Properties

Label 64.8.b.c
Level $64$
Weight $8$
Character orbit 64.b
Analytic conductor $19.993$
Analytic rank $0$
Dimension $8$
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] = [64,8,Mod(33,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, 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("64.33");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 64.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: \(19.9926416310\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2705346343547136.10
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 301x^{6} + 68101x^{4} - 6772500x^{2} + 506250000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{42} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{4}) q^{3} + \beta_1 q^{5} - \beta_{7} q^{7} + (5 \beta_{2} - 617) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{4}) q^{3} + \beta_1 q^{5} - \beta_{7} q^{7} + (5 \beta_{2} - 617) q^{9} + ( - 96 \beta_{6} - 75 \beta_{4}) q^{11} + (3 \beta_{3} + 45 \beta_1) q^{13} + ( - 10 \beta_{7} + 11 \beta_{5}) q^{15} + ( - 21 \beta_{2} + 11370) q^{17} + ( - 208 \beta_{6} + 225 \beta_{4}) q^{19} + (49 \beta_{3} - 358 \beta_1) q^{21} + (20 \beta_{7} + 115 \beta_{5}) q^{23} + (54 \beta_{2} + 45389) q^{25} + ( - 1735 \beta_{6} + 770 \beta_{4}) q^{27} + (192 \beta_{3} - 449 \beta_1) q^{29} + ( - 179 \beta_{7} + 357 \beta_{5}) q^{31} + ( - 459 \beta_{2} + 223740) q^{33} + ( - 7470 \beta_{6} - 2976 \beta_{4}) q^{35} + (361 \beta_{3} + 1135 \beta_1) q^{37} + ( - 387 \beta_{7} + 576 \beta_{5}) q^{39} + (1890 \beta_{2} - 104694) q^{41} + ( - 11429 \beta_{6} - 2259 \beta_{4}) q^{43} + (325 \beta_{3} - 1987 \beta_1) q^{45} + ( - 43 \beta_{7} + 295 \beta_{5}) q^{47} + ( - 2520 \beta_{2} + 450185) q^{49} + (25251 \beta_{6} + 14730 \beta_{4}) q^{51} + ( - 345 \beta_{3} - 2665 \beta_1) q^{53} + (855 \beta_{7} - 804 \beta_{5}) q^{55} + ( - 607 \beta_{2} - 353780) q^{57} + ( - 3219 \beta_{6} - 3435 \beta_{4}) q^{59} + ( - 1541 \beta_{3} + 4537 \beta_1) q^{61} + (2422 \beta_{7} - 2615 \beta_{5}) q^{63} + (1602 \beta_{2} - 1503648) q^{65} + (52624 \beta_{6} - 28791 \beta_{4}) q^{67} + ( - 2705 \beta_{3} + 950 \beta_1) q^{69} + ( - 1856 \beta_{7} - 5347 \beta_{5}) q^{71} + (2349 \beta_{2} - 1912030) q^{73} + (9695 \beta_{6} + 36749 \beta_{4}) q^{75} + ( - 3759 \beta_{3} + 31050 \beta_1) q^{77} + ( - 2406 \beta_{7} - 4572 \beta_{5}) q^{79} + (4765 \beta_{2} - 1905259) q^{81} + (159705 \beta_{6} - 13095 \beta_{4}) q^{83} + ( - 1365 \beta_{3} + 17124 \beta_1) q^{85} + (8522 \beta_{7} + 245 \beta_{5}) q^{87} + ( - 15315 \beta_{2} + 1889298) q^{89} + ( - 319914 \beta_{6} - 96480 \beta_{4}) q^{91} + (3416 \beta_{3} - 83360 \beta_1) q^{93} + ( - 85 \beta_{7} + 2908 \beta_{5}) q^{95} + (14067 \beta_{2} + 148570) q^{97} + (317187 \beta_{6} + 133155 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4936 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4936 q^{9} + 90960 q^{17} + 363112 q^{25} + 1789920 q^{33} - 837552 q^{41} + 3601480 q^{49} - 2830240 q^{57} - 12029184 q^{65} - 15296240 q^{73} - 15242072 q^{81} + 15114384 q^{89} + 1188560 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 301x^{6} + 68101x^{4} - 6772500x^{2} + 506250000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 130784\nu^{6} - 39770984\nu^{4} + 5841976184\nu^{2} - 438311070000 ) / 383068125 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 32\nu^{6} + 111254416 ) / 68101 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30784\nu^{6} - 8815984\nu^{4} + 1539231184\nu^{2} - 109304820000 ) / 34824375 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 408457\nu^{7} - 92413057\nu^{5} + 17090422657\nu^{3} - 686981250000\nu ) / 114920437500 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 104732\nu^{7} - 349494332\nu^{5} + 56165073932\nu^{3} - 8572919940000\nu ) / 28730109375 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -1204\nu^{7} + 272404\nu^{5} - 41403604\nu^{3} + 2025000000\nu ) / 190265625 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 256468\nu^{7} + 267773132\nu^{5} - 31566992732\nu^{3} + 4287965940000\nu ) / 28730109375 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} + 7\beta_{6} - \beta_{5} + 16\beta_{4} ) / 128 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 73\beta_{3} - 8\beta_{2} - 178\beta _1 + 38528 ) / 512 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1357\beta_{6} + 2416\beta_{4} ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 10723\beta_{3} + 2408\beta_{2} - 31078\beta _1 - 5836928 ) / 512 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 206183\beta_{7} + 2007656\beta_{6} + 115883\beta_{5} + 2937728\beta_{4} ) / 1024 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 68101\beta_{2} - 111254416 ) / 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 33193583\beta_{7} - 360044456\beta_{6} + 12763283\beta_{5} - 449376128\beta_{4} ) / 1024 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(63\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
−11.0484 6.37883i
11.0484 6.37883i
10.1824 5.87883i
−10.1824 5.87883i
−10.1824 + 5.87883i
10.1824 + 5.87883i
11.0484 + 6.37883i
−11.0484 + 6.37883i
0 69.0306i 0 232.201i 0 −1504.06 0 −2578.22 0
33.2 0 69.0306i 0 232.201i 0 1504.06 0 −2578.22 0
33.3 0 29.0306i 0 107.493i 0 534.108 0 1344.22 0
33.4 0 29.0306i 0 107.493i 0 −534.108 0 1344.22 0
33.5 0 29.0306i 0 107.493i 0 −534.108 0 1344.22 0
33.6 0 29.0306i 0 107.493i 0 534.108 0 1344.22 0
33.7 0 69.0306i 0 232.201i 0 1504.06 0 −2578.22 0
33.8 0 69.0306i 0 232.201i 0 −1504.06 0 −2578.22 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.8
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 64.8.b.c 8
3.b odd 2 1 576.8.d.f 8
4.b odd 2 1 inner 64.8.b.c 8
8.b even 2 1 inner 64.8.b.c 8
8.d odd 2 1 inner 64.8.b.c 8
12.b even 2 1 576.8.d.f 8
16.e even 4 1 256.8.a.j 4
16.e even 4 1 256.8.a.p 4
16.f odd 4 1 256.8.a.j 4
16.f odd 4 1 256.8.a.p 4
24.f even 2 1 576.8.d.f 8
24.h odd 2 1 576.8.d.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.8.b.c 8 1.a even 1 1 trivial
64.8.b.c 8 4.b odd 2 1 inner
64.8.b.c 8 8.b even 2 1 inner
64.8.b.c 8 8.d odd 2 1 inner
256.8.a.j 4 16.e even 4 1
256.8.a.j 4 16.f odd 4 1
256.8.a.p 4 16.e even 4 1
256.8.a.p 4 16.f odd 4 1
576.8.d.f 8 3.b odd 2 1
576.8.d.f 8 12.b even 2 1
576.8.d.f 8 24.f even 2 1
576.8.d.f 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 5608T_{3}^{2} + 4016016 \) acting on \(S_{8}^{\mathrm{new}}(64, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 5608 T^{2} + 4016016)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 65472 T^{2} + \cdots + 623001600)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 2547456 T^{2} + \cdots + 645335875584)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 36480168 T^{2} + \cdots + 77526545767056)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 150607296 T^{2} + \cdots + 55\!\cdots\!04)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 22740 T + 61426404)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 418474472 T^{2} + \cdots + 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 10219718400 T^{2} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 61019054016 T^{2} + \cdots + 91\!\cdots\!64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 122830319616 T^{2} + \cdots + 29\!\cdots\!64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 282477395904 T^{2} + \cdots + 33\!\cdots\!04)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 209388 T - 538628183964)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 253862622248 T^{2} + \cdots + 10\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 49913723904 T^{2} + \cdots + 34\!\cdots\!04)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 668149656000 T^{2} + \cdots + 99\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 64366187688 T^{2} + \cdots + 60\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 4369584928704 T^{2} + \cdots + 42\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 12222871225448 T^{2} + \cdots + 45\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 31396418854656 T^{2} + \cdots + 13\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 3824060 T + 2806911930244)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 33350493920256 T^{2} + \cdots + 23\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 56321797732200 T^{2} + \cdots + 74\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 3778596 T - 32517358628796)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 297140 T - 30423027470684)^{4} \) Copy content Toggle raw display
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