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Properties

Label 16.3.f.a
Level $16$
Weight $3$
Character orbit 16.f
Analytic conductor $0.436$
Analytic rank $0$
Dimension $6$
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] = [16,3,Mod(3,16)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(16, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("16.3");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 16.f (of order \(4\), degree \(2\), 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: \(0.435968422976\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{5} - \beta_{2} - 1) q^{3} + ( - \beta_{5} - \beta_{3} + \beta_1 - 1) q^{4} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{5}+ \cdots + ( - 2 \beta_{4} - \beta_{3} + 4 \beta_{2} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{5} - \beta_{2} - 1) q^{3} + ( - \beta_{5} - \beta_{3} + \beta_1 - 1) q^{4} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{5}+ \cdots + (13 \beta_{5} + 4 \beta_{4} + \cdots - 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} - 2 q^{3} - 8 q^{4} - 2 q^{5} - 8 q^{6} - 4 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} - 2 q^{3} - 8 q^{4} - 2 q^{5} - 8 q^{6} - 4 q^{7} + 4 q^{8} + 36 q^{10} - 18 q^{11} + 52 q^{12} - 2 q^{13} + 12 q^{14} - 40 q^{16} - 4 q^{17} - 74 q^{18} + 30 q^{19} - 84 q^{20} - 20 q^{21} - 52 q^{22} + 60 q^{23} + 48 q^{24} + 96 q^{26} + 64 q^{27} + 56 q^{28} - 18 q^{29} + 52 q^{30} + 8 q^{32} - 4 q^{33} - 76 q^{34} - 100 q^{35} - 52 q^{36} + 46 q^{37} + 40 q^{38} - 196 q^{39} + 40 q^{40} - 24 q^{42} - 114 q^{43} + 20 q^{44} + 66 q^{45} + 28 q^{46} - 24 q^{48} - 46 q^{49} + 46 q^{50} + 156 q^{51} + 100 q^{52} + 78 q^{53} + 32 q^{54} + 252 q^{55} - 168 q^{56} - 176 q^{58} + 206 q^{59} - 160 q^{60} + 30 q^{61} - 144 q^{62} + 64 q^{64} + 12 q^{65} + 196 q^{66} - 226 q^{67} + 112 q^{68} - 116 q^{69} - 16 q^{70} - 260 q^{71} + 52 q^{72} - 92 q^{74} - 238 q^{75} - 188 q^{76} - 212 q^{77} - 84 q^{78} + 232 q^{80} + 86 q^{81} + 304 q^{82} + 318 q^{83} + 232 q^{84} - 212 q^{85} + 268 q^{86} + 444 q^{87} - 8 q^{88} - 160 q^{90} + 188 q^{91} - 168 q^{92} - 32 q^{93} + 48 q^{94} - 80 q^{96} - 4 q^{97} + 10 q^{98} - 226 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 3\nu^{3} + 6\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 3\nu^{3} + 2\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 3\nu^{3} - 4\nu^{2} + 2\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{5} + 4\nu^{4} - 9\nu^{3} + 12\nu^{2} - 10\nu + 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{5} + 2\nu^{4} - 5\nu^{3} + 8\nu^{2} - 4\nu + 14 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{3} - \beta_{2} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{4} + 6\beta_{3} + \beta_{2} + \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{5} + 3\beta_{4} - 3\beta_{3} + 2\beta _1 + 4 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(-\beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
1.40680 + 0.144584i
−0.671462 + 1.24464i
0.264658 − 1.38923i
1.40680 − 0.144584i
−0.671462 − 1.24464i
0.264658 + 1.38923i
−1.55139 − 1.26222i 2.10278 − 2.10278i 0.813607 + 3.91638i −4.62721 + 4.62721i −5.91638 + 0.608056i 3.04888 3.68111 − 7.10278i 0.156674i 13.0192 − 1.33804i
3.2 −0.573183 + 1.91611i 0.146365 − 0.146365i −3.34292 − 2.19656i 3.68585 − 3.68585i 0.196558 + 0.364346i −9.66442 6.12494 − 5.14637i 8.95715i 4.94981 + 9.17513i
3.3 1.12457 − 1.65389i −3.24914 + 3.24914i −1.47068 − 3.71982i −0.0586332 + 0.0586332i 1.71982 + 9.02760i 4.61555 −7.80605 − 1.75086i − 12.1138i 0.0310355 + 0.162910i
11.1 −1.55139 + 1.26222i 2.10278 + 2.10278i 0.813607 − 3.91638i −4.62721 − 4.62721i −5.91638 − 0.608056i 3.04888 3.68111 + 7.10278i − 0.156674i 13.0192 + 1.33804i
11.2 −0.573183 − 1.91611i 0.146365 + 0.146365i −3.34292 + 2.19656i 3.68585 + 3.68585i 0.196558 − 0.364346i −9.66442 6.12494 + 5.14637i − 8.95715i 4.94981 − 9.17513i
11.3 1.12457 + 1.65389i −3.24914 − 3.24914i −1.47068 + 3.71982i −0.0586332 − 0.0586332i 1.71982 − 9.02760i 4.61555 −7.80605 + 1.75086i 12.1138i 0.0310355 − 0.162910i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.3.f.a ✓ 6
3.b odd 2 1 144.3.m.a 6
4.b odd 2 1 64.3.f.a 6
5.b even 2 1 400.3.r.c 6
5.c odd 4 1 400.3.k.c 6
5.c odd 4 1 400.3.k.d 6
8.b even 2 1 128.3.f.b 6
8.d odd 2 1 128.3.f.a 6
12.b even 2 1 576.3.m.a 6
16.e even 4 1 64.3.f.a 6
16.e even 4 1 128.3.f.a 6
16.f odd 4 1 inner 16.3.f.a ✓ 6
16.f odd 4 1 128.3.f.b 6
24.f even 2 1 1152.3.m.b 6
24.h odd 2 1 1152.3.m.a 6
32.g even 8 2 1024.3.c.j 12
32.g even 8 2 1024.3.d.k 12
32.h odd 8 2 1024.3.c.j 12
32.h odd 8 2 1024.3.d.k 12
48.i odd 4 1 576.3.m.a 6
48.i odd 4 1 1152.3.m.b 6
48.k even 4 1 144.3.m.a 6
48.k even 4 1 1152.3.m.a 6
80.j even 4 1 400.3.k.d 6
80.k odd 4 1 400.3.r.c 6
80.s even 4 1 400.3.k.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.3.f.a ✓ 6 1.a even 1 1 trivial
16.3.f.a ✓ 6 16.f odd 4 1 inner
64.3.f.a 6 4.b odd 2 1
64.3.f.a 6 16.e even 4 1
128.3.f.a 6 8.d odd 2 1
128.3.f.a 6 16.e even 4 1
128.3.f.b 6 8.b even 2 1
128.3.f.b 6 16.f odd 4 1
144.3.m.a 6 3.b odd 2 1
144.3.m.a 6 48.k even 4 1
400.3.k.c 6 5.c odd 4 1
400.3.k.c 6 80.s even 4 1
400.3.k.d 6 5.c odd 4 1
400.3.k.d 6 80.j even 4 1
400.3.r.c 6 5.b even 2 1
400.3.r.c 6 80.k odd 4 1
576.3.m.a 6 12.b even 2 1
576.3.m.a 6 48.i odd 4 1
1024.3.c.j 12 32.g even 8 2
1024.3.c.j 12 32.h odd 8 2
1024.3.d.k 12 32.g even 8 2
1024.3.d.k 12 32.h odd 8 2
1152.3.m.a 6 24.h odd 2 1
1152.3.m.a 6 48.k even 4 1
1152.3.m.b 6 24.f even 2 1
1152.3.m.b 6 48.i odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 2 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{6} + 2 T^{5} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} + \cdots + 8 \) Copy content Toggle raw display
$7$ \( (T^{3} + 2 T^{2} + \cdots + 136)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 18 T^{5} + \cdots + 587528 \) Copy content Toggle raw display
$13$ \( T^{6} + 2 T^{5} + \cdots + 1286408 \) Copy content Toggle raw display
$17$ \( (T^{3} + 2 T^{2} + \cdots - 1544)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} - 30 T^{5} + \cdots + 13448 \) Copy content Toggle raw display
$23$ \( (T^{3} - 30 T^{2} + \cdots + 968)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 18 T^{5} + \cdots + 19046792 \) Copy content Toggle raw display
$31$ \( T^{6} + 1920 T^{4} + \cdots + 16777216 \) Copy content Toggle raw display
$37$ \( T^{6} - 46 T^{5} + \cdots + 42632 \) Copy content Toggle raw display
$41$ \( T^{6} + 4992 T^{4} + \cdots + 67108864 \) Copy content Toggle raw display
$43$ \( T^{6} + 114 T^{5} + \cdots + 42632 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 6056574976 \) Copy content Toggle raw display
$53$ \( T^{6} - 78 T^{5} + \cdots + 783752 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 8410007432 \) Copy content Toggle raw display
$61$ \( T^{6} - 30 T^{5} + \cdots + 151449608 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 87233303432 \) Copy content Toggle raw display
$71$ \( (T^{3} + 130 T^{2} + \cdots - 391864)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 7310934016 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 1550483193856 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 105636303368 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 25681985536 \) Copy content Toggle raw display
$97$ \( (T^{3} + 2 T^{2} + \cdots + 519928)^{2} \) Copy content Toggle raw display
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