Properties

Label 429.1.v.c
Level $429$
Weight $1$
Character orbit 429.v
Analytic conductor $0.214$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
CM discriminant -39
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 429.v (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.214098890420\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{10}\)
Projective field Galois closure of 10.0.1487719872058563.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( -\zeta_{20} + \zeta_{20}^{5} ) q^{2} -\zeta_{20}^{4} q^{3} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{6} ) q^{4} + ( \zeta_{20} + \zeta_{20}^{3} ) q^{5} + ( \zeta_{20}^{5} - \zeta_{20}^{9} ) q^{6} + ( \zeta_{20} - \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{8} + \zeta_{20}^{8} q^{9} +O(q^{10})\) \( q + ( -\zeta_{20} + \zeta_{20}^{5} ) q^{2} -\zeta_{20}^{4} q^{3} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{6} ) q^{4} + ( \zeta_{20} + \zeta_{20}^{3} ) q^{5} + ( \zeta_{20}^{5} - \zeta_{20}^{9} ) q^{6} + ( \zeta_{20} - \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{8} + \zeta_{20}^{8} q^{9} + ( -\zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{10} + \zeta_{20}^{7} q^{11} + ( -1 + \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{12} -\zeta_{20}^{8} q^{13} + ( -\zeta_{20}^{5} - \zeta_{20}^{7} ) q^{15} + ( 1 - \zeta_{20}^{2} + \zeta_{20}^{4} + \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{16} + ( -\zeta_{20}^{3} - \zeta_{20}^{9} ) q^{18} + ( -\zeta_{20} + \zeta_{20}^{5} - \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{20} + ( -\zeta_{20}^{2} - \zeta_{20}^{8} ) q^{22} + ( \zeta_{20} - \zeta_{20}^{5} + \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{24} + ( \zeta_{20}^{2} + \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{25} + ( \zeta_{20}^{3} + \zeta_{20}^{9} ) q^{26} + \zeta_{20}^{2} q^{27} + ( 1 + \zeta_{20}^{2} + \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{30} + ( -\zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{32} + \zeta_{20} q^{33} + ( -1 + \zeta_{20}^{4} - \zeta_{20}^{8} ) q^{36} -\zeta_{20}^{2} q^{39} + ( -1 + \zeta_{20}^{2} - 2 \zeta_{20}^{6} ) q^{40} + ( \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{43} + ( \zeta_{20}^{3} - \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{44} + ( -\zeta_{20} + \zeta_{20}^{9} ) q^{45} + ( -\zeta_{20} - \zeta_{20}^{7} ) q^{47} + ( 1 - \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{48} -\zeta_{20}^{2} q^{49} + ( -\zeta_{20} - \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{9} ) q^{50} + ( 1 - \zeta_{20}^{4} + \zeta_{20}^{8} ) q^{52} + ( -\zeta_{20}^{3} + \zeta_{20}^{7} ) q^{54} + ( -1 + \zeta_{20}^{8} ) q^{55} + ( -\zeta_{20}^{3} - \zeta_{20}^{9} ) q^{59} + ( -\zeta_{20} - \zeta_{20}^{3} + \zeta_{20}^{5} - \zeta_{20}^{9} ) q^{60} + ( -1 - \zeta_{20}^{4} ) q^{61} + ( \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{8} ) q^{64} + ( \zeta_{20} - \zeta_{20}^{9} ) q^{65} + ( -\zeta_{20}^{2} + \zeta_{20}^{6} ) q^{66} + ( \zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{9} ) q^{72} + ( 1 - \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{75} + ( \zeta_{20}^{3} - \zeta_{20}^{7} ) q^{78} + ( -1 + \zeta_{20}^{6} ) q^{79} + ( 2 \zeta_{20} + 2 \zeta_{20}^{7} ) q^{80} -\zeta_{20}^{6} q^{81} + ( -\zeta_{20} - \zeta_{20}^{3} ) q^{83} + ( \zeta_{20} - \zeta_{20}^{5} + \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{86} + ( 1 + \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{8} ) q^{88} + ( 1 + \zeta_{20}^{2} - \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{90} + ( 2 \zeta_{20}^{2} - \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{94} + ( -\zeta_{20} + \zeta_{20}^{3} + \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{96} + ( \zeta_{20}^{3} - \zeta_{20}^{7} ) q^{98} -\zeta_{20}^{5} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{3} - 8q^{4} - 2q^{9} + O(q^{10}) \) \( 8q + 2q^{3} - 8q^{4} - 2q^{9} - 12q^{12} + 2q^{13} + 8q^{16} + 2q^{25} + 2q^{27} + 10q^{30} - 8q^{36} - 2q^{39} - 10q^{40} - 4q^{43} + 12q^{48} - 2q^{49} + 8q^{52} - 10q^{55} - 6q^{61} + 2q^{64} + 8q^{75} - 6q^{79} - 2q^{81} + 10q^{88} + 10q^{90} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(79\) \(287\)
\(\chi(n)\) \(-1\) \(-\zeta_{20}^{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1
0.587785 + 0.809017i
−0.587785 0.809017i
0.951057 + 0.309017i
−0.951057 0.309017i
0.587785 0.809017i
−0.587785 + 0.809017i
0.951057 0.309017i
−0.951057 + 0.309017i
−0.587785 1.80902i 0.809017 + 0.587785i −2.11803 + 1.53884i −0.363271 + 1.11803i 0.587785 1.80902i 0 2.48990 + 1.80902i 0.309017 + 0.951057i 2.23607
38.2 0.587785 + 1.80902i 0.809017 + 0.587785i −2.11803 + 1.53884i 0.363271 1.11803i −0.587785 + 1.80902i 0 −2.48990 1.80902i 0.309017 + 0.951057i 2.23607
311.1 −0.951057 + 0.690983i −0.309017 0.951057i 0.118034 0.363271i 1.53884 + 1.11803i 0.951057 + 0.690983i 0 −0.224514 0.690983i −0.809017 + 0.587785i −2.23607
311.2 0.951057 0.690983i −0.309017 0.951057i 0.118034 0.363271i −1.53884 1.11803i −0.951057 0.690983i 0 0.224514 + 0.690983i −0.809017 + 0.587785i −2.23607
350.1 −0.587785 + 1.80902i 0.809017 0.587785i −2.11803 1.53884i −0.363271 1.11803i 0.587785 + 1.80902i 0 2.48990 1.80902i 0.309017 0.951057i 2.23607
350.2 0.587785 1.80902i 0.809017 0.587785i −2.11803 1.53884i 0.363271 + 1.11803i −0.587785 1.80902i 0 −2.48990 + 1.80902i 0.309017 0.951057i 2.23607
389.1 −0.951057 0.690983i −0.309017 + 0.951057i 0.118034 + 0.363271i 1.53884 1.11803i 0.951057 0.690983i 0 −0.224514 + 0.690983i −0.809017 0.587785i −2.23607
389.2 0.951057 + 0.690983i −0.309017 + 0.951057i 0.118034 + 0.363271i −1.53884 + 1.11803i −0.951057 + 0.690983i 0 0.224514 0.690983i −0.809017 0.587785i −2.23607
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 389.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
11.c even 5 1 inner
13.b even 2 1 inner
33.h odd 10 1 inner
143.n even 10 1 inner
429.v odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.1.v.c 8
3.b odd 2 1 inner 429.1.v.c 8
11.c even 5 1 inner 429.1.v.c 8
13.b even 2 1 inner 429.1.v.c 8
33.h odd 10 1 inner 429.1.v.c 8
39.d odd 2 1 CM 429.1.v.c 8
143.n even 10 1 inner 429.1.v.c 8
429.v odd 10 1 inner 429.1.v.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.1.v.c 8 1.a even 1 1 trivial
429.1.v.c 8 3.b odd 2 1 inner
429.1.v.c 8 11.c even 5 1 inner
429.1.v.c 8 13.b even 2 1 inner
429.1.v.c 8 33.h odd 10 1 inner
429.1.v.c 8 39.d odd 2 1 CM
429.1.v.c 8 143.n even 10 1 inner
429.1.v.c 8 429.v odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 5 T_{2}^{6} + 10 T_{2}^{4} + 25 \) acting on \(S_{1}^{\mathrm{new}}(429, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 25 + 10 T^{4} + 5 T^{6} + T^{8} \)
$3$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$5$ \( 25 + 25 T^{2} + 10 T^{4} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$13$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( ( -1 + T + T^{2} )^{4} \)
$47$ \( 25 + 25 T^{2} + 10 T^{4} + T^{8} \)
$53$ \( T^{8} \)
$59$ \( 25 + 25 T^{2} + 10 T^{4} + T^{8} \)
$61$ \( ( 1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4} )^{2} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( T^{8} \)
$79$ \( ( 1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4} )^{2} \)
$83$ \( 25 + 25 T^{2} + 10 T^{4} + T^{8} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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