Atlas › Modular forms › Weight 2 › Level 609 › 609.2.a

Newspace 609.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(609))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
609.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} + q^{7} + O(q^{8})$	+ − −	$-1$	609b1 rank 1
609.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{6} + q^{7} + O(q^{8})$	+ − −	$-1$	609a1 rank 1
609.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 4q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$
609.2.a.d	3	degree 3	$3q - 3q^{2} + 3q^{3} + 5q^{4} - 6q^{5} - 3q^{6} + 3q^{7} + O(q^{8})$	− − +	$-1$
609.2.a.e	3	degree 3	$3q - q^{2} - 3q^{3} + q^{4} + 4q^{5} + q^{6} - 3q^{7} + O(q^{8})$	+ + +	$-1$
609.2.a.f	4	degree 4	$4q - 2q^{2} - 4q^{3} + 4q^{4} - 2q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	+ − +	$+1$
609.2.a.g	4	degree 4	$4q + 2q^{2} + 4q^{3} + 4q^{4} + 4q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	− − −	$+1$
609.2.a.h	4	degree 4	$4q + 4q^{2} + 4q^{3} + 8q^{4} + 4q^{5} + 4q^{6} - 4q^{7} + O(q^{8})$	− + +	$+1$
609.2.a.i	5	degree 5	$5q - q^{2} - 5q^{3} + 7q^{4} - 4q^{5} + q^{6} - 5q^{7} + O(q^{8})$	+ + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
609.2.a.a	$x + 39$
609.2.a.b	$x + 43$
609.2.a.c	$x^{2} + 66x + 289$
609.2.a.d	$x^{3} + 153x^{2} - 2325x - 589025$
609.2.a.e	$x^{3} + 61x^{2} + 167x - 24833$
609.2.a.f	$x^{4} - 184x^{3} + 9378x^{2} - 167256x + 730593$
609.2.a.g	$x^{4} - 38x^{3} - 3308x^{2} + 141478x - 1360577$
609.2.a.h	$x^{4} - 52x^{3} - 7206x^{2} + 319492x - 3174791$
609.2.a.i	$x^{5} + 69x^{4} - 2202x^{3} - 100690x^{2} + 1132661x + 10691393$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(609))$	27
Old forms	50
Cusp forms $S_{2}(\Gamma_0(609))$	77
Eisenstein series $E_{2}(\Gamma_0(609))$	7
Modular forms $M_{2}(\Gamma_0(609))$	84

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(609)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(87))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(203))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.421 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, 7, 21, 29, 87, 203, 609]: [0, 1, 1, 4, 3, 12, 22, 84] (certified at ell = 2147483353)
T = [(2, 1), (5, 3), (13, 5), (17, 7), (19, 9)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 7 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(609, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 609:2 1000

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