Properties

Label 512.3.d.c
Level $512$
Weight $3$
Character orbit 512.d
Analytic conductor $13.951$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [512,3,Mod(255,512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(512, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("512.255");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 512.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.9509895352\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 2) q^{3} + \beta_{3} q^{5} + (2 \beta_{3} - \beta_{2}) q^{7} + (4 \beta_1 + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 2) q^{3} + \beta_{3} q^{5} + (2 \beta_{3} - \beta_{2}) q^{7} + (4 \beta_1 + 5) q^{9} + (\beta_1 - 6) q^{11} + (\beta_{3} - 4 \beta_{2}) q^{13} + ( - 2 \beta_{3} + 5 \beta_{2}) q^{15} + (4 \beta_1 + 8) q^{17} + (5 \beta_1 - 14) q^{19} + ( - 6 \beta_{3} + 12 \beta_{2}) q^{21} + (2 \beta_{3} - 5 \beta_{2}) q^{23} + 5 q^{25} + ( - 4 \beta_1 - 32) q^{27} + ( - 5 \beta_{3} - 8 \beta_{2}) q^{29} + ( - 4 \beta_{3} - 10 \beta_{2}) q^{31} + (4 \beta_1 + 2) q^{33} + ( - 4 \beta_1 - 40) q^{35} + (11 \beta_{3} + 4 \beta_{2}) q^{37} + ( - 10 \beta_{3} + 13 \beta_{2}) q^{39} + ( - 16 \beta_1 - 14) q^{41} + ( - \beta_1 - 50) q^{43} + (5 \beta_{3} - 20 \beta_{2}) q^{45} + (8 \beta_{3} + 16 \beta_{2}) q^{47} + ( - 16 \beta_1 - 39) q^{49} + ( - 16 \beta_1 - 56) q^{51} + ( - \beta_{3} + 20 \beta_{2}) q^{53} + ( - 6 \beta_{3} - 5 \beta_{2}) q^{55} + (4 \beta_1 - 22) q^{57} + (19 \beta_1 - 10) q^{59} + (9 \beta_{3} - 12 \beta_{2}) q^{61} + (18 \beta_{3} - 45 \beta_{2}) q^{63} + ( - 16 \beta_1 - 20) q^{65} + (25 \beta_1 - 22) q^{67} + ( - 14 \beta_{3} + 20 \beta_{2}) q^{69} + (14 \beta_{3} + 13 \beta_{2}) q^{71} + (4 \beta_1 - 28) q^{73} + ( - 5 \beta_1 - 10) q^{75} + ( - 10 \beta_{3} - 4 \beta_{2}) q^{77} + ( - 4 \beta_{3} + 42 \beta_{2}) q^{79} + (4 \beta_1 + 59) q^{81} + (11 \beta_1 + 38) q^{83} + (8 \beta_{3} - 20 \beta_{2}) q^{85} + ( - 6 \beta_{3} - 9 \beta_{2}) q^{87} + ( - 12 \beta_1 + 36) q^{89} + ( - 36 \beta_1 - 72) q^{91} - 12 \beta_{3} q^{93} + ( - 14 \beta_{3} - 25 \beta_{2}) q^{95} + 4 \beta_1 q^{97} + ( - 19 \beta_1 + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{3} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{3} + 20 q^{9} - 24 q^{11} + 32 q^{17} - 56 q^{19} + 20 q^{25} - 128 q^{27} + 8 q^{33} - 160 q^{35} - 56 q^{41} - 200 q^{43} - 156 q^{49} - 224 q^{51} - 88 q^{57} - 40 q^{59} - 80 q^{65} - 88 q^{67} - 112 q^{73} - 40 q^{75} + 236 q^{81} + 152 q^{83} + 144 q^{89} - 288 q^{91} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 7\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{3} + 2\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/512\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(511\)
\(\chi(n)\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
1.58114 0.707107i
1.58114 + 0.707107i
−1.58114 + 0.707107i
−1.58114 0.707107i
0 −5.16228 0 4.47214i 0 11.7727i 0 17.6491 0
255.2 0 −5.16228 0 4.47214i 0 11.7727i 0 17.6491 0
255.3 0 1.16228 0 4.47214i 0 6.11584i 0 −7.64911 0
255.4 0 1.16228 0 4.47214i 0 6.11584i 0 −7.64911 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 512.3.d.c 4
4.b odd 2 1 512.3.d.e 4
8.b even 2 1 512.3.d.e 4
8.d odd 2 1 inner 512.3.d.c 4
16.e even 4 2 512.3.c.f 8
16.f odd 4 2 512.3.c.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.3.c.f 8 16.e even 4 2
512.3.c.f 8 16.f odd 4 2
512.3.d.c 4 1.a even 1 1 trivial
512.3.d.c 4 8.d odd 2 1 inner
512.3.d.e 4 4.b odd 2 1
512.3.d.e 4 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 4T_{3} - 6 \) acting on \(S_{3}^{\mathrm{new}}(512, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 4 T - 6)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 176T^{2} + 5184 \) Copy content Toggle raw display
$11$ \( (T^{2} + 12 T + 26)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 296 T^{2} + 11664 \) Copy content Toggle raw display
$17$ \( (T^{2} - 16 T - 96)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 28 T - 54)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 560 T^{2} + 14400 \) Copy content Toggle raw display
$29$ \( T^{4} + 2024T^{2} + 144 \) Copy content Toggle raw display
$31$ \( T^{4} + 2240 T^{2} + 230400 \) Copy content Toggle raw display
$37$ \( T^{4} + 5096 T^{2} + \cdots + 5253264 \) Copy content Toggle raw display
$41$ \( (T^{2} + 28 T - 2364)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 100 T + 2490)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 6656 T^{2} + 589824 \) Copy content Toggle raw display
$53$ \( T^{4} + 6440 T^{2} + \cdots + 10112400 \) Copy content Toggle raw display
$59$ \( (T^{2} + 20 T - 3510)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 5544 T^{2} + 219024 \) Copy content Toggle raw display
$67$ \( (T^{2} + 44 T - 5766)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 10544 T^{2} + \cdots + 6594624 \) Copy content Toggle raw display
$73$ \( (T^{2} + 56 T + 624)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 28864 T^{2} + \cdots + 190219264 \) Copy content Toggle raw display
$83$ \( (T^{2} - 76 T + 234)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 72 T - 144)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 160)^{2} \) Copy content Toggle raw display
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