Properties

Label 256.7.c.d
Level $256$
Weight $7$
Character orbit 256.c
Analytic conductor $58.894$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 256.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(58.8938454067\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 23 \beta q^{3} - 1387 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 23 \beta q^{3} - 1387 q^{9} + 1169 \beta q^{11} - 1726 q^{17} - 1241 \beta q^{19} - 15625 q^{25} - 15134 \beta q^{27} - 107548 q^{33} - 134642 q^{41} + 37457 \beta q^{43} + 117649 q^{49} - 39698 \beta q^{51} + 114172 q^{57} - 152479 \beta q^{59} - 298313 \beta q^{67} + 593134 q^{73} - 359375 \beta q^{75} + 381205 q^{81} + 339463 \beta q^{83} + 357262 q^{89} + 1822754 q^{97} - 1621403 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2774 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2774 q^{9} - 3452 q^{17} - 31250 q^{25} - 215096 q^{33} - 269284 q^{41} + 235298 q^{49} + 228344 q^{57} + 1186268 q^{73} + 762410 q^{81} + 714524 q^{89} + 3645508 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
1.00000i
1.00000i
0 46.0000i 0 0 0 0 0 −1387.00 0
255.2 0 46.0000i 0 0 0 0 0 −1387.00 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
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 256.7.c.d 2
4.b odd 2 1 inner 256.7.c.d 2
8.b even 2 1 inner 256.7.c.d 2
8.d odd 2 1 CM 256.7.c.d 2
16.e even 4 1 8.7.d.a 1
16.e even 4 1 32.7.d.a 1
16.f odd 4 1 8.7.d.a 1
16.f odd 4 1 32.7.d.a 1
48.i odd 4 1 72.7.b.a 1
48.i odd 4 1 288.7.b.a 1
48.k even 4 1 72.7.b.a 1
48.k even 4 1 288.7.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.7.d.a 1 16.e even 4 1
8.7.d.a 1 16.f odd 4 1
32.7.d.a 1 16.e even 4 1
32.7.d.a 1 16.f odd 4 1
72.7.b.a 1 48.i odd 4 1
72.7.b.a 1 48.k even 4 1
256.7.c.d 2 1.a even 1 1 trivial
256.7.c.d 2 4.b odd 2 1 inner
256.7.c.d 2 8.b even 2 1 inner
256.7.c.d 2 8.d odd 2 1 CM
288.7.b.a 1 48.i odd 4 1
288.7.b.a 1 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{2} + 2116 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2116 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 5466244 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 1726)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 6160324 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T + 134642)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 5612107396 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 92999381764 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 355962583876 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 593134)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 460940513476 \) Copy content Toggle raw display
$89$ \( (T - 357262)^{2} \) Copy content Toggle raw display
$97$ \( (T - 1822754)^{2} \) Copy content Toggle raw display
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