LMFDB - Space of modular forms of level 5, weight 10, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	5.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 2 $
Sturm bound:			$5$
Trace bound:			$1$
Distinguishing $T_p$:			$2$

Dimensions

The following table gives the dimensions of various subspaces of $M_{10}(\Gamma_0(5))$.

	Total	New	Old
Modular forms	5	3	2
Cusp forms	3	3	0
Eisenstein series	2	0	2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$5$	Dim.
$+$	$1$
$-$	$2$

Trace form

$ 3q - 18q^{2} + 146q^{3} + 596q^{4} + 625q^{5} - 4424q^{6} + 5942q^{7} - 12600q^{8} - 4181q^{9} + O(q^{10}) $

\( 3q - 18q^{2} + 146q^{3} + 596q^{4} + 625q^{5} - 4424q^{6} + 5942q^{7} - 12600q^{8} - 4181q^{9} - 1250q^{10} - 22224q^{11} + 227152q^{12} - 914q^{13} - 474288q^{14} + 233750q^{15} - 144112q^{16} + 907662q^{17} - 1008394q^{18} - 1305260q^{19} + 932500q^{20} + 601116q^{21} + 4083944q^{22} - 1581774q^{23} - 3754080q^{24} + 1171875q^{25} - 2375964q^{26} + 313100q^{27} + 3305504q^{28} + 529410q^{29} - 3905000q^{30} - 1751884q^{31} - 4067808q^{32} + 717232q^{33} + 19920572q^{34} - 1588750q^{35} + 14797508q^{36} - 36053298q^{37} + 3821400q^{38} + 33625748q^{39} - 17475000q^{40} - 15901014q^{41} - 61126176q^{42} + 46492906q^{43} - 5121168q^{44} + 5745625q^{45} + 48651696q^{46} + 22853022q^{47} - 38239424q^{48} - 9204929q^{49} - 7031250q^{50} - 54632204q^{51} + 139262232q^{52} + 703446q^{53} - 72617360q^{54} + 43870000q^{55} - 10570560q^{56} + 87911000q^{57} - 231081500q^{58} - 140181180q^{59} + 78130000q^{60} - 228831074q^{61} + 331039104q^{62} + 198326886q^{63} - 213219264q^{64} + 144346250q^{65} + 430656992q^{66} - 45604738q^{67} - 265631256q^{68} - 153977772q^{69} - 254010000q^{70} - 197098404q^{71} - 139728600q^{72} + 533029126q^{73} + 639347892q^{74} + 57031250q^{75} + 354443280q^{76} - 996146736q^{77} - 794572208q^{78} + 101918360q^{79} - 299990000q^{80} - 486443657q^{81} - 268068116q^{82} + 1664055066q^{83} + 1378576512q^{84} - 51263750q^{85} - 161213784q^{86} - 523405300q^{87} - 368023200q^{88} + 810150030q^{89} - 697116250q^{90} + 189838876q^{91} - 700560288q^{92} - 31047288q^{93} - 1063675648q^{94} + 445137500q^{95} + 1404784256q^{96} + 618891222q^{97} - 621046626q^{98} - 1654739152q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of $S_{10}^{\mathrm{new}}(\Gamma_0(5))$ into newform subspaces

Label	Dim.	$A$	Field	CM	Traces				A-L signs	$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	5	$q$-expansion
5.10.a.a	$1$	$2.575$	$\Q$	None	$-8$	$-114$	$-625$	$4242$	$+$	$q-8q^{2}-114q^{3}-448q^{4}-5^{4}q^{5}+\cdots$
5.10.a.b	$2$	$2.575$	$\Q(\sqrt{1009}) $	None	$-10$	$260$	$1250$	$1700$	$-$	$q+(-5-\beta )q^{2}+(130+2\beta )q^{3}+(522+\cdots)q^{4}+\cdots$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	($ 1 + 8 T + 512 T^{2} $)($ 1 + 10 T + 40 T^{2} + 5120 T^{3} + 262144 T^{4} $)
$3$	($ 1 + 114 T + 19683 T^{2} $)($ 1 - 260 T + 52230 T^{2} - 5117580 T^{3} + 387420489 T^{4} $)
$5$	($ 1 + 625 T $)($ ( 1 - 625 T )^{2} $)
$7$	($ 1 - 4242 T + 40353607 T^{2} $)($ 1 - 1700 T + 35221550 T^{2} - 68601131900 T^{3} + 1628413597910449 T^{4} $)
$11$	($ 1 + 46208 T + 2357947691 T^{2} $)($ 1 - 23984 T + 1217213446 T^{2} - 56553017420944 T^{3} + 5559917313492231481 T^{4} $)
$13$	($ 1 + 115934 T + 10604499373 T^{2} $)($ 1 - 115020 T + 22672043710 T^{2} - 1219729517882460 T^{3} + $$11\!\cdots\!29$$ T^{4} $)
$17$	($ 1 - 494842 T + 118587876497 T^{2} $)($ 1 - 412820 T + 113016614470 T^{2} - 48955447175491540 T^{3} + $$14\!\cdots\!09$$ T^{4} $)
$19$	($ 1 + 1008740 T + 322687697779 T^{2} $)($ 1 + 296520 T + 659218232758 T^{2} + 95683356145429080 T^{3} + $$10\!\cdots\!41$$ T^{4} $)
$23$	($ 1 + 532554 T + 1801152661463 T^{2} $)($ 1 + 1049220 T + 3497852029390 T^{2} + 1889805395460208860 T^{3} + $$32\!\cdots\!69$$ T^{4} $)
$29$	($ 1 - 4196390 T + 14507145975869 T^{2} $)($ 1 + 3666980 T + 20832571957438 T^{2} + 53197414150592105620 T^{3} + $$21\!\cdots\!61$$ T^{4} $)
$31$	($ 1 + 3365028 T + 26439622160671 T^{2} $)($ 1 - 1613144 T + 29382323902526 T^{2} - 42650917850753459624 T^{3} + $$69\!\cdots\!41$$ T^{4} $)
$37$	($ 1 + 14931358 T + 129961739795077 T^{2} $)($ 1 + 21121940 T + 328931801286510 T^{2} + $$27\!\cdots\!80$$ T^{3} + $$16\!\cdots\!29$$ T^{4} $)
$41$	($ 1 - 11056262 T + 327381934393961 T^{2} $)($ 1 + 26957276 T + 811945448362966 T^{2} + $$88\!\cdots\!36$$ T^{3} + $$10\!\cdots\!21$$ T^{4} $)
$43$	($ 1 + 6396794 T + 502592611936843 T^{2} $)($ 1 - 52889700 T + 1703843788760950 T^{2} - $$26\!\cdots\!00$$ T^{3} + $$25\!\cdots\!49$$ T^{4} $)
$47$	($ 1 + 35559158 T + 1119130473102767 T^{2} $)($ 1 - 58412180 T + 2814913257457630 T^{2} - $$65\!\cdots\!60$$ T^{3} + $$12\!\cdots\!89$$ T^{4} $)
$53$	($ 1 - 39738586 T + 3299763591802133 T^{2} $)($ 1 + 39035140 T + 5675030678030830 T^{2} + $$12\!\cdots\!20$$ T^{3} + $$10\!\cdots\!89$$ T^{4} $)
$59$	($ 1 + 85185620 T + 8662995818654939 T^{2} $)($ 1 + 54995560 T + 15674484224932678 T^{2} + $$47\!\cdots\!40$$ T^{3} + $$75\!\cdots\!21$$ T^{4} $)
$61$	($ 1 - 45748642 T + 11694146092834141 T^{2} $)($ 1 + 274579716 T + 41753623519328446 T^{2} + $$32\!\cdots\!56$$ T^{3} + $$13\!\cdots\!81$$ T^{4} $)
$67$	($ 1 + 45286158 T + 27206534396294947 T^{2} $)($ 1 + 318580 T + 48520062064444070 T^{2} + $$86\!\cdots\!60$$ T^{3} + $$74\!\cdots\!09$$ T^{4} $)
$71$	($ 1 + 189967468 T + 45848500718449031 T^{2} $)($ 1 + 7130936 T + 51935375688707086 T^{2} + $$32\!\cdots\!16$$ T^{3} + $$21\!\cdots\!61$$ T^{4} $)
$73$	($ 1 - 412170946 T + 58871586708267913 T^{2} $)($ 1 - 120858180 T + 42707263689423190 T^{2} - $$71\!\cdots\!40$$ T^{3} + $$34\!\cdots\!69$$ T^{4} $)
$79$	($ 1 - 95040840 T + 119851595982618319 T^{2} $)($ 1 - 6877520 T - 115982362290712162 T^{2} - $$82\!\cdots\!80$$ T^{3} + $$14\!\cdots\!61$$ T^{4} $)
$83$	($ 1 - 261706326 T + 186940255267540403 T^{2} $)($ 1 - 1402348740 T + 857904310704391270 T^{2} - $$26\!\cdots\!20$$ T^{3} + $$34\!\cdots\!09$$ T^{4} $)
$89$	($ 1 + 19938630 T + 350356403707485209 T^{2} $)($ 1 - 830088660 T + 692293619421117718 T^{2} - $$29\!\cdots\!40$$ T^{3} + $$12\!\cdots\!81$$ T^{4} $)
$97$	($ 1 + 19503358 T + 760231058654565217 T^{2} $)($ 1 - 638394580 T + 1615411126351062630 T^{2} - $$48\!\cdots\!60$$ T^{3} + $$57\!\cdots\!89$$ T^{4} $)
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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:	\( N \)	\(=\)	\( 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	5.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(5\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Introduction

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Modular forms

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(5))\) 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	5.10.a

Level	$5$
Weight	$10$
Character orbit	5.a
Rep. character	$\chi_{5}(1,\cdot)$
Character field	$\Q$
Dimension	$3$
Newform subspaces	$2$
Sturm bound	$5$
Trace bound	$1$

Label	Dim.	\(A\)	Field	CM	Traces				A-L signs	$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	5	$q$-expansion
5.10.a.a	\(1\)	\(2.575\)	\(\Q\)	None	\(-8\)	\(-114\)	\(-625\)	\(4242\)	\(+\)	\(q-8q^{2}-114q^{3}-448q^{4}-5^{4}q^{5}+\cdots\)
5.10.a.b	\(2\)	\(2.575\)	\(\Q(\sqrt{1009}) \)	None	\(-10\)	\(260\)	\(1250\)	\(1700\)	\(-\)	\(q+(-5-\beta )q^{2}+(130+2\beta )q^{3}+(522+\cdots)q^{4}+\cdots\)

$p$	$F_p(T)$
$2$	(\( 1 + 8 T + 512 T^{2} \))(\( 1 + 10 T + 40 T^{2} + 5120 T^{3} + 262144 T^{4} \))
$3$	(\( 1 + 114 T + 19683 T^{2} \))(\( 1 - 260 T + 52230 T^{2} - 5117580 T^{3} + 387420489 T^{4} \))
$5$	(\( 1 + 625 T \))(\( ( 1 - 625 T )^{2} \))
$7$	(\( 1 - 4242 T + 40353607 T^{2} \))(\( 1 - 1700 T + 35221550 T^{2} - 68601131900 T^{3} + 1628413597910449 T^{4} \))
$11$	(\( 1 + 46208 T + 2357947691 T^{2} \))(\( 1 - 23984 T + 1217213446 T^{2} - 56553017420944 T^{3} + 5559917313492231481 T^{4} \))
$13$	(\( 1 + 115934 T + 10604499373 T^{2} \))(\( 1 - 115020 T + 22672043710 T^{2} - 1219729517882460 T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \))
$17$	(\( 1 - 494842 T + 118587876497 T^{2} \))(\( 1 - 412820 T + 113016614470 T^{2} - 48955447175491540 T^{3} + \)\(14\!\cdots\!09\)\( T^{4} \))
$19$	(\( 1 + 1008740 T + 322687697779 T^{2} \))(\( 1 + 296520 T + 659218232758 T^{2} + 95683356145429080 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} \))
$23$	(\( 1 + 532554 T + 1801152661463 T^{2} \))(\( 1 + 1049220 T + 3497852029390 T^{2} + 1889805395460208860 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))
$29$	(\( 1 - 4196390 T + 14507145975869 T^{2} \))(\( 1 + 3666980 T + 20832571957438 T^{2} + 53197414150592105620 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))
$31$	(\( 1 + 3365028 T + 26439622160671 T^{2} \))(\( 1 - 1613144 T + 29382323902526 T^{2} - 42650917850753459624 T^{3} + \)\(69\!\cdots\!41\)\( T^{4} \))
$37$	(\( 1 + 14931358 T + 129961739795077 T^{2} \))(\( 1 + 21121940 T + 328931801286510 T^{2} + \)\(27\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!29\)\( T^{4} \))
$41$	(\( 1 - 11056262 T + 327381934393961 T^{2} \))(\( 1 + 26957276 T + 811945448362966 T^{2} + \)\(88\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} \))
$43$	(\( 1 + 6396794 T + 502592611936843 T^{2} \))(\( 1 - 52889700 T + 1703843788760950 T^{2} - \)\(26\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!49\)\( T^{4} \))
$47$	(\( 1 + 35559158 T + 1119130473102767 T^{2} \))(\( 1 - 58412180 T + 2814913257457630 T^{2} - \)\(65\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} \))
$53$	(\( 1 - 39738586 T + 3299763591802133 T^{2} \))(\( 1 + 39035140 T + 5675030678030830 T^{2} + \)\(12\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!89\)\( T^{4} \))
$59$	(\( 1 + 85185620 T + 8662995818654939 T^{2} \))(\( 1 + 54995560 T + 15674484224932678 T^{2} + \)\(47\!\cdots\!40\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))
$61$	(\( 1 - 45748642 T + 11694146092834141 T^{2} \))(\( 1 + 274579716 T + 41753623519328446 T^{2} + \)\(32\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} \))
$67$	(\( 1 + 45286158 T + 27206534396294947 T^{2} \))(\( 1 + 318580 T + 48520062064444070 T^{2} + \)\(86\!\cdots\!60\)\( T^{3} + \)\(74\!\cdots\!09\)\( T^{4} \))
$71$	(\( 1 + 189967468 T + 45848500718449031 T^{2} \))(\( 1 + 7130936 T + 51935375688707086 T^{2} + \)\(32\!\cdots\!16\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))
$73$	(\( 1 - 412170946 T + 58871586708267913 T^{2} \))(\( 1 - 120858180 T + 42707263689423190 T^{2} - \)\(71\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!69\)\( T^{4} \))
$79$	(\( 1 - 95040840 T + 119851595982618319 T^{2} \))(\( 1 - 6877520 T - 115982362290712162 T^{2} - \)\(82\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} \))
$83$	(\( 1 - 261706326 T + 186940255267540403 T^{2} \))(\( 1 - 1402348740 T + 857904310704391270 T^{2} - \)\(26\!\cdots\!20\)\( T^{3} + \)\(34\!\cdots\!09\)\( T^{4} \))
$89$	(\( 1 + 19938630 T + 350356403707485209 T^{2} \))(\( 1 - 830088660 T + 692293619421117718 T^{2} - \)\(29\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} \))
$97$	(\( 1 + 19503358 T + 760231058654565217 T^{2} \))(\( 1 - 638394580 T + 1615411126351062630 T^{2} - \)\(48\!\cdots\!60\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} \))
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