Properties

Label 4598.2.a.a
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
Analytic rank $1$
Dimension $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
0
−1.00000 −1.00000 1.00000 −4.00000 1.00000 −3.00000 −1.00000 −2.00000 4.00000
\(n\): 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.a 1
11.b odd 2 1 38.2.a.b 1
33.d even 2 1 342.2.a.d 1
44.c even 2 1 304.2.a.d 1
55.d odd 2 1 950.2.a.b 1
55.e even 4 2 950.2.b.c 2
77.b even 2 1 1862.2.a.f 1
88.b odd 2 1 1216.2.a.n 1
88.g even 2 1 1216.2.a.g 1
132.d odd 2 1 2736.2.a.w 1
143.d odd 2 1 6422.2.a.b 1
165.d even 2 1 8550.2.a.u 1
209.d even 2 1 722.2.a.b 1
209.g even 6 2 722.2.c.f 2
209.h odd 6 2 722.2.c.d 2
209.p even 18 6 722.2.e.d 6
209.q odd 18 6 722.2.e.c 6
220.g even 2 1 7600.2.a.h 1
627.b odd 2 1 6498.2.a.y 1
836.h odd 2 1 5776.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.b 1 11.b odd 2 1
304.2.a.d 1 44.c even 2 1
342.2.a.d 1 33.d even 2 1
722.2.a.b 1 209.d even 2 1
722.2.c.d 2 209.h odd 6 2
722.2.c.f 2 209.g even 6 2
722.2.e.c 6 209.q odd 18 6
722.2.e.d 6 209.p even 18 6
950.2.a.b 1 55.d odd 2 1
950.2.b.c 2 55.e even 4 2
1216.2.a.g 1 88.g even 2 1
1216.2.a.n 1 88.b odd 2 1
1862.2.a.f 1 77.b even 2 1
2736.2.a.w 1 132.d odd 2 1
4598.2.a.a 1 1.a even 1 1 trivial
5776.2.a.d 1 836.h odd 2 1
6422.2.a.b 1 143.d odd 2 1
6498.2.a.y 1 627.b odd 2 1
7600.2.a.h 1 220.g even 2 1
8550.2.a.u 1 165.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3} + 1 \)
\( T_{5} + 4 \)
\( T_{7} + 3 \)
\( T_{13} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T \)
$5$ \( 4 + T \)
$7$ \( 3 + T \)
$11$ \( T \)
$13$ \( -1 + T \)
$17$ \( 3 + T \)
$19$ \( -1 + T \)
$23$ \( 1 + T \)
$29$ \( -5 + T \)
$31$ \( 8 + T \)
$37$ \( 2 + T \)
$41$ \( -8 + T \)
$43$ \( 4 + T \)
$47$ \( -8 + T \)
$53$ \( 1 + T \)
$59$ \( -15 + T \)
$61$ \( 2 + T \)
$67$ \( -3 + T \)
$71$ \( -2 + T \)
$73$ \( 9 + T \)
$79$ \( -10 + T \)
$83$ \( -6 + T \)
$89$ \( T \)
$97$ \( 2 + T \)
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