Properties

Label 1352.1.n.a
Level 1352
Weight 1
Character orbit 1352.n
Analytic conductor 0.675
Analytic rank 0
Dimension 4
Projective image \(D_{3}\)
CM discriminant -104
Inner twists 8

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Newspace parameters

Level: \( N \) \(=\) \( 1352 = 2^{3} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1352.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.674735897080\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.104.1
Artin image $C_{12}\times S_3$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{72} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{12} q^{2} -\zeta_{12}^{4} q^{3} + \zeta_{12}^{2} q^{4} -\zeta_{12}^{3} q^{5} + \zeta_{12}^{5} q^{6} -\zeta_{12}^{5} q^{7} -\zeta_{12}^{3} q^{8} +O(q^{10})\) \( q -\zeta_{12} q^{2} -\zeta_{12}^{4} q^{3} + \zeta_{12}^{2} q^{4} -\zeta_{12}^{3} q^{5} + \zeta_{12}^{5} q^{6} -\zeta_{12}^{5} q^{7} -\zeta_{12}^{3} q^{8} + \zeta_{12}^{4} q^{10} + q^{12} - q^{14} -\zeta_{12} q^{15} + \zeta_{12}^{4} q^{16} -\zeta_{12}^{2} q^{17} -\zeta_{12}^{5} q^{20} -\zeta_{12}^{3} q^{21} -\zeta_{12} q^{24} + q^{27} + \zeta_{12} q^{28} + \zeta_{12}^{2} q^{30} + 2 \zeta_{12}^{3} q^{31} -\zeta_{12}^{5} q^{32} + \zeta_{12}^{3} q^{34} -\zeta_{12}^{2} q^{35} -\zeta_{12} q^{37} - q^{40} + \zeta_{12}^{4} q^{42} -\zeta_{12}^{2} q^{43} + \zeta_{12}^{3} q^{47} + \zeta_{12}^{2} q^{48} - q^{51} -\zeta_{12} q^{54} -\zeta_{12}^{2} q^{56} -\zeta_{12}^{3} q^{60} -2 \zeta_{12}^{4} q^{62} - q^{64} -\zeta_{12}^{4} q^{68} + \zeta_{12}^{3} q^{70} + \zeta_{12}^{5} q^{71} + \zeta_{12}^{2} q^{74} + \zeta_{12} q^{80} -\zeta_{12}^{4} q^{81} -\zeta_{12}^{5} q^{84} + \zeta_{12}^{5} q^{85} + \zeta_{12}^{3} q^{86} + 2 \zeta_{12} q^{93} -\zeta_{12}^{4} q^{94} -\zeta_{12}^{3} q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + 2q^{4} + O(q^{10}) \) \( 4q + 2q^{3} + 2q^{4} - 2q^{10} + 4q^{12} - 4q^{14} - 2q^{16} - 2q^{17} + 4q^{27} + 2q^{30} - 2q^{35} - 4q^{40} - 2q^{42} - 2q^{43} + 2q^{48} - 4q^{51} - 2q^{56} + 4q^{62} - 4q^{64} + 2q^{68} + 2q^{74} + 2q^{81} + 2q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1185\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
315.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 1.00000i −0.866025 0.500000i 0.866025 + 0.500000i 1.00000i 0 −0.500000 0.866025i
315.2 0.866025 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 1.00000i 0.866025 + 0.500000i −0.866025 0.500000i 1.00000i 0 −0.500000 0.866025i
867.1 −0.866025 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 1.00000i −0.866025 + 0.500000i 0.866025 0.500000i 1.00000i 0 −0.500000 + 0.866025i
867.2 0.866025 + 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 1.00000i 0.866025 0.500000i −0.866025 + 0.500000i 1.00000i 0 −0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
104.h odd 2 1 CM by \(\Q(\sqrt{-26}) \)
8.d odd 2 1 inner
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner
104.n odd 6 1 inner
104.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1352.1.n.a 4
8.d odd 2 1 inner 1352.1.n.a 4
13.b even 2 1 inner 1352.1.n.a 4
13.c even 3 1 1352.1.g.a 2
13.c even 3 1 inner 1352.1.n.a 4
13.d odd 4 1 1352.1.p.a 2
13.d odd 4 1 1352.1.p.b 2
13.e even 6 1 1352.1.g.a 2
13.e even 6 1 inner 1352.1.n.a 4
13.f odd 12 1 104.1.h.a 1
13.f odd 12 1 104.1.h.b yes 1
13.f odd 12 1 1352.1.p.a 2
13.f odd 12 1 1352.1.p.b 2
39.k even 12 1 936.1.o.a 1
39.k even 12 1 936.1.o.b 1
52.l even 12 1 416.1.h.a 1
52.l even 12 1 416.1.h.b 1
65.o even 12 1 2600.1.b.a 2
65.o even 12 1 2600.1.b.b 2
65.s odd 12 1 2600.1.o.b 1
65.s odd 12 1 2600.1.o.d 1
65.t even 12 1 2600.1.b.a 2
65.t even 12 1 2600.1.b.b 2
104.h odd 2 1 CM 1352.1.n.a 4
104.m even 4 1 1352.1.p.a 2
104.m even 4 1 1352.1.p.b 2
104.n odd 6 1 1352.1.g.a 2
104.n odd 6 1 inner 1352.1.n.a 4
104.p odd 6 1 1352.1.g.a 2
104.p odd 6 1 inner 1352.1.n.a 4
104.u even 12 1 104.1.h.a 1
104.u even 12 1 104.1.h.b yes 1
104.u even 12 1 1352.1.p.a 2
104.u even 12 1 1352.1.p.b 2
104.x odd 12 1 416.1.h.a 1
104.x odd 12 1 416.1.h.b 1
156.v odd 12 1 3744.1.o.a 1
156.v odd 12 1 3744.1.o.b 1
208.be odd 12 1 3328.1.c.a 2
208.be odd 12 1 3328.1.c.e 2
208.bf even 12 1 3328.1.c.a 2
208.bf even 12 1 3328.1.c.e 2
208.bk even 12 1 3328.1.c.a 2
208.bk even 12 1 3328.1.c.e 2
208.bl odd 12 1 3328.1.c.a 2
208.bl odd 12 1 3328.1.c.e 2
312.bo even 12 1 3744.1.o.a 1
312.bo even 12 1 3744.1.o.b 1
312.bq odd 12 1 936.1.o.a 1
312.bq odd 12 1 936.1.o.b 1
520.ci odd 12 1 2600.1.b.a 2
520.ci odd 12 1 2600.1.b.b 2
520.cv odd 12 1 2600.1.b.a 2
520.cv odd 12 1 2600.1.b.b 2
520.cz even 12 1 2600.1.o.b 1
520.cz even 12 1 2600.1.o.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.1.h.a 1 13.f odd 12 1
104.1.h.a 1 104.u even 12 1
104.1.h.b yes 1 13.f odd 12 1
104.1.h.b yes 1 104.u even 12 1
416.1.h.a 1 52.l even 12 1
416.1.h.a 1 104.x odd 12 1
416.1.h.b 1 52.l even 12 1
416.1.h.b 1 104.x odd 12 1
936.1.o.a 1 39.k even 12 1
936.1.o.a 1 312.bq odd 12 1
936.1.o.b 1 39.k even 12 1
936.1.o.b 1 312.bq odd 12 1
1352.1.g.a 2 13.c even 3 1
1352.1.g.a 2 13.e even 6 1
1352.1.g.a 2 104.n odd 6 1
1352.1.g.a 2 104.p odd 6 1
1352.1.n.a 4 1.a even 1 1 trivial
1352.1.n.a 4 8.d odd 2 1 inner
1352.1.n.a 4 13.b even 2 1 inner
1352.1.n.a 4 13.c even 3 1 inner
1352.1.n.a 4 13.e even 6 1 inner
1352.1.n.a 4 104.h odd 2 1 CM
1352.1.n.a 4 104.n odd 6 1 inner
1352.1.n.a 4 104.p odd 6 1 inner
1352.1.p.a 2 13.d odd 4 1
1352.1.p.a 2 13.f odd 12 1
1352.1.p.a 2 104.m even 4 1
1352.1.p.a 2 104.u even 12 1
1352.1.p.b 2 13.d odd 4 1
1352.1.p.b 2 13.f odd 12 1
1352.1.p.b 2 104.m even 4 1
1352.1.p.b 2 104.u even 12 1
2600.1.b.a 2 65.o even 12 1
2600.1.b.a 2 65.t even 12 1
2600.1.b.a 2 520.ci odd 12 1
2600.1.b.a 2 520.cv odd 12 1
2600.1.b.b 2 65.o even 12 1
2600.1.b.b 2 65.t even 12 1
2600.1.b.b 2 520.ci odd 12 1
2600.1.b.b 2 520.cv odd 12 1
2600.1.o.b 1 65.s odd 12 1
2600.1.o.b 1 520.cz even 12 1
2600.1.o.d 1 65.s odd 12 1
2600.1.o.d 1 520.cz even 12 1
3328.1.c.a 2 208.be odd 12 1
3328.1.c.a 2 208.bf even 12 1
3328.1.c.a 2 208.bk even 12 1
3328.1.c.a 2 208.bl odd 12 1
3328.1.c.e 2 208.be odd 12 1
3328.1.c.e 2 208.bf even 12 1
3328.1.c.e 2 208.bk even 12 1
3328.1.c.e 2 208.bl odd 12 1
3744.1.o.a 1 156.v odd 12 1
3744.1.o.a 1 312.bo even 12 1
3744.1.o.b 1 156.v odd 12 1
3744.1.o.b 1 312.bo even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1352, [\chi])\):

\( T_{3}^{2} - T_{3} + 1 \)
\( T_{11} \)

Hecke characteristic polynomials

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