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Properties

Label 945.2.ce.a
Level $945$
Weight $2$
Character orbit 945.ce
Analytic conductor $7.546$
Analytic rank $0$
Dimension $176$
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] = [945,2,Mod(118,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([4, 9, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.118");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.ce (of order \(12\), degree \(4\), not 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(44\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 315)
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.

\(\operatorname{Tr}(f)(q) = \) \( 176 q + 4 q^{2} - 2 q^{7} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q + 4 q^{2} - 2 q^{7} + 32 q^{8} + 12 q^{11} + 56 q^{16} + 12 q^{22} + 12 q^{23} - 4 q^{25} - 32 q^{28} - 48 q^{32} + 8 q^{35} - 16 q^{37} - 4 q^{43} - 80 q^{46} + 76 q^{50} - 64 q^{53} + 52 q^{56} - 44 q^{58} - 20 q^{65} - 4 q^{67} + 18 q^{70} + 64 q^{71} - 26 q^{77} - 4 q^{85} - 80 q^{86} - 60 q^{88} - 16 q^{91} + 68 q^{92} - 40 q^{95} + 120 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1 −2.50181 − 0.670357i 0 4.07761 + 2.35421i −2.08579 − 0.805916i 0 −1.52828 + 2.15971i −4.96033 − 4.96033i 0 4.67798 + 3.41447i
118.2 −2.50181 − 0.670357i 0 4.07761 + 2.35421i 2.08579 + 0.805916i 0 −0.243675 − 2.63451i −4.96033 − 4.96033i 0 −4.67798 − 3.41447i
118.3 −2.33594 − 0.625914i 0 3.33281 + 1.92420i −0.810739 − 2.08391i 0 −1.56453 − 2.13360i −3.16081 − 3.16081i 0 0.589489 + 5.37536i
118.4 −2.33594 − 0.625914i 0 3.33281 + 1.92420i 0.810739 + 2.08391i 0 −2.42172 + 1.06549i −3.16081 − 3.16081i 0 −0.589489 − 5.37536i
118.5 −2.18730 − 0.586086i 0 2.70875 + 1.56390i −0.784352 + 2.09399i 0 0.841996 + 2.50820i −1.80584 − 1.80584i 0 2.94287 − 4.12049i
118.6 −2.18730 − 0.586086i 0 2.70875 + 1.56390i 0.784352 − 2.09399i 0 1.98329 − 1.75116i −1.80584 − 1.80584i 0 −2.94287 + 4.12049i
118.7 −1.97738 − 0.529836i 0 1.89724 + 1.09537i −1.92292 + 1.14122i 0 1.55462 − 2.14083i −0.276101 − 0.276101i 0 4.40699 − 1.23779i
118.8 −1.97738 − 0.529836i 0 1.89724 + 1.09537i 1.92292 − 1.14122i 0 0.275927 + 2.63132i −0.276101 − 0.276101i 0 −4.40699 + 1.23779i
118.9 −1.59692 − 0.427892i 0 0.634997 + 0.366616i −1.54329 − 1.61810i 0 −2.20873 − 1.45654i 1.48088 + 1.48088i 0 1.77213 + 3.24434i
118.10 −1.59692 − 0.427892i 0 0.634997 + 0.366616i 1.54329 + 1.61810i 0 −2.64109 + 0.157037i 1.48088 + 1.48088i 0 −1.77213 − 3.24434i
118.11 −1.41025 − 0.377876i 0 0.113968 + 0.0657996i −1.50611 − 1.65277i 0 2.52369 + 0.794356i 1.92889 + 1.92889i 0 1.49945 + 2.89994i
118.12 −1.41025 − 0.377876i 0 0.113968 + 0.0657996i 1.50611 + 1.65277i 0 2.58276 + 0.573911i 1.92889 + 1.92889i 0 −1.49945 − 2.89994i
118.13 −1.13556 − 0.304273i 0 −0.535130 − 0.308958i −0.149537 − 2.23106i 0 1.03311 + 2.43571i 2.17625 + 2.17625i 0 −0.509043 + 2.57901i
118.14 −1.13556 − 0.304273i 0 −0.535130 − 0.308958i 0.149537 + 2.23106i 0 2.11255 − 1.59283i 2.17625 + 2.17625i 0 0.509043 − 2.57901i
118.15 −1.06014 − 0.284064i 0 −0.688847 − 0.397706i −2.11673 + 0.720733i 0 −1.74443 − 1.98922i 2.16945 + 2.16945i 0 2.44876 − 0.162792i
118.16 −1.06014 − 0.284064i 0 −0.688847 − 0.397706i 2.11673 − 0.720733i 0 −2.50533 + 0.850498i 2.16945 + 2.16945i 0 −2.44876 + 0.162792i
118.17 −0.768362 − 0.205882i 0 −1.18406 − 0.683616i −1.66577 + 1.49171i 0 0.574501 + 2.58262i 1.89400 + 1.89400i 0 1.58703 − 0.803226i
118.18 −0.768362 − 0.205882i 0 −1.18406 − 0.683616i 1.66577 − 1.49171i 0 1.78884 − 1.94937i 1.89400 + 1.89400i 0 −1.58703 + 0.803226i
118.19 −0.376757 − 0.100952i 0 −1.60030 − 0.923931i −0.922867 + 2.03674i 0 −1.42222 − 2.23099i 1.06126 + 1.06126i 0 0.553309 − 0.674192i
118.20 −0.376757 − 0.100952i 0 −1.60030 − 0.923931i 0.922867 − 2.03674i 0 −2.34717 + 1.22098i 1.06126 + 1.06126i 0 −0.553309 + 0.674192i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
9.c even 3 1 inner
35.f even 4 1 inner
45.k odd 12 1 inner
63.l odd 6 1 inner
315.cb even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.ce.a 176
3.b odd 2 1 315.2.cb.a ✓ 176
5.c odd 4 1 inner 945.2.ce.a 176
7.b odd 2 1 inner 945.2.ce.a 176
9.c even 3 1 inner 945.2.ce.a 176
9.d odd 6 1 315.2.cb.a ✓ 176
15.e even 4 1 315.2.cb.a ✓ 176
21.c even 2 1 315.2.cb.a ✓ 176
35.f even 4 1 inner 945.2.ce.a 176
45.k odd 12 1 inner 945.2.ce.a 176
45.l even 12 1 315.2.cb.a ✓ 176
63.l odd 6 1 inner 945.2.ce.a 176
63.o even 6 1 315.2.cb.a ✓ 176
105.k odd 4 1 315.2.cb.a ✓ 176
315.cb even 12 1 inner 945.2.ce.a 176
315.cf odd 12 1 315.2.cb.a ✓ 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.cb.a ✓ 176 3.b odd 2 1
315.2.cb.a ✓ 176 9.d odd 6 1
315.2.cb.a ✓ 176 15.e even 4 1
315.2.cb.a ✓ 176 21.c even 2 1
315.2.cb.a ✓ 176 45.l even 12 1
315.2.cb.a ✓ 176 63.o even 6 1
315.2.cb.a ✓ 176 105.k odd 4 1
315.2.cb.a ✓ 176 315.cf odd 12 1
945.2.ce.a 176 1.a even 1 1 trivial
945.2.ce.a 176 5.c odd 4 1 inner
945.2.ce.a 176 7.b odd 2 1 inner
945.2.ce.a 176 9.c even 3 1 inner
945.2.ce.a 176 35.f even 4 1 inner
945.2.ce.a 176 45.k odd 12 1 inner
945.2.ce.a 176 63.l odd 6 1 inner
945.2.ce.a 176 315.cb even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(945, [\chi])\).