Properties

Label 230.4.a.f
Level $230$
Weight $4$
Character orbit 230.a
Self dual yes
Analytic conductor $13.570$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
Defining polynomial: \(x^{2} - x - 18\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{73})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{2} + ( -1 - \beta ) q^{3} + 4 q^{4} + 5 q^{5} + ( 2 + 2 \beta ) q^{6} + ( -8 - \beta ) q^{7} -8 q^{8} + ( -8 + 3 \beta ) q^{9} +O(q^{10})\) \( q -2 q^{2} + ( -1 - \beta ) q^{3} + 4 q^{4} + 5 q^{5} + ( 2 + 2 \beta ) q^{6} + ( -8 - \beta ) q^{7} -8 q^{8} + ( -8 + 3 \beta ) q^{9} -10 q^{10} + ( 4 + 10 \beta ) q^{11} + ( -4 - 4 \beta ) q^{12} + ( -9 + 9 \beta ) q^{13} + ( 16 + 2 \beta ) q^{14} + ( -5 - 5 \beta ) q^{15} + 16 q^{16} + ( -34 - 11 \beta ) q^{17} + ( 16 - 6 \beta ) q^{18} + ( 12 + 10 \beta ) q^{19} + 20 q^{20} + ( 26 + 10 \beta ) q^{21} + ( -8 - 20 \beta ) q^{22} -23 q^{23} + ( 8 + 8 \beta ) q^{24} + 25 q^{25} + ( 18 - 18 \beta ) q^{26} + ( -19 + 29 \beta ) q^{27} + ( -32 - 4 \beta ) q^{28} + ( -55 - 2 \beta ) q^{29} + ( 10 + 10 \beta ) q^{30} + ( -33 - 26 \beta ) q^{31} -32 q^{32} + ( -184 - 24 \beta ) q^{33} + ( 68 + 22 \beta ) q^{34} + ( -40 - 5 \beta ) q^{35} + ( -32 + 12 \beta ) q^{36} + ( -242 - 7 \beta ) q^{37} + ( -24 - 20 \beta ) q^{38} + ( -153 - 9 \beta ) q^{39} -40 q^{40} + ( -129 - 74 \beta ) q^{41} + ( -52 - 20 \beta ) q^{42} + ( -166 - 22 \beta ) q^{43} + ( 16 + 40 \beta ) q^{44} + ( -40 + 15 \beta ) q^{45} + 46 q^{46} + ( -297 - 5 \beta ) q^{47} + ( -16 - 16 \beta ) q^{48} + ( -261 + 17 \beta ) q^{49} -50 q^{50} + ( 232 + 56 \beta ) q^{51} + ( -36 + 36 \beta ) q^{52} + ( -156 + 7 \beta ) q^{53} + ( 38 - 58 \beta ) q^{54} + ( 20 + 50 \beta ) q^{55} + ( 64 + 8 \beta ) q^{56} + ( -192 - 32 \beta ) q^{57} + ( 110 + 4 \beta ) q^{58} + ( 126 + 105 \beta ) q^{59} + ( -20 - 20 \beta ) q^{60} + ( -92 + 12 \beta ) q^{61} + ( 66 + 52 \beta ) q^{62} + ( 10 - 19 \beta ) q^{63} + 64 q^{64} + ( -45 + 45 \beta ) q^{65} + ( 368 + 48 \beta ) q^{66} + ( -286 + 41 \beta ) q^{67} + ( -136 - 44 \beta ) q^{68} + ( 23 + 23 \beta ) q^{69} + ( 80 + 10 \beta ) q^{70} + ( 679 - 104 \beta ) q^{71} + ( 64 - 24 \beta ) q^{72} + ( -183 + 23 \beta ) q^{73} + ( 484 + 14 \beta ) q^{74} + ( -25 - 25 \beta ) q^{75} + ( 48 + 40 \beta ) q^{76} + ( -212 - 94 \beta ) q^{77} + ( 306 + 18 \beta ) q^{78} + ( -16 - 56 \beta ) q^{79} + 80 q^{80} + ( -287 - 120 \beta ) q^{81} + ( 258 + 148 \beta ) q^{82} + ( 578 + 117 \beta ) q^{83} + ( 104 + 40 \beta ) q^{84} + ( -170 - 55 \beta ) q^{85} + ( 332 + 44 \beta ) q^{86} + ( 91 + 59 \beta ) q^{87} + ( -32 - 80 \beta ) q^{88} + ( 562 - 18 \beta ) q^{89} + ( 80 - 30 \beta ) q^{90} + ( -90 - 72 \beta ) q^{91} -92 q^{92} + ( 501 + 85 \beta ) q^{93} + ( 594 + 10 \beta ) q^{94} + ( 60 + 50 \beta ) q^{95} + ( 32 + 32 \beta ) q^{96} + ( -1038 - 164 \beta ) q^{97} + ( 522 - 34 \beta ) q^{98} + ( 508 - 38 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 3 q^{3} + 8 q^{4} + 10 q^{5} + 6 q^{6} - 17 q^{7} - 16 q^{8} - 13 q^{9} + O(q^{10}) \) \( 2 q - 4 q^{2} - 3 q^{3} + 8 q^{4} + 10 q^{5} + 6 q^{6} - 17 q^{7} - 16 q^{8} - 13 q^{9} - 20 q^{10} + 18 q^{11} - 12 q^{12} - 9 q^{13} + 34 q^{14} - 15 q^{15} + 32 q^{16} - 79 q^{17} + 26 q^{18} + 34 q^{19} + 40 q^{20} + 62 q^{21} - 36 q^{22} - 46 q^{23} + 24 q^{24} + 50 q^{25} + 18 q^{26} - 9 q^{27} - 68 q^{28} - 112 q^{29} + 30 q^{30} - 92 q^{31} - 64 q^{32} - 392 q^{33} + 158 q^{34} - 85 q^{35} - 52 q^{36} - 491 q^{37} - 68 q^{38} - 315 q^{39} - 80 q^{40} - 332 q^{41} - 124 q^{42} - 354 q^{43} + 72 q^{44} - 65 q^{45} + 92 q^{46} - 599 q^{47} - 48 q^{48} - 505 q^{49} - 100 q^{50} + 520 q^{51} - 36 q^{52} - 305 q^{53} + 18 q^{54} + 90 q^{55} + 136 q^{56} - 416 q^{57} + 224 q^{58} + 357 q^{59} - 60 q^{60} - 172 q^{61} + 184 q^{62} + q^{63} + 128 q^{64} - 45 q^{65} + 784 q^{66} - 531 q^{67} - 316 q^{68} + 69 q^{69} + 170 q^{70} + 1254 q^{71} + 104 q^{72} - 343 q^{73} + 982 q^{74} - 75 q^{75} + 136 q^{76} - 518 q^{77} + 630 q^{78} - 88 q^{79} + 160 q^{80} - 694 q^{81} + 664 q^{82} + 1273 q^{83} + 248 q^{84} - 395 q^{85} + 708 q^{86} + 241 q^{87} - 144 q^{88} + 1106 q^{89} + 130 q^{90} - 252 q^{91} - 184 q^{92} + 1087 q^{93} + 1198 q^{94} + 170 q^{95} + 96 q^{96} - 2240 q^{97} + 1010 q^{98} + 978 q^{99} + O(q^{100}) \)

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
4.77200
−3.77200
−2.00000 −5.77200 4.00000 5.00000 11.5440 −12.7720 −8.00000 6.31601 −10.0000
1.2 −2.00000 2.77200 4.00000 5.00000 −5.54400 −4.22800 −8.00000 −19.3160 −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.f 2
3.b odd 2 1 2070.4.a.s 2
4.b odd 2 1 1840.4.a.i 2
5.b even 2 1 1150.4.a.l 2
5.c odd 4 2 1150.4.b.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.f 2 1.a even 1 1 trivial
1150.4.a.l 2 5.b even 2 1
1150.4.b.k 4 5.c odd 4 2
1840.4.a.i 2 4.b odd 2 1
2070.4.a.s 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 3 T_{3} - 16 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(230))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T )^{2} \)
$3$ \( -16 + 3 T + T^{2} \)
$5$ \( ( -5 + T )^{2} \)
$7$ \( 54 + 17 T + T^{2} \)
$11$ \( -1744 - 18 T + T^{2} \)
$13$ \( -1458 + 9 T + T^{2} \)
$17$ \( -648 + 79 T + T^{2} \)
$19$ \( -1536 - 34 T + T^{2} \)
$23$ \( ( 23 + T )^{2} \)
$29$ \( 3063 + 112 T + T^{2} \)
$31$ \( -10221 + 92 T + T^{2} \)
$37$ \( 59376 + 491 T + T^{2} \)
$41$ \( -72381 + 332 T + T^{2} \)
$43$ \( 22496 + 354 T + T^{2} \)
$47$ \( 89244 + 599 T + T^{2} \)
$53$ \( 22362 + 305 T + T^{2} \)
$59$ \( -169344 - 357 T + T^{2} \)
$61$ \( 4768 + 172 T + T^{2} \)
$67$ \( 39812 + 531 T + T^{2} \)
$71$ \( 195737 - 1254 T + T^{2} \)
$73$ \( 19758 + 343 T + T^{2} \)
$79$ \( -55296 + 88 T + T^{2} \)
$83$ \( 155308 - 1273 T + T^{2} \)
$89$ \( 299896 - 1106 T + T^{2} \)
$97$ \( 763548 + 2240 T + T^{2} \)
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