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Properties

Label 966.2.be.a
Level $966$
Weight $2$
Character orbit 966.be
Analytic conductor $7.714$
Analytic rank $0$
Dimension $320$
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] = [966,2,Mod(19,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 55, 45]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("966.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.be (of order \(66\), degree \(20\), 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: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(16\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 16 q^{2} + 16 q^{4} + 32 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q - 16 q^{2} + 16 q^{4} + 32 q^{8} - 16 q^{9} - 22 q^{14} + 16 q^{16} + 66 q^{17} + 16 q^{18} + 40 q^{23} - 48 q^{25} + 12 q^{26} + 44 q^{28} - 24 q^{29} + 24 q^{31} - 16 q^{32} + 98 q^{35} + 32 q^{36} - 22 q^{37} - 66 q^{38} - 8 q^{39} - 88 q^{43} + 4 q^{46} - 144 q^{47} - 24 q^{49} + 80 q^{50} - 22 q^{51} + 12 q^{52} + 44 q^{53} + 44 q^{57} + 10 q^{58} + 12 q^{59} - 32 q^{64} + 108 q^{70} - 16 q^{71} + 16 q^{72} - 180 q^{73} - 22 q^{74} - 12 q^{75} + 18 q^{77} - 16 q^{78} + 44 q^{79} + 16 q^{81} + 36 q^{82} + 22 q^{84} + 68 q^{85} - 22 q^{86} + 48 q^{87} + 22 q^{88} + 8 q^{92} + 8 q^{93} - 12 q^{94} + 66 q^{95} - 90 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 0.995472 + 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i −3.45344 + 1.78037i −0.755750 + 0.654861i −2.24314 + 1.40298i 0.959493 + 0.281733i −0.0475819 − 0.998867i −3.60703 + 1.44404i
19.2 0.995472 + 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i −2.25010 + 1.16001i −0.755750 + 0.654861i 1.37319 − 2.26149i 0.959493 + 0.281733i −0.0475819 − 0.998867i −2.35018 + 0.940871i
19.3 0.995472 + 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i −1.41181 + 0.727837i −0.755750 + 0.654861i 1.71541 + 2.01429i 0.959493 + 0.281733i −0.0475819 − 0.998867i −1.47460 + 0.590341i
19.4 0.995472 + 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i −0.290919 + 0.149979i −0.755750 + 0.654861i −0.738752 − 2.54052i 0.959493 + 0.281733i −0.0475819 − 0.998867i −0.303858 + 0.121646i
19.5 0.995472 + 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i 0.631353 − 0.325485i −0.755750 + 0.654861i 2.64138 − 0.152057i 0.959493 + 0.281733i −0.0475819 − 0.998867i 0.659433 − 0.263997i
19.6 0.995472 + 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i 1.05593 − 0.544372i −0.755750 + 0.654861i −2.51151 + 0.832044i 0.959493 + 0.281733i −0.0475819 − 0.998867i 1.10290 − 0.441534i
19.7 0.995472 + 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i 3.01447 − 1.55407i −0.755750 + 0.654861i 1.02282 − 2.44005i 0.959493 + 0.281733i −0.0475819 − 0.998867i 3.14854 − 1.26049i
19.8 0.995472 + 0.0950560i −0.690079 + 0.723734i 0.981929 + 0.189251i 3.18053 − 1.63968i −0.755750 + 0.654861i 1.02341 + 2.43980i 0.959493 + 0.281733i −0.0475819 − 0.998867i 3.32199 − 1.32992i
19.9 0.995472 + 0.0950560i 0.690079 − 0.723734i 0.981929 + 0.189251i −3.73421 + 1.92512i 0.755750 − 0.654861i −0.120285 − 2.64302i 0.959493 + 0.281733i −0.0475819 − 0.998867i −3.90030 + 1.56144i
19.10 0.995472 + 0.0950560i 0.690079 − 0.723734i 0.981929 + 0.189251i −3.26423 + 1.68283i 0.755750 − 0.654861i 2.33879 + 1.23696i 0.959493 + 0.281733i −0.0475819 − 0.998867i −3.40941 + 1.36492i
19.11 0.995472 + 0.0950560i 0.690079 − 0.723734i 0.981929 + 0.189251i −1.59513 + 0.822347i 0.755750 − 0.654861i −1.53650 + 2.15387i 0.959493 + 0.281733i −0.0475819 − 0.998867i −1.66608 + 0.666996i
19.12 0.995472 + 0.0950560i 0.690079 − 0.723734i 0.981929 + 0.189251i −0.867285 + 0.447116i 0.755750 − 0.654861i 2.46361 − 0.964688i 0.959493 + 0.281733i −0.0475819 − 0.998867i −0.905859 + 0.362651i
19.13 0.995472 + 0.0950560i 0.690079 − 0.723734i 0.981929 + 0.189251i 0.525481 − 0.270904i 0.755750 − 0.654861i −0.753581 − 2.53616i 0.959493 + 0.281733i −0.0475819 − 0.998867i 0.548853 − 0.219728i
19.14 0.995472 + 0.0950560i 0.690079 − 0.723734i 0.981929 + 0.189251i 1.83046 − 0.943666i 0.755750 − 0.654861i −2.58469 − 0.565134i 0.959493 + 0.281733i −0.0475819 − 0.998867i 1.91187 − 0.765397i
19.15 0.995472 + 0.0950560i 0.690079 − 0.723734i 0.981929 + 0.189251i 2.34527 − 1.20907i 0.755750 − 0.654861i 2.38412 − 1.14715i 0.959493 + 0.281733i −0.0475819 − 0.998867i 2.44958 − 0.980664i
19.16 0.995472 + 0.0950560i 0.690079 − 0.723734i 0.981929 + 0.189251i 2.54871 − 1.31395i 0.755750 − 0.654861i −0.0690376 + 2.64485i 0.959493 + 0.281733i −0.0475819 − 0.998867i 2.66207 − 1.06573i
61.1 0.786053 + 0.618159i −0.814576 − 0.580057i 0.235759 + 0.971812i −2.60993 + 0.503023i −0.281733 − 0.959493i −0.258273 − 2.63312i −0.415415 + 0.909632i 0.327068 + 0.945001i −2.36249 − 1.21795i
61.2 0.786053 + 0.618159i −0.814576 − 0.580057i 0.235759 + 0.971812i −2.19052 + 0.422188i −0.281733 − 0.959493i 1.40168 + 2.24395i −0.415415 + 0.909632i 0.327068 + 0.945001i −1.98285 − 1.02223i
61.3 0.786053 + 0.618159i −0.814576 − 0.580057i 0.235759 + 0.971812i −1.06718 + 0.205682i −0.281733 − 0.959493i −1.24125 + 2.33651i −0.415415 + 0.909632i 0.327068 + 0.945001i −0.966005 − 0.498010i
61.4 0.786053 + 0.618159i −0.814576 − 0.580057i 0.235759 + 0.971812i −0.436006 + 0.0840333i −0.281733 − 0.959493i −2.27972 − 1.34272i −0.415415 + 0.909632i 0.327068 + 0.945001i −0.394670 − 0.203466i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
23.d odd 22 1 inner
161.o even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.be.a ✓ 320
7.d odd 6 1 inner 966.2.be.a ✓ 320
23.d odd 22 1 inner 966.2.be.a ✓ 320
161.o even 66 1 inner 966.2.be.a ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.be.a ✓ 320 1.a even 1 1 trivial
966.2.be.a ✓ 320 7.d odd 6 1 inner
966.2.be.a ✓ 320 23.d odd 22 1 inner
966.2.be.a ✓ 320 161.o even 66 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{320} - 16 T_{5}^{318} - 259 T_{5}^{316} - 2970 T_{5}^{315} + 9400 T_{5}^{314} + \cdots + 28\!\cdots\!21 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\). Copy content Toggle raw display