Properties

Label 3920.2.k.a
Level $3920$
Weight $2$
Character orbit 3920.k
Analytic conductor $31.301$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.3013575923\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{3} + \zeta_{16}^{4} q^{5} + ( 2 - \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{3} + \zeta_{16}^{4} q^{5} + ( 2 - \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{9} + ( \zeta_{16}^{2} - 2 \zeta_{16}^{3} + \zeta_{16}^{4} - 2 \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{11} + ( -\zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{4} - \zeta_{16}^{5} - \zeta_{16}^{6} ) q^{13} + ( \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{15} + ( \zeta_{16} - 2 \zeta_{16}^{3} + 3 \zeta_{16}^{4} - 2 \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{17} + ( 2 - \zeta_{16} + 3 \zeta_{16}^{3} - 3 \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{19} + ( -\zeta_{16} + 3 \zeta_{16}^{2} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{4} - 2 \zeta_{16}^{5} + 3 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{23} - q^{25} + ( -1 + 2 \zeta_{16}^{2} - 5 \zeta_{16}^{3} + 5 \zeta_{16}^{5} - 2 \zeta_{16}^{6} ) q^{27} + ( 1 + 2 \zeta_{16} - 3 \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 3 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{29} + ( 6 + \zeta_{16} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{31} + ( -\zeta_{16} - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 3 \zeta_{16}^{4} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{33} + ( 3 \zeta_{16} + \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{37} + ( -2 \zeta_{16} + \zeta_{16}^{2} - 5 \zeta_{16}^{4} + \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{39} + ( -2 \zeta_{16} - 2 \zeta_{16}^{2} - \zeta_{16}^{3} + 2 \zeta_{16}^{4} - \zeta_{16}^{5} - 2 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{41} + ( 3 \zeta_{16} - 2 \zeta_{16}^{2} + 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} - 2 \zeta_{16}^{6} + 3 \zeta_{16}^{7} ) q^{43} + ( 2 \zeta_{16} - \zeta_{16}^{2} + 2 \zeta_{16}^{4} - \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{45} + ( 3 + \zeta_{16} + 3 \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 3 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{47} + ( -4 \zeta_{16} + 2 \zeta_{16}^{2} + 4 \zeta_{16}^{3} - 7 \zeta_{16}^{4} + 4 \zeta_{16}^{5} + 2 \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{51} + ( 4 - 2 \zeta_{16} + \zeta_{16}^{2} + 3 \zeta_{16}^{3} - 3 \zeta_{16}^{5} - \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{53} + ( -1 + 2 \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{55} + ( -4 + 5 \zeta_{16} + 4 \zeta_{16}^{2} - 7 \zeta_{16}^{3} + 7 \zeta_{16}^{5} - 4 \zeta_{16}^{6} - 5 \zeta_{16}^{7} ) q^{57} + ( 2 + 4 \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{5} - 4 \zeta_{16}^{6} ) q^{59} + ( \zeta_{16} - 4 \zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{5} - 4 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{61} + ( -1 + \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{65} + ( -4 \zeta_{16} - 3 \zeta_{16}^{2} - \zeta_{16}^{3} - 2 \zeta_{16}^{4} - \zeta_{16}^{5} - 3 \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{67} + ( 3 \zeta_{16} - 8 \zeta_{16}^{2} + 4 \zeta_{16}^{4} - 8 \zeta_{16}^{6} + 3 \zeta_{16}^{7} ) q^{69} + ( -4 \zeta_{16} + 7 \zeta_{16}^{2} + 2 \zeta_{16}^{3} + 4 \zeta_{16}^{4} + 2 \zeta_{16}^{5} + 7 \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{71} + ( -\zeta_{16} + 5 \zeta_{16}^{2} + 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} + 5 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{73} + ( 1 - \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{75} + ( 2 \zeta_{16}^{2} - 2 \zeta_{16}^{3} - 5 \zeta_{16}^{4} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{6} ) q^{79} + ( -1 - 4 \zeta_{16} - 5 \zeta_{16}^{2} + 6 \zeta_{16}^{3} - 6 \zeta_{16}^{5} + 5 \zeta_{16}^{6} + 4 \zeta_{16}^{7} ) q^{81} + ( -6 + 4 \zeta_{16} - 4 \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 4 \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{83} + ( -3 + 2 \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{85} + ( -3 + 8 \zeta_{16}^{2} - 5 \zeta_{16}^{3} + 5 \zeta_{16}^{5} - 8 \zeta_{16}^{6} ) q^{87} + ( -4 \zeta_{16} - 2 \zeta_{16}^{2} - 2 \zeta_{16}^{3} - 8 \zeta_{16}^{4} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{89} + ( -4 + 4 \zeta_{16} + 5 \zeta_{16}^{2} + 5 \zeta_{16}^{3} - 5 \zeta_{16}^{5} - 5 \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{93} + ( 3 \zeta_{16} - \zeta_{16}^{3} + 2 \zeta_{16}^{4} - \zeta_{16}^{5} + 3 \zeta_{16}^{7} ) q^{95} + ( -6 \zeta_{16} + \zeta_{16}^{2} + 5 \zeta_{16}^{3} + \zeta_{16}^{4} + 5 \zeta_{16}^{5} + \zeta_{16}^{6} - 6 \zeta_{16}^{7} ) q^{97} + ( 2 \zeta_{16} - 3 \zeta_{16}^{2} - 3 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{3} + 16q^{9} + O(q^{10}) \) \( 8q - 8q^{3} + 16q^{9} + 16q^{19} - 8q^{25} - 8q^{27} + 8q^{29} + 48q^{31} + 24q^{47} + 32q^{53} - 8q^{55} - 32q^{57} + 16q^{59} - 8q^{65} + 8q^{75} - 8q^{81} - 48q^{83} - 24q^{85} - 24q^{87} - 32q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2351.1
−0.382683 0.923880i
−0.382683 + 0.923880i
0.382683 + 0.923880i
0.382683 0.923880i
−0.923880 + 0.382683i
−0.923880 0.382683i
0.923880 0.382683i
0.923880 + 0.382683i
0 −3.17958 0 1.00000i 0 0 0 7.10973 0
2351.2 0 −3.17958 0 1.00000i 0 0 0 7.10973 0
2351.3 0 −1.64885 0 1.00000i 0 0 0 −0.281305 0
2351.4 0 −1.64885 0 1.00000i 0 0 0 −0.281305 0
2351.5 0 −1.43355 0 1.00000i 0 0 0 −0.944947 0
2351.6 0 −1.43355 0 1.00000i 0 0 0 −0.944947 0
2351.7 0 2.26197 0 1.00000i 0 0 0 2.11652 0
2351.8 0 2.26197 0 1.00000i 0 0 0 2.11652 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2351.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.2.k.a 8
4.b odd 2 1 3920.2.k.c yes 8
7.b odd 2 1 3920.2.k.c yes 8
28.d even 2 1 inner 3920.2.k.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3920.2.k.a 8 1.a even 1 1 trivial
3920.2.k.a 8 28.d even 2 1 inner
3920.2.k.c yes 8 4.b odd 2 1
3920.2.k.c yes 8 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 4 T_{3}^{3} - 2 T_{3}^{2} - 20 T_{3} - 17 \) acting on \(S_{2}^{\mathrm{new}}(3920, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( -17 - 20 T - 2 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$5$ \( ( 1 + T^{2} )^{4} \)
$7$ \( T^{8} \)
$11$ \( 289 + 492 T^{2} + 262 T^{4} + 44 T^{6} + T^{8} \)
$13$ \( 289 + 332 T^{2} + 130 T^{4} + 20 T^{6} + T^{8} \)
$17$ \( 1 + 212 T^{2} + 1442 T^{4} + 76 T^{6} + T^{8} \)
$19$ \( ( -136 + 128 T - 16 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$23$ \( 334084 + 92608 T^{2} + 5628 T^{4} + 128 T^{6} + T^{8} \)
$29$ \( ( -383 + 324 T - 62 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$31$ \( ( 578 - 624 T + 196 T^{2} - 24 T^{3} + T^{4} )^{2} \)
$37$ \( ( -2 - 136 T - 56 T^{2} + T^{4} )^{2} \)
$41$ \( 24964 + 12592 T^{2} + 1876 T^{4} + 88 T^{6} + T^{8} \)
$43$ \( 18496 + 106880 T^{2} + 8016 T^{4} + 176 T^{6} + T^{8} \)
$47$ \( ( -289 + 204 T - 2 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$53$ \( ( -1186 + 328 T + 40 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$59$ \( ( 578 + 272 T - 44 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$61$ \( 602176 + 115840 T^{2} + 6736 T^{4} + 144 T^{6} + T^{8} \)
$67$ \( 334084 + 212224 T^{2} + 14076 T^{4} + 224 T^{6} + T^{8} \)
$71$ \( 74580496 + 9281440 T^{2} + 127768 T^{4} + 616 T^{6} + T^{8} \)
$73$ \( 795664 + 256160 T^{2} + 21384 T^{4} + 280 T^{6} + T^{8} \)
$79$ \( 277729 + 137828 T^{2} + 8230 T^{4} + 164 T^{6} + T^{8} \)
$83$ \( ( -4624 - 992 T + 72 T^{2} + 24 T^{3} + T^{4} )^{2} \)
$89$ \( 295936 + 645120 T^{2} + 44992 T^{4} + 448 T^{6} + T^{8} \)
$97$ \( 12271009 + 2819052 T^{2} + 74050 T^{4} + 500 T^{6} + T^{8} \)
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