Properties

Label 966.2.y.d
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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 2 q^{5} - 16 q^{6} + 6 q^{7} - 16 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 2 q^{5} - 16 q^{6} + 6 q^{7} - 16 q^{8} + 8 q^{9} - 2 q^{10} - 4 q^{11} + 8 q^{12} - 4 q^{13} + 5 q^{14} + 26 q^{15} + 8 q^{16} - 13 q^{17} + 8 q^{18} + 4 q^{19} + 26 q^{20} + 11 q^{21} - 124 q^{22} + 26 q^{23} - 80 q^{24} + 14 q^{25} + 2 q^{26} - 16 q^{27} - 22 q^{28} + 54 q^{29} - 2 q^{30} + 15 q^{31} + 8 q^{32} - 4 q^{33} + 4 q^{34} - 2 q^{35} - 16 q^{36} + 7 q^{37} - 7 q^{38} + 2 q^{39} + 9 q^{40} - 30 q^{41} + 6 q^{42} + 36 q^{43} - 4 q^{44} + 20 q^{45} + 4 q^{46} - 6 q^{47} - 16 q^{48} + 14 q^{49} - 28 q^{50} + 9 q^{51} + 2 q^{52} - 24 q^{53} + 8 q^{54} - 14 q^{55} + 6 q^{56} - 30 q^{57} - 5 q^{58} + 4 q^{59} - 2 q^{60} + 32 q^{61} - 8 q^{62} - 6 q^{63} - 16 q^{64} + 38 q^{65} - 4 q^{66} + 6 q^{67} + 42 q^{68} - 30 q^{69} - 8 q^{70} - 180 q^{71} + 8 q^{72} - 32 q^{73} + 7 q^{74} + 14 q^{75} - 8 q^{76} + 61 q^{77} + 18 q^{78} - 66 q^{79} - 2 q^{80} + 8 q^{81} - 7 q^{82} + 130 q^{83} + 5 q^{84} - 70 q^{85} - 40 q^{86} + 6 q^{87} - 15 q^{88} - 4 q^{89} + 4 q^{90} - 6 q^{91} + 58 q^{92} - 18 q^{93} + 16 q^{94} + 19 q^{95} + 8 q^{96} + 40 q^{97} + 43 q^{98} + 8 q^{99}+O(q^{100}) \) 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 −0.786833 3.24337i 0.415415 0.909632i −2.25109 1.39018i −0.142315 + 0.989821i 0.723734 + 0.690079i −3.27714 + 0.631616i
25.2 0.0475819 0.998867i 0.928368 + 0.371662i −0.995472 0.0950560i −0.781948 3.22324i 0.415415 0.909632i 2.33065 1.25222i −0.142315 + 0.989821i 0.723734 + 0.690079i −3.25679 + 0.627695i
25.3 0.0475819 0.998867i 0.928368 + 0.371662i −0.995472 0.0950560i −0.363961 1.50027i 0.415415 0.909632i −0.661406 + 2.56175i −0.142315 + 0.989821i 0.723734 + 0.690079i −1.51589 + 0.292163i
25.4 0.0475819 0.998867i 0.928368 + 0.371662i −0.995472 0.0950560i −0.167957 0.692328i 0.415415 0.909632i 2.14841 1.54413i −0.142315 + 0.989821i 0.723734 + 0.690079i −0.699536 + 0.134824i
25.5 0.0475819 0.998867i 0.928368 + 0.371662i −0.995472 0.0950560i −0.130603 0.538354i 0.415415 0.909632i −2.29097 + 1.32342i −0.142315 + 0.989821i 0.723734 + 0.690079i −0.543959 + 0.104840i
25.6 0.0475819 0.998867i 0.928368 + 0.371662i −0.995472 0.0950560i 0.112949 + 0.465581i 0.415415 0.909632i 1.20408 + 2.35588i −0.142315 + 0.989821i 0.723734 + 0.690079i 0.470428 0.0906676i
25.7 0.0475819 0.998867i 0.928368 + 0.371662i −0.995472 0.0950560i 0.415021 + 1.71074i 0.415415 0.909632i −0.322897 2.62597i −0.142315 + 0.989821i 0.723734 + 0.690079i 1.72855 0.333151i
25.8 0.0475819 0.998867i 0.928368 + 0.371662i −0.995472 0.0950560i 0.968835 + 3.99359i 0.415415 0.909632i 2.64295 0.121825i −0.142315 + 0.989821i 0.723734 + 0.690079i 4.03517 0.777715i
121.1 0.580057 + 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i −0.182773 3.83688i −0.654861 + 0.755750i −1.41671 + 2.23449i −0.959493 + 0.281733i −0.888835 + 0.458227i 3.01941 2.37449i
121.2 0.580057 + 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i −0.180250 3.78390i −0.654861 + 0.755750i 1.94204 + 1.79680i −0.959493 + 0.281733i −0.888835 + 0.458227i 2.97772 2.34171i
121.3 0.580057 + 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i −0.0817448 1.71603i −0.654861 + 0.755750i −2.45748 0.980189i −0.959493 + 0.281733i −0.888835 + 0.458227i 1.35042 1.06199i
121.4 0.580057 + 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i −0.0809367 1.69907i −0.654861 + 0.755750i 1.60083 2.10650i −0.959493 + 0.281733i −0.888835 + 0.458227i 1.33707 1.05149i
121.5 0.580057 + 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i 0.0478985 + 1.00551i −0.654861 + 0.755750i −0.794478 2.52365i −0.959493 + 0.281733i −0.888835 + 0.458227i −0.791284 + 0.622272i
121.6 0.580057 + 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i 0.0802489 + 1.68463i −0.654861 + 0.755750i 1.50657 + 2.17491i −0.959493 + 0.281733i −0.888835 + 0.458227i −1.32571 + 1.04255i
121.7 0.580057 + 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i 0.132422 + 2.77988i −0.654861 + 0.755750i −1.83001 + 1.91077i −0.959493 + 0.281733i −0.888835 + 0.458227i −2.18761 + 1.72035i
121.8 0.580057 + 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i 0.140981 + 2.95955i −0.654861 + 0.755750i 2.61692 0.389498i −0.959493 + 0.281733i −0.888835 + 0.458227i −2.32900 + 1.83155i
151.1 −0.327068 + 0.945001i −0.888835 + 0.458227i −0.786053 0.618159i −4.06127 + 0.387805i −0.142315 0.989821i −0.441487 2.60866i 0.841254 0.540641i 0.580057 0.814576i 0.961837 3.96475i
151.2 −0.327068 + 0.945001i −0.888835 + 0.458227i −0.786053 0.618159i −2.76706 + 0.264222i −0.142315 0.989821i 2.35671 + 1.20246i 0.841254 0.540641i 0.580057 0.814576i 0.655327 2.70130i
151.3 −0.327068 + 0.945001i −0.888835 + 0.458227i −0.786053 0.618159i −0.688242 + 0.0657192i −0.142315 0.989821i −0.529942 + 2.59213i 0.841254 0.540641i 0.580057 0.814576i 0.162997 0.671884i
151.4 −0.327068 + 0.945001i −0.888835 + 0.458227i −0.786053 0.618159i −0.527443 + 0.0503647i −0.142315 0.989821i −2.62119 0.359704i 0.841254 0.540641i 0.580057 0.814576i 0.124915 0.514906i
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.d 160
7.c even 3 1 inner 966.2.y.d 160
23.c even 11 1 inner 966.2.y.d 160
161.m even 33 1 inner 966.2.y.d 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.y.d 160 1.a even 1 1 trivial
966.2.y.d 160 7.c even 3 1 inner
966.2.y.d 160 23.c even 11 1 inner
966.2.y.d 160 161.m even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{160} + 2 T_{5}^{159} - 25 T_{5}^{158} - 78 T_{5}^{157} + 193 T_{5}^{156} + 2045 T_{5}^{155} + 4012 T_{5}^{154} - 31174 T_{5}^{153} - 178941 T_{5}^{152} - 181654 T_{5}^{151} + 2659035 T_{5}^{150} + 12314925 T_{5}^{149} + \cdots + 10\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\). Copy content Toggle raw display