Properties

Label 1476.1.z.a
Level $1476$
Weight $1$
Character orbit 1476.z
Analytic conductor $0.737$
Analytic rank $0$
Dimension $4$
Projective image $D_{10}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1476 = 2^{2} \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1476.z (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.736619958646\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 164)
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.83809775204854016.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{10}^{3} q^{2} -\zeta_{10} q^{4} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{5} -\zeta_{10}^{4} q^{8} +O(q^{10})\) \( q + \zeta_{10}^{3} q^{2} -\zeta_{10} q^{4} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{5} -\zeta_{10}^{4} q^{8} + ( \zeta_{10} - \zeta_{10}^{2} ) q^{10} + ( -\zeta_{10}^{2} + \zeta_{10}^{4} ) q^{13} + \zeta_{10}^{2} q^{16} + ( -\zeta_{10} - \zeta_{10}^{2} ) q^{17} + ( 1 + \zeta_{10}^{4} ) q^{20} + ( -\zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{25} + ( 1 - \zeta_{10}^{2} ) q^{26} - q^{32} + ( 1 - \zeta_{10}^{4} ) q^{34} + ( -1 - \zeta_{10}^{2} ) q^{37} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{40} + \zeta_{10}^{3} q^{41} -\zeta_{10}^{4} q^{49} + ( -1 + \zeta_{10} - \zeta_{10}^{4} ) q^{50} + ( 1 + \zeta_{10}^{3} ) q^{52} + ( -\zeta_{10}^{3} - \zeta_{10}^{4} ) q^{53} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{61} -\zeta_{10}^{3} q^{64} + ( -1 + \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{65} + ( \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{68} + ( \zeta_{10} - \zeta_{10}^{4} ) q^{73} + ( 1 - \zeta_{10}^{3} ) q^{74} + ( 1 - \zeta_{10} ) q^{80} -\zeta_{10} q^{82} + ( \zeta_{10} + \zeta_{10}^{4} ) q^{85} + ( -1 + \zeta_{10}^{2} ) q^{97} + \zeta_{10}^{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - q^{4} - 2 q^{5} + q^{8} + O(q^{10}) \) \( 4 q + q^{2} - q^{4} - 2 q^{5} + q^{8} + 2 q^{10} - q^{16} + 3 q^{20} - 3 q^{25} + 5 q^{26} - 4 q^{32} + 5 q^{34} - 3 q^{37} + 2 q^{40} + q^{41} + q^{49} - 2 q^{50} + 5 q^{52} - 2 q^{61} - q^{64} - 5 q^{65} + 2 q^{73} + 3 q^{74} + 3 q^{80} - q^{82} - 5 q^{97} - q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1476\mathbb{Z}\right)^\times\).

\(n\) \(739\) \(821\) \(1441\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−0.309017 + 0.951057i
0.809017 + 0.587785i
−0.309017 0.951057i
0.809017 0.587785i
0.809017 0.587785i 0 0.309017 0.951057i −0.500000 + 1.53884i 0 0 −0.309017 0.951057i 0 0.500000 + 1.53884i
271.1 −0.309017 + 0.951057i 0 −0.809017 0.587785i −0.500000 0.363271i 0 0 0.809017 0.587785i 0 0.500000 0.363271i
523.1 0.809017 + 0.587785i 0 0.309017 + 0.951057i −0.500000 1.53884i 0 0 −0.309017 + 0.951057i 0 0.500000 1.53884i
1171.1 −0.309017 0.951057i 0 −0.809017 + 0.587785i −0.500000 + 0.363271i 0 0 0.809017 + 0.587785i 0 0.500000 + 0.363271i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
41.f even 10 1 inner
164.l odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1476.1.z.a 4
3.b odd 2 1 164.1.l.a 4
4.b odd 2 1 CM 1476.1.z.a 4
12.b even 2 1 164.1.l.a 4
24.f even 2 1 2624.1.bx.a 4
24.h odd 2 1 2624.1.bx.a 4
41.f even 10 1 inner 1476.1.z.a 4
123.l odd 10 1 164.1.l.a 4
164.l odd 10 1 inner 1476.1.z.a 4
492.v even 10 1 164.1.l.a 4
984.bh even 10 1 2624.1.bx.a 4
984.bk odd 10 1 2624.1.bx.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.1.l.a 4 3.b odd 2 1
164.1.l.a 4 12.b even 2 1
164.1.l.a 4 123.l odd 10 1
164.1.l.a 4 492.v even 10 1
1476.1.z.a 4 1.a even 1 1 trivial
1476.1.z.a 4 4.b odd 2 1 CM
1476.1.z.a 4 41.f even 10 1 inner
1476.1.z.a 4 164.l odd 10 1 inner
2624.1.bx.a 4 24.f even 2 1
2624.1.bx.a 4 24.h odd 2 1
2624.1.bx.a 4 984.bh even 10 1
2624.1.bx.a 4 984.bk odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1476, [\chi])\).

Hecke characteristic polynomials

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