Space of modular forms of level 10, weight 32, and trivial character

• Label: 10.32.a
• Level: $10$
• Weight: $32$
• Character orbit: 10.a
• Rep. character: $\chi_{10}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $4$
• Sturm bound: $48$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 32 \)
Character orbit:	\([\chi]\)	\(=\)	10.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(48\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{32}(\Gamma_0(10))\).

	Total	New	Old
Modular forms	49	9	40
Cusp forms	45	9	36
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
\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(2\)
\(-\)	\(-\)	\(+\)	\(3\)
Plus space		\(+\)	\(5\)
Minus space		\(-\)	\(4\)

Trace form

9 * q + 32768 * q^2 + 62917228 * q^3 + 9663676416 * q^4 + 30517578125 * q^5 + 546512961536 * q^6 + 40452679234344 * q^7 + 35184372088832 * q^8 + 773625920806333 * q^9 + 1000000000000000 * q^10 - 33954123949698492 * q^11 + 67556859153743872 * q^12 + 18506974808973798 * q^13 - 693723698380931072 * q^14 - 2228026489257812500 * q^15 + 10376293541461622784 * q^16 + 512816064594633474 * q^17 + 82038791749239996416 * q^18 - 83265722267870979780 * q^19 + 32768000000000000000 * q^20 - 664147178102312553712 * q^21 + 61601576950766370816 * q^22 - 815148792745897217832 * q^23 + 586813824159306481664 * q^24 + 8381903171539306640625 * q^25 + 15347440807969206501376 * q^26 - 47153562783358673669480 * q^27 + 43435733586771450003456 * q^28 - 63985963480975615843770 * q^29 - 3149972000000000000000 * q^30 + 8845929521136813751488 * q^31 + 37778931862957161709568 * q^32 + 666451202483194012968336 * q^33 + 294491157480489814130688 * q^34 + 1188193533775146484375000 * q^35 + 830674507300271546171392 * q^36 - 3217833821465536391118066 * q^37 - 4007743966567191033282560 * q^38 + 723804281743185039360296 * q^39 + 1073741824000000000000000 * q^40 + 31315238250952116916761978 * q^41 - 48350057258151359858868224 * q^42 + 72759356030870164088278308 * q^43 - 36457962982071343050129408 * q^44 + 70410238551234039306640625 * q^45 - 71919625682178513046339584 * q^46 - 252920624458381337032724736 * q^47 + 72538625171452041550102528 * q^48 + 409806378207482822166532737 * q^49 + 30517578125000000000000000 * q^50 - 919986720209615317399344552 * q^51 + 19871712888109577432727552 * q^52 + 904524338277503938631511678 * q^53 + 509607085312145762008432640 * q^54 - 762197610143755737304687500 * q^55 - 744880149251566776067555328 * q^56 + 2554582325203459139142212240 * q^57 - 5888848454979048632756797440 * q^58 + 11132073747017330971690493460 * q^59 - 2392325226496000000000000000 * q^60 + 1993675490509094573030855958 * q^61 + 6147867149415889064795570176 * q^62 - 11795411115036385449220253272 * q^63 + 11141460353568422474092118016 * q^64 + 12382825027125653015136718750 * q^65 + 47848504448887215404267077632 * q^66 - 25607325075061211743465970676 * q^67 + 550632056574343566984216576 * q^68 - 45661887545718012966991091664 * q^69 + 28491911738744000000000000000 * q^70 + 37875851952241079660398967448 * q^71 + 88088481891585104365469302784 * q^72 + 84505411594301660500923972618 * q^73 + 25355215551445594331290206208 * q^74 + 58596234768629074096679687500 * q^75 - 89405888504581202425644318720 * q^76 + 24709739675746271361232508928 * q^77 - 206011860783189316397696811008 * q^78 - 43417621810453201976055029520 * q^79 + 35184372088832000000000000000 * q^80 + 86295870833743824932365855489 * q^81 - 76784047850925072269297516544 * q^82 + 2602128461163099209651437067388 * q^83 - 713122602420029940040820850688 * q^84 - 720937761565669889099121093750 * q^85 - 1169619189633424407570757320704 * q^86 + 681575015886860063728937546760 * q^87 + 66144189596392241197832208384 * q^88 + 218935145233301899926225117690 * q^89 + 893887851988837000000000000000 * q^90 + 3423133534685566904098805667408 * q^91 - 875259351554377647191457005568 * q^92 - 5778120217738832488013252891104 * q^93 + 8519027416094085335033137594368 * q^94 - 4757863036541982776489257812500 * q^95 + 630086545901229008196925915136 * q^96 - 13461378010917022697870537651886 * q^97 - 4606577489797374657476047896576 * q^98 - 34245383552813541347612177081004 * q^99

Decomposition of \(S_{32}^{\mathrm{new}}(\Gamma_0(10))\) 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
10.32.a.a	$10$	$32$	10.a	1.a	$1$	$2$	$2$	$60.877$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	10.32.a.a	$1$	$1$	\(-65536\)	\(-5904756\)	\(61035156250\)	\(18\!\cdots\!12\)		$+$	$-$	$-$	$2^{2}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{15}q^{2}+(-2952378-\beta )q^{3}+2^{30}q^{4}+\cdots\)
10.32.a.b	$10$	$32$	10.a	1.a	$1$	$2$	$2$	$60.877$	\(\Q(\sqrt{337159}) \)	None	✓	✓	✓	✓	10.32.a.b	$1$	$0$	\(-65536\)	\(29024244\)	\(-61035156250\)	\(12\!\cdots\!12\)		$+$	$+$	$+$	$2^{4}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{15}q^{2}+(14512122+3^{3}\beta )q^{3}+\cdots\)
10.32.a.c	$10$	$32$	10.a	1.a	$1$	$2$	$2$	$60.877$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	10.32.a.c	$1$	$1$	\(65536\)	\(38938356\)	\(-61035156250\)	\(-12\!\cdots\!12\)		$-$	$+$	$-$	$2^{4}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{15}q^{2}+(19469178+\beta )q^{3}+2^{30}q^{4}+\cdots\)
10.32.a.d	$10$	$32$	10.a	1.a	$1$	$3$	$3$	$60.877$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	10.32.a.d	$1$	$0$	\(98304\)	\(859384\)	\(91552734375\)	\(21\!\cdots\!32\)		$-$	$-$	$+$	$2^{9}\cdot 3\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+2^{15}q^{2}+(286461-\beta _{1})q^{3}+2^{30}q^{4}+\cdots\)

Decomposition of \(S_{32}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{32}^{\mathrm{old}}(\Gamma_0(10)) \simeq \) \(S_{32}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{32}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{32}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)