Atlas › Modular forms › Weight 6 › Level 106 › 106.6.a

Newspace 106.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(106))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
106.6.a.a	4	degree 4	$4q + 16q^{2} - 8q^{3} + 64q^{4} - 136q^{5} - 32q^{6} - 210q^{7} + O(q^{8})$	− −	$-1$
106.6.a.b	6	degree 6	$6q - 24q^{2} - 17q^{3} + 96q^{4} - 86q^{5} + 68q^{6} + 74q^{7} + O(q^{8})$	+ +	$-1$
106.6.a.c	6	degree 6	$6q - 24q^{2} + 10q^{3} + 96q^{4} + 39q^{5} - 40q^{6} - 24q^{7} + O(q^{8})$	+ −	$+1$
106.6.a.d	7	degree 7	$7q + 28q^{2} + 19q^{3} + 112q^{4} + 89q^{5} + 76q^{6} + 84q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
106.6.a.a	$x^{4} + 144x^{3} + 4062x^{2} - 81090x + 280835$
106.6.a.b	$x^{6} + 103x^{5} - 8240x^{4} - 588246x^{3} + 2588603x^{2} + 336521963x + 2872066508$
106.6.a.c	$x^{6} - 49x^{5} - 10932x^{4} + 527832x^{3} + 18282693x^{2} - 960677775x + 9530914230$
106.6.a.d	$x^{7} - 108x^{6} - 15833x^{5} + 1674030x^{4} + 57050161x^{3} - 6782736930x^{2} + 34476336939x + 2430502212540$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(106))$	23
Old forms	42
Cusp forms $S_{6}(\Gamma_0(106))$	65
Eisenstein series $E_{6}(\Gamma_0(106))$	4
Modular forms $M_{6}(\Gamma_0(106))$	69

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(106)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(53))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.281 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, 53, 106]: [1, 2, 23, 69] (certified at ell = 2147483053)
T = [(3, 1), (5, 1)]; h of degree 23 = 1 x 23 (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(106, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 106:6 1000

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