Atlas › Modular forms › Weight 2 › Level 897 › 897.2.a

Newspace 897.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(897))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
897.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} + O(q^{8})$	+ − −	$-1$	897c1 rank 1
897.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + 2q^{5} + q^{6} + 4q^{7} + O(q^{8})$	+ − +	$+1$	897b1 rank 0
897.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - 4q^{5} - q^{6} + 2q^{7} + O(q^{8})$	− − +	$-1$	897f1 rank 1
897.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} + 2q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− − +	$-1$	897e1 rank 1
897.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - 4q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$	897a1 rank 0
897.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} + q^{6} - 2q^{7} + O(q^{8})$	− − +	$-1$	897d1 rank 1
897.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} - 2q^{4} + 2q^{6} + 4q^{7} + O(q^{8})$	+ + −	$+1$
897.2.a.h	3	degree 3	$3q - 3q^{2} + 3q^{3} + 11q^{4} - 8q^{5} - 3q^{6} - 5q^{7} + O(q^{8})$	− − +	$-1$
897.2.a.i	3	degree 3	$3q - q^{2} - 3q^{3} + 3q^{4} + 4q^{5} + q^{6} - 7q^{7} + O(q^{8})$	+ − −	$-1$
897.2.a.j	3	degree 3	$3q - q^{2} + 3q^{3} - q^{4} - 4q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− + −	$-1$
897.2.a.k	4	degree 4	$4q - 4q^{3} + 10q^{4} - 2q^{5} + 3q^{7} + O(q^{8})$	+ − +	$+1$
897.2.a.l	5	degree 5	$5q - q^{2} - 5q^{3} + q^{4} + q^{6} - 5q^{7} + O(q^{8})$	+ + +	$-1$
897.2.a.m	5	degree 5	$5q + q^{2} + 5q^{3} + 5q^{4} + 6q^{5} + q^{6} + 5q^{7} + O(q^{8})$	− + +	$+1$
897.2.a.n	5	degree 5	$5q + 3q^{2} - 5q^{3} + 9q^{4} - 2q^{5} - 3q^{6} + 3q^{7} + O(q^{8})$	+ + −	$+1$
897.2.a.o	7	degree 7	$7q + 3q^{2} + 7q^{3} + 7q^{4} + 6q^{5} + 3q^{6} + 3q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{17}$ 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 $T$
897.2.a.a	$x - 47$
897.2.a.b	$x - 43$
897.2.a.c	$x + 43$
897.2.a.d	$x - 11$
897.2.a.e	$x - 3$
897.2.a.f	$x + 105$
897.2.a.g	$x^{2} + 54x - 2799$
897.2.a.h	$x^{3} + 51x^{2} + 314x - 6847$
897.2.a.i	$x^{3} + 95x^{2} - 686x - 57183$
897.2.a.j	$x^{3} + 147x^{2} + 6776x + 95459$
897.2.a.k	$x^{4} - 16x^{3} - 6563x^{2} + 219795x - 1279367$
897.2.a.l	$x^{5} + 73x^{4} - 607x^{3} - 75552x^{2} + 160430x + 16554197$
897.2.a.m	$x^{5} - 115x^{4} + 3619x^{3} - 39910x^{2} + 113090x + 13967$
897.2.a.n	$x^{5} - 7x^{4} - 9979x^{3} + 16068x^{2} + 16619882x - 269834219$
897.2.a.o	$x^{7} - 131x^{6} + 3144x^{5} + 83907x^{4} - 1526669x^{3} + 594161x^{2} + 45418020x - 88235289$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(897))$	43
Old forms	66
Cusp forms $S_{2}(\Gamma_0(897))$	109
Eisenstein series $E_{2}(\Gamma_0(897))$	7
Modular forms $M_{2}(\Gamma_0(897))$	116

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(897)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(69))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(299))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.559 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, 13, 23, 39, 69, 299, 897]: [0, 1, 1, 3, 6, 10, 30, 116] (certified at ell = 2147483053)
T = [(2, 1), (5, 3), (7, 5), (11, 7), (17, 9)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 11 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(897, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 897:2 1000

← 896.2.a · 898.2.a →