# Properties

 Label 1400.2.a.j Level $1400$ Weight $2$ Character orbit 1400.a Self dual yes Analytic conductor $11.179$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 - q^7 - 2 * q^9 $$q + q^{3} - q^{7} - 2 q^{9} - q^{11} - q^{13} - 3 q^{17} - 4 q^{19} - q^{21} + 2 q^{23} - 5 q^{27} - q^{29} - 6 q^{31} - q^{33} + 2 q^{37} - q^{39} - 10 q^{41} + 9 q^{47} + q^{49} - 3 q^{51} - 14 q^{53} - 4 q^{57} + 6 q^{59} - 4 q^{61} + 2 q^{63} + 10 q^{67} + 2 q^{69} - 16 q^{71} + 10 q^{73} + q^{77} - 11 q^{79} + q^{81} + 4 q^{83} - q^{87} + 12 q^{89} + q^{91} - 6 q^{93} - 19 q^{97} + 2 q^{99}+O(q^{100})$$ q + q^3 - q^7 - 2 * q^9 - q^11 - q^13 - 3 * q^17 - 4 * q^19 - q^21 + 2 * q^23 - 5 * q^27 - q^29 - 6 * q^31 - q^33 + 2 * q^37 - q^39 - 10 * q^41 + 9 * q^47 + q^49 - 3 * q^51 - 14 * q^53 - 4 * q^57 + 6 * q^59 - 4 * q^61 + 2 * q^63 + 10 * q^67 + 2 * q^69 - 16 * q^71 + 10 * q^73 + q^77 - 11 * q^79 + q^81 + 4 * q^83 - q^87 + 12 * q^89 + q^91 - 6 * q^93 - 19 * q^97 + 2 * q^99

## 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
 0
0 1.00000 0 0 0 −1.00000 0 −2.00000 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.j 1
4.b odd 2 1 2800.2.a.k 1
5.b even 2 1 1400.2.a.d 1
5.c odd 4 2 280.2.g.a 2
7.b odd 2 1 9800.2.a.p 1
15.e even 4 2 2520.2.t.a 2
20.d odd 2 1 2800.2.a.u 1
20.e even 4 2 560.2.g.d 2
35.c odd 2 1 9800.2.a.bb 1
35.f even 4 2 1960.2.g.a 2
40.i odd 4 2 2240.2.g.b 2
40.k even 4 2 2240.2.g.a 2
60.l odd 4 2 5040.2.t.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.g.a 2 5.c odd 4 2
560.2.g.d 2 20.e even 4 2
1400.2.a.d 1 5.b even 2 1
1400.2.a.j 1 1.a even 1 1 trivial
1960.2.g.a 2 35.f even 4 2
2240.2.g.a 2 40.k even 4 2
2240.2.g.b 2 40.i odd 4 2
2520.2.t.a 2 15.e even 4 2
2800.2.a.k 1 4.b odd 2 1
2800.2.a.u 1 20.d odd 2 1
5040.2.t.a 2 60.l odd 4 2
9800.2.a.p 1 7.b odd 2 1
9800.2.a.bb 1 35.c odd 2 1

## Hecke kernels

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

 $$T_{3} - 1$$ T3 - 1 $$T_{11} + 1$$ T11 + 1 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T + 1$$
$13$ $$T + 1$$
$17$ $$T + 3$$
$19$ $$T + 4$$
$23$ $$T - 2$$
$29$ $$T + 1$$
$31$ $$T + 6$$
$37$ $$T - 2$$
$41$ $$T + 10$$
$43$ $$T$$
$47$ $$T - 9$$
$53$ $$T + 14$$
$59$ $$T - 6$$
$61$ $$T + 4$$
$67$ $$T - 10$$
$71$ $$T + 16$$
$73$ $$T - 10$$
$79$ $$T + 11$$
$83$ $$T - 4$$
$89$ $$T - 12$$
$97$ $$T + 19$$