Properties

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

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 1352 = 2^{3} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1352.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.674735897080\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
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 $C_4\times S_3$
Artin field Galois closure of 12.0.43436029431808.1

$q$-expansion

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

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

Character values

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

\(n\) \(677\) \(1015\) \(1185\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
339.1
1.00000i
1.00000i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 1.00000i 1.00000i 0 1.00000
339.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 1.00000i 1.00000i 0 1.00000
\(n\): 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

Twists

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

Hecke kernels

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

\( T_{3} + 1 \)
\( T_{11} \)

Hecke characteristic polynomials

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