Properties

Label 912.2.ch.d
Level $912$
Weight $2$
Character orbit 912.ch
Analytic conductor $7.282$
Analytic rank $0$
Dimension $6$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(47,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 0, 9, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.ch (of order \(18\), degree \(6\), 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: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

$q$-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 primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{18}^{4} + 2 \zeta_{18}) q^{3} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 3 \zeta_{18}^{2} - \zeta_{18}) q^{7} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{18}^{4} + 2 \zeta_{18}) q^{3} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 3 \zeta_{18}^{2} - \zeta_{18}) q^{7} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{2}) q^{9} + ( - \zeta_{18}^{4} - \zeta_{18}^{3} - 3 \zeta_{18} + 4) q^{13} + ( - 3 \zeta_{18}^{3} + 5) q^{19} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} - 4 \zeta_{18}^{2} - 5) q^{21} - 5 \zeta_{18}^{5} q^{25} + ( - 3 \zeta_{18}^{3} + 6) q^{27} + (5 \zeta_{18}^{5} - 5 \zeta_{18}^{4} - 6 \zeta_{18}^{2} - \zeta_{18}) q^{31} + ( - 7 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 3 \zeta_{18}^{2} + 3 \zeta_{18}) q^{37} + (2 \zeta_{18}^{5} - 5 \zeta_{18}^{4} - 7 \zeta_{18}^{2} + 7 \zeta_{18}) q^{39} + (\zeta_{18}^{5} - 7 \zeta_{18}^{3} - 7 \zeta_{18}^{2} + 1) q^{43} + (8 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 7 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 8 \zeta_{18}) q^{49} + ( - 8 \zeta_{18}^{4} + 7 \zeta_{18}) q^{57} + (9 \zeta_{18}^{4} - 4 \zeta_{18}^{3} - 4 \zeta_{18} + 9) q^{61} + (6 \zeta_{18}^{4} - 6 \zeta_{18}^{3} - 9 \zeta_{18} - 3) q^{63} + (7 \zeta_{18}^{4} + 9 \zeta_{18}^{3} + 2 \zeta_{18} - 2) q^{67} + (8 \zeta_{18}^{5} + 9 \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{73} + ( - 10 \zeta_{18}^{3} + 5) q^{75} + ( - 3 \zeta_{18}^{5} + 10 \zeta_{18}^{3} - 7 \zeta_{18}^{2} - 7) q^{79} + ( - 9 \zeta_{18}^{4} + 9 \zeta_{18}) q^{81} + (18 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + 6 \zeta_{18}^{3} - 9 \zeta_{18}^{2} - 6 \zeta_{18} + 5) q^{91} + ( - 4 \zeta_{18}^{5} + 4 \zeta_{18}^{3} - 7 \zeta_{18}^{2} - 11) q^{93} + 5 \zeta_{18} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 21 q^{13} + 21 q^{19} - 27 q^{21} + 27 q^{27} - 15 q^{43} + 21 q^{49} + 42 q^{61} - 36 q^{63} + 15 q^{67} + 21 q^{73} - 12 q^{79} + 48 q^{91} - 54 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(\zeta_{18}^{2}\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−0.173648 0.984808i
0.939693 + 0.342020i
0.939693 0.342020i
−0.766044 + 0.642788i
−0.766044 0.642788i
−0.173648 + 0.984808i
0 −1.11334 1.32683i 0 0 0 −0.0714517 0.0412527i 0 −0.520945 + 2.95442i 0
479.1 0 1.70574 0.300767i 0 0 0 −3.93242 2.27038i 0 2.81908 1.02606i 0
575.1 0 1.70574 + 0.300767i 0 0 0 −3.93242 + 2.27038i 0 2.81908 + 1.02606i 0
671.1 0 −0.592396 + 1.62760i 0 0 0 4.00387 + 2.31164i 0 −2.29813 1.92836i 0
719.1 0 −0.592396 1.62760i 0 0 0 4.00387 2.31164i 0 −2.29813 + 1.92836i 0
815.1 0 −1.11334 + 1.32683i 0 0 0 −0.0714517 + 0.0412527i 0 −0.520945 2.95442i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
76.l odd 18 1 inner
228.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.ch.d yes 6
3.b odd 2 1 CM 912.2.ch.d yes 6
4.b odd 2 1 912.2.ch.c 6
12.b even 2 1 912.2.ch.c 6
19.e even 9 1 912.2.ch.c 6
57.l odd 18 1 912.2.ch.c 6
76.l odd 18 1 inner 912.2.ch.d yes 6
228.v even 18 1 inner 912.2.ch.d yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.ch.c 6 4.b odd 2 1
912.2.ch.c 6 12.b even 2 1
912.2.ch.c 6 19.e even 9 1
912.2.ch.c 6 57.l odd 18 1
912.2.ch.d yes 6 1.a even 1 1 trivial
912.2.ch.d yes 6 3.b odd 2 1 CM
912.2.ch.d yes 6 76.l odd 18 1 inner
912.2.ch.d yes 6 228.v even 18 1 inner

Hecke kernels

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

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{6} - 21T_{7}^{4} + 441T_{7}^{2} + 63T_{7} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 9T^{3} + 27 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 21 T^{4} + 441 T^{2} + 63 T + 3 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 21 T^{5} + 186 T^{4} + \cdots + 7921 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( (T^{2} - 7 T + 19)^{3} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - 93 T^{4} + 8649 T^{2} + \cdots + 35643 \) Copy content Toggle raw display
$37$ \( (T^{3} - 111 T + 433)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + 15 T^{5} + 204 T^{4} + \cdots + 116427 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} - 42 T^{5} + 771 T^{4} + \cdots + 811801 \) Copy content Toggle raw display
$67$ \( T^{6} - 15 T^{5} + 276 T^{4} + \cdots + 1186923 \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} - 21 T^{5} + 366 T^{4} + \cdots + 844561 \) Copy content Toggle raw display
$79$ \( T^{6} + 12 T^{5} + 285 T^{4} + \cdots + 1719147 \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} - 125 T^{3} + 15625 \) Copy content Toggle raw display
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