Properties

Label 3200.2.f.o
Level 3200
Weight 2
Character orbit 3200.f
Analytic conductor 25.552
Analytic rank 0
Dimension 4
CM discriminant -8
Inner twists 8

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{3} + 5 q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{3} + 5 q^{9} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{11} -6 \zeta_{8}^{2} q^{17} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{19} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{27} -8 \zeta_{8}^{2} q^{33} -6 q^{41} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{43} + 7 q^{49} + ( -12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{51} -24 \zeta_{8}^{2} q^{57} + ( 10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{59} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{67} -2 \zeta_{8}^{2} q^{73} + q^{81} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{83} + 18 q^{89} + 10 \zeta_{8}^{2} q^{97} + ( -10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 20q^{9} + O(q^{10}) \) \( 4q + 20q^{9} - 24q^{41} + 28q^{49} + 4q^{81} + 72q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
0 −2.82843 0 0 0 0 0 5.00000 0
449.2 0 −2.82843 0 0 0 0 0 5.00000 0
449.3 0 2.82843 0 0 0 0 0 5.00000 0
449.4 0 2.82843 0 0 0 0 0 5.00000 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
20.d odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.2.f.o 4
4.b odd 2 1 inner 3200.2.f.o 4
5.b even 2 1 inner 3200.2.f.o 4
5.c odd 4 1 128.2.b.a 2
5.c odd 4 1 3200.2.d.c 2
8.b even 2 1 inner 3200.2.f.o 4
8.d odd 2 1 CM 3200.2.f.o 4
15.e even 4 1 1152.2.d.c 2
20.d odd 2 1 inner 3200.2.f.o 4
20.e even 4 1 128.2.b.a 2
20.e even 4 1 3200.2.d.c 2
40.e odd 2 1 inner 3200.2.f.o 4
40.f even 2 1 inner 3200.2.f.o 4
40.i odd 4 1 128.2.b.a 2
40.i odd 4 1 3200.2.d.c 2
40.k even 4 1 128.2.b.a 2
40.k even 4 1 3200.2.d.c 2
60.l odd 4 1 1152.2.d.c 2
80.i odd 4 1 256.2.a.e 2
80.i odd 4 1 6400.2.a.by 2
80.j even 4 1 256.2.a.e 2
80.j even 4 1 6400.2.a.by 2
80.s even 4 1 256.2.a.e 2
80.s even 4 1 6400.2.a.by 2
80.t odd 4 1 256.2.a.e 2
80.t odd 4 1 6400.2.a.by 2
120.q odd 4 1 1152.2.d.c 2
120.w even 4 1 1152.2.d.c 2
160.u even 8 1 1024.2.e.a 2
160.u even 8 1 1024.2.e.f 2
160.v odd 8 1 1024.2.e.a 2
160.v odd 8 1 1024.2.e.f 2
160.ba even 8 1 1024.2.e.a 2
160.ba even 8 1 1024.2.e.f 2
160.bb odd 8 1 1024.2.e.a 2
160.bb odd 8 1 1024.2.e.f 2
240.z odd 4 1 2304.2.a.t 2
240.bb even 4 1 2304.2.a.t 2
240.bd odd 4 1 2304.2.a.t 2
240.bf even 4 1 2304.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.a 2 5.c odd 4 1
128.2.b.a 2 20.e even 4 1
128.2.b.a 2 40.i odd 4 1
128.2.b.a 2 40.k even 4 1
256.2.a.e 2 80.i odd 4 1
256.2.a.e 2 80.j even 4 1
256.2.a.e 2 80.s even 4 1
256.2.a.e 2 80.t odd 4 1
1024.2.e.a 2 160.u even 8 1
1024.2.e.a 2 160.v odd 8 1
1024.2.e.a 2 160.ba even 8 1
1024.2.e.a 2 160.bb odd 8 1
1024.2.e.f 2 160.u even 8 1
1024.2.e.f 2 160.v odd 8 1
1024.2.e.f 2 160.ba even 8 1
1024.2.e.f 2 160.bb odd 8 1
1152.2.d.c 2 15.e even 4 1
1152.2.d.c 2 60.l odd 4 1
1152.2.d.c 2 120.q odd 4 1
1152.2.d.c 2 120.w even 4 1
2304.2.a.t 2 240.z odd 4 1
2304.2.a.t 2 240.bb even 4 1
2304.2.a.t 2 240.bd odd 4 1
2304.2.a.t 2 240.bf even 4 1
3200.2.d.c 2 5.c odd 4 1
3200.2.d.c 2 20.e even 4 1
3200.2.d.c 2 40.i odd 4 1
3200.2.d.c 2 40.k even 4 1
3200.2.f.o 4 1.a even 1 1 trivial
3200.2.f.o 4 4.b odd 2 1 inner
3200.2.f.o 4 5.b even 2 1 inner
3200.2.f.o 4 8.b even 2 1 inner
3200.2.f.o 4 8.d odd 2 1 CM
3200.2.f.o 4 20.d odd 2 1 inner
3200.2.f.o 4 40.e odd 2 1 inner
3200.2.f.o 4 40.f even 2 1 inner
6400.2.a.by 2 80.i odd 4 1
6400.2.a.by 2 80.j even 4 1
6400.2.a.by 2 80.s even 4 1
6400.2.a.by 2 80.t odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3200, [\chi])\):

\( T_{3}^{2} - 8 \)
\( T_{7} \)
\( T_{11}^{2} + 8 \)
\( T_{13} \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 2 T^{2} + 9 T^{4} )^{2} \)
$5$ 1
$7$ \( ( 1 - 7 T^{2} )^{4} \)
$11$ \( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 + 13 T^{2} )^{4} \)
$17$ \( ( 1 + 2 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 2 T + 19 T^{2} )^{2}( 1 + 2 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 23 T^{2} )^{4} \)
$29$ \( ( 1 - 29 T^{2} )^{4} \)
$31$ \( ( 1 + 31 T^{2} )^{4} \)
$37$ \( ( 1 + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 + 14 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 47 T^{2} )^{4} \)
$53$ \( ( 1 + 53 T^{2} )^{4} \)
$59$ \( ( 1 - 6 T + 59 T^{2} )^{2}( 1 + 6 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 + 62 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{4} \)
$73$ \( ( 1 - 142 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 158 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 18 T + 89 T^{2} )^{4} \)
$97$ \( ( 1 - 94 T^{2} + 9409 T^{4} )^{2} \)
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