Properties

Label 4032.2.v.c
Level $4032$
Weight $2$
Character orbit 4032.v
Analytic conductor $32.196$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4032,2,Mod(1583,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.1583");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.v (of order \(4\), 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: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.653473922154496.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 1008)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{3}) q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{3}) q^{5} + q^{7} + (\beta_{5} + \beta_{3} + \beta_1) q^{11} + ( - \beta_{8} - \beta_{2} + 1) q^{13} + ( - \beta_{10} - \beta_{7} - 3 \beta_{3} - \beta_1) q^{17} - \beta_{10} q^{23} + \beta_{2} q^{25} + ( - \beta_{10} - \beta_{6} + \beta_1) q^{29} + (\beta_{8} + \beta_{4} + 2 \beta_{2}) q^{31} + ( - \beta_{5} + \beta_{3}) q^{35} + ( - \beta_{2} - 1) q^{37} + ( - \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_1) q^{41} + (\beta_{11} + 2 \beta_{8} - 2 \beta_{2} + 2) q^{43} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{47} + q^{49} + (\beta_{10} + \beta_{7} - \beta_{6} - 3 \beta_{5} + 3 \beta_{3}) q^{53} + (\beta_{11} + \beta_{9} - \beta_{8} + \beta_{4} - 4) q^{55} + ( - \beta_{10} - \beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_1) q^{59} + ( - \beta_{8} + 3 \beta_{2} - 3) q^{61} + (2 \beta_{10} + 2 \beta_{3}) q^{65} + (\beta_{4} - 4 \beta_{2} - 4) q^{67} + ( - 3 \beta_{10} - \beta_{7} + 2 \beta_{3} - \beta_1) q^{71} + ( - \beta_{11} + \beta_{9} - 4 \beta_{2}) q^{73} + (\beta_{5} + \beta_{3} + \beta_1) q^{77} + ( - 2 \beta_{8} - 2 \beta_{4} - 4 \beta_{2}) q^{79} + (\beta_{10} - \beta_{6} + 5 \beta_{5} - 5 \beta_{3}) q^{83} + ( - 2 \beta_{9} + 6 \beta_{2} + 6) q^{85} + ( - \beta_{7} + 3 \beta_{6} - 7 \beta_{5} + \beta_1) q^{89} + ( - \beta_{8} - \beta_{2} + 1) q^{91} + ( - \beta_{11} - \beta_{9} - 3 \beta_{8} + 3 \beta_{4} - 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{7} + 16 q^{13} - 12 q^{37} + 20 q^{43} + 12 q^{49} - 32 q^{55} - 32 q^{61} - 44 q^{67} + 64 q^{85} + 16 q^{91} - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{11} + 6\nu^{9} - 5\nu^{7} + 46\nu^{5} - 32\nu^{3} + 32\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{10} + 10\nu^{8} - 27\nu^{6} + 34\nu^{4} - 64\nu^{2} + 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 4\nu^{9} + 5\nu^{7} - 12\nu^{5} + 12\nu^{3} - 16\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{10} + 18\nu^{8} - 55\nu^{6} + 138\nu^{4} - 256\nu^{2} + 352 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{11} + 6\nu^{9} - 19\nu^{7} + 30\nu^{5} - 24\nu^{3} ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} - 18\nu^{9} + 49\nu^{7} - 90\nu^{5} + 136\nu^{3} - 192\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{11} + 10\nu^{9} - 27\nu^{7} + 98\nu^{5} - 128\nu^{3} + 224\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -7\nu^{10} + 18\nu^{8} - 31\nu^{6} + 74\nu^{4} - 160\nu^{2} + 96 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -11\nu^{10} + 10\nu^{8} - 67\nu^{6} + 98\nu^{4} - 160\nu^{2} + 96 ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3\nu^{11} - 12\nu^{9} + 31\nu^{7} - 68\nu^{5} + 116\nu^{3} - 80\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -13\nu^{10} + 54\nu^{8} - 117\nu^{6} + 286\nu^{4} - 320\nu^{2} + 416 ) / 64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - 2\beta_{8} - \beta_{4} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{10} + \beta_{7} - \beta_{6} + 5\beta_{5} - \beta_{3} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{11} + \beta_{9} - \beta_{8} - 10\beta_{2} - 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{10} + 3\beta_{7} + 3\beta_{6} + 7\beta_{5} + 3\beta_{3} + 5\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{11} - 2\beta_{9} + 8\beta_{8} - \beta_{4} - 16\beta_{2} - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3\beta_{10} + 3\beta_{7} + 9\beta_{6} - 13\beta_{5} - 15\beta_{3} + 3\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2\beta_{11} - 13\beta_{9} + 5\beta_{8} - 4\beta_{4} + 26\beta_{2} + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -\beta_{10} - 11\beta_{7} - 3\beta_{6} - 23\beta_{5} - 51\beta_{3} + 3\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -\beta_{11} - 14\beta_{9} - 24\beta_{8} + 17\beta_{4} + 32\beta_{2} - 46 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3\beta_{10} - 19\beta_{7} - 25\beta_{6} - 19\beta_{5} + 31\beta_{3} + 29\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1583.1
1.35489 + 0.405301i
−1.16947 0.795191i
−0.892524 + 1.09700i
0.892524 1.09700i
1.16947 + 0.795191i
−1.35489 0.405301i
1.35489 0.405301i
−1.16947 + 0.795191i
−0.892524 1.09700i
0.892524 + 1.09700i
1.16947 0.795191i
−1.35489 + 0.405301i
0 0 0 −1.41421 1.41421i 0 1.00000 0 0 0
1583.2 0 0 0 −1.41421 1.41421i 0 1.00000 0 0 0
1583.3 0 0 0 −1.41421 1.41421i 0 1.00000 0 0 0
1583.4 0 0 0 1.41421 + 1.41421i 0 1.00000 0 0 0
1583.5 0 0 0 1.41421 + 1.41421i 0 1.00000 0 0 0
1583.6 0 0 0 1.41421 + 1.41421i 0 1.00000 0 0 0
3599.1 0 0 0 −1.41421 + 1.41421i 0 1.00000 0 0 0
3599.2 0 0 0 −1.41421 + 1.41421i 0 1.00000 0 0 0
3599.3 0 0 0 −1.41421 + 1.41421i 0 1.00000 0 0 0
3599.4 0 0 0 1.41421 1.41421i 0 1.00000 0 0 0
3599.5 0 0 0 1.41421 1.41421i 0 1.00000 0 0 0
3599.6 0 0 0 1.41421 1.41421i 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1583.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.v.c 12
3.b odd 2 1 inner 4032.2.v.c 12
4.b odd 2 1 1008.2.v.c 12
12.b even 2 1 1008.2.v.c 12
16.e even 4 1 1008.2.v.c 12
16.f odd 4 1 inner 4032.2.v.c 12
48.i odd 4 1 1008.2.v.c 12
48.k even 4 1 inner 4032.2.v.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.v.c 12 4.b odd 2 1
1008.2.v.c 12 12.b even 2 1
1008.2.v.c 12 16.e even 4 1
1008.2.v.c 12 48.i odd 4 1
4032.2.v.c 12 1.a even 1 1 trivial
4032.2.v.c 12 3.b odd 2 1 inner
4032.2.v.c 12 16.f odd 4 1 inner
4032.2.v.c 12 48.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5}^{4} + 16 \) Copy content Toggle raw display
\( T_{11}^{12} + 1056T_{11}^{8} + 53504T_{11}^{4} + 65536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{4} + 16)^{3} \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 1056 T^{8} + 53504 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$13$ \( (T^{6} - 8 T^{5} + 32 T^{4} + 16 T^{3} + \cdots + 128)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 104 T^{4} + 2960 T^{2} + \cdots + 15488)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( (T^{6} + 30 T^{4} + 188 T^{2} + 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 7200 T^{8} + \cdots + 4294967296 \) Copy content Toggle raw display
$31$ \( (T^{6} + 80 T^{4} + 576 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T + 2)^{6} \) Copy content Toggle raw display
$41$ \( (T^{6} - 128 T^{4} + 656 T^{2} - 512)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 10 T^{5} + 50 T^{4} + \cdots + 412232)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 128 T^{4} + 4352 T^{2} + \cdots - 32768)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 18704 T^{8} + \cdots + 16777216 \) Copy content Toggle raw display
$59$ \( T^{12} + 36992 T^{8} + \cdots + 136651472896 \) Copy content Toggle raw display
$61$ \( (T^{6} + 16 T^{5} + 128 T^{4} + 464 T^{3} + \cdots + 128)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 22 T^{5} + 242 T^{4} + 1376 T^{3} + \cdots + 2888)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 238 T^{4} + 17916 T^{2} + \cdots + 412232)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 156 T^{4} + 7664 T^{2} + \cdots + 118336)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 320 T^{4} + 9216 T^{2} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 52096 T^{8} + \cdots + 68719476736 \) Copy content Toggle raw display
$89$ \( (T^{6} - 488 T^{4} + 58256 T^{2} + \cdots - 359552)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 14 T^{2} - 324 T - 4664)^{4} \) Copy content Toggle raw display
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