Properties

Label 1368.1.bz.a
Level $1368$
Weight $1$
Character orbit 1368.bz
Analytic conductor $0.683$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1368.bz (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.682720937282\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.2888.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - q^{8} + q^{11} - \zeta_{6} q^{16} + \zeta_{6} q^{17} + \zeta_{6}^{2} q^{19} + \zeta_{6} q^{22} + \zeta_{6}^{2} q^{25} - \zeta_{6}^{2} q^{32} + 2 \zeta_{6}^{2} q^{34} - q^{38} - \zeta_{6} q^{41} - \zeta_{6} q^{43} + \zeta_{6}^{2} q^{44} + q^{49} - q^{50} - \zeta_{6} q^{59} + q^{64} - \zeta_{6}^{2} q^{67} - 2 q^{68} + \zeta_{6} q^{73} - \zeta_{6} q^{76} - \zeta_{6}^{2} q^{82} + q^{83} - 2 \zeta_{6}^{2} q^{86} - q^{88} - \zeta_{6}^{2} q^{89} + \zeta_{6} q^{97} + \zeta_{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} - 2 q^{8} + 2 q^{11} - q^{16} + 2 q^{17} - q^{19} + q^{22} - q^{25} + q^{32} - 2 q^{34} - 2 q^{38} - q^{41} - 2 q^{43} - q^{44} + 2 q^{49} - 2 q^{50} - q^{59} + 2 q^{64} + q^{67} - 4 q^{68} + q^{73} - q^{76} + q^{82} + 2 q^{83} + 2 q^{86} - 2 q^{88} + 2 q^{89} + q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{6}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 −1.00000 0 0
235.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 −1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
19.c even 3 1 inner
152.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.1.bz.a 2
3.b odd 2 1 152.1.k.a 2
8.d odd 2 1 CM 1368.1.bz.a 2
12.b even 2 1 608.1.o.a 2
15.d odd 2 1 3800.1.bd.c 2
15.e even 4 2 3800.1.bn.b 4
19.c even 3 1 inner 1368.1.bz.a 2
24.f even 2 1 152.1.k.a 2
24.h odd 2 1 608.1.o.a 2
57.d even 2 1 2888.1.k.a 2
57.f even 6 1 2888.1.f.a 1
57.f even 6 1 2888.1.k.a 2
57.h odd 6 1 152.1.k.a 2
57.h odd 6 1 2888.1.f.b 1
57.j even 18 6 2888.1.u.d 6
57.l odd 18 6 2888.1.u.c 6
120.m even 2 1 3800.1.bd.c 2
120.q odd 4 2 3800.1.bn.b 4
152.k odd 6 1 inner 1368.1.bz.a 2
228.m even 6 1 608.1.o.a 2
285.n odd 6 1 3800.1.bd.c 2
285.v even 12 2 3800.1.bn.b 4
456.l odd 2 1 2888.1.k.a 2
456.s odd 6 1 2888.1.f.a 1
456.s odd 6 1 2888.1.k.a 2
456.u even 6 1 152.1.k.a 2
456.u even 6 1 2888.1.f.b 1
456.x odd 6 1 608.1.o.a 2
456.bt odd 18 6 2888.1.u.d 6
456.bu even 18 6 2888.1.u.c 6
2280.co even 6 1 3800.1.bd.c 2
2280.dj odd 12 2 3800.1.bn.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.k.a 2 3.b odd 2 1
152.1.k.a 2 24.f even 2 1
152.1.k.a 2 57.h odd 6 1
152.1.k.a 2 456.u even 6 1
608.1.o.a 2 12.b even 2 1
608.1.o.a 2 24.h odd 2 1
608.1.o.a 2 228.m even 6 1
608.1.o.a 2 456.x odd 6 1
1368.1.bz.a 2 1.a even 1 1 trivial
1368.1.bz.a 2 8.d odd 2 1 CM
1368.1.bz.a 2 19.c even 3 1 inner
1368.1.bz.a 2 152.k odd 6 1 inner
2888.1.f.a 1 57.f even 6 1
2888.1.f.a 1 456.s odd 6 1
2888.1.f.b 1 57.h odd 6 1
2888.1.f.b 1 456.u even 6 1
2888.1.k.a 2 57.d even 2 1
2888.1.k.a 2 57.f even 6 1
2888.1.k.a 2 456.l odd 2 1
2888.1.k.a 2 456.s odd 6 1
2888.1.u.c 6 57.l odd 18 6
2888.1.u.c 6 456.bu even 18 6
2888.1.u.d 6 57.j even 18 6
2888.1.u.d 6 456.bt odd 18 6
3800.1.bd.c 2 15.d odd 2 1
3800.1.bd.c 2 120.m even 2 1
3800.1.bd.c 2 285.n odd 6 1
3800.1.bd.c 2 2280.co even 6 1
3800.1.bn.b 4 15.e even 4 2
3800.1.bn.b 4 120.q odd 4 2
3800.1.bn.b 4 285.v even 12 2
3800.1.bn.b 4 2280.dj odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1368, [\chi])\).

Hecke characteristic polynomials

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