Properties

Label 255.2.o.a
Level $255$
Weight $2$
Character orbit 255.o
Analytic conductor $2.036$
Analytic rank $0$
Dimension $8$
CM discriminant -51
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [255,2,Mod(152,255)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(255, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("255.152");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 255 = 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 255.o (of order \(4\), degree \(2\), 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: \(2.03618525154\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.443364212736.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + 2 \beta_{3} q^{4} + (\beta_{4} + \beta_{2}) q^{5} - 3 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + 2 \beta_{3} q^{4} + (\beta_{4} + \beta_{2}) q^{5} - 3 \beta_{3} q^{9} + (\beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_{2}) q^{11} + 2 \beta_{5} q^{12} + ( - \beta_{6} + \beta_1 - 1) q^{13} + (\beta_{6} - \beta_{3}) q^{15} - 4 q^{16} + (\beta_{4} + 2 \beta_{2}) q^{17} + ( - 2 \beta_1 + 1) q^{19} + (2 \beta_{7} + 2 \beta_{5}) q^{20} + ( - 2 \beta_{7} - \beta_{5}) q^{23} + (\beta_{6} + 4 \beta_{3}) q^{25} - 3 \beta_{5} q^{27} + ( - 2 \beta_{7} - 3 \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{29} + ( - \beta_{6} - 5 \beta_{3} + \beta_1 + 4) q^{33} + 6 q^{36} + (3 \beta_{7} + 2 \beta_{5} + \beta_{4} + 3 \beta_{2}) q^{39} + (\beta_{7} - 3 \beta_{5} - 4 \beta_{4} - \beta_{2}) q^{41} + ( - \beta_{6} + 5 \beta_{3} + \beta_1 - 6) q^{43} + ( - 2 \beta_{7} + 2 \beta_{5} - 4 \beta_{4} - 2 \beta_{2}) q^{44} + ( - 3 \beta_{7} - 3 \beta_{5}) q^{45} - 4 \beta_{4} q^{48} + 7 \beta_{3} q^{49} + (2 \beta_{6} + \beta_{3}) q^{51} + ( - 2 \beta_{6} - 2 \beta_{3} - 2 \beta_1) q^{52} + (\beta_{6} - 6 \beta_{3} + 2 \beta_1 - 3) q^{55} + ( - 3 \beta_{4} - 6 \beta_{2}) q^{57} + (2 \beta_1 + 2) q^{60} - 8 \beta_{3} q^{64} + (2 \beta_{7} - 3 \beta_{5} - 4 \beta_{4} + \beta_{2}) q^{65} + ( - 2 \beta_{6} + 3 \beta_{3} - 2 \beta_1 + 5) q^{67} + (4 \beta_{7} + 2 \beta_{5}) q^{68} + ( - 2 \beta_1 + 1) q^{69} + ( - 4 \beta_{7} - 3 \beta_{5} + \beta_{4} + 4 \beta_{2}) q^{71} + ( - 3 \beta_{7} + 2 \beta_{5}) q^{75} + (4 \beta_{6} + 2 \beta_{3}) q^{76} + ( - 4 \beta_{4} - 4 \beta_{2}) q^{80} - 9 q^{81} + (\beta_{6} + 9 \beta_{3}) q^{85} + ( - 2 \beta_{6} - 7 \beta_{3} - 2 \beta_1 - 5) q^{87} + (2 \beta_{4} + 4 \beta_{2}) q^{92} + (7 \beta_{4} - 3 \beta_{2}) q^{95} + (3 \beta_{7} - 3 \beta_{5} + 6 \beta_{4} + 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{13} - 32 q^{16} + 36 q^{33} + 48 q^{36} - 44 q^{43} - 8 q^{52} - 16 q^{55} + 24 q^{60} + 32 q^{67} - 72 q^{81} - 48 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{4} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} + 3 ) / 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 24\nu ) / 35 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 24\nu^{2} ) / 175 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 11\nu ) / 35 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} + 127\nu^{3} ) / 875 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{6} + 79\nu^{2} ) / 175 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - \nu^{3} ) / 125 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 7\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -24\beta_{4} + 11\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -24\beta_{6} + 79\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 127\beta_{7} + 7\beta_{5} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/255\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(86\) \(241\)
\(\chi(n)\) \(\beta_{3}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
152.1
−2.07011 0.845366i
0.845366 + 2.07011i
−0.845366 2.07011i
2.07011 + 0.845366i
−2.07011 + 0.845366i
0.845366 2.07011i
−0.845366 + 2.07011i
2.07011 0.845366i
0 −1.22474 + 1.22474i 2.00000i −2.07011 0.845366i 0 0 0 3.00000i 0
152.2 0 −1.22474 + 1.22474i 2.00000i 0.845366 + 2.07011i 0 0 0 3.00000i 0
152.3 0 1.22474 1.22474i 2.00000i −0.845366 2.07011i 0 0 0 3.00000i 0
152.4 0 1.22474 1.22474i 2.00000i 2.07011 + 0.845366i 0 0 0 3.00000i 0
203.1 0 −1.22474 1.22474i 2.00000i −2.07011 + 0.845366i 0 0 0 3.00000i 0
203.2 0 −1.22474 1.22474i 2.00000i 0.845366 2.07011i 0 0 0 3.00000i 0
203.3 0 1.22474 + 1.22474i 2.00000i −0.845366 + 2.07011i 0 0 0 3.00000i 0
203.4 0 1.22474 + 1.22474i 2.00000i 2.07011 0.845366i 0 0 0 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 152.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
51.c odd 2 1 CM by \(\Q(\sqrt{-51}) \)
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner
17.b even 2 1 inner
85.g odd 4 1 inner
255.o even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 255.2.o.a 8
3.b odd 2 1 inner 255.2.o.a 8
5.c odd 4 1 inner 255.2.o.a 8
15.e even 4 1 inner 255.2.o.a 8
17.b even 2 1 inner 255.2.o.a 8
51.c odd 2 1 CM 255.2.o.a 8
85.g odd 4 1 inner 255.2.o.a 8
255.o even 4 1 inner 255.2.o.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
255.2.o.a 8 1.a even 1 1 trivial
255.2.o.a 8 3.b odd 2 1 inner
255.2.o.a 8 5.c odd 4 1 inner
255.2.o.a 8 15.e even 4 1 inner
255.2.o.a 8 17.b even 2 1 inner
255.2.o.a 8 51.c odd 2 1 CM
255.2.o.a 8 85.g odd 4 1 inner
255.2.o.a 8 255.o even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(255, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{4} + 625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 44 T^{2} + 25)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} + 2 T^{2} - 50 T + 625)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 289)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 51)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 289)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 116 T^{2} + 100)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 164 T^{2} + 4225)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 22 T^{3} + 242 T^{2} + 770 T + 1225)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 16 T^{3} + 128 T^{2} + 1120 T + 4900)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 284 T^{2} + 16900)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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