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 $4$

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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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

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

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:

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

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