Properties

Label 430.2.a.h
Level 430
Weight 2
Character orbit 430.a
Self dual yes
Analytic conductor 3.434
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 430.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + q^{4} - q^{5} + ( 1 - \beta_{1} + \beta_{2} ) q^{6} + ( -2 - \beta_{2} ) q^{7} - q^{8} + ( 3 - 2 \beta_{1} ) q^{9} +O(q^{10})\) \( q - q^{2} + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + q^{4} - q^{5} + ( 1 - \beta_{1} + \beta_{2} ) q^{6} + ( -2 - \beta_{2} ) q^{7} - q^{8} + ( 3 - 2 \beta_{1} ) q^{9} + q^{10} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} ) q^{12} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{13} + ( 2 + \beta_{2} ) q^{14} + ( 1 - \beta_{1} + \beta_{2} ) q^{15} + q^{16} + ( 3 - \beta_{1} - \beta_{2} ) q^{17} + ( -3 + 2 \beta_{1} ) q^{18} + ( 2 \beta_{1} - \beta_{2} ) q^{19} - q^{20} + ( 5 - \beta_{1} + \beta_{2} ) q^{21} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{22} + ( 5 - \beta_{1} - \beta_{2} ) q^{23} + ( 1 - \beta_{1} + \beta_{2} ) q^{24} + q^{25} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{26} + ( -4 + 4 \beta_{1} ) q^{27} + ( -2 - \beta_{2} ) q^{28} + ( 3 - 3 \beta_{1} + 4 \beta_{2} ) q^{29} + ( -1 + \beta_{1} - \beta_{2} ) q^{30} + ( 1 - 3 \beta_{1} ) q^{31} - q^{32} + ( 8 - 4 \beta_{1} - 2 \beta_{2} ) q^{33} + ( -3 + \beta_{1} + \beta_{2} ) q^{34} + ( 2 + \beta_{2} ) q^{35} + ( 3 - 2 \beta_{1} ) q^{36} + ( -3 + \beta_{1} + 3 \beta_{2} ) q^{37} + ( -2 \beta_{1} + \beta_{2} ) q^{38} + ( 3 - 7 \beta_{1} + 3 \beta_{2} ) q^{39} + q^{40} + ( 4 + 4 \beta_{1} - \beta_{2} ) q^{41} + ( -5 + \beta_{1} - \beta_{2} ) q^{42} + q^{43} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{44} + ( -3 + 2 \beta_{1} ) q^{45} + ( -5 + \beta_{1} + \beta_{2} ) q^{46} + ( -1 + \beta_{1} - \beta_{2} ) q^{47} + ( -1 + \beta_{1} - \beta_{2} ) q^{48} + ( 1 + 2 \beta_{1} + 3 \beta_{2} ) q^{49} - q^{50} + ( -2 + 6 \beta_{1} - 4 \beta_{2} ) q^{51} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{52} + ( 8 - 2 \beta_{1} + 4 \beta_{2} ) q^{53} + ( 4 - 4 \beta_{1} ) q^{54} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{55} + ( 2 + \beta_{2} ) q^{56} + ( 7 - 3 \beta_{1} - \beta_{2} ) q^{57} + ( -3 + 3 \beta_{1} - 4 \beta_{2} ) q^{58} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( 1 - \beta_{1} + \beta_{2} ) q^{60} + ( -1 + \beta_{1} - 4 \beta_{2} ) q^{61} + ( -1 + 3 \beta_{1} ) q^{62} + ( -4 + 6 \beta_{1} - \beta_{2} ) q^{63} + q^{64} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{65} + ( -8 + 4 \beta_{1} + 2 \beta_{2} ) q^{66} + ( -7 - 3 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 3 - \beta_{1} - \beta_{2} ) q^{68} + ( -4 + 8 \beta_{1} - 6 \beta_{2} ) q^{69} + ( -2 - \beta_{2} ) q^{70} + ( -3 + 3 \beta_{1} - \beta_{2} ) q^{71} + ( -3 + 2 \beta_{1} ) q^{72} + ( -1 - 7 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{74} + ( -1 + \beta_{1} - \beta_{2} ) q^{75} + ( 2 \beta_{1} - \beta_{2} ) q^{76} + ( -1 - 7 \beta_{1} - 3 \beta_{2} ) q^{77} + ( -3 + 7 \beta_{1} - 3 \beta_{2} ) q^{78} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{79} - q^{80} + ( 3 - 6 \beta_{1} + 4 \beta_{2} ) q^{81} + ( -4 - 4 \beta_{1} + \beta_{2} ) q^{82} + ( 2 - 2 \beta_{1} + 4 \beta_{2} ) q^{83} + ( 5 - \beta_{1} + \beta_{2} ) q^{84} + ( -3 + \beta_{1} + \beta_{2} ) q^{85} - q^{86} + ( -21 + 5 \beta_{1} + \beta_{2} ) q^{87} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{88} + ( -7 + 5 \beta_{1} + 3 \beta_{2} ) q^{89} + ( 3 - 2 \beta_{1} ) q^{90} + ( -2 - 8 \beta_{1} - \beta_{2} ) q^{91} + ( 5 - \beta_{1} - \beta_{2} ) q^{92} + ( -7 + 7 \beta_{1} - \beta_{2} ) q^{93} + ( 1 - \beta_{1} + \beta_{2} ) q^{94} + ( -2 \beta_{1} + \beta_{2} ) q^{95} + ( 1 - \beta_{1} + \beta_{2} ) q^{96} + ( -3 + \beta_{1} + 3 \beta_{2} ) q^{97} + ( -1 - 2 \beta_{1} - 3 \beta_{2} ) q^{98} + ( -13 + 9 \beta_{1} - 7 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} - 2q^{3} + 3q^{4} - 3q^{5} + 2q^{6} - 6q^{7} - 3q^{8} + 7q^{9} + O(q^{10}) \) \( 3q - 3q^{2} - 2q^{3} + 3q^{4} - 3q^{5} + 2q^{6} - 6q^{7} - 3q^{8} + 7q^{9} + 3q^{10} + 6q^{11} - 2q^{12} - 4q^{13} + 6q^{14} + 2q^{15} + 3q^{16} + 8q^{17} - 7q^{18} + 2q^{19} - 3q^{20} + 14q^{21} - 6q^{22} + 14q^{23} + 2q^{24} + 3q^{25} + 4q^{26} - 8q^{27} - 6q^{28} + 6q^{29} - 2q^{30} - 3q^{32} + 20q^{33} - 8q^{34} + 6q^{35} + 7q^{36} - 8q^{37} - 2q^{38} + 2q^{39} + 3q^{40} + 16q^{41} - 14q^{42} + 3q^{43} + 6q^{44} - 7q^{45} - 14q^{46} - 2q^{47} - 2q^{48} + 5q^{49} - 3q^{50} - 4q^{52} + 22q^{53} + 8q^{54} - 6q^{55} + 6q^{56} + 18q^{57} - 6q^{58} + 16q^{59} + 2q^{60} - 2q^{61} - 6q^{63} + 3q^{64} + 4q^{65} - 20q^{66} - 24q^{67} + 8q^{68} - 4q^{69} - 6q^{70} - 6q^{71} - 7q^{72} - 10q^{73} + 8q^{74} - 2q^{75} + 2q^{76} - 10q^{77} - 2q^{78} - 4q^{79} - 3q^{80} + 3q^{81} - 16q^{82} + 4q^{83} + 14q^{84} - 8q^{85} - 3q^{86} - 58q^{87} - 6q^{88} - 16q^{89} + 7q^{90} - 14q^{91} + 14q^{92} - 14q^{93} + 2q^{94} - 2q^{95} + 2q^{96} - 8q^{97} - 5q^{98} - 30q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
−1.81361
2.34292
0.470683
−1.00000 −3.10278 1.00000 −1.00000 3.10278 −2.28917 −1.00000 6.62721 1.00000
1.2 −1.00000 −1.14637 1.00000 −1.00000 1.14637 −4.48929 −1.00000 −1.68585 1.00000
1.3 −1.00000 2.24914 1.00000 −1.00000 −2.24914 0.778457 −1.00000 2.05863 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 430.2.a.h 3
3.b odd 2 1 3870.2.a.bn 3
4.b odd 2 1 3440.2.a.n 3
5.b even 2 1 2150.2.a.bf 3
5.c odd 4 2 2150.2.b.t 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.a.h 3 1.a even 1 1 trivial
2150.2.a.bf 3 5.b even 2 1
2150.2.b.t 6 5.c odd 4 2
3440.2.a.n 3 4.b odd 2 1
3870.2.a.bn 3 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(43\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(430))\):

\( T_{3}^{3} + 2 T_{3}^{2} - 6 T_{3} - 8 \)
\( T_{7}^{3} + 6 T_{7}^{2} + 5 T_{7} - 8 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( 1 + 2 T + 3 T^{2} + 4 T^{3} + 9 T^{4} + 18 T^{5} + 27 T^{6} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( 1 + 6 T + 26 T^{2} + 76 T^{3} + 182 T^{4} + 294 T^{5} + 343 T^{6} \)
$11$ \( 1 - 6 T + 11 T^{2} + 4 T^{3} + 121 T^{4} - 726 T^{5} + 1331 T^{6} \)
$13$ \( 1 + 4 T + 12 T^{2} - 2 T^{3} + 156 T^{4} + 676 T^{5} + 2197 T^{6} \)
$17$ \( 1 - 8 T + 57 T^{2} - 228 T^{3} + 969 T^{4} - 2312 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 2 T + 42 T^{2} - 44 T^{3} + 798 T^{4} - 722 T^{5} + 6859 T^{6} \)
$23$ \( 1 - 14 T + 119 T^{2} - 652 T^{3} + 2737 T^{4} - 7406 T^{5} + 12167 T^{6} \)
$29$ \( 1 - 6 T - 4 T^{2} + 194 T^{3} - 116 T^{4} - 5046 T^{5} + 24389 T^{6} \)
$31$ \( 1 + 54 T^{2} - 16 T^{3} + 1674 T^{4} + 29791 T^{6} \)
$37$ \( 1 + 8 T + 53 T^{2} + 300 T^{3} + 1961 T^{4} + 10952 T^{5} + 50653 T^{6} \)
$41$ \( 1 - 16 T + 148 T^{2} - 978 T^{3} + 6068 T^{4} - 26896 T^{5} + 68921 T^{6} \)
$43$ \( ( 1 - T )^{3} \)
$47$ \( 1 + 2 T + 135 T^{2} + 180 T^{3} + 6345 T^{4} + 4418 T^{5} + 103823 T^{6} \)
$53$ \( 1 - 22 T + 223 T^{2} - 1644 T^{3} + 11819 T^{4} - 61798 T^{5} + 148877 T^{6} \)
$59$ \( 1 - 16 T + 201 T^{2} - 1536 T^{3} + 11859 T^{4} - 55696 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 2 T + 84 T^{2} - 118 T^{3} + 5124 T^{4} + 7442 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 24 T + 350 T^{2} + 3340 T^{3} + 23450 T^{4} + 107736 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 6 T + 191 T^{2} + 868 T^{3} + 13561 T^{4} + 30246 T^{5} + 357911 T^{6} \)
$73$ \( 1 + 10 T + 68 T^{2} - 42 T^{3} + 4964 T^{4} + 53290 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 4 T + 202 T^{2} + 640 T^{3} + 15958 T^{4} + 24964 T^{5} + 493039 T^{6} \)
$83$ \( 1 - 4 T + 157 T^{2} - 168 T^{3} + 13031 T^{4} - 27556 T^{5} + 571787 T^{6} \)
$89$ \( 1 + 16 T + 121 T^{2} + 444 T^{3} + 10769 T^{4} + 126736 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 8 T + 233 T^{2} + 1260 T^{3} + 22601 T^{4} + 75272 T^{5} + 912673 T^{6} \)
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