Atlas › Modular forms › Weight 2 › Level 84 › 84.2.a

Newspace 84.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(84))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
84.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + 4q^{5} - q^{7} + O(q^{8})$	+ +			84b1 rank 0
84.2.a.b	1	$\mathbb{Q}$	$q + q^{3} + q^{7} + O(q^{8})$	− −			84a1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{11} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
84.2.a.a	$x + 84$
84.2.a.b	$x + 44$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(84))$	2
Old forms	9
Cusp forms $S_{2}(\Gamma_0(84))$	11
Eisenstein series $E_{2}(\Gamma_0(84))$	11
Modular forms $M_{2}(\Gamma_0(84))$	22

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(84)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(42))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.104 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, 6, 7, 12, 14, 21, 28, 42, 84]: [0, 1, 1, 2, 3, 1, 5, 4, 4, 7, 12, 22] (certified at ell = 2147483647)
T = [(5, 1), (11, 3), (13, 5), (17, 7), (19, 9)]; h of degree 2 = 1 x 2 (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(84, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 84:2 1000

← 83.2.a · 85.2.a →