# Properties

 Label 1400.2.g.i 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: $$\Q(i, \sqrt{33})$$ Defining polynomial: $$x^{4} + 17 x^{2} + 64$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{7} + ( -6 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{7} + ( -6 + \beta_{3} ) q^{9} + ( 3 + \beta_{3} ) q^{11} + ( \beta_{1} + 2 \beta_{2} ) q^{13} + ( \beta_{1} - 2 \beta_{2} ) q^{17} + ( -2 + 2 \beta_{3} ) q^{19} + ( 1 - \beta_{3} ) q^{21} -2 \beta_{1} q^{23} + ( -3 \beta_{1} + 8 \beta_{2} ) q^{27} + ( 1 + \beta_{3} ) q^{29} -8 q^{31} + ( 3 \beta_{1} + 8 \beta_{2} ) q^{33} + 2 \beta_{2} q^{37} + ( -7 - \beta_{3} ) q^{39} + 2 \beta_{3} q^{41} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{43} + 3 \beta_{1} q^{47} - q^{49} + ( -11 + 3 \beta_{3} ) q^{51} + ( 2 \beta_{1} + 6 \beta_{2} ) q^{53} + ( -2 \beta_{1} + 16 \beta_{2} ) q^{57} -8 q^{59} + ( 4 - 2 \beta_{3} ) q^{61} + ( \beta_{1} - 5 \beta_{2} ) q^{63} + 4 \beta_{2} q^{67} + ( 18 - 2 \beta_{3} ) q^{69} + 8 q^{71} -6 \beta_{2} q^{73} + ( \beta_{1} + 4 \beta_{2} ) q^{77} + ( -5 - 3 \beta_{3} ) q^{79} + ( 17 - 8 \beta_{3} ) q^{81} -4 \beta_{1} q^{83} + ( \beta_{1} + 8 \beta_{2} ) q^{87} + ( -8 - 2 \beta_{3} ) q^{89} + ( -1 - \beta_{3} ) q^{91} -8 \beta_{1} q^{93} + ( 5 \beta_{1} - 2 \beta_{2} ) q^{97} + ( -10 - 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 22q^{9} + O(q^{10})$$ $$4q - 22q^{9} + 14q^{11} - 4q^{19} + 2q^{21} + 6q^{29} - 32q^{31} - 30q^{39} + 4q^{41} - 4q^{49} - 38q^{51} - 32q^{59} + 12q^{61} + 68q^{69} + 32q^{71} - 26q^{79} + 52q^{81} - 36q^{89} - 6q^{91} - 44q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 17 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 9 \nu$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 9$$ $$\nu^{3}$$ $$=$$ $$8 \beta_{2} - 9 \beta_{1}$$

## 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
 − 3.37228i − 2.37228i 2.37228i 3.37228i
0 3.37228i 0 0 0 1.00000i 0 −8.37228 0
449.2 0 2.37228i 0 0 0 1.00000i 0 −2.62772 0
449.3 0 2.37228i 0 0 0 1.00000i 0 −2.62772 0
449.4 0 3.37228i 0 0 0 1.00000i 0 −8.37228 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.i 4
4.b odd 2 1 2800.2.g.r 4
5.b even 2 1 inner 1400.2.g.i 4
5.c odd 4 1 280.2.a.c 2
5.c odd 4 1 1400.2.a.r 2
15.e even 4 1 2520.2.a.x 2
20.d odd 2 1 2800.2.g.r 4
20.e even 4 1 560.2.a.h 2
20.e even 4 1 2800.2.a.bk 2
35.f even 4 1 1960.2.a.s 2
35.f even 4 1 9800.2.a.bu 2
35.k even 12 2 1960.2.q.r 4
35.l odd 12 2 1960.2.q.t 4
40.i odd 4 1 2240.2.a.bk 2
40.k even 4 1 2240.2.a.bg 2
60.l odd 4 1 5040.2.a.by 2
140.j odd 4 1 3920.2.a.bt 2

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

 $$T_{3}^{4} + 17 T_{3}^{2} + 64$$ $$T_{11}^{2} - 7 T_{11} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$64 + 17 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( 4 - 7 T + T^{2} )^{2}$$
$13$ $$36 + 21 T^{2} + T^{4}$$
$17$ $$4 + 29 T^{2} + T^{4}$$
$19$ $$( -32 + 2 T + T^{2} )^{2}$$
$23$ $$1024 + 68 T^{2} + T^{4}$$
$29$ $$( -6 - 3 T + T^{2} )^{2}$$
$31$ $$( 8 + T )^{4}$$
$37$ $$( 4 + T^{2} )^{2}$$
$41$ $$( -32 - 2 T + T^{2} )^{2}$$
$43$ $$576 + 84 T^{2} + T^{4}$$
$47$ $$5184 + 153 T^{2} + T^{4}$$
$53$ $$64 + 116 T^{2} + T^{4}$$
$59$ $$( 8 + T )^{4}$$
$61$ $$( -24 - 6 T + T^{2} )^{2}$$
$67$ $$( 16 + T^{2} )^{2}$$
$71$ $$( -8 + T )^{4}$$
$73$ $$( 36 + T^{2} )^{2}$$
$79$ $$( -32 + 13 T + T^{2} )^{2}$$
$83$ $$16384 + 272 T^{2} + T^{4}$$
$89$ $$( 48 + 18 T + T^{2} )^{2}$$
$97$ $$34596 + 453 T^{2} + T^{4}$$