Atlas › Modular forms › Weight 4 › Level 169 › 169.4.a

Newspace 169.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(169))$

Newforms

Label	Dim	Coefficient field	Trace form
169.4.a.a	1	$\mathbb{Q}$	$q - 4q^{2} + 2q^{3} + 8q^{4} - 17q^{5} - 8q^{6} - 20q^{7} + O(q^{8})$
169.4.a.b	1	$\mathbb{Q}$	$q - 3q^{2} - q^{3} + q^{4} + 9q^{5} + 3q^{6} - 15q^{7} + O(q^{8})$
169.4.a.c	1	$\mathbb{Q}$	$q + 3q^{2} - q^{3} + q^{4} - 9q^{5} - 3q^{6} + 15q^{7} + O(q^{8})$
169.4.a.d	1	$\mathbb{Q}$	$q + 4q^{2} + 2q^{3} + 8q^{4} + 17q^{5} + 8q^{6} + 20q^{7} + O(q^{8})$
169.4.a.e	1	$\mathbb{Q}$	$q + 5q^{2} - 7q^{3} + 17q^{4} + 7q^{5} - 35q^{6} + 13q^{7} + O(q^{8})$
169.4.a.f	2	$\mathbb{Q}(\sqrt{17})$	$2q - 5q^{2} + 5q^{3} + 5q^{4} - 15q^{5} - 38q^{6} + 15q^{7} + O(q^{8})$
169.4.a.g	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} + 5q^{3} - 7q^{4} + 3q^{5} + 23q^{6} + 9q^{7} + O(q^{8})$
169.4.a.h	2	$\mathbb{Q}(\sqrt{3})$	$2q - 14q^{3} + 8q^{4} + O(q^{8})$
169.4.a.i	2	$\mathbb{Q}(\sqrt{3})$	$2q + 4q^{3} - 10q^{4} + O(q^{8})$
169.4.a.j	2	$\mathbb{Q}(\sqrt{17})$	$2q + 5q^{2} + 5q^{3} + 5q^{4} + 15q^{5} + 38q^{6} - 15q^{7} + O(q^{8})$
169.4.a.k	9	degree 9	$9q - 5q^{2} + q^{3} + 37q^{4} - 30q^{5} - 48q^{6} - 38q^{7} + O(q^{8})$
169.4.a.l	9	degree 9	$9q + 5q^{2} + q^{3} + 37q^{4} + 30q^{5} + 48q^{6} + 38q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
169.4.a.a	$x - 2$
169.4.a.b	$x + 6$
169.4.a.c	$x$
169.4.a.d	$x - 10$
169.4.a.e	$x + 16$
169.4.a.f	$x^{2} - 10x - 247$
169.4.a.g	$x^{2} - 14x - 376$
169.4.a.h	$x^{2} + 42x + 429$
169.4.a.i	$x^{2} - 12x + 33$
169.4.a.j	$x^{2} - 20x - 325$
169.4.a.k	$x^{9} + 2x^{8} - 1299x^{7} + 4757x^{6} + 471879x^{5} - 3349008x^{4} - 44738526x^{3} + 460072563x^{2} - 792625119x - 397062289$
169.4.a.l	$x^{9} - 8x^{8} - 1557x^{7} + 16039x^{6} + 786555x^{5} - 9938046x^{4} - 127939212x^{3} + 2030821923x^{2} - 277590105x - 52422860867$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(169))$	33
Old forms	6
Cusp forms $S_{4}(\Gamma_0(169))$	39
Eisenstein series $E_{4}(\Gamma_0(169))$	14
Modular forms $M_{4}(\Gamma_0(169))$	53

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(169)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.125 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, 13, 169]: [1, 5, 47] (certified at ell = 2147483053)
T = [(2, 1), (3, 3)]; h of degree 33 = 1 x 33 (phi(ord eps) x dim S_k^new); 6 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(169, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 169:4 1000

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