Properties

Label 3200.2.d.c
Level $3200$
Weight $2$
Character orbit 3200.d
Analytic conductor $25.552$
Analytic rank $1$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.5521286468\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} -5 q^{9} +O(q^{10})\) \( q + \beta q^{3} -5 q^{9} -\beta q^{11} -6 q^{17} + 3 \beta q^{19} -2 \beta q^{27} + 8 q^{33} -6 q^{41} -3 \beta q^{43} -7 q^{49} -6 \beta q^{51} -24 q^{57} -5 \beta q^{59} -3 \beta q^{67} + 2 q^{73} + q^{81} + \beta q^{83} -18 q^{89} + 10 q^{97} + 5 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 10q^{9} + O(q^{10}) \) \( 2q - 10q^{9} - 12q^{17} + 16q^{33} - 12q^{41} - 14q^{49} - 48q^{57} + 4q^{73} + 2q^{81} - 36q^{89} + 20q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1601.1
1.41421i
1.41421i
0 2.82843i 0 0 0 0 0 −5.00000 0
1601.2 0 2.82843i 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
8.b 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.d.c 2
4.b odd 2 1 inner 3200.2.d.c 2
5.b even 2 1 128.2.b.a 2
5.c odd 4 2 3200.2.f.o 4
8.b even 2 1 inner 3200.2.d.c 2
8.d odd 2 1 CM 3200.2.d.c 2
15.d odd 2 1 1152.2.d.c 2
16.e even 4 2 6400.2.a.by 2
16.f odd 4 2 6400.2.a.by 2
20.d odd 2 1 128.2.b.a 2
20.e even 4 2 3200.2.f.o 4
40.e odd 2 1 128.2.b.a 2
40.f even 2 1 128.2.b.a 2
40.i odd 4 2 3200.2.f.o 4
40.k even 4 2 3200.2.f.o 4
60.h even 2 1 1152.2.d.c 2
80.k odd 4 2 256.2.a.e 2
80.q even 4 2 256.2.a.e 2
120.i odd 2 1 1152.2.d.c 2
120.m even 2 1 1152.2.d.c 2
160.y odd 8 2 1024.2.e.a 2
160.y odd 8 2 1024.2.e.f 2
160.z even 8 2 1024.2.e.a 2
160.z even 8 2 1024.2.e.f 2
240.t even 4 2 2304.2.a.t 2
240.bm odd 4 2 2304.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.a 2 5.b even 2 1
128.2.b.a 2 20.d odd 2 1
128.2.b.a 2 40.e odd 2 1
128.2.b.a 2 40.f even 2 1
256.2.a.e 2 80.k odd 4 2
256.2.a.e 2 80.q even 4 2
1024.2.e.a 2 160.y odd 8 2
1024.2.e.a 2 160.z even 8 2
1024.2.e.f 2 160.y odd 8 2
1024.2.e.f 2 160.z even 8 2
1152.2.d.c 2 15.d odd 2 1
1152.2.d.c 2 60.h even 2 1
1152.2.d.c 2 120.i odd 2 1
1152.2.d.c 2 120.m even 2 1
2304.2.a.t 2 240.t even 4 2
2304.2.a.t 2 240.bm odd 4 2
3200.2.d.c 2 1.a even 1 1 trivial
3200.2.d.c 2 4.b odd 2 1 inner
3200.2.d.c 2 8.b even 2 1 inner
3200.2.d.c 2 8.d odd 2 1 CM
3200.2.f.o 4 5.c odd 4 2
3200.2.f.o 4 20.e even 4 2
3200.2.f.o 4 40.i odd 4 2
3200.2.f.o 4 40.k even 4 2
6400.2.a.by 2 16.e even 4 2
6400.2.a.by 2 16.f odd 4 2

Hecke kernels

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

\( T_{3}^{2} + 8 \)
\( T_{7} \)
\( T_{11}^{2} + 8 \)
\( T_{13} \)
\( T_{17} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 2 T + 3 T^{2} )( 1 + 2 T + 3 T^{2} ) \)
$5$ 1
$7$ \( ( 1 + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} ) \)
$13$ \( ( 1 - 13 T^{2} )^{2} \)
$17$ \( ( 1 + 6 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 2 T + 19 T^{2} )( 1 + 2 T + 19 T^{2} ) \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( ( 1 - 29 T^{2} )^{2} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 37 T^{2} )^{2} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 10 T + 43 T^{2} )( 1 + 10 T + 43 T^{2} ) \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( ( 1 - 6 T + 59 T^{2} )( 1 + 6 T + 59 T^{2} ) \)
$61$ \( ( 1 - 61 T^{2} )^{2} \)
$67$ \( ( 1 - 14 T + 67 T^{2} )( 1 + 14 T + 67 T^{2} ) \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 2 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 79 T^{2} )^{2} \)
$83$ \( ( 1 - 18 T + 83 T^{2} )( 1 + 18 T + 83 T^{2} ) \)
$89$ \( ( 1 + 18 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 10 T + 97 T^{2} )^{2} \)
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