Properties

Label 4312.2.a.u
Level $4312$
Weight $2$
Character orbit 4312.a
Self dual yes
Analytic conductor $34.431$
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] = [4312,2,Mod(1,4312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4312, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4312.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4312 = 2^{3} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4312.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: \(34.4314933516\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 616)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} - \beta q^{5} + 2 \beta q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} - \beta q^{5} + 2 \beta q^{9} + q^{11} + ( - 2 \beta - 1) q^{13} + ( - \beta - 2) q^{15} - 2 q^{17} + ( - \beta - 4) q^{19} + ( - \beta + 4) q^{23} - 3 q^{25} + ( - \beta + 1) q^{27} + q^{29} + (4 \beta - 4) q^{31} + (\beta + 1) q^{33} + (5 \beta - 2) q^{37} + ( - 3 \beta - 5) q^{39} + ( - 3 \beta + 2) q^{41} - 8 q^{43} - 4 q^{45} + ( - 2 \beta + 2) q^{47} + ( - 2 \beta - 2) q^{51} + (\beta + 4) q^{53} - \beta q^{55} + ( - 5 \beta - 6) q^{57} + ( - \beta + 3) q^{59} + ( - 2 \beta - 11) q^{61} + (\beta + 4) q^{65} + ( - 5 \beta + 1) q^{67} + (3 \beta + 2) q^{69} + ( - 3 \beta - 10) q^{71} + (\beta - 2) q^{73} + ( - 3 \beta - 3) q^{75} + ( - 5 \beta + 1) q^{79} + ( - 6 \beta - 1) q^{81} + (6 \beta - 2) q^{83} + 2 \beta q^{85} + (\beta + 1) q^{87} + ( - 2 \beta - 12) q^{89} + 4 q^{93} + (4 \beta + 2) q^{95} + (2 \beta - 13) q^{97} + 2 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{11} - 2 q^{13} - 4 q^{15} - 4 q^{17} - 8 q^{19} + 8 q^{23} - 6 q^{25} + 2 q^{27} + 2 q^{29} - 8 q^{31} + 2 q^{33} - 4 q^{37} - 10 q^{39} + 4 q^{41} - 16 q^{43} - 8 q^{45} + 4 q^{47} - 4 q^{51} + 8 q^{53} - 12 q^{57} + 6 q^{59} - 22 q^{61} + 8 q^{65} + 2 q^{67} + 4 q^{69} - 20 q^{71} - 4 q^{73} - 6 q^{75} + 2 q^{79} - 2 q^{81} - 4 q^{83} + 2 q^{87} - 24 q^{89} + 8 q^{93} + 4 q^{95} - 26 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
0 −0.414214 0 1.41421 0 0 0 −2.82843 0
1.2 0 2.41421 0 −1.41421 0 0 0 2.82843 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(11\) \(-1\)

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 4312.2.a.u 2
4.b odd 2 1 8624.2.a.bg 2
7.b odd 2 1 4312.2.a.m 2
7.c even 3 2 616.2.q.b 4
28.d even 2 1 8624.2.a.cd 2
28.g odd 6 2 1232.2.q.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.q.b 4 7.c even 3 2
1232.2.q.h 4 28.g odd 6 2
4312.2.a.m 2 7.b odd 2 1
4312.2.a.u 2 1.a even 1 1 trivial
8624.2.a.bg 2 4.b odd 2 1
8624.2.a.cd 2 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4312))\):

\( T_{3}^{2} - 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 7 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 2 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 7 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
$23$ \( T^{2} - 8T + 14 \) Copy content Toggle raw display
$29$ \( (T - 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 46 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
$43$ \( (T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T + 14 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 7 \) Copy content Toggle raw display
$61$ \( T^{2} + 22T + 113 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T - 49 \) Copy content Toggle raw display
$71$ \( T^{2} + 20T + 82 \) Copy content Toggle raw display
$73$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$79$ \( T^{2} - 2T - 49 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$89$ \( T^{2} + 24T + 136 \) Copy content Toggle raw display
$97$ \( T^{2} + 26T + 161 \) Copy content Toggle raw display
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