Properties

Label 2352.1.dj.a
Level $2352$
Weight $1$
Character orbit 2352.dj
Analytic conductor $1.174$
Analytic rank $0$
Dimension $12$
Projective image $D_{42}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2352.dj (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.17380090971\)
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 \(D_{42}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{42} - \cdots)\)

$q$-expansion

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-\zeta_{42}^{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
0.955573 0.294755i
0.826239 0.563320i
−0.733052 + 0.680173i
−0.733052 0.680173i
0.365341 0.930874i
0.0747301 + 0.997204i
0.955573 + 0.294755i
0.0747301 0.997204i
0.826239 + 0.563320i
−0.988831 + 0.149042i
−0.988831 0.149042i
0.365341 + 0.930874i
0 −0.365341 + 0.930874i 0 0 0 0.955573 + 0.294755i 0 −0.733052 0.680173i 0
143.1 0 0.733052 + 0.680173i 0 0 0 0.826239 + 0.563320i 0 0.0747301 + 0.997204i 0
383.1 0 0.988831 + 0.149042i 0 0 0 −0.733052 0.680173i 0 0.955573 + 0.294755i 0
479.1 0 0.988831 0.149042i 0 0 0 −0.733052 + 0.680173i 0 0.955573 0.294755i 0
719.1 0 −0.0747301 0.997204i 0 0 0 0.365341 + 0.930874i 0 −0.988831 + 0.149042i 0
1055.1 0 −0.955573 + 0.294755i 0 0 0 0.0747301 0.997204i 0 0.826239 0.563320i 0
1151.1 0 −0.365341 0.930874i 0 0 0 0.955573 0.294755i 0 −0.733052 + 0.680173i 0
1487.1 0 −0.955573 0.294755i 0 0 0 0.0747301 + 0.997204i 0 0.826239 + 0.563320i 0
1727.1 0 0.733052 0.680173i 0 0 0 0.826239 0.563320i 0 0.0747301 0.997204i 0
1823.1 0 −0.826239 + 0.563320i 0 0 0 −0.988831 0.149042i 0 0.365341 0.930874i 0
2063.1 0 −0.826239 0.563320i 0 0 0 −0.988831 + 0.149042i 0 0.365341 + 0.930874i 0
2159.1 0 −0.0747301 + 0.997204i 0 0 0 0.365341 0.930874i 0 −0.988831 0.149042i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2159.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
196.p even 42 1 inner
588.bf odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.1.dj.a 12
3.b odd 2 1 CM 2352.1.dj.a 12
4.b odd 2 1 2352.1.dj.b yes 12
12.b even 2 1 2352.1.dj.b yes 12
49.h odd 42 1 2352.1.dj.b yes 12
147.o even 42 1 2352.1.dj.b yes 12
196.p even 42 1 inner 2352.1.dj.a 12
588.bf odd 42 1 inner 2352.1.dj.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.1.dj.a 12 1.a even 1 1 trivial
2352.1.dj.a 12 3.b odd 2 1 CM
2352.1.dj.a 12 196.p even 42 1 inner
2352.1.dj.a 12 588.bf odd 42 1 inner
2352.1.dj.b yes 12 4.b odd 2 1
2352.1.dj.b yes 12 12.b even 2 1
2352.1.dj.b yes 12 49.h odd 42 1
2352.1.dj.b yes 12 147.o even 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{19}^{12} + \cdots\) acting on \(S_{1}^{\mathrm{new}}(2352, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
$5$ \( T^{12} \)
$7$ \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
$11$ \( T^{12} \)
$13$ \( 1 - 7 T + 16 T^{2} - 14 T^{3} + 4 T^{4} + 21 T^{5} + 15 T^{6} - 7 T^{7} + 9 T^{8} + 7 T^{9} - 3 T^{10} + T^{12} \)
$17$ \( T^{12} \)
$19$ \( 1 + 8 T + 56 T^{2} + 76 T^{3} + 118 T^{4} + 49 T^{5} + 78 T^{6} + 28 T^{7} + 34 T^{8} + 6 T^{9} + 7 T^{10} + T^{11} + T^{12} \)
$23$ \( T^{12} \)
$29$ \( T^{12} \)
$31$ \( 1 - 8 T + 56 T^{2} - 76 T^{3} + 118 T^{4} - 49 T^{5} + 78 T^{6} - 28 T^{7} + 34 T^{8} - 6 T^{9} + 7 T^{10} - T^{11} + T^{12} \)
$37$ \( 1 + 6 T - 118 T^{3} + 349 T^{4} - 518 T^{5} + 519 T^{6} - 392 T^{7} + 230 T^{8} - 104 T^{9} + 35 T^{10} - 8 T^{11} + T^{12} \)
$41$ \( T^{12} \)
$43$ \( 1 - 7 T + 30 T^{2} - 70 T^{3} + 88 T^{4} - 42 T^{5} - 13 T^{6} + 14 T^{7} + 2 T^{8} - 3 T^{10} + T^{12} \)
$47$ \( T^{12} \)
$53$ \( T^{12} \)
$59$ \( T^{12} \)
$61$ \( 49 - 147 T + 196 T^{2} - 245 T^{3} + 343 T^{4} - 392 T^{5} + 357 T^{6} - 266 T^{7} + 161 T^{8} - 77 T^{9} + 28 T^{10} - 7 T^{11} + T^{12} \)
$67$ \( 1 + 6 T + 10 T^{2} - 12 T^{3} - 24 T^{4} + 21 T^{5} + 36 T^{6} - 42 T^{7} + 2 T^{8} + 12 T^{9} - T^{10} - 3 T^{11} + T^{12} \)
$71$ \( T^{12} \)
$73$ \( 1 - T + 3 T^{2} + 23 T^{3} + 25 T^{4} + 8 T^{6} - 7 T^{7} + 2 T^{8} - 9 T^{9} + 6 T^{10} - 3 T^{11} + T^{12} \)
$79$ \( 1 - 6 T + 10 T^{2} + 12 T^{3} - 24 T^{4} - 21 T^{5} + 36 T^{6} + 42 T^{7} + 2 T^{8} - 12 T^{9} - T^{10} + 3 T^{11} + T^{12} \)
$83$ \( T^{12} \)
$89$ \( T^{12} \)
$97$ \( ( 7 + 14 T^{2} + 7 T^{4} + T^{6} )^{2} \)
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