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Properties

Label 1470.3.h.a
Level $1470$
Weight $3$
Character orbit 1470.h
Analytic conductor $40.055$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1470,3,Mod(979,1470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1470, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1470.979");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1470.h (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: \(40.0545988610\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 64 q^{4} + 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 64 q^{4} + 96 q^{9} - 96 q^{11} + 24 q^{15} + 128 q^{16} - 144 q^{25} + 176 q^{29} + 48 q^{30} - 192 q^{36} + 144 q^{39} + 192 q^{44} - 64 q^{46} + 192 q^{50} + 48 q^{51} - 48 q^{60} - 256 q^{64} - 344 q^{65} - 144 q^{71} + 832 q^{74} - 256 q^{79} + 288 q^{81} - 736 q^{85} - 608 q^{86} - 472 q^{95} - 288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
979.1 − 1.41421i −1.73205 −2.00000 −4.84337 + 1.24167i 2.44949i 0 2.82843i 3.00000 1.75599 + 6.84956i
979.2 1.41421i −1.73205 −2.00000 −4.84337 − 1.24167i − 2.44949i 0 − 2.82843i 3.00000 1.75599 − 6.84956i
979.3 − 1.41421i 1.73205 −2.00000 3.01203 − 3.99095i − 2.44949i 0 2.82843i 3.00000 −5.64406 − 4.25965i
979.4 1.41421i 1.73205 −2.00000 3.01203 + 3.99095i 2.44949i 0 − 2.82843i 3.00000 −5.64406 + 4.25965i
979.5 − 1.41421i −1.73205 −2.00000 −3.01203 + 3.99095i 2.44949i 0 2.82843i 3.00000 5.64406 + 4.25965i
979.6 1.41421i −1.73205 −2.00000 −3.01203 − 3.99095i − 2.44949i 0 − 2.82843i 3.00000 5.64406 − 4.25965i
979.7 − 1.41421i −1.73205 −2.00000 −2.84576 − 4.11116i 2.44949i 0 2.82843i 3.00000 −5.81406 + 4.02452i
979.8 1.41421i −1.73205 −2.00000 −2.84576 + 4.11116i − 2.44949i 0 − 2.82843i 3.00000 −5.81406 − 4.02452i
979.9 − 1.41421i 1.73205 −2.00000 −3.48251 + 3.58777i − 2.44949i 0 2.82843i 3.00000 5.07388 + 4.92501i
979.10 1.41421i 1.73205 −2.00000 −3.48251 − 3.58777i 2.44949i 0 − 2.82843i 3.00000 5.07388 − 4.92501i
979.11 − 1.41421i 1.73205 −2.00000 2.14085 + 4.51849i − 2.44949i 0 2.82843i 3.00000 6.39011 − 3.02761i
979.12 1.41421i 1.73205 −2.00000 2.14085 − 4.51849i 2.44949i 0 − 2.82843i 3.00000 6.39011 + 3.02761i
979.13 − 1.41421i 1.73205 −2.00000 0.813216 + 4.93342i − 2.44949i 0 2.82843i 3.00000 6.97692 − 1.15006i
979.14 1.41421i 1.73205 −2.00000 0.813216 − 4.93342i 2.44949i 0 − 2.82843i 3.00000 6.97692 + 1.15006i
979.15 − 1.41421i −1.73205 −2.00000 −0.813216 − 4.93342i 2.44949i 0 2.82843i 3.00000 −6.97692 + 1.15006i
979.16 1.41421i −1.73205 −2.00000 −0.813216 + 4.93342i − 2.44949i 0 − 2.82843i 3.00000 −6.97692 − 1.15006i
979.17 − 1.41421i 1.73205 −2.00000 −2.48947 − 4.33619i − 2.44949i 0 2.82843i 3.00000 −6.13230 + 3.52064i
979.18 1.41421i 1.73205 −2.00000 −2.48947 + 4.33619i 2.44949i 0 − 2.82843i 3.00000 −6.13230 − 3.52064i
979.19 − 1.41421i −1.73205 −2.00000 3.48251 − 3.58777i 2.44949i 0 2.82843i 3.00000 −5.07388 − 4.92501i
979.20 1.41421i −1.73205 −2.00000 3.48251 + 3.58777i − 2.44949i 0 − 2.82843i 3.00000 −5.07388 + 4.92501i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 979.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.3.h.a 32
5.b even 2 1 inner 1470.3.h.a 32
7.b odd 2 1 inner 1470.3.h.a 32
7.c even 3 1 210.3.p.a ✓ 32
7.d odd 6 1 210.3.p.a ✓ 32
21.g even 6 1 630.3.bc.b 32
21.h odd 6 1 630.3.bc.b 32
35.c odd 2 1 inner 1470.3.h.a 32
35.i odd 6 1 210.3.p.a ✓ 32
35.j even 6 1 210.3.p.a ✓ 32
35.k even 12 1 1050.3.p.g 16
35.k even 12 1 1050.3.p.h 16
35.l odd 12 1 1050.3.p.g 16
35.l odd 12 1 1050.3.p.h 16
105.o odd 6 1 630.3.bc.b 32
105.p even 6 1 630.3.bc.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.p.a ✓ 32 7.c even 3 1
210.3.p.a ✓ 32 7.d odd 6 1
210.3.p.a ✓ 32 35.i odd 6 1
210.3.p.a ✓ 32 35.j even 6 1
630.3.bc.b 32 21.g even 6 1
630.3.bc.b 32 21.h odd 6 1
630.3.bc.b 32 105.o odd 6 1
630.3.bc.b 32 105.p even 6 1
1050.3.p.g 16 35.k even 12 1
1050.3.p.g 16 35.l odd 12 1
1050.3.p.h 16 35.k even 12 1
1050.3.p.h 16 35.l odd 12 1
1470.3.h.a 32 1.a even 1 1 trivial
1470.3.h.a 32 5.b even 2 1 inner
1470.3.h.a 32 7.b odd 2 1 inner
1470.3.h.a 32 35.c odd 2 1 inner