Properties

Label 1476.2.k.a
Level $1476$
Weight $2$
Character orbit 1476.k
Analytic conductor $11.786$
Analytic rank $0$
Dimension $6$
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] = [1476,2,Mod(73,1476)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1476, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([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("1476.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1476 = 2^{2} \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1476.k (of order \(4\), degree \(2\), 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: \(11.7859193383\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.5089536.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 164)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4}) q^{5} + ( - \beta_{5} + \beta_{2} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{4}) q^{5} + ( - \beta_{5} + \beta_{2} + \beta_1) q^{7} + (\beta_{4} - \beta_1 + 1) q^{11} + ( - \beta_{5} + \beta_{2}) q^{13} + (2 \beta_{5} - \beta_{4} + 2 \beta_{2} + 1) q^{17} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{19} + 2 \beta_{2} q^{23} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{25} + (\beta_{4} + 2 \beta_1 + 1) q^{29} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{31} + ( - 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 3) q^{35} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 - 5) q^{37} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{41} + ( - 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 3 \beta_1) q^{43} + (\beta_{5} - 5 \beta_{4} + \beta_{3} + \beta_{2} + 5) q^{47} + ( - 3 \beta_{5} + 2 \beta_{4}) q^{49} + (2 \beta_{5} + 3 \beta_{4} - 2 \beta_{2} - 4 \beta_1 + 3) q^{53} + (\beta_{4} - 1) q^{55} - 2 \beta_{2} q^{59} + (2 \beta_{4} - \beta_{3} - \beta_1) q^{61} + ( - \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - \beta_{2} - 3) q^{65} + (2 \beta_{5} - 5 \beta_{4} + 5 \beta_{3} + 2 \beta_{2} + 5) q^{67} + ( - 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{2} + \beta_1 - 4) q^{71} + (5 \beta_{5} - 7 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{73} + ( - \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{77} + ( - 3 \beta_{5} + \beta_{4} + 3 \beta_{2} - \beta_1 + 1) q^{79} + (4 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 4) q^{83} + ( - 3 \beta_{5} + 7 \beta_{4} + 3 \beta_{2} + 4 \beta_1 + 7) q^{85} + ( - \beta_{5} + \beta_{2} - 6 \beta_1) q^{89} + ( - 2 \beta_{5} + 6 \beta_{4} + \beta_{3} + \beta_1) q^{91} + ( - \beta_{5} + 4 \beta_{4} + \beta_{2} + 2 \beta_1 + 4) q^{95} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{11} - 2 q^{13} + 2 q^{17} + 2 q^{19} - 4 q^{23} + 6 q^{25} + 10 q^{29} + 8 q^{31} - 18 q^{35} - 36 q^{37} - 6 q^{41} + 26 q^{47} + 14 q^{53} - 6 q^{55} + 4 q^{59} - 20 q^{65} + 16 q^{67} - 26 q^{71} - 2 q^{79} - 36 q^{83} + 44 q^{85} - 14 q^{89} + 26 q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 24\nu^{4} - 6\nu^{3} - \nu^{2} + 6\nu + 285 ) / 131 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} + 95\nu - 138 ) / 131 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23\nu^{5} - 28\nu^{4} + 7\nu^{3} + 154\nu^{2} + 386\nu - 267 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -23\nu^{5} + 28\nu^{4} - 7\nu^{3} - 23\nu^{2} - 386\nu + 267 ) / 131 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 4\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 6\beta_{2} - \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{5} - 3\beta_{4} + 7\beta_{2} - 18\beta _1 - 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1476\mathbb{Z}\right)^\times\).

\(n\) \(739\) \(821\) \(1441\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
0.675970 0.675970i
−1.33641 + 1.33641i
1.66044 1.66044i
1.66044 + 1.66044i
−1.33641 1.33641i
0.675970 + 0.675970i
0 0 0 3.08613i 0 2.76210 2.76210i 0 0 0
73.2 0 0 0 0.428007i 0 −1.90841 + 1.90841i 0 0 0
73.3 0 0 0 1.51414i 0 −0.853695 + 0.853695i 0 0 0
829.1 0 0 0 1.51414i 0 −0.853695 0.853695i 0 0 0
829.2 0 0 0 0.428007i 0 −1.90841 1.90841i 0 0 0
829.3 0 0 0 3.08613i 0 2.76210 + 2.76210i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1476.2.k.a 6
3.b odd 2 1 164.2.f.a 6
12.b even 2 1 656.2.l.f 6
41.c even 4 1 inner 1476.2.k.a 6
123.f odd 4 1 164.2.f.a 6
123.i even 8 2 6724.2.a.e 6
492.l even 4 1 656.2.l.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.f.a 6 3.b odd 2 1
164.2.f.a 6 123.f odd 4 1
656.2.l.f 6 12.b even 2 1
656.2.l.f 6 492.l even 4 1
1476.2.k.a 6 1.a even 1 1 trivial
1476.2.k.a 6 41.c even 4 1 inner
6724.2.a.e 6 123.i even 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 12T_{5}^{4} + 24T_{5}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1476, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 12 T^{4} + 24 T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{6} + 18 T^{3} + 144 T^{2} + \cdots + 162 \) Copy content Toggle raw display
$11$ \( T^{6} - 4 T^{5} + 8 T^{4} + 6 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$13$ \( T^{6} + 2 T^{5} + 2 T^{4} - 8 T^{3} + \cdots + 72 \) Copy content Toggle raw display
$17$ \( T^{6} - 2 T^{5} + 2 T^{4} + 96 T^{3} + \cdots + 72 \) Copy content Toggle raw display
$19$ \( T^{6} - 2 T^{5} + 2 T^{4} - 18 T^{3} + \cdots + 1458 \) Copy content Toggle raw display
$23$ \( (T^{3} + 2 T^{2} - 20 T - 24)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} - 10 T^{5} + 50 T^{4} + \cdots + 2312 \) Copy content Toggle raw display
$31$ \( (T^{3} - 4 T^{2} - 40 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 18 T^{2} + 72 T - 82)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 6 T^{5} + 63 T^{4} + \cdots + 68921 \) Copy content Toggle raw display
$43$ \( T^{6} + 204 T^{4} + 12864 T^{2} + \cdots + 258064 \) Copy content Toggle raw display
$47$ \( T^{6} - 26 T^{5} + 338 T^{4} + \cdots + 7442 \) Copy content Toggle raw display
$53$ \( T^{6} - 14 T^{5} + 98 T^{4} + \cdots + 20808 \) Copy content Toggle raw display
$59$ \( (T^{3} - 2 T^{2} - 20 T + 24)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + 24 T^{4} + 48 T^{2} + 16 \) Copy content Toggle raw display
$67$ \( T^{6} - 16 T^{5} + 128 T^{4} + \cdots + 2068578 \) Copy content Toggle raw display
$71$ \( T^{6} + 26 T^{5} + 338 T^{4} + \cdots + 13122 \) Copy content Toggle raw display
$73$ \( T^{6} + 456 T^{4} + 65280 T^{2} + \cdots + 2910436 \) Copy content Toggle raw display
$79$ \( T^{6} + 2 T^{5} + 2 T^{4} - 354 T^{3} + \cdots + 8978 \) Copy content Toggle raw display
$83$ \( (T^{3} + 18 T^{2} - 36 T - 1304)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 14 T^{5} + 98 T^{4} + \cdots + 267912 \) Copy content Toggle raw display
$97$ \( T^{6} + 6 T^{5} + 18 T^{4} + \cdots + 1352 \) Copy content Toggle raw display
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