# Properties

 Label 124.2.d.a 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: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} - \beta_{3} q^{3} - 2 \beta_1 q^{4} + q^{5} + (\beta_{3} - \beta_{2}) q^{6} - 3 \beta_1 q^{7} + (2 \beta_1 + 2) q^{8} + 3 q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 - b3 * q^3 - 2*b1 * q^4 + q^5 + (b3 - b2) * q^6 - 3*b1 * q^7 + (2*b1 + 2) * q^8 + 3 * q^9 $$q + (\beta_1 - 1) q^{2} - \beta_{3} q^{3} - 2 \beta_1 q^{4} + q^{5} + (\beta_{3} - \beta_{2}) q^{6} - 3 \beta_1 q^{7} + (2 \beta_1 + 2) q^{8} + 3 q^{9} + (\beta_1 - 1) q^{10} + 2 \beta_{3} q^{11} + 2 \beta_{2} q^{12} - \beta_{2} q^{13} + (3 \beta_1 + 3) q^{14} - \beta_{3} q^{15} - 4 q^{16} - 3 \beta_{2} q^{17} + (3 \beta_1 - 3) q^{18} + \beta_1 q^{19} - 2 \beta_1 q^{20} + 3 \beta_{2} q^{21} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{22} - \beta_{3} q^{23} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{24} - 4 q^{25} + (\beta_{3} + \beta_{2}) q^{26} - 6 q^{28} + \beta_{2} q^{29} + (\beta_{3} - \beta_{2}) q^{30} + (\beta_{3} - 5 \beta_1) q^{31} + ( - 4 \beta_1 + 4) q^{32} - 12 q^{33} + (3 \beta_{3} + 3 \beta_{2}) q^{34} - 3 \beta_1 q^{35} - 6 \beta_1 q^{36} + 2 \beta_{2} q^{37} + ( - \beta_1 - 1) q^{38} + 6 \beta_1 q^{39} + (2 \beta_1 + 2) q^{40} + 7 q^{41} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{42} - \beta_{3} q^{43} - 4 \beta_{2} q^{44} + 3 q^{45} + (\beta_{3} - \beta_{2}) q^{46} + 2 \beta_1 q^{47} + 4 \beta_{3} q^{48} - 2 q^{49} + ( - 4 \beta_1 + 4) q^{50} + 18 \beta_1 q^{51} - 2 \beta_{3} q^{52} + 4 \beta_{2} q^{53} + 2 \beta_{3} q^{55} + ( - 6 \beta_1 + 6) q^{56} - \beta_{2} q^{57} + ( - \beta_{3} - \beta_{2}) q^{58} + 11 \beta_1 q^{59} + 2 \beta_{2} q^{60} - 5 \beta_{2} q^{61} + ( - \beta_{3} + \beta_{2} + 5 \beta_1 + 5) q^{62} - 9 \beta_1 q^{63} + 8 \beta_1 q^{64} - \beta_{2} q^{65} + ( - 12 \beta_1 + 12) q^{66} - 8 \beta_1 q^{67} - 6 \beta_{3} q^{68} + 6 q^{69} + (3 \beta_1 + 3) q^{70} - 5 \beta_1 q^{71} + (6 \beta_1 + 6) q^{72} + 4 \beta_{2} q^{73} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{74} + 4 \beta_{3} q^{75} + 2 q^{76} - 6 \beta_{2} q^{77} + ( - 6 \beta_1 - 6) q^{78} - 5 \beta_{3} q^{79} - 4 q^{80} - 9 q^{81} + (7 \beta_1 - 7) q^{82} + 4 \beta_{3} q^{83} + 6 \beta_{3} q^{84} - 3 \beta_{2} q^{85} + (\beta_{3} - \beta_{2}) q^{86} - 6 \beta_1 q^{87} + (4 \beta_{3} + 4 \beta_{2}) q^{88} + \beta_{2} q^{89} + (3 \beta_1 - 3) q^{90} - 3 \beta_{3} q^{91} + 2 \beta_{2} q^{92} + (5 \beta_{2} - 6) q^{93} + ( - 2 \beta_1 - 2) q^{94} + \beta_1 q^{95} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{96} + 13 q^{97} + ( - 2 \beta_1 + 2) q^{98} + 6 \beta_{3} q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 - b3 * q^3 - 2*b1 * q^4 + q^5 + (b3 - b2) * q^6 - 3*b1 * q^7 + (2*b1 + 2) * q^8 + 3 * q^9 + (b1 - 1) * q^10 + 2*b3 * q^11 + 2*b2 * q^12 - b2 * q^13 + (3*b1 + 3) * q^14 - b3 * q^15 - 4 * q^16 - 3*b2 * q^17 + (3*b1 - 3) * q^18 + b1 * q^19 - 2*b1 * q^20 + 3*b2 * q^21 + (-2*b3 + 2*b2) * q^22 - b3 * q^23 + (-2*b3 - 2*b2) * q^24 - 4 * q^25 + (b3 + b2) * q^26 - 6 * q^28 + b2 * q^29 + (b3 - b2) * q^30 + (b3 - 5*b1) * q^31 + (-4*b1 + 4) * q^32 - 12 * q^33 + (3*b3 + 3*b2) * q^34 - 3*b1 * q^35 - 6*b1 * q^36 + 2*b2 * q^37 + (-b1 - 1) * q^38 + 6*b1 * q^39 + (2*b1 + 2) * q^40 + 7 * q^41 + (-3*b3 - 3*b2) * q^42 - b3 * q^43 - 4*b2 * q^44 + 3 * q^45 + (b3 - b2) * q^46 + 2*b1 * q^47 + 4*b3 * q^48 - 2 * q^49 + (-4*b1 + 4) * q^50 + 18*b1 * q^51 - 2*b3 * q^52 + 4*b2 * q^53 + 2*b3 * q^55 + (-6*b1 + 6) * q^56 - b2 * q^57 + (-b3 - b2) * q^58 + 11*b1 * q^59 + 2*b2 * q^60 - 5*b2 * q^61 + (-b3 + b2 + 5*b1 + 5) * q^62 - 9*b1 * q^63 + 8*b1 * q^64 - b2 * q^65 + (-12*b1 + 12) * q^66 - 8*b1 * q^67 - 6*b3 * q^68 + 6 * q^69 + (3*b1 + 3) * q^70 - 5*b1 * q^71 + (6*b1 + 6) * q^72 + 4*b2 * q^73 + (-2*b3 - 2*b2) * q^74 + 4*b3 * q^75 + 2 * q^76 - 6*b2 * q^77 + (-6*b1 - 6) * q^78 - 5*b3 * q^79 - 4 * q^80 - 9 * q^81 + (7*b1 - 7) * q^82 + 4*b3 * q^83 + 6*b3 * q^84 - 3*b2 * q^85 + (b3 - b2) * q^86 - 6*b1 * q^87 + (4*b3 + 4*b2) * q^88 + b2 * q^89 + (3*b1 - 3) * q^90 - 3*b3 * q^91 + 2*b2 * q^92 + (5*b2 - 6) * q^93 + (-2*b1 - 2) * q^94 + b1 * q^95 + (-4*b3 + 4*b2) * q^96 + 13 * q^97 + (-2*b1 + 2) * q^98 + 6*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 4 q^{5} + 8 q^{8} + 12 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + 4 * q^5 + 8 * q^8 + 12 * q^9 $$4 q - 4 q^{2} + 4 q^{5} + 8 q^{8} + 12 q^{9} - 4 q^{10} + 12 q^{14} - 16 q^{16} - 12 q^{18} - 16 q^{25} - 24 q^{28} + 16 q^{32} - 48 q^{33} - 4 q^{38} + 8 q^{40} + 28 q^{41} + 12 q^{45} - 8 q^{49} + 16 q^{50} + 24 q^{56} + 20 q^{62} + 48 q^{66} + 24 q^{69} + 12 q^{70} + 24 q^{72} + 8 q^{76} - 24 q^{78} - 16 q^{80} - 36 q^{81} - 28 q^{82} - 12 q^{90} - 24 q^{93} - 8 q^{94} + 52 q^{97} + 8 q^{98}+O(q^{100})$$ 4 * q - 4 * q^2 + 4 * q^5 + 8 * q^8 + 12 * q^9 - 4 * q^10 + 12 * q^14 - 16 * q^16 - 12 * q^18 - 16 * q^25 - 24 * q^28 + 16 * q^32 - 48 * q^33 - 4 * q^38 + 8 * q^40 + 28 * q^41 + 12 * q^45 - 8 * q^49 + 16 * q^50 + 24 * q^56 + 20 * q^62 + 48 * q^66 + 24 * q^69 + 12 * q^70 + 24 * q^72 + 8 * q^76 - 24 * q^78 - 16 * q^80 - 36 * q^81 - 28 * q^82 - 12 * q^90 - 24 * q^93 - 8 * q^94 + 52 * q^97 + 8 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 3\nu ) / 3$$ (v^3 + 3*v) / 3 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 3\nu ) / 3$$ (-v^3 + 3*v) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$3\beta_1$$ 3*b1 $$\nu^{3}$$ $$=$$ $$( -3\beta_{3} + 3\beta_{2} ) / 2$$ (-3*b3 + 3*b2) / 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
 1.22474 − 1.22474i −1.22474 + 1.22474i 1.22474 + 1.22474i −1.22474 − 1.22474i
−1.00000 1.00000i −2.44949 2.00000i 1.00000 2.44949 + 2.44949i 3.00000i 2.00000 2.00000i 3.00000 −1.00000 1.00000i
123.2 −1.00000 1.00000i 2.44949 2.00000i 1.00000 −2.44949 2.44949i 3.00000i 2.00000 2.00000i 3.00000 −1.00000 1.00000i
123.3 −1.00000 + 1.00000i −2.44949 2.00000i 1.00000 2.44949 2.44949i 3.00000i 2.00000 + 2.00000i 3.00000 −1.00000 + 1.00000i
123.4 −1.00000 + 1.00000i 2.44949 2.00000i 1.00000 −2.44949 + 2.44949i 3.00000i 2.00000 + 2.00000i 3.00000 −1.00000 + 1.00000i
 $$n$$: 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.a 4
3.b odd 2 1 1116.2.g.d 4
4.b odd 2 1 inner 124.2.d.a 4
8.b even 2 1 1984.2.h.d 4
8.d odd 2 1 1984.2.h.d 4
12.b even 2 1 1116.2.g.d 4
31.b odd 2 1 inner 124.2.d.a 4
93.c even 2 1 1116.2.g.d 4
124.d even 2 1 inner 124.2.d.a 4
248.b even 2 1 1984.2.h.d 4
248.g odd 2 1 1984.2.h.d 4
372.b odd 2 1 1116.2.g.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.2.d.a 4 1.a even 1 1 trivial
124.2.d.a 4 4.b odd 2 1 inner
124.2.d.a 4 31.b odd 2 1 inner
124.2.d.a 4 124.d even 2 1 inner
1116.2.g.d 4 3.b odd 2 1
1116.2.g.d 4 12.b even 2 1
1116.2.g.d 4 93.c even 2 1
1116.2.g.d 4 372.b odd 2 1
1984.2.h.d 4 8.b even 2 1
1984.2.h.d 4 8.d odd 2 1
1984.2.h.d 4 248.b even 2 1
1984.2.h.d 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 $$T_{5} - 1$$ T5 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T + 2)^{2}$$
$3$ $$(T^{2} - 6)^{2}$$
$5$ $$(T - 1)^{4}$$
$7$ $$(T^{2} + 9)^{2}$$
$11$ $$(T^{2} - 24)^{2}$$
$13$ $$(T^{2} + 6)^{2}$$
$17$ $$(T^{2} + 54)^{2}$$
$19$ $$(T^{2} + 1)^{2}$$
$23$ $$(T^{2} - 6)^{2}$$
$29$ $$(T^{2} + 6)^{2}$$
$31$ $$T^{4} + 38T^{2} + 961$$
$37$ $$(T^{2} + 24)^{2}$$
$41$ $$(T - 7)^{4}$$
$43$ $$(T^{2} - 6)^{2}$$
$47$ $$(T^{2} + 4)^{2}$$
$53$ $$(T^{2} + 96)^{2}$$
$59$ $$(T^{2} + 121)^{2}$$
$61$ $$(T^{2} + 150)^{2}$$
$67$ $$(T^{2} + 64)^{2}$$
$71$ $$(T^{2} + 25)^{2}$$
$73$ $$(T^{2} + 96)^{2}$$
$79$ $$(T^{2} - 150)^{2}$$
$83$ $$(T^{2} - 96)^{2}$$
$89$ $$(T^{2} + 6)^{2}$$
$97$ $$(T - 13)^{4}$$