# Properties

 Label 1400.2.bh.b 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: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{3} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{7} - 2 \zeta_{12}^{2} q^{9} +O(q^{10})$$ q + z * q^3 + (-2*z^3 - z) * q^7 - 2*z^2 * q^9 $$q + \zeta_{12} q^{3} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{7} - 2 \zeta_{12}^{2} q^{9} + (2 \zeta_{12}^{2} - 2) q^{11} - 4 \zeta_{12} q^{17} - 2 \zeta_{12}^{2} q^{19} + ( - 3 \zeta_{12}^{2} + 2) q^{21} + (\zeta_{12}^{3} - \zeta_{12}) q^{23} - 5 \zeta_{12}^{3} q^{27} - 9 q^{29} + (4 \zeta_{12}^{2} - 4) q^{31} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{33} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{37} + q^{41} - 9 \zeta_{12}^{3} q^{43} + (5 \zeta_{12}^{2} - 8) q^{49} - 4 \zeta_{12}^{2} q^{51} - 10 \zeta_{12} q^{53} - 2 \zeta_{12}^{3} q^{57} + (10 \zeta_{12}^{2} - 10) q^{59} - 9 \zeta_{12}^{2} q^{61} + (6 \zeta_{12}^{3} - 4 \zeta_{12}) q^{63} - 5 \zeta_{12} q^{67} - q^{69} + 14 q^{71} + 12 \zeta_{12} q^{73} + ( - 2 \zeta_{12}^{3} + 6 \zeta_{12}) q^{77} + 14 \zeta_{12}^{2} q^{79} + (\zeta_{12}^{2} - 1) q^{81} - 11 \zeta_{12}^{3} q^{83} - 9 \zeta_{12} q^{87} - 15 \zeta_{12}^{2} q^{89} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{93} - 18 \zeta_{12}^{3} q^{97} + 4 q^{99} +O(q^{100})$$ q + z * q^3 + (-2*z^3 - z) * q^7 - 2*z^2 * q^9 + (2*z^2 - 2) * q^11 - 4*z * q^17 - 2*z^2 * q^19 + (-3*z^2 + 2) * q^21 + (z^3 - z) * q^23 - 5*z^3 * q^27 - 9 * q^29 + (4*z^2 - 4) * q^31 + (2*z^3 - 2*z) * q^33 + (-4*z^3 + 4*z) * q^37 + q^41 - 9*z^3 * q^43 + (5*z^2 - 8) * q^49 - 4*z^2 * q^51 - 10*z * q^53 - 2*z^3 * q^57 + (10*z^2 - 10) * q^59 - 9*z^2 * q^61 + (6*z^3 - 4*z) * q^63 - 5*z * q^67 - q^69 + 14 * q^71 + 12*z * q^73 + (-2*z^3 + 6*z) * q^77 + 14*z^2 * q^79 + (z^2 - 1) * q^81 - 11*z^3 * q^83 - 9*z * q^87 - 15*z^2 * q^89 + (4*z^3 - 4*z) * q^93 - 18*z^3 * q^97 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} - 4 q^{11} - 4 q^{19} + 2 q^{21} - 36 q^{29} - 8 q^{31} + 4 q^{41} - 22 q^{49} - 8 q^{51} - 20 q^{59} - 18 q^{61} - 4 q^{69} + 56 q^{71} + 28 q^{79} - 2 q^{81} - 30 q^{89} + 16 q^{99}+O(q^{100})$$ 4 * q - 4 * q^9 - 4 * q^11 - 4 * q^19 + 2 * q^21 - 36 * q^29 - 8 * q^31 + 4 * q^41 - 22 * q^49 - 8 * q^51 - 20 * q^59 - 18 * q^61 - 4 * q^69 + 56 * q^71 + 28 * q^79 - 2 * q^81 - 30 * q^89 + 16 * q^99

## 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 −0.866025 0.500000i 0 0 0 0.866025 + 2.50000i 0 −1.00000 1.73205i 0
249.2 0 0.866025 + 0.500000i 0 0 0 −0.866025 2.50000i 0 −1.00000 1.73205i 0
849.1 0 −0.866025 + 0.500000i 0 0 0 0.866025 2.50000i 0 −1.00000 + 1.73205i 0
849.2 0 0.866025 0.500000i 0 0 0 −0.866025 + 2.50000i 0 −1.00000 + 1.73205i 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.b 4
5.b even 2 1 inner 1400.2.bh.b 4
5.c odd 4 1 280.2.q.a 2
5.c odd 4 1 1400.2.q.e 2
7.c even 3 1 inner 1400.2.bh.b 4
15.e even 4 1 2520.2.bi.a 2
20.e even 4 1 560.2.q.h 2
35.f even 4 1 1960.2.q.k 2
35.j even 6 1 inner 1400.2.bh.b 4
35.k even 12 1 1960.2.a.e 1
35.k even 12 1 1960.2.q.k 2
35.k even 12 1 9800.2.a.bc 1
35.l odd 12 1 280.2.q.a 2
35.l odd 12 1 1400.2.q.e 2
35.l odd 12 1 1960.2.a.i 1
35.l odd 12 1 9800.2.a.r 1
105.x even 12 1 2520.2.bi.a 2
140.w even 12 1 560.2.q.h 2
140.w even 12 1 3920.2.a.m 1
140.x odd 12 1 3920.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.a 2 5.c odd 4 1
280.2.q.a 2 35.l odd 12 1
560.2.q.h 2 20.e even 4 1
560.2.q.h 2 140.w even 12 1
1400.2.q.e 2 5.c odd 4 1
1400.2.q.e 2 35.l odd 12 1
1400.2.bh.b 4 1.a even 1 1 trivial
1400.2.bh.b 4 5.b even 2 1 inner
1400.2.bh.b 4 7.c even 3 1 inner
1400.2.bh.b 4 35.j even 6 1 inner
1960.2.a.e 1 35.k even 12 1
1960.2.a.i 1 35.l odd 12 1
1960.2.q.k 2 35.f even 4 1
1960.2.q.k 2 35.k even 12 1
2520.2.bi.a 2 15.e even 4 1
2520.2.bi.a 2 105.x even 12 1
3920.2.a.m 1 140.w even 12 1
3920.2.a.y 1 140.x odd 12 1
9800.2.a.r 1 35.l odd 12 1
9800.2.a.bc 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 $$S_{2}^{\mathrm{new}}(1400, [\chi])$$:

 $$T_{3}^{4} - T_{3}^{2} + 1$$ T3^4 - T3^2 + 1 $$T_{11}^{2} + 2T_{11} + 4$$ T11^2 + 2*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 11T^{2} + 49$$
$11$ $$(T^{2} + 2 T + 4)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4} - 16T^{2} + 256$$
$19$ $$(T^{2} + 2 T + 4)^{2}$$
$23$ $$T^{4} - T^{2} + 1$$
$29$ $$(T + 9)^{4}$$
$31$ $$(T^{2} + 4 T + 16)^{2}$$
$37$ $$T^{4} - 16T^{2} + 256$$
$41$ $$(T - 1)^{4}$$
$43$ $$(T^{2} + 81)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4} - 100 T^{2} + 10000$$
$59$ $$(T^{2} + 10 T + 100)^{2}$$
$61$ $$(T^{2} + 9 T + 81)^{2}$$
$67$ $$T^{4} - 25T^{2} + 625$$
$71$ $$(T - 14)^{4}$$
$73$ $$T^{4} - 144 T^{2} + 20736$$
$79$ $$(T^{2} - 14 T + 196)^{2}$$
$83$ $$(T^{2} + 121)^{2}$$
$89$ $$(T^{2} + 15 T + 225)^{2}$$
$97$ $$(T^{2} + 324)^{2}$$