Properties

Label 912.2.f.d
Level $912$
Weight $2$
Character orbit 912.f
Analytic conductor $7.282$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(113,912)]
 
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([0, 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("912.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.f (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: \(7.28235666434\)
Analytic rank: \(0\)
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 114)
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 1) q^{3} + \beta q^{5} + 4 q^{7} + ( - 2 \beta - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 1) q^{3} + \beta q^{5} + 4 q^{7} + ( - 2 \beta - 1) q^{9} - 4 \beta q^{11} + 3 \beta q^{13} + (\beta + 2) q^{15} - 2 \beta q^{17} + (3 \beta - 1) q^{19} + ( - 4 \beta + 4) q^{21} - \beta q^{23} + 3 q^{25} + ( - \beta - 5) q^{27} + 6 q^{29} - 3 \beta q^{31} + ( - 4 \beta - 8) q^{33} + 4 \beta q^{35} + 3 \beta q^{37} + (3 \beta + 6) q^{39} - 2 q^{43} + ( - \beta + 4) q^{45} - \beta q^{47} + 9 q^{49} + ( - 2 \beta - 4) q^{51} - 6 q^{53} + 8 q^{55} + (4 \beta + 5) q^{57} + 12 q^{59} + 2 q^{61} + ( - 8 \beta - 4) q^{63} - 6 q^{65} + ( - \beta - 2) q^{69} - 12 q^{71} - 4 q^{73} + ( - 3 \beta + 3) q^{75} - 16 \beta q^{77} - 3 \beta q^{79} + (4 \beta - 7) q^{81} + 2 \beta q^{83} + 4 q^{85} + ( - 6 \beta + 6) q^{87} - 6 q^{89} + 12 \beta q^{91} + ( - 3 \beta - 6) q^{93} + ( - \beta - 6) q^{95} - 6 \beta q^{97} + (4 \beta - 16) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 8 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 8 q^{7} - 2 q^{9} + 4 q^{15} - 2 q^{19} + 8 q^{21} + 6 q^{25} - 10 q^{27} + 12 q^{29} - 16 q^{33} + 12 q^{39} - 4 q^{43} + 8 q^{45} + 18 q^{49} - 8 q^{51} - 12 q^{53} + 16 q^{55} + 10 q^{57} + 24 q^{59} + 4 q^{61} - 8 q^{63} - 12 q^{65} - 4 q^{69} - 24 q^{71} - 8 q^{73} + 6 q^{75} - 14 q^{81} + 8 q^{85} + 12 q^{87} - 12 q^{89} - 12 q^{93} - 12 q^{95} - 32 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
1.41421i
1.41421i
0 1.00000 1.41421i 0 1.41421i 0 4.00000 0 −1.00000 2.82843i 0
113.2 0 1.00000 + 1.41421i 0 1.41421i 0 4.00000 0 −1.00000 + 2.82843i 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
57.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.f.d 2
3.b odd 2 1 912.2.f.b 2
4.b odd 2 1 114.2.b.a 2
12.b even 2 1 114.2.b.d yes 2
19.b odd 2 1 912.2.f.b 2
57.d even 2 1 inner 912.2.f.d 2
76.d even 2 1 114.2.b.d yes 2
228.b odd 2 1 114.2.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.b.a 2 4.b odd 2 1
114.2.b.a 2 228.b odd 2 1
114.2.b.d yes 2 12.b even 2 1
114.2.b.d yes 2 76.d even 2 1
912.2.f.b 2 3.b odd 2 1
912.2.f.b 2 19.b odd 2 1
912.2.f.d 2 1.a even 1 1 trivial
912.2.f.d 2 57.d even 2 1 inner

Hecke kernels

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

\( T_{5}^{2} + 2 \) Copy content Toggle raw display
\( T_{7} - 4 \) Copy content Toggle raw display
\( T_{29} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + 2 \) Copy content Toggle raw display
$7$ \( (T - 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 32 \) Copy content Toggle raw display
$13$ \( T^{2} + 18 \) Copy content Toggle raw display
$17$ \( T^{2} + 8 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + 2 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 18 \) Copy content Toggle raw display
$37$ \( T^{2} + 18 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 18 \) Copy content Toggle raw display
$83$ \( T^{2} + 8 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 72 \) Copy content Toggle raw display
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