Atlas › Modular forms › Weight 2 › Level 297 › 297.2.a

Newspace 297.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(297))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
297.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} - 2q^{5} + q^{7} + O(q^{8})$	−	297a1 rank 1
297.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + 2q^{5} - 5q^{7} + O(q^{8})$	+	297b1 rank 1
297.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - 2q^{5} - 5q^{7} + O(q^{8})$	−	297c1 rank 1
297.2.a.d	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + 2q^{5} + q^{7} + O(q^{8})$	+	297d1 rank 0
297.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} + 4q^{4} - 2q^{5} - 4q^{7} + O(q^{8})$	+
297.2.a.f	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{2} + 4q^{4} + 2q^{5} - 4q^{7} + O(q^{8})$	−
297.2.a.g	3	degree 3	$3q - q^{2} + 5q^{4} + 2q^{5} + 7q^{7} + O(q^{8})$	−
297.2.a.h	3	degree 3	$3q + q^{2} + 5q^{4} - 2q^{5} + 7q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{5} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ 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 $T$
297.2.a.a	$x + 91$
297.2.a.b	$x + 72$
297.2.a.c	$x - 26$
297.2.a.d	$x - 93$
297.2.a.e	$x^{2} + 108x + 2673$
297.2.a.f	$x^{2} - 52x - 1199$
297.2.a.g	$x^{3} + 12x^{2} - 4764x - 112082$
297.2.a.h	$x^{3} - 110x^{2} - 2304x + 91206$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(297))$	14
Old forms	17
Cusp forms $S_{2}(\Gamma_0(297))$	31
Eisenstein series $E_{2}(\Gamma_0(297))$	11
Modular forms $M_{2}(\Gamma_0(297))$	42

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(297)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(99))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.24 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, 3, 9, 11, 27, 33, 99, 297]: [0, 1, 2, 2, 4, 6, 14, 38] (certified at ell = 2147482801)
T = [(2, 1), (5, 2), (7, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 14 = 1 x 14 (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(297, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 297:2 1000

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