Properties

Label 4598.2.a.bu
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.33452.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + 7x - 1 \) 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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{3} + \beta_{2} + 1) q^{5} + \beta_1 q^{6} + \beta_{3} q^{7} + q^{8} + ( - \beta_{3} + \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{3} + \beta_{2} + 1) q^{5} + \beta_1 q^{6} + \beta_{3} q^{7} + q^{8} + ( - \beta_{3} + \beta_{2} + 1) q^{9} + (\beta_{3} + \beta_{2} + 1) q^{10} + \beta_1 q^{12} + (\beta_{3} + \beta_{2} - 2) q^{13} + \beta_{3} q^{14} + (\beta_{2} + \beta_1 + 1) q^{15} + q^{16} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{3} + \beta_{2} + 1) q^{18} - q^{19} + (\beta_{3} + \beta_{2} + 1) q^{20} + ( - 2 \beta_1 + 1) q^{21} + ( - \beta_{3} + \beta_1 - 1) q^{23} + \beta_1 q^{24} + (2 \beta_{3} - \beta_{2} + \beta_1 + 4) q^{25} + (\beta_{3} + \beta_{2} - 2) q^{26} + (\beta_{2} + 2 \beta_1 - 1) q^{27} + \beta_{3} q^{28} + ( - 2 \beta_{3} - \beta_{2} + 3 \beta_1) q^{29} + (\beta_{2} + \beta_1 + 1) q^{30} + ( - \beta_{2} - \beta_1 + 7) q^{31} + q^{32} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{34} + (3 \beta_{3} - \beta_1 + 4) q^{35} + ( - \beta_{3} + \beta_{2} + 1) q^{36} + (2 \beta_{2} - 2 \beta_1 + 2) q^{37} - q^{38} + (\beta_{2} - 2 \beta_1 + 1) q^{39} + (\beta_{3} + \beta_{2} + 1) q^{40} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{41} + ( - 2 \beta_1 + 1) q^{42} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{43} + ( - 4 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{45} + ( - \beta_{3} + \beta_1 - 1) q^{46} + (2 \beta_{3} + 2 \beta_{2} + 6) q^{47} + \beta_1 q^{48} + (2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{49} + (2 \beta_{3} - \beta_{2} + \beta_1 + 4) q^{50} + ( - \beta_{3} - \beta_1 + 3) q^{51} + (\beta_{3} + \beta_{2} - 2) q^{52} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{53} + (\beta_{2} + 2 \beta_1 - 1) q^{54} + \beta_{3} q^{56} - \beta_1 q^{57} + ( - 2 \beta_{3} - \beta_{2} + 3 \beta_1) q^{58} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{59} + (\beta_{2} + \beta_1 + 1) q^{60} + (2 \beta_{3} + 2) q^{61} + ( - \beta_{2} - \beta_1 + 7) q^{62} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 8) q^{63} + q^{64} + ( - \beta_{3} - 4 \beta_{2} + \beta_1 + 6) q^{65} + ( - \beta_{3} - 2 \beta_{2} + 4) q^{67} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{68} + ( - \beta_{3} + \beta_{2} + \beta_1 + 3) q^{69} + (3 \beta_{3} - \beta_1 + 4) q^{70} + ( - \beta_{3} + \beta_{2} - 5) q^{71} + ( - \beta_{3} + \beta_{2} + 1) q^{72} + (\beta_{3} + \beta_{2} + \beta_1 + 5) q^{73} + (2 \beta_{2} - 2 \beta_1 + 2) q^{74} + ( - \beta_{3} - 2 \beta_1 + 6) q^{75} - q^{76} + (\beta_{2} - 2 \beta_1 + 1) q^{78} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{79} + (\beta_{3} + \beta_{2} + 1) q^{80} + (\beta_{3} + \beta_1 + 5) q^{81} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{82} + ( - 3 \beta_{3} + \beta_{2} + 2 \beta_1 - 7) q^{83} + ( - 2 \beta_1 + 1) q^{84} + ( - 2 \beta_{3} + 2 \beta_{2} - 8) q^{85} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{86} + ( - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 10) q^{87} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{89} + ( - 4 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{90} + ( - \beta_1 + 4) q^{91} + ( - \beta_{3} + \beta_1 - 1) q^{92} + (\beta_{3} - 2 \beta_{2} + 5 \beta_1 - 4) q^{93} + (2 \beta_{3} + 2 \beta_{2} + 6) q^{94} + ( - \beta_{3} - \beta_{2} - 1) q^{95} + \beta_1 q^{96} + (4 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{97} + (2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + q^{3} + 4 q^{4} + 3 q^{5} + q^{6} - q^{7} + 4 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + q^{3} + 4 q^{4} + 3 q^{5} + q^{6} - q^{7} + 4 q^{8} + 5 q^{9} + 3 q^{10} + q^{12} - 9 q^{13} - q^{14} + 5 q^{15} + 4 q^{16} - 2 q^{17} + 5 q^{18} - 4 q^{19} + 3 q^{20} + 2 q^{21} - 2 q^{23} + q^{24} + 15 q^{25} - 9 q^{26} - 2 q^{27} - q^{28} + 5 q^{29} + 5 q^{30} + 27 q^{31} + 4 q^{32} - 2 q^{34} + 12 q^{35} + 5 q^{36} + 6 q^{37} - 4 q^{38} + 2 q^{39} + 3 q^{40} - 5 q^{41} + 2 q^{42} + q^{43} + 11 q^{45} - 2 q^{46} + 22 q^{47} + q^{48} - 7 q^{49} + 15 q^{50} + 12 q^{51} - 9 q^{52} - 2 q^{54} - q^{56} - q^{57} + 5 q^{58} + 5 q^{60} + 6 q^{61} + 27 q^{62} - 30 q^{63} + 4 q^{64} + 26 q^{65} + 17 q^{67} - 2 q^{68} + 14 q^{69} + 12 q^{70} - 19 q^{71} + 5 q^{72} + 20 q^{73} + 6 q^{74} + 23 q^{75} - 4 q^{76} + 2 q^{78} + 12 q^{79} + 3 q^{80} + 20 q^{81} - 5 q^{82} - 23 q^{83} + 2 q^{84} - 30 q^{85} + q^{86} + 45 q^{87} + 16 q^{89} + 11 q^{90} + 15 q^{91} - 2 q^{92} - 12 q^{93} + 22 q^{94} - 3 q^{95} + q^{96} + 8 q^{97} - 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 8x^{2} + 7x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 8\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 8\nu + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 8\beta _1 - 1 \) 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
−2.79836
0.180986
0.669883
2.94749
1.00000 −2.79836 1.00000 0.116104 −2.79836 −2.35735 1.00000 4.83081 0.116104
1.2 1.00000 0.180986 1.00000 4.08332 0.180986 3.52528 1.00000 −2.96724 4.08332
1.3 1.00000 0.669883 1.00000 −3.56566 0.669883 −0.507202 1.00000 −2.55126 −3.56566
1.4 1.00000 2.94749 1.00000 2.36624 2.94749 −1.66073 1.00000 5.68769 2.36624
\(n\): 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.bu yes 4
11.b odd 2 1 4598.2.a.br 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.br 4 11.b odd 2 1
4598.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{4} - T_{3}^{3} - 8T_{3}^{2} + 7T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{4} - 3T_{5}^{3} - 13T_{5}^{2} + 36T_{5} - 4 \) Copy content Toggle raw display
\( T_{7}^{4} + T_{7}^{3} - 10T_{7}^{2} - 19T_{7} - 7 \) Copy content Toggle raw display
\( T_{13}^{4} + 9T_{13}^{3} + 14T_{13}^{2} - 15T_{13} - 13 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} - 8 T^{2} + 7 T - 1 \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} - 13 T^{2} + 36 T - 4 \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} - 10 T^{2} - 19 T - 7 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 9 T^{3} + 14 T^{2} - 15 T - 13 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} - 19 T^{2} - 38 T + 28 \) Copy content Toggle raw display
$19$ \( (T + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} - 15 T^{2} - 20 T + 4 \) Copy content Toggle raw display
$29$ \( T^{4} - 5 T^{3} - 86 T^{2} + \cdots + 2023 \) Copy content Toggle raw display
$31$ \( T^{4} - 27 T^{3} + 249 T^{2} + \cdots + 644 \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} - 68 T^{2} + 224 T - 128 \) Copy content Toggle raw display
$41$ \( T^{4} + 5 T^{3} - 73 T^{2} - 152 T + 796 \) Copy content Toggle raw display
$43$ \( T^{4} - T^{3} - 65 T^{2} + 308 T - 392 \) Copy content Toggle raw display
$47$ \( T^{4} - 22 T^{3} + 116 T^{2} + \cdots - 1408 \) Copy content Toggle raw display
$53$ \( T^{4} - 107 T^{2} + 564 T - 784 \) Copy content Toggle raw display
$59$ \( T^{4} - 79 T^{2} + 72 T + 1088 \) Copy content Toggle raw display
$61$ \( T^{4} - 6 T^{3} - 28 T^{2} + 32 \) Copy content Toggle raw display
$67$ \( T^{4} - 17 T^{3} + 58 T^{2} - 33 T - 23 \) Copy content Toggle raw display
$71$ \( T^{4} + 19 T^{3} + 103 T^{2} + \cdots + 28 \) Copy content Toggle raw display
$73$ \( T^{4} - 20 T^{3} + 121 T^{2} + \cdots + 112 \) Copy content Toggle raw display
$79$ \( T^{4} - 12 T^{3} - 108 T^{2} + \cdots - 2464 \) Copy content Toggle raw display
$83$ \( T^{4} + 23 T^{3} + 43 T^{2} + \cdots - 4048 \) Copy content Toggle raw display
$89$ \( T^{4} - 16 T^{3} - 68 T^{2} + \cdots - 736 \) Copy content Toggle raw display
$97$ \( T^{4} - 8 T^{3} - 216 T^{2} + \cdots + 10624 \) Copy content Toggle raw display
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