# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.990144985064$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{6}, \sqrt{-7})$$ Defining polynomial: $$x^{4} - 2x^{3} - 7x^{2} + 8x + 58$$ x^4 - 2*x^3 - 7*x^2 + 8*x + 58 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + \beta_{3} q^{3} + ( - \beta_1 - 2) q^{4} - 2 q^{5} + (\beta_{3} + \beta_{2}) q^{6} + (\beta_1 - 2) q^{8} + 3 q^{9}+O(q^{10})$$ q - b1 * q^2 + b3 * q^3 + (-b1 - 2) * q^4 - 2 * q^5 + (b3 + b2) * q^6 + (b1 - 2) * q^8 + 3 * q^9 $$q - \beta_1 q^{2} + \beta_{3} q^{3} + ( - \beta_1 - 2) q^{4} - 2 q^{5} + (\beta_{3} + \beta_{2}) q^{6} + (\beta_1 - 2) q^{8} + 3 q^{9} + 2 \beta_1 q^{10} + \beta_{3} q^{11} + ( - \beta_{3} + \beta_{2}) q^{12} + ( - \beta_{3} - 2 \beta_{2}) q^{13} - 2 \beta_{3} q^{15} + (3 \beta_1 + 2) q^{16} - 3 \beta_1 q^{18} + ( - 4 \beta_1 - 2) q^{19} + (2 \beta_1 + 4) q^{20} + (\beta_{3} + \beta_{2}) q^{22} - 2 \beta_{3} q^{23} + ( - 3 \beta_{3} - \beta_{2}) q^{24} - q^{25} + (3 \beta_{3} - \beta_{2}) q^{26} + (\beta_{3} + 2 \beta_{2}) q^{29} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{30} + (2 \beta_{3} + 2 \beta_1 + 1) q^{31} + (\beta_1 + 6) q^{32} + 6 q^{33} + ( - 3 \beta_1 - 6) q^{36} + ( - \beta_{3} - 2 \beta_{2}) q^{37} + ( - 2 \beta_1 - 8) q^{38} + (12 \beta_1 + 6) q^{39} + ( - 2 \beta_1 + 4) q^{40} - 8 q^{41} + \beta_{3} q^{43} + ( - \beta_{3} + \beta_{2}) q^{44} - 6 q^{45} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{46} + ( - 8 \beta_1 - 4) q^{47} + ( - \beta_{3} - 3 \beta_{2}) q^{48} + 7 q^{49} + \beta_1 q^{50} + (5 \beta_{3} + 3 \beta_{2}) q^{52} + (\beta_{3} + 2 \beta_{2}) q^{53} - 2 \beta_{3} q^{55} + (2 \beta_{3} + 4 \beta_{2}) q^{57} + ( - 3 \beta_{3} + \beta_{2}) q^{58} + (4 \beta_1 + 2) q^{59} + (2 \beta_{3} - 2 \beta_{2}) q^{60} + (\beta_{3} + 2 \beta_{2}) q^{61} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{62} + ( - 5 \beta_1 + 2) q^{64} + (2 \beta_{3} + 4 \beta_{2}) q^{65} - 6 \beta_1 q^{66} + ( - 4 \beta_1 - 2) q^{67} - 12 q^{69} + ( - 4 \beta_1 - 2) q^{71} + (3 \beta_1 - 6) q^{72} + ( - 2 \beta_{3} - 4 \beta_{2}) q^{73} + (3 \beta_{3} - \beta_{2}) q^{74} - \beta_{3} q^{75} + (6 \beta_1 - 4) q^{76} + (6 \beta_1 + 24) q^{78} - 4 \beta_{3} q^{79} + ( - 6 \beta_1 - 4) q^{80} - 9 q^{81} + 8 \beta_1 q^{82} + 5 \beta_{3} q^{83} + (\beta_{3} + \beta_{2}) q^{86} + ( - 12 \beta_1 - 6) q^{87} + ( - 3 \beta_{3} - \beta_{2}) q^{88} + ( - 2 \beta_{3} - 4 \beta_{2}) q^{89} + 6 \beta_1 q^{90} + (2 \beta_{3} - 2 \beta_{2}) q^{92} + ( - \beta_{3} - 2 \beta_{2} + 12) q^{93} + ( - 4 \beta_1 - 16) q^{94} + (8 \beta_1 + 4) q^{95} + (5 \beta_{3} - \beta_{2}) q^{96} + 4 q^{97} - 7 \beta_1 q^{98} + 3 \beta_{3} q^{99}+O(q^{100})$$ q - b1 * q^2 + b3 * q^3 + (-b1 - 2) * q^4 - 2 * q^5 + (b3 + b2) * q^6 + (b1 - 2) * q^8 + 3 * q^9 + 2*b1 * q^10 + b3 * q^11 + (-b3 + b2) * q^12 + (-b3 - 2*b2) * q^13 - 2*b3 * q^15 + (3*b1 + 2) * q^16 - 3*b1 * q^18 + (-4*b1 - 2) * q^19 + (2*b1 + 4) * q^20 + (b3 + b2) * q^22 - 2*b3 * q^23 + (-3*b3 - b2) * q^24 - q^25 + (3*b3 - b2) * q^26 + (b3 + 2*b2) * q^29 + (-2*b3 - 2*b2) * q^30 + (2*b3 + 2*b1 + 1) * q^31 + (b1 + 6) * q^32 + 6 * q^33 + (-3*b1 - 6) * q^36 + (-b3 - 2*b2) * q^37 + (-2*b1 - 8) * q^38 + (12*b1 + 6) * q^39 + (-2*b1 + 4) * q^40 - 8 * q^41 + b3 * q^43 + (-b3 + b2) * q^44 - 6 * q^45 + (-2*b3 - 2*b2) * q^46 + (-8*b1 - 4) * q^47 + (-b3 - 3*b2) * q^48 + 7 * q^49 + b1 * q^50 + (5*b3 + 3*b2) * q^52 + (b3 + 2*b2) * q^53 - 2*b3 * q^55 + (2*b3 + 4*b2) * q^57 + (-3*b3 + b2) * q^58 + (4*b1 + 2) * q^59 + (2*b3 - 2*b2) * q^60 + (b3 + 2*b2) * q^61 + (2*b3 + 2*b2 + b1 + 4) * q^62 + (-5*b1 + 2) * q^64 + (2*b3 + 4*b2) * q^65 - 6*b1 * q^66 + (-4*b1 - 2) * q^67 - 12 * q^69 + (-4*b1 - 2) * q^71 + (3*b1 - 6) * q^72 + (-2*b3 - 4*b2) * q^73 + (3*b3 - b2) * q^74 - b3 * q^75 + (6*b1 - 4) * q^76 + (6*b1 + 24) * q^78 - 4*b3 * q^79 + (-6*b1 - 4) * q^80 - 9 * q^81 + 8*b1 * q^82 + 5*b3 * q^83 + (b3 + b2) * q^86 + (-12*b1 - 6) * q^87 + (-3*b3 - b2) * q^88 + (-2*b3 - 4*b2) * q^89 + 6*b1 * q^90 + (2*b3 - 2*b2) * q^92 + (-b3 - 2*b2 + 12) * q^93 + (-4*b1 - 16) * q^94 + (8*b1 + 4) * q^95 + (5*b3 - b2) * q^96 + 4 * q^97 - 7*b1 * q^98 + 3*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 6 q^{4} - 8 q^{5} - 10 q^{8} + 12 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 6 * q^4 - 8 * q^5 - 10 * q^8 + 12 * q^9 $$4 q + 2 q^{2} - 6 q^{4} - 8 q^{5} - 10 q^{8} + 12 q^{9} - 4 q^{10} + 2 q^{16} + 6 q^{18} + 12 q^{20} - 4 q^{25} + 22 q^{32} + 24 q^{33} - 18 q^{36} - 28 q^{38} + 20 q^{40} - 32 q^{41} - 24 q^{45} + 28 q^{49} - 2 q^{50} + 14 q^{62} + 18 q^{64} + 12 q^{66} - 48 q^{69} - 30 q^{72} - 28 q^{76} + 84 q^{78} - 4 q^{80} - 36 q^{81} - 16 q^{82} - 12 q^{90} + 48 q^{93} - 56 q^{94} + 16 q^{97} + 14 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 - 6 * q^4 - 8 * q^5 - 10 * q^8 + 12 * q^9 - 4 * q^10 + 2 * q^16 + 6 * q^18 + 12 * q^20 - 4 * q^25 + 22 * q^32 + 24 * q^33 - 18 * q^36 - 28 * q^38 + 20 * q^40 - 32 * q^41 - 24 * q^45 + 28 * q^49 - 2 * q^50 + 14 * q^62 + 18 * q^64 + 12 * q^66 - 48 * q^69 - 30 * q^72 - 28 * q^76 + 84 * q^78 - 4 * q^80 - 36 * q^81 - 16 * q^82 - 12 * q^90 + 48 * q^93 - 56 * q^94 + 16 * q^97 + 14 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 7x^{2} + 8x + 58$$ :

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{3} + 3\nu^{2} - 16 ) / 31$$ (-2*v^3 + 3*v^2 - 16) / 31 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 14\nu^{2} - 31\nu - 54 ) / 31$$ (v^3 + 14*v^2 - 31*v - 54) / 31 $$\beta_{3}$$ $$=$$ $$( -2\nu^{3} + 3\nu^{2} + 31\nu - 16 ) / 31$$ (-2*v^3 + 3*v^2 + 31*v - 16) / 31
 $$\nu$$ $$=$$ $$\beta_{3} - \beta_1$$ b3 - b1 $$\nu^{2}$$ $$=$$ $$2\beta_{3} + 2\beta_{2} - \beta _1 + 4$$ 2*b3 + 2*b2 - b1 + 4 $$\nu^{3}$$ $$=$$ $$3\beta_{3} + 3\beta_{2} - 17\beta _1 - 2$$ 3*b3 + 3*b2 - 17*b1 - 2

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
123.1
 −1.94949 − 1.32288i 2.94949 − 1.32288i −1.94949 + 1.32288i 2.94949 + 1.32288i
0.500000 1.32288i −2.44949 −1.50000 1.32288i −2.00000 −1.22474 + 3.24037i 0 −2.50000 + 1.32288i 3.00000 −1.00000 + 2.64575i
123.2 0.500000 1.32288i 2.44949 −1.50000 1.32288i −2.00000 1.22474 3.24037i 0 −2.50000 + 1.32288i 3.00000 −1.00000 + 2.64575i
123.3 0.500000 + 1.32288i −2.44949 −1.50000 + 1.32288i −2.00000 −1.22474 3.24037i 0 −2.50000 1.32288i 3.00000 −1.00000 2.64575i
123.4 0.500000 + 1.32288i 2.44949 −1.50000 + 1.32288i −2.00000 1.22474 + 3.24037i 0 −2.50000 1.32288i 3.00000 −1.00000 2.64575i
 $$n$$: 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.b odd 2 1 inner
124.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.2.d.b 4
3.b odd 2 1 1116.2.g.b 4
4.b odd 2 1 inner 124.2.d.b 4
8.b even 2 1 1984.2.h.e 4
8.d odd 2 1 1984.2.h.e 4
12.b even 2 1 1116.2.g.b 4
31.b odd 2 1 inner 124.2.d.b 4
93.c even 2 1 1116.2.g.b 4
124.d even 2 1 inner 124.2.d.b 4
248.b even 2 1 1984.2.h.e 4
248.g odd 2 1 1984.2.h.e 4
372.b odd 2 1 1116.2.g.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.2.d.b 4 1.a even 1 1 trivial
124.2.d.b 4 4.b odd 2 1 inner
124.2.d.b 4 31.b odd 2 1 inner
124.2.d.b 4 124.d even 2 1 inner
1116.2.g.b 4 3.b odd 2 1
1116.2.g.b 4 12.b even 2 1
1116.2.g.b 4 93.c even 2 1
1116.2.g.b 4 372.b odd 2 1
1984.2.h.e 4 8.b even 2 1
1984.2.h.e 4 8.d odd 2 1
1984.2.h.e 4 248.b even 2 1
1984.2.h.e 4 248.g odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(124, [\chi])$$:

 $$T_{3}^{2} - 6$$ T3^2 - 6 $$T_{5} + 2$$ T5 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 2)^{2}$$
$3$ $$(T^{2} - 6)^{2}$$
$5$ $$(T + 2)^{4}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 6)^{2}$$
$13$ $$(T^{2} + 42)^{2}$$
$17$ $$T^{4}$$
$19$ $$(T^{2} + 28)^{2}$$
$23$ $$(T^{2} - 24)^{2}$$
$29$ $$(T^{2} + 42)^{2}$$
$31$ $$T^{4} - 34T^{2} + 961$$
$37$ $$(T^{2} + 42)^{2}$$
$41$ $$(T + 8)^{4}$$
$43$ $$(T^{2} - 6)^{2}$$
$47$ $$(T^{2} + 112)^{2}$$
$53$ $$(T^{2} + 42)^{2}$$
$59$ $$(T^{2} + 28)^{2}$$
$61$ $$(T^{2} + 42)^{2}$$
$67$ $$(T^{2} + 28)^{2}$$
$71$ $$(T^{2} + 28)^{2}$$
$73$ $$(T^{2} + 168)^{2}$$
$79$ $$(T^{2} - 96)^{2}$$
$83$ $$(T^{2} - 150)^{2}$$
$89$ $$(T^{2} + 168)^{2}$$
$97$ $$(T - 4)^{4}$$