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Properties

Label 768.4.f.c
Level $768$
Weight $4$
Character orbit 768.f
Analytic conductor $45.313$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,4,Mod(383,768)]
 
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([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.383");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.f (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: \(45.3134668844\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 12)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + \beta_1 q^{5} + \beta_{2} q^{7} + ( - 3 \beta_{7} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + \beta_1 q^{5} + \beta_{2} q^{7} + ( - 3 \beta_{7} + 3) q^{9} + ( - 5 \beta_{4} - 5 \beta_{3}) q^{11} - 5 \beta_{5} q^{13} + ( - 5 \beta_{6} - 4 \beta_{2}) q^{15} - 4 \beta_{7} q^{17} + (9 \beta_{4} - 9 \beta_{3}) q^{19} + ( - 15 \beta_{5} + 3 \beta_1) q^{21} + 14 \beta_{6} q^{23} - 45 q^{25} + ( - 27 \beta_{4} - 6 \beta_{3}) q^{27} + 17 \beta_1 q^{29} - 29 \beta_{2} q^{31} + ( - 15 \beta_{7} - 120) q^{33} + (10 \beta_{4} + 10 \beta_{3}) q^{35} + 65 \beta_{5} q^{37} + ( - 5 \beta_{6} + 5 \beta_{2}) q^{39} - 14 \beta_{7} q^{41} + ( - 29 \beta_{4} + 29 \beta_{3}) q^{43} + (120 \beta_{5} + 3 \beta_1) q^{45} - 28 \beta_{6} q^{47} + 283 q^{49} + ( - 36 \beta_{4} - 4 \beta_{3}) q^{51} + 61 \beta_1 q^{53} - 40 \beta_{2} q^{55} + ( - 27 \beta_{7} + 270) q^{57} + ( - 25 \beta_{4} - 25 \beta_{3}) q^{59} - 221 \beta_{5} q^{61} + ( - 30 \beta_{6} + 3 \beta_{2}) q^{63} + 10 \beta_{7} q^{65} + ( - 95 \beta_{4} + 95 \beta_{3}) q^{67} + ( - 168 \beta_{5} - 42 \beta_1) q^{69} - 150 \beta_{6} q^{71} - 410 q^{73} + 45 \beta_{3} q^{75} + 30 \beta_1 q^{77} + 11 \beta_{2} q^{79} + ( - 18 \beta_{7} - 711) q^{81} + ( - 181 \beta_{4} - 181 \beta_{3}) q^{83} + 160 \beta_{5} q^{85} + ( - 85 \beta_{6} - 68 \beta_{2}) q^{87} + 94 \beta_{7} q^{89} + ( - 10 \beta_{4} + 10 \beta_{3}) q^{91} + (435 \beta_{5} - 87 \beta_1) q^{93} - 90 \beta_{6} q^{95} + 770 q^{97} + ( - 135 \beta_{4} + 105 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{9} - 360 q^{25} - 960 q^{33} + 2264 q^{49} + 2160 q^{57} - 3280 q^{73} - 5688 q^{81} + 6160 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{6} + 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -4\nu^{6} + 12\nu^{4} - 28\nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} - 6\nu^{6} + 16\nu^{5} + 16\nu^{4} - 44\nu^{3} - 48\nu^{2} + 31\nu + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{7} - 6\nu^{6} - 16\nu^{5} + 16\nu^{4} + 44\nu^{3} - 48\nu^{2} - 31\nu + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 8\nu^{5} - 20\nu^{3} + \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\nu^{7} + 8\nu^{5} - 20\nu^{3} + 14\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 6\nu^{7} - 16\nu^{5} + 44\nu^{3} - 2\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} + 2\beta_{5} - 2\beta_{4} + 2\beta_{3} ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{4} - 3\beta_{3} + 2\beta_{2} - \beta _1 + 12 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 4\beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{4} - 7\beta_{3} + 6\beta_{2} + 3\beta _1 - 28 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} + 11\beta_{6} + 22\beta_{5} + 10\beta_{4} - 10\beta_{3} ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{7} + 29\beta_{6} - 58\beta_{5} + 26\beta_{4} - 26\beta_{3} ) / 16 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
383.1
1.40126 − 0.809017i
0.535233 − 0.309017i
1.40126 + 0.809017i
0.535233 + 0.309017i
−1.40126 + 0.809017i
−0.535233 + 0.309017i
−1.40126 − 0.809017i
−0.535233 − 0.309017i
0 −3.87298 − 3.46410i 0 −8.94427 0 − 7.74597i 0 3.00000 + 26.8328i 0
383.2 0 −3.87298 − 3.46410i 0 8.94427 0 7.74597i 0 3.00000 + 26.8328i 0
383.3 0 −3.87298 + 3.46410i 0 −8.94427 0 7.74597i 0 3.00000 − 26.8328i 0
383.4 0 −3.87298 + 3.46410i 0 8.94427 0 − 7.74597i 0 3.00000 − 26.8328i 0
383.5 0 3.87298 − 3.46410i 0 −8.94427 0 − 7.74597i 0 3.00000 − 26.8328i 0
383.6 0 3.87298 − 3.46410i 0 8.94427 0 7.74597i 0 3.00000 − 26.8328i 0
383.7 0 3.87298 + 3.46410i 0 −8.94427 0 7.74597i 0 3.00000 + 26.8328i 0
383.8 0 3.87298 + 3.46410i 0 8.94427 0 − 7.74597i 0 3.00000 + 26.8328i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 383.8
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
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.4.f.c 8
3.b odd 2 1 inner 768.4.f.c 8
4.b odd 2 1 inner 768.4.f.c 8
8.b even 2 1 inner 768.4.f.c 8
8.d odd 2 1 inner 768.4.f.c 8
12.b even 2 1 inner 768.4.f.c 8
16.e even 4 1 12.4.b.a ✓ 4
16.e even 4 1 192.4.c.b 4
16.f odd 4 1 12.4.b.a ✓ 4
16.f odd 4 1 192.4.c.b 4
24.f even 2 1 inner 768.4.f.c 8
24.h odd 2 1 inner 768.4.f.c 8
48.i odd 4 1 12.4.b.a ✓ 4
48.i odd 4 1 192.4.c.b 4
48.k even 4 1 12.4.b.a ✓ 4
48.k even 4 1 192.4.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.b.a ✓ 4 16.e even 4 1
12.4.b.a ✓ 4 16.f odd 4 1
12.4.b.a ✓ 4 48.i odd 4 1
12.4.b.a ✓ 4 48.k even 4 1
192.4.c.b 4 16.e even 4 1
192.4.c.b 4 16.f odd 4 1
192.4.c.b 4 48.i odd 4 1
192.4.c.b 4 48.k even 4 1
768.4.f.c 8 1.a even 1 1 trivial
768.4.f.c 8 3.b odd 2 1 inner
768.4.f.c 8 4.b odd 2 1 inner
768.4.f.c 8 8.b even 2 1 inner
768.4.f.c 8 8.d odd 2 1 inner
768.4.f.c 8 12.b even 2 1 inner
768.4.f.c 8 24.f even 2 1 inner
768.4.f.c 8 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} - 80 \) Copy content Toggle raw display
\( T_{19}^{2} - 4860 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 6 T^{2} + 729)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 80)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1200)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 1280)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 4860)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 9408)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 23120)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 50460)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 16900)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 15680)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 50460)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 37632)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 297680)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 30000)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 195364)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 541500)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 1080000)^{4} \) Copy content Toggle raw display
$73$ \( (T + 410)^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} + 7260)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 1572528)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 706880)^{4} \) Copy content Toggle raw display
$97$ \( (T - 770)^{8} \) Copy content Toggle raw display
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