↑

Properties

Label 816.2.bd.e
Level $816$
Weight $2$
Character orbit 816.bd
Analytic conductor $6.516$
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] = [816,2,Mod(625,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.625");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.bd (of order \(4\), degree \(2\), 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: \(6.51579280494\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.836829184.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{7} + \beta_{4} + \beta_{3} - 1) q^{5} + (\beta_{7} + \beta_{6} - \beta_{3} + \cdots + \beta_1) q^{7}+ \cdots - \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + (\beta_{7} + \beta_{4} + \beta_{3} - 1) q^{5} + (\beta_{7} + \beta_{6} - \beta_{3} + \cdots + \beta_1) q^{7}+ \cdots + ( - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{5} + 4 q^{7} - 4 q^{13} - 4 q^{17} - 8 q^{21} + 16 q^{23} + 4 q^{29} + 8 q^{31} + 12 q^{33} - 8 q^{35} + 8 q^{37} - 8 q^{39} + 28 q^{41} + 4 q^{45} + 8 q^{47} + 4 q^{51} + 4 q^{55} + 16 q^{61} + 4 q^{63} + 16 q^{65} - 8 q^{67} - 12 q^{69} - 24 q^{71} + 20 q^{73} + 16 q^{75} - 20 q^{79} - 8 q^{81} - 32 q^{85} - 8 q^{89} + 36 q^{91} - 8 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + \nu^{4} + 11\nu^{3} + 7\nu^{2} + 26\nu + 6 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 11\nu^{3} + 7\nu^{2} - 26\nu + 6 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + \nu^{5} + 11\nu^{4} + 7\nu^{3} + 26\nu^{2} + 6\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - \nu^{5} + 11\nu^{4} - 7\nu^{3} + 26\nu^{2} - 6\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 11\nu^{4} + 30\nu^{2} + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 12\nu^{5} + 41\nu^{3} + 38\nu ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} - \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{6} + 7\beta_{5} + 7\beta_{4} + 4\beta_{3} + 4\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{5} + 11\beta_{4} + 7\beta_{3} - 7\beta_{2} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 51\beta_{6} - 47\beta_{5} - 47\beta_{4} - 44\beta_{3} - 44\beta_{2} - 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} + 91\beta_{5} - 91\beta_{4} - 43\beta_{3} + 43\beta_{2} - 181\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/816\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(511\) \(545\) \(613\)
\(\chi(n)\) \(-\beta_{7}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
625.1
1.65222i
− 2.06644i
− 0.222191i
2.63640i
− 1.65222i
2.06644i
0.222191i
− 2.63640i
0 −0.707107 − 0.707107i 0 −2.87540 − 2.87540i 0 −0.652223 + 0.652223i 0 1.00000i 0
625.2 0 −0.707107 − 0.707107i 0 −0.245916 − 0.245916i 0 3.06644 − 3.06644i 0 1.00000i 0
625.3 0 0.707107 + 0.707107i 0 −0.450006 − 0.450006i 0 1.22219 − 1.22219i 0 1.00000i 0
625.4 0 0.707107 + 0.707107i 0 1.57133 + 1.57133i 0 −1.63640 + 1.63640i 0 1.00000i 0
769.1 0 −0.707107 + 0.707107i 0 −2.87540 + 2.87540i 0 −0.652223 − 0.652223i 0 − 1.00000i 0
769.2 0 −0.707107 + 0.707107i 0 −0.245916 + 0.245916i 0 3.06644 + 3.06644i 0 − 1.00000i 0
769.3 0 0.707107 − 0.707107i 0 −0.450006 + 0.450006i 0 1.22219 + 1.22219i 0 − 1.00000i 0
769.4 0 0.707107 − 0.707107i 0 1.57133 − 1.57133i 0 −1.63640 − 1.63640i 0 − 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 625.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 816.2.bd.e 8
3.b odd 2 1 2448.2.be.x 8
4.b odd 2 1 51.2.e.a ✓ 8
12.b even 2 1 153.2.f.b 8
17.c even 4 1 inner 816.2.bd.e 8
51.f odd 4 1 2448.2.be.x 8
68.d odd 2 1 867.2.e.g 8
68.f odd 4 1 51.2.e.a ✓ 8
68.f odd 4 1 867.2.e.g 8
68.g odd 8 1 867.2.a.k 4
68.g odd 8 1 867.2.a.l 4
68.g odd 8 2 867.2.d.f 8
68.i even 16 4 867.2.h.i 16
68.i even 16 4 867.2.h.k 16
204.l even 4 1 153.2.f.b 8
204.p even 8 1 2601.2.a.be 4
204.p even 8 1 2601.2.a.bf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.e.a ✓ 8 4.b odd 2 1
51.2.e.a ✓ 8 68.f odd 4 1
153.2.f.b 8 12.b even 2 1
153.2.f.b 8 204.l even 4 1
816.2.bd.e 8 1.a even 1 1 trivial
816.2.bd.e 8 17.c even 4 1 inner
867.2.a.k 4 68.g odd 8 1
867.2.a.l 4 68.g odd 8 1
867.2.d.f 8 68.g odd 8 2
867.2.e.g 8 68.d odd 2 1
867.2.e.g 8 68.f odd 4 1
867.2.h.i 16 68.i even 16 4
867.2.h.k 16 68.i even 16 4
2448.2.be.x 8 3.b odd 2 1
2448.2.be.x 8 51.f odd 4 1
2601.2.a.be 4 204.p even 8 1
2601.2.a.bf 4 204.p even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 4T_{5}^{7} + 8T_{5}^{6} - 16T_{5}^{5} + 53T_{5}^{4} + 92T_{5}^{3} + 72T_{5}^{2} + 24T_{5} + 4 \) acting on \(S_{2}^{\mathrm{new}}(816, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{8} + 48 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} - 23 T^{2} + \cdots - 64)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{7} + \cdots + 83521 \) Copy content Toggle raw display
$19$ \( T^{8} + 50 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{8} - 16 T^{7} + \cdots + 33856 \) Copy content Toggle raw display
$29$ \( T^{8} - 4 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( T^{8} - 8 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$37$ \( T^{8} - 8 T^{7} + \cdots + 595984 \) Copy content Toggle raw display
$41$ \( T^{8} - 28 T^{7} + \cdots + 2775556 \) Copy content Toggle raw display
$43$ \( T^{8} + 162 T^{6} + \cdots + 1364224 \) Copy content Toggle raw display
$47$ \( (T^{4} - 4 T^{3} - 52 T^{2} + \cdots - 64)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 152 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$59$ \( T^{8} + 324 T^{6} + \cdots + 21827584 \) Copy content Toggle raw display
$61$ \( T^{8} - 16 T^{7} + \cdots + 126736 \) Copy content Toggle raw display
$67$ \( (T^{4} + 4 T^{3} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 24 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$73$ \( T^{8} - 20 T^{7} + \cdots + 107584 \) Copy content Toggle raw display
$79$ \( T^{8} + 20 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$83$ \( T^{8} + 344 T^{6} + \cdots + 27541504 \) Copy content Toggle raw display
$89$ \( (T^{4} + 4 T^{3} + \cdots + 1088)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 12 T^{7} + \cdots + 135424 \) Copy content Toggle raw display
show more
show less