Space of modular forms of level 10, weight 34, and trivial character

• Label: 10.34.a
• Level: $10$
• Weight: $34$
• Character orbit: 10.a
• Rep. character: $\chi_{10}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $4$
• Sturm bound: $51$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 34 \)
Character orbit:	\([\chi]\)	\(=\)	10.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(51\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{34}(\Gamma_0(10))\).

	Total	New	Old
Modular forms	51	11	40
Cusp forms	47	11	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
\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(-\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(3\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(5\)
Minus space		\(-\)	\(6\)

Trace form

11 * q - 65536 * q^2 + 74568256 * q^3 + 47244640256 * q^4 - 152587890625 * q^5 - 11300222795776 * q^6 + 22560918015772 * q^7 - 281474976710656 * q^8 + 11600305489343143 * q^9 - 10000000000000000 * q^10 + 25673795810039532 * q^11 + 320268220839755776 * q^12 - 8874325095364822094 * q^13 - 2553183225590054912 * q^14 + 13608040771484375000 * q^15 + 202914184810805067776 * q^16 + 292948424559249424662 * q^17 + 344984409678544044032 * q^18 - 4376360335694897738900 * q^19 - 655360000000000000000 * q^20 + 36108769558551913249352 * q^21 + 11364797013025828896768 * q^22 + 6370309914738581109876 * q^23 - 48534087345371606941696 * q^24 + 256113708019256591796875 * q^25 + 13872810805198299725824 * q^26 + 2061553675044024442919440 * q^27 + 96898405045477952192512 * q^28 + 4495771058501403713239290 * q^29 - 2899158440000000000000000 * q^30 + 640614209589648603228472 * q^31 - 1208925819614629174706176 * q^32 + 32887585740938981039977872 * q^33 - 54471746073788513627799552 * q^34 + 118095575801391601562500 * q^35 + 49822932700338075706851328 * q^36 - 58155933769018527442357238 * q^37 - 422915893532945301772697600 * q^38 - 26214558840247836502027504 * q^39 - 42949672960000000000000000 * q^40 - 89106014157910099846560618 * q^41 + 1983383128835802691690037248 * q^42 + 40321834485560811883229176 * q^43 + 110268113368301618407145472 * q^44 - 2547730138069429473876953125 * q^45 + 5375551262161136681816162304 * q^46 + 6076370083323837895466025972 * q^47 + 1375541534454856714547101696 * q^48 + 27076946391202393831079179947 * q^49 - 1525878906250000000000000000 * q^50 + 34562557222269489053579800992 * q^51 - 38114936058663992082588237824 * q^52 - 154273000350213599237533444854 * q^53 - 89344492849369561579076976640 * q^54 - 49115754832116487426757812500 * q^55 - 10965838454605076149884157952 * q^56 + 389758793131386406683302765600 * q^57 - 135342231695310834344564490240 * q^58 - 314717712032127685530035827020 * q^59 + 58446090076160000000000000000 * q^60 - 38482533611929622903999116478 * q^61 - 669787211377795339952819535872 * q^62 + 1932755266001120526055248755036 * q^63 + 871509787656907713528983453696 * q^64 + 276216662068944046936035156250 * q^65 + 3530957666704933081795959717888 * q^66 - 383128478684788296357963882968 * q^67 + 1258203902896699493230105853952 * q^68 - 74987381525068270239628223784 * q^69 + 969347727580720000000000000000 * q^70 - 11512893539423329553419503921648 * q^71 + 1481696757199212542013023977472 * q^72 + 11676491506595252901310632876766 * q^73 - 8752992863702764977761798193152 * q^74 + 1736177504062652587890625000000 * q^75 - 18796324517321167222639847014400 * q^76 + 9118295992261211952191322084864 * q^77 + 17794894803607201493766443106304 * q^78 + 19104385717416635754603908668960 * q^79 - 2814749767106560000000000000000 * q^80 + 120709572987866231339990248117051 * q^81 - 21121060322229989613340902162432 * q^82 - 79778192833820459197066044083904 * q^83 + 155085984352780824524195933192192 * q^84 + 20717909761740254037780761718750 * q^85 + 85700588354317409157291112136704 * q^86 + 161485066194342813500375380667040 * q^87 + 48811431496624421114910320099328 * q^88 - 281399490538109431940508468133410 * q^89 - 243105960887968130000000000000000 * q^90 + 125637798118916646171769464722552 * q^91 + 27360272749186754256360802615296 * q^92 - 154490173504452527787707803140688 * q^93 - 30959444603327079225205705408512 * q^94 + 559941878480689900375366210937500 * q^95 - 208452317889578508781550898774016 * q^96 - 752155855759423954859168212154858 * q^97 - 576650432265367709700223374262272 * q^98 + 1453530816991199357634870367630716 * q^99

Decomposition of \(S_{34}^{\mathrm{new}}(\Gamma_0(10))\) 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
10.34.a.a	$10$	$34$	10.a	1.a	$1$	$2$	$2$	$68.983$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	10.34.a.a	$1$	$1$	\(131072\)	\(-74648412\)	\(305175781250\)	\(20\!\cdots\!56\)		$-$	$-$	$+$	$2^{2}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{16}q^{2}+(-37324206-3\beta )q^{3}+\cdots\)
10.34.a.b	$10$	$34$	10.a	1.a	$1$	$3$	$3$	$68.983$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	10.34.a.b	$1$	$1$	\(-196608\)	\(-33025382\)	\(-457763671875\)	\(39\!\cdots\!66\)		$+$	$+$	$+$	$2^{12}\cdot 3^{2}\cdot 5^{3}\cdot 7\cdot 11$	$\mathrm{SU}(2)$	\(q-2^{16}q^{2}+(-11008461-\beta _{1})q^{3}+\cdots\)
10.34.a.c	$10$	$34$	10.a	1.a	$1$	$3$	$3$	$68.983$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	10.34.a.c	$1$	$0$	\(-196608\)	\(156523368\)	\(457763671875\)	\(-86\!\cdots\!84\)		$+$	$-$	$-$	$2^{9}\cdot 3^{3}\cdot 5^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q-2^{16}q^{2}+(52174456-\beta _{1})q^{3}+2^{32}q^{4}+\cdots\)
10.34.a.d	$10$	$34$	10.a	1.a	$1$	$3$	$3$	$68.983$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	10.34.a.d	$1$	$0$	\(196608\)	\(25718682\)	\(-457763671875\)	\(-28\!\cdots\!66\)		$-$	$+$	$-$	$2^{12}\cdot 3^{3}\cdot 5^{3}\cdot 11$	$\mathrm{SU}(2)$	\(q+2^{16}q^{2}+(8572894-\beta _{1})q^{3}+2^{32}q^{4}+\cdots\)

Decomposition of \(S_{34}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{34}^{\mathrm{old}}(\Gamma_0(10)) \simeq \) \(S_{34}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{34}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{34}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)