# Properties

 Label 124.1.i.a Level $124$ Weight $1$ Character orbit 124.i Analytic conductor $0.062$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

This is the first weight $1$ newform whose projective image is not a dihedral group (it is $A_4$).

## Newspace parameters

 Level: $$N$$ $$=$$ $$124 = 2^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 124.i (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.0618840615665$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$A_{4}$$ Projective field: Galois closure of 4.0.15376.1 Artin image: $\SL(2,3):C_2$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{12}^{3} q^{2} - \zeta_{12} q^{3} - q^{4} - \zeta_{12}^{2} q^{5} + \zeta_{12}^{4} q^{6} + \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ q - z^3 * q^2 - z * q^3 - q^4 - z^2 * q^5 + z^4 * q^6 + z * q^7 + z^3 * q^8 $$q - \zeta_{12}^{3} q^{2} - \zeta_{12} q^{3} - q^{4} - \zeta_{12}^{2} q^{5} + \zeta_{12}^{4} q^{6} + \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{5} q^{10} - \zeta_{12}^{5} q^{11} + \zeta_{12} q^{12} + \zeta_{12}^{2} q^{13} - \zeta_{12}^{4} q^{14} + \zeta_{12}^{3} q^{15} + q^{16} + \zeta_{12}^{4} q^{17} - \zeta_{12} q^{19} + \zeta_{12}^{2} q^{20} - \zeta_{12}^{2} q^{21} - \zeta_{12}^{2} q^{22} - \zeta_{12}^{4} q^{24} - \zeta_{12}^{5} q^{26} + \zeta_{12}^{3} q^{27} - \zeta_{12} q^{28} + q^{30} + \zeta_{12}^{3} q^{31} - \zeta_{12}^{3} q^{32} - q^{33} + \zeta_{12} q^{34} - \zeta_{12}^{3} q^{35} + \zeta_{12}^{4} q^{37} + \zeta_{12}^{4} q^{38} - \zeta_{12}^{3} q^{39} - \zeta_{12}^{5} q^{40} - \zeta_{12}^{2} q^{41} + \zeta_{12}^{5} q^{42} - \zeta_{12} q^{43} + \zeta_{12}^{5} q^{44} - \zeta_{12} q^{48} - \zeta_{12}^{5} q^{51} - \zeta_{12}^{2} q^{52} + \zeta_{12}^{2} q^{53} + q^{54} - \zeta_{12} q^{55} + \zeta_{12}^{4} q^{56} + \zeta_{12}^{2} q^{57} + \zeta_{12} q^{59} - \zeta_{12}^{3} q^{60} + q^{62} - q^{64} - \zeta_{12}^{4} q^{65} + \zeta_{12}^{3} q^{66} - \zeta_{12}^{5} q^{67} - \zeta_{12}^{4} q^{68} - q^{70} + \zeta_{12}^{5} q^{71} - \zeta_{12}^{2} q^{73} + \zeta_{12} q^{74} + \zeta_{12} q^{76} + q^{77} - q^{78} + \zeta_{12} q^{79} - \zeta_{12}^{2} q^{80} - \zeta_{12}^{4} q^{81} + \zeta_{12}^{5} q^{82} + \zeta_{12}^{5} q^{83} + \zeta_{12}^{2} q^{84} + q^{85} + \zeta_{12}^{4} q^{86} + \zeta_{12}^{2} q^{88} + \zeta_{12}^{3} q^{91} - \zeta_{12}^{4} q^{93} + \zeta_{12}^{3} q^{95} + \zeta_{12}^{4} q^{96} +O(q^{100})$$ q - z^3 * q^2 - z * q^3 - q^4 - z^2 * q^5 + z^4 * q^6 + z * q^7 + z^3 * q^8 + z^5 * q^10 - z^5 * q^11 + z * q^12 + z^2 * q^13 - z^4 * q^14 + z^3 * q^15 + q^16 + z^4 * q^17 - z * q^19 + z^2 * q^20 - z^2 * q^21 - z^2 * q^22 - z^4 * q^24 - z^5 * q^26 + z^3 * q^27 - z * q^28 + q^30 + z^3 * q^31 - z^3 * q^32 - q^33 + z * q^34 - z^3 * q^35 + z^4 * q^37 + z^4 * q^38 - z^3 * q^39 - z^5 * q^40 - z^2 * q^41 + z^5 * q^42 - z * q^43 + z^5 * q^44 - z * q^48 - z^5 * q^51 - z^2 * q^52 + z^2 * q^53 + q^54 - z * q^55 + z^4 * q^56 + z^2 * q^57 + z * q^59 - z^3 * q^60 + q^62 - q^64 - z^4 * q^65 + z^3 * q^66 - z^5 * q^67 - z^4 * q^68 - q^70 + z^5 * q^71 - z^2 * q^73 + z * q^74 + z * q^76 + q^77 - q^78 + z * q^79 - z^2 * q^80 - z^4 * q^81 + z^5 * q^82 + z^5 * q^83 + z^2 * q^84 + q^85 + z^4 * q^86 + z^2 * q^88 + z^3 * q^91 - z^4 * q^93 + z^3 * q^95 + z^4 * q^96 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 2 q^{5} - 2 q^{6}+O(q^{10})$$ 4 * q - 4 * q^4 - 2 * q^5 - 2 * q^6 $$4 q - 4 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{13} + 2 q^{14} + 4 q^{16} - 2 q^{17} + 2 q^{20} - 2 q^{21} - 2 q^{22} + 2 q^{24} + 4 q^{30} - 4 q^{33} - 2 q^{37} - 2 q^{38} - 2 q^{41} - 2 q^{52} + 2 q^{53} + 4 q^{54} - 2 q^{56} + 2 q^{57} + 4 q^{62} - 4 q^{64} + 2 q^{65} + 2 q^{68} - 4 q^{70} - 2 q^{73} + 4 q^{77} - 4 q^{78} - 2 q^{80} + 2 q^{81} + 2 q^{84} + 4 q^{85} - 2 q^{86} + 2 q^{88} + 2 q^{93} - 2 q^{96}+O(q^{100})$$ 4 * q - 4 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^13 + 2 * q^14 + 4 * q^16 - 2 * q^17 + 2 * q^20 - 2 * q^21 - 2 * q^22 + 2 * q^24 + 4 * q^30 - 4 * q^33 - 2 * q^37 - 2 * q^38 - 2 * q^41 - 2 * q^52 + 2 * q^53 + 4 * q^54 - 2 * q^56 + 2 * q^57 + 4 * q^62 - 4 * q^64 + 2 * q^65 + 2 * q^68 - 4 * q^70 - 2 * q^73 + 4 * q^77 - 4 * q^78 - 2 * q^80 + 2 * q^81 + 2 * q^84 + 4 * q^85 - 2 * q^86 + 2 * q^88 + 2 * q^93 - 2 * q^96

## Character values

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

 $$n$$ $$63$$ $$65$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
1.00000i −0.866025 0.500000i −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i 0.866025 + 0.500000i 1.00000i 0 −0.866025 + 0.500000i
67.2 1.00000i 0.866025 + 0.500000i −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i −0.866025 0.500000i 1.00000i 0 0.866025 0.500000i
87.1 1.00000i 0.866025 0.500000i −1.00000 −0.500000 + 0.866025i −0.500000 0.866025i −0.866025 + 0.500000i 1.00000i 0 0.866025 + 0.500000i
87.2 1.00000i −0.866025 + 0.500000i −1.00000 −0.500000 + 0.866025i −0.500000 0.866025i 0.866025 0.500000i 1.00000i 0 −0.866025 0.500000i
 $$n$$: 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
4.b odd 2 1 inner
31.c even 3 1 inner
124.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.1.i.a 4
3.b odd 2 1 1116.1.x.a 4
4.b odd 2 1 inner 124.1.i.a 4
5.b even 2 1 3100.1.z.a 4
5.c odd 4 1 3100.1.t.a 4
5.c odd 4 1 3100.1.t.b 4
8.b even 2 1 1984.1.s.a 4
8.d odd 2 1 1984.1.s.a 4
12.b even 2 1 1116.1.x.a 4
20.d odd 2 1 3100.1.z.a 4
20.e even 4 1 3100.1.t.a 4
20.e even 4 1 3100.1.t.b 4
31.b odd 2 1 3844.1.i.d 4
31.c even 3 1 inner 124.1.i.a 4
31.c even 3 1 3844.1.b.d 2
31.d even 5 4 3844.1.n.e 16
31.e odd 6 1 3844.1.b.c 2
31.e odd 6 1 3844.1.i.d 4
31.f odd 10 4 3844.1.n.f 16
31.g even 15 4 3844.1.l.d 8
31.g even 15 4 3844.1.n.e 16
31.h odd 30 4 3844.1.l.c 8
31.h odd 30 4 3844.1.n.f 16
93.h odd 6 1 1116.1.x.a 4
124.d even 2 1 3844.1.i.d 4
124.g even 6 1 3844.1.b.c 2
124.g even 6 1 3844.1.i.d 4
124.i odd 6 1 inner 124.1.i.a 4
124.i odd 6 1 3844.1.b.d 2
124.j even 10 4 3844.1.n.f 16
124.l odd 10 4 3844.1.n.e 16
124.n odd 30 4 3844.1.l.d 8
124.n odd 30 4 3844.1.n.e 16
124.p even 30 4 3844.1.l.c 8
124.p even 30 4 3844.1.n.f 16
155.j even 6 1 3100.1.z.a 4
155.o odd 12 1 3100.1.t.a 4
155.o odd 12 1 3100.1.t.b 4
248.m odd 6 1 1984.1.s.a 4
248.p even 6 1 1984.1.s.a 4
372.p even 6 1 1116.1.x.a 4
620.o odd 6 1 3100.1.z.a 4
620.be even 12 1 3100.1.t.a 4
620.be even 12 1 3100.1.t.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.1.i.a 4 1.a even 1 1 trivial
124.1.i.a 4 4.b odd 2 1 inner
124.1.i.a 4 31.c even 3 1 inner
124.1.i.a 4 124.i odd 6 1 inner
1116.1.x.a 4 3.b odd 2 1
1116.1.x.a 4 12.b even 2 1
1116.1.x.a 4 93.h odd 6 1
1116.1.x.a 4 372.p even 6 1
1984.1.s.a 4 8.b even 2 1
1984.1.s.a 4 8.d odd 2 1
1984.1.s.a 4 248.m odd 6 1
1984.1.s.a 4 248.p even 6 1
3100.1.t.a 4 5.c odd 4 1
3100.1.t.a 4 20.e even 4 1
3100.1.t.a 4 155.o odd 12 1
3100.1.t.a 4 620.be even 12 1
3100.1.t.b 4 5.c odd 4 1
3100.1.t.b 4 20.e even 4 1
3100.1.t.b 4 155.o odd 12 1
3100.1.t.b 4 620.be even 12 1
3100.1.z.a 4 5.b even 2 1
3100.1.z.a 4 20.d odd 2 1
3100.1.z.a 4 155.j even 6 1
3100.1.z.a 4 620.o odd 6 1
3844.1.b.c 2 31.e odd 6 1
3844.1.b.c 2 124.g even 6 1
3844.1.b.d 2 31.c even 3 1
3844.1.b.d 2 124.i odd 6 1
3844.1.i.d 4 31.b odd 2 1
3844.1.i.d 4 31.e odd 6 1
3844.1.i.d 4 124.d even 2 1
3844.1.i.d 4 124.g even 6 1
3844.1.l.c 8 31.h odd 30 4
3844.1.l.c 8 124.p even 30 4
3844.1.l.d 8 31.g even 15 4
3844.1.l.d 8 124.n odd 30 4
3844.1.n.e 16 31.d even 5 4
3844.1.n.e 16 31.g even 15 4
3844.1.n.e 16 124.l odd 10 4
3844.1.n.e 16 124.n odd 30 4
3844.1.n.f 16 31.f odd 10 4
3844.1.n.f 16 31.h odd 30 4
3844.1.n.f 16 124.j even 10 4
3844.1.n.f 16 124.p even 30 4

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(124, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4} - T^{2} + 1$$
$11$ $$T^{4} - T^{2} + 1$$
$13$ $$(T^{2} - T + 1)^{2}$$
$17$ $$(T^{2} + T + 1)^{2}$$
$19$ $$T^{4} - T^{2} + 1$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$(T^{2} + 1)^{2}$$
$37$ $$(T^{2} + T + 1)^{2}$$
$41$ $$(T^{2} + T + 1)^{2}$$
$43$ $$T^{4} - T^{2} + 1$$
$47$ $$T^{4}$$
$53$ $$(T^{2} - T + 1)^{2}$$
$59$ $$T^{4} - T^{2} + 1$$
$61$ $$T^{4}$$
$67$ $$T^{4} - T^{2} + 1$$
$71$ $$T^{4} - T^{2} + 1$$
$73$ $$(T^{2} + T + 1)^{2}$$
$79$ $$T^{4} - T^{2} + 1$$
$83$ $$T^{4} - T^{2} + 1$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$