Properties

Label 3200.2.d.b
Level $3200$
Weight $2$
Character orbit 3200.d
Analytic conductor $25.552$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{3} -4 q^{7} + 2 q^{9} +O(q^{10})\) \( q + i q^{3} -4 q^{7} + 2 q^{9} + 3 i q^{11} + q^{17} -7 i q^{19} -4 i q^{21} -4 q^{23} + 5 i q^{27} -8 i q^{29} -4 q^{31} -3 q^{33} -4 i q^{37} + 3 q^{41} -8 i q^{43} + 9 q^{49} + i q^{51} + 12 i q^{53} + 7 q^{57} + 8 i q^{59} + 4 i q^{61} -8 q^{63} -9 i q^{67} -4 i q^{69} + 16 q^{71} -11 q^{73} -12 i q^{77} -4 q^{79} + q^{81} -i q^{83} + 8 q^{87} + 13 q^{89} -4 i q^{93} + 14 q^{97} + 6 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{7} + 4q^{9} + O(q^{10}) \) \( 2q - 8q^{7} + 4q^{9} + 2q^{17} - 8q^{23} - 8q^{31} - 6q^{33} + 6q^{41} + 18q^{49} + 14q^{57} - 16q^{63} + 32q^{71} - 22q^{73} - 8q^{79} + 2q^{81} + 16q^{87} + 26q^{89} + 28q^{97} + O(q^{100}) \)

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
1.00000i
1.00000i
0 1.00000i 0 0 0 −4.00000 0 2.00000 0
1601.2 0 1.00000i 0 0 0 −4.00000 0 2.00000 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.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.b yes 2
4.b odd 2 1 3200.2.d.h yes 2
5.b even 2 1 3200.2.d.g yes 2
5.c odd 4 1 3200.2.f.a 2
5.c odd 4 1 3200.2.f.e 2
8.b even 2 1 inner 3200.2.d.b yes 2
8.d odd 2 1 3200.2.d.h yes 2
16.e even 4 1 6400.2.a.i 1
16.e even 4 1 6400.2.a.v 1
16.f odd 4 1 6400.2.a.c 1
16.f odd 4 1 6400.2.a.p 1
20.d odd 2 1 3200.2.d.a 2
20.e even 4 1 3200.2.f.b 2
20.e even 4 1 3200.2.f.f 2
40.e odd 2 1 3200.2.d.a 2
40.f even 2 1 3200.2.d.g yes 2
40.i odd 4 1 3200.2.f.a 2
40.i odd 4 1 3200.2.f.e 2
40.k even 4 1 3200.2.f.b 2
40.k even 4 1 3200.2.f.f 2
80.k odd 4 1 6400.2.a.h 1
80.k odd 4 1 6400.2.a.w 1
80.q even 4 1 6400.2.a.b 1
80.q even 4 1 6400.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.a 2 20.d odd 2 1
3200.2.d.a 2 40.e odd 2 1
3200.2.d.b yes 2 1.a even 1 1 trivial
3200.2.d.b yes 2 8.b even 2 1 inner
3200.2.d.g yes 2 5.b even 2 1
3200.2.d.g yes 2 40.f even 2 1
3200.2.d.h yes 2 4.b odd 2 1
3200.2.d.h yes 2 8.d odd 2 1
3200.2.f.a 2 5.c odd 4 1
3200.2.f.a 2 40.i odd 4 1
3200.2.f.b 2 20.e even 4 1
3200.2.f.b 2 40.k even 4 1
3200.2.f.e 2 5.c odd 4 1
3200.2.f.e 2 40.i odd 4 1
3200.2.f.f 2 20.e even 4 1
3200.2.f.f 2 40.k even 4 1
6400.2.a.b 1 80.q even 4 1
6400.2.a.c 1 16.f odd 4 1
6400.2.a.h 1 80.k odd 4 1
6400.2.a.i 1 16.e even 4 1
6400.2.a.p 1 16.f odd 4 1
6400.2.a.q 1 80.q even 4 1
6400.2.a.v 1 16.e even 4 1
6400.2.a.w 1 80.k odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 1 \)
\( T_{7} + 4 \)
\( T_{11}^{2} + 9 \)
\( T_{13} \)
\( T_{17} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 4 + T )^{2} \)
$11$ \( 9 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -1 + T )^{2} \)
$19$ \( 49 + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( 64 + T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 16 + T^{2} \)
$41$ \( ( -3 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 144 + T^{2} \)
$59$ \( 64 + T^{2} \)
$61$ \( 16 + T^{2} \)
$67$ \( 81 + T^{2} \)
$71$ \( ( -16 + T )^{2} \)
$73$ \( ( 11 + T )^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( 1 + T^{2} \)
$89$ \( ( -13 + T )^{2} \)
$97$ \( ( -14 + T )^{2} \)
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