Properties

Label 3364.1.h.c
Level $3364$
Weight $1$
Character orbit 3364.h
Analytic conductor $1.679$
Analytic rank $0$
Dimension $12$
Projective image $D_{7}$
CM discriminant -4
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.67885470250\)
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q -\zeta_{28}^{11} q^{2} -\zeta_{28}^{8} q^{4} + ( -1 - \zeta_{28}^{8} ) q^{5} -\zeta_{28}^{5} q^{8} -\zeta_{28}^{12} q^{9} +O(q^{10})\) \( q -\zeta_{28}^{11} q^{2} -\zeta_{28}^{8} q^{4} + ( -1 - \zeta_{28}^{8} ) q^{5} -\zeta_{28}^{5} q^{8} -\zeta_{28}^{12} q^{9} + ( -\zeta_{28}^{5} + \zeta_{28}^{11} ) q^{10} + ( -1 - \zeta_{28}^{4} ) q^{13} -\zeta_{28}^{2} q^{16} + ( \zeta_{28} + \zeta_{28}^{13} ) q^{17} -\zeta_{28}^{9} q^{18} + ( -\zeta_{28}^{2} + \zeta_{28}^{8} ) q^{20} + ( 1 - \zeta_{28}^{2} + \zeta_{28}^{8} ) q^{25} + ( -\zeta_{28} + \zeta_{28}^{11} ) q^{26} + \zeta_{28}^{13} q^{32} + ( \zeta_{28}^{10} - \zeta_{28}^{12} ) q^{34} -\zeta_{28}^{6} q^{36} + ( \zeta_{28}^{3} + \zeta_{28}^{7} ) q^{37} + ( \zeta_{28}^{5} + \zeta_{28}^{13} ) q^{40} + ( -\zeta_{28}^{5} - \zeta_{28}^{9} ) q^{41} + ( -\zeta_{28}^{6} + \zeta_{28}^{12} ) q^{45} + \zeta_{28}^{12} q^{49} + ( \zeta_{28}^{5} - \zeta_{28}^{11} + \zeta_{28}^{13} ) q^{50} + ( \zeta_{28}^{8} + \zeta_{28}^{12} ) q^{52} + ( -\zeta_{28}^{10} + \zeta_{28}^{12} ) q^{53} + ( \zeta_{28}^{5} - \zeta_{28}^{7} ) q^{61} + \zeta_{28}^{10} q^{64} + ( 1 + \zeta_{28}^{4} + \zeta_{28}^{8} + \zeta_{28}^{12} ) q^{65} + ( \zeta_{28}^{7} - \zeta_{28}^{9} ) q^{68} -\zeta_{28}^{3} q^{72} + ( -\zeta_{28}^{9} + \zeta_{28}^{11} ) q^{73} + ( 1 + \zeta_{28}^{4} ) q^{74} + ( \zeta_{28}^{2} + \zeta_{28}^{10} ) q^{80} -\zeta_{28}^{10} q^{81} + ( -\zeta_{28}^{2} - \zeta_{28}^{6} ) q^{82} + ( -\zeta_{28} + \zeta_{28}^{7} - \zeta_{28}^{9} - \zeta_{28}^{13} ) q^{85} + ( \zeta_{28} - \zeta_{28}^{7} ) q^{89} + ( -\zeta_{28}^{3} + \zeta_{28}^{9} ) q^{90} + ( \zeta_{28}^{3} - \zeta_{28}^{13} ) q^{97} + \zeta_{28}^{9} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 2q^{4} - 10q^{5} + 2q^{9} + O(q^{10}) \) \( 12q + 2q^{4} - 10q^{5} + 2q^{9} - 10q^{13} - 2q^{16} - 4q^{20} + 8q^{25} + 4q^{34} - 2q^{36} - 4q^{45} - 2q^{49} - 4q^{52} - 4q^{53} + 2q^{64} + 6q^{65} + 10q^{74} + 4q^{80} - 2q^{81} - 4q^{82} + O(q^{100}) \)

Character values

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

\(n\) \(1683\) \(2525\)
\(\chi(n)\) \(-1\) \(-\zeta_{28}^{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
−0.974928 0.222521i
0.974928 + 0.222521i
−0.974928 + 0.222521i
0.974928 0.222521i
0.781831 0.623490i
−0.781831 + 0.623490i
0.781831 + 0.623490i
−0.781831 0.623490i
0.433884 + 0.900969i
−0.433884 0.900969i
0.433884 0.900969i
−0.433884 + 0.900969i
−0.781831 + 0.623490i 0 0.222521 0.974928i −0.777479 0.974928i 0 0 0.433884 + 0.900969i 0.900969 0.433884i 1.21572 + 0.277479i
63.2 0.781831 0.623490i 0 0.222521 0.974928i −0.777479 0.974928i 0 0 −0.433884 0.900969i 0.900969 0.433884i −1.21572 0.277479i
267.1 −0.781831 0.623490i 0 0.222521 + 0.974928i −0.777479 + 0.974928i 0 0 0.433884 0.900969i 0.900969 + 0.433884i 1.21572 0.277479i
267.2 0.781831 + 0.623490i 0 0.222521 + 0.974928i −0.777479 + 0.974928i 0 0 −0.433884 + 0.900969i 0.900969 + 0.433884i −1.21572 + 0.277479i
651.1 −0.433884 + 0.900969i 0 −0.623490 0.781831i −1.62349 0.781831i 0 0 0.974928 0.222521i 0.222521 + 0.974928i 1.40881 1.12349i
651.2 0.433884 0.900969i 0 −0.623490 0.781831i −1.62349 0.781831i 0 0 −0.974928 + 0.222521i 0.222521 + 0.974928i −1.40881 + 1.12349i
1111.1 −0.433884 0.900969i 0 −0.623490 + 0.781831i −1.62349 + 0.781831i 0 0 0.974928 + 0.222521i 0.222521 0.974928i 1.40881 + 1.12349i
1111.2 0.433884 + 0.900969i 0 −0.623490 + 0.781831i −1.62349 + 0.781831i 0 0 −0.974928 0.222521i 0.222521 0.974928i −1.40881 1.12349i
2719.1 −0.974928 + 0.222521i 0 0.900969 0.433884i −0.0990311 0.433884i 0 0 −0.781831 + 0.623490i −0.623490 0.781831i 0.193096 + 0.400969i
2719.2 0.974928 0.222521i 0 0.900969 0.433884i −0.0990311 0.433884i 0 0 0.781831 0.623490i −0.623490 0.781831i −0.193096 0.400969i
2759.1 −0.974928 0.222521i 0 0.900969 + 0.433884i −0.0990311 + 0.433884i 0 0 −0.781831 0.623490i −0.623490 + 0.781831i 0.193096 0.400969i
2759.2 0.974928 + 0.222521i 0 0.900969 + 0.433884i −0.0990311 + 0.433884i 0 0 0.781831 + 0.623490i −0.623490 + 0.781831i −0.193096 + 0.400969i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2759.2
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.b even 2 1 inner
29.d even 7 1 inner
29.e even 14 1 inner
116.d odd 2 1 inner
116.h odd 14 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.h.c 12
4.b odd 2 1 CM 3364.1.h.c 12
29.b even 2 1 inner 3364.1.h.c 12
29.c odd 4 1 3364.1.j.b 6
29.c odd 4 1 3364.1.j.e 6
29.d even 7 1 3364.1.d.a 6
29.d even 7 1 inner 3364.1.h.c 12
29.d even 7 2 3364.1.h.d 12
29.d even 7 2 3364.1.h.e 12
29.e even 14 1 3364.1.d.a 6
29.e even 14 1 inner 3364.1.h.c 12
29.e even 14 2 3364.1.h.d 12
29.e even 14 2 3364.1.h.e 12
29.f odd 28 2 116.1.j.a 6
29.f odd 28 1 3364.1.b.b 3
29.f odd 28 1 3364.1.b.c 3
29.f odd 28 2 3364.1.j.a 6
29.f odd 28 1 3364.1.j.b 6
29.f odd 28 2 3364.1.j.c 6
29.f odd 28 2 3364.1.j.d 6
29.f odd 28 1 3364.1.j.e 6
87.k even 28 2 1044.1.bb.a 6
116.d odd 2 1 inner 3364.1.h.c 12
116.e even 4 1 3364.1.j.b 6
116.e even 4 1 3364.1.j.e 6
116.h odd 14 1 3364.1.d.a 6
116.h odd 14 1 inner 3364.1.h.c 12
116.h odd 14 2 3364.1.h.d 12
116.h odd 14 2 3364.1.h.e 12
116.j odd 14 1 3364.1.d.a 6
116.j odd 14 1 inner 3364.1.h.c 12
116.j odd 14 2 3364.1.h.d 12
116.j odd 14 2 3364.1.h.e 12
116.l even 28 2 116.1.j.a 6
116.l even 28 1 3364.1.b.b 3
116.l even 28 1 3364.1.b.c 3
116.l even 28 2 3364.1.j.a 6
116.l even 28 1 3364.1.j.b 6
116.l even 28 2 3364.1.j.c 6
116.l even 28 2 3364.1.j.d 6
116.l even 28 1 3364.1.j.e 6
145.o even 28 2 2900.1.bd.a 12
145.s odd 28 2 2900.1.bj.a 6
145.t even 28 2 2900.1.bd.a 12
232.u odd 28 2 1856.1.bh.a 6
232.v even 28 2 1856.1.bh.a 6
348.v odd 28 2 1044.1.bb.a 6
580.bd odd 28 2 2900.1.bd.a 12
580.be even 28 2 2900.1.bj.a 6
580.bm odd 28 2 2900.1.bd.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.1.j.a 6 29.f odd 28 2
116.1.j.a 6 116.l even 28 2
1044.1.bb.a 6 87.k even 28 2
1044.1.bb.a 6 348.v odd 28 2
1856.1.bh.a 6 232.u odd 28 2
1856.1.bh.a 6 232.v even 28 2
2900.1.bd.a 12 145.o even 28 2
2900.1.bd.a 12 145.t even 28 2
2900.1.bd.a 12 580.bd odd 28 2
2900.1.bd.a 12 580.bm odd 28 2
2900.1.bj.a 6 145.s odd 28 2
2900.1.bj.a 6 580.be even 28 2
3364.1.b.b 3 29.f odd 28 1
3364.1.b.b 3 116.l even 28 1
3364.1.b.c 3 29.f odd 28 1
3364.1.b.c 3 116.l even 28 1
3364.1.d.a 6 29.d even 7 1
3364.1.d.a 6 29.e even 14 1
3364.1.d.a 6 116.h odd 14 1
3364.1.d.a 6 116.j odd 14 1
3364.1.h.c 12 1.a even 1 1 trivial
3364.1.h.c 12 4.b odd 2 1 CM
3364.1.h.c 12 29.b even 2 1 inner
3364.1.h.c 12 29.d even 7 1 inner
3364.1.h.c 12 29.e even 14 1 inner
3364.1.h.c 12 116.d odd 2 1 inner
3364.1.h.c 12 116.h odd 14 1 inner
3364.1.h.c 12 116.j odd 14 1 inner
3364.1.h.d 12 29.d even 7 2
3364.1.h.d 12 29.e even 14 2
3364.1.h.d 12 116.h odd 14 2
3364.1.h.d 12 116.j odd 14 2
3364.1.h.e 12 29.d even 7 2
3364.1.h.e 12 29.e even 14 2
3364.1.h.e 12 116.h odd 14 2
3364.1.h.e 12 116.j odd 14 2
3364.1.j.a 6 29.f odd 28 2
3364.1.j.a 6 116.l even 28 2
3364.1.j.b 6 29.c odd 4 1
3364.1.j.b 6 29.f odd 28 1
3364.1.j.b 6 116.e even 4 1
3364.1.j.b 6 116.l even 28 1
3364.1.j.c 6 29.f odd 28 2
3364.1.j.c 6 116.l even 28 2
3364.1.j.d 6 29.f odd 28 2
3364.1.j.d 6 116.l even 28 2
3364.1.j.e 6 29.c odd 4 1
3364.1.j.e 6 29.f odd 28 1
3364.1.j.e 6 116.e even 4 1
3364.1.j.e 6 116.l even 28 1

Hecke kernels

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

\( T_{3} \)
\( T_{5}^{6} + 5 T_{5}^{5} + 11 T_{5}^{4} + 13 T_{5}^{3} + 9 T_{5}^{2} + 3 T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
$3$ \( T^{12} \)
$5$ \( ( 1 + 3 T + 9 T^{2} + 13 T^{3} + 11 T^{4} + 5 T^{5} + T^{6} )^{2} \)
$7$ \( T^{12} \)
$11$ \( T^{12} \)
$13$ \( ( 1 + 3 T + 9 T^{2} + 13 T^{3} + 11 T^{4} + 5 T^{5} + T^{6} )^{2} \)
$17$ \( ( 1 + 6 T^{2} + 5 T^{4} + T^{6} )^{2} \)
$19$ \( T^{12} \)
$23$ \( T^{12} \)
$29$ \( T^{12} \)
$31$ \( T^{12} \)
$37$ \( 1 - 9 T^{2} + 25 T^{4} - T^{6} + 9 T^{8} + 3 T^{10} + T^{12} \)
$41$ \( ( 1 + 6 T^{2} + 5 T^{4} + T^{6} )^{2} \)
$43$ \( T^{12} \)
$47$ \( T^{12} \)
$53$ \( ( 1 + 4 T + 9 T^{2} + 8 T^{3} + 4 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$59$ \( T^{12} \)
$61$ \( 1 - 9 T^{2} + 25 T^{4} - T^{6} + 9 T^{8} + 3 T^{10} + T^{12} \)
$67$ \( T^{12} \)
$71$ \( T^{12} \)
$73$ \( 1 - 2 T^{2} + 25 T^{4} + 6 T^{6} + 2 T^{8} - 4 T^{10} + T^{12} \)
$79$ \( T^{12} \)
$83$ \( T^{12} \)
$89$ \( 1 - 9 T^{2} + 25 T^{4} - T^{6} + 9 T^{8} + 3 T^{10} + T^{12} \)
$97$ \( 1 - 2 T^{2} + 25 T^{4} + 6 T^{6} + 2 T^{8} - 4 T^{10} + T^{12} \)
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