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Properties

Label 32.19.d.b
Level $32$
Weight $19$
Character orbit 32.d
Analytic conductor $65.724$
Analytic rank $0$
Dimension $16$
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] = [32,19,Mod(15,32)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(32, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.15");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 19 \)
Character orbit: \([\chi]\) \(=\) 32.d (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: \(65.7235640671\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 2385848546028 x^{14} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{241}\cdot 3^{13}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 8)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 204) q^{3} - \beta_{2} q^{5} + \beta_{4} q^{7} + ( - \beta_{3} + 1452 \beta_1 + 144616011) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 204) q^{3} - \beta_{2} q^{5} + \beta_{4} q^{7} + ( - \beta_{3} + 1452 \beta_1 + 144616011) q^{9} + ( - \beta_{5} + 47 \beta_1 + 154473204) q^{11} + (\beta_{7} - 13 \beta_{4} - 281 \beta_{2}) q^{13} + ( - \beta_{11} - 20 \beta_{4} + 5192 \beta_{2}) q^{15} + (17 \beta_{6} - 34 \beta_{3} + \cdots - 11027456334) q^{17}+ \cdots + (526635 \beta_{10} + \cdots + 51\!\cdots\!48) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3264 q^{3} + 2313856176 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 3264 q^{3} + 2313856176 q^{9} + 2471571264 q^{11} - 176439301344 q^{17} + 833365634368 q^{19} - 15320509140080 q^{25} - 11565649473408 q^{27} - 900457491648 q^{33} - 20487495736320 q^{35} - 594931562445024 q^{41} + 25\!\cdots\!92 q^{43}+ \cdots + 81\!\cdots\!68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 2385848546028 x^{14} + \cdots + 71\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13\!\cdots\!02 \nu^{15} + \cdots + 22\!\cdots\!00 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\!\cdots\!02 \nu^{15} + \cdots - 22\!\cdots\!00 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 58\!\cdots\!34 \nu^{15} + \cdots + 36\!\cdots\!00 ) / 28\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 44\!\cdots\!88 \nu^{15} + \cdots - 52\!\cdots\!60 ) / 16\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 83\!\cdots\!36 \nu^{15} + \cdots + 16\!\cdots\!00 ) / 59\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 54\!\cdots\!86 \nu^{15} + \cdots + 11\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\!\cdots\!35 \nu^{15} + \cdots - 33\!\cdots\!20 ) / 23\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 85\!\cdots\!18 \nu^{15} + \cdots - 19\!\cdots\!00 ) / 11\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 53\!\cdots\!02 \nu^{15} + \cdots - 53\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12\!\cdots\!82 \nu^{15} + \cdots + 41\!\cdots\!40 ) / 47\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 62\!\cdots\!12 \nu^{15} + \cdots - 17\!\cdots\!20 ) / 12\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 84\!\cdots\!50 \nu^{15} + \cdots + 58\!\cdots\!40 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 18\!\cdots\!42 \nu^{15} + \cdots + 17\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 34\!\cdots\!96 \nu^{15} + \cdots + 82\!\cdots\!20 ) / 30\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 75\!\cdots\!30 \nu^{15} + \cdots - 16\!\cdots\!20 ) / 12\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{11} - 5 \beta_{10} - 24 \beta_{9} - 124 \beta_{8} - 178 \beta_{6} - 46 \beta_{5} + \cdots - 4771697091992 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 132190 \beta_{15} + 129262 \beta_{14} - 2595508 \beta_{13} + 10181718 \beta_{12} + \cdots - 71\!\cdots\!80 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 25732618860 \beta_{15} + 119251636798 \beta_{14} - 25432407642 \beta_{13} + 145844296362 \beta_{12} + \cdots + 20\!\cdots\!56 ) / 128 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 17\!\cdots\!25 \beta_{15} + \cdots + 11\!\cdots\!56 ) / 512 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 57\!\cdots\!60 \beta_{15} + \cdots - 19\!\cdots\!16 ) / 2048 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 28\!\cdots\!45 \beta_{15} + \cdots - 21\!\cdots\!80 ) / 8192 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 12\!\cdots\!80 \beta_{15} + \cdots + 24\!\cdots\!92 ) / 4096 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 23\!\cdots\!50 \beta_{15} + \cdots + 23\!\cdots\!16 ) / 8192 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 23\!\cdots\!00 \beta_{15} + \cdots - 33\!\cdots\!04 ) / 8192 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 18\!\cdots\!75 \beta_{15} + \cdots - 22\!\cdots\!20 ) / 8192 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 20\!\cdots\!00 \beta_{15} + \cdots + 23\!\cdots\!24 ) / 8192 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 14\!\cdots\!80 \beta_{15} + \cdots + 20\!\cdots\!96 ) / 8192 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 46\!\cdots\!40 \beta_{15} + \cdots - 41\!\cdots\!16 ) / 2048 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 11\!\cdots\!25 \beta_{15} + \cdots - 18\!\cdots\!40 ) / 8192 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
8911.24 + 869199.i
8911.24 − 869199.i
5609.53 + 28287.0i
5609.53 − 28287.0i
4762.71 + 225443.i
4762.71 − 225443.i
1433.94 + 570392.i
1433.94 − 570392.i
−2545.09 + 751432.i
−2545.09 − 751432.i
−3216.82 + 671356.i
−3216.82 − 671356.i
−6531.92 + 475784.i
−6531.92 − 475784.i
−8419.59 + 109070.i
−8419.59 − 109070.i
0 −35846.9 0 − 3.47680e6i 0 − 1.08296e7i 0 8.97583e8 0
15.2 0 −35846.9 0 3.47680e6i 0 1.08296e7i 0 8.97583e8 0
15.3 0 −22640.1 0 − 113148.i 0 3.55480e7i 0 1.25155e8 0
15.4 0 −22640.1 0 113148.i 0 − 3.55480e7i 0 1.25155e8 0
15.5 0 −19252.8 0 − 901770.i 0 6.41362e7i 0 −1.67489e7 0
15.6 0 −19252.8 0 901770.i 0 − 6.41362e7i 0 −1.67489e7 0
15.7 0 −5937.76 0 − 2.28157e6i 0 − 3.15838e7i 0 −3.52164e8 0
15.8 0 −5937.76 0 2.28157e6i 0 3.15838e7i 0 −3.52164e8 0
15.9 0 9978.36 0 − 3.00573e6i 0 5.44315e7i 0 −2.87853e8 0
15.10 0 9978.36 0 3.00573e6i 0 − 5.44315e7i 0 −2.87853e8 0
15.11 0 12665.3 0 − 2.68543e6i 0 − 4.44900e7i 0 −2.27011e8 0
15.12 0 12665.3 0 2.68543e6i 0 4.44900e7i 0 −2.27011e8 0
15.13 0 25925.7 0 − 1.90314e6i 0 9.33592e6i 0 2.84720e8 0
15.14 0 25925.7 0 1.90314e6i 0 − 9.33592e6i 0 2.84720e8 0
15.15 0 33476.4 0 − 436282.i 0 − 5.56713e7i 0 7.33247e8 0
15.16 0 33476.4 0 436282.i 0 5.56713e7i 0 7.33247e8 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.19.d.b 16
4.b odd 2 1 8.19.d.b ✓ 16
8.b even 2 1 8.19.d.b ✓ 16
8.d odd 2 1 inner 32.19.d.b 16
12.b even 2 1 72.19.b.b 16
24.h odd 2 1 72.19.b.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.19.d.b ✓ 16 4.b odd 2 1
8.19.d.b ✓ 16 8.b even 2 1
32.19.d.b 16 1.a even 1 1 trivial
32.19.d.b 16 8.d odd 2 1 inner
72.19.b.b 16 12.b even 2 1
72.19.b.b 16 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 1632 T_{3}^{7} - 2126814288 T_{3}^{6} - 1123288083072 T_{3}^{5} + \cdots + 10\!\cdots\!20 \) acting on \(S_{19}^{\mathrm{new}}(32, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + \cdots + 10\!\cdots\!20)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 43\!\cdots\!84)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots - 19\!\cdots\!80)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 60\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 80\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 30\!\cdots\!76)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 83\!\cdots\!20)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 45\!\cdots\!40)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 87\!\cdots\!20)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 55\!\cdots\!36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 14\!\cdots\!20)^{2} \) Copy content Toggle raw display
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