Properties

Label 256.7.c.l
Level $256$
Weight $7$
Character orbit 256.c
Analytic conductor $58.894$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [256,7,Mod(255,256)]
 
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([1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.255");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 256.c (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: \(58.8938454067\)
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: \( x^{8} - 11x^{6} + 516x^{4} - 2816x^{2} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{42} \)
Twist minimal: no (minimal twist has level 8)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{5} q^{5} + \beta_{4} q^{7} + (3 \beta_{3} + 165) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{5} q^{5} + \beta_{4} q^{7} + (3 \beta_{3} + 165) q^{9} + ( - 29 \beta_{2} - 28 \beta_1) q^{11} + ( - \beta_{6} + 2 \beta_{5}) q^{13} + ( - \beta_{7} - 2 \beta_{4}) q^{15} + ( - 25 \beta_{3} + 1042) q^{17} + (13 \beta_{2} + 52 \beta_1) q^{19} + ( - \beta_{6} - 9 \beta_{5}) q^{21} + ( - 3 \beta_{7} + 4 \beta_{4}) q^{23} + ( - 110 \beta_{3} + 5975) q^{25} + (207 \beta_{2} + 399 \beta_1) q^{27} + (8 \beta_{6} - 37 \beta_{5}) q^{29} + (\beta_{7} + 57 \beta_{4}) q^{31} + ( - 55 \beta_{3} + 21012) q^{33} + ( - 690 \beta_{2} - 110 \beta_1) q^{35} + (7 \beta_{6} + 186 \beta_{5}) q^{37} + (8 \beta_{7} - 53 \beta_{4}) q^{39} + ( - 138 \beta_{3} + 29486) q^{41} + ( - 2338 \beta_{2} + 217 \beta_1) q^{43} + ( - 15 \beta_{6} + 240 \beta_{5}) q^{45} + (11 \beta_{7} - 319 \beta_{4}) q^{47} + ( - 712 \beta_{3} + 529) q^{49} + ( - 1725 \beta_{2} + 5167 \beta_1) q^{51} + ( - 7 \beta_{6} - 1032 \beta_{5}) q^{53} + (28 \beta_{7} + 201 \beta_{4}) q^{55} + (143 \beta_{3} - 31668) q^{57} + ( - 4112 \beta_{2} + 4439 \beta_1) q^{59} + ( - 41 \beta_{6} - 722 \beta_{5}) q^{61} + ( - 3 \beta_{7} + 654 \beta_{4}) q^{63} + (1570 \beta_{3} - 51360) q^{65} + ( - 8411 \beta_{2} + 2444 \beta_1) q^{67} + ( - 55 \beta_{6} + 2817 \beta_{5}) q^{69} + ( - 85 \beta_{7} - 488 \beta_{4}) q^{71} + (1067 \beta_{3} - 110978) q^{73} + ( - 7590 \beta_{2} + 24125 \beta_1) q^{75} + (173 \beta_{6} + 1093 \beta_{5}) q^{77} + ( - 52 \beta_{7} - 390 \beta_{4}) q^{79} + (3177 \beta_{3} - 142011) q^{81} + ( - 25912 \beta_{2} + 29001 \beta_1) q^{83} + (125 \beta_{6} - 4417 \beta_{5}) q^{85} + ( - 85 \beta_{7} + 382 \beta_{4}) q^{87} + (1491 \beta_{3} - 190306) q^{89} + ( - 41538 \beta_{2} + 1570 \beta_1) q^{91} + ( - 40 \beta_{6} - 1464 \beta_{5}) q^{93} + ( - 52 \beta_{7} - 169 \beta_{4}) q^{95} + ( - 5337 \beta_{3} - 231694) q^{97} + ( - 24936 \beta_{2} + 9675 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 1320 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 1320 q^{9} + 8336 q^{17} + 47800 q^{25} + 168096 q^{33} + 235888 q^{41} + 4232 q^{49} - 253344 q^{57} - 410880 q^{65} - 887824 q^{73} - 1136088 q^{81} - 1522448 q^{89} - 1853552 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 11x^{6} + 516x^{4} - 2816x^{2} + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -9\nu^{7} + 995\nu^{5} - 164\nu^{3} + 293120\nu ) / 26624 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 71\nu^{7} - 1933\nu^{5} + 30876\nu^{3} - 300800\nu ) / 26624 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 11\nu^{4} + 260\nu^{2} - 1408 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 629\nu^{4} + 6268\nu^{2} - 162304 ) / 208 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - \nu^{5} + 684\nu^{3} + 1408\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17\nu^{7} + 165\nu^{5} - 732\nu^{3} + 126080\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -179\nu^{6} - 6095\nu^{4} - 49484\nu^{2} - 1576448 ) / 208 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} - 11\beta_{5} + 56\beta_{2} + 72\beta_1 ) / 1024 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5\beta_{7} + 63\beta_{4} - 32\beta_{3} + 5632 ) / 2048 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{6} + 15\beta_{5} + 760\beta_{2} + 1928\beta_1 ) / 1024 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -51\beta_{7} - 23\beta_{4} - 352\beta_{3} - 466432 ) / 2048 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -275\beta_{6} + 3665\beta_{5} - 18232\beta_{2} + 2232\beta_1 ) / 1024 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 739\beta_{7} - 16633\beta_{4} + 20832\beta_{3} - 3711488 ) / 2048 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2403\beta_{6} + 46655\beta_{5} - 205640\beta_{2} - 472632\beta_1 ) / 1024 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
−3.26433 2.31174i
3.26433 2.31174i
2.84502 2.81174i
−2.84502 2.81174i
2.84502 + 2.81174i
−2.84502 + 2.81174i
−3.26433 + 2.31174i
3.26433 + 2.31174i
0 32.4939i 0 −199.084 0 19.6656i 0 −326.854 0
255.2 0 32.4939i 0 199.084 0 19.6656i 0 −326.854 0
255.3 0 8.49390i 0 −59.7107 0 483.584i 0 656.854 0
255.4 0 8.49390i 0 59.7107 0 483.584i 0 656.854 0
255.5 0 8.49390i 0 −59.7107 0 483.584i 0 656.854 0
255.6 0 8.49390i 0 59.7107 0 483.584i 0 656.854 0
255.7 0 32.4939i 0 −199.084 0 19.6656i 0 −326.854 0
255.8 0 32.4939i 0 199.084 0 19.6656i 0 −326.854 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 255.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.7.c.l 8
4.b odd 2 1 inner 256.7.c.l 8
8.b even 2 1 inner 256.7.c.l 8
8.d odd 2 1 inner 256.7.c.l 8
16.e even 4 1 8.7.d.b 4
16.e even 4 1 32.7.d.b 4
16.f odd 4 1 8.7.d.b 4
16.f odd 4 1 32.7.d.b 4
48.i odd 4 1 72.7.b.b 4
48.i odd 4 1 288.7.b.b 4
48.k even 4 1 72.7.b.b 4
48.k even 4 1 288.7.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.7.d.b 4 16.e even 4 1
8.7.d.b 4 16.f odd 4 1
32.7.d.b 4 16.e even 4 1
32.7.d.b 4 16.f odd 4 1
72.7.b.b 4 48.i odd 4 1
72.7.b.b 4 48.k even 4 1
256.7.c.l 8 1.a even 1 1 trivial
256.7.c.l 8 4.b odd 2 1 inner
256.7.c.l 8 8.b even 2 1 inner
256.7.c.l 8 8.d odd 2 1 inner
288.7.b.b 4 48.i odd 4 1
288.7.b.b 4 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{4} + 1128T_{3}^{2} + 76176 \) Copy content Toggle raw display
\( T_{5}^{4} - 43200T_{5}^{2} + 141312000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1128 T^{2} + 76176)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 43200 T^{2} + \cdots + 141312000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 234240 T^{2} + \cdots + 90439680)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2848232 T^{2} + \cdots + 1703139841936)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 14312640 T^{2} + \cdots + 28641121812480)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2084 T - 15714236)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 3813992 T^{2} + \cdots + 2696177136016)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 380025600 T^{2} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 969574080 T^{2} + \cdots + 19\!\cdots\!80)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 791654400 T^{2} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 2141703360 T^{2} + \cdots + 10\!\cdots\!80)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 58972 T + 357521476)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 8652467432 T^{2} + \cdots + 29\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 29538094080 T^{2} + \cdots + 10\!\cdots\!80)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 46462898880 T^{2} + \cdots + 29\!\cdots\!80)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 36814656488 T^{2} + \cdots + 33\!\cdots\!56)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 45212594880 T^{2} + \cdots + 33\!\cdots\!80)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 107958699368 T^{2} + \cdots + 57\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 348036514560 T^{2} + \cdots + 11\!\cdots\!80)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 221956 T - 18286467836)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 144092482560 T^{2} + \cdots + 31\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1508800268648 T^{2} + \cdots + 55\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 380612 T - 23540043644)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 463388 T - 711956225084)^{4} \) Copy content Toggle raw display
show more
show less