# Properties

 Label 124.4.e.a Level $124$ Weight $4$ Character orbit 124.e Analytic conductor $7.316$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$124 = 2^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 124.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.31623684071$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - 5 \zeta_{6} + 5) q^{3} + 15 \zeta_{6} q^{5} + (29 \zeta_{6} - 29) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + (-5*z + 5) * q^3 + 15*z * q^5 + (29*z - 29) * q^7 + 2*z * q^9 $$q + ( - 5 \zeta_{6} + 5) q^{3} + 15 \zeta_{6} q^{5} + (29 \zeta_{6} - 29) q^{7} + 2 \zeta_{6} q^{9} + 51 \zeta_{6} q^{11} - 53 \zeta_{6} q^{13} + 75 q^{15} + (51 \zeta_{6} - 51) q^{17} + ( - 145 \zeta_{6} + 145) q^{19} + 145 \zeta_{6} q^{21} + 108 q^{23} + (100 \zeta_{6} - 100) q^{25} + 145 q^{27} - 198 q^{29} + (186 \zeta_{6} - 155) q^{31} + 255 q^{33} - 435 q^{35} + ( - 55 \zeta_{6} + 55) q^{37} - 265 q^{39} - 39 \zeta_{6} q^{41} + ( - 217 \zeta_{6} + 217) q^{43} + (30 \zeta_{6} - 30) q^{45} + 504 q^{47} - 498 \zeta_{6} q^{49} + 255 \zeta_{6} q^{51} - 237 \zeta_{6} q^{53} + (765 \zeta_{6} - 765) q^{55} - 725 \zeta_{6} q^{57} + ( - 597 \zeta_{6} + 597) q^{59} - 466 q^{61} - 58 q^{63} + ( - 795 \zeta_{6} + 795) q^{65} - 749 \zeta_{6} q^{67} + ( - 540 \zeta_{6} + 540) q^{69} + 321 \zeta_{6} q^{71} + 793 \zeta_{6} q^{73} + 500 \zeta_{6} q^{75} - 1479 q^{77} + (1217 \zeta_{6} - 1217) q^{79} + ( - 671 \zeta_{6} + 671) q^{81} + 555 \zeta_{6} q^{83} - 765 q^{85} + (990 \zeta_{6} - 990) q^{87} - 378 q^{89} + 1537 q^{91} + (775 \zeta_{6} + 155) q^{93} + 2175 q^{95} + 1070 q^{97} + (102 \zeta_{6} - 102) q^{99} +O(q^{100})$$ q + (-5*z + 5) * q^3 + 15*z * q^5 + (29*z - 29) * q^7 + 2*z * q^9 + 51*z * q^11 - 53*z * q^13 + 75 * q^15 + (51*z - 51) * q^17 + (-145*z + 145) * q^19 + 145*z * q^21 + 108 * q^23 + (100*z - 100) * q^25 + 145 * q^27 - 198 * q^29 + (186*z - 155) * q^31 + 255 * q^33 - 435 * q^35 + (-55*z + 55) * q^37 - 265 * q^39 - 39*z * q^41 + (-217*z + 217) * q^43 + (30*z - 30) * q^45 + 504 * q^47 - 498*z * q^49 + 255*z * q^51 - 237*z * q^53 + (765*z - 765) * q^55 - 725*z * q^57 + (-597*z + 597) * q^59 - 466 * q^61 - 58 * q^63 + (-795*z + 795) * q^65 - 749*z * q^67 + (-540*z + 540) * q^69 + 321*z * q^71 + 793*z * q^73 + 500*z * q^75 - 1479 * q^77 + (1217*z - 1217) * q^79 + (-671*z + 671) * q^81 + 555*z * q^83 - 765 * q^85 + (990*z - 990) * q^87 - 378 * q^89 + 1537 * q^91 + (775*z + 155) * q^93 + 2175 * q^95 + 1070 * q^97 + (102*z - 102) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{3} + 15 q^{5} - 29 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 5 * q^3 + 15 * q^5 - 29 * q^7 + 2 * q^9 $$2 q + 5 q^{3} + 15 q^{5} - 29 q^{7} + 2 q^{9} + 51 q^{11} - 53 q^{13} + 150 q^{15} - 51 q^{17} + 145 q^{19} + 145 q^{21} + 216 q^{23} - 100 q^{25} + 290 q^{27} - 396 q^{29} - 124 q^{31} + 510 q^{33} - 870 q^{35} + 55 q^{37} - 530 q^{39} - 39 q^{41} + 217 q^{43} - 30 q^{45} + 1008 q^{47} - 498 q^{49} + 255 q^{51} - 237 q^{53} - 765 q^{55} - 725 q^{57} + 597 q^{59} - 932 q^{61} - 116 q^{63} + 795 q^{65} - 749 q^{67} + 540 q^{69} + 321 q^{71} + 793 q^{73} + 500 q^{75} - 2958 q^{77} - 1217 q^{79} + 671 q^{81} + 555 q^{83} - 1530 q^{85} - 990 q^{87} - 756 q^{89} + 3074 q^{91} + 1085 q^{93} + 4350 q^{95} + 2140 q^{97} - 102 q^{99}+O(q^{100})$$ 2 * q + 5 * q^3 + 15 * q^5 - 29 * q^7 + 2 * q^9 + 51 * q^11 - 53 * q^13 + 150 * q^15 - 51 * q^17 + 145 * q^19 + 145 * q^21 + 216 * q^23 - 100 * q^25 + 290 * q^27 - 396 * q^29 - 124 * q^31 + 510 * q^33 - 870 * q^35 + 55 * q^37 - 530 * q^39 - 39 * q^41 + 217 * q^43 - 30 * q^45 + 1008 * q^47 - 498 * q^49 + 255 * q^51 - 237 * q^53 - 765 * q^55 - 725 * q^57 + 597 * q^59 - 932 * q^61 - 116 * q^63 + 795 * q^65 - 749 * q^67 + 540 * q^69 + 321 * q^71 + 793 * q^73 + 500 * q^75 - 2958 * q^77 - 1217 * q^79 + 671 * q^81 + 555 * q^83 - 1530 * q^85 - 990 * q^87 - 756 * q^89 + 3074 * q^91 + 1085 * q^93 + 4350 * q^95 + 2140 * q^97 - 102 * q^99

## Character values

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

 $$n$$ $$63$$ $$65$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 2.50000 4.33013i 0 7.50000 + 12.9904i 0 −14.5000 + 25.1147i 0 1.00000 + 1.73205i 0
25.1 0 2.50000 + 4.33013i 0 7.50000 12.9904i 0 −14.5000 25.1147i 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
31.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.4.e.a 2
3.b odd 2 1 1116.4.i.a 2
4.b odd 2 1 496.4.i.a 2
31.c even 3 1 inner 124.4.e.a 2
93.h odd 6 1 1116.4.i.a 2
124.i odd 6 1 496.4.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.e.a 2 1.a even 1 1 trivial
124.4.e.a 2 31.c even 3 1 inner
496.4.i.a 2 4.b odd 2 1
496.4.i.a 2 124.i odd 6 1
1116.4.i.a 2 3.b odd 2 1
1116.4.i.a 2 93.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 5T_{3} + 25$$ acting on $$S_{4}^{\mathrm{new}}(124, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 5T + 25$$
$5$ $$T^{2} - 15T + 225$$
$7$ $$T^{2} + 29T + 841$$
$11$ $$T^{2} - 51T + 2601$$
$13$ $$T^{2} + 53T + 2809$$
$17$ $$T^{2} + 51T + 2601$$
$19$ $$T^{2} - 145T + 21025$$
$23$ $$(T - 108)^{2}$$
$29$ $$(T + 198)^{2}$$
$31$ $$T^{2} + 124T + 29791$$
$37$ $$T^{2} - 55T + 3025$$
$41$ $$T^{2} + 39T + 1521$$
$43$ $$T^{2} - 217T + 47089$$
$47$ $$(T - 504)^{2}$$
$53$ $$T^{2} + 237T + 56169$$
$59$ $$T^{2} - 597T + 356409$$
$61$ $$(T + 466)^{2}$$
$67$ $$T^{2} + 749T + 561001$$
$71$ $$T^{2} - 321T + 103041$$
$73$ $$T^{2} - 793T + 628849$$
$79$ $$T^{2} + 1217 T + 1481089$$
$83$ $$T^{2} - 555T + 308025$$
$89$ $$(T + 378)^{2}$$
$97$ $$(T - 1070)^{2}$$