Properties

Label 416.2.k.d
Level $416$
Weight $2$
Character orbit 416.k
Analytic conductor $3.322$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_1 q^{5} + \beta_1 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_1 q^{5} + \beta_1 q^{7} + 2 q^{9} + (\beta_{2} - 2 \beta_1 + 1) q^{11} + ( - 2 \beta_{2} - \beta_1 + 2) q^{13} + \beta_{3} q^{15} + (2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{17} + (2 \beta_{3} + 2 \beta_{2} - 2) q^{19} + \beta_{3} q^{21} + (\beta_{3} - \beta_1 + 4) q^{23} + 5 \beta_{2} q^{27} + ( - \beta_{3} + \beta_1 + 2) q^{29} + ( - 2 \beta_{3} - 2 \beta_{2} + 2) q^{31} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{33} + 5 \beta_{2} q^{35} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{37} + ( - \beta_{3} + 2 \beta_{2} + 2) q^{39} + (2 \beta_{2} - 2 \beta_1 + 2) q^{41} - 5 q^{43} + 2 \beta_1 q^{45} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{47} - 2 \beta_{2} q^{49} + (2 \beta_{3} - 2 \beta_1 + 1) q^{51} + (3 \beta_{3} - 3 \beta_1) q^{53} + (\beta_{3} - 10 \beta_{2} + \beta_1) q^{55} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{57} + ( - 4 \beta_{2} - 4) q^{59} + 2 q^{61} + 2 \beta_1 q^{63} + ( - 2 \beta_{3} - 5 \beta_{2} + 2 \beta_1) q^{65} + ( - 4 \beta_{3} + \beta_{2} - 1) q^{67} + ( - \beta_{3} + 4 \beta_{2} - \beta_1) q^{69} - \beta_{3} q^{71} + (6 \beta_{2} - 6) q^{73} + (\beta_{3} - 10 \beta_{2} + \beta_1) q^{77} + ( - 3 \beta_{3} - 6 \beta_{2} - 3 \beta_1) q^{79} + q^{81} + ( - 2 \beta_{3} - 9 \beta_{2} + 9) q^{83} + ( - \beta_{3} + 10 \beta_{2} - 10) q^{85} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{87} + (2 \beta_{3} + 6 \beta_{2} - 6) q^{89} + ( - 2 \beta_{3} - 5 \beta_{2} + 2 \beta_1) q^{91} + (2 \beta_{2} + 2 \beta_1 + 2) q^{93} + (2 \beta_{3} - 2 \beta_1 - 10) q^{95} + ( - 7 \beta_{2} - 2 \beta_1 - 7) q^{97} + (2 \beta_{2} - 4 \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{9} + 4 q^{11} + 8 q^{13} - 8 q^{19} + 16 q^{23} + 8 q^{29} + 8 q^{31} - 4 q^{33} - 8 q^{37} + 8 q^{39} + 8 q^{41} - 20 q^{43} - 8 q^{47} + 4 q^{51} - 8 q^{57} - 16 q^{59} + 8 q^{61} - 4 q^{67} - 24 q^{73} + 4 q^{81} + 36 q^{83} - 40 q^{85} - 24 q^{89} + 8 q^{93} - 40 q^{95} - 28 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−1.58114 + 1.58114i
1.58114 1.58114i
−1.58114 1.58114i
1.58114 + 1.58114i
0 1.00000i 0 −1.58114 + 1.58114i 0 −1.58114 + 1.58114i 0 2.00000 0
31.2 0 1.00000i 0 1.58114 1.58114i 0 1.58114 1.58114i 0 2.00000 0
255.1 0 1.00000i 0 −1.58114 1.58114i 0 −1.58114 1.58114i 0 2.00000 0
255.2 0 1.00000i 0 1.58114 + 1.58114i 0 1.58114 + 1.58114i 0 2.00000 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
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.d yes 4
4.b odd 2 1 416.2.k.c 4
8.b even 2 1 832.2.k.e 4
8.d odd 2 1 832.2.k.f 4
13.d odd 4 1 416.2.k.c 4
52.f even 4 1 inner 416.2.k.d yes 4
104.j odd 4 1 832.2.k.f 4
104.m even 4 1 832.2.k.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.k.c 4 4.b odd 2 1
416.2.k.c 4 13.d odd 4 1
416.2.k.d yes 4 1.a even 1 1 trivial
416.2.k.d yes 4 52.f even 4 1 inner
832.2.k.e 4 8.b even 2 1
832.2.k.e 4 104.m even 4 1
832.2.k.f 4 8.d odd 2 1
832.2.k.f 4 104.j odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 25 \) Copy content Toggle raw display
\( T_{11}^{4} - 4T_{11}^{3} + 8T_{11}^{2} + 72T_{11} + 324 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 25 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$13$ \( T^{4} - 8 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 82T^{2} + 1521 \) Copy content Toggle raw display
$19$ \( T^{4} + 8 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$23$ \( (T^{2} - 8 T + 6)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 8 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$37$ \( T^{4} + 8 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$41$ \( T^{4} - 8 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$43$ \( (T + 5)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} + \cdots + 1369 \) Copy content Toggle raw display
$53$ \( (T^{2} - 90)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 8 T + 32)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 4 T^{3} + \cdots + 6084 \) Copy content Toggle raw display
$71$ \( T^{4} + 25 \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T + 72)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 252T^{2} + 2916 \) Copy content Toggle raw display
$83$ \( T^{4} - 36 T^{3} + \cdots + 20164 \) Copy content Toggle raw display
$89$ \( T^{4} + 24 T^{3} + \cdots + 2704 \) Copy content Toggle raw display
$97$ \( T^{4} + 28 T^{3} + \cdots + 6084 \) Copy content Toggle raw display
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