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Properties

Label 1296.5.e.e
Level $1296$
Weight $5$
Character orbit 1296.e
Analytic conductor $133.967$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1296,5,Mod(161,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1296.161");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1296.e (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: \(133.967472157\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.221456830464.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 38x^{6} - 100x^{5} + 449x^{4} - 736x^{3} + 1900x^{2} - 1548x + 2307 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{18} \)
Twist minimal: no (minimal twist has level 18)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{5} + ( - \beta_1 - 6) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{5} + ( - \beta_1 - 6) q^{7} + (\beta_{7} - \beta_{4} - 2 \beta_{2}) q^{11} + (\beta_{3} - 3 \beta_1 - 1) q^{13} + (\beta_{5} - 5 \beta_{4}) q^{17} + (\beta_{6} - \beta_{3} - \beta_1 - 12) q^{19} + (2 \beta_{7} + 7 \beta_{4} + 5 \beta_{2}) q^{23} + ( - 2 \beta_{6} - 7 \beta_1 - 195) q^{25} + (2 \beta_{7} + 2 \beta_{5} + \cdots - 12 \beta_{2}) q^{29}+ \cdots + (6 \beta_{6} - 26 \beta_{3} + \cdots - 7861) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 52 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 52 q^{7} - 20 q^{13} - 100 q^{19} - 1588 q^{25} - 2956 q^{31} - 32 q^{37} - 136 q^{43} - 4884 q^{49} + 3996 q^{55} + 8956 q^{61} + 15008 q^{67} + 20716 q^{73} - 12100 q^{79} + 32184 q^{85} + 45868 q^{91} - 62672 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 38x^{6} - 100x^{5} + 449x^{4} - 736x^{3} + 1900x^{2} - 1548x + 2307 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{4} + 6\nu^{3} - 33\nu^{2} + 30\nu + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -38\nu^{7} + 133\nu^{6} - 1025\nu^{5} + 2230\nu^{4} - 6077\nu^{3} + 6952\nu^{2} - 1749\nu - 213 ) / 705 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 15\nu^{4} - 30\nu^{3} + 273\nu^{2} - 258\nu + 864 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 132\nu^{7} - 462\nu^{6} + 4080\nu^{5} - 9045\nu^{4} + 35358\nu^{3} - 44223\nu^{2} + 85926\nu - 35883 ) / 470 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -656\nu^{7} + 2296\nu^{6} - 13910\nu^{5} + 29035\nu^{4} - 55484\nu^{3} + 55339\nu^{2} + 29472\nu - 23046 ) / 705 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27\nu^{6} - 81\nu^{5} + 780\nu^{4} - 1425\nu^{3} + 6150\nu^{2} - 5451\nu + 12123 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2242 \nu^{7} + 7847 \nu^{6} - 73165 \nu^{5} + 163295 \nu^{4} - 663103 \nu^{3} + 835283 \nu^{2} + \cdots + 615588 ) / 1410 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 10\beta_{7} + 4\beta_{5} + 62\beta_{4} - 41\beta_{2} + 486 ) / 972 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 10\beta_{7} + 4\beta_{5} + 62\beta_{4} + 9\beta_{3} - 41\beta_{2} + 90\beta _1 - 7335 ) / 972 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -18\beta_{7} - 10\beta_{5} - 98\beta_{4} + 3\beta_{3} + 193\beta_{2} + 30\beta _1 - 2499 ) / 216 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -86\beta_{7} - 47\beta_{5} - 472\beta_{4} - 36\beta_{3} + 889\beta_{2} - 684\beta _1 + 31689 ) / 486 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1282\beta_{7} + 1078\beta_{5} + 6962\beta_{4} - 405\beta_{3} - 20153\beta_{2} - 7290\beta _1 + 354537 ) / 1944 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 262\beta_{7} + 8\beta_{6} + 206\beta_{5} + 1426\beta_{4} + 15\beta_{3} - 3855\beta_{2} + 1690\beta _1 - 62243 ) / 108 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4808 \beta_{7} + 252 \beta_{6} - 5822 \beta_{5} - 25360 \beta_{4} + 1197 \beta_{3} + 92170 \beta_{2} + \cdots - 2594295 ) / 972 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.500000 − 1.74753i
0.500000 − 3.16175i
0.500000 − 3.61825i
0.500000 − 2.20403i
0.500000 + 2.20403i
0.500000 + 3.61825i
0.500000 + 3.16175i
0.500000 + 1.74753i
0 0 0 − 42.7277i 0 −39.6847 0 0 0
161.2 0 0 0 − 37.5315i 0 −2.70916 0 0 0
161.3 0 0 0 − 7.40592i 0 60.3765 0 0 0
161.4 0 0 0 − 2.20976i 0 −43.9826 0 0 0
161.5 0 0 0 2.20976i 0 −43.9826 0 0 0
161.6 0 0 0 7.40592i 0 60.3765 0 0 0
161.7 0 0 0 37.5315i 0 −2.70916 0 0 0
161.8 0 0 0 42.7277i 0 −39.6847 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.5.e.e 8
3.b odd 2 1 inner 1296.5.e.e 8
4.b odd 2 1 162.5.b.c 8
9.c even 3 1 144.5.q.b 8
9.c even 3 1 432.5.q.b 8
9.d odd 6 1 144.5.q.b 8
9.d odd 6 1 432.5.q.b 8
12.b even 2 1 162.5.b.c 8
36.f odd 6 1 18.5.d.a ✓ 8
36.f odd 6 1 54.5.d.a 8
36.h even 6 1 18.5.d.a ✓ 8
36.h even 6 1 54.5.d.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.5.d.a ✓ 8 36.f odd 6 1
18.5.d.a ✓ 8 36.h even 6 1
54.5.d.a 8 36.f odd 6 1
54.5.d.a 8 36.h even 6 1
144.5.q.b 8 9.c even 3 1
144.5.q.b 8 9.d odd 6 1
162.5.b.c 8 4.b odd 2 1
162.5.b.c 8 12.b even 2 1
432.5.q.b 8 9.c even 3 1
432.5.q.b 8 9.d odd 6 1
1296.5.e.e 8 1.a even 1 1 trivial
1296.5.e.e 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 3294T_{5}^{6} + 2765097T_{5}^{4} + 154472184T_{5}^{2} + 688747536 \) acting on \(S_{5}^{\mathrm{new}}(1296, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 3294 T^{6} + \cdots + 688747536 \) Copy content Toggle raw display
$7$ \( (T^{4} + 26 T^{3} + \cdots - 285500)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 53\!\cdots\!89 \) Copy content Toggle raw display
$13$ \( (T^{4} + 10 T^{3} + \cdots + 738398896)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{4} + 50 T^{3} + \cdots + 8154234820)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 44\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( (T^{4} + 1478 T^{3} + \cdots - 727726946432)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 16 T^{3} + \cdots - 305304165104)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 80\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( (T^{4} + 68 T^{3} + \cdots + 150128761105)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 62\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 317147769769024)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 27967816823305)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 55\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 13267734129308)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 851414599190300)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 799415366370601)^{2} \) Copy content Toggle raw display
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