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Properties

Label 111.2.o.a
Level $111$
Weight $2$
Character orbit 111.o
Analytic conductor $0.886$
Analytic rank $0$
Dimension $12$
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] = [111,2,Mod(4,111)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(111, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("111.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 111 = 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 111.o (of order \(18\), degree \(6\), 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: \(0.886339462436\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: 12.0.15342238784889.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 12x^{10} + 48x^{8} + 77x^{6} + 48x^{4} + 12x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{9} q^{3} + (\beta_{9} + \beta_{7} + \beta_{5} + \cdots - 1) q^{4}+ \cdots + (\beta_{10} + \beta_{8}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + \beta_{9} q^{3} + (\beta_{9} + \beta_{7} + \beta_{5} + \cdots - 1) q^{4}+ \cdots + ( - 2 \beta_{11} - \beta_{9} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 3 q^{4} - 3 q^{5} - 6 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{2} - 3 q^{4} - 3 q^{5} - 6 q^{7} - 9 q^{8} - 6 q^{11} - 6 q^{12} - 6 q^{13} + 27 q^{14} - 3 q^{15} + 3 q^{16} - 6 q^{17} - 3 q^{18} - 12 q^{19} + 9 q^{20} + 6 q^{21} - 21 q^{22} - 6 q^{24} - 15 q^{25} + 6 q^{27} + 39 q^{28} + 27 q^{29} + 12 q^{30} - 33 q^{32} - 12 q^{33} + 15 q^{34} + 36 q^{35} + 15 q^{37} - 48 q^{38} - 6 q^{39} + 15 q^{40} + 12 q^{41} + 18 q^{42} - 21 q^{44} - 15 q^{46} + 9 q^{47} - 18 q^{49} - 18 q^{50} + 9 q^{51} + 6 q^{52} - 12 q^{53} - 3 q^{54} - 39 q^{55} - 18 q^{56} - 3 q^{57} - 15 q^{58} - 15 q^{61} + 39 q^{62} + 12 q^{63} - 9 q^{64} - 24 q^{65} + 18 q^{66} + 6 q^{67} - 12 q^{69} - 15 q^{70} - 3 q^{71} - 6 q^{72} + 90 q^{73} + 72 q^{74} - 24 q^{75} - 6 q^{76} + 27 q^{77} + 9 q^{78} + 6 q^{79} + 27 q^{82} - 3 q^{83} + 6 q^{84} - 27 q^{85} - 12 q^{86} + 9 q^{87} + 63 q^{88} + 39 q^{89} - 12 q^{90} - 27 q^{91} + 6 q^{92} - 24 q^{93} + 30 q^{94} - 36 q^{95} - 33 q^{96} - 18 q^{97} - 42 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 12x^{10} + 48x^{8} + 77x^{6} + 48x^{4} + 12x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 12\nu^{8} + 48\nu^{6} + 76\nu^{4} + 42\nu^{2} + \nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 2 \nu^{10} + 12 \nu^{9} + 23 \nu^{8} + 48 \nu^{7} + 85 \nu^{6} + 77 \nu^{5} + 116 \nu^{4} + \cdots + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} + 2 \nu^{10} - 12 \nu^{9} + 23 \nu^{8} - 48 \nu^{7} + 85 \nu^{6} - 77 \nu^{5} + 116 \nu^{4} + \cdots + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{11} + 35\nu^{9} + 132\nu^{7} + 184\nu^{5} + 76\nu^{3} + \nu^{2} + 8\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{11} - 35\nu^{9} - 132\nu^{7} - 184\nu^{5} - 76\nu^{3} + \nu^{2} - 8\nu + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} + 3 \nu^{10} + 12 \nu^{9} + 35 \nu^{8} + 48 \nu^{7} + 132 \nu^{6} + 77 \nu^{5} + 184 \nu^{4} + \cdots + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{11} + 3 \nu^{10} - 12 \nu^{9} + 35 \nu^{8} - 48 \nu^{7} + 132 \nu^{6} - 77 \nu^{5} + 184 \nu^{4} + \cdots + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5 \nu^{11} + \nu^{10} + 58 \nu^{9} + 12 \nu^{8} + 217 \nu^{7} + 48 \nu^{6} + 300 \nu^{5} + 77 \nu^{4} + \cdots + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5 \nu^{11} + \nu^{10} - 58 \nu^{9} + 12 \nu^{8} - 217 \nu^{7} + 48 \nu^{6} - 300 \nu^{5} + 77 \nu^{4} + \cdots + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6\nu^{11} + 71\nu^{9} + 276\nu^{7} + 414\nu^{5} + 212\nu^{3} + 30\nu + 1 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{7} + \beta_{4} - \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{9} - 5\beta_{6} - 5\beta_{5} - 2\beta_{2} + \beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{11} + 5\beta_{8} - 5\beta_{7} - \beta_{6} + \beta_{5} - 8\beta_{4} + 8\beta_{3} + 13\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 8 \beta_{10} - 8 \beta_{9} - \beta_{8} - \beta_{7} + 26 \beta_{6} + 26 \beta_{5} + \beta_{4} + \cdots - 39 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 18 \beta_{11} - \beta_{10} + \beta_{9} - 26 \beta_{8} + 26 \beta_{7} + 11 \beta_{6} - 11 \beta_{5} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 52 \beta_{10} + 52 \beta_{9} + 11 \beta_{8} + 11 \beta_{7} - 142 \beta_{6} - 142 \beta_{5} - 12 \beta_{4} + \cdots + 206 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 122 \beta_{11} + 12 \beta_{10} - 12 \beta_{9} + 142 \beta_{8} - 142 \beta_{7} - 84 \beta_{6} + \cdots - 23 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 316 \beta_{10} - 316 \beta_{9} - 84 \beta_{8} - 84 \beta_{7} + 794 \beta_{6} + 794 \beta_{5} + \cdots - 1130 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 754 \beta_{11} - 96 \beta_{10} + 96 \beta_{9} - 794 \beta_{8} + 794 \beta_{7} + 557 \beta_{6} + \cdots + 180 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(38\) \(76\)
\(\chi(n)\) \(1\) \(-\beta_{7} - \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.545983i
− 1.83156i
1.39889i
− 0.714852i
− 0.545983i
1.83156i
− 1.39889i
0.714852i
0.418704i
− 2.38832i
− 0.418704i
2.38832i
−0.537688 − 0.0948090i −0.173648 − 0.984808i −1.59927 − 0.582085i −1.80373 − 2.14961i 0.545983i −3.69621 + 3.10149i 1.75039 + 1.01059i −0.939693 + 0.342020i 0.766044 + 1.32683i
4.2 1.80373 + 0.318047i −0.173648 − 0.984808i 1.27291 + 0.463303i 0.537688 + 0.640792i − 1.83156i −0.541619 + 0.454472i −1.02371 − 0.591038i −0.939693 + 0.342020i 0.766044 + 1.32683i
25.1 −0.899190 − 1.07161i −0.766044 − 0.642788i 0.00748452 − 0.0424468i −0.459498 − 1.26246i 1.39889i −0.723796 + 0.263440i −2.47517 + 1.42904i 0.173648 + 0.984808i −0.939693 + 1.62760i
25.2 0.459498 + 0.547608i −0.766044 − 0.642788i 0.258560 − 1.46637i 0.899190 + 2.47051i − 0.714852i 2.71652 − 0.988733i 2.15996 − 1.24705i 0.173648 + 0.984808i −0.939693 + 1.62760i
28.1 −0.537688 + 0.0948090i −0.173648 + 0.984808i −1.59927 + 0.582085i −1.80373 + 2.14961i − 0.545983i −3.69621 − 3.10149i 1.75039 − 1.01059i −0.939693 − 0.342020i 0.766044 − 1.32683i
28.2 1.80373 − 0.318047i −0.173648 + 0.984808i 1.27291 − 0.463303i 0.537688 − 0.640792i 1.83156i −0.541619 − 0.454472i −1.02371 + 0.591038i −0.939693 − 0.342020i 0.766044 − 1.32683i
40.1 −0.899190 + 1.07161i −0.766044 + 0.642788i 0.00748452 + 0.0424468i −0.459498 + 1.26246i − 1.39889i −0.723796 − 0.263440i −2.47517 − 1.42904i 0.173648 − 0.984808i −0.939693 − 1.62760i
40.2 0.459498 − 0.547608i −0.766044 + 0.642788i 0.258560 + 1.46637i 0.899190 − 2.47051i 0.714852i 2.71652 + 0.988733i 2.15996 + 1.24705i 0.173648 − 0.984808i −0.939693 − 1.62760i
58.1 −0.143205 + 0.393453i 0.939693 − 0.342020i 1.39779 + 1.17289i −0.816854 − 0.144033i 0.418704i 0.0446807 − 0.253397i −1.38686 + 0.800707i 0.766044 − 0.642788i 0.173648 − 0.300767i
58.2 0.816854 − 2.24429i 0.939693 − 0.342020i −2.83748 − 2.38093i 0.143205 + 0.0252510i − 2.38832i −0.799581 + 4.53465i −3.52461 + 2.03493i 0.766044 − 0.642788i 0.173648 − 0.300767i
67.1 −0.143205 − 0.393453i 0.939693 + 0.342020i 1.39779 − 1.17289i −0.816854 + 0.144033i − 0.418704i 0.0446807 + 0.253397i −1.38686 − 0.800707i 0.766044 + 0.642788i 0.173648 + 0.300767i
67.2 0.816854 + 2.24429i 0.939693 + 0.342020i −2.83748 + 2.38093i 0.143205 − 0.0252510i 2.38832i −0.799581 − 4.53465i −3.52461 − 2.03493i 0.766044 + 0.642788i 0.173648 + 0.300767i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 111.2.o.a ✓ 12
3.b odd 2 1 333.2.bl.b 12
37.h even 18 1 inner 111.2.o.a ✓ 12
37.i odd 36 2 4107.2.a.p 12
111.n odd 18 1 333.2.bl.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.2.o.a ✓ 12 1.a even 1 1 trivial
111.2.o.a ✓ 12 37.h even 18 1 inner
333.2.bl.b 12 3.b odd 2 1
333.2.bl.b 12 111.n odd 18 1
4107.2.a.p 12 37.i odd 36 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 3 T_{2}^{11} + 6 T_{2}^{10} - 6 T_{2}^{9} - 3 T_{2}^{8} - 15 T_{2}^{7} + 29 T_{2}^{6} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(111, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 3 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 3 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{11} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( T^{12} + 6 T^{11} + \cdots + 3249 \) Copy content Toggle raw display
$13$ \( T^{12} + 6 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( T^{12} + 6 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$19$ \( T^{12} + 12 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$23$ \( T^{12} - 51 T^{10} + \cdots + 1203409 \) Copy content Toggle raw display
$29$ \( T^{12} - 27 T^{11} + \cdots + 3845521 \) Copy content Toggle raw display
$31$ \( T^{12} + 192 T^{10} + \cdots + 938961 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 2565726409 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 217061289 \) Copy content Toggle raw display
$43$ \( T^{12} + 174 T^{10} + \cdots + 10169721 \) Copy content Toggle raw display
$47$ \( T^{12} - 9 T^{11} + \cdots + 19158129 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 217238121 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 5160841921 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1081686321 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 20555817129 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 930677049 \) Copy content Toggle raw display
$73$ \( (T^{6} - 45 T^{5} + \cdots - 201153)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 253669329 \) Copy content Toggle raw display
$83$ \( T^{12} + 3 T^{11} + \cdots + 9162729 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 714347826481 \) Copy content Toggle raw display
$97$ \( T^{12} + 18 T^{11} + \cdots + 95004009 \) Copy content Toggle raw display
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