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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,2,Mod(47,416)]
 
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, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.u (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: \(3.32177672409\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{26})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_1 q^{5} - \beta_{3} q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_1 q^{5} - \beta_{3} q^{7} - 2 q^{9} + ( - \beta_{2} - 1) q^{11} + \beta_1 q^{13} + \beta_1 q^{15} - 3 \beta_{2} q^{17} + (2 \beta_{2} - 2) q^{19} - \beta_{3} q^{21} + (\beta_{3} - \beta_1) q^{23} + 8 \beta_{2} q^{25} - 5 q^{27} + ( - \beta_{3} - \beta_1) q^{29} + 2 \beta_1 q^{31} + ( - \beta_{2} - 1) q^{33} + 13 q^{35} + \beta_{3} q^{37} + \beta_1 q^{39} + ( - 6 \beta_{2} + 6) q^{41} + \beta_{2} q^{43} - 2 \beta_1 q^{45} - \beta_{3} q^{47} - 6 \beta_{2} q^{49} - 3 \beta_{2} q^{51} + (\beta_{3} + \beta_1) q^{53} + ( - \beta_{3} - \beta_1) q^{55} + (2 \beta_{2} - 2) q^{57} + ( - 8 \beta_{2} - 8) q^{59} + 2 \beta_{3} q^{63} + 13 \beta_{2} q^{65} + (3 \beta_{2} - 3) q^{67} + (\beta_{3} - \beta_1) q^{69} - 3 \beta_1 q^{71} + ( - 6 \beta_{2} - 6) q^{73} + 8 \beta_{2} q^{75} + (\beta_{3} - \beta_1) q^{77} + (\beta_{3} + \beta_1) q^{79} + q^{81} + (5 \beta_{2} - 5) q^{83} - 3 \beta_{3} q^{85} + ( - \beta_{3} - \beta_1) q^{87} + ( - 2 \beta_{2} - 2) q^{89} + 13 q^{91} + 2 \beta_1 q^{93} + (2 \beta_{3} - 2 \beta_1) q^{95} + (7 \beta_{2} - 7) q^{97} + (2 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 8 q^{9} - 4 q^{11} - 8 q^{19} - 20 q^{27} - 4 q^{33} + 52 q^{35} + 24 q^{41} - 8 q^{57} - 32 q^{59} - 12 q^{67} - 24 q^{73} + 4 q^{81} - 20 q^{83} - 8 q^{89} + 52 q^{91} - 28 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 13 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 13\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 13\beta_{3} \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−2.54951 + 2.54951i
2.54951 − 2.54951i
−2.54951 − 2.54951i
2.54951 + 2.54951i
0 1.00000 0 −2.54951 + 2.54951i 0 −2.54951 − 2.54951i 0 −2.00000 0
47.2 0 1.00000 0 2.54951 − 2.54951i 0 2.54951 + 2.54951i 0 −2.00000 0
239.1 0 1.00000 0 −2.54951 − 2.54951i 0 −2.54951 + 2.54951i 0 −2.00000 0
239.2 0 1.00000 0 2.54951 + 2.54951i 0 2.54951 − 2.54951i 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
8.d odd 2 1 inner
13.d odd 4 1 inner
104.m 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.u.a 4
4.b odd 2 1 104.2.m.a ✓ 4
8.b even 2 1 104.2.m.a ✓ 4
8.d odd 2 1 inner 416.2.u.a 4
12.b even 2 1 936.2.w.g 4
13.d odd 4 1 inner 416.2.u.a 4
24.h odd 2 1 936.2.w.g 4
52.f even 4 1 104.2.m.a ✓ 4
104.j odd 4 1 104.2.m.a ✓ 4
104.m even 4 1 inner 416.2.u.a 4
156.l odd 4 1 936.2.w.g 4
312.y even 4 1 936.2.w.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.m.a ✓ 4 4.b odd 2 1
104.2.m.a ✓ 4 8.b even 2 1
104.2.m.a ✓ 4 52.f even 4 1
104.2.m.a ✓ 4 104.j odd 4 1
416.2.u.a 4 1.a even 1 1 trivial
416.2.u.a 4 8.d odd 2 1 inner
416.2.u.a 4 13.d odd 4 1 inner
416.2.u.a 4 104.m even 4 1 inner
936.2.w.g 4 12.b even 2 1
936.2.w.g 4 24.h odd 2 1
936.2.w.g 4 156.l odd 4 1
936.2.w.g 4 312.y even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(416, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 169 \) Copy content Toggle raw display
$7$ \( T^{4} + 169 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 26)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 26)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2704 \) Copy content Toggle raw display
$37$ \( T^{4} + 169 \) Copy content Toggle raw display
$41$ \( (T^{2} - 12 T + 72)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 169 \) Copy content Toggle raw display
$53$ \( (T^{2} + 26)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 16 T + 128)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 13689 \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T + 72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 26)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 14 T + 98)^{2} \) Copy content Toggle raw display
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