Properties

Label 3200.2.f.a
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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

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

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.a 2
4.b odd 2 1 3200.2.f.f 2
5.b even 2 1 3200.2.f.e 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.e 2
8.d odd 2 1 3200.2.f.b 2
20.d odd 2 1 3200.2.f.b 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.f 2
40.f even 2 1 inner 3200.2.f.a 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.q 1
80.i odd 4 1 6400.2.a.v 1
80.j even 4 1 6400.2.a.p 1
80.j even 4 1 6400.2.a.w 1
80.s even 4 1 6400.2.a.c 1
80.s even 4 1 6400.2.a.h 1
80.t odd 4 1 6400.2.a.b 1
80.t odd 4 1 6400.2.a.i 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 1.a even 1 1 trivial
3200.2.f.a 2 40.f even 2 1 inner
3200.2.f.b 2 8.d odd 2 1
3200.2.f.b 2 20.d odd 2 1
3200.2.f.e 2 5.b even 2 1
3200.2.f.e 2 8.b even 2 1
3200.2.f.f 2 4.b odd 2 1
3200.2.f.f 2 40.e odd 2 1
6400.2.a.b 1 80.t odd 4 1
6400.2.a.c 1 80.s even 4 1
6400.2.a.h 1 80.s even 4 1
6400.2.a.i 1 80.t odd 4 1
6400.2.a.p 1 80.j even 4 1
6400.2.a.q 1 80.i odd 4 1
6400.2.a.v 1 80.i odd 4 1
6400.2.a.w 1 80.j even 4 1

Hecke kernels

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

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 9 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{31} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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