Properties

Label 116.2.c.a
Level $116$
Weight $2$
Character orbit 116.c
Analytic conductor $0.926$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [116,2,Mod(57,116)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(116, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("116.57");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 116 = 2^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 116.c (of order \(2\), degree \(1\), 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: \(0.926264663447\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(\beta = \sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + q^{5} - 2 q^{7} - 4 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + q^{5} - 2 q^{7} - 4 q^{9} - \beta q^{11} + 5 q^{13} - \beta q^{15} + 2 \beta q^{17} + 2 \beta q^{19} + 2 \beta q^{21} + 6 q^{23} - 4 q^{25} + \beta q^{27} + ( - 2 \beta - 1) q^{29} + \beta q^{31} - 7 q^{33} - 2 q^{35} + 4 \beta q^{37} - 5 \beta q^{39} - 2 \beta q^{41} + 3 \beta q^{43} - 4 q^{45} - 3 \beta q^{47} - 3 q^{49} + 14 q^{51} + 5 q^{53} - \beta q^{55} + 14 q^{57} - 14 q^{59} - 2 \beta q^{61} + 8 q^{63} + 5 q^{65} - 4 q^{67} - 6 \beta q^{69} - 8 q^{71} - 4 \beta q^{73} + 4 \beta q^{75} + 2 \beta q^{77} + \beta q^{79} - 5 q^{81} + 2 q^{83} + 2 \beta q^{85} + (\beta - 14) q^{87} - 2 \beta q^{89} - 10 q^{91} + 7 q^{93} + 2 \beta q^{95} + 2 \beta q^{97} + 4 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 4 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 4 q^{7} - 8 q^{9} + 10 q^{13} + 12 q^{23} - 8 q^{25} - 2 q^{29} - 14 q^{33} - 4 q^{35} - 8 q^{45} - 6 q^{49} + 28 q^{51} + 10 q^{53} + 28 q^{57} - 28 q^{59} + 16 q^{63} + 10 q^{65} - 8 q^{67} - 16 q^{71} - 10 q^{81} + 4 q^{83} - 28 q^{87} - 20 q^{91} + 14 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).

\(n\) \(59\) \(89\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
0.500000 + 1.32288i
0.500000 1.32288i
0 2.64575i 0 1.00000 0 −2.00000 0 −4.00000 0
57.2 0 2.64575i 0 1.00000 0 −2.00000 0 −4.00000 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
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 116.2.c.a 2
3.b odd 2 1 1044.2.h.c 2
4.b odd 2 1 464.2.e.b 2
5.b even 2 1 2900.2.d.a 2
5.c odd 4 2 2900.2.f.b 4
8.b even 2 1 1856.2.e.a 2
8.d odd 2 1 1856.2.e.c 2
12.b even 2 1 4176.2.o.e 2
29.b even 2 1 inner 116.2.c.a 2
29.c odd 4 2 3364.2.a.f 2
87.d odd 2 1 1044.2.h.c 2
116.d odd 2 1 464.2.e.b 2
145.d even 2 1 2900.2.d.a 2
145.h odd 4 2 2900.2.f.b 4
232.b odd 2 1 1856.2.e.c 2
232.g even 2 1 1856.2.e.a 2
348.b even 2 1 4176.2.o.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.2.c.a 2 1.a even 1 1 trivial
116.2.c.a 2 29.b even 2 1 inner
464.2.e.b 2 4.b odd 2 1
464.2.e.b 2 116.d odd 2 1
1044.2.h.c 2 3.b odd 2 1
1044.2.h.c 2 87.d odd 2 1
1856.2.e.a 2 8.b even 2 1
1856.2.e.a 2 232.g even 2 1
1856.2.e.c 2 8.d odd 2 1
1856.2.e.c 2 232.b odd 2 1
2900.2.d.a 2 5.b even 2 1
2900.2.d.a 2 145.d even 2 1
2900.2.f.b 4 5.c odd 4 2
2900.2.f.b 4 145.h odd 4 2
3364.2.a.f 2 29.c odd 4 2
4176.2.o.e 2 12.b even 2 1
4176.2.o.e 2 348.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(116, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 7 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 7 \) Copy content Toggle raw display
$13$ \( (T - 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 28 \) Copy content Toggle raw display
$19$ \( T^{2} + 28 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 2T + 29 \) Copy content Toggle raw display
$31$ \( T^{2} + 7 \) Copy content Toggle raw display
$37$ \( T^{2} + 112 \) Copy content Toggle raw display
$41$ \( T^{2} + 28 \) Copy content Toggle raw display
$43$ \( T^{2} + 63 \) Copy content Toggle raw display
$47$ \( T^{2} + 63 \) Copy content Toggle raw display
$53$ \( (T - 5)^{2} \) Copy content Toggle raw display
$59$ \( (T + 14)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 28 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 112 \) Copy content Toggle raw display
$79$ \( T^{2} + 7 \) Copy content Toggle raw display
$83$ \( (T - 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 28 \) Copy content Toggle raw display
$97$ \( T^{2} + 28 \) Copy content Toggle raw display
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