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Properties

Label 396.3.d.a
Level $396$
Weight $3$
Character orbit 396.d
Analytic conductor $10.790$
Analytic rank $0$
Dimension $48$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [396,3,Mod(395,396)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(396, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("396.395");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 396.d (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: \(10.7902184687\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 88 q^{16} + 48 q^{22} - 208 q^{25} - 168 q^{34} + 496 q^{49} + 304 q^{58} + 240 q^{64} - 408 q^{70} - 24 q^{82} + 112 q^{88} - 160 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
395.1 −1.92521 − 0.541817i 0 3.41287 + 2.08622i − 2.65823i 0 −3.52991 −5.44014 − 5.86556i 0 −1.44027 + 5.11765i
395.2 −1.92521 − 0.541817i 0 3.41287 + 2.08622i 2.65823i 0 3.52991 −5.44014 − 5.86556i 0 1.44027 − 5.11765i
395.3 −1.92521 + 0.541817i 0 3.41287 − 2.08622i − 2.65823i 0 3.52991 −5.44014 + 5.86556i 0 1.44027 + 5.11765i
395.4 −1.92521 + 0.541817i 0 3.41287 − 2.08622i 2.65823i 0 −3.52991 −5.44014 + 5.86556i 0 −1.44027 − 5.11765i
395.5 −1.90247 − 0.616950i 0 3.23875 + 2.34745i − 6.91403i 0 −10.0323 −4.71334 − 6.46409i 0 −4.26561 + 13.1537i
395.6 −1.90247 − 0.616950i 0 3.23875 + 2.34745i 6.91403i 0 10.0323 −4.71334 − 6.46409i 0 4.26561 − 13.1537i
395.7 −1.90247 + 0.616950i 0 3.23875 − 2.34745i − 6.91403i 0 10.0323 −4.71334 + 6.46409i 0 4.26561 + 13.1537i
395.8 −1.90247 + 0.616950i 0 3.23875 − 2.34745i 6.91403i 0 −10.0323 −4.71334 + 6.46409i 0 −4.26561 − 13.1537i
395.9 −1.42860 − 1.39968i 0 0.0818169 + 3.99916i − 8.75431i 0 8.30577 5.48065 − 5.82774i 0 −12.2532 + 12.5064i
395.10 −1.42860 − 1.39968i 0 0.0818169 + 3.99916i 8.75431i 0 −8.30577 5.48065 − 5.82774i 0 12.2532 − 12.5064i
395.11 −1.42860 + 1.39968i 0 0.0818169 − 3.99916i − 8.75431i 0 −8.30577 5.48065 + 5.82774i 0 12.2532 + 12.5064i
395.12 −1.42860 + 1.39968i 0 0.0818169 − 3.99916i 8.75431i 0 8.30577 5.48065 + 5.82774i 0 −12.2532 − 12.5064i
395.13 −1.33029 − 1.49343i 0 −0.460637 + 3.97339i − 0.351365i 0 12.9268 6.54674 − 4.59785i 0 −0.524738 + 0.467419i
395.14 −1.33029 − 1.49343i 0 −0.460637 + 3.97339i 0.351365i 0 −12.9268 6.54674 − 4.59785i 0 0.524738 − 0.467419i
395.15 −1.33029 + 1.49343i 0 −0.460637 − 3.97339i − 0.351365i 0 −12.9268 6.54674 + 4.59785i 0 0.524738 + 0.467419i
395.16 −1.33029 + 1.49343i 0 −0.460637 − 3.97339i 0.351365i 0 12.9268 6.54674 + 4.59785i 0 −0.524738 − 0.467419i
395.17 −0.821085 − 1.82368i 0 −2.65164 + 2.99480i − 6.30259i 0 −1.59698 7.63878 + 2.37677i 0 −11.4939 + 5.17496i
395.18 −0.821085 − 1.82368i 0 −2.65164 + 2.99480i 6.30259i 0 1.59698 7.63878 + 2.37677i 0 11.4939 − 5.17496i
395.19 −0.821085 + 1.82368i 0 −2.65164 − 2.99480i − 6.30259i 0 1.59698 7.63878 − 2.37677i 0 11.4939 + 5.17496i
395.20 −0.821085 + 1.82368i 0 −2.65164 − 2.99480i 6.30259i 0 −1.59698 7.63878 − 2.37677i 0 −11.4939 − 5.17496i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 395.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
11.b odd 2 1 inner
12.b even 2 1 inner
33.d even 2 1 inner
44.c even 2 1 inner
132.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 396.3.d.a ✓ 48
3.b odd 2 1 inner 396.3.d.a ✓ 48
4.b odd 2 1 inner 396.3.d.a ✓ 48
11.b odd 2 1 inner 396.3.d.a ✓ 48
12.b even 2 1 inner 396.3.d.a ✓ 48
33.d even 2 1 inner 396.3.d.a ✓ 48
44.c even 2 1 inner 396.3.d.a ✓ 48
132.d odd 2 1 inner 396.3.d.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
396.3.d.a ✓ 48 1.a even 1 1 trivial
396.3.d.a ✓ 48 3.b odd 2 1 inner
396.3.d.a ✓ 48 4.b odd 2 1 inner
396.3.d.a ✓ 48 11.b odd 2 1 inner
396.3.d.a ✓ 48 12.b even 2 1 inner
396.3.d.a ✓ 48 33.d even 2 1 inner
396.3.d.a ✓ 48 44.c even 2 1 inner
396.3.d.a ✓ 48 132.d odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(396, [\chi])\).