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Properties

Label 4032.2.h.h
Level $4032$
Weight $2$
Character orbit 4032.h
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(575,4032)]
 
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, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.h (of order \(2\), degree \(1\), 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: \(12\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: no (minimal twist has level 252)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} + \beta_{3} q^{7} + ( - \beta_{6} - \beta_{5}) q^{11} + \beta_{10} q^{13} + (2 \beta_{4} + \beta_{2}) q^{17} + (\beta_{7} - \beta_{3} + \beta_1) q^{19} + ( - \beta_{9} - 3 \beta_{6}) q^{23} + ( - \beta_{10} - 1) q^{25} + \beta_{4} q^{29} + ( - \beta_{7} - \beta_{3} + \beta_1) q^{31} - \beta_{5} q^{35} + ( - \beta_{10} + 2 \beta_{8}) q^{37} + (\beta_{11} - 2 \beta_{4}) q^{41} + ( - 2 \beta_{3} + 2 \beta_1) q^{43} + (\beta_{9} - 3 \beta_{5}) q^{47} - q^{49} + ( - \beta_{11} - \beta_{4} + \beta_{2}) q^{53} + (\beta_{7} - 7 \beta_{3} - \beta_1) q^{55} + (\beta_{9} - 4 \beta_{6} + \beta_{5}) q^{59} + ( - \beta_{10} - \beta_{8} - 5) q^{61} + ( - \beta_{11} + 2 \beta_{4} - 3 \beta_{2}) q^{65} + ( - 2 \beta_{7} - 3 \beta_{3} - \beta_1) q^{67} + (\beta_{9} - 5 \beta_{6}) q^{71} + (2 \beta_{10} - 3 \beta_{8} - 1) q^{73} + ( - \beta_{4} + \beta_{2}) q^{77} + (\beta_{3} + 3 \beta_1) q^{79} + ( - \beta_{9} - 2 \beta_{6} + \beta_{5}) q^{83} + (\beta_{10} - 2 \beta_{8} + 4) q^{85} + ( - \beta_{11} - 2 \beta_{4} + 2 \beta_{2}) q^{89} - \beta_{7} q^{91} + ( - 4 \beta_{6} + 4 \beta_{5}) q^{95} + (3 \beta_{8} + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{25} - 8 q^{37} - 12 q^{49} - 56 q^{61} + 56 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( \nu^{10} - 14\nu^{8} + 9\nu^{6} - 22\nu^{4} - 64\nu^{2} + 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} - 2\nu^{7} + 9\nu^{5} - 10\nu^{3} + 16\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{10} + 10\nu^{8} - 27\nu^{6} + 34\nu^{4} - 64\nu^{2} + 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 4\nu^{9} + 5\nu^{7} - 12\nu^{5} + 12\nu^{3} - 16\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{11} + 2\nu^{9} - 11\nu^{7} + 26\nu^{5} - 48\nu^{3} + 96\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{11} + 6\nu^{9} - 19\nu^{7} + 30\nu^{5} - 24\nu^{3} ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{10} + 10\nu^{8} - 17\nu^{6} + 50\nu^{4} - 72\nu^{2} + 96 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} + 4\nu^{8} - 13\nu^{6} + 28\nu^{4} - 36\nu^{2} + 40 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -5\nu^{11} + 38\nu^{9} - 109\nu^{7} + 206\nu^{5} - 320\nu^{3} + 480\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{10} + 2\nu^{8} - 5\nu^{6} + 10\nu^{4} - 12\nu^{2} + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3\nu^{11} - 10\nu^{9} + 27\nu^{7} - 50\nu^{5} + 96\nu^{3} - 48\nu ) / 16 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{9} - 2\beta_{6} + 3\beta_{5} - 2\beta_{4} - \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} - \beta_{7} - 2\beta_{3} - 2\beta _1 + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{11} + \beta_{9} + 10\beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{10} + 2\beta_{8} - 11\beta_{3} - \beta _1 - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{11} - 3\beta_{9} + 14\beta_{6} - \beta_{5} + 6\beta_{4} + 15\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4\beta_{10} - 5\beta_{8} + 5\beta_{7} - 14\beta_{3} + 2\beta _1 - 7 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3\beta_{11} - 9\beta_{9} - 26\beta_{6} + 9\beta_{5} - 30\beta_{4} + 15\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -\beta_{10} - 10\beta_{8} + 8\beta_{7} + 19\beta_{3} - 7\beta _1 - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -\beta_{11} + 3\beta_{9} - 46\beta_{6} - 31\beta_{5} - 102\beta_{4} - 15\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -28\beta_{10} + 13\beta_{8} + 3\beta_{7} + 22\beta_{3} - 10\beta _1 - 33 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3\beta_{11} + 25\beta_{9} - 38\beta_{6} - 121\beta_{5} + 62\beta_{4} + 17\beta_{2} ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
1.35489 + 0.405301i
−1.35489 + 0.405301i
0.892524 − 1.09700i
−0.892524 − 1.09700i
−1.16947 − 0.795191i
1.16947 − 0.795191i
1.16947 + 0.795191i
−1.16947 + 0.795191i
−0.892524 + 1.09700i
0.892524 + 1.09700i
−1.35489 − 0.405301i
1.35489 − 0.405301i
0 0 0 − 3.31339i 0 − 1.00000i 0 0 0
575.2 0 0 0 − 3.31339i 0 1.00000i 0 0 0
575.3 0 0 0 − 2.56483i 0 − 1.00000i 0 0 0
575.4 0 0 0 − 2.56483i 0 1.00000i 0 0 0
575.5 0 0 0 − 0.665647i 0 − 1.00000i 0 0 0
575.6 0 0 0 − 0.665647i 0 1.00000i 0 0 0
575.7 0 0 0 0.665647i 0 − 1.00000i 0 0 0
575.8 0 0 0 0.665647i 0 1.00000i 0 0 0
575.9 0 0 0 2.56483i 0 − 1.00000i 0 0 0
575.10 0 0 0 2.56483i 0 1.00000i 0 0 0
575.11 0 0 0 3.31339i 0 − 1.00000i 0 0 0
575.12 0 0 0 3.31339i 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 575.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.h.h 12
3.b odd 2 1 inner 4032.2.h.h 12
4.b odd 2 1 inner 4032.2.h.h 12
8.b even 2 1 252.2.e.a ✓ 12
8.d odd 2 1 252.2.e.a ✓ 12
12.b even 2 1 inner 4032.2.h.h 12
24.f even 2 1 252.2.e.a ✓ 12
24.h odd 2 1 252.2.e.a ✓ 12
56.e even 2 1 1764.2.e.g 12
56.h odd 2 1 1764.2.e.g 12
168.e odd 2 1 1764.2.e.g 12
168.i even 2 1 1764.2.e.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.e.a ✓ 12 8.b even 2 1
252.2.e.a ✓ 12 8.d odd 2 1
252.2.e.a ✓ 12 24.f even 2 1
252.2.e.a ✓ 12 24.h odd 2 1
1764.2.e.g 12 56.e even 2 1
1764.2.e.g 12 56.h odd 2 1
1764.2.e.g 12 168.e odd 2 1
1764.2.e.g 12 168.i even 2 1
4032.2.h.h 12 1.a even 1 1 trivial
4032.2.h.h 12 3.b odd 2 1 inner
4032.2.h.h 12 4.b odd 2 1 inner
4032.2.h.h 12 12.b even 2 1 inner

Hecke kernels

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

\( T_{5}^{6} + 18T_{5}^{4} + 80T_{5}^{2} + 32 \) Copy content Toggle raw display
\( T_{11}^{6} - 28T_{11}^{4} + 132T_{11}^{2} - 128 \) Copy content Toggle raw display
\( T_{13}^{3} - 28T_{13} + 16 \) 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
$5$ \( (T^{6} + 18 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( (T^{6} - 28 T^{4} + \cdots - 128)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - 28 T + 16)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 34 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 64 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 140 T^{4} + \cdots - 67712)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2)^{6} \) Copy content Toggle raw display
$31$ \( (T^{6} + 128 T^{4} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 2 T^{2} + \cdots + 128)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + 114 T^{4} + \cdots + 53792)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 160 T^{4} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 272 T^{4} + \cdots - 524288)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 134 T^{4} + \cdots + 2312)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 208 T^{4} + \cdots - 8192)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 14 T^{2} + 4 T - 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 216 T^{4} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 268 T^{4} + \cdots - 36992)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 172 T - 352)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + 312 T^{4} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 128 T^{4} + \cdots - 32768)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 226 T^{4} + \cdots + 170528)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 156 T + 128)^{4} \) Copy content Toggle raw display
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