# Properties

 Label 775.1.w.a Level $775$ Weight $1$ Character orbit 775.w Analytic conductor $0.387$ Analytic rank $0$ Dimension $4$ Projective image $D_{5}$ CM discriminant -31 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$775 = 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 775.w (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.386775384791$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.375390625.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{10} + 1) q^{2} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{4} + \zeta_{10}^{2} q^{5} + (\zeta_{10}^{4} - \zeta_{10}) q^{7} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{8} + \zeta_{10}^{2} q^{9}+O(q^{10})$$ q + (-z + 1) * q^2 + (z^2 - z + 1) * q^4 + z^2 * q^5 + (z^4 - z) * q^7 + (-z^3 + z^2 - z + 1) * q^8 + z^2 * q^9 $$q + ( - \zeta_{10} + 1) q^{2} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{4} + \zeta_{10}^{2} q^{5} + (\zeta_{10}^{4} - \zeta_{10}) q^{7} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{8} + \zeta_{10}^{2} q^{9} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{10} + (\zeta_{10}^{4} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{14} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{16} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{18} + ( - \zeta_{10}^{3} + 1) q^{19} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{20} + \zeta_{10}^{4} q^{25} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{28} + \zeta_{10}^{4} q^{31} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{32} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{35} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{36} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} + 1) q^{38} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{40} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{41} + \zeta_{10}^{4} q^{45} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{47} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{49} + (\zeta_{10}^{4} + 1) q^{50} + (2 \zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{56} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{59} + (\zeta_{10}^{4} + 1) q^{62} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{63} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10} - 1) q^{64} + ( - \zeta_{10}^{3} + 1) q^{67} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{70} - \zeta_{10} q^{71} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{72} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{76} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{80} + \zeta_{10}^{4} q^{81} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{82} + (\zeta_{10}^{4} + 1) q^{90} + (2 \zeta_{10}^{4} - \zeta_{10}^{3} + 1) q^{94} + (\zeta_{10}^{2} + 1) q^{95} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{97} + (\zeta_{10}^{4} - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{98} +O(q^{100})$$ q + (-z + 1) * q^2 + (z^2 - z + 1) * q^4 + z^2 * q^5 + (z^4 - z) * q^7 + (-z^3 + z^2 - z + 1) * q^8 + z^2 * q^9 + (-z^3 + z^2) * q^10 + (z^4 + z^2 - z + 1) * q^14 + (z^4 - z^3 + z^2 - z + 1) * q^16 + (-z^3 + z^2) * q^18 + (-z^3 + 1) * q^19 + (z^4 - z^3 + z^2) * q^20 + z^4 * q^25 + (z^4 - z^3 + z^2 - z + 1) * q^28 + z^4 * q^31 + (z^4 - z^3 + z^2 - z + 2) * q^32 + (-z^3 - z) * q^35 + (z^4 - z^3 + z^2) * q^36 + (z^4 - z^3 - z + 1) * q^38 + (z^4 - z^3 + z^2 + 1) * q^40 + (-z^3 - z) * q^41 + z^4 * q^45 + (z^4 - z^3) * q^47 + (-z^3 + z^2 + 1) * q^49 + (z^4 + 1) * q^50 + (2*z^4 - z^3 + z^2 - z + 1) * q^56 + (-z^3 - z) * q^59 + (z^4 + 1) * q^62 + (-z^3 - z) * q^63 + (z^4 - z^3 + z^2 - 2*z - 1) * q^64 + (-z^3 + 1) * q^67 + (z^4 - z^3 + z^2 - z) * q^70 - z * q^71 + (z^4 - z^3 + z^2 + 1) * q^72 + (z^4 - z^3 + z^2 - z + 1) * q^76 + (z^4 - z^3 + z^2 - z + 1) * q^80 + z^4 * q^81 + (z^4 - z^3 + z^2 - z) * q^82 + (z^4 + 1) * q^90 + (2*z^4 - z^3 + 1) * q^94 + (z^2 + 1) * q^95 + (z^4 - z^3) * q^97 + (z^4 - 2*z^3 + z^2 - z + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} + q^{8} - q^{9}+O(q^{10})$$ 4 * q + 3 * q^2 + 2 * q^4 - q^5 - 2 * q^7 + q^8 - q^9 $$4 q + 3 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} + q^{8} - q^{9} - 2 q^{10} + q^{14} - 2 q^{18} + 3 q^{19} - 3 q^{20} - q^{25} - q^{28} - q^{31} + 4 q^{32} - 2 q^{35} - 3 q^{36} + q^{38} + q^{40} - 2 q^{41} - q^{45} - 2 q^{47} + 2 q^{49} + 3 q^{50} - 3 q^{56} - 2 q^{59} + 3 q^{62} - 2 q^{63} + 3 q^{64} + 3 q^{67} - 4 q^{70} - 2 q^{71} + q^{72} + 4 q^{76} - q^{81} - 4 q^{82} + 3 q^{90} + q^{94} + 3 q^{95} - 2 q^{97} - q^{98}+O(q^{100})$$ 4 * q + 3 * q^2 + 2 * q^4 - q^5 - 2 * q^7 + q^8 - q^9 - 2 * q^10 + q^14 - 2 * q^18 + 3 * q^19 - 3 * q^20 - q^25 - q^28 - q^31 + 4 * q^32 - 2 * q^35 - 3 * q^36 + q^38 + q^40 - 2 * q^41 - q^45 - 2 * q^47 + 2 * q^49 + 3 * q^50 - 3 * q^56 - 2 * q^59 + 3 * q^62 - 2 * q^63 + 3 * q^64 + 3 * q^67 - 4 * q^70 - 2 * q^71 + q^72 + 4 * q^76 - q^81 - 4 * q^82 + 3 * q^90 + q^94 + 3 * q^95 - 2 * q^97 - q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/775\mathbb{Z}\right)^\times$$.

 $$n$$ $$251$$ $$652$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 −0.309017 + 0.951057i −0.309017 − 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i
1.30902 0.951057i 0 0.500000 1.53884i −0.809017 0.587785i 0 0.618034 −0.309017 0.951057i −0.809017 0.587785i −1.61803
216.1 1.30902 + 0.951057i 0 0.500000 + 1.53884i −0.809017 + 0.587785i 0 0.618034 −0.309017 + 0.951057i −0.809017 + 0.587785i −1.61803
371.1 0.190983 0.587785i 0 0.500000 + 0.363271i 0.309017 + 0.951057i 0 −1.61803 0.809017 0.587785i 0.309017 + 0.951057i 0.618034
681.1 0.190983 + 0.587785i 0 0.500000 0.363271i 0.309017 0.951057i 0 −1.61803 0.809017 + 0.587785i 0.309017 0.951057i 0.618034
 $$n$$: 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 $$\Q(\sqrt{-31})$$
25.d even 5 1 inner
775.w odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.1.w.a 4
5.b even 2 1 3875.1.w.a 4
5.c odd 4 2 3875.1.z.c 8
25.d even 5 1 inner 775.1.w.a 4
25.e even 10 1 3875.1.w.a 4
25.f odd 20 2 3875.1.z.c 8
31.b odd 2 1 CM 775.1.w.a 4
155.c odd 2 1 3875.1.w.a 4
155.f even 4 2 3875.1.z.c 8
775.w odd 10 1 inner 775.1.w.a 4
775.z odd 10 1 3875.1.w.a 4
775.ca even 20 2 3875.1.z.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.1.w.a 4 1.a even 1 1 trivial
775.1.w.a 4 25.d even 5 1 inner
775.1.w.a 4 31.b odd 2 1 CM
775.1.w.a 4 775.w odd 10 1 inner
3875.1.w.a 4 5.b even 2 1
3875.1.w.a 4 25.e even 10 1
3875.1.w.a 4 155.c odd 2 1
3875.1.w.a 4 775.z odd 10 1
3875.1.z.c 8 5.c odd 4 2
3875.1.z.c 8 25.f odd 20 2
3875.1.z.c 8 155.f even 4 2
3875.1.z.c 8 775.ca even 20 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 3T_{2}^{3} + 4T_{2}^{2} - 2T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(775, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$7$ $$(T^{2} + T - 1)^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$37$ $$T^{4}$$
$41$ $$T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1$$
$43$ $$T^{4}$$
$47$ $$T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1$$
$53$ $$T^{4}$$
$59$ $$T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1$$
$61$ $$T^{4}$$
$67$ $$T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1$$
$71$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1$$