Properties

Label 8030.2.a.o
Level $8030$
Weight $2$
Character orbit 8030.a
Self dual yes
Analytic conductor $64.120$
Analytic rank $1$
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] = [8030,2,Mod(1,8030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8030, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8030.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8030.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: \(64.1198728231\)
Analytic rank: \(1\)
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.

\(f(q)\) \(=\) \( q + q^{2} - \beta q^{3} + q^{4} - q^{5} - \beta q^{6} + ( - 2 \beta + 2) q^{7} + q^{8} + (\beta - 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta q^{3} + q^{4} - q^{5} - \beta q^{6} + ( - 2 \beta + 2) q^{7} + q^{8} + (\beta - 2) q^{9} - q^{10} + q^{11} - \beta q^{12} + \beta q^{13} + ( - 2 \beta + 2) q^{14} + \beta q^{15} + q^{16} - 4 q^{17} + (\beta - 2) q^{18} + (\beta - 5) q^{19} - q^{20} + 2 q^{21} + q^{22} + 2 q^{23} - \beta q^{24} + q^{25} + \beta q^{26} + (4 \beta - 1) q^{27} + ( - 2 \beta + 2) q^{28} + (2 \beta - 2) q^{29} + \beta q^{30} + (6 \beta - 2) q^{31} + q^{32} - \beta q^{33} - 4 q^{34} + (2 \beta - 2) q^{35} + (\beta - 2) q^{36} + (4 \beta + 2) q^{37} + (\beta - 5) q^{38} + ( - \beta - 1) q^{39} - q^{40} - 2 q^{41} + 2 q^{42} + (\beta - 5) q^{43} + q^{44} + ( - \beta + 2) q^{45} + 2 q^{46} + ( - 7 \beta - 1) q^{47} - \beta q^{48} + ( - 4 \beta + 1) q^{49} + q^{50} + 4 \beta q^{51} + \beta q^{52} + (4 \beta + 8) q^{53} + (4 \beta - 1) q^{54} - q^{55} + ( - 2 \beta + 2) q^{56} + (4 \beta - 1) q^{57} + (2 \beta - 2) q^{58} + (9 \beta - 4) q^{59} + \beta q^{60} - 5 \beta q^{61} + (6 \beta - 2) q^{62} + (4 \beta - 6) q^{63} + q^{64} - \beta q^{65} - \beta q^{66} + (9 \beta - 8) q^{67} - 4 q^{68} - 2 \beta q^{69} + (2 \beta - 2) q^{70} + (3 \beta - 3) q^{71} + (\beta - 2) q^{72} - q^{73} + (4 \beta + 2) q^{74} - \beta q^{75} + (\beta - 5) q^{76} + ( - 2 \beta + 2) q^{77} + ( - \beta - 1) q^{78} + ( - 6 \beta + 2) q^{79} - q^{80} + ( - 6 \beta + 2) q^{81} - 2 q^{82} + (\beta - 14) q^{83} + 2 q^{84} + 4 q^{85} + (\beta - 5) q^{86} - 2 q^{87} + q^{88} + ( - 11 \beta + 4) q^{89} + ( - \beta + 2) q^{90} - 2 q^{91} + 2 q^{92} + ( - 4 \beta - 6) q^{93} + ( - 7 \beta - 1) q^{94} + ( - \beta + 5) q^{95} - \beta q^{96} + ( - 4 \beta - 4) q^{97} + ( - 4 \beta + 1) q^{98} + (\beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9} - 2 q^{10} + 2 q^{11} - q^{12} + q^{13} + 2 q^{14} + q^{15} + 2 q^{16} - 8 q^{17} - 3 q^{18} - 9 q^{19} - 2 q^{20} + 4 q^{21} + 2 q^{22} + 4 q^{23} - q^{24} + 2 q^{25} + q^{26} + 2 q^{27} + 2 q^{28} - 2 q^{29} + q^{30} + 2 q^{31} + 2 q^{32} - q^{33} - 8 q^{34} - 2 q^{35} - 3 q^{36} + 8 q^{37} - 9 q^{38} - 3 q^{39} - 2 q^{40} - 4 q^{41} + 4 q^{42} - 9 q^{43} + 2 q^{44} + 3 q^{45} + 4 q^{46} - 9 q^{47} - q^{48} - 2 q^{49} + 2 q^{50} + 4 q^{51} + q^{52} + 20 q^{53} + 2 q^{54} - 2 q^{55} + 2 q^{56} + 2 q^{57} - 2 q^{58} + q^{59} + q^{60} - 5 q^{61} + 2 q^{62} - 8 q^{63} + 2 q^{64} - q^{65} - q^{66} - 7 q^{67} - 8 q^{68} - 2 q^{69} - 2 q^{70} - 3 q^{71} - 3 q^{72} - 2 q^{73} + 8 q^{74} - q^{75} - 9 q^{76} + 2 q^{77} - 3 q^{78} - 2 q^{79} - 2 q^{80} - 2 q^{81} - 4 q^{82} - 27 q^{83} + 4 q^{84} + 8 q^{85} - 9 q^{86} - 4 q^{87} + 2 q^{88} - 3 q^{89} + 3 q^{90} - 4 q^{91} + 4 q^{92} - 16 q^{93} - 9 q^{94} + 9 q^{95} - q^{96} - 12 q^{97} - 2 q^{98} - 3 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
1.00000 −1.61803 1.00000 −1.00000 −1.61803 −1.23607 1.00000 −0.381966 −1.00000
1.2 1.00000 0.618034 1.00000 −1.00000 0.618034 3.23607 1.00000 −2.61803 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(11\) \(-1\)
\(73\) \(1\)

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 8030.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.o 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\( T_{3}^{2} + T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$17$ \( (T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 9T + 19 \) Copy content Toggle raw display
$23$ \( (T - 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 9T + 19 \) Copy content Toggle raw display
$47$ \( T^{2} + 9T - 41 \) Copy content Toggle raw display
$53$ \( T^{2} - 20T + 80 \) Copy content Toggle raw display
$59$ \( T^{2} - T - 101 \) Copy content Toggle raw display
$61$ \( T^{2} + 5T - 25 \) Copy content Toggle raw display
$67$ \( T^{2} + 7T - 89 \) Copy content Toggle raw display
$71$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$73$ \( (T + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$83$ \( T^{2} + 27T + 181 \) Copy content Toggle raw display
$89$ \( T^{2} + 3T - 149 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
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