Properties

Label 109.2.e.a
Level $109$
Weight $2$
Character orbit 109.e
Analytic conductor $0.870$
Analytic rank $0$
Dimension $2$
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] = [109,2,Mod(46,109)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(109, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("109.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 109 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 109.e (of order \(6\), 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: \(0.870369382032\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/109\mathbb{Z}\right)^\times\).

\(n\) \(6\)
\(\chi(n)\) \(\zeta_{6}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
0.500000 0.866025i
0.500000 + 0.866025i
1.73205i 0.500000 + 0.866025i −1.00000 −1.50000 2.59808i 1.50000 0.866025i 0.500000 + 0.866025i 1.73205i 1.00000 1.73205i −4.50000 + 2.59808i
64.1 1.73205i 0.500000 0.866025i −1.00000 −1.50000 + 2.59808i 1.50000 + 0.866025i 0.500000 0.866025i 1.73205i 1.00000 + 1.73205i −4.50000 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
109.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 109.2.e.a 2
3.b odd 2 1 981.2.k.b 2
109.e even 6 1 inner 109.2.e.a 2
327.j odd 6 1 981.2.k.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
109.2.e.a 2 1.a even 1 1 trivial
109.2.e.a 2 109.e even 6 1 inner
981.2.k.b 2 3.b odd 2 1
981.2.k.b 2 327.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(109, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$13$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$17$ \( T^{2} + 48 \) Copy content Toggle raw display
$19$ \( T^{2} + 12 \) Copy content Toggle raw display
$23$ \( T^{2} + 12 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$37$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$41$ \( T^{2} + 48 \) Copy content Toggle raw display
$43$ \( (T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$53$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$59$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$79$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$83$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$89$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$97$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
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