Properties

Label 416.1.bl.a
Level $416$
Weight $1$
Character orbit 416.bl
Analytic conductor $0.208$
Analytic rank $0$
Dimension $4$
Projective image $D_{12}$
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.bl (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.207611045255\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of 12.0.469804094334435328.7

$q$-expansion

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

\(f(q)\) \(=\) \( q + (\zeta_{12}^{5} + \zeta_{12}^{4}) q^{5} + \zeta_{12}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{5} + \zeta_{12}^{4}) q^{5} + \zeta_{12}^{2} q^{9} + \zeta_{12} q^{13} + \zeta_{12}^{5} q^{17} + ( - \zeta_{12}^{4} - \zeta_{12}^{3} - \zeta_{12}^{2}) q^{25} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{29} + (\zeta_{12}^{3} - \zeta_{12}^{2}) q^{37} + ( - \zeta_{12}^{5} - 1) q^{41} + ( - \zeta_{12} - 1) q^{45} + \zeta_{12} q^{49} + q^{53} - \zeta_{12}^{2} q^{61} + (\zeta_{12}^{5} - 1) q^{65} + ( - \zeta_{12}^{5} + \zeta_{12}^{4}) q^{73} + \zeta_{12}^{4} q^{81} + ( - \zeta_{12}^{4} - \zeta_{12}^{3}) q^{85} + ( - \zeta_{12}^{4} + \zeta_{12}) q^{89} + (\zeta_{12}^{5} - \zeta_{12}^{2}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{5} + 2 q^{9} - 2 q^{37} - 4 q^{41} - 4 q^{45} + 4 q^{53} - 2 q^{61} - 4 q^{65} - 2 q^{73} - 2 q^{81} + 2 q^{85} + 2 q^{89} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{12}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0 0 0.366025 + 0.366025i 0 0 0 0.500000 + 0.866025i 0
97.1 0 0 0 −1.36603 + 1.36603i 0 0 0 0.500000 + 0.866025i 0
193.1 0 0 0 −1.36603 1.36603i 0 0 0 0.500000 0.866025i 0
353.1 0 0 0 0.366025 0.366025i 0 0 0 0.500000 0.866025i 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.f odd 12 1 inner
52.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.1.bl.a 4
3.b odd 2 1 3744.1.gs.c 4
4.b odd 2 1 CM 416.1.bl.a 4
8.b even 2 1 832.1.bl.a 4
8.d odd 2 1 832.1.bl.a 4
12.b even 2 1 3744.1.gs.c 4
13.f odd 12 1 inner 416.1.bl.a 4
16.e even 4 1 3328.1.bv.a 4
16.e even 4 1 3328.1.bv.b 4
16.f odd 4 1 3328.1.bv.a 4
16.f odd 4 1 3328.1.bv.b 4
39.k even 12 1 3744.1.gs.c 4
52.l even 12 1 inner 416.1.bl.a 4
104.u even 12 1 832.1.bl.a 4
104.x odd 12 1 832.1.bl.a 4
156.v odd 12 1 3744.1.gs.c 4
208.be odd 12 1 3328.1.bv.b 4
208.bf even 12 1 3328.1.bv.b 4
208.bk even 12 1 3328.1.bv.a 4
208.bl odd 12 1 3328.1.bv.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.1.bl.a 4 1.a even 1 1 trivial
416.1.bl.a 4 4.b odd 2 1 CM
416.1.bl.a 4 13.f odd 12 1 inner
416.1.bl.a 4 52.l even 12 1 inner
832.1.bl.a 4 8.b even 2 1
832.1.bl.a 4 8.d odd 2 1
832.1.bl.a 4 104.u even 12 1
832.1.bl.a 4 104.x odd 12 1
3328.1.bv.a 4 16.e even 4 1
3328.1.bv.a 4 16.f odd 4 1
3328.1.bv.a 4 208.bk even 12 1
3328.1.bv.a 4 208.bl odd 12 1
3328.1.bv.b 4 16.e even 4 1
3328.1.bv.b 4 16.f odd 4 1
3328.1.bv.b 4 208.be odd 12 1
3328.1.bv.b 4 208.bf even 12 1
3744.1.gs.c 4 3.b odd 2 1
3744.1.gs.c 4 12.b even 2 1
3744.1.gs.c 4 39.k even 12 1
3744.1.gs.c 4 156.v odd 12 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + 2 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} + 5 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T - 1)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} + 2 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
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