Space of modular forms of level 10, weight 26, and trivial character

• Label: 10.26.a
• Level: $10$
• Weight: $26$
• Character orbit: 10.a
• Rep. character: $\chi_{10}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $4$
• Sturm bound: $39$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 26 \)
Character orbit:	\([\chi]\)	\(=\)	10.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(39\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{26}(\Gamma_0(10))\).

	Total	New	Old
Modular forms	39	7	32
Cusp forms	35	7	28
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(9\)	\(2\)	\(7\)		\(8\)	\(2\)	\(6\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(10\)	\(2\)	\(8\)		\(9\)	\(2\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(10\)	\(2\)	\(8\)		\(9\)	\(2\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(10\)	\(1\)	\(9\)		\(9\)	\(1\)	\(8\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(19\)	\(3\)	\(16\)		\(17\)	\(3\)	\(14\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(20\)	\(4\)	\(16\)		\(18\)	\(4\)	\(14\)		\(2\)	\(0\)	\(2\)

Trace form

7 * q - 4096 * q^2 + 172396 * q^3 + 117440512 * q^4 - 244140625 * q^5 - 167329792 * q^6 - 29178793288 * q^7 - 68719476736 * q^8 + 1158717865771 * q^9 - 1000000000000 * q^10 + 14141739881364 * q^11 + 2892324929536 * q^12 - 179593027428814 * q^13 - 156865797914624 * q^14 - 176719726562500 * q^15 + 1970324836974592 * q^16 - 278192702833218 * q^17 - 18024426694955008 * q^18 + 2662636993077740 * q^19 - 4096000000000000 * q^20 + 140219964236834384 * q^21 - 31904915250266112 * q^22 - 57464588148196344 * q^23 - 2807328063619072 * q^24 + 417232513427734375 * q^25 + 1535535280260456448 * q^26 - 1492578851865494120 * q^27 - 489538917612126208 * q^28 - 3764573232098380590 * q^29 + 1243756000000000000 * q^30 + 11710470057462450464 * q^31 - 1152921504606846976 * q^32 - 23721594650263551888 * q^33 - 4091844322050514944 * q^34 - 18076217521484375000 * q^35 + 19440059917099073536 * q^36 + 155457717802810003802 * q^37 - 17790383732468695040 * q^38 + 77247582049582866632 * q^39 - 16777216000000000000 * q^40 - 616454417673702104586 * q^41 + 19951777482209886208 * q^42 + 1010218220561249182436 * q^43 + 237259024605458202624 * q^44 + 501012207008544921875 * q^45 + 620118219001881919488 * q^46 - 828989199456009681648 * q^47 + 48525160085010251776 * q^48 - 537667981646800428801 * q^49 - 244140625000000000000 * q^50 + 11173034010793814326104 * q^51 - 3013071013267137101824 * q^52 - 3595408668331726645494 * q^53 + 4172553796231080673280 * q^54 + 504608894411132812500 * q^55 - 2631771374625996406784 * q^56 + 4204922984141094699440 * q^57 + 2713638168327776378880 * q^58 - 4310816447663264747580 * q^59 - 2964865024000000000000 * q^60 - 46117408232534380843486 * q^61 + 19279033604719908487168 * q^62 - 4245583819863456185224 * q^63 + 33056565380087516495872 * q^64 - 16483069035179199218750 * q^65 + 1576463167932418031616 * q^66 + 24383692289330109140012 * q^67 - 4667299065056710361088 * q^68 - 103302157856295027118608 * q^69 + 71112720232000000000000 * q^70 + 165985679043215198559864 * q^71 - 302399699937426279497728 * q^72 + 79974919621594950676166 * q^73 - 207417649437471442755584 * q^74 + 10275602340698242187500 * q^75 + 44671635962455748771840 * q^76 - 1568720493209389904209536 * q^77 - 9799023502584299749376 * q^78 + 381648174445280844499760 * q^79 - 68719476736000000000000 * q^80 - 660486416486408743044833 * q^81 + 1501915684254291799400448 * q^82 + 389721063496013168985276 * q^83 + 2352500627513645616594944 * q^84 - 1951961685963754394531250 * q^85 + 756088228261559864147968 * q^86 + 1376374965251964003154920 * q^87 - 535275654615408618504192 * q^88 + 6164073392704937932263030 * q^89 - 2093424774893000000000000 * q^90 - 393445264691285323698896 * q^91 - 964095807713330073698304 * q^92 + 4622114231236669578643232 * q^93 + 1176095827461639971536896 * q^94 + 816436363566440429687500 * q^95 - 47099149306198912663552 * q^96 - 31710595169851658894946418 * q^97 + 24127501086126701924610048 * q^98 + 7893709751864437212207492 * q^99

Decomposition of \(S_{26}^{\mathrm{new}}(\Gamma_0(10))\) 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
10.26.a.a	$10$	$26$	10.a	1.a	$1$	$1$	$1$	$39.600$	\(\Q\)	None	✓	✓	✓	✓	10.26.a.a	$1$	$1$	\(4096\)	\(162864\)	\(244140625\)	\(-17600893492\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{12}q^{2}+162864q^{3}+2^{24}q^{4}+\cdots\)
10.26.a.b	$10$	$26$	10.a	1.a	$1$	$2$	$2$	$39.600$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	10.26.a.b	$1$	$0$	\(-8192\)	\(-438588\)	\(488281250\)	\(-34008596636\)		$+$	$-$	$-$	$2^{2}\cdot 3^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{12}q^{2}+(-219294-\beta )q^{3}+2^{24}q^{4}+\cdots\)
10.26.a.c	$10$	$26$	10.a	1.a	$1$	$2$	$2$	$39.600$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	10.26.a.c	$1$	$1$	\(-8192\)	\(545212\)	\(-488281250\)	\(38567856964\)		$+$	$+$	$+$	$2^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{12}q^{2}+(272606+\beta )q^{3}+2^{24}q^{4}+\cdots\)
10.26.a.d	$10$	$26$	10.a	1.a	$1$	$2$	$2$	$39.600$	\(\Q(\sqrt{95351}) \)	None	✓	✓	✓	✓	10.26.a.d	$1$	$0$	\(8192\)	\(-97092\)	\(-488281250\)	\(-16137160124\)		$-$	$+$	$-$	$2^{4}\cdot 3^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{12}q^{2}+(-48546+\beta )q^{3}+2^{24}q^{4}+\cdots\)

Decomposition of \(S_{26}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{26}^{\mathrm{old}}(\Gamma_0(10)) \simeq \) \(S_{26}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)