Properties

Label 32.21.d.b
Level $32$
Weight $21$
Character orbit 32.d
Analytic conductor $81.124$
Analytic rank $0$
Dimension $18$
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] = [32,21,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, 21, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.15");
 
S:= CuspForms(chi, 21);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 21 \)
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: \(81.1244048329\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 66008406614424 x^{16} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: multiple of \( 2^{309}\cdot 3^{15}\cdot 5^{4}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 6346) q^{3} - \beta_{2} q^{5} + ( - \beta_{3} - 2 \beta_{2}) q^{7} + (\beta_{4} + 21347 \beta_1 + 566536126) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 6346) q^{3} - \beta_{2} q^{5} + ( - \beta_{3} - 2 \beta_{2}) q^{7} + (\beta_{4} + 21347 \beta_1 + 566536126) q^{9} + (\beta_{7} - \beta_{4} + 64218 \beta_1 - 1547910835) q^{11} + (\beta_{6} - 97 \beta_{3} + 718 \beta_{2}) q^{13} + ( - \beta_{9} - 62 \beta_{3} - 11526 \beta_{2}) q^{15} + ( - \beta_{10} - \beta_{8} + 6 \beta_{7} + 132 \beta_{4} + \cdots + 241329174441) q^{17}+ \cdots + ( - 2177802 \beta_{14} + 38538252 \beta_{11} + \cdots - 34\!\cdots\!47) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 114228 q^{3} + 10197650262 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 114228 q^{3} + 10197650262 q^{9} - 27862395020 q^{11} + 4343925139172 q^{17} + 360681653556 q^{19} - 395727477008910 q^{25} + 12\!\cdots\!64 q^{27}+ \cdots - 62\!\cdots\!84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 66008406614424 x^{16} + \cdots + 53\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 14\!\cdots\!51 \nu^{17} + \cdots + 77\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14\!\cdots\!51 \nu^{17} + \cdots + 77\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!13 \nu^{17} + \cdots - 32\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\!\cdots\!61 \nu^{17} + \cdots - 36\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!89 \nu^{17} + \cdots - 20\!\cdots\!00 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 82\!\cdots\!79 \nu^{17} + \cdots - 95\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 54\!\cdots\!56 \nu^{17} + \cdots + 10\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 55\!\cdots\!33 \nu^{17} + \cdots - 36\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14\!\cdots\!44 \nu^{17} + \cdots + 33\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 55\!\cdots\!39 \nu^{17} + \cdots + 53\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 53\!\cdots\!35 \nu^{17} + \cdots - 18\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 24\!\cdots\!91 \nu^{17} + \cdots + 50\!\cdots\!00 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 74\!\cdots\!31 \nu^{17} + \cdots - 43\!\cdots\!00 ) / 97\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13\!\cdots\!38 \nu^{17} + \cdots + 10\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 54\!\cdots\!03 \nu^{17} + \cdots - 22\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 13\!\cdots\!67 \nu^{17} + \cdots - 36\!\cdots\!00 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 48\!\cdots\!47 \nu^{17} + \cdots + 14\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 11 \beta_{14} + 24 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 20 \beta_{8} + 282 \beta_{7} + 36 \beta_{5} - 2650 \beta_{4} - 124 \beta_{3} - 10360 \beta_{2} - 150936296 \beta _1 - 117348278426595 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1456056 \beta_{17} + 1242046 \beta_{16} + 6005492 \beta_{15} - 1051533 \beta_{14} + 15734932 \beta_{13} - 12389229 \beta_{12} - 1861341 \beta_{11} - 9031140 \beta_{10} + \cdots + 18\!\cdots\!36 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 112089641628 \beta_{17} + 313974434408 \beta_{16} - 246725085228 \beta_{15} + \cdots + 24\!\cdots\!82 ) / 256 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 57\!\cdots\!60 \beta_{17} + \cdots - 88\!\cdots\!56 ) / 1024 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 39\!\cdots\!66 \beta_{17} + \cdots - 29\!\cdots\!26 ) / 2048 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 94\!\cdots\!04 \beta_{17} + \cdots + 17\!\cdots\!25 ) / 8192 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 89\!\cdots\!38 \beta_{17} + \cdots + 19\!\cdots\!14 ) / 8192 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 72\!\cdots\!60 \beta_{17} + \cdots - 17\!\cdots\!04 ) / 32768 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 34\!\cdots\!75 \beta_{17} + \cdots - 31\!\cdots\!64 ) / 8192 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 67\!\cdots\!08 \beta_{17} + \cdots + 21\!\cdots\!98 ) / 16384 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 86\!\cdots\!03 \beta_{17} + \cdots + 54\!\cdots\!46 ) / 8192 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 24\!\cdots\!00 \beta_{17} + \cdots - 98\!\cdots\!12 ) / 32768 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 46\!\cdots\!62 \beta_{17} + \cdots - 23\!\cdots\!54 ) / 2048 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 10\!\cdots\!30 \beta_{17} + \cdots + 54\!\cdots\!57 ) / 8192 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 37\!\cdots\!80 \beta_{17} + \cdots + 15\!\cdots\!50 ) / 8192 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 76\!\cdots\!68 \beta_{17} + \cdots - 46\!\cdots\!96 ) / 32768 \) Copy content Toggle raw display

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
22670.7 + 1.21630e6i
22670.7 1.21630e6i
16622.8 + 2.05933e6i
16622.8 2.05933e6i
14907.4 + 4.14330e6i
14907.4 4.14330e6i
7663.46 + 3.49192e6i
7663.46 3.49192e6i
−1372.90 + 1.21540e6i
−1372.90 1.21540e6i
−2443.39 + 2.11221e6i
−2443.39 2.11221e6i
−13803.2 + 714588.i
−13803.2 714588.i
−20011.7 + 4.20507e6i
−20011.7 4.20507e6i
−24233.2 + 2.60731e6i
−24233.2 2.60731e6i
0 −84336.7 0 4.86521e6i 0 1.26224e8i 0 3.62590e9 0
15.2 0 −84336.7 0 4.86521e6i 0 1.26224e8i 0 3.62590e9 0
15.3 0 −60145.3 0 8.23733e6i 0 5.26995e8i 0 1.30668e8 0
15.4 0 −60145.3 0 8.23733e6i 0 5.26995e8i 0 1.30668e8 0
15.5 0 −53283.8 0 1.65732e7i 0 1.30215e8i 0 −6.47621e8 0
15.6 0 −53283.8 0 1.65732e7i 0 1.30215e8i 0 −6.47621e8 0
15.7 0 −24307.8 0 1.39677e7i 0 2.84328e8i 0 −2.89591e9 0
15.8 0 −24307.8 0 1.39677e7i 0 2.84328e8i 0 −2.89591e9 0
15.9 0 11837.6 0 4.86160e6i 0 3.96260e8i 0 −3.34666e9 0
15.10 0 11837.6 0 4.86160e6i 0 3.96260e8i 0 −3.34666e9 0
15.11 0 16119.5 0 8.44883e6i 0 6.17349e7i 0 −3.22694e9 0
15.12 0 16119.5 0 8.44883e6i 0 6.17349e7i 0 −3.22694e9 0
15.13 0 61558.6 0 2.85835e6i 0 1.54680e8i 0 3.02682e8 0
15.14 0 61558.6 0 2.85835e6i 0 1.54680e8i 0 3.02682e8 0
15.15 0 86392.9 0 1.68203e7i 0 3.64653e8i 0 3.97695e9 0
15.16 0 86392.9 0 1.68203e7i 0 3.64653e8i 0 3.97695e9 0
15.17 0 103279. 0 1.04292e7i 0 3.68005e8i 0 7.17976e9 0
15.18 0 103279. 0 1.04292e7i 0 3.68005e8i 0 7.17976e9 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.18
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 32.21.d.b 18
4.b odd 2 1 8.21.d.b 18
8.b even 2 1 8.21.d.b 18
8.d odd 2 1 inner 32.21.d.b 18
12.b even 2 1 72.21.b.b 18
24.h odd 2 1 72.21.b.b 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.21.d.b 18 4.b odd 2 1
8.21.d.b 18 8.b even 2 1
32.21.d.b 18 1.a even 1 1 trivial
32.21.d.b 18 8.d odd 2 1 inner
72.21.b.b 18 12.b even 2 1
72.21.b.b 18 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} - 57114 T_{3}^{8} - 16608937872 T_{3}^{7} + 676542462219936 T_{3}^{6} + \cdots - 68\!\cdots\!00 \) acting on \(S_{21}^{\mathrm{new}}(32, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( (T^{9} - 57114 T^{8} + \cdots - 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{9} + 13931197510 T^{8} + \cdots - 80\!\cdots\!92)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{9} - 2171962569586 T^{8} + \cdots + 75\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{9} - 180340826778 T^{8} + \cdots + 31\!\cdots\!08)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{9} + \cdots + 83\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{9} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{9} + \cdots + 87\!\cdots\!32)^{2} \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{9} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots + 61\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{9} + \cdots + 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots - 17\!\cdots\!28)^{2} \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
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