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Properties

Label 256.6.e.b
Level $256$
Weight $6$
Character orbit 256.e
Analytic conductor $41.058$
Analytic rank $0$
Dimension $16$
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] = [256,6,Mod(65,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.65");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 256.e (of order \(4\), degree \(2\), 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: \(41.0582578721\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{14} + 68094 x^{12} - 631280 x^{10} + 1162609411 x^{8} - 7070628176 x^{6} + \cdots + 25477586689 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{40} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{3} + ( - \beta_{3} + 10 \beta_{2} + 10) q^{5} + (\beta_{14} + 5 \beta_{10}) q^{7} + (\beta_{7} + \beta_{6} + \cdots - 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{3} + ( - \beta_{3} + 10 \beta_{2} + 10) q^{5} + (\beta_{14} + 5 \beta_{10}) q^{7} + (\beta_{7} + \beta_{6} + \cdots - 3 \beta_1) q^{9}+ \cdots + (322 \beta_{15} + \cdots + 5115 \beta_{8}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 160 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 160 q^{5} + 576 q^{13} - 384 q^{17} + 1120 q^{21} + 14624 q^{29} - 78208 q^{33} + 39552 q^{37} + 190336 q^{45} - 196112 q^{49} + 26432 q^{53} + 274944 q^{61} - 630624 q^{65} + 315040 q^{69} + 426912 q^{77} - 1012880 q^{81} + 451488 q^{85} + 1128512 q^{93} - 833152 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{14} + 68094 x^{12} - 631280 x^{10} + 1162609411 x^{8} - 7070628176 x^{6} + \cdots + 25477586689 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 20\!\cdots\!30 \nu^{14} + \cdots + 12\!\cdots\!50 ) / 56\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 88\!\cdots\!24 \nu^{14} + \cdots - 55\!\cdots\!24 ) / 16\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 48\!\cdots\!50 \nu^{14} + \cdots - 78\!\cdots\!38 ) / 62\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12\!\cdots\!72 \nu^{14} + \cdots - 32\!\cdots\!16 ) / 84\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 33283663680400 \nu^{14} + 156525658789092 \nu^{12} + \cdots - 15\!\cdots\!44 ) / 93\!\cdots\!03 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31\!\cdots\!88 \nu^{14} + \cdots + 25\!\cdots\!56 ) / 84\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 99\!\cdots\!16 \nu^{14} + \cdots - 62\!\cdots\!22 ) / 48\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 24\!\cdots\!49 \nu^{15} + \cdots + 91\!\cdots\!22 \nu ) / 26\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 35\!\cdots\!76 \nu^{15} + \cdots + 27\!\cdots\!39 \nu ) / 29\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 22\!\cdots\!96 \nu^{15} + \cdots + 42\!\cdots\!80 \nu ) / 15\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 20\!\cdots\!69 \nu^{15} + \cdots + 69\!\cdots\!26 \nu ) / 51\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 70\!\cdots\!08 \nu^{15} + \cdots + 57\!\cdots\!67 \nu ) / 15\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 14\!\cdots\!04 \nu^{15} + \cdots + 85\!\cdots\!40 \nu ) / 14\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 52\!\cdots\!73 \nu^{15} + \cdots - 98\!\cdots\!93 \nu ) / 15\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 20\!\cdots\!47 \nu^{15} + \cdots - 12\!\cdots\!41 \nu ) / 89\!\cdots\!02 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} + 8\beta_{9} ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} - 2\beta_{6} + \beta_{5} - 6\beta_{3} + 760\beta_{2} + 6\beta _1 + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 36 \beta_{15} + 12 \beta_{14} - 553 \beta_{13} - 18 \beta_{12} + 30 \beta_{11} + \cdots - 2914 \beta_{8} ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12\beta_{7} - 213\beta_{6} - 561\beta_{5} - 362\beta_{4} + 1122\beta_{3} - 2184\beta_{2} + 1182\beta _1 - 136172 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2396 \beta_{15} - 9676 \beta_{14} - 5413 \beta_{13} + 3486 \beta_{12} + 5790 \beta_{11} + \cdots + 20990 \beta_{8} ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 134566 \beta_{7} + 273263 \beta_{6} - 77713 \beta_{5} + 9114 \beta_{4} + 427044 \beta_{3} + \cdots + 2153552 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1168384 \beta_{15} - 213668 \beta_{14} + 22119671 \beta_{13} + 738906 \beta_{12} + \cdots + 49977664 \beta_{8} ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 701673 \beta_{7} + 3387597 \beta_{6} + 15693057 \beta_{5} + 6261918 \beta_{4} - 18556485 \beta_{3} + \cdots + 2314445374 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 71410644 \beta_{15} + 533755956 \beta_{14} + 79691053 \beta_{13} + 45704814 \beta_{12} + \cdots - 2229946554 \beta_{8} ) / 32 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 4660494238 \beta_{7} - 13891473505 \beta_{6} + 2235431537 \beta_{5} - 777572358 \beta_{4} + \cdots - 220132757752 ) / 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 117580484604 \beta_{15} + 10790362044 \beta_{14} - 2359102930501 \beta_{13} + \cdots - 3431613021094 \beta_{8} ) / 32 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 50289019308 \beta_{7} - 84262905681 \beta_{6} - 748654579473 \beta_{5} - 216569715022 \beta_{4} + \cdots - 78792480664180 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 662330232376 \beta_{15} - 12623639988512 \beta_{14} + 7890308106673 \beta_{13} + \cdots + 78098935746520 \beta_{8} ) / 16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 80375591316836 \beta_{7} + 316533844615063 \beta_{6} - 21238369903382 \beta_{5} + 24788954767374 \beta_{4} + \cdots + 75\!\cdots\!72 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 53\!\cdots\!84 \beta_{15} - 72328500458476 \beta_{14} + \cdots + 11\!\cdots\!14 \beta_{8} ) / 32 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(5\) \(255\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
9.61927 − 10.3264i
9.57718 − 8.87007i
2.78199 − 2.07489i
−0.0885219 − 0.618585i
0.0885219 + 0.618585i
−2.78199 + 2.07489i
−9.57718 + 8.87007i
−9.61927 + 10.3264i
9.61927 + 10.3264i
9.57718 + 8.87007i
2.78199 + 2.07489i
−0.0885219 + 0.618585i
0.0885219 − 0.618585i
−2.78199 − 2.07489i
−9.57718 − 8.87007i
−9.61927 − 10.3264i
0 −20.6527 − 20.6527i 0 16.6911 − 16.6911i 0 − 158.994i 0 610.072i 0
65.2 0 −17.7401 − 17.7401i 0 71.5303 − 71.5303i 0 152.118i 0 386.425i 0
65.3 0 −4.14977 − 4.14977i 0 −57.6828 + 57.6828i 0 143.701i 0 − 208.559i 0
65.4 0 −1.23717 − 1.23717i 0 9.46137 − 9.46137i 0 217.226i 0 − 239.939i 0
65.5 0 1.23717 + 1.23717i 0 9.46137 − 9.46137i 0 − 217.226i 0 − 239.939i 0
65.6 0 4.14977 + 4.14977i 0 −57.6828 + 57.6828i 0 − 143.701i 0 − 208.559i 0
65.7 0 17.7401 + 17.7401i 0 71.5303 − 71.5303i 0 − 152.118i 0 386.425i 0
65.8 0 20.6527 + 20.6527i 0 16.6911 − 16.6911i 0 158.994i 0 610.072i 0
193.1 0 −20.6527 + 20.6527i 0 16.6911 + 16.6911i 0 158.994i 0 − 610.072i 0
193.2 0 −17.7401 + 17.7401i 0 71.5303 + 71.5303i 0 − 152.118i 0 − 386.425i 0
193.3 0 −4.14977 + 4.14977i 0 −57.6828 − 57.6828i 0 − 143.701i 0 208.559i 0
193.4 0 −1.23717 + 1.23717i 0 9.46137 + 9.46137i 0 − 217.226i 0 239.939i 0
193.5 0 1.23717 − 1.23717i 0 9.46137 + 9.46137i 0 217.226i 0 239.939i 0
193.6 0 4.14977 − 4.14977i 0 −57.6828 − 57.6828i 0 143.701i 0 208.559i 0
193.7 0 17.7401 − 17.7401i 0 71.5303 + 71.5303i 0 152.118i 0 − 386.425i 0
193.8 0 20.6527 − 20.6527i 0 16.6911 + 16.6911i 0 − 158.994i 0 − 610.072i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.6.e.b yes 16
4.b odd 2 1 inner 256.6.e.b yes 16
8.b even 2 1 256.6.e.a ✓ 16
8.d odd 2 1 256.6.e.a ✓ 16
16.e even 4 1 256.6.e.a ✓ 16
16.e even 4 1 inner 256.6.e.b yes 16
16.f odd 4 1 256.6.e.a ✓ 16
16.f odd 4 1 inner 256.6.e.b yes 16
32.g even 8 1 1024.6.a.d 8
32.g even 8 1 1024.6.a.e 8
32.h odd 8 1 1024.6.a.d 8
32.h odd 8 1 1024.6.a.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
256.6.e.a ✓ 16 8.b even 2 1
256.6.e.a ✓ 16 8.d odd 2 1
256.6.e.a ✓ 16 16.e even 4 1
256.6.e.a ✓ 16 16.f odd 4 1
256.6.e.b yes 16 1.a even 1 1 trivial
256.6.e.b yes 16 4.b odd 2 1 inner
256.6.e.b yes 16 16.e even 4 1 inner
256.6.e.b yes 16 16.f odd 4 1 inner
1024.6.a.d 8 32.g even 8 1
1024.6.a.d 8 32.h odd 8 1
1024.6.a.e 8 32.g even 8 1
1024.6.a.e 8 32.h odd 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{16} + 1125104T_{3}^{12} + 289653982816T_{3}^{8} + 344707832528640T_{3}^{4} + 3204757315051776 \) Copy content Toggle raw display
\( T_{5}^{8} - 80 T_{5}^{7} + 3200 T_{5}^{6} + 154080 T_{5}^{5} + 57225672 T_{5}^{4} - 3258319680 T_{5}^{3} + \cdots + 6793133300496 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots + 6793133300496)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 56\!\cdots\!44)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 80\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 24\!\cdots\!36)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 96 T^{3} + \cdots - 195965784432)^{4} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 24\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 48\!\cdots\!36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 53\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 18\!\cdots\!96)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 22\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 30\!\cdots\!04)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 33\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 13\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 22\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 12\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 38\!\cdots\!04)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 32\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 69\!\cdots\!84)^{4} \) Copy content Toggle raw display
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