Properties

 Label 124.4.d.a Level $124$ Weight $4$ Character orbit 124.d Analytic conductor $7.316$ Analytic rank $0$ Dimension $2$ CM discriminant -31 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$124 = 2^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 124.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$7.31623684071$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-31})$$ Defining polynomial: $$x^{2} - x + 8$$ x^2 - x + 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{-31})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (\beta - 8) q^{4} - 2 q^{5} + ( - 12 \beta + 6) q^{7} + ( - 7 \beta - 8) q^{8} - 27 q^{9}+O(q^{10})$$ q + b * q^2 + (b - 8) * q^4 - 2 * q^5 + (-12*b + 6) * q^7 + (-7*b - 8) * q^8 - 27 * q^9 $$q + \beta q^{2} + (\beta - 8) q^{4} - 2 q^{5} + ( - 12 \beta + 6) q^{7} + ( - 7 \beta - 8) q^{8} - 27 q^{9} - 2 \beta q^{10} + ( - 6 \beta + 96) q^{14} + ( - 15 \beta + 56) q^{16} - 27 \beta q^{18} + ( - 20 \beta + 10) q^{19} + ( - 2 \beta + 16) q^{20} - 121 q^{25} + (90 \beta + 48) q^{28} + ( - 62 \beta + 31) q^{31} + (41 \beta + 120) q^{32} + (24 \beta - 12) q^{35} + ( - 27 \beta + 216) q^{36} + ( - 10 \beta + 160) q^{38} + (14 \beta + 16) q^{40} - 278 q^{41} + 54 q^{45} + (68 \beta - 34) q^{47} - 773 q^{49} - 121 \beta q^{50} + (138 \beta - 720) q^{56} + (188 \beta - 94) q^{59} + ( - 31 \beta + 496) q^{62} + (324 \beta - 162) q^{63} + (161 \beta - 328) q^{64} + ( - 308 \beta + 154) q^{67} + (12 \beta - 192) q^{70} + ( - 236 \beta + 118) q^{71} + (189 \beta + 216) q^{72} + (150 \beta + 80) q^{76} + (30 \beta - 112) q^{80} + 729 q^{81} - 278 \beta q^{82} + 54 \beta q^{90} + (34 \beta - 544) q^{94} + (40 \beta - 20) q^{95} + 1906 q^{97} - 773 \beta q^{98} +O(q^{100})$$ q + b * q^2 + (b - 8) * q^4 - 2 * q^5 + (-12*b + 6) * q^7 + (-7*b - 8) * q^8 - 27 * q^9 - 2*b * q^10 + (-6*b + 96) * q^14 + (-15*b + 56) * q^16 - 27*b * q^18 + (-20*b + 10) * q^19 + (-2*b + 16) * q^20 - 121 * q^25 + (90*b + 48) * q^28 + (-62*b + 31) * q^31 + (41*b + 120) * q^32 + (24*b - 12) * q^35 + (-27*b + 216) * q^36 + (-10*b + 160) * q^38 + (14*b + 16) * q^40 - 278 * q^41 + 54 * q^45 + (68*b - 34) * q^47 - 773 * q^49 - 121*b * q^50 + (138*b - 720) * q^56 + (188*b - 94) * q^59 + (-31*b + 496) * q^62 + (324*b - 162) * q^63 + (161*b - 328) * q^64 + (-308*b + 154) * q^67 + (12*b - 192) * q^70 + (-236*b + 118) * q^71 + (189*b + 216) * q^72 + (150*b + 80) * q^76 + (30*b - 112) * q^80 + 729 * q^81 - 278*b * q^82 + 54*b * q^90 + (34*b - 544) * q^94 + (40*b - 20) * q^95 + 1906 * q^97 - 773*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 15 q^{4} - 4 q^{5} - 23 q^{8} - 54 q^{9}+O(q^{10})$$ 2 * q + q^2 - 15 * q^4 - 4 * q^5 - 23 * q^8 - 54 * q^9 $$2 q + q^{2} - 15 q^{4} - 4 q^{5} - 23 q^{8} - 54 q^{9} - 2 q^{10} + 186 q^{14} + 97 q^{16} - 27 q^{18} + 30 q^{20} - 242 q^{25} + 186 q^{28} + 281 q^{32} + 405 q^{36} + 310 q^{38} + 46 q^{40} - 556 q^{41} + 108 q^{45} - 1546 q^{49} - 121 q^{50} - 1302 q^{56} + 961 q^{62} - 495 q^{64} - 372 q^{70} + 621 q^{72} + 310 q^{76} - 194 q^{80} + 1458 q^{81} - 278 q^{82} + 54 q^{90} - 1054 q^{94} + 3812 q^{97} - 773 q^{98}+O(q^{100})$$ 2 * q + q^2 - 15 * q^4 - 4 * q^5 - 23 * q^8 - 54 * q^9 - 2 * q^10 + 186 * q^14 + 97 * q^16 - 27 * q^18 + 30 * q^20 - 242 * q^25 + 186 * q^28 + 281 * q^32 + 405 * q^36 + 310 * q^38 + 46 * q^40 - 556 * q^41 + 108 * q^45 - 1546 * q^49 - 121 * q^50 - 1302 * q^56 + 961 * q^62 - 495 * q^64 - 372 * q^70 + 621 * q^72 + 310 * q^76 - 194 * q^80 + 1458 * q^81 - 278 * q^82 + 54 * q^90 - 1054 * q^94 + 3812 * q^97 - 773 * q^98

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
 0.5 − 2.78388i 0.5 + 2.78388i
0.500000 2.78388i 0 −7.50000 2.78388i −2.00000 0 33.4066i −11.5000 + 19.4872i −27.0000 −1.00000 + 5.56776i
123.2 0.500000 + 2.78388i 0 −7.50000 + 2.78388i −2.00000 0 33.4066i −11.5000 19.4872i −27.0000 −1.00000 5.56776i
 $$n$$: 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.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 124.4.d.a 2
4.b odd 2 1 inner 124.4.d.a 2
31.b odd 2 1 CM 124.4.d.a 2
124.d even 2 1 inner 124.4.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.d.a 2 1.a even 1 1 trivial
124.4.d.a 2 4.b odd 2 1 inner
124.4.d.a 2 31.b odd 2 1 CM
124.4.d.a 2 124.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(124, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5} + 2$$ T5 + 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 8$$
$3$ $$T^{2}$$
$5$ $$(T + 2)^{2}$$
$7$ $$T^{2} + 1116$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 3100$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 29791$$
$37$ $$T^{2}$$
$41$ $$(T + 278)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 35836$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 273916$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 735196$$
$71$ $$T^{2} + 431644$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T - 1906)^{2}$$