Properties

Label 4096.2.a.e
Level $4096$
Weight $2$
Character orbit 4096.a
Self dual yes
Analytic conductor $32.707$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4096 = 2^{12} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4096.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.7067246679\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -\beta_{1} - 2 \beta_{3} ) q^{5} -\beta_{2} q^{7} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -\beta_{1} - 2 \beta_{3} ) q^{5} -\beta_{2} q^{7} + ( -1 + \beta_{2} ) q^{9} + ( -\beta_{1} + 2 \beta_{3} ) q^{11} + \beta_{1} q^{13} + ( -2 - 3 \beta_{2} ) q^{15} + 2 \beta_{2} q^{17} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{19} + ( -\beta_{1} - \beta_{3} ) q^{21} + ( -4 + 3 \beta_{2} ) q^{23} + ( 5 + \beta_{2} ) q^{25} + ( -3 \beta_{1} + \beta_{3} ) q^{27} + ( -2 \beta_{1} + \beta_{3} ) q^{29} + 4 q^{31} + ( -2 + \beta_{2} ) q^{33} + ( 3 \beta_{1} - \beta_{3} ) q^{35} -\beta_{1} q^{37} + ( 2 + \beta_{2} ) q^{39} + ( -4 - 3 \beta_{2} ) q^{41} + ( 3 \beta_{1} + 4 \beta_{3} ) q^{43} + ( -2 \beta_{1} + 3 \beta_{3} ) q^{45} + ( -6 - 4 \beta_{2} ) q^{47} -5 q^{49} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{51} + ( -4 \beta_{1} - \beta_{3} ) q^{53} + ( -6 + 5 \beta_{2} ) q^{55} + ( 4 + 5 \beta_{2} ) q^{57} + ( -4 \beta_{1} + \beta_{3} ) q^{59} + \beta_{3} q^{61} + ( -2 + \beta_{2} ) q^{63} + ( -2 - 3 \beta_{2} ) q^{65} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{67} + ( -\beta_{1} + 3 \beta_{3} ) q^{69} + ( -4 + 3 \beta_{2} ) q^{71} -7 \beta_{2} q^{73} + ( 6 \beta_{1} + \beta_{3} ) q^{75} + ( -\beta_{1} + 3 \beta_{3} ) q^{77} -6 q^{79} + ( -3 - 5 \beta_{2} ) q^{81} + ( -4 \beta_{1} + \beta_{3} ) q^{83} + ( -6 \beta_{1} + 2 \beta_{3} ) q^{85} + ( -4 - \beta_{2} ) q^{87} + ( -8 + 3 \beta_{2} ) q^{89} + ( -\beta_{1} - \beta_{3} ) q^{91} + 4 \beta_{1} q^{93} + ( -16 - 3 \beta_{2} ) q^{95} + ( 10 - 6 \beta_{2} ) q^{97} + ( 2 \beta_{1} - 5 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{9} - 8 q^{15} - 16 q^{23} + 20 q^{25} + 16 q^{31} - 8 q^{33} + 8 q^{39} - 16 q^{41} - 24 q^{47} - 20 q^{49} - 24 q^{55} + 16 q^{57} - 8 q^{63} - 8 q^{65} - 16 q^{71} - 24 q^{79} - 12 q^{81} - 16 q^{87} - 32 q^{89} - 64 q^{95} + 40 q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.84776
−0.765367
0.765367
1.84776
0 −1.84776 0 3.37849 0 −1.41421 0 0.414214 0
1.2 0 −0.765367 0 −2.93015 0 1.41421 0 −2.41421 0
1.3 0 0.765367 0 2.93015 0 1.41421 0 −2.41421 0
1.4 0 1.84776 0 −3.37849 0 −1.41421 0 0.414214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4096.2.a.e 4
4.b odd 2 1 4096.2.a.f 4
8.b even 2 1 inner 4096.2.a.e 4
8.d odd 2 1 4096.2.a.f 4
64.i even 16 2 32.2.g.a 4
64.i even 16 2 256.2.g.b 4
64.i even 16 2 512.2.g.a 4
64.i even 16 2 512.2.g.d 4
64.j odd 16 2 128.2.g.a 4
64.j odd 16 2 256.2.g.a 4
64.j odd 16 2 512.2.g.b 4
64.j odd 16 2 512.2.g.c 4
192.q odd 16 2 288.2.v.a 4
192.s even 16 2 1152.2.v.a 4
320.bc odd 16 2 800.2.ba.a 4
320.bf even 16 2 800.2.y.a 4
320.bi odd 16 2 800.2.ba.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.a 4 64.i even 16 2
128.2.g.a 4 64.j odd 16 2
256.2.g.a 4 64.j odd 16 2
256.2.g.b 4 64.i even 16 2
288.2.v.a 4 192.q odd 16 2
512.2.g.a 4 64.i even 16 2
512.2.g.b 4 64.j odd 16 2
512.2.g.c 4 64.j odd 16 2
512.2.g.d 4 64.i even 16 2
800.2.y.a 4 320.bf even 16 2
800.2.ba.a 4 320.bc odd 16 2
800.2.ba.b 4 320.bi odd 16 2
1152.2.v.a 4 192.s even 16 2
4096.2.a.e 4 1.a even 1 1 trivial
4096.2.a.e 4 8.b even 2 1 inner
4096.2.a.f 4 4.b odd 2 1
4096.2.a.f 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4096))\):

\( T_{3}^{4} - 4 T_{3}^{2} + 2 \)
\( T_{5}^{4} - 20 T_{5}^{2} + 98 \)
\( T_{7}^{2} - 2 \)
\( T_{23}^{2} + 8 T_{23} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 2 - 4 T^{2} + T^{4} \)
$5$ \( 98 - 20 T^{2} + T^{4} \)
$7$ \( ( -2 + T^{2} )^{2} \)
$11$ \( 2 - 20 T^{2} + T^{4} \)
$13$ \( 2 - 4 T^{2} + T^{4} \)
$17$ \( ( -8 + T^{2} )^{2} \)
$19$ \( 578 - 52 T^{2} + T^{4} \)
$23$ \( ( -2 + 8 T + T^{2} )^{2} \)
$29$ \( 98 - 20 T^{2} + T^{4} \)
$31$ \( ( -4 + T )^{4} \)
$37$ \( 2 - 4 T^{2} + T^{4} \)
$41$ \( ( -2 + 8 T + T^{2} )^{2} \)
$43$ \( 1922 - 100 T^{2} + T^{4} \)
$47$ \( ( 4 + 12 T + T^{2} )^{2} \)
$53$ \( 98 - 68 T^{2} + T^{4} \)
$59$ \( 1058 - 68 T^{2} + T^{4} \)
$61$ \( 2 - 4 T^{2} + T^{4} \)
$67$ \( 578 - 52 T^{2} + T^{4} \)
$71$ \( ( -2 + 8 T + T^{2} )^{2} \)
$73$ \( ( -98 + T^{2} )^{2} \)
$79$ \( ( 6 + T )^{4} \)
$83$ \( 1058 - 68 T^{2} + T^{4} \)
$89$ \( ( 46 + 16 T + T^{2} )^{2} \)
$97$ \( ( 28 - 20 T + T^{2} )^{2} \)
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