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Properties

Label 768.3.e.p
Level $768$
Weight $3$
Character orbit 768.e
Analytic conductor $20.926$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,3,Mod(257,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.257");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.e (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: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.16845963264.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 6x^{6} + 15x^{4} + 8x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3 \)
Twist minimal: no (minimal twist has level 384)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{3} q^{5} + (\beta_{4} + \beta_1) q^{7} + ( - \beta_{6} - \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{3} q^{5} + (\beta_{4} + \beta_1) q^{7} + ( - \beta_{6} - \beta_{3} + 1) q^{9} + \beta_{7} q^{11} + ( - 2 \beta_{6} + \beta_{2}) q^{13} + ( - \beta_{7} + \beta_{5}) q^{15} + (2 \beta_{3} - \beta_{2}) q^{17} + (\beta_{5} - 3 \beta_{4} - \beta_1) q^{19} + ( - 2 \beta_{6} + \beta_{3} + 3 \beta_{2} + 8) q^{21} + (2 \beta_{7} + 2 \beta_{5} + \cdots - 6 \beta_1) q^{23}+ \cdots + (8 \beta_{7} - 3 \beta_{5} + \cdots - 7 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} + 64 q^{21} - 24 q^{25} - 32 q^{33} - 128 q^{37} + 192 q^{45} - 40 q^{49} - 16 q^{57} - 384 q^{61} + 320 q^{69} + 144 q^{73} + 200 q^{81} - 512 q^{85} + 640 q^{93} + 112 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 6x^{6} + 15x^{4} + 8x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6\nu^{7} - 2\nu^{6} - 43\nu^{5} + 7\nu^{4} + 131\nu^{3} - 29\nu^{2} - 37\nu - 6 ) / 33 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12\nu^{7} - 64\nu^{5} + 152\nu^{3} + 212\nu ) / 33 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -14\nu^{7} + 82\nu^{5} - 170\nu^{3} - 240\nu ) / 33 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\nu^{7} + 2\nu^{6} - 75\nu^{5} - 7\nu^{4} + 207\nu^{3} + 29\nu^{2} - 63\nu + 6 ) / 33 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6\nu^{7} + 4\nu^{6} - 54\nu^{5} - 14\nu^{4} + 186\nu^{3} + 58\nu^{2} - 48\nu + 12 ) / 33 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6\nu^{7} - 8\nu^{6} - 32\nu^{5} + 28\nu^{4} + 76\nu^{3} + 16\nu^{2} + 106\nu - 222 ) / 33 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{6} + 26\nu^{4} - 70\nu^{2} - 12 ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 2\beta_{5} - 4\beta_{4} + 3\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{6} + 4\beta_{5} + 4\beta_{4} - 3\beta_{2} - 12\beta _1 + 36 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{5} - 2\beta_{4} + 6\beta_{3} + 7\beta_{2} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{7} + 6\beta_{6} + 26\beta_{5} + 26\beta_{4} - 3\beta_{2} - 78\beta _1 + 36 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{5} + 5\beta_{4} + 45\beta_{3} + 51\beta_{2} - 3\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 7\beta_{7} - 22\beta_{6} + 33\beta_{5} + 33\beta_{4} + 11\beta_{2} - 99\beta _1 - 156 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -154\beta_{5} + 200\beta_{4} + 252\beta_{3} + 291\beta_{2} - 108\beta_1 ) / 24 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.444099 + 0.707107i
0.444099 − 0.707107i
−1.95007 − 0.707107i
−1.95007 + 0.707107i
1.95007 + 0.707107i
1.95007 − 0.707107i
−0.444099 − 0.707107i
−0.444099 + 0.707107i
0 −2.93352 − 0.628052i 0 − 6.51323i 0 −7.64344 0 8.21110 + 3.68481i 0
257.2 0 −2.93352 + 0.628052i 0 6.51323i 0 −7.64344 0 8.21110 − 3.68481i 0
257.3 0 −1.18087 − 2.75782i 0 − 3.68481i 0 5.43855 0 −6.21110 + 6.51323i 0
257.4 0 −1.18087 + 2.75782i 0 3.68481i 0 5.43855 0 −6.21110 − 6.51323i 0
257.5 0 1.18087 − 2.75782i 0 3.68481i 0 −5.43855 0 −6.21110 − 6.51323i 0
257.6 0 1.18087 + 2.75782i 0 − 3.68481i 0 −5.43855 0 −6.21110 + 6.51323i 0
257.7 0 2.93352 − 0.628052i 0 6.51323i 0 7.64344 0 8.21110 − 3.68481i 0
257.8 0 2.93352 + 0.628052i 0 − 6.51323i 0 7.64344 0 8.21110 + 3.68481i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.3.e.p 8
3.b odd 2 1 inner 768.3.e.p 8
4.b odd 2 1 inner 768.3.e.p 8
8.b even 2 1 768.3.e.o 8
8.d odd 2 1 768.3.e.o 8
12.b even 2 1 inner 768.3.e.p 8
16.e even 4 2 384.3.h.g ✓ 16
16.f odd 4 2 384.3.h.g ✓ 16
24.f even 2 1 768.3.e.o 8
24.h odd 2 1 768.3.e.o 8
48.i odd 4 2 384.3.h.g ✓ 16
48.k even 4 2 384.3.h.g ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.h.g ✓ 16 16.e even 4 2
384.3.h.g ✓ 16 16.f odd 4 2
384.3.h.g ✓ 16 48.i odd 4 2
384.3.h.g ✓ 16 48.k even 4 2
768.3.e.o 8 8.b even 2 1
768.3.e.o 8 8.d odd 2 1
768.3.e.o 8 24.f even 2 1
768.3.e.o 8 24.h odd 2 1
768.3.e.p 8 1.a even 1 1 trivial
768.3.e.p 8 3.b odd 2 1 inner
768.3.e.p 8 4.b odd 2 1 inner
768.3.e.p 8 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{4} + 56T_{5}^{2} + 576 \) Copy content Toggle raw display
\( T_{7}^{4} - 88T_{7}^{2} + 1728 \) Copy content Toggle raw display
\( T_{11}^{4} + 320T_{11}^{2} + 432 \) Copy content Toggle raw display
\( T_{19}^{4} - 1000T_{19}^{2} + 139968 \) Copy content Toggle raw display
\( T_{37}^{2} + 32T_{37} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{4} + 56 T^{2} + 576)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 88 T^{2} + 1728)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 320 T^{2} + 432)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 208)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 352 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 1000 T^{2} + 139968)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 2048 T^{2} + 995328)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 1144 T^{2} + 10816)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4120 T^{2} + 3195072)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 32 T + 48)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 4352 T^{2} + 1327104)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 2344 T^{2} + 914112)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 11264 T^{2} + 28311552)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 3832 T^{2} + 2096704)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 2592 T^{2} + 314928)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 96 T + 432)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 360 T^{2} + 15552)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 12800 T^{2} + 31961088)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 36 T - 7164)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 15448 T^{2} + 1453248)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 8192 T^{2} + 10112688)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 20192 T^{2} + 9216)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 28 T - 5004)^{4} \) Copy content Toggle raw display
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