# Properties

 Label 1444.2.e.c Level $1444$ Weight $2$ Character orbit 1444.e Analytic conductor $11.530$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1444 = 2^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1444.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.5303980519$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{3} + ( - \zeta_{6} + 1) q^{5} - 3 q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-2*z + 2) * q^3 + (-z + 1) * q^5 - 3 * q^7 - z * q^9 $$q + ( - 2 \zeta_{6} + 2) q^{3} + ( - \zeta_{6} + 1) q^{5} - 3 q^{7} - \zeta_{6} q^{9} + 5 q^{11} - 4 \zeta_{6} q^{13} - 2 \zeta_{6} q^{15} + ( - 3 \zeta_{6} + 3) q^{17} + (6 \zeta_{6} - 6) q^{21} - 8 \zeta_{6} q^{23} + 4 \zeta_{6} q^{25} + 4 q^{27} - 2 \zeta_{6} q^{29} - 4 q^{31} + ( - 10 \zeta_{6} + 10) q^{33} + (3 \zeta_{6} - 3) q^{35} - 10 q^{37} - 8 q^{39} + ( - 10 \zeta_{6} + 10) q^{41} + (\zeta_{6} - 1) q^{43} - q^{45} + \zeta_{6} q^{47} + 2 q^{49} - 6 \zeta_{6} q^{51} - 4 \zeta_{6} q^{53} + ( - 5 \zeta_{6} + 5) q^{55} + ( - 6 \zeta_{6} + 6) q^{59} + 13 \zeta_{6} q^{61} + 3 \zeta_{6} q^{63} - 4 q^{65} - 12 \zeta_{6} q^{67} - 16 q^{69} + ( - 2 \zeta_{6} + 2) q^{71} + (9 \zeta_{6} - 9) q^{73} + 8 q^{75} - 15 q^{77} + ( - 8 \zeta_{6} + 8) q^{79} + ( - 11 \zeta_{6} + 11) q^{81} - 12 q^{83} - 3 \zeta_{6} q^{85} - 4 q^{87} + 12 \zeta_{6} q^{89} + 12 \zeta_{6} q^{91} + (8 \zeta_{6} - 8) q^{93} + (8 \zeta_{6} - 8) q^{97} - 5 \zeta_{6} q^{99} +O(q^{100})$$ q + (-2*z + 2) * q^3 + (-z + 1) * q^5 - 3 * q^7 - z * q^9 + 5 * q^11 - 4*z * q^13 - 2*z * q^15 + (-3*z + 3) * q^17 + (6*z - 6) * q^21 - 8*z * q^23 + 4*z * q^25 + 4 * q^27 - 2*z * q^29 - 4 * q^31 + (-10*z + 10) * q^33 + (3*z - 3) * q^35 - 10 * q^37 - 8 * q^39 + (-10*z + 10) * q^41 + (z - 1) * q^43 - q^45 + z * q^47 + 2 * q^49 - 6*z * q^51 - 4*z * q^53 + (-5*z + 5) * q^55 + (-6*z + 6) * q^59 + 13*z * q^61 + 3*z * q^63 - 4 * q^65 - 12*z * q^67 - 16 * q^69 + (-2*z + 2) * q^71 + (9*z - 9) * q^73 + 8 * q^75 - 15 * q^77 + (-8*z + 8) * q^79 + (-11*z + 11) * q^81 - 12 * q^83 - 3*z * q^85 - 4 * q^87 + 12*z * q^89 + 12*z * q^91 + (8*z - 8) * q^93 + (8*z - 8) * q^97 - 5*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + q^{5} - 6 q^{7} - q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + q^5 - 6 * q^7 - q^9 $$2 q + 2 q^{3} + q^{5} - 6 q^{7} - q^{9} + 10 q^{11} - 4 q^{13} - 2 q^{15} + 3 q^{17} - 6 q^{21} - 8 q^{23} + 4 q^{25} + 8 q^{27} - 2 q^{29} - 8 q^{31} + 10 q^{33} - 3 q^{35} - 20 q^{37} - 16 q^{39} + 10 q^{41} - q^{43} - 2 q^{45} + q^{47} + 4 q^{49} - 6 q^{51} - 4 q^{53} + 5 q^{55} + 6 q^{59} + 13 q^{61} + 3 q^{63} - 8 q^{65} - 12 q^{67} - 32 q^{69} + 2 q^{71} - 9 q^{73} + 16 q^{75} - 30 q^{77} + 8 q^{79} + 11 q^{81} - 24 q^{83} - 3 q^{85} - 8 q^{87} + 12 q^{89} + 12 q^{91} - 8 q^{93} - 8 q^{97} - 5 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + q^5 - 6 * q^7 - q^9 + 10 * q^11 - 4 * q^13 - 2 * q^15 + 3 * q^17 - 6 * q^21 - 8 * q^23 + 4 * q^25 + 8 * q^27 - 2 * q^29 - 8 * q^31 + 10 * q^33 - 3 * q^35 - 20 * q^37 - 16 * q^39 + 10 * q^41 - q^43 - 2 * q^45 + q^47 + 4 * q^49 - 6 * q^51 - 4 * q^53 + 5 * q^55 + 6 * q^59 + 13 * q^61 + 3 * q^63 - 8 * q^65 - 12 * q^67 - 32 * q^69 + 2 * q^71 - 9 * q^73 + 16 * q^75 - 30 * q^77 + 8 * q^79 + 11 * q^81 - 24 * q^83 - 3 * q^85 - 8 * q^87 + 12 * q^89 + 12 * q^91 - 8 * q^93 - 8 * q^97 - 5 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times$$.

 $$n$$ $$723$$ $$1085$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
429.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.00000 1.73205i 0 0.500000 0.866025i 0 −3.00000 0 −0.500000 0.866025i 0
653.1 0 1.00000 + 1.73205i 0 0.500000 + 0.866025i 0 −3.00000 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1444.2.e.c 2
19.b odd 2 1 1444.2.e.a 2
19.c even 3 1 1444.2.a.a 1
19.c even 3 1 inner 1444.2.e.c 2
19.d odd 6 1 76.2.a.a 1
19.d odd 6 1 1444.2.e.a 2
57.f even 6 1 684.2.a.b 1
76.f even 6 1 304.2.a.a 1
76.g odd 6 1 5776.2.a.p 1
95.h odd 6 1 1900.2.a.b 1
95.l even 12 2 1900.2.c.b 2
133.p even 6 1 3724.2.a.a 1
152.l odd 6 1 1216.2.a.c 1
152.o even 6 1 1216.2.a.q 1
209.g even 6 1 9196.2.a.f 1
228.n odd 6 1 2736.2.a.q 1
380.s even 6 1 7600.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.a.a 1 19.d odd 6 1
304.2.a.a 1 76.f even 6 1
684.2.a.b 1 57.f even 6 1
1216.2.a.c 1 152.l odd 6 1
1216.2.a.q 1 152.o even 6 1
1444.2.a.a 1 19.c even 3 1
1444.2.e.a 2 19.b odd 2 1
1444.2.e.a 2 19.d odd 6 1
1444.2.e.c 2 1.a even 1 1 trivial
1444.2.e.c 2 19.c even 3 1 inner
1900.2.a.b 1 95.h odd 6 1
1900.2.c.b 2 95.l even 12 2
2736.2.a.q 1 228.n odd 6 1
3724.2.a.a 1 133.p even 6 1
5776.2.a.p 1 76.g odd 6 1
7600.2.a.p 1 380.s even 6 1
9196.2.a.f 1 209.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 2T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1444, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T + 4$$
$5$ $$T^{2} - T + 1$$
$7$ $$(T + 3)^{2}$$
$11$ $$(T - 5)^{2}$$
$13$ $$T^{2} + 4T + 16$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 8T + 64$$
$29$ $$T^{2} + 2T + 4$$
$31$ $$(T + 4)^{2}$$
$37$ $$(T + 10)^{2}$$
$41$ $$T^{2} - 10T + 100$$
$43$ $$T^{2} + T + 1$$
$47$ $$T^{2} - T + 1$$
$53$ $$T^{2} + 4T + 16$$
$59$ $$T^{2} - 6T + 36$$
$61$ $$T^{2} - 13T + 169$$
$67$ $$T^{2} + 12T + 144$$
$71$ $$T^{2} - 2T + 4$$
$73$ $$T^{2} + 9T + 81$$
$79$ $$T^{2} - 8T + 64$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} - 12T + 144$$
$97$ $$T^{2} + 8T + 64$$