Properties

Label 430.2.a.f
Level 430
Weight 2
Character orbit 430.a
Self dual yes
Analytic conductor 3.434
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−2.44949
2.44949
1.00000 −2.44949 1.00000 −1.00000 −2.44949 1.00000 1.00000 3.00000 −1.00000
1.2 1.00000 2.44949 1.00000 −1.00000 2.44949 1.00000 1.00000 3.00000 −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.f 2
3.b odd 2 1 3870.2.a.bg 2
4.b odd 2 1 3440.2.a.h 2
5.b even 2 1 2150.2.a.w 2
5.c odd 4 2 2150.2.b.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.a.f 2 1.a even 1 1 trivial
2150.2.a.w 2 5.b even 2 1
2150.2.b.m 4 5.c odd 4 2
3440.2.a.h 2 4.b odd 2 1
3870.2.a.bg 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(430))\):

\( T_{3}^{2} - 6 \)
\( T_{7} - 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( 1 + 9 T^{4} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 - T + 7 T^{2} )^{2} \)
$11$ \( 1 - 4 T + 20 T^{2} - 44 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + T + 13 T^{2} )^{2} \)
$17$ \( 1 + 28 T^{2} + 289 T^{4} \)
$19$ \( 1 - 2 T + 15 T^{2} - 38 T^{3} + 361 T^{4} \)
$23$ \( 1 - 4 T + 44 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( 1 - 14 T + 101 T^{2} - 406 T^{3} + 841 T^{4} \)
$31$ \( 1 + 6 T + 65 T^{2} + 186 T^{3} + 961 T^{4} \)
$37$ \( 1 + 4 T + 72 T^{2} + 148 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 14 T + 107 T^{2} + 574 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( 1 + 88 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 8 T + 26 T^{2} - 424 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 94 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 2 T + 117 T^{2} + 122 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 6 T + 137 T^{2} + 402 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 12 T + 124 T^{2} - 852 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 10 T + 165 T^{2} + 730 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 18 T + 233 T^{2} - 1422 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 12 T + 64 T^{2} - 1068 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 28 T + 384 T^{2} + 2716 T^{3} + 9409 T^{4} \)
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