# Properties

 Label 1400.2.q.c Level $1400$ Weight $2$ Character orbit 1400.q Analytic conductor $11.179$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 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: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{3} + ( - 3 \zeta_{6} + 1) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + (z - 1) * q^3 + (-3*z + 1) * q^7 + 2*z * q^9 $$q + (\zeta_{6} - 1) q^{3} + ( - 3 \zeta_{6} + 1) q^{7} + 2 \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} - 4 q^{13} - 6 \zeta_{6} q^{19} + (\zeta_{6} + 2) q^{21} + 3 \zeta_{6} q^{23} - 5 q^{27} - 3 q^{29} + 2 \zeta_{6} q^{33} - 12 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{39} - 7 q^{41} + 9 q^{43} + (3 \zeta_{6} - 8) q^{49} + (6 \zeta_{6} - 6) q^{53} + 6 q^{57} + ( - 10 \zeta_{6} + 10) q^{59} - 5 \zeta_{6} q^{61} + ( - 4 \zeta_{6} + 6) q^{63} + ( - 11 \zeta_{6} + 11) q^{67} - 3 q^{69} - 10 q^{71} + (8 \zeta_{6} - 8) q^{73} + ( - 2 \zeta_{6} - 4) q^{77} - 6 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 3 q^{83} + ( - 3 \zeta_{6} + 3) q^{87} - 17 \zeta_{6} q^{89} + (12 \zeta_{6} - 4) q^{91} + 2 q^{97} + 4 q^{99} +O(q^{100})$$ q + (z - 1) * q^3 + (-3*z + 1) * q^7 + 2*z * q^9 + (-2*z + 2) * q^11 - 4 * q^13 - 6*z * q^19 + (z + 2) * q^21 + 3*z * q^23 - 5 * q^27 - 3 * q^29 + 2*z * q^33 - 12*z * q^37 + (-4*z + 4) * q^39 - 7 * q^41 + 9 * q^43 + (3*z - 8) * q^49 + (6*z - 6) * q^53 + 6 * q^57 + (-10*z + 10) * q^59 - 5*z * q^61 + (-4*z + 6) * q^63 + (-11*z + 11) * q^67 - 3 * q^69 - 10 * q^71 + (8*z - 8) * q^73 + (-2*z - 4) * q^77 - 6*z * q^79 + (z - 1) * q^81 + 3 * q^83 + (-3*z + 3) * q^87 - 17*z * q^89 + (12*z - 4) * q^91 + 2 * q^97 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - q^3 - q^7 + 2 * q^9 $$2 q - q^{3} - q^{7} + 2 q^{9} + 2 q^{11} - 8 q^{13} - 6 q^{19} + 5 q^{21} + 3 q^{23} - 10 q^{27} - 6 q^{29} + 2 q^{33} - 12 q^{37} + 4 q^{39} - 14 q^{41} + 18 q^{43} - 13 q^{49} - 6 q^{53} + 12 q^{57} + 10 q^{59} - 5 q^{61} + 8 q^{63} + 11 q^{67} - 6 q^{69} - 20 q^{71} - 8 q^{73} - 10 q^{77} - 6 q^{79} - q^{81} + 6 q^{83} + 3 q^{87} - 17 q^{89} + 4 q^{91} + 4 q^{97} + 8 q^{99}+O(q^{100})$$ 2 * q - q^3 - q^7 + 2 * q^9 + 2 * q^11 - 8 * q^13 - 6 * q^19 + 5 * q^21 + 3 * q^23 - 10 * q^27 - 6 * q^29 + 2 * q^33 - 12 * q^37 + 4 * q^39 - 14 * q^41 + 18 * q^43 - 13 * q^49 - 6 * q^53 + 12 * q^57 + 10 * q^59 - 5 * q^61 + 8 * q^63 + 11 * q^67 - 6 * q^69 - 20 * q^71 - 8 * q^73 - 10 * q^77 - 6 * q^79 - q^81 + 6 * q^83 + 3 * q^87 - 17 * q^89 + 4 * q^91 + 4 * q^97 + 8 * q^99

## Character values

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

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
0 −0.500000 0.866025i 0 0 0 −0.500000 + 2.59808i 0 1.00000 1.73205i 0
1201.1 0 −0.500000 + 0.866025i 0 0 0 −0.500000 2.59808i 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
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.c 2
5.b even 2 1 280.2.q.b 2
5.c odd 4 2 1400.2.bh.c 4
7.c even 3 1 inner 1400.2.q.c 2
7.c even 3 1 9800.2.a.z 1
7.d odd 6 1 9800.2.a.o 1
15.d odd 2 1 2520.2.bi.d 2
20.d odd 2 1 560.2.q.e 2
35.c odd 2 1 1960.2.q.d 2
35.i odd 6 1 1960.2.a.l 1
35.i odd 6 1 1960.2.q.d 2
35.j even 6 1 280.2.q.b 2
35.j even 6 1 1960.2.a.c 1
35.l odd 12 2 1400.2.bh.c 4
105.o odd 6 1 2520.2.bi.d 2
140.p odd 6 1 560.2.q.e 2
140.p odd 6 1 3920.2.a.v 1
140.s even 6 1 3920.2.a.q 1

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T + 7$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$(T + 4)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 6T + 36$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$(T + 3)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 12T + 144$$
$41$ $$(T + 7)^{2}$$
$43$ $$(T - 9)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} - 10T + 100$$
$61$ $$T^{2} + 5T + 25$$
$67$ $$T^{2} - 11T + 121$$
$71$ $$(T + 10)^{2}$$
$73$ $$T^{2} + 8T + 64$$
$79$ $$T^{2} + 6T + 36$$
$83$ $$(T - 3)^{2}$$
$89$ $$T^{2} + 17T + 289$$
$97$ $$(T - 2)^{2}$$