Properties

Label 416.1.t.a
Level $416$
Weight $1$
Character orbit 416.t
Analytic conductor $0.208$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 416.t (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.207611045255\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.35152.1
Artin image: $C_4\wr C_2$
Artin field: Galois closure of 8.0.143982592.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + (i + 1) q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (i + 1) q^{5} - q^{9} + i q^{13} - i q^{17} + i q^{25} + ( - i + 1) q^{37} + ( - i - 1) q^{41} + ( - i - 1) q^{45} + i q^{49} - q^{53} - q^{61} + (i - 1) q^{65} + ( - i + 1) q^{73} + q^{81} + ( - 2 i + 2) q^{85} + (i - 1) q^{89} + (i + 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{9} + 2 q^{37} - 2 q^{41} - 2 q^{45} - 4 q^{53} - 4 q^{61} - 2 q^{65} + 2 q^{73} + 2 q^{81} + 4 q^{85} - 2 q^{89} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
1.00000i
1.00000i
0 0 0 1.00000 1.00000i 0 0 0 −1.00000 0
385.1 0 0 0 1.00000 + 1.00000i 0 0 0 −1.00000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
13.d odd 4 1 inner
52.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.1.t.a 2
3.b odd 2 1 3744.1.bd.b 2
4.b odd 2 1 CM 416.1.t.a 2
8.b even 2 1 832.1.t.a 2
8.d odd 2 1 832.1.t.a 2
12.b even 2 1 3744.1.bd.b 2
13.d odd 4 1 inner 416.1.t.a 2
16.e even 4 1 3328.1.j.a 2
16.e even 4 1 3328.1.j.b 2
16.f odd 4 1 3328.1.j.a 2
16.f odd 4 1 3328.1.j.b 2
39.f even 4 1 3744.1.bd.b 2
52.f even 4 1 inner 416.1.t.a 2
104.j odd 4 1 832.1.t.a 2
104.m even 4 1 832.1.t.a 2
156.l odd 4 1 3744.1.bd.b 2
208.l even 4 1 3328.1.j.a 2
208.m odd 4 1 3328.1.j.a 2
208.r odd 4 1 3328.1.j.b 2
208.s even 4 1 3328.1.j.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.1.t.a 2 1.a even 1 1 trivial
416.1.t.a 2 4.b odd 2 1 CM
416.1.t.a 2 13.d odd 4 1 inner
416.1.t.a 2 52.f even 4 1 inner
832.1.t.a 2 8.b even 2 1
832.1.t.a 2 8.d odd 2 1
832.1.t.a 2 104.j odd 4 1
832.1.t.a 2 104.m even 4 1
3328.1.j.a 2 16.e even 4 1
3328.1.j.a 2 16.f odd 4 1
3328.1.j.a 2 208.l even 4 1
3328.1.j.a 2 208.m odd 4 1
3328.1.j.b 2 16.e even 4 1
3328.1.j.b 2 16.f odd 4 1
3328.1.j.b 2 208.r odd 4 1
3328.1.j.b 2 208.s even 4 1
3744.1.bd.b 2 3.b odd 2 1
3744.1.bd.b 2 12.b even 2 1
3744.1.bd.b 2 39.f even 4 1
3744.1.bd.b 2 156.l odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(416, [\chi])\).

Hecke characteristic polynomials

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