Atlas › Modular forms › Weight 2 › Level 629 › 629.2.a

Newspace 629.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(629))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
629.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + 3q^{5} - q^{7} + O(q^{8})$	− −	$-1$	629d1 rank 1
629.2.a.b	1	$\mathbb{Q}$	$q - 3q^{3} - 2q^{4} + 3q^{7} + O(q^{8})$	− −	$-1$	629c1 rank 1
629.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + q^{5} - q^{7} + O(q^{8})$	+ +	$-1$	629a1 rank 1
629.2.a.d	1	$\mathbb{Q}$	$q + 2q^{2} + 3q^{3} + 2q^{4} - 2q^{5} + 6q^{6} + q^{7} + O(q^{8})$	− +	$+1$	629b1 rank 0
629.2.a.e	5	degree 5	$5q + 4q^{4} - 3q^{5} - 5q^{6} - 12q^{7} + O(q^{8})$	− −	$-1$
629.2.a.f	8	degree 8	$8q - 3q^{3} + 4q^{4} - 5q^{5} - 7q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
629.2.a.g	15	degree 15	$15q + 2q^{2} + 22q^{4} + 4q^{5} - 11q^{6} + q^{7} + O(q^{8})$	− +	$+1$
629.2.a.h	17	degree 17	$17q - q^{2} + 7q^{3} + 27q^{4} + 4q^{5} + 5q^{6} + 14q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ 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 $T$
629.2.a.a	$x - 7$
629.2.a.b	$x - 12$
629.2.a.c	$x + 1$
629.2.a.d	$x - 8$
629.2.a.e	$x^{5} + 99x^{4} + 3700x^{3} + 65007x^{2} + 535099x + 1641587$
629.2.a.f	$x^{8} + 69x^{7} + 941x^{6} - 16724x^{5} - 368949x^{4} - 190732x^{3} + 19193125x^{2} + 50421239x - 33653744$
629.2.a.g	$x^{15} - 29x^{14} - 5432x^{13} + 144713x^{12} + 10695479x^{11} - 256830583x^{10} - 9086454853x^{9} + 192625463189x^{8} + 3056682305279x^{7} - 56784524553228x^{6} - 323466639654004x^{5} + 5587350876683173x^{4} + 11546961717786113x^{3} - 171502435196921574x^{2} - 101608000681937099x + 245449439180449631$
629.2.a.h	$x^{17} - 138x^{16} + 4387x^{15} + 245421x^{14} - 20403912x^{13} + 358822008x^{12} + 9656994280x^{11} - 450185268148x^{10} + 3095115953386x^{9} + 122645099039559x^{8} - 2387004693201682x^{7} - 35831885346591x^{6} + 350738356315365656x^{5} - 2442535238507453881x^{4} - 11816998364687857989x^{3} + 170501139890762548558x^{2} - 260912435279216555791x - 1372742734537900969372$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(629))$	49
Old forms	6
Cusp forms $S_{2}(\Gamma_0(629))$	55
Eisenstein series $E_{2}(\Gamma_0(629))$	3
Modular forms $M_{2}(\Gamma_0(629))$	58

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(629)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.245 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, 17, 37, 629]: [0, 2, 3, 58] (certified at ell = 2147482801)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 49 = 1 x 49 (phi(ord eps) x dim S_k^new); 11 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(629, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 629:2 1000

← 628.2.a · 630.2.a →