Properties

Label 45.12.b.a
Level $45$
Weight $12$
Character orbit 45.b
Analytic conductor $34.575$
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] = [45,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.19");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
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: \(34.5754431252\)
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.

\(f(q)\) \(=\) \( q + 23 \beta q^{2} - 597 q^{4} - 3125 \beta q^{5} + 33373 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 23 \beta q^{2} - 597 q^{4} - 3125 \beta q^{5} + 33373 \beta q^{8} + 359375 q^{10} - 5060551 q^{16} + 115454 \beta q^{17} - 18612796 q^{19} + 1865625 \beta q^{20} - 16659892 \beta q^{23} - 48828125 q^{25} - 182093992 q^{31} - 48044769 \beta q^{32} - 13277210 q^{34} - 428094308 \beta q^{38} + 521453125 q^{40} + 1915887580 q^{46} - 1405364308 \beta q^{47} + 1977326743 q^{49} - 1123046875 \beta q^{50} - 785310154 \beta q^{53} - 13027614598 q^{61} - 4188161816 \beta q^{62} - 4838860013 q^{64} - 68926038 \beta q^{68} + 11111839212 q^{76} - 13380631984 q^{79} + 15814221875 \beta q^{80} + 16317816584 \beta q^{83} + 1803968750 q^{85} + 9945955524 \beta q^{92} + 161616895420 q^{94} + 58164987500 \beta q^{95} + 45478515089 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1194 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1194 q^{4} + 718750 q^{10} - 10121102 q^{16} - 37225592 q^{19} - 97656250 q^{25} - 364187984 q^{31} - 26554420 q^{34} + 1042906250 q^{40} + 3831775160 q^{46} + 3954653486 q^{49} - 26055229196 q^{61} - 9677720026 q^{64} + 22223678424 q^{76} - 26761263968 q^{79} + 3607937500 q^{85} + 323233790840 q^{94}+O(q^{100}) \) 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
51.4296i 0 −597.000 6987.71i 0 0 74624.3i 0 359375.
19.2 51.4296i 0 −597.000 6987.71i 0 0 74624.3i 0 359375.
\(n\): 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 45.12.b.a 2
3.b odd 2 1 inner 45.12.b.a 2
5.b even 2 1 inner 45.12.b.a 2
5.c odd 4 2 225.12.a.k 2
15.d odd 2 1 CM 45.12.b.a 2
15.e even 4 2 225.12.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.12.b.a 2 1.a even 1 1 trivial
45.12.b.a 2 3.b odd 2 1 inner
45.12.b.a 2 5.b even 2 1 inner
45.12.b.a 2 15.d odd 2 1 CM
225.12.a.k 2 5.c odd 4 2
225.12.a.k 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2645 \) acting on \(S_{12}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2645 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 48828125 \) 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
$17$ \( T^{2} + 66648130580 \) Copy content Toggle raw display
$19$ \( (T + 18612796)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 13\!\cdots\!20 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 182093992)^{2} \) 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
$47$ \( T^{2} + 98\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{2} + 30\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 13027614598)^{2} \) 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
$79$ \( (T + 13380631984)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 13\!\cdots\!80 \) 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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