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Properties

Label 16.4.e.a
Level $16$
Weight $4$
Character orbit 16.e
Analytic conductor $0.944$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [16,4,Mod(5,16)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("16.5"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(16, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.944030560092\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - x^{8} + 6x^{7} + 14x^{6} - 80x^{5} + 56x^{4} + 96x^{3} - 64x^{2} - 512x + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - \beta_{5} q^{3} + (\beta_{8} + \beta_{6} + \beta_{5} + \cdots + 1) q^{4} + (\beta_{9} - \beta_{8} - \beta_{7} + \cdots - 1) q^{5} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \cdots - 3) q^{6}+ \cdots + (52 \beta_{9} - 20 \beta_{8} + \cdots + 544) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} - 2 q^{3} + 8 q^{4} - 2 q^{5} - 32 q^{6} - 44 q^{8} - 68 q^{10} + 18 q^{11} + 100 q^{12} - 2 q^{13} + 188 q^{14} - 124 q^{15} + 280 q^{16} - 4 q^{17} + 174 q^{18} - 26 q^{19} - 196 q^{20}+ \cdots + 4958 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - x^{8} + 6x^{7} + 14x^{6} - 80x^{5} + 56x^{4} + 96x^{3} - 64x^{2} - 512x + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} - 14\nu^{8} + 7\nu^{7} + 82\nu^{6} - 170\nu^{5} - 120\nu^{4} + 536\nu^{3} + 384\nu^{2} - 2752\nu + 3072 ) / 1280 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3 \nu^{9} - 2 \nu^{8} + 101 \nu^{7} - 114 \nu^{6} + 210 \nu^{5} - 120 \nu^{4} + 8 \nu^{3} + \cdots - 7424 ) / 1280 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7 \nu^{9} - 18 \nu^{8} + 49 \nu^{7} + 14 \nu^{6} - 230 \nu^{5} + 120 \nu^{4} + 872 \nu^{3} + \cdots + 7424 ) / 1280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17 \nu^{9} + 38 \nu^{8} - 39 \nu^{7} + 86 \nu^{6} + 90 \nu^{5} + 520 \nu^{4} - 792 \nu^{3} + \cdots - 4864 ) / 1280 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17 \nu^{9} - 38 \nu^{8} + 39 \nu^{7} + 74 \nu^{6} - 90 \nu^{5} - 680 \nu^{4} + 1432 \nu^{3} + \cdots + 2304 ) / 1280 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5 \nu^{9} + 6 \nu^{8} + 29 \nu^{7} - 58 \nu^{6} + 18 \nu^{5} + 184 \nu^{4} + 136 \nu^{3} + \cdots - 1280 ) / 256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 37 \nu^{9} + 158 \nu^{8} - 179 \nu^{7} + 46 \nu^{6} - 350 \nu^{5} + 1800 \nu^{4} - 4472 \nu^{3} + \cdots + 5376 ) / 1280 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 49 \nu^{9} + 46 \nu^{8} + 297 \nu^{7} - 818 \nu^{6} + 10 \nu^{5} + 2040 \nu^{4} + 616 \nu^{3} + \cdots - 7168 ) / 1280 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - 2\beta_{7} + 2\beta_{6} + 2\beta_{5} - \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 6\beta_{6} + 2\beta_{5} - 4\beta_{3} - 6\beta_{2} + \beta _1 - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{9} - 2\beta_{8} - 8\beta_{6} + 4\beta_{5} + 14\beta_{4} - 2\beta_{3} - 11\beta_{2} - 34 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{9} - 4\beta_{8} + 6\beta_{7} - 6\beta_{6} + 6\beta_{5} + 4\beta_{4} - 18\beta_{2} - 5\beta _1 + 73 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -15\beta_{9} - 2\beta_{8} + 20\beta_{7} - 28\beta_{6} + 14\beta_{4} + 14\beta_{3} + 27\beta_{2} + 24\beta _1 + 34 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 20 \beta_{9} + 4 \beta_{8} - 46 \beta_{7} + 30 \beta_{6} + 50 \beta_{5} + 12 \beta_{4} + 64 \beta_{3} + \cdots + 207 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 7 \beta_{9} + 58 \beta_{8} + 36 \beta_{7} + 292 \beta_{6} + 56 \beta_{5} - 54 \beta_{4} - 22 \beta_{3} + \cdots + 326 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4 \beta_{9} + 28 \beta_{8} - 98 \beta_{7} + 210 \beta_{6} + 94 \beta_{5} + 372 \beta_{4} - 674 \beta_{2} + \cdots - 1111 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(5\) \(15\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
1.28199 − 1.53509i
1.97476 − 0.316760i
−1.62580 − 1.16481i
0.932438 + 1.76934i
−1.56339 + 1.24732i
1.28199 + 1.53509i
1.97476 + 0.316760i
−1.62580 + 1.16481i
0.932438 − 1.76934i
−1.56339 − 1.24732i
−2.81708 − 0.253099i 5.49618 − 5.49618i 7.87188 + 1.42600i −4.66372 − 4.66372i −16.8743 + 14.0921i 24.8965i −21.8148 − 6.00953i − 33.4160i 11.9577 + 14.3185i
5.2 −2.29152 + 1.65800i −5.96513 + 5.96513i 2.50210 − 7.59865i 8.67959 + 8.67959i 3.77903 − 23.5594i 1.63924i 6.86495 + 21.5609i − 44.1656i −34.2802 − 5.49869i
5.3 0.460984 − 2.79061i 0.756776 − 0.756776i −7.57499 − 2.57285i 8.22587 + 8.22587i −1.76300 − 2.46073i 2.67171i −10.6718 + 19.9528i 25.8546i 26.7472 − 19.1632i
5.4 0.836901 + 2.70178i 1.98356 − 1.98356i −6.59919 + 4.52224i −0.596848 − 0.596848i 7.01918 + 3.69910i − 29.0828i −17.7410 − 14.0449i 19.1310i 1.11305 − 2.11205i
5.5 2.81071 − 0.316066i −3.27139 + 3.27139i 7.80020 − 1.77674i −12.6449 − 12.6449i −8.16095 + 10.2289i 13.8754i 21.3626 − 7.45928i 5.59607i −39.5378 − 31.5445i
13.1 −2.81708 + 0.253099i 5.49618 + 5.49618i 7.87188 − 1.42600i −4.66372 + 4.66372i −16.8743 − 14.0921i − 24.8965i −21.8148 + 6.00953i 33.4160i 11.9577 − 14.3185i
13.2 −2.29152 − 1.65800i −5.96513 − 5.96513i 2.50210 + 7.59865i 8.67959 − 8.67959i 3.77903 + 23.5594i − 1.63924i 6.86495 − 21.5609i 44.1656i −34.2802 + 5.49869i
13.3 0.460984 + 2.79061i 0.756776 + 0.756776i −7.57499 + 2.57285i 8.22587 − 8.22587i −1.76300 + 2.46073i − 2.67171i −10.6718 − 19.9528i − 25.8546i 26.7472 + 19.1632i
13.4 0.836901 − 2.70178i 1.98356 + 1.98356i −6.59919 − 4.52224i −0.596848 + 0.596848i 7.01918 − 3.69910i 29.0828i −17.7410 + 14.0449i − 19.1310i 1.11305 + 2.11205i
13.5 2.81071 + 0.316066i −3.27139 − 3.27139i 7.80020 + 1.77674i −12.6449 + 12.6449i −8.16095 − 10.2289i − 13.8754i 21.3626 + 7.45928i − 5.59607i −39.5378 + 31.5445i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 5.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.4.e.a ✓ 10
3.b odd 2 1 144.4.k.a 10
4.b odd 2 1 64.4.e.a 10
8.b even 2 1 128.4.e.b 10
8.d odd 2 1 128.4.e.a 10
12.b even 2 1 576.4.k.a 10
16.e even 4 1 inner 16.4.e.a ✓ 10
16.e even 4 1 128.4.e.b 10
16.f odd 4 1 64.4.e.a 10
16.f odd 4 1 128.4.e.a 10
32.g even 8 2 1024.4.a.n 10
32.g even 8 2 1024.4.b.j 10
32.h odd 8 2 1024.4.a.m 10
32.h odd 8 2 1024.4.b.k 10
48.i odd 4 1 144.4.k.a 10
48.k even 4 1 576.4.k.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.4.e.a ✓ 10 1.a even 1 1 trivial
16.4.e.a ✓ 10 16.e even 4 1 inner
64.4.e.a 10 4.b odd 2 1
64.4.e.a 10 16.f odd 4 1
128.4.e.a 10 8.d odd 2 1
128.4.e.a 10 16.f odd 4 1
128.4.e.b 10 8.b even 2 1
128.4.e.b 10 16.e even 4 1
144.4.k.a 10 3.b odd 2 1
144.4.k.a 10 48.i odd 4 1
576.4.k.a 10 12.b even 2 1
576.4.k.a 10 48.k even 4 1
1024.4.a.m 10 32.h odd 8 2
1024.4.a.n 10 32.g even 8 2
1024.4.b.j 10 32.g even 8 2
1024.4.b.k 10 32.h odd 8 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 2 T^{9} + \cdots + 32768 \) Copy content Toggle raw display
$3$ \( T^{10} + 2 T^{9} + \cdots + 829472 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 202085408 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 1936000000 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 3810412010528 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 11\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( (T^{5} + 2 T^{4} + \cdots - 556317664)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 19\!\cdots\!12 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 71\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( (T^{5} - 184 T^{4} + \cdots + 678952960)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 69\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 54\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( (T^{5} + 472 T^{4} + \cdots + 154359955456)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 63\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 18\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 15\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 42\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 10448447471616)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 16\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 65755091474464)^{2} \) Copy content Toggle raw display
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