Atlas › Modular forms › Weight 2 › Level 446 › 446.2.a

Newspace 446.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(446))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
446.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} + q^{4} - 4q^{5} + 3q^{6} - 4q^{7} + O(q^{8})$	+ −	$+1$	446d1 rank 2
446.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{6} + O(q^{8})$	+ +	$-1$	446a1 rank 1
446.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$	446b1 rank 1
446.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 2q^{6} + O(q^{8})$	− +	$+1$	446c1 rank 0
446.2.a.e	7	degree 7	$7q + 7q^{2} + q^{3} + 7q^{4} + 2q^{5} + q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$
446.2.a.f	8	degree 8	$8q - 8q^{2} + 4q^{3} + 8q^{4} + 4q^{5} - 4q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
446.2.a.a	$x + 15$
446.2.a.b	$x + 1$
446.2.a.c	$x + 7$
446.2.a.d	$x - 2$
446.2.a.e	$x^{7} - 7x^{6} - 224x^{5} + 1692x^{4} + 6968x^{3} - 74928x^{2} + 145792x - 45792$
446.2.a.f	$x^{8} - 16x^{7} - 168x^{6} + 3360x^{5} - 896x^{4} - 153600x^{3} + 365424x^{2} + 1997152x - 6368864$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(446))$	19
Old forms	36
Cusp forms $S_{2}(\Gamma_0(446))$	55
Eisenstein series $E_{2}(\Gamma_0(446))$	3
Modular forms $M_{2}(\Gamma_0(446))$	58

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(446)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(223))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.4 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, 2, 223, 446]: [0, 1, 19, 58] (certified at ell = 2147482591)
T = [(3, 1), (5, 3)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 3 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(446, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 446:2 1000

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