Properties

Label 3364.1.j.a
Level $3364$
Weight $1$
Character orbit 3364.j
Analytic conductor $1.679$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3364 = 2^{2} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3364.j (of order \(14\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.67885470250\)
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: no (minimal twist has level 116)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.38068692544.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{14}^{6} q^{2} -\zeta_{14}^{5} q^{4} + ( -\zeta_{14} + \zeta_{14}^{4} ) q^{5} + \zeta_{14}^{4} q^{8} + \zeta_{14}^{4} q^{9} +O(q^{10})\) \( q + \zeta_{14}^{6} q^{2} -\zeta_{14}^{5} q^{4} + ( -\zeta_{14} + \zeta_{14}^{4} ) q^{5} + \zeta_{14}^{4} q^{8} + \zeta_{14}^{4} q^{9} + ( 1 - \zeta_{14}^{3} ) q^{10} + ( \zeta_{14}^{2} + \zeta_{14}^{4} ) q^{13} -\zeta_{14}^{3} q^{16} + ( -\zeta_{14}^{3} + \zeta_{14}^{4} ) q^{17} -\zeta_{14}^{3} q^{18} + ( \zeta_{14}^{2} + \zeta_{14}^{6} ) q^{20} + ( -\zeta_{14} + \zeta_{14}^{2} - \zeta_{14}^{5} ) q^{25} + ( -\zeta_{14} - \zeta_{14}^{3} ) q^{26} + \zeta_{14}^{2} q^{32} + ( \zeta_{14}^{2} - \zeta_{14}^{3} ) q^{34} + \zeta_{14}^{2} q^{36} + ( -\zeta_{14}^{3} - \zeta_{14}^{5} ) q^{37} + ( -\zeta_{14} - \zeta_{14}^{5} ) q^{40} + ( -\zeta_{14} + \zeta_{14}^{6} ) q^{41} + ( -\zeta_{14} - \zeta_{14}^{5} ) q^{45} + \zeta_{14}^{4} q^{49} + ( 1 - \zeta_{14} + \zeta_{14}^{4} ) q^{50} + ( 1 + \zeta_{14}^{2} ) q^{52} + ( \zeta_{14}^{2} - \zeta_{14}^{3} ) q^{53} + ( -\zeta_{14}^{5} + \zeta_{14}^{6} ) q^{61} -\zeta_{14} q^{64} + ( -\zeta_{14} - \zeta_{14}^{3} - \zeta_{14}^{5} + \zeta_{14}^{6} ) q^{65} + ( -\zeta_{14} + \zeta_{14}^{2} ) q^{68} -\zeta_{14} q^{72} + ( \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{73} + ( \zeta_{14}^{2} + \zeta_{14}^{4} ) q^{74} + ( 1 + \zeta_{14}^{4} ) q^{80} -\zeta_{14} q^{81} + ( 1 - \zeta_{14}^{5} ) q^{82} + ( 1 - \zeta_{14} + \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{85} + ( -\zeta_{14} + \zeta_{14}^{4} ) q^{89} + ( 1 + \zeta_{14}^{4} ) q^{90} + ( \zeta_{14}^{4} + \zeta_{14}^{6} ) q^{97} -\zeta_{14}^{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{2} - q^{4} - 2q^{5} - q^{8} - q^{9} + O(q^{10}) \) \( 6q - q^{2} - q^{4} - 2q^{5} - q^{8} - q^{9} + 5q^{10} - 2q^{13} - q^{16} - 2q^{17} - q^{18} - 2q^{20} - 3q^{25} - 2q^{26} - q^{32} - 2q^{34} - q^{36} - 2q^{37} - 2q^{40} - 2q^{41} - 2q^{45} - q^{49} + 4q^{50} + 5q^{52} - 2q^{53} - 2q^{61} - q^{64} - 4q^{65} - 2q^{68} - q^{72} - 2q^{73} - 2q^{74} + 5q^{80} - q^{81} + 5q^{82} + 3q^{85} - 2q^{89} + 5q^{90} - 2q^{97} - q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3364\mathbb{Z}\right)^\times\).

\(n\) \(1683\) \(2525\)
\(\chi(n)\) \(-1\) \(-\zeta_{14}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
571.1
0.900969 + 0.433884i
0.900969 0.433884i
−0.623490 0.781831i
−0.623490 + 0.781831i
0.222521 + 0.974928i
0.222521 0.974928i
−0.900969 + 0.433884i 0 0.623490 0.781831i −1.12349 + 0.541044i 0 0 −0.222521 + 0.974928i −0.222521 + 0.974928i 0.777479 0.974928i
1031.1 −0.900969 0.433884i 0 0.623490 + 0.781831i −1.12349 0.541044i 0 0 −0.222521 0.974928i −0.222521 0.974928i 0.777479 + 0.974928i
1415.1 0.623490 0.781831i 0 −0.222521 0.974928i −0.277479 + 0.347948i 0 0 −0.900969 0.433884i −0.900969 0.433884i 0.0990311 + 0.433884i
1619.1 0.623490 + 0.781831i 0 −0.222521 + 0.974928i −0.277479 0.347948i 0 0 −0.900969 + 0.433884i −0.900969 + 0.433884i 0.0990311 0.433884i
2287.1 −0.222521 + 0.974928i 0 −0.900969 0.433884i 0.400969 1.75676i 0 0 0.623490 0.781831i 0.623490 0.781831i 1.62349 + 0.781831i
2327.1 −0.222521 0.974928i 0 −0.900969 + 0.433884i 0.400969 + 1.75676i 0 0 0.623490 + 0.781831i 0.623490 + 0.781831i 1.62349 0.781831i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2327.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
29.d even 7 1 inner
116.j odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3364.1.j.a 6
4.b odd 2 1 CM 3364.1.j.a 6
29.b even 2 1 3364.1.j.c 6
29.c odd 4 2 3364.1.h.d 12
29.d even 7 2 116.1.j.a 6
29.d even 7 1 3364.1.b.c 3
29.d even 7 1 inner 3364.1.j.a 6
29.d even 7 2 3364.1.j.b 6
29.e even 14 1 3364.1.b.b 3
29.e even 14 1 3364.1.j.c 6
29.e even 14 2 3364.1.j.d 6
29.e even 14 2 3364.1.j.e 6
29.f odd 28 2 3364.1.d.a 6
29.f odd 28 4 3364.1.h.c 12
29.f odd 28 2 3364.1.h.d 12
29.f odd 28 4 3364.1.h.e 12
87.j odd 14 2 1044.1.bb.a 6
116.d odd 2 1 3364.1.j.c 6
116.e even 4 2 3364.1.h.d 12
116.h odd 14 1 3364.1.b.b 3
116.h odd 14 1 3364.1.j.c 6
116.h odd 14 2 3364.1.j.d 6
116.h odd 14 2 3364.1.j.e 6
116.j odd 14 2 116.1.j.a 6
116.j odd 14 1 3364.1.b.c 3
116.j odd 14 1 inner 3364.1.j.a 6
116.j odd 14 2 3364.1.j.b 6
116.l even 28 2 3364.1.d.a 6
116.l even 28 4 3364.1.h.c 12
116.l even 28 2 3364.1.h.d 12
116.l even 28 4 3364.1.h.e 12
145.n even 14 2 2900.1.bj.a 6
145.p odd 28 4 2900.1.bd.a 12
232.p odd 14 2 1856.1.bh.a 6
232.s even 14 2 1856.1.bh.a 6
348.s even 14 2 1044.1.bb.a 6
580.v odd 14 2 2900.1.bj.a 6
580.bi even 28 4 2900.1.bd.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.1.j.a 6 29.d even 7 2
116.1.j.a 6 116.j odd 14 2
1044.1.bb.a 6 87.j odd 14 2
1044.1.bb.a 6 348.s even 14 2
1856.1.bh.a 6 232.p odd 14 2
1856.1.bh.a 6 232.s even 14 2
2900.1.bd.a 12 145.p odd 28 4
2900.1.bd.a 12 580.bi even 28 4
2900.1.bj.a 6 145.n even 14 2
2900.1.bj.a 6 580.v odd 14 2
3364.1.b.b 3 29.e even 14 1
3364.1.b.b 3 116.h odd 14 1
3364.1.b.c 3 29.d even 7 1
3364.1.b.c 3 116.j odd 14 1
3364.1.d.a 6 29.f odd 28 2
3364.1.d.a 6 116.l even 28 2
3364.1.h.c 12 29.f odd 28 4
3364.1.h.c 12 116.l even 28 4
3364.1.h.d 12 29.c odd 4 2
3364.1.h.d 12 29.f odd 28 2
3364.1.h.d 12 116.e even 4 2
3364.1.h.d 12 116.l even 28 2
3364.1.h.e 12 29.f odd 28 4
3364.1.h.e 12 116.l even 28 4
3364.1.j.a 6 1.a even 1 1 trivial
3364.1.j.a 6 4.b odd 2 1 CM
3364.1.j.a 6 29.d even 7 1 inner
3364.1.j.a 6 116.j odd 14 1 inner
3364.1.j.b 6 29.d even 7 2
3364.1.j.b 6 116.j odd 14 2
3364.1.j.c 6 29.b even 2 1
3364.1.j.c 6 29.e even 14 1
3364.1.j.c 6 116.d odd 2 1
3364.1.j.c 6 116.h odd 14 1
3364.1.j.d 6 29.e even 14 2
3364.1.j.d 6 116.h odd 14 2
3364.1.j.e 6 29.e even 14 2
3364.1.j.e 6 116.h odd 14 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3364, [\chi])\):

\( T_{3} \)
\( T_{5}^{6} + 2 T_{5}^{5} + 4 T_{5}^{4} + 8 T_{5}^{3} + 9 T_{5}^{2} + 4 T_{5} + 1 \)
\( T_{17}^{3} + T_{17}^{2} - 2 T_{17} - 1 \)

Hecke characteristic polynomials

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