Properties

Label 256.7.d.f
Level $256$
Weight $7$
Character orbit 256.d
Analytic conductor $58.894$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(58.8938454067\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{15})\)
Defining polynomial: \(x^{4} - 7 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 4)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} -5 \beta_{1} q^{5} + 5 \beta_{3} q^{7} + 231 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} -5 \beta_{1} q^{5} + 5 \beta_{3} q^{7} + 231 q^{9} + 31 \beta_{2} q^{11} + 733 \beta_{1} q^{13} + 5 \beta_{3} q^{15} -4766 q^{17} + 243 \beta_{2} q^{19} -4800 \beta_{1} q^{21} -169 \beta_{3} q^{23} + 15525 q^{25} + 498 \beta_{2} q^{27} + 12749 \beta_{1} q^{29} -676 \beta_{3} q^{31} -29760 q^{33} + 100 \beta_{2} q^{35} -997 \beta_{1} q^{37} -733 \beta_{3} q^{39} -29362 q^{41} + 695 \beta_{2} q^{43} -1155 \beta_{1} q^{45} + 122 \beta_{3} q^{47} + 21649 q^{49} + 4766 \beta_{2} q^{51} + 96427 \beta_{1} q^{53} -155 \beta_{3} q^{55} -233280 q^{57} + 2531 \beta_{2} q^{59} -5459 \beta_{1} q^{61} + 1155 \beta_{3} q^{63} + 14660 q^{65} -12721 \beta_{2} q^{67} + 162240 \beta_{1} q^{69} + 8589 \beta_{3} q^{71} -288626 q^{73} -15525 \beta_{2} q^{75} + 148800 \beta_{1} q^{77} + 5014 \beta_{3} q^{79} -646479 q^{81} -6589 \beta_{2} q^{83} + 23830 \beta_{1} q^{85} -12749 \beta_{3} q^{87} -310738 q^{89} -14660 \beta_{2} q^{91} + 648960 \beta_{1} q^{93} -1215 \beta_{3} q^{95} -1457086 q^{97} + 7161 \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 924q^{9} + O(q^{10}) \) \( 4q + 924q^{9} - 19064q^{17} + 62100q^{25} - 119040q^{33} - 117448q^{41} + 86596q^{49} - 933120q^{57} + 58640q^{65} - 1154504q^{73} - 2585916q^{81} - 1242952q^{89} - 5828344q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 7 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 3 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\( -2 \nu^{3} + 22 \nu \)
\(\beta_{3}\)\(=\)\( 32 \nu^{2} - 112 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 4 \beta_{1}\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 112\)\()/32\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{2} + 44 \beta_{1}\)\()/16\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.93649 + 0.500000i
1.93649 0.500000i
−1.93649 + 0.500000i
−1.93649 0.500000i
0 −30.9839 0 10.0000i 0 309.839i 0 231.000 0
127.2 0 −30.9839 0 10.0000i 0 309.839i 0 231.000 0
127.3 0 30.9839 0 10.0000i 0 309.839i 0 231.000 0
127.4 0 30.9839 0 10.0000i 0 309.839i 0 231.000 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
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.7.d.f 4
4.b odd 2 1 inner 256.7.d.f 4
8.b even 2 1 inner 256.7.d.f 4
8.d odd 2 1 inner 256.7.d.f 4
16.e even 4 1 4.7.b.a 2
16.e even 4 1 64.7.c.c 2
16.f odd 4 1 4.7.b.a 2
16.f odd 4 1 64.7.c.c 2
48.i odd 4 1 36.7.d.c 2
48.i odd 4 1 576.7.g.h 2
48.k even 4 1 36.7.d.c 2
48.k even 4 1 576.7.g.h 2
80.i odd 4 1 100.7.d.a 4
80.j even 4 1 100.7.d.a 4
80.k odd 4 1 100.7.b.c 2
80.q even 4 1 100.7.b.c 2
80.s even 4 1 100.7.d.a 4
80.t odd 4 1 100.7.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.7.b.a 2 16.e even 4 1
4.7.b.a 2 16.f odd 4 1
36.7.d.c 2 48.i odd 4 1
36.7.d.c 2 48.k even 4 1
64.7.c.c 2 16.e even 4 1
64.7.c.c 2 16.f odd 4 1
100.7.b.c 2 80.k odd 4 1
100.7.b.c 2 80.q even 4 1
100.7.d.a 4 80.i odd 4 1
100.7.d.a 4 80.j even 4 1
100.7.d.a 4 80.s even 4 1
100.7.d.a 4 80.t odd 4 1
256.7.d.f 4 1.a even 1 1 trivial
256.7.d.f 4 4.b odd 2 1 inner
256.7.d.f 4 8.b even 2 1 inner
256.7.d.f 4 8.d odd 2 1 inner
576.7.g.h 2 48.i odd 4 1
576.7.g.h 2 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 960 \) acting on \(S_{7}^{\mathrm{new}}(256, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -960 + T^{2} )^{2} \)
$5$ \( ( 100 + T^{2} )^{2} \)
$7$ \( ( 96000 + T^{2} )^{2} \)
$11$ \( ( -922560 + T^{2} )^{2} \)
$13$ \( ( 2149156 + T^{2} )^{2} \)
$17$ \( ( 4766 + T )^{4} \)
$19$ \( ( -56687040 + T^{2} )^{2} \)
$23$ \( ( 109674240 + T^{2} )^{2} \)
$29$ \( ( 650148004 + T^{2} )^{2} \)
$31$ \( ( 1754787840 + T^{2} )^{2} \)
$37$ \( ( 3976036 + T^{2} )^{2} \)
$41$ \( ( 29362 + T )^{4} \)
$43$ \( ( -463704000 + T^{2} )^{2} \)
$47$ \( ( 57154560 + T^{2} )^{2} \)
$53$ \( ( 37192665316 + T^{2} )^{2} \)
$59$ \( ( -6149722560 + T^{2} )^{2} \)
$61$ \( ( 119202724 + T^{2} )^{2} \)
$67$ \( ( -155350887360 + T^{2} )^{2} \)
$71$ \( ( 283280336640 + T^{2} )^{2} \)
$73$ \( ( 288626 + T )^{4} \)
$79$ \( ( 96538352640 + T^{2} )^{2} \)
$83$ \( ( -41678324160 + T^{2} )^{2} \)
$89$ \( ( 310738 + T )^{4} \)
$97$ \( ( 1457086 + T )^{4} \)
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