Atlas › Modular forms › Weight 2 › Level 739 › 739.2.a

Newspace 739.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(739))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
739.2.a.a	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{7} + O(q^{8})$	−	$+1$	739a1 rank 0
739.2.a.b	3	degree 3	$3q - q^{2} - q^{4} - 8q^{5} + 3q^{7} + O(q^{8})$	+	$-1$
739.2.a.c	23	degree 23	$23q - 6q^{2} - 6q^{3} + 20q^{4} - 24q^{5} - 9q^{6} - 7q^{7} + O(q^{8})$	+	$-1$
739.2.a.d	34	degree 34	$34q + 3q^{2} + 4q^{3} + 35q^{4} + 32q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	−	$+1$

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 $T$
739.2.a.a	$x - 2$
739.2.a.b	$x^{3} + x^{2} - 9x - 1$
739.2.a.c	$x^{23} + 12x^{22} + 11x^{21} - 364x^{20} - 1104x^{19} + 3806x^{18} + 16763x^{17} - 19576x^{16} - 119790x^{15} + 63599x^{14} + 485987x^{13} - 186959x^{12} - 1162965x^{11} + 516131x^{10} + 1548112x^{9} - 881034x^{8} - 977019x^{7} + 638945x^{6} + 285503x^{5} - 198185x^{4} - 37859x^{3} + 25031x^{2} + 2024x - 1003$
739.2.a.d	$x^{34} - 7x^{33} - 106x^{32} + 820x^{31} + 4612x^{30} - 41108x^{29} - 106483x^{28} + 1167935x^{27} + 1397104x^{26} - 21055091x^{25} - 9953917x^{24} + 255767100x^{23} + 24857991x^{22} - 2166101506x^{21} + 148387195x^{20} + 12999469457x^{19} - 1246179806x^{18} - 55299670634x^{17} + 1723497956x^{16} + 163927890558x^{15} + 14452993144x^{14} - 325938779986x^{13} - 74353567770x^{12} + 407658412705x^{11} + 146335227514x^{10} - 292283074525x^{9} - 135004074058x^{8} + 105940838841x^{7} + 55294410648x^{6} - 17989766611x^{5} - 9499265341x^{4} + 1428149124x^{3} + 566489124x^{2} - 51332032x + 1047632$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(739))$	61
Old forms	0
Cusp forms $S_{2}(\Gamma_0(739))$	61
Eisenstein series $E_{2}(\Gamma_0(739))$	1
Modular forms $M_{2}(\Gamma_0(739))$	62

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.242 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, 739]: [0, 62] (certified at ell = 2147483323)
T = [(2, 1), (3, 1)]; h of degree 61 = 1 x 61 (phi(ord eps) x dim S_k^new); 6 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(739, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 739:2 1000

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