↑

Properties

Label 72.3.h.a
Level $72$
Weight $3$
Character orbit 72.h
Analytic conductor $1.962$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,3,Mod(53,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.53");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 72.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: \(1.96185790339\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.33808912384.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 11x^{6} - 18x^{5} + 47x^{4} - 28x^{3} - 44x^{2} + 48x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{5} + 1) q^{4} + ( - \beta_{4} - \beta_{2}) q^{5} + (\beta_{5} - \beta_1 + 1) q^{7} + ( - \beta_{6} + \beta_{4} + \cdots - \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{5} + 1) q^{4} + ( - \beta_{4} - \beta_{2}) q^{5} + (\beta_{5} - \beta_1 + 1) q^{7} + ( - \beta_{6} + \beta_{4} + \cdots - \beta_{2}) q^{8}+ \cdots - 3 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 28 q^{10} - 72 q^{16} - 88 q^{22} + 40 q^{25} + 104 q^{28} - 128 q^{31} + 212 q^{34} - 240 q^{40} - 136 q^{46} + 24 q^{49} + 248 q^{52} + 256 q^{55} + 260 q^{58} - 32 q^{64} - 312 q^{70} - 160 q^{73} + 304 q^{76} - 384 q^{79} - 188 q^{82} - 256 q^{88} - 216 q^{94} - 192 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 11x^{6} - 18x^{5} + 47x^{4} - 28x^{3} - 44x^{2} + 48x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -31\nu^{7} - 2335\nu^{6} + 5047\nu^{5} - 30147\nu^{4} + 48175\nu^{3} - 134417\nu^{2} + 129494\nu + 56970 ) / 19506 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -679\nu^{7} + 3284\nu^{6} - 10895\nu^{5} + 30888\nu^{4} - 59171\nu^{3} + 91534\nu^{2} + 9854\nu - 116964 ) / 39012 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -233\nu^{7} + 1012\nu^{6} - 3595\nu^{5} + 10944\nu^{4} - 21109\nu^{3} + 45860\nu^{2} - 12074\nu + 2208 ) / 9753 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3271 \nu^{7} - 6254 \nu^{6} + 35651 \nu^{5} - 60462 \nu^{4} + 137447 \nu^{3} - 117568 \nu^{2} + \cdots + 71472 ) / 39012 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1895 \nu^{7} - 5761 \nu^{6} + 23713 \nu^{5} - 57405 \nu^{4} + 120697 \nu^{3} - 154439 \nu^{2} + \cdots + 100920 ) / 19506 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3067 \nu^{7} + 6833 \nu^{6} - 36773 \nu^{5} + 65991 \nu^{4} - 176981 \nu^{3} + 149203 \nu^{2} + \cdots - 110994 ) / 19506 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5171 \nu^{7} + 11953 \nu^{6} - 64213 \nu^{5} + 118353 \nu^{4} - 317701 \nu^{3} + 288395 \nu^{2} + \cdots - 209766 ) / 19506 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} - 4\beta_{6} - \beta_{5} + \beta _1 + 1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 4\beta_{3} + \beta _1 - 9 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{7} + 7\beta_{6} + 7\beta_{5} - 9\beta_{4} + 9\beta_{2} - 4\beta _1 + 5 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{7} - 8\beta_{6} - 11\beta_{5} - 8\beta_{4} - 24\beta_{3} - 24\beta_{2} - 13\beta _1 + 19 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{7} - 16\beta_{6} - 175\beta_{5} + 120\beta_{4} + 30\beta_{3} - 240\beta_{2} + 61\beta _1 - 353 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -31\beta_{7} + 58\beta_{6} + 16\beta_{5} + 34\beta_{4} + 52\beta_{3} + 126\beta_{2} + 43\beta _1 + 92 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 338\beta_{7} - 430\beta_{6} + 941\beta_{5} - 462\beta_{4} - 630\beta_{3} + 1974\beta_{2} - 389\beta _1 + 4363 ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−0.651388 − 2.66948i
−0.651388 + 2.66948i
1.15139 + 2.23537i
1.15139 − 2.23537i
1.15139 + 0.593052i
1.15139 − 0.593052i
−0.651388 − 0.158947i
−0.651388 + 0.158947i
−1.77521 − 0.921201i 0 2.30278 + 3.27066i 1.07498 0 7.21110 −1.07498 − 7.92745i 0 −1.90833 − 0.990277i
53.2 −1.77521 + 0.921201i 0 2.30278 − 3.27066i 1.07498 0 7.21110 −1.07498 + 7.92745i 0 −1.90833 + 0.990277i
53.3 −1.16130 − 1.62831i 0 −1.30278 + 3.78190i −7.67101 0 −7.21110 7.67101 − 2.27059i 0 8.90833 + 12.4908i
53.4 −1.16130 + 1.62831i 0 −1.30278 − 3.78190i −7.67101 0 −7.21110 7.67101 + 2.27059i 0 8.90833 − 12.4908i
53.5 1.16130 − 1.62831i 0 −1.30278 − 3.78190i 7.67101 0 −7.21110 −7.67101 − 2.27059i 0 8.90833 − 12.4908i
53.6 1.16130 + 1.62831i 0 −1.30278 + 3.78190i 7.67101 0 −7.21110 −7.67101 + 2.27059i 0 8.90833 + 12.4908i
53.7 1.77521 − 0.921201i 0 2.30278 − 3.27066i −1.07498 0 7.21110 1.07498 − 7.92745i 0 −1.90833 + 0.990277i
53.8 1.77521 + 0.921201i 0 2.30278 + 3.27066i −1.07498 0 7.21110 1.07498 + 7.92745i 0 −1.90833 − 0.990277i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.3.h.a ✓ 8
3.b odd 2 1 inner 72.3.h.a ✓ 8
4.b odd 2 1 288.3.h.a 8
8.b even 2 1 inner 72.3.h.a ✓ 8
8.d odd 2 1 288.3.h.a 8
12.b even 2 1 288.3.h.a 8
16.e even 4 2 2304.3.e.n 8
16.f odd 4 2 2304.3.e.o 8
24.f even 2 1 288.3.h.a 8
24.h odd 2 1 inner 72.3.h.a ✓ 8
48.i odd 4 2 2304.3.e.n 8
48.k even 4 2 2304.3.e.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.h.a ✓ 8 1.a even 1 1 trivial
72.3.h.a ✓ 8 3.b odd 2 1 inner
72.3.h.a ✓ 8 8.b even 2 1 inner
72.3.h.a ✓ 8 24.h odd 2 1 inner
288.3.h.a 8 4.b odd 2 1
288.3.h.a 8 8.d odd 2 1
288.3.h.a 8 12.b even 2 1
288.3.h.a 8 24.f even 2 1
2304.3.e.n 8 16.e even 4 2
2304.3.e.n 8 48.i odd 4 2
2304.3.e.o 8 16.f odd 4 2
2304.3.e.o 8 48.k even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(72, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 60 T^{2} + 68)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 52)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 304 T^{2} + 9792)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 472 T^{2} + 2448)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 836 T^{2} + 171396)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1504 T^{2} + 352512)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 464 T^{2} + 576)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 988 T^{2} + 103428)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 32 T - 212)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1504 T^{2} + 352512)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 932 T^{2} + 133956)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 6400 T^{2} + 10027008)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 4176 T^{2} + 46656)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 2524 T^{2} + 1591812)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 4608 T^{2} + 4456448)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 5472 T^{2} + 3172608)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 5472 T^{2} + 3172608)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 11088 T^{2} + 13483584)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 40 T - 2928)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 96 T - 3988)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 3120 T^{2} + 183872)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 5828 T^{2} + 171396)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 48 T - 256)^{4} \) Copy content Toggle raw display
show more
show less