Properties

Label 45.16.b.a
Level $45$
Weight $16$
Character orbit 45.b
Analytic conductor $64.212$
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] = [45,16,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, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.19");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 16 \)
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: \(64.2120772950\)
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: \( 5 \)
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 \(\beta = 5\sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 29 \beta q^{2} - 72357 q^{4} + 15625 \beta q^{5} - 1148081 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 29 \beta q^{2} - 72357 q^{4} + 15625 \beta q^{5} - 1148081 \beta q^{8} - 56640625 q^{10} + 1790799449 q^{16} - 222666118 \beta q^{17} + 5941047764 q^{19} - 1130578125 \beta q^{20} - 2155991596 \beta q^{23} - 30517578125 q^{25} + 158412509048 q^{31} + 14312865813 \beta q^{32} + 807164677750 q^{34} + 172290385156 \beta q^{38} + 2242345703125 q^{40} + 7815469535500 q^{46} - 586609874764 \beta q^{47} + 4747561509943 q^{49} - 885009765625 \beta q^{50} + 1529168448818 \beta q^{53} - 46029462744598 q^{61} + 4593962762392 \beta q^{62} + 6796777772707 q^{64} + 16111452300126 \beta q^{68} - 429876393059748 q^{76} - 302830978196464 q^{79} + 27981241390625 \beta q^{80} - 42594004868488 \beta q^{83} + 434894761718750 q^{85} + 156001083911772 \beta q^{92} + 21\!\cdots\!00 q^{94} + \cdots + 137679283788347 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 144714 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 144714 q^{4} - 113281250 q^{10} + 3581598898 q^{16} + 11882095528 q^{19} - 61035156250 q^{25} + 316825018096 q^{31} + 1614329355500 q^{34} + 4484691406250 q^{40} + 15630939071000 q^{46} + 9495123019886 q^{49} - 92058925489196 q^{61} + 13593555545414 q^{64} - 859752786119496 q^{76} - 605661956392928 q^{79} + 869789523437500 q^{85} + 42\!\cdots\!00 q^{94}+O(q^{100}) \) 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
324.230i 0 −72357.0 174693.i 0 0 1.28359e7i 0 −5.66406e7
19.2 324.230i 0 −72357.0 174693.i 0 0 1.28359e7i 0 −5.66406e7
\(n\): 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 45.16.b.a 2
3.b odd 2 1 inner 45.16.b.a 2
5.b even 2 1 inner 45.16.b.a 2
15.d odd 2 1 CM 45.16.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.16.b.a 2 1.a even 1 1 trivial
45.16.b.a 2 3.b odd 2 1 inner
45.16.b.a 2 5.b even 2 1 inner
45.16.b.a 2 15.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 105125 \) acting on \(S_{16}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 105125 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 30517578125 \) 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
$17$ \( T^{2} + 61\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T - 5941047764)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 58\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 158412509048)^{2} \) 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
$47$ \( T^{2} + 43\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{2} + 29\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 46029462744598)^{2} \) 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
$79$ \( (T + 302830978196464)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 22\!\cdots\!00 \) 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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