Properties

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

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,2,Mod(337,416)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.e (of order \(2\), degree \(1\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{3} - q^{5} - 3 i q^{7} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} - q^{5} - 3 i q^{7} + 2 q^{9} + 2 q^{11} + (2 i + 3) q^{13} - i q^{15} + 3 q^{17} + 3 q^{21} + 6 q^{23} - 4 q^{25} + 5 i q^{27} + 6 i q^{29} + 2 i q^{33} + 3 i q^{35} - 3 q^{37} + (3 i - 2) q^{39} - 10 i q^{41} - 9 i q^{43} - 2 q^{45} + 7 i q^{47} - 2 q^{49} + 3 i q^{51} - 6 i q^{53} - 2 q^{55} + 10 q^{59} - 10 i q^{61} - 6 i q^{63} + ( - 2 i - 3) q^{65} - 12 q^{67} + 6 i q^{69} + 5 i q^{71} + 6 i q^{73} - 4 i q^{75} - 6 i q^{77} + q^{81} - 16 q^{83} - 3 q^{85} - 6 q^{87} + 4 i q^{89} + ( - 9 i + 6) q^{91} + 18 i q^{97} + 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 4 q^{9} + 4 q^{11} + 6 q^{13} + 6 q^{17} + 6 q^{21} + 12 q^{23} - 8 q^{25} - 6 q^{37} - 4 q^{39} - 4 q^{45} - 4 q^{49} - 4 q^{55} + 20 q^{59} - 6 q^{65} - 24 q^{67} + 2 q^{81} - 32 q^{83} - 6 q^{85} - 12 q^{87} + 12 q^{91} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
0 1.00000i 0 −1.00000 0 3.00000i 0 2.00000 0
337.2 0 1.00000i 0 −1.00000 0 3.00000i 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
104.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.e.a 2
3.b odd 2 1 3744.2.m.c 2
4.b odd 2 1 104.2.e.b yes 2
8.b even 2 1 416.2.e.b 2
8.d odd 2 1 104.2.e.a 2
12.b even 2 1 936.2.m.b 2
13.b even 2 1 416.2.e.b 2
24.f even 2 1 936.2.m.c 2
24.h odd 2 1 3744.2.m.b 2
39.d odd 2 1 3744.2.m.b 2
52.b odd 2 1 104.2.e.a 2
104.e even 2 1 inner 416.2.e.a 2
104.h odd 2 1 104.2.e.b yes 2
156.h even 2 1 936.2.m.c 2
312.b odd 2 1 3744.2.m.c 2
312.h even 2 1 936.2.m.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.e.a 2 8.d odd 2 1
104.2.e.a 2 52.b odd 2 1
104.2.e.b yes 2 4.b odd 2 1
104.2.e.b yes 2 104.h odd 2 1
416.2.e.a 2 1.a even 1 1 trivial
416.2.e.a 2 104.e even 2 1 inner
416.2.e.b 2 8.b even 2 1
416.2.e.b 2 13.b even 2 1
936.2.m.b 2 12.b even 2 1
936.2.m.b 2 312.h even 2 1
936.2.m.c 2 24.f even 2 1
936.2.m.c 2 156.h even 2 1
3744.2.m.b 2 24.h odd 2 1
3744.2.m.b 2 39.d odd 2 1
3744.2.m.c 2 3.b odd 2 1
3744.2.m.c 2 312.b odd 2 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
\( T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 9 \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 13 \) Copy content Toggle raw display
$17$ \( (T - 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 100 \) Copy content Toggle raw display
$43$ \( T^{2} + 81 \) Copy content Toggle raw display
$47$ \( T^{2} + 49 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 10)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 100 \) Copy content Toggle raw display
$67$ \( (T + 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 25 \) Copy content Toggle raw display
$73$ \( T^{2} + 36 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T + 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 16 \) Copy content Toggle raw display
$97$ \( T^{2} + 324 \) Copy content Toggle raw display
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