LMFDB - Space of Modular Forms of Level 403, Weight 2, and Character 403.be

Defining parameters

Level:	$ N $	=	$ 403 = 13 \cdot 31 $
Weight:	$ k $	=	$ 2 $
Character orbit:	$[\chi]$	=	403.be (of order $12$ and degree $4$)
Character conductor:	$\operatorname{cond}(\chi)$	=	$ 403 $
Character field:			$\Q(\zeta_{12})$
Newform subspaces:			$ 3 $
Sturm bound:			$74$
Trace bound:			$6$
Distinguishing $T_p$:			$2$, $7$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(403, [\chi])$.

	Total	New
Modular forms	160	160
Cusp forms	144	144
Eisenstein series	16	16

Trace form

$ 144q - 8q^{2} - 12q^{3} - 2q^{5} - 36q^{6} - 4q^{7} - 4q^{8} + 64q^{9} + O(q^{10}) $

$ 144q - 8q^{2} - 12q^{3} - 2q^{5} - 36q^{6} - 4q^{7} - 4q^{8} + 64q^{9} - 12q^{11} - 6q^{13} + 24q^{14} - 160q^{16} - 40q^{18} - 4q^{19} + 22q^{20} - 42q^{21} + 60q^{22} + 48q^{24} - 18q^{26} + 2q^{28} + 26q^{31} - 16q^{32} - 12q^{33} + 24q^{34} - 8q^{35} + 6q^{37} - 40q^{39} + 12q^{40} - 2q^{41} - 48q^{42} - 72q^{44} + 10q^{50} + 120q^{52} + 24q^{53} - 12q^{55} - 84q^{57} + 14q^{59} + 48q^{63} + 66q^{65} - 176q^{66} + 10q^{67} - 168q^{70} - 38q^{71} + 74q^{72} + 12q^{73} + 72q^{74} + 64q^{76} + 84q^{78} + 96q^{79} + 50q^{80} - 32q^{81} - 18q^{83} - 114q^{84} - 102q^{86} + 64q^{87} - 140q^{93} - 32q^{94} + 192q^{96} + 12q^{97} + 20q^{98} - 24q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Decomposition of $S_{2}^{\mathrm{new}}(403, [\chi])$ into newform subspaces

Label	Dim.	$A$	Field	CM	Traces				$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	$q$-expansion
403.2.be.a	$4$	$3.218$	$\Q(\zeta_{12})$	None	$-2$	$6$	$-2$	$-10$	$q+(-1+\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+\cdots$
403.2.be.b	$4$	$3.218$	$\Q(\zeta_{12})$	None	$-2$	$6$	$-2$	$8$	$q+(-\zeta_{12}-\zeta_{12}^{2})q^{2}+(1-\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{3}+\cdots$
403.2.be.c	$136$	$3.218$		None	$-4$	$-24$	$2$	$-2$

Hecke Characteristic Polynomials

$p$	$F_p(T)$
$2$	($ 1 + 2 T + 2 T^{2} + 2 T^{3} + T^{4} + 4 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} $)($ 1 + 2 T + 2 T^{2} + 2 T^{3} + T^{4} + 4 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} $)
$3$	($ 1 - 6 T + 20 T^{2} - 48 T^{3} + 91 T^{4} - 144 T^{5} + 180 T^{6} - 162 T^{7} + 81 T^{8} $)($ 1 - 6 T + 20 T^{2} - 48 T^{3} + 91 T^{4} - 144 T^{5} + 180 T^{6} - 162 T^{7} + 81 T^{8} $)
$5$	($ ( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} )( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} ) $)($ ( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} )( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} ) $)
$7$	($ ( 1 + 5 T + 7 T^{2} )^{2}( 1 + 2 T^{2} + 49 T^{4} ) $)($ ( 1 - 4 T + 7 T^{2} )^{2}( 1 + 11 T^{2} + 49 T^{4} ) $)
$11$	($ 1 - 6 T + 9 T^{2} + 66 T^{3} - 376 T^{4} + 726 T^{5} + 1089 T^{6} - 7986 T^{7} + 14641 T^{8} $)($ 1 + 9 T^{2} - 48 T^{3} + 20 T^{4} - 528 T^{5} + 1089 T^{6} + 14641 T^{8} $)
$13$	($ 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} $)($ 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} $)
$17$	($ ( 1 - 17 T^{2} + 289 T^{4} )^{2} $)($ ( 1 - 17 T^{2} + 289 T^{4} )^{2} $)
$19$	($ 1 - 6 T + 45 T^{2} - 126 T^{3} + 704 T^{4} - 2394 T^{5} + 16245 T^{6} - 41154 T^{7} + 130321 T^{8} $)($ 1 - 12 T + 45 T^{2} + 108 T^{3} - 1348 T^{4} + 2052 T^{5} + 16245 T^{6} - 82308 T^{7} + 130321 T^{8} $)
$23$	($ ( 1 + 2 T + 20 T^{2} + 46 T^{3} + 529 T^{4} )^{2} $)($ ( 1 - 2 T + 20 T^{2} - 46 T^{3} + 529 T^{4} )^{2} $)
$29$	($ ( 1 - 4 T - 13 T^{2} - 116 T^{3} + 841 T^{4} )( 1 + 4 T - 13 T^{2} + 116 T^{3} + 841 T^{4} ) $)($ ( 1 - 4 T - 13 T^{2} - 116 T^{3} + 841 T^{4} )( 1 + 4 T - 13 T^{2} + 116 T^{3} + 841 T^{4} ) $)
$31$	($ 1 - 13 T^{2} + 961 T^{4} $)($ ( 1 - 7 T + 31 T^{2} )^{2} $)
$37$	($ 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 10656 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8} $)($ 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 10656 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8} $)
$41$	($ 1 + 20 T + 164 T^{2} + 668 T^{3} + 2335 T^{4} + 27388 T^{5} + 275684 T^{6} + 1378420 T^{7} + 2825761 T^{8} $)($ 1 - 16 T + 164 T^{2} - 1204 T^{3} + 8239 T^{4} - 49364 T^{5} + 275684 T^{6} - 1102736 T^{7} + 2825761 T^{8} $)
$43$	($ 1 - 2 T - 56 T^{2} + 52 T^{3} + 1579 T^{4} + 2236 T^{5} - 103544 T^{6} - 159014 T^{7} + 3418801 T^{8} $)($ 1 + 2 T - 56 T^{2} - 52 T^{3} + 1579 T^{4} - 2236 T^{5} - 103544 T^{6} + 159014 T^{7} + 3418801 T^{8} $)
$47$	($ 1 - 20 T + 200 T^{2} - 1460 T^{3} + 9982 T^{4} - 68620 T^{5} + 441800 T^{6} - 2076460 T^{7} + 4879681 T^{8} $)($ 1 - 20 T + 200 T^{2} - 1460 T^{3} + 9982 T^{4} - 68620 T^{5} + 441800 T^{6} - 2076460 T^{7} + 4879681 T^{8} $)
$53$	($ 1 + 30 T + 477 T^{2} + 5310 T^{3} + 44420 T^{4} + 281430 T^{5} + 1339893 T^{6} + 4466310 T^{7} + 7890481 T^{8} $)($ 1 + 30 T + 477 T^{2} + 5310 T^{3} + 44420 T^{4} + 281430 T^{5} + 1339893 T^{6} + 4466310 T^{7} + 7890481 T^{8} $)
$59$	($ 1 + 22 T + 122 T^{2} - 1232 T^{3} - 20033 T^{4} - 72688 T^{5} + 424682 T^{6} + 4518338 T^{7} + 12117361 T^{8} $)($ 1 - 2 T + 122 T^{2} - 608 T^{3} + 8287 T^{4} - 35872 T^{5} + 424682 T^{6} - 410758 T^{7} + 12117361 T^{8} $)
$61$	($ 1 - 182 T^{2} + 15291 T^{4} - 677222 T^{6} + 13845841 T^{8} $)($ 1 - 182 T^{2} + 15291 T^{4} - 677222 T^{6} + 13845841 T^{8} $)
$67$	($ 1 - 8 T + 137 T^{2} - 1224 T^{3} + 11492 T^{4} - 82008 T^{5} + 614993 T^{6} - 2406104 T^{7} + 20151121 T^{8} $)($ 1 + 22 T + 137 T^{2} - 834 T^{3} - 16648 T^{4} - 55878 T^{5} + 614993 T^{6} + 6616786 T^{7} + 20151121 T^{8} $)
$71$	($ 1 - 16 T + 65 T^{2} + 1016 T^{3} - 15428 T^{4} + 72136 T^{5} + 327665 T^{6} - 5726576 T^{7} + 25411681 T^{8} $)($ 1 + 2 T + 65 T^{2} + 818 T^{3} + 2464 T^{4} + 58078 T^{5} + 327665 T^{6} + 715822 T^{7} + 25411681 T^{8} $)
$73$	($ 1 + 6 T + 90 T^{2} - 564 T^{3} - 97 T^{4} - 41172 T^{5} + 479610 T^{6} + 2334102 T^{7} + 28398241 T^{8} $)($ 1 - 18 T + 90 T^{2} + 1356 T^{3} - 21121 T^{4} + 98988 T^{5} + 479610 T^{6} - 7002306 T^{7} + 28398241 T^{8} $)
$79$	($ 1 + 36 T + 694 T^{2} + 9432 T^{3} + 96531 T^{4} + 745128 T^{5} + 4331254 T^{6} + 17749404 T^{7} + 38950081 T^{8} $)($ 1 + 36 T + 694 T^{2} + 9432 T^{3} + 96531 T^{4} + 745128 T^{5} + 4331254 T^{6} + 17749404 T^{7} + 38950081 T^{8} $)
$83$	($ 1 - 38 T + 650 T^{2} - 6944 T^{3} + 62479 T^{4} - 576352 T^{5} + 4477850 T^{6} - 21727906 T^{7} + 47458321 T^{8} $)($ 1 - 34 T + 650 T^{2} - 8464 T^{3} + 86383 T^{4} - 702512 T^{5} + 4477850 T^{6} - 19440758 T^{7} + 47458321 T^{8} $)
$89$	($ 1 + 16 T + 128 T^{2} + 1840 T^{3} + 25774 T^{4} + 163760 T^{5} + 1013888 T^{6} + 11279504 T^{7} + 62742241 T^{8} $)($ 1 - 16 T + 128 T^{2} - 1840 T^{3} + 25774 T^{4} - 163760 T^{5} + 1013888 T^{6} - 11279504 T^{7} + 62742241 T^{8} $)
$97$	($ 1 + 20 T + 200 T^{2} + 2460 T^{3} + 29582 T^{4} + 238620 T^{5} + 1881800 T^{6} + 18253460 T^{7} + 88529281 T^{8} $)($ 1 + 20 T + 200 T^{2} + 2460 T^{3} + 29582 T^{4} + 238620 T^{5} + 1881800 T^{6} + 18253460 T^{7} + 88529281 T^{8} $)
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Introduction	Features
Universe	News

Level:	\( N \)	=	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	403.be (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	=	\( 403 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(74\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\), \(7\)

Label	Dim.	\(A\)	Field	CM	Traces				$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	$q$-expansion
403.2.be.a	\(4\)	\(3.218\)	\(\Q(\zeta_{12})\)	None	\(-2\)	\(6\)	\(-2\)	\(-10\)	\(q+(-1+\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+\cdots\)
403.2.be.b	\(4\)	\(3.218\)	\(\Q(\zeta_{12})\)	None	\(-2\)	\(6\)	\(-2\)	\(8\)	\(q+(-\zeta_{12}-\zeta_{12}^{2})q^{2}+(1-\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
403.2.be.c	\(136\)	\(3.218\)		None	\(-4\)	\(-24\)	\(2\)	\(-2\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(403, [\chi])\) into newform subspaces

Hecke Characteristic Polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	403.2.be

Level	403
Weight	2
Character orbit	be
Rep. character	\(\chi_{403}(57,\cdot)\)
Character field	\(\Q(\zeta_{12})\)
Dimension	144
Newform subspaces	3
Sturm bound	74
Trace bound	6

$p$	$F_p(T)$
$2$	(\( 1 + 2 T + 2 T^{2} + 2 T^{3} + T^{4} + 4 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 + 2 T + 2 T^{2} + 2 T^{3} + T^{4} + 4 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} \))
$3$	(\( 1 - 6 T + 20 T^{2} - 48 T^{3} + 91 T^{4} - 144 T^{5} + 180 T^{6} - 162 T^{7} + 81 T^{8} \))(\( 1 - 6 T + 20 T^{2} - 48 T^{3} + 91 T^{4} - 144 T^{5} + 180 T^{6} - 162 T^{7} + 81 T^{8} \))
$5$	(\( ( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} )( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} ) \))(\( ( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} )( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} ) \))
$7$	(\( ( 1 + 5 T + 7 T^{2} )^{2}( 1 + 2 T^{2} + 49 T^{4} ) \))(\( ( 1 - 4 T + 7 T^{2} )^{2}( 1 + 11 T^{2} + 49 T^{4} ) \))
$11$	(\( 1 - 6 T + 9 T^{2} + 66 T^{3} - 376 T^{4} + 726 T^{5} + 1089 T^{6} - 7986 T^{7} + 14641 T^{8} \))(\( 1 + 9 T^{2} - 48 T^{3} + 20 T^{4} - 528 T^{5} + 1089 T^{6} + 14641 T^{8} \))
$13$	(\( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} \))
$17$	(\( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \))
$19$	(\( 1 - 6 T + 45 T^{2} - 126 T^{3} + 704 T^{4} - 2394 T^{5} + 16245 T^{6} - 41154 T^{7} + 130321 T^{8} \))(\( 1 - 12 T + 45 T^{2} + 108 T^{3} - 1348 T^{4} + 2052 T^{5} + 16245 T^{6} - 82308 T^{7} + 130321 T^{8} \))
$23$	(\( ( 1 + 2 T + 20 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 2 T + 20 T^{2} - 46 T^{3} + 529 T^{4} )^{2} \))
$29$	(\( ( 1 - 4 T - 13 T^{2} - 116 T^{3} + 841 T^{4} )( 1 + 4 T - 13 T^{2} + 116 T^{3} + 841 T^{4} ) \))(\( ( 1 - 4 T - 13 T^{2} - 116 T^{3} + 841 T^{4} )( 1 + 4 T - 13 T^{2} + 116 T^{3} + 841 T^{4} ) \))
$31$	(\( 1 - 13 T^{2} + 961 T^{4} \))(\( ( 1 - 7 T + 31 T^{2} )^{2} \))
$37$	(\( 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 10656 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8} \))(\( 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 10656 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8} \))
$41$	(\( 1 + 20 T + 164 T^{2} + 668 T^{3} + 2335 T^{4} + 27388 T^{5} + 275684 T^{6} + 1378420 T^{7} + 2825761 T^{8} \))(\( 1 - 16 T + 164 T^{2} - 1204 T^{3} + 8239 T^{4} - 49364 T^{5} + 275684 T^{6} - 1102736 T^{7} + 2825761 T^{8} \))
$43$	(\( 1 - 2 T - 56 T^{2} + 52 T^{3} + 1579 T^{4} + 2236 T^{5} - 103544 T^{6} - 159014 T^{7} + 3418801 T^{8} \))(\( 1 + 2 T - 56 T^{2} - 52 T^{3} + 1579 T^{4} - 2236 T^{5} - 103544 T^{6} + 159014 T^{7} + 3418801 T^{8} \))
$47$	(\( 1 - 20 T + 200 T^{2} - 1460 T^{3} + 9982 T^{4} - 68620 T^{5} + 441800 T^{6} - 2076460 T^{7} + 4879681 T^{8} \))(\( 1 - 20 T + 200 T^{2} - 1460 T^{3} + 9982 T^{4} - 68620 T^{5} + 441800 T^{6} - 2076460 T^{7} + 4879681 T^{8} \))
$53$	(\( 1 + 30 T + 477 T^{2} + 5310 T^{3} + 44420 T^{4} + 281430 T^{5} + 1339893 T^{6} + 4466310 T^{7} + 7890481 T^{8} \))(\( 1 + 30 T + 477 T^{2} + 5310 T^{3} + 44420 T^{4} + 281430 T^{5} + 1339893 T^{6} + 4466310 T^{7} + 7890481 T^{8} \))
$59$	(\( 1 + 22 T + 122 T^{2} - 1232 T^{3} - 20033 T^{4} - 72688 T^{5} + 424682 T^{6} + 4518338 T^{7} + 12117361 T^{8} \))(\( 1 - 2 T + 122 T^{2} - 608 T^{3} + 8287 T^{4} - 35872 T^{5} + 424682 T^{6} - 410758 T^{7} + 12117361 T^{8} \))
$61$	(\( 1 - 182 T^{2} + 15291 T^{4} - 677222 T^{6} + 13845841 T^{8} \))(\( 1 - 182 T^{2} + 15291 T^{4} - 677222 T^{6} + 13845841 T^{8} \))
$67$	(\( 1 - 8 T + 137 T^{2} - 1224 T^{3} + 11492 T^{4} - 82008 T^{5} + 614993 T^{6} - 2406104 T^{7} + 20151121 T^{8} \))(\( 1 + 22 T + 137 T^{2} - 834 T^{3} - 16648 T^{4} - 55878 T^{5} + 614993 T^{6} + 6616786 T^{7} + 20151121 T^{8} \))
$71$	(\( 1 - 16 T + 65 T^{2} + 1016 T^{3} - 15428 T^{4} + 72136 T^{5} + 327665 T^{6} - 5726576 T^{7} + 25411681 T^{8} \))(\( 1 + 2 T + 65 T^{2} + 818 T^{3} + 2464 T^{4} + 58078 T^{5} + 327665 T^{6} + 715822 T^{7} + 25411681 T^{8} \))
$73$	(\( 1 + 6 T + 90 T^{2} - 564 T^{3} - 97 T^{4} - 41172 T^{5} + 479610 T^{6} + 2334102 T^{7} + 28398241 T^{8} \))(\( 1 - 18 T + 90 T^{2} + 1356 T^{3} - 21121 T^{4} + 98988 T^{5} + 479610 T^{6} - 7002306 T^{7} + 28398241 T^{8} \))
$79$	(\( 1 + 36 T + 694 T^{2} + 9432 T^{3} + 96531 T^{4} + 745128 T^{5} + 4331254 T^{6} + 17749404 T^{7} + 38950081 T^{8} \))(\( 1 + 36 T + 694 T^{2} + 9432 T^{3} + 96531 T^{4} + 745128 T^{5} + 4331254 T^{6} + 17749404 T^{7} + 38950081 T^{8} \))
$83$	(\( 1 - 38 T + 650 T^{2} - 6944 T^{3} + 62479 T^{4} - 576352 T^{5} + 4477850 T^{6} - 21727906 T^{7} + 47458321 T^{8} \))(\( 1 - 34 T + 650 T^{2} - 8464 T^{3} + 86383 T^{4} - 702512 T^{5} + 4477850 T^{6} - 19440758 T^{7} + 47458321 T^{8} \))
$89$	(\( 1 + 16 T + 128 T^{2} + 1840 T^{3} + 25774 T^{4} + 163760 T^{5} + 1013888 T^{6} + 11279504 T^{7} + 62742241 T^{8} \))(\( 1 - 16 T + 128 T^{2} - 1840 T^{3} + 25774 T^{4} - 163760 T^{5} + 1013888 T^{6} - 11279504 T^{7} + 62742241 T^{8} \))
$97$	(\( 1 + 20 T + 200 T^{2} + 2460 T^{3} + 29582 T^{4} + 238620 T^{5} + 1881800 T^{6} + 18253460 T^{7} + 88529281 T^{8} \))(\( 1 + 20 T + 200 T^{2} + 2460 T^{3} + 29582 T^{4} + 238620 T^{5} + 1881800 T^{6} + 18253460 T^{7} + 88529281 T^{8} \))
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