Properties

Label 4598.2.a.bs
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: \(x^{4} - 2 x^{3} - 12 x^{2} + 6 x + 24\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 2 \nu^{2} - 8 \nu + 2 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 4 \nu^{2} - 4 \nu + 14 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + 2 \beta_{1} + 6\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + 4 \beta_{2} + 12 \beta_{1} + 10\)

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.bs 4
11.b odd 2 1 4598.2.a.bv yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.bs 4 1.a even 1 1 trivial
4598.2.a.bv yes 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( 17 + 20 T - 12 T^{2} - 2 T^{3} + T^{4} \)
$5$ \( 24 - 6 T - 12 T^{2} + 2 T^{3} + T^{4} \)
$7$ \( -48 - 67 T - 21 T^{2} + 3 T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 104 + 20 T - 30 T^{2} + T^{3} + T^{4} \)
$17$ \( -132 + 96 T - 6 T^{2} - 7 T^{3} + T^{4} \)
$19$ \( ( 1 + T )^{4} \)
$23$ \( 96 - 33 T - 21 T^{2} + 5 T^{3} + T^{4} \)
$29$ \( -69 + 84 T + 30 T^{2} - 14 T^{3} + T^{4} \)
$31$ \( 272 + 160 T - 48 T^{2} - 4 T^{3} + T^{4} \)
$37$ \( -4569 + 1450 T - 78 T^{2} - 12 T^{3} + T^{4} \)
$41$ \( -972 - 510 T - 18 T^{2} + 12 T^{3} + T^{4} \)
$43$ \( -1376 - 1840 T - 120 T^{2} + 14 T^{3} + T^{4} \)
$47$ \( -351 - 168 T + 18 T^{2} + 12 T^{3} + T^{4} \)
$53$ \( -213 - 204 T - 6 T^{2} + 10 T^{3} + T^{4} \)
$59$ \( 108 + 153 T - 45 T^{2} - 3 T^{3} + T^{4} \)
$61$ \( 2412 - 226 T - 132 T^{2} + 6 T^{3} + T^{4} \)
$67$ \( -7872 + 2128 T - 84 T^{2} - 15 T^{3} + T^{4} \)
$71$ \( -2916 - 1458 T - 54 T^{2} + 16 T^{3} + T^{4} \)
$73$ \( -36 + 32 T + 6 T^{2} - 9 T^{3} + T^{4} \)
$79$ \( -92 - 34 T + 78 T^{2} + 20 T^{3} + T^{4} \)
$83$ \( 492 - 510 T + 168 T^{2} - 22 T^{3} + T^{4} \)
$89$ \( 1452 - 282 T - 78 T^{2} + 8 T^{3} + T^{4} \)
$97$ \( ( 10 + T )^{4} \)
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