Properties

Label 775.2.bj.a
Level $775$
Weight $2$
Character orbit 775.bj
Analytic conductor $6.188$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [775,2,Mod(57,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.57");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.bj (of order \(12\), degree \(4\), 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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{3} - 2 \zeta_{12}^{3} q^{4} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 1) q^{7} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{3} - 2 \zeta_{12}^{3} q^{4} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 1) q^{7} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{9} + ( - \zeta_{12}^{2} + 2) q^{11} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{12} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{13} - 4 q^{16} + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{17} - 2 \zeta_{12} q^{19} + (4 \zeta_{12}^{2} - 8) q^{21} + (\zeta_{12}^{3} + 1) q^{23} + (4 \zeta_{12}^{3} - 4) q^{27} + ( - 4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{28} + ( - 5 \zeta_{12}^{3} + 10 \zeta_{12}) q^{29} + ( - 5 \zeta_{12}^{2} + 6) q^{31} + (2 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} - 2) q^{33} - 10 \zeta_{12}^{2} q^{36} + ( - 6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{37} - 12 \zeta_{12}^{3} q^{39} - 9 \zeta_{12}^{2} q^{41} + ( - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{43} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{44} + (5 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 10 \zeta_{12} + 5) q^{47} + ( - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 8) q^{48} + (\zeta_{12}^{3} - \zeta_{12}) q^{49} + 4 \zeta_{12}^{2} q^{51} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 6 \zeta_{12}) q^{52} + ( - 7 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 7 \zeta_{12}) q^{53} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12}) q^{57} + 3 \zeta_{12} q^{59} + (10 \zeta_{12}^{2} - 5) q^{61} + ( - 5 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 10 \zeta_{12} + 5) q^{63} + 8 \zeta_{12}^{3} q^{64} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 2) q^{67} + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{68} - 4 \zeta_{12} q^{69} - 9 \zeta_{12}^{2} q^{71} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{73} + (4 \zeta_{12}^{2} - 4) q^{76} + ( - 3 \zeta_{12}^{3} + 3) q^{77} + ( - 14 \zeta_{12}^{3} + 7 \zeta_{12}) q^{79} + ( - \zeta_{12}^{2} + 1) q^{81} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{83} + (8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{84} + (20 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 10 \zeta_{12} - 10) q^{87} + (5 \zeta_{12}^{3} - 10 \zeta_{12}) q^{89} + (12 \zeta_{12}^{2} - 6) q^{91} + ( - 2 \zeta_{12}^{3} + 2) q^{92} + (10 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 12 \zeta_{12} - 2) q^{93} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 6 q^{7} + 6 q^{11} - 8 q^{12} + 6 q^{13} - 16 q^{16} + 2 q^{17} - 24 q^{21} + 4 q^{23} - 16 q^{27} - 12 q^{28} + 14 q^{31} - 20 q^{36} + 12 q^{37} - 18 q^{41} + 6 q^{43} + 16 q^{48} + 8 q^{51} + 12 q^{52} - 14 q^{53} + 8 q^{57} + 6 q^{67} - 4 q^{68} - 18 q^{71} + 6 q^{73} - 8 q^{76} + 12 q^{77} + 2 q^{81} - 4 q^{83} - 60 q^{87} + 8 q^{92} + 16 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(251\) \(652\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(\zeta_{12}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0 −2.73205 + 0.732051i 2.00000i 0 0 2.36603 0.633975i 0 4.33013 2.50000i 0
68.1 0 −2.73205 0.732051i 2.00000i 0 0 2.36603 + 0.633975i 0 4.33013 + 2.50000i 0
243.1 0 0.732051 + 2.73205i 2.00000i 0 0 0.633975 + 2.36603i 0 −4.33013 + 2.50000i 0
657.1 0 0.732051 2.73205i 2.00000i 0 0 0.633975 2.36603i 0 −4.33013 2.50000i 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
5.c odd 4 1 inner
155.i odd 6 1 inner
155.p even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.bj.a 4
5.b even 2 1 775.2.bj.b yes 4
5.c odd 4 1 inner 775.2.bj.a 4
5.c odd 4 1 775.2.bj.b yes 4
31.e odd 6 1 775.2.bj.b yes 4
155.i odd 6 1 inner 775.2.bj.a 4
155.p even 12 1 inner 775.2.bj.a 4
155.p even 12 1 775.2.bj.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.bj.a 4 1.a even 1 1 trivial
775.2.bj.a 4 5.c odd 4 1 inner
775.2.bj.a 4 155.i odd 6 1 inner
775.2.bj.a 4 155.p even 12 1 inner
775.2.bj.b yes 4 5.b even 2 1
775.2.bj.b yes 4 5.c odd 4 1
775.2.bj.b yes 4 31.e odd 6 1
775.2.bj.b yes 4 155.p even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3}^{4} + 4T_{3}^{3} + 8T_{3}^{2} + 32T_{3} + 64 \) Copy content Toggle raw display
\( T_{7}^{4} - 6T_{7}^{3} + 18T_{7}^{2} - 36T_{7} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} + 8 T^{2} + 32 T + 64 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 6 T^{3} + 18 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + 18 T^{2} - 108 T + 324 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$19$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 7 T + 31)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 5184 \) Copy content Toggle raw display
$41$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 6 T^{3} + 18 T^{2} - 108 T + 324 \) Copy content Toggle raw display
$47$ \( T^{4} + 22500 \) Copy content Toggle raw display
$53$ \( T^{4} + 14 T^{3} + 98 T^{2} + \cdots + 9604 \) Copy content Toggle raw display
$59$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$61$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 6 T^{3} + 18 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$71$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 6 T^{3} + 18 T^{2} - 108 T + 324 \) Copy content Toggle raw display
$79$ \( T^{4} + 147 T^{2} + 21609 \) Copy content Toggle raw display
$83$ \( T^{4} + 4 T^{3} + 8 T^{2} + 32 T + 64 \) Copy content Toggle raw display
$89$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
show more
show less