Properties

Label 3328.1.c.e
Level 3328
Weight 1
Character orbit 3328.c
Analytic conductor 1.661
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM discriminant -104
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3328.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.66088836204\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 12.4.981348487528448.4

$q$-expansion

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

\(f(q)\) \(=\) \( q + i q^{3} -i q^{5} + q^{7} +O(q^{10})\) \( q + i q^{3} -i q^{5} + q^{7} -i q^{13} + q^{15} - q^{17} + i q^{21} + i q^{27} + 2 q^{31} -i q^{35} -i q^{37} + q^{39} -i q^{43} - q^{47} -i q^{51} - q^{65} + q^{71} - q^{81} + i q^{85} -i q^{91} + 2 i q^{93} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} + O(q^{10}) \) \( 2q + 2q^{7} + 2q^{15} - 2q^{17} + 4q^{31} + 2q^{39} - 2q^{47} - 2q^{65} + 2q^{71} - 2q^{81} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(769\) \(1535\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3327.1
1.00000i
1.00000i
0 1.00000i 0 1.00000i 0 1.00000 0 0 0
3327.2 0 1.00000i 0 1.00000i 0 1.00000 0 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
104.h odd 2 1 CM by \(\Q(\sqrt{-26}) \)
8.b even 2 1 inner
52.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3328.1.c.e 2
4.b odd 2 1 3328.1.c.a 2
8.b even 2 1 inner 3328.1.c.e 2
8.d odd 2 1 3328.1.c.a 2
13.b even 2 1 3328.1.c.a 2
16.e even 4 1 104.1.h.b yes 1
16.e even 4 1 416.1.h.b 1
16.f odd 4 1 104.1.h.a 1
16.f odd 4 1 416.1.h.a 1
48.i odd 4 1 936.1.o.a 1
48.i odd 4 1 3744.1.o.a 1
48.k even 4 1 936.1.o.b 1
48.k even 4 1 3744.1.o.b 1
52.b odd 2 1 inner 3328.1.c.e 2
80.i odd 4 1 2600.1.b.a 2
80.j even 4 1 2600.1.b.b 2
80.k odd 4 1 2600.1.o.d 1
80.q even 4 1 2600.1.o.b 1
80.s even 4 1 2600.1.b.b 2
80.t odd 4 1 2600.1.b.a 2
104.e even 2 1 3328.1.c.a 2
104.h odd 2 1 CM 3328.1.c.e 2
208.l even 4 1 1352.1.g.a 2
208.m odd 4 1 1352.1.g.a 2
208.o odd 4 1 104.1.h.b yes 1
208.o odd 4 1 416.1.h.b 1
208.p even 4 1 104.1.h.a 1
208.p even 4 1 416.1.h.a 1
208.r odd 4 1 1352.1.g.a 2
208.s even 4 1 1352.1.g.a 2
208.be odd 12 2 1352.1.n.a 4
208.bf even 12 2 1352.1.n.a 4
208.bg odd 12 2 1352.1.p.b 2
208.bh even 12 2 1352.1.p.b 2
208.bi odd 12 2 1352.1.p.a 2
208.bj even 12 2 1352.1.p.a 2
208.bk even 12 2 1352.1.n.a 4
208.bl odd 12 2 1352.1.n.a 4
624.v even 4 1 936.1.o.a 1
624.v even 4 1 3744.1.o.a 1
624.bi odd 4 1 936.1.o.b 1
624.bi odd 4 1 3744.1.o.b 1
1040.w odd 4 1 2600.1.b.b 2
1040.y even 4 1 2600.1.b.a 2
1040.be even 4 1 2600.1.o.d 1
1040.cb odd 4 1 2600.1.o.b 1
1040.co odd 4 1 2600.1.b.b 2
1040.cq even 4 1 2600.1.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.1.h.a 1 16.f odd 4 1
104.1.h.a 1 208.p even 4 1
104.1.h.b yes 1 16.e even 4 1
104.1.h.b yes 1 208.o odd 4 1
416.1.h.a 1 16.f odd 4 1
416.1.h.a 1 208.p even 4 1
416.1.h.b 1 16.e even 4 1
416.1.h.b 1 208.o odd 4 1
936.1.o.a 1 48.i odd 4 1
936.1.o.a 1 624.v even 4 1
936.1.o.b 1 48.k even 4 1
936.1.o.b 1 624.bi odd 4 1
1352.1.g.a 2 208.l even 4 1
1352.1.g.a 2 208.m odd 4 1
1352.1.g.a 2 208.r odd 4 1
1352.1.g.a 2 208.s even 4 1
1352.1.n.a 4 208.be odd 12 2
1352.1.n.a 4 208.bf even 12 2
1352.1.n.a 4 208.bk even 12 2
1352.1.n.a 4 208.bl odd 12 2
1352.1.p.a 2 208.bi odd 12 2
1352.1.p.a 2 208.bj even 12 2
1352.1.p.b 2 208.bg odd 12 2
1352.1.p.b 2 208.bh even 12 2
2600.1.b.a 2 80.i odd 4 1
2600.1.b.a 2 80.t odd 4 1
2600.1.b.a 2 1040.y even 4 1
2600.1.b.a 2 1040.cq even 4 1
2600.1.b.b 2 80.j even 4 1
2600.1.b.b 2 80.s even 4 1
2600.1.b.b 2 1040.w odd 4 1
2600.1.b.b 2 1040.co odd 4 1
2600.1.o.b 1 80.q even 4 1
2600.1.o.b 1 1040.cb odd 4 1
2600.1.o.d 1 80.k odd 4 1
2600.1.o.d 1 1040.be even 4 1
3328.1.c.a 2 4.b odd 2 1
3328.1.c.a 2 8.d odd 2 1
3328.1.c.a 2 13.b even 2 1
3328.1.c.a 2 104.e even 2 1
3328.1.c.e 2 1.a even 1 1 trivial
3328.1.c.e 2 8.b even 2 1 inner
3328.1.c.e 2 52.b odd 2 1 inner
3328.1.c.e 2 104.h odd 2 1 CM
3744.1.o.a 1 48.i odd 4 1
3744.1.o.a 1 624.v even 4 1
3744.1.o.b 1 48.k even 4 1
3744.1.o.b 1 624.bi odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3328, [\chi])\):

\( T_{3}^{2} + 1 \)
\( T_{5}^{2} + 1 \)
\( T_{7} - 1 \)
\( T_{29} \)

Hecke characteristic polynomials

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