Properties

Label 1856.2.e.f
Level $1856$
Weight $2$
Character orbit 1856.e
Analytic conductor $14.820$
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] = [1856,2,Mod(1217,1856)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1856.1217");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1856.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: \(14.8202346151\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) 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 29)
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 \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 3 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + 3 q^{5} - 2 q^{7} - 2 q^{9} - \beta q^{11} + q^{13} + 3 \beta q^{15} + 2 \beta q^{17} - 2 \beta q^{21} - 6 q^{23} + 4 q^{25} + \beta q^{27} + (2 \beta + 3) q^{29} + 3 \beta q^{31} + 5 q^{33} - 6 q^{35} + \beta q^{39} + 2 \beta q^{41} + 3 \beta q^{43} - 6 q^{45} + \beta q^{47} - 3 q^{49} - 10 q^{51} + 9 q^{53} - 3 \beta q^{55} + 6 q^{59} + 6 \beta q^{61} + 4 q^{63} + 3 q^{65} + 8 q^{67} - 6 \beta q^{69} + 4 \beta q^{75} + 2 \beta q^{77} - 3 \beta q^{79} - 11 q^{81} - 6 q^{83} + 6 \beta q^{85} + (3 \beta - 10) q^{87} + 2 \beta q^{89} - 2 q^{91} - 15 q^{93} - 6 \beta q^{97} + 2 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} - 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{5} - 4 q^{7} - 4 q^{9} + 2 q^{13} - 12 q^{23} + 8 q^{25} + 6 q^{29} + 10 q^{33} - 12 q^{35} - 12 q^{45} - 6 q^{49} - 20 q^{51} + 18 q^{53} + 12 q^{59} + 8 q^{63} + 6 q^{65} + 16 q^{67} - 22 q^{81} - 12 q^{83} - 20 q^{87} - 4 q^{91} - 30 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1856\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(581\) \(639\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1217.1
2.23607i
2.23607i
0 2.23607i 0 3.00000 0 −2.00000 0 −2.00000 0
1217.2 0 2.23607i 0 3.00000 0 −2.00000 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
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 1856.2.e.f 2
4.b odd 2 1 1856.2.e.g 2
8.b even 2 1 464.2.e.a 2
8.d odd 2 1 29.2.b.a 2
24.f even 2 1 261.2.c.a 2
24.h odd 2 1 4176.2.o.k 2
29.b even 2 1 inner 1856.2.e.f 2
40.e odd 2 1 725.2.c.c 2
40.k even 4 2 725.2.d.a 4
56.e even 2 1 1421.2.b.b 2
116.d odd 2 1 1856.2.e.g 2
232.b odd 2 1 29.2.b.a 2
232.g even 2 1 464.2.e.a 2
232.k even 4 2 841.2.a.b 2
232.p odd 14 6 841.2.e.g 12
232.t odd 14 6 841.2.e.g 12
232.v even 28 12 841.2.d.h 12
696.l even 2 1 261.2.c.a 2
696.n odd 2 1 4176.2.o.k 2
696.r odd 4 2 7569.2.a.i 2
1160.o odd 2 1 725.2.c.c 2
1160.bb even 4 2 725.2.d.a 4
1624.o even 2 1 1421.2.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.b.a 2 8.d odd 2 1
29.2.b.a 2 232.b odd 2 1
261.2.c.a 2 24.f even 2 1
261.2.c.a 2 696.l even 2 1
464.2.e.a 2 8.b even 2 1
464.2.e.a 2 232.g even 2 1
725.2.c.c 2 40.e odd 2 1
725.2.c.c 2 1160.o odd 2 1
725.2.d.a 4 40.k even 4 2
725.2.d.a 4 1160.bb even 4 2
841.2.a.b 2 232.k even 4 2
841.2.d.h 12 232.v even 28 12
841.2.e.g 12 232.p odd 14 6
841.2.e.g 12 232.t odd 14 6
1421.2.b.b 2 56.e even 2 1
1421.2.b.b 2 1624.o even 2 1
1856.2.e.f 2 1.a even 1 1 trivial
1856.2.e.f 2 29.b even 2 1 inner
1856.2.e.g 2 4.b odd 2 1
1856.2.e.g 2 116.d odd 2 1
4176.2.o.k 2 24.h odd 2 1
4176.2.o.k 2 696.n odd 2 1
7569.2.a.i 2 696.r odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1856, [\chi])\):

\( T_{3}^{2} + 5 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 5 \) Copy content Toggle raw display
$5$ \( (T - 3)^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 5 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 20 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 29 \) Copy content Toggle raw display
$31$ \( T^{2} + 45 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 20 \) Copy content Toggle raw display
$43$ \( T^{2} + 45 \) Copy content Toggle raw display
$47$ \( T^{2} + 5 \) Copy content Toggle raw display
$53$ \( (T - 9)^{2} \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 180 \) Copy content Toggle raw display
$67$ \( (T - 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 45 \) Copy content Toggle raw display
$83$ \( (T + 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 20 \) Copy content Toggle raw display
$97$ \( T^{2} + 180 \) Copy content Toggle raw display
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