# Properties

 Label 1400.2.g.d Level $1400$ Weight $2$ Character orbit 1400.g Analytic conductor $11.179$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1400.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1790562830$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} -i q^{7} - q^{9} +O(q^{10})$$ $$q + 2 i q^{3} -i q^{7} - q^{9} + 5 q^{11} -8 i q^{17} + 2 q^{19} + 2 q^{21} -7 i q^{23} + 4 i q^{27} + 3 q^{29} + 4 q^{31} + 10 i q^{33} + i q^{37} -2 q^{41} + 3 i q^{43} -6 i q^{47} - q^{49} + 16 q^{51} + 10 i q^{53} + 4 i q^{57} + 4 q^{59} -6 q^{61} + i q^{63} -13 i q^{67} + 14 q^{69} + 5 q^{71} + 6 i q^{73} -5 i q^{77} + 13 q^{79} -11 q^{81} + 16 i q^{83} + 6 i q^{87} + 8 i q^{93} + 12 i q^{97} -5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 10q^{11} + 4q^{19} + 4q^{21} + 6q^{29} + 8q^{31} - 4q^{41} - 2q^{49} + 32q^{51} + 8q^{59} - 12q^{61} + 28q^{69} + 10q^{71} + 26q^{79} - 22q^{81} - 10q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 1.00000i 0 −1.00000 0
449.2 0 2.00000i 0 0 0 1.00000i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.g.d 2
4.b odd 2 1 2800.2.g.d 2
5.b even 2 1 inner 1400.2.g.d 2
5.c odd 4 1 1400.2.a.b 1
5.c odd 4 1 1400.2.a.m yes 1
20.d odd 2 1 2800.2.g.d 2
20.e even 4 1 2800.2.a.d 1
20.e even 4 1 2800.2.a.bc 1
35.f even 4 1 9800.2.a.l 1
35.f even 4 1 9800.2.a.bo 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.a.b 1 5.c odd 4 1
1400.2.a.m yes 1 5.c odd 4 1
1400.2.g.d 2 1.a even 1 1 trivial
1400.2.g.d 2 5.b even 2 1 inner
2800.2.a.d 1 20.e even 4 1
2800.2.a.bc 1 20.e even 4 1
2800.2.g.d 2 4.b odd 2 1
2800.2.g.d 2 20.d odd 2 1
9800.2.a.l 1 35.f even 4 1
9800.2.a.bo 1 35.f even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ $$T_{11} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( -5 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$64 + T^{2}$$
$19$ $$( -2 + T )^{2}$$
$23$ $$49 + T^{2}$$
$29$ $$( -3 + T )^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$1 + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$9 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$100 + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$( 6 + T )^{2}$$
$67$ $$169 + T^{2}$$
$71$ $$( -5 + T )^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$( -13 + T )^{2}$$
$83$ $$256 + T^{2}$$
$89$ $$T^{2}$$
$97$ $$144 + T^{2}$$