Properties

Label 109.2.a.b
Level $109$
Weight $2$
Character orbit 109.a
Self dual yes
Analytic conductor $0.870$
Analytic rank $1$
Dimension $3$
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] = [109,2,Mod(1,109)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(109, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("109.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 109 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 109.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: \(0.870369382032\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) 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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
1.80194
−1.24698
0.445042
−2.24698 0.246980 3.04892 −3.35690 −0.554958 0.911854 −2.35690 −2.93900 7.54288
1.2 −0.554958 −1.44504 −1.69202 1.04892 0.801938 −4.85086 2.04892 −0.911854 −0.582105
1.3 0.801938 −2.80194 −1.35690 −3.69202 −2.24698 2.93900 −2.69202 4.85086 −2.96077
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(109\) \(1\)

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 109.2.a.b 3
3.b odd 2 1 981.2.a.c 3
4.b odd 2 1 1744.2.a.l 3
5.b even 2 1 2725.2.a.h 3
7.b odd 2 1 5341.2.a.d 3
8.b even 2 1 6976.2.a.z 3
8.d odd 2 1 6976.2.a.s 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
109.2.a.b 3 1.a even 1 1 trivial
981.2.a.c 3 3.b odd 2 1
1744.2.a.l 3 4.b odd 2 1
2725.2.a.h 3 5.b even 2 1
5341.2.a.d 3 7.b odd 2 1
6976.2.a.s 3 8.d odd 2 1
6976.2.a.z 3 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 2T_{2}^{2} - T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(109))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 2T^{2} - T - 1 \) Copy content Toggle raw display
$3$ \( T^{3} + 4 T^{2} + 3 T - 1 \) Copy content Toggle raw display
$5$ \( T^{3} + 6 T^{2} + 5 T - 13 \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 16 T + 13 \) Copy content Toggle raw display
$11$ \( T^{3} + 13 T^{2} + 54 T + 71 \) Copy content Toggle raw display
$13$ \( T^{3} + T^{2} - 16 T + 13 \) Copy content Toggle raw display
$17$ \( T^{3} - 3 T^{2} - 4 T + 13 \) Copy content Toggle raw display
$19$ \( T^{3} + 5 T^{2} - 8 T - 41 \) Copy content Toggle raw display
$23$ \( T^{3} - T^{2} - 58 T - 13 \) Copy content Toggle raw display
$29$ \( T^{3} + 6 T^{2} - 37 T - 181 \) Copy content Toggle raw display
$31$ \( T^{3} + 7 T^{2} - 28 T + 7 \) Copy content Toggle raw display
$37$ \( T^{3} - 7T - 7 \) Copy content Toggle raw display
$41$ \( T^{3} + 6 T^{2} - 51 T + 71 \) Copy content Toggle raw display
$43$ \( T^{3} - 9 T^{2} - 36 T + 351 \) Copy content Toggle raw display
$47$ \( T^{3} + 10 T^{2} - 25 T - 125 \) Copy content Toggle raw display
$53$ \( T^{3} - 9 T^{2} + 20 T - 13 \) Copy content Toggle raw display
$59$ \( T^{3} + 25 T^{2} + 192 T + 461 \) Copy content Toggle raw display
$61$ \( T^{3} + 10 T^{2} - 144 T - 1336 \) Copy content Toggle raw display
$67$ \( T^{3} + 11 T^{2} - 25 T - 43 \) Copy content Toggle raw display
$71$ \( T^{3} + 10 T^{2} - 11 T - 223 \) Copy content Toggle raw display
$73$ \( T^{3} - 20 T^{2} + 131 T - 281 \) Copy content Toggle raw display
$79$ \( T^{3} + 6 T^{2} - 79 T - 461 \) Copy content Toggle raw display
$83$ \( T^{3} + 13 T^{2} - 2 T - 139 \) Copy content Toggle raw display
$89$ \( T^{3} + 21 T^{2} + 84 T + 91 \) Copy content Toggle raw display
$97$ \( T^{3} + 20 T^{2} + 75 T - 125 \) Copy content Toggle raw display
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