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Properties

Label 256.8.b.n
Level $256$
Weight $8$
Character orbit 256.b
Analytic conductor $79.971$
Analytic rank $0$
Dimension $8$
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] = [256,8,Mod(129,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.129");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 256.b (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: \(79.9705665239\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 62x^{6} - 180x^{5} + 841x^{4} + 5760x^{3} + 13084x^{2} + 13680x + 5625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{42}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 128)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1) q^{3} + (\beta_{3} + 2 \beta_{2} + 8 \beta_1) q^{5} + (\beta_{5} + 340) q^{7} + (3 \beta_{6} - 2 \beta_{5} + \cdots - 1289) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1) q^{3} + (\beta_{3} + 2 \beta_{2} + 8 \beta_1) q^{5} + (\beta_{5} + 340) q^{7} + (3 \beta_{6} - 2 \beta_{5} + \cdots - 1289) q^{9}+ \cdots + (2104 \beta_{7} + 9696 \beta_{3} + \cdots + 1093089 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2720 q^{7} - 10312 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2720 q^{7} - 10312 q^{9} + 48544 q^{15} - 3984 q^{17} + 81248 q^{23} - 283832 q^{25} - 623104 q^{31} + 466016 q^{33} + 1159520 q^{39} - 1659472 q^{41} - 4320576 q^{47} + 3898056 q^{49} + 5906592 q^{55} - 10279072 q^{57} - 20079520 q^{63} + 8455968 q^{65} - 328160 q^{71} - 18151024 q^{73} - 36261696 q^{79} + 33173416 q^{81} - 10679328 q^{87} - 9219824 q^{89} - 26350176 q^{95} + 4506480 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 62x^{6} - 180x^{5} + 841x^{4} + 5760x^{3} + 13084x^{2} + 13680x + 5625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 69616 \nu^{7} - 156960 \nu^{6} - 3938672 \nu^{5} - 3744360 \nu^{4} + 65902336 \nu^{3} + \cdots + 188115840 ) / 377985 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 82684 \nu^{7} + 131520 \nu^{6} + 4925668 \nu^{5} + 6972810 \nu^{4} - 80513624 \nu^{3} + \cdots - 303230100 ) / 125995 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 276532 \nu^{7} - 437784 \nu^{6} - 16809620 \nu^{5} - 20996622 \nu^{4} + 278830336 \nu^{3} + \cdots + 841862160 ) / 75597 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 697024 \nu^{7} + 1634560 \nu^{6} + 39185408 \nu^{5} + 34591520 \nu^{4} - 658256704 \nu^{3} + \cdots - 1667684480 ) / 125995 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 709552 \nu^{7} - 1455360 \nu^{6} + 54409664 \nu^{5} + 178701000 \nu^{4} - 858811792 \nu^{3} + \cdots - 4696677600 ) / 125995 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 270912 \nu^{7} - 836544 \nu^{6} - 14288832 \nu^{5} - 4187616 \nu^{4} + 243431552 \nu^{3} + \cdots + 471234624 ) / 25199 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 26266328 \nu^{7} + 65463840 \nu^{6} + 1479963976 \nu^{5} + 980463780 \nu^{4} + \cdots - 51441020280 ) / 377985 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -12\beta_{6} + 4\beta_{5} - 21\beta_{4} + 192\beta_1 ) / 3072 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - 36\beta_{6} + 12\beta_{5} - 111\beta_{4} + 24\beta_{3} + 156\beta_{2} - 32\beta _1 + 47616 ) / 3072 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 18 \beta_{7} - 780 \beta_{6} + 244 \beta_{5} - 1569 \beta_{4} + 216 \beta_{3} + 2556 \beta_{2} + \cdots + 414720 ) / 6144 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 31 \beta_{7} - 540 \beta_{6} + 180 \beta_{5} - 1305 \beta_{4} + 396 \beta_{3} + 2994 \beta_{2} + \cdots + 415104 ) / 768 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1830 \beta_{7} - 31260 \beta_{6} + 9956 \beta_{5} - 69261 \beta_{4} + 21960 \beta_{3} + \cdots + 20736000 ) / 6144 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 7778 \beta_{7} - 99828 \beta_{6} + 32556 \beta_{5} - 229143 \beta_{4} + 97752 \beta_{3} + \cdots + 70080000 ) / 3072 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 60186 \beta_{7} - 663612 \beta_{6} + 214028 \beta_{5} - 1502295 \beta_{4} + 737352 \beta_{3} + \cdots + 456261120 ) / 3072 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
6.79778 + 0.500000i
−1.38740 − 0.500000i
−3.04372 + 0.500000i
−2.36666 − 0.500000i
−2.36666 + 0.500000i
−3.04372 − 0.500000i
−1.38740 + 0.500000i
6.79778 − 0.500000i
0 − 90.5318i 0 482.686i 0 644.627 0 −6009.00 0
129.2 0 − 67.3033i 0 − 50.5583i 0 1686.22 0 −2342.73 0
129.3 0 − 31.1632i 0 − 436.804i 0 384.352 0 1215.86 0
129.4 0 − 14.3917i 0 − 167.560i 0 −1355.20 0 1979.88 0
129.5 0 14.3917i 0 167.560i 0 −1355.20 0 1979.88 0
129.6 0 31.1632i 0 436.804i 0 384.352 0 1215.86 0
129.7 0 67.3033i 0 50.5583i 0 1686.22 0 −2342.73 0
129.8 0 90.5318i 0 − 482.686i 0 644.627 0 −6009.00 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.8
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 256.8.b.n 8
4.b odd 2 1 256.8.b.m 8
8.b even 2 1 inner 256.8.b.n 8
8.d odd 2 1 256.8.b.m 8
16.e even 4 1 128.8.a.e ✓ 4
16.e even 4 1 128.8.a.h yes 4
16.f odd 4 1 128.8.a.f yes 4
16.f odd 4 1 128.8.a.g yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.8.a.e ✓ 4 16.e even 4 1
128.8.a.f yes 4 16.f odd 4 1
128.8.a.g yes 4 16.f odd 4 1
128.8.a.h yes 4 16.e even 4 1
256.8.b.m 8 4.b odd 2 1
256.8.b.m 8 8.d odd 2 1
256.8.b.n 8 1.a even 1 1 trivial
256.8.b.n 8 8.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 \(S_{8}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{8} + 13904T_{3}^{6} + 52321120T_{3}^{4} + 46303554816T_{3}^{2} + 7467583705344 \) Copy content Toggle raw display
\( T_{7}^{4} - 1360T_{7}^{3} - 1696800T_{7}^{2} + 2269384448T_{7} - 566184154880 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 7467583705344 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} - 1360 T^{3} + \cdots - 566184154880)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 40\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 45\!\cdots\!56)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots - 59\!\cdots\!44)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 92\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 99\!\cdots\!40)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 54\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots - 26\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 87\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 74\!\cdots\!20)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 97\!\cdots\!84)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 34\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 40\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 49\!\cdots\!52)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 41\!\cdots\!36)^{2} \) Copy content Toggle raw display
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