# Properties

 Label 124.3.c.a Level $124$ Weight $3$ Character orbit 124.c Analytic conductor $3.379$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$124 = 2^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 124.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.37875527807$$ 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: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} - 6 q^{5} - 10 q^{7} - 3 q^{9} +O(q^{10})$$ q - b * q^3 - 6 * q^5 - 10 * q^7 - 3 * q^9 $$q - \beta q^{3} - 6 q^{5} - 10 q^{7} - 3 q^{9} + 5 \beta q^{11} - 5 \beta q^{13} + 6 \beta q^{15} - 4 \beta q^{17} - 14 q^{19} + 10 \beta q^{21} - 10 \beta q^{23} + 11 q^{25} - 6 \beta q^{27} - 5 \beta q^{29} - 31 q^{31} + 60 q^{33} + 60 q^{35} + \beta q^{37} - 60 q^{39} + 54 q^{41} + 21 \beta q^{43} + 18 q^{45} + 30 q^{47} + 51 q^{49} - 48 q^{51} + 9 \beta q^{53} - 30 \beta q^{55} + 14 \beta q^{57} - 6 q^{59} - 5 \beta q^{61} + 30 q^{63} + 30 \beta q^{65} - 110 q^{67} - 120 q^{69} + 66 q^{71} - 26 \beta q^{73} - 11 \beta q^{75} - 50 \beta q^{77} - 20 \beta q^{79} - 99 q^{81} + 11 \beta q^{83} + 24 \beta q^{85} - 60 q^{87} + 10 \beta q^{89} + 50 \beta q^{91} + 31 \beta q^{93} + 84 q^{95} + 110 q^{97} - 15 \beta q^{99} +O(q^{100})$$ q - b * q^3 - 6 * q^5 - 10 * q^7 - 3 * q^9 + 5*b * q^11 - 5*b * q^13 + 6*b * q^15 - 4*b * q^17 - 14 * q^19 + 10*b * q^21 - 10*b * q^23 + 11 * q^25 - 6*b * q^27 - 5*b * q^29 - 31 * q^31 + 60 * q^33 + 60 * q^35 + b * q^37 - 60 * q^39 + 54 * q^41 + 21*b * q^43 + 18 * q^45 + 30 * q^47 + 51 * q^49 - 48 * q^51 + 9*b * q^53 - 30*b * q^55 + 14*b * q^57 - 6 * q^59 - 5*b * q^61 + 30 * q^63 + 30*b * q^65 - 110 * q^67 - 120 * q^69 + 66 * q^71 - 26*b * q^73 - 11*b * q^75 - 50*b * q^77 - 20*b * q^79 - 99 * q^81 + 11*b * q^83 + 24*b * q^85 - 60 * q^87 + 10*b * q^89 + 50*b * q^91 + 31*b * q^93 + 84 * q^95 + 110 * q^97 - 15*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 12 q^{5} - 20 q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q - 12 * q^5 - 20 * q^7 - 6 * q^9 $$2 q - 12 q^{5} - 20 q^{7} - 6 q^{9} - 28 q^{19} + 22 q^{25} - 62 q^{31} + 120 q^{33} + 120 q^{35} - 120 q^{39} + 108 q^{41} + 36 q^{45} + 60 q^{47} + 102 q^{49} - 96 q^{51} - 12 q^{59} + 60 q^{63} - 220 q^{67} - 240 q^{69} + 132 q^{71} - 198 q^{81} - 120 q^{87} + 168 q^{95} + 220 q^{97}+O(q^{100})$$ 2 * q - 12 * q^5 - 20 * q^7 - 6 * q^9 - 28 * q^19 + 22 * q^25 - 62 * q^31 + 120 * q^33 + 120 * q^35 - 120 * q^39 + 108 * q^41 + 36 * q^45 + 60 * q^47 + 102 * q^49 - 96 * q^51 - 12 * q^59 + 60 * q^63 - 220 * q^67 - 240 * q^69 + 132 * q^71 - 198 * q^81 - 120 * q^87 + 168 * q^95 + 220 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/124\mathbb{Z}\right)^\times$$.

 $$n$$ $$63$$ $$65$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 3.46410i 0 −6.00000 0 −10.0000 0 −3.00000 0
61.2 0 3.46410i 0 −6.00000 0 −10.0000 0 −3.00000 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
31.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.3.c.a 2
3.b odd 2 1 1116.3.h.c 2
4.b odd 2 1 496.3.e.a 2
5.b even 2 1 3100.3.d.a 2
5.c odd 4 2 3100.3.f.a 4
31.b odd 2 1 inner 124.3.c.a 2
93.c even 2 1 1116.3.h.c 2
124.d even 2 1 496.3.e.a 2
155.c odd 2 1 3100.3.d.a 2
155.f even 4 2 3100.3.f.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.3.c.a 2 1.a even 1 1 trivial
124.3.c.a 2 31.b odd 2 1 inner
496.3.e.a 2 4.b odd 2 1
496.3.e.a 2 124.d even 2 1
1116.3.h.c 2 3.b odd 2 1
1116.3.h.c 2 93.c even 2 1
3100.3.d.a 2 5.b even 2 1
3100.3.d.a 2 155.c odd 2 1
3100.3.f.a 4 5.c odd 4 2
3100.3.f.a 4 155.f even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 12$$ acting on $$S_{3}^{\mathrm{new}}(124, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 12$$
$5$ $$(T + 6)^{2}$$
$7$ $$(T + 10)^{2}$$
$11$ $$T^{2} + 300$$
$13$ $$T^{2} + 300$$
$17$ $$T^{2} + 192$$
$19$ $$(T + 14)^{2}$$
$23$ $$T^{2} + 1200$$
$29$ $$T^{2} + 300$$
$31$ $$(T + 31)^{2}$$
$37$ $$T^{2} + 12$$
$41$ $$(T - 54)^{2}$$
$43$ $$T^{2} + 5292$$
$47$ $$(T - 30)^{2}$$
$53$ $$T^{2} + 972$$
$59$ $$(T + 6)^{2}$$
$61$ $$T^{2} + 300$$
$67$ $$(T + 110)^{2}$$
$71$ $$(T - 66)^{2}$$
$73$ $$T^{2} + 8112$$
$79$ $$T^{2} + 4800$$
$83$ $$T^{2} + 1452$$
$89$ $$T^{2} + 1200$$
$97$ $$(T - 110)^{2}$$