Properties

Label 4598.2.a.bv
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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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.258228.1
Defining polynomial: \( x^{4} - 2x^{3} - 12x^{2} + 6x + 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} - \beta_1 q^{5} + ( - \beta_1 + 1) q^{6} + (\beta_{3} + 1) q^{7} + q^{8} + ( - \beta_{3} + \beta_{2} + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} - \beta_1 q^{5} + ( - \beta_1 + 1) q^{6} + (\beta_{3} + 1) q^{7} + q^{8} + ( - \beta_{3} + \beta_{2} + 4) q^{9} - \beta_1 q^{10} + ( - \beta_1 + 1) q^{12} - \beta_{2} q^{13} + (\beta_{3} + 1) q^{14} + ( - \beta_{3} + \beta_{2} + \beta_1 + 6) q^{15} + q^{16} + ( - \beta_{3} - 2) q^{17} + ( - \beta_{3} + \beta_{2} + 4) q^{18} + q^{19} - \beta_1 q^{20} + (3 \beta_{3} - \beta_1 - 1) q^{21} + (\beta_{2} - 1) q^{23} + ( - \beta_1 + 1) q^{24} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{25} - \beta_{2} q^{26} + ( - \beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{27} + (\beta_{3} + 1) q^{28} + (\beta_{3} - \beta_{2} - \beta_1 - 3) q^{29} + ( - \beta_{3} + \beta_{2} + \beta_1 + 6) q^{30} + ( - 2 \beta_1 + 2) q^{31} + q^{32} + ( - \beta_{3} - 2) q^{34} + (2 \beta_{3} - \beta_1 - 2) q^{35} + ( - \beta_{3} + \beta_{2} + 4) q^{36} + (2 \beta_{3} + \beta_1 + 3) q^{37} + q^{38} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{39} - \beta_1 q^{40} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{41} + (3 \beta_{3} - \beta_1 - 1) q^{42} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{43} + ( - 2 \beta_{2} - 6 \beta_1 + 2) q^{45} + (\beta_{2} - 1) q^{46} + (\beta_{3} - \beta_{2} - 3) q^{47} + ( - \beta_1 + 1) q^{48} + (\beta_{2} - 4 \beta_1 + 8) q^{49} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{50} + ( - 3 \beta_{3} + 2 \beta_1) q^{51} - \beta_{2} q^{52} + ( - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{53} + ( - \beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{54} + (\beta_{3} + 1) q^{56} + ( - \beta_1 + 1) q^{57} + (\beta_{3} - \beta_{2} - \beta_1 - 3) q^{58} + ( - \beta_{2} - \beta_1 + 1) q^{59} + ( - \beta_{3} + \beta_{2} + \beta_1 + 6) q^{60} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{61} + ( - 2 \beta_1 + 2) q^{62} + (5 \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{63} + q^{64} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{65} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{67} + ( - \beta_{3} - 2) q^{68} + (2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{69} + (2 \beta_{3} - \beta_1 - 2) q^{70} + (\beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{71} + ( - \beta_{3} + \beta_{2} + 4) q^{72} + (\beta_{3} - 2) q^{73} + (2 \beta_{3} + \beta_1 + 3) q^{74} + (\beta_{3} - 3 \beta_{2} - 5 \beta_1 - 9) q^{75} + q^{76} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{78} + (\beta_{3} + \beta_{2} - \beta_1 + 6) q^{79} - \beta_1 q^{80} + ( - 5 \beta_{3} + \beta_{2} + 2 \beta_1 + 11) q^{81} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{82} + (\beta_1 - 6) q^{83} + (3 \beta_{3} - \beta_1 - 1) q^{84} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{85} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{86} + (2 \beta_{2} + 6 \beta_1 + 1) q^{87} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{89} + ( - 2 \beta_{2} - 6 \beta_1 + 2) q^{90} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{91} + (\beta_{2} - 1) q^{92} + ( - 2 \beta_{3} + 2 \beta_{2} + 14) q^{93} + (\beta_{3} - \beta_{2} - 3) q^{94} - \beta_1 q^{95} + ( - \beta_1 + 1) q^{96} - 10 q^{97} + (\beta_{2} - 4 \beta_1 + 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} + 2 q^{6} + 3 q^{7} + 4 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} + 2 q^{6} + 3 q^{7} + 4 q^{8} + 16 q^{9} - 2 q^{10} + 2 q^{12} + q^{13} + 3 q^{14} + 26 q^{15} + 4 q^{16} - 7 q^{17} + 16 q^{18} + 4 q^{19} - 2 q^{20} - 9 q^{21} - 5 q^{23} + 2 q^{24} + 8 q^{25} + q^{26} + 8 q^{27} + 3 q^{28} - 14 q^{29} + 26 q^{30} + 4 q^{31} + 4 q^{32} - 7 q^{34} - 12 q^{35} + 16 q^{36} + 12 q^{37} + 4 q^{38} + 5 q^{39} - 2 q^{40} + 12 q^{41} - 9 q^{42} + 14 q^{43} - 2 q^{45} - 5 q^{46} - 12 q^{47} + 2 q^{48} + 23 q^{49} + 8 q^{50} + 7 q^{51} + q^{52} - 10 q^{53} + 8 q^{54} + 3 q^{56} + 2 q^{57} - 14 q^{58} + 3 q^{59} + 26 q^{60} + 6 q^{61} + 4 q^{62} - 18 q^{63} + 4 q^{64} + 4 q^{65} + 15 q^{67} - 7 q^{68} - 7 q^{69} - 12 q^{70} - 16 q^{71} + 16 q^{72} - 9 q^{73} + 12 q^{74} - 44 q^{75} + 4 q^{76} + 5 q^{78} + 20 q^{79} - 2 q^{80} + 52 q^{81} + 12 q^{82} - 22 q^{83} - 9 q^{84} + 14 q^{85} + 14 q^{86} + 14 q^{87} - 8 q^{89} - 2 q^{90} - 18 q^{91} - 5 q^{92} + 56 q^{93} - 12 q^{94} - 2 q^{95} + 2 q^{96} - 40 q^{97} + 23 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 12x^{2} + 6x + 24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 8\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 4\nu^{2} - 4\nu + 14 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 2\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 4\beta_{2} + 12\beta _1 + 10 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
4.19461
1.63927
−1.48761
−2.34628
1.00000 −3.19461 1.00000 −4.19461 −3.19461 1.32286 1.00000 7.20555 −4.19461
1.2 1.00000 −0.639273 1.00000 −1.63927 −0.639273 1.54956 1.00000 −2.59133 −1.63927
1.3 1.00000 2.48761 1.00000 1.48761 2.48761 4.90325 1.00000 3.18819 1.48761
1.4 1.00000 3.34628 1.00000 2.34628 3.34628 −4.77567 1.00000 8.19759 2.34628
\(n\): 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.bv yes 4
11.b odd 2 1 4598.2.a.bs 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.bs 4 11.b odd 2 1
4598.2.a.bv 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{4} - 2T_{3}^{3} - 12T_{3}^{2} + 20T_{3} + 17 \) Copy content Toggle raw display
\( T_{5}^{4} + 2T_{5}^{3} - 12T_{5}^{2} - 6T_{5} + 24 \) Copy content Toggle raw display
\( T_{7}^{4} - 3T_{7}^{3} - 21T_{7}^{2} + 67T_{7} - 48 \) Copy content Toggle raw display
\( T_{13}^{4} - T_{13}^{3} - 30T_{13}^{2} - 20T_{13} + 104 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} - 12 T^{2} + 20 T + 17 \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} - 12 T^{2} - 6 T + 24 \) Copy content Toggle raw display
$7$ \( T^{4} - 3 T^{3} - 21 T^{2} + 67 T - 48 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - T^{3} - 30 T^{2} - 20 T + 104 \) Copy content Toggle raw display
$17$ \( T^{4} + 7 T^{3} - 6 T^{2} - 96 T - 132 \) Copy content Toggle raw display
$19$ \( (T - 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 5 T^{3} - 21 T^{2} - 33 T + 96 \) Copy content Toggle raw display
$29$ \( T^{4} + 14 T^{3} + 30 T^{2} - 84 T - 69 \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} - 48 T^{2} + 160 T + 272 \) Copy content Toggle raw display
$37$ \( T^{4} - 12 T^{3} - 78 T^{2} + \cdots - 4569 \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} - 18 T^{2} + \cdots - 972 \) Copy content Toggle raw display
$43$ \( T^{4} - 14 T^{3} - 120 T^{2} + \cdots - 1376 \) Copy content Toggle raw display
$47$ \( T^{4} + 12 T^{3} + 18 T^{2} + \cdots - 351 \) Copy content Toggle raw display
$53$ \( T^{4} + 10 T^{3} - 6 T^{2} - 204 T - 213 \) Copy content Toggle raw display
$59$ \( T^{4} - 3 T^{3} - 45 T^{2} + 153 T + 108 \) Copy content Toggle raw display
$61$ \( T^{4} - 6 T^{3} - 132 T^{2} + \cdots + 2412 \) Copy content Toggle raw display
$67$ \( T^{4} - 15 T^{3} - 84 T^{2} + \cdots - 7872 \) Copy content Toggle raw display
$71$ \( T^{4} + 16 T^{3} - 54 T^{2} + \cdots - 2916 \) Copy content Toggle raw display
$73$ \( T^{4} + 9 T^{3} + 6 T^{2} - 32 T - 36 \) Copy content Toggle raw display
$79$ \( T^{4} - 20 T^{3} + 78 T^{2} + 34 T - 92 \) Copy content Toggle raw display
$83$ \( T^{4} + 22 T^{3} + 168 T^{2} + \cdots + 492 \) Copy content Toggle raw display
$89$ \( T^{4} + 8 T^{3} - 78 T^{2} + \cdots + 1452 \) Copy content Toggle raw display
$97$ \( (T + 10)^{4} \) Copy content Toggle raw display
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