Atlas › Modular forms › Weight 2 › Level 733 › 733.2.a

Newspace 733.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(733))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
733.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{6} + 2q^{7} + O(q^{8})$	−	$+1$	733a1 rank 0
733.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{3} - 2q^{4} - 4q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	+	$-1$
733.2.a.c	25	degree 25	$25q - 11q^{2} - 18q^{3} + 25q^{4} - 6q^{5} + 4q^{6} - 11q^{7} + O(q^{8})$	+	$-1$
733.2.a.d	32	degree 32	$32q + 12q^{2} + 17q^{3} + 38q^{4} + 8q^{5} + 5q^{6} + 3q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
733.2.a.a	$x$
733.2.a.b	$x^{2} - 2$
733.2.a.c	$x^{25} + 29x^{24} + 332x^{23} + 1696x^{22} + 1121x^{21} - 29473x^{20} - 124924x^{19} - 18993x^{18} + 1088921x^{17} + 2189629x^{16} - 2830171x^{15} - 13391105x^{14} - 2899213x^{13} + 36943251x^{12} + 29754047x^{11} - 55263229x^{10} - 64865183x^{9} + 47808627x^{8} + 68789791x^{7} - 25525722x^{6} - 38499528x^{5} + 9511586x^{4} + 10874628x^{3} - 2529219x^{2} - 1223880x + 330790$
733.2.a.d	$x^{32} - 29x^{31} + 295x^{30} - 619x^{29} - 10852x^{28} + 81274x^{27} - 34934x^{26} - 1739281x^{25} + 5897992x^{24} + 10720447x^{23} - 95387053x^{22} + 73143224x^{21} + 669346645x^{20} - 1428198234x^{19} - 2129700606x^{18} + 8932410838x^{17} + 793675370x^{16} - 29883366484x^{15} + 15831434640x^{14} + 58476042103x^{13} - 52219493333x^{12} - 67406584268x^{11} + 76243003734x^{10} + 44253601406x^{9} - 55668066925x^{8} - 15317471191x^{7} + 18447187180x^{6} + 2317998937x^{5} - 1913456802x^{4} - 72678275x^{3} + 39614568x^{2} + 2404080x + 21600$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(733))$	60
Old forms	0
Cusp forms $S_{2}(\Gamma_0(733))$	60
Eisenstein series $E_{2}(\Gamma_0(733))$	1
Modular forms $M_{2}(\Gamma_0(733))$	61

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.247 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 733]: [0, 61] (certified at ell = 2147479381)
T = [(2, 1), (3, 1)]; h of degree 60 = 1 x 60 (phi(ord eps) x dim S_k^new); 6 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(733, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 733:2 1000

← 732.2.a · 734.2.a →