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Properties

Label 780.2.r.a
Level $780$
Weight $2$
Character orbit 780.r
Analytic conductor $6.228$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [780,2,Mod(73,780)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(780, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("780.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 780 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 780.r (of order \(4\), degree \(2\), 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: \(6.22833135766\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 8 q^{11} - 4 q^{13} - 4 q^{15} - 4 q^{17} - 16 q^{19} + 8 q^{21} - 8 q^{23} + 12 q^{25} - 8 q^{33} + 8 q^{39} + 12 q^{41} + 16 q^{43} + 4 q^{45} - 36 q^{49} + 36 q^{53} + 40 q^{55} + 16 q^{59} + 8 q^{61} - 40 q^{65} + 48 q^{67} - 8 q^{69} + 8 q^{71} + 48 q^{73} - 48 q^{77} - 28 q^{81} - 4 q^{85} - 24 q^{87} - 36 q^{89} - 24 q^{91} + 72 q^{95} - 72 q^{97} + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 0 −0.707107 − 0.707107i 0 −2.16868 + 0.544825i 0 0.657863i 0 1.00000i 0
73.2 0 −0.707107 − 0.707107i 0 −1.89326 − 1.18977i 0 0.757795i 0 1.00000i 0
73.3 0 −0.707107 − 0.707107i 0 −0.705075 + 2.12200i 0 3.56717i 0 1.00000i 0
73.4 0 −0.707107 − 0.707107i 0 0.136841 − 2.23188i 0 − 4.69969i 0 1.00000i 0
73.5 0 −0.707107 − 0.707107i 0 1.17614 − 1.90176i 0 4.16665i 0 1.00000i 0
73.6 0 −0.707107 − 0.707107i 0 1.62796 + 1.53289i 0 − 2.13466i 0 1.00000i 0
73.7 0 −0.707107 − 0.707107i 0 1.82608 − 1.29052i 0 0.513301i 0 1.00000i 0
73.8 0 0.707107 + 0.707107i 0 −2.04752 + 0.898701i 0 − 3.12196i 0 1.00000i 0
73.9 0 0.707107 + 0.707107i 0 −1.77237 − 1.36334i 0 1.66430i 0 1.00000i 0
73.10 0 0.707107 + 0.707107i 0 −1.56908 − 1.59311i 0 − 2.82733i 0 1.00000i 0
73.11 0 0.707107 + 0.707107i 0 −0.530704 + 2.17218i 0 1.25764i 0 1.00000i 0
73.12 0 0.707107 + 0.707107i 0 1.79817 − 1.32912i 0 2.80999i 0 1.00000i 0
73.13 0 0.707107 + 0.707107i 0 1.95050 + 1.09341i 0 − 4.81289i 0 1.00000i 0
73.14 0 0.707107 + 0.707107i 0 2.17100 + 0.535496i 0 2.20183i 0 1.00000i 0
577.1 0 −0.707107 + 0.707107i 0 −2.16868 − 0.544825i 0 − 0.657863i 0 − 1.00000i 0
577.2 0 −0.707107 + 0.707107i 0 −1.89326 + 1.18977i 0 − 0.757795i 0 − 1.00000i 0
577.3 0 −0.707107 + 0.707107i 0 −0.705075 − 2.12200i 0 − 3.56717i 0 − 1.00000i 0
577.4 0 −0.707107 + 0.707107i 0 0.136841 + 2.23188i 0 4.69969i 0 − 1.00000i 0
577.5 0 −0.707107 + 0.707107i 0 1.17614 + 1.90176i 0 − 4.16665i 0 − 1.00000i 0
577.6 0 −0.707107 + 0.707107i 0 1.62796 − 1.53289i 0 2.13466i 0 − 1.00000i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 780.2.r.a ✓ 28
3.b odd 2 1 2340.2.u.i 28
5.b even 2 1 3900.2.r.b 28
5.c odd 4 1 780.2.bm.a yes 28
5.c odd 4 1 3900.2.bm.b 28
13.d odd 4 1 780.2.bm.a yes 28
15.e even 4 1 2340.2.bp.i 28
39.f even 4 1 2340.2.bp.i 28
65.f even 4 1 3900.2.r.b 28
65.g odd 4 1 3900.2.bm.b 28
65.k even 4 1 inner 780.2.r.a ✓ 28
195.j odd 4 1 2340.2.u.i 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
780.2.r.a ✓ 28 1.a even 1 1 trivial
780.2.r.a ✓ 28 65.k even 4 1 inner
780.2.bm.a yes 28 5.c odd 4 1
780.2.bm.a yes 28 13.d odd 4 1
2340.2.u.i 28 3.b odd 2 1
2340.2.u.i 28 195.j odd 4 1
2340.2.bp.i 28 15.e even 4 1
2340.2.bp.i 28 39.f even 4 1
3900.2.r.b 28 5.b even 2 1
3900.2.r.b 28 65.f even 4 1
3900.2.bm.b 28 5.c odd 4 1
3900.2.bm.b 28 65.g odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(780, [\chi])\).