Properties

Label 3332.1.be.c
Level $3332$
Weight $1$
Character orbit 3332.be
Analytic conductor $1.663$
Analytic rank $0$
Dimension $12$
Projective image $D_{14}$
CM discriminant -68
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.be (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.66288462209\)
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.

\(f(q)\) \(=\) \( q -\zeta_{28}^{8} q^{2} + ( \zeta_{28}^{7} + \zeta_{28}^{13} ) q^{3} -\zeta_{28}^{2} q^{4} + ( \zeta_{28} + \zeta_{28}^{7} ) q^{6} + \zeta_{28}^{3} q^{7} + \zeta_{28}^{10} q^{8} + ( -1 - \zeta_{28}^{6} - \zeta_{28}^{12} ) q^{9} +O(q^{10})\) \( q -\zeta_{28}^{8} q^{2} + ( \zeta_{28}^{7} + \zeta_{28}^{13} ) q^{3} -\zeta_{28}^{2} q^{4} + ( \zeta_{28} + \zeta_{28}^{7} ) q^{6} + \zeta_{28}^{3} q^{7} + \zeta_{28}^{10} q^{8} + ( -1 - \zeta_{28}^{6} - \zeta_{28}^{12} ) q^{9} + ( \zeta_{28}^{7} + \zeta_{28}^{9} ) q^{11} + ( \zeta_{28} - \zeta_{28}^{9} ) q^{12} + ( -1 + \zeta_{28}^{2} ) q^{13} -\zeta_{28}^{11} q^{14} + \zeta_{28}^{4} q^{16} + \zeta_{28}^{12} q^{17} + ( -1 - \zeta_{28}^{6} + \zeta_{28}^{8} ) q^{18} + ( -\zeta_{28}^{2} + \zeta_{28}^{10} ) q^{21} + ( \zeta_{28} + \zeta_{28}^{3} ) q^{22} + ( -\zeta_{28} - \zeta_{28}^{3} ) q^{23} + ( -\zeta_{28}^{3} - \zeta_{28}^{9} ) q^{24} -\zeta_{28}^{6} q^{25} + ( \zeta_{28}^{8} - \zeta_{28}^{10} ) q^{26} + ( \zeta_{28}^{5} - \zeta_{28}^{7} + \zeta_{28}^{11} - \zeta_{28}^{13} ) q^{27} -\zeta_{28}^{5} q^{28} + ( \zeta_{28}^{3} - \zeta_{28}^{11} ) q^{31} -\zeta_{28}^{12} q^{32} + ( -1 - \zeta_{28}^{2} - \zeta_{28}^{6} - \zeta_{28}^{8} ) q^{33} + \zeta_{28}^{6} q^{34} + ( -1 + \zeta_{28}^{2} + \zeta_{28}^{8} ) q^{36} + ( -\zeta_{28} - \zeta_{28}^{7} + \zeta_{28}^{9} - \zeta_{28}^{13} ) q^{39} + ( \zeta_{28}^{4} + \zeta_{28}^{10} ) q^{42} + ( -\zeta_{28}^{9} - \zeta_{28}^{11} ) q^{44} + ( \zeta_{28}^{9} + \zeta_{28}^{11} ) q^{46} + ( -\zeta_{28}^{3} + \zeta_{28}^{11} ) q^{48} + \zeta_{28}^{6} q^{49} - q^{50} + ( -\zeta_{28}^{5} - \zeta_{28}^{11} ) q^{51} + ( \zeta_{28}^{2} - \zeta_{28}^{4} ) q^{52} + ( -\zeta_{28}^{6} + \zeta_{28}^{12} ) q^{53} + ( -\zeta_{28} + \zeta_{28}^{5} - \zeta_{28}^{7} - \zeta_{28}^{13} ) q^{54} + \zeta_{28}^{13} q^{56} + ( -\zeta_{28}^{5} - \zeta_{28}^{11} ) q^{62} + ( \zeta_{28} - \zeta_{28}^{3} - \zeta_{28}^{9} ) q^{63} -\zeta_{28}^{6} q^{64} + ( -1 - \zeta_{28}^{2} + \zeta_{28}^{8} + \zeta_{28}^{10} ) q^{66} + q^{68} + ( 1 + \zeta_{28}^{2} - \zeta_{28}^{8} - \zeta_{28}^{10} ) q^{69} + ( -\zeta_{28}^{5} + \zeta_{28}^{13} ) q^{71} + ( \zeta_{28}^{2} + \zeta_{28}^{8} - \zeta_{28}^{10} ) q^{72} + ( \zeta_{28}^{5} - \zeta_{28}^{13} ) q^{75} + ( \zeta_{28}^{10} + \zeta_{28}^{12} ) q^{77} + ( -\zeta_{28} + \zeta_{28}^{3} - \zeta_{28}^{7} + \zeta_{28}^{9} ) q^{78} + ( -\zeta_{28} + \zeta_{28}^{13} ) q^{79} + ( 1 - \zeta_{28}^{4} + \zeta_{28}^{6} - \zeta_{28}^{10} + \zeta_{28}^{12} ) q^{81} + ( \zeta_{28}^{4} - \zeta_{28}^{12} ) q^{84} + ( -\zeta_{28}^{3} - \zeta_{28}^{5} ) q^{88} + ( \zeta_{28}^{2} + \zeta_{28}^{10} ) q^{89} + ( -\zeta_{28}^{3} + \zeta_{28}^{5} ) q^{91} + ( \zeta_{28}^{3} + \zeta_{28}^{5} ) q^{92} + ( -\zeta_{28}^{2} + \zeta_{28}^{4} + 2 \zeta_{28}^{10} ) q^{93} + ( \zeta_{28}^{5} + \zeta_{28}^{11} ) q^{96} + q^{98} + ( \zeta_{28} + \zeta_{28}^{5} - \zeta_{28}^{9} - \zeta_{28}^{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 2 q^{4} + 2 q^{8} - 12 q^{9} + O(q^{10}) \) \( 12 q + 2 q^{2} - 2 q^{4} + 2 q^{8} - 12 q^{9} - 10 q^{13} - 2 q^{16} - 2 q^{17} - 16 q^{18} - 2 q^{25} - 4 q^{26} + 2 q^{32} - 14 q^{33} + 2 q^{34} - 12 q^{36} + 2 q^{49} - 12 q^{50} + 4 q^{52} - 4 q^{53} - 2 q^{64} - 14 q^{66} + 12 q^{68} + 14 q^{69} - 2 q^{72} + 12 q^{81} + 4 q^{89} + 12 q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(-\zeta_{28}^{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
407.1
0.974928 + 0.222521i
−0.974928 0.222521i
0.974928 0.222521i
−0.974928 + 0.222521i
0.433884 + 0.900969i
−0.433884 0.900969i
0.781831 0.623490i
−0.781831 + 0.623490i
0.781831 + 0.623490i
−0.781831 0.623490i
0.433884 0.900969i
−0.433884 + 0.900969i
0.222521 0.974928i −0.974928 + 1.22252i −0.900969 0.433884i 0 0.974928 + 1.22252i 0.781831 + 0.623490i −0.623490 + 0.781831i −0.321552 1.40881i 0
407.2 0.222521 0.974928i 0.974928 1.22252i −0.900969 0.433884i 0 −0.974928 1.22252i −0.781831 0.623490i −0.623490 + 0.781831i −0.321552 1.40881i 0
1359.1 0.222521 + 0.974928i −0.974928 1.22252i −0.900969 + 0.433884i 0 0.974928 1.22252i 0.781831 0.623490i −0.623490 0.781831i −0.321552 + 1.40881i 0
1359.2 0.222521 + 0.974928i 0.974928 + 1.22252i −0.900969 + 0.433884i 0 −0.974928 + 1.22252i −0.781831 + 0.623490i −0.623490 0.781831i −0.321552 + 1.40881i 0
1835.1 0.900969 0.433884i −0.433884 + 1.90097i 0.623490 0.781831i 0 0.433884 + 1.90097i −0.974928 0.222521i 0.222521 0.974928i −2.52446 1.21572i 0
1835.2 0.900969 0.433884i 0.433884 1.90097i 0.623490 0.781831i 0 −0.433884 1.90097i 0.974928 + 0.222521i 0.222521 0.974928i −2.52446 1.21572i 0
2311.1 −0.623490 0.781831i −0.781831 + 0.376510i −0.222521 + 0.974928i 0 0.781831 + 0.376510i −0.433884 0.900969i 0.900969 0.433884i −0.153989 + 0.193096i 0
2311.2 −0.623490 0.781831i 0.781831 0.376510i −0.222521 + 0.974928i 0 −0.781831 0.376510i 0.433884 + 0.900969i 0.900969 0.433884i −0.153989 + 0.193096i 0
2787.1 −0.623490 + 0.781831i −0.781831 0.376510i −0.222521 0.974928i 0 0.781831 0.376510i −0.433884 + 0.900969i 0.900969 + 0.433884i −0.153989 0.193096i 0
2787.2 −0.623490 + 0.781831i 0.781831 + 0.376510i −0.222521 0.974928i 0 −0.781831 + 0.376510i 0.433884 0.900969i 0.900969 + 0.433884i −0.153989 0.193096i 0
3263.1 0.900969 + 0.433884i −0.433884 1.90097i 0.623490 + 0.781831i 0 0.433884 1.90097i −0.974928 + 0.222521i 0.222521 + 0.974928i −2.52446 + 1.21572i 0
3263.2 0.900969 + 0.433884i 0.433884 + 1.90097i 0.623490 + 0.781831i 0 −0.433884 + 1.90097i 0.974928 0.222521i 0.222521 + 0.974928i −2.52446 + 1.21572i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3263.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)
4.b odd 2 1 inner
17.b even 2 1 inner
49.e even 7 1 inner
196.k odd 14 1 inner
833.r even 14 1 inner
3332.be odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.be.c 12
4.b odd 2 1 inner 3332.1.be.c 12
17.b even 2 1 inner 3332.1.be.c 12
49.e even 7 1 inner 3332.1.be.c 12
68.d odd 2 1 CM 3332.1.be.c 12
196.k odd 14 1 inner 3332.1.be.c 12
833.r even 14 1 inner 3332.1.be.c 12
3332.be odd 14 1 inner 3332.1.be.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.be.c 12 1.a even 1 1 trivial
3332.1.be.c 12 4.b odd 2 1 inner
3332.1.be.c 12 17.b even 2 1 inner
3332.1.be.c 12 49.e even 7 1 inner
3332.1.be.c 12 68.d odd 2 1 CM
3332.1.be.c 12 196.k odd 14 1 inner
3332.1.be.c 12 833.r even 14 1 inner
3332.1.be.c 12 3332.be odd 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 7 T_{3}^{10} + 21 T_{3}^{8} + 35 T_{3}^{6} + 49 T_{3}^{4} - 49 T_{3}^{2} + 49 \) acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\).

Hecke characteristic polynomials

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