Atlas › Modular forms › Weight 2 › Level 603 › 603.2.a

Newspace 603.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(603))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
603.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	−	603f1 rank 1
603.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	−	603b1 rank 0
603.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + 3q^{5} - 3q^{7} + O(q^{8})$	+	603c1 rank 0
603.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	−	603a1 rank 0
603.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + q^{5} - 5q^{7} + O(q^{8})$	−	603e1 rank 1
603.2.a.f	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + O(q^{8})$	+	603d1 rank 0
603.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{4} - 4q^{5} + q^{7} + O(q^{8})$	−
603.2.a.h	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} + 3q^{4} + 6q^{5} - q^{7} + O(q^{8})$	+
603.2.a.i	3	degree 3	$3q - 3q^{2} + 5q^{4} - q^{5} + q^{7} + O(q^{8})$	−
603.2.a.j	4	degree 4	$4q - 8q^{7} + O(q^{8})$	+
603.2.a.k	5	degree 5	$5q + 6q^{4} + 3q^{5} + 7q^{7} + O(q^{8})$	+
603.2.a.l	6	degree 6	$6q + 12q^{4} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23}$ 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 $T$
603.2.a.a	$x + 27$
603.2.a.b	$x + 11$
603.2.a.c	$x - 46$
603.2.a.d	$x - 111$
603.2.a.e	$x + 42$
603.2.a.f	$x - 61$
603.2.a.g	$x^{2} + 114x + 3169$
603.2.a.h	$x^{2} + 28x - 5249$
603.2.a.i	$x^{3} + 120x^{2} - 3682x - 322286$
603.2.a.j	$x^{4} + 152x^{3} + 5022x^{2} + 31148x - 125954$
603.2.a.k	$x^{5} - 158x^{4} + 8135x^{3} - 115530x^{2} - 2180622x + 53652958$
603.2.a.l	$x^{6} - 142x^{5} + 4443x^{4} + 91544x^{3} - 2970136x^{2} - 56057208x - 210647058$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(603))$	28
Old forms	37
Cusp forms $S_{2}(\Gamma_0(603))$	65
Eisenstein series $E_{2}(\Gamma_0(603))$	7
Modular forms $M_{2}(\Gamma_0(603))$	72

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(603)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(67))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(201))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.352 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, 67, 201, 603]: [0, 1, 2, 6, 24, 70] (certified at ell = 2147483647)
T = [(2, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 8 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(603, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 603:2 1000

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