Properties

Label 430.2.e.c
Level 430
Weight 2
Character orbit 430.e
Analytic conductor 3.434
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 430.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/430\mathbb{Z}\right)^\times\).

\(n\) \(87\) \(261\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 0 1.00000 0.500000 0.866025i 0 −2.00000 3.46410i 1.00000 1.50000 + 2.59808i 0.500000 0.866025i
251.1 1.00000 0 1.00000 0.500000 + 0.866025i 0 −2.00000 + 3.46410i 1.00000 1.50000 2.59808i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.e.c 2
43.c even 3 1 inner 430.2.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.e.c 2 1.a even 1 1 trivial
430.2.e.c 2 43.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \)
$5$ \( 1 - T + T^{2} \)
$7$ \( ( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \)
$11$ \( ( 1 - 5 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( 1 - 8 T + 35 T^{2} - 232 T^{3} + 841 T^{4} \)
$31$ \( 1 - 31 T^{2} + 961 T^{4} \)
$37$ \( ( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \)
$41$ \( ( 1 + 7 T + 41 T^{2} )^{2} \)
$43$ \( 1 + 5 T + 43 T^{2} \)
$47$ \( ( 1 - 6 T + 47 T^{2} )^{2} \)
$53$ \( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} \)
$59$ \( ( 1 + 3 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \)
$67$ \( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 6 T - 35 T^{2} + 426 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 6 T - 43 T^{2} - 474 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 7 T + 97 T^{2} )^{2} \)
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