Properties

Label 3332.1.be.a
Level $3332$
Weight $1$
Character orbit 3332.be
Analytic conductor $1.663$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -68
Inner twists $4$

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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: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
Defining polynomial: \(x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{7} - \cdots)\)

$q$-expansion

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

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

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(-\zeta_{14}^{3}\) \(-1\)

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
0.900969 + 0.433884i
0.900969 0.433884i
−0.623490 + 0.781831i
0.222521 0.974928i
0.222521 + 0.974928i
−0.623490 0.781831i
−0.222521 + 0.974928i −0.777479 + 0.974928i −0.900969 0.433884i 0 −0.777479 0.974928i −0.623490 + 0.781831i 0.623490 0.781831i −0.123490 0.541044i 0
1359.1 −0.222521 0.974928i −0.777479 0.974928i −0.900969 + 0.433884i 0 −0.777479 + 0.974928i −0.623490 0.781831i 0.623490 + 0.781831i −0.123490 + 0.541044i 0
1835.1 −0.900969 + 0.433884i −0.0990311 + 0.433884i 0.623490 0.781831i 0 −0.0990311 0.433884i 0.222521 0.974928i −0.222521 + 0.974928i 0.722521 + 0.347948i 0
2311.1 0.623490 + 0.781831i −1.62349 + 0.781831i −0.222521 + 0.974928i 0 −1.62349 0.781831i 0.900969 0.433884i −0.900969 + 0.433884i 1.40097 1.75676i 0
2787.1 0.623490 0.781831i −1.62349 0.781831i −0.222521 0.974928i 0 −1.62349 + 0.781831i 0.900969 + 0.433884i −0.900969 0.433884i 1.40097 + 1.75676i 0
3263.1 −0.900969 0.433884i −0.0990311 0.433884i 0.623490 + 0.781831i 0 −0.0990311 + 0.433884i 0.222521 + 0.974928i −0.222521 0.974928i 0.722521 0.347948i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3263.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)
49.e even 7 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.a 6
4.b odd 2 1 3332.1.be.b yes 6
17.b even 2 1 3332.1.be.b yes 6
49.e even 7 1 inner 3332.1.be.a 6
68.d odd 2 1 CM 3332.1.be.a 6
196.k odd 14 1 3332.1.be.b yes 6
833.r even 14 1 3332.1.be.b yes 6
3332.be odd 14 1 inner 3332.1.be.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.be.a 6 1.a even 1 1 trivial
3332.1.be.a 6 49.e even 7 1 inner
3332.1.be.a 6 68.d odd 2 1 CM
3332.1.be.a 6 3332.be odd 14 1 inner
3332.1.be.b yes 6 4.b odd 2 1
3332.1.be.b yes 6 17.b even 2 1
3332.1.be.b yes 6 196.k odd 14 1
3332.1.be.b yes 6 833.r even 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 5 T_{3}^{5} + 11 T_{3}^{4} + 13 T_{3}^{3} + 9 T_{3}^{2} + 3 T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\).

Hecke characteristic polynomials

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