Properties

Label 4598.2.a.bd
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
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] = [4598,2,Mod(1,4598)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4598, 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("4598.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.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: \(36.7152148494\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
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.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + ( - \beta - 1) q^{5} - q^{6} + \beta q^{7} + q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} + ( - \beta - 1) q^{5} - q^{6} + \beta q^{7} + q^{8} - 2 q^{9} + ( - \beta - 1) q^{10} - q^{12} + 3 \beta q^{13} + \beta q^{14} + (\beta + 1) q^{15} + q^{16} + (2 \beta - 3) q^{17} - 2 q^{18} - q^{19} + ( - \beta - 1) q^{20} - \beta q^{21} - q^{23} - q^{24} + (3 \beta - 3) q^{25} + 3 \beta q^{26} + 5 q^{27} + \beta q^{28} + ( - 8 \beta + 4) q^{29} + (\beta + 1) q^{30} + (6 \beta - 1) q^{31} + q^{32} + (2 \beta - 3) q^{34} + ( - 2 \beta - 1) q^{35} - 2 q^{36} + ( - \beta + 11) q^{37} - q^{38} - 3 \beta q^{39} + ( - \beta - 1) q^{40} + ( - 5 \beta + 2) q^{41} - \beta q^{42} + ( - 6 \beta + 2) q^{43} + (2 \beta + 2) q^{45} - q^{46} + ( - 6 \beta + 1) q^{47} - q^{48} + (\beta - 6) q^{49} + (3 \beta - 3) q^{50} + ( - 2 \beta + 3) q^{51} + 3 \beta q^{52} + ( - 4 \beta + 11) q^{53} + 5 q^{54} + \beta q^{56} + q^{57} + ( - 8 \beta + 4) q^{58} + ( - 3 \beta - 1) q^{59} + (\beta + 1) q^{60} + 9 \beta q^{61} + (6 \beta - 1) q^{62} - 2 \beta q^{63} + q^{64} + ( - 6 \beta - 3) q^{65} + ( - \beta - 4) q^{67} + (2 \beta - 3) q^{68} + q^{69} + ( - 2 \beta - 1) q^{70} + ( - 2 \beta + 13) q^{71} - 2 q^{72} + ( - \beta + 2) q^{73} + ( - \beta + 11) q^{74} + ( - 3 \beta + 3) q^{75} - q^{76} - 3 \beta q^{78} + 15 q^{79} + ( - \beta - 1) q^{80} + q^{81} + ( - 5 \beta + 2) q^{82} + (11 \beta - 9) q^{83} - \beta q^{84} + ( - \beta + 1) q^{85} + ( - 6 \beta + 2) q^{86} + (8 \beta - 4) q^{87} + (12 \beta - 1) q^{89} + (2 \beta + 2) q^{90} + (3 \beta + 3) q^{91} - q^{92} + ( - 6 \beta + 1) q^{93} + ( - 6 \beta + 1) q^{94} + (\beta + 1) q^{95} - q^{96} + (10 \beta - 7) q^{97} + (\beta - 6) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + q^{7} + 2 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + q^{7} + 2 q^{8} - 4 q^{9} - 3 q^{10} - 2 q^{12} + 3 q^{13} + q^{14} + 3 q^{15} + 2 q^{16} - 4 q^{17} - 4 q^{18} - 2 q^{19} - 3 q^{20} - q^{21} - 2 q^{23} - 2 q^{24} - 3 q^{25} + 3 q^{26} + 10 q^{27} + q^{28} + 3 q^{30} + 4 q^{31} + 2 q^{32} - 4 q^{34} - 4 q^{35} - 4 q^{36} + 21 q^{37} - 2 q^{38} - 3 q^{39} - 3 q^{40} - q^{41} - q^{42} - 2 q^{43} + 6 q^{45} - 2 q^{46} - 4 q^{47} - 2 q^{48} - 11 q^{49} - 3 q^{50} + 4 q^{51} + 3 q^{52} + 18 q^{53} + 10 q^{54} + q^{56} + 2 q^{57} - 5 q^{59} + 3 q^{60} + 9 q^{61} + 4 q^{62} - 2 q^{63} + 2 q^{64} - 12 q^{65} - 9 q^{67} - 4 q^{68} + 2 q^{69} - 4 q^{70} + 24 q^{71} - 4 q^{72} + 3 q^{73} + 21 q^{74} + 3 q^{75} - 2 q^{76} - 3 q^{78} + 30 q^{79} - 3 q^{80} + 2 q^{81} - q^{82} - 7 q^{83} - q^{84} + q^{85} - 2 q^{86} + 10 q^{89} + 6 q^{90} + 9 q^{91} - 2 q^{92} - 4 q^{93} - 4 q^{94} + 3 q^{95} - 2 q^{96} - 4 q^{97} - 11 q^{98}+O(q^{100}) \) 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 −1.00000 1.00000 −2.61803 −1.00000 1.61803 1.00000 −2.00000 −2.61803
1.2 1.00000 −1.00000 1.00000 −0.381966 −1.00000 −0.618034 1.00000 −2.00000 −0.381966
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(1\)
\(19\) \(1\)

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 4598.2.a.bd 2
11.b odd 2 1 4598.2.a.u 2
11.c even 5 2 418.2.f.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.c 4 11.c even 5 2
4598.2.a.u 2 11.b odd 2 1
4598.2.a.bd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 3T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} - 1 \) Copy content Toggle raw display
\( T_{13}^{2} - 3T_{13} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 80 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 41 \) Copy content Toggle raw display
$37$ \( T^{2} - 21T + 109 \) Copy content Toggle raw display
$41$ \( T^{2} + T - 31 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 41 \) Copy content Toggle raw display
$53$ \( T^{2} - 18T + 61 \) Copy content Toggle raw display
$59$ \( T^{2} + 5T - 5 \) Copy content Toggle raw display
$61$ \( T^{2} - 9T - 81 \) Copy content Toggle raw display
$67$ \( T^{2} + 9T + 19 \) Copy content Toggle raw display
$71$ \( T^{2} - 24T + 139 \) Copy content Toggle raw display
$73$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$79$ \( (T - 15)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 7T - 139 \) Copy content Toggle raw display
$89$ \( T^{2} - 10T - 155 \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 121 \) Copy content Toggle raw display
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