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Properties

Label 8064.2.j.h
Level $8064$
Weight $2$
Character orbit 8064.j
Analytic conductor $64.391$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8064,2,Mod(575,8064)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8064, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8064.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8064 = 2^{7} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8064.j (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: \(64.3913641900\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{11} + 8x^{10} - 32x^{8} + 28x^{7} + 82x^{6} - 72x^{5} + 52x^{4} - 36x^{3} + 20x^{2} - 8x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{9} + 1) q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} + 1) q^{5} - \beta_{2} q^{7} + (\beta_{2} + \beta_1) q^{11} + (\beta_{10} - \beta_{8} - \beta_{2}) q^{13} + ( - \beta_{10} + \beta_{8} + \cdots + \beta_1) q^{17}+ \cdots + (\beta_{11} + \beta_{9} + \beta_{7} + \cdots + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{5} + 8 q^{23} + 12 q^{25} + 16 q^{29} - 32 q^{43} + 16 q^{47} - 12 q^{49} + 32 q^{67} + 8 q^{71} + 8 q^{77} - 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{11} + 8x^{10} - 32x^{8} + 28x^{7} + 82x^{6} - 72x^{5} + 52x^{4} - 36x^{3} + 20x^{2} - 8x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 25494172 \nu^{11} - 374880970 \nu^{10} + 1489599974 \nu^{9} - 3061999619 \nu^{8} + \cdots - 4315063439 ) / 1221640485 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 67250 \nu^{11} + 76912 \nu^{10} + 94186 \nu^{9} - 1021870 \nu^{8} + 1182602 \nu^{7} + \cdots - 179539 ) / 1105557 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 444864716 \nu^{11} + 1262786740 \nu^{10} - 1415877308 \nu^{9} - 4305106717 \nu^{8} + \cdots - 2791980337 ) / 1221640485 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 181035386 \nu^{11} - 587781400 \nu^{10} + 1009126413 \nu^{9} + 690282897 \nu^{8} + \cdots - 1018420883 ) / 407213495 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 491258 \nu^{11} + 1852060 \nu^{10} - 3492129 \nu^{9} - 893706 \nu^{8} + 15762022 \nu^{7} + \cdots + 1590174 ) / 1015495 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 668992 \nu^{11} - 2510480 \nu^{10} + 4673821 \nu^{9} + 1342814 \nu^{8} - 21394708 \nu^{7} + \cdots - 1762396 ) / 1329315 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 617044066 \nu^{11} - 2081699000 \nu^{10} + 3332132773 \nu^{9} + 3203535377 \nu^{8} + \cdots - 2793360403 ) / 1221640485 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 802819982 \nu^{11} + 2894147860 \nu^{10} - 5334945251 \nu^{9} - 1892660989 \nu^{8} + \cdots + 3701648291 ) / 1221640485 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 301677618 \nu^{11} + 1077240830 \nu^{10} - 1880435719 \nu^{9} - 1055090321 \nu^{8} + \cdots + 920551159 ) / 407213495 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1622486692 \nu^{11} + 6115192970 \nu^{10} - 11531908771 \nu^{9} - 2952015689 \nu^{8} + \cdots + 5314423486 ) / 1221640485 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1693947514 \nu^{11} - 6352501790 \nu^{10} + 11814539707 \nu^{9} + 3428768933 \nu^{8} + \cdots - 4434411292 ) / 1221640485 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - 2\beta_{5} + \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{10} - 3\beta_{6} - 3\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7 \beta_{11} - \beta_{10} + 4 \beta_{9} - 2 \beta_{8} - \beta_{7} - 14 \beta_{6} + 2 \beta_{5} + \cdots - 9 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -8\beta_{10} + 5\beta_{8} + 12\beta_{5} - 7\beta_{4} - 28\beta_{2} + 6\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 25 \beta_{11} - 6 \beta_{10} - 12 \beta_{9} - \beta_{8} + 6 \beta_{7} + 51 \beta_{6} + 10 \beta_{5} + \cdots + 36 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 27 \beta_{11} + 27 \beta_{10} - 10 \beta_{9} - 10 \beta_{8} + 4 \beta_{7} + 60 \beta_{6} - 60 \beta_{5} + \cdots + 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 109 \beta_{11} + 363 \beta_{10} + 24 \beta_{9} - 162 \beta_{8} - 145 \beta_{7} - 186 \beta_{6} + \cdots - 447 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 238\beta_{11} + 104\beta_{9} - 152\beta_{7} - 442\beta_{6} - 127\beta_{3} - 595 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 450 \beta_{11} - 1329 \beta_{10} + 153 \beta_{9} + 575 \beta_{8} - 429 \beta_{7} - 804 \beta_{6} + \cdots - 1454 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2835 \beta_{11} - 2835 \beta_{10} - 1191 \beta_{9} + 1191 \beta_{8} + 786 \beta_{7} + 5920 \beta_{6} + \cdots + 4105 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 9780 \beta_{11} + 3556 \beta_{10} - 4175 \beta_{9} - 1365 \beta_{8} + 3556 \beta_{7} + 19851 \beta_{6} + \cdots + 16664 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8064\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(1793\) \(4609\) \(7813\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
−1.52908 + 0.633364i
−1.52908 − 0.633364i
0.182527 + 0.440658i
0.182527 − 0.440658i
2.06835 − 0.856741i
2.06835 + 0.856741i
1.04282 + 2.51759i
1.04282 − 2.51759i
−0.225347 − 0.544037i
−0.225347 + 0.544037i
0.460720 − 0.190837i
0.460720 + 0.190837i
0 0 0 −2.99309 0 − 1.00000i 0 0 0
575.2 0 0 0 −2.99309 0 1.00000i 0 0 0
575.3 0 0 0 −0.855119 0 − 1.00000i 0 0 0
575.4 0 0 0 −0.855119 0 1.00000i 0 0 0
575.5 0 0 0 −0.246999 0 − 1.00000i 0 0 0
575.6 0 0 0 −0.246999 0 1.00000i 0 0 0
575.7 0 0 0 1.01701 0 − 1.00000i 0 0 0
575.8 0 0 0 1.01701 0 1.00000i 0 0 0
575.9 0 0 0 3.25232 0 − 1.00000i 0 0 0
575.10 0 0 0 3.25232 0 1.00000i 0 0 0
575.11 0 0 0 3.82587 0 − 1.00000i 0 0 0
575.12 0 0 0 3.82587 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 575.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8064.2.j.h yes 12
3.b odd 2 1 8064.2.j.c ✓ 12
4.b odd 2 1 8064.2.j.g yes 12
8.b even 2 1 8064.2.j.d yes 12
8.d odd 2 1 8064.2.j.c ✓ 12
12.b even 2 1 8064.2.j.d yes 12
24.f even 2 1 inner 8064.2.j.h yes 12
24.h odd 2 1 8064.2.j.g yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8064.2.j.c ✓ 12 3.b odd 2 1
8064.2.j.c ✓ 12 8.d odd 2 1
8064.2.j.d yes 12 8.b even 2 1
8064.2.j.d yes 12 12.b even 2 1
8064.2.j.g yes 12 4.b odd 2 1
8064.2.j.g yes 12 24.h odd 2 1
8064.2.j.h yes 12 1.a even 1 1 trivial
8064.2.j.h yes 12 24.f 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 \(S_{2}^{\mathrm{new}}(8064, [\chi])\):

\( T_{5}^{6} - 4T_{5}^{5} - 10T_{5}^{4} + 40T_{5}^{3} + 12T_{5}^{2} - 32T_{5} - 8 \) Copy content Toggle raw display
\( T_{19}^{6} - 56T_{19}^{4} - 32T_{19}^{3} + 736T_{19}^{2} + 384T_{19} - 2176 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} - 4 T^{5} - 10 T^{4} + \cdots - 8)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + 56 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{12} + 104 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{12} + 116 T^{10} + \cdots + 341056 \) Copy content Toggle raw display
$19$ \( (T^{6} - 56 T^{4} + \cdots - 2176)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 4 T^{5} + \cdots - 752)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 8 T^{5} + \cdots - 136)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 176 T^{10} + \cdots + 10240000 \) Copy content Toggle raw display
$37$ \( T^{12} + 256 T^{10} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( T^{12} + 212 T^{10} + \cdots + 48776256 \) Copy content Toggle raw display
$43$ \( (T^{6} + 16 T^{5} + \cdots - 512)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 8 T^{5} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 134 T^{4} + \cdots + 15032)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 205520896 \) Copy content Toggle raw display
$61$ \( T^{12} + 288 T^{10} + \cdots + 82591744 \) Copy content Toggle raw display
$67$ \( (T^{6} - 16 T^{5} + \cdots - 3200)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 4 T^{5} + \cdots - 11632)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 232 T^{4} + \cdots - 323712)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 464 T^{10} + \cdots + 1327104 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 81622204416 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 2563599424 \) Copy content Toggle raw display
$97$ \( (T^{6} - 264 T^{4} + \cdots - 20608)^{2} \) Copy content Toggle raw display
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