Properties

Label 430.2.g.a
Level 430
Weight 2
Character orbit 430.g
Analytic conductor 3.434
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 430.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(87\) \(261\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i −1.41421 1.41421i 1.00000i −0.707107 2.12132i 2.00000 −2.82843 + 2.82843i 0.707107 + 0.707107i 1.00000i 2.00000 + 1.00000i
257.2 0.707107 0.707107i 1.41421 + 1.41421i 1.00000i 0.707107 + 2.12132i 2.00000 2.82843 2.82843i −0.707107 0.707107i 1.00000i 2.00000 + 1.00000i
343.1 −0.707107 0.707107i −1.41421 + 1.41421i 1.00000i −0.707107 + 2.12132i 2.00000 −2.82843 2.82843i 0.707107 0.707107i 1.00000i 2.00000 1.00000i
343.2 0.707107 + 0.707107i 1.41421 1.41421i 1.00000i 0.707107 2.12132i 2.00000 2.82843 + 2.82843i −0.707107 + 0.707107i 1.00000i 2.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
43.b odd 2 1 inner
215.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.g.a 4
5.c odd 4 1 inner 430.2.g.a 4
43.b odd 2 1 inner 430.2.g.a 4
215.g even 4 1 inner 430.2.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.g.a 4 1.a even 1 1 trivial
430.2.g.a 4 5.c odd 4 1 inner
430.2.g.a 4 43.b odd 2 1 inner
430.2.g.a 4 215.g even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 16 \) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} \)
$3$ \( ( 1 - 4 T + 8 T^{2} - 12 T^{3} + 9 T^{4} )( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4} ) \)
$5$ \( 1 + 8 T^{2} + 25 T^{4} \)
$7$ \( 1 - 94 T^{4} + 2401 T^{8} \)
$11$ \( ( 1 + 4 T + 11 T^{2} )^{4} \)
$13$ \( ( 1 - 4 T + 8 T^{2} - 52 T^{3} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 8 T + 17 T^{2} )^{2}( 1 - 2 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 20 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 10 T + 50 T^{2} + 230 T^{3} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 40 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 8 T + 31 T^{2} )^{4} \)
$37$ \( 1 - 1294 T^{4} + 1874161 T^{8} \)
$41$ \( ( 1 + 12 T + 41 T^{2} )^{4} \)
$43$ \( 1 + 12 T + 72 T^{2} + 516 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 + 10 T + 50 T^{2} + 470 T^{3} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 2809 T^{4} )^{2} \)
$59$ \( ( 1 - 59 T^{2} )^{4} \)
$61$ \( ( 1 - 104 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 4 T + 8 T^{2} + 268 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 110 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 24 T + 288 T^{2} - 1752 T^{3} + 5329 T^{4} )( 1 + 24 T + 288 T^{2} + 1752 T^{3} + 5329 T^{4} ) \)
$79$ \( ( 1 - 58 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 12 T + 72 T^{2} - 996 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 22 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 18 T + 162 T^{2} - 1746 T^{3} + 9409 T^{4} )^{2} \)
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