Properties

Label 3724.1.dj.a
Level $3724$
Weight $1$
Character orbit 3724.dj
Analytic conductor $1.859$
Analytic rank $0$
Dimension $12$
Projective image $D_{21}$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3724.dj (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.85851810705\)
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{42} - \zeta_{42}^{3} ) q^{5} -\zeta_{42}^{9} q^{7} -\zeta_{42}^{11} q^{9} +O(q^{10})\) \( q + ( -\zeta_{42} - \zeta_{42}^{3} ) q^{5} -\zeta_{42}^{9} q^{7} -\zeta_{42}^{11} q^{9} + ( -\zeta_{42}^{7} - \zeta_{42}^{13} ) q^{11} + ( -\zeta_{42}^{9} + \zeta_{42}^{14} ) q^{17} + \zeta_{42}^{14} q^{19} + ( -\zeta_{42}^{9} + \zeta_{42}^{10} ) q^{23} + ( \zeta_{42}^{2} + \zeta_{42}^{4} + \zeta_{42}^{6} ) q^{25} + ( \zeta_{42}^{10} + \zeta_{42}^{12} ) q^{35} + ( \zeta_{42}^{4} + \zeta_{42}^{20} ) q^{43} + ( \zeta_{42}^{12} + \zeta_{42}^{14} ) q^{45} + ( -\zeta_{42}^{5} + \zeta_{42}^{8} ) q^{47} + \zeta_{42}^{18} q^{49} + ( \zeta_{42}^{8} + \zeta_{42}^{10} + \zeta_{42}^{14} + \zeta_{42}^{16} ) q^{55} + ( -\zeta_{42}^{5} - \zeta_{42}^{11} ) q^{61} + \zeta_{42}^{20} q^{63} + 2 \zeta_{42}^{4} q^{73} + ( -\zeta_{42} + \zeta_{42}^{16} ) q^{77} -\zeta_{42} q^{81} + ( -\zeta_{42}^{17} - \zeta_{42}^{19} ) q^{83} + ( \zeta_{42}^{10} + \zeta_{42}^{12} - \zeta_{42}^{15} - \zeta_{42}^{17} ) q^{85} + ( -\zeta_{42}^{15} - \zeta_{42}^{17} ) q^{95} + ( -\zeta_{42}^{3} + \zeta_{42}^{18} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - q^{5} - 2q^{7} + q^{9} + O(q^{10}) \) \( 12q - q^{5} - 2q^{7} + q^{9} - 5q^{11} - 8q^{17} - 6q^{19} - q^{23} - q^{35} + 2q^{43} - 8q^{45} + 2q^{47} - 2q^{49} - 3q^{55} + 2q^{61} + q^{63} + 2q^{73} + 2q^{77} + q^{81} + 2q^{83} - 2q^{85} - q^{95} - 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3724\mathbb{Z}\right)^\times\).

\(n\) \(1863\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-\zeta_{42}^{11}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−0.988831 0.149042i
0.0747301 0.997204i
0.365341 0.930874i
0.826239 0.563320i
0.955573 0.294755i
−0.988831 + 0.149042i
0.826239 + 0.563320i
0.955573 + 0.294755i
0.0747301 + 0.997204i
−0.733052 0.680173i
−0.733052 + 0.680173i
0.365341 + 0.930874i
0 0 0 −1.88980 0.582926i 0 −0.222521 0.974928i 0 0.0747301 0.997204i 0
417.1 0 0 0 −0.147791 0.0222759i 0 0.623490 0.781831i 0 −0.733052 + 0.680173i 0
1101.1 0 0 0 −0.535628 0.496990i 0 −0.222521 + 0.974928i 0 0.826239 0.563320i 0
1481.1 0 0 0 0.603718 1.53825i 0 0.623490 + 0.781831i 0 0.955573 0.294755i 0
1633.1 0 0 0 1.57906 1.07659i 0 −0.900969 0.433884i 0 −0.988831 + 0.149042i 0
2013.1 0 0 0 −1.88980 + 0.582926i 0 −0.222521 + 0.974928i 0 0.0747301 + 0.997204i 0
2165.1 0 0 0 0.603718 + 1.53825i 0 0.623490 0.781831i 0 0.955573 + 0.294755i 0
2545.1 0 0 0 1.57906 + 1.07659i 0 −0.900969 + 0.433884i 0 −0.988831 0.149042i 0
2697.1 0 0 0 −0.147791 + 0.0222759i 0 0.623490 + 0.781831i 0 −0.733052 0.680173i 0
3077.1 0 0 0 −0.109562 1.46200i 0 −0.900969 0.433884i 0 0.365341 0.930874i 0
3229.1 0 0 0 −0.109562 + 1.46200i 0 −0.900969 + 0.433884i 0 0.365341 + 0.930874i 0
3609.1 0 0 0 −0.535628 + 0.496990i 0 −0.222521 0.974928i 0 0.826239 + 0.563320i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3609.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
49.g even 21 1 inner
931.bq odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3724.1.dj.a 12
19.b odd 2 1 CM 3724.1.dj.a 12
49.g even 21 1 inner 3724.1.dj.a 12
931.bq odd 42 1 inner 3724.1.dj.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3724.1.dj.a 12 1.a even 1 1 trivial
3724.1.dj.a 12 19.b odd 2 1 CM
3724.1.dj.a 12 49.g even 21 1 inner
3724.1.dj.a 12 931.bq odd 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{12} + \cdots\) acting on \(S_{1}^{\mathrm{new}}(3724, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( T^{12} \)
$5$ \( 1 + 15 T + 70 T^{2} + 104 T^{3} + 90 T^{4} + 35 T^{5} + 43 T^{6} + 7 T^{7} + 6 T^{8} + 6 T^{9} + T^{11} + T^{12} \)
$7$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$11$ \( 1 + 3 T + T^{3} + 31 T^{4} + 56 T^{5} + 57 T^{6} + 56 T^{7} + 47 T^{8} + 29 T^{9} + 14 T^{10} + 5 T^{11} + T^{12} \)
$13$ \( T^{12} \)
$17$ \( 1 - 6 T + 118 T^{3} + 349 T^{4} + 518 T^{5} + 519 T^{6} + 392 T^{7} + 230 T^{8} + 104 T^{9} + 35 T^{10} + 8 T^{11} + T^{12} \)
$19$ \( ( 1 + T + T^{2} )^{6} \)
$23$ \( 1 + 15 T + 70 T^{2} + 104 T^{3} + 90 T^{4} + 35 T^{5} + 43 T^{6} + 7 T^{7} + 6 T^{8} + 6 T^{9} + T^{11} + T^{12} \)
$29$ \( T^{12} \)
$31$ \( T^{12} \)
$37$ \( T^{12} \)
$41$ \( T^{12} \)
$43$ \( 1 - 9 T + 52 T^{2} - 46 T^{3} + 96 T^{4} - 97 T^{5} + 49 T^{6} + T^{7} - 9 T^{8} + 3 T^{9} + 3 T^{10} - 2 T^{11} + T^{12} \)
$47$ \( 1 + 3 T + 7 T^{2} + 8 T^{3} + 3 T^{4} - 28 T^{5} + T^{6} + 7 T^{7} + 12 T^{8} - 6 T^{9} - 2 T^{11} + T^{12} \)
$53$ \( T^{12} \)
$59$ \( T^{12} \)
$61$ \( 1 + 3 T + 7 T^{2} + 8 T^{3} + 3 T^{4} - 28 T^{5} + T^{6} + 7 T^{7} + 12 T^{8} - 6 T^{9} - 2 T^{11} + T^{12} \)
$67$ \( T^{12} \)
$71$ \( T^{12} \)
$73$ \( 4096 - 2048 T + 512 T^{3} - 256 T^{4} + 64 T^{6} - 16 T^{8} + 8 T^{9} - 2 T^{11} + T^{12} \)
$79$ \( T^{12} \)
$83$ \( 1 + 5 T + 52 T^{2} + 94 T^{3} + 54 T^{4} - 6 T^{5} + 7 T^{6} - 6 T^{7} + 12 T^{8} - 4 T^{9} + 3 T^{10} - 2 T^{11} + T^{12} \)
$89$ \( T^{12} \)
$97$ \( T^{12} \)
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