Properties

Label 4598.2.a.bb
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
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] = [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: \(1\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} - 2 q^{3} + q^{4} + \beta q^{5} - 2 q^{6} - 3 \beta q^{7} + q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - 2 q^{3} + q^{4} + \beta q^{5} - 2 q^{6} - 3 \beta q^{7} + q^{8} + q^{9} + \beta q^{10} - 2 q^{12} + (4 \beta - 4) q^{13} - 3 \beta q^{14} - 2 \beta q^{15} + q^{16} + ( - \beta + 5) q^{17} + q^{18} + q^{19} + \beta q^{20} + 6 \beta q^{21} + ( - 3 \beta - 3) q^{23} - 2 q^{24} + (\beta - 4) q^{25} + (4 \beta - 4) q^{26} + 4 q^{27} - 3 \beta q^{28} + (4 \beta + 4) q^{29} - 2 \beta q^{30} + 2 q^{31} + q^{32} + ( - \beta + 5) q^{34} + ( - 3 \beta - 3) q^{35} + q^{36} + 4 q^{37} + q^{38} + ( - 8 \beta + 8) q^{39} + \beta q^{40} + (2 \beta - 8) q^{41} + 6 \beta q^{42} + ( - 5 \beta - 1) q^{43} + \beta q^{45} + ( - 3 \beta - 3) q^{46} + ( - \beta + 3) q^{47} - 2 q^{48} + (9 \beta + 2) q^{49} + (\beta - 4) q^{50} + (2 \beta - 10) q^{51} + (4 \beta - 4) q^{52} + (2 \beta - 8) q^{53} + 4 q^{54} - 3 \beta q^{56} - 2 q^{57} + (4 \beta + 4) q^{58} + (4 \beta - 10) q^{59} - 2 \beta q^{60} + ( - 9 \beta + 5) q^{61} + 2 q^{62} - 3 \beta q^{63} + q^{64} + 4 q^{65} + ( - 2 \beta + 6) q^{67} + ( - \beta + 5) q^{68} + (6 \beta + 6) q^{69} + ( - 3 \beta - 3) q^{70} + (8 \beta - 8) q^{71} + q^{72} - 14 q^{73} + 4 q^{74} + ( - 2 \beta + 8) q^{75} + q^{76} + ( - 8 \beta + 8) q^{78} + (6 \beta - 6) q^{79} + \beta q^{80} - 11 q^{81} + (2 \beta - 8) q^{82} + ( - 11 \beta + 4) q^{83} + 6 \beta q^{84} + (4 \beta - 1) q^{85} + ( - 5 \beta - 1) q^{86} + ( - 8 \beta - 8) q^{87} + (2 \beta + 8) q^{89} + \beta q^{90} - 12 q^{91} + ( - 3 \beta - 3) q^{92} - 4 q^{93} + ( - \beta + 3) q^{94} + \beta q^{95} - 2 q^{96} + ( - 2 \beta - 6) q^{97} + (9 \beta + 2) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} + q^{5} - 4 q^{6} - 3 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} + q^{5} - 4 q^{6} - 3 q^{7} + 2 q^{8} + 2 q^{9} + q^{10} - 4 q^{12} - 4 q^{13} - 3 q^{14} - 2 q^{15} + 2 q^{16} + 9 q^{17} + 2 q^{18} + 2 q^{19} + q^{20} + 6 q^{21} - 9 q^{23} - 4 q^{24} - 7 q^{25} - 4 q^{26} + 8 q^{27} - 3 q^{28} + 12 q^{29} - 2 q^{30} + 4 q^{31} + 2 q^{32} + 9 q^{34} - 9 q^{35} + 2 q^{36} + 8 q^{37} + 2 q^{38} + 8 q^{39} + q^{40} - 14 q^{41} + 6 q^{42} - 7 q^{43} + q^{45} - 9 q^{46} + 5 q^{47} - 4 q^{48} + 13 q^{49} - 7 q^{50} - 18 q^{51} - 4 q^{52} - 14 q^{53} + 8 q^{54} - 3 q^{56} - 4 q^{57} + 12 q^{58} - 16 q^{59} - 2 q^{60} + q^{61} + 4 q^{62} - 3 q^{63} + 2 q^{64} + 8 q^{65} + 10 q^{67} + 9 q^{68} + 18 q^{69} - 9 q^{70} - 8 q^{71} + 2 q^{72} - 28 q^{73} + 8 q^{74} + 14 q^{75} + 2 q^{76} + 8 q^{78} - 6 q^{79} + q^{80} - 22 q^{81} - 14 q^{82} - 3 q^{83} + 6 q^{84} + 2 q^{85} - 7 q^{86} - 24 q^{87} + 18 q^{89} + q^{90} - 24 q^{91} - 9 q^{92} - 8 q^{93} + 5 q^{94} + q^{95} - 4 q^{96} - 14 q^{97} + 13 q^{98}+O(q^{100}) \) 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
−0.618034
1.61803
1.00000 −2.00000 1.00000 −0.618034 −2.00000 1.85410 1.00000 1.00000 −0.618034
1.2 1.00000 −2.00000 1.00000 1.61803 −2.00000 −4.85410 1.00000 1.00000 1.61803
\(n\): 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.bb 2
11.b odd 2 1 4598.2.a.t 2
11.c even 5 2 418.2.f.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.d 4 11.c even 5 2
4598.2.a.t 2 11.b odd 2 1
4598.2.a.bb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 3T_{7} - 9 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$17$ \( T^{2} - 9T + 19 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T - 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$43$ \( T^{2} + 7T - 19 \) Copy content Toggle raw display
$47$ \( T^{2} - 5T + 5 \) Copy content Toggle raw display
$53$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$59$ \( T^{2} + 16T + 44 \) Copy content Toggle raw display
$61$ \( T^{2} - T - 101 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$73$ \( (T + 14)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 6T - 36 \) Copy content Toggle raw display
$83$ \( T^{2} + 3T - 149 \) Copy content Toggle raw display
$89$ \( T^{2} - 18T + 76 \) Copy content Toggle raw display
$97$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
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