Properties

Label 3920.1.di.a
Level $3920$
Weight $1$
Character orbit 3920.di
Analytic conductor $1.956$
Analytic rank $0$
Dimension $12$
Projective image $D_{14}$
CM discriminant -20
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3920.di (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: \(\Q(\zeta_{28})\)
Defining polynomial: \(x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{14}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{14} + \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\zeta_{28}^{7} + \zeta_{28}^{11} ) q^{3} + \zeta_{28}^{2} q^{5} + \zeta_{28}^{13} q^{7} + ( -1 + \zeta_{28}^{4} - \zeta_{28}^{8} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{28}^{7} + \zeta_{28}^{11} ) q^{3} + \zeta_{28}^{2} q^{5} + \zeta_{28}^{13} q^{7} + ( -1 + \zeta_{28}^{4} - \zeta_{28}^{8} ) q^{9} + ( -\zeta_{28}^{9} + \zeta_{28}^{13} ) q^{15} + ( \zeta_{28}^{6} - \zeta_{28}^{10} ) q^{21} + ( \zeta_{28} + \zeta_{28}^{11} ) q^{23} + \zeta_{28}^{4} q^{25} + ( -\zeta_{28} + \zeta_{28}^{5} + \zeta_{28}^{7} - \zeta_{28}^{11} ) q^{27} + ( 1 - \zeta_{28}^{2} ) q^{29} -\zeta_{28} q^{35} + ( \zeta_{28}^{6} - \zeta_{28}^{12} ) q^{41} + ( \zeta_{28} - \zeta_{28}^{9} ) q^{43} + ( -\zeta_{28}^{2} + \zeta_{28}^{6} - \zeta_{28}^{10} ) q^{45} + ( \zeta_{28} - \zeta_{28}^{5} ) q^{47} -\zeta_{28}^{12} q^{49} + ( \zeta_{28}^{6} + \zeta_{28}^{10} ) q^{61} + ( -\zeta_{28}^{3} + \zeta_{28}^{7} - \zeta_{28}^{13} ) q^{63} + ( \zeta_{28}^{4} - 2 \zeta_{28}^{8} + \zeta_{28}^{12} ) q^{69} + ( -\zeta_{28} - \zeta_{28}^{11} ) q^{75} + ( 1 - \zeta_{28}^{2} - \zeta_{28}^{4} + \zeta_{28}^{8} - \zeta_{28}^{12} ) q^{81} + ( \zeta_{28}^{3} + \zeta_{28}^{5} ) q^{83} + ( -\zeta_{28}^{7} + \zeta_{28}^{9} + \zeta_{28}^{11} - \zeta_{28}^{13} ) q^{87} + ( -\zeta_{28}^{10} + \zeta_{28}^{12} ) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 2q^{5} - 12q^{9} + O(q^{10}) \) \( 12q + 2q^{5} - 12q^{9} - 2q^{25} + 10q^{29} + 4q^{41} - 2q^{45} + 2q^{49} + 4q^{61} + 12q^{81} - 4q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{28}^{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
0.974928 + 0.222521i
−0.974928 0.222521i
−0.433884 0.900969i
0.433884 + 0.900969i
−0.433884 + 0.900969i
0.433884 0.900969i
0.974928 0.222521i
−0.974928 + 0.222521i
−0.781831 0.623490i
0.781831 + 0.623490i
−0.781831 + 0.623490i
0.781831 0.623490i
0 −0.781831 0.376510i 0 0.900969 + 0.433884i 0 −0.974928 + 0.222521i 0 −0.153989 0.193096i 0
239.2 0 0.781831 + 0.376510i 0 0.900969 + 0.433884i 0 0.974928 0.222521i 0 −0.153989 0.193096i 0
799.1 0 −0.974928 + 1.22252i 0 −0.623490 + 0.781831i 0 0.433884 0.900969i 0 −0.321552 1.40881i 0
799.2 0 0.974928 1.22252i 0 −0.623490 + 0.781831i 0 −0.433884 + 0.900969i 0 −0.321552 1.40881i 0
1359.1 0 −0.974928 1.22252i 0 −0.623490 0.781831i 0 0.433884 + 0.900969i 0 −0.321552 + 1.40881i 0
1359.2 0 0.974928 + 1.22252i 0 −0.623490 0.781831i 0 −0.433884 0.900969i 0 −0.321552 + 1.40881i 0
1919.1 0 −0.781831 + 0.376510i 0 0.900969 0.433884i 0 −0.974928 0.222521i 0 −0.153989 + 0.193096i 0
1919.2 0 0.781831 0.376510i 0 0.900969 0.433884i 0 0.974928 + 0.222521i 0 −0.153989 + 0.193096i 0
2479.1 0 −0.433884 1.90097i 0 0.222521 + 0.974928i 0 0.781831 0.623490i 0 −2.52446 + 1.21572i 0
2479.2 0 0.433884 + 1.90097i 0 0.222521 + 0.974928i 0 −0.781831 + 0.623490i 0 −2.52446 + 1.21572i 0
3599.1 0 −0.433884 + 1.90097i 0 0.222521 0.974928i 0 0.781831 + 0.623490i 0 −2.52446 1.21572i 0
3599.2 0 0.433884 1.90097i 0 0.222521 0.974928i 0 −0.781831 0.623490i 0 −2.52446 1.21572i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3599.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
49.e even 7 1 inner
196.k odd 14 1 inner
245.p even 14 1 inner
980.ba odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.1.di.a 12
4.b odd 2 1 inner 3920.1.di.a 12
5.b even 2 1 inner 3920.1.di.a 12
20.d odd 2 1 CM 3920.1.di.a 12
49.e even 7 1 inner 3920.1.di.a 12
196.k odd 14 1 inner 3920.1.di.a 12
245.p even 14 1 inner 3920.1.di.a 12
980.ba odd 14 1 inner 3920.1.di.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3920.1.di.a 12 1.a even 1 1 trivial
3920.1.di.a 12 4.b odd 2 1 inner
3920.1.di.a 12 5.b even 2 1 inner
3920.1.di.a 12 20.d odd 2 1 CM
3920.1.di.a 12 49.e even 7 1 inner
3920.1.di.a 12 196.k odd 14 1 inner
3920.1.di.a 12 245.p even 14 1 inner
3920.1.di.a 12 980.ba odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3920, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 49 - 49 T^{2} + 49 T^{4} + 35 T^{6} + 21 T^{8} + 7 T^{10} + T^{12} \)
$5$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
$7$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
$11$ \( T^{12} \)
$13$ \( T^{12} \)
$17$ \( T^{12} \)
$19$ \( T^{12} \)
$23$ \( 49 + 98 T^{2} + 49 T^{4} - 14 T^{6} + 14 T^{8} + T^{12} \)
$29$ \( ( 1 - 3 T + 9 T^{2} - 13 T^{3} + 11 T^{4} - 5 T^{5} + T^{6} )^{2} \)
$31$ \( T^{12} \)
$37$ \( T^{12} \)
$41$ \( ( 1 - 4 T + 9 T^{2} - 8 T^{3} + 4 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$43$ \( 49 + 98 T^{2} + 49 T^{4} - 14 T^{6} + 14 T^{8} + T^{12} \)
$47$ \( 49 + 49 T^{2} + 98 T^{4} + 35 T^{6} + T^{12} \)
$53$ \( T^{12} \)
$59$ \( T^{12} \)
$61$ \( ( 1 - 4 T + 9 T^{2} - 8 T^{3} + 4 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$67$ \( T^{12} \)
$71$ \( T^{12} \)
$73$ \( T^{12} \)
$79$ \( T^{12} \)
$83$ \( 49 + 98 T^{2} + 49 T^{4} - 14 T^{6} + 14 T^{8} + T^{12} \)
$89$ \( ( 1 + 4 T + 9 T^{2} + 8 T^{3} + 4 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$97$ \( T^{12} \)
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