Properties

Label 912.2.bn.j
Level $912$
Weight $2$
Character orbit 912.bn
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(65,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bn (of order \(6\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 456)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{3} - 1) q^{5} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{7} + (\beta_{3} + 3 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{3} - 1) q^{5} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{7} + (\beta_{3} + 3 \beta_{2}) q^{9} + (4 \beta_{2} - 2) q^{11} + (\beta_{2} + 2 \beta_1 - 1) q^{13} + (\beta_{3} + 3) q^{15} + ( - \beta_{3} - 1) q^{17} + ( - 2 \beta_{2} + 5) q^{19} + (\beta_{3} + 3 \beta_{2} + 3) q^{21} + ( - 3 \beta_{2} - 5 \beta_1 + 2) q^{23} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{25} + (2 \beta_{3} - 2 \beta_1 - 3) q^{27} + (2 \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{29} + (\beta_{3} - 7 \beta_{2} - \beta_1 + 4) q^{31} + (4 \beta_{3} - 2 \beta_1) q^{33} + (2 \beta_{3} - 2 \beta_{2} + 6) q^{35} + (\beta_{3} - 11 \beta_{2} - \beta_1 + 6) q^{37} + ( - \beta_{3} - 6 \beta_{2}) q^{39} + (\beta_{3} - 4 \beta_{2} - 2 \beta_1 + 1) q^{41} + ( - 2 \beta_{3} + 7 \beta_{2} + 4 \beta_1 - 2) q^{43} + ( - \beta_{3} - 2 \beta_1 - 3) q^{45} + (\beta_{2} - \beta_1 + 2) q^{47} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{49} + (\beta_{3} + 3) q^{51} + ( - 2 \beta_{3} - 9 \beta_{2} + \beta_1 + 8) q^{53} + (2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 2) q^{55} + ( - 2 \beta_{3} - 3 \beta_1) q^{57} + (\beta_{3} + 4 \beta_{2} - 2 \beta_1 + 1) q^{59} + ( - \beta_{2} + 1) q^{61} + (2 \beta_{3} - 5 \beta_1 - 3) q^{63} + ( - \beta_{3} - \beta_{2} - \beta_1 - 5) q^{65} + ( - \beta_{2} + 4 \beta_1 - 5) q^{67} + (2 \beta_{3} + 15 \beta_{2} + \beta_1) q^{69} + ( - \beta_{3} + 10 \beta_{2} + 2 \beta_1 - 1) q^{71} + (2 \beta_{3} - \beta_{2} - 4 \beta_1 + 2) q^{73} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 6) q^{75} + (6 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 4) q^{77} + (8 \beta_{3} + 3 \beta_{2} + 2) q^{79} + (6 \beta_{2} + 5 \beta_1 - 6) q^{81} + ( - 4 \beta_{2} + 2) q^{83} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{85} + (2 \beta_{3} + 3 \beta_{2} + \beta_1 - 6) q^{87} + ( - 2 \beta_{3} - 5 \beta_{2} + \beta_1 + 4) q^{89} + ( - 6 \beta_{2} - \beta_1 - 5) q^{91} + ( - 7 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 3) q^{93} + ( - 5 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 5) q^{95} + ( - 3 \beta_{3} - 6 \beta_{2} + 9) q^{97} + ( - 2 \beta_{3} + 6 \beta_{2} + 4 \beta_1 - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} - 3 q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} - 3 q^{5} - 2 q^{7} + 5 q^{9} + 11 q^{15} - 3 q^{17} + 16 q^{19} + 17 q^{21} - 3 q^{23} - 3 q^{25} - 16 q^{27} - 5 q^{29} - 6 q^{33} + 18 q^{35} - 11 q^{39} - 7 q^{41} + 12 q^{43} - 13 q^{45} + 9 q^{47} + 6 q^{49} + 11 q^{51} + 17 q^{53} - 6 q^{55} - q^{57} + 9 q^{59} + 2 q^{61} - 19 q^{63} - 22 q^{65} - 18 q^{67} + 29 q^{69} + 19 q^{71} - 18 q^{75} + 6 q^{79} - 7 q^{81} + 7 q^{85} - 19 q^{87} + 9 q^{89} - 33 q^{91} + 5 q^{93} - 9 q^{95} + 27 q^{97} - 30 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 2\beta _1 + 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(1 - \beta_{2}\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.68614 0.396143i
−1.18614 + 1.26217i
1.68614 + 0.396143i
−1.18614 1.26217i
0 −1.68614 + 0.396143i 0 −2.18614 1.26217i 0 −3.37228 0 2.68614 1.33591i 0
65.2 0 1.18614 1.26217i 0 0.686141 + 0.396143i 0 2.37228 0 −0.186141 2.99422i 0
449.1 0 −1.68614 0.396143i 0 −2.18614 + 1.26217i 0 −3.37228 0 2.68614 + 1.33591i 0
449.2 0 1.18614 + 1.26217i 0 0.686141 0.396143i 0 2.37228 0 −0.186141 + 2.99422i 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.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bn.j 4
3.b odd 2 1 912.2.bn.i 4
4.b odd 2 1 456.2.bf.a 4
12.b even 2 1 456.2.bf.b yes 4
19.d odd 6 1 912.2.bn.i 4
57.f even 6 1 inner 912.2.bn.j 4
76.f even 6 1 456.2.bf.b yes 4
228.n odd 6 1 456.2.bf.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.bf.a 4 4.b odd 2 1
456.2.bf.a 4 228.n odd 6 1
456.2.bf.b yes 4 12.b even 2 1
456.2.bf.b yes 4 76.f even 6 1
912.2.bn.i 4 3.b odd 2 1
912.2.bn.i 4 19.d odd 6 1
912.2.bn.j 4 1.a even 1 1 trivial
912.2.bn.j 4 57.f even 6 1 inner

Hecke kernels

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

\( T_{5}^{4} + 3T_{5}^{3} + T_{5}^{2} - 6T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} - 8 \) Copy content Toggle raw display
\( T_{17}^{4} + 3T_{17}^{3} + T_{17}^{2} - 6T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} - 2 T^{2} + 3 T + 9 \) Copy content Toggle raw display
$5$ \( T^{4} + 3 T^{3} + T^{2} - 6 T + 4 \) Copy content Toggle raw display
$7$ \( (T^{2} + T - 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 11T^{2} + 121 \) Copy content Toggle raw display
$17$ \( T^{4} + 3 T^{3} + T^{2} - 6 T + 4 \) Copy content Toggle raw display
$19$ \( (T^{2} - 8 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 3 T^{3} - 65 T^{2} + \cdots + 4624 \) Copy content Toggle raw display
$29$ \( T^{4} + 5 T^{3} + 27 T^{2} - 10 T + 4 \) Copy content Toggle raw display
$31$ \( T^{4} + 79T^{2} + 1156 \) Copy content Toggle raw display
$37$ \( T^{4} + 187T^{2} + 7744 \) Copy content Toggle raw display
$41$ \( T^{4} + 7 T^{3} + 45 T^{2} + 28 T + 16 \) Copy content Toggle raw display
$43$ \( T^{4} - 12 T^{3} + 141 T^{2} - 36 T + 9 \) Copy content Toggle raw display
$47$ \( T^{4} - 9 T^{3} + 31 T^{2} - 36 T + 16 \) Copy content Toggle raw display
$53$ \( T^{4} - 17 T^{3} + 225 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$59$ \( T^{4} - 9 T^{3} + 69 T^{2} - 108 T + 144 \) Copy content Toggle raw display
$61$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 18 T^{3} + 91 T^{2} + \cdots + 289 \) Copy content Toggle raw display
$71$ \( T^{4} - 19 T^{3} + 279 T^{2} + \cdots + 6724 \) Copy content Toggle raw display
$73$ \( T^{4} + 33T^{2} + 1089 \) Copy content Toggle raw display
$79$ \( T^{4} - 6 T^{3} - 161 T^{2} + \cdots + 29929 \) Copy content Toggle raw display
$83$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 9 T^{3} + 69 T^{2} - 108 T + 144 \) Copy content Toggle raw display
$97$ \( T^{4} - 27 T^{3} + 279 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
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