Properties

Label 4032.2.p.k
Level $4032$
Weight $2$
Character orbit 4032.p
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(1567,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 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("4032.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.p (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: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 15x^{10} + 90x^{8} - 247x^{6} + 270x^{4} + 21x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 1344)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} - \beta_{10} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} - \beta_{10} q^{7} + \beta_{11} q^{11} + 2 q^{13} + ( - \beta_{5} + \beta_{3}) q^{17} + (\beta_{10} + \beta_{9}) q^{19} + (\beta_{10} + \beta_{9} - \beta_{8} - \beta_{6}) q^{23} + (\beta_{4} + 2 \beta_{2} + 3) q^{25} + (2 \beta_{3} - \beta_1) q^{29} + (2 \beta_{11} + \beta_{7}) q^{31} + (\beta_{11} - 2 \beta_{9} + \beta_{8} + \beta_{7}) q^{35} + ( - \beta_{3} - \beta_1) q^{37} + ( - \beta_{5} + \beta_{3} - 2 \beta_1) q^{41} + (2 \beta_{11} + \beta_{10} - \beta_{9}) q^{43} + ( - 2 \beta_{11} - 2 \beta_{7}) q^{47} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{49} + ( - 2 \beta_{5} - \beta_1) q^{53} + (2 \beta_{11} - \beta_{10} + \beta_{9} - \beta_{7}) q^{55} + 2 \beta_{6} q^{59} + ( - 2 \beta_{4} - 2 \beta_{2} - 6) q^{61} - 2 \beta_{2} q^{65} + ( - 3 \beta_{10} + 3 \beta_{9} + 2 \beta_{7}) q^{67} + (\beta_{10} + \beta_{9} - \beta_{8} - 3 \beta_{6}) q^{71} + (2 \beta_{5} + 2 \beta_{3}) q^{73} + ( - 2 \beta_{5} - \beta_{2} + \beta_1) q^{77} + (\beta_{10} + \beta_{9} - 2 \beta_{8} + 2 \beta_{6}) q^{79} + ( - 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 2 \beta_{6}) q^{83} + ( - 4 \beta_{5} - \beta_{3} - \beta_1) q^{85} + ( - \beta_{5} + \beta_{3} + 2 \beta_1) q^{89} - 2 \beta_{10} q^{91} + (2 \beta_{10} + 2 \beta_{9} - 2 \beta_{8}) q^{95} + (2 \beta_{5} + 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 24 q^{13} + 36 q^{25} - 12 q^{49} - 72 q^{61}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 15x^{10} + 90x^{8} - 247x^{6} + 270x^{4} + 21x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8\nu^{10} - 332\nu^{8} + 3624\nu^{6} - 15496\nu^{4} + 23800\nu^{2} + 126 ) / 2947 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -27\nu^{10} + 489\nu^{8} - 2969\nu^{6} + 6410\nu^{4} + 928\nu^{2} - 7056 ) / 2947 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -54\nu^{10} + 557\nu^{8} - 1728\nu^{6} - 1073\nu^{4} + 6908\nu^{2} + 623 ) / 2947 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 58\nu^{10} - 723\nu^{8} + 3540\nu^{6} - 6675\nu^{4} - 902\nu^{2} + 14175 ) / 2947 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -149\nu^{10} + 2184\nu^{8} - 12767\nu^{6} + 34329\nu^{4} - 38694\nu^{2} + 1337 ) / 2947 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -346\nu^{11} + 5939\nu^{9} - 40121\nu^{7} + 121218\nu^{5} - 123779\nu^{3} - 65863\nu ) / 20629 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} - 15\nu^{9} + 97\nu^{7} - 317\nu^{5} + 501\nu^{3} - 161\nu ) / 49 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 501\nu^{11} - 7109\nu^{9} + 40029\nu^{7} - 98967\nu^{5} + 88545\nu^{3} + 45465\nu ) / 20629 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -151\nu^{11} + 2267\nu^{9} - 13673\nu^{7} + 38203\nu^{5} - 41697\nu^{3} - 14903\nu ) / 5894 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -1137\nu^{11} + 16663\nu^{9} - 94903\nu^{7} + 232931\nu^{5} - 169503\nu^{3} - 135051\nu ) / 41258 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -834\nu^{11} + 13140\nu^{9} - 83523\nu^{7} + 252681\nu^{5} - 346259\nu^{3} + 118006\nu ) / 20629 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{10} - 2\beta_{9} + \beta_{8} - \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{4} - 2\beta_{3} + \beta _1 + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{11} + 10\beta_{10} - 4\beta_{9} + 8\beta_{8} - 5\beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{5} - 8\beta_{4} - 16\beta_{3} - 4\beta_{2} + 11\beta _1 + 30 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -8\beta_{11} + 42\beta_{10} + 8\beta_{9} + 51\beta_{8} - 18\beta_{7} - 10\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 17\beta_{5} - 9\beta_{4} - 46\beta_{3} - 9\beta_{2} + 41\beta _1 + 22 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -10\beta_{11} + 148\beta_{10} + 160\beta_{9} + 265\beta_{8} - 15\beta_{7} - 98\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 208\beta_{5} + 60\beta_{4} - 444\beta_{3} + 8\beta_{2} + 469\beta _1 - 290 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 168\beta_{11} + 312\beta_{10} + 1218\beta_{9} + 1168\beta_{8} + 405\beta_{7} - 594\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 978\beta_{5} + 1098\beta_{4} - 1792\beta_{3} + 738\beta_{2} + 2123\beta _1 - 3670 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1956\beta_{11} - 1050\beta_{10} + 6968\beta_{9} + 4135\beta_{8} + 4364\beta_{7} - 2574\beta_{6} ) / 4 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
1.75780 + 0.500000i
1.75780 0.500000i
−1.75780 + 0.500000i
−1.75780 0.500000i
2.23871 + 0.500000i
2.23871 0.500000i
−2.23871 + 0.500000i
−2.23871 0.500000i
−0.385124 0.500000i
−0.385124 + 0.500000i
0.385124 0.500000i
0.385124 + 0.500000i
0 0 0 −3.88448 0 −2.62383 0.339877i 0 0 0
1567.2 0 0 0 −3.88448 0 −2.62383 + 0.339877i 0 0 0
1567.3 0 0 0 −3.88448 0 2.62383 0.339877i 0 0 0
1567.4 0 0 0 −3.88448 0 2.62383 + 0.339877i 0 0 0
1567.5 0 0 0 1.11575 0 −1.37268 2.26180i 0 0 0
1567.6 0 0 0 1.11575 0 −1.37268 + 2.26180i 0 0 0
1567.7 0 0 0 1.11575 0 1.37268 2.26180i 0 0 0
1567.8 0 0 0 1.11575 0 1.37268 + 2.26180i 0 0 0
1567.9 0 0 0 2.76873 0 −0.480901 2.60168i 0 0 0
1567.10 0 0 0 2.76873 0 −0.480901 + 2.60168i 0 0 0
1567.11 0 0 0 2.76873 0 0.480901 2.60168i 0 0 0
1567.12 0 0 0 2.76873 0 0.480901 + 2.60168i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.p.k 12
3.b odd 2 1 1344.2.p.d yes 12
4.b odd 2 1 inner 4032.2.p.k 12
7.b odd 2 1 4032.2.p.j 12
8.b even 2 1 4032.2.p.j 12
8.d odd 2 1 4032.2.p.j 12
12.b even 2 1 1344.2.p.d yes 12
21.c even 2 1 1344.2.p.c 12
24.f even 2 1 1344.2.p.c 12
24.h odd 2 1 1344.2.p.c 12
28.d even 2 1 4032.2.p.j 12
56.e even 2 1 inner 4032.2.p.k 12
56.h odd 2 1 inner 4032.2.p.k 12
84.h odd 2 1 1344.2.p.c 12
168.e odd 2 1 1344.2.p.d yes 12
168.i even 2 1 1344.2.p.d yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.2.p.c 12 21.c even 2 1
1344.2.p.c 12 24.f even 2 1
1344.2.p.c 12 24.h odd 2 1
1344.2.p.c 12 84.h odd 2 1
1344.2.p.d yes 12 3.b odd 2 1
1344.2.p.d yes 12 12.b even 2 1
1344.2.p.d yes 12 168.e odd 2 1
1344.2.p.d yes 12 168.i even 2 1
4032.2.p.j 12 7.b odd 2 1
4032.2.p.j 12 8.b even 2 1
4032.2.p.j 12 8.d odd 2 1
4032.2.p.j 12 28.d even 2 1
4032.2.p.k 12 1.a even 1 1 trivial
4032.2.p.k 12 4.b odd 2 1 inner
4032.2.p.k 12 56.e even 2 1 inner
4032.2.p.k 12 56.h odd 2 1 inner

Hecke kernels

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

\( T_{5}^{3} - 12T_{5} + 12 \) Copy content Toggle raw display
\( T_{11}^{6} - 36T_{11}^{4} + 96T_{11}^{2} - 48 \) Copy content Toggle raw display
\( T_{13} - 2 \) 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
$5$ \( (T^{3} - 12 T + 12)^{4} \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{10} - 33 T^{8} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{6} - 36 T^{4} + 96 T^{2} - 48)^{2} \) Copy content Toggle raw display
$13$ \( (T - 2)^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} + 72 T^{4} + 1392 T^{2} + \cdots + 8112)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 48 T^{4} + 576 T^{2} + 256)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 60 T^{4} + 1152 T^{2} + \cdots + 7056)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 132 T^{4} + 4272 T^{2} + \cdots + 9408)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 132 T^{4} + 5424 T^{2} + \cdots - 69312)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 96 T^{4} + 1536 T^{2} + 768)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 120 T^{4} + 1104 T^{2} + 48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 180 T^{4} + 7728 T^{2} + \cdots - 9408)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 192 T^{4} + 10752 T^{2} + \cdots - 150528)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 132 T^{4} + 5424 T^{2} + \cdots + 69312)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 96 T^{4} + 2304 T^{2} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 18 T^{2} + 12 T - 712)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} - 324 T^{4} + 22896 T^{2} + \cdots - 292032)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 204 T^{4} + 6048 T^{2} + \cdots + 7056)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 288 T^{4} + 20736 T^{2} + \cdots + 248832)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 240 T^{4} + 4992 T^{2} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 240 T^{4} + 18432 T^{2} + \cdots + 451584)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 312 T^{4} + 24720 T^{2} + \cdots + 241968)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 192 T^{4} + 10752 T^{2} + \cdots + 150528)^{2} \) Copy content Toggle raw display
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