Space of modular forms of level 10, weight 30, and trivial character

• Label: 10.30.a
• Level: $10$
• Weight: $30$
• Character orbit: 10.a
• Rep. character: $\chi_{10}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $4$
• Sturm bound: $45$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 30 \)
Character orbit:	\([\chi]\)	\(=\)	10.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(45\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{30}(\Gamma_0(10))\).

	Total	New	Old
Modular forms	45	11	34
Cusp forms	41	11	30
Eisenstein series	4	0	4

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

\(2\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(-\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(3\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(5\)
Minus space		\(-\)	\(6\)

Trace form

11 * q - 16384 * q^2 - 15889676 * q^3 + 2952790016 * q^4 - 6103515625 * q^5 + 68507009024 * q^6 - 3301442406392 * q^7 - 4398046511104 * q^8 + 415441897106023 * q^9 - 100000000000000 * q^10 - 54708336104988 * q^11 - 4265352422752256 * q^12 - 9994627384813646 * q^13 + 30697561292996608 * q^14 - 10968481445312500 * q^15 + 792633534417207296 * q^16 - 521553325153684122 * q^17 - 2702288166880755712 * q^18 + 5322385630114350220 * q^19 - 1638400000000000000 * q^20 - 1084098985350501688 * q^21 - 62231466094223228928 * q^22 + 139454904901997495784 * q^23 + 18389710206553554944 * q^24 + 409781932830810546875 * q^25 - 811847000758034333696 * q^26 - 255539613774060065960 * q^27 - 886224197817573834752 * q^28 + 3359672993275822274730 * q^29 - 107357600000000000000 * q^30 + 16228581997876424826592 * q^31 - 1180591620717411303424 * q^32 - 46119560272399813118592 * q^33 + 25148420596135423868928 * q^34 - 17152386121777343750000 * q^35 + 111519335091160364351488 * q^36 + 5288864962332411063418 * q^37 - 63027690681444729159680 * q^38 - 133340964393361077445144 * q^39 - 26843545600000000000000 * q^40 + 755428571660158547266182 * q^41 - 273000181132563988348928 * q^42 - 644710088125413611633156 * q^43 - 14685657149343717654528 * q^44 - 823119543566241455078125 * q^45 - 606691272881957885116416 * q^46 + 758395963760620776590688 * q^47 - 1144971822602206614388736 * q^48 + 11688396330830584531795227 * q^49 - 610351562500000000000000 * q^50 - 15783828624305995419406008 * q^51 - 2682912359592538539032576 * q^52 - 3618019055610234593494326 * q^53 + 29064342116501018885488640 * q^54 + 27629055739677172851562500 * q^55 + 8240313863773494074933248 * q^56 + 48857138949841582726922480 * q^57 + 69910241023124646729646080 * q^58 - 78604809610824772873123740 * q^59 - 2944329318400000000000000 * q^60 + 51491971200474867414638962 * q^61 - 304121863678682287742517248 * q^62 + 214624422386392230749827544 * q^63 + 212770944252174734748286976 * q^64 + 209445666264053674316406250 * q^65 + 83511753139624770950135808 * q^66 + 1036411244249033650454359828 * q^67 - 140003404665945467369029632 * q^68 - 1210197367072378349486672424 * q^69 - 624191122269200000000000000 * q^70 + 2165037055980880864002868152 * q^71 - 725389956320039757175324672 * q^72 - 4219319739837575148050226866 * q^73 + 4416918972787504075273043968 * q^74 - 591936558485031127929687500 * q^75 + 1428717013627592933449400320 * q^76 + 4709185236928164902483369136 * q^77 + 3398075965103144930186100736 * q^78 - 5935686124736243674534947920 * q^79 - 439804651110400000000000000 * q^80 - 8653948977635964630702364949 * q^81 - 15607726626299770160571383808 * q^82 + 31454671470146203377996238164 * q^83 - 291010605481699240447049728 * q^84 + 4655078890625522937011718750 * q^85 - 6947802758949398089059270656 * q^86 - 24199876236602858842852655880 * q^87 - 16705131978551351423080071168 * q^88 - 27727553991436576120433677410 * q^89 + 8475230454510700000000000000 * q^90 + 43765674618104033367901024352 * q^91 + 37434640988804333091636117504 * q^92 - 118904172807683916536560285072 * q^93 + 34880754085150026641515020288 * q^94 - 48286550959569646118164062500 * q^95 + 4936450245004057709813694464 * q^96 - 193954048978928548292661906362 * q^97 - 136572537443999123729646501888 * q^98 + 240712056943611484542838784916 * q^99

Decomposition of \(S_{30}^{\mathrm{new}}(\Gamma_0(10))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5
10.30.a.a	$10$	$30$	10.a	1.a	$1$	$2$	$2$	$53.278$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	10.30.a.a	$1$	$1$	\(32768\)	\(-3644748\)	\(12207031250\)	\(-26\!\cdots\!16\)		$-$	$-$	$+$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{14}q^{2}+(-1822374-3\beta )q^{3}+\cdots\)
10.30.a.b	$10$	$30$	10.a	1.a	$1$	$3$	$3$	$53.278$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	10.30.a.b	$1$	$0$	\(-49152\)	\(-5198628\)	\(18310546875\)	\(-435852143976\)		$+$	$-$	$-$	$2^{9}\cdot 3^{5}\cdot 5^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q-2^{14}q^{2}+(-1732876+\beta _{1})q^{3}+\cdots\)
10.30.a.c	$10$	$30$	10.a	1.a	$1$	$3$	$3$	$53.278$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	10.30.a.c	$1$	$1$	\(-49152\)	\(-4836878\)	\(-18310546875\)	\(-21\!\cdots\!26\)		$+$	$+$	$+$	$2^{11}\cdot 3\cdot 5^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q-2^{14}q^{2}+(-1612293-\beta _{1})q^{3}+\cdots\)
10.30.a.d	$10$	$30$	10.a	1.a	$1$	$3$	$3$	$53.278$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	10.30.a.d	$1$	$0$	\(49152\)	\(-2209422\)	\(-18310546875\)	\(19\!\cdots\!26\)		$-$	$+$	$-$	$2^{11}\cdot 3^{5}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+2^{14}q^{2}+(-736474+\beta _{1})q^{3}+2^{28}q^{4}+\cdots\)

Decomposition of \(S_{30}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{30}^{\mathrm{old}}(\Gamma_0(10)) \simeq \) \(S_{30}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{30}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{30}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)