↑

Properties

Label 45.22.b.d
Level $45$
Weight $22$
Character orbit 45.b
Analytic conductor $125.765$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,22,Mod(19,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.19");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 45.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: \(125.764804929\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 31014725 x^{18} + 402351996349620 x^{16} + \cdots + 63\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: multiple of \( 2^{51}\cdot 3^{88}\cdot 5^{39}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 15)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 1004321) q^{4} + ( - \beta_{4} - \beta_{3} + \cdots - 1247238) q^{5}+ \cdots + (\beta_{9} + \beta_{8} - 36 \beta_{4} + \cdots - 25) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 1004321) q^{4} + ( - \beta_{4} - \beta_{3} + \cdots - 1247238) q^{5}+ \cdots + (2560517496 \beta_{19} + \cdots - 838665308490) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20086410 q^{4} - 24944730 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 20086410 q^{4} - 24944730 q^{5} - 1467444110 q^{10} - 11952443580 q^{11} + 3210119749140 q^{14} + 12123723095650 q^{16} + 152310844352560 q^{19} + 228448186221420 q^{20} + 141329092021400 q^{25} - 573110662319340 q^{26} - 51\!\cdots\!20 q^{29}+ \cdots - 23\!\cdots\!80 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 31014725 x^{18} + 402351996349620 x^{16} + \cdots + 63\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3101473 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 28\!\cdots\!65 \nu^{19} + \cdots + 17\!\cdots\!68 \nu ) / 16\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 38\!\cdots\!67 \nu^{19} + \cdots + 16\!\cdots\!56 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 38\!\cdots\!67 \nu^{19} + \cdots - 12\!\cdots\!56 ) / 96\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 70\!\cdots\!23 \nu^{19} + \cdots + 19\!\cdots\!64 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23\!\cdots\!77 \nu^{19} + \cdots + 11\!\cdots\!36 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!61 \nu^{19} + \cdots + 22\!\cdots\!52 ) / 90\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 27\!\cdots\!57 \nu^{19} + \cdots - 64\!\cdots\!24 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 27\!\cdots\!17 \nu^{19} + \cdots - 21\!\cdots\!56 ) / 72\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 69\!\cdots\!97 \nu^{19} + \cdots + 19\!\cdots\!84 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 30\!\cdots\!29 \nu^{19} + \cdots + 34\!\cdots\!08 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 97\!\cdots\!83 \nu^{19} + \cdots + 54\!\cdots\!76 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 98\!\cdots\!01 \nu^{19} + \cdots - 28\!\cdots\!52 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 25\!\cdots\!95 \nu^{19} + \cdots - 12\!\cdots\!32 ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 31\!\cdots\!87 \nu^{19} + \cdots + 18\!\cdots\!60 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 24\!\cdots\!99 \nu^{19} + \cdots - 83\!\cdots\!72 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 28\!\cdots\!39 \nu^{19} + \cdots + 17\!\cdots\!08 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 66\!\cdots\!59 \nu^{19} + \cdots + 37\!\cdots\!88 ) / 28\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3101473 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} - 36\beta_{4} + 9\beta_{3} + 12\beta_{2} - 5068853\beta _1 - 25 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 22 \beta_{18} + 9 \beta_{17} + 30 \beta_{16} + 8 \beta_{15} + 23 \beta_{14} - 17 \beta_{13} + \cdots + 15720920897133 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5196 \beta_{19} - 28162 \beta_{18} + 28369 \beta_{17} + 10379 \beta_{16} - 10379 \beta_{15} + \cdots + 233948831 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 93097150 \beta_{18} - 98837639 \beta_{17} - 295544756 \beta_{16} - 202447606 \beta_{15} + \cdots - 90\!\cdots\!41 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 43066776348 \beta_{19} + 341168876586 \beta_{18} - 381982697077 \beta_{17} - 126747719767 \beta_{16} + \cdots - 19\!\cdots\!91 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 96684637817410 \beta_{18} + 889154910234703 \beta_{17} + \cdots + 55\!\cdots\!73 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 24\!\cdots\!32 \beta_{19} + \cdots + 15\!\cdots\!19 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 58\!\cdots\!98 \beta_{18} + \cdots - 35\!\cdots\!89 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 98\!\cdots\!64 \beta_{19} + \cdots - 11\!\cdots\!59 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 73\!\cdots\!82 \beta_{18} + \cdots + 23\!\cdots\!49 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 14\!\cdots\!28 \beta_{19} + \cdots + 87\!\cdots\!71 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 71\!\cdots\!62 \beta_{18} + \cdots - 15\!\cdots\!13 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 28\!\cdots\!52 \beta_{19} + \cdots - 64\!\cdots\!83 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 61\!\cdots\!70 \beta_{18} + \cdots + 10\!\cdots\!29 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 45\!\cdots\!96 \beta_{19} + \cdots + 47\!\cdots\!99 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 50\!\cdots\!50 \beta_{18} + \cdots - 73\!\cdots\!89 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 49\!\cdots\!00 \beta_{19} + \cdots - 34\!\cdots\!27 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
− 2706.03i
− 2387.58i
− 2338.38i
− 2241.21i
− 1728.04i
− 1468.87i
− 1083.03i
− 998.710i
− 359.921i
− 238.651i
238.651i
359.921i
998.710i
1083.03i
1468.87i
1728.04i
2241.21i
2338.38i
2387.58i
2706.03i
− 2706.03i 0 −5.22544e6 −1.54317e7 + 1.54499e7i 0 − 9.90426e7i 8.46523e9i 0 4.18078e10 + 4.17587e10i
19.2 − 2387.58i 0 −3.60339e6 −9.00948e6 + 1.98914e7i 0 8.76258e8i 3.59626e9i 0 4.74922e10 + 2.15109e10i
19.3 − 2338.38i 0 −3.37088e6 −1.49989e7 − 1.58705e7i 0 − 9.66595e8i 2.97845e9i 0 −3.71112e10 + 3.50730e10i
19.4 − 2241.21i 0 −2.92587e6 5.03127e6 − 2.12491e7i 0 1.14262e8i 1.85733e9i 0 −4.76236e10 − 1.12761e10i
19.5 − 1728.04i 0 −888953. 2.17446e7 + 2.00256e6i 0 − 3.15759e8i − 2.08781e9i 0 3.46050e9 − 3.75754e10i
19.6 − 1468.87i 0 −60422.4 1.68381e7 + 1.39038e7i 0 1.30137e9i − 2.99169e9i 0 2.04229e10 − 2.47329e10i
19.7 − 1083.03i 0 924202. −2.09150e7 − 6.27705e6i 0 − 7.89823e8i − 3.27221e9i 0 −6.79822e9 + 2.26515e10i
19.8 − 998.710i 0 1.09973e6 2.52916e6 − 2.16896e7i 0 1.20991e9i − 3.19276e9i 0 −2.16617e10 − 2.52590e9i
19.9 − 359.921i 0 1.96761e6 1.88731e7 − 1.09838e7i 0 − 2.25801e8i − 1.46299e9i 0 −3.95332e9 − 6.79281e9i
19.10 − 238.651i 0 2.04020e6 −1.71335e7 + 1.35381e7i 0 6.18210e8i − 9.87381e8i 0 3.23088e9 + 4.08892e9i
19.11 238.651i 0 2.04020e6 −1.71335e7 − 1.35381e7i 0 − 6.18210e8i 9.87381e8i 0 3.23088e9 − 4.08892e9i
19.12 359.921i 0 1.96761e6 1.88731e7 + 1.09838e7i 0 2.25801e8i 1.46299e9i 0 −3.95332e9 + 6.79281e9i
19.13 998.710i 0 1.09973e6 2.52916e6 + 2.16896e7i 0 − 1.20991e9i 3.19276e9i 0 −2.16617e10 + 2.52590e9i
19.14 1083.03i 0 924202. −2.09150e7 + 6.27705e6i 0 7.89823e8i 3.27221e9i 0 −6.79822e9 − 2.26515e10i
19.15 1468.87i 0 −60422.4 1.68381e7 − 1.39038e7i 0 − 1.30137e9i 2.99169e9i 0 2.04229e10 + 2.47329e10i
19.16 1728.04i 0 −888953. 2.17446e7 − 2.00256e6i 0 3.15759e8i 2.08781e9i 0 3.46050e9 + 3.75754e10i
19.17 2241.21i 0 −2.92587e6 5.03127e6 + 2.12491e7i 0 − 1.14262e8i − 1.85733e9i 0 −4.76236e10 + 1.12761e10i
19.18 2338.38i 0 −3.37088e6 −1.49989e7 + 1.58705e7i 0 9.66595e8i − 2.97845e9i 0 −3.71112e10 − 3.50730e10i
19.19 2387.58i 0 −3.60339e6 −9.00948e6 − 1.98914e7i 0 − 8.76258e8i − 3.59626e9i 0 4.74922e10 − 2.15109e10i
19.20 2706.03i 0 −5.22544e6 −1.54317e7 − 1.54499e7i 0 9.90426e7i − 8.46523e9i 0 4.18078e10 − 4.17587e10i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.22.b.d 20
3.b odd 2 1 15.22.b.a ✓ 20
5.b even 2 1 inner 45.22.b.d 20
15.d odd 2 1 15.22.b.a ✓ 20
15.e even 4 1 75.22.a.m 10
15.e even 4 1 75.22.a.n 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.22.b.a ✓ 20 3.b odd 2 1
15.22.b.a ✓ 20 15.d odd 2 1
45.22.b.d 20 1.a even 1 1 trivial
45.22.b.d 20 5.b even 2 1 inner
75.22.a.m 10 15.e even 4 1
75.22.a.n 10 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 31014725 T_{2}^{18} + 402351996349620 T_{2}^{16} + \cdots + 63\!\cdots\!24 \) acting on \(S_{22}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 63\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 60\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots - 61\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots - 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 79\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 78\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 17\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 32\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 37\!\cdots\!76 \) Copy content Toggle raw display
show more
show less