Properties

Label 3200.2.f.e
Level $3200$
Weight $2$
Character orbit 3200.f
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.f (of order \(2\), degree \(1\), not 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: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q + q^{3} + 4 i q^{7} -2 q^{9} +O(q^{10})\) \( q + q^{3} + 4 i q^{7} -2 q^{9} -3 i q^{11} -i q^{17} -7 i q^{19} + 4 i q^{21} -4 i q^{23} -5 q^{27} -8 i q^{29} -4 q^{31} -3 i q^{33} + 4 q^{37} + 3 q^{41} -8 q^{43} -9 q^{49} -i q^{51} + 12 q^{53} -7 i q^{57} + 8 i q^{59} -4 i q^{61} -8 i q^{63} + 9 q^{67} -4 i q^{69} + 16 q^{71} -11 i q^{73} + 12 q^{77} + 4 q^{79} + q^{81} - q^{83} -8 i q^{87} -13 q^{89} -4 q^{93} -14 i q^{97} + 6 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{9} - 10q^{27} - 8q^{31} + 8q^{37} + 6q^{41} - 16q^{43} - 18q^{49} + 24q^{53} + 18q^{67} + 32q^{71} + 24q^{77} + 8q^{79} + 2q^{81} - 2q^{83} - 26q^{89} - 8q^{93} + O(q^{100}) \)

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
1.00000i
1.00000i
0 1.00000 0 0 0 4.00000i 0 −2.00000 0
449.2 0 1.00000 0 0 0 4.00000i 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
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.e 2
4.b odd 2 1 3200.2.f.b 2
5.b even 2 1 3200.2.f.a 2
5.c odd 4 1 3200.2.d.b yes 2
5.c odd 4 1 3200.2.d.g yes 2
8.b even 2 1 3200.2.f.a 2
8.d odd 2 1 3200.2.f.f 2
20.d odd 2 1 3200.2.f.f 2
20.e even 4 1 3200.2.d.a 2
20.e even 4 1 3200.2.d.h yes 2
40.e odd 2 1 3200.2.f.b 2
40.f even 2 1 inner 3200.2.f.e 2
40.i odd 4 1 3200.2.d.b yes 2
40.i odd 4 1 3200.2.d.g yes 2
40.k even 4 1 3200.2.d.a 2
40.k even 4 1 3200.2.d.h yes 2
80.i odd 4 1 6400.2.a.b 1
80.i odd 4 1 6400.2.a.i 1
80.j even 4 1 6400.2.a.c 1
80.j even 4 1 6400.2.a.h 1
80.s even 4 1 6400.2.a.p 1
80.s even 4 1 6400.2.a.w 1
80.t odd 4 1 6400.2.a.q 1
80.t odd 4 1 6400.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.a 2 20.e even 4 1
3200.2.d.a 2 40.k even 4 1
3200.2.d.b yes 2 5.c odd 4 1
3200.2.d.b yes 2 40.i odd 4 1
3200.2.d.g yes 2 5.c odd 4 1
3200.2.d.g yes 2 40.i odd 4 1
3200.2.d.h yes 2 20.e even 4 1
3200.2.d.h yes 2 40.k even 4 1
3200.2.f.a 2 5.b even 2 1
3200.2.f.a 2 8.b even 2 1
3200.2.f.b 2 4.b odd 2 1
3200.2.f.b 2 40.e odd 2 1
3200.2.f.e 2 1.a even 1 1 trivial
3200.2.f.e 2 40.f even 2 1 inner
3200.2.f.f 2 8.d odd 2 1
3200.2.f.f 2 20.d odd 2 1
6400.2.a.b 1 80.i odd 4 1
6400.2.a.c 1 80.j even 4 1
6400.2.a.h 1 80.j even 4 1
6400.2.a.i 1 80.i odd 4 1
6400.2.a.p 1 80.s even 4 1
6400.2.a.q 1 80.t odd 4 1
6400.2.a.v 1 80.t odd 4 1
6400.2.a.w 1 80.s even 4 1

Hecke kernels

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

\( T_{3} - 1 \)
\( T_{7}^{2} + 16 \)
\( T_{11}^{2} + 9 \)
\( T_{13} \)
\( T_{31} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( 9 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 1 + T^{2} \)
$19$ \( 49 + T^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( 64 + T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( ( -4 + T )^{2} \)
$41$ \( ( -3 + T )^{2} \)
$43$ \( ( 8 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( ( -12 + T )^{2} \)
$59$ \( 64 + T^{2} \)
$61$ \( 16 + T^{2} \)
$67$ \( ( -9 + T )^{2} \)
$71$ \( ( -16 + T )^{2} \)
$73$ \( 121 + T^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( ( 1 + T )^{2} \)
$89$ \( ( 13 + T )^{2} \)
$97$ \( 196 + T^{2} \)
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