Properties

Label 4598.2.a.bl
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
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] = [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: \(3\)
Coefficient field: 3.3.404.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) 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
−0.210756
−1.65544
2.86620
−1.00000 −2.74483 1.00000 3.32331 2.74483 −1.53407 −1.00000 4.53407 −3.32331
1.2 −1.00000 1.39593 1.00000 −3.70682 −1.39593 4.05137 −1.00000 −1.05137 3.70682
1.3 −1.00000 2.34889 1.00000 4.38350 −2.34889 0.482696 −1.00000 2.51730 −4.38350
\(n\): 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.bl 3
11.b odd 2 1 4598.2.a.bp yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.bl 3 1.a even 1 1 trivial
4598.2.a.bp yes 3 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{3} - T_{3}^{2} - 7T_{3} + 9 \) Copy content Toggle raw display
\( T_{5}^{3} - 4T_{5}^{2} - 14T_{5} + 54 \) Copy content Toggle raw display
\( T_{7}^{3} - 3T_{7}^{2} - 5T_{7} + 3 \) Copy content Toggle raw display
\( T_{13}^{3} - 2T_{13}^{2} - 20T_{13} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 7T + 9 \) Copy content Toggle raw display
$5$ \( T^{3} - 4 T^{2} - 14 T + 54 \) Copy content Toggle raw display
$7$ \( T^{3} - 3 T^{2} - 5 T + 3 \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 2 T^{2} - 20 T - 8 \) Copy content Toggle raw display
$17$ \( T^{3} + 2 T^{2} - 56 T - 196 \) Copy content Toggle raw display
$19$ \( (T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 9 T^{2} - 5 T - 37 \) Copy content Toggle raw display
$29$ \( T^{3} + 5 T^{2} - 17 T - 3 \) Copy content Toggle raw display
$31$ \( (T - 2)^{3} \) Copy content Toggle raw display
$37$ \( T^{3} - 19 T^{2} + 113 T - 201 \) Copy content Toggle raw display
$41$ \( T^{3} - 12T^{2} + 202 \) Copy content Toggle raw display
$43$ \( T^{3} + 8 T^{2} - 48 T - 288 \) Copy content Toggle raw display
$47$ \( T^{3} - T^{2} - 17 T + 21 \) Copy content Toggle raw display
$53$ \( T^{3} - 7 T^{2} - 117 T + 641 \) Copy content Toggle raw display
$59$ \( T^{3} - 19 T^{2} + 9 T + 891 \) Copy content Toggle raw display
$61$ \( T^{3} - 2 T^{2} - 58 T - 18 \) Copy content Toggle raw display
$67$ \( (T - 12)^{3} \) Copy content Toggle raw display
$71$ \( T^{3} - 8 T^{2} - 120 T + 666 \) Copy content Toggle raw display
$73$ \( T^{3} - 26 T^{2} + 208 T - 516 \) Copy content Toggle raw display
$79$ \( T^{3} + 24 T^{2} + 144 T + 202 \) Copy content Toggle raw display
$83$ \( T^{3} + 6 T^{2} - 26 T + 18 \) Copy content Toggle raw display
$89$ \( T^{3} - 8 T^{2} - 88 T + 222 \) Copy content Toggle raw display
$97$ \( T^{3} - 2 T^{2} - 236 T + 616 \) Copy content Toggle raw display
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