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Properties

Label 315.2.ce.a
Level $315$
Weight $2$
Character orbit 315.ce
Analytic conductor $2.515$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,2,Mod(53,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.ce (of order \(12\), degree \(4\), 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: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 8 q^{7} + 8 q^{10} + 32 q^{16} - 48 q^{22} - 16 q^{25} + 88 q^{28} + 32 q^{31} - 16 q^{37} - 40 q^{40} - 16 q^{43} - 80 q^{52} - 32 q^{55} - 88 q^{58} + 48 q^{61} - 32 q^{67} - 112 q^{70} - 88 q^{73} - 320 q^{76} - 56 q^{82} + 16 q^{85} + 120 q^{88} - 128 q^{91} + 208 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −0.695956 + 2.59734i 0 −4.52979 − 2.61528i 1.27254 − 1.83865i 0 −1.50727 − 2.17442i 6.14253 − 6.14253i 0 3.88997 + 4.58485i
53.2 −0.604483 + 2.25596i 0 −2.99191 − 1.72738i −0.254820 + 2.22150i 0 −2.19240 + 1.48100i 2.40251 − 2.40251i 0 −4.85759 − 1.91772i
53.3 −0.543545 + 2.02854i 0 −2.08747 − 1.20520i −1.61611 − 1.54538i 0 1.07923 − 2.41563i 0.609443 − 0.609443i 0 4.01329 − 2.43835i
53.4 −0.494677 + 1.84616i 0 −1.43155 − 0.826508i 0.541988 + 2.16939i 0 2.64533 + 0.0471386i −0.468944 + 0.468944i 0 −4.27315 − 0.0725500i
53.5 −0.364480 + 1.36026i 0 0.0145936 + 0.00842564i −2.23601 − 0.0163814i 0 −2.58285 − 0.573503i −2.00834 + 2.00834i 0 0.837263 − 3.03558i
53.6 −0.271660 + 1.01385i 0 0.777962 + 0.449157i 0.310035 − 2.21447i 0 −1.26800 + 2.32211i −2.15109 + 2.15109i 0 2.16091 + 0.915911i
53.7 −0.0991680 + 0.370100i 0 1.60491 + 0.926596i 1.78935 + 1.34098i 0 2.39134 + 1.13202i −1.04395 + 1.04395i 0 −0.673742 + 0.529258i
53.8 −0.0362581 + 0.135317i 0 1.71505 + 0.990187i −1.87468 + 1.21884i 0 0.702570 − 2.55076i −0.394292 + 0.394292i 0 −0.0969571 − 0.297870i
53.9 0.0362581 − 0.135317i 0 1.71505 + 0.990187i 1.87468 − 1.21884i 0 0.702570 − 2.55076i 0.394292 − 0.394292i 0 −0.0969571 − 0.297870i
53.10 0.0991680 − 0.370100i 0 1.60491 + 0.926596i −1.78935 − 1.34098i 0 2.39134 + 1.13202i 1.04395 − 1.04395i 0 −0.673742 + 0.529258i
53.11 0.271660 − 1.01385i 0 0.777962 + 0.449157i −0.310035 + 2.21447i 0 −1.26800 + 2.32211i 2.15109 − 2.15109i 0 2.16091 + 0.915911i
53.12 0.364480 − 1.36026i 0 0.0145936 + 0.00842564i 2.23601 + 0.0163814i 0 −2.58285 − 0.573503i 2.00834 − 2.00834i 0 0.837263 − 3.03558i
53.13 0.494677 − 1.84616i 0 −1.43155 − 0.826508i −0.541988 − 2.16939i 0 2.64533 + 0.0471386i 0.468944 − 0.468944i 0 −4.27315 − 0.0725500i
53.14 0.543545 − 2.02854i 0 −2.08747 − 1.20520i 1.61611 + 1.54538i 0 1.07923 − 2.41563i −0.609443 + 0.609443i 0 4.01329 − 2.43835i
53.15 0.604483 − 2.25596i 0 −2.99191 − 1.72738i 0.254820 − 2.22150i 0 −2.19240 + 1.48100i −2.40251 + 2.40251i 0 −4.85759 − 1.91772i
53.16 0.695956 − 2.59734i 0 −4.52979 − 2.61528i −1.27254 + 1.83865i 0 −1.50727 − 2.17442i −6.14253 + 6.14253i 0 3.88997 + 4.58485i
107.1 −0.695956 − 2.59734i 0 −4.52979 + 2.61528i 1.27254 + 1.83865i 0 −1.50727 + 2.17442i 6.14253 + 6.14253i 0 3.88997 − 4.58485i
107.2 −0.604483 − 2.25596i 0 −2.99191 + 1.72738i −0.254820 − 2.22150i 0 −2.19240 − 1.48100i 2.40251 + 2.40251i 0 −4.85759 + 1.91772i
107.3 −0.543545 − 2.02854i 0 −2.08747 + 1.20520i −1.61611 + 1.54538i 0 1.07923 + 2.41563i 0.609443 + 0.609443i 0 4.01329 + 2.43835i
107.4 −0.494677 − 1.84616i 0 −1.43155 + 0.826508i 0.541988 − 2.16939i 0 2.64533 − 0.0471386i −0.468944 − 0.468944i 0 −4.27315 + 0.0725500i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.c even 3 1 inner
15.e even 4 1 inner
21.h odd 6 1 inner
35.l odd 12 1 inner
105.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.ce.a ✓ 64
3.b odd 2 1 inner 315.2.ce.a ✓ 64
5.c odd 4 1 inner 315.2.ce.a ✓ 64
7.c even 3 1 inner 315.2.ce.a ✓ 64
15.e even 4 1 inner 315.2.ce.a ✓ 64
21.h odd 6 1 inner 315.2.ce.a ✓ 64
35.l odd 12 1 inner 315.2.ce.a ✓ 64
105.x even 12 1 inner 315.2.ce.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.ce.a ✓ 64 1.a even 1 1 trivial
315.2.ce.a ✓ 64 3.b odd 2 1 inner
315.2.ce.a ✓ 64 5.c odd 4 1 inner
315.2.ce.a ✓ 64 7.c even 3 1 inner
315.2.ce.a ✓ 64 15.e even 4 1 inner
315.2.ce.a ✓ 64 21.h odd 6 1 inner
315.2.ce.a ✓ 64 35.l odd 12 1 inner
315.2.ce.a ✓ 64 105.x even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(315, [\chi])\).