Atlas › Modular forms › Weight 2 › Level 600 › 600.2.a

Newspace 600.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(600))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
600.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 4q^{7} + O(q^{8})$	+	600f1 rank 0
600.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 3q^{7} + O(q^{8})$	+	600b1 rank 1
600.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + O(q^{8})$	+	600a1 rank 1
600.2.a.d	1	$\mathbb{Q}$	$q - q^{3} + 2q^{7} + O(q^{8})$	+	600c1 rank 0
600.2.a.e	1	$\mathbb{Q}$	$q - q^{3} + 5q^{7} + O(q^{8})$	+	600g1 rank 0
600.2.a.f	1	$\mathbb{Q}$	$q + q^{3} - 5q^{7} + O(q^{8})$	−	600e1 rank 1
600.2.a.g	1	$\mathbb{Q}$	$q + q^{3} - 2q^{7} + O(q^{8})$	−	600h1 rank 0
600.2.a.h	1	$\mathbb{Q}$	$q + q^{3} + O(q^{8})$	−	600d1 rank 0
600.2.a.i	1	$\mathbb{Q}$	$q + q^{3} + 3q^{7} + O(q^{8})$	−	600i1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47} + 13T_{53} + 14T_{59} + 15T_{61} + 16T_{67}$ 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 $T$
600.2.a.a	$x - 174$
600.2.a.b	$x + 413$
600.2.a.c	$x + 134$
600.2.a.d	$x - 282$
600.2.a.e	$x - 267$
600.2.a.f	$x + 147$
600.2.a.g	$x + 150$
600.2.a.h	$x - 98$
600.2.a.i	$x + 75$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(600))$	9
Old forms	88
Cusp forms $S_{2}(\Gamma_0(600))$	97
Eisenstein series $E_{2}(\Gamma_0(600))$	47
Modular forms $M_{2}(\Gamma_0(600))$	144

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.027 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, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600]: [0, 1, 1, 2, 1, 3, 3, 3, 5, 4, 6, 8, 3, 10, 10, 9, 18, 12, 18, 32, 34, 34, 66, 128] (certified at ell = 2147482921)
T = [(7, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12), (53, 13), (59, 14), (61, 15), (67, 16)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 4 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(600, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 600:2 1000

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