Atlas › Modular forms › Weight 2 › Level 859 › 859.2.a

Newspace 859.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(859))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
859.2.a.a	29	degree 29	$29q - 10q^{2} - 5q^{3} + 20q^{4} - 21q^{5} - 7q^{6} - 4q^{7} + O(q^{8})$	+	$-1$
859.2.a.b	42	degree 42	$42q + 10q^{2} + q^{3} + 48q^{4} + 21q^{5} + q^{6} + 2q^{7} + O(q^{8})$	−	$+1$

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 $T$
859.2.a.a	$x^{29} + 15x^{28} + 34x^{27} - 516x^{26} - 2664x^{25} + 5390x^{24} + 55739x^{23} + 8086x^{22} - 588644x^{21} - 625192x^{20} + 3655232x^{19} + 5851324x^{18} - 14271315x^{17} - 28253059x^{16} + 36351296x^{15} + 81947056x^{14} - 61888660x^{13} - 148509533x^{12} + 71245976x^{11} + 167920254x^{10} - 54220410x^{9} - 115411437x^{8} + 25348152x^{7} + 45833264x^{6} - 6495539x^{5} - 9550014x^{4} + 763307x^{3} + 871269x^{2} - 19903x - 21820$
859.2.a.b	$x^{42} - 11x^{41} - 101x^{40} + 1493x^{39} + 3175x^{38} - 87871x^{37} + 21114x^{36} + 2965913x^{35} - 4291763x^{34} - 64195522x^{33} + 141206361x^{32} + 947965270x^{31} - 2584851664x^{30} - 9973197039x^{29} + 30984291365x^{28} + 77534652253x^{27} - 259133038819x^{26} - 460480238670x^{25} + 1560466864259x^{24} + 2146823606298x^{23} - 6866489938230x^{22} - 7962840175250x^{21} + 22144517921528x^{20} + 23322353792882x^{19} - 52025456485951x^{18} - 52487265483775x^{17} + 87950061960604x^{16} + 87328619255832x^{15} - 105473690650035x^{14} - 103041899891215x^{13} + 88915492740075x^{12} + 82758948837835x^{11} - 52866960396020x^{10} - 43326353157739x^{9} + 22378858105551x^{8} + 13838458968800x^{7} - 6635770705258x^{6} - 2309093523052x^{5} + 1243251376360x^{4} + 95369447776x^{3} - 106598773344x^{2} + 15343523712x - 652551296$

Orbit

Characteristic polynomial of

T

859.2.a.a

x^{29} + 15x^{28} + 34x^{27} - 516x^{26} - 2664x^{25} + 5390x^{24} + 55739x^{23} + 8086x^{22} - 588644x^{21} - 625192x^{20} + 3655232x^{19} + 5851324x^{18} - 14271315x^{17} - 28253059x^{16} + 36351296x^{15} + 81947056x^{14} - 61888660x^{13} - 148509533x^{12} + 71245976x^{11} + 167920254x^{10} - 54220410x^{9} - 115411437x^{8} + 25348152x^{7} + 45833264x^{6} - 6495539x^{5} - 9550014x^{4} + 763307x^{3} + 871269x^{2} - 19903x - 21820

859.2.a.b

x^{42} - 11x^{41} - 101x^{40} + 1493x^{39} + 3175x^{38} - 87871x^{37} + 21114x^{36} + 2965913x^{35} - 4291763x^{34} - 64195522x^{33} + 141206361x^{32} + 947965270x^{31} - 2584851664x^{30} - 9973197039x^{29} + 30984291365x^{28} + 77534652253x^{27} - 259133038819x^{26} - 460480238670x^{25} + 1560466864259x^{24} + 2146823606298x^{23} - 6866489938230x^{22} - 7962840175250x^{21} + 22144517921528x^{20} + 23322353792882x^{19} - 52025456485951x^{18} - 52487265483775x^{17} + 87950061960604x^{16} + 87328619255832x^{15} - 105473690650035x^{14} - 103041899891215x^{13} + 88915492740075x^{12} + 82758948837835x^{11} - 52866960396020x^{10} - 43326353157739x^{9} + 22378858105551x^{8} + 13838458968800x^{7} - 6635770705258x^{6} - 2309093523052x^{5} + 1243251376360x^{4} + 95369447776x^{3} - 106598773344x^{2} + 15343523712x - 652551296

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(859))$	71
Old forms	0
Cusp forms $S_{2}(\Gamma_0(859))$	71
Eisenstein series $E_{2}(\Gamma_0(859))$	1
Modular forms $M_{2}(\Gamma_0(859))$	72

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.272 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, 859]: [0, 72] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 71 = 1 x 71 (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(859, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 859:2 1000

← 858.2.a · 860.2.a →