Properties

Label 3200.2.f.p
Level $3200$
Weight $2$
Character orbit 3200.f
Analytic conductor $25.552$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(\Q(i, \sqrt{10})\)
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 640)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_{2} q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_{2} q^{7} + 7 q^{9} - 6 q^{13} - \beta_1 q^{17} - 2 \beta_{2} q^{19} - 5 \beta_1 q^{21} + \beta_{2} q^{23} + 4 \beta_{3} q^{27} - 2 \beta_1 q^{29} + 2 \beta_{3} q^{31} - 2 q^{37} - 6 \beta_{3} q^{39} + \beta_{3} q^{43} - 3 \beta_{2} q^{47} - 3 q^{49} - 2 \beta_{2} q^{51} + 6 q^{53} - 10 \beta_1 q^{57} - 2 \beta_{2} q^{59} - \beta_1 q^{61} - 7 \beta_{2} q^{63} + 3 \beta_{3} q^{67} + 5 \beta_1 q^{69} - 2 \beta_{3} q^{71} + 7 \beta_1 q^{73} + 4 \beta_{3} q^{79} + 19 q^{81} - \beta_{3} q^{83} - 4 \beta_{2} q^{87} - 10 q^{89} + 6 \beta_{2} q^{91} + 20 q^{93} + \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 28 q^{9} - 24 q^{13} - 8 q^{37} - 12 q^{49} + 24 q^{53} + 76 q^{81} - 40 q^{89} + 80 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 5\nu ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{3} + 5\beta_{2} ) / 2 \) 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
−1.58114 + 1.58114i
−1.58114 1.58114i
1.58114 + 1.58114i
1.58114 1.58114i
0 −3.16228 0 0 0 3.16228i 0 7.00000 0
449.2 0 −3.16228 0 0 0 3.16228i 0 7.00000 0
449.3 0 3.16228 0 0 0 3.16228i 0 7.00000 0
449.4 0 3.16228 0 0 0 3.16228i 0 7.00000 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
4.b 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.p 4
4.b odd 2 1 inner 3200.2.f.p 4
5.b even 2 1 3200.2.f.q 4
5.c odd 4 1 640.2.d.a 4
5.c odd 4 1 3200.2.d.j 4
8.b even 2 1 3200.2.f.q 4
8.d odd 2 1 3200.2.f.q 4
15.e even 4 1 5760.2.k.t 4
20.d odd 2 1 3200.2.f.q 4
20.e even 4 1 640.2.d.a 4
20.e even 4 1 3200.2.d.j 4
40.e odd 2 1 inner 3200.2.f.p 4
40.f even 2 1 inner 3200.2.f.p 4
40.i odd 4 1 640.2.d.a 4
40.i odd 4 1 3200.2.d.j 4
40.k even 4 1 640.2.d.a 4
40.k even 4 1 3200.2.d.j 4
60.l odd 4 1 5760.2.k.t 4
80.i odd 4 1 1280.2.a.l 2
80.i odd 4 1 6400.2.a.ca 2
80.j even 4 1 1280.2.a.h 2
80.j even 4 1 6400.2.a.bz 2
80.s even 4 1 1280.2.a.l 2
80.s even 4 1 6400.2.a.ca 2
80.t odd 4 1 1280.2.a.h 2
80.t odd 4 1 6400.2.a.bz 2
120.q odd 4 1 5760.2.k.t 4
120.w even 4 1 5760.2.k.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.d.a 4 5.c odd 4 1
640.2.d.a 4 20.e even 4 1
640.2.d.a 4 40.i odd 4 1
640.2.d.a 4 40.k even 4 1
1280.2.a.h 2 80.j even 4 1
1280.2.a.h 2 80.t odd 4 1
1280.2.a.l 2 80.i odd 4 1
1280.2.a.l 2 80.s even 4 1
3200.2.d.j 4 5.c odd 4 1
3200.2.d.j 4 20.e even 4 1
3200.2.d.j 4 40.i odd 4 1
3200.2.d.j 4 40.k even 4 1
3200.2.f.p 4 1.a even 1 1 trivial
3200.2.f.p 4 4.b odd 2 1 inner
3200.2.f.p 4 40.e odd 2 1 inner
3200.2.f.p 4 40.f even 2 1 inner
3200.2.f.q 4 5.b even 2 1
3200.2.f.q 4 8.b even 2 1
3200.2.f.q 4 8.d odd 2 1
3200.2.f.q 4 20.d odd 2 1
5760.2.k.t 4 15.e even 4 1
5760.2.k.t 4 60.l odd 4 1
5760.2.k.t 4 120.q odd 4 1
5760.2.k.t 4 120.w even 4 1
6400.2.a.bz 2 80.j even 4 1
6400.2.a.bz 2 80.t odd 4 1
6400.2.a.ca 2 80.i odd 4 1
6400.2.a.ca 2 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 \(S_{2}^{\mathrm{new}}(3200, [\chi])\):

\( T_{3}^{2} - 10 \) Copy content Toggle raw display
\( T_{7}^{2} + 10 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} + 6 \) Copy content Toggle raw display
\( T_{31}^{2} - 40 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 10)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T + 6)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 40)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 10)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 90)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 40)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 90)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 160)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$89$ \( (T + 10)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
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