Properties

Label 4598.2.a.y
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: \(\Q(\sqrt{21}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta q^{3} + q^{4} + ( - \beta + 2) q^{5} + \beta q^{6} + ( - \beta + 3) q^{7} - q^{8} + (\beta + 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta q^{3} + q^{4} + ( - \beta + 2) q^{5} + \beta q^{6} + ( - \beta + 3) q^{7} - q^{8} + (\beta + 2) q^{9} + (\beta - 2) q^{10} - \beta q^{12} + ( - \beta - 3) q^{13} + (\beta - 3) q^{14} + ( - \beta + 5) q^{15} + q^{16} + ( - 2 \beta + 4) q^{17} + ( - \beta - 2) q^{18} - q^{19} + ( - \beta + 2) q^{20} + ( - 2 \beta + 5) q^{21} + (2 \beta + 2) q^{23} + \beta q^{24} + ( - 3 \beta + 4) q^{25} + (\beta + 3) q^{26} - 5 q^{27} + ( - \beta + 3) q^{28} + ( - \beta - 4) q^{29} + (\beta - 5) q^{30} + (\beta - 9) q^{31} - q^{32} + (2 \beta - 4) q^{34} + ( - 4 \beta + 11) q^{35} + (\beta + 2) q^{36} + 8 q^{37} + q^{38} + (4 \beta + 5) q^{39} + (\beta - 2) q^{40} + ( - \beta - 1) q^{41} + (2 \beta - 5) q^{42} + (3 \beta - 2) q^{43} + ( - \beta - 1) q^{45} + ( - 2 \beta - 2) q^{46} + ( - 4 \beta + 2) q^{47} - \beta q^{48} + ( - 5 \beta + 7) q^{49} + (3 \beta - 4) q^{50} + ( - 2 \beta + 10) q^{51} + ( - \beta - 3) q^{52} + (2 \beta + 2) q^{53} + 5 q^{54} + (\beta - 3) q^{56} + \beta q^{57} + (\beta + 4) q^{58} + ( - \beta + 5) q^{60} - 2 q^{61} + ( - \beta + 9) q^{62} + q^{63} + q^{64} + (2 \beta - 1) q^{65} + (\beta - 3) q^{67} + ( - 2 \beta + 4) q^{68} + ( - 4 \beta - 10) q^{69} + (4 \beta - 11) q^{70} + (3 \beta - 6) q^{71} + ( - \beta - 2) q^{72} + (2 \beta - 6) q^{73} - 8 q^{74} + ( - \beta + 15) q^{75} - q^{76} + ( - 4 \beta - 5) q^{78} + ( - 2 \beta + 2) q^{79} + ( - \beta + 2) q^{80} + (2 \beta - 6) q^{81} + (\beta + 1) q^{82} + ( - \beta + 2) q^{83} + ( - 2 \beta + 5) q^{84} + ( - 6 \beta + 18) q^{85} + ( - 3 \beta + 2) q^{86} + (5 \beta + 5) q^{87} + ( - 2 \beta - 8) q^{89} + (\beta + 1) q^{90} + (\beta - 4) q^{91} + (2 \beta + 2) q^{92} + (8 \beta - 5) q^{93} + (4 \beta - 2) q^{94} + (\beta - 2) q^{95} + \beta q^{96} + 8 q^{97} + (5 \beta - 7) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} + 3 q^{5} + q^{6} + 5 q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} + 3 q^{5} + q^{6} + 5 q^{7} - 2 q^{8} + 5 q^{9} - 3 q^{10} - q^{12} - 7 q^{13} - 5 q^{14} + 9 q^{15} + 2 q^{16} + 6 q^{17} - 5 q^{18} - 2 q^{19} + 3 q^{20} + 8 q^{21} + 6 q^{23} + q^{24} + 5 q^{25} + 7 q^{26} - 10 q^{27} + 5 q^{28} - 9 q^{29} - 9 q^{30} - 17 q^{31} - 2 q^{32} - 6 q^{34} + 18 q^{35} + 5 q^{36} + 16 q^{37} + 2 q^{38} + 14 q^{39} - 3 q^{40} - 3 q^{41} - 8 q^{42} - q^{43} - 3 q^{45} - 6 q^{46} - q^{48} + 9 q^{49} - 5 q^{50} + 18 q^{51} - 7 q^{52} + 6 q^{53} + 10 q^{54} - 5 q^{56} + q^{57} + 9 q^{58} + 9 q^{60} - 4 q^{61} + 17 q^{62} + 2 q^{63} + 2 q^{64} - 5 q^{67} + 6 q^{68} - 24 q^{69} - 18 q^{70} - 9 q^{71} - 5 q^{72} - 10 q^{73} - 16 q^{74} + 29 q^{75} - 2 q^{76} - 14 q^{78} + 2 q^{79} + 3 q^{80} - 10 q^{81} + 3 q^{82} + 3 q^{83} + 8 q^{84} + 30 q^{85} + q^{86} + 15 q^{87} - 18 q^{89} + 3 q^{90} - 7 q^{91} + 6 q^{92} - 2 q^{93} - 3 q^{95} + q^{96} + 16 q^{97} - 9 q^{98}+O(q^{100}) \) 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
2.79129
−1.79129
−1.00000 −2.79129 1.00000 −0.791288 2.79129 0.208712 −1.00000 4.79129 0.791288
1.2 −1.00000 1.79129 1.00000 3.79129 −1.79129 4.79129 −1.00000 0.208712 −3.79129
\(n\): 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.y 2
11.b odd 2 1 418.2.a.f 2
33.d even 2 1 3762.2.a.s 2
44.c even 2 1 3344.2.a.l 2
209.d even 2 1 7942.2.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.f 2 11.b odd 2 1
3344.2.a.l 2 44.c even 2 1
3762.2.a.s 2 33.d even 2 1
4598.2.a.y 2 1.a even 1 1 trivial
7942.2.a.w 2 209.d even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 5 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T - 3 \) Copy content Toggle raw display
$7$ \( T^{2} - 5T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 7 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T - 12 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 6T - 12 \) Copy content Toggle raw display
$29$ \( T^{2} + 9T + 15 \) Copy content Toggle raw display
$31$ \( T^{2} + 17T + 67 \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 3T - 3 \) Copy content Toggle raw display
$43$ \( T^{2} + T - 47 \) Copy content Toggle raw display
$47$ \( T^{2} - 84 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T - 12 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 5T + 1 \) Copy content Toggle raw display
$71$ \( T^{2} + 9T - 27 \) Copy content Toggle raw display
$73$ \( T^{2} + 10T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 2T - 20 \) Copy content Toggle raw display
$83$ \( T^{2} - 3T - 3 \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 60 \) Copy content Toggle raw display
$97$ \( (T - 8)^{2} \) Copy content Toggle raw display
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