Properties

Label 3920.1.fc.a
Level $3920$
Weight $1$
Character orbit 3920.fc
Analytic conductor $1.956$
Analytic rank $0$
Dimension $24$
Projective image $D_{42}$
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.fc (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{42})\)
Coefficient field: \(\Q(\zeta_{84})\)
Defining polynomial: \(x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{84}^{15} + \zeta_{84}^{35} ) q^{3} -\zeta_{84}^{32} q^{5} + \zeta_{84}^{19} q^{7} + ( \zeta_{84}^{8} - \zeta_{84}^{28} + \zeta_{84}^{30} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{84}^{15} + \zeta_{84}^{35} ) q^{3} -\zeta_{84}^{32} q^{5} + \zeta_{84}^{19} q^{7} + ( \zeta_{84}^{8} - \zeta_{84}^{28} + \zeta_{84}^{30} ) q^{9} + ( -\zeta_{84}^{5} + \zeta_{84}^{25} ) q^{15} + ( -\zeta_{84}^{12} - \zeta_{84}^{34} ) q^{21} + ( \zeta_{84} - \zeta_{84}^{9} ) q^{23} -\zeta_{84}^{22} q^{25} + ( -\zeta_{84} + \zeta_{84}^{3} + \zeta_{84}^{21} - \zeta_{84}^{23} ) q^{27} + ( \zeta_{84}^{4} - \zeta_{84}^{14} ) q^{29} + \zeta_{84}^{9} q^{35} + ( \zeta_{84}^{10} + \zeta_{84}^{26} ) q^{41} + ( -\zeta_{84}^{11} - \zeta_{84}^{37} ) q^{43} + ( -\zeta_{84}^{18} + \zeta_{84}^{20} - \zeta_{84}^{40} ) q^{45} + ( \zeta_{84}^{17} + \zeta_{84}^{23} ) q^{47} + \zeta_{84}^{38} q^{49} + ( \zeta_{84}^{6} - \zeta_{84}^{40} ) q^{61} + ( \zeta_{84}^{5} - \zeta_{84}^{7} + \zeta_{84}^{27} ) q^{63} + ( -\zeta_{84}^{21} - \zeta_{84}^{35} ) q^{67} + ( \zeta_{84}^{2} - \zeta_{84}^{16} + \zeta_{84}^{24} + \zeta_{84}^{36} ) q^{69} + ( \zeta_{84}^{15} + \zeta_{84}^{37} ) q^{75} + ( -\zeta_{84}^{14} + \zeta_{84}^{16} - \zeta_{84}^{18} - \zeta_{84}^{36} + \zeta_{84}^{38} ) q^{81} + ( \zeta_{84}^{13} - \zeta_{84}^{17} ) q^{83} + ( \zeta_{84}^{7} - \zeta_{84}^{19} + \zeta_{84}^{29} + \zeta_{84}^{39} ) q^{87} + ( \zeta_{84}^{24} - \zeta_{84}^{34} ) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q - 2q^{5} + 18q^{9} + O(q^{10}) \) \( 24q - 2q^{5} + 18q^{9} + 6q^{21} + 2q^{25} - 10q^{29} - 4q^{41} - 4q^{45} - 2q^{49} + 2q^{61} - 12q^{69} - 12q^{81} - 2q^{89} + O(q^{100}) \)

Character values

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

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{84}^{8}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
0.294755 0.955573i
−0.294755 + 0.955573i
0.294755 + 0.955573i
−0.294755 0.955573i
−0.930874 + 0.365341i
0.930874 0.365341i
0.997204 0.0747301i
−0.997204 + 0.0747301i
−0.149042 + 0.988831i
0.149042 0.988831i
−0.149042 0.988831i
0.149042 + 0.988831i
−0.930874 0.365341i
0.930874 + 0.365341i
0.680173 0.733052i
−0.680173 + 0.733052i
0.563320 0.826239i
−0.563320 + 0.826239i
0.997204 + 0.0747301i
−0.997204 0.0747301i
0 −0.108903 0.277479i 0 0.988831 + 0.149042i 0 0.563320 + 0.826239i 0 0.667917 0.619736i 0
319.2 0 0.108903 + 0.277479i 0 0.988831 + 0.149042i 0 −0.563320 0.826239i 0 0.667917 0.619736i 0
639.1 0 −0.108903 + 0.277479i 0 0.988831 0.149042i 0 0.563320 0.826239i 0 0.667917 + 0.619736i 0
639.2 0 0.108903 0.277479i 0 0.988831 0.149042i 0 −0.563320 + 0.826239i 0 0.667917 + 0.619736i 0
879.1 0 −0.0841939 + 1.12349i 0 −0.826239 0.563320i 0 −0.680173 + 0.733052i 0 −0.266310 0.0401398i 0
879.2 0 0.0841939 1.12349i 0 −0.826239 0.563320i 0 0.680173 0.733052i 0 −0.266310 0.0401398i 0
1199.1 0 −1.29991 + 0.400969i 0 0.733052 + 0.680173i 0 0.149042 0.988831i 0 0.702749 0.479126i 0
1199.2 0 1.29991 0.400969i 0 0.733052 + 0.680173i 0 −0.149042 + 0.988831i 0 0.702749 0.479126i 0
1759.1 0 −1.64786 1.12349i 0 −0.0747301 + 0.997204i 0 0.294755 + 0.955573i 0 1.08786 + 2.77183i 0
1759.2 0 1.64786 + 1.12349i 0 −0.0747301 + 0.997204i 0 −0.294755 0.955573i 0 1.08786 + 2.77183i 0
1999.1 0 −1.64786 + 1.12349i 0 −0.0747301 0.997204i 0 0.294755 0.955573i 0 1.08786 2.77183i 0
1999.2 0 1.64786 1.12349i 0 −0.0747301 0.997204i 0 −0.294755 + 0.955573i 0 1.08786 2.77183i 0
2319.1 0 −0.0841939 1.12349i 0 −0.826239 + 0.563320i 0 −0.680173 0.733052i 0 −0.266310 + 0.0401398i 0
2319.2 0 0.0841939 + 1.12349i 0 −0.826239 + 0.563320i 0 0.680173 + 0.733052i 0 −0.266310 + 0.0401398i 0
2559.1 0 −1.84095 + 0.277479i 0 −0.365341 + 0.930874i 0 −0.997204 0.0747301i 0 2.35654 0.726897i 0
2559.2 0 1.84095 0.277479i 0 −0.365341 + 0.930874i 0 0.997204 + 0.0747301i 0 2.35654 0.726897i 0
2879.1 0 −0.432142 + 0.400969i 0 −0.955573 0.294755i 0 0.930874 + 0.365341i 0 −0.0487597 + 0.650653i 0
2879.2 0 0.432142 0.400969i 0 −0.955573 0.294755i 0 −0.930874 0.365341i 0 −0.0487597 + 0.650653i 0
3119.1 0 −1.29991 0.400969i 0 0.733052 0.680173i 0 0.149042 + 0.988831i 0 0.702749 + 0.479126i 0
3119.2 0 1.29991 + 0.400969i 0 0.733052 0.680173i 0 −0.149042 0.988831i 0 0.702749 + 0.479126i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3679.2
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.g even 21 1 inner
196.o odd 42 1 inner
245.t even 42 1 inner
980.bq odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.1.fc.a 24
4.b odd 2 1 inner 3920.1.fc.a 24
5.b even 2 1 inner 3920.1.fc.a 24
20.d odd 2 1 CM 3920.1.fc.a 24
49.g even 21 1 inner 3920.1.fc.a 24
196.o odd 42 1 inner 3920.1.fc.a 24
245.t even 42 1 inner 3920.1.fc.a 24
980.bq odd 42 1 inner 3920.1.fc.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3920.1.fc.a 24 1.a even 1 1 trivial
3920.1.fc.a 24 4.b odd 2 1 inner
3920.1.fc.a 24 5.b even 2 1 inner
3920.1.fc.a 24 20.d odd 2 1 CM
3920.1.fc.a 24 49.g even 21 1 inner
3920.1.fc.a 24 196.o odd 42 1 inner
3920.1.fc.a 24 245.t even 42 1 inner
3920.1.fc.a 24 980.bq odd 42 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{24} \)
$3$ \( 1 + 16 T^{2} + 126 T^{4} + 62 T^{6} + 957 T^{8} - 84 T^{10} - 713 T^{12} + 378 T^{14} + 108 T^{16} - 134 T^{18} + 49 T^{20} - 10 T^{22} + T^{24} \)
$5$ \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \)
$7$ \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \)
$11$ \( T^{24} \)
$13$ \( T^{24} \)
$17$ \( T^{24} \)
$19$ \( T^{24} \)
$23$ \( 1 - 5 T^{2} + 105 T^{4} + 363 T^{6} + 663 T^{8} - 728 T^{10} + 610 T^{12} - 427 T^{14} + 24 T^{16} + 83 T^{18} - 14 T^{20} - 3 T^{22} + T^{24} \)
$29$ \( ( 1 + 12 T + 45 T^{2} + 10 T^{3} + 61 T^{4} + 92 T^{5} + 105 T^{6} + 92 T^{7} + 68 T^{8} + 38 T^{9} + 17 T^{10} + 5 T^{11} + T^{12} )^{2} \)
$31$ \( T^{24} \)
$37$ \( T^{24} \)
$41$ \( ( 1 - 5 T + 52 T^{2} - 94 T^{3} + 54 T^{4} + 6 T^{5} + 7 T^{6} + 6 T^{7} + 12 T^{8} + 4 T^{9} + 3 T^{10} + 2 T^{11} + T^{12} )^{2} \)
$43$ \( 1 - 11 T^{2} + 96 T^{4} + 234 T^{6} + 1284 T^{8} + 1978 T^{10} + 1519 T^{12} + 716 T^{14} + 258 T^{16} + 38 T^{18} + 13 T^{20} + 6 T^{22} + T^{24} \)
$47$ \( 2401 - 2401 T^{2} - 2401 T^{4} - 1372 T^{6} + 7889 T^{8} - 3430 T^{10} + 1323 T^{12} - 49 T^{14} - 98 T^{16} + 70 T^{18} + T^{24} \)
$53$ \( T^{24} \)
$59$ \( T^{24} \)
$61$ \( ( 1 - 8 T + 28 T^{2} + 71 T^{3} + 48 T^{4} - 21 T^{5} + 22 T^{6} - 15 T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
$67$ \( ( 9 + 3 T^{2} + T^{4} )^{6} \)
$71$ \( T^{24} \)
$73$ \( T^{24} \)
$79$ \( T^{24} \)
$83$ \( 1 - 11 T^{2} + 96 T^{4} + 234 T^{6} + 1284 T^{8} + 1978 T^{10} + 1519 T^{12} + 716 T^{14} + 258 T^{16} + 38 T^{18} + 13 T^{20} + 6 T^{22} + T^{24} \)
$89$ \( ( 1 + 8 T + 28 T^{2} - 71 T^{3} + 48 T^{4} + 21 T^{5} + 22 T^{6} - 15 T^{8} - T^{9} + T^{11} + T^{12} )^{2} \)
$97$ \( T^{24} \)
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