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Properties

Label 111.2.e.b
Level $111$
Weight $2$
Character orbit 111.e
Analytic conductor $0.886$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [111,2,Mod(10,111)] 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("111.10"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(111, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 111 = 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 111.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.886339462436\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
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: \( x^{10} + 9x^{8} - 2x^{7} + 67x^{6} - 12x^{5} + 127x^{4} - 40x^{3} + 199x^{2} - 42x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (\beta_{6} - 1) q^{3} + ( - \beta_{8} + 2 \beta_{6} - 2) q^{4} - \beta_{9} q^{5} + ( - \beta_{3} - \beta_1) q^{6} + (\beta_{7} + \beta_{4} + \beta_1) q^{7} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + \cdots + 1) q^{8}+ \cdots + ( - \beta_{8} + \beta_{6} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 5 q^{3} - 8 q^{4} + 2 q^{5} + 6 q^{8} - 5 q^{9} + 10 q^{10} - 6 q^{11} - 8 q^{12} + 3 q^{13} + 22 q^{14} + 2 q^{15} - 18 q^{16} + 5 q^{17} - 7 q^{19} - 2 q^{20} + 3 q^{22} - 14 q^{23} - 3 q^{24}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 9x^{8} - 2x^{7} + 67x^{6} - 12x^{5} + 127x^{4} - 40x^{3} + 199x^{2} - 42x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 777 \nu^{9} + 14742 \nu^{8} - 6318 \nu^{7} + 111041 \nu^{6} - 71604 \nu^{5} + 994032 \nu^{4} + \cdots + 6161881 ) / 1536868 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1601 \nu^{9} + 777 \nu^{8} - 333 \nu^{7} + 3116 \nu^{6} - 3774 \nu^{5} + 52392 \nu^{4} + \cdots + 330792 ) / 1536868 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2833 \nu^{9} - 50092 \nu^{8} + 21468 \nu^{7} - 453567 \nu^{6} + 243304 \nu^{5} - 3377632 \nu^{4} + \cdots - 6528107 ) / 1536868 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6773 \nu^{9} - 45430 \nu^{8} + 19470 \nu^{7} - 434871 \nu^{6} + 220660 \nu^{5} - 3063280 \nu^{4} + \cdots - 6080223 ) / 1536868 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 110264 \nu^{9} + 4803 \nu^{8} - 990045 \nu^{7} + 219529 \nu^{6} - 7378340 \nu^{5} + \cdots + 4568151 ) / 4610604 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 195107 \nu^{9} - 5415 \nu^{8} - 1808988 \nu^{7} + 412939 \nu^{6} - 13585958 \nu^{5} + \cdots + 9626169 ) / 2305302 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 438725 \nu^{9} - 25014 \nu^{8} - 3941226 \nu^{7} + 544993 \nu^{6} - 29298548 \nu^{5} + \cdots - 213039 ) / 4610604 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 154917 \nu^{9} - 51426 \nu^{8} - 1405052 \nu^{7} - 124643 \nu^{6} - 10288740 \nu^{5} + \cdots - 117819 ) / 1536868 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - 4\beta_{6} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} - 6\beta_{3} - 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - 8\beta_{8} + \beta_{7} + 24\beta_{6} + \beta_{4} + \beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{9} + \beta_{8} - 9\beta_{7} - 10\beta_{6} - 9\beta_{5} + 40\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{5} - 10\beta_{4} - 12\beta_{3} - 58\beta_{2} - 12\beta _1 + 161 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -66\beta_{9} - 14\beta_{8} + 68\beta_{7} + 88\beta_{6} + 68\beta_{4} + 277\beta _1 - 88 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -52\beta_{9} + 411\beta_{8} - 82\beta_{7} - 1120\beta_{6} + 52\beta_{5} + 116\beta_{3} + 411\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 463\beta_{5} - 493\beta_{4} - 1942\beta_{3} - 146\beta_{2} - 1942\beta _1 + 741 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/111\mathbb{Z}\right)^\times\).

\(n\) \(38\) \(76\)
\(\chi(n)\) \(1\) \(-1 + \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−1.35095 − 2.33992i
−0.728588 − 1.26195i
0.108743 + 0.188348i
0.676981 + 1.17257i
1.29382 + 2.24096i
−1.35095 + 2.33992i
−0.728588 + 1.26195i
0.108743 − 0.188348i
0.676981 − 1.17257i
1.29382 − 2.24096i
−1.35095 + 2.33992i −0.500000 − 0.866025i −2.65014 − 4.59018i −0.580820 − 1.00601i 2.70190 −2.02675 − 3.51043i 8.91704 −0.500000 + 0.866025i 3.13864
10.2 −0.728588 + 1.26195i −0.500000 − 0.866025i −0.0616818 − 0.106836i 0.178383 + 0.308968i 1.45718 2.41750 + 4.18723i −2.73459 −0.500000 + 0.866025i −0.519871
10.3 0.108743 − 0.188348i −0.500000 − 0.866025i 0.976350 + 1.69109i 1.88667 + 3.26782i −0.217485 −1.92524 − 3.33462i 0.859655 −0.500000 + 0.866025i 0.820649
10.4 0.676981 − 1.17257i −0.500000 − 0.866025i 0.0833940 + 0.144443i −1.64818 − 2.85474i −1.35396 0.00433002 + 0.00749982i 2.93375 −0.500000 + 0.866025i −4.46316
10.5 1.29382 − 2.24096i −0.500000 − 0.866025i −2.34792 − 4.06672i 1.16395 + 2.01602i −2.58763 1.53016 + 2.65032i −6.97585 −0.500000 + 0.866025i 6.02374
100.1 −1.35095 − 2.33992i −0.500000 + 0.866025i −2.65014 + 4.59018i −0.580820 + 1.00601i 2.70190 −2.02675 + 3.51043i 8.91704 −0.500000 − 0.866025i 3.13864
100.2 −0.728588 − 1.26195i −0.500000 + 0.866025i −0.0616818 + 0.106836i 0.178383 − 0.308968i 1.45718 2.41750 − 4.18723i −2.73459 −0.500000 − 0.866025i −0.519871
100.3 0.108743 + 0.188348i −0.500000 + 0.866025i 0.976350 − 1.69109i 1.88667 − 3.26782i −0.217485 −1.92524 + 3.33462i 0.859655 −0.500000 − 0.866025i 0.820649
100.4 0.676981 + 1.17257i −0.500000 + 0.866025i 0.0833940 − 0.144443i −1.64818 + 2.85474i −1.35396 0.00433002 − 0.00749982i 2.93375 −0.500000 − 0.866025i −4.46316
100.5 1.29382 + 2.24096i −0.500000 + 0.866025i −2.34792 + 4.06672i 1.16395 − 2.01602i −2.58763 1.53016 − 2.65032i −6.97585 −0.500000 − 0.866025i 6.02374
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 111.2.e.b ✓ 10
3.b odd 2 1 333.2.f.c 10
4.b odd 2 1 1776.2.q.q 10
37.c even 3 1 inner 111.2.e.b ✓ 10
37.c even 3 1 4107.2.a.j 5
37.e even 6 1 4107.2.a.k 5
111.i odd 6 1 333.2.f.c 10
148.i odd 6 1 1776.2.q.q 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.2.e.b ✓ 10 1.a even 1 1 trivial
111.2.e.b ✓ 10 37.c even 3 1 inner
333.2.f.c 10 3.b odd 2 1
333.2.f.c 10 111.i odd 6 1
1776.2.q.q 10 4.b odd 2 1
1776.2.q.q 10 148.i odd 6 1
4107.2.a.j 5 37.c even 3 1
4107.2.a.k 5 37.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 9T_{2}^{8} - 2T_{2}^{7} + 67T_{2}^{6} - 12T_{2}^{5} + 127T_{2}^{4} - 40T_{2}^{3} + 199T_{2}^{2} - 42T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(111, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 9 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{9} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{10} + 32 T^{8} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( (T^{5} + 3 T^{4} - 17 T^{3} + \cdots - 48)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} - 3 T^{9} + \cdots + 484 \) Copy content Toggle raw display
$17$ \( T^{10} - 5 T^{9} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( T^{10} + 7 T^{9} + \cdots + 65536 \) Copy content Toggle raw display
$23$ \( (T^{5} + 7 T^{4} + \cdots + 192)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + 5 T^{4} - 53 T^{3} + \cdots - 48)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 6 T^{4} + \cdots + 776)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} - 21 T^{9} + \cdots + 69343957 \) Copy content Toggle raw display
$41$ \( T^{10} + 18 T^{9} + \cdots + 17424 \) Copy content Toggle raw display
$43$ \( (T^{5} - 3 T^{4} + \cdots + 2794)^{2} \) Copy content Toggle raw display
$47$ \( (T^{5} + 6 T^{4} + \cdots - 264)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 20 T^{9} + \cdots + 20736 \) Copy content Toggle raw display
$59$ \( T^{10} - 7 T^{9} + \cdots + 9216 \) Copy content Toggle raw display
$61$ \( T^{10} + 3 T^{9} + \cdots + 28047616 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 2931789316 \) Copy content Toggle raw display
$71$ \( T^{10} + 6 T^{9} + \cdots + 69755904 \) Copy content Toggle raw display
$73$ \( (T^{5} - 20 T^{4} + \cdots + 2152)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 2472675076 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 603979776 \) Copy content Toggle raw display
$89$ \( T^{10} - 31 T^{9} + \cdots + 82944 \) Copy content Toggle raw display
$97$ \( (T^{5} + 8 T^{4} + \cdots - 107)^{2} \) Copy content Toggle raw display
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