# Properties

 Label 775.1.c.a Level $775$ Weight $1$ Character orbit 775.c Analytic conductor $0.387$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -31 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [775,1,Mod(774,775)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(775, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("775.774");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$775 = 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 775.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$0.386775384791$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 31) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.31.1 Artin image: $C_4\times S_3$ Artin field: Galois closure of 12.0.1803751953125.3

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

 $$f(q)$$ $$=$$ $$q - i q^{2} - i q^{7} - i q^{8} - q^{9} +O(q^{10})$$ q - z * q^2 - z * q^7 - z * q^8 - q^9 $$q - i q^{2} - i q^{7} - i q^{8} - q^{9} - q^{14} - q^{16} + i q^{18} + q^{19} + q^{31} - i q^{38} - q^{41} + i q^{47} - q^{56} + q^{59} - i q^{62} + i q^{63} - q^{64} + i q^{67} - q^{71} + i q^{72} + q^{81} + i q^{82} + 2 q^{94} - i q^{97} +O(q^{100})$$ q - z * q^2 - z * q^7 - z * q^8 - q^9 - q^14 - q^16 + z * q^18 + q^19 + q^31 - z * q^38 - q^41 + z * q^47 - q^56 + q^59 - z * q^62 + z * q^63 - q^64 + z * q^67 - q^71 + z * q^72 + q^81 + z * q^82 + 2 * q^94 - z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 2 q^{14} - 2 q^{16} + 2 q^{19} + 2 q^{31} - 2 q^{41} - 2 q^{56} + 2 q^{59} - 2 q^{64} - 2 q^{71} + 2 q^{81} + 4 q^{94}+O(q^{100})$$ 2 * q - 2 * q^9 - 2 * q^14 - 2 * q^16 + 2 * q^19 + 2 * q^31 - 2 * q^41 - 2 * q^56 + 2 * q^59 - 2 * q^64 - 2 * q^71 + 2 * q^81 + 4 * q^94

## Character values

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

 $$n$$ $$251$$ $$652$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
774.1
 1.00000i − 1.00000i
1.00000i 0 0 0 0 1.00000i 1.00000i −1.00000 0
774.2 1.00000i 0 0 0 0 1.00000i 1.00000i −1.00000 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
31.b odd 2 1 CM by $$\Q(\sqrt{-31})$$
5.b even 2 1 inner
155.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.1.c.a 2
5.b even 2 1 inner 775.1.c.a 2
5.c odd 4 1 31.1.b.a 1
5.c odd 4 1 775.1.d.b 1
15.e even 4 1 279.1.d.b 1
20.e even 4 1 496.1.e.a 1
31.b odd 2 1 CM 775.1.c.a 2
35.f even 4 1 1519.1.c.a 1
35.k even 12 2 1519.1.n.a 2
35.l odd 12 2 1519.1.n.b 2
40.i odd 4 1 1984.1.e.a 1
40.k even 4 1 1984.1.e.b 1
45.k odd 12 2 2511.1.m.e 2
45.l even 12 2 2511.1.m.a 2
55.e even 4 1 3751.1.d.b 1
55.k odd 20 4 3751.1.t.c 4
55.l even 20 4 3751.1.t.a 4
155.c odd 2 1 inner 775.1.c.a 2
155.f even 4 1 31.1.b.a 1
155.f even 4 1 775.1.d.b 1
155.o odd 12 2 961.1.e.a 2
155.p even 12 2 961.1.e.a 2
155.r even 20 4 961.1.f.a 4
155.s odd 20 4 961.1.f.a 4
155.w odd 60 8 961.1.h.a 8
155.x even 60 8 961.1.h.a 8
465.m odd 4 1 279.1.d.b 1
620.m odd 4 1 496.1.e.a 1
1085.o odd 4 1 1519.1.c.a 1
1085.cf even 12 2 1519.1.n.b 2
1085.cp odd 12 2 1519.1.n.a 2
1240.s odd 4 1 1984.1.e.b 1
1240.y even 4 1 1984.1.e.a 1
1395.cg odd 12 2 2511.1.m.a 2
1395.cm even 12 2 2511.1.m.e 2
1705.m odd 4 1 3751.1.d.b 1
1705.dk odd 20 4 3751.1.t.a 4
1705.dt even 20 4 3751.1.t.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 5.c odd 4 1
31.1.b.a 1 155.f even 4 1
279.1.d.b 1 15.e even 4 1
279.1.d.b 1 465.m odd 4 1
496.1.e.a 1 20.e even 4 1
496.1.e.a 1 620.m odd 4 1
775.1.c.a 2 1.a even 1 1 trivial
775.1.c.a 2 5.b even 2 1 inner
775.1.c.a 2 31.b odd 2 1 CM
775.1.c.a 2 155.c odd 2 1 inner
775.1.d.b 1 5.c odd 4 1
775.1.d.b 1 155.f even 4 1
961.1.e.a 2 155.o odd 12 2
961.1.e.a 2 155.p even 12 2
961.1.f.a 4 155.r even 20 4
961.1.f.a 4 155.s odd 20 4
961.1.h.a 8 155.w odd 60 8
961.1.h.a 8 155.x even 60 8
1519.1.c.a 1 35.f even 4 1
1519.1.c.a 1 1085.o odd 4 1
1519.1.n.a 2 35.k even 12 2
1519.1.n.a 2 1085.cp odd 12 2
1519.1.n.b 2 35.l odd 12 2
1519.1.n.b 2 1085.cf even 12 2
1984.1.e.a 1 40.i odd 4 1
1984.1.e.a 1 1240.y even 4 1
1984.1.e.b 1 40.k even 4 1
1984.1.e.b 1 1240.s odd 4 1
2511.1.m.a 2 45.l even 12 2
2511.1.m.a 2 1395.cg odd 12 2
2511.1.m.e 2 45.k odd 12 2
2511.1.m.e 2 1395.cm even 12 2
3751.1.d.b 1 55.e even 4 1
3751.1.d.b 1 1705.m odd 4 1
3751.1.t.a 4 55.l even 20 4
3751.1.t.a 4 1705.dk odd 20 4
3751.1.t.c 4 55.k odd 20 4
3751.1.t.c 4 1705.dt even 20 4

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(775, [\chi])$$.

## Hecke characteristic polynomials

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