Properties

Label 912.2.k.f
Level $912$
Weight $2$
Character orbit 912.k
Analytic conductor $7.282$
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] = [912,2,Mod(607,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("912.607");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.k (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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + 3 q^{5} - \beta q^{7} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 3 q^{5} - \beta q^{7} + q^{9} + 3 \beta q^{11} + 4 \beta q^{13} + 3 q^{15} + 3 q^{17} + (\beta - 4) q^{19} - \beta q^{21} - 2 \beta q^{23} + 4 q^{25} + q^{27} + 4 q^{31} + 3 \beta q^{33} - 3 \beta q^{35} - 4 \beta q^{37} + 4 \beta q^{39} - 4 \beta q^{41} - 5 \beta q^{43} + 3 q^{45} - 5 \beta q^{47} + 4 q^{49} + 3 q^{51} + 4 \beta q^{53} + 9 \beta q^{55} + (\beta - 4) q^{57} - 12 q^{59} + 7 q^{61} - \beta q^{63} + 12 \beta q^{65} - 8 q^{67} - 2 \beta q^{69} - 12 q^{71} - 5 q^{73} + 4 q^{75} + 9 q^{77} + 8 q^{79} + q^{81} + 2 \beta q^{83} + 9 q^{85} - 4 \beta q^{89} + 12 q^{91} + 4 q^{93} + (3 \beta - 12) q^{95} - 4 \beta q^{97} + 3 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 6 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 6 q^{5} + 2 q^{9} + 6 q^{15} + 6 q^{17} - 8 q^{19} + 8 q^{25} + 2 q^{27} + 8 q^{31} + 6 q^{45} + 8 q^{49} + 6 q^{51} - 8 q^{57} - 24 q^{59} + 14 q^{61} - 16 q^{67} - 24 q^{71} - 10 q^{73} + 8 q^{75} + 18 q^{77} + 16 q^{79} + 2 q^{81} + 18 q^{85} + 24 q^{91} + 8 q^{93} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
607.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.00000 0 3.00000 0 1.73205i 0 1.00000 0
607.2 0 1.00000 0 3.00000 0 1.73205i 0 1.00000 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
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.k.f yes 2
3.b odd 2 1 2736.2.k.a 2
4.b odd 2 1 912.2.k.c 2
8.b even 2 1 3648.2.k.a 2
8.d odd 2 1 3648.2.k.d 2
12.b even 2 1 2736.2.k.b 2
19.b odd 2 1 912.2.k.c 2
57.d even 2 1 2736.2.k.b 2
76.d even 2 1 inner 912.2.k.f yes 2
152.b even 2 1 3648.2.k.a 2
152.g odd 2 1 3648.2.k.d 2
228.b odd 2 1 2736.2.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.k.c 2 4.b odd 2 1
912.2.k.c 2 19.b odd 2 1
912.2.k.f yes 2 1.a even 1 1 trivial
912.2.k.f yes 2 76.d even 2 1 inner
2736.2.k.a 2 3.b odd 2 1
2736.2.k.a 2 228.b odd 2 1
2736.2.k.b 2 12.b even 2 1
2736.2.k.b 2 57.d even 2 1
3648.2.k.a 2 8.b even 2 1
3648.2.k.a 2 152.b even 2 1
3648.2.k.d 2 8.d odd 2 1
3648.2.k.d 2 152.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{31} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 3 \) Copy content Toggle raw display
$11$ \( T^{2} + 27 \) Copy content Toggle raw display
$13$ \( T^{2} + 48 \) Copy content Toggle raw display
$17$ \( (T - 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + 12 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 48 \) Copy content Toggle raw display
$41$ \( T^{2} + 48 \) Copy content Toggle raw display
$43$ \( T^{2} + 75 \) Copy content Toggle raw display
$47$ \( T^{2} + 75 \) Copy content Toggle raw display
$53$ \( T^{2} + 48 \) Copy content Toggle raw display
$59$ \( (T + 12)^{2} \) Copy content Toggle raw display
$61$ \( (T - 7)^{2} \) Copy content Toggle raw display
$67$ \( (T + 8)^{2} \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 5)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 12 \) Copy content Toggle raw display
$89$ \( T^{2} + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 48 \) Copy content Toggle raw display
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