# Properties

 Label 380.2.i.c Level $380$ Weight $2$ Character orbit 380.i Analytic conductor $3.034$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.03431527681$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} + 9x^{6} + 2x^{5} + 65x^{4} - 20x^{3} + 25x^{2} + 6x + 4$$ x^8 - x^7 + 9*x^6 + 2*x^5 + 65*x^4 - 20*x^3 + 25*x^2 + 6*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + \beta_{4} q^{5} + \beta_{7} q^{7} + (\beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10})$$ q - b1 * q^3 + b4 * q^5 + b7 * q^7 + (b6 - b4 + b3 + b2 + b1 - 1) * q^9 $$q - \beta_1 q^{3} + \beta_{4} q^{5} + \beta_{7} q^{7} + (\beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{9} + ( - \beta_{7} + 2 \beta_{3} + 1) q^{11} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1 + 2) q^{13} + (\beta_{3} + \beta_1) q^{15} + (\beta_{6} + \beta_1) q^{17} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{19} + (\beta_{6} - 2 \beta_{4}) q^{21} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{2}) q^{23} + ( - \beta_{4} - 1) q^{25} + ( - \beta_{7} - 2 \beta_{3} - \beta_{2} + 2) q^{27} + (2 \beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 1) q^{29} + (\beta_{7} - 2 \beta_{3} + \beta_{2} - 3) q^{31} + (\beta_{6} - 6 \beta_{4} + \beta_1) q^{33} - \beta_{5} q^{35} + ( - \beta_{7} + 2 \beta_{3} - 6) q^{37} + (3 \beta_{3} + 2 \beta_{2} - 6) q^{39} + (\beta_{5} + 2 \beta_{4}) q^{41} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_1) q^{43} + ( - \beta_{3} - \beta_{2} + 1) q^{45} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} + 4 \beta_{4} - 2 \beta_{2} + 4) q^{47} + ( - \beta_{7} + 2 \beta_{3} + \beta_{2} + 3) q^{49} + ( - \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - \beta_{2} - 4 \beta_1 + 2) q^{51} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{53} + (\beta_{5} + \beta_{4} + 2 \beta_1) q^{55} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} + 3 \beta_1 + 4) q^{57} + (2 \beta_{5} - 3 \beta_{4} - \beta_1) q^{59} + (2 \beta_{7} - 2 \beta_{5} + 3 \beta_{4} + 3) q^{61} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} - 5 \beta_{3} - 5 \beta_1 - 2) q^{63} + (\beta_{7} + \beta_{3} - 2) q^{65} + ( - 2 \beta_{6} - 2 \beta_{2}) q^{67} + ( - \beta_{7} - 3 \beta_{3} - \beta_{2}) q^{69} + ( - \beta_{6} - 3 \beta_{4} + 2 \beta_1) q^{71} + ( - \beta_{6} - \beta_{5} - 4 \beta_1) q^{73} - \beta_{3} q^{75} + (2 \beta_{7} - 2 \beta_{3} - 3 \beta_{2} - 6) q^{77} + ( - 2 \beta_{6} - \beta_{5} - 3 \beta_{4} + \beta_1) q^{79} + (\beta_{5} + 5 \beta_{4} - 4 \beta_1) q^{81} + (3 \beta_{3} + \beta_{2} + 2) q^{83} + ( - \beta_{6} - \beta_{3} - \beta_{2} - \beta_1) q^{85} + (2 \beta_{3} - \beta_{2}) q^{87} + (\beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{89} + (3 \beta_{7} - 3 \beta_{5} - 12 \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 12) q^{91} + ( - \beta_{6} - \beta_{5} + 8 \beta_{4} + 4 \beta_1) q^{93} + (\beta_{6} - \beta_{3} - 1) q^{95} + ( - 2 \beta_{6} - \beta_1) q^{97} + ( - 4 \beta_{7} - \beta_{6} + 4 \beta_{5} + 5 \beta_{4} - 4 \beta_{3} - \beta_{2} - 4 \beta_1 + 5) q^{99}+O(q^{100})$$ q - b1 * q^3 + b4 * q^5 + b7 * q^7 + (b6 - b4 + b3 + b2 + b1 - 1) * q^9 + (-b7 + 2*b3 + 1) * q^11 + (-b7 + b5 + 2*b4 - b3 - b1 + 2) * q^13 + (b3 + b1) * q^15 + (b6 + b1) * q^17 + (b4 + b3 + b2 + b1 + 1) * q^19 + (b6 - 2*b4) * q^21 + (b7 + b6 - b5 + b2) * q^23 + (-b4 - 1) * q^25 + (-b7 - 2*b3 - b2 + 2) * q^27 + (2*b7 - 2*b5 + b4 - b3 - b1 + 1) * q^29 + (b7 - 2*b3 + b2 - 3) * q^31 + (b6 - 6*b4 + b1) * q^33 - b5 * q^35 + (-b7 + 2*b3 - 6) * q^37 + (3*b3 + 2*b2 - 6) * q^39 + (b5 + 2*b4) * q^41 + (-b6 + b5 - 2*b4 - b1) * q^43 + (-b3 - b2 + 1) * q^45 + (-b7 - 2*b6 + b5 + 4*b4 - 2*b2 + 4) * q^47 + (-b7 + 2*b3 + b2 + 3) * q^49 + (-b7 - b6 + b5 + 2*b4 - 4*b3 - b2 - 4*b1 + 2) * q^51 + (-2*b7 - b6 + 2*b5 + 2*b4 + 3*b3 - b2 + 3*b1 + 2) * q^53 + (b5 + b4 + 2*b1) * q^55 + (-b5 + 2*b4 - b2 + 3*b1 + 4) * q^57 + (2*b5 - 3*b4 - b1) * q^59 + (2*b7 - 2*b5 + 3*b4 + 3) * q^61 + (2*b7 - 2*b5 - 2*b4 - 5*b3 - 5*b1 - 2) * q^63 + (b7 + b3 - 2) * q^65 + (-2*b6 - 2*b2) * q^67 + (-b7 - 3*b3 - b2) * q^69 + (-b6 - 3*b4 + 2*b1) * q^71 + (-b6 - b5 - 4*b1) * q^73 - b3 * q^75 + (2*b7 - 2*b3 - 3*b2 - 6) * q^77 + (-2*b6 - b5 - 3*b4 + b1) * q^79 + (b5 + 5*b4 - 4*b1) * q^81 + (3*b3 + b2 + 2) * q^83 + (-b6 - b3 - b2 - b1) * q^85 + (2*b3 - b2) * q^87 + (b7 + 2*b6 - b5 + b4 - b3 + 2*b2 - b1 + 1) * q^89 + (3*b7 - 3*b5 - 12*b4 - 2*b3 - 2*b1 - 12) * q^91 + (-b6 - b5 + 8*b4 + 4*b1) * q^93 + (b6 - b3 - 1) * q^95 + (-2*b6 - b1) * q^97 + (-4*b7 - b6 + 4*b5 + 5*b4 - 4*b3 - b2 - 4*b1 + 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - q^{3} - 4 q^{5} - 5 q^{9}+O(q^{10})$$ 8 * q - q^3 - 4 * q^5 - 5 * q^9 $$8 q - q^{3} - 4 q^{5} - 5 q^{9} + 4 q^{11} + 9 q^{13} - q^{15} + q^{17} + 3 q^{19} + 8 q^{21} - 4 q^{25} + 20 q^{27} + 5 q^{29} - 20 q^{31} + 25 q^{33} - 52 q^{37} - 54 q^{39} - 8 q^{41} + 7 q^{43} + 10 q^{45} + 16 q^{47} + 20 q^{49} + 12 q^{51} + 5 q^{53} - 2 q^{55} + 27 q^{57} + 11 q^{59} + 12 q^{61} - 3 q^{63} - 18 q^{65} + 6 q^{69} + 14 q^{71} - 4 q^{73} + 2 q^{75} - 44 q^{77} + 13 q^{79} - 24 q^{81} + 10 q^{83} + q^{85} - 4 q^{87} + 5 q^{89} - 46 q^{91} - 28 q^{93} - 6 q^{95} - q^{97} + 24 q^{99}+O(q^{100})$$ 8 * q - q^3 - 4 * q^5 - 5 * q^9 + 4 * q^11 + 9 * q^13 - q^15 + q^17 + 3 * q^19 + 8 * q^21 - 4 * q^25 + 20 * q^27 + 5 * q^29 - 20 * q^31 + 25 * q^33 - 52 * q^37 - 54 * q^39 - 8 * q^41 + 7 * q^43 + 10 * q^45 + 16 * q^47 + 20 * q^49 + 12 * q^51 + 5 * q^53 - 2 * q^55 + 27 * q^57 + 11 * q^59 + 12 * q^61 - 3 * q^63 - 18 * q^65 + 6 * q^69 + 14 * q^71 - 4 * q^73 + 2 * q^75 - 44 * q^77 + 13 * q^79 - 24 * q^81 + 10 * q^83 + q^85 - 4 * q^87 + 5 * q^89 - 46 * q^91 - 28 * q^93 - 6 * q^95 - q^97 + 24 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 9x^{6} + 2x^{5} + 65x^{4} - 20x^{3} + 25x^{2} + 6x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -304\nu^{7} - 31\nu^{6} - 2546\nu^{5} - 1710\nu^{4} - 23348\nu^{3} - 1178\nu^{2} - 380\nu + 48866 ) / 13629$$ (-304*v^7 - 31*v^6 - 2546*v^5 - 1710*v^4 - 23348*v^3 - 1178*v^2 - 380*v + 48866) / 13629 $$\beta_{3}$$ $$=$$ $$( 536\nu^{7} - 304\nu^{6} + 4489\nu^{5} + 3015\nu^{4} + 36145\nu^{3} + 2077\nu^{2} + 670\nu + 3506 ) / 13629$$ (536*v^7 - 304*v^6 + 4489*v^5 + 3015*v^4 + 36145*v^3 + 2077*v^2 + 670*v + 3506) / 13629 $$\beta_{4}$$ $$=$$ $$( 1753\nu^{7} - 2825\nu^{6} + 16385\nu^{5} - 5472\nu^{4} + 107915\nu^{3} - 107350\nu^{2} + 39671\nu - 18080 ) / 27258$$ (1753*v^7 - 2825*v^6 + 16385*v^5 - 5472*v^4 + 107915*v^3 - 107350*v^2 + 39671*v - 18080) / 27258 $$\beta_{5}$$ $$=$$ $$( 1418 \nu^{7} - 2635 \nu^{6} + 15283 \nu^{5} - 9060 \nu^{4} + 100657 \nu^{3} - 100130 \nu^{2} + 131248 \nu - 16864 ) / 13629$$ (1418*v^7 - 2635*v^6 + 15283*v^5 - 9060*v^4 + 100657*v^3 - 100130*v^2 + 131248*v - 16864) / 13629 $$\beta_{6}$$ $$=$$ $$( 3274 \nu^{7} - 5315 \nu^{6} + 30827 \nu^{5} - 12249 \nu^{4} + 203033 \nu^{3} - 201970 \nu^{2} + 65423 \nu - 34016 ) / 13629$$ (3274*v^7 - 5315*v^6 + 30827*v^5 - 12249*v^4 + 203033*v^3 - 201970*v^2 + 65423*v - 34016) / 13629 $$\beta_{7}$$ $$=$$ $$( -1328\nu^{7} + 821\nu^{6} - 11122\nu^{5} - 7470\nu^{4} - 84061\nu^{3} - 5146\nu^{2} - 1660\nu - 16552 ) / 4543$$ (-1328*v^7 + 821*v^6 - 11122*v^5 - 7470*v^4 - 84061*v^3 - 5146*v^2 - 1660*v - 16552) / 4543
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} - 4\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 4$$ b6 - 4*b4 + b3 + b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 8\beta_{3} + \beta_{2} - 2$$ b7 + 8*b3 + b2 - 2 $$\nu^{4}$$ $$=$$ $$-9\beta_{6} + \beta_{5} + 32\beta_{4} - 13\beta_1$$ -9*b6 + b5 + 32*b4 - 13*b1 $$\nu^{5}$$ $$=$$ $$-9\beta_{7} - 14\beta_{6} + 9\beta_{5} + 36\beta_{4} - 72\beta_{3} - 14\beta_{2} - 72\beta _1 + 36$$ -9*b7 - 14*b6 + 9*b5 + 36*b4 - 72*b3 - 14*b2 - 72*b1 + 36 $$\nu^{6}$$ $$=$$ $$-14\beta_{7} - 150\beta_{3} - 81\beta_{2} + 278$$ -14*b7 - 150*b3 - 81*b2 + 278 $$\nu^{7}$$ $$=$$ $$164\beta_{6} - 81\beta_{5} - 466\beta_{4} + 671\beta_1$$ 164*b6 - 81*b5 - 466*b4 + 671*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/380\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$191$$ $$\chi(n)$$ $$-1 - \beta_{4}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 1.58253 + 2.74101i 0.354609 + 0.614201i −0.176725 − 0.306096i −1.26041 − 2.18309i 1.58253 − 2.74101i 0.354609 − 0.614201i −0.176725 + 0.306096i −1.26041 + 2.18309i
0 −1.58253 2.74101i 0 −0.500000 0.866025i 0 −1.53315 0 −3.50877 + 6.07738i 0
121.2 0 −0.354609 0.614201i 0 −0.500000 0.866025i 0 3.11079 0 1.24850 2.16247i 0
121.3 0 0.176725 + 0.306096i 0 −0.500000 0.866025i 0 −4.30507 0 1.43754 2.48989i 0
121.4 0 1.26041 + 2.18309i 0 −0.500000 0.866025i 0 2.72743 0 −1.67727 + 2.90511i 0
201.1 0 −1.58253 + 2.74101i 0 −0.500000 + 0.866025i 0 −1.53315 0 −3.50877 6.07738i 0
201.2 0 −0.354609 + 0.614201i 0 −0.500000 + 0.866025i 0 3.11079 0 1.24850 + 2.16247i 0
201.3 0 0.176725 0.306096i 0 −0.500000 + 0.866025i 0 −4.30507 0 1.43754 + 2.48989i 0
201.4 0 1.26041 2.18309i 0 −0.500000 + 0.866025i 0 2.72743 0 −1.67727 2.90511i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 201.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.i.c 8
3.b odd 2 1 3420.2.t.w 8
4.b odd 2 1 1520.2.q.m 8
5.b even 2 1 1900.2.i.d 8
5.c odd 4 2 1900.2.s.d 16
19.c even 3 1 inner 380.2.i.c 8
19.c even 3 1 7220.2.a.r 4
19.d odd 6 1 7220.2.a.p 4
57.h odd 6 1 3420.2.t.w 8
76.g odd 6 1 1520.2.q.m 8
95.i even 6 1 1900.2.i.d 8
95.m odd 12 2 1900.2.s.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.c 8 1.a even 1 1 trivial
380.2.i.c 8 19.c even 3 1 inner
1520.2.q.m 8 4.b odd 2 1
1520.2.q.m 8 76.g odd 6 1
1900.2.i.d 8 5.b even 2 1
1900.2.i.d 8 95.i even 6 1
1900.2.s.d 16 5.c odd 4 2
1900.2.s.d 16 95.m odd 12 2
3420.2.t.w 8 3.b odd 2 1
3420.2.t.w 8 57.h odd 6 1
7220.2.a.p 4 19.d odd 6 1
7220.2.a.r 4 19.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + T_{3}^{7} + 9T_{3}^{6} - 2T_{3}^{5} + 65T_{3}^{4} + 20T_{3}^{3} + 25T_{3}^{2} - 6T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(380, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + T^{7} + 9 T^{6} - 2 T^{5} + 65 T^{4} + \cdots + 4$$
$5$ $$(T^{2} + T + 1)^{4}$$
$7$ $$(T^{4} - 19 T^{2} + 11 T + 56)^{2}$$
$11$ $$(T^{4} - 2 T^{3} - 35 T^{2} + 21 T + 267)^{2}$$
$13$ $$T^{8} - 9 T^{7} + 86 T^{6} + \cdots + 2704$$
$17$ $$T^{8} - T^{7} + 35 T^{6} + 52 T^{5} + \cdots + 36864$$
$19$ $$T^{8} - 3 T^{7} - 43 T^{6} + \cdots + 130321$$
$23$ $$T^{8} + 39 T^{6} + 198 T^{5} + \cdots + 2916$$
$29$ $$T^{8} - 5 T^{7} + 84 T^{6} + \cdots + 74529$$
$31$ $$(T^{4} + 10 T^{3} - 29 T^{2} - 303 T - 277)^{2}$$
$37$ $$(T^{4} + 26 T^{3} + 217 T^{2} + 609 T + 414)^{2}$$
$41$ $$T^{8} + 8 T^{7} + 59 T^{6} + 106 T^{5} + \cdots + 324$$
$43$ $$T^{8} - 7 T^{7} + 81 T^{6} + \cdots + 31684$$
$47$ $$T^{8} - 16 T^{7} + 281 T^{6} + \cdots + 1382976$$
$53$ $$T^{8} - 5 T^{7} + 143 T^{6} + \cdots + 1483524$$
$59$ $$T^{8} - 11 T^{7} + 176 T^{6} + \cdots + 2518569$$
$61$ $$T^{8} - 12 T^{7} + 166 T^{6} + \cdots + 841$$
$67$ $$T^{8} + 124 T^{6} + \cdots + 10863616$$
$71$ $$T^{8} - 14 T^{7} + 197 T^{6} + \cdots + 5049009$$
$73$ $$T^{8} + 4 T^{7} + 153 T^{6} + \cdots + 311364$$
$79$ $$T^{8} - 13 T^{7} + 297 T^{6} + \cdots + 9096256$$
$83$ $$(T^{4} - 5 T^{3} - 82 T^{2} + 669 T - 1302)^{2}$$
$89$ $$T^{8} - 5 T^{7} + 147 T^{6} + \cdots + 1382976$$
$97$ $$T^{8} + T^{7} + 123 T^{6} + \cdots + 13351716$$