# Properties

 Label 1400.2.a.n Level $1400$ Weight $2$ Character orbit 1400.a Self dual yes Analytic conductor $11.179$ Analytic rank $0$ 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - q^{7} + 6 q^{9}+O(q^{10})$$ q + 3 * q^3 - q^7 + 6 * q^9 $$q + 3 q^{3} - q^{7} + 6 q^{9} - 5 q^{11} + 5 q^{13} + 7 q^{17} - 2 q^{19} - 3 q^{21} + 2 q^{23} + 9 q^{27} + 7 q^{29} + 4 q^{31} - 15 q^{33} + 6 q^{37} + 15 q^{39} - 12 q^{41} + 2 q^{43} - q^{47} + q^{49} + 21 q^{51} - 6 q^{57} - 4 q^{59} + 4 q^{61} - 6 q^{63} - 8 q^{67} + 6 q^{69} - 6 q^{73} + 5 q^{77} - 3 q^{79} + 9 q^{81} + 4 q^{83} + 21 q^{87} - 5 q^{91} + 12 q^{93} - 13 q^{97} - 30 q^{99}+O(q^{100})$$ q + 3 * q^3 - q^7 + 6 * q^9 - 5 * q^11 + 5 * q^13 + 7 * q^17 - 2 * q^19 - 3 * q^21 + 2 * q^23 + 9 * q^27 + 7 * q^29 + 4 * q^31 - 15 * q^33 + 6 * q^37 + 15 * q^39 - 12 * q^41 + 2 * q^43 - q^47 + q^49 + 21 * q^51 - 6 * q^57 - 4 * q^59 + 4 * q^61 - 6 * q^63 - 8 * q^67 + 6 * q^69 - 6 * q^73 + 5 * q^77 - 3 * q^79 + 9 * q^81 + 4 * q^83 + 21 * q^87 - 5 * q^91 + 12 * q^93 - 13 * q^97 - 30 * 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 3.00000 0 0 0 −1.00000 0 6.00000 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.n 1
4.b odd 2 1 2800.2.a.c 1
5.b even 2 1 280.2.a.a 1
5.c odd 4 2 1400.2.g.a 2
7.b odd 2 1 9800.2.a.a 1
15.d odd 2 1 2520.2.a.i 1
20.d odd 2 1 560.2.a.f 1
20.e even 4 2 2800.2.g.b 2
35.c odd 2 1 1960.2.a.o 1
35.i odd 6 2 1960.2.q.a 2
35.j even 6 2 1960.2.q.o 2
40.e odd 2 1 2240.2.a.a 1
40.f even 2 1 2240.2.a.z 1
60.h even 2 1 5040.2.a.a 1
140.c even 2 1 3920.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.a 1 5.b even 2 1
560.2.a.f 1 20.d odd 2 1
1400.2.a.n 1 1.a even 1 1 trivial
1400.2.g.a 2 5.c odd 4 2
1960.2.a.o 1 35.c odd 2 1
1960.2.q.a 2 35.i odd 6 2
1960.2.q.o 2 35.j even 6 2
2240.2.a.a 1 40.e odd 2 1
2240.2.a.z 1 40.f even 2 1
2520.2.a.i 1 15.d odd 2 1
2800.2.a.c 1 4.b odd 2 1
2800.2.g.b 2 20.e even 4 2
3920.2.a.c 1 140.c even 2 1
5040.2.a.a 1 60.h even 2 1
9800.2.a.a 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1400))$$:

 $$T_{3} - 3$$ T3 - 3 $$T_{11} + 5$$ T11 + 5 $$T_{13} - 5$$ T13 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T + 5$$
$13$ $$T - 5$$
$17$ $$T - 7$$
$19$ $$T + 2$$
$23$ $$T - 2$$
$29$ $$T - 7$$
$31$ $$T - 4$$
$37$ $$T - 6$$
$41$ $$T + 12$$
$43$ $$T - 2$$
$47$ $$T + 1$$
$53$ $$T$$
$59$ $$T + 4$$
$61$ $$T - 4$$
$67$ $$T + 8$$
$71$ $$T$$
$73$ $$T + 6$$
$79$ $$T + 3$$
$83$ $$T - 4$$
$89$ $$T$$
$97$ $$T + 13$$