# Properties

 Label 1386.2.e.b Level $1386$ Weight $2$ Character orbit 1386.e Analytic conductor $11.067$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1386.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0672657201$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.12745506816.1 Defining polynomial: $$x^{8} + 23x^{4} + 1$$ x^8 + 23*x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 154) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} + (\beta_{7} + \beta_{3} - \beta_1) q^{7} + \beta_1 q^{8}+O(q^{10})$$ q - b1 * q^2 - q^4 - b2 * q^5 + (b7 + b3 - b1) * q^7 + b1 * q^8 $$q - \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} + (\beta_{7} + \beta_{3} - \beta_1) q^{7} + \beta_1 q^{8} + \beta_{3} q^{10} + ( - \beta_{5} - \beta_1 - 1) q^{11} - \beta_{7} q^{13} + (\beta_{6} + \beta_{2} - 1) q^{14} + q^{16} + ( - \beta_{7} + \beta_{3}) q^{17} + ( - 2 \beta_{7} - \beta_{3}) q^{19} + \beta_{2} q^{20} - \beta_{4} q^{22} - 4 q^{23} + (\beta_{5} - \beta_{4} + 1) q^{25} - \beta_{6} q^{26} + ( - \beta_{7} - \beta_{3} + \beta_1) q^{28} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{29} + ( - \beta_{6} - 3 \beta_{2}) q^{31} - \beta_1 q^{32} + ( - \beta_{6} + \beta_{2}) q^{34} + (\beta_{5} + \beta_{4} + \beta_{3} + 4 \beta_1) q^{35} + ( - \beta_{5} + \beta_{4} - 2) q^{37} + ( - 2 \beta_{6} - \beta_{2}) q^{38} - \beta_{3} q^{40} + ( - \beta_{7} + 3 \beta_{3}) q^{41} + (2 \beta_{5} + 2 \beta_{4}) q^{43} + (\beta_{5} + \beta_1 + 1) q^{44} + 4 \beta_1 q^{46} + ( - \beta_{6} + 3 \beta_{2}) q^{47} + (2 \beta_{6} + 2 \beta_{2} + 5) q^{49} + (\beta_{5} + \beta_{4} + \beta_1) q^{50} + \beta_{7} q^{52} + ( - \beta_{5} + \beta_{4} + 6) q^{53} + (\beta_{7} - \beta_{6} + 3 \beta_{3} - 2 \beta_{2}) q^{55} + ( - \beta_{6} - \beta_{2} + 1) q^{56} + ( - \beta_{5} + \beta_{4} - 4) q^{58} - \beta_{6} q^{59} - 3 \beta_{7} q^{61} + (\beta_{7} + 3 \beta_{3}) q^{62} - q^{64} + 2 \beta_1 q^{65} + (\beta_{5} - \beta_{4} - 2) q^{67} + (\beta_{7} - \beta_{3}) q^{68} + ( - \beta_{5} + \beta_{4} + \beta_{2} + 2) q^{70} - 2 q^{71} + ( - \beta_{7} + 5 \beta_{3}) q^{73} + ( - \beta_{5} - \beta_{4}) q^{74} + (2 \beta_{7} + \beta_{3}) q^{76} + ( - 2 \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_{2}) q^{77} + 4 \beta_1 q^{79} - \beta_{2} q^{80} + ( - \beta_{6} + 3 \beta_{2}) q^{82} + ( - 4 \beta_{7} - 5 \beta_{3}) q^{83} + (\beta_{5} + \beta_{4} + 8 \beta_1) q^{85} + ( - 2 \beta_{5} + 2 \beta_{4} - 4) q^{86} + \beta_{4} q^{88} + ( - 4 \beta_{6} - 4 \beta_{2}) q^{89} + ( - \beta_{6} - \beta_{5} + \beta_{4} - 4) q^{91} + 4 q^{92} + (\beta_{7} - 3 \beta_{3}) q^{94} + ( - \beta_{5} - \beta_{4} - 2 \beta_1) q^{95} + ( - 6 \beta_{6} - 4 \beta_{2}) q^{97} + ( - 2 \beta_{7} - 2 \beta_{3} - 5 \beta_1) q^{98}+O(q^{100})$$ q - b1 * q^2 - q^4 - b2 * q^5 + (b7 + b3 - b1) * q^7 + b1 * q^8 + b3 * q^10 + (-b5 - b1 - 1) * q^11 - b7 * q^13 + (b6 + b2 - 1) * q^14 + q^16 + (-b7 + b3) * q^17 + (-2*b7 - b3) * q^19 + b2 * q^20 - b4 * q^22 - 4 * q^23 + (b5 - b4 + 1) * q^25 - b6 * q^26 + (-b7 - b3 + b1) * q^28 + (b5 + b4 - 2*b1) * q^29 + (-b6 - 3*b2) * q^31 - b1 * q^32 + (-b6 + b2) * q^34 + (b5 + b4 + b3 + 4*b1) * q^35 + (-b5 + b4 - 2) * q^37 + (-2*b6 - b2) * q^38 - b3 * q^40 + (-b7 + 3*b3) * q^41 + (2*b5 + 2*b4) * q^43 + (b5 + b1 + 1) * q^44 + 4*b1 * q^46 + (-b6 + 3*b2) * q^47 + (2*b6 + 2*b2 + 5) * q^49 + (b5 + b4 + b1) * q^50 + b7 * q^52 + (-b5 + b4 + 6) * q^53 + (b7 - b6 + 3*b3 - 2*b2) * q^55 + (-b6 - b2 + 1) * q^56 + (-b5 + b4 - 4) * q^58 - b6 * q^59 - 3*b7 * q^61 + (b7 + 3*b3) * q^62 - q^64 + 2*b1 * q^65 + (b5 - b4 - 2) * q^67 + (b7 - b3) * q^68 + (-b5 + b4 + b2 + 2) * q^70 - 2 * q^71 + (-b7 + 5*b3) * q^73 + (-b5 - b4) * q^74 + (2*b7 + b3) * q^76 + (-2*b7 - b6 - b4 + b3 + 2*b2) * q^77 + 4*b1 * q^79 - b2 * q^80 + (-b6 + 3*b2) * q^82 + (-4*b7 - 5*b3) * q^83 + (b5 + b4 + 8*b1) * q^85 + (-2*b5 + 2*b4 - 4) * q^86 + b4 * q^88 + (-4*b6 - 4*b2) * q^89 + (-b6 - b5 + b4 - 4) * q^91 + 4 * q^92 + (b7 - 3*b3) * q^94 + (-b5 - b4 - 2*b1) * q^95 + (-6*b6 - 4*b2) * q^97 + (-2*b7 - 2*b3 - 5*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{4}+O(q^{10})$$ 8 * q - 8 * q^4 $$8 q - 8 q^{4} - 4 q^{11} - 8 q^{14} + 8 q^{16} - 4 q^{22} - 32 q^{23} - 8 q^{37} + 4 q^{44} + 40 q^{49} + 56 q^{53} + 8 q^{56} - 24 q^{58} - 8 q^{64} - 24 q^{67} + 24 q^{70} - 16 q^{71} - 4 q^{77} - 16 q^{86} + 4 q^{88} - 24 q^{91} + 32 q^{92}+O(q^{100})$$ 8 * q - 8 * q^4 - 4 * q^11 - 8 * q^14 + 8 * q^16 - 4 * q^22 - 32 * q^23 - 8 * q^37 + 4 * q^44 + 40 * q^49 + 56 * q^53 + 8 * q^56 - 24 * q^58 - 8 * q^64 - 24 * q^67 + 24 * q^70 - 16 * q^71 - 4 * q^77 - 16 * q^86 + 4 * q^88 - 24 * q^91 + 32 * q^92

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 23x^{4} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{6} + 24\nu^{2} ) / 5$$ (v^6 + 24*v^2) / 5 $$\beta_{2}$$ $$=$$ $$( \nu^{7} + 24\nu^{3} + 5\nu ) / 5$$ (v^7 + 24*v^3 + 5*v) / 5 $$\beta_{3}$$ $$=$$ $$( \nu^{7} + 24\nu^{3} - 5\nu ) / 5$$ (v^7 + 24*v^3 - 5*v) / 5 $$\beta_{4}$$ $$=$$ $$( -3\nu^{6} - \nu^{4} - 67\nu^{2} - 9 ) / 5$$ (-3*v^6 - v^4 - 67*v^2 - 9) / 5 $$\beta_{5}$$ $$=$$ $$( -3\nu^{6} + \nu^{4} - 67\nu^{2} + 9 ) / 5$$ (-3*v^6 + v^4 - 67*v^2 + 9) / 5 $$\beta_{6}$$ $$=$$ $$( -5\nu^{7} - \nu^{5} - 115\nu^{3} - 24\nu ) / 5$$ (-5*v^7 - v^5 - 115*v^3 - 24*v) / 5 $$\beta_{7}$$ $$=$$ $$( -5\nu^{7} + \nu^{5} - 115\nu^{3} + 24\nu ) / 5$$ (-5*v^7 + v^5 - 115*v^3 + 24*v) / 5
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + \beta_{4} + 6\beta_1 ) / 2$$ (b5 + b4 + 6*b1) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{7} + \beta_{6} + 5\beta_{3} + 5\beta_{2} ) / 2$$ (b7 + b6 + 5*b3 + 5*b2) / 2 $$\nu^{4}$$ $$=$$ $$( 5\beta_{5} - 5\beta_{4} - 18 ) / 2$$ (5*b5 - 5*b4 - 18) / 2 $$\nu^{5}$$ $$=$$ $$( 5\beta_{7} - 5\beta_{6} + 24\beta_{3} - 24\beta_{2} ) / 2$$ (5*b7 - 5*b6 + 24*b3 - 24*b2) / 2 $$\nu^{6}$$ $$=$$ $$-12\beta_{5} - 12\beta_{4} - 67\beta_1$$ -12*b5 - 12*b4 - 67*b1 $$\nu^{7}$$ $$=$$ $$( -24\beta_{7} - 24\beta_{6} - 115\beta_{3} - 115\beta_{2} ) / 2$$ (-24*b7 - 24*b6 - 115*b3 - 115*b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$1135$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 1.54779 + 1.54779i 0.323042 + 0.323042i −0.323042 − 0.323042i −1.54779 − 1.54779i −1.54779 + 1.54779i −0.323042 + 0.323042i 0.323042 − 0.323042i 1.54779 − 1.54779i
1.00000i 0 −1.00000 3.09557i 0 −2.44949 1.00000i 1.00000i 0 −3.09557
307.2 1.00000i 0 −1.00000 0.646084i 0 2.44949 1.00000i 1.00000i 0 −0.646084
307.3 1.00000i 0 −1.00000 0.646084i 0 −2.44949 1.00000i 1.00000i 0 0.646084
307.4 1.00000i 0 −1.00000 3.09557i 0 2.44949 1.00000i 1.00000i 0 3.09557
307.5 1.00000i 0 −1.00000 3.09557i 0 2.44949 + 1.00000i 1.00000i 0 3.09557
307.6 1.00000i 0 −1.00000 0.646084i 0 −2.44949 + 1.00000i 1.00000i 0 0.646084
307.7 1.00000i 0 −1.00000 0.646084i 0 2.44949 + 1.00000i 1.00000i 0 −0.646084
307.8 1.00000i 0 −1.00000 3.09557i 0 −2.44949 + 1.00000i 1.00000i 0 −3.09557
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.e.b 8
3.b odd 2 1 154.2.c.a 8
7.b odd 2 1 inner 1386.2.e.b 8
11.b odd 2 1 inner 1386.2.e.b 8
12.b even 2 1 1232.2.e.e 8
21.c even 2 1 154.2.c.a 8
21.g even 6 2 1078.2.i.b 16
21.h odd 6 2 1078.2.i.b 16
33.d even 2 1 154.2.c.a 8
77.b even 2 1 inner 1386.2.e.b 8
84.h odd 2 1 1232.2.e.e 8
132.d odd 2 1 1232.2.e.e 8
231.h odd 2 1 154.2.c.a 8
231.k odd 6 2 1078.2.i.b 16
231.l even 6 2 1078.2.i.b 16
924.n even 2 1 1232.2.e.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.c.a 8 3.b odd 2 1
154.2.c.a 8 21.c even 2 1
154.2.c.a 8 33.d even 2 1
154.2.c.a 8 231.h odd 2 1
1078.2.i.b 16 21.g even 6 2
1078.2.i.b 16 21.h odd 6 2
1078.2.i.b 16 231.k odd 6 2
1078.2.i.b 16 231.l even 6 2
1232.2.e.e 8 12.b even 2 1
1232.2.e.e 8 84.h odd 2 1
1232.2.e.e 8 132.d odd 2 1
1232.2.e.e 8 924.n even 2 1
1386.2.e.b 8 1.a even 1 1 trivial
1386.2.e.b 8 7.b odd 2 1 inner
1386.2.e.b 8 11.b odd 2 1 inner
1386.2.e.b 8 77.b 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 $$S_{2}^{\mathrm{new}}(1386, [\chi])$$:

 $$T_{5}^{4} + 10T_{5}^{2} + 4$$ T5^4 + 10*T5^2 + 4 $$T_{13}^{4} - 10T_{13}^{2} + 4$$ T13^4 - 10*T13^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{4}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 10 T^{2} + 4)^{2}$$
$7$ $$(T^{4} - 10 T^{2} + 49)^{2}$$
$11$ $$(T^{4} + 2 T^{3} + 2 T^{2} + 22 T + 121)^{2}$$
$13$ $$(T^{4} - 10 T^{2} + 4)^{2}$$
$17$ $$(T^{2} - 14)^{4}$$
$19$ $$(T^{4} - 34 T^{2} + 100)^{2}$$
$23$ $$(T + 4)^{8}$$
$29$ $$(T^{4} + 60 T^{2} + 144)^{2}$$
$31$ $$(T^{4} + 76 T^{2} + 100)^{2}$$
$37$ $$(T^{2} + 2 T - 20)^{4}$$
$41$ $$(T^{4} - 124 T^{2} + 2500)^{2}$$
$43$ $$(T^{4} + 176 T^{2} + 6400)^{2}$$
$47$ $$(T^{4} + 124 T^{2} + 2500)^{2}$$
$53$ $$(T^{2} - 14 T + 28)^{4}$$
$59$ $$(T^{4} + 10 T^{2} + 4)^{2}$$
$61$ $$(T^{4} - 90 T^{2} + 324)^{2}$$
$67$ $$(T^{2} + 6 T - 12)^{4}$$
$71$ $$(T + 2)^{8}$$
$73$ $$(T^{4} - 300 T^{2} + 10404)^{2}$$
$79$ $$(T^{2} + 16)^{4}$$
$83$ $$(T^{4} - 250 T^{2} + 13924)^{2}$$
$89$ $$(T^{2} + 96)^{4}$$
$97$ $$(T^{4} + 328 T^{2} + 18496)^{2}$$