Properties

Label 1352.1.p.b
Level 1352
Weight 1
Character orbit 1352.p
Analytic conductor 0.675
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM discriminant -104
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.674735897080\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
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 $C_6\times S_3$
Artin field Galois closure of 12.0.36138776487264256.6

$q$-expansion

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

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

Character values

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

\(n\) \(677\) \(1015\) \(1185\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
147.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i −0.500000 0.866025i −1.00000 0 0.500000 0.866025i
699.1 0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i −1.00000 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 \(\Q(\sqrt{-26}) \)
13.c even 3 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.p.b 2
8.d odd 2 1 1352.1.p.a 2
13.b even 2 1 1352.1.p.a 2
13.c even 3 1 104.1.h.a 1
13.c even 3 1 inner 1352.1.p.b 2
13.d odd 4 2 1352.1.n.a 4
13.e even 6 1 104.1.h.b yes 1
13.e even 6 1 1352.1.p.a 2
13.f odd 12 2 1352.1.g.a 2
13.f odd 12 2 1352.1.n.a 4
39.h odd 6 1 936.1.o.a 1
39.i odd 6 1 936.1.o.b 1
52.i odd 6 1 416.1.h.a 1
52.j odd 6 1 416.1.h.b 1
65.l even 6 1 2600.1.o.b 1
65.n even 6 1 2600.1.o.d 1
65.q odd 12 2 2600.1.b.b 2
65.r odd 12 2 2600.1.b.a 2
104.h odd 2 1 CM 1352.1.p.b 2
104.m even 4 2 1352.1.n.a 4
104.n odd 6 1 104.1.h.b yes 1
104.n odd 6 1 1352.1.p.a 2
104.p odd 6 1 104.1.h.a 1
104.p odd 6 1 inner 1352.1.p.b 2
104.r even 6 1 416.1.h.a 1
104.s even 6 1 416.1.h.b 1
104.u even 12 2 1352.1.g.a 2
104.u even 12 2 1352.1.n.a 4
156.p even 6 1 3744.1.o.a 1
156.r even 6 1 3744.1.o.b 1
208.bg odd 12 2 3328.1.c.e 2
208.bh even 12 2 3328.1.c.e 2
208.bi odd 12 2 3328.1.c.a 2
208.bj even 12 2 3328.1.c.a 2
312.ba even 6 1 936.1.o.b 1
312.bg odd 6 1 3744.1.o.a 1
312.bh odd 6 1 3744.1.o.b 1
312.bn even 6 1 936.1.o.a 1
520.bx odd 6 1 2600.1.o.b 1
520.cd odd 6 1 2600.1.o.d 1
520.cm even 12 2 2600.1.b.a 2
520.cs even 12 2 2600.1.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.1.h.a 1 13.c even 3 1
104.1.h.a 1 104.p odd 6 1
104.1.h.b yes 1 13.e even 6 1
104.1.h.b yes 1 104.n odd 6 1
416.1.h.a 1 52.i odd 6 1
416.1.h.a 1 104.r even 6 1
416.1.h.b 1 52.j odd 6 1
416.1.h.b 1 104.s even 6 1
936.1.o.a 1 39.h odd 6 1
936.1.o.a 1 312.bn even 6 1
936.1.o.b 1 39.i odd 6 1
936.1.o.b 1 312.ba even 6 1
1352.1.g.a 2 13.f odd 12 2
1352.1.g.a 2 104.u even 12 2
1352.1.n.a 4 13.d odd 4 2
1352.1.n.a 4 13.f odd 12 2
1352.1.n.a 4 104.m even 4 2
1352.1.n.a 4 104.u even 12 2
1352.1.p.a 2 8.d odd 2 1
1352.1.p.a 2 13.b even 2 1
1352.1.p.a 2 13.e even 6 1
1352.1.p.a 2 104.n odd 6 1
1352.1.p.b 2 1.a even 1 1 trivial
1352.1.p.b 2 13.c even 3 1 inner
1352.1.p.b 2 104.h odd 2 1 CM
1352.1.p.b 2 104.p odd 6 1 inner
2600.1.b.a 2 65.r odd 12 2
2600.1.b.a 2 520.cm even 12 2
2600.1.b.b 2 65.q odd 12 2
2600.1.b.b 2 520.cs even 12 2
2600.1.o.b 1 65.l even 6 1
2600.1.o.b 1 520.bx odd 6 1
2600.1.o.d 1 65.n even 6 1
2600.1.o.d 1 520.cd odd 6 1
3328.1.c.a 2 208.bi odd 12 2
3328.1.c.a 2 208.bj even 12 2
3328.1.c.e 2 208.bg odd 12 2
3328.1.c.e 2 208.bh even 12 2
3744.1.o.a 1 156.p even 6 1
3744.1.o.a 1 312.bg odd 6 1
3744.1.o.b 1 156.r even 6 1
3744.1.o.b 1 312.bh odd 6 1

Hecke kernels

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

\( T_{3}^{2} - T_{3} + 1 \)
\( T_{5} - 1 \)

Hecke characteristic polynomials

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