Properties

 Label 124.2.f.b 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 1) q^{3} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 1) q^{5} + (2 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 2 \zeta_{10}) q^{7} - 2 \zeta_{10}^{3} q^{9} +O(q^{10})$$ q + (-z^3 - z^2 + z + 1) * q^3 + (-z^3 + z^2 - 1) * q^5 + (2*z^3 - 3*z^2 + 2*z) * q^7 - 2*z^3 * q^9 $$q + ( - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 1) q^{3} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 1) q^{5} + (2 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 2 \zeta_{10}) q^{7} - 2 \zeta_{10}^{3} q^{9} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{11} + (4 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 4) q^{13} + ( - \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 1) q^{15} + ( - \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{17} + (6 \zeta_{10}^{2} - 3 \zeta_{10} + 6) q^{19} + ( - 4 \zeta_{10}^{2} + 7 \zeta_{10} - 4) q^{21} + (3 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{23} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 3) q^{25} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + 2 \zeta_{10}) q^{27} + ( - \zeta_{10}^{2} - 4 \zeta_{10} - 1) q^{29} + (2 \zeta_{10}^{3} + 3 \zeta_{10} - 6) q^{31} + (2 \zeta_{10}^{2} - 6 \zeta_{10} + 2) q^{33} + ( - 5 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 5 \zeta_{10}) q^{35} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 1) q^{37} + (6 \zeta_{10}^{3} + 7 \zeta_{10} - 7) q^{39} + ( - 4 \zeta_{10}^{2} - 4 \zeta_{10} - 4) q^{41} + (3 \zeta_{10}^{2} - 6 \zeta_{10} + 3) q^{43} + (2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{45} + ( - 3 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} + 3) q^{47} + ( - 2 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} + 2) q^{49} - 5 \zeta_{10}^{2} q^{51} + (3 \zeta_{10}^{3} + 10 \zeta_{10} - 10) q^{53} + (4 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 4 \zeta_{10}) q^{55} + 15 q^{57} + ( - 9 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 9) q^{59} + 2 q^{61} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 2) q^{63} + ( - 3 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 3) q^{65} + 3 q^{67} + (8 \zeta_{10}^{3} - 9 \zeta_{10}^{2} + 8 \zeta_{10}) q^{69} + ( - 10 \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{71} + ( - 3 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 3 \zeta_{10}) q^{73} + (9 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 9) q^{75} + ( - 4 \zeta_{10}^{3} + 10 \zeta_{10}^{2} - 10 \zeta_{10} + 4) q^{77} + 11 \zeta_{10} q^{81} + ( - 3 \zeta_{10}^{2} - 3 \zeta_{10} - 3) q^{83} + ( - \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{85} + (9 \zeta_{10}^{3} - 9 \zeta_{10}^{2} - 7) q^{87} + ( - 13 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 13 \zeta_{10}) q^{89} + ( - 5 \zeta_{10}^{2} + 7 \zeta_{10} - 5) q^{91} + (4 \zeta_{10}^{3} + 10 \zeta_{10}^{2} - 2 \zeta_{10} - 3) q^{93} + ( - 9 \zeta_{10}^{2} + 12 \zeta_{10} - 9) q^{95} + (6 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 6 \zeta_{10}) q^{97} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2}) q^{99} +O(q^{100})$$ q + (-z^3 - z^2 + z + 1) * q^3 + (-z^3 + z^2 - 1) * q^5 + (2*z^3 - 3*z^2 + 2*z) * q^7 - 2*z^3 * q^9 + (-2*z^3 + 2*z^2 - 2*z) * q^11 + (4*z^3 - 3*z^2 + 3*z - 4) * q^13 + (-z^3 + 4*z^2 - 4*z + 1) * q^15 + (-z^3 - 2*z + 2) * q^17 + (6*z^2 - 3*z + 6) * q^19 + (-4*z^2 + 7*z - 4) * q^21 + (3*z^3 - 2*z + 2) * q^23 + (3*z^3 - 3*z^2 - 3) * q^25 + (2*z^3 - z^2 + 2*z) * q^27 + (-z^2 - 4*z - 1) * q^29 + (2*z^3 + 3*z - 6) * q^31 + (2*z^2 - 6*z + 2) * q^33 + (-5*z^3 + 8*z^2 - 5*z) * q^35 + (-6*z^3 + 6*z^2 + 1) * q^37 + (6*z^3 + 7*z - 7) * q^39 + (-4*z^2 - 4*z - 4) * q^41 + (3*z^2 - 6*z + 3) * q^43 + (2*z^3 - 2*z + 2) * q^45 + (-3*z^3 - 6*z^2 + 6*z + 3) * q^47 + (-2*z^3 - 6*z^2 + 6*z + 2) * q^49 - 5*z^2 * q^51 + (3*z^3 + 10*z - 10) * q^53 + (4*z^3 - 6*z^2 + 4*z) * q^55 + 15 * q^57 + (-9*z^3 + 3*z^2 - 3*z + 9) * q^59 + 2 * q^61 + (-4*z^3 + 4*z^2 - 2) * q^63 + (-3*z^3 + 5*z^2 - 5*z + 3) * q^65 + 3 * q^67 + (8*z^3 - 9*z^2 + 8*z) * q^69 + (-10*z^3 - 5*z + 5) * q^71 + (-3*z^3 + 5*z^2 - 3*z) * q^73 + (9*z^3 - 6*z^2 + 6*z - 9) * q^75 + (-4*z^3 + 10*z^2 - 10*z + 4) * q^77 + 11*z * q^81 + (-3*z^2 - 3*z - 3) * q^83 + (-z^3 + 3*z - 3) * q^85 + (9*z^3 - 9*z^2 - 7) * q^87 + (-13*z^3 + 5*z^2 - 13*z) * q^89 + (-5*z^2 + 7*z - 5) * q^91 + (4*z^3 + 10*z^2 - 2*z - 3) * q^93 + (-9*z^2 + 12*z - 9) * q^95 + (6*z^3 - 5*z^2 + 6*z) * q^97 + (4*z^3 - 4*z^2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 5 q^{3} - 6 q^{5} + 7 q^{7} - 2 q^{9}+O(q^{10})$$ 4 * q + 5 * q^3 - 6 * q^5 + 7 * q^7 - 2 * q^9 $$4 q + 5 q^{3} - 6 q^{5} + 7 q^{7} - 2 q^{9} - 6 q^{11} - 6 q^{13} - 5 q^{15} + 5 q^{17} + 15 q^{19} - 5 q^{21} + 9 q^{23} - 6 q^{25} + 5 q^{27} - 7 q^{29} - 19 q^{31} - 18 q^{35} - 8 q^{37} - 15 q^{39} - 16 q^{41} + 3 q^{43} + 8 q^{45} + 21 q^{47} + 18 q^{49} + 5 q^{51} - 27 q^{53} + 14 q^{55} + 60 q^{57} + 21 q^{59} + 8 q^{61} - 16 q^{63} - q^{65} + 12 q^{67} + 25 q^{69} + 5 q^{71} - 11 q^{73} - 15 q^{75} - 8 q^{77} + 11 q^{81} - 12 q^{83} - 10 q^{85} - 10 q^{87} - 31 q^{89} - 8 q^{91} - 20 q^{93} - 15 q^{95} + 17 q^{97} + 8 q^{99}+O(q^{100})$$ 4 * q + 5 * q^3 - 6 * q^5 + 7 * q^7 - 2 * q^9 - 6 * q^11 - 6 * q^13 - 5 * q^15 + 5 * q^17 + 15 * q^19 - 5 * q^21 + 9 * q^23 - 6 * q^25 + 5 * q^27 - 7 * q^29 - 19 * q^31 - 18 * q^35 - 8 * q^37 - 15 * q^39 - 16 * q^41 + 3 * q^43 + 8 * q^45 + 21 * q^47 + 18 * q^49 + 5 * q^51 - 27 * q^53 + 14 * q^55 + 60 * q^57 + 21 * q^59 + 8 * q^61 - 16 * q^63 - q^65 + 12 * q^67 + 25 * q^69 + 5 * q^71 - 11 * q^73 - 15 * q^75 - 8 * q^77 + 11 * q^81 - 12 * q^83 - 10 * q^85 - 10 * q^87 - 31 * q^89 - 8 * q^91 - 20 * q^93 - 15 * q^95 + 17 * q^97 + 8 * 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 1.80902 1.31433i 0 −0.381966 0 0.0729490 + 0.224514i 0 0.618034 1.90211i 0
97.1 0 0.690983 2.12663i 0 −2.61803 0 3.42705 2.48990i 0 −1.61803 1.17557i 0
101.1 0 0.690983 + 2.12663i 0 −2.61803 0 3.42705 + 2.48990i 0 −1.61803 + 1.17557i 0
109.1 0 1.80902 + 1.31433i 0 −0.381966 0 0.0729490 0.224514i 0 0.618034 + 1.90211i 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.b 4
3.b odd 2 1 1116.2.m.c 4
4.b odd 2 1 496.2.n.a 4
31.d even 5 1 inner 124.2.f.b 4
31.d even 5 1 3844.2.a.g 2
31.f odd 10 1 3844.2.a.f 2
93.l odd 10 1 1116.2.m.c 4
124.l odd 10 1 496.2.n.a 4

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 5T_{3}^{3} + 15T_{3}^{2} - 25T_{3} + 25$$ acting on $$S_{2}^{\mathrm{new}}(124, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 5 T^{3} + 15 T^{2} - 25 T + 25$$
$5$ $$(T^{2} + 3 T + 1)^{2}$$
$7$ $$T^{4} - 7 T^{3} + 19 T^{2} - 3 T + 1$$
$11$ $$T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16$$
$13$ $$T^{4} + 6 T^{3} + 16 T^{2} + 11 T + 121$$
$17$ $$T^{4} - 5 T^{3} + 15 T^{2} - 25 T + 25$$
$19$ $$T^{4} - 15 T^{3} + 135 T^{2} + \cdots + 2025$$
$23$ $$T^{4} - 9 T^{3} + 31 T^{2} + 11 T + 121$$
$29$ $$T^{4} + 7 T^{3} + 24 T^{2} + 38 T + 361$$
$31$ $$T^{4} + 19 T^{3} + 151 T^{2} + \cdots + 961$$
$37$ $$(T^{2} + 4 T - 41)^{2}$$
$41$ $$T^{4} + 16 T^{3} + 96 T^{2} + \cdots + 256$$
$43$ $$T^{4} - 3 T^{3} + 54 T^{2} + 108 T + 81$$
$47$ $$T^{4} - 21 T^{3} + 306 T^{2} + \cdots + 9801$$
$53$ $$T^{4} + 27 T^{3} + 379 T^{2} + \cdots + 14641$$
$59$ $$T^{4} - 21 T^{3} + 171 T^{2} + \cdots + 81$$
$61$ $$(T - 2)^{4}$$
$67$ $$(T - 3)^{4}$$
$71$ $$T^{4} - 5 T^{3} + 150 T^{2} + \cdots + 625$$
$73$ $$T^{4} + 11 T^{3} + 46 T^{2} - 4 T + 1$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 12 T^{3} + 54 T^{2} - 27 T + 81$$
$89$ $$T^{4} + 31 T^{3} + 636 T^{2} + \cdots + 43681$$
$97$ $$T^{4} - 17 T^{3} + 139 T^{2} + \cdots + 1681$$