Atlas › Modular forms › Weight 2 › Level 849 › 849.2.a

Newspace 849.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(849))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
849.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - 4q^{5} + q^{6} + q^{7} + O(q^{8})$	+ +	$-1$	849a1 rank 1
849.2.a.b	6	degree 6	$6q - 2q^{2} + 6q^{3} - 2q^{4} - 5q^{5} - 2q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
849.2.a.c	10	degree 10	$10q - q^{2} - 10q^{3} + 7q^{4} - q^{5} + q^{6} - 6q^{7} + O(q^{8})$	+ +	$-1$
849.2.a.d	12	degree 12	$12q + 2q^{2} - 12q^{3} + 12q^{4} + 7q^{5} - 2q^{6} + q^{7} + O(q^{8})$	+ −	$+1$
849.2.a.e	18	degree 18	$18q - q^{2} + 18q^{3} + 27q^{4} + q^{5} - q^{6} + 11q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
849.2.a.a	$x + 5$
849.2.a.b	$x^{6} + 7x^{5} + 13x^{4} - 16x^{2} - 7x + 1$
849.2.a.c	$x^{10} + 2x^{9} - 28x^{8} - 37x^{7} + 208x^{6} + 117x^{5} - 470x^{4} - 109x^{3} + 298x^{2} + 52x - 40$
849.2.a.d	$x^{12} - 9x^{11} - 11x^{10} + 240x^{9} - 67x^{8} - 2428x^{7} + 1183x^{6} + 11123x^{5} - 4007x^{4} - 20533x^{3} + 2741x^{2} + 6311x - 1673$
849.2.a.e	$x^{18} - 99x^{16} + 69x^{15} + 3831x^{14} - 5228x^{13} - 72326x^{12} + 147117x^{11} + 664353x^{10} - 1907242x^{9} - 2252578x^{8} + 11267716x^{7} - 3598036x^{6} - 24211609x^{5} + 28847420x^{4} - 418287x^{3} - 12547402x^{2} + 2322012x + 1774728$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(849))$	47
Old forms	46
Cusp forms $S_{2}(\Gamma_0(849))$	93
Eisenstein series $E_{2}(\Gamma_0(849))$	3
Modular forms $M_{2}(\Gamma_0(849))$	96

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(849)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(283))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.25 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, 3, 283, 849]: [0, 1, 24, 96] (certified at ell = 2147480197)
T = [(2, 1), (5, 1)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 5 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(849, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 849:2 1000

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