Properties

Label 966.2.y.a
Level $966$
Weight $2$
Character orbit 966.y
Analytic conductor $7.714$
Analytic rank $0$
Dimension $160$
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(25,966)]
 
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, 44, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("966.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.y (of order \(33\), 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: \(160\)
Relative dimension: \(8\) over \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 16 q^{6} - 2 q^{7} + 16 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 16 q^{6} - 2 q^{7} + 16 q^{8} + 8 q^{9} - 2 q^{10} + 4 q^{11} - 8 q^{12} + 4 q^{13} - 9 q^{14} - 18 q^{15} + 8 q^{16} + 9 q^{17} - 8 q^{18} + 18 q^{20} + 7 q^{21} + 52 q^{22} + 22 q^{23} - 80 q^{24} + 42 q^{25} + 2 q^{26} + 16 q^{27} - 18 q^{28} - 54 q^{29} + 2 q^{30} - 15 q^{31} - 8 q^{32} + 18 q^{33} + 84 q^{34} + 82 q^{35} - 16 q^{36} + 3 q^{37} + 11 q^{38} + 2 q^{39} - 13 q^{40} + 22 q^{41} - 2 q^{42} - 96 q^{43} + 4 q^{44} - 20 q^{45} - 26 q^{47} + 16 q^{48} - 2 q^{49} - 4 q^{50} + 13 q^{51} - 2 q^{52} + 16 q^{53} + 8 q^{54} - 34 q^{55} + 2 q^{56} - 22 q^{57} - 5 q^{58} - 6 q^{59} - 2 q^{60} + 40 q^{61} - 8 q^{62} - 2 q^{63} - 16 q^{64} - 82 q^{65} + 4 q^{66} + 10 q^{67} - 46 q^{68} - 22 q^{69} + 100 q^{70} + 92 q^{71} - 8 q^{72} + 68 q^{73} - 3 q^{74} + 2 q^{75} - 7 q^{77} - 18 q^{78} - 78 q^{79} + 2 q^{80} + 8 q^{81} + 55 q^{82} - 42 q^{83} - 9 q^{84} + 50 q^{85} + 40 q^{86} + 6 q^{87} + 7 q^{88} - 80 q^{89} + 4 q^{90} - 62 q^{91} + 66 q^{92} + 26 q^{93} + 4 q^{94} - 17 q^{95} + 8 q^{96} - 8 q^{97} - 3 q^{98} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.0475819 + 0.998867i −0.928368 0.371662i −0.995472 0.0950560i −0.740155 3.05096i 0.415415 0.909632i 0.385204 2.61756i 0.142315 0.989821i 0.723734 + 0.690079i 3.08272 0.594146i
25.2 −0.0475819 + 0.998867i −0.928368 0.371662i −0.995472 0.0950560i −0.678320 2.79607i 0.415415 0.909632i −0.304541 + 2.62817i 0.142315 0.989821i 0.723734 + 0.690079i 2.82518 0.544509i
25.3 −0.0475819 + 0.998867i −0.928368 0.371662i −0.995472 0.0950560i −0.582238 2.40002i 0.415415 0.909632i −0.0849128 + 2.64439i 0.142315 0.989821i 0.723734 + 0.690079i 2.42500 0.467381i
25.4 −0.0475819 + 0.998867i −0.928368 0.371662i −0.995472 0.0950560i 0.0760992 + 0.313685i 0.415415 0.909632i 2.64554 0.0334955i 0.142315 0.989821i 0.723734 + 0.690079i −0.316951 + 0.0610872i
25.5 −0.0475819 + 0.998867i −0.928368 0.371662i −0.995472 0.0950560i 0.373397 + 1.53917i 0.415415 0.909632i −2.64405 0.0947728i 0.142315 0.989821i 0.723734 + 0.690079i −1.55519 + 0.299738i
25.6 −0.0475819 + 0.998867i −0.928368 0.371662i −0.995472 0.0950560i 0.445036 + 1.83446i 0.415415 0.909632i 2.08936 1.62314i 0.142315 0.989821i 0.723734 + 0.690079i −1.85356 + 0.357245i
25.7 −0.0475819 + 0.998867i −0.928368 0.371662i −0.995472 0.0950560i 0.534809 + 2.20451i 0.415415 0.909632i −2.08203 1.63252i 0.142315 0.989821i 0.723734 + 0.690079i −2.22746 + 0.429308i
25.8 −0.0475819 + 0.998867i −0.928368 0.371662i −0.995472 0.0950560i 0.779910 + 3.21483i 0.415415 0.909632i 0.591550 + 2.57877i 0.142315 0.989821i 0.723734 + 0.690079i −3.24830 + 0.626059i
121.1 −0.580057 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i −0.150385 3.15698i −0.654861 + 0.755750i −2.56837 + 0.635207i 0.959493 0.281733i −0.888835 + 0.458227i −2.48437 + 1.95373i
121.2 −0.580057 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i −0.128045 2.68799i −0.654861 + 0.755750i 0.622376 + 2.57151i 0.959493 0.281733i −0.888835 + 0.458227i −2.11530 + 1.66349i
121.3 −0.580057 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i −0.0861989 1.80954i −0.654861 + 0.755750i 1.04990 2.42852i 0.959493 0.281733i −0.888835 + 0.458227i −1.42401 + 1.11985i
121.4 −0.580057 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i 0.00766190 + 0.160843i −0.654861 + 0.755750i −0.825321 + 2.51373i 0.959493 0.281733i −0.888835 + 0.458227i 0.126575 0.0995393i
121.5 −0.580057 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i 0.0236187 + 0.495818i −0.654861 + 0.755750i 2.48735 0.901724i 0.959493 0.281733i −0.888835 + 0.458227i 0.390181 0.306842i
121.6 −0.580057 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i 0.106503 + 2.23577i −0.654861 + 0.755750i −2.31212 1.28612i 0.959493 0.281733i −0.888835 + 0.458227i 1.75943 1.38363i
121.7 −0.580057 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i 0.122200 + 2.56530i −0.654861 + 0.755750i −0.168303 + 2.64039i 0.959493 0.281733i −0.888835 + 0.458227i 2.01875 1.58756i
121.8 −0.580057 0.814576i −0.235759 0.971812i −0.327068 + 0.945001i 0.170819 + 3.58593i −0.654861 + 0.755750i 2.64569 + 0.0174095i 0.959493 0.281733i −0.888835 + 0.458227i 2.82193 2.21919i
151.1 0.327068 0.945001i 0.888835 0.458227i −0.786053 0.618159i −4.11027 + 0.392484i −0.142315 0.989821i −2.51634 + 0.817338i −0.841254 + 0.540641i 0.580057 0.814576i −0.973442 + 4.01258i
151.2 0.327068 0.945001i 0.888835 0.458227i −0.786053 0.618159i −2.84187 + 0.271366i −0.142315 0.989821i 2.39273 + 1.12909i −0.841254 + 0.540641i 0.580057 0.814576i −0.673045 + 2.77433i
151.3 0.327068 0.945001i 0.888835 0.458227i −0.786053 0.618159i −2.66247 + 0.254235i −0.142315 0.989821i 1.24716 + 2.33337i −0.841254 + 0.540641i 0.580057 0.814576i −0.630556 + 2.59919i
151.4 0.327068 0.945001i 0.888835 0.458227i −0.786053 0.618159i −1.00054 + 0.0955401i −0.142315 0.989821i −2.50267 0.858289i −0.841254 + 0.540641i 0.580057 0.814576i −0.236960 + 0.976761i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
23.c even 11 1 inner
161.m even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.y.a 160
7.c even 3 1 inner 966.2.y.a 160
23.c even 11 1 inner 966.2.y.a 160
161.m even 33 1 inner 966.2.y.a 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.y.a 160 1.a even 1 1 trivial
966.2.y.a 160 7.c even 3 1 inner
966.2.y.a 160 23.c even 11 1 inner
966.2.y.a 160 161.m even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{160} - 2 T_{5}^{159} - 39 T_{5}^{158} + 2 T_{5}^{157} + 775 T_{5}^{156} + 2105 T_{5}^{155} - 8790 T_{5}^{154} - 69004 T_{5}^{153} + 51413 T_{5}^{152} + 1348378 T_{5}^{151} - 37757 T_{5}^{150} - 15719491 T_{5}^{149} + \cdots + 47\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\). Copy content Toggle raw display