Atlas › Modular forms › Weight 2 › Level 851 › 851.2.a

Newspace 851.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(851))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
851.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} - 2q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$	851a1 rank 1
851.2.a.b	2	$\mathbb{Q}(\sqrt{41})$	$2q + 2q^{3} - 4q^{4} + q^{5} - q^{7} + O(q^{8})$	+ −	$+1$
851.2.a.c	3	degree 3	$3q - q^{2} - q^{4} - 4q^{5} + q^{7} + O(q^{8})$	− −	$-1$
851.2.a.d	3	degree 3	$3q + 3q^{2} + 3q^{4} - 6q^{6} - 9q^{7} + O(q^{8})$	− −	$-1$
851.2.a.e	5	degree 5	$5q - 2q^{2} - 2q^{3} + 6q^{4} - 3q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$
851.2.a.f	12	degree 12	$12q - q^{2} - 3q^{3} + 5q^{4} - 5q^{5} - 10q^{6} + q^{7} + O(q^{8})$	+ +	$-1$
851.2.a.g	20	degree 20	$20q + 2q^{2} + 26q^{4} + 13q^{5} + 5q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$
851.2.a.h	21	degree 21	$21q + 2q^{2} + 2q^{3} + 36q^{4} + 4q^{5} + 8q^{6} + 7q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ 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 $T$
851.2.a.a	$x + 11$
851.2.a.b	$x^{2} - 49x + 508$
851.2.a.c	$x^{3} + 77x^{2} + 1134x - 203$
851.2.a.d	$x^{3} + 87x^{2} + 2070x + 7543$
851.2.a.e	$x^{5} + 46x^{4} - 972x^{3} - 29516x^{2} + 226584x + 3449701$
851.2.a.f	$x^{12} + 10x^{11} - 4362x^{10} - 55243x^{9} + 6388324x^{8} + 91543091x^{7} - 3629687015x^{6} - 52128300110x^{5} + 839282071128x^{4} + 10825494449291x^{3} - 75009481743127x^{2} - 657659522650410x + 2642808494879047$
851.2.a.g	$x^{20} - 208x^{19} + 3388x^{18} + 2132340x^{17} - 151223713x^{16} - 4307020487x^{15} + 781449666982x^{14} - 14616638876400x^{13} - 1370079060656415x^{12} + 65923972828060245x^{11} + 208551791575238647x^{10} - 78038478148583406276x^{9} + 1628425631604100366267x^{8} + 14811156290629794589215x^{7} - 1094965198852843770315877x^{6} + 17104554305115730920860275x^{5} - 87872852528449154733026908x^{4} - 386970594944749584080327831x^{3} + 5205632545788902883224700174x^{2} - 6295679483102233910618200277x - 47130982232540886955321971478$
851.2.a.h	$x^{21} - 5x^{20} - 20115x^{19} + 142492x^{18} + 147075845x^{17} - 1766753130x^{16} - 517190339564x^{15} + 8970056264495x^{14} + 939057029273209x^{13} - 20832961327570560x^{12} - 871835533180274433x^{11} + 23183463352905247982x^{10} + 403284968956375852896x^{9} - 13068303960213411632928x^{8} - 79669721863614419235717x^{7} + 3715153658917693003220147x^{6} + 1249228437129639146488684x^{5} - 490393151169770673315813352x^{4} + 1333751053220262583455651451x^{3} + 22610787600350211405475807086x^{2} - 90416196041291670360131952525x - 98003514715360978794045061239$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(851))$	67
Old forms	8
Cusp forms $S_{2}(\Gamma_0(851))$	75
Eisenstein series $E_{2}(\Gamma_0(851))$	3
Modular forms $M_{2}(\Gamma_0(851))$	78

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(851)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.332 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, 23, 37, 851]: [0, 3, 3, 78] (certified at ell = 2147483053)
T = [(2, 1), (3, 3), (5, 5), (7, 7), (11, 9)]; h of degree 67 = 1 x 67 (phi(ord eps) x dim S_k^new); 16 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(851, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 851:2 1000

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