Properties

Label 4598.2.a.bt
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.7488.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 2x + 1 \) 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( - \beta_{3} - 1) q^{3} + q^{4} + (\beta_1 - 1) q^{5} + ( - \beta_{3} - 1) q^{6} - q^{7} + q^{8} + (2 \beta_{3} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + ( - \beta_{3} - 1) q^{3} + q^{4} + (\beta_1 - 1) q^{5} + ( - \beta_{3} - 1) q^{6} - q^{7} + q^{8} + (2 \beta_{3} - \beta_{2} + 1) q^{9} + (\beta_1 - 1) q^{10} + ( - \beta_{3} - 1) q^{12} - 2 \beta_1 q^{13} - q^{14} + (\beta_{3} + 2) q^{15} + q^{16} + (\beta_{2} - 1) q^{17} + (2 \beta_{3} - \beta_{2} + 1) q^{18} + q^{19} + (\beta_1 - 1) q^{20} + (\beta_{3} + 1) q^{21} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{23} + ( - \beta_{3} - 1) q^{24} + (\beta_{2} - 2) q^{25} - 2 \beta_1 q^{26} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 4) q^{27} - q^{28} + (\beta_{3} + 4 \beta_1 - 1) q^{29} + (\beta_{3} + 2) q^{30} + (3 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{31} + q^{32} + (\beta_{2} - 1) q^{34} + ( - \beta_1 + 1) q^{35} + (2 \beta_{3} - \beta_{2} + 1) q^{36} + ( - \beta_{2} - 3 \beta_1 - 1) q^{37} + q^{38} - 2 q^{39} + (\beta_1 - 1) q^{40} + ( - 3 \beta_{2} - \beta_1) q^{41} + (\beta_{3} + 1) q^{42} + (\beta_{3} + \beta_{2} + \beta_1 + 4) q^{43} + ( - 3 \beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{45} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{46} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{47} + ( - \beta_{3} - 1) q^{48} - 6 q^{49} + (\beta_{2} - 2) q^{50} + (3 \beta_{3} + \beta_1 + 1) q^{51} - 2 \beta_1 q^{52} + (\beta_{3} - \beta_{2} - 2 \beta_1) q^{53} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 4) q^{54} - q^{56} + ( - \beta_{3} - 1) q^{57} + (\beta_{3} + 4 \beta_1 - 1) q^{58} + (3 \beta_{3} + 4 \beta_1 + 5) q^{59} + (\beta_{3} + 2) q^{60} + (4 \beta_{3} - 2 \beta_{2} + 5 \beta_1 - 1) q^{61} + (3 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{62} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{63} + q^{64} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{65} + ( - 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 6) q^{67} + (\beta_{2} - 1) q^{68} + (4 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{69} + ( - \beta_1 + 1) q^{70} + (\beta_{3} - 4 \beta_{2} - 2) q^{71} + (2 \beta_{3} - \beta_{2} + 1) q^{72} + (2 \beta_{3} - 3 \beta_{2} + 1) q^{73} + ( - \beta_{2} - 3 \beta_1 - 1) q^{74} + (4 \beta_{3} + \beta_1 + 2) q^{75} + q^{76} - 2 q^{78} + ( - 5 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{79} + (\beta_1 - 1) q^{80} + (4 \beta_{3} + \beta_{2} + 2 \beta_1 + 6) q^{81} + ( - 3 \beta_{2} - \beta_1) q^{82} + (2 \beta_{2} - 3 \beta_1 - 7) q^{83} + (\beta_{3} + 1) q^{84} + (\beta_{3} - \beta_{2} + \beta_1) q^{85} + (\beta_{3} + \beta_{2} + \beta_1 + 4) q^{86} + (\beta_{2} + 2) q^{87} + (2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 2) q^{89} + ( - 3 \beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{90} + 2 \beta_1 q^{91} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{92} + ( - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 12) q^{93} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{94} + (\beta_1 - 1) q^{95} + ( - \beta_{3} - 1) q^{96} + ( - \beta_{3} + \beta_{2} + 7 \beta_1 - 6) q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{7} + 4 q^{8} - 2 q^{10} - 2 q^{12} - 4 q^{13} - 4 q^{14} + 6 q^{15} + 4 q^{16} - 4 q^{17} + 4 q^{19} - 2 q^{20} + 2 q^{21} - 8 q^{23} - 2 q^{24} - 8 q^{25} - 4 q^{26} - 14 q^{27} - 4 q^{28} + 2 q^{29} + 6 q^{30} + 4 q^{32} - 4 q^{34} + 2 q^{35} - 10 q^{37} + 4 q^{38} - 8 q^{39} - 2 q^{40} - 2 q^{41} + 2 q^{42} + 16 q^{43} - 8 q^{45} - 8 q^{46} - 2 q^{48} - 24 q^{49} - 8 q^{50} - 4 q^{52} - 6 q^{53} - 14 q^{54} - 4 q^{56} - 2 q^{57} + 2 q^{58} + 22 q^{59} + 6 q^{60} - 2 q^{61} + 4 q^{64} - 20 q^{65} - 20 q^{67} - 4 q^{68} - 2 q^{69} + 2 q^{70} - 10 q^{71} - 10 q^{74} + 2 q^{75} + 4 q^{76} - 8 q^{78} + 6 q^{79} - 2 q^{80} + 20 q^{81} - 2 q^{82} - 34 q^{83} + 2 q^{84} + 16 q^{86} + 8 q^{87} - 2 q^{89} - 8 q^{90} + 4 q^{91} - 8 q^{92} - 40 q^{93} - 2 q^{95} - 2 q^{96} - 8 q^{97} - 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 4x^{2} + 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 4\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 8\beta _1 + 3 \) 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
−0.326909
−1.43091
3.05896
0.698857
1.00000 −3.05896 1.00000 −1.32691 −3.05896 −1.00000 1.00000 6.35723 −1.32691
1.2 1.00000 −0.698857 1.00000 −2.43091 −0.698857 −1.00000 1.00000 −2.51160 −2.43091
1.3 1.00000 0.326909 1.00000 2.05896 0.326909 −1.00000 1.00000 −2.89313 2.05896
1.4 1.00000 1.43091 1.00000 −0.301143 1.43091 −1.00000 1.00000 −0.952503 −0.301143
\(n\): 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.bt yes 4
11.b odd 2 1 4598.2.a.bq 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.bq 4 11.b odd 2 1
4598.2.a.bt yes 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{4} + 2T_{3}^{3} - 4T_{3}^{2} - 2T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{4} + 2T_{5}^{3} - 4T_{5}^{2} - 8T_{5} - 2 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{13}^{4} + 4T_{13}^{3} - 16T_{13}^{2} - 16T_{13} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} - 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} - 4 T^{2} - 8 T - 2 \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} - 16 T^{2} - 16 T + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} - 4 T^{2} - 16 T + 4 \) Copy content Toggle raw display
$19$ \( (T - 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 8 T^{3} - 30 T^{2} - 328 T - 587 \) Copy content Toggle raw display
$29$ \( T^{4} - 2 T^{3} - 72 T^{2} - 62 T - 11 \) Copy content Toggle raw display
$31$ \( T^{4} - 88 T^{2} + 72 T + 1336 \) Copy content Toggle raw display
$37$ \( T^{4} + 10 T^{3} - 16 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} - 88 T^{2} + \cdots + 1606 \) Copy content Toggle raw display
$43$ \( T^{4} - 16 T^{3} + 80 T^{2} + \cdots - 104 \) Copy content Toggle raw display
$47$ \( T^{4} - 46 T^{2} + 96 T + 97 \) Copy content Toggle raw display
$53$ \( T^{4} + 6 T^{3} - 28 T^{2} - 42 T - 11 \) Copy content Toggle raw display
$59$ \( T^{4} - 22 T^{3} + 104 T^{2} + \cdots - 107 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} - 168 T^{2} + \cdots + 1126 \) Copy content Toggle raw display
$67$ \( T^{4} + 20 T^{3} + 32 T^{2} + \cdots - 6512 \) Copy content Toggle raw display
$71$ \( T^{4} + 10 T^{3} - 120 T^{2} + \cdots + 4174 \) Copy content Toggle raw display
$73$ \( T^{4} - 100 T^{2} - 24 T + 1924 \) Copy content Toggle raw display
$79$ \( T^{4} - 6 T^{3} - 208 T^{2} + \cdots - 2882 \) Copy content Toggle raw display
$83$ \( T^{4} + 34 T^{3} + 332 T^{2} + \cdots - 5402 \) Copy content Toggle raw display
$89$ \( T^{4} + 2 T^{3} - 160 T^{2} + \cdots + 1678 \) Copy content Toggle raw display
$97$ \( T^{4} + 8 T^{3} - 280 T^{2} + \cdots - 4232 \) Copy content Toggle raw display
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