Properties

Label 416.2.a.c
Level $416$
Weight $2$
Character orbit 416.a
Self dual yes
Analytic conductor $3.322$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,2,Mod(1,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.a (trivial)

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: yes
Analytic conductor: \(3.32177672409\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + (\beta - 2) q^{5} + (\beta - 2) q^{7} + (\beta + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + (\beta - 2) q^{5} + (\beta - 2) q^{7} + (\beta + 1) q^{9} - 2 q^{11} - q^{13} + (\beta - 4) q^{15} + ( - \beta + 2) q^{17} - 6 q^{19} + (\beta - 4) q^{21} + ( - 3 \beta + 3) q^{25} + (\beta - 4) q^{27} + ( - 4 \beta + 2) q^{29} + ( - 2 \beta - 2) q^{31} + 2 \beta q^{33} + ( - 3 \beta + 8) q^{35} + ( - 3 \beta - 2) q^{37} + \beta q^{39} + (2 \beta + 2) q^{41} + (5 \beta - 4) q^{43} + 2 q^{45} + ( - 3 \beta + 6) q^{47} + ( - 3 \beta + 1) q^{49} + ( - \beta + 4) q^{51} + (2 \beta - 10) q^{53} + ( - 2 \beta + 4) q^{55} + 6 \beta q^{57} + 6 q^{59} + (6 \beta - 2) q^{61} + 2 q^{63} + ( - \beta + 2) q^{65} - 6 q^{67} + (3 \beta - 6) q^{71} + 10 q^{73} + 12 q^{75} + ( - 2 \beta + 4) q^{77} - 12 q^{79} - 7 q^{81} + (6 \beta + 2) q^{83} + (3 \beta - 8) q^{85} + (2 \beta + 16) q^{87} + ( - 4 \beta + 2) q^{89} + ( - \beta + 2) q^{91} + (4 \beta + 8) q^{93} + ( - 6 \beta + 12) q^{95} - 6 q^{97} + ( - 2 \beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} - 4 q^{11} - 2 q^{13} - 7 q^{15} + 3 q^{17} - 12 q^{19} - 7 q^{21} + 3 q^{25} - 7 q^{27} - 6 q^{31} + 2 q^{33} + 13 q^{35} - 7 q^{37} + q^{39} + 6 q^{41} - 3 q^{43} + 4 q^{45} + 9 q^{47} - q^{49} + 7 q^{51} - 18 q^{53} + 6 q^{55} + 6 q^{57} + 12 q^{59} + 2 q^{61} + 4 q^{63} + 3 q^{65} - 12 q^{67} - 9 q^{71} + 20 q^{73} + 24 q^{75} + 6 q^{77} - 24 q^{79} - 14 q^{81} + 10 q^{83} - 13 q^{85} + 34 q^{87} + 3 q^{91} + 20 q^{93} + 18 q^{95} - 12 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.56155
−1.56155
0 −2.56155 0 0.561553 0 0.561553 0 3.56155 0
1.2 0 1.56155 0 −3.56155 0 −3.56155 0 −0.561553 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(13\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.a.c 2
3.b odd 2 1 3744.2.a.x 2
4.b odd 2 1 416.2.a.e yes 2
8.b even 2 1 832.2.a.o 2
8.d odd 2 1 832.2.a.l 2
12.b even 2 1 3744.2.a.y 2
13.b even 2 1 5408.2.a.p 2
16.e even 4 2 3328.2.b.ba 4
16.f odd 4 2 3328.2.b.u 4
24.f even 2 1 7488.2.a.cf 2
24.h odd 2 1 7488.2.a.ce 2
52.b odd 2 1 5408.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.a.c 2 1.a even 1 1 trivial
416.2.a.e yes 2 4.b odd 2 1
832.2.a.l 2 8.d odd 2 1
832.2.a.o 2 8.b even 2 1
3328.2.b.u 4 16.f odd 4 2
3328.2.b.ba 4 16.e even 4 2
3744.2.a.x 2 3.b odd 2 1
3744.2.a.y 2 12.b even 2 1
5408.2.a.p 2 13.b even 2 1
5408.2.a.bd 2 52.b odd 2 1
7488.2.a.ce 2 24.h odd 2 1
7488.2.a.cf 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(416))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$19$ \( (T + 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 68 \) Copy content Toggle raw display
$31$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$37$ \( T^{2} + 7T - 26 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$43$ \( T^{2} + 3T - 104 \) Copy content Toggle raw display
$47$ \( T^{2} - 9T - 18 \) Copy content Toggle raw display
$53$ \( T^{2} + 18T + 64 \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 2T - 152 \) Copy content Toggle raw display
$67$ \( (T + 6)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 9T - 18 \) Copy content Toggle raw display
$73$ \( (T - 10)^{2} \) Copy content Toggle raw display
$79$ \( (T + 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 10T - 128 \) Copy content Toggle raw display
$89$ \( T^{2} - 68 \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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