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Properties

Label 966.2.y.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{2} + 8 q^{3} + 8 q^{4} + 6 q^{5} + 16 q^{6} - 10 q^{7} + 16 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{2} + 8 q^{3} + 8 q^{4} + 6 q^{5} + 16 q^{6} - 10 q^{7} + 16 q^{8} + 8 q^{9} - 6 q^{10} + 8 q^{11} + 8 q^{12} + 4 q^{13} - 13 q^{14} + 10 q^{15} + 8 q^{16} + 9 q^{17} - 8 q^{18} - 8 q^{19} + 10 q^{20} - 3 q^{21} - 28 q^{22} + 22 q^{23} + 80 q^{24} + 46 q^{25} + 2 q^{26} - 16 q^{27} - 14 q^{28} + 62 q^{29} - 6 q^{30} + 7 q^{31} - 8 q^{32} - 14 q^{33} - 4 q^{34} + 24 q^{35} - 16 q^{36} + 15 q^{37} + 19 q^{38} - 2 q^{39} + 5 q^{40} - 14 q^{41} + 10 q^{42} - 56 q^{43} + 8 q^{44} - 16 q^{45} + 66 q^{47} - 16 q^{48} + 42 q^{49} + 4 q^{50} + 9 q^{51} - 2 q^{52} + 12 q^{53} - 8 q^{54} - 14 q^{55} + 10 q^{56} - 6 q^{57} - 13 q^{58} + 62 q^{59} + 6 q^{60} + 16 q^{61} - 8 q^{62} + 2 q^{63} - 16 q^{64} + 70 q^{65} - 8 q^{66} - 14 q^{67} - 46 q^{68} + 22 q^{69} + 24 q^{70} - 108 q^{71} - 8 q^{72} + 40 q^{73} - 15 q^{74} + 2 q^{75} + 16 q^{76} + 29 q^{77} + 18 q^{78} + 22 q^{79} + 6 q^{80} + 8 q^{81} - 7 q^{82} - 18 q^{83} - 9 q^{84} + 26 q^{85} - 6 q^{86} + 2 q^{87} + 3 q^{88} + 112 q^{89} + 12 q^{90} + 18 q^{91} - 66 q^{92} - 26 q^{93} + 23 q^{95} - 8 q^{96} + 8 q^{97} + 9 q^{98} + 28 q^{99}+O(q^{100}) \) 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 −0.495624 − 2.04299i −0.415415 + 0.909632i −2.46554 + 0.959758i 0.142315 − 0.989821i 0.723734 + 0.690079i 2.06426 − 0.397853i
25.2 −0.0475819 + 0.998867i 0.928368 + 0.371662i −0.995472 − 0.0950560i −0.494800 − 2.03959i −0.415415 + 0.909632i 2.60154 + 0.481642i 0.142315 − 0.989821i 0.723734 + 0.690079i 2.06083 − 0.397192i
25.3 −0.0475819 + 0.998867i 0.928368 + 0.371662i −0.995472 − 0.0950560i −0.365274 − 1.50568i −0.415415 + 0.909632i 0.990231 + 2.45346i 0.142315 − 0.989821i 0.723734 + 0.690079i 1.52136 − 0.293217i
25.4 −0.0475819 + 0.998867i 0.928368 + 0.371662i −0.995472 − 0.0950560i 0.0842779 + 0.347398i −0.415415 + 0.909632i 0.863387 − 2.50091i 0.142315 − 0.989821i 0.723734 + 0.690079i −0.351015 + 0.0676526i
25.5 −0.0475819 + 0.998867i 0.928368 + 0.371662i −0.995472 − 0.0950560i 0.149969 + 0.618179i −0.415415 + 0.909632i −1.95013 + 1.78801i 0.142315 − 0.989821i 0.723734 + 0.690079i −0.624615 + 0.120385i
25.6 −0.0475819 + 0.998867i 0.928368 + 0.371662i −0.995472 − 0.0950560i 0.152691 + 0.629402i −0.415415 + 0.909632i −1.69725 − 2.02961i 0.142315 − 0.989821i 0.723734 + 0.690079i −0.635954 + 0.122570i
25.7 −0.0475819 + 0.998867i 0.928368 + 0.371662i −0.995472 − 0.0950560i 0.666702 + 2.74818i −0.415415 + 0.909632i 1.08121 + 2.41474i 0.142315 − 0.989821i 0.723734 + 0.690079i −2.77679 + 0.535183i
25.8 −0.0475819 + 0.998867i 0.928368 + 0.371662i −0.995472 − 0.0950560i 0.902348 + 3.71953i −0.415415 + 0.909632i 1.54544 − 2.14747i 0.142315 − 0.989821i 0.723734 + 0.690079i −3.75825 + 0.724344i
121.1 −0.580057 − 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i −0.204977 − 4.30300i 0.654861 − 0.755750i −0.384103 − 2.61772i 0.959493 − 0.281733i −0.888835 + 0.458227i −3.38622 + 2.66296i
121.2 −0.580057 − 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i −0.116162 − 2.43853i 0.654861 − 0.755750i 2.62255 − 0.349625i 0.959493 − 0.281733i −0.888835 + 0.458227i −1.91899 + 1.50911i
121.3 −0.580057 − 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i −0.0822082 − 1.72576i 0.654861 − 0.755750i −1.65083 + 2.06755i 0.959493 − 0.281733i −0.888835 + 0.458227i −1.35808 + 1.06801i
121.4 −0.580057 − 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i −0.0296258 − 0.621921i 0.654861 − 0.755750i −2.62091 − 0.361724i 0.959493 − 0.281733i −0.888835 + 0.458227i −0.489417 + 0.384882i
121.5 −0.580057 − 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i 0.0297398 + 0.624314i 0.654861 − 0.755750i 2.47488 − 0.935389i 0.959493 − 0.281733i −0.888835 + 0.458227i 0.491301 − 0.386363i
121.6 −0.580057 − 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i 0.0618096 + 1.29754i 0.654861 − 0.755750i −1.05626 − 2.42576i 0.959493 − 0.281733i −0.888835 + 0.458227i 1.02109 − 0.802998i
121.7 −0.580057 − 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i 0.121343 + 2.54731i 0.654861 − 0.755750i 0.313232 + 2.62714i 0.959493 − 0.281733i −0.888835 + 0.458227i 2.00459 − 1.57643i
121.8 −0.580057 − 0.814576i 0.235759 + 0.971812i −0.327068 + 0.945001i 0.161616 + 3.39274i 0.654861 − 0.755750i 1.77884 + 1.95851i 0.959493 − 0.281733i −0.888835 + 0.458227i 2.66990 − 2.09963i
151.1 0.327068 − 0.945001i −0.888835 + 0.458227i −0.786053 − 0.618159i −3.90496 + 0.372879i 0.142315 + 0.989821i 2.64365 + 0.105392i −0.841254 + 0.540641i 0.580057 − 0.814576i −0.924818 + 3.81215i
151.2 0.327068 − 0.945001i −0.888835 + 0.458227i −0.786053 − 0.618159i −3.73412 + 0.356565i 0.142315 + 0.989821i −2.47167 − 0.943859i −0.841254 + 0.540641i 0.580057 − 0.814576i −0.884357 + 3.64537i
151.3 0.327068 − 0.945001i −0.888835 + 0.458227i −0.786053 − 0.618159i −1.62160 + 0.154844i 0.142315 + 0.989821i −1.14236 + 2.38642i −0.841254 + 0.540641i 0.580057 − 0.814576i −0.384046 + 1.58306i
151.4 0.327068 − 0.945001i −0.888835 + 0.458227i −0.786053 − 0.618159i −1.13751 + 0.108619i 0.142315 + 0.989821i 1.15912 + 2.37833i −0.841254 + 0.540641i 0.580057 − 0.814576i −0.269397 + 1.11047i
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.b ✓ 160
7.c even 3 1 inner 966.2.y.b ✓ 160
23.c even 11 1 inner 966.2.y.b ✓ 160
161.m even 33 1 inner 966.2.y.b ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.y.b ✓ 160 1.a even 1 1 trivial
966.2.y.b ✓ 160 7.c even 3 1 inner
966.2.y.b ✓ 160 23.c even 11 1 inner
966.2.y.b ✓ 160 161.m even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{160} - 6 T_{5}^{159} - 25 T_{5}^{158} + 258 T_{5}^{157} - 203 T_{5}^{156} - 3099 T_{5}^{155} + \cdots + 13\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\). Copy content Toggle raw display