Properties

Label 430.2.e.d
Level 430
Weight 2
Character orbit 430.e
Analytic conductor 3.434
Analytic rank 0
Dimension 4
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{10})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 10 x^{2} + 100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/10\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/10\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(10 \beta_{2}\)
\(\nu^{3}\)\(=\)\(10 \beta_{3}\)

Character values

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

\(n\) \(87\) \(261\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\)

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
−1.58114 2.73861i
1.58114 + 2.73861i
−1.58114 + 2.73861i
1.58114 2.73861i
−1.00000 1.00000 1.73205i 1.00000 −0.500000 + 0.866025i −1.00000 + 1.73205i −1.58114 2.73861i −1.00000 −0.500000 0.866025i 0.500000 0.866025i
221.2 −1.00000 1.00000 1.73205i 1.00000 −0.500000 + 0.866025i −1.00000 + 1.73205i 1.58114 + 2.73861i −1.00000 −0.500000 0.866025i 0.500000 0.866025i
251.1 −1.00000 1.00000 + 1.73205i 1.00000 −0.500000 0.866025i −1.00000 1.73205i −1.58114 + 2.73861i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
251.2 −1.00000 1.00000 + 1.73205i 1.00000 −0.500000 0.866025i −1.00000 1.73205i 1.58114 2.73861i −1.00000 −0.500000 + 0.866025i 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.d 4
43.c even 3 1 inner 430.2.e.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.e.d 4 1.a even 1 1 trivial
430.2.e.d 4 43.c even 3 1 inner

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} )^{2} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( 1 - 4 T^{2} - 33 T^{4} - 196 T^{6} + 2401 T^{8} \)
$11$ \( ( 1 + 2 T + 13 T^{2} + 22 T^{3} + 121 T^{4} )^{2} \)
$13$ \( 1 - 16 T^{2} + 87 T^{4} - 2704 T^{6} + 28561 T^{8} \)
$17$ \( 1 + 2 T - 21 T^{2} - 18 T^{3} + 268 T^{4} - 306 T^{5} - 6069 T^{6} + 9826 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 2 T - 25 T^{2} - 18 T^{3} + 404 T^{4} - 342 T^{5} - 9025 T^{6} + 13718 T^{7} + 130321 T^{8} \)
$23$ \( ( 1 - 23 T^{2} + 529 T^{4} )^{2} \)
$29$ \( 1 + 8 T + 30 T^{2} - 192 T^{3} - 1541 T^{4} - 5568 T^{5} + 25230 T^{6} + 195112 T^{7} + 707281 T^{8} \)
$31$ \( 1 - 12 T + 56 T^{2} - 312 T^{3} + 2319 T^{4} - 9672 T^{5} + 53816 T^{6} - 357492 T^{7} + 923521 T^{8} \)
$37$ \( ( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 10 T + 67 T^{2} + 410 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( 1 + 6 T + 55 T^{2} + 258 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 - 4 T + 58 T^{2} - 188 T^{3} + 2209 T^{4} )^{2} \)
$53$ \( 1 - 8 T - 18 T^{2} + 192 T^{3} + 523 T^{4} + 10176 T^{5} - 50562 T^{6} - 1191016 T^{7} + 7890481 T^{8} \)
$59$ \( ( 1 - 14 T + 157 T^{2} - 826 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( 1 - 4 T - 100 T^{2} + 24 T^{3} + 8759 T^{4} + 1464 T^{5} - 372100 T^{6} - 907924 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 2 T + 29 T^{2} - 318 T^{3} - 4132 T^{4} - 21306 T^{5} + 130181 T^{6} + 601526 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 16 T + 60 T^{2} + 864 T^{3} + 15199 T^{4} + 61344 T^{5} + 302460 T^{6} + 5726576 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 6 T - 109 T^{2} - 6 T^{3} + 13068 T^{4} - 438 T^{5} - 580861 T^{6} + 2334102 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 148 T^{2} + 15663 T^{4} - 923668 T^{6} + 38950081 T^{8} \)
$83$ \( ( 1 + 6 T - 47 T^{2} + 498 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 14 T + 233 T^{2} - 1358 T^{3} + 9409 T^{4} )^{2} \)
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