Properties

Label 430.2.a.e
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: \(\Q(\sqrt{3}) \)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
−1.00000 −0.732051 1.00000 1.00000 0.732051 4.46410 −1.00000 −2.46410 −1.00000
1.2 −1.00000 2.73205 1.00000 1.00000 −2.73205 −2.46410 −1.00000 4.46410 −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.e 2
3.b odd 2 1 3870.2.a.bk 2
4.b odd 2 1 3440.2.a.g 2
5.b even 2 1 2150.2.a.x 2
5.c odd 4 2 2150.2.b.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.a.e 2 1.a even 1 1 trivial
2150.2.a.x 2 5.b even 2 1
2150.2.b.l 4 5.c odd 4 2
3440.2.a.g 2 4.b odd 2 1
3870.2.a.bk 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(430))\):

\( T_{3}^{2} - 2 T_{3} - 2 \)
\( T_{7}^{2} - 2 T_{7} - 11 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( 1 - 2 T + 4 T^{2} - 6 T^{3} + 9 T^{4} \)
$5$ \( ( 1 - T )^{2} \)
$7$ \( 1 - 2 T + 3 T^{2} - 14 T^{3} + 49 T^{4} \)
$11$ \( 1 - 2 T + 20 T^{2} - 22 T^{3} + 121 T^{4} \)
$13$ \( 1 - 2 T + 15 T^{2} - 26 T^{3} + 169 T^{4} \)
$17$ \( 1 - 10 T + 56 T^{2} - 170 T^{3} + 289 T^{4} \)
$19$ \( 1 + 6 T + 35 T^{2} + 114 T^{3} + 361 T^{4} \)
$23$ \( 1 - 6 T + 28 T^{2} - 138 T^{3} + 529 T^{4} \)
$29$ \( 1 - 16 T + 119 T^{2} - 464 T^{3} + 841 T^{4} \)
$31$ \( 1 + 16 T + 123 T^{2} + 496 T^{3} + 961 T^{4} \)
$37$ \( 1 + 2 T + 48 T^{2} + 74 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 2 T + 71 T^{2} + 82 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( 1 - 14 T + 140 T^{2} - 658 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 4 T + 98 T^{2} + 212 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 20 T + 206 T^{2} + 1180 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 119 T^{2} + 3721 T^{4} \)
$67$ \( 1 + 107 T^{2} + 4489 T^{4} \)
$71$ \( 1 + 10 T + 164 T^{2} + 710 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 8 T + 135 T^{2} - 584 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 4 T + 15 T^{2} + 316 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 154 T^{2} + 6889 T^{4} \)
$89$ \( 1 + 6 T + 40 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 6 T - 40 T^{2} + 582 T^{3} + 9409 T^{4} \)
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