Properties

Label 3584.2.m.d
Level $3584$
Weight $2$
Character orbit 3584.m
Analytic conductor $28.618$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3584.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.6183840844\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (i - 1) q^{3} + ( - 2 i - 2) q^{5} - i q^{7} + i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (i - 1) q^{3} + ( - 2 i - 2) q^{5} - i q^{7} + i q^{9} + ( - 2 i - 2) q^{11} + 4 q^{15} - 2 q^{17} + ( - 3 i + 3) q^{19} + (i + 1) q^{21} + 4 i q^{23} + 3 i q^{25} + ( - 4 i - 4) q^{27} + ( - 7 i + 7) q^{29} + 6 q^{31} + 4 q^{33} + (2 i - 2) q^{35} + (3 i + 3) q^{37} - 2 i q^{41} + ( - 6 i - 6) q^{43} + ( - 2 i + 2) q^{45} - 10 q^{47} - q^{49} + ( - 2 i + 2) q^{51} + (5 i + 5) q^{53} + 8 i q^{55} + 6 i q^{57} + ( - 7 i - 7) q^{59} + ( - 2 i + 2) q^{61} + q^{63} + (8 i - 8) q^{67} + ( - 4 i - 4) q^{69} - 4 i q^{71} + 14 i q^{73} + ( - 3 i - 3) q^{75} + (2 i - 2) q^{77} + 4 q^{79} + 5 q^{81} + (3 i - 3) q^{83} + (4 i + 4) q^{85} + 14 i q^{87} - 10 i q^{89} + (6 i - 6) q^{93} - 12 q^{95} - 6 q^{97} + ( - 2 i + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 4 q^{5} - 4 q^{11} + 8 q^{15} - 4 q^{17} + 6 q^{19} + 2 q^{21} - 8 q^{27} + 14 q^{29} + 12 q^{31} + 8 q^{33} - 4 q^{35} + 6 q^{37} - 12 q^{43} + 4 q^{45} - 20 q^{47} - 2 q^{49} + 4 q^{51} + 10 q^{53} - 14 q^{59} + 4 q^{61} + 2 q^{63} - 16 q^{67} - 8 q^{69} - 6 q^{75} - 4 q^{77} + 8 q^{79} + 10 q^{81} - 6 q^{83} + 8 q^{85} - 12 q^{93} - 24 q^{95} - 12 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
897.1
1.00000i
1.00000i
0 −1.00000 + 1.00000i 0 −2.00000 2.00000i 0 1.00000i 0 1.00000i 0
2689.1 0 −1.00000 1.00000i 0 −2.00000 + 2.00000i 0 1.00000i 0 1.00000i 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
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3584.2.m.d 2
4.b odd 2 1 3584.2.m.s yes 2
8.b even 2 1 3584.2.m.y yes 2
8.d odd 2 1 3584.2.m.j yes 2
16.e even 4 1 inner 3584.2.m.d 2
16.e even 4 1 3584.2.m.y yes 2
16.f odd 4 1 3584.2.m.j yes 2
16.f odd 4 1 3584.2.m.s yes 2
32.g even 8 2 7168.2.a.d 2
32.h odd 8 2 7168.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.m.d 2 1.a even 1 1 trivial
3584.2.m.d 2 16.e even 4 1 inner
3584.2.m.j yes 2 8.d odd 2 1
3584.2.m.j yes 2 16.f odd 4 1
3584.2.m.s yes 2 4.b odd 2 1
3584.2.m.s yes 2 16.f odd 4 1
3584.2.m.y yes 2 8.b even 2 1
3584.2.m.y yes 2 16.e even 4 1
7168.2.a.d 2 32.g even 8 2
7168.2.a.p 2 32.h odd 8 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3584, [\chi])\):

\( T_{3}^{2} + 2T_{3} + 2 \) Copy content Toggle raw display
\( T_{5}^{2} + 4T_{5} + 8 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} + 8 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$31$ \( (T - 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$41$ \( T^{2} + 4 \) Copy content Toggle raw display
$43$ \( T^{2} + 12T + 72 \) Copy content Toggle raw display
$47$ \( (T + 10)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$59$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$67$ \( T^{2} + 16T + 128 \) Copy content Toggle raw display
$71$ \( T^{2} + 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 196 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$89$ \( T^{2} + 100 \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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