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Properties

Label 2009.2
Level 2009
Weight 2
Dimension 155912
Nonzero newspaces 32
Sturm bound 658560
Trace bound 3

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Defining parameters

Level: \( N \) = \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 32 \)
Sturm bound: \(658560\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(2009))\).

Total New Old
Modular forms 167040 159694 7346
Cusp forms 162241 155912 6329
Eisenstein series 4799 3782 1017

Trace form

\( 155912 q - 584 q^{2} - 586 q^{3} - 592 q^{4} - 590 q^{5} - 602 q^{6} - 692 q^{7} - 1052 q^{8} - 604 q^{9} + O(q^{10}) \) \( 155912 q - 584 q^{2} - 586 q^{3} - 592 q^{4} - 590 q^{5} - 602 q^{6} - 692 q^{7} - 1052 q^{8} - 604 q^{9} - 614 q^{10} - 602 q^{11} - 634 q^{12} - 606 q^{13} - 720 q^{14} - 1070 q^{15} - 640 q^{16} - 614 q^{17} - 656 q^{18} - 618 q^{19} - 662 q^{20} - 734 q^{21} - 1094 q^{22} - 626 q^{23} - 698 q^{24} - 640 q^{25} - 662 q^{26} - 658 q^{27} - 776 q^{28} - 1082 q^{29} - 762 q^{30} - 662 q^{31} - 774 q^{32} - 734 q^{33} - 736 q^{34} - 762 q^{35} - 1236 q^{36} - 658 q^{37} - 654 q^{38} - 672 q^{39} - 646 q^{40} - 619 q^{41} - 1398 q^{42} - 1046 q^{43} - 614 q^{44} - 604 q^{45} - 510 q^{46} - 650 q^{47} - 616 q^{48} - 608 q^{49} - 1888 q^{50} - 614 q^{51} - 536 q^{52} - 622 q^{53} - 606 q^{54} - 512 q^{55} - 636 q^{56} - 1098 q^{57} - 590 q^{58} - 614 q^{59} - 662 q^{60} - 604 q^{61} - 686 q^{62} - 804 q^{63} - 1132 q^{64} - 756 q^{65} - 946 q^{66} - 774 q^{67} - 930 q^{68} - 850 q^{69} - 930 q^{70} - 1246 q^{71} - 1148 q^{72} - 806 q^{73} - 926 q^{74} - 986 q^{75} - 1158 q^{76} - 846 q^{77} - 1342 q^{78} - 818 q^{79} - 1064 q^{80} - 842 q^{81} - 860 q^{82} - 1278 q^{83} - 608 q^{84} - 1240 q^{85} - 748 q^{86} - 562 q^{87} - 676 q^{88} - 670 q^{89} - 548 q^{90} - 706 q^{91} - 1184 q^{92} - 410 q^{93} - 668 q^{94} - 562 q^{95} - 484 q^{96} - 644 q^{97} - 384 q^{98} - 1856 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(2009))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
2009.2.a \(\chi_{2009}(1, \cdot)\) 2009.2.a.a 2 1
2009.2.a.b 2
2009.2.a.c 2
2009.2.a.d 2
2009.2.a.e 2
2009.2.a.f 2
2009.2.a.g 3
2009.2.a.h 3
2009.2.a.i 3
2009.2.a.j 3
2009.2.a.k 3
2009.2.a.l 5
2009.2.a.m 5
2009.2.a.n 5
2009.2.a.o 6
2009.2.a.p 7
2009.2.a.q 7
2009.2.a.r 17
2009.2.a.s 17
2009.2.a.t 20
2009.2.a.u 20
2009.2.c \(\chi_{2009}(491, \cdot)\) n/a 138 1
2009.2.e \(\chi_{2009}(165, \cdot)\) n/a 268 2
2009.2.f \(\chi_{2009}(50, \cdot)\) n/a 278 2
2009.2.h \(\chi_{2009}(344, \cdot)\) n/a 552 4
2009.2.j \(\chi_{2009}(655, \cdot)\) n/a 272 2
2009.2.l \(\chi_{2009}(288, \cdot)\) n/a 1128 6
2009.2.m \(\chi_{2009}(342, \cdot)\) n/a 544 4
2009.2.o \(\chi_{2009}(148, \cdot)\) n/a 552 4
2009.2.s \(\chi_{2009}(214, \cdot)\) n/a 544 4
2009.2.u \(\chi_{2009}(204, \cdot)\) n/a 1164 6
2009.2.w \(\chi_{2009}(18, \cdot)\) n/a 1088 8
2009.2.y \(\chi_{2009}(197, \cdot)\) n/a 1112 8
2009.2.z \(\chi_{2009}(247, \cdot)\) n/a 2232 12
2009.2.bb \(\chi_{2009}(68, \cdot)\) n/a 1088 8
2009.2.bd \(\chi_{2009}(155, \cdot)\) n/a 2328 12
2009.2.bg \(\chi_{2009}(312, \cdot)\) n/a 1088 8
2009.2.bh \(\chi_{2009}(57, \cdot)\) n/a 4656 24
2009.2.bj \(\chi_{2009}(48, \cdot)\) n/a 2176 16
2009.2.bl \(\chi_{2009}(81, \cdot)\) n/a 2328 12
2009.2.bo \(\chi_{2009}(27, \cdot)\) n/a 4656 24
2009.2.bp \(\chi_{2009}(128, \cdot)\) n/a 2176 16
2009.2.bt \(\chi_{2009}(64, \cdot)\) n/a 4656 24
2009.2.bu \(\chi_{2009}(9, \cdot)\) n/a 4656 24
2009.2.bw \(\chi_{2009}(16, \cdot)\) n/a 9312 48
2009.2.bx \(\chi_{2009}(19, \cdot)\) n/a 4352 32
2009.2.bz \(\chi_{2009}(8, \cdot)\) n/a 9312 48
2009.2.cb \(\chi_{2009}(3, \cdot)\) n/a 9312 48
2009.2.cd \(\chi_{2009}(4, \cdot)\) n/a 9312 48
2009.2.cg \(\chi_{2009}(6, \cdot)\) n/a 18624 96
2009.2.cj \(\chi_{2009}(2, \cdot)\) n/a 18624 96
2009.2.cl \(\chi_{2009}(12, \cdot)\) n/a 37248 192

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(2009))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(2009)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(41))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(49))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(287))\)\(^{\oplus 2}\)