Properties

Label 8.6.b.a
Level $8$
Weight $6$
Character orbit 8.b
Analytic conductor $1.283$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8,6,Mod(5,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.5");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 8.b (of order \(2\), degree \(1\), 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: \(1.28307055850\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.218489.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 8x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + (\beta_{3} - \beta_1) q^{3} + ( - \beta_{3} + \beta_{2} + \beta_1 + 5) q^{4} + ( - 2 \beta_{2} + 4 \beta_1 + 2) q^{5} + ( - 6 \beta_{3} - 2 \beta_{2} - 32) q^{6} + (4 \beta_{3} + 4 \beta_{2} + 20 \beta_1 + 36) q^{7} + (10 \beta_{3} + 6 \beta_{2} + \cdots - 58) q^{8}+ \cdots + ( - 8 \beta_{3} - 8 \beta_{2} + \cdots - 65) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + (\beta_{3} - \beta_1) q^{3} + ( - \beta_{3} + \beta_{2} + \beta_1 + 5) q^{4} + ( - 2 \beta_{2} + 4 \beta_1 + 2) q^{5} + ( - 6 \beta_{3} - 2 \beta_{2} - 32) q^{6} + (4 \beta_{3} + 4 \beta_{2} + 20 \beta_1 + 36) q^{7} + (10 \beta_{3} + 6 \beta_{2} + \cdots - 58) q^{8}+ \cdots + ( - 1821 \beta_{3} + 3456 \beta_{2} + \cdots - 3456) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 20 q^{4} - 116 q^{6} + 96 q^{7} - 248 q^{8} - 164 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 20 q^{4} - 116 q^{6} + 96 q^{7} - 248 q^{8} - 164 q^{9} + 632 q^{10} + 1576 q^{12} - 2384 q^{14} - 416 q^{15} - 3312 q^{16} + 200 q^{17} + 4754 q^{18} + 4624 q^{20} - 5636 q^{22} + 2336 q^{23} - 7792 q^{24} + 1556 q^{25} + 5608 q^{26} + 5152 q^{28} - 2128 q^{30} - 12928 q^{31} + 5408 q^{32} - 2352 q^{33} - 4772 q^{34} - 10164 q^{36} + 15980 q^{38} + 35104 q^{39} + 16032 q^{40} - 4568 q^{41} - 26144 q^{42} - 29112 q^{44} + 29200 q^{46} - 54720 q^{47} + 35616 q^{48} + 9828 q^{49} - 47498 q^{50} - 36560 q^{52} + 23288 q^{54} + 85472 q^{55} + 40768 q^{56} - 2032 q^{57} + 3784 q^{58} + 2592 q^{60} + 34496 q^{62} - 153440 q^{63} - 41920 q^{64} - 19520 q^{65} + 43224 q^{66} + 10344 q^{68} - 68928 q^{70} + 206688 q^{71} - 83272 q^{72} + 39976 q^{73} + 17464 q^{74} + 99944 q^{76} - 174064 q^{78} - 247872 q^{79} - 35520 q^{80} + 29684 q^{81} + 161132 q^{82} + 196672 q^{84} - 18500 q^{86} + 307872 q^{87} - 167216 q^{88} - 84632 q^{89} + 142280 q^{90} - 49056 q^{92} - 98784 q^{94} - 259744 q^{95} - 115648 q^{96} - 99576 q^{97} - 117042 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 8x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 2\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{2} + 6\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 7\nu^{2} + 6\nu - 20 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta _1 + 3 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{3} - \beta_{2} + 3\beta _1 + 13 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{3} + \beta_{2} - 27\beta _1 + 51 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
2.38600 1.51888i
2.38600 + 1.51888i
−1.88600 2.10784i
−1.88600 + 2.10784i
−4.77200 3.03776i 23.6095i 13.5440 + 28.9924i 1.38521i −71.7200 + 112.665i 160.704 23.4400 179.495i −314.408 4.20793 6.61022i
5.2 −4.77200 + 3.03776i 23.6095i 13.5440 28.9924i 1.38521i −71.7200 112.665i 160.704 23.4400 + 179.495i −314.408 4.20793 + 6.61022i
5.3 3.77200 4.21569i 3.25452i −3.54400 31.8031i 73.9600i 13.7200 + 12.2760i −112.704 −147.440 105.021i 232.408 311.792 + 278.977i
5.4 3.77200 + 4.21569i 3.25452i −3.54400 + 31.8031i 73.9600i 13.7200 12.2760i −112.704 −147.440 + 105.021i 232.408 311.792 278.977i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.6.b.a 4
3.b odd 2 1 72.6.d.b 4
4.b odd 2 1 32.6.b.a 4
5.b even 2 1 200.6.d.a 4
5.c odd 4 2 200.6.f.a 8
8.b even 2 1 inner 8.6.b.a 4
8.d odd 2 1 32.6.b.a 4
12.b even 2 1 288.6.d.b 4
16.e even 4 2 256.6.a.k 4
16.f odd 4 2 256.6.a.n 4
20.d odd 2 1 800.6.d.a 4
20.e even 4 2 800.6.f.a 8
24.f even 2 1 288.6.d.b 4
24.h odd 2 1 72.6.d.b 4
40.e odd 2 1 800.6.d.a 4
40.f even 2 1 200.6.d.a 4
40.i odd 4 2 200.6.f.a 8
40.k even 4 2 800.6.f.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.b.a 4 1.a even 1 1 trivial
8.6.b.a 4 8.b even 2 1 inner
32.6.b.a 4 4.b odd 2 1
32.6.b.a 4 8.d odd 2 1
72.6.d.b 4 3.b odd 2 1
72.6.d.b 4 24.h odd 2 1
200.6.d.a 4 5.b even 2 1
200.6.d.a 4 40.f even 2 1
200.6.f.a 8 5.c odd 4 2
200.6.f.a 8 40.i odd 4 2
256.6.a.k 4 16.e even 4 2
256.6.a.n 4 16.f odd 4 2
288.6.d.b 4 12.b even 2 1
288.6.d.b 4 24.f even 2 1
800.6.d.a 4 20.d odd 2 1
800.6.d.a 4 40.e odd 2 1
800.6.f.a 8 20.e even 4 2
800.6.f.a 8 40.k even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$3$ \( T^{4} + 568T^{2} + 5904 \) Copy content Toggle raw display
$5$ \( T^{4} + 5472 T^{2} + 10496 \) Copy content Toggle raw display
$7$ \( (T^{2} - 48 T - 18112)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 5520765456 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 7999305984 \) Copy content Toggle raw display
$17$ \( (T^{2} - 100 T - 72252)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 120994976016 \) Copy content Toggle raw display
$23$ \( (T^{2} - 1168 T - 2817216)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 535633608132864 \) Copy content Toggle raw display
$31$ \( (T^{2} + 6464 T + 7754752)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 306881230162176 \) Copy content Toggle raw display
$41$ \( (T^{2} + 2284 T - 85109148)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( (T^{2} + 27360 T + 132648192)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 76\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 41\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{2} - 103344 T + 2609278272)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 19988 T - 1604540316)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 123936 T + 3701816576)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 72\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{2} + 42316 T - 6875717724)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 49788 T - 783309052)^{2} \) Copy content Toggle raw display
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