# Properties

 Label 1400.2.q.h Level $1400$ Weight $2$ Character orbit 1400.q 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.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1790562830$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{9} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{11} -2 q^{13} + ( -2 - 4 \beta_{1} - 2 \beta_{2} ) q^{17} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{19} + ( -4 + 2 \beta_{1} - 3 \beta_{2} ) q^{21} + ( -\beta_{1} + 7 \beta_{2} - \beta_{3} ) q^{23} + ( 1 - \beta_{3} ) q^{27} + ( -5 + 2 \beta_{3} ) q^{29} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{31} + ( 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{33} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{37} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{39} + ( 3 - 2 \beta_{3} ) q^{41} + ( -3 - 7 \beta_{3} ) q^{43} + ( -4 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{47} + ( -2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{49} + ( 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{51} -4 \beta_{1} q^{53} + ( 8 + 4 \beta_{3} ) q^{57} + ( 4 + 4 \beta_{2} ) q^{59} + ( 4 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{61} + ( 4 + 8 \beta_{2} - 2 \beta_{3} ) q^{63} + ( -3 - 7 \beta_{1} - 3 \beta_{2} ) q^{67} + ( 9 + 8 \beta_{3} ) q^{69} -12 q^{71} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{73} + ( 8 - 4 \beta_{1} + 6 \beta_{2} ) q^{77} -4 \beta_{2} q^{79} + ( 1 - 6 \beta_{1} + \beta_{2} ) q^{81} + ( 9 + 3 \beta_{3} ) q^{83} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{87} + ( 4 \beta_{1} - 11 \beta_{2} + 4 \beta_{3} ) q^{89} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{91} + ( 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{93} -6 q^{97} + ( -8 - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} + 2q^{7} + O(q^{10})$$ $$4q - 2q^{3} + 2q^{7} + 4q^{11} - 8q^{13} - 4q^{17} - 10q^{21} - 14q^{23} + 4q^{27} - 20q^{29} + 4q^{31} + 12q^{33} + 4q^{39} + 12q^{41} - 12q^{43} + 12q^{47} + 10q^{49} + 12q^{51} + 32q^{57} + 8q^{59} - 2q^{61} - 6q^{67} + 36q^{69} - 48q^{71} - 4q^{73} + 20q^{77} + 8q^{79} + 2q^{81} + 36q^{83} + 2q^{87} + 22q^{89} - 4q^{91} + 12q^{93} - 24q^{97} - 32q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 −1.20711 2.09077i 0 0 0 2.62132 0.358719i 0 −1.41421 + 2.44949i 0
401.2 0 0.207107 + 0.358719i 0 0 0 −1.62132 + 2.09077i 0 1.41421 2.44949i 0
1201.1 0 −1.20711 + 2.09077i 0 0 0 2.62132 + 0.358719i 0 −1.41421 2.44949i 0
1201.2 0 0.207107 0.358719i 0 0 0 −1.62132 2.09077i 0 1.41421 + 2.44949i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.q.h 4
5.b even 2 1 280.2.q.d 4
5.c odd 4 2 1400.2.bh.g 8
7.c even 3 1 inner 1400.2.q.h 4
7.c even 3 1 9800.2.a.bz 2
7.d odd 6 1 9800.2.a.br 2
15.d odd 2 1 2520.2.bi.k 4
20.d odd 2 1 560.2.q.j 4
35.c odd 2 1 1960.2.q.q 4
35.i odd 6 1 1960.2.a.t 2
35.i odd 6 1 1960.2.q.q 4
35.j even 6 1 280.2.q.d 4
35.j even 6 1 1960.2.a.p 2
35.l odd 12 2 1400.2.bh.g 8
105.o odd 6 1 2520.2.bi.k 4
140.p odd 6 1 560.2.q.j 4
140.p odd 6 1 3920.2.a.bz 2
140.s even 6 1 3920.2.a.bp 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.d 4 5.b even 2 1
280.2.q.d 4 35.j even 6 1
560.2.q.j 4 20.d odd 2 1
560.2.q.j 4 140.p odd 6 1
1400.2.q.h 4 1.a even 1 1 trivial
1400.2.q.h 4 7.c even 3 1 inner
1400.2.bh.g 8 5.c odd 4 2
1400.2.bh.g 8 35.l odd 12 2
1960.2.a.p 2 35.j even 6 1
1960.2.a.t 2 35.i odd 6 1
1960.2.q.q 4 35.c odd 2 1
1960.2.q.q 4 35.i odd 6 1
2520.2.bi.k 4 15.d odd 2 1
2520.2.bi.k 4 105.o odd 6 1
3920.2.a.bp 2 140.s even 6 1
3920.2.a.bz 2 140.p odd 6 1
9800.2.a.br 2 7.d odd 6 1
9800.2.a.bz 2 7.c even 3 1

## Hecke kernels

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

 $$T_{3}^{4} + 2 T_{3}^{3} + 5 T_{3}^{2} - 2 T_{3} + 1$$ $$T_{11}^{4} - 4 T_{11}^{3} + 20 T_{11}^{2} + 16 T_{11} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$49 - 14 T - 3 T^{2} - 2 T^{3} + T^{4}$$
$11$ $$16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4}$$
$13$ $$( 2 + T )^{4}$$
$17$ $$784 - 112 T + 44 T^{2} + 4 T^{3} + T^{4}$$
$19$ $$1024 + 32 T^{2} + T^{4}$$
$23$ $$2209 + 658 T + 149 T^{2} + 14 T^{3} + T^{4}$$
$29$ $$( 17 + 10 T + T^{2} )^{2}$$
$31$ $$16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4}$$
$37$ $$1024 + 32 T^{2} + T^{4}$$
$41$ $$( 1 - 6 T + T^{2} )^{2}$$
$43$ $$( -89 + 6 T + T^{2} )^{2}$$
$47$ $$16 - 48 T + 140 T^{2} - 12 T^{3} + T^{4}$$
$53$ $$1024 + 32 T^{2} + T^{4}$$
$59$ $$( 16 - 4 T + T^{2} )^{2}$$
$61$ $$961 - 62 T + 35 T^{2} + 2 T^{3} + T^{4}$$
$67$ $$7921 - 534 T + 125 T^{2} + 6 T^{3} + T^{4}$$
$71$ $$( 12 + T )^{4}$$
$73$ $$784 - 112 T + 44 T^{2} + 4 T^{3} + T^{4}$$
$79$ $$( 16 - 4 T + T^{2} )^{2}$$
$83$ $$( 63 - 18 T + T^{2} )^{2}$$
$89$ $$7921 - 1958 T + 395 T^{2} - 22 T^{3} + T^{4}$$
$97$ $$( 6 + T )^{4}$$