# Properties

 Label 124.2.d.c Level $124$ Weight $2$ Character orbit 124.d Analytic conductor $0.990$ Analytic rank $0$ Dimension $6$ CM discriminant -31 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: $$6$$ Coefficient field: 6.0.21717639.1 Defining polynomial: $$x^{6} - x^{3} + 8$$ x^6 - x^3 + 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} - \beta_{3} q^{4} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{7} + ( - \beta_{2} + 1) q^{8} - 3 q^{9}+O(q^{10})$$ q - b5 * q^2 - b3 * q^4 + (-b5 + b4 + b3) * q^5 + (-b5 - b4 + b3) * q^7 + (-b2 + 1) * q^8 - 3 * q^9 $$q - \beta_{5} q^{2} - \beta_{3} q^{4} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{7} + ( - \beta_{2} + 1) q^{8} - 3 q^{9} + ( - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{10} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{14} + ( - \beta_{5} + \beta_{4} - \beta_1) q^{16} + 3 \beta_{5} q^{18} + (3 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1) q^{19} + ( - \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 - 3) q^{20} + (3 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 + 5) q^{25} + (3 \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 + 5) q^{28} + ( - 2 \beta_{2} + 1) q^{31} + ( - 2 \beta_{4} - \beta_{3} - 2 \beta_1) q^{32} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 2) q^{35} + 3 \beta_{3} q^{36} + (\beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{38} + (3 \beta_{5} + 3 \beta_{4} - \beta_{3} + \beta_1) q^{40} + ( - \beta_{5} - \beta_{4} - 3 \beta_{3} + 2 \beta_1) q^{41} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3}) q^{45} + ( - 4 \beta_{2} + 2) q^{47} + ( - 5 \beta_{5} + 3 \beta_{4} + \beta_{3} + 2 \beta_1 - 7) q^{49} + ( - 5 \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_1 - 7) q^{50} + ( - 5 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + \beta_1) q^{56} + ( - 5 \beta_{5} - 3 \beta_{4} + \beta_{3} - 2 \beta_1) q^{59} + ( - \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{62} + (3 \beta_{5} + 3 \beta_{4} - 3 \beta_{3}) q^{63} + ( - \beta_{2} - 7) q^{64} + (4 \beta_{2} - 2) q^{67} + (2 \beta_{5} - 7 \beta_{4} - \beta_{3} - 3 \beta_{2} + \beta_1 + 1) q^{70} + ( - 5 \beta_{5} - \beta_{4} - 3 \beta_{3} - 4 \beta_1) q^{71} + (3 \beta_{2} - 3) q^{72} + ( - 5 \beta_{5} - \beta_{4} + 3 \beta_{2} + \beta_1 + 1) q^{76} + ( - 2 \beta_{4} + 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 9) q^{80} + 9 q^{81} + (3 \beta_{4} - \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 5) q^{82} + (3 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{90} + ( - 2 \beta_{5} + 4 \beta_{4} - 4 \beta_1) q^{94} + (7 \beta_{5} + 3 \beta_{4} + \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 2) q^{95} + ( - \beta_{5} + 3 \beta_{4} + 5 \beta_{3} - 2 \beta_1) q^{97} + (7 \beta_{5} - \beta_{4} - 5 \beta_{3} + \beta_{2} - \beta_1 + 9) q^{98}+O(q^{100})$$ q - b5 * q^2 - b3 * q^4 + (-b5 + b4 + b3) * q^5 + (-b5 - b4 + b3) * q^7 + (-b2 + 1) * q^8 - 3 * q^9 + (-b4 - b3 + b2 - b1 + 1) * q^10 + (b4 - b3 + b2 + b1 - 3) * q^14 + (-b5 + b4 - b1) * q^16 + 3*b5 * q^18 + (3*b5 + b4 + b3 + 2*b1) * q^19 + (-b5 - b4 - b2 + b1 - 3) * q^20 + (3*b5 - b4 + b3 - 2*b1 + 5) * q^25 + (3*b5 - b4 - b2 + b1 + 5) * q^28 + (-2*b2 + 1) * q^31 + (-2*b4 - b3 - 2*b1) * q^32 + (-b5 + b4 - 3*b3 + 4*b2 - 2*b1 - 2) * q^35 + 3*b3 * q^36 + (b4 + 3*b3 + b2 + b1 + 5) * q^38 + (3*b5 + 3*b4 - b3 + b1) * q^40 + (-b5 - b4 - 3*b3 + 2*b1) * q^41 + (3*b5 - 3*b4 - 3*b3) * q^45 + (-4*b2 + 2) * q^47 + (-5*b5 + 3*b4 + b3 + 2*b1 - 7) * q^49 + (-5*b5 - b4 + 3*b3 + b2 - b1 - 7) * q^50 + (-5*b5 + 3*b4 + 3*b3 + b1) * q^56 + (-5*b5 - 3*b4 + b3 - 2*b1) * q^59 + (-b5 + 2*b4 - 2*b1) * q^62 + (3*b5 + 3*b4 - 3*b3) * q^63 + (-b2 - 7) * q^64 + (4*b2 - 2) * q^67 + (2*b5 - 7*b4 - b3 - 3*b2 + b1 + 1) * q^70 + (-5*b5 - b4 - 3*b3 - 4*b1) * q^71 + (3*b2 - 3) * q^72 + (-5*b5 - b4 + 3*b2 + b1 + 1) * q^76 + (-2*b4 + 3*b3 - b2 - 2*b1 + 9) * q^80 + 9 * q^81 + (3*b4 - b3 - 3*b2 + 3*b1 + 5) * q^82 + (3*b4 + 3*b3 - 3*b2 + 3*b1 - 3) * q^90 + (-2*b5 + 4*b4 - 4*b1) * q^94 + (7*b5 + 3*b4 + b3 - 4*b2 + 4*b1 + 2) * q^95 + (-b5 + 3*b4 + 5*b3 - 2*b1) * q^97 + (7*b5 - b4 - 5*b3 + b2 - b1 + 9) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{8} - 18 q^{9}+O(q^{10})$$ 6 * q + 3 * q^8 - 18 * q^9 $$6 q + 3 q^{8} - 18 q^{9} + 9 q^{10} - 15 q^{14} - 21 q^{20} + 30 q^{25} + 27 q^{28} + 33 q^{38} - 42 q^{49} - 39 q^{50} - 45 q^{64} - 3 q^{70} - 9 q^{72} + 15 q^{76} + 51 q^{80} + 54 q^{81} + 21 q^{82} - 27 q^{90} + 57 q^{98}+O(q^{100})$$ 6 * q + 3 * q^8 - 18 * q^9 + 9 * q^10 - 15 * q^14 - 21 * q^20 + 30 * q^25 + 27 * q^28 + 33 * q^38 - 42 * q^49 - 39 * q^50 - 45 * q^64 - 3 * q^70 - 9 * q^72 + 15 * q^76 + 51 * q^80 + 54 * q^81 + 21 * q^82 - 27 * q^90 + 57 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{3} + 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu$$ v^2 + v $$\beta_{2}$$ $$=$$ $$\nu^{3}$$ v^3 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - \nu ) / 2$$ (v^4 - v) / 2 $$\beta_{4}$$ $$=$$ $$-\nu^{2} + \nu$$ -v^2 + v $$\beta_{5}$$ $$=$$ $$( \nu^{5} - \nu^{2} ) / 4$$ (v^5 - v^2) / 4
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_1 ) / 2$$ (b4 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{4} + \beta_1 ) / 2$$ (-b4 + b1) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{2}$$ b2 $$\nu^{4}$$ $$=$$ $$( \beta_{4} + 4\beta_{3} + \beta_1 ) / 2$$ (b4 + 4*b3 + b1) / 2 $$\nu^{5}$$ $$=$$ $$( 8\beta_{5} - \beta_{4} + \beta_1 ) / 2$$ (8*b5 - b4 + 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
 −1.18073 + 0.778374i −1.18073 − 0.778374i −0.0837246 + 1.41173i −0.0837246 − 1.41173i 1.26446 + 0.633359i 1.26446 − 0.633359i
−1.18073 0.778374i 0 0.788267 + 1.83811i −3.93800 0 5.23296i 0.500000 2.78388i −3.00000 4.64974 + 3.06524i
123.2 −1.18073 + 0.778374i 0 0.788267 1.83811i −3.93800 0 5.23296i 0.500000 + 2.78388i −3.00000 4.64974 3.06524i
123.3 −0.0837246 1.41173i 0 −1.98598 + 0.236394i 3.80451 0 3.29625i 0.500000 + 2.78388i −3.00000 −0.318531 5.37095i
123.4 −0.0837246 + 1.41173i 0 −1.98598 0.236394i 3.80451 0 3.29625i 0.500000 2.78388i −3.00000 −0.318531 + 5.37095i
123.5 1.26446 0.633359i 0 1.19771 1.60171i 0.133492 0 1.93671i 0.500000 2.78388i −3.00000 0.168795 0.0845483i
123.6 1.26446 + 0.633359i 0 1.19771 + 1.60171i 0.133492 0 1.93671i 0.500000 + 2.78388i −3.00000 0.168795 + 0.0845483i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 123.6 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.b odd 2 1 CM by $$\Q(\sqrt{-31})$$
4.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.c 6
3.b odd 2 1 1116.2.g.f 6
4.b odd 2 1 inner 124.2.d.c 6
8.b even 2 1 1984.2.h.f 6
8.d odd 2 1 1984.2.h.f 6
12.b even 2 1 1116.2.g.f 6
31.b odd 2 1 CM 124.2.d.c 6
93.c even 2 1 1116.2.g.f 6
124.d even 2 1 inner 124.2.d.c 6
248.b even 2 1 1984.2.h.f 6
248.g odd 2 1 1984.2.h.f 6
372.b odd 2 1 1116.2.g.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.2.d.c 6 1.a even 1 1 trivial
124.2.d.c 6 4.b odd 2 1 inner
124.2.d.c 6 31.b odd 2 1 CM
124.2.d.c 6 124.d even 2 1 inner
1116.2.g.f 6 3.b odd 2 1
1116.2.g.f 6 12.b even 2 1
1116.2.g.f 6 93.c even 2 1
1116.2.g.f 6 372.b odd 2 1
1984.2.h.f 6 8.b even 2 1
1984.2.h.f 6 8.d odd 2 1
1984.2.h.f 6 248.b even 2 1
1984.2.h.f 6 248.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 $$S_{2}^{\mathrm{new}}(124, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5}^{3} - 15T_{5} + 2$$ T5^3 - 15*T5 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - T^{3} + 8$$
$3$ $$T^{6}$$
$5$ $$(T^{3} - 15 T + 2)^{2}$$
$7$ $$T^{6} + 42 T^{4} + 441 T^{2} + \cdots + 1116$$
$11$ $$T^{6}$$
$13$ $$T^{6}$$
$17$ $$T^{6}$$
$19$ $$T^{6} + 114 T^{4} + 3249 T^{2} + \cdots + 3100$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$(T^{2} + 31)^{3}$$
$37$ $$T^{6}$$
$41$ $$(T^{3} - 123 T + 278)^{2}$$
$43$ $$T^{6}$$
$47$ $$(T^{2} + 124)^{3}$$
$53$ $$T^{6}$$
$59$ $$T^{6} + 354 T^{4} + 31329 T^{2} + \cdots + 273916$$
$61$ $$T^{6}$$
$67$ $$(T^{2} + 124)^{3}$$
$71$ $$T^{6} + 426 T^{4} + 45369 T^{2} + \cdots + 431644$$
$73$ $$T^{6}$$
$79$ $$T^{6}$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$(T^{3} - 291 T - 1906)^{2}$$