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Properties

Label 45.22.a.e
Level $45$
Weight $22$
Character orbit 45.a
Self dual yes
Analytic conductor $125.765$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,22,Mod(1,45)]
 
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, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 45.a (trivial)

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: yes
Analytic conductor: \(125.764804929\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 125326x + 2416960 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 767) q^{2} + ( - 16 \beta_{2} + 1324 \beta_1 + 421908) q^{4} - 9765625 q^{5} + ( - 6045 \beta_{2} + 207601 \beta_1 + 155291491) q^{7} + ( - 36800 \beta_{2} - 16176 \beta_1 + 1279953456) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 767) q^{2} + ( - 16 \beta_{2} + 1324 \beta_1 + 421908) q^{4} - 9765625 q^{5} + ( - 6045 \beta_{2} + 207601 \beta_1 + 155291491) q^{7} + ( - 36800 \beta_{2} - 16176 \beta_1 + 1279953456) q^{8} + ( - 9765625 \beta_1 - 7490234375) q^{10} + ( - 696322 \beta_{2} - 2618198 \beta_1 + 55777904786) q^{11} + ( - 480497 \beta_{2} - 441155851 \beta_1 - 182004063243) q^{13} + ( - 9221536 \beta_{2} + 619117248 \beta_1 + 523159339968) q^{14} + ( - 2103552 \beta_{2} + 613994176 \beta_1 + 85293983040) q^{16} + ( - 65391013 \beta_{2} - 4644285575 \beta_1 + 2699926733873) q^{17} + ( - 281344799 \beta_{2} + 13595209403 \beta_1 + 1317098650077) q^{19} + (156250000 \beta_{2} - 12929687500 \beta_1 - 4120195312500) q^{20} + ( - 637719104 \beta_{2} + 94427715700 \beta_1 + 38097512561804) q^{22} + ( - 2775396889 \beta_{2} + \cdots + 52092913016007) q^{23}+ \cdots + ( - 46\!\cdots\!68 \beta_{2} + \cdots + 29\!\cdots\!27) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2300 q^{2} + 1264400 q^{4} - 29296875 q^{5} + 465666872 q^{7} + 3839876544 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2300 q^{2} + 1264400 q^{4} - 29296875 q^{5} + 465666872 q^{7} + 3839876544 q^{8} - 22460937500 q^{10} + 167336332556 q^{11} - 545571033878 q^{13} + 1568858902656 q^{14} + 255267954944 q^{16} + 8104424487194 q^{17} + 3937700740828 q^{19} - 12347656250000 q^{20} + 114198109969712 q^{22} + 156235274730744 q^{23} + 286102294921875 q^{25} - 29\!\cdots\!56 q^{26}+ \cdots + 87\!\cdots\!32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 125326x + 2416960 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{2} + 246\nu + 166985 ) / 105 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6\nu^{2} + 1782\nu - 501900 ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 45\beta _1 + 135 ) / 360 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 41\beta_{2} - 4455\beta _1 + 10024635 ) / 120 \) Copy content Toggle raw display

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−362.802
344.462
19.3401
−999.805 0 −1.09754e6 −9.76562e6 0 9.82313e7 3.19407e9 0 9.76372e9
1.2 904.285 0 −1.27942e6 −9.76562e6 0 −5.27665e8 −3.05338e9 0 −8.83091e9
1.3 2395.52 0 3.64136e6 −9.76562e6 0 8.95100e8 3.69919e9 0 −2.33937e10
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.22.a.e 3
3.b odd 2 1 15.22.a.b ✓ 3
15.d odd 2 1 75.22.a.g 3
15.e even 4 2 75.22.b.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.22.a.b ✓ 3 3.b odd 2 1
45.22.a.e 3 1.a even 1 1 trivial
75.22.a.g 3 15.d odd 2 1
75.22.b.g 6 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 2300T_{2}^{2} - 1132928T_{2} + 2165809152 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(45))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 2300 T^{2} + \cdots + 2165809152 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T + 9765625)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 465666872 T^{2} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{3} - 167336332556 T^{2} + \cdots + 24\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( T^{3} + 545571033878 T^{2} + \cdots - 16\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{3} - 8104424487194 T^{2} + \cdots - 41\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{3} - 3937700740828 T^{2} + \cdots - 10\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{3} - 156235274730744 T^{2} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{3} - 930273612785494 T^{2} + \cdots + 56\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 36\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 40\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 34\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 87\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 23\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 51\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 82\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 24\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 27\!\cdots\!68 \) Copy content Toggle raw display
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