Properties

Label 3200.2.f.j
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: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + 3 \beta_{2} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + 3 \beta_{2} q^{7} - q^{9} - 4 \beta_{2} q^{11} + 2 q^{13} - 3 \beta_1 q^{17} + 2 \beta_{2} q^{19} - 3 \beta_1 q^{21} + 5 \beta_{2} q^{23} + 4 \beta_{3} q^{27} + 2 \beta_1 q^{29} - 2 \beta_{3} q^{31} + 4 \beta_1 q^{33} - 2 q^{37} - 2 \beta_{3} q^{39} - 8 q^{41} - \beta_{3} q^{43} + \beta_{2} q^{47} - 11 q^{49} + 6 \beta_{2} q^{51} - 2 q^{53} - 2 \beta_1 q^{57} + 2 \beta_{2} q^{59} - 7 \beta_1 q^{61} - 3 \beta_{2} q^{63} - 3 \beta_{3} q^{67} - 5 \beta_1 q^{69} + 2 \beta_{3} q^{71} - 3 \beta_1 q^{73} + 24 q^{77} - 12 \beta_{3} q^{79} - 5 q^{81} + 9 \beta_{3} q^{83} - 4 \beta_{2} q^{87} + 6 q^{89} + 6 \beta_{2} q^{91} + 4 q^{93} - 5 \beta_1 q^{97} + 4 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} + 8 q^{13} - 8 q^{37} - 32 q^{41} - 44 q^{49} - 8 q^{53} + 96 q^{77} - 20 q^{81} + 24 q^{89} + 16 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) 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
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0 −1.41421 0 0 0 4.24264i 0 −1.00000 0
449.2 0 −1.41421 0 0 0 4.24264i 0 −1.00000 0
449.3 0 1.41421 0 0 0 4.24264i 0 −1.00000 0
449.4 0 1.41421 0 0 0 4.24264i 0 −1.00000 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.j 4
4.b odd 2 1 inner 3200.2.f.j 4
5.b even 2 1 3200.2.f.i 4
5.c odd 4 1 640.2.d.d 4
5.c odd 4 1 3200.2.d.s 4
8.b even 2 1 3200.2.f.i 4
8.d odd 2 1 3200.2.f.i 4
15.e even 4 1 5760.2.k.o 4
20.d odd 2 1 3200.2.f.i 4
20.e even 4 1 640.2.d.d 4
20.e even 4 1 3200.2.d.s 4
40.e odd 2 1 inner 3200.2.f.j 4
40.f even 2 1 inner 3200.2.f.j 4
40.i odd 4 1 640.2.d.d 4
40.i odd 4 1 3200.2.d.s 4
40.k even 4 1 640.2.d.d 4
40.k even 4 1 3200.2.d.s 4
60.l odd 4 1 5760.2.k.o 4
80.i odd 4 1 1280.2.a.j 2
80.i odd 4 1 6400.2.a.bl 2
80.j even 4 1 1280.2.a.f 2
80.j even 4 1 6400.2.a.bn 2
80.s even 4 1 1280.2.a.j 2
80.s even 4 1 6400.2.a.bl 2
80.t odd 4 1 1280.2.a.f 2
80.t odd 4 1 6400.2.a.bn 2
120.q odd 4 1 5760.2.k.o 4
120.w even 4 1 5760.2.k.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.d.d 4 5.c odd 4 1
640.2.d.d 4 20.e even 4 1
640.2.d.d 4 40.i odd 4 1
640.2.d.d 4 40.k even 4 1
1280.2.a.f 2 80.j even 4 1
1280.2.a.f 2 80.t odd 4 1
1280.2.a.j 2 80.i odd 4 1
1280.2.a.j 2 80.s even 4 1
3200.2.d.s 4 5.c odd 4 1
3200.2.d.s 4 20.e even 4 1
3200.2.d.s 4 40.i odd 4 1
3200.2.d.s 4 40.k even 4 1
3200.2.f.i 4 5.b even 2 1
3200.2.f.i 4 8.b even 2 1
3200.2.f.i 4 8.d odd 2 1
3200.2.f.i 4 20.d odd 2 1
3200.2.f.j 4 1.a even 1 1 trivial
3200.2.f.j 4 4.b odd 2 1 inner
3200.2.f.j 4 40.e odd 2 1 inner
3200.2.f.j 4 40.f even 2 1 inner
5760.2.k.o 4 15.e even 4 1
5760.2.k.o 4 60.l odd 4 1
5760.2.k.o 4 120.q odd 4 1
5760.2.k.o 4 120.w even 4 1
6400.2.a.bl 2 80.i odd 4 1
6400.2.a.bl 2 80.s even 4 1
6400.2.a.bn 2 80.j even 4 1
6400.2.a.bn 2 80.t odd 4 1

Hecke kernels

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

\( T_{3}^{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} + 18 \) Copy content Toggle raw display
\( T_{11}^{2} + 32 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{31}^{2} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$13$ \( (T - 2)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 50)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{4} \) Copy content Toggle raw display
$41$ \( (T + 8)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$53$ \( (T + 2)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 288)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
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