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Properties

Label 32.18.b.a
Level $32$
Weight $18$
Character orbit 32.b
Analytic conductor $58.631$
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] = [32,18,Mod(17,32)]
 
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([0, 1]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.17");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 32.b (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: \(58.6310679503\)
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: \( x^{16} - 8 x^{15} + 83403052 x^{14} - 583821224 x^{13} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{240}\cdot 3^{14}\cdot 7 \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{9} + 4 \beta_1) q^{5} + ( - \beta_{2} - 720600) q^{7} + (\beta_{3} - 2 \beta_{2} - 37665881) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{9} + 4 \beta_1) q^{5} + ( - \beta_{2} - 720600) q^{7} + (\beta_{3} - 2 \beta_{2} - 37665881) q^{9} + ( - \beta_{10} - 157 \beta_{9} + 2151 \beta_1) q^{11} + ( - \beta_{11} + 176 \beta_{9} + 6871 \beta_1) q^{13} + (\beta_{4} - 8 \beta_{3} + \cdots + 624580136) q^{15}+ \cdots + ( - 255024 \beta_{15} + \cdots - 1429358873670 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 11529600 q^{7} - 602654096 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 11529600 q^{7} - 602654096 q^{9} + 9993282176 q^{15} - 7489125600 q^{17} - 746845345920 q^{23} - 1809682431664 q^{25} + 318979758592 q^{31} + 5633526177600 q^{33} + 18457706051456 q^{39} + 7482251536032 q^{41} + 376698804821760 q^{47} + 127691292101520 q^{49} - 22\!\cdots\!52 q^{55}+ \cdots + 95\!\cdots\!40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 83403052 x^{14} - 583821224 x^{13} + \cdots + 21\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 22\!\cdots\!17 \nu^{14} + \cdots + 17\!\cdots\!40 ) / 40\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 22\!\cdots\!17 \nu^{14} + \cdots + 51\!\cdots\!20 ) / 20\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 81\!\cdots\!73 \nu^{14} + \cdots + 28\!\cdots\!40 ) / 13\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 39\!\cdots\!81 \nu^{14} + \cdots - 29\!\cdots\!80 ) / 61\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 32\!\cdots\!83 \nu^{14} + \cdots + 31\!\cdots\!80 ) / 12\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 84\!\cdots\!25 \nu^{14} + \cdots - 20\!\cdots\!80 ) / 61\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23\!\cdots\!64 \nu^{14} + \cdots + 38\!\cdots\!60 ) / 16\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 50\!\cdots\!82 \nu^{15} + \cdots + 65\!\cdots\!60 ) / 24\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 32\!\cdots\!14 \nu^{15} + \cdots - 15\!\cdots\!80 ) / 62\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 34\!\cdots\!42 \nu^{15} + \cdots - 82\!\cdots\!40 ) / 26\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 25\!\cdots\!86 \nu^{15} + \cdots + 74\!\cdots\!00 ) / 47\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 51\!\cdots\!58 \nu^{15} + \cdots + 29\!\cdots\!80 ) / 84\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 12\!\cdots\!10 \nu^{15} + \cdots + 17\!\cdots\!80 ) / 18\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13\!\cdots\!46 \nu^{15} + \cdots - 25\!\cdots\!20 ) / 18\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 2\beta_{2} + 4\beta _1 - 166806040 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 8 \beta_{15} - 3 \beta_{14} - \beta_{12} - 134 \beta_{11} + 274 \beta_{10} + 64286 \beta_{9} + \cdots - 1000836256 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 32 \beta_{15} - 12 \beta_{14} - 4 \beta_{12} - 536 \beta_{11} + 1096 \beta_{10} + \cdots + 24\!\cdots\!92 ) / 128 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2566225536 \beta_{15} + 836634771 \beta_{14} + 246342348 \beta_{13} + 217858872 \beta_{12} + \cdots + 24\!\cdots\!76 ) / 512 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 30794707712 \beta_{15} + 10039617732 \beta_{14} + 2956108176 \beta_{13} + 2614306624 \beta_{12} + \cdots - 87\!\cdots\!28 ) / 2048 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 13\!\cdots\!96 \beta_{15} + \cdots - 12\!\cdots\!60 ) / 8192 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 26\!\cdots\!60 \beta_{15} + \cdots + 44\!\cdots\!32 ) / 4096 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 40\!\cdots\!04 \beta_{15} + \cdots + 40\!\cdots\!80 ) / 8192 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 20\!\cdots\!96 \beta_{15} + \cdots - 24\!\cdots\!40 ) / 8192 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 12\!\cdots\!00 \beta_{15} + \cdots - 13\!\cdots\!60 ) / 8192 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 72\!\cdots\!84 \beta_{15} + \cdots + 68\!\cdots\!72 ) / 8192 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 35\!\cdots\!60 \beta_{15} + \cdots + 44\!\cdots\!96 ) / 8192 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 61\!\cdots\!88 \beta_{15} + \cdots - 49\!\cdots\!48 ) / 2048 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 10\!\cdots\!64 \beta_{15} + \cdots - 14\!\cdots\!88 ) / 8192 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.500000 + 5420.67i
0.500000 + 4000.58i
0.500000 + 3466.80i
0.500000 + 3446.66i
0.500000 + 3370.46i
0.500000 + 1208.53i
0.500000 + 1062.13i
0.500000 + 409.736i
0.500000 − 409.736i
0.500000 − 1062.13i
0.500000 − 1208.53i
0.500000 − 3370.46i
0.500000 − 3446.66i
0.500000 − 3466.80i
0.500000 − 4000.58i
0.500000 − 5420.67i
0 − 21682.7i 0 465248.i 0 −2.24795e7 0 −3.40998e8 0
17.2 0 − 16002.3i 0 − 1.27253e6i 0 −5.50569e6 0 −1.26934e8 0
17.3 0 − 13867.2i 0 − 665059.i 0 7.57536e6 0 −6.31593e7 0
17.4 0 − 13786.7i 0 96356.3i 0 1.47728e7 0 −6.09318e7 0
17.5 0 − 13481.8i 0 1.59197e6i 0 1.66055e7 0 −5.26193e7 0
17.6 0 − 4834.14i 0 524871.i 0 −1.57495e7 0 1.05771e8 0
17.7 0 − 4248.51i 0 − 663971.i 0 1.66742e7 0 1.11090e8 0
17.8 0 − 1638.94i 0 1.21254e6i 0 −1.76580e7 0 1.26454e8 0
17.9 0 1638.94i 0 − 1.21254e6i 0 −1.76580e7 0 1.26454e8 0
17.10 0 4248.51i 0 663971.i 0 1.66742e7 0 1.11090e8 0
17.11 0 4834.14i 0 − 524871.i 0 −1.57495e7 0 1.05771e8 0
17.12 0 13481.8i 0 − 1.59197e6i 0 1.66055e7 0 −5.26193e7 0
17.13 0 13786.7i 0 − 96356.3i 0 1.47728e7 0 −6.09318e7 0
17.14 0 13867.2i 0 665059.i 0 7.57536e6 0 −6.31593e7 0
17.15 0 16002.3i 0 1.27253e6i 0 −5.50569e6 0 −1.26934e8 0
17.16 0 21682.7i 0 − 465248.i 0 −2.24795e7 0 −3.40998e8 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.18.b.a 16
4.b odd 2 1 8.18.b.a ✓ 16
8.b even 2 1 inner 32.18.b.a 16
8.d odd 2 1 8.18.b.a ✓ 16
12.b even 2 1 72.18.d.b 16
24.f even 2 1 72.18.d.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.18.b.a ✓ 16 4.b odd 2 1
8.18.b.a ✓ 16 8.d odd 2 1
32.18.b.a 16 1.a even 1 1 trivial
32.18.b.a 16 8.b even 2 1 inner
72.18.d.b 16 12.b even 2 1
72.18.d.b 16 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{18}^{\mathrm{new}}(32, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 90\!\cdots\!84 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 10\!\cdots\!76)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots - 41\!\cdots\!36)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 53\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 94\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 17\!\cdots\!04)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 95\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 21\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 18\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 13\!\cdots\!36)^{2} \) Copy content Toggle raw display
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