# Properties

 Label 1400.2.q.a 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: $$\Z[a_1, \ldots, a_{9}]$$ 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{3} + (2 \zeta_{6} - 3) q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (2*z - 2) * q^3 + (2*z - 3) * q^7 - z * q^9 $$q + (2 \zeta_{6} - 2) q^{3} + (2 \zeta_{6} - 3) q^{7} - \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{11} + 3 q^{13} + (2 \zeta_{6} - 2) q^{17} + 5 \zeta_{6} q^{19} + ( - 6 \zeta_{6} + 2) q^{21} + 7 \zeta_{6} q^{23} - 4 q^{27} - 6 q^{29} + (4 \zeta_{6} - 4) q^{31} + 2 \zeta_{6} q^{33} - 5 \zeta_{6} q^{37} + (6 \zeta_{6} - 6) q^{39} - 5 q^{41} - 6 q^{43} - 9 \zeta_{6} q^{47} + ( - 8 \zeta_{6} + 5) q^{49} - 4 \zeta_{6} q^{51} + ( - 11 \zeta_{6} + 11) q^{53} - 10 q^{57} + (8 \zeta_{6} - 8) q^{59} + 12 \zeta_{6} q^{61} + (\zeta_{6} + 2) q^{63} + (4 \zeta_{6} - 4) q^{67} - 14 q^{69} - 4 q^{71} + ( - 12 \zeta_{6} + 12) q^{73} + (3 \zeta_{6} - 1) q^{77} - 14 \zeta_{6} q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + 4 q^{83} + ( - 12 \zeta_{6} + 12) q^{87} - 6 \zeta_{6} q^{89} + (6 \zeta_{6} - 9) q^{91} - 8 \zeta_{6} q^{93} - 6 q^{97} - q^{99} +O(q^{100})$$ q + (2*z - 2) * q^3 + (2*z - 3) * q^7 - z * q^9 + (-z + 1) * q^11 + 3 * q^13 + (2*z - 2) * q^17 + 5*z * q^19 + (-6*z + 2) * q^21 + 7*z * q^23 - 4 * q^27 - 6 * q^29 + (4*z - 4) * q^31 + 2*z * q^33 - 5*z * q^37 + (6*z - 6) * q^39 - 5 * q^41 - 6 * q^43 - 9*z * q^47 + (-8*z + 5) * q^49 - 4*z * q^51 + (-11*z + 11) * q^53 - 10 * q^57 + (8*z - 8) * q^59 + 12*z * q^61 + (z + 2) * q^63 + (4*z - 4) * q^67 - 14 * q^69 - 4 * q^71 + (-12*z + 12) * q^73 + (3*z - 1) * q^77 - 14*z * q^79 + (-11*z + 11) * q^81 + 4 * q^83 + (-12*z + 12) * q^87 - 6*z * q^89 + (6*z - 9) * q^91 - 8*z * q^93 - 6 * q^97 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 4 q^{7} - q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 4 * q^7 - q^9 $$2 q - 2 q^{3} - 4 q^{7} - q^{9} + q^{11} + 6 q^{13} - 2 q^{17} + 5 q^{19} - 2 q^{21} + 7 q^{23} - 8 q^{27} - 12 q^{29} - 4 q^{31} + 2 q^{33} - 5 q^{37} - 6 q^{39} - 10 q^{41} - 12 q^{43} - 9 q^{47} + 2 q^{49} - 4 q^{51} + 11 q^{53} - 20 q^{57} - 8 q^{59} + 12 q^{61} + 5 q^{63} - 4 q^{67} - 28 q^{69} - 8 q^{71} + 12 q^{73} + q^{77} - 14 q^{79} + 11 q^{81} + 8 q^{83} + 12 q^{87} - 6 q^{89} - 12 q^{91} - 8 q^{93} - 12 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 4 * q^7 - q^9 + q^11 + 6 * q^13 - 2 * q^17 + 5 * q^19 - 2 * q^21 + 7 * q^23 - 8 * q^27 - 12 * q^29 - 4 * q^31 + 2 * q^33 - 5 * q^37 - 6 * q^39 - 10 * q^41 - 12 * q^43 - 9 * q^47 + 2 * q^49 - 4 * q^51 + 11 * q^53 - 20 * q^57 - 8 * q^59 + 12 * q^61 + 5 * q^63 - 4 * q^67 - 28 * q^69 - 8 * q^71 + 12 * q^73 + q^77 - 14 * q^79 + 11 * q^81 + 8 * q^83 + 12 * q^87 - 6 * q^89 - 12 * q^91 - 8 * q^93 - 12 * q^97 - 2 * 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 −1.00000 1.73205i 0 0 0 −2.00000 1.73205i 0 −0.500000 + 0.866025i 0
1201.1 0 −1.00000 + 1.73205i 0 0 0 −2.00000 + 1.73205i 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
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.a 2
5.b even 2 1 280.2.q.c 2
5.c odd 4 2 1400.2.bh.e 4
7.c even 3 1 inner 1400.2.q.a 2
7.c even 3 1 9800.2.a.bi 1
7.d odd 6 1 9800.2.a.g 1
15.d odd 2 1 2520.2.bi.e 2
20.d odd 2 1 560.2.q.c 2
35.c odd 2 1 1960.2.q.c 2
35.i odd 6 1 1960.2.a.m 1
35.i odd 6 1 1960.2.q.c 2
35.j even 6 1 280.2.q.c 2
35.j even 6 1 1960.2.a.a 1
35.l odd 12 2 1400.2.bh.e 4
105.o odd 6 1 2520.2.bi.e 2
140.p odd 6 1 560.2.q.c 2
140.p odd 6 1 3920.2.a.bf 1
140.s even 6 1 3920.2.a.i 1

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4T + 7$$
$11$ $$T^{2} - T + 1$$
$13$ $$(T - 3)^{2}$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2} - 5T + 25$$
$23$ $$T^{2} - 7T + 49$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$T^{2} + 5T + 25$$
$41$ $$(T + 5)^{2}$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} + 9T + 81$$
$53$ $$T^{2} - 11T + 121$$
$59$ $$T^{2} + 8T + 64$$
$61$ $$T^{2} - 12T + 144$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$(T + 4)^{2}$$
$73$ $$T^{2} - 12T + 144$$
$79$ $$T^{2} + 14T + 196$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$(T + 6)^{2}$$