Properties

Label 230.4.a.b
Level $230$
Weight $4$
Character orbit 230.a
Self dual yes
Analytic conductor $13.570$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.5704393013\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 4q^{3} + 4q^{4} - 5q^{5} - 8q^{6} + 3q^{7} - 8q^{8} - 11q^{9} + O(q^{10}) \) \( q - 2q^{2} + 4q^{3} + 4q^{4} - 5q^{5} - 8q^{6} + 3q^{7} - 8q^{8} - 11q^{9} + 10q^{10} - 2q^{11} + 16q^{12} - 38q^{13} - 6q^{14} - 20q^{15} + 16q^{16} - 45q^{17} + 22q^{18} - 74q^{19} - 20q^{20} + 12q^{21} + 4q^{22} + 23q^{23} - 32q^{24} + 25q^{25} + 76q^{26} - 152q^{27} + 12q^{28} + 283q^{29} + 40q^{30} - 303q^{31} - 32q^{32} - 8q^{33} + 90q^{34} - 15q^{35} - 44q^{36} + 79q^{37} + 148q^{38} - 152q^{39} + 40q^{40} - 407q^{41} - 24q^{42} - 328q^{43} - 8q^{44} + 55q^{45} - 46q^{46} + 360q^{47} + 64q^{48} - 334q^{49} - 50q^{50} - 180q^{51} - 152q^{52} - 561q^{53} + 304q^{54} + 10q^{55} - 24q^{56} - 296q^{57} - 566q^{58} + 101q^{59} - 80q^{60} - 268q^{61} + 606q^{62} - 33q^{63} + 64q^{64} + 190q^{65} + 16q^{66} - 69q^{67} - 180q^{68} + 92q^{69} + 30q^{70} - 641q^{71} + 88q^{72} + 994q^{73} - 158q^{74} + 100q^{75} - 296q^{76} - 6q^{77} + 304q^{78} - 884q^{79} - 80q^{80} - 311q^{81} + 814q^{82} + 503q^{83} + 48q^{84} + 225q^{85} + 656q^{86} + 1132q^{87} + 16q^{88} + 1608q^{89} - 110q^{90} - 114q^{91} + 92q^{92} - 1212q^{93} - 720q^{94} + 370q^{95} - 128q^{96} + 1082q^{97} + 668q^{98} + 22q^{99} + O(q^{100}) \)

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
−2.00000 4.00000 4.00000 −5.00000 −8.00000 3.00000 −8.00000 −11.0000 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(23\) \(-1\)

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 230.4.a.b 1
3.b odd 2 1 2070.4.a.n 1
4.b odd 2 1 1840.4.a.b 1
5.b even 2 1 1150.4.a.f 1
5.c odd 4 2 1150.4.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.b 1 1.a even 1 1 trivial
1150.4.a.f 1 5.b even 2 1
1150.4.b.c 2 5.c odd 4 2
1840.4.a.b 1 4.b odd 2 1
2070.4.a.n 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 4 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(230))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( -4 + T \)
$5$ \( 5 + T \)
$7$ \( -3 + T \)
$11$ \( 2 + T \)
$13$ \( 38 + T \)
$17$ \( 45 + T \)
$19$ \( 74 + T \)
$23$ \( -23 + T \)
$29$ \( -283 + T \)
$31$ \( 303 + T \)
$37$ \( -79 + T \)
$41$ \( 407 + T \)
$43$ \( 328 + T \)
$47$ \( -360 + T \)
$53$ \( 561 + T \)
$59$ \( -101 + T \)
$61$ \( 268 + T \)
$67$ \( 69 + T \)
$71$ \( 641 + T \)
$73$ \( -994 + T \)
$79$ \( 884 + T \)
$83$ \( -503 + T \)
$89$ \( -1608 + T \)
$97$ \( -1082 + T \)
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