Space of modular forms of level 2, weight 42, and trivial character

• Label: 2.42.a
• Level: $2$
• Weight: $42$
• Character orbit: 2.a
• Rep. character: $\chi_{2}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $10$
• Trace bound: $1$

Defining parameters

Level:	$N$	$=$	$2$
Weight:	$k$	$=$	$42$
Character orbit:	$[\chi]$	$=$	2.a (trivial)
Character field:			$\Q$
Newform subspaces:			$2$
Sturm bound:			$10$
Trace bound:			$1$
Distinguishing $T_p$:			$3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{42}(\Gamma_0(2))$.

	Total	New	Old
Modular forms	11	3	8
Cusp forms	9	3	6
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.

$2$		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
$+$		$5$	$1$	$4$		$4$	$1$	$3$		$1$	$0$	$1$
$-$		$6$	$2$	$4$		$5$	$2$	$3$		$1$	$0$	$1$

Trace form

3 * q + 1048576 * q^2 + 13906864044 * q^3 + 3298534883328 * q^4 + 49094989194930 * q^5 + 4005383123238912 * q^6 + 97714356676438488 * q^7 + 1152921504606846976 * q^8 + 30936992086866501639 * q^9 + 153200497216695828480 * q^10 + 3264403333766748924036 * q^11 + 15290758722277966086144 * q^12 + 161882807152022753465994 * q^13 + 352845035543974457114624 * q^14 - 3539918996489522338057080 * q^15 + 3626777458843887524118528 * q^16 - 37011726883142944282175562 * q^17 + 55583825877405288820113408 * q^18 + 274230100811445606374810460 * q^19 + 53980511485362616066375680 * q^20 + 2665691359852998167875660896 * q^21 - 3191041552072698539506925568 * q^22 + 1420347666919737243157853064 * q^23 + 4403965317698934946463219712 * q^24 + 54412888097875513082262174525 * q^25 + 193675559200226950852527521792 * q^26 + 293544196044657269619833966520 * q^27 + 107438071366395535287216242688 * q^28 - 1238427538159622404128372563910 * q^29 - 3198844223373791442948618977280 * q^30 - 565073962679875259305481759904 * q^31 + 1267650600228229401496703205376 * q^32 + 2233551046463051372662468702608 * q^33 + 17234179508571566375244140642304 * q^34 - 97248489612709189284759665330160 * q^35 + 34015582527923818408303461924864 * q^36 - 101592630862331863544766032309742 * q^37 + 144996736400908851639068697559040 * q^38 + 640160633638918340067446036904168 * q^39 + 168445728070821787776313699860480 * q^40 + 768192138006768072331689412248126 * q^41 + 4057996358704299679930133010972672 * q^42 - 9163911628959853913209536056311836 * q^43 + 3589249423227279134970318931623936 * q^44 - 30457369032504081176750813707625910 * q^45 + 29811534820383674194267711022825472 * q^46 - 12260960588935821767780288588908272 * q^47 + 16812367012661916406114713279135744 * q^48 - 25596406892660163029921184065783829 * q^49 + 147489600972651489691044136380006400 * q^50 - 182678347564418655807945350847940584 * q^51 + 177992028800668832354384608801849344 * q^52 - 625375082980902027370788377449928766 * q^53 + 810306378590012747923651603590021120 * q^54 + 550426134204771570148170929066219160 * q^55 + 387957219383635932970666138454196224 * q^56 - 442953519980269300084782666393665040 * q^57 - 1585298523526682507321181181449338880 * q^58 + 177563983295764361517342239221260180 * q^59 - 3892182098025379135645853384001454080 * q^60 - 5463201135704660666208213208341875814 * q^61 - 7012792278966542721658673238982524928 * q^62 + 26310155382295499484506243647288915704 * q^63 + 3987683987354747618711421180841033728 * q^64 + 12473259455640304370441775300271028220 * q^65 - 31015853645488527189477610831170502656 * q^66 + 71683283838926171050853516149998220908 * q^67 - 40694824072085237602584124037227610112 * q^68 - 11480840010959325121257435993672114912 * q^69 - 114117111792639301686346513623086530560 * q^70 + 68110973123647230466528954560382296216 * q^71 + 61115062868483640529874307679002820608 * q^72 - 294908420905755625150102017496226933346 * q^73 + 358933845813325666940616726553416433664 * q^74 - 149075468778282886564381918940741001900 * q^75 + 301519184528369137116851211238071336960 * q^76 - 796259039077942095596584187532209589984 * q^77 + 791944053109205052642588176234877812736 * q^78 - 1467129308592083502686260888480869532560 * q^79 + 59352200051452113588758453977538887680 * q^80 + 1825606948109165162789233850315793931563 * q^81 + 1858246690007324705570778546446744420352 * q^82 + 665984279782956439537682608316709433884 * q^83 + 2930958646220388991634913623732750647296 * q^84 + 907701952915589923696197022820240971140 * q^85 - 3069134989805620595871083685892449107968 * q^86 - 17068192400013102750887440771225829791320 * q^87 - 3508587291220306237862197575696953376768 * q^88 + 14607841393604595000567679910075985192270 * q^89 - 33059449148344087288488905373318654197760 * q^90 + 18031837917819409202012303003569888867536 * q^91 + 1561688775262764164166701740916069105664 * q^92 + 55858944591888014581432169202537944899968 * q^93 - 40445670686694062392590229249526392160256 * q^94 + 90543084377116752054454370649641547666600 * q^95 + 4842211075132204945687306119655049920512 * q^96 - 95565478890036662057547911249774079434202 * q^97 + 36731367279780021226536126103678601920512 * q^98 - 158014553036426914776149850133844786493132 * q^99

Decomposition of $S_{42}^{\mathrm{new}}(\Gamma_0(2))$ 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
2.42.a.a	$2$	$42$	2.a	1.a	$1$	$1$	$1$	$21.294$	$\Q$	None	✓	✓	✓	✓	2.42.a.a	$1$	$1$	$-1048576$	$5043516516$	$-48\!\cdots\!50$	$-11\!\cdots\!68$		$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2^{20}q^{2}+5043516516q^{3}+2^{40}q^{4}+\cdots$
2.42.a.b	$2$	$42$	2.a	1.a	$1$	$2$	$2$	$21.294$	$\mathbb{Q}[x]/(x^{2} - \cdots)$	None	✓	✓	✓	✓	2.42.a.b	$1$	$0$	$2097152$	$8863347528$	$97\!\cdots\!80$	$21\!\cdots\!56$		$-$	$-$	$2^{7}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	$q+2^{20}q^{2}+(4431673764-\beta )q^{3}+\cdots$

Decomposition of $S_{42}^{\mathrm{old}}(\Gamma_0(2))$ into lower level spaces

$S_{42}^{\mathrm{old}}(\Gamma_0(2)) \simeq$ $S_{42}^{\mathrm{new}}(\Gamma_0(1))$$^{\oplus 2}$