# Properties

 Label 1400.2.g.k Level $1400$ Weight $2$ Character orbit 1400.g Analytic conductor $11.179$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{7} + ( -2 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{7} + ( -2 + \beta_{3} ) q^{9} + ( -1 + \beta_{3} ) q^{11} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{13} + ( \beta_{1} + 6 \beta_{2} ) q^{17} + ( 2 + 2 \beta_{3} ) q^{19} + ( 1 - \beta_{3} ) q^{21} -2 \beta_{1} q^{23} + ( \beta_{1} + 4 \beta_{2} ) q^{27} + ( -3 + \beta_{3} ) q^{29} + ( -4 + 4 \beta_{3} ) q^{31} + ( -\beta_{1} + 4 \beta_{2} ) q^{33} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{37} + ( 13 - \beta_{3} ) q^{39} + ( 4 - 2 \beta_{3} ) q^{41} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{43} + ( -\beta_{1} + 4 \beta_{2} ) q^{47} - q^{49} + ( 1 - 5 \beta_{3} ) q^{51} + ( 2 \beta_{1} + 10 \beta_{2} ) q^{53} + ( 2 \beta_{1} + 8 \beta_{2} ) q^{57} + 4 q^{59} + ( -12 + 2 \beta_{3} ) q^{61} + ( \beta_{1} - \beta_{2} ) q^{63} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{67} + ( 10 - 2 \beta_{3} ) q^{69} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{73} + \beta_{1} q^{77} + ( 7 - 3 \beta_{3} ) q^{79} -7 q^{81} -12 \beta_{2} q^{83} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{87} -2 \beta_{3} q^{89} + ( -1 + 3 \beta_{3} ) q^{91} + ( -4 \beta_{1} + 16 \beta_{2} ) q^{93} + ( -3 \beta_{1} + 6 \beta_{2} ) q^{97} + ( 6 - 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{9} + O(q^{10})$$ $$4q - 6q^{9} - 2q^{11} + 12q^{19} + 2q^{21} - 10q^{29} - 8q^{31} + 50q^{39} + 12q^{41} - 4q^{49} - 6q^{51} + 16q^{59} - 44q^{61} + 36q^{69} + 22q^{79} - 28q^{81} - 4q^{89} + 2q^{91} + 20q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} - 5 \beta_{1}$$

## 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
 − 2.56155i − 1.56155i 1.56155i 2.56155i
0 2.56155i 0 0 0 1.00000i 0 −3.56155 0
449.2 0 1.56155i 0 0 0 1.00000i 0 0.561553 0
449.3 0 1.56155i 0 0 0 1.00000i 0 0.561553 0
449.4 0 2.56155i 0 0 0 1.00000i 0 −3.56155 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.k 4
4.b odd 2 1 2800.2.g.u 4
5.b even 2 1 inner 1400.2.g.k 4
5.c odd 4 1 280.2.a.d 2
5.c odd 4 1 1400.2.a.p 2
15.e even 4 1 2520.2.a.w 2
20.d odd 2 1 2800.2.g.u 4
20.e even 4 1 560.2.a.g 2
20.e even 4 1 2800.2.a.bn 2
35.f even 4 1 1960.2.a.r 2
35.f even 4 1 9800.2.a.by 2
35.k even 12 2 1960.2.q.u 4
35.l odd 12 2 1960.2.q.s 4
40.i odd 4 1 2240.2.a.be 2
40.k even 4 1 2240.2.a.bi 2
60.l odd 4 1 5040.2.a.bq 2
140.j odd 4 1 3920.2.a.bu 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.d 2 5.c odd 4 1
560.2.a.g 2 20.e even 4 1
1400.2.a.p 2 5.c odd 4 1
1400.2.g.k 4 1.a even 1 1 trivial
1400.2.g.k 4 5.b even 2 1 inner
1960.2.a.r 2 35.f even 4 1
1960.2.q.s 4 35.l odd 12 2
1960.2.q.u 4 35.k even 12 2
2240.2.a.be 2 40.i odd 4 1
2240.2.a.bi 2 40.k even 4 1
2520.2.a.w 2 15.e even 4 1
2800.2.a.bn 2 20.e even 4 1
2800.2.g.u 4 4.b odd 2 1
2800.2.g.u 4 20.d odd 2 1
3920.2.a.bu 2 140.j odd 4 1
5040.2.a.bq 2 60.l odd 4 1
9800.2.a.by 2 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 $$S_{2}^{\mathrm{new}}(1400, [\chi])$$:

 $$T_{3}^{4} + 9 T_{3}^{2} + 16$$ $$T_{11}^{2} + T_{11} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 + 9 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( -4 + T + T^{2} )^{2}$$
$13$ $$1444 + 77 T^{2} + T^{4}$$
$17$ $$676 + 69 T^{2} + T^{4}$$
$19$ $$( -8 - 6 T + T^{2} )^{2}$$
$23$ $$256 + 36 T^{2} + T^{4}$$
$29$ $$( 2 + 5 T + T^{2} )^{2}$$
$31$ $$( -64 + 4 T + T^{2} )^{2}$$
$37$ $$( 68 + T^{2} )^{2}$$
$41$ $$( -8 - 6 T + T^{2} )^{2}$$
$43$ $$64 + 52 T^{2} + T^{4}$$
$47$ $$256 + 49 T^{2} + T^{4}$$
$53$ $$4096 + 196 T^{2} + T^{4}$$
$59$ $$( -4 + T )^{4}$$
$61$ $$( 104 + 22 T + T^{2} )^{2}$$
$67$ $$1024 + 208 T^{2} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$2704 + 168 T^{2} + T^{4}$$
$79$ $$( -8 - 11 T + T^{2} )^{2}$$
$83$ $$( 144 + T^{2} )^{2}$$
$89$ $$( -16 + 2 T + T^{2} )^{2}$$
$97$ $$324 + 189 T^{2} + T^{4}$$