Properties

Label 8018.2.a.c
Level $8018$
Weight $2$
Character orbit 8018.a
Self dual yes
Analytic conductor $64.024$
Analytic rank $0$
Dimension $2$
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] = [8018,2,Mod(1,8018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8018, 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("8018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8018 = 2 \cdot 19 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8018.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: \(64.0240523407\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta + 1) q^{3} + q^{4} + (2 \beta - 1) q^{5} + ( - \beta - 1) q^{6} + 2 q^{7} - q^{8} + (3 \beta - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta + 1) q^{3} + q^{4} + (2 \beta - 1) q^{5} + ( - \beta - 1) q^{6} + 2 q^{7} - q^{8} + (3 \beta - 1) q^{9} + ( - 2 \beta + 1) q^{10} + ( - \beta - 4) q^{11} + (\beta + 1) q^{12} + 5 q^{13} - 2 q^{14} + (3 \beta + 1) q^{15} + q^{16} + ( - 2 \beta + 1) q^{17} + ( - 3 \beta + 1) q^{18} + q^{19} + (2 \beta - 1) q^{20} + (2 \beta + 2) q^{21} + (\beta + 4) q^{22} + (2 \beta - 7) q^{23} + ( - \beta - 1) q^{24} - 5 q^{26} + (2 \beta - 1) q^{27} + 2 q^{28} + (3 \beta - 3) q^{29} + ( - 3 \beta - 1) q^{30} + (9 \beta - 4) q^{31} - q^{32} + ( - 6 \beta - 5) q^{33} + (2 \beta - 1) q^{34} + (4 \beta - 2) q^{35} + (3 \beta - 1) q^{36} + ( - 6 \beta + 4) q^{37} - q^{38} + (5 \beta + 5) q^{39} + ( - 2 \beta + 1) q^{40} + (4 \beta + 1) q^{41} + ( - 2 \beta - 2) q^{42} + (3 \beta - 5) q^{43} + ( - \beta - 4) q^{44} + (\beta + 7) q^{45} + ( - 2 \beta + 7) q^{46} + ( - 2 \beta + 7) q^{47} + (\beta + 1) q^{48} - 3 q^{49} + ( - 3 \beta - 1) q^{51} + 5 q^{52} + (4 \beta + 4) q^{53} + ( - 2 \beta + 1) q^{54} + ( - 9 \beta + 2) q^{55} - 2 q^{56} + (\beta + 1) q^{57} + ( - 3 \beta + 3) q^{58} + ( - 3 \beta + 9) q^{59} + (3 \beta + 1) q^{60} + 3 q^{61} + ( - 9 \beta + 4) q^{62} + (6 \beta - 2) q^{63} + q^{64} + (10 \beta - 5) q^{65} + (6 \beta + 5) q^{66} + ( - 6 \beta + 11) q^{67} + ( - 2 \beta + 1) q^{68} + ( - 3 \beta - 5) q^{69} + ( - 4 \beta + 2) q^{70} + ( - \beta + 5) q^{71} + ( - 3 \beta + 1) q^{72} + (6 \beta - 1) q^{73} + (6 \beta - 4) q^{74} + q^{76} + ( - 2 \beta - 8) q^{77} + ( - 5 \beta - 5) q^{78} + ( - 6 \beta + 8) q^{79} + (2 \beta - 1) q^{80} + ( - 6 \beta + 4) q^{81} + ( - 4 \beta - 1) q^{82} + (5 \beta - 1) q^{83} + (2 \beta + 2) q^{84} - 5 q^{85} + ( - 3 \beta + 5) q^{86} + 3 \beta q^{87} + (\beta + 4) q^{88} + ( - 2 \beta + 7) q^{89} + ( - \beta - 7) q^{90} + 10 q^{91} + (2 \beta - 7) q^{92} + (14 \beta + 5) q^{93} + (2 \beta - 7) q^{94} + (2 \beta - 1) q^{95} + ( - \beta - 1) q^{96} + 13 q^{97} + 3 q^{98} + ( - 14 \beta + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 3 q^{6} + 4 q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 3 q^{6} + 4 q^{7} - 2 q^{8} + q^{9} - 9 q^{11} + 3 q^{12} + 10 q^{13} - 4 q^{14} + 5 q^{15} + 2 q^{16} - q^{18} + 2 q^{19} + 6 q^{21} + 9 q^{22} - 12 q^{23} - 3 q^{24} - 10 q^{26} + 4 q^{28} - 3 q^{29} - 5 q^{30} + q^{31} - 2 q^{32} - 16 q^{33} + q^{36} + 2 q^{37} - 2 q^{38} + 15 q^{39} + 6 q^{41} - 6 q^{42} - 7 q^{43} - 9 q^{44} + 15 q^{45} + 12 q^{46} + 12 q^{47} + 3 q^{48} - 6 q^{49} - 5 q^{51} + 10 q^{52} + 12 q^{53} - 5 q^{55} - 4 q^{56} + 3 q^{57} + 3 q^{58} + 15 q^{59} + 5 q^{60} + 6 q^{61} - q^{62} + 2 q^{63} + 2 q^{64} + 16 q^{66} + 16 q^{67} - 13 q^{69} + 9 q^{71} - q^{72} + 4 q^{73} - 2 q^{74} + 2 q^{76} - 18 q^{77} - 15 q^{78} + 10 q^{79} + 2 q^{81} - 6 q^{82} + 3 q^{83} + 6 q^{84} - 10 q^{85} + 7 q^{86} + 3 q^{87} + 9 q^{88} + 12 q^{89} - 15 q^{90} + 20 q^{91} - 12 q^{92} + 24 q^{93} - 12 q^{94} - 3 q^{96} + 26 q^{97} + 6 q^{98} - 12 q^{99}+O(q^{100}) \) 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
−0.618034
1.61803
−1.00000 0.381966 1.00000 −2.23607 −0.381966 2.00000 −1.00000 −2.85410 2.23607
1.2 −1.00000 2.61803 1.00000 2.23607 −2.61803 2.00000 −1.00000 3.85410 −2.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(19\) \(-1\)
\(211\) \(1\)

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 8018.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8018.2.a.c 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8018))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 5 \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 9T + 19 \) Copy content Toggle raw display
$13$ \( (T - 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 5 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 12T + 31 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$31$ \( T^{2} - T - 101 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$43$ \( T^{2} + 7T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 31 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$59$ \( T^{2} - 15T + 45 \) Copy content Toggle raw display
$61$ \( (T - 3)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 16T + 19 \) Copy content Toggle raw display
$71$ \( T^{2} - 9T + 19 \) Copy content Toggle raw display
$73$ \( T^{2} - 4T - 41 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T - 20 \) Copy content Toggle raw display
$83$ \( T^{2} - 3T - 29 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 31 \) Copy content Toggle raw display
$97$ \( (T - 13)^{2} \) Copy content Toggle raw display
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