LMFDB - Space of modular forms of level 13, weight 4, and Character 13.e

Defining parameters

Level:	$ N $	$=$	$ 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	13.e (of order $6$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$ 13 $
Character field:			$\Q(\zeta_{6})$
Newform subspaces:			$ 2 $
Sturm bound:			$4$
Trace bound:			$2$
Distinguishing $T_p$:			$2$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(13, [\chi])$.

	Total	New
Modular forms	8	8
Cusp forms	4	4
Eisenstein series	4	4

Trace form

$ 4q - 3q^{2} + 5q^{3} - q^{4} - 48q^{6} + 15q^{7} + q^{9} + O(q^{10}) $

$ 4q - 3q^{2} + 5q^{3} - q^{4} - 48q^{6} + 15q^{7} + q^{9} + 51q^{10} - 15q^{11} + 76q^{12} + 65q^{13} - 204q^{14} + 174q^{15} + 79q^{16} - 144q^{17} - 351q^{19} - 111q^{20} + 102q^{22} + 21q^{23} + 246q^{24} + 110q^{25} + 351q^{26} - 130q^{27} + 276q^{28} + 210q^{29} - 330q^{30} - 603q^{32} - 321q^{33} + 336q^{35} + 203q^{36} - 318q^{37} + 216q^{38} + 325q^{39} - 462q^{40} - 210q^{41} - 498q^{42} - 19q^{43} + 597q^{45} + 576q^{46} - 562q^{48} + 13q^{49} + 768q^{50} + 90q^{51} - 494q^{52} + 1038q^{53} - 510q^{54} + 336q^{55} + 624q^{56} + 9q^{58} - 525q^{59} - 128q^{61} - 522q^{62} - 1410q^{63} - 742q^{64} - 1209q^{65} + 996q^{66} - 1077q^{67} - 477q^{68} + 555q^{69} + 2841q^{71} + 1425q^{72} - 111q^{74} - 713q^{75} + 378q^{76} - 1398q^{77} - 858q^{78} + 64q^{79} + 1923q^{80} + 418q^{81} + 1833q^{82} + 852q^{84} + 297q^{85} - 201q^{87} + 624q^{88} - 1161q^{89} - 1974q^{90} - 741q^{91} - 1236q^{92} - 1422q^{93} - 1710q^{94} - 1026q^{95} + 2487q^{97} - 1437q^{98} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Decomposition of $S_{4}^{\mathrm{new}}(13, [\chi])$ 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
13.4.e.a	$2$	$0.767$	$\Q(\sqrt{-3}) $	None	$-6$	$7$	$0$	$39$	$q+(-2-2\zeta_{6})q^{2}+(7-7\zeta_{6})q^{3}+4\zeta_{6}q^{4}+\cdots$
13.4.e.b	$2$	$0.767$	$\Q(\sqrt{-3}) $	None	$3$	$-2$	$0$	$-24$	$q+(1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-5\zeta_{6}q^{4}+\cdots$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	($ 1 + 6 T + 20 T^{2} + 48 T^{3} + 64 T^{4} $)($ 1 - 3 T + 11 T^{2} - 24 T^{3} + 64 T^{4} $)
$3$	($ 1 - 7 T + 22 T^{2} - 189 T^{3} + 729 T^{4} $)($ 1 + 2 T - 23 T^{2} + 54 T^{3} + 729 T^{4} $)
$5$	($ 1 - 58 T^{2} + 15625 T^{4} $)($ 1 - 247 T^{2} + 15625 T^{4} $)
$7$	($ 1 - 39 T + 850 T^{2} - 13377 T^{3} + 117649 T^{4} $)($ 1 + 24 T + 535 T^{2} + 8232 T^{3} + 117649 T^{4} $)
$11$	($ 1 + 39 T + 1838 T^{2} + 51909 T^{3} + 1771561 T^{4} $)($ 1 - 24 T + 1523 T^{2} - 31944 T^{3} + 1771561 T^{4} $)
$13$	($ 1 + 26 T + 2197 T^{2} $)($ 1 - 91 T + 2197 T^{2} $)
$17$	($ 1 + 27 T - 4184 T^{2} + 132651 T^{3} + 24137569 T^{4} $)($ 1 + 117 T + 8776 T^{2} + 574821 T^{3} + 24137569 T^{4} $)
$19$	($ 1 + 153 T + 14662 T^{2} + 1049427 T^{3} + 47045881 T^{4} $)($ 1 + 198 T + 19927 T^{2} + 1358082 T^{3} + 47045881 T^{4} $)
$23$	($ 1 + 57 T - 8918 T^{2} + 693519 T^{3} + 148035889 T^{4} $)($ 1 - 78 T - 6083 T^{2} - 949026 T^{3} + 148035889 T^{4} $)
$29$	($ 1 - 69 T - 19628 T^{2} - 1682841 T^{3} + 594823321 T^{4} $)($ 1 - 141 T - 4508 T^{2} - 3438849 T^{3} + 594823321 T^{4} $)
$31$	($ 1 - 54290 T^{2} + 887503681 T^{4} $)($ ( 1 - 308 T + 29791 T^{2} )( 1 + 308 T + 29791 T^{2} ) $)
$37$	($ 1 + 69 T + 52240 T^{2} + 3495057 T^{3} + 2565726409 T^{4} $)($ 1 + 249 T + 71320 T^{2} + 12612597 T^{3} + 2565726409 T^{4} $)
$41$	($ 1 + 681 T + 223508 T^{2} + 46935201 T^{3} + 4750104241 T^{4} $)($ 1 - 471 T + 142868 T^{2} - 32461791 T^{3} + 4750104241 T^{4} $)
$43$	($ 1 - 85 T - 72282 T^{2} - 6758095 T^{3} + 6321363049 T^{4} $)($ 1 + 104 T - 68691 T^{2} + 8268728 T^{3} + 6321363049 T^{4} $)
$47$	($ 1 - 90034 T^{2} + 10779215329 T^{4} $)($ 1 - 116818 T^{2} + 10779215329 T^{4} $)
$53$	($ ( 1 - 426 T + 148877 T^{2} )^{2} $)($ ( 1 - 93 T + 148877 T^{2} )^{2} $)
$59$	($ 1 + 33 T + 205742 T^{2} + 6777507 T^{3} + 42180533641 T^{4} $)($ 1 + 492 T + 286067 T^{2} + 101046468 T^{3} + 42180533641 T^{4} $)
$61$	($ 1 - 17 T - 226692 T^{2} - 3858677 T^{3} + 51520374361 T^{4} $)($ 1 + 145 T - 205956 T^{2} + 32912245 T^{3} + 51520374361 T^{4} $)
$67$	($ 1 - 285 T + 327838 T^{2} - 85717455 T^{3} + 90458382169 T^{4} $)($ 1 + 1362 T + 919111 T^{2} + 409639206 T^{3} + 90458382169 T^{4} $)
$71$	($ 1 - 1011 T + 698618 T^{2} - 361848021 T^{3} + 128100283921 T^{4} $)($ 1 - 1830 T + 1474211 T^{2} - 654977130 T^{3} + 128100283921 T^{4} $)
$73$	($ 1 + 231166 T^{2} + 151334226289 T^{4} $)($ 1 - 567359 T^{2} + 151334226289 T^{4} $)
$79$	($ ( 1 + 1244 T + 493039 T^{2} )^{2} $)($ ( 1 - 1276 T + 493039 T^{2} )^{2} $)
$83$	($ 1 - 962026 T^{2} + 326940373369 T^{4} $)($ 1 - 519766 T^{2} + 326940373369 T^{4} $)
$89$	($ 1 - 531 T + 798956 T^{2} - 374338539 T^{3} + 496981290961 T^{4} $)($ 1 + 1692 T + 1659257 T^{2} + 1192807548 T^{3} + 496981290961 T^{4} $)
$97$	($ 1 - 2139 T + 2437780 T^{2} - 1952207547 T^{3} + 832972004929 T^{4} $)($ 1 - 348 T + 953041 T^{2} - 317610204 T^{3} + 832972004929 T^{4} $)
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Overview	Features
Universe	Knowledge

Degree 1	Degree 2
Degree 3	Degree 4
$\zeta$ zeros

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	13.e (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(4\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Introduction

L-functions

Modular forms

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Groups

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{4}^{\mathrm{new}}(13, [\chi])\) 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	13.4.e

Level	13
Weight	4
Character orbit	e
Rep. character	\(\chi_{13}(4,\cdot)\)
Character field	\(\Q(\zeta_{6})\)
Dimension	4
Newform subspaces	2
Sturm bound	4
Trace bound	2

Label	Dim.	\(A\)	Field	CM	Traces				$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	$q$-expansion
13.4.e.a	\(2\)	\(0.767\)	\(\Q(\sqrt{-3}) \)	None	\(-6\)	\(7\)	\(0\)	\(39\)	\(q+(-2-2\zeta_{6})q^{2}+(7-7\zeta_{6})q^{3}+4\zeta_{6}q^{4}+\cdots\)
13.4.e.b	\(2\)	\(0.767\)	\(\Q(\sqrt{-3}) \)	None	\(3\)	\(-2\)	\(0\)	\(-24\)	\(q+(1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-5\zeta_{6}q^{4}+\cdots\)

$p$	$F_p(T)$
$2$	(\( 1 + 6 T + 20 T^{2} + 48 T^{3} + 64 T^{4} \))(\( 1 - 3 T + 11 T^{2} - 24 T^{3} + 64 T^{4} \))
$3$	(\( 1 - 7 T + 22 T^{2} - 189 T^{3} + 729 T^{4} \))(\( 1 + 2 T - 23 T^{2} + 54 T^{3} + 729 T^{4} \))
$5$	(\( 1 - 58 T^{2} + 15625 T^{4} \))(\( 1 - 247 T^{2} + 15625 T^{4} \))
$7$	(\( 1 - 39 T + 850 T^{2} - 13377 T^{3} + 117649 T^{4} \))(\( 1 + 24 T + 535 T^{2} + 8232 T^{3} + 117649 T^{4} \))
$11$	(\( 1 + 39 T + 1838 T^{2} + 51909 T^{3} + 1771561 T^{4} \))(\( 1 - 24 T + 1523 T^{2} - 31944 T^{3} + 1771561 T^{4} \))
$13$	(\( 1 + 26 T + 2197 T^{2} \))(\( 1 - 91 T + 2197 T^{2} \))
$17$	(\( 1 + 27 T - 4184 T^{2} + 132651 T^{3} + 24137569 T^{4} \))(\( 1 + 117 T + 8776 T^{2} + 574821 T^{3} + 24137569 T^{4} \))
$19$	(\( 1 + 153 T + 14662 T^{2} + 1049427 T^{3} + 47045881 T^{4} \))(\( 1 + 198 T + 19927 T^{2} + 1358082 T^{3} + 47045881 T^{4} \))
$23$	(\( 1 + 57 T - 8918 T^{2} + 693519 T^{3} + 148035889 T^{4} \))(\( 1 - 78 T - 6083 T^{2} - 949026 T^{3} + 148035889 T^{4} \))
$29$	(\( 1 - 69 T - 19628 T^{2} - 1682841 T^{3} + 594823321 T^{4} \))(\( 1 - 141 T - 4508 T^{2} - 3438849 T^{3} + 594823321 T^{4} \))
$31$	(\( 1 - 54290 T^{2} + 887503681 T^{4} \))(\( ( 1 - 308 T + 29791 T^{2} )( 1 + 308 T + 29791 T^{2} ) \))
$37$	(\( 1 + 69 T + 52240 T^{2} + 3495057 T^{3} + 2565726409 T^{4} \))(\( 1 + 249 T + 71320 T^{2} + 12612597 T^{3} + 2565726409 T^{4} \))
$41$	(\( 1 + 681 T + 223508 T^{2} + 46935201 T^{3} + 4750104241 T^{4} \))(\( 1 - 471 T + 142868 T^{2} - 32461791 T^{3} + 4750104241 T^{4} \))
$43$	(\( 1 - 85 T - 72282 T^{2} - 6758095 T^{3} + 6321363049 T^{4} \))(\( 1 + 104 T - 68691 T^{2} + 8268728 T^{3} + 6321363049 T^{4} \))
$47$	(\( 1 - 90034 T^{2} + 10779215329 T^{4} \))(\( 1 - 116818 T^{2} + 10779215329 T^{4} \))
$53$	(\( ( 1 - 426 T + 148877 T^{2} )^{2} \))(\( ( 1 - 93 T + 148877 T^{2} )^{2} \))
$59$	(\( 1 + 33 T + 205742 T^{2} + 6777507 T^{3} + 42180533641 T^{4} \))(\( 1 + 492 T + 286067 T^{2} + 101046468 T^{3} + 42180533641 T^{4} \))
$61$	(\( 1 - 17 T - 226692 T^{2} - 3858677 T^{3} + 51520374361 T^{4} \))(\( 1 + 145 T - 205956 T^{2} + 32912245 T^{3} + 51520374361 T^{4} \))
$67$	(\( 1 - 285 T + 327838 T^{2} - 85717455 T^{3} + 90458382169 T^{4} \))(\( 1 + 1362 T + 919111 T^{2} + 409639206 T^{3} + 90458382169 T^{4} \))
$71$	(\( 1 - 1011 T + 698618 T^{2} - 361848021 T^{3} + 128100283921 T^{4} \))(\( 1 - 1830 T + 1474211 T^{2} - 654977130 T^{3} + 128100283921 T^{4} \))
$73$	(\( 1 + 231166 T^{2} + 151334226289 T^{4} \))(\( 1 - 567359 T^{2} + 151334226289 T^{4} \))
$79$	(\( ( 1 + 1244 T + 493039 T^{2} )^{2} \))(\( ( 1 - 1276 T + 493039 T^{2} )^{2} \))
$83$	(\( 1 - 962026 T^{2} + 326940373369 T^{4} \))(\( 1 - 519766 T^{2} + 326940373369 T^{4} \))
$89$	(\( 1 - 531 T + 798956 T^{2} - 374338539 T^{3} + 496981290961 T^{4} \))(\( 1 + 1692 T + 1659257 T^{2} + 1192807548 T^{3} + 496981290961 T^{4} \))
$97$	(\( 1 - 2139 T + 2437780 T^{2} - 1952207547 T^{3} + 832972004929 T^{4} \))(\( 1 - 348 T + 953041 T^{2} - 317610204 T^{3} + 832972004929 T^{4} \))
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