Atlas › Modular forms › Weight 2 › Level 797 › 797.2.a

Newspace 797.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(797))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
797.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{6} + O(q^{8})$	+	$-1$	797a1 rank 1
797.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{3} - 2q^{4} + 4q^{5} - 2q^{6} + 8q^{7} + O(q^{8})$	−	$+1$
797.2.a.c	25	degree 25	$25q - 6q^{2} - 16q^{3} + 14q^{4} - 4q^{5} - 25q^{7} + O(q^{8})$	+	$-1$
797.2.a.d	38	degree 38	$38q + 9q^{2} + 13q^{3} + 53q^{4} - 2q^{5} + 3q^{6} + 19q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
797.2.a.a	$x$
797.2.a.b	$x^{2} - 2$
797.2.a.c	$x^{25} + 22x^{24} + 163x^{23} + 189x^{22} - 3510x^{21} - 15407x^{20} + 11607x^{19} + 195057x^{18} + 200865x^{17} - 1107633x^{16} - 2165101x^{15} + 3302944x^{14} + 9490776x^{13} - 5453893x^{12} - 22722862x^{11} + 5674827x^{10} + 31330491x^{9} - 5870520x^{8} - 23625899x^{7} + 6239971x^{6} + 7706154x^{5} - 3190757x^{4} - 127689x^{3} + 142418x^{2} - 3345x - 840$
797.2.a.d	$x^{38} - 22x^{37} + 93x^{36} + 1345x^{35} - 12975x^{34} - 11965x^{33} + 526883x^{32} - 1088106x^{31} - 10383263x^{30} + 42631083x^{29} + 102744619x^{28} - 757826731x^{27} - 270869333x^{26} + 8067832110x^{25} - 5337021730x^{24} - 56130872192x^{23} + 74267878253x^{22} + 265421818683x^{21} - 491647251729x^{20} - 868104027906x^{19} + 2049223274196x^{18} + 1972575595598x^{17} - 5770439440696x^{16} - 3103745021446x^{15} + 11198670300861x^{14} + 3375232255865x^{13} - 14894442034658x^{12} - 2591363981983x^{11} + 13199857347391x^{10} + 1505528107177x^{9} - 7357131544947x^{8} - 689967135631x^{7} + 2316473842503x^{6} + 205023072547x^{5} - 338868266135x^{4} - 23824511954x^{3} + 18590952329x^{2} + 900207304x - 137292782$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(797))$	66
Old forms	0
Cusp forms $S_{2}(\Gamma_0(797))$	66
Eisenstein series $E_{2}(\Gamma_0(797))$	1
Modular forms $M_{2}(\Gamma_0(797))$	67

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.369 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, 797]: [0, 67] (certified at ell = 2147483029)
T = [(2, 1), (3, 1)]; h of degree 66 = 1 x 66 (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(797, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 797:2 1000

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