↑

Properties

Label 1064.2.r.h
Level $1064$
Weight $2$
Character orbit 1064.r
Analytic conductor $8.496$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1064,2,Mod(505,1064)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1064, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1064.505");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1064 = 2^{3} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1064.r (of order \(3\), 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: \(8.49608277506\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.3098592225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 10x^{6} + 29x^{4} + 23x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + ( - \beta_{7} - \beta_{4}) q^{5} - q^{7} + ( - \beta_{5} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + ( - \beta_{7} - \beta_{4}) q^{5} - q^{7} + ( - \beta_{5} - \beta_{2}) q^{9} + ( - \beta_{3} - 2) q^{11} + (2 \beta_{6} + \beta_{5} + \cdots - 2 \beta_1) q^{13}+ \cdots + (\beta_{6} + 2 \beta_{5} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} + q^{5} - 8 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} + q^{5} - 8 q^{7} - 6 q^{9} - 12 q^{11} - 6 q^{13} - q^{17} - 28 q^{19} - 2 q^{21} - 4 q^{23} - 13 q^{25} - 10 q^{27} - 8 q^{29} - 14 q^{31} + 5 q^{33} - q^{35} - 6 q^{37} - 58 q^{39} + 14 q^{41} - 8 q^{43} - 8 q^{45} + 10 q^{47} + 8 q^{49} - 9 q^{51} - 29 q^{53} + q^{55} - 16 q^{57} + q^{59} - 7 q^{61} + 6 q^{63} + 4 q^{65} + 12 q^{67} + 46 q^{69} + 6 q^{71} + 6 q^{73} - 116 q^{75} + 12 q^{77} - 15 q^{79} + 16 q^{81} - 46 q^{83} + 36 q^{85} + 24 q^{87} - 26 q^{89} + 6 q^{91} + 14 q^{93} - 8 q^{95} - 5 q^{97} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 10x^{6} + 29x^{4} + 23x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 10\nu^{5} + 27\nu^{3} + 13\nu + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{4} - 6\nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + \nu^{5} + 9\nu^{4} + 8\nu^{3} + 21\nu^{2} + 14\nu + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} + 28\nu^{5} + 2\nu^{4} + 71\nu^{3} + 12\nu^{2} + 41\nu + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} - 28\nu^{5} - 71\nu^{3} + 2\nu^{2} - 35\nu + 6 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{6} - 9\nu^{4} - 21\nu^{2} - 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 2\beta_{6} + 2\beta_{5} + \beta_{3} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 10\beta_{6} - 8\beta_{5} + 2\beta_{4} - 4\beta_{3} - 6\beta_{2} + 5\beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{3} - 6\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{7} + 52\beta_{6} + 36\beta_{5} - 10\beta_{4} + 18\beta_{3} + 48\beta_{2} - 26\beta _1 - 24 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} + 9\beta_{3} + 33\beta _1 - 62 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 23\beta_{7} - 276\beta_{6} - 170\beta_{5} + 46\beta_{4} - 85\beta_{3} - 306\beta_{2} + 138\beta _1 + 153 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1064\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(799\) \(913\) \(1009\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
505.1
2.31840i
− 1.87187i
− 0.937901i
0.491369i
− 2.31840i
1.87187i
0.937901i
− 0.491369i
0 −1.18750 + 2.05681i 0 −0.0732113 + 0.126806i 0 −1.00000 0 −1.32030 − 2.28683i 0
505.2 0 −0.251952 + 0.436394i 0 0.947736 − 1.64153i 0 −1.00000 0 1.37304 + 2.37818i 0
505.3 0 1.06017 − 1.83627i 0 −2.09466 + 3.62805i 0 −1.00000 0 −0.747925 − 1.29544i 0
505.4 0 1.37928 − 2.38898i 0 1.72013 − 2.97935i 0 −1.00000 0 −2.30482 − 3.99206i 0
729.1 0 −1.18750 − 2.05681i 0 −0.0732113 − 0.126806i 0 −1.00000 0 −1.32030 + 2.28683i 0
729.2 0 −0.251952 − 0.436394i 0 0.947736 + 1.64153i 0 −1.00000 0 1.37304 − 2.37818i 0
729.3 0 1.06017 + 1.83627i 0 −2.09466 − 3.62805i 0 −1.00000 0 −0.747925 + 1.29544i 0
729.4 0 1.37928 + 2.38898i 0 1.72013 + 2.97935i 0 −1.00000 0 −2.30482 + 3.99206i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 505.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1064.2.r.h ✓ 8
19.c even 3 1 inner 1064.2.r.h ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.r.h ✓ 8 1.a even 1 1 trivial
1064.2.r.h ✓ 8 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1064, [\chi])\):

\( T_{3}^{8} - 2T_{3}^{7} + 11T_{3}^{6} - 8T_{3}^{5} + 64T_{3}^{4} - 49T_{3}^{3} + 170T_{3}^{2} + 77T_{3} + 49 \) Copy content Toggle raw display
\( T_{11}^{4} + 6T_{11}^{3} - T_{11}^{2} - 15T_{11} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{8} - T^{7} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T + 1)^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 6 T^{3} - T^{2} + \cdots - 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + \cdots + 77284 \) Copy content Toggle raw display
$17$ \( T^{8} + T^{7} + \cdots + 16900 \) Copy content Toggle raw display
$19$ \( (T^{2} + 7 T + 19)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 4 T^{7} + \cdots + 2116 \) Copy content Toggle raw display
$29$ \( T^{8} + 8 T^{7} + \cdots + 10000 \) Copy content Toggle raw display
$31$ \( (T^{4} + 7 T^{3} - 52 T^{2} + \cdots - 50)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3 T^{3} + \cdots + 3622)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 14 T^{7} + \cdots + 46225 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$47$ \( T^{8} - 10 T^{7} + \cdots + 1893376 \) Copy content Toggle raw display
$53$ \( T^{8} + 29 T^{7} + \cdots + 3786916 \) Copy content Toggle raw display
$59$ \( T^{8} - T^{7} + \cdots + 69172489 \) Copy content Toggle raw display
$61$ \( T^{8} + 7 T^{7} + \cdots + 160000 \) Copy content Toggle raw display
$67$ \( T^{8} - 12 T^{7} + \cdots + 6558721 \) Copy content Toggle raw display
$71$ \( T^{8} - 6 T^{7} + \cdots + 306916 \) Copy content Toggle raw display
$73$ \( T^{8} - 6 T^{7} + \cdots + 38809 \) Copy content Toggle raw display
$79$ \( T^{8} + 15 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$83$ \( (T^{4} + 23 T^{3} + \cdots - 473)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 26 T^{7} + \cdots + 2930944 \) Copy content Toggle raw display
$97$ \( T^{8} + 5 T^{7} + \cdots + 3996001 \) Copy content Toggle raw display
show more
show less