Properties

Label 416.1.h.a
Level 416
Weight 1
Character orbit 416.h
Self dual yes
Analytic conductor 0.208
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM discriminant -104
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 416.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.207611045255\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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 $D_6$
Artin field Galois closure of 6.0.173056.2

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
207.1
0
0 1.00000 0 −1.00000 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 \(\Q(\sqrt{-26}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.1.h.a 1
3.b odd 2 1 3744.1.o.b 1
4.b odd 2 1 104.1.h.b yes 1
8.b even 2 1 104.1.h.a 1
8.d odd 2 1 416.1.h.b 1
12.b even 2 1 936.1.o.a 1
13.b even 2 1 416.1.h.b 1
16.e even 4 2 3328.1.c.a 2
16.f odd 4 2 3328.1.c.e 2
20.d odd 2 1 2600.1.o.b 1
20.e even 4 2 2600.1.b.a 2
24.f even 2 1 3744.1.o.a 1
24.h odd 2 1 936.1.o.b 1
39.d odd 2 1 3744.1.o.a 1
40.f even 2 1 2600.1.o.d 1
40.i odd 4 2 2600.1.b.b 2
52.b odd 2 1 104.1.h.a 1
52.f even 4 2 1352.1.g.a 2
52.i odd 6 2 1352.1.p.b 2
52.j odd 6 2 1352.1.p.a 2
52.l even 12 4 1352.1.n.a 4
104.e even 2 1 104.1.h.b yes 1
104.h odd 2 1 CM 416.1.h.a 1
104.j odd 4 2 1352.1.g.a 2
104.r even 6 2 1352.1.p.b 2
104.s even 6 2 1352.1.p.a 2
104.x odd 12 4 1352.1.n.a 4
156.h even 2 1 936.1.o.b 1
208.o odd 4 2 3328.1.c.a 2
208.p even 4 2 3328.1.c.e 2
260.g odd 2 1 2600.1.o.d 1
260.p even 4 2 2600.1.b.b 2
312.b odd 2 1 936.1.o.a 1
312.h even 2 1 3744.1.o.b 1
520.p even 2 1 2600.1.o.b 1
520.bg odd 4 2 2600.1.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.1.h.a 1 8.b even 2 1
104.1.h.a 1 52.b odd 2 1
104.1.h.b yes 1 4.b odd 2 1
104.1.h.b yes 1 104.e even 2 1
416.1.h.a 1 1.a even 1 1 trivial
416.1.h.a 1 104.h odd 2 1 CM
416.1.h.b 1 8.d odd 2 1
416.1.h.b 1 13.b even 2 1
936.1.o.a 1 12.b even 2 1
936.1.o.a 1 312.b odd 2 1
936.1.o.b 1 24.h odd 2 1
936.1.o.b 1 156.h even 2 1
1352.1.g.a 2 52.f even 4 2
1352.1.g.a 2 104.j odd 4 2
1352.1.n.a 4 52.l even 12 4
1352.1.n.a 4 104.x odd 12 4
1352.1.p.a 2 52.j odd 6 2
1352.1.p.a 2 104.s even 6 2
1352.1.p.b 2 52.i odd 6 2
1352.1.p.b 2 104.r even 6 2
2600.1.b.a 2 20.e even 4 2
2600.1.b.a 2 520.bg odd 4 2
2600.1.b.b 2 40.i odd 4 2
2600.1.b.b 2 260.p even 4 2
2600.1.o.b 1 20.d odd 2 1
2600.1.o.b 1 520.p even 2 1
2600.1.o.d 1 40.f even 2 1
2600.1.o.d 1 260.g odd 2 1
3328.1.c.a 2 16.e even 4 2
3328.1.c.a 2 208.o odd 4 2
3328.1.c.e 2 16.f odd 4 2
3328.1.c.e 2 208.p even 4 2
3744.1.o.a 1 24.f even 2 1
3744.1.o.a 1 39.d odd 2 1
3744.1.o.b 1 3.b odd 2 1
3744.1.o.b 1 312.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(416, [\chi])\).

Hecke characteristic polynomials

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