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Properties

Label 966.2.y.c
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
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} + 16 q^{6} - 2 q^{7} - 16 q^{8} + 8 q^{9} + 2 q^{10} + 8 q^{11} - 8 q^{12} - 4 q^{13} + 9 q^{14} - 18 q^{15} + 8 q^{16} - 21 q^{17} + 8 q^{18} + 4 q^{19} + 18 q^{20} - 15 q^{21} + 116 q^{22} + 14 q^{23} + 80 q^{24} + 2 q^{25} + 2 q^{26} + 16 q^{27} - 18 q^{28} - 46 q^{29} - 2 q^{30} + q^{31} + 8 q^{32} - 8 q^{33} - 68 q^{34} - 84 q^{35} - 16 q^{36} + 19 q^{37} - 7 q^{38} - 2 q^{39} - 9 q^{40} + 38 q^{41} + 2 q^{42} + 28 q^{43} + 8 q^{44} + 24 q^{45} - 8 q^{46} - 26 q^{47} + 16 q^{48} - 6 q^{49} - 4 q^{50} + 21 q^{51} + 2 q^{52} - 20 q^{53} - 8 q^{54} + 46 q^{55} - 2 q^{56} - 14 q^{57} - 21 q^{58} + 36 q^{59} - 2 q^{60} + 40 q^{61} - 24 q^{62} - 2 q^{63} - 16 q^{64} - 10 q^{65} - 8 q^{66} + 6 q^{67} + 34 q^{68} + 6 q^{69} - 20 q^{70} + 132 q^{71} + 8 q^{72} - 36 q^{73} + 19 q^{74} - 2 q^{75} - 8 q^{76} - 75 q^{77} - 18 q^{78} + 34 q^{79} + 2 q^{80} + 8 q^{81} + 47 q^{82} + 74 q^{83} + 13 q^{84} + 2 q^{85} + 30 q^{86} + 10 q^{87} - 3 q^{88} - 12 q^{89} - 4 q^{90} - 30 q^{91} - 50 q^{92} + 10 q^{93} - 4 q^{94} + 27 q^{95} - 8 q^{96} + 24 q^{97} + 7 q^{98} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0.0475819 − 0.998867i −0.928368 − 0.371662i −0.995472 − 0.0950560i −0.835382 − 3.44349i −0.415415 + 0.909632i 0.912603 − 2.48338i −0.142315 + 0.989821i 0.723734 + 0.690079i −3.47934 + 0.670588i
25.2 0.0475819 − 0.998867i −0.928368 − 0.371662i −0.995472 − 0.0950560i −0.748398 − 3.08494i −0.415415 + 0.909632i 2.13670 + 1.56029i −0.142315 + 0.989821i 0.723734 + 0.690079i −3.11705 + 0.600763i
25.3 0.0475819 − 0.998867i −0.928368 − 0.371662i −0.995472 − 0.0950560i −0.228734 − 0.942856i −0.415415 + 0.909632i −1.88597 + 1.85556i −0.142315 + 0.989821i 0.723734 + 0.690079i −0.952672 + 0.183612i
25.4 0.0475819 − 0.998867i −0.928368 − 0.371662i −0.995472 − 0.0950560i −0.211998 − 0.873870i −0.415415 + 0.909632i 2.39932 + 1.11502i −0.142315 + 0.989821i 0.723734 + 0.690079i −0.882967 + 0.170178i
25.5 0.0475819 − 0.998867i −0.928368 − 0.371662i −0.995472 − 0.0950560i −0.167516 − 0.690511i −0.415415 + 0.909632i −2.01446 − 1.71521i −0.142315 + 0.989821i 0.723734 + 0.690079i −0.697700 + 0.134471i
25.6 0.0475819 − 0.998867i −0.928368 − 0.371662i −0.995472 − 0.0950560i 0.244232 + 1.00674i −0.415415 + 0.909632i 0.601440 − 2.57648i −0.142315 + 0.989821i 0.723734 + 0.690079i 1.01722 − 0.196053i
25.7 0.0475819 − 0.998867i −0.928368 − 0.371662i −0.995472 − 0.0950560i 0.751462 + 3.09757i −0.415415 + 0.909632i 2.38356 − 1.14832i −0.142315 + 0.989821i 0.723734 + 0.690079i 3.12982 − 0.603223i
25.8 0.0475819 − 0.998867i −0.928368 − 0.371662i −0.995472 − 0.0950560i 0.853588 + 3.51854i −0.415415 + 0.909632i −1.36069 + 2.26904i −0.142315 + 0.989821i 0.723734 + 0.690079i 3.55517 − 0.685202i
121.1 0.580057 + 0.814576i −0.235759 − 0.971812i −0.327068 + 0.945001i −0.163478 − 3.43182i 0.654861 − 0.755750i 0.873534 − 2.49739i −0.959493 + 0.281733i −0.888835 + 0.458227i 2.70065 − 2.12382i
121.2 0.580057 + 0.814576i −0.235759 − 0.971812i −0.327068 + 0.945001i −0.161073 − 3.38134i 0.654861 − 0.755750i 2.36003 + 1.19594i −0.959493 + 0.281733i −0.888835 + 0.458227i 2.66093 − 2.09258i
121.3 0.580057 + 0.814576i −0.235759 − 0.971812i −0.327068 + 0.945001i −0.142165 − 2.98442i 0.654861 − 0.755750i −2.19584 + 1.47590i −0.959493 + 0.281733i −0.888835 + 0.458227i 2.34857 − 1.84694i
121.4 0.580057 + 0.814576i −0.235759 − 0.971812i −0.327068 + 0.945001i −0.0167727 − 0.352103i 0.654861 − 0.755750i −1.88460 + 1.85696i −0.959493 + 0.281733i −0.888835 + 0.458227i 0.277085 − 0.217902i
121.5 0.580057 + 0.814576i −0.235759 − 0.971812i −0.327068 + 0.945001i −0.00337646 − 0.0708807i 0.654861 − 0.755750i 1.92801 + 1.81184i −0.959493 + 0.281733i −0.888835 + 0.458227i 0.0557792 − 0.0438652i
121.6 0.580057 + 0.814576i −0.235759 − 0.971812i −0.327068 + 0.945001i 0.00889795 + 0.186791i 0.654861 − 0.755750i 2.27500 − 1.35069i −0.959493 + 0.281733i −0.888835 + 0.458227i −0.146994 + 0.115597i
121.7 0.580057 + 0.814576i −0.235759 − 0.971812i −0.327068 + 0.945001i 0.0874971 + 1.83679i 0.654861 − 0.755750i 0.484830 − 2.60095i −0.959493 + 0.281733i −0.888835 + 0.458227i −1.44545 + 1.13672i
121.8 0.580057 + 0.814576i −0.235759 − 0.971812i −0.327068 + 0.945001i 0.141678 + 2.97420i 0.654861 − 0.755750i −2.12708 − 1.57338i −0.959493 + 0.281733i −0.888835 + 0.458227i −2.34053 + 1.84061i
151.1 −0.327068 + 0.945001i 0.888835 − 0.458227i −0.786053 − 0.618159i −3.08451 + 0.294535i 0.142315 + 0.989821i −0.562151 + 2.58534i 0.841254 − 0.540641i 0.580057 − 0.814576i 0.730508 − 3.01119i
151.2 −0.327068 + 0.945001i 0.888835 − 0.458227i −0.786053 − 0.618159i −2.78783 + 0.266206i 0.142315 + 0.989821i 2.46897 − 0.950880i 0.841254 − 0.540641i 0.580057 − 0.814576i 0.660246 − 2.72157i
151.3 −0.327068 + 0.945001i 0.888835 − 0.458227i −0.786053 − 0.618159i −2.45803 + 0.234713i 0.142315 + 0.989821i −0.472531 − 2.60321i 0.841254 − 0.540641i 0.580057 − 0.814576i 0.582139 − 2.39961i
151.4 −0.327068 + 0.945001i 0.888835 − 0.458227i −0.786053 − 0.618159i −0.150887 + 0.0144079i 0.142315 + 0.989821i −2.59084 + 0.536227i 0.841254 − 0.540641i 0.580057 − 0.814576i 0.0357347 − 0.147300i
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.c ✓ 160
7.c even 3 1 inner 966.2.y.c ✓ 160
23.c even 11 1 inner 966.2.y.c ✓ 160
161.m even 33 1 inner 966.2.y.c ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.y.c ✓ 160 1.a even 1 1 trivial
966.2.y.c ✓ 160 7.c even 3 1 inner
966.2.y.c ✓ 160 23.c even 11 1 inner
966.2.y.c ✓ 160 161.m even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{160} - 2 T_{5}^{159} - 19 T_{5}^{158} + 130 T_{5}^{157} - 53 T_{5}^{156} - 1587 T_{5}^{155} + \cdots + 18\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\). Copy content Toggle raw display