Properties

Label 256.9.d.g
Level $256$
Weight $9$
Character orbit 256.d
Analytic conductor $104.289$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [256,9,Mod(127,256)]
 
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, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 256.d (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: \(104.288924176\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 9x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 32)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 20) q^{3} + (6 \beta_{3} + 133 \beta_1) q^{5} + (21 \beta_{3} + 196 \beta_1) q^{7} + (40 \beta_{2} - 1297) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 20) q^{3} + (6 \beta_{3} + 133 \beta_1) q^{5} + (21 \beta_{3} + 196 \beta_1) q^{7} + (40 \beta_{2} - 1297) q^{9} + ( - 87 \beta_{2} - 8012) q^{11} + ( - 78 \beta_{3} - 11573 \beta_1) q^{13} + (253 \beta_{3} + 31844 \beta_1) q^{15} + ( - 1032 \beta_{2} + 62130) q^{17} + ( - 1299 \beta_{2} + 60420) q^{19} + (616 \beta_{3} + 106064 \beta_1) q^{21} + (999 \beta_{3} - 177428 \beta_1) q^{23} + ( - 6384 \beta_{2} - 380547) q^{25} + ( - 7058 \beta_{2} + 37400) q^{27} + (6522 \beta_{3} + 61339 \beta_1) q^{29} + ( - 84 \beta_{3} + 371312 \beta_1) q^{31} + ( - 9752 \beta_{2} - 583408) q^{33} + ( - 15876 \beta_{2} - 2555728) q^{35} + (13038 \beta_{3} - 397611 \beta_1) q^{37} + ( - 13133 \beta_{3} - 610852 \beta_1) q^{39} + ( - 8304 \beta_{2} + 1741022) q^{41} + ( - 1311 \beta_{2} + 5655188) q^{43} + ( - 2462 \beta_{3} + 994859 \beta_1) q^{45} + ( - 44070 \beta_{3} + 67016 \beta_1) q^{47} + ( - 32928 \beta_{2} - 2968959) q^{49} + (41490 \beta_{2} - 3777048) q^{51} + ( - 18114 \beta_{3} - 290763 \beta_1) q^{53} + ( - 59643 \beta_{3} - 3604604 \beta_1) q^{55} + (34440 \beta_{2} - 5109936) q^{57} + ( - 60699 \beta_{2} - 6670364) q^{59} + ( - 26190 \beta_{3} - 4872821 \beta_1) q^{61} + ( - 19397 \beta_{3} + 3831548 \beta_1) q^{63} + (319248 \beta_{2} + 15262244) q^{65} + ( - 456159 \beta_{2} + 7847956) q^{67} + ( - 157448 \beta_{3} + 1310576 \beta_1) q^{69} + (23949 \beta_{3} + 1328804 \beta_1) q^{71} + (169800 \beta_{2} - 7444338) q^{73} + ( - 508227 \beta_{2} - 38662716) q^{75} + ( - 185304 \beta_{3} - 10456880 \beta_1) q^{77} + (50310 \beta_{3} - 18892232 \beta_1) q^{79} + ( - 366200 \beta_{2} - 25072495) q^{81} + (503085 \beta_{2} + 34801540) q^{83} + (235524 \beta_{3} - 21854598 \beta_1) q^{85} + (191779 \beta_{3} + 32949788 \beta_1) q^{87} + (427272 \beta_{2} + 11811726) q^{89} + (1033284 \beta_{2} + 40942160) q^{91} + (369632 \beta_{3} + 7017664 \beta_1) q^{93} + (189753 \beta_{3} - 29874156 \beta_1) q^{95} + (504216 \beta_{2} - 98994830) q^{97} + ( - 207641 \beta_{2} - 6535156) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 80 q^{3} - 5188 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 80 q^{3} - 5188 q^{9} - 32048 q^{11} + 248520 q^{17} + 241680 q^{19} - 1522188 q^{25} + 149600 q^{27} - 2333632 q^{33} - 10222912 q^{35} + 6964088 q^{41} + 22620752 q^{43} - 11875836 q^{49} - 15108192 q^{51} - 20439744 q^{57} - 26681456 q^{59} + 61048976 q^{65} + 31391824 q^{67} - 29777352 q^{73} - 154650864 q^{75} - 100289980 q^{81} + 139206160 q^{83} + 47246904 q^{89} + 163768640 q^{91} - 395979320 q^{97} - 26140624 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 9x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} - 8\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -16\nu^{3} + 224\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 64\nu^{2} - 288 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 8\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 288 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} + 28\beta_1 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−2.17945 + 0.500000i
−2.17945 0.500000i
2.17945 0.500000i
2.17945 + 0.500000i
0 −49.7424 0 570.909i 0 2537.18i 0 −4086.70 0
127.2 0 −49.7424 0 570.909i 0 2537.18i 0 −4086.70 0
127.3 0 89.7424 0 1102.91i 0 3321.18i 0 1492.70 0
127.4 0 89.7424 0 1102.91i 0 3321.18i 0 1492.70 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.9.d.g 4
4.b odd 2 1 256.9.d.c 4
8.b even 2 1 256.9.d.c 4
8.d odd 2 1 inner 256.9.d.g 4
16.e even 4 1 32.9.c.b 4
16.e even 4 1 64.9.c.e 4
16.f odd 4 1 32.9.c.b 4
16.f odd 4 1 64.9.c.e 4
48.i odd 4 1 288.9.g.a 4
48.k even 4 1 288.9.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.9.c.b 4 16.e even 4 1
32.9.c.b 4 16.f odd 4 1
64.9.c.e 4 16.e even 4 1
64.9.c.e 4 16.f odd 4 1
256.9.d.c 4 4.b odd 2 1
256.9.d.c 4 8.b even 2 1
256.9.d.g 4 1.a even 1 1 trivial
256.9.d.g 4 8.d odd 2 1 inner
288.9.g.a 4 48.i odd 4 1
288.9.g.a 4 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 40T_{3} - 4464 \) acting on \(S_{9}^{\mathrm{new}}(256, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 40 T - 4464)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 396471715600 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 71004756250624 \) Copy content Toggle raw display
$11$ \( (T^{2} + 16024 T + 27376528)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( (T^{2} - 124260 T - 1320139836)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 120840 T - 4556942064)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{2} + \cdots + 2695753597060)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots + 31972791456400)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{2} + \cdots + 26572987017232)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 66\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots - 950515732500848)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 84820874301756)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( (T^{2} + \cdots - 19904545410800)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots - 748461593591100)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 85\!\cdots\!16)^{2} \) Copy content Toggle raw display
show more
show less