Properties

Label 45.10.b.a
Level $45$
Weight $10$
Character orbit 45.b
Analytic conductor $23.177$
Analytic rank $0$
Dimension $2$
CM discriminant -15
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.19");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(23.1766126274\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 19 \beta q^{2} - 1293 q^{4} - 625 \beta q^{5} - 14839 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 19 \beta q^{2} - 1293 q^{4} - 625 \beta q^{5} - 14839 \beta q^{8} + 59375 q^{10} + 747689 q^{16} + 277822 \beta q^{17} + 1036316 q^{19} + 808125 \beta q^{20} - 1180124 \beta q^{23} - 1953125 q^{25} + 8247368 q^{31} + 6608523 \beta q^{32} - 26393090 q^{34} + 19690004 \beta q^{38} - 46371875 q^{40} + 112111780 q^{46} + 3364516 \beta q^{47} + 40353607 q^{49} - 37109375 \beta q^{50} - 16738658 \beta q^{53} + 197894882 q^{61} + 156699992 \beta q^{62} - 244992917 q^{64} - 359223846 \beta q^{68} - 1339956588 q^{76} + 421557104 q^{79} - 467305625 \beta q^{80} + 58072888 \beta q^{83} + 868193750 q^{85} + 1525900332 \beta q^{92} - 319629020 q^{94} - 647697500 \beta q^{95} + 766718533 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2586 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2586 q^{4} + 118750 q^{10} + 1495378 q^{16} + 2072632 q^{19} - 3906250 q^{25} + 16494736 q^{31} - 52786180 q^{34} - 92743750 q^{40} + 224223560 q^{46} + 80707214 q^{49} + 395789764 q^{61} - 489985834 q^{64} - 2679913176 q^{76} + 843114208 q^{79} + 1736387500 q^{85} - 639258040 q^{94}+O(q^{100}) \) Copy content Toggle raw display

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
2.23607i
2.23607i
42.4853i 0 −1293.00 1397.54i 0 0 33181.0i 0 59375.0
19.2 42.4853i 0 −1293.00 1397.54i 0 0 33181.0i 0 59375.0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
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 45.10.b.a 2
3.b odd 2 1 inner 45.10.b.a 2
5.b even 2 1 inner 45.10.b.a 2
5.c odd 4 2 225.10.a.i 2
15.d odd 2 1 CM 45.10.b.a 2
15.e even 4 2 225.10.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.10.b.a 2 1.a even 1 1 trivial
45.10.b.a 2 3.b odd 2 1 inner
45.10.b.a 2 5.b even 2 1 inner
45.10.b.a 2 15.d odd 2 1 CM
225.10.a.i 2 5.c odd 4 2
225.10.a.i 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1805 \) acting on \(S_{10}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1805 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1953125 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 385925318420 \) Copy content Toggle raw display
$19$ \( (T - 1036316)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6963463276880 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 8247368)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 56599839571280 \) Copy content Toggle raw display
$53$ \( T^{2} + 14\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 197894882)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T - 421557104)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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