Atlas › Modular forms › Weight 2 › Level 300 › 300.2.a

Newspace 300.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(300))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
300.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 4q^{7} + O(q^{8})$	+	300d1 rank 1
300.2.a.b	1	$\mathbb{Q}$	$q - q^{3} + q^{7} + O(q^{8})$	+	300a1 rank 0
300.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - q^{7} + O(q^{8})$	−	300b1 rank 0
300.2.a.d	1	$\mathbb{Q}$	$q + q^{3} + 4q^{7} + O(q^{8})$	−	300c1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 3T_{11}$ 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 $T$
300.2.a.a	$x + 16$
300.2.a.b	$x - 19$
300.2.a.c	$x - 17$
300.2.a.d	$x + 8$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(300))$	4
Old forms	39
Cusp forms $S_{2}(\Gamma_0(300))$	43
Eisenstein series $E_{2}(\Gamma_0(300))$	35
Modular forms $M_{2}(\Gamma_0(300))$	78

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(300)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(50))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(75))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(100))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(150))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.246 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, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300]: [0, 1, 1, 2, 1, 3, 3, 5, 4, 6, 3, 10, 9, 18, 12, 18, 34, 66] (certified at ell = 2147482921)
T = [(7, 1), (11, 3)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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(300, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 300:2 1000

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