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Properties

Label 32.16.b.a
Level $32$
Weight $16$
Character orbit 32.b
Analytic conductor $45.662$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [32,16,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, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.17");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 16 \)
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: \(45.6619216320\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 7 x^{13} + 8071283 x^{12} - 48427607 x^{11} + 24279249501785 x^{10} - 121395803589361 x^{9} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{182}\cdot 3^{6}\cdot 5^{4}\cdot 31^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{8} - 3 \beta_1) q^{5} + (\beta_{3} + 117649) q^{7} + (\beta_{2} - 4099688) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{8} - 3 \beta_1) q^{5} + (\beta_{3} + 117649) q^{7} + (\beta_{2} - 4099688) q^{9} + (\beta_{9} + 62 \beta_{8} - 567 \beta_1) q^{11} + (\beta_{10} - 2 \beta_{8} + 8717 \beta_1) q^{13} + ( - \beta_{4} + 10 \beta_{3} + \cdots - 50866810) q^{15}+ \cdots + ( - 14256 \beta_{13} + \cdots - 25766716110 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 1647088 q^{7} - 57395630 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 1647088 q^{7} - 57395630 q^{9} - 712135312 q^{15} + 728554812 q^{17} + 35548816080 q^{23} - 75899954794 q^{25} + 105758138816 q^{31} - 150458001384 q^{33} + 2251546247120 q^{39} - 53229185940 q^{41} - 12527998446432 q^{47} + 8427385380990 q^{49} + 30557833792176 q^{55} + 18277230892472 q^{57} - 36142362113776 q^{63} + 5437123965600 q^{65} + 173249927708016 q^{71} - 182057837882196 q^{73} + 294370273271392 q^{79} + 256903428263798 q^{81} - 14\!\cdots\!12 q^{87}+ \cdots - 672574291859236 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 7 x^{13} + 8071283 x^{12} - 48427607 x^{11} + 24279249501785 x^{10} - 121395803589361 x^{9} + \cdots + 29\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 16\nu^{2} - 16\nu + 18448599 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16\!\cdots\!89 \nu^{12} + \cdots + 12\!\cdots\!02 ) / 35\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 38\!\cdots\!73 \nu^{12} + \cdots + 54\!\cdots\!36 ) / 85\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 35\!\cdots\!04 \nu^{12} + \cdots + 11\!\cdots\!28 ) / 22\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 41\!\cdots\!84 \nu^{12} + \cdots + 33\!\cdots\!87 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 49\!\cdots\!07 \nu^{12} + \cdots - 14\!\cdots\!74 ) / 22\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\!\cdots\!02 \nu^{13} + \cdots - 47\!\cdots\!68 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 78\!\cdots\!34 \nu^{13} + \cdots + 65\!\cdots\!56 ) / 49\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29\!\cdots\!34 \nu^{13} + \cdots - 87\!\cdots\!56 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14\!\cdots\!54 \nu^{13} + \cdots - 38\!\cdots\!36 ) / 15\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 31\!\cdots\!78 \nu^{13} + \cdots - 98\!\cdots\!52 ) / 33\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 31\!\cdots\!02 \nu^{13} + \cdots + 97\!\cdots\!68 ) / 20\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 4\beta _1 - 18448591 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 8 \beta_{13} - 3 \beta_{12} + \beta_{11} - 109 \beta_{10} - 95 \beta_{9} - 8371 \beta_{8} + \cdots - 110691562 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 32 \beta_{13} - 12 \beta_{12} + 4 \beta_{11} - 436 \beta_{10} - 380 \beta_{9} + \cdots + 303305092022943 ) / 128 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 221055360 \beta_{13} + 111452499 \beta_{12} - 5210904 \beta_{11} + 2242758414 \beta_{10} + \cdots + 30\!\cdots\!66 ) / 512 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2652665600 \beta_{13} + 1337430468 \beta_{12} - 62531008 \beta_{11} + 26913118408 \beta_{10} + \cdots - 11\!\cdots\!13 ) / 2048 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 10\!\cdots\!20 \beta_{13} + \cdots - 16\!\cdots\!86 ) / 8192 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 10\!\cdots\!48 \beta_{13} + \cdots + 28\!\cdots\!08 ) / 2048 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 13\!\cdots\!96 \beta_{13} + \cdots + 25\!\cdots\!38 ) / 4096 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 34\!\cdots\!60 \beta_{13} + \cdots - 72\!\cdots\!27 ) / 2048 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 74\!\cdots\!48 \beta_{13} + \cdots - 15\!\cdots\!14 ) / 8192 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 11\!\cdots\!72 \beta_{13} + \cdots + 18\!\cdots\!38 ) / 2048 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 24\!\cdots\!92 \beta_{13} + \cdots + 60\!\cdots\!90 ) / 1024 \) 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 + 1634.35i
0.500000 + 1589.49i
0.500000 + 1225.16i
0.500000 + 895.864i
0.500000 + 595.041i
0.500000 + 464.740i
0.500000 + 6.81923i
0.500000 − 6.81923i
0.500000 − 464.740i
0.500000 − 595.041i
0.500000 − 895.864i
0.500000 − 1225.16i
0.500000 − 1589.49i
0.500000 − 1634.35i
0 − 6537.39i 0 116500.i 0 2.94378e6 0 −2.83885e7 0
17.2 0 − 6357.96i 0 − 267219.i 0 −120583. 0 −2.60748e7 0
17.3 0 − 4900.64i 0 174283.i 0 −3.39868e6 0 −9.66732e6 0
17.4 0 − 3583.46i 0 82632.2i 0 153730. 0 1.50775e6 0
17.5 0 − 2380.16i 0 − 9943.39i 0 −1.24365e6 0 8.68373e6 0
17.6 0 − 1858.96i 0 − 290191.i 0 −1.26027e6 0 1.08932e7 0
17.7 0 − 27.2769i 0 − 212327.i 0 3.74922e6 0 1.43482e7 0
17.8 0 27.2769i 0 212327.i 0 3.74922e6 0 1.43482e7 0
17.9 0 1858.96i 0 290191.i 0 −1.26027e6 0 1.08932e7 0
17.10 0 2380.16i 0 9943.39i 0 −1.24365e6 0 8.68373e6 0
17.11 0 3583.46i 0 − 82632.2i 0 153730. 0 1.50775e6 0
17.12 0 4900.64i 0 − 174283.i 0 −3.39868e6 0 −9.66732e6 0
17.13 0 6357.96i 0 267219.i 0 −120583. 0 −2.60748e7 0
17.14 0 6537.39i 0 − 116500.i 0 2.94378e6 0 −2.83885e7 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.14
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 32.16.b.a 14
4.b odd 2 1 8.16.b.a ✓ 14
8.b even 2 1 inner 32.16.b.a 14
8.d odd 2 1 8.16.b.a ✓ 14
12.b even 2 1 72.16.d.b 14
24.f even 2 1 72.16.d.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.16.b.a ✓ 14 4.b odd 2 1
8.16.b.a ✓ 14 8.d odd 2 1
32.16.b.a 14 1.a even 1 1 trivial
32.16.b.a 14 8.b even 2 1 inner
72.16.d.b 14 12.b even 2 1
72.16.d.b 14 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(32, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 77\!\cdots\!68 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{7} + \cdots - 10\!\cdots\!84)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots - 21\!\cdots\!52)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 17\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots + 28\!\cdots\!48)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots + 25\!\cdots\!52)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 39\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 12\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots + 69\!\cdots\!68)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 19\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 64\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 52\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 30\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots + 11\!\cdots\!36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots - 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 48\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 29\!\cdots\!92)^{2} \) Copy content Toggle raw display
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