Properties

Label 588.1.z.a
Level $588$
Weight $1$
Character orbit 588.z
Analytic conductor $0.293$
Analytic rank $0$
Dimension $12$
Projective image $D_{21}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 588.z (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.293450227428\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
Defining polynomial: \(x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{21}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{21} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{42}^{10} q^{3} + \zeta_{42}^{8} q^{7} + \zeta_{42}^{20} q^{9} +O(q^{10})\) \( q + \zeta_{42}^{10} q^{3} + \zeta_{42}^{8} q^{7} + \zeta_{42}^{20} q^{9} + ( \zeta_{42}^{4} - \zeta_{42}^{5} ) q^{13} + ( -\zeta_{42} + \zeta_{42}^{6} ) q^{19} + \zeta_{42}^{18} q^{21} -\zeta_{42}^{13} q^{25} -\zeta_{42}^{9} q^{27} + ( \zeta_{42}^{2} + \zeta_{42}^{12} ) q^{31} + ( -\zeta_{42}^{3} - \zeta_{42}^{7} ) q^{37} + ( \zeta_{42}^{14} - \zeta_{42}^{15} ) q^{39} + ( -\zeta_{42}^{17} - \zeta_{42}^{19} ) q^{43} + \zeta_{42}^{16} q^{49} + ( -\zeta_{42}^{11} + \zeta_{42}^{16} ) q^{57} + ( \zeta_{42}^{14} - \zeta_{42}^{17} ) q^{61} -\zeta_{42}^{7} q^{63} + ( -\zeta_{42}^{3} - \zeta_{42}^{11} ) q^{67} + ( -\zeta_{42}^{11} - \zeta_{42}^{15} ) q^{73} + \zeta_{42}^{2} q^{75} + ( -\zeta_{42}^{9} - \zeta_{42}^{19} ) q^{79} -\zeta_{42}^{19} q^{81} + ( \zeta_{42}^{12} - \zeta_{42}^{13} ) q^{91} + ( -\zeta_{42} + \zeta_{42}^{12} ) q^{93} + ( \zeta_{42}^{6} - \zeta_{42}^{15} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + q^{3} + q^{7} + q^{9} + O(q^{10}) \) \( 12q + q^{3} + q^{7} + q^{9} + 2q^{13} - q^{19} - 2q^{21} + q^{25} - 2q^{27} - q^{31} - 8q^{37} - 8q^{39} + 2q^{43} + q^{49} + 2q^{57} - 5q^{61} - 6q^{63} - q^{67} - q^{73} + q^{75} - q^{79} + q^{81} - q^{91} - q^{93} - 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{42}^{20}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.0747301 0.997204i
−0.988831 0.149042i
0.365341 + 0.930874i
−0.733052 + 0.680173i
−0.733052 0.680173i
0.0747301 + 0.997204i
0.955573 + 0.294755i
0.826239 + 0.563320i
−0.988831 + 0.149042i
0.955573 0.294755i
0.826239 0.563320i
0.365341 0.930874i
0 −0.733052 0.680173i 0 0 0 0.826239 + 0.563320i 0 0.0747301 + 0.997204i 0
65.1 0 0.0747301 + 0.997204i 0 0 0 0.365341 + 0.930874i 0 −0.988831 + 0.149042i 0
137.1 0 0.826239 0.563320i 0 0 0 −0.988831 0.149042i 0 0.365341 0.930874i 0
149.1 0 0.365341 0.930874i 0 0 0 0.955573 + 0.294755i 0 −0.733052 0.680173i 0
221.1 0 0.365341 + 0.930874i 0 0 0 0.955573 0.294755i 0 −0.733052 + 0.680173i 0
233.1 0 −0.733052 + 0.680173i 0 0 0 0.826239 0.563320i 0 0.0747301 0.997204i 0
305.1 0 −0.988831 + 0.149042i 0 0 0 −0.733052 + 0.680173i 0 0.955573 0.294755i 0
317.1 0 0.955573 0.294755i 0 0 0 0.0747301 0.997204i 0 0.826239 0.563320i 0
389.1 0 0.0747301 0.997204i 0 0 0 0.365341 0.930874i 0 −0.988831 0.149042i 0
401.1 0 −0.988831 0.149042i 0 0 0 −0.733052 0.680173i 0 0.955573 + 0.294755i 0
473.1 0 0.955573 + 0.294755i 0 0 0 0.0747301 + 0.997204i 0 0.826239 + 0.563320i 0
485.1 0 0.826239 + 0.563320i 0 0 0 −0.988831 + 0.149042i 0 0.365341 + 0.930874i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 485.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
49.g even 21 1 inner
147.n odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.1.z.a 12
3.b odd 2 1 CM 588.1.z.a 12
4.b odd 2 1 2352.1.cv.a 12
12.b even 2 1 2352.1.cv.a 12
49.g even 21 1 inner 588.1.z.a 12
147.n odd 42 1 inner 588.1.z.a 12
196.o odd 42 1 2352.1.cv.a 12
588.bb even 42 1 2352.1.cv.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.1.z.a 12 1.a even 1 1 trivial
588.1.z.a 12 3.b odd 2 1 CM
588.1.z.a 12 49.g even 21 1 inner
588.1.z.a 12 147.n odd 42 1 inner
2352.1.cv.a 12 4.b odd 2 1
2352.1.cv.a 12 12.b even 2 1
2352.1.cv.a 12 196.o odd 42 1
2352.1.cv.a 12 588.bb even 42 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(588, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
$5$ \( T^{12} \)
$7$ \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
$11$ \( T^{12} \)
$13$ \( 1 - 9 T + 52 T^{2} - 46 T^{3} + 96 T^{4} - 97 T^{5} + 49 T^{6} + T^{7} - 9 T^{8} + 3 T^{9} + 3 T^{10} - 2 T^{11} + T^{12} \)
$17$ \( T^{12} \)
$19$ \( 1 + 8 T + 56 T^{2} + 76 T^{3} + 118 T^{4} + 49 T^{5} + 78 T^{6} + 28 T^{7} + 34 T^{8} + 6 T^{9} + 7 T^{10} + T^{11} + T^{12} \)
$23$ \( T^{12} \)
$29$ \( T^{12} \)
$31$ \( 1 + 8 T + 56 T^{2} + 76 T^{3} + 118 T^{4} + 49 T^{5} + 78 T^{6} + 28 T^{7} + 34 T^{8} + 6 T^{9} + 7 T^{10} + T^{11} + T^{12} \)
$37$ \( 1 - 6 T + 118 T^{3} + 349 T^{4} + 518 T^{5} + 519 T^{6} + 392 T^{7} + 230 T^{8} + 104 T^{9} + 35 T^{10} + 8 T^{11} + T^{12} \)
$41$ \( T^{12} \)
$43$ \( 1 + 5 T + 52 T^{2} + 94 T^{3} + 54 T^{4} - 6 T^{5} + 7 T^{6} - 6 T^{7} + 12 T^{8} - 4 T^{9} + 3 T^{10} - 2 T^{11} + T^{12} \)
$47$ \( T^{12} \)
$53$ \( T^{12} \)
$59$ \( T^{12} \)
$61$ \( 1 + 3 T + T^{3} + 31 T^{4} + 56 T^{5} + 57 T^{6} + 56 T^{7} + 47 T^{8} + 29 T^{9} + 14 T^{10} + 5 T^{11} + T^{12} \)
$67$ \( 1 + 8 T + 56 T^{2} + 76 T^{3} + 118 T^{4} + 49 T^{5} + 78 T^{6} + 28 T^{7} + 34 T^{8} + 6 T^{9} + 7 T^{10} + T^{11} + T^{12} \)
$71$ \( T^{12} \)
$73$ \( 1 - 13 T + 49 T^{2} - 29 T^{3} + 69 T^{4} - 20 T^{6} - 21 T^{7} + 6 T^{8} - T^{9} + T^{11} + T^{12} \)
$79$ \( 1 + 8 T + 56 T^{2} + 76 T^{3} + 118 T^{4} + 49 T^{5} + 78 T^{6} + 28 T^{7} + 34 T^{8} + 6 T^{9} + 7 T^{10} + T^{11} + T^{12} \)
$83$ \( T^{12} \)
$89$ \( T^{12} \)
$97$ \( ( -1 - 2 T + T^{2} + T^{3} )^{4} \)
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