Properties

Label 3364.1.d.a
Level $3364$
Weight $1$
Character orbit 3364.d
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.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.67885470250\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 5 x^{4} + 6 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 3 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} + 3 \nu^{2} + 1 \)
\(\beta_{5}\)\(=\)\( \nu^{5} + 4 \nu^{3} + 3 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} - 3 \beta_{2} + 5\)
\(\nu^{5}\)\(=\)\(\beta_{5} - 4 \beta_{3} + 9 \beta_{1}\)

Character values

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

\(n\) \(1683\) \(2525\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3363.1
1.24698i
0.445042i
1.80194i
1.24698i
0.445042i
1.80194i
1.00000i 0 −1.00000 −1.24698 0 0 1.00000i −1.00000 1.24698i
3363.2 1.00000i 0 −1.00000 0.445042 0 0 1.00000i −1.00000 0.445042i
3363.3 1.00000i 0 −1.00000 1.80194 0 0 1.00000i −1.00000 1.80194i
3363.4 1.00000i 0 −1.00000 −1.24698 0 0 1.00000i −1.00000 1.24698i
3363.5 1.00000i 0 −1.00000 0.445042 0 0 1.00000i −1.00000 0.445042i
3363.6 1.00000i 0 −1.00000 1.80194 0 0 1.00000i −1.00000 1.80194i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3363.6
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
116.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3364.1.d.a 6
4.b odd 2 1 CM 3364.1.d.a 6
29.b even 2 1 inner 3364.1.d.a 6
29.c odd 4 1 3364.1.b.b 3
29.c odd 4 1 3364.1.b.c 3
29.d even 7 2 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 2 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 2 3364.1.j.a 6
29.f odd 28 2 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 2 3364.1.j.e 6
87.k even 28 2 1044.1.bb.a 6
116.d odd 2 1 inner 3364.1.d.a 6
116.e even 4 1 3364.1.b.b 3
116.e even 4 1 3364.1.b.c 3
116.h odd 14 2 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 2 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 2 3364.1.j.a 6
116.l even 28 2 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 2 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.c odd 4 1
3364.1.b.b 3 116.e even 4 1
3364.1.b.c 3 29.c odd 4 1
3364.1.b.c 3 116.e even 4 1
3364.1.d.a 6 1.a even 1 1 trivial
3364.1.d.a 6 4.b odd 2 1 CM
3364.1.d.a 6 29.b even 2 1 inner
3364.1.d.a 6 116.d odd 2 1 inner
3364.1.h.c 12 29.d even 7 2
3364.1.h.c 12 29.e even 14 2
3364.1.h.c 12 116.h odd 14 2
3364.1.h.c 12 116.j odd 14 2
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.f odd 28 2
3364.1.j.b 6 116.l even 28 2
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.f odd 28 2
3364.1.j.e 6 116.l even 28 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3364, [\chi])\).

Hecke characteristic polynomials

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