# Properties

 Label 760.2.d.c Level $760$ Weight $2$ Character orbit 760.d Analytic conductor $6.069$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$760 = 2^{3} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 760.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.06863055362$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{3} + ( - 2 \zeta_{8}^{3} + \zeta_{8}) q^{5} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{7} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{9}+O(q^{10})$$ q + (z^3 + z^2 + z) * q^3 + (-2*z^3 + z) * q^5 + (z^3 + z^2 + z) * q^7 + (2*z^3 - 2*z) * q^9 $$q + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{3} + ( - 2 \zeta_{8}^{3} + \zeta_{8}) q^{5} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{7} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{9} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{11} + (2 \zeta_{8}^{3} - \zeta_{8}^{2} + 2 \zeta_{8}) q^{13} + (\zeta_{8}^{3} + 3 \zeta_{8}^{2} + 2 \zeta_{8} + 1) q^{15} - \zeta_{8}^{2} q^{17} + q^{19} + (2 \zeta_{8}^{3} - 2 \zeta_{8} - 3) q^{21} + (3 \zeta_{8}^{3} + \zeta_{8}^{2} + 3 \zeta_{8}) q^{23} + ( - 3 \zeta_{8}^{2} + 4) q^{25} + (\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8}) q^{27} + (2 \zeta_{8}^{3} - 2 \zeta_{8} - 1) q^{29} + ( - \zeta_{8}^{3} + \zeta_{8} + 2) q^{31} + (\zeta_{8}^{3} + 2 \zeta_{8}^{2} + \zeta_{8}) q^{33} + (\zeta_{8}^{3} + 3 \zeta_{8}^{2} + 2 \zeta_{8} + 1) q^{35} + (2 \zeta_{8}^{3} - 8 \zeta_{8}^{2} + 2 \zeta_{8}) q^{37} + (\zeta_{8}^{3} - \zeta_{8} - 3) q^{39} + (5 \zeta_{8}^{3} - 5 \zeta_{8}) q^{41} + (3 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 3 \zeta_{8}) q^{43} + (2 \zeta_{8}^{2} - 6) q^{45} - 8 \zeta_{8}^{2} q^{47} + (2 \zeta_{8}^{3} - 2 \zeta_{8} + 4) q^{49} + ( - \zeta_{8}^{3} + \zeta_{8} + 1) q^{51} + ( - 2 \zeta_{8}^{3} + \zeta_{8}^{2} - 2 \zeta_{8}) q^{53} + ( - \zeta_{8}^{2} + 3) q^{55} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{57} + (\zeta_{8}^{3} - \zeta_{8} - 13) q^{59} + ( - \zeta_{8}^{3} + \zeta_{8} + 2) q^{61} + ( - 2 \zeta_{8}^{3} - 4 \zeta_{8}^{2} - 2 \zeta_{8}) q^{63} + ( - \zeta_{8}^{3} + 6 \zeta_{8}^{2} - 2 \zeta_{8} + 2) q^{65} + (5 \zeta_{8}^{3} - \zeta_{8}^{2} + 5 \zeta_{8}) q^{67} + (4 \zeta_{8}^{3} - 4 \zeta_{8} - 7) q^{69} + ( - 5 \zeta_{8}^{3} + 5 \zeta_{8} - 4) q^{71} + (2 \zeta_{8}^{3} + 11 \zeta_{8}^{2} + 2 \zeta_{8}) q^{73} + (\zeta_{8}^{3} + 4 \zeta_{8}^{2} + 7 \zeta_{8} + 3) q^{75} + (\zeta_{8}^{3} + 2 \zeta_{8}^{2} + \zeta_{8}) q^{77} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} - 2) q^{79} + (6 \zeta_{8}^{3} - 6 \zeta_{8} - 1) q^{81} + ( - 6 \zeta_{8}^{3} + 6 \zeta_{8}^{2} - 6 \zeta_{8}) q^{83} + ( - \zeta_{8}^{3} - 2 \zeta_{8}) q^{85} + ( - 3 \zeta_{8}^{3} - 5 \zeta_{8}^{2} - 3 \zeta_{8}) q^{87} + (3 \zeta_{8}^{3} - 3 \zeta_{8} + 8) q^{89} + (\zeta_{8}^{3} - \zeta_{8} - 3) q^{91} + (3 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + 3 \zeta_{8}) q^{93} + ( - 2 \zeta_{8}^{3} + \zeta_{8}) q^{95} + ( - 4 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 4 \zeta_{8}) q^{97} - 4 q^{99} +O(q^{100})$$ q + (z^3 + z^2 + z) * q^3 + (-2*z^3 + z) * q^5 + (z^3 + z^2 + z) * q^7 + (2*z^3 - 2*z) * q^9 + (-z^3 + z) * q^11 + (2*z^3 - z^2 + 2*z) * q^13 + (z^3 + 3*z^2 + 2*z + 1) * q^15 - z^2 * q^17 + q^19 + (2*z^3 - 2*z - 3) * q^21 + (3*z^3 + z^2 + 3*z) * q^23 + (-3*z^2 + 4) * q^25 + (z^3 - z^2 + z) * q^27 + (2*z^3 - 2*z - 1) * q^29 + (-z^3 + z + 2) * q^31 + (z^3 + 2*z^2 + z) * q^33 + (z^3 + 3*z^2 + 2*z + 1) * q^35 + (2*z^3 - 8*z^2 + 2*z) * q^37 + (z^3 - z - 3) * q^39 + (5*z^3 - 5*z) * q^41 + (3*z^3 - 2*z^2 + 3*z) * q^43 + (2*z^2 - 6) * q^45 - 8*z^2 * q^47 + (2*z^3 - 2*z + 4) * q^49 + (-z^3 + z + 1) * q^51 + (-2*z^3 + z^2 - 2*z) * q^53 + (-z^2 + 3) * q^55 + (z^3 + z^2 + z) * q^57 + (z^3 - z - 13) * q^59 + (-z^3 + z + 2) * q^61 + (-2*z^3 - 4*z^2 - 2*z) * q^63 + (-z^3 + 6*z^2 - 2*z + 2) * q^65 + (5*z^3 - z^2 + 5*z) * q^67 + (4*z^3 - 4*z - 7) * q^69 + (-5*z^3 + 5*z - 4) * q^71 + (2*z^3 + 11*z^2 + 2*z) * q^73 + (z^3 + 4*z^2 + 7*z + 3) * q^75 + (z^3 + 2*z^2 + z) * q^77 + (-2*z^3 + 2*z - 2) * q^79 + (6*z^3 - 6*z - 1) * q^81 + (-6*z^3 + 6*z^2 - 6*z) * q^83 + (-z^3 - 2*z) * q^85 + (-3*z^3 - 5*z^2 - 3*z) * q^87 + (3*z^3 - 3*z + 8) * q^89 + (z^3 - z - 3) * q^91 + (3*z^3 + 4*z^2 + 3*z) * q^93 + (-2*z^3 + z) * q^95 + (-4*z^3 - 2*z^2 - 4*z) * q^97 - 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 4 q^{15} + 4 q^{19} - 12 q^{21} + 16 q^{25} - 4 q^{29} + 8 q^{31} + 4 q^{35} - 12 q^{39} - 24 q^{45} + 16 q^{49} + 4 q^{51} + 12 q^{55} - 52 q^{59} + 8 q^{61} + 8 q^{65} - 28 q^{69} - 16 q^{71} + 12 q^{75} - 8 q^{79} - 4 q^{81} + 32 q^{89} - 12 q^{91} - 16 q^{99}+O(q^{100})$$ 4 * q + 4 * q^15 + 4 * q^19 - 12 * q^21 + 16 * q^25 - 4 * q^29 + 8 * q^31 + 4 * q^35 - 12 * q^39 - 24 * q^45 + 16 * q^49 + 4 * q^51 + 12 * q^55 - 52 * q^59 + 8 * q^61 + 8 * q^65 - 28 * q^69 - 16 * q^71 + 12 * q^75 - 8 * q^79 - 4 * q^81 + 32 * q^89 - 12 * q^91 - 16 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/760\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$381$$ $$401$$ $$457$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
609.1
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
0 2.41421i 0 2.12132 + 0.707107i 0 2.41421i 0 −2.82843 0
609.2 0 0.414214i 0 −2.12132 + 0.707107i 0 0.414214i 0 2.82843 0
609.3 0 0.414214i 0 −2.12132 0.707107i 0 0.414214i 0 2.82843 0
609.4 0 2.41421i 0 2.12132 0.707107i 0 2.41421i 0 −2.82843 0
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.d.c 4
4.b odd 2 1 1520.2.d.d 4
5.b even 2 1 inner 760.2.d.c 4
5.c odd 4 1 3800.2.a.l 2
5.c odd 4 1 3800.2.a.p 2
20.d odd 2 1 1520.2.d.d 4
20.e even 4 1 7600.2.a.x 2
20.e even 4 1 7600.2.a.bc 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.d.c 4 1.a even 1 1 trivial
760.2.d.c 4 5.b even 2 1 inner
1520.2.d.d 4 4.b odd 2 1
1520.2.d.d 4 20.d odd 2 1
3800.2.a.l 2 5.c odd 4 1
3800.2.a.p 2 5.c odd 4 1
7600.2.a.x 2 20.e even 4 1
7600.2.a.bc 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 6T_{3}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(760, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 6T^{2} + 1$$
$5$ $$T^{4} - 8T^{2} + 25$$
$7$ $$T^{4} + 6T^{2} + 1$$
$11$ $$(T^{2} - 2)^{2}$$
$13$ $$T^{4} + 18T^{2} + 49$$
$17$ $$(T^{2} + 1)^{2}$$
$19$ $$(T - 1)^{4}$$
$23$ $$T^{4} + 38T^{2} + 289$$
$29$ $$(T^{2} + 2 T - 7)^{2}$$
$31$ $$(T^{2} - 4 T + 2)^{2}$$
$37$ $$T^{4} + 144T^{2} + 3136$$
$41$ $$(T^{2} - 50)^{2}$$
$43$ $$T^{4} + 44T^{2} + 196$$
$47$ $$(T^{2} + 64)^{2}$$
$53$ $$T^{4} + 18T^{2} + 49$$
$59$ $$(T^{2} + 26 T + 167)^{2}$$
$61$ $$(T^{2} - 4 T + 2)^{2}$$
$67$ $$T^{4} + 102T^{2} + 2401$$
$71$ $$(T^{2} + 8 T - 34)^{2}$$
$73$ $$T^{4} + 258 T^{2} + 12769$$
$79$ $$(T^{2} + 4 T - 4)^{2}$$
$83$ $$T^{4} + 216T^{2} + 1296$$
$89$ $$(T^{2} - 16 T + 46)^{2}$$
$97$ $$T^{4} + 72T^{2} + 784$$