Properties

Label 256.4.a.l
Level $256$
Weight $4$
Character orbit 256.a
Self dual yes
Analytic conductor $15.104$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 256.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.1044889615\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 2 \beta q^{5} + 8 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + 2 \beta q^{5} + 8 q^{7} + q^{9} + 3 \beta q^{11} + 10 \beta q^{13} + 56 q^{15} - 14 q^{17} - 7 \beta q^{19} + 8 \beta q^{21} + 152 q^{23} - 13 q^{25} - 26 \beta q^{27} - 30 \beta q^{29} + 224 q^{31} + 84 q^{33} + 16 \beta q^{35} - 46 \beta q^{37} + 280 q^{39} + 70 q^{41} + 83 \beta q^{43} + 2 \beta q^{45} + 336 q^{47} - 279 q^{49} - 14 \beta q^{51} - 6 \beta q^{53} + 168 q^{55} - 196 q^{57} - 101 \beta q^{59} + 18 \beta q^{61} + 8 q^{63} + 560 q^{65} + 33 \beta q^{67} + 152 \beta q^{69} + 72 q^{71} + 294 q^{73} - 13 \beta q^{75} + 24 \beta q^{77} - 464 q^{79} - 755 q^{81} - 103 \beta q^{83} - 28 \beta q^{85} - 840 q^{87} - 266 q^{89} + 80 \beta q^{91} + 224 \beta q^{93} - 392 q^{95} + 994 q^{97} + 3 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{7} + 2 q^{9} + 112 q^{15} - 28 q^{17} + 304 q^{23} - 26 q^{25} + 448 q^{31} + 168 q^{33} + 560 q^{39} + 140 q^{41} + 672 q^{47} - 558 q^{49} + 336 q^{55} - 392 q^{57} + 16 q^{63} + 1120 q^{65} + 144 q^{71} + 588 q^{73} - 928 q^{79} - 1510 q^{81} - 1680 q^{87} - 532 q^{89} - 784 q^{95} + 1988 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
−2.64575
2.64575
0 −5.29150 0 −10.5830 0 8.00000 0 1.00000 0
1.2 0 5.29150 0 10.5830 0 8.00000 0 1.00000 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 256.4.a.l 2
3.b odd 2 1 2304.4.a.bn 2
4.b odd 2 1 256.4.a.j 2
8.b even 2 1 inner 256.4.a.l 2
8.d odd 2 1 256.4.a.j 2
12.b even 2 1 2304.4.a.v 2
16.e even 4 2 8.4.b.a 2
16.f odd 4 2 32.4.b.a 2
24.f even 2 1 2304.4.a.v 2
24.h odd 2 1 2304.4.a.bn 2
48.i odd 4 2 72.4.d.b 2
48.k even 4 2 288.4.d.a 2
80.i odd 4 2 200.4.f.a 4
80.j even 4 2 800.4.f.a 4
80.k odd 4 2 800.4.d.a 2
80.q even 4 2 200.4.d.a 2
80.s even 4 2 800.4.f.a 4
80.t odd 4 2 200.4.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.b.a 2 16.e even 4 2
32.4.b.a 2 16.f odd 4 2
72.4.d.b 2 48.i odd 4 2
200.4.d.a 2 80.q even 4 2
200.4.f.a 4 80.i odd 4 2
200.4.f.a 4 80.t odd 4 2
256.4.a.j 2 4.b odd 2 1
256.4.a.j 2 8.d odd 2 1
256.4.a.l 2 1.a even 1 1 trivial
256.4.a.l 2 8.b even 2 1 inner
288.4.d.a 2 48.k even 4 2
800.4.d.a 2 80.k odd 4 2
800.4.f.a 4 80.j even 4 2
800.4.f.a 4 80.s even 4 2
2304.4.a.v 2 12.b even 2 1
2304.4.a.v 2 24.f even 2 1
2304.4.a.bn 2 3.b odd 2 1
2304.4.a.bn 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(256))\):

\( T_{3}^{2} - 28 \) Copy content Toggle raw display
\( T_{5}^{2} - 112 \) Copy content Toggle raw display
\( T_{7} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 28 \) Copy content Toggle raw display
$5$ \( T^{2} - 112 \) Copy content Toggle raw display
$7$ \( (T - 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 252 \) Copy content Toggle raw display
$13$ \( T^{2} - 2800 \) Copy content Toggle raw display
$17$ \( (T + 14)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 1372 \) Copy content Toggle raw display
$23$ \( (T - 152)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 25200 \) Copy content Toggle raw display
$31$ \( (T - 224)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 59248 \) Copy content Toggle raw display
$41$ \( (T - 70)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 192892 \) Copy content Toggle raw display
$47$ \( (T - 336)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 1008 \) Copy content Toggle raw display
$59$ \( T^{2} - 285628 \) Copy content Toggle raw display
$61$ \( T^{2} - 9072 \) Copy content Toggle raw display
$67$ \( T^{2} - 30492 \) Copy content Toggle raw display
$71$ \( (T - 72)^{2} \) Copy content Toggle raw display
$73$ \( (T - 294)^{2} \) Copy content Toggle raw display
$79$ \( (T + 464)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 297052 \) Copy content Toggle raw display
$89$ \( (T + 266)^{2} \) Copy content Toggle raw display
$97$ \( (T - 994)^{2} \) Copy content Toggle raw display
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