Properties

Label 416.2.k.f
Level $416$
Weight $2$
Character orbit 416.k
Analytic conductor $3.322$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,2,Mod(31,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.k (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: \(3.32177672409\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 17x^{6} + 84x^{4} + 100x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} - \beta_{6} q^{5} + (\beta_{6} - \beta_{2}) q^{7} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} - \beta_{6} q^{5} + (\beta_{6} - \beta_{2}) q^{7} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 3) q^{9} + ( - \beta_{2} - \beta_1 - 1) q^{11} + ( - \beta_{6} - \beta_{4} + \cdots - \beta_1) q^{13}+ \cdots + (2 \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{7} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{7} - 20 q^{9} - 6 q^{11} - 2 q^{13} + 20 q^{15} - 6 q^{19} - 4 q^{21} + 16 q^{23} + 4 q^{29} - 26 q^{31} + 16 q^{33} - 8 q^{39} - 24 q^{41} + 44 q^{43} + 8 q^{45} - 14 q^{47} - 44 q^{51} + 28 q^{57} + 22 q^{59} - 24 q^{61} + 38 q^{63} - 8 q^{65} - 2 q^{67} + 14 q^{71} + 20 q^{73} - 76 q^{75} + 32 q^{81} + 6 q^{83} + 20 q^{85} + 8 q^{89} - 42 q^{91} - 28 q^{93} - 12 q^{95} - 8 q^{97} + 66 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 17x^{6} + 84x^{4} + 100x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 15\nu^{5} + 62\nu^{3} + 48\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 11\nu^{4} + 26\nu^{2} + 8\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 11\nu^{4} - 26\nu^{2} + 8\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 19\nu^{5} + 106\nu^{3} + 136\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{4} + 9\nu^{2} + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 19\nu^{5} + 11\nu^{4} - 98\nu^{3} + 18\nu^{2} - 80\nu - 24 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - \nu^{6} - 19\nu^{5} - 11\nu^{4} - 98\nu^{3} - 18\nu^{2} - 80\nu + 24 ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} - \beta_{3} + \beta_{2} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} + 4\beta_{4} - 7\beta_{3} - 7\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -18\beta_{7} + 18\beta_{6} + 4\beta_{5} + 9\beta_{3} - 9\beta_{2} + 56 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -22\beta_{7} - 22\beta_{6} - 36\beta_{4} + 55\beta_{3} + 55\beta_{2} - 8\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 146\beta_{7} - 146\beta_{6} - 44\beta_{5} - 81\beta_{3} + 81\beta_{2} - 424 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 206\beta_{7} + 206\beta_{6} + 292\beta_{4} - 439\beta_{3} - 439\beta_{2} + 152\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
2.65328i
1.20241i
2.88540i
0.434529i
0.434529i
2.88540i
1.20241i
2.65328i
0 2.89949i 0 −2.84658 + 2.84658i 0 0.193303 0.193303i 0 −5.40706 0
31.2 0 1.46091i 0 1.87831 1.87831i 0 −0.675897 + 0.675897i 0 0.865736 0
31.3 0 1.19225i 0 −0.720056 + 0.720056i 0 3.60545 3.60545i 0 1.57854 0
31.4 0 3.16816i 0 1.68833 1.68833i 0 −2.12286 + 2.12286i 0 −7.03721 0
255.1 0 3.16816i 0 1.68833 + 1.68833i 0 −2.12286 2.12286i 0 −7.03721 0
255.2 0 1.19225i 0 −0.720056 0.720056i 0 3.60545 + 3.60545i 0 1.57854 0
255.3 0 1.46091i 0 1.87831 + 1.87831i 0 −0.675897 0.675897i 0 0.865736 0
255.4 0 2.89949i 0 −2.84658 2.84658i 0 0.193303 + 0.193303i 0 −5.40706 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.k.f yes 8
4.b odd 2 1 416.2.k.e 8
8.b even 2 1 832.2.k.i 8
8.d odd 2 1 832.2.k.g 8
13.d odd 4 1 416.2.k.e 8
52.f even 4 1 inner 416.2.k.f yes 8
104.j odd 4 1 832.2.k.g 8
104.m even 4 1 832.2.k.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.k.e 8 4.b odd 2 1
416.2.k.e 8 13.d odd 4 1
416.2.k.f yes 8 1.a even 1 1 trivial
416.2.k.f yes 8 52.f even 4 1 inner
832.2.k.g 8 8.d odd 2 1
832.2.k.g 8 104.j odd 4 1
832.2.k.i 8 8.b even 2 1
832.2.k.i 8 104.m even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(416, [\chi])\):

\( T_{3}^{8} + 22T_{3}^{6} + 153T_{3}^{4} + 356T_{3}^{2} + 256 \) Copy content Toggle raw display
\( T_{7}^{8} - 2T_{7}^{7} + 2T_{7}^{6} + 48T_{7}^{5} + 281T_{7}^{4} + 246T_{7}^{3} + 98T_{7}^{2} - 56T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{8} + 6T_{11}^{7} + 18T_{11}^{6} - 24T_{11}^{5} + 84T_{11}^{4} + 312T_{11}^{3} + 648T_{11}^{2} - 288T_{11} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 22 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{8} - 16 T^{5} + \cdots + 676 \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{8} + 2 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 38 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T^{4} - 8 T^{3} + \cdots - 256)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 2 T^{3} + \cdots + 464)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 26 T^{7} + \cdots + 43264 \) Copy content Toggle raw display
$37$ \( T^{8} + 8 T^{5} + \cdots + 669124 \) Copy content Toggle raw display
$41$ \( T^{8} + 24 T^{7} + \cdots + 652864 \) Copy content Toggle raw display
$43$ \( (T^{4} - 22 T^{3} + \cdots - 1424)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 14 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$53$ \( (T^{4} - 30 T^{2} + \cdots + 104)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 22 T^{7} + \cdots + 1048576 \) Copy content Toggle raw display
$61$ \( (T^{4} + 12 T^{3} + \cdots - 4112)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 2 T^{7} + \cdots + 43983424 \) Copy content Toggle raw display
$71$ \( T^{8} - 14 T^{7} + \cdots + 204304 \) Copy content Toggle raw display
$73$ \( T^{8} - 20 T^{7} + \cdots + 43264 \) Copy content Toggle raw display
$79$ \( T^{8} + 208 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$83$ \( T^{8} - 6 T^{7} + \cdots + 10816 \) Copy content Toggle raw display
$89$ \( T^{8} - 8 T^{7} + \cdots + 7744 \) Copy content Toggle raw display
$97$ \( T^{8} + 8 T^{7} + \cdots + 35344 \) Copy content Toggle raw display
show more
show less