Atlas › Modular forms › Weight 2 › Level 240 › 240.2.a

Newspace 240.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(240))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
240.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - q^{5} - 4q^{7} + O(q^{8})$	+ +	240c1 rank 1
240.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - q^{5} + 4q^{7} + O(q^{8})$	+ +	240b1 rank 0
240.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + q^{5} + O(q^{8})$	+ −	240a1 rank 0
240.2.a.d	1	$\mathbb{Q}$	$q + q^{3} + q^{5} + O(q^{8})$	− −	240d1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31}$ 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 $T$
240.2.a.a	$x + 44$
240.2.a.b	$x + 52$
240.2.a.c	$x - 72$
240.2.a.d	$x + 24$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(240))$	4
Old forms	33
Cusp forms $S_{2}(\Gamma_0(240))$	37
Eisenstein series $E_{2}(\Gamma_0(240))$	23
Modular forms $M_{2}(\Gamma_0(240))$	60

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(240)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(40))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(48))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(80))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(120))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.341 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, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240]: [0, 1, 1, 2, 1, 3, 3, 3, 5, 4, 4, 6, 8, 10, 10, 12, 18, 16, 32, 56] (certified at ell = 2147483629)
T = [(7, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 2 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(240, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 240:2 1000

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