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

 $$f(q)$$ $$=$$ $$q + 3q^{3} - q^{7} + 6q^{9} + O(q^{10})$$ $$q + 3q^{3} - q^{7} + 6q^{9} - 5q^{11} + 5q^{13} + 7q^{17} - 2q^{19} - 3q^{21} + 2q^{23} + 9q^{27} + 7q^{29} + 4q^{31} - 15q^{33} + 6q^{37} + 15q^{39} - 12q^{41} + 2q^{43} - q^{47} + q^{49} + 21q^{51} - 6q^{57} - 4q^{59} + 4q^{61} - 6q^{63} - 8q^{67} + 6q^{69} - 6q^{73} + 5q^{77} - 3q^{79} + 9q^{81} + 4q^{83} + 21q^{87} - 5q^{91} + 12q^{93} - 13q^{97} - 30q^{99} + O(q^{100})$$

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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1400))$$:

 $$T_{3} - 3$$ $$T_{11} + 5$$ $$T_{13} - 5$$

Hecke characteristic polynomials

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