Space of modular forms of level 2, weight 34, and trivial character

• Label: 2.34.a
• Level: $2$
• Weight: $34$
• Character orbit: 2.a
• Rep. character: $\chi_{2}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $8$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 2 \)
Weight:	\( k \)	\(=\)	\( 34 \)
Character orbit:	\([\chi]\)	\(=\)	2.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(8\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{34}(\Gamma_0(2))\).

	Total	New	Old
Modular forms	9	3	6
Cusp forms	7	3	4
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.

\(2\)	Dim
\(+\)	\(1\)
\(-\)	\(2\)

Trace form

3 * q + 65536 * q^2 - 124649076 * q^3 + 12884901888 * q^4 + 533466655890 * q^5 + 9264303439872 * q^6 + 99371784824088 * q^7 + 281474976710656 * q^8 + 18852405815679399 * q^9 - 35660209141186560 * q^10 - 243987735048109404 * q^11 - 535363704896618496 * q^12 + 6127452524374998954 * q^13 + 10883328044408766464 * q^14 - 540011606028256440 * q^15 + 55340232221128654848 * q^16 + 163351956857193229878 * q^17 - 354578147647536955392 * q^18 + 2316286200640058928060 * q^19 + 2291221840554035773440 * q^20 - 10966882762375461396384 * q^21 - 4733220146206274224128 * q^22 - 85544934143271027955896 * q^23 + 39789880294470542426112 * q^24 + 225370380364752475573725 * q^25 + 251615281586057289531392 * q^26 - 743752090411160780985480 * q^27 + 426798565964607073026048 * q^28 - 1885570868915532180931110 * q^29 + 9357659564213453452738560 * q^30 - 6214164899357651844971424 * q^31 + 1208925819614629174706176 * q^32 - 28646531485170550011011952 * q^33 + 28939341541930727759806464 * q^34 - 82084709605255936704167280 * q^35 + 80970466429263222725935104 * q^36 + 19582065655883864004622578 * q^37 + 141223277382323304263843840 * q^38 + 110743367844616772680078248 * q^39 - 153159432029916521834741760 * q^40 + 474990906888239155557354366 * q^41 - 1300079482701292075135008768 * q^42 - 854701774198330191704131836 * q^43 - 1047919342656742876810051584 * q^44 + 7113005491199426978344868970 * q^45 - 3755487355665331416256217088 * q^46 - 3359713735806216538266732912 * q^47 - 2299369603996371501308706816 * q^48 + 1027325841837036133332384171 * q^49 - 8022129800291472803432038400 * q^50 + 34605359459142380076105389016 * q^51 + 26317208199983263347464208384 * q^52 - 25719867322108276836703217886 * q^53 + 65835402527390881086630789120 * q^54 - 203329095921753105019451578920 * q^55 + 46743538022375287618581561344 * q^56 - 246920391711219760108249543440 * q^57 + 90426765572487936244912619520 * q^58 + 486574758853958227127412969780 * q^59 - 2319332187351797861701386240 * q^60 + 8583862334335294799056153146 * q^61 - 158990793629680529653626830848 * q^62 - 91891357726791003768354017736 * q^63 + 237684487542793012780631851008 * q^64 + 1570508277569118621577450564860 * q^65 - 3374590807726680306939142864896 * q^66 + 422874132479815900981844675148 * q^67 + 701591312439247864678620069888 * q^68 + 455702719474062148458385981728 * q^69 - 3024467071475429121489009377280 * q^70 + 5198977231352455543531112459736 * q^71 - 1522901548022430558360050860032 * q^72 - 8030146104418329689734403612706 * q^73 + 13924466895002049051747468836864 * q^74 - 23292509902333321603477068569100 * q^75 + 9948373479925147363530516725760 * q^76 + 27691751685039233175366988812576 * q^77 + 27202320153878090082468855545856 * q^78 - 25638830707110254946114730329360 * q^79 + 9840722873060510167738865418240 * q^80 - 26510678888959286274847485768597 * q^81 - 52988841827294806721870070939648 * q^82 - 53261726130515111552883263441796 * q^83 - 47102402803468745970379772657664 * q^84 - 16465362595617182300874127067580 * q^85 + 51199962022156216766968491671552 * q^86 + 401367777050797863744672651928680 * q^87 - 20329025732724286262637534117888 * q^88 + 99483015972296495211736491705870 * q^89 - 390580869706777575871691707514880 * q^90 + 75577323879611270472141157040976 * q^91 - 367412694483822843534875050377216 * q^92 + 194910148274720557866176228573568 * q^93 + 596297622405425787577101230014464 * q^94 - 943613522324582126565259935432600 * q^95 + 170896234576505829355531536433152 * q^96 + 294906994984199222096704110296358 * q^97 + 934885914868789332728527017541632 * q^98 - 1902504845231598113297079280905132 * q^99

Decomposition of \(S_{34}^{\mathrm{new}}(\Gamma_0(2))\) 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
2.34.a.a	$2$	$34$	2.a	1.a	$1$	$1$	$1$	$13.797$	\(\Q\)	None	✓	✓	✓	✓	2.34.a.a	$1$	$1$	\(-65536\)	\(-133005564\)	\(538799132550\)	\(-33\!\cdots\!68\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{16}q^{2}-133005564q^{3}+2^{32}q^{4}+\cdots\)
2.34.a.b	$2$	$34$	2.a	1.a	$1$	$2$	$2$	$13.797$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	2.34.a.b	$1$	$0$	\(131072\)	\(8356488\)	\(-5332476660\)	\(13\!\cdots\!56\)		$-$	$-$	$2^{7}\cdot 3\cdot 5\cdot 11$	$\mathrm{SU}(2)$	\(q+2^{16}q^{2}+(4178244-\beta )q^{3}+2^{32}q^{4}+\cdots\)

Decomposition of \(S_{34}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

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