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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8064,2,Mod(4607,8064)]
 
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, 0, 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("8064.4607");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8064 = 2^{7} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8064.h (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: \( x^{12} + 16x^{10} + 98x^{8} + 290x^{6} + 425x^{4} + 274x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{5} + \beta_{3} q^{7} - \beta_{8} q^{11} + ( - \beta_{8} + \beta_1 - 1) q^{13} + ( - \beta_{5} - 2 \beta_{3}) q^{17} + (\beta_{11} + \beta_{3} + \beta_{2}) q^{19} + (\beta_{4} + 2 \beta_1 - 1) q^{23} + ( - \beta_{10} - \beta_{8} + \beta_{6} + \cdots - 1) q^{25}+ \cdots + ( - \beta_{10} + 2 \beta_{8} + \cdots + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{13} - 8 q^{23} - 12 q^{25} + 8 q^{37} + 16 q^{47} - 12 q^{49} + 48 q^{59} + 40 q^{71} - 16 q^{73} + 48 q^{83} + 72 q^{85} - 48 q^{95} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 16x^{10} + 98x^{8} + 290x^{6} + 425x^{4} + 274x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 8\nu^{4} + 13\nu^{2} - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{11} - 25\nu^{9} - 105\nu^{7} - 167\nu^{5} - 52\nu^{3} + 89\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{11} - 25\nu^{9} - 105\nu^{7} - 167\nu^{5} - 66\nu^{3} + 19\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{8} + 21\nu^{6} + 66\nu^{4} + 63\nu^{2} + 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{11} - 41\nu^{9} - 196\nu^{7} - 380\nu^{5} - 218\nu^{3} + 67\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{10} + 14\nu^{8} + 71\nu^{6} + 159\nu^{4} + 148\nu^{2} + 38 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{11} + 41\nu^{9} + 203\nu^{7} + 450\nu^{5} + 428\nu^{3} + 115\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -2\nu^{10} - 26\nu^{8} - 117\nu^{6} - 216\nu^{4} - 145\nu^{2} - 19 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -9\nu^{11} - 130\nu^{9} - 679\nu^{7} - 1539\nu^{5} - 1354\nu^{3} - 212\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3\nu^{10} + 42\nu^{8} + 211\nu^{6} + 461\nu^{4} + 406\nu^{2} + 86 ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( \nu^{11} + 14\nu^{9} + 70\nu^{7} + 151\nu^{5} + 132\nu^{3} + 32\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{7} - 3\beta_{5} + 3\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} + 3\beta_{6} - 2\beta _1 - 15 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{9} - 5\beta_{7} + 15\beta_{5} - 6\beta_{3} - 9\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{10} + \beta_{8} - 7\beta_{6} - \beta_{4} + 2\beta _1 + 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3\beta_{11} + 29\beta_{9} + 26\beta_{7} - 75\beta_{5} + 48\beta_{3} + 33\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -59\beta_{10} - 24\beta_{8} + 129\beta_{6} + 24\beta_{4} - 10\beta _1 - 231 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -15\beta_{11} - 83\beta_{9} - 65\beta_{7} + 195\beta_{5} - 150\beta_{3} - 69\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 118\beta_{10} + 51\beta_{8} - 252\beta_{6} - 49\beta_{4} - 10\beta _1 + 365 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 219\beta_{11} + 934\beta_{9} + 649\beta_{7} - 2082\beta_{5} + 1764\beta_{3} + 636\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -2050\beta_{10} - 915\beta_{8} + 4332\beta_{6} + 831\beta_{4} + 472\beta _1 - 5523 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -1413\beta_{11} - 5207\beta_{9} - 3284\beta_{7} + 11289\beta_{5} - 10152\beta_{3} - 3135\beta_{2} ) / 6 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4607.1
− 0.527279i
2.35771i
− 1.16284i
1.47078i
1.94451i
1.69313i
− 1.69313i
− 1.94451i
− 1.47078i
1.16284i
− 2.35771i
0.527279i
0 0 0 − 3.84946i 0 − 1.00000i 0 0 0
4607.2 0 0 0 − 3.61868i 0 1.00000i 0 0 0
4607.3 0 0 0 − 2.33034i 0 1.00000i 0 0 0
4607.4 0 0 0 − 1.18344i 0 − 1.00000i 0 0 0
4607.5 0 0 0 − 1.10465i 0 − 1.00000i 0 0 0
4607.6 0 0 0 − 0.188517i 0 1.00000i 0 0 0
4607.7 0 0 0 0.188517i 0 − 1.00000i 0 0 0
4607.8 0 0 0 1.10465i 0 1.00000i 0 0 0
4607.9 0 0 0 1.18344i 0 1.00000i 0 0 0
4607.10 0 0 0 2.33034i 0 − 1.00000i 0 0 0
4607.11 0 0 0 3.61868i 0 − 1.00000i 0 0 0
4607.12 0 0 0 3.84946i 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4607.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b 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.h.a ✓ 12
3.b odd 2 1 8064.2.h.c yes 12
4.b odd 2 1 8064.2.h.c yes 12
8.b even 2 1 8064.2.h.f yes 12
8.d odd 2 1 8064.2.h.h yes 12
12.b even 2 1 inner 8064.2.h.a ✓ 12
24.f even 2 1 8064.2.h.f yes 12
24.h odd 2 1 8064.2.h.h yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8064.2.h.a ✓ 12 1.a even 1 1 trivial
8064.2.h.a ✓ 12 12.b even 2 1 inner
8064.2.h.c yes 12 3.b odd 2 1
8064.2.h.c yes 12 4.b odd 2 1
8064.2.h.f yes 12 8.b even 2 1
8064.2.h.f yes 12 24.f even 2 1
8064.2.h.h yes 12 8.d odd 2 1
8064.2.h.h yes 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(8064, [\chi])\):

\( T_{5}^{12} + 36T_{5}^{10} + 436T_{5}^{8} + 2032T_{5}^{6} + 3424T_{5}^{4} + 1920T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{6} - 28T_{11}^{4} + 24T_{11}^{3} + 188T_{11}^{2} - 288T_{11} + 72 \) Copy content Toggle raw display
\( T_{13}^{6} + 4T_{13}^{5} - 36T_{13}^{4} - 112T_{13}^{3} + 128T_{13}^{2} + 128T_{13} - 64 \) 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
$5$ \( T^{12} + 36 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( (T^{6} - 28 T^{4} + \cdots + 72)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 4 T^{5} - 36 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 60 T^{10} + \cdots + 3136 \) Copy content Toggle raw display
$19$ \( T^{12} + 160 T^{10} + \cdots + 200704 \) Copy content Toggle raw display
$23$ \( (T^{6} + 4 T^{5} + \cdots + 1544)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 220 T^{10} + \cdots + 4981824 \) Copy content Toggle raw display
$31$ \( T^{12} + 192 T^{10} + \cdots + 200704 \) Copy content Toggle raw display
$37$ \( (T^{6} - 4 T^{5} - 52 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 364 T^{10} + \cdots + 12446784 \) Copy content Toggle raw display
$43$ \( T^{12} + 248 T^{10} + \cdots + 200704 \) Copy content Toggle raw display
$47$ \( (T^{6} - 8 T^{5} + \cdots + 512)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 2000593984 \) Copy content Toggle raw display
$59$ \( (T^{6} - 24 T^{5} + \cdots + 860672)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 288 T^{4} + \cdots - 326592)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 232 T^{10} + \cdots + 32444416 \) Copy content Toggle raw display
$71$ \( (T^{6} - 20 T^{5} + \cdots - 7672)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 8 T^{5} + \cdots - 688192)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 24566173696 \) Copy content Toggle raw display
$83$ \( (T^{6} - 24 T^{5} + \cdots - 3584)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 428 T^{10} + \cdots + 254016 \) Copy content Toggle raw display
$97$ \( (T^{6} - 8 T^{5} + \cdots - 318528)^{2} \) Copy content Toggle raw display
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