3} + 442 T^{4} - 4320 T^{5} + 15824 T^{6} + 58560 T^{7} - 349805 T^{8} + 995520 T^{9} + 4573136 T^{10} - 21224160 T^{11} + 36916282 T^{12} + 272612544 T^{13} - 675851932 T^{14} + 6975757441 T^{16}$$)($$1 - 28 T^{2} - 192 T^{3} + 442 T^{4} + 4320 T^{5} + 15824 T^{6} - 58560 T^{7} - 349805 T^{8} - 995520 T^{9} + 4573136 T^{10} + 21224160 T^{11} + 36916282 T^{12} - 272612544 T^{13} - 675851932 T^{14} + 6975757441 T^{16}$$)
$19$ ($$( 1 + 12 T + 92 T^{2} + 528 T^{3} + 2487 T^{4} + 10032 T^{5} + 33212 T^{6} + 82308 T^{7} + 130321 T^{8} )^{2}$$)($$1 + 58 T^{2} + 1881 T^{4} - 1872 T^{5} + 44906 T^{6} - 71184 T^{7} + 869156 T^{8} - 1352496 T^{9} + 16211066 T^{10} - 12840048 T^{11} + 245133801 T^{12} + 2728661098 T^{14} + 16983563041 T^{16}$$)($$1 + 58 T^{2} + 1881 T^{4} - 1872 T^{5} + 44906 T^{6} - 71184 T^{7} + 869156 T^{8} - 1352496 T^{9} + 16211066 T^{10} - 12840048 T^{11} + 245133801 T^{12} + 2728661098 T^{14} + 16983563041 T^{16}$$)
$23$ ($$( 1 + 28 T^{2} + 255 T^{4} + 14812 T^{6} + 279841 T^{8} )^{2}$$)($$( 1 - 6 T + 57 T^{2} - 270 T^{3} + 1772 T^{4} - 6210 T^{5} + 30153 T^{6} - 73002 T^{7} + 279841 T^{8} )^{2}$$)($$( 1 + 6 T + 57 T^{2} + 270 T^{3} + 1772 T^{4} + 6210 T^{5} + 30153 T^{6} + 73002 T^{7} + 279841 T^{8} )^{2}$$)
$29$ ($$( 1 - 62 T^{2} + 1995 T^{4} - 52142 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 36 T^{2} + 470 T^{4} - 30276 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 36 T^{2} + 470 T^{4} - 30276 T^{6} + 707281 T^{8} )^{2}$$)
$31$ ($$( 1 + 6 T + 23 T^{2} + 66 T^{3} - 468 T^{4} + 2046 T^{5} + 22103 T^{6} + 178746 T^{7} + 923521 T^{8} )^{2}$$)($$1 + 100 T^{2} + 5706 T^{4} + 237200 T^{6} + 7904915 T^{8} + 227949200 T^{10} + 5269610826 T^{12} + 88750368100 T^{14} + 852891037441 T^{16}$$)($$1 + 100 T^{2} + 5706 T^{4} + 237200 T^{6} + 7904915 T^{8} + 227949200 T^{10} + 5269610826 T^{12} + 88750368100 T^{14} + 852891037441 T^{16}$$)
$37$ ($$( 1 - 8 T - 8 T^{2} + 16 T^{3} + 1447 T^{4} + 592 T^{5} - 10952 T^{6} - 405224 T^{7} + 1874161 T^{8} )^{2}$$)($$1 + 24 T + 216 T^{2} + 1680 T^{3} + 21025 T^{4} + 178920 T^{5} + 986040 T^{6} + 7528032 T^{7} + 58723920 T^{8} + 278537184 T^{9} + 1349888760 T^{10} + 9062834760 T^{11} + 39404235025 T^{12} + 116497847760 T^{13} + 554196904344 T^{14} + 2278365051192 T^{15} + 3512479453921 T^{16}$$)($$1 + 24 T + 216 T^{2} + 1680 T^{3} + 21025 T^{4} + 178920 T^{5} + 986040 T^{6} + 7528032 T^{7} + 58723920 T^{8} + 278537184 T^{9} + 1349888760 T^{10} + 9062834760 T^{11} + 39404235025 T^{12} + 116497847760 T^{13} + 554196904344 T^{14} + 2278365051192 T^{15} + 3512479453921 T^{16}$$)
$41$ ($$( 1 + 20 T^{2} - 1146 T^{4} + 33620 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 + 16 T + 232 T^{2} + 2000 T^{3} + 15591 T^{4} + 82000 T^{5} + 389992 T^{6} + 1102736 T^{7} + 2825761 T^{8} )^{2}$$)($$( 1 - 16 T + 232 T^{2} - 2000 T^{3} + 15591 T^{4} - 82000 T^{5} + 389992 T^{6} - 1102736 T^{7} + 2825761 T^{8} )^{2}$$)
$43$ ($$( 1 + 8 T + 84 T^{2} + 344 T^{3} + 1849 T^{4} )^{4}$$)($$( 1 - 8 T + 180 T^{2} - 1000 T^{3} + 11750 T^{4} - 43000 T^{5} + 332820 T^{6} - 636056 T^{7} + 3418801 T^{8} )^{2}$$)($$( 1 - 8 T + 180 T^{2} - 1000 T^{3} + 11750 T^{4} - 43000 T^{5} + 332820 T^{6} - 636056 T^{7} + 3418801 T^{8} )^{2}$$)
$47$ ($$1 - 152 T^{2} + 13198 T^{4} - 834176 T^{6} + 42212419 T^{8} - 1842694784 T^{10} + 64402029838 T^{12} - 1638440730008 T^{14} + 23811286661761 T^{16}$$)($$1 + 8 T - 78 T^{2} - 240 T^{3} + 6305 T^{4} - 3648 T^{5} - 344398 T^{6} - 95176 T^{7} + 11426244 T^{8} - 4473272 T^{9} - 760775182 T^{10} - 378746304 T^{11} + 30766388705 T^{12} - 55042801680 T^{13} - 840778795662 T^{14} + 4052984963704 T^{15} + 23811286661761 T^{16}$$)($$1 - 8 T - 78 T^{2} + 240 T^{3} + 6305 T^{4} + 3648 T^{5} - 344398 T^{6} + 95176 T^{7} + 11426244 T^{8} + 4473272 T^{9} - 760775182 T^{10} + 378746304 T^{11} + 30766388705 T^{12} + 55042801680 T^{13} - 840778795662 T^{14} - 4052984963704 T^{15} + 23811286661761 T^{16}$$)
$53$ ($$1 + 158 T^{2} + 13753 T^{4} + 883694 T^{6} + 47672164 T^{8} + 2482296446 T^{10} + 108517785193 T^{12} + 3501969058382 T^{14} + 62259690411361 T^{16}$$)($$1 - 24 T + 362 T^{2} - 4080 T^{3} + 37881 T^{4} - 336960 T^{5} + 2978650 T^{6} - 25456152 T^{7} + 197950532 T^{8} - 1349176056 T^{9} + 8367027850 T^{10} - 50165593920 T^{11} + 298899310761 T^{12} - 1706237611440 T^{13} + 8023498728698 T^{14} - 28193067356088 T^{15} + 62259690411361 T^{16}$$)($$1 + 24 T + 362 T^{2} + 4080 T^{3} + 37881 T^{4} + 336960 T^{5} + 2978650 T^{6} + 25456152 T^{7} + 197950532 T^{8} + 1349176056 T^{9} + 8367027850 T^{10} + 50165593920 T^{11} + 298899310761 T^{12} + 1706237611440 T^{13} + 8023498728698 T^{14} + 28193067356088 T^{15} + 62259690411361 T^{16}$$)
$59$ ($$1 - 38 T^{2} - 4727 T^{4} + 30058 T^{6} + 20937316 T^{8} + 104631898 T^{10} - 57278765447 T^{12} - 1602860278358 T^{14} + 146830437604321 T^{16}$$)($$1 - 24 T + 224 T^{2} - 1392 T^{3} + 9358 T^{4} - 16056 T^{5} - 613888 T^{6} + 6806952 T^{7} - 48734957 T^{8} + 401610168 T^{9} - 2136944128 T^{10} - 3297565224 T^{11} + 113394264238 T^{12} - 995174624208 T^{13} + 9448439535584 T^{14} - 59727635635656 T^{15} + 146830437604321 T^{16}$$)($$1 + 24 T + 224 T^{2} + 1392 T^{3} + 9358 T^{4} + 16056 T^{5} - 613888 T^{6} - 6806952 T^{7} - 48734957 T^{8} - 401610168 T^{9} - 2136944128 T^{10} + 3297565224 T^{11} + 113394264238 T^{12} + 995174624208 T^{13} + 9448439535584 T^{14} + 59727635635656 T^{15} + 146830437604321 T^{16}$$)
$61$ ($$( 1 - 12 T + 176 T^{2} - 1536 T^{3} + 15591 T^{4} - 93696 T^{5} + 654896 T^{6} - 2723772 T^{7} + 13845841 T^{8} )^{2}$$)($$1 + 124 T^{2} + 6186 T^{4} + 30240 T^{5} + 293552 T^{6} + 4411200 T^{7} + 18080963 T^{8} + 269083200 T^{9} + 1092306992 T^{10} + 6863905440 T^{11} + 85650372426 T^{12} + 6388526420764 T^{14} + 191707312997281 T^{16}$$)($$1 + 124 T^{2} + 6186 T^{4} + 30240 T^{5} + 293552 T^{6} + 4411200 T^{7} + 18080963 T^{8} + 269083200 T^{9} + 1092306992 T^{10} + 6863905440 T^{11} + 85650372426 T^{12} + 6388526420764 T^{14} + 191707312997281 T^{16}$$)
$67$ ($$( 1 + 10 T + 33 T^{2} + 670 T^{3} + 4489 T^{4} )^{4}$$)($$1 - 24 T + 332 T^{2} - 3696 T^{3} + 30922 T^{4} - 206568 T^{5} + 963056 T^{6} - 1103736 T^{7} - 11820413 T^{8} - 73950312 T^{9} + 4323158384 T^{10} - 62128011384 T^{11} + 623112963562 T^{12} - 4990062395472 T^{13} + 30032182880108 T^{14} - 145457078527752 T^{15} + 406067677556641 T^{16}$$)($$1 - 24 T + 332 T^{2} - 3696 T^{3} + 30922 T^{4} - 206568 T^{5} + 963056 T^{6} - 1103736 T^{7} - 11820413 T^{8} - 73950312 T^{9} + 4323158384 T^{10} - 62128011384 T^{11} + 623112963562 T^{12} - 4990062395472 T^{13} + 30032182880108 T^{14} - 145457078527752 T^{15} + 406067677556641 T^{16}$$)
$71$ ($$( 1 - 176 T^{2} + 15234 T^{4} - 887216 T^{6} + 25411681 T^{8} )^{2}$$)($$1 - 280 T^{2} + 41532 T^{4} - 4371368 T^{6} + 352979654 T^{8} - 22036066088 T^{10} + 1055397935292 T^{12} - 35868079497880 T^{14} + 645753531245761 T^{16}$$)($$1 - 280 T^{2} + 41532 T^{4} - 4371368 T^{6} + 352979654 T^{8} - 22036066088 T^{10} + 1055397935292 T^{12} - 35868079497880 T^{14} + 645753531245761 T^{16}$$)
$73$ ($$( 1 + 12 T + 182 T^{2} + 1608 T^{3} + 16131 T^{4} + 117384 T^{5} + 969878 T^{6} + 4668204 T^{7} + 28398241 T^{8} )^{2}$$)($$1 + 156 T^{2} + 12202 T^{4} + 230256 T^{6} - 10850829 T^{8} + 1227034224 T^{10} + 346515336682 T^{12} + 23608139301084 T^{14} + 806460091894081 T^{16}$$)($$1 + 156 T^{2} + 12202 T^{4} + 230256 T^{6} - 10850829 T^{8} + 1227034224 T^{10} + 346515336682 T^{12} + 23608139301084 T^{14} + 806460091894081 T^{16}$$)
$79$ ($$( 1 - 14 T + 7 T^{2} - 434 T^{3} + 13996 T^{4} - 34286 T^{5} + 43687 T^{6} - 6902546 T^{7} + 38950081 T^{8} )^{2}$$)($$1 + 24 T + 252 T^{2} + 1584 T^{3} + 5050 T^{4} + 17832 T^{5} - 542160 T^{6} - 18801768 T^{7} - 226869549 T^{8} - 1485339672 T^{9} - 3383620560 T^{10} + 8791871448 T^{11} + 196697909050 T^{12} + 4874057336016 T^{13} + 61258038791292 T^{14} + 460893815667816 T^{15} + 1517108809906561 T^{16}$$)($$1 + 24 T + 252 T^{2} + 1584 T^{3} + 5050 T^{4} + 17832 T^{5} - 542160 T^{6} - 18801768 T^{7} - 226869549 T^{8} - 1485339672 T^{9} - 3383620560 T^{10} + 8791871448 T^{11} + 196697909050 T^{12} + 4874057336016 T^{13} + 61258038791292 T^{14} + 460893815667816 T^{15} + 1517108809906561 T^{16}$$)
$83$ ($$( 1 + 278 T^{2} + 32811 T^{4} + 1915142 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 + 8 T + 148 T^{2} + 1192 T^{3} + 18438 T^{4} + 98936 T^{5} + 1019572 T^{6} + 4574296 T^{7} + 47458321 T^{8} )^{2}$$)($$( 1 - 8 T + 148 T^{2} - 1192 T^{3} + 18438 T^{4} - 98936 T^{5} + 1019572 T^{6} - 4574296 T^{7} + 47458321 T^{8} )^{2}$$)
$89$ ($$( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} )^{2}$$)($$1 - 16 T - 48 T^{2} + 1824 T^{3} + 2078 T^{4} - 135504 T^{5} + 38912 T^{6} + 7970384 T^{7} - 72309597 T^{8} + 709364176 T^{9} + 308221952 T^{10} - 95526119376 T^{11} + 130378376798 T^{12} + 10185324434976 T^{13} - 23855101966128 T^{14} - 707701358328464 T^{15} + 3936588805702081 T^{16}$$)($$1 + 16 T - 48 T^{2} - 1824 T^{3} + 2078 T^{4} + 135504 T^{5} + 38912 T^{6} - 7970384 T^{7} - 72309597 T^{8} - 709364176 T^{9} + 308221952 T^{10} + 95526119376 T^{11} + 130378376798 T^{12} - 10185324434976 T^{13} - 23855101966128 T^{14} + 707701358328464 T^{15} + 3936588805702081 T^{16}$$)
$97$ ($$( 1 - 190 T^{2} + 20643 T^{4} - 1787710 T^{6} + 88529281 T^{8} )^{2}$$)($$1 + 24 T^{2} + 19036 T^{4} - 653400 T^{6} + 157960902 T^{8} - 6147840600 T^{10} + 1685243393116 T^{12} + 19991328118296 T^{14} + 7837433594376961 T^{16}$$)($$1 + 24 T^{2} + 19036 T^{4} - 653400 T^{6} + 157960902 T^{8} - 6147840600 T^{10} + 1685243393116 T^{12} + 19991328118296 T^{14} + 7837433594376961 T^{16}$$)