# Properties

 Label 1400.2.bh.d Level $1400$ Weight $2$ Character orbit 1400.bh Analytic conductor $11.179$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1400.bh (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1790562830$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{12} q^{3} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + 2 \zeta_{12} q^{3} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} + ( -4 + 4 \zeta_{12}^{2} ) q^{11} + 2 \zeta_{12}^{3} q^{13} + 3 \zeta_{12} q^{17} + ( 2 + 4 \zeta_{12}^{2} ) q^{21} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{23} -4 \zeta_{12}^{3} q^{27} + 6 q^{29} + ( -9 + 9 \zeta_{12}^{2} ) q^{31} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{33} + ( -4 + 4 \zeta_{12}^{2} ) q^{39} + 5 q^{41} + 6 \zeta_{12}^{3} q^{43} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{47} + ( 5 + 3 \zeta_{12}^{2} ) q^{49} + 6 \zeta_{12}^{2} q^{51} -6 \zeta_{12} q^{53} + ( 8 - 8 \zeta_{12}^{2} ) q^{59} -8 \zeta_{12}^{2} q^{61} + ( \zeta_{12} + 2 \zeta_{12}^{3} ) q^{63} -14 \zeta_{12} q^{67} -6 q^{69} + 11 q^{71} -2 \zeta_{12} q^{73} + ( -8 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{77} + 9 \zeta_{12}^{2} q^{79} + ( 11 - 11 \zeta_{12}^{2} ) q^{81} + 6 \zeta_{12}^{3} q^{83} + 12 \zeta_{12} q^{87} + 11 \zeta_{12}^{2} q^{89} + ( -4 + 6 \zeta_{12}^{2} ) q^{91} + ( -18 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{93} -11 \zeta_{12}^{3} q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{9} + O(q^{10})$$ $$4q + 2q^{9} - 8q^{11} + 16q^{21} + 24q^{29} - 18q^{31} - 8q^{39} + 20q^{41} + 26q^{49} + 12q^{51} + 16q^{59} - 16q^{61} - 24q^{69} + 44q^{71} + 18q^{79} + 22q^{81} + 22q^{89} - 4q^{91} - 16q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
249.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 −1.73205 1.00000i 0 0 0 −2.59808 0.500000i 0 0.500000 + 0.866025i 0
249.2 0 1.73205 + 1.00000i 0 0 0 2.59808 + 0.500000i 0 0.500000 + 0.866025i 0
849.1 0 −1.73205 + 1.00000i 0 0 0 −2.59808 + 0.500000i 0 0.500000 0.866025i 0
849.2 0 1.73205 1.00000i 0 0 0 2.59808 0.500000i 0 0.500000 0.866025i 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
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.bh.d 4
5.b even 2 1 inner 1400.2.bh.d 4
5.c odd 4 1 1400.2.q.b 2
5.c odd 4 1 1400.2.q.f yes 2
7.c even 3 1 inner 1400.2.bh.d 4
35.j even 6 1 inner 1400.2.bh.d 4
35.k even 12 1 9800.2.a.k 1
35.k even 12 1 9800.2.a.bl 1
35.l odd 12 1 1400.2.q.b 2
35.l odd 12 1 1400.2.q.f yes 2
35.l odd 12 1 9800.2.a.j 1
35.l odd 12 1 9800.2.a.bm 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.q.b 2 5.c odd 4 1
1400.2.q.b 2 35.l odd 12 1
1400.2.q.f yes 2 5.c odd 4 1
1400.2.q.f yes 2 35.l odd 12 1
1400.2.bh.d 4 1.a even 1 1 trivial
1400.2.bh.d 4 5.b even 2 1 inner
1400.2.bh.d 4 7.c even 3 1 inner
1400.2.bh.d 4 35.j even 6 1 inner
9800.2.a.j 1 35.l odd 12 1
9800.2.a.k 1 35.k even 12 1
9800.2.a.bl 1 35.k even 12 1
9800.2.a.bm 1 35.l odd 12 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 - 4 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$49 - 13 T^{2} + T^{4}$$
$11$ $$( 16 + 4 T + T^{2} )^{2}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$81 - 9 T^{2} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$81 - 9 T^{2} + T^{4}$$
$29$ $$( -6 + T )^{4}$$
$31$ $$( 81 + 9 T + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$( -5 + T )^{4}$$
$43$ $$( 36 + T^{2} )^{2}$$
$47$ $$6561 - 81 T^{2} + T^{4}$$
$53$ $$1296 - 36 T^{2} + T^{4}$$
$59$ $$( 64 - 8 T + T^{2} )^{2}$$
$61$ $$( 64 + 8 T + T^{2} )^{2}$$
$67$ $$38416 - 196 T^{2} + T^{4}$$
$71$ $$( -11 + T )^{4}$$
$73$ $$16 - 4 T^{2} + T^{4}$$
$79$ $$( 81 - 9 T + T^{2} )^{2}$$
$83$ $$( 36 + T^{2} )^{2}$$
$89$ $$( 121 - 11 T + T^{2} )^{2}$$
$97$ $$( 121 + T^{2} )^{2}$$