# Properties

 Label 1400.2.bh.f 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: no (minimal twist has level 56) 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.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{12} q^{3} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + 6 \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + 3 \zeta_{12} q^{3} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + 6 \zeta_{12}^{2} q^{9} + ( 1 - \zeta_{12}^{2} ) q^{11} -2 \zeta_{12}^{3} q^{13} -3 \zeta_{12} q^{17} + 5 \zeta_{12}^{2} q^{19} + ( -3 + 9 \zeta_{12}^{2} ) q^{21} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{23} + 9 \zeta_{12}^{3} q^{27} + 6 q^{29} + ( 1 - \zeta_{12}^{2} ) q^{31} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{33} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{37} + ( 6 - 6 \zeta_{12}^{2} ) q^{39} -10 q^{41} + 4 \zeta_{12}^{3} q^{43} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{47} + ( -5 + 8 \zeta_{12}^{2} ) q^{49} -9 \zeta_{12}^{2} q^{51} -9 \zeta_{12} q^{53} + 15 \zeta_{12}^{3} q^{57} + ( 3 - 3 \zeta_{12}^{2} ) q^{59} -3 \zeta_{12}^{2} q^{61} + ( -6 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{63} -11 \zeta_{12} q^{67} + 9 q^{69} + 16 q^{71} + 7 \zeta_{12} q^{73} + ( 3 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{77} -11 \zeta_{12}^{2} q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + 4 \zeta_{12}^{3} q^{83} + 18 \zeta_{12} q^{87} -9 \zeta_{12}^{2} q^{89} + ( 6 - 4 \zeta_{12}^{2} ) q^{91} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{93} + 6 \zeta_{12}^{3} q^{97} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} + 2q^{11} + 10q^{19} + 6q^{21} + 24q^{29} + 2q^{31} + 12q^{39} - 40q^{41} - 4q^{49} - 18q^{51} + 6q^{59} - 6q^{61} + 36q^{69} + 64q^{71} - 22q^{79} - 18q^{81} - 18q^{89} + 16q^{91} + 24q^{99} + O(q^{100})$$

## 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 −2.59808 1.50000i 0 0 0 −1.73205 2.00000i 0 3.00000 + 5.19615i 0
249.2 0 2.59808 + 1.50000i 0 0 0 1.73205 + 2.00000i 0 3.00000 + 5.19615i 0
849.1 0 −2.59808 + 1.50000i 0 0 0 −1.73205 + 2.00000i 0 3.00000 5.19615i 0
849.2 0 2.59808 1.50000i 0 0 0 1.73205 2.00000i 0 3.00000 5.19615i 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.f 4
5.b even 2 1 inner 1400.2.bh.f 4
5.c odd 4 1 56.2.i.a 2
5.c odd 4 1 1400.2.q.g 2
7.c even 3 1 inner 1400.2.bh.f 4
15.e even 4 1 504.2.s.e 2
20.e even 4 1 112.2.i.c 2
35.f even 4 1 392.2.i.f 2
35.j even 6 1 inner 1400.2.bh.f 4
35.k even 12 1 392.2.a.a 1
35.k even 12 1 392.2.i.f 2
35.k even 12 1 9800.2.a.bp 1
35.l odd 12 1 56.2.i.a 2
35.l odd 12 1 392.2.a.f 1
35.l odd 12 1 1400.2.q.g 2
35.l odd 12 1 9800.2.a.b 1
40.i odd 4 1 448.2.i.f 2
40.k even 4 1 448.2.i.a 2
60.l odd 4 1 1008.2.s.e 2
105.k odd 4 1 3528.2.s.o 2
105.w odd 12 1 3528.2.a.k 1
105.w odd 12 1 3528.2.s.o 2
105.x even 12 1 504.2.s.e 2
105.x even 12 1 3528.2.a.r 1
140.j odd 4 1 784.2.i.a 2
140.w even 12 1 112.2.i.c 2
140.w even 12 1 784.2.a.a 1
140.x odd 12 1 784.2.a.j 1
140.x odd 12 1 784.2.i.a 2
280.bp odd 12 1 3136.2.a.a 1
280.br even 12 1 448.2.i.a 2
280.br even 12 1 3136.2.a.bc 1
280.bt odd 12 1 448.2.i.f 2
280.bt odd 12 1 3136.2.a.b 1
280.bv even 12 1 3136.2.a.bb 1
420.bp odd 12 1 1008.2.s.e 2
420.bp odd 12 1 7056.2.a.bi 1
420.br even 12 1 7056.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.a 2 5.c odd 4 1
56.2.i.a 2 35.l odd 12 1
112.2.i.c 2 20.e even 4 1
112.2.i.c 2 140.w even 12 1
392.2.a.a 1 35.k even 12 1
392.2.a.f 1 35.l odd 12 1
392.2.i.f 2 35.f even 4 1
392.2.i.f 2 35.k even 12 1
448.2.i.a 2 40.k even 4 1
448.2.i.a 2 280.br even 12 1
448.2.i.f 2 40.i odd 4 1
448.2.i.f 2 280.bt odd 12 1
504.2.s.e 2 15.e even 4 1
504.2.s.e 2 105.x even 12 1
784.2.a.a 1 140.w even 12 1
784.2.a.j 1 140.x odd 12 1
784.2.i.a 2 140.j odd 4 1
784.2.i.a 2 140.x odd 12 1
1008.2.s.e 2 60.l odd 4 1
1008.2.s.e 2 420.bp odd 12 1
1400.2.q.g 2 5.c odd 4 1
1400.2.q.g 2 35.l odd 12 1
1400.2.bh.f 4 1.a even 1 1 trivial
1400.2.bh.f 4 5.b even 2 1 inner
1400.2.bh.f 4 7.c even 3 1 inner
1400.2.bh.f 4 35.j even 6 1 inner
3136.2.a.a 1 280.bp odd 12 1
3136.2.a.b 1 280.bt odd 12 1
3136.2.a.bb 1 280.bv even 12 1
3136.2.a.bc 1 280.br even 12 1
3528.2.a.k 1 105.w odd 12 1
3528.2.a.r 1 105.x even 12 1
3528.2.s.o 2 105.k odd 4 1
3528.2.s.o 2 105.w odd 12 1
7056.2.a.s 1 420.br even 12 1
7056.2.a.bi 1 420.bp odd 12 1
9800.2.a.b 1 35.l odd 12 1
9800.2.a.bp 1 35.k even 12 1

## Hecke kernels

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

 $$T_{3}^{4} - 9 T_{3}^{2} + 81$$ $$T_{11}^{2} - T_{11} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 - 9 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$49 + 2 T^{2} + T^{4}$$
$11$ $$( 1 - T + T^{2} )^{2}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$81 - 9 T^{2} + T^{4}$$
$19$ $$( 25 - 5 T + T^{2} )^{2}$$
$23$ $$81 - 9 T^{2} + T^{4}$$
$29$ $$( -6 + T )^{4}$$
$31$ $$( 1 - T + T^{2} )^{2}$$
$37$ $$625 - 25 T^{2} + T^{4}$$
$41$ $$( 10 + T )^{4}$$
$43$ $$( 16 + T^{2} )^{2}$$
$47$ $$1 - T^{2} + T^{4}$$
$53$ $$6561 - 81 T^{2} + T^{4}$$
$59$ $$( 9 - 3 T + T^{2} )^{2}$$
$61$ $$( 9 + 3 T + T^{2} )^{2}$$
$67$ $$14641 - 121 T^{2} + T^{4}$$
$71$ $$( -16 + T )^{4}$$
$73$ $$2401 - 49 T^{2} + T^{4}$$
$79$ $$( 121 + 11 T + T^{2} )^{2}$$
$83$ $$( 16 + T^{2} )^{2}$$
$89$ $$( 81 + 9 T + T^{2} )^{2}$$
$97$ $$( 36 + T^{2} )^{2}$$