Properties

Label 111.2.a.b
Level $111$
Weight $2$
Character orbit 111.a
Self dual yes
Analytic conductor $0.886$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [111,2,Mod(1,111)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(111, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("111.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 111 = 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 111.a (trivial)

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: yes
Analytic conductor: \(0.886339462436\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.6224.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - 2x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{3} - \beta_{2} - 1) q^{5} - \beta_1 q^{6} + ( - 2 \beta_{3} + 2) q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{3} - \beta_{2} - 1) q^{5} - \beta_1 q^{6} + ( - 2 \beta_{3} + 2) q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{8} + q^{9} + (2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{10} + (2 \beta_{2} + 2 \beta_1) q^{11} + (\beta_{2} + \beta_1 + 1) q^{12} + (2 \beta_{3} - 2 \beta_{2}) q^{13} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{14} + (\beta_{3} - \beta_{2} - 1) q^{15} + (2 \beta_1 - 1) q^{16} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{17} - \beta_1 q^{18} + (2 \beta_{2} + 2) q^{19} + (\beta_{3} - \beta_{2} - 3) q^{20} + ( - 2 \beta_{3} + 2) q^{21} + ( - 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{22} + ( - 3 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{23} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{24} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{25} + (4 \beta_{3} - 2 \beta_{2} - 4) q^{26} + q^{27} + 2 \beta_{2} q^{28} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{29} + (2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{30} + 2 \beta_{3} q^{31} + (2 \beta_{3} + \beta_1 - 4) q^{32} + (2 \beta_{2} + 2 \beta_1) q^{33} + ( - 3 \beta_{2} - 3 \beta_1 - 4) q^{34} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{35} + (\beta_{2} + \beta_1 + 1) q^{36} - q^{37} + ( - 2 \beta_{3} - 4 \beta_1 + 4) q^{38} + (2 \beta_{3} - 2 \beta_{2}) q^{39} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 2) q^{40} + ( - 2 \beta_{3} + 2 \beta_1 + 4) q^{41} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{42} + (2 \beta_{3} - 4 \beta_1) q^{43} + (2 \beta_{2} + 6 \beta_1 + 8) q^{44} + (\beta_{3} - \beta_{2} - 1) q^{45} + ( - 2 \beta_{3} + 5 \beta_{2} + 7 \beta_1 + 4) q^{46} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{47} + (2 \beta_1 - 1) q^{48} + ( - 4 \beta_{2} - 4 \beta_1 + 5) q^{49} + ( - 4 \beta_{3} + 4 \beta_{2} + \beta_1 + 10) q^{50} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{51} + (2 \beta_{3} + 2 \beta_1 - 4) q^{52} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{53} - \beta_1 q^{54} - 4 q^{55} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{56} + (2 \beta_{2} + 2) q^{57} + (\beta_{2} + \beta_1 + 8) q^{58} + (5 \beta_{3} - 3 \beta_{2} - 4 \beta_1 - 5) q^{59} + (\beta_{3} - \beta_{2} - 3) q^{60} + (4 \beta_1 - 2) q^{61} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{62} + ( - 2 \beta_{3} + 2) q^{63} + (2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 1) q^{64} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 10) q^{65} + ( - 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{66} + (2 \beta_{3} - 2) q^{67} + (\beta_{3} + \beta_{2} + 6 \beta_1 + 5) q^{68} + ( - 3 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{69} + (2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 10) q^{70} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{71} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{72} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{73} + \beta_1 q^{74} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{75} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 8) q^{76} + (4 \beta_{3} + 4 \beta_{2} - 4) q^{77} + (4 \beta_{3} - 2 \beta_{2} - 4) q^{78} + (2 \beta_{2} + 4 \beta_1 - 2) q^{79} + ( - 5 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 5) q^{80} + q^{81} + ( - 2 \beta_{3} - 4 \beta_1 - 6) q^{82} + (2 \beta_{3} - 2 \beta_{2} - 6) q^{83} + 2 \beta_{2} q^{84} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{85} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 12) q^{86} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{87} + (2 \beta_{3} - 2 \beta_{2} - 8 \beta_1 - 10) q^{88} + ( - 5 \beta_{3} + \beta_{2} + 9) q^{89} + (2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{90} + ( - 4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 4) q^{91} + ( - \beta_{3} - 3 \beta_{2} - 10 \beta_1 - 9) q^{92} + 2 \beta_{3} q^{93} + ( - 2 \beta_{3} + 4 \beta_{2} + 6 \beta_1 + 6) q^{94} + (6 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 10) q^{95} + (2 \beta_{3} + \beta_1 - 4) q^{96} + (6 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 4) q^{97} + (4 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 4) q^{98} + (2 \beta_{2} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{4} - 2 q^{5} + 4 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{4} - 2 q^{5} + 4 q^{7} - 6 q^{8} + 4 q^{9} - 4 q^{10} + 4 q^{12} + 4 q^{13} - 4 q^{14} - 2 q^{15} - 4 q^{16} - 2 q^{17} + 8 q^{19} - 10 q^{20} + 4 q^{21} - 12 q^{22} - 10 q^{23} - 6 q^{24} - 8 q^{26} + 4 q^{27} - 2 q^{29} - 4 q^{30} + 4 q^{31} - 12 q^{32} - 16 q^{34} - 16 q^{35} + 4 q^{36} - 4 q^{37} + 12 q^{38} + 4 q^{39} + 4 q^{40} + 12 q^{41} - 4 q^{42} + 4 q^{43} + 32 q^{44} - 2 q^{45} + 12 q^{46} - 12 q^{47} - 4 q^{48} + 20 q^{49} + 32 q^{50} - 2 q^{51} - 12 q^{52} + 8 q^{53} - 16 q^{55} + 20 q^{56} + 8 q^{57} + 32 q^{58} - 10 q^{59} - 10 q^{60} - 8 q^{61} + 4 q^{62} + 4 q^{63} + 36 q^{65} - 12 q^{66} - 4 q^{67} + 22 q^{68} - 10 q^{69} - 36 q^{70} - 12 q^{71} - 6 q^{72} + 12 q^{73} + 28 q^{76} - 8 q^{77} - 8 q^{78} - 8 q^{79} + 10 q^{80} + 4 q^{81} - 28 q^{82} - 20 q^{83} + 4 q^{85} + 52 q^{86} - 2 q^{87} - 36 q^{88} + 26 q^{89} - 4 q^{90} - 24 q^{91} - 38 q^{92} + 4 q^{93} + 20 q^{94} - 28 q^{95} - 12 q^{96} - 4 q^{97} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.44579
0.796815
−1.37033
−1.87228
−2.44579 1.00000 3.98187 −0.670719 −2.44579 0.269264 −4.84724 1.00000 1.64044
1.2 −0.796815 1.00000 −1.36509 0.845635 −0.796815 4.63253 2.68135 1.00000 −0.673815
1.3 1.37033 1.00000 −0.122207 1.78220 1.37033 −4.06064 −2.90812 1.00000 2.44220
1.4 1.87228 1.00000 1.50542 −3.95712 1.87228 3.15885 −0.925994 1.00000 −7.40882
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(37\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 111.2.a.b 4
3.b odd 2 1 333.2.a.g 4
4.b odd 2 1 1776.2.a.u 4
5.b even 2 1 2775.2.a.x 4
7.b odd 2 1 5439.2.a.u 4
8.b even 2 1 7104.2.a.cc 4
8.d odd 2 1 7104.2.a.cf 4
12.b even 2 1 5328.2.a.bs 4
15.d odd 2 1 8325.2.a.bv 4
37.b even 2 1 4107.2.a.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.2.a.b 4 1.a even 1 1 trivial
333.2.a.g 4 3.b odd 2 1
1776.2.a.u 4 4.b odd 2 1
2775.2.a.x 4 5.b even 2 1
4107.2.a.i 4 37.b even 2 1
5328.2.a.bs 4 12.b even 2 1
5439.2.a.u 4 7.b odd 2 1
7104.2.a.cc 4 8.b even 2 1
7104.2.a.cf 4 8.d odd 2 1
8325.2.a.bv 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 6T_{2}^{2} + 2T_{2} + 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(111))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 6 T^{2} + 2 T + 5 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} - 8 T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} - 16 T^{2} + 64 T - 16 \) Copy content Toggle raw display
$11$ \( T^{4} - 32 T^{2} - 32 T + 64 \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} - 32 T^{2} + 144 T - 80 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} - 24 T^{2} - 72 T - 28 \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} - 8 T^{2} + 144 T - 224 \) Copy content Toggle raw display
$23$ \( T^{4} + 10 T^{3} - 32 T^{2} + \cdots + 652 \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} - 56 T^{2} - 40 T + 724 \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} - 16 T^{2} + 16 T + 32 \) Copy content Toggle raw display
$37$ \( (T + 1)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + 304 T - 400 \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} - 128 T^{2} + \cdots + 3424 \) Copy content Toggle raw display
$47$ \( T^{4} + 12 T^{3} + 16 T^{2} + \cdots - 128 \) Copy content Toggle raw display
$53$ \( T^{4} - 8 T^{3} - 56 T^{2} + 320 T + 464 \) Copy content Toggle raw display
$59$ \( T^{4} + 10 T^{3} - 176 T^{2} + \cdots - 7156 \) Copy content Toggle raw display
$61$ \( T^{4} + 8 T^{3} - 72 T^{2} - 480 T + 656 \) Copy content Toggle raw display
$67$ \( T^{4} + 4 T^{3} - 16 T^{2} - 64 T - 16 \) Copy content Toggle raw display
$71$ \( T^{4} + 12 T^{3} - 48 T^{2} + \cdots + 1664 \) Copy content Toggle raw display
$73$ \( T^{4} - 12 T^{3} - 8 T^{2} + 176 T - 32 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} - 56 T^{2} + \cdots - 1504 \) Copy content Toggle raw display
$83$ \( T^{4} + 20 T^{3} + 112 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$89$ \( T^{4} - 26 T^{3} + 128 T^{2} + \cdots - 5452 \) Copy content Toggle raw display
$97$ \( T^{4} + 4 T^{3} - 272 T^{2} + \cdots + 17008 \) Copy content Toggle raw display
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