Properties

Label 912.2.k.h
Level $912$
Weight $2$
Character orbit 912.k
Analytic conductor $7.282$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - \beta_{3} q^{5} + \beta_1 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta_{3} q^{5} + \beta_1 q^{7} + q^{9} + \beta_1 q^{11} + (2 \beta_{2} - 2 \beta_1) q^{13} - \beta_{3} q^{15} + (\beta_{3} + 2) q^{17} + (\beta_{2} + 4) q^{19} + \beta_1 q^{21} + (4 \beta_{2} - 2 \beta_1) q^{23} + (\beta_{3} + 3) q^{25} + q^{27} + ( - 4 \beta_{2} + 4 \beta_1) q^{29} + (2 \beta_{3} + 2) q^{31} + \beta_1 q^{33} + (2 \beta_{2} + \beta_1) q^{35} + (2 \beta_{2} - 2 \beta_1) q^{37} + (2 \beta_{2} - 2 \beta_1) q^{39} - 4 \beta_{2} q^{41} + (4 \beta_{2} - 3 \beta_1) q^{43} - \beta_{3} q^{45} + (2 \beta_{2} - \beta_1) q^{47} + (\beta_{3} + 3) q^{49} + (\beta_{3} + 2) q^{51} - 4 \beta_1 q^{53} + (2 \beta_{2} + \beta_1) q^{55} + (\beta_{2} + 4) q^{57} - 4 q^{59} + (\beta_{3} + 2) q^{61} + \beta_1 q^{63} + ( - 8 \beta_{2} + 4 \beta_1) q^{65} + ( - 4 \beta_{3} + 4) q^{67} + (4 \beta_{2} - 2 \beta_1) q^{69} - 4 q^{71} + ( - 3 \beta_{3} + 6) q^{73} + (\beta_{3} + 3) q^{75} + (\beta_{3} - 4) q^{77} + (2 \beta_{3} - 2) q^{79} + q^{81} + 2 \beta_{2} q^{83} + ( - 3 \beta_{3} - 8) q^{85} + ( - 4 \beta_{2} + 4 \beta_1) q^{87} + (4 \beta_{2} - 8 \beta_1) q^{89} + 4 q^{91} + (2 \beta_{3} + 2) q^{93} + ( - 4 \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{95} + ( - 4 \beta_{2} + 8 \beta_1) q^{97} + \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 2 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 2 q^{5} + 4 q^{9} - 2 q^{15} + 10 q^{17} + 16 q^{19} + 14 q^{25} + 4 q^{27} + 12 q^{31} - 2 q^{45} + 14 q^{49} + 10 q^{51} + 16 q^{57} - 16 q^{59} + 10 q^{61} + 8 q^{67} - 16 q^{71} + 18 q^{73} + 14 q^{75} - 14 q^{77} - 4 q^{79} + 4 q^{81} - 38 q^{85} + 16 q^{91} + 12 q^{93} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + \nu^{2} - \nu + 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 5\nu + 3 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} + \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} - 2\beta _1 + 4 \) 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
1.68614 + 0.396143i
1.68614 0.396143i
−1.18614 + 1.26217i
−1.18614 1.26217i
0 1.00000 0 −3.37228 0 0.792287i 0 1.00000 0
607.2 0 1.00000 0 −3.37228 0 0.792287i 0 1.00000 0
607.3 0 1.00000 0 2.37228 0 2.52434i 0 1.00000 0
607.4 0 1.00000 0 2.37228 0 2.52434i 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.h yes 4
3.b odd 2 1 2736.2.k.o 4
4.b odd 2 1 912.2.k.g 4
8.b even 2 1 3648.2.k.g 4
8.d odd 2 1 3648.2.k.h 4
12.b even 2 1 2736.2.k.n 4
19.b odd 2 1 912.2.k.g 4
57.d even 2 1 2736.2.k.n 4
76.d even 2 1 inner 912.2.k.h yes 4
152.b even 2 1 3648.2.k.g 4
152.g odd 2 1 3648.2.k.h 4
228.b odd 2 1 2736.2.k.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.k.g 4 4.b odd 2 1
912.2.k.g 4 19.b odd 2 1
912.2.k.h yes 4 1.a even 1 1 trivial
912.2.k.h yes 4 76.d even 2 1 inner
2736.2.k.n 4 12.b even 2 1
2736.2.k.n 4 57.d even 2 1
2736.2.k.o 4 3.b odd 2 1
2736.2.k.o 4 228.b odd 2 1
3648.2.k.g 4 8.b even 2 1
3648.2.k.g 4 152.b even 2 1
3648.2.k.h 4 8.d odd 2 1
3648.2.k.h 4 152.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 \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{2} + T_{5} - 8 \) Copy content Toggle raw display
\( T_{31}^{2} - 6T_{31} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + T - 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 7T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{4} + 7T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{4} + 28T^{2} + 64 \) Copy content Toggle raw display
$17$ \( (T^{2} - 5 T - 2)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 8 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 76T^{2} + 256 \) Copy content Toggle raw display
$29$ \( T^{4} + 112T^{2} + 1024 \) Copy content Toggle raw display
$31$ \( (T^{2} - 6 T - 24)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 28T^{2} + 64 \) Copy content Toggle raw display
$41$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 87T^{2} + 36 \) Copy content Toggle raw display
$47$ \( T^{4} + 19T^{2} + 16 \) Copy content Toggle raw display
$53$ \( T^{4} + 112T^{2} + 1024 \) Copy content Toggle raw display
$59$ \( (T + 4)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 5 T - 2)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 4 T - 128)^{2} \) Copy content Toggle raw display
$71$ \( (T + 4)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 9 T - 54)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 2 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 176)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 176)^{2} \) Copy content Toggle raw display
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