# Properties

 Label 1400.2.a.s Level $1400$ Weight $2$ Character orbit 1400.a Self dual yes Analytic conductor $11.179$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.1790562830$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10})$$ q - b1 * q^3 + q^7 + (b2 + 2*b1 + 1) * q^9 $$q - \beta_1 q^{3} + q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9} + ( - \beta_{2} + 2) q^{11} + (2 \beta_{2} + \beta_1 + 2) q^{13} + ( - \beta_{2} - 2 \beta_1) q^{17} + (\beta_{2} - \beta_1 + 2) q^{19} - \beta_1 q^{21} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{23} + ( - \beta_{2} - 2 \beta_1 - 6) q^{27} - 3 \beta_{2} q^{29} + ( - 2 \beta_1 + 4) q^{31} + ( - \beta_{2} - 2 \beta_1 - 2) q^{33} + 6 q^{37} + (\beta_{2} - 4 \beta_1) q^{39} + (2 \beta_{2} + 2 \beta_1 + 6) q^{41} + (2 \beta_{2} + 2 \beta_1) q^{43} + (3 \beta_{2} - 2) q^{47} + q^{49} + (\beta_{2} + 4 \beta_1 + 6) q^{51} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{53} + (2 \beta_{2} + 6) q^{57} + ( - 3 \beta_{2} - 3 \beta_1 + 2) q^{59} + ( - \beta_{2} - \beta_1 + 8) q^{61} + (\beta_{2} + 2 \beta_1 + 1) q^{63} + (2 \beta_{2} + 4 \beta_1 - 4) q^{67} + (6 \beta_1 + 4) q^{69} + ( - 2 \beta_1 + 2) q^{71} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{73} + ( - \beta_{2} + 2) q^{77} + ( - \beta_{2} + 4 \beta_1 + 4) q^{79} + ( - 2 \beta_{2} + 4 \beta_1 + 3) q^{81} + ( - \beta_{2} + 5 \beta_1 - 6) q^{83} + ( - 3 \beta_{2} - 6) q^{87} + (2 \beta_{2} - 4 \beta_1 + 2) q^{89} + (2 \beta_{2} + \beta_1 + 2) q^{91} + (2 \beta_{2} + 8) q^{93} + ( - \beta_{2} + 2 \beta_1 - 4) q^{97} + (4 \beta_{2} + 6 \beta_1) q^{99}+O(q^{100})$$ q - b1 * q^3 + q^7 + (b2 + 2*b1 + 1) * q^9 + (-b2 + 2) * q^11 + (2*b2 + b1 + 2) * q^13 + (-b2 - 2*b1) * q^17 + (b2 - b1 + 2) * q^19 - b1 * q^21 + (-2*b2 - 2*b1 - 2) * q^23 + (-b2 - 2*b1 - 6) * q^27 - 3*b2 * q^29 + (-2*b1 + 4) * q^31 + (-b2 - 2*b1 - 2) * q^33 + 6 * q^37 + (b2 - 4*b1) * q^39 + (2*b2 + 2*b1 + 6) * q^41 + (2*b2 + 2*b1) * q^43 + (3*b2 - 2) * q^47 + q^49 + (b2 + 4*b1 + 6) * q^51 + (-2*b2 + 2*b1 + 2) * q^53 + (2*b2 + 6) * q^57 + (-3*b2 - 3*b1 + 2) * q^59 + (-b2 - b1 + 8) * q^61 + (b2 + 2*b1 + 1) * q^63 + (2*b2 + 4*b1 - 4) * q^67 + (6*b1 + 4) * q^69 + (-2*b1 + 2) * q^71 + (-2*b2 - 2*b1 - 2) * q^73 + (-b2 + 2) * q^77 + (-b2 + 4*b1 + 4) * q^79 + (-2*b2 + 4*b1 + 3) * q^81 + (-b2 + 5*b1 - 6) * q^83 + (-3*b2 - 6) * q^87 + (2*b2 - 4*b1 + 2) * q^89 + (2*b2 + b1 + 2) * q^91 + (2*b2 + 8) * q^93 + (-b2 + 2*b1 - 4) * q^97 + (4*b2 + 6*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} + 3 q^{7} + 4 q^{9}+O(q^{10})$$ 3 * q - q^3 + 3 * q^7 + 4 * q^9 $$3 q - q^{3} + 3 q^{7} + 4 q^{9} + 7 q^{11} + 5 q^{13} - q^{17} + 4 q^{19} - q^{21} - 6 q^{23} - 19 q^{27} + 3 q^{29} + 10 q^{31} - 7 q^{33} + 18 q^{37} - 5 q^{39} + 18 q^{41} - 9 q^{47} + 3 q^{49} + 21 q^{51} + 10 q^{53} + 16 q^{57} + 6 q^{59} + 24 q^{61} + 4 q^{63} - 10 q^{67} + 18 q^{69} + 4 q^{71} - 6 q^{73} + 7 q^{77} + 17 q^{79} + 15 q^{81} - 12 q^{83} - 15 q^{87} + 5 q^{91} + 22 q^{93} - 9 q^{97} + 2 q^{99}+O(q^{100})$$ 3 * q - q^3 + 3 * q^7 + 4 * q^9 + 7 * q^11 + 5 * q^13 - q^17 + 4 * q^19 - q^21 - 6 * q^23 - 19 * q^27 + 3 * q^29 + 10 * q^31 - 7 * q^33 + 18 * q^37 - 5 * q^39 + 18 * q^41 - 9 * q^47 + 3 * q^49 + 21 * q^51 + 10 * q^53 + 16 * q^57 + 6 * q^59 + 24 * q^61 + 4 * q^63 - 10 * q^67 + 18 * q^69 + 4 * q^71 - 6 * q^73 + 7 * q^77 + 17 * q^79 + 15 * q^81 - 12 * q^83 - 15 * q^87 + 5 * q^91 + 22 * q^93 - 9 * q^97 + 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 4$$ v^2 - 2*v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 4$$ b2 + 2*b1 + 4

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.12489 −0.363328 −1.76156
0 −3.12489 0 0 0 1.00000 0 6.76491 0
1.2 0 0.363328 0 0 0 1.00000 0 −2.86799 0
1.3 0 1.76156 0 0 0 1.00000 0 0.103084 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.a.s 3
4.b odd 2 1 2800.2.a.br 3
5.b even 2 1 1400.2.a.t 3
5.c odd 4 2 280.2.g.b 6
7.b odd 2 1 9800.2.a.cg 3
15.e even 4 2 2520.2.t.g 6
20.d odd 2 1 2800.2.a.bq 3
20.e even 4 2 560.2.g.f 6
35.c odd 2 1 9800.2.a.cd 3
35.f even 4 2 1960.2.g.c 6
40.i odd 4 2 2240.2.g.l 6
40.k even 4 2 2240.2.g.m 6
60.l odd 4 2 5040.2.t.y 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.g.b 6 5.c odd 4 2
560.2.g.f 6 20.e even 4 2
1400.2.a.s 3 1.a even 1 1 trivial
1400.2.a.t 3 5.b even 2 1
1960.2.g.c 6 35.f even 4 2
2240.2.g.l 6 40.i odd 4 2
2240.2.g.m 6 40.k even 4 2
2520.2.t.g 6 15.e even 4 2
2800.2.a.bq 3 20.d odd 2 1
2800.2.a.br 3 4.b odd 2 1
5040.2.t.y 6 60.l odd 4 2
9800.2.a.cd 3 35.c odd 2 1
9800.2.a.cg 3 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1400))$$:

 $$T_{3}^{3} + T_{3}^{2} - 6T_{3} + 2$$ T3^3 + T3^2 - 6*T3 + 2 $$T_{11}^{3} - 7T_{11}^{2} + 8T_{11} + 8$$ T11^3 - 7*T11^2 + 8*T11 + 8 $$T_{13}^{3} - 5T_{13}^{2} - 22T_{13} + 106$$ T13^3 - 5*T13^2 - 22*T13 + 106

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + T^{2} - 6T + 2$$
$5$ $$T^{3}$$
$7$ $$(T - 1)^{3}$$
$11$ $$T^{3} - 7 T^{2} + 8 T + 8$$
$13$ $$T^{3} - 5 T^{2} - 22 T + 106$$
$17$ $$T^{3} + T^{2} - 24 T + 20$$
$19$ $$T^{3} - 4 T^{2} - 14 T - 8$$
$23$ $$T^{3} + 6 T^{2} - 28 T - 136$$
$29$ $$T^{3} - 3 T^{2} - 72 T + 108$$
$31$ $$T^{3} - 10 T^{2} + 8 T + 80$$
$37$ $$(T - 6)^{3}$$
$41$ $$T^{3} - 18 T^{2} + 68 T + 88$$
$43$ $$T^{3} - 40T + 64$$
$47$ $$T^{3} + 9 T^{2} - 48 T - 232$$
$53$ $$T^{3} - 10 T^{2} - 44 T + 472$$
$59$ $$T^{3} - 6 T^{2} - 78 T - 44$$
$61$ $$T^{3} - 24 T^{2} + 182 T - 440$$
$67$ $$T^{3} + 10 T^{2} - 64 T - 512$$
$71$ $$T^{3} - 4 T^{2} - 20 T + 64$$
$73$ $$T^{3} + 6 T^{2} - 28 T - 136$$
$79$ $$T^{3} - 17 T^{2} - 32 T + 548$$
$83$ $$T^{3} + 12 T^{2} - 142 T - 824$$
$89$ $$T^{3} - 172T - 464$$
$97$ $$T^{3} + 9 T^{2} - 16 T - 44$$