# Properties

 Label 124.2.f.a Level $124$ Weight $2$ Character orbit 124.f Analytic conductor $0.990$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$124 = 2^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 124.f (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.990144985064$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{10}^{3} - 1) q^{3} + 2 q^{5} + (\zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10}) q^{7} + (\zeta_{10}^{3} - \zeta_{10} + 1) q^{9}+O(q^{10})$$ q + (z^3 - 1) * q^3 + 2 * q^5 + (z^3 + z^2 + z) * q^7 + (z^3 - z + 1) * q^9 $$q + (\zeta_{10}^{3} - 1) q^{3} + 2 q^{5} + (\zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10}) q^{7} + (\zeta_{10}^{3} - \zeta_{10} + 1) q^{9} + (\zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10}) q^{11} + ( - \zeta_{10}^{3} + 1) q^{13} + (2 \zeta_{10}^{3} - 2) q^{15} + ( - 4 \zeta_{10}^{3} - \zeta_{10} + 1) q^{17} + ( - 3 \zeta_{10}^{2} - 3) q^{19} + ( - 2 \zeta_{10}^{2} - \zeta_{10} - 2) q^{21} + ( - 6 \zeta_{10}^{3} + \zeta_{10} - 1) q^{23} - q^{25} + ( - 4 \zeta_{10}^{3} + \zeta_{10}^{2} - 4 \zeta_{10}) q^{27} + ( - 3 \zeta_{10}^{2} + 4 \zeta_{10} - 3) q^{29} + (6 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{31} + ( - 2 \zeta_{10}^{2} - \zeta_{10} - 2) q^{33} + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{35} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 2) q^{37} + (2 \zeta_{10}^{3} + \zeta_{10} - 1) q^{39} + (5 \zeta_{10}^{2} + 4 \zeta_{10} + 5) q^{41} + ( - 3 \zeta_{10}^{2} - 4 \zeta_{10} - 3) q^{43} + (2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{45} + (\zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 1) q^{47} + ( - 2 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 2) q^{49} + (4 \zeta_{10}^{3} + \zeta_{10}^{2} + 4 \zeta_{10}) q^{51} + ( - 8 \zeta_{10}^{3} - \zeta_{10} + 1) q^{53} + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{55} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 6) q^{57} + ( - 7 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 7) q^{59} + (12 \zeta_{10}^{3} - 12 \zeta_{10}^{2} - 6) q^{61} - q^{63} + ( - 2 \zeta_{10}^{3} + 2) q^{65} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4) q^{67} + (6 \zeta_{10}^{3} - \zeta_{10}^{2} + 6 \zeta_{10}) q^{69} + (6 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{71} + (\zeta_{10}^{3} + 11 \zeta_{10}^{2} + \zeta_{10}) q^{73} + ( - \zeta_{10}^{3} + 1) q^{75} + (5 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 5) q^{77} + ( - 2 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{79} + (6 \zeta_{10}^{2} - 2 \zeta_{10} + 6) q^{81} + ( - 3 \zeta_{10}^{2} + 8 \zeta_{10} - 3) q^{83} + ( - 8 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{85} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 2) q^{87} + ( - 7 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 7 \zeta_{10}) q^{89} + (2 \zeta_{10}^{2} + \zeta_{10} + 2) q^{91} + (\zeta_{10}^{3} - 4 \zeta_{10}^{2} - 7) q^{93} + ( - 6 \zeta_{10}^{2} - 6) q^{95} + (\zeta_{10}^{3} + 15 \zeta_{10}^{2} + \zeta_{10}) q^{97} - q^{99}+O(q^{100})$$ q + (z^3 - 1) * q^3 + 2 * q^5 + (z^3 + z^2 + z) * q^7 + (z^3 - z + 1) * q^9 + (z^3 + z^2 + z) * q^11 + (-z^3 + 1) * q^13 + (2*z^3 - 2) * q^15 + (-4*z^3 - z + 1) * q^17 + (-3*z^2 - 3) * q^19 + (-2*z^2 - z - 2) * q^21 + (-6*z^3 + z - 1) * q^23 - q^25 + (-4*z^3 + z^2 - 4*z) * q^27 + (-3*z^2 + 4*z - 3) * q^29 + (6*z^2 - 2*z + 3) * q^31 + (-2*z^2 - z - 2) * q^33 + (2*z^3 + 2*z^2 + 2*z) * q^35 + (4*z^3 - 4*z^2 - 2) * q^37 + (2*z^3 + z - 1) * q^39 + (5*z^2 + 4*z + 5) * q^41 + (-3*z^2 - 4*z - 3) * q^43 + (2*z^3 - 2*z + 2) * q^45 + (z^3 - 4*z^2 + 4*z - 1) * q^47 + (-2*z^3 + 5*z^2 - 5*z + 2) * q^49 + (4*z^3 + z^2 + 4*z) * q^51 + (-8*z^3 - z + 1) * q^53 + (2*z^3 + 2*z^2 + 2*z) * q^55 + (-3*z^3 + 3*z^2 + 6) * q^57 + (-7*z^3 + 4*z^2 - 4*z + 7) * q^59 + (12*z^3 - 12*z^2 - 6) * q^61 - q^63 + (-2*z^3 + 2) * q^65 + (4*z^3 - 4*z^2 + 4) * q^67 + (6*z^3 - z^2 + 6*z) * q^69 + (6*z^3 - 3*z + 3) * q^71 + (z^3 + 11*z^2 + z) * q^73 + (-z^3 + 1) * q^75 + (5*z^3 - 2*z^2 + 2*z - 5) * q^77 + (-2*z^3 - 3*z + 3) * q^79 + (6*z^2 - 2*z + 6) * q^81 + (-3*z^2 + 8*z - 3) * q^83 + (-8*z^3 - 2*z + 2) * q^85 + (z^3 - z^2 + 2) * q^87 + (-7*z^3 + 3*z^2 - 7*z) * q^89 + (2*z^2 + z + 2) * q^91 + (z^3 - 4*z^2 - 7) * q^93 + (-6*z^2 - 6) * q^95 + (z^3 + 15*z^2 + z) * q^97 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{3} + 8 q^{5} + q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q - 3 * q^3 + 8 * q^5 + q^7 + 4 * q^9 $$4 q - 3 q^{3} + 8 q^{5} + q^{7} + 4 q^{9} + q^{11} + 3 q^{13} - 6 q^{15} - q^{17} - 9 q^{19} - 7 q^{21} - 9 q^{23} - 4 q^{25} - 9 q^{27} - 5 q^{29} + 4 q^{31} - 7 q^{33} + 2 q^{35} - q^{39} + 19 q^{41} - 13 q^{43} + 8 q^{45} + 5 q^{47} - 4 q^{49} + 7 q^{51} - 5 q^{53} + 2 q^{55} + 18 q^{57} + 13 q^{59} - 4 q^{63} + 6 q^{65} + 24 q^{67} + 13 q^{69} + 15 q^{71} - 9 q^{73} + 3 q^{75} - 11 q^{77} + 7 q^{79} + 16 q^{81} - q^{83} - 2 q^{85} + 10 q^{87} - 17 q^{89} + 7 q^{91} - 23 q^{93} - 18 q^{95} - 13 q^{97} - 4 q^{99}+O(q^{100})$$ 4 * q - 3 * q^3 + 8 * q^5 + q^7 + 4 * q^9 + q^11 + 3 * q^13 - 6 * q^15 - q^17 - 9 * q^19 - 7 * q^21 - 9 * q^23 - 4 * q^25 - 9 * q^27 - 5 * q^29 + 4 * q^31 - 7 * q^33 + 2 * q^35 - q^39 + 19 * q^41 - 13 * q^43 + 8 * q^45 + 5 * q^47 - 4 * q^49 + 7 * q^51 - 5 * q^53 + 2 * q^55 + 18 * q^57 + 13 * q^59 - 4 * q^63 + 6 * q^65 + 24 * q^67 + 13 * q^69 + 15 * q^71 - 9 * q^73 + 3 * q^75 - 11 * q^77 + 7 * q^79 + 16 * q^81 - q^83 - 2 * q^85 + 10 * q^87 - 17 * q^89 + 7 * q^91 - 23 * q^93 - 18 * q^95 - 13 * q^97 - 4 * q^99

## Character values

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

 $$n$$ $$63$$ $$65$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 0.809017 + 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 − 0.587785i
0 −1.30902 + 0.951057i 0 2.00000 0 0.809017 + 2.48990i 0 −0.118034 + 0.363271i 0
97.1 0 −0.190983 + 0.587785i 0 2.00000 0 −0.309017 + 0.224514i 0 2.11803 + 1.53884i 0
101.1 0 −0.190983 0.587785i 0 2.00000 0 −0.309017 0.224514i 0 2.11803 1.53884i 0
109.1 0 −1.30902 0.951057i 0 2.00000 0 0.809017 2.48990i 0 −0.118034 0.363271i 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.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.2.f.a 4
3.b odd 2 1 1116.2.m.a 4
4.b odd 2 1 496.2.n.c 4
31.d even 5 1 inner 124.2.f.a 4
31.d even 5 1 3844.2.a.h 2
31.f odd 10 1 3844.2.a.e 2
93.l odd 10 1 1116.2.m.a 4
124.l odd 10 1 496.2.n.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.2.f.a 4 1.a even 1 1 trivial
124.2.f.a 4 31.d even 5 1 inner
496.2.n.c 4 4.b odd 2 1
496.2.n.c 4 124.l odd 10 1
1116.2.m.a 4 3.b odd 2 1
1116.2.m.a 4 93.l odd 10 1
3844.2.a.e 2 31.f odd 10 1
3844.2.a.h 2 31.d even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 3T_{3}^{3} + 4T_{3}^{2} + 2T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(124, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1$$
$5$ $$(T - 2)^{4}$$
$7$ $$T^{4} - T^{3} + 6 T^{2} + 4 T + 1$$
$11$ $$T^{4} - T^{3} + 6 T^{2} + 4 T + 1$$
$13$ $$T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1$$
$17$ $$T^{4} + T^{3} + 16 T^{2} + 66 T + 121$$
$19$ $$T^{4} + 9 T^{3} + 36 T^{2} + 54 T + 81$$
$23$ $$T^{4} + 9 T^{3} + 46 T^{2} + \cdots + 1681$$
$29$ $$T^{4} + 5 T^{3} + 40 T^{2} + 50 T + 25$$
$31$ $$T^{4} - 4 T^{3} + 46 T^{2} - 124 T + 961$$
$37$ $$(T^{2} - 20)^{2}$$
$41$ $$T^{4} - 19 T^{3} + 136 T^{2} + \cdots + 121$$
$43$ $$T^{4} + 13 T^{3} + 64 T^{2} + \cdots + 361$$
$47$ $$T^{4} - 5 T^{3} + 40 T^{2} - 50 T + 25$$
$53$ $$T^{4} + 5 T^{3} + 60 T^{2} + \cdots + 3025$$
$59$ $$T^{4} - 13 T^{3} + 64 T^{2} + \cdots + 361$$
$61$ $$(T^{2} - 180)^{2}$$
$67$ $$(T^{2} - 12 T + 16)^{2}$$
$71$ $$T^{4} - 15 T^{3} + 90 T^{2} + \cdots + 2025$$
$73$ $$T^{4} + 9 T^{3} + 136 T^{2} + \cdots + 17161$$
$79$ $$T^{4} - 7 T^{3} + 34 T^{2} - 88 T + 121$$
$83$ $$T^{4} + T^{3} + 76 T^{2} - 434 T + 961$$
$89$ $$T^{4} + 17 T^{3} + 184 T^{2} + \cdots + 3721$$
$97$ $$T^{4} + 13 T^{3} + 244 T^{2} + \cdots + 57121$$